Magnetoelectricity induced by rippling of magnetic nanomembranes and wires

Magnetoelectric crystals have the interesting property that they allow electric fields to induce magnetic polarizations, and vice versa, magnetic fields to generate ferroelectric polarizations. Having such a magnetoelectric coupling usually requires complex types of magnetic textures, e.g., of spiraling type. Here, we establish a previously unknown approach to generate linear magnetoelectric coupling in ferromagnetic insulators with intrinsic Dzyaloshinskii-Moriya interaction (DMI). We show that the effect of nanoscale curved geometries combined with the intrinsic DMI of the magnetic shell lead to a reorganization of the magnetic texture that spontaneously breaks inversion symmetry and thereby induces macroscopic magnetoelectric multipoles. Specifically, we prove that structural deformation in the form of controlled ripples activates a magnetoelectric monopole in the recently synthesized two-dimensional magnets. We also demonstrate that in zigzag-shaped ferromagnetic wires in planar architectures, a magnetic toroidal moment triggers direct linear magnetoelectric coupling.

Figure 1

(a) Sketch of a two-dimensional rippled surface with the Euclidean and the local curvilinear coordinates. $\mathrm{\Theta }$ is the canting angle from the normal direction. (b) Behavior of the local curvature and of its first derivative for a ripple with a sinusoidal shape of height $h=0.1\lambda$ as a function $\widehat{x}$. (c) The magnetic ground state excluding curvature effects (top); with curvature effects but in the absence of an intrinsic DMI interaction (middle); and in the presence of DMI-induced curvature effects. The density plot shows the local canting angle along the rippled surface. (d) Behavior of the local canting angle of the magnetic texture in the absence ($d/\lambda =0$) and presence ($d/\lambda =0.05,0.1$) of intrinsic DMI coupling. The exchange length $l/\lambda =0.15$.

Figure 2

(a) The $\widehat{x}$ and $\widehat{z}$ components of the compensated magnetic texture $\mathbf{m}\left(x\right)-\overline{m}$ by artificially neglecting curvature effects (top panels), including only exchange-induced curvature effects with $l/\lambda =0.1$ (middle panels), and in the presence of the intrinsic DMI with $d/\lambda =l/\lambda$ (bottom panels). The ripple height $h=0.1\lambda$. The gray dashed line indicates the inversion center of the ripple while the orange lines are the mirror points located at the crests and valleys of the ripple. The same for the phase dependence of the toroidization component $T_y$ (b) and monopolization generated by the $\widehat{x}$ component of the magnetization (c). The phase on the sinusoidal ripple profile has been imposed as $z\left(x\right)=hsin(2\pi x/\lambda +\phi )$. (d) Behavior of the total monopolization as a function of the intrinsic DMI length $d/\lambda$ for ripples of different heights $h$. The exchange length has been set to $l/\lambda =0.1$.

Figure 3

(a)–(c) Tangential, quasitangential, i.e., including only exchange-induced curvature effects, and DMI-induced out-of-plane canted magnetic state for a zigzag magnetic nanowire. (d) shows the local behavior of the Euclidean components of the compensated normalized magnetic texture. We also show the inversion and rotation axis centers discussed in the main text.

Figure 4

(a)–(c) Phase dependence of the monopolization and toroidization components $T_{z,y}$ measured in units of $M_s\lambda$ with $M_s$ the saturated magnetization. We have chosen $l/\lambda =d/\lambda =0.1$. (d) Behavior of the toroidization as a function of the sinusoidal height $h/\lambda$.