Cu $K$-edge resonant inelastic x-ray scattering in edge-sharing cuprates

We present calculations for resonant inelastic x-ray scattering (RIXS) in edge-shared copper oxide systems, such as ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$ and $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$, appropriate for hard x-ray scattering, such as the copper $K$ edge. We perform exact diagonalizations of the multiband Hubbard model and determine the energies, orbital character, and resonance profiles of excitations which can be probed via RIXS. We find excellent agreement with recent results on ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$ and $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$ in the $2–7\mathrm{eV}$ photon energy loss range.

Figure 1

(Color online) Top: Edge-shared ${\mathrm{Cu}}_5{\mathrm{O}}_{12}$ cluster. Bottom: XPS spectrum as a function of photon energy transfer and incident photon energy, for corner and edge-sharing copper-oxide clusters.

Figure 2

(Color online) (a) RIXS spectra for ${\mathrm{Cu}}_3{\mathrm{O}}_8$ cluster as a function of photon energy transfer $\Omega$ and incident photon energy ${\omega }_I$, measured relative to the binding energy ${ϵ}_{4p-1s}$. The inset shows a contour map over large frequencies. Matrix elements for ${\mathrm{Cu}}_3{\mathrm{O}}_8$ are shown in (b) and (c), for the main peaks in RIXS at $({\omega }_i-{ϵ}_{4p-1s},\Omega )=(2.2,2.2)$ and (3.1,5.07), eV respectively. Here, the left and right eigenvectors is the core hole and final states, respectively, and the dot and the circle represent a singlet state formed by a hole on the copper site (dot) and a hole delocalized on the oxygens having the ZR symmetry (circle).

Figure 3

(Color online) (a) $\mathrm{Cu}4p$ partial density of states for ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$. (b) RIXS spectrum for parameters corresponding to ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$, convoluted with the $p_z$-DOS. (c) RIXS spectrum for parameters corresponding to ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$, convoluted with the $p_x$-DOS. For both (b) and (c), we took a core-hole lifetime of $\Gamma =1.5\mathrm{eV}$.

Figure 4

(a) Experimental data for ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$ by Kim et al. in Ref. 8. (b) Calculated RIXS spectrum for parameters corresponding to ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$ convoluted with the $4p\mathrm{DOS}$.

Figure 5

(Color online) RIXS spectrum as a function of photon energy transfer and incident photon energy for parameters corresponding to $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$, as defined in the text.

Figure 6

(Color online) $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$ structure.

Figure 7

(Color online) (a) $\mathrm{Cu}4p$ partial density of states in $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$ calculated in WIEN2k. (b) RIXS spectrum for parameters corresponding to $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$, convoluted with the $p_z$-DOS. (c) RIXS spectrum for parameters corresponding to $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$, convoluted with the $p_z$-DOS.

Figure 8

(Color online) (Left) Experimental RIXS spectrum for $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$. (Right) Theoretical spectrum obtained for the same polarization.

Figure 9

(Color online) (Left) Experimental RIXS spectrum for $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$ with polarization dependence. (Right) Exact-diagonalization results obtained after convolution with the $4p$ projected DOS of Fig. 7a (see Sec. 3C).