Self-consistent theory of magnetoconductance in two-dimensional Anderson localized systems

The conductivity of noninteracting electrons in a two-dimensional disordered system with weak impurity scattering is studied by using the diagrammatic Green's-function method. Starting from the current-current correlation functions, a self-consistent equation can be constructed for the conductivity $\sigma \left(\omega \right)$. In the absence of a magnetic field, we show that $\mathrm{Re}\sigma \left(\omega \right)\sim 1-{\left(2\mathrm{}\pi E_F\tau \right)}^{-1\mathrm{}}\ln \left(\frac{1}{\tau \omega }\right)$ for large $\omega$ and $\mathrm{Re}\sigma \left(\omega \right)\sim {\omega }^2$ as $\omega \to 0$. This result is in agreement with that of Vollhardt and Wölfle. In the presence of a weak magnetic field, we show that the dc magnetoconductance $\sigma \mathrm{}\left(0\right)\mathrm{}\sim 1-{\left(2\mathrm{}\pi E_F\tau \right)}^{-1\mathrm{}}\ln \left(\frac{1}{H}\right)$ for $l_L\gg l_B\gg l$ and $\sigma \mathrm{}\left(0\right)=0\mathrm{}$ for $l_B\gg l_L\gg l$ where $l_L$, $l_B$, and $l$ are, respectively, the localization length, radius of the lowest Landau orbit, and mean free path.