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

p-type
and energy:
neutral
n-type
donors
++++++
-Wp
acceptors
neutral
x
-
Wn
– Galvani potential
y– Volta potential (electrostatic)
d – surface dipole changes
– charge density
Δ – LaPlace operator
Poisson´s equation connects charge and potential:
  
 y  d

 0
Here, since d  0, which holds for homojunctions, we have set Y   and continue to use the latter from now on.
Wn,p - spatial limit of charged areas
p-type
n-type
donors
neutral
acceptors

p  

 qN A
0
++++++
-Wp
neutral
-
 Wp  x  0
  x  Wp
x
Wn

qN D

0
n  
0  x  Wn
Wn  x  
First integration of φ with respect to x
rp
dj
(-qN A )
=
=
, r p = qN A
2
dx
e pe0
e pe0
2
with E’ as electric field:
d 2j
rn
qN D
=
=
, r = -qN D
2
dx
ene0
ene0
E '   grad 
d
' ( x )  
dx
d 2 d  d 
 

2
dx
dx  dx 
d 2
dE '


dx 2
dx
The first integral yields the electric field since E’= -grad φ
p-type
n-type
-Wp
dE ' qN A
=
dx e pe 0
-
dE '
qN
=- A
dx
e pe 0
E '(x) = - ò
E '(x) = -
qN A
e pe0
qN A
e pe 0
Wn
dx
x +C'
for - Wp £ x £ 0
dE '
qN
=- D
dx
e ne 0
dE ' qN D
=
dx e ne 0
qN D
E '(x) = ò
dx
e ne 0
E '(x) =
qN D
e ne 0
x +C
for 0 £ x £ Wn
p-type
-Wp
for x = 0
E '(x) = -
n-type
qN A
e pe 0
E '(x) = C '
x +C'
Wn
forx = 0
qN D
E '(x) =
x +C
e ne 0
E '(x) = C
For x=0 the electric field attains its maximum value.
p-type
n-type
-Wp
for x  W p  E ' ( x)  0
E ' ( x)  
0
qN A
 p 0
 p 0
C'  
qN A
x  C'
Wp  C '
qN A
 p 0
Wp
Wn
for x  Wn  E ' ( x)  0
E ' ( x) 
0
qN D
 n 0
C
qN D
 n 0
xC
Wn  C
qN D
 n 0
Wn
The integration constant is determined by the boundary condition that E’(x) vanishes outside the charged region
p-type
n-type
-Wp
for  W p  x  0
E ' ( x)  
C'  
qN A
 p 0
qN A
 p 0
for 0  x  Wn
x  C'
E ' ( x) 
C
Wp
 qN A


E ' ( x)  
x 
Wp 
  

 p 0
p
0


qN A
qN A
E ' ( x)  
x
Wp
qN A
E ' ( x)  
 p 0
 p 0
qN A
x  W 
 p 0
Wn
p
qN D
 n 0
qN D
 n 0
xC
Wn
 qN D

E ' ( x) 
x   
Wn 
 n 0
  n 0

qN D
qN D
E ' ( x) 
x
Wn
qN D
E ' ( x) 
Electric field is given by
 n 0
 n 0
qN D
x  Wn 
 n 0
Graphic integration for semiconductor pair
p-type
 qN A
E '(x) = -
qN A
e pe0
( x +Wp )
n-type
qN D
E '(x) =
qN D
ene0
( x -Wn )
n-type
p-type
for x  0
E ' (0)  
E ' (0)  
qN A
 p 0
0  W 
qN AW p
 p 0
p
for x  0
qN A
0  Wn 
E ' (0) 
 n 0
E ' (0) 
qN DWn
 n 0
For x=0 the electric field attains its maximum value.
D  dielectric displaceme nt at the sufrace x  0
D p (0)   p E ' p (0)
Dn (0)   n E 'n (0)
E ' p (0) p  
D p (0)  
qN AW p
E 'n (0) n  
0
qN AW p
0
Dn (0)  
qN DWn
0
qN DWn
0
D p (0)  Dn (0)

qN AW p
0

qN DWn
0
N AW p  N DWn
Extension of space charge layer is inversely proportional to the respective doping layer.
higher relative doping –smaller the space charge layer
second derivative to know φ (electrostatic potential)
n-type
p-type
d
E ' ( x)  
dx
E ( x)dx   d
    E ' ( x)dx
d
dx
E ( x)dx  d
 ( x)    E ' ( x)dx
    E ' ( x)dx
E ' ( x)  
qN A
 p 0
 ( x)    
 ( x)  
 ( x)  
 ( x) 
 p 0
 p 0
 ( x)    E ' ( x)dx
( x  Wp )
qN A
qN A
E ' ( x)  
( x  W p )dx
( x  W p )dx
qN A
 p 0
x
qN A
 p 0
Wp
1 qN A 2 qN A
x 
W p x  D'
2  p 0
 p 0
E ' ( x) 
qN D
 n 0
 ( x)   
 ( x)   
 ( x)  
( x  Wn )
qN D
 n 0
( x  Wn )dx
qN D
 n 0
x
qN D
 n 0
Wn
1 qN D 2 qN D
x 
Wn x  D
2  n 0
 n 0
at the surface (x=0) Galvani potential is equal zero (φ=0)
n-type
p-type
  0 for x  0
  0 for x  0
0  0  0  D'
0  00 D
D'  0
D0
D  D' 0
qN A  1 2

 n ( x) 
 x  Wp x 
 p 0  2

qN D  1 2

 n ( x) 
 x  Wn x 
 n 0  2

The energetic position of the band edges at the surface of each material remains unaltered.
Graphic integration for semiconductor pair
p-type
 qN A
electric field
 ( x)  
qN A
 p 0
x  W 
p
galvani potential
qN A 1 2
 ( x) 
( x  W p x)
 p 0 2
energy
n-type
qN D
 ( x) 
qN D
 n 0
 ( x)  
x  Wn 
qN D 1 2
( x  Wn x)
 n 0 2
E = ej = -qj
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
diffusion potential defined by the electric potential difference
p-type
Vn = f (Wn ) - f (0)
ö
qN æ 1
fn (x) = - D ç x 2 - Wn x ÷
ø
e ne 0 è 2
ö
ö æ qN A
qN D æ 1 2
Vn = (0 + 0)÷÷
ç Wn - (Wn ×Wn )÷ - çç ø è e pe 0
e ne 0 è 2
ø
qN D æ 1 2
2ö
W
W
ç
n
n ÷
ø
e ne 0 è 2
qN D æ 1 ö
Vn = -Wn2
ç -1÷
ene 0 è 2 ø
Vn = -
æ 1 ö 2 qN D
Vn = - ç - ÷Wn
è 2ø
e ne 0
qN DWn2
Vn =
2e ne0
n-type
Vp = f (0) - f (-Wp )
ö
qN A æ 1 2
f p (x) =
ç x + Wp x ÷
ø
e pe0 è 2
Vp =
qN A
e pe0
Vp = -
(0 + 0) -
ö
qN A æ 1 2
ç Wp + (Wp × (-Wp )÷
ø
e pe0 è 2
qN A æ 1 2
2ö
ç Wp - W p ÷
ø
e pe 0 è 2
Vp = -Wp2
qN A æ 1 ö
ç -1÷
e pe 0 è 2 ø
æ 1ö
qN A
Vp = - ç - ÷Wp2
è 2ø
e pe 0
Vp =
qN AWp2
2e pe0
2 p 0
N D pW
Vn qN DW



2
Vp
2 n 0 qN AW p
N A nW
2
n
N A Wn
because

N D Wp
 pWn
Vn W p pW


V p Wn nW
 nW p
2
n
2
p
N A p
Vn N D p N


V p N A n N
N D n
2
A
2
D
2
n
2
p
n-type
p-type
Vp 
qN AW
2
p
2 p 0
\ 2 p 0
2 p 0V p  qN AW \ qN A
2
p
Wn 
2 p 0V p
qN A
qN DWn2
Vn  
\ 2 n 0
2 n 0
2 n 0Vn  qN DWn2 \ qN D
2 n 0Vn
Wn 
qN D
Graphic integration for semiconductor pn junctions
Important relations for pn junctions
(to memorize)
Wn 
2 n  0Vn
qN D
Wn 
2 n  0Vn
qN D
Electroneutrality condition
N AW p  N DWn
Wn N A

Wp N D
Diffusion voltage relations
Vn  p N A

Vp  n N D
V n  pW n

V p  nW p
The width of the space charge layer depends on:
•
doping level
•
voltage drop
n-type
p-type
Eg=1.12 eV
Eg=1.12 eV
NCB=3.2 1019 cm-3
NVB=1.8 1019 cm-3
ND=1017 cm-3
NA=1015 cm-3
kT=26 meV
n-type
p-type
Position of nEF before contact
EF = ECB - kT ln
NCB
ND
N
ECB - EF = kT ln CB
ND
Position of pEF before contact
EF = EVB - kT ln
NVB
NA
EF - EVB = kT ln
NVB
NA
3.2 ×1019 é
cm -3 ù
ECB - EF = 26 ln
êmeV -3 ú
1017 ë
cm û
1.8×1019 é
cm -3 ù
EF - EVB = 26 ln
êmeV -3 ú
15
10
cm û
ë
ECB - EF = 26 ln3.2 ×10 2
EF - EVB = 26 ln1.8×10 4
ECB - EF = 26 × 5.7
ECB - EF = 150meV
ECB - EF = 0.15meV
EF - EVB = 26 × 9.8
EF - EVB = 254.8meV
EF - EVB = 0.25meV
ECB
ECB
EF
0.15 eV
E
0.25 eV
EVB
F
EVB
Contact potential difference
ECB
ECB
EF
0.15 eV
eVC
E
0.25 eV
EVB
F
EVB
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
nEF ® eVn
eVc = nEF - pEF = eVn + eVp
Vn N A
=
Vp N D
N A 1015
a=
= 17 = 10 -2 = 0.01
N D 10
Vn =
NA
Vp
ND
Vn = aVp
VC = Vn + Vp
VC = aVp + Vp = Vp (a +1)
Vp =
VC
(a +1)
pEF ® eVp
Vn = VC -Vp
Vn = 0.71- 0.703 = 0.007
VC
Vp =
(a +1)
0.71
Vp =
= 0.703
0.01+1
2e 0e pVp
2e 0e nVn
Wn =
qN D
Wp =
e n = 11.7
e p = 11.7
N D = 1017 cm -3
N A = 1015 cm-3
Vn = 0.007eV
Vp = 0.703eV
Wn =
Wn =
-14
2 ×8.85×10 ×11.7× 0.007
1.6 ×10 -19 ×1017
-14
1.45×10
1.6 ×10-2
Wn = 0.9 ×10
Wn = 9.5×10 -7
-12
qN A
2 ×8.85×10-14 ×11.7× 0.703
Wp =
1.6 ×10 -19 ×1015
145×10 -14
Wp =
1.6 ×10-4
Wp = 90.6 ×10-10
Wp = 9.5×10-5
e0 = 8.85×10-14 [ F cm]
q =1.6 ×10 -19 [C]
éF
A×s ù
e0 ê =
ú
ë cm V × cm û
-3
N D [cm ]
q[C = A × s]
A×s
V
A×s
-3
V
×
cm
W=
=
A
×
s
×
cm
= cm
-3
A × s × cm
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 ECBp
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 th n(Vd ) = en th n0 e
eVd
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
Eg
jng = qn th NVB e kT = qn th n p
np – e- concentration in the neutral region of ECB of p-type sc
eVd + ECB - E << Eg
C
F
jnr ¹ jng
Applying negative voltage (forward) to the n-type side:
- decrease of band bending
- jnr increase
- jng is not influenced
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 (Va ) = en th n0 e
e(Vd -Va )
kT
= jnr (0)e
eVa
kT
jnr (0) = en th n0 e
eVd
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 th n0 e
-
e(Vd +Va )
kT
= jnr (0)e
-
eVa
kT
jnr (0) = en th n0 e
eVd
kT
Total e- dark current: sum of generation and recombination currents (opposite sign)
jn (Va ) = jnr (Va ) - jng (Va )
using:
jnr (Va ) = en th n0 e
-
e(Vd +Va )
kT
= jnr (0)e
-
eVa
kT
jng (Va ) = jng (0) = - jnr (0)
jng (Va ) = jnr (0) = j0 e
jn (Va ) = j0 e
Total current:
eVd
kT
eVd
kT
æ eVa ö
æ eVa ö
ç e kT -1÷ = jng ç e kT -1÷
è
ø
è
ø
jD (Va ) = jn (Va )+ j p (Va )
Diode relationship by Shockley
æ eV ö
jD = ( jng + j pg ) ç e kT -1÷
è
ø
jng+ jpg – diffusion constants and minority
carrier diffusion lengths
eDp p0 eDn n0
js = jng + j pg =
+
Lp
Ln
æ eV ö
jD = js ç e kT -1÷
è
ø
js – reverse saturation current described by
metal glow emission properties
- 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)
Constant illumination- number of absorbed photons per second and cm2 mulitiled by
elementary charge
Light induced photocurrent:
jL = enph (Eg )(1- R)
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)
æ eV ö
j ph = jD - jL = js ç e kT -1÷ - jL
è
ø
- 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
The approximation for the light induced current (jL)
jL  enPh (h  Eg )  (1  R)
Photocurrent dependence follows dark-current-voltage behavior
æ eV ö
j ph = jD - jL = js ç e kT -1÷ - jL
è
ø
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
45
Semiconductor/Metal Schottky
type junctions
Dr. Katarzyna Skorupska
1
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
Ec
Electron affinity
Work function
Evac
4.05 eV
0.2-0.3 eV
EF
Eg
Ev
1.12 eV
2
semiconductor – metal
Schottky contact
Thermionic interaction
- Contact formation based on energetic considerations
- Interfacial effects neglected
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
EFsc
EFm
mobile nature of charges
under contact formation
development of electrical field
Potential drop across interface
EFsc
EFm
redistribution of charges on
the metal side
4
Metal - semiconductor Schottky contact
(rectifying semiconductor-metal junction)
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:
The junction is characterized by
• the semiconductor and metal
work function
(ΦSC- given by doping)
• the semiconductor electron
affinity and its energy gap.
Definition: contact potential difference
∆Εc = EFSC – EFM = ΦM - ΦSC
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:
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 EFM 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 EFM (after contact
formation EFM=EFSC) and the band edge ECB
ΦBh = eVbb + ECB - EFn
Barrier height defines in Schottky (photo)diodes the reverse saturation current as will be shown below.
Schottky Junctions
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 electron depletion of the
semiconductor during contact
formation leads to a charged
region near the surface;
• the relative distribution of VC follows
where CM and CSC denote metal and semiconductor capacitance, respectively
8
Schottky barrier: Decreasing the gap to zero:
• 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)
Origin of the spatial dependence of the energy bands and the
vacuum level: Poisson´s equ. in 1dimension
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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
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
metal
interface
barrier
forward current – from n-SC to metal
reverse current – from metal to n-SC
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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
15
• 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:
νth- thermal velocity
16
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:
17
18
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
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Current from metal to semiconductor (reverse current)
In the equilibrium situation considered for V=0
The forward current (SCM) must be equal and opposite in sign to the reverse current (MSC)
Therefore:
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To t a l c u r r e n t
js
The pre-factor called reverse saturation current (js)
• contains material properties
• temperature
• and gives the current at V=0
js = A * T 2e
−
Φ bh
kT
DIODE CHARACTERISATION
21
DIODE CHARACTERISATION
question: which sign for voltage and current for an n-type semiconductor-metal junction?
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P h o to c u r re nt
The approximation for the light induced current (jL)
jL = enPh (hν > E g ) ⋅ (1 − R )
Photocurrent dependence follows dark-current-voltage behavior
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
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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
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At open-circuit voltage (VOC) the photocurrent is equal zero IPh=0
Photovoltage (Vph) is given by jph=0
VPh = VOC
kT  jL 
=
ln + 1
q  js

the photovoltage changes
logarithmically with the light intensity
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Example:
VPh = VOC
kT  jL 
=
ln + 1
q  js

VPh = 0.74V
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