Spontaneous 2-Dimensional Carrier Confinement at the $n$-Type ${\mathrm{SrTiO}}_3/{\mathrm{LaAlO}}_3$ Interface

We describe the intrinsic mechanism of 2-dimensional electron confinement at the $n$-type ${\mathrm{SrTiO}}_3/{\mathrm{LaAlO}}_3$ interface as a function of the sheet carrier density $n_s$ via advanced first-principles calculations. Electrons localize spontaneously in Ti $3d_{xy}$ levels within a thin ($\lesssim 2\mathrm{nm}$) interface-adjacent ${\mathrm{SrTiO}}_3$ region for $n_s$ lower than a threshold value $n_c\sim {10}^{14}{\mathrm{cm}}^{-2}$. For $n_s>n_c$ a portion of charge flows into Ti $3d_{xz}\mathrm{\text{−}}{d}_{yz}$ levels extending farther from the interface. This intrinsic confinement can be attributed to the interface-induced symmetry breaking and localized nature of Ti $3d$ $t_{2g}$ states. The sheet carrier density directly controls the binding energy and the spatial extension of the conductive region. A direct, quantitative relation of these quantities with $n_s$ is provided.

Figure 1

Layer-projected and orbital-resolved density of states of the Ti $3d$ $t_{2g}$ orbitals in the symmetric STO/LAO supercell calculated by pSIC. Black lines: $d_{xy}$ singlet; red lines: ($d_{xz}$, $d_{yz}$) doublet; panels (a), (b), and (c) refer, respectively, to $n_s=0.5\mathrm{electrons}/A$ (i.e., $3.3\times {10}^{14}{\mathrm{cm}}^{-2}$), $0.15\mathrm{electrons}/A$ (${10}^{14}{\mathrm{cm}}^{-2}$), and $0.03\mathrm{electrons}/A$ ($0.2\times {10}^{14}{\mathrm{cm}}^{-2}$). ${\mathrm{Ti}}^0$ is the Ti atom at the interface, ${\mathrm{Ti}}^i$ the Ti atom of the $i$th layer below it. The $e_g$ submanifold is empty and well above this energy range.

Figure 2

Top: pSIC-calculated band energies for (a) $n_s=0.5\mathrm{electrons}/A$ ($3.3\times {10}^{14}{\mathrm{cm}}^{-2}$), and (b) $n_s=0.15\mathrm{electrons}/A$ (${10}^{14}{\mathrm{cm}}^{-2}$). Bottom: Panel (c) calculated Fermi surfaces for $n_s=0.5\mathrm{electrons}/A$ in the $1\times 1$ Brillouin zone of edge $2\pi /a_{\mathrm{STO}}$; labels identify the dominant $t_{2g}$ orbital character and the ${\mathrm{TiO}}_2$ layer they belong to (same notations as in Fig. 1); notice that $x{z}^i$, $y{z}^i$, with $i\ge 1$ label the (nearly degenerate) occupied doublet orbitals located on ${\mathrm{Ti}}^1$, ${\mathrm{Ti}}^2$, etc. Panel (d) sketch of the calculated extremal Fermi surfaces (dotted black line) divided up into three contributions, the circular $x{y}^0$ and two cigar-shaped $x{z}^i$ and $y{z}^i$ bands.

Figure 3

Left: sketch of the band alignment at the LAO/STO interface as calculated in pSIC; yellow (light gray) and red (dark gray) areas indicate $d_{xy}$ and $d_{xz}+d_{yz}$ contributions, respectively. Right: total $t_{2g}$ (dashed) and $d_{xy}$ (solid) charge densities per unit area as a function of chemical potential, calculated from the interface with $1/2\mathrm{electrons}/A$ assuming a rigid-band behavior ($E_F=0$ corresponds to occupancy $1/2\mathrm{electrons}/A$ or $3.3\times {10}^{14}{\mathrm{cm}}^{-2}$). Yellow (light gray) and red (dark gray) areas are contributions of planar $d_{xy}$ and orthogonal ($d_{xz}$, $d_{yz}$) orbitals, respectively. On the right $y$ axis, ${\mathrm{Ti}}^i$ indicates up to which Ti layer the $d_{xy}$ charge (indicated by the dashed horizontal line) spreads.