Vogel-Fulcher-Tammann freezing of a thermally fluctuating artificial spin ice probed by x-ray photon correlation spectroscopy

We report on the crossover from the thermal to the athermal regime of an artificial spin ice formed from a square array of magnetic islands whose lateral size, 30 nm $\times$ 70 nm, is small enough that they are dynamic at room temperature. We used resonant magnetic soft x-ray photon correlation spectroscopy as a method to observe the time-time correlations of the fluctuating magnetic configurations of spin ice during cooling, which are found to slow abruptly as a freezing temperature of $T_0=178\pm 5$ K is approached. This slowing is well described by a Vogel-Fulcher-Tammann law, implying that the frozen state is glassy, with the freezing temperature being commensurate with the strength of magnetostatic interaction energies in the array. The activation temperature, $T_{\mathrm{A}}=40\pm 10$ K, is much less than that expected from a Stoner-Wohlfarth coherent rotation model. Zero-field-cooled/field-cooled magnetometry reveals a freeing up of fluctuations of states within islands above this temperature, caused by variation in the local anisotropy axes at the oxidised edges. This Vogel-Fulcher-Tammann behavior implies that the system enters a glassy state upon freezing, which is unexpected for a system with a well-defined ground state.

Figure 1

Scanning electron micrograph of an artificial spin ice with a lattice spacing of 240 nm, with Permalloy islands of lateral size 30 nm $\times$ 70 nm and 8 nm in thickness.

Figure 2

Coherent soft x-ray scattering measurements. (a) Schematic of the experimental XPCS setup, showing the incoming x-ray beam and scattered beams from the ASI array, which is represented by an atomic force microscopy image. The three main diffraction spots in the row above the specular reflection are shown here. Since the diffraction spots arise from a small, disordered region of the sample that has been coherently illuminated, they contain speckle contrast. (b) Fraunhofer diffraction obtained using the 10-$\mu \mathrm{m}$ pinhole and straight-through beam on to the CCD detector.

Figure 3

XPCS results. (a) Normalized $g_2\left(\tau \right)$ functions at various temperatures. The lines are fits to Eq. (2). (b) Relaxation times $t_{\mathrm{r}}$ as a function of temperature, fitted by a Vogel-Fulcher-Tammann (VFT) law. Also plotted is the superparamagnetic Néel-Brown (NB) law and the critical slowing down (CSD) law near a phase transition, which fit poorly. Data points for $T=180$ K (where $t_{\mathrm{r}}\gg {10}^5$ s) and $T=295$ K (where $t_{\mathrm{r}}\lesssim 5$ s) are not plotted since the experimental constraints mean that determining values within these bounds was not possible.

Figure 4

ZFC (open circles) and FC (solid circles) magnetization measurements under different applied magnetic probe fields of (a) 50 mT, (b) 35 mT, (c) 25 mT, and (d) 5 mT. The average blocking temperature, $T_{\mathrm{B}}$, is also defined by the maximum in the ZFC curve and is indicated by the black arrows ($T_{\mathrm{B}}>400$ K at 5 mT). The gray arrow indicates the $\sim 40$ K feature in the ZFC data. The field was applied along the $\left[01\right]$ direction of the ASI lattice.

Figure 5

The temperature dependence of the coercivity measured along [01]. The line is a fit to Sharrock's equation, the departure from which is clear below 40 K (which is indicated with the dotted line).