Direct Visualization of the Magnetic Monopole Field in a 3D Artificial Spin Ice

TL;DR

This study directly visualizes the three-dimensional stray magnetic fields of a diamond-bond 3D artificial spin ice (3DASI) using scanning NV magnetometry, revealing antivortex textures above ice-rule vertices and highly divergent monopole fields upon pair creation. By benchmarking with micromagnetic simulations and performing a multipole expansion up to $\ell_{\max}=6$, the authors show that monopoles in 3DASI are micromagnetic objects carrying both magnetic charge and an intrinsic moment, leading to anisotropic, geometry-dependent interactions that can be tuned by lattice topology. The work demonstrates that 3DASI is a programmable magnetic metamaterial, where nanoscale geometry controls monopole energetics, propagation, and collective behavior, enabling controlled studies of emergent magnetic charges in three dimensions.

Abstract

Magnetic monopoles, long hypothesised as fundamental particles carrying isolated magnetic charge, emerge in spin-ice systems as fractionalised excitations governed by the ice rule. Yet their three-dimensional field structure has never been directly visualised. Here, we use two-photon lithography and processing to fabricate a fully three-dimensional artificial spin-ice lattice with diamond-bond geometry. We then use scanning nitrogen-vacancy magnetometry to directly measure the stray magnetic fields of both charge-neutral and monopole vertices. We find that ice-rule vertices produce antivortex textures directly above their vertices, stabilised by the local frustrated two-in/two out ordering principle. Direct imaging of the monopole stray field shows a highly divergent profile. By correlating experiment with micromagnetic simulations and performing a multipole expansion of the reconstructed magnetisation, we reveal that monopoles in 3DASI are non-trivial micromagnetic entities, carrying both magnetic charge and an intrinsic moment, giving rise to anisotropic interactions that are dependent upon the quasiparticles position on the lattice. Results suggest that as monopoles separate under an applied field, the dipolar contribution to their interaction reorients relative to the underlying Coulombic field, revealing that monopole coupling is tunable through geometry, being set by the local vertex topology. These findings establish 3DASI as a programmable magnetic metamaterial in which nanoscale geometry governs the energetics and dynamics of emergent magnetic charges.

Figures (4)

• Figure 1: A 3D Artificial Spin-ice (a) A diamond probe with a single nitrogen vacancy is scanned across a 3D artificial spin-ice sample and fluorescence is measured with a small antenna. The process measures the component of the sample field along the NV axis indicated by a red arrow. Inset: Schematic of a diamond-bond 3D Artificial Spin-ice lattice with sublattices coloured red (L1), green (L2), blue (L3) and magenta (L4). (b) Scanning electron microscopy image of representative experimental lattice measured at a 45 degree tilt. Scale bar indicates 20 µm. (c) High-magnification top-down SEM of the 3D artificial spin-ice lattice with sublattices marked according to the colours in panel b. Scale bar indicates 1 µm. (d) Schematic showing different vertex types and corresponding magnetic charge, black arrows indicate each wire’s magnetisation, green arrows indicate the net magnetic moment of the vertex.
• Figure 2: NV magnetometry imaging of a 3D artificial spin ice lattice in a polarised state (a) NV magnetometry image measured in contact mode (left) with measured topography (right). (b) NV magnetometry image measured at 200 nm lift height (left) with measured topography (right). Key features observed in the measured contrast are superimposed on corresponding topography to identify their location. (c) Simulated NV magnetometry contrast in contact mode and (d) at a 200 nm lift height. Key features in the simulation are superimposed on corresponding topography measurements for direct comparison to experiment. Dotted lines in panels a-d indicate the locations of the sublattices to guide the eye, coloured according to the schematic in figure 1a, inset. (e) Tomographic representation of the simulated sample field over the L1 coordination two vertex. Cyan panels show cross-sections of the field in the x-z plane and magenta panels show cross-sections of the field in the y-z plane. In both cases, a circulating texture is present directly above the apex. (f) Tomographic representation of the simulated sample field above the L1/L2 vertex. Cyan panels show cross-sections of the field in the x-z plane and magenta panels show cross-sections of the field in the y-z plane. An antivortex can be seen at the central part of the vertex and arises due to the ice-rule configuration in surrounding nanowires.
• Figure 3: NV magnetometry of magnetic monopoles in 3DASI (a) NV magnetometry measurement in contact mode (left) with measured topography (right). A pair of monopoles with bright contrast can be clear seen in the centre of the image. (b) NV magnetometry measurement at 50 nm lift height (left) with measured topography (right). Key features superimposed upon corresponding topography measurements shown in right panels. (c) Simulated NV magnetometry measurements in contact mode (d) Simulated NV magnetometry measurements at a 200 nm lift height with contours of key features superimposed upon corresponding simulated topography measurements (right). Dotted lines in panels a-d indicate the locations of the sublattices to guide the eye, coloured according to the schematic in the inset of figure \ref{['Fig1']}a.
• Figure 4: Monopole interactions in 3DASI. (a–d) Multipole expansion of a Type IIIa–Type IIIb monopole pair. (a) Schematic showing monopole positions on the lattice; insets show reconstructed magnetization at a radius of $180 \pm 100$ nm for vertices with $\mathcal{Q} = \pm 2q$. Colour indicates the $z$-component of magnetization, while vectors show the reconstructed in-plane components; vector contrast reflects cosine similarity between reconstructed and raw magnetization. (b) Amplitude of each multipole order with inset showing charge alignment per order. (c) Nematic order $S$ of magnetization for charged (cyan, magenta) and uncharged (black) vertices as a function of radius, where $S=1$ denotes full anisotropy. (d) Low-order multipole moments of the magnetic charges, with real-space orientations indicated. The inter-charge distance is normalised to a unit vector (black line); surface opacity represents the amplitude of multipole order $\ell$. (e–h) Equivalent analysis for a Type IIIa–Type IIIa monopole pair.