Semiconductors physics and devices basic principles 4th edition,

Tai ngay!!! Ban co the xoa dong chu nay!!! Semiconductor Physics and Devices Basic Principles Fourth Edition Donald A Neamen University of New Mexico TM nea29583_fm_i-xxiv.indd i 12/11/10 1:01 PM TM SEMICONDUCTOR PHYSICS & DEVICES: BASIC PRINCIPLES, FOURTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2012 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2003, 1997 and 1992 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper DOC/DOC ISBN MHID 978-0-07-352958-5 0-07-352958-3 Vice President & Editor-in-Chief: Marty Lange Vice President EDP/Central Publishing Services: Kimberly Meriwether David Publisher: Raghu Srinivasan Sponsoring Editor: Peter E Massar Marketing Manager: Curt Reynolds Development Editor: Lora Neyens Project Manager: Melissa M Leick Design Coordinator: Brenda A Rolwes Cover Designer: Studio Montage, St Louis, Missouri Cover Image: © Getty Images RF Buyer: Sherry L Kane Media Project Manager: Balaji Sundararaman Compositor: MPS Limited, a Macmillan Company Typeface: 10/12 Times Roman Printer: RR Donnelley, Crawfordsville All credits appearing on page or at the end of the book are considered to be an extension of the copyright page Library of Congress Cataloging-in-Publication Data Neamen, Donald A Semiconductor physics and devices : basic principles / Donald A Neamen — 4th ed p cm Includes index ISBN 978-0-07-352958-5 (alk paper) Semiconductors I Title QC611.N39 2011 537.6'22—dc22 2010045765 www.mhhe.com nea29583_fm_i-xxiv.indd ii 12/11/10 1:01 PM ABOUT THE AUTHOR Donald A Neamen is a professor emeritus in the Department of Electrical and Computer Engineering at the University of New Mexico where he taught for more than 25 years He received his Ph.D from the University of New Mexico and then became an electronics engineer at the Solid State Sciences Laboratory at Hanscom Air Force Base In 1976, he joined the faculty in the ECE department at the University of New Mexico, where he specialized in teaching semiconductor physics and devices courses and electronic circuits courses He is still a part-time instructor in the department He also recently taught for a semester at the University of Michigan-Shanghai Jiao Tong University (UM-SJTU) Joint Institute in Shanghai, China In 1980, Professor Neamen received the Outstanding Teacher Award for the University of New Mexico In 1983 and 1985, he was recognized as Outstanding Teacher in the College of Engineering by Tau Beta Pi In 1990, and each year from 1994 through 2001, he received the Faculty Recognition Award, presented by graduating ECE students He was also honored with the Teaching Excellence Award in the College of Engineering in 1994 In addition to his teaching, Professor Neamen served as Associate Chair of the ECE department for several years and has also worked in industry with Martin Marietta, Sandia National Laboratories, and Raytheon Company He has published many papers and is the author of Microelectronics Circuit Analysis and Design, 4th edition, and An Introduction to Semiconductor Devices nea29583_fm_i-xxiv.indd iii 12/11/10 1:01 PM CONTENTS Preface x 2.2 Prologue—Semiconductors and the Integrated Circuit xvii PART I—Semiconductor Material Properties CHAPTER 2.2.1 2.2.2 2.2.3 2.3 Preview Semiconductor Materials Types of Solids Space Lattices 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.5 *1.6 2.4 Primitive and Unit Cell Basic Crystal Structures Crystal Planes and Miller Indices Directions in Crystals The Diamond Structure 10 Atomic Bonding 12 Imperfections and Impurities in Solids 1.6.1 1.6.2 *1.7 14 Imperfections in Solids 14 Impurities in Solids 16 Growth of Semiconductor Materials 17 1.7.1 Growth from a Melt 17 1.7.2 Epitaxial Growth 19 1.8 3.0 3.1 Preview 58 Allowed and Forbidden Energy Bands 59 Electrical Conduction in Solids 72 3.2.1 The Energy Band and the Bond Model 72 3.2.2 Drift Current 74 3.2.3 Electron Effective Mass 75 3.2.4 Concept of the Hole 78 3.2.5 Metals, Insulators, and Semiconductors 80 Introduction to Quantum Mechanics 25 2.1.1 Energy Quanta 26 2.1.2 Wave–Particle Duality 27 2.1.3 The Uncertainty Principle 30 3.1.1 Formation of Energy Bands 59 *3.1.2 The Kronig–Penney Model 63 3.1.3 The k-Space Diagram 67 Preview 25 Principles of Quantum Mechanics Summary 51 Problems 52 Introduction to the Quantum Theory of Solids 58 3.2 2.0 2.1 The One-Electron Atom 46 The Periodic Table 50 CHAPTER Summary 20 Problems 21 CHAPTER Electron in Free Space 35 The Infinite Potential Well 36 The Step Potential Function 39 The Potential Barrier and Tunneling 44 Extensions of the Wave Theory to Atoms 46 2.4.1 2.4.2 2.5 31 The Wave Equation 31 Physical Meaning of the Wave Function 32 Boundary Conditions 33 Applications of Schrodinger’s Wave Equation 34 2.3.1 2.3.2 2.3.3 2.3.4 The Crystal Structure of Solids 1.0 1.1 1.2 1.3 Schrodinger’s Wave Equation 26 3.3 Extension to Three Dimensions 3.3.1 3.3.2 83 The k-Space Diagrams of Si and GaAs 83 Additional Effective Mass Concepts 85 iv nea29583_fm_i-xxiv.indd iv 12/11/10 1:01 PM Contents 3.4 Density of States Function 4.7 85 3.4.1 Mathematical Derivation 85 3.4.2 Extension to Semiconductors 88 3.5 Statistical Mechanics 91 3.6 Preview 106 Charge Carriers in Semiconductors 4.1.1 4.1.2 4.1.3 4.1.4 4.2 107 Equilibrium Distribution of Electrons and Holes 107 The n0 and p0 Equations 109 The Intrinsic Carrier Concentration 113 The Intrinsic Fermi-Level Position 116 Dopant Atoms and Energy Levels The Extrinsic Semiconductor 123 4.3.1 Equilibrium Distribution of Electrons and Holes 123 4.3.2 The n0 p0 Product 127 *4.3.3 The Fermi–Dirac Integral 128 4.3.4 Degenerate and Nondegenerate Semiconductors 130 4.4 5.2 5.3 Statistics of Donors and Acceptors 131 Charge Neutrality 135 *5.4 5.5 4.6 Position of Fermi Energy Level nea29583_fm_i-xxiv.indd v Graded Impurity Distribution 176 Induced Electric Field 176 The Einstein Relation 178 The Hall Effect 180 Summary 183 Problems 184 Nonequilibrium Excess Carriers in Semiconductors 192 6.0 6.1 Preview 192 Carrier Generation and Recombination 6.1.1 6.1.2 6.2 198 6.2.1 Continuity Equations 198 6.2.2 Time-Dependent Diffusion Equations Ambipolar Transport 6.3.1 6.3.2 6.3.3 6.3.4 *6.3.5 193 The Semiconductor in Equilibrium 193 Excess Carrier Generation and Recombination 194 Characteristics of Excess Carriers 141 4.6.1 Mathematical Derivation 142 4.6.2 Variation of EF with Doping Concentration and Temperature 144 4.6.3 Relevance of the Fermi Energy 145 172 Diffusion Current Density 172 Total Current Density 175 CHAPTER 6.3 4.5.1 Compensated Semiconductors 135 4.5.2 Equilibrium Electron and Hole Concentrations 136 Carrier Diffusion 5.3.1 5.3.2 4.4.1 Probability Function 131 4.4.2 Complete Ionization and Freeze-Out 132 4.5 Preview 156 Carrier Drift 157 5.2.1 5.2.2 118 4.2.1 Qualitative Description 118 4.2.2 Ionization Energy 120 4.2.3 Group III–V Semiconductors 122 4.3 5.0 5.1 5.1.1 Drift Current Density 157 5.1.2 Mobility Effects 159 5.1.3 Conductivity 164 5.1.4 Velocity Saturation 169 The Semiconductor in Equilibrium 106 4.0 4.1 Carrier Transport Phenomena 156 Summary 98 Problems 100 CHAPTER Summary 147 Problems 149 CHAPTER 3.5.1 Statistical Laws 91 3.5.2 The Fermi–Dirac Probability Function 91 3.5.3 The Distribution Function and the Fermi Energy 93 v 199 201 Derivation of the Ambipolar Transport Equation 201 Limits of Extrinsic Doping and Low Injection 203 Applications of the Ambipolar Transport Equation 206 Dielectric Relaxation Time Constant 214 Haynes–Shockley Experiment 216 12/11/10 1:01 PM vi 6.4 *6.5 Contents 6.5.1 6.5.2 *6.6 8.1.4 Minority Carrier Distribution 283 8.1.5 Ideal pn Junction Current 286 8.1.6 Summary of Physics 290 8.1.7 Temperature Effects 292 8.1.8 The “Short” Diode 293 Quasi-Fermi Energy Levels 219 Excess Carrier Lifetime 221 Shockley–Read–Hall Theory of Recombination 221 Limits of Extrinsic Doping and Low Injection 225 Surface Effects 8.2 227 6.6.1 Surface States 227 6.6.2 Surface Recombination Velocity 229 6.7 Summary 231 Problems 233 PART 8.2.1 Generation–Recombination Currents 296 8.2.2 High-Level Injection 302 8.3 *8.4 Preview 241 Basic Structure of the pn Junction Zero Applied Bias 243 242 7.2.1 Built-in Potential Barrier 243 7.2.2 Electric Field 246 7.2.3 Space Charge Width 249 7.3 Reverse Applied Bias 251 7.3.1 Space Charge Width and Electric Field 251 7.3.2 Junction Capacitance 254 7.3.3 One-Sided Junctions 256 7.4 *7.5 Junction Breakdown 258 Nonuniformly Doped Junctions *8.5 8.6 CHAPTER 9.0 9.1 9.2 8.1.1 9.3 277 Qualitative Description of Charge Flow in a pn Junction 277 8.1.2 Ideal Current–Voltage Relationship 278 8.1.3 Boundary Conditions 279 nea29583_fm_i-xxiv.indd vi 318 Preview 331 The Schottky Barrier Diode 332 Metal–Semiconductor Ohmic Contacts 349 9.2.1 Ideal Nonrectifying Barrier 349 9.2.2 Tunneling Barrier 351 9.2.3 Specific Contact Resistance 352 Preview 276 pn Junction Current The Tunnel Diode Summary 321 Problems 323 9.1.1 Qualitative Characteristics 332 9.1.2 Ideal Junction Properties 334 9.1.3 Nonideal Effects on the Barrier Height 338 9.1.4 Current–Voltage Relationship 342 9.1.5 Comparison of the Schottky Barrier Diode and the pn Junction Diode 345 The pn Junction Diode 276 8.0 8.1 314 Metal–Semiconductor and Semiconductor Heterojunctions 331 262 Summary 267 Problems 269 304 The Turn-off Transient 315 The Turn-on Transient 317 CHAPTER 7.5.1 Linearly Graded Junctions 263 7.5.2 Hyperabrupt Junctions 265 7.6 Charge Storage and Diode Transients 8.4.1 8.4.2 The pn Junction 241 7.0 7.1 7.2 Small-Signal Model of the pn Junction 8.3.1 Diffusion Resistance 305 8.3.2 Small-Signal Admittance 306 8.3.3 Equivalent Circuit 313 II—Fundamental Semiconductor Devices CHAPTER Generation–Recombination Currents and High-Injection Levels 295 Heterojunctions 9.3.1 9.3.2 9.3.3 *9.3.4 *9.3.5 354 Heterojunction Materials 354 Energy-Band Diagrams 354 Two-Dimensional Electron Gas 356 Equilibrium Electrostatics 358 Current–Voltage Characteristics 363 12/11/10 1:01 PM Contents 9.4 11.1.2 Channel Length Modulation 446 11.1.3 Mobility Variation 450 11.1.4 Velocity Saturation 452 11.1.5 Ballistic Transport 453 Summary 363 Problems 365 CHAPTER 10 Fundamentals of the Metal–Oxide– Semiconductor Field-Effect Transistor 371 10.0 10.1 Preview 371 The Two-Terminal MOS Structure Capacitance–Voltage Characteristics The Basic MOSFET Operation Frequency Limitations 10.4.1 10.4.2 *10.5 10.6 11.4 CHAPTER 394 422 Additional Electrical Characteristics Radiation and Hot-Electron Effects 11.5.1 11.5.2 11.5.3 11.6 464 475 Radiation-Induced Oxide Charge 475 Radiation-Induced Interface States 478 Hot-Electron Charging Effects 480 Summary 481 Problems 483 CHAPTER 12 The Bipolar Transistor 491 12.0 12.1 Preview 491 The Bipolar Transistor Action 12.1.1 12.1.2 427 12.1.3 12.1.4 12.2 492 The Basic Principle of Operation 493 Simplified Transistor Current Relation— Qualitative Discussion 495 The Modes of Operation 498 Amplification with Bipolar Transistors 500 Minority Carrier Distribution 501 12.2.1 Forward-Active Mode 502 12.2.2 Other Modes of Operation 508 11 Preview 443 Nonideal Effects 444 11.1.1 Subthreshold Conduction 444 nea29583_fm_i-xxiv.indd vii *11.5 403 Metal–Oxide–Semiconductor Field-Effect Transistor: Additional Concepts 443 11.0 11.1 457 11.4.1 Breakdown Voltage 464 *11.4.2 The Lightly Doped Drain Transistor 470 11.4.3 Threshold Adjustment by Ion Implantation 472 Small-Signal Equivalent Circuit 422 Frequency Limitation Factors and Cutoff Frequency 425 The CMOS Technology Summary 430 Problems 433 Threshold Voltage Modifications 11.3.1 Short-Channel Effects 457 11.3.2 Narrow-Channel Effects 461 10.3.1 MOSFET Structures 403 10.3.2 Current–Voltage Relationship—Concepts 404 *10.3.3 Current–Voltage Relationship— Mathematical Derivation 410 10.3.4 Transconductance 418 10.3.5 Substrate Bias Effects 419 10.4 MOSFET Scaling 455 11.2.1 Constant-Field Scaling 455 11.2.2 Threshold Voltage—First Approximation 456 11.2.3 Generalized Scaling 457 11.3 10.2.1 Ideal C–V Characteristics 394 10.2.2 Frequency Effects 399 10.2.3 Fixed Oxide and Interface Charge Effects 400 10.3 11.2 372 10.1.1 Energy-Band Diagrams 372 10.1.2 Depletion Layer Thickness 376 10.1.3 Surface Charge Density 380 10.1.4 Work Function Differences 382 10.1.5 Flat-Band Voltage 385 10.1.6 Threshold Voltage 388 10.2 vii 12.3 Transistor Currents and Low-Frequency Common-Base Current Gain 509 12.3.1 12.3.2 Current Gain—Contributing Factors 509 Derivation of Transistor Current Components and Current Gain Factors 512 12/11/10 1:01 PM viii Contents 12.3.3 12.3.4 12.4 Nonideal Effects 12.4.1 12.4.2 12.4.3 12.4.4 *12.4.5 12.4.6 12.5 Summary 517 Example Calculations of the Gain Factors 517 *13.3 13.3.1 13.3.2 13.3.3 522 Base Width Modulation 522 High Injection 524 Emitter Bandgap Narrowing 526 Current Crowding 528 Nonuniform Base Doping 530 Breakdown Voltage 531 Equivalent Circuit Models *13.4 *13.5 536 Frequency Limitations 12.7 Large-Signal Switching 13.6 549 Other Bipolar Transistor Structures 551 552 14.0 14.1 14.2.3 14.2.4 14.2.5 13.1.1 13.1.2 13.2 14.3 572 The Device Characteristics 578 13.2.1 Internal Pinchoff Voltage, Pinchoff Voltage, and Drain-to-Source Saturation Voltage 578 13.2.2 Ideal DC Current–Voltage Relationship— Depletion Mode JFET 582 13.2.3 Transconductance 587 13.2.4 The MESFET 588 nea29583_fm_i-xxiv.indd viii 14.4 619 624 The pn Junction Solar Cell 624 Conversion Efficiency and Solar Concentration 627 Nonuniform Absorption Effects 628 The Heterojunction Solar Cell 629 Amorphous Silicon Solar Cells 630 Photodetectors 14.3.1 14.3.2 14.3.3 14.3.4 14.3.5 Basic pn JFET Operation 572 Basic MESFET Operation 576 618 Photon Absorption Coefficient 619 Electron–Hole Pair Generation Rate 622 Solar Cells 14.2.1 14.2.2 13 Preview 571 JFET Concepts 14 Preview 618 Optical Absorption 14.1.1 14.1.2 The Junction Field-Effect Transistor 571 13.0 13.1 III—Specialized Semiconductor Devices Optical Devices 14.2 Summary 558 Problems 560 CHAPTER 602 Summary 609 Problems 611 CHAPTER 12.8.1 Polysilicon Emitter BJT 552 12.8.2 Silicon–Germanium Base Transistor 554 12.8.3 Heterojunction Bipolar Transistors 556 12.9 High Electron Mobility Transistor PART 546 12.7.1 Switching Characteristics 549 12.7.2 The Schottky-Clamped Transistor *12.8 Small-Signal Equivalent Circuit 598 Frequency Limitation Factors and Cutoff Frequency 600 13.5.1 Quantum Well Structures 603 13.5.2 Transistor Performance 604 545 12.6.1 Time-Delay Factors 545 12.6.2 Transistor Cutoff Frequency 593 Channel Length Modulation 594 Velocity Saturation Effects 596 Subthreshold and Gate Current Effects 596 Equivalent Circuit and Frequency Limitations 598 13.4.1 13.4.2 *12.5.1 Ebers–Moll Model 537 12.5.2 Gummel–Poon Model 540 12.5.3 Hybrid-Pi Model 541 12.6 Nonideal Effects 633 Photoconductor 633 Photodiode 635 PIN Photodiode 640 Avalanche Photodiode 641 Phototransistor 642 Photoluminescence and Electroluminescence 643 14.4.1 Basic Transitions 644 14.4.2 Luminescent Efficiency 645 14.4.3 Materials 646 12/13/10 6:09 PM Contents 14.5 Light Emitting Diodes 14.6 Laser Diodes 654 15.7 Summary 701 Problems 703 APPENDIX 14.6.1 Stimulated Emission and Population Inversion 655 14.6.2 Optical Cavity 657 14.6.3 Threshold Current 658 14.6.4 Device Structures and Characteristics 660 14.7 15.6.3 SCR Turn-Off 697 15.6.4 Device Structures 697 648 14.5.1 Generation of Light 648 14.5.2 Internal Quantum Efficiency 649 14.5.3 External Quantum Efficiency 650 14.5.4 LED Devices 652 Summary 661 Problems 664 ix A Selected List of Symbols 707 APPENDIX B System of Units, Conversion Factors, and General Constants 715 APPENDIX C The Periodic Table 719 CHAPTER 15 Semiconductor Microwave and Power Devices 670 15.0 15.1 15.2 15.3 15.4 Preview 670 Tunnel Diode 671 Gunn Diode 672 Impatt Diode 675 Power Bipolar Transistors Power MOSFETs The Thyristor 15.6.1 15.6.2 nea29583_fm_i-xxiv.indd ix E “Derivation” of Schrodinger’s Wave Equation 722 677 684 15.5.1 Power Transistor Structures 684 15.5.2 Power MOSFET Characteristics 685 15.5.3 Parasitic BJT 689 15.6 D Unit of Energy—The Electron Volt 720 APPENDIX 15.4.1 Vertical Power Transistor Structure 677 15.4.2 Power Transistor Characteristics 678 15.4.3 Darlington Pair Configuration 682 15.5 APPENDIX 691 The Basic Characteristics 691 Triggering the SCR 694 APPENDIX F Effective Mass Concepts 724 APPENDIX G The Error Function 729 APPENDIX H Answers to Selected Problems 730 Index 738 12/11/10 1:01 PM 13.2 The Device Characteristics 585 Equation (13.27) can also be written as _ ID1  G01 VDS  _ where V [(V  V  V )32  (V  V )3 2] _ DS bi GS bi GS p0 n(eNd)2 Wa3 en Nd Wa 3IP1 G01  _   _ L Vp0 2s LVp0 (13.30) (13.31) The channel conductance is defined as  ID1 gd  _ VDS V DS (13.32) →0 Taking the derivative of Equation (13.30) with respect to VDS , we obtain  ID1 gd  _ VDS V DS →0   Vbi  VGS  G01  Vp0   12 (13.33) We may note from Equation (13.33) that G01 would be the conductance of the channel if both Vbi and VGS were zero This condition would exist if no space charge regions existed in the channel We may also note, from Equation (13.33), that the channel conductance is modulated or controlled by the gate voltage This channel conductance modulation is the basis of the field-effect phenomenon We have shown that the drain becomes pinched off, for the n-channel JFET, when VDS  VDS (sat)  Vp0  (Vbi  VGS) (13.34) In the saturation region, the saturation drain current is determined by setting VDS  VDS(sat) in Equation (13.29) so that ID1  ID1(sat)  IP1  Vbi  VGS  Vp0  1_ Vbi  VGS Vp0  (13.35) The ideal saturation drain current is independent of the drain-to-source voltage Figure 13.12 shows the ideal current–voltage characteristics of a silicon n-channel JFET Objective: Calculate the maximum current in an n-channel JFET EXAMPLE 13.3 Consider a silicon n-channel JFET at T  300 K with the following parameters: Na  1018 cm3, Nd  1016 cm3, a  0.75 m, L  10 m, W  30 m, and n  1000 cm2 /V-s ■ Solution The pinchoff current from Equation (13.28) becomes (1000)[(1.6 1019) (1016)]2 (30 104)(0.75 104)3 IP1  _  0.522 mA 6(11.7) (8.85 1014) (10 104) We also have from Example 13.1 that Vbi  0.814 V and Vp0  4.35 V The maximum current occurs when VGS  0, so from Equation (13.35) ID1(max)  IP1 nea29583_ch13_571-617.indd 585 V   Vbi 13 _ p0 _ 1_ V V  bi _ (13.36) p0 12/11/10 12:47 PM 586 CHAPTER 13 The Junction Field-Effect Transistor or  0.814 ID1(max)  (0.522)  _ 4.35  _ 1_  0.313 mA 4.35  _ 0.814 ■ Comment The maximum current through the JFET is less than the pinchoff current IP1 ■ EXERCISE PROBLEM Ex 13.3 Consider an n-channel silicon pn JFET with parameters Na  1018 cm3, Nd  1016 cm3, a  0.40 m, L  m, W  50 m, and n  900 cm2/V-s Calculate the pinchoff current IP1 and the maximum drain current ID1 (sat) for VGS  [Ans IP1  0.237 mA, ID1 (sat)  22.13 ] The maximum saturation current calculated in this example is considerably less than that shown in Figure 13.12 because of the big difference in the width-to-length ratios Once the pinchoff voltage of JFET has been designed, the channel width W is the primary design variable for determining the current capability of a device Summary Equations (13.29) and (13.35) are rather cumbersome to use in any hand calculations We may show that, in the saturation region, the drain current is given to a good approximation by Equation (13.14), stated at the beginning of this section as  VGS ID  IDSS  _ Vp  The current IDSS is the maximum drain current and is the same as ID1(max ) in Equation (13.36) The parameter VGS is the gate-to-source voltage and Vp is the pinchoff VDS (sat)  (Vp0  Vbi )  VGS 28 Nonsaturation region 24 Saturation region ID (mA) 20 16 VGS  12 1 V 2 V 4 V 0 VDS (V) 3 V Figure 13.12 | Ideal current–voltage characteristics of a silicon n-channel JFET with a  1.5 m, WL  170, and Nd  2.5 1015 cm3 (From Yang [22].) nea29583_ch13_571-617.indd 586 12/11/10 12:47 PM 13.2 The Device Characteristics 587 1.0 0.8 0.6 Eq [13.14] Approximation 0.4 Ideal Eq [13.35] 4.0 3.0 ID1 ID10 0.2 2.0 1.0 VGS (V) Figure 13.13 | Comparison of Equations (13.14) and (13.35) for the ID versus VGS characteristics of a JFET biased in the saturation region voltage We may note that, for n-channel depletion mode JFET, both VGS and Vp are negative and, for the p-channel depletion mode device, both are positive Figure 13.13 shows the comparison between Equations (13.14) and (13.35) 13.2.3 Transconductance The transconductance is the transistor gain of the JFET; it indicates the amount of control the gate voltage has on the drain current The transconductance is defined as ID gm  _ VGS (13.37) Using the expressions for the ideal drain current derived in the last section, we can write the expressions for the transconductance The drain current for an n-channel depletion mode device in the nonsaturation region is given by Equation (13.29) We can then determine the transconductance of the transistor in the same region as gmL ID1 3IP1 _ _ VGS Vp0 Vbi  VGS Vp0   V V V     DS bi (13.38) GS Taking the limit as VDS becomes small, the transconductance becomes VDS 3IP1 _ gmL  _ · 2Vp0 Vp0(Vbi  VGS) (13.39) We can also write Equation (13.39) in terms of the conductance parameter G01 as VDS G01 _ gmL  _ · Vp0(Vbi  VGS) nea29583_ch13_571-617.indd 587 (13.40) 12/11/10 12:47 PM 588 CHAPTER 13 The Junction Field-Effect Transistor The ideal drain current in the saturation region for the JFET is given by Equation (13.35) The transconductance in the saturation region is then found to be VGS Vp0  ID1(sat) _ 3I gms   P1   Vp0  V V V  (13.41a) Vbi  VGS  G01  bi GS p0 Using the current–voltage approximation given by Equation (13.14), we can also write the transconductance as 2IDSS VGS gms  1_ (13.41b) Vp Vp   Since Vp is negative for the n-channel JFET, gms is positive EXAMPLE 13.4 Objective: Determine the maximum transconductance of an n-channel depletion mode JFET biased in the saturation region Consider the silicon JFET described in Example 13.3 We had calculated IP1  0.522 mA, Vbi  0.814 V, and Vp0  4.35 V ■ Solution The maximum transconductance occurs when VGS  Then Equation (13.41a) can be written as _ _ 3(0.522) Vbi 3IP1 0.814  0.204 mA/V gms(max)  _ 1 _   _ Vp0 Vp0 4.35 4.35