Observations on the resonant inelastic x-ray scattering cross section in copper oxide compounds

We report properties of the resonant inelastic x-ray scattering (RIXS) cross section in a number of cuprates. We find that the observed spectra have the periodicity of the underlying crystal lattice, that is, the excitations are a function of reduced momentum $\mathbf{q}$ and not the total momentum $\mathbf{Q}=\mathbf{G}+\mathbf{q}$, where $\mathbf{G}$ is a reciprocal lattice vector. Measurements with two different incident photon polarizations show that the line shape and momentum dependence of the observed spectra are independent of the particular state the $4p$ photoelectron resides in the intermediate state. These latter results support theoretical approaches in which the $4p$ is taken to be a spectator in the RIXS process. In addition, a discussion is presented of what might be expected to be generic features of the RIXS cross section with the aid of some crude models, and a comparison is made between RIXS and electron energy loss data for a number of cuprates both at $\mathbf{q}=0$ and at finite $\mathbf{q}$. Taken together, there is a suggestion in the data that RIXS measures an $S(\mathbf{q},\omega )$-like response.

Figure 1

Schematic diagram of the experimental setup described in Sec. 2. The scattering plane is vertical, that is, perpendicular to the electron orbit of the synchrotron. The polarization of incident photon $\left(ϵ\right)$ is perpendicular to the scattering plane.

Figure 2

Comparison of RIXS spectra taken at different total $\mathbf{Q}$ positions but with an equivalent reduced wave vector $\mathbf{q}$ for three different $\mathbf{q}$’s corresponding to $\mathbf{q}=(0,0),(\pi ,0)$, and $(\pi ,\pi )$, from top to bottom. The two-dimensional reciprocal space net is shown in the top panel. The same symbols are used both to denote the $\mathbf{Q}$ position in reciprocal space and to show the spectra. All data were obtained with $ϵ\Vert c$ and ${\omega }_1=8991\mathrm{eV}$.

Figure 3

Comparison of RIXS scans obtained with different incident photon polarizations. The open and solid symbols denote the data taken with $ϵ\Vert c$, ${\omega }_1=8991\mathrm{eV}$ and $ϵ\perp c$, ${\omega }_1=8995\mathrm{eV}$, respectively. The $\mathbf{Q}$ positions correspond to (1 0), (1.5 0), and (2.5 0.5), from top to bottom. Each scan is scaled to the intensity of $K{\beta }_5$ emission line.

Figure 4

(Color online) The incident energy-energy transfer profile for a four peaked scattering function, showing the effect of the weight factor on each excitation. The scattering function has four equal weight Gaussians of width $0.3\mathrm{eV}$ at 2, 3, 4, and $5\mathrm{eV}$ energy loss. The core-hole lifetime is taken to be $1.5\mathrm{eV}$. The lower-energy features are enhanced more than the higher-energy features, while the latter exhibit a double resonance.

Figure 5

(Color online) Same as Fig. 4 but for a single excitation at $3\mathrm{eV}$ energy loss, with a width $\left(1.5\mathrm{eV}\right)$ that is larger than the core-hole lifetime $\left(0.8\mathrm{eV}\right)$. The resonant weight now acts to give an apparent dispersion in peak position with incident photon energy.

Figure 6

For each sample, the RIXS scans are compared with EELS results at equivalent $\mathbf{q}$ positions. The $\mathbf{q}\approx 0$ data are used unless noted otherwise. The EELS results of $\mathrm{Sr}\mathrm{Cu}{\mathrm{O}}_2$, ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$, ${\mathrm{Li}}_2\mathrm{Cu}{\mathrm{O}}_2$, and $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$ are taken from Refs. 37, 38, 39, and 40, respectively.