Siegfried Selberherr

Analysis and Simulation of Semiconductor Devices

Springer-Verlag Wien New York

Dipl.-Ing. Dr. Siegfried Selberherr Institut flir Allgemeine Elektrotechnik und Elektronik, Technische Universitiit Wien, Austria

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 1984 by Springer-VerlagfWien Softcover reprint of the hardcover 1st edition 1984

With 126 Figures

ISBN-13:978-3-7091-8754-8 e-ISBN-13:978-3-7091-8752-4 DOl: 10.1007/978-3-7091-8752-4

To Margit

Preface

The invention of semiconductor devices is a fairly recent one, considering classical time scales in human life. The bipolar transistor was announced in 1947, and the MOS transistor, in a practically usable manner, was demonstrated in 1960. From these beginnings the semiconductor device field has grown rapidly. The first integrated circuits, which contained just a few devices, became commercially available in the early 1960s. Immediately thereafter an evolution has taken place so that today, less than 25 years later, the manufacture of integrated circuits with over 400.000 devices per single chip is possible. Coincident with the growth in semiconductor device development, the literature concerning semiconductor device and technology issues has literally exploded. In the last decade about 50.000 papers have been published on these subjects. The advent of so called Very-Large-Scale-Integration (VLSI) has certainly revealed the need for a better understanding of basic device behavior. The miniaturization of the single transistor, which is the major prerequisite for VLSI, nearly led to a breakdown of the classical models of semiconductor devices. The characteristic feature of early (classical) device modeling is primarily the separation of the interior of the device into different regions, treated by closed form solutions based on restrictive and sometimes drastic assumptions. The solutions in the independently treated regions are simply connected and matched at boundaries to produce a global solution. Any other approach is obviously prohibitive if results with an analytic appearance are intended. For the purpose of analysis, however, this classical approach has turned out to be only limitedly applicable, particularly when a technically acceptable prediction of device performance is desired. As a consequence numerical analysis and simulation based on comparatively fundamental differential equations has become necessary and populac. This trend has been supported considerably by the enormous progress in technology and performance of digital computers. Contemporary modeling of semiconductor devices has attained such a high level of sophistication that two-dimensional simulation of the static behavior is almost standard in the development stage of device prototypes. Even three-dimensional transient simulations have been re­ported very recently, but, due to a still too extensive consumption of computer resources, these are at the moment more of academic importance than of practical relevance. Numerical analysis of semiconductor devices can be expected to become a basic methodology of research and development engineers. However, one must not expect

VIII Preface

that people using computer programs as numerical analysis tools are specialists considering the complexity of the assumptions, algorithms and implementation details of the programs they use. In particular, this book has been written with two primary objectives kept in view: First, the interested device engineer should be introduced to the physical and mathematical problems to be solved by an analysis program. This category of readers should gain a more fundamental understanding of the applicability of device simulation programs. Secondly, this book will benefit authors of device simulation programs by providing a compact reference with many citations and a critical overview of the various physical and mathematical approaches which are used worldwide today. The chapters in this book are arranged in a logical sequence without many crossreferences. Each chapter is more or less self-contained. Readers who are only interested in particular subjects will be able to extract easily the information they require. In preparing the material for this book many people have assisted me to a considerable extent. I am extremely grateful to Prof. H. Potzl for reviewing my manuscripts and the resulting endless discussions and suggestions. I am indebted to my colleagues at the university for many discussions and the friendly atmosphere: Drs. W. Agler, J. Demel, A. Franz, G. Franz, W. Griebel, E. Guerrero, W. Jlingling, M.Kowatsch, H.Lafferl, E.Langer, W.Mader, P.Markowich, Prof.F.Paschke, P. Pichler, C. Ringhofer, A. Schlitz, Prof. F. Seifert, Prof. H. Stetter, F. Straker, Doz. Ch. Oberhuber, Prof. R. WeiB. I would like to express my sincere appreciation to Dr.S.E.Laux, IBM T.J. Watson Research Center, for proofreading my manu­script. I would like to thank the Austrian "Fonds zur Forderung der wissenschaft­lichen Forschung" and the Research Laboratory of Siemens AG, Munich, FRG, for supporting many projects which have evolved into much of the material presented in this book. Last but not least I would like to gratefully acknowledge the generous amount of computer resources provided by Dipl.-Ing. D. Schornbock, and the excellent computer access made possible by the whole staff ofthe local computer center. I hope my book will be used by many engineers and scientists who wish to gain insight into the subject of numerical device modeling. It is my sincere wish that this book will contribute to bridging the gaps between solid-state physicists, numerical analysists, computer scientists and device engineers.

Vienna, April 1984 Siegfried Selberherr

Notation XI

1. Introduction 1

1.1 The Goal of Modeling 1 1.2 The History of Numerical Device Modeling 2 1.3 References 4

2. Some Fundamental Properties 8

2.1 Poisson's Equation 8 2.2 Continuity Equations 10 2.3 Carrier Transport Equations 11 2.4 Carrier Concentrations 23 2.5 Heat Flow Equation 40 2.6 The Basic Semiconductor Equations 41 2.7 References 42

3. Process Modeling 46

3.1 Ion Implantation 46 3.2 Diffusion 63 3.3 Oxidation 72 3.4 References 76

4. The Physical Parameters 80

4.1 Carrier Mobility Modeling 80 4.2 Carrier Generation-Recombination Modeling 103 4.3 Thermal Conductivity Modeling 118 4.4 Thermal Generation Modeling 120 4.5 References 121

Contents

5. Analytical Investigations About the Basic Semiconductor Equations 127

5.1 Domain and Boundary Conditions 128 5.2 Dependent Variables 134

X Contents

5.3 The Existence of Solutions 140 5.4 Uniqueness or Non-Uniqueness of Solutions 141 5.5 Scaling 141 5.6 The Singular Perturbation Approach 144 5.7 References 147

6. The Discretization of the Basic Semiconductor Equations 149

6.1 Finite Differences 150 6.2 Finite Boxes 175 6.3 Finite Elements 181 6.4 The Transient Problem 191 6.5 Designing a Mesh 197 6.6 References 199

7. The Solution of Systems of Nonlinear Algebraic Equations 202

7.1 Newton's Method and Extensions 203 7.2 Iterative Methods 208 7.3 References 212

8. The Solution of Sparse Systems of Linear Equations 214

8.1 Direct Methods 214 8.2 Ordering Methods 216 8.3 Relaxation Methods 239 8.4 Alternating Direction Methods 245 8.5 Strongly Implicit Methods 246 8.6 Convergence Acceleration of Iterative Methods 249 8.7 References 254

9. A Glimpse on Results 258

9.1 Breakdown Phenomena in MOSFET's 258 9.2 The Rate Effect in Thyristors 270 9.3 References 284

Author Index 286

Subject Index 291

Table Index 294

Notation

A vector potential Jj magnetic induction vector C net ionized impurity concentration CI total ionized impurity concentration CN total neutral impurity concentration C~PT optical capture coefficient CC i electrically inactive concentration of i-th impurity C1;,u Auger capture coefficient C~~H Shockley-Read-Hall capture coefficient C~PT optical emission coefficient C:,u Auger emission coefficient C:~H Shockley-Read-Hall emission coefficient Cti total concentration of i-th species 15 electric displacement vector Dt diffusivity of i-th impurity due to singly positive charged vacancies Di- diffusivity of i-th impurity due to singly negative charged vacancies Di= diffusivity of i-th impurity due to doubly negative charged vacancies D? diffusivity of i-th impurity due to neutral vacancies Di effective diffusivity of i-th impurity D~ thermal diffusion coefficient D, effective diffusivity E electric field vector E, effective field E energy E electric field Eac acoustic deformation potential of conduction band Eav acoustic deformation potential of valence band Ec conduction band energy Eco conduction band edge Ef , quasi-Fermi energy Eg band gap Ei ionization energy Ei intrinsic Fermi energy E~ril critical field

XII

Ev

E, Ev Evo EJ. Ell F Fve FVi F1/2 i1 H J Ji

Jv

NA N~ Ne Nd Nt Nv

R R AU

RII

ROPT

RSRH

RSURF

Rp Sv T Tv Ut v+ v­v= Zi a aB

(Xv

P2 Pc C

C

bEe bEv df B

driving force average energy loss per high energetic collision valence band energy valence band edge electric field component perpendicular to current flow direction electric field component parallel to current flow direction force vector external force internal force Fermi integral of order 1/2 magnetic field vector thermal generation total electric current density flux of i-th impurity carrier current density concentration of singly ionized acceptors concentration of singly ionized donors effective density of states in conduction band implantation dose concentration of traps effective density of states in valence band net carrier generation/recombination net Auger generation/recombination net impact ionization generation rate net optical generation/recombination net Shockley-Read-Hall generation/recombination net surface generation/recombination projected range scattering probability lattice temperature carrier temperature thermal voltage (=k. T/q) normalized concentration of singly positive charged vacancies normalized concentration of singly negative charged vacancies normalized concentration of doubly negative charged vacancies charge state of i-th impurity crystal lattice constant Bohr radius (=5.2917706.10- 11 m) ionization rate kurtosis equilibrium cluster coefficient speed of light in vacuum ( = 2.99792458 . 108 ms -1)

specific heat shift energy for conduction band edge shift energy for valence band edge field enhancement factor absolute permittivity

Notation

Notation

eo e, Iv ({)v

It Y1 h k k k kc kd A. A.

ni

nie

no P Po q

P P PA PD Pc Pv (fDA

(fev

(fp

t

t mask

'tv

'tv

itv

l/I l/Ib Vv v~at

X

XIII

permittivity constant in vacuum (= 8.854187818.10- 12 As y- 1 m -1)

relative permittivity distribution function quasi-Fermi potential fraction of occupied traps skewness Planck constant (=6.626176.1O- 34 YAs2)

momentum vector thermal conductivity Boltzmann constant (= 1.380662.10- 23 Y As K -1) clustering rate declustering rate mean free path between high energetic collisions screening length cluster size i-th central moment effective mass carrier mobility permeability constant in vacuum ( = 4· n) electron rest mass (=9.109534.10- 31 Y As3 m- 2 )

electron concentration intrinsic carrier concentration effective intrinsic carrier concentration equilibrium concentration of electrons hole concentration equilibrium concentration of holes elementary charge (= 1.6021892.10- 19 As) specific mass density space charge density of states in acceptor band density of states in donor band density of states in conduction band density of states in valence band standard deviation for donor and acceptor band standard deviation for conduction and valence band tails standard deviation time mask thickness relaxation time lifetime group velocity electrostatic potential built-in potential drift velocity saturation velocity space vector oxide thickness

XIV Notation

Subscript "v" stands for "n" or "p" denoting the respective quantity for electrons or holes.

Superscript "L:" in the carrier mobility stands for any combination of the following list.

C carrier-carrier impurity scattering E velocity saturation I ionized impurity scattering L lattice scattering N neutral impurity scattering S surface scattering

A superscript "*,, or no superscript indicates the effective mobility which is comprised of all above given effects.

Landau Symbols

A) f(x)=O(g(x)) as x~xo means that

I f(x) I <const. g(x)

for x sufficiently close to Xo'

B) f(x)=o(g(x)) as X~Xo means that

lim f(x) =0 X--+Xo g (x)

C) Sometimes we say (sloppily) that "a quantity f is 0 (g)" which means that If I is of approximate order of magnitude I g I.