Understanding magnetoelectric switching in BiFeO$_3$ thin films

In this work we use a phenomenological theory of ferroelectric switching in BiFeO$_3$ thin films to uncover the mechanism of the two-step process that leads to the reversal of the weak magnetization of these materials. First, we introduce a realistic model of a BiFeO$_3$ film, including the Landau energy of isolated domains as well as the constraints that account for the presence of the substrate and the multidomain configuration found experimentally. We use this model to obtain statistical information about the switching behavior - by running dynamical simulations based on the Landau-Khalatnikov time-evolution equation, including thermal fluctuations - and we thus identify the factors that drive the two-step polarization reversal observed in the experiments. Additionally, we apply our model to test potential strategies for optimizing the switching characteristics.