Quantum Hall Effect and Fractional Quantum Hall Effect

Quantum Hall Effect and Fractional Quantum Hall Effect

Hall effect and magnetoresistance Edwin Herbert Hall (1879): discovery of 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 the Lorentz force in equilibrium jy = 0 → the transverse field (the Hall field) Ey due to the accumulated charges balances the Lorentz force quantities of interest: magnetoresistance resistivity (transverse magnetoresistance) Hall (off-diagonal) resistance Hall resistivity the Hall coefficient RH → measurement of the sign of the carrier charge RH is positive for positive charges and negative for negative charges

force acting on electron equation of motion for the momentum per electron in the steady state px and py satisfy cyclotron frequency of revolution of a free electron in the magnetic field H multiply by the Drude model DC conductivity at H=0 at H = 0. 1 T the resistance does not depend on H RH → measurement of the density 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 f to E f is the Hall angle tan f = wct

Higher Mobility= fewer localized states

Single electron in the lowest Landau level Filled lowest Landau level

Modulation doping and high mobility heterostructures

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”