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 $H_c=H_{c0}+\alpha ln(n+1)$. 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.

Figure 1

Top (a) and side (b) views of the ${\mathrm{Ni}}_{26}$ IOLS sample and images of the surface of ${\mathrm{Ni}}_{0.5}$ (c) and ${\mathrm{Ni}}_{\mathrm{film}}$ (d).

Figure 2

Unit cell of the IOLS surrounded by the spheres of the initial opal template (a) and basic element of the IOLS (b).

Figure 3

Geometry of the experiment with the directions of the magnetic field H and $\left[111\right],\left[20\overline{2}\right]$ axes.

Figure 4

Magnetization reversal curves for ${\mathrm{Ni}}_{26}$ IOLS for different angles $\theta$ plotted as a function of the external (a) and the internal (b) magnetic field. Critical fields derived from the remagnetization curve for $\theta =0^{\circ }$ (c).

Figure 5

Thickness dependencies of (a) $H_c$, (b) $H_{ir}$, and (c) $H_s$ for the nickel based IOLS.

Figure 6

Angular dependence of the coercivity $H_c$ for a Ni-based IOLS with different thicknesses (a). The main crystallographic axes of IOLS corresponding to the field orientation are shown on the top of the graph. Angular dependence of the coercivity $H_c$ for ${\mathrm{Ni}}_{0.5}$ and the Ni film (b).

Figure 7

Thickness dependence of the orientation dependent fit parameters $H_c^0$ and $\lambda$ for Ni-based IOLS (empty symbols) and for the film (solid symbols). The solid line shows the fit by the logarithmic function (a). Thickness dependence of the critical angle corresponding to the maximum coercive field for Ni IOLS (empty symbols) and for the film (solid symbols) (b).