Continuous magnetic phase transition in artificial square ice

Critical behavior is very common in many fields of science and a wide variety of many-body systems exhibit emergent critical phenomena. The beauty of critical phase transitions lies in their scale-free properties, such that the temperature dependence of physical parameters of systems differing at the microscopic scale can be described by the same generic power laws. In this work we establish the critical properties of the antiferromagnetic phase transition in artificial square ice, showing that it belongs to the two-dimensional Ising universality class, which extends the applicability of such concepts from atomistic to mesoscopic magnets. Combining soft x-ray resonant magnetic scattering experiments and Monte Carlo simulations, we characterize the transition to the low-temperature long-range order expected for the artificial square-ice system. By measuring the critical scattering, we provide direct quantitative evidence of a continuous magnetic phase transition, obtaining critical exponents which are compatible with those of the two-dimensional Ising universality class. In addition, by varying the blocking temperature relative to the phase transition temperature, we demonstrate its influence on the out-of-equilibrium dynamics due to critical slowing down at the phase transition.

Figure 1

(a) Scanning electron microscopy image of a representative part of the artificial square-ice array. The red arrows represent the structural lattice vectors of length $a$ and the black arrows indicate the magnetic lattice vectors of the ground-state ordering of length $a\sqrt{2}$. (b) Schematic of the antiferromagnetic ground state reflecting the long-range order of the low-temperature phase (left panel), which consists of two two-dimensional Ising systems with orthogonal moments (indicated by the red and blue arrows). This in turn may be viewed as a tile of alternating microvortices. The right panel lists the vertex-type convention, permutations of which produce the 16 possible vertex configurations.

Figure 2

Schematic illustration of the experimental geometry for soft x-ray resonant magnetic scattering experiments with circularly polarized $C_{+}$ synchrotron x rays. $k_i$ ($k_f$) and ${\theta }_i$ (${\theta }_f$) are the incident (final) x-ray wave vector and angle. $Q_x$ and $Q_y$ are the momentum transfer along the $x$ and $y$ directions, respectively.

Figure 3

Critical scattering profiles of an antiferromagnetic Bragg peak at $(Q_x,Q_y)=(\frac{3}{2},-\frac{1}{2})$ taken along the $Q_x$ direction in reciprocal space. The profile from array 1 (a) at 119 K was fitted with the sum of a Lorentzian and Voigt profile and (b) at 158 K was fitted with a Lorentzian profile only. The insets are the corresponding scattering patterns measured on the CCD camera. The (blue) line represents the portion of reciprocal space used to obtain the profiles shown in the main figure.

Figure 4

Intensity of the Bragg peak (red circles) and critical scattering (black circles) extracted by fitting the peak profiles from array 1. The transition temperature was determined to be $T_N=119\pm 0.1\text{K}$ using a power-law fit (red line). Due to the weakness of the critical scattering just below $T_N$, it was not possible to unambiguously separate the two components in the scattered intensities, so that only the Bragg contribution was estimated in the fits.

Figure 5

Temperature dependence of the critical correlation length in the direction perpendicular to the scattering plane for array 1 (black circles), array 2 (blue circles), and array 3 (red circles and diamonds) of artificial square ice. For array 3, data were collected in two different temperature runs, indicated by circles and diamonds, respectively. Solid lines with the corresponding color are fits to a power law giving the critical correlation length exponent $\nu$. The fit was performed using the data points indicated by closed symbols.

Figure 6

(a) Staggered magnetization per site, (b) magnetic susceptibility, and (c) specific-heat scaling collapse assuming critical exponents of a two-dimensional Ising system.

Figure 7

Temperature dependence of the coercive field $H_c$ for array 1 (black circles), array 2 (red up-pointing triangles), and array 3 (blue down-pointing triangles) obtained from magneto-optical Kerr effect (MOKE) measurements, displayed using a square-root scale. Solid lines correspond to the fits performed using Eq. (3) to obtain the single-particle blocking temperature.

Figure 8

Binder cumulant as a function of temperature for a range of system sizes. Solid lines are guides to the eye.

Figure 9

Population percentage of vertex types as a function of temperature for a system size of $n=40$.