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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.

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  1. Frenkel, D. Order through entropy. Nat. Mater. 14, 9–12 (2015).

    Article  ADS  Google Scholar 

  2. Percus, J. K. (ed.) The Many-Body Problem (Interscience, 1963).

  3. Lin, K.-H. et al. Entropically driven colloidal crystallization on patterned surfaces. Phys. Rev. Lett. 85, 1770–1773 (2000).

    Article  ADS  Google Scholar 

  4. Onsager, L. The effects of shape on the interaction of colloidal particles. Ann. N. Y. Acad. Sci. 51, 627–659 (1949).

    Article  ADS  Google Scholar 

  5. Fraden, S., Maret, G., Caspar, D. L. D. & Meyer, R. B. Isotropic-nematic phase transition and angular correlations in isotropic suspensions of tobacco mosaic virus. Phys. Rev. Lett. 63, 2068–2071 (1989).

    Article  ADS  Google Scholar 

  6. van der Beek, D. & Lekkerkerker, H. N. W. Nematic ordering vs. gelation in suspensions of charged platelets. Europhys. Lett. 61, 702–707 (2003).

    Article  ADS  Google Scholar 

  7. Dussi, S. & Dijkstra, M. Entropy-driven formation of chiral nematic phases by computer simulations. Nat. Commun. 7, 11175 (2016).

    Article  ADS  Google Scholar 

  8. Kil, K. H., Yethiraj, A. & Kim, J. S. Nematic ordering of hard rods under strong confinement in a dense array of nanoposts. Phys. Rev. E 101, 032705 (2020).

    Article  ADS  Google Scholar 

  9. Filion, L. et al. Self-assembly of a colloidal interstitial solid with tunable sublattice doping. Phys. Rev. Lett. 107, 168302 (2011).

    Article  ADS  Google Scholar 

  10. Sciortino, F. Entropy in self-assembly. Riv. Nuovo Cim. 42, 511–548 (2019).

  11. Pusey, P. N. & van Megen, W. Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. Nature 320, 340–342 (1986).

    Article  ADS  Google Scholar 

  12. Barry, E. & Dogic, Z. Entropy driven self-assembly of nonamphiphilic colloidal membranes. Proc. Natl Acad. Sci. USA 107, 10348–10353 (2010).

    Article  ADS  Google Scholar 

  13. Damasceno, P. F., Engel, M. & Glotzer, S. C. Predictive self-assembly of polyhedra into complex structures. Science 337, 453–457 (2012).

    Article  ADS  Google Scholar 

  14. Zhu, G., Huang, Z., Xu, Z. & Yan, L.-T. Tailoring interfacial nanoparticle organization through entropy. Acc. Chem. Res. 51, 900–909 (2018).

    Article  Google Scholar 

  15. Zhang, Y. et al. Microstructures and properties of high-entropy alloys. Prog. Mater. Sci. 61, 1–93 (2014).

    Article  Google Scholar 

  16. Villain, J., Bidaux, R., Carton, J.-P. & Conte, R. Order as an effect of disorder. J. Phys. 41, 1263–1272 (1980).

    Article  MathSciNet  Google Scholar 

  17. Henley, C. L. Ordering due to disorder in a frustrated vector antiferromagnet. Phys. Rev. Lett. 62, 2056–2059 (1989).

    Article  ADS  Google Scholar 

  18. Zhitomirsky, M. E., Gvozdikova, M. V., Holdsworth, P. C. W. & Moessner, R. Quantum order by disorder and accidental soft mode in Er2Ti2O7. Phys. Rev. Lett. 109, 077204 (2012).

    Article  ADS  Google Scholar 

  19. Plat, X., Fuji, Y., Capponi, S. & Pujol, P. Selection of factorizable ground state in a frustrated spin tube: order by disorder and hidden ferromagnetism. Phys. Rev. B 91, 064411 (2015).

    Article  ADS  Google Scholar 

  20. Guruciaga, P. C. et al. Field-tuned order by disorder in frustrated Ising magnets with antiferromagnetic interactions. Phys. Rev. Lett. 117, 167203 (2016).

    Article  ADS  Google Scholar 

  21. Ross, K. A., Qiu, Y., Copley, J. R. D., Dabkowska, H. A. & Gaulin, B. D. Order by disorder spin wave gap in the XY pyrochlore magnet Er2Ti2O7. Phys. Rev. Lett. 112, 057201 (2014).

    Article  ADS  Google Scholar 

  22. Green, A. G., Conduit, G. & Krüger, F. Quantum order-by-disorder in strongly correlated metals. Annu. Rev. Condens. Matter Phys. 9, 59–77 (2018).

    Article  ADS  Google Scholar 

  23. Okuma, R. et al. Fermionic order by disorder in a van der Waals antiferromagnet. Sci. Rep. 10, 15311 (2020).

    Article  ADS  Google Scholar 

  24. Schiffer, P. & Nisoli, C. Artificial spin ice: paths forward. Appl. Phys. Lett. 118, 110501 (2021).

    Article  ADS  Google Scholar 

  25. Skjærvø, S. H., Marrows, C. H., Stamps, R. L. & Heyderman, L. J. Advances in artificial spin ice. Nat. Rev. Phys. 2, 13–28 (2020).

    Article  Google Scholar 

  26. Morrison, M. J., Nelson, T. R. & Nisoli, C. Unhappy vertices in artificial spin ice: new degeneracies from vertex frustration. N. J. Phys. 15, 045009 (2013).

    Article  Google Scholar 

  27. Gilbert, I. et al. Emergent reduced dimensionality by vertex frustration in artificial spin ice. Nat. Phys. 12, 162–165 (2016).

    Article  Google Scholar 

  28. Baxter, R. J. Spontaneous staggered polarization of the F-model. J. Stat. Phys. 9, 145–182 (1973).

    Article  ADS  Google Scholar 

  29. Nisoli, C., Kapaklis, V. & Schiffer, P. Deliberate exotic magnetism via frustration and topology. Nat. Phys. 13, 200–203 (2017).

    Article  Google Scholar 

  30. Vansteenkiste, A. et al. The design and verification of MuMax3. AIP Adv. 4, 107133 (2014).

    Article  ADS  Google Scholar 

  31. Farhan, A. et al. Direct observation of thermal relaxation in artificial spin ice. Phys. Rev. Lett. 111, 057204 (2013).

    Article  ADS  Google Scholar 

  32. Lao, Y. et al. Classical topological order in the kinetics of artificial spin ice. Nat. Phys. 14, 723–727 (2018).

    Article  Google Scholar 

  33. Zhang, X. et al. String phase in an artificial spin ice. Nat. Commun. 12, 6514 (2021).

    Article  ADS  Google Scholar 

  34. Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zero-point entropy in ‘spin ice’. Nature 399, 333–335 (1999).

    Article  ADS  Google Scholar 

  35. King, A. D., Nisoli, C., Dahl, E. D., Poulin-Lamarre, G. & Lopez-Bezanilla, A. Qubit spin ice. Science 373, 576–580 (2021).

    Article  ADS  Google Scholar 

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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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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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