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Entropy-driven order in an array of nanomagnets


Long-range ordering is typically associated with a decrease in entropy. Yet, it can also be driven by increasing entropy in certain special cases. Here we demonstrate that artificial spin-ice arrays of single-domain nanomagnets can be designed to produce such entropy-driven order. We focus on the tetris artificial spin-ice structure, a highly frustrated array geometry with a zero-point Pauling entropy, which is formed by selectively creating regular vacancies on the canonical square ice lattice. We probe thermally active tetris artificial spin ice both experimentally and through simulations, measuring the magnetic moments of the individual nanomagnets. We find two-dimensional magnetic ordering in one subset of these moments, which we demonstrate to be induced by disorder (that is, increased entropy) in another subset of the moments. In contrast with other entropy-driven systems, the discrete degrees of freedom in tetris artificial spin ice are binary and are both designable and directly observable at the microscale, and the entropy of the system is precisely calculable in simulations. This example, in which the system’s interactions and ground-state entropy are well defined, expands the experimental landscape for the study of entropy-driven ordering.

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Fig. 1: Tetris artificial spin ice.
Fig. 2: Entropic interactions in tetris ice.
Fig. 3: Two-dimensional ordering in tetris ice.
Fig. 4: Longitudinal and transverse moment correlations.
Fig. 5: Two-dimensional ordering in tetris ice simulation.

Data availability

Underlying data are available at


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We thank I.-A. Chioar for fruitful discussions and A. Scholl for assistance with the early XMCD-PEEM measurements. Work at Yale University and the University of Illinois at Urbana-Champaign was funded by the US Department of Energy (DOE), Office of Basic Energy Sciences, Materials Sciences and Engineering Division under grant nos. DE-SC0010778 and DE-SC0020162 to H.S., A.K., N.H., X.Z., N.S.B., Y.L., I.G., J.S. and P.S. This research used resources of the Advanced Light Source, a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231 to R.V.C. Work at the University of Minnesota was supported by NSF through grant nos. DMR-1807124 and DMR-2103711 to J.R., J.D.W. and C.L. Work at the University of Liverpool was supported by the UK Royal Society through grant no. RGS\R2\180208 to D.B. and L.O. Work at Los Alamos National Laboratory was carried out under the auspices of the US DOE through LANL, operated by Triad National Security, LLC under contract no. 892333218NCA000001 and financed by DOE LDRD (A.D. and C.N.).

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Authors and Affiliations



J.R. and J.D.W. performed film depositions under the guidance of C.L., and D.B. prepared other samples under the guidance of L.O., with H.S., X.Z., I.G., Y.L., J.S. and N.S.B. overseeing the lithography. H.S., X.Z., I.G., Y.L., J.S., N.S.B. and R.V.C. performed the XMCD-PEEM characterization of the thermally active samples, and H.S., A.K. and N.H. analysed the data. H.S. performed micromagnetic calculations. A.D. performed Monte Carlo simulations, under the guidance of C.N. C.N. and P.S. supervised the entire project. All authors contributed to the discussion of results and to the finalization of the manuscript.

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Correspondence to Cristiano Nisoli or Peter Schiffer.

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Nature Physics thanks Erik Folven, Alan Farhan and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Saglam, H., Duzgun, A., Kargioti, A. et al. Entropy-driven order in an array of nanomagnets. Nat. Phys. 18, 706–712 (2022).

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