The integral & fractional quantum hall effect

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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