Theoretical Demonstration of How the Dispersion of Magnetic Excitations in Cuprate Compounds can be Determined Using Resonant Inelastic X-Ray Scattering

We show that in resonant inelastic x-ray scattering (RIXS) at the copper $L$ and $M$ edge direct spin-flip scattering is in principle allowed. We demonstrate how this possibility can be exploited to probe the dispersion of magnetic excitations, for instance magnons, of cuprates such as the high $T_c$ superconductors. We compute the relevant local and momentum dependent magnetic scattering amplitudes, which we compare to the elastic and $dd$-excitation scattering intensities. For cuprates these theoretical results put RIXS as a technique on the same footing as neutron scattering.

Figure 1

$L_3$ RIXS cross section for a single ${\mathrm{Cu}}^{2+}$ ion with $\sigma$ and $\pi$ polarization of the incident beam. (a) The geometry: the scattering plane is (100), scattering angle 90°, the incident photons impinge at an angle ${\theta }_{\mathrm{IN}}$ to the [001] direction ($c$ axis). (b) Scattering intensities to different orbital and spin orientations, starting from a $(x^2\mathrm{\text{−}}{y}^2){\downarrow }_z$ state (left panels) and a $(x^2\mathrm{\text{−}}{y}^2){\downarrow }_{xy}$ ground state (right panels) as a function of ${\theta }_{\mathrm{IN}}$ or alternatively the in-plane transferred momentum $q_{\parallel }$. For ${\downarrow }_z$, i.e., spin along [001], the spin-flip cross section vanishes (bottom left), not so for ${\downarrow }_{xy}$, with spin along [110] (right).

Figure 2

The dependence of scattering cross section for spin-flip scattering to $(x^2\mathrm{\text{−}}{y}^2)\uparrow$ final states on the atomic spin orientation for selected cases of scattering geometry given by varying ${\theta }_{\mathrm{IN}}$ and fixed ${\varphi }_{\mathrm{IN}}=0$.

Figure 3

Momentum dependence of the magnetic RIXS spectral weight at the copper $L_3$ edge (red) and the magnon dispersion (green) for a simple 2D Heisenberg model (solid lines) and an extended model relevant for ${\mathrm{La}}_2{\mathrm{CuO}}_4$ [24] (dashed).