Bilinear electric-field characteristics in the problem of the galvanomagnetic properties of composite materials
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ONIC PROPERTIES OF SOLID
Bilinear ElectricField Characteristics in the Problem of the Galvanomagnetic Properties of Composite Materials B. Ya. Balagurov Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, ul. Kosygina 4, Moscow, 119334 Russia email: [email protected], [email protected] Received August 21, 2014
Abstract—Some general properties of the effective conductivity tensor σˆ e of a composite material placed in magnetic field H are considered. It is shown that the derivatives of the tensor σˆ e components with respect to their arguments can be expressed in terms of various bilinear characteristics, namely, the mean products of the electric fields in an initial system and the “transposed” system (which differs from the initial system in the –H). At H 0, the general formulas derived in this work yield the wellknown expres substitution H sions for the Hall coefficient and magnetoresistance that are valid for the case of weak magnetic fields. DOI: 10.1134/S106377611501001X
1. INTRODUCTION To study the galvanomagnetic properties of hetero geneous media (in particular, composite materials) is of deep interest from both applied and fundamental standpoints. Although an analytical approach to this problem encounters significant difficulties, its solu tion in the twodimensional (2D) case has certain progress. For example, Dykhne [1] used symmetry transformation and found effective conductivity ten ˆ e of a 2D twocomponent system with a critical sor σ composition in an explicit form. In [2, 3], we revealed the isomorphism of the problems of the galvanomag netic properties of a 2D binary system and its conduc tivity in the absence of a magnetic field. The isomor phism relations found in [2, 3] made it possible to ˆ e components in terms of the gal express the tensor σ vanomagnetic characteristics of the system compo nents and the effective conductivity of the system at H = 0. Thus, the problem of the galvanomagnetic properties of a 2D twocomponent system with an arbitrary structure was completely solved. The symmetry transformation proposed in [1] and used in [3] cannot be transferred to the 3D case; there fore, no exact results that are as general as the results in the 2D case were found. Hence, analytical results for 3D systems were only obtained in certain limiting cases. For example, in the case of arbitrary magnetic ˆ e was calculated fields, effective conductivity tensor σ for a weakly heterogeneous medium [4–6] and in the approximation that is linear in the concentration of inclusions (of an arbitrary shape) [6]. The Hall coeffi cient in a weak magnetic field was studied analytically [7, 8] and numerically [9, 10]. A consistent theory of the galvanomagnetic properties of binary composite materials in the case of a weak magnetic field was
developed in [11], where general expressions were derived for the Hall coefficient and magnetoresis tance. Moreover, in [11], we paid attention to the importance of studying various partial quadratic effec tive characteristics, n
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