Ferroelectricity in [111]-oriented epitaxially strained ${\mathrm{SrTiO}}_3$ from first principles

We use first-principles density-functional theory calculations to investigate the effect of biaxial strain in the low-temperature structural and ferroelectric properties of [111]-oriented ${\mathrm{SrTiO}}_3$. We find that [111] biaxial strain, achievable by coherent epitaxial growth along the [111] direction, induces structural distortions in ${\mathrm{SrTiO}}_3$ that are not present in either bulk or [001]-oriented ${\mathrm{SrTiO}}_3$. Under [111] biaxial strain, ${\mathrm{SrTiO}}_3$ displays ferroelectricity at tensile strain, and paraelectricity at compressive strain. We compute the phonon spectrum and macroscopic polarization of ${\mathrm{SrTiO}}_3$ as a function of [111] biaxial strain, and relate our results to the predictions of the free-energy phenomenological model of Pertsev, Tagantsev, and Setter [Phys. Rev. B 61, 825 (2000); Phys. Rev. B 65, 219901 (2002)].

Figure 1

(a) Cubic unit cell displaying a (111) plane and cubic lattice vectors ([100], [010], [001]). (b) Relationship between hexagonal (${\vec{a}}_1, {\vec{a}}_2, {\vec{a}}_3$) and Cartesian ($x, y, z$) lattice vectors. Sr, Ti, and O atoms are denoted by red, green, and blue spheres, respectively.

Figure 2

Phonon dispersion for ${\mathrm{SrTiO}}_3$ under (a) $0\%$, (b) $-4\%$, and (c) $+4\%$ [111] biaxial strain computed with DFT-LDA. Colors denote the atomic displacement magnitude for Sr (red), Ti (green), O (blue) at a given phonon frequency. We label longitudinal optical (LO) modes and phonon degeneracies. For easier comparison, we use the symmetry point labels $\mathrm{\Gamma }=(0,0,0), X=(1/2,0,0), M=(1/2,1/2,0), M^{\prime }=(1/2,-1/2,0), R=(1/2,1/2,1/2)$, and $R^{\prime }=(1/2,-1/2,1/2)$. (d) Brillouin zone for the reference structure at $-4\%$ (left) and $+4\%$ (right) strain [44]. Nonzero [111] biaxial strain lowers the symmetry of the Brillouin zone and leads to an additional symmetry line around the cubic $R$ point.

Figure 3

Energy (meV/f.u.) versus [111] biaxial strain (%) diagram for ${\mathrm{SrTiO}}_3$. For each strain value, relative energy stability is computed with respect to the reference $R\overline{3}m$ structure. Negative (positive) values represent compressive (tensile) [111] biaxial strain. The legend shows the symmetry modes for the low-symmetry structures. $C{\mathit{2}}^{*}$ denotes the structure obtained from C2 by freezing in the $R_5^{-}$[111] mode.

Figure 4

Polarization ($P$) versus [111] biaxial strain along the Cartesian coordinates ($x,y,z$). The relation between Cartesian and cubic directions is given in Fig. 1. Polar structures are separated among those without: (a) R3m, (b) C2, (c) $\mathit{Cm}$, and those with: (d) R3c, (e) C2${}^{*}$, (f) $\mathit{Cc}$, oxygen octahedral rotations.