Magnetic Excitation Spectra of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors

We used resonant inelastic x-ray scattering to reveal the nature of magnetic interactions in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$, a $5d$ transition-metal oxide with a spin-orbit entangled ground state and $J_{\mathrm{eff}}=1/2$ magnetic moments. The magnon dispersion in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ is well-described by an antiferromagnetic Heisenberg model with an effective spin one-half on a square lattice, which renders the low-energy effective physics of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ much akin to that in superconducting cuprates. This point is further supported by the observation of exciton modes in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$, whose dispersion is strongly renormalized by magnons, which can be understood by analogy to hole propagation in the background of antiferromagnetically ordered spins in the cuprates.

Figure 1

(a) Because of a staggered in-layer rotation of oxygen octahedra, ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ has four ${\mathrm{IrO}}_2$ layers in the unit cell [9], which coincides with the magnetic unit cell. (b) $J_{\mathrm{eff}}=1/2$ moments lie and are canted in the ${\mathrm{IrO}}_2$ plane [8].

Figure 2

(a) Blue dots with error bars show the single magnon dispersion extracted by fitting the energy loss curves shown in Fig. 3b [13]. The magnons disperse up to $\approx 205\mathrm{meV}$ at $(\pi ,0)$ and 110 meV at $(\pi /2,\pi /2)$. The solid purple line shows the best fit to the data with $J=60$, $J^{\prime }=-20$, and $J^{\prime \prime }=15\mathrm{meV}$. (b) Momentum dependence of the intensities showing diverging intensity at $(\pi ,\pi )$ and vanishing intensity at (0,0). The inset shows the Brillouin zone of the undistorted tetragonal ($\mathrm{I}4/\mathrm{mmm}$) unit cell (black square) and the magnetic cell (blue diamond), and the notation follows the convention for the tetragonal unit cell, as, for instance, in ${\mathrm{La}}_2{\mathrm{CuO}}_4$.

Figure 3

(a) Energy loss spectra recorded at $T=15\mathrm{K}$, well below the $T_N\approx 240\mathrm{K}$ [8, 12], along a path in the constant $L=34$ plane. The path was chosen to avoid the magnetic Bragg peaks, which appear at two of the four corners of the unfolded unit cell (black square) shown in the inset (where the same conventions as in Fig. 2 are used). (b) Image plot of the data shown in (a). (c) Schematic of the three representative features in the data. (d) A real space description of the spin-orbit exciton mode.

Figure 4

(a) Second derivative of the data shown in Fig. 3b, overlaid with the calculation of the dispersion (red solid and dashed lines). $W=63\mathrm{meV}$ was chosen in the calculation, which is within the estimated range [13]. (b) Comparison of the experimental and theoretical integrated spectral weight for the two spin-orbit exciton modes normalized to the single magnon mode at $(\pi ,0)$. (c) Spectrum at $(\pi ,0)$ fitted with five Gaussian peaks. From the right, the peaks (red dashed lines) correspond to elastic, single magnon, two magnon, and two branches of spin-orbit excitons. (d) Schematic of the spin-orbit exciton hopping in the hole representation. By the RIXS process, the hole in the $i$ site is excited to the $J_{\mathrm{eff}}=3/2$ quartet. This excited hole hops ($t_{3/2}$ process) to the neighboring site $j$ with the intermediate energy of $U^{\prime }$. The other hole in the $j$ site hops back ($t_{1/2}$ process) to the $i$ site, thereby completing the spin-orbit exciton hopping processes and also creating a magnon (blue wavy line).