Property control from polyhedral connectivity in $AB{\mathrm{O}}_3$ oxides

We investigate the effect of the degree of metal-oxygen octahedral facesharing on the mechanical and electronic properties of $d^0 {\mathrm{BaTiO}}_3$ and $d^3 {\mathrm{BaMnO}}_3$. We find that increased facesharing softens the elastic constants of both materials due to the increased volume per atom, with polar distortions also contributing to the reduction in the bulk modulus. Owing to orbital filling in the $d$ manifold, we find the electronic band gap of ${\mathrm{BaTiO}}_3$ is relatively unaffected by changes in percent facesharing whereas the band gap of ${\mathrm{BaMnO}}_3$ increases by more than 200% as the percent facesharing increases from 0% (cubic perovskite) to 100% (hexagonal ${\mathrm{BaNiO}}_3$ perovskite). We identify that the trigonal distortions present in the face-connected polymorphs represent useful atomistic structural knobs to tune band structure in hexagonal perovskites. Our results indicate that facesharing hexagonal polymorphs provide an expanded oxides arena with additional structural flexibility beyond the usual fully corner-connected perovskites for property control.

Figure 1

Changes in octahedral connectivity and orbital structure in $AB{\mathrm{O}}_3$ perovskites. Perovskites with different percentages of facesharing. (Right) Local coordination of a transition metal $B$ cation in a trigonally distorted ${\mathrm{O}}_6$ octahedron. An ideal undistorted octahedron with equal bond lengths exhibit a trigonal angular distortion $\theta =\arccos (1/\sqrt{3})$ and a trigonal octahedral strain $\eta ={\ell }_1/{\ell }_2=1$. Crystal field splitting of the degenerate $B\left(1\right)t_{2g}$ levels (${\mathrm{\Delta }}_1$) is due to these trigonal distortions and the contribution from the neighboring $B\left(2\right)$ cations to the crystal field. The sign of ${\mathrm{\Delta }}_1$ reverses if the trigonal distortion is compressive rather than tensile as depicted here.

Figure 2

Schematic illustration of polar and antipolar displacements present in various polymorphs of ${\mathrm{BaTiO}}_3$ and ${\mathrm{BaMnO}}_3$. The 100% facesharing polymorph of ${\mathrm{BaMnO}}_3$ in this study does not have polar displacments.

Figure 3

Average (a) $A$-O and (b) $B$-O bond lengths in ${\mathrm{BaTiO}}_3$ and ${\mathrm{BaMnO}}_3$ as a function of percentage facesharing with their standard deviations. (c) shows the trigonal octahedral strain $\eta ={\ell }_1/{\ell }_2$ as defined in Fig. 1. (d) and (e) show the evolution in the metal-oxygen-metal bond angles for the corner-connected and faceshared octahedra, respectively, with percentage facesharing.

Figure 4

Relative energy differences with percentage facesharing given with respect to the lowest energy polymorph of ${\mathrm{BaTiO}}_3$ and ${\mathrm{BaMnO}}_3$, respectively.

Figure 5

Evolution in the (a) computed bulk moduli with percentage facesharing. Variation in elastic constants with percentage facesharing for (b) ${\mathrm{BaMnO}}_3$ and (c) ${\mathrm{BaTiO}}_3$. In these panels, each elastic stiffness coefficient at a fixed facesharing percentage is represented as a bar and the corresponding index of the tensor from left to right is $11\to 22\to 33\to 44\to 55\to 66\to 12\to 23\to 31$ as shown in the legend of (c).

Figure 6

Evolution in the bulk modulus with unit-cell volume per atom for centrosymmetric (CS, empty symbols) and polar perovskite polymorphs (filled symbols). The unit cell volume per atom increases with increasing percentage facesharing (from left to right). The 0% facesharing structures (solid symbols) exhibit the smallest volumes per atom and are mechanically softer owing to the polar transition metal displacements present in the structures. The arrowed open symbols are for a reference CS cubic phase and illustrate the sensitivity of the modulus to both the higher density and presence of inversion symmetry.

Figure 7

Orbital energies as a function of trigonal angular distortion $\theta$ from a point charge model. Ideal octahedra exhibit $\theta \approx 54.72$ as indicated by the broken line. Values of $\theta \gtrsim 54.72$ ($\theta \lesssim 54.72$) corresponds to compression (elongation) of the octahedra along the trigonal axis. The insets depict the (upper) $e_g^{\sigma }$ orbitals which are oriented along $M$-O bonds, (center) $e_g^{\pi }$ orbitals, and (lower) $a_{1g}$ orbital which overlaps with a similar orbital on a neighboring metal ion.

Figure 8

Electronic density of states (DOS) for 100%, 50%, 33%, and 0% percent facesharing phases of ${\mathrm{BaTiO}}_3$ (left column, descending) and ${\mathrm{BaMnO}}_3$ (center column, descending). Black lines are the total DOS, red lines represent O $2p$ states from shared octahedral corners, green represents O $2p$ states from a common face, and blue represents metal $3d$ states. The minority spin channel for ${\mathrm{BaMnO}}_3$ is not shown but is equivalent symmetry. (Right column) Evolution in the electronic band gaps with percentage facesharing calculated at the DFT-PBEsol level. The overall trend is that the electronic gap increases as the number of faceshared octahedra increase.

Figure 9

Band structures of 100% facesharing ${\mathrm{BaTiO}}_3$ and ${\mathrm{BaMnO}}_3$. The Fermi level is set to the top of the valence band (0 eV, broken line). The majority and minority spin bands overlap in ${\mathrm{BaMnO}}_3$ owing to the antiferromagnetic arrangement of spins and so only the majority spin-channel is presented. See the main text for a description of the labeled bands.

Figure 10

(a) Projection of the Wannier orbital with $a_{1g}$ character onto the manifold of conduction band states. (b) Schematic $a_{1g}$ bonding and $a_{1g}^{*}$ antibonding orbital combinations at the $\mathrm{\Gamma }$ and $\mathrm{A}$ points illustrate why the bonding (antibonding) bands run “uphill” (“downhill”) along the trajectory $\mathrm{\Gamma }\to \mathrm{A}$. The band degeneracy at the $\mathrm{A}$ point is due to the equal proportion of mixed-bonding character. (c) The DFT partial charge density, supporting the assignment of the bonding and antibonding character at the zone center, derived from the metal-metal bonding with $a_{1g}$ and $a_{1g}^{*}$ symmetry. The isosurface levels for the density contours correspond to $0.128e{\AA }^{-3}$ and $0.162e{\AA }^{-3}$, respectively.

Figure 11

Evolution in the electronic band gap for the 100% facesharing ${\mathrm{BaTiO}}_3$ and ${\mathrm{BaMnO}}_3$ phases with respect to uniaxial tension and compression. The mechanical constraints induce trigonal octahedral strains, $\eta$, that alter the transition metal distances and thus the $d-d$ interactions. The band gap of ${\mathrm{BaTiO}}_3$ is more strongly affected than that of ${\mathrm{BaMnO}}_3$ because of the $a_{1g}$ character at the conduction band edge. The equilibrium trigonal octahedral strains for ${\mathrm{BaTiO}}_3$ and ${\mathrm{BaMnO}}_3$ are indicated by the broken vertical lines.