NCN Summer School: July 2011
Solar Cell Physics:
recombination and generation
Prof. Mark Lundstrom
lundstro@purdue.edu
Electrical and Computer Engineering
Purdue University
West Lafayette, Indiana USA
copyright 2011
This material is copyrighted by Mark Lundstrom
under the following Creative Commons license.
Conditions for using these materials is described at
http://creativecommons.org/licenses/by-nc-sa/2.5/
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acknowledgement
Dionisis Berdebes, Jim Moore, and Xufeng Wang
played key roles in putting together this tutorial.
Their assistance is much appreciated.
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solar cell physics
A solar cell is a simple device – just a pn junction with
light shining on it.
To maximize efficiency, we must maximize the
generation of e-h pairs and minimize the recombination
of e-h pairs.
This lecture is a short introduction to the physics of
crystalline solar cells – specifically Si.
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outline
1)
2)
3)
4)
5)
Introduction
Recombination at short circuit
Recombination at open circuit
Discussion
Summary
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dark current and recombination
+
N
P
ID
s.s. excess
holes
s.s. excess
electrons
electron-injecting
contact
hole-injecting
contact
− VA +
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recombination in the N-type QNR
-
-
N
P
+
electron-injecting
contact
ID
hole-injecting
contact
− VA +
Anytime an electron and hole recombine anywhere within the diode, one
electron flows in the external circuit.
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Shockley-Read-Hall recombination
minority carriers injected across junction
ET
Fn
qVA FP
SRH
recombination
ID
− VA +
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recombination at a contact
minority carriers injected across junction
Fn
qVA FP
ID
− VA +
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light-current and generation
Vbi − VA
EF
“base”
(absorbing layer)
“emitter”
− VA +
10
ID < 0
Every time a minority electron is generated and collected, one
electron flows in the external
current.
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2011
light-current and recombination
3 e-h pairs generated
“emitter”
1 e in external circuit
Every time a minority electron is generated and recombines before being
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collected, the solar cell current suffers.
solar cells and recombination
• Carrier recombination lowers the short-circuit current and
reduces the open-circuit voltage.
• To optimize solar cell performance, we need a clear
understanding of how many carriers are recombining and
where they are recombining.
• Then we need to establish a quantitative relation between
recombination and solar cell performance.
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solar cells and recombination
J p ( 0)
N
Jn ( L)
=
J D (VA ) q ( RTOT (VA ) − GTOT )
ID
L
RTOT=
∫ R ( x )dx −
J p ( 0)
0
P
q
Jn ( L)
−
q
L
GTOT = ∫ Gop ( x )dx
0
0
L
x
For a formal derivation of this result, see the appendix.
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outline
1)
2)
3)
4)
5)
Introduction
Recombination at short circuit
Recombination at open circuit
Discussion
Summary
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generic crystalline Si solar cell
SF = 1000 cm/s
n+
“emitter” (0.3 μm)
base doping: NA = 1016 /cm3
p-type “base”
200 um
key device
parameters
emitter doping ND = 6 x 1019 /cm3
(198.9 μm)
minority carrier lifetime
(base)
p+ “Back Surface Field” (BSF)
(0.8 μm)
τn = 34 μs
base thickness W = 198.9 μm
front junction depth xjf = 0.3 μm
back junction depth xjb = 0.8 μm
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light-generated current
=
J D ( 0 ) q ( RTOT ( 0 ) − GTOT )
SF = 1000 cm/s
n+ “emitter” (0.3 μm)
1) What is GTOT?
200 um
p-type “base”
2) How is GTOT spatially distributed?
3) What is RTOT?
(198.9 μm)
4) How is RTOT spatially distributed?
p+ “Back Surface Field” (BSF)
(0.8 μm)
5) How do things change if we
remove the BSF?
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light-generated current: numbers
J SC= J D (V=
0=
) q ( RTOT − GTOT )
A
n+ “emitter” (0.3 μm)
∞
WD ≈ 0.3 µ m
GMAX
=
17
-2 -1
2.97
10
cm
s
G
x
=
dx
×
(
)
op
∫
200 um
0
2L
p-type “base”
(198.9 μm)
G=
TOT
0
Ln ≈ 320 µ m
p+ “Back Surface Field” (BSF)
(0.8 μm)
17
-2 -1
G
x
=
dx
2.79
×
10
cm
s
(
)
∫ op
J SC 39.4 mA/cm 2
=
= 2.46 ×1017 cm -2s -1
q
q
RTOT (=
0 ) 3.31×1016 cm -2s -1
17
CE = 0.88
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light-generated current: understanding
entire device
near surface
xj
18
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x j + WD
light-generated current: summary
∞
GMAX
=
)dx 2.97 ×10 cm s
∫ Gop ( x =
17
-2 -1
2L
G=
TOT
0
low lifetime (Auger recombination)
surface recombination
19
good
collection
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-2 -1
G
x
=
dx
2.79
×
10
cm
s
(
)
∫ op
0
minority carrier lifetime
BSF
recombination at short circuit
entire device
near surface
xj
20
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x j + WD
recombination at short circuit: summary
J SC 39.4 mA/cm 2
=
= 2.46 ×1017 cm -2s -1
q
q
RTOT (=
0 ) 3.31×1016 cm -2s -1
(0.37)
(0.49)
(0.14)
low lifetime (Auger recombination)
surface recombination
21
good
collection
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minority carrier lifetime
BSF
about recombination in the base
expect: R ( x ) ≈
d 2 ∆n ∆n
−
=
0
2
dx
Ln
∆n ( x )
τn
We find the excess minority
electron profile by solving the
minority carrier diffusion
equation:
∆n
Jn =
( L′ ) q sback ∆n ( L′ )
d
−R
( J n −q ) =
dx
d ∆n
J n ≈ qDn
dx
′ ) q s j ∆n ( 0′ )
J n ( 0=
0=′ x j + W
L′= L − xBSF
x
xj +W
22
Ln = Dnτ n
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L
Adept simulation results
R ( x) ≈
∆n ( x )
τn
∆n ( x )
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the BSF
∆E =
0.13 eV
EC
Sback ≈ υ th e− ∆E kBT
EI
EF

; 0.6 × 10 7 cm s
EV
What happens if we
remove the BSF?
EC
Sback ≈ υ th
EI
EF

EV
24
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; 1 × 10 7 cm s
without the BSF
BSF
no BSF
Without BSF
With BSF
25
J SC = 39.4 mA/cm 2
J SC = 38.2 mA/cm 2
qRTOT = 5.3 mA/cm 2
qRTOT = 6.5 mA/cm 2
CE = 0.88
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CE = 0.85
internal quantum efficiency
With BSF
No BSF
J D (V = 0, λ )
IQE =
Finc ( λ )
26
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questions
1) Can you determine a way to find the actual back surface
recombination velocity from the Adept simulation results.
(Hint: Use plots of n(x) and Jn(x).)
2) How much could the performance improve if the back
surface recombination velocity could be reduced to zero?
3) With the original BSF, how much would the performance
increase if the minority carrier lifetime was 10 times longer?
4) In the original design, how would the short-circuit current
change if the base was twice as thick?
5) Since most of the recombination loss occurs in the emitter,
why not just make the emitter junction depth a lot smaller?
27
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2D effects
ID
VD
I ( x)
V ( x ) < VD
xj
dR = ρ S
ρ=
S
dx
W
ρ
1
=
x j N D qµn x j
distributed series resistance
28
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outline
1)
2)
3)
4)
5)
Introduction
Recombination at short circuit
Recombination at open circuit
Discussion
Summary
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dark I-V
=
J D (VA ) q ( RTOT (VA ) − GTOT )
=
0 q ( RTOT =
(VA VOC ) − GTOT )
Under open circuit conditions:
RTOT =
GTOT
(VA V=
OC )
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superposition
JD
=
J D (VA ) q ( RTOT (VA ) − GTOT )
dark IV
J SC
dark:
J
dark
D
(VA ) = q R
dark
TOT
(VA )
=
J D J 0 ( e qVD
− 1)
VA
VOC
illuminated:
light
=
J Dlight (VA ) q ( RTOT
(VA ) − GTOT )
− J SC
JL < 0
superposition:
illuminated at VOC:
light
RTOT
(VOC ) = GTOT
nk B T
?
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J Ddark (VOC ) = J SC
dark
RTOT
(VOC ) = J SC q
31
dark current characteristics (sketch)
=
J Ddark (VA ) J 0 ( e qVA
J Ddark
=
(VA ) J 01 ( eqVA
nk B T
kBT
− 1)
− 1) + J 02 ( e qVA
2 kBT
− 1)
series
resistance
or…
n=1
log10 J Ddark
shunt
resistance
or…
n=2
VA
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dark current characteristics (Adept)
=
J Ddark (VA ) J 0 ( e qVA
(
J Ddark
=
(VA ) J 01 eqVA
nk B T
kBT
− 1)
)
(
− 1 + J 02 e qVA
2 kBT
)
−1
n >1
n =1
n≈2
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what determines J0 and n?
(
=
J Ddark (VA ) J 0 e qVA
nk B T
)
−1
dark
J Adark (VA ) = q RTOT
(VA )
Answer:
Electron-hole recombination determines I0.
The location of recombination within the solar cell
determines the ideality factor, n.
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recombination in the dark (VA = 0.7 V)
Emitter
Base
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recombination summary: (VA = 0.7 V)
Short-circuit recombination
qRTOT ( 0 ) = 5.3 mA/cm
light
VA = 0.7 V recombination
dark
qRTOT
( 0.7 ) = 465 mA/cm 2
2
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what happens if we remove the BSF? (VA = 0.7 V)
Without BSF
With BSF
~70%
~85%
J D ( 0.7 ) = 1372 mA/cm 2
J D ( 0.7 ) = 644 mA/cm 2
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dark current physics (n = 1)
I D (VA ) = qRTOT (VA )
FB: minority carriers injected across junction
1) Recombination in QNRs:
Fn
ID > 0
qVA
FP
2) Electrons and holes can also
recombine within the SCR of
the junction.
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n = 1 device physics
I D (VA ) = qRTOT (VA )
nP ( 0′ ) ≈ n0 P e qVA
kBT
q (Vbi − VA )
Qn
qRTOT (VA ) =
tn
ni2 qVA
Qn ∝
e
NA
(
Fn
FP
kBT
)
−1
tn : minority carier lifetime
n0P ≈ ni2 N A
or base transit time
Recombination in quasi-neutral regions gives rise to n = 1 currents.
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dark current characteristics (sketch)
=
J Ddark (VA ) J 0 ( e qVA
J Ddark
=
(VA ) J 01 ( eqVA
nk B T
kBT
− 1)
− 1) + J 02 ( e qVA
2 kBT
− 1)
series
resistance
or…
n=1
log10 J Ddark
shunt
resistance
or…
n=2
VA
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recombination in the dark (VA = 0.2 V)
base region
emitter region
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recombination summary: (VA = 0.2 V)
VA = 0.7 V recombination
VA = 0.2 V recombination
dark
qRTOT
( 0.7 ) = 465 mA/cm 2
dark
qRTOT
( 0.7=) 8.4 ×10−6 mA/cm 2
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dark current physics
I D (VA ) = qRTOT (VA )
FB: minority carriers injected across junction
1) Recombination in QNRs:
Fn
ID > 0
qVA
FP
2) Electrons and holes can also
recombine within the SCR of
the junction.
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recombination in SCRs
dark
J D (VA ) = qRTOT
(VA )
q (Vbi − VA )
Fn
np = ni2 e qVA
Maximum recombination
occurs when n(x) ≈ p(x)
n ( x ) p ( x ) = ni2 e qVA
FP
nˆ ≈ pˆ ∝ ni e qVA
dark
qRTOT
(VA ) ∝
kBT
kBT
2 kBT
qni e qVA
2 kBT
τ eff
Recombination in space-charge regions gives rise to n = 2 currents.
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recombination in SCR
J D (VA ) = qRTOT (VA )
nˆ ≈ pˆ ∝ ni e qVA
Rˆ (V=
A)
2 kBT
ni e qVA / 2 kBT
nˆ
=
τ eff
τ eff
J D (VA ) = q Rˆ Weff
Weff =
E ˆ = 2.3 × 104 V cm
=
Weff
k BT q
≈ 11 nm
ˆ
E
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k BT q
Eˆ
45
dark IV
(
=
J D (VA ) J 02 e qVA
2 kBT
)
(
)
(
1k B T
− 1 + J 01 e qVA =
− 1 J 0 e qVA
Recombination in
depletion regions
Recombination in
neutral regions
J 02 ∝ ni ∝ e − EG / 2 kBT
J 01 ∝ ni2 ∝ e − EG / kBT
large bandgaps and
low temperatures
nk B T
)
−1
small bandgaps and
high temperatures
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questions
1) What do you expect to happen if the BSF were removed?
Run an Adept simulation to confirm.
2) What do you expect to happen if the minority carrier lifetime
were reduced to 0.1 microseconds? Run an Adept
simulation.
3) Why is recombination in the emitter so important under shortcircuit conditions, but not under FB in the dark?
4) How much could VOC be increased if a BSF with near-zero
surface recombination velocity could be achieved?
5) Series resistance affects the dark current, but it has no effect
at open-circuit. What are the implications?
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outline
1)
2)
3)
4)
5)
Introduction
Recombination at short circuit
Recombination at open circuit
Discussion
Summary
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reducing recombination
higher material quality (longer lifetimes)
thinner base layer (but optically thick)
J D (VA )
q ( RTOT (VA ) − GTOT )
built-in fields
back-surface-fields / minority carrier mirrors
reducing contact areas
….
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high-efficiency Si solar cells
24.5% at 1 sun
Martin Green Group UNSW – Zhao, et al, 1998
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how good is superposition?
V = 0.62 V - Dark
VOC = 0.62 V - Illuminated
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how good is superposition? (ii)
dark
JJDdark
J Dlight
J Ddark + J Dlight (V = 0 )
superposition
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outline
1)
2)
3)
4)
5)
Introduction
Recombination at short circuit
Recombination at open circuit
Discussion
Summary
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summary
1) Diode current = q times (total recombination – total
generation)
2) At VOC, recombination = optical generation
3) At V = 0, recombination lowers the collection efficiency
4) Dark current tells us much about the internal
recombination mechanisms
5) Solar cell design is all about maximizing total generation
and minimizing total recombination.
6) Simulations can be useful for understanding –especially
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54
if you look “inside” andLundstrom
not just
at the IV.
questions
1)
2)
3)
4)
5)
Introduction
Recombination at short circuit
Recombination at open circuit
Discussion
Summary
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Appendices
1) Formal derivation of the relation between current and
recombination/generation.
2) Mathematical justification of superposition
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Appendix 1: current and recombination
Formal derivation of the relation between current and
recombination/generation.
J p ( 0)
N
Jn ( L)
J=
q ( RTOT − GTOT )
D (V )
ID
L
RTOT=
∫ R ( x )dx −
J p ( 0)
0
P
q
Jn ( L)
−
q
L
GTOT = ∫ Gop ( x )dx
0
0
L
x
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continuity equation for electrons
Wabash
River
Rate of increase of
water level in lake
∂n
∂t
=
= (in flow - outflow) + rain - evaporation

−∇ • J n −q
(
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)+ G − R
58
solar cell physics
“semiconductor equations”
Conservation Laws:

∇• D =ρ
) (G

q
∇• Jp =
(



D=
κε 0 E =
−κε 0∇V
ρ= q( p − n + N D+ − N A− )

q
∇ • J n −=
(
Relations:
op
) (G
(steady-state)
op
− R)



J=
nq µn E + qDn∇n
n



J=
pq µ p E − qD p ∇p
p
− R)
R = f (n, p )
Gop = optical generation rate
etc.
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diode current and recombination

∇ • J n −=
q
(
) (G
op
− R)
d
( J n −q ) = Gop − R
dx
L
L
0
0
ID
(1D)
ID
N
P
0
L
=
∫ dJ n q ∫  R ( x ) − Gop ( x ) dx
L
J n ( L ) − J n ( 0=
) q ∫  R ( x ) − Gop ( x ) dx
0
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x
current and recombination-generation
L
J n ( L ) − J n ( 0=
) q ∫  R ( x ) − Gop ( x ) d + J p (x0 ) − J p ( 0 )
0
L
− { J n ( 0 ) + J p ( 0 )} = J D (V ) = q ∫  R ( x ) − Gop ( x )  dx − J n ( L ) − J p ( 0 )
0
J=
q ( RTOT − GTOT )
D (V )
ID
L
qRTOT
= q ∫ R ( x )dx − J n ( L ) − J p ( 0 )
0
L
GTOT = ∫ Gop ( x )dx
N
P
0
L
0
61 2011
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x
61
current and generation-recombination
=
J D (VA ) q ( RTOT (VA ) − GTOT )
The diode current is q times the total recombination minus the total
generation.
The total recombination is the integrated recombination rate within
the device plus the flux of minority carriers into each contact.
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62
Appendix 2: justifying superposition
=
J D (VA ) q ( RTOT (VA ) − GTOT )
(valid in light or dark)
dark
J Ddark (VA ) = qRTOT
(VA )
(dark current)
light
=
J Dlight ( 0 ) q ( RTOT
( 0 ) − GTOT )
(short circuit current)
J Dsuper (=
VA ) J Ddark + J Dlight ( 0 )
(principle of superposition)
dark
light
J Dsuper (VA ) = qRTOT
(VA ) + q ( RTOT
( 0 ) − GTOT )
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(How does this compare to
the exact answer?)
63
mathematical justification for superposition
( ) (
( )
J D V A = q RTOT V A − GTOT
( ) (
( )
( )
( ) (
)
light
J Dlight V A = q RTOT
V A − GTOT
(valid in light or dark)
)
()
dark
light
V A + q RTOT
0 − GTOT
J Dsuper V A = qRTOT
( )
( )
()
light
dark
light
RTOT
V A = RTOT
V A + RTOT
0 ??
)
(principle of superposition)
(criterion to justify superposition)
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