Single-Electron Circuit Design

I n t r o d u c t i o n t o

Nanoelectronic

Jaap HoekstraDelft University of Technology, The Netherlands

Single-Electron Circuit Design

I n t r o d u c t i o n t o

Nanoelectronic

Published by

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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Introduction to Nanoelectronic Single-Electron Circuit Design Copyright © 2009 by Pan Stanford Publishing Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4241-93-9 (Hardcover) ISBN 978-981-4267-63-2 (eBook) Printed in Singapore.

Preface

In Introduction to Nanoelectronic Single-Electron Circuit Design single­electron circuits are studied as an introduction to the fast expanding field of nanoelectronics. In nanoelectronics, single-electron circuits are those cir­cuits that process information and signals by making use of time-dependent currents and voltages due to charge transport by just one single or only a few electrons.

This textbook follows an unconventional approach to explaining the op­eration and design of single-electron circuits. In general, the conventional approach to this subject is to begin with a brief introduction to the quantum physics of the nanodevices followed by modeling the devices by mathemat­ical means. It is the author's opinion that an alternative approach with an emphasis on experiments to obtain a characterization of the devices will enhance the reader's comprehension. Therefore after the introduction, first some landmark experiments are reviewed and discussed. Then a brief in­troduction to the relevant (quantum) physics of the nanodevices is given. Subsequently, the characterization of devices is used to obtain equivalent circuit diagrams. To ease the discussions on the characteristics and the equivalent circuits some topics from linear and nonlinear circuit theory are briefly reviewed. Knowing this, a circuit theoretical framework will be built. Devices and (small) circuits are modeled both by mathematical means and by circuit simulations. Also currents in classical and quantum physics are reviewed. Simple circuits including single-electron devices are treated. After this, circuit design methodologies are discussed as well as typical electronics' topics as signal amplification, biasing, coupling, noise, and circuit simulation. When looking forward to dealing with systems em­phasis is placed on redundant and fault-tolerant architectures to cope with the uncertainties related to the critical nanometer-sized device dimensions

v

vi Introduction to Nanoelectronic Single-Electron Circuit Design

and the "probabilistic" nature of quantum physics. The book ends with a brief discussion on potential applications and challenges.

Due to their simplicity, this monograph mainly considers metallic single­electron tunneling junctions (metallic SET junctions). For an introduction to single-electron circuit design, circuits with these devices are already com­plex enough.

The basic physical phenomena under consideration are the quantum mechanical tunneling of electrons through a small insulating gap between two metal leads, the Coulomb blockade, and the associated phenomenon of Coulomb oscillations- the last two resulting from the quantization of charge. The metal-insulator-metal structure through which the electrons may tunnel is called a tunnel(ing) junction. This tunneling is considered to be stochastic, that is, successive tunneling events across a tunnel junc­tion are uncorrelated, and is described by a Poisson process. Throughout the text tunneling through a potential barrier is considered to be non­dissipative (the tunneling process through the barrier is considered to be elastic), unless explicitly stated differently.

Electron transport in the nanoelectronic devices can best be described by quantum physics; nanoelectronic circuits can best be described by Kirch­hoff's voltage and current laws, which have a firm basis in classical physics. This tension between quantum physics and classical physics is taken for granted; experiments with circuits will have to approve whether we can successfully include the quantum character of charge in a circuit theory for single-electron electronics.

As quantum mechanics is described in terms of energy, it seems obvi­ous to describe, that is to analyze, the behavior of SET devices and SET circuits with energies. This is what the so-called orthodox theory of single electronics does. In this semiclassical-physics theory an electron will tunnel if the free (electrostatic) energy in the system after tunneling is lower than the free energy in the system before tunneling. However, as we will see, this energy loss cannot always be modeled as dissipation by heat or by radiation without violating the Kirchhoff laws. Especially, when the tunnel event of single electrons is considered the dissipation cannot be modeled by a finite resistance. To design, that is to synthesize circuits with these devices, we need a circuit theory. It must be based on Kirchhoff's voltage and current laws. In contrast, these laws ensure energy conservation in circuits: any energy dissipation in the circuit is delivered by sources; and vice versa, all energy delivered by sources is either stored either dissipated in circuit ele­ments. It is because of these arguments that the orthodox theory of single

Preface vii

electronics is not followed for designing circuits. This text provides a circuit theoretical model of the single-electron tun­

neling junction to analyze and synthesize nanoelectronic circuits. In the absence of tunneling, the metallic junction is modeled as a capacitor. Tun­neling is explained by considering the (matter) wave nature of the electron, expressed by the De Broglie wave length AF = h/v'2mEF • Two metallic junctions in series form an island. To model the two junctions as capaci­tors (for example, In case of predicting Coulomb bloekatie) Lhe i:;lallU lllu:;L be large compared with AF; that is, unless explicitly stated quantum-dot systems are not considered. The main results of this approach are that instead of attributing the blockade phenomenon to the existence of islands, the Coulomb blockade is found as a property of the tunnel junction for non quantum-dot systems, and tunneling must be modeled by an impulsive current source.

In chapter 1 nanoelectronics and single-electron electronics are defined and the scope ofthis text is presented. A bird's eye view is presented, with many pointers to later chapters, in order to familiarize the reader with the kind of possibilities and challenges this book is dealing with. Chapter 2 dis­cusses landmark experiments that form a chain from the first experiments showing quantum mechanical tunneling to experiments showing Coulomb blockade, that is, no tunneling is observed while tunnel events were ex­pected to happen. Chapter 3 consists of a brief review of the modeling of currents in classical physics and introduces circuit theory using lumped circuit elements, the chapter is essential for understanding of what can be or what cannot be modeled in a (electronic) circuit theory. Chapter 4 fo­cusses on the quantum mechanical description of free electrons. In quantum mechanics electrons are described both as particles and as waves. Typical quantum mechanical phenomena as energy quantization and tunneling are possible due the wave nature of the electron. This chapter is the first of two that introduce the quantum mechanics needed for the understanding of the nanoelectronic devices. The second chapter, chapter 5 treats the quan­tum mechanical descriptions of currents in general, and the tunnel current in particular. Ballistic transport and quantized resistance are presented briefly. Chapter 6 the relation between lumped circuits, Kirchhoff's laws and energy in circuit theory is examined. It focusses on the conservation of energy based on Tellegen's theorem. The concepts bounded and un­bounded currents are introduced in chapter 7, where energy conservation in the switched two-capacitor circuit is discussed. Also the initial charge mod­els for the capacitor are presented. Knowing the circuit theoretical basics

viii Introduction to Nanoelectronic Single-Electron Circuit Design

of capacitor circuits, the impulse circuit model for single-electron tunneling is presented in two chapters. First in chapter 8, based on energy conser­vation and a hot-electron model, the impulse circuit model is derived in case of zero-tunneling time. In chapter 9 the model is extended to nonzero tunneling times, to circuit including resistors, and to circuits excited by nonideal energy sources. Also, tunneling of many electrons in the same time interval is considered leading to the definition of the tunnel resistance. In chapter 10 the theory is generalized to multi-junction <.:ircuiLt:>. Espe­cially, answers are discussed to the following question: How much energy is needed to tunnel onto a metallic island? Chapter 11 applies the impulse circuit model to the most basic single-electron tunneling circuits. It treats the electron-box, SET transistor, three-junction structures, and the SET inverter. Knowing how to analyze single-electron tunneling circuits chap­ter 12 starts the discussion on nanoelectronic circuit design methodologies and SET circuit design issues. As examples of possible circuit solutions to coping with uncertainties and inaccuracies SET based artificial neural network building blocks are described. The last regular chapter, chapter 13 gives an outlook to potential applications and challenges. At the end of this book an epilogue is added, especially for those readers who are already familiar to the orthodox theory of single electronics for circuit design.

I must thank all my colleagues, Ph.D. and master students that in many ways contributed to this book. Especially, I wish to thank Mar­tijn Goossens, Chris Verhoeven, Jose Camargo da Costa, and Arthur van Roermund for introducing me to the field of nanoelectronics, and for the many discussions on electronic design methodologies; Roelof Klunder for the discussions on the SET circuit design issues and the development of the electron-box logic; and Rudie van de Haar for developing the Spice simula­tion environment, and his design of the neural node. For the more general discussions on nanoelectronic architectural issue I thank Eelco Rouw. Their Ph.D. theses can be downloaded from the library site ofthe Delft University of Technology. My special thanks must go to my wife, Judy, and my sons Tom, Jeroen, and Peter. They always encouraged me to write this book.

The textbook is based on a nanoelectronics course at the Delft Univer­sity of Technology and is intended for senior undergraduate and graduate students. The prerequisites for understanding the material are the basic principles of solid-state and semiconductor physics and devices, and the basic principles of linear circuit analysis.

J aap Hoekstra

Contents

Preface v

1. Introduction 1

1.1 Scope....... . ............... 1 1.1.1 N anoelectronic circuit design issues . . . 2 1.1.2 Levels in modeling, a top-down approach 3 1.1.3 Overview of tunneling capacitor circuit models 5 1.1.4 Important quantum mechanical phenomena. 8

1.2 Electron Tunneling 9 1.2.1 Free electrons 9 1.2.2 Tunneling 10 1.2.3 Hot electrons 11 1.2.4 Tunneling time and transition time 12

1.3 Tunneling Capacitors and Island Charges 12 1.3.1 Two-junction circuit in Coulomb blockade 14

1.4 Energy in Simple Capacitor Circuits, Bounded and Unbounded Currents. . . . . . . . . . . . . . . . . 16 1.4.1 Switching circuits: energy in a resistive circuit 17 1.4.2 Charging a capacitor: bounded current . . 20 1.4.3 Charging a capacitor: unbounded current 21 1.4.4 Energy calculation with the generalized

delta-function .. 23 1.5 Operational Temperature 1.6 Research Questions Problems and Exercises .... .

2. Tunneling Experiments in Nanoelectronics

ix

25 27 28

29

x

2.2

Introduction to Nanoelectronic Single-Electron Circuit Design

'I\mneIing in the 'I\mnel Diode . . . . . . . . . . . . 2.1.1 Energy-band diagram for the p - n diode 2.1.2 Experiments of Esaki on the (tunnel) diode 2.1.3 Nonlinear voltage-current characteristic of the

tunnel diode . . 2.1.4 Tunnel current 2.1.5 Energy paradox 2.1.6 Equivalent circuit 2.1. 7 Resonant tunneling diode Tunneling Capacitor . . . . . . . . 2.2.1 Tunneling between plates and the hot electron 2.2.2 2.2.3

Tunnel capacitor and electron-box Single-electron tunneling transistor and Coulomb blockade ...................... .

29 30 36

37 42 44 45 46 48 48 53

54 2.2.4 Array of tunnel junctions and Coulomb oscillations 56 2.2.5 Single-electron tunneling junction and Coulomb

blockade 57 Problems and Exercises ... 57

3. Current in Electrodynamics and Circuit Theory 59

59 3.1 Charges in Electrodynamics 3.2 Conservation of Charge and Continuity Equation 62 3.3 Electromagnetics' Field Equations in Vacuum 63 3.4 Equations in the Presence of Charges and Currents. 65 3.5 Conservation of Energy and Poynting's Theorem 67 3.6 Steady-State and Constant Currents 70

3.6.1 Kirchhoff's current law 71 3.6.2 Vector potential A 71 3.6.3 Ohm's law ...... . 3.6.4 Kirchhoff's voltage law

3.7 Time-Dependent Current Flow 3.8 Towards Circuit Theory Problems and Exercises . . . . . . . .

4. Free Electrons in Quantum Mechanics

72

73 75 77 78

79

4.1 Particles, Fields, Wave Packets, and Uncertainty Relations 79 4.2 Schrodinger's Equation .. . .. . ... ..... 82

4.2.1 Time-independent Schrodinger equation 83

Contents

4.3 Free Electrons ....... . 4.3.1 Electron as a particle 4.3.2 4.3.3 4.3.4

Electron as a wave A beam of free electrons Electron as a wave packet

4.3.5 Phase- and group velocity 4.4 Free Electrons Meeting a Boundary

4.4.1 Step potential: E < Eo 4.4.2 Step potential: E > Eo ...

4.5 Electrons in Potential Wells 4.5.1 Infinite well: standing waves 4.5.2 Finite well: periodic boundary conditions 4.5.3 Quantization of energy . . . . . . . . 4.5.4 Free-electron model . . . . . . . . . . 4.5.5 Quantum cellular automata (QCAs)

Problems and Exercises .............. .

5. Current and Tunnel Current in Quantum Physics

xi

84 84 85 87 88 88 89 90 93 94 94 97 98 99

101 102

105

5.1 Electrical Conductivity in Metals . . . . .. . . .. . 105 5.1.1 Drude model ...... ........... 106 5.1.2 Electrical conductivity in quantum mechanics 109

5.2 Current in Quantum Physics ....... . 111 5.2.1 Current density in quantum physics 112 5.2.2 Current of free electrons 113 5.2.3 Purely real waves . . . . . . . . . .

5.3 Tunneling and Tunnel Current ...... . 5.3.1 Tunneling through a rectangular barrier 5.3.2 Tunnel current .... . ... . ... . .

5.4 Shrinking Dimensions and Quantized Conductance 5.4.1 Two-dimensions .............. . 5.4.2 One-dimension and the quantum wire 5.4.3 Ballistic Transport and the Landauer formula

Problems and Exercises.

6. Energy in Circuit Theory

6.1 Lumped Circuits . 6.1.1 Kirchhoff's laws 6.1.2 Circuit elements

114 114 114 118 119 119 120 121 123

125

125 125 126

xii Introduction to Nanoelectronic Single-Electron Circuit Design

6.1.3 Energy considerations: passive and active elements 131 6.1.4 Linear elements and superposition 133 6.1.5 Affine linear and nonlinear elements

6.2 Circuit Theorems ......... . . . . 136 139 139 143 145

6.2.1 Tellegen's theorem . .... . . . 6.2.2 Thevenin and Norton equivalents

Problems and Exercises . ... .. ...... .

7. Energy in the Switched Two-Capacitor Circuit 149

149 7.1 Problem Statement 7.2 Continuity Property in Linear Networks . . . . . . . . .. 150

7.2.1 Continuity property of bounded capacitor currents 151 7.3 Unbounded Currents . . . . . . . . . . . . . . . . . . 152

7.3.1 Voltages in circuits with unbounded currents 152 7.4 Zero Initial Capacitor Voltage (Zero State) 152

7.4.1 p-Operator notation . . . . . . . . . . . . 152 7.4.2 Impedance and admittance operators of

the capacitor . . . . . 154 7.4.3 Generalized functions . 155

7.5 Initial Charge Models ..... 155 7.6 Solution A: Bounded Currents 159 7.7 Solution B: Unbounded Currents 159

7.7.1 Energy generation and absorption in circuits with unbounded currents . . . . . . . . . . . . . . 160

7.7.2 Energy conservation. . . . . . . . . . . . . . 161 7.8 Unbounded or Bounded Currents Through Circuits. 162 Problems and Exercises . . . . . . . . . . . . . . . . . . . 162

8. Impulse Circuit Model for Single-Electron Thnneling-Zero Thnneling Time 165

8.1 SET Junction Excited by an Ideal Current Source-Zero Thnneling Time. . . . . . . . . . . . . . . . . . . . . 167 8.1.1 Coulomb oscillations ............. 167 8.1.2 Thnneling of a single electron modeled by an

impulsive current .. . . . . . . . . . . . . 167 8.1.3 Energy is conserved: critical voltage. . . . 171

8.2 SET Junction Excited by an Ideal Voltage Source . 173 8.2.1 Critical voltage 174

8.3 Basic Assumptions . . . . . . . . . . . . . . . . . . 174

Contents

12.4 Circuit Simulation ..................... . 12.5 Random Background Charges ............... .

12.5.1 Put the information in the amplitude or frequency component of the signal. . . . . . . . . . . . . . .

12.5.2 Use compensation (circuits) to control the charge among the islands . . . . . . . . . . . . . . . . . .

12.5.3 Use redundancy on a higher level (system level) . 12.6 An outlook to System Design: Fuzzy Logic and Neural

Networks ....... . 12.6.1 Fuzzy logic ...... . 12.6.2 Neural networks ... . 12.6.3 SET Perceptron examples

Pro blems and Exercises . . . . . . . . . .

13. More Potential Applications and Challenges

13.1 Logic Circuits ...... . 13.1.1 Electron-box logic 13.1.2 Memory elements

13.2 Analog Functionality ... 13.2.1 Voltage controlled variable capacitor. 13.2.2 Charge detection. . . . . . . 13.2.3 Electron pump in metrology

Problems and Exercises .

Epilogue

Bibliography

Index

xv

251 255

256

256 257

257 258 258 262 266

267

267 268 277 279 279 280 281 281

283

289

293