Theory for equilibrium 180° stripe domains in PbTiO3 films

A thermodynamic theory is developed for equilibrium 180° stripe domains in ferroelectric thin films on insulating substrates. Such stripe domains form to minimize the energy of the depolarizing field, and lead to a suppression of $T_C$ in thin films. Expressions including depolarizing field and domain wall energy are developed and applied to coherently strained $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ films on $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ substrates, with an upper boundary condition of either a dielectric $\left(\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3\right)$ , a conductor, or vacuum. An elastic solution appropriate for epitaxially strained stripe domains and 180° domain walls is presented. We minimize the full nonlinear free energy using a numerical technique to obtain equilibrium polarization and field distributions, and determine the equilibrium stripe period as a function of temperature and film thickness for each upper boundary condition. While the stripe periods found agree reasonably well with the existing analytical solution using a linearized free energy, the suppression of $T_C$ as film thickness decreases is as much as a factor of 10 smaller than that given by the linear solution.