  The finite screening length by real metallic electrodes, albeit very small
(<1A), results in finite depolarizing field that tends to split the film into
domains. In very thin ferroelectric films the domain structure reduces to
sinusoidal distribution of polarization considered first in the 1980s. We
discuss the phase transition between this structure and a single domain state
and show that it is first order, if it exists at all. The alternative
possibility is that the single domain state at zero bias voltage would be
metastable for all temperatures in most systems. This scenario defines a path
towards solution to a problem of finding parameters of a system that can
sustain the ferroelectric memory over a desired period of time.
