  We describe ferroelectric thin films with circular electrodes and develop a
thermodynamic theory that explains previously mysterious experiments. It is
found to be especially useful for restricted geometries such as microstructures
for which boundary conditions are well known to play an important role in
ferroelectric properties. We have explored a new switching mechanism, which
consists of an inhomogeneous rotational motion of the polarization and leads to
an extended vortex state. The vortex appearance exhibits characteristic
properties of a first-order field-induced phase transition: three critical
electric fields and the possibility of hysteresis behavior.
