Figure 1

(a), (b) Decomposition of the total valence-charge density according to the “dipole-free unit-cell” picture. Light and dark shadings indicate the portions that belong to the BO${}_2$ and AO layers, respectively. We impose both distributions to be symmetric around the respective atomic-layer locations and to contain a number of electrons equal to the total valence charge of the ions. The unit cell indicated by the arrow and the dashed lines have zero dipole moment by construction in both LaAlO${}_3$ (a) and SrTiO${}_3$ (b). (c), (d) Decomposition of the total valence charge based on maximally localized Wannier functions. The total Wannier densities of the AO and BO${}_2$ layers are shown by thick solid and thick dashed curves, respectively. The total charge density (obtained by the superposition of the periodically repeated Wannier densities) is shown as a thin solid line.Reuse & Permissions

Figure 2

Relationship between ${\mathbf{P}}_{\mathrm{bulk}}$ and the dipole moment of the bulk unit cell. Large red (dark gray) circles represent O ions, with charge $Q_{\mathrm{O}}=-2e$. Medium-size gold (light gray) circles represent the A-site cation, either Sr ($Q_{\mathrm{Sr}}=+2e$) or La ($Q_{\mathrm{La}}=+2e$); small green (light gray) circles superimposed to O are the B-site cation, either Ti ($Q_{\mathrm{Ti}}=+4e$) or Al ($Q_{\mathrm{Al}}=+3e$). The sketches (a)–(d) represent the projection of the atomic positions onto a (100)-oriented plane. On the top and the bottom are shown the values of ${\mathbf{P}}_{\mathrm{bulk}}$ (in units of $e/a_0^2$, where $a_0$ is the lattice parameter in either material), resulting in LaAlO${}_3$ and SrTiO${}_3$, respectively, from each cell arrangement.Reuse & Permissions

Figure 3

Lattices of allowed values of ${\mathbf{P}}_{\mathrm{bulk}}$ in LaAlO${}_3$ (left) and SrTiO${}_3$ (right). In the left panel, we show four possible surface-plane orientations. The polar orientations (dashed red lines) have no intersections with the ${\mathbf{P}}_{\mathrm{bulk}}$ lattice. The contrary is true for the nonpolar orientation (solid black lines).Reuse & Permissions

Figure 4

Slab models for the vicinal LAO surfaces described in the text. Top: (011). Center: (013). Bottom: (015). Color code for atoms is the same as in Fig. 2. Thick dashed lines indicate the supercells used in the simulations. Thin lines are guides to the eye. Shaded areas highlight the basic primitive unit that was used to construct the slab model. Thick arrows indicate the dipole moment of the basic unit, parallel to the surface plane. By displacing a surface O atom to a neighboring step edge, one can change the surface type from “A” to “B” (curved gray arrow in the bottom panel) and, hence, obtain pure AA or BB centrosymmetric slabs.Reuse & Permissions

Figure 5

Total density of states of the LAO (013) slab models. Surface A (black curve), B (red dashed curve), and bulk (shaded area) are shown.Reuse & Permissions

Figure 6

Primitive bulk SrTiO${}_3$ unit cells for the model A (left) and B (right) (111) slabs. Sr atoms (large green circle) lie at the corners of the cube; O atoms (small red circles) lie at the face-centered sites; Ti atoms lie at the center of the cube. Both choices of the primitive unit have zero dipole moment along any direction.Reuse & Permissions

Figure 7

Relaxed geometries of the two (111) slabs described in the text. The left structure corresponds to model A and the right one to model B.Reuse & Permissions

Figure 8

Local density of states (LDOS) on the Ti atoms for the two slab models described in the text. The Ti atom closest to top surface (type 1) corresponds to the red dashed curve; that lying closest to the bottom (type 2) surface is indicated as a solid blue curve. The average LDOS on the two central Ti atoms of either slab is shown as a shaded gray area. Top panel: model A; the inset shows a blowup of the spectrum in a neighborhood of the Fermi level (the LDOS of the Sr atom closest to the bottom A2 surface is also plotted as a thin solid curve). Bottom panel: model B; the inset shows the atomiclike spectrum of the topmost O atom (solid red curve with white shading), as compared with the bulklike spectrum of an O atom lying far from the surfaces (dark areas). Note that in the A case (top panel), we used a finer $(16\times 16\times 1)$ $k$-point grid to compute the LDOS in order to better describe the dispersive surface state at the A2 termination.Reuse & Permissions

Figure 9

(a), (c): Total DOS of the (100) LaAlO${}_3$ slabs, highlighting the population of the states involved in the compensation of the surface polarity (shaded area, indicated with an arrow). (b), (d): Plane-averaged density of compensating charge, related to the shaded portion of the DOS in (a) and (c). Top [(a) and (b)] and bottom [(c) and (d)] panels refer to the LaO- and AlO${}_2$-terminated slabs, respectively.Reuse & Permissions

Figure 10

Relaxed structure of the LaAlO${}_3$(100) slab with LaO (top) and AlO${}_2$ (bottom) terminations, compensated with OH($-$) and H(+) groups, respectively.Reuse & Permissions