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

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).