Magnetic-Field Control of Emergent Order in a 3D Dipolar Pyramid Artificial Spin Ice

Anja Weber; Flavien Museur; Gavin M. Macauley; Laura J. Heyderman; Luca Berchialla

REVIEW 2 major objections 5 minor 35 references

Magnetic-Field Control of Emergent Order in a 3D Dipolar Pyramid Artificial Spin Ice

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper reports the first 3D artificial spin ice with separated, purely dipolarly coupled nanomagnets, patterned on square pyramids, and shows that a strong out-of-plane field makes the net vertex moments form an emergent coarse-grained

desk verdict Genuinely new separated-dipolar 3D ASI platform with an emergent square-ice phase; main risk is uncalibrated MFM readout of the tilted magnets. read the letter →

arxiv 2509.01534 v1 pith:25HSEWDF submitted 2025-09-01 cond-mat.mes-hall

Luca Berchialla , Gavin M. Macauley , Flavien Museur , Anja Weber , Laura J. Heyderman This is my paper

classification cond-mat.mes-hall

keywords artificialspinicethree-dimensionalnanomagnetismdipolarcouplingsquarepyramidlatticeemergentcoarse-grainingmagneticforcemicroscopyMonteCarlosimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reports the first artificial spin ice built from physically separated nanomagnets in a genuinely three-dimensional arrangement: the magnets sit on the faces of square pyramids and on the flat substrate between them, so their interactions are purely dipolar rather than exchange-mediated. Using Monte Carlo simulations the authors map out a phase diagram controlled by the pyramid face angle and an out-of-plane magnetic field, which selectively acts on the tilted pyramid magnets and leaves the in-plane magnets untouched. At high out-of-plane field, the net moments of the mixed vertices order into head-to-tail loops with twice the lattice periodicity — an emergent, coarse-grained square ice of vertex-level effective spins. Experiments with tailored demagnetization protocols reach this emergent phase and, at zero field, large patches of the conventional square-ice ground state, establishing a platform for studying emergent order in three-dimensional frustrated magnetism.

What carries the argument

The central object is the Pyramid ASI lattice: square-based pyramids whose sloped faces carry 'tilted' nanomagnets and whose intervening flat areas carry 'in-plane' nanomagnets, with all magnet ends spaced equally from each vertex. This geometry lets an out-of-plane magnetic field switch only the tilted nanomagnets, and it makes the net magnetic moment of a mixed vertex act as a new, larger-scale Ising spin. The argument is carried by a point-dipolar Monte Carlo Hamiltonian: each nanomagnet is a fixed-length macrospin, and the phase diagram, vertex energetics, and coarse-grained ordering are all computed from this Hamiltonian and compared with magnetic force microscopy images of demagnetized

What would settle it

Compare the MFM-determined macrospin orientations in a demagnetized Pyramid ASI, vertex by vertex, with an independent vector magnetization measurement (e.g., X-ray magnetic vector tomography) of the same sample; a misassignment rate above a few percent would change the vertex populations enough to shift the claimed match with simulation.

Watch

Extended reading notes

Core claim

The paper claims to realize a 3D artificial spin ice composed of disconnected, single-domain nanomagnets whose only mutual interaction is the magnetic dipole interaction. The design places half the nanomagnets on the sloped faces of square pyramids (tilted) and half on the planar spaces between pyramids (in-plane), so the lattice looks square from above but is non-planar in three dimensions. Because an out-of-plane field couples only to the tilted magnets, the system supports field-tunable states. Monte Carlo simulations yield a five-regime phase diagram as the pyramid face angle θP and out-of-plane field HZ vary; for strong HZ all tilted magnets align, and the net moments of the mixed verti

Load-bearing premise

The vertex readout, and therefore every population and phase shown, assumes that MFM contrast gives the true magnetization direction of each nanomagnet, especially the tilted ones, despite the slight pyramid-face curvature and thickness variations that the paper notes in the Supplemental Material and sets aside as negligible without a quantitative calibration.

Editorial extensions

If this is right

The Pyramid ASI becomes a testbed for purely dipolar three-dimensional frustration, where every interaction channel is magnetostatic and can be tuned locally.
Out-of-plane fields provide a control knob unavailable in 2D artificial spin ice: they selectively address the tilted population and switch the system between distinct ordered phases.
The emergent square ice realizes coarse-graining in a magnetic metamaterial, with vertex-level net moments playing the role of macrospins at twice the lattice spacing.
Because the pyramid angle and base shape can be varied within a single sample, different phases and types of geometrical frustration can be patterned on one chip.
This opens routes to magnetic charge ordering and charge propagation along the third dimension, as well as 3D magnonic and logic devices built from dipolarly coupled nanomagnets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same selective-switching mechanism could be used on any lattice where one sublattice is tilted out of plane; the coarse-grained square ice may then be a generic consequence of pinning one sublattice, not a special feature of pyramids.
The authors note the best demagnetization angle (45°) deviates from the geometrically expected ~30°, which suggests that the states reached are controlled by dynamic coercivity and nucleation rather than static energy minima; this could be tested with micromagnetic simulations of the demagnetization protocol.
The near-50° crossover in mixed-vertex energy hierarchy implies that a sample with pyramid angles straddling 50° would contain coexisting phases, letting one study interfaces between different artificial spin ice orders on a single substrate.
A quantitative MFM calibration against a bulk-sensitive or vector magnetometry technique would test whether the residual curvature and thickness variation truly are negligible, and would strengthen the vertex-population measurements.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper delivers what it claims: the first 3D artificial spin ice with physically separated nanomagnets interacting purely by dipolar coupling. The pyramid geometry is new, the fabrication looks careful, and the Monte Carlo phase diagram is built from a standard point-dipole Hamiltonian without fitting to the experimental data. The emergent coarse-grained square ice at high out-of-plane field is a genuinely interesting idea, and the simulations support it independently of the experiments. The demagnetization protocols are described in enough detail to be reproduced. That is real progress. The soft spots are mostly around the experimental readout. The stress-test note is right that the MFM-to-macrospin mapping for tilted nanomagnets is asserted rather than calibrated. The main text says the apex-end contrast is the same for all tilted magnets, but the supplemental admits curvature of the pyramid faces and thickness variations, then dismisses them as negligible without quantitative support. Since vertex populations and the experimental demonstration of the emergent square ice depend on that mapping, this is a load-bearing assumption. It is not fatal - the platform itself does not depend on the exact vertex classification, and the simulations carry much of the weight - but a careful referee should ask for a sensitivity analysis or an independent check of the contrast-to-orientation relationship. Without that, the experimental match to the simulated phase diagram is weaker than it appears. The other concerns are minor. The effective temperature is fitted, not predicted, which is fine but should be labeled as a consistency check. The 37% ground-state population is low, but the paper is honest about the difficulty and the patches of ordered ground state are visible. The surprising optimal field angle is noted but not explained quantitatively; that is a minor gap. Data are not yet open but are promised to referees on request, which is standard. Overall, this is a solid experimental and simulation study. The new platform is significant if the readout concern can be addressed. I would send this to peer review rather than desk reject it. The referee should push for a quantitative calibration of the MFM readout or at least an error analysis that bounds the misclassification rate. If that comes back clean, this will be a well-cited paper. I would also bring it to a reading group, because the emergent square ice and the pyramid geometry are worth discussing.

Referee Report

2 major / 5 minor

Summary. The manuscript introduces a three-dimensional artificial spin ice ('Pyramid ASI') in which Permalloy nanomagnets are patterned on square-pyramid faces and on the flat substrate between pyramids. The nanomagnets are physically separated and interact via dipolar fields. Monte Carlo simulations based on a point-dipole Hamiltonian map out a phase diagram as a function of pyramid angle θP and out-of-plane field HZ, including a high-field phase in which the tilted spins are fully polarized and mixed vertices host Type III configurations whose net moments form an emergent square ice. Experimentally, the authors use bipolar and fixed-polarity rotating-field demagnetization protocols and MFM imaging to extract vertex populations. They report partial access to the zero-field ground state (maximum Type I population 37%) and observe structures attributed to the emergent square ice after a fixed-polarity protocol, estimating an effective temperature slightly above the ordering transition.

Significance. If the central claims hold, this is a notable advance: it would be the first 3D ASI with separated, purely dipolar nanomagnets and the first experimental realization of vertex-level coarse-graining. The simulation approach is standard and transparent, and the fabrication and demagnetization protocols are described in unusual detail. The MFM readout idea, exploiting the continuous pyramid topography to avoid valley artifacts, is clever and potentially enabling. However, the load-bearing experimental steps—readout calibration and quantitative demonstration of the emergent order—are not yet supported to the standard required for the paper's strongest claims.

major comments (2)

[Section II and Supplemental S5] The MFM readout for tilted nanomagnets is the foundation of all experimental vertex populations. The method uses the average signal in boxes near the pyramid apex, justified by the statement that 'the strength of the magnetic signal from all nanomagnets at the apex being the same.' Yet Supplemental S5 admits 'a slight curvature of the pyramid faces, which leads to minor variations in the thickness of the deposited nanomagnets across each face,' and asserts a 'negligible impact' without quantitative calibration or error analysis. Since MFM contrast depends on tip-sample distance and on the magnetization direction relative to the tip, a systematic variation across faces or along the tilted magnets could bias the classification of mixed and pyramid vertices and hence every population reported in Figs. 3 and 4. The authors should provide a calibration linking measured contrast to macrospin d
[Section IV, Fig. 4] The claim of 'the first experimental realization of coarse-graining in ASI' rests on Fig. 4(b), which shows 'small domains' and 'short head-to-tail loops,' and on the effective-temperature fit in Fig. 4(c). This demonstrates that some mixed vertices are in Type III configurations, but it does not quantitatively establish that the vertex-level degrees of freedom order as an emergent square ice. The authors should report the fraction of mixed vertices in Type III in the experimental images, compare spatial correlation functions or loop-length/domain-size distributions with the Monte Carlo results, and provide a goodness-of-fit for the effective-temperature estimate. Without such quantitative measures, the emergent square ice remains a suggestive but not fully supported interpretation.

minor comments (5)

[Supplemental S1] The statement that simulations were performed on a 10×10 lattice 'with no boundary conditions' is ambiguous. Please clarify whether these are free (open) boundaries and discuss possible finite-size effects on the phase boundaries, especially near θP ≈ 50°.
[Section IV, Fig. 3(a)] The maximum ground-state Type I population is 37%. Since the random expectation for Type I vertices is approximately 12.5% for a four-vertex system, the observed ordering is significant, but the text should state this comparison or another baseline to make the 'relatively large patches' claim quantitative.
[Section IV, Fig. 4(c)] The effective-temperature estimate is obtained by matching coarse-grained vertex populations, which are not fully independent (the four populations sum to one). Please report the uncertainty in the fitted temperature and the number of independent data points used.
[Section II and Fig. 1(d)] The terms 'Type I(coarse-grained)' and 'Type IIIa/IIIb' are defined only in the Supplemental Material. A brief definition or visual example in the main text would improve readability.
[Introduction and Conclusions] The claims 'first realization of a 3D ASI with physically-separated nanomagnets' and 'first experimental realization of coarse-graining in ASI' should be carefully worded with explicit comparison to the closest prior work, including multilayered planar ASIs, to avoid overclaiming if any prior separated-nanomagnet 3D arrangement exists.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: MC phase diagram and emergent square ice are derived from an independent dipolar Hamiltonian, not from fit or self-citation.

full rationale

Walked the claimed derivation chain. The phase diagram and emergent square ice are obtained from point-dipolar Monte Carlo simulations (Supplemental Material S1) whose Hamiltonian contains no experimental fitted values; experimental MFM configurations are then compared with these independently simulated states. The only fitted quantity is the effective temperature in Sec. IV, Fig. 4(c): 'We estimate the effective temperature of the experimentally obtained states by finding the best match between the measured coarse-grained vertex populations to those in the simulations.' That is a post hoc comparison/parameter estimate, not a prediction, and no central claim is derived from it. The coarse-grained square-ice classification is definitional (effective spins are the net moments of Type III mixed vertices, with non-Type-III vertices contributing no spin), but the ordering of that emergent lattice is tested through the simulated heat capacity and measured vertex populations, so it is not circular. The only self-citation is Ref. [7], a review authored by the present authors, used in the introduction as a general pointer to 3D ASI concepts; it is not load-bearing. Supplemental Material S5 acknowledges 'a slight curvature of the pyramid faces... minor variations in the thickness of the deposited nanomagnets across each face' and asserts these have negligible impact; this is an experimental readout-robustness limitation, not a circularity, because a readout error would degrade the experiment-simulation agreement but would not make the simulation-derived predictions equal to their inputs. Verdict: no significant circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central platform claim rests on standard physical idealizations (point dipoles, single-domain macrospins) and on the reliability of MFM readout. The only explicitly fitted number is the effective temperature of the experimental states. The emergent square ice is a conceptual layer added on top of the simulated and measured vertex configurations, with its own falsifiable signatures.

free parameters (1)

Effective temperature of experimental states
The effective temperature is estimated by fitting experimental coarse-grained vertex populations to simulated curves (Fig. 4(c)). The value is not given numerically, only described as 'slightly above the ordering transition.' This is a post-hoc fit used for comparison, not a predictive constraint.

assumptions (5)

domain assumption Point-dipole approximation for inter-nanomagnet interactions
The Hamiltonian in Supplemental Material S1 treats each nanomagnet as a point dipole at its center, neglecting finite-size and multipole effects. This is standard in ASI simulations but is an idealization that could affect the finer energy balances in the phase diagram.
domain assumption Each nanomagnet behaves as a single-domain, Ising-like macrospin
Assumed from the stadium shape (450x120 nm) and shape anisotropy; the paper states 'the elongated shape of the nanomagnets ensures robust Ising-like behavior' (S5). No direct measurement of single-domain behavior in the pyramid sample is provided.
domain assumption MFM contrast maps monotonically to macrospin orientation
The vertex extraction procedure (Section II) relies on averaging MFM signals in boxes to assign spin directions. This assumes a reliable, uniform correspondence between magnetic contrast and magnetization direction, despite acknowledged curvature and thickness non-uniformity (S5).
domain assumption The demagnetization protocol drives the system toward low-energy equilibrium states
The rotating oscillating field is used as a proxy for thermal annealing. The paper acknowledges it only partially succeeds (max Type I population 37%), and the optimal angle deviates from expectation, indicating this assumption is only weakly fulfilled.
domain assumption The pyramid geometry is accurately described by the face angle θP with flat nanomagnet deposition
The phase diagram varies θP, but the experimental sample has θP ~ 45° with slight face curvature (S5). The paper asserts negligible impact, but this is not quantitatively verified against the simulation geometry.

invented entities (1)

Emergent square ice of vertex-level effective spins independent evidence
purpose: Describes the coarse-grained degrees of freedom at mixed vertices in the high-field phase, where net moments form head-to-tail loops.
The emergent square ice is a coarse-grained description, not a new physical object. It has independent evidence in the form of a simulated heat-capacity peak and vertex population behavior that is compared to experiment (Fig. 4(c)), so it is not pulled from a hat. However, it is a re-description of the vertex configurations, not a new entity with separate physical consequences beyond those already in the model.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Magnetic-Field Control of Emergent Order in a 3D Dipolar Pyramid Artificial Spin Ice." pith.science (2026). https://pith.science/paper/25HSEWDF

@misc{pith2026250901534,
  author       = {Pith},
  title        = {Pith review of: Magnetic-Field Control of Emergent Order in a 3D Dipolar Pyramid Artificial Spin Ice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25HSEWDF}},
  note         = {Machine review of arXiv:2509.01534}
}

read the original abstract

We realize a three-dimensional artificial spin ice of disconnected nanomagnets interacting solely via dipolar coupling, patterned on square pyramids. This Pyramid artificial spin ice, with both tilted and in-plane nanomagnets, supports tunable states. Monte Carlo simulations reveal a rich phase diagram and an emergent square ice of vertex-level effective spins. Tailored demagnetization protocols and magnetic force microscopy allow experimental access to low-energy states, establishing a platform for exploring three-dimensional artificial spin ices.

Figures

Figures reproduced from arXiv: 2509.01534 by the authors.

**Figure 2.** FIG 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG 3. Effectiveness of the demagnetization protocol [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (c), the experimental data are consistent with a temperature slightly above the ordering transition, with a partially ordered configuration having small domains of Type I(coarse-grained) vertices and short head-to-tail loops of effective macrospins, as seen in the expe…

**Figure 5.** Figure 5: FIG 5. (a) Isotropic etching of a silicon substrate [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG 6. (a) Schematic of the demagnetization setup. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

[1]

Fernández-Pacheco, R

A. Fernández-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fischer, and R. P. Cowburn, Three-dimensional nanomagnetism, Nat. Commun. 8, 15756 (2017)

work page 2017
[2]

Gubbiotti et al., 2025 roadmap on 3D nanomagnetism, J

G. Gubbiotti et al., 2025 roadmap on 3D nanomagnetism, J. Phys.: Condens. Matter 37, 143502 (2025)

work page 2025
[3]

Donnelly, M

C. Donnelly, M. Guizar-Sicairos, V. Scagnoli, S. Gliga, M. Holler, J. Raabe, and L. J. Heyderman, Three-dimensional magnetization structures revealed with X-ray vector nanotomography, Nature 547, 328 (2017)

work page 2017
[4]

Sanz-Hernández et al., Artificial Double- Helix for Geometrical Control of Magnetic Chirality, ACS Nano 14, 8084 (2020)

D. Sanz-Hernández et al., Artificial Double- Helix for Geometrical Control of Magnetic Chirality, ACS Nano 14, 8084 (2020)

work page 2020
[5]

Hierro-Rodriguez, C

A. Hierro-Rodriguez, C. Quirós, A. Sorrentino, L. M. Alvarez-Prado, J. I. Martín, J. M. Alameda, S. McVitie, E. Pereiro, M. Vélez, and S. Ferrer, Revealing 3D magnetization of thin films with soft X-ray tomography: magnetic singularities and topological charges, Nat. Commun. 11, 6382 (2020)

work page 2020
[6]

Kent et al., Creation and observation of Hopfions in magnetic multilayer systems, Nat

N. Kent et al., Creation and observation of Hopfions in magnetic multilayer systems, Nat. Commun. 12, 1562 (2021)

work page 2021
[7]

Berchialla, G

L. Berchialla, G. M. Macauley, and L. J. Heyderman, Focus on three-dimensional artificial spin ice, Appl. Phys. Lett. 125, 220501 (2024)

work page 2024
[8]

Farhan, M

A. Farhan, M. Saccone, C. F. Petersen, S. Dhuey, K. Hofhuis, R. Mansell, R. V. Chopdekar, A. Scholl, T. Lippert, and S. van Dijken, Geometrical Frustration and Planar Triangular Antiferromagnetism in Quasi- Three-Dimensional Artificial Spin Architecture, Phys. Rev. Lett. 125, 267203 (2020)

work page 2020

Show all 35 references

[9]

Möller and R

G. Möller and R. Moessner, Artificial Square Ice and Related Dipolar Nanoarrays, Phys. Rev. Lett. 96, 237202 (2006)

2006
[10]

L. A. S. Mól, W. A. Moura-Melo, and A. R. Pereira, Conditions for free magnetic monopoles in nanoscale square arrays of dipolar spin ice, Phys. Rev. B 82, 5 (2010)

2010
[11]

Chern, C

G.-W. Chern, C. Reichhardt, and C. Nisoli, Realizing three-dimensional artificial spin ice by stacking planar nano-arrays, Appl. Phys. Lett. 104, 013101 (2014)

2014
[12]

Perrin, B

Y. Perrin, B. Canals, and N. Rougemaille, Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice, Nature 540, 7633 (2016)

2016
[13]

Farhan et al., Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice, Sci

A. Farhan et al., Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice, Sci. Adv. 5, 2 (2019)

2019
[14]

de Rojas, D

J. de Rojas, D. Atkinson, and A. O. Adeyeye, Tuning magnon spectra via interlayer coupling in pseudo-3D nanostructured artificial spin ice arrays, Appl. Phys. Lett. 123, 232407 (2023)

2023
[15]

de Rojas, D

J. de Rojas, D. Atkinson, and A. O. Adeyeye, Tailoring magnon modes by extending square, kagome, and trigonal spin ice lattices vertically via interlayer coupling of trilayer nanomagnets, J. Phys.: Condens. Matter 36, 415805 (2024)

2024
[16]

Dion et al., Ultrastrong magnon-magnon coupling and chiral spin-texture control in a dipolar 3D multilayered artificial spin-vortex ice, Nat

T. Dion et al., Ultrastrong magnon-magnon coupling and chiral spin-texture control in a dipolar 3D multilayered artificial spin-vortex ice, Nat. Commun. 15, 4077 (2024)

2024
[17]

Donnelly et al., Element-Specific X-Ray Phase Tomography of 3D Structures at the Nanoscale, Phys

C. Donnelly et al., Element-Specific X-Ray Phase Tomography of 3D Structures at the Nanoscale, Phys. Rev. Lett. 114, 115501 (2015)

2015
[18]

Williams et al., Two-photon lithography for 3D magnetic nanostructure fabrication, Nano Res

G. Williams et al., Two-photon lithography for 3D magnetic nanostructure fabrication, Nano Res. 11, 845 (2018)

2018
[19]

A. May, M. Hunt, A. Van Den Berg, A. Hejazi, and S. Ladak, Realisation of a frustrated 3D magnetic nanowire lattice, Commun. Phys. 2, 13 (2019). *Contact author: l.berchialla@gmail.com *Contact author: laura.heyderman@psi.ch

2019
[20]

A. May, M. Saccone, A. van den Berg, J. Askey, M. Hunt, and S. Ladak, Magnetic charge propagation upon a 3D artificial spin- ice, Nat. Commun. 12, 3217 (2021)

2021
[21]

Pip et al., X-ray imaging of the magnetic configuration of a three-dimensional artificial spin ice building block, APL Materials 10, 101101 (2022)

P. Pip et al., X-ray imaging of the magnetic configuration of a three-dimensional artificial spin ice building block, APL Materials 10, 101101 (2022)

2022
[22]

Saccone, A

M. Saccone, A. Van den Berg, E. Harding, S. Singh, S. R. Giblin, F. Flicker, and S. Ladak, Exploring the phase diagram of 3D artificial spin-ice, Commun. Phys. 6, 217 (2023)

2023
[23]

Harding et al., Imaging the magnetic nanowire cross section and magnetic ordering within a suspended 3D artificial spin-ice, APL Materials 12, 021116 (2024)

E. Harding et al., Imaging the magnetic nanowire cross section and magnetic ordering within a suspended 3D artificial spin-ice, APL Materials 12, 021116 (2024)

2024
[24]

Zhang, I.-A

X. Zhang, I.-A. Chioar, G. Fitez, A. Hurben, M. Saccone, N. S. Bingham, J. Ramberger, C. Leighton, C. Nisoli, and P. Schiffer, Artificial Magnetic Tripod Ice, Phys. Rev. Lett. 131, 126701 (2023)

2023
[25]

Ladak, D

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, Direct observation of magnetic monopole defects in an artificial spin-ice system, Nat. Phys. 6, 359 (2010)

2010
[26]

Askey, M

J. Askey, M. O. Hunt, L. Payne, A. van den Berg, I. Pitsios, A. Hejazi, W. Langbein, and S. Ladak, Direct visualization of domain wall pinning in sub-100 nm 3D magnetic nanowires with cross-sectional curvature, Nanoscale 16, 17793 (2024)

2024
[27]

R. F. Wang et al., Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands, Nature 439, 303 (2006)

2006
[28]

See Supplemental Material at [URL will be inserted by publisher] for additional information and figures., (n.d.)
[29]

E. O. Wollan and W. C. Koehler, Neutron Diffraction Study of the Magnetic Properties of the Series of Perovskite-Type Compounds [(1−x)La, xCa]MnO 3, Phys. Rev. 100, 545 (1955)

1955
[30]

Olsen, Antiferromagnetism in two- dimensional materials: progress and computational challenges, 2D Mater

T. Olsen, Antiferromagnetism in two- dimensional materials: progress and computational challenges, 2D Mater. 11, 033005 (2024)

2024
[31]

F. Ott, T. Maurer, G. Chaboussant, Y. Soumare, J.-Y. Piquemal, and G. Viau, Effects of the shape of elongated magnetic particles on the coercive field, J. Appl. Phys. 105, 013915 (2009)

2009
[32]

Zhang, Y

X. Zhang, Y. Lao, J. Sklenar, N. S. Bingham, J. T. Batley, J. D. Watts, C. Nisoli, C. Leighton, and P. Schiffer, Understanding thermal annealing of artificial spin ice, APL Materials 7, 111112 (2019)

2019
[33]

Cheenikundil and R

R. Cheenikundil and R. Hertel, Switchable magnetic frustration in buckyball nanoarchitectures, Appl. Phys. Lett. 118, 212403 (2021)

2021
[34]

A. Imre, G. Csaba, L. Ji, A. Orlov, G. H. Bernstein, and W. Porod, Majority Logic Gate for Magnetic Quantum-Dot Cellular Automata, Science 311, 205 (2006)

2006
[35]

Alatteili, A

G. Alatteili, A. Roxburgh, and E. Iacocca, Ferromagnetic resonance in three-dimensional tilted-square artificial spin ices, Phys. Rev. B 110, 144406 (2024). *Contact author: l.berchialla@gmail.com *Contact author: laura.heyderman@psi.ch END MATTER A. Sample Fabrication Pyramid...

2024

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Fernández-Pacheco, R

A. Fernández-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fischer, and R. P. Cowburn, Three-dimensional nanomagnetism, Nat. Commun. 8, 15756 (2017)

work page 2017

[2] [2]

Gubbiotti et al., 2025 roadmap on 3D nanomagnetism, J

G. Gubbiotti et al., 2025 roadmap on 3D nanomagnetism, J. Phys.: Condens. Matter 37, 143502 (2025)

work page 2025

[3] [3]

Donnelly, M

C. Donnelly, M. Guizar-Sicairos, V. Scagnoli, S. Gliga, M. Holler, J. Raabe, and L. J. Heyderman, Three-dimensional magnetization structures revealed with X-ray vector nanotomography, Nature 547, 328 (2017)

work page 2017

[4] [4]

Sanz-Hernández et al., Artificial Double- Helix for Geometrical Control of Magnetic Chirality, ACS Nano 14, 8084 (2020)

D. Sanz-Hernández et al., Artificial Double- Helix for Geometrical Control of Magnetic Chirality, ACS Nano 14, 8084 (2020)

work page 2020

[5] [5]

Hierro-Rodriguez, C

A. Hierro-Rodriguez, C. Quirós, A. Sorrentino, L. M. Alvarez-Prado, J. I. Martín, J. M. Alameda, S. McVitie, E. Pereiro, M. Vélez, and S. Ferrer, Revealing 3D magnetization of thin films with soft X-ray tomography: magnetic singularities and topological charges, Nat. Commun. 11, 6382 (2020)

work page 2020

[6] [6]

Kent et al., Creation and observation of Hopfions in magnetic multilayer systems, Nat

N. Kent et al., Creation and observation of Hopfions in magnetic multilayer systems, Nat. Commun. 12, 1562 (2021)

work page 2021

[7] [7]

Berchialla, G

L. Berchialla, G. M. Macauley, and L. J. Heyderman, Focus on three-dimensional artificial spin ice, Appl. Phys. Lett. 125, 220501 (2024)

work page 2024

[8] [8]

Farhan, M

A. Farhan, M. Saccone, C. F. Petersen, S. Dhuey, K. Hofhuis, R. Mansell, R. V. Chopdekar, A. Scholl, T. Lippert, and S. van Dijken, Geometrical Frustration and Planar Triangular Antiferromagnetism in Quasi- Three-Dimensional Artificial Spin Architecture, Phys. Rev. Lett. 125, 267203 (2020)

work page 2020