Dynamic Structure Factor of Liquid and Amorphous Ge From Ab Initio Simulations

We calculate the dynamic structure factor $S(k,\ensuremath{\omega})$ of liquid Ge $(l$-Ge) at temperature $T=1250\mathrm{}\mathrm{K},$ and of amorphous Ge $(a$-Ge) at $T=300\mathrm{K},$ using ab initio molecular dynamics. The electronic energy is computed using density-functional theory, primarily in the generalized gradient approximation, together with a plane-wave representation of the wave functions and ultrasoft pseudopotentials. We use a 64-atom cell with periodic boundary conditions, and calculate averages over runs of up to about 16 ps. The calculated liquid $S(k,\ensuremath{\omega})$ agrees qualitatively with that obtained by Hosokawa et al. [Phys. Rev. B 63, 134205 (2001)] using inelastic x-ray scattering. In a-Ge, we find that the calculated $S(k,\ensuremath{\omega})$ is in qualitative agreement with that obtained experimentally by Maley et al. [Phys. Rev. Lett. 56, 1720 (1986)]. Our results suggest that the ab initio approach is sufficient to allow approximate calculations of $S(k,\ensuremath{\omega})$ in both liquid and amorphous materials.