Lattice distortion and magnetism of $3d-t_{2g}$ perovskite oxides

Several puzzling aspects of interplay of the experimental lattice distortion and the magnetic behavior of four narrow $t_{2g}$-band perovskite oxides (${\mathrm{YTiO}}_3$, ${\mathrm{LaTiO}}_3$, ${\mathrm{YVO}}_3$, and ${\mathrm{LaVO}}_3$) are clarified using results of first-principles electronic structure calculations. First, we derive parameters of the effective Hubbard-type Hamiltonian for the isolated $t_{2g}$ bands using newly developed downfolding method for the kinetic-energy part and a hybrid approach, based on the combination of the random-phase approximation and the constraint local-density approximation, for the screened Coulomb interaction part. Apart from the above-mentioned approximation, the procedure of constructing the model Hamiltonian is totally parameter free. The results are discussed in terms of the Wannier functions localized around transition-metal sites. The obtained Hamiltonian was solved using a number of techniques, including the mean-field Hartree-Fock (HF) approximation, the second-order perturbation theory for the correlation energy, and a variational superexchange theory, which takes into account the multiplet structure of the atomic states. We argue that the crystal distortion has a profound effect not only on the values of the crystal-field (CF) splitting, but also on the behavior of transfer integrals and even the screened Coulomb interactions. Even though the CF splitting is not particularly large to fully quench the orbital degrees of freedom (ODF), the crystal distortion imposes a severe constraint on the form of the possible orbital states, which favor the formation of the experimentally observed magnetic structures in ${\mathrm{YTiO}}_3$, ${\mathrm{YVO}}_3$, and ${\mathrm{LaVO}}_3$ even at the level of mean-field HF approximation. Particularly, ${\mathrm{LaVO}}_3$ presents an interesting example of the system where the ODF are well quenched only in one of the monoclinic planes and remain relatively flexible in the second plane, leaving some room for the orbital fluctuations. It is also remarkable that for all three compounds, the main results of all-electron calculations can be successfully reproduced in our minimal model derived for the isolated $t_{2g}$ bands. We confirm that such an agreement is possible only when the nonsphericity of the Madelung potential is explicitly included into the model. Beyond the HF approximation, the correlation effects systematically improve the agreement with the experimental data and additionally stabilize the experimentally observed $G$- and $C$-type antiferromagnetic states in ${\mathrm{YVO}}_3$ and ${\mathrm{LaVO}}_3$. Using the same type of approximations we could not obtain the correct magnetic ground state for ${\mathrm{LaTiO}}_3$. However, we expect that the situation may change by systematically improving the level of approximations for treating the correlation effects.

Figure 1

(Color online) Total and partial densities of states for ${\mathrm{YTiO}}_3$, ${\mathrm{LaTiO}}_3$, ${\mathrm{YVO}}_3$ (orthorhombic phase), and ${\mathrm{LaVO}}_3$ in the local-density approximation. The shaded area shows the contributions of the transition-metal $3d$ states. Other symbols show the positions of the main bands. The Fermi level is at zero energy.

Figure 2

(Color online) Crystal structure of orthorhombic ${\mathrm{LaTiO}}_3$ (left), ${\mathrm{YTiO}}_3$ (middle), and ${\mathrm{YVO}}_3$ (right). The La and Y atoms are indicated by the big (blue) dark spheres, the Ti and V atoms are indicated by the medium (red) dark gray spheres, and the oxygen atoms are indicated by the small (green) light gray spheres. The symbols $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ stand for the orthorhombic translations. The symbols $1–4$ denote the transition-metal sites, which form the unit cell of the distorted perovskite oxides.

Figure 3

(Color online) Contour plot of Wannier functions for ${\mathrm{LaTiO}}_3$, in three orthorhombic planes: $\mathbf{ab}$ (left), $\mathbf{ac}$ (center), and $\mathbf{bc}$ (right). The solid and dashed lines correspond to the positive and negative values of the Wannier functions. The projections of different atoms on the planes are denoted by the following symbols: $+$ (La), $엯$ (Ti), and $\times$ (O). Around each atomic site, the Wannier function increases or decreases with the step $0.04$ from the values indicated on the graph.

Figure 4

(Color online) Spacial extension of the Wannier functions in the case of ${\mathrm{LaTiO}}_3$: the weight of the Wannier function accumulated around the central Ti site after adding every new sphere of neighboring atomic sites (also denoted as the “accumulated charge” in Ref. 34).

Figure 5

(Color online) Left panel shows LDA energy bands for ${\mathrm{LaTiO}}_3$ obtained in LMTO calculations and after the tight-binding (TB) parametrization using two different schemes of the downfolding method. In the scheme I, the local coordinate frame has been obtained from the diagonalization of the local density matrix. In the scheme II, the local coordinate frame has been obtained from the diagonalization of the site-diagonal part of a more general $40\times 40$ $\mathrm{Ti}\left(3d\right)\text{−}\mathrm{La}\left(5d\right)$ tight-binding Hamiltonian. Notations of the high-symmetry points of the Brillouin zone are taken from Ref. 42. Right panel shows distance dependence of averaged parameters of the kinetic energy ${\overline{h}}_{{\mathbf{RR}}^{\prime }}\left(d\right)=({\sum }_{\alpha \beta }{h}_{{\mathbf{RR}}^{\prime }}^{\alpha \beta }{h}_{{\mathbf{R}}^{\prime }\mathbf{R}}^{\beta \alpha }{)}^{1∕2}$, where $d$ is the distance between transition-metal sites $\mathbf{R}$ and ${\mathbf{R}}^{\prime }$. The data for $d=0$ correspond to the crystal-field splitting of the covalent type, and $d\sim 4\mathrm{\AA }$ is the distance between nearest transition-metal sites. Two schemes generate very similar electronic structure in the reciprocal space, which is well consistent with results of original LMTO calculations. However, the real-space parameters appear to be different. The scheme II yields larger crystal-field splitting. However, the transfer integrals are less localized and spread beyond the nearest neighbors.

Figure 6

(Color online) Distance dependence of averaged parameters of the kinetic energy for various compounds. In the orthorhombic $\left(o\right)$ structure, all sublattices of the transition-metal sites are equivalent and shown by a single symbol. In the monoclinic $\left(m\right)$ structure, the results obtained for two different sublattices are shown by closed and open symbols. For other notations, see Fig. 5.

Figure 7

Convergence of the crystal-field splitting as the function of the cutoff radius for the real-space summation in Eq. (2).

Figure 8

Summary of the $t_{2g}$-level splitting in various compounds. The notations “site 1” and “site 3” correspond to two nonequivalent transition-metal sites in the monoclinic phase (shown in Fig. 2).

Figure 9

The multiplet structure of the excited atomic configuration $t_{2g}^2$ in ${\mathrm{YTiO}}_3$ and ${\mathrm{LaTiO}}_3$.

Figure 10

(Color online) Distribution of the charge density around V sites in various magnetic phases of orthorhombically distorted ${\mathrm{YVO}}_3$ $(T<77\mathrm{K})$, as obtained in the Hartree-Fock calculations. Different magnetic sublattices are shown by dark gray (blue) and light gray (yellow) colors.

Figure 11

(Color online) Orbital ordering in the $G$-type antiferromagnetic phase of orthorhombically distorted ${\mathrm{YVO}}_3$ computed without crystal field (left), including the crystal field of the covalent type (middle), and the crystal field of both covalent and Madelung type (right).

Figure 12

(Color online) Distribution of the charge density around V sites in various magnetic phases of monoclinically distorted ${\mathrm{YVO}}_3$, as obtained in the Hartree-Fock calculations. Different magnetic sublattices are shown by dark gray (blue) and light gray (yellow) colors.

Figure 13

(Color online) Distribution of the charge density around Ti sites in various magnetic phases of ${\mathrm{YTiO}}_3$, as obtained in the Hartree-Fock calculations. Different magnetic sublattices are shown by dark gray (blue) and light gray (yellow) colors.

Figure 14

(Color online) Orbital ordering in the ferromagnetic phase of ${\mathrm{YTiO}}_3$ computed with (left) and without (right) crystal-field splitting.

Figure 15

(Color online) Distribution of the charge density around V sites in various magnetic phases of ${\mathrm{LaVO}}_3$, as obtained in the Hartree-Fock calculations. Different magnetic sublattices are shown by dark gray (blue) and light gray (yellow) colors.

Figure 16

(Color online) Distribution of the charge density around Ti sites in various magnetic phases of ${\mathrm{LaTiO}}_3$, as obtained in the Hartree-Fock calculations. Different magnetic sublattices are shown by dark gray (blue) and light gray (yellow) colors.

Figure 17

(Color online) Total energies measured relative to the $G$-type antiferromagnetic state as obtained in the Hartree-Fock calculations after scaling the Madelung contribution to the crystal-field splitting.