Hall effect in a three-dimensional percolation system

Abstract

We have studied the Hall effect in random mixtures of Al-Ge. The observed variation of the Hall coefficient over a wide range of concentration was found to be in full agreement with recent theoretical predictions for three-dimensional percolation systems. For the first time the critical behavior was studied near both percolation thresholds, namely, near the critical-volume concentrations of Al and of Ge. We observed a strong divergence of the Hall coefficient with exponent 3.8~0.2 near the threshold of Al, and a much slower divergence with exponent 0.38~0.05 close to the threshold of Ge.The Hall eff'ect in percolation systems near the transition threshold has been studied during the last ten years by several authors. ' In a two-dimensional (2D) system, it is well understood that the Hall coefficient does not diverge at the threshold, ' a prediction that has recently been confirmed experimentally.In 3D systems, it has been predicted that the Hall coefficient diverges near the threshold as (xx, )~, values of the critical index g ranging from 0.9 to 8.3. Only some incomplete results are available, and therefore the critical behavior of the Hall coefficient in 3D has not yet been fully established.According to the theory of Bergman and Stroud the effective Hall coefficient of a percolating mixture consisting of a good and a poor conductor is described by two terms: 2 R, =AR (xx, ) g+BR;~m (xx, ) " (I) R (R;) and cr (cr;) are the Hall coefficient and the conductivity of the good (poor) conductor, respectively. A and B are constants of order unity, and t is the conductivity critical index.In this paper we report for the first time experimental results that confirm the existence of the two diverging terms in Eq. (1). Measurements of the Hall effect in an Al-Ge compound system allowed us to observe the variation of the eAective Hall eA'ect in agreement with the second term of Eq. (I). Furthermore, by chemical etching we have converted this compound into a one element (Ge) 3D percolation system, and measured its Hall coefficient. The results of these measurements are consistent with the first term in Eq. (1). Equation (1) is based on an important result of Bergman, Kantor, Stroud, and Webman. The Hall conductivities of two components X, X; and that of the mixture k, in low field satisfy the relation for the Ohmic conductivity above x, predicts a divergence of the Ohmic conductivity as (xx, )'. Using the relation Rq =X~/o) (for k=m, t e) we can obtain Eq. (1), and also the relation g=2t -7;. A more general expression for the second term of the right-hand side (rhs) of Eq. (1) is given by BR;(cr;/o, ), where a, is the effective conductivity of the medium. Equation (1) is obtained under the approximation o, =o (xx, )'. Practically, this term becomes negligible when the ratio o;/a, is smaller than 10 . Then the effective Hall coefficient diverges more slowly than (a;/cr ) with the exponent g =2tr. Samples typically 2000 A thick were prepared by vacuum coevaporation of Al and Ge on a hot substrate. ' Figure l shows the diffrac...

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