Set of supplementary notes.

7.2. INTRINSIC CARRIER CONCENTRATION 101
and valence band dispersions are thus (see Fig. 7.2)
E c (k) = E c +
h¯ 2k 2
2m ∗ e
; E v (k) = E v −
h¯ 2k 2
2m ∗ h
(7.1)
We shall need the densities of states for the conduction band
g e (E) = 1 ( ) 2m
∗ 3/2
e
(E − E
2π 2 h¯ 2
c ) 1/2 (7.2)
and for the valence band
g h (E) = 1 ( ) 2m
∗ 3/2
h
(E
2π 2 h¯ 2
v − E) 1/2 (7.3)
We can calculate the carrier density once the chemical potential µ is known. For electrons
in the conduction band
with f the Fermi function
f(E) =
n =
∫ ∞
E c
dE g e (E)f(E) (7.4)
1
e (E−µ)/(k BT )
+ 1 ≈ e−(E−µ)/(k BT )
with the latter approximation valid when E − µ >> k B T (non-degenerate Fermi gas). This
gives
( ) m
∗ 3/2
n ≈ 2 e k B T
e − Ec−µ
k B T
(7.6)
2πh¯ 2
(7.5)
A similar calculation determines the concentration of holes
( ) m
∗ 3/2
p ≈ 2 h k B T
e − µ−Ev
k B T
(7.7)
2πh¯ 2
Note that the prefactors to the Boltzmann factors e − Ec−µ
k B T
absorbed into temperature-dependent concentrations
and e − µ−Ev
k B T
can conveniently be
( ) m
∗ 3/2
n c (T ) = 2 e k B T
(7.8)
2πh¯ 2
( ) m
∗ 3/2
n v (T ) = 2 h k B T
(7.9)
2πh¯ 2
These functions express the (temperature dependent) number of states within range k B T
of the band edge for the conduction and valence band, respectively. The resulting expression
for the number of electrons in the conduction band
n = n c (T )e − Ec−µ
k B T
(7.10)