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° and 180° domain walls, as well as for polar skyrmions, in ${\mathrm{BaTiO}}_3$, using a combination of density functional theory and phenomenological modeling. We find that the magnetic field reaches the order of several $\mu \mathrm{T}$ at the center of the wall, and we propose two experiments to measure the effect with nitrogen-vacancy center magnetometry.

Figure 1

(a) Schematic of a 90° domain wall in the $xy$ plane of ${\mathrm{BaTiO}}_3$ with thickness $2w$ and velocity $v$ separating the two domains $A$ and $B$. At the bottom, the rearrangement of the titanium and oxygen ions in the moving domain wall is depicted, as well as the change of ferroelectric polarization in time (thick arrows). The displacements of the oxygen and titanium ions are exaggerated for illustration purposes. Note that atomic radii, not ionic, are shown. (b) Illustration of the polarization evolution across 90° and 180° Néel-type domain walls, and in a polar skyrmion.

Figure 2

Time evolution of domain walls. Shown are the dimensionless amplitudes $Q_x$ (solid lines) and $Q_y$ (dashed lines) for the (a) 90° domain wall, (b) 180° domain wall, and (c) polar skyrmion. In (c), the amplitudes are shown only for $v=4000\mathrm{m}/\mathrm{s}$ for clarity. The resulting magnetic moments $m$ in units of ${10}^{-6}{\mu }_B$ and internal magnetic fields $B$ in units of $\mu \mathrm{T}$ are shown in (d)–(f).

Figure 3

Proposed experiments to measure the dynamical magnetic field induced by the motion of a ferroelectric domain wall. (a) A time-dependent electric field induces a back-and-forth motion (shivering) of a single domain wall, generating an oscillating magnetic field. (b) A time-dependent electric field in combination with a sawtooth potential provided by the substrate induces unidirectional motion of multiple domain walls as in Ref. [60], generating a magnetic field pulse train with alternating sign.