Abstract
Artificial spinices consist of lithographic arrays of singledomain magnetic nanowires organised into frustrated lattices. These geometries are usually twodimensional, allowing a direct exploration of physics associated with frustration, topology and emergence. Recently, threedimensional geometries have been realised, in which transport of emergent monopoles can be directly visualised upon the surface. Here we carry out an exploration of the threedimensional artificial spinice phase diagram, whereby dipoles are placed within a diamondbond lattice geometry. We find a rich phase diagram, consisting of a doublecharged monopole crystal, a singlecharged monopole crystal and conventional spinice with pinch points associated with a Coulomb phase. In experimental demagnetised systems, broken symmetry forces formation of ferromagnetic stripes upon the surface, forbidding the lower energy doublecharged monopole crystal. Instead, we observe crystallites of single magnetic charge, superimposed upon an ice background. The crystallites are found to form due to the distribution of magnetic charge around the 3D vertex, which locally favours monopole formation.
Introduction
The manybody interaction of dipoles is crucial to understanding a diverse range of phenomena across physics, with its longrange anisotropic nature yielding a wealth of fascinating phenomena. For example, dipolar interactions can yield novel vortex stripes in an ultracold quantum gas^{1}, a low temperature residual entropy in frustrated condensed matter systems^{2} and Rosensweig instabilities in ferrofluids^{3}, yielding selforganised surface structures. The pioneering work of Luttinger and Tisza^{4} provided a foundation for understanding dipolar ordering in simple lattice geometries, but this was only extended recently to arbitrary geometries^{5}. To date, the experimental placement of dipoles into complex 3D arrangements has been lacking, with scientists mainly relying upon arrangements provided by condensed matter systems. One such model system, known as spinice^{6}, has been studied intensively and has allowed systematic study of frustration and associated emergence^{7}. These systems consist of rare earth moments on corner sharing tetrahedra. The Hamiltonian consists of dipolar and exchange terms and the ~10 Bohr magneton moment means that dipolar interactions are important in determining the nature of the ground state. Since all pairwise interactions within a single tetrahedra cannot be simultaneously satisfied, the system is geometrically frustrated. This yields a local ordering principle known as the icerule, in which two spins point into the centre of a tetrahedron and two spins point out, yielding a macroscopically degenerate ground state and a residual entropy measured at low temperature. Interestingly, MonteCarlo (MC) simulations which encompass sufficient dynamics via a loop algorithm find an ordered phase in spinice at very low temperatures, which consists of stripes of antiparallel spins^{8} but so far this has not been measured experimentally.
A new framework to understand the physics of spinice was later proposed which treated each spin as a dimer consisting of equal and opposite charges^{9,10}. Within this framework local excitations above the icemanifold are magnetic monopoles in the vector fields M and H, since once the chemical potential has been surpassed, they interact via a magnetic equivalent of Coulomb’s law. Subsequent studies provided experimental evidence of magnetic charge transport in bulk spinice materials^{11,12}. The ground state of spinice and the associated dynamic route can then be considered within the framework of magnetic charge, where the ratio of the chemical potential to the magnetic Coulomb energy of a nucleated pair is an important quantity^{13}. When this effective chemical potential approaches a value of half the Madelung constant (M/2 = 0.819 for a diamond lattice), a magnetic charge crystal is expected, whereby charges of alternating polarity are tiled throughout the structure^{13}. For the canonical spinice materials, the effective chemical potential is 1.42, suggesting the monopoles are free to propagate through the system, yielding a disordered spinice phase. To observe chargeordered states in bulk solidstate systems, one needs to find systems with specific material properties. One example, the spinice candidate Nd_{2}Zr_{2}O_{7} has recently shown charge crystal behaviour, combined with disordered spin background, a signature of magnetic fragmentation whereby the local magnetic moment splits into divergencefull and divergencefree parts^{14}. A tuneable, engineered system has the capability to explore this phase space systematically.
Artificial spinice materials are arrays of lithographically patterned singledomain nanomagnets^{15,16}. As such they are a powerful means to explore ordering in dipolar systems by design. Initial studies focussed upon simple square^{15} and Kagome arrays^{17}, which has subsequently been extended to a wide range of 2D geometries providing a means to explore a variety of model spin systems in statistical physics and more exotic phenomenon such as topological frustration in the Shatki lattice^{18} and superferromagnetism in pinwheel lattices^{19}. To date most ASI studies have focussed upon 2D systems due to ease of fabrication but interest has spanned into layered systems^{20,21} with both theoretical and experimental studies investigating how these can be used to realise a range of ground states including model vertex systems^{22} and superlattice structures^{23}.
The advent of threedimensional lithography now allows the creation of lattices that directly mimic bulk spinice geometries^{24,25,26,27}, but with tunability to control factors such as magnetic moment and lattice spacing. Such 3D artificial spinice (3DASI) systems within a diamondbond geometry and which have a Hamiltonian governed purely by dipolar energetics, have been the focus of nanofabrication efforts using focussed electron beam induced deposition (FEBID)^{27} and using twophoton lithography (TPL)^{24}. Recent work with the TPL methodology has produced systems with the required geometry and degeneracy and through simple linear field driving protocols, magnetic charge has been directly observed across the 3DASI surface^{24}. Theoretical work upon 3DASI geometries has further demonstrated tensionless Dirac strings and mobile magnetic monopoles that can be steered using an applied magnetic field^{28}. Another novel 3D structure which has relevance to frustration and ASI is the buckyball^{29}, which has been fabricated using TPL and theoretical work indicates tuneable magnonic properties^{30}.
In this article we first use finite temperature MC simulations to carry out a detailed mapping of ordering in idealised 3DASI systems within a diamondbond lattice geometry. We find a rich phase diagram consisting of a doublecharged monopole crystal, singlecharged monopole crystal and a spinice phase. We move on to measure the demagnetised state in an experimental 3DASI system and find evidence of an outofequilibrium state, whereby crystallites of magnetic charge are superimposed upon an ice background.
Results and discussion
Simulating the phase diagram of an idealised 3D artificial spinice
Figure 1a,b shows a schematic of the simulated unitcell geometry. Compass needle dipoles are placed upon a diamondbond lattice, which has a lateral extent of 15 × 15 unit cells and a thickness of a singleunitcell. To aid in discussion, we define a series of sublattices which are labelled L1–L4. The upper surface terminates in coordinationtwo vertices (L1), below which two layers of coordinationfour vertices are found (L2, L3). Finally, the lower lattice surface again terminates in coordinationtwo vertices (L4). This geometry matches our experimental 3DASI system. The compass needle model (see Methods), is equivalent to treating magnetic dipoles as two magnetic monopoles with a small variable separation. We use a metropolis algorithm to determine the ground state of the system as a function of the dipole length (b), with a fixed lattice spacing (a = 1), over a range of temperatures. We note that in experimental systems, previous work^{24} has shown that complex domain walls form at vertices in order to minimise the total micromagnetic energy, consisting of magnetostatic and exchange contributions. The net result of this is a reduction in the uniform, Isinglike part of the nanowire to some fraction of the lattice spacing. Hence, even in connected 3DASI systems it is appropriate to explore the phase diagram for b < 1.
Figure 1c,d show an overview of the phase diagram as a function of b over a range of temperatures, whilst Fig. 1e,f show the specific heat C_{v} and corresponding entropy per site s for four values of b. To facilitate interpretation, we define an order parameter (M_{c}, see Methods) which quantifies the extent to which a magnetic charge crystal has formed. For lower temperatures, a high b lattice yields strong local Coulomb interactions upon vertices, forcing charge neutrality and a spinice ground state as can be seen in Fig. 1c,d. A representative arrow map of the spinice state is shown in Fig. 1g. Ice vertices dominate the microstate occurring at frequencies reflecting underlying vertex probabilities (ergodic balance). The surface L1 layer forms short ferromagnetic strings as seen in previous theoretical studies^{31}. The magnetic structure factor (Fig. 1h) shows pinch points associated with a Coulomb phase and signatures of shortrange magnetic strings with diagonal lines seen along q = [1,1] and q = [−1,−1]. At b = 1, low temperature, the ground state entropy s_{0} of spinice is evident (Fig. 1f). In the Methods we calculate s_{0} analytically using two models: first, using Pauling’s method of independent tetrahedra which is well tested in bulk spinice. Second, by assuming that the surfaces order first and constrain subsequent layers. Figure 1f shows a closer agreement with the latter model, a fundamental difference between the bulk and slab geometries. As b decreases, the frustration and ground state entropy disappear.
Reducing b lowers the chemical potential and in the low temperature regime this yields a phase transition to a doublecharge crystal (CII). Of particular interest is how such a crystal forms whilst constrained to an odd number of charge layers. The state is characterised by ±2q charges upon surface coordinationtwo vertices (L1 and L4) and ±4q charges upon L1/L2 and L3/L4 coordinationfour vertices, as portrayed in Fig. 1i,d. The order parameter (M_{c}) of this CII state is found to be greater than 0.8, as seen by the yellow region in Fig. 1c. A neutral layer is found in the centre, consisting of type I vertices. Notably, the sheet geometry produces a coarsegrained field that is approximately constant with respect to distance. This makes the inclusion of a neutral spacing layer more negligible. The magnetic structure factor (Fig. 1j) shows clear Bragg peaks due to antiferromagnetic order and associated charge ordering. With intermediate values of b, and at higher temperature, one of the coordinationfour doublecharged sheets “spreads” into the neutral layer, creating two consecutive single charge sheets, as depicted in Fig. 1k, crosssectional view shown in Fig. 1d, rightpanel and the associated structure factor shown in Fig. 1l. This state is named CI. This increases the entropy of the system while maintaining a relatively favourable environment for charges. As temperature is further increased, a peak in specific heat (Fig. 1e), corresponding increase in entropy (Fig. 1f) and decrease in M_{c} (Fig. 1c) indicates a phase transition to a paramagnetic state. Overall, the phase diagram described by MC simulations is also captured analytically with a simple mean field analysis (See Methods).
Exploring the ordering in experimental 3D artificial spinice systems
A 3DASI system was fabricated to explore the extent to which the idealised theoretical phase diagram can be captured experimentally. The system was fabricated using a combination of twophoton lithography and evaporation (See Methods)^{24,32}. Figure 2a shows a scanning electron microscopy (SEM) image of the array which takes a diamondbond lattice geometry and has a lateral extent of 50 μm × 50 μm. Figure 2b shows a zoomed topview, falsecolour SEM image with the upper four sublattices labelled (L1–L4). As in the simulated systems, the lattice terminates in coordinationtwo vertices upon the surface, with typical coordinationfour vertices found below at the L1/L2 and L2/L3 junctions. The lower L4 sublattice, again terminates in coordinationtwo vertices.
Our previous work has shown that individual nanowires are singledomain and magnetic force microscopy (MFM) can be used to determine the contrast for different vertex types^{24,32}. We now exploit this to determine the demagnetised configuration obtained in 3DASI systems. Note, due to the limited resolution of MFM with lift height, we are only able to measure contrast upon the upper three layers, L1–L3. MFM was performed over large portions of the lattice after planar demagnetisation protocols (See Methods). All vertex types observed in previous experiments^{24}, including icerule vertices with zero magnetic charge and monopole states with magnetic charge Q = ±2q are again observed (Fig. 2c, d). The demagnetised array also contains previously unseen monopole states of charge Q = ±4q, as can be seen in Fig. 2e.
Figure 3a shows an experimental magnetic charge map of a 30 μm × 30 μm region of the lattice, determined by MFM, with experimental images shown in Supplementary Fig. 1. Three distinct phases are measured and can be readily identified in the charge map with detailed configuration shown in Fig. 3b–d. Magnetic charge crystallites can be seen with ±2q tiling, as highlighted by the green box in Fig. 3a. An arrow map of a typical charge crystallite region is shown in Fig. 3b, which shows that it arises due to two types of distinct ordering. The L1 sublattice that consists of alternating coordination two and coordinationfour vertices is found to order into ferromagnetic stripes (MFM contrast shown in Supplementary Fig. 1a). Analysis of the L1 sublattice, shows that this is the case for the entire measured area, with coordinationtwo monopoles being very rare and only observed upon <1% of vertices consistent with previous work^{24}. Over large regions of the measured area (~20 %), including in the charge crystallite regions, the L2 sublattice is found to host antiferromagnetic ordering. Typical MFM contrast of such antiferromagnetic ordering is shown in Supplementary Fig. 1b. Experimental MFM images of the three different phases are shown in Supplementary Fig. 1c–f, with example charge crystal patches from additional samples shown in Supplementary Fig. 1g–j. Arrow maps depicting L1 and L2 magnetisation separately are shown in Supplementary Fig. 2a, b. Breaks in the antiferromagnetic ordering upon L2, via short ferromagnetic strings occurs frequently, with frequency decaying with string length (Supplementary Fig. 2c, d). Interestingly, we find that breaks in the antiferromagnetic ordering often occurs to mitigate the formation of ±4q charges. We note that since the configuration of the charge crystallites observed experimentally (CI_{E}) has ferromagnetic stripes on L1, it is distinct to the CI charge crystal seen in simulations.
Between areas of magnetic charge crystallite, large patches of the ice phase are observed, as shown by the orange region in Fig. 3a, with full representative arrow map shown in Fig. 3c. These ice regions are largely composed of type II vertices, which due to a subtle broken symmetry in 3D geometry, are the lowest energy vertex type according to micromagnetic simulations^{24}. Finally, only very small regions of the doublecharge (CII) crystallite are observed as shown by purple region in Fig. 3a and associated arrow map in Fig. 3d. The full measured region is shown in Supplementary Fig 3a, with associated vertex types shown in Supplementary Fig. 3b and vertex charge shown in Supplementary Fig. 3c. The vertex statistics show a strong preference for type III vertices (61.2%), followed by type II vertices (29.8%). Both low energy type I vertices and high energy type IV vertices are only observed occasionally at 5.3% and 3.6% respectively. As would be expected, our measurements indicate charge neutrality, within error as shown in Supplementary Fig. 3c. Overall, the charge order parameter as calculated for simulations takes a value of 0.31, for this experimental system (See Methods).
The magnetic structure factor of the entire measured data is shown in Fig. 4a, with subsets corresponding to individual sublattices shown in Fig. 4(b–d). Focussing upon the data for all layers (Fig. 4a), the presence of intense Bragg peaks can be seen, superimposed upon weaker diagonal lines along q = [1,1]. In order to further interpret this data, we deconvolve the layers. The L1 structure factor (Fig. 4b) consists of a peak upon q = [0,0], indicative of ferromagnetic order on the surface. Weaker split peaks about q = [1/2,1/2] come about due to presence of longer period domains upon L1, as demonstrated in Supplementary Fig. 4. The L2 structure factor (Fig. 4c) shows peaks due to both type II tiling as well as the magnetic charge crystallite regions as demonstrated in Supplementary Fig. 5. Finally, the L3 structure factor (Fig. 4d) shows a diffuse signal, with weak Bragg peaks superimposed. This is consistent with the full arrow map (Supplementary Fig. 3a), which shows a mixture of charge ordered and ice states upon L3. Further breakdown of the structure factor via layer and region can be found in Supplementary Fig. 6.
Magnetic charge crystallite formation
We now discuss the observed experimental configuration in terms of the states predicted by MC simulations. For the real experimental systems studied here, the scaled needle length (b) depends upon the vertex type (Supplementary Fig. 7), due to the presence of domain walls close to the vertex. When considering all icerule vertices, an average b of 0.89 is obtained, suggesting a Q = ±4q monopole crystal would be expected as the ground state. However, a set of Q = ±2q crystallites form, superimposed upon an ice background. A number of factors may account for this discrepancy. Previous work has suggested that in experimental 3DASI systems, magnetic charges upon surface coordinationtwo vertices are very unfavourable with micromagnetic calculations of single vertices suggesting such excitations cost a factor of three larger than coordinationfour monopoles^{24}. The immediate implication of this is that ferromagnetic stripes upon L1 will forbid the formation of a Q = ±4q monopole crystal, apart from regions with local disorder. This is reinforced by the deterministic demagnetisation protocol which favours the formation of ferromagnetic stripes upon the surface. Given this constraint, the system can only form a single charge crystal. However, the formation of a charge crystallite via a demagnetisation routine remains surprising and has not been seen previously in either pristine, traditional 2DASI or more exotic layered 2.5D systems. In the former case, charge crystals can be formed in modified square ASI by utilising an MFM tip to selectively switch islands^{33} but demagnetisation of conventional square systems yields a low magnetisation, disordered ice phase with low frequencies of monopole excitations^{15}. An interesting question is whether conventional (planar) square ASI has some region of parameter space where charge ordering may occur naturally after demagnetisation or thermalisation. We note that in our system, it is the 3D geometry of the vertex that allows like charges to separate sufficiently, reducing the monopole energy and allowing the stable formation of charge crystallites. Furthermore, the charge crystal in the 3DASI lattice yields 3D shells of attractive charges, with a reduced average separation, lowering the energy. It is possible, that within a connected regime that such charge ordering may be accessed by designing the vertex geometry so as to control degeneracy and monopole energy. However, we note that previous extensive work has not found any evidence of this^{34,35}. For pristine Kagome systems, demagnetisation yields a 2in/1out icerule throughout the lattice^{17} with only thermally annealed systems yielding some degree of charge ordering^{36}. Modifications of the Kagome geometry, either by tuning island lengths within a singleunitcell^{37}, or by placing exotic nanobridges at vertices, can also yield charge ordering^{38}.
Considering the dynamics of the demagnetising protocol and starting in saturating fields, the system becomes uniformly tiled in type II vertices. Though these are the lowest energy state for single vertices^{24}, the net magnetisation makes these less favourable globally. The effective chemical potential upon L1^{24} as modified by surface energetics (μ* = 1.22) means that deconfined monopoles nucleate and propagate for each 180° rotation of the field. At threshold fields, nucleation events upon L1 become less likely, leaving long ferromagnetic strings as observed in the experimental data. The effective chemical potential upon the L2 sublattice^{24}, within a simple dipolar approximation is lower (μ* = 1.03) and favours the local production of correlated charge pairs (type III vertices). Quenched disorder in the lattice, means that this will occur more favourably at positions where an L2 nanowire has a slightly reduced coercive field, with respect to the mean. Such initial nucleation events seed the formation of charge crystallites. It is interesting to note that type III vertices have a slightly lower b value (0.78) due to the complex 3D distribution of magnetic charge around the vertex. Specifically, the equilibrium distance between like charges across the vertex is increased, whilst the distance between opposing charges on the nanowire are decreased. When taking this into account, the effective chemical potential is reduced and has value (See methods) of μ* = 0.91, approaching the critical value of M/2 = 0.819. The implication of this is that nucleated monopole pairs on L2 are particularly stable, as supported by previous experiments^{24}.Once a single monopole pair is formed, it is energetically favourable for a charge crystallite to grow by minimising local vertexvertex Coulomb interactions and tiling charges of opposing sign. The residual icerule regions reflect regions which have not yet equilibrated. It is possible that longer or more complex 3D demagnetisation protocols will promote more efficient exploration of the energy landscape, allowing such ice regions to be further minimised.
In summary, returning artificial spinice to its threedimensional origins unlocks previously inaccessible exploration of phase space. We anticipate that fine control of 3D vertex geometry and NiFe thickness will allow suppression of surface energetics and together with an exploration of more complex demagnetisation protocols, or thermal relaxation will allow a realisation of the doublecharged crystal. It is also expected that more sophisticated synchrotron techniques^{39} may allow imaging of systems greater than oneunitcell in thickness.
Methods
Entropy calculations
In this section we provide the details of our analytical calculation of the ground state entropy of our spinice model. We employ the following conventions.

We adopt the graph theory terminology of vertices connected by edges. In our system each edge hosts a single, Isinglike magnetisation.

Two edges meet at each vertex in layers L_{1} and L_{5}. Four edges meet at each vertex in layers L_{2}, L_{3}, L_{4} ; each of these vertices shares two edges with vertices in each of its neighbouring layers.

There are the same number of vertices, N, in each layer.

Let S_{n} be the entropy in layer n, in units where K_{B} = 1, and s_{n} = S_{n}/N is the entropy per vertex in layer n.

Let S be the total entropy of the system, and s = S/(5 N).
Paramagnet
we use the hightemperature paramagnetic phase to constrain the entropy in our numerical calculations. Each domain’s orientation is independent. In layers 1 and 5 there are two edges per vertex, giving s_{1,5} = log 2^{2}, and in the other layers there are 4 edges per vertex giving s_{2,3,4} = log 2^{4} . Therefore,
Spinice
for the ice state without field annealing (the physically relevant case around b/a≈1) we have a net charge of zero at every vertex. Each L_{1} vertex therefore has a net magnetisation (1in 1out means both domains align). It appears the L_{1} vertices are completely uncorrelated, even along a single L_{1} line. Therefore, there are 2 choices per vertex, and s_{1} = s_{5} = (1/5) log(2) . Each L_{2} vertex now has two of its domains fixed by L_{1} vertices, giving no freedom in these two domains. There are two remaining independent choices per vertex, giving s_{2,4} = (1/5)log(2). L_{3} is then completely constrained by L_{2} and L_{4} . Therefore S_{3} = 0 . Overall,
This agrees with our numerically calculated result to within standard error. The value differs from the Pauling estimate in bulk spinice; this is because surface energetics dominate in our singleunitcell slabs.
MonteCarlo simulations
The interaction energy between two artificial nanomagnets may accurately account for their finite size through the compass needle model. That is, the energy, \({E}_{{ij}}\), between magnets \(i\) and \(j\) is approximated by considering two point charges at the end of each nanomagnet that interact with Coulomb attraction or repulsion:
Here \({\mu }_{0}\) is the permeability of free space, \(m\) is the nanomagnet’s magnetic moment, and \(L\) is the nanomagnet length. \({{{{{{\boldsymbol{r}}}}}}}_{{{{{{{\rm{ai}}}}}}}}\) is the position of a magnetic charge, the first index \(a\) referring to it being positive and the second index \(i\) denoting to which magnet it belongs. \({\alpha }_{{{{{{{\rm{ij}}}}}}}}\) is the surface energy factor, which for data presented in this publication was set to 1. Since nanomagnet length wildly influences energy scales, all computational energies were normalised by their strongest interactions, such that \({\widetilde{E}}_{{{{{{{\rm{ij}}}}}}}}={E}_{{{{{{{\rm{ij}}}}}}}}/{E}_{\max }\,\).
From this energy we can see that increasing length of the magnets increases nearest neighbour dominance. It’s worth noting that the exact distribution of charge and, therefore, precisely what the “length” of the magnets is, depends largely on the details of the nanomagnet’s geometry and domain wall arrangement. In this study, this energy is used in the evaluation of a metropolis method MonteCarlo analysis.
Effective chemical potential calculations
The chemical potential of a coordinationfour vertex, upon a diamondbond geometry has previously been calculated within a dipolar framework. The energy between any pair of dipoles can be written as:
Where m represents the magnetic moment unit vector, r is the moment separation, a is the lattice constant and u is the Coulomb energy between charges:
with \(Q=2m/a\) . One can then simply write the chemical potential as the energy difference between a monopole and an icerule state, offset by the magnetic Coulomb interaction, between created charges:
with
Assuming a perfect dipolar model whereby the charges are separated by a singlelattice spacing yields a chemical potential \(\mu\) of 1.03 u. The effective chemical potential is therefore \({\mu }^{* }=\frac{\mu }{u}=1.03\)
However, in our real experimental system the charge separation in the monopole state is reduced, with \({r}_{{charge}}\approx\)0.8a, yielding a reduced \({\mu }^{* }=0.91\). This locally promotes the formation of charge crystallites.
Magnetic charge crystal order parameter
In chargeordered systems, twofold degenerate patterns emerge as the ground states. To measure similarity to these states, we can calculate a charge crystal order parameter defined as:
\({\triangle }_{i}\) is a template of +1’s, −1’s, and 0’s representing a ground state. This was used to calculate the order parameter for both MonteCarlo simulations, as shown in Fig. 1c and for experiments.
Magnetic structure factor
In the canonical spinice materials, spinflip neutron scattering^{12} provides what is probably the clearest evidence of spinice behaviour. Neutron scattering probes the magnetic structure factor projected along the direction of neutron propagation. In artificial spinice there is a similar tradition of calculating the structure factors, although neutron scattering is not used as a probe. Instead, the structure factor can be inferred directly by Fourier transforming the MFM image^{16,40}. In this work we calculated the magnetic structure factor for spin configurations modelling those in our real lattices, as well as those generated in our MonteCarlo simulations. We calculated the full 3D structure factors before taking the q_{Z} = 0 slice, suitable for modelling what would be seen when Fourier transforming a surface MFM arrow map.
Mean field analysis
Considering the system in the dumbbell model approximation,
where Q_{i} = ±2, ±1, 0 is the value of the charge on the ith vertex, \({K}_{{ij}}\) is the interaction strength between charges, and \(\mu\) is the chemical potential of a charge. We can calculate the MaxwellBoltzmann distribution in the mean field approximation and motivate how charge ordering differs from spinice ground states. Taking the change of variables Q_{i} = Δ_{i} X_{i}, where \({\Delta }_{i}\) is a general chargeordered ground state, and introducing a perturbative “field” to this variable \(h\) which will later be set to zero,
The variable is approximated by deviations from its mean value, \({X}_{i}=\langle X\rangle +\delta {X}_{i}\). The energy gained by a chargeordered state is called the Madelung constant, which can be written as \(\alpha =\frac{1}{N}{\sum }_{{ij}}{K}_{{ij}}\,{\Delta }_{i}\,{\Delta }_{j}\) and \(\alpha ={\sum }_{j}{K}_{{ij}}{\Delta }_{i}{\Delta }_{j}\). Substituting then yields
from this we can calculate the partition function of a single variable and, because they are independent, \(Z={\left({Z}_{1}\right)}^{N}\). For a pyrochlore lattice, \({Q}_{i}=\pm 2\,(\Omega =1)\), \(\pm 1\,(\Omega =4)\), and \(0\,(\Omega =6)\) where \(\Omega\) is the degeneracy. Substituting \(k=\beta (\alpha +2\mu )\),
the expectation value of the charge ordering variable is then obtained self consistently from the partition function:
This selfconsistency equation is relatively standard for mean field theories. At high values of \(k\), effectively equivalent to low temperatures, \(\left\langle X\right\rangle =\pm 2\). As \(k\) becomes closer to zero, these values gradually drop until the system is no longer ordered. This ordering transition may be characterised for small \(k\left\langle X\right\rangle\) through the first term of the Taylor series:
This is true when \(\left\langle X\right\rangle =\,0\) and \(k\le 1\), meaning below a critical temperature, the system will transition to a nonzero order parameter, corresponding to a charge crystal. That critical temperature is
This agrees with the previous experimental results that found a spinice ground state in systems with a reduced chemical potential greater than \(\frac{\alpha }{2}\) and a lack of discrete transition in this regime. The critical temperature also decreases with chemical potential as previously observed. Also, since as temperature approaches zero, the order parameter approaches 2, a doublecharged crystal is the anticipated ground state. One can justify this by considering the lower entropy of the doubly charged state. Since the experimental system is limited to single charges on the surface, the maximum order parameter we predict for the charge crystal ground state of the pyrochlore thin film with 5 charge sites is \({M}_{c}^{s}=1.5\).
Fabrication of 3DASI lattices
Threedimensional artificial spinice lattices were fabricated using twophoton lithography followed by thermal evaporation of Ni_{81}Fe_{19}. The coverslips were cleaned in acetone in an ultrasonic cleaner and then washed by isopropyl alcohol (IPA), after which samples were gently dried using compressed air. The coverslip was prepared for TPL with a droplet of immersion oil on one side and Nanoscribe negativetone photoresist (IPL780) on the reverse side. The coverslip was then loaded into a Nanoscribe TPL apparatus, and a fabrication script created a number of diamondbond lattice geometries, each with varying power and scan speed settings. The dimensions of each created lattice are 50 μm × 50 μm × 10 μm. The completed sample was developed in propyl glycol monomethyl ether acetate (PGMEA) and then rinsed in IPA. An air gun was then again used to remove excess IPA. The sample was then subject to a 50 nm Ni_{81}Fe_{19} evaporation, at a base pressure of 1 × 10^{−6} mBar. Approximately 0.06 g of evaporated permalloy was used to achieve this thickness based on previous depositions. A crystal quartz monitor (QCM) present during evaporation measured the deposited thickness; this was later confirmed with atomic force microscopy measurements. The resultant structure has a diamondbond geometry polymer scaffold with magnetic material upon the upper surface of nanowires forming a crescentshaped crosssectional geometry. Due to lineofsight deposition, the magnetic coating creates a 3DASI lattice which is oneunitcell in thickness, as described previously^{28}. Individual nanowires are single domain and have a crescentshaped crosssection with effective width of 200 nm and length of 866 nm.
Experimental demagnetisation of lattices
We used a demagnetising protocol akin to method 1 in a previous publication^{41} with a sample rotating at ~1000 revolutions per minute, with axis perpendicular to substrate plane. This effectively yields a rotating magnetic field in the substrate plane. The magnetic field starts at 0 mT and ramps up to 75 mT at 2.5 Ts^{−1} where it is held for 1 s. After this, the field ramps down at 2.5 T s^{−1} to −75 mT and is held for a second. The field then oscillates, whilst the magnitude decreases stepwise to zero over a period of five days.
Magnetic force microscopy (MFM)
MFM data was captured using a Bruker (Dimension Icon) scanning probe microscope in tapping mode. Ultralow moment probes were magnetised along the tip axis using a 0.5 T permanent magnet. The samples were placed with the L1 sublattice parallel to the probe cantilever with a 45degree scan angle to the L1 sublattice. MFM data were captured using a 65 nm liftheight. Separate scans with reversed tip magnetisation were performed to verify consistency of the contrast, and separate scans with the sample rotated 180 degrees were performed to control for artefacts in the scans. Nominally identical behaviour with charge crystallites superimposed upon a type II ice background have been observed in four independent samples.
Data availability
Information on the data presented here, including how to access them, can be found in the Cardiff University data catalogue at https://doi.org/10.17035/d.2023.0256753033.
Code availability
All codes utilised within this study is available upon reasonable request to the corresponding author.
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Acknowledgements
S.L. gratefully acknowledges funding from the Engineering and Physics Research Council (EP/L006669/1, EP/R009147) and Leverhulme Trust (RPG2021139). S.R.G. acknowledges funding from the EPSRC (EP/S016554/1) and Leverhulme Trust (RPG2021139). The work of M.S. was carried out under the NNSA of the U.S., DoE at LANL, Contract No. DEAC5206NA25396 (LDRD grant  PRD20190195).
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S.L. conceived of the study, supervised experimental work, micromagnetic simulations and wrote the first draft of the manuscript. A.V. carried out sample fabrication, magnetic force microscopy (MFM), micromagnetic simulations and analysed experimental data. E.H. carried out MFM, analysed the experimental data and analysed micromagnetic simulation data. M.S. wrote code to carry out the MonteCarlo simulations, derived the mean field theory and with A.V., performed simulations to determine the phase diagram of 3D artificial spinice. S.S. wrote the code to calculate magnetic structure factors and together with A.V., S.G., F.F. and S.L., interpreted the data. F.F. carried out entropy calculations of the 3DASI system in paramagnetic and spinice phases. All authors contributed to the editing of the final manuscript.
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Saccone, M., Van den Berg, A., Harding, E. et al. Exploring the phase diagram of 3D artificial spinice. Commun Phys 6, 217 (2023). https://doi.org/10.1038/s42005023013382
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DOI: https://doi.org/10.1038/s42005023013382
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