Quantum Hall Effect and Fractional Quantum Hall Effect

Hall effect and magnetoresistanceEdwin Herbert Hall (1879): discovery of the Hall effect

HvF ce

Lthe 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 (transverse magnetoresistance)

Hall (off-diagonal) resistance

( ) xxx

x

yyx

x

yH

x

EHj

EjE

Rj H

RH → measurement of the sign of the carrier charge RH is positive for positive charges and negative for negative charges

the Hall effect is the electric field developed across twofaces of a conductorin the direction j×Hwhen a current j flows across a magnetic field H

( ) xxx

x

yyx

x

VR H RI

VR

I

resistivity

Hall resistivity

the Hall coefficient

mceH

ppeE

ppeE

mce

dtd

ce

c

yxcy

xycx

0

0

1

1

pHpEp

HvEfforce acting on electron

equation of motionfor the momentum per electron

in the steady state px and py satisfy

cyclotron frequencyfrequency of revolutionof a free electron in the magnetic field H

at H = 0.1 T0yj xx

cy j

necHjE

0

multiply by

yxcy

xycx

jjE

jjE

0

0

mne

mpnej

mne

2

0

/

the Drudemodel DCconductivityat H=0

Hrcerm cc 2

weak magnetic fields – electrons can complete only a small part of revolution between collisionsstrong magnetic fields – electrons can complete many revolutions between collisions1

1

c

c

1cj is at a small angle f to E f is the Hall angle tan f c

RH → measurement of the density nec

RH1

xx Ej 0 the resistance does not depend on H

GHzcc 1~

2

TGHzcc /10~

2

Higher Mobility= fewer localized states

Single electron in thelowest 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 orbitssmaller than electron separation

new quantum Hall state found atfractional filling factor 1/3

Even higher mobilitiesresult in even morefractional quantum Hallstates

Uncorrelated ? = 1/3 state

Correlated ? = 1/3 state

Whole new concept of a “Composite Fermion”