Mixing of $t_{2g}-e_g$ orbitals in $4d$ and $5d$ transition metal oxides

Using exact diagonalization, we study the spin-orbit coupling and interaction-induced mixing between $t_{2g}$ and $e_{g}d$-orbital states in a cubic crystalline environment, as commonly occurs in transition metal oxides. We make a direct comparison with the widely used $t_{2g}$-only or $e_g$-only models, depending on electronic filling. We consider all electron fillings of the $d$ shell and compute the total magnetic moment, the spin, the occupancy of each orbital, and the effective spin-orbit coupling strength (renormalized through interaction effects) in terms of the bare interaction parameters, spin-orbit coupling, and crystal-field splitting, focusing on the parameter ranges relevant to $4d$ and $5d$ transition metal oxides. In various limits, we provide perturbative results consistent with our numerical calculations. We find that the $t_{2g}-e_g$ mixing can be large, with up to 20% occupation of orbitals that are nominally “empty,” which has experimental implications for the interpretation of the branching ratio in experiments, and can impact the effective local moment Hamiltonian used to study magnetic phases and magnetic excitations in transition metal oxides. Our results can aid the theoretical interpretation of experiments on these materials, which often fall in a regime of intermediate coupling with respect to electron-electron interactions.

Figure 1

Symmetry lowering and level splitting in a cubic crystal-field environment. A transition metal ion in free space has a full rotational SO(3) symmetry reduced to octahedral symmetry $O_h$. The fivefold-degenerate $d$ levels in the vacuum split into a lower-lying triply degenerate $t_{2g}$, and a higher-lying doubly degenerate $e_g$ set of levels, with an energy difference $\mathrm{\Delta }$ (called the crystal-field splitting) between them.

Figure 2

The $t_{2g}$ wave functions have electron clouds pointing in-between the point charges of the ligands, thus they repel less and have lower energy, compared to the $e_g$ states which point towards the oxygen ligands.

Figure 3

Evolution of $d$-orbital states under a cubic crystal field and spin-orbit coupling. The green lines correspond to the commonly used T-P equivalence that neglects $t_{2g}-e_g$ mixing by spin-orbit coupling. The red lines indicate an extra mixing contribution going beyond TP equivalence, of the $J_{\mathrm{eff}}=\frac{3}{2}$ by a factor of $\pm i\sqrt{\frac{3}{2}}\frac{\zeta }{\zeta /2+\mathrm{\Delta }}|e_g\rangle$, and to the upper quartet $e_g$ by the same factor of $J_{\mathrm{eff}}=\frac{3}{2}$ states, as shown in Eqs. (16) and (17). The energies of the lower quartet is shifted down by $-\frac{3}{2}\frac{{\zeta }^2}{\zeta /2+\mathrm{\Delta }}$, and of the upper quartet is shifted up by $+\frac{3}{2}\frac{{\zeta }^2}{\zeta /2+\mathrm{\Delta }}$. Notice that the $J_{\mathrm{eff}}=\frac{1}{2}$ states are not affected.

Figure 4

Exact diagonalization one-electron results. (a) Total magnetic moment $M_{\mathrm{tot}}$, (b) effective spin-orbit coupling $\overline{\zeta }$, (c) single $S_5$, zero $Z_5$, and double $D_5$ occupancies of the $e_g{d}_{x^2-y^2}$ orbital, for different crystal-field values $a_1:\mathrm{\Delta }=1\mathrm{eV},a_2:\mathrm{\Delta }=2$ eV, and $a_3:\mathrm{\Delta }=3$ eV. Note there is substantial enhancement of the total magnetic moment and effective spin-orbit coupling in the $t_{2g}-e_g$ model relative to the $t_{2g}$-only model.

Figure 5

Exact diagonalization two-electron results for crystal-field splitting $\mathrm{\Delta }=3$ eV. (a) Total magnetic moment $M_{\mathrm{tot}}$, (b) spin quantum number $S$, (c) single $S_i$, zero $Z_i$, double $D_i$ occupancy per $e_g$ orbital, (d) effective spin-orbit coupling $\overline{\zeta }$. Different Hund's coupling parameters $a_1:J_H=0.1\text{eV},a_2:J_H=0.5$ eV are used.

Figure 6

Exact diagonalization three-electron results for crystal-field splitting $\mathrm{\Delta }=3$ eV. (a) Total magnetic moment $M_{\mathrm{tot}}$. (b) Spin quantum number $S$. (c) Single $S_4$, zero $Z_4$, and double $D_4$ occupancies of the $d_{3z^2-r^2}$ orbitals. (d) Single $S_5$, zero $Z_5$, and double $D_5$ occupancies of the $d_{x^2-y^2}$ orbitals. (e) Effective spin-orbit coupling $\overline{\zeta }$. Different Hund's coupling parameters $a_1:J_H=0.1\text{eV},a_2:J_H=0.5$ eV.

Figure 7

Exact diagonalization four-electron results for crystal-field splitting $\mathrm{\Delta }=3$ eV. (a) Single $S_4$, zero $Z_4$, and double $D_4$ occupancies of the $d_{3z^2-r^2}$ orbitals. (b) Single $S_5$, zero $Z_5$, and double $D_5$ occupancies of the $d_{x^2-y^2}$ orbitals. (c) Spin quantum number $S$. (d) Effective spin-orbit coupling $\overline{\zeta }$. Different Hund's coupling parameters $a_1:J_H=0.1\text{eV},a_2:J_H=0.5\text{eV},a_3:J_H=0.7$ eV.

Figure 8

Exact diagonalization five-electron results for crystal-field splitting $\mathrm{\Delta }=2.7$ eV. (a) Spin quantum number $S$. (b) Single $S_i$, zero $Z_i$, and double $D_i$ occupancies per $e_g$ orbital. (c) Total magnetic moment $M_{\mathrm{tot}}$. (d) Effective spin-orbit coupling $\overline{\zeta }$. Different Hund's coupling parameters $a_1:J_H=0.1\text{eV},a_2:J_H=0.5\text{eV},a_3:J_H=0.6$ eV.

Figure 9

Exact diagonalization six-electron results. (a) Spin quantum number $S$. (b) Total magnetic moment $M_{\mathrm{tot}}$. (c) Single $S_i$, zero $Z_i$, and double $D_i$ occupancy per $e_g$ orbital. (d) Effective spin-orbit coupling $\overline{\zeta }$. Parameters for predominately low-spin configurations: $a_1:\mathrm{\Delta }=3\text{eV},J_H=0.5\text{eV},a_2:\mathrm{\Delta }=2.5\text{eV},J_H=0.5\text{eV},a_3:\mathrm{\Delta }=3\text{eV},J_H=0.7\text{eV},a_4:\mathrm{\Delta }=3\text{eV},J_H=0.1\text{eV}$. Parameters for predominately high-spin-configurations $\beta :\mathrm{\Delta }=2.5\text{eV},J_H=0.7\text{eV}$.

Figure 10

Exact diagonalization seven-electron results. (a) Spin quantum number $S$. (b) Angular momentum quantum number $L$. (c) Single, double, zero occupancies of the $d_{3z^2-r^2}$ orbital $\left(S_4,D_4,Z_4\right)$. (d) Single, double, zero occupancies of the $d_{x^2-y^2}$ orbital $\left(S_5,D_5\right)$. (e) Total magnetic moment $M_{\mathrm{tot}}$. (f) Effective spin-orbit coupling $\overline{\zeta }$ for ${\alpha }_1:\mathrm{\Delta }=3\text{eV},J_H=0.1\text{eV},{\alpha }_2:\mathrm{\Delta }=2.5\text{eV},J_H=0.5\text{eV},\beta :\mathrm{\Delta }=2.5\text{eV},J_H=0.7\text{eV}$ configurations.

Figure 11

Exact diagonalization eight-electron results. (a) Spin quantum number $S$. (b) Angular momentum quantum number $L$. (c) Total magnetic moment $M_{\mathrm{tot}}$, orbital magnetic moment $M_l$, and spin magnetic moment $M_S$. (d) Single $S_i$, double $D_i$, and zero occupancies $Z_i$ per $t_{2g}$ orbital. (e) Effective spin-orbit coupling $\left(\overline{\zeta }\right)$ for ${\alpha }_1:\mathrm{\Delta }=1\text{eV},J_H=0.5\text{eV},{\alpha }_2:\mathrm{\Delta }=2\text{eV},J_H=0.5\text{eV},{\alpha }_3:\mathrm{\Delta }=3\text{eV},J_H=0.5\text{eV},{\alpha }_4:\mathrm{\Delta }=3\text{eV},J_H=0.1\text{eV}$ configurations.

Figure 12

Exact diagonalization nine-electron results. (a) Total magnetic moment $M_{\mathrm{tot}}$, orbital magnetic moment $M_l$, and spin magnetic moment $M_S$. (b) Effective spin-orbit coupling $\left(\overline{\zeta }\right)$ for ${\alpha }_1:\mathrm{\Delta }=1\text{eV},{\alpha }_2:\mathrm{\Delta }=2\text{eV},{\alpha }_3:\mathrm{\Delta }=3\text{eV}$ configurations.