Heyd-Scuseria-Ernzerhof hybrid functional for calculating the lattice dynamics of semiconductors

We present an ab initio study of the lattice dynamics of group-IV elemental semiconductors and insulators using a finite differences approach. The investigated solids include cubic diamond (C), silicon (Si), germanium (Ge), and the zero-gap semiconductor gray tin $(\ensuremath{\alpha}\text{-Sn})$. The main objective of this work is to examine the performance of the screened hybrid functional (HSE) proposed by Heyd, Scuseria, and Ernzerhof [J. Chem. Phys. 118, 8207 (2003); J. Chem. Phys. 124, 219906(E) (2006)] for calculating phonon-dispersion relations. We find that all local and semilocal functionals tend to underestimate the phonon frequencies, with the errors increasing with increasing atomic mass. For $\ensuremath{\alpha}\text{-Sn}$, semilocal functionals even qualitatively fail to describe the dispersion of the highest optical phonon mode. We show that this is related to semilocal functionals predicting $\ensuremath{\alpha}\text{-Sn}$ to be a metal, whereas experimentally it is a zero-gap semiconductor. The HSE functional yields the correct electronic band structure resulting in qualitatively correct phonon-dispersion relations for all four solids. Quantitatively, the phonon frequencies are slightly overestimated using HSE, in particular for the lighter elements C and Si. Our results are compared to previously reported theoretical findings.