Set of supplementary notes.

Chapter 7
Semiconductors
7.1 Semiconductor band structure
Direct gap semiconductors
The band structure near k = 0 of a diamond-structure (Si, Ge) or zincblende-structure (GaAs)
semiconductor is shown in Fig. 7.1. The conduction band is a simple parabola, but the valence
bands are more complex. The complexity arises because the symmetry of the valence bands is
p − like and there are three degenerate bands (in cubic symmetry) at k = 0. At finite k they
split into light hole and heavy hole bands, so called because of the difference in the electron
masses. Additionally, there is a deeper lying band, split off by spin-orbit interactions from the
others. This is usually not important for thermally excited carriers.
The band masses are quite different from free electron masses, for example, in GaAs m ∗ e =
0.066, m ∗ lh = 0.082, m∗ hh = 0.17 (in units of the free electron mass). The cubic symmetry of the
crystal means that the bands are isotropic (to order k 2 ).
Indirect gap semiconductors
As we remarked earlier, the while there is a local minimum at the origin (Γ−-point), the
conduction bands of Si and Ge do not have their global minima at the Γ-point, but far out in
the zone.
The conduction band minima of Ge are at the eight equivalent L-points 2π/a(0.5 0.5 0.5),
on the surface of the Brillouin zone. Here the band edges have a spheroidal energy surface, and
are not isotropic as near the centre of the zone. In Ge, the longitudinal mass – along (111) –
is m l = 1.59 m, much larger than the tranverse mass m t = 0.082 m.
In Si the conduction band minima are along the six (100) directions, close to the zone
boundary at X [2π/a(100)]. The constant energy surfaces are ellipsoids, m l = 0.92 m, and
m t = 0.19 m.
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