Phonon dispersion relations for unstrained Si$_{1-x}$Ge$_{x}$ via density functional theory

Phonon dispersions for the Si$_{1-x}$Ge$_{x}$ alloy system are computed using localized basis density functional theory. Including interactions up to third-nearest-neighbors and the effect of atomic randomness, phonon dispersion for the full Brillouin zone of a supercell containing 8 atoms is computed for 8 different compositions. Frequencies are found to be in excellent agreement with available experimental results for both crystalline Si and Ge. Results are compared with a 64-atom supercell calculation for optical frequencies to show any possible effect of supercell size on the phonon calculation. The atoms in the calculation are relaxed to a force tolerance of 0.0001eV/{\AA}, which is found to be important to correctly determine the dispersion near the $\Gamma $- and X-points of the Brillouin zone where q-convergence is harder to achieve. The highest optical phonon frequencies are observed to vary nonlinearly with composition, a fact not investigated before using computational methods.