Orbital ordering under reduced symmetry in transition metal perovskites: Oxygen vacancy in SrTiO${}_3$

By using a combination of density function theory and model Hamiltonian analysis, we explain the general electronic structure features induced by an oxygen vacancy (OV) in SrTiO${}_3$. We show that the most important effect caused by an oxygen vacancy is the direct on-site coupling between the $3d_{3z^2-r^2}$ and $4s$,$4p$ orbitals of Ti atoms adjacent to the vacancy caused by lifting of the local cubic symmetry. This would be the case for any transition metal perovskite under symmetry-reduced environments such as an interface, surface, or a defect. We find that the OV-induced localized state is highly one dimensional and is mainly composed of Ti $3d_{3z^2-r^2}$ orbitals along the Ti-OV-Ti axis (defined as the $z$ axis) and Ti $4s$,$4p$ orbitals at the OV site. The oxygen vacancy does not lead to Ti $t_{2g}$-based localized states.

Figure 1

Bulk structure and effects due to one oxygen vacancy. (a) Clean bulk system. All oxygen $2p$ orbitals are occupied and all Ti $3d$ orbitals are empty, leading to a band insulator. (b) When an oxygen vacancy (OV) is presented, the local level at OV site becomes very large, and the Ti $3d$ levels next to the OV are lowered. The blue double arrows represent the direct Ti $3d$-O $2p$ hoppings. We define the Ti-OV-Ti axis as $z$ throughout our discussion. (c) Effects of local orbital character on Ti next to the OV. Left: Under the cubic symmetry, $3d_{3z^2-r^2}$ and $4p_z$ orbitals are local eigenstates. The coupling between $3d_{3z^2-r^2}$ and $4p_z$, induced by $C_4$ symmetry, changes the character of the local eigenstates, pushing $3d_{3z^2-r^2}$-based ($|a\rangle$) and $4p_z$-based ($|b\rangle$) orbitals down and up in energy, respectively. (d) The schematic $3d_{3z^2-r^2}$-based orbitals under cubic and $C_4$ symmetry. Under cubic symmetry, $3d_{3z^2-r^2}$ is an eigenstate, while under the reduced $C_4$ symmetry, it mixes significantly with the $4p_z$ state, making the resulting $|a\rangle$ asymmetric and more extended.

Figure 2

(a) Wave functions of two localized states projected at Ti sites. 0 is the position of oxygen vacancy, integers label O positions, and half integers label Ti positions. They are almost degenerate when $t^{\prime }=0$ and are split to bonding and antibonding states when $t^{\prime }\ne 0$. (b) Total DOS with an OV and LDOS of Ti next to the OV. Here ${\overline{\epsilon }}_d=-2.5$ eV and $t^{\prime }=1$ eV are used. The integrated total DOS (black, solid) is normalized to 2, whereas the integrated LDOS (red, dashed) is normalized to 1. The contributions of total DOS below $-8$ eV are mainly from O $2p$, while those above 0 eV are from Ti $3d$. A broadening of 0.05 eV is used.

Figure 3

(a) LDOS of Ti next to the OV computed by DFT (without $U$) (Ref. 18). The $3d_{3z^2-r^2}$ peaks at $\sim$$0.8$ and $\sim$$2.6$ eV are interpreted as bonding and antibonding localized states. (b) The same Ti LDOS computed by $\mathrm{LDA}+U$ with $U=8.0$ eV with ionic relaxation. (c) The same Ti LDOS computed by LDA without ionic relaxation. Only one of three $t_{2g}$ orbitals is shown since they are close in energy. In all three cases, the $3d_{3z^2-r^2}$ bonding and antibonding peaks (indicated by arrows) are associated with substantial Ti $4s$ and $4p$ contributions. Note the $4s$ and $4p$ components are multiplied by 3 for clarity. (d) Three $t_{2g}$ orbitals LDOS computed by $\mathrm{LDA}+U$ with $U=8.0$ eV with ionic relaxation. The splitting, estimated by taking the energy average, is smaller than 0.3 eV with $3d_{xy}$ higher than the other two. A similar small $t_{2g}$ splitting is observed in the LDA calculations. (e) The charge density plot at $\sim$$0.8$ eV in (a). At the OV site the charge density is extended enough to cover the whole OV site, which is a character of the bonding state. (f) The charge density plot at $\sim$$2.6$ eV in (a). At the OV site the charge density is almost zero but is extended to the neighboring Ti sites, which is a character of the antibonding state.