Quantum Hall Effect and Fractional Quantum Hall Effect
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Transcript of Quantum Hall Effect and Fractional Quantum Hall Effect
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”