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## Quantum Hall Effect and Fractional Quantum Hall Effect

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**Quantum Hall Effect and Fractional**Quantum Hall Effect**the Lorentz force**Hall effect and magnetoresistance Edwin Herbert Hall (1879): discovery of the Hall effect 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 in equilibrium jy= 0 → the transverse field (the Hall field) Ey due to the accumulated charges balances the Lorentz force quantities of interest: magnetoresistance (transverse magnetoresistance) Hall (off-diagonal) resistance resistivity Hall resistivity the Hall coefficient RH→ measurement of the sign of the carrier charge RHis positive for positive charges and negative for negative charges**multiply by**the Drude model DC conductivity at H=0 RH→ measurement of the density force acting on electron equation of motion for the momentum per electron in the steady state px and py satisfy cyclotron frequency frequency of revolution of a free electron in the magnetic field H at H = 0.1 T the resistance does not depend on H 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 fto E f is the Hall angle tan f = wct**Single electron in the**lowest Landau level Filled lowest Landau level**This was just the beginning of high**mobilities**At high magnetic fields, electron orbits**smaller than electron separation**new quantum Hall state found at**fractional filling factor 1/3**Even higher mobilities**result in even more fractional quantum Hall states**Uncorrelated ? = 1/3 state**Correlated ? = 1/3 state Whole new concept of a “Composite Fermion”