Dynamic structure factor and dielectric function of silicon for finite momentum transfer: Inelastic x-ray scattering experiments andab initiocalculations

Abstract

We present a detailed investigation of the dynamic structure factor $S(\mathbf{Q},\ensuremath{\omega})$ as well as of the dielectric function ${\ensuremath{\epsilon}}_{\text{M}}(\mathbf{Q},\ensuremath{\omega})$ of the prototypical semiconductor silicon for finite momentum transfer, combining inelastic x-ray scattering measurements and ab initio calculations. We show that, in contrast to optical spectra, for finite momentum transfer, time-dependent density-functional theory in adiabatic local-density approximation (TDLDA) together with the inclusion of lifetime effects in a modified independent-particle polarizability ${\ensuremath{\chi}}^{0,\text{LT}}$ describes the physics of valence excitations with high precision. This applies to the dynamic structure factor as well as to the dielectric function, which demonstrates that TDLDA contains the short-range many-body effects that are crucial for a correct description of ${\ensuremath{\epsilon}}_{\text{M}}(\mathbf{Q},\ensuremath{\omega})$ in silicon at finite momentum transfer. The form of a nonlocal and energy-dependent exchange-correlation kernel is presented which provides the inclusion of the lifetime effects using the true independent-particle polarizability ${\ensuremath{\chi}}^{0}$. The description of the silicon ${L}_{2,3}$ absorption edge has been possible by including the outer core electrons $2s$ and $2p$ in the valence electrons of the pseudopotential. The energy of the edge is underestimated but a scissor shift of the respective states by the self-energy correction for these states yields good agreement with experiment. Short-range crystal local-field effects and exchange-correlation effects become important with increasing momentum transfer. The inclusion of crystal local-field effects in the random-phase approximation is able to describe the anisotropy of the response well. Our results demonstrate the quantitative predictive power of the first-principles description.