Atomistic growth of two-dimensional quasicrystals

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Abstract
An atomistic growth model based on local growth rules and bond lengths defined by fivefold symmetry is constructed to obtain both perfect and defective structures that exhibit long-range orientational order. The interplay of randomness and energy in the calculation of growth priority is investigated and a proof is given for the condition of perfect structures. Variation of the energy parameters representing the depth of the potential well results in different phases characterized by the dominance of different local clusters. The perfect structure is a special phase resembling a multiple twinning structure made from stretched hexagons and stripes. It can be described as the projection from higher dimension with randomness in the form of random stacking of layers. Defects are illustrated and discussed for various initial clusters to show that the general low-temperature phase is a defective one. All defects found can be traced back from the occurrence of two elementary defects, whose existence arises from the geometric definition of the model. The existence of a new kind of defect not describable by Burgers vectors is illustrated by the growth from the interior surface of an annulus. The ground-state property is investigated and the perfect structure is shown to have a lower energy than structures with only rhombic defects. We conjecture that the perfect structure is indeed the ground state for a wide range of energy.