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Thermal ground-state ordering and elementary excitations in artificial magnetic square ice

Abstract

Recent advances in nanotechnology allow model systems to be constructed, in which frustrated interactions can be tuned at will, such as artificial spin ice. The symmetry of the square ice lattice leads to the emergence of a long-range-ordered ground state from the manifold of frustrated states. However, it is experimentally very difficult to access using the effective thermodynamics of rotating-field demagnetization protocols, because the energy barriers to thermal equilibrium are extremely large. Here we study an as-fabricated sample that approaches the ground state very closely. We identify the small localized departures from the ground state as elementary excitations of the system, at frequencies that follow a Boltzmann law. We therefore identify the state we observe as the frozen-in residue of true thermodynamics that occurred during the fabrication of the sample. The relative proportions of different excitations are suggestive of monopole interactions during thermalization.

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Figure 1: Artificial square ice of single-domain nanobar magnets.
Figure 2: MFM images of a 400-nm-pitch as-grown square ice array.
Figure 3: Elementary excitations above the GS of square ice.
Figure 4: Statistics of square ice excitations.

References

  1. 1

    Vedmedenko, E. Competing Interactions and Pattern Formation in Nanoworld (Wiley, 2007).

    Book  Google Scholar 

  2. 2

    Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935).

    Article  Google Scholar 

  3. 3

    Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7 . Phys. Rev. Lett. 79, 2554–2557 (1997).

    ADS  Article  Google Scholar 

  4. 4

    Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495–1501 (2001).

    ADS  Article  Google Scholar 

  5. 5

    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 

  6. 6

    Remhof, A. et al. Magnetostatic interactions on a square lattice. Phys. Rev. B 77, 134409 (2008).

    ADS  Article  Google Scholar 

  7. 7

    Tanaka, M., Saitoh, E., Miyajima, H., Yamaoka, T. & Iye, Y. Magnetic interaction in a ferromagnetic honeycomb nanoscale network. Phys. Rev. B 73, 052411 (2006).

    ADS  Article  Google Scholar 

  8. 8

    Mengotti, E. et al. Building blocks of an artificial kagome spin ice: Photoemission electron microscopy of arrays of ferromagnetic islands. Phys. Rev. B 78, 144402 (2008).

    ADS  Article  Google Scholar 

  9. 9

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

    ADS  Article  Google Scholar 

  10. 10

    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. Nature Phys. 6, 359–363 (2010).

    ADS  Article  Google Scholar 

  11. 11

    Schumann, A., Sothmann, B., Szary, P. & Zabel, H. Charge ordering of magnetic monopoles in triangular spin ice patterns. Appl. Phys. Lett. 97, 022509 (2010).

    ADS  Article  Google Scholar 

  12. 12

    Mengotti, E. et al. Real-space observation of emergent magnetic monopoles and associated Dirac strings in artificial kagome spin ice. Nature Phys. advance online publication, 10.1038/nphys1794 (17 October 2010).

  13. 13

    Nisoli, C. et al. Ground state lost but degeneracy found: The effective thermodynamics of ‘artificial spin ice’. Phys. Rev. Lett. 98, 217103 (2007).

    Article  Google Scholar 

  14. 14

    Ke, X. et al. Energy minimization and ac demagnetization in a nanomagnet array. Phys. Rev. Lett. 101, 037205 (2008).

    ADS  Article  Google Scholar 

  15. 15

    Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).

    ADS  Article  Google Scholar 

  16. 16

    Jaubert, L. D. C. & Holdsworth, P. C. W. Signatures of magnetic monopole and Dirac string dynamics in spin ice. Nature Phys. 5, 258–261 (2009).

    ADS  Article  Google Scholar 

  17. 17

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

    ADS  Article  Google Scholar 

  18. 18

    Morris, D. J. P. et al. Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7 . Science 326, 411–414 (2009).

    ADS  Article  Google Scholar 

  19. 19

    Kadowaki, H. et al. Observation of magnetic monopoles in spin ice. J. Phys. Soc. Jpn 78, 103706 (2009).

    ADS  Article  Google Scholar 

  20. 20

    Bramwell, S. T. et al. Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956–959 (2009).

    ADS  Article  Google Scholar 

  21. 21

    Möller, G. & Moessner, R. Magnetic multipole analysis of kagome and artificial ice dipolar arrays. Phys. Rev. B 80, 140409(R) (2009).

    ADS  Article  Google Scholar 

  22. 22

    Mól, L. A. et al. Magnetic monopole and string excitations in two-dimensional spin ice. J. Appl. Phys. 106, 063913 (2009).

    ADS  Article  Google Scholar 

  23. 23

    Mól, L. A., Moura-Melo, W. A. & Pereira, A. R. Conditions for free magnetic monopoles in synthetic square ice dipolar nanoarrays. Phys. Rev. B 82, 054434 (2010).

    ADS  Article  Google Scholar 

  24. 24

    Jaeger, H. M., Nagel, S. R. & Behringer, R. P. Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259–1273 (1996).

    ADS  Article  Google Scholar 

  25. 25

    D’Anna, G., Mayor, P., Barrat, A., Loreto, V. & Nori, F. Observing Brownian motion in vibration-fluidized granular matter. Nature 424, 909–912 (2003).

    ADS  Article  Google Scholar 

  26. 26

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

    Article  Google Scholar 

  27. 27

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

    ADS  Article  Google Scholar 

  28. 28

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

    ADS  Article  Google Scholar 

  29. 29

    Li, J. et al. Comparing artificial frustrated magnets by tuning the symmetry of nanoscale permalloy arrays. Phys. Rev. B 81, 092406 (2010).

    ADS  Article  Google Scholar 

  30. 30

    Melko, R. G., den Hertog, B. C. & Gingras, M. P. Long-range order at low temperatures in dipolar spin ice. Phys. Rev. Lett. 87, 067203 (2001).

    ADS  Article  Google Scholar 

  31. 31

    Joseph, R. I. & Schlömann, E. Demagnetizing field in nonellipsoidal bodies. J. Appl. Phys. 36, 1579–1593 (1965).

    ADS  Article  Google Scholar 

  32. 32

    Libál, A., Reichhardt, C. & Olson Reichhardt, C. J. Realizing colloidal artificial ice on arrays of optical traps. Phys. Rev. Lett. 97, 228302 (2006).

    ADS  Article  Google Scholar 

  33. 33

    Libál, A., Olson Reichhardt, C. J. & Reichhardt, C. Creating artificial ice states using vortices in nanostructured superconductors. Phys. Rev. Lett. 102, 237004 (2009).

    ADS  Article  Google Scholar 

  34. 34

    Davidović, D. et al. Correlations and disorder in arrays of magnetically coupled superconducting rings. Phys. Rev. Lett. 76, 815–818 (1996).

    ADS  Article  Google Scholar 

  35. 35

    Davidović, D. et al. Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings. Phys. Rev. B 55, 6518–6540 (1997).

    ADS  Article  Google Scholar 

  36. 36

    Hilgenkamp, H. et al. Ordering and manipulation of the magnetic moments in large-scale superconducting π-loop arrays. Nature 422, 50–53 (2003).

    ADS  Article  Google Scholar 

  37. 37

    Kirtley, J. R., Tsuei, C. C., Ariando, Smilde, H. J. H. & Hilgenkamp, H. Antiferromagnetic ordering in arrays of superconducting π-rings. Phys. Rev. B 72, 214521 (2005).

    ADS  Article  Google Scholar 

  38. 38

    Han, Y. et al. Geometrical frustration in buckled colloidal monolayers. Nature 456, 898–903 (2008).

    ADS  Article  Google Scholar 

  39. 39

    Shokef, Y. & Lubensky, T. C. Stripes, zigzags, and slow dynamics in buckled hard spheres. Phys. Rev. Lett. 102, 048303 (2009).

    ADS  Article  Google Scholar 

  40. 40

    Mengotti, E. et al. Dipolar energy states in clusters of perpendicular magnetic nanoislands. J. Appl. Phys. 105, 113113 (2009).

    ADS  Article  Google Scholar 

  41. 41

    Budrikis, Z., Politi, P. & Stamps, R. L. Vertex dynamics in finite two-dimensional square spin ices. Phys. Rev. Lett. 105, 017201 (2010).

    ADS  Article  Google Scholar 

  42. 42

    Zhu, Y. (ed.) Modern Techniques for Characterizing Magnetic Materials (Springer, 2005).

  43. 43

    Shpyrko, O. G. et al. Direct measurement of antiferromagnetic domain fluctuations. Nature 447, 68–71 (2007).

    ADS  Article  Google Scholar 

  44. 44

    Pierce, M. S. et al. Quasistatic X-ray speckle metrology of microscopic magnetic return-point memory. Phys. Rev. Lett. 90, 175502 (2003).

    ADS  Article  Google Scholar 

  45. 45

    Horcas, I. et al. WSxM: A software for scanning probe microscopy and a tool for nanotechnology. Rev. Sci. Inst. 78, 013705 (2007).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

This work was supported financially by EPSRC and the STFC Centre for Materials Physics and Chemistry. The research was carried out in part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the US Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02-98CH10886.

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Contributions

J.P.M. and A.S. conducted sample fabrication. J.P.M conducted the MFM, data processing, calculations and analysis. C.H.M. and S.L. supervised the work, and contributed to discussion and direction. All authors discussed the results and commented on the manuscript.

Corresponding author

Correspondence to Christopher H. Marrows.

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The authors declare no competing financial interests.

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Morgan, J., Stein, A., Langridge, S. et al. Thermal ground-state ordering and elementary excitations in artificial magnetic square ice. Nature Phys 7, 75–79 (2011). https://doi.org/10.1038/nphys1853

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