The oscillatory amplitude of ρ xx (B) was well fitted by the Shubnikov-de Haas (SdH) theory [21–23], with amplitude given by (1) where μ q represents the quantum mobility, D(B, T) = 2π 2 k B m * T/ℏeB sinh (2π 2 k B m * T/ℏeB), and C is a constant relevant to the value of ρ xx at B = 0 T. The observation of the SdH oscillations suggests the possible existence of a Fermi-liquid metal. It should be pointed out that the SdH theory is derived by considering Landau quantization
in the metallic regime without taking localization effects into account [24, 25]. By observing the T-dependent Hall slope, Cyclopamine order however, the importance of e-e interactions in the metallic regime can be demonstrated . In addition, as reported in , with a long-range
scattering potential, SdH-type oscillations appear to Pritelivir nmr span from the insulating to the QH-like regime when the e-e interaction correction is weak. Recently, the significance of percolation has been revealed both experimentally  and theoretically [29, 30]. Therefore, to fully understand the direct I-QH transition, further studies on e-e interactions in the presence of background disorder are required. At low B, quantum corrections resulting from weak localization (WL) and e-e interactions determine the temperature and magnetic field dependences of the conductivity, and both can lead to insulating behavior. The contribution of e-e interactions can be extracted after the suppression of WL at B > B tr, where the transport magnetic field (B tr) is beta-catenin inhibitor given by with reduced Planck’s constant (ℏ), electron charge (e), diffusion constant (D), and transport relaxation time (τ). In systems with short-range potential fluctuations, the theory of e-e interactions is well established . It is derived
based on the interference of electron waves that follow different paths, one that is scattered off an impurity and another that is scattered by the potential oscillations (Friedel oscillation) created by all remaining electrons. The underlying physics is strongly related to the return probability of a scattered electron. In the diffusion regime (k B Tτ/ℏ < < 1 with Boltzmann constant k B), e-e interactions contribute only to the longitudinal conductivity (σ xx) without modifying the Hall conductivity (σ xy). On the other hand, in the ballistic regime (k B Tτ/ℏ > > 1), e-e interactions contribute both to σ xx and σ xy, and effectively reduce to a renormalization of the transport mobility. However, the situation is different for long-range potential fluctuations, which are usually dominant in high-quality GaAs-based heterostructures in which the dopants are separated from the 2D electron gas by an undoped spacer.