A theory of magnetoconductance in Anderson insulators

Abstract

We present a simple theory to understand the effect ofmagnetic field on the localisation length f in Anderson insulators. For thin wires, we find that f is doubled, a result recently derived through random matrix theories. For fi1nls or bulk samples, new results are obtained. In this case, the localisation length is multiplied by a non universal factor. We discuss quantitatively the fttll dependence off on the magnetic field. Interest in quantum transport phenomena has soared during the past few years (see e-g[1-3]). Among other interesting progress, one may cite the discovery of persistent currents in mesoscopic rings [4], the development of Diflusing Wave Spectroscopy [2] (which takes advantage of weak localisation effects to probe in a new manner e-gconcentrated suspensions), or the unveiling of the peculiar nature of conductance fluctuations in disordered metals (universal amplitude fig m e2 /h, sensitivity to single impurity rearrangments [I,fl, relevance to I/ f noise [6]). The theoretical understanding of these fluctuations was greatly improved after Imry, and Alt'shuler and Shklovskii [7,8] suggested that the random matrix theory of Dyson and Mehta was the natural language to describe these effects: the universal character of the conductance fluctuations is intimately related to the spectral rigidity of random matrices. It was then realized that breaking of a basic sylnmetry (e.g. imposing a magnetic field to break time reversal symmetry) simply changed the relevant ensemble of random matrices, which ultimately leads to the following simple and spectacular prediction [5,9,10,18]: when a magnetic field is imposed, the conductance fluctuations of a metallic sample are reduced by a factor 2 (in absence of spin-orbit coupling) compared to the zero field case. This prediction was confirmed experimentally [10]. Very recently, Pichard and collaborators ill] (see also [12]) pointed out that a similar phenomenon also occurs in the highly disordered, insulating case. Again using random matrix theory, they showed that at least for quasi I D wires the localisation length f should simply be multiplied by 2 when a sufficiently strong magnetic field B is applied (or divided by 2 if strong spin orbit scattermg is present). This should have dramatic consequences on the conductivity of these insulating samples lgiven, in the Mott r6gime, by a « exp(To/T)V~~+~~ with To « f~d], or on the static (*) Associd au CNRS et aux Universit6s de Paris W et WI. 986 JOURNALDEPHYSIQUEI N°7 dielectric constant e « f2. Numerical simulations on ld strips and Idm [13], we examine the stationary situation schematized in figure I. An energy current Jo is injected at O along Ox and eventually (since stationarity is assumed) leaves the sample far from O. The loml equilibrium current J1m h bui~ by all Feynman paths reinjecting energy at O along the initial direction Oz. Following [13], this initiatdirection is in fact, due to the wave character of the problem we consider, not precisely defined: it lieK within a cone of solid angle SD = (6@)d~l with (le IA) 6@ m I. Only the fraction 6Q/Q of the reitljected energy will thus be coherent) backscattered. Jim is then obtained as: Jim = Jo + R(6Q/Q)J~« = JOI(I -R(6Q IQ)] (1) _ ~ ~ Jo / / ~ / la ' sy o /_--~ ~ / /