Research ArticlePHYSICS

Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice

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Science Advances  08 Feb 2019:
Vol. 5, no. 2, eaav6380
DOI: 10.1126/sciadv.aav6380
  • Fig. 1 Thermally activated two-dimensional artificial square ice with height offsets between nanomagnets.

    (A) Tilted-sample scanning electron microscopy (SEM) image of an artificial square ice with an introduced height offset h, which can be varied from sample to sample, until the competing interactions J1 and J2 are equalized and an extensive spin ice degeneracy is achieved. Scale bar, 400 nm. (B) XMCD image of the same artificial square ice array. Nanomagnets with moments pointing toward the incoming x-rays (indicated by a yellow arrow) appear dark, while those opposing the x-ray direction appear with bright contrast. (C) The 16 possible moment configurations on a four-nanomagnet vertex are traditionally listed into four topological types. Without a height offset (h = 0 nm), the ice rule–obeying (two-in-two-out) type I and II configurations have a significantly different energy. Once a critical height offset is introduced, their energies are equalized and spin ice degeneracy is realized. Highlighted with magenta, cyan blue, and yellow frames in (B) and (C) are type I, type II, and type III vertices, respectively.

  • Fig. 2 Vertex populations, magnetic structure factors, and pinch-point analysis.

    (A to C) Low-energy moment configurations achieved after thermal annealing in artificial square ice arrays with height offsets of (A) h = 55 nm, (B) h = 145 nm, and (C) h = 180 nm. Scale bars, 1 μm. (D) Average vertex-type populations of thermalized artificial square ice, plotted as a function of introduced height offsets. Type I vertices dominate the configuration landscape up to an offset of 40 nm but continue to decrease in population with increasing height offset. A turning point is observed at an offset around 80 nm, where type I and II populations reach nearly identical values. The type II population continues to rise with increasing height offset and reaches twice the population of type I vertices at a height offset between 145 and 155 nm. As the height offset is increased beyond this critical value, type II vertices start to fully dominate the moment configuration in the spin ice. (E) Magnetic structure factor of an artificial square ice with a height offset of 145 nm. The structure factor is calculated from magnetic moment configurations recorded with PEEM imaging and exhibits pinch-point singularities, a typical feature of a magnetic Coulomb phase. The line scan through (qx, qy) = (2, 2) is fitted by a Lorentzian function (black curve in inset) from which an average spin-spin correlation length ξ = 10.8a ± 0.1 is derived. r.l.u., reciprocal lattice unit.

  • Fig. 3 Temporal evolution of emergent magnetic monopoles.

    XMCD image sequence (recorded at T = 190 K) highlighting the thermally driven motion of emergent magnetic monopole defects (blue dots: Q = −2q, red dots: Q = +2q) in two-dimensional artificial square ice with a height offset h = 145 nm. Arrows of different colors (magenta, cyan blue, and yellow) indicate sequential changes in moment configurations at each instant of time (7, 14, and 21 s). The green bar and the big white arrow indicate a length of 1 μm and the incoming x-ray direction, respectively.

  • Fig. 4 Debye-Hückel behavior and crystallization of emergent magnetic monopoles.

    (A) Ratio of correlated to uncorrelated monopole defects observed in the h = 145 nm sample (black dots from Eq. 2) compared to the prediction from the Debye-Hückel theory with Bjerrum association corrections (blue stars from Eq. 1). The error bars correspond to real-time thermal fluctuations over observations of approximately 15 min at each temperature. The best fit is obtained for a magnetic charge Q = 9.765 × 10−12Am and a magnetization M = 54 kA/m in the Debye-Hückel analysis. The overall monopole density ρ as a function of temperature is shown as an inset. (B) Crystallization order parameter over the same temperature range.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/2/eaav6380/DC1

    Fig. S1. SEM image of quasi–three-dimensional artificial spin ice.

    Fig. S2. Sample fabrication process.

    Fig. S3. Magnetic structure factors as a function of introduced height offset.

    Fig. S4. Illustrations of correlated and uncorrelated emergent magnetic monopoles.

    Fig. S5. Illustration of possible low-energy configurations, whether being a dilute gas of magnetic charges or a magnetic monopole crystalline ground state.

    Movie S1. XMCD image sequence of a thermally activated extensively degenerate artificial square ice (height offset = 145 nm) recorded at 190 K.

    Movie S2. Emergent magnetic monopole dynamics at 210 K (height offset = 145 nm).

  • Supplementary Materials

    The PDF file includes:

    • Fig. S1. SEM image of quasi–three-dimensional artificial spin ice.
    • Fig. S2. Sample fabrication process.
    • Fig. S3. Magnetic structure factors as a function of introduced height offset.
    • Fig. S4. Illustrations of correlated and uncorrelated emergent magnetic monopoles.
    • Fig. S5. Illustration of possible low-energy configurations, whether being a dilute gas of magnetic charges or a magnetic monopole crystalline ground state.
    • Legends for movies S1 and S2

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    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.avi format). XMCD image sequence of a thermally activated extensively degenerate artificial square ice (height offset = 145 nm) recorded at 190 K.
    • Movie S2 (.avi format). Emergent magnetic monopole dynamics at 210 K (height offset = 145 nm).

    Files in this Data Supplement:

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