I n t r o d u c t i o n t o. Nanoelectronic. Single-Electron Circuit Design

Transcription

1 I n t r o d u c t i o n t o Nanoelectronic Single-Electron Circuit Design

2 I n t r o d u c t i o n t o Nanoelectronic Single-Electron Circuit Design Jaap Hoekstra Delft University of Technology, The Netherlands

3 Published by Pan Stanford Publishing Pte. Ltd. Penthouse Level, Suntec Tower 3 8 Temasek Boulevard Singapore editorial@panstanford.com Web: 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 (Hardcover) ISBN (ebook) Printed in Singapore.

4 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

5 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

6 Preface vii electronics is not followed for designing circuits. This text provides a circuit theoretical model of the single-electron tunneling 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'2me F 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

7 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

8 Contents Preface v 1. Introduction Scope N anoelectronic circuit design issues Levels in modeling, a top-down approach Overview of tunneling capacitor circuit models Important quantum mechanical phenomena Electron Tunneling Free electrons Tunneling Hot electrons Tunneling time and transition time Tunneling Capacitors and Island Charges Two-junction circuit in Coulomb blockade Energy in Simple Capacitor Circuits, Bounded and Unbounded Currents Switching circuits: energy in a resistive circuit Charging a capacitor: bounded current Charging a capacitor: unbounded current Energy calculation with the generalized delta-function Operational Temperature 1.6 Research Questions Problems and Exercises Tunneling Experiments in Nanoelectronics 29 ix

9 x Introduction to Nanoelectronic Single-Electron Circuit Design 2.2 'I\mneIing in the 'I\mnel Diode Energy-band diagram for the p - n diode Experiments of Esaki on the (tunnel) diode Nonlinear voltage-current characteristic of the tunnel diode Tunnel current Energy paradox Equivalent circuit Resonant tunneling diode Tunneling Capacitor Tunneling between plates and the hot electron Tunnel capacitor and electron-box Single-electron tunneling transistor and Coulomb blockade Array of tunnel junctions and Coulomb oscillations Single-electron tunneling junction and Coulomb blockade 57 Problems and Exercises Current in Electrodynamics and Circuit Theory Charges in Electrodynamics Conservation of Charge and Continuity Equation Electromagnetics' Field Equations in Vacuum Equations in the Presence of Charges and Currents Conservation of Energy and Poynting's Theorem Steady-State and Constant Currents Kirchhoff's current law Vector potential A Ohm's law Kirchhoff's voltage law Time-Dependent Current Flow Towards Circuit Theory 77 Problems and Exercises Free Electrons in Quantum Mechanics Particles, Fields, Wave Packets, and Uncertainty Relations Schrodinger's Equation Time-independent Schrodinger equation

10 Contents xi 4.3 Free Electrons Electron as a particle Electron as a wave A beam of free electrons Electron as a wave packet Phase- and group velocity 4.4 Free Electrons Meeting a Boundary Step potential: E < Eo Step potential: E > Eo Electrons in Potential Wells Infinite well: standing waves Finite well: periodic boundary conditions Quantization of energy Free-electron model Quantum cellular automata (QCAs) Problems and Exercises Current and Tunnel Current in Quantum Physics Electrical Conductivity in Metals Drude model Electrical conductivity in quantum mechanics Current in Quantum Physics Current density in quantum physics Current of free electrons Purely real waves Tunneling and Tunnel Current Tunneling through a rectangular barrier Tunnel current Shrinking Dimensions and Quantized Conductance Two-dimensions One-dimension and the quantum wire Ballistic Transport and the Landauer formula 121 Problems and Exercises Energy in Circuit Theory 6.1 Lumped Circuits Kirchhoff's laws Circuit elements

11 xii Introduction to Nanoelectronic Single-Electron Circuit Design Energy considerations: passive and active elements Linear elements and superposition Affine linear and nonlinear elements Circuit Theorems Tellegen's theorem Thevenin and Norton equivalents 143 Problems and Exercises Energy in the Switched Two-Capacitor Circuit Problem Statement Continuity Property in Linear Networks Continuity property of bounded capacitor currents Unbounded Currents Voltages in circuits with unbounded currents Zero Initial Capacitor Voltage (Zero State) p-operator notation Impedance and admittance operators of the capacitor Generalized functions Initial Charge Models Solution A: Bounded Currents Solution B: Unbounded Currents Energy generation and absorption in circuits with unbounded currents Energy conservation Unbounded or Bounded Currents Through Circuits. 162 Problems and Exercises Impulse Circuit Model for Single-Electron Thnneling- Zero Thnneling Time SET Junction Excited by an Ideal Current Source-Zero Thnneling Time Coulomb oscillations Thnneling of a single electron modeled by an impulsive current Energy is conserved: critical voltage SET Junction Excited by an Ideal Voltage Source Critical voltage Basic Assumptions

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14 Contents xv 12.4 Circuit Simulation Random Background Charges Put the information in the amplitude or frequency component of the signal Use compensation (circuits) to control the charge among the islands Use redundancy on a higher level (system level) An outlook to System Design: Fuzzy Logic and Neural Networks Fuzzy logic Neural networks SET Perceptron examples Pro blems and Exercises More Potential Applications and Challenges 13.1 Logic Circuits Electron-box logic Memory elements 13.2 Analog Functionality Voltage controlled variable capacitor Charge detection Electron pump in metrology Problems and Exercises. Epilogue Bibliography Index