Density and spatial distribution of charge carriers in the intrinsic $n$-type ${\text{LaAlO}}_3{\text{-SrTiO}}_3$ interface

In order to establish the density and spatial distribution of charge carriers intrinsic to the $n$-type ${\text{LaAlO}}_3{\text{-SrTiO}}_3$ heterointerface, we carry out first-principles calculations on the ${\left({\text{LaAlO}}_3\right)}_n{\left({\text{SrTiO}}_3\right)}_{15}$ slab model with $n=2–10$. As the thickness of the ${\text{LaAlO}}_3$ layer increases, the charge transfer from ${\text{LaAlO}}_3$ to ${\text{SrTiO}}_3$ converges to half an electron per two-dimensional unit cell. It is found that the electrons in the conduction band of ${\text{SrTiO}}_3$ consist of various types of interface-bound states. The mobile electrons evaluated by excluding those states tightly bound to the interface within 2 nm or having large effective masses are in good agreement with the experimental carrier densities for all ${\text{LaAlO}}_3$ thicknesses, suggesting that the loosely bound states play a major role in the transport property. A large calculation including up to ${\left({\text{LaAlO}}_3\right)}_5{\left({\text{SrTiO}}_3\right)}_{30}$ shows that about 70% of electrons are confined within 3 nm from the interface, which is in good comparison with the experiments. It is found that the transferred electrons decay exponentially at short distances from the interface, but there is a crossover to an algebraically decaying region at $\sim 4\text{nm}$.

Figure 1

(Color online) (a) The unit cell of ${\left(\text{LAO}\right)}_5{\left(\text{STO}\right)}_{15}$. (b) The $xy$-averaged (solid line) and macroscopically averaged (dashed line) local potential. Due to the artificial dipole moment introduced to screen the Coulomb potential from neighboring cells, there is a potential discontinuity in the middle of the vacuum region $\left(z=0\right)$. (c) The partial density of states (PDOS) projected onto layers near the interface. The Fermi level is set to zero. (d) The spatial distributions of free electrons (light gray or yellow) and holes (dark gray or blue) in STO and LAO sides, respectively.

Figure 2

(Color online) (a) Carrier densities of ${\left(\text{LAO}\right)}_n{\left(\text{STO}\right)}_{15}$ with respect to the number of LAO layers $\left(n_{\text{LAO}}\right)$. The experimental data are presented as crosses (Ref. 1). The open and filled triangles indicate the contribution by $d_{\parallel }$ bands omitting the lowest subband alone $\left[d_{\parallel }\left(1\right)\right]$ or several bands localized within $\sim 2\text{nm}$ from the interface $\left[d_{\parallel }\left(2\right)\right]$, respectively. (b) The $xy$-averaged spatial distributions of transferred charges when $n_{\text{LAO}}$ is 5, 7, and 10. The position of the interfacial Ti site is indicated by a vertical dashed line. For the visual purpose, the top position of each peak is linked with a line.

Figure 3

Schematic diagrams to describe the carrier generation mechanism for two representative LAO thicknesses of (a) four and (b) six unit layers. The shaded parts are occupied by electrons. The left side of the figure shows the hypothetical situation where the charge transfer from LAO to STO is prohibited. The right side is the equilibrium state obtained in the self-consistent calculations with the well-defined Fermi level $\left(E_F\right)$. Because of the charge transfer, the internal field inside LAO is smaller in the right sides.

Figure 4

(Color online) (a) Band structure of ${\left(\text{LAO}\right)}_5{\left(\text{STO}\right)}_{15}$ along the special points in the surface Brillouin zone. The Fermi level is set to zero. (b) $xy$ average of squared wave functions for low-lying energy states in the solid circle of (a). $d_{\parallel }$ and $d_{\perp }$ mean that the plane of $\text{Ti}3d$ orbital is parallel and perpendicular to the interface plane, respectively. The vertical dashed line shows the position of interfacial Ti site. The figures in the legend are schematic geometries of $\text{Ti}3d$ orbitals along the interface normal.

Figure 5

(Color online) Spatial distributions of $xy$-averaged carriers with various thicknesses of STO. The figures are aligned at the interfacial Ti site (dashed line). As a visual guide, the top position of each peak is connected with a line. In the inset, the charge distribution for ${\left(\text{LAO}\right)}_5{\left(\text{STO}\right)}_{30}$ is fitted with two functional forms with respect to the distance $x$ from the interfacial ${\text{TiO}}_2$ layer. Fitted values are $a=0.027\text{electrons}/\text{Å}$, $\lambda =18.35\text{Å}$, $b=0.082$ electrons, and $x_0=15.90\text{Å}$