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

TL;DR

This paper extends first-principles calculations to three-dimensional electric displacement fields, mapping the energy landscape of PbTiO3 and revealing stable structures and low barriers that explain its piezoelectric properties.

Contribution

It generalizes the fixed-displacement-field approach to three dimensions, enabling detailed energy surface mapping for complex ferroelectric systems.

Findings

Identifies stable tetragonal, orthorhombic, and rhombohedral structures in PbTiO3.

Finds the energy minimum along [001] is deeper than along [110] or [111].

Discovers small energy barriers between minima, explaining large piezoelectric effects.

Abstract

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 [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.