Diffusion on a Disordered Branched Lattice and its Application to 1/F Noise
Abstract
We propose a model for 1/f noise in certain conducting materials, namely, materials in which conduction takes place by rapid transitions of charge carriers among states localized and randomly located in space. These materials include ceramicmetallic composites (cermets) with sufficiently small volume fraction of metal, discontinuous metal films, and carbon resistors. We propose that the noise results from diffusion of the charge carriers on the network made up of these localized states. Since the transitions are rapid compared with other time scales of interest, we treat the transport as a random walk on the network. To keep the problem manageable, the network is idealized to a one dimensional lattice with side branches of random length attached to it. The diffusion is described by a master equation which allows transitions between nearest neighbors on the lattice. The probability per unit time of a given transition (transition rate) on the side branches is taken as a random variable, in agreement with the randomness of transition rates between states in the real network. The master equation is simplified analytically, and some of the properties of the moments of the random walk are calculated and shown to agree with experiment and reasonable expectations. We find the term quadratic in field in the second moment, which is the term most relevant to 1/f noise, by numerically solving the simplified master equation and averaging over many choices of the random chain lengths and transition rates. It is shown that geometrical considerations lead plausibly to certain types of broad distributions of transition rates. The numerical results demonstrated that with reasonable choices of these distributions, the dependence of the noise in our model on sample size and frequency agrees well with experiment for frequencies within the range of allowed transition rates. They also show that the spectrum is insensitive to the distribution of chain lengths. This suggests that 1/f noise in these materials is principally due to broad random distributions of transition rates.
 Publication:

Ph.D. Thesis
 Pub Date:
 1985
 Bibcode:
 1985PhDT........58H
 Keywords:

 TAILS;
 RANDOM WALKS;
 Physics: Condensed Matter