Screw Dislocations in Chiral Magnets

Helimagnets realize an effective lamellar ordering that supports disclination and dislocation defects. Here, we investigate the micromagnetic structure of screw dislocation lines in cubic chiral magnets using analytical and numerical methods. The far field of these dislocations is universal and classified by an integer strength $\nu$ that quantifies its Burgers vector. We demonstrate that a rich variety of dislocation-core structures can be realized even for the same strength $\nu$. In particular, the magnetization at the core can be either smooth or singular. We present a specific example with $\nu =1$ for which the core is composed of a chain of singular Bloch points. In general, screw dislocations carry a noninteger but finite skyrmion charge so that they can be efficiently manipulated by spin currents and should contribute to the topological Hall effect.

Figure 1

(a) Conical magnetic helix with pitch ${\lambda }_h$ and cone angle ${\theta }_0$ enclosed by the magnetic moments and the direction of the applied magnetic field $\mathit{H}$. (b) Helimagnetic order is characterized by equidistant isosurfaces, the hallmark of lamellar order, where, e.g., the $x$ component of magnetization, $n_x$, assumes a particular value. (c)–(e) Example of a screw dislocation with strength $\nu =1$ illustrated by different vertical cross sections. The core of this specific example contains a chain of magnetic Bloch points.

Figure 2

Core of the screw dislocation ${\mathrm{sd}}_{-1}^{+}$ with strength $\nu =-1$ obtained by micromagnetic simulations for $H=0$. The magnetic moments represented by arrows form within a given $z$ plane an anti-vortex-like structure. The orange isosurface in the main panel is defined by $n_z=1/2$. The magnetization at the core is preferentially pointing in $\widehat{\mathit{z}}$ direction; the configuration with opposite core magnetization, ${\mathrm{sd}}_{-1}^{-}$, is degenerate at $H=0$ (not shown). Note that the periodicity of the core structure along $z$ is characterized by a wavelength $2{\lambda }_h$.

Figure 3

Energy per length of the various screw dislocation lines shown in Figs. 2 and 4, as a function of $H$ with a comparison to the skyrmion string energy (gray lines) whose core magnetization is aligned (${\mathrm{Sk}}^{+}$) or antialigned (${\mathrm{Sk}}^{-}$) with the field. The energies were obtained by micromagnetic simulations for a cylinder-shaped system with radius $R=5{\lambda }_h$. Dashed line shows the logarithmic contribution of the far field in Eq. (6) assuming ${\rho }_{\mathrm{core}}={\lambda }_h/2$ for illustration. The energy of dislocations ${\mathrm{sd}}_{-1}^{+}$ and ${\mathrm{sd}}_1^{+}$ vanishes at $H_{c2}$ where they can be identified as vortex lines of the $XY$-order parameter. The energy of ${\mathrm{sd}}_1^{\mathrm{Sk}-}$ merges at $H_{c2}$ with that of the skyrmion string ${\mathrm{Sk}}^{-}$. Points $A–D$ mark field values where the corresponding configurations became unstable in numerical simulations due to the lattice discreteness.

Figure 4

Core of various screw dislocations with strength $\nu =1$ obtained by micromagnetic simulations. Orange and blue isosurfaces are, respectively, defined by $n_z=1/2$ and $n_z=-1/2$ except in panel (a) where $n_z=1/2$ and $n_z=-1/4$. (a) Dislocation ${\mathrm{sd}}_1^{+}$ with core magnetization aligned with the field for $H=0.21H_{c2}$. (b) Dislocation ${\mathrm{sd}}_1^{\mathrm{Bp}}$ for $H=0$ with an alternating core magnetization separated by Bloch points (yellow spheres) with alternating topological charges $\pm 1$. (c) Dislocation ${\mathrm{sd}}_1^{\mathrm{Sk}-}$ at $H=0$ with an antialigned core magnetization; it smoothly connects to a skyrmion string configuration for $H\to H_{c2}$.