Emergent ice rule and magnetic charge screening from vertex frustration in artificial spin ice

Journal name:
Nature Physics
Year published:
Published online


Artificial spin ice comprises a class of frustrated arrays of interacting single-domain ferromagnetic nanostructures. Previous studies of artificial spin ice have focused on simple lattices based on natural frustrated materials. Here we experimentally examine artificial spin ice created on the shakti lattice, a structure that does not directly correspond to any known natural magnetic material.  On the shakti lattice, none of the near-neighbour interactions is locally frustrated, but instead the lattice topology frustrates the interactions leading to a high degree of degeneracy. We demonstrate that the shakti system achieves a physical realization of the classic six-vertex model ground state. Furthermore, we observe that the mixed coordination of the shakti lattice leads to crystallization of effective magnetic charges and the screening of magnetic excitations, underscoring the importance of magnetic charge as the relevant degree of freedom in artificial spin ice and opening new possibilities for studies of its dynamics.

At a glance


  1. Sketches of several different artificial spin-ice lattices.
    Figure 1: Sketches of several different artificial spin-ice lattices.

    a,b, The square (a) and kagome (b) lattices that have been extensively investigated in previous studies. c,d, The short-island shakti (c) and long-island shakti (d) lattices considered in this work. The darkened islands in c,d define a plaquette of the lattice, which is the basis for understanding its frustration.

  2. Experimental realization of the shakti lattice.
    Figure 2: Experimental realization of the shakti lattice.

    ac, Scanning electron micrographs of artificial square spin ice (a), the short-island shakti lattice (b) and the long-island shakti lattice (c). The island separation is 360 nm for all of these images, and the scale bar in a also applies to all of the images. df, Magnetic force microscope images of the corresponding lattices in ac. The black and white contrast indicates the magnetic poles of the nanomagnets. gi, Representation of the magnetic moments extracted from the corresponding images in df as arrows. Note that the ordered moments of the square lattice demonstrate the effectiveness of our thermalization protocol.

  3. Mapping the shakti lattice to the six-vertex model.
    Figure 3: Mapping the shakti lattice to the six-vertex model.

    a, The vertex types found in the shakti lattice, showing only the short-island shakti lattice. Consistent with the standard literature nomenclature for the vertices in artificial square spin ice, we number the vertices with Roman numerals in order of increasing energy. We distinguish vertices with different numbers of islands by a subscript indicating the number of islands. b, A map of the three-island vertex configurations in the magnetic force microscope image in Fig. 2e. The solid lines indicate the boundaries of the plaquettes, the dashed lines indicate the two-island vertices in the middle of each plaquette, and the directions of the island moments are reproduced as red arrows in the upper-leftmost plaquette for illustrative purposes. The circles indicate the location of the three-moment vertices; the Type I3 vertices are denoted by open circles, and the higher energy Type II3 vertices are denoted by filled circles. c, The six degenerate ground-state configurations for arrangement of the three-moment vertex types on a plaquette.

  4. Vertex population fractions for the shakti lattice.
    Figure 4: Vertex population fractions for the shakti lattice.

    af, Depiction of the vertex fractions for the short-island shakti lattice (ac), the long-island shakti lattice (d,e) and the square lattice (f), all as a function of lattice spacing. These plots show that the system converged to the ground state at small lattice spacing as expected. In the ground states of all three lattices, the four-island vertices will all be in the Type I4 configuration. In the ground state of both the shakti lattices, half of the three-island vertices will be in the (minimum energy) Type I3 configuration and half will be in the (defect) Type II3 configuration. Finally, all of the two-island vertices in the short-island shakti lattice will be Type I2 in the ground state. Error bars correspond to the inverse square root of the number of vertices observed for each type.

  5. Charge ordering in the shakti lattices.
    Figure 5: Charge ordering in the shakti lattices.

    a,b, Magnetic force microscope (MFM) images of the 320 nm short-island (a) and long-island (b) shakti lattices. c,d, Schematics of the charge-ordered three-moment vertices, where the red and blue dots correspond to three-island vertices belonging to the two degenerate antiferromagnetic charge orderings. Note that the charge order of the three-moment vertices coexists in a background of disordered island moments (as evidenced by the absence of order in the MFM images).

  6. Evidence of magnetic charge screening in the long-island shakti lattice.
    Figure 6: Evidence of magnetic charge screening in the long-island shakti lattice.

    a, Average of the relative net charge (Qnn) of an excited Type III4 vertexs nearest neighbours. The blue line (MC 1) represents the Monte Carlo simulation that included only nearest-neighbour vertex interactions, and the red line (MC 2) represents the simulation that also included long-range charge–charge interactions between the magnetically charged vertices; inset: Qnn plotted as a function of the fraction of charge excitations for the shakti long-island lattices and for the kagome lattice, showing opposite behaviour; as temperature decreases, the screening in kagome also decreases. The lines are Monte Carlo simulations. b, The probability of . Again, the blue line (MC 1) represents the Monte Carlo simulation that included only nearest-neighbour vertex interactions, and the red line (MC 2) represents the simulation that also included long-range charge–charge interactions between the magnetically charged vertices. In both a and b, only simulations that include long-range charge–charge interactions can replicate the behaviour of the experimental data. Error bars for each lattice spacing are calculated from the standard deviation of Qnn over all of the excitations for those arrays.


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Author information


  1. Department of Physics and Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

    • Ian Gilbert,
    • Bryce Fore &
    • Peter Schiffer
  2. Theoretical Division, and Center for Nonlinear Studies MS B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

    • Gia-Wei Chern &
    • Cristiano Nisoli
  3. Department of Physics and Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, USA

    • Sheng Zhang
  4. Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA

    • Liam OBrien
  5. Thin Film Magnetism Group, Department of Physics, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

    • Liam OBrien


C.N. and P.S. designed this study and supervised the experiments, simulations and data analysis. Artificial spin-ice samples were fabricated by I.G. and S.Z. Thin-film deposition and thermal annealing was done by L.OB. Magnetic force microscopy and data analysis was performed by I.G. with assistance from B.F. Simulations and theoretical interpretation was given by G-W.C. and C.N. The paper was written by I.G., G-W.C., C.N. and P.S. with input from all of the co-authors.

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