Observation of end-vortex nucleation in individual ferromagnetic nanotubes

The reversal of uniform axial magnetization in a ferromagnetic nanotube (FNT) has been predicted to occur through the nucleation and propagation of vortex domains forming at the ends. We provide experimental evidence for this behavior through dynamic cantilever magnetometry measurements of individual FNTs. In particular, we identify the nucleation of the vortex end domains as a function of applied magnetic field and show that they mark the onset of magnetization reversal. We find that the nucleation field depends sensitively on the angle between the end surface of the FNT and the applied field. Micromagnetic simulations substantiate the experimental results and highlight the importance of the ends in determining the reversal process. The control over end-vortex nucleation enabled by our findings is promising for the production of FNTs with tailored reversal properties.

Figure 1

Schematic diagram of the measurement: Si cantilever (gray) and CoFeB FNT (blue) with GaAs core (red). The cantilever oscillates about $\widehat{y}$ and the FNT axis is parallel to $\widehat{z}$. The applied magnetic field $\mathbf{H}$ can be rotated in the $xz$ plane by an angle ${\theta }_H$ with respect to $\widehat{z}$. The unit vector ${\widehat{n}}_T$ (${\widehat{n}}_B$) defines a perpendicular plane, in which the top (bottom) end of the FNT lies. The angle ${\alpha }_T$ (${\alpha }_B$) refers to the angle of this vector with respect to the FNT axis ($\widehat{z}$).

Figure 2

Magnetic reversal of (a) a 2.2-$\mu \mathrm{m}$-long and (b) a 0.6-$\mu \mathrm{m}$-long FNT measured by DCM at 280 K. $\mathbf{H}$ is applied approximately along $\widehat{z}$. As in all following figures, color-coded arrows denote the direction, in which the magnetic field is stepped.

Figure 3

Simulated and measured reversal of a 2.2-$\mu \mathrm{m}$-long FNT. (a) Simulated magnetization configurations for $\mathrm{\Delta }f\left(H\right)$ corresponding to the labels. (b) Simulated (lines) and measured (points) DCM signal. Squares highlight those simulated vortex domain nucleation/annihilation features, which are difficult to see. (c) A detailed view of DCM signature corresponding to the nucleation (annihilation) of the first vortex domain [29]. The simulation, which is offset for clarity, uses ${\alpha }_T=6.5^{\circ }, {\alpha }_B=10.5^{\circ }$, and ${\theta }_H=11.0^{\circ }$.

Figure 4

Simulated reversal of a 0.6-$\mu \mathrm{m}$-long FNT. Equilibrium magnetization configurations corresponding to the labeled points in $\mathrm{\Delta }f\left(H\right)$ for a FNT initialized with vortex end domains of (a) opposing and (b) matching circulation sense. (c) Plots of the simulated $\mathrm{\Delta }f\left(H\right)$ in purple (red) and $E_m\left(H\right)$ in gray (black) for end domains of opposing (matching) circulation sense. For the simulation, ${\alpha }_T=6.0^{\circ }, {\alpha }_B=10.0^{\circ }$, and ${\theta }_H=10.0^{\circ }$.

Figure 5

Simulated and measured reversal of a 0.6-$\mu \mathrm{m}$-long FNT. (a) Calculated magnetization configurations for $\mathrm{\Delta }f\left(H\right)$ corresponding to the labels. (b) Simulated (lines) and measured (points) DCM response. Squares highlight those simulated vortex nucleation/annihilation features, which are difficult to see. For the simulation, ${\alpha }_T=4.0^{\circ }, {\alpha }_B=6.5^{\circ }$, and ${\theta }_H=10.0^{\circ }$.

Figure 6

Simulated and measured dependence of vortex nucleation field on field angle of a 0.6-$\mu \mathrm{m}$-long FNT. Black (red) points show the measured $H_n$ as a function of ${\theta }_H$ for the top (bottom) end vortex in a single FNT. The black (red) solid line shows the corresponding simulations for ${\alpha }_T=6.0^{\circ }$ (${\alpha }_B=6.0^{\circ }$). The schematic diagram depicts the FNT, its angled ends, and $\mathbf{H}$.

Figure 7

Scanning electron micrographs (SEMs) of the FIB milled 2.2-$\mu \mathrm{m}$-long FNT (a) placed on a Si surface and (b) attached to the tip of an ultra-soft Si cantilever.

Figure 8

SEMs of the 0.6-$\mu \mathrm{m}$-long FNT attached to the tip of a Si cantilever. The FNT was shortened in a second FIB step after the FNT was already attached to the cantilever.

Figure 9

Magnetic reversal of the three FNTs of different lengths measured by DCM at 280 K: (a) the 2.9-$\mu \mathrm{m}$-long, including a zoom of the low field region; (b) the 2.2-$\mu \mathrm{m}$-long; and (c) the 0.6-$\mu \mathrm{m}$-long FNT.

Figure 10

Magnetic reversal of the three FNTs of different lengths measured by DCM at 4 K: (a) the 2.9-$\mu \mathrm{m}$-long, including a zoom of the low field region; (b) the 2.2-$\mu \mathrm{m}$-long; and (c) the 0.6-$\mu \mathrm{m}$-long FNT.

Figure 11

(a) A detailed view of the simulated (line) and measured (points) DCM signal at low field, where the magnetization of the 2.2-$\mu \mathrm{m}$-long FNT irreversibly switches. The green line shows the simulated DCM response of a magnetization configuration, shown in (b) and (c), which is initialized and calculated to be stable at $H=0$. The configuration includes two vortices each residing in a hexagonal facet of the FNT and produces a $\mathrm{\Delta }f\left(H\right)$, which matches the plateau appearing in the measured response.

Figure 12

(a) Simulated and (b) measured segments of $\mathrm{\Delta }f\left(H\right)$ showing the vortex formation in a 0.6-$\mu \mathrm{m}$-long NT for different values of ${\theta }_H$ as labeled in the plots. Arrows highlight the specific features corresponding to end-vortex nucleation. Here, $H$ is stepped from positive to negative values.

Figure 13

(a) Schematic diagram of an FNT with a notch imperfection at its end. The notch cuts through the full 30-$\mathrm{nm}$ thickness of the FNT, is 40-$\mathrm{nm}$-wide along the FNT circumference, and has length $l_n$ along the FNT axis. (b) Simulated segments of $\mathrm{\Delta }f\left(H\right)$ showing vortex end-domain nucleation in a 0.6-$\mu \mathrm{m}$-long NT. Simulations are shown for an FNT without a notch in its end and with a notch of various lengths $l_n$ as shown in the legend. Here, $H$ is stepped from positive to negative values.

Figure 14

Simulated dependence of the nucleation field $H_n$ of the top vortex on the slant angle of the top end ${\alpha }_T$ for the 0.6-$\mu \mathrm{m}$-long FNT. The magnetic field is applied parallel to $\widehat{z}$, i.e., ${\theta }_H=0$.