Nonlinear geometric scaling of coercivity in a three-dimensional nanoscale analog of spin ice

Magnetization hysteresis loops of a three-dimensional nanoscale analog of spin ice based on the nickel inverse opal-like structure (IOLS) have been studied at room temperature. The samples are produced by filling nickel into the voids of artificial opal-like films. The spin ice behavior is induced by tetrahedral elements within the IOLS, which have the same arrangement of magnetic moments as a spin ice. The thickness of the films vary from a two-dimensional, i.e., single-layered, antidot array to a three-dimensional, i.e., multilayered, structure. The coercive force, the saturation, and the irreversibility field have been measured in dependence of the thickness of the IOLS for in-plane and out-of-plane applied fields. The irreversibility and saturation fields change abruptly from the antidot array to the three-dimensional IOLS and remain constant upon further increase of the number of layers $n$. The coercive force ${H}_{c}$ seems to increase logarithmically with increasing $n$ as ${H}_{c}={H}_{c0}+\ensuremath{\alpha}ln(n+1)$. The logarithmic law implies the avalanchelike remagnetization of anisotropic structural elements connecting tetrahedral and cubic nodes in the IOLS. We conclude that the ``ice rule'' is the base of mechanism regulating this process.