A tunable magnetic metamaterial based on the dipolar four-state Potts model

Abstract

Metamaterials, tunable artificial materials, are useful playgrounds to investigate magnetic systems. So far, artificial Ising spin systems have revealed features such as emergent magnetic monopoles1,2 and charge fragmentation3. Here we present a metasystem composed of a lattice of dipolarly coupled nanomagnets. The magnetic spin of each nanomagnet is constrained to lie along a body diagonal, which yields four possible spin states. We show that the magnetic ordering of this metasystem (antiferromagnetic, ferromagnetic or spin ice like) is determined by the spin states orientation relative to the underlying lattice. The dipolar four-state Potts model explains our experimental observations and sheds light on the role of symmetry, as well as short- and long-range dipolar magnetic interactions, in such non-Ising spin systems.

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Fig. 1: Four-state Potts model based on an artificial spin system.
Fig. 2: Distribution of different magnetic orders in an artificial four-state Potts model.
Fig. 3: Modelling of the dependence of spin configurations on angle between spins and underlying lattice α.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. 1.

    Ladak, S., Read, D. E., Perkins, G. K., Cohen, L. F. & Branford, W. R. Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys. 6, 359–363 (2010).

    CAS  Article  Google Scholar 

  2. 2.

    Mengotti, E. et al. Real-space observation of emergent magnetic monopoles and associated Dirac strings in artificial kagome spin ice. Nat. Phys. 7, 68–74 (2011).

    CAS  Article  Google Scholar 

  3. 3.

    Canals, B. et al. Fragmentation of magnetism in artificial kagome dipolar spin ice. Nat. Commun. 7, 11446 (2016).

    CAS  Article  Google Scholar 

  4. 4.

    Schrefl, T., Schmidts, H. F., Fidler, J. & Kronmüller, H. The role of exchange and dipolar coupling at grain boundaries in hard magnetic materials. J. Magn. Magn. Mater. 124, 251 (1993).

    CAS  Article  Google Scholar 

  5. 5.

    Panissod, P. & Drillon, M. in Magnetism: Molecules to Materials IV: Nanosized Magnetic Materials (eds Miller, J. S. & Drillon, M.) Ch 7 (Wiley, Weinheim, 2003).

  6. 6.

    Majetich, S. A. & Sachan, M. Magnetostatic interactions in magnetic nanoparticle assemblies: energy, time and length scales. J. Phys. D 39, 407 (2006).

    Article  Google Scholar 

  7. 7.

    Anghinolfi, L. et al. Thermodynamic phase transitions in a frustrated magnetic metamaterial. Nat. Commun. 6, 9278 (2015).

    Article  Google Scholar 

  8. 8.

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

    CAS  Article  Google Scholar 

  9. 9.

    Perrin, Y., Canals, B. & Rougemaille, N. Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice. Nature 540, 410–413 (2016).

    CAS  Article  Google Scholar 

  10. 10.

    Wu, F. Y. The Potts model Rev. Mod. Phys. 54, 235-268 (1982).

    Article  Google Scholar 

  11. 11.

    Schelling, T. C. Dynamic models of segregation. J. Math. Sociol. 1, 143–186 (1971).

    Article  Google Scholar 

  12. 12.

    Sun, L., Chang, Y. F. & Cai, X. A discrete simulation of tumor growth concerning nutrient influence. Int. J. Mod. Phys. B 18, 2651–2657 (2004).

    CAS  Article  Google Scholar 

  13. 13.

    Sanyal, S. & Glazier, J. A. Viscous instabilities in flowing foams: a cellular Potts model approach J. Stat. Mech. 2006, 10008 (2006).

  14. 14.

    Rougemaille, N. et al. Chiral nature of magnetic monopoles in artificial spin ice. New J. Phys. 15, 035026 (2013).

    Article  Google Scholar 

  15. 15.

    Gliga, S., Kakay, A., Heyderman, L. J., Hertel, R. & Heinonen, O. G. Broken vertex symmetry and finite zero-point entropy in the artificial square ice ground state. Phys. Rev. B 92, 060413 (2016).

    Article  Google Scholar 

  16. 16.

    Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006).

    CAS  Article  Google Scholar 

  17. 17.

    Qi, Y., Brintlinger, T. & Cumings, J. Direct observation of the ice rule in an artificial kagome spin ice. Phys. Rev. B 77, 094418 (2008).

    Article  Google Scholar 

  18. 18.

    Arnalds, U. B. et al. A new look on the two-dimensional Ising model: thermal artificial spins. New J. Phys. 18, 023008 (2016).

    Article  Google Scholar 

  19. 19.

    Zhang, S. et al. Perpendicular magnetization and generic realization of the Ising model in artificial spin ice. Phys. Rev. Lett. 109, 087201 (2012).

    Article  Google Scholar 

  20. 20.

    Chioar, I. A. et al. Nonuniversality of artificial frustrated spin systems. Phys. Rev. B 90, 064411 (2014).

    Article  Google Scholar 

  21. 21.

    Louis, D. et al. Interfaces anisotropy in single crystal V/Fe/V trilayer. J. Magn. Magn. Mater. 372, 233–235 (2014).

    CAS  Article  Google Scholar 

  22. 22.

    Vaz, C. A. F. et al. Direct observation of remanent magnetic states in epitaxial fcc Co small disks. Phys. Rev. B 67, 140405(R) (2003).

    Article  Google Scholar 

  23. 23.

    Mitsuzuka, K., Lacour, D., Hehn, M., Andrieu, S. & Montaigne, F. Magnetic vortices in single crystalline Fe–V disks with four folds magnetic anisotropy. Appl. Phys. Lett. 100, 192406 (2012).

    Article  Google Scholar 

  24. 24.

    Li, J. et al. Stabilizing a magnetic vortex/antivortex array in single crystalline Fe/Ag(001) microstructures. Appl. Phys. Lett. 104, 262409 (2014).

    Article  Google Scholar 

  25. 25.

    Louis, D. et al. in Spintronics IX 99311M (eds. Drouhin, H. J., Wegrowe, J. E. & Razeghi, M.) 9931-57 (Spie-Int Soc Optical Engineering, Bellingham, 2016).

  26. 26.

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

    CAS  Article  Google Scholar 

  27. 27.

    Puntes, V. F., Gorostiza, P., Aruguete, D. M., Bastus, N. G. & Alivisatos, A. P. Collective behaviour in two-dimensional cobalt nanoparticle assemblies observed by magnetic force microscopy. Nat. Mater. 3, 263–268 (2004).

    CAS  Article  Google Scholar 

  28. 28.

    Yamamoto, K. et al. Direct visualization of dipolar ferromagnetic domain structures in Co nanoparticle monolayers by electron holography. Appl. Phys. Lett. 93, 082502 (2008).

    Article  Google Scholar 

  29. 29.

    Sun, S. H., Murray, C. B., Weller, D., Folks, L. & Moser, A. Monodisperse FePt nanoparticles and ferromagnetic FePt nanocrystal superlattices. Science 287, 1989–1992 (2000).

    CAS  Article  Google Scholar 

  30. 30.

    Yang, W., Yu, Y., Wang, L., Yang, C. & Li, H. Controlled synthesis and assembly into anisotropy arrays of magnetic cobalt-substituted magnetite nanocubes. Nanoscale 7, 2877–2882 (2015).

    CAS  Article  Google Scholar 

Download references

Acknowledgements

The authors thank N. Rougemaille and B. Canals for fruitful discussions. This work was supported by the Agence Nationale de la Recherche through project number ANR12-BS04-009 ‘Frustrated’ and partially supported by the French PIA project ‘Lorraine Université d’Excellence’ ANR15-IDEX-04-LUE.

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D.Louis, T.H., M.H. and F.M. prepared the samples. MFM measurements were carried out by D.Louis, D.Lacour and M.H., and D.Louis, D.Lacour and F.M. analysed the data. D.Louis, F.M. and V.L. carried out theory and simulations. All authors discussed the results and implications at all stages and prepared the manuscript.

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Correspondence to F. Montaigne.

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Louis, D., Lacour, D., Hehn, M. et al. A tunable magnetic metamaterial based on the dipolar four-state Potts model. Nature Mater 17, 1076–1080 (2018). https://doi.org/10.1038/s41563-018-0199-x

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