Modeling the Motion of Ferroelectric Domain Walls with the Classical Stefan Problem

With advances in nanotechnology, ferroelectric switching by individual domain walls (DWs) has become a subject of broad interest. Conventional models consider DW motion in a fixed homogeneous or inhomogeneous electric field. However, it is clear that the electric field commonly evolves in time due to the redistribution of bound charges and screening free charges on the ferroelectric surface, particularly due to surface conductance. Taking this effect into account remains a serious challenge. Here we propose a simple concept to describe simultaneously the evolution of the electric field and the DW motion in a ferroelectric sample. The approach is based on a full analogy between charge transport during ferroelectric switching and heat transport in a moving melting front: the classical Stefan problem. The analogy helps in the establishment of control of DW motion in thin films. Experimentally, DWs are displaced by voltage pulses under electrodes with a limited conductivity and detected using piezoresponse force microscopy. In the frame of a combined theoretical and experimental approach, DW-friction mechanisms and the impact of the DW energy on its shape are discussed. The study facilitates the development of nanoelectronic devices.