Physics of Semiconductor Devices

Massimo Rudan

Physics of SemiconductorDevices

Second Edition

123

Massimo RudanDEIUniversity of BolognaBologna, Italy

ISBN 978-3-319-63153-0 ISBN 978-3-319-63154-7 (eBook)DOI 10.1007/978-3-319-63154-7

Library of Congress Control Number: 2017947902

© Springer International Publishing AG 2015, 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

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To Rossella and Morgana

Foreword to the First Edition

We are delighted to write the foreword of this comprehensive book by MassimoRudan, entitled Physics of Semiconductor Devices, a treatise that goes to the heartof the physics involved in the study of electron devices. While the literature on thistopic is very wide and diversified, it typically covers only part of the whole subject:either semiconductor physics or device theory. This book bridges the gap betweenthese two aspects of the discipline, as it thoroughly covers both of them and evenaddresses process technology.

Why do we need a book on physics written by an engineer for engineers? Arenot physics books written by physicists good enough? The answer is rooted in thedifference of motivation of science as opposed to engineering. The scientist wantsto know why and investigates the cause or the underlying fundamental laws fromobserved effects. The engineer wants to know what for and pursues a purpose, likebuilding a device, based on such effects. The engineer tries to arrange physicalobjects in such a way that a desired and foreseeable action is achieved as can beexpected by virtue of the laws of physics. The engineer goes from the known causesto the desired effects. In this sense, the engineer’s objective is the inverse of thescientist’s endeavor.

Is there common ground between physics and engineering? Yes, and that is whyphysicists and engineers should be friends. Obviously, knowledge of the appropriatepart of physics (but not all physics) is indispensable for the engineer to buildsomething useful. That is what this book aims at in the area of electronic devices. Onthe other hand, physicists build sophisticated laboratory equipment (sometimes withthe help of engineers) for the purpose of investigating new physical phenomena.The builders of such unique tools do not have to worry about mass manufacturingin contrast to engineering, where low-cost production and reliability are majorconcerns, yet another difference of motivation.

The complexity of the physical background required for a deep understandingof device behavior makes it difficult to pursue a deductive teaching methodology,where every new concept is justified on fundamental physical principles. The limitednumber of lecture hours, and the need to comply with the program and objectives ofengineering courses, is often such that many teachers, as well as textbooks, tend to

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ignore such a background and to use an inductive approach instead. So doing, somefundamental concepts are delivered to students intuitively, in the form of physicalmodels with some degree of simplification and no proof behind them. While theinductive approach may be useful for practical purposes, it bears the drawback ofhiding assumptions and limitations of the model, thereby leading to an insufficientdepth of understanding and lack of criticism on the validity of the obtained results.

This book is fully functional to a deductive teaching approach and, as such,represents a very useful tool not just for students but for researchers and scientists aswell. The reader can thus find within the same volume both a rigorous justificationfor the semiclassical and quantum-mechanical models adopted in practice to clarifythe device operation, such as the most popular drift-diffusion transport model,and detailed descriptions of the operating principles of the most important semi-conductor devices. The evolution of microelectronics toward nanoscale structuresmakes it imperative to master quantum mechanics and solid-state physics in orderto understand the operation of miniaturized devices. This book is therefore timelyand meets a need felt by many students as well as teachers.

Roughly speaking, this volume addresses four main areas: (i) a review ofanalytical mechanics and electromagnetism, (ii) an introduction to statistical andquantum mechanics, (iii) a treatment of solid-state physics and carrier trans-port in semiconductors, and (iv) a description of the basic device concepts andtheir operating principles. The Appendices highlight mathematical concepts andproofs of theorems cited somewhere throughout the volume. For example, Chap. 6,where the Boltzmann transport equation is derived and discussed; Chap. 17, wherethe fundamental concepts of crystalline solid-state physics are introduced; andChap. 18, with the description of the basic properties of electrons and holes insemiconductors under equilibrium conditions, highlight the rigorous, yet elegant,approach to the description of delicate physical concepts, such as the Bloch theorem,the band structure of crystalline solids in the first Brillouin zone, and the phononvibrational spectra. The author’s deep mathematical background shows up in severaldemonstrations, where the use of sophisticated mathematical tools leads to simple,albeit rigorous, proofs of the concepts addressed in the treatment. An importantspecificity of this book is also that simplifying assumptions are widely discussedand their impact on final results is typically highlighted.

In conclusion, Physics of Semiconductor Devices by Massimo Rudan representsa remarkable piece of work that is likely to provide a valuable learning toolto electrical engineering students willing to deepen their understanding of thisfascinating field. It will also be a useful consultation tool for researchers, scientists,and engineers involved in the fundamental aspects of material science and devicedesign.

Bologna, Italy Giorgio BaccaraniZürich, Switzerland Henry BaltesSeptember 2014

Preface to the Second Edition

The second edition of this book has been enriched mostly in the parts devotedto semiconductors and devices (Chaps. 20 through 25). The analysis of the netrecombination rate in polycrystalline materials, relevant for, e.g., thin-film tran-sistors, is carried out in Chap. 20. The derivation of the characteristics of thebipolar transistor, photodiode, and solar cell has been added to Chap. 21, alongwith some considerations about the scaling limits of the bipolar architecture. Thenew material of Chap. 22 features the analysis of the n-substrate MOS capacitor,MOS photo-capacitor, and p-channel MOSFET, followed by the description of theCMOS architecture. A more general theory of the MOSFET, including the diffusivecontribution to the channel current, is given; a brief outline about the scaling rules ofsemiconductor devices and the design steps of integrated circuits completes the newmaterial of this chapter. Chapter 23 has been extended substantially by including thedescription of ion implantation. As for the measurement techniques, Chap. 25 hasbeen enriched with the theory of the Hall effect in the nonideal case and with theresistivity measurement based on the van der Pauw method.

More material has sparsely been added to other chapters: the derivation of theSchrödinger equation as the Euler equation of a constrained functional is extendedto more than one dimension in Chap. 1; the theory of the Green function in thetop half plane, instrumental to the van der Pauw measurement scheme, has beenadded to Chap. 4; a full derivation of the field generated by a point-like charge isnow included in Chap. 5. The demonstration of the Boltzmann H-theorem and adiscussion about the Kac-ring paradox have been added to Chap. 6. In the chaptersdevoted to quantum mechanics, the discussion of Chap. 13 about the propertiesof the one-dimensional Schrödinger equation is more extended, and the treatmentof the time-dependent perturbation of Chap. 14 is now embracing the harmonicperturbation of a general form and the spatially periodic case; the latter analysisprovides in turn the selection rules that are exploited in the discussion of theelectron-phonon interaction, added to Chap. 17. The transport model worked out inChap. 19 is used in the complements of the chapter to derive the Onsager relations.

The Appendices have also been enriched with new material, with the aim ofcompleting a number of topics tackled in the book’s chapters; several sections about

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matrix analysis have been added to Appendix A, followed by other sections showingthe application to the numerical solution of the semiconductor equations in onedimension. Appendix B has been enriched with the theories of Lagrange multipliers,conformal mapping, and contraction mappings; finally, the approximations to theFermi integrals, the illustration of the Bernoulli numbers and polynomials, thecalculation of a special integral using Cauchy’s residue theorem, the treatmentof the random walk problem, and the central limit theorem have been added toAppendix C.

About 50 problems, 50 figures, and 3 tables have been added to those of the firstedition and about 20 figures reworked; several new figures are in color. A subjectindex featuring about 2;600 entries has been inserted at the end of the volume.

Bologna, Italy Massimo RudanJune 2017

Preface to the First Edition

This volume originates from the lectures on solid-state electronics and microelec-tronics that I have been giving since 1978 at the School of Engineering of theUniversity of Bologna. Its scope is to provide the reader with a book that, startingfrom the elementary principles of classical mechanics and electromagnetism,introduces the concepts of quantum mechanics and solid-state theory and describesthe basic physics of semiconductors including the hierarchy of transport models,ending up with the standard mathematical model of semiconductor devices andthe analysis of the behavior of basic devices. The ambition of the work has beento write a book, self-contained as far as possible, that would be useful for bothstudents and researchers; to this purpose, a strong effort has been made to elucidatephysical concepts, mathematical derivations, and approximation levels, withoutbeing verbose.

The book is divided into eight parts. Part I deals with analytical mechanics andelectromagnetism; purposedly, the material is not given in the form of a resumé:quantum mechanics and solid-state theory concepts are so richly intertwined withthe classical ones that presenting the latter in an abridged form may make thereading unwieldy and the connections more difficult to establish. Part II providesthe introductory concepts of statistical mechanics and quantum mechanics, followedby the description of the general methods of quantum mechanics. The problem ofbridging the classical concepts with the quantum ones is first tackled using thehistorical perspective, covering the years from 1900 to 1926. The type of statisticaldescription necessary for describing the experiments, and the connection with thelimiting case of the same experiments involving massive bodies, is related to theproperties of the doubly stochastic matrices. Part III illustrates a number of appli-cations of the Schrödinger equation: elementary cases, solutions by factorization,and time-dependent perturbation theory. Part IV analyzes the properties of systemsof particles, with special attention to those made of identical particles, and themethods for separating the equations. The concepts above are applied in Part Vto the analysis of periodic structures, with emphasis to crystals of the cubic typeand to silicon in particular, which, since the late 1960s, has been and still is themost important material for the fabrication of integrated circuits. Part VI illustrates

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the single-electron dynamics in a periodic structure and derives the semiclassicalBoltzmann transport equation; from the latter, the hydrodynamic and drift-diffusionmodels of semiconductor devices are obtained using the moments expansion. Thedrift-diffusion model is used in Part VII to work out analytically the electricalcharacteristics for the basic devices of the bipolar and MOS type. Finally, Part VIIIpresents a collection of items which, although important per se, are not in the book’smainstream: some of the fabrication-process steps of integrated circuits (thermaldiffusion, thermal oxidation, layer deposition, epitaxy) and methods for measuringthe semiconductor parameters.

In the preparation of the book, I have been helped by many colleagues. I wishto thank, in particular, Giorgio Baccarani, Carlo Jacoboni, and Rossella Brunetti,who gave me important suggestions about the matter’s distribution in the book, readthe manuscript, and, with their observations, helped me to clarify and improve thetext; I wish also to thank, for reading the manuscript and giving me their comments,Giovanni Betti Beneventi, Fabrizio Buscemi, Gaetano D’Emma, Antonio Gnudi,Elena Gnani, Enrico Piccinini, Susanna Reggiani, and Paolo Spadini.

Last, but not least, I wish to thank the students, undergraduate, graduate, andpostdocs, who for decades have accompanied my teaching and research activity withstimulating curiosity. Many comments, exercises, and complements of this book arethe direct result of questions and comments that came from them.

Bologna, Italy Massimo RudanSeptember 2014

Contents

Part I A Review of Analytical Mechanics and Electromagnetism

1 Analytical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Lagrangian Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Force Deriving from a Potential Energy . . . . . . . . . . . . . . . . 71.3.2 Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.4 Hamilton Principle—Synchronous Trajectories . . . . . . . . 10

1.4 Generalized Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Hamiltonian Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Hamilton Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Time–Energy Conjugacy—Hamilton-Jacobi Equation . . . . . . . . . . . 151.8 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.9 Phase Space and State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.10 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.10.1 Higher-Order Variational Calculus . . . . . . . . . . . . . . . . . . . . . . 201.10.2 Lagrangian Invariance and Gauge Invariance . . . . . . . . . . 201.10.3 Variational Calculus with Constraints . . . . . . . . . . . . . . . . . . 211.10.4 An Interesting Example of Extremum Equation . . . . . . . 221.10.5 Constant-Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Coordinate Transformations and Invariance Properties . . . . . . . . . . . . . . . 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 An Application of the Canonical Transformation . . . . . . . . . . . . . . . . . 292.4 Separation—Hamilton’s Characteristic Function . . . . . . . . . . . . . . . . . 302.5 Phase Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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2.6 Invariance Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.1 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.2 Translation of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.3 Translation of the Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6.4 Rotation of the Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 Maupertuis Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.8 Spherical Coordinates—Angular Momentum . . . . . . . . . . . . . . . . . . . . . 372.9 Linear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.10 Action-Angle Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.11 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.11.1 Infinitesimal Canonical Transformations . . . . . . . . . . . . . . . 422.11.2 Constants of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Applications of the Concepts of Analytical Mechanics . . . . . . . . . . . . . . . . . 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Particle in a Square Well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Linear Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4 Central Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Two-Particle Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6 Energy Exchange in the Two-Particle Collision . . . . . . . . . . . . . . . . . . . 513.7 Central Motion in the Two-Particle Interaction . . . . . . . . . . . . . . . . . . . 533.8 Coulomb Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.9 System of Particles Near an Equilibrium Point . . . . . . . . . . . . . . . . . . . . 563.10 Diagonalization of the Hamiltonian Function . . . . . . . . . . . . . . . . . . . . . 583.11 Periodic Potential Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.12 Energy-Momentum Relation in a Periodic Potential Energy . . . . . 633.13 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.13.1 Comments on the Linear Harmonic Oscillator . . . . . . . . . 643.13.2 Degrees of Freedom and Coordinate Separation . . . . . . . 653.13.3 Comments on the Normal Coordinates . . . . . . . . . . . . . . . . . 663.13.4 Areal Velocity in the Central-Motion Problem . . . . . . . . . 673.13.5 Initial Conditions in the Central-Motion Problem . . . . . 673.13.6 The Coulomb Field in the Attractive Case. . . . . . . . . . . . . . 683.13.7 Dynamic Relations of Special Relativity . . . . . . . . . . . . . . . 713.13.8 Collision of Relativistic Particles . . . . . . . . . . . . . . . . . . . . . . . 723.13.9 Energy Conservation in Charged-Particles’

Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Extension of the Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Lagrangian Function for the Wave Equation . . . . . . . . . . . . . . . . . . . . . . 784.4 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5 Potentials and Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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4.6 Lagrangian Density for the Maxwell Equations . . . . . . . . . . . . . . . . . . . 834.7 Helmholtz Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.8 Helmholtz Equation in a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.9 Solution of the Helmholtz Equation in an Infinite Domain . . . . . . . 884.10 Solution of the Wave Equation in an Infinite Domain. . . . . . . . . . . . . 884.11 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.12 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.12.1 Invariance of the Euler Equations . . . . . . . . . . . . . . . . . . . . . . . 904.12.2 Wave Equations for the E and B Fields . . . . . . . . . . . . . . . . . 904.12.3 Comments on the Boundary-Value Problem . . . . . . . . . . . 914.12.4 Green Function for the Upper Half Plane. . . . . . . . . . . . . . . 92

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Applications of the Concepts of Electromagnetism . . . . . . . . . . . . . . . . . . . . . 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Potentials Generated by a Point-Like Charge . . . . . . . . . . . . . . . . . . . . . 955.3 Energy Continuity—Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4 Momentum Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.5 Modes of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.6 Energy of the Electromagnetic Field in Terms of Modes . . . . . . . . . 1025.7 Momentum of the Electromagnetic Field in Terms of Modes . . . . 1045.8 Modes of the Electromagnetic Field in an Infinite Domain . . . . . . . 1055.9 Eikonal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.10 Fermat Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.11 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.11.1 Field Generated by a Point-Like Charge. . . . . . . . . . . . . . . . 1085.11.2 Power Radiated by a Point-Like Charge . . . . . . . . . . . . . . . . 1095.11.3 Decay of Atoms According to the Classical Model . . . . 1115.11.4 Comments About the Field’s Expansion into Modes . . 1135.11.5 Finiteness of the Total Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.11.6 Analogies Between Mechanics and Geometrical

Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Part II Introductory Concepts to Statistical and QuantumMechanics

6 Classical Distribution Function and Transport Equation . . . . . . . . . . . . . . 1216.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3 Statistical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4 Maxwell-Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.5 Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.6 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.6.1 Momentum and Angular Momentum atEquilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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6.6.2 Averages Based on the Maxwell-BoltzmannDistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.6.3 Boltzmann’s H-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.6.4 Paradoxes—Kac-Ring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.6.5 Equilibrium Limit of the Boltzmann Transport

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7 From Classical Mechanics to Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . 1437.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.2 Planetary Model of the Atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.3 Experiments Contradicting the Classical Laws. . . . . . . . . . . . . . . . . . . . 148

7.3.1 Stability of the Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.3.2 Spectral Lines of Excited Atoms . . . . . . . . . . . . . . . . . . . . . . . . 1497.3.3 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.3.4 Spectrum of the Black-Body Radiation . . . . . . . . . . . . . . . . . 1527.3.5 Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.4 Quantum Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.4.1 Planck’s Solution of the Black-Body Problem . . . . . . . . . 1567.4.2 Einstein’s Solution of the Photoelectric Effect . . . . . . . . . 1577.4.3 Explanation of the Compton Effect . . . . . . . . . . . . . . . . . . . . . 1577.4.4 Bohr’s Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.4.5 De Broglie’s Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.5 Heuristic Derivation of the Schrödinger Equation . . . . . . . . . . . . . . . . 1627.6 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.6.1 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.6.2 Massive Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.6.3 Need of a Description of Probabilities . . . . . . . . . . . . . . . . . . 169

7.7 Born’s Interpretation of the Wave Function . . . . . . . . . . . . . . . . . . . . . . . 169Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8 Time-Independent Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.2 Properties of the Time-Independent Schrödinger Equation . . . . . . . 171

8.2.1 Schrödinger Equation for a Free Particle . . . . . . . . . . . . . . . 1738.2.2 Schrödinger Equation for a Particle in a Box. . . . . . . . . . . 1748.2.3 Lower Energy Bound in the Schrödinger Equation . . . . 176

8.3 Norm of a Function—Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.3.1 Adjoint Operators and Hermitean Operators . . . . . . . . . . . 178

8.4 Eigenvalues and Eigenfunctions of an Operator . . . . . . . . . . . . . . . . . . . 1798.4.1 Eigenvalues of Hermitean Operators. . . . . . . . . . . . . . . . . . . . 1808.4.2 Gram-Schmidt Orthogonalization. . . . . . . . . . . . . . . . . . . . . . . 1818.4.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828.4.4 Parseval Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.5 Hamiltonian Operator and Momentum Operator . . . . . . . . . . . . . . . . . . 185

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8.6 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.6.1 Examples of Hermitean Operators . . . . . . . . . . . . . . . . . . . . . . 1868.6.2 A Collection of Operators’ Definitions and

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1878.6.3 Examples of Commuting Operators . . . . . . . . . . . . . . . . . . . . 1908.6.4 Momentum and Energy of a Free Particle . . . . . . . . . . . . . . 190

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9 Time-Dependent Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939.2 Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939.3 Time-Dependent Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1969.4 Continuity Equation and Norm Conservation . . . . . . . . . . . . . . . . . . . . . 1979.5 Hamiltonian Operator of a Charged Particle. . . . . . . . . . . . . . . . . . . . . . . 1999.6 Approximate Form of the Wave Packet for a Free Particle . . . . . . . 2009.7 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.7.1 About the Units of the Wave Function . . . . . . . . . . . . . . . . . . 2019.7.2 An Application of the Semiclassical Approximation . . 2029.7.3 Polar Form of the Schrödinger Equation . . . . . . . . . . . . . . . 2029.7.4 Effect of a Gauge Transformation on the Wave

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

10 General Methods of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.2 General Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.3 Separable Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20910.4 Eigenfunctions of Commuting Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 21110.5 Expectation Value and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21210.6 Heisenberg Uncertainty Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21410.7 Time Derivative of the Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . . 21510.8 Ehrenfest Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21610.9 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.9.1 Minimum-Uncertainty Wave Function . . . . . . . . . . . . . . . . . 217Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Part III Applications of the Schrödinger Equation

11 Elementary Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22111.2 Step-Like Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

11.2.1 Case A: 0 < E < V0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22211.2.2 Case B: E > V0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

11.3 Energy Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22611.3.1 Case A: 0 < E < V0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22611.3.2 Case B: 0 < V0 < E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

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11.4 Energy Barrier of a General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23011.5 Energy Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

12 Cases Related to the Linear Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 23712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23712.2 Linear Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23712.3 Quantization of the Electromagnetic Field’s Energy . . . . . . . . . . . . . . 24112.4 Quantization of the Electromagnetic Field’s Momentum . . . . . . . . . 24312.5 Quantization of a Diagonalized Hamiltonian Function . . . . . . . . . . . 24412.6 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

12.6.1 Comments About the Linear Harmonic Oscillator . . . . . 246

13 Other Examples of the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 24713.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24713.2 Properties of the One-Dimensional Schrödinger Equation . . . . . . . 24813.3 Localized States—Operator’s Factorization . . . . . . . . . . . . . . . . . . . . . . . 251

13.3.1 Factorization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25213.3.2 First-Order Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25313.3.3 The Eigenfunctions Corresponding to l < n . . . . . . . . . . . . 25413.3.4 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

13.4 Schrödinger Equation with a Periodic Coefficient . . . . . . . . . . . . . . . . 25613.5 Schrödinger Equation for a Central Force . . . . . . . . . . . . . . . . . . . . . . . . . 258

13.5.1 Angular Part of the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26013.5.2 Radial Part of the Equation in the Coulomb Case . . . . . . 262

13.6 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26313.6.1 Operators Associated with Angular Momentum . . . . . . . 26313.6.2 Eigenvalues of the Angular Equation . . . . . . . . . . . . . . . . . . . 26513.6.3 Eigenfunctions of the Angular Equation. . . . . . . . . . . . . . . . 26713.6.4 Eigenvalues of the Radial Equation—Coulomb

Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27013.6.5 Eigenfunctions of the Radial

Equation—Coulomb Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27113.6.6 Transmission Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

14 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27714.2 Discrete Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27814.3 First-Order Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27914.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28114.5 Degenerate Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28214.6 Continuous Energy Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28314.7 Screened Coulomb Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28514.8 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

14.8.1 Perturbation Constant in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 286

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14.8.2 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28814.8.3 Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28914.8.4 Transitions from Discrete to Continuous Levels . . . . . . . 29014.8.5 Harmonic Perturbation—General Case . . . . . . . . . . . . . . . . . 29014.8.6 Spatially Oscillating Harmonic Perturbation . . . . . . . . . . . 292

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Part IV Systems of Interacting Particles: Quantum Statistics

15 Many-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30115.2 Wave Function of a Many-Particle System . . . . . . . . . . . . . . . . . . . . . . . . 30115.3 Symmetry of Functions and Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30315.4 Conservation of Symmetry in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30515.5 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

15.5.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30815.6 Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30915.7 Conservative Systems of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31015.8 Equilibrium Statistics in the Quantum Case . . . . . . . . . . . . . . . . . . . . . . . 313

15.8.1 Fermi-Dirac Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31615.8.2 Bose-Einstein Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

15.9 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31915.9.1 Connection with Thermodynamic Functions . . . . . . . . . . . 31915.9.2 Density of States for a Particle in a

Three-Dimensional Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32015.9.3 Density of States for a Two- or One-Dimensional

Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32215.9.4 Density of States for Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . 32315.9.5 Derivation of Planck’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

16 Separation of Many-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32716.2 System of Interacting Electrons and Nuclei . . . . . . . . . . . . . . . . . . . . . . . 32816.3 Adiabatic Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32916.4 Hartree Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33216.5 Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33416.6 Schrödinger Equation for the Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33416.7 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

16.7.1 Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Part V Applications to Semiconducting Crystals

17 Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34117.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34117.2 Bravais Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

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17.3 Reciprocal Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34517.4 Wigner-Seitz Cell—Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34817.5 Translation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

17.5.1 Bloch Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35017.5.2 Periodic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35217.5.3 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 352

17.6 Schrödinger Equation in a Periodic Lattice . . . . . . . . . . . . . . . . . . . . . . . . 35517.6.1 Wave Packet in a Periodic Potential. . . . . . . . . . . . . . . . . . . . . 35817.6.2 Parabolic-Band Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 35917.6.3 Density of States in the Parabolic-Band

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36317.6.4 Crystals of Si, Ge, and GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36417.6.5 Band Structure of Si, Ge, and GaAs . . . . . . . . . . . . . . . . . . . . 36617.6.6 Further Comments About the Band Structure . . . . . . . . . . 37317.6.7 Subbands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37517.6.8 Subbands in a Periodic Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 378

17.7 Calculation of Vibrational Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38217.7.1 Labeling the Degrees of Freedom—Dynamic Matrix. . 38317.7.2 Application of the Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . 38617.7.3 Properties of the Eigenvalues and Eigenvectors . . . . . . . . 388

17.8 Interaction of an Electron with the Lattice. . . . . . . . . . . . . . . . . . . . . . . . . 39017.8.1 Rigid Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39317.8.2 Energy Exchange Between Electron and Lattice . . . . . . . 394

17.9 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39617.9.1 Crystal Planes and Directions in Cubic Crystals . . . . . . . 39617.9.2 Examples of Translation Operators . . . . . . . . . . . . . . . . . . . . . 39817.9.3 Symmetries of the Hamiltonian Operator. . . . . . . . . . . . . . . 39817.9.4 Kronig-Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40117.9.5 Linear, Monatomic Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40517.9.6 Linear, Diatomic Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40817.9.7 Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

18 Electrons and Holes in Semiconductors at Equilibrium . . . . . . . . . . . . . . . 41518.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41518.2 Equilibrium Concentration of Electrons and Holes . . . . . . . . . . . . . . . 41618.3 Intrinsic Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42018.4 Uniform Distribution of Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

18.4.1 Donor-Type Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42618.4.2 Acceptor-Type Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43318.4.3 Compensation Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

18.5 Nonuniform Distribution of Dopants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44018.6 Band-Gap Narrowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44218.7 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

18.7.1 Si, Ge, GaAs in the Manufacturing of IntegratedCircuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Contents xxi

18.7.2 Qualitative Analysis of the Impurity Levels . . . . . . . . . . . . 44618.7.3 Position of the Impurity Levels . . . . . . . . . . . . . . . . . . . . . . . . . 447

Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Part VI Transport Phenomena in Semiconductors

19 Mathematical Model of Semiconductor Devices . . . . . . . . . . . . . . . . . . . . . . . . 45319.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45319.2 Equivalent Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

19.2.1 Electron Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45519.2.2 Expectation Values—Crystal Momentum . . . . . . . . . . . . . . 45719.2.3 Dynamics in the Parabolic-Band Approximation . . . . . . 460

19.3 Dynamics in the Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46119.3.1 Collision Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46419.3.2 Point-Like Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46619.3.3 Perturbative Form of the BTE . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

19.4 Moments Expansion of the BTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47019.4.1 Time Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47119.4.2 Diffusion Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47219.4.3 Drift Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47219.4.4 Collision Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47319.4.5 Moment Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47319.4.6 Hierarchical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

19.5 Hydrodynamic and Drift-Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . 48119.5.1 HD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48119.5.2 DD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48219.5.3 DD Model for the Valence Band . . . . . . . . . . . . . . . . . . . . . . . . 48519.5.4 Coupling with Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . 48819.5.5 Semiconductor-Device Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 49019.5.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49219.5.7 Quasi-Fermi Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49419.5.8 Poisson Equation in a Semiconductor . . . . . . . . . . . . . . . . . . 496

19.6 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49719.6.1 Comments on the Equivalent Hamiltonian Operator . . . 49719.6.2 Special Cases of Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49719.6.3 ˛-Moment at Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49819.6.4 Closure Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49819.6.5 Matthiessen’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50019.6.6 Order of Magnitude of Mobility and Conductivity. . . . . 50119.6.7 Onsager Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50219.6.8 A Resumé of the Transport Model’s Derivation . . . . . . . . 505

Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

xxii Contents

20 Generation-Recombination and Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50720.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50720.2 Net Thermal Recombinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

20.2.1 Direct Thermal Recombinations . . . . . . . . . . . . . . . . . . . . . . . . 50820.2.2 Trap-Assisted Thermal Recombinations . . . . . . . . . . . . . . . . 51120.2.3 Shockley-Read-Hall Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51420.2.4 Thermal Recombination with Tail and Deep States . . . . 519

20.3 Auger Recombination and Impact Ionization . . . . . . . . . . . . . . . . . . . . . 52120.3.1 Strong Impact Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

20.4 Optical Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52520.5 Macroscopic Mobility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

20.5.1 Example of Phonon Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . 52920.5.2 Example of Ionized-Impurity Collision. . . . . . . . . . . . . . . . . 53120.5.3 Bulk and Surface Mobilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53220.5.4 Beyond Analytical Modeling of Mobility . . . . . . . . . . . . . . 534

20.6 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53620.6.1 Transition Rates in the SRH Recombination

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53620.6.2 Coefficients of the Auger and Impact-Ionization

Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53820.6.3 Total Recombination-Generation Rate . . . . . . . . . . . . . . . . . . 54020.6.4 Screened Coulomb Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

Part VII Basic Semiconductor Devices

21 Bipolar Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54521.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54521.2 P-N Junction in Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

21.2.1 Built-In Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54721.2.2 Space-Charge and Quasi-Neutral Regions. . . . . . . . . . . . . . 550

21.3 Shockley Theory of the P-N Junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55421.3.1 Derivation of the I.V/ Characteristic . . . . . . . . . . . . . . . . . . . 557

21.4 Depletion Capacitance of the Abrupt P-N Junction . . . . . . . . . . . . . . . 56021.5 Avalanche Due to Impact Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56321.6 Photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

21.6.1 Photodiode—Continuous Mode. . . . . . . . . . . . . . . . . . . . . . . . . 56821.6.2 Photodiode—Storage Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

21.7 Solar Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57321.7.1 Current of the Solar Cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

21.8 Bipolar Junction Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57721.8.1 Current-Voltage Characteristics of the P-N-P BJT . . . . . 58021.8.2 Equivalent Circuit of the P-N-P BJT . . . . . . . . . . . . . . . . . . . 584

21.9 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58821.9.1 Weak-Injection Limit of the Drift-Diffusion

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

Contents xxiii

21.9.2 Shockley’s Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 59021.9.3 Depletion Capacitance—Arbitrary Doping Profile. . . . . 59121.9.4 Order of Magnitude of Junction’s Parameters . . . . . . . . . . 59221.9.5 Solar Cell’s Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59321.9.6 Equivalent Circuit of the P-N-P BJT (Single

Generator) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59521.9.7 Comment on the Diffusion Length . . . . . . . . . . . . . . . . . . . . . . 597

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

22 MOS Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60122.2 Metal-Insulator-Semiconductor Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . 602

22.2.1 Surface Potential—P-Type Substrate . . . . . . . . . . . . . . . . . . . 60522.2.2 Relation Between Surface Potential and Gate

Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60822.3 Capacitance of the MOS Structure—P-Type Substrate . . . . . . . . . . . 61322.4 Simplified Expression of the Inversion Charge . . . . . . . . . . . . . . . . . . . . 615

22.4.1 Flat-Band Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61722.4.2 Quantitative Relations in the MOS Capacitor . . . . . . . . . . 619

22.5 MOS Photocapacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62022.6 MOS Capacitor—N-Type Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62622.7 Insulated-Gate Field-Effect Transistor—MOSFET . . . . . . . . . . . . . . . 62922.8 N-Channel MOSFET—Current-Voltage Characteristics . . . . . . . . . . 630

22.8.1 Gradual-Channel Approximation . . . . . . . . . . . . . . . . . . . . . . . 63322.8.2 Differential Conductances and Drain Current,

N-Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63422.9 P-Channel MOSFET—Current-Voltage Characteristics . . . . . . . . . . 639

22.9.1 Differential Conductances and Drain Current,P-Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642

22.10 CMOS Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64522.10.1 I-O Characteristic of the CMOS Inverter . . . . . . . . . . . . . . . 648

22.11 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65122.11.1 Poisson’s Equation in the MOSFET Channel . . . . . . . . . . 65122.11.2 Inversion-Layer Charge and Mobility Degradation . . . . 65422.11.3 Comments About the CMOS Inverter . . . . . . . . . . . . . . . . . . 65522.11.4 Exact Charge Partitioning in the MOS Capacitor . . . . . . 65722.11.5 MOSFET Theory Including the Subthreshold

Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65822.11.6 Scaling Rules for MOSFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66122.11.7 A Brief Account on IC Design . . . . . . . . . . . . . . . . . . . . . . . . . . 666

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

xxiv Contents

Part VIII Miscellany

23 Thermal Diffusion—Ion Implantation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67323.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67423.3 Diffusive Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67723.4 Diffusion Equation—Model Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67823.5 Predeposition and Drive-in Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680

23.5.1 Predeposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68023.5.2 Drive-in Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684

23.6 Generalization of the Model Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68523.7 Ion Implantation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

23.7.1 Ion Implanter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68823.7.2 Ion Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69023.7.3 Implanted Profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69423.7.4 Deviations from the Gaussian Profile . . . . . . . . . . . . . . . . . . . 69523.7.5 Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698

23.8 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69823.8.1 Generation and Destruction of Particles . . . . . . . . . . . . . . . . 69823.8.2 Balance Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69923.8.3 Lateral Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69923.8.4 Alternative Expression of the Dose . . . . . . . . . . . . . . . . . . . . . 70023.8.5 The Initial Condition of the Predeposition Step . . . . . . . . 700

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700

24 Thermal Oxidation—Layer Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70324.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70324.2 Silicon Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70524.3 Oxide-Growth Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70624.4 Linear-Parabolic Model of the Oxide Growth . . . . . . . . . . . . . . . . . . . . . 70824.5 Layer Deposition and Selective Oxide Growth . . . . . . . . . . . . . . . . . . . . 71124.6 Epitaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71324.7 Kinetics of Epitaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71424.8 Complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

24.8.1 An Apparent Contradiction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71624.8.2 Elementary Contributions to the Layer’s Volume . . . . . . 71724.8.3 Features of the Oxide Growth and Epitaxial Growth . . 71724.8.4 Reaction Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71824.8.5 Molecular Beam Epitaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71824.8.6 Secondary Reaction in the Epitaxial Growth . . . . . . . . . . . 719

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

25 Measuring the Semiconductor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72325.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72325.2 Lifetime Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724

25.2.1 Thermal Velocity and Capture Cross-Section . . . . . . . . . . 726

Contents xxv

25.3 Mobility Measurement—Haynes-Shockley Experiment . . . . . . . . . . 72725.4 Hall-Voltage Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73025.5 Hall Voltage—Arbitrary Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732

25.5.1 Solution of the Stream-Function Equation . . . . . . . . . . . . . 73425.5.2 Local Hall Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

25.6 Measurement of Doping Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73725.7 Van der Pauw Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738

25.7.1 Solution over the Upper Half Plane . . . . . . . . . . . . . . . . . . . . . 73825.7.2 Solution over the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742

A Vector and Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745A.1 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745A.2 Schwarz Inequality and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 746A.3 Nabla Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747A.4 Dyadic Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748A.5 Divergence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749A.6 Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750A.7 Mixed Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750A.8 Rotational of a Vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751A.9 Rotational Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752A.10 Helmholtz Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752A.11 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

A.11.1 Eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755A.11.2 Properties of Hermitean Matrices . . . . . . . . . . . . . . . . . . . . . . . 758A.11.3 Similarity Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763A.11.4 Doubly Stochastic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764A.11.5 Diagonally Dominant Matrices and Irreducible

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764A.11.6 Properties of Diagonally Dominant Matrices. . . . . . . . . . . 766A.11.7 Solution of a Tridiagonal Algebraic System. . . . . . . . . . . . 768

A.12 Wronskian Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772A.13 Numerical Solution of the Semiconductor Equations . . . . . . . . . . . . . 773

A.13.1 Decoupled/Coupled Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774A.13.2 Discretization Scheme in One Dimension . . . . . . . . . . . . . . 776A.13.3 The Numerov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

B Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785B.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785B.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787B.3 Coordinate Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787B.4 Differential Operators Under Coordinate Transformations . . . . . . . 788B.5 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789B.6 Constrained Extrema—Lagrange Method . . . . . . . . . . . . . . . . . . . . . . . . . 793B.7 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

xxvi Contents

B.8 Contraction Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797B.8.1 Determining the Zero of a Function . . . . . . . . . . . . . . . . . . . . 799B.8.2 Solving an Algebraic System. . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

C Special Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803C.1 Sine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803C.2 Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805C.3 Gauss Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806C.4 Dirac’s ı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808C.5 Some Properties of Dirac’s ı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813C.6 Moments Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814C.7 Error Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815C.8 Parametrized Gaussian Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816C.9 Euler’s Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817C.10 Euler’s Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820C.11 Gamma Function’s Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 821C.12 Integrals Related to the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 822C.13 Fermi Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822C.14 Hölder’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824C.15 Integrals Related to the Electromagnetic Modes . . . . . . . . . . . . . . . . . . 825C.16 Riemann’s Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826C.17 Bernoulli Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828C.18 Random Walk Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831C.19 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834

D Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893

About the Author

Massimo Rudan received a degree in electricalengineering in 1973 and a degree in physics in 1976,both from the University of Bologna, Italy. His researchinterests are in the field of physics of carrier transportand numerical analysis of semiconductor devices. In1986, he was a visiting scientist, on a one-year assign-ment, at the IBM Thomas J. Watson Research Center atYorktown Heights, NY, studying the discretization tech-niques for the higher-order moments of the Boltzmanntransport equation.

From 1979, he has been teaching annual courses in the Faculty of Engineeringof the University of Bologna, firstly as lecturer and then as associate professor. In1990, he became full professor of microelectronics at the University of Bologna.An IEEE fellow (2008) and life fellow (2014), M.R. is author, coauthor, or editor of12 books and has authored or coauthored about 250 technical papers published inmajor journals or presented in international conferences, in the fields of electronics,solid-state physics, and solid-state sensors.

M.R. has coordinated several research projects funded by the European Commis-sion, international companies and foundations, the National Council of Research,and the National Ministry of University and Research. In 2001, he was one of thefounders of the Advanced Research Center for Electronic Systems (ARCES) of theUniversity of Bologna.

xxvii

Acronyms

AM Air mass. Length of the optical path, through Earth’s atmosphere, oflight rays coming from an extraterrestrial source. The AM value at sealevel, when the source is at the zenith, is set equal to unity and used asnormalization parameter.

BJT Bipolar junction transistor. A transistor whose operation is obtained bya suitable arrangement of two p-n junctions. The term “bipolar” is usedbecause both electrons and holes are involved in the device functioning.

BTE Boltzmann transport equation. The equation expressing the continuityof the distribution function in the phase space.

CVD Chemical vapor deposition. A deposition process in which the materialto be deposited is the product of a chemical reaction that takes place onthe surface of the substrate or in its vicinity.

DD Drift diffusion. The term indicates a transport model for semiconduc-tors made, for each energy band, of the balance equations for the carriernumber and average velocity. Such equations contain the electric fieldand the magnetic induction; as a consequence, their solution must becalculated consistently with that of the Maxwell equations. Comparewith the HD model.

DRAM Dynamic random access memory. A type of random-access, solid-state memory where each bit is stored in a separate capacitor. Thecharged/discharged states of the latter correspond to the logic statesof the memory bit. Due to the leakage of the circuits connected tothe capacitor, the stored datum tends to fade unless it is periodicallyrefreshed (whence the designation “dynamic”). DRAM memories arevolatile, namely, they lose the data when the power supply is removed.

HD Hydro dynamic. The term indicates a transport model for semiconduc-tors made, for each energy band, of the balance equation for the carriernumber, average velocity, average kinetic energy, and average flux ofthe kinetic energy. Such equations contain the electric field and themagnetic induction; as a consequence, their solution must be calculated

xxix

xxx Acronyms

consistently with that of the Maxwell equations. Compare with the DDmodel.

IC Integrated circuit. Also called chip or microchip. An assembly of elec-tronic circuits on the same plate of a semiconductor material. The ideawas proposed in the early 1950s and demonstrated in 1958; it providedan enormous improvement, both in cost and performance, with respectto the manual assembly of circuits using discrete components.

IGFET Insulated gate field effect transistor. A device architecture demon-strated in the early 1930s. Its first implementation (1960) using athermally oxidized silicon layer gave rise to the MOSFET architecture.

LASER Light amplification by stimulated emission of radiation. A device madeof a material having a high probability of radiative emission, in whichthe electron concentration of high-energy states is artificially kepthigh by an external power source. The material produces a coherentemission, and a suitable feedback makes it to oscillate; this results in anearly monochromatic light.

LOCOS Local oxidation. The technological process consisting in depositingand patterning a layer of silicon nitride over the areas where thesubstrate’s oxidation must be prevented.

MBE Molecular beam epitaxy. A low-temperature epitaxial process based onevaporation.

MIS Metal insulator semiconductor. Structure made of the superposition ofa metal contact, an insulator, and a semiconductor.

MOS Metal oxide semiconductor. Structure made of the superposition of ametal contact, an oxide that acts as an insulator, and a semiconductor.

MOSFET Metal oxide semiconductor, field effect transistor. A transistor whoseactive region is an MOS structure. In last-generation devices, theinsulator may be deposited instead of being obtained by oxidizing thesemiconductor underneath. The MOSFET has been for decades, andstill is, the fundamental device of the integrated-circuit architecture.

PCM Phase change memory. A solid-state memory whose logic states areassociated with a high-resistance or low-resistance condition of thematerial; such conditions are in turn associated to the material’s phases,amorphous or crystalline, respectively. The material is forced to switchfrom one phase to the other by an externally applied electric signal.

PDE Partial differential equation.PV Photo voltaic. The adjective refers to physical processes that convert

the energy of a radiation into electric energy (e.g., photovoltaic effect)or to devices where such processes occur (e.g., photovoltaic cell).

PVD Physical vapor deposition. A deposition process in which the materialto be deposited does not react chemically with other substances.

SGOI Silicon germanium on insulator. A technology analogous to SOI. SGOIincreases the speed of the transistors by straining the material under thegate, thus making the electron mobility higher.

Acronyms xxxi

SOI Silicon on insulator. A technology introduced in 1998 for semiconduc-tor manufacturing, in which the standard silicon substrate is replacedwith a layered structure of the silicon-insulator-silicon type. SOIreduces the parasitic capacitances and the short-channel effect in MOStransistors.

SOS Silicon on sapphire. A technological process that consists in growing athin layer of silicon on a wafer made of sapphire (Al2O3).

SS Subthreshold slope. The inverse of d log10.I/=dVG in a device where aconductive channel is controlled by a gate; I is the channel current, VG

the gate voltage. A smaller SS corresponds to a faster transition of thedevice from the on to the off state or vice versa.

TFT Thin film transistor. A type of field-effect transistor fabricated bydepositing the active semiconductor layer, the dielectric layer, andthe metallic contacts over a nonconducting substrate. An importantapplication of TFTs is in liquid-crystal displays; for this reason, atypical type of nonconducting substrate of TFTs is glass.

List of Figures

Fig. 2.1 Example of potential energy discussed in Sect. 2.9 . . . . . . . . . . . . . . . 39

Fig. 3.1 The example of the square well analyzed in Sect. 3.2. Onlythe case 0 � E � V0 is shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Fig. 3.2 The example of the linear harmonic oscillator analyzed inSect. 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Fig. 3.3 Graphic representation of the vector relation (3.20) . . . . . . . . . . . . . . 51Fig. 3.4 Graphic representation of the trajectory (3.35) for different

values of the angular momentum. The curves have beenobtained by setting the parameters’ values to s0 D 1,'0 D 0, � D 0:5, and � D 0:01; : : : ; 0:6 (the units arearbitrary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Fig. 3.5 Graphic representation of (3.36) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Fig. 3.6 Graphic representation of (3.69) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Fig. 3.7 Definition of the angles used in Sects. 3.6 and 3.13.5 . . . . . . . . . . . . . 68Fig. 3.8 Dependence of Ve on the distance s from the center of

force, as given by (3.74) in arbitrary units . . . . . . . . . . . . . . . . . . . . . . . . . 69Fig. 3.9 The elliptical trajectory described by (3.79) with '0 D 0 . . . . . . . . 71

Fig. 4.1 The domain V used for the solution of the Helmholtzequation (4.43). The three possible positions of point r areshown: external to V , internal to V , or on the boundary S . . . . . . . . 85

Fig. 5.1 The domain used for the expansion of the vector potentialinto a Fourier series (Sect. 5.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Fig. 6.1 Schematic picture used for discussing the issue of the totalmomentum of identical molecules within a container . . . . . . . . . . . . 132

Fig. 6.2 Schematic picture used for discussing the issue of thetotal angular momentum of identical molecules within acontainer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xxxiii

xxxiv List of Figures

Fig. 6.3 Kac-ring model: computer calculation of the time evolutionof the number of “0” states in two samples made ofN D 4;000 objects, which at time t D 0 were all set to“0.” The markers of the two samples are n D 4 and n D 8,respectively, and the number of time steps is much smallerthan N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Fig. 6.4 Kac-ring model: computer calculation of the time evolutionof the number of “0” states in two samples made ofN D 4;000 objects, which at time t D 0 were all set to “0.”The markers of the two samples are n D 16 and n D 32,respectively, and the number of time steps is much smallerthan N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Fig. 6.5 Graph of Pr log Pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Fig. 6.6 Kac-ring model: computer calculation of the time evolution

of the number of “0” states in two samples made ofN D 4;000 objects, which at time t D 0 were all set to “0.”The markers of the two samples are n D 16 and n D 32,respectively, and the number of time steps is larger than N . . . . . . 139

Fig. 7.1 Classical description of the electron’s orbit for E � 0 . . . . . . . . . . . . 145Fig. 7.2 Classical description of the electron’s orbit for E < 0. For

simplicity, a circular orbit is considered . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Fig. 7.3 Schematic description of the potential energy in a linear

monatomic chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Fig. 7.4 The same structure of Fig. 7.3, where the peaks are

replaced with the envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Fig. 7.5 Designation of the lower series of spectral lines (7.4) . . . . . . . . . . . . 149Fig. 7.6 Schematic cross-section of the apparatus used for

measuring the photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Fig. 7.7 The I D I.VAK/ curves, in arbitrary units, obtained from

the photoelectric effect at constant frequency of theradiation, with the spectral power used as a parameter . . . . . . . . . . . 151

Fig. 7.8 The I D I.VAK/ curves, in arbitrary units, obtained fromthe photoelectric effect at constant spectral power of theradiation, with frequency used as a parameter . . . . . . . . . . . . . . . . . . . . 151

Fig. 7.9 The approximation to a black body consisting in asmall hole in the wall of an enclosure kept at constanttemperature. If a thermometer (represented by the shadedarea) was suspended within the enclosure, it wouldindicate the same temperature T as the walls, irrespectiveof its position or orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Fig. 7.10 Spectral energy density of the black body at differenttemperatures. The value T D 5;800 K corresponds to thesurface temperature of the sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

List of Figures xxxv

Fig. 7.11 Scheme of the experimental setup for measuring theCompton effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Fig. 8.1 The first eigenfunctions of the Schrödinger equation in thecase of a particle in a box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Fig. 11.1 The example of the step-like potential energy analyzed inSect. 11.2. Only the case 0 � E � V0 is shown . . . . . . . . . . . . . . . . . . . 222

Fig. 11.2 Another example of the step-like potential energy analyzedin Sect. 11.2, with V0 < 0 and jV0j � E . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Fig. 11.3 The example of the one-dimensional energy barrieranalyzed in Sect. 11.3. Only the case 0 � E � V0 is shown . . . . . . 229

Fig. 11.4 The example of the one-dimensional energy well analyzedin Sect. 11.5. Only the case V0 < E < 0 is shown . . . . . . . . . . . . . . . . 233

Fig. 11.5 Graphic solution of (11.50) using the auxiliary variable �.The solutions �1; : : : ; �5 are the intercepts of the left-handside (thicker line) with the branches of the right-hand side.The data are given in Prob. 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Fig. 11.6 The smooth potential energy considered in Prob. 11.1, withV0 D 2 and E D 2:5 (arbitrary units) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Fig. 12.1 The potential energy of the linear harmonic oscillator(Sect. 12.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Fig. 13.1 Form of the potential energy that gives rise to localizedstates (Sect. 13.3). Only one state E is shown . . . . . . . . . . . . . . . . . . . . . 251

Fig. 13.2 Geometrical construction showing the relation between Mand Mz. The l D 3 is case considered, whence one findsm D �3; : : : ; 0; : : : ; 3 and

pl .l C 1/ ' 3:46 . . . . . . . . . . . . . . . . . . . . . 262

Fig. 13.3 Illustration of the concepts of node, element, and cell(Sect. 13.6.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Fig. 13.4 Example of a potential energy V.x/ replaced with apiecewise-constant function Vi (Sect. 13.6.6) . . . . . . . . . . . . . . . . . . . . . 275

Fig. 14.1 Form of f .!rs/=tP, with f given by the second expressionin (14.36), for different values of tP (in arbitrary units) . . . . . . . . . . . 287

Fig. 14.2 Form of F.!rs/=tP, with F given by the second expressionin (14.41), with tP D 1, !0 D 5 (in arbitrary units) . . . . . . . . . . . . . . . 289

Fig. 14.3 Plot of sin2.� �s Ns/=.� �s/2 with Ns D 10. The function

is appreciable only around integer values of �s, even whenthe value of Ns is rather small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

Fig. 15.1 Schematic description of a system made of two identicalparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Fig. 15.2 Placement of three identical particles into equally spacedenergy states. The particles’ total energy equals threeenergy units �. Different graphic symbols are used for theparticles to make the classical treatment clearer . . . . . . . . . . . . . . . . . . 314

xxxvi List of Figures

Fig. 15.3 The Fermi-Dirac statistics as a function of energy fordifferent values of the system’s temperature. For simplicitythe temperature dependence of the Fermi level EF is notconsidered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Fig. 15.4 Constant-energy sphere of the k space illustrating theprocedure for determining the density of states . . . . . . . . . . . . . . . . . . . 324

Fig. 17.1 Schematic description of a two-dimensional Bravais latticeof the oblique type. Three atoms have been removed tobetter show the characteristic vectors. The latter are notorthogonal to each other, and their lengths are different . . . . . . . . . . 343

Fig. 17.2 Examples of cells in a two-dimensional Bravais lattice ofthe oblique type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Fig. 17.3 Schematic description of a three-dimensional Bravaislattice of the FCC type. Four atoms have been removedto better show the characteristic vectors. The latter areorthogonal to each other and of equal length . . . . . . . . . . . . . . . . . . . . . . 344

Fig. 17.4 A Wigner-Seitz cell in a two-dimensional, oblique lattice . . . . . . . . 348Fig. 17.5 The first Brillouin zone of the FCC lattice . . . . . . . . . . . . . . . . . . . . . . . . 349Fig. 17.6 A finite block of material obtained by sectioning a crystal

by means of three pairs of parallel crystal planes . . . . . . . . . . . . . . . . . 353Fig. 17.7 A one-dimensional example of the periodic factor �n 0 of

(17.56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Fig. 17.8 A one-dimensional example of the envelope function

A.r � un t I k0/ of (17.56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Fig. 17.9 Product of the two functions shown in Figs. 17.7 and 17.8 . . . . . . . 360Fig. 17.10 The function of Fig. 17.9 squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Fig. 17.11 Tetrahedral organization of the elementary, body-centered

cubic block of silicon or germanium. The side of the cubeis one half the lattice constant a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Fig. 17.12 Diamond structure. The top and bottom halves are shownseparately . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Fig. 17.13 Diamond structure obtained by joining together the top andbottom halves shown separately in Fig. 17.12 . . . . . . . . . . . . . . . . . . . . . 366

Fig. 17.14 Calculation of the particles’ population in the conductionand valence bands of a semiconductor. To make themmore visible, the products g.E/ P.E/ and g.E/ Œ1 � P.E/�

have been amplified with respect to g.E/ alone. The gap’sextension is arbitrary and does not refer to any specificmaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Fig. 17.15 Schematic view of the two branches of the valence band ofSi, Ge, or GaAs in the Œ100� direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Fig. 17.16 Schematic view of the conduction band of GaAs in theŒ100� direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

List of Figures xxxvii

Fig. 17.17 Schematic view of the conduction band of Si in the Œ100�

direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371Fig. 17.18 Constant-energy surfaces of the conduction band of silicon . . . . . . 372Fig. 17.19 Schematic view of the conduction band of Ge in the Œ111�

direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373Fig. 17.20 Normalized, two-dimensional density of states (17.109) for

the 1; 2; 4; 5 valleys of silicon, as a function of E=Et, in theparabolic-band approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Fig. 17.21 Normalized, one-dimensional density of states for the1; 4 valleys of silicon, as a function of E=E11, in theparabolic-band approximation and with d2 D d3 . . . . . . . . . . . . . . . . . 382

Fig. 17.22 Definition of the labels used to identify the degrees offreedom in a periodic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

Fig. 17.23 Vectors used for determining the Bragg relation . . . . . . . . . . . . . . . . . . 393Fig. 17.24 Example of node labeling in the cubic lattice . . . . . . . . . . . . . . . . . . . . . 397Fig. 17.25 Schematic representation of the .111/ plane (top left) and

of the .001/ and .010/ planes (bottom right) in a cubiccrystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Fig. 17.26 Potential energy in the Kronig-Penney model . . . . . . . . . . . . . . . . . . . . . 402Fig. 17.27 Graphic solution of 17.210, with # D 10. The two vertical

lines mark the values of ˛ a delimiting the lowest band . . . . . . . . . . 404Fig. 17.28 Normalized dispersion relation of a linear, monatomic

chain. The vertical lines, placed at a q=2 D ˙�=2, are thelimits of the first Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

Fig. 17.29 Normalized dispersion relation of a linear, diatomic chainwith �1 D �2 D � and 1 D 3 2. The vertical lines,placed at a q=2 D ˙�=2, are the limits of the first Brillouinzone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Fig. 18.1 Description of the particles’ population in the conductionand valence bands of an insulator. To make them morevisible, the products g.E/ P.E/ and g.E/ Œ1 � P.E/� havebeen amplified, with respect to g.E/ alone, by a factor2 � 1031 (compare with Figs. 17.14 and 18.2). The gap’sextension is arbitrary and does not refer to any specificmaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

Fig. 18.2 Description of the electron population in the conductionband of a conductor. The product g.E/ P.E/ is drawn in thesame scale as g.E/ alone (compare with Figs. 17.14 and18.1). The gap’s extension is arbitrary and does not refer toany specific material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Fig. 18.3 Plot of the gap as a function of temperature for Ge, Si, andGaAs. The vertical line marks T D 300 K .. . . . . . . . . . . . . . . . . . . . . . . 421

Fig. 18.4 Arrhenius plot of the intrinsic concentration in Ge, Si, andGaAs. The vertical line marks T D 300 K .. . . . . . . . . . . . . . . . . . . . . . . 423

xxxviii List of Figures

Fig. 18.5 Two-dimensional representation of the intrinsic siliconlattice. The upper-left part of the figure shows the T ! 0

limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425Fig. 18.6 Two-dimensional representation of the n-doped silicon

lattice. The upper-left part of the figure shows the T ! 0

limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426Fig. 18.7 Density of states in an n-doped semiconductor. The gap’s

extension is arbitrary and does not refer to any specificmaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Fig. 18.8 Schematic representation of the donor states . . . . . . . . . . . . . . . . . . . . . . 429Fig. 18.9 Arrhenius plot of n.T/ for an n-type semiconductor, in

arbitrary units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432Fig. 18.10 Two-dimensional representation of the p-doped silicon

lattice. The upper-left part of the figure shows the T ! 0

limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434Fig. 18.11 Density of states in a p-doped semiconductor. The gap’s

extension is arbitrary and does not refer to any specificmaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Fig. 18.12 Schematic representation of the acceptor states . . . . . . . . . . . . . . . . . . . 436Fig. 18.13 Schematic representation of a semiconductor with both

donor and acceptor states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Fig. 18.14 Density of states in an n-doped semiconductor, where the

high concentration of the dopant produces the band-gapnarrowing. The gap’s extension is arbitrary and does notrefer to any specific material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

Fig. 18.15 Band-gap narrowing as a function of the total dopingconcentration, in normalized form, using the experimentalexpression (18.67) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Fig. 18.16 Potential energy in the Kronig-Penney model modified toaccount for impurity atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

Fig. 19.1 Example of the time evolution of a phase-space domainin a one-dimensional case. The situation with no externalforce is considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Fig. 19.2 Qualitative picture of a collision between an electron anda negatively ionized impurity. The latter is schematicallyrepresented by the black circle, whereas the gray areaindicates the screening region. The initial and final stateof the electron are indicated with .r; k/ and .r0; k0/,respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Fig. 19.3 MOS structure used to discuss the boundary conditions forthe mathematical model of semiconductor devices. Onlythe conducting boundaries are shown. Note that the verticalscale of the drawing is not realistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

List of Figures xxxix

Fig. 19.4 The same structure as in Fig. 19.3, to which the insulatingboundaries have been added (dash-dotted lines). Theupper part of the figure shows the correct placement ofthe insulating boundaries, the lower part shows a wrongplacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Fig. 20.1 A graphic example of direct thermal recombination (a)and generation (b). The edges of the conduction andvalence bands are indicated with the same symbolsused in Sect. 18.2. The same drawing applies also tothe description of the direct optical recombinations andgenerations (Sect. 20.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

Fig. 20.2 Different types of trap-assisted transitions . . . . . . . . . . . . . . . . . . . . . . . . 511Fig. 20.3 Auger recombinations initiated by electrons (a) and holes

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522Fig. 20.4 Impact-ionization transitions initiated by electrons (b) and

holes (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523Fig. 20.5 Sketch of photon absorption in a material layer . . . . . . . . . . . . . . . . . . . 527Fig. 20.6 Graph of the theoretical mobility curve (20.80), normalized

to its maximum, for different values of b, with b0 D 0.Each curve has a flex at r D rflex D �b0=b and takes thevalue 0:5 there. The slope at the flex is �b=4 . . . . . . . . . . . . . . . . . . . . . 533

Fig. 20.7 Electron mobility in silicon calculated with thespherical-harmonics expansion method (HARM) as afunction of the total ionized-dopant concentration NI , usingthe lattice temperature T as parameter. The calculations arecompared with measurements by Lombardi [91], Klaassen[80], and Arora [2] (courtesy of S. Reggiani) . . . . . . . . . . . . . . . . . . . . 535

Fig. 20.8 Hole mobility in silicon calculated with thespherical-harmonics expansion method (HARM) as afunction of the total ionized-dopant concentration NI , usingthe lattice temperature T as parameter. The calculations arecompared with measurements by Lombardi [91], Klaassen[80], and Arora [2] (courtesy of S. Reggiani) . . . . . . . . . . . . . . . . . . . . 535

Fig. 20.9 Electron surface mobility in silicon calculated with thespherical-harmonics expansion method (HARM) methodat room temperature, using the acceptor concentrationNA as parameter. The calculations are compared withmeasurements by Takagi [131] (courtesy of S. Reggiani) . . . . . . . 536

Fig. 20.10 Hole surface mobility in silicon calculated with thespherical-harmonics expansion method (HARM) atroom temperature, using the donor concentration ND

as parameter. The calculations are compared withmeasurements by Takagi [131] (courtesy of S. Reggiani) . . . . . . . 536

Fig. 21.1 Schematic example of a one-dimensional p-n junction . . . . . . . . . . . 546

xl List of Figures

Fig. 21.2 Solution of the one-dimensional Poisson equation (21.8)in an abrupt p-n junction at equilibrium, with NA D 1016

cm�3, ND D 1015 cm�3. The continuous vertical linemarks the position of the metallurgical junction, the dashedvertical lines mark the edges of the space-charge region . . . . . . . . . 551

Fig. 21.3 Form of the bands for the same device as in Fig. 21.2 . . . . . . . . . . . . 551Fig. 21.4 Electron and hole concentrations in a one-dimensional,

abrupt p-n junction at equilibrium, with NA D 1016 cm�3,ND D 1015 cm�3. The figure is drawn in a logarithmicscale. The continuous vertical line marks the position ofthe metallurgical junction, the dashed vertical lines markthe edges of the space-charge region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

Fig. 21.5 The same concentrations as in Fig. 21.4, drawn in a linearscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

Fig. 21.6 Electric potential for the same device as in Fig. 21.2,including the built-in potentials of the contacts . . . . . . . . . . . . . . . . . . . 554

Fig. 21.7 Symbol and typical I, V reference for the p-n junction . . . . . . . . . . . 555Fig. 21.8 Electric potential ' in a p-n junction in forward-biased

(V > 0, thick-red lines) and equilibrium (thin-bluelines) conditions. When V > 0, the extension l of thespace-charge region is smaller than in equilibrium. Thedrawing is not in the same scale as that of Fig. 21.6 . . . . . . . . . . . . . . 555

Fig. 21.9 Electric potential ' in a p-n junction in reverse-biased(V < 0, thick-red lines) and equilibrium (thin-bluelines) conditions. When V < 0, the extension l of thespace-charge region is larger than in equilibrium. Thedrawing is not in the same scale as that of Fig. 21.6 . . . . . . . . . . . . . . 556

Fig. 21.10 Charge density in a reverse-biased p-n junction using theASCE approximation, in arbitrary units. The ratio NA=ND

is the same as in Fig. 21.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560Fig. 21.11 Electric field consistent with the charge density of

Fig. 21.10, in arbitrary units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561Fig. 21.12 Schematic description of the avalanche phenomenon. The

details are given in the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565Fig. 21.13 Continuous mode photodiode. The applied bias V is

negative and constant; the resistor mimics the inputresistance of the circuit that measures the photodiode’scurrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Fig. 21.14 Storage mode photodiode. The applied bias V0 is negativeand constant, while V , VD vary with time depending onthe action of the switch; the resistor mimics the inputresistance of the circuit that measures the photodiode’scurrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

Fig. 21.15 Equivalent circuit used to analyze the functioning of thestorage-mode photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

List of Figures xli

Fig. 21.16 Time evolution of the voltage VD across a storage-modephotodiode. The integration time Ti ranges from t D 0 tot D 15 ms (vertical line); the sampling time Ts ranges fromt D 15 ms to t D 20 ms. The calculations are shown inProbs. 21.10 and 21.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

Fig. 21.17 Solar cell. The resistor mimics the input resistance of theload to which the energy is delivered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Fig. 21.18 Equivalent circuit of the solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577Fig. 21.19 Structure of the bipolar junction transistor of the n-p-n type . . . . . 578Fig. 21.20 Functioning regimes and symbol of the n-p-n type transistor . . . . 579Fig. 21.21 Functioning regimes and symbol of the p-n-p type transistor . . . . 580Fig. 21.22 Structure and effective dopant concentration in the intrinsic

transistor of the n-p-n type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580Fig. 21.23 Structure and effective dopant concentration in the intrinsic

transistor of the p-n-p type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581Fig. 21.24 Symbols used to indicate the positions of the two junctions,

the edges of the space-charge regions, and the width of thequasi-neutral base region. The same nomenclature appliesto both the n-p-n and p-n-p types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

Fig. 21.25 The Ebers and Moll equivalent circuit for the BJT of thep-n-p type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

Fig. 21.26 Output characteristics of a BJT of the p-n-p type, drawnwith hFB D 0:95. The units of the voltage axis arenormalized to kB T=q, those of the current axis are arbitrary . . . . . 588

Fig. 21.27 I.V/ characteristic of the solar cell, in arbitrary units(black line), and maximum-power load characteristicI D �V=RMP (red line). The absolute value of the shadedarea shows the power delivered by the cell . . . . . . . . . . . . . . . . . . . . . . . 594

Fig. 21.28 The upper part of the figure shows the transformationfrom the two-generator equivalent circuit of Fig. 21.25to the single-generator circuit, still in the common-baseconfiguration. The lower part shows the transformation ofthe latter into the common-emitter configuration . . . . . . . . . . . . . . . . . 596

Fig. 22.1 Cross-section of a metal-insulator-semiconductorcapacitor. The thickness of the insulator layer is notrealistic: in real devices the layer is much thinner than thecontacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

Fig. 22.2 The three materials forming the MOS capacitor shownseparately. The symbols’ meaning is illustrated in the text . . . . . . . 603

Fig. 22.3 The three materials forming the MOS capacitor after beingbrought into contact. The symbols’ meaning is illustratedin the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

Fig. 22.4 The cylinder used to calculate the relation between electricdisplacement and charge per unit area across an interface . . . . . . . . 607

xlii List of Figures

Fig. 22.5 Normalized surface potential us in an MOS capacitor witha p-type substrate (NA D 1016 cm�3), as a function of thenormalized gate voltage u0

G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609Fig. 22.6 Schematic representation of the charge density and

electric potential in a p-substrate MOS capacitor in theaccumulation regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

Fig. 22.7 Schematic representation of the charge density and electricpotential in a p-substrate MOS capacitor in the mid-gapcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

Fig. 22.8 Schematic representation of the charge density and electricpotential in a p-substrate MOS capacitor at threshold . . . . . . . . . . . . 612

Fig. 22.9 Normalized capacitance C=Cox as a function of thenormalized gate voltage u0

G, in a p-substrate MOScapacitor with NA D 1016 cm�3, for different values ofr D "sc tox=."ox

p2 LA/. The details of the calculations are

in Prob. 22.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614Fig. 22.10 Normalized charge per unit area as a function of the

normalized surface potential, in a p-substrate MOScapacitor with NA D 1016 cm�3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

Fig. 22.11 Individual contributions of electrons, holes, and bulkcharge to F2 D ŒQsc=Q.1/

sc �2, as a function of the normalizedsurface potential us, in a p-substrate MOS capacitor withNA D 1016 cm�3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

Fig. 22.12 The same materials as in Fig. 22.3, with W � A < EC � EF.The semiconductor’s bands curve downwards near thesemiconductor–oxide interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

Fig. 22.13 Normalized semiconductor charge Qsc=Q.1/sc as a function

of the normalized gate voltage u0G, for a p-substrate MOS

capacitor with NA D 1016 cm�3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620Fig. 22.14 Normalized concentrations n=pp0 and .NA � p/=pp0 as a

function of position x=LA, for a p-substrate MOS capacitorwith NA D 1016 cm�3 in strong inversion (us D 2:5 uF) . . . . . . . . . 621

Fig. 22.15 One-dimensional sketch of the MOS photocapacitor with auniform, p-type substrate. The relative thicknesses of thedifferent layers shown in the figure are not realistic . . . . . . . . . . . . . . 621

Fig. 22.16 One-dimensional sketch of the MOS photocapacitor builtin a p-type epitaxial layer. The relative thicknesses of thedifferent layers shown in the figure are not realistic . . . . . . . . . . . . . . 624

Fig. 22.17 Normalized charge per unit area as a function of thenormalized surface potential, in a n-substrate MOScapacitor with ND D 1016 cm�3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

Fig. 22.18 Cross-section of an n-channel MOSFET. The black areasare the metal contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

List of Figures xliii

Fig. 22.19 Low frequency, small-signal circuit of an n-channelMOSFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

Fig. 22.20 Output characteristics of an n-type MOSFET obtainedfrom the linear-parabolic model, with VT D 1 V, ˇ D 0:3

A V�2. The dashed curve represents (22.94) . . . . . . . . . . . . . . . . . . . . . . 639Fig. 22.21 Symbol of the n-channel MOSFET of the enhancement type . . . . 639Fig. 22.22 Symbol of the p-channel MOSFET of the enhancement type . . . . 640Fig. 22.23 The CMOS inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646Fig. 22.24 Cross-section of an n-well CMOS inverter . . . . . . . . . . . . . . . . . . . . . . . . 647Fig. 22.25 The black, continuous line shows the input-output curve

of the CMOS inverter. The curve has been drawn usingVSS D 6 V, VTp D VTn D 1 V, and ˇp D ˇn . . . . . . . . . . . . . . . . . . . . . . . 651

Fig. 22.26 Illustration of the electric field’s components at the channelends in the saturation condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

Fig. 22.27 The inverter made of an n-channel MOSFET with aresistive load. It is Vin D VGS ; Vout D VDS . . . . . . . . . . . . . . . . . . . . . . . . 656

Fig. 22.28 The inverter made of a p-channel MOSFET with a resistiveload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656

Fig. 22.29 The us D us.n/ relation calculated from (22.147)for different values of u0

GB. The constant acceptorconcentration of the substrate is NA D 1016 cm�3,corresponding to 2 uF ' 27:6. The dashed line shows theus D n C 2 uF approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

Fig. 22.30 Pictorial scheme of the top-down and bottom-up design ofintegrated circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

Fig. 22.31 Stick diagram of the CMOS inverter of Fig. 22.23 . . . . . . . . . . . . . . . . 667

Fig. 23.1 Illustration of the symbols used in the calculation of the flux . . . . 676Fig. 23.2 Normalized profiles N=C produced at different instants

by a predeposition, using the first of (23.15) as initialcondition with arbitrary units for the x coordinate. Theoutcome is a set of complementary error functions whoseexpression is the first of (23.16). The legends show thevalue of 4 a for each curve, also in arbitrary units, witha D a.t/ given by the second of (23.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

Fig. 23.3 Diffused profiles calculated at t1 and t2 D 16 t1 when twodifferent materials are involved. The calculation is basedon (23.17), (23.18) as described at the end of Sect. 23.5.1.The legends show the .4 D t/1=2 value for each curve . . . . . . . . . . . . . 683

Fig. 23.4 Schematic cross-section of an ion implanter . . . . . . . . . . . . . . . . . . . . . . 687Fig. 23.5 Detail of the analyzing system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688Fig. 23.6 Sketch of the trajectory of implanted ions . . . . . . . . . . . . . . . . . . . . . . . . . 691Fig. 23.7 The nuclear and electronic stopping power. The vertical

line marks the critical energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

xliv List of Figures

Fig. 23.8 Example of the random walk of an ion within thesemiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

Fig. 23.9 For the trajectory on the left (green line) a large number ofcollisions occur. The central trajectory (blue line) describesa channeling ion. The trajectory on the right (red line)describes an ion that soon after entering the semiconductorsuffers a nuclear collision in which most of its kineticenergy is lost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

Fig. 24.1 Structure of quartz. Silicon atoms are represented ingray, oxygen atoms in white. Within the tetrahedron, thedistance between two oxygen atoms is about 0:227 nm,that between the silicon atom and an oxygen atom is about0:160 nm. The schematic representation in two dimensionsis shown in the lower-right part of the figure . . . . . . . . . . . . . . . . . . . . . . 704

Fig. 24.2 Furnace for silicon oxidation. The intake of the carrier gas(O2 or H2O) is on the left end of the furnace, the exhauston the right end. The tube, end cap, and boat are made offused quartz to avoid contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

Fig. 24.3 The left part of the figure shows the position of the originalsilicon surface (prior to oxidation). The right part showsthe position of the oxide’s surface on of the silicon–oxideinterface after an oxide layer of thickness s has been grown . . . . . 706

Fig. 24.4 Parabolic coefficient cp as a function of 1;000=T . The unitsare m2 h�1. The activation energy of the steam case is0:71 eV, that of the dry case is 1:24 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710

Fig. 24.5 Linear coefficient cl as a function of 1;000=T . The units arem h�1. The activation energy of the steam case is 2:05

eV, that of the dry case is 2:0 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710Fig. 24.6 The linear-parabolic model (24.11) is drawn using

dimensionless variables (blue line). The linear (blackcurve) and parabolic (red curve) limiting cases are alsoshown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

Fig. 24.7 Schematic cross-section of the transition from field oxideto gate oxide, showing the bird’s beak profile . . . . . . . . . . . . . . . . . . . . . 713

Fig. 24.8 Schematic description of the thinning of a metal layerdeposited over a step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713

Fig. 24.9 Normalized growth velocity as a function of the normalizedinverse temperature, as given by (24.23) and (24.24), atdifferent values of the rv D vG=vr0 ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 716

Fig. 24.10 Oxidant concentration within the oxide at two differentinstants, t1 and t2 > t1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

List of Figures xlv

Fig. 24.11 Typical growth velocity cl of an epitaxial process,expressed in microns per minute, as a function of the molefraction of tetrachloride. The shaded area shows the typicaloperating range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

Fig. 25.1 Measurement scheme for the minority-carrier lifetime . . . . . . . . . . . 724Fig. 25.2 Time dependence of the current flowing in the sample of

Fig. 25.1 when the minority-carrier lifetime (�p in thiscase) is measured . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

Fig. 25.3 Measurement scheme for mobility (Haynes-Shockleyexperiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

Fig. 25.4 Scheme of a Hall-voltage measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 730Fig. 25.5 Scheme for the combined conductivity and Hall-voltage

measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733Fig. 25.6 Form of the D �!= logŒ1 � exp.�!/� relation, with

D %dcab=%ad

bc and ! D %dcab � , found in the analysis of the

van der Pauw method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742

Fig. A.1 Rotational theorem (Sect. A.9): orientation of the unitvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752

Fig. A.2 Illustration of the concepts of node, element, and cell usedin the discretization of the semiconductor equations . . . . . . . . . . . . . . 776

Fig. B.1 Cartesian (x; y; z) and spherical (r; #; ') coordinates . . . . . . . . . . . . . 785

Fig. C.1 Generation of a Dirac ı using a barrier-like function. Thepeak’s width is equal to a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

Fig. C.2 Generation of a Dirac ı using a Lorentzian function. Thepeak’s width is proportional to a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

Fig. C.3 Generation of a Dirac ı using a parameterized Gaussianfunction. The peak’s width is proportional to a . . . . . . . . . . . . . . . . . . . 811

Fig. C.4 Generation of a Dirac ı using a Fermi function. The peak’swidth is proportional to a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812

Fig. C.5 Integration path of (C.87) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819Fig. C.6 Approximations to the Fermi integrals: the black line

represents (C.112), applicable to the Fermi integrals of anyorder for � �1; the red line represents (C.117) thatrenders ˚1=2 within ˙3% when < 1:3; the blue linerepresents (C.118) that renders ˚1=2 within ˙3% when � 0:7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

Fig. C.7 Bernoulli function (thick line) and its asymptote forx ! �1 (thin line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

Fig. D.1 Constant-energy curves of the linear harmonic oscillatordiscussed in Prob. 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840

Fig. D.2 Constant-energy curves of the nonlinear harmonicoscillator discussed in Prob. 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840

xlvi List of Figures

Fig. D.3 State trajectory of the linear harmonic oscillator . . . . . . . . . . . . . . . . . . 841Fig. D.4 State trajectory of the harmonic oscillator of the general

form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842Fig. D.5 Form of the potential energy cjxjs=s for c D 1 and different

values of s (Prob. 3.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844Fig. D.6 Form of the potential energy �kjxj�s=s for k D 1 and

different values of s (problem 3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845Fig. D.7 Normalized loss of energy c .T1a � T1b/=˛ as a function of

the normalized initial energy c T1a=˛ (problem 3.3), fordifferent values of the ratio m1=m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

Fig. D.8 Example of charge density such that M0 D 0 and M1 D 0 . . . . . . . 847Fig. D.9 The blue line is the inverse function of (21.141), drawn

with pn0 D 1, Ap D 2, Lp D 2 in arbitrary units. Thetangent to p.x/ at x D 0 (dashed line) intersects theasymptotic value pn0 at x D Lp. The area of the rectanglemarked in red is equal to the area between p.x/ and theasymptotic value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872

Fig. D.10 Input-output curve of the inverter made of an n-channelMOSFET with a resistive load (Probs. 22.7 and 22.8) . . . . . . . . . . . . 876

Fig. D.11 Input-output curve of the inverter made of an p-channelMOSFET with a resistive load (Probs. 22.9 and 22.10) . . . . . . . . . . . 878

Fig. D.12 Normalized profiles h N=Q resulting from the drive-indiffusion of problem 23.8. The coordinate is � D x=h.Each profile corresponds to the value of b.t/ shown in thelegend. The parameter is defined by b D 4 a=h2, whilea D a.t/ is defined by the second of (23.10). As explainedin Sect. 23.5.2, only the profile’s portion on the right of theorigin must be considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882

List of Tables

Table 13.1 The lowest-order spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261Table 13.2 Symbols and names for the states corresponding to

l D 0; 1; 2; 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Table 14.1 Possible coefficients of g� when k C kd D k0 C g� in (14.67) . . 296Table 14.2 Possible coefficients of g� when k � kd D k0 C g� in (14.67) . . . 297

Table 17.1 Crystal constants of silicon and germanium . . . . . . . . . . . . . . . . . . . . . . 344Table 17.2 Crystal constants of some III-V semiconductors . . . . . . . . . . . . . . . . . 345Table 17.3 Normalized effective masses of the valence band of Si, Ge,

and GasAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370Table 17.4 Normalized effective masses of the conduction band of Si,

Ge, and GasAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Table 18.1 Gap and average effective masses of silicon, germanium,and gallium arsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Table 18.2 Intrinsic concentrations of silicon, germanium, and galliumarsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

Table 21.1 BJT: nomenclature of the functioning regimes . . . . . . . . . . . . . . . . . . . 579

Table 22.1 MOS capacitor, p substrate—functioning regimes . . . . . . . . . . . . . . . 609Table 22.2 MOS capacitor, n substrate—functioning regimes . . . . . . . . . . . . . . . 627Table 22.3 CMOS inverter—functioning conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 650

Table 23.1 Example of parameters of an ion implanter . . . . . . . . . . . . . . . . . . . . . . . 692

Table 24.1 Examples of CVD reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712

Table A.1 FORTRAN program for solving a tridiagonal system with(A.95) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

Table D.1 Fundamental constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837Table D.2 Greek alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838

xlvii