Pressure-induced antiferrodistortive phase transition and phonon softening in SrTiO$_3$

SrTiO$_3$, at room temperature, undergoes an antiferrodistortive transition under pressure with a critical pressure of P$_c \sim 9.6$ GPa. This transition is accompanied by a cubic--to--tetragonal structural distortion, and the same distortion can be induced at ambient pressure by lowering the sample temperature to below T$_c \sim 105$ K. The temperature-induced transition is known to involve a soft phonon at the R point in the Brillouin zone based on neutron scattering, inelastic x-ray scattering, and thermal diffuse scattering studies. The same soft mode is expected for the pressure induced transition, and we report herein the first direct measurement using inelastic x-ray scattering and a diamond-anvil pressure cell. The phonon softening behavior follows a power law and is accompanied by a central peak. The results are analyzed theoretically and correlated with those for temperature-induced transition.