Consequences of Oxygen-Vacancy Correlations at the ${\mathrm{SrTiO}}_3$ Interface

The Kondo effect and ferromagnetism are the two many-body phenomena that emerge at the ${\mathrm{SrTiO}}_3$ interfaces with polar materials, but do not occur in bulk ${\mathrm{SrTiO}}_3$. By regarding the oxygen vacancy (OV) in ${\mathrm{SrTiO}}_3$ as a magnetic impurity, we show that these two interface-specific phenomena can be attributed to the vacancies residing in the top ${\mathrm{TiO}}_2$ plane of ${\mathrm{SrTiO}}_3$. We identify three crucial ingredients: the local orbital mixing caused by an OV, reduced symmetry at the interface, and a strong in-plane stray electric field of the polar material. All three factors combine to result in the coupling between the impurity and conduction band at the interface, and can lead to both emergent phenomena. An OV-based Anderson impurity model is derived and solved using the numerical renormalization group method. The Kondo and Curie temperatures are estimated. Several experiments are discussed based on this interpretation.

Figure 1

(a) Without vacancies, the local $|3d_{3x^2-r^2}⟩$ orbital is symmetric with respect to the Ti site. A vacancy lowers the local symmetry to $C_{4v}$, and a hybrid state involving a $4p$ component forms. The OV-induced bonding state is the lowest single-particle level induced by an OV [Eq. (1)]. The blue (gray) and red (white) circles indicate the signs of wave functions. (b) The inhomogeneous surface charge from the ionic liquid gate or polar materials can cause a local strong electric field. The coupling between the impurity and conduction band becomes nonzero when an external electric field is present. The arrows indicate the directions of the electric fields at two Ti sites.

Figure 2

(a) The bath orbitals are composed of Ti $3d_{xy}$, O $2p_x$, O $2p_y$ orbitals (of energies ${ϵ}_d$, ${ϵ}_p$). The OV-induced bonding state $|b⟩$ can couple to two Ti $3d_{xy}$ orbitals across the OV. The coupling $g$ can be nonzero near the interface. The inset illustrates relative energies of the conduction band and the impurity levels. (b) The tight-binding band composed of Ti $3d_{xy}$ orbitals only. The OV-induced orbital, represented by a local spin, is localized between two adjacent lattice sites (the bond).

Figure 3

The scaled magnetic susceptibility $4T_K{\chi }_{\text{imp}}/[w(g{\mu }_B{)}^2]$ ($w$ the Wilson number, $\sim 0.413$), specific heat $C_{\text{imp}}/k_B$ (red dashed line), and the entropy $S_{\text{imp}}/(k_B\log 2)$ (blue dotted line) for a single OV impurity as a function of temperature. All quantities are dimensionless and computed using the numerical renormalization group method, as illustrated in the inset.

Figure 4

The RKKY magnetic coupling in Kelvin between two local impurities for two electron fillings (per spin per site): (a) $n_{2\mathrm{D}}=0.015$ and (b) $n_{2\mathrm{D}}=0.05$. The distance is in the unit of lattice constant. The negative values favor FM coupling. The inset of (a) indicates the three scanning directions: (1, 0), (0, 1), (1, 1) are represented by square, diamond, and circle ligands, respectively.