Semiconductor/ Semiconductor
p-n junctions
Dr. Katarzyna Skorupska
Space charge regions in semiconductors
flatband Depletion Inversion Accumulation
1. semiconductor – metal
Schottky contact
2. semiconductor – semiconductor
p-n junction
homojunction (p-Si : n-Si) , heterojunction
3. semiconductor - electrolyte
Schottky like contact
Space charge layer
Leads to spatial separation of charges minority carriers are driven to the surface by
electric field
Field acceleration impacts excess energy to both carriers
semiconductor – semiconductor
p-n junction
homojunction (p-Si : n-Si) , heterojunction
Contact potentials and space charge layers
With the Ansatz that the charge is distributed evenly with x (homogenous doping) one considers the relation of :
charge,
electric field,
electrostatic potential
and energy:
Poisson´s equation connects charge and potential:
0
Here, since d 0, which holds for homojunctions, we have set Y and continue to use the latter from now on.
+ + + + + +
-
-
-
-
-
x
donors
acceptors
neutral neutral -Wp
Wn
p-type n-type
– Galvani potential
y– Volta potential (electrostatic)
d – surface dipole changes
– charge density
Δ – LaPlace operator
dy
+ + + + + +
-
-
-
-
-
x
donors
acceptors
neutral neutral -Wp
Wn
p-type
pAqN 0 xWp
0 pWx
nDqN nWx 0
0 xWn
n-type
Wn,p - spatial limit of charged areas
First integration of φ with respect to x
d2j
dx2= -
rnene0
= -qND
ene0
, r = -qNDd2j
dx2= -
rp
epe0
= -(-qNA )
epe0
, rp = qNA
with E’ as electric field:
dx
dE
dx
d
dx
d
dx
d
dx
d
dx
dx
gradE
'
)('
'
2
2
2
2
The first integral yields the electric field since E’= -grad φ
p-type n-type
-dE '
dx=qNA
e pe0
dE '
dx= -qNA
e pe0
E '(x) = -qNA
e pe0
ò dx
E '(x) = -qNA
e pe0
x +C '
for -Wp £ x £ 0
-dE '
dx= -qND
ene0
dE '
dx=qND
ene0
E '(x) =qND
ene0
ò dx
E '(x) =qND
ene0
x +C
for 0 £ x £Wn
-Wp Wn
for x = 0
E '(x) = -qNA
epe0
x +C '
E '(x) =C '
forx = 0
E '(x) =qND
ene0
x +C
E '(x) =C
-Wp Wn
p-type n-type
For x=0 the electric field attains its maximum value.
p-type n-type
p
p
A
p
p
A
p
A
p
WqN
C
CWqN
CxqN
xE
xEWxfor
0
0
0
'
'0
')('
0)('
n
n
D
n
n
D
n
D
n
WqN
C
CWqN
CxqN
xE
xEWxfor
0
0
0
0
)('
0)('
-Wp Wn
The integration constant is determined by the boundary condition that E’(x) vanishes outside the charged region
p-type n-type
p
p
A
p
p
A
p
A
p
p
A
p
A
p
p
A
p
A
p
WxqN
xE
WqN
xqN
xE
WqN
xqN
xE
WqN
C
CxqN
xE
xWfor
0
00
00
0
0
)('
)('
)('
'
')('
0
n
n
D
n
n
D
n
D
n
n
D
n
D
n
n
D
n
D
n
WxqN
xE
WqN
xqN
xE
WqN
xqN
xE
WqN
C
CxqN
xE
Wxfor
0
00
00
0
0
)('
)('
)('
)('
0
-Wp Wn
Electric field is given by
Graphic integration for semiconductor pair
DqN
E '(x) =qND
ene0
x -Wn( )E '(x) = -qNA
epe0
x+Wp( )
AqN
p-type n-type
p-type n-type
0
0
)0('
0)0('
0
p
pA
p
p
A
WqNE
WqN
E
xfor
0
0
)0('
0)0('
0
n
nD
n
n
A
WqNE
WqN
E
xfor
For x=0 the electric field attains its maximum value.
nDpA
nDpA
np
nDn
pA
p
nDnn
pA
pp
nnnppp
WNWN
WqNWqN
DD
WqND
WqND
WqNE
WqNE
EDED
xsufracetheatntdisplacemedielectricD
00
00
00
)0()0(
)0()0(
)0(')0('
)0(')0()0(')0(
0
Extension of space charge layer is inversely proportional to the respective doping layer.
higher relative doping –smaller the space charge layer
p-type n-type
second derivative to know φ (electrostatic potential)
'2
1)(
)(
)()(
)()(
)()('
)(')(
)('
)(
)('
0
2
0
00
0
0
0
DxWqN
xqN
x
WqN
xqN
x
dxWxqN
x
dxWxqN
x
WxqN
xE
dxxEx
dxxE
ddxxE
dx
dxE
p
p
A
p
A
p
p
A
p
A
p
p
A
p
p
A
p
p
A
DxWqN
xqN
x
WqN
xqN
x
dxWxqN
x
WxqN
xE
dxxEx
dxxE
ddxxE
dx
dxE
n
n
D
n
D
n
n
D
n
D
n
n
D
n
n
D
0
2
0
00
0
0
2
1)(
)(
)()(
)()('
)(')(
)('
)(
)('
p-type n-type
at the surface (x=0) Galvani potential is equal zero (φ=0)
0'
'000
00
D
D
xfor
0
000
00
D
D
xfor
0' DD
xWx
qNx p
p
An
2
0 2
1)(
xWx
qNx n
n
Dn
2
0 2
1)(
The energetic position of the band edges at the surface of each material remains unaltered.
Graphic integration for semiconductor pair
DqN
n
n
D WxqN
x 0
)(
p
p
A WxqN
x 0
)(
AqN
p-type n-type
)2
1()( 2
0
xWxqN
x p
p
A
)
2
1()( 2
0
xWxqN
x n
n
D
E = ej = -qj
electric field
galvani potential
energy
Graphic integration for semiconductor pn junctions
Junction geometry and charge distribution (which material has a higher doping concentration?)
The charge profile
The electrical field across the contact (E = - d/dx)
Second integration: Galvani or electrostatic
potential
Energy E = e = -q sign change
p-type n-type
diffusion potential defined by the electric potential difference
Vp = f(0)-f(-Wp )
fp(x) =qNA
e pe0
1
2x2 +Wpx
æ
èç
ö
ø÷
Vp =qNA
e pe0
(0 + 0)-qNA
e pe0
1
2Wp
2 + (Wp × (-Wp )æ
èç
ö
ø÷
Vp = -qNA
e pe0
1
2Wp
2 -Wp
2æ
èç
ö
ø÷
Vp = -Wp
2 qNA
epe0
1
2-1
æ
èç
ö
ø÷
Vp = - -1
2
æ
èç
ö
ø÷Wp
2 qNA
e pe0
Vp =qNAWp
2
2epe0
Vn = f(Wn )-f(0)
fn (x) = -qND
ene0
1
2x2 -Wnx
æ
èç
ö
ø÷
Vn = -qND
ene0
1
2Wn
2 - (Wn ×Wn )æ
èç
ö
ø÷- -
qNA
epe0
(0 + 0)æ
èçç
ö
ø÷÷
Vn = -qND
ene0
1
2Wn
2 -Wn
2æ
èç
ö
ø÷
Vn = -Wn
2 qND
ene0
1
2-1
æ
èç
ö
ø÷
Vn = - -1
2
æ
èç
ö
ø÷Wn
2 qND
ene0
Vn =qNDWn
2
2ene0
nD
pA
DnA
ApD
p
n
pn
np
pnn
npp
p
n
p
n
D
A
pnA
npD
pA
p
n
nD
p
n
N
N
NN
NN
V
V
W
W
WW
WW
V
V
W
W
N
Nbecause
WN
WN
WqN
WqN
V
V
2
2
2
2
2
2
2
0
0
2 2
2
p-type n-type
D
nnn
DnDnn
n
n
nDn
qN
VW
qNWqNV
WqNV
0
2
0
0
0
2
2
\2
2\2
A
pp
n
ApApp
p
p
pA
p
qN
VW
qNWqNV
WqNV
0
2
0
0
0
2
2
\2
2\2
Graphic integration for semiconductor pn junctions
D
nn
nqN
VW 02
D
nn
nqN
VW 02
Important relations for pn junctions
(to memorize)
nDpA WNWN
Electroneutrality condition
D
A
p
n
N
N
W
W
Diffusion voltage relations
Dn
Ap
p
n
N
N
V
V
pn
np
p
n
W
W
V
V
The width of the space charge layer depends on:
• doping level
• voltage drop
Eg=1.12 eV
NCB=3.2 1019 cm-3
ND=1017 cm-3
Eg=1.12 eV
NVB=1.8 1019 cm-3
NA=1015 cm-3
p-type n-type
kT=26 meV
n-type p-type
Position of nEF before contact Position of pEF before contact
EF = EVB - kT lnNVB
NAEF = ECB - kT ln
NCB
ND
EF -EVB = kT lnNVB
NAECB -EF = kT lnNCB
ND
ECB -EF = 26 ln3.2 ×1019
1017meV
cm-3
cm-3
é
ëê
ù
ûú
ECB -EF = 26 ln3.2 ×102
ECB -EF = 26 ×5.7
ECB -EF =150meV
ECB -EF = 0.15meV
EF -EVB = 26 ln1.8 ×1019
1015meV
cm-3
cm-3
é
ëê
ù
ûú
EF -EVB = 26 ln1.8 ×104
EF -EVB = 26 ×9.8
EF -EVB = 254.8meV
EF -EVB = 0.25meV
ECB ECB
EVB EVB
EF
E
F 0.25 eV
0.15 eV
Contact potential difference
ECB ECB
EVB EVB
EF
E
F 0.25 eV
0.15 eV
eVC
eVC = nEF - pEF = eVn -eVp = e(Vn -Vp ) =
= Eg - (nEF + pEF ) =1.12 - (0.15+ 0.26) = 0.71eV
Changes of position of nEF and pEF after contact formation
pEF® eVpnEF® eVn eVc = nEF - pEF = eVn + eVp
Vn
Vp=NA
ND
a =NA
ND=
1015
1017=10-2 = 0.01
Vn =NA
NDVp
Vn = aVp
VC =Vn +Vp
VC = aVp +Vp =Vp(a +1)
Vp =VC
(a +1)
Vn =VC -Vp
Vn = 0.71- 0.703 = 0.007
Vp =VC
(a +1)
Vp =0.71
0.01+1= 0.703
Wn =2e0enVnqND
en =11.7
ND =1017cm-3
Vn = 0.007eV
Wp =2e0e pVp
qNA
e p =11.7
NA =1015cm-3
Vp = 0.703eV
Wn =2 ×8.85 ×10-14 ×11.7 ×0.007
1.6 ×10-19 ×1017
Wn =1.45 ×10-14
1.6 ×10-2
Wn = 0.9 ×10-12
Wn = 9.5 ×10-7
Wp =2 ×8.85 ×10-14 ×11.7 ×0.703
1.6 ×10-19 ×1015
Wp =145 ×10-14
1.6 ×10-4
Wp = 90.6 ×10-10
Wp = 9.5 ×10-5
e0 = 8.85 ×10-14[Fcm
]
q =1.6 ×10-19[C]
e0
F
cm=A × s
V ×cm
é
ëê
ù
ûú
ND[cm-3]
q[C = A × s]
W =
A × s
V ×cmV
A × s ×cm-3=A × s
cmA × s ×cm-3 = cm
Current voltage characteristic at p-n junction
For simplicity we consider:
- homojunction
- electron current
- voltage dependence of n-type side of the junctions
Absence of generation and recombination of carriers within the space charge layer
Electron current (from n-type to p-type) jnr
– number of e- on the n-type side that can thermally overcome the barrier given by
energetic distance between ECBn and ECB
p
Majority carriers (e-) on the n-type side become minority carriers on the p-type side
where they recombine.
Electron current (from p-type to n-type) jng
- thermal generation of e- in the neutral region of the p-type junction
- Drift to the n-type side
- Minority carriers (e-) on the p-type side become majority carriers on the n-type
side
r – recombination
g - generation
The recombination current jnr from n-type to p-type at the equilibrium:
-by contact potential difference Vd
jnr (Va = 0) = jnr (Vd ) = en thn(Vd ) = en thn0eeVd
kT
Va – applied potential
Vd – potential difference
vth - thermal velocity
n(Vd)- carrier concentration
n0 – concentration of e- at the bottom of conduction band (given by doping level)
Thermal excitation of e- at the p-type side
in the EVB across the Eg
jng = qn thNVBeEg
kT = qn thnp
np – e- concentration in the neutral region of ECB of p-type sc
eVd +ECB -EFC << Eg
jnr ¹ jng
Applying negative voltage (forward) to the n-type side:
- decrease of band bending
- jnr increase
- jng is not influenced
jnr (Va ) = en thn0e
e Vd-Va( )kT = jnr (0)e
eVa
kT
Va – applied potential
Vd – potential difference
vth - thermal velocity
n(Vd)- carrier concentration
n0 – concentration of e- at the bottom of conduction band (given by doping level)
jnr (0) = en thn0eeVd
kT
Applying positive voltage (reverse) to the n-type side:
- increase of band bending
- Jnr decrease exponentially with the increase barrier height
- jng is not influenced
jnr (Va ) = en thn0e-e Vd+Va( )kT = jnr (0)e
-eVa
kT jnr (0) = en thn0eeVd
kT
jng(Va ) = jnr (0) = j0eeVd
kT
jn(Va ) = j0eeVd
kT eeVa
kT -1æ
èç
ö
ø÷ = jng e
eVa
kT -1æ
èç
ö
ø÷
jD(Va ) = jn(Va )+ jp(Va )Total current:
Total e- dark current: sum of generation and recombination currents (opposite sign)
jn(Va ) = jnr (Va )- jng(Va )
jnr (Va ) = en thn0e-e Vd+Va( )kT = jnr (0)e
-eVa
kT
using:
jng(Va ) = jng(0) = - jnr (0)
jD = jng + jpg( ) eeV
kT -1æ
èç
ö
ø÷
Diode relationship by Shockley
jD = js eeV
kT -1æ
èç
ö
ø÷
js – reverse saturation current described by
metal glow emission properties jng+ jpg – diffusion constants and minority
carrier diffusion lengths
js = jng + jpg =eDpp0
Lp+eDnn0
Ln
Constant illumination- number of absorbed photons per second and cm2 mulitiled by
elementary charge
Light induced photocurrent: jL = enph(Eg )(1-R)
- p-type – photoactive part
- positive dark current under forward bias
from p-type absorber to n-type emitter
- photocurrent is opposite sign
- photocurrent does not exhibit voltage
dependent (simple approach)
Where:
jPh- photocurrent
jD- dark current
Js- dark saturation current
jL- light-induced current
nPh- number of absorbed photons per second and
cm2
R- sample reflectivity
Photocurrent – dark- and light induced current (having opposite sign)
jph = jD - jL = js eeV
kT -1æ
èç
ö
ø÷- jL
- p-type – photoactive part
- positive dark current under forward bias
from p-type absorber to n-type emitter
- photocurrent is opposite sign
- photocurrent does not exhibit voltage
dependent (simple approach)
Where:
jPh- photocurrent
jD- dark current
Js- dark saturation current
jL- light-induced current
nPh- number of absorbed photons per second and
cm2
R- sample reflectivity
45
Photocurrent
The approximation for the light induced current (jL)
jph = jD - jL = js eeV
kT -1æ
èç
ö
ø÷- jL
Photocurrent dependence follows dark-current-voltage behavior
)1()( REhenj gPhL
Where:
jPh- photocurrent
jD- dark current
Js- dark saturation current
jL- light-induced current
nPh- number of absorbed photons per second and
cm2
R- sample reflectivity
Short circuit current jL (Rext ~ 0)
Open circuit voltage VOC (R ∞)
Maximum power point MPP (largest area under jPh curve)
Current and voltage at Maximum power point jMP , VMP
Output power Pout = jMP x VMP
Solar Cell efficiency h = Pout / Pin , Pin : light intensity
Semiconductor/Metal Schottky type junctions
Dr. Katarzyna Skorupska
1
4.05 eV
Evac
Ev
Ec
EF
Wor
k fu
nctio
n
Elec
tron
affin
ity
0.2-0.3 eV
1.12 eVEg
ECB-energy of conduction band lowest unoccupied level EVB- energy of valence band highest occupied level Eg- band gap energy distance between EVB and ECB EF- Fermi level
2
Work function -is the minimum energy (usually measured in electronvolts) needed to remove an electron from a solid to a point immediately outside the solid surface (or energy needed to move an electron from the Fermi level into vacuum). Electron affinity - is the energy difference between the vacuum energy and the conduction band minimum
semiconductor – metal Schottky contact
Thermionic interaction - Contact formation based on energetic considerations - Interfacial effects neglected
4
semiconductor metal
EFsc
EFm
mobile nature of charges
under contact formation development of electrical field Potential drop across interface
EFsc EF
m
redistribution of charges on the metal side
Metal - semiconductor Schottky contact (rectifying semiconductor-metal junction)
Definition: contact potential difference
∆Εc = EFSC – EF
M = ΦM - ΦSC
The junction is characterized by • the semiconductor and metal
work function (ΦSC- given by doping) • the semiconductor electron
affinity and its energy gap.
Contact formation (ideal case: absence of surface states) Consider a neutral but doped (n-type) semiconductor and a metal with higher work function before contact:
5
Schottky junction formation
Consider a n-type semiconductor-metal contact where the work function of the metal is higher: a macroscopic gap between the phases decreases successively until contact;
connected by a conductive wire : equilibrium formation
Metal (high e- concentration -> electrostatic field at top most layer (0.1Å) - the potential drop can be neglected
Electrostatic effects are restricted to the SC side
contact energy difference eVc drops exclusively across the interlayer gap d (the vacuum level course)
Schottky junction formation cont´d Definitions: barrier height and band bending; relation between them:
Barrier height defines in Schottky (photo)diodes the reverse saturation current as will be shown below.
Lowered distance d - lowered energetic drop across the interlayer eVc
(1) - Partial contact energy difference in the SC eVc
(2)
Distance d=0 - difference in EF
M and EFSC drops completely in the SC space
charge region
Barrier height – energetic barrier e- have to overcome to enter the other phase. ΦBh- energetic distance between EF
M (after contact formation EF
M=EFSC) and the band edge ECB
ΦBh = eVbb + ECB - EF
n
Short circuiting semiconductor and metal and decreasing their distance: • electrons flow from the semiconductor to the metal the metal becomes negatively charged,
the semiconductor positively
• at small, finite distance, the contact potential VC drops across the air gap and the
semiconductor surface region
• the relative distribution of VC follows where CM and CSC denote metal and semiconductor capacitance, respectively
Schottky Junctions
The electron depletion of the semiconductor during contact formation leads to a charged region near the surface;
8
Schottky barrier: Decreasing the gap to zero:
Origin of the spatial dependence of the energy bands and the vacuum level: Poisson´s equ. in 1dimension
• the contact potential drops almost exclusively across the semiconductor near surface region
(depending on doping and contact potential difference, i.e. extension of the charged region). • BARRIER HEIGHT (Φbh): energetic
barrier which metal electrons have to overcome (thermally) to reach the semiconductor.
• Band banding (eVbb)
9
C - capacitance; Q - charges on the plates; V - the voltage between the plates; A - area of overlap of the two plates; εr - relative static permittivity (sometimes called the dielectric constant) of the material between the plates (for a vacuum, εr = 1); ε0 - electric constant (ε0 ≈ 8.854×10−12 F m–1); d - separation between the plates.
Space charge regions in semiconductors
flatband Depletion
Space charge regions in semiconductors II
Inversion
Space charge regions in semiconductors III
Inversion Accumulation
forward current – from n-SC to metal reverse current – from metal to n-SC
Dark current – influence of applied voltage to determine electron currents from: • semiconductor to metal (forward current) • metal to semiconductor (reverse current) to find the expressions for the currents based on the thermionic emission model for an applied voltage (forward and reverse currents)
n-semiconductor
interface barrier
metal
14
15
The thermionic emission model
(i) the barrier height is much larger than the thermal energy (Φbh >>kT),
(ii) thermal equilibrium exists in the plane of emission (x =0) and (i) non-degenerate semiconductors
• the band edge positions at the surface (x=0) remain unaltered hence the barrier height does not change
• the (cathodic) voltage reduces the band bending • the Fermi levels on both sides of the junction are different
The dark current from semiconductor is given:
16 νth- thermal velocity
17
Expression for the forward current (SC to M)
using the Boltzmann exponential term
The voltage dependence of the current density is given by the energetic shift of the Fermi level EF(0) to EF(V) using EF(V) = EF(0) + eVc one obtains for the (increased) carrier concentration at the semiconductor surface:
the forward current is given:
The expression ECB-EF(0) represents the barrier height of the junction (Φbh). The forward dark current density can then be expressed in terms of the barrier height and applied voltage:
18
19
The expression of the thermal velocity (vth) and the effective density of states (DOS) at the conduction band edge (ECB) by their dependence on temperature and effective electron mass (m*
e)
thermal velocity effective DOS
using the expression for the effective Richardson constant which describes the glow-emission properties of a material
one obtains the equation for the dark current in forward direction
20
Current from metal to semiconductor (reverse current)
In the equilibrium situation considered for V=0 The forward current (SCM) must be equal and opposite in sign to the reverse current (MSC) Therefore:
21
To ta l c u r r e nt
js
The pre-factor called reverse saturation current (js) • contains material properties • temperature • and gives the current at V=0
kTs
bh
eTAjΦ
−= 2*
DIODE CHARACTERISATION
22
DIODE CHARACTERISATION
question: which sign for voltage and current for an n-type semiconductor-metal junction?
23
Photocurrent
The approximation for the light induced current (jL)
Photocurrent dependence follows dark-current-voltage behavior
)1()( REhenj gPhL −⋅>= ν
Where: jPh- photocurrent jD- dark current Js- dark saturation current jL- light-induced current nPh- number of absorbed photons per second and cm2 R- sample reflectivity Short circuit current jL (Rext ~ 0)
Open circuit voltage VOC (R ∞)
Maximum power point MPP (largest area under jPh curve)
Current and voltage at Maximum power point jMP , VMP
Output power Pout = jMP x VMP
Solar Cell efficiency η = Pout / Pin , Pin : light intensity
24
Illumination of the semiconductor with photons of energy greater than Eg, - accumulates the electrons in semiconductor side and - holes in the metal side of the depletion region. There occurs an electron-hole pair generation. The light splits the Fermi level and creates a photovoltage V, equal to the difference in the Fermi levels of semiconductor and metal far from the junction.
Vph
At open-circuit voltage (VOC) the photocurrent is equal zero IPh=0
26
Photovoltage (Vph) is given by jph=0
+== 1ln
s
LOCPh j
jq
kTVV
the photovoltage changes logarithmically with the light intensity
27
VPh = 0.74V
+== 1ln
s
LOCPh j
jq
kTVV
Example: