Table of Contents

Chapter 1: Introduction

1.1. Introduction and preliminary warning

1.2. Bibliography

Chapter 2: Some Useful Concepts and Reminders

2.1. Quantum mechanics and the Schrödinger equation

2.2. Energy band structure in a periodic lattice

2.3. Semi-classical approximation

2.4. Electrons and holes

2.5. Semiconductor heterostructure

2.6. Quantum well

2.7. Tight-binding approximation

2.8. Effective mass approximation

2.9. How good is the effective mass approximation in a confined structure?

2.10. Density of states

2.11. Fermi-Dirac statistics

2.12. Examples of 2D systems

2.13. Characteristic lengths and mesoscopic nature of electron transport

2.14. Mobility: Drude model

2.15. Conduction in degenerate materials

2.16. Einstein relationship

2.17. Low magnetic field transport

2.18. High magnetic field transport

2.19. Exercises

2.20. Bibliography

Chapter 3: Ballistic Transport and Transmission Conductance

3.1. Conductance of a ballistic conductor

3.2. Connection between 2D and 1D systems

3.3. A classical analogy

3.4. Transmission conductance: Landauer’s formula

3.5. What if the device length really does go down to zero?

3.6. A smart experiment which shows you everything

3.7. Relationship between the Landauer formula and Ohm’s law

3.8. Dissipation with a scatterer

3.9. Voltage probe measurements

3.10. Comment about the assumption that T is constant

3.11. Generalization of Landauer’s formula: Büttiker’s formula

3.12. Non-zero temperature

3.13. The integer quantum Hall effect

3.14. Exercises

3.15. Bibliography

Chapter 4: S-matrix Formalism

4.1. Scattering matrix or S-matrix

4.2. S-matrix combination rules

4.3. A simple example: the S-matrix of a Y-junction

4.4. A more involved example: a quantum ring

4.5. A final more complex example: solving the 2D Schrödinger equation

4.6. Exercises

4.7. Bibliography

Chapter 5: Tunneling and Detrapping

5.1. Introduction

5.2. Single barrier tunneling

5.3. Two coherent devices in series: resonant tunneling

5.4. Physical meaning of the terms appearing in the resonant transmission probability

5.5. Tunneling current

5.6. Resonant tunneling in the real world

5.7. Discrete state coupled to a continuum

5.8. Fano resonance

5.9. Fano resonance in a quantum-coherent device

5.10. Fano resonance in the real world

5.11. Exercises

5.12. Bibliography

Chapter 6: An Introduction to Current Noise in Mesoscopic Devices

6.1. Introduction

6.2. Ergodicity and stationarity

6.3. Spectral noise density and Wiener-Khintchine theorem

6.4. Measured power spectral density

6.5. Shot noise in the classical case

6.6. Why the shot noise formula is not valid in a macroscopic conductor

6.7. Classical example 1: a game with cannon balls

6.8. Classical example 2: cars and anti-cars

6.9. Quantum shot noise

6.10. Bibliography

Chapter 7: Coulomb Blockade Effect

7.1. Introduction

7.2. Energy balance when charging capacitors

7.3. Coulomb blockade in a two-terminal device

7.4. Coulomb blockade in a single-electron transistor

7.5. Single-electron turnstile

7.6. Coulomb blockade in the real world

7.7. Exercises

7.8. Bibliography

Chapter 8: Specific Interference Effects

8.1. Classical Lagrangian with a magnetic field

8.2. Classical Lagrangian without a magnetic field

8.3. Phase shift due to a magnetic field

8.4. Aharonov-Bohm effect in mesoscopic rings

8.5. 1D localization

8.6. Weak localization

8.7. Universal conductance fluctuations

8.8. Bibliography

Chapter 9: Graphene and Carbon Nanotubes

9.1. Introduction

9.2. Graphene band structure

9.3. Integer quantum hall effect in graphene

9.4. Carbon nanotube band structure

9.5. Carbon nanotube bandgap

9.6. Carbon nanotube density of states and effective mass

9.7. Electron transport in and quantum dots from carbon nanotubes

9.8. Exercises

9.9. Bibliography

Chapter 10: Appendices

10.1. The uncertainty principle

10.2. Crystalline lattice; some definitions and theorems

10.3. The harmonic oscillator

10.4. Stationary perturbation theory

10.5. Method of Lagrange multipliers

10.6. Variational principle

10.7. Wiener-Khintchine theorem

10.8. Binomial probability law

10.9. Random Poisson process

10.10. Transformation of the Cartesian wavevector coordinates into transverse and parallel components

10.11. Useful physical constants

Solutions to Exercises

Exercise 2.19.1

Exercise 2.19.2

Exercise 2.19.3

Exercise 2.19.4

Exercise 3.14.1

Exercise 3.14.2

Exercise 3.14.3

Exercise 3.14.4

Exercise 3.14.5

Exercise 5.11.1

Exercise 5.11.2

Exercise 5.11.3

Exercise 5.11.4

Exercise 5.11.5

Exercise 5.11.6

Exercise 7.7.1

Exercise 7.7.2

Exercise 7.7.3

Exercise 9.8.1

Exercise 9.8.2

Exercise 9.8.3

Index

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Thierry Ouisse to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Ouisse, Thierry.

Electron transport in nanostructures and mesoscopic devices / Thierry Ouisse.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-050-9

1. Electron transport. 2. Nanostructured materials--Electric properties. 3. Nanostructures--Electric properties. 4. Mesoscopic phenomena (Physics) I. Title.

QC176.8.E4O95 2008

530.4’1--dc22

2008008768

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-050-9

Chapter 1

Introduction

1.1. Introduction and preliminary warning

Matter stability and the way in which rigid crystalline or amorphous arrays of atoms can be formed are ruled y two pillars of physics: electromagnetism and quantum mechanics; nothing else, provided that we admit the existence of elementary constituents such as atom nuclei without having to derive their internal structure from the first principles (then we need to add nuclear forces to our bunch of tools). The postulates and basic equations of these two theories can be written on a couple of pages, and everything can be derived from them¹. If the world was ruled by classical mechanics, it would simply be impossible to obtain stable atoms² or stable chemical bonding to ensure the existence of matter as we all experience it in our everyday life. Thus, it is something of a misnomer to say that we are going to study quantum devices as opposed to devices which would not be quantum. Everything is ruled by quantum mechanics, from the insulating or conducting character to the color of any piece of matter or object that you can see inside the room where you are now reading this introduction (see also Figure 1.1). To understand our macroscopic world, we often feel that once we admit the existence of stable matter, we can content ourselves with using the second Newton’s law of motion and classical gravitational forces. An aeronautics engineer does not put too much quantum mechanics in his calculations, but this is certainly no longer the case if we want to justify the way in which electrons and therefore the electrical current behaves in a bulk semiconductor. Without a periodic atomic lattice and quantum mechanics, we could not find free electrons able to carry a current in a p-n junction, or in the channel of transistors which form the integrated circuits inside our computers. Thus, the reason why the devices under study in this book are called quantum is that we can straightforwardly apply to them the basic quantum effects that students are accustomed to calculating in an introductory quantum mechanics course.

Figure 1.1. The ubiquitous character of quantum mechanics

In nanostructures, electrons can be confined in potential wells narrow enough to obtain energy quantization along the confining direction. Their dimension is small enough for probing the dual wave-particle nature of the electron in a straightforward manner, because the electron wave function phase can be kept coherent over the whole device length. Thus, it becomes possible to observe wave interference effects just by measuring the average current which can be passed through such components, and particle-like properties from current noise data. As once stated by the physicist Esaki, this looks like some kind of “do-it-yourself” quantum mechanics: you are not required to become a specialist in group theory and irreducible representations, or of field-theoretic methods to get in touch with the essence of the topic (see also Figure 1.2). In addition, other specific effects, although not quantum-mechanical, are also due to reduced dimensions: if you can inject a few electrons into a nanostructure and if the capacitance between this nanostructure and the rest of the world is very small, we can probe effects which are due to charge granularity (we cannot divide the electron charge), and which are known as Coulomb blockade. Such effects are the subject of intensive research in R&D laboratories, because many people hope to put them to good use to produce new types of memories and devices that are smaller, faster and require a smaller amount of operating power. The aim of this book is to give an introduction to the basic concepts which govern the conduction mechanisms taking place in such small devices.

Figure 1.2. The quantum garage

Many (not to say most) of the phenomena described in this book usually take place at quite low temperatures, or in devices not yet (and for some of them never) used in the industry. The physics described here is not useful for understanding how industrial semiconductor devices behave in most applications right now, with the notable exception of resonant tunneling. Nevertheless, “today’s” silicon (Si) Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFETs) definitely exhibit non-stationary and ballistic transport effects. Explaining these effects requires us to use some of the concepts developed in this book, even the high electric fields involved in MOSFET operation make the application of such concepts much more complicated than what is described in this introduction. At room temperature, the electron mean free path in silicon is in the 5-10 nm range, not far from the 45 nm channel length of the current CMOS technology, and integrated chips using a 32 nm process technology have already been demonstrated by the INTEL corporation in 2007. Figure 1.3 shows the picture of a 20 nm channel length prototype MOSFET produced in 2006 by LETI-CEA. Thus, even at room temperature some commercial electronic devices are close to the ballistic regime. Those industrial MOSFET’s are fabricated with an incredibly high reproducibility in order to form extremely complex integrated circuits (and as a side note such precision and reproducibility are actually far from being achieved in most research laboratories working in the realm of mesoscopic physics and nanostructures, or with semiconductors more exotic and physically more appealing than silicon). Device-modeling based on ballistic properties has thus become an active research field, even in the case of silicon devices (see, e.g., [NAT 94] for one of the pioneering Si papers).

In addition, mesoscopic effects are important in four respects:

(i) they are often of great physical significance, and give a deep and straightforward insight into some of the most striking implications of quantum mechanics (for instance, they provide unambiguous and clear demonstrations of the dual electron nature, particle and wave);

(ii) although often obtained at low temperatures or high magnetic fields they are very useful for extracting physical parameters dealing with (nano)structures actually used in applications;

(iii) some of the effects are already used in (e.g. resonant tunneling) or potentially useful for (e.g. Coulomb blockade) applications;

(iv) although still difficult to engineer, devices made from graphene or carbon nanotubes exhibit truly ballistic and quantum-coherent effects even at room temperature. Thus, it is quite possible that not only ballistic, but also quantum-coherent effects may be present in electronic applications in the near future.

Figure 1.3. A transmission electron microscope view of a planar double-gate MOSFET fabricated by LETI-CEA with a 20 nm channel length; reproduced by permission after J. Widiez et al., IEEE Transactions on nanotechnology, vol. 5, p. 643 (2006),copyright ©2006IEEE ([WID 06])

As a consequence, in most of the largest semiconductor companies, and in a very large number of university labs, intensive research work is devoted to such structures. Scarcely applied though it may seem at first sight, this field of activity is in fact the leading edge of semiconductor research.

This book is designed to be accessible to the independent reader, and to students not having a strong background in solid-state physics (e.g. issued from engineering disciplines). As a matter of fact, this book is an attempt to answer the following question: what must be taught to students starting from scratch to make them understand the bases of electron transport in mesoscopic devices? A professor placed in such a situation soon realizes that a good deal of solid-state physics and quantum mechanics is required. This explains the incorporation of chapters which are usually absent from the more specialized, already-existing books, and marks the difference between them and this. In addition, to follow the classification once given by J.M. Ziman, this book does not fall into the category of a “treatise” but into that of a “textbook”, with the purpose of introducing and explaining concepts. The text has been written with the aim of being as self-contained as possible, and is based on an oral course delivered at an international European master’s degree course involving three technical universities (GrenobleINP, EPFLausanne and Polit’oTorino). It is a deliberate choice of the author to keep in the book the spirit of the oral course, and this is the reason why the reader should not be surprised to be sometimes interpellated or hailed in a somewhat familiar way³.

Assimilating the quantum-mechanical rules summarized at the very beginning of the book suffices to derive any subsequent result, but should by no means be considered as enough to master quantum mechanics itself. Hence, and despite the fact that the text remains at an introductory level, a complete understanding of the course probably requires a minimum prior knowledge and self-maturation of the basic quantum-mechanical concepts. A reader not acquainted with this field will certainly feel the need to consult more authoritative manuals, due to the innumerable number of questions, either technical or fundamental, that a concise and incomplete presentation of quantum mechanics must arouse in any normally constituted mind. Some knowledge of solid-state and semiconductor physics certainly help as well, but all concepts useful for understanding the book can in principle be found in the book itself, and since this book is an introduction dedicated to a broad audience, maybe some of you are probably already acquainted with the required solid-state physics notions. For those who are experienced in solid-state physics it is possible to simply skip most of the reminders which make up Chapter 2. Besides, many of those reminders are not always quite rigorously demonstrated. All undemonstrated or heuristically-derived quantum-mechanical formulae can be found and are rigorously derived in a self-contained, encyclopedic textbook: [COH 77]. Solid-state physics has its self-contained book too: [ASH 76]. For bulk semiconductor physics and transport, an advanced and quite remarkable and complete textbook was written by [RID 82], but it is not essential for understanding this book. Eventually, we can find books specifically devoted to mesoscopic electron transport, which can be of great support for a better understanding or for gaining more information (the list below is not exhaustive): [BEE 91], [KEL 95], [DAT 95] and [FER 97]. The book which is the closest in spirit to this course is the one by Datta. It includes many exercises and also contains more advanced formalisms (e.g. Green’s functions) and discussions,which are not necessarily required at this introductory level. The book by Kelly presents a very large amount of data and also deals with aspects which are either more related to technological aspects or closer to the applications.

This book is an introduction and as such a number of important aspects have been omitted, mainly those which imply the use of mathematical concepts too involved to be developed in front of an audience new to the field. In particular, the reader will not find here a rigorous description of Green’s function formalism, which is necessary to include electron-electron interactions in transport modeling. A general discussion and study of many-body effects is also absent, which would be mandatory to understand a physical phenomenon such as the fractional quantum Hall effect, metal-based mesoscopic devices, carbon nanotubes operating in the 1D form of a Luttinger liquid and many others. Justice has not been done to the electron spin and its possible applications. This book could thus be given a second title: how far can we go using only independent electrons and the exclusion Pauli principle (see also Figure 1.4)? Surprising though it may seem, a good deal of nanostructure physics can still be grasped that way, but the reader will not find in this book a wealth of phenomena associated with electron-electron interactions. If they are not discouraged by this introductory text their study should constitute the next step, to be achieved by studying more specialized treatises and articles. Thus, if after studying the various chapters the student decides to read further and deeper, the main objective of this book will have been fulfilled. In the same spirit, we shall skip some difficult demonstrations which would be required for a rigorous derivation of some important solid-state physics results⁴. However, even if difficult theoretical techniques have been deliberately banished from the text, “the language of physics is mathematics”, and none of the chapters escape from the rule.

Figure 1.4. The quantum society and Pauli’s exclusion principle

Most exercises proposed at the end of each chapter are easy and their purpose is to provide the reader with a means of checking that they have correctly assimilated the chapter content and concepts. However, some of them require more time, and have been inserted to complete points not detailed in the main text.

Not all the sections were dealt with during the original oral course. I have put indicators at the beginning of each section:

This section is a reminder. Thus it can be skipped if the reader is already familiar with the corresponding field.

This section is essential to the book (and, quite accessorily, it may be helpful to prepare an exam). Some reminders belong to this category.

This section is not a reminder, but is not considered as essential to understand the other parts.

This section can be skipped at first reading.

1.2. Bibliography

[ASH 76] ASHCROFT N.W. and MERMIN N.D., Solid State Physics, Wiley, New York, 1976.

[BEE 91] BEENAKER C.W.J. and VAN HOUTEN H., Quantum Transport in Semiconductor Nanostructures, Solid State Physics 44, Academic Press, 1991.

[COH 77] COHEN-TANNOUDJI C., DIU B. and LALOË F., Quantum Mechanics, Wiley, New York, 1977.

[DAT 95] DATTA S., Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995.

[FER 97] FERRY D.K. and GOODNICK S.M., Transport in Nanostructures, Cambridge University Press, 1997.

[KEL 95] KELLY M.J., Low-dimensional Semiconductors, Oxford University Press, 1995.

[NAT 94] NATORI K., “Ballistic metal-oxide-semiconductor field effect transistor”, Journal of Applied Physics, vol. 76, no. 8, 1994, p. 4879-4890.

[RID 82] RIDLEY, B.K. Quantum Processes in Semiconductors, Clarendon Press, Oxford, 1982.

[WID 06] WIDIEZ J., POIROUX T., VINET M., MOUIS M., DELEONIBUS S., “Experimental comparison between sub-0.1µm ultrathin SOI single- and double-gate MOSFETs: performance and mobility”, IEEE Transactions on Nanotechnology, vol. 5, no. 6, 2006, p. 643-648.

1 Of course with a substantial amount of hard work and mathematics, and adding some thermodynamics. Note also that if quantum mechanical predictions can be verified with an astonishingly high precision, their interpretation was (and is) the source of thousands of scientific articles and books.

2 Classical electrons accelerated over orbits radiate electromagnetic waves and thus lose energy. Thus, bound electrons would collapse onto the atoms.

3 As you may have already noticed, the familiar way of addressing the reader began in the very first lines of this introduction.

4 Whenever this occurs, the unsatisfied reader will always be left with the possibility of consulting the more advanced textbooks or specialized articles mentioned in the bibliography.

Chapter 2

Some Useful Concepts and Reminders

2.1. Quantum mechanics and the Schrödinger equation

2.1.1. A more than brief introduction

The following is only a summary which includes the basic quantum-mechanical (QM) equations required for understanding the book. It is by no means a rigorous introduction to the topics, and if you want to go further, a wise thing to do would be to immerse yourself, e.g., in the introductory textbook by R.P. Feynman [FEY 65], and then in the book by Cohen-Tannoudji et al. [COH 77] for a while¹. Besides, several formulations can be used to describe quantum mechanics, and here we shall not really make the effort of differentiating them from one another. A concise description of those different formulations can be found in [STY 02].

In classical mechanics the elementary constituents of matter are massive point particles whose movement is controlled by electromagnetic or gravitational forces. At any instant we can precisely define the particle position and, provided that at a time t we are given the position and velocity of all the system particles, we can calculate everything at any other time, and obtain well-defined trajectories (with a powerful enough computer if the particles are numerous, etc.), even if the system remains isolated. Thus, the whole picture is in principle perfectly deterministic. In quantum mechanics the situation is far more subtle. Experimentally, it appears that if we let a system evolve isolated for a while, the maximum information concerning this system that is physically accessible to human knowledge does not allow us to predict in a deterministic and unique way the result which will be obtained once we act on this system to measure some of its properties.

Figure 2.1. Quantum-mechanical interference experiment illustrating the dual wave-particle nature of the electron

The celebrated double-slit interference experiment is probably one of the more striking and meaningful illustrations of the quantum nature of matter. This effect figures in due place in almost any introduction chapter on quantum mechanics, and we shall respect this very justified habit. Interference experiments such as that illustrated by Figure 2.1 reveal that it is no longer possible to consider an entity such as an electron or a proton as a particle, and that it is not possible to consider it as a pure wave either [FEY 65]. “Identically prepared” electrons propagating through double slits exhibit interference patterns like waves [JON 61], but if we put a screen behind the plane of those slits we always obtain localized spots, as for particles [TON 89]. It is the statistical collection of a large number of such individual events which forms the interference pattern. Thus, in quantum mechanics (and in the real world) we have to assign a dual nature to electrons, whose behavior can be modeled only as a combination of both a particle and a wave. Suppress one slit and we lose the interference pattern. The wave really passes through both slits. Try to detect the electron at one of the two slits and we also lose the pattern, because the particle-like detection at one slit instantaneously reduces the extended propagating wave.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!