PHY.F20 Molecular and Solid State Physics

Empty lattice approximation

The energy-momentum dispersion relation for a free electron gas is a paraboloid.

If the electrons move through a crystal, they will diffract if their k vector falls on a Brillouin zone boundary. In this process their crystal momentum can be changed by a reciprocal lattice vector k → k + G without changing their energy. For a weak periodic potential, the dispersion relation will be nearly a paraboloid except near the Brillouin zone boundaries where it will bend to strike the boundaries at 90°. The dispersion relation in a reduced zone scheme can be constructed by drawing a collection of paraboloids, each centered around a reciprocal lattice point.

Cross sections of this collections of paraboliods are taken in the high symmetry directions of the Brillouin zone to produce the dispersion relation. The resulting band structures for simple cubic, face centered cubic, body centered cubic, hexagonal, and tetragonal crystals are ploted below in various high symmetry directions of k-space. The Brilluoin zones with the symmetery points are shown to the right of the plots.

1-D

2-D square

2-D hexagonal

Simple cubic

Face centered cubic

Body centered cubic

Hexagonal

Tetragonal

Orthorhombic

Some examples of the utility of the empty lattice approximation are shown below.


On the left is the empty lattice approximation for an fcc crystal and on the right is the calculated band structure of aluminum (an fcc metal). The right image was taken from W.A. Harrison, Physical Review, vol. 118 pp. 1182-1189 (1960).