Phonon in SrTiO$_3$ under finite electric fields

We propose an efficient approach within the density-functional theory to determine the phonon structure of infinite solids under finite electric fields. We apply this approach to technological SrTiO$_3$, predicting many unusual properties--including a giant frequency shift, anomalous piezoelectric response, as well as striking dielectric tunability. A microscopic understanding for individual phonon modes under finite electric fields is also provided.