II. Conversion efficiency of solar cells

In the first part of this course we have seen that solar cells are based on the photovoltaic

effect, in which a potential difference is generated by the separation of electrons and holes in

the built-in electric field of a p-n junction in a semiconductor material. These electron-hole

pairs form by absorption of photons with higher energy than the energy gap of the

semiconductor. The built-in electric field drifts the minority charge carriers that reach the

junction as a result of diffusion from one part of the junction to the other (where they become

in majority), and forbids the injection of majority carriers from one part to the other (where

they become in minority). We start this part with an estimation of the solar power and then

discuss means to enhance the conversion efficiency of solar cells.

Incident solar power The solar cells convert with higher or smaller efficiencies (typically, between 15–30%), the

solar radiation that reaches the Earth surface. We can estimate the total power irradiated by

Sun by modeling it as a black body. A black body absorbs all the incident electromagnetic

radiation, without reflecting or transmitting any part of it, and emits a thermal radiation with a

temperature-dependent spectrum. The power emitted by a black body at temperature T per

unit surface and per unit energy is

1)/exp(12),( 32

3

−=

TkEhcETEI

B

π . (1)

This power density has the same form for any temperature, but its maximum shifts with T

according to Wien’s law , or (see the figure below, left) TkE B82.2max =

Km109.2 3max ⋅⋅= −Tλ (2)

Problem: Find the temperature of a black body which has a maximum of the spectral

irradiance at a) 550 nm (in the green region of the visible spectrum) and b) 700 nm (in the red

region).

The total power radiated per unit surface (called also power density or irradiance) is given by

the Stefan-Boltzmann law,

2

Soare

Pamant

D

RS

RP

4

0

34

320 1)exp(

)(2),(/)( TxdxxTk

hcdETEIATP B σπ

=−

== ∫∫∞∞

, (3)

where = 5.6·10–8 W/m2⋅K4 is Stefan’s constant. If the Sun temperature is

taken as = 5762 K, the corresponding power density is 6·107 W/m2. From it, as can be

seen from the figure above, right, only the fraction

)15/(2 3245 hckBπσ =

ST

=APP / =22 )/()/)(/( PSPS RRDRAP

1366 W/m2 reaches the Earth, where = 696·103 km is the Sun radius

(109 times larger than the average radius of the Earth, = 6371 km) and D = 149.6·106 km

is the average Earth-Sun distance, used as unit of measure (1 AU) in astronomy.

≅2)/)(/( DRAP SS SR

PR

100%

70% 7%

3%

20% atmosfera

Pamant

3

Because the Earth-Sun distance is not constant, the radiated solar power the reaches

the Earth has a weak variation with the season. From this power, 20% is absorbed in the upper

layers of the atmosphere and 3% is scattered back in Cosmos, so that (taking into account also

the average angle of incidence) only 70% reaches directly the Earth surface and 7% is

scattered toward it (see the figure above, left). Obviously, the usable power depends on the

location on Earth, weather, and the incident angle on the solar cell, the absorption of solar

power being maximum for normal incidence. Moreover, the spectrum of the radiation that

reaches Earth is slightly modified compared to the light spectrum emitted by Sun (see the

figure above, right) due to absorption on the atoms and molecules in the atmosphere (CO2,

H2O, N2, etc.) or non-uniform scattering that takes place predominantly in certain spectral

intervals.

To quantify the solar radiation that reaches the earth we define the Air Mass AM,

which represents the ratio between the pathlength of light through the atmosphere and the

shortest possible path (corresponding to normal incidence). AM is a measure of the decrease

of light power as it passes through the atmosphere. We define AM1 (see the figure below), if

the Sun is exactly above, AM1.5, or conventional solar radiation, for an incidence angle of

48.2°, AM0 above the atmosphere (1366 W/m2), which is a parameter used in space

applications, and AM2 for an incidence angle of 60°. The conventional solar radiation can be

AM1.5D (if only the direct solar radiation, of 900 W/m2, is accounted) or AM1.5G (1000

W/m2 if the scattered radiation is also considered).

Problem: Defining the power as energy per unit time, find the energy received from the Sun,

at the upper limit of the atmosphere in one hour, over the whole surface of the Earth.

Compare this energy with that spent on Earth in one year, for a used power (in 2009) of 15

TW.

4

The output power of a photovoltaic modulus To increase the generated power, the solar cells are connected in series and/or parallel and

encapsulated to form a photovoltaic modulus. Several modules form a panel. The output

power of a photovoltaic modulus is defined as the generated power when the incident solar

radiation is 1000 W/m2 (AM1.5G) and the temperature is of 25 Celsius. This is atypical solar

radiation value in the mid of a sunny summer day.

Example: A modulus with a surface of 1 m2, with an efficiency of 15%, produces an output

power of 150 W at noon in a sunny day.

Problem: If I have o photovoltaic array with a power of 2.2 kW and the incident solar

radiation (irradiance) is 4 kWh/m2·day, what is the generated energy in a year? (The energy is

equal to the product of power and time.)

Solution: To find the energy generated per day, we must divide the incident solar radiation at

AM1.5G, and to find how many “Suns” per day illuminate the photovoltaic array, i.e. (4

kWh/m2·day)/1000 W/m2 = 4/day, and then we calculate the energy generated per day as E =

(2200 W)×4/day = 8.8 kWh/day = 3200 kWh/year.

To calculate the area of the panel that produces a certain amount of energy per year for

a given conversion efficiency, we use the formula:

necessary energy (kWh/year) area (m2) = —–————————————————————— (4) [solar resource (kWh/m2/year)×conversion efficiency]

A typical solar cell has a diameter of 5 cm, a thickness of 1 mm, and (if fabricated from Si)

generates 0.5 V.

Example: Aria of the panel that should produce an energy of 3.5·1013 kWh/year using an

array of solar cells with a conversion efficiency of 12% for an irradiance of 6 kWh/m2·day

should be 1.3·105 km2.

In the solar cell, the voltage )/1ln()/( sLBoc IIeTkV += β , which corresponds to I = 0,

i.e. to the case when there is no current flow (the circuit is open), is generated by separating

the photogenerated electrons and holes in the built-in electric field of a p-n junction. The

open-circuit voltage cannot be higher than and is determined by the difference between gE

5

the Fermi quasi-levels in the n and p regions, which are no longer the same at illumination. If

the junction is connected in a circuit which contains a load , a photocurrent appears and an

electric power is generated on . An illuminated p-n junction and its

sR

sR VI − characteristic are

represented in the figures below, on left and center, and right, respectively.

VVcd

I Isc

Vm

Im

P

eVoc

l

Although in the characteristic VI −

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−= 1exp

TkeVIII

BsL β

(5)

the photocurrent is negative, it is common in practice to consider it as positive (as in the

figure above); the sign of the current is reversed. The p-type region is called base and the n-

type is referred to as emitter; the electrons are transported by the built-in electric field

(oriented from the n to the p region) from the n region to the p-type region. At illumination

the current is negative since it has an opposite sign to the current induced by a voltage V; the

electrons and holes drift in an opposite direction to that imposed by diffusion. The ideality

parameter 21 ≤≤ β describes the current mechanism in the junction: β = 1 corresponds to the

case in which the current is dominated by diffusion (see the demonstration in the first part of

the course), and β = 2 describes the situation in which the transport of charge carriers is

mainly due to drift. The short-circuit current in the solar cell Lsc II −= (at V = 0) depends on

the number of electrons excited in the conduction band, which is proportional to the incident

solar radiation, while the open-circuit voltage is almost independent of the incident energy.

To increase the generated power (equal to the product between the current and voltage

in the working point), one can use photovoltaic modules that produce the desired voltage and

current (see the figure below). The current increases if the area of the solar cell increases or if

the incident of the solar radiation is enhanced (is concentrated) and, if the solar cells are

6

connected in series, the current remains the same and the voltage increases. On the other

hand, for solar cells connected in parallel the voltage is the same and the current increases.

The characteristic is still exponential, at least in the first approximation, also for solar

cells based on heterojunctions.

VI −

Example: If a solar cell generates 0.5 V (a typical value for Si-based cells), to obtain a

module of 12 V we need to connect in series 24 solar cells.

An array of solar cells (or photovoltaic array) can contain up to several thousands

photovoltaic modules, organized in solar panels (a solar panel is a group of photovoltaic

modules). Such panels can generate direct currents with powers between several watts to tens

of megawatts, and can be used to recharge a load that can be anything from a computer

battery to a communication system in a building or a town. When a photovoltaic array is

connected to a utility network, this array must first be connected to an inverter that transforms

the direct current into an alternative current. Such inverters have efficiencies of 90% and

generate sinusoidal currents with very few distortions or higher harmonics.

The maximum conversion efficiency of solar cells The maximum generated power or conversion efficiency of a single solar cell containing a p-n

junction is estimated assuming that the solar cell is in thermodynamic equilibrium with the

environment and taking into account the detailed balance of the possible processes (this is the

Shockley-Queisser theory). To obtain a maximum conversion efficiency we suppose perfect

absorption of photons with energy , which create exactly one electron-hole pair,

perfect collection of the photogenerated carriers (i.e. equality between the collected electron-

hole pairs at electrodes and the number of generated electron-hole pairs), and only radiative

recombinations (in general, non-radiative recombinations exist also) of electrons and holes,

gEE >

7

which generate photons. Under these circumstances, the photocurrent density (current per unit

area) generated at illumination with a flux of photons with energy E is

∫∞

Φ==gE

SLL dEEeAIj )(/ (6)

where

1)/exp(12/),( 32

2

−==Φ

SBSS TkEhc

EETEI π (7)

is the photon flux at the temperature of the Sun. It was considered that only photons with

energy higher than generate electron-hole pairs in a semiconductor with direct energy

bands in which the dominant absorption mechanism is the band-to-band absorption.

gE

At thermodynamic equilibrium, each process in the solar cell is in equilibrium with the

inverse process. More precisely, the photon flux to and from the solar cell is equal if the cell

and the environment have the same temperature. Modeling a solar cell as a blackbody at

temperature T, the flux of photons with energy E emitted by a cell at voltage V is given by the

Planck law:

)/exp()(1]/)exp[(

12),( 23

2

TkeVETkeVEch

EEV BcB

Φ≅−−

=Φπ (8)

where it was assumed that and TkeVE B>>−

)/exp(21)/exp(

12)( 23

2

23

2

TkEchE

TkEchEE B

Bc −≅

−=Φ

ππ (9)

is the cell radiation at V = 0. This photon emission must be caused by a recombination current

with density

∫∞

Φ=gE

rec dEEVej ),( . (10)

8

The photons created as a result of recombination of electrons in the conduction band with

holes in the valence band of a semiconductor with bandgap have energies higher than

. But in thermodynamic equilibrium the total current should vanish, so that the

recombination current must be equal to the photocurrent caused by absorbing the blackbody

radiation of the environment. Thus, the density of dark current in the presence of a voltage is

gE

gE

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛Φ= ∫

∞

1exp1exp)(Tk

eVjTk

eVdEEejB

sBE

cg

(11)

and, at illumination

LB

s jTk

eVjj −⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1exp . (12)

The characteristics, although deduced from different considerations, has the same

exponential form calculated previously. This exponential form is therefore much more general

and encountered not only at p-n junctions but also at heterojunctions and p-i-n structures.

VI −

The maximum voltage that can be obtained is that at which the solar cell emits as

many photons as it absorbs, case in which there is no net energy transfer and no net current,

i.e. the solar cell is in an open circuit. So, the maximum voltage is given by

, and the conversion efficiency of the solar cell is )1/ln()/( += sLBoc jjeTkV

APFFVj

PP

rad

ocsc

rad

m

/==η (13)

where FF is the filling fraction and is the power of the incident solar radiation. From the

above formulas it follows that the conversion efficiency depends on the bandgap and has a

maximum value of about 33% for AM1.5G, which corresponds to = 1.4 eV. (The

maximum value is obtained by varying the temperature T of the solar cell.) The influence of

the bandgap on the conversion efficiency is weak in the interval 1−1.45 eV. For higher

values the open-circuit voltage increases (the built-in electric field increases) but the fraction

of non-absorbed photons (with energies smaller than the bandgap) becomes too large, while

for smaller the photocurrent increases, since a larger number of photons can generate

radP

gE

gE

gE

9

electron-hole pairs, but the open-circuit voltage and the efficiency drop because the built-in

field in the junction diminishes. Extrapolating, all photons could participate at photocurrent if

= 0, but the conversion efficiency would vanish because the photoinduced voltage is zero.

Also, η = 0 if = ∞ (no current can pass through the junction). In the figure below we have

represented the dependence of the maximum efficiency, of the open-circuit voltage and of the

short-circuit current on the semiconductor bandgap at T = 300 K and AM1.5G.

gE

gE

Mechanisms of power loss in solar cells Typical solar cells fabricated from p-n junctions in a single material, for example Si, have

conversion efficiencies of solar energy of only 15−20%, although the maximum efficiency

should be of about 30%. What are the causes of losses in energy conversion and how can

these be minimized?

The main cause of losses for solar cells containing a p-n junction in a single material,

which limits theoretically the conversion efficiency of solar energy, is the narrow spectral

band of photons that generate electron-hole pairs. More exactly, for Si, for which = 1.12

eV, for an incident solar radiation with a power of 100 mW, 21 mW are not absorbed

(because ) and 31 mW is the excess energy corresponding to photons with ,

which is transformed in heat (see the figures below).

gE

gEE < gEE >

10

As discussed in the previous section, in a solar cell from a single material the optimum

situation corresponds to a bandgap of = 1.4 eV, for which the maximum efficiency is 31%

for one sun (with no concentrators).

gE

In addition, the conversion efficiency decreases because not all incident photons with

high enough energy generate electron-hole pairs. The generation efficiency is characterized

by the internal quantum efficiency of the material, 1<ciη , defined as the ratio between the

output electrons and the absorbed photons, which depends strongly on the wavelength of the

incident radiation (see the figures below: left for GaInP, with = 1.8 eV, right for CIGS,

with = 2.4 eV); this parameter has typical values of 80−95%. The photocurrent density for

an incident radiation with broad spectrum and a flux of photons with energy E denoted as

is

gE

gE

)(EΦ

∫Φ= dEEEej ci )()( η . (14)

Other loss mechanisms, which reduce the conversion efficiency, are thermalization of

the crystalline lattice (process 1 in the figure below), voltage losses at the junction (process

2), voltage losses at contacts (process 3), and recombination losses (process 4 in the figure

below).

11

The thermalization of the crystalline lattice, i.e. the temperature increase due to collisions of

energetic carriers with phonons (quanta of vibrational motion of the crystalline lattice),

reduces the conversion efficiency with (typically) 0.1%/°C, so that, if the efficiency is 16% at

25°C, it becomes 9% at 100°C. An increase in temperature leads to a slight increase in the

short-circuit current and to significant decrease of the open-circuit voltage. The recombination

losses can be minimized by limiting the thicknesses of the n and p layers to the diffusion

lengths of the carriers. The surface recombinations can be reduced by introducing a

passivation layer (see the figure below), which decreases the concentration of minority

carriers at the surface. (In the first part of the course we have seen that in a p-n junction the

current is due to minority carriers in excess!) The passivation layer can be a strongly doped

layer, another semiconductor with a higher energy bandgap, or both. Irrespective of the case,

the energy bands bend.

Example of energetic balance for a Si solar cell: from the incident power of 100 mW,

21 mW are not absorbed, and 31 mW constitute the excess energy. We are left with 48 mW.

If the open-circuit voltage is 0.6 V (0.7 V), the maximum current that can be generated is

given by 48 mW/1.12 V = 44 mA. This maximum current corresponds to the case when all

photons, with a number of 48 mW/( gE=ωh ) = 48 mW/1.12 eV, are converted in electricity.

Because of photon reflection at the surface, shadow of electrodes, incomplete absorption, and

incomplete charge collection, the short-circuit current is of only 28 mA (41 mA). For a filling

ratio of 75% (80%), the generated power is then only 14 mW (24 mW).

12

From the example above it follows that it is desirable to have an as-large-as-possible

filling ratio. For the same and , the filling fraction increases if the characteristic

has a larger plat region (the characteristic in the figure below, left, has a higher filling

ratio than the device in the figure below, right).

scI ocV VI −

VI −

I

V

Isc

Voc

Im

Vm

I

V

Isc

Voc

Im

Vm

The form of the characteristic depends on the series and shunt resistances of the

device. The series resistance must be small and the shunt resistance must be high for

efficient devices. The series resistance includes the contribution of all resistive elements in the

device, of the ohmic contacts and of the semiconductor/contact interface. The shunt resistance

represents the fraction of generated electric energy that is lost through recombination at the p-

n junction or at the surface of the solar cell. The

VI −

sR shR

VI − characteristic in the presence of these

resistances and for an ideality factor β is

Lsh

s

B

sS I

RIRV

TkIRVeII −

−+⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −= 1)(exp

β (15)

The equivalent circuit of the non-ideal solar cell (with finite and ) is illustrated in the

figure below, left. is the dark current. The influence of and on the

sR shR

intI sR shR VI −

characteristic is represented in the figure below, center and right, respectively.

Rs

Rsh IL Iint V

+

-

I

V

Rs increases

I

V

Rsh decreases

13

Methods of optimizing the conversion efficiency From the previous example it follows that some losses can be minimized by an adequate

design of solar cells. Because the power density of the incident solar radiation is constant, to

enhance the conversion efficiency

APFFVj

PP

rad

ocsc

rad

m

/==η (16)

we must increase the short-circuit current, the open-circuit voltage and/or the filling ratio.

Because

)/ln()/()1/ln()/( sLBsLBoc jjeTkjjeTkV ββ ≅+= , (17)

The open-circuit voltage increases if the short-circuit current (the photocurrent) increases or if

the saturation current decreases. For predominantly diffusive transport (see the first part of the

course) this last parameter is given by

)//( pnpnpns LpeDLneDj += . (18)

For instance, if β = 1, for each 10 times increase of the ratio , increases with 60

mV = . If increases twice, the open-circuit voltage enhances with 18 mV

= . We will discuss further below methods for increasing the photocurrent. To

decrease the saturation current we must have large diffusion lengths

sL jj / ocV

)10ln()/( eTkB Lj

)2ln()/( eTkB

nnn DL τ= ,

ppp DL τ= and small diffusion constants , , i.e. large carrier lifetimes nD pD nτ , pτ . On

the other hand, the concentrations of minority carriers should be small, requirement that

implies a large diffusion potential, easily implemented in heterostructures.

The series and shunt resistances influence the filling ratio. The first one can be

decreased and the second increased by reducing the losses in the solar cell and by using ohmic

contacts, which affects especially the series resistance. Besides improving the filling ratio, a

small series resistance, minimizes also the (Joule) heat dissipated in the material, increasing

the conversion efficiency.

14

Types of electric contacts There are two types of contacts between a semiconductor material and a metal: ohmic and

Schottky. At the metal/semiconductor interface, as at the interfaces between n and p regions

in a junction, the energy bands bend due to charge redistribution at the interface. The

curvature is more significant in the semiconductor (see the figure below, left), which has a

smaller carrier concentration. If an electron from the conduction band of the semiconductor

has to pass over a potential barrier in order to reach the Fermi level in the metal (which is the

same as in semiconductor, at equilibrium), the contact is called Schottky. The VI −

characteristics of a Schottky contact is exponential (see the figure below, right), as in a p-n

junction, in which the carriers must overcome the diffusion potential. On the contrary, if the

electrons in semiconductor do not encounter a potential barrier, the contact is ohmic and the

corresponding characteristic is linear. A small series resistance of the solar cell implies

ohmic contacts, except the case in which the contacts are used for generating a built-in

electric field that separates the carriers. For a given semiconductor, a metal forms an ohmic or

a Schottky contact depending on its workfunction

VI −

meφ (defined as the energy necessary for an

electron on the Fermi level to reach the vacuum state, i.e. to extract it out of the metal). If this

workfunction is higher than the energy needed to extract an electron in vacuum, out of the

conduction band of the semiconductor, χe , the contact is Schottky. Otherwise, it is ohmic. A

metal can have ohmic contacts with some semiconductors and Schottky with others.

15

Methods to increase absorption The photocurrent is produced by electron-hole pairs generated after photon absorption and

collected by the electrical contacts. To increase the photocurrent we can increase the number

of absorbed photons or decrease the number of recombination processes.

From the expression of light intensity inside the semiconductor

)exp()1()( 0 xIRxI α−−= (19)

where α is the absorption coefficient, it follows that the absorption increases if the reflection

coefficient R decreases, the incident radiation increases, or the pathlength of light inside

the semiconductor increases.

0I

To decrease the reflection coefficient one may use antireflection coatings, which

consists from successions of thin layers with thicknesses (usually) of 4/λ from materials

with different refractive indices. These coatings are necessary because Si, for example, with a

refractive index of = 3.46, reflects almost 30% at normal incidence. The reflection

coefficient at the air/semiconductor interface at normal incidence is given by

sn

2

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=saer

saer

nnnnR (20)

where is the refractive index of the air, and of the semiconductor. In the figure

below we have illustrated the decrease of reflectivity with the increase in the number of

antireflection coatings. Note that the decrease of R occurs only in a spectral interval.

1≅aern sn

16

Problem: Find the reflection coefficient at normal incidence at the interface between air and a

semiconductor with an index of refraction of a) 2.5, b) 3, c) 3.5.

In addition, to allow the penetration in the neighborhood of the solar cell junction of a

larger fraction of the incident light, the electrical contact must be interdigitated, as in the

figure below, left, or transparent, as in the figure below, right. Transparent contacts are

usually fabricated from oxides with a high electrical conductivity, such as ZnO, SnO2, or ITO

(indium tin oxide). The top illuminated electrode is not metallic, because metals reflect

strongly in the visible part of the spectrum; the bottom (not illuminated) contact is, however,

metallic, its high reflectivity increasing the light absorption in the material because the

reflected photons are redirected toward the active region. The contact electrodes could be

placed also on the sides of the device, but in this case the electrons should travel along longer

distance to reach the contacts, which worsens the conversion efficiency due to recombination

processes.

Another method for increasing the number of absorbed photons is the concentration of the

solar energy, i.e. the increase of the intensity. This method (see the figure below) is based

on the focalization of the incident solar energy on a smaller surface, on which the solar cell is

placed. The concentrators are characterized by the ratio

0I

incc IIC /= (21)

17

where is the intensity on the solar cell and is the incident solar intensity. For weak

concentrators of the solar energy (2−10 suns) the device need not be cooled and one can use

standard Si cells, the cooling being mandatory for intermediate (10−100 suns) and strong

concentrators (> 100 suns). In the last case, a perfect alignment of the system perpendicular to

the incident solar radiation is required and multijunction solar cells (with efficiencies up to

41% and high currents) are generally used. Although the concentrators increase the power

generated by the solar cell, a decrease of associated with a temperature increase is

observed for strong concentrators. Thus, an optimum concentration factor of (typically) few

hundred suns exists for each solar cell.

cI incI

ocV

Light concentration can be achieved through several methods. In the figure below, left,

a parabolic mirror is represented, which can concentrate light in its focal point. Such a

parabolic mirror with a height of 8-storey building has been built in 1969 at Odeillo in the

French Pyrenees (see the figure below, center) and used as solar furnace in which

temperatures can reach 3800°C. Such temperatures are rather useful for direct conversion of

solar energy in thermal energy. For light concentration on solar cells the parabolic mirrors are

arranged as arrays, the solar cells being placed in the focal points (see the figure below, right)

18

Light concentration can be accomplished also with the help of planar Fresnel lenses

(see the figure below, left; at right it is represented a common convergent lens). Because the

light rays in a common lens are refracted especially by the concave surface, the same

focalization effect can be achieved if layers with thicknesses equal to a multiple of πλ2 are

eliminated from the lens. In this way one obtains planar Fresnel lenses, which are lighter and

in which the losses due to light absorption in the lens material are significantly reduced.

Planar Fresnel lenses are used also in lighthouses.

Another method to increase the number of absorbed photons is based on the

enhancement of the distance traveled by light in the material. This can be achieved in

luminescent concentrators (see the figure below) that consist from transparent layers of

polymers doped with luminescent atoms or molecules, or with quantum dots.

The incident solar light is absorbed by the luminescent substances and re-radiated

subsequently in arbitrary directions (preferable, with high efficiency) such that a fraction of

the incident light is captured in the layer (it cannot exit the layer due to total internal

reflection at the polymer/air interface, which occurs if the incidence angle on the surface is

higher than a critical value). In this way, the pathlength of light to the solar cell placed on the

side increases and the spectrum of the incident light can be better adapted to the energy gap of

the semiconductor in the solar cell since the re-directed photons have a smaller energy than

19

the incident ones. The planar luminescent concentrator is especially efficient for diffuse light,

which is particularly hard/impossible to concentrate with concentrators based on light

reflection or refraction.

Alternatively, the pathlength of photons in the absorbing material is increased by

structured surfaces (see the figure below, left) in which the absorption is significant even in

thin layers due to the trapping effect caused by multiple reflections. In the figure below, right,

it is illustrated a nanostructured Si layer. This solution is especially suited in materials in

which the probability of electron-hole pair recombination is high and it is therefore necessary

to generate photocarriers as close as possible to the p-n junction.

The absorption coefficient can increase also in materials with nanometer scales

(nanosized wires or particles) in which the spatial confinement effects modify the electrical

and optical properties of these structures with respect to the properties of bulk (macroscopic)

materials. In nanometer materials the absorption coefficient is with at least one order of

magnitude higher than in the bulk material. In addition, the photon absorption increases also

due to the enhancement of the photon/material interaction surface as well as due to the

trapping effect of photons caused by multiple reflections on the parallel nanometer wires or,

respectively, by scattering processes at the air/nanometer material surface that increase the

light pathlength. Moreover, the absorption spectrum of nanowires (see the figure below, left)

and nanodots (the figure below, right) can be modified by changing their diameter.

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Influence of the semiconductor material

In general, semiconductor materials with high electron and hole mobilities (the mobility μ is

the ratio between the average carrier velocity v and the applied electric field E) are desirable

for solar cells, since the charge carriers travel along longer distances before recombination

(have higher velocities!). The average distance until recombination is given by τμτ Ev =

where τ is the carrier’s lifetime, i.e. the average time interval between the creation of the

electron-hole pair and recombination. In applications, a compromise must be reached between

the requirement that the pathlength of photons from the surface of the solar cell to the junction

is high and the requirement that this parameter is low. In the first case, the absorption is high

but also the losses caused by the recombination of the electron-hole pairs generated outside

the barrier layer are high, whereas in the second case both photon absorption and

recombination losses decrease.

The probability of collecting the electron-hole pairs is significant only in the so-called

active region, which extends on both sides of the interface across distances equal to the

respective diffusion lengths. The photocarriers generated outside the active region do not

reach the space charge region before recombination and, therefore, do not contribute to the

current. In other words, if the thickness of the absorbing layer is larger than the active region

the conversion efficiency of the solar cell decreases. The mobility is related to the diffusion

coefficient D through the relation TkeD B/=μ , and the diffusion length is defined as

τDLdif = . The lifetime τ has typical values of tens of nanoseconds for semiconductors

with direct energy bands and of few milliseconds for semiconductors with indirect bands. The

optimal thickness of a layer in which both photon absorption and recombination losses occur

depends of the product μτ and of the absorption coefficient value.

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To avoid as much as possible recombination losses, in many cases the top (illuminated) layer

has smaller charge carrier mobility. Because, in general, the holes have a smaller mobility

than the electrons, the illuminated layer is the p layer such that the holes can reach the space

charge region before recombination. On the other hand, in a heterostructure consisting of two

semiconductors with energy gaps and the layer with a higher energy gap (layer 1 in

the figure above) is illuminated to allow photons with energy

1gE 2gE

12 gg EEE << to penetrate

deeper in the solar cell before generating electron-hole pairs. (As can be seen from the figure

above, unlike in p-n junctions, in heterojunctions both conduction and valence bands are

discontinuous at the interface.) The semiconductor with a higher energy gap acts as window

for photons and as emitter for electrons. The recombination of charge carriers is lower (the

solar cell is more efficient) in heterojunctions between materials with the same doping type (n

or p). To increase photon absorption in heterostructures, the emitter/window is thin and the

base (the layer in which the absorption takes place, layer 2 in our case) is thick.

Thus, the solar cell performance is influenced by material parameters and, in

particular, by the μτ product. The table below summarizes the parameters of some

semiconductor materials used for solar cell fabrication measured at room temperature, and the

maximum efficiency of the corresponding solar cells. The most efficient solar cells with a

single p-n junction have been fabricated from III-V semiconductor compounds (for example,

GaAs), but these materials have a much higher cost compared to Si, and are therefore utilized

only together with light concentrators.

22

23

Configurations of solar cells with inorganic materials The mobility and lifetime values determine the configuration of the solar cell. The p-n

junctions, used in solar cells based on crystalline Si, or the heterojunctions between

semiconductor materials with different properties, such as those used in thin film solar cells

(for example Cu(In,Ga)Se2 or CdTe) do not exhaust the possible configurations. Another

configuration commonly used in solar cells is the p-i-n structure, consisting of a succession of

three layers: the first p doped, the second undoped/intrinsic i, and the third n doped.

Irrespective of the configurations of the semiconductor layers, the solar cells are in the

superlayer configuration if the light penetrates first the substrate.

A solar cell consists in general from layers that differ in energy bandgap , doping

and carrier concentrations. Irrespective of the configuration, the solar cell must fulfill three

functions:

gE

(1) at least one layer must absorb photons and generate free or bound electron-hole pairs (in

the last case the pairs are called excitons and are especially encountered in organic

semiconductors). This layer must have a suitable and a large absorption coefficient. gE

(2) the carriers must be collected, so that the electrons and holes reach separate contacts in

order to produce a photocurrent. In other words, we need selective contacts.

(3) capacitive elements or regions with net electric charges must exist in the solar cell in order

to have a built-in potential difference.

In solar cells with p-n junctions or heterojunctions the light is absorbed in one layer

and the junction has a double role: as selective element for electrons and holes, and as

generator of the built-in electric field. The gathering of charges is limited by the diffusive

transport of the minority carriers toward the junction. In this case, the system is linear in the

first approximation and the collection efficiency of minority charges does not significantly

depend on voltage.

In the p-i-n structure, the absorbing layer is i, and the selectivity of the contacts is

realized through very thin and usually highly doped p and n regions, which generate a built-in

electric field. The charge transport is no longer diffusive, but controlled by the built-in field.

This configuration is suitable for materials with small mobility. In this case the thickness of

the i layer is determined by the μτ product and by the absorption coefficient, typical

optimum values of the thickness of this layer being of few hundreds of nm, to 1 μm. In the

presence of an external voltage the built-in field decreases and the field-dependent charge

transport depends on the voltage, the p-i-n type solar cell being strongly nonlinear. In the

24

latter case the charge gathering efficiency and the photocurrent depend on the voltage if the

mobilities are small enough. In the figure below, left, we have represented the p-n junction,

while the p-i-n configuration is illustrated in the center. (The structures are called p-i-n or n-i-

p depending on the deposition order of the layers.) In the first approximation the VI −

characteristic of the p-i-n structure is also exponential, but the saturation current has another

expression.

There is also the flat-band configuration, in which the charge separation occurs due to

sufficiently-high discontinuities in the energy bands (see the figure above, right). There is no

need for a built-in field V , but charge gathering is poor. In this case there is no capacitive

element with an associated voltage difference, which does not affect charge gathering.

bi

As will be seen in the third part of the course, both cost-related considerations and the

conversion efficiency lead to the development of solar cell types based on materials or

configurations different from the simple p-n junction, p-i-n, or heterojunctions considered so

far (see the figure below). In the next part we will discuss briefly these types of solar cells.