Arnulf Materny- Quantum Hall Effect

8/3/2019 Arnulf Materny- Quantum Hall Effect

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April 5, 2002 International University Bremen 1

Quantum Hall

EffectModern PhysicsSpring 2002

Arnulf Materny


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Some Keywords

Resistance

Electric Charge

"Hall Effect"

"Integer Quantum Hall Effect"

"Fractional Quantum Hall Effect"


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Introduction

Why are faces so different, while e.g. oxygen atoms

are about the same in air (O2) and water (H2O)?


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Introduction

Consider a digital photo!On the large scale you see smooth

contrastHowever, on a microscopic scale ...



Introduction

Microscopic picture: quantization

E.g. electrical charge is quantized

...,3,2,1C10602.1

19

=

==

nnenQ

However ...

Under certain conditions electrons behave like

a fluid and fractional charges can be observed



What is Resistance?

Analogy:

Water flows down a stream

Resistance = scattering by stones



What is Resistance?

Electrical conduction:

IUR =

Ohm`s Law

Scattering of single

electrons in a

conductorMomentum of

electron is changedOverall flow is

reduced



Is Zero Resistance Possible?

Yes, it is!If the moving "particle" does not fit to the "obstacle"

This is the case if electrons perform "linkedmotions" the obstacles are just ignored!



Zero Resistance

Electrons form a "quantum fluid"

Condensation only under extreme conditions

(very low temperatures)Reason:

weak interactionamong electrons

becomes significant

0but0 = IV

Wh H If W Add



What Happens If We Add a

Magnetic Field?

Consider an electron moving in an magnetic field

A force acts on the electron: "Lorentz Force"

BvqF

r

r

r

=

vr

Fr

Br



The Hall Effect

Edwin Hall in 1879

(discovery made as

graduate student of

Johns Hopkins Univ.)

tenBIVH

=

(n = density of mobile charges)



Experimental Verification

For a given currentItheHall Voltage VH is found

to be proportional to the

magnetic fieldB

For the sameI, t, andB,

the Hall Voltage is stilldifferent for different

materials!


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The "Quantum Hall Effect"

Experiments investigating the Hall

Effect under extreme conditions

Klaus von Klitzing

Nobel Prize 1985


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Conditions:

t few nm

T about 1 K

B several T

Hall Resistance:

RH= VH/ I

and first derivative

Hall Resistance:

RH= VH/ I

and first derivative


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Plateaus are found

always for the

same values ofRH!

B / T

RH

/

Independent of materialsIndependent of materials


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Explanation of the QHE

What was so special in von Klitzing`s

experimental arrangement?

Extreme conditions (low T, highB, small t)

Macroscopic microscopic picture

Quantum effects become important!

Single atoms change properties in quantumsteps (defined by Planck's constant h)

Energies in a Thin Crystal Layer


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Energies in a Thin Crystal Layer

in a Perpendicular Magnetic Field

Electrons only below

the Fermi energy

Fermi energy drops

with risingBDrop from one

Landau level to the

next results in new

plateau seen in the

QHE experiments

Landau

level

Landau

level

Landau

level

Ideal crystalIdeal crystal + impurities+ impurities + increase ofB+ increase ofB

Fermi

energy Fermi

energy

energy

Quantized Energies Result in


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Quantized Energies Result in

QHEAnother picture explaining the QHEMotion only along discrete circles possible

which would not be the case if

LowB no circles

High T energies higher thenEFermi allowed

Large t circles not in the plane, washing out of

discrete energy steps


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The "Integer QHE"

Connection with constants of nature:

22222

charge

timeenergy

/ timecharge

imeenergy / t

e

h

I

IV

I

VR

==

==

B-1 / T-1

RH

/

he-2

"Integer QHE":

K

,3,2,1with

1

2==

iie

h

RH

Experimental accuracy

better than 1 in 107 !


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There is More ...

Nobel Prize 1998 for the discovery of

the "Fractional Quantum Hall Effect"

Daniel C. TsuiPrinceton University

Horst L. StrmerColumbia Universityand Bell Labs

Robert B. LaughlinStanford University


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The "Fractional QHE"

Even more extreme conditions:

T < 1 K

B > 10 T

RH

/

he-

2

B / T

Fractional QHE:

( )

K,

5

1,

5

2,

3

1with

1

2

=

==

i

eie

h

ie

h

RH


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Explanation A Difficult Task

Electrons behave like particles carrying n lines of flux

Theory: n must be a odd number

Motion of flux induces a transverse voltage

Contribution by one "particle" (boson): n

RHappears as "fractional multiple" ofh / e

2


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"Composite Fermions Model"

P.A.M. Dirac:

"The main object of physical systems is not the provision of pictures,

but the formulation of laws governing phenomena, and the discovery of

these laws to the discovery of new phenomena. If a picture exists, so

much the better, but whether a picture exists or not is a matter of only

secondary importance."

P.A.M. Dirac:

"The main object of physical systems is not the provision of pictures,

but the formulation of laws governing phenomena, and the discovery of

these laws to the discovery of new phenomena. If a picture exists, so

much the better, but whether a picture exists or not is a matter of onlysecondary importance."

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Arnulf Materny- Quantum Hall Effect