Environmental Engineering Reference
In-Depth Information
Formally, this would be
2
=
3
E
FN
E
G
þðh
2
=
8
mÞð
3
N
D
=pÞ
:
ð
3
:
59
Þ
So, heavily doped InAs is really a metal, and we would expect that the number of
free electrons will not change appreciably if this sample is cooled to a low
temperature.
Finally, to complete the discussion of heavily doped InAs, making use of the mass
action law
N
e
N
h
¼N
i
at 300 K, we
nd that
N
h
¼
(1.22
10
20
)
2
/10
24
¼
1.49
10
16
/
m
3
¼
1.49
10
10
/cc. This minority concentration of holes ismuch smaller than the
electron concentration, 10
18
/cc. The minority concentration will change rapidly with
varying temperature, according to exp(
0.33
eV
/
k
B
T
).
3.7
The PN Junction, Diode I
-
V Characteristic, Photovoltaic Cell
To begin thinking about a PN junction in Si, we represent the semiconductor by two
horizontal lines, the conduction band edge at
E¼E
c
and the valence band edge
E
V
¼
0, at a distance
E
G
¼
1.1 eV below. The Fermi energy is indicated by a dashed
line, and for a pure Si sample it will be near the middle of the gap at 0.55 eV, as
sketched in Figure 3.16a.
To introduce a PN junction, imagine at the center of our picture of the two
horizontal lines, we make a transition to P-type material on the left and to N-type
material on the right.
In the N-type region, using Equation 3.54, we can see that the Fermi level is given
by
E
FN
¼ E
C
k
B
T
ln
ðN
C
=
N
e
Þ;
ð
3
:
60
Þ
while on the P-type side we have
E
FP
¼ k
B
T
ln
ðN
V
=N
h
Þ:
ð
3
:
61
Þ
In such an imagined
abrupt junction
, electrons will quickly
flow from donor levels
on the right to negatively ionize acceptor levels on the left until the bands have shifted
to align the Fermi level as constant across the structure. The shift will be
eV
B
, called
the
band bending
, and
V
B
the
built-in potential
. The shift will be
E
C
k
B
T½
ln
ðN
C
=
N
e
Þþ
ln
ðN
V
=
N
h
Þ ¼ eV
B
:
ð
3
:
62
Þ
This will have the effect of shifting the bands on the left upward and shifting the bands on
the right downward
. The voltage
V
B
called the built-in potential is typically near 0.6 V
for Si, depending on doping levels. The width
W
over which the shift occurs is called
the
depletion region width
:
1
=
2
W ¼½
2
ee
o
ðV
B
VÞðN
D
þN
A
Þ=
eðN
D
N
A
Þ
;
ð
3
:
63
Þ
