Low temperature electronic transports in the presence of a density gradient

In this paper, we review low temperature electronic transport results in high quality two-dimensional electron systems. We discuss the quantization of the diagonal resistance, R_{x x}, at the edges of several quantum Hall states. Each quantized R_{x x} value turns out to be close to the difference between the two adjacent Hall plateaus in the off-diagonal resistance, R_{x y}. Moreover, peaks in R_{x x} occur at different positions in positive and negative magnetic fields. All three R_{x x} features can be explained quantitatively by a ∼1% cm electron density gradient. Furthermore, based on this observation, the well known but still enigmatic resistivity rule, relating R_{x x} to d R_{x y} / d B, finds a simple interpretation in terms of this gradient. In another sample, at 1.2 K, R_{x x} we observe a strongly linear magnetic field dependence. Surprisingly, this linear magnetoresistance also originates from the density gradient. Our findings throw an unexpected light on the relationship between the experimentally measured R_{x x} and the diagonal resistivity ρ_{x x}.