the integral & fractional quantum hall effect
TRANSCRIPT
THE INTEGRAL &FRACTIONAL QUANTUM
HALL EFFECT
SUDIPTO DAS
15PH40041
Presented by
IIT Kharagpur
Outline
Classical to quantum hall effect
2 Dimensional Electron gas (2DEG) in B-field
Integral Quantum Hall Effect
Fractional Quantum Hall Effect
Composite Fermion
Application
From Classical to Quantum hall effect
xx
LWRne
xyBRne
Electrical resistance
Hall Resistance
xy
xx
Low temperature T<4KHigh Magnetic FieldHigh mobility > 2 x 104 cm2/Vs
From Classical to Quantum hall effect
xy
xx
xx
LWRne
xyBRne
Electrical resistance
Hall Resistance
2DEG in quantizing B-field
2
*
12
H p eAm
Landau Gauge,
(0, )A xB
2
* *
( );
2 2yx
y y
eBx kpH p km m
2* 2 2
* *1 ( )
2 2;k c
xc
pH m x Bx
mem yk
k eBx
,
( 1/ 2)
( , ) ( ) y
n
ik yn k n k
E n
x y xx e
xk centre coordinate
xq= xk + (h/eBL)
Thus flux between neighbouring states, ɸ0 = B.{ (xq-xk)L } = h/e Dirac flux
quanta
Degeneracy of States
Integral Quantum Hall Effect
The Nobel Prize in Physics 1985 was awarded to Klaus von Klitzing "for the discovery of the quantized Hall effect”
Integral Quantum Hall Effect
RH =
RH = hall Resistance i = filling factor (having values1,2,…. for Integer QHE)
xy
xx
Fractional Quantum Hall Effect
From where this fractional terms came !!
Electrons in B-field
Flux quantum:
ɸ0 = BA = Bπr2 = h/e
Degeneracy:
nɸ = B/ɸ0 = eB/h
Filling factor:
v = ne/ nɸ
Fractional QHE
H. Stormer, Physics B 177 (1992)
Flux Attachment Transformation
Fermionic:
1, Z2) = - (Z2, Z1) Fermionic Antisymmetry
Bosonic:
1, Z2) = - (Z2, Z1)eiπ Fermionic Antisymmetry1, Z2) = + (Z2, Z1) +2π phase shift
Composite FermionsA new kind of quasi-
particlsJ. Jain
PRL. (1989)
V = 1/2B* CF = 0
B = magnetic fieldv = ne/nɸ = filling factor
B* = B - 2ɸ0ne = effective field v*CF = nCF/nɸ = effective v
Composite Fermions
v = 1
v*CF = 1
v*CF = 2
v = 1/3 v*CF = p v =
v = 2/3 v*CF = p v =
q* CF= 2(e/3) + (-e) = - e/3 q*CF =
q* CF= 2(2e/3) + (-e) = + e/3 q*CF =
Charge of flux quantumΡbkgnd = +ene q+ = A Ρbkgnd = +ene/nɸ = +ev
Fractional QHE
Applications Used as Magnetometers, i.e. to measure magnetic field.
Hall effect sensor is also used as Current Sensor.
Magnetic Position Sensing in Brushless DC Electric Motors
Automotive fuel level indicator.
Spacecraft propulsion.
Quantum hall effect can be used for a determination of h/e2 or as a resistance standard.
Since the inverse fine structure constant α-1 is more or less identical h/e2 , high-precision measurements of the quantized hall resistance are important for all areas in physics that are connected with the fine structure constant.
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