Excitation pathways in resonant inelastic x-ray scattering of solids

Resonant inelastic x-ray scattering (RIXS) is a powerful spectroscopic technique that offers an elemental- and orbital-selective probe of the electronic excitations over a huge energy range. We present a many-body approach to determine RIXS spectra in solids, yielding an intuitive expression for the RIXS cross section in terms of pathways between intermediate many-body states containing a core hole, and final many-body states containing a valence hole. Explicit excited many-body states are obtained from the diagonalization of the Bethe-Salpeter equation in an all-electron framework. For the paradigmatic example of the fluorine $K$ edge of LiF, we show how the excitation pathways determine the spectral shape of the emission, and demonstrate the nontrivial role of electron-hole correlation in the RIXS spectra.

Figure 1

Scheme of the RIXS process: A core excitation yields the intermediate many-body state $|{\lambda }_c\rangle$, where blue circles represent the distribution of the core hole (open circles) and the excited electron (solid circles). Cyan arrows indicate dipole transitions. The deexcitation from $|{\lambda }_c\rangle$ yields the final many-body state $|{\lambda }_o\rangle$. The final state is represented by green circles, where the valence hole distribution is shown in open green circles, and the distribution of the excited electron in solid ones.

Figure 2

Calculated (red) and experimental (black) [34] RIXS spectrum of the fluorine $K$ edge in LiF. Both spectra are normalized for each absorption energy. A Lorentzian broadening of 0.15 eV is employed in the calculated spectra.

Figure 3

F $K$ edge RIXS cross section calculated from the BSE (center top) and within the IPA (center bottom), together with the optical (top left) and the F $K$ edge (right) absorption spectrum. The calculated spectra (red BSE, gray IPA) are compared to experimental results (black) for the optical [35] and core [36] excitations. Scissor operators are applied to the calculated spectra (compare SM [33]) to correct the DFT band gap. The excitation pathways $|t^{\left(2\right)}{|}^2$ between the first 500 core and 1000 optical excitations are shown on the top right.