24/10/14

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The quantum Hall effect (QHE) Presentation created for the Theoretisches-Physikalisches Seminar on problems in quantum mechanics

Daniel Issing

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Overview

1. Introduction

Classical Hall effect & quantum mechanical changes

2. Creating a 2-dimensional electron gas (2DEG)

3. Properties of the 2DEG: Classical & qm description

4. The integer quantum Hall effect( IQHE)

Explaining the existence of plateaus

5. A look ahead: The fractional QHE

6. Summary

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Introduction

Classical Hall Effect

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Introduction

However (von Klitzing et al., 1980), for

high magnetic fields and low

temperatures, one observes the

following:

Magnetic field B

Carrier

Concentratio

n n_1

Hal

l res

ista

nce

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Introduction

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Fabrication of the 2DEG

Several possibilities:

MOSFETs,

Heterojunctions,

surface of liquid

helium,....

Here: Only MOSFET

(Metal Oxid

Semiconductor Field

Effect Transitor)

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Fabrication of the 2DEG

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Fabrication of the 2DEG

The Fermi-Dirac

distribution

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Fabrication of the 2DEG

Band structure of a MOSFET

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Beschreibung des 2DEG

Classical Analysis. Homogenous B-Field

Equation of motion:

Solution:

Lagrange function

Hamilton operator

Angular momentum

(z-component)

for symmetric

gauge

Cyclotron rotation

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Description of the 2DEG

Additional electrical field (homogenous)

EoM

Solution

Drift velocity

current density

-> Classical Hall effect!

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Description of the 2DEG

Quantum mechanical analysis (magnetic field only)

:

Hamilton operator

Dynamic

momentum

Larmor length

Pseudo

momentum

Landau levels

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Description of the 2DEG

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Description of the 2DEG

Additional electric field

with Landau gauge

Energy eigenvalues

Expectation value for

the velocity

Entries of the resistance

tensor

Quantized resistance at the Hall plateaus Klitzing constant

with

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Integer QHE

Density of states without

external magnetic field (in

2D):

Separation (splitting) of the continuous

density of states into Landau levels

http://demonstrations.wolfram.com/IntegerQuantumHallEffect/

High field

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Integer QHE

“Landau fan”

Localized and

extended states

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Integer QHE

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Fractionaler QHE

Plateaus also exists for fractions

of integers!

-> Interactions between different

electrons can no longer be

ignored

2 approaches:

wave function ansatz

(Laughlin 1982)

“Composite Particles”

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Sources for the images

The number in square brackets refers to the page where the image appeared

[1]http://www.physik.uni-regensburg.de/aktuell/KollSS05/Schueller-Vortrag-Dateien/image004.gif(16.10.2014)

[3] http://www.leifiphysik.de/sites/default/files/medien/hall05_beweggeladteilch_ver.gif (16.10.2014)

[5,18]

http://www.fkp.uni-hannover.de/fileadmin/institut/Praktikum/Anleitung_Praktikum/Anleitung_Quanten-Hall-Effekt.pdf

(16.10.2014)

[5] http://kouroshziabari.com/wp-content/uploads/2013/01/Klaus-VonKlitzing.jpg (16.10.2014)

[6] http://www.pit.physik.uni-tuebingen.de/PIT-II/teaching/ExPhys-V_WS03-04/ExP-V(3)-Kap1_8-Halbleiter-QHE.pdf

(16.10.2014)

[7] http://upload.wikimedia.org/wikipedia/commons/c/c0/Halbleiter1.PNG (16.10.2014)

[8]

http://upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Fermi-Verteilung_(Temperatur).svg/220px-Fermi-

Verteilung_(Temperatur).svg.png (16.10.2014)

[9] D. Yokoshika, The Quantum Hall Effekt, Springer 2004

[10,11,13] http://www.physik.tu-dresden.de/~becker/hauptseminar/Laubrich_Skript.pdf (16.10.2014)

[15] http://www.pit.physik.uni-tuebingen.de/PIT-II/teaching/ExPhys-V_WS03-04/ExP-V(3)-Kap1_8-Halbleiter-QHE.pdf

(16.10.2014)

[16,4]http://www.nano.physik.uni-muenchen.de/education/praktika/f1_quanten-hall-effekt.pdf (16.10.2014)

[17] MarkO. Goerbig, Quantum Hall Effekt. [cond-mat] 0909.1998v2

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References

K.v. Klitzing, G. Dorda, M. Pepper, Physical Review Letter, Volume 45, Page 494

D.Yoshioka: The Quantum Hall Effekt, Springer 2004

K.v. Klitzing, Der Quanten Hall Effekt, Spektrum der Wissenschaft 1986, S.46

http://www.fkp.uni-hannover.de/fileadmin/institut/Praktikum/Anleitung_Praktikum/

Anleitung_Quanten-Hall-Effekt.pdf (16.10.2014)

http://www.fkp.uni-hannover.de/fileadmin/institut/Praktikum/Anleitung_Praktikum/

Anleitung_Quanten-Hall-Effekt.pdf

(16.10.2014)