Magnetic Charge Propagation upon a 3D Artificial Spin-ice

Magnetic charge propagation in bulk frustrated materials has yielded a paradigm-shift in science, allowing the symmetry between electricity and magnetism to be studied. Recent work is now suggesting magnetic charge dynamics upon the spin-ice surface may have important implications in determining the ordering and associated phase space. Here we detail a 3D artificial spin-ice, a 3D nanostructured array of magnetic islands which captures the exact geometry of bulk systems, allowing field-driven dynamics of magnetic charge to be directly visualized upon the surface. Using magnetic microscopy, we observe vastly different magnetic charge dynamics along two principle directions. These striking differences are found to be due to the surface-termination and associated coordination which yields different energetics and interaction strengths for magnetic charges upon the surface.


Introduction
The concept of magnetic monopole transport within a condensed matter setting has captivated scientists, allowing established theory [1] to become an experimental realization [2][3][4] within the bulk pyrochlore systems known as spin-ice [5]. In these three-dimensional (3D) systems, rare earth spins are located upon corner-sharing tetrahedra, and energy minimisation yields a local ordering principle known as the ice-rule, where two spins point into the centre of a tetrahedron and two spins point out. Representing each spin as a dimer, consisting of two equal and opposite magnetic charges (± ), is a powerful means to understand the physics of spin-ice [5]. Using this description, known as the dumbbell model [1], the ice-rule is a result of charge minimisation, yielding a net magnetic charge of zero in the tetrahedra centre ( = ∑ ! ! = 0).
Then the simplest excitation within the manifold produces a pair of magnetic charges (∑ ! ! = ±2 ) which, once created, can propagate thermally and only at an energy cost equivalent to a magnetic analogue of Coulomb's law. The energy scale for the production of monopoles upon the spin-ice lattice is controlled by the chemical potential (µ), which is governed by properties intrinsic to the material such as lattice constant and magnetic moment [6]. Canonical spin-ice materials have a chemical potential that places them in a weakly correlated regime where only a small fraction of bound monopole-antimonopole pairs are found. Recent theoretical work has studied the ordering of magnetic charges upon cleaved spin-ice surfaces, perpendicular to the [001] direction [7]. In such systems, the orphan bonds upon the surface are found to order in either a magnetic charge crystal or magnetic charge vacuum, depending upon the scales of exchange and dipolar energies [7]. Recent experimental work is now hinting at the presence of a surface-driven phase transition [8] but the transport of magnetic charge across such surfaces has not been considered previously.
The arrangement of magnetic nanowires into two-dimensional lattices has recently shown to be a powerful means to explore the physics of frustration and associated emergent physics.
These artificial spin-ice (ASI) systems [9][10][11][12][13][14][15], where each magnetic nanowire behaves as an effective Ising spin, have recently yielded an experimental realisation of the square ice model [16] and have also been used to study the thermal dynamics of monopoles in the context of Debye-Hückel theory [17]. Controlled formation of magnetic charge is an exotic means to realise advanced multistate memory devices. Such concepts have been shown in simple 2D lattices using magnetic force microscopy (MFM) [18]. The extension of artificial spin-ice into true 3D lattices that capture the exact underlying geometry of bulk systems is paradigm- The surface L1 layer (red) consists of an alternating sequence of coordination two and coordination four vertices. Below this, the L2 (blue) and L3 (green) layers can be seen. Within these layers, only vertices of coordination four are present. The L4 layer (grey) is the lower surface termination which again has an alternating sequence of coordination two and coordination four vertices. Inset: Cross-section of Ni81Fe19 (grey) upon the polymer scaffold (yellow). (b) A false colour scanning electron microscopy image of the 3D artificial spin-ice lattice. Scale bar is 20µm. (c) A false colour scanning electron microscopy image showing the L1 (red) and L2 (blue) sub-lattices, viewed at a 45° tilt with respect to the substrate plane. Scale bar is 1µm. (d) Atomic force microscopy image of the 3D artificial spin-ice system. Scale bar is 2µm. Coordinate system for field application is shown in top-right of image. (e) Possibilities for creating magnetic charge upon L1. (f) The possible states and associated magnetic charge that can be realised at vertices of coordination two and coordination four. shifting, allowing the exploration of ground state ordering and magnetic charge formation in the bulk as well as upon the surface. The production of 3DASI systems harbouring magnetic charge also allows marriage with advanced racetrack device concepts [19,20].
In this study, we use state-of-the-art 3D nanofabrication and processing in order to realise a 3DASI in a diamond-bond 3D lattice geometry, producing an artificial experimental analogue of the originally conceived dumbbell model [1]. MFM is then harnessed to image the formation and propagation of magnetic charge upon the 3D nanowire lattice. Figure 1a shows a schematic of the 3DASI, which is composed of four distinct layers, labelled by colour. The system is fabricated by using two-photon lithography [21][22][23][24] to define a polymer lattice in a diamond-bond geometry, upon which 50nm Ni81Fe19 is evaporated (See methods for further details). This yields NiFe nanowires within a diamond lattice geometry as shown previously [24]. Each nanowire has a crescent shaped cross-section (Fig 1a inset), is single domain and exhibits Ising-like behaviour [24]. The L1 layer, which is coloured red in Fig 1a, is the upper surface termination and consists of an alternating sequence of coordination two (bipods) and coordination four vertices (tetrapods). The L2 and L3 layers, coloured blue and green respectively are ice-like with only vertices of coordination four. Finally, the L4 layer (Grey) is the lower surface termination of the lattice and consists of vertices which alternate between coordination two and coordination four.

Results
The overall array size is approximately 50 µm x 50 µm x 10 µm as seen in the scanning electron microscopy (SEM) image (Fig. 1b). Analysis of SEM data indicates the long-axis of L1 wires is orientated at θ = (33.11 ± 2.94)° from the substrate plane, matching within error the angle of 35.25° which is expected for an idealised diamond-bond geometry [5].  bottom-right). Note that a field applied in either direction with a projection along the L2 sublattice produces only magnetic charges at four-way junctions ( Fig S1). Overall, the 3DASI surface can realise effective magnetic charge of ±4q, ±2q and 0 as summarised in Fig 1f.

Imaging the magnetic configuration of a 3DASI
MFM is a convenient method to deduce the magnetization configuration of the 3DASI during field-driven experiments. This imaging technique is sensitive to the second derivative of the stray field with respect to z (d 2 Hz/dz 2 ) which makes it ideal for imaging magnetic charge [25] upon the 3DASI lattice. In the present study, we focus upon the field-driven transport of magnetic charge upon the L1 and L2 layers. The volume of the individual nanowires is sufficiently high that the 3DASI system is frozen at room temperature and thus thermal energies are negligible when compared to the energy required to switch a wire.
It is insightful to first study the simplest scenarios where each sub-lattice is saturated. Optical magnetometry (see Fig S2) indicates 30 mT is well above the saturating field for each sublattice. Figure  with negative contrast seen in bottom left. Overall, the vertex configuration is consistent with a type 2 ice-rule configuration produced by the applied field protocol. We note that near the bottom left of the L2 nanowires, faint positive contrast is seen (Labelled A). A previous investigation, which took images in reversed tip configurations identified this as an artifact [24], due to the abrupt upwards change in topography experienced by the tip at this point. Since the signal originating from the artifact is approximately a factor of two smaller than the signal  Vector maps illustrating the magnetisation configuration and associated monopole excitations in five snapshots during a reversal sequence upon the L1 sub-lattice. Here a saturating field was first applied along the unit vector (1,-1,0) after which a field of 8mT was applied along the unit vector (-1,1,0). Successive images were then captured at remanence following 0.25 mT increments. Each island represents a bipod, coloured with the local in-plane magnetization, as determined by key. Each monopole excitation is assigned a unique index to track propagation between images. (f-j) Vector maps illustrating an equivalent reversal of the L2 sub-lattice. Here the samples was first saturated along the unit vector (-1,-1,0) after which a field of 6.50mT was applied along (1,1,0). Successive images were then captured at 0.25mT increments. Full datasets, including raw MFM images can be found in the supplementary information. charge upon the wire ends closest to L1-L2, smears over the vertex area, the absolute magnitude of the phase is increased, when compared to an ice-rule state. We have used all three criteria simultaneously to identify monopoles at the L1-L2 vertex. Interestingly, so long as a well defined field protocol is used, it is also possible to infer the presence of monopoles at the L2-L3 vertex. Here, so long as L3 has been saturated, we expect this sub-lattice to be uniformly magnetised. However, if the extremities of two adjacent L2 nanowires both have positive or negative contrast, a monopole is implied at the L2-L3 vertex.
In intermediate states (Fig 3a,c), the sub-lattice that extends along the field direction is effectively demagnetised (M < 0.1MS), so it is intriguing that a vast difference in the density of monopoleexcitations is seen between the two images.  direction. Though much of the array remains saturated, six wires (three bipods) have switched yielding two monopole states, each with charge -2q (Monopoles 1,2). In both cases, the monopoles are found at the intersection between L1 and L2. Further field increments yield additional chains of wires switching (Fig. 4b-d), with a further two negative monopoles (Monopoles 3, 4) residing at the L1-L2 junction, after which L1 reaches saturation within the sampled area (Fig. 4e). Additional field increments lead to the creation of further monopoles (Monopoles 12-18), whilst others move along the L2 nanowires or propagate out of the measured area (Fig 4g-4j).

Tracking monopole propagation on the lattice surface and sub-surface
The differences in monopole formation upon the L1 and L2 sub-lattices is striking. Application of an external field with component along L1 yields few uncorrelated magnetic charges ( Fig   S10a) within the measured region, which seem to only be observed within a narrow field window (8 mT -10.5 mT). We note that whilst this yields a net charge locally in the measured area, charge neutrality is expected across the full lattice. Analysis of the switching also shows a distinct absence of magnetic charges upon surface vertices with coordination two. On the contrary, the L2 switching leads to nucleation of many correlated pairs yielding almost equal numbers of positive and negative magnetic charges (Fig S10b), meaning the net charge within the measured area is close to zero throughout the field range ( Fig S10c). The magnetic charges are also formed at a lower field (6.5mT) for the L2 sub-lattice and remain for a wider field range (6.5mT -10.75mT).

Modelling the 3DASI system
Calculating the total energy density of every possible vertex state, within a micromagnetic framework (Fig S11) is an insightful exercise and provides some initial understanding of the system. Here it can be seen that the energy density to create a magnetic charge upon a coordination two, surface vertex is 3.2 times higher than that of a monopole at a coordination four vertex suggesting surface charges will be very unfavourable. To understand the significance of this within the context of switching the entire array, we carry out Monte-Carlo   1,0). Upon the threshold of switching (Fig   5b), chains of islands switch upon the L1 sub-lattice producing uncorrelated monopoles and long Dirac strings as seen in the experimental data, before the majority of the array becomes saturated (Fig 5c). Critically, charges upon surface coordination two vertices are now very rare, which is in agreement with experiment. Fig 5d-f shows MC simulations for the field aligned along unit vector (1,1,0). Here, a low field immediately produces large numbers of correlated monopole-antimonopole pairs (Fig 5e), separated by a single lattice spacing, closely aligned with the experimental data. Fig 5g summarises

Discussion
As in all ferromagnetic materials, the 3DASI studied here passes through a field-driven state whereby the component along the field is effectively demagnetised. It is interesting to identify two main ways that this can be achieved in this novel 3D nanostructured system. The first possibility is that of local demagnetization upon each vertex, whereby the production of monopole/anti-monopole pairs locally yield a net magnetization of zero upon the relevant sublattice. A second possibility is the production of stripes of alternating magnetization direction, yielding complete demagnetization upon a given sub-lattice. Here magnetic charges can only be found at the stripe ends. A key quantity which will be important in determining the means of demagnetization is that of the monopole effective chemical potential, which quantifies the extent to which monopoles remain closely correlated. This is defined as µ * = µ/u, where µ is the chemical potential of a monopole and u = µ $ Q " /4πa.  Fig S12,S13) must be cleared to produce monopoles upon the L1 sub-lattice.
A key question that remains is the magnitude of surface energetic factor (a) and why such large values are required in MC simulations (a=6.4) when compared to the magnitude implied by micro-magnetics. The surface energetics in these systems arise due to a difference in how the magnetic charge is distributed for two-way and four-way junctions [24]. In both cases this will be dictated by a balance between exchange and dipolar energies. For coordination two vertices, the reduced effective dimensionality and resulting confinement produces an unfavourably large energy for monopoles upon the vertex. In contrast, the coordination four system allows the magnetic charge to spread across the vertex area, overall reducing the energy and yielding a stable monopole configuration. It is important to note that even when a=3.2 (value indicated by MM simulations) the resulting MC simulations still bear a far closer resemblance to experiments than when enhanced surface energetics are not considered (a=1), in terms of string length, monopole density, and density of charges upon surface coordination two vertices.
However, increasing a beyond the value predicted by MM simulations yields an even closer resemblance to experiments, due to fundamental differences in the two methods. In particular, the MC simulations use a compass needle model, where the magnetic charge associated with each wire is distributed evenly across each needle, effectively reducing the energy barrier for surface charges to form. Therefore, a greater value of a is required to suppress surface charges and hence approximate the experimental observations.
In conclusion, we have demonstrated the fabrication of a 3DASI system, where the magnetic configuration upon the upper two nanowire layers can be determined. We find a striking difference in the field-driven magnetic monopole transport along two principle axes. With a field applied along the projection of surface termination, magnetic imaging shows a low number of uncorrelated monopoles during the switching, which are always found at coordination four vertices. Applying a field along the projection of L2 yields large numbers of correlated monopoles. Micromagnetic and Monte Carlo simulations, supported by simple calculations within a dipolar framework, suggest it is the difference in effective chemical potential, as well as the energy landscape experienced during surface monopole dynamics, which accounts for the measured differences. We anticipate that our study will inspire a new generation in artificial spin-ice study whereby the ground state in these 3DASI systems are explored as a function of key parameters such as magnetic moment and lattice spacing.
Ultimately, this may also yield the realisation of monopole crystals as predicted in bulk spinice [26] or bespoke spin-ice ground states only possible in artificial systems of novel 3D geometry. By utilizing a full suite of magnetic imaging techniques including MFM, nanoscale ballistic sensing [27] and novel synchrotron-based methods [28], it is hoped that full 3D characterization of the bulk and surface will soon be possible.

Fabrication
Diamond-bond lattice structures were fabricated upon glass coverslips via two-photon lithography (TPL). Substrates were first cleaned in acetone, followed by isopropyl alcohol (IPA), and dried with a compressed air. Next, droplets of Immersol 518 F immersion oil and IPL-780 photoresist are applied to the lower and upper substrate surfaces respectively. Using a Nanoscribe Photonic Professional GT system, a polymer scaffold in the diamond-bond lattice geometry was defined within the negative tone photoresist, of dimensions 50 µm × 50 µm × 10 µm. Samples were developed in propylene glycol monomethyl ether acetate for 20 minutes, then 2 minutes in IPA, to remove any unexposed photoresist. Once again, the samples were dried with a compressed air gun.
Using a thermal evaporator, a uniform 50 nm film of Permalloy (Ni81Fe19) was deposited on the samples from above, yielding a magnetic nanowire lattice upon the polymer scaffold. This deposition requires a 0.06 g ribbon of Ni81Fe19, washed in IPA, and evaporated in an alumina coated molybdenum boat. A base pressure of 10 -6 mBar is achieved prior to evaporation, the deposition rate is 0.2 nm/s, as measured by a crystal quartz monitor.

Scanning electron microscopy
Imaging was performed using a Hitachi SU8230 SEM with an accelerating voltage of 10kV.
Images were taken from top view as well as at a 45° tilt with respect to the substrate plane.

Magnetic force microscopy
MFM measurements were performed in tapping mode using a Bruker Dimension 3100 Atomic Force Microscope. Commercial low moment MFM tips were magnetised along the tip axis with a 0.5 T permanent magnet. Once mounted, uniform magnetic fields could be applied parallel to each sub-lattice using a bespoke quadrupole electromagnet, which was fixed upon the surface of the AFM stage. During the application of a field, the MFM tip was positioned several mm above the scanning height, such that the tip magnetisation was not influenced.
MFM data was taken at a lift height of 100 nm. Prior to capturing MFM images, feedback settings were carefully optimised to ensure sample topography was being accurately measured on the three uppermost lattice layers (L1, L2, L3).
In order to probe the transport of magnetic charge upon the 3DASI surface, the system was placed into a well-defined state by saturating the array along a principal direction (Hsat obtained via optical magnetometry, see Fig. S2). MFM images were then obtained after successive field increments in the reverse direction. MFM measures the stray field gradient d 2 Hz/dz 2 due to magnetic charges and hence is an ideal methodology to visualise such transport across the surface [25].

Finite element simulations
Micro-magnetic simulations of bipod and tetrapod structures were carried out with NMAG 28 , using finite element method discretisation. Geometries possessing wires with a crescent-shaped cross-section were designed such that the arcs subtend a 160° angle. The inner arc is defined from a circle with 80 nm radius corresponding to the 160 nm lateral feature size of the TPL system. The outer arc is based on an ellipse with an 80 nm minor radius and 130 nm major radius, yielding a thickness gradient with a peak of 50 nm. The length of all wires is set to 780 nm, due to computational restraints. All geometries were meshed using adaptive mesh spacing Coulomb's law with corrections for experimental considerations: The resulting arrow maps were plotted in Fig. 5a-f and Fig. S13. Additionally, the number of excited states was recorded from these final arrow maps and plotted as a function of field in Fig. 5g.

Dipolar approximation calculations
One can define an effective chemical potential * = / , where = $ " /4 . Here is calculated within the dipolar approximation. Magnetic moments are located upon a diamondbond lattice. The energy of interaction between moments can then be approximated as: For our experimentally confirmed = 6.4, this implies a transition state costing !%& * = 4.331.

Magneto-optical Kerr Effect Magnetometry
A 0.5 mW, 637 nm wavelength laser was expanded to a diameter of 1 cm, passed through a Glan-Taylor polarizer to obtain an s-polarized beam, then focused onto the sample using an achromatic doublet (f = 10 cm), to obtain a spot size of approximately 10 μm 2 . During the source-to-sample path the laser is attenuated, approximately reducing the power by a factor of 4. The reflected beam was also collected using an achromatic doublet (f = 10 cm) and passed through a second Glan-Taylor polarizer, from which the transmitted signal was directed onto an amplified Si photodetector, yielding the Kerr signal. After magneto-optical Kerr effect data was captured from the nanowire lattice, a second dataset was obtained from the substrate film.
The film data is scaled to the lattice data and subtracted off. This corrected for any Kerr signal contributions originating from the film, during the lattice measurements.