Domain formation and dielectric response in PbTiO$_3$: A first-principles free energy landscape analysis

We determine the relative thermodynamic stability of competing homogeneously and inhomogeneously ordered ferroelectric phases of PbTiO$_3$ using its free energy landscape, obtained from a newly developed method based on a combination of constrained polarization molecular dynamics simulations with a first-principles effective Hamiltonian and thermodynamic integration. While we find that the tetragonal structure is thermodynamical ly most stable at temperatures below the ferroelectric transition temperature ($T_0=660 K$), free energy of an ``orthorhombic-like'' 90$^{\circ}$ domain phase relative to the tetragonal phase almost vanishes at $T=540 K$, and remains small at all temperatures below $T_0$. In contrast to BaTiO$_3$, 90$^{\circ}$ domain walls are an order of magnitude lower in energy than 180$^{\circ}$ domain walls. We show that the computed dielectric response of the ``orthorhombic-like'' phase includes contributions from domain walls, and thus is significantly larger than that of the unifor mly polarized tetragonal phase of PbTiO$_3$.