Crossover behavior of conductivity in a discontinuous percolation model

Abstract

When conducting bonds are occupied randomly in a two-dimensional square lattice, the conductivity of the system increases continuously as the density of those conducting bonds exceeds the percolation threshold. Such a behavior is well known in percolation theory; however, the conductivity behavior has not been studied yet when the percolation transition is discontinuous. Here we investigate the conductivity behavior through a discontinuous percolation model evolving under a suppressive external bias. Using effective medium theory, we analytically calculate the conductivity behavior as a function of the density of conducting bonds. The conductivity function exhibits a crossover behavior from a drastically to a smoothly increasing function beyond the percolation threshold in the thermodynamic limit. The analytic expression fits well our simulation data. The concept of percolation transition has played a central role as a model for the formation of a spanning cluster connecting two opposite edges of a system in Euclidean space as a control parameter p is increased beyond a certain threshold p c [1]. This model has been used to study many phenomena such as metal-insulator transitions and sol-gel transitions. The order parameter P ∞ of percolation transition is defined as the probability that a bond belongs to a spanning cluster, which increases in the form, where p is a control parameter indicating the fraction of occupied bonds and β is the critical exponent related to the order parameter. As an application of percolation model, one can construct a random resistor network in which each occupied bond is regarded as a resistor with unit resistance, and the system is in contact with two bus bars at the opposite edges of the system. When a voltage difference is applied between these two bus bars, the system is in a insulating state for p < p c , but changes to conducting state for p > p c , due to the formation of several conducting paths at p c . Above p c , the conductivity increases continuously as g ∼ (p − p c ) µ , where µ is the conductivity exponent [2].Recently the subject of discontinuous percolation transition (DPT) has been a central issue [3][4][5][6][7][8][9][10][11][12] with, for example, applicability to cascading failures in complex networks [13]. Among others [14][15][16][17][18][19], a model called spanning cluster avoiding (SCA) was introduced [20] aiming to generate a DPT. The DPT of the SCA model is rather trivial, for the percolation threshold is placed at p c = 1 in the thermodynamic limit, but for finite-sized systems p c < 1. Here, we study the conductivity as a function of p in finite-sized systems for the SCA model. Indeed, we find that the conductivity increases drastically just after the percolation threshold and then exhibits a crossover to a smoothly increasing behavior. Such crossover has never been reported, though it is meaningful, as, a drastic change of conductivity in random resistor networks can find application, for example, on resistance switching phenomena in non-volatile memory...

Effective resistances of two-dimensional resistor networks

“…Now we further note that α(z, L y ) approaches z as L y → ∞, where z = 3 is the lattice coordination number for a hexagonal lattice. Since α(z, L y ) has to be dimensionless to keep equation (39) physically consistent, a convenient guess could be where c is a constant. Now figure 8(c) plots ln α against L y −1 for various L x and we can see when L y −1 approaches zero (thermodynamic limit), ln α approaches ln z = ln 3 1.1 vindicating our guessed formula for α in equation (41).…”

“…Conventional equilibrium-based thermodynamics cannot be applied to a small finite size system and hence the limit is called thermodynamic[38]. The limit has significance to the study of percolation-based conductivity through a resistor network[39].…”