Quasiclassical Theory of Shubnikov-de Haas Effect in 2D Electron Gas

Abstract

A new approach to the theory of magneto-transport in 2D gas is developed. We make use of Keldysh technique and introduce a modified Green′s function which is translationally invariant and gauge invariant. The modification simplifies the calculation of diagrams and allows us to obtain a new helpful addition theorem for the electron wave functions in magnetic field. The modified Green′s function is very convenient to follow the transition between quantum magnetic transport and transport in zero magnetic field. For the calculation of conductivity we use the self-consistent Born approximation (SCBA). We carefully check its validity and figure out the physical meaning of the corresponding conditions. All specific calculations are made for the case when the separation between Landau levels, ħΩ, is much smaller than the Fermi energy, without limiting the magnitude of ħΩ relative to the width of the levels (induced by scattering), Γ. For the first time the case of a long-range scattering potential is carefully studied. We study magneto-oscillation effects with the help of an evolution equation for the Green′s function. In this equation two relaxation times, the single particle relaxation time and the transport relaxation time, naturally come about. Analytical results for the conductivity tensor are obtained for both the Shubnikov-de Haas effect, when ħΩ ⪡ Γ, and the quantum Hall regime, ħΩ ⪢ Γ. In the latter the temperature dependence of the conductivity depends on the relation between the temperature, ħΩ, and Γ. Although SCBA does not describe localization, it, however, allows one to separate the Shubnikov-de Haas effect from the quantum Hall effect.