Dynamic and static structure factor of electrons in Si: Inelastic x-ray scattering results

The dynamic structure factor S(q,ω) of electrons in single-crystal Si was measured with 1.6-eV resolution by means of inelastic x-ray scattering spectroscopy for ${\mathbf{q}}_{\mathrm{\parallel }}$ [100], [110], and [111] with 0.37<q<2.06 a.u. using synchrotron radiation from the DORIS storage ring. By utilizing the f-sum rule, S(q,ω) could be brought on to an absolute scale, so that also the static structure factor S(q) could be obtained. The orientation-averaged features of the dynamic and the static structure factor, such as dispersion, width, and shape of the spectra, could be brought in reasonable agreement with jellium calculation, when we went beyond the random-phase approximation by taking into account both exchange corrections via a static local-field factor and momentum-dependent lifetime in an on-shell approximation of the self-energy. Results of fitting of the static local-field factor to the experiment are presented. The rich q-orientation-dependent fine structure found in the experimental S(q,ω) spectra either could be attributed to the enhanced density of states on zone boundaries in the extended zone scheme, or could be understood in terms of plasmon-Fano resonances, which are the result of a plasmon-band-induced coupling between the continuum of electron-hole excitations and the discrete plasmon resonances. This interpretation is supported by pseudopotential calculations within the limits of the two-plasmon-band model.