Set of supplementary notes.

76 CHAPTER 5. BANDSTRUCTURE OF REAL MATERIALS
• You will also often see particular bands labelled either along lines or at points by greek
or latin capital letters with a subscript. These notations label the group representation
of the state (symmetry) and we won’t discuss them further here.
Density of states
We have dealt earlier with the density of states of a free electron band in 2. The maxima
E max and minima E min of all bands must have a locally quadratic dispersion with respect to
momenta measured from the minima or maxima. Hence the density of states (in 3D) near the
minima will be the same
g(E > ∼ E min ) = V (
m ∗ 2m ∗ (E − E min )
π 2 h¯ 2 h¯ 2
) 1
2
. (5.10)
as before, with now however the replacement of the bare mass by an effective mass m ∗ =
(m ∗ xm ∗ ym ∗ z) 1/3 averaging the curvature of the bands in the three directions 3 . A similar form
must apply near the band maxima, but with now g(E) ∝ (E max − E) 1 2 . Notice that the flatter
the band, the larger the effective mass, and the larger the density of states 4 .
Since every band is a surface it will have saddle points (in two dimensions or greater) which
are points where the bands are flat but the curvature is of opposite signs in different directions.
Examples of the generic behaviour of the density of states in one, two and three dimensions
are shown in Fig. 5.1. The saddle points give rise to cusps in the density of states in 3D, and
a logarithmic singularity in 2D.
For any form of E(k), the density of states is
g(E) = ∑ n
g n (E) = ∑ n
∫
dk
4π 3 δ(E − E n(k)) , (5.11)
Because of the δ-function in (5.11), the momentum integral is actually over a surface in k-space S n
which depends on the energy E; S n (E F ) is the Fermi surface. We can separate the integral in k into
a two-dimensional surface integral along a contour of constant energy, and an integral perpendicular to
this surface dk ⊥ (see Fig. 5.2). Thus
g n (E) =
=
∫
∫
S n(E)
S n(E)
∫
dS
4π 3
dk ⊥ (k) δ(E − E n (k))
where ∇ ⊥ E n (k) is the derivative of the energy in the normal direction. 5
dS
4π 3 1
|∇ ⊥ E n (k)| , (5.12)
Notice the appearance of the gradient term in the denominator of (5.12), which must vanish at the
edges of the band, and also at saddle points, which exist generically in two and three dimensional bands.
Maxima, minima, and saddle points are all generically described by dispersion (measured relative to the
3 Since the energy E(k) is a quadratic form about the minimum, the effective masses are defined by h¯ 2
m ∗ α
∣
along the principal axes α of the ellipsoid of energy.
∂ 2 E(k)
∂k
∣
α
2 kmin
4 The functional forms are different in one and two dimensions.
5 We are making use of the standard relation δ(f(x) − f(x 0 )) = δ(x − x 0 )/|f ′ (x 0 )|
=