1. Quantum Hall Effect and Fractional Quantum Hall Effect

2. Hall effect and magnetoresistance Edwin Herbert Hall (1879): discovery of the Hall effect the Lorentz force in equilibrium j y = 0 → the transverse field (the Hall field) E y due to the accumulated charges balances the Lorentz force quantities of interest: magnetoresistance (transverse magnetoresistance) Hall (off-diagonal) resistance R H → measurement of the sign of the carrier charge R H is positive for positive charges and negative for negative charges the Hall effect is the electric field developed across two faces of a conductor in the direction j×H when a current j flows across a magnetic field H resistivity Hall resistivity the Hall coefficient

3. force acting on electron equation of motion for the momentum per electron in the steady state p x and p y satisfy cyclotron frequency frequency of revolution of a free electron in the magnetic field H  at H = 0.1 T multiply by the Drude model DC conductivity at H=0 weak magnetic fields – electrons can complete only a small part of revolution between collisions strong magnetic fields – electrons can complete many revolutions between collisions j is at a small angle  to E  is the Hall angle tan  c  R H → measurement of the density the resistance does not depend on H

11. Higher Mobility= fewer localized states

12. Single electron in the lowest Landau level Filled lowest Landau level

13. Modulation doping and high mobility heterostructures

15. This was just the beginning of high mobilities

17. At high magnetic fields, electron orbits smaller than electron separation

18. new quantum Hall state found at fractional filling factor 1/3

19. Even higher mobilities result in even more fractional quantum Hall states

20. Uncorrelated ? = 1/3 state Correlated ? = 1/3 state Whole new concept of a “Composite Fermion”