# 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
the Hall effect is the electric
field developed across two
faces of a conductor
in the direction j&times;H
when a current j flows across
a magnetic field H
e
the Lorentz force FL   v  H
c
in equilibrium jy = 0 → the transverse field (the Hall field) Ey due to the accumulated charges
balances the Lorentz force
quantities of interest:
magnetoresistance
R ( H )  Rxx 
(transverse magnetoresistance)
Hall (off-diagonal) resistance R yx 
Vy
Ix
Vx
Ix
resistivity
 ( H )   xx 
Hall resistivity
 yx 
Ey
the Hall coefficient RH 
Ey
RH → measurement of the sign of the carrier charge
RH is positive for positive charges and negative for negative charges
jx
jx H
Ex
jx
1


f   e E  v  H 
c


dp
1

 p
equation of motion
  e E 
p  H 
mc
for the momentum per electron dt

 
px
0


eE


p

x
c y
in the steady state px and py

satisfy
p
0   eE y  c p x  y
force acting on electron

c 
eH
mc
cyclotron frequency
frequency of revolution
of a free electron in the
magnetic field H
e
2
mc r  c r H
c
multiply by  ne / m
the Drude
model DC
conductivity
at H=0
j   ne
ne 
m
jy  0
0 
jx   0 E x
p
m
2
 0 E x  cj y  jx
 0 E y  cjx  j y
  
 H 
E y   c  jx  
 jx

nec


 0 
the resistance does not
depend on H
RH  
1
nec
c 
c
~ 1GHz
2
at H = 0.1 T
RH → measurement of the density
c  1 weak magnetic fields – electrons can complete only a small part of revolution between collisions
c  1 strong magnetic fields – electrons can complete many revolutions between collisions
c  1
j is at a small angle f to E
f is the Hall angle
tan f  c
c 
c
~ 10GHz / T
2
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”
```