Characteristic dynamic modes and domain-wall motion in magnetic nanotubes excited by resonant rotating magnetic fields

We performed micromagnetic numerical calculations to explore a cylindrical nanotube's magnetization dynamics and domain-wall (DW) motions driven by eigen-circular-rotating magnetic fields of different frequencies. We discovered the presence of two different localized DW oscillations as well as asymmetric ferromagnetic resonance precession and azimuthal spin-wave modes at the corresponding resonant frequencies of the circular-rotating fields. Associated with these intrinsic modes, there exist very contrasting DW motions of different speed and propagation direction for a given DW chirality. The direction and speed of the DW propagation were found to be controllable according to the rotation sense and frequency of noncontact circular-rotating fields. Furthermore, spin-wave emissions from the moving DW were observed at a specific field frequency along with their Doppler effect. This work furthers the fundamental understanding of soft magnetic nanotubes’ intrinsic dynamic modes and spin-wave emissions and offers an efficient means of manipulating the speed and direction of their DW propagations.

Figure 1

(a) Spin configurations in the Py nanotube of indicated dimensions. The DW at the nanotube center is the head-to-head type and in the spiral configuration with the opposite two chiralities CCW and CW, as indicated. The gray arrows on the surface correspond to the local magnetization directions, while the color bar colors correspond to the $m_x(=M_x/M_{\mathrm{s}})$ components of the local magnetizations. (b) $m_x$ profile along the $x$ axis (longitudinal axis). (c) Local magnetizations on a cross section of the nanotube at $x=0$, indicating CCW and CW rotational senses of DWs.

Figure 2

(a) Log-scale FFT power with respect to the longitudinal axis and frequency (on the $x\text{−}f$ plane), as obtained from FFTs of $m_y(=M_y/M_{\mathrm{s}})$ oscillations averaged over a cross section of the nanotube (on the $y\text{−}z$ plane). (b) Log-scale FFT power spectrum averaged over the entire nanotube.

Figure 3

(a) Same as Fig. 2. (b) and (c) plot the speed of the DW propagation versus the rotating-field frequency $f_{\mathrm{H}}$, as averaged over its motion in an interval of $t=0-2.0\mathrm{ns}$ for the CCW and CW DW chiralities, respectively, in response to each of the ${\mathbf{H}}_{\mathrm{CCW}}$ (green open squares) and ${\mathbf{H}}_{\mathrm{CW}}$ (blue open circles) applications for a given strength of $H_0=50\mathrm{Oe}$.

Figure 4

DW oscillations in response to ${\mathbf{H}}_{\mathrm{CCW}}$ of (a) $f_{\mathrm{H}}=5.9\mathrm{GHz}$ and (b) $f_{\mathrm{H}}=9.8\mathrm{GHz}$, as represented by serial snapshot images of local magnetization distributions. The colors on the surface indicate the $m_x$ components. The black dashed line indicates the cross section of the nanotube at $x=0$. The gray ribbons are displayed on the isosurfaces of $m_x=-0.5$ and 0.5. In the right panel, the first shows the rotating fields, the second the $m_y$ component, the third ${\tau }_x^p$, and the last ${\tau }_z^d$ on the cross section of the nanotube at $x=0$. The outer and inner surfaces are marked by the black and green lines, respectively. The colors correspond to the local values of $m_y,{\tau }_x^p$, and ${\tau }_z^d$, as indicated by the corresponding color bars. The large green arrows show the directions of the rotating fields at the indicated time (first column), and the white arrows represent the orientations of the local magnetizations (second column). In the third and fourth columns, the white and black arrows represent the directions of the effective local torques for precession and damping, respectively.

Figure 5

Trajectories of orientations of local magnetizations averaged on the $y\text{−}z$ plane (orthogonal to the nanotube axis) at both $x=-0.5\mu \mathrm{m}$ (left) and $+0.5\mu \mathrm{m}$ (right) in the period of $t=0-0.5\mathrm{ns}$ for FMR (nonresonant precession) in the right (left) domain by circular-rotating fields of (a) ${\mathbf{H}}_{\mathrm{CW}}$ and (b) ${\mathbf{H}}_{\mathrm{CCW}}$ at $f_{\mathrm{H}}=12.9\mathrm{GHz}$. The insets show the trajectories of the local magnetization vectors for the nonresonant precession. (c) Temporal evolution of magnetostatic, exchange, and Zeeman energy densities for the entire nanotube and their sum in response to ${\mathbf{H}}_{\mathrm{CW}}$.

Figure 6

(a) Snapshot images of DW motion and spin-wave emission excited by ${\mathbf{H}}_{\mathrm{CW}}$ of $f_{\mathrm{H}}=20.8\mathrm{GHz}$, as represented by local $m_y$ distributions on the surface of the nanotube. (b) Local $\mathrm{\Delta }{m}_y\left[=m_y\left(t\right)-m_y\left(0\right)\right]$ images on a cross section (on the $y\text{−}z$ plane) at $x=-0.5\mu \mathrm{m}$ (left column) and $+0.5\mu \mathrm{m}$ (right). (c) Temporal evolution of magnetostatic, exchange, and Zeeman energy densities for the entire nanotube and their sum in response to ${\mathbf{H}}_{\mathrm{CW}}$.

Figure 7

(a) Perspective view of $m_y$ oscillations on the $x\text{−}t$ plane during DW motion and spin-wave emission in response to ${\mathbf{H}}_{\mathrm{CW}}$ of $f_{\mathrm{H}}=20.5\mathrm{GHz}$. The position of the moving DW is represented by the black line. (b) FFT power plot on the $x\text{−}f$ plane as obtained from FFTs of temporal $m_y$ oscillations at each $x$ value. (c) Dispersion on the $k_x\text{−}f$ plane as obtained from FFTs of spatiotemporal $m_y$ oscillations shown in (a).