Spin-wave spectrum of the Jahn-Teller system ${\mathrm{LaTiO}}_3$

We present an analytical calculation of the spin-wave spectrum of the Jahn-Teller system ${\mathrm{LaTiO}}_3$. The calculation includes all superexchange couplings between nearest-neighbor Ti ions allowed by the space-group symmetries: The isotropic Heisenberg couplings and the antisymmetric (Dzyaloshinskii-Moriya) and symmetric anisotropies. The calculated spin-wave dispersion has four branches, two nearly degenerate branches with small zone-center gaps and two practically indistinguishable high-energy branches having large zone-center gaps. The two lower-energy modes are found to be in satisfying agreement with neutron-scattering experiments. In particular, the experimentally detected approximate isotropy in the Brillouin zone and the small zone-center gap are well reproduced by the calculations. The higher-energy branches have not been detected yet by neutron scattering but their zone-center gaps are in satisfying agreement with recent Raman data.

Figure 1

The crystallographic structure of ${\mathrm{LaTiO}}_3$. The ten Ti ions, which constitute the twelve inequivalent nearest-neighbor Ti-Ti bonds are enumerated. For simplicity, only the oxygen octahedra around four Ti sites are shown. La ions from two layers are depicted as small spheres. We use orthorhombic coordinates, in which the $x,y,z$ axes are oriented along the crystallographic $a,b,c$ directions.

Figure 2

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.

Figure 3

The spin-wave dispersion along selected directions in the magnetic Brillouin zone. We use pseudocubic coordinates, in which the Ti ions no. 1 and 2 are located along the $x$ axis. Panels (a)–(c) show the four branches ${\Omega }_i\left(\mathbf{q}\right)$ of the calculated dispersion (solid curves) and the single branch $\Omega \left(\mathbf{q}\right)$ which has been fitted onto neutron scattering experiments (dashed curves), Eq. (23). The acoustic branches ${\Omega }_1\left(\mathbf{q}\right)$ and ${\Omega }_2\left(\mathbf{q}\right)$ are quasidegenerate, such that away from the zone center no splitting between them can be seen. The optical branches ${\Omega }_3\left(\mathbf{q}\right)$ and ${\Omega }_4\left(\mathbf{q}\right)$ are practically indistinguishable over the entire Brillouin zone. (a) The dispersion along (1,1,1) [$\mathbf{q}=\frac{\pi }{2}\left(\mathrm{1,\; 1,\; 1}\right)\xi$]. This direction is chosen because the experimental paper on the neutron scattering contains a plot along this direction where the measured points of the dispersion are shown (Ref. 1). Though the calculated acoustic branches of the dispersion give slightly lower energies at the zone-center than the fitted function and slightly higher energies at the zone edge, these deviations are within the uncertainty of the measurement and hence, the agreement of our calculated dispersion with the the measured points and with the fitted function is satisfying. The splitting of the calculated acoustic branches at the zone center is too small to be resolved in the experiment. From panels (b) and (c) one can see that the tetragonal anisotropy of the calculated acoustic branches is rather small. The agreement between the acoustic branches and the neutron scattering data is satisfying also along the (1,0,0) [panel (b), $\mathbf{q}=\pi \left(\mathrm{1,\; 0,\; 0}\right)\xi$] and (0,0,1) [panel (c), $\mathbf{q}=\frac{\pi }{2}\left(\mathrm{0,\; 0,\; 1}\right)\xi$] directions.