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PMP_7300_Doucot_Titelei 15.6.2005 15:58 Uhr Seite 1 PMP_7300_Doucot_Titelei 15.6.2005 15:58 Uhr Seite 2 Progress in Mathematical Physics Volume 45 Editors-in-Chief Anne Boutet de Monvel,Université Paris VII Denis Diderot,France Gerald Kaiser,Center for Signals and Waves,Austin,TX,USA Editorial Board Sir M.Berry,University of Bristol, UK C.Berenstein,University of Maryland,College Park,USA P.Blanchard,University of Bielefeld, Germany A.S.Fokas,University of Cambridge, UK D.Sternheimer,Université de Bourgogne, Dijon, France C.Tracy,University of California,Davis,USA PMP_7300_Doucot_Titelei 15.6.2005 15:58 Uhr Seite 3 The Quantum Hall Effect Poincaré Seminar 2004 Benoît Douçot Bertrand Duplantier Vincent Pasquier Vincent Rivasseau Editors Birkhäuser Verlag Basel Boston Berlin • • PMP_7300_Doucot_Titelei 15.6.2005 15:58 Uhr Seite 4 Editors: Benoît Douçot Bertrand Duplantier LPTHE Service de Physique Théorique CNRS et Universités Paris 6 et 7 Orme des Merisiers Tour 24-14 5e étage CEA –Saclay 4 place Jussieu F-91191 Gif-sur-Yvette Cedex F-75252 Paris Cedex 05 e-mail:[email protected] e-mail:[email protected] Vincent Pasquier Vincent Rivasseau Service de Physique Théorique Laboratoire de Physique Théorique Orme des Merisiers Université Paris XI CEA –Saclay F-91405 Orsay Cedex F-91191 Gif sur Yvette Cedex e-mail:[email protected] e-mail:[email protected] 2000 Mathematics Subject Classification 81V70 A CIP catalogue record for this book is available from the Library of Congress,Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-7300-8 Birkhäuser Verlag,Basel – Boston – Berlin This work is subject to copyright.All rights are reserved,whether the whole or part of the material is concerned,specifically the rights of translation,reprinting,re-use of illustrations,broadcasting, reproduction on microfilms or in other ways,and storage in data banks.For any kind of use whatsoever,permission from the copyright owner must be obtained. © 2005 Birkhäuser Verlag,P.O.Box 133,CH-4010 Basel,Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp.TCF (cid:39) Printed in Germany ISBN-10:3-7643-7300-8 ISBN-13:978-3-7643-7300-9 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Foreword.....................................................................vii Klaus von Klitzing 25 Years of Quantum Hall Effect (QHE) A Personal View on the Discovery, Physics and Applications of this Quantum Effect...................................................1 Benoˆıt Douçot and Vincent Pasquier Physics in a Strong Magnetic Field......................................23 Beat Jeckelmann and Blaise Jeanneret The Quantum Hall Effect as an Electrical Resistance Standard...........55 Steven M. Girvin Introduction to the Fractional Quantum Hall Effect.....................133 D. Christian Glattli Tunneling Experiments in the Fractional Quantum Hall Effect Regime.............................163 We thank the Institute of Physics Publishing for its authorizationto repro- duce the paper by B. Jeckelmann and B. Jeanneret. Foreword This book is the third in a series of lectures of the Séminaire Poincaré, which is directed towards a large audience of physicists and of mathematicians. The goal of this seminar is to provide up to date information about general topics of great interest in physics. Both the theoretical and experimental aspects are covered, with some historical background. Inspired by the Bourbaki seminar in mathematics in its organization, hence nicknamed “Bourbaphi”, this Poincaré SeminarisheldtwiceayearattheInstitutHenriPoincaréinParis,withcontribu- tions prepared in advance. A particular care is devoted to the pedagogicalnature ofthe presentationsoas tofulfill the goalofbeing readablebya largeaudience of scientists. ThisvolumecontainsthesixthsuchSeminar,heldin2004.Itisdevotedtothe Quantum Hall Effect. After a historical and general presentation by Nobel prize Klaus von Klitzing, discovererof this effect, the volume proceeds with reviews on the mathematics and physics of both the integer and fractionalcase,and includes up to date presentations of the tunneling and metrology experiments related to the Quantum Hall Effect. Wehopethatthepublicationofthisserieswillservethecommunityofphysi- cists and mathematicians at professional or graduate student level. We thank the Commissariat à l’Eńergie Atomique (Division des Sciences de laMatière),theCentreNationaldelaRechercheScientifique(SciencesPhysiqueet Mathématiques), and the Daniel Iagolnitzer Foundation for sponsoring the Semi- nar. Special thanks are due to Chantal Delongeas for the preparationof the man- uscript. Benoˆıt Douçot Bertrand Duplantier Vincent Pasquier Vincent Rivasseau TheQuantumHallEffect,1–21 (cid:1)c BirkhäuserVerlag,Basel,2005 Poincaré Seminar 2004 25 Years of Quantum Hall Effect (QHE) A Personal View on the Discovery, Physics and Applications of this Quantum Effect Klaus von Klitzing 1 Historical Aspects The birthday of the quantum Hall effect (QHE) can be fixed very accurately. It was the night of the 4th to the 5th of February 1980 at around 2 a.m. during an experiment at the High Magnetic Field Laboratory in Grenoble. The research topicincludedthecharacterizationoftheelectronictransportofsiliconfieldeffect transistors. How can one improve the mobility of these devices? Which scattering processes (surface roughness, interface charges, impurities etc.) dominate the mo- tionoftheelectronsintheverythinlayerofonlyafewnanometersattheinterface betweensiliconandsilicondioxide?Forthisresearch,Dr.Dorda(SiemensAG)and Dr. Pepper (Plessey Company) provided specially designed devices (Hall devices) as shown in Fig.1, which allow direct measurements of the resistivity tensor. For the experiments, low temperatures (typically 4.2 K) were used in order to suppress disturbing scattering processes originating from electron-phonon in- teractions. The application of a strong magnetic field was an established method toget more informationabout microscopic details of the semiconductor. A review articlepublishedin1982byT.Ando,A.Fowler,andF.Sternabouttheelectronic properties of two-dimensional systems summarizes nicely the knowledge in this field at the time of the discovery of the QHE [1]. Since 1966 it was known, that electrons, accumulated at the surface of a silicon single crystal by a positive voltage at the gate (= metal plate parallel to the surface), form a two-dimensional electron gas [2]. The energy of the electrons for a motion perpendicular to the surface is quantized (“particle in a box”) and even the free motion of the electrons in the plane of the two-dimensional system becomes quantized (Landau quantization), if a strong magnetic field is applied perpendicular to the plane. In the ideal case, the energy spectrum of a 2DEG in strong magnetic fields consists of discrete energy levels (normally broadened due to impurities) with energy gaps between these levels. The quantum Hall effect is observed, if the Fermi energy is located in the gap of the electronic spectrum and if the temperature is so low, that excitations across the gap are not possible. The experimental curve, which led to the discovery of the QHE, is shown in Fig. 2. The blue curve is the electrical resistance of the silicon field effect tran- 2 K.vonKlitzing Figure 1: Typical silicon MOSFET device used for measurements of the xx- and xy-components of the resistivity tensor. For a fixed source-drain current between the contacts S and D, the potential drops between the probes P −P and H−H are directly proportional to the resistivities ρ and ρ . A positive gate voltage xx xy increases the carrier density below the gate. sistor as a function of the gate voltage. Since the electron concentration increases linearly with increasing gate voltage, the electrical resistance becomes monotoni- cally smaller. Also the Hall voltage (if a constant magnetic field of e.g. 19.8 Tesla is applied) decreases with increasing gate voltage, since the Hall voltage is basi- cally inversely proportional to the electron concentration. The black curve shows the Hall resistance, which is the ratio of the Hall voltage divided by the current through the sample. Nice plateaus in the Hall resistance (identical withthe trans- verse resistivity ρ ) are observed at gate voltages, where the electrical resistance xy (which is proportional to the longitudinal resistivity ρ ) becomes zero. These ze- xx rosareexpectedforavanishing density ofstateof (mobile)electronsatthe Fermi energy. The finite gate voltage regions where the resistivities ρ and ρ remain xx xy unchangedindicate,thatthegatevoltageinducedelectronsintheseregionsdonot contributetotheelectronictransport-theyarelocalized.Theroleoflocalizedelec- trons in Hall effect measurements was not clear. The majority of experimentalists believed, that the Hall effect measures only delocalized electrons. This assump- tion was partly supported by theory [3] and formed the basis of the analysis of QHE data published already in 1977 [4]. These experimental data, available to 25YearsofQuantumHallEffect(QHE) 3 Figure2:Hallresistanceandlongitudinalresistance(atzeromagneticfieldandat B =19.8 Tesla) of a silicon MOSFET at liquid helium temperature as a function of the gate voltage. The quantized Hall plateau for filling factor 4 is enlarged. thepublic 3yearsbeforethediscoveryofthequantumHalleffect,containalready all information of this new quantum effect so that everyone had the chance to make a discovery that led to the Nobel Prize in Physics 1985. The unexpected findinginthenightof4./5.2.1980wasthefact,thattheplateauvaluesintheHall resistance ρ are not influenced by the amount of localized electrons and can be xy expressed with high precision by the equation ρ = h/ie2 (h=Planck constant, xy e=elementary charge and i the number of fully occupied Landau levels). Also it becameclear,thatthecomponentρ oftheresistivitytensorcanbemeasureddi- xy rectlywithavolt-andamperemeter(afactoverlookedbymanytheoreticians)and that for the plateau values no information about the carrier density, the magnetic field, and the geometry of the device is necessary. 4 K.vonKlitzing Figure 3: Copy of the original notes, which led to the discovery of the quantum Halleffect.ThecalculationsfortheHallvoltageU foronefullyoccupiedLandau H level show, that the Hall resistance U /I depends exclusively on the fundamental H constant h/e2.