Polarization Dependence of $L$- and $M$-Edge Resonant Inelastic X-Ray Scattering in Transition-Metal Compounds

The resonant inelastic x-ray scattering (RIXS) cross section at the $L$ and $M$ edges of transition-metal compounds is studied using an effective scattering operator. The intensities of the elastic peak and for spin-flip processes are derived. It is shown how the polarization dependence can be used to select transitions. Comparisons are made with experiment. A detailed analysis of the polarization and angular dependence of $L$- and $M$-edge RIXS for divalent copper compounds, such as the high-$T_c$ superconductors, is given.

Figure 1

(a) Possible changes in $\Delta m$ and $\Delta s_z$ for a total angular momentum $q=\Delta m+\Delta s_z$. (b) Typical energies of a hole in a $3d$ orbital in a crystal field in $O_h$ and $D_{4h}$ symmetry. (c),(d) The effective transition operator $W_{Qq}^{\pm }$ scatters holes from a state with orbital $\alpha$ and spin $s_z$ to a state with index ${\alpha }_{\Delta m}$ (for this diagram $\Delta m\ne 0$) and spin $s_z+\Delta s_z$. To obtain the corresponding index ${\alpha }_{\Delta m}$, we first determine what atomic orbitals are contained in $\alpha$. Note that $\alpha =0,\pm 1$ are atomic orbitals with $m=0,\pm 1$ and that $\alpha =2^{\pm }$ is a combination of atomic orbitals with $m=\pm 2$. Addition of $\Delta m$ gives an orbital moment of $m+\Delta m$. Therefore, the final state indices always correspond to atomic orbitals, i.e., ${\alpha }_{\Delta m}=m+\Delta m$. Note, however, that if ${\alpha }_{\Delta m}=\pm 2$, the hole ends up in the $2^{\pm }$ orbitals since $d_{\pm 2,s_z}^{†}=\frac{1}{\sqrt{2}}(d_{2^{+}{s}_z}^{†}\pm d_{2^{-}{s}_z}^{†})$.

Figure 2

(a) A typical 90° scattering geometry. (b) RIXS intensities as a function of the angle $\theta$ of the incoming radiation with the surface normal and the angle $\phi$ of the incoming polarization vector with the scattering plane for a ground state of a hole in the $x^2-y^2\uparrow$ orbital.

Figure 3

Density plot of the RIXS intensities at the $L_3$ or $M_3$ edge for a 90° scattering geometry, see Fig. 2a, as a function of the angle $\theta$ of the incoming radiation with the surface normal and the angle $\phi$ between $\mathbf{e}$ and the scattering plane. The ground state has a hole in the $x^2-y^2\uparrow$ orbital. The intensities for scattering into $x^2-y^2\downarrow$ and $xy\downarrow$ are zero.