Controlled evolution of three-dimensional magnetic states in strongly coupled cylindrical nanowire pairs

During the simulated fabrication of the nanowire pairs, we expect that the magnetic state that forms at each growth step depends strongly upon the geometry of the system, i.e. how tall the nanowires have been grown and the separation between them. Figure 2(a) shows a vector representation of the six main states obtained from simulations for the ranges of separation (0–60 nm) and height (0–240 nm) under investigation. Four of these states ('Planar', 'Vortex', 'Parallel' and 'anti-parallel') are as discussed the previous sections, appearing either for low heights or for extremes values of separation. The anti-parallel state occurs for separated nanowires where the uniform axial magnetic states form anti-parallel to each other due to magnetostatic coupling (top right of figure 1(a)). All simulations show the general trend of planar –> vortex-like –> axial, as the height of the wires increase. Two less conventional states are observed for intermediate degrees of overlap. We denote these the 'Landau pattern' and 'Helical domain-wall' states. These two magnetic configurations feature net anti-parallel wires, but with a complex emergent 3D domain structure. The classification of these 6 magnetic states is based on the observation of sharp changes in the energy profiles at the transitions between them (further discussed in section S1 of the supplementary documentation). These transitions are used in figure 2(b) to generate a magnetic phase diagram as a function of height and separation. Figure 2(c) shows a map of the difference between exchange and magnetostatic energy density on the same axes, allowing the competition between energies to be evaluated for each state.

When the wires are fully overlapped (separation = 0 nm), they form a single cylindrical wire. During growth, the magnetic state evolves as previously reported in the literature for these geometries [19–22]: Planar—Vortex—Parallel (axial). Vortex caps of the same chirality are observed at the top and bottom of the Parallel state to reduce surface charges.

For the next simulation (separation = 10 nm), a new state appears—the 'Helical domain wall (DW) state'—evolving from the initial Planar and Vortex states (see figure 2(b)). This state is characterised by anti-parallel axial domains in the nanowires, with Néel caps at both ends and a DW in between them. In the bulk of the structure, a Bloch-type DW is favoured over a Néel-type DW to reduce magnetic volume charges. As the Néel caps at both ends are antiparallel to each other, the spins forming the wall spiral continuously for 180° around the z-axis with height, forming a helix (a vector profile is shown in the examples of the helical DW state figure 2(a)). The name 'Helical DW' is chosen from the profile of the spins in the region of overlap between the wires (this DW profile is reminiscent of both the 'Twisted DW' and 'Hybrid chiral DW' seen in [43] and [44] respectively). For this nanowire separation, and as the height increases during growth, the shape anisotropy of the long wire axis becomes increasingly dominant so that one of the axial domains is pushed out in favour of a Vortex state, finally followed by the Parallel state. For the next value of separation (20 nm), the growth will begin in a similar fashion. However, now the system transitions from a Helical DW state to the Parallel state, not through a Vortex, but instead via what we denominate a 3D 'Landau pattern' (named due to its resemblance to a Landau–Lifshitz pattern [48]). This state is similar to that previously described for rectangular magnetic systems, and consists of two anti-parallel domains [49, 50]. Here, however, the magnetisation in one wire is strongly axial (red, net +M_z domain in figure 2(a)), and takes the form of a vortex in the other (blue, net −M_z domain). Due to their complex nature, both the Landau pattern and Helical DW states will be discussed in more detail in section 3.2.

For the subsequent simulations with larger separation between the wires, the final state of the structure (up to the full height that was simulated) consists of anti-parallel domains. The domain structure is defined by the thickness of the region of overlap between the wires. At a separation of 30 nm, the Landau pattern state is the final state achieved. For the next two values of separation (40 and 50 nm, figure 2(b)), the final state is the Helical DW. It can also be noted that at these separations, the Planar state transitions directly into the Helical DW state without going through a Vortex, due to the larger separation favouring anti-parallel magnetostatic coupling between the wires.

Finally, when the wires are fully separated (60 nm), the two individual wires follow a similar growth pattern to the fully overlapped simulations (separation of 0 nm, an isolated nanowire). However, for separated wires, the inter-wire magnetostatic coupling means that the vortex cores in each wire form anti-parallel to each other. Additionally, the formation of single domain axial states also occurs at lower heights than for the single wire, and the caps at the tops of the wires become reminiscent of Néel caps (due in part to their rounded geometry).

Figure 2(c) shows the difference between the exchange and magnetostatic energy densities, in order to evaluate which one dominates for each geometry. As expected, for colinear states (Planar or Parallel) the exchange energy is lower than the magnetostatic energy. Conversely, more complex magnetic states (Vortex, Helical DW, Landau pattern) correspond to a higher exchange energy density, due to the formation of non-collinear spin configurations. Out of all the states found in the simulations, the Landau pattern is the most costly in exchange energy density, followed by the Helical DW and Vortex states, which have similar values. These results show that for the design of coupled geometries capable of stabilising complex spin textures, care must be taken to ensure that the coupling between the structures has an appropriate intermediate value to allow the magnetic energies to compete with each other (i.e. not allowing one to dominate over the other). In the case of this study, this is observed when the overlapped wires form a net magnetic anti-parallel alignment to each other.

For intermediate separation, the coupled nanowire system tends to evolve towards states where the magnetostatic energy is minimized by aligning the overall magnetisation of both wires anti-parallel to each other. However, in doing so, an increase in exchange energy takes place due to the formation of complex 3D domain structures. The type of configuration that forms is defined by the thickness of the overlapping region between the wires. A comparison between the two states emerging for intermediate overlapping is shown in figure 3 with views along each axis. The viewing directions for figures 3(d)–(f) are shown in 3a-c respectively. Figure 3(d) shows a side view for both states on the left and right: a colour plot of the M_x component on a slice in the overlapping region of the wires and an isosurface of spins within 10% of the −M_y component to visualise the domain wall. The top central inset of figure 3(d) corresponds to the M_x component as a function of height along the z-axis for the middle overlapping area, as marked by coloured dashed lines overlaid on the simulation plots. The bottom inset shows a detailed zoom-in of the magnetic configuration for the Landau pattern where the DW meets the surface. Figures 3(e) and (f) complement figure 3(d) by plotting a front view of the M_z isosurfaces (figure 3(e)) and xy projections of the magnetisation vector coloured by M_z (figure 3(f)) for seven representative heights along the z axis (locations of which are marked with grey dashed lines in figure 3(e)).

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We first focus on the Landau pattern state. In this case, the overlapping region is nearly as thick as the wires themselves (in the y-direction, shown in figure 3(d)), and thus allows for the extension of the Néel caps along the surface of the structure (shown as yellow on the left and purple on the right in figure 3(d)). This magnetic state reduces surface charges and creates an S-shaped DW (grey isosurfaces in figure 3(d)) between the wires that only has one point on each of its edges where it meets the surface of the structure (see left panel in figure 3(d) and the corresponding bottom inset) [49, 50]. A notable feature of this magnetic state is that the anti-parallel domains are not equivalent: whereas in one wire (the net +M_z domain, red in figure 3(e)) the magnetisation is strongly aligned with the z-axis, the domain in the other wire (the net −M_z domain, blue in figure 3(e)) takes the form of a vortex. In fact, this structure contains one single vortex tube that begins and ends at the two points where the Landau domain meets the surface of the structure. However, the vortex does not follow the Landau domain within the bulk of the structure (in-between cross-sections iii and iv in figure 3(e)). Instead, it extends through the net −M_z domain, resulting in a vortex extending along the length of this wire (distinguishing this state from a Landau–Lifshitz pattern in a rectangular geometry [50]). This is due to the vortex being able to adapt easily to the cylindrical geometry of one of the wires. Cross-section iv of the Landau pattern in figure 3(f) highlights this vortex-axial configuration in the net structure of anti-parallel domains. The other cross-sections in this figure show the transition between the ends, where Néel caps are formed at top and bottom, and the body of the structure, where the magnetisation evolves towards a vortex-axial configuration.

The second emergent 3D magnetic state observed in the simulations for intermediate overlap is the Helical DW. In comparison with the Landau pattern, the two anti-parallel domains of the wires in any Helical DW state are equivalent to each other. When the state first forms, the anti-parallel domains are strongly axial. However, with increasing height, they both become more vortex-like (with opposite cores and circulations, and hence, the same chirality). Cross-section iv (for the Helical DW) in figure 3(f) shows this double vortex configuration of the domains in the mid-point of the structure.

For the Helical DW, the region in-between the wires is typically smaller than in the previous case discussed (s = 30 nm), inhibiting the formation of a Landau pattern due to the high cost in exchange energy. Instead, a Bloch-like DW is formed between the wires. As mentioned in section 3.1, the DW observed here bears resemblance to DWs reported in [48, 51, 52]. For low heights, the Helical DW displays the same continuous rotation of spins (when going from the bottom to the top of the structure) as reported in those references. However, with further growth, our Helical DW develops a sinusoidal modulation (as seen when plotting the M_x component across the Néel caps and DW, top inset in figure 3(d)). We understand that the modulation forms to oppose the charges created by the Néel caps (and is therefore bound to it). The formation of the modulation in the Helical DW is strongly dependant on the volume magnetic charges. Figure 4 plots the divergence of magnetisation as a way to visualise the magnetic charges in the system. Figure 4(a) shows how the charge distribution evolves during the simulation with s = 40 nm (for a front view of the magnetisation). For a vortex state, positive and negative magnetic charges appear at the top and bottom of the structure, respectively. When the Helical DW state forms however, we see 8 alternating regions of positive and negative magnetic charges arrange themselves in a swirl-like pattern to reduce magnetostatic energy. This pattern is very reminiscent of the swirls and helical patterns of magnetisation discussed in [53]. As the structure continues to grow, we see a region of zero divergence split the top four and bottom four areas of alternating charge, creating a pure Bloch DW in-between the anti-parallel domains.

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Figure 4(b) shows a comparison between two final magnetic states (s = 30 nm for the Landau pattern and s = 40 nm for the Helical DW state). On the right hand side, we have a fully grown Helical DW, which shows 4 regions of magnetic charge appearing in the body of the structure (in addition to the 8 charges at the Néel caps). It is these areas of charge that lead to the appearance of the modulation in the DW (top inset in figure 3(d)). They are formed to reduce magnetostatic energy and oppose the 4 regions of charge that are confined in both Néel caps. We have thus regions of pure Bloch character appearing, where the divergence drops to zero, and deviations from a pure Bloch DW where the magnetic charges appear (how the Helical DW evolves with continued growth is discussed in the section S2 of the supplementary information). In the top and side view (orange and blue boxes to the right hand side of figure 4(b)), we see large regions of positive and negative magnetic charge on the outside of the indent between the wires. The left hand side of figure 4(b) shows the same representations for a fully grown Landau pattern state. In the front view, a similar magnetic charge distribution to what is seen in the Helical DW is observed. The main difference is that the regions of charge in the body of the structure are stronger in the left wire. This is because the net M_z domain on the right hand side is very strongly axial and so has a lower divergence. In the top view to the left (left orange box), we see four regions of alternating magnetic charge that are not centred around the vortex in the net axial domain of the left wire, but are instead centred around the indent between the wires. In the side view of figure 4(b) (left blue box), we see the strongest regions of charge appearing where the Landau domain meets the surface of the structure (with regions of opposite magnetic charge in the centre to offset those).

The emergence of both the Helical DW and the Landau pattern show the influence that competing magnetic coupling between wires can have on the magnetic states available to the system. We also demonstrate how the exchange and magnetostatic energies can compete in more subtle ways and how this can influence the structure of a DW. However, as discussed in the introduction, we also want to investigate the effect of the growth history of the system. Therefore, in the next section we examine the impact of the parallel growth mechanism on the states formed.

Given that we are simulating a parallel growth mechanism, (i.e. two wires growing at the same time in small sequential steps) the equilibrium magnetic state reached may not necessarily be the ground state. To investigate the effect of the growth mechanism, the regions of stability for each of the fully grown overlapped states (i.e. Parallel, Landau pattern and Helical DW) were investigated between separations of 10 and 50 nm (as these were the values of separation that marked out the limits of structures consisting of two overlapped nanowires). The output of a growth simulation was selected for each state: s = 10 nm for Parallel, 30 nm for Landau pattern and 50 nm for Helical DW. The separation of the wires was then increased or reduced by one cell at a time. Any new cells introduced were given a random magnetisation. At this new value of separation, the system was allowed to evolve dynamically to ascertain if the state in question would be stable (to the same criterion as the growth simulations described in section 2). If the state remained stable, the energy was evaluated and the separation was altered by one cell again. This process was repeated until either the state was no longer stable or the full range of separation had been probed.

Figure 5 shows the energy of each of these states at every value of separation where they remained stable. From this protocol, it is observed that the Parallel state remains metastable for all degrees of overlap, despite not forming during our simulated growth procedure for separation $\geqslant$30 nm. The energy density of the Parallel state increases with separation due to magnetostatic energy becoming more costly (this relationship varies from a linear trend for separations above 45 nm, due to subtle changes in the magnetic configuration of the caps of the structure). The Helical DW state however, shows the opposite trend, with energy density increasing with decreasing separation until it collapses into the Landau pattern below separations of 31.25 nm. The trends for these two states show the expected trade-off between the magnetostatic and exchange energies, with a cross-over in ground state as magnetostatic energy dominates at higher separations and exchange dominates at lower separations (figure 1(a)). The Landau pattern shows a similar energy trend to that of the Helical DW, and is stable between separations of 22.5 and 38.75 nm. Below 22.5 nm, the Parallel state is the only stable configuration of the system, and above 38.75 nm the overlapping region between the wires becomes too small and prevents the Landau pattern from forming (as discussed in section 3.2).

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As shown in figure 5, there are three regions where multiple states coexist. Above 38.75 nm the Parallel and Helical DW states can form. However, the saving in magnetostatic energy from having an anti-parallel alignment between the wires means that the Helical DW is the ground state. This fact lines up with the growth simulations, as the Helical DW was achieved as the final state for separations of 40 and 50 nm. Between 31.35 and 38.75 nm, all three states are metastable with each other. The energy densities of the Landau pattern and Helical DW have very similar values in this region as their respective savings of magnetostatic and exchange energy over each other effectively cancel out in this region. The ground state switches in this region from an anti-parallel alignment of wires to a parallel one below 34 nm, which remains the case for any nanowire pair with a smaller value of separation.

The most revealing aspect of this method of simulation is highlighted in the final region of metastability. Between 22.5 and 31.25 nm only the Landau pattern and the parallel state can exist, but the most energetically favourable state is the parallel state, due to its savings in exchange energy. Nevertheless, for separations of 30 nm, it is the Landau pattern that forms during our growth simulations. In the parallel growth mechanism utilised here, the small aspect ratio seen at the start of growth favours an anti-parallel alignment between the wires. As the aspect ratio increases, the parallel state will become more energetically favourable [15–18]. However, an energy barrier may form, preventing the annihilation of one of the domains. For lower separations, this energy barrier can be overcome (e.g. for a separation of 20 nm, figure 2(b)). Conversely, for a separation of 30 nm, the energy barrier is high enough to prevent this transition. A common method to explore the magnetic state of a nanowire involves relaxing the system from a saturated state [48, 49]. In this case however, due to the metastability of the Parallel state at all separations, this method would not return the overlapped anti-parallel states presented here. Consequently, we would be building an inaccurate picture of the effect of the magnetic coupling. Contrastingly, the controlled-evolution method used here gives a more comprehensive depiction of this type of system.

The results presented here highlight the influence and control that the fabrication method can exert on the magnetic state of the system. In particular, a direct-write technique such as FEBID allows for significantly different growth strategies to be employed to realize the same geometry (e.g. fully growing one wire before starting on the second may produce a different as-grown magnetic remanent state).