Magnetic structure of the Jahn-Teller system $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$

We investigate the effect of the experimentally observed Jahn-Teller distortion of the oxygen octahedra in $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ on the magnetic exchange. We present a localized model for the effective hopping between nearest-neighbor Ti ions and the intrasite Coulomb interactions, based on a nondegenerate orbital ground state due to the static crystal field. The latter corresponds to an orbital order which has recently been confirmed experimentally. Using perturbation theory we calculate, in addition to the Heisenberg coupling, antisymmetric (Dzyaloshinskii-Moriya) and symmetric anisotropy terms of the superexchange spin Hamiltonian, which are caused by the spin-orbit interaction. Employing this spin Hamiltonian, we deduce that at low temperatures the spins have predominantly a G-type antiferromagnetic ordering along the crystallographic $a$ axis, accompanied by a weak ferromagnetic moment along the $c$ axis and by a weak A-type antiferromagnetic moment along the $b$ axis. The first two components are found to be in good agreement with experiment.

Figure 1

The crystallographic structure of $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$. The ten Ti ions, which constitute the 12 inequivalent nearest-neighbor $\mathrm{Ti}\text{−}\mathrm{Ti}$ bonds are enumerated. For simplicity, oxygen octahedra are only shown around four Ti sites. La ions from two layers are shown as small spheres. For example, the sites 2 and 6 are crystallographically equivalent but the bond 61 emerges from the 12 bond by a glide reflection, so that the effective hopping matrix for the bond 16 is different than that of the 12-bond: It is the transposed one, see Table .

Figure 2

The orbital order of the Ti ions in the ground state [Eq. (2)], resulting from the calculated crystal field (the energy scale of the superexchange, which could in principle also affect the orbital degree of freedom, is about one order of magnitude smaller than the crystal-field gap). The order is ferro-orbital in the $ab$ planes and canted antiferro-orbital between the planes.

Figure 3

The magnetic order of the Ti ions in the classical ground state of the effective spin Hamiltonian of the lattice. The ions are enumerated according to the sublattice to which they belong. The $x$ components of the spins order antiferromagnetically in the G-type configuration, the $y$ components order antiferromagnetically in the A-type one, and the $z$ components order ferromagnetically.

Figure 4

In cubic symmetry, there is one $t_{2g}$ orbital per each $\mathrm{Ti}\text{−}\mathrm{O}\text{−}\mathrm{Ti}$ bond, which cannot participate in the hopping between the Ti and the O ions because of the parity of the $\mathrm{O}\text{−}2p$ orbitals. This is shown in panels (a)–(c), for each of the three $t_{2g}$ orbitals, respectively. (a) The hopping between the orbitals $\mid y^{\prime }z⟩$ at the Ti sites 1 and 2 is not possible, since on the intermediate oxygen site there is no $p$ orbital with a parity that would allow such hopping. For the same reason, an electron cannot hop from the $p$ orbital $\mid x^{\prime }⟩$, which is shown between the Ti sites 1 and 2, to any of the $t_{2g}$ orbitals of the Ti sites. On the other hand, the hopping between the orbitals $\mid y^{\prime }z⟩$ at the Ti sites 1 and 3, which is mediated by the orbital $\mid y^{\prime }⟩$ on the intermediate oxygen site, is possible. (b) Analogously to the case (a), the hopping between the orbitals $\mid x^{\prime }{y}^{\prime }⟩$ at the Ti sites 1 and 3 is not permitted, because on the intermediate oxygen site there is no appropriate $p$ orbital. Likewise, an electron cannot hop from the $p$ orbital $\mid z⟩$, which is shown between the Ti sites 1 and 3, to any of the $t_{2g}$ orbitals of the Ti sites. The (allowed) hopping between the orbitals $\mid x^{\prime }{y}^{\prime }⟩$ at the Ti sites 1 and 2 is mediated by the orbital $\mid y^{\prime }⟩$ on the intermediate oxygen site. (c) The hopping between the orbitals $\mid x^{\prime }z⟩$ at the Ti sites 1 and 2 is mediated by the orbital $\mid z⟩$ on the intermediate oxygen site. The hopping between the orbitals $\mid x^{\prime }z⟩$ at the Ti sites 1 and 3 is mediated by the orbital $\mid x^{\prime }⟩$ on the intermediate oxygen site.