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3.2.9 Minority-carrier Diffusion Equation
In a uniformly doped semiconductor, the band gap and electric permittivity are inde-
pendent of position. Since the doping is uniform, the carrier mobilities and diffusion
coefficients are also independent of position. As we are mainly interested in the steady
state operation of the solar cell, the semiconductor equations reduce to
d
E
d
x
=
q
ε
(p
−
n
+
N
D
−
N
A
)
(3.75)
qµ
p
d
d
x
(p
E)
−
qD
p
d
2
p
d
x
2
=
q(G
−
R)
(3.76)
and
qµ
n
d
d
x
(n
E)
+
qD
n
d
2
n
d
x
2
=
q(R
−
G)
(
3
.
77
)
In regions sufficiently far from the
pn
-junction of the solar cell (quasi-neutral regions),
the electric field is very small. When considering the minority carrier (holes in the
n
-
type material and electrons in the
p
-type material) and low-level injection (
p
=
n
N
D
, N
A
), the drift current can be neglected with respect to the diffusion current. Under
low-level injection,
R
simplifies to
R
=
n
P
−
n
P o
τ
n
=
n
P
τ
n
(
3
.
78
)
in the
p
-type region and to
R
=
p
N
−
p
N o
τ
p
=
p
N
τ
p
(
3
.
79
)
P N
-JUNCTION DIODE ELECTROSTATICS
83
in the
n
-type region.
p
N
and
n
P
are the excess minority-carrier concentrations. The
minority-carrier lifetimes,
τ
n
and
τ
p
, are given by equation (3.48). For clarity, the cap-
italized subscripts, “
P
” and “
N
”, are used to indicate quantities in
p
-type and
n
-type
regions, respectively, when it may not be otherwise apparent. Lower-case subscripts, “
p
”
and “
n
”, refer to quantities associated with minority holes and electrons, respectively.
Equations (3.76) and (3.77) thus each reduce to what is commonly referred to as the
minority-carrier diffusion equation
. It can be written as
D
p
d
2
p
N
d
x
2
−
p
N
τ
p
= −
G(x)
(
3
.
80
)
in
n
-type material and as
D
n
d
2
n
P
d
x
2
−
n
P
τ
n
= −
G(x)
(
3
.
81
)
in
p
-type material. For example,
n
P
is the minority electron concentration in the
p
-type
material. The minority-carrier diffusion equation is often used to analyze the operation of
semiconductor devices, including solar cells.
3.3
PN
-JUNCTION DIODE ELECTROSTATICS
Where an
n
-type semiconductor comes into contact with a
p
-type semiconductor, a
pn
-
junction is formed. In thermal equilibrium there is no net current flow and by definition the
Fermi energy must be independent of position. Since there is a concentration difference
of holes and electrons between the two types of semiconductors, holes diffuse from the
p
-
type region into the
n
-type region and, similarly, electrons from the
n
-type material diffuse
into the
p
-type region. As the carriers diffuse, the charged impurities (ionized acceptors
in the
p
-type material and ionized donors in the
n
-type material) are uncovered – that is,
no longer screened by the majority carrier. As these impurity charges are uncovered, an
electric field (or electrostatic potential difference) is produced, which limits the diffusion
of the holes and electrons. In thermal equilibrium, the diffusion and drift currents for
each carrier type exactly balance, so there is no net current flow. The transition region
between the
n
-type and the
p
-type semiconductors is called the
space-charge region
. It
is also often called the
depletion region
, since it is effectively depleted of both holes
and electrons. Assuming that the
p
-type and the
n
-type regions are sufficiently thick, the
regions on either side of the depletion region are essentially charge-neutral (often termed
quasi-neutral
). The electrostatic potential difference resulting from the junction formation
is called the
built-in voltage
,
V
bi
. It arises from the electric field created by the exposure
of the positive and the negative space charge in the depletion region.
The electrostatics of this situation (assuming a single acceptor and a single donor
level) are governed by Poisson’s equation
∇
2
φ
=
q
ε
(n
o
−
p
o
+
N
−
A
−
N
+
D
)
(
3
.
82
)
84
THE PHYSICS OF THE SOLAR CELL
0
p
-type
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
− −
− −
− −
− −
− −
− −
− −
− −
− −
− −
− −
Depletion
region
n
+
−
W
N
W
P
−
x
N
x
P
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