  The calculation of the polarization in ferroelectric thin films is performed
using an analytical solution of the Euler-Lagrange differential equation with
boundary conditions with different extrapolation lengths of positive sign on
the surfaces. The depolarization field effect is taken into account in the
model for a short-circuited single domain film, that is a perfect insulator. It
is shown that the calculation of the polarization and other properties profiles
and average values can be reduced to the minimization of the free energy
expressed as a power series of the average polarization with a renormalized
coefficient which depends on temperature, film thickness, extrapolation
lengths, and a coefficient for the polarization gradient term in the free
energy functional, the depolarization field being also included into the
renormalized coefficient. The function defining the space distribution
properties is calculated as well and its amplitude is shown to coincide with
the average polarization. The detailed calculations of the spontaneous
polarization, dielectric susceptibility and pyrocoefficient is performed. The
divergence of the dielectric susceptibility and pyrocoefficient for critical
parameters of the thickness induced ferroelectric phase transition, namely at
temperature T(cl) and critical length l(c), is shown to exist with and without
the depolarization field contribution, although the values of T(cl) and l(c)
are different in both cases. The detailed analysis of the depolarization field
space distribution and of this field dependence on temperature and film
thickness is performed.
