A dynamical magnetic field accompanying the motion of ferroelectric domain walls

The recently proposed dynamical multiferroic effect describes the generation of magnetization from temporally varying electric polarization. Here, we show that the effect can lead to a magnetic field at moving ferroelectric domain walls, where the rearrangement of ions corresponds to a rotation of ferroelectric polarization in time. We develop an expression for the dynamical magnetic field, and calculate the relevant parameters for the example of 90$^circ$ and 180$^circ$ domain walls in BaTiO$_3$ using a combination of density functional theory and phenomenological modeling. We find that the magnetic field reaches the order of several $mu$T at the center of the wall, and we propose two experiments to measure the effect with nitrogen-vacancy center magnetometry.