Direct entropy determination and application to artificial spin ice

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From thermodynamic origins, the concept of entropy has expanded to a range of statistical measures of uncertainty, which may still be thermodynamically significant1,2. However, laboratory measurements of entropy continue to rely on direct measurements of heat. New technologies that can map out myriads of microscopic degrees of freedom suggest direct determination of configurational entropy by counting in systems where it is thermodynamically inaccessible, such as granular3,4,5,6,7,8 and colloidal9,10,11,12,13 materials, proteins14 and lithographically fabricated nanometre-scale arrays. Here, we demonstrate a conditional-probability technique to calculate entropy densities of translation-invariant states on lattices using limited configuration data on small clusters, and apply it to arrays of interacting nanometre-scale magnetic islands (artificial spin ice)15. Models for statistically disordered systems can be assessed by applying the method to relative entropy densities. For artificial spin ice, this analysis shows that nearest-neighbour correlations drive longer-range ones.

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Figure 1: Artificial spin-ice arrays.
Figure 2: Entropy density upper bounds for honeycomb artificial spin ice at four lattice constants as a function of the demagnetization step size ΔHext.
Figure 3: Entropy density upper bounds for square-lattice artificial spin ice.


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We acknowledge the financial support from the Army Research Office and the National Science Foundation MRSEC program (DMR-0820404) and the National Nanotechnology Infrastructure Network. We thank C. Leighton and M. Erickson for permalloy growth.

Author information

P.S. and V.H.C. conceived the initial idea of this project and kept it on track. X.K., J.L. and D.M.G. carried out the experiments and collected data. C.N. made theoretical contributions. P.E.L. analysed data and developed theory.

Correspondence to Paul E. Lammert.

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The authors declare no competing financial interests.

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Lammert, P., Ke, X., Li, J. et al. Direct entropy determination and application to artificial spin ice. Nature Phys 6, 786–789 (2010) doi:10.1038/nphys1728

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