Polarization dependence and symmetry analysis in indirect $K$-edge RIXS

We present a study of the charge-transfer excitations in undoped ${\text{Nd}}_2{\text{CuO}}_4$ using resonant inelastic x-ray scattering (RIXS) at the $\text{Cu}K$-edge. At the Brillouin zone center, azimuthal scans that rotate the incident-photon polarization within the ${\text{CuO}}_2$ planes reveal weak fourfold oscillations. A comparison of spectra taken in different Brillouin zones reveals a spectral weight decrease at high-energy loss from forward- to back-scattering. We show that these are scattered-photon polarization effects related to the properties of the observed electronic excitations. Each of the two effects constitutes about 10% of the inelastic signal while the “$4p$-as-spectator” approximation describes the remaining 80%. Raman selection rules can accurately model our data, and we conclude that the observed polarization-dependent RIXS features correspond to $E_g$ and $B_{1g}$ charge-transfer excitations to non-bonding oxygen $2p$ bands, above 2.5 eV energy-loss, and to an $E_g$ $d\to d$ excitation at 1.65 eV.

Figure 1

(Color online) (a) Side and (b) top view of the scattering geometry used throughout this paper. The polarization of the incident beam $\left(\sigma \right)$ and the two possible polarization conditions for the scattered beam ($\sigma$ and $\pi$) are emphasized in both panels. Note that $\psi$-scans are collected at $\omega =0$ only. (b) Inset: two-dimensional (2D) Brillouin zone: the red line shows the full region of integration arising from the momentum resolution whereas the blue line shows its half width. (c) Energy scans taken at the 2D zone center with incident energy $E_i=8997\text{eV}$ (circles) and 9001 eV (squares). The superposed colored circles and square show where the $\psi$ scans of corresponding color and energy are measured in (d) are measured. The charge-transfer gap measured with optical conductivity is indicated for comparison (Ref. 26). (d) Azimuthal scans taken in a full circle around the 2D zone center at 1.625, 2, 3.5, 5 eV energy-loss (for $E_i=8997\text{eV}$) and at 6 eV energy loss for $\left(E_i=9001\text{eV}\right)$ for $\mathit{Q}=\left(009.1\right)$ and symmetrized with respect to the $90°$ rotations and mirror planes of the underlying tetragonal structure. (e) Energy scans with elastic tail subtracted (see text) taken at $\mathit{Q}=\left(00.511.1\right)$ for different values of the azimuthal angle $\left(\psi \right)$. The scans indicated by circles were taken at $E_i=8997\text{eV}$ whereas those indicated by squares were taken at $E_i=9001\text{eV}$. Energy scans with $\psi =45°$ are corrected for changes in self-absorption $\left(+5.2\mathrm{\%}\right)$ and incident polarization $\left(+0.8\mathrm{\%}\right)$ compared to $\psi =10.7°$. (f) Fourier components of the raw (unsymmetrized) $\psi$ scan at 3.5 eV energy loss with amplitude given in percent of 0-fold (DC) component (not shown). The fourfold component is well above the statistical noise level (gray region). There is also a twofold component present in the raw data.

Figure 2

(Color online) (a) Two-dimensional zone-center spectra collected for a wide range of scattering geometries. Data sets are separated vertically by 0.3 units. The solid lines through the data are smooth interpolations and they are also overlaid at the bottom of the figure to highlight the evolution of the measured spectra. The data sets are scaled using the $\text{FM}+\text{SA}$ method (see text for details). (b-c) Fits of the Brillouin-zone dependence to the linear form $y={\sigma }_1+{\sigma }_2{\text{cos}}^2\left(\theta -\omega \right)$ at 1.65, 3, 4, and 6.5 eV, comparing the $\text{FM}+\text{SA}$ normalization procedure with (b) FM and (c) SW2 normalizations. (d-e) Fitted coefficients ${\sigma }_1$ and ${\sigma }_2$ as a function of energy loss, comparing the $\text{FM}+\text{SA}$ normalization procedure with (d) FM and (e) SW2 normalizations.

Figure 3

(Color online) Symmetry interpretation of the photon-polarization dependent and independent RIXS spectral weight for tetragonal ${\text{Nd}}_2{\text{CuO}}_4$. While composed of many different symmetry channels, the blue spectrum is most likely dominated by the $A_{1g}$ symmetry channel. See text for details.

Figure 4

(Color online) (a) Position of the fluorescence monitor relative to the analyzer crystal (b) Self-absorption parameters in reflection geometry. (c) X-ray absorption spectroscopy (XAS) data (by partial fluorescence yield of the $\text{Cu}{K}_{\alpha 1}$ emission line) in the $\text{Cu}K$-edge region. The black lines correspond to the calculated x-ray absorption coefficients above and below the $\text{Cu}K$ edge for ${\text{Nd}}_2{\text{CuO}}_4$. The measured XAS curves are scaled to follow the published curves by Tranquada et al. (Ref. 56). (d) ${\text{Nd}}_2{\text{CuO}}_4$ absorption curve at the $\text{Cu}K$ edge measured with the fluorescence monitor as described in the text. The black arrows indicate incident energies of 8997 and 9001 eV.