Kinetic pathways to the magnetic charge crystal in artificial dipolar spin ice

We experimentally investigate magnetic frustration effects in thermally active artificial kagome spin ice. Starting from a paramagnetic state, the system is cooled down below the Curie temperature of the constituent material. The resulting magnetic configurations show that our arrays are locally brought into the so-called spin ice 2 phase, predicted by at-equilibrium Monte Carlo simulations and characterized by a magnetic charge crystal embedded in a disordered kagome spin lattice. However, by studying our arrays on a larger scale, we find the unambiguous signature of an out-of-equilibrium physics. Comparing our findings with numerical simulations, we interpret the efficiency of our thermalization procedure in terms of kinetic pathways that the system follows upon cooling and which drive the arrays into degenerate low-energy manifolds that are hardly accessible otherwise.

Figure 1

(a) A typical XMCD-PEEM image of our arrays. Black and white contrasts give the local direction of the magnetization of each individual nanomagnet. (b) Schematics of the magnetic charge crystallites deduced from (a). Red and blue dots correspond to $\pm 1$ magnetic charges, respectively. Charge domains are colored in white and green. This image is characterized by a mean nearest-neighbor charge correlator of $-0.3$.

Figure 2

Average values of the pairwise spin correlators defined up to the seventh neighbor and computed within the charge domains (green diamonds, full line) and at the network scale (black diamonds, dashed line). For comparison, values expected for the ground-state manifolds of the nearest-neighbor (NN) SI model and DSI model are represented as red and blue spheres, respectively. The purple spheres correspond to the values given by Monte Carlo simulations for a temperature of $T/J_{\alpha \beta }$ $\simeq 0.034$, which best fits the experimental data. The error bars represent the numerical standard deviations of each correlation type at a given temperature and are computed over the set of sampling Monte Carlo snapshots (see note [33]). The lines linking the experimental data points have no physical meaning and are just guides for the eyes.

Figure 3

Pairwise spin and charge (insets) correlators predicted by (a) Monte Carlo and (c) kinetic simulations. In both cases, long-range dipolar interactions between spins are taken into account. The colored spheres are the averaged experimental data points computed on the network scale and further averaged over the 18 different arrays. (b) The definition of the first seven pairwise spin correlations. The color code is the same for all three images.

Figure 4

Snapshots of the kinetic algorithm at a low temperature. The nanomagnets that are magnetic are represented by a black arrow. The numbers indicate the ordering sequence. Red and blue dots correspond to $\pm 1$ magnetic charges, respectively, whereas black dots are associated with charges that have not yet been fully defined.