Dzyaloshinskii-Moriya domain walls in magnetic nanotubes

We present an analytic study of domain-wall statics and dynamics in ferromagnetic nanotubes with spin-orbit-induced Dzyaloshinskii-Moriya interaction (DMI). Even at the level of statics, dramatic effects arise from the interplay of space curvature and DMI: the domains become chirally twisted, leading to more compact domain walls. The dynamics of these chiral structures exhibits several interesting features. Under weak applied currents, they propagate without distortion. The dynamical response is further enriched by the application of an external magnetic field: the domain-wall velocity becomes chirality dependent and can be significantly increased by varying the DMI. These characteristics allow for enhanced control of domain-wall motion in nanotubes with DMI, increasing their potential as information carriers in future logic and storage devices.

Figure 1

Domain-wall profile in a thin nanotube with Dzyaloshinskii-Moriya interaction. The magnetization lies tangent to the nanotube surface.

Figure 2

Mechanical analogy: The profile $\theta \left(\zeta \right)$ may be regarded as the instanton orbit of a particle $\theta$ taking infinite time $\eta$ to move between consecutive local maxima of $V\left(\theta \right)-h_{e}cos\theta$ $(\kappa =1$ throughout). Dashed curve: With no DMI or applied field, the local maxima are $\theta =0$ and $\theta =\pi$, corresponding to domains aligned along the nanotube axis. The instanton orbit, indicated by the arrows, describes a tail-to-tail DW with negative chirality. Dotted-dashed curve: With DMI parameter $\eta =1$ but no applied field, the local maxima are shifted by $\delta =-\pi /8$, corresponding to twisted domains. The instanton orbit corresponds to a head-to-head DW with positive chirality. Solid curve: With $\eta =1$ and applied field $h_e=1$, the values of $V-h_{e}cos\theta$ at consecutive maxima are no longer equal; a specific value $j+v$, the coefficient of linear damping in Eq. (15), is required to ensure that particle reaches the second maximum without overshooting. Hashed curve: For large enough $h_e$, a maximum and minimum of $V-h_{e}cos\theta$ coalesce, and the instanton orbit is destroyed.

Figure 3

DW velocity $v$ vs DMI parameter $\eta$ for different chiralities $\chi$ and polarities $\sigma$ $(h_e=\kappa =1$ and $j=0)$. Changing the sign of $\chi$ and $\eta$ leaves $v$ unchanged, while changing the sign of $\sigma$ and $\eta$ alters the sign of $v$.

Figure 4

DW velocity $v$ vs DMI parameter $\eta$ for different values of the external field $h_e$ $(\sigma =+,\chi =+,\kappa =1$, and $j=0$ throughout). The solid curves are obtained from the numerical solution of Eq. (18); the dotted curves are given by the approximate analytical formula (19).

Figure 5

DW velocity $v$ vs applied field $h_e$ for different values of the DMI parameter $\eta$ and chirality $\chi$ $(\sigma =+,\kappa =1$, and $j=0)$. For $\eta =0,v$ is given by the exact linear relation $v=\sigma h_e/a$. For $\eta \ne 0,v$ is obtained by solving Eq. (18) numerically. Curves are computed up to the critical field $h_c$.

Figure 6

Critical applied field $h_c$ vs DMI parameter $\eta$.

Figure 7

Domain-wall profiles in a thin nanotube with Dzyaloshinskii-Moriya interaction and polarity $\sigma$ and chirality $\chi$. The magnetization lies tangent to the nanotube surface.