Mapping the energy surface of PbTiO₃ in multidimensional electric-displacement space

In recent years, methods have been developed that allow first-principles electronic-structure calculations to be carried out under conditions of fixed electric field. For some purposes, however, it is more convenient to work at fixed electric displacement field. Initial implementations of the fixed-displacement-field approach have been limited to constraining the field along one spatial dimension only. Here, we generalize this approach to treat the full three-dimensional displacement field as a constraint and compute the internal-energy landscape as a function of this multidimensional displacement-field vector. Using PbTiO₃ as a prototypical system, we identify stable or metastable tetragonal, orthorhombic, and rhombohedral structures as the displacement field evolves along the [001], [110], and [111] directions, respectively. The energy minimum along [001] is found to be deeper than that along [110] or [111], as expected for a system having a tetragonal ground state. The barriers connecting these minima are found to be quite small, consistent with the current understanding that the large piezoelectric effects in PbTiO₃ arise from the easy rotation of the polarization vector.