Diffusion in 2 D Quasi - Crystals

Self-diffusion induced by phasonic flips is studied in an octagonal model quasi-crystal. To determine the temperature dependence of the diffusion coefficient, we apply a Monte Carlo simulation with specific energy values of local configurations. We compare the results of the ideal quasi-periodic tiling and a related periodic approximant and comment on possible implications to real quasi-crystals. Recently, Kalugin and Katz [1] proposed a possible mechanism for bulk self-diffusion in quasi-crystals. Their model is based on specific geometric properties of the quasi-crystalline structure. In ordinary crystals, the diffusion process depends on the presence of vacancies in the lattice and is activated by the vacancy production rate and the hopping of atoms in the neighbourhood. Both lead to a Brownian-like motion, and the temperature dependence of the diffusion coefficient thus obeys an Arrhenius law [2]. Due to additional degrees of freedom of quasi-periodic systems-the so-called phasons-one can construct an additional process for bulk diffusion in quasi-crystals. Based on general arguments, Kalugin and Katz conclude that the phason degree of freedom leads to a deviation of the diffusion coefficient from the Arrhenius law. Unfortunately, because of the general nature of their considerations, they were neither able to determine at which temperature these deviations are to be expected nor to estimate their order of magnitude. In this article, we calculate the diffusion coefficient of the eightfold symmetric Ammann-Beenker tiling [3] quantitatively. We start from a mean-field model which takes the additional degrees of freedom into account and we apply a Monte Carlo method to estimate the phason contribution to the bulk diffusion in the quasi-periodic plane. We discuss general properties and similarities with the ansatz of Kalugin and Katz and compare the quasi-periodic tiling with a periodic approximant. The Arnmann-Beenker tiling is chosen because, on the one hand, it provides generic quasi-periodic properties while, on the other hand, it is as simple as possible. Nevertheless, all results stated below can easily be extended to other two-dimensional quasi-periodic tilings like the Penrose tiling [4] or the Ti.ibingen triangle tiling [5]. The Ammann-Beenker tiling consists of squares and 45-degree rhombi which form (modulo operations of the dihedral group d8 ) 6 different vertex configurations. Combinatorially, another 10 ( + 3 mirror inverted) vertices are possible without gaps or overlaps, which were introduced and 452 EUROPHYSICS LETTERS Fig. 1. The simpleton flip in the octagonal Ammann-Beenker tiling. investigated earlier as defects of the first kind [6]. One can easily show that all of them can be inserted locally in an ideal tiling. Regarding the ideal tiling as energetically favoured, one can follow a mean-field approach [6, 7], which assigns an average defect energy to each of the 10 ( + 3) forbidden vertex configurations due to nearest-neighbour interactions (1 ). This approach has only one free parameter, compare eq. (3.1) and the following discussion in [6], which scales the measure of the elastic energy and plays the role of an activation energy. We fix this parameter to be E0 = 2kfJ, k the Boltzmann constant, and f) the Debye temperature. The value is chosen to match specific-heat contributions of these defects with experimental fmdings [6, 7]. Taking a typical Debye temperature, E 0 corresponds to approximately 0.05 eV. The basic mechanism of the extraordinary diffusion (also called self-diffusion) is the simpleton flip [8], cf. fig. 1, the simplest representation of a phason degree of freedom. It can be seen as the geometric realization of local two-level systems with small excitation threshold. In fact, the rotating octagons used by Kalugin and Katz [1] can be realized as a sequence of eight simpleton flips. Starting from an ideal, defect-free Ammann-Beenker tiling we use the simpleton flips to randomize the tiling, cf. fig. 2. Gahler [9] has given an argument that the simpleton flip randomization should be ergodic (though not necessarily uniformly ergodic) on the tiling ensemble in this case. As a consistency check, we estimated the configurational entropy per tile S by integrating Cv JT over T and adding the ground-state contribution. This resulted in a value of S = 0.41(5) which is consistent with previous calculations [10]. During the randomization we keep the edges of the patch fixed. One can easily see that all 16 ( +3) vertex configurations can be reached this way. Each new defect of the first kind corresponds to an energy cost in the mean-field approach above. Therefore, we can judge for each simpleton flip energetically whether it should be accepted or not in an importance sampling [11] by the classical Metropolis algorithm [12]. To determine the diffusion coefficient, we have to calculate the expectation value of the