[Paper Review] On calculation of effective galvanomagnetic characteristics of inhomogeneous metals. Exact solution for the longitudinal effective conductivity of polycrystals of metals in high magnetic fields

Inna M. Kaganova, M. I. Kaganov|arXiv (Cornell University)|Feb 16, 2004

Magnetic Properties and Applications1 references4 citations

TL;DR

This paper presents an exact analytical solution for the longitudinal effective conductivity of polycrystalline metals with closed Fermi surfaces in high magnetic fields, using perturbation theory under the condition $ r_c \ll l $. The key result is a closed-form expression for the leading-order longitudinal conductivity that accounts for grain orientation effects, showing significant deviation from isotropic behavior even in polycrystals.

ABSTRACT

In the framework of the perturbation theory an expression suitable for calculation of the effective conductivity of 3-D inhomogeneous metals in uniform magnetic field $H$ is derived. For polycrystals of metals with closed Fermi surfaces in high magnetic fields the perturbation series defining the longitudinal and the hall elements of the perturbation series can be summed allowing us to obtain the exact expression for the leading terms of all these elements of the effective conductivity tensor.

Motivation & Objective

• To derive an exact expression for the longitudinal effective conductivity of 3D polycrystalline metals in high magnetic fields.
• To extend perturbation theory to include the effects of random grain orientations and high-field electron dynamics.
• To determine whether the perturbation series for the effective conductivity tensor can be summed exactly under specific high-field conditions.
• To assess the validity of assuming isotropic behavior in polycrystalline metals under high magnetic fields.

Proposed method

• Derives a perturbation theory framework for the effective conductivity tensor (ECT) in a uniform magnetic field $\mathbf{H}$, applicable to stochastically inhomogeneous metals.
• Applies the method to polycrystals with closed Fermi surfaces, assuming isotropy in average and small spatial fluctuations of the local conductivity tensor.
• Uses the condition $ r_c \ll l $ (cyclotron radius much smaller than mean free path) to simplify the electron dynamics and enable summation of the perturbation series.
• Focuses on the longitudinal and Hall components of the ECT, deriving exact expressions by summing the leading-order terms of the perturbation series.
• Relies on the structure of the local conductivity tensor in high fields, particularly for ellipsoidal Fermi surfaces, to obtain closed-form results.
• Validates the approach by comparing with known results for single crystals and isotropic limits, confirming consistency in asymptotic regimes.

Experimental results

Research questions

• RQ1 Can the perturbation series for the effective conductivity tensor be summed exactly in high magnetic fields for polycrystalline metals with closed Fermi surfaces?
• RQ2 How does the effective longitudinal conductivity of a polycrystal deviate from the isotropic limit when grain orientations are considered?
• RQ3 What is the exact form of the leading-order longitudinal conductivity in high magnetic fields under the $ r_c \ll l $ condition?
• RQ4 Does the Hall component of the effective resistivity retain the single-crystal form in polycrystalline samples under high fields?
• RQ5 To what extent do grain boundary effects and structural inhomogeneities influence the effective conductivity beyond orientation averaging?

Key findings

• The perturbation series for the longitudinal effective conductivity can be summed exactly in the limit $ r_c \ll l $, yielding a closed-form expression for polycrystals with closed Fermi surfaces.
• For an ellipsoidal Fermi surface, the exact longitudinal effective conductivity is $ \sigma_{||}^{\text{ef}} = \frac{3\sigma_z(1+\nu)}{3+2\nu} $, where $ \nu $ is the anisotropy parameter.
• This result deviates significantly from the isotropic value $ \sigma_0 $, with the ratio $ \sigma_{||}^{\text{ef}} / \langle \sigma(0) \rangle = \frac{9(1+\nu)}{(3+2\nu)(3+\nu)} $, showing that polycrystals are not isotropic in high fields.
• The effective Hall resistivity remains $ \rho_{12}^{\text{ef}} = R_\infty H $, identical to the single-crystal form, indicating that the Hall effect is insensitive to polycrystalline averaging.
• The transverse effective conductivity cannot be obtained exactly with current methods due to higher-order pole contributions from scattering processes.
• The model assumes grain boundaries do not significantly affect transport, justifying the focus on orientation-induced inhomogeneity alone.