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ANALYSIS AND DESIGNOF ANALOG INTEGRATEDCIRCUITSFourth EditionPAUL R. GRAYUniversity of California, BerkeleyPAUL J. HURSTUniversity of California, DavisSTEPHEN H. LEWISUniversity of California, DavisROBERT G. MEYERUniversity of California, BerkeleyJOHN WlLEY & SONS, INC.New York / Chichester / Weinheim / Brisbane /Singapore / Toronto -7 CL-,
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ACQUISITIONS EDITOR William ZobristEDITORIAL ASSISTANT Susannah BarrSENIOR MARKETING MANAGER Katherine HepburnPRODUCTION SERVICES MANAGER Jeanine FurinoPRODUCTION EDITOR Sandra RussellDESIGN DIRECTOR Madelyn LesurePRODUCTION MANAGEMENT SERVICES Publication Services, Inc.Cover courtesy of Dr. Kenneth C. Dyer and Melgar Photography.This book was set in 10i12 Times Roman by Publication Services, Inc. and printed and bound byHamilton Printing Company. The cover was printed by Lehigh Press, Inc.This book was printed on acid-free paper. @Copyright 2001 O John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-mail: PERMREQ@WILEY.COM. To order books or for customer service please call 1-800-CALL-WILEY (255-5945).Library of Congress Cataloging-in-Publication DataAnalysis and design of analog integrated circuits 1Paul R. Gray. . .[et al.]. - 4th ed. p. cm. lncludes bibliographical references and index. ISBN 0-471-32 168-0 (cloth: alk. paper) 1. Linear integrated circuits-Computer-aided design. 2. Metal oxide semiconductors-Computer-aided design. 3. Bipolar transistors-Computer-aided design. I. Gray, Paul R., 1942- TK7874.A588 2000 621.38154~21 00-043583 Printed in the United States of America 10987654321
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PrefaceIn the 23 years since the publication of the first edition of this book, the field df analogintegrated circuits has developed and matured. The initial groundwork was laid in bipolartechnology, followed by a rapid evolution of MOS analog integrated circuits. Further-more, BiCMOS technology (incorporating both bipolar and CMOS devices on one chip)has emerged as a serious contender to the original technologies. A key issue is that CMOStechnologies have become dominant in building digital circuits because CMOS digitalcircuits are smaller and dissipate less power than their bipolar counterparts. To reducesystem cost and power dissipation, analog and digital circuits are now often integratedtogether, providing a strong economic incentive to use CMOS-compatible analog circuits.As a result, an important question in many applications is whether to use pure CMOS or aBiCMOS technology. Although somewhat more expensive to fabricate, BiCMOS allowsthe designer to use both bipolar and MOS devices to their best advantage, and also al-lows innovative combinations of the characteristics of both devices. In addition, BiCMOScan reduce the design time by allowing direct use of many existing cells in realizing agiven analog circuit function. On the other hand, the main advantage of pure CMOS isthat it offers the lowest overall cost. Twenty years ago, CMOS technologies were only fastenough to support applications at audio frequencies. However, the continuing reduction ofthe minimum feature size in integrated-circuit (IC) technologies has greatly increased themaximum operating frequencies, and CMOS technologies have become fast enough formany new applications as a result. For example, the required bandwidth in video appli-cations is about 4 MHz, requiring bipolar technologies as recently as 15 years ago. Now,however, CMOS can easily accommodate the required bandwidth for video and is evenbeing used for radio-frequency applications. In this fourth edition, we have combined the consideration of MOS and bipolar cir-cuits into a unified treatment that also includes MOS-bipolar connections made possibleby BiCMOS technology. We have written this edition so that instructors can easily se-lect topics related to only CMOS circuits, only bipolar circuits, or a combination of both.We believe that it has become increasingly important for the analog circuit designer tohave a thorough appreciation of the similarities and differences between MOS and bipolardevices, and to be able to design with either one where this is appropriate. Since the SPICE computer analysis program is now readily available to virtuallyall electrical engineering students and professionals, we have included extensive use of ,SPICE in this edition, particularly as an integral part of many problems. We have usedcomputer analysis as it is most commonly employed in the engineering design process-both as a more accurate check on hand calculations, and also as a tool to examine complexcircuit behavior beyond the scope of hand analysis. In the problem sets, we have also in-cluded a number of open-ended design problems to expose the reader to real-world situa-tions where a whole range of circuit solutions may be found to satisfy a given performancespecification. This book is intended to be useful both as a text for students and as a reference bookfor practicing engineers. For class use, each chapter includes many worked problems; theproblem sets at the end of each chapter illustrate the practical applications of the materialin the text. All the authors have had extensive industrial experience in IC design as well
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viii Preface as in the teaching of courses on this subject, and this experience is reflected in the choice of text material and in the problem sets. Although this book is concerned largely with the analysis and design of ICs, a consid- erable amount of material is also included on applications. In practice, these two subjects are closely linked, and a knowledge of both is essential for designers and users of ICs. The latter compose the larger group by far, and we believe that a working knowledge of IC design is a great advantage to an IC user. This is particularly apparent when the user mdst choose from among a number of competing designs to satisfy a particular need. An understanding of the IC structure is then useful in evaluating the relative desirability of the different designs under extremes of environment or in the presence of variations in supply voltage. In addition, the IC user is in a much better position to interpret a manufacturers data if he or she has a working knowledge of the internal operation of the integrated circuit. The contents of this book stem largely from courses on analog integrated circuits given at the University of California at the Berkeley and Davis campuses. The courses are un- dergraduate electives and first-year graduate courses. The book is structured so that it can be used as the basic text for a sequence of such courses. The more advanced mate- rial is found at the end of each chapter or in an appendix so that a first course in analog integrated circuits can omit this material without loss of continuity. An outline of each chapter is given below together with suggestions for material to be covered in such a first course. It is assumed that the course consists of three hours of lecture per week over a 15-week semester and that the students have a working knowledge of Laplace transforms and frequency-domain circuit analysis. It is also assumed that the students have had an introductory course in electronics so that they are familiar with the principles of transistor operation and with the functioning of simple analog circuits. Unless otherwise stated, each chapter requires three to four lecture hours to cover. Chapter 1 contains a summary of bipolar transistor and MOS transistor device physics. We suggest spending one week on selected topics from this chapter, the choice of topics depending on the background of the students. The material of Chapters 1 and 2 is quite important in IC design because there is significant interaction between circuit and device design, as will be seen in later chapters. A thorough understanding of the influence of device fabrication on device characteristics is essential. Chapter 2 is concerned with the technology of IC fabrication and is largely descriptive. One lecture on this material should suffice if the students are assigned to read the chapter. Chapter 3 deals with the characteristics of elementary transistor connections. The ma- terial on one-transistor amplifiers should be a review for students at the senior and gradu- ate levels and can be assigned as reading. The section on two-transistor amplifiers can be covered in about three hours, with greatest emphasis on differential pairs. The material on device mismatch effects in differential amplifiers can be covered to the extent that time allows. In Chapter 4, the important topics of current mirrors and active loads are considered. These configurations are basic building blocks in modern analog IC design, and this ma- terial should be covered in full, with the exception of the material on band-gap references and the material in the appendices. Chapter 5 is concerned with output stages and methods of delivering output power to a load. Integrated-circuit realizations of Class A, Class B, and Class AB output stages are described, as well as methods of output-stage protection. A selection of topics from this chapter should be covered. Chapter 6 deals with the design of operational amplifiers (op amps). Illustrative exam- ples of dc and ac analysis in both MOS and bipolar op amps are performed in detail, and the limitations of the basic op amps are described. The design of op amps with improved
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Preface ixcharacteristics in both MOS and bipolar technologies is considered. This key chapter onamplifier design requires at least six hours. In Chapter 7, the frequency response of amplifiers is considered. The zero-value time-constant technique is introduced for the calculations of the -3-dB frequency of complexcircuits. The material of this chapter should be considered in full. Chapter 8 describes the analysis of feedback circuits. Two different types of analysisare presented: two-port and return-ratio analyses. Either approach should be covered infull with the section on voltage regulators assigned as reading. Chapter 9 deals with the frequency response and stability of feedback circuits andshould be covered up to the section on root locus. Time may not pennit a detailed discussionof root locus, but some introduction to this topic can be given. In a 15-week semester, coverage of the above material leaves about two weeks forChapters 10, 11, and 12. A selection of topics from these chapters can be chosen as follows.Chapter 10 deals with nonlinear analog circuits, and portions of this chapter up to Section10.3 could be covered in a first course. Chapter 11 is a comprehensive treatment of noisein integrated circuits, and material up to and including Section 11.4 is suitable. Chapter 12describes fully differential operational amplifiers and common-mode feedback and maybe best suited for a second course. We are grateful to the following colleagues for their suggestions for andlor eval-uation of this edition: R. Jacob Baker, Bemhard E. Boser, A. Paul Brokaw, John N.Churchill, David W. Cline, Ozan E. Erdogan, John W. Fattaruso, Weinan Gao, Edwin W.Greeneich, Alex Gros-Balthazard, Tiinde Gyurics, Ward J. Helms, Timothy H. Hu, ShafiqM. Jamal, John P. Keane, Haideh Khorramabadi, Pak-Kim Lau, Thomas W. Matthews,Krishnaswamy Nagaraj, Khalil Najafi, Borivoje NikoliC, Robert A. Pease, Lawrence T.Pileggi, Edgar Shnchez-Sinencio, Bang-Sup Song, Richard R. Spencer, Eric J. Swanson,Andrew Y. J. Szeto, Yannis P. Tsividis, Srikanth Vaidianathan, T. R. Viswanathan, Chomg-Kuang Wang, and Dong Wang. We are also grateful to Kenneth C. Dyer for allowing us touse on the cover of this book a die photograph of an integrated circuit he designed and toZoe Marlowe for her assistance with word processing. Finally, we would like to thank thepeople at Wiley and Publication Services for their efforts in producing this fourth edition. The material in this book has been greatly influenced by our association with Donald0. Pederson, and we acknowledge his contributions.Berkeley and Davis, CA, 2001 Paul R. Gray Paul J. Hurst Stephen H. Lewis Robert G. Meyer
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ContentsCHAPTER 1 1.5.2 Comparison of Operating RegionsModels for Integrated-Circuit Active of Bipolar and MOS TransistorsDevices 1 45 1.5.3 Decomposition of Gate-Source1.1 Introduction 1 Voltage 471.2 Depletion Region of a pn Junction 1 1.5.4 Threshold Temperature Dependence 47 1.2.1 Depletion-Region Capacitance 5 1.5.5 MOS Device Voltage Limitations 1.2.2 JunctionBreakdown 6 481.3 Large-Signal Behavior of Bipolar 1.6 Small-Signal Models of the MOS Transistors 8 /Transistors 49 1.3.1 Large-Signal Models in the 1.6.1 Transconductance 50 Forward-Active Region 9 1.6.2 Intrinsic Gate-Source and 1.3.2 Effects of Collector Voltage on Gate-Drain Capacitance 51 Large-Signal Characteristics in the Forward-Active Region 14 1.6.4 Output Resistance 52 1.3.3 Saturation and Inverse Active Regions 16 1.6.5 Basic Small-Signal Model of the MOS Transistor 52 1.3.4 Transistor Breakdown Voltages 20 1.6.6 Body Transconductance 53 1.3.5 Dependence of Transistor Current 1.6.7 Parasitic Elements in the Gain PF on Operating Conditions Small-Signal Model 54 23 1.6.8 MOS Transistor Frequency1.4 Small-Signal Models of Bipolar Response 55 Transistors 26 1.7 Short-Channel Effects in MOS 1.4.1 Transconductance 27 Transistors 58 1.4.2 Base-Charging Capacitance 28 1.7.1 Velocity Saturation from the Horizontal Field 59 1.4.3 Input Resistance 29 1.7.2 Transconductance and Transition 1.4.4 Output Resistance 29 Frequency 63 1.4.5 Basic Small-Signal Model of the 1.7.3 Mobility Degradation from the Bipolar Transistor 30 Vertical Field 65 1.4.6 Collector-Base Resistance 30 1.8 Weak Inversion in MOS Transistors 1.4.7 Parasitic Elements in the 65 Small-Signal Model 3 1 1.8.1 Drain Current in Weak Inversion 1.4.8 Specification of Transistor 66 Frequency Response 34 1.8.2 Transconductance and Transition1.5 Large Signal Behavior of Frequency in Weak Inversion 68 Metal-Oxide-Semiconductor 1.9 Substrate Current Flow in MOS Field-Effect Transistors 38 Transistors 7 1 1.5.1 Transfer Characteristics of MOS A. 1.1 Summary of Active-Device Devices 38 Parameters 73
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Contents xi CHAPTER 2 2.9.1 n-Channel Transistors 131 Bipolar, MOS, and BiCMOS Integrated-Circuit Technology 78 2.9.2 p-Channel Transistors 141 2.1 Introduction 78 2.9.3 Depletion Devices 142 2.2 Basic Processes in Integrated-Circuit 2.9.4 Bipolar Transistors 142 Fabrication 79 2.10 Passive Components in MOS 2.2.1 Electrical Resistivity of Silicon Technology 144 79 2.10.1 Resistors 144 2.2.2 Solid-State Diffusion 80 2.10.2 Capacitors in MOS Technology 2.2.3 Electrical Properties of Diffused 145 Layers 82 2.10.3 Latchup in CMOS Technology 2.2.4 Photolithography 84 148 2.2.5 Epitaxial Growth 85 2.11 BiCMOS Technology 150 2.2.6 Ion Implantation 87 2.12 Heterojunction Bipolar Transistors 152 2.2.7 Local Oxidation 87 2.13 Interconnect Delay 153 2.2.8 Polysilicon Deposition 87 2.14 Economics of Integrated-Circuit 2.3 High-Voltage Bipolar Fabrication 154 Integrated-Circuit Fabrication 88 2.14.1 Yield Considerations in 2.4 Advanced Bipolar Integrated-Circuit Integrated-Circuit Fabrication Fabrication 92 154 2.5 Active Devices in Bipolar Analog 2.14.2 Cost Considerations in Integrated Circuits 95 Integrated-Circuit Fabrication 2.5.1 Integrated-Circuit npn Transistor 157 96 2.15 Packaging Considerations for 2.5.2 Integrated-Circuit pnp Transistors Integrated Circuits 159 107 2.15.1 Maximum Power Dissipation 159 2.6 Passive Components in Bipolar 2.15.2 Reliability Considerations inI Integrated Circuits 115 2.6.1 Diffused Resistors 115 Integrated-Circuit Packaging 162 A.2.1 SPICE Model-Parameter Files 163 2.6.2 Epitaxial and Epitaxial Pinch Resistors 119 CHAPTER 3 2.6.3 Integrated-Circuit Capacitors 120 Single-Transistor and Multiple-Transistor 2.6.4 Zener Diodes 121 Amplifiers 170 2.6.5 Junction Diodes 122 3.1 Device Model Selection forI 2.7 Modifications to the Basic Bipolar Approximate Analysis of AnalogI Process 123 Circuits 171 2.7.1 Dielectric Isolation 123 3.2 Two-Port Modeling of Amplifiers 172 2.7.2 Compatible Processing for 3.3 Basic Single-Transistor Amplifier High-Performance Active Devices 124 Stages 174 2.7.3 High-Performance Passive 3.3.1 Common-Emitter Configuration Components 127 175 2.8 MOS Integrated-Circuit Fabrication 3.3.2 Common-Source Configuration 127 179 3.3.3 Common-Baseconfiguration 183 2.9 Active Devices in MOS Integrated 4 Circuits 131 3.3.4 Common-Gate Configuration 186
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xii Contents 3.3.5 Common-Base and Common-Gate 3.5.6.5 Input Offset Current of Configurations with Finite r, 188 the Emitter-Coupled Pair 3.3.5.1 Common-Base and 235 Common-Gate Input 3.5.6.6 Input Offset Voltage of the Resistance 188 Source-Coupled Pair 236 3.3.5.2 Common-Base and 3.5.6.7 Offset Voltage of the Common-Gate Output Source-Coupled Pair: Ap- Resistance 190 proximate Analysis 236 3.3.6 Common-Collector Configuration 3.5.6.8 Offset Voltage Drift in the (Emitter Follower) 191 Source-Coupled Pair 238 3.3.7 Common-Drain Configuration 3.5.6.9 Small-Signal (Source Follower) 195 Characteristics of 3.3.8 Common-Emitter Amplifier with Unbalanced Differential Emitter Degeneration 197 Amplifiers 238 3.3.9 Common-Source Amplifier with A.3.1 Elementary Statistics and the Source Degeneration 200 Gaussian Distribution 2463.4 Multiple-Transistor Amplifier Stages 202 CHAPTER 4 3.4.1 The CC-CE, CC-CC, and Current Mirrors, Active Loads, and Darlington Configurations 202 References 253 3.4.2 The Cascode Configuration 206 4.1 Introduction 253 3.4.2.1 The Bipolar Cascode 206 4.2 Current Mirrors 253 3.4.2.2 The MOS Cascode 208 4.2.1 General Properties 253 3.4.3 The Active Cascode 211 4.2.2 Simple Current Mirror 255 3.4.4 The Super Source Follower 213 4.2.2.1 Bipolar 2553.5 Differential Pairs 215 4.2.2.2 MOS 257 3.5.1 The dc Transfer Characteristic of 4.2.3 Simple Current Mirror with Beta an Emitter-Coupled Pair 2 15 Helper 260 3.5.2 The dc Transfer Characteristic with 4.2.3.1 Bipolar 260 Emitter Degeneration 2 17 4.2.3.2 MOS 262 3.5.3 The dc Transfer Characteristic of a 4.2.4 Simple Current Mirror with Source-CoupledPair 2 18 Degeneration 262 3.5.4 Introduction to the Small-Signal 4.2.4.1 Bipolar 262 Analysis of Differential Amplifiers 22 1 4.2.4.2 MOS 263 3.5.5 Small-Signal Characteristics of 4.2.5 Cascode Current Mirror 263 Balanced Differential Amplifiers 224 4.2.5.1 Bipolar 263 3.5.6 Device Mismatch Effects in 4.2.5.2 MOS 266 Differential Amplifiers 23 1 4.2.6 Wilson Current Mirror 274 3.5.6.1 Input Offset Voltage and Current 23 1 4.2.6.1 Bipolar 274 3.5.6.2 Input Offset Voltage of 4.2.6.2 MOS 277 the Emitter-Coupled Pair 232 4.3 Active Loads 278 3.5.6.3 Offset Voltage of the 4.3.1 Motivation 278 Emitter-Coupled Pair: 4.3.2 Common-Emitter/Common-Source ~~~roximate Analysis Amplifier with Complementary 232 Load 279 3.5.6.4 Offset Voltage Drift in 4.3.3 Common-Emitter/Common-Source the Emitter-Coupled Pair Amplifier with Depletion Load 234 282
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Contents xiii 4.3.4 Common-Emitter/Common-Source 5.2 The Emitter Follower As an Output Amplifier with Diode-Connected Stage 344 Load 284 5.2.1 Transfer Characteristics of the 4.3.5 Differential Pair with Emitter-Follower 344 Current-Mirror Load 287 4.3.5.1 Large-Signal Analysis 5.2.2 Power Output and Efficiency 347 287 5.2.3 Emitter-Follower Drive 4.3.5.2 Small-Signal Analysis Requirements 354 288 5.2.4 Small-Signal Properties of the 4.3.5.3 Common-Mode Rejection Emitter Follower 355 Ratio 293 5.3 The Source Follower As an Output4.4 Voltage and Current References 299 Stage 356 4.4.1 Low-Current Biasing 299 5.3.1 Transfer Characteristics of the 4.4.1.1 Bipolar Widlar Current Source Follower 356 Source 299 5.3.2 Distortion in the Source Follower 4.4.1.2 MOS Widlar Current 358 Source 302 4.4.1.3 Bipolar Peaking Current 5.4 Class B Push-Pull Output Stage 362 Source 303 5.4.1 Transfer Characteristic of the 4.4.1.4 MOS Peaking Current Class B Stage 363 Source 304 5.4.2 Power Output and Efficiency of the 4.4.2 Supply-Insensitive Biasing 306 Class B Stage 365 4.4.2.1 Widlar Current Sources 5.4.3 Practical Realizations of Class B 306 Complementary Output Stages 4.4.2.2 Current Sources Using 369 Other Voltage Standards 5.4.4 All-npn Class B Output Stage 307 376 4.4.2.3 Self Biasing 309 5.4.5 Quasi-Complementary Output 4.4.3 Temperature-Insensitive Biasing Stages 379 317 5.4.6 Overload Protection 3 80 4.4.3.1 Band-Gap-Referenced Bias Circuits in Bipolar 5.5 CMOS Class AB Output Stages 382 Technology 3 17 5.5.1 Common-Drain Configuration 4.4.3.2 Band-Gap-Referenced 3 83 Bias Circuits in CMOS 5.5.2 Common-Source Configuration Technology 323 with Error Amplifiers 384A.4.1 Matching Considerations in Current 5.5.3 Alternative Configurations 391 Mirrors 327 5.5.3.1 Combined Common-Drain Common-Source A.4.1.1 Bipolar 327 Configuration 39 1 A.4.1.2 MOS 329 5.5.3.2 Combined Common-Drain Common-SourceA.4.2 Input Offset Voltage of Configuration with High Differential Pair with Swing 393 Active Load 332 5 S.3.3 Parallel Common-Source A.4.2.1 Bipolar 332 Configuration 394 A.4.2.2 MOS 334 CHAPTER 6 Operational Amplifiers with CHAPTER 5 Single-Ended Outputs 404 Output Stages 344 6.1 Applications of Operational Amplifiers 5.1 Introduction 344 405
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xiv Contents 6.6 MOS Folded-Cascode Operational 6.1.1 Basic Feedback Concepts 405 Amplifiers 446 6.1.2 Inverting Amplifier 406 6.7 MOS Active-Cascode Operational 6.1.3 Noninverting Amplifier 408 Amplifiers 450 6.1.4 Differential Amplifier 408 6.8 Bipolar Operational Amplifiers 453 6.1.5 Nonlinear Analog Operations 409 6.8.1 The dc Analysis of the 741 6.1.6 Integrator, Differentiator 4 10 Operational Amplifier 456 6.1.7 Internal Amplifiers 41 1 6.8.2 Small-Signal Analysis of the 741 6.1.7.1 Switched-Capacitor Operational Amplifier 461 Amplifier 411 6.8.3 Input Offset Voltage, Input 6.1.7.2 Switched-Capacitor Offset Current, and Integrator 416 Common-Mode Rejection Ratio of the 741 4706.2 Deviations from Ideality in Real Oper- 6.9 Design Considerations for Bipolar ational Amplifiers 419 Monolithic Operational Amplifiers 6.2.1 Input Bias Current 419 472 6.2.2 Input Offset Current 420 6.9.1 Design of Low-Drift Operational 6.2.3 Input Offset Voltage 421 Amplifiers 474 / -,/6.2.4 Common-Mode Input Range 421 6.9.2 Design of Low-Input-Current Operational Amplifiers 476 $.2.5 Common-Mode Rejection Ratio / (CMRR) 421 / $2.6 Power-Supply Rejection Ratio CHAPTER 7 I 1/ (PSRR) 422 Frequency Response of Integrated 6.2.7 Input Resistance 424 Circuits 488 6.2.8 Output Resistance 424 7.1 Introduction 488 6.2.9 Frequency Response 424 7.2 Single-Stage Amplifiers 488 6.2.10 Operational-Amplifier Equivalent 7.2.1 Single-Stage Voltage Amplifiers Circuit 424 and The Miller Effect 4886.3 Basic Two-Stage MOS Operational 7.2.1.1 The Bipolar Differential Amplifiers 425 Amplifier: Differential- Mode Gain 493 6.3.1 Input Resistance, Output 7.2.1.2 The MOS Differential Resistance, and Open-Circuit Amplifier: Differential- Voltage Gain 426 Mode Gain 496 6.3.2 Output Swing 428 7.2.2 Frequency Response of the Common-Mode Gain for a 6.3.3 Input Offset Voltage 428 Differential Amplifier 499 6.3.4 Common-Mode Rejection Ratio 7.2.3 Frequency Response of Voltage 431 Buffers 502 6.3.5 Common-Mode Input Range 432 7.2.3.1 Frequency Response of the 6.3.6 Power-Supply Rejection Ratio Emitter Follower 503 (PSRR) 434 7.2.3.2 Frequency Response of the 6.3.7 Effect of Overdrive Voltages 439 Source Follower 509 7.2.4 Frequency Response of Current 6.3.8 Layout Considerations 439 Buffers 511 6.4 Two-Stage MOS Operational 7.2.4.1 Common-Base-Amplifier Amplifiers with Cascodes 442 Frequency Response 5 14 7.2.4.2 Common-Gate-Amplifier 6.5 MOS Telescopic-Cascode Operational Frequency Response 5 15 Amplifiers 444
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Contents xv7.3 Multistage Amplifier Frequency 8.6.2 Local Shunt Feedback 591 Response 516 8.7 The Voltage Regulator as a Feedback 7.3.1 Dominant-Pole Approximation Circuit 593 5 16 7.3.2 Zero-Value Time Constant 8.8 Feedback Circuit Analysis Using Analysis 517 Return Ratio 599 7.3.3 Cascode Voltage-Amplifier 8.8.1 Closed-Loop Gain Using Return Frequency Response 522 Ratio 601 7.3.4 Cascode Frequency Response 8.8.2 Closed-Loop Impedance Formula 525 Using Return Ratio 607 7.3.5 Frequency Response of a Current 8.8.3 Summary-Return-Ratio Analysis Mirror Loading a Differential Pair 612 532 8.9 Modeling Input and Output Ports in 7.3.6 Short-Circuit Time Constants 533 Feedback Circuits 6 137.4 Analysis of the Frequency Response of the 741 Op Amp 537 CHAPTER 9 7.4.1 High-Frequency Equivalent Circuit Frequency Response and Stability of of the741 537 Feedback Amplifiers 624 7.4.2 Calculation of the -3-dB Frequency of the 741 538 9.1 Introduction 624 7.4.3 Nondominant Poles of the 741 9.2 Relation Between Gain and 540 Bandwidth in Feedback Amplifiers7.5 Relation Between Frequency 624 Response and Time Response 542 9.3 Instability and the Nyquist Criterion 626CHAPTER 8 9.4 Compensation 633Feedback 553 9.4.1 Theory of Compensation 6338.1 Ideal Feedback Equation 553 9.4.2 Methods of Compensation 6378.2 Gain Sensitivity 555 9.4.3 Two-Stage MOS Amplifier8.3 Effect of Negative Feedback on Compensation 644 Distortion 555 9.4.4 Compensation of Single-Stage8.4 Feedback Configurations 557 CMOS OP Amps 652 8.4.1 Series-Shunt Feedback 557 9.4.5 Nested Miller Compensation 656 8.4.2 Shunt-Shunt Feedback 560 9.5 Root-Locus Techniques 664 8.4.3 Shunt-SeriesFeedback 561 9.5.1 Root Locus for a Three-Pole 8.4.4 Series-Series Feedback 562 Transfer Function 664 9.5.2 Rules for Root-Locus Construction8.5 Practical Configurations and the Effect 667 of Loading 563 9.5.3 Root Locus for Dominant-Pole 8.5.1 Shunt-Shunt Feedback 563 Compensation 675 8.5.2 Series-Series Feedback 569 9.5.4 Root Locus for Feedback-Zero Compensation 676 8.5.3 Series-Shunt Feedback 579 9.6 Slew Rate 680 8.5.4 Shunt-Series Feedback 583 9.6.1 Origin of Slew-Rate Limitations 8.5.5 Summary 587 6808.6 Single-Stage Feedback 587 9.6.2 Methods of Improving Slew-Rate 8.6.1 Local Series Feedback 587 684
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xvi Contents 9.6.3 Improving Slew-Rate in Bipolar 11.2.1 Shot Noise 748 Op Amps 685 9.6.4 Improving Slew-Rate in MOS Op 11.2.2 Thermal Noise 752 Amps 686 11.2.3 Flicker Noise (11f Noise) 753 9.6.5 Effect of Slew-Rate Limitations on 11.2.4 Burst Noise (Popcorn Noise) 754 Large-Signal Sinusoidal Performance 690 11.2.5 Avalanche Noise 755A.9.1 Analysis in Terms of Return-Ratio 11.3 Noise Models of Integrated-Circuit Parameters 69 1 Components 756 11.3.1 Junction Diode 756A.9.2 Roots of a Quadratic Equation 692 11.3.2 Bipolar Transistor 757 11.3.3 MOS Transistor 758CHAPTER 10 11.3.4 Resistors 759Nonlinear Analog Circuits 702 11.3.5 Capacitors and Inductors 75910.1 Introduction 70210.2 Precision Rectification 702 11.4 Circuit Noise Calculations 760 11-4.1 Bipolar Transistor Noise10.3 Analog Multipliers Employing the Performance 762 Bipolar Transistor 708 11.4.2 Equivalent Input Noise and the 10.3.1 The Emitter-Coupled Pair as a Minimum Detectable Signal 766 Simple Multiplier 708 11.5 Equivalent Input Noise Generators 10.3.2 The dc Analysis of the Gilbert 768 Multiplier Cell 7 10 11.5.1 Bipolar Transistor Noise 10.3.3 The Gilbert Cell as an Analog Generators 768 Multiplier 7 12 11.5.2 MOS Transistor Noise Generators 10.3.4 A Complete Analog Multiplier 773 715 11.6 Effect of Feedback on Noise 10.3.5 The Gilbert Multiplier Cell as a Performance 776 Balanced Modulator and Phase Dectector 7 16 11.6.1 Effect of Ideal Feedback on Noise Performance 77610.4 Phase-Locked Loops (PLL) 720 11.6.2 Effect of Practical Feedback on 10.4.1 Phase-Locked Loop Concepts Noise Performance 776 720 11.7 Noise Performance of Other Transistor 10.4.2 The Phase-Locked Loop in the Configurations 7 83 Locked Condition 722 10.4.3 Integrated-Circuit Phase-Locked 11.7.1 Common-Base Stage Noise Loops 731 Performance 783 10.4.4 Analysis of the 560B Monolithic 11.7.2 Emitter-Follower Noise Phase-Locked Loop 735 Performance 784 11.7.3 Differential-Pair Noise10.5 Nonlinear Function Symbols 743 Performance 785 11.8 Noise in Operational Amplifiers 788CHAPTER 1 1 11.9 Noise Bandwidth 794Noise in Integrated Circuits 748 11.10 Noise Figure and Noise Temperature11.1 Introduction 748 79911.2 Sources of Noise 748 11.lo. 1 Noise Figure 799 11.10.2 Noise Temperature 802
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Contents xvii CHAPTER 12 12.5.3 CMFB Using Transistors in the Fully DifferentialOperational Amplifiers Triode Region 830 808 12.5.4 Switched-CapacitorCMFB 832 12.1 Introduction 808 12.6 Fully Differential Op Amps 835 12.2 Properties of Fully Differential 12.6.1 A Fully Differential Two-Stage Op Amplifiers 808 Amp 835 12.3 Small-Signal Models for Balanced 12.6.2 Fully Differential Telescopic Differential Amplifiers 811 Cascode Op Amp 845 12.6.3 Fully Differential Folded-CascodeI 12.4 Common-Mode Feedback 8 16 Op Amp 846 12.4.1 Common-Mode Feedback at Low 12.6.4 A Differential Op Amp with Two Frequencies 8 17 Differential Input Stages 847 12.4.2 Stability and Compensation 12.6.5 Neutralization 849 Considerations in a CMFB Loop 822 12.7. Unbalanced Fully Differential Circuits 850 12.5 CMFB Circuits 823 12.5.1 CMFB Using Resistive Divider 12.8 Bandwidth of the CMFB Loop 856 and Amplifier 824 12.5.2 CMFB Using Two Differential Index 865 Pairs 828
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xviii Symbol Convention Symbol Convention Unless otherwise stated, the following symbol convention is used in this book. Bias or dc quantities, such as transistor collector current Ic and collector-emitter voltage V C Eare , represented by uppercase symbols with uppercase subscripts. Small-signal quantities, such as the incremental change in transistor collector current i,, are represented by lowercase symbols with lowercase subscripts. Elements such as transconductance g, in small-signal equivalent circuits are represented in the same way. Finally, quantities such as total collector current I,, which represent the sum of the bias quantity and the signal quantity, are represented by an uppercase symbol with a lowercase subscript.
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Modelsfor Integrated-CircuitActive Devices1.1 IntroductionThe analysis and design of integrated circuits depend heavily on the utilization of suitablemodels for integrated-circuit components. This is true in hand analysis, where fairly simplemodels are generally used, and in computer analysis, where more complex models areencountered. Since any analysis is only as accurate as the model used, it is essential thatthe circuit designer have a thorough understanding of the origin of the models commonlyutilized and the degree of approximation involved in each. This chapter deals with the derivation of large-signal and small-signal models forintegrated-circuit devices. The treatment begins with a consideration of the properties ofpn junctions, which are basic parts of most integrated-circuit elements. Since this book isprimarily concerned with circuit analysis and design, no attempt has been made to producea comprehensive treatment of semiconductor physics. The emphasis is on summarizing thebasic aspects of semiconductor-device behavior and indicating how these can be modeledby equivalent circuits.1.2 Depletion Region of a pn JunctionThe properties of reverse-biased pn junctions have an important influence on the charac-teristics of many integrated-circuit components. For example, reverse-biased p n junctionsexist between many integrated-circuit elements and the underlying substrate, and thesejunctions all contribute voltage-dependent parasitic capacitances. In addition, a numberof important characteristics of active devices, such as breakdown voltage and output re-sistance, depend directly on the properties of the depletion region of a reverse-biased pnjunction. Finally, the basic operation of the junction field-effect transistor is controlled bythe width of the depletion region of a p n junction. Because of its importance and applica-tion to many different problems, an analysis of the depletion region of a reverse-biased p njunction is considered below. The properties of forward-biased pn junctions are treated in Section 1.3 when bipolar-transistor operation is described. Consider a pn junction under reverse bias as shown in Fig. 1.1. Assume constantdoping densities of ND atoms/cm3 in the n-type material and NA atoms/cm3 in the p-type material. (The characteristics of junctions with nonconstant doping densities will be described later.) Due to the difference in carrier concentrations in the p-type and n-typeregions, there exists a region at the junction where the mobile holes and electrons havebeen removed, leaving the fixed acceptor and donor ions. Each acceptor atom carries anegative charge and each donor atom carries a positive charge, so that the region near thejunction is one of significant space charge and resulting high electric field. This is called
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2 Chapter 1 Models for Integrated-Circuit Active Devices VR ),( Applied external Charge density p t reverse bias I Potential 4 I I Figure 1.1 The abrupt junc- tion under reverse bias V R .( a ) Schematic. (b)Charge density. (c) Electric field. (d) Electro- static potential. the depletion region or space-charge region. It is assumed that the edges of the depletion region are sharply defined as shown in Fig. 1.1, and this is a good approximation in most cases. For zero applied bias, there exists a voltage $0 across the junction called the built-in potential. This potential opposes the diffusion of mobile holes and electrons across the junction in equilibrium and has a value1 where the quantity ni is the intrinsic carrier concentration in a pure sample of the semiconductor and ni = 1.5 X 1 0 ~ c m -at 300°K for silicon. ~ In Fig. 1.1 the built-in potential is augmented by the applied reverse bias, V R ,and the + total voltage across the junction is ($0 V R ) . the depletion region penetrates a distance If W1 into the p-type region and Wz into the n-type region, then we require because the total charge per unit area on either side of the junction must be equal in mag- nitude but opposite in sign.
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1.2 Depletion Region of a pn Junction 3 Poissons equation in one dimension requires that =2 v- - = - q N ~ for d - wl < x < o dx2 E Ewhere p is the charge density, q is the electron charge (1.6 X 10-l9 coulomb), and E is thepermittivity of the silicon (1.04 X 10-l2 faradcm). The permittivity is often expressed aswhere Ks is the dielectric constant of silicon and € 0 is the permittivity of free space (8.86 Xl o w 4 Flcm). Integration of (1.3) giveswhere C1 is a constant. However, the electric field % is given bySince there is zero electric field outside the depletion region, a boundary condition is . % =0 for x = -W1and use of this condition in (1.6) gives 5 dx for -Thus the dipole of charge existing at the junction gives rise to an electric field that varieslinearly with distance. Integration of (1.7) givesIf the zero for potential is arbitrarily taken to be the potential of the neutral p-type region,then a second boundary condition is V = O for x = -W1and use of this in (1.8) givesAt x = 0, we define V = V 1 ,and then (1.9) givesIf the potential difference from x = 0 to x = W 2 is V2,then it follows thatand thus the total voltage across the junction is
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4 Chapter 1 . Models for Integrated-CircuitActive Devices Substitution of (1.2) in (1.12) gives From (1.13), the penetration of the depletion layer into the p-type region is Similarly Equations 1.14 and 1.15 show that the depletion regions extend into the p-type and n-type regions in inverse relation to the impurity concentrations and in proportion to d m . If either No or N A is much larger than the other, the depletion region exists almost entirely in the lightly doped region. EXAMPLE An abrupt pn junction in silicon has doping densities N A = 1015 atoms/cm3 and ND = 1016 atoms/cm3. Calculate the junction built-in potential, the depletion-layer depths, and the maximum field with 10 V reverse bias. From (1.1) From (1.14) the depletion-layer depth in the p-type region is = 3.5 p m (where 1 p m = 1 micrometer = m) The depletion-layer depth in the more heavily doped n-type region is Finally, from (1.7) the maximum field that occurs for x = 0 is Note the large magnitude of this electric field.
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1.2 Depletion Region of a pn Junction 51.2.1 Depletion-Region CapacitanceSince there is a voltage-dependent charge Q associated with the depletion region, we cancalculate a small-signal capacitance C, as follows:Nowwhere A is the cross-sectional area of the junction. Differentiation of (1.14) givesUse of (1.17) and (1.18) in (1.16) gives The above equation was derived for the case of reverse bias VR applied to the diode.However, it is valid for positive bias voltages as long as the forward current flow is small.Thus, if VD represents the bias on the junction (positive for forward bias, negative forreverse bias), then (1.19) can be written aswhere Cjo is the value of Cj for VD = 0. Equations 1.20 and 1.21 were derived using the assumption of constant doping inthe p-type and n-type regions. However, many practical diffused junctions more closelyapproach a graded doping profile as shown in Fig. 1.2. In this case a similar calculationyieldsNote that both (1.21) and (1.22) predict values of Cj approaching infinity as VD ap-proaches $0. However, the current flow in the diode is then appreciable and the equationsno longer valid. A more exact analysis2v3of the behavior of Cj as a function of VD givesthe result shown in Fig. 1.3. For forward bias voltages up to about $012, the values of Cjpredicted by (1.21) are very close to the more accurate value. As an approximation, somecomputer programs approximate Cj for VD > $012 by a linear extrapolation of (1.21) or(1.22).
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6 Chapter 1 rn Models for Integrated-Circuit Active Devices Charge density p p=m . - . I FX Distance ,.-- ..-I .I Figure 1.2 Charge density versus dis- tance in a graded junction. Reverse bias 1 Foward bias Figure 1.3 Behavior of p n junction depletion-layer capacitance C j as a function of bias voltage V D .rn EXAMPLE If the zero-bias capacitance of a diffused junction is 3 pF and $o = 0.5 V, calculate the capacitance with 10 V reverse bias. Assume the doping profile can be approximated by an abrupt junction. From (1.21) 2 1.2.2 Junction Breakdown From Fig. 1.lc it can be seen that the maximum electric field in the depletion region occurs at the junction, and for an abrupt junction (1.7) yields a value
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1.2 Depletion Region of a pn Junction 7Substitution of (1.14) in (1.23) gives I- max - [ ~ ~ N A N D 1V" R E (NA + No)where I!,~ has been neglected. Equation 1.24 shows that the maximum field increases asthe doping density increases and the reverse bias increases. Although useful for indicat-ing the functional dependence of %,, on other variables, this equation is strictly validfor an ideal plane junction only. Practical junctions tend to have edge effects that causesomewhat higher values of %,, due to a concentration of the field at the curved edgesof the junction. Any reverse-biased pn junction has a small reverse current flow due to the presenceof minority-carrier holes and electrons in the vicinity of the depletion region. These areswept across the depletion region by the field and contribute to the leakage current of thejunction. As the reverse bias on the junction is increased, the maximum field increases andthe carriers acquire increasing amounts of energy between lattice collisions in the depletionregion. At a critical field %,., the carriers traversing the depletion region acquire sufficientenergy to create new hole-electron pairs in collisions with silicon atoms. This is called theavalanche process and leads to a sudden increase in the reverse-bias leakage current sincethe newly created carriers are also capable of producing avalanche. The value of ,% is ,, about 3 X lo5 V/cm for junction doping densities in the range of 1015 to 1016 atoms/cm3,but it increases slowly as the doping density increases and reaches about lo6 V/cm for doping densities of 1018 atoms/cm3. A typical I-V characteristic for a junction diode is shown in Fig. 1.4, and the effect of avalanche breakdown is seen by the large increase in reverse current, which occurs as the reverse bias approaches the breakdown voltage BV. This corresponds to the maximum field %,, approaching %". It has been found empirically4 that if the normal reverse bias ,, current of the diode is IR with no avalanche effect, then the actual reverse current near thebreakdown voltage is Figure 1.4 Typical I-V characteristic of a junction diode showing avalanche breakdown.
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8 Chapter 1 . Models for Integrated-CircuitActive Devices where M is the multiplication factor defined by In this equation, V R is the reverse bias on the diode and n has a value between 3 and 6. The operation of a pn junction in the breakdown region is not inherently destruc- tive. However, the avalanche current flow must be limited by external resistors in order to prevent excessive power dissipation from occurring at the junction and causing dam- age to the device. Diodes operated in the avalanche region are widely used as voltage references and are called Zener diodes. There is another, related process called Zener breakdown, which is different from the avalanche breakdown described above. Zener breakdown occurs only in very heavily doped junctions where the electric field becomes large enough (even with small reverse-bias voltages) to strip electrons away from the valence bonds. This process is called tunneling, and there is no multiplication effect as in avalanche breakdown. Although the Zener breakdown mechanism is important only for breakdown voltages below about 6 V, all breakdown diodes are commonly referred to as Zener diodes. The calculations so far have been concerned with the breakdown characteristic of plane abrupt junctions. Practical diffused junctions differ in some respects from these results and the characteristics of these junctions have been calculated and tabulated for use by designers. In particular, edge effects in practical diffused junctions can result in breakdown voltages as much as 50 percent below the value calculated for a plane junction. EXAMPLE An abrupt plane pn junction has doping densities NA = 5 X 10 atoms/cm3 and ND = 1016 atoms/cm3. Calculate the breakdown voltage if %,*, = 3 X 10 Vlcm. The breakdown voltage is calculated using, % , , = % in (1.24) to give * , 1.3 Large-Signal Behavior of Bipolar Transistors In this section, the large-signal or dc behavior of bipolar transistors is considered. Large- signal models are developed for the calculation of total currents and voltages in transistor circuits, and such effects as breakdown voltage limitations, which are usually not included in models, are also considered. Second-order effects, such as current-gain variation with collector current and Early voltage, can be important in many circuits and are treated in detail. The sign conventions used for bipolar transistor currents and voltages are shown in Fig. 1.5. All bias currents for both npn and pnp transistors are assumed positive going into the device.
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1.3 Large-Signal Behavior of Bipolar Transistors 9 -A VBE E Figure 1.5 Bipolar transistor sign PnP convention.1.3.1 Large-SignalModels in the Forward-ActiveRegionA typical npn planar bipolar transistor structure is shown in Fig. 1.6a, where collector,base, and emitter are labeled C , B, and E, respectively. The method of fabricating suchtransistor structures is described in Chapter 2. It is shown there that the impurity dopingdensity in the base and the emitter of such a transistor is not constant but varies withdistance from the top surface. However, many of the characteristics of such a device canbe predicted by analyzing the idealized transistor structure shown in Fig. 1.6b. In thisstructure the base and emitter doping densities are assumed constant, and this is sometimescalled a uniform-base transistor. Where possible in the following analyses, the equationsfor the uniform-base analysis are expressed in a form that applies also to nonuniform-basetransistors. A cross section AA is taken through the device of Fig. 1.6b and carrier concentrationsalong this section are plotted in Fig. 1 . 6 ~Hole concentrations are denoted by p and elec- .tron concentrations by n with subscripts p or n representing p-type or n-type regions. Then-type emitter and collector regions are distinguished by subscripts E and C, respectively.The carrier concentrations shown in Fig. 1 . 6 ~ apply to a device in the forward-active re-gion. That is, the base-emitter junction is forward biased and the base-collector junction isreverse biased. The minority-carrier concentrations in the base at the edges of the depletionregions can be calculated from a Boltzmann approximation to the Fermi-Dirac distributionfunction to give6 VBE np(0) = npoexp - VT np(WB) = npoexp - VBC 0 VT -where WB is the width of the base from the base-emitter depletion layer edge to the base-collector depletion layer edge and n,, is the equilibrium concentration of electrons in thebase. Note that VBC is negative for an npn transistor in the forward-active region andthus n,(WB) is very small. Low-level injection conditions are assumed in the derivation of(1.27) and (1.28). This means that the minority-carrier concentrations are always assumedmuch smaller than the majority-carrier concentration. If recombination of holes and electrons in the base is small, it can be shown that7the minority-carrier concentration np(x) in the base varies linearly with distance. Thus astraight line can be drawn joining the concentrations at x = 0 and x = WB in Fig. 1 . 6 ~ . For charge neutrality in the base, it is necessary that
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10 Chapter 1 a Models for Integrated-Circuit Active Devices , Carrier concentration ! I Depletion region II I A f e x A Y=o Emitter Base Collector (4 Figure 1.6 (a) Cross section of a typical npn planar bipolar transistor structure. (b) Idealized tran- sistor structure. (c) Carrier concentrations along the cross section AA of the transistor in (b). Uni- form doping densities are assumed. (Not to scale.) and thus where p,(x) is the hole concentration in the base and NA is the base doping density that is assumed constant. Equation 1.30 indicates that the hole and electron concentrations are separated by a constant amount and thus p p ( x ) also varies linearly with distance. Collector current is produced by minority-carrier electrons in the base diffusing in the direction of the concentration gradient and being swept across the collector-base depletion region by the field existing there. The diffusion current density due to electrons in the base is
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1.3 Large-Signal Behavior of Bipolar Transistors 11where D, is the diffusion constant for electrons. From Fig. 1 . 6 ~If Ic is the collector current and is taken as positive flowing into the collector, it followsfrom (1.32) thatwhere A is the cross-sectional area of the emitter. Substitution of (1.27) into (1.33) gives qAD n VBE zc = exp - WB VT = VBE Is exp - VTwhereand Is is a constant used to describe the transfer characteristic of the transistor in theforward-active region. Equation 1.36 can be expressed in terms of the base doping densityby noting thats (see Chapter 2 )and substitution of (1.37) in (1.36) giveswhere QB = WBNA is the number of doping atoms in the base per unit area of theemitter and ni is the intrinsic carrier concentration in silicon. In this form (1.38) appliesto both uniform- and nonuniform-base transistors and D, has been replaced by D,,which is an average effective value for the electron diffusion constant in the base. Thisis necessary for nonuniform-base devices because the diffusion constant is a functionof impurity concentration. Typical values of Is as given by (1.38) are from 10-l4 to10-l6 A. Equation 1.35 gives the collector current as a function of base-emitter voltage. Thebase current IB is also an important parameter and, at moderate current levels, consists oftwo major components. One of these ( I B l )represents recombination of holes and electronsin the base and is proportional to the minority-carrier charge Qe in the base. From Fig.1.6c, the minority-carrier charge in the base isand we have
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12 Chapter 1 4 Models for Integrated-CircuitActive Devices where rb is the minority-carrier lifetime in the base. IB1 represents a flow of majority holes from the base lead into the base region. Substitution of (1.27) in (1.40) gives 1 n,oW~qA VBE IBl = - exp - 2 Tb VT The second major component of base current (usually the dominant one in integrated- circuit npn devices) is due to injection of holes from the base into the emitter. This current component depends on the gradient of minority-carrier holes in the emitter and is9 where D, is the diffusion constant for holes and L, is the diffusion length (assumed small) for holes in the emitter. pnE(0) is the concentration of holes in the emitter at the edge of the depletion region and is If ND is the donor atom concentration in the emitter (assumed constant), then The emitter is deliberately doped much more heavily than the base, making ND large and p n ~small, so that the base-current component, IB2,is minimized. o Substitution of (1.43) and (1.44) in (1.42) gives qAD n2 VBE IB2 -- exp - = The total base current, IB, is the sum of ZB1 and IB2: Although this equation was derived assuming uniform base and emitter doping, it gives the correct functional dependence of IB device parameters for practical double-diffused on nonuniform-base devices. Second-order components of ZB, which are important at low current levels, are considered later. Since Ic in (1.35) and IBin (1.46) are both proportional to exp(VBE/VT) this anal- in ysis, the base current can be expressed in terms of collector current as where PF is the forward current gain. An expression for PF can be calculated by substi- tuting (1.34) and (1.46) in (1.47) to give where (1.37) has been substituted for n,,. Equation 1.48 shows that PF is maximized by minimizing the base width WB and maximizing the ratio of emitter to base doping
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1.3 Large-Signal Behavior of Bipolar Transistors 13densities NDINA.Typical values of PF for npn transistors in integrated circuits are 50 to500, whereas lateral p n p transistors (to be described in Chapter 2) have values 10 to 100.Finally, the emitter current iswhere The value of ( Y F can be expressed in terms of device parameters by substituting (1.48)in (1.50) to obtainwhereThe validity of (1.51) depends on WiI27bDn < 1 and (Dp/Dn)(WB/L,)(NA/N~) < 1, < <and this is always true if PF is large [see (1.48)]. The term y in (1.51) is called the emitterinjection efJiciency and is equal to the ratio of the electron current (npn transistor) injectedinto the base from the emitter to the total hole and electron current crossing the base-emitterjunction. Ideally y + 1, and this is achieved by making NDINAlarge and WB small. Inthat case very little reverse injection occurs from base to emitter. The term a~ in (1.51) is called the base transport factor and represents the fraction ofcarriers injected into the base (from the emitter) that reach the collector. Ideally cur + 1and this is achieved by making WB small. It is evident from the above development thatfabrication changes that cause a r and y to approach unity also maximize the value of PFof the transistor. The results derived above allow formulation of a large-signal model of the transis- tor suitable for bias-circuit calculations with devices in the forward-active region. One such circuit is shown in Fig. 1.7 and consists of a base-emitter diode to model (1.46) and a controlled collector-current generator to model (1.47). Note that the collector volt- age ideally has no influence on the collector current and the collector node acts as ahigh-impedance current source. A simpler version of this equivalent circuit, which isoften useful, is shown in Fig. 1.7b, where the input diode has been replaced by a bat- tery with a value VBE(,,,,),which is usually 0.6 to 0.7 V. This represents the fact that inthe forward-active region the base-emitter voltage varies very little because of the steep slope of the exponential characteristic. In some circuits the temperature coefficient of VBE(on) important, and a typical value for this is -2 mVPC. The equivalent circuits of is Fig. 1.7 apply for npn transistors. For p n p devices the corresponding equivalent circuits are shown in Fig. 1.8.
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14 Chapter 1 . Models for Integrated-CircuitActive Devices Figure 1.7 Large-signal models of npn transistors for use in bias calcula- E E tions. (a) Circuit incor- 1s VBE porating an input diode. zB = - exp - (b) Simplified circuit PF VT with an input voltage (b) (a) source. Figure 1.8 Large-signal models of pnp transistors corresponding to the (b) circuits of Fig. 1.7. 1.3.2 Effects of Collector Voltage on Large-Signal Characteristics in the Forward-Active Region In the analysis of the previous section, the collector-base junction was assumed reverse biased and ideally had no effect on the collector currents. This is a useful approximation for first-order calculations, but is not strictly true in practice. There are occasions where the influence of collector voltage on collector current is important, and this will now be investigated. The collector voltage has a dramatic effect on the collector current in two regions of de- vice operation. These are the saturation (VCE approaches zero) and breakdown (VCE very large) regions that will be considered later. For values of collector-emittervoltage VCE be- tween these extremes, the collector current increases slowly as VCE increases. The reason for this can be seen from Fig. 1.9, which is a sketch of the minority-carrier concentration in the base of the transistor. Consider the effect of changes in VCE on the carrier concen- tration for constant VBE. Since VBE is constant, the change in VcB equals the change in VCE and this causes an increase in the collector-base depletion-layer width as shown. The change in the base width of the transistor, AWB, equals the change in the depletion-layer width and causes an increase AIc in the collector current. From (1.35) and (1.38) we have q ~ ~ , n : VBE Ic = - - exp - - - QB VT Differentiation of (1.52) yields arc - - - - - qADnn; ~VCE Qi ( ) VBE ~ Q B exp - - VT ~ V C E and substitution of (1.52) in (1.53) gives
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1.3 Large-Signal Behavior of Bipolar Transistors 15 Carrier concentration A I I Collector depletion I I I Figure 1.9 Effect of in- DX Emitter I Base creases in VCEon the collector depletion re- gion and base width of a bipolar transistor.For a uniform-base transistor QB = W E N A and (1.54) becomes ,Note that since the base width decreases as VCE increases, d WBldVcEin (1.55) is negativeand thus dIcldVCE is positive. The magnitude of d WBldVcEcan be calculated from (1.18)for a uniform-base transistor. This equation predicts that dWB/dVcE is a function of thebias value of V C E , the variation is typically small for a reverse-biased junction and butdWsldVcE is often assumed constant. The resulting predictions agree adequately withexperimental results. Equation 1.55 shows that dIcldVcs is proportional to the collector-bias current andinversely proportional to the transistor base width. Thus narrow-base transistors showa greater dependence of Ic on VCEin the forward-active region. The dependence ofdIcldVcE on Ic results in typical transistor output characteristics as shown in Fig. 1.10.In accordance with the assumptions made in the foregoing analysis, these characteristicsare shown for constant values of VBE. However, in most integrated-circuit transistors thebase current is dependent only on VBEand not on V C E ,and thus constant-base-currentcharacteristics can often be used in the following calculation. The reason for this is thatthe base current is usually dominated by the IB2 component of (1.45), which has no de-pendence on VCE. Extrapolation of the characteristics of Fig. 1.10 back to the VCEaxisgives an intercept V Acalled the Early voltage, whereSubstitution of (1.55) in (1.56) giveswhich is a constant, independent of Ic. Thus all the characteristics extrapolate to the samepoint on the VCEaxis. The variation of Ic with VCEis called the Early effect, and V A isa common model parameter for circuit-analysis computer programs. Typical values of V A
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16 Chapter 1 rn Models for integrated-Circuit Active Devices Figure 1.10 Bipolar transistor output characteristics showing the Early voltage, V A . for integrated-circuit transistors are 15 to 100 V. The inclusion of Early effect in dc bias calculations is usually limited to computer analysis because of the complexity introduced into the calculation. However, the influence of the Early effect is often dominant in small- signal calculations for high-gain circuits and this point will be considered later. Finally, the influence of Early effect on the transistor large-signal characteristics in the forward-active region can be represented approximately by modifying (1.35) to Ic = ( + v:,.) 7: Is 1 - exp - This is a common means of representing the device output characteristics for computer simulation. 1.3.3 Saturation and Inverse-Active Regions Saturation is a region of device operation that is usually avoided in analog circuits because the transistor gain is very low in this region. Saturation is much more commonly encoun- tered in digital circuits, where it provides a well-specified output voltage that represents a logic state. In saturation, both emitter-base and collector-base junctions are forward biased. Con- sequently, the collector-emitter voltage V C Eis quite small and is usually in the range 0.05 to 0.3 V.The carrier concentrations in a saturated npn transistor with uniform base doping are shown in Fig. 1.11. The minority-carrier concentration in the base at the edge of the depletion region is again given by (1.28) as but since VBCis now positive, the value of n p ( W B ) no longer negligible. Consequently, is changes in VCEwith VBEheld constant (which cause equal changes in V B C directly affect ) np(We).Since the collector current is proportional to the slope of the minority-carrier con- centration in the base [see (1.3 I ) ] ,it is also proportional to [ n p ( 0 - nP(WB)] ) from Fig. 1.11. Thus changes in n p ( W B )directly affect the collector current, and the collector node of the transistor appears to have a low impedance. As V C Eis decreased in saturation with V B E held constant, VBCincreases, as does n p ( W B from (1.59).Thus from Fig. 1.1 1 the collector )
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1.3 Large-Signal Behavior o f Bipolar Transistors 17 nnE j Carrier concentration l I P,(x) , I ! ! I Pnc P~E ,X I Emitter Base Collector ! Figure 1.1 1 Carrier concentrations in a saturated npn transistor. (Not to scale.)current decreases because the slope of the carrier concentration decreases. This gives riseto the saturation region of the Ic - VCEcharacteristic shown in Fig. 1.12. The slope ofthe Ic - VCE characteristic in this region is largely determined by the resistance in serieswith the collector lead due to the finite resistivity of the n-type collector material. A usefulmodel for the transistor in this region is shown in Fig. 1.13 and consists of a fixed voltagesource to represent VB,qon), a fixed voltage source to represent the collector-emitter andvoltage VCqsat). more accurate but more complex model includes a resistor in series Awith the collector. This resistor can have a value ranging from 20 to 500 a, depending onthe device structure. An additional aspect of transistor behavior in the saturation region is apparent fromFig. 1.11. For a given collector current, there is now a much larger amount of stored chargein the base than there is in the forward-active region. Thus the base-current contributionrepresented by (1.41) will be larger in saturation. In addition, since the collector-base junc-tion is now forward biased, there is a new base-current component due to injection ofcarriers from the base to the collector. These two effects result in a base current IB in sat-uration, which is larger than in the forward-active region for a given collector current Ic.Ratio Ic/IB in saturation is often referred to as the forced P and is always less than PF.As the forced / is made lower with respect to PF, the device is said to be more heavily 3saturated. The minority-carrier concentration in saturation shown in Fig. 1.11 is a straight linejoining the two end points, assuming that recombination is small. This can be representedas a linear superposition of the two dotted distributions as shown. The justification for thisis that the terminal currents depend linearly on the concentrations np(0) and np(WB).Thispicture of device carrier concentrations can be used to derive some general equations de-scribing transistor behavior. Each of the distributions in Fig. 1.11 is considered separatelyand the two contributions are combined. The emitter current that would result from npl (x)above is given by the classical diode equation ( ? ) : IEF = -IES exp - - 1where IESis a constant that is often referred to as the saturation current of the junction (noconnection with the transistor saturation previously described). Equation 1.60 predicts thatthe junction current is given by IEF = IESwith a reverse-bias voltage applied. However,
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18 Chapter 1 rn Models for Integrated-Circuit Active Devices Figure 1.12 Typical Ic-VcE characteristics for an npn bipolar transistor. Note the different scales for positive and negative currents and voltages. V B(on) ~ Vet. (sat) 0 - 0 E E (a) nPn (b) Figure 1.13 Large-signal models for bipolar transistors in the saturation region. in practice (1.60) is applicable only in the forward-bias region, since second-order effects dominate under reverse-bias conditions and typically result in a junction current several orders of magnitude larger than IES. The junction current that flows under reverse-bias conditions is often called the leakage current of the junction. Returning to Fig. 1.11, we can describe the collector current resulting from nP2(x) alone as ICR = -ICS ( exp - - 1 7 ) : where Ics is a constant. The total collector current Ic is given by ICR plus the fraction of IEF that reaches the collector (allowing for recombination and reverse emitter injection). Thus (?," ) Ic = a F I E s exp- - 1 - (7: Ics exp- - ) 1
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1.3 Large-Signal Behavior of Bipolar Transistors 19where a~ has been defined previously by (1.51). Similarly, the total emitter current iscomposed of IEF plus the fraction of ICR that reaches the emitter with the transistor actingin an inverted mode. Thus IE = (7:) ( " V " , " ) -IES exp-+ - 1 CYRICSexp- - 1where ( Y R is the ratio of emitter to collector current with the transistor operating inverted(i.e., with the collector-base junction forward biased and emitting carriers into the baseand the emitter-base junction reverse biased and collecting carriers). Typical values of f f Rare 0.5 to 0.8. An inverse current gain PR is also definedand has typical values 1 to 5. This is the current gain of the transistor when operatedinverted and is much lower than PF because the device geometry and doping densitiesare designed to maximize PF. The inverse-active region of device operation occurs forV C E negative in an n p n transistor and is shown in Fig. 1.12. In order to display thesecharacteristics adequately in the same figure as the forward-active region, the negativevoltage and current scales have been expanded. The inverse-active mode of operation israrely encountered in analog circuits. Equations 1.62 and 1.63 describe n p n transistor operation in the saturation regionwhen V B E and VBC are both positive, and also in the forward-active and inverse-activeregions. These equations are the Ebers-Moll equations. In the forward-active region, theydegenerate into a form similar to that of (1.33, (1.47), and (1.49) derived earlier. This canbe shown by putting V B E positive and VBc negative in (1.62) and (1.63) to obtain IE = ( ? ) : -IES exp- - 1 - ffRIcsEquation 1.65 is similar in form to (1.35) except that leakage currents that were previ-ously neglected have now been included. This minor difference is significant only at hightemperatures or very low operating currents. Comparison of (1.65) with (1.35) allows usto identify Is = a F I E S ,and it can be shown1° in general thatwhere this expression represents a reciprocity condition. Use of (1.67) in (1.62) and (1.63)allows the Ebers-Moll equations to be expressed in the general form ( ; : Ic = Is exp - - 1 - ) - ieXp - I)This form is often used for computer representation of transistor large-signal behavior. The effect of leakage currents mentioned above can be further illustrated as follows.In the forward-active region, from (1.66) ( 2 ) IES exp - - 1 = -IE - QRICS
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20 Chapter 1 .Models for Integrated-Circuit Active Devices Substitution of (1.68) in (1.65) gives where Ico = Ics(1 - ~ R ~ F I (1.69a) and Ico is the collector-base leakage current with the emitter open. Although Ico is given theoretically by (1.69a), in practice, surface leakage effects dominate when the collector- base junction is reverse biased and Ico is typically several orders of magnitude larger than the value given by (1.69a). However, (1.69) is still valid if the appropriate measured value for Ico is used. Typical values of Ico are from 10-lo to 10-l2 A at 25OC, and the magnitude doubles about every 8°C. As a consequence, these leakage terms can become very significant at high temperatures. For example, consider the base current IB. From Fig. 1.5 this is r, = -(lc + I E ) (1.70) If IE is calculated from (1.69) and substituted in (1.70), the result is But from (1SO) and use of (1.72) in (1.7 1) gives Since the two terms in (1.73) have opposite signs, the effect of Ico is to decrease the magnitude of the external base current at a given value of collector current. EXAMPLE If Ico is 10-lo A at 24"C, estimate its value at 120°C. Assuming that Ico doubles every 8"C, we have IC0(12O0C) = 10-lo x 212 = 0.4 pA 1.3.4 Transistor Breakdown Voltages In Section 1.2.2 the mechanism of avalanche breakdown in a pn junction was described. Similar effects occur at the base-emitter and base-collector junctions of a transistor and these effects limit the maximum voltages that can be applied to the device. First consider a transistor in the common-base configuration shown in Fig. 1 . 1 4 ~and supplied with a constant emitter current. Typical Ic - V C Bcharacteristics for an npn tran- sistor in such a connection are shown in Fig. 1.14b. For IE = 0 the collector-base junction breaks down at a voltage BVcBo, which represents collector-base breakdown with the emitter open. For finite values of IE, the effects of avalanche multiplication are apparent for values of VcB below BVcBo. In the example shown, the effective common-base current
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1.3 Large-Signal Behavior of Bipolar Transistors 21 A I,= 0 I v~~ 0 20 40 60 80 1 100 "Olts Figure 1.14 Common-base I I transistor connection. (a) Bvc~o Test circuit. (b) Zc - V C B (b) characteristics.gain a F = IC/IEbecomes larger than unity for values of VCBabove about 60 V. Operationin this region (but below BVcso) can, however, be safely undertaken if the device powerdissipation is not excessive. The considerations of Section 1.2.2 apply to this situation, andneglecting leakage currents, we can calculate the collector current in Fig. 1 . 1 4 as ~where M is defined by (1.26) and thusOne further point to note about the common-base characteristics of Fig. 1.14b is that forlow values of VcB where avalanche effects are negligible, the curves show very little of theEarly effect seen in the common-emitter characteristics. Base widening still occurs in thisconfiguration as VcB is increased, but unlike the common-emitter connection, it produceslittle change in Ic. This is because IEis now fixed instead of VBEor IB, and in Fig. 1.9,this means the slope of the minority-carrier concentration at the emitter edge of the baseis fixed. Thus the collector current remains almost unchanged. Now consider the effect of avalanche breakdown on the common-emitter characteris-tics of the device. Typical characteristics are shown in Fig. 1.12, and breakdown occurs ata value BVcEo, which is sometimes called the sustaining voltage LVcEo. As in previouscases, operation near the breakdown voltage is destructive to the device only if the current(and thus the power dissipation) becomes excessive. The effects of avalanche breakdown on the common-emitter characteristics are morecomplex than in the common-base configuration. This is because hole-electron pairs areproduced by the avalanche process and the holes are swept into the base, where they ef-fectively contribute to the base current. In a sense the avalanche current is then amplijied
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22 Chapter 1 .Models for Integrated-CircuitActive Devices by the transistor. The base current is still given by IB = -(Ic + IE) Equation 1.74 still holds, and substitution of this in (1.76) gives where 1 M = Equation 1.77 shows that Ic approaches infinity as M a F approaches unity. That is, the effective p approaches infinity because of the additional base-current contribution from the avalanche process itself. The value of BVcEo can be determined by solving If we assume that VcB = VCE, gives this and this results in and thus Equation 1.81 shows that BVcEo is less than BVcBo by a substantial factor. However, the value of BVcBo, which must be used in (1.81), is the plane junction breakdown of the collector-base junction, neglecting any edge effects. This is because it is only collector- base avalanche current actually under the emitter that is amplified as described in the pre- vious calculation. However, as explained in Section 1.2.2, the measured value of BVcBo is usually determined by avalanche in the curved region of the collector, which is remote from the active base. Consequently, for typical values of PF = 100 and n = 4, the value of BVcEo is about one-half of the measured BVcBo and not 30 percent as (1.81) would indicate. Equation 1.8 1 explains the shape of the breakdown characteristics of Fig. 1.12 if the dependence of PF on collector current is included. As VCEis increased from zero with IB = 0, the initial collector current is approximately PFICofrom (1.73);since Ico is typ- ically several picoamperes, the collector current is very small. As explained in the next section, PF is small at low currents, and thus from (1.81) the breakdown voltage is high. However, as avalanche breakdown begins in the device, the value of Ic increases and thus PF increases. From (1.81) this causes a decrease in the breakdown voltage and the characteristic bends back as shown in Fig. 1.12 and exhibits a negative slope. At higher collector currents, PF approaches a constant value and the breakdown curve with IB = 0 becomes perpendicular to the VCEaxis. The value of V C Ein this region of the curve is
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1.3 Large-SignalBehavior of Bipolar Transistors 23 usually defined to be BVcEo, since this is the maximum voltage the device can sustain. The value of PF to be used to calculate BVcEo in (1.81) is thus the peak value of PF. Note from (1.81) that high-/3 transistors will thus have low values of BVcEo. The base-emitter junction of a transistor is also subject to avalanche breakdown. How- ever, the doping density in the emitter is made very large to ensure a high value of PF[ND is made large in (1.45) to reduce IB2].Thus the base is the more lightly doped side of the junction and determines the breakdown characteristic. This can be contrasted with the collector-base junction, where the collector is the more lightly doped side and results in typical values of BVcBo of 20 to 80 V or more. The base is typically an order of magni- tude more heavily doped than the collector, and thus the base-emitter breakdown voltage is much less than BVcBo and is typically about 6 to 8 V. This is designed BVEBO. The breakdown voltage for inverse-active operation shown in Fig. 1.12 is approximately equal to this value because the base-emitter junction is reverse biased in this mode of operation. The base-emitter breakdown voltage of 6 to 8 V provides a convenient reference volt- age in integrated-circuit design, and this is often utilized in the form of a Zener diode. However, care must be taken to ensure that all other transistors in a circuit are protected against reverse base-emitter voltages sufficient to cause breakdown. This is because, un- like collector-base breakdown, base-emitter breakdown is damaging to the device. It can cause a large degradation in PF, depending on the duration of the breakdown-current flow and its magnitude. If the device is used purely as a Zener diode, this is of no consequence, but if the device is an amplifying transistor, the PF degradation may be serious.w EXAMPLE If the collector doping density in a transistor is 2 X 1015 atoms/cm3 and is much less than the base doping, calculate BVcEo for P = 100 and n = 4. Assume = 3 X lo5 V/cm. The plane breakdown voltage in the collector can be calculated from (1.24) using Zmax = gait: Since ND < N A , we have < From (1.81) 1.3.5 Dependence of Transistor Current Gain P,con Operating Conditions Although most first-order analyses of integrated circuits make the assumption that PF is constant, this parameter does in fact depend on the operating conditions of the transistor. It was shown in Section 1.3.2, for example, that increasing the value of VCE increases Ic while producing little change in le,and thus the effective PF of the transistor increases. In Section 1.3.4 it was shown that as VCE approaches the breakdown voltage, BVcEo, the collector current increases due to avalanche multiplication in the collector. Equation 1.77 shows that the effective current gain approaches infinity as VCEapproaches BVcEo.
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24 Chapter 1 rn Models for Integrated-Circuit Active Devices In addition to the effects just described, PF also varies with both temperature and transistor collector current. This is illustrated in Fig. 1.15, which shows typical curves of PF versus Ic at three different temperatures for an npn integrated circuit transistor. It is evident that PF increases as temperature increases, and a typical temperature coefficient for PF is +7000 ppmI0C (where ppm signifies parts per million). This temperature de- pendence of PF is due to the effect of the extremely high doping density in the emitter,12 which causes the emitter injection efficiency y to increase with temperature. The variation of PF with collector current, which is apparent in Fig. 1.15, can be divided into three regions. Region I is the low-current region, where PF decreases as Ic decreases. Region I1 is the midcurrent region, where PF is approximately constant. Region 111 is the high-current region, where PF decreases as Ic increases. The reasons for this behavior of PF with IC can be better appreciated by plotting base current IB and collector current Ic on a log scale as a function of V B E . This is shown in Fig. 1.16, and because of the log scale on the vertical axis, the value of In PF can be obtained directly as the distance between the two curves. At moderate current levels represented by region I1 in Figs. 1.15 and 1.16, both Ic and IB follow the ideal behavior, and VBE Ic =Is exp - VT IB - Is exp -VBE PFM VT where PFM is the maximum value of PF and is given by (1.48). At low current levels, Ic still follows the ideal relationship of (1.82), and the decrease in PF is due to an additional component in IB, which is mainly due to recombination of carriers in the base-emitter depletion region and is present at any current level. However, at higher current levels the base current given by (1.83) dominates, and this additional component has little effect. The base current resulting from recombination in the depletion region is5 VBE IBX = Isx exp - ~ V T where PF t Region I I Region 11 I Region 1 1 1 Figure 1.15 Typical curves of PI. versus Zc for an npn integrated- 0 1 1 - circuit transistor with 6 pm2 0.1 pA 1 pA 10 pA 100 pA 1 mA 10 mA emitter area.
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1.3 Large-Signal Behavior of Bipolar Transistors 25 Figure 1.16 Base and collector currents of a bipolar transistor plotted on a log scale versus VBEon a linear scale. The distance between I * "BE the curves is a direct (linearscale) measure of In p F .At very low collector currents, where (1.84) dominates the base current, the current gaincan be calculated from (1.82) and (1.84) asSubstitution of (13 2 ) in (1.85) givesIf rn = 2, then (13 6 ) indicates that PF is proportional to &at very low collector currents. At high current levels, the base current IB tends to follow the relationship of (1.83),and the decrease in PF in region I11 is due mainly to a decrease in Ic below the valuegiven by (1 22). (In practice the measured curve of IB versus VBEin Fig. 1.16 may alsodeviate from a straight line at high currents due to the influence of voltage drop across thebase resistance.) The decrease in Ic is due partly to the effect of high-level injection, andat high current levels the collector current approaches7 VBE Ic = ISHexp - ~ V TThe current gain in this region can be calculated from (13 7 ) and (13 3 ) asSubstitution of (13 7 ) in (13 8 ) givesThus PF decreases rapidly at high collector currents.
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26 Chapter 1 .Models for Integrated-Circuit Active Devices In addition to the effect of high-level injection, the value of PF at high currents is also decreased by the onset of the Kirk effect,13 which occurs when the minority-carrier con- centration in the collector becomes comparable to the donor-atom doping density. The base region of the transistor then stretches out into the collector and becomes greatly enlarged. 1.4 Small-Signal Models of Bipolar Transistors Analog circuits often operate with signal levels that are small compared to the bias currents and voltages in the circuit. In these circumstances, incremental or small-signal models can be derived that allow calculation of circuit gain and terminal impedances without the ne- cessity of including the bias quantities. A hierarchy of models with increasing complexity can be derived, and the more complex ones are generally reserved for computer analysis. Part of the designers skill is knowing which elements of the model can be omitted when performing hand calculations on a particular circuit, and this point is taken up again later. Consider the bipolar transistor in Fig. 1 . 1 7 ~ with bias voltages V B E and Vcc applied as shown. These produce a quiescent collector current, Ic, and a quiescent base current, IB, and the device is in theforward-active region. A small-signal input voltage vi is applied in series with V B E and produces a small variation in base current ib and a small variation in collector current i,. Total values of base and collector currents are Ib and I,, respectively, + and thus Ib = ( I B ib) and I, = ( I c + i,). The carrier concentrations in the base of the transistor corresponding to the situation in Fig. 1 . 1 7 ~ shown in Fig. 1.17b. With only are -Ic= I c + ic - I,, = In "BE + i, Vi d-vc (4 Carrier concentration I Collector I I depletion I 1. Emitter deplet~on region (b) Figure 1.17 Effect of a small-signal input voltage applied to a bipolar transistor. (a) Circuit schematic. (b) Corresponding changes in carrier concentrations in the base when the device is in the forward-active region.
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1.4 Small-Signal Models of Bipolar Transistors 27bias voltages applied, the carrier concentrations are given by the solid lines. Applicationof the small-signal voltage vi causes n,(O) at the emitter edge of the base to increase, andproduces the concentrations shown by the dotted lines. These pictures can now be used toderive the various elements in the small-signal equivalent circuit of the bipolar transistor.1.4.1 TransconductanceThe transconductance is defined asSincewe can write A I c = grnAVBEand thus = gmvi (1.90)The value of g, can be found by substituting (1.35) in (13 9 ) to giveThe transconductance thus depends linearly on the bias current Ic and is 38 mAN forIc = 1 mA at 25°C for any bipolar transistor of either polarity (npn or pnp), of any size,and made of any material (Si, Ge, GaAs). To illustrate the limitations on the use of small-signal analysis, the foregoing relationwill be derived in an alternative way. The total collector current in Fig. 1 . 1 7 ~ be cancalculated using (1.35) as VBE+ vi VBE vi I, = Is exp = Is exp -exp - VT VT VTBut the collector bias current is Ic = VBE Is exp - VTand use of (1.93) in (1.92) gives Vi Ic Ic exp - = (1.94) VTIf vi < V T ,the exponential in (1.94) can be expanded in a power series,Now the incremental collector current is ic = I, - Ic
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28 Chapter 1 Models for Integrated-Circuit Active Devices and substitution of (1.96) in (1.95) gives If vi < Vr, (1.97) reduces to (1.90), and the small-signal analysis is valid. The cri- < terion for use of small-signal analysis is thus vi = AVBE < 26 mV at 25°C. In practice, < if AVBE is less than 10 mV, the small-signal analysis is accurate within about 10 percent. 1.4.2 Base-Charging Capacitance Figure 1.17b shows that the change in base-emitter voltage AVBE = vi has caused a change AQ, = q, in the minority-carrier charge in the base. By charge-neutrality require- ments, there is an equal change AQh = qh in the majority-carrier charge in the base. Since majority carriers are supplied by the base lead, the application of voltage vi requires the supply of charge qh to the base, and the device has an apparent input capacitance The value of Cb can be related to fundamental device parameters as follows. If (1.39) is divided by (1.33), we obtain The quantity TF has the dimension of time and is called the base transit time in the forward direction. Since it is the ratio of the charge in transit (Q,) to the current flow ( I c ) , it can be identified as the average time per carrier spent in crossing the base. To a first order it is independent of operating conditions and has typical values 10 to 500 ps for integrated npn transistors and 1 to 40 ns for lateral pnp transistors. Practical values of TF tend to be somewhat lower than predicted by (1.99)for diffused transistors that have nonuniform base doping.14 However, the functional dependence on base width WB and diffusion constant D, is as predicted by (1.99). From (1.99) But since AQ, = AQh, we have and this can be written qh = 7Fic Use of (1.102) in (1.98) gives and substitution of (1.90) in (1.103) gives Cb = TFgm
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1.4 Small-Signal Models of Bipolar Transistors 29Thus the small-signal, base-charging capacitance is proportional to the collector biascurrent. In the inverse-active mode of operation, an equation similar to (1.99) relates storedcharge and current via a time constant TR. This is typically orders of magnitude larger thanr~ because the device structure and doping are optimized for operation in the forward-active region. Since the saturation region is a combination of forward-active and inverse-active operation, inclusion of the parameter rR in a SPICE listing will model the largecharge storage that occurs in saturation.1.4.3 Input ResistanceIn the forward-active region, the base current is related to the collector current by (1.47) asSmall changes in IB and Ic can be related using (1.47):and thuswhere Po is the small-signal current gain of the transistor. Note that if PF is constant, thenPF = Po. Typical values of Po are close to those of PF, and in subsequent chapters littledifferentiation is made between these quantities. A single value of /3 is often assumed fora transistor and then used for both ac and dc calculations. Equation 1.107 relates the change in base current ib to the corresponding change incollector current i,, and the device has a small-signal input resistance given bySubstitution of (1.107) in (1.108) givesand use of (1.90) in (1.109) givesThus the small-signal input shunt resistance of a bipolar transistor depends on the currentgain and is inversely proportional to Zc.1.4.4 Output ResistanceIn Section 1.3.2 the effect of changes in collector-emitter voltage VCEon the large-signalcharacteristics of the transistor was described. It follows from that treatment that smallchanges AVcE in VCEproduce corresponding changes AIc in Ic, where
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30 Chapter 1 Models for Integrated-CircuitActive Devices Substitution of (1.55) and (1.57) in (1.11 1 ) gives where V Ais the Early voltage and r, is the small-signal output resistance of the transistor. Since typical values of V A are 50 to 100 V, corresponding values of r, are 50 to 100 ki2 for Ic = 1mA. Note that r, is inversely proportional to Ic, and thus r, can be related to g,, as are many of the other small-signal parameters. where If V A = 100 V, then 77 = 2.6 X l o p 4 at 25OC. Note that llr, is the slope of the output characteristics of Fig. 1.10. 1.4.5 Basic Small-Signal Model of the Bipolar Transistor Combination of the above small-signal circuit elements yields the small-signal model of the bipolar transistor shown in Fig. 1.18. This is valid for both npn and pnp devices in the forward-active region and is called the hybrid-.rr model. Collector, base, and emitter nodes are labeled C, B and E, respectively. The elements in this circuit are present in the equivalent circuit of any bipolar transistor and are specified by relatively few parameters (P, TF,7, IC).Note that in the evaluation of the small-signal parameters for pnp transistors, the magnitude only of Ic is used. In the following sections, further elements are added to this model to account for parasitics and second-order effects. 1.4.6 Collector-Base Resistance Consider the effect of variations in VCE the minority charge in the base as illustrated in on Fig. 1.9. An increase in VCE causes an increase in the collector depletion-layer width and consequent reduction of base width. This causes a reduction in the total minority-carrier charge stored in the base and thus a reduction in base current IB due to a reduction in IB1 given by (1.40).Since an increase AVcE in VCE causes a decrease AIB in Ie, this effect can be modeled by inclusion of a resistor r , from collector to base of the model of Fig. 1.18. If VBEis assumed held constant, the value of this resistor can be determined as follows. P r I r = g , , r o = -1 g , = , qIc --,C,=T,~, Figure 1.18 Basic bipolar transistor 78, kT small-signal equivalent circuit.
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1.4 Small-Signal Models of Bipolar Transistors 31Substitution of (1.112) in (1.115) gives If the base current IB is composed entirely of component I B ~then (1.107) can be used ,in (1.116) to giveThis is a lower limit for r,. In practice, IB1 is typically less than 10 percent of IB [compo-nent IB2 from (1.42) dominates] in integrated npn transistors, and since IB1 is very small,the change AIBl in lBlfor a given AVCE and AIc is also very small. Thus a typical valuefor r, is greater than 10Por,. For lateral pnp transistors, recombination in the base is moresignificant, and r, is in the range 2Por, to 5Poro1.4.7 Parasitic Elements in the Small-Signal ModelThe elements of the bipolar transistor small-signal equivalent circuit considered so far maybe considered basic in the sense that they arise directly from essential processes in the de-vice. However, technological limitations in the fabrication of transistors give rise to anumber of parasitic elements that must be added to the equivalent circuit for mostintegrated-circuit transistors. A cross section of a typical npn transistor in a junction-isolated process is shown in Fig. 1.19. The means of fabricating such devices is describedin Chapter 2. As described in Section 1.2, all pn junctions have a voltage-dependent capacitance as-sociated with the depletion region. In the cross section of Fig. 1.19, three depletion-regioncapacitances can be identified. The base-emitter junction has a depletion-region capaci-tance Cj, and the base-collector and collector-substrate junctions have capacitances C ,and C,., respectively. The base-emitter junction closely approximates an abrupt junctiondue to the steep rise of the doping density caused by the heavy doping in the emitter.Thus the variation of Cj, with bias voltage is well approximated by (1.21). The collector-base junction behaves like a graded junction for small bias voltages since the doping den-sity is a function of distance near the junction. However, for larger reverse-bias values(more than about a volt), the junction depletion region spreads into the collector, which is @ SubstrateFigure 1.19 Integrated-circuit npn bipolar transistor structure showing parasitic elements. (Notto scale.)
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32 Chapter 1 Models for Integrated-Circuit Active Devices uniformly doped, and thus for devices with thick collectors the junction tends to behave like an abrupt junction with uniform doping. Many modem high-speed processes, how- ever, have very thin collector regions (of the order of one micron), and the collector deple- tion region can extend all the way to the buried layer for quite small reverse-bias voltages. When this occurs, both the depletion region and the associated capacitance vary quite slowly with bias voltage. The collector-base capacitance C, thus tends to follow (1.22) for very small bias voltages and (1.21) for large bias voltages in thick-collector devices. In practice, measurements show that the variation of C , with bias voltage for most devices can be approximated by where V is the forward bias on the junction and n is an exponent between about 0.2 and 0.5. The third parasitic capacitance in a monolithic n p n transistor is the collector-substrate capacitance C,,, and for large reverse bias voltages this varies according to the abrupt junction equation (1.2 1)for junction-isolated devices. In the case of oxide-isolated devices, however, the deep p diffusions used to isolate the devices are replaced by oxide. The sidewall component of C,, then consists of a fixed oxide capacitance. Equation 1.117a may then be used to model C,,, but a value of n less than 0.5 gives the best approximation. In general, (1.117a) will be used to model all three parasitic capacitances with subscripts e, c , and s on n and +o used to differentiate emitter-base, collector-base, and collector-substrate capacitances, respectively. Typical zero-bias values of these parasitic capacitances for a - minimum-size npn transistor in a modern oxide-isolated process are Cjeo-- 10 fF, CPo 10 fF, and CCso 20 fF. Values for other devices are summarized in Chapter 2. As described in Chapter 2, lateral pnp transistors have a parasitic capacitance Cbs 21 from base to substrate in place of C,,. Note that the substrate is always connected to the most negative voltage supply in the circuit in order to ensure that all isolation regions are separated by reverse-biased junctions. Thus the substrate is an ac ground, and all parasitic capacitance to the substrate is connected to ground in an equivalent circuit. The final elements to be added to the small-signal model of the transistor are resis- tive parasitics. These are produced by the finite resistance of the silicon between the top contacts on the transistor and the active base region beneath the emitter. As shown in Fig. 1.19, there are significant resistances rb and r, in series with the base and collector con- tacts, respectively. There is also a resistance re, of several ohms in series with the emitter lead that can become important at high bias currents. (Note that the collector resistance r,-, r, is actually composed of three parts labeled r , ~ , and rC3.)Typical values of these parameters are rb = 50 to 500 a, re, = 1 to 3 a , and r, = 20 to 500 a.The value of rb varies significantly with collector current because of current crowding.15 This occurs at high collector currents where the dc base current produces a lateral voltage drop in the base that tends to forward bias the base-emitter junction preferentially around the edges of the emitter. Thus the transistor action tends to occur along the emitter periphery rather than under the emitter itself, and the distance from the base contact to the active base region is reduced. Consequently, the value of rb is reduced, and in a typical npn transistor, rb may decrease 50 percent as lc increases from 0.1 mA to 10 mA. The value of these parasitic resistances can be reduced by changes in the device struc- ture. For example, a large-area transistor with multiple base and emitter stripes will have a smaller value of rb. The value of r, is reduced by inclusion of the low-resistance buried n+ layer beneath the collector.
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1.4 Small-Signal Models of Bipolar Transistors 33 Figure 1.20 Complete bipolar transistor small-signal equivalent circuit. The addition of the resistive and capacitive parasitics to the basic small-signal circuit of Fig. 1.18 gives the complete small-signal equivalent circuit of Fig. 1.20. The internal base node is labeled B to distinguish it from the external base contact B. The capaci- tance C, contains the base-charging capacitance Co and the emitter-base depletion layer capacitance C j , . Note that the representation of parasitics in Fig. 1.20 is an approximation in that lumped elements have been used. In practice, as suggested by Fig. 1.19, C, is distributed across r b and C,, is distributed across r,. This lumped representation is adequate for most purposes but can introduce errors at very high frequencies. It should also be noted that while the parasitic resistances of Fig. 1.20 can be very important at high bias currents or for high-frequency operation, they are usually omitted from the equivalent circuit for low-frequency calculations, particularly for collector bias currents less than 1 mA.I EXAMPLE Derive the complete small-signal equivalent circuit for a bipolar transistor at lc = 1 mA, VcB = 3 V, and Vcs = 5 V. Device parameters are Cjeo = 10 fF, n, = 0.5, $0, = 0.9 V, CPo = lOfF, n, = 0.3,+0, = 0.5V, Ccso = 20fF, n, = 0.3,$0, = 0.65V,po = 1 0 0 ,=~~ lops, V A = 20V,rb = 3 0 0 0 , r , = 50Cl,r,, = 5Cl,r, = 10Poro. Since the base-emitter junction is forward biased, the value of Cje is difficult to deter- mine for reasons described in Section 1.2.1. Either a value can be determined by computer or a reasonable estimation is to double CjeO.Using the latter approach, we estimate Using (1.117a) gives, for the collector-base capacitance, The collector-substrate capacitance can also be calculated using (1.1 17a)
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34 Chapter 1 Models for Integrated-CircuitActive Devices From (1.9 1) the transconductance is From (1.104) the base-charging capacitance is Cb = v g m = 10 X 10-l2 X 38 X F = 0.38 pF The value of C, from (1.118) is C, = 0.38 + 0.02 pF = 0.4 pF The input resistance from (1.110) is The output resistance from (1.112) is and thus the collector-base resistance is The equivalent circuit with these parameter values is shown in Fig. 1.2 1. 1.4.8 Specification of Transistor Frequency Response The high-frequency gain of the transistor is controlled by the capacitive elements in the equivalent circuit of Fig. 1.20. The frequency capability of the transistor is most often spec- ified in practice by determining the frequency where the magnitude of the short-circuit, common-emitter current gain falls to unity. This is called the transition frequency, fT, and is a measure of the maximum useful frequency of the transistor when it is used as an ampli- fier. The value of fT can be measured as well as calculated, using the ac circuit of Fig. 1.22. A small-signal current ii is applied to the base, and the output current i, is measured with the collector short-circuited for ac signals. A small-signal equivalent circuit can be formed for this situation by using the equivalent circuit of Fig. 1.20 as shown in Fig. 1.23, where re, and r, have been neglected. If r, is assumed small, then r, and C,, have no 0 1 0 E Figure 1.21 Complete small-signal equivalent circuit for a bipolar transistor at Ic = 1 mA, Vcs = 3 V, and Vcs = 5 V. Device parameters are Cieo = 10 fF, n, = 0.5, rCloe = 0.9 V, CpO= 10fF,n, = 0.3,$0, = 0.5V,C,,To= 20fF,n, = 0.3,1,!~~, = 0.65V,po = 1 0 0 , ~ ~ = lops, V A = 20 V, rb = 300 a, a, r, = 50 re.r = 5 Cl, rP = 10Poro.
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1.4 Small-Signal Models of Bipolar Transistors 35 Figure 1.22 Schematic of ac circuit for measurement - of f ~ . +- Figure 1.23 Small-signal equivalent circuit for the calculation of f T .influence, and we have ~1 r, ii 1 + r,(C, + C,)s %If the current fed forward through C, is neglected, 10 gmV1Substitution of (1.119) in (1.120) givesand thus 10 T (jm) = Po li 1 + Pocr + cp jw gmusing (1.110). Now if iolii(jo) is written as p ( j w ) (the high-frequency, small-signal current gain),thenAt high frequencies the imaginary part of the denominator of (1.122) is dominant, and wecan writeFrom (1.123), IP(jm)l = 1 when
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36 Chapter 1 Models for Integrated-Circuit Active Devices 1000 - . p, (low frequency value) 100 - Figure 1.24 Magnitude of small-signal ac cur- rent gain Ip(jco)l versus log scale frequency for a typical bipolar transistor. and thus The transistor behavior can be illustrated by plotting IP(jo)l using (1.122) as shown in Fig. 1.24. The frequency wg is defined as the frequency where P ( j o ) is equal to Pol, h (3 dB down from the low-frequency value). From (1.122) we have From Fig. 1.24 it can be seen that can be determined by measuring IP(jw)l at some frequency w, where IP(jo)l is falling at 6 dB1octave and using This is the method used in practice, since deviations from ideal behavior tend to occur as Ip (jw )I approaches unity. Thus IP (j o ) ( is typically measured at some frequency where its magnitude is about 5 or 10, and (1.127) is used to determine w ~ . It is interesting to examine the time constant, T T , associated with w ~ This is de- . fined as and use of (1.124) in (1.128) gives Substitution of (1.118) and (1.104) in (1.129) gives Equation 1.130 indicates that 71.is dependent on Ic (through gm)and approaches a constant value of r~ at high collector bias currents. At low values of Ic, the terms involving C;, and C, dominate, and they cause TT to rise and fT tofall as Ic is decreased. This behavior is illustrated in Fig. 1.25, which is a typical plot of fT versus Ic for an integrated-circuit npn transistor. The decline in fT at high collector currents is not predicted by this simple theory and is due to an increase in TF caused by high-level injection and Kirk effect at high
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1.4 Small-Signal Models of Bi~olar Transistors 37 fr GHz Figure 1.25 Typical curve of f~ver- sus Ic for an npn integrated-circuit transistor with 6 p,m2 emitter area in a high-speed process. currents. These are the same mechanisms that cause a decrease in PF at high currents as described in Section 1.3.5.I EXAMPLE A bipolar transistor has a short-circuit, common-emitter current gain at 1 GHz of 8 with Ic = 0.25 mA and 9 with Ic = 1 mA. Assuming that high-level injection effects are negligible, calculate C j , and TF, assuming both are constant. The measured value of C, is 10 f E From the data, values of fT are f ~= ,8 X 1 = 8GHz at Ic = 0.25 rnA fT2 = 9 X 1 = 9 GHz at Ic = 1 mA Corresponding values of 77. are Using these data in (1.130), we have 19.9 X 10-l2 = 7~ f 104(Cp + Cje) at Ic = 0.25 mA. At Ic = 1 mA we have 17.7 X 10-l2 + 26(Cp + Cje) = . r ~ Subtraction of (1.132) from (1.13 1) yields C, + Cj, = 28.2 fF Since C , was measured as 10 fF, the value of Cje is given by C j , = 18.2 fF
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38 Chapter 1 . Models for Integrated-Circuit Active Devices Substitution in (1.13 1) gives This is an example of how basic device parameters can be determined from high-frequency current-gain measurements. Note that the assumption that Cj, is constant is a useful approximation in practice because VBEchanges by only 36 mV as Ic increases from 0.25 mA to 1 mA. 1.5 Large-Signal Behavior of Metal-Oxide-Semiconductor Field-Effect Transistors Metal-oxide-semiconductor field-effect transistors (MOSFETs) have become dominant in the area of digital integrated circuits because they allow high density and low power dis- sipation. In contrast, bipolar transistors still prbvide many advantages in stand-alone ana- log integrated circuits. For example, the transconductance per unit bias current in bipolar transistors is usually much higher than in MOS transistors. So in systems where analog techniques are used on some integrated circuits and digital techniques on others, bipolar technologies are often preferred for the analog integrated circuits and MOS technologies for the digital. To reduce system cost and increase portability, both increased levels of integration and reduced power dissipation are required, forcing the associated analog cir- cuits to use MOS-compatible technologies. One way to achieve these goals is to use a processing technology that provides both bipolar and MOS transistors, allowing great de- sign flexibility. However, all-MOS processes are less expensive than combined bipolar and MOS processes. Therefore, economic considerations drive integrated-circuit manu- facturers to use all-MOS processes in many practical cases. As a result, the study of the characteristics of MOS transistors that affect analog integrated-circuit design is important. 1.5.1 Transfer Characteristics of MOS Devices A cross section of a typical enhancement-mode n-channel MOS (NMOS) transistor is shown in Fig. 1.26. Heavily doped n-type source and drain regions are fabricated in a p-type substrate (often called the body). A thin layer of silicon dioxide is grown over the Metal or poly silicon gate contract source. Drain, Figure 1.26 Typical enhancement-mode NMOS structure.
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1.5 Large-SignalBehavior of Metal-Oxide-SemiconductorField-EffectTransistors 39substrate material and a conductive gate material (metal or polycrystalline silicon) coversthe oxide between source and drain. Note that the gate is horizontal in Fig. 1.26, and wewill use this orientation in all descriptions of the physical operation of MOS devices. Inoperation, the gate-source voltage modifies the conductance of the region under the gate,allowing the gate voltage to control the current flowing between source and drain. Thiscontrol can be used to provide gain in analog circuits and switching characteristics indigital circuits. The enhancement-mode NMOS device of Fig. 1.26 shows significant conduction be-tween source and drain only when an n-type channel exists under the gate. This obser-vation is the origin of the n-channel designation. The term enhancement mode refers tothe fact that no conduction occurs for Vcs = 0. Thus, the channel must be enhanced tocause conduction. MOS devices can also be made by using an n-type substrate with ap-type conducting channel. Such devices are called enhancement-mode p-channel MOS(PMOS) transistors. In complementary MOS (CMOS) technology, both device types arepresent. The derivation of the transfer characteristics of the enhancement-mode NMOS deviceof Fig. 1.26 begins by noting that with VGS = 0, the source and drain regions are separatedby back-to-back p n junctions. These junctions are formed between the n-type source anddrain regions and the p-type substrate, resulting in an extremely high resistance (about a)1012 between drain and source when the device is off. Now consider the substrate, source, and drain grounded with a positive voltage VGSapplied to the gate as shown in Fig. 1.27. The gate and substrate then form the plates ofa capacitor with the SiOz as a dielectric. Positive charge accumulates on the gate andnegative charge in the substrate. Initially, the negative charge in the p-type substrateis manifested by the creation of a depletion region and the exclusion of holes underthe gate as described in Section 1.2 for a pn-junction. The depletion region is shown inFig. 1.27. The results of Section 1.2 can now be applied. Using (1.lo), the depletion-layerwidth X under the oxide iswhere 4 is the potential in the depletion layer at the oxide-silicon interface, NA is thedoping density (assumed constant) of the p-type substrate in atoms/cm3, and E is the per-mittivity of the silicon. The charge per area in this depletion region is Induced n-type channel SiO, Figure 1.27 Idealized NMOS device cross section with positive Vc5 applied, showing depletion regions - - and the induced channel.
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40 Chapter 1 8 Models for Integrated-Circuit Active Devices When the surface potential in the silicon reaches a critical value equal to twice the Fermi level r$f, a phenomenon known as inversion occur^.^ The Fermi level 4f is defined as where k is Boltzmanns constant. Also, ni is the intrinsic carrier concentration, which is where E, is the band gap of silicon at T = O°K, Nc is the density of allowed states near the edge of the conduction band, and Nv is the density of allowed states near the edge of the valence band, respectively. The Fermi level 4f is usually about 0.3 V. After the potential in the silicon reaches 2 4 f , further increases in gate voltage produce no further changes in the depletion-layer width but instead induce a thin layer of electrons in the depletion layer at the surface of the silicon directly under the oxide. Inversion produces a continuous n-type region with the source and drain regions and forms the conducting channel between source and drain. The conductivity of this channel can be modulated by increases or decreases in the gate-source voltage. In the presence of an inversion layer, and without substrate bias, the depletion region contains a fixed charge density If a substrate bias voltage VSB(positive for n-channel devices) is applied between the source and substrate, the potential required to produce inversion becomes (24f + Vss), and the charge density stored in the depletion region in general is The gate-source voltage VGS required to produce an inversion layer is called the threshold voltage Vt and can now be calculated. This voltage consists of several com- ponents. First, a voltage [24 + (Qb/Cox)]is required to sustain the depletion-layer charge Qb, where Cox is the gate oxide capacitance per unit area. Second, a work-function dif- ference r$, exists between the gate metal and the silicon. Third, positive charge density Qs, always exists in the oxide at the silicon interface. This charge is caused by crystal dis- continuities at the Si - Si02 interface and must be compensated by a gate-source voltage contribution of - Q,s/Co,. Thus we have a threshold voltage Qb Qss Vt = 4 m s +2$f + Cox cox where (1.137) and (1.138) have been used, and Vtois the threshold voltage with Vss = 0. The parameter y is defined as
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1.5Large-Signal Behavior of Metal-Oxide-SemiconductorField-Effect Transistors 41andwhere E ox and to, are the permittivity and the thickness of the oxide, respectively. A typicalvalue of y is 0.5 vli2, and Cox = 3.45 fFlPrn2 for to, = 100 angstroms. In practice, the value of Vrois usually adjusted in processing by implanting additionalimpurities into the channel region. Extra p-type impurities are implanted i~ the channel toset Vto between 0.5 V and 1.5 V for n-channel enhancement devices. By implanting n-typeimpurities in the channel region, a conducting channel can be formed even for VGS = 0,forming a depletion device with typical values of Vro in the range - 1 V to -4 V. If Q,is the charge density due to the implant, then the threshold voltage given by (1.139) isshifted by approximately Qi/Cox4 The preceding equations can now be used to calculate the large-signal characteris-tics of an n-channel MOSFET. In this analysis, the source is assumed grounded and biasvoltages VGS, VDS, and VSB are applied as shown in Fig. 1.28. If VGS > Vr, inversionoccurs and a conducting channel exists. The channel conductivity is determined by thevertical electric field, which is controlled by the value of (VGS- Vr). If VDs = 0, the cur-rent ID that flows from drain to source is zero because the horizontal electric field is zero.Nonzero VDs produces a horizontal electric field and causes current ID to flow. The valueof the current depends on both the horizontal and the vertical electric fields, explaining thetemjeld-efSect transistor. Positive voltage VDs causes the reverse bias from the drain tothe substrate to be larger than from the source to substrate, and thus the widest depletionregion exists at the drain. For simplicity, however, we assume that the voltage drop alongthe channel itself is small so that the depletion-layer width is constant along the channel. The drain current ID iswhere d Q is the incremental channel charge at a distance y from the source in an incre-mental length d y of the channel, and d t is the time required for this charge to cross lengthdy. The charge d Q iswhere W is the width of the device perpendicular to the plane of Fig. 1.28 and Qr is theinduced electron charge per unit area of the channel. At a distance y along the channel, thevoltage with respect to the source is V ( y )and the gate-to-channel voltage at that point is1 ~e~letion rein Y Y + ~ Y Y p-type substrate 4- - "? Figure 1.28 NMOS device with bias voltages applied.
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42 Chapter 1 rn Models for Integrated-CircuitActive Devices VGS- V(y). We assume this voltage exceeds the threshold voltage Vt. Thus the induced electron charge per unit area in the channel is - QI(Y) = COX[VGS V(Y)- Vt1 Also, where vd is the electron drift velocity at a distance y from the source. Combining (1.144) and (1.146) gives The drift velocity is determined by the horizontal electric field. When the horizontal elec- tric field %(y)is small, the drift velocity is proportional to the field and where the constant of proportionality p, is the average electron mobility in the channel. In practice, the mobility depends on both the temperature and the doping level but is almost constant for a wide range of normally used doping levels. Also, pn is sometimes called the surface mobility for electrons because the channel forms at the surface of the silicon. Typical values range from about 500 cm2/(v-s) to about 700 cm2/(v-s), which are much less than the mobility of electrons in the bulk of the silicon (about 1400 cm2N-s) because surface defects not present in the bulk impede the flow of electrons in MOS transistors.17 The electric field %(y)is where dV is the incremental voltage drop along the length of channel dy at a distance y from the source. Substituting (1.145), (1.148), and (1.149) into (1.147) gives Separating variables and integrating gives Carrying out this integration gives where When VDs < 2(VGs - Vr), (1.152) predicts that ID approximately proportional to VDs. < is This result is reasonable because the average horizontal electric field in this case is VDs/L, and the average drift velocity of electrons is proportional to the average field when the field is small. Equation 1.152 is important and describes the I-V characteristics of an MOS transistor, assuming a continuous induced channel. A typical value of k for to, = 100 angstroms is about 200 for an n-channel device.
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1.5 Large-SignalBehavior of Metal-Oxide-SemiconductorField-Effect Transistors 43 As the value of VDs is increased, the induced conducting channel narrows at the drainend and (1.145) indicates that Q, at the drain end approaches zero as VDs approaches(VGS- Vt). That is, the channel is no longer connected to the drain when VDs > VGS- Vt.This phenomenon is called pinch-off and can be understood by writing a KVL equationaround the transistor: VDS = VDG+ VGS (1.154)Therefore, when VDs > VGS - Vt,Rearranging (1.155) gives VGD < Vr Equation 1.156 shows that when drain-source voltage is greater than (VGS - Vt), thegate-drain voltage is less than a threshold, which means that the channel no longer exists atthe drain. This result is reasonable because we know that the gate-to-channel voltage at thepoint where the channel disappears is equal to Vt by the definition of the threshold voltage.Therefore, at the point where the channel pinches off, the channel voltage is (VGS - Vt).As a result, the average horizontal electric field across the channel in pinch-off does notdepend on the the drain-source voltage but instead on the voltage across the channel, whichis (VGs - Vr). Therefore, (1.152) is no longer valid if VDs > VGS - Vt. The value of IDin this region is obtained by substituting VDs = VGS - Vt in (1.152), giving Equation 1.157 predicts that the drain current is independent of VDs in the pinch-offregion. In practice, however, the drain current in the pinch-off region varies slightly as thedrain voltage is varied. This effect is due to the presence of a depletion region between thephysical pinch-off point in the channel at the drain end and the drain region itself. If thisdepletion-layer width is X d , then the effective channel length is given byIf Leff is used in place of L in (1.157), we obtain a more accurate formula for current inthe pinch-off regionBecause Xd (and thus Leff)are functions of the drain-source voltage in the pinch-off region,I D varies with VDs. This effect is called channel-length modulation. Using (1.158) and(1.159), we obtainand thusThis equation is analogous to (1.55) for bipolar transistors. Following a similar procedure,the Early voltage can be defined as
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44 Chapter 1 Models for Integrated-Circuit Active Devices and thus For MOS transistors, a commonly used parameter for the characterization of channel- length modulation is the reciprocal of the Early voltage, As in the bipolar case, the large-signal properties of the transistor can be approximated by assuming that A and VA are constants, independent of the bias conditions. Thus we can include the effect of channel-length modulation in the I-V characteristics by modifying (1.157) to In practical MOS transistors, variation of Xd with voltage is complicated by the fact that the field distribution in the drain depletion region is not one-dimensional. As a result, the calculation of A from the device structure is quite difficult,18and developing effective values of A from experimental data is usually necessary. The parameter A is inversely proportional to the effective channel length and a decreasing function of the doping level in the channel. Typical values of A are in the range 0.05 V- to 0.005 V-. Plots of IDversus VDs with VGSas a parameter are shown in Fig. 1.29 for an NMOS transistor. The device operates in the pinch-off region when VDs > (VGS- Vt). The pinch-off region for MOS devices is often called the saturation region. In saturation, the output characteristics are almost flat, which shows that the current depends mostly on the gate-source voltage and only to a small extent on the drain-source voltage. On the other hand, when VDs < (VGS- Vt), the device operates in the Ohmic or triode region, where the device can be modeled as a nonlinear voltage-controlled resistor connected between the drain and source. The resistance of this resistor is nonlinear because the v; term , in (1.152) causes the resistance to depend on VDs. Since this tern is small when VDs is small, however, the nonlinearity is also small when VDs is small, and the triode region is also sometimes called the linear region. The boundary between the triode and satura- tion regions occurs when VDs = (VGS- Vt). On this boundary, both (1.152) and (1.157) correctly predict ID.Since VDs = (VGS- Vt) along the boundary between triode and sat- uration, (1.157) shows that the boundary is ID= ( ~ I ~ ) ( w I L )This~ . v ~ parabolic function ID Ohmic or I Active or triode region , pinch-off region increases - v~~ Figure 1.29 NMOS device characteristics.
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1.5 Large-Signal Behavior of Metal-Oxide-SemiconductorField-Effect Transistors 45of VDs is shown in Fig. 1.29. For depletion n-channel MOS devices, V , is negative, andID is nonzero even for VGS = 0. For PMOS devices, all polarities of voltages and currentsare reversed. The results derived above can be used to form a large-signaI model of the NMOStransistor in saturation. The model topology is shown in Fig. 1.30, where ID is given by(1.152) in the triode region and (1.157) in saturation, ignoring the effect of channel-lengthmodulation. To include the effect of channel-length modulation, (1.159) or (1.165) shouldbe used instead of (1.157) to find the drain current in saturation.1.5.2 Comparison of Operating Regions of Bipolar and MOS TransistorsNotice that the meaning of the word saturation for MOS transistors is quite different thanfor bipolar transistors. Saturation in bipolar transistors refers to the region of operationwhere both junctions are forward biased and the collector-emitter voltage is approximatelyconstant or saturated. On the other hand, saturation in MOS transistors refers to the regionof operation where the channel is attached only to the source but not to the drain and thecurrent is approximately constant or saturated. To avoid confusion, the term active regionwill be used in this book to describe the flat region of the MOS transistor characteristics, asshown in Fig. 1.29. This wording is selected to form a link between the operation of MOSand bipolar transistors. This link is summarized in the table of Fig. 1.31, which reviewsthe operating regions of npn bipolar and n-channel MOS transistors. When the emitter junction is forward biased and the collector junction is reverse bi-ased, bipolar transistors operate in the forward-active region. They operate in the reverse-active region when the collector junction is forward biased and the emitter junction isreverse biased. This distinction is important because integrated-circuit bipolar transistorsare typically not symmetrical in practice; that is, the collector operates more efficientlyas a collector of minority carriers than as an emitter. Similarly, the emitter operates moreefficiently as an emitter of minority carriers than as a collector. One reason for this asym-metry is that the collector region surrounds the emitter region in integrated-circuit bipolartransistors, as shown in Fig. 1.19. A consequence of this asymmetry is that the currentgain in the forward-active region PF is usually much greater than the current gain in thereverse-active region PR. - o 1 J Figure 1.30 Large-signal model for the NMOS S transistor. npn Bipolar Transistor n-channel MOS Transistor Region VBE VBC Region VGS VGD Cutoff < < VBC(~~) vB~(on) Cutoff < vl < vt Forward Active 2 VBECon) < VBC(on) Saturation(Active) 2 V, < vt Reverse Active < VBE(on) 2 V R ~ ( ~ Saturation(Active) ~ ) < V, m V, Saturation 2 VBE(O~) 2 VBC(O~) Triode r V, 2 V, Figure 1.31 Operating regions of npn bipolar and n-channel MOS transistors.
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46 Chapter 1 . Models for Integrated-Circuit Active Devices In contrast, the source and drain of MOS transistors are completely interchangeable based on the preceding description. (In practice, the symmetry is good but not perfect.) Therefore, distinguishing between the forward-active and reverse-active regions of oper- ation of an MOS transistor is not necessary. Figure 1.31 also shows that npn bipolar transistors operate in cutoff when both junc- tions are reversed biased. Similarly, MOS transistors operate in cutoff when the gate is biased so that inversion occurs at neither the source nor the drain. Furthermore, npn tran- sistors operate in saturation when both junctions are forward biased, and MOS transistors operate in the triode region when the gate is biased so that the channel is connected to both the source and the drain. Therefore, this comparison leads us to view the voltage required to invert the surface of an MOS transistor as analogous to the voltage required to forward bias a pn junction in a bipolar transistor. To display this analogy, we will use the circuit symbols in Fig. 1 . 3 2 ~ represent MOS transistors. These symbols are inten- to tionally chosen to appear similar to the symbols of the corresponding bipolar transistors. In bipolar-transistor symbols, the arrow at the emitter junction represents the direction of current flow when the emitter junction is forward biased. In MOS transistors, the pn junc- tions between the source and body and the drain and body are reverse biased for normal operation. Therefore, the arrows in Fig. 1 . 3 2 ~ not indicate pn junctions. Instead they do indicate the direction of current flow when the terminals are biased so that the terminal labeled as the drain operates as the drain and the terminal labeled as the source operates as the source. In NMOS transistors, the source is the source of electrons; therefore, the source operates at a lower voltage than the drain, and the current flows in a direction op- posite that of the electrons in the channel. In PMOS transistors, the source is the source of holes; therefore, the source operates at a higher voltage than the drain, and the current flows in the same direction as the holes in the channel. In CMOS technology, one device type is fabricated in the substrate, which is common to all devices, invariably connected to a dc power-supply voltage, and usually not shown on the circuit diagram. The other device type, however, is fabricated in separate isolation S S n-channel p-channel (4 n-channel p-channel n-channel p-channel Figure 1.32 (a) NMOS and PMOS symbols used in CMOS circuits. (b) NMOS and PMOS sym- bols used when the substrate connection is nonstandard. (c) Depletion MOS device symbols.
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1.5 Large-SignalBehavior of Metal-Oxide-SemiconductorField-Effect Transistors 47regions called wells, which may or may not be connected together and which may or maynot be connected to a power-supply voltage. If these isolation regions are connected tothe appropriate power supply, the symbols of Fig. 1 . 3 2 ~ will be used, and the substrateconnection will not be shown. On the other hand, if the individual isolation regions areconnected elsewhere, the devices will be represented by the symbols of Fig. 1.32b, wherethe substrate is labeled B. Finally, symbols for depletion-mode devices, for which a channelforms for VGs = 0 , are shown in Fig. 1 . 3 2 ~ .1S.3 Decompositionof Gate-Source VoltageThe gate-source voltage of a given MOS transistor is usually separated into two parts: thethreshold, Vl, and the voltage over the threshold, VGS - Vt. We will refer to this latter partof the gate-source voltage as the overdrive. This decomposition is used because these twocomponents of the gate-source voltage have different properties. Assuming square-lawbehavior as in (1.157), the overdrive isSince the transconductance parameter kt is proportional to mobility, and since mobil-ity falls with increasing temperature, the overdrive rises with temperature. In contrast,the next section shows that the threshold falls with increasing temperature. Furthermore,(1.140) shows that the threshold depends on the source-body voltage, but not on the cur-rent; (1.166) shows that the overdrive depends directly on the current, but not on thesource-body voltage.1.5.4 Threshold Temperature DependenceAssume that the source-body voltage is zero. Substituting (1.138) into (1.139) gives vl = J~~NA.(W~) Qss + 24f + +ms - - Cx o Cx oAssume that + , Qss, and C,, are independent of temperature.19 Then differentiating(1.167) givesSubstituting (1.136) into (1.135) givesAssume both N , and N , are independent of temperature.20 Then differentiating (1.169)gives
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48 Chapter 1 Models for Integrated-Circuit Active Devices Substituting (1.169) into (1.170) and simplifying gives Substituting (1.141) and (1.171) into (1.168) gives Equation 1.172 shows that the threshold voltage falls with increasing temperature if + < Egl(2q). The slope is usually in the range of -0.5 mVPC to -4 ~ v I O C . ~ ~ EXAMPLE Assume T = 300°K, NA = 1015 ~ m - and to, ~ , = 100 A. Find dV,ldT. From (1.135), Also Substituting (1.173) and (1.174) into (1.17 1) gives From (1.142), 3.9 (8.854 x 10-14 Flcm) fF Cx = o 100 x cm = 3.45 - km2 Also, Substituting (1.173) - (1.177) into (1.172) gives 1.5.5 MOS Device Voltage Limitations The main voltage limitations in MOS transistors are described next.22v23Some of these limitations have a strong dependence on the gate length L; others have little dependence on L. Also, some of the voltage limitations are inherently destructive; others cause no damage as long as overheating is avoided.
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1.6 Small-Signal Models of MOS Transistors 49Junction Breakdown. For long channel lengths, the drain-depletion region has littleeffect on the channel, and the I ~ - v e r s u s - V ~ ~ closely follow the ideal curves of curvesFig. 1.29. For increasing VDs, however, eventually the drain-substrate pn-junctionbreakdown voltage is exceeded, and the drain current increases abruptly by ava-lanche breakdown as described in Section 1.2.2. This phenomenon is not inherentlydestructive.Punchthrough. If the depletion region around the drain in an MOS transistor touches thedepletion region around the source before junction breakdown occurs, increasing the drain-source voltage increases the drain current by reducing the barrier to electron flow betweenthe source and drain. This phenomenon is called punchthrough. Since it depends on thetwo depletion regions touching, it also depends on the gate length. Punchthrough is not in-herently destructive and causes a more gradual increase in the drain current than is causedby avalanche breakdown. Punchthrough normally occurs below the surface of the siliconand is often prevented by an extra ion implantation below the surface to reduce the size ofthe depletion regions.Hot Carriers. With sufficient horizontal or vertical electric fields, electrons or holes mayreach sufficient velocities to be injected into the oxide, where most of them increase thegate current and some of them become trapped. Such carriers are called hot because therequired velocity for injection into the oxide is usually greater than the random thermalvelocity. Carriers trapped in the oxide shift the threshold voltage and may cause a tran-sistor to remain on when it should turn off or vice versa. In this sense, injection of hotcarriers into the oxide is a destructive process. This process is most likely to be prob-lematic in short-channel technologies, where horizontal electric fields are likely to behigh.Oxide Breakdown. In addition to VDs limitations, MOS devices must also be protectedagainst excessive gate voltages. Typical gate oxides break down with an electric field ofabout 6 X lo6 Vlcm to 7 X lo6 v / c ~ , which * ~ ~ ~ ~ corresponds to 6 to 7 V applied fromgate to channel with an oxide thickness of 100 angstroms. Since this process depends onthe vertical electrical field, it is independent of channel length. However, this process isdestructive to the transistor, resulting in resistive connections between the gate and thechannel. Oxide breakdown can be caused by static electricity and can be avoided by us-ing pn diodes and resistors to limit the voltage range at sensitive nodes internal to theintegrated circuit that connect to bonding pads.1.6 Small-Signal Models of MOS TransistorsAs mentioned in Section 1.5, MOS transistors are often used in analog circuits. To simplifythe calculation of circuit gain and terminal impedances, small-signal models can be used.As in the case for bipolar transistors, a hierarchy of models with increasing complexity canbe derived, and choosing the simplest model required to do a given analysis is importantin practice. Consider the MOS transistor in Fig. 1.33 with bias voltages VGS and VDD appliedas shown. These bias voltages produce quiescent drain current ID. If VGS > V , andVDD > (VGS- Vt), the device operates in the saturation or active region. A small-signalinput voltage vi is applied in series with VGSand produces a small variation in drain currentid. The total value of the drain current is Id = (ID + id).
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50 Chapter 1 Models for Integrated-Circuit Active Devices t Id = ID + id -- - -VDD -- Figure 1.33 Schematic of an MOS transistor + with biasing. 1.6.1 Transconductance Assuming square-law operation, the transconductance from the gate can be determined from (1.165) by differentiating. If AVDs < 1, (1.179) simplifies to < Unlike the bipolar transistor, the transconductance of the MOS transistor is propor- tional to the square root of the bias current and depends on device geometry (oxide thick- ness via k and WIL). Another key difference between bipolar and MOS transistors can be seen by calculating the ratio of the transconductance to the current. Using (1.157) and (1.180) for MOS transistors shows that Also, for bipolar transistors, (1.91) shows that At room temperature, the thermal voltage VT is about equal to 26 mV. In contrast, the over- drive V,, for MOS transistors in many applications is chosen to be approximately several hundred mV so that MOS transistors are fast enough for the given application. (Section 1.6.8 shows that the transition frequency fr of an MOS transistor is proportional to the overdrive.) Under these conditions, the transconductance per given current is much higher for bipolar transistors than for MOS transistors. One of the key challenges in MOS analog circuit design is designing high-quality analog circuits with a low transconductance-to- current ratio. The transconductance calculated in (1.180) is valid for small-signal analysis. To de- termine the limitation on the use of small-signal analysis, the change in the drain current resulting from a change in the gate-source voltage will be derived from a large-signal standpoint. The total drain current in Fig. 1.33 can be calculated using (1.157) as
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1.6 Small-Signal Models of M S Transistors O 51Substituting (1.157) in (1.183) givesRearranging (1.184) givesIf the magnitude of the small-signal input IviI is much less than twice the overdrive definedin (1.166), substituting (1.180) into (1.185) givesIn particular, if lvil = lAvGsl is less than 20 percent of the overdrive, the small-signalanalysis is accurate within about 10 percent.1.6.2 Intrinsic Gate-Source and Gate-DrainCapacitanceIf Coxis the oxide capacitance per unit area from gate to channel, then the total capacitanceunder the gate is CoxW L . This capacitance is intrinsic to the device operation and mod-els the gate control of the channel conductance. In the triode region of device operation,the channel exists continuously from source to drain, and the gate-channel capacitance isusually lumped into two equal parts at the drain and source with In the saturation or active region, however, the channel pinches off before reaching thedrain, and the drain voltage exerts little influence on either the channel or the gate charge.As a consequence, the intrinsic portion of Cgdis essentially zero in the saturation region.To calculate the value of the intrinsic part of C,, in the saturation or active region, we mustcalculate the total charge QT stored in the channel. This calculation can be carried out bysubstituting (1.145) into (1.144) and integrating to obtain L QT = wcOxlo[VGS - V ( Y )- vtldy (1.188)Solving (1.150) for d y and substituting into (1.188) giveswhere the limit y = L corresponds to V = (VGS- V t )in the saturation or active region.Solution of (1.189) and use of (1.153) and (1.157) gives 2 QT = - vt) SwLcox(v~~ (1.190)Therefore, in the saturation or active region, ~ Q T 2 C,, = - = -WLC,, ~VGS 3and
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52 Chapter 1 . Models for Integrated-Circuit Active Devices 1.6.3 Input Resistance The gate of an MOS transistor is insulated from the channel by the Si02 dielectric. As a result, the low-frequency gate current is essentially zero and the input resistance is essentially infinite. This characteristic is important in some circuits such as sample-and- hold amplifiers, where the gate of an MOS transistor can be connected to a capacitor to sense the voltage on the capacitor without leaking away the charge that causes that voltage. In contrast, bipolar transistors have small but nonzero base current and finite input resistance looking into the base, complicating the design of bipolar sample-and- hold amplifiers. 1.6.4 Output Resistance In Section 1.5.1, the effect of changes in drain-source voltage on the large-signal char- acteristics of the MOS transistor was described. Increasing drain-source voltage in an n-channel MOS transistor increases the width of the depletion region around the drain and reduces the effective channel length of the device in the saturation or active re- gion. This effect is called channel-length modulation and causes the drain current to increase when the drain-source voltage is increased. From that treatment, we can calcu- late the change in the drain current AID arising from changes in the drain-source voltage AVDs as Substitution of (1.161), (1.163), and (1.164) in (1.193) gives where VA is the Early voltage, A is the channel-length modulation parameter, ID is the drain current without channel-length modulation given by (1.157), and r, is the small- signal output resistance of the transistor. 1.6.5 Basic Small-Signal Model of the MOS Transistor Combination of the preceding small-signal circuit elements yields the small-signal model of the MOS transistor shown in Fig. 1.34. This model was derived for n-channel transistors in the saturation or active region and is called the hybrid-.rr model. Drain, gate, and source nodes are labeled D, G, and S, respectively. When the gate-source voltage is increased, the model predicts that the incremental current id flowing from drain to source increases. Since the dc drain current ID also flows from drain to source in an n-channel transistor, Figure 1.34 Basic small-signal model of an MOS transistor in the saturation or active region.
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1.6 Small-Signal Models of MOS Transistors 53increasing the gate-source voltage also increases the total drain current Id. This result isreasonable physically because increasing the gate-source voltage in an n-channel transistorincreases the channel conductivity and drain current. The model shown in Fig. 1.34 is also valid forp-channel devices. Therefore, the modelagain shows that increasing the gate-source voltage increases the incremental current idflowing from drain to source. Unlike in the n-channel case, however, the dc current ID ina p-channel transistor flows from source to drain because the source acts as the source ofholes. Therefore, the incremental drain current flows in a direction opposite to the dc draincurrent when the gate-source voltage increases, reducing the total drain current Id. Thisresult is reasonable physically because increasing the gate-source voltage in a p-channeltransistor reduces the channel conductivity and drain current.1.6.6 Body TransconductanceThe drain current is a function of both the gate-source and body-source voltages. On the onehand, the gate-source voltage controls the vertical electric field, which controls the channelconductivity and therefore the drain current. On the other hand, the body-source voltagechanges the threshold, which changes the drain current when the gate-source voltage isfixed. This effect stems from the influence of the substrate acting as a second gate and iscalled the body effect. Note that the body of an MOS transistor is usually connected to aconstant power-supply voltage, which is a small-signal or ac ground. However, the sourceconnection can have a significant ac voltage impressed on it, which changes the body-source voltage when the body voltage is fixed. Therefore, when the body-source voltageis not constant, two transconductance terms are required to model MOS transistors: oneassociated with the main gate and the other associated with the body or second gate. Using (1.165), the transconductance from the body or second gate isFrom (1.140)This equation defines a factor X , which is the rate of change of threshold voltage with bodybias voltage. Substitution of (1.141) in (1.196) and use of (1.20) giveswhere Cj, is the capacitance per unit area of the depletion region under the channel, assum-ing a one-sided step junction with a built-in potential $o = 24f . Substitution of (1.196)in (1.195) givesIf hVDs < 1, we have <
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54 Chapter 1 . Models for Integrated-CircuitActive Devices The ratio gmb/gmis an important quantity in practice. From (1.179) and (1.198), we find The factor x is typically in the range 0.1 to 0.3; therefore, the transconductance from the main gate is typically a factor of about 3 to 10 times larger than the transconductance from the body or second gate. 1.6.7 Parasitic Elements in the Small-Signal Model The elements of the small-signal model for MOS transistors described above may be con- sidered basic in the sense that they arise directly from essential processes in the device. As in the case of bipolar transistors, however, technological limitations in the fabrication of the devices give rise to a number of parasitic elements that must be added to the equiva- lent circuit for most integrated-circuit transistors. A cross section and top view of a typical n-channel MOS transistor are shown in Fig. 1.35. The means of fabricating such devices is described in Chapter 2. All pn junctions in the MOS transistor should be reverse biased during normal op- eration, and each junction exhibits a voltage-dependent parasitic capacitance associated with its depletion region. The source-body and drain-body junction capacitances shown in Fig. 1 . 3 5 are Csband Cdb,respectively. If the doping levels in the source, drain, and ~ body regions are assumed to be constant, (1.21) can be used to express these capacitances as follows: Source Gate Drain Body
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1.6 Small-Signal Models of MOS Transistors 55These capacitances are proportional to the source and drain region areas (including side-walls). Since the channel is attached to the source in the saturation or active region, CXbalso includes depletion-region capacitance from the induced channel to the body. A de-tailed analysis of the channel-body capacitance is given in T ~ i v i d i s . ~ ~ In practice, C,, and Cgd,given in (1.187) for the triode region of operation and in(1.191) and (1.192) for the saturation or active region, are increased due to parasitic oxidecapacitances arising from gate overlap of the source and drain regions. These overlapcapacitances COlare shown in Fig. 1.35a, and their values are calculated in Chapter 2. Capacitance Cgbbetween gate and body or substrate models parasitic oxide capaci-tance between the gate-contact material and the substrate outside the active-device area.This capacitance is independent of the gate-body voltage and models coupling frompolysilicon and metal interconnects to the underlying substrate, as shown by the shadedregions in the top view of Fig. 1.35b. Parasitic capacitance of this type underlies allpolysilicon and metal traces on integrated circuits. Such parasitic capacitance should betaken into account when simulating and calculating high-frequency circuit and deviceperformance. Typical values depend on oxide thicknesses. With a silicon dioxide thick-ness of 100 A , the capacitance is about 3.45 fF per square micron. Fringing capacitancebecomes important for lines narrower in width than several microns. Parasitic resistance in series with the source and drain can be used to model thenonzero resistivity of the contacts and diffusion regions. In practice, these resistances areoften ignored in hand calculations for simplicity but included in computer simulations.These parasitic resistances have an inverse dependence on channel width W. Typical val-ues of these resistances are 50 fl to 100 fl for devices with W of about 1 Fm. Similarparasitic resistances in series with the gate and body terminals are sometimes includedbut often ignored because very little current flows in these terminals, especially at lowfrequencies. The small-signal model including capacitive parasitics but ignoring resistiveparasitics is shown in Fig. 1.36.1.6.8 MOS Transistor Frequency ResponseAs for a bipolar transistor, the frequency capability of an MOS transistor is usually spec-ified by finding the transition frequency f ~ . an MOS transistor, f~ is defined as the For Go II OD I- - -Ff cdb S CXb B Figure 1.36 Small-signal MOS transistor equivalent circuit.
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56 Chapter 1 Models for Integrated-Circuit Active Devices * * (a) (b) Figure 1.37 Circuits for calculating the fT of an MOS transistor: (a) ac schematic and (b) small- signal equivalent. frequency where the magnitude of the short-circuit, common-source current gain falls to unity. Although the dc gate current of an MOS transistor is essentially zero, the high- frequency behavior of the transistor is controlled by the capacitive elements in the small- signal model, which cause the gate current to increase as frequency increases. To calculate f T , consider the ac circuit of Fig. 1.37~ the small-signal equivalent of Fig. 1.37b. Since and Vsb = vds = 0, gmb,ro, Csb,and Cdbhave no effect on the calculation and are ignored. The small-signal input current ii is If the current fed forward through Cgdis neglected, Solving (1.203) for v,, and substituting into (1.204) gives To find the frequency response, we set s = j w . Then The magnitude of the small-signal current gain is unity when Therefore, Assume the intrinsic device capacitance C,, is much greater than (Cgb+ Cgd).Then sub- stituting (1.180) and (1.191) into (1.208) gives Comparison of this equation with the intrinsic fT of a bipolar transistor when parasitic depletion-layer capacitance is neglected leads to an interesting result. From (1.128) and (1.130) with TF > ( C j e+ Cp)lgm, >
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1.6 Small-SignalModels of MOS Transistors 57 Substituting from (1.99) for TF and using the Einstein relationship Dnlpn = kTlq = Vr, we find for a bipolar transistor The similarity in form between (1.21 1 ) and (1.209) is striking. In both cases, the intrinsic device fT increases as the inverse square of the critical device dimension across which carriers are in transit. The voltage VT = 26 mV is fixed for a bipolar transistor, but the fT of an MOS transistor can be increased by operating at high values of (VGS- V r )Note that the base width W B in a bipolar transistor is a vertical dimension determined by dif- fusions or implants and can typically be made much smaller than the channel length L of an MOS transistor, which depends on surface geometry and photolithographic processes. Thus bipolar transistors generally have higher fT than MOS transistors made with compa- rable processing. Finally, (1.209) was derived assuming that the MOS transistor exhibits square-law behavior as in (1.157). However, as described in Section 1.7, submicron MOS transistors depart significantly from square-law characteristics, and we find that for such devices fT is proportional to L- rather than L - ~ .I EXAMPLE Derive the complete small-signal model for an NMOS transistor with ID = 100 FA, VsB = 1 V, VDS = 2 V. Device parameters are c $ ~ = 0.3 V , W = 10 pm, L = 1 pm, y = 0.5 v " ~ , k t = 200 p m 2 , A = 0.02 V - l , to, = 100 angstroms, $0 = 0.6 V , CsbO= CdbO= 10 fF. Overlap capacitance from gate to source and gate to drain is 1 fF. Assume Cgb = 5 fF. From (1.166), Since VDs > Vov,the transistor operates in the saturation or active region. From (1.180), From (1.199), From (1.194), 1 - T O = - - looo kdl = 500 kdl AID 0.02 X 100 Using (1.201) with Vss = 1 V , we find The voltage from drain to body is
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58 Chapter 1 Models for Integrated-CircuitActive Devices and substitution in (1.202) gives 10 Cdb = fF=4fF (l+&:;" From (1.142), the oxide capacitance per unit area is The intrinsic portion of the gate-source capacitance can be calculated from (1.191), giving C,, = 2 - 3 X 10 X 1 x 3.45 fF - 23 fF The addition of overlap capacitance gives C,, = 24 fF Finally, since the transistor operates in the saturation or active region, the gate-drain ca- pacitance consists of only overlap capacitance and is The complete small-signal equivalent circuit is shown in Fig. 1.38. The fT of the device can be calculated from (1.208) as 1 gm - -I 1015 T - - - X 632 X lop6 X Hz = 3.4 GHz - 2 n c,, + + cgb cgd 237 24+5+1 1.7 Short-Channel Effects in MOS Transistors The evolution of integrated-circuit processing techniques has led to continuing reduc- tions in both the horizontal and vertical dimensions of the active devices. (The minimum 1 fF I ! Figure 1.38 Complete small-signal equivalent circuit for an NMOS transistor with ID = 100 pA, f V S B = 1 V , VDS = 2 V . Device parameters are W = 10 pm, L = 1 pm, y = 0.5 V1I2,k = 1 A, 200 p A N 2 , A = 0.02 V-I, to, = 100 $0 = 0.6 V , CsbO= CdhO 10 fF, Cgd = 1 fF, and = Cgb= 5 fF. 1
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1.7 Short-Channel Effects in MOS Transistors 59allowed dimensions of passive devices have also decreased.) This trend is driven primarilyby economics in that reducing dimensions increases the number of devices and circuits thatcan be processed at one time on a given wafer. A second benefit has been that the frequencycapability of the active devices continues to increase, as intrinsic fr values increase withsmaller dimensions while parasitic capacitances decrease. Vertical dimensions such as the base width of a bipolar transistor in production pro-cesses may now be on the order of 0.05 p m or less, whereas horizontal dimensions suchas bipolar emitter width or MOS transistor gate length may be significantly less than 1pm. Even with these small dimensions, the large-signal and small-signal models of bipo-lar transistors given in previous sections remain valid. However, significant short-channeleffects become important in MOS transistors at channel lengths of about 1 p m or less andrequire modifications to the MOS models given previously. The primary effect is to mod-ify the classical MOS square-law transfer characteristic in the saturation or active regionto make the device voltage-to-current transfer characteristic more linear. However, evenin processes with submicron capability, many of the MOS transistors in a given analog cir-cuit may be deliberately designed to have channel lengths larger than the minimum andmay be well approximated by the square-law model.1.7.1 Velocity Saturation from the Horizontal FieldThe most important short-channel effect in MOS transistors stems from velocity satura-tion of carriers in the channel.27 When an MOS transistor operates in the triode region,the average horizontal electric field along the channel is VDsIL.When VDsis small andlorL is large, the horizontal field is low, and the linear relation between carrier velocity andfield assumed in (1.148) is valid. At high fields, however, the carrier velocities approachthe thermal velocities, and subsequently the slope of the carrier velocity decreases withincreasing field. This effect is illustrated in Fig. 1.39, which shows typical measured elec-tron drift velocity vd versus horizontal electric field strength magnitude 8 in an NMOSsurface channel. While the velocity at low field values is proportional to the field, the ve-locity at high field values approaches a constant called the scattering-limited velocity v,,~.A first-order analytical approximation to this curve is Figure 1.39 Typical measured electron drift velocity vd versus hori- zontal electric field 8 in an MOS surface chan- nel (solid plot). Also shown (dashed plot) is the analytical approxi- 5 x 1 0 ~ ~ I I c 1o5 1o6 1o7 mation of Eq. 1.212 with 8, = 1.5 x 106 V/m Electric field %: (V/m) and p n = 0.07 m2N-s.
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60 Chapter 1 Models for Integrated-CircuitActive Devices where %, = 1.5 X lo6 Vlm and p n = 0.07 m2/v-s is the low-field mobility close to the gate. Equation 1.212 is also plotted in Fig. 1.39. From (1.212), as % - w, vd -+ v , , ~ = . pn%,.At the critical field value % the carrier velocity is a factor of 2 less than the low- , field formula would predict. In a device with a channel length L = 0.5 pm, we need a voltage drop of only 0.75 V along the channel to produce an average field equal to %, and this condition is readily achieved in short-channel MOS transistors. Similar results are found for PMOS devices. Substituting (1.212) and (1.149) into (1.147) and rearranging gives Note that as %, + and velocity saturation becomes negligible, (1.213) approaches the original equation (1.147). Integrating (1.213) along the channel, we obtain and thus In the limit as 8,--+ a,(1.215) is the same as (1.152), which gives the drain current in the triode region without velocity saturation. The quantity VDs/L in (1.215) can be interpreted as the average horizontal electric field in the channel. If this field is comparable to % the
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