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## Main Question or Discussion Point

My only problem with a basic conceptual understanding of the Quantum Hall Effect is the relation between longitudinal conductivity and resistivity when the magnetic field is such that the filling factor is an integer, and the Hall resistance is quantized. I fully understand the splitting of the 2D DOS into Landau levels, and can see why the longitudinal conductivity will go to zero since the Fermi level is now at an energy where the DOS is approximately zero.

My confusion is that the longitudinal resistivity also goes to zero. In reading the literature, one can often see in the same paper, plots of ro_xx and sigma_xx as functions of magnetic field, and both show the zeros at the same fields! Perhaps I do not understand the physics behind:

sigma_xx = ro_xx / (ro_xx^2 + ro_xy^2)

Which clearly explains ro_xx = 0 implying sigma_xx. But fundamentally, how can resistivity AND conductivity be zero? Thanks for any light someone can shed.

My confusion is that the longitudinal resistivity also goes to zero. In reading the literature, one can often see in the same paper, plots of ro_xx and sigma_xx as functions of magnetic field, and both show the zeros at the same fields! Perhaps I do not understand the physics behind:

sigma_xx = ro_xx / (ro_xx^2 + ro_xy^2)

Which clearly explains ro_xx = 0 implying sigma_xx. But fundamentally, how can resistivity AND conductivity be zero? Thanks for any light someone can shed.