Title: A coupled magneto-structural continuum model for multiferroic $\mathrm{BiFeO}_3$

Abstract: A continuum approach to study magnetoelectric multiferroic $\mathrm{BiFeO}_3$ (BFO) is proposed. Our modeling effort marries the ferroelectric (FE) phase field method and micromagnetic simulations in order to describe the entire multiferroic order parameter sector (polarization, oxygen antiphase tilts, strain, and magnetism) self-consistently on the same time and length scale. In this paper, we discuss our choice of ferroelectric and magnetic energy terms and demonstrate benchmarks against known behavior. We parameterize the lowest order couplings of the structural distortions against previous predictions from density functional theory calculations giving access to simulations of the FE domain wall (DW) topology. This allows us to estimate the energetic hierarchy and thicknesses of the numerous structural DWs. We then extend the model to the canted antiferromagnetic order and demonstrate how the ferroelectric domain boundaries influence the resulting magnetic DWs. We also highlight some capabilities of this model by providing two examples relevant for applications. We demonstrate spin wave transmission through the multiferroic domain boundaries which identify rectification in qualitative agreement with recent experimental observations. As a second example of application, we model fully-dynamical magnetoelectric switching, where we find a sensitivity on the Gilbert damping with respect to switching pathways. We envision that this modeling effort will set the basis for further work on properties of arbitrary 3D nanostructures of BFO (and related multiferroics) at the mesoscale.