Table Of ContentNato Science Series B: {closed)
Hermann Grabert
Michel H. Devoret Editors
Single Charge
Tunneling:
Coulomb Blockade
Phenomena in
Nanostructures
Single Charge Tunneling
Coulomb Blockade Phenomena
ln Nanostructures
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edited by Hermann Grabert and Michel H. Devoret
Series B: Ph ysics
Single Charge Tunneling
Coulomb Blockade Phenomena
ln Nanostructures
Edited by
Hermann Grabert
Universität Essen
Essen, Germany
and
Michel H. Devoret
Centre d'Etudes de Saclay
Gif-sur-Yvette, France
This limited facsimile edition has been issued
for the purpese of keeping this title available
to the scientific community.
Springer Science+Business Media, LLC
Proceedings of a NATO Advanced Study Institute
on Single Charge Tunneling,
held March 5-15, 1991,
in Les Houches, France
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Library of Congress Cataloging in Publication Data
Single charge tunneling: Coulomb blockade phenomena in nanostructures I
edited by Hermann Grabert and Michel H. Devoret.
p. cm.-(NATO ASI series. Series B, Physics; v. 294}
"Proceedings of a NATO Advanced Study Institute on Single Charge Tunnel
ing, held March 5-15, 1991, in Les Houches, France"-Verso t.p.
"Published in cooperation with NATO Scientific Affairs Division."
lncludes bibliographical references and index.
1. Tunneling (Physics}-Congresses. 2. Coulomb potentiai-Congresses. 3.
Nanostructures-Congresses. I. Grabert, Hermann, date. II. Devoret, Michel H.
111. NATO Advanced Study Institute on Single Charge Tunneling (1991: Les
Houches, Haute-Savoie, France) IV. North Atlantic Treaty Organization. Scien
tific Affairs Division. V. Series.
QC176.8.T8S56 1992 92-13436
530.4'16-dc20 CIP
ISBN 978-1-4757-2168-3 ISBN 978-1-4757-2166-9 (eBook)
DOI 10.1007/978-1-4757-2166-9
© Springer Science+Business Media New York 1992
Origina11y published by Plenum Press, New York in 1992
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
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Preface
The field of single charge tunneling comprises of phenomena where the tunneling of a
microscopic charge, usually carried by an electron or a Cooper pair, leads to macro
scopically observable effects. The first conference entirely devoted to this new field was
the NATO Advanced Study Institute on Single Charge Tunneling held in Les Hauches,
France, March 5-15, 1991. This book contains a series of tutorial articles based on
lectures presented at the meeting. It was intended to provide both an introduction for
nonexperts and a valuable reference summarizing the state of the art of single charge tun
neling. A complementary publication with contributions by participants of the NATO
Advanced Study Institute is the Special Issue on Single Charge Tunneling of Zeitschrift
für Physik B, Vol. 85, pp. 317-468 (1991 ). That issue with original papers provides a
snapshot af the leading edge of current research in the field.
The success of the meeting and the publicatian of this volume was made possible through
the generaus support af the NATO Scientific A:ffairs Division, Brussels, Belgium. The
Centre de Physique des Hauches has provided a superbly situated conference site and
took care af many lacal arrangements. Both far the preparation of the conference and
the handling af some manuscripts the suppart af the Centre d 'Etudes de Saclay was
essential. The editing of the proceedings volume would not have been passible without
the dedicated efforts of Dr. G.-1. Ingald, who tailared a 1\.TEX style-file to the needs
af this book, and the assistance of J. Ankerhold and P. Wyrowski from the sta:ff of the
Fachbereich Physik of the Universität Essen, who prepared the manuscripts in camera
ready form. The editors are most appreciative of the efforts made by the colleagues
who accepted the burden of writing a chapter covering a specific aspect of single charge
as
tunneling in this volume, and they thank all participants of the conference well as the
supporting staff for their help in making the Advanced Study Institute so successful.
Michel H. Devoret Hermann Grabert
Graupe Quantronique Fachbereich Physik
Centre d 'Etudes de Saclay Universität Essen
France Germany
V
lntroduction
This book is divided into 9 chapters, which, except for chapter 1, deal with a particu
lar aspect of single charge tunneling (SCT) phenomena. Chapter 1 by DEVORET and
GRABERT provides an introduction to the other chapters, each of which is to a large
extent self-contained. In recent years it has become increasingly clear that the more fun
damental objects in the theory of SCT phenomena should be "islands", that is metallic
regions with small total capacitance into which electrons can tunnel, rather than indi
vidual small capacitance tunnel junctions. Chapter 1 is devoted mainly to the discussion
of this idea.
Chapter 2 by IN GOLD and N AZAROV deals with the dynamics of ultrasmall junction
circuits and teaches one how to compute the tunneling rate of electrons and Cooper pairs
across a particular junction in the circuit. Starting with single junction systems, pro
gressively more complex cases are treated. In the Appendix, a microscopic justification
for the starting hypotheses of the calculations is given. Chapter 3 by ESTEVE explains
the basic ideas on which the manipulation of electrons one-by-one in metallic junction
circuits are founded. The primary focus is thus on normal junction circuits, although
the manipulation of Cooper pairs is briefiy discussed. Particular emphasis is put on the
accuracy of devices in which electrons are clocked by an external frequency, such as the
"turnstile" and the "electron pump". These devices may lead one day to experiments
with metrological accuracy.
Chapter 4 by TINKHAM treats the superconducting single junction system. It gives
a survey of phenomena observed in low-capacitance tunnel junctions and also provides
a link v.ith the usual Josephson effects. Chapter 5 by VAN HOUTEN, BEENAKKER, and
STARING deals with semiconductor systems, mostly GaAs/ AlGaA.s heterostructures, in
which the metallic islands consist of quasi-isolated regions of the 2D electron gas. In
these systems, the interplay between the Coulomb effects and both the Quantum Hall
effect and the resonant tunneling effect leads to exciting phenomena. While in the first
five chapters single charge tunneling phenomena are analysed mainly in terms of charge
tunneling across one junction at a time, Chapter 6 by AVERIN and N AZAROV is a detailed
discussion of the process of co-tunneling in which several tunneling events across different
junctions occur coherently. Co-tunneling causes fundamentallimi tations of the accuracy
of the charge transferring devices described in Chapter 3.
Chapters 7 and 8 deal with systems containing a !arge nurober of junctions but which
nevertheless display great simplicity because of their translational symmetry properties.
Chapter 7 by DELSING treats one-dimensional arrays, with a particular emphasis on RF
excitation revealing the self-correlations between tunnel events that exist in sufficiently
long arrays. A link with the turnstile of Chapter 3 is made. Chapter 8 by Moou and
VII
viii lntroduction
SCHÖN treats two-dimensional arrays, both in the vortex and the charge regimes. The
two-dimensional character of these arrays allows for the possibility of a phase transition.
The approximate duality between the vortex and charge regimes is elucidated. Finally,
Chapter 9 by AVERIN and LIKHAREV handles the delicate task of predicting the future of
the fi.eld. In particular, it discusses the bold and tantalizing claim that digital electronics,
in its ultimate stage, will be based on SCT phenomena.
Contents
1 Introduction to Single Charge Tunneling
by M. H. DEVORET and H. GRABERT 1
1. Basic ingredients of single charge tunneling phenomena 1
2. Single current biased junction 4
3. Single island circuits . . . . . 10
4. Circuits with several islands 15
5. Conclusions 16
References . . . . . . . . . . . . 18
2 Charge Tunneling Rates in Ultrasmall J unctions
by G.-L. lNGOLD and Yu. V. NAZAROV 21
1. Introduction . . . . . . . . . . . . . 21
1.1. Ultrasmall tunnel junctions . . . 21
1.2. Voltage-biased tunnel junction . 22
1.3. Charging energy considerations . 23
1.4. Local and global view of a single tunnel junction 24
2. Description of the environment . . . . . . . 25
2.1. Classical charge relaxation . . . . . . 25
2.2. Quantum mechanics of an LC-circuit 26
2.3. Hamiltonian of the environment . . . 28
3. Electron tunneling rates for single tunnel junctions . 30
3.1. Tunneling Hamiltonian . . . . . . . . . . . . . 30
3.2. Calculation of tunneling rates . . . . . . . . . . 31
3.3. Phase-phase correlation function and environmental impedance 36
3.4. General properties of P(E) . . . . . . . . . . . . . . 37
3.5. General properties of current-voltage characteristics 39
3.6. Low impedance environment . . . . . 40
3.7. High impedance environment . . . . . 41
4. Examples of electromagnetic environments 42
4.1. Coupling to a single mode . . . . . . 42
4.2. Ohrnie impedance . . . . . . . . . . 47
4.3. A mode with a finite quality factor . 49
4.4. Description of transmission lines 51
4.5. LC transmission line. 53
4.6. RC transmission line . . . . . . 54
ix
X Contents
5. Tunneling rates in Josephson junctions 56
5.1. Introduction . . . . . . . . . . .. 56
5.2. Tunneling of Cooper pairs . . . .. 57
5.3. Charge-phase duality and incoherent tunneling of the phase 60
5.4. Tunneling of quasiparticles ....... . 61
6. Double junction and single electron transistor . 65
6.1. Island charge . . . . . . . . . . ..... . 65
6.2. Network analysis . . . . . . . . ..... . 68
6.3. Tunneling rates in a double junction system . 72
6.4. Double junction in a low impedance environment 73
6.5. Double junction in a high impedance environment 75
6.6. Current-voltage characteristics of a double junction . 77
6.7. Coulomb staircase .......... . 80
6.8. SET-transistor and SET-electrometer 83
6.9. Other multijunction circuits . 86
A. Microscopic foundation 91
A.l. Introduction ... 91
A.2. General problern . 92
A.3. Time of tunneling 97
AA. One-pi10ton processes: anomalies and :fingerprints . 98
A.5. Diffusive anomalies . . . . . . . . . . 99
A.6. Field moves faster than the electrons . 102
A.7. Junction-localized oscillations . 104
A.8. A gateway into networks 105
References . . . . . . . . . . . . . . . . 106
3 Transferring Electrons One By One
by D. ESTEVE 109
1. Introduction . . . . . . . . . . . . . . 109
2. Basic concepts of small junction circuits . 110
2.1. Electrons and electronics ..... . 110
2.2. Charge con:figurations . . . . . . . . 111
2.3. Tunneling out of an unstable con:figuration 113
2.4. Co-tunneling out of a locally stable con:figuration 115
2.5. Coexistence of tunneling and co-tunneling 116
3. Single electron box .............. . 117
3.1. Averagecharge in the single electron box 117
3.2. Measurement of the junction charge (Q) 118
3.3. Superconducting case .......... . 122
4. Single electron turnstile . . . . . . . . . . . . . 122
4.1. Basic principles of the controlled transfer of single electrons 122
4.2. The trap: an irreversible single electron box . 123
4.3. From the trap to the turnstile . 124
4.4. Experimental results . 125
4.5. Transfer accuracy . . . . . . . 126
