Shubnikov–de Haas oscillations, peaks, and different temperature regimes of the diagonal conductivity in the integer quantum Hall conductor

Abstract

A theory for the Shubnikov-de Haas oscillations in the diagonal conductivity s xx of a 2D conductor is developed for the case when electron states within the broaden Landau levels are localized except the narrow stripe in the center. The standard Shubnikov-de Haas oscillations take place only in the low-field region which at higher magnetic fields crosses over into peaks. In the limit Wt >> 1 peaks in the s xx became sharp and between them s xx ® 0 (W is the cyclotron frequency, t is the electron scattering time). The conductivity peaks display different temperature behavior with the decrease of temperature, T: a thermal activation regime, In spite of more than two decades of intensive experimental research and large theoretical efforts, the quantum magnetic oscillations of the conductivity in 2D conductors still have some open questions. Even in the most studied case of the integer quantum Hall effect (IQHE) [1] a complete picture is missing, in particular, concerning different regimes in temperature and magnetic field dependencies of the Hall, s xy , and longitudinal, s xx , conductivities. Although the origin of the quantized plateaus in the s xy is well understood [2] the transitional regions between them, where localization and scaling [3] play an important role, needs a deeper insight. The scaling properties of diagonal conductivity s xx in the variable-range hopping (VRH) regime of an IQHE were recently established experimentally at low temperatures down to 60 mK in [4]. To explain this universal scaling behavior as well as transitions between different regimes in the IQHE is a theoretical challenge. At the moment there is no coherent analytic description of the quantum magnetic oscillations of the diagonal conductivity s xx which takes into account the localization effects at different temperature regimes in the IQHE. In particular, it is not clear so far why quantum oscillations in s xx do survive in spite of the fact that most states within the broaden Landau levels (LL) are localized (i); Why s xx ® 0 between the peaks in the limit Wt >> 1, if at low fields it displays a standard Shubnikov-de Haas (SdH) oscillations (ii); Why with the decrease of temperature, T, the peaks in s xx display first a thermal activation behavior s xx /T µ -exp( ) D , which then crosses over into the VRH regime at low temperatures withWhy the prefactor 1/T is absent in the conductance (iv).The well established fact is that localization in the IQHE picture plays a crucial role. On the other hand, nonzero conductivity is impossible without the extended states. It is believed that extended states are within the narrow stripe at the center of the broaden LL and all the other states are localized [5]. One can not give a simple physical picture for these localized states in general. At high fields the presence of the localized states in the 2D conductor means that Landau orbits drift along the closed equipotential contours of the impurity potential. At places where contours come close electrons can tunnel from one contour to anoth...