|Periodic and Quasiperiodic Artificial Spin Ices: Magnetic Structure and Dynamics|
Spin ices are rare earth pyrochlores where the crystal geometry leads to frustration of the rare earth moments , which meet in fours at tetrahedra in the lattice. Like water ice, they violate the third law of thermodynamics, as the frustration prevents the formation of a unique ground state. Nanotechnology allows many of the essential features of this physical system to be reproduced in arrays of patterned nanomagnets, known as artificial spin ices . Each nanomagnet is small enough to be a single domain, representing a single Ising spin in the model. When their moments meet at the vertices of a square grid (left panel of Figure 1), a two-dimensional analogue of the pyrochlore system is formed . This nanotechnological approach offers the opportunity to continuously tune the various parameters controlling the magnetic microstate of this model statistical mechanical system, such as the interelement spacing that controls the magnetostatic coupling between islands (leading to artificial spin ices sometimes being described as designer matter), accompanied by the ability to inspect each microstate using advanced magnetic microscopy. A significant difference to the pyrochlore spin-ices is that the change in symmetry gives rise to a true long-range ordered ground state that is antiferromagnetic, although the frustrated interactions in these athermal systems mean that its observation is extremely difficult as they cannot thermally equilibrate .
Methods of reaching this ground state include single-shot annealing during fabrication of the system  or true annealing above the Curie point of the individual nanoislands [6,7]. The recent demonstration that islands with very low volume (and hence anisotropy energy barrier) can undergo thermal fluctuations at or close to room temperature  also permits such systems to slowly relax into their ground state. In that case, the low barrier was achieved by making the nanoislands from atomically thin films. We have achieved the same result by making our islands laterally small, as shown in the left hand panel of Figure 1. By proper selection of island size, we can tune the room temperature fluctuation rates in the range 10s of milliseonds to 100s of seconds (as determined by the frequency dependence of the coercivity of the arrays measured using focused MOKE).
Imaging such islands has been carried out at the XM-1 soft x-ray full-field transmission microscope at the Advanced Light Source. In this case, the islands were fabricated from CoFeB to improve the x-ray magnetic circular dichroism contrast. The arrays were fabricated on x-ray transparent Si3N4 membranes with on-membrane heaters and thermometers. Thermal activity has been observed, with flip rates being dependent on both temperature and field. Larger islands (80 nm × 250 nm) are thermally stable at remanence, but can be made to flip at a rate of about 0.1 islands per second at a temperature of 300 K by tuning the barrier height to approximately 10% of its usual size according to the Bean-Livingston formula E = KV(1-H/H0)2. Raising the temperature to 415 K increased the flip rates by an order of magnitude.
Islands that are too small to image directly have been studied using soft x-ray photon correlation spectroscopy at the BLADE beamline at Diamond. Here, a small part of the sample is illuminated coherently through a pinhole. The resulting Bragg spots have speckle that can be imaged on a CCD camera. As the sample’s magnetic configuration varies with time, so must the details of the speckle. Fluctuations in islands as small as 30 nm × 75 nm have been observed with this method.
More complex behaviour is found in quasiperiodic structures, such as the Penrose ice depicted in the right hand panel of Figure 1. These have a multiplicity of different structural vertex types, each with its own energy spectrum. MFM imaging shows that high-magnetic charge/high-energy configurations are avoided, but long-range magnetic order is difficult to observe. Analysis of the vertex energies suggests that the structure can be divided into two parts: a percolating ‘skeleton’ that possesses a rigid twofold ground state, containing spins (individual or in groups) that are ‘flippable’ and do not affect the overall system energy, a signature of true frustration. Correlations observed in the skeleton part extends to a few islands at most.
Sophie Morley1, Dong Shi1, Aaron Stein2, Diego Albe-Venero3, Aleš Hrabec1, Philippa Shepley1, Susan Riley1, Mi-Young Im4, Peter Fischer4, Mark Rosamond1, Paul Steadman5, Matt Bryan6, Dan Allwood6, Sean Langridge3, Jason Morgan1, and Christopher Marrows1
 S. T. Bramwell and M. J. P. Gingras, Science 294, 1495 (2001).