  The ferroelectric properties of disorder Pb(Zr$_{0.5}$Ti$_{0.5}$)O$_{3}$ thin
films are investigated with Monte Carlo simulations on the basis of a
first-principles-derived Hamitonian. It is found that there exists a critical
thickness of about three unit cells ($\sim$12 \AA) below which the
ferroelectricity disappears under the condition that the in-plane polarizations
are suppressed by sufficient clamping effect. Above the critical thickness,
periodic $180^{\circ}$ stripe domains with out-of-plane polarizations are
formed in the systems in order to minimize the energy of the depolarizaing
field. The stripe period increases with increasing film thickness. The
microscopic mechanism responsible for these phenomena is discussed.
