Kondo Effect and Dephasing in Low-Dimensional Metallic Systems

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IOS Press Kluwer Academic Publishers IOS Press Kluwer Academic Publishers IOS Press

Series II: Mathematics, Physics and Chemistry - Vol. 50

Kondo Effect and Dephasing in Low-Dimensional Metallic Systems

edited by

Venkat Chandrasekhar Department of Physics and Astronomy, Northwestern University, Evanston, U.S.A.

Chris Van Haesendonck Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Leuven, Belgium

and

Alfred Zawadowski Institute of Physics, Budapest University of Technology and Economics, Budapest, Hungary

Springer-Science+Business Media, B.V.

Proceedings of the NATO Advanced Research Workshop on Size Dependent Magnetic Scattering Pecs, Hungary 29 May-1 June 2000

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-0401-8 ISBN 978-94-010-0427-5 (eBook) DOI 10.1007/978-94-010-0427-5

Printed on acid-free paper

All Rights Reserved ©2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

CONTENTS

PREFACEV. Chandrasekhar, e. Van Haesendonck, and A. Zawadowski

INTRODUCTIONN.O. Birge, D. Ralph, V. Chandrasekhar, e. Van Haesendonck, andA. Zawadowski

Finite Size Effects in Kondo Alloys

EFFECT OF DISORDER ON THE KONDO BEHAVIOR OF THIN CU(MN)FILMST.M. Jacobs and N. Giordano

THE KONDO EFFECT AND WEAK LOCALIZATIONP. Phillips and I. Martin

SURFACE MAGNETIC ANISOTROPY OF KONDO IMPURITIESINDUCED BY SPIN-ORBIT SCATTERINGO. Ujsaghy

Finite Size Effects in Spin Glass Alloys

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11

23

THERMOPOWER OF MESOSCOPIC SPIN GLASSESe. Strunk, G. Neuttiens, M. Henny, e. Van Haesendonck, ande. Schonenberger 33

SHAPE-INDUCED MAGNETIC ANISOTROPY IN DILUTE MAGNETICALLOYSV.N. Gladilin, V.M. Fornin, and J.T. Devreese 43

ZERO-BIAS TRANSPORT ANOMALY IN METALLIC NANOBRIDGESMagnetic field dependence and universal conductance fluctuationsH.B. Weber, R. Haussler, H. v. LOhneysen, and J. Kroha 53

CONDUCTANCE NOISE AND IRREVERSIBILITY IN DILUTEDMAGNETIC SEMICONDUCTORSJ. Jaroszynski, J. Wrobel, G. Karczewski, T. Wojtowicz, T. Dietl, E. Karninska,E. Papis, A. Piotrowska, D.K. Maude, P. van der Linden, and J.e. Portal 63

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Point Contact Spectroscopy and Tunneling Spectroscopy ofKondo Impurities

ENHANCEMENT OF KONDO TEMPERATURE IN NANOMETER-SIZEPOINT CONTACTSLK. Yanson, V.V. Fisun, J.A. Mydosh, and J.M. van Ruitenbeek 73

SCANNING TUNNELLING SPECTROSCOPY OF A SINGLE KONDOIMPURITYR. Berndt and W.-D. Schneider 87

Two-Level Systems and Dephasing in Thin Metal Structures

TWO-CHANNEL KONDO EFFECT FROM TUNNELING IMPURITIESG. Zanind 97

ELECTRON DECOHERENCE AT ZERO TEMPERATUREThe phenomenology and associated difficultiesP. Mohanty 107

PROBING INTERACTIONS IN MESOSCOPIC GOLD WIRESF. Pierre, H. Pothier, D. Estyve, M.H. Devoret, A.B. Gougam, and N.O. Birge 119

KONDO EFFECT IN NON-EQUILIBRIUMTheory ojenergy relaxation induced by dynamical dejects in diffusive nanowiresJ.Kroha 133

Kondo Effect in Quantum Dots

TUNNELING THROUGH A QUANTUM DOTThe out-oj-equilibrium Kondo effectA. Schiller

ELECTRON TRANSPORT THROUGH QUANTUM DOTS:AN UNUSUAL KONDO EFFECTS. De Franceschi, S. Sasaki, J.M. E1zerman, W.G. van der Wiel, M. Eto,S. Tarucha, and L.P. Kouwenhoven

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THE KONDO EFFECT IN A SINGLE-ELECTRON TRANSISTORD. Goldhaber-Gordon, J. Gores, H. Shtrikman, D. Mahalu, U. Meirav, andM.A. Kastner

Contributions related to Poster Presentations

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163

FLUX DEPENDENT DIELECTRIC RESPONSE OF STACKEDNANOSCOPIC RINGSK-H. Ahn and P. Fulde 171

HIGH~FREQUENCY RESPONSE OF TWO-LEVEL SYSTEMS IN NIxNBI_XMETALLIC POINT CONTACTSO.P. Ba1kashin, I.K Yanson, A. Halbritter, and G. Mihaly 175

GIANT MAGNETORESISTANCE OF A SINGLE INTERFACEMagnetoresistance ofAg/Fe/Ag trilayers1. Balogh, A. Gabor, D. Kaptas, L.F. Kiss, M. Csontos, A. Halbritter,I. Kezsmarki, and G. Mihaly 181

FERMI EDGE SINGULARITIES IN TRANSPORT THROUGH QUANTUMDOTSE. Bascones, c.P. Herrero, F. Guinea, and G. Schon 185

THEORY OF MAGNETORESISTANCE IN FILMS OF DILUTEMAGNETIC ALLOYSL. Borda 189

CURRENT AND SHOT NOISE IN A FERROMAGNETIC DOUBLETUNNEL JUNCTION WITH AN ATOMIC SIZE SPACERB.R. Bulka, J. Martinek, G. Michalek, and J. Barnas 193

POSITIVE DOMAIN-WALL MAGNETORESISTANCE OFFERROMAGNETIC POINT CONTACTSJ. Caro, SJ.C.H. Theeuwen, K.I. Schreurs, R.P. van Gorkom, KP. Wellock,N.N. Gribov, S. Radelaar, and V.1. Kozub 197

ENHANCEMENT OF KONDO EFFECT DUE TO SPIN-SINGLET-TRIPLET COMPETITION IN QUANTUM DOTSM. Eto and Yu.V. Nazarov 203

A NON-KONDO INTERPRETATION OF THE EXPERIMENTALLYOBSERVED "KONDO RESONANCES" IN QUANTUM DOTSJ. Fransson and I. Sandalov 207

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DIAGRAMMATIC THEORY OF THE ANDERSON IMPURITY MODELWITH FINITE COULOMB INTERACTIONK. Haule, S. Kirchner, J. Kroha, and P. Woltle 211

GENERALIZED CONDUCTANCE SUM RULE IN ATOMIC BREAKJUNCTIONSS. Kirchner, J. Kroha, and E. Scheer 215

IS CENISN A KONDO SEMICONDUCTOR?Break-junction experimentsYu.G. Naidyuk, K. Gloos, and T. Takabatake 219

THE INFLUENCE OF SINGLE MAGNETIC IMPURITIES ON THECONDUCTANCE OF QUANTUM MICROCONSTRICTIONSA. Namiranian, YU.A. Kolesnichenko, A.N. Omelyanchouk 223

STRUCTURAL PROPERTIES OF COLLOIDAL Co NANOPARTICLESF. Pedreschi, J.D. O'Mahony, and C.FJ. Flipse 227

INTERACTING ELECTRONS IN A NEARLY STRAIGHT QUANTUMWIRET. Rejec, A. Ramsak, and J.H. Jefferson 231

RESONANT TUNNELING THROUGH AN IMPURITY LEVEL: A PROBEOF COHERENT STATES IN A DISORDERED METALT. Schmidt, P. Konig, RJ. Haug, E. McCann, and V.I. Fal'ko 237

PAIR BREAKING IN s-WAVE SUPERCONDUCTORS BY TWO-CHANNELKONDO IMPURITIESG. Sellier, S. Kirchner, and J. Kroha 241

THEORY OF SCANNING TUNNELING SPECTROSCOPY OF KONDOIONS ON METAL SURFACESO. Ujsaghy, J. Kroha, L. Szunyogh, and A. Zawadowski 245

TWO-LEVEL SYSTEMS IN ATOMIC-SIZE POINT CONTACTSH.E. van den Brom, Y. Noat, and J.M. van Ruitenbeek 249

IS THE MULTI-CHANNEL KONDO MODEL APPROPRIATE TODESCRIBE SINGLE ELECTRON TRANSISTORS?G. Zarand 253

LIST OF OBSERVERS 257

LIST OF PARTICIPANTS 259

PREFACE

The NATO Advanced Research Workshop took place from 29 May toI June 2000 in the picturesque Hungarian town of Pecs, 220 km south ofBudapest. The main goal of the workshop was to review and promoteexperimental and theoretical research on the problem of Kondo-typescattering of the electrons in systems of reduced dimensionalities. 53 regularparticipants and 7 observers from 17 different countries attended theworkshop.

The Kondo effect has been a topic of intense interest for many years, due inpart to its relevance to a variety of other branches of condensed matterphysics. In addition to the best known example of magnetic impurities innoble metals, the physics of the Kondo effect is important in many areas ofcurrent research, including heavy-fermion physics, correlated electronsystems, and high-temperature superconductivity. Of central importance inthis problem is the interaction of conduction electrons in the metal withindividual magnetic impurities, an interaction which also mediates theinteraction of the impurities with each other.

Despite numerous theoretical as well as experimental efforts, the Kondoproblem is far from being understood, in particular for smaller metallicsystems. In the past few years, the availability of nanolithographicfabrication techniques and new spectroscopic measuring tools (scanningtunneling microscope, mechanically controllable break junctions, ... ) hasopened the possibility of directly probing these interactions on a mesoscopicsize scale. The first experiments on such systems measured the electricaltransport properties of low-dimensional metallic films with dilute magneticimpurities. More recently, there have been a number of beautifulexperiments on a variety of materials systems which probe different aspectsof the Kondo problem. The unifying theme of all these experiments,however, is that they investigate systems whose dimensions are comparableto fundamental length scales of the Kondo effect.

The workshop in Pecs provided the first international forum to discuss themajor experimental and theoretical progress that had been made during thepast few years for low-dimensional Kondo systems. Researchers coveringvarious aspects of Kondo physics could come together to discuss thecommon aspects of their work. During the workshop the attention has beenfocused on the following interrelated topics.

• Finite size effects in Kondo alloys: What is the relevance of the Kondoscreening cloud, the surface-induced anisotropy and disorder to our

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understanding of the size dependence of the Kondo transport propertiesin small metallic samples of dilute Kondo alloys?

• Finite size effects in spin glasses: What happens to the size dependencefor more concentrated alloys, i.e., spin glass alloys, where the interactionbetween magnetic impurities becomes important and time-dependentfluctuations of the frozen spin configuration can no longer be neglected?

• Spectroscopy of Kondo impurities: What can measurements with low-temperature scanning tunneling spectroscopy and with mechanicallycontrollable break junctions teach us about the spatial extent of theKondo effect and the relevance of local fluctuations in the electronicdensity of states?

• Two-level systems: What is the relevance of two-level systems forelectron transport through metallic nanobridges, and how can theinteraction between the conduction electrons and two-level systems bedescribed in terms of two-channel Kondo scattering?

• Dephasing of conduction electron waves in metals: Is the experimentallyobserved saturation of the dephasing time an intrinsic effect or is it dueto the presence of two-level systems or residual magnetic impurities?

• Kondo effect in quantum dots: Can quantum dots be used as perfectlytunable Kondo systems?

Acknowledgments

We would like to thank all speakers, discussion leaders and participants whohave contributed to the success of the workshop in Pees. We would also liketo thank all participants who have prepared a manuscript for the proceedings.

We are much indebted to our collaborators at the Budapest University ofTechnology and Economics and at the University of Leuven who haveassisted us before, during and after the workshop. In particular, we are verygrateful to Dr. Orsolya Ujsaghy (Budapest University of Technology andEconomics) and to Maria Werner (Centre of the Pees Academic Committee).

Finally, we acknowl~dge the generous grant by the Scientific AffairsDivision of the North Atlantic Treaty Organization (NATO) without whichthe workshop would not have been possible.

Venkat ChandrasekharChris Van HaesendonckAlfred Zawadowski

June 2000

INTRODUCTION

Norman O. Birge a , Daniel C. Ralph b, Orsolya UjsaghyC,Gergely Zarand c, Alfred Zawadowski c, Venkat Chandrasekhar d,and Chris Van Haesendonck e

a Department of Physics and Astr'onomy, Michigan State University,

East Lansing, MI48824-1116, USA

b Laboratory of Atomic and Solid State Physics, Cornell University,

Ithaca, NY 14853, USA

C Institute of Physics, Budapest University of Technology and Economics,

Budafoki lit 8, H-1111 Budapest, Hungm'Y

d Department of Physics and Astronomy, Northwestern University,

2145 Sheridan Road, Evanston, IL 60208, USA

e Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven,

Celestijnenlaan 200 D, B-3001 Leuven, Belgium

Since the discovery of the anomalous low temperature resistivity in-crease exhibited by some metallic samples [1] these anomalies have at-tracted considerable interest. The first theoretical work to explain theanomalies was due to Kondo, who demonstrated that the scatteringrate of electrons in metals by magnetic impurities has an anomalousthird order contribution, which increases logarithmically as the ,temper-ature is reduced and leads to the breakdown of perturbation theory [2].Since then this phenomenon is known as the Kondo effect. FollowingKondo's original work a lot of theoretical effort has been devoted tounderstanding this phenomenon in detail. Wilson's numerical renormal-ization group treatment of the strong coupling limit [3] and Nozieres'Fermi liquid theory [4] turned out to be the most important milestonesin this development.Recently, the number of papers related to the Kondo effect showed a

significant increase with broader and broader applications of the model.Various dilute and dense U and Ce based metallic alloys have been sug-

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gested as Kondo systems with both magnetic and orbital features [5].In these systems at low temperature very strong correlations build up,hence they became known as strongly correlated systems. In addition,nanotechnology opens up new perspectives and offers new possibilities tostudy magnetic impurities and strongly correlated systems. New devel-opments have been in the direction of the observation of the Kondo effectin mesoscopic systems such as thin layers and point contacts, and alsoartificial mesoscopic atoms (quantum dots). In the latter nanofabricateddevices the d-Ievel of the magnetic impurity in the metal is mimickedby degenerate states of a quantum dot, which is coupled to metallic orsemiconducting leads.The NATO Advanced Research Workshop on Size Dependent Mag-

netic Scattering dealt with different topics associated with the Kondoeffect, ranging from new avenues of research in traditional Kondo sys-tems, to the exploration of Kondo physics and magnetic scattering inneW materials systems. In addition, the workshop also covered electrondephasing in mesoscopic systems, because the mechanisms proposed toexplain the experimental results include electron scattering from two-level systems which can be described in terms of a two-channel Kondoeffect. The main topics discussed during the workshop are introducedbelow.

1. FINITE SIZE EFFECTS IN KONDO ANDSPIN-GLASS ALLOYS

An appealing physical picture of the Kondo effect in dilute magneticalloys involves the formation of a "screening cloud" of conduction elec-trons around each magnetic impurity. The spatial extent of this screen-ing cloud is determined by the Kondo temperature, and is given by~K = hVF/kBTK in the ballistic regime, and by ~K = JhD/kBTI\ inthe diffusive regime, where VF is the Fermi velocity, and D = iVFfeis the electronic diffusion coefficient (fe is the elastic mean free path).With modern lithographic techniques, this length scale is now in theexperimentally accessible regime, and one might naturally expect thatconfining the screening cloud might result in measurable changes in theKondo effect. In the last decade, many experiments [6, 7, 8, 9] have beenperformed on thin films and narrow wires of dilute magnetic alloys insearch of the Kondo screening cloud. In the pioneering work of Giordanoand coworkers described in this volume (T.M. Jacobs and N. Giordano,this volume, p. 1), a suppression of the Kondo resistivity amplitude wasobserved for small sample sizes. Covering a thin layer of magnetic al-loys by another pure metal layer, a partial recovery of the Kondo signal

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was found [10, 11J which was smaller for more disordered overlayers [12J.The first natural explanation concerning the compensation cloud [13Jwas ruled out both theoretically [14, 15J and experimentally [11J as theKondo singlet is formed whenever the level spacing is small comparedto the Kondo temperature.Two theories have been developed that seem to explain the two lim-

iting cases in these experiments: The first, presented by Philips (P.Phillips and I. Martin, this volume, p. 11), is based on weak localiza-tion, and may be valid in disordered samples [6, 10], where the smallestsystem size is large compared to the elastic mean free path and theKondo anomaly depends on the level of disorder. The other theory, de-scribed in the contribution by Ujsaghy (0. Ujsaghy, this volume, p. 23),depends on the strong surface anisotropy which is developed in sampleswith strong spin-orbit interaction on the non-magnetic host atoms [16J.In this case electrons can mediate information about the geometry ofthe sample, resulting in an anisotropy for the impurity spin nearby thesurfaces, but only in those cases where the angular momenta of the local-ized orbital I f:. 0 (e.g. I = 2). An elegant extension of these calculationsto general geometries was described by Fomin. Fomin and his coworkersinvestigated the dependence of the anisotropy on the roughness of thesurface as well (V.N. Gladilin et aI., this volume, p. 43).In addition to the temperature and magnetic field dependent resis-

tance of Kondo alloys, the thermopower of dilute magnetic alloys alsoshows a pronounced size dependence. In experiments described in thecontribution of Strunk and coworkers (C. Strunk et aI., this volume,p. 33), the thermopower of AuFe wires was found to depend on theirwidth, with the thermopower decreasing with decreasing width. This isin agreement with the theory of spin-orbit induced surface anisotropy,although a detailed theory of the size dependence of the thermopowerin Kondo alloys has not been developed.

It should be pointed out that there are two sets of experiments onAuFe wires (one in the Kondo regime [8] and one in the spin-glass regime[7]) that do not show a size dependence in the resistance as a functionof temperature, although the dimensions of the wires are such that theyshould exhibit the effects of spin-orbit induced surface anisotropy. Thisremains an outstanding issue in this field.A related topic that is discussed in the contribution by Jaroszynski

and coworkers is the effect of the movement of magnetic impurities onthe conductance of mesoscopic samples (J. Jaroszynski et aI., this vol-ume, p. 63). This effect arises from the time dependent modification ofthe interference of electrons by the dynamics of the spins. Jaroszynskiand coworkers describe how their measurements on magnetic field de-

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pendence of the resistance and conductance noise in dilute magneticsemiconductors can be used to obtain information on the motion of im-purity spins.Parallel to the experiments on thin films and wires, a thorough study

of the Kondo effect in ultra small Kondo alloy point contacts (PCs) hasbeen carried out as described in detail in the contribution of Yanson andcoworkers (I. Yanson et ai., this volume, p. 73). Rather surprisingly, inthis case not a suppression, but an increase of both the Kondo signal andthe Kondo temperature has been reported. These anomalies can be wellexplained by the presence of local density of states (LDOS) fluctuations:For a small PC, even a weak channel quantization induces huge LDOSfluctuations [17] which become larger with decreasing contact sizes. AsTJ( depends on the LDOS exponentially, this may produce an extremelywide distribution of the Kondo temperatures for impurities in the contactregion. The zero bias anomaly of the PC, however, turns out to bedominated by magnetic impurities with the largest TJ(, since these arethe ones that show a well-developed Kondo resonance.

2. TUNNELING SPECTROSCOPY OF KONDOIMPURITIES

It has been known for a long time [18] that the local electron densityof states nearby a magnetic Kondo impurity has a specific structure dueto the Kondo resonance. In the early experimental attempts a changein the electron density of states due to a layer of dilute magnetic alloysfabricated inside a metal has been measured [19] by an oxide tunneljunction placed in a few atomic distances from that layer, and the Kondostructure was indeed observed.Recently, several groups have demonstrated using scanning tunneling

microscopy (STM) [20,21,22] that a magnetic Kondo impurity adsorbedon the surface of a normal metal produces a narrow, resonance-like struc-ture in the electronic surface density of states, whose asymmetric lineshape resembles that of a Fano resonance [23]. The experiments wereperformed with single Ce atoms on Ag by measuring the I(V) char-acteristics of the tunneling current through the tip of an STM placedclose to the surface and at a small distance from the magnetic atom,as described in the article by Berndt (R. Berndt and W.-D. Schneider,this volume, p. 87). Further experiments with single Co atoms on Auand Cu surfaces have also been carried out by Madhavan et ai. [21] andManoharan et ai. [22]. These experiments permit the direct measure-ment of the Kondo screening cloud, as the tunneling spectra are foundnot to change once one goes a distance larger than a few nanometers

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away from the center of the impurity. In addition, the experiments canalso provide an indication of the type of magnetic scattering, i.e., s-waveor d-wave, that occurs at the impurity.

3. TWO LEVEL SYSTEMS

It is by now well established that scattering by fast dynamical defectscan produce Kondo-like anomalies [2, 24]. In the simplest model, thedefect atom tunnels between two positions and thus forms a two-levelsystem (TLS). These two levels are typically split due to the spontaneoustunneling between the positions and the asymmetry in energy betweenthem, resulting in a typical splitting ~ "" 1 - 100K. In the TLS Kondomodel, the coordinate of the dynamical impurity is coupled to the angu-lar momentum of the conduction electrons through an effective exchangeinteraction, and the real spins of the conduction electrons act as silentchannel indices. Consequently, in the absence of splitting, the physics ofthe TLS is described by the two-channel Kondo model predicting non-Fermi liquid (NFL) behavior below the Kondo temperature TK. In thismodel, the spin-flip scattering of the original Kondo model is replacedby electron assisted tunneling. The contribution by Zanind (G. Zarand,this volume, p. 97) gives an overview of the theory, including a compar-ison of the standard Kondo effect arising from magnetic impurities, andthe multi-channel Kondo physics arising from the off-diagonal couplingof the conduction electrons to the position of a tunneling defect.Several experiments have been reported where the observed low tem-

perature anomalies were attributed to TLS Kondo defects [25, 26, 27,28, 29, 30]. In all these experiments, the observed anomalies were un-ambiguously due to dynamical structural defects: They disappear underannealing and are not (or only slightly) dependent on magnetic field.The most spectacular experiments were carried out in Cu and Ti pointcontacts where a two-channel Kondo-like temperature and voltage de-pendence (T 1/ 2 and V1/ 2 ) NFL scaling behavior due to non-magneticscatterers has been observed in the contact resistance [27, 28]. Thewidths of the zero bias anomalies were associated with a Kondo temper-ature TK "" 5K.Several puzzles remain. In all these experiments, the estimated Kondo

temperature is in the range of TK "" 10 K, while estimates of the Kondotemperature of a TLS with a heavy atom which tunnels a distance ofabout 40 pm are in the range of 0.01 - 1K [24]. Another interestingquestion is related to the splitting of the two levels, which provides alower cutoff for the NFL scaling. The presence of splitting and thecutoff of NFL behavior has been observed in several experiments. In

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the experiments of Ralph and coworkers [27J the number of TLSs hasbeen estimated to be about 50, a concentration for which a significantdeviation from NFL scaling should appear due to the presence of disordergenerated splitting. However, no such deviation has been reported. Thesolution of this puzzle may be related to the precise microstructure ofthe tunneling impurities.

4. ELECTRON DEPHASING IN THIN METALSTRUCTURES

Recent developments in mesoscopic physics raised an interesting ques-tion about the electronic dephasing time Tq,. This is the time scale foran electron to stay in a given exact one-electron state in the presenceof static impurities. The transitions between these states may be dueto a variety of processes, including scattering of the electron by otherelectrons, by phonons, by magnetic impurities, and by two-level systems.At low temperatures, the electron-phonon interaction freezes out, andthe other scattering processes become increasingly important.

If electron-electron interactions are the dominant scattering mecha-nism at low temperatures, it has long been expected that Tq, ----7 00 whenthe temperature is lowered, since the available phase space for electron-electron scattering gradually vanishes, as shown by Altshuler, Aronovand Khmelnitskii (AAK) [31J. However, as recent work has shown, thisis not what is always found in experiment. Experiments which esti-mate T q, based on weak localization measurements on narrow metallicwires have found that Tq, saturates as the temperature is lowered. Asdescribed by Mohanty in his contribution (P. Mohanty, this volume,p. 107), the temperature and value at which Tq, saturates appears tobe directly related to the material parameters of the Au metal filmsin their experiments. Mohanty and coworkers showed that many sam-ples measured by various groups in the past fifteen years in a variety ofmaterials also showed a saturation in T q" and that this saturation was re-lated to the material parameters of the samples. Pothier and coworkersdescribe in their contribution (F. Pierre et a/., this volume, p. 119) ex-periments which directly measure the energy relaxation rate of electronsin normal metal wires by measuring the scaling of the non-equilibriumelectron distribution function of the electrons. Although they also findthat the energy relaxation rate saturates on occasion, the saturation intheir samples is not intrinsic in the sense that it does not appear to bedetermined only by the material parameters of the metal wire. How-ever, the experimental consensus is that the observed saturation is notan experimental artifact (arising from conduction electron heating, for

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example, or electromagnetic interference), but a real effect that pointsto a clear deviation from the AAK theory.In addition to the intrinsic decoherence mechanisms discussed by Mo-

hanty in this workshop (and developed by Zaikin and coworkers else-where), TLSs can cause dephasing and energy relaxation at a level con-sistent with the experiments. It should be noted first that the standardmodel of tunneling systems, which assumes a constant density of statesper unit energy splitting for the TLS, predicts an electron dephasingrate with a linear temperature dependence, i.e., hence no saturation.Von Delft and coworkers introduced [32J a model for electron dephasingby TLS in the two-channel Kondo regime. They argued that the sumof the dephasing rate from TLS with a distribution of Kondo temper-atures, coupled with the AAK electron-electron scattering rate, couldmimic a saturation of T¢ over an extended but finite temperature range(at still lower temperatures, T¢ -+ 00 again). This two-channel Kondomechanism is non-universal, and could depend on sample fabricationprocedure, metallurgical variables, sample annealing, thermal cycling,etc. Kroha presented a theoretical model for energy relaxation by TLSin the two-channel Kondo regime (J. Kroha, this volume, p. 133). Heshowed that he can fit the data of Pothier and coworkers by assuminga reasonable density of TLS (105 atoms for the eu samples) and witha distribution of Kondo temperatures whose maximum is less than theenergy scale probed in the experiments.One major weakness of the two-channel Kondo picture is that it is dif-

ficult to reconcile a necessarily large density of high-Kondo-temperatureTLS with common assumptions about the distributions of energy asym-metries and tunneling parameters for TLS. There is no easy way to ob-tain direct information about the TLS in a mesoscopic sample, so thatthere is a lack of direct experimental information regarding the param-eters in the model. Traditional measurements (heat capacity, ultrasonicattenuation) are more suitable for macroscopic samples. Measurementsmore easily applied to mesoscopic samples, such as 1/f noise, are typi-cally performed in the mHz - kHz range, far from the GHz frequenciesrelevant to electron dephasing at 1K. Further work is clearly requiredin this area.

5. KONDO EFFECT IN QUANTUM DOTS

Semiconductor quantum dots are in many ways analogous to atomicmagnetic impurities. Both can be understood using the same Hamilto-nian - the Anderson model of a charge trap in tunneling contact to abulk metal. In fact, quantum dots are often called "artificial atoms" be-

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cause of this similarity. In recent years, after approximately a decade ofeffort, a number of groups have observed the Kondo effect using quantumdots, and this is opening up a broad-ranging new research area. Whatis gained by doing experiments with quantum dots is that they can beused to explore Kondo issues that are inaccessible with real atoms. Forinstance, all of the experimental parameters entering into Kondo physicscan be tuned continuously in quantum dots - including the electroniclevel energies, their coupling to the leads, their degeneracies, and thenumber of electrons in the dot. This permits quantitative tests of the-ory. A second advantage of quantum dots over real atoms is that theregime of the Kondo effect out of equilibrium can be readily investigated,using a non-zero source-drain voltage. Third, the fact that quantum dotsare reasonably large ('" 100 nm) will allow them to be controllably in-corporated into new types of experimental geometries.In their contributions, de Franceschi (S. de Franceschi et ai., this vol-

ume, p. 153) and Goldhaber-Gordon (D. Goldhaber-Gordon et ai., thisvolume, p. 163) described experimental observations of Kondo-assistedtunneling in both lateral quantum dots formed from semiconductor two-dimensional electron gases and vertical dots formed by etching a semi-conductor multilayer structure into a pillar geometry. One of the mainmessages to be taken from their work is that the results for the simplestspin S = 1/2 dots in equilibrium (negligible source-drain voltage) are inexcellent quantitative agreement with theory. This is true first for thevalue of the Kondo temperature, which has been measured as a functionof both varying the energy difference between the electron state and theFermi level and also the degree of coupling between the dot and theelectrodes. It is also true for the scaling with temperature and voltageof the tunneling conductance. The newest data from de Franceschi andcoworkers demonstrate impressive agreement with theory even deep intothe unitary regime where the experimental temperature is much lessthan the Kondo temperature.Experiments underway now are turning to more complicated real-

izations of Kondo physics and also the non-equilibrium regime. Thisnon-equilibrium regime is discussed theoretically in the contribution bySchiller (A. Schiller, this volume, p. 143). Whereas the simple S = 1/2Kondo effect involves an odd number of electrons on the dot, with twodegenerate levels (spin up and spin down), de Franceschi and cowork-ers have demonstrated that they can produce a different kind of Kondoeffect with an even number of electrons on a dot, by using a magneticfield to tune 4 states (a spin singlet and a spin triplet) into near degen-eracy. They have also begun examination of the S = 1 Kondo effectat small magnetic fields. Several interesting effects occur as the source-

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drain voltage is increased and the dot is driven out of equilibrium. Forinstance, one signature of a magnetic Kondo effect is a peak in dI / dVwhen the applied voltage is equal to the Zeeman splitting 9JLBH/e. Aspointed out by Schiller, this non-equilibrium regime provides a challengeto many-body theory, and current calculations are not capable of quanti-tative agreement with experiments. However, Schiller also points a wayforward, by highlighting experimentally accessible quantities of partiCll-lar theoretical interest. These include shot noise, scaling functions, pairco-tunneling, and the use of time-dependent fields to probe time scalesand dissipation.

6. PERSPECTIVES

In spite of the substantial progress in understanding the spin Kondoproblem in mesoscopic systems, there remain many questions to be an-swered. With regard to the size dependence of the Kondo effect in thinmetal films and wires, there needs to be further experimental work toclarify the role of interface and materials parameters on the Kondo effect,as well as measurements of new properties to gain a fuller understand-ing of the mechanism of electron scattering. In this respect, the newexperiments on the Kondo thermopower are a promising start, but onemight also consider properties such as the magnetization and the heatcapacity. Although these properties are much more difficult to measure,they may be easier to analyze theoretically. Concerning development ofthe theory, the crossover from the ballistic to the dirty limits and theeffect of disorder on the surface anisotropy should be further clarified. Inaddition, calculation of other experimentally accessible parameters suchas the thermopower should also be carried out. Similar to the surfaceanisotropy, local density of states fluctuations decay as 1/d as a functionof the distance d from the surface. These fluctuations probably give thedominant effect in very thin films and films with a weak spin-orbit in-teraction for alloys with a relatively small TK. Measurements on a hostwith weak spin-orbit scattering could help to clarify these issues.The observation of the Kondo resonance by STM due to a single

ferromagnetic atom on a metallic surface is a very impressive technicalachievement. In the future, it would be worthwhile to study magneticimpurities inside the first few surface layers to establish stronger couplingbetween the spin and the host metals, as well as two magnetic impuritiesplaced close together in order to study the interaction between them. Inorder to understand the data or to make predictions further electroniccalculations are required for the host metal at the surface, the charge

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redistribution due to the impurity and the value of spin at the impurityatom.The issue of electron dephasing and energy relaxation in metal films

and wires remains open, and perhaps the most pressing need is for newwell-defined experiments which can clarify the origins of the differencesin the measurements so far. In this regard, the role of magnetic impuri-ties in electron transport in these systems needs to be further explored.It should be noted that some experimental results on magnetic scatter-ing in mesoscopic metallic devices still require explanation: Apart fromthe size dependence of the Kondo effect, one can also point to the lackof saturation behavior of the temperature dependent resistivity in meso-scopic Kondo alloys at low temperature. In addition, in some of theearliest experiments on quantum interference in disordered metals, theapparently different length scales in the same samples for the observationof weak localization and the h/2e Aharanov-Bohm effect on one hand,and conductance fluctuations and the h/e Aharanov-Bohm effect on theother, demonstrate that our understanding of scattering mechanisms indisordered systems is not complete.The rapidly advancing art for nanofabricating quantum dots should

allow a wide variety of other new experiments. Goldhaber-Gordon sug-gested that the spectral function for a Kondo impurity out of equilibriummight be measured directly by coupling a third electrode to a quantumdot as a weak tunneling probe. Both the group at the Delft Universityof Technology and Heiblum's group in Israel are attempting to incorpo-rate Kondo-coupled quantum dots into Aharonov-Bohm rings to makemeasurements of the Kondo scattering phase shift. The physics of theKondo effect with more than one nearly degenerate electron orbital hasup to now received considerable theoretical attention, and is worth fur-ther experimental exploration. Looking a little farther into the future,all the physics of interacting magnetic impurities should become accessi-ble by fabricating multiple quantum dots sufficiently close together thatthey are not independent. Geometries of this sort might provide modelsystems for understanding the competition between Kondo effects andmagnetic interactions - both RKKY-type interactions mediated by con-duction electrons and also direct tunnel coupling between dots.In summary, it appears that the Kondo effect will continue to be

increasingly important in the physics of mesoscopic systems for sometime to come.

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