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!!

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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)

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Skipping orbits

immune to disorder

(semi-classical) quantization

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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

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Transmission and conductance

⇒

Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988)

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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

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Quantum Hall effectButtiker, Phys. Rev. B38, 9375 (1988)

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Quantized longitudinal resistances

N edge states in the bulk, K edge states reflected

Buttiker, Phys. Rev. B38, 9375 (1988)

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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!!

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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)

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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.

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Gauge and Topology

Response to Aharonov-Bohm flux

Requires global phase coherence

Niu and Thouless, PRB 35, 2188 (1987)Laughlin, PRB 23, 5632 (1981)

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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

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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

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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

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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