First-principles study of ferroelectric and antiferrodistortive instabilities in tetragonal ${\mathrm{SrTiO}}_3$

We carry out first-principles density-functional calculations of the antiferrodistortive (AFD) and ferroelectric (FE) soft-mode instabilities in tetragonal ${\mathrm{SrTiO}}_3,$ with the structural degrees of freedom treated in a classical, zero-temperature framework. In particular, we use frozen-phonon calculations to make a careful study of the anisotropy of the AFD and FE mode frequencies in the tetragonal ground state, in which an R-point AFD soft phonon has condensed. Because of the anharmonic couplings, the presence of this AFD distortion substantially affects both the AFD and FE mode frequencies. The AFD mode is found to be softer for rotations around a perpendicular axis ${(E}_g$ mode) than for rotations about the tetragonal axis ${(A}_{1g}$ mode), in agreement with experimental results. The FE mode, on the other hand, is found to be softer when polarized perpendicular to the tetragonal axis ${(E}_u$ mode) than parallel to it ${(A}_{2u}$ mode). The sign of this frequency splitting is consistent with the experimentally reported anisotropy of the dielectric susceptibility and other evidence. Finally, we present a discussion of the influence of various types of structural distortions on the FE instability and its anisotropy.