Crystal-field splitting for low symmetry systems in ab initio calculations

In the framework of the $\mathrm{LDA}+\mathrm{U}$ approximation we propose the direct way of calculation of crystal-field excitation energy and apply it to La and Y titanates. The method developed can be useful for comparison with the results of spectroscopic measurements because it takes into account fast relaxations of electronic system. For titanates these relaxation processes reduce the value of crystal-field splitting by $\sim 30\%$ as compared with the difference of LDA one electron energies. However, the crystal-field excitation energy in these systems is still large enough to make an orbital liquid formation rather unlikely and experimentally observed isotropic magnetism remains unexplained.

Figure 1

(Color online) Band structure for $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ obtained within the LDA approximation in the framework of two methods: The full potential linearized augmented plane waves—FP-LAPW (left) and linear muffin-tin orbitals—LMTO (right). The energy region is chosen to illustrate the behavior of $t_{2g}$ bands. The Fermi level corresponds to zero energy.

Figure 2

(Color online) Total and partial DOS of $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ (top panel) and $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$ (bottom panel) calculated within the $\mathrm{LDA}+\mathrm{U}$ approximation for AFM-G and FM magnetic structures, respectively. The Fermi level corresponds to zero energy.

Figure 3

(Color online) $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$. $\mathrm{Ti}\text{−}3d$ partial DOS calculated within $\mathrm{LDA}+\mathrm{U}$ approach. The inset shows $\mathrm{Ti}\text{−}3d$ partial DOS in AFM-G (solid line) and FM (dashed line) configurations. Parts of plots with positive (negative) ordinates denote majority (minority) spin DOS. The Fermi level corresponds to zero energy.

Figure 4

(Color online) Orbital ordering in $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ (a) and $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$ (b) in $ab$-plane obtained in the framework of the $\mathrm{LDA}+\mathrm{U}$ calculations for AFM-G and FM structures, respectively.

Figure 5

(Color online) Orbital ordering in $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ (a) and $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$ (b) in $c$ direction obtained in the framework of the $\mathrm{LDA}+\mathrm{U}$ calculations for AFM-G and FM structures, respectively.