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

Nature volume 540, pages 410–413 (2016)Cite this article

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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.

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Figure 1: Role of the nearest-neighbour coupling strength.
Figure 2: Experimental results.
Figure 3: Magnetic structure factors and pinch-point analysis.
Figure 4: Magnetic monopoles in square-ice systems.

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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

Authors and Affiliations

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

    Yann Perrin, Benjamin Canals & Nicolas Rougemaille

  2. Université Grenoble Alpes, Institut NÉEL, Grenoble, F-38000, France

    Yann Perrin, Benjamin Canals & Nicolas Rougemaille

Authors
  1. Yann Perrin
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  2. Benjamin Canals
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  3. Nicolas Rougemaille
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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.

Corresponding author

Correspondence to Nicolas Rougemaille.

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Competing interests

The authors declare no competing financial interests.

Additional information

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 figures and tables

Extended Data Figure 1 Dumbbell description of the nanomagnets.

Map of J1/J2 as a function of l/a and h/a for an isolated vertex. The condition J1 = J2 is indicated by the dark line. Our results perfectly reproduce those reported in ref. 29. The white dots indicate the values that correspond to the different samples studied here.

Extended Data Figure 2 Illustration of the two-step electron-beam lithography process.

a, Schematic of the gold bases subsequently used to shift the vertical sublattice. b, Schematic of the permalloy magnets on the vertical and horizontal sublattices.

Extended Data Figure 3 Magnetic structure factor of the square-ice model.

a, Sketch of the vectors involved in equation (1). b, Magnetic structure factor for an ideal square-ice model, computed for 1,000 low-energy states made of N = 840 spins. Red circles indicate the regions of interest for the intensity profiles in Fig. 3f and Extended Data Fig. 5.

Extended Data Figure 4 Loop flips in the square lattice.

Schematic illustrating the open (red arrows) and closed (green arrows) spin loops used to generate a low-energy configuration that is representative of the massively degenerate ground-state manifold of the square-ice model19. The lattice contains 840 spins and the number of loops that are flipped between two decorrelated configurations is set to N = 840. L corresponds to the linear size of the square lattices.

Extended Data Figure 5 Analysis of the pinch points.

a–d, Maps of the pinch points indicated by red circles in Extended Data Fig. 3b (left) and associated intensity profiles along the qv = 0 direction (right), for different lattice sizes L: L = 10 (a), L = 20 (b), L = 40 (c), L = 80 (d). The colour scale refers to the intensity at a given point of reciprocal space. The coordinates (qu, qv) are relative to the intensity profile and do not correspond to the real axes of reciprocal space. The red curves are single-peaked Lorentzian fits; the points represent the mean and the error bars represent the standard deviation calculated from 1,000 random ice-rule configurations.

Extended Data Figure 6 Magnetic monopoles in artificial square ice.

Experimental spin configuration for h = 100 nm. Type-I and -II vertices appear as blue and red squares, respectively. Monopoles appear as red and blue circles. Their associated pairing is represented by black ellipses.

Extended Data Table 1 Correlation lengths extracted from the intensity profiles
Full size table

Supplementary information

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. (MP4 1036 kb)

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. (MP4 5713 kb)

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Perrin, Y., Canals, B. & Rougemaille, N. Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice. Nature 540, 410–413 (2016). https://doi.org/10.1038/nature20155

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