Coupling between octahedral rotations and local polar displacements in $\mathrm{W}{\mathrm{O}}_3$/$\mathrm{Re}{\mathrm{O}}_3$ superlattices

We model short-period superlattices of $\mathrm{W}{\mathrm{O}}_3$ and $\mathrm{Re}{\mathrm{O}}_3$ with first-principles calculations. In fully relaxed superlattices we observe that octahedral tilts about an axis in the planes of the superlattices do not propagate from one material, despite the presence of the corner-shared oxygen atoms. However, we find that octahedral rotation is enhanced within $\mathrm{W}{\mathrm{O}}_3$ layers in cases in which strain couples with native antiferroelectric displacements of tungsten within their octahedral cages. Resulting structures remain antiferroelectric with low net global polarization. Thermodynamic analysis reveals that superlattices with sufficiently thick $\mathrm{Re}{\mathrm{O}}_3$ layers, the absolute number being three or more layers and the Re fraction $\ge 50\%$, tend to be more stable than the separated material phases and also show enhanced octahedral rotations in the $\mathrm{W}{\mathrm{O}}_3$ layers.

Figure 1

Octahedral tilt and rotation angles in each plane along the long axis of $2\times 2\times 6$ supercells of CaZrO${}_3$, WO${}_3$, and ReO${}_3$. The layer zero equatorial oxygen atoms were held fixed to maintain an excess octahedral tilt [(a), (c), and (e)] or octahedral rotation [(b), (d), and (f)] alone. The equatorial oxygen atoms in layer five were held fixed to maintain calculated bulk tilt and rotation angles. All other O atoms and all $A$ and $B$ cation locations were relaxed. The characteristic decay lengths $\lambda$ are computed by fitting to Eq. (1) and given in units of the layer index. (g) Tilt angles are the angles between the supercell $c$ axis and the polar axis of the octahedra. (h) Rotation angles evaluated by averaging the angles between the supercell $a$ and $b$ axes and $ab$-projected line of the nearest octahedral O-O axis. The differences between the lattice directions and the Cartesian axes in this work are small and are ignored.

Figure 2

The rotation [(a)–(e)] and tilt [(f)–(j)] angles of six-layer WO${}_3$/ReO${}_3$ fully relaxed superlattices are presented as a function of layer index. The vertical dashed line indicates the boundary between WO${}_3$ (to the left) and ReO${}_3$ (to the right). ReO${}_3$ layers all have nearly zero rotation. The superlattices with three or fewer WO${}_3$ layers have rotation angles $\approx 8^{\circ }$. The rotation in majority WO${}_3$ superlattices are near the bulk values $\approx 5^{\circ }$. All WO${}_3$ tilt angles are small, nearly zero in majority ReO${}_3$ superlattices.

Figure 3

The average angle to the $c$ axis of polar displacements in WO${}_3$ interface planes vary with the average magnitude of antferroelectric polar displacements induced in the ReO${}_3$ interface planes. The horizontal dashed line indicates the average polar displacement angle in bulk WO${}_3$ ($\approx {31}^{\circ }$). See Table 2 for detailed comparison of the data.

Figure 4

The densities of states for (a) two-layer, (b) four-layer, (c) six-layer, and (d) eight-layer superlattices are compared with the density of states of (a) bulk $B$O${}_3$ materials. In all calculations there were four formula units in each layer. We observe that all the superlattices possess Fermi levels within bands. In the cases with lowest Re fraction, there appears to be a gap opening below the Fermi energy.

Figure 5

The densities of states projected onto O-$p$ and metal-$d$ states in the (WO${}_3$)${}_3$/(ReO${}_3$)${}_3$ superlattice shows that the states just around the Fermi energy are made up of hybridized Re-$d$ and O-$p$ states.

Figure 6

Projected densities of states for the (WO${}_3$)${}_7$/(ReO${}_3$)${}_1$ superlattice are displayed for three layers within the superlattice. (a) Projection on the ReO${}_6$ octahedra, (b) projection on the WO${}_6$ octahedra of the second layer after rhenium layer, and (c) projection on the WO${}_6$ octahedra on the layer farthest from the rhenium layers.

Figure 7

The formation energies $E_f$ relative to separated WO${}_3$ and ReO${}_3$ phases [calculated from Eq. (2)] for all superlattice calculations are plotted as a function of Re concentration. Generally superlattices consisting of larger numbers of layers of ReO${}_3$ are more stable than those with fewer layers. The straight lines are energies calculated from the model [Eq. (3)] using the values in Table 3.