Three-dimensional nanomagnetism

Magnetic nanostructures are being developed for use in many aspects of our daily life, spanning areas such as data storage, sensing and biomedicine. Whereas patterned nanomagnets are traditionally two-dimensional planar structures, recent work is expanding nanomagnetism into three dimensions; a move triggered by the advance of unconventional synthesis methods and the discovery of new magnetic effects. In three-dimensional nanomagnets more complex magnetic configurations become possible, many with unprecedented properties. Here we review the creation of these structures and their implications for the emergence of new physics, the development of instrumentation and computational methods, and exploitation in numerous applications.

characterization techniques available to create (Box 1) and probe (Box 2 and Box 3) these systems, and the latest advances in computational methods for their modelling (Box 4). We also highlight some of the major challenges still associated with these studies, and the great potential of this field to impact areas such as sensing, data storage, nanoelectronics, the Internet-of-Things and medical biology.
Emerging physics of 3D nanomagnetism New types of domain walls. The extension of 2D nanostructures into the 3D world brings with it the emergence of unconventional spin textures, where novel physical effects comprising geometry, topology and chirality are involved. A paradigmatic example regards domain walls (DWs) in magnetic nanowires (NWs). NWs are subject to extensive investigation in nanomagnetism because of their potential use as DW conduits for memory and sensing applications [12][13][14] . Considering in detail NWs formed by soft magnetic materials, magnetization tends to be aligned parallel to their long direction, due to shape anisotropy. In this case, DWs are of either head-to-head or tail-to-tail type, holding a magnetostatic charge 2SM s (with S the NW section and M s the spontaneous magnetization of the material), irrespective of the spin texture within the wall. This gives rise to magnetostatic energy, such that competition with exchange determines a complex panorama of DW types, in a manner depending on dimensionality and strip size. The size is compared with the characteristic dipolar exchange length D d ¼ (2A/m 0 M s 2 ) 1/2 (ref. 15), where A is the exchange stiffness of the material. We first describe their static features, before highlighting some peculiar aspects of their dynamics.
2D NWs (from now on denoted nanostrips to distinguish them from their 3D counterparts) have been widely studied in theory, simulation and experiments, because of their ease of fabrication and integration into devices. These are patterned with standard top-down lithography methods from thin films, having widths much larger than D d and thicknesses comparable or not much larger than a few times this length. Thus, any significant variation of magnetization across thickness is prohibitively expensive energetically, such that magnetization textures can be faithfully described by a 2D field of a 3D magnetization vector M(x,y). For those nanostrips with moderate widths, significant variation of magnetization across the nanostrip would also be energetically unfavourable. Therefore, such DWs may be described using a 1D model by M(y), where magnetization in the core of the wall is aligned transverse to the strip long axis y; this is the so-called transverse DW (TDW, see Fig. 2a). Obviously, the transverse component imparts additional dipolar energy due to the edge charges that are created. For sufficiently wider strips, this energy can be reduced by the emergence of a different type of magnetic boundary, the vortex DW (VDW): here, magnetization becomes more parallel to the strip edge by curling around a vortex with core perpendicular to its surface (Fig. 2a). These two types of DWs share a large metastability region in the phase space, consisting of thickness (t) versus width (w), around an iso-energy line, determined by simulations 16 as twC61D d 2 . To extend this already well understood phase diagram (see dashed region in Fig. 2b) to an arbitrary cross-section, the thickness of a nanostrip can be progressively increased towards a NW of square cross-section. DWs initially transverse and vortex in nature end up in magnetization textures with a tube of magnetization going through the wire along a transverse direction, x and z respectively. Remarkably, these two types of walls, characterized now by a full 3D magnetization field M(x,y,z), are identical upon rotation by p/2 around the wire axis. For large lateral wire dimensions, they display simultaneously transverse and vortex characteristics. They may, therefore, be denoted transverse-vortex domain walls (TVDWs) 15 , whereas the denotation of either transverse or vortex is sufficient in nanostrips (or p/2-rotated: very narrow and thick NWs), where one of these features is largely dominant (Fig. 2a). Being able to transform from one to the other using only continuous transformations from the flat nanostrips reveals that they share the same topology. This topological equivalence remains valid for any NW cross-sectional shape, for instance either square or circular.
Until now in our discussion, 3D NWs were considered as if the 2D DW magnetization was simply extruded along the third direction. However, considering 3D NWs from the start allows for the emergence of novel spin textures. The DWs so far described display two areas where magnetization is perpendicular to the wire surface: the entry and outlet points of the vortex core. New synthesis, characterization and computational methods have the potential to make the leap to 3D. The combination of more complex magnetic states and additional degrees of freedom in 3D nanomagnets leads to the emergence of new physical phenomena, which may find applications in multiple areas. (a) Examples of 3D nanomagnets, from left to right: magnetic sphere with vortex configuration. Magnetic thin film element with a skyrmion. Symmetry breaking is caused by bulk or interfacial Dzyaloshinskii-Moriya interaction. Möbius strip with perpendicular magnetization; a DW is present in the ground state due to the object's topology. Cylindrical NW with modulated diameter, with different magnetic configurations depending on the diameter. Antiferromagnetic (AF) superlattice (interlayers not shown for clarity) with a wide soliton in the middle. (b) Examples of 2D nanomagnets, from left to right: Single-domain magnet. Magnetic multi-layered element with perpendicular anisotropy. Nanostrip with protrusions for DW trapping. Bi-layered magnet with AF coupling due to indirect exchange via an interlayer (not shown for clarity).
Since magnetization pointing perpendicular to a surface implies an increase in magnetostatic energy, DW configurations with a more efficient magnetic flux closure (that is, with magnetization mostly parallel to the wire surface at any point) should be expected. Such a texture is possible for cylindrical symmetry, leading to a curling of the magnetization parallel to the wire axis. For a solid cylindrical NW, and with these boundary conditions, the continuity of a vector field of uniform magnitude becomes impossible 17 . Hence, there must exist one point where M cancels, that is, a local singularity where ferromagnetism is quenched. This very peculiar object, called a Bloch point, was theorised decades ago, based on similar external boundary conditions in bubble media 18,19 . It refers to a micromagnetic object of zero dimensions (0D); it is the counterpart of Bloch walls as 2D objects separating 3D domains. This wall, shown in Fig. 2a, is characterized by an axial vortex, that is, its axis is parallel to the wire, and with the Bloch point lying on the axis. It can accordingly be named a Bloch point DW (BPDW) 20 . Its existence was first predicted by means of micromagnetic simulations 21,22 . Taking these considerations into account, a general thicknessversus-width phase diagram reflecting the energetics of TVDWs and BPDWs in 3D NWs can be constructed (Fig. 2b). In the white and green regions of the diagram, the TVDW is the DW ground state, with either transverse or vortex features, respectively, becoming more evident for each area. The blue region, on the contrary, refers to the parameter space where the BPDW is the ground state, which occurs for lateral dimensions typically above 7D d . It is worth noting that there exist large areas of this phase diagram where two or even three of these walls may coexist as (meta)stable states 15 . The first experimental imaging of TVDWs 23,24 and BPDWs 24 in NWs has recently been reported.
The above description of DWs in NWs illustrates characteristics of 3D spin textures displayed by other non-planar nanomagnetic objects. For instance, nanotubes (NTs) exhibit a rather similar phenomenology 25 , with just qualitative differences of the iso-energy lines, and notably with the absence of the Bloch points seen in the case of DWs with an axial vortex. The ends of elongated objects are also worthy of consideration, as they constitute the loci of nucleation for magnetization reversal. With the same physics (magnetostatics versus exchange) at play, these loci display end states called 'C' or curling, which may be viewed as a fraction of a TDW or BPDW, respectively 22 .
Additionally, unlike planar surfaces, curved surfaces lack space inversion symmetry. This can be the source of remarkable effects 26 .
2D nanomagnetism has been mostly developed via the combination of physical vapour deposition (PVD) and top-down lithography. Progress in 3D nanomagnetism relies however on new synthesis methods and strategies. The three main routes currently used for this purpose are physical, chemical and 3D nano-printing methods (see figure).
PVD can be extended to 3D if the deposition is done onto previously patterned 3D scaffolds, instead of onto flat substrates. Such scaffolds include those created by two-photon direct-write optical lithography 125,126 and self-assembled methods 127,128 . However, this method faces challenges concerning conformal deposition and material shadowing. The anisotropic character of PVD has nevertheless been exploited to create ultra-small nano-helices 129,130 presenting a strong magneto-chiral dichroic effect 131 . A complementary approach for 3D manufacturing based on thin films is rolled-up nanotechnology 132 . Here, differential strain acting on films provides a means to engineer nanomembranes of virtually any material 31 .
Chemical synthesis techniques are generally more suited to deliver 3D nanostructures. Using heterogeneous nucleation methods, assemblies of rods, dumbbells or branched networks can be produced 133 , with potential applications as rare-earth-free magnets 134 and magnetic recording media 135 . However, the most common approach uses 3D-templates, exploiting the isotropic character and conformal growth achieved in electroplating, electroless plating and atomic layer deposition. Anodised alumina matrices are the most common templates, producing arrays of high vertical aspect-ratio NWs 136 , NTs 137 , core-shell 138 and multi-layered NWs 139 , formed by a wide variety of materials. The remarkable control achieved over interfaces between materials during co-deposition has made possible spin-transport effect studies 140,141 . However, atomic-scale control over the thickness is not possible yet, hindering the synthesis of perpendicular materials based on surface anisotropy. Truly 3D template manufacturing is becoming available thanks to new methods to modulate template diameters along the pore length 142,143 , and latest advances in block-copolymer 144 and other self-assembly methods. Complex 3D geometrical textures such as inverse opal lattices 145 and gyroids 146 have been realised, for applications in magnetic collective phenomena studies, for example, 3D spin ice.
The third route comprises of 3D nano-printing via focused electron beam induced deposition (FEBID) 147 . FEBID is an emerging direct-write patterning technique able to prototype individual complex 3D systems with resolution of a few tens of nanometre, in times ranging from seconds to a few minutes. Unlike in the past, FEBID is not restricted to the growth of carbonaceous materials, but polycrystalline metallic cobalt and iron can be synthesized 148,149 . Progress towards the growth of magnetic alloys 150,151 and functional spintronic interfaces 152 have been recently reported. 3D geometries grown by FEBID include vertical NWs for MFM 153 , spintronic applications 82,154 and 3D actuators 91 , as well as nano-spheres 83 , nano-helices 154 , nano-ellipsoids 155 and nano-bridges 156 . Future works should scale the growth to larger areas 157 , and will exploit new purification stategies [158][159][160] , as well as recently developed simulation growth algorithms combining continuum models and Monte Carlo methods 161 .

Chemical methods
For instance, micromagnetic simulations have shown that the surface curvature in cylindrical NTs and NWs can give rise to a chiral symmetry breaking that results in different properties, depending on the rotation sense of the magnetization. This effect resembles the Dzyaloshinskii-Moriya interaction (DMI) 26 . To address this phenomenon of curvature-induced DMI, a 3D theory has recently been proposed 27 , consisting of reducing magnetostatic effects to an effective anisotropy. Under this assumption, curvature and torsion in NWs and shells split up the existing exchange interaction into scalar Heisenberg exchange and two effective magnetic interactions, namely curvature-induced effective anisotropy (magnetization patterning) and curvatureinduced effective DMI (chirality selection) 27,28 . Systematic studies of magnetic NTs have proven the existence of an exchange energy that manifests as an effective easy-axis anisotropy along the tube axis 25,29 , which induce a gap in the dispersion relation of magnonic spin excitations in NTs 30 . Remarkably, the strength of the curvature-induced chiral effects could be tuned mechanically by bending flexible magnetic nanomembranes 31 , thereby providing a new method to manipulate magnetic properties at the nanoscale.
Dynamic effects. Dynamic magnetic properties may also differ significantly between 2D and 3D nanomagnets, as recently demonstrated in the case of DW propagation and NWs. The dynamic motion of DWs in NWs is of paramount importance for their exploitation in applications. This has led to a large amount of works in nanostrips in the last years 20,32-34 , where a complex scenario including several regimes of DW propagation, can be observed. In particular, DWs under external fields usually become unstable when propagating above a critical speed, an effect called Walker breakdown 35 . This effect originates from a torque exerted by the driving field on the DW. If this torque is sufficiently strong, the wall magnetic structure changes significantly in a periodic fashion, causing the motion to become oscillatory, as described by the Landau-Lifshitz-Gilbert equation 20 . Recent studies using micromagnetic simulations 36,37 carried out in cylindrical NWs and NTs, reveal that DWs with an axial vortex configuration (BPDWs in the case of NWs) can propagate smoothly at very high speed (B1 km s À 1 ) when driven by an external magnetic field, overcoming the Walker breakdown 36,38 . Axial VDWs in NTs and NWs can surpass this limit due to the radial component of the magnetization 36,37 , caused by magnetostatic field distributions. This stability against the Walker breakdown in NTs and cylindrical NWs, described by micromagnetics, can also be understood in the context of the aforementioned chiral break of symmetry due to the surface curvature 36 . The effective dynamic-DMI counteracts this torque and thereby preserves the magnetic structure of the DW, even at large fields. Since this is a chiral effect, only one of the two possible vortex configurations of the DW is stabilised. These ultra-stable DWs can even reach the phase velocity of spin waves and thereby give rise to the Spin-Cherenkov Effect 36 (Fig. 2c,d). This effect describes the spontaneous emission of spin waves Magnetic characterization techniques at the nanoscale have been mostly developed for and are suitable to probe planar systems. Their extension to the study of 3D nanomagnets, presenting complex vectorial spin textures and geometries is far from trivial, and requires new methodologies and multi-technique approaches. In particular, magnetometry techniques able to probe single planar nanostructures [162][163][164] has been an essential factor for the progress of nanomagnetism in recent years. These studies have been extended to 3D for the simplest geometries, such as NWs and NTs, following an intermediate micromanipulation step, which places them flat at a particular position on a substrate. Using this approach, the switching field angular dependence of cylindrical NWs has been measured, to determine the mechanism behind magnetization reversal. This has been done by detecting the emanating magnetic flux using nano-SQUID 162 and their magneto-optical response using Kerr effect 154 . Analogously, dynamic-cantilever magnetometry has been employed in NWs and NTs glued at the edge of a cantilever, detecting the frequency shift of oscillation caused by the magnetic torque under external fields 165,166 . Switching probability measurements also allow a method to probe the thermally activated nature of the switching process 162,167 . For the advance of the area, new magnetometry methods for measurements of as grown 3D suspended nanostructures, which do not require such complicated micromanipulation intermediate steps are desired. This has been recently reported using magneto-optical Kerr effect 154 , where the switching fields of 100 nm-diameter, several mm-long cobalt NWs grown tilted with respect to the substrate were measured. The NW angle creates a projection on the substrate of the order of the laser diameter (5 mm), making the magneto-optical signal measurable.
A traditional way to probe magnetism in relatively large volumes consists of using neutron scattering techniques 168 . For instance, polarized neutron reflectivity has been used to determine the magnetic state of superlattices during soliton propagation 64,66 , and small-angle scattering has revealed the formation of skyrmion lattices in non-centrosymmetric bulk crystals 169,170 . Recently developed neutron tomography methods 171 enable the vectorial reconstruction of magnetic domains with resolution of B100 mm. In the case of spatially localized 3D objects, magnetic arrays such as wires and nanoparticles can be investigated in a small-angle scattering geometry 145 . However, the analysis soon reaches its limits when many parameters contribute to the recorded signal, such as distributions, imperfect texturing or correlations. A powerful approach consists of probing a highly ordered array of textured objects, where the scattering pattern can be analysed in a similar way as crystallography makes use of the reciprocal space, to extract detailed information of the structure of the lattice cell 172 .
Magnetometry measurements on 3D cylindrical NWs. In (a), the wire is placed on top of a nano-SQUID using a micromanipulator. Scale bar, 1 mm. Reprinted with permission from ref. 162. Copyright (1996) by the American Physical Society. In (b), the magnetic switching of free-standing NWs forming 45 o with the substrate plane is detected using spatially resolved magneto-optical Kerr effect. Additional sources of noise result from the mechanical motion via vibrations and coupling with the external field, and the large amount of diffusive scattering produced by a non-planar surface. occurring when a perturbation in a magnetic system moves faster than a threshold velocity, a phenomenon similar to sound waves generated by supersonic aircraft crossing the sound barrier. At these elevated velocities, DWs emit bichromatic spin waves of velocity-dependent wave length 36 .
Moreover, while the static structure of BPDWs in cylindrical NWs is now relatively well understood, the dynamic structure remains largely unexplored. This is because BPDWs are difficult to access experimentally and because the singularities of the magnetization field are characterized by a pronounced inhomogeneity of the magnetization that persists down to the atomistic scale, thereby making their simulation unreliable with standard micromagnetic codes. To resolve this problem, a hybrid atomistic-micromagnetic method has been recently developed with which the structure and the dynamics of BPDWs in NWs have been simulated at the atomic scale 39 , predicting a peculiar oscillation in the dynamics during field-driven motion 40 .
3D spin textures. 3D magnetic spin textures with non-trivial topological charges (vortices, skyrmions and chiral bubbles) possess an exceptional stability under external perturbations against transitions into trivial states, for example, to collinear magnetization. Enhanced stability, particle-like properties and high susceptibility to electric, spin and heat currents, and magnonic excitations have put these chiral structures into the spotlight of fundamental sciences, as well as application-oriented research. A paradigmatic 3D spin texture is the magnetic vortex with its nanometric core magnetization (Fig. 3a). Experimentally observed for the first time almost two decades ago 41 , it is typically present in patterned thin films formed by soft magnetic materials. The spin structure is defined by both in-plane magnetic circulation and out-of-plane core polarity, with the core expanding laterally B10 nm 42 , as set by D d . This strong confinement has allowed the quantification of pinning site potentials in thin films by probing Barkhausen noise 43 , generate nano-and micro-oscillators with tuneable frequency 44 and design of magnetic vortex memories 45 .
While the chirality of vortices (defined as the product of circulation and core polarity) and other objects such as Bloch points is intrinsically degenerate in common 2D nanomagnets, 3D geometries with inherent new types of interactions can break this symmetry. As mentioned above, this is the case of 3D curved nanomagnets, which reveal a chirality selection due to curvaturedriven DMI 46 . Similarly, heterostructures formed by ultrathin magnetic thin films and heavy-element materials with large spin-orbit coupling, result in interfacial antisymmetric spin couplings due to spin-polarized scattering from conduction electrons in the heavy-element layer. This antisymmetric Imaging 3D magnetic textures requires techniques with nanoscale spatial resolution, quantitative information and sensitivity to the vector character of the spin. Transmission electron microscopies (TEM) and X-ray microscopies (XM) are currently the techniques most suitable for these studies 173 . Magnetic lateral spatial resolution ranges from tens of nm for XM, down to sub-10 nm for TEM. Recent advances in Lorentz lens aberration corrections have enabled magnetic imaging at zero field down to 0.5 nm (ref. 174). XM has some other advantages, such as the straightforward application of electromagnetic fields, and the possibility to perform time-resolved experiments exploiting the temporal structure of pulsed X-ray sources.
TEM and XM are sensitive to either the projection of the magnetic induction vector B onto the plane perpendicular to the electron path (TEM), or to the component of the magnetization M along the photon direction (XM). The standard methodology to investigate 3D magnetic textures thus consists of comparing the information of a 2D projection for a given angle, with the one obtained by micromagnetic simulations. For instance, the magnetic configuration of a skyrmion lattice in FeCoSi and FeGe thin films has been determined using Lorentz TEM 175,176 . Using off-axis Electron Holography, multi-layered NWs reveal different magnetic configurations depending on their microstructure and composition 174 . Also, remanent states of Ni cylindrical NWs 23 and multi-layered Co/Ni NWs 177 give experimental evidence of some of the types of DWs discussed in the main text.
The increasing demand on 3D imaging techniques is leading to novel concepts that go beyond the aforementioned approach, by extending nano-tomography of scalar quantities 178 to magnetic investigations. By rotating the stage along an axis and acquiring 2D projections of B at several angles, both chirality and core orientation of vortices in magnetic disks have been determined 179 , and the magnetic structure of NWs and nano-spirals has been imaged [180][181][182] . Whereas in special symmetries full vector-field tomographic reconstructions of B is feasible by one simple projection 183 , in general projection series need to be recorded while rotating the sample about at least two axes. This has been applied to reconstruct the vectorial magnetic configuration of Permalloy squares 184 and of magnetic vortices in two stacked disks 185 .
In the realm of XMs, new 3D imaging methodologies have been reported so far for transmission-X-ray photoemission electron microscopy (XPEEM) 128,186 , ptychography 125 and magnetic X-ray tomography (MXT) 187 . In XPEEM, the 3D magnetic configuration of core-shell NWs has been visualized, by collecting both secondary photoelectrons emanating from the NW surface (conventional XPEEM) and from the substrate (XMCD shadow contrast) 186 . This same method has been employed to prove the existence of Bloch point DWs 24 and to image magnetization reversal processes in buried layers of curved magnetic nanomembranes 188  exchange interaction, referred to as interfacial DMI (IDMI) 47 , favours non-collinear chiral spin configurations. The competition of IDMI with standard symmetric exchange interactions (that is, Heisenberg and Ruderman-Kittel-Kasuya-Yosida (RKKY)), as well as magnetic anisotropies, results in a very rich magnetic phenomenology 48 , including in some cases the presence of magnetic skyrmions (Fig. 3b). Skyrmions are topologically protected 3D spin textures which subtend the whole 4p steradians of the spin space with a resulting non-trivial integer topological charge. They may exist as truly localized solitary objects with continuous magnetization rotation. For the formation of skyrmion lattices, large IDMI values, and typically magnetic bias fields are needed. The magnitude of IDMI depends significantly on the electron orbit overlap, and thus on the film elements, their arrangement, temperature and strain. Except in some special cases 49 , IDMI is typically orders of magnitude lower than the symmetric exchange interactions, with its effective value decreasing with magnetic film thickness 50 . In films with smaller IDMI, this is nevertheless enough to break the inversion symmetry and select one of the two possible DW chiralities 51 , forming, for example, chiral bubbles (Fig. 3c).
Following the first theoretical predictions and experimental evidence of skyrmions in single-phase bulk magnetic materials with non-centrosymmetric crystallographic structures, research activities are now mostly focused on heterostructures with IDMI, due to their potential for applications 52 . Crucial aspects, such as creation, detection, manipulation and deletion of individual nanoscale skyrmions were first shown in Pd/Fe bilayers on Ir(111) utilizing spin-polarized scanning tunnelling microscopy 53 . The strong IDMI in single Fe monolayers on Ir(111) stabilizes a non-reconfigurable BOX 4 | COMPUTATIONAL METHODS.
Micromagnetic simulations, rooted in a rigorous implementation of the fundamental equations of the theory of nanomagnetism, describe the spatial and temporal evolution of the continuous vector field of the magnetization and of the magnetostatic field, and can incorporate spin-torque effects. This powerful computational framework has provided a deep understanding of experimental observations and promoted research toward new effects 190 .
The experimental realization of 3D magnetic nanostructures necessitates micromagnetic investigations at a large scale. Progress in computer hardware and code development has made it possible to simulate micron-sized samples with an ordinary desktop computer. Owing to enhanced numerical methods and massively parallel computations, the upper limit in size is no longer a critical issue. Similarly, memory limits, which used to be hit easily several years ago, no longer represent a serious concern. Today, micromagnetic modelling requires high-performance computing for other reasons, such as the simulation of large arrays of interacting nanomagnets or hybrid atomistic-continuum simulations. Programmable Graphical Processing Units (GPUs) represent an impressively powerful approach to high-performance computing for these purposes [191][192][193] . In the past years, GPU-based personal supercomputing has outpaced the use of traditional large-scale supercomputing facilities, since they are inexpensive and require much less energy. As a result, post-processing, visualization and storage of huge amounts of simulated data have emerged as new challenges.
Another important point is which of the two possible branches, either finite difference methods (FDM) or finite element methods (FEM), are employed. FEM can model arbitrary shapes with high accuracy, making it more suitable for simulating complex 3D shapes, especially if curved surfaces are involved. On the other hand, FDM generally has higher speed for calculating magnetization dynamics and is easier to use and implement. In the past, the regular grid required by FDM (allowing for the use of Fast-Fourier Transform methods for the calculation of magnetostatic fields) allowed for shorter computational times and lower memory requirements. However, current matrix compression schemes employed now by FEM have largely removed these differences. Still, owing to the regular discretization grid of the FDM, the time integration of the Landau-Lifshitz-Gilbert is generally less critical than in FEM, which allows for larger time steps and thus faster integration.
Although micromagnetic simulations studies of 3D structures have proven to be generally very reliable, the continuum theory behind these algorithms ceases to be valid in certain cases 194 . Strongly inhomogeneous magnetic structures 195 , laser-induced magnetization dynamics, and sample sizes approaching the atomic limit call for extensions to the standard approach. In such cases, atomistic magnetic models 196,197 can be introduced. These models may either replace a micromagnetic code, or complement it in the form of a hybrid model (see figure) where atomistic magnetic moments are coupled to a surrounding continuum 39 .
Multiscale micromagnetic and atomistic models. The magnetic sphere combines both models to simulate the structure and dynamics of Bloch points. The central blue and outer orange areas correspond to the space where either a purely atomistic or a purely micromagnetic model is used to calculate the exchange interaction between spins, with the green area in between as the interface between both. Reprinted with permission from ref. 39. Copyright (2014) by the American Physical Society. skyrmion lattice that exists for temperatures up to 30 K and magnetic fields up to 9 T. The stabilization of skyrmions at room temperature has been achieved by increasing the film thickness of the transition metal, providing interlayer exchange interaction, and engineering asymmetric multilayer stacks with distinct heavyelement films for additive IDMI at top and bottom interfaces of the magnetic layer 54 . The corresponding skyrmion core/bubble size at room temperature is on the order of 10-100 nm 54,55 , and hence 10-100 times larger than those in Fe/Ir (111).
Owing to their small size, highly non-collinear spin texture and topological protection, they can act as localized mobile data for spintronic applications. To write, read and delete these skyrmions/ chiral bubbles, several mechanisms have been proposed and some demonstrated. These include spin-transfer/spin-orbit torque via spin-polarized current, magnons, spin Hall effect and electric fielddriven magnetoelectric coupling 53,56-58 . The potentially low pinning and small scattering cross-section of skyrmions originating from their spatial confinement implies both small on-set current density and large lateral deflection due to an accompanying Magnus force 59 . Current-independent unidirectional motion along nanostrips may be enforced via strain, exchange or anisotropy modulations 58 , or eventually considering antiferromagnetic (AF) skyrmions 60 . In practice, pinning/scattering at grain boundaries caused by structural and magnetic imperfection in polycrystalline and amorphous films results in larger current densities required for coherent motion 54 .
Vertical data motion using magnetic solitons. The extension of magnetic data mobility to the vertical direction has been recently realised by means of magnetic solitons in AF superlattices; this opens a new route to 3D magnetic random access memory devices with out-of-plane functionality. The term soliton refers here to a (topological) kink in a discrete chain of spins 61 , and is equivalent to an AF wall. The idea is analogous to the one employed in Nano-Magnetic Logic 62 , but instead of using soliton motion in dipolarcoupled nanomagnets positioned on the substrate plane, solitons are moved perpendicularly to the substrate plane in magnetic superlattices, where the coupling between magnetic thin films is provided by non-magnetic spacers via AF RKKY interactions. The system can be represented in its simple form by a 1D chain of spins (Fig. 3d), following a macrospin approximation, where layer thickness (t), anisotropy (K) and surface coupling (J) energies rule the physics of the system. Depending on the ratio between RKKY coupling and anisotropy fields (H J ¼ |J|/(m 0 M s t) and H u ¼ 2 K/(m 0 M s ) respectively) three regimes regarding magnetic texture and motion of solitons can be distinguished, as schematically shown in Fig. 3d.
For high RKKY couplings, solitons are wide, comprising a large number of layers, with those spins forming the soliton deviating a great amount from the anisotropy direction. In systems with very high coupling/anisotropy ratios, the rotation results in helical spin structures 63 . For soliton nucleation, the surface-spin flop transition can be exploited 64 , and reliable unidirectional propagation can be achieved by creating asymmetric superlattices, where thickness and anisotropy of a gate edge layer is tuned differently from the others. Following this approach, asynchronous motion under external magnetic fields using gates formed by single 65 and highly coupled multiple layers 66  REVIEW achieved. As RKKY coupling decreases, becoming comparable to anisotropy, solitons get narrower, acquiring a well-defined magnetic moment. Interestingly, macrospin and micromagnetic simulations predict how in this case, external rotating magnetic fields can couple to the soliton magnetic configuration, resulting in synchronous propagation with the field, where the direction of motion is defined by the soliton chirality and sense of rotation of the field. This could be exploited to create bi-directional multi-bit vertical shift registers 67 . Moreover, in spite of losing their chirality, synchronous motion is also possible for sharp solitons (when anisotropy becomes much larger than coupling). This has been realised under external oscillating magnetic fields, in Ising systems with strong perpendicular magnetic anisotropy, where the intrinsic layer coercivity (H c ) substitutes H u as relevant parameter for operation. By carefully engineering the thickness of the layers and the exchange coupling between them, it is possible to break inversion symmetry, which induces a unidirectional ratchet action 68 . Figure 3e shows schematically the structure of a ratchet superlattice, where layer thickness and interlayer RKKY coupling oscillate periodically between two possible values. Such a scheme makes possible the synchronous propagation of a soliton with perpendicular oscillating magnetic fields 68 . Remarkably, and following this approach, a unidirectional soliton shift register with vertical functionality equivalent to B20 transistors has been achieved within a thickness of 2 nm (ref. 68). Simple logic operations involving soliton-soliton annihilation have been realised as well 69 . Analogous ratchet schemes are possible; for example, a similar behaviour has been achieved by designing a superlattice with same thickness for all magnetic layers, and a combination of AF and ferromagnetic interlayer couplings 70 .
The aforementioned results refer to studies in extended films. Future studies are expected in laterally patterned nanostructures, to investigate the effect of rotating fields and dipolar interactions on the domain formation and soliton operation 71,72 . The use of different mechanisms for propagation, such as spin-transfer torque 73 , microwave excitations 74 or all optical switching 75 , are also anticipated. Additionally, a new energy storage concept based on the continuous injection of solitons via the rotation of one edge spin in a cork-screw fashion, has been recently proposed 63 .

Future applications of 3D nanomagnetism
Sensing and actuation. The use of 3D nanomagnets for sensing and actuation applications has a huge potential in many scientific and technological areas. Some of the novel functionalities foreseen in the field of 3D magnetic nano-sensing are the substantial increase in surface-to-volume ratio with respect to planar systems, their capability to probe vectorial fields, an ad hoc geometrical design to match a sensed structure, and the exploitation of mechanical and thermal effects present in suspended structures which may couple to their magnetic response.
In particular, 3D magnetic nanomembranes (forming part of the large family of emerging flexible devices, for example, electronics displays and solar cells) which are transforming rigid electronics into light and shapeable elements, have already proven to bear great universal potential as flexible magnetic sensors. Remarkably, their magnetic sensitivity, harnessing giant magnetoresistance or giant magnetoimpedance, is similar (or in some cases more than one order of magnitude larger 76 ) to devices produced on Si wafers, even after demanding manipulations 4,77 . This is due to the continuity of the magnetization along their closed surface, that is, due to their topology 78 . These features have stimulated desire for their application in industry as precise and flexible positioning control of magnetic bearings in electric motors 79 , and as printable sensing paste for simple but versatile magnetic switches (for example, electric postcards 80 ). Further potential applications in life sciences and diagnostics include light, mobile and affordable devices with great spatial localization so as to facilitate magnetoencephalography 76 for early stage disease detection.
Additionally, 3D nanomagnets could be incorporated as sensors in a new generation of scanning probe microscopy methods for advanced imaging and spectroscopy. In fact, high aspect-ratio NWs have already been employed as ultra-sharp MFM tips, improving magnetic resolution down to 10 nm 81 and overcoming limitations of standard triangular tips in probing 3D nanostructures 82 . Furthermore, nano-spheres grown on top of cantilevers have been exploited to perform ferro-magnetic-resonant force microscopy, measuring magnetization dynamics with high spatial resolution 83 . The development of new tip designs incorporating new geometries and materials can exploit high-order resonant cantilever modes to measure field gradient components along several directions in space 84 . This and other approaches 85 could make possible the realization of scanning probe vectorial microscopy: mapping the three components of stray fields emanating from a magnetic system at nanoscale resolution.
Moreover, 3D nanostructures often make very good actuators: their larger volume compared to planar 2D structures increases the torque and force generated by electromagnetic fields and strain. They are often free to deform reversibly, with interest for the NEMS industry, nanoelectronics, sensing, robotics and studies in the quantum regime [86][87][88][89][90] . In the case of 3D nanomagnets, additionally to standard electrical driving methods, remote magnetic actuation is possible via external magnetic fields 91 . This can be exploited to create nanomotors in fluids, where chiral helical structures 92 or Janus particles 93 have been controllably moved under rotating and oscillating magnetic fields, respectively. We foresee future works exploiting dipolar interactions, integrating different magnetic materials and more complex geometries, incorporating optical and electrical readout methods, and further integration into biological environments.
In particular, 3D nano-and micro-scale mechanical actuators are a key part of the tool-kit of mechano-biology 94,95 , an emerging field which tries to understand the role of cell and tissue mechanics in diseases. Single 3D magnetic nanostructures subjected to laboratory-scale magnetic fields can generate tens of nanonewton of force locally and controllably to cells. This is sufficient to probe the phenotype of cells 96 , mechanically induce differentiation of bone stem cells 97 , burst payload-carrying neural stem cells 98 and destroy cancer cells 99 . Additionally, mammals use motile cilia (flagella-like microtubules) beating together to sweep objects through organs (such as removing dirt from the lungs or transporting eggs through female Fallopian tubes). Magnetic NWs could perhaps form artificial cilia 100 , transporting reagents through on-chip microfluidic channels.
Towards data storage and the Internet-of-Things. 3D magnetic nanostructures could form the basis of a new storage class of non-volatile memories, in which multiple data bits are stored magnetically above a single electronic memory cell on an integrated circuit. This is sustained by the impressive progress on 'magnetic data mobility' in the last few years, in which physical effects such as spin-transfer torque 101,102 allow digital information to be propagated through nanomagnets via magnetic entities such as DWs, skyrmions or solitons. The realization of 3D magnetic memories could be implemented using perhaps the original racetrack architecture 103 . Alternatively, existing magnetic random access memory technology could be enhanced by increasing the number of data storing magnetic layers in the stack, such that each cell can store a data word instead of a data bit 104 . In both of these cases, digital shift-register action is an essential ingredient 68 , allowing data bits in magnetic form to be sequentially pumped into the nanomagnet during writing and then pumped back out during reading. Also, a recent trend in microelectronics is to create 2.5-dimensional devices 105 by stacking multiple thin substrates on top of each other in a single package, with wire interconnects running between layers. Magnetic NWs could use these ideas to enable 2.5-dimensional spintronic devices (that is, chips which use both electronic and magnetic components), allowing chip designers to connect electronic parts using electrical wires and magnetic parts using magnetic NWs.
Magnetic interconnect is a particularly interesting idea because it can host both magnetic DWs 106 and magnetic spin waves 107 . The former would carry digital information in a very compact form but at low speed, while the latter would offer long-range transmission. Intriguingly, the magnetic hysteresis and non-linearity present in many magnetic NWs mean that the interconnect is potentially also both a memory element and a logic gate 108 , opening up new computing architectures in which the traditional boundaries between memory, logic and interconnect are eliminated. Such chips could even go beyond conventional Boolean logic and implement neuromorphic computing architectures 109 in which 3D networks of magnetic NWs mimic the neurons and synapses in living brains.
However, significant challenges still remain before such applications become reality. In particular, modern microelectronics makes great use of precision interfaces between materials; many of the growth techniques used for 3D structures do not currently offer sufficient purity and control to engineer interfaces with single atomic layer precision. Also, while planar microchips successfully integrate and connect 10 8 transistors on a single chip, complex interconnectivity in 3D is still to be developed. A human brain may connect up to 10 4 synapses to each neuron 110 ; there are currently no known 3D fabrication techniques able to achieve this.
The integrated circuits of the future will increasingly incorporate a wider range of technologies onto a single die to form full systems (Fig. 4). The inclusion of physical sensors and energy harvesters into the menu of devices available to designers is particularly important for the Internet-of-Things. 3D magnetic nanostructures may find a role in both of these, where large surface areas, the ability to easily strain, and the possibility of sustaining temperature differentials are often important and difficult to achieve in 2D planar geometries. Recent advances in spin caloritronics 111 and multiferroic materials 112 will provide the necessary interconversion between the electronic, magnetic and thermal worlds.

Perspectives
The move from 2D to 3D is not only a current trend within nanomagnetism, but is part of a wider theme within nanotechnology in general. Nanoelectronics, nanophotonics, energy storage and harvesting, and nanomedicine, all stand to benefit from a new era of greener, more capable, multi-functional technologies brought about by moving into the third dimension. Despite the great challenges ahead, recent advances in bottom-up lithography, microscopy and computational techniques make its future realization feasible.
3D nanomagnetism, still in its infancy, needs to fulfil key milestones in the future. The experimental realization of spintronic devices requires high-quality materials, for example, DW motion may be severely affected by the presence of defects. Achieving such a quality with unconventional synthesis methods is still work in progress. Also, next-generation 3D nanomagnets will require advanced magnetic materials such as Heusler alloys, multi-layered and epitaxial thin films, and highly functional interfaces. These barriers could be overcome using fabrication strategies which combine multiple synthesis techniques, including innovative ways to exploit differential strain 113 . The low throughput currently achievable in 3D nano-printing could be significantly enhanced by moving from the gas phase to the liquid phase 114 . This may enable the development of complex architectures, mass fabrication and the creation of multifunctional designs. Future years will also see rapid progress in the synthesis of novel magnetic materials such as 3D nanocomposites 115 , where strong electric fields at interfaces or magnetostrictive effects could be used for energy harvesting. Magnetic tools able to probe 3D nanomagnets are essential as well. Vectorial magnetometry and magnetic microscopy of nanostructures require a phenomenal combined effort comprising instrumentation, electronics and data analysis.
Recent progress in magnetic microscopy methods [116][117][118][119] , pushing further spatial and temporal resolution, is expected to bring new opportunities in this area. An essential cornerstone still missing due to its technical complexity is the development of methodologies 120 to electrically contact individual 3D nanogeometries.
Advances in these areas are essential to study 3D spin textures further. Recent experimental work has confirmed theoretical and computational predictions about their magnetic structure and static behaviour. The next step is now to focus on the experimental realization of specific dynamic features driven by different mechanisms. These are often related to the interplay between the chirality and/or curvature of the boundaries of the texture itself and the chiral nature of the Landau-Lifshitz equation. Interest is also growing in composite spin textures such as coupled bilayers 121 , where flux closure and moment compensation can enhance magnetic DW motion within the plane, as well as leading to motion between horizontal planes. The search for and realization of ever more complex geometries will continue, for example, by connecting planar and vertical structures, creating core-shell 3D objects, or exploiting topological effects via the interplay between 3D shapes and spin configurations 122 . The possibility of creating complex 3D networks of nanomagnets could lead to new computational paradigms 123 , and is also expected to elucidate open questions regarding effects involving numerous interacting magnetic particles, for example, artificial spin-ice lattices 124 . Threedimensional nanomagnetism, with its vast amount of unexplored science and huge potential to impact society, is a fascinating new research area, which will flourish in the years to come.