Magnetic Quantum Oscillations of the Conductivity in Two-dimensional Conductors with Localization

Abstract

An analytic theory is developed for the diagonal conductivity ${\sigma }_{xx}$ of a 2D conductor which takes account of the localized states in the broaden Landau levels. In the low-field region ${\sigma }_{xx}$ display the Shubnikov-de Haas oscillations which in the limit $\Omega \tau \gg 1$ transforms into the sharp peaks ($\Omega$ is the cyclotron frequency, $\tau$ is the electron scattering time). Between the peaks ${\sigma }_{xx}\to 0$. With the decrease of temperature, $T$, the peaks in ${\sigma }_{xx}$ display first a thermal activation behavior ${\sigma }_{xx}\propto \exp (-\Delta /T)$, which then crosses over into the variable-range-hopping regime at lower temperatures with ${\sigma }_{xx}\propto 1/T\exp (-\sqrt{T_0/T})$ (the prefactor 1/T is absent in the conductance).