Lattice screening of the polar catastrophe and hidden in-plane polarization in KNbO${}_3$/BaTiO${}_3$ interfaces

We have carried out first-principles simulations, based on density functional theory, to obtain the atomic and electronic structure of (001) BaTiO${}_3$/KNbO${}_3$ interfaces in an isolated slab geometry. We tried different types of structures including symmetric and asymmetric configurations and variations in the thickness of the constituent materials. The spontaneous polarization of the layer-by-layer non-neutral material (KNbO${}_3$) in these interfaces cancels out almost exactly the “built-in” polarization responsible for the electronic reconstruction. As a consequence, the so-called polar catastrophe is quenched and all the simulated interfaces are insulating. A model, based on the modern theory of polarization and basic electrostatics, allows an estimation of the critical thickness for the formation of the two-dimensional electron gas between 33 and 36 KNbO${}_3$ unit cells. We also demonstrate the presence of an unexpected in-plane polarization in BaTiO${}_3$ localized at the $p$-type TiO${}_2$/KO interface, even under in-plane compressive strains. We expect this in-plane polarization to remain hidden due to angular averaging during quantum fluctuations unless the symmetry is broken with small electric fields.

Figure 1

(a) Schematic representation of a pristine polar interface in an isolated slab geometry. Atoms are represented by balls: O (small size, red), Ba (large size, yellow), Ti (medium size, blue), K (large size, green), and Nb (medium size, gray), with the corresponding atomic layer printed on top. The dots at the left of the BaTiO${}_3$ indicates that we assume a thick layer. At the other side, a free KO-terminated surface of the polar KNbO${}_3$ is assumed. Numbers below each layer indicate the formal ionic charge. The interface is marked with a red dashed line. Our choice for the unit cells that tile the entire crystal are represented by black dashed boxes. $a$ stands for the out-of-plane lattice constant of the polar material unit cell. (b) Electrostatic potential of the polar interface in the isolated slab geometry. Within our electrostatic boundary conditions, the macroscopic electric field in BaTiO${}_3$ is forced to be zero. Note that the field within the nonneutral layers points away from the interface. (c) Induced polarizations $\Delta P$ due to the macroscopic electric fields. (d) Schematic representation of the energy bands of the polar interface.

Figure 2

Schematic representation about the procedure to calculate (a) the out-of-plane rumpling ${\eta }^z$ and (b) the in-plane rumplings ${\eta }^x\text{and}{\eta }^y$. Solid lines represent the position of the atoms in the ideal unrelaxed structure. After relaxation, the atoms move in the directions indicated by the arrows. Meaning of the balls as in Fig. 1.

Figure 3

Out-of-plane (solid blue) and in-plane (long-dashed red) lattice polarization calculated for (a) symmetric-$p$ [$l$ fixed to 4.5 and $m=2$ (diamonds), $m=3$ (squares), $m=4$ (circles), $m=5$ (crosses)], (b) symmetric-$n$ ($m=5$, $l=4.5$), and (c) asymmetric ($m=4$, $l=8$) BaTiO${}_3$/KNbO${}_3$ slabs. Short-dashed (dot-dashed) lines represent the rumplings of the NbO${}_2$ (KO) layers in bulk KNbO${}_3$ under the same epitaxial constraint. The arrows point along the direction of the “built-in” polarization. Vertical dashed lines indicate the position of the interfaces. The in-plane rumplings plotted here correspond to ${\eta }_{xy}=\sqrt{{\eta }_x^2+{\eta }_y^2}$ defined in Fig. 2.

Figure 4

Macroscopically averaged internal electrostatic potential (black solid line) for (a) symmetric-$p$ ($m=5$, $l=4.5$), (b) symmetric-$n$ ($m=5$, $l=4.5$), and (c) asymmetric ($m=4$, $l=8$) BaTiO${}_3$/KNbO${}_3$ slabs. The red dashed lines indicates the region used to extract numerically the macroscopic field.

Figure 5

Schematic view of bulk unit cells of (a) BaTiO${}_3$ and (b) KNbO${}_3$, together with the ideal atomic structure at (c) $p$-type TiO${}_2$/KO, and (d) $n$-type BaO/NbO${}_2$ interfaces. Atoms are represented by balls following the same conventions as in Fig. 3. Numbers indicate the nominal charge of the different ions. Solid blue (dashed red) arrows represent an enhancement (depletion) of the in-plane electrostatic repulsion between ions. The schema shows how the replacement of ions favors the movement of the transition metal B-cation in TiO${}_2$/KO interfaces and hinders it in the BaO/NbO${}_2$ interfaces.

Figure 6

Two-dimensional energy surface as a function of the in-plane distortions of the BaTiO${}_3$ atoms, as indicated in the main text. The dashed line follows the minimum of the valley. Units in milli-electron-volts.