Measurement of Magnetic Excitations in the Two-Dimensional Antiferromagnetic ${\mathrm{Sr}}_2{\mathrm{CuO}}_2{\mathrm{Cl}}_2$ Insulator Using Resonant X-Ray Scattering: Evidence for Extended Interactions

We measured the momentum dependence of magnetic excitations in the model spin-$1/2$ 2D antiferromagnetic insulator ${\mathrm{Sr}}_2{\mathrm{CuO}}_2{\mathrm{Cl}}_2$ (SCOC). We identify a single-spin-wave feature and a multimagnon continuum, with different polarization dependences. The spin waves display a large (70 meV) dispersion between the zone-boundary points ($\pi$, 0) and ($\pi /2$, $\pi /2$). Employing an extended $t\mathrm{\text{−}}{t}^{\prime }\mathrm{\text{−}}{t}^{\prime \prime }\mathrm{\text{−}}U$ one-band Hubbard model, we find significant electronic hopping beyond nearest-neighbor Cu ions, indicative of extended magnetic interactions. The spectral line shape at ($\pi$, 0) indicates sizable quantum effects in SCOC and probably more generally in the cuprates.

Figure 1

(a) Schematics of the scattering geometry. (b) Cu $L_3$ RIXS spectra ($T=15\mathrm{K}$) along the (100) direction. The incident energy is set at the maximum of the absorption (XAS) (inset). (c) Intensity map extracted from (b). The red line is the spin-wave dispersion for the NN Heisenberg model and $J=130\mathrm{meV}$.

Figure 2

(a) (top) RIXS spectra for $\sigma$ and $\pi$ polarization and $Q_{\parallel }=0.58{\AA }^{-1}$ along (100). The dashed line is the difference between the $\pi$ spectrum and the Gaussian line shape ($M$, solid blue line) representing the single-magnon contribution. (bottom) Comparison of Cu $L$-edge and O $K$-edge RIXS for $Q_{\parallel }=0.19{\AA }^{-1}$. (b) RIXS spectra at $0.92(-\pi ,0)$ and ($-\pi /2$, $\pi /2$) measured with $\sigma$ polarization. The two magnons ($2M$) to single-magnon ($M$) intensity ratio is 0.26 at ($-\pi /2$, $\pi /2$), and increases to 0.49 at $0.92(-\pi ,0)$.

Figure 3

(a) Magnon energies extracted from the RIXS data. Open and closed symbols stem from 2 independent measurements on different samples. The dot-dashed line is a NN Heisenberg model with $J=130\mathrm{meV}$. The red dashed line is a NN Hubbard model fit for $t=0.261\pm 0.004\mathrm{eV}$ and $U=1.59\pm 0.04\mathrm{eV}$. Blue lines are the further-neighbor Hubbard fits. (b) same as (a) with NN Heisenberg dispersion subtracted to better visualize details of the dispersion. The blue band shows the spread in dispersions obtained for fits with $1.9\mathrm{eV}<U<4\mathrm{eV}$. (c) The fit to the neutron data on ${\mathrm{La}}_2{\mathrm{CuO}}_4$ [7], shifted by ($\pi$, $\pi$) for ease of comparison.

Figure 4

Results of extended Hubbard model fits for SCOC (left) and LCO (right). (a),(b) ${\chi }^2$ after fitting $t$ and $t^{\prime }$ for fixed $U$ and $t^{\prime \prime }$. Dashed lines mark the minima versus $U$, giving (c),(d) and (e),(f). (c),(d) ${\chi }^2$ versus $U$, showing the range of $U$ that our data can support. (e),(f) extracted hopping parameters $t$, $t^{\prime }$, and $t^{\prime \prime }$ as function of $U$. For given $U$, the uncertainty on the $t$s is around 5 meV. In all panels, thick red lines are for $t^{\prime \prime }>0$ and thin black lines for $t^{\prime \prime }<0$.