solar energy v1.2 - markups

8Semiconductor Junctions

Almost all solar cells contain junctions between (differ-ent) materials of different doping. Since these junctionsare crucial to the operation of the solar cell, we will dis-cuss their physics in this chapter.

A p-n junction fabricated in the same semiconductormaterial such as c-Si is an example of an p-n homojunc-tion. There are also other types of junctions: A p-n junc-tion that is formed by two chemically different semi-conductors is called a p-n heterojunction. In a p-i-n junc-tions, the region of the internal electric field is exten-ded by inserting an intrinsic,i, layer between the p-typeand the n-type layers. The i-layer behaves like a capa-citor and it stretches the electric field formed by the p-njunction across itself. Another type of the junction is ajunction between a metal and a semiconductor, a so-called

MS junction. The Schottky barrier formed at the metal-semiconductor interface is a typical example of the MSjunction.

8.1 p-n homojunctions

8.1.1 Formation of a space-charge regionin the p-n junction

Figure 8.1 shows schematically isolated pieces of a p-type and an n-type semiconductor and their corres-ponding band diagrams. In both isolated pieces thecharge neutrality is maintained. In the n-type semicon-

83

Copyright Delft University of Technology, 2014 This copy is provided for free, for personal use only.

,

DES

DES 10/05/2014: Second sentence pg 83 after "," change to their physics is discussed in this chapter.

84 8. Semiconductor Junctions

ductor the large concentration of negatively-chargedfree electrons is compensated by positively-charged ion-ised donor atoms. In the p-type semiconductor holesare the majority carriers and the positive charge ofholes is compensated by negatively-charged ionised ac-ceptor atoms. For the isolated n-type semiconductor wecan write

n = nn0 ≈ ND, (8.1a)

p = pn0 ≈ n2i

/ND . (8.1b)

For the isolated p-type semiconductor we have

p = pp0 ≈ NA, (8.2a)

n = np0 ≈ n2i

/NA . (8.2b)

When a p-type and an n-type semiconductor arebrought together, a very large difference in electronconcentration between n- and p-type regions causes adiffusion current of electrons from the n-type materialacross the metallurgical junction into the p-type material.The term “metallurgical junction” denotes the interfacebetween the n- and p-type regions. Similarly, the dif-ference in hole concentration causes a diffusion currentof holes from the p- to the n-type material. Due to thisdiffusion process the region close to the metallurgicaljunction becomes almost completely depleted of mobilecharge carriers. The gradual depletion of the charge car-riers gives rise to a space charge created by the charge

----

----

----

----

++++

++++

++++

++++

Band diagrams:

n-type region p-type region

electronsholes

Figure 8.1: Schematic representation of an isolated n-type andp-type semiconductor and corresponding band diagrams.


8.1. p-n homojunctions 85

----

----

----

----

++++

++++

++++

++++

+ –

n-type region p-type region

holeselectrons

space charge region

E-eld

Figure 8.2: Formation of a space-charge region, when n-typeand p-type semiconductors are brought together to form ajunction. The coloured part represents the space-charge region.

of the ionised donor and acceptor atoms that is notcompensated by the mobile charges any more. This re-gion of the space charge is called the space-charge regionor depleted region and is schematically illustrated in Fig.8.2. Regions outside the depletion region, in which thecharge neutrality is conserved, are denoted as the quasi-neutral regions.

The space charge around the metallurgical junction res-ults in the formation of an internal electric field whichforces the charge carriers to move in the opposite dir-

ection than the concentration gradient. The diffusioncurrents continue to flow until the forces acting on thecharge carriers, namely the concentration gradient andthe internal electrical field, compensate each other. Thedriving force for the charge transport does not exist anymore and no net current flows through the p-n junction.

8.1.2 The p-n junction under equilibrium

The p-n junction represents a system of chargedparticles in diffusive equilibrium in which the electro-chemical potential is constant and independent of pos-ition. The electro-chemical potential describes an aver-age energy of electrons and is represented by the Fermienergy. It means that under equilibrium conditions theFermi level has constant position in the band diagramof the p-n junction. Figure 8.3 shows the energy-banddiagram of a p-n junction under equilibrium. The dis-tance between the Fermi level and the valence and/orconduction bands does not change in the quasi-neutralregions and is the same as in the isolated n- and p-typesemiconductors. Inside the space-charge region, theconduction and valence bands are not represented bystraight horizontal lines any more but they are curved.This indicates the presence of an electric field in thisregion. Due to the electric field a difference in the elec-trostatic potential is created between the boundariesof the space-charge region. Across the depletion re-



gion the changes in the carriers concentration are com-pensated by changes in the electrostatic potential. Theelectrostatic-potential profile is also drawn in Fig. 8.3

The concentration profile of charge carriers in a p-njunction is schematically presented in Fig. 8.4. In thequasi-neutral regions the concentration of electrons andholes is the same as in the isolated doped semiconduct-ors. In the space-charge region the concentrations ofmajority charge carriers decrease very rapidly. This factallows us to use the assumption that the space-chargeregion is depleted of mobile charge carriers. This as-sumption means that the charge of the mobile carriersrepresents a negligible contribution to the total spacecharge in the depletion region. The space charge in thisregion is fully determined by the ionised dopant atomsfixed in the lattice.

The presence of the internal electric field inside the p-n junction means that there is an electrostatic poten-tial difference, ψ0, across the space-charge region. Weshall determine a profile of the internal electric fieldand electrostatic potential in the p-n junction. First weintroduce an approximation, which simplifies the cal-culation of the electric field and electrostatic-potential.This approximation (the depletion approximation) assumesthat the space-charge density, ρ, is zero in the quasi-neutral regions and it is fully determined by the con-centration of ionised dopants in the depletion region.In the depletion region of the n-type semiconductor it

n-type Si p-type Siψ

ψ0

0x-ℓn ℓp-eψ0

EC

EF

EV

E1

EG

E2

-eψ0

χe

E

Figure 8.3: Energy-band diagram of the p-n junction. Theelectrostatic potential profile (red curve) is also presented in thefigure.



p=pp0=NAn=nn0=ND

p-type silicon

ln n

ln p

Position

n-type silicon

p=pn0=ni2/ND n=np0=pi

2/NA

Figure 8.4: Concentrations profile of mobile charge carriers ina p-n junction under equilibrium.

is the concentration of positively charged donor atoms,ND, which determines the space charge in this region.In the p-type semiconductor, the concentration of negat-ively charged acceptor atoms, NA, determines the spacecharge in the depletion region. This is illustrated inFig. 8.5. Further, we assume that the p-n junction is astep junction; it means that there is an abrupt changein doping at the metallurgical junction and the dopingconcentration is uniform both in the p-type and the n-type semiconductors.

According to Fig. 8.5 the position of the metallurgicaljunction is placed at zero, the width of the space-chargeregion in the n-type material is denoted as `n and thewidth of the space-charge region in the p-type materialis denoted as `p. The space-charge density is describedby

ρ(x) = eND for −`n ≤ x ≤ 0, (8.3a)ρ(x) = −eNA for 0 ≤ x ≤ `p, (8.3b)

where ND and NA is the concentration of donor and ac-ceptor atoms, respectively. Outside the space-charge re-gion the space-charge density is zero. The electric fieldis can be calculated from the Poisson’s equation, in onedimension can be written as

d2 ψ

dx2 = −d ξ

dx= − ρ

εr ε0, (8.4)

where ψ is the electrostatic potential, ξ is the electricfield, ρ is the space-charge density, εr is the semicon-



ℓp

ξmax

eND

ξ

ρ

ψ

ψ0

x

x

xℓn

eNA

p-typen-type

0

Figure 8.5: The space-charge density ρ(x), the electric fieldξ(x), and the electrostatic potential ψ(x) across the depletionregion of a p-n junction under equilibrium.

ductor dielectric constant and ε0 is the vacuum permit-tivity. The vacuum permittivity is ε0 = 8.854 · 10−14

F/cm and for crystalline silicon εr = 11.7.. The elec-tric field profile can be found by integrating the space-charge density across the space-charge region,

ξ =1

εr ε0

∫ρ dx. (8.5)

Substituting the space-charge density with Eqs. (8.3)and using the boundary conditions

ξ (−`n) = ξ(`p)= 0, (8.6)

we obtain as solution for the electric field

ξ (x) =e

εr ε0ND (`n + x) for −`n ≤ x ≤ 0, (8.7a)

ξ (x) =e

εr ε0NA

(`p − x

)for 0 ≤ x ≤ `p. (8.7b)

At the metallurgical junction, x = 0, the electric field iscontinuous, which requires that the following conditionhas to be fulfilled

NA`p = ND`n. (8.8)

Outside the space-charge region the material is electric-ally neutral and therefore the electric field is zero there.



The profile of the electrostatic potential is calculatedby integrating the electric field throughout the space-charge region and applying the boundary conditions,

ψ = −∫

ξdx. (8.9)

We define the zero electrostatic potential level at theoutside edge of the p-type semiconductor. Since we as-sume no potential drop across the quasi-neutral regionthe electrostatic potential at the boundary of the space-charge region in the p-type material is also zero,

ψ(`p) = 0. (8.10)

Using Eqs. (8.7) for describing the electric field in then-type and p-type parts of the space-charge region, re-spectively, and taking into account that at the metallur-gical junction the electrostatic potential is continuous,the solution for the electrostatic potential can be writtenas

ψ (x) =− e2εr ε0

ND (x + `n)2

+e

2εr ε0

(ND `2

n + NA `2p

) for − `n ≤ x ≤ 0,

(8.11a)

ψ (x) =e

2εr ε0NA

(x− `p

)2 for 0 ≤ x ≤ `p.

(8.11b)

Under equilibrium a difference in electrostatic poten-tial, ψ0, develops across the space-charge region. The

electrostatic potential difference across the p-n junc-tion is an important characteristic of the junction andis denoted as the built-in voltage or diffusion potential ofthe p-n junction. We can calculate ψ0 as the differencebetween the electrostatic potential at the edges of thespace-charge region,

ψ0 = ψ (−`n)− ψ(`p)= ψ (−`n) . (8.12)

Using Eq. (8.11a) we obtain for the built-in voltage

ψ0 =e

2εr ε0

(ND `2

n + NA `2p

)(8.13)

Another way to determine the built-in potential ψ0 is touse the energy-band diagram presented in Fig. 8.3.

eψ0 = EG − E1 − E2. (8.14)

Using Eqs. (6.1) and (6.20), which determine the bandgap, and the positions of the Fermi energy in the n- andp-type semiconductor, respectively,

EG = EC − EV ,E1 = EC − EF = kT ln (NC/ND) ,E2 = EF − EV = kT ln (NV/NA) .

We can write

eψ0 = EG − kT ln(

NVNA

)− kT ln

(NCND

)= EG − kT ln

(NV NCNA ND

). (8.15)



Using the relationship between the intrinsic concentra-tion, ni and the band gap, EG [see Eq. (6.7)],

n2i = NC NV exp

[−EG

kT

],

we can rewrite Eq. (8.15) and obtain

ψ0 =kTe

ln

(NA ND

n2i

)(8.16)

This equation allows us to determine the built-in poten-tial of a p-n junction from the standard semiconductorparameters, such as doping concentrations and the in-trinsic carrier concentration. With the built-in potentialwe can calculate the width of the space charge regionof the p-n junction in the thermal equilibrium. Substi-tuting ψ0 using Eq. (8.16) into Eq. (8.12) and taking theboundary condition [Eq. (8.6) into account, the resultingexpressions for `n and `p are obtained. The full deriva-tion can be found for example Ref. [20].

`n =

√2εr ε0

qψ0

NAND

(1

NA + ND

), (8.17a)

`p =

√2εr ε0

qψ0

NDNA

(1

NA + ND

). (8.17b)

The total space-charge width, W, is the sum of the par-tial space-charge widths in the n- and p-type semicon-

ductors and can be calculated with

W = `n + `p =

√2εr ε0

eψ0

(1

NA+

1ND

). (8.18)

The space-charge region is not uniformly distributed inthe n- and p- regions. The widths of the space-chargeregion in the n- and p-type semiconductor are determ-ined by the doping concentrations as illustrated by Eqs.(8.17). Knowing the expressions for `n and `p we candetermine the maximum value of the internal electricfield, which is at the metallurgical junction. By substi-tuting `p from expressed by Eq. (8.17b) into Eq. (8.7b)we obtain the expression for the maximum value of theinternal electric field,

ξmax =

√2e

εr ε0ψ0

(NA ND

NA + ND

). (8.19)



Example

A crystalline silicon wafer is doped with 1016 acceptor atoms per cubic centimetre. A 1 micrometer thick emitter layer is formed at the sur-face of the wafer with a uniform concentration of 1018 donors per cubic centimetre. Assume a step p-n junction and that all doping atomsare ionised. The intrinsic carrier concentration in silicon at 300 K is 1.5 · 1010 cm−3.

Let us calculate the electron and hole concentrations in the p- and n-type quasi-neutral regions at thermal equilibrium. We shall use Eqs.(8.1) and (8.2) to calculate the charge carrier concentrations.

P-type region: p = pp0 ≈ NA = 1016 cm−3.

n = np0 = n2i

/pp0 =

(1.5 · 1010

)2/

1016 = 2.25 · 104 cm−3

N-type region: n = nn0 ≈ NA = 1018 cm−3.

p = pn0 = n2i

/nn0 =

(1.5 · 1010

)2/

1018 = 2.25 · 102 cm−3

We can calculate the position of the Fermi energy in the quasi-neutral n-type and p-type regions, respectively, using Eq. (6.20a). Weassume that the reference energy level is the bottom of the conduction band, EC = 0 eV.

N-type region: EF − EC = −kT ln (NC/n) = −0.0258 ln(

3.32 · 1019/

1018)= −0.09 eV.

P-type region: EF − EC = −kT ln (NC/n) = −0.0258 ln(

3.32 · 1019/

2.24 · 104)= −0.90 eV.

The minus sign tells us that the Fermi energy is positioned below the conduction band.

The built-in voltage across the p-n junction is calculated using Eq. (8.16),

ψ0 =kTe

ln

(NA ND

n2i

)= 0.0258 V

[1016 1018(1.5 · 1010

)2

]= 0.81 V.


Dave van Dongen

Dave van Dongen 09/23/2014: Should this not be kT/q, so q instead of e...?


The width of the depletion region is calculated from Eq. (8.18),

W =

√2εr ε0

eψ0

(1

NA+

1ND

)=

√2 · 11.7 · 8.854 · 10−14

1.602 · 10−19 · 0.81(

11016 +

11018

)= 3.25 · 10−5 cm = 0.325 µm.

A typical thickness of c-Si wafers is 300 µm. The depletion region is 0.3 µm which represents 0.1% of the wafer thickness. It is importantto realise that almost the whole bulk of the wafer is a quasi-neutral region without an internal electrical field.

The maximum electric field is at the metallurgical junction and is calculated from Eq. (8.19).

ξmax =

√2 e

εr ε0ψ0

(NA ND

NA + ND

)=

2 · 1.602 · 10−19

11.7 · 8.854 · 10−14 · 0.81

(1016 1018

1016 + 1018

)= 5 · 104 V cm−1.


anonymous

anonymous 09/22/2014: ??


8.1.3 The p-n junction under appliedvoltage

When an external voltage, Va, is applied to a p-n junc-tion the potential difference between the n- and p-typeregions will change and the electrostatic potential acrossthe space-charge region will become (ψ0 −Va). Remem-ber that under equilibrium the built-in potential is neg-ative in the p-type region with respect to the n-typeregion. When the applied external voltage is negativewith respect to the potential of the p-type region, theapplied voltage will increase the potential differenceacross the p-n junction. We refer to this situation as p-njunction under reverse-bias voltage. The potential bar-rier across the junction is increased under reverse-biasvoltage, which results in a wider space-charge region.The band diagram of the p-n junction under reverse-biased voltage is presented in Fig. 8.6 (a). Under ex-ternal voltage the p-n junction is not under equilibriumany more and the concentrations of electrons and holesare described by the quasi-Fermi energy for electrons,EFC, and the quasi-Fermi energy for holes, EFV , respect-ively. When the applied external voltage is positivewith respect to the potential of the p-type region, theapplied voltage will decrease the potential differenceacross the p-n junction. We refer to this situation as p-njunction under forward-bias voltage. The band diagramof the p-n junction under forward-biased voltage ispresented in Fig. 8.6 (b). The potential barrier across the

junction is decreased under forward-bias voltage andthe space charge region becomes narrower. The balancebetween the forces responsible for diffusion (concen-tration gradient) and drift (electric field) is disturbed.Lowering the electrostatic potential barrier leads to ahigher concentration of minority carriers at the edgesof the space-charge region compared to the situationin equilibrium. This process is referred to as minority-carrier injection. This gradient in concentration causesthe diffusion of the minority carriers from the edge intothe bulk of the quasi-neutral region.

The diffusion of minority carriers into the quasi-neutralregion causes a so-called recombination current, Jrec,since the diffusing minority carriers recombine with themajority carriers in the bulk. The recombination cur-rent is compensated by the so-called thermal generationcurrent, Jgen, which is caused by the drift of minoritycarriers, which are present in the corresponding dopedregions (electrons in the p-type region and holes in then-type region), across the junction. Both, the recombin-ation and generation currents have contributions fromelectrons and holes. When no voltage is applied to thep-n junction, the situation inside the junction can beviewed as the balance between the recombination andgeneration currents,

J = Jrec − Jgen = 0 for Va = 0 V. (8.20)

It is assumed that when a moderate forward-biasvoltage is applied to the junction the recombination cur-




+Va

0x

-eψ

ECEFC

EV

EG-e(ψ0+Va)

χe

E

EFV


-Va

0x

-eψ

EC

EFV

EV

EG

-e(ψ0-Va)

χe

E

EFC

(a) (b)

Figure 8.6: Energy band diagram and potential profile (in red colour) of a p-n junction under (a) reverse bias, and (b) forward bias.



rent increases with the Boltzmann factor exp(eVa/kT),

Jrec (Va) = Jrec (Va = 0) exp(

eVa

kT

). (8.21)

This assumption is called the Boltzmann approximation.

The generation current, on the other hand, is almostindependent of the potential barrier across the junctionand is determined by the availability of the thermally-generated minority carriers in the doped regions,

Jgen (Va) ≈ Jgen (Va = 0) . (8.22)

The external net-current density can be expressed as

J (Va) = Jrec (Va) − Jgen (Va)

= J0

[exp

(eVa

kT

)− 1]

,(8.23)

where J0 is the saturation-current density of the p-njunction, given by

J0 = Jgen (Va = 0) . (8.24)

Equation (8.23) is known as the Shockley equation thatdescribes the current-voltage behaviour of an ideal p-ndiode. It is a fundamental equation for microelectron-ics device physics. The detailed derivation of the dark-current density of the p-n junction is carried out in Ap-pendix B.1. The saturation-current density is given by

J0 = e n2i

(DN

LN NA+

DPLP ND

). (8.25)

The saturation-current density depends in a complexway on the fundamental semiconductor parameters.Ideally the saturation-current density should be as lowas possible and this requires an optimal and balanceddesign of the p-type and n-type semiconductor prop-erties. For example, an increase in the doping concen-tration decreases the diffusion length of the minoritycarriers, which means that the optimal product of thesetwo quantities requires a delicate balance between thesetwo properties.

The recombination of the majority carriers due to thediffusion of the injected minority carriers into the bulkof the quasi-neutral regions results in a lowering of theconcentration of the majority carriers compared to theone under equilibrium. The drop in the concentrationof the majority carriers is balanced by the flow of themajority carriers from the electrodes into the bulk. Inthis way the net current flows through the p-n junc-tion under forward-bias voltage. For high reverse-biasvoltage, the Boltzmann factor in Eq. (8.23) becomes verysmall and can be neglected. The net current density isgiven by

J(Va) = −J0, (8.26)

and represents the flux of thermally generated minoritycarriers across the junction. The current density-voltage(J-V) characteristic of an ideal p-n junction is schematic-ally shown in Fig. 8.7.



ln J-J

Forward bias

lnJ

V

J

V

Reverse bias

Slope e/kT

00

Forward bias

Reverse bias

Figure 8.7: J-V characteristic of a p-n junction; (a) linear plotand (b) semi-logarithmic plot.

8.1.4 The p-n junction under illumination

When a p-n junction is illuminated the additionalelectron-hole pairs are generated in the semiconductor.The concentration of minority carriers (electrons in thep-type region and holes in the n-type region) stronglyincreases. This increase in the concentration of minor-ity carriers leads to the flow of the minority carriersacross the depletion region into the quasi-neutral re-gions. Electrons flow from the p-type into the n-typeregion and holes from the n-type into the p-type re-gion. The flow of the photo-generated carriers causesthe so-called photo-generation current, Jph, which addsto the thermal-generation current, Jgen. When no ex-ternal contact between the n-type and the p-type re-

gions is established, which means that the junction isin the open-circuit condition, no net current can flowinside the p-n junction. It means that the current res-ulting from the flux of photo-generated and thermally-generated carriers has to be balanced by the oppositerecombination current. The recombination current willincrease through lowering of the electrostatic poten-tial barrier across the depletion region. This situationof the illuminated p-n junction under open-circuit con-dition using the band diagram is presented in Fig. 8.8(a). The electrostatic-potential barrier across the junctionis lowered by an amount of Voc. We refer to Voc as theopen-circuit voltage. Under non-equilibrium conditionsthe concentrations of electrons and holes are describedby the quasi-Fermi energy levels. It is illustrated in Fig.8.8 (a) that the electrochemical potential of electrons,denoted by EFC, is higher in the n-type region than inthe p-type region by an amount of eVoc. This meansthat a voltmeter will measure a voltage difference ofVoc between the contacts of the p-n junction. Underillumination, when the n-type and p-type regions areshort circuited, the photo-generated current will alsoflow through the external circuit. This situation is illus-trated in Fig. 8.8 (b). Under the short-circuit conditionthe electrostatic-potential barrier is not changed, butfrom a strong variation of the quasi-Fermi levels insidethe depletion region one can determine that the currentis flowing inside the semiconductor.

When a load is connected between the electrodes of the




-Voc

0x

EC

EFV

EV

-e(ψ0-Voc)

χe

E

EFC


0x

-eψ

EC

EFV

EV

-eψ0

χe

E

EFC

(a) (b)

-eVoc

-eψ

Figure 8.8: Energy band diagram and electrostatic-potential (in red colour) of an illuminated p-n junction under the (a) open-circuitand (b) short-circuit conditions.



illuminated p-n junction, only a fraction of the photo-generated current will flow through the external circuit.The electro-chemical potential difference between then-type and p-type regions will be lowered by a voltagedrop over the load. This in turn lowers the electrostaticpotential over the depletion region which results inan increase of the recombination current. In the super-position approximation, the net current flowing throughthe load is determined as the sum of the photo- andthermal generation currents and the recombination cur-rent. The voltage drop at the load can be simulated byapplying a forward-bias voltage to the junction, there-fore Eq. (8.23), which describes the behaviour of thejunction under applied voltage, is included to describethe net current of the illuminated p-n junction,

J (Va) = Jrec (Va) − Jgen (Va)− Jph

= J0

[exp

(eVa

kT

)− 1]− Jph.

(8.27)

Both the dark and illuminated J-V characteristics ofthe p-n junction are represented in Fig. 8.9. Note, thatin the figure the superposition principle is reflected.The illuminated J-V characteristic of the p-n junctionis the same as the dark J-V characteristic, but it is shif-ted down by the photo-generated current density Jph.The detailed derivation of the photo-generated currentdensity of the p-n junction is carried out in Appendix

B.2. and its value under uniform generation rate, G, is

Jph = eG (LN + W + LP) , (8.28)

where LN and LP is the minority-carrier-diffusionlength for electrons and holes, respectively, and W isthe width of the depletion region. It means only carri-ers generated in the depletion region and in the regionsup to the minority-carrier-diffusion length from the de-pletion region contribute to the photo-generated cur-rent. When designing the thickness of a solar cell, Eq.(8.28) must be considered. The thickness of the absorbershould not be thicker than the region from which thecarriers contribute to the photo-generated current.

8.2 p-n heterojunctions

8.3 Metal-semiconductor junctions


8.3. Metal-semiconductor junctions 99

J

V

Jph

Jsc

Voc

Vmp

Jmp

Dark

Illuminated

Peak power

PP=J×V

Pmax

Figure 8.9: J-V characteristics of a p-n junction in the darkand under illumination.


9Solar Cell Parameters and Equivalent Circuit

9.1 External solar cell parameters

The main parameters that are used to characterise theperformance of solar cells are the peak power Pmax, theshort-circuit current density Jsc, the open-circuit voltage Voc,and the fill factor FF. These parameters are determinedfrom the illuminated J-V characteristic as illustrated inFig. 8.9. The conversion efficiency η can be determinedfrom these parameters.

9.1.1 Short-circuit current density

The short-circuit current Isc is the current that flowsthrough the external circuit when the electrodes of the

solar cell are short circuited. The short-circuit current ofa solar cell depends on the photon flux density incidenton the solar cell, which is determined by the spectrumof the incident light. For a standard solar cell measure-ments, the spectrum is standardised to the AM1.5 spec-trum. The Isc depends on the area of the solar cell. Inorder to remove the dependence of the solar cell areaon Isc, often the short-circuit current density is used to de-scribe the maximum current delivered by a solar cell.The maximum current that the solar cell can deliverstrongly depends on the optical properties of the solarcell, such as absorption in the absorber layer and reflec-tion.

In the ideal case, Jsc is equal to Jph as can be easily de-rived from Eq. (8.27). Jph can be approximated by Eq.

101


102 9. Solar Cell Parameters and Equivalent Circuit

(8.28), which shows that in case of an ideal diode (forexample no surface recombination) and uniform gen-eration, the critical material parameters that determineJph are the diffusion lengths of minority carriers. Crys-talline silicon solar cells can deliver under an AM1.5spectrum a maximum possible current density of 46mA/cm2. In laboratory c-Si solar cells the measured Jscis above 42 mA/cm2, while commercial solar cell havean Jsc exceeding 35 mA/cm2.

9.1.2 Open-circuit voltage

The open-circuit voltage is the voltage at which no cur-rent flows through the external circuit. It is the max-imum voltage that a solar cell can deliver. Voc corres-ponds to the forward bias voltage, at which the darkcurrent compensates the photocurrent. Voc depends onthe photo-generated current density and can be calcu-lated from Eq. (8.27) assuming that the net current iszero,

Voc =kTe

ln( Jph

J0+ 1)

. (9.1)

This equation shows that Voc depends on the saturationcurrent of the solar cell and the photo-generated cur-rent. While Jph typically has a small variation, the keyeffect is the saturation current, since this may vary byorders of magnitude. The saturation current density, J0,

depends on the recombination in the solar cell. There-fore, Voc is a measure of the amount of recombinationin the device. Laboratory crystalline silicon solar cellshave a Voc of up to 720 mV under the standard AM1.5conditions, while commercial solar cells typically haveVoc exceeding 600 mV.

9.1.3 Fill factor

The fill factor is the ratio between the maximum power(Pmax = JmpVmp) generated by a solar cell and theproduct of Voc with Jsc (see Fig. 8.9),

FF =JmpVmp

JscVoc. (9.2)

Assuming that the solar cell behaves as an ideal diodethe fill factor can be expressed as a function of open-circuit voltage Voc

1,

FF =voc − ln (voc + 0.72)

voc + 1, (9.3)

where voc = Voc · e/(kT) is a normalised voltage. Eq.(9.3) is a good approximation of the ideal value of FFfor voc > 10. The FF as a function of Voc is illustratedin Fig. 9.1. This figure shows that FF does not change

1M.A. Green, Solar Cells; Operating Principles, Technology and System Applica-tions, Prentice-Hall, 1982.


Giusva

Giusva 09/16/2014: i suggest the use of q instead of e

Giusva

Giusva 09/16/2014: i suggest the use of q instead of e

9.1. External solar cell parameters 103

0.4 0.6 0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Voc (V)

Fill

Fact

or (−

)

n=1n=1.5n=2

Figure 9.1: The FF as a function of Voc for a solar cell withideal diode behaviour.

drastically with a change in Voc. For a solar cell witha particular absorber, large variations in Voc are notcommon. For example, at standard illumination condi-tions, the difference between the maximum open-circuitvoltage measured for a silicon laboratory device and atypical commercial solar cell is about 120 mV, givinga maximal FF of 0.85 and 0.83, respectively. However,the variation in maximum FF can be significant forsolar cells made from different materials. For example,a GaAs solar cell may have a FF approaching 0.89.

However, in practical solar cells the dark diode currentEq. (8.23) does not obey the Boltzmann approximation.The non-ideal diode is approximated by introducing anideality factor n, into the Boltzmann factor,

expeVa

nkT.

Fig. 9.1 also demonstrates the importance of the diodeideality factor when introduced into the normalisedvoltage in Eq. (9.3). The ideality factor is a measure ofthe junction quality and the type of recombination in asolar cell. For the ideal junction where the recombina-tion is represented by the recombination of the minor-ity carriers in the quasi-neutral regions the n is equalto 1. However, when other recombination mechanismsoccur, the n can have a value of 2. A high n value notonly lowers the FF, but since it signals a high recom-bination, it leads to a low Voc. Eq. 9.3) describes a max-imum achievable FF. In practice the FF is often lowerdue to the presence of parasitic resistive losses.

9.1.4 Conversion efficiency

The conversion efficiency is calculated as the ratiobetween the maximal generated power and the incidentpower. The irradiance value Pin of 1000 W/m2 for theAM1.5 spectrum has become a standard for measuring



the conversion efficiency of solar cells,

η =Pmax

Pin=

Jmp Vmp

Pin=

Jsc Voc FFPin

. (9.4)

Typical external parameters of a crystalline silicon solarcell as shown are; Jsc ≈ 35 mA/cm2, Voc up to 0.65 Vand FF in the range 0.75 to 0.80. The conversion effi-ciency lies in the range of 17 to 18%.


9.1. External solar cell parameters 105

Example

A crystalline silicon solar cell generates a photo-current density of Jph = 35 mA/cm2. The wafer is doped with 1017 acceptor atoms percubic centimetre and the emitter layer is formed with a uniform concentration of 1019 donors per cubic centimetre. The minority-carrierdiffusion length in the p-type region and n-type region is 500 · 10−6 m and 10 · 10−6 m , respectively. Further, the intrinsic carrierconcentration in silicon at 300 K is 1.5 · 1010 cm−3, the mobility of electrons in the p-type region is µn = 1000 cm2V−1s−1 and holesin the n-type region is µp = 100 cm2V−1s−1. Assume that the solar cell behaves as an ideal diode. Calculate the built-in voltage, theopen-circuit voltage and the conversion efficiency of the cell.

Jph = 350 Am−2.

NA = 1017 cm−3 = 1023 m−3.

ND = 1019 cm−3 = 1025 m−3.

LN = 500 · 10−6 m.

LP = 10 · 10−6 m.

DN = (kT/e)µn = 0.0258 V · 1000 · 10−4 cm2V−1s−1 = 2.58 · 10−3 m2s−1.

DP = (kT/e)µp = 0.0258 V · 100 · 10−4 cm2V−1s−1 = 2.58 · 10−4 m2s−1.

Using Eq. (8.16) we calculate the built-in voltage of the cell,

ψ0 =kTe

ln

(NA ND

n2i

)= 0.0258 V · ln

[10231025

(1.5 · 1016)2

]= 0.93 V.



According to the assumption that the solar cell behaves as an ideal diode, the Shockley equation describing the J-V characteristic is applic-able. Using Eq. (8.25) we determine the saturation-current density,

J0 = e n2i

(DN

LN NA+

DPLP ND

)= 1.602 · 10−19 C (1.5 · 1016)2 m−6

(2.58 · 10−3 m2s−1

500 · 10−6 m 1023 m−3 +2.58 · 10−4 m2s−1

100 · 10−6 m 1025 m−3

)

= 1.95 · 10−9 Cm2s

= 1.95 · 10−9 Am2 .

Using Eq. (9.1) we determine the open-circuit voltage,

Voc =kTe

ln( Jph

J0+ 1)= 0.0258 V ln

(350 Am−2

1.95 · 10−9 Am−2 + 1

)= 0.67 V.

The fill factor of the cell can be calculated from Eq. (9.3). First, we normalise Voc,

voc = Voc

/kTe

=0.67 V

0.0258 V= 26.8.

Hence,


voc + 1= 0.84.

Finally, the conversion efficiency is determined using Eq. (9.4),

η =Jsc Voc FF

Pin=

350 Am−2 0.67 V 0.841000 Wm−2 = 0.197 = 19.7%.


9.2. The external quantum efficiency 107

Wavelength (nm)

Ext.

Qua

ntum

Efficie

ncy

(–)

12001000800600400

1

0.8

0.6

0.4

0.2

0

Figure 9.2: The external quantum efficiency of a high-qualitycrystalline silicon based solar cell.

9.2 The external quantum effi-ciency

The external quantum efficiency EQE(λ0) is the fractionof photons incident on the solar cell that create electron-hole pairs in the absorber, which are successfully col-lected. It is wavelength dependent and is usually meas-ured by illuminating the solar cell with monochromaticlight of wavelength λ0 and measuring the photocurrentIph through the solar cell. The external quantum effi-

ciency is then determined as

EQE(λ0) =Iph(λ0)

e Φph(λ0), (9.5)

where e is the elementary charge and Φph is the photonflux incident on the solar cell. Since Iph is dependenton the bias voltage, the bias voltage must be fixed. Thephoton flux is usually determined by measuring theEQE of a calibrated photodiode under the same lightsource.

Figure 9.2 illustrates a typical EQE for a high qualitycrystalline silicon based solar cell. We can identify themajor optical loss mechanisms for such a solar cell: Forshort wavelengths only a small fraction of the light isconverted into electron-hole pairs. Most photons arealready absorbed in the layers that the light traversesprior to the absorber layer. For long wavelengths, thepenetration depth2 of the light exceeds the optical thick-ness of the absorber. Then the absorber itself becomestransparent so that most of the light leaves the solarcell before it can be absorbed. We can see that for thistype of solar cells the EQE is close to 1 for a broadwavelength band. Hence, in this band almost all ab-sorbed photons are converted into electron-hole pairsthat can leave the solar cell.

2According to Lambert-Beer’s law, the intensity of light in an absorbing layerdecays exponentially, I(z)∝ exp (−αz), where α is the absorption coefficient.The penetration depth is then defined as dpen(λ0)=1/α(λ0). The absorption coef-ficient α is related to the imaginary part k of the complex refractive index viaα(λ0)=4πk/λ0.



When a bias voltage of 0 V is applied, the measuredphotocurrent density equals the short circuit currentdensity. When applying a sufficiently large reversedbias voltage, it can be assured that nearly all photo-generated charge carriers in the intrinsic layer are col-lected. Thus, this measurement can be used to studythe optical effectiveness of the design, i.e. light trappingand light absorption in inactive layers, such as the TCOlayer, doped layers and the back reflector.

Measuring the EQE

EQE spectra are measured using an EQE-setup that alsocalled spectral response setup. For this measurement, usu-ally a wavelength selective light source, a calibratedlight detector and a current meter are required. Usually,the used light sours is a xenon gas discharge lamp that asa very broad spectrum covering all the wavelengths im-portant for the solar cell performance. With the help offilters and monochromators a very narrow wavelengthband of photon energies can be selected that then canbe incident on the solar cell.

As already seen in Eq. (9.5), EQE(λ) is proportionalto the the current divided by the photon flux. Whilethe current can be easily determined using an Amperemeter, the photon flux must be determined indirectly.This is done by performing a measurement with acalibrated photodetector (or solar cell), whose EQE is

known. Via this measurement we find

Φph(λ0) =Irefph (λ0)

e EQEref(λ0), (9.6)

By combining Eqs. (9.5) and (9.6) we therefore find

EQE(λ0) = EQEref(λ0)Iph(λ0)

Irefph (λ0)

. (9.7)

The EQE therefore can be determined by performingtwo current measurements. Of course it is very import-ant that the light source is sufficiently stable during thewhole measurement as we assume that the photon fluxin the reference measurement and the actual measure-ment is unchanged.

If we perform the EQE measurement under short circuitconditions, the measurement can be used to determ-ine the short circuit current density Jsc. Determining Jscvia the EQE has the advantage that it is independentof the spectral shape of the used light source, in con-trast to determining the Jsc via an JV measurement.Secondly, on lab scale the real contact area of solar cellsis not accurately determined during JV measurements.When using shading masks, the EQE measurement isindependent on the contact area. Hence, for accuratelymeasuring the sort circuit current density, it is not suffi-cient to rely on JV measurements only, but EQE setupshave to be used.


9.3. The equivalent circuit 109

For determining Jsc we combine that the photon flux ata certain wavelength with the EQE at this wavelength,leading to the flow of electrons leaving the solar cellat this wavelength. Jsc then is obtained by integratingacross all the relevant wavelength,

Jsc = −e∫ λ2

λ1

EQE(λ)ΦAM1.5(λ)dλ. (9.8)

For crystalline silicon, the important range would befrom 300 to 1200 nm.

9.3 The equivalent circuit

The J-V characteristic of an illuminated solar cell thatbehaves as the ideal diode is given by Eq. (8.27),

J (Va) = Jrec (Va) − Jgen (Va)− Jph

= J0

[exp

(eVa

kT

)− 1]− Jph.

This behaviour can be described by a simple equival-ent circuit, illustrated in Fig. 9.3 (a), in which a diodeand a current source are connected in parallel. The di-ode is formed by a p-n junction. The first term in Eq.(8.27) describes the dark diode current density whilethe second term describes the photo-generated currentdensity. In practice the FF is influenced by a series res-istance Rs, and a shunt resistance Rp. The influence of

Id

Id

Iph

IphV

+

−

I

Rs

Rp

V

+

−

I

(a)

(b)

Figure 9.3: The equivalent circuit of an (a) ideal solar cell anda (b) solar cell with a series resistance Rs and a shunt resistanceRp.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-500

-400

-300

-200

-100

0

100

200

Voltage (V)

Cur

rent

Den

sity

(A/m

2 )

Rp = 104 Ω

Voc

Rs = 0.0 ΩRs = 2.5 ΩRs = 5.0 ΩRs = 7.5 ΩRs = 10. Ω

Rs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-500

-400

-300

-200

-100

0

100

200

Voltage (V)

Cur

rent

Den

sity

(A/m

2 )

Voc

Rp = 0.001 ΩRp = 0.005 ΩRp = 0.010 ΩRp = 0.030 ΩRp = 10400 Ω

RpRs = 0 Ω

(a) (b)

Figure 9.4: Effect of the (a) series resistance and (b) parallel resistance on the J-V characteristic of a solar cell.


9.3. The equivalent circuit 111

these parameters on the J-V characteristic of the solarcell can be studied using the equivalent circuit presen-ted in Fig. 9.3 (b). The J-V characteristic of the one-diode equivalent circuit with the series resistance andthe shunt resistance is given by

J = J0

exp

[e (V − AJRs)

kT

]− 1+

V − AJRs

Rp− Jph,

(9.9)where A is the area of the solar cell. The effect of Rsand Rp on the J-V characteristic is illustrated in Fig. 9.4.

In real solar cells the FF is influenced by additional re-combination occurring in the p-n junction. This non-ideal diode is often represented in the equivalent cir-cuit by two diodes, an ideal one with an ideality factorequal to unity and a non-ideal diode with an idealityfactor larger than one. The equivalent circuit of a realsolar cell is presented in Fig. 9.5. The J-V characteristicof the two-diode equivalent circuit is given by

J = J01

[exp

(e (V − AJRs)

n1kT

)− 1]

+ J02

[exp

(e (V − AJRs)

n2kT

)− 1]

+V − AJRs

Rp− Jph,

(9.10)

where J01 and J02 are the saturation currents of the twodiodes, respectively. n1 and n2 are the ideality factors ofthe two diodes.

n1=1 n2>1Rs

Rs

Id2Id11 2 V

+I

Iph

−

Figure 9.5: The equivalent circuit of a solar cell based on thetwo-diode model.


10Losses and Efficiency Limits

In the previous chapters we have learned the basicphysical principles of solar cells. In this chapter we willbring the different building blocks together and analyse,how efficient a solar cell theoretically can be.

After discussing different efficiency limits and the majorloss mechanisms, we will finalise this chapter with theformulation of three design rules that should always bekept in mind when designing solar cells.

It is very important to understand, why a solar cell can-not convert 100% of the incident light into electricity.Different efficiency limits can be formulated, each tak-ing different effects into account.

10.1 The thermodynamic limit

The most general efficiency limit is the thermodynamicefficiency limit. In this limit, the photovoltaic device isseen as a thermodynamic heat engine, as illustrated inFig. 10.1. Such a heat engine operates between two heatreservoirs; a hot one with temperature TH and a coldone temperature TC. For the heat engine, three energyflows are relevant. First, the heat flow QH from thehot reservoir to the engine. Secondly, the work W thatis performed by the engine and thirdly, heat flowingfrom the engine to the cold reservoir that serves as aheat sink, QC. Clearly, the third energy flow is a loss andconsequently, the efficiency of the heat engine is given

113


114 10. Losses and Efficiency Limits

hot reservoircold reservoir

(heat sink)HQ CQ

W

engine

Figure 10.1: Illustrating the major heat flows in a generic heatengine.

by

η =WQh

. (10.1)

The second law of thermodynamics teaches us that the en-tropy of an independent system never decreases. It onlyincreases or stays the same. While the heat flows QHand QC carry entropy, the performed work W is anentropy-free form of energy. Thermodynamics teachesus that there is an efficiency limit for the transformationof heat into entropy-free energy. An (ideal) engine thathas this maximal efficiency is called a Carnot engine andits efficiency is given by

ηCarnot = 1− TCTH

. (10.2)

For a Carnot engine, the entropy does not increase.Note that all the temperatures must be given in a tem-perature scale where the absolute zero takes the value

0. For example, the Kelvin scale is such a scale. FromEq. (10.2) we can already see two important trends thatare basically true for every heat engine, i.e. also steamengines of combustion engines. The efficiency increases,if the higher temperature TH is increased and/or thelower temperature TC is decreased.

Let us now look at a solar cell that we imagine as aheat engine operating between an absorber of temper-ature TA (this is our hot reservoir) and a cold reservoir,which is given by the surroundings and that we assumeto be of temperature TC = 300 K. What this heat engineactually does is that it converts the energy stored in theheat of the absorber into entropy-less chemical energythat is stored in the electron-hole pairs. We may hereassume that the transformation of chemical energy intoelectrical energy happens lossless, i.e. with an efficiencyof 1. Clearly, the efficiency of this thermodynamic heatengine is given by

ηTD = 1− TCTA

. (10.3)

The absorber will be heated by absorbing sunlight. Aswe look at the ideal situation, we assume the absorberto be a black body that absorbs all incident radiation.Further, we assume the sun to be a blackbody of tem-perature TS = 6000 K. As we have seen in Chapter 5,the solar irradiance incident onto the absorber is given


10.2. The Shockley-Queisser Limit 115

byISe = σT4

SΩinc, (10.4)

where Ωinc is the solid angle covered by the incidentsunlight. As the absorber is a black body of temperat-ure TA it also will emit radiation. The emittance of theabsorber is given by

EAe = σT4

AΩemit. (10.5)

Ωemit is the solid angle into that the absorber can emit.

We can easily see that the efficiency of the absorptionprocess is given by

ηA =ISe − EA

e

ISe

= 1− EAe

ISe

= 1− Ωemit

Ωinc

T4A

T4S

. (10.6)

The absorber efficiency can be increased by increasingΩinc, which can be achieved by concentrating the in-cident sunlight. Under maximal concentration sunlightwill be incident onto the absorber from all angles of thehemisphere, i.e. Ωmax

inc = 2π. We assume the absorberto be open towards the surroundings and hence the sunon the top side. Its bottom side is connected to the heatengine such that radiative loss only can happen via thetop side. Therefore, also Ωemit = 2π. The maximal ab-sorber efficiency is therefore achieved under maximalconcentration and it is given by

ηmaxA = 1−

T4A

T4S

. (10.7)

Note that ηA is the larger the lower TA while the effi-ciency of the heat engine ηTD is the larger the higherTA.

For the total efficiency of the ideal solar cell we com-bine Eq. (10.3) with Eq. (10.7) and obtain

ηSC =

(1−

T4A

T4S

)(1− TC

TA

). (10.8)

Figure 10.2 shows the absorber efficiency, the thermo-dynamic efficiency and the solar cell efficiency. Wesee that the solar cell efficiency reaches its maximumof about 85% for an absorber temperature of 2480 K.Please note that the solar cell model presented in thissection does not resemble a real solar cell but that it isonly intended to discuss the physical limit of convert-ing solar radiation into electricity. Several much moredetailed studies on the thermodynamic limit have beenperformed. We want to refer the interested reader toworks by Würfel [21] and Markvart et al. [30–32].

10.2 The Shockley-Queisser Limit

We now will take a look at the the theoretical limit forsingle-junction solar cells. This limit is usually referredto as the Shockley-Queisser (SQ) limit, as they where the



ηSC

ηTDηA

Absorber temperature TA (K)

Efficie

ncy

(–)

6000500040003000200010000

1.0

0.8

0.6

0.4

0.2

0.0

Figure 10.2: The absorber efficiency ηA, the thermodynamicefficiency ηTD and the combined solar cell efficiency ηSC underfull concentration for a solar temperature of 5800 K and anambient temperature of 300 K.

first ones to formulate this limit based purely on phys-ical assumptions and without using empirically determ-ined constants [23]. We will derive the SQ limit in atwo-step approach. First, we will discuss the losses dueto spectral mismatch. Secondly, we also will take into ac-count that the solar cell will have a temperature differ-ent from 0 K which means that it emits electromagneticradiation according to Planck’s law. Just like Shockleyand Queisser, we will do this with the detailed balanceapproach.

10.2.1 Spectral mismatch

There are two principal losses that strongly reduce theenergy conversion efficiency of single-junction solarcells. As discussed in Chapter 8, an important part ofa solar cell is the absorber layer, in which the photonsof the incident radiation are efficiently absorbed result-ing in a creation of electron-hole pairs. In most cases,the absorber layer is formed by a semiconductor ma-terial, which we characterise by its bandgap energy Eg.In principle, only photons with energy higher than theband gap energy of the absorber can generate electron-hole pairs. Since the electrons and holes tend to oc-cupy energy levels at the bottom of the conductionband and the top of the valence band, respectively,the extra energy that the electron-hole pairs receivefrom the photons is released as heat into the semicon-



ductor lattice in the thermalisation process. Photons withenergy lower than the band gap energy of the semi-conductor absorber are in principle not absorbed andcannot generate electron-hole pairs. Therefore thesephotons are not involved in the energy conversion pro-cess. The non-absorption of photons carrying less en-ergy than the semiconductor band gap and the excessenergy of photons, larger than the band gap, are thetwo main losses in the energy conversion process us-ing solar cells. Both of these losses are thus related tothe spectral mismatch between the energy distributionof photons in the solar spectrum and the band gap of asemiconductor material.

Shockley and Queisser call the efficiency that is ob-tained when taking the spectral mismatch losses intoaccount the ultimate efficiency, that is given accordingto the hypothesis that ‘each photon with energy greaterthan hνg produces one electronic charge e at a voltage ofVg = hνg/e’ [23].

Let us now determine the fraction of energy of theincident radiation spectrum that is absorbed by asingle-junction solar cell. When we denote λg as thewavelength of photons that corresponds to the bandgap energy of the absorber of the solar cell, only thephotons with λ ≤ λg are absorbed. The fraction pabs ofthe incident power that is absorbed by a solar cell and

used for energy conversion can be expressed as

pabs =

∫ λg0

hcλ Φph, λ dλ∫ ∞

0hcλ Φph, λ dλ

, (10.9)

where Φph, λ is the spectral photon flux of the incidentlight as defined in Chapter 5. The fraction of the ab-sorbed photon energy exceeding the bandgap energy islost because of thermalisation. The fraction of the ab-sorbed energy that the solar can deliver as useful en-ergy is then given by

puse =Eg∫ λg

0 Φph, λ dλ∫ λg0

hcλ Φph, λ dλ

. (10.10)

By combining Eqs. (10.9) and (10.10), we can determinethe ultimate conversion efficiency,

ηult = pabs puse =Eg∫ λg

0 Φph, λ dλ∫ ∞0

hcλ Φph, λ dλ

. (10.11)

Figure 10.3 illustrates the fraction of the AM1.5 spec-trum that can be converted into a usable energy by acrystalline silicon solar cell. Figure 10.4 shows the ul-timate conversion efficiency of a solar cells band gapof a semiconductor absorber for three different radi-ation spectra, black-body radiation at 6000 K, AM0and AM1.5 solar radiation spectra. The figure demon-strates that in the case of a crystalline silicon solar cell



Wavelength (nm)

Spec

tral

irrad

ianc

e[ W

/( m

2 nm

)]

25002000150010005000

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Figure 10.3: The fraction of the AM1.5 spectrum that can beconverted into a usable energy by a crystalline silicon solar cellwith Eg = 1.12 eV.

(Eg = 1.12 eV) the losses due to the spectral mismatchaccount for almost 50%. It also shows that an absorbermaterial for a single junction solar cell has an optimalband gap of 1.1 eV and 1.0 eV for the AM0 and AM1.5spectra, respectively. Note that the maximum conver-sion efficiency for the AM1.5 spectrum is higher thanthat for AM0, while the AM0 spectrum has a higheroverall power density. This is because of the fact thatthe AM1.5 spectrum has a lower power density in partsof the spectrum that are not contributing to the energyconversion process as can be seen in Fig. 10.3. The dipsin the AM1.5 spectrum also result in the irregular shapeof the conversion efficiency as function of the band gap.

10.2.2 Detail balance limit of the effi-ciency

Similar to Shockley and Queisser we now will formulatethe detail balance limit of the efficiency. But before we startwe briefly will discuss the reason that the ultimate ef-ficiency formulated above is not physical for solar cellswith temperatures higher than 0 K.

Let us estimated that the solar cell is embedded in anenvironment of ambient temperature of 300 K and thatthe solar cell temperature also is 300 K. As the solar cellwill be in thermal equilibrium with its surroundings,it will absorb thermal radiation according to the ambi-ent temperature and it will also emit the same amount


estimate


6000 KAM 1.5AM 0

Bandgap energy (eV)

Efficie

ncy

(–)

32.521.510.5

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 10.4: The ultimate conversion efficiency for the blackbody spectrum at 6000 K, the AM0 and AM1.5 solar radiationspectra, limited only by the spectral mismatch as a function ofthe band gap of a semiconductor absorber in single junctionsolar cells.

of radiation. Therefore recombination of electron-holepairs will be present in the semiconductor leading toa recombination current different from zero. As wehave seen in Eq. (9.1), the open-circuit voltage will bereduced with increasing recombination current, whichis an efficiency loss.

We start the derivation of the detailed balance limitwith recalling the definition of the efficiency from Eq.(9.4),

η =JphVoc FF

Pin. (10.12)

For calculating ηult we made the assumption that ‘eachphoton with energy greater than hνg produces one electroniccharge e at a voltage of Vg = hνg/e’. Under the same as-sumption, we obtain for the short circuit current density

Jph(Eg) = −e∫ λg

0Φph, λ dλ (10.13)

with λg = hc/Eg. Note that we here implicitly assumedthat the photo-generated current Jph is equivalent to theshort circuit current. This assumption is valid as the re-combination current originating from thermal emissionis orders of magnitude lower than the photo-generatedcurrent. By combining Eqs. (10.11) and (10.13) we find

Jph = − eEg

Pinηult = −Pinηult

Vg. (10.14)

Let us now define the bandgap utilisation efficiency ηV



that is given by

ηV =Voc

Vg(10.15)

and tells us the fraction of the bandgap that can beused as open-circuit voltage (Shockley and Queisser usethe letter v for this efficiency). We now combine Eqs.(10.12), (10.12) and (10.15) and find

η = ηultηV FF. (10.16)

For determining the efficiency in the detailed balancelimit, we therefore must determine the bandgap utilisa-tion efficiency and the fill factor. Let us start with ηV .

According to Eq. (9.1), the open circuit voltage is givenas

Voc =kTe

ln( Jph

J0+ 1)

. (10.17)

The only unknown in this equation is the dark cur-rent J0. We assume the solar cell to be in thermal equi-librium with its surroundings at an ambient temperat-ure of Ta = 300 K. Further, we assume that the solarcell absorbs and emits as a black body for wavelengthsshorter than the bandgap wavelength of the solar cellabsorber. For wavelengths longer than the bandgap weassume the solar cell to be completely transparent thusto neither absorb nor emit. This is the same assumptionthat we already used for the absorption of sunlight.

Using the equation for the blackbody radiance LBBeλ as

given in Eq. (5.18a) we find for the radiative recombina-tion current

J0(Eg) = −2e∫ λg

0

∫2π

LBBeλ (λ; Ta) cos θ dΩ dλ

= −2eπ∫ λg

0

2hc2

λ5

[exp

(hc

λkBTa

)− 1]−1

dλ,

(10.18)where the factor 2 arises from the fact that we assumethe solar cell to emit thermal radiation both at its frontand back sides.

Combining Eqs. (10.15) with (10.17) we find

ηV(Eg) = kT/Eg ln

[Jph(Eg)

J0(Eg)+ 1

]. (10.19)

Figure 10.5 shows the bandgap utilisation efficiency forthree different spectra of the incident sunlight. For abandgap of 1.12 eV this efficiency is about ηV ≈ 77%.

For the fill factor we take the empirical but very accur-ate approximation


voc + 1(10.20)

with voc = eVoc/kT. We already discussed this approx-imation in Eq. (9.3).

Figure 10.6 finally shows the Shockley-Queisser effi-ciency limit for three different spectra of the incident



6000 KAM 1.5AM 0

Bandgap energy (eV)

Efficie

ncy

(–)

32.521.510.5

0.9

0.8

0.7

0.6

0.5

0.4

Figure 10.5: The bandgap utilisation efficiency ηV for theblack body spectrum at 6000 K, and the AM0 and AM1.5 solarspectra.

6000 KAM 1.5AM 0

Bandgap energy (eV)

Efficie

ncy

(–)

32.521.510.5

0.4

0.3

0.2

0.1

0

Figure 10.6: The Shockley-Queisser efficiency limit for theblack body spectrum at 6000 K, and the AM0 and AM1.5 solarspectra.

light. For the AM1.5 spectrum the limit is about 33.1%at 1.34 eV. For AM0 it is 30.1% at 1.26 eV.

The major loss mechanisms that are taken into accountin the Shockley-Queisser limit are illustrated in Fig.10.7. The major losses are non-absorbed photons belowthe bandgap and thermalised energy of photons abovethe bandgap. The other losses are due to the voltageloss because of thermal radiation and the fill factor be-ing different from 100%.



below bandgap losseslossesthermalisation

other losses

SQ limit

Bandgap energy (eV)

Efficie

ncy

(–)

32.521.510.5

10.90.80.70.60.50.40.30.20.1

0

Figure 10.7: The major loss mechanisms in the ShockleyQueisser limit. For this calculation the AM1.5 spectrum wasused as incident light.

10.2.3 Efficiency limit for silicon solarcells

It is very important to note that the Shockley-Queisser(SQ) limit is not directly applicable to solar cells madefrom crystalline silicon. The reason for this is that sil-icon is a so-called indirect bandgap semiconductor as wewill discuss in detail in Chapter 12. This means thatAuger recombination, which is a non-radiative recom-bination mechanism, is dominant. For the derivation ofthe SQ limit we assumed that only radiative recombina-tion is present. Clearly, this assumption cannot be validfor crystalline silicon solar cells. Several attempts to cal-culate the efficiency limit while taking radiative recom-bination mechanisms into account were performed inthe past. A study from 2013 by Richter et al. derives anefficiency limits of 29.43% for silicon solar cells .

As the Shockley-Queisser limit only considers radiat-ive recombination, it is most valid for direct band gapmaterials such as GaAs. Because of its direct band gap,radiative recombination is the limiting recombinationmechanism for GaAs.

10.3 Other Losses

The Shockley-Queisser limit is a very idealised model.For example all optical losses are neglected. Now we


10.3. Other Losses 123

will discuss several other loss mechanisms that werenot taken account above. At the end of this section wewill derive at an equation for the efficiency were allthese important losses are taken into account.

10.3.1 Optical losses

For the Shockley-Queisser limit, we only took thebandgap energy Eg into account for deriving the ef-ficiency limit. However, the real performance is alsostrongly influenced by the optical properties given asthe complex refractive index n = n − ik, which is afunction of the wavelength.

As we already discussed in Section 4.3, a part of thelight is reflected from and the other part is transmit-ted when light arrives on an interface between twomedia. The interface is therefore characterised by thewavelength dependent reflectance R(λ) and transmit-tance T(λ). All the reflections and transmissions at thedifferent interfaces in the solar cell result in a total re-flectance between the solar cell and the surrounding air.Hence, a part of the incident energy that can be con-verted into a usable energy by the solar cell is lost byreflection. We shall denote the total effective reflectancein the wavelength range of interest as R∗.

As we will discuss in more detail in Chapter 12, inmost c-Si solar cells thin metal strips are placed on the

front side of the solar cell that serve as front electrode.The metal-covered area does not allow the light to enterthe solar cell because it reflects or slightly absorbs theincident light. The area that is covered by the electrodeeffectively decreases the active area of the solar cell.When we denote the total area of the cell as Atot andthe cell area that is not covered by the electrode as A f ,the fraction of the active area of the cell is determinedby the ratio

C f =A f

Atot, (10.21)

which is called the coverage factor C f . These loss iscalled the shading loss. The design of the front electrodeis of great importance since it should one the one handminimise losses due to the series resistance of the frontelectrode, i.e. should be designed with sufficient cross-section. The optimal design of the front electrode istherefore a trade-off between a high coverage factor anda sufficiently low series resistance of the front electrode.

When light penetrates into a material, it will be par-tially absorbed as it propagates through the material.The absorption of light in the material depends on itsabsorption coefficient and the layer thickness, as wehave seen in Section 4.4. In general, light is absorbedin all layers of the solar cell. All the absorption in lay-ers different from the absorber layer is loss. It is calledthe parasitic absorption. Further, due to the limited thick-ness of the absorber layer, not all the light entering theabsorber layer is absorbed. Incomplete absorption in



the absorber due to its limited thickness is an addi-tional loss that lowers the energy conversion efficiency.The incomplete absorption loss can be described bythe internal optical quantum efficiency IQEop, whichis defined as the probability of a photon being absorbedin the absorber material. Since there is a chance thata highly energetic photon can generate more than oneelectron-hole pair, we also define the quantum efficiencyfor carrier generation ηg which represents the number ofelectron-hole pairs generated by one absorbed photon.Usually ηg is assumed to be unity.

10.3.2 Solar cell collection losses

Not all charge carries that are generated in a solar cellare collected at the electrodes. The photo-generated car-ries are the excess carriers with respect to the thermalequilibrium and are subjected to the recombination. Thecarriers recombine in the bulk, at the interfaces, and/orat the surfaces of the junction. The recombination is de-termined by the electronic properties of materials thatform the junction, such as density of states introducedinto the band gap by the R-G centers. The concentra-tion of R-G centers strongly influences the minority-carrier lifetimes as discussed in Chapter 7.

The contributions of both the electronic and opticalproperties of the solar cell materials to the photovol-taic performance are taken into account in the absolute

external quantum efficiency that we already defined inChapter 9. The EQE can be approximated by

EQE(λ) = (1− R∗)IQEop(λ)ηg(λ)IQEel(λ), (10.22)

where the IQEel denotes the electrical quantum effi-ciency and is defined as the probability that a photo-generated carrier is collected.

When we take the shading losses and the EQE into ac-count, we find the short-circuit current density to be

Jsc = Jph(1− R∗)IQE∗opη∗gIQE∗elC f , (10.23)

where the ∗ denote averages across the relevantwavelength range and Jph is as defined in Eq. (10.13).

10.3.3 Additional limiting factors

We have seen in Section 10.2 that the Voc of a solar celldepends on the saturation current J0 and the photo-generated current Jsc of the solar cell. The saturationcurrent density depends on the recombination in thesolar cell. Recombination cannot be avoided and de-pends on the doping of the different regions (n-typeand p-type regions) of a junction and the electronicquality of materials forming the junction. The dopinglevels and the recombination determine the bandgaputilisation efficiency ηV that we already defined in Sec-tion 10.2.


10.3. Other Losses 125

For determining the Shockley-Queisser limit we as-sumed for the FF that the solar cell behaves as an idealdiode. In a real solar cell, however, the FF is lower thanthe ideal value because of the following reasons:

• The voltage drop due to the series resistance Rs ofa solar cell, which is introduced by the resistanceof the main current path through which the photo-generated carriers arrive to the external circuit. Thecontributions to the series resistance come from thebulk resistance of the junction, the contact resist-ance between the junction and electrodes, and theresistance of the electrodes themselves.

• The voltage drop due to leakage currents, which ischaracterised by the shunt resistance Rp of a solarcell. The leakage current is caused by the currentthrough local defects in the junction or due to theshunts at the edges of solar cells.

• The recombination in a non-ideal solar cell results ina decrease of the FF.

10.3.4 Conversion efficiency

Again, we start with the expression for the efficiency

η =JscVoc FF

Pin. (10.24)

Using Eqs. (10.23) and (10.14) and we find

η =ηultVoc FF

Vg(1− R∗)IQE∗opη∗gIQE∗elC f

= pabs puse(1− R∗)IQE∗opη∗gIQE∗elC f ηV FF,(10.25)

where we used Eqs. (10.9) and (10.10). By filling in thedefinitions for pabs, puse, ηV and C f , we obtain [33].

η =

∫ λg0

hcλ Φph, λ dλ∫ ∞

0hcλ Φph, λ dλ︸︷︷︸

1

Eg∫ λg

0 Φph, λ dλ∫ λg0

hcλ Φph, λ dλ︸︷︷︸

2

(1− R∗)︸︷︷︸3

IQE∗opη∗g︸︷︷︸4

IQE∗el︸︷︷︸5

A f

Atot︸︷︷︸6

eVoc

Eg︸︷︷︸7

FF︸︷︷︸8

. (10.26)

This describes the conversion efficiency of a solar cellin terms of components that represent particular lossesin energy conversion.

1. Loss due to non-absorption of long wavelengths,

2. Loss due to thermalisation of the excess energy of



photons,

3. Loss due to the total reflection,

4. Loss by incomplete absorption due to the finitethickness,

5. Loss due to recombination,

6. Loss by metal electrode coverage, shading losses,

7. Loss due to voltage factor,

8. Loss due to fill factor.

10.4 Design Rules for Solar Cells

Now, as we extensively have discussed all the factorsthat limit the efficiency of a solar cell we are able todistill three design rules. We will use this design rules inPart III when we discuss different PV technologies. Thethree design rules are

1. Utilisation of the band gap energy,

2. Spectral utilisation,

3. Light trapping.

We now will take a closer look at each of these designrules.

pn

EFn

EFp

eVoc

Figure 10.8: The open circuit voltage of a solar cell is determ-ined by the splitting of the quasi-Fermi levels.

10.4.1 Bandgap utilisation

As we have seen earlier in this chapter, the open cir-cuit voltage Voc is always below the voltage Vg corres-ponding to the bandgap, which we characterised withthe bandgap utilisation efficiency ηV . The open circuitvoltage is determined by the extent to which the quasi-Fermi levels are able to split, which is limited by thecharge carrier recombination mechanisms. We will dis-cuss that various PV materials have different recombin-ation mechanism that limit the utilisation of the bandgap energy.

Let us now discuss the first design rule, bandgap utilisa-tion. Figure 10.8 shows a p-n-junction, with the p-dopedregion on the left and the n-doped region right. Fur-


10.4. Design Rules for Solar Cells 127

ther the quasi-Fermi levels are depicted. The extent ofsplitting between the quasi-Fermi levels determines theopen circuit voltage.

The open circuit voltage, for example expressed in Eq.(10.17) also can be expressed in terms of the generationrate GL, the life time τ0 of the minority charge carriersand the intrinsic density of the charge carriers ni in thesemiconductor material,

Voc =2kT

eln(

GLτ0

ni

). (10.27)

The derivation of this equation is out of the scope ofthis book.

Let us now take a closer look on Eq. (10.27). If we in-crease the irradiance, or in other words, the generationrate of charge carriers, the open circuit voltage is in-creased. This is a welcome effect which is utilised inconcentrator photovoltaics, which we will briefly dis-cuss in Section 13.2. Secondly, we see that the lifetimeτ0 plays an important role. The larger the lifetime ofthe minority charge carrier, the larger the open circuitvoltage can be. Or in other words, the longer the life-time, the larger the possible splitting between the quasi-Fermi levels and the larger the fraction of the bandgapenergy that can be utilised.

The lifetime of the minority charge carrier is de-termined by the recombination rate. As discussed in

Chapter 7, we have to consider three different recom-bination mechanisms: radiative, Shockley-Read-Hall,and Auger recombination. While radiative and Augerrecombination depend on the semiconductor itself, SRHrecombination is proportional to the density of trapsor impurities in the semiconductor. In the three recom-bination mechanisms energy and momentum are trans-ferred from charge carriers to phonons or photons.

The efficiency of the different recombination processesdepends on the nature of the band gap of the usedsemiconductor material used. We distinguish betweendirect and indirect bandgap semiconductors. Crystallinesilicon is an indirect band gap material. The radiativerecombination in an indirect band gap material is in-efficient and recombination will be dominated by theAuger mechanism. For direct band gap materials suchas GaAs under moderate illumination conditions, radi-ative recombination will be the dominant loss mech-anism of charge carriers. For very high illuminationconditions, Auger recombination starts to play a roleas well.

To summarise, we find that in the defect rich solar cells,the open circuit voltage is limited by the SRH recom-bination. In low-defect solar cells based on indirectband gap materials, the open circuit voltage is limitedby Auger recombination. In low-defect solar cells basedon direct band gap materials, the open circuit voltage islimited by radiative recombination.



Besides band gap utilisation, it also is important to dis-cuss the relationship between the maximum thicknessfor the absorber layer of a solar cell and the dominantrecombination mechanism. As we have seen in Chapter7, the recombination mechanism also affects the diffu-sion length of the minority charge carrier. The diffusionlength Ln of minority electrons is given by

Ln =√

Dnτn, (10.28)

where Dn is the diffusion coefficient and τ is the life-time of the minority charge carrier. Similarly, we canformulate the diffusion length for minority holes, Lp.

It is important to realise that the thickness of the ab-sorber layer should not exceed the diffusion length. Tounderstand this requirement, we consider photons thatwould penetrate far into the absorber layer being ab-sorbed. We want these charge carriers to be separatedat the p-n junction or at the back contact. If the dis-tance of these charge carriers from the p-n junction orthe back contact is exceeding the diffusion length, theexcited charge carriers will recombine within the typicaldiffusion length before arriving at the p-n junction orback contact. In other words, their lifetime is too short.This means that all charge carriers generated at a dis-tance larger than the diffusion length from the p-n junc-tion or the back contact cannot be collected and henceare lost. If the charge carriers are generated with diffu-sion length, they can be collected. This means that the

diffusion length of the minority charge carrier, limitsthe maximum thickness of the solar cell.

To summarise this section, the open circuit voltage islimited by the dominant recombination mechanism. Thedominance of radiative, Auger or Shockley-Reed-Hallrecombination depends on the type of semiconductormaterials used in the solar cell and the illuminationconditions. We will discuss several different cases onPart III on PV technology.

10.4.2 Spectral utilisation

The spectral utilisation is mainly determined by thechoice of materials from which the solar cell is made of.As we have seen in Section 10.2, and mainly Eq. (10.13),the photocurrent is determined by the bandgap of thematerial. For a bandgap of 0.62 eV corresponding to awavelength of 2000 nm, we could theoretically generatea short circuit current density of 62 mA/cm2. If we con-sider c-Si, having a band gap of 1.12 eV (1107 nm), wearrive at a theoretical current density of 44 mA/cm2.

The optimal bandgap for single-junction solar cells isdetermined by the Shockley-Queisser limit, as illus-trated in Fig. 10.6. For single junction solar cells, semi-conductor material as such silicon, gallium arsenideand cadmium telluride have a band gap close to theoptimum.



In Part III we will discuss various concepts that allowto surpass the Shockley-Queisser limit. Here, we willbriefly discuss the concept of multijunction solar cells.In this devices, solar cells with different bandgaps arestacked on top of each other. As illustrated in 10.9,the excess energy can be reduced significantly, and thespectral utilisation will improve.

10.4.3 Light trapping

The third and last design rule that we discuss is lighttrapping. In an ideal solar cell, all light that is incidenton the solar cell should be absorbed in the absorberlayer. As we have discussed in Section 4.4, the intensityof light decreases exponentially as it travels through anabsorptive medium. This is described by the Lambert-Beer law that we formulated in Eq. (4.25),

I(d) = I0 exp(−αd). (10.29)

From the Lambert-Beer law it follows that at the side,at which the light is entering the film, more light is ab-sorbed in reference to the back side. The total fractionof the incident light absorbed in the material is equal tothe light intensity entering the absorber layer minus theintensity transmitted through the absorber layer,

Iabs(d) = I0[1− exp(−αd)]. (10.30)

Excess energy

Excess energy

Excess energy

(a)

(b)

Figure 10.9: Illustrating the lost excess energy in (a) a single-juntion and (b) a multi-junction solar cells.



107

106

105

104

103

102

101

100

200

GaAsInPGeSi

Abs

orpt

ion

coe

cien

t α (c

m-1

)

400 600 800 1000 1200 1400 1600 1800 2000

Wavelength λ (nm)

Figure 10.10: Absorption coefficients of different semicon-ductors.

Ideally, we would like to absorb a solar cell 100% of theincident light. Such an absorber is called optically thickand has a transmissivity very close to 0. As we can seefrom the Lambert-Beer law, this can be achieved byeither absorbers with a large thickness d or with verylarge absorption coefficients α.

Figure 10.10 shows the absorption coefficients for fourdifferent semiconductor materials: germanium (Ge), sil-icon (Si), gallium arsenide (GaAs) and indium phos-phide (InP). We notice that germanium has the low-est band gap. It starts to absorb at long wavelengths,

which corresponds to a low photon energy. GaAs hasthe highest band gap, as it starts to absorb light atthe smallest wavelength, or highest photon energy.Secondly, if we focus on the visible spectral part from300 nm up 700 nm, we see that the absorption coeffi-cients of InP and GaAs are significantly higher than forsilicon. This is related to the fact that InP and GaAs aredirect band gap materials as discussed earlier. Materialswith an indirect band gap have smaller absorption coef-ficients. Only in the very blue and ultraviolet part be-low 400 nm, Si has a direct band gap transition. Siliconis a relatively poor absorber. Therefore for the samefraction of light thicker absorber layers are required incomparison to GaAs.

In general, for all semiconductor materials the ab-sorption coefficient in the blue is orders of magnitudehigher than in the red. Therefore the penetration depthof blue light into the absorber layer is rather small. Incrystalline silicon, the blue light is already fully ab-sorbed within a few nanometers. The red light requiresan absorption path length of 60 µm to be fully ab-sorbed. The infrared light is hardly absorbed, and afteran optical path length of 100 µm only about 10% of thelight intensity is absorbed.

As the absorption of photons generates excited chargecarriers, the wavelength dependence of the absorptioncoefficient determines the local generation profile of thecharge carriers. At the front side where the light enters



the absorbing film, the generation of charge carriersis significantly higher than at the back side. It followsthat the EQE values measured in the blue correspond tocharge carriers generated close to the front of the solarcell, whereas the EQE in the red part represents chargecarriers generated throughout the entire absorber layer.

Further, it is important to reduce the optical loss mech-anisms such as shading losses, reflection, and parasitic ab-sorption that we already discussed in Section 10.3.1.

For reducing the reflection, anti-reflective coatings(ARC) can be used. Light that is impinging onto a sur-face between two media with different refractive indiceswill always be partly reflected and partly transmitted.In order to reduce losses, it is important to minimisethese reflective losses.

The first method is based on a clever utilisation of theFresnel equations that we discussed in Section 4.3. Forunderstanding how this can work, we first will take alook at interfaces with silicon, the most common usedmaterial for solar cells. Let us consider light of 500 nmwavelengths falling onto an air-silicon interface per-pendicularly. At 500 nm, the refractive index of air isn0 = 1 and that of silicon is ns = 4.3. With the Fresnelequations we hence find that the optical losses due toreflection are significant with 38.8%.

The reflection can be significantly reduced by intro-ducing an interlayer with a refractive index n1 with a

0.8

0.6

0.4

0.2

0.0

Refractive index n1

Tota

l re

ectiv

ity

1.0 1.5 2.0 2.5 3.0 3.5 4.0

λ = 500 nmn0 = 1.0ns = 4.3

opt1 0 s 2.1n n n= »

Figure 10.11: Illustrating the effect of an interlayer withrefractive index n1 in between n0 and ns on the reflectivity.



value in between that of n0 and ns. If no multiple re-flection or interference is taken into account, it can eas-ily be shown that the reflectivity becomes minimal if

n1 =√

n0ns. (10.31)

This is also seen in Fig. (10.11), where n1 takes all thevalues in between n0 and ns. In this example, includ-ing a single interlayer can reduce the reflection at theinterface from 38.8% down to 22.9%. If more than oneinterlayers are used, the reflection can be reduced evenfurther. This technique is called refractive index grading.

In another approach constructive and destructive in-terference of light is utilised. In Chapter 4 we alreadydiscussed that light can be considered as an electromag-netic wave. Waves have the interesting properties thatthey can interfere with each-other, they can be super-imposed. For understanding this we look at two wavesA and B that have the same wavelength and cover thesame portion of space,

A(x, t) = A0eikx−iωt, (10.32a)

B(x, t) = B0eikx−iωt+iφ. (10.32b)

The letter φ denotes the phase shift between A and Band is very important. We can superimpose the two

waves by simply adding them

C(x, t) = A(x, t) + B(x, t)

= A0eikx−iωt + B0eikx−iωt+iφ

= A0eikx−iωt + B0eikx−iωteiφ

=(

A0 + B0eiφ)

eikx−iωt.

(10.33)

The amplitude of the superimposed wave is thus

C0 = A0 + B0eiφ. (10.34)

Depending of the phase shift, the superimposed wavewill be stronger or weaker than A and B. If φ is inphase, i.e. a multiple of 2π, i.e. 0, 2π, 4π, . . ., we willhave maximal amplification of waves,

C0 = A0 + B0 · 1 = A0 + B0. (10.35)

This situation is called constructive interference.

But if φ is in antiphase, i.e. from the set π, 3π, 5π, . . . wehave maximal attenuation of the waves, or destructiveinterference,

C0 = A0 + B0 · (−1) = A0 − B0. (10.36)

Based on this principle, we can design an anti-reflectioncoating, as illustrated in Fig. 10.12. The green waveshows the reflection back from the first interface andthe red wave shows the wave which is reflected back



(a) (b)

dark fringe

bright fr

inge

Destructiveinterference

Reected waves 180° out of phase

Constructiveinterference

Reectedwavesin phase

Path lengthdierence =even multipleof λ/2

Path lengthdierence = odd multipleof λ/2

Figure 10.12: Illustrating the working principle of an anti-reflective coating based on interference [34].

from the second interface. If we look at the two wavescoupled out of this system, they appear to be in anti-phase. As a result the total amplitude of the electricfield of the outgoing wave is smaller and hence thetotal irradiance coupled out of the system is smalleras well.

It can be shown easily that the have antiphase, whenthe product of the refractive index and thickness of theinterlayer is equal to the wavelength divided by four,

n1d =λ0

4. (10.37)

Using an antireflection coating based on interferencedemands that the typical length scale of the interlayerthickness must be in the order of the wavelength.

The last approach that we discuss for realising anti-reflective coating is using textured interfaces. Here, weconsider the case where the typical length scales of thesurface features are larger than the typical wavelengthof light. In this case, which is also called the geomet-rical limit, the reflection and transmission of the lightrays are fully determined by the Fresnel equations andSnell’s law.

The texturing helps to enhance the coupling of lightinto the layer. For example, for light that is perpendic-ular incident, light that is reflected at one part of thetextured surface can be reflected into angles in whichthe trajectory of the light ray is incident a second time



re.

Incidentlight

reected36°

Figure 10.13: Illustrating the effect of texturing.

somewhere else on the interfaces, as illustrated in Fig.10.13. Here, another fraction of the light will be trans-mitted into the layer and effectively less light will bereflected, when compared to a flat interface in-betweenthe same materials.

In summary, we have discussed three types of anti-reflective coatings: Rayleigh films with intermediaterefractive indexes, anti-reflection coatings based on de-structive interference and enhanced in-coupling of lightdue to scattering at textured interfaces.

At the end of this chapter, we want to discuss some-thing that is very important for thin-film solar cells. Be-cause of reducing production cost but also because ofreducing bulk recombination it is desirable to have theabsorber layer as thin as possible. On the other hand,

it should be optically thick in order to absorb as muchlight as possible. In principle, if the light can be reflec-ted back and force inside the absorber until everythingis absorbed. However, at every internal reflection partof the light is transmitted out of the film.

But if the light would travel through the layer at anangle larger than the critical angles of the front andback interfaces of the absorber, it could stay there un-til everything is absorbed without any loss. Unfortu-nately, it is very difficult if not impossible to designsuch a solar cell.

Note that textured interfaces do not help in this case.For the same reason more light can be coupled into theabsorber using such a texturisation, also more light iscoupled out of the absorber!


Part III

PV Technology

135


11A Little History of Solar Cells

We will start our discussion on PV technology with abrief summary of the history of solar energy in generaland of photovoltaics in particular. Already in the sev-enth century BCE, humans used magnifying glasses toconcentrate sunlight and hence to make fire. Later, theancient Greeks and Romans used concentrating mirrorsfor the same purpose.

In the 18th century the Swiss physicist Horace-Bénédictde Saussure build heat traps, which are a kind of mini-ature green houses. He constructed hot boxes, con-sisting of a glass box, within another bigger glass box,with a total number of up to five boxes. When exposedto direct solar irradiation, the temperature in the in-nermost box could rise up to values of 108°C; warmenough to boil water and cook food. These boxes can

be considered as the World’s first solar collectors.

In 1839, the French phycisist Alexandre-EdmondBecquerel, discovered the photovoltaic effect at an ageof only 19 years. He observed this effect in an electro-lytic cell, which was made out of two platinum elec-trodes, placed in an electrolyte. An electrolyte is anelectrically conducting solution; Becquerel used silverchloride dissolved in an acidic solution. Becquerel ob-served that the current of the cell was enhanced whenhis setup was irradiated with sunlight.

In the 1860s and 1870s, the French inventor AugustinMouchot developed solar powered steam engines usingthe World’s first parabolic trough solar collector, that wewill discuss in Chapter 20. Mouchot’s motivation was

137


138 11. A Little History of Solar Cells

his believe that the coal resources were limited. At thattime, coal was the energy source for driving steam en-ginges. However, as coal became cheaper, the Frenchgovernment decided that solar energy was too expens-ive and stopped funding Mouchet’s research.

In 1876, the British natural philosopher William GryllsAdams together with his student Richard Evans Daydemonstrated the photovoltaic effect in a junction basedon platinum and the semiconductor selenium, howeverwith a very poor performance. Seven years later, theAmerican inventor Charles Fritts managed to make aPV-device based on a gold-selenium junction. The en-ergy conversion efficiency of that device was 1%.

In 1887, the German physicist Heinrich Hertz dis-covered the photo-electric effect, already briefly men-tioned in Chapter 3. In this effect, electrons are emit-ted from a material that has absorbed light with awavelength shorter than a material-dependent thresholdfrequency. In 1905 Albert Einstein published a paperin which he explained the photoelectric effect with as-suming that light energy is being carried with quant-ised packages of energy [19], which we nowadays callphotons.

In 1918 the Polish chemist Jan Czochralski inventeda method to grow high-quality crystalline materials.This technique nowadays is very important for grow-ing monocrystalline silicon used for high-quality siliconsolar cells that we will study in detail in Chapter 12.

The development of the c-Si technology started in thesecond half of the 20th century.

In 1953, the American chemist Dan Trivich was the firstone to perform theoretical calculations on the solar cellperformance for materials with different bandgaps.

The real development of solar cells as we know themtoday, started at the Bell Laboratories in the United states.In 1954, their scientists Daryl M. Chapin, Calvin S.Fuller, and Gerald L. Pearson, made a silicon-basedsolar cell with an efficiency of about 6% [35]. In thesame year, Reynolds et al. reported on the photovoltaiceffect for cadmium sulfide (CdS), a II-VI semiconductor[36].

In the mid and late 1950s several companies and labor-atories started to develop silicon-based solar cells in or-der to power satellites orbiting the Earth. Among thesewere RCA Corporation, Hoffman Electronics Corpor-ation but also the Unites States Army Signal Corps.In these days, research on PV technology was mainlydriven by supplying space applications with energy.For example, the American satellite Vanguard 1, whichwas launched by the U.S. Navy in 1958, was poweredby solar cells from Hoffman Electronics. It was thefourth artificial Earth satellite and the first one to bepowered with solar cells. It was operating until 1964and still is orbiting Earth. In 1962 Bell Telephone Labor-atories launched the first solar powered telecommunic-ations satellite and in 1966 NASA launched the first Or-


139

biting Astronomical Observatory, which was poweredby a 1 kW photovoltaic solar array.

In 1968, the Italien scientist Giovanni Francia built thefirst concentrated-solar power plant near Genoa, Italy.The plant was able to produce 1 MW with superheatedsteam at 100 bar and 500°C.

In 1970, the Soviet physicist Zhores Alferov developedsolar cells based on a gallium arsenide heterojunction.This was the first demonstrator of a solar cell basedon III-V semiconductor materials that we will discussin Section 13.2. In 1976, Dave E. Carlson and ChrisR. Wronski developed the first thin-film photovoltaicdevices based on amorphous silicon at RCA Laborat-ories. We will discuss this technology in Section 13.3.In 1978, the Japanese companies SHARP and TokyoElectronic Application Laboratory bring the first solarpowered calculators on the market.

Because of the oil crisis, induced by the OPEC oil em-bargo in 1973 and as a consequence a sharply rising oilprice, the public interest in photovoltaic technology forterrestrial application was increasing in the 1970s. Inthat time, PV technology moved from a niche techno-logy for space application to a technology applicablefor terrestrial applications. In the late 1970s and 1980smany companies started to develop PV modules andsystem for terrestrial applications.

In 1980 the first thin film solar cells based on a copper-

sulfide/cadmium-sulfide junction was demonstratedwith a conversion efficiency above 10% at the Univer-sity of Delaware. In 1985, crystalline silicon solar cellswith efficiencies above 20% were demonstrated at theUniversity of New South Wales in Australia.

From 1984 through 1991 the World’s largest solarthermal energy generating facility in the world wasbuilt in the Mojave Desert in California. It consits of9 plants with a combined capacity of 354 Megawatts.

In 1991 the first high efficiency Dye-sensitized solarcell was published by the École polytechnique fédéralede Lausanne in Switzerland by Michael Grätzel andcoworkers. The Dye-sensitized solar cell is a kind ofphoto-electrochemical system, in which a semicon-ductor material based on molecular sensitizers, isplaced between a photoanode and an electrolyte. Wewill introduce this technology in Section 13.6.

In 1994, the U.S. National Renewable Energy Laborat-ory placed in Golden, Colorado, demonstrated a con-centrator solar cell based on III-V semiconductormaterials. Their cell based on a indium-gallium-phosphide/gallium-arsenide tandem junction exceededthe 30% conversion limit.

In 1999, the total global installed photovoltaic powerpassed 1 GWp. Starting from about 2000, environmentalissues and economic issues started to become moreand more important in the public discussion, which re-


140 11. A Little History of Solar Cells

newed the public interest in solar energy. Since 2000,the PV market therefore transformed from a regionalmarket to a global market, as discussed in Chapter 2.Germany took the lead with a progressive feed-in tar-iff policy, leading to a large national solar market andindustry [15].

Since about 2008, the Chinese government has beenheavily investing in their PV industry. As a result,China has been the dominant PV module manufacturerfor several years now. In 2012 the world-wide solar en-ergy capacity surpassed the magic barrier of 100 GWp[13]. Between 1999 and 2012, the installed PV capacityhence has grown with a factor 100. In other words, inthe last 13 years, the average annual growth of the in-stalled PV capacity was about 40%.


12Crystalline Silicon Solar Cells

As we already discussed in Chapter 6, most semicon-ductor materials have a crystalline lattice structure. As astarting point for our discussion on crystalline siliconPV technology, we will take a closer look at some prop-erties of the crystal lattice.

12.1 Crystalline silicon

In such a lattice, the atoms are arranged in a certainpattern that repeats itself. The lattice thus has long-rangeorder and symmetry. However, the pattern is not thesame in every direction. In other words, if we make alarge cut through the lattice, the various planes you canmake do not look the same.

Figure 12.1: A unit cell of a diamond cubic crystal [25].

141


142 12. Crystalline Silicon Solar Cells

(a) (b)

[100]

(100)

(111)[111]

Figure 12.2: The (a) 100 and the (b) 111 surfaces of a siliconcrystal.

Crystalline silicon has a density of 2.3290 g/cm3 and adiamond cubic crystal structure with a lattice constant of543.07 pm, as illustrated in Fig. 12.1. Figure 12.2 showstwo different sections through a crystalline silicon lat-tice, which originally consisted out of three by three bythree unit cells. The first surface shown in Fig. 12.2 (a)is the 100 surface, whose surface normal is the 100 direc-tion. At a 100 surface, each Si atom has two back bondsand two valence electrons pointing to the front. Thesecond surface, shown in Fig. 12.2 (b), is the 111 sur-face. Here, every Si atom has three back bonds and onevalence electron pointing towards the plane normal.

To understand the importance of these directions, welook at the electronic band dispersion diagram for silicon,

0

L X U,K

wavevector k

ΣΛ ∆Γ Γ

EC

Ener

gy E

(eV) EV

2

4

–6

–4

–2

–10

–8

1.12 eV3.4 eV

Figure 12.3: The band diagram of crystalline silicon.


12.1. Crystalline silicon 143

shown in Fig. 12.3. On the vertical axis, the energy po-sition of the valence and conduction bands is shown.The horizontal axis shows the crystal momentum, i.e.the momentum of the charge carriers. The white arearepresents the energy levels in the forbidden band gap.The band gap of silicon is determined by the lowestenergy point of the conduction band at X, which cor-responds to the 100 direction, and the highest energyvalue of the valence band, at Γ. The band gap energy isthe difference between those two levels and is equal1.12 eV, or 1107 nm, when expressed in wavelengths.1107 nm is in the infrared part of the spectrum of light.This bandgap is an indirect bandgap, because the chargecarriers must change in energy and momentum to beexcited from the valence to the conduction band. Aswe can see, crystalline silicon has a direct transition aswell. This transition has an energy of 3.4 eV, which isequivalent to a wavelength of 364 nm, which is in theblue spectral part.

Because of the required change in momentum, for foran indirect band gap material it is less likely that aphoton with an energy exceeding the bandgap can ex-cite the electron, with respect to a direct bandgap ma-terial like gallium arsenide (GaAs) or indium phos-phite (InP). Consequently, the absorption coefficientof crystalline silicon is significantly lower than that ofdirect band gap materials, as we can see in Fig. 12.4.While in the visible part of the spectrum c-Si absorbsless than the GaAs and InP, below 364 nm, it absorbs

107

106

105

104

103

102

101

100

200

GaAsInPGeSi

Abs

orpt

ion

coe

cien

t α (c

m-1

)

400 600 800 1000 1200 1400 1600 1800 2000

Wavelength λ (nm)

Figure 12.4: Absorption coefficients of different semiconduct-ors.



just as much as GaAs and InP, because there silicon hasa direct band to band transition as well. Germanium(Ge), also indicated in the figure, is an indirect bandgapmaterial, just like silicon. The bandgap of Ge is 0.67eV, which means it already starts to absorb light atwavelengths shorter than 1850 nm. In the visible partof the spectrum, germanium has some direct transitionsas well.

Let us now take another look at the design rules forsolar cells that we introduced in Section 10.4. First, welook at spectral utilisation. The c-Si band gap of 1.12 eVmeans that in theory we can generate a maximum shortcircuit current density of 45 mA per square centimeter.Let us now consider the second design rule, i.e. lighttrapping. First, we look at a wavelength around 800nm, where c-Si has an absorption coefficient of 1000cm−1. Using the Lambert-Beer law [Eq. (4.25)], we eas-ily can calculate that for realising an an absorption of90% of the incident light at 800 nm, the required ab-sorption path length is 23 µm. Secondly, we look at 970nm wavelength, where c-Si has an absorption coefficientof 100 cm−1. Hence, an absorption path length of 230µm is required to absorb 90% of the light. 230 µm isa typical thickness for silicon wafers. This calculationdemonstrates that the light trapping techniques becomeimportant for crystalline silicon absorber layers above awavelength of about 900 nm.

Let us now consider the design rule of bandgap utilisa-

Monocrystalline Multicrystalline

(a) (b)

Figure 12.5: Illustrating a (a) monocrystalline and a (b) mul-ticrystalline silicon wafer.

tion, which is determined by the recombination losses.As silicon is an indirect band gap material, only Augerrecombination and Shockley-Read-Hall (SRH) recom-bination will determine the open circuit voltage, whileradiative recombination can be neglected. Consider-ing SRH recombination, the recombination rate of thecharge carriers is related to the electrons trapped at de-fect states. When looking at the defect density in thebulk of silicon, we can differentiate between two ma-jor types of silicon wafers: monocrystalline silicon andmulticrystalline silicon, which is also called polycrystallinesilicon.

Monocrystalline silicon, also known as called single-crystalline silicon, is a crystalline solid, in which thecrystal lattice is continuous and unbroken without any


12.2. Production of silicon wafers 145

grain boundaries over the entire bulk, up to the edges.In contrast, polycrystalline silicon, often simply ab-breviated with polysilicon, is a material that consistsof many small crystalline grains, with random ori-entations. Between these grains grain boundaries arepresent. Figure 12.5 shows two pictures of monocrystal-line and multicrystalline wafers. While a monocrystal-line silicon wafer has one uniform color, in multicrys-talline silicon, the various grains are clearly visible forthe human eye. At the grain boundaries we find latticemismatches, resulting in many defects at these bound-aries. As a consequence, the charge carrier lifetime forpolycrystalline silicon is shorter than for monocrystal-line silicon, because of the SRH recombination. Themore grain boundaries in the material, the shorter thelifetime of the charge carriers. Hence, the grain sizeplays an important role in the recombination rate.

Figure 12.6 shows the relationship between the opencircuit voltage and the average grain size for varioussolar cells developed around the world, based on themulticrystalline wafers [37]. The larger the grain size,the longer the charge carrier lifetimes and the larger theband gap utilisation and hence the open circuit voltagewill be. On the right hand side of the graph the opencircuit voltages of various solar cells, based on mono-crystalline wafers, is shown. As monocrystalline sil-icon has no grain boundary, much larger open circuitvoltages can be obtained.

IMECBP PHASE

Tonen

DaidoISE

Mitsubishi

IMEC ETL

ASE/ISFHAstropower

Ope

n ci

rcui

t vol

tage

Voc

(mV)

750

700

650

600

550

500

450

400

35010-2 10-1 100 101 102 103 104

UNSWMPIISEIpe

SONYZAE

Canon

Average grain size (μm)

Figure 12.6: The relationship between the open circuit voltageand the average grain size [37]

12.2 Production of silicon wafers

After the initial considerations on designing c-Si solarcells, we now will discuss how monocrystalline andmulticrystalline silicon wafers can be produced. In Fig.12.7 we illustrate the production process of monocrys-talline silicon wafers.

The lowest quality of silicon is the so-called metallur-gical silicon, which is made from quartzite. Quartziteis a rock consisting of almost pure silicon dioxide(SiO2). For producing silicon the quartzite is moltenin a submerged-electrode arc furnace by heating it up



SiO2+2CSi+2CO

CO

Si+3HClH2+HSiCl3

HSiCl3

H2, Clcarbon

quartzite metallurgicalsilicon powder

H2HCl

arc furnace chemical reactor distillation

HSiCl3

chemical vapour deposition

polysiliconChozralski processoat zone processsiliconingot

sawing

silicon wafers

Siemens process

Figure 12.7: Illustrating the production process of monocrystalline silicon wafers.



to around 1900°C, as illustrated in Fig. 12.7. Then, themolten quartzite is mixed with carbon. As a carbonsource, a mixture of coal, coke and wood chips is used.The carbon then starts reacting with the SiO2. Sincethe reactions are rather complex, we will not discussthem in detail here. The overall reaction how ever canbe written as

SiO2 + 2 C→ Si + 2 CO (12.1)

As a result, carbon monoxide (CO) is formed, whichwill leave the furnace in the gas phase. In this way, thequartzite is purified from the silicon. After the reactionsare finished, the molten silicon that was created duringthe process is drawn off the furnace and solidified. Thepurity of metallurgic silicon, shown as a powder in Fig.12.7, is around 98% to 99%.

About 70% of the worldwide produced metallurgicalsilicon is used in the aluminium casting industry formake aluminium silicon alloys, which are used in auto-motive engine blocks. Around 30% are being used formake a variety of chemical products like silicones. Onlyaround 1% of metallurgical silicon is used as a rawproduct for making electronic grade silicon.

The silicon material with the next higher level of purityis called polysilicon. It is made from a powder of me-tallurgical silicon in the Siemens process. In the process,the metallurgical silicon is brought into a reactor and

exposed hydrogen chloride (HCl) at elevated temper-atures in presence of a catalyst. The silicon reacts withthe hydrogen chloride,

Si + 3 HCl→ H2 + HSiCl3, (12.2)

leading to the creation of trichlorosilane (HSiCl3). Thisis a molecule that contains one silicon atom, threechlorine atoms and one hydrogen atoms. Then, the tri-chlorosilane gas is cooled and liquified. Using distil-lation, impurities with boiling points higher or lowerthan HSiCl3 are removed. The purified trichlorosilaneis evaporated again in another reactor and mixed withhydrogen gas. There, the trichlorosilane is decomposedat hot rods of highly purified Si, which are at a hightemperature in between around 850°C and 1050°C. TheSi atoms are deposited on the rod whereas the chlorineand hydrogen atoms are desorbed from the rod sur-face back in to the gas phase. As a result a pure sil-icon material is grown. This method of depositing sil-icon on the rod is one example of chemical vapour depos-ition (CVD). As the exhaust gas still contains chlorosil-anes and hydrogen, these gasses are recycled and usedagain: Chlorosilane is liquified, distilled and reused.The hydrogen is cleaned and thereafter recycled back into the reactor. The Siemens process consumes a lot ofenergy.

Another method for producing poly silicon granules isusing Fluid Bed Reactors (not shown in Fig. 12.7). Thisprocess is operated at lower temperatures and con-



sumes much less energy. Polycrystalline silicon canhave an purity as high as 99.9999%, or in other wordsonly one out of million atoms is an atom different fromSi.

The last approach we briefly mention is that of upgradedmetallurgical silicon (not shown in Fig. 12.7). In this pro-cess metallurgical silicon is chemically refined by blow-ing gasses through the silicon melt removing the impur-ities. Although processing is cheap, the silicon purity islower than that achieved with the Siemens or the FluidBed reactor approaches.

Now we introduce two methods that are industriallyused for making monocrystalline silicon ingots, i.e. largecylinders of silicon that consist of one crystal only. Thismeans that inside the ingot no grain boundaries arepresent. Such a monocrystalline ingot and both themethods are sketched in Fig. 12.7.

The first method we discuss is based on the Czochral-ski process that was discovered by the Polish scientistJan Czochralski in 1916. In this method, highly purifiedsilicon is melted in a crucible at typical temperature of1500°c. Intentionally boron or phosphorus can be ad-ded for making p-doped or n-doped silicon, respect-ively. A seed crystal that is mounted on rotating shaft isdipped in to the molten silicon. The orientation of thisseed crystal is well defined; it has either is either 100or 111 oriented. The melt solidifies at the seed crystaland adopts the orientation of the crystal. The crystal is

rotating and pulled upwards slowly, allowing the form-ation of a large, single-crystal cylindrical column fromthe melt — the ingot. For successfully conducting theprocess, temperature gradients, the rate of pulling theshaft upwards and the rotational speed must be wellcontrolled. Due to improved process control through-out the years, nowadays ingots of diameters of 200 mmor even 300 mm with lengths of up to two meters canbe fabricated. To prevent the incorporation of impurit-ies, the Czochralski this process takes place in an inertatmosphere, like argon gas. The crucible is made fromquartz, which partly dissolves in the melt as well. Con-sequently, monocrystalline silicon made with the Czo-chralski method has a relatively high oxygen level.

The second method to make monocrystalline silicon isthe float zone process, which allows fabricating ingotswith extremely low densities of impurities like oxygenand carbon. As a source material, a polycrystalline rodmade with the Siemens process is used. The end of therod is heated up and melted using an induction coiloperating at radio frequency (RF). The molten part isthen brought in contact with the seed crystals, where itsolidifies again and adopts the orientation of the seedcrystal. Again 100 or 111 orientations are being used.As the molten zone is moved along the polysilicon rod,the single-crystal ingot is growing as well. Many im-purities remain in and move along with the moltenzone. Nowadays, during the process nitrogen is inten-tionally added in order to improve the control over mi-



crodefects and the mechanical strength of the wafers.One advantage of the float—zone technique is that themolten silicon is not in contact with other materials likequartz, as this is the case when using the Czochralskimethod. In the float-zone process the molten silicon isonly in contact with the inert gas like argon. The siliconcan be doped by adding doping gasses like diborane(B2H6) and phosphine (PH3) to the inert gas to get p-doped and n-doped silicon, respectively. The diameterof float-zone processed ingots generally is not largerthan 150 mm, as the size is limited by surface tensionsduring the growth.

Not only monocrystalline silicon ingots, but also mul-ticrystalline silicon ingots, which consist of many smallcrystalline grains, can be fabricated (not shown in Fig.12.7). This can be made by melting highly purified sil-icon in a dedicated crucible and pouring the moltensilicon in a cubic shaped growth crucible. There, the mol-ten silicon solidifies in to multi-crystalline ingot in aprocess called silicon casting. If both melting and solidi-fication is done in the same crucible it is referred to asdirectional solidification. Such multicrystalline ingots canhave a front surface area of up to 70 × 70 cm2 and aheight of up tp 25 cm.

Now, as we know how to produce monocrystalline andmulticrystalline ingots we will discuss how to makewafers out of them. The process that is used to make thewafers is sawing, as illustrated in Fig. 12.7. Logically,

sawing will damage the surface of the wafers. There-fore, this processing step is followed by an polishingstep. The biggest disadvantage of sawing is that a sig-nificant fraction of the silicon is lost as kerf loss, whichusually is determined by the thickness of the wire orsaw used for sawing. Usually, it is in the order of 100µm. As typical wafers used in modern solar cells havethicknesses in the order of 150 µm up to 200 µm, thekerf loss is very significant.

A completely different process for making siliconwafers is the silicon ribbon technique (not shown in Fig.Fig. 12.7). As this technique does not include any saw-ing step, no kerf loss occurs. In the silicon ribbon tech-nique a string is used that is resistant agains high tem-peratures. This string is pulled up from a silicon melt.The silicon solidifies on the string and hence a sheet ofcrystalline silicon is pulled out of the melt. Then, theribbon is cut into wafers. Before the wafers can be pro-cessed further in order to make solar cells, some surfacetreatments are required. The electronic quality of ribbonsilicon is not as high as that of monocrystalline.

To summarise, we discussed how to make metallurgicalsilicon out of quartzite and how to fabricate polysilicon.We have seen that monocrystalline ingots are madeusing either the Czochralski or the float-zone process,while multicrystalline ingots are made using a castingmethod. Wafers are fabricated by sawing these ingots.A method that does not have any kerf losses in the rib-



≈ 20

0 μm

front contact (metal grid)

serial connections (to the back contact of the next cell)

n+-type emitterp+-type layer

antireectivecoating

p-type wafer

back contact

Figure 12.8: Scheme of a modern crystalline silicon cell.

bon silicon approach.

12.3 Fabricating solar cells

Before we start with the actual discussion on how c-Sisolar cells are fabricated, we briefly discuss the operat-ing principles of c-Si solar cells. Especially, we discussseveral technical aspects that play an important role inthe collection of the light, excitation of charge carriersand the reduction of optical losses.

In Chapter 8 we discussed how an illuminated p-njunction can operate as a solar cell. In the illustrations

Cell depth x (µm)

Gen

erat

ion

rate

(m-3

)

0 50 100 150 200 250

1022

1021

1020

1019

1018

1017

300

Figure 12.9: The generation profile in crystalline silicon.

used there, both the p-doped and n-doped regionshave the same thickness. This is not the case in realc-Si devices. For example, the most conventional typeof c-Si solar cells is built from a p-type silicon wafer,as sketched in Fig. 12.8. However, the n-type layer onthe top of the p-wafer is much thinner than the wafer;it typically has a thickness of around 1 µm. Often, thislayer is called the emitter layer. As mentioned before,the whole wafer has typically thicknesses in between100 and 300 µm.

Because of the Lambert-Beer law [Eq. (4.25)], the intens-ity if light inside the silicon bulk decays exponentially


12.3. Fabricating solar cells 151

with the depth. Hence also the generation profile willshow an exponential decay as discussed in Section 7.2.1and shown in Fig. 12.9. Therefore, the largest fractionof the light is absorbed close to the front surface of thesolar cell. In the first 10 µm by far the most charge car-riers are generated. By making the front emitter layervery thin, a large fraction of the light excited chargecarriers generated by the incoming light are createdwithin the diffusion length of the p-n junction.

The emitter layer can be created with a solid state diffu-sion process. In this process, the wafers are placed in afurnace at around 850°C, where the dopant atoms arepresent in atmosphere in the form of phosphine (PH3).These atoms react at these high temperatures with thesurface. Further, at these temperatures the phosphineatoms are mobile in the silicon crystal. Based on Fick’slaw, the therefore can diffuse into the wafer,

J(x) = −Ddndx

. (12.3)

This law states that the particle flux J is proportionalto the negative gradient of the particle density n. D isthe diffusion constant. The diffusion process has to becontrolled such that the dopants penetrate into the solidto establish the desired emitter thickness.

Now we are discuss the collection of charger carriersin a crystalline silicon solar cell. The crucial compon-ents that play a role in charge collection are the emit-ter layer, the metal front contacts and the back contact.

First, the emitter layer: At the p-n junction the minor-ity charge carriers, which are excited by the light, areseparated at the p-n junction: the minority electrons inthe p-layer drift to the n-layer, where they have to becollected. Since the silicon n-emitter is not sufficientlyconductive we have to use the much more conductivemetal contacts, which are placed on top of the emitterlayer. Very often, the metal contacts are made of thecheap metal aluminium.

This means that the electrons have to diffuse laterallythrough the emitter layer to the electric front contactto be collected. Which factors are important for a goodtransport of the electrons to the contact? First, the life-time of the charge carriers needs to be high. A highlifetime guarantees large open circuit voltages, or inother words the optimal utilisation of the band gap en-ergy. For increasing the lifetime, recombination lossesmust be reduced as much as possible.

Recombination not only reduces the Voc, it also limitsthe collected current. As mentioned earlier, in silicontwo recombination mechanisms are present: Shockley-Read-Hall recombination and Auger recombination.First, let us take a look at Shockley-Read-Hall recom-bination. A bare c-Si surface contains many defects,because the surface silicon atoms have some valenceelectrons that cannot make molecular orbitals due tothe absence of neighbouring atoms. These valence or-bitals containing only one electron at the surface act


Felipe

Felipe 10/01/2014: we are going to discuss - we will discuss


like defects. They are also called dangling bonds. Atthe dangling bonds, the charge carriers can recombinethrough the SRH process. The probability and speedat which charge carriers can recombine is usually ex-pressed in terms of the surface recombination velocity.Since a large fraction of the charge carriers are gener-ated close to the front surface, a high surface recom-bination velocity at the emitter front surface will leadto significant charge carriers losses and consequentlylower short-circuit current densities. In high-qualitymonocrystalline silicon wafers, for example, no defect-rich boundaries are present in the bulk. Thus, the life-time of charge carriers is limited by the recombinationprocesses at the wafer surface.

In order to reduce the surface recombination two ap-proaches are used. First, the defect concentration onthe surface is reduced by depositing a thin layer of adifferent material on top of the surface. This materialpartially restores the bonding environment of the sil-icon atoms. In addition, the material must be an insu-lator, it must force the electrons to remain in and movethrough the emitter layer. Typical materials used forthis chemical passivation layers are silicon oxides (SiOx)and silicon nitride (SixNy).1 A silicon oxide layer can beformed by heating up the silicon surface in an oxygen-rich atmosphere, leading to the oxidation of the surface

1Stoichiometric silicon oxide and silicon nitride have the chemical formula SiO2and Si3N4, respectively. As the layers used for passivation are not stoichiometric, weindicate the elementary fractions with x and y.

silicon atoms. Si3N4 can be deposited using plasma-enhanced chemical vapour deposition (PE-CVD) thatwe will discuss in more detail in Chapter 13.

A second approach for reducing the surface recombin-ation velocity is to reduce the minority charge carrierdensity near the surface. As the surface recombinationvelocity is limited by the minority charge carriers dens-ity, it is beneficial to have the minority charge carrierdensity at the surface as low as possible. By increas-ing the doping of the emitter layer, the density of theminority charge carriers can be reduced. This results inlower surface recombination velocities. However, this isin competition with the diffusion length of the minor-ity charge carriers. The blue part of the solar spectrumleads to generation of many charge carrier very closeto the surface, i.e. in the emitter layer. For utilisingthese light excited minority charge carriers, the diffu-sion length of the holes has to be large enough to reachthe depletion zone at the p-n junction as indicated bythe blue arrow. However, increasing the doping levelsleads to a decreasing diffusion length of the minorityholes in the emitter. Therefore, too high doping levelsor too thick emitter layers would result in a poor blueresponse or – in other words – low External QuantumEfficiency (see Chapter 9) values in the blue part of thespectrum. Such an emitter layer could be called “deadlayer” as the light excited minority charge carriers cannot be collected.



As a next step, we take a closer look at the metal-emitter interface. Because electrons must be easily con-ducted from the emitter to the metal, insulating pas-sivation layers such as SiOx or SixNy cannot be used.Therefore, the metal-semiconductor interface has moredefects and hence an undesirably high interface recom-bination velocity. Additionally, a metal-semiconductorjunction induces a barrier for the majority charge car-riers. It is, however, out of the scope of this book todiscuss the physics of this barrier in more detail. Wejust must keep in mind that this high barrier will giverise to a higher contact resistance. Again high dop-ing levels can reduce the recombination velocity at themetal-semiconductor interface and reduce the contactresistance. In order to minimise the recombination atthe interface defects as much as possible, the area of themetal-semiconductor interface must be minimised andthe emitter directly below the interface should be heav-ily doped, which is indicated with n++. The sides ofthe metal contacts are buried in the insulting passiva-tion layer. The area below the contact has been heavilydoped. These two approaches reduce the recombinationand collection losses at the metal contact.

The solar cell shown in Fig. 12.8 has a classic metal gridpattern on top. We see two high ways for the electronsin the middle of top surface of the solar cell. They arecalled busbars. The small stripes going from the busbarsto the edges of the solar cell are called the fingers. Letnow R be the resistance of such a finger. If L, W, and H

are the length, width and height of the fingers, respect-ively, and ρ is the resistivity of the metal, R is given by

R = ρL

WH. (12.4)

This equation shows that the longer the path length foran electron, the larger the resistance the electrons exper-ience. Further, he smaller the cross-section (W · H) ofthe finger, the larger the resistance will be. Note, thatthe resistance of the contacts will act as a series resist-ance in the equivalent electric circuit. Larger series res-istance will result in lower fill factors of the solar cell asdiscussed in Chapter 9. Hence, electrically large cross-sections of the fingers are desirable.

To arrive at a metallic contact, electrons in the emitterhave to travel laterally through the emitter. Because ofthe resistivity of the n-type silicon, the charge carriersin the emitter layer also experience a resistance. It canbe shown that the power loss due to the resistivity ofthe emitter layer scales with the spacing between twofingers to the power 3. EXERCISE???

As the metal contacts are at the front surface, they actas unwelcome shading objects, or in other words lightincident on the metallic front contact area cannot beabsorbed in the PV-active layers. Therefore, the con-tact area should be kept as small as possible, whichis in competition with the fact that the finger crosssection should be maximised. So basically, the finger


anonymous

anonymous 10/10/2014: Is there an exercise here?


Pow

er lo

ss

Finger spacing

Shadow

Total

Resistivecontact

Resistiveemitter

Finger width

Resistive

Total

Shadow

Pow

er lo

ss

(a) (b)

Figure 12.10: Power loss due to (a) spacing and (b) widths ofthe metallic fingers on top of a c-Si solar cell.

height should be as high as possible while the fingerwidth should you would like to have a small as pos-sible width and a high as possible height to complywith these requirements.

We can see that several effects compete with other.Fig. 12.10 (a) shows the relationship between powerlosses and finger spacing. With increasing finger spa-cing the power losses of the solar cell decreases becauseof less shading. On the other hand, the losses due tothe increased resistivity in the emitter layer increase.Hence, there is an optimal spacing distance at that thepower loss is minimal. A similar plot can be made forthe power loss in dependence of the finger width; it is

shown in Fig. 12.10 (b). The larger W the larger shad-ing losses will be. But with increasing W the resistancedecreases. Again, here an optimum exists at which thepower losses are minimal. We see that optimising thefront contact pattern is a complex interplay between thefinger width and spacing.

For designing the back contact we find similar issues.Just as the electrons are to be collected in the front n-type layer, the holes are to be collected at the back con-tact. Electrons are the only charge carriers that exist inmetal. Therefore the holes have to recombine with theelectrons at the back semiconductor-metal interface. Ifthe distance between the p-n interface and the backcontact is smaller than the typical diffusion length ofthe minority electrons, the minority electrons can belost at the defects of the back contact interface becauseof SRH recombination

Several methods can be used to reduce this loss. First,the area between the metal contact and the semicon-ductor can be reduced, just as for the front contact. Todo this, point contacts can be used, while the rest of therear surface is passivated by an insulating passivationlayer, similar as we already discussed for the emitterfront surface. The recombination loss of electrons at theback contact can be further reduced by a introducinga back surface field. Above the point contacts a highlydoped p-doped region is placed, which is indicated byp+.



AlAl p+

p-typen-type

p+-type

Figure 12.11: Effect of the back surface field illustrated in aband diagram.

To understand how the back surface field works, wetake a look at the band diagram shown in Fig. 12.11.The interface between the normally doped p-regionand the highly doped p+-region acts like an n-p junc-tion. Here, this junctions acts as a barrier that preventsminority electrons in the p-region from diffusing to theback surface. Further, the space charged field behaveslike a passivation layer for the defects at the back con-tact interface and allows to have higher minority carrierdensities in the p-doped bulk.

After this thorough discussion on managing the chargecarriers, we now take a closer look on managing thephotons in a crystalline silicon solar cell. Several opticalloss mechanisms must be addressed. These are shading,reflection losses, parasitic absorption losses in the non-PV active layers, and transmission through the back ofthe solar cell. As already mentioned, shading losses arecaused by the metallic front contact grid.

Secondly, reflection from the front surface is an im-portant loss mechanisms. We briefly mention two ap-proaches to design anti-reflective coatings (ARC) for re-ducing these losses. First, reflection can be minim-ised based on the Rayleigh film principle: by putting afilm with a refractive index smaller than that of siliconwafer between the cell and the wafer losses can be re-duced. The optimal value for the refractive index of theintermediate layer equals the square root of the product



of refractive indexes of the two other media,

nopt =√

nairnSi. (12.5)

At a wavelength of 500 nm, the optimal refractive indexfor a layer in-between air and silicon is 2.1. Note thatin practice a solar cell in encapsulated under a glass orpolymer plate, which will have a beneficial effect on therefractive index grading as well, reducing the reflectionlosses further. Secondly, using the concept of destruct-ive interference, the thickness and refractive index ofan antireflection coating can be chosen such that in acertain wavelength range the reflection is minimised.This happens when the light reflected from the air-ARCinterface is in anti-phase with the light reflected fromthe ARC-Si interface, as discussed in Chapter 4. We findthat the thickness of such a layer should be a quarter ofthe wavelength in the layer,

dARC =λ0

4n, (12.6)

where λ0 denotes the wavelength in vacuo and n is therefractive index of the antireflective layer. For a refract-ive index of 2.1, a layer thickness of 60 nm would leadto destructive interference at 500 nm. As mentionedearlier, a typical material for passivation is silicon ni-tride. we have discussed earlier, one of the typical pas-sivation layers of standard crystalline silicon is siliconnitride. Fig. 12.12 (a) shows a multicrystalline waferwithout any ARC. It appears silverish, which means

(a) (b)

Figure 12.12: A multicrystalline silicon wafer (a) without and(b) with an antireflective coating from silicon nitride.

that it is highly reflective. In Fig. 12.12 (b) a similarwafer is shown after it was passivated with SixNy. Wesee that it has a dark-blue appearance, hence its reflec-tion is much lower. Interestingly enough, the refract-ive index of SixNy at 500 nm is in the range of 2 to 2.2,close to the optimum mentioned earlier. The blue ap-pearance indicates that reflection in the blue is strongerthan at other wavelengths.

The reflection losses also can be minimised by textur-ing the wafer surface. Light that is reflected at the tex-tured surface can be reflected at angles in which thetrajectory of the light ray is incident somewhere else onthe surface, where it still can be coupled into the sil-icon. Additionally, the scattering at textured surfaceswill couple the light under angles different from the


12.4. High-efficiency concepts 157

(a) (b)

Figure 12.13: (a) A textured multicrystalline Si wafer and (b) awafer with a black silicon surface.

interface normal into to the wafer. Therefore, the av-erage path length of the light in the absorber will beincreased, which leads to stronger absorption of thelight. For realising the textures, usually wet etching ap-proaches are used. Interestingly enough, 100 surfacesare etched much faster than the 111 surfaces. Such a be-haviour is called unisotropic etching. Thus, when a 100wafer is etched, the 100 surface will be etched away,while the (slanted) 111 surface facets will remain. There-fore, we obtain a wafer with a pyramid structure con-sisting of 111 facets only, as illustrated in Fig. 12.13(a) that shows such a wafer that was processed in theDIMES Research Center in Delft. With a different etch-ing approach, it is even possible to make silicon to ap-pear completely black, as shown in Fig. 12.13 (b). Thisis called black silicon.

n+ np-type wafer

oxide

p+

oxiderear contact

“ inverted ” pyramidsnger

p+

p+

p+

Figure 12.14: Illustrating the structure of a PERL solar cell.

12.4 High-efficiency concepts

In the last section of this chapter we discuss three ex-amples of high-efficiency concepts based on crystallinesilicon technology. As already discussed earlier, differ-ent types of silicon wafers with different qualities canbe used. Naturally, for achieving the highest efficien-cies, the bulk recombination must be as small as pos-sible. Therefore the high efficiency concepts are basedon monocrystalline wafers.



12.4.1 The PERL concept

The first high efficiency concept was developed in thelate 1980s and the early 1990s at the University of NewSouth Wales in the group of Martin Green. in the lateeighties and early nineties. Figure 12.14 shows an illus-tration of the PERL concept, which uses a p-type floatzone silicon wafer. With this concept, conversion effi-ciencies of 25% were achieved [38]. The abbreviationPERL stands for Passivated Emitter Rear Locally. This ab-breviation indicates two important concepts that havebeen integrated in this technology: First, the opticallosses of the PERL solar cell at the front side are min-imised using three techniques:

1. The top surface of the solar cell is textured withinverted-pyramid structures. This microscopic tex-ture allows a fraction of the reflected light to be in-cident on the front surface for a second time, whichenhances the total amount of light coupled in tothe solar cell.

2. The inverted pyramid structure is covered with adouble-layer anti-reflection coating (ARC), whichresults in an extremely low top surface reflection.Often a double layer coating of magnesium flu-oride (MgF2) and zinc-sulfide (ZnS) is used as anantireflection coating.

3. The contact area at the front side has to be as smallas possible, to reduce the shading losses. In the

PERL concept the very thin and fine metal fingersare processed using photolithography technology.

Secondly, the emitter layer is smartly designed. As dis-cussed in the previous sections, the emitter should behighly doped underneath the contacts, which in thePERL concept is achieved by heavily phosphorus dif-fused regions. The rest of the emitter is moderatelydoped, or in other words lightly diffused, to preservean excellent blue response. The emitter is passivatedwith a silicon oxide layer on top of the emitter to sup-press surface recombination as much as possible. Withthe PERL concept, the surface recombination velocitycould be reduced so far that open circuit voltages withvalues of above 700 mV could be obtained.

At the rear surface of the solar cell, point contacts havebeen used in combination with thermal oxide passiva-tion layer, which passivates the non-contacted area, andhence reduces the undesirable surface recombination.A highly doped boron region, created by local Borondiffusion, operates as a local back surface field, to limitthe recombination of the minority electrons at the metalback contact.

The PERL concept includes some expensive processingsteps. Therefore, the Chinese company Suntech de-veloped in collaboration with University of New SouthWales a more commercially viable crystalline siliconwafer technology, which is inspired on the PERL cellconfiguration.



12.4.2 The interdigitated back contact(IBC) solar cell

A second successful cell concept is the interdigitatedback contact (IBC) solar cell. The main idea of the IBCconcept is to have no shading losses at the front metalcontact grid at all. All the contacts responsible for col-lecting of charge carriers at the n- and p-side are posi-tioned at the back of the crystalline wafer solar cell. Asketch of such a solar cell is shown in Fig. 12.15 (a).

An advantage of the IBC concept is that monocrystal-line float-zone n-type wafers can be used. This is in-teresting because n-type wafers have some interestingadvantages with respect to p-type wafers. First, then-type wafers do not suffer from light induced de-gradation. In p-type wafers both boron and oxygenare present, which under light exposure start to makecomplexes that act like defects. This light induced de-gradation causes a reduction of the power output with2-3% after the first weeks of installation. No such ef-fect is present in n-type wafers. The second advantageis that n-type silicon is not that sensitive for impuritieslike for example iron impurities. As a result, less ef-forts have to be made to fabricate high quality n-typesilicon, and thus high quality n-type silicon can be pro-cessed cheaper than p-type silicon. On the other hand,p-doped wafers have the advantage that the boron dop-ing is more homogeneously distributed across the waferas this is possible for n-type wafers. This means that

n+-dopedfront diusion

PassivatingSiO2 layer

Backside gridlines

Localizedcontacts

n-type silicon (240 μm)

Backsidemirror

p+ p+ p+n+n+ n+

ARCPassivating SiO2 layer

(a)

(b)

Figure 12.15: (a) Illustrating the structure of an IBC solar celland (b) the contacts on its back.



within one n-type wafer the electronic properties canvary, which lowers the yield of solar cell productionbased on n-type monocrystalline wafers.

Besides that IBC cells are made from n-type wafers,they lack one large p-n junction. Instead, IBC cells havemany localised junctions. The holes are separated at ajunction between the p+ silicon and the n-type silicon,whereas the electrons are collected using n+-type sil-icon. The semiconductor-metal interfaces are kept assmall as possible in order to reduce the undesired re-combination at this defect-rich interface. Another ad-vantage is that the cross section of the metal fingers canbe made much larger, because they are at the back andtherefore do not cause any shading losses. Thus, resist-ive losses at the metallic contacts can be reduced. Sinceboth electric contacts are on the back side, it containstwo metal grids, as illustrated in Fig. 12.15 (b). The pas-sivation layer should made from a low-refractive-indexmaterial such that it operates like a backside mirror.It will reflect the light above 900 nm, which is not ab-sorbed during the first pass back in to the absorberlayer. Thus, this layer enhances the absorption pathlength.

At the front side of the IBC cell losses of light-excitedcharge carriers due to surface recombination is sup-pressed by a front surface field similar to the back sur-face field discussed earlier. This field is created witha highly doped n+ region at the front of the surface.

Thus, a n+-n junction is created that acts like a n-pjunction. It will act as a barrier that prevents the light-excited minority holes in the n-region from diffusing to-wards the front surface. The front surface field behaveslike a passivation for the defects at the front interfaceand allows to have higher levels for the hole minoritydensity in the n-doped bulk.

Reflective losses at the front side are reduced in a sim-ilar way as for PERL solar cells: Deposition of doublelayered antireflection coatings and texturing of the frontsurface.

The IBC concept is commercialised by the U.S. com-pany SunPower Corp. They have achieved high solar cellefficiencies of 24.2%.

12.4.3 Hetero junction (HIT) solar cells

The third high-efficiency concept are heterojunction withintrinsic thin layer (HIT) solar cells. Before we discussthe technological details, we briefly recall the principlesof heterojunctions, that already were discussed in Sec-tion 8.2.

Homojunctions, which are present in all the c-Si solarcell types discussed so far in this chapter, are fabricatedby different doping types within the same semicon-ductor material. Hence, the band gap in the p- and n-regions is the same. A junction consisting of a p-doped



EgP

EVP

EFP

ΔEC

eVbin

n-type FZ c-Si p-type a-Si:H

ECP

EVn

ECn

EgnΔEC

EFn

Figure 12.16: Illustrating the band diagram of a heterojunc-tion.

semiconductor material and an n-doped semiconductormade from another material is called a heterojunction.In HIT cells, the heterojunction is formed in betweentwo different silicon-based semiconductor materials: Onthe one hand, we use a n-type float zone monocrystal-line silicon wafer. The other material is hydrogenatedamorphous silicon (a-Si:H), that we will discussed inmore detail in Section 13.3. a-Si:H is a silicon mater-ial in which the atoms are not ordered in a crystallinelattice but in a disordered lattice. For the moment weonly must keep in mind that a-Si:H has a band gap ofaround 1.7 eV which is considerably higher than that ofc-Si (1.12 eV).

In Fig. 12.16 a band diagram of a heterojunction in

p/i a-Si:Htextured n-type c-Si wafer

i/n a-Si:H

ngers

Figure 12.17: Illustrating the structure of a HIT solar cell.

between n-doped crystalline silicon and p-dopedamorphous silicon in the dark and thermal equilib-rium is sketched. We see that next to the induced fieldbecause of the space charge region, some local energysteps are introduced. These steps are caused by the twodifferent bandgaps for the p and n regions. The valenceband is higher positioned in the p-type amorphous sil-icon than in the n-type crystalline silicon. This will al-low the minority charge carriers in the n-type c-Si, theholes, to drift to the p-type silicon. However, the holesexperience a small barrier. While they could not travelacross such a barrier in classical mechanics, quantummechanics allows them to tunnel across this barrier.

Let us now take a closer look to HIT solar cells. TheHIT concept was developed by the Japanese company



Sanyo, that is currently a part of the Japanese PanasonicCorp. As we can see in Fig. 12.17, the HIT cell configur-ation has two junctions: The junction at the front sideis formed using a thin layer of only 5 nanometers ofintrinsic amorphous silicon which is indicated by thecolor red. A thin layer of p-doped amorphous siliconis deposited on top and here indicated with the colorblue. The heterojunction forces the holes to drift to thep-layer. At the rear surface a similar junction is made:First, a thin layer of intrinsic amorphous silicon is de-posited on the wafer surface, indicated by red. On topof the intrinsic layer an n-doped amorphous silicon isdeposited indicated, which is indicated by the yellowcolor.

As discussed earlier, for high quality wafers, like thisn-type float zone monocrystalline silicon wafer, the re-combination of charge carriers at the surface determ-ines the charge carrier lifetime. The advantage of theHIT concept is that the amorphous silicon acts as a verygood passivation layer. With this approach the highestpossible charge carrier lifetimes are accomplished.Thus, c-Si wafer based heterojunction solar cells havethe highest achieved open circuit voltage among thedifferent crystalline silicon technologies. The current re-cord cell2 has an open circuit voltage of 0.74 V and enefficiency of 25.6%.

How do the charge carriers travel to the contacts? The

2As of May 2014.

conductive properties of the p-doped amorphous siliconare relatively poor. While in the homojunction solarcells the lateral diffusion to the contacts takes place inthe emitter layer, in a HIT solar cell this occurs througha transparent conducting oxide (TCO) material, like in-dium tinoxide (ITO), which is deposited on top of thep-doped layer. TCOs are discussed in more detail inSection 13.1.

The same contacting scheme is applied at the n-typeback side. This means that this solar cell can be usedin a bifacial configuration: it can collect light from thefront, and scattered and diffuse light falling on theback-side of the solar cell. Another important advant-age of the HIT technology is that the amorphous sil-icon layers are deposited using cheapplasma enhancedchemical vapour deposition (PE-CVD) technology atlow temperatures, not higher than 200°C. Therefore,making the front surface and back surface fields in HITsolar cells is very cheap.

In this chapter we discussed many aspects of crystallinesilicon solar cell technology. Before the solar cells canbe installed, they must be packed as a PV module. Howsuch modules are made is discussed in Part IV on PVsystems in Section 17.1.


13Thin-film solar cells

In Chapter 12 we have discussed the PV technologybased on c-Si wafers, which currently is by far the dom-inant PV technology. It is very likely that it will staydominant for a long time to go.

In this chapter we will look at an alternative family oftechnologies, namely thin film technologies, also referredto as the second generation PV technology. These solarcells are made from films that are much thinner thanthe wafers that form the base for first generation PV.According to Chopra et al. [40], ‘a thin film is a film thatis created ab initio by the random nucleation process ofindividually condensing/reacting atomic/ionic/ mo-lecular species on a substrate. The structural, chemical,metallurgical and physical properties of such a materialare strongly dependent on a large number of deposition

parameters and may also be thickness dependent.’

Thin-film solar cells were expected to become cheaperthan first generation solar cells. However, due to thecurrent price decline in wafer based solar cells thin-filmsolar cells have not yet become interesting from an eco-nomic point of view yet.1 In general thin-film cells havea lower efficiency than c-Si solar cells. GaAs is an ex-ception to this rule of thumb [38]. In contrast to waferbased silicon solar cells that are self-supporting, thin-film solar cells require a carrier that gives them mech-anical stability. Usual carrier materials are glass, stain-less steel or polymer foils. It is thus possible to produceflexible thin-film solar cells.

1According to the PHOTON module price index, the price for wafer-basedmodules has decreased around 40% within one year as of 25 May 2012 [41].

163


164 13. Thin-film solar cells

0 10 20 30 40 50 60 70 80 9010-6

10-3

100

103

106

109

Abu

ndan

ce (a

tom

s of

ele

men

t per

106 a

tom

s of

Si)

Atomic number Z

H

Li

BeB

N

C

O

F

Major industrial metals in redPrecious metals in purpleRare earth elements in blue

Na

Si

MgPS

Cl

Al

KCa

Sc

Ti

VCr

Mn

Fe

Rock-forming elements

Rarest "metals"

Co Ni

Cu ZnGa

GeAs

Se

Br

Rb

Sr

Y

Zr

Nb

Mo

TePd

AgSbCd I

In

Sn

Rh

Ru

Ba

Cs

LaNd

Ce

Pr

Re

TmHo

Yb

Lu

IrOs

HfErGd

Eu

PtAu

TaDy

Tb

Sm

Hg

W Tl

Pb

Bi

ThU

Figure 13.1: The abundance of elements (as atomic fraction)in the Earth’s upper crust. Data provided by the USGS [39].

In thin-film solar cells the active semiconductor layersare sandwiched between a transparent conductive ox-ide (TCO) layer and the electric back contact. Often aback reflector is introduced at the back of the cell in or-der to minimise transmissive solar cell losses. As wewill see in this chapter, many different semiconductorsare used for thin-film solar cells. Figure 13.1 shows theabundance of metals in the Earth’s crust. Some semi-conductors require very rare elements such as indium(In), selenium (Se), or tellurium (Te). For terawatt scalephotovoltaics, solar cells should be based on abundantelements only.

Thin-film PV technologies easily can fill a book on itsown, see for example the book edited by Poortmansand Arkhipov [42]. In this chapter we therefore onlycan give a general introduction into the different thin-film technologies. We will focus on the working prin-ciples of the various devices, the current status and thefuture challenges of the various technologies. But beforewe start with this discussion, we will begin with a shortintroduction on transparent conducting oxides (TCOs).

13.1 Transparent conductingoxides

Due to the paramount importance of the TCO layer forthe solar cell performance we briefly discuss its main


13.1. Transparent conducting oxides 165

properties. The TCO layer acts as electric front con-tact of the solar cell. Furthermore, it guides the incid-ent light to the active layers. It therefore should be bothhighly conductive and highly transparent in the activewavelength range. The first resistance measurements onthin-films of what we nowadays call TCOs were pub-lished by Bädeker in 1907 [43].

Typical TCO layers are made from fluor-doped tin ox-ide (SnO2:F), aluminium-doped zinc oxide (ZnO:Al),boron-doped zinc oxide (ZnO:B), hydrogen-doped (hy-drogenated) indium oxide (In2O3:H) [44], and indiumtin oxide, which is a mixture of about 90% indiumoxide (In2O3) and 10% tin oxide (SnO2). These filmsare processed using either sputtering (ZnO:Al), low-pressure-chemical vapour deposition (LP-CVD, used e.g.for ZnO:B), metal-organic chemical vapour deposition(MO-CVD), or atmospheric-pressure chemical vapourdeposition (AP-CVD, used e.g. for SnO2:F).

Figure 13.2 shows the transmission, reflection and ab-sorption spectra of a flat ZnO:Al layer. Following Kluth,we divide this spectrum into three parts [45]: For shortwavelength, the transmission is very low due to thehigh absorption of light with energies higher than thebandgap. For longer wavelength, with photon-energiesbelow the bandgap, the transmission is very high. Wehere see interference fringes that can be used to determ-ine the film thickness. After a broad highly transmissivewavelength band, the absorption increases again. This

transmission reflection

absorption

carrier absorption

increasedfree

fringesinterference

plas

ma

wav

elen

gth

λ p

due

toTC

O-b

andg

appa

rasi

ticab

sorp

tion

Wavelength (nm)

T,R,

A (–

)

2500200015001000500

1

0.8

0.6

0.4

0.2

0

Figure 13.2: Transmission, reflection and absorption of aZnO:Al layer (d = 880 nm).



absorption is called free carrier absorption an can beexplained with the Drude theory of metals that was de-veloped in 1900 [46, 47]. In this model, the frequency-dependent electric permittivity is given by

ε(ω) =[n(ω)− ik(ω)

]2= 1 + χ(ω) = 1−

ω2p

ω2 + i ωt

,(13.1)

where χ(ω) is the dielectric susceptibility, t is the relax-ation time,2 n − ik is the complex refractive index andωp denotes the plasma frequency that is given by

ωp =Ne2

ε0m∗2e. (13.2)

Here, N is the density of free charge carriers, e is theelementary charge, ε0 is the permittivity of vacuum andm∗e is the effective electron mass in the TCO layer.

The real and imaginary part of the susceptibility aregiven by

(<χ)(ω) = −ω2p

t2

ω2t2 + 1, (13.3a)

(=χ)(ω) = ω2p

t/ω

ω2t2 + 1. (13.3b)

2The relaxation time denotes the average time between two collisions, i.e. twoabrupt changes of velocity, of the electrons.

If ωt 1, ε can be simplified to

ε(ω) ≈ 1−ω2

p

ω2 , (13.4)

while the imaginary part is negligible.

If this approximation is valid around ωp, the material istransparent for ω > ωp (ε > 0). For ω < ωp, ε becomesnegative, i.e. the refractive index is purely imaginaryand the material therefore has a reflectivity of 1. In thiscase, the material changes dramatically from transpar-ent to reflective, as ωp is crossed.

If the approximation ωt 1 is not valid, =χ cannotbe neglected. The imaginary part will increase withdecreasing frequency, i.e. with increasing wavelengthsthe absorption increases. For wavelengths longer thanthe plasma wavelength the material becomes more re-flective, what we also see in Fig. 13.2. For applicationin solar cells, the TCO should be highly transparent inthe active region of the absorber. Therefore the plasma-wavelength should at least be longer than the bandgapwavelength of the absorber. On the other hand theplasma frequency is proportional to the free carrierdensity N. A longer plasma wavelength therefore cor-responds to a lower N. Finding an optimum betweenhigh transparency and high carrier densities is an im-portant issue in designing TCOs for solar cell applica-tions.

Even though the Drude model gives a good approxima-


13.1. Transparent conducting oxides 167

tion of the free carrier related phenomena in TCOs, thismodel often is too simple. Therefore several authorsused extended Drude models with more parameters[48–50].

Of all TCO materials currently available, the trade offbetween transparency and conductivity is best for in-dium tin oxide [51]. However, indium is a rare earthelement with a very low abundance of 0.05 ppm in theEarth’s crust, similar to the abundance of silver (0.07ppm) and mercury (0.04 ppm) [52], which makes it lesspreferable for cheap large-scale PV applications. There-fore other TCO materials are thoroughly investigatedand used in industry. Amongst them are aluminiumdoped zinc oxide, boron doped zinc oxide and fluorinedoped tin oxide. The abundances of the used elementsare: aluminium: 7.96%, zinc: 65 ppm, boron: 11 ppm,fluorine: 525 ppm, and tin: 2.3 ppm [52].

The morphology of selected TCO samples

As we have briefly mentioned in Chapter 10 and willdiscuss in more detail in Chapter ??, light managementis crucial for designing solar cells. Very often, the TCOlayers in thin-film solar cells have nano-textured sur-faces. The incident light is scattered at these interfacessuch that the average photon path length in the ab-sorber layer is enhanced. Thus, more light can be ab-sorbed and the photocurrent density can be increased.

However, the texture may also influence the growth oflayers that are deposited onto the TCO and thus alterthe electrical properties, which may lead to a reductionin Voc and the fill factor. Further, the increased interfacearea due to the texture may lead to more surface recom-bination and also reduce the voltage.

Figure 13.3 shows the surface morphology, rms rough-ness σr and correlation length `c of some nano-texturedTCO samples made from three different materials.Fluorine-doped AP-CVD tin oxide (SnO2:F) of Asahi U-type [54] and boron-doped LP-CVD zinc-oxide (ZnO:B)of so-called ‘B-type’ from PV-LAB of the École polytech-nique fédérale de Lausanne (EPFL), Switzerland [55],obtain their nano-structure already during the depos-ition process. The nano-structure is due to the crystalgrowth of the TCO layers and has a pyramid-like shape.

In contrast, RF-sputtered, aluminium-doped zinc oxide(ZnO:Al) is flat after deposition (σr ≈ 3 nm, depend-ing on the deposition condition als higher). To obtain anano-texture, ZnO:Al is etched in a 0.5% HCl solution[56, 57]. This etching process leads to crater-like features.The lateral size of these craters is influenced by thegrain size of the zinc oxide crystals. Fig. 13.3 (c) and(d) shows etched ZnO:Al with broad craters. Fig. 13.3(e) and (f) shows etched ZnO:Al with narrow craters.

The surface morphologies shown in Fig. 13.3 were ob-tained with atomic force microscopy (AFM). Also the twostatistical parameters rms roughness σr and correlation



0

290

z (n

m)

2 μm

760

0

z (n

m)

2 μm0.0

1.3z

(μm

)

2 μm

390

0

z (n

m)

2 μm

510

0

z (n

m)

2 μm

290

0

z (n

m)

2 μm

(a) (c) (e)

(b) (d) (f)

SnO2:F (Asahi-U) σr ≈ 40 nm, ℓc ≈ 175 nm

ZnO:B σr ≈ 220 nm, ℓc ≈ 470 nm

ZnO:Al (40 s, broad craters) σr ≈ 60 nm, ℓc ≈ 520 nm

ZnO:Al (60 s, broad craters) σr ≈ 90 nm, ℓc ≈ 625 nm

ZnO:Al (20 s, narrow craters) σr ≈ 50 nm, ℓc ≈ 215 nm

ZnO:Al (40 s, narrow craters) σr ≈ 100 nm, ℓc ≈ 375 nm

Figure 13.3: The morphology, rms roughness, and correlation length for some selected TCO samples [53].


13.2. The III-V PV technology 169

length `c were obtained with AFM scans of 256×256points over an area of 20×20 µm2. AFM and the de-termination of the statistical parameters are explainedin Appendix C.

13.2 The III-V PV technology

Of all the thin-film technologies, the III-V (speak three-five) PV technology results in the highest conversionefficiencies under both one sun standard test conditionsand concentrated sun conditions. Therefore, it is mainlyused in space technology and in solar concentrator tech-nology.

As some concepts use a crystalline germanium or anGaAs wafer as substrate, it might not be consideredas a real thin film PV technology, in contrast to thin-film silicon, CdTe, CIGS or organic PV, which we willdiscuss later in this chapter. However, the III-V basedabsorber layers themselves can be considered as thincompared to the thickness of crystalline silicon wafers.

The III-V materials are based on the elements with threevalence electrons like aluminium (Al), gallium (Ga) or in-dium (In) and elements with five valence electrons likephosporus (P) or arsenic (As). Various different semi-conductor materials such as gallium arsenide (GaAs),gallium phosphide (GaP), indium phosphide (InP), indium

Figure 13.4: A unit cell of a gallium arsenide crystal in thezinc blend structure [58].

arsenide (InAs), and more complex alloys like GaInAs,GaInP, AlGaInAs and AlGaInP have been explored.

We now will take a closer look on GaAs, which is theIII-V semiconductor most used for solar cells. The GaAshas a zinc blend crystal structure, illustrated in Fig. 13.4.It is very similar to the diamond cubic crystal structurewith the difference that it has atoms of alternating ele-ments at its lattice sites. In contrast to c-Si, where everySi atom has four neighbours of the same kind, in GaAs,every Ga has four As neighbours, while every As atomhas four Ga neighbours. When compared to silicon,GaAs has a slightly larger lattice constant of 565.35 pm,



EX

EL

L-valley[111]

Wavevector

Heavy holes

1.42

eV

X-valley[100]

Ener

gy

Eg

Γ-valley

T = 300K

1.71

eV

1.90

eV

Figure 13.5: The electronic band dispersion diagram of gal-lium arsenide.

and significant denser than silicon with a density of5.3176 g/cm3 (Si: 543.07 pm, 2.3290 g/cm3). Note thatboth Ga and As are are roughly twice as dense as sil-icon. In contrast to silicon, which is highly abundant,the abundance of gallium in the Earth’s crust is onlyabout 14 ppm [59]. GaAs therefore is a very expensivematerial. Arsenic is highly toxic; it is strongly suggestedthat GaAs is carcinogenic for humans [60].

Figure 13.5 shows the electronic band dispersion dia-gram of gallium arsenide. We can easily see that GaAs

is a direct band gap material, i.e. , the highest energylevel in the valence band is vertically aligned with thelowest energy level in the conduction band. Hence,only transfer of energy is required to excite an electronfrom the valence to the conduction band, but no trans-fer of momentum is required. The band gap of GaAsis 1.424 eV [20]. Here we look again at the absorptioncoefficient versus the wavelength. Consequently, as youcan see in Fig. 12.4, the absorption coefficient of GaAsis significantly larger than that of silicon. The same istrue for InP, another III-V material that also is shownin Fig. 12.4. Because of the high absorption coefficient.Because of the high absorption coefficient, the sameamount of light can be absorbed in a film more thanone order of magnitude as thin when compared to sil-icon. Another advantage of the direct III-V semicon-ductor materials their sharp band gap. Above Eg, theabsorption coefficient increases quickly.

Let us now take a look at the utilisation of the bandgap energy. Since GaAs is a direct band gap material,radiative recombination processes become important. Onthe other hand, Shockley-Read-Hall recombination canbe kept at a low level because III-V films can be depos-ited tih epitaxy processes that result in high purity films.


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13.2.1 Multi-junction cells

III-V PV devices can reach very high efficiencies be-cause they are often based on the multi-junctionconcept, which means that more than one band gap isused. As discussed in Chapter 10, the maximum the-oretical efficiency of single-junction cells is describedby the Shockley-Queisser limit. A large fraction of theenergy of the energetic photons are lost as heat, whilephotons with energies below the band gap are lost asthey are not absorbed.

For example, if we use a low bandgap material, a largefraction of the energy carried by the photons is notused. However, if we use more band gaps, the sameamount of photons can be used but less energy iswasted as heat. Thus, large parts of the solar spectrumand larges part of the energy in the solar spectrum canbe utilised at the same time, if more than one p-n junc-tions are used.

In Fig. 13.6 a typical III-V triple junction cell is illus-trated. As substrate, a germanium (Ge) wafer ie used.From this wafer, the bottom cell is created. Germaniumhas a band gap of 0.67 eV. The middle cell is based onGaAs and has a band gap of about 1.4 eV. The top cellis based on GaInP with a band gap in the order of 1.86eV.

Let now take a closer look on how a multi-junctionsolar cell works. Light will enter the device from

AR

n+-GaAsn-AllnP

n-GalnP

n-GalnP

p-AlGalnP

Middle Cell Window

n-GaAs

p-GaAs

p-GalnPp++

n++

buer

nucleationn-Ge

p-Ge substrate

contact

Contact=

Top cell base/BSF

Wide-Eg tunnel junction

Middle cell window/emitter

Middle cell base/BSF

p++

n++

TC & MC crystal quality:Nucleation, buer,Interface control,Lattice-matching

Top cell window/emitter

Middle Cell

Window

Middle Cell

Bottom Cell

Top Cell

Tunnel junction

Wide Eg tunnel

p++

n++

Figure 13.6: Illustrating a typical III-V triple junction solarcell.



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2Voltage (V)

3

2

1 1+2+3J (m

A/c

m2 )

0

5

10

30

15

25

20

Figure 13.7: The J-V curves of the three junctions used inthe III-V triple junction cell and J-V curves of the III-V triplejunction cell (in black).

the top. As the spectral part with the most energeticphotons like blue light has the smallest penetrationdepth in materials, the junction with the highest bandgap always acts as the top cell. On the other hand, asthe near infrared light outside the visible spectrum hasthe longest penetration depth, the bottom cell is the cellwith the lowest band gap.

Figure 13.7 shows the J-V curve of the three single p-n junctions. We observe p-n junction 1 has the highest

IphI

I

I

Iph

Iph

ID

ID

ID

Voc,1

Voc,2

Voc,3

Figure 13.8: The equivalent circuit of the three junctionsconnected in series.

open circuit voltage and the lowest short circuit cur-rent density, which means that this p-n junction has thehighest band gap. In contrast, p-n junction 3 has a lowopen circuit voltage and a high current density, con-sequently it has the lowest band gap. p-n junction 2has a band gap in between. Hence, if we are designinga triple-junction cell out of these three junctions, junc-tion 1 will act as the top cell, junction 2 will act as themiddle cell and junction 3 will act as the bottom cell.



For understanding how the J-V curve of the triple junc-tion looks like, we take a look at the equivalent circuit.Every p-n junction in the multijunction cell can be rep-resented by the circuit of a single-junction cell, as dis-cussed in chapter 9. As the three junctions are stackedonto each other, they are connected to each other inseries, as illustrated in Fig. 13.8. In a series connection,the voltages of the individual cell add up in the triplejunction cell. Further, the current density in a seriesconnection is equal over the entire solar cell, hence thecurrent density is determined by the p-n junction gener-ating the lowest current. The resulting J-V curve is alsoillustrated in Fig. 13.7. We see that the voltages add upand the current is determined by the cell delivering thelowest current.

Figure 13.9 (a) shows a typical band diagram of suchtriple junction. The top cell with high band gap isshown at the left hand side and the bottom cell is at theright hand side. However, this band diagram does notrepresent reality. If we would place three p-n junctionsin series, for example the p-layer of the top cell and then-layer of the middle cell would form a p-n junctionin the reverse direction than the p-n junctions of thethree single junction solar cells. These reverse junctionswould significantly lower the voltage of the total triplejunction.

For preventing the creation of such reverse junctions,so-called tunnel junctions are included. These tunnel

(a)

(b)

pn pn pn

pn pn pn

Figure 13.9: The band diagram of the III-V triple junction cell(a) without and (b) with tunnel junctions.



junctions align the valence band at one side with theconduction band at the other side of the tunnel junc-tion, as illustrated in Fig. 13.9 (b). They have a highband gap to prevent any parasitic absorption losses.Further, tunnel junctions are relatively thin and havean extremely narrow depletion zone. As a result, theslopes of the valence and conduction band are so steepthat the electrons from n-layer can tunnel through thesmall barrier to the p-layer, where they recombine withthe holes. It is important to have the tunnel junctionswith a low resistance such that the voltage loss acrossthem is low.

In the triple junction cell, two tunnel junctions arepresent. First, a tunnel junction via that the holes inthe p-layer of the top cell have to recombine with theelectrons of the n-layer of the middle cell. Secondly, atunnel junction where the holes in the p-layer of themiddle cell recombine with the electrons of the n-layerof the bottom cell. The electrons in the top cell n-layerare collected at the front contact and the hole in the p-layer of the bottom cell are collected at the back contact.It is important to realise that the recombination currentat the tunnel junctions represents the current density ofthe complete triple junction.

Let us take a look at the external parameters of a typ-ical lattice matched triple junction solar cell from Spec-troLab, which is a subsidiary of The Boeing Company.Because this cell was developed for space applications,

it is not tested under AM1.5 conditions, but under AM0conditions with an irradiance of 135 mW/cm2. Thetotal open circuit voltage is 2.6 Volts. The short circuitcurrent density is 17.8 mA/cm2, which corresponds to aspectral utilisation up to the lowest band gap of 0.67 eVof

Jtot = 3 · 17.8 mA/cm2 = 53.4 mA/cm2. (13.5)

The maximal theoretical current that could be generatedwhen absorbing AM0 for all wavelength shorter than1850 nm, which corresponds to 0.67 eV, is 62 mA/cm2.The average EQE of the solar cell thus is 86%, which isan very impressive value. Further, this cell has a veryhigh fill factor of 0.85, which means that it has a con-version efficiency of 29.5% under AM0 illumination.

Figure 13.10 shows the EQE of the subcells of a typ-ical III-V triple junction cell. We note that the shapeof the spectral utilisation of all the curves approachesthe shape of block functions, which would be the mostideal shape. These block shapes are possible becausethe III-V materials have very sharp band gap edges andhigh absorption coefficients. We see that the bottom cellgenerates much more current than the middle and topcells. Hence, a lot of current is lost in the bottom cell.

This ineffective use of the near infrared part can bereduced using quadruple junctions instead of triplejunctions. In these cell, an additional cell is placed in-between the middle and bottom cells of the triple junc-



0

10

20

30

40

50

60

70

80

90

100

300

Wavelength (nm)

Exte

rnal

Qua

ntum

E

cien

cy (

%)

500 700 900 1100 1300 1500 1700 1900

GaInP GaInAs Ge

Figure 13.10: The external quantum efficiency of a three-junction III-V solar cell (Data from SPECTROLAB).

tion. The spectral utilisation can be even increased fur-ther by moving to multi-junction solar cells consist-ing of 5 or even 6 junctions. The major challenge forthese cells is that lattice matching can not be guaran-teed anymore. Lattice-mismatched multi-junctions arecalled metamorphic multijunctions. They require bufferlayers that have a profiling in the lattice constant, go-ing from the lattice constant of one p-n junction to thelattice constant of the next p-n junction. Using this tech-nology, Spectrolab could demonstrated an impressive38.8% conversion efficiency of a 5-junction solar cell un-der 1-sun illumination [38].

The III-V PV technology is very expensive. Hence, suchcells are mainly used for space-applications and in con-centrator technology, where high performance is moreimportant that the cost. For reducing the cost per Wattpeak, a solar concentrator can be used. In such a concen-trator, the irradiance of a large area is concentrated ontothe small solar device. One advantage of concentratedsunlight is that the open circuit voltage is increasingwith the increasing irradiance as well, as long as Augerrecombination of light excited charge carriers does notbecome dominant. It is, however, very important thatthe small solar cell devices are actively cooled, as theirperformance decreases with increasing temperatures.This temperature dependence is caused by stronger re-combination and hence higher dark currents and is dis-cussed in more detail in Section 18.3. Additionally, con-centrator systems require sun-tracking systems, as only



direct sunlight can be concentrated. The tracking sys-tems ensure that the optical concentrator system tracksthe sun and guarantees the optimal light concentrationon the small PV device during the entire day. Hence,the non-modular costs of concentrator systems are lar-ger than that of conventional PV system based on c-Sior other thin-film technologies.

The current world record conversion efficiency for allsolar cell technologies on lab scale is 44.4% for a three-junction III-V solar cell under 302-fold concentrated sunlight conditions. This result was achieved by Sharp [38].

13.2.2 Processing of III-V semiconductormaterials

As already mentioned above, high quality III-V semi-conductor materials can be deposited using epitaxy de-position methods. In this method, crystalline overlayersare deposited on a crystalline substrate, such that theyadopt the crystal lattice structure of the substrate. Theprecursor atoms from that the layers are grown areprovided by various elemental sources. For example,if GaAs is deposited with epitaxy, Ga and As atoms aredirected to a growth surface under ultra-high vacuumconditions. For growing III-V semiconductors, ger-manium substrates usually are used. On this substratethe GaAs crystalline lattice is grown layer-by-layer andadopts the structure of the crystalline substrate.

As the layer-by-layer growth process is very slow, it al-lows the deposition of compact materials without anyvacancy defects. Furthermore, processing at high va-cuum conditions prevents the incorporations of impur-ities. Hence, III-V semiconductors can be deposited upto a very high degree of purity. Dopants can be addedto make it n- or p-type.

Typically, III-V semiconductor layers are depos-ited by using metal-organic chemical vapour deposition(MOCVD). Typical precursor gasses are trimethylgal-lium [Ga(CH3)3], trimethylindium [In(CH3)3], trime-htylaluminium [Al2(CH3)6] , arsine gas (AsH3) andphosphine gas (PH3). Surface reactions of the metal-organic compounds and hydrides, which contain therequired metallic chemical elements, create the rightconditions for the epitaxial crystalline growth. Epitaxialgrowth is a very expensive process. Similar techniquesare also used in the microchip production process.

A big challenge of depositing III-V semiconductor ma-terials is that the lattice constants of the various materi-als are different, as seen in Fig. 13.11. We see that everyIII-V semiconductor has a unique bandgap-lattice con-stant combination. Hence, interfaces between differentIII-V materials show a lattice mismatch, as illustrated inFig. 13.12. Because of this mismatch, not every valenceelectron is able to make a bond with a neighbour. Thisproblem can be solved by lattice matching, as this alsois done in the triple junction cell discussed in Section



4.0

3.6

3.2

2.8

2.4

2.0

1.6

1.2

0.8

0.4

0.04.5 5.0 5.5 6.0 6.5

GaN

BP

BAs

InN

Si

GeInAs

GaSb

Band

gap

(eV)

ZnSMgSe

ZnScAlP CdS

AlSb

CdTeCdSeInPGaAs

GaP ZnTe

InSb

AlAs

Lattice Constant (Å)

Figure 13.11: The bandgap and lattice constant of variousIII-V semiconductor materials.

lattice constant a1

lattice constant a0

dangling bond

Figure 13.12: Illustrating the lattice mismatch at an interfacebetween two different III-V materials.

13.2.1. For understanding what this means, we take an-other look at the phase diagram shown in Fig. 13.12.The triple junction is processed on a p-type germaniumsubstrate, the bottom cell is a Ge cell. One top it is bestto place a junction that has a higher band gap but thesame lattice constant. As we can see in the plot, GaAshas exactly the same lattice constant as Germanium.Therefore, between GaAs and Germanium interfacescan be made without any coordination defects relatedto mismatched lattices. For reasonable current matchingthe desired band gap of the top cell should be around1.8 eV. However, we see that no alloys, based on solelytwo elements, exist. But if we take a mixture of gallium,indium and phosphorus, we can make a III-V alloys



with a band gap of 1.8 eV material and with matchinglattice constant. Consequently, triple junction cells basedon GaInP, GaAs and Germanium can be fully latticematched.

13.3 Thin-film silicon technology

In this section we are going to discuss the thin-film sil-icon PV. An advantage of thin-film silicon solar cells isthat they can be deposited on glass substrates and evenon flexible substrates.

13.3.1 Thin-film silicon alloys

Thin-film silicon materials usually are deposited withchemical vapour deposition (CVD) processes that we willdiscuss in more detail in Section 13.3.3. In chemical va-pour deposition different precursor gasses are broughtinto the reaction chamber. Due to chemical reactions,layer is formed on the substrate. Depending on theused precursors and other deposition parameters suchas the gas flow rate, pressure, and temperature, vari-ous different alloys with different electrical and opticalparameters can be deposited. We will discuss the mostimportant alloys in the following paragraphs.

We start with two alloys consisting of silicon and hy-drogen: hydrogenated amorphous silicon (a-Si:H) and hy-drogenated nanocrystalline silicon (nc-Si:H), which is alsoknown as microcrystalline silicon. Usually in discus-sions the term ‘hydrogenated’ is left out for simplicity.Hydrogenated means that some of the valence electronsin the silicon lattice are passivated by hydrogen, whichis indicated by the ‘:H’ in the abbreviation. The typicalatomic hydrogen content of these alloys is from 5% upto about 15%. The hydrogen passivates most defects inthe material, resulting in a defect density around 1016

cm−3 [61], which is suitable for PV applications. Pureamorphous silicon (a-Si) would have an extremely highdefect density (> 1019 cm−3) [42], which would resultin fast recombination of photo-excited excess carriers.Similarly, we can make alloys from germanium and hy-drogen: hydrogenated amorphous germanium (a-Ge:H) andhydrogenated nanocrystalline germanium (nc-Ge:H).

Let us now take a look at alloys of silicon with fourvalence electrons with other elements with four valenceelectrons, carbon and germanium. In thin-film silicon solarcells, both hydrogenated amorphous and nanocrystal-line silicon-germanium alloys (a-SiGe:H and nc-SiGe:H)are being used. Silicon is also mixed using the fourvalence electron material, carbon, leading to hydrogen-ated amorphous silicon carbide (a-SiC:H).

Another interesting alloy is obtained when oxygen withsix valence electrons is incorporated into the lattice:


13.3. Thin-film silicon technology 179

1

2

3

4Si

H

Figure 13.13: Illustrating the atomic structure of amorphoussilicon with four typical defects: (1) monovacancies, (2) diva-cancies, (3) nanosized voids with mono- and (4) dihydrides.With kind permission of M. A. Wank [62].

hydrogenated nanocrystalline silicon oxide is often used inthin-film silicon solar cells.

All these alloys can be doped, usually boron is used asa p dopant while phosphorus is the most common ndopant.

Many of the alloys mentioned above are present asamorphous materials. It is hence important to discuss thestructure of amorphous lattices. In this discussion wewill limit ourselves to amorphous silicon, since it is thebest studied amorphous semiconductor and the gen-eral properties of the other amorphpous alloys are sim-ilar. In Chapter 12 we thoroughly discussed crystalline

silicon, which has an ordered lattice in which the ori-entation and structure is repeated in all directions. Foramorphous materials this is not the case, but the latticeis disordered as it is illustrated in Fig. 13.13. This fig-ure shows a so-called continuous random network (CRN).On atomic length scales, or also called short-range order,the atoms still have a tetrahedral coordination structure,just like crystalline silicon. But the silicon bond anglesand silicon-silicon bond lengths are slightly distortedwith respect to a crystalline silicon network. However,at larger length scales, or also referred to as long range,the lattice does not look crystalline anymore. Not allsilicon atoms have four silicon neighbours, but somevalence electrons form dangling bonds, similar to theunpassivated surface electrons already discussed inChapter 12. Recent studies show that these danglingbonds are not distributed homogeneously throughoutthe amorphous lattice, but that they group in divacan-cies, multivacancies or even nanosized voids smets:2007.The surfaces of these volume deficiencies are passivatedwith hydrogen.

Another phase of hydrogenated silicon alloys is thenanocrystalline lattice. It is a heterogeneous materialwith a structure that even is more complex than thatof amorphous materials. Nanocrystalline silicon con-sists of small grains that have a crystalline lattice andare a few tens of nanometres big. These grains are em-bedded in a tissue of hydrogenated amorphous silicon.Figure 13.14 shows the various phases of thin-film sil-



Figure 13.14: The various phases of thin-film silicon with ahighly crystalline phase on the left and amorphous silicon onthe right[63].

icon [63]. On the left-hand side, a fully crystalline phaseis shown, which is close to that of poly-crystalline sil-icon, except that is has much more cracks and poresin it. On the right hand side, the phase represents theamorphous lattice. Please note again, that the termsmicrocrystalline and nanocrystalline refer to the samematerial. Going from right to left, the amorphous phaseis changing in to a mixed phase with a few small crys-talline grains to a phase which is dominated by largecrystalline grains and a small fraction of amorphous tis-sue. Research has shown that the best nanocrystallinebulk materials used in solar cells has a network close tothe amorphous-nanocrystalline silicon transition region andits crystalline volume fraction is in the order of 60%.

2.48 1.24 0.83 0.62 0.50 0.41

105

104

103

102

101

100

0.5 1.0 1.5 2.0 2.5 3.0

Wavelength (μm)

Photon energy (eV)

Abs

orpt

ion

coe

cien

t (cm

-1)

a-Si Ge:H

a-Si:H

c-Si

Figure 13.15: The absorption coefficients of different thin-filmsilicon materials.


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The band gap of nanocrystalline silicon is close to thatof crystalline silicon (≈ 1.12 eV) due to the crystallinenetwork in the grains. The band gap of amorphous sil-icon is in the order of 1.6 up to 1.8 eV, which can betuned by the amount of hydrogen incorporated in tothe silicon network. It is larger than that of crystal-line silicon because of the distortions in bond anglesand bond lengths. It is out of the scope of this bookto discuss the reasons for this increase in band gap inmore detail. An important consequence of a disorderedamorphous lattice is that the electron momentum ispoorly defined in contrast to crystalline silicon. As wediscussed in Chapter 12, both energy and momentumtransfer are needed to excite and electron from thevalence band to the conduction band. Hence, crystal-line silicon is an indirect band gap material. This is nottrue for amorphous silicon, which is a direct band gapmaterial. Therefore the absorptivity of a-Si:H is muchhigher than that of c-Si, as we can see in Fig. 13.15. Wesee that the absorption coefficient for amorphous sil-icon in the visible spectrum is much larger that thatof crystalline silicon. In some wavelength regions it isabout two orders of magnitude larger, which meansthat much thinner silicon films can be used in refer-ence to the typical wafers in crystalline silicon solarcells. In the figure, also data for amorphous silicon-germanium are shown. a-SiGe:H has lower band gapand even higher absorption coefficient in the visible. Itsband gaps are in the range of 1.4 up to 1.6 eV. Amorph-

ous silicon carbide alloys have band gaps of 1.9 eV andlarger. Finally, nanocrystalline silicon oxides have bandgaps exceeding 2 eV.

13.3.2 The design of thin-film silicon solarcells

The first successful a-Si:H solar cell with an efficiencyof 2.4% was reported by Carlson and Wronski in 1976[64]. We now will discuss how such solar cells can bedesigned and what are the issues in designing them.

When compared to c-Si, hydrogenated amorphous andnanocrystalline silicon films have a relatively high de-fect density of around 10−16 cm−3. As mentioned be-fore, because of the disordered structure not all valenceelectrons are able to make bonds with the neighbour-ing atoms. The dangling bonds can act as defects thatlimit the lifetime of the light excited charge carriers.This is described with Shockley-Read-Hall recombina-tion, which controls the diffusion length. Because of thehigh defect density the diffusion length of charge carri-ers in hydrogenated amorphous silicon is only 100 nmup to 300 nm. Hence, the transport of charge carriers ina thick absorber can not rely on diffusion.

Therefore, amorphous silicon solar cells are not basedon a p-n junction like wafer based c-Si solar cells.Instead, they are based on a p-i-n junction, which



glass

transparent conducting oxide (TCO)

back contact

silicon p-i-n junction

glass

back contact

silicon p-i-n junction


glass

(a) (c)

EFermi

intrinsic a-Si:H n-a-Si:Hp-a-SiC:H

conduction band

valence band

(b)

ZnO or SiOx back reector

Figure 13.16: Illustrating the (a) layer structure and the (b)band diagram of an amorphous silicon solar cell. (c) Thin-filmsilicon solar cells hava nanotextured interfaces.

means that an intrinsic (undoped) layer is sandwichedbetween thin p-doped and n-doped layers, as illustratedin Fig. 13.16 (a). While the i-layer is several hundredsof nanometres thick, the doped layers are only about 10nm thick. Between the p- and n-doped layers a built-inelectric field across the intrinsic absorber layer is cre-ated. This is illustrated in the electronic band diagramshown in Fig. 13.16 (b). If the layers are not connectedto each other, the Fermi level in the p- n layers is closerto the valence and the conduction bands, respectively.For the intrinsic layer, it is in the middle of the bandgap. When the p, i, and n layers are connected to eachother, the Fermi level has to be the same throughout thejunction, if it is in the dark and under thermal equilib-rium. This creates a slope over the electronic band inthe intrinsic film as we can see in the illustration. Thisslope reflects the built-in electric field.

Another way to look at the field is to consider the i-layer as an dielectric in between to charged plates (thedoped layers). Neglecting any edge effects, the electricfield in the i layer is given by

E =σ

εε0, (13.6)

where σ is the charge density on the doped layers andε is the electric permittivity of the i-layer material. Notethat the electric field in the i layer is constant. Follow-



ing Eq. (4.29), we find the voltage in the i layer to be

U(x) = −∫

E(x)dx = U0 −σ

εε0x. (13.7)

We see that the voltage and hence the energy in the ilayer is linear, just as shown in the band diagram ofFig. 13.16 (b).

Because of the electric field, the light excited chargecarrier will move through the intrinsic layer. As dis-cussed in Section 6.5.1, the holes move up the slopein the valence band towards the p layer and the elec-trons move down the slope in the conduction bandtowards the n layer. Such a device, where electronicdrift because of an electric field is the dominant trans-port mechanism is called a drift device. In contrast, awafer based crystalline silicon solar cell, as discussedin Chapter 12, can be considered as a diffusion device.

Note, that because of the intrinsic nature of the ab-sorber layer the hole and electron density is in the sameorder of magnitude. On the other hand, in the p layerthe holes are the majority charge carriers and the dom-inant transport mechanism is diffusion. Similarly, in then layer the electrons are the majority charge carriersand again diffusion in the dominant transport mechan-ism. Because of the low diffusion length, both p and nlayers must be very thin thin.

Now we take another look at Fig. 13.16 (a). The cellsketched there is deposited in superstrate configuration.

This means that the layer that is passed first by the in-cident light in the solar cell is also deposited first in theproduction process. Also the term p-i-n layer refers tothis superstrate configuration, as it indicates the orderof the depositions: because in thin-film silicon holeshave a significantly lower mobility than electrons, thep layer is in front of the n layer and hence the p layeris deposited first. As then the generation rate is higherclose to the p layer, more holes can reach it.

In superstrate thin-film silicon solar cells, usually glassis used as a superstrate because it is highly transpar-ent and can easily handle all the chemical and physicalconditions in that all the depositions are carried out.Before the p-i-n layers can be deposited, a transparentfront contact has to be deposited. Usually transparent con-ducting oxides are used, that are discussed in more detailin Section 13.1.

Another possible configuration is the substrate configur-ation. There, either the substrate acts like a back con-tact or the back contact is deposited on the substrate.Consequently, no light will pass through the substrate.Thin-film silicon solar cells deposited in the substrateconfiguration are also called n-i-p cells as the n layer isdeposited before the i layer and the p layer.

Usually, thin-film silicon solar cells have no flat inter-faces, as shown in Fig. 13.16 (a), but nano-textured in-terfaces, as illustrated in Fig. 13.16 (c). These texturedinterfaces scatter the incident light and hence prolong


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the average path length of the light through the ab-sorber layer. Therefore, more light can be absorbed andthe photocurrent can be increased. This is an exampleof light management that is discussed in more detail inChapter ??.

For the p layer, often not amorphous silicon but higherband gap materials such as silicon carbide or silicon ox-ides are used in order to minimise parasitic absorptionmainly in the blue part of the spectrum. Usually, boronis used as a dopant. Also for the n layer nowadays of-ten silicon oxides are used, but sometimes still a-Si:His used. Phosphorus is the main dopant used here. Then-SiOx:H is very transparent. Therefore it also can beused as a back reflector structure when its thickness ischosen such that destructive interference occurs at thei-n interface minimising the electric field strength andhence parasitic absorption in the n layer. Additionally,also several tenth of nm thick TCO can be used withthe same purpose. Further, a metallic back reflector thatalso acts as the electric back contact is used. Because ofits attractive cost, mainly Al is used for this purpose.However, also silver that has a higher reflectivity can beused, yet is it more expensive.

The band gap of hydrogenated amorphous silicon isin the order of 1.75 eV, hence it only is absorptive forwavelength shorter than 700 nm. The highest currentdensities achieved in single junction amorphous sil-icon solar cell are 17 up to 18 mA·cm−2, whereas the

maximum theoretic current that could be achieved upto 700 nm is in the order of 23 mA·cm−2. Thus, theEQE averaged over the spectrum is in the order of 74to 77%.

The highest open circuit voltages that have been exper-imentally achieved are in the order of 1.0 eV. With re-spect to a band gap of 1.75 eV the band gap utilisationis quite low because of the high levels of SRH recom-bination and the relative broad valence and conduction-band tails. The highest stabilised efficiencies of a singlejunction solar cells is 10.1% [38]. It was obtained by theresearch Oerlikon Solar Lab in Switzerland, which cur-rently is a subsidiary of the Japanese Tokyo ElectronLtd.

Besides a-Si:H, also nanocrystalline silicon films areused for the intrinsic absorber layers in p-i-n solar cellas well. The spectral utilisation of the nc-Si:H is bet-ter than that of amorphous silicon because of the lowerband gap of nc-Si:H. However, to utilise the spectralpart from 700 up to 950 nm, thicker films are required,because of the indirect band gap of the silicn crystal-lites. Typical intrinsic film thicknesses are in between1 µm and 3 µm. The current nc-Si:H record cell has ashort circuit current density of 28.8 mA·cm−2, an opencircuit voltage of 523 mV and an efficiency of 10.8%[38]. This result was achieved by the Japanese Instituteof Advanced Industrial Science and Technology (AIST).

Neither a-Si:H nor nc-Si:H has an optimal spectral util-


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metal back contact

glass

a-Si:H p-i-n junction


(a)

EFermi

n-nc-Si:H

p-a-SiC:H

conduction band

valenceband

(b)

nc-Si:H p-i-n junction

1000-2000 nm 150-300 nm

n-a-SiC:Hi-a-SiC:H p-nc-Si:H

i-nc-Si:H

ZnO or SiOx intermediate reector

TCO or SiOx back reector

Figure 13.17: Illustrating the (a) layer structure and the (b)band diagram of an micromorph silicon solar cell.

isation. Therefore also in thin-film technology the mul-tijunction approach is used, just like for III-V solar cells.Probably the most studied concept is the micromorphconcept illustrated in Fig. 13.17 (a), which is a doublejunction concept consisting of an a-Si:H and a nc-Si:Hjunction. Like before, the solar cell with the highestbandgap is used as a top cell that converts the most en-ergetic photons into electricity, while the lower bandgapmaterial is used for the bottom cell and converts thelower energetic photons.

Figure 13.17 (b) shows the typical band diagram ofsuch a micromorph solar cell, that is also called a tan-dem solar cell. On the left hand side the electronic banddiagram of the amorphous silicon top cell is shown;on the right hand side the electronic band diagram ofnanocrystalline silicon bottom cell is shown. The blueand green short-wavelength light is absorbed in the topcell, where electron-hole pairs are generated. Similarly,the red and infrared long-wavelength light is absorbedin the bottom cell, where also electron-hole pairs aregenerated. Let us take a closer look on the two electron-hole pairs excited in the top and bottom cells, respect-ively. The hole generated in the amorphous top cellmoves to the top p layer and the electron excited in thebottom cell drifts to the bottom n layer. Both can becollected at the front and back contacts. However, theelectron excited in the top cell drifts to the top n layerand the hole generated in the bottom cell drifts to thep layer. Just as for the III-V multijunction devices, the



0.0 0.3 0.6 0.9 1.2 1.5Voltage (V)

nc-Si:H

a-Si:H a-Si:H/nc-Si:Htandem

J (m

A/c

m2 )

0

5

10

30

15

25

20

Figure 13.18: The J-V curve of a micromorph solar cell andits isolated subcells.

electrons and holes have to recombine at a tunnel recom-bination junction between the n layer of the top cell andthe p layer of the bottom cell. Often a very thin and de-fect rich layer is used for this purpose. Just as for theIII-V multijunction solar cells, the total current densityis equal to that of the junction with the lowest currentdensity. Therefore, for an optimised multijunction cellall current densities in the various sub cells have to bematched in order to achieve the best spectral utilisation.

Figure 13.18 shows the J-V curves of a single junction

a-Si:H solar cell and of a single junction nc-Si:H solarcell. Let us assume that the Voc of the high band gapa-Si:H top cell has an open circuit voltage of 0.9 V anda relatively low short circuit density of 15 mA·cm−2,whereas the low band gap material of nc-Si:H has alower open circuit voltage of 0.5 Volts and a highershort circuit current density of 25 mA·cm−2. If wemake a tandem of both junctions, the resulting cur-rent density of the double junction is lower than thecurrents in both bottom cells. Because the open circuitvoltage is approximately proportional to ln(Jph/J0),the open circuit voltages of the junctions in a tandemcell will be slightly lower than for similar single junc-tion cells. The total current utilisation of the tandemcell is determined by the bottom cell, i.e. 25 mA·cm−2.Given the examples of the single junctions here, thebest current matching of both cells would deliver 12.5mA·cm−2. The current tandem record cell has an effi-ciency of 12.3% and was manufactured by the Japanesecompany Kaneka [38].

Just as for the III-V technology, also with thin-film tech-nology multijunction cells with more than two junc-tions can be made. For example the former US com-pany United Solar Ovonic LLC made a thin-film siliconbased triple junction device with an a-Si:H top cell, ana-SiGe:H middle cell and a nc-Si:H bottom cell, illus-trated in Fig. 13.19 (a). As illustrated in this figure, alsovarious other combinations for triple junctions can bemade, for example a-Si:H/nc-Si:H/nc-Si:H. Figure 13.19



metal back contact

glass



(a)



metal back contact

glass




a-SiGe:H p-i-n junction

(b)

1

2

3

1 2 3

AM 1.5

400 500 600 700 800 900 1000 1100 1200Wavelength (nm)

180

160

140

120

100

80

60

40

20

0

spec

tral

irra

dian

ce[m

W/(

cm2 μm

)]Figure 13.19: (a) Two possible structures and (b) the typical spectral utilisation (EQE·AM1.5) of a triple-junction thin-film siliconcell.



(b) shows the spectral utilisation3 of the three junctionsand the total cell of the record device by United Solar.In difference to the EQE of the multijunction III-V cell(Fig. 13.10), here the individual cells have various over-laps instead of being block functions as for the III-Vdevice. Here, light with wavelengths below 450 nm isutilised by the top cell only, around 550 nm light is util-ised by the top cell and middle cell, wavelengths at 650nm are utilised by all three junctions, and wavelengthsabove 900 nm are utilised by the bottom cell only. Con-sequently, optimising thin film silicon multijunctionsolar cells is a complex interplay between the variousthicknesses and light trapping concepts used in thesolar cell. The initial efficiency of the current record triplejunction cell by Uni Solar Ovonic is 16.3%.

The term initial refers to a big issue for amorphousthin-film silicon alloys: they suffer from light induceddegradation which leads to a reduction of the efficiency.For example, for the initial 16.3% record cell the effi-ciency drops below the 13.4% of the current stabilisedrecord cell. This record cell consists of an a-Si:H/nc-Si:H/nc-Si:H stack and was produced by the SouthKorean LG Corp. [38].

Light induced degradation, which is also referred to asthe Staebler-Wronski effect (SWE), is one of the biggestchallenges for the thin film solar cells. It was alreadydiscovered one year after the first a-Si:H solar cells

3The spectral utilisation is given as EQE·AM1.5.

were made in 1977 [65]. Because of the recombinationof light excited charge carriers, meta-stable defects arecreated in the absorber layers. The increased defectdensity leads to increased bulk charge carrier recom-bination, which mainly affects the performance of theamorphous solar cells. After about 1000 hours of illu-mination, the efficiency of amorphous thin-film solarcells stabilises at around 85% - 90% of the initial effi-ciency and stays stable for the rest of its lifetime. If theSWE could be tackled, thin-film silicon devices easilycould achieve stable efficiencies well above 16%.

Just as for III-V cells, current matching is very import-ant for thin-film silicon multijunction cells. First, nan-otextured surfaces scatter the incident light in order toenhance the average photon path length and hence toincrease the absorption in the various absorber layers ofthe multijunction cell. Scattering becomes more import-ant for the bottom cell because its nc-Si:H film is thethickest layer in the device has to absorb most of thered and infrared, where nc-Si:H is not that absorptive.Secondly, intermediate reflector layers are used as a toolto redistribute the light between the junctions aboveand below them. More specifically, the top and bottomjunctions are separated with a low reflective index ma-terial, such as nc-SiOx:H. Because of the refractive indexdifference of the top absorber and the nc-SiOx:H, morelight is reflected back into the top cell. Thus, the a-Si:Htop cell can be made thinner, which makes it less sensit-ive to light-induced degradation.



13.3.3 Making thin-film silicon solar cells

Let us now discuss how lab-scale thin-film silicon solarcells are made in the DIMES Laboratory in Delft, theNetherlands. Before the deposition can start, the samplehas to be cleaned in a ultrasonic cleaning bath. Duringcleaning, dirt and dust particles are removed, whichcould lead to a shunt between the front and back con-tact in the final solar cell. Now, the TCO layer can bedeposited. In Delft, we can deposit ZnO:Al or ITOwith sputtering processes. After sputtering, ZnO:Al canbe etched with acids in order to achieve a crater-likestructure for light scattering. Alternatively, the samplecan already be covered with a TCO layer, for exampleSnO2:F from the Japanese Asahi company, which alreadyhas a pyramid-like structure for light scattering be-cause of its deposition process. Also on SnO2:F a 5-10nm thick ZnO:Al layer is deposited that protects theSnO2:F from being reduced by hydrogen-rich plasmapresent during the deposition of the thin-film siliconlayers. During sputtering, the zinc oxide target is bom-barded using an ionised noble gas like Argon. The gen-erated aluminium zinc oxide species are sputtered intothe chamber and deposited on to the substrate.

Before the thin-film silicon layers are deposited, a thinAl strip is deposited on the side of the sample, that willact as electric front contact when the cells are measured.

Processing of the different thin-film silicon layers often

happens in multi-chamber setups that allow to processeach layer in a separate chamber and therefore can pre-vent cross-contamination, e.g. from p and n dopants,which would reduce the quality of the layers. Afterthe sample is mounted on a suitable substrate holder, itenters the setup via a load lock, in which the substrate isbrought under low pressure before it’s moved into theprocessing chambers. This avoids the processing cham-bers to be contaminated with various unwelcome atomsand molecules present in ambient air. Then the sampleis brought into a central chamber, from where all pro-cess chambers can be accessed.

A typical precursor gas for depositing thin-film Si lay-ers is silane (SiH4), which often is diluted with hydro-gen (H2). The precursor gasses are brought into theprocess chamber at low pressure. A radio-frequency(RF) or very-high-frequency (VHF) bias voltage betweentwo electrodes is used to generate a plasma. Thisplasma leads to dissociation of the SiH4 atoms into rad-icals such as SiH3, SiH2 or SiH. This radicals react withthe substrate, where a layer is deposited. With increas-ing the H2 content, the material becomes more nano-crystalline, as sketched in Fig. 13.14.

As dopants, mainly diborane (B2H6) is used for p lay-ers and phosphine (PH3) is used for the n layers. Forsilicon carbide layers, additionally to the silane, meth-ane (CH4) is used. Silicon oxide layers can be made bycombining silane with carbon dioxide (CO2).



After the p-i-n junction is deposited, the sample iscovered with a mask, which defines the areas on thatthe metallic back contacts are deposited. In Delft, wedeposit silver with evaporation. Little pieces of silver areput in a boat that is heated by a very high current thatis flowing through it. The silver melts and evaporatedsilver particles can move freely through a vacuum untilthey hit the sample, where a silver layer is deposited.

Now the solar cells are ready, every metallic squaredefines a little solar cell. When they are measured, thesmall metal strip is used as front contact and the backof the metallic square is the electric back contact.

Of course, such a configuration is not suitable for large-scale thin-film PV modules. How such a module ismade, is discussed in Section 17.1 during our generaldiscussion on PV modules.

13.3.4 Crystalline silicon thin-film solarcells

At the end of our discussion on thin-film silicon tech-nology we briefly discuss crystalline silicon thin-film solarcells, where the advantages of crystalline silicon techno-logy and thin-film silicon technology should be com-bined [66]. This means, that high-quality crystallinefilms of only several tenths of micrometers that are po-sitioned on a substrate or a superstrate are used as ab-

sorbers.

Different approaches are investigated for reaching thisgoal: Kerf-less wafering techniques are investigatedwhere thin wafers can be prepared without any kerfloss. The wafers then are transferred on a glass sub-strate. With this technique, efficiencies of 10.0% wereobtained with a 50 µm thick absorber [66].

Alternatively, films of large-grain nanocrystalline siliconor amorphous silicon can be deposited. The amorphousfilms can be crystallised after deposition, for example ina thermal annealing step. Because of the low electricalquality of the defect-rich silicon films, the maximallyachieved efficiency is 10.5% and was achieved by theChinese company CSG Solar [66].

The highest efficiencies are reached when the thin crys-talline layers are prepared in an epitaxy process, just asfor the high-performance III-V solar cells. The epitaxyfilms then are transferred on glass. The current recordcell made with this method has an efficiency of 20.1%on a 43 µm thick substrate and was made by the Amer-ican company SOLEXEL [38].

13.4 Chalcogenide solar cells

The third class of thin-film solar cells that we discussare the large class of chalcogenide solar cells, where our


13.4. Chalcogenide solar cells 191

focus mainly will be on CIGS and cadmium telluride(CdTe) solar cells. The term chalcogenides refers to allchemical compounds consisting of at least one chalco-gen anion from the group 16 (also known as group VI)with at least one or more electropositive elements. Fiveelements belong to group 16: oxygen (O), sulphur (S),selenium (Se), tellurium (Te), and the radioactive po-lonium (Po). Typically, oxides are not included in dis-cussions of chalcogenides. Because of its radioactivity,compounds with Po are of only very limited relevancefor semiconductor physics.

13.4.1 Chalcopyrite solar cells

The first group of chalcogenide solar cells that we dis-cuss are chalcopyrite solar cells. The name of this class ofmaterials is based on chalcopyrite (copper iron disulfide,CuFeS2). Like all the chalcopyrites, its forms tetragonalcrystals, as illustrated in Fig. 13.20.

Many chalcopyrites are semiconductors. As they con-sist of elements from groups I, III, and VI, they are alsocalled I-III-VI semiconductors or ternary semiconduct-ors. In principle, all these combinations can be used:Cu

AgAu

AlGaIn

SSeTe

2

Figure 13.20: The crystal structure of copper indium diselenide,a typical chalcopyrite. The colors indicate copper (red), selen-ium (yellow) and indium (blue). For copper gallium diselenide,the In atoms are replaced by Ga atoms [67].



3.0

2.5

2.0

1.5

1.0

5.4 5.6 5.8 6.0 6.2

Band

gap

(eV)

Lattice Constant (Å)

CuAlSe2

CuGaS2

CuInS2

CuInSe2

AgInSe2

CuInTe2

CuGaSe2

Figure 13.21: The bandgap vs. the lattice constant for severalchalcopyrite materials. Data taken from [42].

Figure 13.21 shows the bandgap vs. the lattice structurefor several chalcopyrite materials.

The most common chalcopyrite used for solar cells is amixture copper indium diselenide (CuInSe2, CIS) and cop-per gallium diselenide (CuGaSe2, CGS). This mixture iscalled copper indium gallium diselenide [Cu(InxGa1-x)Se2,CIGS], where the x can vary between 0 and 1. Severalresearch groups and companies use a compound thatalso contains sulfur; it is called copper indium galliumdiselenide/disulfide [Cu(InxGa1-x)(SeyS1-y)2, CIGSS], where

y is a number in between 0 and 1.

The physical properties of CIGS(S) are rather complexand many different views exist on these propertiesamong scientists. CuInSe2 has a band gap of 1.0 eV, thebandgap of CuInS2 is 1.5 eV and CuGaSe2 has a bandgap of 1.7 eV. By tuning the In:Ga ratio x and the Se:Sratio y, the band gap of CIGS can be tuned from 1.0eV to 1.7 eV. As CIGS(S) is a a direct band gap semi-conductor material, it has a large absorption coefficient,hence an absorber thickness of 1-2 µm is sufficient forabsorbing a large fraction of the fraction of the lightwith energies above the band gap energy. Also thetypical electron diffusion length is the order of a fewmicrometers. CIGS(S) is a p-type semiconductor, thep-type character resulting from intrinsic defects in thematerial that amongst others are related to Cu deficien-cies. The many different types of defects in CIGS(S) andtheir properties are a topic of ongoing research.

Figure 13.22 (a) illustrates a typical CIGS solar cellstructure deposited on glass, which acts as a substrate.On top of the glass a molybdenum layer (Mo) of typic-ally 500 nm thick is deposited, which acts as the elec-tric back contact. Then, the p-type CIGS absorber layeris deposited with a thickness up to 2 µm. Onto the p-CIGS layer, a thin n-CIGS layer is deposited, for ex-ample an indium/gallium rich Cu(InxGa1-x)3Se5 alloy.The pn-junction is formed by stacking a thin cadmiumsulfide (CdS) buffer layer of around 50 nm thickness onto



n-ZnO:Al

i-ZnO

glass substrate

p-CIGS

(a) (b)

Mo back contact

p-CdS buer

3.3 eV

2.5 eV

1.1 eV

p-CIGS

n-Cd

S

i-ZnO

n-Zn

O:A

l

EC

EV

EF

n-CIGS buriedjunction

Figure 13.22: The (a) layer structure and (b) band diagram ofa typical CIGS solar cell.

the CIGS layers. The n-type region is extended with theTCO layer, that also is of n type: First an intrinsic zincoxide (ZnO) layer deposited followed by a layer of Aldoped ZnO. The Al is used as an n dopant for the ZnO.Similar to thin film silicon technology, the n-type TCOacts as the transparent front contact for the solar cell.

Figure 13.22 (b) shows the electronic band diagram ofa CIGS solar cell. The light enters the cell from the left,via the ZnO. The p-type CIGS absorber layers used inindustrial modules typically has a band gap of 1.1-1.2eV, which is achieved using Cu(InxGa1-x)Se2 with x ≈0.3 [68]. The n-type CdS buffer layer has a band gapof 2.5 eV. The band gaps of the n- and p-type materialsare different, which means that such CIGS cells can be

can be considered as heterojunctions. The band gap ofZnO is very large with values of 3.2 eV or even higher,which minimises the parasitic absorption losses in thisdevice [69].

The defect density at the surface is higher than in thebulk, which could be a loss mechanism for the minor-ity charge carriers. This recombination can be reducedby placing an n-type CIGS layer between the p-typeCIGS and the n-type CdS layers, as already mentionedabove. The pn-junction within the CIGS is called a bur-ied junction. There the electron-hole pairs are separated.In p-CIGS, which is Cu-deficient, the dominant recom-bination mechanism is Shockley-Read-Hall recombina-tion in the bulk. In contrast, in Cu rich CIGS films theSRH recombination at the CIGS/CdS interface becomesdominant.

A very important issue in the development of CIGSsolar cells is the role of sodium (Na). A low contamin-ation with sodium appears to reduce the recombinationin p-type CIGS materials because of better recombin-ation of the grain boundaries. The reduced recombin-ation rate results in a higher band gap utilisation andthus a higher open circuit voltages. Typically, the op-timal concentration of sodium in the CIGS layers isabout 0.1%. Often, as a Na source soda-lime glass isused, that is also the substrate for the solar cell. If nosoda-lime glass is used, the Na has to be intentionallyadded during the deposition process. The scientific



question why Na significantly improves several prop-erties of the CIGS films is still not completely solved.Currently, also the influence of potassium (K) is heavilyinvestigated.

Fabrication of CIGS solar cells

CIGS films can be deposited with various differenttechnologies. Because many of these activities are de-veloped within companies, not much detailed inform-ation is available on many of these processing tech-niques. One example of a processing technique is co-evaporation under vacuum conditions. Using variouscrucibles of copper, gallium, selenium and indium, theprecursors in are co-evaporated onto a substrate invarious steps. Two approaches can be used: First, co-evaporation on a heated substrate. During the processan additional selenium source is used, such that a CIGSfilm can be formed. The second approach is sputter-ing onto a substrate at room temperature. After that,the substrate is thermally annealed under presence ofselenium vapour, such that the CIGS structure can beformed. Alternatively, a selenium rich layer can be de-posited on top of the initially deposited alloy, followedby an annealing step. Because of the variety and com-plexity of the reactions taking place during the ‘selenisa-tion’ process, the properties of CIGS are difficult to con-trol. Companies that use or have used co-evaporationprocess are Würth Solar, Global Solar and Ascent Solar

Technologies. Among CIGS companies using sputterapproaches are Solar Frontier, Avancis, MiaSolé andHonda Soltec.

An alternative approach to produce CIGS layers isbased on a wafer bonding technique. Two differentfilms are deposited onto a substrate and a superstrate,repspectively. Then, the films are pressed together onhigh pressure. During annealing, the film is releasedfrom the superstrate and a CIGS film remains on a sub-strate.

Non-vacuum techniques can be based on depositingnanoparticles of the precursor materials on a substrateafter which the film is sintered. During the sintering pro-cess films are made out of powder. For achieving this,the powder is heated up to a temperature below themelting point. At such high temperatures, atoms in theparticles can diffuse across the particle boundaries. Theparticles thus fuse together forming one solid. This pro-cess has to be followed by a selenisation step.

An important advantage of the CIGS PV technologyis that on it has achieved the highest conversion effi-ciencies among the different thin-film techologies ex-cept the III-V technology. The current record for lab-scale CIGS solar cells processed on glass is 20.8% andwas achieved by the German research institute ZSW[38]. This record cell has a Voc of 0.757 V, a Jsc of 34.77mA/cm2, and a FF of 79.2%. The world record on flex-ible substrates has been obtained at the Swiss Fed-



eral Laboratories for Materials Science and Technology(EMPA). On flexible polymer foil they achieved a con-version efficiency of 20.4% [70].

For making CIGS modules, the interconnection is donesimilar as for other thin-film technologies and is dis-cussed in detail in Section 17.1. As generally true forthe different PV technologies, the record efficiencies ofmodules are significantly lower than that of lab-scalecells. The record efficiencies of 1 m2 modules are in theorder of 13%, whereas the aperture-area efficiencies arejust above 14% as confirmed by NREL. The Germanmanufacturer Manz AG has presented an 15.9% aper-ture area efficiency and a total area efficiency of 14.6%.The Japanese company Solar Frontier claims a 17.8%aperture area efficiency on a small module of 900 cm2

size.

Despite the very high conversion efficiencies, the CIGStechnology faces several challenges. As CIGS is a de-posited in a complex deposition processes, it is challen-ging to perform large area deposition with a high pro-duction yield. The production yield is the percentage ofall the modules coming out of the production line thatfulfil the product specifications.

Kesterites

Figure 13.1 shows the abundance in the Earth’s crustfor several elements. Is we can see, indium is a very

rare element. As we have seen above, In is a crucialelement of CIGS solar cells. Because of its scarcity, Inmight be the limiting step in the upscaling of the CIGSPV technology to future Terawatt scales. In addition,the current thin-film display industry depends on In aswell, as ITO is integrated in many display screens.

As a consequence, other chalcogenic semiconductorsare investigated that do not contain rare elements. Ainteresting class of materials are the kesterites whichare quarternary or pentary semiconductors consistingof four or five elements, respectively. While the namegiving mineral kesterite [Cu2(ZnFe)SnS4], where zincand iron atoms can substitute each other, is not used asa semiconductor, kesterite without iron (Cu2ZnSnS4) isused. It also is known as copper zinc tin sulfide (CZTS)and is a I2-II-IV-VI4 semiconductor. Other kesteritesare for example copper zinc tin selenide (Cu2ZnSnSe4,CZTSe), or ones using a mixture of sulphur and sel-enium, Cu2ZnSn(SSe)4 (CZTSS).

In difference to CIGS, CZTS is based on non-toxic andabundantly available elements. The current record effi-ciency is 12%. It is achieved with an CZTSS solar cellson lab-scale by IBM [38].



13.4.2 Cadmium telluride solar cells

In this section we will discuss the cadmium telluride(CdTe) technology, which currently is the thin-film tech-nology with the lowest demonstrated cost per Wp. Westart with discussing the physical properties of CdTe,which is a II-VI semiconductor because it consists of theII valence electron element cadmium (Cd) and the VIvalence electron element tellurium (Te). Like the III-Vsemiconductors discussed in Section 13.2, CdTe forms azincblende lattice structure where every Cd atom is bon-ded to four Te atoms and vice versa.

The band gap of CdTe is 1.44 eV, a value which is closeto the optimal band gap for single junction solar cell.CdTe is a direct band gap material, consequently onlya few micrometres of CdTe are required to absorb allthe photons with an energy higher than the band gapenergy. If the light-excited charge carriers should beefficiently collected at the contacts, their diffusion has tobe in the order of the thickness.

CdTe can be n-doped by replacing the II-valence elec-tron atom Cd with a III-valence electron atom like alu-minium, gallium or indium. n-doping can be achievedas well by replacing a VI-valence tellurium atom witha VII-valence electron element like fluorine, chlorine,bromine and iodine atoms. The III- and VII-valenceatoms act as shallow donors. Also a tellurium vacancyacts like a donor.

p-doping of CdTe can be achieved by replacing Cd witha I-valence electron atom like copper, silver or gold. Itcan be achieved as well by replacing Te atoms with V-valence electron elements such as nitrogen, phosphorusor arsenicum. These elements act as shallow acceptors.But also a Cd vacancy acts as an acceptor. In solar cells,p-doped CdTe is used. However, it is difficult to obtainCdTe with a high doping level.

Figure 13.23 (a) shows the structure of a typical CdTesolar cell. First, transparent front contact is depositedonto the glass superstrate. This can be tin oxide or cad-mium stannate, which is a Cd-Sn-oxide alloy. On top ofthat the n-layer is deposited, which is a cadmium sulf-ide layer, similar to the n-uffer layer in CIGS solar cells(Section 13.4.1). Then, the p-type CdTe absorber layeris deposited with a typical thickness of a few micro-metres. Making a good back contact on CdTe is ratherchallenging because the material properties of CdTe donot allow a large choice of acceptable metals. Heavilydoping the contact area with a semiconductor materialimproves the contact qualities, however, achieving highdoping levels in CdTe is problematic. Copper contain-ing contacts have been used as back contacts, however,on the long term they may face instability problemsdue to diffusion of Cu through the CdTe layer up to theCdS buffer layer. Nowadays, stable antimony telluridelayers in combination with molybdenum are used.

The band diagram of a CdTe solar cell is shown on Fig.



front glass

n-doped SnO2

metal back contact

p-CdTe

(a)

(b)

n-CdS buer

ΔEV = 1.2 ± 0.2 eV

ITO SnO2 CdS CdTe

1.45 eV2.4 eV

3.7 eV

3.6 eV

EC

EV

ΔEV = 0.95 ± 0.1 eVΔEV= 0.6 ±0.1 eV

0.33 eV

TeSb2Te3 NiV

EF

Figure 13.23: The (a) layer structure and (b) band diagram ofa typical CIGS solar cell. Band diagram data taken from [71].

13.23 (b). The p-type semiconductor CdTe has a bandgap of 1.44 eV whereas the n-type CdS has a band gapof 2.4 eV. Consequently, the junction is a heterojunction,similar to what we have seen for CIGS PV devices. Thelight excited minority electrons in the p-layer are sep-arated at the heterojunction and collected at the TCObased front contact. The holes are collected at the backcontact.

Manufacturing CdTe solar cells

Usually, the CdS/CdTe layers are processed usingthe closed space sublimation method. In this method, thesource and the substrate are placed at a short distancefrom each other, ranging from a few millimetres upto several cm, under vacuum conditions.The sourcecan either consist of granulates or powders of CdTe.Both, the source and the substrate are heated, where thesource is kept at a higher temperature than the sub-strate. This temperature difference induces a drivenforce on the precursors, which are deposited on thesubstrate. Such, the bulk p-type CdTe is formed. In theprocess chamber, an inert carrier gas like argon or nitro-gen can be used.

The US company First Solar Inc. is the leading com-pany in the CdTe technology [72]. Like the Germancompany Antec Solar, they use the closed space sub-limation method. Another company producing CdTe



solar cells is the German Calyxo. First Solar is by farthe largest CdTe manufacturer. In 2008, First Solar hadan annual production rate of 500 MW. In 2006 and 2007it already was one of the biggest solar module manu-facturers in the world.

The record conversion efficiency of lab-scale solar CdTesolar cells is 19.6% [38] and was obtained by GE GlobalResearch in 2013. The open circuit voltage of this cellis 857.3 mV, the short circuit current density is 28.59mA/cm2, and the FF is 80.0%. The current record mod-ule based on CdTe technology has a conversion effi-ciency of 16.1% at an area of 7200 cm2.

The current cost of modules from First Solar product isin the order of 0.68 to 0.70 USD/Wp and is expected todrop to 0.40 USD/Wp, hence keeping the cost per Wplower than that of modules based on crystalline siliconwafers.

An important aspect that needs to be addressed isthe toxicity of cadmium. However, insoluble Cd com-pounds like CdTe and CdS are much less toxic than theelementary Cd. Nonetheless, it is very important to pre-vent cadmium entering into the ecosystem. It is an im-portant question whether, CdTe modules could becomea major source of Cd pollution. AS OF»», First Solarhas an installed production capacity of 2 GW/year.With this capacity, about 2% of the total industrial Cdconsumption would be taken by First Solar. Neverthe-less, recycling schemes have been set up for installed

CdTe solar modules. For example, First Solar has a re-cycling scheme in which a deposit of 5 dollar cents perWp is included to cover the cost for the recycling at theend of the module lifetime.

Maybe an even bigger challenge for the CdTe techno-logy is the supply with tellurium. Let us take anotherlook at Fig. 13.1 that shows the abundance of the vari-ous elements in the Earth’s crust. Tellurium is one ofthe rarest stable solid elements in the Earth’s crust withan abundance of about 1 µg/kg, which is comparableto that of platinum [73]. Because of its scarcity, the sup-ply with Te might be the limiting step to upscale theCdTe PV technology to future Terawatt scales. On theother hand, Tellurium as source material has only hada few users so far. Thus, for Te no dedicated mininghas been explored so far. In addition, new supplies oftellurium-rich ores were found in Xinju, China. There-fore at this moment it remains unclear to which extentthe CdTe PV technology might be limited by the supplywith tellurium.

13.5 Organic Photovoltaics

So far we discussed inorganic thin-film semiconductormaterials such as III-V semiconductors, amorphous andnanocrystalline silicon, CIGS and CdTe solar cells. Inthis section we will take a look at organic photovoltaics.


13.5. Organic Photovoltaics 199

The used absorber materials are either conductive organicpolymers or organic molecules that are based on carbon,which may form a cyclic, a-cyclic, linear or mixed com-pound structure.

Figure 13.24 shows some example of organic materialsthat can be used for PV applications: P3HT, Phtalocy-anine, PCBM and Ruthenium Dye N3. All these ma-terials can be considered as large conjugated systems,which means that carbon atoms in the chain have analternating single or a double bond and every atomin the chain has a p-orbital available. In such conjug-ated compounds, the p-orbitals are delocalised, whichmeans that they can form one big mixed orbital. Hence,the valence electron of the original p-orbital is sharedover all the orbitals. A classical example would be thebenzene molecule, which is a cyclic conjugated com-pound. As we can see in Fig. 13.25 (a), This moleculehas 6 carbon atoms and six p-orbitals, which mix andfrom two circular orbitals that contain three electronseach. These electrons do not belong to one single atombut to a group of atoms.

In contrast, a methane molecule (CH4) is tretrahedrallycoordinated, which means that it has 4 equivalent sp3

hydride bonds with a bond angle of 109.5°. An ethene(C2H4) molecule has three equivalent sp2 hybrid bondswith a bond angle of 120° plus an electron in a pz or-bital. Two neighbouring pz orbitals form a so-called πorbital, as illustrated in 13.25 (b).

In Chapter 6 we have discussed that two individualsp3 hybrid orbitals can make an anti-bonding and abonding state. The same is valid for the two pz orbitalsforming a molecular π orbital. They can make bond-ing and anti-bonding π states. Therefore, conjugatedmolecules can have similar properties as semiconductormaterials.

At room temperature, most electrons are in the bondingstate, which is also called the highest occupied molecu-lar orbital (HUMO). The anti-bonding state can be con-sidered as the lowest unoccupied molecular orbital (LUMO).As the conjugated molecules are getting longer, theHOMO and LUMO will broaden and act similar tovalence and conduction band in conventional semicon-ductors. The energy difference between the HOMO andLUMO levels can be considered as the band gap of thepolymer material.

To discuss whether an organic material is p-type or n-type, we first have to introduce the concept of the va-cuum level, shown in Fig. 13.26 (a). The vacuum level isdefined as the energy of a free stationary electron thatis outside of any material, or in other words, in vacuo.This level often is used as the level of alignment forthe energy levels of two different materials. The ionisa-tion energy is the energy required to excite an electronfrom the valence band or HOMO to the vacuum state.The electron affinity is the energy obtained by movingan electron form the vacuum just outside the semicon-


anonymous

anonymous 10/12/2014: The Acronym acording to the definition is HOMO not HUMO


O

O

N

N

COOH

COO-

N

N

COOH

COO-

Ru

N

SC

NC

S

PCBM-C61

P3HT Phtalocyanine

Rhutenium dye N3

Figure 13.24: Some examples of organic molecules used for organic photovoltaics.


13.5. Organic Photovoltaics 201

Benzene6 pz orbitals

Benzenedelocalisedπ orbital

H

H

H

HH

H

H

H

H

HH

H

(a) (b)

Ethenedelocalisedπ orbital

H

H

H

H

H

H

H

H

Figure 13.25: The chemical structure and π orbitals of (a) benzene and (b) ethene.

ductor or conjugated polymer to the bottom of the con-duction band or LUMO. When a material has a lowionisation potential, it can release an electron out ofthe material relatively easy, i.e. it can act as an electrondonor. On the other hand, when a material has a highelectron affinity, it can easily accept an additional elec-tron in the LUMO or conduction band, it thus acts asan electron acceptor.

As we have discussed earlier, in inorganic semiconduct-ors an electron can be excited from the valence bandto the conduction band leaving a hole in the valenceband. Such an electron-hole pair is only weakly boundand both entities are easily separated and can diffuseaway from each other. In organic materials this is notthe case. Absorption of a photon with sufficient en-

ergy results in the creation of an exciton, illustrated inFig. 13.26 (b) ¶. An exciton is an excited electron-holepair that is still in bound state because of the mutualCoulomb forces between the particles. Such excitonscan diffuse through the material, but they have a lowlife time in organic materials, recombining back to theground state within a few nanoseconds. Hence, the dif-fusion length of such excitons is in the order of only 10nm.

If an electron donor and an electron acceptor materialare brought together, an interface is formed betweenthose two. The HOMO and LUMO of both polymerscan be aligned considering their energy levels with ref-erence to the vacuum level, as illustrated in Fig. 13.26(b). At the interface we see a difference in the HOMO



HOMO

LUMO

electron donor electron acceptor

vacuum levelelectron anityionisation energy

HOMO

LUMO

HOMO

LUMO

electron donor electron acceptor

vacuum level

HOMO

LUMO

exciton

(a)

(b)

Figure 13.26: Illustrating (a) the energy levels in organic PVmaterials and (b) the separation of electrons and holes in anexciton at the acceptor-donor interface..

(a) (b)

Electrode 1 (e.g. ITO)

Electrode 2 (Al, Mg, Ca)

electron acceptor

electron donor

diffusion length Electrode 1 (e.g. ITO)

Electrode 2 (Al, Mg, Ca)

bulkhetero-junction

Figure 13.27: Illustrating (a) the layer structure of organicsolar cells and (b) an organic solar cell with a bulk heterojunction.

and LUMO levels. Because of this difference, an elec-trostatic force exists between the two materials. If thematerials are chosen such that the difference is largeenough, these local electric fields can break up the ex-citon ·. The electron then can be injected into the elec-tron acceptor and a hole remains in the electron donormaterial ¸.

Figure 13.27 (a) shows the structure of a typical organicsolar cell. Here, we consider a solar cell consisting ofboth organic acceptor-type and donor-type materials.Similar as for semiconductor materials, a heterojunctionbased on two different materials or conjugated com-pounds can be constructed. As mentioned before, the


13.6. Hybrid organic-inorganic solar cells 203

typical diffusion length in the organic materials is onlyabout 10 nm. Hence, the thickness of the solar cell inprinciple is strongly limited by the diffusion length,while it has to be at least 100 nm to absorb a suffi-cient fraction of the light. Therefore, organic solar cellsare based on bulk heterojunction photovoltaic devicesthat is illustrated in Fig. 13.27 (b), where the electron-donor and the electron-acceptor materials are mixedtogether and form a blend. In this way, typical lengthscales in the order of the exciton diffusion lengths canbe achieved. Hence, a large fraction of the excitonscan reach an interface, where they are separated intoan electron and a hole. The electrons move though theacceptor material to the electrode and the holes movethrough the donor material to be collected at the otherelectrode. Usually, the holes are usually collected at aTCO electrode, for example indium tin oxide (ITO). Theelectrons are collected at a metal back electrode.

The record organic solar cells are based on double junc-tions. Heliatek achieved a 12.0% solar cell efficiency onlab-scale, with unknown stability.

An advantage of organic solar cells is that they havelow production costs. With chemical engineering theband gap can be tuned. Organic solar cell can be integ-rated into flexible substrates. Important disadvantagesare a low efficiency, low stability and low strength com-pared to inorganic PV cells.

Although organic solar cells themselves can be cheaply,

the need of highly expensive encapsulation materialsto protec the organic materials from humidity, moistureand air limits the industrial application. To my know-ledge, no company is producing organic PV modules atthe moment. Konarka Technologies has been active inthe past. Their small PV modules had efficiencies in therange of 3 up to 5% and worked for a couple of yearsonly.

13.6 Hybrid organic-inorganicsolar cells

In this section we discuss two hybrid concepts, wherethe junction consists of both inorganic and organic com-pounds. These concepts are dye-sensitised solar cells andperovskite solar cells.

13.6.1 Dye-sensitised solar cells

An alternative solar cell concept based on organic ma-terials is the dye-sensitised solar cell (DSSC), which isa photo-electrochemical system. It contains titanium di-oxide (TiO2) nanoparticles, organic dye particles, an elec-trolyte and platinum contact. Figure 13.28 (a) shows anillustration of a DSSC. The dye sensitiser, which actslike an electron donor, is the photo-active component



Transparent Electrode

Counter Electrode

Electrolyte

Light

TiO2

l3 3l

Valence band

Conduction band

HOMO

LUMO

DyeTiO2

S *

SI3

-

I-

Electrolyte Pt

2Dye+ + 3I- → 2Dye + I3-

I3- + 2e - → 3I-

+

(a)

(b)

Figure 13.28: (a) A sketch of a dye sensitised solar cell and (b)the relevant energy levels in its operation.

of the solar cell. The second major component are theTiO2 nanoparticles that act as the electron–acceptor.Similar to organic bulk heterojunction cells, the dye ismixed with TiO2.

As a photoactive dye sensitiser, ruthenium polypyridineis used. The operation is illustrated in 13.28 (b). If aphoton is absorbed by the ruthenium polypyridine, itcan excite an electron from its ground state, the S state,to an excited state, referred to as S*. In this case the Scan be considered as a HOMO and the S* state can beconsider as LUMO. The S* energy level is above theenergy level of the conduction band of the TiO2. As aresult, the light excited electrons are injected into theTiO2 nanoparticles, while the dye-sensitiser molecule re-mains positively charged. The electrons move throughthe TiO2 to the TCO based back contact in a diffusionbased transport mechanism. Via the electric circuit, theelectrons move to the front contact. The front contact iselectrically connected to the dye via an electrolyte. Elec-trolytes are solutions or compounds that contain ionisedentities that can conduct electricity. The typical elec-trolyte used for DSSC contains iodine. The positivelycharged oxidised dye molecule is neutralised by a neg-atively charge iodide. Three negatively charged iodides,neutralise two dye molecules and create one negativelycharged tri-iodide. This negatively charged tri-iodidemoves to the counter electrode where it is reduced us-ing two electrons into three negatively charged iodine.To facilitate the chemical reactions, these photoelectro-


13.6. Hybrid organic-inorganic solar cells 205

chemical cells require a platinum back contact.

The performance of a DSSC depends on the HOMOand LUMO level of the dye material, the Fermi level ofTiO2 nanoparticles and the redox potential of the iodideand triodide reactions. The current record efficiency ofdye-sensitised PV devices on lab-scale is 11.9% and wasachieved by Sharp [38].

The major advantage of DSSC are the low productioncosts. A disadvantage is the stability of the electrolyteunder various weather conditions: At low temperaturesthe electrolyte can freeze, which stops the device fromgenerating power and might even result in physicaldamage. High temperatures result in thermal expansionof the electrolyte, which make encapsulating modulesmore complicated. Another challenge is the high cost ofthe platinum electrodes, hence replacing the platinumwith cheaper materials is a topic of ongoing research.Secondly, more stable and resistive electrolyte mater-ials must be developed. Thirdly, research is done onimproved dyes that enhance the spectral and band gaputilisation of the solar cells.

Yet, no dye-sensitised PV products are available com-mercially.


14Third Generation Concepts

The term third generation photovoltaics refers to all novelapproaches that aim to overcome the Shockley-Queisser(SQ) single bandgap limit, preferably at a low cost. TheSQ limit that we discussed extensively in Chapter 10 isa thermodynamic approach to estimate the maximumefficiency of a single junction solar cell in dependenceof the bandgap of its semiconductor material. For itsderivation we assumed that the AM1.5 spectrum is in-cident on a solar cell. Further, we do not allow the solarcell to increase in temperature but we force it to keepthe ambient temperature of 300 K. This means that allenergy absorbed by the solar cell can escape the solarcell by either the generated current density or by ra-diative recombination of charge carriers. Under theseassumptions the efficiency limit is around 33% in the

band gap range from 1.0 eV up to 1.8 eV.

Now we are going to look at some very fundamentallimitations of classical single-junction solar cells.

First, in single-junction solar cells only on band gap ma-terial is used. Hence, a large fraction of the energy ofthe most energetic photons is lost as heat as illustratedin Fig. 10.7.

Secondly, most solar cells concepts are based on an in-cident irradiance level of 1 sun. However, higher irra-diance means more current generation and also highervoltage levels, resulting in a higher overall efficiency.

Thirdly, every photon only excites one electron in theconduction band creating only one electron-hole pair.

207


208 14. Third Generation Concepts

The energy of highly energetic photons could be util-ised better if they would create more than one excitedelectron in the conduction band.

Fourthly, the photons with energies below the bandgapare not used. Hence, they do not result in charge carrierexcitation, as illustrated in Fig. 10.7.

If these fundamental limitations could be solved, PVconcepts with conversion efficiencies exceeding theShockley-Queisser limit could be developed. In thischapter, we will discuss several third generation con-cepts: Multi-junction solar cells, concentrator photovol-taics, spectral up and down conversion, multi-excitongeneration, intermediate band-gap solar cells and hotcarrier solar cells. Note, that besides multi-junctionand the concentrator approach, none of these conceptshave resulted in high efficiency solar cells or even beendemonstrated yet. These other concepts are still in fun-damental research phase and it is not clear whetherthey will ever become a large scale PV technology [74].

14.1 Multi-junction solar cells

The first limitation discussed in the list above can beattacked by using multi-junction solar cells. Dependingon the author, these solar cells are seen as part of thesecond or the third generation, and indeed we alreadybriefly discussed them in Chapter 13 on thin-film solar

cells when looking at the III-V technology (Section 13.2)and thin-film silicon technology (Section 13.3). For com-pleteness, we here will summarise this technology.

In multi-junction cells, several cell materials with dif-ferent bandgaps are combined in order to maximise theamount of the sun light that can be converted into elec-tricity, as we already illustrated in Fig. 10.9. To realisethis, two or more cells are stacked onto each other. Thetop cell has the highest bandgap, in order to absorb andconvert the short wavelength (blue) light. Light withwavelengths longer than the bandgap-wavelength cantraverse the top cell and be absorbed in the cells belowwith lower bandgaps. The bottom cell has the lowestbandgap to absorb the long wavelength (red and nearinfrared) light. In order to optimise the performance ofmulti-junction solar cells with two electrical terminals,matching the currents of all the subcells (current match-ing) is crucial. Multi-junction cells with more termin-als do not have this restriction, but their production ismore complicated.

In thin-film silicon tandem cells, an a-Si:H top cell isstacked onto a nc-Si:H bottom cell. In order to achievecurrent matching, the top cell is much thinner than thebottom cell. The cell can be further optimised by usingan intermediate reflector between the top and the bot-tom cell in order to reflect the blue light back into thetop cell while letting the red light pass to the bottomcell. The reported record efficiency of a-Si:H/nc-Si:H


14.2. Spectral conversion 209

tandem cells is 12.3% and for a a-Si:H/nc-Si:H/nc-Si:Hit is 13.4% [38].

Multi-junction cells containing III-V semiconductors areat present the most efficient solar cells. The currentworld record efficiency is 44.4% for a triple-junctionGaInP/GaAs/GaInNAs cell that is used in a concen-trated PV-system [38]. Because of the high concentra-tion of 302 the overall efficiency increases and hencewe also tackle the second limitation mentioned in thetable above. As a result the SQ limit can be exceededby more than 10%.

14.2 Spectral conversion

Single-junction solar cells have the limitation that everyphoton can only generate one collected electron at amaximum. In theory, this limitation can be tackled byspectral conversion. The main idea of spectral conversionto use an additional layer consisting of a “magical” ma-terial such that the incident spectrum can be altered.Materials that are investigated for spectral conversionare organic dyes, quantum dots, lanthanide ions, andtransition metal ion systems [75].

h+

vs.

e-C

V

C

V

upconversion

Figure 14.1: Illustrating the principle of spectral up-conversion.

14.2.1 Spectral upconversion

In spectral upconversion, two or more low-energyphotons excite electron-hole pairs in several steps, asillustrated in Fig. 14.1. If these electron-hole pairs re-combine radiatively, they emit a photon with a higherenergy. As a result, two or more low-energy photonsare converted in one high-energy photon, which can beabsorbed in the PV-active material.

The up-converting layers should be placed at theback of the solar cell as low-energy photons can passthrough the solar cell absorber to the up-convertinglayer and can be converted there to high energyphotons that are absorbed by the solar cell in a secondpass [75]. In this way, also up-converting layers with



h+

vs.

e-C

V

C

V

downconversion

h+

e-

h+

e-

Figure 14.2: Illustrating the principle of spectral down-conversion.

low efficiencies become interesting. Due to parasitic ab-sorption it is not advised to put these layers at the frontof the solar cell.

14.2.2 Spectral downconversion

The idea of spectral down-conversion is to split one high-energy photon in multiple lower energetic photons [76],as illustrated in Fig 14.2. A high-energy photon is ab-sorbed at the front of the solar cell and converted intoat least two photons with lower energies. If the energyof the initial photon is Eph > 2Eg and the energy theresulting two photons is still larger than that of theband gap of the absorber material, both photons canbe absorbed and used for exciting charge carriers. As aresult, a high energetic photon, for example in the blue,

can result in two excited electrons in the blue part. Inother words, the maximum theoretical EQE of 100% atthe wavelength of the blue photon can be increased to200%. If the photon had sufficient energy to be splitinto three photons with an energy higher than the bandgap, a theoretical EQE of 300% could be obtained. Indifference to upconversion, a down-converting layerhas to be at the front of the solar cell, as highly ener-getic photons are always absorbed in the absorber layer.Hence, parasitic absorption might be a problem in thistechnology.

One possibility that is investigated for realising spec-tral down-conversion is to use so-called quantum dots(QDs). These are small spherical nanoparticles made ofsemiconductor materials with typical diameters of a fewnanometers, as illustrated in Fig. 14.3 (a). These semi-conductor particles still behave like a semiconductormaterial; however, due to so-called quantum confinementthe band gap of the semiconductor quantum dots canbe larger than that of the same semiconductor in a bulkconfiguration. The band gap of the QDs can be tunedby varying their size. The smaller the particles, the lar-ger the band gap. This enables interesting opportunitiesfor bandgap engineering, such as a multijunction solarcells based on junctions with different QDs of differentsizes in every junction.

If QDs should be used for down-conversion, an en-semble of nanoparticles is embedded in a host mater-


14.2. Spectral conversion 211

CB

VB

1Pe

1Se

1Sh

1Ph

Eg

CB

VB

2 nm 6 nm

CB

1 3 3

2

Si NCSi NC

CB

VBVB

3

3

2

1

Process:

(a)

(b)

Figure 14.3: Illustrating (a) the effect of the quantum dot(QD) size on the bandgap and (b) spectral down-conversionwith QDs.

ial, where the particles are in very close proximity ofeach other. Figure 14.3 (b) shows the electronic bandgap diagram of two nanoparticles. Now, a high-energyphoton is absorbed by one QD and hence one electronis excited into the conduction band of the particle. Indifference to a bulk semiconductor, the excess energyof the photon is not necessarily lost as heat, but it canbe transferred as a quantised energy package to a neigh-bouring quantum dot. Here a second electron is excitedinto conduction band of the second quantum dot. As aresult, two electron-hole pairs have been created out ofone photon. If non-radiative recombination mechanismslike Auger recombination and SRH recombination aresufficiently suppressed, the electron-hole pairs in bothquantum dots can radiatively recombine such that eachof the two QDs emits one reddish photon. In summary,one incident bluish photon is converted in to two red-dish photons, which can be absorbed by a PV material.

Figure 14.4 shows some experimental results byJurbergs et al. on down conversion based on siliconquantum dots in a narrow spectral range [77]. The hori-zontal axis represents the photon emission wavelength.At around 790 nm a down-conversion efficiency of 60%achieved. The EQE exceeds 100% in the blue regionfrom 3.1 up to 3.4 eV. The major challenge is have QDlayers with a spectral response exceeding 100% at lowerphoton energies, because the solar spectrum containsfar more photons in that spectral range.



Figure 14.4: Down conversion at Si quantum dots [77].

CB

1

2

Si NCSi NC

CB

VBVB

2

1

Process:

Figure 14.5: Illustrating multi exciton generation withquantum dots.

In practice, only small enhancements in efficiency dueto up/down-converters have been reported [74].

14.3 Multi-exciton generation

Another approach to enhance the charge carrier excit-ation by a single energetic photon is called multiple ex-citon generation (MEG). Here, more than one electron-hole pairs is generated from high energy photons. Indifference to spectral down-conversion here all thefundamental energy conversion steps occur in the PV-active layers.

Like down conversion, in principle MEG can be realisedwith quantum dots, as illustrated in Fig. 14.5. Again, in


14.4. Intermediate band solar cells 213

E mutnau

Q lanretxE

)%( ycneiciff

120

110

100

90

80

70

603.63.43.23.02.82.6

Photon Energy (eV)

Bandgap: 0.73 eV 0.72 eV 0.71 eV AR coated

Au

PbSe QDs

ITO

ZnO

Glass

Figure 14.6: Multi-exciton generation using PbSe quantumdots with EQEs exceeding 100% [78].

one particle an electron is excited into the conductionband and the excess energy is transferred to a neigh-bouring QD, where a second electron is excited in toconduction band of the second quantum dot. However,here the charge carriers in the two electron-hole pairsare separated before they can recombine such that oneincident photons results in more then one generatedelectrons [79]. Just as for spectral down-conversion, alsohere quantum efficiencies exceeding 100% are theoretic-ally possible when one incident photon creates statist-ically more than charge carriers. Figure 14.6 shows thatEQEs exceeding 100% can be achieved as demonstratedby Semonin et al. [78]. Here, the absorber layer consistsPbSe quantum dots.

eV

IB material np

+

(a) (b)

+

+

Figure 14.7: Band diagram (a) with the intermediate band de-picted and (b) with the quasi-Fermi levels exceeding the energyof the low-energy photons.

14.4 Intermediate band solar cells

The concept of intermediate band solar cells (IB) tries totackle the problem photons with energies below thebandgap cannot be utilised for current generation. Asshown in Fig. 14.7 (a), in intermediate band cells energylevels are created artificially in the bandgap of the ab-sorber material [80]. As in conventional single-junctionsolar cells, photons with a sufficient energy can excitean electron from the valence band in to the conductionband. However, in difference to conventional semicon-ductors photons with energies below the bandgap canexcite an electron from the valence band in to the inter-



mediate band. A second low-energy photon is requiredto excite the electron from the intermediate band intothe conduction band. As illustrated in 14.7 (a), thereforetwo photons with small energies can result in quasi-Fermi level splitting larger than the energy of each ofthese photons.

Various studies have been performed on how interme-diate band cells can be realised, as for example sum-marised in Reference [80]. In such solar cells, a layerwith the intermediate states is placed in between p-and n-layers. For example quantum dots can be usedto realise the intermediate states. Further, various bulkmaterials are studied for realising the intermediatestates. One major problem of experimental IB cells isthat the voltages are lower than the voltages of refer-ence cells without IB structures. Yet, the efficiency lossdue to the lower voltage cannot be overcome by sub-bandgap current density originating from the IB trans-itions.

14.5 Hot carrier solar cells

The idea of hot carrier solar cells is to reduce the energylosses due to relaxation and hence thermalisation. Asillustrated in Fig. 14.8, this should be achieved by col-lect electron-hole pairs of high energy photons just afterlight excitation before they have a chance to relax back

Eg

electroncontact

hot electron and holeenergy distributions

holecontact

Selective energycontacts

eV

V > Eg

Figure 14.8: Illustrating the working principle of a hot carriersolar cell.


14.5. Hot carrier solar cells 215

to the edges of the electronic bands. In the figure, thepopulation of the charge carrier levels reflects the situ-ation just after the excitation by the absorption of aphoton. This distribution is not in thermal equilibriumas many electrons are excited into position further up inthe conduction band and the holes are excited down tolower levels in the valence band. These charge carriersare called hot electrons and holes [22, 81].

It takes only a few picoseconds (10−12 s) for the hotcharge carriers to relax back to the edges of the elec-tronic bands. The idea of hot carrier cells it to collectthe charge carriers as long as they are still hot. Hence,an energy larger than the band gap energy can be util-ised per excited charge carrier and the average bandgaputilisation would exceed the bandgap.

The fundamental challenge is to collect the hot carri-ers before they relax back to the edges of the electronicbands. Such a concept would require selective contacts,which only select electrons above a particular energylevel in the conduction band and holes below a certainenergy level in the valence band, respectively. At themoment the main challenge is to increase the lifetimeof the hot charge carriers, such that they have time tomove from the absorber material to the selective con-tacts.


Part IV

PV Systems

217


15Introduction to PV systems

15.1 Introduction

After we have discussed the fundamental scientific the-ories required for solar cells in Part II and have taken alook at modern PV technology in Part III, we now willuse the gained knowledge to discuss complete PV sys-tems. A PV system contains many different componentsbesides the PV modules. For successfully planning aPV system it is crucial to understand the function of thedifferent components and to know their major specific-ations. Further, it is important to know the effect on thelocation of the (expected) performance of a PV system.

15.2 Types of PV systems

PV systems can be very simple, consisting of just a PVmodule and load, as in the direct powering of a wa-ter pump motor, which only needs to operate when thesun shines. However, when for example a whole houseshould be powered, the system must be operationalday and night. It also may have to feed both AC andDC loads, have reserve power and may even include aback-up generator. Depending on the system configur-ation, we can distinguish three main types of PV sys-tems: stand-alone, grid-connected, and hybrid. The ba-sic PV system principles and elements remain the same.Systems are adapted to meet particular requirements byvarying the type and quantity of the basic elements. A

219


220 15. Introduction to PV systems

DCDCDC

PV Modules

Water pump

Water reservoirDC

(a) (b)

PV Modules

Charge controller

Inverter

Batteries

Loads

DC

AC=

~

Figure 15.1: Schematic representation of (a) a simple DC PV system to power a water pump with no energy storage and (b) acomplex PV system including batteries, power conditioners, and both DC and AC loads.


15.2. Types of PV systems 221

modular system design allows easy expansion, whenpower demands change.

15.2.1 Stand-alone systems

Stand-alone systems rely on solar power only. Thesesystems can consist of the PV modules and a load onlyor they can include batteries for energy storage. Whenusing batteries charge regulators are included, whichswitch off the PV modules when batteries are fullycharged, and may switch off the load to prevent thebatteries from being discharged below a certain limit.The batteries must have enough capacity to store theenergy produced during the day to be used at nightand during periods of poor weather. Figure 15.1 showsschematically examples of stand-alone systems; (a) asimple DC PV system without a battery and (b) a largePV system with both DC and AC loads.

15.2.2 Grid-connected systems

Grid-connected PV systems have become increasinglypopular for building integrated applications. As illus-trated in Fig. 15.2, they are connected to the grid viainverters, which convert the DC power into AC electri-city. In small systems as they are installed in residen-tial homes, the inverter is connected to the distribution

PV Modules

AC

Inverter

Loads

=~

Distribution board

Electricity grid

Figure 15.2: Schematic representation of a grid-connected PVsystem.



board, from where the PV-generated power is trans-ferred into the electricity grid or to AC appliances inthe house. These systems do not require batteries, sincethey are connected to the grid, which acts as a bufferinto that an oversupply of PV electricity is transportedwhile the grid also supplies the house with electricity intimes of insufficient PV power generation.

Large PV fields act as power stations from that all thegenerated PV electricity is directly transported to theelectricity grid. They can reach peak powers of up toseveral hundreds of MWp. Figure 15.3 shows a 25.7MWp system installed in Germany.

15.2.3 Hybrid systems

Hybrid systems consist of combination of PV modulesand a complementary method of electricity generationsuch as a diesel, gas or wind generator. A schematic ofan hybrid system shown in Fig. 15.4. In order to optim-ise the different methods of electricity generation, hy-brid systems typically require more sophisticated con-trols than stand-alone or grid-connected PV systems.For example, in the case of an PV/diesel system, thediesel engine must be started when the battery reachesa given discharge level and stopped again when batteryreaches an adequate state of charge. The back-up gener-ator can be used to recharge batteries only or to supplythe load as well.

Figure 15.3: The 25.7 MWp Lauingen Energy Park in Bav-arian Swabia, Germany [82].


15.3. Components of a PV system 223

PV Modules

Charge controller

Inverter

Batteries

DC

AC

=~

Back-up generator

Figure 15.4: Schematic representation of a hybrid PV systemthat has a diesel generator as alternative electricity source..

15.3 Components of a PV system

As we have seen earlier in this book, a solar cell canconvert the energy contained in the solar radiation intoelectrical energy. Due to the limited size of the solarcell it only delivers a limited amount of power underfixed current-voltage conditions that are not practicalfor most applications. In order to use solar electricityfor practical devices, which require a particular voltageand/or current for their operation, a number of solarcells have to be connected together to form a solar panel,also called a PV module. For large-scale generation ofsolar electricity solar panels are connected together intoa solar array.

Although, the solar panels are the heart of a PV sys-tem, many other components are required for a workingsystem, that we already discussed very briefly above.Together, these components are called the Balance of Sys-tem (BOS). Which components are required depends onwhether the system is connected to the electricity gridor whether it is designed as a stand-alone system. Themost important components belonging to the BOS are:

• A mounting structure is used to fix the modules andto direct them towards the sun.

• Energy storage is a vital part of stand-alone sys-tems because it assures that the system can de-liver electricity during the night and in periods of



bad wheather. Usually, batteries are used as energy-storage units.

• DC-DC converters are used to convert the moduleoutput, which will have a variable voltage depend-ing on the time of the day and the weather con-ditions, to a fixed voltage output that e. g. can beused to charge a battery or that is used as input foran inverter in a grid-connected system.

• Inverters or DC-AC converters are used in grid-connected systems to convert the DC electricityoriginating from the PV modules into AC electri-city that can be fed into the electricity grid.

• Cables are used to connect the different compon-ents of the PV system with each other and to theelectrical load. It is important to choose cables ofsufficient thickness in order to minimise resistivelosses.

Even though not a part of the PV system itself, the elec-tric load, i.e. all the electric appliances that are connec-ted to it have to be taken into account during the plan-ning phase. Further, it has to be considered whether theloads are AC or DC loads.

The different compontents of a PV system are schem-atically presented in Fig. 15.5 and will be discussed indetail in Chapter 17.

=~ DC

AC

PV Modules Battery Inverter DC load

Charge controller

Back-up generator

AC load

Figure 15.5: A schematic of the different components of a PVsystem.


16Location issues

In Chapter 5 we discussed solar radiation on earth andintroduced the AM1.5 spectrum, normalised to a totalirradiance of 1000 W/m2. This spectrum is used toevaluate the performance of solar cells and modulesin laboratories and industry. The AM1.5 spectrum rep-resents the solar irradiance if the centre of the solar discis at an angle of 48.2° off the zenith (or 41.2° above thehorizon).

Of course, the sun not always is at this position, butthe position is dependent on the time of the day andthe year and of course it is dependent on the locationon earth. In this section we will discuss how to calcu-late the position of the sun at every location on earthat an arbitrary time and date. Furthermore we willdiscuss scattering of sunlight when it traverses the at-

mosphere and how this influences the direct and dif-fuse spectrum. We also will discuss the influence of themounting angle and position of a PV module on theirradiance at the module.

16.1 The position of the sun

For planning a PV system it is crucial to know the po-sition of the sun in the sky at the location of the solarsystem at a given time. In this section we explain howthis position can be calculated.

Since celestial objects like the sun, the moon and thestars are very far away from the earth it is convenientto describe their motion projected on a sphere with ar-

225


I

226 16. Location issues

Observer

S180°

Zenith

Celestial sphere

Object

N0°

W270°

E90°

Azimuth A Horizon

Merid

ian

Alti

tude

a

ξ

ζυ

Figure 16.1: Illustrating the definition of the altitude a and theazimuth A in the horizontal coordinate system. Note that Northis at the bottom of the figure.

bitrary radius and concentric to the earth. This sphereis called the celestial sphere. The position of every ce-lestial object thus can be parameterised by two angles.For photovoltaic applications it is most convenient touse the horizontal coordinate system, where the horizonof the observer constitutes the fundamental plane. In thiscoordinate system, the position of the sun is expressedby two angles that are illustrated in Fig. 16.1: The alti-tude a that is the angular elevation of the centre of thesolar disc above the horizontal plane. Its angular rangeis a ∈ [−90, 90], where negative angles correspondto the object being below the horizon and thus not vis-ible. The azimuth A that is the angle between the line ofsight projected on the horizontal plane and due North.It is usually counted eastward, such that A = 0, 90,180, 270 correspond to due North, East, South andWest, respectively. Its angular range is A ∈ [0, 360]. Ina different convention also used by the PV community,due South corresponds to 0 and is counted westward,the angles then are in between −180 and 180. Figure16.1 also shows the meridian, which is great circle on thecelestial sphere passing through the celestial North andSouth poles as well as the zenith.

Instead of using the spherical coordinates a and A, wealso could use Cartesian coordinates, that we here callξ (xi), υ (upsilon) and ζ (zeta) and that are also depic-ted in Fig. 16.1. The principal direction is parallel to ξ.The Cartesian coordinates are connected to the spherical


16.1. The position of the sun 227

coordinates via ξυζ

=

cos a cos Acos a sin A

sin a

. (16.1)

Note that ξ2 + υ2 + ζ2 = 1 for all points on the celestialsphere.

Earth orbits the sun in an elliptic orbit at an averagedistance of about 150 million kilometres. Due to the el-liptic orbit the speed of Earth is not constant. This isbecause of Kepler’s second law that states that “A linejoining a planet and the Sun sweeps out equal areasduring equal intervals of time.” On the celestial spherethe sun seems to move on a circular path with one re-volution per year. This path is called the ecliptic andillustrated in Fig. 16.2. For describing the apparentmovement of the sun on the celestial sphere is is con-venient to use coordinates in that the ecliptic lies in thefundamental plane. These coordinates are called the ec-liptic coordinates. As principal direction, the position ofthe sun at the spring (vernal) equinox (thus around 21March) is used, which is indicated by the sign of Aries,à. As obvious from Fig. 16.2, à lies both in the eclipticplane as well as in the equatorial plane. The ecliptic co-ordinate system is sketched in Fig. 16.3 (a). The two an-gular coordinates are called the ecliptic longitude λ andthe ecliptic latitude β. Note, that in this coordinate sys-tem the rotation of the earth around its axis is not takeninto account.

North celestialpole

21 Dec

21 Mar

21 Jun

23 Sep EclipticCelestial equator

ε

Vernal equinox

Axial tilt

Northecliptic

pole

EARTH

Celestial s

phere

a

rctic circle

Figure 16.2: Illustrating the ecliptic, i.e. the apparent move-ment of the sun around earth. Further, the celestial equator andthe direction of the vernal equinox are indicated. The sizes of Sunand Earth are not in scale.



In ecliptic coordinates approximate position the pos-ition of the sun can be expressed easily. The approx-imation presented here has an accuracy of about 1 ar-cminute within two centuries of 2000 and is publishedby the the Astronomical Applications Departement of theU.S. Naval Observatory.

To express the position of the sun we first have to ex-press the time D elapsed since Greenwich noon, Ter-restrial Time, on 1 January 2000, in days. For astro-nomic purposes it may be convenient to relate D to theJulian date JD via

D = JD− 2451545.0. (16.2)

The Julian Date1 that is defined as the number of dayssince 1 January 4713 BC in a proleptic2 Julian calendaror since 24 November 4717 BC in a proleptic Gregoriancalendar.

Now, the mean longitude of the sun corrected to the ab-erration of the light is given by

q = 280.459 + 0.98564736D (16.3)

Because of the elliptic orbit of earth and hence a vary-ing speed throughout the year, we have to correct withthe so-called mean anomaly of the Sun,

g = 357.529 + 0.98560028D. (16.4)1Calculating the Julian Date is implemented in MatLab.2Proleptic means that a calendar is applied to dates before its introduction.

It may be convenient to normalise q and g to the range[0, 360) by adding or subtracting multiples of 360.

Now the ecliptic longitude of the sun is given by

λS = q + 1.915 sin g + 0.020 sin 2g. (16.5)

The ecliptic latitude can be approximated by

βS = 0. (16.6)

For estimating the radiation it might also be convenientto approximate the distance of the Sun from the Earth.In astronomical units (AU) this is given by

R = 1.00014− 0.01671 cos g− 0.00014 cos 2g. (16.7)

As stated above, for PV applications it is convenient touse horizontal coordinates. We therefore have to trans-form from ecliptic coordinates into the horizontal co-ordinates. This is done via three rotations that are to beperformed after each other consecutively:

First, we have to transform from ecliptic coordinatesinto equatorial coordinates. As illustrated in Fig. 16.2,the fundamental plane of these coordinates is tilted tothe ecliptic with an angle ε,

ε = 23.429 − 0.00000036D (16.8)

The principal direction is again given by the vernalequinox à. In Fig. 16.3 (b) the equatorial coordinate



Eclipticlongitude λ

Object

Cele

stial

Sphere

Centre of the Earth

(a)North ecliptic pole

Ecliptic

Ecliptic

altitude β

Object

North celestial pole

Cele

stial

Sphere

Celestial equator

Declination δ Rightascension α

Centre of the Earth

(b)

Figure 16.3: Illustrating (a) the ecliptic coordinate system and (b) the equatorial coordinate system.



system is sketched. The two coordinates are called theright ascension α and the declination δ. The transforma-tion from ecliptic coordinates to equatorial coordinatesis a rotation by the angle ε about the vernal equinox asrotational axis. Mathematically this is expressed bycos δ cos α

cos δ sin αsin δ

=

1 0 00 cos ε − sin ε0 sin ε cos ε

cos β cos λcos β sin λ

sin β

.

(16.9)

Secondly, we have to take the rotation of the eartharound its axis into account. We do this by using theso-called hour angle h instead of the right ascension α.Those two angles are connected to each other via

h = θL − α, (16.10)

where θL is the local mean sidereal time, i.e. the anglebetween the vernal equinox and the meridian. All theseangles are illustrated in Fig. 16.4. A sidereal day isthe duration between two passes of the vernal equi-nox through the meridian and it is slightly shorterthan a solar day. We can understand this by realisingthat the earth has to rotate by 360 and approximately360/365.25 between two passes through the meridian.The duration of mean sidereal day is approximately 23h, 56 m and 4 s.

For calculating θL we first have to determine the Green-wich Mean Sidereal Time (GMST), which is (approxim-

GMST

GREENWICH

OBSERVER

N

longitude λ0

right ascension αLM

ST θL

local hour angle h

EARTH

SunVernal equinox

Figure 16.4: Illustrating the right ascension α, the local hourangle h, the Greenwich Mean Sidereal Time GMST and the localmean sidereal time θL.



ately) given by

GMST =18.697374558 h+ 24.06570982441908 h · D+ 0.000026 h · T2,

(16.11)

where D is as defined above and T is the number ofcenturies past since Greenwich noon, Terrestrial Time,on 1 January 2000,

T =D

36525. (16.12)

For many applications, the quadratic term may be omit-ted. GMST is given in hours and has to be normalisedto the range [0 h, 24 h). We then can obtain the localmean sidereal time in degrees with

θL = GMST15

hour+ λ0, (16.13)

where λ0 is the longitude of the observer.

We thus have to the following transform about the rota-tional axis of the earth,cos δ cos h

cos δ sin hsin δ

=

cos θL sin θL 0sin θL − cos θL 0

0 0 1

cos δ cos αcos δ sin α

sin δ

.

(16.14)Note that this transform is no rotation but a reflectionat an angle of θL/2. We thus now transformed the prin-cipal direction of the coordinate system from vernalequinox to the local mean sidereal time.

Thirdly we transform to the horizontal coordinate sys-tem by rotating with the latitude angle φ0 of the ob-server. While the ecliptic and equatorial coordinate sys-tems use the centre of the earth as origin, the horizontalcoordinate system uses the actual position on the sur-face of Earth as origin. However, because of the dis-tance of celestial objects in general and the Sun in par-ticular being much larger than the radius of the Earth,we may neglect this translational shift of the origin ofthe coordinate systems.

The rotational axis of the third transform is the axis thatis normal to both the principal direction θL and the ro-tational axis of the earth (δ = 90).ξ ′

υ′

ζ

=

sin φ0 0 − cos φ00 1 0

cos φ0 0 sin φ0

cos δ cos hcos δ sin h

sin δ

. (16.15)

However, here the directions ξ ′ and υ′ point to dueSouth and West, respectively. We thus must applyξ ′ → −ξ and υ′ → −υ in order to get the directionssuch as they are defined in Fig. 16.1 (ξ and υ pointingto due North and East, respectively). We do this withthe matrix transformξ

υζ

=

−1 0 00 −1 00 0 1

ξ ′

υ′

ζ

. (16.16)

By combining Eqs. (16.9), (16.14)–(16.16) we directly cantransform from ecliptic to horizontal coordinates,



ξυζ

=

−1 0 00 −1 00 0 1

sin φ0 0 − cos φ00 1 0

cos φ0 0 sin φ0

cos θL sin θL 0sin θL − cos θL 0

0 0 1

1 0 00 cos ε − sin ε0 sin ε cos ε

cos β cos λcos β sin λ

sin β

. (16.17)

Please note that matrix multiplications do not commute, i.e. the order in which the rotations are applied must not bealtered. Now, we apply that the ecliptic latitude of the sun βS = 0. By calculating Eq. (16.32) we find

ξS = cos aScos AS = − sin φ0 cos θL cos λS − (sin φ0 sin θL cos ε− cos φ0 sin ε) sin λS, (16.18a)υS = cos aSsin AS = − sin θL cos λS + cos θL cos ε sin λS, (16.18b)ζS = sin aS = cos φ0 cos θL cos λS + (cos φ0 sin θL cos ε + sin φ0 sin ε) sin λS, (16.18c)

where we also used the the relationship between Cartesian and spherical horizontal coordinates from Eq. (16.1). Di-viding Eq. (16.18b) by Eq. (16.18a) and leaving Eq. (16.18c) unchanged leads to the final expressions for the solar po-sition,

tan AS =υSξS

=− sin θL cos λS + cos θL cos ε sin λS

− sin φ0 cos θL cos λS − (sin φ0 sin θL cos ε− cos φ0 sin ε) sin λS, (16.19a)

sin aS = ζS = cos φ0 cos θL cos λS + (cos φ0 sin θL cos ε + sin φ0 sin ε) sin λS. (16.19b)

AS and aS now can be derived by applying inverse trigonometric functions. While arcsin uniquely delivers an alti-tude in between −90 and 90, applying arctan leads to ambiguities. For deriving an azimuth in between 0 and360, we have to look in which quadrant is lying. Therefore we use ξ and υ from Eqs. (16.18a) and (16.18b), respect-ively. We find

ξ > 0 ∧ υ > 0⇒ AS = arctan f (. . .), (16.20a)ξ < 0 ⇒ AS = arctan f (. . .) + 180, (16.20b)ξ > 0 ∧ υ < 0⇒ AS = arctan f (. . .) + 360. (16.20c)



f (. . .) denotes the function at the right hand side of Eq. (16.19a). Note that an altitude aS < 0 corresponds to thesun being below the horizon. This means that the Sun is not visible and no solar energy can be harvested.

The approximations presented on the previous pages are accurate within arcminutes for 200 centuries of 2000. Sev-eral years ago, NREL presented a much more complicated model, the so-called Solar Position Algorithm (ASP), withuncertainties of only ±0.0003 in the period from 2000 BC to 6000 AD [83].

Example

As an example we will calculate the position of the Sun in Delft on 14 April 2014 at 11:00 local time.

For determining the solar position we need next to date and time (in UTC) the latitude and longitude. Since the time zone in Delft on 14April is the CEST, the Central European Summer Time, the time difference with UTC is +2 hours, such that 11:00 CEST corresponds to9:00 UTC. According to Google Maps, the latitude and longitude of the Markt in the centre of Delft are given by

φ0 = 52.01N = +52.01,

λ0 = 4.36 E = + 4.36.

For the calculation we first have to express date and time as the time elapsed since 1 January 2000 noon UTC.

D = 4 · 366 + 10 · 365 + 2 · 31 + 28 + 13− 0.5 + 924 = 5216.875.

Now we can calculate the mean longitude q and the mean anomaly g of the sun according to Eqs. (16.3) and (16.4),

q = 280.459 + 0.98564736D = 22.4580712,

g = 357.529 + 0.98560028D = 99.28246073,

where the values were normalised to [0, 360). From Eq. (16.5) we thus obtain for the latitude of the Sun in ecliptic coordinates

λS = q + 1.915 sin g + 0.020 sin 2g = 24.34162696.



For the axial tilt ε of the Earth we obtain from Eq. (16.8)

ε = 23.429 − 0.00000036D = 23.42712193.

The Greewich Mean Sidereal Time (GMST) is given by Eq. (16.11),

GMST = 18.697374558 h + 24.06570982441908 h · D + 0.000026 h · T2 = 22.49731535 h,

where we used T = D/36525 and normalised to [0 h, 24 h). We then find for the local mean sidereal time θL

θL = GMST15

hour+ λ0 = 341.8197303,

Now we have all variables required to calculate the solar position. From Eqs. (16.19) and (16.20) we thus find

tan AS =− sin θL cos λS + cos θL cos ε sin λS

− sin φ0 cos θL cos λS − (sin φ0 sin θL cos ε− cos φ0 sin ε) sin λS= −1.318180633.

sin aS = cos φ0 cos θL cos λS + (cos φ0 sin θL cos ε + sin φ0 sin ε) sin λS = 0.589415473,

which leads to the solar altitude aS=36.1 and the solar azimuth AS=127.2.


16.3. The equation of time 235

16.2 The sun path at different loc-ations

In this section we discuss the solar paths throughoutthe year at several locations around the earth. Figures16.5-16.8 shows four examples: Delft, the Netherlands(φ0 = 52.01 N), the North Cape, Norway (φ0 = 71.17

N), Cali, Colombia (φ0 = 3.42 N), and Sydney, Aus-tralia (φ0 = 33.86 S). Note that all the times are givenin the apparent solar time (AST). While in Delft and onthe North Cape, the Sun at noon always is South of thezenith, in Sydney it is always North. In Cali, close tothe equator, the Sun is either South or North, depend-ing on the time of the year. Since the North Cape isnorth of the arctic circle, the Sun does not set around 21June. This phenomenon is called the midnight sun. Onthe other hand, the sun always stays below the horizonaround 21 December - this is called the polar night.

16.3 The equation of time

Figure 16.9 shows the position of the sun throughoutthe year in Delft at 8:30, 12:00 and 15:30 mean solartime (MST), i.e. UTC + λ0, where the longitude λ0 is ex-pressed in hours. We see that the Sun not only changesfrom altitude but also from azimuth in the course of a

year, such that it seems to run along the shape of anEight. This closed curve is called the analemma.

The difference between the apparent solar time (AST),i.e. the timescale where the sun really is highest at noonevery day, and mean solar time is described by the so-called equation of time, which is defined as

EoT = AST−MST. (16.21)

The equation of time is given as the difference betweenthe mean longitude q, as defined in Eq. (16.3), and theright ascendent αS of the sun in the equatorial coordin-ate system,

EoT(D) = [q(D)− αS(D)]1

15hourdeg

. (16.22)

The right ascendent is connected to the ecliptic longit-ude of the sun λS, as given in Eq. (16.5) via

tan αs = cos ε tan λS, (16.23)

and ε is the axial tilt as given in Eq. (16.8).

The equation of time has two major contributors: theanomaly due to the elliptic orbit of the Earth aroundthe Sun and the Axial tilt of the rotational axis ofthe Earth with respect to the ecliptic. Both effects areshown in Fig. 16.10. In this figure, also the total EoT isshown, which is nearly the sum of the two contributors(The maximal deviation is less than a minute).



Figure 16.5: The sun path in apparent solar time in Delft, the Netherlands (φ0 = 52.01 N). The sun path was calculated with theSun path chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].



Figure 16.6: The sun path in apparent solar time on the North Cape, Norway (φ0 = 71.17 N). The sun path was calculated withthe Sun path chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].



Figure 16.7: The sun path in apparent solar time in Cali, Colombia (φ0 = 3.42 N). The sun path was calculated with the Sun pathchart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].



Figure 16.8: The sun path in apparent solar time in Sydney, Australia (φ0 = 33.86 S). The sun path was calculated with the Sunpath chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].



3 Nov

1 Sep

13 Jun

15 Apr

11 Feb

25 Dec

15:3008:30

12:00

Solar Azimuth

Solar

Altit

ude

WSWSSEE

60

50

40

30

20

10

0

Figure 16.9: The analemma, i.e. the apparent curve of thesun throughout the year when observed at the same meansolar time every day. The analemma is shown for Delft (52 Nlatitude) at three points in time during the day.

Equation of Time

effect of anomaly

effect of axial tilt

Appa

rent

−M

ean

Solar

Tim

e(m

in)

01 Dec01 Oct01 Aug01 Jun01 Apr01 Feb

15

10

5

0

-5

-10

-15

Figure 16.10: The effect of the anomaly of the terrestrial orbitand the axial tilt of the Earth rotation axis on the differencebetween apparent and mean solar time. The equation of timenearly is the sum of these two effects.

We see that the largest negative shift is on 11 February,where, the apparent noon is about 14 min 12 s priorto the mean solar noon. The largest positive shift is on3 November, when the apparent solar noon is about16 min 25 s past the mean solar noon. These pointsare also marked in Fig. 16.9. There, also the zeros ofthe EoT are shown, which are on 15 April, 13 June, 1September, and 25 December.


16.4. Irradiance on a PV module 241

Module

Zenith

Celestial sphere

N0°

Azimuth A M

Horizon

Alti

tude

a M

θM

θ M

Module

normal n

M

Figure 16.11: Illustrating the angles used to describe theorientation of a PV module installed on a horizontal plane.

16.4 Irradiance on a PV module

After having discussed how to calculate the positionof the Sun everywhere on the Earth and having lookedat several examples, it now is time to discuss the im-plications for the irradiance present on solar modules.For this discussion we assume that the solar module ismounted on a horizontal plane and that it is tilted un-der an angle θM, as illustrated in Fig. 16.11. The anglebetween the projection of the normal of the moduleonto the horizontal plane and due north is AM. We

then can describe the position of the module by the dir-ection of the module normal in horizontal coordinates(AM, aM), where the altitude is given by aM = 90 − θ.Let now the sun be at the position (AS, aS). Then thedirect irradiance on the module GM is given by theequation

GdirM = Idir

e cos γ, (16.24)

where γ = ^(AM, aM)(AS, aS) is the angle betweenthe surface normal and the incident direction of thesunlight. Now we know that the scalar product of twounit vectors is equal to the cosine of the enclosed angle.Thus, we can write,

cos γ = nM · nS. (16.25)

The normal vectors are given by

nM =

ξMυMζM

=

cos aM cos AMcos aM sin AM

sin aM

, (16.26)

nS =

ξ Sυ Sζ S

=

cos a S cos A Scos a S sin A S

sin a S,

. (16.27)

where we used the relationship between Cartesian andspherical horizontal coordinates given in Eq. (16.1).



Hence, we find

cos γ =nM · nS

= cos aM cos AM cos aS cos AS

+ cos aM sin AM cos aS sin AS + sin aM sin aS

= cos aM cos aS (cos AM cos AS + sin AM sin AS)

+ sin aM sin aS

= cos aM cos aS cos (AM − AS) + sin aM sin aS.(16.28)

Thus we obtain for the irradiance

GdirM = Idir

e [cos aM cos aS cos (AM − AS) + sin aM sin aS]

= Idire [sin θ cos aS cos (AM − AS) + cos θ sin aS] .

(16.29)Note that this equation only holds when the sun isabove the horizon (aS > 0) and the azimuth of the sunis within ±90 of AM, AS ∈ [AS − 90, AS + 90]. Oth-erwise, Gdir

M = 0.

If the azimuth of the solar position is the same as theazimuth of the module normal AM = AS, Eq. (16.29)becomes

GdirM = Idir

e [cos aM cos aS + sin aM sin aS]

= Idire cos (aM − aS) .

(16.30)

When using the tilt angle θ = 90 − aM we find

GdirM = Idir

e sin (θ + aS) . (16.31)

Modules mounted on a tilted roof

When a module is mounted on a horizontal plane, itis easy to determine its normal nM. However, when amodule is to be mounted on an arbitrarily tilted roofthings become more complicated. We thus will derivehow to calculate the normal of the module in horizontalcoordinates when the coordinates with respect to theroof are given. In fact, we thus have to transform themodule normal from the roof coordinate system to the ho-rizontal coordinate system.

As illustrated in Fig. 16.12 (a), the orientation of theroof in horizontal coordinates is characterised by theazimuth AR and the altitude aR of its normal nR. Themodule is installed on the roof, and its orientation withrespect to the roof is best described in the roof coordin-ate system, where the fundamental plane is parallel tothe roof and the principal direction is along the gradi-ent of the roof, as illustrated in Fig. 16.12 (b). In thissystem, the module normal is given by the azimuth φMand the altitude is given by δM. The coordinate trans-form itself is transformed by combining two rotations:

First, we rotate with the angle 90 − aR around the axisthat is perpendicular to both nR and the gradient dir-ection of the roof. Secondly, we rotate with the angleAR + 180 along the zenith. We thus obtain


16.4. Irradiance on a PV module 243

ξMυMζM

=


sin aM

=

− cos AR sin AR 0− sin AR − cos AR 0

0 0 1

sin aR 0 − cos aR0 1 0

cos aR 0 sin aR

cos δM cos φMcos δM sin φM

sin δM

. (16.32)

The coordinates of the module in the horizontal coordinate system then are given by

ξM = cos aMcos AM = − cos AR sin aR cos δM cos φM + sin AR cos δM sin φM + cos AR cos aR sin δM, (16.33a)υM = cos aMsin AM = − sin AR sin aR cos δM cos φM − cos AR cos δM sin φM + sin AR cos aR sin δM, (16.33b)ζM = sin aM = cos aR cos δM cos φM + sin aR sin δM. (16.33c)

Dividing Eq. (16.33b) by Eq. (16.33a) and leaving Eq. (16.33c) unchanged leads to the final expressions for the mod-ule orientation in horizontal coordinates,

tan AM =− sin AR sin aR cos δM cos φM − cos AR cos δM sin φM + sin AR cos aR sin δM− cos AR sin aR cos δM cos φM + sin AR cos δM sin φM + cos AR cos aR sin δM

, (16.34a)

sin aM = cos aR cos δM cos φM + sin aR sin δM. (16.34b)

Finally, the cosine of the angle between the module orientation and the solar position is given by

cos γ = nM · nS = cos aS cos(AR − AS) (cos aR sin δM − sin aR cos δM cos φM)

+ cos aS sin(AR − AS) cos δM sin φM + sin aS (cos aR cos δM cos φM + sin aR sin δM) .(16.35)



We will try to understand these results by discussingeasy examples:

First, we look at a roof that faces eastward and has atilt angle θR. Then, aR = 90 − θR and AR = 90.On this roof a solar module is installed under a tilt-ing angle θM with respect to the roof. The modules aremounted parallel to the gradient of the roof. We thushave δM = 90 − θM and φM = 270. From Eqs. (16.34)we thus obtain

sin aM = cos aR cos δM · 0 + sin aR sin δM.= sin aR sin δM

(16.36a)

tan AM =−0− 0 + 1 · cos aR sin δM

0 + sin AR cos δM sin ·1 + 0= cos aR tan δM.

(16.36b)

In the second example, the roof is facing southwardsand tilted under an angle θR. We thus have aR =90 − θR and AR = 180. Now the module tilted un-der an angle θM with respect to the roof and moun-ted perpendicular to the gradient of the roof. Hence,δM = 90 − θM and φM = 180. Using Eqs. (16.34) wefind

sin aM = cos aR cos δM · (−1) + sin aR sin δM.= cos(aR + etaM) = sin(aR − θM).

(16.37a)

tan AM =−0− 0 + 0

− sin aR cos δM + 0 + cos AR sin δM= 0.

(16.37b)

Roof

Zenith

Celestial sphere

Azimuth A R

Horizon

Alti

tude

a R

θR

θ R

Roofnormal n

R

Gradient N0°

Module

Roof normal nR

Roof gradient

Azimuth φ M

Roof

Alti

tude

δ M

θM

θ M

Module

normal n

M

(a)

(b)

Figure 16.12: Illustrating the angles used to describe (a) theorientation of a roof on a horizontal plane and (b) the orienta-tion of a module mounted on a roof.


16.5. Direct and diffuse irradiance 245

16.5 Direct and diffuse irradiance

As sunlight traverses the atmosphere, it is partiallyscattered, leading to an attenuation of the direct beamcomponent. On the other hand,the scattered light alsopartially will arrive at on the earths surface as diffuselight. For PV applications it is important to be able toestimate the strength of the direct and diffuse compon-ents.

First, we discuss a simple model that allows to estimatethe irradiance on a cloudless sky in dependence of theair mass and hence the altitude of the sun. As we haveseen in section 5.5, the air mass is defined as

AM =1

cos θ=

1sin aS

, (16.38)

where we used that the angle between the sun and thezenith θ is connected to the solar altitude via θ = 90 −aS. This equation, however, does not take the curvatureof the earth into account. If the curvature is taken intoaccount, we find [85]

AM(aS) =1

sin aS + 0.50572(6.07995 + aS)−1.6364 . (16.39)

To estimate the direct irradiance at a certain solar alti-tude aS and altitude of the observer h, we can use thefollowing empirical equation [86]

Idire = I0

e

[(1− ch) · 0.7(AM0.678) + ch

], (16.40)

with the constant c = 0.14. The solar constant is givenas I0

e = 1361 Wm-2. In a first approximation, the diffuseirradiance is about 10% of the direct irradiance. For theglobal irradiance we hence obtain [87]

Iglobale ≈ 1.1 · Idir

e . (16.41)

A more accurate model was developed in the frame-work of the European Solar Radiation Atlas [88]. In thatmodel the direct irradiance for clear sky is given by

Idire = I0

e ε exp [−0.8662 TL(AM2)m δR(m)] . (16.42)

I0 is the solar constant that takes a value of 1361 W/m2.The factor ε allows to correct for deviations of the sun-earth distance from its mean value. aS is the solar alti-tude angle. TL(AM2) is the Linke turbidity factor withthat the haziness of the atmosphere is taken into ac-count. In this equation its value at an Air Mass 2 isused. m is the relative optical air mass, and finallyδR(m) is the integral Rayleigh optical thickness. Thedifferent components can be evaluated as follows:

The correction factor ε is given by

ε =Ie(R)

I0e

=R2

AU2 . (16.43)

The distance between earth and sun as a multiple ofastronomic units (AU) is given in Eq. (16.7). We thus



obtain

ε = (1.00014− 0.01671 cos g− 0.00014 cos 2g)2 , (16.44)

which leads to annual variations of about ±3.3%.

The Linke Turbidity factor approximates absorption andscattering in the atmosphere and takes both absorptionby water vapour and scattering by aerosol particles intoaccount. It is a unit-less number and typically takes val-ues between 2 for very clear skies and 7 for heavily pol-luted skies.

The relative optical air mass m expresses the ratio ofthe optical path length of the solar beam through theatmosphere to the optical path through a standard at-mosphere at sea level with the Sun at zenith. In can beapproximated as a function of the solar altitude aS by

m(aS) =exp(−z/zh)

sin aS + 0.50572(aS + 6.07995)−1.6364 . (16.45)

Here, z is the site elevation and zh is the scale height ofthe Rayleigh atmosphere near the Earth surface, givenby 8434.5 m.

Finally, the Rayleigh optical thickness δR(m) is given by

1δR(m)

=6.62960 + 1.75130 m− 0.12020 m2

+ 0.00650 m3 − 0.00013 m4.(16.46)

In their paper, Rigollier et al. also take the effect of ofrefraction at very low altitudes into account [88]. This,however is not relevant for our application.

They also present and expression for the diffuse irradi-ance of the light, which is given by

Idife = I0

e εTrd[TL(AM2)]Fd[aS, TL(AM2)], (16.47)

where Trd is the diffuse transmission function at zenith,which is a second-order polynomial of TL(AM2) Trdhas typical values in between 0.05 for very clear skiesand 0.22 for a very turbid atmosphere. Fd is a diffuseangular function, given as a second-order polynomial ofsin aS. For more details we refer to the paper [88].

16.6 Shadowing

Shadowing had to be kept in mind when planning a PVsystem that consists of several rows of PV modules,which are placed behind each other. In this section wewill determine how far behind the module the shadowreaches in dependence of the solar position, the mod-ule orientation and the length l of the module. Figure16.13 shows the important notions that we use in thisderivation.

For the determination we look at a module that is tiltedat an angle θM. Its normal angle has an azimuth AM.


16.6. Shadowing 247

Zenith

Celestial sphere

N0°

Azimuth AM

Alti

tude

a M

θM

θ M

nM

Azimuth A S

n S A

ltitude aS

E

F

PP’d

ℓ

h

Figure 16.13: Derivation of the length d of a shadow behind aPV module with ground length l and height h. The module hasthe normal nM while the position of the sun is given by the dir-ection nS. The length of the shadow d is given as the distancebetween the line connecting the lower module corners E and Fand the projection of the upper corner P on the ground (P′).

This module touches the ground at two corner pointsthat we call E and F. Without loss of generality, wemay assume that E is at the origin of our horizontalcoordinate system, E = (0, 0, 0). Further, P is the cornerpoint of top the module lying above E. The lengthof the module, i.e. the distance between E and P is l,EP = l. The shadow of this point on the horizontalplane, we denote by P′. Then, the length of the shadow dis defined as the shortest distance between P′ and theline g, which connects E with F.

For the determination of d we first must derive P, andthen P′. The normal of the module nM is given by

nM =


sin aM

=

sin θM cos AMsin θM sin AM

cos θM

. (16.48)

The direction vector r of the line g connecting E with Fis then given as

r =

sin AM− cos AM

0

. (16.49)

Then, we can calculate the direction vector h of the line



that connects E with P with the vector product

h = −nM × r

= −

sin θM cos AMsin θM sin AM

cos θM

× sin AM− cos AM

0

=

− cos θM cos AM− cos θM sin AM

sin θM

,

(16.50)

where the − sign is because of the fact that the ho-rizontal coordinate system is left-hand. Since h haslength 1, i.e. it is a unit vector, we easily can derive theposition of the point P with

P = E + l · h = l


sin θM

. (16.51)

For calculating the position of P′ we define the line s,which goes through P and points towards the sun,

s(t) = P + t · nS. (16.52)

When the position of the sun is described by its altitudeaS and azimuth AS, we find

s(t) = l


sin θM

+ t

cos aS cos AScos aS sin AS

sin aS,

(16.53)

We find the shadow P′ of the point P as the intersectionof the line s with the horizontal plane, z = 0,

l sin θM + t sin aS = 0. (16.54)

Hence,

t = −lsin θMsin aS

. (16.55)

The coordinates of P′ = (P′x, P′y, 0) then are given as

P′x = −l (cos θM cos AM + sin θM cot aS cos AS) , (16.56a)

P′y = −l (cos θM sin AM + sin θM cot aS sin AS) . (16.56b)

As stated already earlier, the length of the shadow d isgiven as the shortest distance between P′ and the line gconnecting E and F. Let g′ be the line through P′ that isperpendicular to g, g ⊥ g′. Since E = (0, 0, 0) we findfor g and g′

g(u) = u · r, (16.57a)

g′(v) = P′+v · r′, (16.57b)

where the direction vector r′ is given as

r′ =

cos AMsin AM

0

. (16.58)

d is the distance between P′ and the intersection of g


is it Cos or not?

anonymous

anonymous 11/19/2014: is it Cos or not?

16.6. Shadowing 249

with g′. At this intersection we have

g(u) = g′(v),

u

sin AM− cos AM

0

= P′ + v

cos AMsin AM

0

u sin AM = P′x + v cos AM

−u cos AM = P′y + v sin AM

u sin AM cos AM = P′x cos AM + v cos2 AM

−u cos AM sin AM = P′y sin AM + v sin2 AM

By adding the last two equations we find

P′x cos AM + v cos2 AM + P′y sin AM + v sin2 AM = 0,(16.59)

and hence

v = −P′x cos AM − P′y sin AM. (16.60)

Using Eqs. (16.56), we derive

v =l[(cos θM cos AM + sin θM cot aS cos AS) cos AM

+ (cos θM sin AM + sin θM cot aS sin AS) sin AM]

=l(cos θM cos2 AM + sin θM cot aS cos AS cos AM

+ cos θM sin2 AM + sin θM cot aS sin AS sin AM).(16.61)

Because the direction vector r′ of the line g′ is a unitvector, the length of the shadow d is equal to v. There-fore we obtain from Eq. (16.61) with some trigonometric

operations

d = l [cos θM + sin θM cot aS cos(AM − AS)] . (16.62)

As a rule of thumb, d should be at least three times l,d > 3l.

Example

A PV system should be installed on a flat roof in Naples (Italy).The area of the roof that can be utilized for installing the PVsystem is 10×10 m2. The roof is oriented such that the sidesare parallel to the East-West and North-South directions,respectively.

The owner of the roof decides to use Yingli PANDA 60 mod-ules with dimensions of 1650×990×40 mm3. The modules areinstalled facing south with a tilt of 30.

He wants to install as many modules as possible underthe condition that on the shortest day of the year no mutualshading must occur for the duration of 6 hours.

Should the modules be mounted with the long or short sidetouching the ground? How many modules can be mounted inthis case?

Answer: The shortest day of course is 21 December. The solarposition on this day at 9:00 and 15:00 is

Time Altitude () Azimuth ()9:00 13.59 138.55

15:00 13.13 222.17



Because of the Equation of Time, the sun is not at its highestpoint at exactly 12:00 noon. We see, that the solar altitude at15:00 is just slightly lower than at 9:00. Thus, when using9:00 for calculating the length of the shadow, the durationwithout mutual shading will be slightly shorter than 6 hours.Thus, we use the position at 15:00 for the calculation.

The length of the shadow can be calculated with equation(16.62)

d = l [cos θM + sin θM cot aS cos(AM − AS)] .

We have θM=30, AM=180, aS=13.13, and AS=222.17.

If the module is mounted to the ground on the short side, wehave l=1650 mm. Hence, we find d=4050 mm. The areadirectly beneath the module at the last row is

d′ = l cos θM = 1429 mm.

Thus, we can mount three rows behind each other, because

2d + d′ = 2 · 4050 + 1429 = 9529 mm, (16.63)

which is less than 10 m. In one row fit 10 modules because 10 ·990 = 9900 mm. Thus, we can place 30 modules.

If the module is mounted to the ground on the long side, wehave l=990 mm. Hence, we find d=2430 mm. The areadirectly beneath the module at the last row is

d′ = l cos θM = 857 mm.

Thus, we can mount four rows behind each other, because

3d + d′ = 3 · 2430 + 857 = 8147 mm,

which is lower than 10 m. In one row fit 6 modules because6 · 1650 = 9900 mm. Thus, we can place 24 modules.


17Components of PV Systems

17.1 PV modules

In this section we will discuss PV modules (or solarmodules), their fabrication and how to to determinetheir performance.

Before we start with the actual treatment of PV mod-ules, we briefly want to introduce different terms. Fig-ure 17.1 (a) shows a crystalline solar cell, which we dis-cussed in Chapter 12. For the moment we will consideronly modules that are made from this type of solarcells. A PV module, is a larger device in which manysolar cells are connected, as illustrated in Fig. 17.1 (b).The names PV module and solar module are often usedinterchangeably. A solar panel, as illustrated in Fig. 17.1

(c), consists of several PV modules that are electricallyconnected and mounted on a supporting structure. Fi-nally, a PV array consists of several solar panels. An ex-ample of such an array is shown in Fig. 17.1 (d). Thisarray consists of two strings of two solar panels each,where string means that these panels are connected inseries.

17.1.1 Series and parallel connections inPV modules

If we make a solar module out of an ensemble ofsolar cells, we can connect the solar cells in differentways: first, we can connect them in a series connec-

251


252 17. Components of PV Systems

(a)

(c)

(b) (d)

solar cell

PV module

PV panel

series connection

series connection

string of panels string of panels

PV array

Figure 17.1: Illustrating (a) a solar cell, (b) a PV module, (c) a solar panel, and (d) a PV array.


17.1. PV modules 253

1.8 V

0.6 V

0.6 V

(a)

32

1

0 0.3 0.6 0.9 1.2 1.5

Isc

3Voc

Curr

ent I

(A)

Voltage V (V) 1.8

65

4

98

7 2Isc

3Isc

Voc 2Voc

series

parallel

(b)

(c) (d)

Figure 17.2: Illustrating (a) a series connection of three solar cells and (b) realisation of such a series connection for cells with aclassical front metal grid. (c) Illustrating a parallel connection of three solar cells. (d) I-V curves of solar cells connected in series andparallel.



tion as shown in Fig. 17.2 (a). In a series connectionthe voltages add up. For example, if the open circuitvoltage of one cell is equal to 0.6 V, a string of threecells will deliver an open circuit voltage of 1.8 V. Forsolar cells with a classical front metal grid, a series con-nection can be established by connecting the bus bars atthe front side with the back contact of the neighbouringcell, as illustrated in Fig. 17.2 (b). For series connectedcells, the current does not add up but is determinedby the photocurrent in each solar cell. Hence, the totalcurrent in a string of solar cells is equal to the currentgenerated by one single solar cell.

Figure Fig. 17.2 (d) shows the I-V curve of solar cellsconnected in series. If we connect two solar cells inseries, the voltages add up while the current stays thesame. The resulting open circuit voltage is two timesthat of the single cell. If we connect three solar cells inseries, the open circuit voltage becomes three times aslarge, whereas the current still is that of one single solarcell.

Secondly, we can connect solar cells in parallel as il-lustrated in Fig. 17.2 (c), which shows three solar cellsconnected in parallel. If cells are connected in paral-lel, the voltage is the same over all solar cells, whilethe currents of the solar cells add up. If we connect e.g.three cells in parallel, the current becomes three timesas large, while the voltage is the same as for a singlecell, as illustrated in Fig. 17.2 (d).

The reader may have noticed that we used I-V curves,i.e. the current-voltage characteristics, in the previousparagraphs. This is different to Parts II and III, wherewe used I-V curves instead, i.e. the current density -voltage characteristics. The reason for this switch fromJ to I is that on module level, the total current that themodule can generate is of higher interest than the cur-rent density. As the area of a module is a constant, theshapes of the I-V and J-V curves of a module are sim-ilar.

For a total module, therefore the voltage and currentoutput can be partially tuned via the arrangements ofthe solar cell connections. Figure 17.3 (a) shows a typ-ical PV module that contains 36 solar cells connectedin series. If a single junction solar cell would have ashort circuit current of 5 A, and an open circuit voltageof 0.6 V, the total module would have an output ofVoc = 36 · 0.6 V = 21.6 V and Isc = 5 A. However,if two strings of 18 series-connected cells are connec-ted in parallel, as illustrated in Fig. 17.3 (b), the outputof the module will be Voc = 18 · 0.6 V = 10.8 V andIsc = 2× 5 A = 10 A. In general, for the I-V characterist-ics of a module consisting of m identical cells in seriesand n identical cells in parallel the voltage multipliesby a factor m while the current multiplies by a factor n.Modern PV modules often contain 60 (10× 6), 72 (9× 8)or 96 (12× 8) solar cells that are usually all connectedin series in order to minimise resistive losses.



(a) (b)

Figure 17.3: Illustrating a PV module consisting (a) of a stringof 36 solar cells connected in series and (b) of two strings of 18solar cells each that are connected in parallel.

17.1.2 PV module parameters

In a nutshell, for a PV module a set of parameters canbe defined, similar than for solar cells. The most com-mon parameters are the open circuit voltage Voc, the shortcircuit current Isc and the module fill factor FFM. On mod-ule level, we have to distinguish between the aperturearea efficiency and the module efficiency. The aperture areais defined as the area of the PV-active parts only. Thetotal module area is given as the aperture area plusthe dead area consisting of the interconnections andthe edges of the module. Clearly, the aperture area effi-ciency is larger than the module efficiency.

Determining the the efficiency and the fill factor of aPV module is less straight-forward than determiningvoltage and current. In an ideal world with perfectlymatched solar cells and no losses, one would expectthat the efficiency and fill factor at both the moduleand cell levels to be the same. This is not the case inreal life. As mentioned above. The cells are connectedwith each other using interconnects that induce resist-ive losses. Further, there might be small mismatchesin the interconnected cells. For example, if m × n cellsare interconnected, the cell with the lowest current in astring of m cells in series determines the module cur-rent. Similarly, the string with the lowest voltage inthe n strings that are connected in parallel dictates themodule voltage. The reason for mismatch between in-dividual cells are inhomogeneities that occur during



the production process. Hence, in practice PV moduleperform a little less than what one would expect fromideally matched and interconnected solar cells. This lossin performance translates to a lower fill factor and ef-ficiency at module level. If the illumination across themodule is not constant or if the module is heats upnon-uniformly, the module performance reduces evenfurther.

Often, differences between cell and module perform-ance are mentioned in datasheets that are provide bythe module manufacturers. For example, the datasheetof a Sanyo HIT-N240SE10 module gives a cell level ef-ficiency of 21.6%, but a module level efficiency of only19%. Despite all the technological advancements beingmade at solar cell level for improving the efficiency, stilla lot must be done at the PV systems level to ensure ahealthy PV yield. For the performance of a PV system,not only the module performance is important, but alsothe yield of the PV system.

17.1.3 Partial shading and bypass diodes

PV modules have so-called bypass diodes integrated. Tounderstand the reason for using such diodes, we haveto consider modules in real-life conditions, where theycan be partially shaded, as illustrated in Fig. 17.4 (a). Theshade can be from an object nearby, like a tree, a chim-ney or a neighbouring building. It also can be caused

by a leaf that has fallen from a tree. Partial shadingcan have significant consequences for the output of thesolar module. To understand this, we consider the situ-ation in which one solar cell in the module shaded fora large part shaded. For simplicity, we assume that allsix cells are connected in series. This means that thecurrent generated in the shaded cell is significantlyreduced. In a series connection the current is limitedby the cell that generates the lowest current, this cellthus dictates the maximum current flowing through themodule.

In Fig. 17.4 (b) the theoretical I-V curve of the five un-shaded solar cells and the shaded solar cell is shown. Ifthe cells are connected to a constant load R, the voltageacross the module is dropping due to the lower currentgenerated. However, since the five unshaded solar cellsare forced to produce high voltages, they act like a re-verse bias source on the shaded solar cell. The dashedline in Fig. 17.4 (b) represents the reverse bias loadput on the shaded cell, which is the I-V curve of thefive cells, reflected across the vertical axis equal to 0 V.Hence, the shaded solar cell does not generate energy,but starts to dissipate energy and heats up. The temper-ature can increase to such a critical level, that the en-capsulation material cracks, or other materials wear out.Further, high temperatures generally lead to a decreaseof the PV output as well.

These problems occurring from partial shading can be



(a)

(c)

(b)

R

R

current

current

-1

1

0

-2

-3

Voltage (V)

Curr

ent (

A)

-5 -4 -3 0 5-2 -1 1 2 3 4

5Voc

Jsc

J < Jsc

Current from 5 cells

Current fromshaded6th cellDissipated

energy

Figure 17.4: Illustrating (a) string of six solar cells of which one is partially shaded, which (b) has dramatic effects on the I-V curveof this string. (c) Bypass diodes can solve the problem of partial shading.



prevented by including bypass diodes in the module,as illustrated in 17.4 (c). As discussed in Chapter 8,a diode blocks the current when it is under negativevoltage, but conducts a current when it is under posit-ive voltage. If no cell is shaded, no current is flowingthrough the bypass diodes. However, if one cell is (par-tially) shaded, the bypass diode starts to pass currentthrough because of the biasing from the other cells. Asa result current can flow around the shaded cell and themodule can still produce the current equal to that of aunshaded single solar cell.

For cells that are connected in parallel, partial shad-ing is less of a problem, because the currents gener-ated in the others cells do not need to travel throughthe shaded cell. However, a module consisting of 36cells in parallel have very high currents (above 100 A)combined with a very low voltage (approx. 0.6 V). Thiscombination would lead to very high resistive losses inthe cables; further an inverter that has only 0.6 V as in-put will not be very efficient, as we will see in Section17.3. Therefore, combining the cells in series and usingbypass diodes is much better an option to do.

17.1.4 Fabrication of PV modules

As discussed in the subsection 17.1.5, a PV modulemust withstand various influences in order to survivea lifetime of 25 years or even longer. In order to ensure

front glass

front encapsulant

solar cells

encapsulantback layers

junction box

Al frame

back

Al frame

Figure 17.5: The components of a typical c-Si PV module.

a long lifetime, the components of that a PV module isbuilt must be well chosen. Fig. 17.5 shows the typicalcomponents of a usual crystalline silicon PV module.Of course, the layer stack may consist of different ma-terials dependent on the manufacturer. The major com-ponents are [89]:

• Soda-lime glass with a thickness of several mil-limetres, which provides mechanical stabilitywhile being transparent for the incident light. Itis important the glass has a low iron content be-cause iron leads to absorption of light in the glasswhich can lead to losses. Further, the glass mustbe tempered in order to increase its resistance to im-pacts.



• The solar cells are sandwiched in between two lay-ers of encapsulants. The most common materialis ethylene-vinyl-acetate (EVA), which is a thermo-plastic polymer (plastic). This means that it goesinto shape when it is heated but that these changesare reversible.

• The back layer acts as a barrier against humidityand other stresses. Depending on the manufac-turer, it can be another glass plate or a compositepolymer sheet. A material combination that is oftenused is PVF-polyester-PVF, where PVF stands forpolyvinyl fluoride, which is often known by its brandname Tedlar®. PVF has a low permeability for va-pours and is very resistive against weathering. Atypical polyester is polyethylene terephthalate (PET)

• A frame usually made from aluminium is putaround the whole module in order to enhance themechanical stability.

• A junction box usually is placed at the back of themodule. In it the electrical connections to the solarcell are connected with the wires that are used toconnect the module to the other components of thePV system.

One of the most important steps during module pro-duction is laminating, which we briefly will explain forthe case that EVA is used as encapsulant [89]. For lam-ination, the whole stack consisting of front glass, the

encapsulants, the interconnected solar cells, and theback layer are brought together in a laminator, whichis heated above the melting point of EVA, which isaround 120°C. This process is performed in vacuo inorder to ensure that air, moisture and other gasses areremoved from within the module stack. After someminutes, when the EVA is molten, pressure is appliedand the temperature is increased to about 150°C. Nowthe curing process starts, i.e. a curing agent, which ispresent in the EVA layer, starts to cross-link the EVAchains, which means that transverse bonds between theEVA molecules are formed. As a result, EVA has elasto-meric, rubberlike properties.

The choice of the layers that light traverses before en-tering the solar cell is also very important from an op-tical point of view. If this layers have an increasingrefractive index, they act as antireflective coating andthus can enhance the amount of light that is in-coupledin the solar cell and finally absorbed, which increasesthe current produced by the solar cell.

17.1.5 Lifetime testing of PV Modules

The typical lifetime of PV systems is about 25 years.In these as little maintenance as possible should be re-quired on the system components, especially the PVmodules are required to be maintenance free. Further-more, degradation in the different components of that



the module is made should be little: manufacturers typ-ically guarantee a power between 80% and 90% of theinitial power after 25 years. During the lifetime of 25years or more, PV modules are exposed to various ex-ternal stress from various sources [90]:

• temperature changes between night and day as wellas between winter and summer,

• mechanical stress for example from wind, snow andhail,

• stress by agents transported via the atmosphere suchas dust, sand, salty mist and other agents,

• moisture originating from rain, dew and frost,

• humidity originating from the atmosphere,

• irradiance consisting of direct and indirect irradi-ance from the sun; mainly the highly-energetic UVradiation is challenging for many materials.

Before PV modules are brought to the market, they areusually tested extensively in order to assure their stabil-ity against these various stresses. The required tests areextensively defined in the standards IEC 61215 for mod-ules based on crystalline silicon solar cells and in IEC61646 for thin-film modules. Since the modules cannotbe tested during a period of 25 years, accelerated stresstesting must be performed. The required tests are [91]:

• Thermal cycles for studying whether thermal stress

leads to broken interconnects, broken cells, elec-trical bond failure, adhesion of the junction box,. . .

• Damp heat testing to see whether the modules suf-fer from corrosion, delamination, loss of adhesionand elasticity of the encapsulant, adhesion of thejunction box,. . .

• Humidity freeze testing in order to test delamination,adhesion of the junction box, . . .

• UV testing, because UV light can lead to delamin-ation, loss of adhesion and elasticity of the encap-sulant, ground fault due to backsheet degradation.Mainly, UV light can lead to a discoloration of theencapsulant and back sheet, which means that theyget yellow. This can lead to losses in the amount oflight that reaches the solar cells.

• Static mechanical loads in order to test whetherstrong winds or heavy snow loads lead to struc-tural failures, broken glass, broken interconnect rib-bons or broken cells.

• Dynamic mechanical load, which can lead to rokenglass, broken interconnect ribbons or broken cells.

• Hot spot testing in order to see whether hot spotsdue to shunts in cells or inadequate bypass diodeprotection are present.

• Hail testing to see whether the module can handlethe mechanical stress induced by hail.



• Bypass diode thermal testing to study whether over-heating of these diodes causes degradation of theencapsulant, backsheet or the junction box.

• Salt spray testing to see whether salt that is presentin salty mist or that is used in salty water for snowand ice removal leads to corrosion of PV modulecomponents.

How these tests are to be performed is defined in otherstandards, for example IEC 61345 for UV testing andIV 61701 for salt-mist corrosion testing Usually thesetests are carried out by organisations like TÜV Rhein-land. Refining the test requirements and understand-ing which accelerated tests are required to guarantee alifetime of 25 years and more is subject to ongoing re-search and development.

17.1.6 Thin-film modules

Making thin-film modules is very different from mak-ing modules from c-Si solar cells. While for c-Si tech-nology producing solar cells and producing PV mod-ules are two distinct steps, in thin-film technology pro-ducing cells and modules cannot be separated fromeach other. To illustrate this we look at a PV modulewhere the thin-films are deposited in superstrate config-uration on glass, as illustrated in Fig. 17.6. For makingsuch a module, a transparent front contact, a stack of

(photo)active layers that also contain one or more semi-conductor junctions, and a metallic back contact are de-posited onto each other. In industrial production, theglass plates on that these layers are deposited can bevery large, with sizes significantly exceeding 1× 1 m2.

Such a stack of layers deposited onto a large glassplate in principle forms one very large solar cell thatwill produce a very high current. Since all the currentwould have to be transported across the front and backcontacts, which are very thin, resistive losses in themodule is even a bigger problem than for c-Si mod-ules. Therefore, the module is produced such that itconsists of many very narrow cells of about 1 cm widthand the length being equal to the module length. Thesecells then are connected in series across the width ofthe module. On the very left and right of the modulemetallic busbars collect the current and conduct it tothe bottom of the module where they are connectedwith external cables.

The series connection is established with laser scribing.In total, three laser scribes are required for separatingtwo cells from each other and establishing a series con-nection between them. The first laser scribe, called P1,is performed after the transparent front contact is de-posited, as shown in Fig. 17.6 (a). The wavelength ofthe laser is such that the laser light is absorbed in thefront contact and the material is evaporated, leavinga “gap” in the front contact, as shown in Fig. 17.6 (b).



Then the photoactive layers are deposited onto the frontcontact and also fill the gaps. Then, the second laserscribe, called P2, is performed, as illustrated in Fig.17.6 (c). The laser wavelength has to be chosen suchthat it is not absorbed in the transparent front contactbut in the absorber layer. For example, if the absorberconsists of amorphous silicon, green laser light can beused. The P2 scribe leaves a gap in the absorber layer,as illustrated in Fig. 17.6 (d). The next step is the de-position of the metallic back contact that also fills theP2 gap. Finally, the third laser scribe (P3) is performedas illustrated in Fig. 17.6 (e). The wavelength for thisscribe has to be chosen such that it is neither absorbedin the front contact nor in the absorber stack, so it is,for example, infrared. The P3 scribe shoots a gap intothe back contact, as shown in Fig. 17.6 (f).

To understand the action of the laser scribes, we takea look at Fig. 17.6 (f): the P1 scribe filled with absorbermaterial forms an barrier, since the absorber is orders ofmagnitudes less conductive than the transparent frontcontact. Similar, the P3 scribes forms an insulating gapin the metallic back contact. However, the P2 scribe thatalso is filled with metal forms a highly conducting con-nection between the front and back contacts – here theactual series connection is performed.

For example, for making CIGS solar cells, first the mo-lybdenum back contact is deposited on top of the glasssubstrate and the cell areas are defined by P1 laser

(a)

(c)

(b)

(d)

(e) (f)

P1

P2

P3

glassfront contact

glassfront contact

absorber

back metalcurrent

interconnect width

Figure 17.6: Schematic of creating an interconnect in thin-filmmodule. (Explanation given in the text).



scribes. Then the CIGS p layer and the CdS n layer aredeposited including a P2 laser scribe step. Finally theintrinsic and n-doped zinc oxide is deposited, followedby a final P3 laser scribe step. Now the front TCO elec-trode is connected with the Molybdenum back contactof the next solar cell.

The performance of such an interconnect established vialaser scribes and hence the total module performanceis determined by several things. First, the P2 scribe hasto be highly conductive, This means that it has to bewide enough and that there must be no barrier at theinterface between the front contact and the metal of theP2 scribe. Further, the P1 and P3 scribes must performgood barriers to effectively separate the cells from eachother. Thirdly, the region between the P1 and P3 scribesdoes not contribute to the the current generated by themodule. Therefore, the ratio between this width andthe total cell width (including the scribes) has to beas small as possible. Another issue is the fact that thethree laser scribes are performed in different steps ofproduction and thus often in different machines. Fur-ther, the distance between the scribes might be differentat the different processes when they are performed atdifferent temperatures. This, aligning the glass platesin all the production steps is extremely important formanufacturing high-quality thin-film modules.

The production steps and also the exact processing ofthe laser scribes is of course dependent on which thin-

film technology is used and even on the manufactureritself. However, the basic principles and the action be-hind these processes is valid in general.

One advantage of thin-film PV technology is that theycan be deposited onto flexible substrates. For example,the Dutch company HyET Solar developed a techno-logy, where thin-film silicon layers are deposited ontoa temporary aluminium substrate [92]. After the solarcell layers are encapsulated on the back side, the tem-porary substrate is etched away, and the front side isencapsulated. This results in a very low weight flex-ible substrate, which can be integrated for example incurved roof top elements. A very big advantage is thatsuch very light modules can be installed on simple rooftop constructions that only can handle little ballast. Onsuch roofs, heave PV panels with glass cannot be in-stalled. Further, if such flexible modules are directlyintegrated into roofing elements, installation costs canbe reduces significantly. Often, installation costs are thelargest contributor to the non-modular costs of a PVsystem. Currently, only thin-film silicon technologieshave demonstrated flexible modules with reasonableefficiencies.

17.1.7 Some examples

Table 17.1 shows some parameters of PV modules usingdifferent PV technologies:



Table 17.1: Specifications of different PV modules.

SunPower Avancis Kaneka First SolarX21-345 PowerMax140 U-EA120 FS-392

Technology c-Si CIS a-Si/nc-Si CdTeRated power Pmpp (Wp) 345 140 114 92.5Rated current Impp (A) 6.02 2.98 2.18 1.94Rated voltage Vmpp (V) 57.3 47 55 47.7Short circuit current Isc (A) 6.39 3.31 2.6 2.11Open circuit voltage Voc (V) 68.2 61.5 71 60.5Dimensions (m×m) 1.56×1.05 1.6×0.67 1.2×1.00 1.2×0.60Max warranty on Pmpp (years) 25 25 25 25

• A SunPower module made based on monocrystal-line silicon solar cells,

• an Avancis module based on copper indium disel-enide (CIS) technology,

• a Kaneka amorphous silicon /nanocrystalline sil-icon tandem (a-Si:H/nc-Si:H) module, and

• a module of First Solar based on cadmium telluride(CdTe) technology.

17.2 Maximum power point track-ing

In this section we discuss the concept of Maximumpower point tracking (MPPT). This concept is very uniqueto the field of PV Systems, and hence brings a veryspecial application of power electronics to the field ofphotovoltaics. The concepts discussed in this section areequally valid for cells, modules, and arrays, althoughMPPT usually is employed at PV module/array level.

As discussed earlier, the behaviour of an illuminatedsolar cell can be characterised by an I-V curve. Inter-connecting several solar cells in series or in parallelmerely increases the overall voltage and/or current, but


17.2. Maximum power point tracking 265

Curr

ent I

Voltage V

highertemperature T

higher irradiance GM

Figure 17.7: Effect of increased temperature T or irradianceGM on the I-V curve.

does not change the shape of the I-V curve. Therefore,for understanding the concept of MPPT, it is sufficientto consider the I-V curve of a solar cell. The I-V curveis dependent on the module temperature on the irra-diance, as we will discuss in detail in Section 18.3. Forexample, an increasing irradiance leads to an increasedcurrent and slightly increased voltage, as illustrated inFig. 17.7. The same figure shows that an increasing tem-perature has a detrimental effect on the voltage.

Now we take a look at the concept of the operatingpoint, which is the defined as the particular voltage and

current, at that the PV module operates at any givenpoint in time. For a given irradiance and temperature,the operating point corresponds to a unique (I, V) pairwhich lies onto the I-V curve. The power output at thisoperating point is given by

P = I ·V. (17.1)

The operating point (I, V) corresponds to a point onthe power-voltage (P-V) curve, shown in Fig. 17.8. Forgenerating the highest power output at a given irradi-ance and temperature, the operating point should suchcorrespond to the maximum of the (P-V) curve, whichis called the maximum power point (MPP).

If a PV module (or array) is directly connected to anelectrical load, the operating point is dictated by thatload. For getting the maximal power out of the module,it thus is imperative to force the module to operate atthe maximum power point. The simplest way of forcingthe module to operate at the MPP, is either to force thevoltage of the PV module to be that at the MPP (calledVmpp) or to regulate the current to be that of the MPP(called Impp).

However, the MPP is dependent on the ambient con-ditions. If the irradiance or temperature change, the I-V and the P-V characteristics will change as well andhence the position of the MPP will shift. Therefore,changes in the I-V curve have to be tracked continu-ously such that the operating point can be adjusted to



MPP

Power P

Curr

ent I

Voltage V Voc

IscPmpp

Vmpp

Impp

Figure 17.8: A generic I-V curve and the associated P-Vcurve. The maximum power point (MPP) is indicated.

be at the MPP after changes of the ambient conditions.

This process is called Maximum Power Point Tracking orMPPT. The devices that perform this process are calledMPP trackers. We can distinguish between two categor-ies of MPP tracking:

• Indirect MPP tracking, for example performed withthe Fractional Open Circuit Voltage method.

• Direct MPP tracking, for example performed withthe Perturb and Observe method or the the Incre-mental Conductance method.

All the MPPT algorithms that we discuss in this sectionare based on finding the and tuning the voltage untilVMPP is found. Other algorithms, which are not dis-cussed in this section, work with the power instead andaim to find IMPP.

17.2.1 Indirect MPPT

First, we discuss indirect MPP Tracking, where simpleassumptions are made for estimating the MPP based ona few measurements.

Fixed voltage method

For example, in the fixed voltage method (also calledconstant voltage method), the operating voltage of the



solar module is adjusted only on a seasonal basis. Thismodel is based on the assumption that for the samelevel of irradiance higher MPP voltages are expectedduring winter than during summer. It is obvious thatthis method is not very accurate. It works best at loc-ations with minimal irradiance fluctuations betweendifferent days.

Fractional open circuit voltage method

One of the most common indirect MPPT techniques isthe fractional open circuit voltage method. This methodexploits the fact that – in a very good approximation –the Vmpp is given by

Vmpp = k ·Voc, (17.2)

where k is a constant. For crystalline silicon, k usu-ally takes values in between 0.7 and 0.8. In general,k of course is dependent on the type of solar cells.As changes in the open circuit voltage can be easilytracked, changes in the Vmpp can be easily estimatedjust by multiplying with k. This method thus can be im-plemented easily. However, there are also certain draw-backs.

First, using a constant factor k only allows to roughlyestimate the position of the MPP. Therefore, the oper-ating point usually will not be exactly on the MPP butin its proximity, with is called the MPP region. Secondly,

every time the system needs to respond to a change inillumination conditions, the Voc must be measured. Forthis measurement, the PV module needs to be discon-nected from the load for a short while, which will leadto a reduced total output of the PV system. The moreoften the Voc is determined, the larger the loss in outputwill be. This drawback can be overcome my slightlymodifying the method. For this modification a pilotPV cell is required, which is highly matched with therest of the cells in the module. The pilot cell receivesthe same irradiance as the rest of the PV module, anda measurement of the pilot PV cell’s Voc also gives anaccurate representation of that of the PV module, henceit can be used for estimating Vmpp. Therefore, the op-erating point of the module can be adjusted withoutneeding to disconnect the PV module.

17.2.2 Direct MPPT

Now we discuss direct MPP tracking, which is more in-volved than indirect MPPT, because current, voltageor power measurements are required. Further, the sys-tem must response more accurately and faster than inindirect MPPT. We shall look at a couple of the mostpopular kind of algorithms.



Table 17.2: A summary of the possible options in the P&Oalgorithm.

Prior Change in NextPerturbation Power Perturbation

Positive Positive PositivePositive Negative NegativeNegative Positive NegativeNegative Negative Positive

Perturb and observe (P&O) algorithm

The first algorithm that we discuss is the Perturb andObserve (P&O) algorithm, which also is known as "hillclimbing" algorithm. In this algorithm, a perturbationis provided to the voltage at that the module is cur-rently driven. This perturbation in voltage will lead to achange in the power output. If an increasing voltageleads to an increasing in power, the operating pointis at a lower voltage than the MPP, and hence furthervoltage perturbation towards higher voltages is re-quired to reach the MPP. In contrast, if an increasingvoltage leads to a decreasing power, further perturb-ation towards lower voltages is required in order toreach the MPP. Hence, the algorithm will converge to-wards the MPP over several perturbations. This prin-ciple is summarised in Table 17.2.

A problem with this algorithm is that the operat-

Pow

er P

Voltage V

A

B

C

PMPP

C’

PMPP’

VMPPVMPP’

Figure 17.9: The perturb&observe algorithm struggles fromrapidly changing illumination conditions.



ing point is never steady at the MPP but meander-ing around the MPP. If very small perturbation stepsare used around the MPP, this meandering, however,can be minimised. Additionally, the P&O algorithmstruggles from rapidly changing illuminations. For ex-ample, if the illumination (and hence the irradiance)changes in between two sampling instants in the pro-cess of convergence, then the algorithm essentially failsin its convergence efforts, as illustrated in Fig. 17.9 Inthe latest perturbation, the algorithm has determinedthat the MPP lies to the at a higher voltage than ofpoint B, and hence the next step is a perturbation toconverge towards the MPP accordingly. If the illumin-ation was constant, it would end up at C and the al-gorithm would conclude that the MPP is at still highervoltages, which is correct. However, as the illuminationchanges rapidly before the next perturbation, the nextperturbation shifts the operating point to C’ instead toC, such that

PC’ < PB (17.3)

While the MPP still lies to the right of C’, the P&O al-gorithm thinks that it is on the left of C’. This wrongassumption is detrimental to the speed of conver-gence of the P&O algorithm, which is one of the crit-ical figures of merit for MPPT techniques. Thus, drasticchanges in weather conditions severely affect the effic-acy of the P&O algorithm’s.

Incremental conductance method

Next, we look at the Incremental Conductance Method.The conductance G1 of an electrical component isdefined as

G =IV

(17.4)

At the MPP, the slope of the P-V curve is zero, hence

dPdV

= 0. (17.5)

We can write

dPdV

=d(IV)

dV= I + V

dIdV

. (17.6)

If the sampling steps are small enough, the approxima-tion

dIdV≈ ∆I

∆V(17.7)

can be used. We call ∆I/∆V the incremental conductanceand I/V the constanteneous conductance. Hence, we have

∆I∆V

= − IV

if V = Vmpp, (17.8a)

∆I∆V

> − IV

if V < Vmpp, (17.8b)

∆I∆V

< − IV

if V > Vmpp. (17.8c)

1The electrical conductance G must not be confused with the irradiance incid-ent on the module GM .



Increase><Decrease

Maintain

RETURN

ref( ), ( ), ( )V t I t V t

?I IV V

D-

DrefV refV

refV

=

( ) ( )( ) ( )

I I t I t tV V t V t tD = - -DD = - -D

Figure 17.10: A conceptual flowchart of the incremental con-ductance algorithm.

These relationships are exploited by the incrementalconductance algorithm.

Figure 17.10 shows a conceptual flowchart. Note thatthis flowchart is not exhaustive. While both, the instant-aneous voltage and current are the observable paramet-ers, the instantaneous voltage is also the controllableparameter. Vref is the voltage value forced on the PVmodule by the MPPT device. It is the latest approx-imation of the Vmpp. For any change of the operatingpoint, the algorithm compares the instantaneous withthe incremental conductance values. If the incrementalconductance is larger than the negative of the instantan-

eous conductance, the current operating point is to theleft of the MPP; consequently, Vref must be incremented.In contrast, if the incremental conductance is lower thanthe negative of the instantaneous conductance, the cur-rent operating point is to the left of the MPP and the isconsequently decremented. This process is iterated untilthe incremental conductance is the same as the negativeinstantaneous conductance, in which case Vref = Vmpp.

The incremental conductance algorithm can be more ef-ficient than the P&O algorithm as it does not meanderaround the MPP under steady state conditions. Fur-ther, small sampling intervals make it less susceptibleto changing illumination conditions. However, underconditions that are strongly varying and under partialshading, the incremental conductance method mightalso become less efficient. The major drawback of thisalgorithm is the complexity of its hardware implement-ation. Not only currents and voltages must be meas-ured, but also the instantaneous and incremental con-ductances must be calculated and compared. How sucha hardware design can look like however is beyond thescope of this book.

17.2.3 Some remarks

While a MPPT is used to find the MPP by changing thevoltage, it does not perform changes of the operatingvoltage. This is usually done by a DC-DC converter


17.3. Photovoltaic Converters 271

Power P

Curr

ent I

Voltage V

global MPP

local MPP

local MPP

Figure 17.11: The P-V curve of partially shaded system thatexhibits several local maxima.

that will be discussed in section 17.3.2.

In modern PV systems, the MPPT is often implementedwithin other system components like the inverters orcharge controllers. The list of techniques presented inthis section not exhaustive, we just discussed the mostcommon ones. The development of more advancedMPPT techniques is going on rapidly and many sci-entific papers as well as patents are published in thisarea. Furthermore, manufacturers usually use propriet-ary techniques.

Up to now we only looked at situations the total I-V

curve is similar to that of a single cell. Let us now con-sider a system that is partially shaded, as illustratedin Fig. 17.11. Then, the P-V curve will have differentlocal maxima. Depending on the used MPPT algorithm,it is not sure at all that the algorithm finds the globalmaximum. Different companies use proprietary solu-tions to tackle this issue. Alternatively, each string canbe connected to a separate MPPT. There are invertersthat have connections for several strings (usually two).

17.3 Photovoltaic Converters

A core technology associated with PV systems is thepower electronic converter. An ideal PV convertershould draw the maximum power from the PV paneland supply it to the load side. In case of grid connectedsystems, this should be done with the minimum har-monic content in the current and at a power factor closeto unity. For stand-alone systems the output voltageshould also be regulated to the desired value. In thissection a short review of different topologies often as-sociated with PV systems is given. The semiconductorswitches in the following are assumed to be ideal.



17.3.1 System configurations

Before digging into details about different converter to-pologies used for power conversion in PV systems, ageneral overview of different system architecture willbe presented. The system architecture determines howPV modules are interconnected and how the interfacewith the grid is established. Which of these system ar-chitectures will be employed in a particular PV plantdepends on many factors such as environment of theplant (whether the plant is situated in an urban envir-onment or at an open area), scalability, costs etc. Figure17.12 an overview of different system architectures isgiven. The main advantages and disadvantages of thedifferent architectures are discussed below.

In general solar inverters should have the followingcharacteristics [93]:

• Inverters should be highly efficient because theowner of the solar system requires the absolutemaximum possible generated energy to be de-livered to the grid/load.

• Special demands regarding the potential betweensolar generator and earth (depending on the solarmodule type).

• Special safety features like active islanding detec-tion capability.

• Low limits for harmonics of the line currents. This

requirement is enforced by law in most countriessince the harmonic limits of both sources and loadsconnected to the grid are regulated.

• Special demands on electromagnetic interference(EMI), which are regulated by law in most coun-tries. The goal of these minimise the unwantedinfluence of EMI on other equipment in the vicin-ity or connected to the same supply. Think for ex-ample of the influence of a mobile phone on an oldradio.

• In many instances the solar system is to be in-stalled outdoors and inverters should adhere tocertain specification regarding temperature and hu-midity conditions, e.g. IP 54.

• Design for high ambient temperatures.

• Designed for 20 years operation under harsh envir-onmental conditions.

• Silent operation (no audible noise).

We have to distinguish between single-phase and three-phase inverters. For low powers, as they are commonin small residential PV systems, single-phase invertersare used. They are connected to one phase of the grid.For higher powers, three-phase inverters are used thatare connected to all phases of the grid. If a high powerwould be delivered to one phase, the currents flowingacross the three phases would become very asymmetric



Multi-string inverterCentral inverter String inverterModule inverter

(a) (b) (c) (d)

Figure 17.12: Different system architectures employed in PV systems.



leading to several problems in the electricity grid.

Note that the term inverter is often used for two differ-ent things: First, it is used for the actual inverter, whichis the electronic building block that performs the DC-to-AC conversion, as described in section 17.3.3. Secondly,the term inverter also is used for the total unit pro-duced by manufacturers, that nowadays usually con-tains, an MPP tracker, a DC-DC converter, an DC-ACconverter and possibly also a charge controller of also abattery is connected.

Central inverters

This is the simplest architecture employed in PV sys-tems. Here, PV modules are connected in strings lead-ing to an increased system voltage. Many strings arethen connected in parallel forming a PV array, which isconnected to one central inverter. The inverter performsmaximum power point tracking and power conversionas shown in Fig. 17.12 (a), where a three-phase inverteris depicted. This configuration is mostly employed invery-large scale PV production, with central inverterusually being DC to three phase [94].

Many different inverter topologies are utilised as athree phase inverter. Sometimes they are organised asa single DC to three phase unit but sometimes as athree separate DC-to-AC single phase units workingwith a phase displacement of 120 degrees each. Having

all the PV modes connected in a single array in such acentralised configuration offers the lowest specific cost(cost per kWpp of installed power). Since central invert-ers only use a few components, they are very reliablewhat makes them the preferred option in large scale PVpower plants.

In spite of their simplicity and low specific cost, centralinverters suffer from the following disadvantages:

1. Due to the layout of the system, a large amount ofpower is carried over considerable distances usingDC wiring. This can cause safety issues becausefault DC currents are difficult to interrupt. Spe-cial precaution measures must to be taken suchas ticker insulation on the DC cabling and specialcircuit breakers, which can increase the costs.

2. All strings operate at the same maximum powerpoint, which leads to mismatch losses in the mod-ules. This is significant disadvantage. Mismatchlosses increase even more with ageing and withpartial shading of sections of the array. Mismatchbetween the different strings may significantly re-duce the overall system output.

3. Low flexibility and expandability of the system.Due to the high ratings a system is normally de-signed as a unit and hence difficult to extend. Inother words the system design is not very flexible.

4. Power losses in the string diodes, which are put in



series with each string to prevent current circula-tion inside strings.

Module Integrated or module oriented inverters

A very different architecture is that of the module integ-rated inverters, as shown in 17.12 (b). These invertersoperate directly at one or several PV modules and havepower ratings of several hundreds of watts. Becauseof the low voltage rating of the PV module, these in-verters require often require a two stage power conver-sion. In a first stage boosts the DC voltage is boostedto the required value while it is inverted to AC in thesecond stage. Often, a high frequency transformer isincorporated providing full galvanic isolation, whichenhances the system flexibility even further. As theirname suggests, these inverters are usually integratedwith PV panel (so called ‘AC PV panels’). In this waythe highest flexibility and the expandability of systemis obtained. One of the most distinguishing features ofthis system is the “plug and play” characteristic, whichallows to build a complete (and readily expandable) PVsystem at a low investment cost. Another advantage ofthese inverters, is minimisation of the mismatch lossesthat can occur because of non-optimal MPPT.

All these advantages come at certain expenses. Becausethese inverters are be mounted on a PV module, theymust operate in harsh environment such as high tem-

perature and large daily and seasonal temperature vari-ations. Also, the specific are the highest of all the in-verter topologies. Many topologies for module integ-rated inverters have been proposed, with some of thembeing already implemented in commercially availableinverters.

String Inverters

String inverter, as illustrated in 17.12 (c), combine theadvantages of central and module integrated inverterconcepts with little tradeoffs. A number of PV mod-ules that are connected in series form a PV string witha power rating of up to 5 kWp and with the open cir-cuit voltage of up to 1 kV. Now, a number of smallerinverters can be used to connect the PV system to thegrid.

One disadvantage of the topology is the fact that thehigh DC voltage requires special consideration, simil-arly as this already was the case for the central inverterarchitecture. Here, this issue is even more importantbecause string inverters are usually being installed inhouseholds or on office buildings, without designatedsupport structure or increased safety requirements. Ingeneral a qualified electrician is needed to perform theinterconnections between the modules and the inverter.The protection of the system also requires special con-sideration, with emphasis on proper DC cabling.



Although partial shading of the string will influencethe overall efficiency of the system, each string can in-dependently be operated at its MPP. Also, because nostrings are connected in parallel, there is no need forseries diodes as in the case of PV arrays with manyparallel strings. This reduces losses associated withthese diodes. However, it still is a risk that within astring hot-spot occurs because of unequal current andpower sharing inside the string.

Multi String Inverters

The Multi String inverter concept, illustrated in Fig.17.12 (d), has been developed to combine the advantageof the higher energy yield of a string inverter with thelower costs of a central inverter [95]. Lower power DC-DC converters are connected to individual PV stringseach having its own MPP tracking, which independ-ently optimises the energy output from each PV string.To expand the system within a certain power rangeonly a new string with a DC-DC converter has to beincluded. All DC-DC converters are connected via a DCbus through a central inverter to the grid.

Team Concept

Aside of the four system architecture already described,many other concepts are also present in literature.

Teamconnection

Figure 17.13: The team concept of inverters.



However, these concepts are less widely utilised thanthe ones already presented. One of the alternative con-cepts is so called team concept, which combines thestring technology with the master-slave concept. Acombination of several string inverters working withthe team concept is shown in Figure 17.13. At verylow irradiation the complete PV array is connectedto a single inverter. This reduces the overall losses asany power electronic converter is designed such that ithas maximum efficiency near full load. With increasingsolar radiation more inverters are being connected di-viding the PV array into smaller units until every stringinverter operates close to its rated power. In this modeevery string operates independently with its own MPPtracker. At low solar radiation the inverters are con-trolled in a master-slave fashion.

In literature many other concepts are discussed. Forfurther reading on this subject, we refer for example toRef. [96].

17.3.2 DC-DC converters

DC-DC converters fulfil multiple purposes. In an in-verter, DC power is transformed into AC power. TheDC input voltage of the inverter often is constant whilethe output voltage of the modules at MPP is not. There-fore a DC-DC converter is used to transform the vari-able voltage from the panels into stable voltage used

by the DC-AC inverter. Additionally, as already statedin section 17.2, the MPP Tracker controls the operatingpoint of the modules, but it cannot set it. This also isdone by the DC-DC converter. Further, in a stand-alonesystem the MPP voltage of the modules might differfrom that required by the batteries and the load. Alsohere, a DC-DC converter is useful. Three topologies areused for DC-DC converters: buck, boost, and buck-boostconverters. They are described below.

Step-Down (Buck) Converter

Figure 17.14 (a) illustrates the simplest version of abuck DC-DC converter. The unfiltered output voltagewaveform of such a converter operated with pulse-width-modulation (PWM) is shown in Fig. 17.14 (b).If the switch is on, the input voltage Vd is applied to theload. When the switch is off, the voltage across the loadis zero. From the figure we see that the average DCoutput voltage is denoted as Vo. From the unfilteredvoltage, the average output voltage is given as

Vo =1Ts

∫ Ts

0vo(t)dt =

1Ts

(tonVd + toff · 0) =ton

TsVd.

(17.9)The different variables are defined in Fig. 17.14 (b). Tosimplify the discussion we define a new term, the duty



Load

Switch control signal

LR

+

−

+

−

oV

dV

ov

0

dVoV

ont otsT

t

(a)

(b)

Figure 17.14: (a) A basic buck converter without any filtersand (b) the unfiltered switched waveform generated by thisconverter.

cycle D, as

D :=ton

Ts(17.10)

and henceVo = D ·Vd. (17.11)

In general the output voltage with such a high har-monic content is undesirable, and some filtering is re-quired. Figure 17.15 (a) shows a more complex modelof a step-down converter that has output filters in-cluded and supplies a purely resistive load. As filterelements an inductor L and a capacitor C are used. Therelation between the input and the output voltages,as given in Equ. 17.11, is valid in continuous conduc-tion mode, i.e. when the current through the inductornever reaches zero value but flows continuously. Wecan change the ratio between the voltages on the inputand output side by changing the duty cycle D. A de-tailed discussion about different modes of operation ofa buck converter can be found in Reference [97].

In steady-state operation the time integral of the voltageacross the inductor vL taken during one switching cycleis equal to zero. If this is not the case, the circuit is notin steady state. Thus, in steady state, we obtain the fol-lowing inductor volt-second balance:

∫ Ts

0vL dt =

∫ ton

0vL dt +

∫ Ts

tonvL dt = 0. (17.12)



Load

+

−

+

−

oV

dV

(a)

oiLi

di

Lv+ −

L

(b)

Load

−

+

−

oV

dV

oi

Li

Lv+ −L

C

C

Figure 17.15: (a) A buck converter with filters and (b) a boostconverter.

Solving this equation leads to

Vo = DVd, (17.13)

which is the same result as in Eq. 17.11.

Step-Up (Boost) Converter

In a boost converter, illustrated in Fig. 17.15 (b), an in-put DC voltage Vd is boosted to a higher DC voltageVo. By applying the inductor volt-second balance acrossthe inductor as explained in Eq. 17.12, we find

Vdton + (Vd −V0)toff = 0. (17.14)

Using the definition for the duty cycle we find

V0

Vd=

11− D

. (17.15)

The above relation is valid in the continuous conduc-tion mode. The principle of operation is that energystored in the inductor (during the switch is on) is laterreleased against higher voltage Vo. In this way theenergy is transferred from lower voltage (solar cellvoltage) to the higher voltage (load voltage).

Buck-Boost Converter

In a buck-boost converter the output voltage can be bothhigher or lower than the input voltage. The simpli-



Load

+

− +

−

oV

dV

oi

Li

di

Lv

+

− LC

Figure 17.16: A buck-boost converter.

fied schematic of a buck-boost converter is depicted inFig. 17.16. Using inductor volt second balance as in Eq.17.12, we find

Vdton + (−Vo)toff = 0 (17.16)

and henceVo

Vd=

D1− D

(17.17)

in the continuous conduction mode.

The above described topologies are only the most basicDC-DC converter topologies. The interested reader canfind more in-depth information in Reference [97].

1V 2V

+

−measR

1I

R

2I

MPP Tracker

Figure 17.17: A combination of a unit performing an MPPTalgorithm and a DC-DC converter (adapted from [98]).

MPP Tracking

In section 17.2 we extensively discussed MaximumPower Point Tracking. Or – more specific we discusseddifferent algorithms that are used for performing MPPT.In these algorithms, usually the operating point of themodule is set such that its power output becomes max-imal. However, the MPPT algorithm itself cannot actu-ally adjust the voltage or current of the operating point.For this purpose a DC-DC converter is needed. Figure17.17 shows such a combination of the unit performingthe MPPT and DC-DC converter. As illustrated in thefigure, this MPPT unit measures the voltage or the cur-



rent on the load side and can very those by adaptingthe duty cycle of the DC-DC converter. In this illustra-tion, current and voltage on the load side are measured,but they also can be measured on the PV side.

Example

Assume a PV module has its MPP at VPV = 17 V andIPV = 6 A at a given level of solar irradiance. The module hasto power a load with a resistance RL = 10 Ω. Calculate theduty cycle of the DC-DC converter, if a buck-boost converter isused.

The maximum power from the module is PMPP = VMPP ·IMPP = 102 W. If this power should be dissipated at the res-istor, we have to use the relation

PR = U2R/R

and hence find for the voltage at the resistor

VR =√

PMPPRL = 31.94 V

Using Eq. (17.17),Vo

Vd=

D1− D

with Vo = VR and Vd = VPV we find D = 0.65.

17.3.3 DC-AC converters (inverters)

Earlier in this section we have discussed different archi-tectures that are used for the power conversion in PVsystems. Further, we looked at DC-DC converters thatare mainly used in combination with MPPTs in orderto push the variable output from the PV modules to alevel of constant voltage.

As nowadays most appliances are designed for thestandard AC grids, for most PV systems a DC-AC con-verter is required. As already stated earlier, the terminverter is used both for the DC-AC converter and thecombination of all the components that form the actualpower converter.

The H-bridge inverter

Figure 17.18 shows a very simple example of a so calledH-bridge or full-bridge inverter. On the left, the DC in-put is situated. The load (or in our case the AC output)is situated in between four switches. During usual op-eration we can distinguish between three situations:

A All four switches open: No current flows across theload.

B S1 and S4 closed, S2 and S3 open: Now a current isflowing to through the load, where + is connected



+

−

dV

di

Load

1S

3S

2S

4S

+ −

+−

Figure 17.18: A simple representation of an H-bridge.

to the left- and − is connected to the right-handside of the load.

C S1 and S4 open, S2 and S3 closed: Now a current isflowing to through the load, where − is connectedto the left- and + is connected to the right-handside of the load.

WARNING It must be assured that S1 and S2 never areopen at the same time because this would lead toshort-circuiting. The same is true for S3 and S4.

From this list we see that the H-bridge configurationallows to put the load on three different levels, whichare +Vd, 0, and −Vd. Continuously switching between

positive and negative voltages is exactly what is hap-pening in AC. In the easiest operation mode, the H-bridge switches between situations B and C continu-ously, which will provide a square wave. Note thatthere will always be a dead time in the order of µs inorder to prevent short-circuiting. While such a squarewave might be useful for some applications, it is notsuited at all for grid-connected installations. The reasonfor this are harmonic distortions. To understand this welook at a Fourier transform of a square wave, which isgiven as

Vsquare(t) =4π

[sin(2πνt) +

13

sin(6πνt)

+15

sin(10πνt) + · · ·]

.(17.18)

Thus, such a square wave contains not only the prin-cipal sine function with frequency ν, but also all thehigher harmonics with frequencies 3ν, 5ν, and so on.These higher harmonics can lead to distortions of theelectricity grid and thus must be reduced as much aspossible. One principle how to do this is pulse-widthmodulation (PWM) that we already discussed in the sec-tion on DC-DC conversion (17.3.2). In this configura-tion, each leg (the one via S1 and S4 and the one via S2and S3) in fact acts as a buck converter.

If a low-pass filter consisting of capacitors and induct-ors is used, as in Fig. 17.19, the high frequency com-ponents are filtered out and hence a very smooth sine



C

+

−

DCV

1S

3S

2S

4S

ACVPWMV

Figure 17.19: Illustration of an H-bridge containing a low-pass filter for removing the high-frequency components of thesignal.

curve can be obtained that complies to the regulationsfor grid-connected systems. Figure 17.20 shows the un-filtered PWM output and the filtered sine output.

Note that in Fig. 17.19 diodes are connected in paral-lel to the switches. The reason for placing these di-odes is the following: If the switch goes from closedto open very fast, no current can flow through the in-ductor anymore, meaning that the change of currentflowing through the inductor is very high. This inducesa voltage given by

Vinduced = −LdIdt

, (17.19)

Mean voltage output

DCV+

DCV-

0

PMW output of bridge

Figure 17.20: The unfiltered PWM signal and the sine signalthat is obtained with a low-pass filter.

which is the higher the faster the current changes. Ashigh induced voltages will damage the electric circuitry,they must be prevented. The diode ensures that currentalso can flow after the switch opens. Hence current isflowing always and no high induced voltages appear.

Since this configuration is grid-connected, we easily candetermine the required DC input voltage, which mustbe minimal the peak voltage of the AC voltage. For aneffective AC voltage of 240 V, as used in large parts ofthe world, the peak voltage is Vpeak = 1.1

√2 · 240 V =

373 V, where we also took a 10% tolerance into account.

If the PV array delivers a lower voltage, thus a boostconverter would be required prior in addition to theinverter. Alternatively, a transformer can be used thattransforms a lower-voltage AC signal to the 240 V ACsignal. Such a system is sketched in Fig. 17.23. This sys-tem has the advantage of galvanic separation with allthe advantages discussed below, but on the other hand



+

−

DCV1L

2L

3L

Figure 17.21: Illustrating a three-phase inverter.

the transformer reduces the overall efficiency.

Figure 17.21 shows a three-phase inverter. As it can beseen, it is very similar to the single-phase inverter dis-cussed above, just it contains three legs of that eachcreates a sine-wave output with a phase shift of 120°between them.

Half-bridge inverters

A simpler inverter topology, the so-called half-bridgeinverter, is shown in Fig. 17.22. In contrast to the H-bridge configuration, here, two switches are replacedby capacitors and the midpoint in-between the two ca-

+

−

DCV

2S

4S

ACVLoad

Figure 17.22: Illustrating a half-bridge inverter.

pacitors is directly connected to the ground. The half-bridge configuration is much simpler than the H-bridgeconfiguration, but it has some drawbacks.

The main drawback is requirement for a high DC linkvoltage, which needs to be two times higher than thatof a full bridge inverter. For a effective AC voltageof 240 V this would be 746 V. Because the topologyprovides two-levels in the output voltage (in differenceto three levels for the full bridge configuration, a highercurrent ripple is present in the output filter inductor.Hence, larger value of the output filters are required.



17.3.4 Some remarks

Switches

All the DC-DC transformers and DC-AC transformersdiscussed above contain switches. Traditionally, in line-commutated inverters, thyristors are used as switches. Athyristor is a electronic component consistent of pnpnlayers. It thus contains 3 p-n junctions. One disadvant-age is that thyristors cannot be turned off, but onlyturned on. Thus, one has to wait for the next zero passof the grid signal [98]. The current flow thus is rect-angular which leads to a very high harmonic contentrequiring additional filters in order to make the outputcompatible with the electricity grid. Nowadays, thyris-tors only are used for inverters with a power of 100 kWand higher.

All other inverters are self-commutated inverters thatgenerate an output with very little harmonic contentas described above and in Fig. 17.20. The switches thereare fully-controllable such that pulse-width modula-tion becomes possible. As switches, GTOs (gate turn-offthyristors), IGBTs (insulated-gate bipolar transistors) orMOSFETs (metal-oxide-semiconductor field-effect tran-sistors) are used. More information can be found forexample in Reference [20].

Overall configuration

Figure 17.23 shows an example of a transformer-lessabsorber as it could be sold for household systems. Be-sides the actual DC-AC converter, which is realised asan H-bridge, it also contains a DC-DC boost converterand an MPPT that uses the voltage and current meas-ured on the PV side as input. The system sketched alsocontains several electronic components for increasedsafety: a residual current measurement to detect leakagecurrents above a certain threshold and to shut-downthe inverter if these currents appear. Further, a grid-monitoring unit to prevent islanding (see below) [98].

Potential induced degradation (PID)

As mentioned earlier, in PV systems that have atransformer-less inverter, no galvanic separationbetween the DC- and the AC-parts of the system isgiven. Because of this lack of galvanic isolation, a po-tential of -500 V or more between the PV modules andthe ground can occur, which can lead to potential in-duced degradation (PID). Many thin-film contain a TCOfront contact that is deposited in superstrate configura-tion on the glass top plate. Positively charged sodiumions than can travel into the TCO because of this po-tential. This leads to corrosion and consequently to per-formance loss of the module. Also, for crystalline mod-



Earth-faultMonitoring

MPPRegulation

Central ControlResidual Current

Measurement

GridMonitoring

PV Array EMCFilter

EMCFilter

BoostConverter

DCBus

PWMBridge

L

N

230 V ~

Data Interface

Figure 17.23: Illustrating an example of a transformer-less inverter unit as it is sold for residential PV systems. As switches, MOS-FETs are used. (Adapted from [98]).



ules, PID can be a problem [98]. Therefore, for systemscontaining thin-film modules the inverter must have atransformer.

Islanding

A potential danger of grid-connected systems is island-ing. Imagine that a potential PV system is installed ina street where the electricity grid is shut-down in orderto do maintenance work on the electricity cables. If itis a sunny day, the PV system will produce power and– without protection – would deliver the power to thegrid. The electricity worker thus can be in danger. Thisphenomenon is called islanding and due to its danger itmust be prevented.

The inverter therefore must be able to detect, when theelectricity grid is shut-down. If this is the case, also theinverter must stop delivering power to the grid.

Efficiency of power converters

For planning a PV system it is very important to knowthe efficiency of the power converters. This efficiency ofDC-DC and DC-AC converters is defined as

η =Po

Po + Pd, (17.20)

i.e. the fraction of the output power Po to the sum ofPo with the dissipated power Pd. For estimating theefficiency we thus must estimate the dissipated (lost)power, which can be seen as a sum of several compon-ents,

Pd = PL + Pswitch + other losses, (17.21)

i.e. the power lost in the inductor PL and the power lostin the switch. PL is given as

PL = I2L, rmsRL, (17.22)

where IL, rms is the mean current flowing through theinductor. The losses in the switch strongly depend onthe type of switch that is used. Other losses are for ex-ample resistive losses in the circuitry in-between theswitches.

For a complete inverter unit it is convenient to definethe efficiency as

ηinv =PAC

PDC, (17.23)

which is the ratio of the output AC power to the DC in-put power. Figure 17.24 shows the efficiency of a com-mercially available inverter for different input voltages.As we can see, in general the lower the output power,the less efficient the inverter. This efficiency charac-teristic must be taken into account when planning aPV system. Further, the efficiency is lower if the inputvoltage deviates from the nominal value.



85

90

95

100

Inve

rter

E

cien

cy (%

)

0 10 20 30 40 50 60 70 80 90 100

Fraction of Rated Output Power (%)

120 V DC260 V DC335 V DC

Figure 17.24: The power-dependent efficiency for severalinput voltages of a Fronius Galvo 1.5-1 208-240 inverter at 240 VAC. The nominal input voltage is 260 V DC. (Data taken fromRef. [99]).

17.4 Batteries

In this section we discuss a vital component not only ofPV systems but of renewable energy systems in general.Energy storage is very important at both small and largescales in order to tackle the intermittency of renewableenergy sources. In the case of PV systems, the inter-mittency of the electricity generation is of two kinds:first, diurnal fluctuations, i.e. the difference of irradianceduring the 24 hour period. Secondly, the seasonal fluc-tuations, i.e. the difference of irradiance between thesummer and winter months. There are several technolo-gical options for realising storage of energy. Therefore itis important to make an optimal choice.

Fig. 17.25 shows a Ragone chart, where the specific energyis plotted against the peak power density. Because it usesa double-logarithmic chart storage technologies withvery different storage properties can be compared inone plot. Solar energy application require a high energydensity and – depending on the application – also areasonably high power density. For example, we cannotuse capacitors because of their very poor energy dens-ity. For short term to medium term storage, the mostcommon storage technology of course is the battery.Batteries have both the right energy density and powerdensity to meet the daily storage demand in small andmedium-size PV systems. However, the seasonal stor-age problem at large scales is yet to be solved.


17.4. Batteries 289

Li-ionbat.

100

Energy density (Wh/kg)

Pow

er d

ensi

ty (W

/kg)

10

double layer cap.

elektrolytecap.

Leadacid

NiMHNiCd

ultracap. 104

103

102

1010310210-1

CH4

Li-ion cap.

gasoline

H2,combustion

H2, fuel cell

Figure 17.25: A Ragone chart of different energy storagemethods. Capacitors are indicated with “cap”.

The ease of implementation and efficiency of the bat-teries is still superior to that of other technologies, likepumping water to higher levels, compressed air energystorage, conversion to hydrogen, flywheels, and others.Therefore we will focus on battery technology as a vi-able storage option for PV systems.

Batteries are electrochemical devices that convert chem-ical energy into electrical energy. We can distinguishbetween primary or secondary batteries. Primary batter-ies convert chemical energy to electrical energy irrevers-ibly. For example, zinc carbon and alkaline batteries areprimary batteries.

Secondary batteries or rechargeable batteries, as they aremore commonly called, convert chemical energy to elec-trical energy reversibly. This means that they can be re-charged when an over-potential is used. In other words,excess electrical energy is stored in these secondary bat-teries in the form of chemical energy. Typical examplesfor rechargeable batteries are lead acid or lithium ion bat-teries. For PV systems only secondary batteries are ofinterest.

17.4.1 Types of batteries

Lead-acid batteries are the oldest and most mature tech-nology available. They will be discussed in detail laterin this section.

Other examples are nickel-metal hydride (NiMH) andnickel cadmium (NiCd) batteries. NiMH batteries havea high energy density, which is comparable to thatof lithium-ion batteries (discussed below). However,NiMH batteries suffer from a high rate of self dis-charge. On the other hand, NiCd batteries have muchlower energy density than lithium ion batteries. Fur-thermore, because of the toxicity of cadmium, NiCdbatteries are widely banned in the European Union forconsumer use. Additionally, NiCd batteries suffer fromwhat is called the memory effect: the batteries loose theirusable energy capacity if they are repeatedly chargedafter only being partially discharged. These disadvant-



Positive electrolyte tank

Negative electrolyte tank

Pump Pump

22VO VO+ +« 2 3V V+ +«

+ −

proton exchangemembrane

Figure 17.26: Schematic on a vanadium redox-flow batterythat employs vanadium ions.

ages make NiMH and NiCD batteries unsuitable can-didates for PV storage systems.

Lithium ion batteries (LIB) and lithium-ion polymer bat-teries, which are often referred to as lithium polymer(LiPo) batteries, have been heavily investigated in re-cent years. Their high energy density already has madethem the favourite technology for light-weight storageapplications for example in mobile telephones. How-ever, these technologies still suffer from high costs andlow maturity.

The last and the most recent category of batteries thatwe will discuss in this treatise are redox-flow batteries.

Lead-acid batteries and LIB, the two main storage op-tions for PV systems, are similar in the sense that theirelectrodes undergo chemical conversion during char-ging and discharging, which makes their electrodes todegenerate with time, leading to inevitable “ageing” ofthe battery. In contrast, redox flow batteries combineproperties of both batteries and fuel cells, as illustratedin Fig. 17.26. Two liquids, a positive electrolyte and a neg-ative electrolyte are brought together only separated bya membrane, which only is permeable for protons. Thecell thus can be charged and discharged without the re-actants being mixed, which in principle prevents the li-quids from ageing. The chemical energy in a redox flowbattery is stored in its 2 electrolytes, which are storedin two separate tanks. Since it is easy to make the tankslarger, the maximal energy that can be stored in sucha battery thus is not restricted. Further, the maximaloutput power can easily be increased by increasing thearea of the membrane, for example by using more cellsat the same time. The major disadvantages is that sucha battery system requires additional components suchas pumps, which makes them more complicated thanother types of batteries.

Figure 17.27 shows a Ragone plot as it is used to com-pare different battery technologies. In contrast to Fig.17.25, here the the gravimetric energy density and volu-metric energy density are plotted against each other.The gravimetric energy density is the amount of energystored per mass of the battery; it typically is meas-


17.4. Batteries 291

Volu

met

ric e

nerg

y de

nsity

(Wh/

l)

smal

ler

lighterPb-acid

NiMHLi-ion

Li-polymer

Thin-lm Li

1000

4003002001000

200

400

600

800

Gravimetric energy density (Wh/kg)

Figure 17.27: A Ragone chart for comparing different second-ary battery technologies with each other.

ured in Wh/kg. The volumetric energy density is theamount of energy stored per volume of battery; it isgiven in Wh/l. The higher the gravimetric energy dens-ity, the lighter the battery can be. The higher the volu-metric energy density, the smaller the battery can be.

Figure 17.27 shows that lead acid batteries have boththe lowest volumetric and gravimetric energy densitiesamong the different battery technologies. Lithium ionbatteries show ideal material properties for using themas storage devices. Redox-flow batteries are very prom-ising. However, both LIB and redox-flow batteries arestill in a development phase which makes these techno-

logies still very expensive. Thus, because of to the un-equalled maturity and hence low cost of the lead-acidbatteries, they are still the storage technology of choicefor PV systems.

Figure 17.28 shows a sketch of a lead-acid battery. Atypical battery is composed of several individual cells,of which each has a nominal cell voltage around 2 V.Different methods of assembly are used. In the block as-sembly, the individual cells share the housing and areinterconnected internally. For example, to get the typ-ical battery voltage of 12 V, six cells are connected inseries. As the name suggests, lead acid batteries usean acidic electrolyte. More specifically they use dilutedsulphuric acid H2SO4. Two plates of opposite polar-ity are inserted in the electrolyte solution, which act asthe electrodes. The electrodes contain grid shaped leadcarrier and porous active material. This porous activematerial has a sponge-like structure, which providessufficient surface area for the electrochemical reaction.The active mass in the negative electrode is lead, whilein the positive electrode lead dioxide (PbO2) is used.

When the battery is discharged, electrons flow from thenegative to the positive electrode through the externalcircuit, causing a chemical reaction between the platesand the electrolyte. This forward reaction also depletesthe electrolyte, affecting its state of charge (SoC). Whenthe battery is recharged, the flow of electrons is re-versed, as the external circuit does not have a load, but



Container

Sediment space Element rests

Negative plate

Envelopeseparators

Positive plate

Plate lugsPost straps

Cover

Tapered terminal postThrough partitionconectors

Vent plugs

Positive plate Negative plateH2SO4

external circuit

+ −

Figure 17.28: Schematic of a lead-acid battery.


17.4. Batteries 293

a source with a voltage higher than that of the batteryenables the reverse reaction. In the PV systems, thissource is nothing but the PV module or array provid-ing clean solar power.

17.4.2 Battery parameters

Let us now discuss some parameters that are used tocharacterise batteries.

Voltage

First, we will discuss the voltage rating of the battery.The voltage at that the battery is rated is the nominalvoltage at which the battery is supposed to operate. Theso called solar batteries or lead acid grid plate batteriesare usually rated at 12 V, 24 V or 48 V.

Capacity

When talking about batteries, the term capacity refersto the amount of charge that the battery can deliverat the rated voltage. The capacity is directly propor-tional to the amount of electrode material in the bat-tery. This explains why a small cell has a lower capacitythan a large cell that is based on the same chemistry,even though the open circuit voltage across the cell will

be the same for both the cells. Thus, the voltage of thecell is more chemistry based, while the capacity is morebased on the quantity of the active materials used.

The capacity Cbat is measured in ampere-hours (Ah).Note that charge usually is measured in coulomb (C).As the electric current is defined as the rate of flow ofelectric charge, Ah is another unit of charge. Since 1 C= 1 As, 1 Ah = 3600 C. For batteries, Ah is the moreconvenient unit, because in the field of electricity theamount of energy usually is measured in watt-hours(Wh). The energy capacity of a battery is simply givenby multiplying the rated battery voltage measured inVolt by the battery capacity measured in Amp-hours,

Ebat = CbatV, (17.24)

which results in the battery energy capacity in Watt-hours.

C-rate

A brand new battery with 10 Ah capacity theoreticallycan deliver 1 A current for 10 hours at room temperat-ure. Of course, in practice this is seldom the case due toseveral factors. Therefore, the C-rate is used, which is ameasure of the rate of discharge of the battery relativeto its capacity. It is defined as the multiple of the cur-rent over the discharge current that the battery can sus-tain over one hour. For example, a C-rate of 1 for a 10



Ah battery corresponds to a discharge current of 10 Aover 1 hour. A C-rate of 2 for the same battery wouldcorrespond to a discharge current of 20 A over half anhour. Similarly, a C-rate of 0.5 implies a discharge cur-rent of 5 A over 2 hours. In general, it can be said thata C-rate of n corresponds to the battery getting fullydischarged in 1/n hours, irrespective of the battery ca-pacity.

Battery efficiency

For designing PV systems it is very importent to knowthe efficiency of the storage system. For storage systems,usually the round-trip efficiency is used, which is givenas the ratio of total storage system input to the totalstorage system output,

ηbat =Eout

Ein100%. (17.25)

For example, if 10 kWh is pumped into the storage sys-tem during charging, but only 8 kWh can be retrievedduring discharging, the round trip efficiency of the stor-age system is 80%. The round-trip efficiency of batteriescan be broken down into two efficiencies: first, the vol-taic efficiency, which is the ratio of the average dischar-ging voltage to the average charging voltage,

ηV =Vdischarge

Vcharge100%. (17.26)

This efficiency covers the fact that the charging voltageis always a little above the rated voltage in order todrive the reverse chemical (charging) reaction in thebattery.

Secondly, we have the coulombic efficiency (or Faradayefficiency), which is defined as the ratio of the totalcharge got out of the battery to the total charge put intothe battery over a full charge cycle,

ηC =Qdischarge

Qcharge100%. (17.27)

The battery efficiency then is defined as the product ofthese two efficiencies,

ηbat = ηV · ηC =Vdischarge

Vcharge

Qdischarge

Qcharge100%. (17.28)

When comparing different storage devices, usuallythis round-trip efficiency is considered. It includes allthe effects of the different chemical and electrical non-idealities occurring in the battery.

State of charge and depth of discharge

At another important battery parameter is the State ofCharge (SoC), which is defined as the percentage of thebattery capacity available for discharge,

SoC =Ebat

CbatV. (17.29)


anonymous

anonymous 10/26/2014: typo, must say: important

17.4. Batteries 295

Thus, a 10 Ah rated battery that has been drained by2 Ah is said to have a SoC of 80%. Also the Depth ofDischarge (DoD) is an important parameter. It is definedas the percentage of the battery capacity that has beendischarged,

DoD =CbatV − Ebat

CbatV. (17.30)

For example, a 10 Ah battery that has been drainedby 2 Ah has a DoD of 20%. The SoC and the DoD arecomplimentary to each other.

Cycle lifetime

The cycle lifetime is a very important parameter. It isdefined as the number of charging and dischargingcycles after that the battery capacity drops below 80%of the nominal value. Usually, the cycle lifetime is spe-cified by the battery manufacturer as an absolute num-ber. However, stating the battery lifetime as a singlenumber is a oversimplification because the differentbattery parameters discussed so far are not only relatedto each other but are also dependent on the temperat-ure.

Figure 17.29 (a) shows the cycle lifetime as a function ofthe DoD for different temperatures. Clearly, colder op-erating temperatures mean longer cycle lifetimes. Fur-thermore, the cycle lifetime depends strongly on the

(a) (b)

Figure 17.29: (a) Qualitative illustration of the cycle lifetimeof a lead-acid battery in dependence of the DoD and temperat-ure [100]. (b) Effect of the temperature on the battery capacity[100].

DoD. The smaller the DoD, the higher the cycle life-time. Thus, that the battery will last longer if the av-erage DoD can be reduced during the lifetime of thebattery. Also, battery overheating should be strictly con-trolled. Overheating can because of overcharging andsubsequent over-voltage of the lead acid battery. To pre-vent this, charge controllers are used that we address inthe next section.

Temperature effects

While the battery lifetime is increased at lower tem-peratures, one more effect must to be considered. The


anonymous

anonymous 10/26/2014: Typo. Must say: Celsius

anonymous

anonymous 10/26/2014: Typo. Must say: lifetime


temperature affects the battery capacity during regularuse too. As seen from Fig. 17.29 (b), the lower the tem-perature, the lower the battery capacity. This is because,the chemicals in the battery are more active at highertemperatures, and the increased chemical activity leadsto increased battery capacity. It is even possible to reachan above rated battery capacity at high temperatures.However, such high temperatures are severely detri-mental to the battery health.

Ageing

The major cause for ageing of the battery is sulphation.If the battery is insufficiently recharged after being dis-charged, sulphate crystals start to grow, which cannotbe completely transformed back into lead or lead oxide.Thus the battery slowly loses its active material massand hence its discharge capacity. Corrosion of the leadgrid at the electrode is another common ageing mech-anism. Corrosion leads to increased grid resistance dueto high positive potentials. Further, the electrolyte candry out. At high charging voltages, gassing can occur,which results in the loss of water. Thus, demineralisedwater should be used to refill the battery from time totime.

17.4.3 Charge controllers

After having discussed different types of batteries anddifferent battery characteristics, it is now time to dis-cuss charge controllers. Charge controllers are used inPV systems that use batteries, which are stand-alonesystems in most cases. As we have seen before, it isvery important to charge and discharge batteries atthe right voltage and current levels in order to ensurea long battery lifetime. A battery is an electrochem-ical device that requires a small over-potential to becharged. However, batteries have strict voltage limits,which are necessary for their optimal functioning. Fur-ther, the amount of current sent to the battery by thePV array and the current flowing through the batterywhile being discharged have to be within well-definedlimits for proper functioning of the battery. We haveseen before that lead acid batteries suffer from over-charge and over-discharge. On the other hand, the PVarray responds dynamically to ambient conditions likeirradiance, temperature and other factors like shading.Thus, directly coupling the battery to the PV array andthe loads is detrimental to the battery lifetime.

Therefore a device is needed that controls the currentsflowing between the battery, the PV array and the loadand that ensures that the electrical parameters presentat the battery are kept within the specifications givenby the battery manufacturer. These tasks are done bya charge controller, that nowadays has several different


17.4. Batteries 297

PV Modules Batteries

DC

Chargecontroller

Load

Figure 17.30: Illustrating the position of the charge controllerin a generic PV system with batteries.

functionalities, which also depend on the manufacturer.We will discuss the most important functionalities. Aschematic of its location in a PV system is shown inFig. 17.30.

When the sun is shining at peak hours during summer,the generated PV power excesses the load. The excessenergy is sent to the battery. When the battery is fullycharged, and the PV array is still connected to the bat-tery, the battery might overcharge, which can cause sev-eral problems like gas formation, capacity loss or over-heating. Here, the charge controller plays a vital roleby de-coupling the PV array from the battery. Similarly,

during severe winter days at low irradiance, the loadexceeds the power generated by the PV array, such thatthe battery is heavily discharged. Over-discharging thebattery has a detrimental effect on the cycle lifetime, asdiscussed above. The charge controller prevents the bat-tery from being over-discharged by disconnecting thebattery from the load.

For optimal performance, the battery voltage has to bewithin specified limits. The charge controller can helpin maintaining an allowed voltage range in order toensure a healthy operation. Further, the PV array willhave its Vmpp at different levels, based on the temper-ature and irradiance conditions. Hence, the charge con-troller needs to perform appropriate voltage regulationto ensure the battery operates in the specified voltagerange, while the PV array is operating at the MPP.Modern charge controllers often have an MPP trackerintegrated.

As we have seen above, certain C-rates are used as bat-tery specifications. The higher the charge/dischargerates, the lower the coulombic efficiency of the battery.The optimal charge rates, as specified by the manufac-turer, can be reached by manipulating the current flow-ing into the battery. A charge controller that containsa proper current regulation is also able to control theC-rates. Finally, the charge controller can impose thelimits on the maximal currents flowing into and fromthe battery.



If no blocking diodes are used, it is even possible thatthe battery can “load” the PV array, when the PV ar-ray is operating at a very low voltage. This means thatthe battery will impose a forward bias on the PV mod-ules and make them consume the battery power, whichleads to heating up the solar cells. Traditionally, block-ing diodes are used at the PV panel or string level toprevent this back discharge of the battery through the PVarray. However, this function is also easily integrated inthe charge controller.

We distinguish between series and shunt controllers,as illustrated in 17.31. In a series controller, overchar-ging is prevented by disconnecting the PV array until aparticular voltage drop is detected, at which point thearray is connected to the battery again. On the otherhand, in a parallel or shunt controller, overcharging isprevented by short-circuiting the PV array. This meansthat the PV modules work under short circuit mode,and that no current flows into the battery. These topolo-gies also ensure over-discharge protection using powerswitches for the load connection, which are appropri-ately controlled by the algorithms implemented into thecharge controller algorithm. As charge controllers area necessary component of of stand-alone PV systems,they should have a very high efficiency.

As we have seen above, temperature plays a crucialrole in the functioning of the battery. Not only doestemperature affect the lifespan of the battery, but it also

changes its electrical parameters significantly. Thus,modern charge controllers have a temperature sensorincluded, which allows the charge controller to adjustthe electrical parameters of the battery, like the operat-ing voltage, to the temperature. The charge controllerthus keeps the operating range of the battery withinthe optimal range of voltages. The charge controller isusually kept in close proximity to the battery, such thatthe operating temperature of the battery is close to thatof the charge controller. However, when the battery isheavily loaded, the battery might heat up, leading todifferences between the temperature expected by thecharge controller and the actual temperature of the bat-tery. Therefore, high end charge controllers also taketemperature effects due to high currents into account.

17.5 Cables

The overall performance of PV systems also is stronglydependent on the correct choice of the cables. We there-fore will discuss how to choose suitable cables. But westart our discussion with color conventions.

PV systems usually contain DC and AC parts. For cor-rectly installing a PV system, it is important to knowthe color conventions. For DC cables,

• red is used for connecting the +-contacts of thedifferent system components with each other while


17.5. Cables 299

Chargecontroller

core

ChargecontrollerLoad Load

corePVArray

PVArray

batVbatV

batIbatI

PVI PVILoadI LoadI(a) (b)

Figure 17.31: Basic wiring scheme of (a) a series and (b) a shunt charge controller.



• black is used for interconnecting the −-contacts.

For AC wiring, different colour conventions are usedaround the world.

• For example, in the European Union,blue is used for neutral,green-yellow is used for the protective earth andbrown (or another color) is used for the phase.

• In the United States and Canada,silver is used for neutral,green-yellow, green or a bare conductor is usedfor the protective earthand black (or another color) is used for the phase.

• In India and Pakistan,black is used for neutral,green is used for the protective earth andblue, red, or yellow is used for the phase.

Therefore it is very important to check the standardsof the country in that the PV system is going to be in-stalled.

The cables have to be chosen such that resistive losses areminimal. For estimating these losses, we look at a verysimple system that is illustrated in Fig. 17.32 The sys-tem consists of a power source and a load with resist-ance RL. Also the cables have a resistance Rcable, whichalso is sketched. The power loss at the cables is given

RL Rcable

Figure 17.32: Illustrating a circuit with a load RL and cableresistance Rcable. The connections drawn in between the com-ponents are lossless.

asPcable = I · ∆Vcable, (17.31)

where ∆Vcable is the voltage drop across the cable,which is given as

∆Vcable = VRcable

RL + Rcable. (17.32)

UsingV = I(RL + Rcable) (17.33)

we findPcable = I2Rcable. (17.34)

Hence, as the current doubles, four times as much heatwill be dissipated at the cables. It now is obvious whymodern modules have connected all cells in series.


17.5. Cables 301

Let us now calculate the resistance of a cable withlength ` and cross section A. It is clear, that if ` isdoubled, also Rcable doubles. In contrast, if A doubles,Rcable decreases to half. The resistance thus is given by

Rcable = ρ`

A=

1σ

`

A, (17.35)

where ρ is the specific resistance or resistivity and σ is thespecific conductance or conductivity. If both, ` and A aregiven in metres, their units are [ρ] = Ω ·m and [σ] =S/m, where S denotes the unit for conductivity, whichis Siemens.

For electrical cables it is convenient to have ` in metresand A in mm2. When using this convention, we findthe following units for ρ and σ:

1 Ω ·m = 1 Ωm2

m= 1 Ω

106 mm2

m= 106 Ω

mm2

m,

1Sm

= 1 Smm2 = 1 Ω

m106 mm2 = 10−6 Ω

mmm2 .

The most widely used metals used for electrical cablesare copper and aluminium. Their resistances and con-

ductivities are

ρCu = 1.68 · 10−8 Ω ·m = 1.68 · 10−2 Ωmm2

m,

σCu = 5.96 · 107 Sm

= 59.6 Sm

mm2 ,

ρAl = 2.82 · 10−8 Ω ·m = 2.82 · 10−2 Ωmm2

m,

σAl = 3.55 · 107 Sm

= 35.5 Sm

mm2 .

Usual thicknesses for cables are 0.75 mm2, 1.5 mm2,2.5 mm2, 4 mm2, 6 mm2, 10 mm2, 16 mm2, 25 mm2,35 mm2, et cetera. Since DC circuits are driven at lowervoltages than AC currents, the currents are higher, re-quiring thicker cables.


18PV System Design

In the previous chapters we discussed how to estimatethe irradiance on a PV module in dependence of theposition on Earth, date and time (Chapter 16). Furtherwe introduced and discussed all the different compon-ents of PV Systems in Chapter 17.

In this chapter, we will put the things together and usethem such that we can design complete PV systems. Ofcourse, PV system designs can be made with differentlevels of complexity. For a first approximation the STCperformance of the PV modules and the performanceof the other components (like the inverter) at STC con-ditions and the number of Equivalent Sun Hours (ESH)at the location of the PV system are sufficient. The no-tion of ESH will be discussed below. In a more detailedapproach, performance changes of the modules and

the other components due to changing irradiance andweather conditions are taken into account. Since theseperformance changes can be quite high, they can alterthe optimal system design considerably.

There are two main paradigms for designing PV sys-tems. First, the system can be designed such that thegenerated power and the loads, i.e. the consumedpower, match. A second way to design a PV systemis to base the design on economics. We must distin-guish between grid-connected and off-grid systems. Aswe will see, grid-connected systems have very differentdemands than off-grid systems.

This chapter is organised as follows: First, as an ex-ample, we will discuss a design of a simple off-grid

303


304 18. PV System Design

system in Section 18.1. After that we will take a moredetailed look on load profiles in Section 18.2. In Section18.3 and on how weather and irradiance conditionsaffect the performance of PV modules and BOS com-ponents, mainly inverters. Finally, in Sections 18.4 and18.5 we learn how to design grid-connected and off-grid systems, respectively.

18.1 A simple approach for design-ing off-grid systems

In this section, we will design a simple off-grid system,as depicted in Fig. 18.1. The design presented here isbased on very simple assumptions and does not takeany weather-dependent performance changes into ac-count. Nonetheless, we will see the major steps that arenecessary for designing a system. Such a simple designcan be performed in a six step plan:

1. Determine the total load current and operationaltime

2. Add system losses

3. Determine the solar irradiation in daily equivalentsun hours (ESH)

4. Determine total solar array current requirements

PV Modules

Charge controller

Inverter

Batteries

Loads

DC

AC=

~

Figure 18.1: Illustrating a simple off-grid PV system with ACand DC loads (see also Fig. 15.1 b).


18.1. A simple approach for designing off-grid systems 305

5. Determine optimum module arrangement for solararray

6. Determine battery size for recommended reservetime

1. Determine the total load current and operationaltime

Before starting determining the current requirementsof loads of a PV system one has to decide the nominaloperational voltage of the PV system. Usual nominalvoltages are 12 V or 24 V. When knowing the voltage,the next step is to express the daily energy require-ments of loads in terms of current and average oper-ational time expressed in Ampere-hours [Ah]. In caseof DC loads the daily energy [Wh] requirement is cal-culated by multiplying the power rating [W] of an in-dividual appliance with the average daily operationaltime [h]. Dividing the Wh by the nominal PV systemoperational voltage, the required Ah of the appliance isobtained.

Example

A 12 V PV system has two DC appliances A and B requiring15 and 20 W respectively. The average operational time per dayis 6 hours for device A and 3 hours for device B. The daily en-

ergy requirements of the devices expressed in Ah are calculatedas follows:

Device A: 15 W · 6 h = 90 Wh

Device B: 20 W · 3 h = 60 Wh

Total: 90 Wh + 60 Wh = 150 Wh

150 Wh/12 V = 12.5 Ah

In case of AC loads the energy use has to be expressedas a DC energy requirement since PV modules gener-ate DC electricity. The DC equivalent of the energy useof an AC load is determined by dividing the AC loadenergy use by the efficiency of the inverter, which typ-ically can be assumed to be 85%. By dividing the DCenergy requirement by the nominal PV system voltagethe Ah is determined.

Example

An AC computer (device C) and TV set (device D) are connec-ted to the PV system. The computer, which has rated power40 W, runs 2 hours per day and the TV set with rated power60 W is 3 hours per day in operation. The daily energy re-quirements of the devices expressed in DC Ah are calculated asfollows:



Device C: 40 W · 2 h = 80 Wh

Device D: 60 W · 3 h = 180 Wh

Total: 80 Wh + 180 Wh = 260 Wh

DC requirement: 260 Wh/0.85 = 306 Wh

306 Wh/12 V = 25.5 Ah

2. Add system losses

Some components of the PV system, such as charge reg-ulators and batteries require energy to perform theirfunctions. We denote the use of energy by the systemcomponents as system energy losses. Therefore, thetotal energy requirements of loads, which were determ-ined in step 1, are increase with 20 to 30% in order tocompensate for the system losses.

Example

The total DC requirements of loads plus the system losses(20%) are determined as follows:

(12.5 Ah + 25.5 Ah) · 1.2 = 45.6 Ah

3. Determine the solar irradiation in daily equival-ent sun hours (EHS)

How much energy a PV module delivers depends onseveral factors, such as local weather conditions, sea-sonal changes, and installation of modules. PV modulesshould be installed under the optimal tilt-angle in orderto achieve best year-round performance. It is also im-portant to know whether a PV system is expected to beused all-year round or only during a certain period ofa year. The power output during winter is much lessthan the yearly average and in the summer months thepower output will be above the average.

In the PV community, 1 equivalent sun means a solarirradiance of 1000 W/m2. This value corresponds tothe standard, at which the performance of solar cellsand modules is determined. The rated parameters ofmodules (see for example Table 17.1) are determined atsolar irradiance of 1 sun.

When solar irradiation data are available for a partic-ular location than the equivalent sun hours can be de-termined. For example, in the Netherlands the averageannual solar irradiation is 1000 kWh/m2. Accordingto the AM1.5 spectrum, 1 sun delivers 1000 W/m2 = 1kW/m2. Hence, the Dutch average annual solar irradi-ation can be expressed as

1000 kWh ·m−2

1 kW ·m−2 = 1000 h,


Mycophagist

Mycophagist 12/02/2014: Should be ESH, not EHS


Figure 18.2: The average global horizontal insolation of the World given in kWh/year and PSH/day [101]. Note that this map onlyshows latitudes in between 60° N and 60° S.



i.e. 1000 equivalent sun hours, which corresponds to2.8 h per day. Figures 18.2 and 18.3 show the globalhorizontal irradiation given in units of daily equivalentsun hours (ESH) and in kWh/year. Note that these fig-ures show horizontal values, meaning that a PV mod-ule would get this irradiance if it would be flat. For atilted module, the annual irradiance can be consider-ably higher, as we have seen in Chapter 16.

4. Determine total solar array current requirements

The current that has to be generated by the solar ar-ray is determined by dividing the total DC energy re-quirement of the PV system including loads and systemlosses (calculated in step 2 and expressed in Ah) by thedaily equivalent sun hours (determined in step 3).

Example

The total DC requirements of loads plus the system lossesare 45.6 Ah. The daily EHS for the Netherlands is about 3hours. The required total current generated by the solar arrayis 45.6 Ah / 3 h = 15.2 A.

Figure 18.3: The average global horizontal insolation of theNetherlands given in kWh/year and PSH/day [101].



5. Determine optimum module arrangement forsolar array

Usually, PV manufacturers produce modules in a wholeseries of different output powers. In the optimum ar-rangement of modules the required total solar arraycurrent (as determined in step 4) is obtained with theminimum number of modules. Modules can be eitherconnected in series or in parallel to form an array.When modules are connected in series, the nominalvoltage of the PV system is increased, while the parallelconnection of modules results in a higher current.

The required number of modules in parallel is calcu-lated by dividing the total current required from thesolar array (determined in step 4) by the current gen-erated by module at peak power (rated current in thespecification sheet). The number of modules in series isdetermined by dividing the nominal PV system voltagewith the nominal module voltage (in the specificationsheet under configuration). The total number of mod-ules is the product of the number of modules requiredin parallel and the number required in series.

Example

The required total current generated by the solar array is15.2 A. We have Shell SM50-H modules available. The spe-cification of these modules is given in Table 17.1. The rated

current of a module is 3.15 A. The number of modules in par-allel is 15.2 A / 3.15 A = 4.8 < 5 modules. The nominalvoltage of the PV system is 12V and the nominal modulevoltage is 12 V. The required number of modules in series thusis 12 V / 12 V = 1 module. Therefore, the total number ofmodules in the array is 5 · 1 = 5 modules.

6. Determine battery size for recommended reservetime

Batteries are a major component of stand-alone PV sys-tems. The batteries provide load operation at night orin combination with the PV modules during periodsof limited sunlight. For a safe operation of the PV sys-tem one has to anticipate periods with cloudy weatherand plan a reserve energy capacity stored in the bat-teries. This reserve capacity is referred to as PV systemautonomy, which means the period of time that the sys-tem is not dependent on energy generated by PV mod-ules. It is given in days. The system autonomy dependson the type of loads. For critical loads such as telecom-munications components the autonomy can be 10 daysand more, for residential use it is usually 5 days or less.The capacity [Ah] of the batteries is calculated by mul-tiplying the daily total DC energy requirement of thePV system including loads and system losses (calcu-lated in step 2 and expressed in Ah) by the number ofdays of recommended reserve time. In order to prolong



Table 18.1: Worksheet for designing a simple off-grid PV system based on rough assumptions.

1 Daily DC energy use (DC loads)

DC load W × h = Wh 1 Daily DC energy use (AC loads) +

1 Daily DC energy use (all loads) =

PV system nominal voltage /

Daily Ah requirements (all loads) =

2 Add PV system losses ×

Daily Ah requirements (system) =

3 Design EHS /

4 Total solar array current =

5 Select module type

Module rated current /

DC load W × h = Wh Number of modules in parallel =

PV system nominal voltage

Modules nominal voltage /

Number of modules in series =

Number of modules in parallel ×

Total number of modules =

6 Determine battery capacity

Daily Ah requirements (system)

Recommended reserve time ×

Usable battery capacity /

Minimum battery capacity =

Daily DC loads requirements

Total DC loads energy use:

Daily AC loads requirements

Total AC loads energy use:

/0.85 = DC energy requirement


18.2. Load profiles 311

(see Figure 4(c) and (d), respectively). There may also be load profiles

requiring a longer period of time than 24 hours that need to be

described; in this case, an average and a maximum daily load should be

given.

This information allows the derivation of the load profile, or at least

the typical load profile for a 24-hour-period load. This is performed by

multiplying each load current by the duration and then adding all the

products to find the daily ampere-hour load. If the duration of the

momentary load is not known, then 1 minute should be taken as recom-

mended in reference [3]. The load profile also provides the necessary

information about the maximum discharge rates that need to be sustained

by the battery.

2.1 Electricity Consumption by Lighting and ElectricalAppliancesTable 1 gives a summary of the load values for the main domestic appli-

ances. The principal results on energy consumption for lighting in the

Figure 4 (a) Individual load having a momentary and running currents and a num-ber n of occurrences, (b) load used a number of hours per day, (c) coincident loads,and (d) noncoincident loads. Elaborated from IEEE standard 1144-1996.

664 Luis Castañer et al.




given.








by the battery.








given.








by the battery.








given.








by the battery.





(a) (b) (c) (d)

Figure 18.4: Illustrating different load profiles.

the life of the battery it is recommended to operate thebattery using only 80% of its capacity. Therefore, theminimal capacity of the batteries is determined by di-viding the required capacity by a factor of 0.8.

Example

The total DC requirements of loads plus the system lossesare 45.6 Ah. The recommended reserve time capacity forthe installation side in the Netherlands is 5 days. Batterycapacity required by the system is 45.6 Ah · 5 = 228 Ah.The minimal battery capacity for a safe operation therefore is228 Ah/0.8 = 285 Ah.

Designing a simple PV system as described in this sec-tion can be carried out using a worksheet as in Table18.1, where the PV system design rules are summarised.

18.2 Load profiles

Now we take a look at the load profile. Figure 18.4 il-lustrates different shapes that loads can have. (a) Asimple load draws a constant amount of power for acertain time. (b) However, the consumed power doesnot need to be constant but can also show peaks thatcorrespond to switching electrical appliances on or off.A household of course has several different loads that(c) can be switched on at the same time (coincident) or(d) at different times (non-coincident).

Analysing load profiles can be performed with in-creasing complexity and hence accuracy. The simplestmethod is to determine the loads on a 24-hour basis. Todo this, an arbitrary day can be taken and the electri-city consumption can be monitored. However, severalloads do not fit in such a scheme. Several examples for



this already were treated in Section 18.1. For example,washing machines and dishwashers do not fit in a 24-hour scheme because typically they are only used sev-eral times in a week. Additionally, several loads areseasonal in nature, for example, air conditioning orheating, in case this is performed with a heat pump.Therefore it is advisable to look at load profiles for awhole year.

The total energy consumed in a year is given by

EYL =

∫year

PL(t)dt. (18.1)

It is expressed in kWh/year.

18.3 Meteorological effects

Standard Test Conditions (STC) of Photovoltaic (PV)modules are generally not representative of the realworking conditions of a solar module. For example,high levels of incident irradiation, may cause the tem-perature of a module to rise many degrees above theSTC temperature of 25°C, therefore lowering the mod-ule performances. For example, in the in a climate suchas the one of the Netherlands, real operating conditionsfor PV systems correspond to relatively low levels of ir-radiance combined with a cold and windy weather. Inorder to effectively estimate real time energy production

from a given set of ambient parameters, it is essential todevelop an accurate thermal model of the solar module,which evaluates the effects of the major meteorologicalparameters on the final temperature of a solar module.

In this section, we will develop an accurate thermalmodel for estimating the PV module working temper-ature as a function of meteorological parameters willbe developed. The model is based on a detailed energybalance between the module itself and the surround-ing environment. Both the installed configuration of thearray together with external parameters such as directincident solar irradiance on the panels, wind speed andcloud cover will be taken into account. From the calcu-lated equilibrium temperature, the overall efficiency ofthe PV array will be calculated by separately assessingthe temperature and irradiance effect on the efficiencyat STC.

18.3.1 Simplified thermal model for aphotovoltaic array

The temperature strongly influences the performancesof a PV module [102]. While the level of incident irra-diation can be easily determined by measuring it witha pyranometer, the temperature reached inside the cellis much harder evaluate. In order to give an estimateof the average operating temperature of the module,olar manufacturers provide, together with rated per-


18.3. Meteorological effects 313

Figure 18.5: Rise in temperature above the ambient level withrespect of increasing solar irradiance [103].

formances at STC, also the so called Nominal OperatingCell Temperature (NOCT). This value corresponds to theperformances of a cell under an irradiance level of 800W/m2, ambient temperature of 20°C and an externalwind speed of 1 m/s [102].

Simplified steady state models use a linear relation-ship between the solar irradiance GM and the differ-

ence between the module temperature and the ambienttemperature (TM − Ta), where the NOCT is used as areference point, [103]

TM = Ta +TNOCT − 20

800GM. (18.2)

Equation (18.2) is however too simplistic and thus leadsto significant errors in the modules temperature eval-uations. The reason is that it does not take environ-mental conditions such as wind speed and mountingconfiguration of the array into account [103].

In order to take into mounting configuration of themodule into account, the Installed Nominal Operat-ing Conditions Temperature (INOCT) has been defined[104]. This value is described as the cell temperatureof an installed array at NOCT conditions. Its value cantherefore be obtained from the NOCT and the mount-ing configuration. The evaluation of how the INOCTvaries with the module mounting configuration hasbeen experimentally determined by measuring theNOCT at various mounting heights. The results canbe found in literature and are here summarised in theTable 18.2.

Evaluating the influence of external meteorologicalparameters on the the module temperature is morecomplex and has been approached by developing a de-tailed thermal energy balance between the module andthe surroundings. Here, the module is assumed to be a



Table 18.2: Derivation of the INOCT from the NOCT forvarious mounting configurations [104].

Rack Mount INOCT = NOCT− 3CDirect Mount INOCT = NOCT+ 18CStandoff INOCT = NOCT+ X

where X is given by

W (inch) X (°C)1 113 26 − 1

single uniform mass at temperature TM. The three typesof heat transfer between the module itself with the sur-roundings are conduction, convection and radiation. Thecontributions considered in the model are:

• Heat received from the Sun in the form of insola-tion ϕGM, where ϕ is the absorptivity of the mod-ule.

• Convective heat exchange with surrounding airfrom the front and rear side of the module

hc(TM − Ta),

where hc denotes the overall convective heat trans-fer coefficient of the module.

• Radiative heat exchange between the upper mod-

ule surface and the sky

εtopσ(

T4M − T4

sky

),

where εtop = 0.84 is the emissivity of the modulefront glass and σ is the Stefan-Boltzmann constantas defined in Eq. (5.20). and between the rear sur-face and the ground

εbackσ(

T4M − T4

gr.

),

where the emissivitiy of the back is assumed to beεback = 0.89.

• Conductive heat transfer between the module andthe mounting structure. We neglect this contribu-tion due to the small area of the contact points[105].

By separately considering each of the above mentionedcontributions, we write down the balance for the heattransfer, [104]

mcdTM

dt=ϕGM − hc(TM − Ta)− εbackσ

(T4

M − T4gr.

)− εtopσ

(T4

M − T4sky

).

(18.3)Before providing a solution for this differential equa-tion, it is appropriate to remark the fact that we areconsidering the entire module as a uniform piece at a



a

a + θ

θTgr.

Tsky

qsun

qrad, sky

qconvection

qrad, ground

TM

qconvection

Ta

Figure 18.6: Representation of heat exchange between a tiltedmodule surface and the surroundings.

temperature TM. However, this is not entirely realisticsince modules are made of various layers of differentmaterials surrounding the actual solar cells. It is thepurpose of this section to evaluate the temperature ofthe inner cell, which is the place where absorption ofsolar radiation effectively place. This temperature willbe to some extent higher than the surface module tem-perature TM due to the heat produced in the cell dueto light absorption. The approximation of consideringthe temperature uniform throughout the module layershowever is justified because of the relatively low thick-ness of the active cell together with the low heat capa-city of the cell material compared to the other layers[105]. This results into a very low thermal resistance of

the cell to heat flow and therefore justifies the uniformtemperature approximation.

In addition to this, a steady state approach will here beconsidered, meaning that the module temperature willnot change over time for each of the 10-minute timesteps. In reality, the temperature follows an exponentialdecay lagging behind variations in irradiation level. Itis defined as Time Constant of a module, which is ‘thetime it takes for the module to reach 63% of the totalchange in temperature resulting from a step change inirradiance’ [105]. Time constants for PV modules aregenerally of the order of approximately 7 minutes. Fortime steps greater that the Time Constant, as it is in ourcase, the module can be approximated as being in asteady state condition. In light of this assumption, theterm on the left hand side of Eq. (18.3) vanishes.

It is now possible to proceed with the solution of thethermal energy balance equation. The formula can belinearised by noticing that(

a4 − b4)=(

a2 + b2)(a + b)(a− b). (18.4)

Since (T2

M + T2sky

)(TM + Tsky)

changes less than 5% for a 10°C variation in TM, wecan consider this term to be constant when TM varies[104]. Therefore the energy balance can be simplified



becoming linear with respect to TM. By defining

hr, sky = εtopσ(

T2M + T2

sky

)(TM + Tsky), (18.5a)

hr, gr. = εbackσ(

T2M + T2

gr.

)(TM + Tgr.), (18.5b)

we can rewrite Eq. (18.3) and find

ϕGM−hc(TM − Ta)− hr, sky(TM − Tsky)

−hr, gr.(TM − Tgr.) = 0.(18.6)

By rearranging the terms, the formula can be explicitlyexpressed as a function of TM,

TM =ϕG + hcTa + hr, skyTsky + hr, gr.Tgr.

hc + hr, sky + hr, gr.(18.7)

However, since hr, gr. and hr, sky are also function of themodule temperature, the equation needs to be solvediteratively: an initial module temperature is assignedand by hr, gr. and hr, sky are updated each iterations. Anearly exact solution can be obtained after 5 iterations.

Before solving iteratively Eq. (18.7) there are still manyunknown variables that need to be determined, whichwe will do in the following sections.

18.3.2 Calculating the convective transfercoefficients

Convection is a form of energy transfer from one placeto another caused by the movement of a fluid. Convect-ive heat transfer can be either free or forced dependingon the cause of the fluid motion.

In free convection, heat transfer is caused by temper-ature differences which affect the density of the fluiditself. Air starts circulating due to difference in buoy-ancy between hot (less dense) fluid and cold (denser)fluid. A circular motion is therefore initiated with risinghot fluid and sinking cold fluid. Free convection onlytakes place in a gravitational field [106].

Forced convection, on the other hand, is caused by afluid flow due to external forces which therefore en-hance the convective heat exchange. The heat transferdepends very much on whether the induced flow overa solid surface is laminar or turbulent. In the case ofturbulent flow, an increased heat transfer is expectedwith respect to the laminar situation. This fact is dueto an increased heat transport across the main directionof the flow. On the contrary, in laminar flow regime,only conduction is responsible for transport in the crossdirection. For this reason, forced convection is alwaysstudied separately in the laminar and turbulent regime[106]. The overall convection transfer is made of thetwo relative contribution for free and forced compon-



ents. Mixed convective coefficient can be obtained bytaking the cube root of the cubes of the forced and con-vective coefficients according to the equation [102]

h3mixed = h3

forced + h3free. (18.8)

Since the convective heat transfer coefficients will bedifferent on the top and rear surface of the module, de-termining the total heat transfer coefficient has to bedecoupled between the top hT

c and rear hBc surfaces. The

overall heat transfer will eventually be determined bythe sum of the two components.

Convective heat transfer on the top surface

Convective heat transfer has to be distinguished in freeand forced components. For the forced component wefurther have to distinguish between laminar and tur-bulent flow. We obtain for the laminar and turbulentconvective heat transfer coefficients

hlam.forced =

0.86 Re−0.5

Pr0.67 ρcairw, (18.9a)

hturb.forced =

0.028 Re−0.2

Pr0.4 ρcairw. (18.9b)

Re is the Reynolds number that expresses the ratio of ofthe inertial forces to viscous forces,

Re =wDh

ν, (18.10)

where w is the wind speed at the height of the PV array,Dh is the hydraulic diameter of the module, which is usedas relevant length scale, and ν is the kinematic viscosityof air. Pr is the Prandtl number which is the ratio of themomentum diffusivity to the thermal diffusivity. It isconsidered to be 0.71 for air. Finally, ρ and cair are thedensity and heat capacity of air, respectively. The hy-draulic diameter of a rectangle of length L and widthW, and thus of the PV module, is given as

Dh =2LW

L + W. (18.11)

Figure 18.7 shows the variation of the forced heat trans-fer coefficient at various wind speeds. We notice anoverall increase in the heat transfer with increasingwind speeds due to the increased force transfer com-ponent. There are two different regions in the graphwhich represent the laminar and turbulent regimes,respectively. The laminar flow extends till around 3m/s and is characterised by a lower convective heatexchange compared to the turbulent regime.

In a good approximation, hforcedand w are proportional



Wind speed @ 10 m (m/s)

h for

ced

(Wm

-2K

-1)

876543210

14

12

10

8

6

4

2

0

Figure 18.7: Forced convective heat transfer coefficient withincreasing wind speeds

to each other and we may write

hlam.forced ' w0.5, (18.12a)

hturb.forced ' w0.8. (18.12b)

To determine the free heat transfer coefficient we onlyneed to utilise the dimensionless Nusselt number Nu,which expresses the ratio between the convective andconductive heat transfer [104]

Nu =hfreeDh

k= 0.21(Gr · Pr)0.32, (18.13)

where k is the heat conductivity of air and Gr is theGrasshoff number that is the ratio between the buoyancyand viscous forces,

Gr =gβ(T − Ta)D3

hν2 . (18.14)

Here, g is the acceleration due to gravity on Earth and βis the volumetric thermal expansion coefficient of air,which can be approximated to be β = 1/T.

With the value of both free and forced coefficient wecan calculate the total mixed heat convective masstransfer coefficient using Eq. (18.8).

hmixed = hTc = 3

√h3

forced + h3free. (18.15)



Convective heat transfer coefficient on the rear sur-face of the module.

Convection on the back side of the module will belower than on the top because of the mounting struc-ture and the relative vicinity to the ground. For ex-ample, a rack mount configuration, which is approx-imately installed at 1 m height, will achieve a largerheat exchange rate than a standoff mounted array thatis mounted 20 cm above the ground. We model the ef-fect of the different mounting configurations by scalingthe convection coefficient obtained for the top of themodule. We determine the scaling factor by performingan energy balance at the INOCT conditions [104],

ϕGM − hTc (TINOCT − Ta)− hr, sky(TINOCT − Tsky) =

hBc (TINOCT − Ta) + hr, gr.(TINOCT − Tgr.)

(18.16)We define R as the ratio of the actual to the ideal heatloss from the back side,

R =hB

c (TINOCT − Ta) + εbackσ(T4INOCT − T4

gr.)

hTc (TINOCT − Ta) + εtopσ(T4

INOCT − T4sky)

. (18.17)

Substituting this into Eq. (18.16) at INOCT conditionsyields

R =ϕGM − hT

c (TINOCT − Ta)− εtopσ(T4INOCT − T4

sky)

hTc (TINOCT − Ta) + εtopσ(T4

INOCT − T4sky)

.

(18.18)

The back side convection is therefore given by

hBc = R · hT

c . (18.19)

We therefore find the overal convective heat transfercoefficient to be

hc = hTc + hB

c . (18.20)

18.3.3 Other parameters

Sky temperature evaluation

The sky temperature can be expressed as a function ofthe measured ambient temperature, humidity, cloudcover and cloud elevation [104]. On a cloudy day, usu-ally when the cloud cover is above 6 okta1, the skytemperature will approach the ambient temperature,Tsky = Ta [105]. However, on a clear day the sky tem-perature can drop below Ta and can be estimated by

Tsky = 0.0552 · T3/2a . (18.21)

Wind speed at module height evaluation

Since the anemometer used for the evaluation of thewind speed is at a higher height than the module array,

1Okta is a measure for the cloud cover, where 0 is clear sky and 8 is a com-pletely cloudy sky.



the real wind speed experienced by the module will bescaled down with

w = wr

(yMyr

) 15

, (18.22)

where yM and yr denote the module and anemometerheights, respectively [104].

The elevation factor of 1/5 is determined by the land-scape surrounding the installation, which is supposedto be open country.

Absorptivity and emissivity of the module

In the thermal model developed, absorptivity is definedas the fraction of the incident radiation that is conver-ted into thermal energy in the module. This value islinked to the reflectivity R and efficiency of the moduleby the equation [104]

ϕ = (1− R)(1− η). (18.23)

Typical value for reflectivity of solar modules are 0.1[104]. As far as the emissivity is concerned, a value of0.84 has been used for the front glass surface and 0.89for the back surface [107].

Now that all the unknown variables have been determ-ined it is possible to the make iterations to calculate the


Mod

ule

tem

pera

ture

(°C)

876543210

56

52

48

44

40

Figure 18.8: Influence of the wind speed on the temperat-ure of a solar module at a fixed radiation of 1000 W/m2 andambient temperature of 25°C on a clear day.

final temperature of the module as a function of thelevel of irradiance, wind speed and ambient temperat-ure.

18.3.4 Evaluation of the thermal model

In the previous section, all the variables that are re-quired to solve the thermal model have been derived.Therefore, the thermal model now can be solved via it-



eration.

Figure 18.8 below shows the effect of the wind speedon the module temperature at a fixed level of irradiance(1000 W/m2) and an ambient temperature of 25 °C ona clear day. It is obvious that the module temperatureis significantly above the value of 25 °C for all wind-speeds. Thus, STC are not representative for real oper-ating cell conditions. Two different regions are visible inthe graph corresponding to laminar and turbulent flow,respectively.

In Fig. 18.9 the thermal model and the NOCT modelpresented in Eq. (18.2) are compared to each other independence of the incident radiation for 1 m/s windspeed. We see that for 1000 W/m2 the NOCT modelpredicts a module temperature of 54.3°C regardless ofthe wind speed, which is much above the values ob-tained with the thermal model. The contribution offree convective and radiative heat exchange lower themodule temperature with respect to the NOCT model.The difference between the two models is more pro-nounced at low levels of radiation, but for irradianceshigher than 1000 W/m2 the two curves start divergingagain. This can be understood when we consider thatthe NOCT model has been developed as a linearisa-tion of the temperature/irradiation dependency aroundthe NOCT conditions [103]. The accuracy of the modeltherefore decreases with the increasing distance fromthe NOCT.

NOCT modelThermal model @ 1 m/s

Incident irradiance (Wm-2)

Mod

ule

tem

pera

ture

(°C)

10008006004002000

56

52

48

44

40

36

32

28

24

20

Figure 18.9: The module temperature calculated with theNOCT and the thermal model in dependence of the incidentirradiance for 1 m/s wind speed.



18.3.5 Effect of temperature on the solarcell performance

The effect of a module temperature deviating from the25°C of STC is expressed by the temperature coeffi-cients that given on the data sheet provided by solarmanufacturers. When knowing the temperature coeffi-cient of a certain parameter, its value at a certain tem-perature TM can be estimated with

Voc(TM, GSTC) = Voc +∂Voc

∂T(STC)(TM − TSTC),

(18.24)

Isc(TM, GSTC) = Isc +∂Isc

∂T(STC)(TM − TSTC), (18.25)

Pmpp(TM, GSTC) = Pmpp +∂Pmpp

∂T(STC)(TM − TSTC),

(18.26)

η(TM, GSTC) =Pmpp(TM, GSTC)

GSTC. (18.27)

If the efficiency temperature coefficient ∂η/∂T is notgiven in the datasheet, it can be obtained by rearran-ging

η(TM, GSTC) = η(STC) +∂η

∂T(STC)(TM − 25C). (18.28)

An increase in the solar cell temperature will shift theI-V curve as shown in Fig. 18.10. The slight increase

Curr

ent I

Voltage V

highertemperature T

higher irradiance GM

Figure 18.10: Effect of a temperature increase on the I-V solarcell characteristic.



in the short circuit current at higher temperatures iscompletely outweighed by the decrease in open circuitvoltage. The overall effect is of a general linear decreasein the maximum achievable power and therefore a de-crease in the system efficiency and fill factor. This effectis due to an increase of the intrinsic carrier concentra-tion at higher temperatures which in turns leads to anincrease of the reverse saturation current I0, which rep-resents a measure of the leakage of carriers across thesolar cell junctions, as we have seen in Chapter 8. Theexponential dependence of I0 from the temperature isthe main cause of the linear reduction of Vocwith thetemperature,

Voc =kTe

ln(

Isc

I0

). (18.29)

On the other hand, the slight increase in the generatedcurrent is due to a moderate increase in the photo-generated current resulting from an increased numberof thermally generated carriers. The overall reduction ofpower at high temperature shows that cold and sunnyclimates are the best environment where to place a solarsystem.

18.3.6 Effect of light intensity on the solarcell performance

Intuitively, performances of a solar cell decrease con-siderably with decreasing light intensity incident on themodule with respect to STC. The evaluation of the ex-tent of this reduction is however less straightforwardthan for the case of the temperature since solar man-ufacturer often do not explicitly provide a reductionfactor of the efficiency at every light intensity level.

By definition the efficiency is given by

η =IscVocFF

GM(18.30)

The maximum variation of the FF for light intensitybetween 1 and 1000 W/m2 is about 2% for CdTe, 5%for a-Si:H, 22% for poly-cristalline silicon, and 23% formono-crystalline silicon [108].

The short circuit current of a solar cell is directly pro-portional to the incoming radiation,

Isc ' λGM, (18.31)

where λ is simply a constant of proportionality. By ex-pressing Voc as in Eq. (18.29), the efficiency can be writ-ten as

η ' FF λkTe(ln GM + ln λ− ln I0). (18.32)



By defining

a = FF λkTe

, (18.33a)

b = FF λkTe(ln λ− ln I0), (18.33b)

the efficiency can be finally written as

η(25C, GM) = a ln GM + b (18.34)

The values of the coefficients a and b are device specificparameters and are rarely given by the manufacturer.The overall trend of the efficiency is represented by astraight line on a logarithmic scale [108].

From this model the values of Isc, Voc and the efficiencyat a irradiance level GM can be determined from theSTC as follows

Voc(25°C, GM) = Voc(STC)ln GM

ln GSTC, (18.35)

Isc(25°C, GM) = Isc(STC)GM

GSTC, (18.36)

PMPP(25°C, GM) = FF Voc(25°C, GM)Isc(25°C, GM),(18.37)

η(25°C, GM) =PMPP(25°C, GM)

AMGM, (18.38)

where AM is the module area.

18.3.7 Overall module performance

By combining the two effects of temperature and lightintensity, the final efficiency of the module at everylevel of irradiance and temperature can be determinedas [109]

η(TM, GM) = η(25°C, GM) [1 + κ(TM − 25°C)] , (18.39)

whereκ =

∂η

∂T1

η(SCT). (18.40)

Typical values for κ are −0.0025/C for CdTe,−0.0030/C for CIS, and −0.0035/C for c-Si [109].

All the other parameters such as Isc, Voc, and Pmpp canalso be evaluated at every level of irradiance and mod-ule temperature by simply adapting Eq. (18.39) to thecorresponding coefficients and parameters.

In Fig. 18.11 the difference between η(25°C, GM) andη(TM, GM), which takes both irradiance GM and mod-ule temperature TM into account, is shown. The graphhas been derived by keeping the wind speed constantat a value of 1 m/s and by determining TM iteratingthe heat thermal model at each level of irradiance fora constant ambient temperature of 25°C. At low levelsof irradiance no effects of the module temperature areobserved. At higher level of incident light intensity thedifference between the two curves becomes more pro-nounced. since only ηmpp(GM, TM) takes into account



DC efficiency @ Tmod

DC efficiency @ Tmod ≡ 25C

Incident irradiance (Wm-2)

Mod

ule

efficie

ncy

(%)

10008006004002000

16

14

12

10

8

6

4

2

0

Figure 18.11: Comparison of η(25°C, GM) and η(TM, GM)in dependence of the light irradiance at a wind speed of 1 m/sand ambient temperature of Ta = 25C for a c-Si solar cell.

1000 Wm-2400 Wm-2

GM = 200 Wm-2


Mod

ule

efficie

ncy

(%)

876543210

14.013.813.613.413.213.012.812.612.412.212.0

Figure 18.12: Overall efficiency of the module with increasingwind speed at various irradiance levels.

the marked reduction in efficiency resulting from therising TM.

Figure 18.12 shows the overall efficiency at variouslight intensities in dependence of the wind speed. Oncemore, the beneficial effect of turbulent motion in cool-ing the module is reflected by an increase of efficiencywith the wind speed.

Finally, the power output of a module before the BOS isgiven by

PDC = η(GM, TM) · GM · AM, (18.41)



and the power output at STC is given by

PSTC = η(GSTC, 25°C) · GSTC · AM, (18.42)

The energy yield of the DC side then is defined as

YDC =PDC

PSTC· 100% =

η(GM, TM) · GMη(GSTC, 25°C) · GSTC

· 100%.

(18.43)

18.4 Designing grid-connected PV-system

Now we can use all the things learned in this and theother chapters for actually designing a PV system. Fordesigning the PV system we use the energy balance ap-proach, meaning that we design the system such thatthe generated energy and the consumed energy duringone year match. Of course, there are also other waysof designing systems for example based on economicarguments.

For the energy balance we first need to calculate theannual load, which already happened in section 18.2.The energy yield at the DC side is given by

EYDC = Atot

∫year

GM(t)η(t)dt, (18.44)

where Atot is the total module area. It is related to thearea of one module AM via

Atot = NT · AM, (18.45)

where NT is the number of modules. The energy bal-ance now writes as

EYDC = EY

L · SF, (18.46)

where SF is a sizing factor that usually is assumed to be1.1. We therefore can calculate the required number ofmodules,

NT =

⌈EY

L · SFAM ·

∫year GM(t)η(t)dt

⌉, (18.47)

where dxe denotes the ceiling function, i.e. the lowestinteger that is greater or equal than x.

Now it is important to decide how many modules areto be connected in series (NS) and in parallel (NP). Ofcourse,

NT = NS · NP. (18.48)

Such a PV array hence consists of P strings of S mod-ules each. The NT determined in Eq. (18.86) not ne-cessarily needs to be an even number. For example, ifNT = 11, you might want to choose for NT = 12 panels,because you can install them as S × P= 12 × 1, 6 × 2,4 × 3, 3 × 4, 2 × 6 or 1 × 12 strings. In principle, it is


18.4. Designing grid-connected PV-system 327

preferable to connect as many modules as possible inseries since then the currents on the DC side and hencethe cable losses stay low. Further, thinner cables can bechosen. However, many modern inverters can connecttwo or even more individual strings, which can be im-portant if the installation contains two or more areaswith different shading, for example on two differentsides of a roof. Further, it can be chosen to connect dif-ferent strings to different inverters as well, such that thesystem uses several string inverters. NS and hence thevoltage at the inverter is also restricted by the choseninverter typ — or, if seen the other way round, the in-verter has to be chosen such that its operating voltagefits well to the string voltage.

In a conservative assumption, the power on the DCside at STC now is given as

PSTCDC = NT × PSTC

MPP. (18.49)

The inverter must be chosen such that its maximalpower Pinv

DC, max is above the maximal PV output,

PinvDC, max > PSTC

DC . (18.50)

Further, the nominal DC power of the inverter shouldbe approximately equal to the PV power,

PDC0 ≈ PSTCDC . (18.51)

Usually, for PDC0 < 5 kWp, single-phase inverters areused while for PDC0 > 5 three-phase inverters are ad-vised.

Also the inverter efficiency is dependent on the inputpower and voltage. A model discussing the inverterefficiency is presented below.

18.4.1 Inverter efficiency

As we already discussed in Chapter 17, modern invert-ers fulfil two major functions: First, maximum power pointtracking (MPPT), and secondly, the actual inverter func-tion, ie converting the incoming direct current (DC) toalternating current (AC) that can be fed into the electri-city grid.

In order not to waste electricity produced by the PV ar-ray, an inverter should always work as close as possibleat its maximum achievable efficiency. However, the in-verter efficiency of an inverter strongly depends on theDC input voltage as well as the total DC input poweras well as on the DC input voltage of the system. Theinverter efficiency ηinv with respect to the input DCpower at various DC voltage level is usually given atthe data sheet, at least for some values. Often, only thepeak efficiency is given as a single value.

Weighted Efficiencies

A more reliable way of expressing the inverter effi-ciency in a single number is to use weighted efficiencies,



which combine the inverter efficiencies over a widerange of solar resource regimes [110]. Two differentweighted efficiencies are commonly used. First, theEuropean Efficiency, which represents a low-insolationclimate such as in central Europe, and the California En-ergy Commission (CEC) efficiency, which represents thePV system performance in high-insolation regions suchas in the southwest of the United States [110]. They aregiven by

ηEuro = 0.03 η 5% + 0.06 η10% + 0.13 η20%+

0.10 η30% + 0.48 η50% + 0.20 η100%,(18.52a)

ηCEC = 0.04 η10% + 0.05 η20% + 0.12 η30%+

0.21 η50% + 0.53 η75% + 0.05 η100%,(18.52b)

where ηx% denotes the efficiency at x% of nominalpower of the inverter. Note that the CEC efficiency con-tains a 75% value that is not present in the Europeanefficiency.

Even though the weighted efficiencies represents amore accurate approximation of the effective yearlyworking performance of the inverter compared to themere peak efficiency, it still only is an approximation ofthe average performance of a system in the Europeanclimate.

If a better estimate of the real time energy yield of an ex-tended PV system is needed, a more accurate represent-ation of the instantaneous inverter performance at every

Figure 18.13: Field test results for a 2.5 kW Solectria PVI2500inverter recorded during a period of 13 days for system opera-tion at Sandia Laboratories [111].

level of input power and voltage must be developed,for example the model discussed below.

Sandia National Laboratories (SNL) model

Due to a lack of detailed data from inverters manufac-turers, many research institutes around the world havepublished extended data, which are publicly availableonline. These data present efficiency curves for a large



range of inverters as a function of a several character-istic parameters. One database that can be used is theone provided by the Sandia National Laboratories [112].

Figure 18.13 shows an example of an inverter efficiencycurve CITE. For the graph, field test measurementswere taken during a period of 13 days with changingweather condition. While the relationship between theAC and DC power appears to be linear at first, a closerlook to the graph reveals that this is not entirely thecase, as shown in Fig. 18.14. The power consumptionof the inverter itself together with the electrical char-acteristics of the switching modes and circuits at dif-ferent power levels results in a degree of non linear-ity between AC and DC power at a given DC voltagelevel. Assuming that the inverter efficiency is a constantvalue throughout the whole DC power range is equival-ent to assuming a linear relationship between DC andAC power, which has been shown not to be the case.

The dependency of the inverter efficiency on the DCinput voltage is very complex phenomenon that stilllacks a full physical explanation [113]. The differencesbetween the different inverter types can partially be ex-plained with different types of switches used. Figure18.15 shows the voltage dependent inverter efficiencyfor different inverter types. These curves were determ-ined with the Sandia model that is described in the nextparagraph.

In the following, we will introduce a mathematical

Figure 18.14: A closer look on to the relationship between ACoutput and DC input power for an inverter [111].



VDC > Vnom

VDC = Vnom

VDC < Vnom

[CEC 2011]Solaron 500 HE [3159502-XXXX] (480 V)

DC power (kW)

Inve

rter

efficie

ncy

(%)

400350300250200150100500

100

95

90

85

80

VDC > Vnom

VDC = Vnom

VDC < Vnom

[CEC 2013]Samil Power SolarRiver9000TL-US (208 V)

DC power (kW)

Inve

rter

efficie

ncy

(%)

15129630

100

95

90

85

80

Figure 18.15: Variation in the inverter efficiency with DCinput voltage for two different inverters.

model developed at Sandia Research Institute that hasbeen chosen due to its accuracy of both the effect ofDC input power and voltage on the AC power output.In this model the relationship between PAC and PDC isgiven by [111]

PAC =

[PAC0

A− B− C(A− B)

]· (PDC − B)− C(PDC − B)2,

(18.53)where the coefficients A, B, and C are given by

A = PDC0

[1 + C1

(VDC −VDC0

)], (18.54a)

B = PS0

[1 + C2

(VDC −VDC0

)], (18.54b)

C = C0[1 + C3

(VDC −VDC0

)]. (18.54c)

Several of the parameters are depicted in Fig. 18.14. Theparameters are [111]:

PAC: AC-power output from inverter based on inputpower and voltage, (W).

PDC: DC-power input to inverter, typically assumed tobe equal to the PV array maximum power, (W).

VDC: DC-voltage input, typically assumed to be equalto the PV array maximum power voltage, (V).

PAC0 : Maximum AC-power “rating” for inverter at ref-erence or nominal operating condition, assumed tobe an upper limit value, (W).

PDC0 : DC-power level at which the ac-power rating isachieved at the reference operating condition, (W).



VDC0 : DC-voltage level at which the ac-power rating isachieved at the reference operating condition, (V).

PS0 : DC-power required to start the inversion process,or self-consumption by inverter, strongly influencesinverter efficiency at low power levels, (W).

C0: Parameter defining the curvature (parabolic) of therelationship between AC-power and DC-power atthe reference operating condition, default value ofzero gives a linear relationship, (1/W).

Ci: Empirical coefficient allowing PDC0 to vary linearlywith DC-voltage input, default value is zero, (i =1, 2, 3, 1/V).

This model takes the following losses into account[113]:

• Self consumption of the inverter. This value corres-ponds to the DC power required to start the inver-sion process.

• Losses proportional to the output power due tofixed voltage drops in semiconductors and switch-ing losses.

• Ohmic losses

The accuracy of the model depends on the data avail-able for determining the performance parameters. Aninitial estimate can be performed using the little inform-ation provided by the manufacturer. Using all the re-

quired parameters will provide a model with an errorof approximately 0.1% between the modeled and meas-ured inverter efficiency [113].

All the parameters required for handling Eq. (18.53) aregiven in the Sandia laboratory database where a fulllist of a wide range of inverters with nominal powersfrom 200 W up to 1 MW are covered [112]. This modeltherefore uses the instantaneous value of PDC and VDCproduced by the entire PV array to evaluate the ACpower output and inverter efficiency.

Maximum Power Point Tracker and additional losses

The efficiency of the MPPT has not been included ex-plicitely in the Sandia performance model. This is be-cause the efficiency of most inverters ranges between98% and nearly 100% at every level of input power andthe voltage provided is within the accepted minimumand maximum window for the MPPT to function cor-rectly. A decrease of 1% in the system performance hastherefore been used to take the MPPT losses into ac-count.

An an additional decrease of 3% in the system perform-ance can be considered to cover losses caused by mis-match between modules (−1.5%), ohmic cable losses(−0.5%) and soiling (−1%) [109], if the cable lossesare not determined via the method described in section17.5.



18.4.2 Performance Analysis

Now we will put all things together that we discussedearlier in this section. The instantaneous power output onthe AC side can be described with

PAC(t) = AM GM(t) ηM(t) ηinverter(t) ηMPPT(t) ηother.(18.55)

The system efficiency then is given as

ηsystem(t) =PAC(t)

Atot GM(t)· 100%, (18.56)

which leads us to the instantaneous AC-side yield (alsoknown as performance ratio),

YAC(t) =PAC(t)PSTC

· 100%. (18.57)

Then, the yearly energy yield at the AC side can be calcu-lated with

EYAC =

∫year

PAC(t)dt; (18.58)

it is given in Wh/year. Another important parameter isthe annual efficiency of the system

ηYsystem =

EYAC

EYi, sys· 100%, (18.59)

where EYi, sys is the solar energy incident on the PV sys-

tem throughout the year. It can be calculated with

EYi, sys = Atot

∫year

GM(t)dt. (18.60)

The last parameter we look at is the yearly electricityyield

YE =EY

ACNSNP · PSTC

, (18.61)

which is given by Wh/(year kWp).

At the end of the design phase it is important to checkwhether the system really fulfils the requirements. Ifthe yearly energy yield exceeds the annual load, thesystem is well designed. Otherwise, another iterationhas to be done in order to scale up the system. How-ever, as stated earlier, for a grid-connected system italso can be a choice not the require the whole load toby covered by PV electricity.

18.5 Designing off-grid PV systems

In this last section of this chapter on PV system designwe take a closer look on the design of off-grid PV sys-tems (also called stand-alone systems). Choosing agood design is more critical for off-grid systems thanfor grid-connected systems. The reason for this is that


18.5. Designing off-grid PV systems 333

off-grid systems cannot fall back on the electricity grid,which increases the requirements on the reliability ofthe off-grid system.

A major component of off-grid systems is the storagecomponent, which can store energy in times when thePV modules generate more electricity than required andit can deliver energy to the electric appliances when theelectricity generated by the PV modules in not suffi-cient. A major design parameter for off-grid systems isthe required number of autonomous days, i.e. the num-ber of days a fully charged storage must be able to de-liver energy to the system until discharged.

The sizing of both the PV array and the storage com-ponent, usually a battery bank, is interconnected. Here,two parameters arise: first, Efail, which is the energy re-quired by the electric load that cannot be delivered bythe PV system, for example if the batteries are emptiedafter several cloudy days. Secondly, Edump, which is theenergy produced by the PV array that neither is usedfor driving a load nor is stored in the battery, for ex-ample, if the batteries already are full after a number ofsunny days. Now, we can define the Loss of Load Probab-ility,

LLP =Efail∫

year PL(t)dt. (18.62)

Of course, the lower LLP, the more stable and reliablethe PV system. Table 18.3 shows recommended LLPsfor different applications.

Table 18.3: Recommended Loss of Load Probability (LLP) forsome Applications

Application Recommended LLPDomestic illumination 10−2

Appliances 10−1

Telecommunications 10−4

Figure 18.16 (a) shows a schematic of an off-grid sys-tem with all the required components, i.e. the PV array,a maximum power point tracker (MPPT), a charge con-troller (CC), a discharge controller (DC), a battery bank(BB), an inverter, and the load. The CC prevents theBB from being overcharged by the PV system, whilethe DC prevents the battery to be discharged belowthe minimal allowed SoC. In the simplest case, they(dis)connect the PV system and load from the batteryby switches. For optimal charging, however, nowadaysoften CC with pulse-width modulation (PWM) areused. By far not all off-grid systems contain an MPPT,thus it is represented with a dashed line. However, ifan MPPT is present it usually is delivered in one unittogether with the CC. Usually, the DC and inverter arecombined in one battery inverter unit. Especially at largersystems the inverter currents may become very large.For example, if a 2400 W load is present for a shorttime, this means 100 A on the DC side in a 24 V sys-tem. Therefore the battery inverter usually is directlyconnected to the BB with thick cables. In very small



VMPP VBB

I MPP I inv

VBB

I CC

I BBVBB

VAC

I AC

MPPT + CC

MPPT + CC + DC + Inverter

Controller

InverterLOAD

MPPT

Battery Bank (BB)

VOC-BB

VBB-ch

VBB-dis

+

–

IBB-ch

IBB-dis

Curr

ent I

V

chargingregime

dischargingregime

I BB

Ri

+

–VBB

VOC-BB

(a) (b)

Charge

Controller

Discharge

DC + Inverter

Figure 18.16: (a) PV topology of an off-grid system. (b) Simplified I-V curve and schematic of the battery bank.



systems with maximal powers of several hundreds ofWatt, the CC and DC may be combined in one unit.

18.5.1 A closer look at the battery

The battery bank is the workhorse of any off-grid sys-tem, because it is its stable power source. It is thus veryimportant to understand how the battery bank will actin the PV system. Therefore, we will discuss it in moredetail in this section.

When we want to understand how a battery works wehave to consider the net effect of all the forces that try tocharge or discharge the battery. In Fig. 18.16 (a) the mostimportant currents and voltages that we need and thefollowing discussion are depicted. In Fig. 18.16 (b) aschematic and an I-V curve of an idealised batterybank is shown. As we can see, this battery bank isconsidered to consist of an voltage source with a totalvoltage VOC-BB and an internal resistance Ri. In reality,VOC-BB will not be constant, but a function of the stateof charge, the ambient temperature, and others,

VOC-BB = f (SoC, T, . . .). (18.63)

In the following discussion, however, we will neglectthis dependence and assume VOC-BB to be constant. Be-cause of Ri, the voltage VBB will differ from VOC-BB andalso is dependent on the current IBB flowing through

the battery, as seen by the equation

IBB =1Ri

(VBB −VOC-BB) . (18.64)

Note that we use the convention, where IBB is positivewhen the battery is charged and negative when it isdischarged. Let us now derive an expression for thevoltage of the battery bank (VBB) as a function of theother PV system parameters. We start with the poweron the left and right-hand sides of the MPPT,

ηMPP IMPPVMPP = ICCVBB =: β, (18.65)

and hence

ICC =ηMPP IMPPVMPP

VBB=

β

VBB. (18.66)

Consequently, β = 0 means that the PV system is notactive, for example during night.

In a similar manner, we look at the power on the leftand right hand side of the inverter:

PL = IACVAC = ηinv IinvVBB =: ηInvα, (18.67)

and consequently

Iinv =PL

ηinvVBB=

α

VBB. (18.68)

Combining Eqs. (18.66) and (18.68), we find

IBB = ICC − Iinv =β− α

VBB. (18.69)



Clearly, α = 0 indicates that no load is present.

Combining Eqs. (18.64) and (18.69), multiplying withRiVBB and rearranging leads to a quadratic equation,

V2BB −VOC-BBVBB − Ri(β− α) = 0. (18.70)

with the solutions

V±BB =VOC-BB

2±

√(VOC-BB

2

)2+ Ri(β− α). (18.71)

The correct solution is the "+" solution as we can checkwith

V+BB(β− α = 0) = VOC-BB. (18.72)

Hence, the final solution is

VBB =VOC-BB

2+

√(VOC-BB

2

)2+ Ri(β− α). (18.73)

Let us now take a short look on this solution. If thebattery is charged, ICC is higher than Iinv and hence(β− α) and IBB are positive. From Eq. (18.73) it followsthat in this case VBB is higher than VOC-BB. On the otherhand, if the battery is discharged, ICC is lower than Iinvand hence (β− α) and IBB are negative. Therefore, VBBwill be lower than VOC-BB.

Note that only the net current IBB = ICC flows in or outof the battery. Therefore, only this current determines

Table 18.4: Recommended number of autonomous days dA atseveral latitudes.

Latitude () Recommended dA0-30 5-6

20-50 10-1250-60 15

the power loss in the battery,

PBB(loss) = I2BBRi. (18.74)

This power always is lost, irrespective of the sign ofIBB.

18.5.2 Designing a system with energybalance

Now we discuss how to design a PV system based onthe principle of energy balance. For determining the ad-equate components, the analysis for the load side andthe PV side is performed separately. Let us begin withthe load side. We can determine the annual load EY

L asdiscussed in section 18.2 and Eq. (18.1),

EYL =

∫year

PL(t)dt. (18.75)



From that we can estimate the average daily load with

EDL =

1365

EYL . (18.76)

Next, we have to choose an adequate number of daysof autonomy dA. Some values given in table 18.4. Now,we can calculate the required energy of the batterybank,

EBB = dAED

L · SFbat

DoDmax, (18.77)

where SFbat is the sizing factor of the battery, whichis similar to the sizing factor for the PV array alreadydefined in section 18.4. DoDmax is the maximally al-lowed depth of discharge of the batteries. The ratedenergy of the chosen batteries is

Ebat = VOC-batCbat, (18.78)

where Cbat is the battery capacity (unit Ampere-hours).Hence, the required number of batteries is

Nbat =

⌈EBB

Ebat

⌉. (18.79)

Similarly as for grid-connected systems we now need tochoose a suitable inverter. Therefore we must considerthe maximal load power Pmax

L . This we can do, for ex-ample, by looking at the appliance with the maximalpower consumption, or by adding up the power of all

appliances. It may be more beneficial to choose a sys-tem design, such that not all appliances can be used atthe same time. The final decision of course is up to thedesigner of the system. Similar as for grid-connectedsystems, the inverter must fulfil several requirements:First, its maximally allowed power output must exceedthe maximal power required by the appliances,

PinvDC, max > Pmax

L . (18.80)

Secondly, the nominal power of the inverter should beapproximately equal to the maximal load power,

PDC, 0 ≈ PmaxL . (18.81)

Thirdly, the nominal inverter input voltage should beapproximately equal to the nominal voltage of the bat-tery back,

VDC, inv ≈ VOC-BB. (18.82)

For a more detailed analysis of the inverter perform-ance the Sandia model can be used that was discussedin section 18.4.

Typical voltages for the battery bank are 12 V, 24 V, 48V or 96 V. It can be adjusted by the number of batteriesthat are connected in series,

NSbat =

VOC-BB

VOC-bat. (18.83)



From that we also can determine the number of batter-ies that must be connected in parallel,

NPbat =

⌈Nbat

NSbat

⌉. (18.84)

Now, after designing the load side is completed withchoosing inverter and batteries, we look at the PV sideof the system. Sizing the PV array is very similar to theprocedure used for grid-connected systems in section18.4. The energy balance can be written down as Theenergy balance now writes as

EYDC = EY

L · SF, (18.85)

where SF is a sizing factor that usually is assumed to be1.1. We therefore can calculate the required number ofmodules,

NT =

⌈EY

L · SFAM ·

∫year GM(t)η(t)dt

⌉. (18.86)

For minimising losses, the MPP voltage of the PV arrayand the nominal voltages of the inverter and the batterypack should be approximately equal, since otherwisethe losses of the DC-DC converter that is included inthe MPPT-CC unit will be higher. The number of PVmodules that are connected in series in the PV array isgiven by

NS =

⌊VOC-BB

Vmod-MPP

⌋, (18.87)

where VMPP-mod denotes the annual average of theMPP voltage of the PV modules. Of course, the max-imally allowed input voltage of the MPPT-CC unit mustnot be exceeded by the PV array,

VMPP ≥ NS ·Vmaxmod-MPP. (18.88)

The number of required parallel PV strings is given by

NP =

⌈NTNS

⌉. (18.89)

18.5.3 Performance Analysis

Like for grid-connected systems, also for off-grid sys-tems, a performance analysis should be done to evalu-ate the chosen design. For this analysis, it is useful touse an algorithm that can simulate the performance ofthe PV system throughout the year. Conceptually, thisalgorithm can look as follows

1. Set a starting state of charge (SoC) of the batterybank.

2. Then calculate the SoC of the battery throughoutthe year for time steps ∆t,

• Depending on the actual load and PV arrayoutput, determine the battery current IBB(t).

• Determine the actualised SoC.



Of course, the function of the charge controller, i.e.its switching behaviour, must be accurately mim-icked by the algorithm.

3. Determine when the system cannot deliver the re-quired load and hence Efail.

4. Now, calculate the Loss of Load Probability of thesystem (LLP).

5. Finally, determine the yearly energy yield on theAC-side of the system, EY

AC.

Mathematically, the annual energy yield on the AC sidecan be expressed by

EYAC =

∫year

PL(t)dt− Efail + ∆EBB. (18.90)

In contrast to the expression used for determining theAC yield for grid-connected systems in Eq. (18.58), thisequation contains two additional components: Efail, theenergy that is required by the load but cannot be de-livered and ∆EBB is the difference in energy stored inthe battery bank between the beginning and end of theyear,

∆EBB =(

SoCend − SoCbeginning

)CbatVBB. (18.91)

If ∆EBB < 0, the system might not be sustainable. Onthe other hand, if it is > 0, Edump will increase in thefollowing, if the average meteorological conditions areunchanged.

Because of Efail > 0,

EYAC <

∫year

PL(t)dt. (18.92)

As already stated earlier, the Loss of Load Probability isgiven by

LLP =Efail∫

year PL(t)dt. (18.93)

For evaluating the design, it is very important to lookat LLP:

• LLP acceptable:

– Edump low: The system design is OK.

– Edump high: Resize the PV array.

• LLP not acceptable:

– Increase the size of the PV array.

– Increase the capacity of the battery bank.


19PV System Economics and Ecology

19.1 PV System Economy

We conclude our discussions on PV systems with look-ing on several important topics on the economics ofPV systems. Note that the economics of PV can be dis-cussed at several levels, such as the consumer level, themanufacturing level, the level of PV installers, and thetechnology level where PV is compared to other electri-city generation technologies on the scale of the electri-city grid.

We will start this discussion with the definition of thepayback time, which in finance is defined as the amountof time required to recover the cost of an investment. It

can be calculated with

payback time =initial investment

annual return. (19.1)

Translated to the consumer level, the payback time isthe time it takes to recover the initial investment of thePV system as the system continuously reduces the elec-tricity bill. Please note that the financial payback time isdifferent from the energy payback time that we will dis-cuss in Section 19.2 Let us look at an example:

Example

Let us assume that family Smith have installed a PV systemwith a power of 1 kWp on their rooftop. The initial investment

341


342 19. PV System Economics and Ecology

was e 8000. Family Smith has an annual electricity bill ofe 2000. The installation of the PV system leads to an averageannual reduction of the electricity bill of e 800. As a part oftheir consumed electricity is provided by their PV system,the electricity bill is therefore constantly reduced. Hence, theaverage annual return on their PV system is e 800. As a con-sequence, the Smith’s have earned the final investment backafter 5 years, the payback time hence is 5 years.

The payback time is strongly influenced by the annualsolar radiation on the PV system. As we have seen inChapter 16, this is dependent on the orientation of thePV modules and on the location of the PV systems.In general we can say that the sunnier the location,the greater the PV yield and the shorter the paybacktime. Another factor that influences the payback timeis the grid electricity costs: the higher these costs, theshorter the payback time. Finally, the payback time alsois strongly dependent on the initial costs of the PV sys-tem.

In practice, often more factors must be taken into ac-count than in the simple example above. This will in-crease the complexity of calculating the payback time.For instance, if we are considering a significant periodof time, also the change of the value of money has tobe taken into account, which is due to inflation. Forexample today e 1000 will have a different purchasingpower than in ten years time. Another factor that shouldbe considered are policies regarding renewable energy.

For example, subsidies and feed-in tariffs can affect theinitial investments and savings.

Let us briefly discuss the concept of feed-in tariffs. At theconsumer level, the feed-in tariff is the price at which aconsumer can sell renewable electricity to the electricityprovider. We distinguish between two kinds of feed-in tariffs, gross and net. Gross feed-in tariffs are paidfor all the electricity the panels produce, irrespectiveof the consumer’s electricity consumption. In contrast,net feed-in tariffs promise a higher rate for the surpluselectricity fed into the grid after domestic use of theconsumers is subtracted. For implementing feed-in tar-iff schemes at consumer level, the facility of net meteringis pivotal.

Another very important concept is the Levelized Cost ofElectricity (LCoE), which is defined as the cost per kWhof electricity produced by a power generation facility. Itis usually used to compare the lifetime costs of differentelectricity generation technologies. To be able to estim-ate the effective price per kWH, the concept of LCoEallocates the costs of an energy plant across its full li-fecycle. It is somehow similar to averaging the upfrontcosts of production over a long period of time. Depend-ing on the number of variables that are to be taken intoaccount, calculating the LCoE can become very com-


19.1. PV System Economy 343

plex. In a simple case the LCoE can be determined with

LCoE =∑n

t=1It+Mt+Ft(1+r)t

∑nt=1

Et(1+r)t

. (19.2)

The sums expand across the whole lifetime of the sys-tem n, every year accounts for one summand. It are theinvestment expenditures in the year t, Mt are the oper-ational and maintenance expenditures in the year t, andFt are the fuel expenditures in the year t. Of course, forPV Ft ≡ 0. Further, Et is the electricity yield in the yeart. Finally, r is the discount rate which is a factor usedto discount future costs and translating them into thepresent value.

Figure 19.1 shows the LCoE for different methods ofelectricity generation. We see that the LCoE of wind en-ergy is lowest, it is even below the LCoE of fossil fueland nuclear based electricity. The LCoE of PV gener-ated electricity is still slightly above that of the non-renewable technologies.

Depending on the location of a PC system and the ini-tial investment required for the PV system, the LCoEfor PV can vary a lot between different projects. Addi-tionally, the discount rate r used for the calculation willstrongly influence the LCoE.

For the electricity supplier, the LCoE is a valuable in-dicator of the cost competitiveness of a certain energytechnology. It is also a good indicator for determining

Estim

ated

LCo

E (U

S$/M

Wh)

0

50

100

150

200

250

300Transmission Investment

Variable O&M

Levelized capital cost

Fixed O&M

Solar therm

alPV

Wind (osh

ore)Wind

Nuclear

Fossil fu

els

Figure 19.1: The levelised cost of electricity for differentelectricity generation technologies.



the electricity price: for making profit this price mustbe above the LCoE. Surely, the supplier cannot determ-ine the electricity price independently, as it is stronglyinfluenced by policy factors such as feed-in tariffs, sub-sidies and other incentives.

The final concept that we discuss is that of grid parity.For solar electricity this is the situation at which it canbe generated at an LCoE that is equal to the electricityprice from the conventional electric grid. Of course, ifthe PV electricity price is below the grid price, the situ-ation becomes even better,

LCOEPV ≤ LCoEconventional. (19.3)

In principle, the concept of grid parity can be general-ised to the other renewable technologies as well. How-ever, there is one significant difference between PVand other renewable technologies such as wind andhydro electricity. Wind and hydro electricity installa-tions usually only can be financed by companies butare no option for a single consumer. In contrast, PV canbe scaled down to the level of a single module, suchthat a house owner can become an electricity produ-cer with his small scalable PV installation on his roof.As a consequence, the consumer offering PV electri-city to the grid now is effectively competing with theretail grid price, which conventionally includes othercosts like transmission, distribution, and so on. As forthe consumer these costs are no part of the PV electri-

Sola

r ele

ctric

ity p

rice

Typical sites

Best sites

Volume of PV systems installed

Grid powerprices

Wholesale

Grid parity whenlines cross

EarliestLatest

Figure 19.2: Grid parity for PV systems [114].

city price, PV grid parity for retail grid prices can bereached faster, as shown in Fig. 19.2.

The graph is showing the volume of installed PV sys-tems versus the price of PV electricity. The installedvolume can be directly correlated with time, as the pastdecade has seen the implemented PV volume rise tre-mendously. As capital costs decline with increasingvolumes, the price of PV generated electricity is ex-pected to decrease in the future. On the other hand,the price of fossil fuels is expected to rice because ofincreasing scarcity and cost linked to the right to emitCO2 emissions into the atmosphere costs. These trends


19.2. PV System Ecology 345

will lead to increasing prices for electricity generatedwith combusting fossil fuels. Grid parity is reachedwhen the prices of PV electricity and grid electricityare crossing. Naturally, with incentives and subsidiesgrid parity can be reached even faster, however gridparity reached with public financial support often isdisregarded. Therefore it is said that true grid parity isreached when the PV electricity prices fall below thegrid prices without any subsidies. Real grid parity forexample already is reached in the sunny country ofSpain.

To conclude, grid parity is a very useful concept to in-dicate the feasibility of a renewable energy technology.The closer a technology is to grid parity, the easier itcan be integrated in the electricity mix. With the ad-vancements in technology and the maturity of manufac-turing processes, grid parity for solar is expected to bereached at many locations around the World in the nextyears.

19.2 PV System Ecology

Besides discussing the economics of PV systems, it alsois very important to consider their ecological and envir-onmental aspects. The main reason for that is that theaim of photovoltaics is to generate electricity withoutany considerable effect on the environment. It is there-

fore very important to check the ecological aspects ofthe different PV technologies. In this section we willdiscuss different concepts to quantify the environmentalimpact of PV systems.

The concept of the carbon footprint estimates the emis-sions of CO2 caused by manufacturing the PV modulesand compares them with the reduction of CO2 emis-sions due to the electricity generated with PV insteadof combusting fossile fuels. A more analytical approachis to look at the total energy required to produce eitherthe PV modules or all the components of a PV system.As the different PV technologies vary considerably inthe required production processes, the energy consump-tion for producing 1 kWp varies considerably betweenthe different technologies. If a complete life cycle assess-ment (LCA)is performed, it is tried to trace the energyand carbon footprints of the PV panels throughout theirlifetime. Therefore LCA also is known as cradle-to-graveanalysis.

We now are going to introduce several indicators thatare used to judge the different ecological aspects. TheEnergy Yield Ratio is defined as the ratio of the total en-ergy yield of a PV module or system throughout itslifetime with all the energy that has to be invested inthe PV system in that time. This invested energy notonly contains the energy for producing the components,transporting them to the location and installing thembut also the energy that is required to recycle the differ-



ent components at the end of their lifecycle.

As the energy required for producing a PV system de-pends strongly on the PV technology and also on thequality of the panels, the energy yield ratio for the dif-ferent technologies varies a lot. While the energy yieldratio for PV modules can be as large as 10 to 15, PVsystems usually have a lower ratio because of the en-ergy invested in the components other than the mod-ules.

A very important concept is the Energy Payback Time,which is defined as the total required energy invest-ment over the lifetime divided by the average annualenergy yield of the system,

Energy Payback Time =totally invested energy

average annual energy yield.

(19.4)Note that the energy payback time is different from theeconomic payback time introduced in Section 19.1.

The energy payback time of typical PV systems isbetween 1 and 7 years and it also depends on locationissues such as the orientation of the PV array as well asthe solar irradiance throughout the year.

Figure 19.3 shows the specific energy required for pro-ducing PV modules with different technologies. As wecan see, the differences between the technologies arelarge. The specific energy required for producing thin-film modules from materials such as amorphous silicon,

0 4 000 8 000 12 000 16 000

Primary energy use for manufacturing PV systems (kWh/kWp)

With frame, min

With frame, max

Without frame, min

Without frame, max

CIGS modules

CdTe modules

Amorphous modules

Polycrystalline modules

Monocrystalline modules

20 000

Figure 19.3: Specific energy used to produce PV modules ofdifferent technologies (Data from [115]).


19.2. PV System Ecology 347

0

1

2

3

4

5

6

7

Ener

gy P

ayba

ck ti

me

(yea

rs) High irradiation

(2200 kWh/m2/year) Medium irradiation

(1700 kWh/m2/year) Low irradiation

(1100 kWh/m2/year)

Roof Ground Roof Ground Roof Groundc-Si a-Si c-Si a-Si c-Si a-Si c-Si a-Si c-Si a-Si c-Si a-Si

Figure 19.4: The energy payback time for the different com-ponents constituting PV systems (Data from [114]).

cadmium telluride and CIGS is significantly below thatof modules made from polycrystalline and monocrys-talline silicon, where the specific energy can reach val-ues up to 12000-18000 kWh/kWp. Because of furtherimprovements in the module efficiency and the manu-facturing process we however may expect that that thespecific energy follows a decreasing trend.

For a PV system it is more difficult to allocate the en-ergy that was used for its production, as all the com-ponents constituting the balance of system have to betaken into account as well. For example, for compon-ents like batteries and inverters the technologies andmanufacturing processes may vary a lot between the

different products available on the market. Nonethe-less, studies were carried out that estimated the energyrequired by whole PV systems. Generally, as we cansee in Fig. 19.4 it is found that the energy required forthe BOS is significantly below that used for manufac-turing the modules. In the figure, amorphous and crys-talline silicon modules are compared. As expected, wesee that the energy payback time in regions with highsolar irradiance have significantly shorter energy pay-back times than regions with low irradiance. While a-Si:H based modules have a shorter energy payback timethan c-Si based modules, the energy payback time forthe module frame and the BOS of a-Si:H based systemscan be significantly higher than that of c-Si based sys-tems. This can be explained with the lower efficiency ofa-Si:H that increases the required framing material perWp.

The irradiance strongly influences the energy paybacktime and varies between two years (high irradiance)and six years (low irradiance). Roof-mounted systemsalways have a shorter energy payback time than sys-tems mounted on the ground, mainly because of theBOS that is more energy extensive for ground-mountedsystems.

No matter which PV technology is chosen, the energypayback time always is far below the expected systemlifetime, which usually is between 25 and 30 years. Forthe PV systems discussed in Fig. 19.4, the energy yield



ratio is between 4 and 10. Hence, the energy investedin the PV system is paid back several times through-out the life cycle of the PV system. The urban legendthat PV modules require more energy to be producedthan they will ever produced thus is not backed by anydata. In contrast, the net energy produced is much lar-ger than the energy required for PV production.

However, a lot of work still needs to be done and canbe done. Some studies indicate that the energy requiredfor producing PV modules can be reduced by up to80%. Further, as the amount of installed PV systemsbecomes larger and larger, recycling of the componentsat the end of the lifecycle becomes very important. Forexample, the European Union introduced already sev-eral directives that induced recycling schemes for c-Sibased PV modules.

The last environmental issue that we want to mentionis pollution caused by the production of PV modules.As many sometimes toxic chemicals are required forproducing PV modules, this can be a serious threat tothe environment. Therefore it is very important to havestrong legislation in order to prevent pollution of thesurroundings of PV factories. Especially in countrieswith weak environmental legislation pollution can be asevere problem that affects the environment and peopleliving in the surroundings negatively.


Part V

Alternative Methods of Solar EnergyUtilisation

349


20Solar Thermal Energy

Solar thermal energy is an application of solar energythat is very different from photovoltaics. In differenceto photovoltaics, where we used electrodynamics andsolid state physics for explaining the underlying prin-ciples, solar thermal energy is mainly based on the lawsof thermodynamics. In this chapter we give a brief in-troduction to that field. After introducing some basicsin Section 20.1, we will discuss Solar Thermal Heatingin Section 20.2 and Concentrated Solar (electric) Power(CSP) in Section 20.3.

20.1 Solar thermal basics

We start this section with the definition of heat, whichsometimes also is called thermal energy. The moleculesof a body with a temperature different from 0 K ex-hibit a disordered movement. The kinetic energy of thismovement is called heat. The average of this kinetic en-ergy is related linearly to the temperature of the body.1

Usually, we denote heat with the symbol Q. As it is aform of energy, its unit is Joule (J).

If two bodies with different temperatures are broughttogether, heat will flow from the hotter to the cooler

1The interested reader will find more details in every textbook on physics orthermodynamics. This definition is taken from Ref. [116].

351


352 20. Solar Thermal Energy

body and as a result the cooler body will be heated.Dependent on its physical properties and temperature,this heat can absorbed in the cooler body in two forms,sensible heat and latent heat.

Sensible heat is that form of heat that results in changesin temperature. It is given as

Q = mCp(T2 − T1), (20.1)

where Q is the amount of heat that is absorbed by thebody, m is its mass, Cp is its heat capacity and (T2 − T1)is the temperature difference. On the other hand, if abody absorbs or releases latent heat, the temperaturestays constant but the phase changes. This happens forexample when ice is melting: When its temperature isequivalent to its melting point, heat that is absorbed bythe ice will not result in increasing temperature but intransformation into from the solid to the liquid phase,which is water. Mathematically, this is expressed as

Q = mL, (20.2)

where L is the specific latent heat.

The two forms of heat are illustrated in Fig. 20.1, whichshows what happens when a body absorbs heat. In thebeginning, the body is solid and has temperature T1. Itthen heats up and the heat is stored as solid sensibleheat. When its melting point T∗ is achieved, its tem-perature will not increase any more but the phase willchange from solid to liquid. After everything is molten,

T1

T*

T2

Solid sensibleheat

Liquidsensibleheat

Phase-changelatent heat

Tem

pera

ture

(T)

Storage heat (Qs)

Figure 20.1: Illustrating the difference of sensible and latentheat.

the liquid will heat up further, the heat now is stored asliquid sensible heat.

Now we will take a look at the three basic mechanismsof heat transfer: conduction, convection and radiation.

Conduction is the transfer of heat in a medium due toa temperature gradient. For a better understanding welook at the heating of a house during winter, as illus-trated in Fig. 20.2. The inside of the house is warmat 20°C, but the outside is cold with a temperature of0°C. Because of this heat gradient, heat will be trans-ferred from the inside through the wall to the outside.Mathematically, heat conduction can be described by


20.1. Solar thermal basics 353

20°C

0°C

x

outside wall inside

Figure 20.2: Illustrating conductive heat transfer through awall.

the Fourier law,

dQconddt

= −kAdTdx

, (20.3)

where dQ/dt is the heat flow in W, k is the thermalconductivity of the wall [given in W/(Km)], A is thecontact area given in m2 and dT/dx is the temperaturegradient in the x direction given in K/m. If we assumethe wall to be made from a uniform material, we canassume that dT/dx is constant throughout the wall andwe can replace it by ∆T/∆x, where ∆x is the thicknessof the wall.

Convection is the second possible mechanism for heattransfer. It is the transfer of heat by the movement of a

(a) (b)

Figure 20.3: Illustrating (a) natural and (b) forced conductiveheat between a solid and a surrounding fluid.

fluid. We distinguish between two forms of convection,as sketched in Fig. 20.3: In forced convection the move-ment of the fluid is caused by external variables, whilein natural convection the movement of the fluid is causedby density differences due to temperature gradients. Inboth cases, the heat transfer from a medium of temper-ature T1 into a fluid of temperature T2 can be describedby Newton’s law

dQconv

dt= −hA(T1 − T2), (20.4)

where h is the heat transfer coefficient which is given inW/(m2K). We will not discuss the calculation of h indetail but we want to mention that it depends on vari-ous factors such as the velocity of the fluid, the shapeof the surface or the kind of flow that is present, i.e.whether it is laminar or turbulent flow.

The third heat transfer mechanism is radiative heat trans-



fer, which is the most important mechanism of heattransfer for solar thermal systems. As we already dis-cussed in Chapter 5, thermal radiation is electromag-netic radiation propagated through space at the speedof light. Thermal radiation is emitted by bodies de-pending on their temperature. This is caused when ex-cited electrons fall back on their ground level and emita photon and hence electromagnetic radiation.

As discussed in Chapter 5, a black body is an idealisedconcept of a body, which is a perfect absorber of ra-diation, independent on the wavelength or directionof the incident light. Further, it is a perfect emitterof thermal radiation. The Stefan-Boltzmann law [Eq.(20.5)] describes the total irradiant emittance of a blackbody of temperature T,

MBBe (T) = σT4, (20.5)

where M is given in W/m2 and σ ≈ 5.670 ·10−8 W/(m2K4) is the Stefan-Boltzmann constant. Innature, no black bodies exist. However we can describethem as so-called grey bodies. The energy emitted by agrey body still can be described by Plank’s law [Eq.(5.18a)] when it is multiplied with a wavelength de-pendent emission coefficient ε(λ). For a black body wewould have ε(λ) ≡ 1.

collector array

collector circuit

secondary heat supply

heat storagefresh water

warm water

pump

Figure 20.5: Illustrating the main components of a solar waterheating system.

20.2 Solar thermal heating

About half of the world’s energy consumption is in theform of heat. About two thirds of the heat demand iscovered by coal, oil, and natural gas, as we can see inFig. 20.4 (a). As shown in Fig. 20.4 (b), heat is mainlyused in the industrial sector for facilitating for examplechemical processes and in the residential sector forheating and warm water supply.

Figure 20.4 (c) shows the total energy demand of a typ-ical household in the United States. We see that spaceheating and water heating represent 59% of the totalenergy consumption. If also the demand for cooling istaken into account, about two third of the energy con-sumption are related to the use of heat.


20.2. Solar thermal heating 355

Industry 43.5%

Residential 38.3%

Services 12.3%

Agriculture 1.3% Other 4.6%

Coal 37.9%

Oil 6.2%

Gas 47.7%

Biofuels 2.8%

Waste 2.3%

Other 2.9%

Space heating41%

Air conditioning6%

Water heating

18%

Appliances, electronics, and lighting

35%

(a) (b) (c)

Figure 20.4: (a) The used primary energy suppliers for the generation of heat and (a) the demand of heat by sector [9]. (c) Energyconsumption of U.S. homes [117]. All data is for 2009.



The residential demand for heat can at least partiallycovered with a solar water heater, which is a combinationof a solar collector array, an energy transfer system anda storage tank, as sketched in Fig. 20.5. The main partof a solar water heater is the collector array, which ab-sorbs solar radiation and converts it into heat. This heatis absorbed by a heat transfer fluid that passes throughthe collector. The heat can be stored or used directly.The amount of the hot water produced by a solar waterheater throughout a year depends on the type and sizeof the solar collector array, the size of the water stor-age, the amount of sunshine available at the site andthe seasonal hot water demand pattern.

There are several ways to classify solar water heatingsystems. The first way is by the fluid heated in the col-lector. When the fluid used in the application is thesame that is heated in the collector it is called a director open loop. In contrast, when the fluid heated in thecollector goes to a heat exchanger to heat up the utilityfluid, it is called an indirect or closed loop.

The second way to classify the systems is by the waythe heat transfer fluid is transported. This can either bepassive, where no pumps are required, or by forced cir-culation, using a pump. The passive solar water heatingsystems, uses natural convection to transport the fluidfrom the collector to the storage. This happens becausethe density of the fluid drops when the temperature in-creases, such that the fluid rises from the bottom to the

top of the collector – this is the same as natural con-vection that we discussed in Section 20.1. The advant-age of passive systems is that they do not require anypumps or controllers, which makes them very reliableand durable. However, depending on the quality of theused water, pipes can be clogged, which considerablyreduced the flow rate.

On the other hand, active systems like the one sketchedin Fig. 20.5 require pumps that force the fluid to circu-late from the collector to the storage tank and the restof the circuit. These systems usually are more expensivethan passive systems. However, they have the advant-age that the flow rates can be tuned more easily.

20.2.1 Solar Thermal Collectors

Now we will take a closer look at the solar collector,in which the working fluid is heated by the solar ra-diation. The collector determines how efficiently thethe incident light is used. It usually consists of a blacksurface, called the absorber, and a transparent cover.The absorber is able to absorb most of the incident en-ergy from the sun, Qsun, raising its temperature andtransferring that heat to a working fluid. Hence, the ab-sorber can be cooled and the heat can be transferredelsewhere.

Not all of Qsun can be used, as there are some losses,



Transparent cover

Qsun

Qcol

Qre

Qconv

Qrad

Figure 20.6: Illustrating the major energy fluxes in a coveredsolar collector.

as illustrated in Fig. 20.6. A part Qrefl is lost as reflec-tion either in the encapsulation or in the absorber itself.Other losses are related to the heat exchanged with thesurrounding air by the convection mechanism, Qconvand radiation from the hot absorber, Qrad. When weput all these energies in an energy balance, we find forthe heat Qcol that can be collected by the collector

Qcol = Qsun −Qrefl −Qconv −Qrad. (20.6)

The efficiency of the collector depends mainly on twofactors: the extent to which the sunlight is convertedinto heat by the absorber and the heat losses to thesurroundings. It will therefore depend on the weatherconditions and the characteristics of the collector itself.To reduce losses, insulation from the surroundings isimportant, especially when the temperature differencebetween the absorber and the ambient is high.

Usually, collectors are classified in three categories: un-covered, covered and vacuum, as shown in Fig. 20.7.Uncovered collectors do not have a transparent cover, sothe sun strikes directly the absorber surface, hence thereflection losses are minimised. This collector type onlyis used for applications where the temperature differ-ences between the absorber and the surroundings aresmall, for example for swimming pools. Covered collect-ors are covered by a transparent material, providing ex-tra insulation but also increasing the reflection losses.These collectors are used for absorber temperatures ofup to 100°C. Finally, in vacuum collectors the absorber is



Absorber

Insolation

Transparent cover

Absorber Heat pipe

Vacuum chamber

Absorber

(a) (b) (c)

Figure 20.7: Illustrating (a) an uncovered, (b) a covered and (c) a vacuum collector.

encapsulated in vacuum tubes. In that case, little heat islost to the surroundings. The manufacturing process ofthese collectors in more complicated and expensive, butthe collector can be used for high temperature applica-tions since the convection losses to the surroundings aremuch lower than for the other types.

Modern solar collectors often use light trapping tech-niques just as solar cells. For example, transparentconducting oxide layers at the front window, that wediscussed in Section 13.1, can be used. If they havethe plasma frequency in the infrared, almost all thesolar radiation can enter the window, but the long-wavelength infrared radiation originating from the hotparts in the collector cannot leave the collector as it isreflected back at the window.

Collectors also can be classified by their shape. We candistinguish between flat-plate collectors and concen-trating collectors. Flat-plate collectors consist of flat ab-sorbers that are oriented towards the sun. They can

deliver moderate temperatures, up to around 100°C.They use both direct and diffuse solar radiation andno tracking systems are required. Their main applica-tions are solar water heating, heating of building, airconditioning and industrial processes heat. In contrast,concentrating collectors are suited for systems that requirea higher temperature than achievable with flat collect-ors. The performance of concentrating collectors can beoptimised by decreasing the area of heat loss. This isdone by placing an optical device in between the sourceof radiation and the energy absorbing surface. becauseof this optical device the absorber will be smaller andhence will have a lower heat loss compared to a flat-plate collector at the same absorber temperature. Onedisadvantage of concentrator systems is that they re-quire a tracking system to maximise the incident radi-ation at all times. This increases the cost and leads toadditional maintenance requirements.



Like for PV systems,2 also for solar heat collector ar-rays it is important to decide whether the collectorsshould be connected in series or in parallel. Connect-ing collectors in parallel means that all collectors havethe same input temperature, while for collectors connec-ted in series the outlet temperature of one collector isthe input temperature of the next collector. Most com-mercial and industrial systems require a large numberof collectors to satisfy the heating demand. Thereforemost of the time, a combination of collectors in seriesand in parallel is used. Parallel flow is used more fre-quently because it is inherently balanced and minim-ises the pressure drop. In the end, the choice of seriesor parallel arrangement will ultimately depend on thetemperature required by the application the system.

20.2.2 Heat storage

Now we will discuss the heat storage, which is an ex-tremely important component as it has an enormousinfluence on the overall system cost, performance andreliability. Its design affects the other basic elementssuch as the collector or the thermal distribution system.The task of the storage is twofold. First, it improvesthe utilisation of the collected solar energy. Secondly, itimproves the system efficiency by preventing the fluid

2See Section 17.1.

flowing through the collectors from quickly reachinghigh temperatures.

Several storage technologies are available; some of themcan even be combined to cover daily and seasonal fluc-tuations. In general, heat can be stored in liquids, solidsor phase?change materials, abbreviated as PCM. Wateris the most frequently used storage medium for liquidsystems, because it is inexpensive, non-toxic, and it hasa high specific heat capacity. In addition, the energy canbe transported by the storage water itself, without theneed for additional heat exchangers.

The usable energy stored in a water tank can be calcu-lated with

Qstored = VρCp∆T, (20.7)

where V is the volume of the tank, ρ is the density ofwater, Cp is the specific heat capacity of water and ∆Tis the temperature range of operation. The lower tem-perature limit is often set by external boundaries, suchas the temperature of the cold water, or by specific pro-cess requirements. The upper limit may be determinedby the process, the vapour pressure of the liquid or theheat loss of the water storage. For example, for resid-ential water heating systems the maximally allowedtemperature is set to 80°C because at higher temperat-ures calcium carbonate will be released from the water,clogging the warm water tubes [118].



The heat loss of the tank, Qloss, can be determined with

Qloss = UA∆T, (20.8)

where A is the area of the heat storage tank. U is theglobal heat exchange coefficient and is a measure forthe quality of the insulation. Usually it varies between2 and 10 W/K. Further, U is also a function of the dif-ferent media between which the heat exchange takesplace.

The same principles can be applied to small and bigstorage systems. Small water energy storage can coverdaily fluctuations and is usually in the form of watertanks with volumes from several hundreds of litres upto several thousands of litres. Large storage systems canbe used as a seasonal storage. Often they are realised asunderground reservoirs.

Another type of energy storage are so-called packed beds,which are based on heat storage in solids. They use theheat capacity of a bed of loosely packed particulate ma-terial to store the heat. A fluid, usually air, is circulatedthrough the bed to add or remove energy. A varietyof solids can be used, rock being the most common. Inoperation, flow is maintained through the bed in onedirection during addition of heat and in the oppositedirection during removal. The bed is in general heatedduring the day with hot air from the collector. In theevening and during night energy is removed with airtemperatures around 20°C flowing upward.

Another way to store heat in buildings is to use thesolids of that the walls and roofs are built as a storagematerial. A case of particular interest is the collector-storage wall, which is arranged such that the solar radi-ation is transmitted through a glazing and absorbed inone side of the wall. As a result, the temperature of thewall increases and the energy can be transferred fromthe wall to the room by radiation and convection. Someof these walls are vented to transfer more energy to theroom via forced convection.

The last method to store heat that we will discuss isto use phase change materials (PCM). While in all thestorage methods discussed so far energy is stored assensible heat, in PCM storage takes place as latent heat,which is used for a phase change, without any changein the material temperature. PCM used for energy stor-age must have a high specific latent heat L, such that alarge amount of energy can be stored. In addition, thephase change must be reversible, and survive cycleswithout significant degradation. The stored amount ofheat can be calculated with

Qstored = m [Cs(T∗ − T1) + L + Cl(T2 − T∗)] , (20.9)

where the different temperatures are also sketched inFig. 20.1. We consider the specific heat capacity in thesolid phase, Cs, to be constant from the initial temperat-ure T1 up to the phase change temperature T∗, the lat-ent heat of the material L and the specific heat capacityin the liquid phase Cl from T∗ up to the final temper-



ature T2. Materials that are commonly used s PCM aremolten salts, such as Na2SO4, CaCl2 or MgCl2. PCMstorage in general is used for high temperature applica-tions.

20.2.3 System design

Often, at least at larger latitudes with large differencesbetween summer and winter, solar thermal systemsinclude a boiler as a backup. Its main function is toprovide the necessary energy when the solar power isnot sufficient. It is basically a normal heater that addsthe remaining heat needed to achieve the desired tem-perature in the storage tank. As energy source, usuallynatural gas, oil, or biomass is used. Sometimes, the ad-ditional heat also is provided with electricity or a heatpump.

Up to now we discussed how to collect and store theenergy, but we did not consider yet how to transportit from the collector to the storage system. The trans-port of heat is done with a collector circuit, which usu-ally transports heat using either a liquid or gas. If a li-quid is used, it is important that it neither should freezenor boil, even at the most extreme operating conditions.Further, the medium should have a large specific heatcapacity, a low viscosity, and it should be non-toxic,cheap and abundant. The most common fluids are wa-ter, oils or air.

As mentioned above, the flux can be caused either nat-urally by the temperature gradients, forced by a pump,or by a heat pipe, in which the fluid is allowed to boiland condense again. The optimal choice will depend onthe specific system design. Further, heat losses in thecollector circuit must be taken into account, especiallywhen the pipes are very long. During the planningphase it therefore is important to minimise the circuitlength.

In systems with a pump often a controller is present thatregulates the fluxes of fluid through the collector, thestorage and the boiler in order to assure the desiredtemperature of the. This can for example be done bycalculating the optimal flux in dependence of the fluidtemperature at the collectors and in the storage suchthat the heat transfer is maximised.

20.2.4 Solar air conditioning

Another interesting application is solar air conditioning(also called solar cooling), which seems a bit contradict-ory on first sight. Before we start with the actual dis-cussion on solar air conditioning, we briefly recap theprinciple of a compressor-driven air conditioning sys-tem, which is sketched in Fig. 20.8. Similar to every heatpump, the task of an air conditioning system is to trans-port heat from a cool reservoir B to a warmer reser-voir A. Such a system has typically four components:



cold side Awarm side B

compressor

thermal expansionvalve

cond

ense

r evaporator

Figure 20.8: A compressor-driven cooling circuit.

a compressor, a condenser, a thermal expansion valve(throttle) and an evaporator.

In the compressor (1), vaporised coolant is compressedto a higher pressure. Under release of its latent heat tothe surroundings, it is condensed to its liquid state inthe condenser (2). Then the coolant passes a thermalexpansion valve (3), such that its pressure on the otherside of this valve is so low that it can evaporated again.For the evaporation, it must absorb latent heat fromthe cool reservoir A. Then, the coolant containing thelatent heat from A is compressed again. We see thatthis cooling circuit utilises the latent heat stored and

released during phase changes.

Naturally, heat only flows from warm to cold reser-voires. For reversing the heat flow, as it happens in aheat engine, additional energy must be added to thesystem. This happens in the compressor and tradition-ally is supplied as electric energy. Probably the simplestoption to realise solar cooling is to generate this elec-tricity by a conventional PV system or a solar thermalpower system as discussed in Section 20.3. Further, it isalso possible to drive the compressor with mechanic en-ergy obtained from solar energy directly with a Rankinecycle, which is discussed in 20.3.

The choice of the coolant is based on the temperatureregime in which the air conditioner is supposed to op-erate. However, also their effect on the environmenthas to be taken into account. For example the use, chlo-rofluorocarbons that were widely used as coolants wasstrongly banned in the 1980s because of their destruct-ive effect on the ozone layer [119].

Another concept that that directly can be driven bysolar heat is that of an absorption cooling machine. In dif-ference to compressor-driven cooling, no mechanicalenergy is required for absorption cooling, but the cool-ing process is directly driven by heat that can be sup-plied by solar collectors. Instead of one coolant, heretwo substancesare used: a refrigerant and an absorbant. Of-ten, ammonia (NH3) is used as refrigerant, while wateris used as an absorbant. Under atmospheric pressure,



Figure 20.9: Illustrating an absorption cooling circuit asinvented by Einstein and Szilard [120]. (Annotations by P.Brandon Malloy.)

ammonia has a boiling point of -33°C. Further, it is verysoluble in water, a property that is utilised in the cool-ing process. Figure 20.9 sketches the cooling circuit ofan absorption cooler. In the boiler (1), a mixture of bothsubstances is heated. Because of its low boiling pointthe refrigerant will leave the mixture in the gas phases.Absorbant that is still present in this gas is separated(2) and brought back into the boiler. Then, the refriger-ant is then is condensed in a condenser (3). Then, thecondensed refrigerant is brought to a thermal expansionvalve (4). Similar to the compressor circuit, latent heatis used to evaporate the expanded refrigerant, such thatcooling takes place. In the last step, the refrigerant isabsorbed by the absorbant and the circuit starts again.Instead of using a throttle valve to expand the refriger-ant, it also can be mixed with a third gas, for examplehydrogen (H2), such that the partial pressure of the re-frigerant is reduced allowing it to evaporate. In thatversion, the whole system operates at one pressure.

The last option of cooling that we discuss is that ofsolar desiccant cooling, illustrated in Fig. 20.10. In sucha system, air air is dried when passing through a desic-cant, like silica. Then water is sprayed into the air. As itevaporates, the air is cooled and humidified, such thatit guarantees optimal interior conditions. Air that leavesthe interior is heated up using a solar collector. The hotair streams through the desiccant. Hence, the desiccantis tried and can be reused for adsorbing humidity.



solar collector

exterior interior

heat exchanger

desiccant (rotating)

air heater for desiccant regeneration

humidier

Figure 20.10: Desiccant cooling.

20.3 Concentrated solar power(CSP)

In this section we take a closer look at concentrated solarpower (CSP), where high temperature fluids are used insteam turbines to produce electricity. Much of the earlyattention on CSP systems was on small-scale applica-tions, mainly for water pumping. However, since about1985 several large-scale power systems with a poweroutput up to 80 MW have been built.

The CSP technology is especially interesting for desertregions, where almost all the solar radiation is incid-ent as direct radiation. As illustrated in Fig. 20.11, thesesystems consist basically a collector, where the solarenergy is absorbed, a storage system, usually water orphase-change storage, a boiler that acts as a heat ex-changer between the operational fluids of the collectorand the heat engine, and the heat engine itself, whichconverts the thermal energy into mechanical energy.In large CSP plants, the heat engine is a steam turbine,where the energy stored in the hot steam is partiallyconverted to rotational (mechanic) energy as the steamexpands along the turbine. This thermodynamic processis called Rankine cycle, called after the Scottish engineerWilliam John Macquorn Rankine (1820-1872).

The mechanical energy can be further used to drive anelectrical generator. The collectors are built as concen-


20.3. Concentrated solar power (CSP) 365

collector array

electricitygrid

cooling tower

generatorsteam

turbine

evaporator

heat storage

pump

condenser

feedwater pump

Figure 20.11: Sketching a solar thermal electric power sys-tem.

trator systems, in order to reach the high temperaturesof several hundreds of degree Celsius, which are re-quired to operate the heat engine.

One problem of CSP systems is that the efficiency of thecollector diminishes as its operating temperature rises,while the efficiency of the engine increases with tem-perature as we already have seen when discussing thethermodynamic efficiency limit in Section 10.1. Hence,a compromise between the two has to be found whenchoosing the operating temperature. Current systemshave efficiencies up to about 30% [121],

Now we will take a closer look at the solar concentratorsystem, illustrated in Fig. 20.12. First concepts alreadywere developed by the ancient scientist Archimedes(287 BC - 212 BC); Leonardo da Vinci (1452 - 1519) de-signed some concentrators. Different types of concen-trators produce different peak temperatures and hencethermodynamic efficiencies, due to the different ways oftracking the sun and focusing light. Innovations in thisfield are leading to more and more energy efficient andcost effective systems.

The most common concentrator type with about 96%of all installed CSP installations is the parabolic trough[126], which consists of a linear parabolic reflector thatconcentrates the light onto an absorber tube located inthe middle of the parabolic mirror, in which the work-ing fluid is located. The fluid is heated to 150 to 350degrees Celsius, and then used in a heat engine. This



(a) Parabolic trough

(b) Fresnel concentrator (c) Dish Stirling

(d) Solar power tower

Figure 20.12: Different types of concentrators used in CSPsystems [122–125].

technology is far developed, well-known examples arethe Nevada Solar One power plant and the power plantthat the Plataforma Solar de Almeria (PSA) in Spain.

Another concentrator concept is that of Fresnel concen-trators, where flat mirrors are used, as illustrated in20.12. Flat mirrors allow more reflection in the sameamount of space as parabolic mirrors, reflect more sun-light, and are much cheaper.

Other important concentrator systems is the dish Stirl-ing or dish engine system. It consists of a parabolic re-flector that concentrates light to the focal point, wherethe working fluid absorbs the energy and is heated upto about 500°C. The heat is then converted to mechan-ical energy using a Stirling engine. With about 31.25%efficiency demonstrated at Sandia National Laborator-ies, this design has currently the highest demonstratedefficiency [121].

The last concentrator type is that of solar power towerplants. Here, an array of dual-axis tracking reflectors,commonly named heliostats, are arranged around atower. There the concentrated sunlight hits a central re-ceiver, which contains the working fluid, which can beheated to 500 or even 1000°C. Examples for solar powertowers are Solar One and Solar Two in the Unites Starsand the Eureka project in Spain [126].

CSP systems can be combined with heat storage, suchas phase-change materials discussed in Section 20.2.


20.3. Concentrated solar power (CSP) 367

This allows CSP systems to operate during day andnight.


21Solar fuels

In this chapter we discuss how to store solar energyin the form of chemical energy in so-called solar fuels.As we have seen in previous chapters, we are able toconvert solar energy to electricity with efficiencies of upto 44%. Also, we are able to convert solar energy easilyin heat, which we can use for preparing warm water,heating and even cooling.

The biggest obstacle towards a economy that is drivenby renewable energy is not the generation of sustain-able energy sources but the storage the renewable en-ergy. As we have seen in Chapter 17, storing electri-city with batteries is very difficult. While day-nightstorage that makes solar electricity generated duringdaytime available in the night is easily done, seasonalstorage that allows to store electricity generated in the

summer until winter would require very large bat-tery systems that usually are not feasible. This prob-lem becomes more severe in regions that are far awayfrom the equator such that the differences in solar ir-radiance between summer and winter are significant.The same also is true for solar heat that was discussedin Chapter 20: while a hot water storage tank with avolume of several hundreds of litres is sufficient as day-night storage, seasonal storage that would allow to usesolar-heated water throughout the year would requirestorage tanks with a volume of tens of thousands litres.This tank also would need to be heavily isolated. If wemanage to store solar energy as chemical energy, we donot have this problem as chemicals can be easily stored.

Let us take another look at the Ragone chart in Fig.

369


F

370 21. Solar fuels

Li-ionbat.

100

Energy density (Wh/kg)

Pow

er d

ensi

ty (W

/kg)

10

double layer cap.

elektrolytecap.

Leadacid

NiMHNiCd

ultracap. 104

103

102

1010310210-1

CH4

Li-ion cap.

gasoline

H2,combustion

H2, fuel cell

Figure 21.1: A Ragone chart of different energy storage meth-ods. Capacitors are indicated with “cap”.

21.1 that we already analysed in Section 17.4. Thischart shows the amount of energy per mass stored ina certain storage technology versus the specific powerprovided by the technology. As we know, by far themost used form of stored energy are fossil fuels. Aswe can see from the graph, fossil fuels have high en-ergy density properties, are stable and reliable. But wealso know that fossil fuels are depleting and their useis detrimental to our environment. Hydrogen (H2) hasan even better gravimetric energy density compared tofossil fuels, but it is a very light gas, so the volumetricenergy density, i.e. the energy per unit volume, is muchlower. This is one of the reasons it has not been widely

used until now. When we compare batteries and hydro-gen as energy storage for solar electricity, we see thatbatteries have a lower energy density. Further, they re-quire a much higher initial investment than hydrogen.Hence, it seems an interesting option to use hydrogenas a storage material, i.e. as a solar fuel.

As we see in this chapter, we can produce hydrogen us-ing solar energy by utilising electrochemistry. In nature,photosynthesis takes place, where sunlight is used toconvert carbon dioxide and water into oxygen and sug-ars, which can be seen as nature has been convertingthe energy from the sun into chemical energy for a longtime by converting carbon dioxide and water into oxy-gen and sugars. Here, we try to mimic nature with in-organic semiconductor materials that re able to split awater molecule into oxygen and hydrogen using the en-ergy of sunlight. This process sometimes is referred toas artificial photosynthesis.

For storing solar energy as chemical energy in the formof hydrogen, water splitting can be used,

2H2O→ 2H2 + O2. (21.1)

The energy required for this reaction is given by theGibbs free energy and it has a value of G = 237.2 kJ/mol.In solar water splitting this energy is provided by thesun.

Another product that seems very promising for energystorage is methane (CH4), as it is easier to store and


21.1. Electrolysis of water 371

has less hazardous problems than H2. It can be pro-duced from hydrogen and carbon dioxide (CO2) withthe Fischer-Tropsch process. In this process, first the hy-drogen is combined with carbon dioxide in a water gasshift reaction to obtain carbon monoxide and water inthe form of steam,

H2 + CO2 → 2H2O + CO. (21.2)

This gas mixture, known as synthesis gas, can be thenrefined to finally obtain methane,

CO + 3H2 → CH4 + H2O. (21.3)

When the stored energy is required, the methane can beburned in a combustion reaction,

CH4 + 2O2 → CO2 + 2H2O (21.4)

that will give water and carbon dioxide as byproducts.The water can be reused for the electrolysis and thecarbon dioxide can be used for the water gas shift re-action, hence, the cycle is closed. Each reaction in thecycle has a certain efficiency; the overall efficiency canbe obtained by combining all those efficiencies.

Many different methods can be used for water splitting.In this chapter we will discuss two methods: electrolysisof water and photoelectrochemical water splitting (PEC).

O2 H2O 2H2

= =– +

– +

Anode Cathode

Control electronics

PV array

Figure 21.2: Illustrating a lab-scale Hofmann voltameter thatis used to split water. In the sketch the voltameter is connectedto a PV system via some control electronics.


372 21. Solar fuels

21.1 Electrolysis of water

Figure 21.2 shows a typical lab-scale setup for the hy-drolysis of water. It is called a Hofmann voltameter afterthe German chemist August Wilhelm von Hofmann(1818-1892). It contains of three upright cylinders thatare joined at the bottom. While electrodes are placedin the left and the right cylinder, the central cylinderis used to refill the device with water. For starting theelectrolysis, a voltage has to be applied between thetwo electrodes.

Water splitting is a reduction-oxidation reaction, which iscommonly abbreviated as redox reaction. In such redoxreactions, the reaction happens due to the exchangeof electrons between atoms or molecules. They can bedivided in two half reactions, the oxidation and the re-duction. For water splitting these half reactions look asfollows: During the oxidation, which happens at theanode, water is split giving oxygen and protons to thesolutions and electrons to the anode,

H2O→ 12 O2 + 2H+ + 2e−. (21.5)

In the reduction, electrons originating from the cathodereact with the protons to hydrogen,

2H+ + 2e− → H2. (21.6)

The total reaction therefore can be written as

H2O→ 12 O2 + H2. (21.7)

Each half reaction has an associated potential and thesum of the potentials of each half reaction is the poten-tial for the whole redox reaction. Potentials always aredefined with respect to a reference, i.e. a zero point. Forredox reactions, the zero is defined as the hydrogen halfreaction that happens at the cathode. For calculatingthe potential at the anode, ie of the oxygen productionreaction, we can apply the equation

V0 =GnF

, (21.8)

where G is the Gibbs free energy already mentionedabove, n is the number of electrons transferred betweenthe anode and the cathode for the production of one H2molecule, and F = 96 485 C/mol is the Faraday constantthat tells us the amount of charge per mole of electrons.Using G = 237.2 kJ/mol and n = 2, as two electrons aretransferred per H2 molecule, we find V0 = 1.23 V.

In reality, an applied voltage of 1.23 V is not enough todrive the redox reaction because of some voltage losses.These voltage losses are mainly because of the activa-tion energy at the electrodes as well as Ohmic losses inthe cables, electrodes and also in the solution. For in-creasing the conductivity of the water, a base or an acidcan be added. For the classic Hofmann voltameter, asmall amount of sulphuric acid (H2SO4) is used.

The potential difference between V0 and the potentialused in the real device is called overpotential ∆V. The


21.2. Photoelectrochemical (PEC) water splitting 373

overpotential due to the activation energy required atthe electrodes is strongly related to the electrode mater-ial and the gas that is produced at that electrode. Forhydrogen production at the cathode, platinized (black)platinum is one of the best materials with an overpo-tential of −0.07 V. For the oxygen production at the an-ode, nickel with an overpotential of +0.56 V is very wellsuited. Combining these two voltages and the overpo-tentials we can understand that the typical overpoten-tial is usually around 0.8V.

It is important to realise that all the energy consumedby the hydrolysis process that is related to the overpo-tential is a loss that will lead to heating up of the hy-drolysis device. Only a fraction

ηWS =V0

V0 + ∆V(21.9)

of the supplied energy is actually stored in hydrogen. Itis clear that it is a very important task for science andindustry to reduce the required overpotential. Further,the current requirement for platinum electrodes makeshydrolysis not feasible from an economic point of view.

For an overpotential of 0.8 V, we would get an effi-ciency of around 60%. If we would use a good PV sys-tem with an overall efficiency of ηPV = 18% we wouldhave a total maximal solar-to-hydrogen efficiency of

ηSTH = ηPVηWS = 10.9%. (21.10)

21.2 Photoelectrochemical (PEC)water splitting

Hydrogen can also be produced utilising solar energyby using a photoelectrode, which uses light to produce anelectrochemical reaction, in which water is split intooxygen and hydrogen. In this process, the photonsreach the surface of the photoelectrode, which is madeof a photoactive semiconductor. As in any other semi-conductor, photons with an energy equal or largerthan the semiconductor band gap energy will create anelectron-hole pair. The electrons and holes will be sep-arated by an electric field, and both will be used in thetwo half reactions involved in the overall water split-ting process. To generate the required electrical field,we need a voltage source, for example a solar cell. Thephotoelectrode can be either an anode or a cathode.

If the solar-splitting device is made of a solar cell thatis placed behind the photoanode, the solar cell willreceive the light transmitted through the photoanode.This light creates another electron-hole pair and electricfield that brings the electrons to the photoanode andthe holes to the photocathode with enough potential todrive the redox reaction in the electrolyte. As a result,the water molecule is split into oxygen and hydrogen.

When the photoelectrode is made from an n-type semi-conductor it acts as electron donor; if it is p-type it


374 21. Solar fuels

acts as electron acceptor. As an acceptor, the materialwill attract more holes to the interface, which will en-hance the reduction half reaction. Hence it acts as thephotocathode and produces hydrogen. If the semicon-ductor is n-type, it acts as the photoanode. Electronsare moved to the interface by the internal electric field,and those electrons are involved in the oxidation halfreaction, such that oxygen is produced.

The semiconductor material has to fulfil several require-ments. First, it has to absorb the light that is incidenton its surface. Secondly, charge carrier transport in-side the material and separation into the two electrodesmust be efficient. Thirdly, a bandgap of 1.23 eV is notenough to drive the reaction because of the overpoten-tial already discussed above. It has been estimated thatmaterials with an energy band gap close to 2.1 eV havethe potential to split water. Fourthly, the energy levelsof the material have to be adequate to couple with theenergy needed for the reaction. In particular, the energylevels of the reactions have to be located somewherein the energy band gap of the semiconductor, which iscalled the favourable position. To further enhance thereaction, a catalyst may be added to the semiconductorsurface. Fifthly, on the practical side, it is important thatthe used materials are photochemically stable and rel-atively cheap. From these criteria we can conclude thatthe main technical challenges to be addressed are lightabsorption, the separation of charges and the catalysisof the reaction.

Figure 21.3: The solar-to-hydrogen efficiency and bandgapof different potential photocathode materials. Also the photonflux is shown [127].



The absorption and catalysis problems can be tackledby carefully choosing the semiconductor material andits corresponding catalyst. Several materials can beconsidered for solar water splitting. Some of the mostpopular materials studied for this application are ti-tanium dioxide (TiO2), tungsten oxide (WO3), bismuthvanadate (BiVO4), hematite (Fe2O3) or amorphous sil-icon carbide (a-SiC). As shown in Fig. 21.3, BiVO4,Fe2O3, and a-SiC have the most promising potentialsolar-to-hydrogen efficiency. , as shown in the graph.Hematite and silicon carbide are the best choice forthe optimal absorption of light, since they have bandgap energies closest to the optimal bandgap of 2.1 eV.However, if also other factors such as the band posi-tion or stability are considered, materials like bismuthvanadate may also be a viable option. Because of theirlarge bandgap energies, the other materials are not con-sidered.

The overall efficiency of water splitting depends on thecatalytic efficiency and the separation efficiency of thephotoelectrode. The catalytic efficiency can be improvedby placing a catalyst on the semiconductor surface. Forexample, for a ceBiVO4 photoanode the inclusion of acobalt phosphate [Co3(PO4)2] catalyst on the surfacewill ease the oxidation reaction and hence increase thewater splitting efficiency. The separation efficiency canbe improved by introducing an electric field inside thematerial. One way to do this is to introduce gradientdoping, starting from no doping at the surface to 1%

0.5 1.0 1.5 2.00

2

4

BiVO4+Gradient doping +Catalyst

Curr

ent d

ensit

y (m

A/cm

2 )

Potential vs. RHE (V)

BiVO4

BiVO4+ Catalyst

dark

Figure 21.4: The effect on using a catalyst and gradient dop-ing on the performance of a BiVO4 photoanode [127].


376 21. Solar fuels

doping close to the back of the electrode. With gradi-ent doping a depletion region is created in between thesemiconductor and the electrolyte. As a result, elec-trons will move more easily from the electrolyte intothe semiconductor. Combining both effects can highlyimprove the efficiency of the overall device. Both the ef-fects of the catalyst and the gradient doping are shownin Fig. 21.4, where the current density of the illumin-ated photoanode is shown in dependence of the appliedvoltage with respect to the reversible hydrogen electrode(RHE). The RHE is a special electrode with the propertythat the measured potential does not changed when thepH value of the solution is changed.

As we already mentioned above, for the photoelec-trochemical (PEC) water splitting process, a potentialdifference of at least 1.23 V plus the overpotential ∆Vmust be present. The value of ∆V will depend on thematerials and electrolyte used, but it is usually around0.8V. Both potentials added will result in the total po-tential difference required to drive the redox reaction.This voltage will be partially covered by the potentialdifference created within the photoelectrode when lightshines on it. However, depending on the material, thePEC only creates 0.6 V of the required voltage. For theextra potential that is required to allow water splitting,the photoelectrode can be combined with solar cellsthat provide the extra potential needed for the reactionto happen. The combination of a photoelectrode and asolar cell forms a photoelectrochemical device.

In Fig. 21.5 (a) such a PEC device is sketched. The pho-toelectrode (in this case a photoanode) is connected to asolar cell in series. The photoanode is connected to thepositive contact of the solar cell. The negative contactof the solar cell is connected to another electrode viaan external circuit. This counter electrode may or maynot be photoactive. The electric circuit is closed by theelectrolyte. The band diagram of this device is shownin Fig. 21.5 (b).

Since the photoelectrode and the solar cell are connec-ted in series, the same current will go through boththe devices, similar as for multi-junction solar cells. Inthis device, the light is utilised better then it wouldbe in the solar cell alone. In the photoelectrode, thephotons with energies exceeding that of the photoelec-trode bandgap will be absorbed. The fraction of thelight that has not been absorbed or reflected, called thetransmitted spectrum, reaches the solar cell where it canbe absorbed to generate the extra potential differencerequired for water splitting. The solar cell must be op-timised for the transmitted spectrum, which is differentfrom the standard AM1.5 spectrum that is usually usedfor solar cell optimisation.

As any semiconductor device, the photoelectrode hasits own characteristic J-V curve. When a solar cell anda photoelectrode are combined, the conditions at whichthey will work can be estimated by studying their J-Vcurve characteristics. Figure 21.6 shows the J-V curves



Electrolyte

Electrolyte

H2O/H 2

H2O/O 2

Pt coil Co-Pi

Gradient-dopedBiVO 4

Gradient-dopedBiVO 4

p–i–n

Co-Pi

p–i–n

–

––

–

–

–

+

++

–

–

–

–

–

+

+

+

Glass ITO

kcaBOCTcontact

Tunneljunction

Semiconductorelectrolyteinterface

Metalcathode

FTO

Transmitted

GraphiteO2

A

H2

AM 1.5

2-jn a-Si Ag/Cr/Alcontact

(a) (b)

Figure 21.5: (a) Illustrating a photoelectrochemical (PEC) device consisting of a BiVO4 photoanode and a tandem a-Si:H/a-Si:Hsolar cell. (b) The band diagram of this PEC device [127].


378 21. Solar fuels

Curr

ent d

ensit

y (m

A/cm

2 )

Potential

solar cell

photoanode

Voc

Jsc

Voperational

Jphoto

Figure 21.6: The J-V characteristics of the photoelectrode andthe solar cell. The operating point is at the crossing of the two.

of the solar cell and of the photoanode in this case.Since both elements are connected in series, the cur-rent of both the solar cell and the photoelectrode mustbe the same. Hence, the operational point will be whereboth J-V curves cross.

Now we will estimate the solar-to-hydrogen efficiencyηSTH that is related to the amount of produced hydro-gen. This amount is directly calculated by the currentdensity measured at the operating point, which we canunderstand by realising that current is the flow of elec-tric charge per time.

When we assume that all generated charges producedare involved in the production of hydrogen, the overallsolar-to-hydrogen conversion efficiency, is described by

ηSTH =Jph × 1.23 V

Pin, (21.11)

where Jph is the operational current density, Pin is theirradiance arriving at the PEC device, usually it will be1000 W/m2 from the AM 1.5 spectrum.

The only free variable in Eq. (21.11) is the current dens-ity: the higher the operational current, the more hydro-gen is produced and the higher the device efficiency.The major focus of current academic research there-fore is to improve the current density. Besides utilisinggradient doping water-oxidation catalysts also other im-provements can be done to increase the performanceof PEC devices. These are optimising the materials and



layer thicknesses in the used solar cell and texturing ofthe photoanode for better light trapping.

All these concepts were applied at the Delft Univer-sity of Technology, where a bismuth vanadate photoan-ode was combined with a double junction amorphoussilicon solar cell and a platinum cathode. This deviceachieved a solar-to-hydrogen efficiency of ηSTH = 4.9%,which is actually the highest efficiency reported forsuch devices based on a metal oxide photoanode [127].


380 21. Solar fuels


Appendix

381


ADerivations in Electrodynamics

A.1 The Maxwell equations

The four Maxwell equations couple the electric andmagnetic fields to their sources, i.e. electric charges andcurrent densities, and to each other. They are given by

∇ ·D(r, t) = ρF(r), (A.1a)

∇× E(r, t) = −∂B(r, t)∂t

, (A.1b)

∇ · B(r, t) = 0, (A.1c)

∇×H(r, t) = +∂D(r, t)

∂t+ JF(r), (A.1d)

where r and t denote location and time, respectively.D is the electric displacement, E the electric field, B is themagnetic induction and H is the magnetic field. ρF is thefree charge density and JF is the free current density.

The electric displacement and field are related to eachother via

D = εε0E, (A.2a)

where ε is the relative permittivity of the medium inthat the fields are observed and ε0 = 8.854 × 10−12

As/(Vm) is the permittivity in vacuo. Similarly, the mag-netic field and induction are related to each other via

B = µµ0H, (A.2b)

where µ is the relative permeability of the medium in

383


384 A. Derivations in Electrodynamics

that the fields are observed and µ0 = 4π · 10−7 Vs/(Am)is the permeability of vacuum. Eqs. (A.2) are only validif the medium is isotropic, i.e. ε and µ are independentof the direction. We may assume all the materials im-portant for solar cells to be nonmagnetic, i.e. µ ≡ 1.

A.2 Derivation of the electromag-netic wave equation

We now derive de electromagnetic wave equations insource-free space, ρF ≡ 0 and jF ≡ 0. For the derivationof the electromagnetic wave equations we start with ap-plying the rotation operator ∇× to the second Maxwellequation, Eq. (A.1b),

∇× (∇× E) = −∇×(

∂B∂t

). (A.3)

Now we take the fourth Maxwell equation, Eq. A.1d,with jF = 0 and Eqs. (A.2),

1µµ0∇× B = εε0

∂E∂t

,

and substitute it into Eq. (A.3),

∇× (∇× E) = −εε0µµ0

(∂2E∂t2

). (A.4)

By using the relation

∇× (∇× E) = ∇(∇ · E)− ∆E = −∆E, (A.5)

we find

∆E = εε0µµ0

(∂2E∂t2

). (A.6)

In Eq. (A.5) we used that we are in source free space,i.e. ∇E = 0. Equation (A.6) is the wave equation for theelectric field. Note that the factor

1εε0µµ0

has the unit of (m/s)2, i.e. a speed to the square. Ineasy terms, it is the squared propagation speed of thewave.1 We now set

c20 :=

1ε0µ0

(A.7)

andn2 = ε, (A.8)

where c0 is the speed of light vacuo and n is the refract-ive index of the medium. Since we also assume µ ≡ 1,we finally obtain for the wave equation for the electricfield

∆E− n2

c20

(∂2E∂t2

)= 0. (A.9a)

1In reality, if the medium is absorbing, things are getting much more complex.


A.3. Properties of electromagnetic waves 385

λ E

B kk

Figure A.1: Illustrating the E and B field and the k vector of aplane electromagnetic wave.

In a similar manner we can derive the wave equationfor the magnetic field,

∆H− n2

c20

(∂2H∂t2

)= 0. (A.9b)

A.3 Properties of electromagneticwaves

In section A.2, we found that plane waves can be de-scribed by

E(x, t) = E0 · eikzz−iωt, (A.10a)

H(x, t) = H0 · eikzz−iωt. (A.10b)

In this section we study some general properties ofplane electromagnetic waves, illustrated in Fig. A.1.

Substituting Eq. (A.10a) into the first Maxwell equation,Eq. (A.1a), yields

∇D = εε0∇E = 0,∇E = 0,

∇(

E0eikzz−iωt)= 0,

Ex,0∂

∂x eikzz−iωt︸︷︷︸=0

+Ey,0∂

∂y eikzz−iωt︸︷︷︸=0

+

+Ez,0∂∂z eikzz−iωt = 0,

iEz,0kzeikzz−iωt = 0,Ez,0 = 0,

(A.11a)

where we used the notation E0 =(Ex,0, Ey,0, Ez,0

). In a

similar manner, by substituting Eq. (A.10b) into the firstMaxwell equation, Eq. (A.1c), we obtain

Hz,0 = 0. (A.11b)

Thus, neither the electric nor the magnetic fields havecomponents in the propagation direction but only com-ponents perpendicular to the propagation direction (thex- and y-directions in our case).

Without loss of generality we now assume that theelectric field only has an x-component, E0 = (Ex,0, 0 0).Substituting this electric field into the left hand sideof the second Maxwell equation, Eq. (A.1b), yields



∇× E =

∂

∂x∂

∂x∂

∂x

× Ex

Ey

Ez

=

∂

∂y Ez − ∂∂z Ey

∂∂z Ex − ∂

∂x Ez∂

∂x Ey − ∂∂y Ex

=

0∂∂z Ex

0

=

0Ex,0

∂∂z eikzz−iωt

0

=

0iEx,0kzeikzz−iωt

0

. (A.12)

The right hand side of Eq. (A.1b) yields

− ∂B(r, t)∂t

= −µ0∂H(r, t)

∂t= −µ0

∂∂t Hx∂∂t Hy∂∂t Hz

= −µ0

Hx,0∂∂t eikzz−iωt

Hy,0∂∂t eikzz−iωt

0

= −µ0

−iωHx,0eikzz−iωt

−iωHy,0eikzz−iωt

0

. (A.13)

Thus, we obtain for Eq. (A.1b) 0iEx,0kzeikzz−iωt

0

= −µ0

−iωHx,0eikzz−iωt

−iωHy,0eikzz−iωt

0

, (A.14)

i.e. only the y-component of the magnetic field survives.The electric and magnetic components are related toeach other via

Ex,0kz = µ0ωHy,0 (A.15)By substituting Eq. (4.4) into Eq. (A.15), we find

Hy,0 =n

cµ0Ex,0 =

nZ0

Ex,0, (A.16)

where

Z0 = cµ0 =

√µ0

ε0= 376.7 Ω (A.17)

is the impedance of free space.

In summary, we found the following properties of theelectromagnetic field.

• The electric and magnetic field vectors are perpen-dicular to each other and also perpendicular to thepropagation vector,

k · E0 = k ·H0 = H0·E0 = 0. (A.18)

• The electric and magnetic fields are proportionalto the propagation direction, hence electromagneticwaves are transverse waves.

• The electric and magnetic vectors have a constant,material dependent ratio. If the electric field isalong the x-direction and the magnetic field is


A.3. Properties of electromagnetic waves 387

along the y-direction, this ratio is given by

Hy,0 =n

cµ0Ex,0 =

nZ0

Ex,0, (A.19)

where

Z0 = cµ0 =

√µ0

ε0= 376.7 Ω (A.20)

is the impedance of free space.

The derivations in this section were done for planewaves. However, it can be shown that the propertiesof electromagnetic waves summarised in the itemisationabove are valid for electromagnetic waves in general.


BDerivation of homojunction J-V curves

In this appendix we present the derivations for the J-Vcurves in dark and under illumination for p-n homo-junctions, as they were discussed in Section

B.1 Derivation of the J-V character-istic in dark

When an external voltage, Va, is applied to a p-n junc-tion the potential difference between the n-type and p-type regions will change and the electrostatic potentialacross the space-charge region will become (ψ0 − Va).Under the forward-bias condition an applied externalvoltage decreases the electrostatic-potential difference

across the p-n junction. The concentration of the minor-ity carriers at the edge of the space-charge region in-creases exponentially with the applied forward-biasvoltage but it is still much lower that the concentra-tion of the majority carriers (low-injection conditions).The concentration of the majority carriers in the quasi-neutral regions do not change significantly under for-ward bias. The concentration of charge carriers in a p-njunction under forward bias is schematically presentedin Fig. B.1

The concentrations of the minority carriers at the edgesof the space-charge region, electrons in the p-type semi-conductor and holes in the n-type semiconductor after

389


CH

390 B. Derivation of homojunction J-V curves

p=pp0=NAn=nn0=ND

p-type silicon

ln n

ln p

Position

n-type silicon

p=pn0=ni2/ND n=np0=pi

2/NA

np(a)=np0eeV/(kT)

pn(b)=pn0eeV/(kT)

ab

Figure B.1: Concentration profiles of mobile charge carriersin a p-n junction under forward bias (red lines). Concentrationprofiles of carriers under thermal equilibrium are shown forcomparison (black lines).

applying forward-bias voltage are described by

np(a) = np0 eeVakT =

n2i

NAe

eVakT , (B.1a)

pn(b) = pn0 eeVakT =

n2i

NDe

eVakT . (B.1b)

Since we assume that there is no electric field in thequasi-neutral region the current-density equations ofcarriers reduce to only diffusion terms and are notcoupled by the electric field. The current is based onthe diffusive flows of carriers in the quasi-neutral re-gions and is determined by the diffusion of the minor-ity carriers. The minority-carriers concentration can becalculated separately for both quasi-neutral regions. Theelectron-current density in the quasi-neutral region ofthe p-type semiconductor and the hole-current densityin the quasi-neutral region of the n-type semiconductorare described by

Jn = eDNdndx

, (B.2a)

Jp = −eDPdpdx

. (B.2b)

The continuity equations (4.34) for electrons and holesin steady-state (∂n/∂t = 0 and ∂p/∂t = 0) can be writ-


B.1. Derivation of the J-V characteristic in dark 391

ten as1e

dJNdx− RN + GN = 0, (B.3a)

−1e

dJPdx− RP + GP = 0. (B.3b)

Under low-injection conditions, a change in the con-centration of the majority carriers due to generationand recombination can be neglected. However, therecombination-generation rate of minority carriers de-pends strongly on the injection and is proportional tothe excess of minority carriers at the edges of the de-pletion region. The recombination-generation rate ofelectrons, Rn, in the p-type semiconductor and holes,Rp, in the n-type semiconductor is described by Eqs. (),

Rn =∆nτn

, (B.4a)

Rp =∆pτp

, (B.4b)

where ∆n is the excess concentration of electrons in thep-type semiconductor with respect to the equilibriumconcentration np0, and τn is the electrons (minority car-riers) lifetime. ∆p is the excess concentration of holesin the n-type semiconductor with respect to the equi-librium concentration pn0 and τp is the holes (minoritycarriers) lifetime. ∆n and ∆p are given by

∆n = np (x)− np0, (B.5a)

∆ p = pn (x)− pn0. (B.5b)

By combining Eqs. (B.2a), (B.3a) and (B.4a) we obtain

DNd2np (x)

dx2 =∆nτn− GN . (B.6a)

This equation describes the diffusion of electrons inthe p-type semiconductor. Similarly, by combining Eqs.(B.2b), (B.3b) and (B.4b) we obtain

DPd2 pn (x)

dx2 =∆ pτp− GP, (B.6b)

that describes the diffusion of holes in the n-type semi-conductor.

Now we substitute np(x) from Eq. (B.5a) and pn(x)from Eq. (B.5b) into Eqs. (B.6a) and (B.6b), respectively.By using that d2np0/dx2 = d2 pn0/dx2 = 0, and in darkGN = GP = 0 we obtain

d2∆ndx2 =

∆nDN τn

, (B.7a)

d2∆pdx2 =

∆pDP τp

. (B.7b)

The electron-concentration profile in the quasi-neutralregion of the p-type semiconductor is given by the gen-eral solution to Eq. (B.7a),

∆n (x) = A exp(

xLN

)+ B exp

(− x

LN

),



where LN =√

DNτn [see Eq. (7.28a)] is the electronminority-carrier diffusion length. The origin of the xaxis is set to the edge of the depletion region in the p-type semiconductor and denoted as a in Fig. B.1. Forthe p-type semiconductor infinite thickness is assumed(approximation of the infinite thickness). The constants Aand B can be determined from the boundary conditions

1. At x = 0, np(a) = np0 exp(qVa/kT),

2. np is finite at x → ∞, therefore A = 0.

With these boundary conditions the solution for theconcentration profile of electrons in the p-type quasi-neutral region is found to be

np (x) = np0 + np0

(e

eVakT − 1

)e−

xLN . (B.8a)

The hole concentration profile in the quasi-neutral re-gion of the n-type semiconductor is given by the gen-eral solution to Eq. (B.7a),

∆ p(x′)= A′ exp

(x′

LP

)+ B′ exp

(− x′

LP

),

where LP =√

DPτp [see Eq. (7.28b)] is the holeminority-carrier diffusion length. The origin of the x′-axis (x′ := −x) is set at the edge of the depletion regionin the n-type semiconductor and denoted as b in Fig.B.1. Here, we use the approximation of infinite thick-ness for the n-type semiconductor. The constants A′

and B′ can be determined from the boundary condi-tions

1. At x′ = 0, pn(b) = pn0 exp(qVa/kT),

2. p− n is finite at x′ → ∞, therefore A′ = 0.

We thus find the concentration profile of holes in thequasi-neutral region of the n-type semiconductor to be

pn(x′)= pn0 + pn0

(e

eVakT − 1

)e−

xLP . (B.8b)

When substituting the corresponding concentration pro-files of minority carriers (Eqs.B.8) into Eqs. (B.2) we ob-tain for the current densities

JN (x) =e DN np0

LN

(e

eVakT − 1

)e−

xLN , (B.9a)

JP(x′)=

e DP pn0

LP

(e

eVakT − 1

)e−

xLP . (B.9b)

Under the assumptions that the effect of recombina-tion and thermal generation of carriers in the deple-tion region can be neglected, which means that the elec-tron and hole current densities are essentially constantacross the depletion region, one can write for the cur-rent densities at the edges of the depletion region

JN |x=0 = JN |x′=0 =e DNnp0

LN

(e

eVakT − 1

), (B.10a)

JP |x′=0 = JP |x=0 =e DP pn0

LP

(e

eVakT − 1

). (B.10b)


B.2. Derivation of the J-V characteristic under illumination 393

The total current density flowing through the p-n junc-tion at the steady state is constant across the devicetherefore we can determine the total current density asthe sum of the electron and hole current densities at theedges of the depletion region,

J (Va) = JN |x=0 + JP |x=0 (B.11)

=

(e DNnp0

LN+

e DP pn0

LP

)(e

eVakT − 1

).

Using Eqs. (8.1b) and (8.2b), Eq. (B.11) can be rewrittenas

J (Va) = J0

[exp

(eVa

kT

)− 1]

, (B.12)

where J0 is the saturation-current density of the p-njunction which is given by

J0 =

(e DNn2

iLN NA

+e DP n2

iLP ND

). (B.13)

Equation (B.12) is known as the Shockley equation thatdescribes the current-voltage behaviour of an ideal p-ndiode. It is a fundamental equation for microelectronicsdevice physics.

p=pp0=NAn=nn0=ND

p-type silicon

ln n

ln p

Position

n-type silicon

p=pn0=ni2/ND

n=np0=pi2/NA

ab

GτpGτn

Figure B.2: Concentration profiles of mobile charge carriersin an illuminated p-n junction with uniform generation rateG (red line). Concentration profiles of charge carriers underequilibrium conditions are shown for comparison (black line).



B.2 Derivation of the J-V character-istic under illumination

When a p-n junction is illuminated the additionalelectron-hole pairs are generated through the junc-tion. In case of moderate illumination the concentra-tion of majority carriers does not change significantlywhile the concentration of minority carriers (electronsin the p-type region and holes in the n-type region) willstrongly increase. In the following section it is assumedthat the photo-generation rate, G, is uniform through-out the p-n junction, this is called the uniform generation-rate approximation. This assumption reflects the situationwhere the device is illuminated with a long-wavelengthlight, which is weakly absorbed by the semiconductor.The concentration of charge carriers in a p-n junctionwith uniform photo-generation rate is schematicallypresented in Fig. B.2.

Eqs. (B.6a) described have seen the steady-state situ-ation for minority carriers. In section B.1 we then pro-ceeded with assuming that the generation rate is zero.Now we analyse the case where the junction is illumin-ated, i.e. the generation rate is not zero. The equations

thus can be rewritten to

d2∆ndx2 =

∆nDN τn

− GDN

, (B.14a)

d2∆pdx2 =

∆pDP τp

− GDP

. (B.14b)

Under the assumption that G/DN and G/DP are con-stant, the general solution to Eq. (B.14) is

∆n (x) = G τn + C ex

LN + D e−x

LN , (B.15a)

∆ p(x′)= G τp + C′ e

x′LP + D′ e−

x′LP . (B.15b)

The constants in Eqs. (B.15) can be determined fromthe same boundary conditions as were used in the ana-lysis of the p-n junction in dark. The particular solutionfor the concentration profile of electrons in the quasi-neutral region of the p-type semiconductor and holes inthe quasi-neutral region of the n-type semiconductor isdescribed by

np (x) = np0 + G τn +[np0

(e

eVkT − 1

)− G τn

]e−

xLN ,

(B.16a)

pn(x′)= pn0 + G τp +

[pn0

(e

eVkT − 1

)− G τp

]e−

x′LP .

(B.16b)

By substituting these concentration profiles of minoritycarriers into Eqs. (B.2), we obtain for the current densit-


B.2. Derivation of the J-V characteristic under illumination 395

ies

JN (x) =e DN np0

LN

(e

qVkT − 1

)e−

xLN − e G LNe−

xLN ,

(B.17a)

JP(

x′)=

e DP pn0

LP

(e

qVkT − 1

)e−

x′LP − e G LPe−

x′LP .

(B.17b)

In case of an ideal p-n junction the effect of recombin-ation in the depletion region was neglected. However,the contribution of photo-generated charge carriers tothe current in the depletion region has to be taken intoaccount. The contribution of optical generation from thedepletion region to the current density is given by

JN |x=0 = e0∫

−W

−G dx = −e GW, (B.18a)

JP |x′=0 = e0∫

−W

−G dx′ = −e GW. (B.18b)

The total current density flowing through the p-n junc-tion in the steady state is constant across the junction.Under the superposition approximation we thus can de-termine the total current density as the sum of the elec-tron and hole current densities at the edges of the de-

pletion region,

J (Va) =JN |x=0 + JP |x=0

=

(e DNnp0

LN+

e DP pn0

LP

)(e

eVakT − 1

)− e G (LN + LP + W)

(B.19)

Eq. (B.19) can be rewritten as

J (Va) = J0

[exp

(qVa

kT

)− 1]− Jph, (B.20)

where Jph is the photo-current given by

Jph = e G (LN + LP + W) . (B.21)

A number of approximations have been made in or-der to derive the analytical expressions for the current-voltage characteristics of an ideal p-n junction in darkand under illumination. These approximations are sum-marised below:

• The depletion-region approximation

• The Boltzmann approximation

• Low-injection conditions

• The superposition principle

• Infinite thickness of doped regions

• Uniform generation rate



The derived expressions describe the behaviour of anideal p-n junction and help us understand the basicprocesses behind the operation of the p-n junction.However, they do not fully and correctly describe realp-n junctions. For example, the thickness of a real p-njunction is limited, which means that the recombina-tion at the surface of the doped regions has to be takeninto account. The thinner the p-n junction, the more im-portant surface recombination becomes. The surfacerecombination modifies the value of the saturation-current density. Further it was assumed that there areno recombination-generation processes in the depletionregion. However, in real p-n junctions, the recombina-tion in the depletion region represents a substantial lossmechanism. These and other losses in a solar cell arediscussed in Chapter 10.


CAtomic force microscopy and statistical parameters

C.1 Atomic force microscopy (AFM)

AFM1 is a powerful technique with which the mor-phology of nano-textured surfaces can be determined[128]. During an AFM measurement a tiny probe witha tip radius of several nanometers is brought so closeto the surface that atomic forces between the surfaceand the probe become important. Due to these forcesthe surface morphology can be determined. Under amp-litude modulation the amplitude and the phase of the tipwill change during the scan depending on the Van derWaals forces, i.e. the distance between the tip and thesample. Several modes can be used. We used the so

1This appendix is taken from the PhD thesis of Klaus Jäger [53].

called tapping mode in which the probe is oscillating atits resonance frequency. From the obtained amplitudeand phase pictures the height profile of the investigatedsample can be obtained.

The height profile is not the real height profile of thesample but a convolution of the height profile and theshape of the tip. A blunt tip therefore will lead to ablurry picture. If the tip is broken and therefore con-sists of two or more peaks, the features on the scan willbe doubled. Further, scanning the surface too fast willlead to contact loss between the tip and the sample atsteep features. Therefore the AFM measurements haveto be performed very carefully.

Figure C.1 shows a picture of a typical tip of AFM

397


F M S P

398 C. Atomic force microscopy and statistical parameters

Figure C.1: The NSG 10 gold-coated single-crystal silicon tipused throughout this project [129].

probes. This so-called NSG 10 tip is made from anti-mony doped single-crystal silicon and coated with gold.Its tip radius is typically 6 nm, with a guaranteed ra-dius of maximal 10 nm [129].

C.2 Statistical parameters

The AFM scan reveals three dimensional data of thenano-textured surface via the height function z(x, y),which can be used to extract statistical properties ofthe samples. To compare the different samples we usetwo different parameters, the root-mean-squared (rms)roughness σr and the correlation length `c.

The rms roughness in principle is the standard devi-ation of the height profile. It is defined as

σr =

√√√√ 1N − 1

N

∑i=1

(zi − z)2, (C.1)

where N is the number of data points, zi is the heightof the ith datapoint and z is the average height. It be-comes clear from the definition that σr is insensitiveto the lateral feature sizes. Samples with very smallor large lateral features both can have the same rmsroughness.

The correlation length gives an indication of the lateralfeature sizes. Its derivation is less straightforward thenthat of σr: it has to be extracted from autocorrelationfunction (ACF) and/or the height-height correlationfunction (HHCF) [130]. For a discrete set of data, thetwo-dimensional ACF is given by

ACF(τx, τy) =

1(N − n)(M−m)

·N−n

∑l=1

M−m

∑k=1

z(kδ + τx, lδ + τy) z(kδ, lδ),

(C.2)

where δ is the distance between two data points,m = τx/δ and n = τy/δ. For AFM scans usually theone-dimensional ACF along the fast scanning axis (x) is


C.2. Statistical parameters 399

fitHHCF

(b)

τx (µm)

HH

CF(n

m2 )

1010.1

3000

2400

1800

1200

600

0

fitACF

(a)

τx (µm)

ACF

(nm

2 )

1010.1

1500

1200

900

600

300

0

Figure C.2: The ACF, HHCF and fitted Gaussian functions forSnO2:F.

used:

ACFx(τx) = ACF(τx, 0) =

1N(M−m)

N

∑l=1

M−m

∑k=1

z(kδ + τx, lδ) z(kδ, lδ). (C.3)

The one-dimensional HHCF is given by

HHCFx(τx) =

1N(M−m)

N

∑l=1

M−m

∑k=1

[z(kδ + τx, lδ)− z(kδ, lδ)]2 . (C.4)

To determine `c, Gaussian functions can be fitted to theACF and the HHCF. They are given by

ACFfitx (τx) = σ2

r exp(−τ2

x`2

c

), (C.5)

HHCFfitx (τx) = 2σ2

r

[1− exp

(−τ2

x`2

c

)], (C.6)

respectively. The correlation length then is the length atwhich the (fitted) ACF has decayed to 1/e of its highestvalue. Instead of applying Eq. (C.1) directly, the rmsroughness also can be obtained by fitting Gaussianfunctions to the ACF and/or the HHCF.

Figure C.2 shows the ACF and the HHCF and fittedGaussian functions for the SnO2:F sample depicted inFig. 13.3 (a). Instead of fitting to Gaussian functions,some authors also use exponential functions for fitting.For this sample, Gaussians lead to better fits.


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solar energy v1.2 - markups

Engineering

p type

ptype material

ntype material

ptype regions

pn homojunctions

ptype semiconductors

pn heterojunction

ntype layers