144
6 Band Structure
Fig. 6.8 Schematic band structures with (a) topologically trivial wave functions and c topologically non-trivial wave
functions where within one band the character changes from s (blue, ‘+’ for positive parity) to p (red, ‘−’ for negative
parity). In b the crossing point is visualized
states are separated by a gap. For a three-dimensional crystal and k-space several topological invariants
exist but a Chern number can be assigned to the Fermi surface or surface states (cmp. Sect. 11.6.3).
As an schematic example we show Fig. 6.8 for a topological trivial and non-trivial band structure.
In the trivial bandstructure (as most semiconductors), the phase character of the wavefunction changes
only little within the Brillouin zone, mostly p-type (negative parity) for the valence band and mostly stype (positive parity) for the conduction band (Fig. 6.8a). In the topologically non-trivial bandstructure,
band inversion takes place and the character of the wave function changes within a band (Fig. 6.8c).
This sketch should be compared to Fig. 5.8 where a similar situation had been discussed for the lattice
vibrations. An example for a semiconductor with band inversion is HgTe while CdTe or MnTe have
trivial topology. Alloying leads at the transition from trivial to non-trivial (Fig. 6.8b) to zero-gap
semiconductors (cf. Sect. 6.11).
6.3 Band Structures of Selected Semiconductors
In the following, the band structures of various important and prototype semiconductors are discussed.
The band below the energy gap is called the valence band; the band above the gap is the conduction
band. The band gap cv , mostly denoted as E g , is the energy separation between the highest valenceband state and the lowest conduction-band state. The maximum of the valence band is for most
semiconductors at the point.
6.3.1 Silicon
For silicon, an elemental semiconductor, (Fig. 6.9a) the minimum of the conduction band is located
close to the X-point at 0.85π/a in the 100 direction. Thus, it is not at the same point in k space as the
top of the valence band. Such a band structure is called indirect. Since there are six equivalent 100
directions, there are six equivalent minima of the conduction band.
6 Band Structure
Fig. 6.8 Schematic band structures with (a) topologically trivial wave functions and c topologically non-trivial wave
functions where within one band the character changes from s (blue, ‘+’ for positive parity) to p (red, ‘−’ for negative
parity). In b the crossing point is visualized
states are separated by a gap. For a three-dimensional crystal and k-space several topological invariants
exist but a Chern number can be assigned to the Fermi surface or surface states (cmp. Sect. 11.6.3).
As an schematic example we show Fig. 6.8 for a topological trivial and non-trivial band structure.
In the trivial bandstructure (as most semiconductors), the phase character of the wavefunction changes
only little within the Brillouin zone, mostly p-type (negative parity) for the valence band and mostly stype (positive parity) for the conduction band (Fig. 6.8a). In the topologically non-trivial bandstructure,
band inversion takes place and the character of the wave function changes within a band (Fig. 6.8c).
This sketch should be compared to Fig. 5.8 where a similar situation had been discussed for the lattice
vibrations. An example for a semiconductor with band inversion is HgTe while CdTe or MnTe have
trivial topology. Alloying leads at the transition from trivial to non-trivial (Fig. 6.8b) to zero-gap
semiconductors (cf. Sect. 6.11).
6.3 Band Structures of Selected Semiconductors
In the following, the band structures of various important and prototype semiconductors are discussed.
The band below the energy gap is called the valence band; the band above the gap is the conduction
band. The band gap cv , mostly denoted as E g , is the energy separation between the highest valenceband state and the lowest conduction-band state. The maximum of the valence band is for most
semiconductors at the point.
6.3.1 Silicon
For silicon, an elemental semiconductor, (Fig. 6.9a) the minimum of the conduction band is located
close to the X-point at 0.85π/a in the 100 direction. Thus, it is not at the same point in k space as the
top of the valence band. Such a band structure is called indirect. Since there are six equivalent 100
directions, there are six equivalent minima of the conduction band.