Highly Confined Spin-Polarized Two-Dimensional Electron Gas in ${\mathrm{SrTiO}}_3/{\mathrm{SrRuO}}_3$ Superlattices

We report first-principles characterization of the structural and electronic properties of $({\mathrm{SrTiO}}_3{)}_5/({\mathrm{SrRuO}}_3{)}_1$ superlattices. We show that the system exhibits a spin-polarized two-dimensional electron gas, extremely confined to the $4d$ orbitals of Ru in the ${\mathrm{SrRuO}}_3$ layer. Every interface in the superlattice behaves as a minority-spin half-metal ferromagnet, with a magnetic moment of $\mu =2.0{\mu }_B/{\mathrm{SrRuO}}_3$ unit. The shape of the electronic density of states, half-metallicity, and magnetism are explained in terms of a simplified tight-binding model, considering only the $t_{2g}$ orbitals plus (i) the bidimensionality of the system and (ii) strong electron correlations.

Figure 1

Schematic representation of the relaxed structure of $\mathrm{STO}/\mathrm{SRO}$ superlattices with $\mathrm{LDA}+U$. (a) Unit cell periodically repeated in space. O atoms (in red) are located at the vertex of the octahedra, while Sr (in green) are at the interstices between them and Ti (in blue) and Ru (in gray) are situated at the center according to the layer labeling. (b) Rotation angle of the oxygen octahedra, $\theta$, around the [001] direction ($z$ axis). Squares represent $\theta$ for the corresponding octahedra at the same height as in (a). Dotted lines are the theoretical value of $\theta$ for bulk STO. (c) Layer-by-layer rumpling, ${\eta }_i=[z(M_i)-z({\mathrm{O}}_i)]/2$, where $z(M_i)$ is the position of the cation and $z({\mathrm{O}}_i)$ is the position of O, in a given layer $i$. Atoms in layer $i$ move in the direction indicated by the arrows. Black lines represent the mean position of each atomic layer. Numbers inside the structure are the change in the interplanar distance between consecutive planes with respect to the ideal unrelaxed structure (half of the bulk value of STO, $a=3.874\AA$). Similar results are obtained within the B1-WC functional (see Discussion 3 of the Supplemental Material [25]). Units in Å.

Figure 2

Layer-by-layer PDOS on the atoms at the $\mathrm{Sr}B{\mathrm{O}}_3$ layers ($B=\mathrm{Ti}$ or Ru), within (a) the $\mathrm{LSDA}+U$ and (b) the B1-WC hybrid functional, for the corresponding relaxed $\mathrm{STO}/\mathrm{SRO}$ superlattice. Each panel represents the PDOS of a unit cell, numbered as in Fig. 1a. Corresponding layers in the upper half of the structure are equivalent by symmetry. Majority (minority) spin is represented in the upper (lower) half of each panel.

Figure 3

PDOS on the atoms at the central SRO layer, projected on the orbitals of (a) an apical O atom, (b) the Ru atom, and (c) an equatorial O atom. In the DOS projections corresponding to oxygen, the $2p$ orbital directed towards the transition metal ions is denoted by $\mathrm{O}(\sigma )$, while the two quasidegenerate orbitals perpendicular to this bond direction are $\mathrm{O}(\pi )$. The arrows in (b) mark the position of the peaks for the Ru-$d_{xz,yz}$ orbitals (see text).

Figure 4

Result of the tight-binding model used to interpret the DOS of $t_{2g}$ $d$ orbitals. Our model shows the transformation from (a) the three-dimensional bulk to (b) a two-dimensional structure. Refinements of the model to include, first, covalent bonding effects in plane and out of plane (c), and second, shifts of the bands due to strong electronic correlation via a Hubbard-$U$ term (d), are required to reproduce the first-principles band structure. Dotted black lines in (a) represent the $d_{xy,xz,yz}$ manifold, degenerated in energy. Once the symmetry is broken, the $d_{xy}$ state is represented by a solid green line and the degenerated $d_{xz,yz}$ orbitals by dotted red lines. Panel (e) illustrates the orbitals used in the tight-binding model and differences of in-plane and out-of-plane bonding for $d_{xz}$ orbitals (identical sketch can be done for the $d_{yz}$ orbitals).