p/n junction
FE
AE
iE
FE
VE
CEDE
vacE
Isolated p, n regions: no electric contact, not in equilibrium
p n
iE
VE
CE
p/n junctionIn equilibrium, the Fermi level must be constant. Shift the energy levels in p and n regions up/down to align EF:
AE
iE
FE
VE
CE
DE
vacE
VE
iE
vacE
CE
p nelectronenergy
Electrons tend to lowest available energy:The junction must have an electric field pointing from n to p.
p
n
p/n junction
vac F vac C C F
F V C V
V
Aln
p p
p g
E E E E E E
E E E ENkT EN
vac F vac C C F
Dln
n n
Cn
E E E E E E
NkTN
C V
A D
2
A D
A D2
ln
eln
ln
g
p n g
E kTi
g
i
N NqV E kTN N
nE kTN N
N NqV kTn
2C V
2C V
ee
g
g
E kTi
E kTi
n N NN N n
Find potential difference (voltage):
//p-side work function//n-side work function
vac vac F Fp n p np np nqV E E E E
Equilibrium: F F Fp nE E E
Built-in voltage: A D2lnbi
i
kT N NVq n
using:
Depletion approximation: space-charge region
xnwpw
Assume the regions directly adjacent to the junction on either sideare completely depleted of free carriers.
metallurgicaljunction
p nionized
acceptorsionizeddonors
depletion regionor
space-charge region
electronsholes
0
neutral region neutral region
This is called the depletion region or space-charge region (SCR).
Depletion approximation: charge density p x n x
D AN x N x
D Ax p x n x N x N x
np
---
+ ++
+ ++++ +
--- ---
---
+ ++
netcharge density
ionizeddopants
freecarriers
A
D
0,, 0, 0
0,
p
p
n
n
x wN w x
xN x w
w x
Depletion approximation orabrupt space-charge approx.
AN
DN
DN
AN
We are left with ionized acceptors and donors in the SCR.
DN
AN
Depletion approximation: electric field
A
D
0,
, 0Ε
, 0
0,
p
p p
xn n
n
x wqN x w w xq x dxqN w x x w
w x
Ε q xddx
Ε x
np
x
A Dp nN w N w
No surface charge:
0 0E Ex x Ε
maxΕ
maxΕ a p d nqN w qN w
//Poisson equation
No net charge: Ε 0 outside of SCR
Depletion approximation: potential
2A
22 2A D
2 2A D
0,
, 02Ε
, 02
,2
p
p p
xp n n n
p n n
x wqN x w w x
V x x dx q N w N w w x x w
q N w N w w x
ΕdVdx
V x np
biV
pw nwx
0
2 2A D2bi p n
qV N w N w
So:
Depletion approximation: depletion width
2 2A D
A D A D
1 1 1 12 2bi p nq qV N w N wN N N N
D
Ap n
Nw wN A
Dn p
Nw wN
D
A D
1 121 1
bin Vw N qN N
A
A D
1 121 1
bip Vw N qN N
A D
2 1 1bip n
Vw w w q N N The depletion width is:
Depletion approximation: energy diagram
AEFEDE
vacE
VE
iE
CE
p nelectron
energy
biVp
n
p/n junction: Non-equilibrium (I)External voltage (e.g., applied bias or photovoltage):
2 2A DΕ
2
n
p
w
p n bix w
qx dx N w N w V V
A D
2 1 1bip n
V Vw w w
q N N
A
D
e
e
i Fp
F in
E E kT
i
E E kTi
p N n
n N n
In the neutral regions:
F F2 2A D e ei in p bi
E E E E kT q V V kTi iN N n n
F Fn p biE E q V V A D
2lni
N NkTn
qV
The quasi-Fermi level splitting in the SCR equals the external voltage.
In the SCR:
p p
n n
F F
F F
E E
E E
p/n junction: Non-equilibrium (II)
AE
p nEelectronenergy
biV V
p n
FpE
Forward bias: + -V
2 1 1bi
a d
V Vw q N N
FnEDE
vacE
VE
iE
CE
p/n Junction: Non‐equilibrium (III)
AE
FnEFpEqV
FE
FE
pwnw
First, analyze without photogeneration:
DE
VE
iE
CE
p/n junction: non-equilibrium (IV)
2 e kTin p n In the SCR:
A
0 e
p
p qV kTp
p w N
n w n
The depletion approximation gives:
D
0 e
n
n qV kTn
n w N
p w p
qV
Far from the junction:
2
0A
2
0D
p i
n i
nn nN
np pN
0
0 e 1
pp p
p qV kT
n w n w n
n
0
0 e 1
nn n
n qV kT
p w p w p
p
where:
Excess minority-carrier concentrations at the edges of the SCR:
First, analyze without photogeneration:
Photogeneration RateAbsorption:
, , ,d b E x E x b E xdx
//separation of variables
0 0
b x x
b b x
db x dx xb
//treat each E separately
uniform
ln0
b x xb
, ,0 e E xb E x b E
Absorptioncoefficient
Carrier generation rate:
, ,
,
, 1 e
E xs
dg E x b E xdxE b E x
g E x R E E b E
,0 1 sb E R E b E
Spectral photon flux density in
material
p/n junction solar cell (I)
In steady-state:
0
p pg
n nn n
n n nG U G
g nn G
0
n ng
p pp p
p p pG U G
g pp G
n pG G G
, 1 1sg E x R E E b E g E
e 1E x no attenuation
0E
G x dE g E G
//uniform generation
//band-to-band absorption
p-side n-side
//Assume weak absorption
Photogeneration:
0
p pgn n n n
0n n
gp p p p
p/n junction solar cell (II)
FnE pFEqV
pwnw
With uniform photogeneration,there are excess minority carriers in the neutral regions,so the quasi-Fermi levels are split.
FE
DE
VE
iE
CE
AE
FE
p/n junction solar cell (III)
pp pn w n w n
0 0
0
e
e 1
n qV kT nn g
n qV kTg
p w p p p
p p
pn x n x n np x p x p
Consider generation in the quasi-neutral regions (assuming uniform generation).p-side n-side
Outside the SCR:
0 0
0
e
e 1
p pqV kTp g
p qV kTg
n w n n n
n n
0 ep qV kT
pn w n
The carrier concentrations are still related to the quasi Fermi-level spitting:
0 en qV kT
np w p
nn np w p w p
At the edges of the SCR:
So the excess minority-carrier concentration at the edges of the SCR are:
p/n junction solar cell (IV)
n pJ J x J x
Total current density:
sign convention for PV
//must be constant at any point in circuit
Positive J when device is delivering power
Calculate using: n p p pJ J w J w
n n p nJ J w J w
Depletion approx: All the potential difference occurs across the SCR.So, E=0 at the SCR edges: , p nx w x w
Εn n n ndn dnJ q q D q Ddx dx
Εp p p pdp dpJ q q D q Ddx dx
or:
p/n junction solar cell (V)
e e
e e
p n p n
p pn p
x w L x w Ln n
x w Lx w Lp p
n x A B
p x A B
e
e
p n
n p
x w Lp
x w Ln
n x n w
p x p w
In the neutral regions, we have diffusion only:p
n
x ww x
00
n
p
AB
nn n
n
pp p
p
d n q DJ x q D n xdx Ld p q DJ x q D p x
dx L
nn p p
n
pp n n
p
q DJ w n wL
q DJ w p wL
p
n
x ww x
The boundary conditions give:
0 e 1p qV kTp gn w n n
0n
0 e 1n qV kTn gp w p p
0p
0 e 1n p qV kTn p g
n
q DJ w n nL 0 e 1p n qV kT
p n gp
q DJ w p pL
Assume infinitely thick neutral regions.
The currents are:
Using previous results:
p/n Junction solar cell (VI)
or
,SCR
,SCR
p p p n p p
n n n p n n
J w J w J w
J w J w J wwe need either:
,SCR
,rec ,gen
n
p
wn n
x w
n n
J w q U x G x dx
J J
n pJ w p nJ wWe have and
n p p p n n p nJ J w J w J w J w To find
,SCR
,rec ,gen
n
p
wp p
x w
p p
J w q U x G x dx
J Jwhere
p/n Junction solar cell (VII)
Assume: ,rec 0pJ
,gen ,gen
n
p
wn p
x wJ J q G dx q G w
,SCR 0 e 1 n p qV kT
n n n p n n gn
q DJ w J w J w q G w n n q G wL
//no recombination in SCR
//uniform generation in SCR
,rec 0nJ
,SCR ,SCR p p n nJ w J w q G w
Just need one or the other. Let's find p n n nJ J w J w
We know: 0 e 1p n qV kTp n g
p
q DJ w p pL
,SCR n n n p n nJ w J w J w 0 e 1n p qV kTn p g
n
q DJ w n nL
,SCR n nJ w q G w
p/n Junction solar cell (VIII)
0 e 1p n qV kTp n p
p
q DJ w p q G LL
p n n nJ J w J w
n ng n n
n n
q D q Dn G q G LL L
0 e 1n p qV kTn n n
n
q DJ w n q G L wL
0 e 1n p qV kTn n g
n
q DJ w q G w n n q G wL 0 e 1p n qV kT
p n gp
q DJ w p pL
p pg p p
p n
q D q Dp G q G LL L
Use:
p/n Junction solar cell (IX)
photo n pJ q G w L L
20 00
A D
n p n pp ni
n p n p
D D D DJ q n p q nL L L N L N
//photocurrent
//dark current
photo 0 e 1qV kTJ V J J
00
photo 0
e 1
e 1
n pp n qV kTn p
n p
qV kT
q D q DJ q G w L L n pL L
J J J
SCR transit timeCan we ignore recombination in the SCR?Estimate transit time across SCR:
Ev max1E = E2 max
1 E2
v max
2E
w wv
A DmaxE
p nqN w qN w
p nw w w A
A D
11 1
pN w w
N Nmax
1E 1 1
a d
q w
N N
max
A D
E 11 1
qw
N N
A D
2 1 1 q N N
0
2
16 3A D
10
cm100 V s
10 cm
N N
112.2 10 s Short compared to typicalminority carrier lifetimes.
Assume:
Plots: depletion approx. (I)equilibrium (V=0 V, G=0):
no quasi-Fermilevel splittingin SCR
no minority-carrierconc. gradientsin neutral regions
Plots: depletion approx. (II)forward bias, no illumination (V=0.5 V, G=0):
minority-carrierconc. gradientsin neutral regions→diffusion awayfrom SCR
positivequasi-Fermi levelsplitting in SCR
Plots: depletion approx. (III)short circuit, illuminated (V=0 V, G=5e19 1/cm3/s):
minority-carrierconc. gradientsin neutral regions→diffusion towards SCR
no quasi-Fermilevel splitting
Plots: depletion approx. (IV)operating, illuminated (V=0.5 V, G=5e19 1/cm3/s):
minority-carrierconc. gradientsin neutral regions→diffusion towards SCR
positivequasi-Fermi levelsplitting in SCR
2
A
eqV kTinN
2
D
eqV kTinN
2
A
inN
2
D
inN
ANDNn
2
A
eqV kTinN
2
D
eqV kTinN
2
A
inN
2
D
inN
p
ANDNnp
pw nw
pw nw
forward bias,no generation
forward biaswith generation
Plots: carrier concentrations in depletion approx.
electron diffusion
electron diffusion
hole diffusion
hole diffusion
2
A
ig
n nN
2
D
ig
n pN
Solving p/n homojunction characteristics without the depletion approx.
Fnn n
dEJ ndx
1 nn n
dJ G Uq dx
1 pp p
dJ G Uq dx
2
2q xd V
dx
Fpp p
dEJ pdx
//Poisson equation
//electron current
//hole current
//hole continuity
//electron continuity
Plots: no depletion approx. (I)equilibirium (V=0 V, G=0):
CE
VE
iEFn
E
tot 0p nJ J J
p n
p n
FpE
0U G
Plots: no depletion approx. (II)forward bias, no generation (V=0.7 V, G=0):
U
G-U
CE
VE
iE FnE
pJ nJtotJ
p n
p n
FpE
Plots: no depletion approx. (III)
UG-U
CE
VE
iEFn
E
FpE
pJ nJ
totJ
p n
p n
G
operating, illuminated (V=0.7 V, G=1e23 1/cm3/s):
depl. approx.: dashed linesfull calc.: solid lines
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