Fingerprints of spin-orbital physics in cubic Mott insulators: Magnetic exchange interactions and optical spectral weights

The temperature dependence and anisotropy of optical spectral weights associated with different multiplet transitions is determined by the spin and orbital correlations. To provide a systematic basis to exploit this close relationship between magnetism and optical spectra, we present and analyze the spin-orbital superexchange models for a series of representative orbital-degenerate transition metal oxides with different multiplet structure. For each case we derive the magnetic exchange constants, which determine the spin wave dispersions, as well as the partial optical sum rules. The magnetic and optical properties of early transition metal oxides with degenerate $t_{2g}$ orbitals (titanates and vanadates with perovskite structure) are shown to depend only on two parameters, viz. the superexchange energy $J$ and the ratio $\eta$ of Hund’s exchange to the intraorbital Coulomb interaction, and on the actual orbital state. In $e_g$ systems important corrections follow from charge transfer excitations, and we show that ${\mathrm{KCuF}}_3$ can be classified as a charge transfer insulator, while ${\mathrm{LaMnO}}_3$ is a Mott insulator with moderate charge transfer contributions. In some cases orbital fluctuations are quenched and decoupling of spin and orbital degrees of freedom with static orbital order gives satisfactory results for the optical weights. On the example of cubic vanadates we describe a case where the full quantum spin-orbital physics must be considered. Thus information on optical excitations, their energies, temperature dependence, and anisotropy, combined with the results of magnetic neutron scattering experiments, provides an important consistency test of the spin-orbital models, and indicates whether orbital and/or spin fluctuations are important in a given compound.

Figure 1

Energies of $d_i^m{d}_j^m\to d_i^{m+1}{d}_j^{m-1}$ charge excitations in selected cubic transition metal oxides, as obtained from Eq. (2.5) for (a) $e_g$ excitations of ${\mathrm{Cu}}^{3+}\left(d^8\right)$ and ${\mathrm{Mn}}^{2+}\left(d^5\right)$ ions [in the $d^5$ case the spectrum[41] was obtained from Eq. (2.2)] and (b) $t_{2g}$ excitations of ${\mathrm{Ti}}^{2+}\left(d^2\right)$ and ${\mathrm{V}}^{2+}\left(d^3\right)$ ions. The splittings between different states are due to Hund’s exchange element $J_H$ which refers to a pair of $e_g$ electrons in (a) and to a pair of $t_{2g}$ electrons in (b), respectively.

Figure 2

Schematic representation of $e_g$ orbitals as obtained for different angle $\theta$ in Eq. (3.15). Pairs of orthogonal orbitals, forming a basis in the orbital space, differ by angle $\theta =\pi$. Directional $e_g$ orbitals $\mid \zeta ⟩$ along $a$ and $b$ axes are obtained from the $\{\mid 3z^2-r^2⟩,\mid x^2-y^2⟩\}$ basis (3.16) by the transformation (3.15) with angle $\theta =\pm 2\pi ∕3$. Dashed lines for $\theta =\pm \pi ∕3$ indicate the possible OO of $\mid y^2-z^2⟩∕\mid x^2-z^2⟩$ orbitals occupied by holes in ${\mathrm{KCuF}}_3$, suggested for $T<T_N$ in Ref. 7.

Figure 3

(Color online) Exchange interactions (3.18): $J_c$ (3.18) and $J_{ab}$ (3.19) for the copper fluoride model as functions of the orbital angle $\theta$ which describes the OO [see Eq. (3.14)]. The dashed lines show the $U$ term alone, while the solid lines include the CT contributions as well. The OO induced by the JT distortions $(\theta \simeq 70°)$ is indicated by dotted line. Parameters: $J\simeq 33.3\mathrm{meV},$ $\eta =0.12,R=1.2$.

Figure 4

Ratio of the exchange interactions $\mid J_{ab}\mid ∕J_c$ in ${\mathrm{KCuF}}_3$, given by Eqs. (3.18, 3.19), for increasing charge transfer parameter $R$, obtained for a few values of $\eta$ and for the OO induced by the JT effect ($\theta \simeq 70°$, solid lines). The dashed line shows $\mid J_{ab}\mid ∕J_c$ obtained for $\eta =0.12$ and for alternating $(x^2-z^2)∕(y^2-z^2)$ orbitals $(\theta =60°)$.

Figure 5

(Color online) Kinetic energy terms per bond (2.13), as obtained for the $c$ axis with the OO (3.14) suggested by the JT distortions $(\theta \simeq 70°)$ in ${\mathrm{KCuF}}_3$ (in units of $J$): high-spin $K_1^{\left(c\right)}$ (solid line) and low-spin $K_{2,3}^{\left(c\right)}$ (dashed lines), for increasing temperature $T∕J_c$. Parameters as in Fig. 3.

Figure 6

(Color online) Superexchange constants for ${\mathrm{LaMnO}}_3$: (a) $J_c$ along the $c$ axis and (b) $J_{ab}$ in the $ab$ plane, as functions of orbital angle $\theta$ which defines the $C$-type OO, see Eqs. (3.17). Dashed lines for increasing $R=0$, 0.15 and 0.30 from bottom to top; solid lines for $R=0.6$. Note that the dependence of $J_{ab}$ on $R$ is nonmonotonic. Horizontal dotted lines indicate the experimental values of Ref. 62 for ${\mathrm{LaMnO}}_3$; diamond and circle show the experimental values of $J_c$ and $J_{ab}$ for $\theta =102°$. Parameters: $J=150\mathrm{meV},\eta =0.181$.

Figure 7

(Color online) Superexchange interactions $-J_{ab}$ (4.22) and $J_c$ (4.23) as functions of the orbital angle $\theta$, obtained within the superexchange model (4.1) for ${\mathrm{LaMnO}}_3$ (solid lines). Contributions due to the $U$ term and due to the CT term to both exchange constants ($-J_{ab}$ and $J_c$) are shown by dashed and dashed-dotted lines, respectively. Isotropic and orbital independent $t_{2g}$ terms superexchange terms $J_{tU}$ and $J_{t\Delta }$ (4.20) are 1.45 and 0.83 meV. Experimental values (Ref. 62) of $-J_{ab}$ and $J_c$ (indicated by diamond and circle) are well reproduced at angle $\theta =102°$. Parameters $J=150\mathrm{meV},$ $\eta =0.181,R=0.6$.

Figure 8

(Color online) Anisotropy of the low-energy spectral weights in ${\mathrm{LaMnO}}_3$, given by the ratio $K_1^{\left(ab\right)}∕K_1^{\left(c\right)}$, for increasing temperature $T$, as obtained for $\theta =102°$ (solid line), $\theta =108°$ (dashed line), $\theta =98°$ (dashed-dotted line). The arrows indicate the Néel temperature $T_N$ derived from the exchange constants in each case (see text). Parameters: $J=150\mathrm{meV},\eta =0.181$.

Figure 9

(Color online) Kinetic energies per bond (in meV) as functions of temperature $T$, obtained for ${\mathrm{LaMnO}}_3$ for the high-spin (solid lines, $n=1$) and for the low-spin (dashed lines, $n>1$) excitations at the Mn ions for polarization: (a) in the $ab$ planes $K_n^{\left(ab\right)}$ and (b) along the $c$ axis $K_n^{\left(c\right)}$. The total kinetic energies $K^{\left(\gamma \right)}$ are given by the long dashed-dotted lines, while the vertical dotted lines indicate the value of $T_N$ derived within the present model (see text). Filled circles show the experimental intensities (Ref. 28) at low energy $(n=1)$. Parameters: $J=150\mathrm{meV}$, $\eta \simeq 0.181$, $R=0.6$, and $\theta \simeq 102°$.

Figure 10

(Color online) Magnetic and optical properties of ${\mathrm{LaTiO}}_3$, as obtained for the classical wave functions (5.13) with isotropic orbital amplitudes (5.22), for increasing Hund’s exchange $\eta =J_H∕U$: (a) exchange interactions $J_c=J_{ab}$, and (b) kinetic energy terms $K_n^{\left(\gamma \right)}$ at $T=0$: high-spin ($n=1$, solid line), and low-spin intermediate-energy ($n=2$, dashed line), and high-energy excitation ($n=3$, long-dashed line). The total kinetic energy $K^{\left(\gamma \right)}$ is shown by the dashed-dotted line.

Figure 11

Exchange interactions $J_c$ and $J_{ab}$, Eqs. (5.12) (solid and dashed line) for the titanite model for different types of orbital order given by Eqs. (5.13). The coefficients in the wave functions (5.13) vary between the isotropic state $\alpha =\beta =\gamma =1∕\sqrt{3}$ and the state with linear combinations of $\mid a⟩∕\mid b⟩$ and $\mid c⟩$ orbitals, with the coefficients given by: $\alpha =\gamma =1∕\sqrt{3}+\delta (1∕\sqrt{2}-1∕\sqrt{3})$ and $\beta ={(1-{\alpha }^2-{\gamma }^2)}^{1∕2}$. Parameters: $\eta =0.123$.

Figure 12

(Color online) Kinetic energy terms (in units of $J$) per bond (2.13), as obtained for the $C$-AF phase of the cubic vanadates: high-spin $K_1^{\left(\gamma \right)}$ (solid lines) and low-spin $K_2^{\left(\gamma \right)}$ (dashed lines) and $K_3^{\left(\gamma \right)}$ (long-dashed lines), for increasing temperature $T∕J$: (a) along the FM $c$ axis $(\gamma =c)$; (b) within the AF $ab$ plane $(\gamma =ab)$. In part (a) the experimental points from Ref. 91 for the low-energy spectral weight were reproduced, after proper rescaling to match the value of $K_1^{\left(\gamma \right)}$ at $T\to 0$. Parameters: $\eta =0.13$.

Figure 13

(Color online) Orbital transition temperature $T_o$ for increasing JT interaction $J_{\mathrm{JT}}$ in: orbital-only model [Eq. (6.24), dashed line], and spin-orbital model [Eqs. (6.24, 6.25), solid line], compared with the Néel temperature $T_N$ (dashed-dotted line). Both $T_o$ and $T_N$ in units of $J_c^{\tau }$. The inset shows intersite orbital correlations due to JT distortions in ${\mathrm{LaVO}}_3$.

Figure 14

(Color online) Temperature dependence of (a) optical spectral weight of the high-spin transition $K_1^{\left(c\right)}$ in units of $J$ ($J$ is the principal energy scale of the spin-orbital model) and (b) orbital correlations $⟨{\vec{\tau }}_i\bullet {\vec{\tau }}_{i+1}⟩$ along the $c$ axis (bottom), as obtained for the spin-orbital model Eqs. (6.26), as given in Ref. 16, with small $J_{\mathrm{JT}}=0.2J_c^{\tau }$ (solid lines), and the orbital model Eq. (6.24) with $J_{\mathrm{JT}}=1.6J_c^{\tau }$, as analyzed in Ref. 26 (dashed lines).

Figure 15

(Color online) Superexchange interactions $-J_{ab}$ (4.22) and $J_c$ (4.23) as functions of the orbital angle $\theta$ (solid lines), obtained within the effective superexchange model ${\mathcal{H}}_U\left(d^4\right)(R=0)$. Experimental values (Ref. 62) of $-J_{ab}$ and $J_c$ for ${\mathrm{LaMnO}}_3$ (indicated by circle and diamond) are nearly reproduced for the OO with $\theta =94°$. Parameters: $J=170\mathrm{meV},\eta =0.16$.

Figure 16

(Color online) Temperature dependence of (a) spin-spin correlations ${⟨{\vec{S}}_i\bullet {\vec{S}}_j⟩}_{\gamma }$ for $\gamma =a,c$, and the order parameter $⟨S^z⟩$ (dashed line) and (b) kinetic energies $K_1^{\left(\gamma \right)}$ for the high-spin excitations, standing for the optical intensities (2.15) at low energy along $\gamma =a,c$ axes (solid lines), and total kinetic energies $K^{\left(\gamma \right)}$ (long-dashed lines), as obtained for ${\mathrm{LaMnO}}_3$ within the effective superexchange model ${\mathcal{H}}_U\left(d^4\right)$ for the OO given by an angle $\theta \simeq 94°$. Parameters as in Fig. 15.