Pressure-induced anomalous enhancement of piezoelectricity and polarization rotation via an unexpected monoclinic phase of PbTiO$_3$

Pressure-induced phase transitions and piezoelectricity of PbTiO$_3$ were studied using the {\it ab initio\ } density functional perturbation theory (ABINIT4.3.3). A tetragonal ($P4mm$) to monoclinic ($Pm$) phase transition occurs at 9 GPa, while this monoclinic phase transforms into paraelectric cubic phase at 22 GPa. The monoclinic ($Pm$) phase acts as the pressure-induced structural bridge between the tetragonal ($P4mm$) and rhombohedral ($R3m$) phases since its polarization rotates continuously in the pseudocubic ($\bar{1}$10) plane from the [001] towards the [111] pseudocubic directions. Under hydrostatic pressure, the enthalpy ($H = E + PV$) difference between the tetragonal and rhombohedral phases becomes tiny, and the polarization rotation via the monoclinic phase is possible, which results in huge enhancement of the piezelectric constant $e_{15}$, very similar to relaxor PZT. At 9 GPa, the spontaneous polarization is roughly half of that at 0 GPa, while the piezoelectric coefficient $d_{15}$ is comparable in magnitude to $d_{33}$ of relaxors PMN-PT and PZN-PT.