Magneto-Conductance Anisotropy and Interference Effects in Variable Range Hopping

TL;DR

This paper studies how quantum interference affects magneto-conductance anisotropy in three-dimensional variable range hopping, revealing conditions under which anisotropy appears or averages out, with implications for thin films and strong electric fields.

Contribution

It provides a detailed analysis of MC anisotropy in 3D variable range hopping, including scaling laws, the role of spin-orbit scattering, and the effects of sample geometry and electric fields.

Findings

Scaling variables for magnetic field and hopping length are identified.

Anisotropy is prominent in single-hop conduction but averages out in bulk samples.

Thin films exhibit persistent anisotropy due to restricted hop orientations.

Abstract

We investigate the magneto-conductance (MC) anisotropy in the variable range hopping regime, caused by quantum interference effects in three dimensions. When no spin-orbit scattering is included, there is an increase in the localization length (as in two dimensions), producing a large positive MC. By contrast, with spin-orbit scattering present, there is no change in the localization length, and only a small increase in the overall tunneling amplitude. The numerical data for small magnetic fields $B$, and hopping lengths $t$, can be collapsed by using scaling variables $B_\perp t^{3/2}$, and $B_\parallel t$ in the perpendicular and parallel field orientations respectively. This is in agreement with the flux through a `cigar'--shaped region with a diffusive transverse dimension proportional to $\sqrt{t}$. If a single hop dominates the conductivity of the sample, this leads to a characteristic orientational `finger print' for the MC anisotropy. However, we estimate that many hops contribute to conductivity of typical samples, and thus averaging over critical hop orientations renders the bulk sample isotropic, as seen experimentally. Anisotropy appears for thin films, when the length of the hop is comparable to the thickness. The hops are then restricted to align with the sample plane, leading to different MC behaviors parallel and perpendicular to it, even after averaging over many hops. We predict the variations of such anisotropy with both the hop size and the magnetic field strength. An orientational bias produced by strong electric fields will also lead to MC anisotropy.