Fourier projection algorithm for calculating phonon dispersion relations of crystals

We propose a Fourier projection algorithm and demonstrate its superiority
for calculation of the phonon dispersion relations of 2D and 3D crystals
based on the atomic coordinate trajectories from supercell molecular
dynamics/first-principle molecular dynamics simulations. The lattice
vibration states are described in a six-dimensional phase space composed
of lattice and wave vectors. Phonon dispersion information in the first
Brillouin zone sampled with a k-mesh grid of N×N×N is generated by Fourier
projection of the lattice vibration frequencies in the supercell by N×N×N
primitive cells, the dispersion relations along high-symmetry paths are
retrieved therefrom. Our algorithm is physically intuitive,
computationally convenient, and universally applicable. As examples of
application, the successful dispersion relation calculations of 2D
graphene, 3D BCC-Fe and FCC-Cu are presented. This algorithm shows the
correspondence between real and reciprocal lattices, that is, the spatial
information in the states of the lattice vibration were preserved. The
high-symmetry paths of the first Brillouin zone in the reciprocal lattice
primitive cells of the hexagonal, BCC, and FCC lattice were also given in
detail. In combination with first-principles molecular dynamics, as long
as the interatomic force is accurately known, reliable and accurate
dispersion relation can be achieved by this Fourier projection algorithm.