Thermal activation, long-range ordering, and topological frustration of artificial spin ice

Frustrated systems, typically characterized by competing interactions that cannot all be simultaneously satisfied, are ubiquitous in nature and display many rich phenomena and novel physics. Artificial spin ices (ASIs), arrays of lithographically patterned Ising-like single-domain magnetic nanostructures, are highly tunable systems that have proven to be a novel method for studying the effects of frustration and associated properties. The strength and nature of the frustrated interactions between individual magnets are readily tuned by design and the exact microstate of the system can be determined by a variety of characterization techniques. Recently, thermal activation of ASI systems has been demonstrated, introducing the spontaneous reversal of individual magnets and allowing for new explorations of novel phase transitions and phenomena using these systems. In this work, we introduce a new, robust material with favorable magnetic properties for studying thermally active ASI and use it to investigate a variety of ASI geometries. We reproduce previously reported perfect ground-state ordering in the square geometry and present studies of the kagome lattice showing the highest yet degree of ordering observed in this fully frustrated system. We consider theoretical predictions of long-range order in ASI and use both our experimental studies and kinetic Monte Carlo simulations to evaluate these predictions. Next, we introduce controlled topological defects into our square ASI samples and observe a new, extended frustration effect of the system. When we introduce a dislocation into the lattice, we still see large domains of ground-state order, but, in every sample, a domain wall containing higher energy spin arrangements originates from the dislocation, resolving a discontinuity in the ground-state order parameter. Locally, the magnets are unfrustrated, but frustration of the lattice persists due to its topology. We demonstrate the first direct imaging of spin configurations resulting from topological frustration in any system and make predictions on how dislocations could affect properties in numerous materials systems.

Multiscale Modeling and Simulation of Stepped Crystal Surfaces

A primary goal of this dissertation is to understand the links between mathematical models that describe crystal surfaces at three fundamental length scales: The scale of individual atoms, the scale of collections of atoms forming crystal defects, and macroscopic scale. Characterizing connections between different classes of models is a critical task for gaining insight into the physics they describe, a long-standing objective in applied analysis, and also highly relevant in engineering applications. The key concept I use in each problem addressed in this thesis is coarse graining, which is a strategy for connecting fine representations or models with coarser representations. Often this idea is invoked to reduce a large discrete system to an appropriate continuum description, e.g. individual particles are represented by a continuous density. While there is no general theory of coarse graining, one closely related mathematical approach is asymptotic analysis, i.e. the description of limiting behavior as some parameter becomes very large or very small. In the case of crystalline solids, it is natural to consider cases where the number of particles is large or where the lattice spacing is small. Limits such as these often make explicit the nature of links between models capturing different scales, and, once established, provide a means of improving our understanding, or the models themselves. Finding appropriate variables whose limits illustrate the important connections between models is no easy task, however. This is one area where computer simulation is extremely helpful, as it allows us to see the results of complex dynamics and gather clues regarding the roles of different physical quantities. On the other hand, connections between models enable the development of novel multiscale computational schemes, so understanding can assist computation and vice versa. Some of these ideas are demonstrated in this thesis. The important outcomes of this thesis include: (1) a systematic derivation of the step-flow model of Burton, Cabrera, and Frank, with corrections, from an atomistic solid-on-solid-type models in 1+1 dimensions; (2) the inclusion of an atomistically motivated transport mechanism in an island dynamics model allowing for a more detailed account of mound evolution; and (3) the development of a hybrid discrete-continuum scheme for simulating the relaxation of a faceted crystal mound. Central to all of these modeling and simulation efforts is the presence of steps composed of individual layers of atoms on vicinal crystal surfaces. Consequently, a recurring theme in this research is the observation that mesoscale defects play a crucial role in crystal morphological evolution.