Abstract
Conventional approaches to studying nanomagnets often focus on the energetic relaxation from an initialized state to a final low-energy state, typically assuming a single spin-flip regime. This relaxation process inherently involves intermediate states with variable magnetostatic energies, which influence the probability of reaching specific final configurations. Here, we investigate the nature of intermediate states in a simple nanomagnet model consisting of four nanomagnets arranged onto a square plaquette. Through systematic exploration, we demonstrate how geometry influences energy relaxation pathways via multipolar analysis. We elaborate on the nature of energy relaxation pathways, with direct consequences for understanding magnetic frustration and metastability in artificial spin ice structures. Our theoretical models are supported by experimental results from field-induced relaxation using Magnetic Force Microscopy measurements. This work provides an intuitive framework for understanding energy relaxation in artificial spin ice structures.
Data availability
The raw data supporting the findings in the study can be made available upon reasonable request to the corresponding author.
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Acknowledgements
H.A, J.S.W., T.C., and J.F. were funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Science and Engineering Division. Work performed at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, was supported by the U.S.DOE, Office of Basic Energy Sciences, under Contract No.DE-AC02-06CH11357. P.M. and I.T. were funded by the Fondecyt Regular Grant 1210083 from the Chilean Government.
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H.A. conceived the idea and supervised the experimental part of the research. P.M. supervised the theoretical part of the research. I.T. and P.M. carried out all the numerical simulations, along with multipolar analysis and Monte Carlo simulations. F.B. provided support to I.T. and P.M. with theoretical analysis. J.S.W. prepared the samples. T.C. and H.A. performed the MFM characterization. J.F. performed the SQUID VSM measurements. The manuscript was prepared by H.A. and P.M. with input from all co-authors.
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Arava, H., Tapia, I., Cote, T. et al. Geometry driven intermediate states in artificial square ice structures. Commun Mater (2026). https://doi.org/10.1038/s43246-026-01147-4
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DOI: https://doi.org/10.1038/s43246-026-01147-4
