Optical magnons with dominant bond-directional exchange interactions in the honeycomb lattice iridate $\alpha -{\mathrm{Li}}_2{\mathrm{IrO}}_3$

We have used resonant inelastic x-ray scattering to reveal optical magnons in a honeycomb lattice iridate $\alpha -{\mathrm{Li}}_2{\mathrm{IrO}}_3$. The spectrum in the energy region 20–25 meV exhibits momentum dependence, of which energy is highest at the location of the magnetic Bragg peak, $(h,k)=(\pm 0.32,0)$, and lowered toward $(0,0)$ and $(\pm 1,0)$. We compare our data with a linear spin-wave theory based on a generic nearest-neighbor spin model. We find that a dominant bond-directional Kitaev interaction of order 20 meV is required to explain the energy scale observed in our study. The observed excitations are understood as stemming from optical magnon modes whose intensity is modulated by a structure factor, resulting in the apparent momentum dependence. We also observed diffuse magnetic scattering arising from the short-range magnetic correlation well above $T_N$. In contrast to ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, this diffuse scattering lacks the $C_3$ rotational symmetry of the honeycomb lattice, suggesting that the bond anisotropy is far from negligible in $\alpha -{\mathrm{Li}}_2{\mathrm{IrO}}_3$.

Figure 1

(a) Honeycomb lattice of $\alpha -{\mathrm{Li}}_2{\mathrm{IrO}}_3$ formed by ${\mathrm{Ir}}^{4+}$ ions (spheres) inside the edge-shared oxygen octahedra. The other ions are not shown for clarity. The $x, y$, and $z$ bonds denoted with green, blue, and red lines are normal to the local $x, y$, and $z$ axes of the octahedron, respectively. (b) Counter-rotating spiral magnetic order. The magnetic moments on each dashed line ($\parallel a$ axis) rotate about the line. The rotation directions are opposite between the adjacent lines. (c) The $l$-scans at the magnetic Bragg peak positions for three possible magnetic domains. Each pair of the same color-coded circles (red) and crosses (black and cyan) inside the hexagonal Brillouin zone depict the positions for the different domains (inset). (d) Temperature dependence of the integrated intensity at ${\mathit{Q}}_{\mathbf{mag}}$ stemming from the magnetic order below ${\mathit{T}}_N$ = 12 K.

Figure 2

(a) The $l$-dependence of RIXS spectra at -${\mathit{q}}_{\mathbf{mag}} +$ (−2, −2) at 4 K. The spectrum of the magnetic Bragg spot is reduced by 50 for comparison with those of the other $l$ values. (b) The RIXS spectra of $l$ = 6 and 6.42 below and slightly above ${\mathit{T}}_N$, i.e. 4 K and 19 K, respectively. (c), (d) The RIXS spectra at ($h, k$) = (−0.32, −2) and (−0.32, −1.6) marked with the circles inside another Brillouin zone (inset) below and exceedingly above ${\mathit{T}}_N$, i.e. 4 K and 260 K.

Figure 3

(a) RIXS spectra at $\mathit{q}=(0,0)$ (top), -${\mathit{q}}_{\mathbf{mag}}$ (middle), and (-0.66, 0) (bottom). The raw spectrum and elastic background are shown with the squares and black line, respectively. The circle and pink line depict the spectrum subtracted by the elastic background and fitting result from a damped harmonic oscillator (DHO) model, respectively. Intensity maps of the raw RIXS spectra (b) and the spectra subtracted by the elastic background (c) at 4 K along ($h$, 0). Red dashed lines correspond to $\pm {\mathit{q}}_{\mathbf{mag}}$. Open triangles depict the peak centers obtained by fitting with the DHO model. (d) RIXS spectra along ($h$, 0) in another Brillouin zone centered at (−2, −2). The $l$ value ranges from 6.42 to 6.36. The dotted line is a guide to the eye. (e) Intensity map of the calculated magnon dispersion after convolution with the energy and momentum instrumental functions. (f) Magnon dispersion relations along ($h$, 0) simulated by the linear spin wave theory based on the $K-\mathrm{\Gamma }-{\mathrm{\Gamma }}^{\prime }-J$ model.

Figure 4

(a) Intensity map of diffuse magnetic scattering on the $\mathit{hk}$ plane at 20 K. (b) Illustration of the horizontal scattering geometry. The scattering plane is defined by the incident (${\mathit{k}}_{\mathit{i}}$) and outgoing (${\mathit{k}}_{\mathit{f}}$) wavevectors. The incident ($\pi$) and outgoing x-ray polarizations (${\pi }^{\prime }, {\sigma }^{\prime }$) are shown with gray arrows. The azimuthal $\mathrm{\Psi }$ is defined as the angle between the $a$ axis and the scattering plane (the curved black arrow). (c) Temperature($T$)-dependence of the diffuse scattering along ($h$, 0, 6). The data are vertically shifted for clarity. The magnetic Bragg peak (blue) is scaled down for comparison. (d) $T$-dependence of the RIXS spectra at ${\mathit{Q}}_{\mathbf{mag}}$. (e) RIXS spectra at ${\mathit{Q}}_{\mathbf{mag}}$ (red line) and $\pm {120}^{\circ }$ rotated positions about the $l$ direction (black and cyan lines) at 20 K and $\mathrm{\Psi }$ = 150${}^{\circ }$. The same color-coded spots are denoted inside the Brillouin zone (inset). (f) Elastic spectral weight at ${\mathit{Q}}_{\mathbf{mag}}$ for selected $\mathrm{\Psi }$ (circles). Solid and dashed curves display the simulated $\mathrm{\Psi }$-dependence of summed $\pi -{\pi }^{\prime }$ and $\pi -{\sigma }^{\prime }$ channels for the counter-rotating spiral orders at the same color-coded positions in the inset of (e). The average of these three curves is denoted as gray line.