We calculated the three components of polarization in phenomenological theory framework by consideration of three Euler-Lagrange equations allowing for mismatch effect and influence of misfit dislocations, surface piezoelectric effect caused by broken symmetry on the film surface, surface tension, and depolarization field. The equations were solved with the help of variational method proposed earlier [M. D. Glinchuk et al., Physica B 332, 356 (2002)]. This approach leads to the free energy in the form of algebraic expression for different powers of polarization components with the coefficients dependent on film thickness, misfit strain, temperature, etc. The odd powers of polarization related to built-in electric field normal to the surface originated from misfit strain and piezoelectricity in the vicinity of the surface which appeared in the free energy expansion. The obtained free energy opens the way for the calculation of the ferroelectric film properties by conventional procedure of minimization. As an example we calculated phase diagrams of PbZr0.5Ti0.5O3 [PZT(50∕50)] compressive- or tensile-strained films. The temperature behavior of pyroelectric coefficient and dielectric permittivity lead to the forecast of the electretlike polar state, i.e., existence of pyroelectricity below the critical thickness of ferroelectric-paraelectric phase transitions. The mismatch-induced electric field was shown to be the physical nature of the self-polarization phenomenon in ferroelectric thin films.

Glinchuk, M. D., Morozovska, A. N., & Eliseev, E. A. (2006). Ferroelectric thin films phase diagrams with self-polarized phase and electret state. Journal of Applied Physics, 99(11).