Excitons on a microscopic level: The mixed dynamic structure factor

The dynamic structure factor of materials is proportional to their linear electronic response and it displays their excitation spectra. Usually the response is measured on the same length scale as the perturbation. Here, we illustrate that much can be gained by studying also the mixed dynamic structure factor, which connects different spatial components of perturbation and response. We extend state-of-the-art ab initio calculations to access the mixed dynamic structure factor, including excitonic effects. Using bulk silicon and lithium fluoride as prototype examples, we show that these effects play a crucial role above and below the quasiparticle gap, and are needed in order to explain coherent inelastic x-ray scattering experiments. Our approach also yields important information concerning the microscopic structure of time-dependent density functional theory.

Figure 1

Dynamic structure factor for $\mathbf{q}=(-1/2,-1/2,-1/2), \mathbf{G}=(0,0,0), {\mathbf{G}}^{\prime }=(1,1,1)$ in reciprocal-lattice units for (a) Si and (b) LiF, obtained from the BSE (blue lines for diagonal and black lines for off-diagonal elements) and within $\mathrm{RPA}+GW$ (red lines for diagonal and magenta lines for off-diagonal) compared to the experiments (diamonds for diagonal and squares for off-diagonal): Si CIXS [60] and IXS [22, 61]; LiF EELS [62].

Figure 2

Real (top panel) and imaginary (bottom panel) parts of exchange-correlation kernels of LiF for $\mathbf{q}\to \mathit{0}$: $\mathbf{G}={\mathbf{G}}^{\prime }=\mathit{0}$ element of $f_{\mathrm{xc}}$ (red dotted line) and of $f_{\mathrm{xc}}^{\text{mb}}$ (green dotted-dashed line), together with the effective scalar kernels $f_{\mathrm{xc}}^{\text{eff}}$ (blue dashed line) and $f_{\mathrm{xc}}^{\text{mb}\text{eff}}$ (magenta solid line) that simulate absorption spectra (see text for definitions).

Figure 3

LiF, absorption spectrum. The reference spectrum in black is the result of a BSE calculation, which is by definition, and numerically, identical to results obtained in TDDFT using $f_{\mathrm{xc}}^{\text{eff}}$. Results obtained by using only the head of the matrix $f_{\mathrm{xc}}$, and of the matrix $f_{\mathrm{xc}}^{\text{mb}}$ are shown in red dotted and green dotted-dashed lines, respectively.