Introduction to Electronic Devices 3031271955, 9783031271953

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Introduction to Electronic Devices

Corrado Di Natale

Introduction to Electronic Devices

Corrado Di Natale Department of Electronic Engineering University of Rome Tor Vergata Rome, Italy

ISBN 978-3-031-27195-3 ISBN 978-3-031-27196-0 (eBook) https://doi.org/10.1007/978-3-031-27196-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

With love and gratitude to L.E.D. that constantly shine my life.

Preface

The optimal communication of a subject requires a personal elaboration of all the concepts and principles underlying the discipline. In order to account for all the elements of a subject, moreover, a narrative flow needs to be created, so as to enable all the involved parts to follow a logical sequence. In this respect, even in a field such as that of electronic devices, characterized by countless textbooks, the authors personal viewpoint makes each textbook in some way unique. This book stems from my teaching experience in the undergraduate courses of electronic engineering at the University of Rome Tor Vergata. The basic principles of electronic devices are presented in a simple but rigorous form, and, wherever possible, a formal explanation of device behaviors has been provided. In other cases, intuitive arguments have been introduced to provide theoretical justifications to factual observations. The physical principles at the basis of solid-state devices are reviewed in the first chapter, where the discussion is limited to those arguments necessary for the understanding the main working mechanisms of the devices. The presentation of the devices begins with the metal–semiconductor junction and continues with the PN junction, the bipolar junction transistors (BJT), and the fieldeffect transistors. The decision to start with the metal–semiconductor device may be unusual, given that this is the case where simple theory is strongly challenged by reality, not least due to the presence of surface states. On the other hand, I considered it necessary to begin the book with the junction between a semiconductor and a metal on the grounds that it is the basic ingredient for the connection of semiconductors to electric circuits. The metal–semiconductor junction is also aimed at introducing the common properties of junctions among materials, either metals or semiconductors. Namely, junctions among materials result in space charge regions and built-in potentials, and the difference between semiconductors and metals is merely the dimension of the space charge region: large enough, in semiconductors, to influence the macroscopic properties, and small enough in metals, where, thanks to the tunnel effect, it is transparent to charge motions.

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To preserve the narrative continuity, the generation and recombination phenomena, as well as the recombination function, are discussed in a separate chapter before the PN junction. For the same reason, the heterojunctions are introduced after the chapter on bipolar junction transistors, in order to promptly show their role in the improvement of the BJT characteristics. The final chapters are dedicated to the MOS, the MOSFET, and, in general, to the field-effect devices. At the end of each chapter, a list of further readings is provided. In these reading lists, the textbooks that most inspired the author are mentioned. The suggested textbooks are nonetheless good references for those readers that want to expand their knowledge. Besides the textbooks, at the end of each chapter, a list of journal papers is also reported. Journal papers are a sort of specialized literature hardly accessible to students. Among the papers, I privileged the historical papers, reporting the crucial discoveries and breakthroughs. The reading of these papers is suggested, so as to check the progresses and get how each discovery is related to the previous ones. Furthermore, it is an occasion to make a list of the names of those people who created the field of electronic devices, together with the procedure they implemented to achieve those results that dramatically shaped the contemporary age. In the recent years, the interest in electronic devices has expanded to include not only solid-state physics and electronic engineering, but also other fields. The progresses in molecular electronics, particularly the interface of biomolecules with electronic devices, have increased the number of students of other disciplines interested in getting the fundamentals of electronic devices. This book, indeed, has also been written in such a way that any readers with a sufficient knowledge of basic physics, including rudiments of quantum mechanics, could find it accessible. This book is a testimony of the teachings of the late Professor Arnaldo D’Amico, who taught this subject for decades at the University of Rome Tor Vergata. His ideas and intuitions about the teaching methodologies to be used for best conveying these topics to students are scattered throughout this book and constitute such a great part of my experience that I can hardly distinguish between my original visions and those gained out of our mutual discussions. Probably, this is the dynamics of an intellectual legacy, and, ultimately, this is the reason why I wrote this book. I am in debt to the many students who, over the last ten years, have provided feedbacks which contributed to refine the quality of my lectures on these topics. Finally, I am obliged to Maria Cristina Teodorani, for her skillful and competent revision of the text. Rome, Italy December 2022

Corrado Di Natale

Contents

1 The Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Phenomenology of Semiconductors . . . . . . . . . . . . . . . . . . . . . . . 1.3 Electrons in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Orbitals Splitting and Coupled Oscillators . . . . . . . . . . . . . . . 1.3.2 Crystals, Periodic Potentials and Energy Gaps . . . . . . . . . . . . 1.3.3 Electron Distributions in Energy Levels . . . . . . . . . . . . . . . . . 1.3.4 The Band Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Statistics of Electrons and Holes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The Intrinsic Fermi Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Charge Transport: The Drift-Diffusion Model . . . . . . . . . . . . . . . . . . 1.5.1 Thermal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Drift Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Diffusion Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Non-uniform Distribution of Dopant Atoms and Built-in Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Quasi-neutrality Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 6 7 8 11 17 23 24 26 27 33 35 35 40 43 46 47 48

2 The Metal-Semiconductor Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The General Rule of Junctions at the Equilibrium . . . . . . . . . 2.2 The Metal-Semiconductor Junction at the Equilibrium . . . . . . . . . . . 2.3 Biased Metal-Semiconductor Junction . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Capacitance of the Junction . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The I/V Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Barrier Height Lowering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Non-rectifying Metal-Semiconductor Contact . . . . . . . . . . . . . . . . . .

49 49 50 51 59 61 63 69 71

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2.4.1 Ohmic Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Tunnel Ohmic Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Space Charge Limited Current . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Surface States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76 77 79 80 81 82

3 Generation and Recombination Processes . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Generation and Recombination Phenomena . . . . . . . . . . . . . . . . . . . . 3.3.1 Generation and Recombination Rates . . . . . . . . . . . . . . . . . . . 3.3.2 Traps and Recombination Centers . . . . . . . . . . . . . . . . . . . . . . 3.4 The Shockley-Hall-Read Generation-Recombination Model . . . . . . 3.4.1 Example of Application of the SHR Model: The Dynamics of Generation-Recombination Phenomena . . . . . . 3.4.2 The Generation-Recombination Function for Direct Band Gap Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84 85 86 89 92 94 97 98 98

4 PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 PN Junction at the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Removal of Charge Discontinuity at the Depletion Layer Border . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Physical Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Current in the PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Ideal Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Generation and Recombination Current . . . . . . . . . . . . . . . . . 4.4 Capacitive Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Minority Charge Storage and Diffusion Capacitance Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Breakdown Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Avalanche Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Zener Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 101

130 135 135 136 138 138

5 Negative Differential Resistance Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Tunnel Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 NDR Behavior in GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Gunn Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 141 144 147 149 149

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6 Bipolar Junction Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Ideal Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Electron Current in the Active Zone . . . . . . . . . . . . . . . . . . . . 6.3 Current Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Base Recombination Current . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Forward Hole Current in the Emitter . . . . . . . . . . . . . . . . . . . . 6.3.3 Numerical Comparison of αT and γ . . . . . . . . . . . . . . . . . . . . 6.3.4 Total Current Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 BJT Operative Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Non Ideal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Early Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Emitter Band-Gap Narrowing . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Small Base Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 High Injection Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Physical Effects in Real BJT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Junction and Diffusion Capacitances . . . . . . . . . . . . . . . . . . . . 6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 152 154 159 161 163 164 165 166 168 168 169 170 171 173 175 175 179 179

7 Heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Band Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Staggered Bandgaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Straddled Bandgaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Electric Field and Built-In Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Quasi-electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Current-Voltage Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Thermionic Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Heterojunction Bipolar Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Graded Band Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 183 184 184 187 189 190 193 193 196 199 199

8 Metal-Oxide-Semiconductor Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Band Diagram and Electrostatic Quantities at the Equilibrium . . . . . 8.2.1 Relation Between Potential and Charge Carrier Concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The MOS Under Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The C/V Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Minority Charges Generation in the Depletion Layer . . . . . . 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 203 207 208 212 221 222 222

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9 Field Effect Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Channel Charge Modulation and the Threshold Voltage . . . . . . . . . . 9.3 Metal Oxide Semiconductor Field Effect Transistor . . . . . . . . . . . . . 9.3.1 Channel Length Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Body Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Subthreshold Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Short Channel MOSFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Threshold Voltage Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Drain Induced Barrier Lowering . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Velocity Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.5 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 CMOS Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Metal Semiconductor Field Effect Transistor (MESFET) . . . . . . . . . 9.7 High Electron Mobility Transistor: HEMT . . . . . . . . . . . . . . . . . . . . . 9.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 226 228 236 237 237 239 240 241 241 243 246 247 249 252 256 259 260

Appendix A: Elements of Classic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Appendix B: Basic Principles of Quantum Mechanics . . . . . . . . . . . . . . . . . 265 Appendix C: The Chua Formalism of Electric Network Elements . . . . . . 273 Appendix D: The SHR Generation-Recombination Function . . . . . . . . . . 275 Appendix E: Numerical Examples of Rectifying Junctions . . . . . . . . . . . . 277 Appendix F: Majority Current in a BJT with a Linearly Doped Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Chapter 1

The Physical Background

1.1 Introduction Electronics has been defined as the “Science and technology which deals with the supplementing of human capabilities by devices which collect and process information, transmit it to the point needed, and there either control machines or presents the processed information to humans” (Everitt, W.L.: ‘Let us re-define electronics’, Proc. I.R.E., 1952, 40). The most efficient available tool to accomplish these functions is the electric charge carried by electrons. Hence, electronics can also be defined as the “Study of the physics of the electron and its motion under different conditions” (Gatti E., Enciclopedia del Novecento, 1989). The functions of electronics correspond to the motion and the accumulation of electrons inside matter. The state of electrons is altered by electromagnetic forces, so that the electric and magnetic fields are the tools we use to modify the state of electrons, while the Coulomb and Lorentz forces define the quantitative relationships between the forces acting on the electrons and applied fields (F = q · E + q · v × B). In most of electronic devices, magnetic fields are neglected and the electric field is considered as the only source of electric phenomena. In practice, it is useful to consider an additional variable: the potential (φ). It is defined as the integral of the electric field between two points in space (φ = x − x12 Ed x). Thus, the potential affects the state of the charge, namely the total charge and the charge variation with time. The relationship between voltage and charge can be described by a second order differential equation: dQ d2 Q + γ Q. (1.1) φ =α 2 +β dt dt The first derivative of the total charge defines the current, namely the stream of charged particles moving inside matter. Current is a physically observable quantity, so that the above relation is normally written as a function of current: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_1

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φ=α

di + βi + γ dt

 idt.

(1.2)

The coefficients define the phenomena due to the motion and accumulation of charges. They express three fundamental quantities: the inductance (α = L), the resistance (β = R) and the capacitance (γ = 1/C). It is worth noting that an analog equation is also valid in mechanics. Where, voltage and charge are replaced by force and position, and the three parameters are the mass, and the damping and elastic coefficients. The theory of circuits defines the rules to predict the behavior of networks of different materials under the action of electric fields elicited by voltage sources. In Eq. 1.2 the coefficients are constant. However, the above mentioned functions of electronics are achieved when also the coefficients of Eq. 1.2 are functions of the potential. More precisely, electronics needs non-linear components (resistors and capacitors), and controlled generators (namely, devices where voltage and current are controlled by applied voltages and currents flowing through given locations in the circuit). These components lead to the operations of rectification and amplification, which are among the basic elements of electronics. The first technological support enabling the functions of electronics has been offered by the exploitation of the thermionic effect. Thermionic devices have been the basic building blocks for the development of a large part of those analog electronics circuits and concepts that are still valid nowadays. The thermionic effect (discovered in 1873) consists in the emission of electrons outside a material, when it is heated beyond a material-dependent temperature. It is a manifestation of the fact that temperature can provide electrons with enough energy to enable them to cross the energy barriers that keep them inside. This effect is of crucial importance also for semiconductor devices, where the equations governing the thermionic effect remain valid. The thermionic effect occurs at the interface between solid and vacuum, and it is intrinsically not reversible. Namely, the electrons can only flow from the material towards the outside. This intrinsic, unidirectional charge flow led to the development of the diode: a device where current can only flow in one direction (Fleming, 1904). The emission of electrons requires an ancillary current, which is used to heat the metallic electrode (cathode). The emitted electrons are collected at another metallic electrode (anode) kept at a positive potential with respect to the anode. To facilitate current flow, the device, enclosed in a glass container, is kept under vacuum (Fig. 1.1). In 1906, De Forest modified the diode so as to obtain a device, the triode, where the charge flow could have been controlled by an additional voltage. The modification consisted in a grid placed between the electrons path. The grid voltage could actually accelerate or decelerate, the electrons modulating the current collected at the anode. Such a device is a three-terminal component, connected to two loops in such a way that the current flowing in one loop depends on the voltage applied to the other one (Fig. 1.2).

1.1 Introduction

3

Fig. 1.1 Schematic view of a thermionic diode and rectifying I/V characteristics

Fig. 1.2 Schematic view of a thermionic triode, and its I/V characteristics. In the direct bias region the I/V curve is almost linear

Thermionic valves were instrumental for the development of electronics. The characteristics of diodes and triodes made possible the development of the basic configurations of analog electronics (such as amplifiers, feedback amplifiers, active filters, operational amplifiers and oscillators), which resulted in radio-communications, radar, and even digital circuits. However, the temperature to extract electrons from thermionic valves is of the order of thousands of centigrades, so that the excessive energy consumption limited the realization of circuits with many devices. Since the beginning, it had been clear that the development of electronics would have required devices where electrons could have been emitted at lowest temperature, possibly at room temperature. The solution to this problem was going to change the interface from a solid-vacuum to a solid-solid one, by exploiting the peculiar properties of a class of materials called semiconductors. The development of quantum mechanics provided the means to study such properties, and thus to engineer the interfaces between materials. Once introduced, solid-state semiconductor devices opened the modern era of electronics. Solid-state electronics exploits the properties of semiconductors, or, more precisely, the properties of the junctions between a semiconductor and any other materials: a metal, a semiconductor or an insulator. In order to study solid-state devices, it is necessary to understand the properties of charge carriers. In particular, the space-energy distribution of electrons is responsible for the relationships between applied voltage, accumulated charge and current flow. Most of the phenomena in electronic devices can be adequately described by models built at the border between classical and quantum physics. In practice, we will see that quantum mechanics mainly provides a description of the energy distribution of electrons, while the transport phenomena can be described by classical concepts. The range of validity of quantum mechanics with respect to classical physics can

4

1 The Physical Background

be defined with qualitative criteria considering the fundamental constant of quantum mechanics: the Planck constant. This constant has the dimensions of a length multiplied by momentum, and this product defines a physical quantity called action. A rule of thumb to determine whether a system can be modeled by classic physics assumptions is thus consider if the product of the occurring lengths and momenta is larger than the Planck constant. In electronic devices, voltages of the order of 1 V are applied across distances (L) of the order of 1 µm. The momentum can be estimated considering that in such a space√the potential energy is converted into kinetic energy: q V = 21 mv 2 → p = mv = 2mq V . Replacing the mass (m = 9.1 · 10−31 kg) and the charge (q = 1.6 · 10−19 C) of the electron, the estimated action is: L · p = 5.4 · 10−31 J s, which is quite larger than the Planck constant (h = 6.6 · 10−34 J s). Thus, we can conclude that the classic laws of physics may adequately account for the charge transport for most of electronic devices. As the size of the devices becomes smaller, the classical concepts loose their validity, and a full quantum description becomes necessary. However, it is always important to keep in mind that theories in sciences are plausible as far as they can predict and explain the experimental observations.

1.2 The Phenomenology of Semiconductors Nomen omen, the most evident property of semiconductors is the resistivity, which lies between those of conductors and insulators. The resistivity of materials (ρ) is one of the physical quantities with the largest variability. The resistivity of semiconductors occurs in a range from ρ = 106 to 10−2 m. The resistivity of metals lies in the interval from ρ = 10−4 to 10−8 m. As for insulators, it goes from ρ = 1010 up to 1018 m. The inverse of resistivity is the conductivity (σ ). It is worth to remind that resistance, defined as the ratio of voltage to current, and measured in Ohms (), is a combination of resistivity (a property of the material) and geometrical shape. In longitudinal configurations, this combination is expressed by the relationship: R = ρ · l/A, where l is the length of the conductor and A is the section through which current flows. The limited conductivity makes semiconductors quite uninteresting for electric applications. However, they exhibit other peculiar features (Fig. 1.3), whose explanation was made possible by quantum mechanics. The relationship between conductivity and temperature, for instance, is characterized by a negative temperature coefficient. Namely, resistance decreases with temperature. On the contrary, metals, in general, show an opposite behavior (positive temperature coefficient). This peculiar property of semiconductors is used to measure temperature with a class of sensors called thermistors. Furthermore, semiconductors are photoconductors. Namely, conductivity increases when the material is shined with photons with a wavelength λ smaller than a λ0 , which is specific of each material.

1.2 The Phenomenology of Semiconductors

5

Fig. 1.3 Resistivity of metals and semiconductors as a function of temperature and light. Note that the resistance of metals is orders of magnitude smaller than that of semiconductors

Fig. 1.4 Simple picture of nuclei and electrons arrangement in solids held together by covalent and metal bonds

Another important characteristics of semiconductors is the dependence of conductivity on the chemical purity of the material. Hence, conductivity is altered by the addition of impurities either in the bulk or on the surface of the resistor. The first case is technologically exploited, as it will be discussed thereafter, to prepare semiconductors for electronics, whereas the second case provides the basis for chemical sensors. All of these differences are ultimately related to the nature of atoms and the bonds that keep them together. Metal atoms are held together by metal bonds, whereas semiconductors are made up of covalent bonds. As for metal bonds, the electrons participating to the bonds (valence electrons) are not localized, but equally distributed in the space around the atoms. This leads to the formation of a population of weakly bonded electrons, which, at room temperature, are free to move inside the solid. In the case of covalent bonds, each valence electron remains localized in a molecular orbital shared only by two adjacent atoms. As a consequence, valence electrons are strongly bonded to their own atoms, so that a non negligible amount of energy is required to set them free to move. In semiconductors, the energy provided by temperature is sufficient to break a limited amount of such bonds (Fig. 1.4).

6

1 The Physical Background

1.3 Electrons in Solids In quantum mechanics, the energy of particles confined in closed spaces is quantized into a finite number of energy levels. The confinement is provided by the forces acting on the particle and the confinement in space is equivalent to a potential well. An introduction to the basic principles of quantum mechanics can be found in the Appendix B. Besides the quantization of energy levels, quantum mechanics introduces additional rules to accommodate particles onto energy levels. Electrons, and in general all the particles with a semi-integer spin, are subject to the Pauli principle, which states that each quantum state can contain only one electron. Since the spin can take two values (s = ± 21 ), no more than two electrons are allowed per each energy level. Thus, N levels can accommodate 2N electrons, and N electrons require the existence of at least N/2 levels. Energy levels are calculated with the Schrödinger equation, once the field electrons are subjected to is known. The solution can be exactly calculated only for simple configurations. In the case of the simplest atom, the hydrogen atom, the equation can be solved under the hypothesis of a still nucleus (the proton). More complex cases require simplifications and numerical solutions (Fig. 1.5). Complex structures, such as molecules, liquids and solids, are aggregates of atoms kept together by interatomic forces. Solids, in particular, can aggregate either into ordered (crystals) or disordered (amorphous) structures. The interactions among atoms involve the electrons of the outer atomic shells. The interaction with these electrons (valence electrons) provides the “glue” that binds the atoms together. Electrons in identical and non-interacting atoms are expected to have the same energy levels (orbitals). When atoms interact, electrons “feel” the presence of the electrons in the adjacent atoms. However, Pauli’s principle does not allow these electrons to occupy the same energy level. Thus, in order to to obey

Fig. 1.5 The hydrogen molecule offers the simplest example of orbitals multiplication. The ground state of the hydrogen atoms splits into two molecular orbitals. The lowest one is the molecular ground level, and the highest one is the excited state. The energies of the molecular orbitals are smaller or greater than those of the atomic orbitals. Electrons differ according to their arrow spin directions (note that to accommodate the two electrons in the ground state, the spin of the electrons is forced to be different). The energy to change the spin is a part of the energy necessary to form the molecule

1.3 Electrons in Solids

7

Fig. 1.6 The figure qualitatively illustrates the process of orbitals multiplication and the merging of distinct energy levels into continuous bands of energies. In the figure, the orbitals are plotted as a function of distance. The distance is related to the strength of the interaction. According to the interatomic distance, excited and ground states may either form a continuous band or split into bands, separated by an energy gap. The first is the case of metals (x M ), the latter is the case of semiconductors (x S )

the Pauli principle, the interacting electrons must slightly change their energy. This leads to a multiplication of energy levels. In practice, the original atomic levels split into a number of orbitals roughly equal to the number of the atoms involved in the interaction. In the case of molecules, the atoms are few, so that the orbital multiplication still leads to discontinuous energy levels. On the contrary, in the case of solids, where the number of atoms is very large (e.g. the density of atoms in silicon is about 5 · 1022 cm−3 ), the multiplication of energy levels leads to a quasi-continuum distribution of energy levels known as energy band. The way energy levels combine together depends on the nature of atoms. In some cases, the separation between orbitals, occurring at the atomic energy levels, is preserved. In other cases, it is canceled, so that all the orbitals merge into continuous band of energies (Fig. 1.6).

1.3.1 Orbitals Splitting and Coupled Oscillators Orbitals splitting is similar to the frequency splitting observed in coupled oscillators. Let us consider, for instance, two identical circuits made of an inductor (L) and a capacitor (C). The resonant frequencies of the isolated circuits are obviously coincident, but in the coupled circuits the resonant frequencies split into two distinct values. One of them is smaller and the other one is larger than the unperturbed resonant frequency. The analogy between particles and oscillators is supported by the fact that, in quantum mechanics, electrons are described by waves and stable orbitals are similar to steady oscillators (Fig. 1.7).

8

1 The Physical Background

Fig. 1.7 The resonance frequency of identical LC circuit splits into two frequencies when the resonators are coupled. The phenomenon is analog to energy levels splitting in interacting atoms

1.3.2 Crystals, Periodic Potentials and Energy Gaps Crystals are a class of solids where atoms are arranged into regular periodic patterns. The form of the crystal is fundamental in determining the properties of the material. Semiconductors are characterized by more complex arrangements with respect to metals. Their crystals are generally made by a tetrahedral structure, where each atom is bond to four surrounding atoms. In pure semiconductors (such as germanium and silicon) the tetrahedra are arranged into a diamond lattice, whereas in composite semiconductors, such as GaAs, each atom of one species form a tetrahedron with the atoms of the other species. Tetrahedra arrangements form a more complex structure known as zincblende. In ideal crystals the pattern of atoms is infinitely repeated. In real crystals, the perfect periodicity of the pattern is disturbed by defects, such as atoms dislocation and impurities (namely, alien atoms included during crystals growth). An obvious deviation from ideality always occurs at the surface, where the periodic pattern stops abruptly. Atomic nuclei are positively charged, so that the regular pattern of atoms in the crystal gives rise to a periodic potential for the electrons. Figure 1.8 shows a simplified view of the potential energy in a monodimensional crystal. The potential energy of the electrons of the last atom decays on the outside. Close to the surface, it gives rise to the so-called surface potential. Space periodicity is defined by the Wiener cell. This is the elementary cell made by the smallest arrangement of atoms that is infinitely repeated. The space periodicity is complemented by the periodic behavior in the reciprocal space. The reciprocal space is the Fourier transform of the real space. The elementary cell in the reciprocal space is the Brillouin zone. In quantum mechanics, particles are represented by wave functions, which are periodic in time and space. The frequency ω defines the periodicity in time and the wavelength λ the space period. The latter is conveniently represented in the reciprocal space by the wavenumber: 2π . k= λ

1.3 Electrons in Solids

9

Fig. 1.8 Periodic electric potential due to a periodic sequence of atoms. The potential energy of the last atom vanishes at infinite distance from the surface. The surface occurs when the density of mass drops to zero

For this reason, the reciprocal space is also called k-space. A traveling particle, such as an electron, corresponds to a wave (ex p(i(ωt − kx)), where frequency and wavenumber are proportional to energy (E = ω) and momentum ( p = k) respectively -where  = h/2π and h is the Planck constant (h ≈ 6.62 · 10−34 J/s). It is important to observe that only ideal sinusoids are characterized by a unique frequency and a unique wavenumber. Actually, real particles are confined both in space (e.g. solids) and time. Thus, instead of single values of frequency and wavenumber, real particles are characterized by distributions of frequencies ( ω) and wavenumbers ( k). Eventually, real particles correspond to the superposition of many waves, known as λ wave-packet. Energy and momentum are connected by the dispersion relationship. This function defines the propagation conditions of the wave packet and depends on the forces acting upon the particle. The equation of motion, in quantum mechanics, is the Schrödinger equation: −

∂ψ(x, t) 2 ∂ 2 ψ(x, t) + V (x)ψ(x, t) = i 2m ∂x2 ∂t

(1.3)

For a free particle, when the potential is null (V = 0), the solution of the Schrödinger equation is a plane wave: ψ(x, t) = A · ex p(i(kx − ωt))

(1.4)

A free electron can assume any energy values. The relationship between energy (ω) and momentum (k) is:  2 k (1.5) E= 2m A different picture emerges when the particle is exposed to a potential. The simplest case is when the particle is confined in an infinite potential well (particle in a box). In this case, the wave vectors depend on the size of the box (L), and the longest possible wavelength is λ = 2L, so that all the waves for which λ = 2L/n, where n is an integer, are allowed. Hence, p = k = 2π/λ = nπ/L. From the energy/momentum relationship (E = ω = 2 k 2 /2m), the energy levels are found:

10

1 The Physical Background

En =

n 2 π 2 2 . 2m L 2

The dispersion relation of Eq. 1.5 still holds, but since a limited set of energy is allowed, the dispersion relation is not continuous. It is worth noting that when the potential is not zero, the total energy of the particle is E = T + V , so that the relationship E = p 2 /2m is no longer valid. The quantity k is not the actual momentum, but rather it still describes the number of peaks of the wave inside the box, namely the wavenumber. In a solid, besides to be confined, the electrons undergo the action of the potential generated by the atoms. In the case of a crystal, this potential is periodic. An important theorem of quantum mechanics (Bloch’s theorem) states that a particle in a periodic potential is described by a periodic wave function. Thus, the properties of the electrons are periodic in the crystal, so that the behavior in the elementary cell can be extended to the entire material. Exact solutions require the knowledge of the actual potential. However, some simple toy-models can elucidate the general properties. A simple case in point is the Kronig-Penney model, where the potential is made up of an infinite sequence of pulses. The Schrödinger equation applied to such a potential results in a dispersion relationship that is discontinuous in energy. In practice, energy gaps appear. The shape of the dispersion relation approximates the free particle around k = 0, and deviates from the free particle as k approaches the border of the elementary cell (Fig. 1.9). Potentials in real solids are obviously more complex than they are in the PenneyKronig model. Indeed, atoms are arranged in tridimensional structures and, in general, more than one atomic species is present. As a consequence, the shape of the bands may be rather complex. Figure 1.10 shows the calculated dispersion relationship of silicon. The continuous branches of these plots form the band energies, so that such a diagram is also known as band diagram. The lower band identifies the ground state, the valence band, whereas the upper band is the excited state, the conduction band. The electrons in the conduction band are quasi-free particles that can be kept in movement by an applied electric field.

Fig. 1.9 In the Kronig-Penney model, the potential (V) is a periodic sequence of pulses in the real space. The solution of the Schrödinger equations results in a discontinuous dispersion interaction E(k). The plot on the right shows the first two bands in the reduced zone schemes

1.3 Electrons in Solids

11

Fig. 1.10 Calculated band diagram of silicon. The coordinates in the k-space are given as Miller indexes. This is a system of coordinates used to indicate the main directions in crystals

The passage from the valence band to the conduction band is possible if the electrons receive, from an external source, an extra amount of energy and momentum necessary to displace one electron from the top of the valence band to the bottom of the conduction band.

1.3.3 Electron Distributions in Energy Levels It is straightforward that without any external inputs of energy, electrons, like any other physical systems, occupy the lowest available states. Temperature is a ubiquitous energy contribution that the electrons, and the whole material, receive. Thermal equilibrium is the condition where all the elements of the system (electrons, atoms, external world) share the same temperature, without any net transfer of energy from one element to another. Let us consider a simplified, though useful for our scopes, band diagram where conduction and valence bands are separated by a gap of energy. In practice, the diagram is restricted to the bottom of the conduction band and the top of the valence band, where electrons and holes are quasi-free particles (E ∝ k 2 ). The distribution of electrons in the available states is ruled by statistical laws. Thus, rather than describing the behavior of each electron, the average behavior of a large population of electrons is considered. It is important to note that statistics is valid for the average. But otherwise, individual electrons can strongly deviate from the collective behavior. The statistical approach is justified by the large number of electrons in the material. . Thus, since each atom In the case of silicon, the density of atoms is 5 · 1022 atoms cm3 of silicon has 4 valence electrons, the density of electrons that must be distributed between the valence and conduction bands is 4N = 2 · 1023 cm−3 .

12

1 The Physical Background

On the other hand, all measurable quantities always involve a large number of electrons. As an example, let us consider a tiny current of 1 pA. This is equivalent to 10−12 C/s across a section of the conductor. Since the electron charge is 1.6 · 10−19 C, 1 pA corresponds to a flow of about 107 electrons per second. The concentration of electrons with an energy between E and E + E is given by the product of the density of available states in the energy intervals and the probability that electrons can actually have energies in that interval: E+ E 

g(E) · f (E)d E.

n=

(1.6)

E

The function g(E) is the density of the allowable states. This quantity depends on the nature of the atoms and the characteristics of their interactions. Namely, on the overlap and multiplication of the atomic orbitals. The function f(E) is the probability function, which depends on the total number of electrons. The probability is a function of temperature. The greater the temperature, the larger the probability to find electrons at high energy. In classical physics, the probability function for non-mutually interacting particles is the Boltzmann partition function. The Boltzmann equation is a direct consequence of the hypothesis of noninteracting particles. Indeed, considering a system made of two states, the probability of occupancy of the total system is the product of the probability of occupancy of each state ( p(1, 2) = p(1) · p(2)), and the total energy is the sum of the energies of the two states (E(1, 2) = E(1) + E(2)). Thus, the probability to find the system at the energy E(1, 2) is: P(E 1 + E 2 ) = P(E 1 ) · P(E 2 ). This condition is fulfilled by the exponential function, where the function of a sum is the product of the functions of the individual arguments (e A+B = e A · e B ). Consequently, the probability function can be written as: P = const · ex p(−β E), where β = 1/k B T and k is the Boltzmann constant (k = 1.38 · 10−23 J K −1 ). Given N = n 1 + n 2 particles distributed in two energy levels (E 1 and E 2 ), the ratio between the number of particles in the two states (n 1 and n 2 ) is:   E2 − E1 n2 . = ex p − n1 kB T

(1.7)

The distribution of particles in the two states is driven by temperature. At T = 0 K, all the particles lie in the lower states; nn 21 = 0, whereas at infinite temperature nn 21 = 1, so that the particles are equally distributed in the two states. The classical statistical theory fails in the case of elementary particles for which quantum concepts hold. In particular, electrons, like any other particles with noninteger spin, obey the Pauli principle of exclusion. Pauli’s principle states that no more than two electrons can be found in the same state, or that, even at T = 0 K, electrons cannot lie in a single ground state, but rather at least N/2 states are necessary

1.3 Electrons in Solids

13

to accomodate N particles. Thus, at T = 0 K, particles pile up the stack of states until a maximum allowable energy level is reached. The statistical law that incorporates the Pauli principle in the Boltzmann equation is the Fermi-Dirac function: f (E) =

1 .  F 1 + ex p E−E kB T

(1.8)

The quantity E F is called Fermi level. It is the highest energy level that can be occupied at T = 0 K. The Fermi level depends on the total number of electrons and it is variable with temperature. It will be shown hereafter that the Fermi level is equivalent to the electrochemical potential of a population of non-interacting charged particles. The shape of the Fermi-Dirac function is qualitatively different in the two cases T = 0 K and T > 0 K . At T = 0 K, f (E ≤ E F ) = 1 and f (E > E F ) = 0. At T > 0K , the function assumes the following values: f (E < E F ) < 1, f (E > E F ) > 0, with the condition: f (E = E F ) = 21 . The Fermi-Dirac function is shown in Fig. 1.11. Noteworthy, when (E − E F )  k B T , the Fermi-Dirac function is approximated by the Boltzmann equation. At room temperature (T = 300 K), k B T is approximately 26 meV. Thus, at room temperature, if E − E F  26 meV , the Fermi- Dirac function can be written as:   E − EF (1.9) f (E > E F ) ≈ ex p − kB T This exponential function happens to be ubiquitous in all of the equations describing the behavior of electronic devices. Its presence reminds of the statistical nature of the principles which the devices are based on.

Fig. 1.11 The Fermi-Dirac function as a function of energy and at different temperatures. As temperature increases, also the probability to find electrons at higher energy increases

14

1 The Physical Background

Fig. 1.12 Comparison between the Fermi-Dirac function and the density of states. Zero energy level is the condition of free electron at infinite distance from any materials. The energy levels of the electrons inside the material are negative (binding energies). The density of states is zero in the band gap

Most of the electric characteristics of materials are a consequence of the position of the Fermi level with respect to the conduction and valence bands. In semiconductors, the Fermi level occurs inside the gap between the valence and conduction bands. Thus, at low temperature, the probability to find electrons in the conduction band is practically zero. Figure 1.12 shows a simplified band diagram, where both the Fermi-Dirac function and the density of states have been superimposed. The density of states is obviously zero in the band gap. The states in their respective bands have a parabolic dependence on energy. The function g(E) will be explicitly calculated hereafter. In pure silicon, the concentration of electrons in the conduction band at room temperature is of the order of 1010 cm−3 . This number determines the small, though non-negligible, conductivity of pure silicon.

1.3.3.1

Electrons in Valence Band: The Concept of Holes

The Fermi-Dirac function describes the probability of finding electrons at a given energy. As temperature increases, the probability to find electrons in the conduction band increases. This means that the electrons engaged in covalent bonds leave their location and can be kept in movement by an applied electric field. Each electron promoted to the conduction band leaves an empty spot in the valence band due to the nature of the covalent bond. This vacancy is localized in energy and space. The total charge surrounding an unperturbed atom is zero, so that when an electron leaves the atom, a fixed positive charge is left behind. Under the influence of an electric field, the electrons engaged in adjacent bonds can be displaced to occupy the empty position. This movement can be represented either considering the displacement of electrons (negative charges) or the displacement of the empty, positively charged, locations. The empty spots are called holes. Indeed, they carry a positive charge, whose value is the absolute value of the electron charge.

1.3 Electrons in Solids

15

Actually, charge transport in semiconductors involves electrons at two energy intervals, those in the conduction band and those in the valence band. In order to distinguish them, it is convenient to introduce the concept of holes and to treat the electron transport in the valence band as the transport of complementary positively charged particles. Obviously, the concept of holes is only valid in semiconductors. When a semiconductor is contacted by metal electrodes (as it always happens, on a practical level), the holes are either annihilated or created at the contact (the correct terms are “recombined” and “generated”) by the electrons of the metal. This ensures that only electrons can circulate in the metallic wire, whereas in the semiconductor the current can be due to both electrons and holes.

1.3.3.2

The Effective Mass and the Free Electron Approximation

Electrons in crystals are subjected to periodic potentials, whose consequence is the dispersion relation between energy and momentum. Consequently, when we study the behavior of an electron inside a crystal, for example the motion of an electron under an applied electric field, it is necessary to include also the internal periodic potentials in the equation of motion. On the other hand, the shape of the dispersion relation E(k) at the bottom of the conduction band and at the top of the valence band is very close to the parabolic behavior, which is typical of free electrons. A rescaling of the properties of the electron could then allow us to treat the electrons and the holes in the crystal as free particles. This approximation is implemented through the concept of effective mass (m ∗ ). This is a very convenient way to embed the potential that keeps the electron in the crystal into the amount of mass, so as to apply the free particle equation of motion to the electron. This gives rise to an abstract entity (a quasi-particle) with charge q and mass m ∗ . Such a quasi-particle is the charge carrier in the semiconductor, so that we will continue to call it electron. The effective mass of the electrons in the crystal is obviously different from the rest mass of actual electrons (m 0 = 9 · 10−31 kg). To calculate the effective mass, it is necessary to consider that in quantum mechanics particles are described by waves characterized by wavelength (λ), angular frequency (ω = 2π f ) and wavenumber (k = 2π/λ). Angular frequency and wavenumber are proportional to energy and momentum respectively (E = ω; p = k). The velocity of propagation of a pure sinusoid is known as phase velocity (v ph ). Considering the relationship between wavelength and frequency, the phase velocity is: 2π 2π ω v → =v → v ph = . (1.10) λ= f k ω k Sinusoids are analytical functions existing from t = −∞ to t = +∞ and moving anywhere in space. Of course, a real particle can be observed only for a limited amount of time, when it is confined in a limited amount of space (e.g. the solid). According

16

1 The Physical Background

to the Fourier transform theorem, waves limited in space and time correspond to a distribution of frequencies and wavenumbers. Thus, instead of a single pure wave, real particles correspond to a superposition of waves: the wave packet. The velocity of propagation of the wave packet is the group velocity (vg ), defined as: vg =

dω . dk

(1.11)

The mass of the particle, defined by a dispersion relation E(k), can be calculated by the laws of dynamics. The force acting on the particle is: F=

dp dk = . dt dt

Force is mass times acceleration, so that: ma = m Thus, reminding that dω = 

dE , 

we can write:

d dω m d dE dk =m = . dt dt dk  dt dk

Multiplying the last expression by 

dv d dω =m . dt dt dk

dk dk

we get:

m d d E dk m d 2 E dk dk = = dt  dt dk dk  dk 2 dt

from which the definition of effective mass is obtained: m∗ =

2 d2 E dk 2

.

(1.12)

The effective mass is inversely proportional to the second derivative of the dispersion relation, namely to the inverse of the curvature of the band. As previously discussed, the bottom of the conduction band and the top of the valence band have a parabolic shape, so that E ∝ k 2 and in this situation the effective mass is constant. The effective mass is usually smaller than the rest mass (Table 1.1). This indicates that the electrons inside the crystal have less inertia than the free electrons. The different effective mass of electrons and holes is a consequence of the separated conditions of motion. It is important to remind that holes are actually electrons whose motion looks like a series of leaps from one atom to another. It is interesting to observe that the concept of holes is a consequence of the definition of effective mass. Indeed, since the actual shapes of the conduction and valence bands are characterized by opposite curvatures (see Fig. 1.10), the effective mass is

1.3 Electrons in Solids

17

Table 1.1 Effective mass of electrons and holes for typical semiconductors Semiconductor Electrons effective mass Holes effective mass Silicon Germanium Gallium arsenide

0.26 0.12 0.068

0.38 0.3 0.5

positive in the conduction band and negative in the valence band. The excitation of electrons in the conduction bands leaves empty states in the valence band, so that charge motion is possible. But since the effective mass is negative, motion occurs in the opposite direction to that of electrons in the conduction band. The physical absurdity of a negative mass is removed by introducing a positive charge for the mobile particles in the valence band. Eventually, the concept of hole emerges as a consequence of the band diagram, so that it is a direct property of crystalline structures.

1.3.4 The Band Diagram The band diagram, even in its simplified form, is the fundamental tool to interpret the electric properties of materials and the behavior of electronic devices. The diagram fixes the relative position of the conduction band, the valence band and the Fermi level. Since the condition E = 0 is not accessible, in order to define the energy values it is necessary to introduce a reference value that can be actually observed. A convenient value, to this regard, is the potential energy of a free electron placed immediately outside the material. This level is the vacuum level and it corresponds to the surface potential of a given solid. When more solids are kept in contact, for instance in junctions, the difference in surface potentials corresponds to the built-in potential. In the case of semiconductors, the band diagram can be drawn considering three fundamental experimental quantities: the affinity, the energy gap and the work function. As for the metal, since there is no band gap, the only meaningful quantity is the work function. Figure 1.13 shows the typical band diagram of semiconductors and metals.

1.3.4.1

Electrons Affinity

Electrons affinity (qχ ) is a material property corresponding to the largest energy necessary to displace an electron from the vacuum level to the inside of the material. In the band diagram qχ is the distance between the vacuum level and the conduction band. Under the assumption that the concentration of electrons in the conduction

18

1 The Physical Background

Fig. 1.13 Simplified band diagram of a semiconductor and a metal. For each material the vacuum level assumes a different value with respect to the absolute energy ladder. Note that, in semiconductors, the affinity and work function are about four times the energy gap. For sake of simplicity, energy gaps are usually out of scale in band diagrams. The interrupted energy axis is introduced to facilitate the comprehension of the diagram Table 1.2 Electron affinities of common semiconductors Semiconductor Affinity (eV) Silicon Germanium Gallium arsenide

4.05 4.00 4.07

band is much smaller than the density of available states, there are always available states at E C , so that the affinity does not depend on the density of electrons. The affinity can be measured with a variety of sophisticated experimental techniques. Inverse photoemission is one of them. With this technique, electrons are delivered at very low kinetic energy towards the surface. Due to the low kinetic energy, electrons are absorbed in highest energy levels close to the vacuum level. From this level they may decade towards the lowest allowable state lying at the bottom of the conduction band. The transition can occur via a number of intermediate steps or via a single step. Each transition may correspond to the emission of a photon. In the case of a single transition, the photon with the largest energy is emitted. This energy is approximately equivalent to the distance between the vacuum level and the bottom of the conduction band, namely the affinity (Table 1.2).

1.3 Electrons in Solids

1.3.4.2

19

Energy Gap

The energy gap is the difference between the bottom of the conduction band and the top of the valence band. The energy gap corresponds to the energy necessary to displace an electron from the valence band to the conduction band. This energy can be provided by an external source, such as a photon releasing its energy to an electron of the valence band. When the energy of the photon is larger than or equal to the energy gap, an increase in conductivity is observed. This phenomenon is known as photoconductivity. In the case of silicon, the energy gap is about 1.1 eV, which corresponds (E gap = hc/λ) to a photon with a wavelength of 1.11 µm. Photons with a wavelength shorter than 1.1 µm can elicit photoconductivity. In particular, photons in the visible range (λ = 400–700 nm) excite the photoconductivity in silicon: this property led to the development of digital cameras. It is worth noting that the energy gap is slightly dependent on temperature, according to the following equation: E gap = E gap0 −

αT 2 . T +β

(1.13)

As for silicon, E gap0 = 1.166 eV, α = 0.473 meV/K and β = 636 K.

1.3.4.3

Work Function

The work function corresponds to the energy necessary to relocate an electron from the inside of a material up to the vacuum level with null kinetic energy. The vacuum level corresponds to that particular energy level at which an electron can be considered as being on the outside of the material. Strictly speaking, the electrostatic potential decays as 1/x and becomes null at infinite distance. In other words, the presence of a material should influence the whole universe, so that locations where the electrons can be considered as free particle should not exist. However, the universe is populated with countless materials, and the concept of vacuum level is important to define one material with respect to the others. To clarify this concept, let us consider two materials at which a voltage difference is applied (e.g. the plates of a capacitor). For sake of simplicity, let us consider the material 1 kept at ground and the material 2 at a potential V A . In this way, the potential energy of the electrons in material 2 is shifted downwards by a quantity q V A (consider that the charge of electrons is negative). Figure 1.14 shows the behavior of the electrostatic potential. Between the materials the electrostatic potential is linear. The energy in proximity of material 1 defines the vacuum energy level of material 2. As shown in Fig. 1.13, the work function fixes the position of the Fermi level with respect to the vacuum level. The relationship between the Fermi level and the work function can be obtained from the following thermodynamical considerations (Fig. 1.15).

20

1 The Physical Background

Fig. 1.14 Behavior of the potential energy between the surfaces of two materials in vacuum. Material 1 is kept at ground, and material 2 at a potential V A . The electrostatic potential energy declines from the last atom towards infinity. Between the materials, the potential shows a linear behavior with a net potential energy drop equal to q V A . Thus, outside the materials the potential energy change its course because of the applied voltage. The energy level at which the potential deviates from the 1/x behavior can be considered as the vacuum energy level

Fig. 1.15 Before the extraction of one particle, the gas has a free energy G N . After the extraction, the number of particles decreases by a unit

Let us suppose to remove one particle from a gas of N particles. The total energy of the particles before and after the extractions are: E be f or e = G N and E a f ter = G N −1 + E vac where G N and G N −1 are the Gibbs free energies of a gas of N and N-1 particles calculated at constant pressure, temperature and volume. The work function (qφ) is defined as the change in energy necessary for the extraction process:

1.3 Electrons in Solids

21

q = E a f ter − E be f or e = E vac + G N −1 − G N G N − G N −1 ∂η = E vac − = E vac − N − (N − 1) ∂N

(1.14)

where η is the chemical potential at constant pressure, temperature, and volume. If the particle is charged, then the electric potential must be added and the chemical potential is replaced with the electrochemical potential: η = η0 − q V.

(1.15)

Hence, the work function is the difference between the vacuum level and the electrochemical potential. The electrochemical potential describes the collective energy of an ensemble of non-interacting charged particles. For an ensemble of electrons, the electrochemical potential is replaced by the Fermi level. This change is required by the Pauli principle. The concept of collective energy is replaced with the energy level, whose probability of occurrence is 21 . Some additional arguments about the relationship between the Fermi level and the electrochemical potential will be provided thereafter. The work function is a statistical quantity. Hence, even if a single electron may be extracted from any levels, the average energy of extraction is the difference between the vacuum level and the Fermi level. More interesting is the observation that in semiconductors the Fermi level lies in the energy gap, so that there are no electrons at the Fermi level. However, since the probability to find electrons at the Fermi level is 21 , the average extraction energy is still the work function. Strictly speaking, in semiconductors the definition does not hold at 0 K, but any applied energies rise the temperature above the absolute zero, so that the definition is always valid. There are several experimental methods to measure the work function. Among them, it is worth mentioning those based on the thermionic effect, discussed in the introduction, and the photoelectric effect. The latter consists in measuring the current of electrons released from a material shined with a radiation of variable wavelength. Since the Fermi level depends on the density of quasi-free electrons in the material, the value of the work function in a material is constant only in the case of metals. In semiconductors, the concentration of electrons can be varied by a technological procedure known as doping, so that even the work function is a variable depending on doping. When the Fermi level lies in the band gap, the value of the work function is in the range of qχ and qχ + E gap . In Table 1.4 the work function of some metals used in microelectronics technology is given. Figure 1.16 shows a practical setup for the measurement of the work function. Photons are the most practical solution to provide the energy for electrons emission. However, it is difficult to finely tune the wavelength of light to achieve a photon with hν = q. Furthermore, by definition, the work function corresponds to the energy to move the electron from the inside to the outside at null kinetic energy. Thus, even though it was possible to provide exactly the right amount of energy, the electron

22

1 The Physical Background

Table 1.3 Energy gaps at room temperature of typical semiconductors Semiconductor Energy gap (eV) Silicon Germanium Gallium arsenide

1.12 0.67 1.42

Table 1.4 Typical work function values for some metals Metal Work function (eV) Silver Gold Copper Titanium Aluminum

4.26–4.74 4.95–5.47 4.53–5.10 4.33 4.06–4.26

The work function depends on the surface electronic states, which can be arranged in a variety of structures, so that in some cases an interval, rather than a single value, is found

Fig. 1.16 Schematic measurement setup of the work function. A monochromatic flux of photons with energy hν > q extracts electrons from the material under investigation. The kinetic energy E kin = hν − q drags the electrons towards an electrode and a current is measured. A variable voltage source is applied in series to compensate the kinetic energy. In conditions of null current the work function corresponds to the difference between the energy of the photons and the applied potential energy

could not be observed. The practical method consists in shining the material with photons with an energy greater than the work function. The excess energy results in a kinetic energy that moves the electrons towards another electrode, so that a current is measured. A variable voltage source is then applied in order to cancel the current. In this condition, the difference between the photon energy and the applied voltage corresponds to the work function of the material.

1.4 The Statistics of Electrons and Holes

23

1.4 The Statistics of Electrons and Holes The electron and hole concentrations in the conduction and valence bands depends on the availability of states and the Fermi-Dirac function. Being the holes a lack of electrons, the probability to find a hole at a certain amount of energy is complementary to the probability to find an electron at the same energy. Thus, the distribution function for holes is 1 − f F D . Eventually, the electron and hole concentrations at the energy E inside the respective bands are (Fig. 1.17): E n=

gc (E) · f F D (E)d E EC

(1.16)

E V gv (E) · (1 − f F D (E))d E.

p= E

Fig. 1.17 Probability functions, density of states and electron and hole concentrations. At room temperature, most of the electrons and holes occupies the energy levels at the bottom of the conduction band and the top of the valence band respectively

24

1 The Physical Background

The total concentration of electrons and holes in their respective band is: ∞ gc (E) · f F D (E)d E

n= EC

(1.17)

E V gv (E) · (1 − f F D (E))d E.

p= ∞

Let us consider the case of the conduction band. In normal conditions, the Fermi level lies in the band gap, which is much larger than 26 meV, so that E − E F  k B T is likely to be valid and the Fermi-Dirac function can be replaced with its first order approximation:   ∞ E − EF d E. (1.18) n = gc (E) · ex p − kB T EC

The numerator of the argument of the exponential can be split into two parts: E − E F = (E C − E F ) + (E − E C ), so that:  ∞    EC − E F E − EC d E. n = ex p − gc (E) · ex p − kB T kB T

(1.19)

EC

The integral now provides a constant value that is independent of the Fermi level. This is indicated by NC , which corresponds to the total density of states in the conduction band. Note that, due to the fast decay of the exponential function, infinity can be conveniently used as the upper limit of integration. The same approximation and calculation hold for the holes in the valence band. The integral in this case gives the total density of states in the valence band, which is indicated as N V . Thus, the total concentration of electrons and holes in their respective bands can be written as:   EC − E F ; n = NC · ex p − kB T

  E F − EV p = N V · ex p − . kB T

(1.20)

The density of states is a fundamental quantity of the material. In the next section, a calculation of the density of states will be outlined.

1.4.1 The Density of States The density of states is the density per unit of volume and per unit of energy of the solutions of the Schrödinger equation. For our scope, the semiconductor can be modeled as an infinite potential well applied to free particles (conveniently called

1.4 The Statistics of Electrons and Holes

25

electrons) with charge q and mass m*. In practice, once the internal potentials are included in the effective mass, the electrons in a solid correspond to the particle-ina-box model. The macroscopic shape of the material does not affect the density of states. Thus, for sake of simplicity, let us consider a cube whose side is L. The free-electron conditions means that the potential inside the well is null (V (x) = 0). Thus, the solution of the Schrödinger equation can be written as a superposition of sine and cosine functions: ψ = A · sin(k x x) + B · cos(k x x).

(1.21)

As discussed above, the boundary conditions are fixed by the potential well, so that  = 0 at the borders of the well: x = 0 and x = L. As a consequence B = 0, and the possible values of k x are: kx =

nπ , n = 1, 2, 3, ... L

(1.22)

The previous analysis must be repeated for the other two dimensions (y and z). Thus, each solution corresponds to a cube in the k-space of volume π/L. The total number of solutions are characterized by positive values of k x , k y , and k z . A modulus k of the wavevector is calculated considering one eighth of the volume of a sphere of radius k divided by the volume of the single solution (π/L): N =2

1 8

 3 L 4 3 πk . π 3

(1.23)

The factor 2 gives account of the fact that each solution accommodates two electrons (two opposite spins). The density per energy unit is obtained using the chain rule for the derivative: d N dk dN = = dE dk d E

 3 L dk . π k2 π dE

As for the dispersion relation of the free particle (E = m∗ dk = 2 ; k= dE  k

2 k 2 ), 2m∗

√ 2 m∗ E . 

(1.24) we obtain: (1.25)

Thus, the density of states per unit of energy, for E > 0, is: √ √ 8π 2 1 dN 3/2 = m ∗ E. g(E) = 3 L dE h3

(1.26)

The density of states is zero at the bottom of the well and even for negative values.

26

1 The Physical Background

As for the electrons in the conduction band, the minimum of energy corresponds to the bottom of the conduction band (E C ). Consequently, the density of states in the conduction band must be written by scaling the energy to the bottom of the conduction band: √  8π 2 3/2 m ∗ E − EC . (1.27) g(E) = h3 An analog calculation leads to the density of states for the holes in the valence band. The integral in Eq. 1.16 and its counterpart for the valence band provides the values for NC and N V :   2π m ∗n k B T 3/2 NC = 2 h2 (1.28)   2π m ∗p k B T 3/2 NV = 2 . h2 NC and N V are quantitatively different because the effective masses of electrons and holes are different. The effective masses previously introduced have been calculated considering the dynamic properties of electrons and holes. The value of the effective mass to be used to calculate the density of states is different. The values for silicon are m ∗n = 1.08 ; m ∗p = 0.81. NC and N V in silicon and at room temperature are: NC ≈ 2.8 · 1019 cm−3 N V ≈ 1.04 · 1019 cm−3 .

(1.29)

These values are almost similar. It is important to remark that all these quantities depend on temperature. Equation 1.20 is among the fundamental tools to study the behavior of semiconductors and their junctions.

1.4.2 The Intrinsic Fermi Level Equation 1.20 connect the density of holes and electrons to the distance of the conduction and valence bands from the Fermi level. The energy of the Fermi level is not experimentally accessible. Indeed, the measure of the work function fixes the Fermi level with respect to the vacuum level. On the other hand, the position of the Fermi level can be calculated once the concentration of holes and electrons is known, as in a doped material. However, the position of the Fermi level with respect to the bands can be calculated for an intrinsic semiconductor. This is a pure material, where electrons and holes are only generated by the ionization of the atoms of the semiconductor. Thus, for each electron in the conduction band there is a hole in the valence band.

1.4 The Statistics of Electrons and Holes

27

As a consequence, the definition of an intrinsic semiconductor is n i = pi , where the subscript i indicates the intrinsic condition. Using the Eq. 1.20 we get:     E C − E Fi E Fi − E V = N V · ex p − NC · ex p − kB T kB T   E Fi − E V NV E C − E Fi + = ex p − kB T kB T NC   NV 2E Fi − E C + E V = ex p − . kB T NC

(1.30)

Hence, the intrinsic Fermi level (E f i ) is: E Fi =

kB T EC + E V + ln 2 2



NV NC

 .

(1.31)

The first term is the center of the band gap, while the second one depends on the ratio between the effective masses. In silicon and at room temperature (T = 300 K) this additive quantity is approximately −13 meV. Since the whole band gap of silicon is 1.12 eV, with a relative error of about 2%, the Fermi level of intrinsic silicon lies at the centre of the band gap. This conclusion applies to most of semiconductors. Consequently, the work function of the intrinsic silicon is: EC + E V = 4.61 eV. (1.32) qi = qχ + 2

1.4.3 Doping The structure of real crystals is far from being perfect. Rather, real crystals contain a number of defects whose existence has a strong impact on the electronic properties of semiconductors. The most important defects are impurities and vacancies. Impurities are involved in the processes of charge transport, whereas vacancies are at the basis of an important technological feature of semiconductors. Crystals can be considered as an abstract pattern defining the location of each atom of the solid. A vacancy occurs when one of these location is actually empty. Around a vacancy, the distances among atoms get altered, so that the lattice is deformed. One of the consequences of the larger distance among adjacent atoms is that the binding forces are weaker. Thus, around a vacancy, the crystal is less rigid and the missing position can be filled with an atom added by purpose (Fig. 1.8). This opportunity is exploited in a process called doping, which is aimed at altering the balance between electrons and holes and at increasing conductivity. A pristine semiconductor is doped via the implantation of impurity atoms. This operation can

28

1 The Physical Background

take place via a physically impacting implantation of accelerated ions, followed by a thermal diffusion of impurities from the surface down to the internal region of the material. Since defects are usually uniformly distributed, then also dopant impurities might be uniformly distributed, at least after a distance from the surface. There are two important categories of impurities: those favoring the increase in electrons concentration and those increasing the concentration of holes. In silicon. these conditions are fulfilled by pentavalent and trivalent atoms respectively (Fig. 1.19). In silicon, the pentavalent atoms (elements of the V group of the periodic table, such as phosphorous and arsenic) are called N-type dopants or donors. When a pentavalent atom replaces the position of a silicon atom in the crystal, only four out of the five available valence electrons are engaged in covalent bonds with the adjacent silicon atoms. The fifth electron remains bonded to its own atom, but at an energy level rather close to the conduction band of the crystal. In practice, the distance between the bottom of the conduction band and the energy level of the unpaired electron is about the 10% of the energy gap. Due to thermal energy, a conspicuous portion of these

Fig. 1.18 In perfect crystals (left), atoms are arranged in regular lattices and all the interactions have the same magnitude. In case of vacancies (right), the atoms around the empty position are displaced, the interatomic distances increase, and the binding energies are less intense

Fig. 1.19 Pictorial representation of the effect of doping with a pentavalent (left) and a trivalent (right) atom. As for the pentavalent case, one of the electrons of the dopant atom is not engaged in a covalent bond, so that it can be promoted to the conduction band. In the case of a trivalent doping, one of the adjacent silicon atoms remains with an unpaired electron that might be filled with an electron from a nearby silicon atom, giving rise to a hole

1.4 The Statistics of Electrons and Holes

29

electrons can leave phosphorous and populate the conduction band. However, even if the difference in energy is small, according to the Boltzmann probability function only a fraction of the phosphorous levels is actually transferred to the conduction band. The loss of one electron changes the total charge around the phosphorous atom, which, instead of being neutral, becomes positively charged. This positive charge is fixed and is not displaced by any applied electric fields. Furthermore, it lies well above the valence band, so that it is impossible for the electrons of the valence band to occupy this state. Eventually, the dopant atom donates an electron to the conduction band, but it does not give rise to any holes. Given a density N D of donors, the statistics allows to calculate the percentage of effectively ionized donors. Let E D be the energy level of the donor states and let us consider that both E C − E F and E D − E F are larger than k B T . Thus, the FermiDirac equation can be replaced with the Boltzmann equation. The concentration of the electrons in the conductance band is dominated by the ionized donors n = F ). In the same way, the concentration of the electrons that occupy NC ex p(− ECk −E BT F the N D donor levels (n D ) is n D = N D ex p(− E Pk −E ). Of course, N D = n + n D , so BT that the percentage of ionized donors is: F NC ex p(− ECk −E ) n BT = . E −E C F F n + nD NC ex p(− k B T ) + N D ex p(− E Dk B−E ) T

(1.33)

F Dividing by ex p( −E ), we get the following expression: kB T

nD = n + nD 1+

1 NC D ex p(− ECk−E ) ND BT

.

(1.34)

The above equation is qualitatively correct, even though a little correction due to the degeneracy of the donor levels should be introduced. The energy level E D depends on the nature of the dopant atom. In any case, the fraction of ionized donors depends on the doping concentration: it decreases as the doping concentration increases. The relationship between the percentage of ionized donors and the concentration of donors is shown in Fig. 1.20. In the case of phosphorous in silicon, the energy level of the donor state occurs at 0.045 eV below the bottom of the conduction band. At room temperature, and for a doping of 1018 cm−3 , roughly the 71% of donors are actually ionized. However, since the exact number of donors is unknown, and the order of magnitude of ionized donors is equal to the order of magnitude of the total number of donors, it is customary to assume that N D donors give rise to N D electrons in the conduction band. These are equilibrium average values and, on a practical level, donors continuously lose and acquire electrons. An opposite behavior is obtained using trivalent atoms (elements of the group III of the periodic table, such as boron and aluminum). Trivalent atoms in silicon are P-

30

1 The Physical Background

Fig. 1.20 Percentage of ionized donors as a function of doping concentration. Calculations are relative to phosphorous in silicon at room temperature. A factor 2 must be introduced in Eq. 1.34 to take into account the degeneracy factor of the donor level. As the doping concentration increases, the distance between the conduction band and the Fermi level decreases, so that the Boltzmann approximation tends to lose its validity

type dopants or acceptors. A boron atom replacing a silicon in the crystal leaves one of the adjacent silicon atoms with an unpaired electron. Since the unpaired electron is not engaged in a covalent bond, its energy level is slightly higher than the top of the valence band. Namely, the electron is less bound to its atom. The magnitude of the distance is comparable with the one observed between pentavalent electrons and conduction band. It is useful to remind that the energy of paired electrons is lower (more negative) than that of unpaired electrons, so that the minimization of this energy leads to the stability of the chemical bonds. The statistics of donors also applies to acceptor states, so that the plot in Fig. 1.20 is also valid for P-type doping. As a consequence of the distribution of electrons in the acceptor states, the total charge around the boron atom, instead of being neutral, becomes negative. This negative charge is fixed and cannot be removed by any applied electric fields. The electron that moves from an adjacent silicon atom to fill the octet of a silicon adjacent to the boron leaves a hole that may be occupied by another electron. The hole moves through the crystal, whereas boron remains negatively charged. The statistics of donors is also valid for acceptors, so that we consider that N A acceptors give rise to N A holes in the valence band. It is important to note that in both cases the creation of a free mobile charge is not compensated by the creation of a mobile charge of opposite sign (as it happens in intrinsic semiconductors). On the contrary, the countercharge is a fixed charge and not a mobile one. Eventually, doping results in an increase in one of the two charge

1.4 The Statistics of Electrons and Holes Table 1.5 Charges produced by doping Type Mobile charges N type P type

Electrons (negative) Holes (positive)

31

Fixed charges Donors (positive) Acceptors (negative)

carriers and, in order to respect the neutrality of the material, in a concentration of fixed charges (Table 1.5). The laws ruling the statistics of electrons and semiconductors are still valid for a doped material. As a consequence, in order to accommodate the increase in electron or hole concentrations with Eq. 1.20, it is necessary that the Fermi level changes its value. If n increases, then the distance between the conduction band and the Fermi level must decrease. On the other hand, if p increases, then the distance between the Fermi level and the valence band decreases. Eventually, the position of the Fermi level defines the doping. It has been shown in the previous section that the Fermi level of an intrinsic semiconductor is approximately at the center of the band gap. For N type materials the Fermi level is close to the conduction band. On the contrary, if the Fermi level is close to the valence band, then the material is P-type (Fig. 1.21). Typical concentrations of dopants in silicon are in the range 1015 –1018 cm−3 . This value must be compared with the concentration of silicon atoms, which is about 1023 cm−3 . Thus, the corresponding doping is of the order of one atom of impurity for every 10 million silicon atoms. Strikingly, this tiny quantity is sufficient to change the electric characteristics. However, it leaves untouched the other parameters of the material, such as energy gap, electron affinity, density, and so on. At such level of concentration, the dopant atoms are sparse within the material. Consequently, there is no interaction among the dopant atoms, so that their atomic orbitals do not degenerate into bands. For this reason, the dopant energy level mentioned before can still be considered as a single energy level, rather than a band.

1.4.3.1

Mass Action Law

According to Eq. 1.20, the Fermi level defines the electron and hole concentrations. In a doped material, the concentration of only one out of the two species increases. Since the material is in equilibrium, this cannot leave the other one unaffected. The product of the concentrations of the two species is ruled by the action mass law. For an intrinsic semiconductor, this product is n i pi = n i2 . Anyway, it can be calculated under any conditions using Eq. 1.20:

32

1 The Physical Background

    EC − E F E F − EV · N V ex p − np = NC ex p − kB T kB T   −E C + E F − E F + E V = NC N V ex p kB T

(1.35)

Thus, the product does not depend on the Fermi level -that is, it does not depend on doping:   E gap 2 np = n i = NC N V ex p − . (1.36) kB T The product np is maintained constant under any kind of doping and depends, besides temperature, on the energy gap. Thus, np = n i pi . 20 cm−6 . Consequently, the intrinsic conIn silicon, at T = 300 K, n i2 ≈ 1.45 · 10 centration of charge carriers is n i = pi = n i2 ≈ 1010 cm−3 . In the case of N-type doping, the concentration of electrons is the concentration of dopants and the concentration of holes is calculated from the mass action law: n = ND ; p =

n i2 . ND

(1.37)

Hence, if N D = 1017 cm−3 , n = 1017 cm−3 and p = 1020 /1017 = 103 cm−3 . This great inequality in concentrations is a striking consequence of doping, whose major effect is to discriminate the charge carriers in majority and minority charges. The doping concentration determines the position of the Fermi level, and thus the work function. In a N-type semiconductor we can write the concentration of electrons with respect to the intrinsic concentration as:     EC − E F EC − Ei − E F + Ei = NC exp − ; (1.38) n = N D = NC exp − kB T kB T n = N D = NC

    E f − Ei EC − Ei exp exp − kB T kB T

where E i is the Fermi level of the intrinsic semiconductor. Finally:

Fig. 1.21 The relative position of the Fermi level with respect to the conduction and valence band energies manifests the kind of doping of the semiconductor

(1.39)

1.5 Charge Transport: The Drift-Diffusion Model

 N D = n i ex p

33

E f − Ei kB T

 .

(1.40)

The concentration of electrons thus depends on the distance between the Fermi level and the intrinsic Fermi level:   ND (1.41) E F = E i + k B T ln ni If N D = 1017 cm−3 , then the distance between the Fermi level and its intrinsic value is 0.40 meV. Hence, the work function of the doped semiconductor is:  q = E vac − E F = E vac − E i − k B T ln

ND ni



 = qi − k B T ln

ND ni

 (1.42)

where qi is the work function of the intrinsic semiconductor. As for silicon: qi = 4.61 eV. Similar equations hold in the case of P-type doping simply replacing E C − E F with E F − E V . Finally, it is important to remind the hypothesis on which these calculations have been performed. In particular they are based on the approximation of the Fermi-Dirac function with the Boltzmann probability function. The approximation is valid if the distance between the Fermi level and either the conduction band or the valence band is of the order of 2–3 times k B T . When the doping is large, the Fermi level lies too close to the conduction band, so that the approximation loses its validity. In silicon, this happens when N D and N A are greater than 1019 cm−3 . Beyond this value, the Fermi level invades the bands and the semiconductor behaves like a metal. Such a semiconductor is said to be degenerate.

1.5 Charge Transport: The Drift-Diffusion Model In the first part of this chapter the properties of electrons and holes at the equilibrium have been illustrated. The equilibrium condition defines the response of the material to the perturbation due to the applied bias. The most important non-equilibrium phenomenon is charge transport, where a net amount of electric charges are kept in movement. In this section, the charge transport phenomena are described in the framework of a classic approach. In this context, the current flowing through a semiconductor is the sum of two components: the drift current and the diffusion current. This approach is valid when the dimensions of the material are larger than few nanometers. A more precise validity requirement about the classical model will be provided thereafter.

34

1 The Physical Background

In order to study charge motion, it is necessary to consider the forces acting on charges, namely the electric force F = qE, where E is the electric field, and their effects on charge acceleration: F = ma. Thus, the relationship between electric field and current depends on two fundamental quantities: charge and mass. In the first part of this chapter it has been shown that, in semiconductors, there are two kinds of mobile particles: electrons and holes. The charge of electrons is negative and the charge of holes is positive. Also, the absolute value is the electron charge (q = 1.6 · 10−19 C). The masses of the particles are the effective masses. Electric current is defined as the amount of charge flowing across a section of the conductor per unit of time (I = Q/T ). It is more convenient to consider current density, which is defined independently of the section of the system. In this way, a monodimensional description of the devices is possible: j=

Q TA



 C . s m2

(1.43)

The current (measured in Amperes) is obtained by simply multiplying the current density by the area of the section. Current density is the macroscopic manifestation of the movement of individual charges. The definition of j can be achieved by considering the instantaneous work done by the applied electric field on a charge: d L = Fd x = qEd x. Replacing the V V ), the work done is d L = q w d x, where w is the length of field with voltage (E = w the conductor. At the equilibrium, this work is equal to the energy dissipated by the current itself Pdt, where P is the electric power (P = V I ). From this equality, the definition of current is obtained: q

q dx qv V d x = i V dt → i = = . w w dt w

(1.44)

The above calculation is derived from a fundamental theorem (the Ramo Shockley theorem), which establishes the relationship between the microscopic motion of a charge and the observed current. The above definition describes the current produced by a single particle. Real currents are due to the motion of a density of particles. Thus, the moving charge must be replaced with qn Aw, namely the product of the density of the charges (nq) and the volume of the conductor. In a semiconductor, where two charge carriers exist, the total amount of current is the sum of the currents of electrons and holes: jn = qnvn ;

j p = qpv p

(1.45)

where q is the elementary charge, n and p are the densities of electrons and holes, and vn and v p are the velocities of electrons and holes respectively.

1.5 Charge Transport: The Drift-Diffusion Model

35

1.5.1 Thermal Velocity The temperature of a gas of non-interacting particles is proportional to the average kinetic energy of the particles. This implies that particles are kept in motion even in absence of an external force. The equipartition theorem assigns a kinetic energy equal to k B T /2 to each degree of freedom. The quantity proportional to temperature is the kinetic energy, namely the square of velocity. Thus, thermal motion is isotropic, so that the average position of the particles does not change with time. To this regard, it is important to remind that velocity is a vector, whose average value can be zero even if speed (the magnitude of velocity) is different from zero: 1 ∗ 2 3 m v = k B T. 2 n th 2

(1.46)

In silicon, the effective mass of electrons is m ∗n = 0.26 · m 0 , so that the thermal . velocity at room temperature (T = 300 K) is of the order of 107 cm s The effect of temperature on the random motion of particles is defined by the Maxwell-Boltzmann distribution, which defines the distribution of the velocity of the particles held at the same temperature, namely with the same energy. The above calculated speed is the most probable speed, which is different from the mean speed given by: 8k B T (1.47) vmean = πm

1.5.2 Drift Current A net displacement of charges is achieved by the application of a voltage drop across the material. The applied voltage changes the potential energy of the electrons inside the material. The potential energy corresponds to the energy of the bottom of the conduction band (for the electrons) and the top of the valence band (for the holes). Thus, in the presence of an applied voltage, the band diagram is altered (Fig. 1.22). According to quantum mechanics, an electron kept in movement in a perfect periodic potential should not undergo any scattering processes. In this condition, the electron does not lose the acquired kinetic energy, so that the velocity, instead of reaching a stable value, grows up to a maximum value that ultimately depends on the length of the conductor. Such a motion, without scatters and energy dissipation, is said ballistic. Actually, in a real material atoms are not arranged in perfect regular lattices because of alterations in their positions and nature, collectively known as lattice defects. The major sources of defects are impurities (doping among them) and vacancies (Fig. 1.23).

36

1 The Physical Background

Fig. 1.22 Applied voltages alter the potential energy of electrons ( E C = −q V A ) and holes ( E V = +q V A ). Electrons and holes move due to the electric field. The energies above (or below, for holes) the bands are the acquired kinetic energies

Fig. 1.23 Atoms different from those of semiconductors have also a different potential, so that their presence perturbs the potential energy periodicity. In the same way, vacancies are missing atoms that elicit an interruption of the periodic potential

Due to thermal motion, moreover, even the atoms fluctuate around their equilibrium positions. The vibration of atoms around their equilibrium positions gives rise to collective modes of oscillation, which are treated as quasi-particles called phonons, endowed with proper energy, momentum and dispersion relations. Defects in the arrangement of atoms alter the profile of the potential, breaking the perfect periodicity. Eventually, defects of any nature play the role of scattering centers. Electrons are able to scatter with these centers, losing the entire, or a part of, kinetic energy acquired during motion. Scatter events, and the consequent transfer of energy from the electrons to the atoms, are at the origin of the Joule effect. In order to derive a simple, but effective, model of voltage-current relationship, let us assume that in each scatter event the electron loses all the kinetic energy and momentum acquired from the acceleration produced by the electric field. In this way, after each scatter, the electron begins to acquire energy again.

1.5 Charge Transport: The Drift-Diffusion Model

37

Let us introduce τc as the average time between two consecutive scatters. This is also known as relaxation time. The average momentum acquired between two consecutive scatters is thus: p = F · τc → m ∗n vd = −qEτc

(1.48)

where vd is the drift velocity, which is the average speed of displacement of the electrons subject to an electric field E. vd = −

qτc E. m ∗n

(1.49)

The above equation establishes a proportion between electric field and average velocity. This is the microscopic version of the Ohm law. Noteworthy, the drift velocity is independent of the size of the material. Note that as the scatter probability approaches zero, τc becomes infinite and the drift velocity diverges to infinity. An important, complementary quantity of the average time between scatters is the free mean path (lc ), which is defined as the average distance traveled between two consecutive scatters. (1.50) lc = τc vd . Note that the above model is valid only if the electrons reach the electrodes after a large number of scatters. When the path is too short, l ≤ lc the scatters do not occur, so that the charge transport is ballistic. All the above considerations are still valid in the case of holes. The quantity reassuming the characteristics of the motion of electrons and holes is known as mobility, which is the proportion between drift velocity and electric field: qτc qτc (1.51) μn = ∗ ; μ p = ∗ . mn mp Drift velocities must be written considering the sign of the charge of the particles: vd = −μn E ; vd = +μ p E.

(1.52)

Mobility depends on the nature of the material and on its purity. This is particularly important for semiconductors, where the mobility depends on the density of defects. Among the defects, doping is particularly important because dopants are artificially added with a purpose (Fig. 1.24). The mobilities of electrons and holes at room temperature for intrinsic silicon are of the order of: cm2 cm2 ; μ p = 500 . (1.53) μn = 1400 Vs Vs From the mobility, which can be experimentally determined, it is possible to estimate the relaxation time. For instance, in the case of electrons in silicon, we get:

38

1 The Physical Background

τ=

1400 · 10−4 · 0.26 · 9.1 · 10−31 m ∗n μn = ≈ 2 · 10−13 s. q 1.6 · 10−19

(1.54)

Eventually, the drift currents due to electrons and holes are: Jn = −qn(−vd ) = qnμn E ; J p = +qp(+vd ) = qpμ p E.

(1.55)

The total drift current is: J = Jn + J p = q(nμn + pμ p )E.

(1.56)

The quantity q(nμn + pμ p ) is the total conductivity (σ ) of the semiconductor, so that the Ohm law can be synthetically written as: j = σ E. Note that σ = 1/ρ. The current I A that flows in a material of length w, section A, and biased with a potential difference V A is: VA 1w · A → VA = I (1.57) IA = j · A = σ · w σ A which is the usual definition of electric resistance assumed at the beginning of this chapter.

Fig. 1.24 Mobility of electrons and holes in silicon as a function of dopant atom concentrations. Electron mobility values are valid for arsenic or phosphorous, whereas holes mobility are related to boron

1.5 Charge Transport: The Drift-Diffusion Model

1.5.2.1

39

Velocity Saturation

For large electric fields, mobility tends to deviate from its constant value. This is why some additional mechanisms intervene to limit velocity. In silicon, the most important of these mechanisms is the increase in the scattering probability with the atoms of the lattice. These phenomena become important when the drift velocity becomes larger than thermal velocity. These electrons are called hot because their temperature (according to the equipartition theorem) is larger than the background temperature. Hot electrons, among the other properties, are able to activate scattering mechanisms with phonons, which are described by a different dispersion relation (optical phonons). The details of these interactions are outside the scope of this textbook. Here, it is important to bear in mind that as the electric field increases, the mobility decreases and velocity reaches a saturation value. The observable consequence is the limitation of current. for electrons, and it is reached for a Saturation velocity in silicon is about 107 cm s V saturation electric field Esat = 104 cm . Saturation may become important when the voltage is applied across short distances (Fig. 1.25). In semiconductors characterized by a different relative position of the conduction and valence bands, the relationship between drift velocity and electric field is more complex and gives rise to peculiar behaviors, which will be discussed thereafter.

Fig. 1.25 Approximated behavior of electron drift velocity versus electric field in silicon

40

1 The Physical Background

1.5.3 Diffusion Current Unlike metals, semiconductors can maintain an internal non-homogeneous distributions of charge carriers. Due to thermal motion, the particles that are nonhomogeneously distributed tend to equate their density. This process is called diffusion and it is observed in any mobile sets of particles, such as gas molecules in the atmosphere. In the case of charged particles, this process gives rise to a current known as diffusion current. Thus, in a semiconductor, it is possible to observe a current even without an applied voltage. The energy necessary to motion is thermal, but the magnitude of current is proportional to the gradient of charge densities. Given a concentration of particles, n(x), its gradient gives rise to a flux of particles. The relationship between flux and gradient is the Fick first law: F = −D ·

dn dx

(1.58)

where D is the diffusion coefficient. The negative sign indicates that the direction of flow is opposite to the direction of concentration growth, as shown in Fig. 1.26. The diffusion coefficient has the dimensions of cm2 /s. In the case of charged particles, such as electrons and holes, the electric current associated to the flow is J = q · F. Electrons and holes are characterized by different diffusion coefficients: Dn and D p . Thus, the diffusion current of electrons and holes is:   dn dn = q · Dn · ; Jn = (−q) · −Dn · dx dx   dn dn = −q · D p · J p = (+q) · −D p · dx dx

(1.59)

Eventually, the total current of electrons and holes is given by the sum of the drift and diffusion currents:

Fig. 1.26 Thermal flow is proportional to the concentration of particles. In the case of a nonuniform concentration, the flow impinging onto a surface from the most populated side is larger than the flow coming from the opposite region. Note that thermal flow is absolutely isotropic and the diffusion current is observed in the line of separation between regions at different concentrations

1.5 Charge Transport: The Drift-Diffusion Model

41

dn dx dp J p = qpμ p E − q D p . dx Jn = qnμn E + q Dn

(1.60)

The diffusion coefficient describes the motion of the charges under the influence of a concentration gradient. This quantity is similar to mobility, which describes the charge motion under the influence of an electric field. In order to calculate the relationship between D and μ, let us consider the case of electron current equilibrium. In the drift-diffusion model, the equilibrium is not merely the absence of current, but rather a situation where the diffusion current is compensated by the drift current and vice-versa. Hence, in a semiconductor, the equilibrium can be also achieved when the electric field and the charge concentration gradient are simultaneously present. The equilibrium condition is: Jdi f f = Jdri f t → qnμn E = −q Dn

dn . dx

(1.61)

Let us replace E = −d V /d x and calculate the mobility considering that, being the voltage a function of x, also the concentration is a function of position: − n(x)μn

Dn dn dV dn = −Dn →μ= . dx dx n(x) d V

(1.62)

In order to calculate dn/d V , let us consider that any potential difference is added to the conduction energy band. Thus, the concentration n(x) can be written as:   E c − q V (x) − E f n(x) = NC ex p − kB T       Ec − E f qV qV ex p = n 0 · ex p = NC ex p − kB T kB T kB T

(1.63)

where n 0 is the concentration of electrons calculated where the voltage is null. Consequently, the mobility is: μ=

Dn q q Dn dn = n(x) = Dn . n(x) d V n(x) k B T kB T

(1.64)

The same calculation applies to the holes diffusion coefficient. Eventually, the relationship between the diffusion coefficients and the mobilities are: kB T μn q kB T Dp = μp q Dn =

(1.65)

42

1 The Physical Background

which are known as Einstein-Smoluchowski relations. The diffusion coefficient is the product between mobility and a quantity that corresponds to voltage-equivalent temperature (VT ). At room temperature (T = 330 K), the thermal voltage is about 26 mV. In practice, temperature provides the energy for the diffusion current, whose magnitude is determined by both concentration gradient and mobility.

1.5.3.1

Total Current, Electrochemical Potential and Fermi Level

The drift and diffusion currents seem to be generated by two different phenomena. The drift current originates from an electric field, namely the gradient of the applied voltage, whereas the diffusion current is due to a gradient of charge density. The Einstein-Smoluchowski equation suggests the existence of a strong relationship between the two currents, which can actually be derived from the gradient of a unique quantity: the electrochemical potential. Indeed, since E = − ddVx and Dn = k BqT μn , the total current of electrons is proportional to the gradient of a unique potential function: Jn = −qnμn

 dV dV dn dη kB T k B T dn = nμn +q μn = nμn −q + (1.66) dx q dx dx n dx dx

where η is the electrochemical potential. Solving the above equation, we obtain: η = η0 + k B T ln(n) − q V

(1.67)

where η0 is a constant. It is easy to show that the above equation corresponds to the Fermi level. Indeed, from Eq. 1.20: E F = E c + k B T ln(n) − k B T ln(Nc ).

(1.68)

The first term is the potential energy equivalent to −q V , whereas the last term is a constant equivalent to η0 . Thus, in the Boltzmann approximation (E C − E F k B T ), the Fermi level and the electrochemical potential of a gas of charged particles are coincident. Consequently, as the gradient of the electrochemical potential determines the current of particles, the gradient of the Fermi level determines the current of the charges in the semiconductor. Hence, the equilibrium condition where the sum of all currents is zero is achieved when the gradient of the Fermi level is zero, namely when the Fermi level is constant throughout the whole material.

1.6 Non-uniform Distribution of Dopant Atoms and Built-in Potential

43

1.6 Non-uniform Distribution of Dopant Atoms and Built-in Potential Contrarily to metals, the electric field inside a semiconductor can be steadily different from zero. Thus, it is possible to maintain a non-uniform distribution of fixed and mobile charges. The uneven distribution of charges is maintained by an internal potential, which is known as built-in potential. To study this situation, let us consider a semiconductor with a continuous distribution of dopant atoms, extended from a region where donors dominate to a region where acceptors are the majority. Let us consider an ideal experiment, where the distribution of dopant atoms is instantaneously created. At the time t0 , the electron and hole concentrations are n = Nd and p = Na , and the electric field is zero everywhere. In terms of band diagram, the above described situation is shown in Fig. 1.28. Since the concentration of electrons and holes is not constant, a diffusion current

Fig. 1.27 Non-uniform distribution of dopant atoms. The plot shows the transition from a region where the N-type are dominant to a region dominated by P-type doping

Fig. 1.28 The consequences of the dopant distribution of Fig. 1.27 are manifested in the band diagram. Left: the band diagram immediately after the doping, before the equilibrium is established. The Fermi level is not constant and its distance from the bands depends on the doping concentration. Right: at the equilibrium the Fermi level becomes constant, so that the bands are curved. The intrinsic Fermi level is also plotted to allow for a straightforward identification of the kind of doping

44

1 The Physical Background

emerges. However, as mobile charges move, the total charge locally changes, giving rise to an electric field that prompts a drift current, which compensates the diffusion current. The system evolves towards the equilibrium, corresponding to a constant Fermi level. This condition is obtained by imposing a curvature to the other energy levels: the vacuum level, the conduction and the valence band. Since the conduction and the valence bands are the potential energies of electrons and holes, the curvature of the bands corresponds to the internal potential energy. To evaluate this energy, let us consider that, at each coordinate x, the concentration of electrons and holes is given by:   E C (x) − E F n(x) = NC ex p − kB T   E F − E V (x) . p(x) = N V ex p − kB T

(1.69)

Let us measure all the energies with respect to the intrinsic Fermi level, whose distance from the bands is constant. E C − E F = (E C − E i ) − (E F − E i ).

(1.70)

Thus, the potential of the electrons can be written as: 1 1 1 φ = − (E C − E F ) = − (E C − E i ) + (E F − E i ) = φ0 + φi . q q q

(1.71)

Since the potential can be defined with respect to any constants, we can consider φi , the distance between the Fermi level and the intrinsic level as the potential of the electrons. φi counteracts the concentration gradient. Consequently, the drift current is equal and opposite to the diffusion current and the system is in equilibrium. The sign of the potential φi defines the kind of doping. If it is positive, then the material is N-type; if negative, it is P-type (Fig. 1.29). At the equilibrium, the total current is zero, this condition determining the relationship between the built-in potential and the concentration of electrons. Jn = qnμn E + q Dn from which: E=−

dn =0 dx

dφ Dn 1 dn k B T 1 dn =− =− dx μn n d x q n dx

where the Einstein relation was used.

(1.72)

(1.73)

1.6 Non-uniform Distribution of Dopant Atoms and Built-in Potential

45

Fig. 1.29 This figure shows a potential φi originated by a non-uniform doping concentration. The related diffusion and drift currents are noted on the right side. The direction of the vector of all the quantities composing the two currents is also shown. Eventually, drift the and diffusion currents balance and a built-in potential keeping the system in equilibrium is formed

The potential can be calculated by integrating the previous equation from point 1 to point 2. φ2 n 2 k B T 1 dn dφ = dx (1.74) q n dx φ1

n1

from which the built-in potential is calculated: φ 2 − φ1 =

kB T ln q



n2 n1

 .

(1.75)

The previous equation allows to calculate the potential across a perturbed region. The analytical behavior of the potential (φ(x)) is calculated from the Poisson equation, which relates the charge distribution to the electric potential: dφ 2 ρ(x) =− . 2 dx s

(1.76)

The charge density ρ is contributed by the four kinds of charges that are found in a semiconductor: (1.77) ρ = p − n + Nd − Na where p and n are the mobile charges and Nd and Na are the densities of donors and acceptors.

46

1 The Physical Background

The concentration of electrons is:   EC − E F n = NC ex p − kB T       E F − Ei EC − Ei qφi ex p = n i ex p . = NC ex p − kB T kB T kB T

(1.78)

A similar expression is found for the holes:   qφi . p = n i ex p − kB T

(1.79)

Hence, the Poisson equation can be rewritten as:

     dφ 2 q qφi qφi n − n − N = ex p ex p − + N i i d a . dx2 s kB T kB T

(1.80)

Introducing the hyperbolic sine (sinh (x) = (e x − e−x )/2), we get:

   q qφi dφ 2 2n i sinh − Nd + Na . = dx2 s kB T

(1.81)

In order to solve the above equation, it is necessary to know the distribution of donors and acceptors. In this textbook, the Poisson equation is solved only under simple assumptions. Charge distribution, electric field and built-in potential are fundamental quantities that characterize the properties of the junctions between materials. They will be thoroughly calculated, even though in ideal conditions, for all the junctions studied in this textbook.

1.6.1 Quasi-neutrality Condition Given the initial situation in Fig. 1.28, the equilibrium is reached with a negligible displacement of charges. In practice, at the equilibrium, the amount of electrons and holes is still given by the concentration of donors and acceptors. This assumption may be justified with a numerical example. Let us consider a N-type silicon (s = 11.7), where the concentration of donors changes from 1016 to 1018 cm−3 at a distance of 0.5 µm. The built-in potential generated by this gradient of concentration is: φ =

kB T ln q



n2 n1

 = 0.026 ln

1018 ≈ 0.12 V. 1016

(1.82)

1.7 Summary

47

The electric field is: E =

φ 0.12 V = . = 0.2 104 −4 x 0.5 10 cm

(1.83)

The derivative of the electric field can be calculated from the Poisson equation n2 considering the charges in a N-type semiconductor, where Na = 0 and p = NiD is negligible: q dE dφ 2 = − [n − Nd ] =− (1.84) dx2 dx s from which n − Nd =

dE s . dx q

(1.85)

Considering the finite differences, the difference between mobile electrons and fixed donors is: n − Nd =

E s 0.2 104 11.7 8.8 10−14 ≈ 1014 cm−3 . = x q 0.5 10−4 1.6 10−19

(1.86)

Thus, n = Nd − 1014 , which is negligible with respect to Nd . Therefore, the local concentration of electrons is always equal to the local concentration of donors, so that the equilibrium condition is reached by just moving a negligible amount of mobile charges.

1.7 Summary The properties of electronic devices mainly descend from the energy distribution of the electrons inside the materials the device is composed by. These properties are compactly represented in the band diagram, which contains the four typical quantities: work function, affinity, energy gap, and Fermi level. The peculiar role of semiconductors in electronics comes from some properties manifested by a number of phenomena, such as photoconductivity and negative temperature coefficients. Relevant to electronic devices is the two-fold current conduction mechanism, involving two kinds of quasi-particles: electrons and holes, which are the mobile charges populating the conduction and valence bands respectively. An additional singular property of semiconductors is the diffusion current, which results from the effects of thermal motion over a non-uniform concentration of mobile charges. The reason for a diffusion current is the dielectric nature of semiconductors, which are characterized by a real electric permittivity that allows, unlike metals, non-uniform concentrations of electrons and holes.

48

1 The Physical Background

The consequence of the existence of two forms of currents leads to an equilibrium condition that is not merely absence of current, but rather the equilibrium between two currents, each non-zero, equal and opposite. Thus, the equilibrium conditions in case of non-uniform concentration of mobile charges implies an electric field. This fact is at the basis of the junction built-in potentials and built-in electric fields, which play fundamental roles in electronic devices.

Further Reading Textbooks C. Kittel, Introduction to Solid State Physics, 8th edn. (Wiley, New York, 2018) R. Muller, T. Kamins, M. Chen, Device Electronics for Integrated Circuits, 3rd edn. (Wiley, New York, 2002) R. Pierret, Advanced Semiconductor Fundamentals, 2nd edn. (Prentice Hall, Hoboken, 2002) S. Sze, K.K. Ng, Physics of Semiconductor Devices, 3rd edn. (Wiley-Interscience, New York, 2006) P. Yu, M. Cardona, Fundamentals of Semiconductors, 3rd edn. (Springer, Berlin, 2001)

Chapter 2

The Metal-Semiconductor Junction

2.1 Introduction Junctions are the building blocks of electronic devices. Ideally, a junction is a plane surface where one material abruptly ends and the other begins. For our scope it is important to understand the effect of the junction on the electron and hole populations. Of course, the interesting case is when the junction is formed between two different materials. Thus, the energy and the density of the states of electrons and holes on the two sides of the junction are obviously different. In this way, the charges tend to leave the states at highest energy so as to occupy free allowable states with neither acquisition nor production of energy. Hence, the currents from one material to the other and vice-versa may appear as soon as the junction is formed. The transfer proceeds until an equilibrium is reached, namely the total current across the junction is zero. Equilibrium does not mean that all the currents are null, but rather that their algebraic sum is zero. We have seen in the previous chapter that, if the Fermi-Dirac function is approximated by the Boltzmann equation, then the null current condition is provided by the constant Fermi level. In the next section, a more general extension of this property will be presented. The differences among the materials forming the junctions confer a variety of properties to the junction itself. On a practical level, the three basic materials of electronics—metals, semiconductors, and insulators—are joint together to give rise to basic structures, which are subsequently exploited by electronic devices. In this chapter, we will start studying the junction considering the case of the interface between a semiconductor and a metal. This necessary element appears anytime a piece of semiconductor is connected to an electric circuit. Depending on the characteristics of the materials, the metal-semiconductor junction can give rise to a manifold of relationships between current and voltage. In the most practical of these cases, the metal-semiconductor junction behaves either as a rectifier (the current flows only in one direction) or as a ohmic contact (the current is independent of the applied voltage direction). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_2

49

50

2 The Metal-Semiconductor Junction

From a didactical point of view, the metal-semiconductor junction is a tool to introduce a methodological approach to study the electric properties of the junctions. In general, the junction elicits a change in the properties of the charge carriers across the interface, and these changes are extended to a region across the physical boundary between the materials. This region is the junction itself, and its properties determine the characteristics of the whole device. We will study the properties of the junctions under the thermal equilibrium and applied voltage conditions. Usually, semiconductors are assumed as being uniform and homogeneous materials. Actually, at the end of the previous chapter we have seen that even tiny inhomogeneities in doping result in internal built-in potentials. In theory, these imperfections could be eliminated by careful technological refinements. However, some inhomogeneities cannot be eliminated. Among them, those occurring at the surface level. On the surface, the regular crystal lattice inside the semiconductor (the bulk) is interrupted. Thus, the interactions among the atoms close to the surface change with respect to those in the bulk. This leads to the formation of additional energy states for the electrons (surface states), extended towards the bulk. The surface states of interests are those that lie in the energy gap that is no longer empty of electrons. Since the metal-semiconductor junction is formed by “adding” metal atoms on the surface of the semiconductor, it is obvious that some surface conditions can influence the properties of the junction. In this chapter, we will be treating the ideal junction neglecting the surface states, whose effects will be nonetheless discussed at the end of the chapter.

2.1.1 The General Rule of Junctions at the Equilibrium In the previous chapter it was demonstrated that the equilibrium is achieved when the Fermi level is constant throughout the material. This conclusion is also true in case of a combination of materials, such as that of junctions. However, this result was based on the assumption of validity of the Boltzmann approximation of the Fermi-Dirac function. Here, a more general approach to the equilibrium condition is derived. For the scope, let us consider a junction of two materials labeled as 1 and 2. The electrons in one materials may occupy free states in the other material, and vice-versa. The charge transfer from one material to the other occurs without the input of any external energy sources but temperature, which is uniform everywhere. It is important to remark that temperature uniformity is a general valid assumption, except where noted, throughout this textbook. Let us call G the rate of transfer (electrons per second) of electrons from one material to the other. As a general rule, the rate of any transitions between states can be written as the product of the density of filled states, from which electrons move, and the density of the empty available states, times the probability of transfer occurrence. In other words, the displacement of electrons from one material to the other requires that the electrons move from a filled state to an empty state, and that this

2.2 The Metal-Semiconductor Junction at the Equilibrium

51

transition is physically possible. The first two terms are provided by statistical laws, but the third one implies that the junction is permeable to electrons. To calculate the rate of transition of the electrons across the junction, let us consider n(E) as the density of filled states, v(E) as the density of empty states and k as the transition probability. The rate of transition from material 1 to material 2 is G 1→2 = n 1 (E) · v2 (E) · k, while in the opposite direction it is G 2→1 = n 2 (E) · v1 (E) · k. At the equilibrium: G 1→2 = G 2→1 → n 1 (E)v2 (E) = n 2 (E)v1 (E).

(2.1)

The densities of filled and empty states are calculated from statistics as n(E) = g(E) f D (E) and v(E) = g(E)(1 − f D (E)), where f D is the Fermi-Dirac function. Since the electrons do not change energy, the latter can be omitted in the equations, so that the equilibrium condition provides: n 1 v2 = n 2 v1 → g1 f 1 g2 (1 − f 2 ) = g2 f 2 g1 (1 − f 1 ) → f 1 = f 2 .

(2.2)

The equilibrium condition does not depend on the density of states, namely it does not depend on the nature of materials, but only on the Fermi-Dirac function. The equilibrium is achieved when the Fermi Dirac function is the same everywhere. In practice, this means that electrons are in equilibrium if the probability to find electrons at a given energy is the same everywhere. Since we assume temperature as being uniform, the Fermi Dirac function is constant only if the Fermi level is uniform. Eventually, a junction at the equilibrium, and any materials in general, requires that the Fermi level be constant everywhere. Since the Fermi levels in the pristine materials are generally different, a uniform Fermi level is achieved after a net transfer of charges from one material to the other. This rule holds for any kind of materials (metal or semiconductor) kept in contact, where electrons can freely flow from one material to another.

2.2 The Metal-Semiconductor Junction at the Equilibrium The behavior of the metal-semiconductor junction depends on the relative magnitude of the work function of the metal (qm ) and the semiconductor (qs ), as well as on the kind of doping (either N-type or P-type). This gives rise to four combinations: Ntype and qm > qs , N-type and qm < qs , P-type and qm > qs , and P-type and qm < qs . Let us first consider the case of a junction made of a N-type semiconductor and a metal, such that qm > qs . As an example, chromium and N-type silicon. The ideal junction is a useful model where the semiconductor is uniform until the surface (no surface states), and it is characterized by an isotropic distribution of

52

2 The Metal-Semiconductor Junction

Fig. 2.1 Band diagrams of chromium and N-type silicon before the formation of the junction. Note that the drawing is not in scale, the affinity being almost 4 times larger than the energy gap

dopant atoms (N D ). The surface of the semiconductor is perfectly planar and the metal grows in the direction parallel to the surface. Most of the properties of the junction can be derived from band diagrams. The only relevant quantity for metals is the work function, which in the case of chromium is about qCr = 4.60 eV. On the other hand, the band diagram of the semiconductor is characterized by three quantities: affinity, energy gap and work function. As for silicon, we have: qχ Si = 4.05 eV; E gapSi = 1.12 e. However, the work function, namely the position of the Fermi level, depends on doping concentration according to Eq. 2.4:   ND (2.3) q = qi − k B T ln ni where qi = qχ + gap = 4.61 eV. In the case of N D = 1016 cm−3 , the work 2 function is q Si = 4.25 eV . All these quantities allow us to design the band diagrams of the two materials before the junction is formed (Fig. 2.1). The band diagram at the equilibrium is achieved as a consequence of a displacement of electrons according to the mutual position of the Fermi levels in the two materials. In this example, the Fermi level of the semiconductor has a higher energy than the Fermi level of the metal, in terms of work function qm > qs . It is worth to remark that the displacement of electrons requires also the availability of the states in the material of destination. As shown in Fig. 2.2, the electrons in the conduction band are less numerous than those in the metal, but they have access to a larger density of empty states. On the other hand, the transfer of the many electrons in the metal is hindered by the scarcity E

2.2 The Metal-Semiconductor Junction at the Equilibrium

53

Fig. 2.2 Fermi-Dirac function and density of states of semiconductor and metal. Electrons can transfer from a material to the other proportionally to the availability of the corresponding free states

of available free states in the valence band of the semiconductor. The empty states in the valence band are the holes, whose number is given by the action mass law: p = n i2 /Nd . In this numerical example, the density of available states in the valence band is p = 104 cm−3 . Immediately after the formation of the junction, the current of electrons from the semiconductor to the metal is larger than the current flowing in the opposite direction. As the electrons leave the semiconductor, the density of holes increases, so that the current from the metal to the semiconductor increases, too. Eventually, the equilibrium (zero total current) is reached after a net amount of charges is transferred from the semiconductor to the metal. Since the electric field inside metals is null, the excess of electrons coming from the semiconductor accumulate on the metal surface at the interface. On the other hand, the electrons that had left the semiconductor leave behind them a region where the total charge is negative, as it is dominated by the fixed donor charges. In this region close to the junction, before the formation of the junction itself, we find n = Nd and, when the junction reaches the equilibrium, n < N D . A volume of the material where the total charge is different from zero is known as space charge region. Charge alteration is limited to a region immediately close to the interface with the metal, whereas the rest of the material (the bulk) is left unchanged. Note that the size of the region depends on the density of dopant atoms.

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2 The Metal-Semiconductor Junction

The electron and hole concentration still abides the law of statistics: in particular, this concentration depends on the difference between the Fermi level and the conduction band.   EC − E F . (2.4) n = NC · ex p − kB T In the space charge region, the difference between the conduction band and the Fermi level changes, so as to account for the decrease in the electron density. Both the affinity and energy gap, being related to the intimate nature of the semiconductor, do not change. Due to the loss of electrons, in the region close to the interface the difference between the conduction band and the Fermi level increases. This band bending indicates that the region is depleted of electrons. For this reason, such a region is also called depletion layer. All these elements contribute to drawing up the equilibrium band diagram of the whole metal-semiconductor system at thermal equilibrium. The floatings in the band diagram of insulated materials are due to the impossibility to measure the energy distance from E = 0. In the junction, the constant Fermi level provides an anchor for the drawing of the band diagram, allowing us to establish the energy variation from one material to the other. The drawing of the equilibrium band diagram is an important tool to understand the properties of junctions. It may be easily accomplished, in any situations, following the six steps listed in Table 2.1 Following these steps, the equilibrium band diagram shown in Fig. 3.3 is obtained (Fig. 2.3). The equilibrium between currents is maintained by an energy barrier applied to the electrons of both the materials. Due to the original differences in work function and the different concentrations of electrons, the equilibrium condition requires two barriers with different heights: one is applied to the electrons of the metal (qφ B ), and the other is applied to the electrons of the semiconductor (φi ), where qφi is smaller than qφ B . The barrier applied to the electrons of the semiconductor (qφi ) is the built-in potential, which is equal to the difference between the work functions of the two materials. It is also the energy difference between the vacuum energy level at the surfaces of the two materials. This quantity is the potential difference measured between the surfaces of

Table 2.1 Steps to draw the equilibrium band diagram 1 Identify the interface and the space charge region in the semiconductor 2 Draw a unique Fermi level 3 Draw the unaltered band diagram of the metal and the bulk of the semiconductor 4 Draw a continuous curve connecting the vacuum level of the metal and the vacuum level of the bulk 5 Draw the conduction band parallel to the vacuum level (constant affinity) 6 Draw the valence band parallel to the conduction band (constant energy gap)

2.2 The Metal-Semiconductor Junction at the Equilibrium

55

Fig. 2.3 Equilibrium band diagram of a metal-semiconductor junction. The coordinate x=0 indicates the interface, while the coordinate xd , at the end of the perturbed region, is the total length of the depletion layer

the two materials. It is called contact potential difference or Volta potential. It is an observable manifestation of the internal potentials. It is interesting to observe that the Fermi level is an absolute quantity defined with respect to the zero energy level, namely to the energy at infinite distance from the material, while the absolute value of the vacuum level is undetermined and becomes evident only as a difference of surface potentials between materials. The equilibrium band diagram is achieved shifting the Fermi levels. In practice, the absolute energy of the electrons increases where electrons are added, and decreases where electrons are subtracted. The difference of the original Fermi levels forms a non-observable inner potential called Galvani potential. The amount and the behavior of potential, electric field and depletion layer size can be calculated solving the Poisson equation (Eq. 1.76). The solution of the Poisson equation requires the knowledge of the distribution of mobile and fixed charges. It has been mentioned above that one of the hypothesis of ideal junction is the uniform doping of semiconductors. As a consequence, in a non-perturbed N-type semiconductor, also the electrons are uniformly distributed, so that the total charge in any volume of the semiconductor is null. However, at the equilibrium, in the semiconductor region from x = 0 to x = xd , the concentration of electrons is less than Nd and it is variable, as shown by the bending of the conduction band. The electron concentration depends on the potential according to Eq. 1.76: The Poisson equation for a N-type semiconductor can be written as:    dφ 2 q qφ N . (2.5) = − − N exp d d dx2 s kB T The previous equation must be solved in the depletion layer, where the electron concentration is variable, so that the derivative of the potential is different from zero. The Poisson equation can be solved numerically, but a practical solution can be obtained making some assumptions about the distribution of mobile charges. In particular, we can assume that in the perturbed region the fixed charge of donors is

56

2 The Metal-Semiconductor Junction

much greater than the mobile charge of electrons (N D  n). This condition is called deep depletion hypothesis. The deep depletion hypothesis is justified by the fact that the concentration of electrons is an exponential function of (E C − E F ), so that small changes in energy bring about great changes in concentration. Of course, in the neighbor of xd , the deep depletion approximation fails. A more realistic description of the behavior around xd will be discussed in Chap. 4. The total charge distribution is shown in Fig. 2.4. The negative charge is accumulated in a thin layer at the surface of the metal, and the positive charge, formed by donor charges, extends to the semiconductor. Obviously, the total charge in the whole device is always zero.

Fig. 2.4 Charge density, electric field, and potential of an ideal metal-semiconductor junction at the equilibrium under the hypothesis of deep depletion and uniform doping

2.2 The Metal-Semiconductor Junction at the Equilibrium

57

Fig. 2.5 The electric field at x = 0 is calculated considering the flux through a close surface surrounding the interface and placed immediately outside the metal. The portion of the surface crossed by the electric field is the area A lying towards the semiconductor

+ Q = q Nd xd ; −Q = −q Nd xd .

(2.6)

Note that we are dealing with charge densities in unit of C · cm−2 . The relationship between charge density and the electric field is given by the Gauss theorem:  − → − → E ·d A (2.7) ϕ= S

− → − → where ϕ is the electric flux, E is the electric field and d A is a vector orthogonal to the infinitesimal surface element. The surface integral is extended to a closed surface S. Using the divergence theorem, the Gauss theorem can be written in a differential − → form: ∇ · E = ρ , which in one dimension is: ρ(x) dE = → E(x) = E0 + dx 

x 0

ρ(x) d x. 

(2.8)

At x < 0, inside the metal the electric field is zero by definition. The electric field at the interface (x = 0) is generated by a surface charge density. This can be calculated by a simple application of the Gauss theorem, considering that the electric field generated by a uniformly distributed sheet of charges is orthogonal to the surface and that the electric field towards the metal is null (Fig. 2.5). The field generated by the charges at the surface of the metal is calculated at the surface of the semiconductor. Thus, the involved dielectric constant is the dielectric constant of the semiconductor (e.g.  Si = 11.70 ): E(0) = −

q Nd x d . s

The electric field in the depletion layer is calculated using Eq. 2.8.

(2.9)

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2 The Metal-Semiconductor Junction

x E(x) = E0 + 0

q Nd x d dx 

(2.10)

where E0 is the electric field at (x = 0). The solution is: E(x) = −

q Nd x d q Nd x q Nd +− → E(x) = (x − xd ). s s s

(2.11)

Finally, in the bulk of the semiconductor (x > xd ) the total charge is zero, so that the electric field is null. Given the electric field, the electric potential can be calculated: x (2.12) φ = φ0 − E(x)d x. 0

. Inside the metal (x < 0) the potential is null, while in the depletion layer it is: x φ(x) = − 0

q Nd q Nd (x − xd )d x = − s s



 1 2 x − x xd . 2

(2.13)

At x = xd the potential reaches its maximum value: φmax =

1 q Nd 2 x = φi . 2 s d

(2.14)

Since the electric field at x > xd is null, the maximum potential value is maintained in the bulk of the semiconductor. This quantity is the built-in potential of the metalsemiconductor junction. The built-in potential is indeed equal to the difference between the work functions, which have been known since the beginning, when the materials for the junctions were chosen. From the built-in potential, we can calculate the size of the depletion layer:  xd =

2φi s . q ND

(2.15)

Following the band diagram in Fig. 2.3, we get Nd = 1016 cm−3 ; qφs = 4.25 eV; qφm = 4.60 eV, from which we calculate φi = 0.25 V and xd = 179 nm. Finally, we can observe from Eq. 2.15 that the extension of the depletion layer depends on doping. In particular, the larger is the doping, the narrower is the depletion layer. To this regard, it is worth to note that also the built-in potential depends on doping, but as a logarithm. Hence, the above stated behavior remains valid.

2.3 Biased Metal-Semiconductor Junction

59

The relationship between doping and depletion layer size is a general rule valid for any junctions. In the case of a very large doping, the semiconductor degenerates into a metal-like behavior and the size of the space charge region approximates the bidimensional layer of the metal.

2.3 Biased Metal-Semiconductor Junction The equilibrium condition illustrated in the previous section is altered by the application of an external voltage. The voltage is applied via two contacts made up of metal wires and the two sides of the metal-semiconductor system. The voltage is applied by means of additional metal-metal and metal-semiconductor junctions. This gives rise to a tautology: in order to study the behavior of a junction, we need another junction of the same kind. For the moment, let us consider that the junctions between the system under study and the circuit are negligible, namely they are ohmic contacts. We will study thereafter the properties of ohmic contacts and the conditions under which they occur. The system under study is formed by three distinct regions: metal, junction and semiconductor. The voltage is thus applied to a series of three different materials, each with a distinct electric property. In particular, the metal is characterized by a very large population of mobile electrons (n ≈ 1022 cm−3 ), whereas the doped semiconductor contains a smaller quantity of electrons (n = Nd ). Finally, the junction is practically depleted of mobile electrons (n  Nd ). Thus, the three regions are characterized by very different conductivities, and the depletion layer is the least conductive element of the series. As a consequence, the applied voltage drops almost completely across the depletion layer. This is an important assumption, considered as valid throughout this textbook. The applied voltage falls across the depletion layer, while the electric field in the bulk of the semiconductor is null. For this reason, the bulk of the semiconductor is also called neutral zone (Fig. 2.6). The external voltage violates the conditions of thermal equilibrium, so that the statistical laws derived in Chap. 1 are no longer valid. However, for modest values of applied voltage, it is still possible to calculate the electron and hole concentrations with statistical laws. This condition is the so-called quasi-equilibrium hypothesis. The applied voltage shown in Fig. 2.6 locally modifies the energy of the electrons. In particular, they acquire an extra potential energy equal to q V . Since the charge of the electron is negative, if V A is positive, then the energy is shifted towards negative values. The electrons in the metal are subject to the same voltage, so that the energy of all the electrons in the metals is translated downward by a quantity q V A . On the other hand, the energy of the electrons in the bulk of the semiconductor remains unchanged. Eventually, the application of an external voltage does not change the barrier applied to the electrons of the metal (qφb ), even though it changes the barrier applied to the electrons of the semiconductor (qφi − q V A ): when V A is positive, the barrier

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2 The Metal-Semiconductor Junction

Fig. 2.6 Behavior of applied voltage across a metal-semiconductor system. The voltage drop in the bulk of the semiconductor is practically negligible

Fig. 2.7 Potential energy of electrons (conduction band) before and after the application of an external voltage V A

(qφi ) decreases, and when V A is negative the barrier increases. These two conditions are called forward bias and reverse bias (Fig. 2.7). The uneven change in the barriers breaks the equilibrium between the currents. The current from the metal is unchanged and maintains the thermal equilibrium value. On the other hand, the current from the semiconductor changes. In the case of forward bias, the barrier decreases and the current increases, whereas under reverse bias the barrier increases and the current decreases. This phenomenon depicts the behavior of a diode and will be quantitatively studied in the next section. The bias also affects the depletion layer size:  xd =

2s (φi − V A ). q ND

(2.16)

In the case of a reverse bias, the depletion layer becomes larger, whereas under forward bias, it becomes thinner. The behavior of the metal-semiconductor system under applied voltage can be conveniently studied by separating the capacitive and the resistive effects according to the classical approach adopted in the network theory.

2.3 Biased Metal-Semiconductor Junction

61

2.3.1 The Capacitance of the Junction The capacitance measures the modulation of the concentration of charges due to applied voltage. dQ . (2.17) C = dV The amount of fixed charges in the depletion region is proportional to xd . Since the length of the depletion layer is modulated by the applied voltage, also the charge therein contained varies according to the applied voltage:  Q = q N D xd = q N D

2s (φi − V A ) = 2q N D s (φi − V A ). q ND

(2.18)

The capacitance is simply calculated by applying the definition. Note that, as many other quantities in this textbook, the following equation actually describes a density of capacitance (F/m 2 ):  dQ

1 q ND 1 = 2q N D s = s C = . d VA 2(φi − V A ) 2s φi − V A Finally: C=

s . xd

(2.19)

(2.20)

The depletion layer behaves as the dielectric of a capacitor with parallel plates. It is interesting to note that the capacitance diverges when V A  φi . In these conditions, however, the quasi-equilibrium hypothesis may be no longer valid, and the calculated quantities do not represent the real device. Capacitance manifested at best when the resistive behavior is negligible, which happens under reverse bias. Equation 2.19 shows that the capacitance is not constant, but rather depends on applied voltage. The behavior of the capacitance as a function of the applied voltage (the C/V curve) is a fundamental tool to measure some important parameters of electronic devices. In the case of metal-semiconductor junction, it provides a measurement of the doping concentration and the built-in potential. To this scope, let us consider the inverse of the square of Eq. 2.19: 2(φi − V A ) 1 = . 2 C qs N D

(2.21)

By measuring C at different values of V A and plotting C12 as a function of V A , we obtain experimental points aligned along a straight line, whose slope contains the

62

2 The Metal-Semiconductor Junction

Fig. 2.8 Plot of C12 versus V A . The relationship between the curve parameters and the built-in potential and donor concentration is also shown. Experimental data are collected under reverse bias, where the current is negligible and the device behaves like an almost pure capacitor

concentration of the dopant atoms, and whose intercept with the horizontal axis is the built-in potential (Fig. 2.8).

2.3.1.1

Experimental Set-Up for the C/V Curve Measurement

In order to measure the C/V curve, it is necessary to bias the device with a d.c. voltage (V A ), and to superimpose an a.c. signal. The time-variable part of the total applied signal allows for the extraction of a current proportional to the capacitance value. Figure 2.9 shows a simple example of a circuit for the measurement of the C/V curve. The circuit is based on a current-to-voltage converter made with an operational amplifier. It is important to consider that V A fixes the value of the capacitance as reported in Eq. 2.19, while the a.c. signal is the probe necessary to measure the capacitance. The condition is such that vt   V A , in order for the probe signal not to alter the capacitance value. The frequency ω of the probe signal is not particularly important for the metalsemiconductor junction. However, it is very important in other systems, such as the metal-oxide-semiconductors (see Chap. 9). From an electronic point of view, it is interesting to note that, in order to properly work, the circuit in Fig. 2.9 must behave as a differentiator amplifier. Due to the frequency response of the operational amplifier,

Fig. 2.9 Electronic circuit for the measurement of the C/V curve

2.3 Biased Metal-Semiconductor Junction

63

Fig. 2.10 Transfer function of the circuit of Fig. 2.9

this is ensured only if ω is smaller than the frequency at which the transfer function of the feedback network meets the transfer function of the amplifier (Fig. 2.10).

2.3.2 The I/V Characteristics The rectifying I/V curve of metal-semiconductor junctions has been known since the beginning of the twentieth century, when diodes made of lead sulfide (PbS) and phosphor bronze were introduced as the first solid-state electronics device. PbS, a mineral also known as galena, is a small band gap semiconductor, whereas phosphor bronze is a tin alloy with a small amount of copper and phosphor. The contact was simply made by leaning the wire on the mineral surface. Such components became popular before the advent of modern devices. The diode made of a metalsemiconductor junction is called Schottky diode, after the German physicist Walter Schottky. The relationship between current and applied voltage is here calculated following two approaches. The first one is a qualitative approach that considers current as the result of the emission of electrons from one material towards the other. In such an approach, the actual shape of the barrier is not relevant, but only its height. In the second, the current is directly calculated from the drift/diffusion model. The calculations are restricted to the depletion layer, where both the electric field and concentration gradient may exist. The two approaches achieve a good description of the diode behavior under direct bias, but with a slight difference in the reverse current. The quest for a more accurate model may be useless due to the presence of surface states, which strongly affect the behavior of real metal-semiconductor junctions. Thus, the thermionic and drift-diffusion models are sufficient to explain most of the characteristics of the Schottky diode. It is important to note that, in both the models, the electron concentration in the conduction band in non-equilibrium condition is still calculated with the statistical law of Eq. 1.20. The use of equilibrium quantities is indeed guaranteed by the quasiequilibrium assumption (the applied voltage introduces a small perturbation of the

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2 The Metal-Semiconductor Junction

equilibrium quantities). A discussion about the validity of this assumption will be done after the study of the PN junction.

2.3.2.1

The Thermionic Current Model

At thermal equilibrium the total current across the device is null. Namely, the absolute value of the current flowing from the metal to the semiconductor and vice-versa are equal: JM S  = JS M . In this model, we consider that the current at the interface (x = 0) is given by the thermal motion of the charges. This is particularly true for the current directed towards the metal, where the electric field is zero. Towards the semiconductor, on the contrary, the charges are subjected to the electric field (E(0)) given by Eq. 2.11. Since at the equilibrium the two currents are equal, we can calculate the current considering only the thermal motion. The thermal flow of a gas of particles impinging onto a surface is studied by the kinetic theory of gases and is given by: F=

nvave . 4

(2.22)

The above equation is called law of Knudsen. The average velocity of the particles (vave ) results from the Maxwell distribution of velocities:  vave =

8k B T . π m∗

(2.23)

As for the electrons in the semiconductor, the mass is replaced by the effective mass. At x = 0 the current due to the flow of electrons is: Jth =

1 qn 0 vave . 4

(2.24)

The thermal current is proportional to the concentration of the electrons at the interface (n 0 ). This quantity can be calculated from Eq. 2.4 considering that φ B = E C (x = 0) − E F :   qφ B . (2.25) n 0 = NC · exp − kB T Thus, the thermal current is: Jth =

    1 qφ B qφ B qvave NC · exp − = K NC · exp − 2 kB T kB T

(2.26)

2.3 Biased Metal-Semiconductor Junction

65

where K = 21 qvave . The barrier (qφ B ) can be written as: qφ B = qφi + (E C − E F )bulk , so that the thermal current is:   qφi + (E C − E F )bulk Jth = −K NC · exp − kB T       qφi (E C − E F )bulk qφi · exp − = K N D · exp − = K NC · exp − kB T kB T kB T (2.27) The current across the interface depends on both the doping concentration and the height of the barrier in the semiconductor. When V A = 0, the equilibrium is no longer valid, but the current JM S remains unchanged because V A does not affect φ B . On the contrary, the current JS M changes, because the barrier for the electrons of the semiconductor is reduced by a quantity q V A . Namely:     qφi q(φi − V A ) ; JS M = K N D · exp − . JM S = K N D · exp − kB T kB T

(2.28)

Noteworthy, the applied voltage modifies only the current originated from the semiconductor -namely, the material with less electrons. The total current is:       q VA qφi · exp −1 . (2.29) J = JS M − JM S = K N D · exp − kB T kB T This is the typical I/V characteristic of the diode J = J0 (exp( kqBVTA ) − 1). The quantity J0 is the reverse current and corresponds to the thermal current from the metal to the semiconductor. The reverse current can be written as:   1 qφ B . (2.30) J0 = qvave NC exp − 2 kB T Replacing in Eq. 2.30 the definition of average thermal velocity and NC (Eq. 1.28), we get:      8k B T 1 2π m ∗n kT 3/2 qφ B (2.31) 2 exp − J0 = q 4 π m ∗n h2 kB T which can be written as



qφ B J0 = RT exp − kB T 2

4πm ∗ k 2 q

 (2.32)

where R = hn3 is the Richardson constant. This is the thermionic current equation ruling the working principles of the thermionic devices, as discussed in the preface. As for the free electrons (m ∗n = m 0 ), R = 120 cmA2 K 2 .

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2 The Metal-Semiconductor Junction

The simple thermionic model has a remarkable agreement with the data fitting the experimentally measured current in a range wider than 5 orders of magnitudes in forward bias. 2.3.2.2

The Drift/diffusion Model

The current can also be directly calculated from the drift/diffusion model. In particular, it can be calculated by integrating the drift/diffusion equation in the depletion layer, where both potential and concentration gradient are different from zero. The straightforward hypothesis for this approach is that the depletion layer is sufficiently large to support the definition of mobility and the diffusion constant. This means that the depletion must be at least longer than a few electrons mean free paths. Since both the electric field and electron concentrations are a function of x, the drift and diffusion currents evolve along the depletion layer, so as to maintain a constant current through the device:   dn . J = q nμn E(x) + Dn dx

(2.33)

Using the definition of electric potential and the Einstein relationship, it becomes:   q dφ dn n + . J = q Dn − kB T d x dx

(2.34)

It is worth to remark that the relationship between diffusion constant and mobility has been derived under the thermal equilibrium conditions, so that we are allowed to use it because of the quasi-equilibrium assumption. The solution of Eq. 2.34 can be obtained by integrating over both the sides of the depletion layer, from x = 0 to x = xd and from φ = 0 to φ = φi − V A . ): The integral can be solved multiplying the equation by exp(− kqφ BT xd J

exp(− 0

⎡

q Dn ⎣

xd

qφ )d x = kB T

⎤     xd qφ dφ q qφ dn ⎦ n exp − d x exp − dx − kB T kB T d x kB T d x

0

(2.35)

0

The first of the two integrals on the right-hand side can be solved using the rule of integration by parts f g = f g − f g, where f = n and g = exp(− kqφ ): BT ⎞   xd xd     xd qφ dn qφ dn ⎠ qφ ⎝ q Dn − exp − d x + exp − dx n exp − kB T kB T d x kB T d x 0 ⎛

 = q Dn

 xd qφ n exp − kB T 0

0

0

(2.36)

2.3 Biased Metal-Semiconductor Junction

67

Fig. 2.11 Potential in the depletion layer and its linear approximation. The dashed area is the difference in the integral between approximated and “exact” solutions

with the following boundary conditions: qφ(0) = 0; qφ(xd ) = qφ B − (E C − F )bulk ). E F )bulk − q V A ; n(0) = NC exp(− kqφB TB ); n(xd ) = NC exp(− (EC −E kB T Replacing the boundary conditions, we get:      qφ B − (E C − E F )bulk − q V A (E C − E F )bulk exp − q Dn NC exp − kB T kB T   qφ B )·1 + q Dn −NC exp(− kB T (2.37)        q VA qφ B qφ B exp − NC exp − q Dn NC exp − kB T kB T kB T      (2.38) q VA qφ B exp −1 = q Dn NC exp − kB T kB T Hence, the total current is given by:     q Dn NC exp(− kqφB TB ) q VA exp −1 J = xd  kB T exp(− kqφ )d x BT

(2.39)

0

which can be immediately recognized as the diode equation J = J0 (exp(− kqBVTA ) − 1). To complete the calculation, it is necessary to solve the integral at the denominator, whose solution depends on the applied voltage. Thus, unlike the thermionic model, the drift/diffusion model predicts that reverse current depends on applied voltage. In order to calculate the integral, the analytical form of the potential derived in Eq. 2.13 is used. A simplified solution is obtained by replacing the potential with its first-order approximation. In this way, the integral is underestimated and, since it is at the denominator, the reverse current is overestimated as well (Fig. 2.11). The integral at the denominator of Eq. 2.39 is approximated as:

68

2 The Metal-Semiconductor Junction

    xd qφ q φi − V A exp − x dx d x ≈ exp − kB T kB T xd 0 0    xd q kB T 1 − exp − = (φi − V A ) q φi − V A kB T

xd

(2.40)

The solution can be further simplified considering that φi − V A  k BqT ≈ 26 meV . This condition is surely fulfilled under reverse bias, when V A < 0. With this approximation, the exponential in bracket is negligible with respect to 1, so that, by replacing xd with the expression of Eq. 2.15, we get: xd 0

   kB T 2s qφ dx ≈ . exp − kB T q q N D (φi − V A )

(2.41)

Eventually, the reverse current is:    q q N D (φi − V A ) qφ B J0 = q Dn NC · exp − . kB T kB T 2s

(2.42)

It can be observed that the electrons in x = 0 are subjected to the electric field Emax , so that the electrons injected from the metal to the semiconductor are accelerated by the electric field at the interface and produce a drift current J0 = qn 0 μn Emax . Considering the expression of the electric field at the interface (Eq. 2.9), the depletion layer width under bias (Eq. 2.16) and the Einstein relation, we obtain:   2q N D (φi − V A ) q qφ B · Dn · NC · exp − . J0 = q · kB T kB T s

(2.43)

Apart from a factor 21 , likely due to the approximations necessary to solve the driftdiffusion model, we can say that the reverse current is the current due to the electrons which, from the metal, manage to cross the barrier  B . Note that the reverse current depends on semiconductors doping because of the electric field at the interface, and not because of electron concentration. With respect to the thermionic model, the drift/diffusion model introduces a slight dependence of the reverse current on applied voltage. It is also important to note that a reverse bias independent of the applied voltage would represent an ideal current source. Thus, a slight dependence on the applied voltage is necessary to abide the fundamental laws of electric networks. To this regard, as will be discussed in the next section,  B has a slight dependence on the applied voltage, so that also the reverse current of the thermionic model depends on applied voltage. In this way, the differences between the two models are attenuated.

2.3 Biased Metal-Semiconductor Junction

69

In conclusion, the junction between a N-type semiconductor and a metal whose work function is greater than the work function of the semiconductor, behaves like a rectifier. The reason for non-linearity of the I/V curve lies in the existence of the depletion layer: a region at the surface of the semiconductor depleted of mobile majority charges. Since the interface is depleted of electrons, the applied voltage drops across the depletion layer, so that the barriers keeping the current in equilibrium are differently affected by the applied voltage. The barrier applied to the electrons of the metal (φ B ) is unchanged, while the barrier applied to the electrons of the semiconductor (φi ) is altered. The applied bias modulates only the current from the semiconductor: the forward bias makes this current increase, whereas the reverse bias leads to a decrease in such a current. The current-voltage relationship is non-linear and depends exponentially on the applied voltage. This result can be obtained following either the thermionic current model or the drift/diffusion current model. Both the models give the same result. With respect to the PN junction diode, which will be discussed in Chap. 4, the Schottky diode is usually faster (only majority charges are involved), and the voltage drop in forward bias is smaller. On the other hand, the reverse current is larger, since it is a current from a metal (a large reservoir of electrons), and the behavior of the device is affected by the surface states, whose effects are discussed below. The same behavior can be obtained by considering the symmetry of a junction between a P-type semiconductor and a metal with a smaller work function: φ S > φm . In this case, the device is still a diode and the charge carriers are holes. Due to the smaller mobility of holes, this configuration is not convenient for real devices. The band diagram and the equilibrium electrostatic quantities are shown in Fig. 2.12.

2.3.3 Barrier Height Lowering The electric potential has been calculated above, considering only the double layer of the charges formed by the electrons at the surface of the metal and the fixed charges distributed in the semiconductor. From the point of view of an electron in the depletion layer, an additional electric field must be considered. This supplementary potential results from the fact that a charge in the semiconductor is actually placed in a dielectric material close to a charged conductor plane. It is known from electrostatics that a charge (q) close to the surface of a metal experiences a Coulomb’s force equivalent to that produced by a virtual charge (image charge) of the same magnitude but opposite in sign, placed at the symmetrical point behind the charged plane. The situation is depicted in Fig. 2.13.

70

2 The Metal-Semiconductor Junction

Fig. 2.12 Band diagram, charge density, electric field, and potential of a metal (P-type) semiconductor junction where the work function of the semiconductor is greater than that of the metal

2.4 Non-rectifying Metal-Semiconductor Contact

71

Fig. 2.13 Coulomb’s force between a conductor plane and a charge −q. The image charge is placed at distance 2× from the real charge. The real charge undergoes the effects of the electric field Ei generated by the virtual image charge

The image charge gives rise to an electric field Ei and a potential: ∞ φi (x) = −

Ei d x = x

q . 16π s x

(2.44)

This potential is additive to the one generated by the double layer. The potential of the image charge is confined at few nanometers from the surface. In this space, the potential of the double layer can be linearly approximated as: φ DL = −Emax x. Note that one potential decreases while the other one increases, so that there exists a coordinate where it reaches a maximum value (see Fig. 2.14): q = − Emax xmax → xmax = 16π s xmax



q . 16π s Emax

(2.45)

The barrier lowering is equal to the product of the maximum electric field and xmax , which also depends on the maximum electric field. Eventually, the barrier lowering is proportional to the fourth root of the built-in potential, so that, under applied voltage, it is inversely proportional to the fourth power of the semiconductor potential barrier: 1

φ B = K (φi − V A ) 4 . (2.46) Hence, the height of the barrier applied to the electrons of both metal and semiconductor slightly depends on the applied voltage.

2.4 Non-rectifying Metal-Semiconductor Contact A junction made of a metal and a N-type semiconductor has a rectifying behavior only when the work function of the metal is greater than the work function of the

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2 The Metal-Semiconductor Junction

Fig. 2.14 The potential generated by the image charge (φi ) is added to the built-in potential generated by the double layer (/ phi dl ). As a consequence, the actual barrier (φb ) is smaller than that calculated from the double layer (φb )

semiconductor. We will see in this section that if the relative magnitude of the work functions is inverted the rectifying property is lost and the curve becomes linear. A non-rectifying junction is obtained when an N-type semiconductor is joint to a metal whose work function is smaller than that of the semiconductor. In this situation, the equilibrium is reached via the electrons displacement from the metal to the semiconductor. In the semiconductor, this leads to a region at the interface with the metal where the concentration of electrons is larger than that in the bulk. Such a region is called an accumulation layer. The electrons that had left the metal leave behind them a distribution of positive charges, which, as usual for a metal, forms a thin layer at the interface with the semiconductor. The situation is depicted in the equilibrium band diagram shown in Fig. 2.15. The double layer of charges is now formed by a thin layer of positive charges in the metal and a distribution of mobile electrons in the semiconductor. This gives rise to a built-in potential, whose sign is opposite to that of the previous case and whose magnitude is still given by the difference between the work functions. Unlike metals, semiconductors allow for a charge distribution in their volume. In this way, the excess of mobile charges is not accumulated at the surface but distributed in the semiconductor (see Fig. 2.15c). Furthermore, these mobile charges are not bounded to fixed donor atoms, so that the distribution of the electron excess cannot be easily predicted. The total charge in the accumulation layer is Q = (−n + N D ) − n , where n = N D is the charge in the non-perturbed semiconductor and n are the charges transferred from the metal. The profile of the accumulation charge distribution depends on the profile of the potential. The charge density at the interface is:       qφi qφ B (E C − E F )bulk = NC exp − exp − n s = NC exp − kB T kB T kB T   qφi = N D exp − kB T

(2.47)

where: qφi = qφm − qφs . The perturbation of the electron concentration extends for a distance xa from the surface. The boundaries of the electron concentration are: n(0) = n s and n(xa ) = N D ,

2.4 Non-rectifying Metal-Semiconductor Contact

73

Fig. 2.15 Band diagram and charge density of a non-rectifying junction. a Band diagrams of pristine materials. b Equilibrium band diagram. The downward bending of the conduction band is an evidence of the increase in the electron concentration. The perturbation is extended for a length xa . c Charge density. The N-type doping concentration is emphasized by a dotted line

while the boundaries for the potential are: φ(0) = 0 and φ(xa ) = −φi . Under these conditions, the electron concentration is an exponential function of the potential:  qφ(x) . n(x) = n s exp kB T 

(2.48)

The analytical behavior of the potential and the related accumulation charge distribution are found solving the Poisson equation:   d 2φ ρ(x) qn(x) qn s qφ(x) . =− =− = exp dx2 s s s kB T

(2.49)

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2 The Metal-Semiconductor Junction

The problem consists in determining the behavior of the charge distribution inside a dielectric, whose value on the surface is considered as known (n s ). = −E, so that: To solve the equation, let us consider that dφ dx d 2φ dE dE dφ dE =− =− =E . 2 dx dx dφ d x dφ

(2.50)

The Poisson equation can thus be written as:   qn s qφ(x) dE = . exp E dφ s kB T

(2.51)

This equation can be integrated with respect to dE from 0 to E and with respect to dφ from −φi to φ: E 0

which gives:

qn s EdE = s



φ exp −φi

qφ kB T

 dφ

(2.52)

     qφ 1 2 ns k B T qφi exp E = − exp − . 2 s kB T kB T

(2.53)

Since qφi  k B T , the electric field produced by the accumulation charges is:  E= and the potential is:

  2n s k B T qφ exp s 2k B T

 dφ − = dx

(2.54)

  2n s k B T qφ . exp s 2k B T

(2.55)

The integration of the equation is easily achieved by a separation of the variables: x  0

2n s k B T dx = − s

φ



qφ exp − 2k B T

 dφ

(2.56)

0

so that the solution is:   2  qφ q ns = x + 1. exp − 2k B T 2s k B T

(2.57)

2.4 Non-rectifying Metal-Semiconductor Contact

75

The above equation depends on an parameter of the material known as the Debye length:  s k B T . (2.58) LD = q 2ns The Debye length is a fundamental quantity ruling the separability of the charges in a material. In practice, it is the length scale at which a dipolar charge distribution can be created by an electric field. As the dielectric constant increases, the Debye length becomes larger and the charges, together with the electric field, are more widely distributed within the material. Using the Debye length, the solution of the Poisson equation can be written as:   x qφ =1+ √ exp − 2k B T 2L D

(2.59)

This relation allows to calculate the distribution of the accumulated charges: ρ = qn = qn s exp



qφ kB T

 = qn s  1+

1 √x 2L D

2 .

(2.60)

The excess of charges decays in the semiconductor as x −2 . The potential, which can be directly calculated from Eq. 2.59, has a logarithmic behavior:   x 2k B T . (2.61) ln 1 + √ φ=− q 2L D The depth of the accumulation region (xa ) can also be calculated from the boundary condition: at φ(xa ) = −φi : xa =

√

    qφi −1 . 2L D exp kB T

(2.62)

The Debye length is a reference distance for the accumulation layer size, approx√ imately half of the accumulated charge lying at a distance 2L D from the interface with the metal. As an example, let us consider a N-Type silicon doped with a concentration N D = 1016 cm−3 of donors. The work function is thus qs = 4.25 eV. The junction is formed with a metal with a work function qm = 4.10 eV. Such a work function can be found in aluminum. At the equilibrium, the built-in potential is φi = (1/q)(qm − qs ) = −0.15 V . The concentration of the charges accumulated at the interface is n s = 3.3 · 1018 cm−3 , the Debye length is L D = 2.3 nm and the depth of the accumulation layer is xa = 55 nm. The total accumulated charge is calculated by integrating Eq. 2.60 from 0 to xa , resulting in approximately 1.17 · 1019 C · cm−3.

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2 The Metal-Semiconductor Junction

Fig. 2.16 Applied voltage distribution in the case of an accumulation layer at the metalsemiconductor interface. The voltage drop in the contact region is practically negligible

2.4.1 Ohmic Contact The presence of an accumulation layer, instead of a depletion layer, drastically changes the distribution of the voltage applied to a metal-semiconductor system. Indeed, while the depletion layer corresponds to a region of negligible conductivity, in the accumulation layer the charge concentration is larger than in the rest of the semiconductor. The accumulation layer conductivity is thus larger than that of the bulk. As a consequence, the applied voltage tends to completely drop in the bulk of the semiconductor, so that no power is dissipated in the contact region (Fig. 2.16). The absence of power dissipation is a practical definition of the ohmic contact. Namely, the contact simply vehicles current and voltage to the material of interest. It is important to note that, as discussed in the case of the Schottky diode, the applied voltage modifies the band bending in the contact region, which is also valid for the non-rectifying contact. Here, for negative applied voltages, the band bending tends to decrease, namely the concentration of the accumulated charges becomes smaller until the flat-band condition is reached. At more negative values, the band may bend upwards, so that the initial accumulation layer turns into a depletion layer. Such a change of character, induced by the applied voltage, is common to all the junctions among semiconductors. In particular, it is fully manifested in metal-oxidesemiconductor junctions. Here, it emphasizes the fact that behaviors predicted from the equilibrium condition strictly hold for perturbations around the equilibrium condition. On the other hand, this is the condition under which the quasi-equilibrium hypothesis holds. The effective ohmic characteristics of the junction, however, is valid only in a limited voltage interval. The ohmic contact is also found in the symmetric case of a P-type semiconductor, where the work function of the metal exceeds that of the semiconductor. Eventu-

2.4 Non-rectifying Metal-Semiconductor Contact

77

Fig. 2.17 Doping and work function differences give rise to four possible metal-semiconductor junctions

ally, Fig. 2.17 reassumes the four metal-semiconductor junctions and their behaviors around the equilibrium. It is nonetheless important to remark that the phenomena outlined in this chapter, in particular the charges double layer and the built-in potential, occur between any couple of materials, both metals and semiconductors. The main difference, in the case of metals, is the dimension of the junction. The fact that no electric field can exist inside a metal, makes only possible for the layer of charges displaced at the equilibrium to lie on the bidimensional sheets located at the interface between the materials. The size of the junction is thus extremely short, so that electrons are able to cross it via the tunnel effect. The built-in potential among metals is of the same order of magnitude of the built-in potentials observed in semiconductors. However, these potentials cancel out in any closed networks, so that they do not affect the currents and voltages across the circuit. Anyway, built-in potentials depend on temperature, so that they only vanish if the temperature of all the network junctions stays the same. In the case of temperature gradients in the circuit, built-in potentials do not cancel out and become observable. Thermoelectric phenomena are indeed based upon non-homogenous temperature distribution in the circuits. They are at the basis of important devices, such as the thermocouple (a sensor used for the measurement of temperature differences) and the Peltier cell (an actuator used for the cooling and heating of small masses).

2.4.2 Tunnel Ohmic Contacts The previous section has shown that in order to obtain a ohmic contact with a semiconductor, we need a metal whose work function is particularly related to the work function of the semiconductor. This condition is not commonly met. The work function of a silicon with 1016 cm−3 donors is roughly 4.25 eV, so that this quantity is smaller than the work function of most of the metals of technological interest (see Table 1.4 of Chap. 1). A more convenient approach to the fabrication of ohmic contacts is offered by the tunnel effect. The tunnel effect is a typical quantum phenomenon contradicting the classical physics beliefs. It states that given an energy barrier, a particle whose energy is

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2 The Metal-Semiconductor Junction

Fig. 2.18 Equilibrium conduction band of a system made of metal-N + −N semiconductor. The depletion layer at the metal-semiconductor interface is narrow enough to be crossed by the tunnel effect, while at the junction between the semiconductors an accumulation layer is formed. Due to a large electro concentration in the N + material, the loss of electrons towards the N semiconductor is negligible

smaller than the height of the barrier has a non-zero probability to pass through the barrier. The probability is proportional to the product of the height and the width of the barrier. In practice, if the barrier is confined in a very short space, of the order of nanometers, the probability for the particle to pass through becomes significant. A very narrow Schottky barrier can be obtained with a highly doped semiconductor. Equation 2.16 shows that the size of the depletion layer is inversely proportional to the concentration of mobile charges. Thus, regardless of the difference between the work functions, the junction between a metal and a heavily doped semiconductor always results in a ohmic contact, because the depletion layer is so narrow that the electrons can be transferred from one material to the other even if their energy is smaller than the energy at the top of the barrier. A tunnel ohmic contact in silicon becomes possible when the doping concentration is larger than 1019 cm−3 . This is roughly the limit of degeneracy beyond which the Fermi level is almost inside the conduction band (in the case of N-type) and of the valence band (in the case of P-type). Note that, with such a doping, the Fermi-Dirac function is no longer approximated by the Boltzmann equation, so that the equation used to calculate the device properties are not valid. The ohmic contact requires a thin layer of heavy doped semiconductor in contact with the metal. In this way, the same metal giving rise to a rectifying junction can be used for a ohmic contact. The junction between a normal semiconductor and an heavily doped layer (indicated with a superscript + or − according to the kind of doping) gives rise to an accumulation layer in the normal semiconductor, so that it behaves as a ohmic contact. The situation in terms of conduction band shape at the equilibrium is shown in Fig. 2.18. Figure 2.19 shows a realistic planar configuration of a Schottky diode where the same metal is used for the diode and the ohmic contact.

2.4 Non-rectifying Metal-Semiconductor Contact

79

Fig. 2.19 Principle scheme of a Schottky diode in planar technology. Deep oxide layers insulate the device from the rest of the wafer

2.4.3 Space Charge Limited Current In the previous discussion about charge transport, it has been assumed that the current in the semiconductor is made up of the mobile charges from both the conduction and valence bands. In this condition, the flow of current does not alter charge equilibrium in the semiconductor. Thus, the total charge density remains zero and, according to the Poisson equation, the derivative of the electric field is null, so that the electric field is constant in the semiconductor. This assumption leads to the Ohm law, where current and voltage are proportional. However, in a structure such as the N + − N junction used for ohmic contacts, the + N layer is more heavily doped, so that it can inject, at a sufficiently high voltage, a concentration of charges larger than the donor density in the N-type material. In this condition, the density of the total charge in the semiconductor is no more null, so that a space charge region takes place in the semiconductor. When the injected charges are much larger than N D , the charge density is: ρ = q(N D − n) ≈ −qn. Hence, the Poisson equation can be written as: qn dE d 2 =− . =− dx2 dx s

(2.63)

Replacing n in the current density definition and in the small electric field regime (so that v = μE), we get: dE (2.64) j = qnv = s μE dx which is integrated as: j s μ

L

E dx =

0

Emax

dE dx

(2.65)

0

where L is the length of the N-type semiconductor and Emax is the largest electric field at the edge of the semiconductor: Emax = − VL . Solving the integrals and replacing Emax with the ratio between voltage and size L we get: j=

1 s μ 2 1 s μ 2 E = V . 2 L max 2 L3

A more accurate calculation results in:

(2.66)

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2 The Metal-Semiconductor Junction

j=

9 s μ 2 V . 8 L3

(2.67)

This is known as the Mott-Guerney law. It expresses the deviation from the linear Ohm’s law in those semiconductors where a mobile charge larger than the equilibrium charge is injected into. This behavior is not limited to the N + − N junctions, being it also possible in the ohmic Schottky contacts where the metal injects charges into the semiconductor, or under a localized injection of photogenerated charges.

2.5 Surface States The metal-semiconductor junction occurs at the surface of the semiconductor, where the metallic layer is deposited. In this portion of the material, we cannot neglect the fact that the semiconductor surface is very different from the bulk. Indeed, disregarding the impurities that can be accumulated on the surface, the regular pattern of atoms arranged according to the crystalline structure is broken at the surface, where a part of the bonds of the last layer of atoms is used to bind the lateral atoms. The nature of these bonds is obviously different from that of the periodic crystal, so that the energy of these states is different with respect to the conduction and valence bands. Thus, a number of additional electronic states appears in proximity of the surface. Actually, the modifications begin a few atomic layers below the surface. The density of these states is roughly equal to the density of the surface atoms. Thus, if n 0 is the atom concentration per unit of volume, then the density of the atoms 2

, at the surface is n 03 . In the case of silicon, there are approximately 5 × 1022 atoms cm3 so that the density of the atoms at the surface level, and the surface states, is about . 1015 atoms cm2 The paramount states are the so-called Tamm-Shockley states, whose energy falls within the energy gap of the semiconductor. The maximum density of surface states occurs at about one third the energy gap. Noteworthy, if the semiconductor is N-type, then the Tamm-Shockley states surely lie below the Fermi level. Hence, at the surface level, the electrons provided by doping, instead of populating the conduction band are segregated to the surface states. This means that, in proximity of the material, the semiconductor is depleted of mobile charges, which gives rise to a bend-bending and a built-in potential that naturally occur on the surface of the semiconductor (Figs. 2.20 and 2.21). Furthermore, the density of the surface states is much greater than the surface , then the surface density of density of the dopant atoms. Indeed, if N D = 1017 atoms cm3 : roughly four orders of magnitude smaller dopant atoms is approximately 1011 atoms cm2 than the density of states. Hence, most of the surface states are empty. When such a semiconductor is used for a Schottky diode, the electrons we need to equilibrate the Fermi level are actually provided by the surface states, so that the electrons from the semiconductor are still subjected to the built-in potential due to the surface states. In practice, the metal does not alter the built-in potential. In this

2.6 Summary

81

Fig. 2.20 Band-bending due to a surface distribution of Tamm-Shockley states

Fig. 2.21 In the case of a metal-semiconductor junction, the built-in potential due to the difference between the metal and semiconductor work functions vanishes on the surface region. Thus, the potential barrier applied to the electrons in the conduction band is not affected by the junction, but rather it depends only on the surface states. In this condition the Fermi level of the semiconductor is pinned by the surface states independently of the work function of the metal

condition, the Fermi level is said to be pinned by the surface states. This condition makes the Schottky diode independent of both the metal work function and the doping concentration. Moreover, since the actual distribution of surface states is unpredictable, the device cannot be properly designed.

2.6 Summary Besides to be a ubiquitous element of the electronic devices, the metal-semiconductor junction has been here introduced as the prototype of junctions, and as a tool to define an approach to the study of any junctions between materials. In this chapter, the rule to design the equilibrium band diagram of the junctions has been introduced. The equilibrium band diagram graphically illustrates the character of the junction, so that most of the device properties can be derived by simply observing and interpreting the band diagram. Additionally, some fundamental approximation and assumptions, such as the deep depletion and the quasi-equilibrium statistics, introduced in this

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2 The Metal-Semiconductor Junction

chapter, are used throughout the textbook to calculate the electrostatic properties of the junctions. Metal-semiconductor junctions can behave as either a ohmic or a rectifier contact. The different behavior depends on the mutual relationship between the work functions of the two materials. The rectifier contact leads to a solid-state diode called Schottky diode. The electric characteristics of junctions are manifested through I/V and C/V curves. They show that junctions do not behave as theoretical electric elements and, more importantly, that resistive and capacitive properties are not constant. On the contrary, they depend on the applied voltage. Finally, in this chapter the notion that the current through the devices is ruled by statistics has been presented. In this view, the main effect of an applied voltage is the modulation of barriers heights and, consequently, the probability of the electrons to migrate from one material to another.

Further Reading Textbooks S. Sze, K.K. Ng, Physics of Semiconductor Devices, 3rd edn. (Wiley-Interscience, 2006) R. Muller, T. Kamins, M. Chen, Device Electronics for Integrated Circuits, 3rd edn. (Wiley, 2002) C.C. Hu, Modern Semiconductor Devices for Integrated Circuits. (Pearson College, 2009) B. Streetman, S. Banerjee, Solid State Electronic Devices. (Prentice Hall, 2006) D. Neamen, Semiconductor Physics and Devices. (McGraw Hill, 2003)

Journal papers F. Braun, Uber die Stromleitung Durch Schewelmetalle. Ann. Phys. Chem. 153, 556 (1874) W. Schottky, Halbleitertheorie der Sperrschicht. Naturwissenschaften 26, 843 (1938) N. Mott, Note on the contact between a metal and an insulator or semiconductor. Proc. Cambridge Philos. Soc. 34, 568 (1938) H. Bethe, Theory of the Boundary Layer of Crystal Rectifier. (MIT Radiat. Lab, Rep, 1942) A. Goodman, Metal-semiconductor barrier height measurement by the differential capacitance method. J. Appl. Phys. 34, 329 (1963) F. Padovani, R. Stratton, Field and thermionic field emission in Schottky barriers. Solid State Electron. 9, 695 (1966) C. Crowell, S. Sze, Current transport in metal-semiconductor barriers. Solid State Electron. 9, 1035 (1966) C. Garrett, W. Brattain, Physical theory of semiconductor surfaces. Phys. Rev. 99, 376 (1955) J. Hilibrand, R.D. Gold, Determination of the impurity distribution junction diodes from capacitance-voltage measurements. RCA Rev. 21, 245 (1960)

Chapter 3

Generation and Recombination Processes

3.1 Introduction The concentrations of mobile charges (electrons and holes) in semiconductors are governed by statistical laws that provide their equilibrium average values. Equilibrium means that the average values are maintained by continuous fluctuations of mobile charges subject to creation and annihilation phenomena. Such events at least involve the transition from the states in the conduction band, the valence band and the doping atoms energy levels. These processes, called generation and recombination, are of outmost importance for the properties of devices. They operate so as to maintain the equilibrium condition, represented by the law mass action, when additional charges are injected into a volume of material. The generation of electrons in the conduction band, and holes in the valence band, occurs in intrinsic and doped semiconductors with an important difference: in the intrinsic semiconductor, the electrons and holes are generated together, whereas in doped semiconductors, to the generation of holes and electrons corresponds the generation of a fixed counter charge. It is straightforward that the process of generation is reversible, so that electrons and holes can recombine together in both intrinsic and doped materials. Hence, the charge density in the volume may change as a consequence of both current and generation/recombination events. The total charge variation in a volume of material is described by the continuity equation, which is a useful tool to determine the charge balance.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_3

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3 Generation and Recombination Processes

Fig. 3.1 Charge balance in a volume of semiconductor. Due to current, charges can enter and leave the volume. Furthermore, they can be generated and recombined

3.2 The Continuity Equation Let us consider a volume of semiconductor extended from x to x + d x, with a section area A. A current (e.g. of electrons) Jn (x) is injected into the volume through the area A at the coordinate x, such that a current Jn (x + d x) leaves the volume at the coordinate x + d x. Inside the volume, generation and recombination phenomena may also occur at rates G n and Rn respectively. The continuity equation describes the behavior in time of the total amount of charge (N ) inside the volume. This is defined as (Fig. 3.1): N=

∂n · Ad x. ∂t

(3.1)

In the case of electrons, the charge balance can be written as: ∂n Jn (x) Jn (x + d x) Ad x = A− A + G n Ad x − Rn Ad x. ∂t −q −q

(3.2)

After replacing Jn (x + d x) − Jn (x) = ∂∂Jxn d x, the equation is normalized by Ad x, so that it does not depend on volume. The continuity equation for electrons is thus: 1 ∂ Jn ∂n = + (G n − Rn ); ∂t q ∂x

(3.3)

symmetrically, the continuity equation for holes is: ∂p 1 ∂ Jp =− + (G p − R p ). ∂t q ∂x

(3.4)

These formulas can be further expanded by replacing the current, which in a semiconductor is the sum of the drift and diffusion currents, with (for electrons): Jn = qμn nE + q Dn ddnx . Thus, the explicit form of the continuity equation is: ∂n ∂ 2n ∂E ∂n = μn n + μn E + Dn 2 + (G n − Rn ). ∂t ∂x ∂x ∂x

(3.5)

3.3 Generation and Recombination Phenomena

85

As for holes: ∂p ∂2 p ∂E ∂p = −μ p p − μpE + Dn 2 + (G p − R p ). ∂t ∂x ∂x ∂x

(3.6)

The previous set of equations describes the charge continuity when mobility and the diffusion coefficient are constant. It is worth to remind that this is the case of a uniformly doped material. A typical situation is the steady-state condition, which corresponds to the time-invariant concentration of charges ( ∂n = 0). ∂t

3.3 Generation and Recombination Phenomena The generation and recombination of mobile charges are peculiar phenomena in semiconductors. These processes are manifested as either the production of a pair of mobile charges (generation) or their disappearance (recombination). Ultimately, this implies a charge transition between the valence and conduction bands. The main action of generation and recombination processes is to restore equilibrium, namely the action mass law. The equilibrium condition is perturbed each time the charges are altered by an external cause. Typical cases are the illumination with a flux of photons and the injection or extraction of a current inside or out of a nearby volume of the material respectively. The transition of an electron from the valence to the conduction band requires an amount of energy at least equal to the energy band gap. A change in momentum is also required in some cases, which is nonetheless irrelevant in those materials where the top of the valence band coincides, in the k-space, with the bottom of the conduction band. But otherwise, when the maximum and minimum of the two bands are not coincident in the k-space, the transition requires a change in momentum (k = 0). k = 0 is the direct band gap case, which can be found in those semiconductors formed by the III-V elements of the periodic table, such as Gallium Arsenide (GaAs) or Aluminum Arsenide (AlAs). On the other hand, k = 0 corresponds to the case of an indirect band gap, which is typical of the semiconductors of the IV group, such as silicon and germanium. The probability of the events where energy and momentum change simultaneously is small. The change in energy and momentum requires a sort of three-body interaction. In practice, the additional momentum necessary for the transition is provided by lattice vibrations, which, in crystals, are represented as quasi-particles endowed with both energy and momentum, known as phonons. The small probability indicates that transitions from one band to the other occur at a slow rate, so that they are not efficient in maintaining the equilibrium in the case of fast events, such as illumination with a pulse of light. In these materials, actually, generation and recombination occur with the assistance of impurity states located in the band gap.

86

3 Generation and Recombination Processes

Fig. 3.2 Processes of capture and emission of electrons and holes in a density of defect states at a generic energy level E T inside the band gap

Doping is a typical source of states in the band gap. Doping states are close to the respective band according to the nature of dopants: either acceptors or donors. Besides dopants, which are intentionally added, a number of naturally occurring impurities are also found in the crystal, which may introduce additional electron energy states in the band gap. Thus, the energy gap is not totally forbidden. Inter-gap states may be found at any energy levels. An example are the surface states. In the bulk of the material, the density of the inter-gap states is smaller with respect to the surface, but it is not negligible. It is important to consider that defects in real materials are inevitable, so that a density of states inside the band gap always exists even in intrinsic semiconductors. In the rest of this section, the generation and recombination phenomena driven by the inter-gap states are discussed.

3.3.1 Generation and Recombination Rates In order to derive a quantitative description of generation and recombination, let us consider a semiconductor characterized by a density of inter-gap states (N T ) at the same energy (E T ). These states are spatially localized. They are the manifestation of material defects (impurities or lattice defects). In acceptable materials, defects are so sparse that their states are non-interacting. Consequently, like the doping states, even defect states do not degenerate into bands (Fig. 3.2). The interaction of defect states with the electrons of the conduction band and the holes of the valence band is detailed by four different processes. events : Each process is described by the rate of occurrence, whose dimensions are time cm 3 r1 : capture of electrons; r2 : emission of electrons; r3 : capture of holes; r4 : emission of holes.

3.3 Generation and Recombination Phenomena

87

Let us calculate the rate of capture and emission in the case of electrons. The same procedure holds to determine the rates of the processes involving the holes.

3.3.1.1

The Rate of Capture

Since defects are spatially localized, capture may occur only after the physical encounter of an electron with the defect. Given a volume of material, let us consider a flux of electrons impinging into the volume. The rate of captured electrons is equal to the product of the flux of the incoming electrons and the density of the empty capture states, times the intrinsic capability that a state could capture an electron. This last quantity (indicated as σ ) can be considered as the affinity between the electron and the defect and depends on the nature of the defect itself. The flux of electrons is the total current. However, since thermal velocity is much greater than the velocity of electrons due to both drift and diffusion, the amount of flux of electrons onto a surface inside the material is due to thermal current. Indeed, at a microscopic level and even under either an electric field or a concentration gradient, the instantaneous speed of electrons is the thermal velocity. The flux of electrons impinging onto a plane is calculated by the classical kinetic theory of gases, where the flux is proportional to thermal velocity according to the Knudsen law : F = 41 vth n, where n is the density of electrons. The averagethermal velocity vth is calculated

BT . from the Maxwell distribution of velocity vth = 8k πm ∗ The density of empty capture states is equal to the product of the total density of capture states and the probability that they are not occupied by electrons. This probability is the complementary Fermi-Dirac function to the energy of the states: N T,empt y = N T (1 − f F D (E T )). Hence, the rate of capture of electrons is:

r1 =

1 vth n · σn · N T (1 − f F D (E T )). 4

(3.7)

The quantity σn is the affinity between the impurity and the electron. It is known as capture cross-section and has the dimensions of an area. Sometimes σn is described as the equivalent area of a target, so that the larger is the area, the easier is the capture. Anyway, it defines the occurrence of the capture event without any relations to the actual area of the defect. The σn of gold atoms, a typical capture state in silicon, is of the order of 10−15 cm 2 . Beryllium has the largest capture cross section in silicon: σn = 10−10 cm 2 . The density of defects, as always in silicon, is of the order of 10−15 cm −3 .

3.3.1.2

The Rate of Emission

Emission is the release, back to the conduction band, of previously captured electrons. The rate of emission is simply proportional to the density of electrons in the capture

88

3 Generation and Recombination Processes

states and the intrinsic probability of emission. Differently from the previous case, since the conduction band has a very large density of free states, the density of states at destination is irrelevant. The density of filled capture states is given by the Fermi-Dirac function: N T, f ill = N T f F D (E T ). The rate of emission of electrons is: r2 = N T f F D (E T ) · en

(3.8)

where en is the intrinsic probability of emission. It has the dimensions of the inverse of time and corresponds to the inverse of the average lifetime of an electron in a capture state. It must be noted that σn and en are not probabilities (they should be dimensionless and bounded between 0 and 1). Rather, they are related the chance of occurrence of the respective phenomena.

3.3.1.3

Equilibrium

Capture cross-section and emission constant are related to each other in such a way that their relationship defines the effect of the capture state on the population of charge carriers. In steady-state condition, the mean densities of electrons and holes is constant, so that the rates of capture and emission are the same: r1 = r2 . We get: 1 vth n · σn · N T (1 − f F D (E T )) = N T f F D (E T ) · en 4

(3.9)

from which en is calculated: 1 en = vth n · σn · 4



 1 −1 . f F D (E T )

(3.10)

Since the capture state is in the band gap, the Boltzmann approximation of the Fermi-Dirac function is not valid and the complete form of the Fermi-Dirac function must be used. This leads to:   1 ET − E F . (3.11) en = vth n · σn · ex p 4 kB T The constant of emission depends on the distance between the energy level of the defect state and the Fermi level. Replacing the Fermi level with: E F = E i + E F − E i (where E i is the intrinsic i ), we obtain: Fermi level), and the density of electrons with n = n i ex p( EkFB−E T 1 en = vth · σn · n i · ex p 4



E T − Ei kB T

 .

(3.12)

3.3 Generation and Recombination Phenomena

89

Namely, emission does not depend on the Fermi level, but rather on the defect energy with respect to the intrinsic Fermi level -that is, with respect to the center of the band gap. Similar results are obtained for the rates of the processes involving the holes: r3 =

1 vth p · σ p · N T f F D (E T ); 4

r4 = N T (1 − f F D (E T )) · e p .

(3.13) (3.14)

The equilibrium condition (r3 = r4 ) leads to the expression of the emission constant of holes:   1 Ei − E T . (3.15) e p = vth · σ p · n i · ex p 4 kB T In conclusion, the chance of electron emission increases as the defect level is close to the conduction band, whereas the chance of hole emission increases as the defect level is close to the valence band. As a consequence, the same capture state, if not located at the centre of the band gap, behaves differently according to electrons and holes.

3.3.2 Traps and Recombination Centers Although the probability of electron and hole capture may be equal, their emission rates are different, because they depend on the position of the capture state with respect to the intrinsic Fermi level. Of course, electrons and holes are only re-emitted with the same rate when E T = E i . At the equilibrium, the electron and hole concentrations are stationary. This condition is achieved when r1 = r2 and r3 = r4 . Let us suppose that, for some reasons, the concentration of holes suddenly increases. Then, the rate of capture of holes (r3 ) increases, too. As a consequence, the rate of emission of holes (r4 ) should increase. However, the hole capture also affects the rate of electron capture. Indeed, an increase in holes in the defect states means an increase in empty states and, consequently, an increase in the rate of electron capture (r1 ). The processes r3 and r1 are exclusive, so that the consequence of the capture of a hole is either the capture of an electron or the release of a hole. Which of them prevails, depends on the magnitude of the related rates. There are two extreme situations: r1  r4 and r4  r1 . In the first case, at the end of the process, the number of electrons in the capture states does not change: the hole in excess is eliminated and one electron disappears from the conduction band. In the second case, at the end of the process, the number of electrons in the capture states does not change: the exceeding hole is only temporarily removed and released

90

3 Generation and Recombination Processes

Fig. 3.3 Reaction of traps and recombination centers to an excess of holes. The black dot in the capture state indicates an electron, whereas the white dot indicates a hole. Both traps and recombination centers react to the perurbation capturing the excess hole. After the capture, traps release the hole, while the recombination center captures an electron from the conduction band to recombine the excess hole. Eventually, traps do not alter the charge density, whereas, due to recombination centers, the amount of hole excess is removed, so that the electron density decreases of the same quantity

back to the valence band, while the electrons in the conduction band are not affected by the process. Thus, if r1  r4 , then the exceeding hole is removed by one electron of the conduction band, and such a state is called recombination center. If r4  r1 , then the exceeding hole is captured and re-emitted after a delay time: such a state is called trap. In other words, traps respond to the variation of the concentration of holes restoring the pristine value. In the case of recombination centers, the concentration of holes is still restored to the equilibrium value, but the concentration of electrons decreases (Fig. 3.3). In order to behave like a trap, the probability of emission of a hole must be larger than the probability of capture of an electron. e p depends on the distance between the energy of the defect state from the intrinsic Fermi level. The largest rate of emission of holes is exhibited by those states whose energy is close to the top of the valence band. Symmetrically, the states closer to the conduction band have the largest probability to emit electrons. These conditions are fulfilled by the acceptor and donor states. Thus, it can be concluded that dopant atoms are traps, whereas recombination centers correspond to different kinds of impurities and defects.

3.3 Generation and Recombination Phenomena

91

Fig. 3.4 The continuity equations of electrons and holes (Eqs. 3.3 and 3.4) describe the effect of the recombination centers on a current of holes injected into the volume. The hole current has a negative gradient, so that it decays inside the volume. The electron current has a positive gradient, so that it grows across the volume

Fig. 3.5 The effect of traps on a hole current injected into the volume can be appraised by means of the continuity equation. The hole current has a null gradient, so that it remains constant

3.3.2.1

Effect of Traps and Recombination Centers on Current

The presence of traps and recombination centers affects charge transport. To illustrate the effect of recombination centers, let us consider the case of a volume of semiconductor populated by a density of recombination centers filled with electrons. When a hole current is injected, the holes in excess are recombined (they disappear), but also the concentration of electrons decreases. As a consequence, a steady current cannot exist unless the disappeared electrons are supplied by the adjacent portion of material. If this process is possible, then the hole current is transformed into a current of electrons flowing in the opposite direction. Since the charges have a different sign, a net current is observed. Figure 3.4 shows the role of the recombination centers in the continuity equation. The case of traps is shown in Fig. 3.5. The hole current herein is not modified. In practice, traps extend the transit time of holes across the volume, so that they play a role in the definition of mobility. Since donors and acceptors are traps, this is consistent with the fact that mobility decreases as doping increases.

92

3 Generation and Recombination Processes

The defects resulting in electronic states inside the band gap are important elements in semiconductors. They have two extreme effects on mobile charge concentrations: traps and recombination centers. Traps affect the transit time of charges across the volume, so that they are important constituents of the mobility of the material. On the other hand, the effect of recombination centers is completely different. Recombination centers, indeed, remove the excess of those specific charges that consume the concentration of the other charge carriers. If the decreased charges can be replaced by an external source (typically a ohmic contact), then the recombination centers are able to transform the current of one charge carrier into the current of the other one. This mechanism of charge transport is fundamental, for instance, in PN junctions, where the current injected from one side of the junction to the other is formed by only one of the two charge carriers. For this reason, it is important to define an analytical tool that allows us to treat the recombination and generation phenomena in a semiconductor.

3.4 The Shockley-Hall-Read Generation-Recombination Model In doped semiconductors, the processes of generation and recombination can be easily modeled. As noted in the above section, the energy level of the recombination centers is distant from the conduction and the valence bands. Thus, in a N-type semiconductor, the energy level of the recombination centers likely lies below the Fermi level, whereas in the case of P-type semiconductors, it lies above the Fermi level. As a consequence, in a N-type semiconductor, at such level f F D ≈ 1, whereas in a P-type semiconductor 1 − f F D ≈ 1. Hence, the centers of recombination are always filled with majority charges. In a N-type semiconductor, the equilibrium conditions (r1  r3 and r2  r4 and r1 = r2 and r3 = r4 ) are simultaneously fulfilled, even if holes are few and their capture is a rare event. In such a condition, any increase in hole concentrations elicits a large increase of r3 , whereas in order to maintain the equilibrium only a modest, practically negligible, change in r1 is required. Eventually, the material is ready to recombine any changes in minority charges. An additional argument can be obtained considering the condition under which a capture state acts like a trap or recombination center for a hole. It has been seen in the previous section that a trap occurs when r4  r1 . Let us evaluate, then, the ratio between the two rates: 1 T v · σ p · n i · ex p( Eki −E ) N T (1 − f F D (E T )) · e p r4 4 th BT = 1 = 1 . (3.16) r1 v n · σn · N T (1 − f F D (E T ))] v n · σn · N T (1 − f F D (E T ))] 4 th 4 th

3.4 The Shockley-Hall-Read Generation-Recombination Model

93

If the semiconductor is doped, then n = N D and the holes are minority charges. When σn = σ p (this condition will be discussed again below), the ratio of hole emission to electron capture is: T n i · ex p( Eki −E ) r4 BT = . r1 ND

(3.17)

Thus, the position of the energy level of the capture state with respect to the intrinsic Fermi level is:   r4 n D . (3.18) E i − E T = k B T · ln r 1 Ni If N D = 1016 cm −3 , then, in order to have r4 > 10 · r1 , it must be E i − E T > 478 meV . In silicon, this means that only those states whose energy is less than 80 meV above the valence band act like traps for holes. This is a negligible portion of the band gap, so that all of the states in the band gap are recombination centers for holes. Of course, the same argument holds for electrons in a P-type material. Thus, we can conclude that the totality of the states in the band gap are recombination centers for the minority charges. The difference between the rates of recombination and generation is the generationrecombination function (U): U = R − G = r1 − r2 = r3 − r4 .

(3.19)

The function U can be calculated from the definition of the four rates (Eqs. 3.7, 3.8, 3.13, and 3.14) and the coefficients of emission (Eqs. 3.12 and 3.15). The calculation introduces the following relevant quantities: τn0 =

1 1 ; τ p0 = Nt vth σn Nt vth σ p

(3.20)

which correspond to recombination time for both electrons and holes. These quantities define the typical capture time scale of excess charges. They depend on temperature (vth ), capture cross-section and recombination center concentration. The calculation results in the recombination function U (for the detailed calculation see the Appendix D): U=

τn0



np − n i2    .  E T −E i T p + n i ex p Eki −E + τ n + n ex p p0 i T k T B B 

(3.21)

The numerator of U is different from zero if np = n i2 , namely when the material is in non-equilibrium conditions. However, the function U reacts to a non-equilibrium condition, in such a way that the denominator expresses the magnitude of the reaction.

94

3 Generation and Recombination Processes

The capture of an electron corresponds to the fact that a wandering electron is bonded to a previously empty orbital of the impurity atom, whereas the capture of a hole is the transfer of an electron, previously bonded to an orbital of the impurity, to a nearby silicon atom. Since the recombination event always involves a pair of electrons and holes, then the probability of the processes are rather similar, so that we can assume σn = σ p = σ and τn0 = τ p0 = τ0 . From Eq. 3.19, considering the typical defects density as approximately 10−15 cm −3 and a cross-section for electrons and holes capture σ = 10−15 cm 2 , we get τ0 ≈ 100 ns. The recombination function can thus be simplified as: U=



np − n i2

τ0 p + n + 2n i cos h



E T −E i kB T



(3.22)

−x

). where the definition of the hyperbolic cosine has been used (cos hx = e +e 2 The function U reaches its maximum value when the hyperbolic cosine reaches its minimum, namely when E T = E i . This means that the maximum efficiency of recombination is achieved when E T is close to E i , namely when the energy level of the recombination centers is around the middle of the band gap. As a consequence, in doped semiconductors, the most efficient states are always filled with majority charges. Typical centers of recombination in silicon are gold and copper atoms, whose energy levels are approximately 0.03 and 0.01 eV above the the intrinsic Fermi level respectively. The function U describes the reaction of the semiconductor to non-equilibrium situations. When np > n i2 → U > 0, so that recombination prevails. On the other hand, when np < n i2 → U < 0, so that generation prevails. x

3.4.1 Example of Application of the SHR Model: The Dynamics of Generation-Recombination Phenomena The generation-recombination function (GR function) of Eq. 3.22 can be further simplified in some particular cases. The most interesting and frequent one is the low injection limit, where the density of excess charges is small with respect to the concentration of majority charges. This condition guarantees that the number of majority charges present in the recombination centers is sufficient to recombine any excesses of minority charges. In order to elucidate the use of the GR function, let us consider two examples of symmetrical and asymmetrical non equilibrium conditions.

3.4 The Shockley-Hall-Read Generation-Recombination Model

3.4.1.1

95

Electron-Hole Pairs Creation

Let us consider a semiconductor where the thermal equilibrium (n 0 p0 = n i2 ) is altered by the simultaneous creation of an excess of electrons and holes. This can be obtained, for instance, by a flash of light whose wavelength be sufficiently small to create electron-hole pairs. Notice that this condition is met when the energy of photons is > E gap ). greater than the energy gap ( hc λ Let n and p be the instantaneous charge excesses with respect to equilibrium concentrations: n 0 and p0 . Then, the total charge concentrations are: n = n 0 + n  and p = p0 + p  . Now, let us quantitatively focus on the fate of electrons when the semiconductor is not biased and the charges are homogeneously created in the volume of the semiconductor. These conditions mean that the drift and diffusion currents are both null (Jn = 0). Applying the continuity equation we find: np − n i2 d(n 0 + n  ) dn  dn   .  = = = G − R = −U = − dt dt dt i τn0 p + n + n i cos h EkT B−E T (3.23) The numerator of the GR function can be written as: np − n i2 = ( p  + p0 )(n  + n 0 ) − n 0 p0 = p  n  + n  p0 + p  n 0 .

(3.24)

Under the low injection limit, p  n  is the negligible product of two small quantities. Low injection limit also means that p’ and n’ are much smaller than p0 and n 0 respectively. Furthermore, since the electron-hole pairs are simultaneously created and the recombination time of electrons and holes is equal, then it is p  = n  every time. The U function can be further simplified considering that the most efficient centers of recombination are located at E T = E i , so that the hyperbolic cosine can be replaced by 1. This assumption is supported by the fast growth of the hyperbolic cosine. As an example, the function is ten times its minimum value when the argument is about 3. Thus, the hyperbolic cosine contribution to the GR function becomes negligible if E T − E i > 3k B T , which is a very small quantity with respect to the whole energy gap. We can thus assume hereafter that all the effective centers of recombination lie at E T ≈ E i and that the hyperbolic cosine is replaced with 1. Hence, the U function is: U=

n  ( p0 + n 0 ) . τ0 ( p0 + n 0 + 2n i )

(3.25)

If the semiconductor is N-type, then: p0  n 0 and n i  n 0 . On the contrary, if the semiconductor is P-type, then: p0  n 0 and n i  p0 . In both cases the GR function for the electrons can be written as:

96

3 Generation and Recombination Processes

Fig. 3.6 A flash of light of intensity Iλ shines the semiconductor at time t0 . The absorption of light immediately generates an excess of charges that, thanks to recombination, decays exponentially with a time scale τ0

U=

n . τ0

(3.26)

Besides than on excess charge density, the U function thus depends only on τ0 , namely on the inverse of the product between the density of the centers of recombination (N T ), the thermal velocity (vth ) and the intrinsic probability of capture (σ0 ). The U function determines, via the continuity equation, the fate of the excess charges created by the flash of light: n dn  = −U = − . dt τ0

(3.27)

The equation can be integrated with respect to dn’ from n  (0) and n  (t) and with respect to dt from 0 to t. This results in:  ln

n  (t) n  (o)

 =−

  t t . → n  (t) = n  (o)ex p − τ0 τ0

(3.28)

The charges are completely eliminated after about four times the recombination time (τ0 ). After this interval, another independent process of creation-recombination can take place. This particular time plays an important role in the response time of photoconductors (Fig. 3.6).

3.4.1.2

Non-symmetrical Charge Injection

Let us now consider the case in which only the density of one of the two charge carriers is altered. Excess charges can be due to the injection of a hole or a electron current from a nearby volume inside the material.

3.4 The Shockley-Hall-Read Generation-Recombination Model

97

As an example, let us consider an injection of holes into a N-type semiconductor. In this case, n = n 0 and p = p0 + p  . The doping conditions the material undergoes to stay the same and the low injection limit still holds ( p  and p0  n 0 and n i  n 0 ). Furthermore, the centers of recombination are located around the band gap center, so that the hyperbolic cosine can be replaced with 1. The generation-recombination function can be thus simplified as in the previous case: p (3.29) U= . τ0 The same expression is also found in the case of excess electrons injected into a P-type semiconductor. Hence, we have introduced a simple and compact version of the GR function that can be used in the continuity equation for any deviations from equilibrium, which is only valid when the low-injection limit is fulfilled. This equation is an important tool to calculate the current in PN junction-based devices, such as the PN diode and the Bipolar Junction Transistor.

3.4.2 The Generation-Recombination Function for Direct Band Gap Materials The GR function previously derived is valid when the processes of generation and recombination are mediated by inter-gap states. This is the typical process occurring in non-direct band-gap materials, where the probability of direct transitions from valence to conduction band and vice-versa is low. On the other hand, in direct band-gap semiconductors, the band-to-band transition is highly probable. Direct band gap means that the top of the valence band is aligned, in the k-space, with the bottom of the conduction band. This is the case of the semiconductors of the III-V group of the periodic table, such as gallium arsenide (GaAs), indium phosphide (inP), or aluminum arsenide (AlAs). At the thermal equilibrium, the recombination rate is simply proportional to the concentration of electrons and holes: R = βn 0 p0 . At the equilibrium, R = G, so that the generation rate can be defined as G = βn 0 p0 , where β is a constant of the phenomenon. In the case of a symmetrical excess of charges, such as the one described in Sect. 3.4.1.1, the recombination process is increased, but generation remains fixed at the equilibrium value. Thus: R = β(n 0 + n  )( p0 + p  ) and G = βn 0 p0 .

98

3 Generation and Recombination Processes

The generation-recombination function is: U = R − G = β(n 0 + n  )( p0 + p  ) − βn 0 p0 .

(3.30)

If the low-injection limit is fulfilled, then n  p  is negligible, so that: U = β(n 0 + p0 )n  =

n . τ0

(3.31)

The above equation holds in a N-type semiconductor, where p0 is negligible. Thus, τo = (βn 0 )−1 is the the recombination time. An identical expression is obtained in the case of a P-type material. Under the low-injection limits, the generation-recombination functions of both indirect and direct band-gap semiconductors are analytically identical. Notice that recombination time depends on the number of recombination centers, which are N T when the recombination is mediated by the inter-gap centers, and n 0 in the case of a band-to-band recombination. Eventually, when the low-injection limit is satisfied, the continuity equation can be solved independently of the kind of semiconductor.

3.5 Summary Generation and recombination phenomena are peculiar of semiconductors. The possibility to generate and recombine charges makes the continuity equation non-trivial and provides the ground to evaluate the propagation of minority charges in semiconductors. In this chapter, an equation for the recombination and generation has been derived. It is valid for indirect bandgap semiconductors, such as silicon. The equation greatly simplifies if the excess of minority charges remains much smaller than the concentration of majority charges. This condition, called “low-injection limit”, will be used in the following chapters. Interestingly, the recombination function of direct band gap semiconductors, such as GaAs, has the same form of the low-limit injection approximation recombination function in silicon. This result makes the derived formula universal, so that all devices produced with different semiconductors can be treated with the same formalism.

Further Reading Textbooks S. Sze, K.N. Kwok, Physics of semiconductor devices 3rd edn. (Wiley-Interscience, 2006)

Further Reading

99

Journal Papers R. Hall, Electron-hole recombination in Germanium. Phys. Rev. 87, 387 (1952) J.R. Haynes, W. Shockley, The mobility and life of injected holes and electrons in Germanium. Phys. Rev. 81, 835 (1951) C.T. Sah, R.N. Noyce, W. Shockley, Carrier Generation and Recombination in p-n Junctions and p- Junction Characteristics. Proc. IRE 45, 1228 (1957) W. Shockley, The theory of p-n junctions in semiconductors and p-n junction transistors. Bell Syst. Tech. J. 28, 435 (1949) W. Shockley, W. Read Jr., Statistics of the recombination of holes and electrons. Phys. Rev. 87, 835 (1952) D.T. Stevenson, R.J. Keyes, Measurement of carrier lifetimes in Germanium and Silicon. J. Appl. Phys. 26, 190 (1955)

Chapter 4

PN Junction

4.1 Introduction The PN junction is one of the principal elements of semiconductor-based devices. In particular, it is paramount in the fabrication of diodes, bipolar junction transistors and a plethora of sensors, not least photodetectors. Moreover, the modeling of a PN junction reveals the peculiar properties of semiconductors. The ideal PN junction is a homogeneous piece of semiconductor characterized by an abrupt change in the character of dopant atoms: at x < 0, the material is P-type doped by a uniform distribution of acceptors N A , whereas at x > 0 it is N-type doped by a uniform distribution of donors N D . Obviously, this is just an ideal approximation of real PN junctions. Actually, the distribution of dopant atoms is not uniform, so that junctions are not so sharp. However, the model derived from the ideal junction is surprisingly accurate in the description of the principal electric properties of the junction on a realistic level. PN junctions can be formed by any couple of P and N semiconductors. To this regard, the noteworthy cases are both homojunctions, formed by the same semiconductors, and heterojunctions, formed by different semiconductors. In this chapter, homojunctions are treated, whereas heterojunctions will be discussed in Chap. 7. In terms of band diagram, a homojunction is made up of two materials sharing the same affinity and band gap, but with different work functions (Fig. 4.1). The properties of materials are thereafter specified with the suffixes p and n, which obviously stand for P-type and N-type respectively.

4.2 PN Junction at the Equilibrium The formation of the ideal junction can be thought of as the instantaneous union of two separated materials kept in equilibrium at the same temperature. Once joint, the combination of materials reaches a new equilibrium condition following the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_4

101

102

4 PN Junction

Fig. 4.1 Band diagrams of the two constituents of an ideal PN junction

same steps outlined in Chap. 2, except that this time both electrons and holes must be considered. Electrons, indeed, migrate from the material with the smallest work function towards the material with the largest work function, and vice versa for the motion of holes. In other words, electrons start off from the N-type material in order to migrate into the P-type one, whereas holes get transferred from the P-type material to the N-type one. Charges transfer proceeds until the Fermi level becomes uniform in the entire device. At the beginning, the charges injected from one material into the other are majority charges. But otherwise, they turn into minority charges as they reach the destination regions, where they are indeed quickly recombined by majority charges. The result of these processes is the depletion of majority charges on both the sides of the junctions. This leaves a density of fixed charges made up of ionized dopant atoms in the proximity of the junction. Eventually, a double layer of oppositely charged regions (positive in the N-type side and negative in the P-type) is formed. The very consequence of this charge density is the formation of an electric field that prevents any further charges from being transferred, so that an equilibrium of the system is established. It is worth to remind, once again, that equilibrium means a balance between opposite currents. The equilibrium band diagram is shown in Fig. 4.2. At the equilibrium, the whole system is formed by two unperturbed bulk regions connected by a space charge region (in particular, a depletion layer). The total charge density in any volume is the sum of the concentrations of the four kinds of available charges: Q = q( p − n + N D − N A ). The charge densities of each zone are listed in Table 4.1. Both the electric field and potential are calculated from the Poisson equation solved in the space charge region, where the total charge is non-zero. Also, in this case the electrostatic characteristics of the junction are calculated under the deep depletion hypothesis, assuming that the total charge in the depletion layer is only contributed by the donor and acceptor atoms. Thus, the Poisson equation is: q d 2φ = − (N D − N A ). dx2 s

(4.1)

4.2 PN Junction at the Equilibrium

103

Fig. 4.2 Band diagram at the equilibrium Table 4.1 Charge condition in the three regions of the whole semiconductor region size holes density electrons density total charge density P neutral zone

x < −x p

pp = NA n i2 ND

N neutral zone

x > xn

pn =

Space charge region

−x p < x < xn

p < NA

n i2 NA

ρ=0

nn = N D

ρ=0

n < ND

ρ = 0

np =

Fig. 4.3 Total charge distribution under the deep depletion approximation

The charge density distribution under the deep depletion hypothesis is shown in Fig. 4.3. The striking feature of this distribution is the sharp transition from the neutral zone to the depletion layer. Let us remark that this is an ideal representation of the real charge distribution. A modification of the model will be introduced thereafter, so as to at least remove the abrupt transitions at the borders of the space charge region.

104

4 PN Junction

The amounts of charge density in the two sides of the depletion layer are: +Q = +q N D xn and −Q = −q Na x p . Since the total charge must be null, the following relationship holds: N D xn = N A x p

(4.2)

Equation 4.2 shows that the size of the depletion layer is inversely proportional to doping. Namely, the depletion layer mostly extends on the less doped side of the junction. Obviously, the two sides of the depletion layers are symmetrical only when the concentrations of the dopant atoms stay the same, as in Fig. 4.3. Charge distribution is the input used to calculate both electric field and potential. The electric field is given by: ρ dE = . dx s

(4.3)

On the N-type side of the depletion layer (0 < x < xn ), under the constant doping hypothesis, the electric field is: E(x  n)

xn

dE = E(x)

x

q ND q ND d x → E(xn ) − E(x) = (xn − x). s s

(4.4)

Given that semiconductor bulks are neutral zones, the boundary condition is E(xn ) = 0, so that the electric field in the N-type side of the depletion layer is: En (x) =

q ND (x − xn ). s

(4.5)

Similarly, in the P-type part of the space charge region, we get: E(x)

x dE = −

E(−x p )

−x p

q NA q NA d x → E(x) − E(x p ) = − (x + x p ). s s

(4.6)

With the boundary condition E(−x p ) = 0, the electric field on the P-type side is: E p (x) = −

q NA (x + x p ). s

(4.7)

Since the material is homogeneous, the electric permittivity is constant and the electric field is continuous at the interface (En (0) = E p (0)), where its absolute value is maximum:

4.2 PN Junction at the Equilibrium

105

Emax = −

q NA q ND xp = − xn s s

(4.8)

It is worth noticing that Eq. (4.8) is similar to Eq. (4.2) and corresponds to an alternative form of the null total charge condition. The potential is calculated from the electric field (dφ = −Ed x) on the two sides of the depletion layer. The boundary conditions for the calculation of the potential are: φ(x ≥ xn ) = φn and φ(x ≤ −x p ) = −φ p , where φn and −φ p are the potentials in the neutral zone as defined by Eqs. (1.73) and (1.74), and where φ = E F − E i : x < 0 → φ(x) = −φ p +

x > 0 → φ(x) = φn −

q NA (x + x p )2 ; 2s

q ND (x − xn )2 . 2s

(4.9)

(4.10)

Note that the potential in x=0 is null only if the concentrations of acceptors and donors stay the same (N A = N D ). The null potential occurs where the semiconductor becomes intrinsic ( p = n = n i ). Noteworthy, in the case of asymmetric doping, the potential becomes null inside the depletion layer of the less doped semiconductor. Thus, on this side of the depletion layer, the region located between the one obeying the condition φ = 0 and the interface is populated with a larger number of minority charges than majority charges. Such a situation is known as inversion and is paramount in metal-oxide-semiconductors junctions. The net potential drop across the entire junction is the built-in potential. It can be calculated from the potentials in the unperturbed neutral zones, φbi = φn − φ p , which depend on the concentration of majority charges in the respective regions (see Sect. 1.6):     ND NA kB T kB T ; φp = − ; (4.11) ln ln φn = q ni q ni from which: φi =

       NA kB T ND NA ND kB T + ln = . ln ln q ni ni q ni

(4.12)

As an example, if N D = N A = 1016 cm−3 , then the built-in potential is approximately 0.72 V. The total size of the depletion layer is a consequence of the condition of the potential continuity in x = 0: φn −

q ND 2 q NA 2 x = φp + x . 2s n 2s p

(4.13)

106

4 PN Junction

Fig. 4.4 Electric field and potential at the equilibrium

Thus, introducing the built-in potential, we obtain: φi = φn − φ p =

q (N D xn2 + N A x 2p ). 2s

(4.14)

Replacing xn with the expression derived from the charge neutrality condition (Eq. 4.2): xn = NNDA x p , the depletion layer in the P region can be calculated:  ND 2φi s  . (4.15) xp = q N N2 + N N2 D

A

A

D

Replacing x p , the depth of the depletion layer in the N region is found:  NA 2φi s  . xn = q N N2 + N N2 D

A

A

(4.16)

D

Eventually, the complete depletion layer expression is:    1 2φi s 1 . + xd = xn + x p = q NA ND

(4.17)

4.2 PN Junction at the Equilibrium

107

4.2.1 Removal of Charge Discontinuity at the Depletion Layer Border Under the hypothesis of deep-depletion, the concentration of majority charges becomes abruptly negligible at the borders of the depletion layer. The relationship between charge density and potential is ruled by an exponential function, so that even a small increase in E C − E F may result in a large change in electron (and, analogously, in hole) concentration. This makes the deep depletion hypothesis inside the depletion layer plausible. However, around the border of the depletion layer a smooth transition must exist. Figure 4.5 shows a qualitatively more realistic behavior of both the mobile total charge. In this section, a quantitative expression of the behavior of the potential around the border of the space charge region is evaluated. It is important to remark that the result is common to any depletion layers, independently of the material. In particular, the same conclusions hold for the metal-semiconductor junctions described in Chap. 2. To calculate the potential around the border of the depletion layer, let us consider the Poisson equation. The calculation herein is for the N-type side of the junction. Of course, on the other side, a similar calculation can be carried out. The total charge in the N-type material is contributed by both donors and electrons. Unlike the donor concentration, which is constant by hypothesis, the electron concentration varies with potential via the Poisson equation. The latter can be written as:    q q qφ d 2φ N . (4.18) = − (N − n) = − − n exp D D i dx2 s s kB T At the border of the depletion layer, the actual potential is smaller than φn , so that it can be written as: φ = φn − φ  . The deep depletion hypothesis results in φ = φn at x = xn . The smooth electron density transition from 0 to N D is represented by the potential φ  . Replacing φ = φn − φ  in the Poisson equation, we get:      d 2 (φn − φ  ) qφ  q qφn N D − n i exp exp − . (4.19) =− dx2 s kB T kB T Considering that φn is a constant and that the first exponential defines the electron concentration in the neutral zone (see Eq. 4.11), namely the donor concentration, we obtain:    qφ  q ND d 2φ 1 − exp − . (4.20) = − − dx2 s kB T Around xn , φ  is a small perturbation, so that the exponential can be replaced with its first order approximation (e−x = 1 − x):

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4 PN Junction

Fig. 4.5 Qualitative behavior of mobile charge (electrons and holes) and total charge made by mobile charges plus acceptors and donors

d 2φ q ND − =− 2 dx s

  qφ  q2 ND  1−1+ =− φ. kB T s k B T

The previous equation contains the Debye length, defined as L D = Thus, the Poisson equation can be written as: d 2φ φ = dx2 L 2D

(4.21) 

s k B T q 2 Nd

(4.22)

whose general solution is: φ  = A exp( LxD ) + B exp(− LxD ). The constants A and B depend on the boundary conditions. The first boundary conditions is φ  (0) = φ0 , which is the deviation with respect to φn at x = xn . For sake of simplicity, the origin of the coordinate x is translated to x = xn . As for the second boundary condition, let us assume that the length of the bulk is much greater than the Debye Length, so that the perturbative potential φ  (x) vanishes inside the semiconductor. The two boundary conditions result in A = 0 and B = φ0 . Eventually, the excess potential due to the smooth behavior of the total charge is φ  = φ0 exp(− LxD ). The additive term for the electric field is:   φ0 dφ  x  . (4.23) = exp − E =− dx LD LD

4.2 PN Junction at the Equilibrium

109

Even the electric field decays exponentially beyond xn and x p towards the bulk of the semiconductor. The Debye length is the length scale of the exponential decay. In practice, the electric field becomes negligible beyond approximately four Debye lengths from the border of the depletion layer calculated under the deep-depletion approximation hypothesis. The Debye length depends on both permittivity and doping. As for silicon, in the case of N A = N D = 1016 cm−3 , the Debye length is L D = 40 nm. This must be compared to the corresponding depletion layer width (Eq. 5.17): xd = 332 nm. The deep-depletion approximation can be nonetheless applied. Anyway, on the outside of the depletion layer, the electric field is small but non-zero. In the description of electronic devices, such a small field can be neglected. But otherwise, it is reasonably relevant in the characterization of, for instance, light detectors, such as photodiodes and solar cells. Both the depletion layer size and the Debye length depend on doping concentrations. It may be of interest to determine the relationship between these two quantities. For this scope, let us consider a symmetric diode (N D = N A ), where the potential at the origin is null, so that φ0 = φn − q2NsD xn2 = 0. Replacing the expression for φn , we obtain:     s k B T ND ND 2 2 = 2L D ln . (4.24) ln xn = 2 2 q ND ni ni Thus, the ratio of the depletion layer size to the Debye length is:    xn ND . = 2ln LD ni

(4.25)

The ratio of the depletion layer size to the Debye length is rather independent of doping concentrations. At N D = 1016 cm−3 , the ratio is 5.25. If N D is 100 times larger, then the value increases up to 6.06.

4.2.2 Physical Configurations Figure 4.6 shows the simplified sketch of a planar PN junction. The actual area of the device is defined by the density of the field lines connecting the two terminals. As a first approximation, the device can be restricted to the separation surface parallel to the surface of the semiconductor. The device can be fabricated in a P-type semiconductor. A circular area is defined through a suitable lithographic step at the surface. In the selected area, a N-type region is created via the diffusion of N-dopant atoms until a donor concentration greater than that of acceptors is reached. The most used methods for doping are ion implantation and gaseous deposition. In the first case, dopant atoms are ionized and accelerated towards the semiconductor

110

4 PN Junction

Fig. 4.6 Principle scheme of a PN junction in planar technology. The basic subtrate is a P-type silicon, so the N-type well is formed in a volume where donors concentration dominates the density of acceptors. Ohmic contacts are made up of N + and P + implantations. As shown in the top view, for sake of symmetry, the P-type contact form a ring around the N-type

surface. In the second method, dopant atoms are vaporized in the gaseous phase and absorbed by the semiconductor through a solubility process. Diffusion enables doping atoms to cover a relevant volume of the semiconductor. Either implanted or absorbed atoms are diffused under controlled temperature. At low concentrations, the dopant profile is properly described by both the diffusion current and the continuity equations, which correspond to the first and the second Fick’s laws: J = −D

∂C(x, t) ; ∂x

∂C(x, t) ∂ 2 C(x, t) . =D ∂t ∂x2

(4.26)

The solution of these equations depends on the boundary conditions. In the case of ion implantation, the amount of atoms is limited, so that as the dopants penetrate the material, the concentration at the surface decreases. The dopant profile is exponential. In the case of absorbed atoms, the concentration at the surface is maintained constant by the atoms in the gaseous phase. In this condition, the concentration inside the material evolves ∞ as2 the complementary error function (erfc), which is defined as: er f c = √2π x e−t dt. In both methods, the implantation of donors in the P-type material results in a non-constant profile. The junction occurs where the added dopant concentration becomes smaller than the background dopant concentration. Figure 4.7 shows an example of the dopant concentration profile in the case of doping from the gaseous phase. Around the junction, the profile of the doping atoms can be approximated by a linear function. Thus, instead of being constant, the concentration of charges in the depletion layer linearly grows from the junction towards the borders of the space charge region. The electric field is proportional to the integral of the charge density, whereas the potential is proportional to the integral of the electric field. Hence, electric field and potential vary with the square and the third power of distance respectively. Qualitatively, this results in a small perturbation in the electrostatic quantities calculated under the hypothesis of constant donor concentrations.

4.3 The Current in the PN Junction

111

Fig. 4.7 Example of solution of the diffusion equations in the case of gaseous deposition. The example shows the diffusion of boron atoms into a silicon wafer doped with 1016 cm−3 donors.The process occurs at about 1100◦ C. At this temperature, boron diffusion constant is D = 2.96 · 20−13 cm2 /s. The concentration of boron atoms at the surface is C0 = 1019 cm3 . The solution of the diffusion equation is C(x, t) = C0 er ˙ f c( 2sqrxt(Dt) ), where t is time. The curves are calculated at t=2 h, 1 h and 20 min. The junction, where the doping character of the material switches from N to P, occurs at a distance from the surface in the range 1 − 2.5 µm approximately

Fig. 4.8 Ideal sketch a PN junction. The device is biased by a voltage V A applied through two ohmic contacts. For sake of simplicity, the N type material is kept grounded. The interface between the two materials fixes the origin of the coordinate x along which current flows

4.3 The Current in the PN Junction In order to calculate the current-to-voltage relationship of a PN junction, let us consider the ideal device sketched in Fig. 4.8. The device is made up of a space charge region and two neutral regions on the P and N type sides. Ohmic contacts are provided at each end of the device. The ideal diode model is monodimensional and the current density J is calculated, so that the measurable current is I = J A, where A is the cross-sectional area of the device. The voltage V A is applied between the P-type material electrode and the N-type one. For sake of simplicity, the N-type material is grounded. As discussed in Chap. 2,

112

4 PN Junction

Fig. 4.9 Band diagram modification due to the voltage V A applied according to the condition in Fig. 4.8. The Fermi level is no more uniform as a consequence of the non equilibrium condition

the applied voltage is almost completely found across the depletion layer. This is a fundamental assumption for the study of the current through the device. As a consequence, the electric fields in both the neutral zones are null, so that no drift current can flow through the semiconductor bulk. As a result of the applied voltage, the charge energy in the P-type semiconductor is shifted by a quantity q V A , while the the charge energy in the N-type semiconductor remains unchanged. This means that the change in electron energy on the P-type side is −q V A . The built-in potential thus becomes qφi = q(φi − V A ), so that all the quantities depending on the built-in potential change as a consequence. For instance, the depletion layer size turns into:    1 2(φi − V A )s 1 . (4.27) . + xd = xn + x p = q NA ND The built-in potential (φi ) keeps the entire system in equilibrium when the total current across each section of the device is zero. In the equilibrium condition, the concentrations of majority and minority charges in the neutral zones obey the mass action law. The change in the built-in potential barrier depends on the sign of the applied voltage. The condition V A > 0 is known as forward bias, which corresponds to the potential barrier lowering case. The change in potential makes the mass action law no more valid. Barrier lowering increases the concentration of minority charges in both the neutral zones. The built-in potential, indeed, prevents majority charges from diffusing in the other region, where their concentration is much smaller. Hence, with respect to the equilibrium condition, an excess of minority charges is observed in the neutral zones: np →

n i2 n2 + n  ; pn → i + p  NA ND

(4.28)

4.3 The Current in the PN Junction

113

where n p is the concentration of electrons and pn is the concentration of holes in the P-type material. The excess of minority charges is responsible for current. However, in the regions between the depletion layer and the electrodes, no electric field is present, so that charge collection at the electrodes is not straightforward. Rather, current is driven by recombination phenomena. Under the opposite bias condition (V A < 0), known as reverse bias, the potential barrier is increased, so that the transfer of minority charges from one material to the other is favored. Due to the different number of majority and minority charges, the two currents are obviously different on a quantitative level. This very difference leads to the rectifier character of the PN junction. As a matter of fact, under direct bias, the large population of majority charges is transferred to the other region, where they give place to an excess of minority charges, which in turn are recombined according to the Shockley-Hall-Read (SHR)recombination law. Under reverse bias, on the contrary, the same transfer towards the other region concerns the small population of minority charges, which results in an almost negligible increase in majority charges. Eventually, the properties of the PN junction depend on the fate of minority charges. The total current through the device is made up of two main contributions. The dominant component is the current due to the increase in minority charges in the neutral zone. The second additional term, which is dominant in reverse bias, is due to the processes of generation and recombination in the space charge region. The first contribution to current is also known as ideal diode current.

4.3.1 Ideal Current The ideal current model is a consequence of the change in minority charge concentration at the borders of the neutral zones due to applied voltage. The calculation is carried out using the quasi-equilibrium and low-injection limit main assumptions. According to the quasi-equilibrium approximation, even under bias, both electron and hole concentrations can be calculated via the statistical equilibrium equations. As for the low-injection limit assumption, on the other hand, the minority charge excess concentration in a region is much smaller than the corresponding majority charge concentration. At the equilibrium, the density of minority charges is approximately 11 orders of magnitude smaller than the density of majority charges. Hence, the low-injection condition is not fulfilled, until after the excess charge reaches this level. This condition allows to consider the simplified version of the generationrecombination function (Eqs. 3.27 and 3.30). At the equilibrium (V A = 0), the number of majority charges at the coordinates xn and −x p is: n n0 (xn ) = N D ; p p0 (−x p ) = N A ;

(4.29)

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4 PN Junction

whereas the number of minority charges is:     n2 qφi qφi = N D exp − = i n p0 (−x p ) = n n0 (xn ) exp − kB T kB T NA

(4.30)

and 

qφi pn0 (xn ) = p p0 (−x p ) exp − kB T



  n2 qφi = N A exp − = i . kB T ND

(4.31)

The subscripts p and n indicate the junction sides, whereas the subscript 0 stands for equilibrium . Under bias (V A = 0), the concentration of minority charges at the borders of the neutral zone is altered into:   n2 q(φi − V A ) > i (4.32) n p (−x p ) = N D exp − kB T NA together with:   n2 q(φi − V A ) > i . pn (xn ) = N A exp − kB T ND

(4.33)

The creation of this excess of minority charges at the borders of the neutral zones is actually the chief effect of applied voltage, which keeps their concentration constant. Thus, even though these charges are removed by recombination, their concentration at xn and −x p remains constant. Excess charges concentration can be written as:     q(φi − V A ) qφi − N D exp − n p (−x p ) = n p − n p0 = N D exp − kB T kB T      (4.34) q VA qφi exp −1 = N D exp − kB T kB T and     q(φi − V A ) qφi − N A exp − pn (xn ) = n p − n p0 = N A exp − kB T kB T      q VA qφi exp −1 . = N A exp − kB T kB T

(4.35)

4.3 The Current in the PN Junction

115

The minority charge excess (electrons in the P-type region and holes in the N-type one) being created at the border of the neutral zones, the electric field in the neutral zone is null. Consequently, the current collected at the electrode is a diffusion current. This means that in order for a constant current to be measured, a constant gradient of minority charges must be established in the neutral zones. The gradient of minority charges is a consequence of both recombination and diffusion. The excess of minority charges activates the recombination processes, which tend to restore equilibrium. Under the low-injection limit hypothesis, the generationrecombination function can be written in a simplified form, so that the U function on the two sides of the junction is: Un = (R − G)n =

n p τn

; U p = (R − G) p =

pn . τp

(4.36)

Let us now calculate the current due to the holes injected onto the N-type side of the junction. Calculations are symmetrical for the electrons on the other side of the device. To calculate the effect of the recombination, let us start from the continuity equation for holes: 1 ∂ Jp ∂ pn =− − U. ∂t q ∂x

(4.37)

The total concentration of holes is pn = pn + pn0 . Since the electric field is null, current is a diffusion current: J p = −q D p ∂∂pxn . Moreover, being doping uniform, the derivatives of pn0 with respect to time and space are null. Replacing the recombination function, diffusion current and total holes concentration in the continuity equation, we obtain: ∂ pn ∂ 2 pn pn = Dp − . ∂t ∂x2 τp

(4.38)

Let us restrict the analysis to the steady-state case, namely to d.c. current. Some considerations about transient phenomena will be given at the end of this chapter. ∂ p Steady state means that ∂tn = 0, so that the continuity equation for the holes injected into the N-type region is reduced to: ∂ 2 pn pn = . ∂x2 L 2p

(4.39)

The quantity L p = D p τ p is known as recombination length. It is a measure of the hole penetration depth deep inside the N-type semiconductor. The solution of the continuity equation is a linear combination of two exponential functions:

116

4 PN Junction



pn (x)

x − xn = A exp

Dpτp





x − xn + B exp −

Dpτp

(4.40)

which can also be written in terms of hyperbolic functions: pn (x) = A∗ sinh



x − xn Lp



+ B ∗ cosh



x − xn Lp

 ,

(4.41)

where A∗ = A−B and B ∗ = A+B . The constants are calculated from the boundary 2 2 conditions. The hole concentration steady-state profile is achieved when the equilibrium between the hole diffusion from the space charge region towards the interior of the semiconductor and recombination, which tends to eliminate the hole excess with respect to thermal equilibrium, is reached. The whole process is ruled by the diffusion length. A short L p is found in materials characterized by an efficient recombination of holes. It is worth to remind that recombination time τ p is inversely proportional to the density of recombination centers and the corresponding hole capture crosssection (see Eq. 3.21). Moreover, D p is proportional to mobility, so that it is inversely proportional to doping. The spatial distribution of holes is paramount in current calculation. It is important to bear in mind that even though this function is steady in time, charges are continuously injected from the depletion layers and recombined by the recombination centers in the neutral zone. The current is directly calculated from the definition of diffusion current. It is proportional to the derivative of the excess hole concentration profile, which is calculated by imposing boundary conditions to Eq. (4.41). The first constraint concerns the excess hole concentration at xn , while the second one is fixed by the electrode position. At the electrode, all the excess charges are eliminated, so that pn (W B ) = 0. This means that: pn (x = xn ) → B ∗ = pn (xn ) p  (xn ) pn (W B ) = 0 → A∗ = − n W B . tanh( L p ) Finally, the distribution of excess holes is given by:

 sinh( x−xn )   x − xn Lp   pn (x) = pn (xn ) cosh − . W Lp tanh( L pB )

(4.42)

(4.43)

The analytical behavior of excess charges depends on the distance between the electrode and the depletion layer with respect to the recombination length. The function approximates an exponential as W B  L p and a linear function as W B L p . Figure 4.10 shows the behavior of the excess holes at different W B /L p ratios.

4.3 The Current in the PN Junction

117

Fig. 4.10 Decay of the normalized concentration of the injected excess holes in a N-type semiconductor calculated from Eq. (4.43) at different W B /L p ratios. Note that, as W  L, the behavior tends to an exponential function and, on the other hand, when W L, it tends to be linear

As a consequence, also current depends on the neutral zone width, defined as the distance between the border of the depletion layer, where excess charges are injected from, and the electrode, where charges are collected by the external circuit. To this regard, it is convenient to discuss the two limit cases, known as long base diode and short base diode, where the solution of the continuity equation assumes two extreme conditions. The terms used to describe the asymptotic conditions are clearly referred to the bipolar transistor, since the PN junction is the fundamental block of this device. An expression for current in each of these extreme situation is hereafter calculated.

118

4 PN Junction

Fig. 4.11 Steady state distribution of excess hole concentration in the neutral region of the N-type semiconductor. The charge excess pn is added to the hole concentration at the equilibrium ( pn 0 )

4.3.1.1

Long Base Diode

The long base diode condition is met when the distance between the electrode and the depletion layer is much larger than the diffusion length. In accordance with the symbols in Fig. 4.8, the long base diode corresponds to the following conditions: W B − xn  L p in the N-type material and −W E + x p  L n in the P-type material. The two zones are independent, so that it is possible to make a diode where the bases are long and short respectively. Since L p is much shorter than the distance that must be traveled to reach the electrode, all the excess holes are recombined before leaving the semiconductor (Fig. 4.11). This condition applied to Eq. (4.40) results in A = 0 and the solution of the continuity equation is limited to the decreasing exponential function. Thus, the constant B in Eq. (4.40) is calculated from the other boundary condition: pn (xn ). Eventually, the hole concentration steady-state profile is: pn (x)

=

pn (xn )



x − xn exp − Lp

 .

(4.44)

Replacing pn (xn ) with its expression (from Eq. 4.35), we have:           n2 q VA q VA qφi pn (xn ) = N A exp − exp exp −1 = i −1 . kB T kB T ND kB T (4.45) Namely: pn (x) =

n i2 ND

      q VA x − xn exp . − 1 exp − kB T Lp

(4.46)

From this profile, the diffusion current is calculated: J p (x) = −q D p

D p n i2 ∂ pn =q ∂x L p ND

      q VA x − xn exp . − 1 exp − kB T Lp

(4.47)

4.3 The Current in the PN Junction

119

Obviously, the excess hole current is not constant in space, but it replicates the excess hole density profile. In particular, as holes travel inside the material, current decreases, and it vanishes well before reaching the electrode. In this condition, no current should be recorded in the external circuit. However, hole recombination is achieved by consuming the concentration of majority charges (electrons), in such a way that the hole concentration gradient is complemented by a specular electron concentration gradient. As a consequence, a corresponding electron current originates from the electrode in order to supply the necessary electrons for the recombination process. Furthermore, it is important to note that these electrons are necessary to keep the total charge in the neutral zone and the electric field at zero. Recombination electron current can be calculated solving the continuity equation for the electrons in the neutral region N. The equation is solved in the steady-state = 0: condition, namely for ∂n ∂t 1 ∂ Jn − Un = 0. q ∂x

(4.48)

Since to each recombined hole corresponds the disappearance of one electron, the rate of recombination of electrons is exactly equal to the rate of recombination of n) holes, so that Un = p (x−x . Thus, replacing p  (x − xn ) with Eq. (4.44), the current τp Jn as a function of x can be calculated by integrating the continuity equation over the neutral zone: 

Jn (x) = q · p (xn ) ·

x exp(− ξ −xn ) Lp xn

τp

dξ

(4.49)

whose solution is: Jn (x) = q · p  (xn ) ·

   Lp x − xn 1 − exp − . τp Lp

From the definition of L p , it is easy to deduce that p  (xn ), the electron recombination current is:

Jn (x) = q

Lp τp

=

Dp . Lp

(4.50) Thus, replacing

      D p n i2 q VA x − xn exp . − 1 1 − exp − L p ND kB T Lp

(4.51)

The behaviors of Jn and J p are complementary. Namely, from the junction towards the contact J p decreases, while Jn increases. Thus, the total current J = Jn + J p is constant throughout the entire material. It is interesting to note that the electrons necessary for the recombination are provided via the ohmic contact by the external circuit, which plays the role of “electron reservoir”.

120

4 PN Junction

Fig. 4.12 The current in a long base diode currents is due to the change in the concentration of the minority charges at the border of the neutral zones. The current of holes and electrons are marked in red and blue respectively. The dashed line represents the current through the depletion layer where additional processes therein occurring must be accounted for a complete description of the currents across the PN junction

Eventually, the total current flowing through the N-type material due to the holes injected from the P-type side of the junction is the sum of Eqs. 4.47 and 4.51:     D p n i2 q VA exp −1 . (4.52) JN = J p (xn ) = q L p ND kB T A similar calculation can be carried out for the electrons injected into the P-type material, which give rise to the current flowing through the P-type side of the junction:     q VA Dn n i2 exp −1 . (4.53) J P = Jn (−x p ) = q Ln NA kB T Finally, the total current is the sum of the two currents. Note that only in the case of equal doping the two currents have the same magnitude:  Jtot = JN + J P = qn i2

Dp Dn + NA Ln ND L p



 exp

q VA kB T



 −1 .

(4.54)

The majority charges transferred from one material to the other are continuously provided by the electrodes, in such a way that the total current is constant throughout the whole device. The ideal current is the result of the change in minority charge concentration at the interface between the space charge region and the neutral zone. Additional phenomena occurring in the depletion layer, however, must be accounted for a complete description (Fig. 4.12).

4.3 The Current in the PN Junction

121

Fig. 4.13 Holes profile in a forward-biased N-type region of a short base diode

4.3.1.2

Short Base Diode

The other geometric limit occurs when the distance between the electrode and the depletion layer is much shorter than the diffusion length. Namely: W B − xn L p in the N-type material and −W E + x p L n in the P-type material. The solution can be directly derived from Eq. (4.40), considering that the short base diode condition, corresponds to a small argument of the exponential function. Thus, the solution of the continuity equation can be linearized as: pn = A + Bx. Applying the boundary  and pn (W B ) = 0 the steady-state profile of holes is: conditions: p  (xn ) = pn0

pn

=

 pn0

       n i2 x − xn q VA x − xn 1− = exp , −1 1− W B ND kB T W B

(4.55)

where W B is the distance between the contact and the border of the depletion layer (Fig. 4.13). The linear profile of holes gives rise to a constant diffusion current, whose expression is:     n i2 q VA − 1 . (4.56) exp Jp = q D p N D W B kB T The short distance between the contact and the junction is such to make the recombination of holes negligible. Hence, unlike the long base diode, the injected holes produce a constant current from the junction to the contact. The same result would have been obtained assuming that the recombination processes are negligible and that all the injected holes recombine at the ohmic contact. On a practical level, this means that the excess holes only recombine with the electrons of the metal contact, so that the hole current through the contact surface is compensated by an electron current flowing from the external circuit to the contact. Under this assumption, U p ≈ 0, whereas the flowing current is still the diffusion current (even when it is small, the semiconductor is a neutral zone):

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4 PN Junction

∂ pn ∂ 2 pn = Dp = 0. ∂t ∂x2

(4.57)

The above equation, set to zero to indicate the steady-state case, leads to a linear pn . Even though recombination is negligible, a recombination current of electrons must exist. This current can be calculated, under the steady-state condition, from the continuity equation (Eq. (4.48)). Replacing Un = p  (x)/τ p and p  (x) as shown in Eq. (4.56), we obtain:    W B 1 − x−x n W B Jn = q · p  (xn ) · dx (4.58) τp xn

whose solution is: Jn = q · p  (xn ) ·

    q VA W B n i2 exp −1 . 2τ p N D kB T

(4.59)

Reminding the recombination length definition, the ratio of injected-hole diffusion current (Eq. 4.57) to electron recombination current is thus: L 2p Jp = 2  2. Jn WB

(4.60)

Since the short-base diode condition is W B L p , then the recombination current is absolutely negligible, so that it can be assumed that the total current is therein given by the injected hole diffusion current. We will reconsider the short diode recombination current in the bipolar junction transistor section. A similar condition is obviously found for the electrons injected in the P-type material, so that the total current in a short base diode is (Fig. 4.14):  Jtot = JN + J P =

qn i2

Dp Dn + N A W E N D W E

    q VA exp −1 . kB T

(4.61)

Note that this equation is formally similar to the long-base diode current, the only difference being the typical length parameter involved in the equation. In the shortbase diode model, the distances between the ohmic contact and the depletion layer (W E and W B ) replace the long-base diode diffusion lengths (L n and L p ). The depletion layer size dependence on applied voltage in turn introduces a slight dependence of the reverse current on V A . Moreover, being W E and W B smaller than L n and L p respectively, the short-base diode reverse current is larger. In both cases, though, the reverse current is more contributed by the less doped material.

4.3 The Current in the PN Junction

123

Fig. 4.14 The current in short-base diode is due to the change in minority charges at the borders of the neutral zones. Currents of holes and currents are marked in red and blue, respectively. The dotted line represents the current through the depletion layer. Additional processes therein occurring must be accounted for a complete description of the currents across the PN junction

However, as it will be shown in the next section, reverse current is dominated by the generation-recombination processes in the depletion layer. 4.3.1.3

Evaluation of the Approximations

A discussion about the validity of the quasi-equilibrium and neutral zone assumptions under which the ideal current has been derived is necessary before calculating the contribution due to the current across the depletion layer. The quasi-equilibrium hypothesis requires that the applied voltage elicits a small perturbation of the equilibrium condition. In this case, the statistical laws used to calculate the electron and hole concentrations are still valid. The neutral zone condition, on the other hand, assumes that the voltage drops in the bulks of the semiconductors are negligible. Numerical examples can help give reasons of the two assumptions. Let us consider a junction between two pieces of silicon, whose doping concentrations are: N D = N A = 1016 cm−3 . With such doping levels, the mobility of holes 2 2 and μ p = 500 cm . From the mobility, and electrons (see Fig. 1.24) is: μn = 1200 cm Vs Vs 2 cm2 the diffusion constants at T = 300 K are calculated: Dn = 31.2 s and D p = 13 cms . Let us consider τn = τ p = τ0 and τ0 = 1/(N T σ vth ), where N T is the density of the recombination centers, σ is their cross-section and vth is the thermal velocity. With N T = 1015 cm−3 , σ = 10−15 cm2 and vth = 107 cm/s, the recombination time is τ0 = 10−7 s. From the diffusion constant and the recombination time we have: L n ≈ 17 µm and L p ≈ 11µm. The built-in potential and the depletion layer are calculated with Eqs. 4.12 and 4.17: φi = 0.71 V and xd = 0.43 µm. At the equilibrium, the total current across the depletion layer is zero. Actually, it is the result of the algebraic sum of the drift and diffusion currents. An estimation of the average hole diffusion current in the depletion layer is obtained considering the net change in concentration across the depletion layer:

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4 PN Junction

J p0 = −q D p

p xd

(4.62)

where p = 1016 − 104 cm−3 . The hole diffusion current is of the order of J p0 ≈ 482 cmA 2 . Under forwards bias, applying the formula of the ideal current with V A = 0.6 V, we find that the forward current is Jtot = 0.65 cmA 2 . Namely, the forward current is more than seven hundred times smaller than the currents kept in balance by the thermal equilibrium. Thus, forward biasing introduces just a small perturbation in the equilibrium state, so that the quasi-equilibrium condition becomes plausible. In order to verify the neutral zone assumption, we must calculate the voltage drop in the bulk of the semiconductor. The resistivity is calculated from both mobility and the doping densities (ρ N = 1/q N D μn and ρ P = 1/q N A μ p ): ρ N ≈ 0.51 cm and ρ P ≈ 1.25 cm. To calculate the actual resistance, it is necessary to introduce the dimensions of the device. In the case of a long-base diode (namely, the worse condition where resistance is larger), we can assume W = 100 µm as large enough to extinguish the recombination processes. As for the area, let us consider A = 10−5 cm2 . Hence, the resistances are: R N = 520 and R P = 1250 and the current is I = Jtot A = 3.3 × 10−5 A, so that the voltage drops are VN = R N I = 0.0029 V and

V P = R P I = 0.0070 V. Less than 2% of the applied voltage falls in the neutral zones, so that more than 98% is found across the space charge region. Of course, as current increases, the voltage drop in the neutral zone cannot be neglected, so that the actual voltage across the space charge region is V A − R I , where R is the total resistance of the two neutral zones. In this condition, the voltage is: V A = R I + k BqT ln( II0 − 1).

4.3.2 Generation and Recombination Current Ideal current depends on the change in minority charge concentration at the borders of the depletion layer due to applied bias. Charge flow through the depletion layer is indeed contemplated by the ideal current model regardless of the processes therein occurring. But nonetheless, both majority and minority charges change the equilibrium concentrations as they cross the layer, when forward and reverse biased respectively. The alteration induced by the injected charges, and the consequent generation-recombination processes that tend to restore equilibrium, affect the total current through the device, even though the density of the recombination centers is constant throughout the material and does not depend on doping. At the equilibrium (V A = 0), the hole and electron concentration is significantly smaller than the corresponding doping values ( p N A and n N D ), but the mass action law is still valid (np = n i2 ), so that the recombination function is null. The change in charge concentration under bias (V A = 0) is such that the recombination function is different from zero.

4.3 The Current in the PN Junction

125

In the case of forward bias, the depletion layer becomes narrow and, due to the injection of majority charges, both the hole and electron concentrations increase. As a consequence, recombination processes prevail. On the other hand, under reverse bias, the size of the depletion layer increases and, due to the direction of the electric field, the residual majority charges in the depletion layer are dragged away from the region. The decrease in mobile charges activates generation phenomena. In order to evaluate these phenomena, it is necessary to calculate the generationrecombination function (U) under these two conditions. Since the hole and electron concentration in the depletion layer is very small, the bias-induced variation violates the low-injection limit in such a way that the complete form of the U function must be considered. But otherwise, the proximity of the efficient recombination centers to the intrinsic Fermi level and the approximation of the hyperbolic cosine to 1 (Eq. 3.22) are still valid assumptions. Moreover, it is still σn = σ p -namely, one single recombination time (τ0 ). As a consequence, the U function stays the same on both sides of the space charge region: U = R−G =

np − n i2 . τ0 [ p + n + 2n i ]

(4.63)

Generation and recombination processes give rise to an additional current, which is calculated solving the continuity equation: 1 ∂ JG R ∂n = − U. ∂t q ∂x

(4.64)

In the case of a steady current, all the derivatives with respect to time are null, so that the current is: xn JG R = q

U d x.

(4.65)

−x p

In order to evaluate U, we first estimate the product np. This can be promptly calculated at the borders of the neutral zones, where charge concentrations are known. At the border of the P-type zone, namely at x = −x p , it is: n(−x p ) p(−x p ) = (n p0 + n p ) p p0 =  2       ni n i2 q VA q VA 2 exp −1 N A = n i exp . + NA NA kB T kB T

(4.66)

Similarly, at the border of the N-type zone: n(xn ) p(xn ) = n n0 ( pn0 + pn ) =  2      ni n2 q VA q VA − 1] N D = n i2 exp . + i exp( ND ND kB T kB T

(4.67)

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4 PN Junction

The product np remains the same at two extremities of the depletion layer. It can take any values inside the region; however, the concentrations of n and p in the depletion layer change in the opposite direction, so that it is plausible to assume that their product remains constant. Replacing np in the recombination function we obtain: U = R−G =

n i2 [exp( kqBVTA ) − 1] τ0 [ p + n + 2n i ]

.

(4.68)

It is easy to see that, as previously anticipated, if V A > 0, then U > 0 and recombination dominates, while if V A < 0, then U < 0 and generation is the dominant phenomenon. The function U can be further approximated considering its maximum value. Since p and n are at the denominator, the recombination and generation phenomena are more efficient where the sum p + n takes its minimum value. We have thus assumed that the product np is constant. The following two relations lead us to calculate the conditions that maximize the function U: d( p + n) =0 dx d( pn) = 0. dx

(4.69)

Expanding the derivatives, we find: dp dn + =0 dx dx d( p) d(n) +n =0 p dx dx

(4.70)

from which we derive that the recombination function is maximum where p = n. The behaviors of p, n and p+n in the depletion layer calculated at the equilibrium are shown in Fig. 4.15, in the case of symmetrical doping. Under bias, the modifications shift both the p and n response curves either upward or downward, their analytical behaviors being nonetheless maintained. The concentrations of holes and electrons where U is maximum are: p = n = √ pn = n i exp( 2kq VB AT ). Replacing all this conditions in the U function we obtain:     n i2 exp( kqBVTA ) − 1 n i2 exp( kqBVTA ) − 1   U= = τ0 [ p + n + 2n i ] τ0 2n i exp( 2kq VB AT ) + 1   (4.71) n i exp( kqBVTA ) − 1  . = 2τ0 exp( 2kq VB AT ) + 1

4.3 The Current in the PN Junction

127

Fig. 4.15 Charge carrier concentrations in the depletion layer in the case of 1017 cm−3 symmetrical doping. Charge distributions are calculated applying Eqs. (2.73) and (2.74), where potential is the built-in potential (Eqs. 4.9 and 4.10). The sum of electrons and holes get its minimum value exactly at p = n. It can be shown that this condition also holds for Na = N D

Under forward bias, if V A  k B T /q, then exp( kqBVTA )  −1 and exp( 2kq VB AT )  +1. Thus: U FB =

ni q VA ). exp( 2τ0 2k B T

(4.72)

Finally, integrating from −x p to xn , we obtain the forward generationrecombination current:   qn i q VA FB . (4.73) xd exp JG R = 2τ0 2k B T The calculations have been performed approximating the U function to its maximum value, which leads to a slight overestimation of the actual generationrecombination current. However, this approximated result correctly describes the exponential dependence on applied voltage. Under reverse bias, if V A k B T /q, then exp( kqBVTA ) −1 and exp( 2kq VB AT ) +1. Thus: U RB = −

ni . 2τ0

(4.74)

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4 PN Junction

Hence, the reverse bias GR current is: JGRRB = −

qn i xd . 2τ0

(4.75)

It is worth to remind that xd depends on the root square of the applied voltage, so that, even if not explicitly indicated, reverse current depends on applied voltage. Forward and reverse currents can be combined together so as to provide the contribution of the generation-recombination processes in the depletion layer to the total current:     q VA qn i −1 . (4.76) xd exp JG R = 2τ0 2k B T Eventually, the total current in the diode is the sum of the ideal and the generationrecombination currents: J = Jideal + JG R . (4.77) On a practical level, the two currents merge in one single expression via a nonideality factor η emphasizing the balance between the two currents. The value of η is between 1 and 2, being 1 when Jideal dominates and 2 when JG R prevails:     q VA −1 . I = I0 exp ηk B T

(4.78)

Figure 4.16 shows a numerical example related to a silicon PN junction, where N D =N A =1018 cm−3 . The device in this example is in the long-base configuration. The recombination lengths are 36 µm in the P-type material and 55 µm on the N-type side. Diffusion current is proportional to n i2 , whereas GR current is proportional to n i , so that the relative magnitude of JG R with respect to Jideal depends on n i , which is a function of the energy gap (see Eq. 1.36). In small band gap semiconductors, such as germanium, n i is large, so that diffusion current dominates over GR current. As a consequence, the ideality factor is close to 1. Further deviations from ideality occurs at large applied voltages, where the lowinjection condition is no more fulfilled. A number of effects occur under high injection. Among them, a decrease in the slope of the I-V curve due to the fact that large currents result in non-negligible voltage drops across the neutral zone, as mentioned in the previous section.

4.4 Capacitive Effects The capacitance associated with the metal-semiconductor junction was discussed in Chap. 2. Similarly, also the PN junction presents a capacitive behavior. The first contribution to the total capacitance is associated with the depletion layer. As in the metal-semiconductor junction, the value of this capacitance is calculated considering the relationship between the charge present in the space charge region

4.4 Capacitive Effects

129

Fig. 4.16 Numerical example of GR and ideal currents in a silicon PN diode. GR current dominates in reverse bias regime and at small forward biases. In order to allow for a logarithmic representation the currents are plotted as absolute values

and the applied voltage. The connection between these two quantities lies in the dependence of the depletion layer width on applied voltage:    1 2s 1 xd = (φi − V A ). + (4.79) q NA ND Since the charge in the two regions is the same (Q =| q N A x p |=| q N D xn |), the capacitance can be calculated considering either the P-type or the N-type side of the depletion layer: C=

xp xn dQ = q ND = q NA . d VA d VA d VA

(4.80)

Let us consider the N-type side. The relationship between the extension of the depletion layer on one side and the total size of the depletion layer is: xp =

NA NA xn ; xd = xn + x p ; xn = xd . ND ND + NA

(4.81)

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4 PN Junction

Thus: C=

N D N A d xd dQ =q . d VA N D + N A d VA

(4.82)

Calculating the derivative of xd and factoring inside the square root all the terms except s , we obtain the same result of the metal-semiconductor junction capacitance (C j ): Cj =

s . xd

(4.83)

It is worth to remind that this is a capacitance density, its units of measurement being Farad/cm2 . The actual device capacitance is obtained multiplying the capacitance density by the area of the device. Such a capacitance is associated with any depletion layer wherever it occurs. It must be noted that, under forward bias, the depletion layer becomes smaller and the capacitance tends to be large. As a consequence, the associated impedance 1 ) becomes negligible. On the other hand, the capacitance value is significant (Z = jωC under reverse bias.

4.4.1 Minority Charge Storage and Diffusion Capacitance Density The peculiar current transport in the PN junction is at the origin of additional capacitances contributing to the total impedance of the device. These capacitances account for the charge accumulated in the neutral zones. In order to achieve a steady current, it is indeed necessary to establish constant electron (on the P-type side) and hole (on the N-type side) profiles. Since charges do not accumulate instantaneously, from a zero current condition to a forward bias one there must be a transit time during which charges accumulate and a charges profile is established. Because the amount of charges is a function of applied voltage, a capacitance effect is found. The time required to accumulate and to deplete these charges determines the PN junction response time. In order to calculate the stored charges, let us consider the total quantity of excess holes in the N-type region: 

W B Qp = q xn

pn (x)d x.

(4.84)

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131

In the case of a long-base diode, pn (x) is given by Eq. 4.44 and the integral is solved  n ≈ 0. applying the long-base diode condition: W B − xn  L p → exp − W BL−x p Thus, the amount of excess holes is:     n i2 q VA Qp = q −1 . (4.85) L p exp ND ηk B T Replacing the hole current (Eq. 4.52), a simple relationship between excess charges and current is found: Q p = Jpτ p.

(4.86)

In other words, the current is given by the ratio of the steady stored amount of charges to recombination time. This equation emphasizes how this charge controls the current flowing through a PN junction and establishes recombination time as being the factor regulating the relationship between injected charges and current. In the case of the short-base diode, the excess charge profile is given by Eq. (4.55) and the amount of excess charges is:     n i2 q VA W B − xn exp −1 . (4.87) Qp = q ND ηk B T 2 Considering the short-base diode current (Eq. 4.56), the following expression for the amount of stored charges is found: Q p = Jp

(W B − xn )2 2D p

(4.88)

B −x n ) where τtr = (W 2D is the time charges take to transit through the neutral zone. In p other words, it is the charges transit time in case of a steady diffusion current. This can be easily demonstrated by applying the definition of electric current to diffusion current (qp ddtx = q D p ddpx ) and considering that a stationary diffusion current corresponds to a linear profile of charge: p(x) = kx: 2

dx dp =p → Dp dx dt

t

1 dt = Dp

0

L 0

t=

1 Dp

L

p dp dx

dx → (4.89) 2

L kx dx = . k 2D p

0

In both short-base and long-base diode, the net charge is proportional to current and the dimension of the proportionality term is time. This is the, recombination time and the transit time for the long-base and short-base diode respectively. A fast

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4 PN Junction

Fig. 4.17 Equivalent circuit of PN junction. The conductance G is derived from the I/V relationship: G = ddVIA = q k B T I0

exp( kqBVTA )

device requires less charge to be accumulated a condition achieved by increasing the recombination centers in a long-base diode and by high doping in a short-base diode. We can thus conclude that in both cases the net charge is proportional to the time charges spend in the neutral zone, either before recombination or before being collected by the contact. Since this charge is modulated by the applied voltage, it defines a capacitance, which is known as diffusion capacitance (Cd ). The above calculations have a significant consequence. Since a steady diffusion current requires a steady charge profile, the diffusion capacitance is a necessary term for the diffusion current. Thus, any time a diffusion current exists, the semiconductor is not a pure resistor. dQ The diffusion capacitance is defined as Cd = d VAp and the charge can be written as Q p = Q p0 (exp( kqBVTA ) − 1), where Q p0 is different in the cases of long and short-base diode. Of course, the same calculations apply to the charges stored on the P-type side of the device. Thus, the diffusion capacitance densities are:     q q VA q q VA p n Q p0 exp ; Cd = Q n0 exp . (4.90) Cd = kB T kB T kB T kB T Since doping can be different, the two capacitances may also be different. Diffusion capacitances are important in forward bias and, as mentioned before, they rule the response time of the device. An equivalent circuit based on lumped elements can be derived considering that conductivity and the three capacitances depend on the same voltage (V A ). Consequently, the equivalent circuit of the PN junction is formed by a resistor and three capacitors in parallel as in Fig. 4.17. The presence of the diffusion capacitances emphasizes the fact that the steadystate is reached only when an amount of charge is steadily present in the neutral zones. Diffusion capacitances rule the diode behavior under forward bias. In particular, the transitory times necessary to turn a diode on or off depend on diffusion capacitances. To illustrate these effects, let us consider a couple of examples related to a long-base diode switch. The first example considers the circuit in Fig. 4.18, where a PN diode is biased by an ideal current generator. The switch is turned on at t = 0. At t = 0− the diode is in the equilibrium condition with I D = 0. In order to obey the Kirchhoff law, at

4.4 Capacitive Effects

133

Fig. 4.18 a: current source bias circuit; b: behavior of the excess charges in the neutral zone. In the figure the holes in the N-type zone are shown. Different curves correspond to the different times after the switch is turned on. The gradient at xn is shown. Note that the gradient is constant over time. As the voltage across the diode increases, xn is shifted to the left. This effect is not represented in figure; c: time behavior of the voltage across the diode; d: diode current vs time.

t = 0+ the current becomes immediately I0 . In this condition, the voltage across the diode can be calculated by inverting the diode characteristics: VD = k BqT ln( II0s ). The former is valid when Is  Io , namely under forward bias. Actually, this is the voltage steady-state value. The ideal current of the diode has indeed been calculated solving the continuity equation under the steady-state assumption ( ddtJ = 0). The charge corresponding to this current cannot be accumulated instantaneously, so that the voltage reaches its steady state value only after a certain time. Apparently, it is not possible to obtain a current that immediately changes from 0 to Is . However, it must be considered that current is not proportional to the amount of charge, but to the derivative of the charge profile, and, in particular, to the derivative calculated at the edge of the space charge region (xn and −x p ). Hence, the total amount of charge can grow with time according to the current/voltage relationship of the capacitance, but nonetheless achieve the profile that accounts for I D = I S in a very short time. Figure 4.18 shows the progression of the charge profile and the behavior of the current and the voltage across the diode. Junction capacitance also elicits peculiar behaviors manifested when the diode is switched off. To this regard, let us consider the example in Fig. 4.19. At t < 0, the switch T1 is off and the switch T2 is on, and the diode is in forward regime. Let us suppose that the diode has been kept in this condition for a time sufficient to establish

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4 PN Junction

Fig. 4.19 a: Circuit emphasizing the transition from forward to reverse bias. b: Excess charge behavior after forward-to-reverse transition. Note that the depletion layer shifts to the right. c: Ideal current behavior. Ideal current is actually superimposed to the reverse current due to generation phenomena in the space charge region

the steady state. At t = 0, T1 is instantaneously closed and T2 is open, which causes the diode to be reverse biased at t > 0. To switch off the diode means to deplete the neutral zones of the accumulated charges until no excess charges are present in the bulk of the material. The excess charges profiles are maintained by means of both minority charge injection through the depletion layer and charge recombination in the neutral zone. By applying voltage to the depletion layer only, the latter is immediately reverse biased as soon as the voltage sign swaps, so that majority charge injection in the neutral zone ends abruptly and charge concentration at the edge of the space charge regions decreases. On the other hand, recombination requires time to consume excess charges. The decrease at the interface is faster than recombination, so that, except for a small region close to the interface, the charges profile changes slowly. The decrease in charges at the interface gives rise to an inversion of the gradient and thus to an inversion of the current direction. As a consequence, immediately after switching, the current reverses its sign and maintains a large value until the interface is depleted of excess charges. After that, the recombination processes eliminate the charges in the bulk and the forward current decreases all the way down to zero, until reverse current is the only circulating current.

4.5 Breakdown Phenomena

135

4.5 Breakdown Phenomena The exponential relationship between current and voltage strongly limits voltage drops in forward biased diodes. On a practical level, in a silicon PN junction, at around 0.7 V, a slight increases in voltage elicits a very large change in current, which might dissipate such a large power to destroy the device. Consequently, the voltage across the forward biased diode remains fixed. On the contrary, reverse current is very small and only weakly variable with voltage, so that reverse voltage is virtually unlimited. However, the voltage applied to the narrow region of the depletion layer can give rise to such a large electric field to trigger non-linear phenomena leading to a sharp increase in the reverse current. These phenomena, generally known as breakdown, find their origin in the avalanche effect and the Zener effect. The avalanche effect is a classical phenomenon occurring when the electrons in the depletion layer are accelerated up to a kinetic energy sufficient to ionize the atoms which the electrons impact upon. The Zener effect, on the other hand, is based on the quantum tunnel effect. Both the effects take place around the interface, where the electric field is maximum (see Fig. 4.4).

4.5.1 Avalanche Effect The avalanche effect can be treated in the frame of classical physics. It depends on the amount of kinetic energy that an electron acquires under the influence of an electric field. This energy is given by: λ Ed x

E = q

(4.91)

0

where λ is the free mean path of the electron defined as the average distance traveled by an electron between two consecutive impacts. In order to activate the avalanche effect, the energy of the impacting electron must be sufficient to ionize the atom (E gap ) and to provide the emerging particles with kinetic energy. This can be approximately calculated considering a free electron impacting upon a lattice atom with a velocity (v0 ) and producing an electron-hole pair. Let us assume that the three particles (two electrons and one hole) have the same mass and, after the impact, also the same kinetic energy. Hence, from the conservations of energy and momentum we have: 1 2 3 mv0 = E gap + mv 2f 2 2 mv0 = 3mv f .

(4.92)

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4 PN Junction

Fig. 4.20 Behavior of the electric field at the equilibrium and under reverse bias. The electric field necessary to activate the avalanche is achieved in a region across the interface

The threshold energy for the avalanche effect is thus E = 21 mv02 = 23 E gap . Despite the very simple model, this is a reasonable value for the expected ionization kinetic energy. As shown in Fig. 4.20, the avalanche effect occurs within a restricted region around the interface. The consequence of ionization is that, for each impacted electron, two electrons and one hole emerge from the impact. These charges can be accelerated enough to liberate other free charges, giving rise to a sudden and large increase in reverse current. The free mean path is essential in the avalanche activation. The larger is the free mean path, indeed, the larger is the energy acquired between two impacts and the more probable is the ionization of the impacted atom. The free mean path is strictly related to mobility and decreases with doping concentration, so that in heavily doped devices it becomes short and the avalanche requires a larger applied voltage. In silicon, where N D and N A are of the order of 1018 cm−3 , the electric field required for the avalanche is comparable to the saturation field, so that the probability of the avalanche effect becomes small.

4.5.2 Zener Effect The Zener effect is the other phenomenon leading to an abrupt increase in reverse current. Although it manifests itself in a similar way to the avalanche effect, it is caused by a typical quantum phenomenon: the tunneling effect. The tunneling probability becomes relevant when the de Broglie wavelength h ) of the electrons approximates the potential barrier width. In heavily doped (λ = mv materials (N D ≈ N A ≈ 1018 cm−3 ), the depletion layer is of the order of few nanometers. In these conditions, under reverse bias, the wavelength of electrons can become small enough to be comparable to that of the depletion layer. As a consequence, the transfer of electrons from the P-type material valence band to the N-type material conduction band becomes possible. This leads to a large increase in reverse current.

4.5 Breakdown Phenomena

137

Fig. 4.21 Band diagram of a PN junction made up of heavily doped semiconductors. At the equilibrium (V A = 0), although the depletion layer may be narrow, a null density of states corresponds to the electrons of the valence band. Under a large reverse bias, the density of states accessible to the valence band electrons becomes large and a tunneling current of electrons can flow from the P-type to the N-type material. Due to the valence band large increase in energy and despite the enlargement of the depletion layer, the barrier width becomes smaller than at the equilibrium. In the inset, the shape of the triangular barrier is shown. Electrons are accelerated in the depletion layer before reaching the barrier

The shape of the barrier is triangular and the transmission probability of electrons l across the barrier is: T ≈ exp( 0 E). Hence, current depends exponentially on applied voltage (V A ) and the Zener effect results in a large reverse current, whose exponential behavior stabilizes the value of the voltage drop across the diode. As shown in Fig. 4.21, although the depletion layer is larger under reverse bias, the large shift in energy of the electrons in the P-type material makes the barrier shorter, favoring the tunneling of electrons. Furthermore, the electric field in the depletion layer accelerates the electrons by increasing their energy, thus increasing the transmission probability. Breakdown phenomena are used to design devices that are able to stabilize the voltage across a passive element. These devices are known as Zener diodes, even though both the avalanche and the Zener effect might coexist. It is important to note that, in a forward biased diode, the applied voltage is also fixed. However, the voltage drop in the forward bias regime is fixed at around 0.7 V (in the case of silicon), whereas the breakdown voltage in a Zener diode is of the order of few Volts. This makes the use of Zener diodes as voltage regulators in practical circuits extremely more flexible.

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4 PN Junction

Avalanche and Zener effects show a different behavior with temperature. In particular, the breakdown voltage the Zener effect occurs at (BVzener ) decreases with temperature. This can be attributed to the fact that the band gap (E gap ) is slightly depen E ), so that the barrier height becomes smaller as dent on temperature ( Tgap ≈ 0.3 mV K temperature increases and a smaller reverse bias is necessary to activate the breakdown. On the other hand, the voltage necessary to ignite the avalanche effect (BVavalanche ) increases with temperature. This can be understood considering that mobility decreases with temperature, so that also the free mean path decreases. As a consequence, a larger reverse bias is necessary to activate the avalanche. In silicon devices with BV ≈ 5 ÷ 6V both the effects are possible. As a consequence, a stabilization of the breakdown voltage with respect to temperature is obtained.

4.6 Summary The comprehension of the properties of junctions between a P and N semiconductor has been fundamental to the development of solid-state semiconductor devices. PN junctions occur in any complex devices, either as major constituents, as in the BJT, or as an ancillary element in more complex designs, as, for example, in MOSFETs. The properties of PN junctions expand those of the metal-semiconductor case, so that, in the case of an asymmetrical junction with one of the sides more doped than the other, the electrostatic quantities approximate those of the metal-semiconductor couple. However, the peculiar properties of the PN junction emerge in the current behavior, where the bipolar character of the device is expressed by the dominance of minority charges. Contrarily to the metal-semiconductor case, the PN junction is always rectifying. Furthermore, the minority charge accumulation in the neutral zone is at the origin of an additional capacitance, which, compared to the Schottky diode, reduces the frequency response of the device.

Further Reading Textbooks C.C. Hu, in Modern Semiconductor Devices for Integrated Circuits (Pearson College, 2009) R. Muller, T. Kamins, M. Chen, in Device Electronics for Integrated Circuits, 3rd edn (Wiley, 2002) D. Neamen, in Semiconductor Physics and Devices (McGraw Hill, 2003) B. Streetman, S. Banerjee, in Solid State Electronic Devices (Prentice Hall, 2006) S.M. Sze, K.K. Ng, in Physics of Semiconductor Devices, 3rd edn (Wiley-Interscience, 2006)

Further Reading

139

Journal Papers H.L. Armstrong, A theory of voltage breakdown of cylindrical p-n junctions, with applications. IRE Trans. Electron Devices 4, 15 (1957) C.R. Crowell, S.M. Sze, Temperature dependence of avalanche multiplication in semiconductors. Appl. Phys. Lett. 9, 242 (1966) H. Lawrence, R. Warner Jr., Diffused junction depletion layer calculation. Bell Syst. Tech. J. 39, 389 (1960) K. McKay, Avalanche breakdown in silicon. Phys. Rev. 94, 877 (1954) R.H. Kingston, Switching time in junction diodes and junction transistors. Proc. IRE 42, 829 (1954) W. Shockley, The theory of p-n junctions in semiconductors and p-n junction transistors. Bell Syst. Tech. J. 28, 435 (1949)

Chapter 5

Negative Differential Resistance Effects

5.1 Introduction The negative resistance (NDR) condition is shown by certain materials under proper circumstances. Furthermore, in some devices, at given applied voltages, the slope of the I/V curve becomes negative. It is straightforward that the definition of negative resistance is only related to the differential resistance ddVI , the value of the resistance ( VI ) being always positive. The behavior of a device characterized by NDR can be appraised considering the intersection with a load curve. As shown in Fig. 5.1, in the case of NDR the presence of more than one quiescent point leads to an unstable condition in the circuit that may give rise to voltage and current oscillatory patterns. The NDR behavior can be observed in a number of different devices, such as the Tunnel diode, and in semiconductors (like the GaAs) where the conduction band exhibits two relative minima.

5.2 Tunnel Diode The tunnel diode is a PN junction made up of heavily doped semiconductors. The doping herein is large enough to displace the Fermi level into the conduction and the valence bands, as shown in Fig. 5.2. Such materials are called degenerate and, on a practical level, they behave like metals. The great difference, however, lies in the fact that charge carriers are still electrons and holes. In these conditions, the Boltzmann approximation of the Fermi-Dirac function is no longer valid, so that a thorough description of the device requires the use of the complete equation. Due to heavy doping, the space charge regions are very narrow and the tunnel effect drives the crossing of the barriers. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_5

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5 Negative Differential Resistance Effects

Fig. 5.1 Positive and negative differential devices and quiescent points of a device characterized by a negarive differential resistance tract

Fig. 5.2 Band diagram of heavily doped P and N type semiconductor. The large doping displaces the E gap ), the ideal current overcomes the tunneling current well beyond V0 , so that the tunneling current dominates at small voltages. Typical values of V0 are of the order of hundreds mV . On the other hand, due to the very narrow depletion layer, the generation-recombination current is negligible and the reverse current is produced by the tunneling current, like a Zener effect with a null breakdown voltage.

5.3 NDR Behavior in GaAs The NDR behavior is spontaneously observed in homogeneous semiconductors made up of the atoms of the elements in groups III and V of the periodic table, such as GaAs and InP. These materials are characterized by a peculiar shape of the bands in the k space, where the conduction band exhibits a structure with a double minima. In GaAs, as shown in Fig. 5.5), the first minimum occurs at 1.42 eV above the top of the valence band with k = 0 (direct band-gap), whereas the second minimum occurs in the k space and is located at about 0.3 eV above the first minimum. The curvature of the conduction band in the second minimum neighborhood is smaller than the curvature of the principal minimum, so that the effective mass is greater and the mobility is smaller. The transition from the two minima requires a change in momentum, which can be achieved when the electrons are accelerated by a large electric field. In practice, as the electric field grows, the concentration of electrons in the second valley becomes

5.3 NDR Behavior in GaAs

145

Fig. 5.5 Simplified band diagram of GaAs. The conduction band is characterized by two valleys with different curvatures. As a consequence, the electrons effective mass in the two valleys, as well as the mobility, is different

numerically important and part of the conduction electrons move with a smaller mobility. The electric field necessary to populate the second valley is large enough that the drift velocity is comparable to the thermal velocity. In other words, the “temperature” of the electrons becomes larger than the temperature of the lattice. Electrons under these conditions are known as hot electrons. The total kinetic energy can be separated in thermal and drift velocity contributions: T =

1 2 m(vth + vd2 ). 2

(5.1)

In order to calculate the relationship between the average velocity and the electric field, let us assume that the electron concentrations in the deepest and the second valley are n 1 and n 2 respectively. The curvatures of the band (the second derivative of the band profile) in the two valleys are different, so that the electrons experience different effective masses. Eventually, the two populations of electrons move with different mobilities μ1 and μ2 . The current is given by the sum of the contributions of the two groups of electrons: J = q(n 1 μ1 + n 2 μ2 )E = q(n 1 + n 2 )v,

(5.2)

from which v=

μ1 (n 1 μ1 + μμ21 n 2 ) n 1 μ1 + n 2 μ2 E= E. n1 + n2 n1 + n2

(5.3)

Let us assume that μ2  μ1 , so that: v=

μ1 E. 1 + nn 21

(5.4)

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5 Negative Differential Resistance Effects

Because the intervals of energy under consideration are well above the Fermi level, the ratio between the electron concentrations in the two valleys can be calculated with the Boltzmann relationship:   E n2 = R exp − n1 kTe

(5.5)

where R is the ratio between the densities of the states in the two valleys and Te is the effective temperature of the electrons. Replacing the last equation in Eq. 5.4 we get:    E −1 v = μ1 E 1 + R exp − kTe

(5.6)

where E is the energy lost by the electrons in the scattering with the lattice atoms, which can be written in terms of electrons effective temperature (Te ) as: E = 23 kTe Thus, recalling the scattering processes outlined in Chap. 1 we can write: E =

3 kTe = qElc 2

(5.7)

where lc = vτc is the mean free path. Finally, the effective temperature can be calculated: Te =

   E −1 2qτc 2 E μ1 1 + R exp − . 3k kTe

(5.8)

Equations 5.6 and 5.8 form a couple of parametric equations. In practice, for each value of Te , the corresponding E and v are calculated and a curve of velocity versus electric field can be drawn. The result is shown in Fig. 5.6. Fig. 5.6 Calculated velocity versus electric field. The calculation has been done considering Te in the range 300–700 K, τc = 10−12 s, R = 103 , and μ = 8800 cm2 V−1 s−1

5.4 Gunn Oscillations

147

The curve is characterized by a NDR tract occurring beyond the peak. In the case of V gallium arsenide, the peak occurs at E ≈ 4 · 104 cm 2 and the corresponding velocity 7 cm is v peak = 2 · 10 s . The NDR tract converges towards a saturation velocity vsat ≈ 107 cm . s The saturation velocity is more than one order of magnitude larger than in silicon, this providing the basis for the largest operating frequency in III-V semiconductor based devices. The NDR tract in III-V semiconductors, and in particular in GaAs, is at the basis of a complex phenomenon called Gunn oscillation, which is used to generate high frequency signals.

5.4 Gunn Oscillations Gunn oscillations appear in homogeneous resistors made of III-V semiconductors (such as GaAs) biased with a voltage such that the electric field exceeds the peak value in the velocity/electric field relationship. In this condition, the material becomes unstable -namely, small fluctuations in the supposed uniform concentration of electrons do not cancel out, but they rather grow and propagate until reaching the electrode, where they give rise to high frequency current components. A simple explanation of the phenomenon can be obtained considering the effects of electron-density fluctuations on the electric field distribution and then on charge carriers velocity. The static situation is depicted in Fig. 5.7. The concentration of electrons is uniform, so that also the electric field is uniform. Its value lies beyond the peak of the velocity/electric field relationship. Namely, the material is in the negative differential resistance regime. Let us suppose that a small fluctuation of concentration occurs in a position close to the negative electrode (see Fig. 5.8). The charge fluctuation corresponds to a double layer of charges that generates a fluctuation in the electric field that is summed up to the electric field due to the applied voltage. Since the applied voltage is constant, the integral of the electric field along the material is also constant, so that a local increase in the electric field is balanced by a decrease in the electric field in the rest

Fig. 5.7 Initial conditions for Gunn oscillations: electrons concentration, electric field, and the corresponding drift velocity

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5 Negative Differential Resistance Effects

Fig. 5.8 Electron density perturbation is shown as a tract of the semiconductor where charges are accumulated. The accumulation of charges lead to a local excess and a global decrase of the electric field. As a consequence, the speed of accumulated charges decreases while the rest of charges increase their velocity

Fig. 5.9 Once the speed of charges is the same everywhere, the perturbation reaches a stable density and electric field profiles

of the semiconductor. In other words, the electric field decreases everywhere, except for the small region involved in the fluctuation, where it increases. The material is biased in the NDR condition, then the electrons involved in the fluctuation decrease their velocity, while the velocity of the rest of the electrons increases. This condition leads to a progressive increase in the density of the electrons involved in the fluctuations. The fluctuation tends to diverge as the electric fields of the perturbed and non-perturbed regions become more different from each other. In particular, the smallest electric field may become smaller than the peak value, so that the electrons outside the perturbation begin to behave according to a positive differential resistance, losing their velocity. Eventually, the velocity of all the electrons (both perturbed and non perturbed) equalize and the density of perturbed electrons stops growing, so that a stable dominion is formed (Fig. 5.9). The dominion moves towards the positive electrode, where it elicits a temporary increase in current. A fluctuation appears spontaneously in the material, then a train of current spikes is observed. Finally, although the resistor is biased with a d.c. voltage, the current is characterized by a stable a.c. component occurring in the microwave spectral region. Gunn oscillations are exploited in a device called Gunn diode, which is used as a microwave signal generator.

Further Reading

149

5.5 Summary Negative differential resistors are important elements in oscillatory circuits. They can be obtained exploiting specific physical effects resulting in portions of the I/V curve characterized by a negative slope. In this chapter, one of such devices, the tunnel diode, has been introduced. The I/V curve has been derived only qualitatively, describing the effect of bias on the band diagram and the tunneling between filled and empty states across the junction. A more intrinsic negative differential conductivity originates from the double valley configuration of the conduction band on III/V semiconductors. This phenomena can be observed as a peculiar effect found in homogeneous resistors made of these semiconductors. The Gunn effect consists in the appearance of a high frequency fluctuation of current superimposed to a constant current. This effect is exploited in a device, the Gunn diode, used to generate high frequency currents from a d.c. bias.

Further Reading Textbooks K. Brennan, A. Brown., Theory of modern electronic semicionductor devices J. Wiley (2002) S. Sze, K.N. Kwok, Physics of Semiconductor Devices, 3rd edn. Wiley-Interscience (2006)

Journal Papers P. Butcher, Theory of stable domain propagation in the Gunn effect. Phys. Lett. 19, 546 (1965) L. Esaki, New Phenomenon in Narrow Germanium p- Junctions. Phys. Rev. 109, 603 (1958) J. Gunn, Microwave oscillations of current in III-V semiconductors. Solid State Commun. 1, 88 (1963) R. Hall, Tunnel diodes. IRE Trans. Electron. Devices ED-7, 1 (1960) Kroemer, Theory of the Gunn effect. Proc. IEEE 52, 1736 (1964) J. Ruch, G.S. Kino, Measurement of the velocity-field characteristic of Gallium Arsenide. Appl. Phys. Lett. 10, 40 (1967)

Chapter 6

Bipolar Junction Transistor

6.1 Introduction The Bipolar Junction Transistor (BJT) brilliantly exploits the properties of the PN junction. It is a three-contact device made out of a cascade of two PN junctions, whose main application field is current amplification. However, it is also used as both switch and digital electronic circuit element, commuting from high to low current states. The BJT concept stems from the observation of the fact that in a PN junction the doping of the two regions modulates the relative intensity of the currents carried by electrons and holes. In practice, by changing doping, it is possible to make current be dominated either by electrons or holes. The PN junction, however, has two terminals and since the collected current is the sum of the currents carried by electrons and holes, the dominating charge carrier remains invisible to the external circuit. The ideal structure of a BJT is shown in Fig. 6.1. It is a sequence of NPN (or PNP) materials defining two junctions. Each portion of the device has its own contact, so that the two junctions can be separately biased. The three regions are called emitter, base and collector, whereas the base-emitter and the base-collector are the two junctions. VB E and VBC are the corresponding voltage drops across each junction respectively. The central part of the device is the base. This region exchanges charges via three contacts: two junctions and one metallic terminal. Let us consider that one of the junctions defines a normal diode, so that the charge injected into the base region can be extracted at two contacts. One of them is the metal electrode, which accepts both electrons and holes, whereas the other one is a PN junction, which accepts either holes or electrons (namely, the base minority or majority charges) according to the sign of the applied voltage. When this junction is reversely biased, it drags the base minority charges across the junction. In this condition, the two base contacts are selective: minority charges are collected at the junction and majority charges are collected at the metal contact. The PN junction inverse current is orders of magnitude smaller than the forward current. There nonetheless exist some conditions under which this current may © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_6

151

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6 Bipolar Junction Transistor

Fig. 6.1 Scheme of principle of a BJT. The three connectors enables the polarization of the three regions: emitter (E), base (B), and collector (C). The sequence of doping gives rise to two complementary devices: NPN (as in figure) and PNP

become large. In a photodiode, for instance, the generated electron-hole pairs are drifted away from the electric field in the space charge region, in such a way that they contribute to the inverse current. In the BJT, a forward biased junction injects an excess of minority charges into the base, which can be collected at the other junction if this is reversely biased. The majority charges are of course collected at the metallic contact. This behavior occurs if the base is short enough to avoid the recombination of the injected minority charges, thus allowing the electrons to reach the opposite junction. We will see that the base geometric width is a fundamental parameter for the device performance. In the following of this chapter, the discussion about the BJT properties is carried out for a NPN configuration. A completely symmetric behavior is obtained with the PNP sequence of materials. The fact that the effective mass, and then the mobility, of electrons and holes in silicon are almost similar makes the characteristics of the devices very similar. This leads to an important degree of freedom in circuit design. The BJT is designed so as to make the holes contribution to the current crossing the device from the emitter to the collector negligible respect to the current carried by the electrons. This condition occurs under any combinations of VB E and VBC . Figure 6.2 shows the hole current in the three possible bias configurations, where the junctions are either forward or reverse biased. The role of electrons and holes is exchanged in a PNP device. Thus, in a BJT, the majority charges of both the emitter and collector regions are the dominant carriers of the device.

6.2 The Ideal Transistor The current flowing from the emitter to the collector can be studied on the basis of the models developed for the PN junction. For this scope, let us consider the ideal structure shown in Fig. 6.1. The emitter and collector regions are more doped than the base (N D > N A ) and the base width is much shorter than the recombination length of the electrons in the base.

6.2 The Ideal Transistor

153

Fig. 6.2 The two junctions of the BJT can be forward (F) or reverse (R) biased. In all cases the current of holes from the emitter to the collector is negligible. When both the junctions are forward biased, two large currents of holes are oppositely injected from the base towards the two adjacent regions. With only one junction in forward bias, the currents towards the forward biased junction is large but the other from the reverse biased junction one is small. Thus, from the emitter to collector : the small one dominates. In the third region, a small hole current is injected from the two adjacent regions towards the base. In all cases the total hole current is small and thus negligible

Fig. 6.3 Left: equilibrium band diagram of symmetric BJT. Right: electrons concentration profile with N D = 1016 cm−3 and N A = 1014 cm−3 .

Under these conditions, the recombination of the charges injected into the base from the forward biased junctions is a rather unlikely event, so that the base region can be treated as a short base diode. Noteworthy, the ideal device is completely symmetric, so that the emitter and the collector can be exchanged. We will see thereafter some reasons leading to a non-symmetrical device, where the collector is differently doped with respect to the emitter. Figure 6.3 shows the equilibrium band diagram and the electrons concentration profile. Figure 6.4 shows the band diagram and the electron concentration profile in case of reverse-reverse, forward-forward, and forward-reverse bias of the emitter-base and base-collector junctions respectively. As usual, the applied voltage is distributed across the space charge regions, so that the bulk of the semiconductors are neutral zones where the electric field is null. In the case of reverse-reverse bias, the base is depleted of electrons and a small electron current flows between the emitter and the collector. In the case of forwardforward bias, the base is overpopulated by electrons. In this condition, a large electron current can be expected. Finally, in the case of forward-reverse bias, the largest gradient of electrons concentration is found in the base. This last case is known as active zone. The electrons injected from the emitterbase junction that is forward biased are completely collected at the base-collector junction, which is reverse biased. The collector junction plays the role of an electric

154

6 Bipolar Junction Transistor

Fig. 6.4 Band diagram and electrons concentration profile under the three possible bias schemes that can be applied to the BJT

contact, with the important difference that it can drag electrons only. The active zone is the configuration where the BJT behaves as a current amplifier.

6.2.1 Electron Current in the Active Zone The active zone corresponds to the condition where the two junctions are oppositely biased. The base-emitter junction (BE) is forward biased and the base-collector junction (BC) is reverse biased. Then, the electrons from the emitter, where they are majority charges, are injected into the base, where they are minority charges. The electrons injected into the base are collected by the BC junction and transferred to the collector. This is the basic transistor mechanism: the direct current of a diode is collected by a reverse biased diode. The reverse biased BC junction transfers the electrons from the base to the collector and the holes from the collectors to the base. Then, from the base viewpoint, the collector in the active zone can only accept electrons. In practice, the bipolar conductivity of the diode is broken into a unipolar

6.2 The Ideal Transistor

155

Fig. 6.5 Excess electrons profile in base consequent to V B E > 0. The coordinate starts from the edge of the depletion layer of the BE junction (x = 0) down to the edge of the depletion layer of the BC junction (x B )

current. In the NPN device, the current flowing from the collector towards the emitter is almost totally due to electrons. Thus, only for the electrons, the BC junction is equivalent to a metal contact. It is important to remark that this mechanism holds only if the base width is shorter than the electron recombination length. In the case of a long base, all the injected excess electrons recombine in the base and the electron current is converted into a hole current. The BC junction being reverse biased, it is not able to provide a large hole current, so that the forward current injected from the BE junction is collected at the base contact and the device merely corresponds to a diode. Thus, the first requirement for the BJT is that the base region must be short, in such a way that the electrons injected from the emitter experience the short-base diode condition. The hole contribution to the total current is discussed thereafter. Let us first calculate the electron current in the active zone. The equations calculated for the short base diode are still valid and the profile of the injected electrons is linear, as shown in Fig. 6.5:      x q VB E  −1 1− . (6.1) n (x) = n p0 ex p kB T xB Under forward bias q VB E  k B T , then:    x q VB E 1− n  (x) = n p0 ex p kB T xB

(6.2)

and the electron diffusion current is: Jn = q Dn

  n2 1 dn  q VB E = −q Dn i . ex p dx NA xB kB T

(6.3)

  Thus: Jn = −JS exp qkVBBTE , with J P ≈ 0. The current at the collector contact is opposite to the electron current: JC = −Jn . This current is controlled by the voltage applied to the BE junction.

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6 Bipolar Junction Transistor

Fig. 6.6 Experimental behavior of the collector current versus the base emitter voltage (at T = 300 K). JS is the extrapolated value of the current at V B E = 0

Figure 6.6 shows an example of the measured JC as a function of VB E . The exponential behavior is valid for about 7 decades of current density values. The extrapolated value of the current at VB E = 0 gives us the experimental estimation of JS : J S = q Dn

n i2 . NAxB

(6.4)

An important quantity in the BJT is the total charge of the holes stored in the base region Q B0 . In case of homogeneous doping, this is simply given by: Q B0 = q N A x B . In practice, the doping is not constant, so that the charge in the base is given by: X B Q B0 = q

pd x.

(6.5)

0

If doping is not constant, also the diffusion constant is not a constant. In order to account for this condition, the diffusion constant is replaced with an average quantity indicated as D˜n . Using the total hole charge and the average diffusion constant, the term JS can be written as: JS =

q 2 D˜n n i2 . Q B0

(6.6)

Q B0 is a characteristic parameter of the BJT, established at the time of the device fabrication. It is proportional to the total amount of holes in the base, namely to the

6.2 The Ideal Transistor

157

Fig. 6.7 Concentration of charges injected in base from the emitter as a function of the base doping and for different values of V B E . Above the dashed line the concentration of injected electrons is larger than the acceptors concentration. The calculation is strictly valid only when the concentration of charges is smaller than the doping, namely under the low injection conditions. However, it gives an idea of the fact that the low injection condition is easily violated if the doping of the base is too small

total doping of the base. The total charge of the majority carriers in the regions of the BJT is known as Gummel Number (GN). The Gummel number of the base is: X B G NB =

N A (x)d x =

q 2 D˜n n i2 Q B0 = . q JS

(6.7)

0

The base Gummel number (G N B ) is a quality factor of the BJT. It defines the magnitude of the current flowing through the device. A small base Gummel number is necessary for a large current. A small G N B can be obtained in two ways. The most immediate method suggests that a small doping N A should be used. However, this method may jeopardize the validity of the low-injection limit condition. Figure 6.7 shows the behavior of the ratio of the injected charges at the interface as a function of the doping concentration. The plot is only quanlitative because when the low-injection condition is violated, the model is no longer valid. In any case, it demonstrates that the low-injection limit is violated at low base doping concentrations. The second method is based upon a variable doping concentration profile, larger in the region close to the emitter and smaller elsewhere. The low-injection condition is critical at the interface, where the concentration of injected electrons is large, whereas the Gummel number is the integral of the hole concentration extended to the entire base region. Thus, with a variable profile, it is possible to maintain the integral of

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6 Bipolar Junction Transistor

Fig. 6.8 The same Gummel number can be obtained with a constant base doping or a variable profile. In this last case, the large doping close to the emitter ensures the low-limit condition (compare with Fig. 6.5)

the doping atoms small and, at the same time, a large doping concentration on the border with the emitter (Fig. 6.8). It is worth to mention that the low-injection condition not only ensures the validity of the models we have calculated so far. Besides the evident model simplification issues, in the high injection regime the performance of the devices are deteriorated, so that the low-injection condition also preserves the optimal working condition of the devices. A non-uniform acceptor profile in the base makes possible to maintain the lowinjection limit within a small Gummel number. However, a non-uniform dopant density gives rise to a non-uniform hole distribution. At the equilibrium, an electric field is necessary to compensate the diffusion current due to the non-zero hole gradient. This electric field keeps at zero the hole current in the base. It is interesting to note that the hole current is always negligible between the emitter and the collector, so that the same electric field that keeps at zero the hole current may act upon the excess electrons in order to create the electron current in the base. The electric field necessary, in the base, to satisfy the condition J P = 0 can be calculated from the equilibrium between drift and diffusion current: qμ p pEx − q D p

dp k B T 1 dp → Ex = . x q p x

(6.8)

The same electric field acts upon the electrons and gives rise to the electron current:   dn dn k B T 1 dp Jn = qμn nEx + q Dn = qμn n + q Dn x q p x x (6.9) n dp dn + q Dn . = q Dn p dx dx This can be written as: Jn = q

Dn p

  dp dn Dn d n +p =q (np). dx dx p dx

(6.10)

6.3 Current Gain

159

The current in the base is then calculated by an integration from x = 0 (at the interface with the emitter) to x B (at the interface with the collector): x B Jn

p dx = q Dn

0

x B d(np) = n(x B ) p(x B ) − n(0) p(0).

(6.11)

0

The product of the concentration of holes and electrons at the edge of the depletion layer has been calculated in Chap. 4 (see Eqs. 4.66 and 4.67): np = n i2 exp(q V A /k B T ). Replacing V A with VB E and VBC respectively, the final expression for the current is calculated:      qn 2 q VB E q VBC − exp . (6.12) exp Jn = x B pi kB T kB T 0 D dx n

The hole profile at the equilibrium is equivalent to the profile of the acceptors (N A ). The uneven concentration of acceptors makes the diffusion constant variable, which is replaced with an equivalent term D˜n . Thus, the integral at the denominator represents the total mobile charge in the base at the equilibrium (Q Bo ). The current can be written as:      qn i2 D˜n q VBC q VBC Jn = exp − . Q Bo kB T kB T

(6.13)

In the case of active bias VB E > 0 and VBC < 0 and Eq. 6.13 is equivalent to Eq. 6.3, where the total charge in the base replaces the product N A x B . The two equations have been calculated in one case under the hypothesis of X B  L n , namely a short base region, and in the other case under the hypothesis of J p = 0. Obviously, in a short base the current in the active zone is totally due to the excess electrons, the recombination current being negligible and, consequently, also the hole current. Equation 6.13 is sometimes called the equation of the transistor, because it establishes the current with any values of VB E and VBC 0.

6.3 Current Gain In the previous section it has been stated that the current due to holes is always negligible. This is particularly true when the device is biased in the active zone. In this condition, the base-collector junction is reverse biased and the base minority charges are dragged towards the collector terminal. However, since the base-emitter junction is forward biased, the electrons, which are the majority charges in the emitter, are injected into the base. Thus, the concentration of the minority charges in the base exceeds of several orders of magnitude the concentration of the minority charges in the collector. Eventually, the forward current of the BE junction is injected into

160

6 Bipolar Junction Transistor

Fig. 6.9 The five hole current sources in active-zone BJT: (1) forward recombination current in the depletion layer of the BE junction; (2) recombination current in base; (3) forward hole current in the base-emitter junction; (4) reverse diffusion current of the base-collector junction; (5) reverse generation current in the depletion layer the BC junction

the BC junction. We demonstrated in the previous section that the collector current depends on the bias applied to the BE junction. The collector current is numerically dominated by electrons. However, holes exist in the device. An important property of semiconductor devices is the fact that all of the quantities in the system are in equilibrium with one another. This means that even if holes are numerically negligible, their current is always proportional to the electron current. The device being built to take advantage of this relation, a ohmic contact is applied to the base not only to have the BE and BC junctions biased, but also in order to inject and extract currents. In active bias, the base is endowed with two terminals with respect to the BE junction: the ohmic contact and the BC reverse biased junction. The two contacts separate the charge carriers in the base. As previously discussed, the BC junction collects electrons and the ohmic contact collects holes. It is possible to identify at least five different hole currents through the device, as illustrated in Fig. 6.9. The components 4 and 5 are reverse currents, whereas the other ones are direct currents. Thus, the hole currents contributing to the reverse current of the BC junction are always negligible with respect to the direct current contributions. The recombination It is propor current in the BE junction is given by Eq. 4.73.   q VB E q VB E tional to exp 2k B T , whereas the ideal current depends on exp k B T . The ratio between the two exponentials is negligible when VB E is greater  than few hundreds q VB E q VB E of millivolts. For instance, the ratio of exp k B T to exp 2k B T is about 50 when VB E ≈ 200 mV. Eventually, the current of the holes collected at the base terminal is almost completely made out of the recombination of the electrons in the base and the direct injection of holes in the emitter. Since the hole current is in equilibrium with the electron current, the ratio of the collector current to the current injected into the base is in equilibrium, too. Thus, a slight increase in the base current results in an increase in the collector current. The ratio between these two quantities defines the current gain.

6.3 Current Gain

161

6.3.1 Base Recombination Current The recombination of minority charges in the neutral zones has been studied in Chap. 4 and the same equations are valid in the BJT base. In Chap. 4 we observed that the magnitude of the base recombination phenomena depends on the length of the base region, or, more precisely, on the base length with respect to the recombination length. In particular, the shortest is the base, the smaller is the recombination current. The calculation of the recombination current in the base requires the exact solution of the continuity equation (Eq. 4.38). Here, we calculate the recombination current in two steps. In the first one, the profile of excess electrons is calculated as in the shortbase diode, by linearizing Eq. 4.40. Subsequently, the recombination contribution to the current is calculated by applying the generation-recombination function to this charge excess. In order to calculate the hole current, let us consider the continuity equation for the stationary condition: 1 ∂ Jp ∂p =0=− − U. ∂t p ∂x

(6.14)

The current is the integral of the recombination function: x B J p = −q

U d x.

(6.15)

0

Under the low-injection hypothesis, using the approximations detailed in chapter 3, the recombination function is the ratio of the excess charges to recombination time. In order to describe the actual current collected at the terminals, it is necessary to introduce the area that the current flows through. Here, the relevant area is that of the emitter contact through which the direct current producing the excess charge originates. The current at the base terminal is thus: x B Ir b = −J p A E = q A E 0

n  (x) d x. τn

Replacing the formula of the excess charge in a short diode, we have:   q Vbe q A E n i2 x B exp −1 . Ir b = τn N A 2 kB T

(6.16)

(6.17)

The recombination current describes the loss of electrons traveling from the emitter to the collector. The loss of electrons is equivalent to the holes injected from the base contact. This is measured by the transport factor in the base (αT ), defined as the ratio of the current leaving the emitter to the current at the base contact.

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6 Bipolar Junction Transistor



Ir b

|Ine − Ir b | αT = = 1 −



. |Ine | Ine The current from the emitter is the short diode forward current:   q Vbe q A E n i2 Dn Ine = exp −1 . τn N A x B kB T

(6.18)

(6.19)

The transport factor is easily calculated replacing Eqs. 6.17 and 6.19 in Eq. 6.18: αT = 1 −

x B2 2L 2n

(6.20)

√ where the definition of L n = Dn τn has been used. This result is fairly predictable because the loss of electrons in the base, and thus the holes current due to recombination, depends on the ratio of the base physical length to the recombination length. The recombination length is the square root of the product of the diffusion constant and the recombination time, so that it depends on both the dopant atoms concentration and the recombination centers. On the other hand, xb is of course limited by the technological capabilities of the device fabrication. x2 It is interesting to note that 2LB2 corresponds to the ratio of the transit time in n the base to recombination time. Indeed, an electron injected into the base is either recombined inside the base by a hole or able to reach the interface with the collector. Thus, the ratio between the typical time constants of these two events is a measure of the efficiency of the electron transfer across the base. It is also important to note, observing the current, that in a diode it is impossible to discriminate between short and long bases. But otherwise, in a BJT, since the base contacts are selective for the charge carriers, the two currents (electron and hole currents) are separated into two distinct networks. Quantitatively, αT is very close to one. In a typical case, where L n = 10µm and x B = 0.3µm, the transport factor is αT = 0.9996, so that Ine ≈ 2500Ir b . This means that one hole injected into the base is able to control the transfer of 2500 electrons from the emitter to the base. To understand the amplification process, it is convenient to turn our attention to holes and focus on the consequences of an extra hole injected into the base from the base contact. The extra charge in the base results in a change in the charge state of the base itself, the process requiring that an extra electron be injected from the emitter contact in order for the charge neutrality to be restored. If we limit the hole current to the recombination current, then the injected electron will work to recombine the extra hole. However, the injected electron also induces an electron diffusion current, so that if the hole recombination time is larger than the transit time, then the electron will reach the collector before recombining the hole. Thus, one electron flows from the collector terminal, while the hole is still in the base, so that another electron is required

6.3 Current Gain

163

from the emitter, the process continuing until an electron eventually recombines the hole. As a consequence, the larger is the ratio of the hole recombination time to the electron transit time, the larger is the increase in the collector current. A similar process takes place in photoconductors, where, although a single photon creates an electron-hole couple, due to the long recombination time of holes a large amount of electrons can be collected at the contacts until the holes are recombined. The transport factor tends to be null in a long-base device. In such a case, the electron current is a recombination current, so that only one electron, injected from the emitter, is necessary to recombine a single hole injected into the base. In other words, due to complete recombination, the current injected from the emitter corresponds to the hole current collected at the base terminal, namely |Ine | = |Ir b | and αT = 0. Such a device in the active zone would merely behave as a diode between the base and the emitter contacts with a negligible reverse current at the collector terminal.

6.3.2 Forward Hole Current in the Emitter In the active zone, the BE junction is a forward biased PN junction, so that the flow of electrons injected from the emitter to the base is complemented by the flow of holes from the base to the emitter. The hole current obviously originates from the base contact, so that it is an additional contribution to the recombination current calculated in the previous section. The hole current is calculated from the theory of the PN junction, so that Eqs. 4.52 and 4.59 hold if the emitter is long or short respectively, this being related to the hole recombination length in the emitter neutral region. Typically, transistor dimensions are such that also the emitter is a short region. The relationship between the hole and electron currents is described by the emitter efficiency γ , which is defined as the ratio of the electron current to the total current at the emitter contact: γ =

|Ine | |Ine | 1

=

. =

Ine + I pe

I

IE 1 + Inepe

In the case of a short emitter region, the hole current of is:     n i2 q VA I pe = q A E D pe exp −1 . ND xE kB T

(6.21)

(6.22)

The electron current has been previously calculated (Eq. 6.19), so that from the two currents the emitter efficiency is: γ =

1 1+

x B N A D pe x E N D Dnb

.

(6.23)

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6 Bipolar Junction Transistor

Thus, the emitter efficiency depends on the doping concentration of the base with respect to the doping concentration of the emitter. The previous equation has been calculated for homogeneous base and emitter doping. The same equation can be extended to the case of variable doping profiles, using the concept of Gummel number. In the case of a uniform doping, the emitter and base Gummel numbers are: G N B = x B · N A ; and G N E = x E · N D . Thus, the emitter efficiency takes the following general form: γ =

1 1+

G N B D pe G N E Dnb

.

(6.24)

In order to maximize the emitter efficiency, it is necessary to make the emitter Gummel number larger than the base Gummel number. Since both the regions are short, this condition is met when the emitter doping is much larger than the base doping. It is worth to mention that, due to the difference in doping concentrations, the base diffusion constant is larger than the emitter diffusion constant.

6.3.3 Numerical Comparison of αT and γ Transport factor and emitter efficiency are two figures of merit describing the ratio of the current due to electrons to the current due to holes. Since electrons are collected at the collector and holes are collected at the base, these quantities actually describe the ratio between the currents at the contacts of the device. Transport factor and emitter efficiency are numerically slightly different. This can be appreciated in a practical case. Let us consider a BJT made of silicon with N De = 1017 cm−3 and N Ab = 1015 cm−3 . Namely, the base is 100 times less doped than the emitter. In these conditions, at room temperature, the recombination lengths are: L pe = 9 µm and L nb = 14 µm, whereas the diffusion coefficients are D pe = 8.25 cm2 s−1 and Dnb = 18.90 cm2 s−1 . In order to satisfy the short neutral zone condition, the neutral zones are chosen as being around 30 times smaller than the respective recombination lengths, so that: x E = 0.30 µm and x B = 0.45 µm. With these values, the transport factor is αT = 0.9997. The emitter and base Gummel numbers are: G Ne = x E N De = 3.02 · 1012 cm−2 and G Nb = x B N Ab = 4.58 · 1010 cm−2 . Then, the emitter efficiency is: γ = 0.9934. Eventually, both the figures of merit are very close to one, but γ is less close to one, its third decimal figure being different from nine.

6.3 Current Gain

165

6.3.4 Total Current Gain According to the previous definitions, the electron current from the emitter is γ times the emitter current: Ine = γ I E . The emitter current is the sum of the electron and hole currents: I E = Ine + I pe . On the other hand, the collector current is αT times the current of electrons injected into the base, with a negative sign due to the conventional current directions: IC = −αT I E . Thus, combining the previous definitions, we have: IC = −αT Ine = −αT γ I E = −α F I E

(6.25)

where α F = αT γ describes the current loss between the emitter and the collector contacts. The current gain, defined as the ratio between the collector and base currents, is obtained considering the current node in the BJT: IC + I E + I B = 0, from which we can write: IC + IB = 0 αF

(6.26)

αF I B = βF I B 1 − αF

(6.27)

IC − which in turn gives: IC =

αF where β F = 1−α is the d.c. amplification of the BJT. Numerically, considering the F above calculated numbers αT = 0.9997 and γ = 0.9934, we get α F = 0.9931 and β F = 144. Both αT and γ contribute to β F , even if with different magnitudes. In order to evaluate the contribution of each term, let us calculate β F for each factor while keeping at 1 the other one -that is, considering the other one as ideal. This gives us the following results: 1 · αT → β F = 3460 γ · 1 → β F = 151 γ · αT → β F = 144 Clearly, the d.c. amplification factor is mostly based on the emitter efficiency, so that it is the difference in doping that builds up the amplification factor. The reasons for the current amplification are to be found in the equilibrium between Jn and J p , as well as in the fact that different doping and a narrow base can greatly change the ratio between the currents, but they are still connected to each other. The relationship between currents balance and doping are yet present in the PN junction. The great advance of the BJT is the double contact of the base, which makes the separation of electrons from holes possible.

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6 Bipolar Junction Transistor

6.4 BJT Operative Conditions The behavior of the BJT depends on the voltages applied to the two junctions. According to the sign of the applied voltages, there are four different operative conditions, usually known as: cut-off, saturation, active zone, and inverse active zone (see Fig. 6.10). So far, we have not found any reasons to make the collector different from the emitter, so that in principle the active zone can be obtained either with a forward-biased BE junction and a reverse-biased BC junction, or vice-versa. We will see thereafter that in order to ensure a proper behavior, it is necessary to make the collector doping different from that of the emitter. In the cut-off condition, both the junctions are reverse biased, so that the base is depleted of electrons and any currents flowing through the device is small. In the saturation condition, both the junctions are forward biased. In this state, the electrons are injected into the base from both sides. Since the base is narrow, the electrons concentration profile is linear and the total current is proportional to the slope of the profile. The situation is depicted in Fig. 6.11. If the applied voltage is exactly the same, then the total current is zero. The current reaches its maximum value when one of the two voltages (VBC in particular) is null or negative (active zone). In general, the current is given by the difference between the currents injected from each junction:   q Vbc q Vbe − . (6.28) Jn = J0 exp kB T kB T In the active zone, the current still depends upon the voltages applied to the two junctions, but also on the current injected into the base contact. To describe the device, let us consider the so-called common emitter configuration shown in Fig. 6.12, where the device is described by two currents (IC and I B ) and two voltages (VB E and VC E ). VC E =VB E -VBC is the difference between the voltages applied to the two junctions. Each quantity is a function of the other ones, such that for example IC = f (I B , VB E , VC E ). Instead of considering a multidimensional function, it is more useful to represent two separate characteristics, which are generally known as input

Fig. 6.10 The sign of > VB E and VBC > defines the four different operative conditions of the BJT

6.4 BJT Operative Conditions

167

Fig. 6.11 The current of electrons in base is the slope of the electrons concentration profile which depends on the relative magnitude of V BC with respect to VB E . For a given VB E , the largest current i achieved when VBC = 0 Fig. 6.12 Input and output currents and voltages in the common emitter configuration

(I B = f (VB E )) and output characteristics (IC = f (VC E )), referring to the two separated input and output networks of the common emitter configuration. The input characteristic has the analytical form of a forward biased diode, but since the current flowing from the base to the emitter is only made of holes, the magnitude of this current is at least two orders smaller than the normal forward current of a PN junction. As for the output characteristics, when VC E = 0, then VB E = VBC . The amounts of electrons injected from the two sides are the same and the current is zero regardless of the VB E value. At VC E > 0, it is VB E > VBC , so that the current grows as VC E increases and stops growing when VC E = VB E , namely when VBC = 0. Beyond this value, the BJT works in the active zone, the collector current remains constant and any further increases in VC E makes the base-collector reverse biasing larger. The value at which the active zone is reached depends on the base current (I B ). Indeed, considering q VB E  k B T , the following condition holds:   q Vbe . (6.29) IC = β F I B = β F I B0 exp kB T The functional behavior of the output characteristics is described by the reverse biasing of the base-collector junction, with the very important difference that the current herein is the current of the electrons injected from the emitter region (and controlled by the base current), so that it is quantitatively large. Large values of VC E can bring about the junction breakdown, as described in section 4. Actually, since the base is lightly doped, the BJT breakdown is dominated by the avalanche effect. It is worth to remind that the input and output quantities are defined by the circuital configurations. For instance, in the common base configuration, the base is

168

6 Bipolar Junction Transistor

the common terminal and the input network quantities are I E and VB E , whereas the output network quantities are IC and VC B . Obviously, in such a configuration, the current gain is a little less than 1.

6.5 Non Ideal Effects The BJT model we have developed so far is a plain derivation from the PN junction ideal current. The main characteristic of the model lies in the fact that the device properties (in particular the current gain) are independent of the operative conditions. However, a number of minor effects must be considered in order for the device description to better represent the real behavior.

6.5.1 Early Effect The simple BJT model predicts that in the active zone the device behaves like an ideal current source. Namely, the current is independent of the voltage and the output characteristics (IC versus VC E ) is flat. However, the ideal current source violates the fundamental principles of the electric networks. Because, in this condition, different BJTs connected in series should be forced to provide the same current even if each device is biased with a different VB E . Consequently, IC must show some dependence on VC E . The simplest evidence of this dependency is represented by the Early effect, which emerges as a consequence of the BC-junction reverse biasing. In the active zone, since VB E is locked at the forward bias value (typically around 0.7 V for silicon devices), any positive increase in VC E results in an increase in the absolute value of VBC , and thus in a deeper reverse bias. Under reverse bias, the space charge region becomes wider in both the directions of the junctions. In the base, the position along the coordinate x, where the excess of injected charge is zero, moves towards the emitter, the slope of the concentration of the excess charges in the base increases and the current increases. The Early effect is manifested in a slight increase in current with VC E , making the output characteristics in the active zone far from being flat. The magnitude of the Early effect is described by a voltage (V A ) called Early voltage. It is defined by the calculation of the slope of the output characteristics. The slope is the derivative of IC with respect to VC E , which corresponds to the derivative with respect to VBC . Since the current changes because of the variation of x B , it is convenient to write: ∂ IC ∂ IC ∂ x B ∂ IC = = . ∂ VC E ∂ VBC ∂ x B ∂ VBC

(6.30)

6.5 Non Ideal Effects

169

The current at the collector depends on 1/x B (see Eq. 6.19), thus differentiating with respect to x B , we get: IC ∂ x B ∂ IC =− . ∂ VBC x B ∂ VBC

(6.31)

Finally, the Early tension can be defined as: Va = −

xB ∂ xB ∂ VBC

.

(6.32)

Graphically, the Early voltage on the V CC E axis of the output characteristics corresponds to the convergence point of the IC = f (VC E ) curves. The BJT is designed with a narrow base region and the BE and BC junctions further decrease the size of the base. With the device in active zone, the base is furtherly narrowed by the expansion of the depletion layer of the BC junction. In extreme cases, the space charge region can occupy all the neutral zone in the base, touching the base-emitter junction. This condition is said “punch-through” and corresponds to a shunt of the base. In order to mitigate the Early effect, it is necessary to limit the expansion of the space charge region, under reverse bias, towards the base. For this scope, the collector region needs to be less doped than the base. This condition breaks the symmetry of the device and makes the inverse active zone different from the active zone. The base is poorly doped in order to maintain the current gain large, and the collector is still less doped in order to mitigate the Early effect. Clearly, the doping of the collector cannot be too small in order to guarantee that the low-injection condition holds.

6.5.2 Emitter Band-Gap Narrowing The emitter efficiency depends on the ratio of the emitter doping to the base. It is thus straightforward that increasing the doping of the emitter leads to an increased ideality of the emitter efficiency. However, the effects of doping increase is limited by an additional phenomenon known as band-gap narrowing. As the doping concentration increases, the dopant atoms begin to sense the each other presence, so that, due to the Pauli principle, the donor level splits into a band. At doping concentrations larger than 1019 cm−3 , the donor levels begin to merge with the conduction or the valence band and the whole band gap of the semiconductor is narrowed. Eventually, the intrinsic concentration (n i ) in the emitter becomes larger and the hole current injected into the emitter increases, so that the emitter efficiency is reduced (n i = Nc Nv exp(−E g /k B T )). The emitter efficiency (Eq. 7.23) can be rewritten, including the intrinsic concentration of the base (n ib ) and the emitter (n ie ), as:

170

6 Bipolar Junction Transistor

Fig. 6.13 Intensity of the band-gap narrowing effect. Left: intrinsic concentration versus doping, Right: emitter efficiency versus doping. The plots are a direct application of equation 7.33 with N A = 1016 cm−3 . The band-gap narrowing is negligible when N D = 1018 cm−3 and it is 50, 95, and 150 meV at N D equal to 1019 , 1020 and 1021 cm−3 respectively

γ =

1 1+

2 n ie x B N A D pe 2 n ib x E N D Dnb

.

(6.33)

Figure 6.13 shows an example of the relationships between the doping concentration, the intrinsic concentration and the emitter efficiency. Clearly, an increase in doping beyond 1019 cm−3 results in a deterioration of the emitter efficiency.

6.5.3 Small Base Current In the active zone, and for VB E  k B T /q, the voltage drop across the base-emitter junction is: VB E =

Ib kB T ln . q Ib0

(6.34)

So, if I B is small, then also VB E is small. As discussed in Sect. 6.3, at small direct bias voltages the recombination current in the depletion layer is not negligible. The recombination current, labeled as 1 in Fig. 6.9, has been ignored so far because we supposed the base-emitter junction to be fully directly biased. Actually, when the exponential term of the ideal current is comparable with the exponential term of  q VB E the recombination current - namely, when exp k B T ≈ exp( q2kVBBTE )—an additional

6.5 Non Ideal Effects

171

quantity of electrons gets lost while traveling from the emitter to the collector. This corresponds to a decrease in α F . Since β F is very sensitive to small variations in α F , the recombination current in the depletion layer of the base-emitter junction results in a reduction the current gain factor.

6.5.4 High Injection Effects As the current increases, the concentration of injected charges may become comparable with the equilibrium charges. This condition is called high-injection and is related to the necessity of extracting large currents from the BJT. The occurrence of high injection jeopardizes the validity of the model that has been developed so far. In particular, the generation-recombination function cannot be approximated with Eq. 4.36. Besides increasing the complexity of the models, high injection is also detrimental to the performance of the device. The effects of high injections can be observed at both the base-emitter and base-collector junctions. High injection effects in the base-emitter junction High injection in the base-emitter junction results in a decrease in the ratio of Vbe to the injected electrons n  (0). This can be easily understood considering the product between electrons and holes at the interface between the base and the depletion layer of the base-emitter junction. This quantity has been calculated in Chap. 4 (Eq. 4.66). In high injection n  (0) ≈ N A , so that n ≈ p. The concentration of injected electrons can be directly calculated from Eq. 4.66:

n  (0) · p = n i2 exp



q Vbe kB T



→ n  (0) =



 n  (0) · p = n i exp

q Vbe 2k B T

 .

(6.35)

In practice, the efficiency of Vbe in controlling the density of injected charges is reduced. High injection effects on the base-collector junction At the base-collector junction level, the high-injection elicits a number of effects collectively known as Kirk effect. Without going into details, it is possible to explain the decrease in performance with a simple consideration about the concentration of the charges in the BC-junction depletion layer. The increase in the collector current consists in a large amount of electrons that transit through the depletion layer of the BC junction. The base and collector doping concentrations are not large enough to increase the current gain and hold back the Early effect respectively. Thus, the concentration of the electrons may easily become comparable to the concentration of the dopant atoms. This affects the total charge in the depletion layer. On the base side: Q = −N A − n, whereas on the collector side: Q = +N D − n. As a consequence, as shown in Fig. 6.14, the total charge increases in the base and decreases in the collector.

172

6 Bipolar Junction Transistor

Fig. 6.14 Variation of the charge concentration profile in the base-collector depletion layer. The solid line is the equilibrium condition, whereas the dashed line represents the concentration in high injection condition

The charge in the depletion layer affects the electric field. In particular, the electric field increases on the base side of the depletion layer, whereas it decreases on the collector side: EB =

q(−N A − n) (x + x B ); S

(6.36)

EC =

q(+N D − n) (x − xC ). S

(6.37)

The voltage drop across the junction is due to the applied VC E . In presence of a d.c. bias, it is practically constant (VBC = VC E − VB E ). The relation between the voltage drop and the electric field is as follows: xC VBC = −

Ed x.

(6.38)

xB

Thus, since the electric field changes, a change in the depletion layer dimensions is necessary to keep the voltage drop constant. Due to the different sign of the charge variation, the depletion layer in the base region is reduced, whereas in the collector region it gets expanded. As in the Early effect, this has a consequence on the current: the neutral zone in the base region increases, so that the current decreases. The Kirk effect thus behaves in an opposite way to the Early effect. Again, at large base currents the total current gain decreases. The qualitative behavior of β f versus VbE is shown in Fig. 6.15. Summarizing, Fig. 6.16 shows the qualitative behavior of the output characteristics of a generic BJT.

6.6 Physical Effects in Real BJT

173

Fig. 6.15 The current gain decreases respect to the calculated value both at small and large V B E because of the low injection and high injection effects respectively

Fig. 6.16 Typical behavior of the output characteristics of a BJT. The dotted lines indicate the slope of the Early effect. The active zone I/V curves converge to a negative VC E corresponding to the Early voltage (−V A ). Breakdown effects at the base-collector junction appear at large VC E

6.6 Physical Effects in Real BJT Figure 6.17 shows a schematic view of a practical BJT implementation in planar technology. Although the design is even more complex on a practical level, the scheme gives us an idea of the geometrical arrangement of the device. In this example, a npn transistor is grown on a p-type substrate and is separated from the rest of the wafer by deep oxide trenches. The core of the BJT are the n+ doped emitter, the narrow p-type base and the n-doped collector. In order to separate the base and the collector contacts, the latter is fabricated behind a shallow oxide trench that splits the collector into two regions connected by a thin n+ layer. This arrangement further insulates the device from the rest of the wafer by creating a deep depletion in the substrate, thus limiting the current leakage through the substrate itself. The base contact is annular around the emitter contact. The emitter area (A E ) is defined by the area below the emitter contact. The particular configuration of the base contact gives rise to undesired effect called base current-crowding. Since the base is poorly doped, its resistivity is high. As a consequence, the current from the base to the emitter tends to follow the shortest path and accumulates at the border of the emitter region. Then, even if the total base current is modest, the density of current at the edges of the emitter tends to become large. A high injection regime in this region can be activated and an asymmetric heating may happen. The current crowding can be avoided by reducing the distance

174

6 Bipolar Junction Transistor

Fig. 6.17 Principle scheme of a NPN BJT made on a p-type substrate. The paths of the base and collector currents are indicated by the arrows. The top view shows the concentric emitter and base contacts, together with the lateral collector contact. The figure shows a principle of implementation and the drawing is not in scale Fig. 6.18 Interdigitated base-emitter contacts. A large equivalent emitter area is obtained reducing the distance between the contacts and consequently the base resistance. It corresponds to a series of parallel base-emitter junctions

between the emitter and the base contact, but this solution in the configuration of Fig. 6.17 also requires a decrease in the emitter area and, consequently, a decrease in current. An advantageous solution to maintain a large emitter area and a short inter-contact distance is offered by the interdigitated contacts, as shown in Fig. 6.18. This geometry allows a reduction of the distance between the base contact and the base-emitter junction, maintaining the same emitter area. In practice, it corresponds to a parallel connection of many base-emitter junctions.

6.7 Dynamic Response

175

Finally, it must be mentioned that the distance between the contacts also affects the actual VB E across the junction, which is given by: VB E = VB E − R B I B , where R B is the total base resistance.

6.7 Dynamic Response The BJT dynamic response depends on three major factors: the capacitance associated to the depletion layer of the reverse biased base-collector junction (C jbc ), the diffusion capacitance due to the accumulation of minority charges in the forward biased base-emitter junction (Cdbe ) and the base resistance. A BJT equivalent circuit for the common emitter configuration is shown in Fig. 6.19. The resistance of the base is split into two parts related to the effective resistance of the base-emitter junction (R B E ) and to an additional resistance due to the distance between the base contact and the base-emitter junction (R B B ). The diffusion capacitance Cdbe is in parallel with (R B E ), while C jbc connects the base and the collector contacts.

6.7.1 Junction and Diffusion Capacitances The junction capacitance between the base and the collector is simply the depletion layer capacitance previously calculated both in Schottky and PN diodes: Cj =

s xd

(6.39)

where xd is the depth of the depletion layer xd = 2qs φi ( N1A + N1D ). The diffusion capacitance of a forward bias PN junction has been calculated in Chap. 4. Here we recall that the diffusion capacitance is the derivative of the total

Fig. 6.19 BJT equivalent circuit. B indicates the physical contact of the base, whereas B’ is the internal contact

176

6 Bipolar Junction Transistor

charge of the injected minority charges (electrons and holes in the respective parts of the junction) with respect to the junction voltage. The capacitance in the p-type base is due to the injected electrons. Cd =

d Q nb . d VB E

(6.40)

The relationship between Q nb and the diffusion current generated by this charge is: d Q nb =

x B2 Jn = τtr Jn 2Dn

(6.41)

where τtr is the time of transit of the charges in the base region. Eventually, the diffusion capacitance in the base is: Cd = τtr

d Jn . d VB E

(6.42)

The transit time can be significantly reduced if the base region (x B ) is small and the diffusion constant is large, this last condition meaning that the base is poorly doped. To provide an example of transit time, let us consider a p-type base, where N A = 1015 cm−3 , L nb = 14 µm and Dnb = 18.9 cm2 /s. For a base region depth x B = 0.45 µm, the transit time is about 50 ps. The transit time can be further reduced if the transport of electrons in the base could be facilitated by an electric field (v = μE). In a previous section we have considered that a non uniform doping profile in base ensures the low-injection limit condition even when the total amount of holes in base is small. However, the consequence of a non uniform doping is a built-in potential (see Sect. 1.5). Since the applied voltage across the base region is null, the built-in potential in base in unaffected by the bias. However, this potential drags the injected charges towards the collector and produce an additional term to the collector current. A linear doping profile results in a linear built-in potential, and then, in a constant electric field. A qualitative comparison between drift and diffusion currents can be achieved considering that the drift current, elicited by the built-in electric field, is proportional to the total amount of injected charges, namely on the integral of the injected charges profile. On the other hand, the diffusion current is proportional to the derivative of the injected charges in the base. Thus, the condition of short base, essential for the BJT, makes the drift current smaller than the diffusion current. A more deeper description of the effect of the built-in potential in base on the collector current is discussed in Appendix F. Here, let us limit the discussion to the effect of the drift current on the transit time of current in base. For the scope, let us separate the effects of the two currents, and compare the transit time of drift respect to that due to the diffusion. The transit time due to drift is:

6.7 Dynamic Response

177

dri f t

τtr

=

xB xB = . v μn E

(6.43)

The electric field is proportional to the voltage drop across the base region: E = V /x B , so that: dri f t

τtr

=

x B2 . μm V

(6.44)

The application of an electric field is advantageous if the transit time due to drift exceeds the transit time due to diffusion. Considering the ratio between the two transit times, a condition for the voltage drop across the base necessary to improve the transit time can be obtained: dri f t

τtr

di f f

τtr

=

2Dn x B2 kB T · 2 = μm V x B qV

(6.45)

where the Einstein relationship has been used to replace the diffusion constant with the mobility. Then, the transit time is dominated by drift if the voltage drop is at least twice the thermal voltage. At room temperature we have: V ≥ 2

kB T 50mV. q

(6.46)

This voltage drop can result from a graded doping profile of the base. Figure 6.20 shows that in the case of a graded doping, the equilibrium (constant Fermi level) is maintained by a voltage drop across the material. Let N AE and N AC be the doping concentration at the borders of the emission and collector regions. The voltage drop at the equilibrium is given by:   N AE kB T . (6.47) ln V = q N AC Thus, the condition V = 2 k BqT is obtained when: N AE = e2 7. N AC

(6.48)

The excursion of the doping concentration, however, must satisfy the conditions to obtain current amplification, namely the Early effect and the low-injection limit. The diffusion capacitance due to the holes in the emitter region is given by: Cdp =

d Q pe d Vbe

(6.49)

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6 Bipolar Junction Transistor

Fig. 6.20 Band diagram of a non uniformly doped p-type region. Left: hypothetical band diagram of slices of P-type regions before to be brought in contact; right: bands in equilibrium . The equilibrium is maintained by a built-in voltage drop

where Q pe = τtr E Jne , τtr E being the hole transit time in the emitter region. The current defining the holes injected into the emitter is the current flowing towards the base. Actually, the concentration of injected holes and electrons is rather different due to amplification, but, disrespectful of their own value, their ratio is always constant. Namely, the equilibrium is always maintained. Consequently, the time necessary for the injected holes to be accumulated is the same time we need to accumulate the injected electrons. The concentration of injected holes is: x E Q pe =

x E



qp (x)d x = 0

=q

n i2 ND

 exp

   n i2 x q Vbe dx 1− exp ND kB T xE

0

q Vbe kB T



(6.50)

xE . 2

Reminding the relationship between electron and hole currents in the forward biased BE junction and the expression for the hole current (Eqs. 6.21 and 6.22), the transit time of the holes in the emitter can be written as: τtr E =

x E2 G nb 2Dnb G ne

(6.51)

where G nb and G ne are the Gummel numbers of the base and emitter respectively. The diffusion capacitance due to the hole injection into the emitter is thus given by: Cd E = τtr E

d Jn x 2 G nb d Jn = E G ne . d VB E 2Dnb G ne d VB E

(6.52)

Hence, with respect to the diffusion capacitance due to the injected electrons, the capacitance due to the hole injection is depressed by a factor corresponding to the ratio of the Gummel numbers. Namely, the more the BJT amplifies, the smaller is this capacitance. Since the two diffusion capacitances are in parallel, the only meaningful

Further Reading Table 6.1 Doping—Performance relationship Current gain Early effect counteraction Kirk effect counteraction Small transit time in the base Small Junction capacitance Small base resistance

179

N DE > N AB and short base C N AB > N D C4 B large N A and N D B small N A or graded N AB (x) C , then narrow BC junction large N AB and N D depletion layer large N AB and/or short base region

capacitance of the base-emitter junction is the diffusion capacitance of the electrons injected into the base.

6.8 Summary The BJT is the most important application of the PN junction. The fundamental idea of the BJT is to configure one of the regions of a PN junction in order to separate the electrons and holes in two different contacts. This region is the base where the minority charges are collected by a reversed bias junction, and the majority charges by a metallic contact. The BJT exploits the characteristics of short-base diodes where the minority charges injected by the emitter, because of the forward bias of the emitter-base junction, travel through the base with a negligible recombination. The resulting current is then conveyed into the collector by the reverse bias of the base-collector junction. The BJT properties derive from a balance between the doping concentrations and the dimensions of the three neutral regions. The optimization of the device requires different and sometimes opposite conditions (Table 6.1). In particular, the conditions leading to a large current gain are in contrast with the conditions necessary for a high frequency operation. In the next chapter we will see that heterostructures - namely, junctions formed by materials differing not only for their doping but also for their band gap and affinity - offer an optimal solution for BJT designs, making the gain independent of the base doping.

Further Reading Textbooks C.C. Hu, Modern Semiconductor Devices for Integrated Circuits (Pearson College, 2009) R. Muller, T. Kamins, M. Chen, Device Electronics for Integrated Circuits, 3rd edn. (Wiley, 2002)

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6 Bipolar Junction Transistor

D. Neamen, Semiconductor Physics and Devices (McGraw Hill, 2003) B. Streetman, S. Banerjee, Solid State Electronic Devices (Prentice Hall, 2006) S.M. Sze, Kwok K. Ng, Physics of Semiconductor Devices, 3rd edn. (Wiley, 2006)

Journal Papers L. Bardeen, Brattain, The transistor, a semiconductor triode. Phys. Rev. 74, 230 (1948) J.M. Early, Effects of space-charge layer widening in junction transistors. Proc. IRE 40, 1401 (1952) H. Gummel, Measurement of the number of impurities in the base layer of a transistor. Proc. IRE 49, 834 (1961) C. Kirk, Jr., A theory of transistor cutoff frequency ( f T ) falloff at high current densities. IEEE Trans. Electron. Devices ED-9 164 (1962) W. Shockley, The theory of p-n Junctions in semiconductors and p-n junction transistors. Bell Syst. Tech. J. 28(435), 5 (1949) W. Shockley, Transistor electronics: imperfections, unipolar and analog transistors. Proc. IRE 40, 1289 (1952) W. Shockley, M. Sparks, G. Teal, P-N junction transistors. Phys. Rev. 83, 151 (1951) W.M. Webster, On the variation of junction-transistor current amplification factor with emitter current. Proc. IRE 42, 914 (1954)

Chapter 7

Heterojunctions

7.1 Introduction In the previous chapters, we studied the properties of single semiconductor devices characterized by a variable distribution of acceptor and donor atoms. In the band diagrams we have drawn so far, the affinity and the energy gaps of the semiconductors were the same in any regions of the device and the work functions, which depend on doping, were the only variables. Although the discussion has been so far mostly limited to the elements of the IV group of the periodic table (silicon and germanium), there is a large variety of available semiconductors. Among them, those obtained by combining elements of the groups III and V, notably gallium arsenide (GaAs) and indium phosphate (InP), are endowed with very interesting properties, which have made them appealing for high frequency and optoelectronic applications. One of these features is the negative differential resistance, which was discussed in Chap. 5. Even ternary mixtures, such as aluminum gallium arsenide (AlGaAs), may result in semiconducting materials. The semiconducting character is also exhibited by the transition metal oxides. Some of these materials are gaining importance for their application in optoelectronic and piezoelectric devices, such as zinc oxide (ZnO) and titanium dioxide (TiO2 ). More recently, semiconductor materials based on organic molecules and polymers have also emerged, giving rise to the so-called organic electronics. With all these materials, it is reasonable to conceive devices where more materials can coexist, forming junctions between different semiconductors. Such junctions are called heterojunctions. On the contrary, all the junctions we have considered so far can be called homojunctions. In the frame of the theory developed in the previous chapters, a heterojunction is defined as the junction between two materials, where not only the work function, but also the affinity or the energy gap, or even the both of them, are different. We will see thereafter that the properties of heterojunctions are peculiarly different than those of homojunctions. In particular, they are advantageous for those BJTs where the base-emitter junction is formed by two different materials, which is what we © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_7

181

182

7 Heterojunctions

will focus on. These properties have been theoretically predicted since the beginning of the semiconducting era. Notably, the idea of an heterojunction-based BJT was patented by W. Shockley in 1951. The fabrication of heterojunctions was nonetheless impossible until the beginning of the seventies. The main technological problem in making heterojunctions is the excess of interface defect density. The interface defects arising in a metalsemiconductor junction were described in Chap. 2. Surface defects are such as to change completely the energy of electrons and the device properties. These defects are much more evident in a junction between two different semiconductors, where the materials should preserve their crystalline structure. The most evident problem comes from the fact that since the crystalline periodicity is different and the atoms of one material must bond the atoms of the other, a crystal strain necessarily occurs at the interface. This gives rise to defects that modify the work function (as seen in Chap. 2), the mobility and the generation-recombination processes. In practice, for many years the concentration of interface defects has been so high that the devices built with heterojunctions have not been able to work properly. Mitigation of the crystalline mismatch occurs in semiconductor alloys where the transition from one material to another can be obtained with gradual addition of one element. Examples of these materials are silicon-germanium (Si1−x Gex ) and aluminum-gallium-arsenide (Alx Ga1−x As). The previous composition formulas are characterized by a variable x, which is a measurement of the alloy composition. For instance, in the case of silicon-germanium, x=0 indicates pure silicon, whereas x=1 is the case of pure germanium. Thus, as a function of x, the properties of the material change. The crucial quantity that is altered by the alloy composition is the energy gap. As for the silicon-germanium case, the energy gap is between 1.12 eV (pure silicon) and 0.67 eV (pure germanium). For small values of x (of the order of x qχ2 , qφ1 < qφ2 and E gap1 > E gap2

7.2.1 Staggered Bandgaps Figure 7.2 shows the bands before the formation of the junction in the case of staggered band gaps. Let us consider a case where qχ1 > qχ2 . Of course, band gaps and work functions are also different. In this example, the material with the largest bandgap is n-type and the other one is p-type. The steps for the correct drawing of the equilibrium band diagram are outlined in Table 2.1. Figure 8.3 shows the band alignment procedure applied to the case of staggered band gaps. As a consequence of the differences in affinities and energy gaps, the conduction and valence bands are not continuos. The changes in energy at the interface between the conduction band and the valence band are: E C = q(χ1 − χ2 ) = qχ ; E V = (qχ1 + E gap1 ) − (qχ2 + E gap2 ) = E g + qχ .

(7.3)

7.2.2 Straddled Bandgaps This case is typical of alloys. For instance, Si versus SiGe. In this arrangement, shown in Fig. 7.4, the band diagram of the material with the smallest bandgap is contained in the bandgap of the material with the largest bandgap. The application of the procedure outlined in Fig. 7.2 leads to the equilibrium band diagram in Fig. 7.5. The main difference between the two cases is the shape of the conduction band discontinuity. In the case of the straddled band gap, the discontinuity has the shape of a spike. The main consequence of the spike is the fact that the electron concentration in the band gap does not gradually decrease from the n-type to the ptype material. One the contrary, at the interface there exists a small region where the electron concentration increases.

7.2 Band Diagram

185

Fig. 7.3 Steps for the construction of the equilibrium band diagram in the case of staggered bands. 1 At the equilibrium the Fermi level is constant. 2 The space charge region is qualitatively identified. The bulk of the material is not modified by the junction and the band diagram of the unperturbed regions is drawn using the Fermi level as reference. 3 The vacuum level is continuous and the affinity is constant, so that in both materials the conduction band is parallel to the vacuum level. Since the affinities are different, at the interface the conduction band is not continuous (E C ). 4 The energy gap is constant, so that the valence bands are parallel to the conduction bands. Since the band gaps are different, alsop the valence band is not continuous at the interface (E V )

In normal conditions, this increase is modest and the interface region is still depleted of electrons. However, under proper conditions, the spike may give rise to a narrow region where the concentration of electrons is very high. This case will be discussed in the last chapter of this textbook. The changes in energy at the interface between the conduction and valence bands are still given by the same formulas of the staggered case: E C = q(χ1 − χ2 ) E V = E g + qχ .

(7.4)

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7 Heterojunctions

Fig. 7.4 Band diagrams of straddled band gaps of n-type and p-type materials. In this example qχ1 < qχ2 , qφ1 < qφ2 and E gap1 > E gap2

Fig. 7.5 Equilibrium band diagram of the straddled band gap is drawn following the steps outlined in Fig. 7.3. Respect to the staggered band gap, here the non continuity of the conduction band takes the form of a spike

However, due to the relative magnitude of the affinities, E C is negative. In Fig. 7.5, qbi is the contact potential difference, which is equal to the difference between the work functions. This quantity can be measured at the surface of the materials and is different from both the built-in potential for the electrons (qbn ) and the built-in potential for the holes (qbp ). Thus, one of the most characteristic features of heterojunctions, regardless of the mutual position of the bands, is the different potential barriers acting upon electrons and holes. This difference is at the basis of the behaviors that will be discussed thereafter. In Figs. 7.3 and 7.5 the curves connecting the non-perturbed band diagram have been only qualitatively drawn. The exact behavior is the result of the Poisson equation. Since the straddled case is more frequent and more interesting, the electrostatic quantities herein are calculated for this case. However, the calculation can be straightforwardly extended to the straddled case.

7.3 Electric Field and Built-In Potential

187

Fig. 7.6 Charge density distribution according to the hypothesis of step junction and deep depletion

7.3 Electric Field and Built-In Potential To solve the Poisson equation, let us follow the same procedure used for the PN homojunction. The different dielectric constants in the two regions, however, make an important difference. Since the amount of charges in the two regions must be the same (charge neutrality condition), the electric fields at the interface are indeed different. The main assumption for the Poisson equation solution concerns the distribution of the dopant atoms and, consequently, the total charge distribution. The step junction and the deep depletion hypotheses are still applied here. Namely, dopants are uniformly distributed until the interface and the space charge region is completely depleted of the majority charges until the border of the neutral zones. The charge distribution is shown in Fig. 7.6. The electric field in the two regions are calculated by integrating the charge density distributions: q ND (x + xn ); (7.5) En (x) = 1 E p (x) =

q NA (x p − x). 2

(7.6)

Since 1 = 2 , the electric field is not continuous at the interface. Actually, the continuous quantity is the electric displacement field D, whose relation with the electric field is: D = E. Thus, at the interface, the continuity of the electric displacement D1 (0) = D2 (0) provides: q ND q NA 1 x n = 2 xp (7.7) 1 2 which still corresponds to the charge neutrality condition: q N D xn = q Na x p . In the case of Si − Si 1−x Gex junction, since  Si = 11.7 and Ge = 16, we have 1 < 2 , so that the electric field in silicon is greater with respect to the germanium side (Fig. 7.7). The potential is calculated by integrating each electric field in its own region: φ1 (x) = φ(−xn ) −

q ND (x + xn )2 ; 21

(7.8)

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7 Heterojunctions

Fig. 7.7 Electric field in the case of 1 > 2

Fig. 7.8 Electric potential in the case of 1 > 2

φ2 (x) = φ(x p ) +

q NA (x p − x)2 . 22

(7.9)

The potential behavior is shown in Fig. 7.8. The potential, of course, is continuous at the interface φ1 (0) = φ2 (0): φ(−xn ) −

q ND 2 q NA 2 x = (x p ) + x . 21 n 22 p

(7.10)

Thus, the built-in potential is: φbi = φ(−xn ) − φ(x p ) =

q ND 2 q NA 2 x − x . 21 n 22 p

(7.11)

It is important to remind that the built-in potential of an heterojunction appears only between the surfaces of the materials. Thus, it does not directly influence the behavior of the device, as we will see in the next section. Rather, it allows for the measurement of the depletion layer size. Following the case of PN homojunction, by combining the charge neutrality condition with the built-in potential we are nonetheless able to find the relationship between the depletion layer size and the material parameters:  xp =

21 2 N D φbi ; xn = q N A (1 N D + 2 Na )



21 2 N A φbi . q N D (1 N D + 2 Na )

(7.12)

7.4 The Quasi-electric Field

189

Note that, as found in the PN homojunction, the size of the depletion layer in each region is proportional to the doping in the opposite region.

7.4 The Quasi-electric Field As mentioned above, electrons and holes in heterojunctions are subject to different potentials. This fact seems a paradox because two different electric fields, one acting on negative charges and the other one on positive ones, cannot exist at the same time and inside the same material. To solve this apparent paradox is sufficient to consider that the energies at the bottom of the conduction band and at the top of the valence band correspond to the potential energy of electrons and holes respectively. The value of these energies with respect to zero, as shown in Fig. 7.9, are: E C (x) = qφ + qχ E V (x) = qφ + qχ + E g

(7.13)

where φ is the vacuum energy, namely the electrostatic potential at the surface of the material. The force acting upon each particle is the derivative of the potential energy. Hence, the forces applied to electrons and holes are: dφ dχ d E C (x) = −q −q dx dx dx d Eg dφ dχ d E V (x) Fp = − = −q −q − . dx dx dx dx Fn = −

(7.14)

In homojunctions, activity and energy gap are constant everywhere, so that electrons and holes are only subjected to the electrostatic force (Fel = qE = −q dφ ). On dx the contrary, in heterojunctions, affinity and energy gap are not constant, so that the forces applied to electrons and holes are not equal either in magnitude or in direction. Furthermore, the electric field is only one component of these forces. However,

Fig. 7.9 Conduction band of a straddled band gap case in equilibrium and under bias

190

7 Heterojunctions

since electric fields are usually thought of as producing forces on charges, the fields applied to the charges are called quasi-electric fields.

7.5 Current-Voltage Relationship The main consequences of heterojunctions are the different barriers that keep electrons and holes in equilibrium. We will see that this difference also implies that the electron and hole currents are different. The calculation of the current/voltage relationship follows the same steps of the PN homojunction case. In particular, the PN junction ideal current is calculated by evaluating the equilibrium charge concentration at the edge of the neutral zones. To this scope, it is necessary to evaluate the barriers for electrons and holes. Let us consider the equilibrium band diagram in Fig. 7.5. The concentration of the charges at the borders of the depletion layer is: n n0 = N D p p0 = N A n p0 pn0

  qφbn = N D ex p − = KBT   qφbp = = N A ex p − KBT

n i2p

(7.15)

NA 2 n in . ND

The minority charge concentration at the equilibrium depend on the barrier heights, but also, via the mass action law, on the concentration of the intrinsic charge carriers. The quantity n i2 is a function of the energy gap: n i2 =NC N V ex p(−E G /K B T ), so that it is different in the two materials. The square of the intrinsic concentrations can be very different. Considering the extreme cases of silicon and germanium, we have: n i2Si = 1020 cm −6 and n i2Ge = 1026 cm −6 . This, indeed, makes the minority charge concentration different in the two semiconductors. From the minority charge equation, we get the relationship between the potential barriers and the energy gaps: NC p N V p ex p(−E Gp /K B T ) qφbn )= ; KBT NA qφbp NCn N V n ex p(−E Gn /K B T ) )= = N A ex p(− , KBT ND

n p0 = N D ex p(− pn0

(7.16)

7.5 Current-Voltage Relationship

191

 NC p N V p ; NA ND   NCn N V n . − K B T ln NA ND 

where:

qφbn = E Gp − K B T ln qφbp = E Gn

(7.17)

The difference between the two barriers is:  qφbp − qφbn = E Gn − E Gp − K B T ln

NCn N V n NC p N V p

 (7.18)

where NC and N V are the density of states in conduction and valence band respectively:   2π m ∗n K B T 3/2 NC = 2 h2 (7.19)   2π m ∗p K B T 3/2 NV = 2 . h2 The density of states depends on the effective mass of electrons and holes, which are determined by the shape of the bands E(K ). Thus, the effective mass is a peculiar characteristics of each material. However, the differences among materials are modest and the density of states is barely similar in different materials. In Table 7.1, the effective masses and the density of states for silicon and germanium are listed. To appreciate the difference due to the density of states, let us consider a case of N D = N A = 1018 cm −3 in this case considering the extreme case  of N-type  NCn N V n silicon and P-type germanium, at room temperature, K B T ln NC p NV p ≈ 40 meV . This small quantity is small with respect to the energy gaps difference, so that the potential barrier applied to the holes can be simply written as: qφbp = qφbn + (E G )

(7.20)

where (E G ) is the difference between the two band gaps. It is clear that the different energy gap makes the intrinsic concentrations different and, if the two materials are doped with the same amount of doping, the concentration of the minority charges may also be very different.   = φbn − V A and φbp = φbp − Under external bias, both the barriers change: φbn V A . Thus, while the density of the majority charges remains constant, the density of the minority charges increases:  qφbn − q V A n p = N D ex p − KBT     E G qφbn − q V A ex p − pn = N A ex p − KBT KBT 

(7.21)

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7 Heterojunctions

Table 7.1 Density of states and effective masses in silicon and germanium Silicon Germanium NC NV m ∗n m ∗p

2.8 · 1019 1.04 · 1019 1.08 · m 0 0.81 · m 0

1.04 · 1019 0.60 · 1019 0.55 · m 0 0.30 · m 0

which gives rise to the excess minority charges: n  = n − n 0 and p  = p − p0 :    qφbn q VA n p = N D ex p − ex p( )−1 KBT KBT      E G q VA qφ bn ex p − ex p( )−1 . pn = N A ex p − KBT KBT KBT

(7.22)

In the discussion about homojunctions, we have seen how the excess charges at the border of the neutral zone diffuse and, during the diffusion, how they get recombined. Under stationary bias, this results in a steady distribution of minority charges that gives rise to a diffusion current, which depends on the size of the neutral zone. In the case of a long-base diode - namely, when the distance between the depletion layer and the electrode is much larger than the recombination length - the recombination process consumes the majority charges, so that a majority charge diffusion current is required to maintain the steady state condition. This majority charge current has the same magnitude, but opposite direction. Eventually, both electrons and holes produce a net current:    q VA Dn qφbn ex p( )−1 ; N D ex p − Jn = q Ln KBT KBT      E G q VA Dn qφbn Jp = q ex p − ex p( )−1 . N D ex p − Ln KBT KBT KBT

(7.23)

(7.24)

As for the short-base diode, the expressions L n and L p are replaced with W B and W E respectively. The ratio between the two currents is:   Dn L p N D E G Jn . (7.25) = ex p Jp Ln Dp NA KBT Except for the last term, this is the same equation we found in the case of the PN homojunction, where the ratio between the currents was different from 1 only if the doping was different. In the case of heterojunctions, the ratio can be very different

7.6 Heterojunction Bipolar Transistor

193

even if the doping is equal. This result emphasizes the fact that the current in the junction is dominated by the minority charges. For example, given a junction between Si and Si 1−x Gex with x=0.3, the ratio E G /K B T = 5.4, whose exponential is approximately 220.

7.5.1 Thermionic Current In the case of straddled band gaps, the discontinuity of the conduction band at the interface takes the shape of a spike, which means that the concentration of electrons inside the depletion layer abruptly increases immediately beyond the interface. If the bottom of the spike is sufficiently above the Fermi level, then the increase in concentration does not interfere with the depletion layer hypothesis, so that the concentration of electrons still remains negligible. The main hypothesis in the calculation of the current is that the minority charge concentration derives from the majority charge concentration in the opposite region weighted by the potential barrier, which represents the change in the conduction band at the border of the space charge region. This hypothesis is valid if the top of the spike lies below the conduction band in the neutral region. As the applied voltage increases, it may happen that the the top of the spike exceeds the conduction band, as shown in Fig. 7.9. In this condition, the actual barrier for the electrons is E s − E C1 , where E s is the energy at the top of the spike. Since V A is applied across the entire depletion layer, the voltage that drops between the top of the spike and the conduction band is lower than V A . Hence, we find a smaller current efficiency: In this condition, the current coincides with the flow of electrons that cross the barrier E s − E C . In practice, it is similar to the thermionic current of the Schottky diode. Eventually, a decrease in the differential conductance is observed for large applied voltages (Fig. 7.10): Finally, it must be remarked that in a heterojunction, even in the case of a graded alloy, the interface region is much more defected compared to a homojunction. Consequently, a more efficient recombination in the depletion layer is expected with respect to a normal homojunction, which will increase the magnitude of the recombination current in the depletion layer under direct bias. Compared to a homojunction, indeed, the effect of the recombination current persists for a larger value of the applied voltage.

7.6 Heterojunction Bipolar Transistor One of the most immediate applications of heterojunctions is the so-called Heterojunction Bipolar Transistor (HBT). On a theoretical level, the advantages offered by heterojunctions have been straightforward since the first development of a BJT. Actu-

194

7 Heterojunctions

Fig. 7.10 Qualitative current-voltage relationship of a p-n heterojunction

Fig. 7.11 Equilibrium band diagram of a n-p-n HBT

ally, the original patent released by the Bell Telephone in 1948 contained the HBT concept. In particular, the design showed a base-emitter junction and a base-collector junction working as a heterojunction and a homojunction respectively. In a n-p-n device, the material the emitter is made of shows a larger band gap than those of both the base and collector. The majority charges of the large-band-gap material (electrons in a n-p-n device) defines the BJT main current. Figure 7.11 shows the equilibrium band diagram of such a device. The most important consequence of the heterojunction is the asymmetry of the electron and hole currents across the base-emitter junction. The ratio between the currents is proportional to the exponential of the change in energy gaps. The ratio between the currents is important for the emitter efficiency, which in turn is the current gain paramount element:

7.6 Heterojunction Bipolar Transistor

195

γ =

1 1+

Jp e Jn e

.

(7.26)

Using the same nomenclature for the BJT in Eq. 18, Chap. 6, the emitter efficiency can be written as: 1 . (7.27) γ = x B N A D pe G 1 + x E N D Dnb ex p(− E ) KBT The above equation holds for uniform doping distributions. In the case of variable doping profiles, the Gummel number replaces the product between the doping and the neutral regions size: γ =

1 1+

G N B D pe G ex p(− E ) G N E Dnb KBT

.

(7.28)

Thanks to the exponential of the change in energy gaps, the emitter efficiency can be very close to 1 even if the condition G N B  G N E is not fulfilled. Thus, in order for large current gain to be obtained, a light base doping is not necessary. With a HBT, on the contrary, the current gain is achieved even if the base if highly doped (Fig. 7.12). A heavy doping of the base is an efficient counteraction against the Kirk effect (by extending the low injection limit of the device) and the Early effect, as it improves the current source output characteristics of the device.

Fig. 7.12 β f as a function of the x percentage of Germanium in a Si − Si 1−x Gex base-emitter junction

196

7 Heterojunctions

Fig. 7.13 Qualitative behavior of β f versus VB E for HBT and BJT. Typically, HBT exhibits a larger β f , but for a shorter range of VB E

Actually, the amplification involves not only the emitter efficiency, but also the base transport factor: x2 αT = 1 − B2 . (7.29) 2L n In a BJT, the base width cannot be too narrow because of the Early effect and the consequent punch-through effect. In a HBT, a heavy base doping strongly reduces the Early effect and the punch-through risk. Thus, since the base length is immune from bias, the base can be made very narrow and αT is practically equal to 1. Eventually, the current amplification factor β F depends only on the emitter efficiency: γ αT γ βf = . (7.30) ≈ 1 − γ αT 1−γ Due to the influence of both the recombination and thermionic currents, the current gain is expected to reach its maximum level for a shorter interval of VB E with respect to a BJT homojunction (Fig. 7.13). Another important advantage of the HBT technology concerns the dynamic response. Since the base may be highly doped, the base resistance could be made very thin. In this way the base length is immune from bias. Figure 7.14 shows a simplified scheme of a HBT made with GaAs technology. The emitter is made of n-AlGaAs and the base is made of a p + − Ga As thin layer. Note that, due to the low conductivity of the intrinsic GaAs (n i ≈ 106 cm −3 , the device is insulated from the substrate.

7.6.1 Graded Band Gap The band-gap engineering technology allows for the preparation of materials where the band gap can be graded by a continuous variation of the x parameter. Band-gap-graded materials may help improve some electronic devices features. For instance, the response time of bipolar transistors is limited by the transit time in

7.6 Heterojunction Bipolar Transistor

197

Fig. 7.14 Simplified scheme of an epitaxial HBT in GaAs technology

Fig. 7.15 Band diagram of a uniformly doped band-gap-graded p-type material

the base. As discussed in Chap. 6, a method to reduce transit time is the creation of an electric field in the base region via graded doping. The band-gap engineering technology offers a more efficient solution to the problem, by avoiding doping reduction at the base-collector interface, which can counteract both the Kirk and the Early effects. In practice, the base is fabricated with a material that presents a graded reduction of the band gap. Also, it is doped with a uniform concentration of acceptors. The band diagrams before the equilibrium and at the equilibrium are shown in Fig. 7.15. This is an extreme case of different quasi-electric field, since there are no forces that act upon the holes when a potential φi is applied to the electrons: φi =

K B T nc ln q ne

(7.31)

where n e and n c are the electron concentrations at the interface with the emitter and the collector respectively.

198

7 Heterojunctions

These concentrations can be calculated from the mass-action law:   n2 NC N V E Ge n e = i1 = ex p − ; NA NA KBT   n2 NC N V E Gc . ex p − n c = i2 = NA NA KBT

(7.32)

Hence, the potential is given by: φi =

   E G KBT E Gc − E Ge = ln ex p − . q KBT q

(7.33)

With such a potential, the transit time due to drift is: dri f t

τtr

=

x B2 q x B2 . = μn V μn E G

(7.34)

It is also important to point out that the graded doping gives rise to an electric field in the material that accelerates both electrons and holes. Consequently, as the transit time of electrons decreases, also the hole current slightly increases. On the other hand, in a band-gap-graded material, only the electrons are accelerated. In order to be efficient, the transit time due to drift must be greater than the x2 di f f transit time due to diffusion: τtr = 2DBn . To compare the two transit times, it is then convenient to express the mobility via the diffusion coefficient, using the Einstein relation. In this way, the ratio between the two transit times can be written as: dri f t

τtr

di f f τtr

=

2K B T . E G

(7.35)

The ratio is smaller than one if E G > 2(K B T /q) ≈ 52 meV. in Si 1−x Gex , this condition is fulfilled with an increase in germanium equal to x = 0.1. Thus, a slight increment of germanium is enough to improve the transit time of the electrons in the base and, consequently, to reduce the diffusion capacitance, as well as to extend the bandwidth of the device. In terms of frequency range, Si/SiGe devices can work up to 300 GHz. The largest operating frequency up to 500 GHz has been achieved by an InP/InGaAs device. Band-gap grading can also be used at the emitter-base junction in order to smooth the electron concentration spike at the interface and thus to reduce the influence of the thermionic current.

Further Reading

199

7.7 Summary This chapter introduced the properties of junctions made out of different semiconductors. The most interesting consequence is the different force acting on the two charge carriers. Different forces that can be explained considering that the electrostatic potential arising in the space charge region is only one of the potentials at which holes and electrons are exposed. Additional forces arise due to differences in affinities and energy gaps. The total forces have been called quasi-electric field to remark the fact that they are applied to charges, but they are different for each of them. Thank to the quasi-electric fields, the asymmetry in PN junctions between the electron and hole currents is not only related to the different doping of the two regions, but also to the different band gaps. This property is exploited in heterojunction bipolar transistors, where a great amplification factor can be obtained with equally doped emitter and base regions. As a consequence, a largely doped base results in a bipolar transistor bearing larger currents and a more extended bandwidth respect to the devices based on homojunctions. It is interesting to remark that the properties of heterojunctions were devised well in advance respect to the actual capability of producing them.

Further Reading Textbooks K. Brennan, A. Brown. Theory of modern electronic semicionductor devices. J. Wiley (2002) C.C. Hu, Modern Semiconductor Devices for Integrated Circuits (Pearson College, 2009) D. Neamen, Semiconductor Physics and Devices (McGraw Hill, 2003)

Journal papers R. Anderson, Experiments on Ge-GaAs heterojunctions. Electron. 5, 341 (1962) H. Kroemer, Der drifttransistor. Naturwiss. 40, 578 (1953) H. Kroemer, Theory of a wide-gap emitter for transistors. Proc. IRE 45, 1535 (1957) H. Kroemer, Heterostructure bipolar transistors and integrated circuits. Proc. IEEE 70, 1325 (1982) K. Ismail, S. Nelson, J. Chu, B. Meyerson, Electron transport properties of Si/SiGe heterostructures: measurements and device implications. Appl. Phys. Lett. 63, 660–62 (1993)

Chapter 8

Metal-Oxide-Semiconductor Junction

8.1 Introduction The electric characteristics of the junctions between metals and doped semiconductors, as seen in Chap. 2, are characterized by conductivity and capacitance, whose value is a non-linear function of the applied voltage. In this chapter we will study the properties of a device where the metal and the semiconductor are separated by an insulator layer. The presence of the insulator, indeed, eliminates the conductivity, so that this device is electrically equivalent to a capacitor, where the capacitance is still a non-linear function of the applied voltage. In such a device, the semiconductor charge states (quantity and sign) are modulated by the applied potential. This effect, known as Field Effect, is of paramount importance in electronics because it is at the origin of the family of Field Effect Transistors. When voltage is applied to a typical metal-insulator-metal capacitor, electrons are accumulated on the surface of one plate and depleted on the other one. Because of the properties of metals, the charge layers are confined to ideal planes at the metal-insulator interfaces. On the other hand, if one of the plates is replaced with a semiconductor, the potential can penetrate the semiconductor and modify the electron and hole concentrations. The metal-insulator-semiconductor structure is a fundamental building block of the silicon technology. Silicon native oxide (SiO2 ) being the obvious electrical insulator, this junction is called metal-oxide-semiconductor (MOS). The formation of silicon dioxide on crystalline silicon is a peculiar process that contributes to the overall performance of the device. Silicon dioxide is formed by oxidation of the crystalline silicon in an atmosphere of either oxygen or water vapor, both kept at a temperature between 850 and 1100 ◦ C. The involved chemical reactions are: Sisolid + O2(gas) → SiO2 Sisolid + 2H2 O → SiO2 + 2H2 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_8

201

202

8 Metal-Oxide-Semiconductor Junction

Fig. 8.1 Structure of an ideal MOS structure

The oxide layer grows while consuming the silicon atoms close to the surface. In order to create a thick oxide film, the oxygen molecules must diffuse through the newborn oxide layer to reach the silicon atoms below the surface and oxidize them. The oxygen diffusion is favored by the high temperature. As the oxide becomes thicker, the diffusion towards the silicon becomes less probable and the growth of the oxide becomes slower. When the process stops, a silicon oxide layer is formed on top of the silicon substrate. Since each silicon oxide unit contains one atom of silicon and two atoms of oxygen, the oxide layer is thicker than the consumed silicon layer. Approximately, 46% of the total height of the silicon oxide layer occupies the pristine silicon layer. Due to the low diffusion of oxygen through the oxide layer, the above described process gives rise to very thin oxide layers of the order of tens of nanometers. Thicker oxide layers, necessary, for instance, for passivation purposes, are formed through a chemical vapor deposition (CVD) process involving the reaction in air of gaseous silane (SiH4 ) and oxygen (O2 ). The oxide molecules then condense onto the solid surface, which is kept at few hundreds of centigrades. The oxide is formed consuming the surface atoms of silicon, so that the silicon surface is displaced towards the interior of the crystal where the the amount of defects is reduced respect to the surface. Eventually, the silicon close to the oxide is less defected than the original surface and the combination Si–SiO2 is, in terms of interface defects, the most favorable case of a junction between different materials. This fact has an extraordinary effect on the properties of the metal-oxide-semiconductor field effect transistors (MOSFETs). In this chapter, the properties of the ideal MOS structure, shown in Fig. 8.1, are studied. The electric contact of the semiconductor is an ideal ohmic contact. The following analysis is valid for the combination of any metal insulators and semiconductors. Clearly, in many practical cases, the defects at the interface between oxide and semiconductor can make the theory very distant from the real behavior.

8.2 Band Diagram and Electrostatic Quantities at the Equilibrium

203

8.2 Band Diagram and Electrostatic Quantities at the Equilibrium To study the MOS at the equilibrium, let us consider a structure made up of aluminium, silicon dioxide, and p-type silicon. The quantities necessary to draw the band diagrams are the following (the subscripts s, m, and ox stand for semiconductor, metal and oxide): affinities: qχs = 4.05 eV; qχox = 0.45 eV work functions: qs = 4.9 eV; qm = 4.1 eV band gaps: E Gs = 1.1 eV; E Gox = 8 eV Figure 8.2 shows the band diagrams of the separated materials. The oxide is characterized by a low affinity and a large band gap. A low affinity means that a little energy is sufficient to add electrons to an insulator, which is why insulators are easily polarizable materials. On the other hand, the band gap is so large that at room temperature the conduction band is empty (exp(−E Gox /k B T ) ≈ 10−134 ). The Fermi level lies in the band gap. Since a large band gap makes inter-band transitions highly improbable, the position of the Fermi level is actually irrelevant. As a consequence, the oxide does not contribute to the charge redistribution that is necessary to establish the equilibrium.

Fig. 8.2 Band diagram of the three MOS separated elements

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8 Metal-Oxide-Semiconductor Junction

Fig. 8.3 Equilibrium band diagram of the MOS structure

Since the work function of the semiconductor is greater than the work function of the metal, the equilibrium is reached through a transfer of electrons from the metal to the semiconductor. Such a charge transfer is straightforward in a metalsemiconductor junction, but in this case the two materials are separated by the insulator, which is supposed to avoid any charge transfers. Then, the ideal structure shown in figure 1 should not allow the establishment of the equilibrium. However, although very large, the resistivity of the silicon oxide is not infinite, so that the charges can be transferred from one material to the other. Furthermore, electrons can be transferred r through the external electric connections. Due to the relocation of charges, a voltage drop appears on the sides of the oxide layer, so that two perturbed regions are found: one in the metal and the other in the semiconductor, in close proximity of the oxide. In the metal, the electrons transferred to the semiconductor leave a surface layer of positive charges, whereas in the p-type semiconductor the electrons transferred from the metal recombine with the holes, forming a space charge region. In Fig. 8.3 the equilibrium band diagram is shown. The built-in potential is the contact potential difference between the metal and the semiconductor surfaces. Due to the presence of the oxide, the built-in potential is split into two parts: one across the oxide (φox ) and the other across the depletion layer.

8.2 Band Diagram and Electrostatic Quantities at the Equilibrium

205

Fig. 8.4 Charge density distribution at the equilibrium

Thus, the built-in potential is larger than the potential (φs ) applied to the charges of the semiconductor: qφbi = qm − qs = qφox + qφs .

(8.1)

In order to determine the electric field and the potential at the equilibrium, it is necessary, to introduce the charge distribution. As in all the previous cases, the deep depletion and the uniform doping hypotheses are assumed. Figure 8.4 shows the charge density distribution, where xox is the thickness of the oxide layer and xd is the depth of the depletion layer in the semiconductor. The electric field in the oxide (Eox ) is constant and due to the charges on the metal surface. The charge neutrality condition holds, so that the charge on the surface of the metal is Q = q N A xd : Eox =

Q q N A xd = . ox ox

(8.2)

Across the interface between the oxide and the semiconductor, the electric displacement vector is continuous: Dox = Ds → ox Eox = s Es , so that the electric field on the surface of the semiconductor (x=0) is: Es (0) =

ox Eox . s

(8.3)

In the case of silicon-silicon dioxide, ox = 3.9 0 and s = 11.7 0 , so that the electric field in the oxide is about three times larger than the electric field on the silicon surface. In the semiconductor, the electric field is directly calculated from the Gauss law: E(x)  x x ρ q NA dE = dx = − d x. s s

E(0)

0

0

(8.4)

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8 Metal-Oxide-Semiconductor Junction

Fig. 8.5 Behavior of the electric field across the MOS. The proportion between Es (0) and Eox is pertinent to the silicon/silicon dioxide case

Replacing E(0), we obtain: E(x) =

q NA (xd − x). s

(8.5)

The electric field vanishes at xd , and its general behavior is shown in Fig. 8.5. The potential is calculated by integrating the electric field. As shown in the band diagram, the potential partially drops in the oxide and partially in the semiconductor across the depletion layer. The integral of the electric fields provides: φox (x) = −

q N A xd (x + xox ); ox

φ(x) = φox + φs = φox −

q NA s

 xd x −

(8.6) x2 2

 .

(8.7)

where φox and φs are the potential in the oxide and semiconductor respectively. The behavior of the potential is shown in Fig. 8.6. In the case of a MOS made of aluminum-silicon dioxide- p-type silicon, the total built-in potential obtained by the sum of φox and φs is given by the difference of the work functions of the metal and the semiconductor: qφbi = q Si − q Al = 0.8 eV . The length of the space charge region (xd ) is calculated from Eq. 8.7 as φs = φ Si (xd ):  2s |φs | , (8.8) xd = q NA where the concentration of acceptors can be derived from the work function, the affinity, and the energy gap:   Ei − E F , (8.9) N A = p = n i exp kB T

8.2 Band Diagram and Electrostatic Quantities at the Equilibrium

207

Fig. 8.6 Behavior of the potential in the MOS structure

where in turn E i − E F = qφ Si − (qχ Si + E2G ) = 0.29 eV , then N A ≈ 1015 cm−3 . As an example, if the built-in potential is equally distributed between the oxide and the semiconductor, then: φox = φ Si = φ2bi = 0.4 eV . The length of the depletion layer and the thickness of the oxide layer are: xd = 865 nm and xox = 114 nm respectively. Thus, the oxide is less thick than the space charge region.

8.2.1 Relation Between Potential and Charge Carrier Concentrations For the scope, it is convenient to express the concentration of the charge carriers as a function of the potential calculated as the difference between the Fermi level and the intrinsic Fermi level. Let us introduce the potential at the surface of the semiconductor φs and the potential in the bulk φ p . In practice, as shown in Fig. 8.7, the potential of Fig. 8.6 is shifted along the vertical axis in such a way that the potential at the interface with the oxide is φs and the potential in the bulk of the semiconductor is φp. From these potentials, the surface and bulk electron and hole concentrations are:  qφs ; = n i exp − kB T   qφ p ; = N A = n i exp − kB T   qφs ; = n i exp kB T   n i2 qφ p . = n i ex p = NA kB T 

ps pb ns nb

(8.10)

Combining these equations, we obtain the charge concentrations at the interface (x = 0):

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8 Metal-Oxide-Semiconductor Junction

Fig. 8.7 Surface and bulk potentials in the MOS semiconductor. Left: potentials are defined as the difference between the Fermi level and the intrinsic Fermi level. Right: If the potential of the metal is kept at zero, the potentials φs and φ p are defined respect to to a constant potential level

 q(φs − φ p ) ; ps = N A exp − kB T   n2 q(φs − φ p ) n s = i exp . NA kB T 

(8.11)

The change in potential between the interface and the oxide modulates the charge concentrations at the interface, while in the bulk they remain constant. As shown in Fig. 8.7, the potentials φs and φ p are defined respect to an arbitrary constant potential level. So the potential φs and the potential in the oxide φox are different.

8.3 The MOS Under Bias In order to study the charge modulation at the oxide-semiconductor interface, let us consider the MOS capacitor previously studied, biased with a d.c. voltage supply (V A ). For sake of simplicity, the semiconductor is grounded. The applied voltage is distributed between the oxide and the depletion layer. At V A < 0, the applied voltage reduces the band bending in the semiconductor, so that the hole concentration at the interface increases with respect to the equilibrium. This behavior is similar to a forward biased Schottky diode. In this case, however, due to the presence of the oxide, no current can be observed. At a particular value of the applied voltage, the depletion layer disappears, so that the hole concentration at the interface equals the concentration in the bulk. This is known as flat-band condition, the corresponding voltage value being the flat-band

8.3 The MOS Under Bias

209

voltage (VF B ), where the effects of the junction actually vanish. The flat-band voltage, moreover, is equal to the built-in potential: VF B = φbi . Beyond the flat-band value, at V A < VF B , the bands bend along the opposite direction so that, instead of being depleted, the interface region starts to be overpopulated with majority charges. This condition is called accumulation and corresponds to the space charge region in a ohmic Schottky contact (see Chap. 3, Sect. 3.4). At V A > VF B , the equilibrium band bending is strengthened: the depletion layer becomes wider, so that the interface region is furtherly depleted of majority charges and more populated by minority charges. As the applied voltage grows, the electron concentration increases and the hole concentration decreases. It is important to remind that since the current is zero, the semiconductor is always at the equilibrium and the mass-action law is always valid. As V A increases, two important conditions are met: the intrinsic condition and the inversion. The first occurs when the Fermi level at the interface equals the intrinsic Fermi level (E Fs = E is ), so that the hole and electron concentrations are equal to the intrinsic semiconductor values. Beyond the intrinsic condition, the interface region is still a depletion layer, but the material is now inverted: although the semiconductor is doped with acceptors, the electron concentration is greater than the hole concentration. The discrepancy between electrons and holes increases until the difference between the intrinsic and doped Fermi levels at the surface is, in absolute value, equal to the condition in the bulk (|E i − E F |s = |E i − E F |b ). Under this condition: n S = pb . The effects of the progressive increase of V A from very negative to very positive values are summarized in Table 8.1. The applied voltage modifies the density of the charges at the silicon-silicon oxide interface, where the semiconductor may experience all the possible conditions. It is important to note that, since no current flows through the device, under the applied voltage the junction is still in thermal equilibrium and the applied voltage is simply added to the built-in potential. In this way, biasing is equivalent to changing the work function of the metal: in practice, the metal becomes a sort of virtual element, whose work function can be gradually changed by the applied voltage. The absence of current implies that the Fermi level in the semiconductor is constant and uniform and that the applied voltage modifies the band bending in the interface region. The potential in the semiconductor is still described as the difference between the Fermi level and the intrinsic Fermi level. The intrinsic Fermi level follows the band bending, while the Fermi level remains constant. Figure 8.8 displays the behaviour of the potential under the critical values of V A . In strong inversion mode, the total charge on the semiconductor surface is the sum of the mobile electrons and the fixed acceptors (Q = Q n + Q A ). The electron concentration is given by Eq. 8.11. Due to the exponential function, the concentration is extremely sensitive to the variations of φs . In practice, when the surface potential reaches the inversion, the potential remains almost fixed at φs . As an example, an increase of only 58 mV results in 10 times the concentration of electrons. The large

210

8 Metal-Oxide-Semiconductor Junction

Table 8.1 MOS cases at different applied voltages VA Sign of φs Relation Condition between φs and φb

Holes concentration

Electrons concentration

< −VF B

–

|φs | > |φ p |

Accumulation ps > N A

ns


Ni2 NA Ni2 NA Ni2 NA

>0

–

|φs | < |φ p |

Depletion

ps < N A

ns >

>>0 >>>0

Null +

|φs | = 0 |φs | < |φ p |

Intrinsic Weak inversion

ps < n i ps < n i

Ni2 NA

ns > ni ns > ni

>>>>0

+

|φs | = |φ p |

Onset of strong inversion

ps
>>>>0

+

|φs | > |φ p |

Strong inversion

ps
0 corresponds to a reverse biasing of the depletion layer, which, as a consequence, becomes deeper and deeper. In strong inversion, the potentials on the surface and in the bulk are equal: φs = −φ p . Thus, since the surface potential is fixed by the exponential, the depletion layer reaches the maximum extension: 2 . Note that since the semiconductor contact φ Si = −φ p − φ p = −2φ p = q2NsA xdmax is grounded, the bulk potential (φ p ) does not change. From the above expression, we 2 : can calculate xdmax  4s |φ p | . (8.12) xdmax = q NA The charge associated to the maximum depletion layer is:  Q dmax = −q N A xdmax = − 4q N A s |φ p |.

(8.13)

In order to draw the total charge distribution at the interface, it is convenient to consider the band diagram in strong inversion mode (see Fig. 8.9). The inversion region occupies a narrow region close to the interface, where the Fermi Level lies above the intrinsic Fermi Level, while the depletion layer, whose charges are the homogeneously distributed acceptors, spreads along a wider distance.

8.3 The MOS Under Bias

211

Fig. 8.8 Potential profiles in the MOS as a function of the applied voltage

Figure 8.10 shows the qualitative behavior of the hole and electron concentrations at the interface with the oxide as a function of the applied voltage.

212

8 Metal-Oxide-Semiconductor Junction

Fig. 8.9 Band diagram in strong inversion and relative distribution of mobile and fixed charges

Fig. 8.10 Modulation of absolute value hole and electron density as a function of applied voltage

8.4 The C/V Curve The conclusions reached in the previous section are valid at the equilibrium with an applied d.c. voltage. It is clear that any changes in V A elicits a transitory charge distribution that evolves towards novel equilibrium conditions. The applied voltage may bring about accumulations of either electrons or holes at the interface and, in principle, the respective accumulation layers should have the same behavior. In a p-type silicon, however, the electrons are always minority charges, so that their generation is an unfavored process. This property is manifested when the electron

8.4 The C/V Curve

213

concentration is required to increase, as it happens when an a.c. signal is applied to the MOS. The variable conditions of the charges on the surface of the semiconductor are visible in the total capacitance of the MOS. Like the other junctions, also the MOS capacitance is a function of the applied voltage, and also in this case the C/V curve is an experimental method that helps appraise the properties of the device. In the Schottky diode, for instance, the C/V curve provides both the doping concentration and the built-in potential. In metal-semiconductor junctions, the large conductivity under forward bias hides the capacitance, so that the C/V curve is limited to the reverse bias condition only. In MOS systems, the oxide layer is not electrically conductive, so that the capacitance can be measured with any applied voltages. The experimental set-up for the C/V curve has been shown in Fig. 2.9. The bias voltage determining the MOS condition, and thus the capacitance value, is fixed by the d.c. voltage supply VG . In order to measure the capacitance, a small a.c. signal (vt ) is added in series. The amplitude of the a.c. signal is much smaller than the d.c. signal, so that it does not interfere with the MOS condition. We will see thereafter that some features of the C/V curve depend on the frequency of the a.c. signal. The MOS is formed by a straightforward capacitance associated with the oxide layer, together with an additional capacitance due to the semiconductor. The most obvious capacitive effect in the semiconductor is the one related to the depletion layer, which exists on a VG interval between the flat-band voltage and the inversion voltage. Beyond these values, additional capacitive effects exist, which will be discussed in this section. VG is applied across the metal and the bulk of the semiconductor, so that it is distributed between the two capacitances forming a series of two capacitors, whose total capacitance is: 1 1 1 = + Ctot Cox CS

(8.14)

where C S is the semiconductor capacitance and Cox is the oxide capacitance. As reported in Table 1, the interface condition changes as a function of VG , while the oxide capacitance remains constant: Cox =

ox xox

(8.15)

where xox is the thickness of the oxide. It is convenient to start describing the C/V curve from the flat-band condition, when VG = −V f b. The flat-band voltage is the difference between the metal and semiconductor work functions: V f b = φm − φs . Under the flat-band condition, the applied voltage eliminates the band bending induced by the work function difference, so that the electron and hole concentration is constant everywhere in the semiconductor and equal to the equilibrium condition: p = N A and n = n i2 /N A .

214

8 Metal-Oxide-Semiconductor Junction

Although, under this condition, the depletion layer does not exist, the semiconductor shows an additional capacitive effect, which is, in some sense, intrinsic of the semiconducting material and does not depend on the junction. In the case of a capacitor made of metal plates, the capacitance is only due to the insulator, because the charge modulation in the metal is confined to the surface in contact with the dielectric. The electric field in the metal is indeed null, so that the voltage drops on the surface. In a semiconductor, the situation is different: the voltage perturbs a non negligible volume of the semiconductor, so that the modulation of the charge due to the applied voltage takes place at a distance from the dielectric surface. This is equivalent to an additional capacitor in series with the oxide capacitor. To calculate this capacitance, it is necessary to evaluate the amount of charges created by the applied voltage. For this scope, let us consider the Poisson equation and the perturbation that the applied voltage induces to the flat-band condition. The semiconductor is P-type doped, so that the total charge is limited to the mobile holes and the fixed acceptors. The Poisson equation is: ρ(x) q d 2φ =− = − ( p − N A ). 2 dx s s

(8.16)

It is worth to remind that the majority charges are mobile in space and also in time. The temporal variation is ensured by the fact that only a portion of the acceptor atoms is actually ionized. Thus, in non-equilibrium conditions, the hole concentration can still change. In order to calculate the capacitance, let us limit the calculation to φ, the portion of modulating voltage (vt ) applied to the semiconductor from the interface with the oxide to the bulk. Thus, φ is the potential with respect to the potential in the bulk ). (φ p ), which elicits a perturbation in the hole concentration: p = N A ex p(− kqφ BT Replacing this in the Poisson equation, we get:     d 2φ qφ q exp − − 1 . (8.17) = − N A dx2 s kB T Considering that vt is a small perturbation and that φ is just a part of it, the exponential can be replaced with the first order Taylor series:   d 2φ q2 NA q qφ − 1 = = − N φ (8.18) 1 − A 2 dx s kB T s which can be written as: d 2φ φ =− 2 dx2 LD where: L D =



s k B T q2 NA

is the Debye length of the semiconductor.

(8.19)

8.4 The C/V Curve

215

The generic solution of Eq. 8.19 is a sum of positive and negative exponentials. The first boundary condition fixes the potential in x=0: φ(0) = φts . For the second condition, let us consider that the length of semiconductor is much greater than the Debye length, so that the second boundary condition can be placed at infinite distance from the interface: φ(∞) = 0. Eventually, the potential decays towards the semiconductor bulk, so that the Poisson equation solution is limited to the negative exponential:   x . (8.20) φ(x) = φts exp − LD The applied a.c. signal creates, in the semiconductor, a charge modulation expiring at about four-five times the Debye length from the surface. The modulated charge density can be calculated directly from the Poisson equation (Eq. 8.16):   d 2φ φts x . (8.21) ρ(x) = −s 2 = −s 2 exp − dx LD LD The corresponding total charge is: ∞ Q=

−s

d 2φ s dx = − φts . dx2 LD

(8.22)

0

Finally, the flat-band semiconductor capacitance can be calculated: Csf b = |

s dQ |= . dφt s LD

(8.23)

This capacitance is equivalent to the capacitance of a metal-insultator-metal system, where the insulator dielectric constant is s and the conductors are separated by L D . Note that L D depends on the concentration of the charge carriers, so that it depends on doping. The more doped is the material, the shorter is the Debye length and the larger is the capacitance. The MOS total capacitance at VG = V f b is the series of the two capacitors: 1 fb Ctot

=

1 1 xox LD + fb = + . Cox ox s Cs

(8.24)

As an example, for a p-type silicon with N A = 1015 cm−3 and a silicon dioxide layer of thickness xox = 150 nm, we have: L D = 130 nm, Cox = 2.30 × 10−8 cmF 2 fb fb and Cs = 7.98 × 10−8 cmF 2 . The total capacitance is then: Ctot = 1.78 × 10−8 cmF 2 . When V A < V f b , the interface region enters the accumulation mode. Namely, the hole concentration on the surface exceeds the hole concentration in the bulk. In order to describe the behavior of Cs , we can consider that the Debye length is inversely proportional to the square of the carrier concentration. Consequently, with respect

216

8 Metal-Oxide-Semiconductor Junction

to the flat-band condition, since the hole concentration increases, the Debye length decreases, so that the capacitance associated with the semiconductor becomes larger. q(φ −φ ) The hole concentration is: p = Na exp(− ksB T p ). At the flat-band, φs − φ p = 0. Following the above mentioned example, we observe that at φs − φ p = 0.1 V the semiconductor capacitance is: Cs = 54.9 × 10−8 cmF 2 and, consequently, the total capacitance is Ctot = 2.21 × 10−8 cmF 2 . If the applied voltage doubles to 0.2 V , the value of the capacitances becomes Cs = 373 × 10−8 cmF 2 and Ctot = 2.28 × 10−8 cmF 2 . Eventually, the total capacitance converges to the oxide capacitance. On the other hand, for VG > V f b , the interface becomes depleted of holes. The capacitance associated with the depletion layer is: Cs = xds , where xd is the depth of the depletion layer, which in turn depends on the voltage drop between the surface and the bulk of the semiconductor:  2s |φs − φ p | . (8.25) xd = q NA As VG grows into positive values, the absolute difference φs − φ p increases, as well as xd . Consequently, the semiconductor capacitance, and the total capacitance, become small. Following, once again, the above numeric example, at φs − φ p  = 0.1 V , we have: xd = 35 nm, Cs = 2.87 × 10−8 cmF 2 , and Ctot = 1.27 × 10−8 cmF 2 . The depletion layer expands with the applied voltage until the strong inversion threshold is met. In this condition, as seen in the previous section, the depletion layer reaches its maximum extension:  4s |φ p | (8.26) xdmax = q NA while the total capacitance reaches its smallest value: 1 fb

Ctot

=

1 1 xox xdmax + fb = + . Cox ox s Cs

(8.27)

The largest extension of the depletion layer in the above example is xdmax = 622 nm, so that Csimi = 1.66 × 10−8 cmF 2 and Ctot,min = 0.96 × 10−8 cmF 2 . Until the strong inversion threshold, the interface is populated by both the electrons and the fixed charge defined by xd . The population of electrons grows rapidly, but, until the inversion, the fixed charge dominates, so that the capacitance is dominated by the depletion layer capacitance. At the inversion threshold, and beyond it, the layer of electrons close to the interface cannot be neglected. In order to affect the capacitance, however, it is necessary that the charge be modulated by the a.c. voltage. The inversion layer is formed by minority charges, which, in a narrow interface region and due to the applied bias, become more numerous than the holes. However,

8.4 The C/V Curve

217

Fig. 8.11 MOS complete equivalent circuit. Cox is the oxide capacitance. The depletion layer is represented by two impedances controlled by the same voltage and thus connected in parallel; Cd and Ci are the capacitances associated with the depletion and inversion layers respectively. RG is used to represent, via the time constant RG Ci , the slow response of the inversion layer charge. Finally, Rs is the semiconductor bulk resistance

the minority charge character of the electrons is still valid and, in particular, the generation of electrons in the depletion layer requires a non-negligible time. The typical time for the minority charges generation is of the order of tenths of seconds. An explicit calculation of the time necessary to create and modulate the electron inversion layer will be explicitly calculated in the next section. The limited speed of the charge variation in the inversion layer makes the capacitance strongly dependent on the frequency of the probing a.c. signal (vt ). At frequencies of few Hertz, the signal variation occurs in a time interval that is compatible with the electron generation time. Thus, at a low frequency, the capacitance in inversion mode is again equal to Cox , which indicates that at the inversion an accumulation layer made of electrons, instead of holes, actually exists. This behavior looks similar to the accumulation condition met at VG < V f b . However, in inversion mode, the situation is different, because the depletion layer still exists and it may respond to the variable vt . However, due to the large electron concentration in the inversion layer, the Debye length is much shorter than xd , so that the applied potential vanishes and does not perturbate the depletion layer charge. Eventually, the total capacitance is the oxide capacitance. On the other hand, if the frequency of vt is much larger than few Hertz, the applied signal changes too fast with respect to the rate of electron generation. Thus, despite the large electron concentration, the inversion layer charges are insensitive to the applied signal, so that the capacitance is only contributed by the depletion layer. In this condition, the presence of the inversion layer is only reflected by the fact that the capacitance reaches its minimum value and is insensitive to any increase in VG . Very low frequency and high frequency are the two asymptotical conditions at which the C/V curves manifest the largest differences. At intermediate frequencies

218

8 Metal-Oxide-Semiconductor Junction

Fig. 8.12 MOS C/V curve

both behaviors coexist, but the capacitance of the inversion layer tends to lose relevance as the frequency increases. Inversion layer and depletion layer define two capacitances both controlled by the voltage applied to the semiconductor. A complete qualitative equivalent circuit of the MOS capacitor is shown in Fig. 8.11. The slow response of the inversion layer charge is accounted by the resistance RG in series with the inversion layer capacitance. For sake of completeness, the set of capacitors is complemented with a series resistance representing the conductivity of the semiconductor bulk. The complete C/V curve is illustrated in Fig. 8.12. The two asymptotic behaviors are shown: as frequency increases, the low-frequency curve approaches the high frequency response. In order to better understand the different observed values of the capacitance, it may be useful to visualize, as shown in Fig. 8.13, the distribution of those charges whose concentration is modulated by the applied a.c. voltage in the different cases. The previous discussion about the C/V curve has been carried out assuming a stepwise-variable d.c. signal and a constant-VG capacitance measurement. In this way, enough time is left for the inversion layer to get formed. Alternatively, VG may be applied as a ramp. In this case, no time is left to generate the minority charge equilibrium concentration, so that the inversion layer is not formed (see Fig. 8.14). The absence of inversion layer, however, does not limit the expansion of the depletion layer, which may continue to grow even beyond xdmax , so that the total capacitance continues to decrease. This condition is known as deep depletion. Figure 8.14 shows the two modalities of voltage application. The complete C/V curve is shown in Fig. 8.15. The C/V curve is an invaluable tool to investigate the oxide layer defects and the interface traps. During the oxidation process, the oxide layer may incorporate impurities either from the silicon or from the atmosphere. These impurities, moreover, can be charged. Thus, since the ions are able to slowly migrate through the oxide with a very low mobility, the latter can be populated with both fixed and mobile charge concentrations. The charges inside the oxide create a further potential, which adds to the builtin potential. The flat-band voltage shift towards either positive or negative values

8.4 The C/V Curve

219

Fig. 8.13 The different cases of the C/V curves correspond to the different locations of the MOS where the charge modulation occurs

according to the sign of the total charge is a macroscopic manifestation of the charge presence in the oxide. Interface states are due to the rearrangement of the silicon atoms at the interface with the oxide. As seen in Chap. 2, these states may sequestrate both electrons and holes by subtracting charges to the inversion and accumulation layers respectively. The presence of the surface states makes the C/V less sharp than expected around the flat-band voltage, whose value remains unchanged. Figure 8.16 shows both the cases.

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8 Metal-Oxide-Semiconductor Junction

Fig. 8.14 The two different voltage application modes in a MOS C/V curve experiment. Left: the stepwise voltage allows the formation of the inversion layer, so that the depletion capacitance reaches a maximum value. Right: the ramp, if sufficiently fast, does not allow the formation of the inversion layer, so that the depletion capacitance continues to decrease, together with the MOS total capacitance

Fig. 8.15 The complete MOS C/V curve complemented with the deep depletion case

Fig. 8.16 C/V curves in the cases of charges in the oxide (left) and in the interface states (right) with respect to the theoretical ideal curve, where V f b is the difference between the work functions of the metal and the semiconductor

8.4 The C/V Curve

221

8.4.1 Minority Charges Generation in the Depletion Layer Let us now estimate the generation time of the inversion layer in a P-type MOS. The MOS is initially in depletion condition. Let us suppose that at the time t = 0 the MOS is suddenly biased at the inversion voltage. The inversion layer is formed by the electrons generated in the depletion layer that reach the interface region. This situation corresponds to the generation current in a reverse biased PN junction, with the important difference that the current flows as the depletion layer increases from xd (at VG = 0) to xd = xd0 , corresponding to the new equilibrium established by the applied voltage. During the transition the depletion layer, as well as the electron concentration, are a function of time. The generation current (Eq. 4.75) does not explicitly depend on the applied voltage, but only on the difference between xd and xd0 : qn i (xd − xd0 ) d Qn =− . (8.28) JG = dt 2τ0 The relationship between the gate charge and the inversion layer charge generation rate can be written considering that the total charge in the semiconductor is Q G = −(Q n − q N A xd ). Note that Q G is also the total charge in the metal. Replacing xd , we obtain:   −qn i Q g + Q n d Qn (8.29) =− − xd0 . dt 2τ0 q NA The above equation can be rearranged as:

Introducing τn =

Qn +

2N A τ0 d Q n = −(Q G − q N A xd0 ). ni dt

2Na τ , ni 0

the equation becomes:

Qn d Qn Q G − q N A xd0 + =− . dt τn τn is:

(8.30)

(8.31)

The solution of the above equation with the boundary condition: Q n (t = 0) = 0    t . (8.32) Q n = −(Q G − q N A xd0 ) 1 − exp − τn

Hence, the equilibrium charge in the inversion layer is reached through an exponential behavior, whose time constant is τn . Note that the generation time does not depend only on the time scale of the generation/recombination processes, but it is proportional to the density of doping, so that the minority charge generation rate decreases with doping. In the case of τ0 = 1 µs and N A = 1016 cm−3 , we obtain τn = 0.13 s. This is the time scale necessary to form, and even to modulate, the charge in the inversion region.

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8 Metal-Oxide-Semiconductor Junction

The charge in the inversion layer follows the modulation of the applied a.c. voltage only if the frequency of the signal does not exceed 1/τn . In the above example, where τn = 0.13 s, this frequency corresponds to about 8 Hz. The time taken for the inversion layer to be formed/modulated is strongly decreased in those cases where the minority charges (electrons in this case) are provided by different sources. Noteworthy cases are the nearby N-type semiconductors, such as the drain and source contacts of a MOSFET (see Chap. 9), and the photo-generated electrons in optical photodetectors.

8.5 Summary The capacitor formed by metal oxide and semiconductor is a fundamental building block in microelectronics. In principle, the device is similar to the metalsemiconductor junction, but due to the presence of the insulator layer, the junction, even under bias is, in the steady-state condition, always in equilibrium. Thus, the applied voltage can modulate the charge concentrations at the interface between the oxide and the semiconductor, creating conditions of depletion, accumulation of majority charges, and accumulation of minority charges. All these conditions are manifested in the capacitance of the MOS structure. The condition of accumulation of majority charges leads to a standard capacitor that depends only on the dielectric constant and the thickness of the oxide layer. The accumulation of minority charges, in the so-called inversion condition, is expected to behave in the same way, but it shows a peculiar behavior related to the properties of minority charges. Namely, the fact that the generation of such charges, in a material doped with charges of opposite sign, is an unlikely event, so that it happens at a long-time scale. As a consequence, the accumulation layer of minority charges follows the applied voltage only if the latter changes slowly. Thus, the capacitance in the inversion condition depends on the frequency of the applied voltage, and the inversion layer is actually manifested only at very low frequencies. It is remarkable to note that the low-frequency limit can be circumvented once the minority charges are generated by an additional source, as it happens in photodetectors and MOSFETs, which are the most practiced MOS applications.

Further Reading Textbooks C.C. Hu, Modern Semiconductor Devices for Integrated Circuits (Pearson College, 2009) R. Muller, T. Kamins, M. Chen, Device Electronics for Integrated Circuits, 3rd edn. (Wiley, 2002) D. Neamen, Semiconductor Physics and Devices (McGraw Hill, 2003) S. Sze, K.K. Ng, Physics of Semiconductor Devices, 3rd edn. (Wiley, 2006)

Further Reading

223

Journal Papers A. Grove, E. Snow, B. Deal, C. Sah, Simple physical model for there space charge capacitance of metal-oxide semiconductor structures. J. Appl. Phys. 35, 2458 (1964) D. Kahng, Silicon-Silicon Dioxide Surface Device (Technical Memorandum of Bell Laboratories, 16 Jan 1961)

Chapter 9

Field Effect Transistors

9.1 Introduction Metal-oxide-semiconductor junctions are at the core of a three-terminal-device family, where the inversion layer conductivity is modulated by the voltage applied to the MOS structure. Such devices are known as Field Effect Transistors (FETs). The basic concept underlying the FET technology, it is based upon the evidence that the mobile charges lying on the conductor side can be either accumulated or depleted via a voltage application in a direction orthogonal to the current flow. The idea of the FET, as an alternative to triodes, arose in the twenties. The first patent was filed in 1925, even though it was the following patent in 1934 that provided a more detailed description of the device. Subsequently, the concept attracted W. Shockley, who, in 1948, developed a prototype that did not properly work because of a too high density of defects at the oxide-semiconductor interface. The failure, however, urged Shockley to direct his attention towards the PN junction. The first working FET was fabricated only in 1957, at the Bell Lab., using silicon. This achievement was the result of the maturity of semiconductors technology, which allowed the fabrication of a sufficiently clean oxide-semiconductor interface. The silicon FET thus began to be known as Metal Oxide Semiconductor Field Effect Transistor (MOSFET). The first commercial devices were produced at the Bell Lab after 1960. In the first part of this chapter, the basic working mechanisms of silicon MOSFETs are illustrated. The second part of this chapter is dedicated to the further development of field effect devices mostly based on materials characterized by large mobilities, such as GaAs and other III–V materials.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0_9

225

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9 Field Effect Transistors

9.2 Channel Charge Modulation and the Threshold Voltage Before discussing the properties of MOSFETs, it is necessary to evaluate the relationship between the charge in the inversion layer, the MOSFETs channel, and the voltage applied to the MOS (VG ). The discussion considers a MOS made of a p-type semiconductor biased in strong inversion. In this condition, the depletion layer reaches its maximum extension:  2s (2φ p ) xd max = . (9.1) q NA As for the charge stored in the depletion layer, it is:  Q d = − 2s q N A (2φ p ).

(9.2)

In order to write the relationship between (VG ) and the total charge in the semiconductor, let us consider the potential profile between the gate and the bulk of the semiconductor (Fig. 9.1). The flat band voltage (VG = V f b ) corresponds to a null potential drop across the semiconductor. V f b is negative and corresponds to the difference between the work functions of the metal gate and the semiconductor. To calculate the charge in the semiconductor, let us first consider the electric field in the oxide: Eox =

VG − VB − V f b − Vsi Vox = . xox xox

Fig. 9.1 Distribution of the potential in the MOS structure in strong inversion mode

(9.3)

9.2 Channel Charge Modulation and the Threshold Voltage

227

At the oxide-semiconductor interface, the continuity of the electric displacement requires that: ox Eox = si Esi0 , where Esi0 is the electric field at the interface on the silicon side. Replacing Eox with Eq. (9.3), the electric field on the silicon surface is: si Esi0 =

ox (VG − VB − V f b − Vsi ) = Cox (VG − VB − V f b − Vsi ). xox

(9.4)

Thanks to the Gauss theorem, the electric field in the semiconductor depends on the total charges in the semiconductor: Esib − Esi0 =

Qn + Qd s

(9.5)

where Esib is the electric field in the bulk, Q n are the mobile electrons and Q d are the fixed acceptors. Since the electric field in the bulk is null, the field at the interface d , from which the mobile charge is calculated: is −Esi0 = Q n +Q s Q n = −s Esi0 − Q d .

(9.6)

Replacing the electric field at the interface in Eq. (9.4) and Q d with Eq. (9.2) (note that Q d is negative), we get: Q n = −Cox [VG − VB − V f b − 2φ p ] +



2s q N A (2φ p ).

(9.7)

The mobile charge at the interface is made of electrons, then, it charge exists when Q n is negative. This condition happens if the first term of Eq. (9.7) is greater than the second one. The condition Q n = 0 is particularly important because it defines the onset of the inversion layer formation. The voltage VG at which this condition is met is the threshold voltage (VT ). Setting Q n = 0 in Eq. (9.7) leads to: VT = V f b + 2φ p  +

1  2s q N A (2φ p ). Cox

(9.8)

Considering that (VG − VB ) − V f b = Vox + Vsi , at the strong inversion Vsi = 2φ p . Thus, VG = VT is equivalent to: VT = V f b + 2φ p  + Vox .

(9.9)

Comparing Eq. (9.8) with Eq. (9.9), we find that the voltage applied to the oxide is also responsible for the depletion layer charge (Vox = Q d /Cox ). Given the definition of VT , a concise expression for the charge in the inversion layer is obtained: Q n = −Cox (VG − VT ).

(9.10)

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9 Field Effect Transistors

This fundamental relationship states that the charge, and thus the conductivity, of the channel is controlled by the voltage applied to the MOS. It is worth noting that all of the MOS characteristics contribute to the threshold voltage, which depends on the difference between the work functions of the metal and the semiconductor (V f b ), on the doping concentration (φ p ) and on the oxide thickness (Cox ).

9.3 Metal Oxide Semiconductor Field Effect Transistor Equation 9.10 describes the control, via the voltage applied to the gate, of the electron concentration in the inversion layer. These charges form a thin layer of mobile electrons surrounded on one side by the oxide and on the other one by a space charge region. The charge layer is then confined between two insulating regions. The charges can be kept in movement if two suitable electrodes are provided at the edges of the layer. These electrodes are two n + regions implanted at the edges of the inversion layer. Eventually, the device is a three-electrode structure, where the inversion layer is turned into a conductive channel, whose resistivity is controlled by the voltage applied to the MOS capacitor. The electrodes at the edges of the channel are called drain and source, while the electrode applied to the metal of the MOS is called gate. The metal electrode is separated from the p-type semiconductor by a thick oxide layer. In this way, the associated capacitance is so small that makes any field effect negligible. The configuration contains a fourth electrode applied to the substrate, known as body. The n + electrodes introduce two PN junctions between the electrodes and the bulk. Being Vd Vs and Vb the voltages at the source, drain abd body electrodes. In normal operative conditions, VD − VB > 0 and VS − VB > 0, so that both the junctions are reversely biased and only a negligible current flows towards the body contact. Such a device is the Metal Oxide Semiconductor Field Effect Transistor (MOSFET). It is interesting to observe that a conductive channel can be obtained by accumulating either majority or minority charges. In both cases, the conductivity between source and drain may be regulated by the gate voltage. However, the inversion case is much more interesting, because the current in the inversion channel flows through a narrow strip confined between two insulators. As a consequence, the conductivity between the contacts is only due to the channel. In the case of accumulation, both source and drain must be ohmic contacts for the majority charges. The bulk of the semiconductor (where p = N A ), however, also contributes to the conduction between source and drain. Since the accumulation layer and the bulk are in parallel, if the semiconductor is much thicker than the accumulation layer, then the conductivity is always dominated by the bulk and the field effect is not observable. Such a configuration is becoming popular in the molecular electronics field, where thin layers of molecular semiconductors are being used. Another important advantage of using the device in inversion mode lies in the intrinsic insulation from the rest of the substrate, which is provided by the depletion

9.3 Metal Oxide Semiconductor Field Effect Transistor

229

Fig. 9.2 Basic configuration of a MOSFET

layer. This is an important issue for the integration of more devices on the same substrate. The MOSFET is used as a trans-amplifier, where a voltage signal applied to the gate electrode is turned into a current signal at the drain-source contacts. In the previous chapter we have seen that, in a MOS, the modulation of the charges in the inversion layer is a very slow process, the generation of minority charges occurring at a very low rate. However, in a MOSFET, the charges in the channels can be varied at high velocity. This difference is due to the n + wells, which act as electron reservoirs, providing, with great efficiency, all of the necessary electrons to modulate the charge in the channel. Figure 9.2 illustrates the basic MOSFET configuration. Most of the device characteristics result from the difference between metal and semiconductor work functions (V f b ), the semiconductor doping (φ p ) and the oxide thickness (Cox ). The combination of these three quantities defines two categories of devices known as enhancement and depletion MOSFETs. In an enhancement MOSFET, at VG = 0 the channel is not formed. In particular, in a p-type semiconductor, the channel is formed and then modulated at VG > 0, whereas the sign of VG is inverted when a n-type semiconductor is considered. In a depletion MOSFET, on the contrary, the channel exists even at VG = 0, so that the gate voltage can increase, decrease or even shut-off the inversion layer. In the following of this section, the enhancement MOSFET is illustrated. VG sets the working condition. When VG < V f b , the interface is in accumulation mode, so that the path from source to drain forms a series of two opposite PN junctions (a configuration called back-to-back). In this condition, the channel is analog to a

230

9 Field Effect Transistors

BJT in cutoff region, so that the current from source to drain is the reverse current of the two PN junctions. As VG increases, the condition such that V f b < VG < VT is met. In this interval, the channel region is depleted of mobile charges. By increasing VG , a weak population of electrons gets formed. But otherwise, this is not the inversion layer. A small current called sub-threshold current flows between source and drain. Finally, when VG > VT the strong inversion condition is met, so that the electron channel is completely formed. In strong inversion mode, the depletion layer remains constant and any increase in VG results in an increase in the electron concentration of the channel. Let us now derive the relationship between the current flowing through the channel and the couple of voltages VDS (applied to the channel) and VG (applied to the gate). In principle, VG sets the electron concentration, and thus the conductivity of the channel, whereas VDS drives the drift current. Actually, the situation is a little more complicated because VDS is distributed along the channel, so that, at each point, an additional voltage VC (y) is found (Fig. 9.3). This voltage is additive to VG , so that it affects the charge density in the channel: Q n (y) = −Cox (VG − V f b − 2φ p  + VC (y)) +



2s q N A (2φ p  + VC (y)). (9.11) Above the threshold voltage, the inversion channel is formed and the junction between the n + regions and the channel is negligible. Thus, the current is equivalent to the current flowing through a homogeneously doped semiconductor, and is made of a drift and diffusion component. The mechanism is analogous to that of the active-zone BJT transport in the base, where in order to decrease the transit time, an electric field is created, either because of graded doping or graded band gap. In both cases, a limited voltage greater than twice the thermal voltage (56 mV) is sufficient to make the drift current contribution greater than that of diffusion. It is thus straightforward to suppose that, since VDS is greater than the thermal voltage, the drift current component prevails. In order to calculate the effect of VDS on the charges in the channel, let us assume that the potential along the coordinate channel (y-coordinate) changes less rapidly than across the channel (x-coordinate direction): 

∂φ ∂φ    . ∂y ∂x

(9.12)

This condition is the graded-channel approximation. Under this approximation, for each y the corresponding x is constant. In practice, the channel can be sliced in portions where the charge is constant. Noteworthy, this condition is met when the length of the channel (L) is much greater than the depletion layer. The main consequence of the graded-channel approximation is that the relationship Q n = −Cox (VG − VT ) is valid in every interval dy. To calculate the distribution of VDS , it is necessary to calculate the channel resistance profile. The differential resistance (d R) of a small slice of the channel (from y

9.3 Metal Oxide Semiconductor Field Effect Transistor

231

Fig. 9.3 Behaviour of V DS along the channel. A system of x-y coordinates is settled to describe the charge behavior along the channel and towards the bulk. Thanks to the voltage added to the channel, the inversion layer under the oxide is not uniform, but becomes thinner as approaching the drain contact. For this reason, even if the mobile charges decrease, the charge concentration is practically constant, so that the diffusion current can be neglected. Eventually, above the threshold, the MOSFET current can be described in terms of drift current only

to dy) is : d R(y) = ρ(y) dy . The resistivity ρ depends on the electron concentrations A at the y position: ρ(y) =

1 1 = σ (y) μn Q(y)

(9.13)

where μn is the channel electron mobility. The channel lies at the interface between the semiconductor and the oxide, in a region overcrowded with defects with respect to the bulk. We can thus expect the mobility at the interface to be smaller than that in the bulk. The actual value of the mobility, however, depends on the conditions under which the device is grown and cannot be theoretically calculated. But otherwise, an approximate estimation is given as μn ≈ 21 μbulk , where μn is an unknown parameter of the device. The unit of measurement for the charge concentration in Eq. (9.13) is C cm−3 , whereas in Eq. (9.10) it is C cm−2 . It is easy to figure out that given a slice of the channel, its volume is Ady and the area below the oxide is wdy, where w is the

232

9 Field Effect Transistors

charge cm3 charge Qn (y) = cm2 Q(y) =

total charge = Q(y) · A · dy = Qn (y) · w · dy Fig. 9.4 Volume and surface charge concentrations is illustrated in a slice of the channel

lateral dimension of the channel and A is the area of the slice. Eventually, we have: Q(y)A = Q n (y)w. The relationship between the volumetric and surface concentrations is shown in Fig. 9.4. The differential resistance of a slice dy of the channel at a y position is: dR =

dy 1 dy = . μn Q(y) A μn Q n (y)w

(9.14)

The current I D through the channel is a drift current: Id =

d VC dR

(9.15)

which is obviously constant along the channel and independent of y. The charge concentration in the channel is given by Eq. (9.11) as a function of VC . However, d R is a function of y. Since VC monotonically grows along y, the variables y and VC can be interchanged by replacing y with VC (y) and Q n (y) with Q n (VC ). Equation 9.15 can thus be integrated with respect to y from 0 to L and with respect to VC from VS to VD : VD w

L Q n (VC )d VC = Id

VS

0

dy . μn

(9.16)

Assuming that μn remains constant along the channel, we obtain the drain current:

9.3 Metal Oxide Semiconductor Field Effect Transistor

μn w Id = L

233

VD Q n (VC )d VC

(9.17)

VS

 where Q n = −Cox [VG − V f b − 2φ p  + VC − VB ] + 2s q N A (2φ p  + VC ). The calculation of the integral can be simplified neglecting the depletion layer charge dependence on VC . Obviously, the size of the depletion layer changes along the channel. However, the uncertainty about the actual value of the mobility in the channel (μn ) makes the approximation tolerable. On a practical level, what is important here is to calculate a plausible functional relationship between currents and voltages, leaving the evaluation of the device actual parameters to experimental calibration. Considering the threshold voltage definition [see Eq. (9.10)], the charge can then be written as: Q n = −Cox (VG − VT − VC ).

(9.18)

The mobile charge decreases as VC increases towards the drain contact. As a consequence, VDS is not equally distributed along the channel, but it is more intense towards the drain contact. The drain current is obtained integrating Eq. 9.17.  2  VDS w (VG − VT )VDS − . (9.19) Id = −μn Cox L 2 The negative sign indicates that the current flows from drain to source. This equation is called the long-channel MOSFET equation. It describes a family of Id / (VDS ) parabolic functions, where VG is a parameter. Noteworthy, the parabolic shape implies the existence of a negative differential resistance tract. Actually, when VC = VG − VT the charge in the channel becomes null and the model, based on the continuity of the charge in the channel, is no longer valid. The condition of null charge implies that the resistance in that tract becomes infinite. In other words, at VDS = VG − VT the channel is interrupted. This condition is met at the top of the parabolic function, when the current reaches its maximum value (Fig. 9.5). Id max = −μn Cox

w (VG − VT )2 . L 2

(9.20)

The drain-source voltage at which the maximum drain current is obtained is called saturation voltage. It corresponds to the condition where Q n is null at the drain contact. At VDS > VDSsat any further increase in voltage drops at the border of the region, where Q n = 0. Then, the rest of the channel remains still biased at VDSsat , so that the current in the channels remains fixed at (Id max ). The region depleted of mobile electrons is the pinch-off region. The pinch-off is very narrow and biased at VDS − VDSsat . Such a voltage across a small region gives rise to a large electric field,

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9 Field Effect Transistors

Fig. 9.5 Drain current versus source-drain voltage with gate voltage as a parameter. The model is correct until the maximum value of the parabola is reached. After that, the conductivity of a channel tract close to the drain contact becomes zero and the model is no longer valid

Fig. 9.6 Left: charge behavior in the channel as a function of the offset voltage. Beyond VG − VT , the model predicts an impossible positive charge in the channel. Right: channel shape for increasing VDS and VDS distribution along the channel

which can drag the current from the end of the channel towards the drain contact (Fig. 9.6). The characteristics of the MOSFET are shown in Fig. 9.7. The Id / VDS output characteristics has VG as a parameter. In Fig. 9.6, the dependence of the I D max on VG is also shown. The MOSFET experiences a number of different working conditions according to the combination of VG and VDS : the cut-off, the sub-threshold condition, the linear-parabolic behavior of the current (the so-called triode region) and, finally, the saturation condition, where the current does not depend on VDS . Figure 9.10 reassumes all the conditions in the VG − VDS plane (Fig. 9.8). The output characteristics is limited by the current breakdown. At large VDS , the increased electric field gives the accelerated electrons a kinetic energy sufficient to

9.3 Metal Oxide Semiconductor Field Effect Transistor

235

Fig. 9.7 Left: MOSFET characteristics. The breakdown is not visible in the plot. However, it must be considered that at large V DS the drain current undergoes a strong increase due to the avalanche effect. Right: dependence on saturation current versus gate voltage. Note that the threshold voltage defines the current onset

Fig. 9.8 The different MOSFET working conditions depend on the combinations of VG and VDS

create an electron-hole couple by scattering. This gives rise to an avalanche mechanism that elicits a sharp increase in current, similar to the avalanche effect observed in the reverse biased PN junction (Fig. 9.9). Before discussing additional details of the MOSFET model, it is worth considering the role of the diffusion current. The model we have developed so far, assumes that the drain current is only the drift current. The diffusion current has been neglected, even though the charge in the channel is variable from source to drain. The diffusion current can be calculated as: Idi f f = jdi f f · A = A · Dn ·

d Q(y) dy

(9.21)

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9 Field Effect Transistors

where A is the area of the channel, variable along y. Considering that Q(y) = Q n (y)w/A, we have: Idi f f = A · Dn ·

w d Q n (Vc ) d Vc . A d Vc dy

(9.22)

Replacing Q n (Vc ) = −Cox (VG − VT − Vc ) and Dn = K B T /qμn , we get: L

VD dy = Dn · wCox

Idi f f 0

d Vc =

w KBT μn Cox VDS . q L

(9.23)

Vs

To get the total current, the diffusion current is subsequently added to the drift current:   2  VDS w KBT VG − VT + + . (9.24) Itot = −μn Cox L q 2 Eventually, under the common condition VG − VT  KqB T , the contribution of the diffusion current to the total drain current in strong inversion regime is negligible.

9.3.1 Channel Length Modulation With the MOSFET in saturation regime, each further increase in VDS falls across the pinch-off region. The size of the pinch-off region thus increases with VDS and, consequently, the electron channel length decreases. Since the drain current is inversely proportional to the channel length, the increase in VDS eventually induces an increase in the drain current. This effect is similar to the Early effect in the BJT. It is interesting to note that in both BJT and MOSFET the working mechanism of the devices avoids the output characteristics from being flat, or the devices from behaving as ideal current sources, which is prohibited by the Kirchhoff network law. The channel length modulation is sometimes expressed by a parameter λ that plays the role of the Early voltage in the BJT. The increase in current due to the channel length modulation is expressed by the factor [1 + λ · (VDS − (VG − VT ))]. In practice, for large VDS , the saturation current may be written as: Idsat = (kVG − VT )2 (1 + λVDS ). The graphical meaning of lambda is given in Fig. 9.11.

(9.25)

9.3 Metal Oxide Semiconductor Field Effect Transistor

237

Fig. 9.9 MOSFET characteristics with channel length modulation. The channel modulation parameter, sometimes called Early voltage after the BJT nomenclature, is at the origin of the slopes of the characteristics

9.3.2 Body Effect The body contact (VB ) provides the reference point for the voltage applied to the device. In the previous derivation of the MOSFET characteristics, it was conveniently set to zero. Actually, the body contact provides a further degree of freedom for the modulation of the threshold voltage. Considering Eq. (9.8), by subtracting the case of VB = 0 and VB = 0 we get the change in threshold voltage due to the body voltage: √

 2s q N A  VT = 2φ p  + VS B − 2φ p  C (9.26)

ox  2φ p  + VS B − 2φ p  . =γ The quantity γ depends on the MOSFET parameters and is known as body effect √ parameter, measured in units of 1/ V .

9.3.3 Subthreshold Current At VG < VT the channel is not formed and the interface with the oxide is depleted of mobile charges, but the electron concentration is larger than the hole concentration. In practice, immediately below the threshold the electron concentration is similar to the concentration found in a lightly doped semiconductor. As a consequence, a small electron current can still flow between drain and source. This current is known as subthreshold current. Although small, it is important because even when the MOSFET is nominally switched off ( VG < VT ), a current still exists. This current is an important component of the leakage current, which tends to discharge the memory cells.

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9 Field Effect Transistors

At VG < VT , the interface region is a much thicker dielectric layer than that of the depletion layer separating the n + -type regions and the p-type substrate. Thus, most of the electric field generated by VDS is distributed towards the substrate. This makes the drift negligible, whereas diffusion is the dominant contribution to the subthreshold current. The current is ruled by the charge concentration at the drain and source junctions. In subthreshold condition, the conduction band in the channel is rather distant from the Fermi level, so that the junctions with the n + regions are not negligible and φs is the equilibrium barrier between the channel and the highly doped contacts. Under bias, the barriers at the drain and source junctions are: φs − VD and φs − VS and the different barrier heights give rise to a charge gradient in the channel. Consequently, . the current due to VDS is j = −q Dn ∂n ∂y The electron concentrations at the edges of the channel (n D at the drain and n S at the source) are calculated as:     q(φs − VD ) q(φs − VS ) ; n S = n i exp . (9.27) n D = n i exp KBT KBT In order to estimate the current, let us assume the behavior of the charges from drain to source as linear:

j = −q Dn

nD − nS L 

Dn n i exp = −q L Usually, VD 

KBT q

qφs KBT



    q VD q VS exp − − exp − . KBT KBT

, so that the first exponential is negligible:   q(φs − VS ) Dn n i exp . j = −q L KBT

(9.28)

(9.29)

In strong inversion mode, φs is independent of VG and it is equal to −φ p . In subthreshold condition, however, φs is proportional to VG . We can thus replace φs with VG , introducing a factor η accounting for the fact that φs is smaller than VG . As a consequence, the subthreshold current can be written as:   Dn q VG S n i exp . (9.30) jst = −q L ηK B T The subthreshold current has an exponential dependence on VG S , whereas the inversion current is proportional to VG2 S . This difference is due to the different nature of the two currents, the inversion current being a drift current and the subthreshold current a diffusion current. Note that the source contact is generally grounded. Here, VS is maintained as a reference for the gate potential.

9.3 Metal Oxide Semiconductor Field Effect Transistor

239

Fig. 9.10 Behavior of the root square of the saturation current versus the gate voltage. The intercept of the graded approximation model provides the threshold voltage. Below the threshold voltage the small subthreshold current is plotted. The transition between the two regimes provides a deviation from both models around VT

It is important to consider that the subthreshold current is also present when VG S > VT . Its amplitude, however, is smaller than that of the drift current, so that it is of course neglected. Indeed, the magnitude of the gradient is independent of the quantity of charges. There may be a large diffusion current with a small amount of charges and, on the other hand, a large drift current with a negligible charge gradient.

9.3.4 Transit Time The response time of the MOSFET mainly depends on the time taken to charge the depletion layer and on the transit time of the electrons in the channel. The time necessary to charge the depletion layer was treated in the PN junction section. As for the transit time, being a peculiar characteristic of the MOSFET, it deserves a little more attention. The velocity of the electrons is variable along the channel because their concentration is variable. A variable velocity, indeed, ensures that the drain current is maintained constant all along the channel. The transit time can be calculated as follows: L Ttr = 0

1 dy = − v(y)

L 0

1 dy μn E(y)

(9.31)

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9 Field Effect Transistors

C (y) where E(y) = d Vdy . The function VC (y) can be calculated from the definition of drain current:

y

VC (y)

Id dy = −wμn Cox 0

(VG − VT − Vc )d VC

(9.32)

2Id y . wμn Cox

(9.33)

0

from which VC (y) is found:  VC (y) = VG − VT −

(VG − VT )2 −

Replacing the drain current with its saturation value (Eq. 9.20), we obtain:  y VC (y) = (VG − VT ) − (VG − VT ) 1 − . (9.34) L Thus, the electric field E(y) is: E(y) =

VG − VT 1 d VC (y) =− dy 2L 1−

y L

.

(9.35)

Replacing the electric field and solving the integral, the transit time is found: Ttr =

L2 4 3 μn (VG − VT )

(9.36)

The transit time strongly depends on the channel length and mobility. The decrease in channel length and the increase in mobility are the possible actions to be taken to reduce the transit time and, consequently, the response time of the device.

9.4 Short Channel MOSFET In the previous section, the so-called long-channel model of the MOSFET has been derived. The long-channel length condition ensures the graded-channel approximation, which is the basis for the derivation of the above model. Furthermore, the current has been calculated assuming a constant mobility in the channel. We will see soon that this is one of the conditions that must be removed when a short-channel MOSFET is considered. The length of the channel is inversely proportional to the drain-source current (Eq. 9.19) and the transit time (Eq. 9.36). Therefore, a decrease in channel length is an optimal choice to increase the device performance. The channel length reduction,

9.4 Short Channel MOSFET

241

however, has a number of additional consequences, not always positive, that must be considered. We will see that in order to maintain the performance, a scaling of the channel length must be paralleled with a scaling of the other MOSFET parameters. Some of the consequences of the channel length scaling are thereafter introduced.

9.4.1 Threshold Voltage Modulation The n + doped regions allow the drain and source contacts to create a PN junction with a p-type substrate. As a consequence, two additional depletion layers, overimposed on the MOS depletion layer, appear at the contact-substrate interfaces. In normal conditions, the PN junctions are reverse biased, so that the additional depletion layers tend to expand towards the substrate and, also, towards the channel. In a longchannel device, the size of these regions can be neglected with respect to the size of the MOS depletion layer. However, when the distance between drain and source is small, the contributions of these two regions to the total depletion layer are no longer negligible. In the depletion layers pertinent to the PN junction, the acceptor charge is not controlled by VG , but rather by VD and VS . As a consequence, the charge of the depletion layer controlled by VG decreases. This charge is the last term of the threshold voltage in Eq. (9.9), so that if the depletion layer charge controlled by VG decreases, then, as shown in the next equation, also the threshold voltage becomes smaller (with respect to a similar device where only the channel length has changed): VT = V f b − 2φ p  +

Q ∗d Cox

(9.37)

where Q ∗d is the portion of the depletion layer charge actually controlled by VG S . Since the source is normally grounded, the threshold voltage modulation occurs at the drain contact. VT has an exponential behavior with the channel length. The decrease in Q ∗d can be compensated, or increased, by the oxide capacitance. Leaving untouched the materials, a reduction in the oxide thickness can properly counteract the threshold voltage decrease. As anticipated in the previous section, in order to maintain the device properties the scaling of the channel length must be complemented with the scaling of other quantities. In this case, to maintain the same threshold voltage the oxide thickness must be reduced as well.

9.4.2 Drain Induced Barrier Lowering The reduction of the distance between the source and drain n + regions makes the merging of the depletion layers across the substrate possible. Figure 9.11 shows

242

9 Field Effect Transistors

Fig. 9.11 Behavior of the threshold voltage with the channel length

Fig. 9.12 Scheme of Drain Induced Barrier Lowering effect. Left: path from source to drain through the substrate. Right: band diagram along the path. Under reverse bias, at the drain contact the depletion layer expands through the substrate until to reduce the source substrate barrier

the band diagram from source to drain across the substrate. The drain contact is positively biased, whereas the source and the substrate are kept grounded. In this condition, the drain-substrate junction is reverse biased, the barrier becomes larger and the depletion layer gets expanded. In a long channel device, this does not alter the conditions of the source-substrate junction, but in a short channel device the depletion layer can become so large to influence the barrier of the source-substrate junction. As a consequence, an additional current of electrons from source to drain is found. This current is summed to the subthreshold current and depends on VDS . This phenomenon is similar to the punchthrough of a BJT base. The Drain Induced Barrier Lowering can be counteracted by increasing the doping of the substrate, in order to limit the expansion of the depletion layer. However, this reduces mobility, and thus current. Alternative solutions consist in making the substrate so thin to be completely depleted also at VG S = 0 (Fig. 9.12).

9.4 Short Channel MOSFET

243

9.4.3 Velocity Saturation When the channel is short, the effects of the non-linear mobility discussed in Chap. 1 become manifested. Indeed, as L decreases the electric field increases and the velocity of the electrons tends to reach the saturation value. The effect is favored by the fact that since the mobility in the channel region is smaller with respect to the bulk, the saturation occurs at smaller electric fields. Of course, also the saturation velocity p n ≈ 6 ÷ 10 × 106 cm/s; vsat ≈ 4 ÷ is smaller. The typical values for silicon are: vsat 6 8 × 10 cm/s for electrons and holes respectively. Obviously, the largest velocities are found in the bulk. The main consequence of the non-linear mobility is that the device may reach saturation before the pinch-off condition, namely at a smaller VDS than the one predicted by the long-channel model. The drain current in the regime of non-linear mobility is calculated considering the behavior of the mobility with the electric field, described in Fig. 1.22. The relationship between electric field and velocity is given by the following piecewise relationship:

E < Esat v =

μe f f E 1 + EEsat

E ≥ Esat v = vsat where μe f f is a parameter calculated from the saturation electric field: Esat = 2

vsat . μe f f

(9.38)

In order to calculate the current, let us consider: Id = qn Av = w Q n v = wCox [VG − VT − Vc (y)]

μe f f E . 1 + EEsat

(9.39)

In the channel, the space coordinate progresses from 0 to y, and the voltage from 0 to VDS . In this situation, the electric field is directed from drain to source, so that it is negative. With a little algebra, the above equation is rearranged into:   Id (−E). (9.40) Id = Cox μe f f w [VG − VT − Vc (y)] − Esat Considering that E = −d Vc /dy, and integrating dy from 0 to L and d Vc from 0 to VDS , we have:

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9 Field Effect Transistors

L

VDS Id dy =

0

VDS Cox μe f f w [VG − VT − Vc (y)] d Vc −

0

0

Id d Vc Esat

whose solution gives the short channel model of the MOSFET:   V2 μe f f Cox w 1 Id = − . (VG − VT )VDS − DS L 2 1 + EVsatDSL

(9.41)

(9.42)

Again, the negative sign means that the current flows from drain to source. This is the same current calculated for the long channel device, divided by a correction term that accounts for the decreasing mobility. The calculation has been carried out using the same approximations of the long-channel MOSFET. Namely, μe f f is constant along the channel and the charge variation in the depletion layer is negligible. This simplified equation is nonetheless sufficiently accurate to describe the device behavior. At Esat  VLDS the equation converges to the long-channel model. Hence, we can consider Eq. (9.42) as the complete model for the MOSFET current. The long channel condition is defined as: L

VDS . Esat

(9.43)

Considering Esat = 4 × 104 V/cm and VDS =5 V, the long channel model requires L  1.25 μm. An obsolete condition for integrated MOSFETs. Due to the velocity saturation, the current reaches its saturation value at VDS = VDsat , which is smaller than the value predicted by the long-channel model. In practice, saturation is a combined effect of mobility reduction and decrease in channel charge. In saturation regime, the charge in the inversion channel is: Q nsat = −Cox (VG − VT − VDsat )

(9.44)

to which corresponds the saturation current: μe f f Esat . 2 (9.45) VDsat occurs at the intersection of saturated current (Eq. 9.45) and short-channel current model (Eq. 9.42): Idsat = wCox (VG − VT − VDsat )vsat = wCox (VG − VT − VDsat )

9.4 Short Channel MOSFET

245

Fig. 9.13 Behavior of channel charge and velocity in long and short-channel MOSFETs. In the case of a short-channel MOSFET, saturation is given by a combination of mobility decrease and charge reduction

  V2 μe f f Cox w 1 (VG − VT )VDsat − Dsat L 2 1 − EVDsatL sat

μe f f Esat = wCox (VG − VT − VDsat ) 2

(9.46)

from which VDsat is calculated: VDsat =

Esat L(VG − VT ) . Esat L + (VG − VT )

(9.47)

At Esat L  VG − VT , VDsat coincides with the one predicted by the long channel model. This gives a further condition for the long-channel MOSFET: L

VG − VT . Esat L

(9.48)

On the contrary, when Esat L  VG − VT , the saturation voltage is simply VDsat = Esat L, and the saturation current is: Idsat = wCox (VG − VT − Esat L)vsat .

(9.49)

Note that the saturation current increases as the channel length decreases (Fig. 9.13). In Fig. 9.14 the behavior of the charge in the channel and the electron velocity in both the models is illustrated. In a short-channel MOSFET, the saturation current depends linearly on VG − VT , whereas in a long-channel MOSFET the dependence is quadratic. Furthermore, the saturation condition occurs at smaller VDS . As the channel length approaches zero, the saturation voltage VDsat becomes also small, because when L is very small each applied voltage may provide an electric

246

9 Field Effect Transistors

Fig. 9.14 Drain current behavior with L in the long and short MOSFET models

field greater than the saturation electric field. On the contrary, in the long-channel MOSFET, the current diverges as L tends to zero. Actually, this effect is expected in any conductors and is necessary for the saturation of velocity. The largest possible current is obtained at L = 0 and is given by: Id max = wCox (VG − VT )vsat .

(9.50)

The ratio of saturation current to the largest current is the ideality factor (α), which provides a measure of the actual current with respect to its largest theoretical value: α=

Idsat (VG − VT ) − VDsat = . Id max VG − VT

(9.51)

For instance, if xox = 40 nm; L = 1 μm; VDsat = 13 V ; VG − VT = 4.3 V , then the ideality factor is k = 0.72. Any other reductions in the dimension of the device results in a current increase of no more than 30 %. Actually, we must also consider that the threshold voltage changes with L.

9.4.4 Transit Time The transit time can be calculated using the same approach of the long-channel case. In saturation condition, the result for a short channel is a linear dependence of the transit time on the channel length: Ttr =

4 L . 3 vsat

(9.52)

This must be compared with the case of the long-channel model (Eq. 9.36), where the transit time depends quadratically on the channel length. Consequently, in short-channel MOSFETs, the decrease in transit time and the increase in cut-off frequency is less dependent on the dimension of the device. This means that in silicon MOSFETs, without considering many other important effects, the device performance cannot be indefinitely increased reducing the dimen-

9.4 Short Channel MOSFET

247

Fig. 9.15 Comparison between the short-channel model (continuous line) and the long-channel model (dotted line). The lines marking the saturation onsets in both the cases are also plotted

sions. In particular, the extension of the cut-off frequency towards larger frequencies requires additional strategies and, in particular, the introduction of semiconductors with larger mobility, such as gallium arsenide. However, since the interface with the insulators of these materials are worse than the case of silicon, a different kind of field-effect device must be introduced (Fig. 9.15).

9.4.5 Scaling The use of the MOSFET technology in electronics and, in particular, digital electronics, has been crucial. The consequent reduction of the MOSFETs dimensions brought about the great technological advances in microelectronics. The self-insulation MOSFET properties in comparison to the rest of the substrate, and the relatively easy implementation of complementary MOSFETs based on n-type and p-type silicon (CMOS technology), are at the basis of the high demand for integration of more and more complex and miniaturized devices. The dramatic reduction in the MOSFET dimensions, as it has emerged over the last decades, can be reassumed by the so-called Moore’s law. Actually, Gordon Moore gave different versions of his law. In 1965, when he was the Fairchild Semiconductor

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9 Field Effect Transistors

Table 9.1 Scaling rule preserving either the voltage or the electric field Parameter Constant voltage Constant field scaling scaling Scaling assumptions

Derived quantities

Gate length Gate width Oxide thickness Semiconductor doping Electric field Oxide capacitance Transit time Voltage Current Power

1/k 1/k 1/k k k k 1/k 2 1 k k

1/k 1/k 1/k k 1 k 1/k 2 1/k 1/k 1/k 2

The scaling factor is k Fig. 9.16 VG conditions to achieve p-mos and n-mos conductive channels

research and development director, he observed that the complexity for minimum component costs was increasing at a rate of roughly a factor of two per year. In 1975, he stated that the circuit density-doubling would have occurred every 24 months. This last sentence was nicknamed as “Moore’s law”. In simple terms, the law fixes a general and constant trend about the doubling of the number of transistors in integrated circuits. In spite of the changes in technologies, the law is still quite valid nowadays. Moore’s law clearly focuses on the increase in performances of those devices directly related to the very number of devices, such as the solid-state memories and the image sensors, whose capabilities increase despite a reduction of their prices. As discussed in the previous sections, the MOSFET scaling can induce a number of non-ideal behaviors that can be adequately corrected with a proper scaling of the dimensions, involving not only the reduction of the gate length and width, but also a proper change in other quantities, such as the oxide thickness and the depletion layer widths, which also implies the scaling of the semiconductors doping. The device can be scaled choosing to maintain constant either the electric field or the voltage. A constant electric field, indeed, avoids the decrease in mobility due to the large electric field, but requires a voltage reduction, which makes the device no longer compatible with the existing circuits. On the other hand, a constant voltage scaling exposes the device to the risk of mobility degradation. In Table 9.1 the scaling rule for constant voltage are shown (Fig. 9.16).

9.5 CMOS Configuration

249

Fig. 9.17 Constructive scheme of the CMOS structure

9.5 CMOS Configuration The MOSFET described in the previous sections has been fabricated on a p-type substrate, where the channel is made of electrons. Of course, the complementary configuration can be also fabricated on a n-type substrate and a channel made of holes. The different devices are indicated as n-mos and p-mos respectively. The pmos characteristics are opposite to those of the n-mos. For instance, the threshold voltage is negative and a more negative gate voltage is necessary to activate the channel. In practice, the channel formation conditions are VG − VT > 0 for a n-mos and VG − VT < 0 for a p-mos, with the consideration that VT , in the case of p-mos, is negative (Fig. 9.17). The two complementary devices can be formed on the same substrate, where a well doped in a way opposite to that of the substrate is created. The well hosts the second device, as shown in Fig. 9.18. This opportunity gives rise to a composed device called CMOS (complementary MOS). The most interesting aspect of the CMOS is the fact that it can be used as an inverter generating logical functions, working as primary building block of logic circuits. Figure 9.19 shows the CMOS circuit and the device connections. The gate voltages pmos = Vin − VD . As Vin increases from of the two transistors are VGnmos = Vin and VG ground to VD D , the charge at the oxide-semiconductor interface changes, in particular considering the conditions under which the conductive channels are formed in the two devices (see Fig. 9.17). For small values of Vin , only the p-mos transistor conducts currents, whereas at higher values only the n-mos transistor conducts. Ideally, when both the transistors have the same VT and the subthreshold current is negligible, the current between VD D and ground is always zero. Actually, since the threshold voltages are different and the subthreshold current is greater than zero, a small current flows as Vin goes from low to high states. Figure 9.19 shows the gate voltages versus Vin and the transfer function of the inverter circuit. It is interesting to note that, considering the inverter transfer function of Fig. 9.20, a large amplification factor,

250

9 Field Effect Transistors

Fig. 9.18 Inverter as function generator made with complementary MOSFETs, and physical implementation with a CMOS structure

Fig. 9.19 (Left) gate voltages as a function of the input voltage. Thicker lines show the open channel conditions. (Right) inverter transfer function. Deviations from ideality result in a gradual change in the output voltage around the transition voltage

defined as d Vout /d Vin , is achieved in the short interval where the current flows through the inverter. The CMOS structure contains a sequence of n and p regions that form a combination of npn and npn bipolar transistors. Due to the reduced dimension of the MOSFETs in the CMOS structure, the base regions of the two transistors are sufficiently short to allow for the transistor effect to take place. As shown in Fig. 9.21, the two transistors are connected with a base-collector loop that may give rise to a positive feedback resulting in a steadily growing current from the power supply to ground. In Fig. 9.21, the large current can be initiated by a sudden increase in the reverse current at the drain-bulk contact in the p-mos. Such an increase can be due to a release of energy provided, for instance, by a background-emitted ionizing particle. This disruptive phenomenon is called latch-up and must be avoided in practical devices. A method to avoid latch-up is the addition of recombination centers in the substrates, for example through the injection of recombinant elements, such as gold atoms, in order to increase the recombination length by reducing the collector current. Another method is the formation of a deep oxide trench, so as to physically insulate the two transistors.

9.5 CMOS Configuration

251

Fig. 9.20 Parasite pnp and npn transistors in the CMOS configuration form a positive feedback loop between the collector and base terminals. Consequently, a small increase in current at the emitter of the pnp transistor may give rise to a large current that can destroy the component

a

b

Fig. 9.21 a CMOS inverter in SOI technology; b FinFET configuration in SOI technology. Right: exploded vision showing the channel structure

252

9 Field Effect Transistors

Recent silicon technologies, such as the silicon on insulator substrates (SOI), have provided a further improvement of the MOSFET structure. A SOI substrate consists of a silicon wafer and a buried oxide layer. The surface silicon can be shaped in such a way to form individual devices electrically separated by the oxide layer (see Fig. 9.21) SOI substrates, moreover, crucially improved the performance of short-channel devices. As an example, since the doped p-type substrate jeopardizes the performance in a n-channel MOS, a completely depleted silicon substrate would eliminate, for instance, the increase in subthreshold current due to drain-induced barrier lowering. Figure 9.21 shows a SOI-based configuration with a tridimensional gate. The latter induces a depletion layer from each side of the silicon layer, which is sufficiently thin to be completely depleted of majority charges. This structure is called FinFET because the transistor emerges from the substrate as the fin of a fish. These structures are at the edge of the current MOSFET miniaturization and allow for channel lengths smaller than 100 nm as well as gate widths below 10 nm.

9.6 Metal Semiconductor Field Effect Transistor (MESFET) The scaling of dimensions is a powerful method to improve the MOSFET performance in terms of device densities, current levels and operative frequency. However, to further enhance the device response time to high frequencies, it is necessary to replace the semiconductor with materials endowed with a larger mobility. Such materials are those made of atoms of the III and the V group of the periodic table (note that silicon and germanium belong to the IV group). Among them, gallium arsenide (GaAs) is the most widely used. The design of III–V element-based devices is nonetheless affected by some technological limitations, particularly due to the lack of a natural oxide whose interface has the same characteristics of the silicon dioxide/silicon system. Indeed, this MOSFET configuration leads to such a defected channel that transistor performances are not adequate for a practical exploitation. Due to this limitations, these devices are designed according to a different architecture. In particular, the oxide layer is omitted and the gate is formed by a direct metal-semiconductor junction (a Schottky diode), which, during the operation, is reverse biased. Such a device is called Metal Semiconductor Field Effect Transistor (MESFET). Despite the non-null gate current, it allows the field effect transistor principle and the high electron mobility of III–V semiconductors to be combined in order to achieve a high operating frequency. The velocity versus electric field behavior of GaAs and other III–V semiconductors has been discussed in Chap. 5. It is characterized by a maximum velocity and a negative differential tract that converges to the saturation velocity. Compared to 2 , the mobility of electrons is about the mobility in silicon, notably μ Si ≈ 1400 cm Vs 2 cm μGa As ≈ 8000 V s . On the other hand, the mobility of holes is lower in GaAs than in

9.6 Metal Semiconductor Field Effect Transistor (MESFET)

253

Fig. 9.22 Schematic drawing of a MESFET. The dotted line indicates the current path from drain to source. The dashed line indicates the intrinsic FET

silicon. This asymmetry hinders the possibility of developing complementary devices as in the CMOS technology. GaAs has a wider band gap than that of silicon (E gap = 1.24 eV), so that the intrinsic electron concentration is also smaller (n i = 9 × 106 cm−3 ). As a consequence, the intrinsic GaAs is less conductive than the intrinsic silicon. This is an interesting property exploited in the design of devices. The MESFET configuration is shown in Fig. 9.22. A n-doped GaAs is formed on a substrate of intrinsic material. The gate contact is made of evaporated metal (as in the Schottky diode) and two n + regions form the drain and gate contacts. The metal is chosen in order to form a rectifying Schottky contact (Schottky diode), whose characteristics have been described in Chap. 2. At the equilibrium, a depletion layer is formed at the interface between the metal and the doped semiconductor. The presence of the depletion layer shrinks the conductive region in the N-type GaAs between the drain and source contacts, giving rise to a conductive structure analogous to the MOSFET channel. The scope of the device is the control of the drain-source current via the voltage applied to the gate contact. This is obtained when the metal-semiconductor junction is under reverse bias. In this condition, the depletion layer becomes wider with respect to the equilibrium. If the thickness of the doped semiconductor layer is small, then the conductivity is largely modulated by the applied voltage. Note that the device behaves like a MOSFET in depletion mode. The great advantage over the MOSFET is that the current flows through the bulk of the device at the interface between the doped and the intrinsic semiconductor. However, the current still flows through a doped material, so that the mobility is lower than the largest mobility expected in the intrinsic material. The voltage between the drain and source contacts is distributed along the path indicated by the dotted line in Fig. 9.22. The voltage acts as a further reverse bias of the metal-semiconductor junction, contributing to increase the size of the depletion layer according to the profile shown in Fig. 9.23. The effect is similar to that observed in

254

9 Field Effect Transistors

Fig. 9.23 The MESFET under bias. Note that the gate voltage is negative and the depletion layer is deformed by the action of VDS

the MOSFET, with the great difference that here the current is formed by the majority charges without neither inversion nor accumulation and flows through the bulk. The doping in the n-type region is of the order of 1017 cm−3 . Thus, the contact resistance between drain and source and the intrinsic FET structure (see Fig. 9.17) is very small, and VDS is applied across the channel. The total voltage drops across the depletion layer is: Vs = φi − VG + φc (y), where φi is the built-in potential of the metal-semiconductor junction, VG is the gate potential and Vc (y) is the voltage due to VDS . The depth of the depletion layer is [see Eq. 2.15]:  2s (9.53) xd (y) = [φi − VG + φc (y)]. q ND As in the MOSFET, VDS is distributed across the channel according to the conductivity, which is modulated by the depletion layer depth. Being t the thickness of the doped layer, the differential resistance in the tract dy of the channel is: dR =

dy 1 qμn N D w(t − xd y)

(9.54)

where w is the transverse dimension of the device. The current Id = dφ/d R is : Id =

qμn N D w[t − xd (y)] dφ. dy

(9.55)

The above equation can be solved replacing xd with Eq. (9.53) and integrating, as usual, dy from 0 to L and Vc from 0 to VDS . The result provides the MESFET characteristics:

9.6 Metal Semiconductor Field Effect Transistor (MESFET)

255



  2s 2  3/2 3/2 (φi − VG + VT ) − (φi − VG ) . VDS q ND 3 (9.56) The I/V curve is nonlinear, with an exponent 3/2 instead of the exponent 2 found in the MOSFET. The current reaches a maximum value when the channel touches the intrinsic layer. Under this condition, the model is no longer valid. As in the MOSFET, the model predicts a physically impossible positive charge. Since the intrinsic layer is quasiinsulating, the case is analogous to the MOSFET pinch-off. The same argument used in Sect. 9.3 can be applied here to justify the fact that after the pinch-off condition the current maintains its maximum value, which is ideally independent of VDS . The saturation condition is xdsat (Vdsat ) = t, and the saturation voltage is: qμn N D wt Id = L

1 − t

Vdsat =

q NDt 2 − φi − VG . 2s

(9.57)

The saturation voltage depends on the thickness of the doped layer and on the doping level (N D and φi ). The first two terms of Eq. (9.57) define a sort of threshold voltage for the MESFET. Differently from the MOSFET, this voltage does not define the occurrence of the channel, but it is the value of VG , at VDS = 0, for which the channel disappears. Replacing Vdsat in Eq. (9.56), the relation between the saturation current and VG is: Idsat ≈ (VG − VT )3/2 . The exact relationship of the saturation current is:

     1 2 φi − VG 2/3 φi − VG + − Idsat = I p (9.58) 3 Vp 3 Vp m q2 N 2 t3w

2

and V p = q N2Dst . where I p = n 2s LD Being the term I p inversely proportional to L, it contains the modulation of the current due to the channel length. The same considerations that have been done in the case of the MOSFET about the channel length modulation, also apply to the MESFET (Fig. 9.24). The MESFET structure also exists in silicon, where it is called Junction Field Effect Transistor (JFET). Due to the non-negligible conductivity of intrinsic silicon, the MESFET configuration cannot work in the case of silicon, where the device is designed so as to confine the conductivity at the center of the semiconductor. The channel is created by a symmetric couple of PN junctions located at both the sides of the semiconductor. A JFET scheme is shown in Fig. 9.25. Besides the different symmetry, the JFET equations are rather similar to the MESFET equations. The JFET has allowed the development of field effect transistors by avoiding the mobility reduction issues occurring at the oxide/semiconductor interface. On the other hand, being a non-planar device, it is not suitable for integrated circuits, where the planar architecture is a fundamental requirement for device integration. The introduction of

256

9 Field Effect Transistors

Fig. 9.24 Output characteristics of a MESFET

Fig. 9.25 Structure of the Junction Field Effect Transistor. In order to confine the depletion layer into the n-type semiconductor, the gate contact is a PN junction with a heavily doped p-silicon. Two gates are used to confine the channel in the bulk of the semiconductor

the silicon-on-insulator technology has made both the growth of thin doped silicon layers on insulators and the fabrication of transistors according to the MESFET architecture possible.

9.7 High Electron Mobility Transistor: HEMT The MESFET offers the possibility to exploit the high mobility of GaAs and similar materials. This makes the extension of the operating frequency up to the microwave region possible. The lack of a good semiconductor-insulator interface brought about the design of a device where the channel gets formed in the bulk of the doped semiconductor and the gate voltage simply regulates the channel width via the modulation of the depletion layer of the metal-semiconductor junction. However, the performances are still limited by the fact that the current flows through a doped material, where the mobility is lower than that of the intrinsic

9.7 High Electron Mobility Transistor: HEMT

257

Fig. 9.26 Band diagram before (above) and after (below) the equilibrium in metal—n-type AlGaAs and intrinsic GaAs structure. Since the Fermi level of the n-type AlGaAs lies above the conduction band of the intrinsic GaAs, at the equilibrium the conduction band of the GaAs lies below the Fermi level

semiconductor. The optimal solution may be the design of a device where the channel is formed in an intrinsic material. Obviously, this is not straightforward, because the electron concentration in the intrinsic material is rather small. In GaAs, the intrinsic electron concentration is about four orders of magnitude smaller than in silicon. The solution to the problem is achieved using a heterojunction between the metal and the intrinsic substrate. As an example, let us consider a device where the gate structure is formed by a sequence of metal-n-type Al x Ga1−x As-GaAs. The band diagram of Al x Ga1−x As is modulated by the percentage of aluminum. In practice: E gap = 1.42 + 1.24 · x and qχ = 4.07 − 1.1 · x for x  0.5, where 1.42 eV and 4.07 eV are the gallium arsenide energy gap and the affinity respectively. At x = 0.3, for instance, the energy gap is E gap = 1.80 eV and the affinity is qχ = 3.74 eV. The methods employed to study heterojunctions have been discussed in Chap. 7. In Fig. 9.26 an example of a band diagrams is shown. The metal is chosen in order to form a Schottky junction with the AlGaAs layer. Furthermore, the AlGaAs layer is sufficiently thin to be completely depleted by the electron transfer necessary to equilibrate the whole structure.

258

9 Field Effect Transistors

The materials are chosen in order for the Fermi level of the AlGaAs layer to lie above the conduction band of the intrinsic GaAs. By applying the rules for the equilibrium band diagram drawing outlined in Chaps. 1 and 7, the bottom of the spike at the interface happens to lie below the Fermi level in a very narrow region close to the interface with AlGaAs. In this region, the GaAs becomes a degenerated semiconductor with a very large electron concentration. In such a narrow region, the electrons form a bi-dimensional gas of free electrons. This region is known as 2-DEG (Fig. 9.26). The 2-DEG is a high electron concentration strip connecting the drain and source contacts, embedded in the intrinsic GaAs. The 2-DEG electrons move with very high mobility. Consequently, this system allows to achieve large currents at a very short transit time, and thus very high operating frequencies. Such a device is called a High Electrons Mobility Transistor (HEMT). The AlGaAs layer is typically highly doped and very narrow, so as to obtain a total depletion of the N-type region at the equilibrium. AlGaAs depletion is necessary to avoid the 2-DEG from being shunted by a parallel conductor. In this condition, the AlGaAs behaves like an insulator between the metal and the 2-DEG. As a consequence, the voltage applied to the metal is able to optimally modulate the 2-DEG depth by changing the concentration of the electrons confined within the 2-DEG. The 2-DEG electrons are hindered to move towards the metal by a large barrier, so that even if their concentration is large, the current towards the metal is negligible and actually smaller than that of the MESFET. However, this leakage current, virtually null in the MOSFET, deteriorates the device performance. Like the MESFET, the HEMT works in a depletion mode, and the 2-DEG usually exists at VG = 0. The application of a VDS between drain and source results in a narrowing of the channel at the drain contact and in a consequent current saturation. Thus, the Id versus VDS characteristics is similar to those found in the other fieldeffect devices. The mobility in the 2-DEG is large, but it is inferior to that found in the bulk of the intrinsic material. To this regard, we must consider that the dimensions of the 2-DEG requires a quantum description of the electrons, so that the solution of the Schrodinger equation results in a wave function that is partially evanescing into the AlGaAs region. This means that the AlGaAs mobility influences the mobility of the electrons in the 2-DEG region. AlGaAs mobility is indeed reduced by high doping. To avoid the influence of doping on mobility, a narrow layer of undoped AlGaAs, called spacer, is created between GaAs and N-AlGaAs. An important parameter in the device is the lattice constant of the two materials forming the heterojunction. The lattice constant is a parameter proportional to the distance between atoms in the crystal. Differences in the lattice constants, the lattice mismatch, strain the crystals at the interface with a consequent increase of defects and mobility deterioration. The density of defects depends on the thickness of the epitaxial layer. Indeed, in very thin layer the bonds between atoms can stretch to match the lattice constant of the substrate without creating additional states or defects. Such junctions are called pseudomorphic, and the device where they are used are called p-HEMT.

9.8 Summary

259

Fig. 9.27 HEMT schematic structure. Typical thickness of the various layers are of the following order of magnitude: intrinsic GaAs: 1000 µm; intrinsic AlGaAs (spacer): 5 nm; N-AlGaAs: 50 nm. N-AlGaAs doping is of the order of 1018 cm−3

The leakage current can also be furtherly decreased by reducing the gate contact without affecting the channel length. For the scope, the gate contact is shaped as a mushroom. Finally, it is important to mention that superior performance in terms of operating speed is obtained when materials with larger mobility, such as InGaAs, replaces the GaAs substrate (Fig. 9.27).

9.8 Summary The current development of microelectronics is dominated by the metal-oxide-silicon (MOS) field-effect transistor (FET). The idea upon which the FET is based, namely controlling the conductivity of a region by varying the surface potential, precedes by many years the practical implementation, which became possible only after the first developments of the microelectronics technology. The strength of the device lies in the low defect density of the interface between silicon and silicon dioxide. Additionally, the MOSFET design leads to a device naturally insulated with respect to any other devices in the substrate. This makes the MOSFET the natural candidate for integrated circuits. Another interesting feature of the MOSFET is that the properties of the device improves as the area occupied on the silicon substrate decreases. Thus, the MOSFET reduced dimensions not only allowed for more complex integrated circuits, but also for devices with improved characteristics. However, the reduced size also arose drawbacks that can now be brilliantly solved by the most advanced technologies, such as the SOI wafer and tridimensional structures like FinFETs. In this chapter, the extension of the Field Effect Transistor to high mobility III– V semiconductor is also discussed. This development has been prompted by the necessity to extend the bandwidth of the device to higher frequency, particularly for telecommunication purposes. In these materials, due to the lack of an equivalent of silicon dioxide, the field effect is actuated by a reverse biased metal-semiconductor junction. This approach led to the exploitation of gallium arsenide in MESFET configurations. Further improvements are obtained by heterojunctions, where the work functions are sized so as to obtain a bidimensional gas of electrons suitable to extend the bandwidth up to hundreds of GHz.

260

9 Field Effect Transistors

In a bidimensional gas, electrons are forced in a narrow space, where quantum physics cannot be neglected. In this situation, the assumptions made in the first chapter are no longer valid, and a full quantum treatise becomes necessary to describe the behavior of the devices.

Further Reading Textbooks S. Sze, K.K. Ng, Physics of Semiconductor Devices, 3rd edn. (Wiley-Interscience, 2006) R. Muller, T. Kamins, M. Chen, Device Electronics for Integrated Circuits, 3rd edn (John Wiley & Sons, 2002) C.C. Hu, Modern Semiconductor Devices for Integrated Circuits (Pearson College, 2009) D. Neamen, Semiconductor Physics and Devices (McGraw Hill, 2003) K. Brennan, A. Brown, Theory of Modern Electronic Semicionductor Devices (John Wiley & Sons, 2002) T. Fieldly, T. Ytterdal, M. Shur, Introduction to Device Modeling and Circuit Simulation (John Wiley & Sons, 1998) J. Davies, The Physics of Low-Dimensional Semiconductors (Cambridge University Press, 1998)

Journal Papers M. Atalla, R. Soshea, Hot carrier triodes with thin film metal base. Solid State Electron. 6, 245 (1963) J. Lindmayer, The metal-gate transistor. Proc. IRE 40, 1361 (1952) H. Ihantola, J. Moll, Theory of a surface filed-effects transistor. Solid State Electron. 7, 423 (1964) S. Hofstein, P. Heilman, The silicon insulated gate field effect transistor. Proc. IEEE 51, 1190 (1963) J. Brews, W. Fichtner, E. Nicollian, S. Sze, Generalized guide for MOSFET miniaturization, in International Electron Devices Meeting, EDL-1, 2 (1980) D. Critchlow, MOSFET scaling: the driver of VLSI technology. Proc. IEEE 87, 659?67 (1999) K. Ismail, B. Meyerson, P. Wang, High electron mobility in modulation-doped Si/SiGe. Appl. Phys. Lett. 58, 2117–2119 (1991)

Appendix A

Elements of Classic Physics

A.1

Newtonian Mechanics

In classical Mechanics, matter is made up of point-like particles reacting to forces according to the laws formulated by Newton. The most important of these laws is the basic law of motion: For ce = mass · acceleration, which establishes the relationship between the cause (force) and the effect (acceleration) and introduces mass as a factor of proportionality between force and acceleration. The essential assumption of classical mechanics is that matter can be treated as being made of particles with finite mass, whose motion is defined by energy and momentum. Mechanical energy consists of potential and kinetic energies. Potential energy depends on the specific phenomena that elicit motion, whereas spatial variations of potential energy generate forces that activate motion. During motion, potential energy converts into kinetic energy, whose value is given by the relationship E k = 21 mv 2 , where v is the velocity of the body. Momentum is another important vector quantity related to the force a particle in motion can impress. It is defined as: p = m · v. The relationships connecting a force to potential energy (V) and momentum are: F = −∇V dp F= dt

(A.1)

where V is potential energy.

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262

A.2

Appendix A: Elements of Classic Physics

Electromagnetism

The other important branch of classical physics is concerned with those phenomena related to the electric charge. Like mass, charge is a property of elementary particles. Differently from mass, which can assume any positive values, the charge of any bodies is an integer multiple of the elementary charge, which is the charge of the basic atomic constituents, namely electrons and protons: q ≈ 1.6 · 10−19 C. Furthermore, unlike mass, charge can be either positive (as in protons) or negative (as in electrons). Thus, in a compounded body, the total charge can be zero, as in neutral atoms, whereas mass can never be nullified. Charges interact via electromagnetic forces, which play a crucial role in nature. Electromagnetic forces are conveniently represented by the electric field E and the magnetic field B, which are generated by the spatial charge distribution ρ and its motion, namely the currents j. The effects of the electric and magnetic fields propagate in space and matter, whereas the magnitude of the propagated fields depends on the electric and magnetic permittivities  and μ, which are characteristics of each media. The relationships between fields, charges, currents and permittivities are reassumed in the Maxwell equations: ρ  ∇ ·B=0 ∇ ·E=

∇ ×E=−

∂B ∂t

∇ ×B=μ·j+

(A.2) ∂E ∂t

The electric field is associated with an electric potential (φ), defined as the energy necessary to displace a unit charge from a reference point to another point in space. Since the electric field is conservative, this energy does not depend on the path but only on the initial and final points. The first Maxwell equation describes the relationship between charge density and electric field. The potential can be derived from the relationship between field and potential: x φ(x) = −

Ed x.

(A.3)

xr e f

Thus, the first Maxwell equation can also be written in terms of the electric potential with the Poisson Equation: ∂ 2φ ρ =− . 2 ∂x 

(A.4)

Appendix A: Elements of Classic Physics

263

Fig. A.1 Relationships among charge density, electric field and potential

The concept of fields and potential is useful to describe the effects of a spatial charge density on a charge small enough not to affect the field and the potential. The relationships between forces and potential energy with the electric field and electric potential are: F = q · E; E = q · φ.

(A.5)

Figure A.1 reassumes the relationships among charge density, electric field and electric potential. The solution of the Maxwell equations in a free space away from the charges and currents leads to those electric and magnetic fields behaving as waves. They are the electromagnetic waves whose spatial and temporal distribution is described by sinusoidal functions that, thanks to the Euler formula, can be expressed via exponential functions of a complex argument: ex p(−i · (ωt − kx))

(A.6)

where ω is the angular frequency and k is the wavevector. These quantities express the time and space periodicity of the traveling wave. The speed of the electromagnetic wave is : c=

ω |k|

(A.7)

The electromagnetic wave propagation speed in vacuum (and, on a very good approximation, also in air) is the speed of light c, which, according to the theory of relativity, is the largest achievable speed: c ≈ 3 · 1010 cm . Furthermore, it does not s depend on the reference frame. Electric and magnetic fields elicit forces acting on charges, so that, according to the laws of mechanics, the fields affect the motion of charged bodies. The forces produced by electric and magnetic fields are called Coulomb and Lorentz forces respectively. The total electromagnetic force is: F(x) = q · (E(x) + v × B(x))

(A.8)

It is important to notice that, in the study of electronic devices, magnetic forces are usually neglected.

Appendix B

Basic Principles of Quantum Mechanics

B.1

Waves and Particles

In spite of the enormous successes of mechanics and electromagnetism, at the end of the nineteenth century a number of theoretical conundrums and novel experiments were challenging the correctness of these theories. In particular, a theoretical problem was challenging the fundamentals of classical physics. It was the black-body radiation spectrum. This is the spectrum of electromagnetic waves in thermal equilibrium with the matter. One of the staples of classical physics has been the continuity of values of the physical quantities. This implies that electromagnetic waves can assume any arbitrary energy values. When applied to the radiation-matter equilibrium, this principle led to absurd results, such as the divergence of the spectrum of radiation at high frequency. In the attempt to solve this problem, in 1901 Planck introduced the concept that radiation can only exchange discrete quantities of energy. He also assumed that such energy is proportional to the frequency of the radiation: E = h · ν =  · ω. The quantity h = 2π  = 6.62 · 10−24 J · s is a universal constant known as the Planck constant. The existence of a discrete amount of exchangeable energy implies that the electromagnetic radiation, besides to be a classic wave, behaves as a flux of particles (later called photons), each of energy E = ω. Countless experimental proofs supported this assumption. The concept of photons was used by Einstein to explain the photoelectric effect, namely the fact that the capability of radiation to extract electrons from a metal does not depend on the intensity of light, but on its frequency. Photons are particles of peculiar properties. They exist only in motion and they only move at the speed of light. Thus, because of the theory of relativity, their rest mass is zero. However, they present the same properties of massive particles, such as momentum.

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Appendix B: Basic Principles of Quantum Mechanics

The light wave-particle duality challenged the physicists into assuming that the same duality should have existed in matter, too. In 1925 De Broglie advanced the hypothesis that a wave could have been associated with massive particles, such as electrons, and proposed that for each particle there was an associated wave with an assigned energy E and a momentum p:   (E · t − k · x) (B.1) ψ = A · exp −i · h In 1927 Davisson and Germer experimentally demonstrated that a beam of electrons impacting on a slit produces a pattern of intensity analogous to that produced by a beam of light. The diffraction of electrons and the photoelectric effect showed the plausibility of the wave-matter duality. Parallel to these findings, a novel physical theory able to explain this evidence and to predict new ones, was emerging: quantum mechanics.

B.2

Operators and State Functions

The concept of observation, namely the measuring operation, is crucial in Physics. Even if it can be conceived as a theory, on a practical level Physics is very different from pure Mathematics, because it describes real objects that can be observed. In other words, experiments provide theories with an input, whereas theories predict the experimental results. Measuring operations always modify the object of a measurement. The application of a voltmeter to a circuit, for instance, subtracts voltage from the circuit, so that the measured quantity is the voltage of the circuit plus the voltmeter. The simplest measurement consists in observing something, in such a way that light could shine on the object and our eyes could detect the reflected light. However, light consists of photons that release their energy to the object, virtually changing its state. Classical Physics assumes that these disturbances could be evaluated and subtracted, or small enough to be neglected. For instance, if the input resistance of the voltmeter is much larger than the equivalent resistance of the circuit which it is applied to, the disturbance introduced by the measuring operation is negligible. The light example above provides us with an explanation about the limits of this approximation. Indeed, when the object of a measurement becomes smaller and smaller, the light wavelength necessary for the observation must be shorter and shorter, so that the photon energy is larger and larger. In this way, the disturbance may become so large to completely hinder the measurement object. Quantum mechanics is an attempt to solve this problem, or how to achieve reliable information about small systems that are largely disturbed by any measurements. The theoretical foundation of this attempt lies in the fact that the result of an observation is the eigenvalue of an operator associated with the measured physical quantity. One of the consequences is that if we perform a sequence of measurements

Appendix B: Basic Principles of Quantum Mechanics

267

of two quantities represented by the operators A and B, then the obtained results are different if the order of the measurements is reversed. In other words: A · B − B · A = 0

(B.2)

The representation of physical quantities with operators is a fundamental postulate of quantum mechanics. The state of the system is described by the state functions (or wave functions) which the operators operate upon. Among the physical meanings of these functions, it is important to remark that the square of a particle state function is the probability density of the particle position in space and time. Among the operators, those representing position and momentum are very important. They can be defined as: xˆ → x pˆ → i

B.3

∂ ∂x

(B.3)

The Schrödinger Equation

Position and momentum are fundamentals in theoretical mechanics, where they define a couple of canonical variables that can describe the motion of any systems. On these basis, the operator of energy can be written as a function of the operators of position and momentum. This operator is called Hamiltonian:   ∂ . (B.4) Hˆ (x, ˆ p) ˆ = Hˆ x, i ∂x The eigenvalue equation of the hamiltonian corresponds to the time independent Schrödinger equation, which provides the space-dependent part of the total wavefunction ψ:   ∂ · ψ E (x) = E · ψ E (x). (B.5) Hˆ x, i ∂x In accordance with De Broglie’s wave function, the total wave function of a particle is: ψ(x, t) = ψ(x) · exp(−iωt).

(B.6)

Once the hamiltonian is known (namely, the potential energy function describing the physics of the system is defined), the solution of the Schrödinger equation results in the spatial distribution of the wavefunction and in all of the possible observable energy values.

268

Appendix B: Basic Principles of Quantum Mechanics

The solution of the differential equation needs boundary conditions. The first of these conditions is the continuity of the wavefunction at the border of the space, where the particle is confined. The second one comes from the relationship between wavefunction and probability. Since the particle exists, the integral of the wavefunction extended to all the space must be equal to one. ∞ |ψ(x)|2 d x = 1.

(B.7)

−∞

For instance, given a particle in rectilinear motion exposed to a potential V(x), the corresponding Hamiltonian contains the sum of the kinetic and potential energies. In terms of momentum operator and considering that p = mv, the kinetic energy operator is: EK =

1 1 2 1 2 2 ∂ 2 m · v2 = p → EˆK = pˆ = − . 2 2m 2m 2m ∂ x 2

Thus, the Schrödinger equation is:   2 ∂ 2 − + V (x) · ψ E (x) = E · ψ E (x)). 2m ∂ x 2

(B.8)

(B.9)

The solution of the Schrödinger equation can be very complex and it can be solved only for simple cases. The simplest one is a constant potential (V (x) = U0 ). Under this condition, the Schrödinger equation assumes a simple form:  ∂ 2 ψ(x) 2m(E − U0 ) 2 = −k · ψ E (x) with: k = . (B.10) 2 ∂x 2 The solution of the equation can be written as: ψ(x) = A · sin(kx) + B · cos(kx)

(B.11)

where A and B are calculated from the specific boundary conditions. The above solution is valid when E > U0 . However, the equation can also be solved under the opposite condition (E < U0 ), which is forbidden by classical Physics. The solution, in this case, is: ψ(x) = A · exp(k  x) + B · exp(−k  x) which, with the Euler formula, can also be written as:

(B.12)

Appendix B: Basic Principles of Quantum Mechanics

269

 





ψ(x) = A · exp(ikx) + B · exp(−ikx) with: k =

2m(U0 − E) . 2

(B.13)

The two cases correspond to very different wavefunctions. In the first case, admitted by classical Physics, the wavefunction is the sum of sinusoidal functions that ensure the presence of the particle from −∞ to +∞. In the second case, forbidden by classical Physics, the wavefunction is made up of exponential functions that asymptotically decay towards −∞ and +∞. In practice, when E < U0 the particle can exist only in a limited space region. The second solution is at the origin of the tunnel effect, which enables electrons to cross the barriers even if their energy is smaller than the barrier itself. The tunnel effect is fundamental in electronic devices, where it is exploited for ohmic contacts, Zener diodes, and tunnel diodes among others.

B.3.1

Potential Well and Discrete Energy Levels

The potential V (x) contains the interactions that pertain to a particular physical case. We have seen in the previous section that a particle can be found in any regions of space, independently of the relative energy intensity with respect to the potential. However, the presence of a particle is forbidden when the potential diverges to infinity. This is the case of the infinite potential well, which defines a free particle confined in a restricted space. This is a very rough one-electron approximation in an atom, or free electron in a solid. The corresponding potential in this case is: V (x) = 0, |x| < a V (x) = ∞, |x| ≥ a For |x| < a the Schrödinger equation is:   2 ∂ 2 · ψ E (x) = E · ψ E (x). − 2m ∂ x 2

(B.14)

(B.15)

The boundary conditions for the potential well are: ψn (x) = 0 for |x| ≥ a. As for the solution, let us introduce the term: kn2 =

2m En 2

So that the equation becomes:  2  ∂ 2 + kn · ψ E (x) = 0 ∂x2

(B.16)

(B.17)

270

Appendix B: Basic Principles of Quantum Mechanics

The solutions that fulfill the boundary condition at x = a, for even and odd indexes, are: ψ2n (x) = A · sin(k2n x) ψ2n+1 (x) = B · cos(k2n+1 x)

(B.18)

where ak2n = 2n

π 2

ak2n+1 = (2n + 1)

π 2

(B.19)

Combining these conditions we obtain: kn =

π n; n = 1, 2, ? 2a

Using the kn2 definition, the allowed energy levels are thus found:   2 π 2 n2 En = 2m 4a 2

(B.20)

(B.21)

Figure B.1 shows the wave functions and probability densities of the first four states in the one-dimensional infinite potential energy well.

Fig. B.1 The wave functions (solid lines) and probability densities (filled areas) of the first four states in the one-dimensional infinite potential energy well

Appendix B: Basic Principles of Quantum Mechanics

271

The discrete sequence of energy levels is a very rough approximation of either the energy of electrons in atoms or the energy of free electrons in solids. In the case of solids, the periodic internal potentials created by the pattern of electrostatic forces generated by nuclei determine the degeneracy of discrete energy levels into bands.

Appendix C

The Chua Formalism of Electric Network Elements

Electronics is interested in studying the relationship between v(t) and i(t) as they occur in materials. The relationship between voltage and current is complex, so that it is usual, in electric network theory, to decompose such a relation in three elements, which emphasize three different phenomena occurring in matter when voltage and current are considered. Voltage and current are actually the observable macroscopic quantities of two other quantities which, in some sense, are even more fundamental: electric charge and magnetic field flux. The relationship between observables and internal quantities is mediated by the integral and derivative operators. The relations between the four electric quantities (internal and observable) are conveniently represented in a diagram originally introduced by Chua (Fig. C.1). The ideal elements connecting the four variables are: Resistance: v = f R (i). In case of linearity f R = R. This is the Ohm law, but in general the relationship is non-linear.

Fig. C.1 Relation between electric quantities and the properties of matter

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274

Appendix C: The Chua Formalism of Electric Network Elements

Capacitance: v = f C (q). Even in this case linearity defines the standard capacitor element. However, devices may show non-linear behaviors. inductance: ϕ = f L (i). In the linear case f L = L. Eventually, in 1971 L. Chua introduced, for symmetry reasons, a fourth element connecting charge and magnetic flow, which he called memristor: ϕ = f M (q). It is easy to observe that in the case of linearity, a memristor is simply a resistor. In case of non linearity, however, it gives rise to a component whose conductivity depends on the amount of charge that has flown through the device, as it happens in electrochemical cells. In some sense, this element preserves a memory of past events: as its name suggests, it is a sort of resistor with a memory. ϕ = f M (q) →

∂ f ∂q dϕ =v= = M(q)i dt ∂q ∂t

(C.1)

Appendix D

The SHR Generation-Recombination Function

The generation-recombination function can be calculated considering that at the equilibrium the rates of generation and recombination of electrons and holes are equal, so that both the conditions define the function U :  U = r1 − r2 (D.1) U = r3 − r4 Replacing Eqs. 3.7 and 3.8 in the first equation, and Eqs. 3.13 and 3.14 in the second one, we obtain:  U = 41 vth n N T (1 − f )σn − N T f en (D.2) U = 41 vth pN T f σ p − N T (1 − f )e p where in both equations en and e p have been replaced with Eqs. 3.11 and 3.15 respectively. The first equation gives:   E T − Ei 1 (D.3) U = vth n N T (1 − f )σn − N T f exp 4 kB T from which the Fermi-Dirac function is calculated: f =

1 v n N T σn 4 th 1 v n N T σn (n 4 th

−U

i + n i exp( EkT B−E )) T

.

(D.4)

Introducing the electron capture time scale: τn0 =

1 1 v N σ 4 th T n

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0

(D.5)

275

276

Appendix D: The SHR Generation-Recombination Function

Equation D.4 becomes: f =

n − U τn0 i n + n i exp( EkT B−E ) T

.

(D.6)

Repeating the same calculation for the second equation in Eq. D.2 , and introducing the hole capture time scale, we obtain: τ p0 =

1

(D.7)

1 v N σ 4 th T p

which leads to a second expression for the Fermi-Dirac function: f =

T U τn0 + n i exp( Eki −E ) BT T p + n i exp( Eki −E ) BT

.

(D.8)

The two expressions of the Fermi-Dirac function are obviously equal: n − U τn0 i n + n i exp( EkT B−E ) T

=

T U τn0 + n i exp( Eki −E ) BT T p + n i exp( Eki −E ) BT

(D.9)

so that the U function is easily calculated: U=

np − n i2

τn0 [ p +

T n i exp( Eki −E BT

i )] + τ p0 [n + n i exp( EkT B−E )] T

(D.10)

Assuming that σn = σ p , also τn0 = τ p0 . Introducing τ0 as the unique characteristic time of hole and electron capture, and considering the definition of hyperbolic cosine x −x ), we obtain the simplified version of the generation-recombination (coshx = e −e 2 function: U=

np − n i2

i τn0 [ p + n + 2n i cosh( EkT B−E )] T

.

(D.11)

Appendix E

Numerical Examples of Rectifying Junctions

In this section, the behavior of a metal-semiconductor and a PN junction are calculated in two numerical cases. For each case, band diagrams, electrostatic quantities, and I/V and C/V curves are plotted. All calculations have been performed in MATLAB.

E.1

Metal-Semconductor Junction

In this section, some of the equations describing the ideal Schottky diode are explicitly calculated in order to provide a quantitative evaluation of the developed model. The example considers a junction formed by chromium and N-type silicon. The work function of chromium is approximately q = 4.95 eV, and silicon is uniformly doped with a density of donors equal to N D = 10−17 cm−3 . Figure E.1 shows the equilibrium band diagram. The electrostatic quantities are plotted in Fig. E.2. The results of the Poisson equation have been used to draw the band diagram. Finally, Fig. E.3 shows the dependence of depletion layer width, current and junction capacitance on applied voltage. For a clearer representation, the current is plotted on a logarithmic scale, so that the reverse current appears to be positive and the origin is not plotted.

E.2

PN Junction

Let us consider a PN junction formed by a P-type and a N-type silicon equally doped with a concentration of 1017 cm−3 of acceptors and donors respectively. Figure E.4 shows the band diagram at the equilibrium. Since the concentration of dopants is the same on both the sides of the junction, the depletion layer is equally distributed. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0

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Appendix E: Numerical Examples of Rectifying Junctions

Fig. E.1 Equilibrium band diagram of a chromium-N-type silicon junction with N D = 10−17 cm−3

Figure E.5 shows the results of the Poisson equation in the case of deep depletion hypothesis. The total width of the depletion layer is about 147 nm, equally distributed between the two sides of the junction. The maximum electric field at x = 0 is 5.68 × 104 V/cm and the built-in potential is 0.83 V. Figure E.6 shows the effects of applied voltage in terms of depletion layer size, current, and capacitance. The ideal diode current and generation-recombination current are calculated and summed up. At V A > 0.5 V the ideal current dominates over the total current of the device. The total capacitance is calculated as the sum of the depletion layer capacitance and the diffusion capacitances due to electrons and holes. At V A < 0.6 V , and in particular in reverse bias, the total capacitance is dominated by the depletion layer contribution. Figures E.7, E.8 and E.9 show the band diagram, the equilibrium electrostatic quantities, and the I/V and C/V curves in the case of an asymmetric PN junction, where N D = 5 · 1017 cm−3 and N A = 1016 cm−3 . The asymmetric doping results in an asymmetric junction, which makes the PN junction rather similar to the metalsemiconductor one. It is important to observe that, in the less doped region, which in this example is the p-type region, the intersection between the intrinsic Fermi level and the Fermi level does not occur at the interface (as in the equally doped diode), but inside the depletion

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279

Fig. E.2 Behavior at the equilibrium of charge density, electric field and potential for a chromiumN-type silicon junction with N D = 10−17 cm−3

layer of the p-type region. This means that there is a region inside the depletion layer where the p-type material is characterized by a dominance of electrons. This effect is known as inversion and is fully exploited in the metal-oxide-semiconductor devices. The intrinsic condition occurs where the potential is null.

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Fig. E.3 Space charge region width vs. applied voltage, current versus applied voltage and junction capacitance versus applied voltage in a gold-N-type silicon junction with N D = 10−16 cm−3 . The current is calculated with the thermionic model

Appendix E: Numerical Examples of Rectifying Junctions

281

Fig. E.4 Equilibrium band diagram of a PN junction equally doped with 10−17 cm−3 dopant atoms

282

Appendix E: Numerical Examples of Rectifying Junctions

Fig. E.5 Charge density, electric field and potential of a PN junction equally doped with 10−17 cm−3 dopant atoms

Appendix E: Numerical Examples of Rectifying Junctions

283

Fig. E.6 Effects of the applied voltage in terms of depletion layer size, currents and capacitances of a PN junction equally doped with 10−17 cm−3 dopant atoms

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Appendix E: Numerical Examples of Rectifying Junctions

Fig. E.7 Equilibrium band diagram of a PN junction asymmetrically doped with N D = 5 · 1017 cm−3 and N A = 1016 cm−3

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285

Fig. E.8 Charge density, electric field and potential of a PN junction asymmetrically doped with N D = 5 · 1017 cm−3 and N A = 1016 cm−3

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Appendix E: Numerical Examples of Rectifying Junctions

Fig. E.9 Effects of applied voltage in terms of depletion layer size, currents and capacitances of a PN junction asymmetrically doped with N D = 5 · 1017 cm−3 and N A = 1016 cm−3

Appendix F

Majority Current in a BJT with a Linearly Doped Base

The non-constant doping profile of a BJT base, besides to be a practical consequence of the doping process, is also required to optimize the device performance. For instance, it is useful to maintain the low-injection limit condition with a limited amount of total majority charges, and to decrease the diffusion capacitance of the base-emitter junction by reducing the transit time of the injected charges in the base. However, the non-constant doping profile elicits a built-in potential in the base (see Fig. 6.20). The associated electric field applied to the injected charges results in a drift current, which is additive compared to the diffusion current calculated in Chap. 6. Since the applied voltage is still distributed only across the depletion layers, the built-in potential in the base remains constant even in presence of a bias. Thus, the total current should be written as: dn . j = q · Q nb · μ˜ nb · E + q · D˜ nb · dx

(F.1)

Due to the non-constant doping profile, both mobility and diffusion coefficient are not constant along the base. In order to emphasize this variability, they are replaced ˜ Q nb is the in the following equations with an average value indicated with μ˜ and D. total excess of the minority charges given by: x B Q nb =

n(x)d x.

(F.2)

0

In the case of a linear doping profile, the built-in potential (φbi ) is also linear, and the electric field is constant: E = φbi /x B , where φbi is the variation of the built-in potential across the base and x B is the length of the base.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Di Natale, Introduction to Electronic Devices, https://doi.org/10.1007/978-3-031-27196-0

287

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Appendix F: Majority Current in a BJT with a Linearly Doped Base

The drift current is proportional to the integral of the excess charge profile, whereas the diffusion current is proportional to the derivative of the excess charge profile. Thus, since the base is short, the diffusion current is expected to dominate over the drift current. However, the drift current should be considered in the continuity equation, so that the excess charge profile is expected to be different from that calculated assuming only the diffusion current. Under the hypothesis of low-injection, the steady-state excess charge profile is given by the continuity equation: n 1 dJ − = 0. q dx τ

(F.3)

Replacing the total current, we get: d 2n φbi dn n D˜ nb · 2 + μnb · − = 0. dx xB dx τ

(F.4)

Dividing each term by Dnb , using the Einstein relations and the definition of recombination length (L n ), we obtain: d 2n n φbi dn − 2 =0 + 2 dx VT · x B d x Ln

(F.5)

where VT = k B T /q. Note that also the recombination length is not constant, so that it should be replaced by its average value. The solution of the continuity equation is an exponential function n = ekx . Thus, 2 replacing in the equation ddnx = k · ekx , dd xn2 = k 2 · ekx , an algebraic equation for the coefficient k is obtained: k2 +

φbi 1 k− 2 =0 VT · x B Ln

(F.6)

whose solutions are: 1 φbi 1 k1 = − − 2 VT · x B 2 1 φbi 1 k2 = − + 2 VT · x B 2

 

2 φbi 4 + 2 Ln VT2 · x B2 2 φbi 4 + 2. 2 2 Ln VT · x B

(F.7)

Hence, the total solution is a linear combination of two exponentials: n(x) = A · ek1 x + B · ek2 x . The constants A and B are calculated from the boundary conditions. n2 When the BJT is biased in active mode, they are: n(0) = NiA exp(q VB E /k B T ) and n(X B ) = 0.

Appendix F: Majority Current in a BJT with a Linearly Doped Base

289

For  bi approaching zero, namely when the doping profile is almost uniform, the solution converges to that found in Chap. 4. The BJT requires a short bases that the profile is linear and it is only defined by the excess charge at x=0 and the size of the base (x B ). However, the condition for which the base is short is no longer x B L n , but x B must be much smaller than k2−1 . The total current due to the excess of injected charges is still derived from the ideal current of the PN junction, with the addition of a drift component. Under the hypothesis of short base with linear doping profile, the drift current can be written as the integral of the injected charges: x B

x B Jdri f t d x = q · μ˜n · E ·

0

n(x)d x.

(F.8)

0

The quantity N A x B should be replaced with the integral of the excess charge. However, for a linear doping profile, it may be replaced with N¯A x B , where N¯A is the average doping concentration. Finally, replacing the electric field with its expression in the case of constant built-in potential, the total integral of injected charges calculated in Eq. 4.87, we get:   n2 1 q VB E (F.9) μ˜ n φbi exp Jdri f t = q i kB T N¯A 2x B whereas the diffusion current is still that calculated in Eq. 4.56: Jdi f f

  n i2 D˜ n q VB E . =q exp kB T N¯A x B

(F.10)

The total current is the sum of the drift and diffusion components, so that using the Einstein relationship the following expression is found: Jtot = q

    n i2 D˜ n q φbi q VB E + 1 exp . kB T N¯A x B k B T 2

(F.11)

Thus, the drift current exceeds the diffusion current if the term in brackets is larger than 1, namely when: φbi > 2 k BqT , which is the condition found in Sect. 6.7.1 to reduce, via a built-in potential, the electron transit time in the base. Finally, note that both the drift and diffusion currents are inversely proportional to the size of the base region (x B ). Hence, both the currents are affected by the inverse bias of the base-collector junction, so that also the drift current contributes to the Early effect.

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Appendix F: Majority Current in a BJT with a Linearly Doped Base

Further Reading L. Chua, Memristors-The missing circuit element. IEEE Trans. Circuit Theory 18, 507 (1971) D. Strutkov, G. Snider, D. Stewart, S. Williams, The missing memristor found. Nature 453, 80 (2008)