Curvilinear magnetic structures often have unique magnetic behavior compared to their rectilinear counterparts. This is due to the unique curvilinear boundary conditions as well as the curvature induced Dzyaloshinskii–Moriya-like interaction and the curvature induced anisotropy. The effects of a curvilinear geometry are best studied in 3D structures, where the curvature can have a significant spatial extent. Of these 3D structures, the simplest structure to study is the cylindrical nanowire. Here, we have simulated the magnetization reversal in cylindrical NiFe nanowires and present in situ Lorentz TEM images to support the findings of the simulations. We studied the domain formation and reversal of nanowires with two distinct diameters that give rise to a different reversal behavior. We have, thus, found that the zero-field magnetization configuration in these wires can take on a double helix type of configuration. The reversal in these structures then proceeds through the winding and unwinding of these helical configurations rather than through domain wall propagation.

Traditionally, studies of magnetism and magnetic materials have largely focused on structures with linear or planar features patterned out of functional 2D thin films. However, in recent years, the study of magnetism has broadened to include structures with a curvilinear geometry and structures that have a non-trivial geometry in the third dimension.1 These more complicated structures are of considerable interest to understanding magnetism from a fundamental physics perspective. Most notably, in magnetic structures, curvature can induce chiral effects, such as a Dzyaloshinskii–Morya (DM)-like interaction and magnetic anisotropy.2 The ability to induce such effects can allow for the stabilization, and theoretically even the creation, of topologically non-trivial defects, such as magnetic skyrmions, without the use of a heavy metal. In addition to these chiral effects, a curvilinear geometry results in boundary conditions that can give rise to unique domain wall structures that are not observed in wires with a rectangular cross section, such as Bloch-point domain walls.3 Furthermore, these structures are also of considerable interest for technological applications. For example, magnetic systems have been proposed for a number of wearable devices, such as wearable magnetoreceptors4 or wearable touchless sensors.5 An understanding of the effects of curvature and a 3D structure on the resulting magnetization configuration and its reversal behavior is necessary to the realization of such devices.

The effect of curvature on magnetic properties can be observed in 2D structures, such as parabolic and spiral nanostripes. These 2D structures are often easier to pattern and characterize than 3D structures. There has been a significant amount of work investigating the effects of curvature in such structures. The effect of curvature on the magnetic properties has been studied in 2D nanostripes shaped like parabolas, in which both simulation and experiment had demonstrated domain wall pinning at the apex of the parabola, suggesting the presence of the curvature induced DM-like interaction.6 Similarly, simulations of magnetization reversal in nanostripes patterned into spirals of various geometries showed that as the curvature of the spiral increased, the hysteresis loop became increasingly stepped, even taking on a “wasp-waist” appearance at the highest levels of curvature.7 This suggests that the curvature in these spirals induces both a DM-like interaction and magnetic anisotropy as predicted. However, the study of the effect of curvature on magnetization in these 2D systems is difficult as regions of high curvature are highly localized, e.g., at the apex of the parabola or at the center of the spiral, and do not extend across the structure.

More complicated curved structures, in which the curvature has a larger spatial extent, can be realized through the creation of 3D structures. The simplest are nanocaps, effectively as indentations in a planar material, and nanospheres. These curved structures have been shown to be able to stabilize chiral structures, such as magnetic skyrmions and magnetic vortexes.8 Micromagnetic simulations of Gaussian indentations on a magnetic film have shown that a skyrmion configuration pinned on an indentation has a lower energy than other possible states.9 Similarly, simulations of spherical shells showed the stabilization of double-vortex states.10 More complicated 3D curved structures have also been studied, such as the topologically non-trivial Möbius strip, which has been shown in simulations to result in topologically protected domain walls.11 

To date, most studies of the curvature effects on the magnetic configuration in 3D nanostructures are theoretical due to challenges in the fabrication of 3D structures. However, in recent years, there has been interest in the realization of 3D magnetic nanostructures. Some of these structures are curved and have the potential to experimentally demonstrate curvature-induced magnetic effects. For example, Phatak et al. demonstrated the creation of 3D cobalt nanospirals and their magnetic characterization.12 Similarly, Donnelly et al. studied nanowires arranged into double-helix configurations and observed how such a geometry results in the formation of locked domain wall pairs.13 Nanotubes are another common 3D magnetic structure, made either by growing a magnetic layer on an inner non-magnetic core or rolling up flat magnetic layers. The magnetization in nanotubes is unique because it typically takes on a vortex configuration where the magnetization points along an azimuthal direction.14,15 In larger microtube structures, this vortex structure can take on a multidomain configuration in which the vortex configurations alternate their sense.16,17

The structures previously described are not as readily suitable for applications involving magnetic transport since they either do not have a significant spatial extent or cannot easily have electrical contacts added to them. Many of these 3D structures are also relatively complicated and labor intensive to fabricate. Nanowires, on the other hand, do not have these shortcomings. Nanowires can be grown relatively easily, and they can be integrated with electrical contacts in a straightforward manner. Most importantly, however, nanowires provide a structure with both regions of high geometric curvature and a significant spatial extent, allowing for the study of the effects of magnetic curvature. These curvature effects can result both from the curvilinear boundary conditions of the wires3,18 and from bends or curves along the length of the wire.19 Nanowires have also exhibited interesting magnetization configurations, particularly during magnetization reversal.20 To date, most nanowire studies use bulk measurements, such as first-order reversal curve measurements, to study magnetization reversal. In this article, we study the magnetization reversal in NiFe nanowires using micromagnetic simulations, which are supported with in situ Lorentz TEM images. We were able to see in the simulations that the zero-field magnetization configuration of a NiFe nanowire takes on a double-helix configuration, and we have found experimental evidence for the stabilization of such a state at zero-field. Simulations also showed that during magnetization reversal, the pitch of the helix increases until the magnetization configuration is uniform along the applied field. The experimental data provide evidence that supports this reversal mechanism.

To study the effects of nanowire curvature on the magnetization configurations visible in nanowires, we studied the magnetization reversal of Ni0.8Fe0.2 nanowires. We were able to grow two sets of nanowires, one with a diameter of 150 nm and the other with a diameter of 30 nm in order to explore the effect of radius and, therefore, the curvature on the resulting magnetization. These nanowires were grown by pulsed DC current electrodeposition of NiFe into anodic aluminum oxide (AAO) membranes using a process similar to that reported by Carignan et al.21 Two AAO volume porosities were used—15% (pore diameter 150 nm, pore period 370 nm) and 20% (pore diameter 30 nm, pore period 65 nm). After deposition, the nanowires were released by dissolving AAO in NaOH water solutions. The magnetization configuration in these nanowires was then observed using a Lorentz TEM, with the magnetic induction reconstructed from phase data obtained using both off-axis electron holography22 and the transport of intensity equation (TIE) method.23 

Figure 1 shows the structural data obtained for the nanowires whose magnetization was studied. Figures 1(a) and 1(b) show bright field TEM images of the 150 nm diameter and 30 nm diameter nanowires, respectively, where the diffraction pattern obtained from each set of nanowires is presented as insets. For both the 150 and 30 nm wires, the wires are polycrystalline and have an FCC crystal structure. The polycrystalline structure of the nanowires suggests that the magnetization configurations observed in the nanowires are independent of the crystallographic orientation. Figures 1(c) and 1(d) show elemental analysis obtained from EDS spectroscopy of the nanowires for the 150 and 30 nm wires, respectively. The areas analyzed in Figs. 1(c) and 1(d) are indicated by the red boxes in Figs. 1(a) and 1(b), respectively. In Figs. 1(c)1(e), the locations of Fe and Ni are depicted by green dots and red dots, respectively. Figure 1(e) shows a 30 nm nanowire away from the edge where the Ni and Fe distributions were largely uniform, which was representative of the 30 nm nanowires as a whole. In both cases, the composition of the nanowires was determined to be around 75 at. % Ni and 20 at. % Fe, with the remaining composition being mostly oxygen from oxidation at the surface of the nanowires. From Fig. 1(c), we can see that the 150 nm wire shows a stratified structure in which iron-rich and nickel-rich layers alternate, and each layer is ∼10 nm thick. This stratified structure is likely the result of the pulsed plating process and the different electrodeposition rates between Ni and Fe. This stratification was absent in the 30 nm wires, although there is still slight composition nonuniformity in these wires.

FIG. 1.

Bright field TEM images of the NiFe nanowires with a diameter of 150 nm (a) and a diameter of 30 nm (b). The upper right-hand corner of each inset shows the diffraction pattern for each nanowire. (c) and (d) EDX analysis of the composition of the nanowires in the red squares outlined in (a) and (b), respectively. (e) EDX map of a 30 nm nanowire away from the edge.

FIG. 1.

Bright field TEM images of the NiFe nanowires with a diameter of 150 nm (a) and a diameter of 30 nm (b). The upper right-hand corner of each inset shows the diffraction pattern for each nanowire. (c) and (d) EDX analysis of the composition of the nanowires in the red squares outlined in (a) and (b), respectively. (e) EDX map of a 30 nm nanowire away from the edge.

Close modal

All simulations were performed using the MuMax3 micromagnetic simulation package.24,25 The saturation magnetization, Msat = 5.8 × 105 A/m, obtained from experimental measurements was used, while the theoretical exchange stiffness of Permalloy (Ni0.8Fe0.2), Aex = 1.2 × 10−11 J/m, was used. All of the nanowires simulated had a cylindrical geometry with a length of 2 μm and a diameter that varied from 30 to 200 nm. A 2 nm cubic cell was used for all simulations. The magnetization was initialized in a random state and then allowed to relax before hysteresis was simulated. The results of these simulations showed two distinct behaviors: that of nanowires with a diameter ≤100 nm and that of nanowires with a diameter ≥150 nm. The remnant magnetization state at various diameters is shown in the supplementary material, Fig. S1. As a result, this paper focuses on the observation of nanowires with a diameter of 30 and 150 nm.

Figure 2 shows the hysteresis loops for 150 [Fig. 2(a)] and 30 nm nanowires [Fig. 2(b)]. In Figs. 2(a) and 2(b), the blue solid line shows experimental data obtained from SQUID measurements and the red dashed lines indicate the results from micromagnetic simulations of 2 μm long wires. The SQUID measurements of the NiFe nanowires were taken after electrodeposition of the wires but before their release from the AAO template. The magnetic field was applied parallel to the nanowires’ long axis direction. From the experimental hysteresis loops, we see that the 150 nm nanowires are significantly magnetically softer than the 30 nm nanowires. The 150 nm nanowires have a coercive field of ∼100 Oe and a remnant magnetization of 0.1⋅Ms. Meanwhile, the 30 nm nanowires have a coercive field of ∼800 Oe and a remnant magnetization of 0.5⋅Ms. Both samples did not show indication of two phase magnetic behavior in the experimental hysteresis loops, indicating that the composition stratification did not significantly affect the magnetic behavior and the entire wire acts as a single magnetic object through magnetic coupling.

FIG. 2.

Hysteresis loops of the (a) 150 nm nanowires and (b) the 30 nm nanowires. The solid blue line shows the experimental results obtained from SQUID measurements, while the dashed red line shows the result of micromagnetic simulations.

FIG. 2.

Hysteresis loops of the (a) 150 nm nanowires and (b) the 30 nm nanowires. The solid blue line shows the experimental results obtained from SQUID measurements, while the dashed red line shows the result of micromagnetic simulations.

Close modal

The hysteresis data obtained from micromagnetic simulations, however, show a much squarer hysteresis loop for both the 150 and the 30 nm nanowires. For the 30 nm nanowires, the remnant magnetization was effectively equivalent to Ms. Additionally, the coercive field that was simulated was lower than the experimentally determined coercive field for both nanowires; the coercive field for the simulated 150 nm wires was <100 Oe and the coercive field for the simulated 30 nm wires was 600 Oe. This discrepancy is likely because the SQUID measurements were done for an entire array of nanowires, each with a varying length that is often >10 μm, instead of for a single 2 μm wire. In the array, the magnetic nanowires are magnetostatically coupled and, therefore, can result in a collective switching behavior, resulting in an earlier onset of magnetization reversal. The high coercive field of the 30 nm nanowires presents a challenge for in situ studies, as the magnetizing holder we use for applying magnetic fields in situ can only apply fields of ±750 Oe. Therefore, it is not possible to reverse the magnetization of the 30 nm nanowires in situ inside the TEM as was confirmed from our experiments.

Figure 3 shows the phase that was obtained using holography experiments. Holography was performed on the aberration-corrected Lorentz TEM equipped with an electrostatic biprism. The biprism voltage was 70 V, resulting in a hologram width of 450 nm, and images were acquired using 2 s exposure to give a fringe contrast of >25%. The phase was reconstructed using the Holoworks software plugin in Digital Micrograph.26Figures 3(a) and 3(b) show the electrostatic and the magnetic phase shifts, respectively, for the 150 nm diameter wire. The two phase shifts were separated by switching the magnetization direction, that is, by taking holograms at +100 and at −100 Oe and subtracting one phase image from the other. Unlike the TIE reconstruction, electron holography allows for the reconstruction of the absolute phase shift. The value of the phase at each point is given by the color bars in the lower left of each image. The mottled appearance in the center of the 150 nm wires is the result of the thickness of the nanowire. Because of the thickness, the signal to noise ratio in the interferogram is extremely low and the phase reconstruction is extremely noisy. Figures 3(c) and 3(d) show the electrostatic and magnetic phase shifts, respectively, for the 30 nm nanowire. In this case, since switching the magnetization was not possible inside the TEM, we separated the phase shifts by flipping the sample upside down and then subtracting one phase image from the other. For both the 150 and 30 nm wires, the change in the electrostatic phase shift across the wire is about twice that of the magnetic phase shift, which is expected. Figures 3(e) and 3(f) show line graphs of the profile of the electrostatic [Fig. 3(e)] and magnetic [Fig. 3(f)] phase shifts across the white dashed lines and averaged over 50 nm shown in Figs. 3(a) and 3(b), respectively. The red lines show the experimental values of the phase shift, while the black lines show values of the phase shift calculated using PyLorentz27 from the magnetization obtained from micromagnetic simulations, which used the saturation magnetization value obtained from experimental SQUID measurements from Fig. 2. For both the electrostatic shift and the magnetic phase shift, the experimental and simulated phase shifts show good agreement even with the noise from the center of the 150 nm wire. Notably, the measurements of the electrostatic phase were able to capture the “shoulders” at the edge of the electrostatic phase shift that correspond to the oxide layer that forms on the outside of the nanowire. This approach enables us to quantify the saturation magnetization, Msat = 7300 Oe, and the mean inner potential, VMIP = 27 V, of the nanowires. The value of the mean inner potential is in agreement with the value determined by Beleggia et al.28 Similarly, the saturation magnetization agrees well with that measured using SQUID.

FIG. 3.

Maps of the electrostatic phase shift (a) and the magnetic phase shift (b) in the 150 nm NiFe nanowires. Maps of the phase shift of the 30 nm nanowires are shown in (c) and (d) for the electrostatic (c) and magnetic (d) phase shifts. (e) and (f) Line profiles of the phase shifts along the white dashed lines in (a) and (b), respectively. The red line shows the experimental values in comparison to the theoretically calculated values given by the black lines.

FIG. 3.

Maps of the electrostatic phase shift (a) and the magnetic phase shift (b) in the 150 nm NiFe nanowires. Maps of the phase shift of the 30 nm nanowires are shown in (c) and (d) for the electrostatic (c) and magnetic (d) phase shifts. (e) and (f) Line profiles of the phase shifts along the white dashed lines in (a) and (b), respectively. The red line shows the experimental values in comparison to the theoretically calculated values given by the black lines.

Close modal

The phase shifts obtained from electron holography can be used to retrieve magnetic information about the nanowires just as in the TIE formalism. We were able to carry out holography experiments in an in situ magnetizing holder that allowed us to take images as we applied an external magnetic field. The field could be swept from 750 to −750 Oe; however, images were shifted outside the field of view of the microscope for values of an applied field outside the range of 200 to −200 Oe. Since retrieval of the magnetic information requires taking the gradient of the magnetic phase, it is extremely sensitive to noise. As a result, it is not possible to obtain an accurate image of the magnetization inside the 150 nm nanowire; however, it is possible to image the magnetic induction outside of the nanowire. From these holography experiments, we were able to determine that, as expected from the data in Fig. 2, our in situ magnetizing holder did not apply a sufficiently strong magnetic field to reverse the magnetization in the 30 nm nanowire, as shown in the supplementary material, Figs. S2 and S3. MuMax simulations had suggested that for a single nanowire, the coercive field was closer to 600 Oe. However, those simulations were for nanowires that were 2 μm long, while our nanowires were on average longer than 10 μm.

The magnetization in the 150 nm nanowires, however, was able to be reversed by the field applied by the in situ magnetizing holder. Figure 4 shows the results obtained from the in situ magnetization reversal of the 150 nm nanowire. The hysteresis loop in orange shown in the background of the image is obtained from MuMax simulations of magnetization reversal. This hysteresis loop was obtained from the same data as in Fig. 2(b). We swept the field from 0 to +750 Oe and then from +750 to −750 Oe. During this sweep, we took holograms of the 150 nm wires at values of −100, 0, and 100 Oe. An image of the magnetic induction inside of and surrounding the nanowire at each of these values is given next to its corresponding point. The image is a composite image of the cosine of the phase, where the density of the lines outside of the nanowire is proportional to the strength of the magnetic induction outside of the nanowire and the color throughout indicates its direction as given by the color wheel in the lower right. Inside the nanowire, the recorded signal is a 2D projection of the in-plane component of the magnetic induction and the stray fields outside the nanowire as well as the demagnetizing field within the nanowire integrated along the beam direction. For a nanowire with a cylindrical shape, this integrated in-plane component of magnetic induction will point along the same direction as the magnetization. As expected, the magnetic induction inside the nanowire is too noisy to form a detailed picture of the magnetization inside. However, the direction of the magnetic induction can be determined from the image. We see that the direction of the field lines reverses as the field is increased from 0 to 100 Oe and reverses once again when the field is decreased from 0 to −100 Oe. From the simulated hysteresis loop, we see that the NiFe nanowire is quite magnetically soft, with a low value of remanence magnetization. This is supported by the holography data as we see that the density of the field lines for the magnetic induction is lower at 0 G than it is for ±100 G.

FIG. 4.

A hysteresis loop for the 150 nm using values obtained from micromagnetic simulations. The net magnetization is given as the applied field is swept from 0 to 400 Oe and then from 400 to −400 Oe. The images show the cosine of the phase as determined from electron holography overlaid with a color map showing the direction of the magnetic induction as per the color wheel on the bottom-right at values −100, 0, and 100 Oe. The white arrows indicate the direction of magnetization in the nanowire.

FIG. 4.

A hysteresis loop for the 150 nm using values obtained from micromagnetic simulations. The net magnetization is given as the applied field is swept from 0 to 400 Oe and then from 400 to −400 Oe. The images show the cosine of the phase as determined from electron holography overlaid with a color map showing the direction of the magnetic induction as per the color wheel on the bottom-right at values −100, 0, and 100 Oe. The white arrows indicate the direction of magnetization in the nanowire.

Close modal

Figure 5 shows simulations of the magnetization of the 150 nm wire at zero field in the ground state. These simulations were performed using the MuMax3 micromagnetic simulation package. The simulation was done for a nanowire that was 150 nm in diameter and 2 μm long. This is considerably shorter than most of the nanowires that we studied experimentally, which often had a length that was >10 μm; however, the magnetization of the 2 μm wire is still likely representative of what would be observed in a longer wire. Figure 5(a) shows the x (top), y (middle), and z (bottom) components of magnetization in the nanowire. The x and y components, which are constant throughout the length of the wire, are shown at various diameter-wise cross sections of the nanowire along its length. The magnitude of each component is given by the red–blue color bar to the right of these cross sections.

FIG. 5.

(a) The x-, y-, and z-components of the magnetization output from MuMax simulations of a 2 µm long NiFe nanowire with a diameter of 150 nm. (b) A simulated magnetic induction map as would be obtained using the TIE using the results obtained in (a). The direction of magnetization is given by the color wheel on the left. (c) A bright field Lorentz TEM image of a 150 nm NiFe wire. (d) An experimental magnetic induction map of the wire from (c). The direction of magnetization is given by the color wheel to the right. The dashed white line indicates the edges of the nanowire.

FIG. 5.

(a) The x-, y-, and z-components of the magnetization output from MuMax simulations of a 2 µm long NiFe nanowire with a diameter of 150 nm. (b) A simulated magnetic induction map as would be obtained using the TIE using the results obtained in (a). The direction of magnetization is given by the color wheel on the left. (c) A bright field Lorentz TEM image of a 150 nm NiFe wire. (d) An experimental magnetic induction map of the wire from (c). The direction of magnetization is given by the color wheel to the right. The dashed white line indicates the edges of the nanowire.

Close modal

Together, the x- and y-components form a vortex state along the length of the wire. Therefore, along the center of the wire, there is a small region of uniform magnetization along the z-direction (along the length of the wire) akin to an elongated vortex core. Outside of this small core region, the x- and y-components of magnetization take a constant vortex configuration, while the z-component varies both along the azimuth and the length of the wire. This spin configuration is similar to what was observed by Ruiz-Gómez et al.29 in NiFe nanowires segmented with Fe-rich barrier layers in which the magnetization occurred at the surface, except that in our NiFe nanowires there is a non-zero component to the magnetization along the length of the wire. The magnetization configuration that we observed is similar to what Ruiz-Gómez et al. observed near the domain walls at the Fe-rich barrier layers. However, in our case, we do not observe the presence of domain walls in the simulated magnetization of the nanowires.

The corkscrew configurations studied by Fernandez-Roldan et al. in the 1 μm long 130 nm diameter regions of nanowires with a modulated diameter are a configuration that is very similar to what we have observed in the longitudinal component of magnetization shown in Fig. 5(a). Although the structures were on a microscale rather than a nanoscale, the microtubes of Fe-rich Permalloy studied by Streubel et al.16 show an effect that is analogous to the double-helix structure we have observed. Rolled up microtubes of Fe-rich Permalloy that were 50 μm long showed a spiral-like magnetization configuration with oblique domain walls, whereas those that were 100 μm exhibited a vortex structure in which the magnetization pointed entirely along the azimuthal direction. This spiral structure observed in the shorter microtubes has some similarities with the double-helix structure we observe, but it still contains distinct transverse domain walls.

For the z-component, an arrow giving the complete vector of the magnetization is shown throughout the length of the wire, with the color of the arrow indicating the magnitude of the z-component as per the yellow–blue color bar on the right. We can see that, based on simulations, the magnetization in the nanowire takes on a “double-helix” structure in which two separate helices with opposite values of magnetization along the z-direction wind around the diameter of the nanowire. The z-component of a specific point in the cross section will rotate as it moves along the wire, repeating with a specific pitch as shown in Fig. 5(a); meanwhile, the x- and y-components will stay constant in a vortex configuration as shown by the x- and y-components of magnetization in Fig. 5(a). In most simulations, these double-helix configurations were free of topological defects in the magnetization. However, every so often, simulations of the remnant state showcased domain walls and Bloch points dividing the nanowire into two double-helices of opposite sense. As a result, we believe that these topological defects are metastable states that may sometimes be present in the remnant state (see the supplementary material, Fig. S4).

Figure 5(b) shows a color map of the projected magnetic induction near the edge of the simulated 2 μm long wire, similar to what one would obtain using a TIE reconstruction, where the direction of magnetization is given by the color wheel. In this color map, due to the double-helix configuration of the magnetization in the ground state, there is a change in the component of magnetization that points across the length of the wire (along the directions indicated by red and cyan in the color maps) and across the width of the wire. Figure 5(d) shows the magnetic induction in the wire obtained from the phases calculated using the TIE approach. The magnetization configuration in the wire in areas away from the edge of the nanowire was determined to be similar to the configuration shown in Fig. 5(d) (see the supplementary material, Fig. S5). The TIE reconstruction in Fig. 5(d) was obtained from a through-focus series of images taken at zero field after the external field had been reduced from 750 to 0 Oe. The state shown in Fig. 5(d) is not exactly what was predicted by micromagnetic simulations, as the wave pattern arising from the winding of the helix, with a reversal in the lengthwise component of magnetization as the width of the wire is crossed, is not present, except at the very edge of the nanowire. However, the experimental data show that the lengthwise component of the magnetization does alternate between the direction indicated by red and the direction indicated by cyan along the length of the wire.

Figure 6 shows the z-component of magnetization in 150 nm diameter 2 μm long nanowires as they are relaxed from three initial states: a vortex state [Fig. 6(a)], a uniformly magnetized state [Fig. 6(b)], and a two-domain state with a longitudinal domain wall [Fig. 6(c)]. The color indicates the direction of the z component of magnetization, which is the vertical direction in the plane of the page. The magnetization resulting from an initial vortex configuration [Fig. 6(a)] remained in the vortex configuration throughout the nanowire. The final energy density is ∼1806 J/m3. This is lower than the energy density of the double helix state that was observed following relaxation from a random configuration, which was ∼1873 J/m3. The uniformly magnetized state [Fig. 6(b)] relaxed into a state that was similar to the double-helix configuration but contained a domain wall with a Bloch point in the center. The energy density of this configuration was ∼1720 J/m3. Finally, the two-domain configuration [Fig. 6(c)] relaxed to the same double-helix configuration observed from the random configuration.

FIG. 6.

Micromagnetic simulations of a 150 nm diameter 2 μm long nanowire relaxed from various initial states: (a) a vortex state, (b) a uniformly magnetized state, and (c) a two-domain state. The color indicates the value of the z component of magnetization.

FIG. 6.

Micromagnetic simulations of a 150 nm diameter 2 μm long nanowire relaxed from various initial states: (a) a vortex state, (b) a uniformly magnetized state, and (c) a two-domain state. The color indicates the value of the z component of magnetization.

Close modal

Figure 7 shows simulations of the 2 μm long 150 nm diameter nanowire as an applied field that varies from 100 to −60 Oe is applied along the length of the nanowire (along z-axis) in steps of 10 Oe. As in Figs. 5(b) and 5(d), the direction of magnetization is given by the color wheel in the top right. The magnetization shown is integrated along the out of plane direction of the image (the y-axis of Fig. 5). Figure 7(a) shows that the nanowire is in a state of uniform magnetization at an applied field of 100 Oe. As the field is lowered to 20 Oe [Fig. 7(b)], the magnetization appears to buckle at the edges, taking on the wave pattern characteristic of the double helix configuration. At zero applied field, [Fig. 7(c)], we see the wave pattern indicative of the double-helix structure shown in Fig. 5. As the field decreases to −20 Oe [Fig. 7(d)], the pitch of the double helix begins to increase. This continues as the field decreases further to −40 Oe [Fig. 7(e)], with the region that points along the positive z-axis (the blue–green region) shrinking and the region that points along the negative z-axis (red region) growing. Finally, at an applied field of −60 Oe, the magnetization in the nanowire is almost completely pointing along the negative z-axis, except for a few conical regions at the edges. Most importantly, these simulations show that magnetization reversal occurs without the formation of transverse or vortex domain walls. This is in agreement with Fresnel contrast images taken during magnetization reversal that did not show the presence of domain walls (see the supplementary material, Figs. S2 and S3).

FIG. 7.

Simulated magnetic induction color maps of the 2 μm long 150 nm diameter NiFe nanowire as the field is swept from 100 (positive direction to the right) to −100 Oe. The direction of magnetization is given by the color wheel to the top-right. Outputs are shown for applied field values of (a) 100, (b) 20, (c) 0, (d) −20, (e) −40, and (f) −60 Oe.

FIG. 7.

Simulated magnetic induction color maps of the 2 μm long 150 nm diameter NiFe nanowire as the field is swept from 100 (positive direction to the right) to −100 Oe. The direction of magnetization is given by the color wheel to the top-right. Outputs are shown for applied field values of (a) 100, (b) 20, (c) 0, (d) −20, (e) −40, and (f) −60 Oe.

Close modal

Figure 8 shows simulations of magnetization reversal for a 2 μm long 30 nm diameter nanowire as the field is varied from 800 [Fig. 8(a)] to −510 Oe [Fig. 8(d)]. The external field is applied along the length of the wire, with the positive direction given by the right (corresponding to cyan on the color wheel). The magnetization of the nanowire stays relatively constant; external applied fields of 800 [Fig. 8(a)], 0 [Fig. 8(b)], and −500 Oe [Fig. 8(c)] all show that the nanowire is largely uniformly magnetized to the right. At −500 Oe, the magnetization does taper off toward the edges where it is forming domain closure structures. Unlike in the domain closure structures seen in the 150 nm nanowires, the magnetization in these domain closure structures is symmetric across the diameter of the nanowire (see the supplementary material, Fig. S1). Figure 8(d) shows that after a relatively small step size of 10 Oe, magnetization has fully reversed. There is no evidence of nucleation or propagation of a domain wall, leaving coherent rotation as the most likely mechanism for reversal.20 

FIG. 8.

Simulated magnetic induction color maps of the 2 μm long 30 nm diameter NiFe nanowire as the field is swept from 800 (positive direction to the right) to −510 Oe. The direction of magnetization is indicated by the color wheel in the upper right. The reversal is shown at field values of (a) 500, (b) 0, (c) −500, and (d) −510 Oe.

FIG. 8.

Simulated magnetic induction color maps of the 2 μm long 30 nm diameter NiFe nanowire as the field is swept from 800 (positive direction to the right) to −510 Oe. The direction of magnetization is indicated by the color wheel in the upper right. The reversal is shown at field values of (a) 500, (b) 0, (c) −500, and (d) −510 Oe.

Close modal

The double-helix zero-field configuration observed in the simulations of NiFe nanowires with a diameter of 150 nm in Fig. 5(a) is similar to the structures observed in tomographic electron holography images of a 60 nm diameter nanowire composed of alternating layers of Cu and Co. In these structures, the magnetization in the Co layers took on a vortex configuration, with the chirality of each vortex often alternating between each layer.30 It is possible that the double-helix configuration observed in the zero-field state of the NiFe nanowires evolves out of this alternating vortex structure as the size of the Cu spacer layer goes to zero. This suggests that this double-helix structure is largely the result of the curvilinear boundary conditions imposed by the shape of the nanowire.

The magnetic induction map that would result from a double-helix configuration, as seen in Fig. 5(b), does not match with the magnetic induction map obtained using the TIE shown in Fig. 5(d), but it does look very similar to the simulated magnetic induction map of the 2 μm wire shown in Fig. 7(b). The most notable difference between the experimentally observed zero-field magnetic configuration [shown in Fig. 5(d)] and the simulated zero field magnetic configuration component of the nanowires [shown in Figs. 5(b) and 6(c)] is that the experimentally observed magnetization is typically constant across the diameter of the wire. Meanwhile, in the double-helix state predicted by simulations, the lengthwise component of magnetization reverses across the width of the wire. However, the two do share important similarities, namely, the magnetization configuration at the edge of the nanowire is antisymmetric along the diameter of the nanowire. This is almost identical to the magnetization configuration of the simulated nanowire under a 20 Oe applied field, as shown in Fig. 7(b). However, this region with the antisymmetric magnetization configuration was shown to have a slight drop in the Fe concentration during EDX analysis. Therefore, it is possible that this change in composition is at least partially responsible for the magnetization configuration observed at the edge of the nanowire in Fig. 5(d).

The most likely reason for the discrepancy between the zero field of the simulated and the experimental images of the zero-field magnetization configuration is that the simulated nanowires had a length of only 2 μm, whereas the experimentally observed nanowires had a length that was on average >10 μm. It is, therefore, possible that longer wires have a higher value of remnant magnetization, which would likely result in a longer pitch of the double helix. The polycrystalline structure could also pin the magnetization, resulting in a magnetization configuration at 0 Oe that is similar to what is observed theoretically at 20 Oe.

While helical structures have been seen in micromagnetic simulations of nanowires, they are typically seen as a metastable state that propagates through the nanowire.31 The simulations of the zero-field states that were relaxed from different initial configurations suggest that the double-helix configuration that is observed is a metastable state. In situ Lorentz TEM observations, which followed the same procedure of applying a field to saturate along one direction and then gradually reducing the applied field to zero, suggest that this configuration may be stable enough to be observed experimentally. Ultimately, the stability of the double helix-configuration is likely a result of a combination of the curvilinear boundary conditions of the nanowire and the dimensions of the structure. The curvilinear boundary conditions allow for the magnetization to wind around the edge while constantly being parallel to the edge of the nanowire. In a rectangular or hexagonal prism structure, such a configuration would result in components of magnetization that are normal to the surface of the structure, increasing the magnetostatic energy. The double-helix structure is likely increasingly stable as the length of the nanowire decreases. As the length of a nanowire increases, the increase in the exchange energy resulting from non-aligned adjacent spins of the double-helix configuration is no longer offset by a decrease in the magnetostatic energy. At that point, a multidomain state with domains separated by a domain wall likely becomes the most stable. Additionally, in nanowires with a diameter >100 nm, there is more space for magnetization to wind without significantly increasing the exchange energy. As a result, the curvilinear boundary conditions and the dimensions of the nanowires give rise to a balance between the competing demands of the magnetostatic energy and the exchange energy in which the double-helix configuration becomes stable.

Examples of magnetization configurations similar to the double-helixes shown in Fig. 5(a) being stabilized by the curvature and the dimensions of the underlying structures exist in other curved systems. Studies of out of plane magnetization in simulated FeGe nanotubes showed that as the curvature and DMI strength increased, the out of plane magnetization took on a spiral configuration similar to the double-helix configuration of Fig. 5(a).32 Although the exact physics of stabilization is likely different, this is an example of how a curvature stabilizes a non-trivial magnetization configuration. Furthermore, the tilted vortex structure observed in Fe-rich Py microtubes,16 which has many similarities to the double helix structure shown here, also transitioned into a more conventional vortex structure as the microtubes increased in length. It is possible that in both circumstances, the shorter length of the nanowires and microtubes results in a winding structure becoming stabilized. However, due to the different shapes of the nanowires and the microtubes, it is likely that the physics of the stabilization of the double-helix configuration of the nanowires and the spiral configuration of the microtubes is different. Similarly, the wider regions of the modulated nanowires observed by Fernandez-Roldan et al. were of a similar length and diameter to our simulated nanowires and exhibited a corkscrew magnetization configuration that was extremely similar to our double-helix structure. Therefore, it is likely that the double-helix structure observed in the simulations is also stabilized by its shorter length.

It is also important to consider defects as a possible cause of the double-helix configuration. Since the nanowire modeled in micromagnetic simulations was compositionally uniform, it is unlikely that variations in Ni and Fe composition, such as the bands seen in Fig. 1(c), contribute to the double-helix state that is seen at zero field. Instead, it is more likely that the discretization of the geometry in the micromagnetic modeling could nucleate vortex defects during the reversal. This discretization may be the result of the cell size; however, the double helix configuration was observed with simulations that used a cubic cell size of both 2 and 1 nm. As a result, it is more likely that defects resulting from the discretization of the curved edges would affect the magnetization. Given that an in-plane component of a cross section of the magnetization of the double-helix configuration along the width of the wire yields a vortex [see Fig. 5(a)], it is possible that this double-helix configuration is the result of several such nucleated vortexes coming together. Experimentally, the nucleation sites most analogous to the discretized geometry would be grain boundaries and other crystalline defects. It is possible that these defects allow for the nucleation of vortexes in the physical nanowire, which can then join to form a double-helix texture. These vortexes would most likely form first near the edges where the reversal begins [Fig. 7(b)] and then as they join with existing helical structures at the edges, the helical structure would spread through the rest of the nanowire [Fig. 7(c)].

Simulations of magnetization reversal in the 150 nm NiFe nanowire, shown in Fig. 7, showed that as a reversal field is applied to a uniformly magnetized nanowire, helical configurations start to appear at the ends of the wire and then propagate through its length. This results in the full double-helix configuration at zero-field. Then, as the reversal continues, the pitch of the double-helix configuration slowly increases, and the width of the helix configuration that points in the direction opposite to the applied field begins to decrease until the wire is in a single domain configuration. This occurs without the formation of any domain walls. This is supported by our in situ Fresnel contrast images (see the supplementary material, Figs. S2 and S3) of magnetization reversal that saw no transverse or vortex domain walls cross the nanowires during magnetization reversal, even though electron holography images were able to confirm that the direction of the stray fields outside the wire, and hence the magnetization, had reversed at an applied field of 100 Oe. Due to the thickness of the wire, any magnetic contrast that results from the gradual variations in the magnetization of the double-helix configuration would be very difficult to see in a Fresnel contrast image. Furthermore, the reversal mechanism seen in Fig. 7 is similar to the “curling” reversal mechanism observed in Fe–Ga nanowires.20 

Further work on these nanowires would seek to understand how the magnetization configuration would respond to current-induced reversal. In particular, it would be interesting to understand the effect of electric current on the double-helix domain structure and how the current-induced reversal would occur along the length of the nanowire. Such a reversal mechanism would have many applications based on controllable change in magnetoresistance, such as for neuromorphic computing. Furthermore, the electrodeposition technique used to fabricate the nanowires allows for easy control of the composition of the nanowire. Therefore, it would be interesting to study the effects of varying the composition of nanowires on the reversal behavior.

In conclusion, we have used micromagnetic simulations supported by in situ Lorentz TEM techniques, namely, electron holography and the TIE, to explore the magnetic state of NiFe nanowires during magnetization reversal. We have observed that these nanowires can take on novel ground state magnetization configurations, such as a double helix configuration. As the double-helix configuration can be repeatedly obtained during simulations of magnetization reversal, it has some degree of stability. These configurations are largely the result of the curvilinear boundary conditions imposed by the nanowire. Furthermore, we were able to discern from simulations that magnetization reversal in these structures does not occur through the propagation of domain walls along the length of the nanowire. Instead, the reversal occurs through the gradual winding and unwinding of these helical configurations. Evidence from experimental observations supported the conclusions reached using micromagnetic simulations. Therefore, the work presented in this article paves the way to a new playground that explores metastable states in magnetic nanowires.

See the supplementary material for magnetization in the zero-field state for various diameters of nanowires ranging from 50 to 200 nm (Fig. S1), images from the in situ magnetization reversal in the 150 nm diameter and 30 nm diameter nanowires, respectively (Figs. S2 and S3), the double-helix configuration in the 150 nm diameter nanowires with and without a domain wall that is sometimes observed in zero-field (Fig. S4), and the Lorentz TEM image of magnetization inside the 150 nm diameter nanowires away from the edge (Fig. S5).

This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. This work was performed, in part, at the Center for Nanoscale Materials, a U.S. Department of Energy, Office of Science User Facility, and supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-06CH11357.

The authors have no conflicts to disclose.

Vuk Brajuskovic: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Arthur R. C. McCray: Methodology (equal); Software (equal). Yuepeng Zhang: Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Charudatta Phatak: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this paper are available from the corresponding author upon reasonable request.

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Supplementary Material