First-principles study of the effect of (111) strain on octahedral rotations and structural phases of $\mathrm{LaAl}{\mathrm{O}}_3$

The structural and electronic response of $\mathrm{LaAl}{\mathrm{O}}_3$ to biaxial strain in the (111) plane is studied by density functional theory (DFT) and compared with strain in the (001) plane and isostatic strain. For (111) strain, in-plane rotations are stabilized by compressive strain and out-of-plane rotations by tensile strain. This is an opposite splitting of the modes compared with (001) strain. Furthermore, for compressive (111) strain, in-plane rotations are degenerate with respect to the rotation axis, giving rise to Goldstone-like modes. We rationalize these changes in octahedral rotations by analyzing the $V_A/V_B$ polyhedral volume ratios. Finally, we investigate how strain affects the calculated band gap, and find a 28% difference between the strain planes under 4% tension. This effect is attributed to different A-site dodecahedral crystal field splitting for (001) and (111) strains.

Figure 1

(a), (b) The definition of the rotation axes for (001) and (111) strains, respectively. For (001) strain, the $x$ and $y$ axes are parallel to the strain plane, while the $z$ axis is perpendicular to the strain plane. For (111) strain, neither the $x, y$, or $z$ axes are parallel or perpendicular to the strain plane. Note that under the assumption of quadratic strain, the assignment of $x,y,$ and $z$ for (111) strain is arbitrary. (c) The 40-atom cell used for (001) strain calculations. (d) The 30-atom unit cell used for (111) strain calculations. For the cells in (c) and (d), the a-b plane spans the (001) and (111) pseudocubic planes, respectively. For quadratic strain, we have $a=b$, which are the locked in-plane lattice parameters, while $c$ is the free to relax out-of-plane lattice parameter, and $a_{\mathrm{pc}}$ denotes the pseudocubic lattice parameter. Atomic structure figures made with VESTA [47].

Figure 2

Illustration of the in-plane and out-of-plane rotational modes under (a) (001) strain and (b) (111) strain. For (001) strain, $\left[100\right]$ and $\left[010\right]$ are in-plane, while $\left[001\right]$ is out-of-plane. For (111) strain, $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$ are in-plane, while $\left[111\right]$ is out-of-plane.

Figure 3

Phonon frequencies for the different rotational modes as a function (a) (001) strain and (b) (111) strain. Illustrations of the different rotational modes are shown in Fig. 2. The dashed lines are guides to the eye.

Figure 4

Energy vs strain for the different phases. (a) (001) strain, (b) (111) strain. The solid lines correspond to fits to parabolic functions, as described in the text. The data points correspond to the strain values where the respective phases were stable under geometry optimization.

Figure 5

Out-of-plane vs in-plane strain for the (001) and (111) planes. The different Poisson's ratios $\nu$ are calculated from the linear fits to the data. For (001) strain, the 0% strain point is from another space group and, hence, is not included in any of the fitted lines. For tensile (111) strain, the different degenerate phases relax to the same out-of-plane lattice parameter; thus, the they have the same $\nu$.

Figure 6

Energy difference with respect to the global minimum as a function of mode amplitude for the in-plane rotational (rot.) modes. (a) One percent tensile (001) strain where the $\left[100\right]$ in-plane rotation mode with $Fmmm$ symmetry is frozen in. (The $\left[010\right]$ mode is degenerate for all amplitudes.) (b) One percent compressive (111) strain, where the $\left[1\overline{1}0\right]$ phonon mode with symmetry $C2/m$ is frozen in. (The $\left[11\overline{2}\right]$ mode with $C2/c$ symmetry is degenerate for all amplitudes.) (c) Logarithmic energy difference for 1% tensile (001) strain when both the degenerate in-plane rotational modes $\left[100\right]$ and $\left[010\right]$ are frozen in simultaneously. A global energy minimum is observed for a single point with equal amplitude of $\left[100\right]$ and $\left[010\right]$ rotations, giving $Imma$ symmetry. (d) Logarithmic energy difference for 1% compressive (111) strain when both the degenerate in-plane rotational modes $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$ are frozen in simultaneously. The global energy minimum is no longer a single point but a circle with constant mode amplitude resembling a Mexican hat potential. For (c) and (d), each pixel represents the energy from one calculation; the black lines are contour plots of the energy landscape, and the arrows show the relationship to other high-symmetry directions.

Figure 7

(a) Pseudocubic octahedral rotation angles (degrees) vs (001) strain. (b)–(d) (111) strain. (e) Isostatic strain. $\alpha ,\beta ,$ and $\gamma$ are defined in Fig. 1. The colors and labels represent which space group has the lowest energy in this strain range. (b)–(d) The response for the three different space groups, which are degenerate under compressive strain in the (111) direction. Note that the response is equal for tensile strain. For the $P\overline{1}$ symmetry in (d), one arbitrary combination of $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$ rotation is shown. The dashed lines are guides to the eye.

Figure 8

Polyhedral volume ratio as a function of strain: (a) (001) strain, (b) (111) strain, (c) isostatic strain. The colors and labels represent which space groups have the lowest energy in this strain range. $V_{\mathrm{A}}/V_{\mathrm{B}}=5$ for the unrotated cubic phase and then decreases from this value as the distortions are increased. The dashed lines are guides to the eye.

Figure 9

Splitting of $d$ states of dodecahedral coordinated sites for (001) and (111) strain. Note that $e_g$ and $t_{2g}$ levels are reversed for dodecahedral sites compared with octahedral sites. The trigonal distortion from (111) strain splits the $t_{2g}$ levels into $a_{1g}$ and $e_g^{\prime },$ which are superpositions of the $d_{xy},d_{yz},$ and $d_{xz}$ states [45].

Figure 10

PBE-sol band gap of the different structures as a function of different strains. The colors and labels corresponds to which space groups are stable for (111) strain in the given strain range. For (001) strain, there is a change in symmetry at 0% strain, while for isostatic strain, there is a change in symmetry at −2% strain. For the $P\overline{1}$ symmetry, we have selected the same arbitrary combination of $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$ rotations, as in Fig. 7. The dashed lines are guides to the eye.

Figure 11

Schematic density of states showing the splitting of the $\mathrm{L}{\mathrm{a}}_{5d}$ states under (001), isostatic, and (111) strains. It is further shown how the splittings are related to the changes in band gap when LAO is strained.