Origin of the Two-Dimensional Electron Gas Carrier Density at the ${\mathrm{LaAlO}}_3$ on ${\mathrm{SrTiO}}_3$ Interface

Transport measurements of the two-dimensional electron gas at the ${\mathrm{LaAlO}}_3\mathrm{\text{−}}{\mathrm{SrTiO}}_3$ interface have found a density of carriers much lower than expected from the “polar catastrophe” arguments. From a detail density-functional study, we suggest how this discrepancy may be reconciled. We find that electrons occupy multiple subbands at the interface leading to a rich array of transport properties. Some electrons are confined to a single interfacial layer and susceptible to localization, while others with small masses and extended over several layers are expected to contribute to transport.

Figure 1

Nominal layer charges at the $n$-type interface with $\mathrm{LaO}\mathrm{\text{−}}{\mathrm{TiO}}_2$ termination. Bottom figure shows the diverging Coulomb potential (“polar catastrophe,” solid line) and an example of how it may be avoided by adding half an electron to the single ${\mathrm{TiO}}_2$ layer at the interface (dashed line).

Figure 2

Computed relaxed atomic positions, consistent with the spread of the interfacial electrons into many ${\mathrm{TiO}}_2$ layers.

Figure 3

Layer-projected densities-of-states indicating the spread of the interface electrons [$0.5e/\mathrm{\text{cell}}$ occupying $\mathrm{Ti}(d)$ bands] into several layers near the interface on the STO side. The Fermi energy is taken as zero.

Figure 4

Computed band offset at the interface. Valence band positions were computed by using the O-1s core state as the reference level. The conduction band edges were inferred by using the experimental band gaps [20].

Figure 5

Charge contours of the occupied states in the conduction bands (Fig. 6) plotted on the $xz$ plane (normal to the interface). Electrons penetrate very little to the LAO side. Note that only the $xz$ orbitals, but not the $xy$ or $yz$ orbitals, show up on such a plane. Nevertheless, the contours faithfully reproduce the spread of some of the electrons (see text) deeper into the STO part. The starting density for the contours is $3.2\times {10}^{-5}e/{\AA }^3$ and successive values increase by the factor of 3. The inset shows the Ti ($d$) occupancy as a function of the distance of the Ti layers from the interface.

Figure 6

Conduction bands near the Fermi energy and their predominant orbital characters. Band dispersions are shown along the interface ($xy$ plane) with $X=\pi /a(1,0,0)$ and $M=\pi /a(1,1,0)$, where “$a$” is the in-plane lattice constant.

Figure 7

Idealized Fermi surface for the isolated interface, extracted from the DFT results of Fig. 6. Hybridization between bands rounds out the sharp crossing points in the Fermi surface, leading to more complex orbits as seen in the inset of Fig. 6.