Magnetization-induced shape transformations in flexible ferromagnetic rings

Flexible ferromagnetic rings are spin-chain magnets, in which the magnetic and mechanical subsystems are coupled. The coupling is achieved through the tangentially oriented anisotropy axis. The possibility to operate the mechanics of the nanomagnets by controlling their magnetization is an important issue for the nanorobotics applications. A minimal model for the deformable curved anisotropic Heisenberg ferromagnetic wire is proposed. An equilibrium phase diagram is constructed for the closed loop geometry: (i) A vortex state with vanishing total magnetic moment is typical for relatively large systems; in this case, the wire has the form of a regular circle. (ii) A topologically trivial onion state with the planar magnetization distribution is realized in small enough systems; magnetic loop is elliptically deformed. By varying geometrical and elastic parameters, a phase transition between the vortex and onion states takes place. The detailed analytical description of the phase diagram is well confirmed by numerical simulations.

Figure 1

Equilibrium state of the flexible ring: (a), (b) Magnetization (green arrows) and magnetic sites (blue dots) distribution obtained from numerical simulations for a chain with $L\approx 10.5$. (c), (d) Comparison of theoretical predictions Eqs. (8), (11) (lines) and results of numerical simulations (markers). The vortex state is obtained for $\beta =2$ and the onion state is obtained for $\beta =0.2$.

Figure 2

Phase diagram of the equilibrium magnetization and shape states for the flexible ferromagnet. Symbols correspond to simulation: Yellow circles correspond to the vortex magnetization distribution with circular shape of the wire; blue diamonds correspond to the onion magnetization distribution with elliptical wire shape. Green solid line describes the boundary $L_b\left(\beta \right)$ between the vortex and the onion states plotted with the prediction Eqs. (14) and the dashed line is its fitting by Eq. (15).

Figure 3

Rotation of the loop plane: (a)–(d) Magnetic and elastic configuration of the flexible system according to numerical simulations with $L\approx 10,\beta =0.5,\alpha =0.01$, and $\nu =0.01{\mu }_s/\left(a^2\left|\gamma \right|\right)$: (a) Initial equilibrium configuration; (b) deformed system under the action of external magnetic field and mechanical stress with temporal profile $f\left(\tau \right)$, described by Eq. (B4) with ${\tau }_1=50,{\tau }_2=550$ and $\delta =5$; (c) intermediate state of the ring during the rotation; (d) relaxed configuration after switching off external influence. Red arrow determines the direction of the normal vector $\mathit{n}$ to the ring plane [in (d) the normal vector $\mathit{n}$ does not lie in the $x0z$ plane]. Black dashed ellipselike ring in (b)–(d) corresponds to the initial shape of the ring (a). Purple and yellow arrows in (b) determine the direction of the external magnetic field $f\left(\tau \right)\mathit{b}$ and deformation $f\left(\tau \right)\left[{\mathit{r}}_i\left(\tau \right)-{\mathit{r}}_i\left(0\right)\right]$, respectively. (e) The comparison of theoretical predictions Eq. (19) and the results of numerical simulations. The corresponding dynamics is illustrated in the Supplemental movie [37].

Figure 4

Flexible ferromagnetic rings with inclusion of the dipole-dipole interaction: (a), (b) Comparison of theoretical prediction Eqs. (8), (11) (lines) and results of numerical simulations (markers). The vortex state is obtained for $\beta =2$ and the onion state is obtained for $\beta =0.2$. (c) Phase diagram of the equilibrium magnetization and shape states for the flexible ferromagnet. Symbols correspond to simulation: yellow circles correspond to the vortex magnetization distribution with circular shape of the wire; blue diamonds to the onion magnetization distribution with elliptical wire shape. Green solid line describes the boundary $L_b\left(\beta \right)$ between the vortex and the onion states plotted with the prediction Eqs. (14) and the dashed line is its fitting by Eq. (15).