Lec 11: Quantum Hall effect I – classical Hall effect, Landau levels and their degeneracy

Corrections: (i) From 06:47-07:11 while describing conductivity, I have said that it reaches its maximum right at the middle of the plateaus of Hall resistivity and is zero otherwise. This is not correct. The conductivity is zero over the plateaus and goes through a maximum when Hall resistivity changes from one plateau to the other.

(ii) At 49:07, I have said that the velocity of electrons is (hbar.k/m) in the negative y direction. That is not correct. The operator py gives the canonical momentum. Therefore the velocity vy = (hbar.k/m+eBx). Given the relation between the centre of harmonic potential and k, this velocity (expectation value) is zero.

Classical Hall effect is revived. It is show that if electric field is applied parallel to the plane of 2D electron gas and magnetic field perpendicular to it, conductivity and resistivity are both zero when there is no scattering. Results are interpreted physically. Schrodinger equation for electron in magnetic field perpendicular to the plane of 2D electron gas is solved in Landau gauge and Landau levels and their energy is obtained. Then with addition electric field in the plane of the gas, it is shown that electrons drift with the same speed as in classical physics.

Reference: (i) Chapter 17 (relevant section): Kittel
(ii) Klitzing, Dorda and Pepper, Phys. Rev. Lett. 45, 494 (1980)
(iii) Klitzing, Phys. Rev. Lett. 122, 200001 (2019)

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