6.13 Density of States
177
Fig. 6.56 Theoretical
calculation for the density
of states of crystalline
(dashed lines, conduction
and valence bands
indicated by greay areas)
and amorphous ZnSnO 3
with different
configurations (solid lines).
States due to
under-coordinated oxygen
(O uc ) and metal-metal
bonds are labelled.
Adapted from [552]
Sn-Sn Sn-Zn 2Sn 1Sn
Density of states (eV )
-1
60
40
20
0
- 1
0
1
2
3
4
5
Energy (eV)
ZnSnO 3
O uc
The factor 2 is for spin degeneracy, the integration runs over M dimensions. The density of states is
the derivative
D(E) =
dN
dE
.
(6.67)
In the following, the density of states for M = 3, 2, 1 and zero dimensions is derived. A visualization
is given in Fig. 14.1.
6.13.3.1 M = 3
This case relates to bulk material in which electrons are free to move in all three dimensions. Performing
the integral (6.66) for M = 3 yields for an isotropic mass,
N
3D
=
V
3π 2 k
3
F =
V
3π 2
2 m E F
2
3/2
.
(6.68)
Therefore, k F and E F are given by
k F =
3π
2 N
V
1/3
(6.69)
E F =
2
2 m ∗
3π
2 N
V
2/3
,
(6.70)
and the density of states in three dimensions is
D
3D
(E) =
V
2π 2
2 m
∗
2
3/2 √
E .
(6.71)
Mostly the density of states is used as density of states per volume, then the factor V in (6.71) is
omitted.
If a conduction-band minimum is degenerate, a factor g v (valley degeneracy) must be included in
the density of states, i.e. g v = 6 for Si and g v = 8 for Ge (g v = 1 for GaAs). This factor is typically
included in the mass used in (6.71) that then becomes the density of states mass m d,e . If the conductionband minimum has cylindrical symmetry in k-space, such as for Si and Ge, the mass that has to be