Phonons in SrTiO$_3$ under finite electric fields

It has been nearly 40 years since Worlock and Fleury discovered by means of Raman scattering that {\it under finite electric fields} the transverse-optic (TO) phonons in SrTiO$_3$ exhibit a striking increase in frequency---by more than 400\% of its zero-field value---when a very small field of 12 kV/cm is applied [1]. Despite its obvious importance, this giant field- induced shift of phonon frequency has not thus far been independently investigated via the density-functional theory (DFT). Here, we propose an approach within the DFT theory to determine the phonon structure of infinite solids under finite electric fields. We applied this approach to SrTiO$_3$ and found a giant shift in TO frequencies, in accordance with the observations [1]. Our calculations further predict other unusual properties that occur under finite electric fields, that is, an anomalous piezoelectric response and large dielectric tunability. (This work was supported by ONR). [1] J.M. Worlock and P.A. Fleury, Phys. Rev. Lett. {\bf 19}, 1176 (1967).