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The Quantum Hall Effect
Markus ButtikerUniversity of Geneva
International School "Quantum Metrology and Fundamental constants“,'Ecole de Physique des Houches" (France) from Octobe r 1st to October 12 th, 2007.
Generalized longitudinal and Hall resistances2DEG pattterened into a multirpobe conductor
Generalized longitudinal resistance
Four-probe resistance
Generalized Hall resistance
Quantum Hall effect
For all (generalized ) Hall and longitudinal resist ances!!
2
Edge states: smooth potential
⇒⇒
⇒
equipotential line
Transition from N=3 to N= 2 edge states
velocity
7
Integer quantum Hall Effect
GaAs/AlGaAs
T = 0.3 K
BJ&BJ, 1992
Von Klitzing, et al, PRL 1980
3
Electron Focusing
skipping orbit
electron focusingvan Houten et al. , Phys. Rev. B39, 8556 (1989)
4
Skipping orbits
immune to disorder
(semi-classical) quantization
5
2DEG
bulk; Landau levels edge states
Halperin, Phys. Rev. B25, 2185 (1982)
6Edge states
Hall Cross
In the two geometrically very different conductors, edge states connect thecontacts in the same way: in the absence of of backscattering the two conductorsare equivalent. ⇒Same R_H and same R_L
8
Transmission and conductance
⇒
Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988)
9
Four-probe resistances
Current contacts
Voltage probes
G has eigenvalue zero!
10Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988)
Strengths of transmission approach
Includes contacts
Treats all contacts on equal footing
Treats longitudinal and Hall conductance on equal f ooting
All conductances (whether longitudianl or Hall )depen d on states at the Fermi energy only
11
Quantum Hall effectButtiker, Phys. Rev. B38, 9375 (1988)
12
Quantized longitudinal resistances
N edge states in the bulk, K edge states reflected
Buttiker, Phys. Rev. B38, 9375 (1988)
13
QHE: Non-ideal contacts 14
Left: Transition from 2DEG to metallic contact
Right: Transmission through a quantum point contact
With ideal contacts between contacts with reflectio n the Hall resistancesremain quantized and the longitudinal resistances r emain zero! However..
QHE: Non-ideal contactsNon-ideal contacts generate a non-equilibrium popul ation of edge states
Ideal contacts populate all edge states equally; th ey relax a non-equilibrium population: this process is inelastic
Conductor with non-ideal contacts separated by ideal contacts which generate inelastic relaxation
Hall resistances are quantized Longitudinal resistances zero
Inelastic relaxation (equilibration among edge chan nels) helps quantization!
Quantum Hall effect does not require global phase c oherence!!
15
Selective Injection and Detection
Selective population and detection of non-equilibiu m populations
Contact 1 reflects K edge states
Contact 2 reflects L edge states
B. Van Wees et al, PRL 62, 1181 (1989); H. van Houte n et al., PRB 39, 8556 (89)
Contact 1: Injector
Contact 2: Detector
Injection into outermost edge state only, successive opening o detector contactB. W. Alphenaar, et al. PRL (1990)
16
Experimental proof of existence of edge states
Selective generation of currents in edge states and their selective detection is the best proof we have for the relevan ce of the edge states. Obviously in a bulk picture (no boundary effects) s uch experiments would seem impossible to explain.
However misunderstandings and misconceptions and si mply lack ofInformation still lead often to objections.
17
Gauge and Topology
Response to Aharonov-Bohm flux
Requires global phase coherence
Niu and Thouless, PRB 35, 2188 (1987)Laughlin, PRB 23, 5632 (1981)
18
Current distribution at equilibrium
Smooth potential
Equilibrium current density pattern (diamagnetic cu rrent)
Equilibrium electrostatic potential
The equilibrium current through any cross-section o f the conductor vanishes
19
µ
U
a y b
Current distribution in the presence of transport
Linear transport regime
Current at the edge
Current in the bulk
Potential must be calculated from Poisson equation
Density of states
Electrochemical capacitance
Edge to bulk ratio
20
T. Christen and M.B. PRB 53, 2064 (1996)
µL µRCg
+ + + + + + + + + + + + + + +
- - - - - - - - - - - - - -
µR
µL
µ
U
a y b
S. Komiyama and H. Hirai , Phys. Rev. B 54, 2067 (1996)
Current distribution: Summary
..the study of the local current distribution dos n ot prove or disproveEdge channel formulation…
C. W. J. Beenakker and H. van Houten, in “Solid Stat e Physics:Advances in Research and Applications, edited by H. Ehrenreichand D. Turnbull (Academic, San Diego, 1991). Vol. 4 4, p. 177
Physical properties of the QHE
Large voltages
Temperature
s = -0.01 and -0.51 after Ref. 88 Tsui
VRH exponent alpha = ½Field dependent hopping model Polyakovand Shklovskii, Phys. Rev. B48, 11167 (1993)
..but also experiments with positive sSee Shklovskii, universal prefactor, PRL 74, 150-3
Low, kT < 1
Hot spots, cyclotron rad. emission
Intermediate 1K < T < 10K
21
Summary
Transmission, conductance and resistance
Treats longitudinal and Hall resistances on an equi valent footing
Quantum Hall effect: electron motion along edge sta tes, absence of backscattering
Contacts, non-equilibrium injection and detection o f edge states
Edge states an experimental reality
Current distribution