Abrupt appearance of the domain pattern and fatigue of thin ferroelectric films

We study the domain structure in ferroelectric thin films with a `passive' layer (material with damaged ferroelectric properties) at the interface between the film and electrodes within a continuous medium approximation. An abrupt transition from a monodomain to a polydomain state has been found with the increase of the `passive' layer thickness $d$. The domain width changes very quickly at the transition (exponentially with $d^{-2}$). We have estimated the dielectric response $dP/dE$ (the slope of the hysteresis loop) in the `fatigued' multidomain state and found that it is in agreement with experiment, assuming realistic parameters of the layer. We derive a simple universal relation for the dielectric response, which scales as $1/d$, involving only the properties of the passive layer. This relation qualitatively reproduces the evolution of the hysteresis loop in fatigued samples and it could be tested with controlled experiments. It is expected that the coercive field should increase with decreasing lateral size of the film. We believe that specific properties of the domain structure under bias voltage in ferroelectrics with a passive layer can resolve the long-standing `paradox of the coercive field'.