|
3.4.2 Generation Rate
For light incident at the front of the solar cell,
x
= −
W
N
, the optical generation rate takes
the form (see equation 3.35)
G(x)
=
(
1
−
s)
λ
(
1
−
r(λ))f (λ)α(λ)
e
−
α(x
+
W
N
)
d
λ.
(
3
.
105
)
Only photons with
λ
≤
hc/E
G
contribute to the generation rate.
3.4.3 Solution of the Minority-carrier Diffusion Equation
Using the boundary conditions defined by equations (3.96), (3.98), (3.103), and (3.104)
and the generation rate given by equation (3.105), the solution to the minority-carrier
diffusion equation, equation (3.80), is easily shown to be
p
N
(x)
=
A
N
sinh[
(x
+
x
N
)/L
p
]
+
B
N
cosh[
(x
+
x
N
)/L
p
]
+
p
N
(x)
(
3
.
106
)
in the
n
-type region and
n
P
(x)
=
A
P
sinh[
(x
−
x
P
)/L
n
]
+
B
P
cosh[
(x
−
x
P
)/L
n
]
+
n
P
(x)
(
3
.
107
)
in the
p
-type region. The particular solutions due to
G(x)
,
p
N
(x)
, and
n
P
(x)
are
given by
p
N
(x)
= −
(
1
−
s)
λ
τ
p
(L
2
p
α
2
−
1
)
[1
−
r(λ)
]
f (λ)α(λ)
e
−
α(x
+
W
N
)
d
λ
(
3
.
108
)
and
n
P
(x)
= −
(
1
−
s)
λ
τ
n
(L
2
n
α
2
−
1
)
[1
−
r(λ)
]
f (λ)α(λ)
e
−
α(x
+
W
N
)
d
λ.
(
3
.
109
)
Using the boundary conditions set above,
A
N
, B
N
, A
P
, and
B
P
are easily obtained.
3.4.4 Terminal Characteristics
The minority-carrier current densities in the quasi-neutral regions are just the diffusion
currents, since the electric field is negligible. Using the active sign convention for the
current (since a solar cell is typically thought of as a battery) gives
J
p,N
(x)
= −
qD
p
d
p
N
d
x
(
3
.
110
)
90
THE PHYSICS OF THE SOLAR CELL
and
J
n,P
(x)
=
qD
n
d
n
P
d
x
(
3
.
111
)
The total current is given by
I
=
A
[
J
p
(x)
+
J
n
(x)
]
(
3
.
112
)
and is true everywhere within the solar cell (
A
is the area of the solar cell).
Equations (3.110) and (3.111) give only the hole current in the
n
-type region and the
electron current in the
p
-type region, not both at the same point. However, integrating
equation (3.72), the electron continuity equation, over the depletion region, gives
x
P
−
x
N
d
J
n
d
x
d
x
d
x
=
J
n
(x
P
)
−
J
n
(
−
x
N
)
=
q
x
P
−
x
N
[
R(x)
−
G(x)
] d
x
(
3
.
113
)
G(x)
is easily integrated and the integral of the recombination rate can be approximated
by assuming that the recombination rate is constant within the depletion region and is
R(x
m
)
where
x
m
is the point at which
p
D
(x
m
)
=
n
D
(x
m
)
and corresponds to the maximum
recombination rate in the depletion region. If recombination via a midgap single level trap
is assumed, then, from equations (3.37), (3.99), (3.100), and (3.102), the recombination
rate in the depletion region is
R
D
=
p
D
n
D
−
n
2
i
τ
n
(p
D
+
n
i
)
+
τ
p
(n
D
+
n
i
)
=
n
2
D
−
n
2
i
(τ
n
+
τ
p
)(n
D
+
n
i
)
=
n
D
−
n
i
(τ
n
+
τ
p
)
=
n
i
(
e
qV /
2
kT
−
1
)
τ
D
(
3
.
114
)
where
τ
D
is the effective lifetime in the depletion region. From equation (3.113),
J
n
(
−
x
N
)
,
the majority carrier current at
x
= −
x
N
, can now be written as
J
n
(
−
x
N
)
=
J
n
(x
P
)
+
q
x
P
−
x
N
G(x)
d
x
−
q
x
P
−
x
N
R
D
d
x
=
J
n
(x
P
)
+
q(
1
−
s)
λ
[1
−
r(λ)
]
f (λ)
[e
−
α(W
N
−
x
N
)
−
e
−
α(W
N
+
x
P
)
] d
λ
−
q
W
D
n
i
τ
D
(
e
qV /
2
kT
−
1
)
(3.115)
where
W
D
=
x
P
+
x
N
. Substituting into equation (3.112), the total current is now
I
=
A
J
p
(
−
x
N
)
+
J
n
(x
P
)
+
J
D
−
q
W
D
n
i
τ
D
(
e
qV /
2
kT
−
1
)
(
3
.
116
)
where
J
D
=
q(
1
−
s)
λ
[1
−
r(λ)
]
f (λ)(
e
−
α(W
N
−
x
N
)
−
e
−
α(W
N
+
x
P
)
)
d
λ
(
3
.
117
)
is the generation current from the depletion region and A is the area of the solar cell. The
last term of equation (3.116) represents recombination in the space-charge region.
SOLAR CELL FUNDAMENTALS
91
The solutions to the minority-carrier diffusion equation, equations (3.106)
and (3.107), can be used to evaluate the minority-carrier current densities, equa-
tions (3.110) and (3.111). These can then be substituted into equation (3.116), which,
with some algebraic manipulation, yields
I
=
I
SC
−
I
o
1
(
e
qV /kT
−
1
)
−
I
o
2
(
e
qV /
2
kT
−
1
)
(
3
.
118
)
where
I
SC
is the short-circuit current and is the sum of the contributions from each of
the three regions: the
n
-type region (
I
SC
N
), the depletion region (
I
SCD
=
AJ
D
), and the
p
-type region (
I
SC
P
)
I
SC
=
I
SC
N
+
I
SCD
+
I
SC
P
(
3
.
119
)
where
I
SC
N
=
qAD
p
p
(
−
x
N
)T
p
1
−
S
F
,
eff
p
(
−
W
N
)
+
D
p
d
p
d
x
x
=−
W
N
L
p
T
p
2
−
d
p
d
x
x
=−
x
N
(3.120)
with
T
p
1
=
D
p
/L
p
sinh[
(W
N
−
x
N
/L
p
]
+
S
F
,
eff
cosh[
(W
N
−
x
N
/L
p
]
(3.121)
T
p
2
=
D
p
/L
p
cosh[
(W
N
−
x
N
)/L
p
]
+
S
F
,
eff
sinh[
(W
N
−
x
N
)/L
p
]
(3.122)
and
I
SC
P
=
qAD
n
n
(x
P
)T
n
1
−
S
BSF
n
(W
P
)
+
D
n
d
n
d
x
x
=
W
P
L
n
T
n
2
+
d
n
d
x
x
=
x
P
(3.123)
with
T
n
1
=
D
n
/L
n
sinh[
(W
P
−
x
P
)/L
n
]
+
S
BSF
cosh[
(W
P
−
x
P
)/L
n
]
(3.124)
T
n
2
=
D
n
/L
n
cosh[
(W
P
−
x
P
)/L
n
]
+
S
BSF
sinh[
(W
P
−
x
P
)/L
n
]
(3.125)
I
o
1
is the dark saturation current due to recombination in the quasi-neutral regions,
I
o
1
=
I
o
1
,p
+
I
o
1
,n
(
3
.
126
)
with
I
o
1
,p
=
qA
n
2
i
N
D
D
p
L
p
D
p
/L
p
sinh[
(W
N
−
x
N
)/L
p
]
+
S
F
,
eff
cosh[
(W
N
−
x
N
/L
p
]
D
p
/L
p
cosh[
(W
N
−
x
N
)/L
p
]
+
S
F
,
eff
sinh[
(W
N
−
x
N
)/L
p
]
(3.127)
92
THE PHYSICS OF THE SOLAR CELL
and
I
o
1
,n
=
qA
n
2
i
N
A
D
n
L
n
×
D
n
/L
n
sinh[
(W
P
−
x
P
)/L
n
]
+
S
BSF
cosh[
(W
P
−
x
P
)/L
n
]
D
n
/L
n
cosh[
(W
P
−
x
P
)/L
n
]
+
S
BSF
sinh[
(W
P
−
x
P
)/L
n
]
(3.128)
These are very general expressions for the dark saturation current and reduce to more
familiar forms when appropriate assumptions are made, as will be seen later.
I
o
2
is the dark saturation current due to recombination in the space-charge region,
I
o
2
=
qA
W
D
n
i
τ
D
(
3
.
129
)
and is bias-dependent since the depletion width,
W
D
, is a function of the applied voltage
(equation 3.89).
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