Consider an electron confined to a potential of the form,

$$V(f_{\text{square}}(x) + f_{\text{square}}(y) + f_{\text{square}}(z)),$$

where $V = V_2 - V_1$ is the amplitude of the potential and $f_{\text{square}}(x)$ is a square wave function.

This potential has the advantage that the density of states can be easily calculated for a given set of parameters $V$ and $b/a$. [1]

The thermodynamic properties depend dramatically on the number of electrons per primitive unit cell $n$. For the same dispersion relation and density of states, the crystal can be a metal, a semiconductor, or an insulator depending on the number of electrons per primitive unit cell.

V = 13.3, b = 0.4951, n = 1

            

Dispersion relation		Density of states
Internal energy density u Helmholtz free energy density f		Chemical potential μ
Specific heat cv		Entropy density s

            

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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