Letter | Published:

Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice

Nature volume 540, pages 410413 (15 December 2016) | Download Citation

Abstract

Artificial spin-ice systems are lithographically patterned arrangements of interacting magnetic nanostructures that were introduced as way of investigating the effects of geometric frustration in a controlled manner1,2,3,4. This approach has enabled unconventional states of matter to be visualized directly in real space5,6,7,8,9,10,11,12,13,14,15,16,17,18, and has triggered research at the frontier between nanomagnetism, statistical thermodynamics and condensed matter physics. Despite efforts to create an artificial realization of the square-ice model—a two-dimensional geometrically frustrated spin-ice system defined on a square lattice—no simple geometry based on arrays of nanomagnets has successfully captured the macroscopically degenerate ground-state manifold of the model19. Instead, square lattices of nanomagnets are characterized by a magnetically ordered ground state that consists of local loop configurations with alternating chirality1,20,21,22,23,24,25,26. Here we show that all of the characteristics of the square-ice model are observed in an artificial square-ice system that consists of two sublattices of nanomagnets that are vertically separated by a small distance. The spin configurations we image after demagnetizing our arrays reveal unambiguous signatures of a Coulomb phase and algebraic spin-spin correlations, which are characterized by the presence of ‘pinch’ points in the associated magnetic structure factor. Local excitations—the classical analogues of magnetic monopoles27—are free to evolve in an extensively degenerate, divergence-free vacuum. We thus provide a protocol that could be used to investigate collective magnetic phenomena, including Coulomb phases28 and the physics of ice-like materials.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1.

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

  2. 2.

    , & Artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013)

  3. 3.

    & Artificial ferroic systems: novel functionality from structure, interactions and dynamics. J. Phys. Condens. Matter 25, 363201 (2013)

  4. 4.

    , , & Focus on artificial frustrated systems. New J. Phys. 16, 075016 (2014)

  5. 5.

    , , , & Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys. 6, 359–363 (2010)

  6. 6.

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

  7. 7.

    et al. Artificial kagome arrays of nanomagnets: a frozen dipolar spin ice. Phys. Rev. Lett. 106, 057209 (2011)

  8. 8.

    et al. Crystallites of magnetic charges in artificial spin ice. Nature 500, 553–557 (2013)

  9. 9.

    et al. Size distribution of magnetic charge domains in thermally activated but out-of-equilibrium artificial spin ice. Sci. Rep. 4, 5702 (2014)

  10. 10.

    , & FePd3 as a material for studying thermally active artificial spin ice systems. Phys. Rev. B 91, 224406 (2015)

  11. 11.

    et al. Thermodynamic phase transitions in a frustrated magnetic metamaterial. Nat. Commun. 6, 8278 (2015)

  12. 12.

    et al. Kinetic pathways to the magnetic charge crystal in artificial dipolar spin ice. Phys. Rev. B 90, 220407(R) (2014)

  13. 13.

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

  14. 14.

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

  15. 15.

    , & Ground-state candidate for the dipolar kagome Ising antiferromagnet. Phys. Rev. B 93, 214410 (2016)

  16. 16.

    et al. Emergent ice rule and magnetic charge screening from vertex frustration in artificial spin ice. Nat. Phys. 10, 670–675 (2014)

  17. 17.

    , , , & Magnetic-moment fragmentation and monopole crystallization. Phys. Rev. X 4, 011007 (2014)

  18. 18.

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

  19. 19.

    Residual entropy of square ice. Phys. Rev. 162, 162–172 (1967)

  20. 20.

    et al. Effective temperature in an interacting vertex system: theory and experiment on artificial spin ice. Phys. Rev. Lett. 105, 047205 (2010)

  21. 21.

    , , & Thermal ground-state ordering and elementary excitations in artificial magnetic square ice. Nat. Phys. 7, 75–79 (2011)

  22. 22.

    , & Diversity enabling equilibration: disorder and the ground state in artificial spin ice. Phys. Rev. Lett. 107, 217204 (2011)

  23. 23.

    et al. Domain dynamics and fluctuations in artificial square ice at finite temperatures. New J. Phys. 14, 035014 (2012)

  24. 24.

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

  25. 25.

    , , & Exploring thermally induced states in square artificial spin-ice arrays. New J. Phys. 15, 055012 (2013)

  26. 26.

    et al. Thermal fluctuations in artificial spin ice. Nat. Nanotechnol. 9, 514–519 (2014)

  27. 27.

    , & Magnetic monopoles in spin ice. Nature 451, 42–45 (2008)

  28. 28.

    The “Coulomb phase” in frustrated systems. Annu. Rev. Condens. Matter Phys. 1, 179–210 (2010)

  29. 29.

    & Artificial square ice and related dipolar nanoarrays. Phys. Rev. Lett. 96, 237202 (2006)

  30. 30.

    , , & Thermal string excitations in artificial spin-ice square dipolar arrays. J. Phys. Condens. Matter 26, 266006 (2014)

  31. 31.

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

  32. 32.

    et al. The non- random walk of chiral magnetic charge carriers in artificial spin ice. Sci. Rep. 3, 1252 (2013)

  33. 33.

    & OOMMF User’s Guide, Version 1.0. Report No. NISTIR 6376 (National Institute of Standards and Technology, 1999)

  34. 34.

    et al. Demagnetization protocols for frustrated interacting nanomagnet arrays. J. Appl. Phys. 101, 09J104 (2007)

  35. 35.

    , , , & Linear field demagnetization of artificial magnetic square ice. Front. Phys. 1, 28 (2013)

  36. 36.

    & Classical spin liquid: exact solution for the infinite-component antiferromagnetic model on the kagomé lattice. Phys. Rev. B 59, 443–456 (1999)

  37. 37.

    et al. Magnetic Coulomb phase in the spin ice Ho2Ti2O7. Science 326, 415–417 (2009)

  38. 38.

    , & Why spin ice obeys the ice rules. Phys. Rev. Lett. 95, 217201 (2005)

  39. 39.

    Classical and Quantum Two-dimensional Ice: Coulomb and Ordered Phases. PhD thesis, Ecole normale supérieure de Lyon (2013)

  40. 40.

    , , , & Nanoscale structure of the magnetic induction at monopole defects in artificial spin-ice lattices. Phys. Rev. B 83, 174431 (2011)

  41. 41.

    et al. Photoemission electron microscopy of three-dimensional magnetization configurations in core-shell nanostructures. Phys. Rev. B 84, 174406 (2011)

  42. 42.

    et al. Observation of Bloch-point domain walls in cylindrical magnetic nanowires. Phys. Rev. B 89, 180405 (2014)

  43. 43.

    et al. Quantitative analysis of shadow x-ray magnetic circular dichroism photoemission electron microscopy. Phys. Rev. B 92, 144428 (2015)

Download references

Acknowledgements

This work was supported by the Agence Nationale de la Recherche through project number ANR12-BS04-009 ‘Frustrated’. We acknowledge support from the Nanofab team at the Institut NÉEL and thank S. Le-Denmat and O. Fruchart for technical help during atomic force microscope and magnetic force microscope measurements.

Author information

Affiliations

  1. CNRS, Institut NÉEL, F-38000 Grenoble, France

    • Yann Perrin
    • , Benjamin Canals
    •  & Nicolas Rougemaille
  2. Université Grenoble Alpes, Institut NÉEL, F-38000 Grenoble, France

    • Yann Perrin
    • , Benjamin Canals
    •  & Nicolas Rougemaille

Authors

  1. Search for Yann Perrin in:

  2. Search for Benjamin Canals in:

  3. Search for Nicolas Rougemaille in:

Contributions

B.C. and N.R. conceived the project. Y.P. was in charge of the sample fabrication and characterization, the magnetic imaging measurements and the analysis of the data. All authors contributed to the preparation of the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Nicolas Rougemaille.

Reviewer Information Nature thanks C. Nisoli, A. Ramirez and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data

Supplementary information

Videos

  1. 1.

    Avalanche Process

    Video showing how magnetization reverses during a modelled field demagnetization protocol. The applied external magnetic field is represented by a rotating black arrow. Small black and white arrows represent point dipole spins on a square lattice, and red/blue/green squares code for type-II, type-I and type-III vertices, respectively.

  2. 2.

    Full Demagnetization shifted array

    Video showing how magnetization reverses during a modelled field demagnetization protocol. The applied external magnetic field is represented by a rotating black arrow. Small black and white arrows represent point dipole spins on a square lattice, and red/blue/green squares code for type-II, type-I and type-III vertices, respectively.

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/nature20155

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.