Phonon dispersions of silicon and germanium from first-principles calculations

We present the calculation of the full phonon spectrum for silicon and germanium with the pseudopotential method and the local-density approximation without using linear-response theory. The interplanar-force constants for three high-symmetry orientations (100), (110), and (111) are evaluated by supercell calculations using the Hellmann-Feynman theorem. By considering the symmetry of the crystal, three-dimensional interatomic-force-constant matrices are determined by a least-squares fit. Interactions up to the eighth nearest neighbors are included. The dynamical matrix, which is the Fourier transform of the force constant matrix, is hence constructed and diagonalized for any arbitrary wave vector in the Brillouin zone, yielding the phonon dispersion. In this paper we will present the calculation details and discuss various aspects of convergence. Phonon dispersions of Si and Ge calculated are in excellent agreement with experiments.