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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Frenkel, D. Order through entropy. Nat. Mater. 14, 9–12 (2015).
Percus, J. K. (ed.) The Many-Body Problem (Interscience, 1963).
Lin, K.-H. et al. Entropically driven colloidal crystallization on patterned surfaces. Phys. Rev. Lett. 85, 1770–1773 (2000).
Onsager, L. The effects of shape on the interaction of colloidal particles. Ann. N. Y. Acad. Sci. 51, 627–659 (1949).
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).
van der Beek, D. & Lekkerkerker, H. N. W. Nematic ordering vs. gelation in suspensions of charged platelets. Europhys. Lett. 61, 702–707 (2003).
Dussi, S. & Dijkstra, M. Entropy-driven formation of chiral nematic phases by computer simulations. Nat. Commun. 7, 11175 (2016).
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).
Filion, L. et al. Self-assembly of a colloidal interstitial solid with tunable sublattice doping. Phys. Rev. Lett. 107, 168302 (2011).
Sciortino, F. Entropy in self-assembly. Riv. Nuovo Cim. 42, 511–548 (2019).
Pusey, P. N. & van Megen, W. Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. Nature 320, 340–342 (1986).
Barry, E. & Dogic, Z. Entropy driven self-assembly of nonamphiphilic colloidal membranes. Proc. Natl Acad. Sci. USA 107, 10348–10353 (2010).
Damasceno, P. F., Engel, M. & Glotzer, S. C. Predictive self-assembly of polyhedra into complex structures. Science 337, 453–457 (2012).
Zhu, G., Huang, Z., Xu, Z. & Yan, L.-T. Tailoring interfacial nanoparticle organization through entropy. Acc. Chem. Res. 51, 900–909 (2018).
Zhang, Y. et al. Microstructures and properties of high-entropy alloys. Prog. Mater. Sci. 61, 1–93 (2014).
Villain, J., Bidaux, R., Carton, J.-P. & Conte, R. Order as an effect of disorder. J. Phys. 41, 1263–1272 (1980).
Henley, C. L. Ordering due to disorder in a frustrated vector antiferromagnet. Phys. Rev. Lett. 62, 2056–2059 (1989).
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).
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).
Guruciaga, P. C. et al. Field-tuned order by disorder in frustrated Ising magnets with antiferromagnetic interactions. Phys. Rev. Lett. 117, 167203 (2016).
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).
Green, A. G., Conduit, G. & Krüger, F. Quantum order-by-disorder in strongly correlated metals. Annu. Rev. Condens. Matter Phys. 9, 59–77 (2018).
Okuma, R. et al. Fermionic order by disorder in a van der Waals antiferromagnet. Sci. Rep. 10, 15311 (2020).
Schiffer, P. & Nisoli, C. Artificial spin ice: paths forward. Appl. Phys. Lett. 118, 110501 (2021).
Skjærvø, S. H., Marrows, C. H., Stamps, R. L. & Heyderman, L. J. Advances in artificial spin ice. Nat. Rev. Phys. 2, 13–28 (2020).
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).
Gilbert, I. et al. Emergent reduced dimensionality by vertex frustration in artificial spin ice. Nat. Phys. 12, 162–165 (2016).
Baxter, R. J. Spontaneous staggered polarization of the F-model. J. Stat. Phys. 9, 145–182 (1973).
Nisoli, C., Kapaklis, V. & Schiffer, P. Deliberate exotic magnetism via frustration and topology. Nat. Phys. 13, 200–203 (2017).
Vansteenkiste, A. et al. The design and verification of MuMax3. AIP Adv. 4, 107133 (2014).
Farhan, A. et al. Direct observation of thermal relaxation in artificial spin ice. Phys. Rev. Lett. 111, 057204 (2013).
Lao, Y. et al. Classical topological order in the kinetics of artificial spin ice. Nat. Phys. 14, 723–727 (2018).
Zhang, X. et al. String phase in an artificial spin ice. Nat. Commun. 12, 6514 (2021).
Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zero-point entropy in ‘spin ice’. Nature 399, 333–335 (1999).
King, A. D., Nisoli, C., Dahl, E. D., Poulin-Lamarre, G. & Lopez-Bezanilla, A. Qubit spin ice. Science 373, 576–580 (2021).
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.).
The authors declare no competing interests.
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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). https://doi.org/10.1038/s41567-022-01555-6