Superferromagnetism and domain-wall topologies in artificial 'pinwheel' spin ice

For over ten years, arrays of interacting single-domain nanomagnets, referred to as artificial spin ices, have been engineered with the aim to study frustration in model spin systems. Here, we use Fresnel imaging to study the reversal process in 'pinwheel' artificial spin ice, a modified square ASI structure obtained by rotating each island by some angle about its midpoint. Our results demonstrate that a simple 45{\deg} rotation changes the magnetic ordering from antiferromagnetic to ferromagnetic, creating a superferromagnet which exhibits mesoscopic domain growth mediated by domain wall nucleation and coherent domain propagation. We observe several domain-wall configurations, most of which are direct analogues to those seen in continuous ferromagnetic films. However, novel charged walls also appear due to the geometric constraints of the system. Changing the orientation of the external magnetic field allows control of the nature of the spin reversal with the emergence of either 1-D or 2-D avalanches. This unique property of pinwheel ASI could be employed to tune devices based on magnetotransport phenomena such as Hall circuits.

systems. This is done by controlling the shape and size of each nanoelement to ensure that they behave as single-domain magnets. One of the most appealing and perhaps well known aspects of ASI systems is their capability to display geometrical frustration. This magnetic topological frustration gives rise to interesting properties 1-6, 6-12 , such as monopole-like defects 5,[11][12][13][14][15] and multifold ground-state degeneracy 1,2,16,17 .
The classic artificial spin ice tiling, that of square ice, has a well known long-range antiferromagnetic ground state arising from its two-fold degenerate 'two-in-two-out' spin configuration of each vertex 7,18,19 . This structure, which obeys the so-called 'ice rule' 20 and possesses four welldefined vertex energies, was initially investigated by Wang et al 1

. Their work ignited great interest
in not only square ASI, but also in several other ASI arrangements. In particular, Morrison et al. 21 pointed out the importance of vertex of interactions and their dependence on geometry. A simply modified square ASI system provides a recent example of emergent dynamics: the 'pinwheel' ice 22,23 . The pinwheel geometry is obtained by rotating each island in square ASI around its centre 23 .
Gliga et. al. have found that thermal relaxation in this system behaves as if it obeyed an intrinsic chirality 22 . Frustration in pinwheel ASI is markedly different than that in square ice in that the energies of the different pinwheel units are found to be nearly degenerate, whereas the energy levels of square ice vertices are well separated 23 .
In this work, we use Lorentz transmission electron microscopy (LTEM) 24 to directly visualise the magnetisation reversal process in a pinwheel ASI array in the presence of a static externally applied magnetic field. Under such conditions, the system behaves as a superferromagnet, i.e. an ensemble of macrospins with collective ferromagnetic behaviour 25 . Our superferromagnets has coherent domain growth and shrinking as opposed to the chain avalanche reversal seen in square ASI 26,27 . The different magnetic domains seen in pinwheel ASI are separated by domain walls, some of which behave much like the classical ferromagnetic Néel domain walls in continuous films. However, new types of charged domain walls are also observed and the magnetic charge ordering of these walls is dependent on the magnetisation alignment of the neighbouring domains.
The behaviour of these walls and domains is significantly affected by the field orientation, which is also investigated. These properties of pinwheel ASI offer a possible avenue to designing functional materials exploiting the emergent magnetic spin textures and controllable reversal dimensionality.

Results
Array edges for pinwheel ASI arrays are typically either 'diamond' or 'lucky-knot' designs 23 , where the array termination edges lie at either 45 • or 90 • to an island long axis, respectively.
Here, we investigate the behaviour of a diamond-edge permalloy (Ni 80 Fe 20

Ising Hysteresis Behaviour
In order to characterise the behaviour of the pinwheel ASI, we first look at the behaviour of the Ising net magnetisation of the individual pinwheel units, where each unit is formed by the four nearest-neighbour islands. This is done by examining the defocused Fresnel LTEM images recorded during a reversal, an example of which is shown in the bottom half of Fig. 1. Magnetic contrast arises through deflection of the electron beam by the induction from the magnetisation of each island. Since the magnetisation lies along the island's long axis, a bright edge will be seen on one side and a dark edge on the other, with the direction dependent on the orientation of the moment. From this, the magnetisation of each unit can be directly measured through its magnetic contrast as shown in the inset of Fig. 1. In this way, it is then possible to follow the moment orientations throughout the entire array as a function of the external field. Example Ising hysteresis loops extracted from these orientations are given in Figs. 2(a)-(c) for various field angles with respect to the array edges, as defined in Fig. 1. Although the coercivity for each field angle is slightly different, the general behaviour is similar. Note that when the external static field lies parallel to the y−axis (θ = 0), the component of the externally applied field along the easy-axis of each of the pinwheel islands is the same. The measured coercive field, H C , agrees very well with that calculated using the field protocols outlined in supplemental (see Fig. S1). However, despite extensive efforts to replicate the precise details of the magnetisation, this model does not adequately capture the richness of the behaviour discussed in the following sections.
For an array of entirely uncoupled islands, the easy anisotropy axis should lie at θ = 0. Inter-island interaction may modify this angle in a real system, so it is important to characterise it experimentally. We can determine the anisotropy axis of our array by examining the net mag-  In order to understand the meaning of these plots, is useful to think of the pinwheel array as a two interpenetating sublattices of collinear islands 22 as shown in Fig. S2(b). With the external field applied parallel to the easy anisotropy axis of the array, the reversal of each sublattice happens simultaneously, causing the polar hysteresis loop to collapse, and the reversal should be described by overlapping lines to and from 90 • and 270 • . For the pinwheel array, this occurs at θ = −6 • as shown in Fig. 2(b). When the field is misaligned with the easy axis, the magnetisation rotates with a sense of direction that reflects the sign of the angle, θ . For example, in Fig. 2(a) for θ = −13 • , the moment rotates clockwise, whereas in Fig. 2(c) for θ = 0, the moment rotates anticlockwise.
Using the width of the polar hysteresis loop as a measure of the angle between the applied field and anisotropy axes, we estimate that the anisotropy axis lies at −5.7 • ± 1.4 • to the array edge (see Supplementary Fig. S3 for further details). Careful measurement of in-focus TEM images of an untilted array confirms that the angles between the sublattices and with respect to the array edges in the realised array are accurate to within ±0.6 • . While this interesting result deserves further investigation, it is beyond the scope of this work and, other than the angle offset, we do not expect it to dramatically affect the reversal process which will be discussed in the following sections.

Magnetisation Reversal and Domain Formation
The hysteresis loops shown in Figs. 2(a)-(c), constructed using the component of magnetisation parallel to the field, M , suggests a ferromagnetic ordering of the array. Due to the mesocopic scale inherent to ASI, we can examine the Ising magnetisation of individual islands as a function of the externally applied field. As is commonly done in ASI, for the reminder of this work we will consider the array as a system of four-island units as shown in the insets of Fig. 1. Here and elsewhere we adopt the unit type names from square ASI vertices 28 , as detailed in Fig. S4. So far we have only discussed the ferromagnetic behaviour associated with small angles of the externally applied static field. We find that the magnetic ordering and reversal behaviour at high angles of applied field are markedly different.

Reversal at Low Applied Field Angles
In Fig. 3, we show examples of field-driven evolution of the four island unit magnetisation configuration composing an entire pinwheel array in the vicinity of coercive field. The colours represent the unit magnetisation direction as defined in the legend. We show snapshots at different field angles in (a) and (b) at various field magnitudes (points marked in the hysteresis loops going from (I) to (V)). Further details can be seen in the supplemental videos of the full reversal and snapshots of individual island reversal (see supplemental Video S1 and supplemental Fig. S5).
Here, we only focus on the general domain reversal behaviour which can be observed from the colour contrast.
When the array is saturated (e.g. as marked as (V) in Fig. 3), a single mesoscopic domain is formed by the so-called Type II units. The Type II unit possesses the largest net moment, as one might expect, and zero net magnetic charge. At the small angles of applied field shown in Fig. 3, reversal starts through a small number of nucleation points, typically located at the edge of the array where the element reversal energy is lower, and progresses by domain growth through domain wall movement perpendicular to the direction of the field. We note that the behaviour of magnetisation reversal at low field angles mimics that observed for easy-axis reversal of continuous ferromagnetic films with uniaxial anisotropy 29 . Interestingly, the reversal appears somewhat more ordered at θ = −6 • than at θ = 0. This angle offset is consistent with the analysis results of the previous section which showed that the easy anisotropy axis for this array lies at around -6 • .

Reversal at High Applied Field Angles
At higher angles of applied field with respect to the array edge, the reversal process is quite different. This is because, as θ is increased from 0 to 45 • , the easy axis of one sublattice and the Field-induced domain growth and domain wall patterns in an entire ASI array. The unit magnetisation orientation is depicted by the colour-coded arrows as defined in the inset to part I of (c).
hard axis of the other one will become more closely aligned with the field. Therefore, one sublattice will switch before the other one during a reversal. This gives rise to a 'ratcheting' behaviour yielding a stepped hysteresis loop as shown in Fig. 4(a) for an applied field angle θ = 30 • . Further details can be seen in the supplemental videos of the full reversal and snapshots of individual island reversal (see supplemental Video S2). We note that there is some discrepancy between coercive fields within the positive and negative range at this field angle. The coercive field varies between all field sweeps and we attribute this to variation in the precise magnetisation configuration during reversal and to possible small sample movements during the measurement changing the applied field angle with respect to the array.
At sufficiently high angles of applied field, the large difference in the applied field angle with respect to the easy axes of the two sublattices causes one sublattice to completely reverse before the other one starts. Because the shape anisotropy of an island only allows the moment to align parallel or antiparallel to the long-axis, the array net magnetisation is constrained to move along 45 • lines. This can be seen for the data in Fig. 4(a) in the polar plot of the same data in Fig 4(b).
Starting with the moment pointing up, when the first sublattice reverses, the moment moves from north to west, and when the second sublattice reverses, the net moment changes from west to south, and so on. Thus, the hysteresis loops at high angles of applied field describe a rotated square with a sense of direction that reflects the misalignment angle between the field and the anisotropy axis.
The behaviour of the hysteresis loops shown at higher angles of applied field can be translated into a reversal process mediated through a different mechanism to that at low field angle.
This can be seen in the texture of the magnetisation across the reversal shown in the snapshots of the magnetisation in Fig. 4(c). In this case, islands in the sublattice with their easy axes more closely aligned with the field are more likely to reverse first, forming diagonal stripe patterns. Examples of this can be seen in panels (I) and (II) of Fig. 4(c). As the nanomagnets do not couple strongly to those in adjacent diagonal lines, reversal of the entire array occurs through many nucleation points, creating a spatially inhomogeneous reversal with scattered stripe domains. When one sublattice completely switches (e.g. at point (III) in the hysteresis plot in Fig. 4(c)), the fully magnetised net magnetisation lies perpendicular to the initial domain direction. The process then repeats for the other sublattice, to complete the reversal. As a consequence of different reversal mechanisms, the critical field angle marking the transition between the square and 'stepped' loops, can be determined from the relative populations of domain walls. This is described in the next section.

Mesoscopic domain-wall topologies
In the reversal processes that were described in the previous section, large domains are seen at low angles of applied field, separated by transition regions, much like domains and domain walls in continuous ferromagnetic films. In pinwheel ASI, the domain walls separate neighbouring mesoscopic domains and each wall type exhibits a discrete macrospin texture. In the reversal regime seen at low applied field angles, the domains are almost entirely formed by Type II units grouped together throughout a reversal. These carry the largest moments which appear in the macroscale as ferromagnetism. Within a domain wall, the macrospin texture is composed by the arrangements of either Type III or a mixture of Type IV and I units.  31,32 , which is uncharged, and the 180X wall resemble a cross-tie wall 33 formed by alternating Type IV and Type I units. The 180NC and 180NCD walls are charged walls not seen continuous ferromagnetic films which lack the reduced degrees of freedom of our pinwheel lattice. We note, however, that analogous domain wall configurations are commonly observed in highly anisotropic continuous structures such as nanowires 34,35 .
The polarity of charged walls depends on the magnetisation orientation of the adjacent domains. For example, the domain wall carries positive or negative net unit charge when the magnetisation directions of neighbouring domains are head-to-head or tail-to-tail, respectively. This is analogous to the characteristic signatures of charged walls in ferroelectric materials 36,37 in which walls carry polarised electrostatic charges.
All 90 • walls ( Figure 5(e)-(g)) separate domains in which the magnetisation directions lie at right angles to one another, and all exhibit Néel rotation. Following the 180 • wall naming system, we denote these as '90NC', '90N' and '90NCD'. The 90N wall, being uncharged, is analogous to a classical Néel wall. As in the 180 • walls, charged Néel walls exist which are not found in natural ferromagnets due to the energetically unfavourable head-to-head (see Figure 5) or tail-to-tail (see Figure S6) alignment. The charge ordering of this wall type is also found to be dependent on the magnetisation orientation of their adjacent domains. These peculiar properties of specific charge ordering in pinwheel ASI are the direct result of the high anisotropy within a system of discrete magnetisation.
At the level of individual islands, the fundamental difference between the formation of 180 • and 90 • walls arises from individual moment reversals of the two sublattices, as illustrated by the top row images of Figure 5(a)-(g). When one goes through the domain wall interface of a 180 • wall, the spins in both sublattices reverse simultaneously, whereas for a 90 • wall, only spins in one of the sublattices flip. This fact can be used to map the transition between the reversal regimes of pinwheel ASI, from ferromagnetic ordering at low angles of applied field to the spatially inhomogeneous reversal at higher angles of applied field. As the applied field angle increases from zero, one sublattice easy axis becomes more aligned with the field, while that of the other lies at a higher angle to the field and, thus, the two sublattices are no long coupled and a transition from

Discussion and Conclusions
Pinwheel artificial spin ice provides an example of how a simple geometry modification can dramatically affect the magnetic properties of a spin ice array. Here, we have shown experimentally the emergence of superferromagnetism in this structure.
We expect array edges to influence aspects of the magnetisation such as array anisotropy and that the extended nature of the island may also have some effect. For instance, recent theoretical work on the pinwheel geometry has shown that the details of the thermal ground state depend on the array edges 23 . In order to probe this experimentally, preliminary measurements comparing magnetisation processes in diamond pinwheel arrays with different edge 'cuts' have been made.
The results are suggestive that there may be an effective magnetic anisotropy axis dependence on array edge geometry (See Supplemental Fig. S2). Definitive conclusions on this aspect require a more detailed and extensive experimental study that goes beyond the scope of the present work.
The reversal process is strongly affected by the direction of the field with respect to the array edges. In pinwheel ASI, the dimensionality of magnetic avalanches in the reversal process is determined by the the field angle θ . In the low-field-angle regime ( 20 • from array edges), the magnetic ordering is ferromagnetic. Reversal of the mesoscopic domains in this regime is through two dimensional avalanches of macrospins (see supplemental Fig. S5), while at applied field angles approaching 45 • the magnetisation is disordered and reversal is through the formation of 1-D stripes. The low applied field angle behaviour is the opposite of the magnetisation processes in square ASI, where 1-D monopole-like defects and dirac-like strings form 5,11,14,15,38,39 , and is a direct result of modification of the inter-island coupling. In square ASI, the strongest coupling is with the nearest neighbours and the coupling strength falls off monotonically with increasing distance 40 . By rotating each island in square ASI by 45 • , the nearest-neighbour coupling that is dominant in square ASI is greatly reduced and the dipolar coupling strength increases with distance, peaking at the fourth nearest neighbour 23 . The pinwheel ASI system is prone to the formation of domain walls analogous to those seen in continuous film natural ferromagnetic materials, such as Néel and cross-tie walls. However, novel and intriguing domain walls types can also be seen. These have specific charge ordering and net moments due to the high anisotropy and constrained degrees of freedom of the system. Furthermore, we have shown that by simply changing the orientation of the externally applied field with respect to the array edges it is possible to completely modify the nature of the domain wall configurations and the field evolution. This unique property of pinwheel ASI could be used to effectively tune devices based on magnetotransport phenomena such as the recently suggested Hall circuits 41 .
Lastly, the work reported here has concentrated on one particular array geometry. The key driving force behind the interest in other ASI systems has been the tunability of key magnetic properties through the geometrical design. We expect this also to be true for the superferromagnetic pinwheel ASI, where the coupling parameters can be varied so that different phases can be achieved or controlled. Our results show that this structure presents an interesting model system for experimental exploration of fundamental magnetisation processes such as magnetic interfaces, exchange bias phenomena, and spin wave propagation. For example, the array geometry may be tailored in such a way as to extend the Ising-like domain walls reported here to spread over several elements, potentially leading to controlling over the mesoscopic wall formation and propagation.  We approximate each extended island as an Ising point dipole with an energy barrier, H C , taken from micromagnetic simulations . In hysteresis calculations, our switching criterion is given by 28,44,45 −(H

Methods
where H dip is the total dipolar field from all other spins acting on spin i 23 and H ext is the externally applied magnetic field which is allowed to rotate with respect to sample by an angle, θ , as defined in Fig.1. This means that the component of the total field lying antiparallel to an island's magnetisation must be greater than the island's intrinsic switching field, H i C , in order for that island to flip.
For the energy barrier calculation we employ the software MuMax3. We use the same island dimensions as those in our experimental array. The cell size in each direction was taken as 2.5 nm; the saturation magnetisation was taken to be 860 × 10 3 A/m; and the exchange stiffness was 13.0 pJ/m with a damping 0.002.

Anisotropy Axis Orientation
It is important to note that whilst we have worked with a diamond-edge array throughout this work, it is possible to 'cut' the pinwheel structure in two different ways: the one discussed in the manuscript, where the centre of both sublattices do not line up (see Fig. S2(b)); and another one where the centres of both sublattices lie in the same position (see Fig. S2(a)). We call these 'asymmetric' and 'symmetric' pinwheel, respectively. Whilst these edges are not expected to significantly impact the hysteresis behaviour of large arrays, they may play some role in determining the anisotropy axis of the array. The results of initial experimental measurements of a symmetric array are shown in Figs. S2(c)-(e). In the symmetric array, the anisotropy axis, as indicated by the angle at which the sublattices reverse simultaneously resulting in a closed polar hysteresis loop, is found to occur at θ = −0.4 • . The precise angle of the anisotropy axis, determined through examining the angular dependence of the polar hysteresis loop width as discussed in the main text, is 0.7±0.4 • as shown in Fig. S3(b). Further work is required to fully understand this observation.

Square ASI Vertices to Pinwheel Units Labeling
The definition of all pinwheel units is shown in Fig. S4. The unit type names in the pinwheel geometry are carried over from the square ASI vertex definition as shown in panels (a) and (b). In (c) we show the net charge of each unit as determined from the dumbbell model. Finally, Fig. S4(d) and (

Domain Wall Population
We note that the domain wall definitions shown in Fig