We have calculated inelastic mean free paths (IMFPs) for 41 elemental solids (Li, Be, graphite, diamond, glassy C, Na, Mg, Al, Si, K, Sc, Ti, V, Cr, Fe, Co, Ni, Cu, Ge, Y, Nb, Mo, Ru, Rh, Pd, Ag, In, Sn, Cs, Gd, Tb, Dy, Hf, Ta, W, Re, Os, Ir, Pt, Au and Bi) for electron energies from 50 eV to 30 keV. The IMFPs were calculated from experimental optical data using the full Penn algorithm for energies up to 300 eV and the simpler single-pole approximation for higher energies. The calculated IMFPs could be fitted to a modified form of the Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 30 keV. The average root-mean-square (RMS) deviation in these fits was 0.48%. The new IMFPs were also compared with IMFPs from the predictive TPP-2M equation; in these comparisons, the average RMS deviation was 12.3% for energies between 50 eV and 30 keV. This RMS deviation is almost the same as that found previously in a similar comparison for the 50 eV–2 keV range. Relatively large RMS deviations were found for diamond, graphite and cesium. If these three elements were excluded in the comparison, the average RMS deviation was 9.2% between 50 eV and 30 keV. We found satisfactory agreement of our calculated IMFPs with IMFPs from recent calculations and from elastic-peak electron-spectroscopy experiments. Copyright © 2010 John Wiley & Sons, Ltd.

Calculations of electron inelastic mean free paths. IX. Data for 41 elemental solids over the 50 eV to 30 keV range

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

/pdf/dynamic-structure-factor-and-dielectric-function-of-silicon-1lzcd5vy8y.pdf

Dynamic structure factor and dielectric function of silicon for finite momentum transfer: Inelastic x-ray scattering experiments and ab initio calculations

We present an investigation of the dynamic structure factor and of the dielectric function ${\ensuremath{\epsilon}}_{M}(\mathbf{Q},\ensuremath{\omega})$ of the prototypical semiconductor silicon for finite momentum transfer, combining inelastic x-ray scattering experiments and ab initio calculations. In contrast with optical spectra, for finite momentum transfer time-dependent density-functional theory in the adiabatic local-density approximation together with lifetime broadening describes the physics of valence excitations correctly. Major structures in the spectra, governed by short-range crystal and exchange-correlation local-field effects, are strongly influenced by a mixing of transitions of positive and negative energies, in striking difference to spectra for vanishing momentum transfer. This mixing gives rise to a pronounced Fano asymmetry.

/pdf/signatures-of-short-range-many-body-effects-in-the-agqs6vdzch.pdf

Signatures of short-range many-body effects in the dielectric function of silicon for finite momentum transfer.

The Liouville-Lanczos approach to linear-response time-dependent density-functional theory is generalized so as to encompass electron energy loss and inelastic x-ray scattering spectroscopies in periodic solids. The computation of virtual orbitals and the manipulation of large matrices are avoided by adopting a representation of response orbitals borrowed from (time-independent) density functional perturbation theory and a suitable Lanczos recursion scheme. The latter allows the bulk of the numerical work to be performed at any given transferred momentum only once, for a whole extended frequency range. The numerical complexity of the method is thus greatly reduced, making the computation of the loss function over a wide frequency range at any given transferred momentum only slightly more expensive than a single standard ground-state calculation and opening the way to computations for systems of unprecedented size and complexity. Our method is validated on the paradigmatic examples of bulk silicon and aluminum, for which both experimental and theoretical results already exist in the literature.

/pdf/electron-energy-loss-and-inelastic-x-ray-scattering-cross-1q3iv2qbq5.pdf

Electron energy loss and inelastic x-ray scattering cross sections from time-dependent density-functional perturbation theory

An atomiclike basis representation is used to analyze the dielectric function $\ensuremath{\epsilon}(\stackrel{P\vec}{q}+\stackrel{P\vec}{G},\stackrel{P\vec}{q}+{\stackrel{P\vec}{G}}^{\ensuremath{'}};\ensuremath{\omega})$ of Si. First, we show that a $s{p}^{3}{d}^{5}$ local basis set yields good results for the electronic band structure of this crystal and, then, we analyze the Si optical properties including local field and excitonic effects. In our formulation, we follow Hanke and Sham [W. Hanke and L. J. Sham, Phys. Rev. B 12, 4501 (1975); Phys. Rev. B 21, 4656 (1980)], and introduce excitonic effects using a many-body formulation that incorporates a static screened electron-hole interaction. Dynamical effects in this interaction are also analyzed and shown to introduce non-negligible corrections in the optical spectrum. Our results are found in reasonable agreement with the experimental evidence and with other theoretical results calculated with the computationally more demanding plane-wave representation. Finally, calculations for the stopping power of Si are also presented.

Si dielectric function in a local basis representation: Optical properties, local field effects, excitons, and stopping power

The inverse microscopic dielectric matrix of crystalline silicon is evaluated within the time-dependent density-functional theory using one-electron energies and wave functions obtained by all-electron modified augmented-plane-wave band-structure calculations. Local fields are taken into account by inverting the dielectric matrix. The dependency of the exchange and correlation on the frequency and momentum is described using different approximations. We find that the polarization of the 2s and 2p core electrons changes the inverse dielectric function by up to 8%. In the long-wavelength limit our results for the region below 5 eV deviate from optical experiments by an almost constant energy shift, which is the same size as the band-gap error in local density approximation. Our results correspond with inelastic x-ray scattering spectroscopy measurements.

https://iopscience.iop.org/article/10.1088/0953-8984/9/6/009/pdf

Calculation of the dielectric matrix of Si

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Calculation of the dielectric matrix of Si