Flicker noise near the percolation threshold

2014 An effective-medium theory, for the flicker (1/f) noise amplitude, is formulated for random resistor networks. Close to the percolation threshold pc, the magnitude of the noise is shown to diverge as (p pc)-03BA, with 03BA = 1 for both site and bond percolation models. The exponent 03BA is also calculated in the framework of two different renormalization group transformations. The possible observation of the predicted behaviour in metal-insulator mixtures and cermets is discussed. Tome46 No 4 15 FEVRIER 1985 LE JOURNAL DE PHYSIQUE-LETTRES J. Physique Lett. 46 (1985) L-129 L-136 15 FEVRIER 1985, .Classification Physics Abstracts 05.40 72.70 72.90 1. Flicker (1//) noise [1] refers to the frequency spectrum of excess voltage fluctuations measured when a constant current is applied to a resistor. That spectrum almost always has a power law form OJ ex with (x close to unity. The origin of this power law has been the subject of innumerable controversies and remains not very well established. Recently [2], a simple model for the magnitude of this noise in random resistor networks has been introduced. This model is based on two well established properties of 1// noise : it is resistance noise [3] and the resistance fluctuations are correlated over microscopic distances only [4]. The simplest version of the model can be formulated as follows. The resistances, all assumed to have the same value r, fluctuate independently in time on each bond with a correlation function ( 5~((u) br~(OJ) > = p2(W) 6 As long as each bond resistance fluctuates independently with the same spectrum, the explicit frequency dependence can be removed and will not be considered anymore. The magnitude of the relative noise for the overall resistor network is defined by SR = 5jR.~ >it , where Rand 6R are respectively the overall resistance and its time fluctuation. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01985004604012900 L-130 JOURNAL DE PHYSIQUE LETTRES A general expression for SR has been obtained [2, 5] : Here ia denotes the current that flows in branch a in the steady-state without resistance fluctuations. Using equation (1), lower and upper bounds for SR may be found (2) : where s = p¿’ /r¿. is the relative noise for the individual resistances and Nb is the total number of conducting branches in the system. The lower bound is obviously reached for regular Euclidean networks : SR ~ s. L d where L denotes the length scale of the network. However, for a selfsimilar network, it was shown that §~ ~ s. L 6, where b denotes a new exponent. This exponent b appears as a member of a new hierarchy of exponents describing the cumulants of the resistance fluctuations and is related to the scaling of the quantities ¿ ¡ex21l (n = positive integer). The