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A two-dimensional phase field simulation of ferroelectric films is used that incorporates Landau-Devonshire energy, gradient energy and depolarization electrical energy. A new intermediate electrical boundary condition is firstly presented to study the effects on domain structures of ferroelectric films. Two-dimensional simulations of domain structures are carried out under the open circuit (OC), short circuit (SC) and intermediate (IM) electrical boundary conditions. The simulation results show that there are multi-vortices domains and 180° multi-stripes domains under OC and SC electrical boundary condition, respectively. And there is a transition from multi-vortices domains to 180° multi-stripes domains under the IM electrical boundary condition due to competition between the elastic energy and depolarization electrical energy in the films. The present IM electrical boundary condition can completely characterize different degrees of compensation for surface charges by the electrodes and further describe the effect on the depolarization electrical energy. It can also be reduced to OC and SC electrical boundary conditions. Hence, for nano-thin ferroelectric films, the IM electrical boundary condition plays an important role in the formation of domain structures.


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