Letter | Published:

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

Nature Materialsvolume 17pages10761080 (2018) | Download Citation

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.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Data availability

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

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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).

  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).

  3. 3.

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

  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).

  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).

  7. 7.

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

  8. 8.

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

  9. 9.

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

  10. 10.

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

  11. 11.

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

  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).

  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).

  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).

  16. 16.

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

  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).

  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).

  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).

  20. 20.

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

  21. 21.

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

  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).

  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).

  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).

  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).

  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).

  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).

  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).

  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).

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.

Author information

Affiliations

  1. Institut Jean Lamour, CNRS—Université de Lorraine, Nancy, France

    • D. Louis
    • , D. Lacour
    • , M. Hehn
    • , T. Hauet
    •  & F. Montaigne
  2. University of California San Diego, Department of Electrical and Computer Engineering, La Jolla, CA, USA

    • V. Lomakin

Authors

  1. Search for D. Louis in:

  2. Search for D. Lacour in:

  3. Search for M. Hehn in:

  4. Search for V. Lomakin in:

  5. Search for T. Hauet in:

  6. Search for F. Montaigne in:

Contributions

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.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to F. Montaigne.

Supplementary information

  1. Supplementary Information

    Text, 6 Figures, 3 references

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/s41563-018-0199-x