Solid state physics an introduction to principles of materials science

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SMART ELECTRONIC MATERIALS Smart materials respond rapidly to external stimuli to alter their physical properties They are used in devices that are driving advances in modern information technology and have applications in electronics, optoelectronics, sensors, memories and other areas This book fully explains the physical properties of these materials, including semiconductors, dielectrics, ferroelectrics, and ferromagnetics Fundamental concepts are consistently connected to their real-world applications It covers structural issues, electronic properties, transport properties, polarization-related properties, and magnetic properties of a wide range of smart materials The book contains carefully chosen worked examples to convey important concepts and has many end-of-chapter problems It is written for first year graduate students in electrical engineering, material sciences, or applied physics programs It is also an invaluable book for engineers working in industry or research laboratories A solution manual and a set of useful viewgraphs are also available for instructors by visiting http://www.cambridge.org/ 0521850274 JASPRIT SINGH obtained his Ph.D in Solid State Physics from the University of Chicago He is currently a professor in the Applied Physics Program and in the Department of Electronic and Computer Science at the University of Michigan, Ann Arbor He has held visiting positions at the University of California in Santa Barbara He has authored over 250 technical articles He has also authored eight textbooks in the area of applied physics and technology His area of expertise is novel materials for applications in intelligent devices SMART ELECTRONIC MATERIALS Fundamentals and Applications JASPRIT SINGH University of Michigan CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK www.cambridge.org Information on this title: www.cambridge.org/9780521850274 © Cambridge University Press 2005 This book is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2005 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this book is available from the British Library Library of Congress Cataloging in Publication data ISBN-13 978-0-521 -85027-4 hardback ISBN-10 0-521 -85027-4 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate CONTENTS PREFACE INTRODUCTION page xi xiii SMART MATERIALS: AN INTRODUCTION xiii INPUT—OUTPUT DECISION ABILITY xiv 2.1 Device based on conductivity changes xiv 2.2 Device based on changes in optical response xv BIOLOGICAL SYSTEMS: NATURE'S SMART MATERIALS xix ROLE OF THIS BOOK STRUCTURAL PROPERTIES xxii 1.1 INTRODUCTION 1.2 CRYSTALINE MATERIALS 1.2.1 Basic lattice types 1.2.2 Some important crystal structures 1.2.3 Notation to denote planes and points in a lattice: Miller indices 1.2.4 Artificial structures: superlattices and quantum wells 1.2.5 Surfaces: ideal versus real 1.2.6 Interfaces 12 16 17 19 1.3 DEFECTS IN CRYSTALS 20 1.4 HETEROSTRUCTURES 23 1.5 NON-CRYSTALLINE MATERIALS 24 1.5.1 1.5.2 1.5.3 1.5.4 1.6 Polycrystalline materials Amorphous and glassy materials Liquid crystals Organic materials SUMMARY 25 26 27 31 31 vi Contents 1.7 PROBLEMS 33 1.8 FURTHER READING 37 QUANTUM MECHANICS AND E L E C T R O N I C LEVELS 2.1 INTRODUCTION 39 2.2 N E E D FOR Q U A N T U M DESCRIPTION 40 2.2.1 Some experiments that ushered in the quantum age 40 SCHRODINGER EQUATION AND PHYSICAL OBSERVABLES 48 2.3.1 Wave amplitude 2.3.2 Waves, wavepackets, and uncertainty 52 54 2.3 2.4 PARTICLES IN AN ATTRACTIVE POTENTIAL: BOUND STATES 57 2.4.1 Electronic levels in a hydrogen atom 2.4.2 Particle in a quantum well 2.4.3 Harmonic oscillator problem 58 62 67 2.5 FROM ATOMS TO MOLECULES: COUPLED WELLS 69 2.6 ELECTRONS IN CRYSTALLINE SOLIDS 77 2.6.1 Electrons in a uniform potential 2.6.2 Particle in a periodic potential: Bloch theorem 2.6.3 Kronig-Penney model for bandstructure U 39 80 85 87 2.7 SUMMARY 93 2.8 PROBLEMS 93 2.9 FURTHER READING 99 E L E C T R O N I C LEVELS IN SOLIDS 100 3.1 INTRODUCTION 100 3.2 OCCUPATION OF STATES: DISTRIBUTION FUNCTION 100 3.3 METALS, INSULATORS, AND SUPERCONDUCTORS 3.3.1 Holes in semiconductors 3.3.2 Bands in organic and molecular semiconductors 3.3.3 Normal and superconducting states 3.4 BANDSTRUCTURE OF SOME IMPORTANT SEMICONDUCTORS 3.4.1 Direct and indirect semiconductors: effective mass 104 104 107 108 110 111 Contents 3.5 VII MOBILE CARRIERS 3.5.1 Electrons in metals 3.5.2 Mobile carriers in pure semiconductors 3.6 DOPING OF SEMICONDUCTORS 3.7 TAILORING ELECTRONIC PROPERTIES 3.7.1 Electronic properties of alloys 3.7.2 Electronic properties of quantum wells 3.8 LOCALIZED STATES IN SOLIDS 3.8.1 Disordered materials: extended and localized states 116 117 120 126 131 131 132 136 138 3.9 SUMMARY 141 3.10 PROBLEMS 141 3.11 FURTHER READING 146 CHARGE TRANSPORT IN MATERIALS 148 4.1 INTRODUCTION 148 4.2 A N OVERVIEW OF ELECTRONIC STATES 149 4.3 TRANSPORT AND SCATTERING 4.3.1 Scattering of electrons 4.4 MACROSCOPIC TRANSPORT PROPERTIES 4.4.1 Velocity-electric field relations in semiconductors 4.5 CARRIER TRANSPORT BY DIFFUSION 4.5.1 Transport by drift and diffusion: Einstein's relation 4.6 IMPORTANT DEVICES BASED ON CONDUCTIVITY CHANGES 4.6.1 Field effect transistor 4.6.2 Bipolar junction devices 4.7 TRANSPORT IN NON-CRYSTALLINE MATERIALS 4.7.1 Electron and hole transport in disordered systems 4.7.2 Ionic conduction 4.8 IMPORTANT NON-CRYSTALLINE ELECTRONIC DEVICES 4.8.1 Thin film transistor 4.8.2 Gas sensors 151 154 162 162 173 175 178 179 184 186 187 191 193 193 195 4.9 SUMMARY 195 4.10 PROBLEMS 199 4.11 FURTHER READING 200 Contents VIII L I G H T A B S O R P T I O N AND EMISSION 202 5.1 INTRODUCTION 202 5.2 IMPORTANT MATERIAL SYSTEMS 204 5.3 OPTICAL PROCESSES IN SEMICONDUCTORS 5.3.1 Optical absorption and emission 5.3.2 Chargei injection, quasi-Fermi levels, and recombination 5.3.3 Optical absorption, loss, and gain 5.4 5.5 OPTICAL PROCESSES IN QUANTUM WELLS IMPORTANT SEMICONDUCTOR OPTOELECTRONIC DEVICES 5.5.1 Light detectors and solar cells 5.5.2 Light emitting diode 5.5.3 Laser diode 5.6 ORGANIC SEMICONDUCTORS: OPTICAL PROCESSES & DEVICES 5.6.1 Excitonic state 207 210 219 225 226 231 231 238 243 251 252 5.7 SUMMARY 255 5.8 PROBLEMS 255 5.9 FURTHER READING 262 DIELECTRIC RESPONSE: POLARIZATION EFFECTS 264 6.1 6.2 INTRODUCTION POLARIZATION IN MATERIALS: DIELECTRIC RESPONSE 6.2.1 Dielectric response: some definitions 264 265 265 6.3 FERROELECTRIC DIELECTRIC RESPONSE 273 6.4 TAILORING POLARIZATION: PIEZOELECTRIC EFFECT 275 6.5 TAILORING POLARIZATION: PYROELECTRIC EFFECT 285 6.6 DEVICE APPLICATIONS OF POLAR MATERIALS 6.6.1 6.6.2 6.6.3 6.6.4 Ferroelectric memory Strain sensor and accelerometer Ultrasound generation Infrared detection using pyroelectric devices 287 287 288 289 289 Contents 6.7 SUMMARY 291 6.8 PROBLEMS 291 6.9 FURTHER READIN G 295 OPTICAL MODULATION AND SWITCHING 296 7.1 INTRODUCTION 296 7.2 LIGHT PROPAGATION IN MATERIALS 297 7.3 MODULATION OF OPTICAL PROPERTIES 302 7.3.1 Electro-optic effect 7.3.2 Electro-absorption modulation 7.4 OPTICAL MODULATION DEVICES 303 309 312 7.4.1 Electro-optic modulators 316 7.4.2 Interferroelectric modulators 318 7.5 SUMMARY 323 7.6 PROBLEMS 325 7.7 FURTHER READING 325 M A G N E T I C EFFECTS IN SOLIDS 326 8.1 INTRODUCTION 326 8.2 MAGNETIC MATERIALS 326 8.3 ELECTROMAGNETIC FIELD MAGNETIC MATERIALS 327 8.4 PHYSICAL BASIS FOR MAGNETIC PROPERTIES 331 8.5 COHERENT TRANSPORT: QUANTUM INTERFERENCE 8.5.1 Aharonov Bohm effect 8.5.2 Quantum interference in superconducting materials 8.6 DlAMAGNETIC AND PARAMAGNETIC EFFECTS 8.6.1 Diamagnetic effect 8.6.2 Paramagnetic effect 8.6.3 Paramagnetism in the conduction electrons in metals 335 335 338 340 340 341 345 396 E.2 Defect scattering and mobility SCREENED COULOMBIC SCATTERING As another example of the use of the Fermi golden rule or Born approximation, we will examine the scattering of an electron from a charged particle The scattering potential is Coulombic in nature This scattering plays a very important role in many important applications Problems that require an understanding of this scattering process include: • Scattering of a particles in matter: When a thin film of metal is bombarded with a particles (He-nuclei), the properties of the outgoing particles are understood on the basis of Coulombic scattering • Mobility in devices: Semiconductor devices have regions that are doped with donors or acceptors These dopants provide excess carriers in the conduction or the valence band Without these carriers most devices will not function When a dopant provides a free carrier, the remaining ion provides a scattering center for the free car- riers This causes scattering which is understood on the basis of electron-ion scattering Additionally, at high densities one can have electron-electron scattering as well as electron-hole scattering, which is also understood for the general problem discussed in this subsection Before starting our study of scattering from a Coulombic interaction, it is important to note that in most materials there is a finite mobile carrier density These carriers can adjust their spatial position in response to a potential and thus screen the potential The screening is due to the dielectric response of the material and includes the effect that the background ions as well as the other free electrons have on the potential A number of formalisms have been developed to describe the dielectric response function We will use a form given by the Thomas-Fermi formalism Let us consider an electron scattering from a charged particle in a crystalline material We will assume that the electron is described by the effective mass theory We also asume that the density of free carriers is UQ In the Thomas-Fermi formalism, the background free carriers modify their carrier concentration near the impurity so that when the scattering electron is far from the impurity it sees a potential much weaker than the Coulombic potential Very close to the impurity the potential is not affected much by the screening The real-space behavior of the screened potential is given by tot(r) = where q is the charge of the impurity and e is the dielectric constant The quantity A, which represents the effect of the background free carriers is given for a non-degenerate carrier gas (i.e., a carrier distribution where the Fermi statistics is reasonably approximated by the Boltzmann statistics) as E.2 Screened Coulombic scattering 397 r(arbitrary units) '3 -1.0 I Unscreened potential energy -2.0 -3.0 Figure E.2: Comparison of screened and unscreened Coulomb potentials of a unit positive charge as seen by an electron The screening length is A"1 When the free carrier density is high so that the carriers are degenerate, _ 3n e ~ 2eEF (E.16) where Ep is the Fermi energy As noted, the effect of screening is to reduce the range of the potential from a 1/r variation to a exp(—Ar)/r variation This is an extremely important effect and is shown schematically in Fig E.2 We now calculate the matrix element for the screened Coulombic potential U(v) = Ze2 e " ; (E.17) where Ze is the charge of the impurity We choose the initial normalized state to be |k) = exp(ik • r)/\/V F and the final state to be |k') = exp(ik' • v)/y/V\ where V is the volume of the crystal The matrix element is then o-\r Ze2 *(k k)r 47T61/ J r dr sin dO d ,2 N(Ek) h \ e J V 32k4 ^ ^ ( c o s °) d(t> x / (1 - cos 0) = F / (1 - cos 6>) =• d(cos 6>) d0 F-2 x \ \ 2~| (E.25) E.3 Ionized impurity limited mobility 401 Finally I = T * fZe2\2N(Ek) 4h\e) Vk4 m*3l2E1'2 (E.26) Note that the spin degeneracy is ignored, since the ionized impurity scattering cannot alter the spin of the electron In terms of the electron energy, Ek, we have l+(h2\2/$m*Ek) (E.27) To calculate the mobility limited by ionized impurity scattering, we have to find the ensemble averaged r To a good approximation, the effect of this averaging is essentially to replace Ek by kBT in the expression for 1/r A careful evaluation of the average ((r)) gives (see Eq E.23) Ze2Y (kBT) In + 3/2 (2Am*kBT\ 2 J V (E.28) nx 24m* kBT If there are N{ impurities per unit volume, and if we assume that they scatter electrons independently, the total relaxation time is simply obtained by multiplying the above results by NiV, ((r)) 128V27T (kBT) (24m*kBT\ In + I +2 >2 j 3/2 H2\2 (E.29) + 24m* The mobility is then m* Mobility limited by ionized impurity scattering has the special fi ~ j " / behavior that is represented in Eq E.29 This temperature dependence (the actual temperature dependence is more complex due to the other T-dependent terms present) is a special Defect scattering and mobility 402 2,000,000 1,000,000 500,000 200,000 MOBILITY uiolri 100,000 50,000 20,000 10,000 5,000 2,000 10 20 50 100 200 400 TEMPERATURE [K] Figure E.5: (a) A typical plot of electron mobility as a function of temperature in a uniformly doped GaAs with ND = 1017 cm" The mobility drops at low temperature due to ionized impurity scattering, becoming very strong In contrast, the curve (b) shows a typical plot of mobility in a modulation-doped structure where ionized impurity is essentially eliminated signature of the ionized impurity scattering One can understand this behavior physically by realizing that at higher temperatures the electrons are traveling faster and are less affected by the ionized impurities Ionized impurity scattering plays a very central role in controlling the mobility of carriers in semiconductor devices This is especially true at low temperatures where the other scattering processes (due to lattice vibrations) are weak To avoid impurity scattering, the concept of modulation doping has been developed In this approach, the device is made from two semiconductors—a large bandgap barrier layer and a smaller bandgap well layer The barrier layer is doped so that the free carriers spill over into the well region where they are physically separated from the dopants This essentially eliminates ionized impurity scattering Fig E.5 compares the mobilities of conventionally doped and modulation doped GaAs channels As can be seen, there is a marked improvement in the mobility, especially at low temperatures Modulation doping forms the basis of the highest performance semiconductor devices in terms of speed and noise In Fig E.6 we show how the mobility in Ge, Si and GaAs varies as a function of doping density The mobility shown includes the effects of lattice scattering, as well as ionized impurity scattering E.4 Alloy scattering limited mobility 403 o on o5 1014 10!5 1016 1017 1018 IMPURITY CONCENTRATION (cm" ) Figure E.6: Drift mobility of Ge, Si, and GaAs as 300 K versus impurity concentration (After S M Sze, Physics of Semiconductor Devices, 2nd ed., John Wiley and Sons, New York, 1981.) E.4 ALLOY SCATTERING LIMITED MOBILITY The ensemble averaged relaxation time for the alloy scattering is quite simple (see Eqs E.12 and E.23):