Environmental Engineering Reference
In-Depth Information
energy entirely to an electron, and in the semiconductor context to create an electron
-
-
hole pair.
The radiation-induced current density of a solar cell is
J
L
¼
L
p
),
where
L
is the diffusion length for the minority carrier and
W
is the junction width.
Looking at Figure 6.1a, when under illumination,
G
will represent the rate at which
electron
-
hole pairs are created by light photons of energy exceeding the local
bandgap energy. To be useful, the carriers have to be generated within a diffusion
length
L
of the junction, where
L
Ge
(
W
þ
L
n
þ
(
D
t
r
)
1/2
. Here t
r
is explicitly the minority carrier
lifetime against recombination.
L
can be rewritten as
L
¼
(t
r
t
s
k
B
T
/
m
)
1/2
making use
¼
of the relations
m¼
e
t
s
/
m
¼
eD
/
k
B
T
. (The scattering lifetime can also be written as
¼
l/
h
v
i
t
s
, using the mean free path and thermal velocity.) A long minority carrier
lifetime is achieved by minimizing defects, which would facilitate recombination of
electrons and holes, with particular attention to recombination at surfaces, as
mentioned above. The total current density is the sum of the light-induced density
and the thermionic current:
J ¼ J
o
½
exp
ðeV=k
B
TÞ
1
Ge ðW þ L
n
þ L
p
Þ:
ð
6
:
1
Þ
The reverse current density
J
o
, as discussed earlier,
is strongly temperature
dependent.
Setting
V
¼
0 (short-circuit condition), the square-bracket term is zero, so we
recognize
J
sc
¼
Ge
ð
W
þ
L
n
þ
L
p
Þ;
ð
:
Þ
6
2
where
W
is the depletion region width.
The open-circuit voltage
V
oc
is obtained by setting
J
¼
0 in Equation 6.1, so that exp
(
eV
oc
/
k
B
T
)
¼
1
J
sc
/
J
o
, and
V
oc
¼ð
k
B
T
=
e
Þ
ln
ð
1
þ
J
sc
=
J
o
Þ:
ð
6
:
3
Þ
From a basic point of view, this value cannot exceed
E
g
/
e
.
The output power per unit area is
JV
, and this is maximized at maximum power
voltage
V
mp
(adjusted by the load), satisfying
q
(
JV
)/
q
V
¼
0, which gives
V
mp
¼
V
oc
ð
k
B
T
=
e
Þ
ln
ð
1
þ
eV
oc
=
k
B
T
Þ;
ð
6
:
4
Þ
and a corresponding current density
J
mp
, slightly reduced from the short-circuit
value.
The ratio between the available power at optimum load and the limiting power is
dened as the lling factor FF
J
mp
V
mp
/
J
sc
V
oc
. In Figure 3.17, the FF is simply the
ratio of the areas of the inner hatched region de
ning the optimum power point and
the larger rectangle, de
ned by the short-circuit current density and the open-circuit
voltage. The basic ef
ciency is
¼
g
¼
J
sc
V
oc
FF
=
P
inc
;
ð
6
:
5
Þ
