Vortex Domain Walls in Helical Magnets

We show that helical magnets exhibit a nontrivial type of domain wall consisting of a regular array of vortex lines, except for a few distinguished orientations. This result follows from topological consideration and is independent of the microscopic models. We used simple models to calculate the shape and energetics of vortex walls in centrosymmetric and noncentrosymmetric crystals. Vortices are strongly anisotropic, deviating from the conventional Berezinskii-Kosterlitz-Thouless form. The width of the domain walls depend only weakly on the magnetic anisotropy, in contrast to ferromagnets and antiferromagnets. We show that vortex walls can be driven by external currents and in multiferroics also by electric fields.

Figure 1

Different types of helical ordering. (a) The magnetization rotates in a plane perpendicular to the helical ($x$) axis as in Tb, Dy, Ho. (b) Conical phase with a nonzero $m_3$ component of the magnetization as in Ho below 19 K. (c) The magnetization rotates in a plane parallel to the helical axis as in${\mathrm{TbMnO}}_3$.

Figure 2

DWs in centrosymmetric helical magnets. Cross section parallel to the $x\mathrm{\text{−}}y$ plane of (a) a Hubert wall, (b) a vortex wall parallel to the helical axis in a system where the magnetization rotates in the $x\mathrm{\text{−}}y$ plane, (c) a vortex wall tilted with respect to the helical axis. The arrows denote the orientation of $\mathbf{m}$. For systems where $\mathbf{m}$ is confined to the $y\mathrm{\text{−}}z$ plane, $\mathbf{m}$ has been rotated by $\pi /2$ for better visibility. The closed (red) contour is described in the text.

Figure 3

DWs in noncentrosymmetric helical magnets. A detail of Fig. 1(g) of Ref. 29 (center) showing two types of DWs in the ferromagnet FeGe; the left one includes vortices, the right one is vortex-free. The panels are theoretically calculated DWs, right without vortices, left with vortices.