Crystalline Interpolation with Applications to Brillouin-Zone Averages and Energy-Band Interpolation

Abstract

The new interpolation procedure of Shankland is applied to interpolation problems found in a periodic lattice. Two problems, the method of averaging the valence electron density over the Brillouin zone and the method of interpolating energy bands throughout the Brillouin zone from values at symmetry points, are discussed in detail. To illustrate the advantages of this procedure, examples are taken from self-consistent cubic ZnS and ZnSe orthogonalized-plane-wave calculations.

Phonon-limited electron mobility in Si, GaAs, and GaP with exact treatment of dynamical quadrupoles

“…where v nk,α is the α-th component of the matrix element v nk of the electron velocity operator. These quantities can be obtained from interpolation methods such as Wannier [58] or Shankland-Koelling-Wood (SKW) [59][60][61][62] whose form allows one to analytically compute the derivatives of the eigenvalues with respect to k. Alternatively, one can obtain v nk using finite differences between shifted grids [55]. In our implementation, we prefer to compute the velocity matrix elements without any approximation using the commutator of the Hamiltonian with the position operator [63]:…”

“…We start with a NSCF calculation on a reasonably dense k mesh to determine the position of the band edges within a certain tolerance. Then we use the SKW method [59][60][61] to interpolate electron energies on a much denser k grid. This step allows us to identify the k points lying inside a predefined energy window around the band edges.…”

BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients

“…where Λ are so-called stars representing a set of symmetryequivalent lattice vectors. BoltzTraP was based on the idea by Shankland [28][29][30] that the coefficients should be obtained by minimizing a roughness function under the constraints that calculated quasi-particle energies should be exactly reproduced. This in turn means that the number of coefficients should be larger than the number of calculated points.…”

Fourier transformation by smooth interpolation

“…As indicated in Bibliography [l], the basic idea of a smooth interpolant has many applications, of which crystalline interpolation is only one. Other special forms [3]- [5] have been worked out and offer similar calculation gains.…”