10.5 Electrons in Semiconductors
561
or
n ce = n D
n c
2n D
1
2
e
−βE D .
(10.5.40)
From this last result we see that the number density of conduction electrons will
be much smaller than the number of donor levels, i.e.,
n ce n D ,
(10.5.41)
and that the Fermi level will be given by
μ F = −
1
2 E D +
1
2 k B T ln
n D
2n c
1
2 ≈ −
1
2 E D .
(10.5.42)
This result shows us that the Fermi level lies approximately at the middle of
the gap between the bottom of the conduction band and the donor levels, thus
corresponding to a slight ionization of the donors.
• High Temperatures: If the temperature is sufficiently high, then e βE D ≈ 1,
8n D
n c
e βE D 1, and we can employ the Maclaurin expansion (1 + x)
1
2 ≈
1 +
1
2 x −
1
8 x 2 + . . ., so that
n ce
n c
4
e
−βE D
−1 + 1 +
1
2
8n D
n c
e
βE D −
1
8
8n D
n c
e
βE D
2
+ . . .
= n D −
2n 2
D
n c
e
βE D + . . .
or
n ce n D
1 −
2n D
n c
e
βE D + . . .
,
(10.5.43)
while the Fermi level becomes
μ F = −E D + k B T ln
1
4
4n D
n c
e
βE D −
1
8
8n D
n c
e
βE D
2
+ . . .
= −E D + k B T ln
n D
n c
+ E D + k B T ln
1 −
2n D
n c
e
βE D + . . .
,
or
μ F k B T
ln
n D
n c
−
2n D
n c
e
βE D + . . .
.
(10.5.44)