Theory of electronic magnetoelectric coupling in $d^5$ Mott insulators

Motivated by recent terahertz (THz) spectroscopy measurements in $\alpha \text{−}{\mathrm{RuCl}}_3$, we develop a theory for magnetoelectric (ME) effects in Mott insulators of $d^5$ transition metal ions in an octahedral crystal field. For $4d$ and $5d$ compounds, the relatively wide-spread orbitals favor charge fluctuations of localized electrons to neighboring ions and a significant ME effect from electronic mechanisms is expected. From a three-orbital Hubbard model with strong spin-orbit coupling, we derive the mechanisms for the electric polarization originating from virtual hopping of the localized holes carrying the spins. We consider the electric polarization generated by pairs of spin operators on nearest-neighbor bonds with either an edge-sharing geometry (i.e., two ligands are shared) or a corner-sharing geometry (i.e., one ligand is shared). The allowed couplings are first derived using a symmetry approach. Then, we explicitly calculate the coupling constants and evaluate the effective polarization operator in the ground state manifold using perturbation theory and exact diagonalization. The results are relevant when considering the THz optical conductivity of magnetic systems such as some perovskite iridates or Kitaev materials. In particular, they help explain the recent THz optical measurements of $\alpha \text{−}{\mathrm{RuCl}}_3$ for which the electric-dipole-induced contribution has been shown to be strong.

Figure 1

(a) Corner-sharing and (b) edge-sharing geometries. The blue spheres represent the TM ions and the smaller red spheres the ligands. Deviations from $\varphi ={90}^{\circ }$ and $\phi ={180}^{\circ }$ are considered in Appendices pp1 and pp2.

Figure 2

Crystal structure of Kitaev materials. The blue spheres are the TM ions (Ir or Ru) and the red spheres are the ligand (O or Cl). Each TM-TM bond is in the edge-sharing geometry.

Figure 3

The different $p\text{−}d$ (top) and $d\text{−}d$ (bottom) polarization integrals as a function of the atomic spacing $d$, in units of $a_0$, the Bohr radius. The cases of chromium and oxygen atoms (left) and ruthenium and chloride atoms (right) are plotted using hydrogenlike atomic orbitals.

Figure 4

Numerical evaluation of ${\mathbb{A}}_1$ in Eq. (39). Results using exact diagonalization (ED) (red line), perturbation theory (PT) exact in $\mathrm{\Delta }$ (blue line), and perturbation theory linear in $\mathrm{\Delta }$ (green dashed line) are plotted in (a) and (b), for typical values of the physical parameters: $U=2310$ meV, $J_H=320$ meV, $\lambda =140$ meV, and we set $t_1=t_3=t_4=0$. It is plotted (a) as a function of $t_2$ with $\mathrm{\Delta }=-50$ meV and (b) as a function of $\mathrm{\Delta }$ with $t_2=150$ meV. (c) shows ED calculation of ${\mathbb{A}}_1$ as a function of $\mathrm{\Delta }$ for different sets of values $t_{1-4}$ taken from the literature [45, 55]. The values are indicated in meV units.

Figure 5

Numerical evaluation of ${\mathbb{A}}_1$, ${\mathbb{A}}_2$, and ${\mathbb{A}}_4$ in Eqs. (37) and (45) using exact diagonalization as a function of (a) $J_H$, (b) $\lambda$, and (c) $\mathrm{\Delta }$. The nonvarying physical parameters are chosen as $U=2310$ meV, $J_H=320$ meV, $\lambda =140$ meV, $\mathrm{\Delta }=-50$ meV, $t_1=50$ meV, $t_2=160$ meV, $t_3=-150$ meV, and $t_4=-20$ meV, typical for $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 6

Numerical evaluation of ${\mathbb{B}}_{2-4}$ in Eq. (37) using exact diagonalization as a function of (a) $J_H$, (b) $\lambda$, and (c) $\mathrm{\Delta }$. The nonvarying physical parameters are chosen as $U=2310$ meV, $J_H=320$ meV, $\lambda =140$ meV, $\mathrm{\Delta }=-50$ meV, $t_1=50$ meV, $t_2=160$ meV, $t_3=-150$ meV, and $t_4=-20$ meV, typical for $\alpha \text{−}{\mathrm{RuCl}}_3$. The coupling constants are plotted in units of $b_5$ [see Eq. (52)] where we assumed $b_4=-b_5$ for simplicity.