178
6 Band Structure
used is
m d,e = g
2/3
v
m
2
t m l
1/3 .
(6.72)
In the case of a degeneracy of the valence band, the states of several bands need to be summed. In bulk
material, typically the heavy and light hole bands are degenerate at the -point. If the split-off band is
not populated because of insufficient temperature, the valence-band edge density of states is expressed
by the density of states hole mass
m d,h =
m
3/2
hh + m
3/2
lh
2/3 .
(6.73)
The density of states (per volume) at the conduction and valence band edges are thus given by
D
3D
e (E) =
1
2π 2
2 m d,e
2
3/2
E − E C , E > E C
(6.74)
D
3D
h (E) =
1
2π 2
2 m d,h
2
3/2
E V − E , E < E V .
(6.75)
6.13.3.2 M = 2
This case is important for thin layers in which the electron motion is confined in one direction and free
in a plane. Such structures are called quantum wells (see Sect. 12.3.2). We find for the 2D density of
states (for each subband over which it is not summed here, including spin degeneracy)
N
2D
=
A
2π
k
2
F =
A
π
m
∗
2 E ,
(6.76)
where A is the area of the layer. The density of states is thus constant and given by
D
2D
(E) =
A
π
m
∗
2 .
(6.77)
6.13.3.3 M = 1
The case M = 1 describes a quantum wire in which the electron motion is confined in two dimensions
and free in only one dimension. For this case, we find for a wire of length L
N
1D
=
2L
π
k F =
2L
π
2m
∗ E
2
1/2
.
(6.78)
The density of states becomes singular at E = 0 and is given by (for one subband)
D
1D
(E) =
L
π
2m
∗
2
1/2 1
√
E
.
(6.79)
6.13.3.4 M = 0
In this case electrons have no degrees of freedom, as, e.g., in a quantum dot (Sect. 14.4), and each state
has a δ-like density of states at each of the quantized levels.