Manipulation of Magnetic Domain Walls by Ferroelectric Switching: Dynamic Magnetoelectricity at the Nanoscale

Controlling magnetism using voltage is highly desired for applications, but remains challenging due to a fundamental contradiction between polarity and magnetism. Here, we propose a mechanism to manipulate magnetic domain walls in ferrimagnetic or ferromagnetic multiferroics using the electric field. Different from those studies based on static domain-level couplings, here the magnetoelectric coupling relies on the collaborative spin dynamics around domain walls. Accompanying the reversal of spin chirality driven by polarization switching, a “rolling-downhill”-like motion of the domain wall is achieved in nanoscale, which tunes the magnetization locally. Our mechanism opens an alternative route to the pursuit of practical and fast converse magnetoelectric functions via spin dynamics.

Figure 1

(a) Schematic of ferrimagnetic domain walls. Magnetic sublattices $A$ and $B$ are distinguished by colors. $\mathbf{m}$: magnetic moments. $\mathbf{n}$: normalized staggered moments defined as ${\mathbf{m}}^A/|{\mathbf{m}}^A|$ and $-{\mathbf{m}}^B/|{\mathbf{m}}^B|$. The chirality of the domain wall is characterized by the azimuthal angle of $\mathbf{n}$ at the center of the wall: clockwise ($\varphi =0$) or counterclockwise ($\varphi =\pi$). (b) The major component of DM vector $\mathbf{D}$ is perpendicular to the $A\text{−}X\text{−}B$ plane, whose sign is determined by the displacement direction of anion $X$. (c) Schematic of the rolling-downhill-like mechanism. The domain wall energy from DM interaction is a function of chirality (i.e., $\varphi$). For a given chirality, the reversal of the DM vector makes the energy minimum to be maximum, which leads to the spin dynamics of the domain wall.

Figure 2

(a)–(c) Snapshots of $\mathbf{n}$ around the domain wall at 0 ps (counterclockwise Néel type), 23 ps (intermediate Bloch type), and 36 ps (clockwise Néel type), since the sudden reversal of $D_y$. Left: top view. The in-plane $xy$ components are represented by arrows and the $z$ components are indicated by the color map. Right: corresponding side view. (d) Motion distance ($|d|$) of domain wall center as a function of time with various $\alpha$’s. Inset: the relationship between final distance $|d_s|$ and $1/\alpha$. (e) The $x$, $y$, and $z$ components of net magnetization (in average of the whole lattice) as a function of time. The change of the $z$ component is due to the resize of the domains, while the reversal of the $x$ component is due to the reversal of chirality. The emergence of the $y$ component is due to the intermediate Bloch-type state. In (d) and (e), the simulation results (dots) agree with the analytical solutions (curves) well.

Figure 3

Dynamic analysis around the domain wall. (a) After the polarization flipping, the DM effective fields (${\mathbf{f}}_D^{A/B}\sim \nabla {\mathbf{m}}^{B/A}\times \mathbf{D}$) are along the $x$ axis and antiparallel between neighbors, since $\nabla {\mathbf{m}}^{A/B}$ are along the $z$ axis and antiparallel between neighbors. The DM torques (${\mathbf{\Gamma }}_D^{A/B}\sim {\mathbf{f}}_D^{A/B}\times {\mathbf{m}}^{A/B}$) are parallel between neighbors, and along the $z$ axis. (b) Such parallel ${\mathbf{\Gamma }}_D^{A/B}$ induce the canting between ${\mathbf{m}}^{A/B}$ and following effective fields (${\mathbf{f}}_J^{A/B}\sim 2J{\mathbf{m}}^{B/A}$) from exchange and thus additional antiparallel in-plane rotational torques (${\mathbf{\Gamma }}_J^{A/B}\sim 2J{\mathbf{m}}^{B/A}\times {\mathbf{m}}^{A/B}$). (c) Despite the identical amplitude of ${\mathbf{\Gamma }}_J^{A/B}$, the different amplitudes of ${\mathbf{m}}^{A/B}$ lead to different rotational angles. Such asynchronous procession further induces additional torques ${\mathbf{\Gamma }}_J^{A/B}$, antiparallel between neighbors and with components along the $z$ axis. (d) Such $xz$-plane rotational torques lead to the domain wall motion. Noting the direction of motion, locked by the azimuthal angular velocity ($\dot{\varphi }$) of moments. The cartoon of wheels is for intuitional understanding. (d) The distributions of three components of DM torques and net exchange torques, taking from above LLG simulation at 23 ps. Indeed, ${\mathbf{\Gamma }}_D$ (${\mathbf{\Gamma }}_J$) are nearly parallel (antiparallel) between nearest neighbors. A movie of dynamics and torques is included in Supplemental Material [25].

Figure 4

Directional control of domain wall motion using (a) a transverse magnetic bias or (b) a longitudinal DM bias which can be driven by an electric field along the $x$ axis via the dielectric polarization. Inset of (a): reversible CME switched by an ac electric field under a fixed transverse magnetic field. (c) Left axis: motion of domain wall as a function of time when $D_y$ changes linearly: $D_y^0(1-2t/t_p)$ from $t=0$ to $t_p$. Various $t_p$’s are tested. Right axis: an example of $D_y$. (d) Simulations of domain wall motion at finite temperatures.