Single Charge Tunneling Coulomb Blockade Phenomena ln Nanostructures

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Single Charge Tunneling Coulomb Blockade Phenomena ln Nanostructures Edited by

Hermann Grabert Universität Essen Essen, Germany

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Michel H. Devoret Centre d'Etudes de Saclay Gif-sur-Yvette, France

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Proceedings of a NATO Advanced Study Institute on Single Charge Tunneling, held March 5-15, 1991, in Les Houches, France

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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. Scien­tific Affairs Division. V. Series. QC176.8.T8S56 1992 530.4'16-dc20

ISBN 978-1-4757-2168-3 ISBN 978-1-4757-2166-9 (eBook) DOI 10.1007/978-1-4757-2166-9

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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 macro­scopically 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 tun­neling. 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 camera­ready 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 tunneling in this volume, and they thank all participants of the conference as well as the supporting staff for their help in making the Advanced Study Institute so successful.

Michel H. Devoret Graupe Quantronique Centre d 'Etudes de Saclay France

Hermann Grabert Fachbereich Physik

Universität Essen Germany

V

lntroduction

This book is divided into 9 chapters, which, except for chapter 1, deal with a particu­lar 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 fun­damental 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 indi­vidual 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, pro­gressively 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 5.1. Introduction . . . . . . . . . . .. 5.2. Tunneling of Cooper pairs . . . .. 5.3. Charge-phase duality and incoherent tunneling of the phase 5.4. Tunneling of quasiparticles ....... .

6. Double junction and single electron transistor . 6.1. Island charge . . . . . . . . . . ..... . 6.2. Network analysis . . . . . . . . ..... . 6.3. Tunneling rates in a double junction system . 6.4. Double junction in a low impedance environment 6.5. Double junction in a high impedance environment 6.6. Current-voltage characteristics of a double junction . 6.7. Coulomb staircase .......... . 6.8. SET-transistor and SET-electrometer 6.9. Other multijunction circuits .

A. Microscopic foundation A.l. Introduction ... A.2. General problern . A.3. Time of tunneling AA. One-pi10ton processes: anomalies and :fingerprints . A.5. Diffusive anomalies . . . . . . . . . . A.6. Field moves faster than the electrons . A.7. Junction-localized oscillations . A.8. A gateway into networks

References . . . . . . . . . . . . . . . .

3 Transferring Electrons One By One by D. ESTEVE

1. Introduction . . . . . . . . . . . . . . 2. Basic concepts of small junction circuits .

2.1. Electrons and electronics ..... . 2.2. Charge con:figurations . . . . . . . . 2.3. Tunneling out of an unstable con:figuration 2.4. Co-tunneling out of a locally stable con:figuration 2.5. Coexistence of tunneling and co-tunneling

3. Single electron box .............. . 3.1. Averagecharge in the single electron box 3.2. Measurement of the junction charge (Q) 3.3. Superconducting case .......... .

4. Single electron turnstile . . . . . . . . . . . . . 4.1. Basic principles of the controlled transfer of single electrons 4.2. The trap: an irreversible single electron box . 4.3. From the trap to the turnstile . 4.4. Experimental results . 4.5. Transfer accuracy . . . . . . .

56 56 57 60 61 65 65 68 72 73 75 77 80 83 86 91 91 92 97 98 99

102 104 105 106

109 109 110 110 111 113 115 116 117 117 118 122 122 122 123 124 125 126

5. Single electron pump . . . . . . . . . . . . . . . . . . . . . . 127 5.1. The principle of the pump: two coupled electron boxes . 127 5.2. A pumping cycle . . . . . . . . . . . . . . . . . . . . 128 5.3. Experimental observation of single electron pumping 129 5.4. Transfer accuracy . . . . . . . 131

6. Is metrological accuracy achievable? . . . . . . . . . . . . 132 6.1. Metrological applications . . . . . . . . . . . . . . . 132 6.2. Improving the accuracy of charge transferring devices 133

7. Conclusion 136 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4 Josephson Effect in Low-Capacitance Tunnel Junctions by M. TINKHAM 139 1. Introduction . . . . . . . . . . . . . . . . . . . 139 2. Classical Josephson junctions . . . . . . . . . . 140

2.1. The RCSJ model of a Josephson junction 140 2.2. Effect ofthermal fluctuations 141 2.3. E:ffect of Iead impedance. . . . . . 144 2.4. The phase diffusion branch . . . . 150 2.5. Recapitulation of classical regime . 152

3. Quantum e:ffects in Josephson junctions 153 3.1. Introduction . . . . . . . . . . . 153 3.2. The isolated junction . . . . . . 155 3.3. Estimation of the critical current 159 3.4. Estimation of Ro 162

References . . . . . . . . . . . . . . . . . 165

5 Coulomb-Blockade Oscillations in Semiconductor Nanostructures by H. VAN HOUTEN, C. W. J. BEENAKKER, and A. A. M. STARING 167 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 167

1.1. Preface . . . . . . . . . . . . . . . . . . . . . . . 167 1.2. Basic properties of semiconductor nanostructures 171

2. Theory of Coulomb-blockade oscillations 173 2.1. Periodicity of the oscillations . . . . . . . 174 2.2. Amplitude and lineshape . . . . . . . . . 178

3. Experiments on Coulomb-blockade oscillations 185 3.1. Quantum dots . . . . . . . . . . . . . . . 185 3.2. Diserdered quantum wires . . . . . . . . . 187 3.3. Relation to earlier work on diserdered quantum wires 190

4. Quantum Hall e:ffect regime. . . . . . . . . . . . . . . 192 4.1. The Aharonov-Bohm effect in a quantum dot . . 192 4.2. Coulomb blockade of the Aharonov-Bohm e:ffect . 200 4.3. Experiments on quantum dots . . . . . . . . . . 203 4.4. Experiments on diserdered quantum wires . . . . 206

A. Conductance of a quantum dot coupled to two electron reservoirs . 208 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

XII Contents

6 Macroscopic Quantum Tunneling of Charge and Co-Tunneling by D. V. AVERIN and Yu. V. NAZAROV 217 1. Introduction . . . . . . . . . . . . . . . . . . . . . 217 2. Inelastic q-mqt . . . . . . . . . . . . . . . . . . . . 221

2.1. Inelastic q-mqt in the double junction system 221 2.2. Multijunction circuits . . . . . . . . . . . . 224 2.3. Experimentalobservation of inelastic q-mqt . 227

3. Elastic q-mqt and virtual electron di:ffusion . . . . 230 4. Transport of electron-hole pairs in coupled arrays of small tunnel junctions 238

4.1. General picture of electron-hole transport 238 4.2. Three-junction arrays . . . . . 240 4.3. Multijunction arrays . . . . . . 243

5. Unsolved problems and perspectives 244 References . . . . . . . . . . . . . . . . 246

7 One-Dimensional Arrays of Small Tunnel Junctions by P. DELSING 249 1. Introduction . . . . . . 249 2. Theoretical background 250 3. Experiments . . . . . . 254

3.1. General properties, I-V curves 254 3.2. Time correlation of tunnel events . 258 3.3. Space correlation of tunnel events 263 3.4. An R-SET transistor using an array as gate resistor 265 3.5. Decoupling of the electromagnetic environment 268

4. Discussion . . . . . . 269 4.1. Self correlation . 269 4.2. Array turnstiles 271 4.3. Coupled arrays 271

5. Conclusions 272 References . . . . . . . 273

8 Single Charges in 2-Dimensional Junction Arrays by J. E. MOOIJ and G. ScHÖN 275 1. Introduction . . . . . . . . 275 2. Charges in junction arrays . . 276

2.1. Normal state . . . . . . . 276 2.2. Two-dimensional phase transition 279 2.3. Simulations of single electron tunneling in normal junction arrays 281 2.4. Infl.uence of quasiparticle tunneling on the charge KTB transition . 282 2.5. Capacitances in real junction systems . . . . . 283 2.6. Chargetransition in the superconducting state 285

3. Vortices in 2D junction arrays . . . . . . . 286 4. Charge-vortex duality . . . . . . . . . . . . . . . 288

4.1. The coupled-Coulomb-gas description . . . 289 4.2. The phase transitions in the junction array 292

Contents xiii

5. Quantum vortices . . . . . . . . . 294 5.1. The vortex mass . . . . . . . 294 5.2. The Aharonov-Casher effect . 297 5.3. Forces acting on vortices . . 298 5.4. Dissipation by quasiparticle tunneling . . 299 .:>.o. Experimental observation of ballistic vortices 301

6. Charge dynamics . . . . . . . . . . . . . . . . . . 303 A. Arrays of normal junctions with arbitrary strength of the tunneling 304 B. Effect of an imposed current 307 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9 Possible Applications of the Single Charge Tunneling by D. V. AVERIN and K. K. LIKHAREV 311 1. Introduction . . . . . . . . . . . . . . . . 311 2. Single electronics: The rules of the game 312 3. Rules of the game: An illustration 313 4. DC current standards . . . . 317 5. Supersensitive electrometry . 320 6. Infrared radiation receivers 323 7. Digital circuits . . . . . . . . 324 8. Discussion . . . . . . . . . . 328

8.1. Backgroundchargerelaxation . 328 8.2. Fabrication technology . 329 8.3. Design automation . 330

References . . . . . . . 331

Index ..... 333