Direct Observation of Thermal Relaxation in Artificial Spin Ice

We study the thermal relaxation of artificial spin ice with photoemission electron microscopy, and are able to directly observe how such a system finds its way from an energetically excited state to the ground state. On plotting vertex-type populations as a function of time, we can characterize the relaxation, which occurs in two stages, namely a string and a domain regime. Kinetic Monte Carlo simulations agree well with the temporal evolution of the magnetic state when including disorder, and the experimental results can be explained by considering the effective interaction energy associated with the separation of pairs of vertex excitations.

Figure 1

Artificial square ice. (a) Scanning electron microscope image ($a=425\mathrm{nm}$) and (b) associated XMCD image resolving vertex-type configurations detailed in (c). Highlighted in orange and yellow are type I vertices, and green and blue are a type II and a type III vertex, respectively. Present in (b) are two type I ground state domains of opposite chirality separated by a domain boundary (pink line) consisting of type II and type III vertices.

Figure 2

XMCD images of thermal relaxation in artificial spin ice: (a) string regime, (b) domain regime, and (c) the ground state (field of view $20\mu \mathrm{m}$; see also the Supplemental Material [20]). Migration of type III vertices (green and yellow dots) in detail: (d) in the string regime, pairs of type III vertices are created and separate, and (e) in the domain regime, a type III vertex travels along various routes within a type II domain boundary (pink line) so that the bigger type I domain expands at the cost of the smaller domain. The path taken by the type III vertex is indicated with colored arrows and $a=425\mathrm{nm}$.

Figure 3

Temporal evolution of vertex-type population. (a) Experimental data obtained at a constant temperature of 350 K, showing a 100% ground state ordering within eight hours. (b) Kinetic Monte Carlo simulation data for a Gaussian disorder in the intrinsic nanomagnet energy barrier with a standard deviation $\sigma =0.05$ and also for the case of no disorder, $\sigma =0$.

Figure 4

Interaction energy as a function of string length connecting two isolated type III vertices. Type I and II vertices are indicated by orange and blue dots, respectively, and the numbers in the schematics indicate the string length as the right-hand type III vertex follows a zigzag path.