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)