Spontaneous deformation of flexible ferromagnetic ribbons induced by Dzyaloshinskii-Moriya interaction

Here, we predict the effect of the spontaneous deformation of a flexible ferromagnetic ribbon induced by Dzyaloshinskii-Moriya interaction (DMI). The geometrical form of the deformation is determined both by the type of DMI and by the equilibrium magnetization of the stripe. We found three different geometrical phases, namely, (i) the DNA-like deformation with the stripe central line in the form of a helix, (ii) the helicoid deformation with the straight central line, and (iii) cylindrical deformation. In the main approximation the magnitude of the DMI-induced deformation is determined by the ratio of the DMI constant and the Young's modulus. It can be effectively controlled by the external magnetic field, which can be utilized for the nanorobotics applications. All analytical calculations are confirmed by numerical simulations.

Figure 1

Equilibrium states of flexible ferromagnetic ribbon with DMI in form ${\mathcal{E}}_{\text{D}}={\mathcal{E}}_{\text{D}}^{\text{B}}$: (a),(b) Shapes of DNA-like (a) and helicoid (b) states. (c) and (d) are phase diagrams of equilibrium states of the flexible ribbon. Symbols show the results of the numerical simulations: circles and triangles correspond to helicoid and DNA-like states, respectively. The thick green line in (c) and (d) describes the boundary between equilibrium states [36]. In all cases we have $\nu =1/3$.

Figure 2

Shapes and geometrical parameters of flexible ferromagnetic ribbons as functions of DMI strength: (a)–(c) are the ribbon shapes obtained by means of numerical simulations. (d) and (e) are the radius and pitch of the DNA-like state plotted for $A/\left(Y{h}^2\right)=1$, respectively. (f) Pitch of the helicoid state for $A/\left(Y{h}^2\right)=0.11$. (g) Radius of the cylindrical state for $A/\left(Y{h}^2\right)=1$. The red lines in (d) and (e) and the green line in (g) represent analytical predictions (2) and (6), respectively. The blue line in (f) is obtained from solution of the cubic Eq. (S26) in [36]. Symbols correspond to data obtained by means of numerical simulations. The length of ribbons in (d)–(f) is $L=50\ell$ and $L=10\ell$ in (g). All data are presented for ribbons with width $w\approx 2\ell$, thickness $h=0.05\ell$, and Poisson ratio $\nu =1/3$. The dynamical process of the DMI-induced deformation is illustrated in the Supplemental Material movie [36].

Figure 3

Influence of the magnetic field on the geometrical parameters of the ribbon: (a),(b) Relative field-induced changes of radius and pitch of the DNA-like state as a function of the applied magnetic field, respectively. (c) Relative field-induced change of the pitch of the helicoid state as a function of the applied field. (d) Relative field-induced change of the radius of the cylindrical state as a function of the applied field. Here $R_{\text{H}}$ and $P_{\text{H}}$ are the radius and pitch of structures in the presence of an external magnetic field, respectively. All data are obtained by means of numerical simulations for ribbons with $w\approx 2\ell , h=0.05\ell , A/\left(Y{h}^2\right)=1$, and $\nu =1/3$. Insets demonstrate the direction of the magnetic field $\mathit{H}$ with respect to the deformed ribbon and magnetization orientation. Field-induced dynamics is illustrated in the Supplemental Material movie [36].