  In present work we calculated the three components of polarization in
phenomenological theory framework by consideration of three Euler-Lagrange
equations, which include mismatch effect and influence of misfit dislocations,
surface piezoelectric effect caused by broken symmetry on the film surface,
surface tension and depolarization field. These equations were solved with the
help of variational method proposed by us earlier. This approach lead to the
free energy in the form of algebraic equation of different powers of
polarization components with the coefficients dependent on film thickness,
mismatch effect, temperature and other parameters. Several new terms
proportional to misfit strain appeared in the free energy expression: built-in
electric field normal to the surface originated from piezoelectricity in
vicinity of surface even for the cubic symmetry of bulk ferroelectrics, odd
powers of normal to the surface component of polarization. The obtained free
energy made it possible to calculate all properties of the film by conventional
procedure of minimization. As an example we calculated phase diagrams of PZT
50/50 films on different substrates that lead to compressive or tensile strain.
The calculations of pyroelectric coefficient and dielectric permittivity
temperature dependencies had shown the electret-like polar state, e.g.
existence of pyroelectricity, below the critical thickness of
ferroelectric-paraelectric phase transitions. Our theory predicts, that
mismatch-induced field could be compatible with thermodynamic coercive field
and thus cause self-polarization in thin ferroelectric films.
