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Field-induced phase coexistence in an artificial spin ice

Abstract

Artificial spin-ice systems are magnetic metamaterials consisting of nanomagnet arrays that can be designed to study exotic magnetic states not found in natural materials. Typically, these arrays are modelled as interacting binary macrospins that can only be in an up or down state and are described by the Ising model. These materials have demonstrated ordering transitions, but only via a spontaneous symmetry-breaking mechanism. We have designed and studied a quadrupole artificial spin-ice system consisting of interacting plaquettes of coupled single-domain nanomagnets that can be interpreted as a composite, ternary variable. After annealing this system in an external magnetic field, we observe both a ferroquadrupolar and an antiferroquadrupolar phase, with an apparent first-order phase boundary and a coexistence regime. The phase diagram of this material is reminiscent of a model used to describe phase coexistence in the superfluid transition of 4He with 3He impurities. These results illustrate how composite magnetic objects realize exotic statistical physics models beyond the Ising model.

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Fig. 1: Artificial quadrupole lattice.
Fig. 2: Two ordered Potts states.
Fig. 3: Phase coexistence in the quadrupole lattice.
Fig. 4: Phase diagram and neighbour correlations.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.

References

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

    ADS  Article  Google Scholar 

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

    Article  Google Scholar 

  3. Gilbert, I., Nisoli, C. & Schiffer, P. Frustration by design. Phys. Today 69, 54–59 (2016).

    Article  Google Scholar 

  4. 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).

    Article  Google Scholar 

  5. 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).

    Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  7. Fisher, M. E. & Nelson, D. R. Spin flop, supersolids, and bicritical and tetracritical points. Phys. Rev. Lett. 32, 1350–1353 (1974).

    ADS  Article  Google Scholar 

  8. Hu, X. Bicritical and tetracritical phenomena and scaling properties of the SO(5) theory. Phys. Rev. Lett. 87, 057004 (2001).

    ADS  Article  Google Scholar 

  9. Eichhorn, A., Mesterházy, D. & Scherer, M. M. Multicritical behavior in models with two competing order parameters. Phys. Rev. E 88, 042141 (2013).

    ADS  Article  Google Scholar 

  10. Wang, Y.-L. et al. Rewritable artificial magnetic charge ice. Science 352, 962–966 (2016).

    ADS  Article  Google Scholar 

  11. Gartside, J. C. et al. Realization of ground state in artificial kagome spin ice via topological defect-driven magnetic writing. Nat. Nanotech. 13, 53–58 (2018).

    ADS  Article  Google Scholar 

  12. Haldar, A., Kumar, D. & Adeyeye, A. O. A reconfigurable waveguide for energy-efficient transmission and local manipulation of information in a nanomagnetic device. Nat. Nanotech. 11, 437–443 (2016).

    ADS  Article  Google Scholar 

  13. Gliga, S., Kakay, A., Hertel, R. & Heinonen, O. G. Spectral analysis of topological defects in an artificial spin-ice lattice. Phys. Rev. Lett. 110, 117205 (2013).

    ADS  Article  Google Scholar 

  14. Iacocca, E., Gliga, S., Stamps, R. L. & Heinonen, O. Reconfigurable wave band structure of an artificial square ice. Phys. Rev. B 93, 134420 (2016).

    ADS  Article  Google Scholar 

  15. Bhat, V. S. et al. Controlled magnetic reversal in permalloy films patterned into artificial quasicrystals. Phys. Rev. Lett. 111, 077201 (2013).

    ADS  Article  Google Scholar 

  16. Jungfleisch, M. B. et al. Dynamic response of an artificial square spin ice. Phys. Rev. B 93, 100401(R) (2016).

    ADS  Article  Google Scholar 

  17. Zhou, X., Chua, G.-L., Singh, N. & Adeyeye, A. O. Large area artificial spin ice and anti–spin ice Ni80Fe20 structures: static and dynamic behavior. Adv. Funct. Mater. 26, 1437–1444 (2016).

    Article  Google Scholar 

  18. Bhat, V. S., Heimbach, F., Stasinopoulos, I. & Grundler, D. Magnetization dynamics of topological defects and the spin solid in a kagome artificial spin ice. Phys. Rev. B 93, 140401(R) (2016).

    ADS  Article  Google Scholar 

  19. Grollier, J., Querlioz, D. & Stiles, M. D. Spintronic nanodevices for bioinspired computing. Proc. IEEE 104, 2024–2039 (2016).

    Article  Google Scholar 

  20. Bhanja, S., Karunaratne, D. K., Panchumarthy, R., Rajaram, S. & Sarkar, S. Non-Boolean computing with nanomagnets for computer vision applications. Nat. Nanotech. 11, 177–183 (2015).

    ADS  Article  Google Scholar 

  21. Arnalds, U. B. et al. Thermal transitions in nano-patterned XY-magnets. Appl. Phys. Lett. 105, 042409 (2014).

    ADS  Article  Google Scholar 

  22. Leo, N. et al. Collective magnetism in an artificial 2D XY spin system. Nat. Commun. 9, 2850 (2018).

    ADS  Article  Google Scholar 

  23. Läuchli, A., Mila, F. & Penc, K. Quadrupolar phases of the S = 1 bilinear-biquadratic Heisenberg model on the triangular lattice. Phys. Rev. Lett. 97, 087205 (2006).

    ADS  Article  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  25. Blume, M., Emery, V. J. & Griffiths, R. B. Ising model for the λ transition and phase separation in He3-He4 mixtures. Phys. Rev. A 4, 1071–1077 (1971).

    ADS  Article  Google Scholar 

  26. Chern, G.-W., Reichhardt, C. & Reichhardt, C. J. O. Frustrated colloidal ordering and fully packed loops in arrays of optical traps. Phys. Rev. E 87, 062350 (2013).

    Article  Google Scholar 

  27. Tierno, P. Geometric frustration of colloidal dimers on a honeycomb magnetic lattice. Phys. Rev. Lett. 116, 038303 (2016).

    ADS  Article  Google Scholar 

  28. Hügli, R. V. et al. Artificial kagome spin ice: dimensional reduction, avalanche control and emergent magnetic monopoles. Phil. Trans. R. Soc. A 370, 5767–5782 (2012).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  30. Farhan, A. et al. Exploring hyper-cubic energy landscapes in thermally active finite artificial spin-ice systems. Nat. Phys. 9, 375–382 (2013).

    Article  Google Scholar 

  31. Porro, J. M., Bedoya-Pinto, A., Berger, A. & Vavassori, P. Exploring thermally induced states in square artificial spin-ice arrays. New J. Phys. 15, 055012 (2013).

    ADS  Article  Google Scholar 

  32. Drisko, J., Daunheimer, S. & Cumings, J. FePd3 as a material for studying thermally active artificial spin ice systems. Phys. Rev. B 91, 224406 (2015).

    ADS  Article  Google Scholar 

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

    Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  35. Fan, C. & Wu, F. Y. Ising model with second-neighbor interaction. I. Some exact results and an approximate solution. Phys. Rev. 179, 560–570 (1969).

    ADS  Article  Google Scholar 

  36. Rohrer, H. & Gerber, C. Bicritical and tetracritical behavior of GdAlO3. Phys. Rev. Lett. 38, 909–912 (1977).

    ADS  Article  Google Scholar 

  37. Tokiwa, Y., Garst, M., Gegenwart, P., Bud’ko, S. L. & Canfield, P. C. Quantum bicriticality in the heavy-fermion metamagnet YbAgGe. Phys. Rev. Lett. 111, 116401 (2013).

    ADS  Article  Google Scholar 

  38. Mydosh, J. A. Spin Glasses: An Experimental Introduction (Taylor & Francis, London, 1993).

    Google Scholar 

  39. Louis, D. et al. A tunable magnetic metamaterial based on the dipolar four-state Potts model. Nat. Mater. 17, 1076–1080 (2018).

    ADS  Article  Google Scholar 

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Acknowledgements

The work of J.S., Y.L. and P.S. was funded by the US Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under grant no. DE-SC0010778. The work of C.N. was carried out under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under contract no. DE-AC52-06NA25396. Work at the University of Minnesota was supported by the National Science Foundation under DMR-1507048.

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Contributions

J.S. conceived the quadrupole geometry and experiment. J.S. and P.S. designed the field annealing capability. Y.L. and J.S. prepared the lithographic patterns and configured the annealing apparatus. A.A. and J.D.W. prepared the permalloy deposition for the samples. J.S. measured and processed the data. G.-W.C. developed connections to statistical physics models, and C.N. developed the theoretical Potts model formalism. Under supervision from G.-W.C. and C.N., J.S. performed Monte Carlo simulations. J.S., G.-W.C., C.N. and P.S wrote the manuscript and all authors read and edited it. P.S. supervised the experimental work and coordinated the entire project.

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Correspondence to Joseph Sklenar.

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Supplementary information

Supplementary Information

Supplementary theoretical details, Supplementary References 1–4, Supplementary Figures 1–14

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Sklenar, J., Lao, Y., Albrecht, A. et al. Field-induced phase coexistence in an artificial spin ice. Nature Phys 15, 191–195 (2019). https://doi.org/10.1038/s41567-018-0348-9

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