Focus on three-dimensional artificial spin ice

Artificial spin ices are arrays of single domain nanomagnets,^1,2 which exhibit novel collective behaviors that depend precisely on how the nanomagnets are arranged. Originally envisaged as an experimentally tractable mesoscopic analog to the magnetically frustrated bulk rare-earth pyrochlores,^3,4 themselves a counterpart to water ice,⁵ artificial spin ices are now a rich field in their own right. The name “artificial spin ice” reflects its origins,⁶ although artificial spin ices are no longer only confined to the study of ice physics, but encompass a variety of periodic and aperiodic lattices. These display a whole host of fascinating behaviors that depend on the lattice geometry, including unusual phase transitions,^7,8 Coulomb spin liquid behavior,⁹ spin glass physics,¹⁰ emergent magnetic monopoles,^11,12 and other forms of collective phenomena.

For the most part, the nanomagnets in artificial spin ices have been arranged on two-dimensional (2D) lattices with either an in-plane^13–15 or out-of-plane^16–19 Ising degree of freedom associated with their magnetization. The nanomagnets have lateral dimensions on the order of a few 10's to a few 100's nm, while the thickness of the magnetic material usually ranges from a few to several 10's nm. The nanomagnets are usually composed of ferromagnetic materials such as Permalloy (NiFe), cobalt, nickel, iron, CoGd, and CoFeB. In general, we would expect that any ferromagnetic material or ferromagnetic alloy can be used as the basis for a 3D artificial spin ice. Designing three-dimensional (3D) systems where layers of nanomagnets are placed on top of each other or even truly 3D artificial spin ices—by which we mean systems where the nanomagnets can be placed at arbitrary positions with any orientation of their associated moment—is now an active and exciting area of research in the field of artificial spin ice. Indeed, extending nanomagnetism into the third dimension, with possibilities to create curved geometries, offers unprecedented opportunities to realize novel magnetic behavior. At the same time, by harnessing the extra degree of freedom offered by the third dimension, the ensuing exotic magnetic states can provide new functionality, with several possibilities for applications including computation.^20–24 By pushing nanomagnetic systems upward, rather than outward, there is the potential to create devices that have a reduced lateral footprint and can be fabricated with higher densities, as well as being both energy efficient and realizable on the wafer-scale using modern lithographic techniques. This drive into the third dimension in artificial spin ice mirrors the general trend in nanomagnetism,^25–29 and other fields, such as photonics,³⁰ electronics,³¹ and metamaterials.³² 

In this perspective, we review the development of research into 3D artificial spin ice, from its inception to the first tentative steps toward the imaging of the magnetic configurations. We address the main methods to fabricate these structures, including the use of conventional electron beam lithography, which is the mainstay of 2D artificial spin ice fabrication and can be used to create layered systems. We will also highlight fabrication techniques that are explicitly geared toward the realization of 3D structures, including two-photon lithography, focused electron beam-induced deposition (often referred to by its acronym, FEBID), and self-assembly. We touch upon the ways in which the magnetic configurations in such systems can be probed using either laboratory-based equipment or large scale facility methods making use of synchrotron x-rays and neutrons. Finally, as an outlook, we will present areas for development and highlight some exciting new frontiers in the field of 3D artificial spin ice.

Artificial spin ices were originally designed to mimic the magnetic frustration in the rare-earth pyrochlores,¹ replacing the frustrated magnetic moments in bulk crystals with those in 2D arrays of coupled elongated single-domain nanomagnets. Indeed, the two most widely studied artificial spin ices—the artificial square ice and artificial kagome spin ice, where the nanomagnets are placed on the sites of the respective lattices, are related to the pyrochlore lattice either by projection of the spin structure or by considering spins in certain crystal planes,³³ respectively. The artificial kagome spin ice is a geometrically frustrated system, since the dipolar interactions between nanomagnets at a vertex can never be completely satisfied. Interestingly, however, the artificial kagome spin ice does not possess a specific residual entropy in the thermodynamic limit, which is another signature of frustration. The artificial square ice was originally designed to mimic the frustration associated with the spins in the rare-earth pyrochlores.^3,4 This, in turn, is related to the frustration in water ice, where incommensurate bonding distances between oxygen centers and hydrogen ions give rise to a residual entropy.⁵ However, the artificial square ice is not a true representation of the rare-earth pyrochlores in its 2D implementation because nanomagnets across a vertex are slightly further away from each other than adjacent nanomagnets at a vertex. It should also be mentioned that there are other non-magnetic mesoscopic systems that display frustration similar to that found in the artificial spin ices. A few recent examples include $C60$ molecules on corrugated surfaces,³⁴ colloids,³⁵ superconductors,³⁶ and mechanical metamaterials.³⁷ In addition, some interesting effects have been discovered in macroscopic systems.³⁸ 

Coming back to our focus on artificial spin ices made up of nanomagnets, their principle experimental advantage is that they are highly amenable to real-space probes of the magnetic configuration, including magnetic force microscopy (MFM),^1,16 x-ray photoemission electron microscopy (XPEEM),^8,12 Lorentz transmission electron microscopy (LTEM),^39,40 and magneto-optical Kerr effect (MOKE) microscopy.⁴¹ These techniques allow the orientation of the magnetic moment—the macrospin—associated with each nanomagnet to be unambiguously determined, so that the microstate of the system can be tracked in response to various stimuli such as magnetic fields and heat. For example, the emergence of long-range magnetic order has been observed using XPEEM in both the artificial square ice⁴² and in a modified form of the artificial kagome ice.⁸ Moreover, imaging with XPEEM has revealed that transitions between particular magnetic configurations are favored⁴³ and that it is possible to engineer a step-by-step magnetization reversal in large arrays of nanomagnets.⁴⁴ In addition to imaging of the magnetic configurations, it is possible to use scattering techniques to determine the critical exponents, transition temperatures, and correlation lengths in artificial spin ice.⁴⁵ In doing so, the universality class of the phase transition can be obtained. In addition, ferromagnetic resonance techniques have been used to probe the response of artificial spin ices to microwave frequency magnetic fields, revealing a rich spectrum of coupled modes. Such resonance spectra provide a fingerprint of the magnetic configurations,^46,47 which can be exploited in neuromorphic computing schemes.⁴⁸ 

Moving into the third dimension with artificial spin ice offers new possibilities in terms of the physics that can be probed, with the macrospins no longer confined to lie in the same plane and with new topologies making full use of 3D space. With the development of 3D nanofabrication methods, in addition to layered systems, one can create fully 3D artificial spin ices, providing a new handle with which to tune the couplings, exploiting both the magnetic and spatial degrees of freedom. Currently, it is possible to arrange subsets of nanomagnets in different planes that are parallel to the substrate,^49–51 and the fabrication of fully 3D magnetic nanostructures using various methods is becoming more widespread.^52,53 Significant work has already been carried out on the realization and characterization of an artificial 3D analog of the pyrochlore lattice.^54,55 This has enabled real-space observations of the magnetic configurations, which cannot be carried out on the bulk crystal counterparts since they can only be probed with reciprocal-space techniques or bulk magnetometry. Indeed, we are now free to explore a broad variety of 3D spin lattices, limited only by our imagination and drawing inspiration from other 3D frustrated systems such as liquid crystals,⁵⁶ metal-organic frameworks,⁵⁷ granular matter,⁵⁸ folded proteins,⁵⁹ and neural networks.⁶⁰ For example, we can imagine realizing a mesoscopic analog of metal-organic framework structures or designing an artificial spin ice whose unusual connectivity imparts a dimensionality different from three, as seen in granular matter.⁵⁸ This is made possible with 3D artificial spin ice structures where one can have the full control of the position, orientation, and connectivity of the nanomagnets.

The simplest extensions of artificial spin ice into the third dimension can be achieved through patterning certain subsets of nanomagnets in different planes, or by creating arrangements in which several planar artificial spin ices are arranged on top of each other. Here, we give an overview of advancements in such multilayer artificial spin ices.

Early on, it was realized that the artificial square ice does not truly represent the pyrochlore spin ice because adjacent nanomagnets at a vertex are closer to each other than those situated across a vertex. This breaks the degeneracy of the lowest-energy magnetic configurations in which the magnetic moments of two of the nanomagnets point into a vertex and those of the other two nanomagnets point out, which is referred to as the ice rule. This means that the system has a well-defined antiferromagnetic ground state, described by a checkerboard pattern of loops of head-to-tail magnetic moments with alternating circulation in the square rings of nanomagnets, which has been experimentally observed.^42,64 Möller and Moessner⁶⁵ first proposed that the inequivalence between ice-rule-obeying vertex configurations could be remedied by adding a height offset between the two sublattices of nanomagnets, predicting the existence of a critical offset at which all interactions at a four-nanomagnet vertex are equivalent. In addition, excitations above the ground state in artificial spin ice often carry a magnetic charge, which are referred to as emergent magnetic monopoles. In artificial square ice, these monopoles are created in pairs with a certain tension between them that inhibits the ability of the individual monopoles to wander the lattice as independent entities.^66,67 The density and thermal stability of the monopoles were predicated to scale with the height offset⁶⁸ and, around the critical height offset, the tension between the emergent magnetic monopoles disappears so that they are able to move more freely.⁶⁶ At other height offsets, it has been shown theoretically that the string tension results in an effective anisotropy directing the chain of reversed nanomagnets between them—referred to as the Dirac string—to lie along nanomagnets with the same height.⁶⁹ 

In the works discussed so far, there are only two planes containing nanomagnets. Instead, one can think of introducing multiple planes, separated by offsets, as in Ref. 70. In this work, Chern et al. theoretically showed that this provides a means to tune the phase diagram and to realize a Coulomb phase,⁷¹ in which the emergent magnetic monopoles are unbound and their interactions follow a Coulomb-like law. This was again achieved by tuning the layer separation to reduce the tension between the monopole pairs to zero in order to decouple them. They referred to this arrangement of nanomagnet layers as a stacked artificial square spin ice and restricted themselves to the case where no two nanomagnets overlap when viewed from above.

In terms of experimental realizations, an artificial square ice with an offset between the sublattices was fabricated by Perrin et al.⁴⁹ who used a multistep electron beam lithography process to manufacture plinths onto which the nanomagnets belonging to one of the sublattices could be deposited [Fig. 1(a); top]. They imaged the frozen magnetic configurations after the application of a demagnetization protocol involving an alternating magnetic field [Fig. 1(a); bottom] and found signatures of the Coulomb phase, including algebraic spin-spin correlations reflected by pinch points in the magnetic structure factor as highlighted by the red circle in Fig. 1(b). By excavating trenches in a silicon substrate, Farhan et al.⁵⁰ also realized such an offset artificial square ice and observed its thermally activated dynamics using XPEEM, providing evidence of the deconfinement of emergent magnetic monopoles near to the critical offset. A snapshot of one of the magnetic configurations is displayed in Fig. 1(c), where the small purple arrows indicate the movement of the monopoles. The authors drew a connection between the motion of the emergent magnetic monopoles and Debye–Hückel theory, which is normally used to model plasmas or ionic solutions and has also been used to explain the behavior of emergent magnetic monopoles at low temperature in the rare-earth pyrochlores.^72,73 In a related work, Farhan et al.⁵¹ showed that, by varying the offset between nanomagnets on a triangular lattice, various magnetic phases can be realized, including a long-range-ordered antiferromagnetic phase and a disordered phase with only nearest-neighbor correlations. The Coulomb phase and the magnetic field driven dynamics of monopoles was also observed in a macroscopic magneto-mechanical version of the artificial square ice where the role of the spins is taken up by magnets that can rotate on 3D-printed rotors.⁷⁴ 

One can also envisage the realization of bilayer artificial spin ices with one layer rotated with respect to the other, inspired by research into twisted graphene, which would lend themselves to fabrication with electron beam lithography. For twisted graphene bilayers, the Moiré patterns lead to interesting properties such as superconductivity⁷⁵ or an insulating state driven by strong electronic interactions,⁷⁶ and one might also expect twist-angle dependent phenomena to arise in twisted bilayers of artificial spin ice. Initial computational work on such bilayer artificial spin ices has already been undertaken, with the coupling between the layers tuned by adjusting the layer separation.⁶¹ In addition, by tuning the twist angle between the two layers, it was found that the magnetization reversal in an applied magnetic field can be precisely engineered,⁶² offering a level of control that is not possible in a single layer of artificial spin ice. Shown in Fig. 1(d) is a small area of a Moiré pattern associated with a twisted bilayer artificial square ice with a twist angle of $38°$ that maximizes the interactions between the nanomagnets in the two layers. Here, the magnetic configuration of each layer is prepared in the conventional artificial square ice ground state, and head-to-tail loops of spins are observed on a length scale larger than the lattice constant. There have also been proposals to use one of the layers as an input layer for the system.⁷⁷ This allows the other artificial spin ice layer to access a much larger portion of the configuration phase space on application of a magnetic field than could be accessed by a single layer of artificial spin ice. Consequently, magnetization reversal processes that are highly improbable in a single-layer artificial spin ice become feasible and can be controlled using the input layer. It has been predicted that such a system can perform active inference,⁷⁷ which may be interesting for device applications.

To fabricate a twisted bilayer artificial spin ice, one could imagine planarizing the first layer before fabrication of the second layer, or patterning an artificial spin ice on either side of a silicon nitride membrane. Both of these approaches are not without challenges and, as a simpler alternative, one could simply fabricate one artificial spin ice directly on top of the other, perhaps separated by a non-magnetic spacer layer. However, this imparts a surface profile—that of the nanomagnets in the first layer—into the next layer. An even simpler approach would be to create an artificial spin ice with nanomagnets consisting of two (or more) magnetic layers separated by non-magnetic spacers in a single step lithographic process. For the Permalloy/aluminum/Permalloy stack schematically shown in Fig. 1(e), this has been shown to give ultra-strong magnon–magnon coupling in artificial spin ices.⁶³ Moreover, the magnetization states and the resonance spectra can be precisely tuned by varying the non-magnetic spacer thickness as has been shown for a Permalloy/copper/Permalloy stack.^78,79 Using this approach allows each nanomagnet to be composed of several layers, both magnetic and non-magnetic, in a single stack. However, the layers are superimposed on top of each other, and this leaves no scope for rotation or translation of the individual layers, or the fabrication of a multilayer system comprising layers with different artificial spin ice lattices.

There are also artificial spin ices where the nanomagnets are coupled to a functional underlayer, such as a heavy-metal to promote the interfacial Dzyaloshinskii–Moriya interaction (DMI)⁸⁰ or a ferromagnetic thin film to boost interactions between the nanomagnets.⁸¹ For the first case, the heavy-metal layer lowers the blocking temperature of the individual nanomagnets, allowing artificial kagome spin ice structures composed of a small number of rings to reach their lowest-energy state.⁸⁰ For the second case, by biasing the ferromagnetic underlayer with a magnetic field, the rotational symmetry of the system is broken, which gives rise to stronger magnetic correlations along certain directions.⁸¹ 

Whether these multilayer artificial spin ices are truly 3D or, at best 2.5D, is open to contention. Nevertheless, they make use of electron beam lithography, which is relatively standard these days and, through layering of different materials, provide a means to create novel phenomena in artificial spin ices. However, characterizing the magnetic state in multilayer artificial spin ices is still quite challenging. In particular, disentangling the magnetic signal is difficult when multiple nanomagnets are superimposed on top of each other, even if only partially. This is certainly true when the artificial spin ices are probed with transmission microscopy methods, which rely on the integrated signal through the total thickness of magnetic material. With a surface-sensitive technique, such as XPEEM, it is possible to access only the top layer of nanomagnets unless the layers are extremely thin, but such thin layers would commensurately give a small dipolar interaction between the nanomagnets. Furthermore, when the individual magnets are manufactured from a multilayer film, the magnetostatic coupling will favor an antiparallel alignment of the magnetization in adjacent layers. This configuration minimizes the magnetic stray field, thus making it difficult to measure with techniques that detect this field, such as magnetic force or nitrogen vacancy microscopy.

The experimental examples given in this section all make use of conventional electron beam lithography for manufacturing artificial spin ice. For these multilayer artificial spin ices, the magnetization in each nanomagnet is in-plane, even if the individual nanomagnets are placed at different heights. As already mentioned, it would be advantageous to realize truly 3D artificial spin structures, which would allow for full flexibility in the position and orientation of each nanomagnet, and we now turn to this topic.

In addition to electron beam lithography, several techniques have been developed to create fully 3D artificial spin ices, and we provide a summary of these approaches in Box 1. One of the first experimentally realized 3D artificial spin ices was based on the buckyball, which was fabricated using two-photon lithography. This consisted of a buckyball polymer scaffold coated with a cobalt thin film using sputter deposition.⁵² The resulting structure was characterized in 3D using hard x-ray tomography exploiting the element specificity provided by synchrotron x-rays. This meant that the signal from the polymer scaffold and the cobalt coating could be distinguished, and it was confirmed that the cobalt layer was not uniformly applied to the structure. However, the magnetic configuration of the lattice was not measured.

Finite-element simulations of such a connected buckyball structure [Fig. 2(a)] predict an intriguing magnetization reversal with the formation of high-energy vertices associated with magnetic charge excitations.¹⁰⁰ Such defects impart a characteristic signature in the ferromagnetic resonance spectrum of the system.¹⁰³ This highlights an advantage offered by truly 3D structures; magnetic charge excitations can be placed into the lattice at specific locations by choosing an appropriate orientation of the applied magnetic field in a way that is not possible with planar geometries.

The thermodynamics of such finite lattices are just starting to be explored. These closed lattices have non-trivial topologies resulting from their curvature and, while they do not exhibit phase transitions in the conventional sense, they do undergo crossovers between different types of magnetic order. For example, the buckyball artificial spin ice, constructed by placing magnetic point dipoles on the midpoints of the edges of a regular buckyball, exhibits a complex five-step ordering process, that is a direct result of the spherical geometry of the buckyball.¹⁰¹ The signatures of the crossovers are given by peaks in the heat capacity on decreasing the temperature, which is determined using Monte Carlo simulations Fig. 2(b). At high temperature, the buckyball displays paramagnetic behavior, with no spin or charge order. On cooling, there is a crossover to a spin-ice sector where the ice-rule (two-spins-in/one-spin-out or vice versa) is obeyed at every vertex. As the temperature is decreased further, the system enters the charge-ice sector, forming an imperfect charge-ordered crystal before system-spanning spin order is established via three distinct steps. The ground state includes a pair of topological defects that are located at the two poles of the buckyball, with loops of head-to-tail spins forming in the pentagonal and hexagonal faces around the equator.

By extending this analysis to the families of Platonic and Archimedean solids, a wide range of exotic forms of magnetic order are expected. Experimental realization and characterization of such 3D artificial spin ices would therefore offer an unparalleled opportunity to observe this.

Up to now, one of the most successful approaches to realize 3D artificial spin ices is to use connected nanowires, implementing geometries that can take inspiration from 2D artificial spin ices,⁸⁷ bulk spin systems,^54,55,102 or other 3D lattices.⁸⁶ In this case, the interaction between the nanowires at a vertex is not only dipolar, so through the stray fields, it also has an exchange component with a domain wall often present at a vertex. The vertices can be studied either as a single building block or as an array of building blocks to form an extended lattice.

Using nanowires to provide an analog to the spins in the rare-earth pyrochlores, it is natural to define the tetrapod as a building block for 3D artificial spin ices. Tetrapod building blocks were first realized by creating a mold with two-photon lithography, which was then filled with cobalt using electrodeposition. These cobalt tetrapods were imaged using spin-polarized SEM to reveal domain-wall pinning at the vertex.⁹⁰ The diameter of each nanowire in the cobalt tetrapods is 435 nm, which is close to the minimum feature size provided by two-photon lithography in positive photoresist. However, using FEBID, it is possible to print fully magnetic CoFe tetrapods with a nanowire diameter of 130 nm. Hysteresis loops of these FEBID-grown CoFe tetrapods, measured using micro-Hall magnetometry, have revealed cascades in the magnetization reversal.⁹⁴ Using a coarse-grained macrospin model, these cascades could be explained by including a distribution in the magnetic properties, and vortex states were observed in the legs of the tetrapod.

Reshaping a vertex of the 2D kagome lattice into a 3D tripod, Pip et al.⁸⁷ realized a 3D artificial spin ice building block using two-photon lithography. The 3D tripod structure was reduced in size using pyrolysis and subsequently sputter coated with GdCo. Using soft x-ray laminography, the full magnetic configuration of the 3D tripod structure was reconstructed, which was the first such characterization of a 3D artificial spin ice building block. It was found that the magnetic configuration of the three legs of the tripod, displayed in Fig. 2(e), consisted of two legs with the magnetization pointing toward the vertex and one leg with the magnetization pointing away from the vertex, in a similar manner to the ice-rule-obeying magnetic configurations at the vertices of the 2D artificial kagome spin ice. However, the data collection and reconstruction for such a measurement is time consuming and requires specialist knowledge and equipment. As a result, such synchrotron x-ray methods are yet to be use to characterize more than the simplest of 3D artificial spin ice building blocks.

Recently, a fully magnetic CoFe structure composed of two tripods joined together to form a cuboid has been realized using FEBID by Volkov et al.⁹⁵ [Fig. 3(a)]. For this work, magnetic imaging was performed using shadow XPEEM, exploiting the fact that the FEBID-produced nanowires are thin, with a diameter below 180 nm, so that soft x-rays can readily pass through them. The dichroic contrast in Fig. 3(b) reveals that the thicker wires with both red and blue contrast host vortices, where the magnetization follows a helical path along the wire, while the thinner wires with a single contrast have a uniform magnetization pointing along the wire. Moreover, the authors confirm with simulations that the diameter of the CoFe nanowires determines whether the vortices are present or not. In particular, for diameters below 60 nm, the nanowires are homogeneously magnetized. For diameters between 80 and 140 nm, a rotation in the magnetization starts to appear at the end of the nanowires. For nanowire diameters larger than 160 nm, the vortices span the length of the nanowire. Even though this study does not focus on artificial spin ice, the authors give an indication of suitable nanowire dimensions for 3D artificial spin ice and demonstrate a useful way to image such nanowire structures.

The 3D artificial spin ice building blocks can be combined to form extended 3D lattices, with which it is possible to investigate the collective behavior of the magnetic struts, as well as effects that originate at the lattice boundaries. May et al.⁵⁴ experimentally observed the magnetic configurations of an extended lattice based on the pyrochlore lattice [Fig. 2(c)], which is a 3D tiling of tetrapods. They fabricated this nanowire structure using two-photon lithography and then used MFM to identify the local magnetic states. For the fabrication, the top layers of a polymer scaffold were coated using line-of-sight thermal evaporation of Permalloy. This means that there were polymer struts within the lattice shadowed by those above so that they were not coated with magnetic material. In addition, imaging such a structure with MFM is challenging because of the non-trivial topography that contributes strongly to the detected signal. Nevertheless, it is possible to measure and distinguish different magnetic configurations, and the magnetic states at individual vertices, which are accessible with the MFM tip, are shown in Fig. 2(d). Such vertex configurations can be sorted into four types with increasing energy going from type 1 to type 4, which differ in the net number of magnetic moments associated with the four bars meeting at a vertex that point toward or away from the vertex. However, the type 4 vertices with all four moments pointing towards or away from the vertex are rarely seen. It is then useful to track the relative populations of these vertex types upon application of a magnetic field. Interestingly, after demagnetization of such a 3D artificial spin in an alternating magnetic field, it was found that chains of bars with magnetic moments pointing in the same direction tend to form at the surface of the structure where there are orphan bonds, and that patches of ice-rule-obeying vertices form in the bulk of the lattice.¹⁰² Moreover, the motion of the emergent magnetic monopoles, on steadily increasing an applied magnetic field, was found to depend strongly on the magnetic field orientation.⁵⁵ It would also be of benefit to observe the magnetization reversal with scanning transmission x-ray microscopy, and initial imaging using this technique has been carried out, with the 3D artificial spin ice suspended over an aperture.⁸⁸ In a simulated 3D artificial spin ice lattice, it has been shown that it is possible to stabilize unbound magnetic monopoles at room temperature,¹⁰⁴ which is not possible in 2D artificial spin ice and only possible in spin ice crystals at cryogenic temperatures.

The fast dynamics of 3D artificial spin ice structures can be probed using micro-Brillouin light scattering. With this technique, coherent spin waves in both tetrapod building blocks and larger structures have been measured.^85,105 Moreover, using micromagnetic simulations, it has been shown that there are two classes of magnon modes, one related to the physical structure of the lattice and the other associated with particular magnetic configurations in which magnetic charges are present.¹⁰⁶ In addition, in a 3D version of the artificial square ice, where the nanomagnets are tilted out of the plane, simulations revealed that the magnon mode frequencies can be tuned using the tilt angle.¹⁰⁷ 

Several of the 3D artificial spin ices described up until now cannot be extended in the vertical direction since they use a line-of-sight method to deposit the magnetic material. To overcome this limitation, Guo et al.⁸⁶ used atomic layer deposition to conformally coat nickel onto a pyrolyzed polymer woodpile structure, which was originally created using two-photon lithography [shown to the right in Fig. 2(f)]. Using micro-Brillouin light scattering, the authors found that the frequency of the magnon modes in the top layer of the woodpile structure was approximately 10 GHz higher than those in the second layer, or in a comparable thin film of the same magnetic material, as shown to the left in Fig. 2(f). The difference in the bulk and surface behavior brings opportunities for creating new magnonic devices based on these effects.

Here, it is worth reiterating that the 3D artificial spin ices created up until now have been lattices of connected nanowires, which means that the physics is primarily driven by the formation and propagation of domain walls within the wires. While not a direct analog to bulk crystal systems, in addition to the fascinating domain wall behavior, such connected networks of domain wall conduits are interesting for computing applications.^108–111 

Gyroid structures created with self-assembled templates provide a route toward realizing fully magnetic extended 3D lattices,^84,113 with a high degeneracy of magnetic states and magnetic frustration similar to that seen in bulk and artificial spin ices. Micromagnetic simulations and electron holography of such gyroid structures [Figs. 3(c) and 3(d)] reveal the presence of multiple low energy magnetic configurations.⁸⁴ Finite-element simulations suggest that, while individual struts in such a structure can be approximated as macrospins, there is an effective DMI, which arises from the curvature and which acts to reduce the number of ice-rule-obeying vertices.¹¹³ 

Inverse opal-like structures (IOLS), such as that shown in Fig. 3(e), provide an analog to the pyrochlore spin ice structure and are created by electrodeposition into a template formed from self-assembled polystyrene spheres. The magnetization reversal in IOLS has been measured using magnetometry,¹¹⁴ which reveals different reversal mechanisms for different orientations of the applied magnetic field. In addition, reciprocal space measurements have been carried out using small angle neutron scattering.¹¹⁵ An example neutron scattering pattern, containing both magnetic and nuclear information, is shown in Fig. 3(f).¹¹² Depending on the aspect ratio of the struts in such an IOLS, which can be controlled by the degree of sintering, the extent to which ice-like rules are obeyed at each vertex can be tuned.¹¹⁶ However, with self-assembly approaches, it is not possible to have complete control of the lattice design and developments in tomography methods are required to have sufficient spatial resolution to capture real-space images of the 3D magnetic configurations. In addition, one should always be mindful that, for connected lattices, there is often a complex magnetic texture present at the vertices. As an example, a micromagnetic simulation of a swirl in the magnetization at an IOLS vertex is displayed in Fig. 3(g). Such micromagnetic simulations are not only required to interpret the small angle neutron scattering data presented in Fig. 3(f), but also reveal how such self-assembled structures allow us to move beyond the Ising paradigm, so bringing new and intriguing properties of their own.

Despite the allure of 3D artificial spin ices, which should host a plethora of new physics, there remain several areas for development, not least in fabricating structures made up of magnetic elements that are small enough to avoid them being multidomain and developing methods to unambiguously identify the magnetic configurations with sufficient spatial and temporal resolution. Here and in Fig. 4, we highlight how progress in this field can be achieved block-by-block.

Starting with the foundations, the block in the middle of the bottom row in Fig. 4 represents design, which is at the heart of all endeavors in the field of artificial spin ice, whether in two or three dimensions. This is because the choice of underlying lattice and magnetic material determines the spectrum of interactions in the system. In turn, this choice will influence the route by which fabrication is attempted and even which measurement techniques are applied.

Given the wide variety of possible lattices, the reader may be wondering which 3D lattices to target first. This is not an easy question to answer since each lattice will have its own unique and fascinating properties. We highlight two desirable properties, which are not particular to 3D artificial spin ice, but which might help guide such a decision. First, in lattices where the vertices have odd coordination, the lowest-energy magnetic configuration for a single vertex possesses a net magnetic charge. This means that there is usually some form of emergent charge-ordered sector in the phase diagram. Second, it can be advantageous for the vertices to have a high degeneracy of low-energy configurations. This can lead to a high level of frustration with a concomitant residual configurational entropy.^117,118 While these characteristics can also be found in 2D artificial spin ice, 3D artificial spin ice offers a radically different connectivity between vertices. This means, for example, that different forms of charge order or frustrated spin motifs can emerge because of the different way that vertices can be connected in three dimensions.

Even with the pyrochlore lattice, there is still much to explore, such as observing the behavior of an extended lattice where all of the struts are magnetic, including the magnetization reversal, fast dynamics, and thermodynamics. More generally, there are many examples of emergent phenomena that have been observed in 2D artificial spin ice but remain to be explored in 3D artificial spin ice. These include vertex frustration, phase transitions, entropy driven effects, and emergent magnetic monopoles.

As to whether future systems should involve connected or disconnected nanomagnets, the macrospins in single-domain nanomagnets provide a closer analog to the uncompensated magnetic moments in the crystals on which artificial spin ices were originally based. Additionally, when such single-domain nanomagnets are elongated, say, into the conventional stadium shape, one can treat them as effective Ising spins, so simplifying how to consider and simulate 3D artificial spin ices. While 3D lattices with disconnected nanomagnets have not yet been realized experimentally, micromagnetic simulations have been used to characterize the magnon modes¹⁰⁷ and the behavior of emergent magnetic monopoles¹⁰⁴ in such systems.

Connected 3D magnetic structures can still be represented with a macrospin model after a careful evaluation of the magnetic texture. Moreover, their connected nature permits magnetoresistance measurements and provides a basis for complex domain wall devices. Indeed, while disconnected nanomagnets interact solely through dipolar coupling, for such connected structures, the exchange interaction between the magnetic elements is important and magnetization reversal occurs via the creation, propagation, and annihilation of domain walls.

Shape anisotropy plays a significant role in both connected and disconnected structures, since it determines the local magnetic easy axis. In addition, an effective curvature-induced DMI can induce a vortex-like magnetic configuration in nanowires with diameters on the order of 100 nm,⁹⁵ which is an effect that depends greatly on the diameter of the nanowires. This possibility to control the twist in the magnetization within the nanowires opens up ways to tune the magnetic configuration at a vertex or the interaction between neighboring nanowires, which could lead to novel types of magnetic order.¹¹⁹ 

Up to now, most 3D artificial spin ices are created with polycrystalline magnetic materials, which are relatively straightforward to coat or electroplate. It may also be of interest in the future to introduce amorphous materials, while the path to creating 3D artificial spin ice from single crystal materials with complex anisotropies is not clear.

In terms of the methods used to fabricate 3D artificial spin ices (bottom-left block in Fig. 4), magnetic material has either been added to a scaffold, as for two-photon lithography, or forms the structure itself as in FEBID. However, it is not yet clear how magnetic materials can be locally added or removed from 3D magnetic structures without compromising their structural integrity. Finding routes to achieve this would allow the creation of 3D artificial spin ices made up of disconnected nanomagnets. This would open the way to realize a true mesoscopic analog of the pyrochlore lattice, where the magnetic moments associated with the individual dipolar-coupled nanomagnets take on the role of the Ising spins in the bulk spin crystals. There exist several innovative approaches for adding magnetic materials to scaffolds, which may help in this respect. These include atomic layer deposition,⁸⁶ electrodeposition,¹²⁰ and electroless deposition.¹²¹ 

We would hope that, in the future, new types of large-volume structures beyond gyroids and IOLS might be realized using self-assembly. Here, expertise in other fields, such as metal-organic frameworks¹²² and crystal growth,¹²³ may provide inspiration on how to make suitably tailored self-assembled templates.

Even without developing new fabrication processes, advances in the field can still be achieved with standard electron beam lithography. Indeed, in terms of the multilayer artificial spin ices, there exist several different geometries that await an experimental realization with this approach.

Another cornerstone in the endeavor to investigate 3D artificial spin ice is numerical simulations, both Monte Carlo and micromagnetic (bottom-right block in Fig. 4). Simulations play a fundamental role in understanding the behavior of artificial spin ices and act as a guide for experiments by giving insights into magnetization reversal, fast dynamics, thermal relaxation, and thermodynamics. Indeed, with the increase in interest in fully 3D artificial spin ices, it will be beneficial for the community to make use of finite-element micromagnetic solvers^124–126 since a mesh-based discretization of space is highly advantageous when approximating curved objects. In some respects, particularly for 3D artificial spin ices, numerical simulations currently outpace experimental work, providing inspiration for the field.

We now turn to the topics that underpin experimental work (middle row in Fig. 4). In terms of characterization (left-hand block in the middle row of Fig. 4), further developments are required to image the magnetic configurations, both at the surface and in the bulk, with sufficient spatial and temporal resolution to allow the observation of magnetization dynamics, including magnetization reversal in an applied magnetic field, thermal relaxation, and thermodynamics. One possibility is to image the magnetic configurations with electron or synchrotron x-ray transmission methods. However, these methods only provide images of the projection of the magnetic induction¹²⁷ or magnetization, respectively. From such projections, it can be difficult to reconstruct the magnetization exactly, so that interpretation of the data often requires extensive micromagnetic simulations. Nevertheless, without such a priori knowledge, it is still possible to reconstruct the magnetic configuration in a 3D mesoscopic system using tomography and laminography techniques, with either electron or synchrotron x-ray probes.^87,93,128 For this, projections need to be taken of the same sample at several different orientations, which is very time-intensive. This necessarily limits the amount of data that can be collected in a single experiment. In addition, the magnetization must remain the same over several hours if not days. The fourth generation of synchrotron x-ray facilities with diffraction-limited x-ray sources should help here because they offer both greater coherence and a significant increase in photon flux.^129,130 With this, measurements that once took several days, might now be completed in a few hours, enabling magnetic tomography with a time resolution sufficient to capture slow magnetization dynamics.

The mainstays for characterizing 2D artificial spin ices are MFM and XPEEM, which were developed to image flat samples. The spatial resolution of these microscopy methods is sufficient to image the magnetic configuration in 2D artificial spin ices with high contrast. However, there are certain limitations when applying MFM and XPEEM to the imaging of 3D magnetic structures. For both techniques, the topographic contribution from rugged surfaces can conceal the magnetic signal. Furthermore, for XPEEM, the incident x-rays may not uniformly illuminate the whole sample due to self-shadowing, resulting in no magnetic contrast from the shadowed regions. Nevertheless, recent efforts have shown that these microscopy methods can be adapted.^54,95 For example, one recent promising step toward imaging 3D artificial spin ices is the use of scanning transmission x-ray microscopy to image a 3D artificial spin ice suspended over an aperture.⁸⁸ By further tailoring these imaging methods, in combination with developments in fabrication, we envisage that, just as for 2D nanomagnet arrays, imaging of the magnetic configurations in 3D artificial spin ices will become standard.

Beyond imaging techniques, we foresee opportunities to use other approaches, such as neutron and x-ray scattering,^45,112 magnetometry,¹³¹ calorimetry,^132,133 or transport measurements⁹⁴ to probe the collective behavior of these systems. Indeed, some of these techniques have already been applied with great success to 3D artificial spin ices.^45,94,112 For most of these techniques, large-volume samples are required, for which self-assembly routes to fabrication are optimal.

Finally, fast and ultrafast dynamics can be probed using ferromagnetic resonance,¹³⁴ Brillouin light scattering,⁸⁶ free electron lasers, and time-resolved magneto-optical Kerr-effect techniques.⁸⁵ All of these characterization techniques can be applied to 3D artificial spin ices while manipulating them with various stimuli, such as applied magnetic and electric fields, spin and orbital currents, heat, light, or mechanical forces^2,135–138 (right-hand block in the middle row of Fig. 4).

Observations of thermally activated dynamics provides a means, for example, to observe how an artificial spin ice relaxes into the ground state or how emergent magnetic monopoles propagate.⁴² For this, the height of the energy barrier to switching must be close to the thermal energy supplied, and the moment switching rates should match the measurement timescale that, for example, is on the order of 1–10 s for XPEEM imaging of 2D artificial spin ices.^42,43 For Permalloy nanomagnets in 2D artificial spin ices, with lateral dimensions of a few 100's nm, this occurs at thicknesses in the range of 2–5 nm, and we expect that the nanomagnet dimensions that give thermally active dynamics in 3D systems will be similar.

Observations of magnetic field-driven processes can reveal avalanche behavior, as seen in 2D artificial spin ice.^12,139 In addition, in 3D artificial spin ice, magnetic fields can induce high-energy all-in/all-out vertex configurations with careful choice of the applied field direction.¹⁰⁰ Seeding such high-energy magnetic configurations at specific locations with a global magnetic field is virtually impossible for a 2D lattice of identical nanomagnets. Electric fields, spin currents, and orbital currents can influence magnetization through various mechanisms such as spin-transfer torques and spin–orbit torques. Charge currents will give rise to Joule heating or magnetic fields, which can also induce interesting magnetic behavior. Laser light can be used to impart thermal energy into artificial spin ices.²⁴ It has also been shown that 2D artificial spin ice can transfer orbital angular momentum to light, which depends on its magnetic configuration.¹⁴⁰ The interaction of light with 3D artificial spin ice, where the magnetization can point out of the plane, may impart more complex orbital angular momentum modes. Mechanical forces applied to 3D artificial spin ices, or to the flexible substrates on which they reside, can be used to anisotropically alter the magnetic interactions. This provides a way to change the energy landscape in 3D artificial spin ices, so modifying the collective behavior during magnetization reversal or thermal relaxation, including the motion of emergent magnetic monopoles and corresponding Dirac string avalanches.^11,12

The uppermost block in Fig. 4 represents the novel physics and applications that will flow from research into 3D artificial spin ice.

In terms of fundamental research, we envisage several important avenues for the future. For example, measuring the magnetization dynamics in a mesoscopic pyrochlore spin structure in real space would provide an opportunity to observe exactly how ordering proceeds on a macrospin-by-macrospin basis. In addition, since interesting effects are predicted to arise at the surfaces as a result of orphan bonds,¹⁴¹ one may be able to manipulate the behavior of such 3D connected nanowire networks by adjusting the way in which the lattice surfaces are terminated. Beyond the ice-like physics to be found in the mesoscopic pyrochlore lattice, we are confident that a range of unexpected physical phenomena can arise in novel 3D geometries, including complex phase diagrams with different classes of phase transitions, new forms of excitations, and topology-driven effects. Interesting geometries for 3D artificial spin ice include tilings where the vertices have either odd coordination or a high degeneracy of low energy states because these lattices often possess a high degree of frustration. It would be interesting to realize mesoscopic magnetic analogs of 3D quasicrystals, which should host new and intriguing physics due to their unique symmetry.^142–144 Even in structures that are limited in extent, such as the buckyball, the curved topology is likely to give rise to unexplored magnetic configurations and complex phase diagrams.

We have discussed how non-trivial spin configurations occur at the vertices or inside the individual elements of connected systems.^{95,99,112,145} This may affect the spin ice physics in interesting ways, for example imparting a chirality to otherwise non-chiral systems. One can also envisage introducing structural chirality through careful design as in the gyroid systems, which may impart a chirality in the magnetic configurations¹⁴⁶ or give rise to a chirality in the magnetization dynamics.¹⁴⁷ 

Finally, combining nanomagnets with micromechanical systems would offer a new degree of freedom to give shape transformations in an applied magnetic field or with other stimuli,^148,149 altering, for example, the level of frustration. This might be as simple as incorporating strain-induced effects from the underlying substrate to modify the magnetic state^137,138 or involve more complex deformations.^148–150 

Concerning applications, 2D artificial spin ices are currently of interest for reservoir computing,⁴⁸ a special form of neuromorphic computing, and also for probabilistic computing¹⁵¹ or for electrically programmable nanomagnetic Ising networks.¹⁵² 3D artificial spin ices would be particularly useful for such applications because they host many more magnetic configurations in a reduced footprint. Artificial spin ices on curved surfaces, such as the buckyball, have the advantage that further-neighbor dipolar interactions are more important than in a comparable 2D system, since the lattice is wrapped in on itself. This would enable more complex artificial spin ice devices that exploit such interactions. In addition, in such shell structures, the nanomagnetic elements are more easily accessible from the outside than those within a dense 3D lattice.

The fast dynamics of 3D artificial spin ices has been experimentally investigated in multilayer systems⁶³ and numerically simulated for buckyball lattices,¹⁰³ and readout of the spin wave spectra has proven to be useful for reservoir computing.⁴⁸ In order to make real-world devices, it would be helpful to be able to electrically read out the fast dynamics. For this, one promising way forward is to record the ferromagnetic resonance using planar microresonators, which have the required sensitivity to measure mesoscopic systems.^153,154

Artificial spin ices are also interesting as reconfigurable magnonic crystals,¹⁵⁵ and, for example, spin wave channels can be created when combining a 2D artificial spin ice with a ferromagnetic thin film.^156,157 Such spin wave channels can then be implemented in programmable magnonic circuits, for example, for data transport and logic devices. In a similar way, one could imagine embedding a 3D artificial spin ice in a ferromagnetic matrix to realize a more complex interconnected spin wave network.

In addition to computation, we envisage that 3D artificial spin ices could be used as ultra-sensitive magnetometers to measure the local magnetic field in 3D.

In general, stable nanomagnetic elements, whose magnetic state does not fluctuate over time, are desirable for data storage. For sensing applications, the energy barrier to switching should be low in order to increase the sensitivity to magnetic fields. For computing, the thermal properties of the artificial spin ice can be tailored for specific applications.¹⁵⁸ 

One can think of Fig. 4 as the mountain that still needs to be climbed to reach the summit of research and development involving 3D artificial spin ices. The individual blocks correspond to the different areas of progress that are needed for this. With the advent of the next generation of ultra-bright synchrotron x-ray sources in conjunction with the widespread adoption of two-photon lithography and FEBID, we foresee several advances in this field in the near future. In particular, harnessing the various advancements to explore a wide variety of 3D artificial spin ices will bring both novel collective behavior and innovative applications. We look forward to seeing how researchers scale these blocks in the future to reach ever new heights along the path to discovery in the mountain range of 3D artificial spin ice.

The authors would like to acknowledge funding from the Swiss National Science Foundation (Project No. 200020_200332).

There were a few key events in 2024 that prompted us to put together a review. L.J.H was invited by Peter Fischer and Adekunle Adeyeye to give a tutorial on three-dimensional artificial spin ice at the SPICE Workshop on “Nanomagnetism in 3D” in Ingelheim. At the same time, she was also preparing a section on 3D nanomagnetic imaging with Claire Donnelly and Valerio Scagnoli for the “2025 Roadmap on 3D Nanomagnetism,” organized by Gianluca Gubbiotti and Anjan Barman. We had also obtained our first results in this area, with L.B. in the last stages of his doctoral project and G.M.M. addressing with Monte Carlo simulations the thermodynamics of lattices wrapped on the surfaces of 3D solids. We are very grateful to the members of Mesoscopic Systems, to all of our collaborators, as well as for all the opportunities that our colleagues and funding agencies have given us. We are watching with great interest the increase in enthusiasm, not just about 3D artificial spin ice, but also concerning other nanomagnetic systems in three dimensions, that has served as inspiration for this review.

The authors have no conflicts to disclose.

Luca Berchialla and Gavin M. Macauley contributed equally to this work.

Luca Berchialla: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Gavin M. Macauley: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Laura J. Heyderman: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.