Wave reversal mode in permalloy wire-tube nanostructures

The wave mode is a new magnetization reversal mechanism recently reported for magnetic nanotubes, which exhibit a characteristic well-defined S-shaped hysteresis loop when an external magnetic field is applied perpendicular to the axis of such nanostructures.^1,2 In addition, the dependence of the wave mode stability with the nanotube diameter has been also investigated by means of micromagnetic simulations, study that evidenced a key tool to identify the appearance of the wave reversal mode.² In fact, this latter work showed that by just measuring the absolute enclosed area of the hysteresis curve, the presence of the wave reversal mode can be noticed for tubular morphologies with external diameters ranging between 40 and 160 nm. Likewise, micromagnetic studies focused on the angular dependence of the quasi-static properties performed for another type of cylindrical nanostructures, such as permalloy (Py) wire-shaped ones, reported that wires with 100 nm diameter exhibit a monotonic increase of coercivity, except when the external field is applied perpendicular to the nanowire axis. In this latter case, the magnetization of the system was found to be along the wire hard axis, where a small external magnetic field variation induces a pronounced change in the magnetization,³ resulting in zero coercivity and with a pseudo-coherent rotation as the operative magnetization reversal mode for the nanowire.

Based on the above-described magnetic properties for both single configurations, wire and tube, an interesting question comes forth motivated by the underlying fundamental physics as well as potential applications that could be developed through mixed morphologies. In fact, recent theoretical and experimental approaches reported that diverse geometry-modulated one-dimensional nanostructures are desirable emerging materials since they can exhibit multifunctionalities when a smart design using combined shapes is performed.^4–7 Thus, the idea behind these studies is to obtain nanostructures from two or more geometries, with the aim of reaching the integration of properties in a single object. In this framework, multisegmented magnetic nanoparticles contribute to the understanding of magnetization reversal processes and the magnetostatic interactions present between the various segments that make up the nanostructure.^8–13 In particular, 1-D continuous materials reduce the possible complexity of the wall compared with extended thin films, easing the interpretation of phenomena related with domain wall motion, such as precessional dynamics and spin-torques.^8,10 In addition, from a technological point of view, domain walls have been proposed as key tools to design and control storage, transport, and process information.⁵ An example of this latter has been noticed by the observation of a step in the hysteresis curve, assigned to a partial pinning of the domain wall at the wire-tube interface zone.^14–16 Nevertheless, such step becomes smaller as the angle between the nanostructure axis and the external applied magnetic field changes. Hence, the nucleation and removal of a domain wall is easily controllable by modifying the geometrical parameters, turning wire-tube nanostructures into an amusing proposal for different nanotechnological applications such as information storage devices.^17–20 

In this work, we have performed micromagnetic simulations in order to look into the evolution of the wave reversal mode in Py wire-tube nanostructures by systematically changing the aspect ratio from a thin nanotube up to reach a nanowire configuration, fixing the external magnetic field perpendicular to the easy axis (x direction), and the external diameter of each obtained nanostructure in 100 nm. As far as we know, this is the first approach that tackles the stability conditions of the wave reversal mode in nanostructures with hybrid morphology.

Magnetic properties of 1 μm long permalloy nanostructures with wire-tube morphology have been investigated by means of micromagnetic simulations using the OOMMF software,²¹ which solves the Landau-Lifshitz-Gilbert equation (LLG). The magnetic parameters used to simulate permalloy^1,2 were exchange stiffness constant A_Py = 13 × 10⁻¹² J/m, and saturation magnetization Ms_Py = 800 × 10³ A/m. Polycrystalline nanostructures were assumed, so magnetocrystalline anisotropy was not considered. In addition, a damping constant of α = 0.5 and a cell size of 2 × 2 × 10 nm³ were used for all cases. Discretization effects are inherent in the methodology used here. Since long wire-tube nanostructures were simulated, a sufficiently large cell size was used along the wire-tube axis with the aim to optimize the computational cost of the simulations, but small enough in the xy plane to reproduce the cylindrical geometry of the wire-tube morphology. This discretization criterion has been drawn up in order to reduce the edge roughness caused by simulating a cylinder with cubic cells and has been used in similar systems as well.^1–3 The contributions of nanotube and nanowire sections to the hybrid nanostructure were described by β and η geometrical parameters, where β = a/R, being R and a the external and internal radii of the nanostructure, respectively, while η = h/L, being h the nanowire height and L the total length of the studied nanostructure, as shown in Fig. 1. Hysteresis curves and magnetization reversal modes were obtained applying an external magnetic field perpendicularly to the easy axis of each simulated nanostructure (θ = 90°), in the x direction, and for different η values, sweeping from η = 0.0 (nanotube) up to η = 1.0 (nanowire). The β value was fixed in 0.72, i.e., nanotubes with 100 nm external diameter and wall thickness of 14 nm, according to previous studies.^1,2 The enclosed area within each hysteresis curve normalized to the maximum value obtained among all calculated areas was depicted as a function of η.

Fig. 2 shows the normalized hysteresis curves for single isolated 1 μm long Py nanostructures as a function of the geometrical parameter η, which has been varied from 0.0 (entire nanotube) up to 1.0 (entire nanowire), moving through intermediate values of η that correspond to the hybrid wire-tube structure. According to previous reports,^1,2 Fig. 2.a clearly shows the characteristic S-type shape of the hysteresis loop, interpreted as the distinctive feature to identify the presence of the wave reversal mode for the above-described geometrical conditions defined for η = 0.0, corresponding to the entire Py nanotube, within of the considerations previously stated. The coercivity as well as the normalized remanence reach values of 23 mT and ∼ 0.36, respectively, although small they are different of zero. However, the hysteresis loops depicted for increasing η values from 0.2 (Fig. 2.b) up to 0.8 (Fig. 2.e) that describe the wire-tube structures, reveal that a noticeable change suddenly appears in the loop shape already starting from η = 0.2, showing a gradual loss of the S-type due to a progressive decrease of the enclosed area within the triangular zones of the curves. Then, for η equal to 0.8 (Fig. 2.e), the S-shape is totally absent and the triangular segments are negligible.

For η = 1.0, case that matches with the complete Py nanowire, the expected linear behavior for this nanostructured is achieved, in good agreement with former works,³ and close the smooth trend observed from increasing η values. Therefore, since the higher the nanowire contribution in the hybrid structure, the lower the definition of the S-type shape in the hysteresis loop. These gradual changes evidence that an attenuation of the wave reversal mode occurs when the wire contribution to the hybrid configuration becomes larger. Besides the progressive shape variation in the hysteresis curves, it is worth to note that the values of coercivity and normalized remanence keep null from η = 0.2 up to η = 1.0. From insets displayed in Fig. 2.b–2.f, no changes in the slope of the hysteresis curve are noticeable.

As concluded in a previous article,¹ neither coercivity nor reduced remanence are good indicators to detect the wave reversal mechanism, so in this article we use the area within the hysteresis curve normalized with respect to the maximum area found among all (see Fig. 3), which allows us to have a better relationship between the shape of the hysteresis curve and the appearance of this new reversal mechanism. From Fig. 3 we can see that the area within the hysteresis curve for a complete nanowire (η = 1.0) is almost zero, reflecting a pseudo-coherent rotation of the magnetic moments. At the other extreme we have that the area within the hysteresis curve for a complete nanotube (η = 0.0) exhibits its maximum value, clearly reflecting that the nanotube reverses its magnetization through the wave reversion mode (see Fig. 4). One would expect that for a hybrid wire-tube nanostructure, the area within the hysteresis curve would be proportional to the length of the wire or tube segment that it contains, however, an abrupt drop in the area within the hysteresis curve is observed between η = 0.0 and η = 0.1, falling almost 70% from its maximum value. This means that a small wire segment is sufficient to completely change the reversal mechanism of the wire-tube nanostructure, being the pseudo-coherent reversal mechanism the dominant one. From η = 0.1 to η = 1.0, the area within the hysteresis curve exhibits a smooth drop.

With the aim of gaining deeper insight into the major wave reversal mode variation observed in Fig. 2, i.e., between η=0.0 and η=0.2, snapshots of such states between 400 and -400 mT were depicted in Fig. 4.

From this figure we can verify that indeed a nanotube (η = 0.0) reverses its magnetization through the wave reversal mechanism, and that a noticeable vanishing of the wave mode is clearly observed already in the next snapshot corresponding to the η = 0.2. In fact, an abrupt change is evident in the behavior trend of the tube towards the wire reversal mechanism, as it was shown in Fig. 2 and 3. This effect is interpreted as the fast interaction dominated by the wire geometry, characterized by the pseudo-coherent rotation, which produces a drastic decrease in the area within the hysteresis curve.

In conclusion, we have carried out a systematic comparison of the hysteresis curves and magnetization reversal mechanisms of nanowires, nanotubes and hybrid wire-tube nanostructures of 1 μm long, when an external magnetic field is applied perpendicular to the axis of the nanostructure (θ = 90°), in the x direction. The nanotube exhibits non-zero coercivity and remanence, as well as a large area within the hysteresis curve, an unequivocal sign that the system reverses its magnetization through the wave reversal mechanism. On the other hand, the nanowire exhibits a hysteresis curve that does not present coercivity, remanence or significant area within the curve, a clear sign that the nanowire reverses its magnetization through pseudo-coherent rotation. Likewise, 1 μm long hybrid wire-tube nanostructures also do not exhibit coercivity or remanence. For this reason, the area within the hysteresis curve was used to investigate the reversion mechanism of the system, concluding that the wire-tube nanostructure for such length, although it starts reversing as two independent segments, the wire segment and the tube segment, the tube segment quickly accommodates the pseudo-coherent rotation of the wire segment as its own reversal mechanism. In fact, considering a wire-tube nanostructure with only 10% wire is enough to make the area within the hysteresis curve fall by 70% compared to that exhibited by a complete nanotube. Our results are in agreement with the literature on magnetic nanowires and nanotubes and provide guidelines for predicting the magnetic behavior of hybrid wire-tube nanostructures, so they will be helpful for their utilization in future applications.

The authors acknowledge the access to Mendieta cluster (CCAD-UNC) and financial support from Fondecyt 1200302, 042131EM_POSTDOC and Basal Project AFB180001.

The authors have no conflicts to disclose.

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