Crossover From Two to Three Dimensional Percolation in Thin Layers

Abstract

Percolation is now well accepted as a paradigm to quantify the structure of random systems and has been utilized in numerous areas of physics and materials science. Singular behavior of percolation clusters at the percolation threshold is known to be universal, that is the critical exponents characterizing the singularity of various properties at the threshold depend only on the dimensionality of the system though the threshold itself depends on the structure of lattice. In recent years, layered structures fabricated by deposition techniques have been attracting wide interests because of their various functionalities and the characterization of the layered structure is one of the most important tasks in the study of thin layers. Needless to say a monolayer is a two dimensional system and an in nitely thick layer is a three dimensional system. Furthermore, if one considers Lx Ly n cubic systems, systems with 1 1 n behave as two dimensional for any nite n and as three dimensional only when n 1⁄4 1. Thus the critical behavior of the percolation properties is well understood in the thermodynamic limit. In many practical appplications including granular materials, however, one has to deal with a nite system, and it is imporatant to know the thickness dependence of various apparent physical quantities near the percolation threshold. The purpose of this short note is to elucidate the crossover of apparent percolation properties when the thickness of thin layers is increased. We utilize the Monte Carlo simulation to determine the percolation threshold and the e ective dimensionality as functions of the thickness. We focus on the regime where Lx 1⁄4 Ly L 1 and 1 n L. We prepare a square lattice of L L and stack n layers of the square lattice to form an L L n cubic lattice. Each site of the cubic lattice is occupied randomly by an object and clusters of the object are constructed as in the standard percolation process. We are interested in percolation of the objects in the horizontal directions. We rst obtained the percolation probability P ðp; L;nÞ for given n, that is the probability of nding a cluster spanning horizontally. According to the idea of Monte Carlo renormalization method, the xed point determined by p 1⁄4 P ðp; L;nÞ is a good estimate of the percolation threshold pcðnÞ. Figure 1 shows the n dependence of pcðnÞ which is determined by the average over 200 samples for L 1⁄4 200. It is clear that pcð1Þ 1⁄4 0:599 agrees with the percolation threshold p c 1⁄4 0:59275 for the square lattice within the error bar ( 1%) of our simulation. We also notice that pcð16Þ is already very close to the percolation threshold p c 1⁄4 0:3117 of the simple cubic lattice. The dependence of pcðnÞ on n is well tted by pcðnÞ 1⁄4 0:310þ 0:384 n 1 0:0950 n ; ð1Þ