The dispersion relation and the density of states for a simple cubic crystal are plotted below. The amplitude $V$ of the periodic potential can be adjusted with a slider. At $V=0$, the dispersion relation is the empty lattice approximation, and the density of states increases like the square root of the energy. As $V$ increases, the dispersion curve strikes the Brillouin zone boundaries at 90° and the density of states develops some structure. For large values of $V$, band gaps appear in the dispersion relation, and the density of states goes to zero for some range of energies.

$V=$ 10.1 [eV] 

Dispersion relation

Density of states

This calculation was done for a separable square wave potential with a simple cubic Bravais lattice. This potential was chosen because it is easy to calculate the density of states for this case. [1] Other three-dimensional periodic potentials exhibit qualitatively the same behavior as the amplitude of the potential is increased from zero.

There are vertical red lines in the density of states plot, which show where the Fermi energy would be for one electron per primitive unit cell up to ten electrons per primitive unit cell. For large amplitudes of the potential, and two electrons per unit cell, the Fermi energy falls in the band gap. In this case, all of the states below the band gap are filled, and all of the states above the band gap are empty. When the filled states are separated from the empty states by a band gap of more than 3 eV, the material is an insulator. For this size band gap, there are no electrons that get thermally activated across the band gap at room temperature. A consequence of this is that electrons have an insignificant contribution to the thermodynamic properties of insulators.

A semiconductor, like an insulator, has a band gap that separates the occupied electron states from the empty electrons, but the band gap of a semiconductor is smaller so that there is some thermal activation of electrons from the occupied band (called the valence band) to the empty band above the band gap (called the conduction band).

Calculations of the thermodynamic properties were calculated using the density of states and the results are plotted at https://lampz.tugraz.at/~hadley/ss1/book/bands/thermo.php. For a given dispersion relation and density of states, the thermodynamic properties can vary dramatically depending on the number of electrons in the primitive unit cell.

• Metal: The chemical potential is in a band so that there are empty electron states with energies just above the occupied electron states.
• Semiconductor: The chemical potential lies in a band gap and the band gap is smaller than 3 eV.
• Insulator: The chemical potential lies in a band gap and the band gap is larger than 3 eV.

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1. B. Kollmitzer, P. Hadley, Thermodynamic properties of separable square-wave potentials, Physica B: Condensed Matter, Volume 406, Issue 23, 2011, Pages 4373-4380, https://doi.org/10.1016/j.physb.2011.08.089