B. Kollmitzer and P. Hadley
Institute of Solid State Physics, Graz University of Technology, Graz, Austria
The electronic band structure of three-dimensional separable potentials can be efficiently calculated to determine the temperature dependence of thermodynamic properties such as the specific heat or the entropy. We have considered 10000 classes of cubic separable potentials. Depending on the amplitude and lattice constant of the potentials, they correspond to metals, semiconductors or insulators. The numerically calculated band structure, density of states, and thermodynamic properties for the 10000 classes of potentials can be viewed by following the link below.
The band structure, density of states, and temperature dependence of the thermodynamic quantities.
The thermodynamic properties of metals can be estimated using the Sommerfeld expansion if the density of states at the Fermi energy and the derivative of the density of states at the Fermi energy are known. Plots of these quantities for the 10000 classes of potentials and ten electron densities can be viewed by following the link below.
The density of states at the Fermi energy and the derivative of the density of states at the Fermi energy.
The plots of the dispersion relations were generated by first determining k(E) for the one-dimensional cases using equation 11 of the main text. The k(E) relationships were fit to smooth functions so that they could be inverted to yield E(k). The three-dimensional dispersion relation was constructed from three one-dimensional dispersion relations, E(kx,ky,kz) = Ex(kx) + Ey(ky) + Ez(kz).
The density of states in one-dimension is given by equation 12 of the main text. The three-dimensional density of states is the convolution of the three one-dimensional densities of states. A fast Fourier transform algorithm was used to perform the convolutions.
The chemical potential μ was calculated as a function of temperature by finding the root of ∫D(E)F(E,μ,T)dE - n using a modified secant method. The internal energy was then calculated directly via its definition (equation 16 of the main text) for the same temperature grid points as the chemical potential. The specific heat is the derivative of the internal energy density with respect to temperature. This derivative was approximated by a difference quotient. The entropy density was then determined by integrating s = ∫(cv/T)dT. Finally the Helmholtz free energy density was determined by the relation f = u - Ts. This was checked by also calculating the Helmholtz free energy density directly from its definition (equation 17 of the main text).
To determine the band gaps, the roots of α = ± 2 (equation 8 of the main text) were determined for each the one-dimensional potential. These roots determine the band edges which separate energies of allowed electron states from energies of forbidden electron states. The band gaps of the one-dimensional potential can be used to construct the band gaps of the three-dimensional potentials.
For semiconductors, the effective masses of the electrons and holes were determined by the slope of α(E) at the band edges (see equation 26 of the main text). This was approximated with the difference quotient at the band edge.
As the amplitude of the potential becomes larger, the bands get narrower and the density of states contains many sharp peaks. This makes it more difficult to numerically determine the thermodynamic quantities. For the semiconducting materials we were able to check our numerical integration of the density of states. The integral of the density of states over the lowest band should result in an electron density of 2 electrons/primitive cell and the integral of the density of states over the first four bands should result in an electron density of 8 electrons/primitive cell. If the numerically determined electron density was more than 10% from the correct value, we do not present the calculations of the thermodynamic properties. This happens for the largest normalized potential amplitudes that we considered.