Symmetry and Correlation Effects on Band Structure Explain the Anomalous Transport Properties of (111) ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$

The interface between the two insulating oxides ${\mathrm{SrTiO}}_3$ and ${\mathrm{LaAlO}}_3$ gives rise to a two-dimensional electron system with intriguing transport phenomena, including superconductivity, which are controllable by a gate. Previous measurements on the (001) interface have shown that the superconducting critical temperature, the Hall density, and the frequency of quantum oscillations, vary nonmonotonically and in a correlated fashion with the gate voltage. In this Letter we experimentally demonstrate that the (111) interface features a qualitatively distinct behavior, in which the frequency of Shubnikov–de Haas oscillations changes monotonically, while the variation of other properties is nonmonotonic albeit uncorrelated. We develop a theoretical model, incorporating the different symmetries of these interfaces as well as electronic-correlation-induced band competition. We show that the latter dominates at (001), leading to similar nonmonotonicity in all observables, while the former is more important at (111), giving rise to highly curved Fermi contours, and accounting for all its anomalous transport measurements.

Figure 1

Gate dependence of transport parameters. (a) The sheet carrier density ($n_{2\mathrm{D}}$) determined from low-field Hall measurements and quantum oscillations as a function of the zero field sheet conductance ($G_{S0}$). (b) Normalized superconducting critical temperature $T_c/T_{\mathrm{c},\max }$ as a function of $G_{S0}$.

Figure 2

Crystal structure of the (111) ${\mathrm{LaAlO}}_3/\mathrm{SrTi}{\mathrm{O}}_3$ interface. (a) Three inequivalent layers of Ti atoms ($\text{green}\equiv {\mathrm{Ti}}_1$, $\text{orange}\equiv {\mathrm{Ti}}_2$, $\text{blue}\equiv {\mathrm{Ti}}_3$) forming one unit cell at the (111) interface of a cubic lattice. (b) Top view of the (111) trilayer. ${\overrightarrow{R}}_{1,2}$ are the lattice vectors of the triangular lattice at the interface. (c),(d) Level structure of the Ti $3d$ orbitals at the (111) and (001) interfaces, respectively. The bulk cubic structure leads to the splitting into $t_{2g}$ and $e_g$ orbitals. (c) The trigonal crystal symmetry at the (111) interface leads to further splitting of $t_{2g}$ into $a_{1g}$ and $e_g^{\prime }$ orbitals. (d) The in-plane crystal symmetry does not change at the (001) interface but the surface confinement lifts the degeneracy.

Figure 3

Results of the theoretical model for the (111) interface. (a), (b) Band structure for two different chemical potentials $\mu$ before ($\mu =0.03\mathrm{eV}$) and after ($\mu =0.12\mathrm{eV}$) the second band starts getting occupied. The pink (green) line corresponds to band 1 (2), which is composed of $a_{1g}$ ($e_g^{\prime }$) orbitals close the $\mathrm{\Gamma }$ point. The dashed blue line marks the Fermi energy. (c), (d) The Fermi contours corresponding to the band structures in (a), (b), respectively. The outer surface (corresponding to band 1) is highly anisotropic at all $\mu$. (e) The carrier and Hall densities as a function of $\mu$. The carrier density is monotonic for both bands, while the Hall density is nonmonotonic. The peak in $n_{\text{Hall}}$ occurs before the second band starts getting populated. This is in accordance with the experiment (Fig. 1) and indicates that it is due to the anisotropic shape of the lowest band. (f) Normalized superconducting critical temperature ($T_c/T_{\mathrm{c},\max }$) as a function of $\mu$ within the single-band BCS model.

Figure 4

Results of the theoretical model for the (001) interface. (a),(b) Band structure for two different chemical potentials $\mu$ before ($\mu =0.025\mathrm{eV}$) and after ($\mu =0.10\mathrm{eV}$) the second band starts getting occupied. The pink (green) line corresponds to band 1 (2), which is composed of $xy$ (other $t_{2g}$) orbitals close to the $\mathrm{\Gamma }$ point. The dashed blue line marks the Fermi energy. (c),(d) The Fermi contours corresponding to the band structures in (a),(b), respectively. The outer surface (corresponding to band 1) is isotropic at low values of $\mu$. When the second band gets populated, the orbital character of the bands switches and the outer band becomes anisotropic. (e) The carrier and Hall densities as a function of $\mu$. The carrier density for the second band is nonmonotonic due to correlation-induced transfer of electrons from the lighter to heavier band. The Hall density is nonmonotonic due to the anisotropy in the Fermi contour of the bands. (f) Normalized superconducting critical temperature ($T_c/T_{\mathrm{c},\max }$) as a function of $\mu$ within the single-band BCS model.