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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 ...
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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
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What is Resistance?
Analogy:
Water flows down a stream
Resistance = scattering by stones
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What is Resistance?
Electrical conduction:
IUR =
Ohm`s Law
Scattering of single
electrons in a
conductorMomentum of
electron is changedOverall flow is
reduced
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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!
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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
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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
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The Hall Effect
Edwin Hall in 1879
(discovery made as
graduate student of
Johns Hopkins Univ.)
tenBIVH
=
(n = density of mobile charges)
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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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The "Quantum Hall Effect"
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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The "Quantum Hall Effect"
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."