Magnetic Couplings, Optical Spectra, and Spin-Orbit Exciton in $5d$ Electron Mott Insulator ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$

Based on the microscopic model including spin-orbit coupling, on-site Coulomb and Hund’s interactions, as well as crystal field effects, we investigate the magnetic and optical properties of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$. Taking into account all intermediate state multiplets generated by virtual hoppings of electrons, we calculate the isotropic, pseudodipolar, and Dzyaloshinsky-Moriya coupling constants, which describe the experiment quite well. The optical conductivity $\sigma (\omega )$ evaluated by the exact diagonalization method shows two peaks at $\sim 0.5$ and $\sim 1.0\mathrm{eV}$ in agreement with experiment. The two-peak structure of $\sigma (\omega )$ arises from the unusual Fano-type overlap between the electron-hole continuum of the $J_{\mathrm{eff}}=1/2$ band and the intrasite spin-orbit exciton observed recently in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$.

Figure 1

(a) Energy levels of the $d$ electron in the presence of the cubic crystal field $10Dq$, the tetragonal crystal field ${\Delta }_{xy}$, and the SO interaction $\lambda$ [29]. The lowest multiplet levels of (b) $d^5$ and (c) $d^4$ configurations including SO $\lambda$ and the Hund’s coupling $J_H$. (d) Two-site and (e) four-site clusters in the $xy$ plane, which were employed in the evaluation of the magnetic couplings and conductivity $\sigma (\omega )$.

Figure 2

Magnetic coupling constants between NN Ir ions (${\overrightarrow{r}}_{12}\parallel x$) with respect to the bonding angle ($\varphi$). $J_{xx}$, $J_{yy}$, and $J_{zz}$ represent diagonal parts of the superexchange tensor $\mathcal{J}$. $J_{xx}=J+\delta J_{xy}$, $J_{yy}=J$, and $J_{zz}=J+\delta J_z$ are deduced from Eq. (5). $D$ is a $z$ component of the DM vector $\mathbf{D}$. Arrows denote the bonding angle in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$. Inset shows $|D/J|$ ratio as a function of the tetragonal splitting ${\Delta }_{xy}$.

Figure 3

(a) Schematic diagrams of possible multiplets included in subsets (${\mathbf{H}}_i$) of the Hilbert space. $D$, $Q$, $S$, and $T$ refer to multiplets of $d^5$ and $d^4$, as labeled in Figs. 1b, 1c, and $A$ denotes a nondegenerate $d^6$ state. ${\mathbf{H}}_5$ includes also diagrams (not shown) with $P$, $P^{\prime }$, $S^{\prime }$ multiplets of the $d^4$ state [Fig. 1c] instead of $T$. (b) Two examples of optically active transitions that contribute to $\sigma (\omega )$ at low energies. The final states ${\mathbf{H}}_3$ and ${\mathbf{H}}_5$ have an overlap with the on-site local SO exciton $Q$ (i.e., with ${\mathbf{H}}_2$ sector), due to intersite hoppings between $J_{\mathrm{eff}}=1/2$ and $J_{\mathrm{eff}}=3/2$ states, resulting in the two-peak structure of $\sigma (\omega )$.

Figure 4

(a) Optical conductivity calculated (solid line) and experimental data [2] (dashed line). Red vertical sticks show relative strengths and positions of optical transitions without broadening. Inset: the result for $\sigma (\omega )$ when the engenstates $|{\psi }_n⟩$ and energies $E_n$ in Eq. (6) are approximated by purely local ionic multiplets. (b) Calculated $L_3$ edge (solid line) and $10\times L_2$ edge (dotted line) RIXS spectra at $\mathbf{q}=(\pi ,\pi )$. Dashed line: experimental data [10]. (c) Projected excitation spectra: ${\Lambda }_1$ represents the magnon band, ${\Lambda }_2$ contains one or more SO excitons, ${\Lambda }_3$ and ${\Lambda }_4$ represent NN and more distantly separated $e\mathrm{\text{−}}h$ excitations derived from the $J_{\mathrm{eff}}=1/2$ states, respectively, and ${\Lambda }_5$ shows the $e\mathrm{\text{−}}h$ continuum of $J_{\mathrm{eff}}=3/2$ states. ${\Lambda }_6$ refers to other excitations, e.g., simultaneous transitions in $e\mathrm{\text{−}}h$ and SO exciton channels.