Hyeong-Chai Jeong

Steps on surfaces: experiment and theory

The properties of steps in thermal equilibrium are described in the context of prediction of the stability and evolution of nanostructures on surfaces. Experimental techniques for measuring the appropriate step parameters are described, and simple lattice models for interpreting the observations are reviewed. The concept of the step chemical potential and its application to the prediction of step motion (and therefore surface mass transport) is presented in depth. Examples of the application of this step-continuum approach to experimental observations of evolution of surface morphology are presented for morphological phase transitions, the decay of metastable structures, and the spontaneous evolution of metastable structure due to kinetic instabilities.

Quasi-Unit-Cell Model for an Al-Ni-Co Ideal Quasicrystal based on Clusters with Broken Tenfold Symmetry

We present new evidence supporting the quasi-unit-cell description of the Al72Ni20Co8 decagonal quasicrystal which shows that the solid is composed of repeating, overlapping decagonal cluster columns with broken tenfold symmetry. We propose an atomic model which gives a significantly improved fit to electron microscopy experiments compared to a previous proposal by us and to alternative proposals with tenfold symmetric clusters.

Optional games on cycles and complete graphs.

We study stochastic evolution of optional games on simple graphs. There are two strategies, A and B, whose interaction is described by a general payoff matrix. In addition, there are one or several possibilities to opt out from the game by adopting loner strategies. Optional games lead to relaxed social dilemmas. Here we explore the interaction between spatial structure and optional games. We find that increasing the number of loner strategies (or equivalently increasing mutational bias toward loner strategies) facilitates evolution of cooperation both in well-mixed and in structured populations. We derive various limits for weak selection and large population size. For some cases we derive analytic results for strong selection. We also analyze strategy selection numerically for finite selection intensity and discuss combined effects of optionality and spatial structure.

Comparing reactive and memory-one strategies of direct reciprocity

Direct reciprocity is a mechanism for the evolution of cooperation based on repeated interactions. When individuals meet repeatedly, they can use conditional strategies to enforce cooperative outcomes that would not be feasible in one-shot social dilemmas. Direct reciprocity requires that individuals keep track of their past interactions and find the right response. However, there are natural bounds on strategic complexity: Humans find it difficult to remember past interactions accurately, especially over long timespans. Given these limitations, it is natural to ask how complex strategies need to be for cooperation to evolve. Here, we study stochastic evolutionary game dynamics in finite populations to systematically compare the evolutionary performance of reactive strategies, which only respond to the co-player’s previous move, and memory-one strategies, which take into account the own and the co-player’s previous move. In both cases, we compare deterministic strategy and stochastic strategy spaces. For reactive strategies and small costs, we find that stochasticity benefits cooperation, because it allows for generous-tit-for-tat. For memory one strategies and small costs, we find that stochasticity does not increase the propensity for cooperation, because the deterministic rule of win-stay, lose-shift works best. For memory one strategies and large costs, however, stochasticity can augment cooperation.

A restricted solid-on-solid model for growth on fractal substrates

The restricted solid-on-solid (RSOS) model for growth on two different fractal substrates having the same fractal dimension df = 2 is studied. The Sierpinski gasket and the checkerboard fractal embedded in three dimensions are considered as the substrates. It is found that the interface width W grows as tβ with β≈0.26 for growth on a Sierpinski gasket and with β≈0.30 for growth on a checkerboard fractal. For the saturated regime, W follows W~Lα, L being the size of the system, with α≈0.50 for a Sierpinski gasket and α≈0.56 for a checkerboard fractal, implying that the growing surfaces of fractal substrates are rougher than that of a regular substrate. The growth exponent is not fully determined by the fractal dimension only, and the dynamic exponent z, obtained from the relation z = α/β, for both fractal lattices does not satisfy the scaling relation α+z = 2 due to the intrinsic fractal nature of the substrate. The RSOS model for growth on a regular lattice is generally believed to be described by the Kardar–Parisi–Zhang (KPZ) equation. However, the RSOS model for fractal substrates does not appear to follow the KPZ type universality. Generalization to the equilibrium RSOS model for growth on the fractal substrates is also investigated.

Noncommutative torus from Fibonacci chains via foliation

We classify the Fibonacci chains (F-chains) by their index sequences and construct an approximately finite-dimensional (AF) C*-algebra on the space of F-chains as Connes did on the space of Penrose tiling. The K-theory on this AF algebra suggests a connection between the noncommutative torus and the space of F-chains. A noncommutative torus, which can be regarded as the C*-algebra of a foliation on the torus, is explicitly embedded into the AF algebra on the space of F-chains. As a counterpart of that, we obtain a relation between the space of F-chains and the leaf space of Kronecker foliation on the torus using the cut-procedure of constructing F-chains. Our embedding of the C*-algebra of the foliation is consistent with the recent result of Landi, Lizzi, and Szabo that the C*-algebra of noncommutative torus can be embedded into an AF algebra.

Effects of step–step interactions on the fluctuations of an individual step on a vicinal surface and its wavelength dependence

We relate properties of an anisotropic continuum model of a two-dimensional vicinal surface to those of a model with fluctuating and interacting steps. We show that analysis of the fluctuations of an individual step in the array provides information about the length scale on which the surface has reached equilibrium and can be used to estimate fundamental step parameters from locally equilibrated surfaces. Monte Carlo simulations of a stable vicinal surface using the terrace–step–kink model agree with the theoretical predictions. We further apply this analysis to steps on an unstable surface during reconstruction-induced faceting and show that it can be used to determine whether the faceting process is in the nucleation or spinodal decomposition regime.

Growing Perfect Decagonal Quasicrystals by Local Rules

A local growth algorithm for a decagonal quasicrystal is presented. We show that a perfect Penrose tiling (PPT) layer can be grown on a decapod tiling layer by a three dimensional (3D) local rule growth. Once a PPT layer begins to form on the upper layer, successive 2D PPT layers can be added on top resulting in a perfect decagonal quasicrystalline structure in bulk with a point defect only on the bottom surface layer. Our growth rule shows that an ideal quasicrystal structure can be constructed by a local growth algorithm in 3D, contrary to the necessity of nonlocal information for a 2D PPT growth.

Dynamics and scaling of one-dimensional surface structures

We study several one-dimensional step flow models. Numerical simulations show that the slope of the profile exhibits scaling in all cases. We apply a scaling ansatz to the various step flow models and investigate their long time evolution. This evolution is described in terms of a continuous step density function, which scales in time according to

Rules for computing symmetry, density, and stoichiometry in a quasi-unit-cell model of quasicrystals

{D(x,t)=F(xt}^{\ensuremath{-}1/\ensuremath{\gamma}}).