Brown’s Equation for the Simulation of Complex Domain Structures in Ferroelectrics

Ferroelectric materials possess spontaneous polarization at room temperature whose direction can be switched by applying an external electric field. The rich domain structures inside ferroelectric materials, which contain polarizations in different directions, often have significant impacts on their physical properties and practical applications. As complex domain walls have been observed in experiments, their numerical simulation will be able to help understand or validate the structures observed in experiments, or provide certain guidance on controling such structures. A popular way is to build an energy model and employ it in Monte-Carlo simulations to find dipole configurations. However, as the Monte Carlo simulation usually reaches the ground state of the system (the state with the lowest global energy), interesting dipole configurations (e.g., topological domain structure) are often missed in the simulations which are usually metastable.Here, in order to simulate complex domain structures, we introduce Brown’s equations, which are originally used for micromagnetic simulation, into the largescale simulation of ferroelectric materials. Using the effective Hamiltonian as the energy model, we derive Brown’s equations with respect to the electric dipoles in ferroelectric materials, and invesitgate perovskites such as BaTiO3 bulk, PbTiO3 bulk, and SrTiO3/PbTiO3/SrTiO3 sandwiched structures. We demonstrate the reliability and feasibility of Brown’s equation in ferroelectrics through the simulation of BaTiO3 bulk and PbTiO3 bulk, which are consistent with the experiments. Then, using the Brown’s equation derived in our work, we discover various domain structures in SrTiO3/PbTiO3/SrTiO3 sandwiched structures, including periodic stripe domains, vortex domains, etc. Their comparation with exprimental results are also discussed in the paper.