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Semiconductors for micro and nanotechnology an introduction for engineers
Semiconductors for micro and nanotechnology an introduction for engineers
Jan G. Korvink, Andreas Greiner
Semiconductors play a major role in modern microtechnology, especially in microelectronics. Since the dimensions of new microelectronic components, e.g. computer chips, now reach nanometer size, semiconductor research moves from microtechnology to nanotechnology. An understanding of the semiconductor physics involved in this new technology is of great importance for every student in engineering, especially electrical engineering, microsystem technology and physics. This textbook emphasizes a systemoriented view of semiconductor physics for applications in microsystem technology. While existing books only cover electronic device physics and are mainly written for physics students, this text gives a more handson approach to semiconductor physics and so avoids overloading engineering students with mathematical formulas not essential for their studies.
Categories:
Technique\\Nanotechnology
Year:
2002
Edition:
1
Publisher:
WileyVCH
Language:
english
Pages:
341
ISBN 10:
3527302573
ISBN 13:
9783527302574
File:
PDF, 9.58 MB
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Semiconductors for Micro and Nanotechnology— An Introduction for Engineers Semiconductors for Micro and Nanotechnology— An Introduction for Engineers Jan G. Korvink and Andreas Greiner Authors: Prof. Dr. Jan G. Korvink IMTEKInstitute for Microsystem Technology Faculty for Applied Sciences Albert Ludwig University Freiburg D79110 Freiburg Germany Dr. Andreas Greiner IMTEKInstitute for Microsystem Technology Faculty for Applied Sciences Albert Ludwig University Freiburg D79110 Freiburg Germany This book was carefully produced. Nevertheless, authors, editors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for. British Library CataloguinginPublication Data: A catalogue record for this book is available from the British Library. Die Deutsche Bibliothek — CIPCataloguinginPublication Data A catalogue record for this book is available from Die Deutsche Bibliothek. ISBN 3527302573 © WILEYVCH Verlag GmbH, Weinheim 2002 Printed on acidfree paper . All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form — by photoprinting, microﬁlm, or any other means — nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Printing: Strauss Offsetdruck GmbH, Mörlenbach Bookbinding: Litges & Dopf Buchbinderei GmbH, Heppenheim Printed in the Federal Republik of Germany Dedicated to Micheline Pﬁster, Sean and Nicolas Vogel, and in fond memory of Gerrit Jörgen Korvink Maria Cristina Vecchi, Sarah Maria Greiner und im Gedenken an Gertrud Maria Greiner Contents Contents Preface Chapter 1 Introduction 7 13 15 The System Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Popular Deﬁnitions and Acronyms . . . . . . . . . . . . . . . . . . 19 Semiconductors versus Conductors and Insulators . . . . . . . . 19 The Diode Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The Transistor Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Passive Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Microsystems: MEMS, MOEMS, NEMS, POEMS, etc. . . . . . 21 Sources of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Summary for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 References for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . 25 Semiconductors for Micro and Nanosystem Technology 7 Chapter 2 The Crystal Lattice System Observed Lattice Property Data 27 . . . . . . . . . . . . . . . . . . . . 29 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Silicon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Silicon Nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Gallium Arsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Symmetries of Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Elastic Properties: The Stressed Uniform Lattice Statics . . . . . . . 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 The Vibrating Uniform Lattice . . . . . . . . . . . . . . . . . . . . . 64 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Phonons, Speciﬁc Heat, Thermal Expansion . . . . . . . . . . . . . 81 Modiﬁcations to the Uniform Bulk Lattice . . . . . . . . . . . . 88 Summary for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 References for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 92 Chapter 3 The Electronic System 95 Quantum Mechanics of Single Electrons . . . . . . . . . . . . . 96 Wavefunctions and their Interpretation . . . . . . . . . . . . . . . . . 97 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Free and Bound Electrons, Dimensionality Effects . . . . 106 Finite and Inﬁnite Potential Boxes . . . . . . . . . . . . . . . . . . . . 106 Continuous Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 115 Potential Barriers and Tunneling . . . . . . . . . . . . . . . . . . . . . 115 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Transitions Between Electronic States . . . . . . . . . . . . . . . . . 127 Fermion number operators and number states . . . . . . . . . . 130 Periodic Potentials in Crystal . . . . . . . . . . . . . . . . . . . . . 132 The Bloch Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Formation of Band Structure . . . . . . . . . . . . . . . . . . . . . . . . 133 Types of Band Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Effective Mass Approximation . . . . . . . . . . . . . . . . . . . . . . . 139 Summary for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 140 References for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 141 8 Semiconductors for Micro and Nanosystem Technology Chapter 4 The Electromagnetic System 143 Basic Equations of Electrodynamics . . . . . . . . . . . . . . . 144 TimeDependent Potentials . . . . . . . . . . . . . . . . . . . . . . . . .149 QuasiStatic and Static Electric and Magnetic Fields . . . . .151 Basic Description of Light . . . . . . . . . . . . . . . . . . . . . . . 158 The Harmonic Electromagnetic Plane Wave . . . . . . . . . . . .158 The Electromagnetic Gaussian Wave Packet . . . . . . . . . . . .160 Light as Particles: Photons . . . . . . . . . . . . . . . . . . . . . . . . .162 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Example: The Homogeneous Glass Fiber . . . . . . . . . . . . . .166 Summary for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 167 References for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 168 Chapter 5 Statistics Systems and Ensembles 169 . . . . . . . . . . . . . . . . . . . . . . . . . 170 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . .171 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . .176 Particle Statistics: Counting Particles . . . . . . . . . . . . . . . 178 MaxwellBoltzmann Statistics . . . . . . . . . . . . . . . . . . . . . . .178 BoseEinstein Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .180 FermiDirac Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181 Quasi Particles and Statistics . . . . . . . . . . . . . . . . . . . . . . . .182 Applications of the BoseEinstein Distributions . . . . . . . 183 Electron Distribution Functions . . . . . . . . . . . . . . . . . . . 184 Intrinsic Semiconductors Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . .184 . . . . . . . . . . . . . . . . . . . . . . . . . . .187 Summary for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 190 References for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 190 Chapter 6 Transport Theory The SemiClassical Boltzmann Transport Equation 191 . . . . 192 The Streaming Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193 The Scattering Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195 The BTE for Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197 Balance Equations for Distribution Function Moments . . .197 Semiconductors for Micro and Nanosystem Technology 9 Relaxation Time Approximation Local Equilibrium Description . . . . . . . . . . . . . . . . . . . . . . 201 . . . . . . . . . . . . . . . . . . . . 204 Irreversible Fluxes and Thermodynamic Forces . . . . . . . . . 205 Formal Transport Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 From Global Balance to Local NonEquilibrium . . . . . . 219 Global Balance Equation Systems . . . . . . . . . . . . . . . . . . . . 220 Local Balance: The Hydrodynamic Equations . . . . . . . . . . 220 Solving the DriftDiffusion Equations . . . . . . . . . . . . . . . . . 222 Kinetic Theory and Methods for Solving the BTE . . . . . . . . 227 Summary for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 231 References for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 231 Chapter 7 Interacting Subsystems PhononPhonon 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Phonon Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Heat Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 ElectronElectron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 The Coulomb Potential (Poisson Equation) . . . . . . . . . . . . . 240 The Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Plasma Oscillations and Plasmons . . . . . . . . . . . . . . . . . . . 243 ElectronPhonon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Acoustic Phonons and Deformation Potential Scattering . . 246 Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Piezoelectric Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Stress Induced Sensor Effects: Piezoresistivity . . . . . . . . . . 260 Thermoelectric Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 ElectronPhoton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Intra and Interband Effects . . . . . . . . . . . . . . . . . . . . . . . . . 268 Semiconductor Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 PhononPhoton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 ElastoOptic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Light Propagation in Crystals: PhononPolaritons . . . . . . . 277 Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Lattice Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Scattering Near Interfaces (Surface Roughness, . . . . . . . . . 281 10 Semiconductors for Micro and Nanosystem Technology Phonons at Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284 The PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300 MetalSemiconductor Contacts . . . . . . . . . . . . . . . . . . . . . .313 Summary for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 324 References for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . 324 Index Semiconductors for Micro and Nanosystem Technology 327 11 Preface This book addresses the engineering student and practising engineer. It takes an engineeringoriented look at semiconductors. Semiconductors are at the focal point of vast number of technologists, resulting in great engineering, amazing products and unheardof capital growth. The work horse here is of course silicon. Explaining how semiconductors like silicon behave, and how they can be manipulated to make microchips that work—this is the goal of our book. We believe that semiconductors can be explained consistently without resorting 100% to the complex language of the solid state physicist. Our approach is more like that of the systems engineer. We see the semiconductor as a set of welldeﬁned subsystems. In an approximately topdown manner, we add the necessary detail (but no more) to get to grips with each subsystem: The physical crystal lattice, and charge carriers in lattice like potentials. This elemental world is dominated by statistics, making strange observations understandable: This is the glue we need to put the systems together and the topic of a further chapter. Next we show the the Semiconductors for Micro and Nanosystem Technology 13 ory needed to predict the behavior of devices made in silicon or other semiconducting materials, the building blocks of modern electronics. Our book wraps up the tour with a practical engineering note: We look at how the various subsystems interact to produce the observable behavior of the semiconductor. To enrich the subject matter, we tie up the theory with concise boxed topics interspersed in the text. There are many people to thank for their contributions, and for their help or support. To the Albert Ludwig University for creating a healthy research environment, and for granting one of us (Korvink) sabbatical leave. To Ritsumeikan University in Kusatsu, Japan, and especially to Prof. Dr. Osamu Tabata, who hosted one of us (Korvink) while on sabbatical and where one chapter of the book was written. To the ETH Zurich and especially to Prof. Dr. Henry Baltes, who hosted one of us (Korvink) while on sabbatical and where the book project was wrapped up. To Prof. Dr. Evgenii Rudnyi, Mr. Takamitsu Kakinaga, Ms. Nicole Kerness and Mr. Sadik Hafizovic for carefully reading through the text and ﬁnding many errors. To the anonymous reviewers for their invaluable input. To Ms. Anne Rottler for inimitable administrative support. To VCHWiley for their deadline tolerance, and especially to Dr. Jörn Ritterbusch and his team for support. To Micheline and Cristina for enduring our distracted glares at home as we fought the clock (the calendar) to ﬁnish, and for believing in us. Jan G. Korvink and Andreas Greiner, Freiburg im Breisgau, February 2002 14 Semiconductors for Micro and Nanosystem Technology Chapter 1 Introduction Semiconductors have complex properties, and in the early years of the twentieth century these were mainly discovered by physicists. Many of these properties have been harnessed, and have been exploited in ingenious microelectronic devices. Over the years the devices have been rendered manufacturable by engineers and technologists, and have spawned off both a multibillion € (or $, or ¥) international industry and a variety of other industrial minirevolutions including software, embedded systems, the internet and mobile communications. Semiconductors still lie at the heart of this revolution, and silicon has remained its champion, warding off the competitors by its sheer abundance, suitability for manufacturing, and of course its tremendous headstart in the ﬁeld. Silicon is the working material of an exciting, competitive world, presenting a seemingly endless potential for opportunities. Chapter Goal The goal of this chapter is to introduce the reader to the ﬁeld of semiconductors, and to the purpose and organization of the book. Semiconductors for Micro and Nanosystem Technology 15 Introduction Chapter Roadmap In this chapter we ﬁrst explain the conceptual framework of the book. Next, we provide some popular deﬁnitions that are in use in the ﬁeld. Lastly, we indicate some of the sources of information on new inventions. 1.1 The System Concept This book is about semiconductors. More precisely, it is about semiconductor properties and how to understand them in order to be exploited for the design and fabrication of a large variety of microsystems. Therefore, this book is a great deal about silicon as a paradigm for semiconductors. This of course implies that it is also about other semiconductor systems, namely for those cases where silicon fails to show the desired effects due to a lack of the necessary properties or structure. Nevertheless, we will not venture far away from the paradigmatic material silicon, with its overwhelming advantage for a wide ﬁeld of applications with low costs for fabrication. To quote the Baltes theorem [1.1]: To prove your idea, put in on silicon. Since you will need circuitry, make it with CMOS. If you want to make it useful, get it packaged. The more expensive fabrication becomes, the less attractive the material is for the design engineer. Designers must always keep in mind the cost and resources in energy and personnel that it takes to handle materials that need additional nonstandard technological treatment. This is not to say that semiconductors other than silicon are unimportant, and there are many beautiful applications. But most of today’s engineers encounter silicon CMOS as a process with which to realize their ideas for microscopic systems. Therefore, most of the emphasis of this book lies in the explanation of the properties and behavior of silicon, or better said, “the semiconductor system silicon”. 16 Semiconductors for Micro and Nanosystem Technology The System Concept A semiconductor can be viewed as consisting of many subsystems. For one, there are the individual atoms, combining to form a chunk of crystalline material, and thereby changing their behavior from individual systems to a composite system. The atomic length scale is still smaller than typical length scales that a designer will encounter, despite the fact that subnanometer features may be accessible through modern experimental techniques. The subsystems that this book discusses emerge when silicon atoms are assembled into a crystal with unique character. We mainly discuss three systems: • the particles of quantized atom vibration, or phonons; • the particles of quantized electromagnetic radiation, or photons; • the particles of quantized charge, or electrons. There are many more, and the curious reader is encouraged to move on to other books, where they are treated more formally. The important feature of these subsystems is that they interact. Each interaction yields effects useful to the design engineer. Why do we emphasize the system concept this much? This has a lot to do with scale considerations. In studying nature, we always encounter scales of different order and in different domains. There are length scales that play a signiﬁcant role. Below the nanometer range we observe single crystal layers and might even resolve single atoms. Thus we become aware that the crystal is made of discrete constituents. Nevertheless, on a micrometer scale—which corresponds to several thousands of monatomic layers—the crystal appears to be a continuous medium. This means that at certain length scales a homogenous isotropic continuum description is sufﬁcient. Modern downsizing trends might force us to take at least anisotropy into account, which is due to crystal symmetry, if not the detailed structure of the crystal lattice including single defects. Nanotechnology changes all of this. Here we are ﬁnally designing at the atomic length scale, a dream that inspired the early twentieth century. Semiconductors for Micro and Nanosystem Technology 17 Introduction Almost everything becomes quantized, from thermal and electrical resistance to interactions such as the Hall effect. Time scale considerations are at least as important as length scales. They are governed by the major processes that take place within the materials. The shortest technological time scales are found in electronelectron scattering processes and are on the order of a few femtoseconds, followed by the interaction process of lattice vibrations and electronic systems with a duration of between a few hundreds of femtoseconds to a picosecond. Direct optical transitions from the conduction band to the valence band lie in the range of a few nanoseconds to a few microseconds. For applications in the MHz (106 Hz) and GHz (109 Hz) regime the details of the electronelectron scattering process are of minor interest and most scattering events may be considered to be instantaneous. For some quantum mechanical effects the temporal resolution of scattering is crucial, for example the intracollisional ﬁeld effect. The same considerations hold for the energy scale. Acoustic electron scattering may be considered elastic, that is to say, it doesn’t consume energy. This is true only if the system’s resolution lies well above the few meV of any scattering process. At room temperature ( 300 K) this is a good approximation, because the thermal energy is of the order of 25.4 meV. The level of the thermal energy implies a natural energy scale, at which the band gap energy of silicon of about 1.1 eV is rather large. For high energy radiation of several keV the band gap energy again is negligible. The above discussion points out the typical master property of a composite system: A system reveals a variety of behavior at different length (time, energy, …) scales. This book therefore demands caution to be able to account for the semiconductor as a system, and to explain its building blocks and their interactions in the light of scale considerations. 18 Semiconductors for Micro and Nanosystem Technology Popular Deﬁnitions and Acronyms 1.2 Popular Deﬁnitions and Acronyms The microelectronic and microsystem world is replete with terminology and acronyms. The number of terms grows at a tremendous pace, without regard to aesthetics and grammar. Their use is ruled by expedience. Nevertheless, a small number have survived remarkably long. We list only a few of the most important, for those completely new to the ﬁeld. 1.2.1 Semiconductors versus Conductors and Insulators A semiconductor such as silicon provides the technologist with a very special opportunity. In its pure state, it is almost electrically insulating. Being in column IV of the periodic table, it is exceptionally balanced, and comfortably allows one to replace the one or other atom of its crystal with atoms from column III or V (which we will term P and N type doping). Doing so has a remarkable effect, for silicon then becomes conductive, and hence the name “semiconductor”. Three important features are easily controlled. The density of “impurity” atoms can vary to give a tremendously wide control over the conductivity (or resistance) of the bulk material. Secondly, we can decide whether electrons, with negative charge, or holes, with positive charge, are the dominant mechanism of current ﬂow, just by changing to an acceptor or donor atom, i.e., by choosing P or N type doping. Finally, we can “place” the conductive pockets in the upper surface of a silicon wafer, and with a suitable geometry, create entire electronic circuits. If silicon is exposed to a hot oxygen atmosphere, it forms amorphous silicon dioxide, which is a very good insulator. This is useful to make capacitor devices, with the SiO 2 as the dielectric material, to form the gate insulation for a transistor, and of course to protect the top surface of a chip. Silicon can also be grown as a doped amorphous ﬁlm. In this state we lose some of the special properties of semiconductors that we will explore in this book. In the amorphous form we almost have metallike Semiconductors for Micro and Nanosystem Technology 19 Introduction behavior, and indeed, semiconductor foundries offer both real metals (aluminium, among others) and polysilicon as “metallic” layers. 1.2.2 The Diode Family The simplest device that one can make using both P and N doping is the diode. The diode is explained in Section 7.6.4. The diode is a oneway valve with two electrical terminals, and allows current to ﬂow through it in only one direction. The diode provides opportunities for many applications. It is used to contact metal wires to silicon substrates as a Shottkey diode. The diode can be made to emit light (LEDs). Diodes can detect electromagnetic radiation as photodetectors, and they form the basis of semiconductor lasers. Not all of these effects are possible using silicon, and why this is so is also explained later on. 1.2.3 The Transistor Family This is the true fame of silicon, for it is possible to make this versatile device in quantities unheard of elsewhere in the engineering world. Imagine selling a product with more than 10 9 working parts! Through CMOS (complimentary metal oxide semiconductor) it is possible to create reliable transistors that require extraordinary little power (but remember that very little times 10 9 can easily amount to a lot). The trend in miniaturization, a reduction in lateral dimensions, increase in operation speed, and reduction in power consumption, is unparalleled in engineering history. Top that up with a parallel manufacturing step that does not essentially depend on the number of working parts, and the stage is set for the revolution that we have witnessed. The transistor is useful as a switch inside the logic gates of digital chips such as the memories, processors and communications chips of modern computers. It is also an excellent ampliﬁer, and hence found everywhere where high quality analog circuitry is required. Other uses include magnetic sensing and chemical sensing. 20 Semiconductors for Micro and Nanosystem Technology Popular Deﬁnitions and Acronyms 1.2.4 Passive Devices In combination with other materials, engineers have managed to miniaturize every possible discrete circuit component known, so that it is possible to create entire electronics using just one process: resistors, capacitors, inductors and interconnect wires, to name the most obvious. For electromagnetic radiation, waveguides, ﬁlters, interferometers and more have been constructed, and for light and other forms of energy or matter, an entirely new industry under the name of microsystems has emerged, which we now brieﬂy consider. 1.2.5 Microsystems: MEMS, MOEMS, NEMS, POEMS, etc. In North America, the acronym MEMS is used to refer to microelectromechanical systems, and what is being implied are the devices at the length scale of microelectronics that include some nonelectrical signal, and very often the devices feature mechanical moving parts and electrostatic actuation and detection mechanisms, and these mostly couple with some underlying electrical circuitry. A highly successful CMOS MEMS, produced by Inﬁneon Technologies, is shown in Figure 1.1. The device, a) Figure 1.1. MEMS device. a) Inﬁneon’s surface micromachined capacitive pressure sensor with interdigitated signal conditioning Type KP120 for automotive BAP and MAP applications. b) SEM photograph of the pressure sensor cells compared with a human hair. Image © Inﬁneon Technologies, Munich [1.2]. b) Semiconductors for Micro and Nanosystem Technology 21 Introduction placed in a lowcost SMD package, is used in MAP and BAP tire pressure applications. With an annual production running to several millions, it is currently sold to leading automotive customers [1.2]. MEMS has to date spawned off two further terms that are of relevance to us, namely MOEMS, for microoptoelectromechanical systems, and NEMS, for the inevitable nanoelectromechanical systems. MOEMS can include entire miniaturized optical benches, but perhaps the most familiar example is the digital light modulator chip sold by Texas Instruments, and used in projection display devices, see Figure 1.2. Figure 1.2. MOEMS device. This 30 µm by 30 µm device is a single pixel on a chip that has as many pixels as a modern computer screen display. Each mirror is individually addressable, and deﬂects light from a source to a systems of lenses that project the pixel onto a screen. Illustration © Texas Instruments Corp., Dallas [1.3]. As for NEMS, the acronym of course refers to the fact that a critical dimension is now no longer the large micrometer, but has become a factor 1000 smaller. The atomic force microscope cantilever [1.4] may appear to be a MEMSlike device, but since it resolves at the atomic diameter scale, it is a clear case of NEMS, see Figure 1.3. Another example is the distributed mirror cavity of solidstate lasers made by careful epitaxial growth of many different semiconductor layers, each layer a few nanometers thick. Of major commercial importance is the submicron microchip electronic device technology. Here the lateral size of a transistor gate is the key size, which we know has dropped to below 100 22 Semiconductors for Micro and Nanosystem Technology Popular Deﬁnitions and Acronyms Figure 1.3. NEMS devices. Depicted are two tips of an atomic force microscope, made in CMOS, and used to visualize the force ﬁeld surrounding individual atoms. Illustration © Physical Electronics Laboratory, ETH Zurich, Switzerland [1.4]. nm in university and industrial research laboratories. Among NEMS we count the quantum wire and the quantum dot, which have not yet made it to the technologicalcommercial arena, and of course any purposefullydesigned and functional molecular monolayer ﬁlm. POEMS, or polymer MEMS, are microstructures made of polymer materials, i.e., they completely depart from the traditional semiconductorbased devices. POEMS are usually made by stereo microlithography through a photopolymerization process, by embossing a polymer substrate, by milling and turning, and by injection moulding. This class of devices will become increasingly important because of their potentially low manufacturing cost, and the large base of materials available. In Japan, it is typical to refer to the whole ﬁeld of microsystems as Micromachines, and manufacturing technology as Micromachining. In Europe, the terms Microsystems, Microtechnology or Microsystem Technology have taken root, with the addition of Nanosystems and the inevitable Nanosystem Technology following closely. The European naming convention is popular since it is easily translated into any of a large number of languages (German: Mikrosystemtechnik, French: Microtechnique, Italian: tecnologia dei microsistemi, etc.). Semiconductors for Micro and Nanosystem Technology 23 Introduction 1.3 Sources of Information We encourage every student to regularly consult the published literature, and in particular the following journals: IOP • Journal of Micromechanics and Microengineering • Journal of Nanotechnology IEEE • Journal of Microelectromechanical Systems • Journal of Electron Devices • Journal of Sensors • Journal of Nanosystems Other • WileyVCH Sensors Update, a journal of review articles • MYU Journal of Sensors and Materials • Elsevier Journal of Sensors and Actuators • Springer Verlag Journal of Microsystem Technology • Physical Review • Journal of Applied Physics Of course there are more sources than the list above, but it is truly impossible to list everything relevant. Additional sources on the worldwideweb are blossoming (see e.g. [1.5]), as well as the emergence of standard texts on technology, applications and theory. A starting point is best taken from the lists of chapter references. Two useful textbook references are Sze’s book on the physics of semiconductor devices [1.6] and Middelhoek’s book on silicon sensors [1.7]. 1.4 Summary for Chapter 1 Silicon is a very important technological material, and understanding its behavior is a key to participating in the largest industry ever created. To 24 Semiconductors for Micro and Nanosystem Technology References for Chapter 1 understand the workings of the semiconductor silicon, it helps to approach it as a system of interacting subsystems. The subsystems comprise the crystal lattice and its quantized vibrations—the phonons, electromagnetic radiation and its quantized form— the photons, and the loosely bound quantized charges—the electrons. The interactions between these systems is a good model with which to understand most of the technologically useful behavior of silicon. To understand the ensuing topics, we require a background in particle statistics. To render the ideas useful for exploitation in devices such as diodes, transistors, sensors and actuators, we require an understanding of particle transport modelling. These topics are now considered in more detail in the six remaining chapters of the book. 1.5 References for Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Prof. Dr. Henry Baltes, Private communication Prof. Dr. Christofer Hierold, Private communication. See e.g. http://www.dlp.com D. Lange, T. Akiyama, C. Hagleitner, A. Tonin, H. R. Hidber, P. Niedermann, U. Staufer, N. F. de Rooij, O. Brand, and H. Baltes, Parallel Scanning AFM with OnChip Circuitry in CMOS Technology, Proc. IEEE MEMS, Orlando, Florida (1999) 447452 See e.g. http://www.memsnet.org/ S. M. Sze, Physics of Semiconductor Devices, 2nd Ed., John Wiley and Sons, New York (1981) Simon Middelhoek and S. A. Audet, Silicon Sensors, Academic Press, London (1989) Semiconductors for Micro and Nanosystem Technology 25 Chapter 2 The Crystal Lattice System In this chapter we start our study of the semiconductor system with the crystals of silicon ( Si ) , adding some detail on crystalline silicon dioxide ( SiO 2 ) and to a lesser extent on gallium arsenide ( GaAs ) . All three are regular latticearrangements of atoms or atoms. For the semiconductors silicon and gallium arsenide, we will consider a model that completely decouple the behavior of the atoms from the valence electrons, assuming that electronic dynamics can be considered as a perturbation to the lattice dynamics, a topic dealt with in Chapter 3. For all the electrons of the ionic crystal silicon dioxide, as well as the bound electrons of the semiconductors, we here assume that they obediently follow the motion of the atoms. We will see that by applying the methods of classical, statistical and quantum mechanics to the lattice, we are able to predict a number of observable constitutive phenomena of interest—i.e., we are able to explain macroscopic measurements in terms of microscopic crystal lattice mechanics. The effects include an approximation for the elastic coef Semiconductors for Micro and Nanosystem Technology 27 The Crystal Lattice System ﬁcients of continuum theory, acoustic dispersion, speciﬁc heat, thermal expansion and heat conduction. In fact, going beyond our current goals, it is possible to similarly treat dielectric, piezoelectric and elastooptic effects. However, the predictions are of a qualitative nature in the majority of cases, mainly because the interatomic potential of covalently bonded atoms is so hard to come by. In fact, in a sense the potential is reverse engineered, that is, using measurements of the crystal, we ﬁt parameters that improve the quality of the models to make them in a sense “predictive”. Chapter Goal Our goal for this chapter is to explain the observed crystal data with preferably a single comprehensive model that accounts for all effects. Chapter Roadmap Our road map is thus as follows. We start by stating some of the relevant observable data for the three materials Si , SiO 2 and GaAs , without more than a cursory explanation of the phenomena. Our next step is to get to grips with the concept of a crystal lattice and crystal structure. Beyond this point, we are able to consider the forces that hold together the static crystal. This gives us a method to describe the way the crystal responds, with stress, to a strain caused by stretching the lattice. Then we progress to vibrating crystal atoms, progressively reﬁning our method to add detail and show that phonons, or quantized acoustic pseudo “particles”, are the natural result of a dynamic crystal lattice. Considering the phonons in the lattice then leads us to a description of heat capacity. Moving away from basic assumptions, we consider the anharmonic crystal and ﬁnd a way to describe the thermal expansion. The section following presents a cursory look at what happens when the regular crystal lattice is locally deformed through the introduction of foreign atoms. Finally, we leave the inﬁnitelyextended crystal model and brieﬂy consider the crystal surface. This is important, because most microsystem devices are build on top of semiconductor wafers, and so are repeat 28 Semiconductors for Micro and Nanosystem Technology Observed Lattice Property Data edly subject to the special features and limitations that the surface introduces. 2.1 Observed Lattice Property Data Geometric Structure The geometric structure of a regular crystal lattice is determined using xray crystallography techniques, by recording the diffraction patterns of xray photons that have passed through the crystal. From such a recorded pattern (see Figure 2.1 (a)), we are able to determine the reﬂection planes formed by the constituent atoms and so reconstruct the relative positions of the atoms. This data is needed to proceed with a geometric (or, strictly speaking, grouptheoretic) characterization of the crystal lattice’s symmetry properties. We may also use an atomic force microscope (AFM) to map out the force ﬁeld that is exerted by the constituent atoms on the surface of a crystal. From such contour plots we can reconstruct the crystal structure and determine the lattice constants. We must be careful, though, because we may observe special surface conﬁgurations in stead of the actual bulk crystal structure, see Figure 2.1 (b). Elastic Properties The relationship between stress and strain in the linear region is via the elastic property tensor, as we shall shortly derive in Section 2.3.1. To measure the elastic parameters that form the entries of the elastic property tensor, it is necessary to form special test samples of exact geometric shape that, upon mechanical loading, expose the relation between stress and strain in such a way that the elastic coefﬁcients can be deduced from the measurement. The correct choice of geometry relies on the knowledge of the crystal’s structure, and hence its symmetries, as we shall see in Section 2.2.1. The most common way to extract the mechanical properties of crystalline materials is to measure the directiondependent velocity of sound inside the crystal, and by diffracting xrays through the crystal (for example by using a synchrotron radiation source). Semiconductors for Micro and Nanosystem Technology 29 The Crystal Lattice System (a) (b) Figure 2.1. (a) The structure of a silicon crystal as mapped out by xray crystallography in a Laue diagram. (b) The famous 〈 111〉 7 × 7 reconstructed surface of silicon as mapped out by an atomic force microscope. The bright dots are the Adatoms that screen the underlying lattice. Also see Figure 2.30. Dispersion Curves Dispersion curves are usually measured by scattering a neutron beam (or xrays) in the crystal and measuring the directiondependent energy lost or gained by the neutrons. The absorption or loss of energy to the crystal is in the form of phonons. Thermal Expansion Thermal expansion measurements proceed as for the stressstrain measurement described above. A thermal strain is produced by heating the sample to a uniform temperature. Armed with the knowledge of the elastic parameters, the inﬂuence of the thermal strain on the velocity of sound may then be determined. The material data presented in the following sections and in Table 2.1 was collected from references [2.5, 2.6]. Both have very complete tables of measured material data, together with reference to the publications where the data was found. 30 Semiconductors for Micro and Nanosystem Technology Crystalline Silicon Semiconductors for Micro and Nanosystem Technology Property Si LPCVD PolySi Atomic Weight Si: 28.0855 Si: 28.0855 . Process Gallium Arsenide α Quartz Si 3 N 4 GaAs Si: 28.0855 Si: 28.0855 Si: 28.0855 O: 15.994 O: 15.994 N: 14.0067 Ga: 69.72 As: 74.9216 Thermal Oxide SiO 2 Carbonlike, Cubic symmetry Polycrystalline Amorphous Trigonal symmetry Amorphous Zincblende Cubic symmetry Density ( kg ⁄ m ) 2330 2330 < 2200 2650 3100 5320 Elastic moduli ( Gpa ) Y: Young modulus B: Biaxial modulus S: Shear modulus C i : Tensor Pure: C 11 : 166 Y: 130 – 174 S: 69 Y: 72 – 75 Y: 87 Y: 97 – 320 B: 249  311 Pure: C 11 : 118.1 Crystal class/ Symmetry 3 coefﬁcients. C 12 : 63.9 C 12 : 53.2 C 44 : 79.6 C 44 : 59.4 pType: C 11 : 80.5 C 12 : 115 C 44 : 52.8 nType: C 11 : 97.1 C 12 : 54.8 C 44 : 172 Observed Lattice Property Data Table 2.1. Lattice properties of the most important microsystem base materialsa. 31 Crystalline Silicon Property Semiconductors for Micro and Nanosystem Technology Hardness ( Gpa ) Process Gallium Arsenide α Quartz Si 3 N 4 GaAs 14.4 – 18 8.2   5.43 , polycrystal Amorphous X i 4.9127 Amorphous 5.65 Si LPCVD PolySi Thermal Oxide SiO 2 〈 100〉 : 10.5 – 12.5 5.1 – 13 ; 〈 111〉 : 11.7 Lattice parameter ( ) X, C: axes 5.43 Melting point ( °C ) 1412 1412 . 1705  1902  Poisson ratio ν̃ 〈 100〉 : 0.28 0.2 – 0.3 0.17  0.22 0.169 0.26 0.31 C 5.4046 〈 111〉 : 0.36 Speciﬁc heat C p ( J ⁄ kgK 702.24 702.24 740 740 750 350 Thermal conductivity ( W ⁄ mK ) 150 150 1.1  1.5 1.4 18 46 ( 12 ) Thermal expansion coefﬁcient 2.33 ×10 XY cut 14.3 Z cut 7.8 2.7 ×10 α (T –1 ) –6 –6 2.33 ×10 –6 0.4 ×10 –6 0.55 ×10  a. The tabulated values for amorphous process materials are foundrydependent and are provided only as an indication of typical values. Also, many of the measurements on crystalline materials are for doped samples and hence should be used with care. Properties depend on the state of the material, and a common choice is to describe them based on the temperature and the pressure during the measurements. For technological work, we require the properties under operating conditions, i.e., at room temperature and at 1 atmosphere of pressure. –6 –6 6.86 ×10 The Crystal Lattice System 32 Table 2.1. Lattice properties of the most important microsystem base materialsa. Observed Lattice Property Data 2.1.1 Silicon A semiconductor quality Silicon ingot is a gray, glassy, facecentered crystal. The element is found in column IV of the periodic table. It has the same crystal structure as diamond, as illustrated in Figure 2.2. Silicon (a) (b) (c) (d) (e) Figure 2.2. The diamondlike structure of the Silicon crystal is caused by the tetrahedral 3 arrangement of the four sp bondforming orbitals of the silicon atom, symbolically shown (a) as stippled lines connecting the balllike atomic nuclei in this tetrahedral repeating unit. (b) When the atoms combine to form a crystal, we observe a structure that may be viewed as a set of two nested cubic lattices, or (c) a single facecentered cubic lat3 tice with a basis. (d) The structure of the sp hybrid bonds. (e) The fcc unit cell can also be viewed as four tetrahedra. has a temperature dependent coefﬁcient of thermal expansion in K described by Semiconductors for Micro and Nanosystem Technology –1 33 The Crystal Lattice System α(T ) –3 –4 = ( 3.725 ( 1 – exp ( – 5.88 ×10 ( T – 124 ) ) ) + 5.548 ×10 T ) ×10 and a lattice parameter (the interatomic distance in temperature as –6 ) that varies with a(T ) –5 –9 (2.1) = 5.4304 + 1.8138 ×10 ( T – 298.15 ) + 1.542 ×10 ( T – 298.15 ) 2 (2.2) We will later take a more detailed look at the thermal strain ε ( T – T 0 ) = α ( T ) ( T – T 0 ) . Both α ( T ) and a ( T ) are plotted in Figure 2.3. Both of 5.440 4 ×10 –6 2 ×10 –6 5.438 5.436 5.434 400 800 0 Temperature in K 0 600 1200 Temperature in K Figure 2.3. The thermal expansion properties of Silicon. Shown on the left is the temperature dependence of the lattice parameter (the size of a unit cell), and to its right is the temperature dependence of the thermal expansion coefﬁcient. The typical engineering temperature range for silicon electronic devices is indicated by the background gray boxes. these properties are also dependent on the pressure experienced by the material, hence we should write α ( T , p ) and a ( T , p ) . It is important to note that “technological” silicon is doped with foreign atoms, and will in general have material properties that differ from the values quoted in Table 2.1, but see [2.6] and the references therein. Silicon’s phonon dispersion diagram is shown in Figure 2.4. 34 Semiconductors for Micro and Nanosystem Technology Observed Lattice Property Data ω(k ) X U W K Γ L k Γ ∆ X U, K Σ Γ Λ k L L K W X Figure 2.4. The measured and computed dispersion diagrams of crystalline silicon. The vertical axis represents the phonon frequency, the horizontal axis represents straightline segments in kspace between the main symmetry points of the Brillouin zone, which is shown as an insert. Figure adapted from [2.5]. Silicon is the carrier material for most of today’s electronic chips and microsystem (or microelectromechanical system (MEMS), or micromachine) devices. The ingot is sliced into wafers, typically 0.5 mm thick and 50 to 300 mm in diameter. Most electronic devices are manufactured in the ﬁrst 10 µ m of the wafer surface. MEMS devices can extend all the way through the wafer. Cleanroom processing will introduce foreign dopant atoms into the silicon so as to render it more conducting. Other processes include subtractive etching steps, additive deposition steps and modiﬁcations such as oxidation of the upper layer of silicon. Apart from certain carefully chosen metal conductors, the most common materials used in conjunction with silicon are ovengrown or deposited thermal silicon oxides and nitrides (see the following sections), as well as polycrystalline silicon. Semiconductors for Micro and Nanosystem Technology 35 The Crystal Lattice System IC process quality LPCVD polycrystalline silicon (PolySi) has properties that depend strongly on the foundry of origin. It is assumed to be isotropic in the plane of the wafer, and is mainly used as a thin ﬁlm thermal and electrical conductor for electronic applications, and as a structural and electrode material for MEMS devices. 2.1.2 Silicon Dioxide Crystalline silicon dioxide is better known as fused quartz. It is unusual to obtain quartz from a siliconbased process, say CMOS, because the production of crystalline quartz usually requires very high temperatures that would otherwise destroy the carefully produced doping proﬁles in the silicon. Semiconductorrelated silicon dioxide is therefore typically amorphous. Since quartz has a noncubic crystal structure, and therefore displays useful properties that are not found in highsymmetry cubic systems such as silicon, yet are of importance to microsystems, we also include it in our discussion. We consider α quartz, one of the variants of quartz that is Figure 2.5. Three perspective views of the trigonal unit cell of α Quartz. From left to right the views are towards the origin along the X 1 , X 2 and the C axes. The six large spheres each with two bonds represent oxygen atoms, the three smaller spheres each with four tetrahedral bonds represent silicon atoms. 36 Semiconductors for Micro and Nanosystem Technology Observed Lattice Property Data o stable below 573 C , with trigonal crystal symmetry. The unit cell of the quartz crystal is formed by two axes, called X 1 and X 2 , at 60° to each other, see Figure 2.5. Quartz is noncentrosymmetric and hence piezoelectric. It also has a handedness as shown in Figure 2.6. Figure 2.6. α Quartz is found as either a right or a lefthanded structure, as indicated by the thick lines in the structure diagram that form a screw through the crystal. In the ﬁgure, 8 unit cells are arranged in a ( 2 × 2 × 2 ) block. 2.1.3 Silicon Nitride Crystalline silicon nitride (correctly known as trisilicon tetranitride) is not found on silicon IC wafers because, as for silicon dioxide, very high temperatures are required to form the pure crystalline state, see Figure 2.7. These temperature are not compatible with silicon foundry processing. In fact, on silicon wafers, silicon nitride is usually found as an amorphous mixture that only approaches the stochiometric relation of Si 3 N 4 , the speciﬁc relation being a strong function of process parameters and hence is ICfoundry speciﬁc. In the industry, it is variously referred to as “nitride”, “glass” or “passivation”, and may also contain amounts of oxygen. 2.1.4 Gallium Arsenide Crystalline gallium arsenide (GaAs) is a “goldgray” glassy material with the zincblende structure. When bound to each other, both gallium and arsenic atoms form tetrahedral bonds. In the industry, GaAs is referred to as a IIIV (threeﬁve), to indicate that it is a compound semiconductor Semiconductors for Micro and Nanosystem Technology 37 The Crystal Lattice System Figure 2.7. Silicon nitride appears in many crystalline conﬁgurations. The structure of α and of β Si 3 N 4 are based on vertical stacks of SiN 4 tetrahedra, as shown in (I) and (II) respectively. (I) (II) whose constituents are taken from the columns III and V of the periodic table. The atoms form into a zincblende crystal, structurally similar to the diamondlike structure of silicon, but with gallium and arsenic atoms alternating, see Figure 2.8. Figure 2.8. The zincblende structure of gallium arsenide. The two atom types are represented by spheres of differing diameter. Also see Figure 2.2. 38 Semiconductors for Micro and Nanosystem Technology Crystal Structure Gallium arsenide is mainly used to make devices and circuits for the allimportant optoelectronics industry, where its raw electronic speed or the ability to act as an optoelectronic lasing device is exploited. It is not nearly as popular as silicon, though, mainly because of the prohibitive processing costs. Gallium arsenide has a number of material features that differ signiﬁcantly from Silicon, and hence a reason why we have included it in our discussion here. Gallium arsenide’s phonon dispersion diagram is shown in Figure 2.9. ω(k ) W K X U Γ L k Γ ∆ X X U, K Σ Γ Λ L Figure 2.9. The measured and computed dispersion diagrams of crystalline gallium arsenide. The vertical axis represents the phonon frequency, the horizontal axis represents straightline segments in kspace between the main symmetry points of the Brillouin zone, which is shown as an insert. Figure adapted from [2.5]. 2.2 Crystal Structure As we have seen, crystals are highly organized regular arrangements of atoms or ions. They differ from amorphous materials, which show no regular lattice, and polycrystalline materials, which are made up of adjacent irregularlyshaped crystal grains, each with random crystal orienta Semiconductors for Micro and Nanosystem Technology 39 The Crystal Lattice System tion. From observations and measurements we ﬁnd that it is the regular crystalline structure that leads to certain special properties and behavior of the associated materials. In this section we develop the basic ideas that enable us to describe crystal structure analytically, so as to exploit the symmetry properties of the crystal in a systematic way. 2.2.1 Symmetries of Crystals Consider a regular rectangular arrangement of points on the plane. The points could represent the positions of the atoms that make up a hypothetical twodimensional crystal lattice. At ﬁrst we assume that the atoms are equally spaced in each of the two perpendicular directions, say by a pitch of a and b . More general arrangements of lattice points are the rule. Translational Invariance Consider a vector a that lies parallel to the horizontal lattice direction and with magnitude equal to the pitch a . Similarly, consider vector b in the other lattice direction with magnitude equal to the pitch b . Then, starting at point p i , we can reach any other lattice point q j with q j = α j a + β j b , where α j and β j are integers. Having reached another interior point q j , the vicinity is the same as for point p i , and hence we say that the lattice is invariant to translations of the form α j a + β j b , see the example in Figure 2.10. Figure 2.10. In this 2dimensional inﬁnitelyextending regular lattice the 2 lattice vectors are neither perpendicular nor of equal length. Given a starting point, all lattice points can be reached through q j = α j a + β j b . The vicinities of p i and q i are similar. 40 q j = 2a + 5b qj pi b a Semiconductors for Micro and Nanosystem Technology b a Crystal Structure Rotational Symmetry If we consider the vicinity of an interior point p i in our lattice, and let us assume that we have very many points in the lattice, we see that by rotating the lattice in the plane about point p i by an angle of 180° , the vicinity of the point p i remains unchanged. We say that the lattice is invariant to rotations of 180° . Clearly, setting a = b and a ⊥ b makes the lattice invariant to rotations of 90° as well. Rotational symmetry in lattices are due to rotations that are multiples of either 60° , 90° or 180° , see Figure 2.11. If the underlying lattice has a rotational symmetry, we will expect the crystal’s material properties to have the same symmetries. 180° 90° 60° Figure 2.11. Illustrations of lattice symmetries w.r.t. rotations. Bravais Lattice With the basic idea of a lattice established, we now use the concept of a Bravais lattice to model the symmetry properties of a crystal’s structure. A Bravais lattice is an inﬁnitely extending regular threedimensional array of points that can be constructed with the parametrized vector q j = α ja + β jb + γ jc (2.3) where a , b and c are the noncoplanar lattice vectors and α j , β j and γ j are arbitrary (positive and negative) integers. Note that we do not Semiconductors for Micro and Nanosystem Technology 41 The Crystal Lattice System assume that the vectors a , b and c are perpendicular. The Bravais lattice is inherently symmetric with respect to translations q j : this is the way we construct it. The remainder of the symmetries are related to rotations and reﬂections. There are 7 crystal systems: hexagonal, trigonal, triclinic, monoclinic, orthorhombic, tetragonal and cubic. In addition, there are 14 Bravais lattice types, see Figure 2.12. These are grouped into the following six lattice systems in decreasing order of geometric generality (or increasing order of symmetry): triclinic, monoclinic, orthorhombic, tetragonal, Hexagonal and cubic. The facecentered cubic (fcc) diamondlike lattice structure of silicon is described by the symmetric arrangement of vectors shown in Figure 2.13 (I). The fcc lattice is symmetric w.r.t. 90o rotations about all three coordinate axes. Gallium arsenide´s zincblende bccstructure is similarly described as for silicon. Because of the presence of two constituent atoms, GaAs does not allow the same translational symmetries as Si. Primitive Unit Cell We associate with a lattice one or more primitive unit cells. A primitive unit cell is a geometric shape that, for singleatom crystals, effectively contains one lattice point. If the lattice point is not in the interior of the primitive cell, then more than one lattice point will lie on the boundary of the primitive cell. If, for the purpose of illustration, we associate a sphere with the lattice point, then those parts of the spheres that overlap with the inside of the primitive cell will all add up to the volume of a single sphere, and hence we say that a single lattice point is enclosed. Primitive cells seamlessly tile the space that the lattice occupies, see Figure 2.14. WignerSeitz Unit Cell The most important of the possible primitive cells is the WignerSeitz cell. It has the merit that it contains all the symmetries of the underlying Bravais lattice. Its deﬁnition is straightforward: The WignerSeitz cell of lattice point p i contains all spatial points that are closer to p i than to any other lattice point q j . Its construction is also straightforward: Considering lattice point p i , connect p i with its neighbor lattice points q j . On each connection line, construct a plane perpendicular to the connecting line at a position halfway along the line. The planes intersect each other 42 Semiconductors for Micro and Nanosystem Technology Crystal Structure TC OP T OI MI MP OC OF HP HR TI TP CF CI CP Figure 2.12. The fourteen Bravais lattice types. CP: Cubic P; CI: Bodycentered cubic; CF: Facecentered cubic; a = b = c , α = β = γ = 90° . TP: Tetragonal; TI: Bodycentered tetragonal; a = b ≠ c , α = β = γ = 90° . HP: Hexagonal; HR: Hexagonal R; a = b ≠ c , α = β = 90° , γ = 120° (this not a separate Bravais lattice type). OP: Orthorhombic; OI: Bodycentered orthorhombic; OF: Facecentered orthorhombic; OC: Corthorhombic; a ≠ b ≠ c , α = β = γ = 90° . MP: Monoclinic; MI: Facecentered monoclinic; a ≠ b ≠ c , α = β = 90° ≠ γ . T: Trigonal; a = b = c , α ≠ β ≠ γ . TC: Triclinic; a ≠ b ≠ c , α ≠ β ≠ γ . a, b, c refer to lattice pitches; α, β, γ to lattice vector angles. Semiconductors for Micro and Nanosystem Technology 43 The Crystal Lattice System b b c c a a Figure 2.13. The equivalent lattice vectors sets that deﬁne the structures of the diamondlike fcc structure of Si and the zincblendelike structure of GaAs . The vectors are either the set: a = ( 2, 0, 0 ) , b = ( 0, 2, 0 ) , c = ( 0.5, 0.5, 0.5 ) or the set: a = ( 0.5, 0.5, – 0.5 ) , b = ( 0.5, – 0.5, 0.5 ) , c = ( 0.5, 0.5, 0.5 ) . Figure 2.14. Illustrations of some of the many primitive unit cells for a 2dimensional lattice. Of these, only (g) is a WignerSeitz cell. Figure adapted from [2.10]. (f) (e) (c) (b) (a) (g) (d) and, taken together, deﬁne a closed volume around the lattice point p i . The smallest of these volumes is the WignerSeitz cell, illustrated in Figure 2.15. Reciprocal Lattice The spatial Bravais crystal lattice is often called the direct lattice, to refer to the fact that we can associate a reciprocal lattice with it. In fact, in studying the properties of the crystal lattice, most data will be referred to 44 Semiconductors for Micro and Nanosystem Technology Crystal Structure (I) (II) (III) Figure 2.15. The WignerSeitz cell for the bcc lattice. (I) shows the cell without the atomic positions. (II) includes the atomic positions so as to facilitate understanding the method of construction. (III) The WignerSeitz cell tiles the space completely. the reciprocal lattice. Let us consider a plane wave (see Box 2.1), traversing the lattice, of the form e ik • r . (2.4) We notice two features of this expression: the vectors k and r are symmetrical in the expression (we can swap their positions without altering the expression); and the crystal is periodic in r space. The vector r is a position in real space; the vector k is called the position vector in reciprocal space, and often also called the wave vector, since it is “deﬁned” using a plane wave. Next, we consider waves that have the same periodicity as the Bravais lattice. The Bravais lattice points lie on the regular grid described by the vectors q , so that a periodic match in wave amplitude is expected if we move from one grid position to another, and therefore e ik • ( r + q ) = e ik • r . (2.5) This provides us with a condition for the wave vector k for waves that have the same spatial period as the atomic lattice, because we can cancel ik • r out the common factor e to obtain e ik • q 0 = 1 = e . Semiconductors for Micro and Nanosystem Technology (2.6) 45 The Crystal Lattice System Equation (2.6) generates the k (wave) vectors from the q (lattice position) vectors, and we see that these are mutually orthogonal. We summarize the classical results of the reciprocal lattice: Reciprocal Lattice Properties • The reciprocal lattice is also a Bravais lattice. • Just as the direct lattice positions are generated with the primitive unit vectors q j = α j a + β j b + γ j c , the reciprocal lattice can also be so generated using k j = δ j d + ε j e + ζ j f . The relation between the two primitive vector sets is b×c d = 2π a ⋅ (b × c) (2.7a) c×a e = 2π a ⋅ (b × c) (2.7b) a×b f = 2π a ⋅ (b × c) (2.7c) • The reciprocal of the reciprocal lattice is the direct lattice. • The reciprocal lattice also has a primitive cell. This cell is called the Brillouin zone after its inventor. • The volume of the direct lattice primitive cell is v = a ⋅ ( b × c ) . • The volume of the reciprocal lattice V = d ⋅ ( e × f ) . From (2.7c) we see that 2π d =  ( b × c ) v 2π e =  ( c × a ) v primitive 2π f =  ( a × b ) v cell is (2.8) and hence that 3 ( 2π ) V = v (2.9) Miller Indices The planes formed by the lattice are identiﬁed using Miller indices. These are deﬁned on the reciprocal lattice, and are deﬁned as the coordinates of the shortest reciprocal lattice vector that is normal to the plane. Thus, if we consider a plane passing through the crystal, the Miller indi 46 Semiconductors for Micro and Nanosystem Technology Crystal Structure Box 2.1. Plane waves and wavevectors. We describe a general plane wave with i ( k ⋅ r – ωt ) ψ ( r, t ) = Ae . (B 2.1.1) Recalling the relation between the trigonometric functions and the exponential function, iθ (B 2.1.2) e = cos ( θ ) + i sin ( θ ) . we see that ψ is indeed a wavelike function, for θ parametrizes an endless circular cycle on the complex plane. I e θ iθ counts the number cycles completed along a spatial segment. If we ﬁx the time t (“freezing” the wave in space and time), then moving along a spatial direction we will experience a wavelike variation of the amplitude of ψ as k ⋅ r . If we now move in a direction perpendicular to the propagation of the wave k , then we will experience no amplitude modulation. Next, staying in the perpendicular direction to the wave propagation at a ﬁxed position, if the time is again allowed to vary, we will now experience an amplitude modulation as the wave moves past us. R Figure B2.1.1: The exponential function describes a cycle in the complex plane. Equation (B 2.1.1) is a powerful way of describing a plane wave. Two vectors, the spatial position vector r and the wave vector, or reciprocal position vector k , are position arguments to the exponential function. The cyclic angle that it the wave has rotated through is the complete factor ( k ⋅ r – ωt ) . Since r measures the distance along an arbitrary spatial direction and k points along the propagation direction of the wave, k ⋅ r gives the component of this parametric angle. k measures rotation angle per distance travelled (a full cycle of 2π is one wavelength), so that k ⋅ r x t Figure B2.1.2: The appearance of a 1dimensional wave plotted for t and x as parameters. If t is kept stationary, moving in the xdirection is accompanied by a wavelike variation. ces are the components of a k space vector (the vector is normal to the plane) that fulﬁl k ⋅ r = Constant , where r lies on the plane of interest. As an example, consider a cubic lattice with the Miller indices ( m, n, p ) of a plane that lies parallel to a face of the cube. The indices determine the planenormal k space vector k = mb 1 + nb 2 + pb 3 . The plane is Semiconductors for Micro and Nanosystem Technology 47 The Crystal Lattice System k ⋅ r = Constant = 2π ( r 1 m + r 2 n + r 3 p ) . ﬁxed by Since k ⋅ a 1 = 2πm , k ⋅ a 2 = 2πn and k ⋅ a 3 = 2πp , the space axis interr 1 = Constant ⁄ 2πm , cepts for the plane are therefore r 2 = Constant ⁄ 2πn and r 3 = Constant ⁄ 2πp . Silicon Reciprocal Lattice Shape We have seen that the silicon crystal can be represented by a facecentered cubic lattice with a basis. This means that its reciprocal lattice is a bodycentered lattice with a basis. This has the following implications. The WignerSeitz cell for the silicon direct lattice is a rhombic dodecahedron, whereas the ﬁrst Brillouin zone (the WignerSeitz cell of the reciprocal lattice) is a truncated dodecahedron. 2.3 Elastic Properties: The Stressed Uniform Lattice In a broad sense the geometry of a crystal’s interatomic bonds represent the “structural girders” of the crystal lattice along which the forces act that keep the crystal intact. The strength of these directional interatomic forces, and the way in which they respond to small geometrical perturbations; these are the keys that give a crystal lattice its tensorial elastic properties and that enable us to numerically relate applied stress to a strain response. In this section we derive the Hooke law for crystalline, amorphous and polycrystalline materials based on lattice considerations. 2.3.1 Statics Atomic Bond Model It is well known that atoms form different types of bonds with each other. The classiﬁcation is conveniently viewed as the interaction between a pair of atoms: • Ionic—a “saturated” bond type that is characterized by the fact that one atom ties up the electrons participating in the bond in its outermost shell. This leaves the two atoms oppositely charged. The cou 48 Semiconductors for Micro and Nanosystem Technology Elastic Properties: The Stressed Uniform Lattice lombic (electrostatic) force between the net charges of the constituent “ions” form the bond. This bond is sometimes termed localized, because the electrons are tightly bound to the participating atoms. • van der Waals—a very weak bonding force that is often termed the ﬂuctuating dipole force because it is proportional to an induced dipole between the constituent atoms, and this effective dipole moment has a nonvanishing time average. • Valency—another localized electronpair bond. Here, the electronpair of the participating atoms form a hybrid orbital that is equally shared by the two atoms, hence the term covalent. Clearly, ionic and covalent bonds are the two limiting cases of a similar phenomena, so that a bond inbetween these limits can also be expected. Valency bonds are quite strong, and account for the hardness and brittle nature of the materials. • Metallic—the electrons participating in the bonding are nonlocalized. Typically, the number of valence electrons at a point is exceeded by the number of nearestneighbor atoms. The electrons are therefore shared by many atoms, making them much more mobile, and also accounts for the ductility of the material. a) b) 1 σ 1 =  ( s + p x + p y + p z ) 2 1 σ 3 =  ( s – p x + p y – p z ) 2 c) 1 σ 2 =  ( s + p x – p y – p z ) 2 1 σ 4 =  ( s – p x – p y + p z ) 2 Figure 2.16. The geometry of a tetrahedral bond for Carbonlike atoms can be repre3 sented by a hybrid sp function as a superposition of 2s and 2porbitals. a) sorbital. b) 3 porbital. c) s p orbital. Semiconductors for Micro and Nanosystem Technology 49 The Crystal Lattice System The semiconductors that we consider are covalently bonded. We now give a qualitative description of the tetrahedral covalent bond of the atoms of an fcc crystal. Solving the Schrödinger equation (see Box 2.2 Box 2.2. The stationary Schrödinger equation and the Hamiltonian of a Solid [2.4]. In principle, all basic calculations of solid state properties are computed with the Schrödinger equation Hϕ = Eϕ (B 2.2.1) The Hamiltonian operator H deﬁnes the dynamics and statics of the model, ϕ represents a state of the model, and E is the scalarvalued energy. Thus the Schrödinger equation is an Eigensystem equations with the energy plying the role of an eigenvalue and the state the role of an eigenfunction. The Hamiltonian is usually built up of contributions from identiﬁable subcomponents of the system. Thus, for a solidstate material we write that H = H e + H i + H ei + H x (B 2.2.2) i.e., the sum of the contributions from the electrons, the atoms, their interaction and interactions with external inﬂuences (e.g., a magnetic ﬁeld). Recall that the Hamiltonian is the sum of kinetic and potential energy terms. Thus, for the electrons we have H e = T e + U ee 2 = pα 1 e 2 +  ′ ∑ 2m 8πε 0 ∑ r α – r β α The second term is the Coulombic potential energy due to the electron charges. Note that the sum is primed: the sum excludes terms where α = β. For the atoms the Hamiltonian looks similar to that of the electron H i = T i + U ii 2 = Pα 1 +  ′V ( R – R β ) ∑ 2M 2 ∑ i α α (B 2.2.4) αβ For the electronatom interaction we associate only a potential energy H ei = U ei = ∑ V ei ( rα – Rβ ) (B 2.2.5) αβ Equation (B 2.2.1) is hardly ever solved in all its generality. The judicious use of approximations and simpliﬁcations have yielded not only tremendous insight into the inner workings of solid state materials, but have also been tremendously successful in predicting complex phenomena. We will return to this topic in the next chapter, where we will calculate the valence band structure of silicon to remarkable accuracy. (B 2.2.3) αβ and Chapter 3) for a single atom yields the orthogonal eigenfunctions that correspond to the energy levels of the atom, also known as the orbitals. The spherical harmonic functions shown in Table 2.2 are such eigenfunctions. The tetrahedral bond structure of the Si atom can be made, 3 through a superposition of basis orbitals, to form the hybrid s p orbital 50 Semiconductors for Micro and Nanosystem Technology Elastic Properties: The Stressed Uniform Lattice m 2 Table 2.2. The spherical harmonic functions Y l ( θ, φ ) , Also see Table 3.1 and Figure 3.9 in Chapter 3. l m Plot l m Plot 0 0 1 l m Plot 0 1 1 2 0 2 3 0 4 0 l m Plot 1 2 2 3 1 3 2 3 3 4 1 4 2 4 3 that we associate with its valence electrons. The construction is illustrated in Figure 2.16. The valence electron orbitals may overlap to form bonds between atoms. The interpretation is straightforward: a valence electron’s orbital represents the probability distribution of ﬁnding that electron in a speciﬁc region of space. In a ﬁrst approximation, we let the orbitals simply overlap and allow them to interfere to form a new shared orbital. Bonding takes place if the new conﬁguration has a lower energy than the two separate atom orbitals. The new, shared, hybrid orbital gives the electrons of Semiconductors for Micro and Nanosystem Technology 51 The Crystal Lattice System the two binding atoms a relatively high probability of occupying the space between the atoms. Figure 2.17 shows what happens when atoms bond to form a diamond a a E(r) r a (I) (II) r=a Figure 2.17. When atoms with a tetrahedral bond structure form a covalentlybonded crystal lattice, the valence electrons are localized about the nearestneighbor atoms. We visualize these (I) as the overlap positions of the sp3hybrid orbitals, shown here for four atoms in a diamondlike lattice. We assume that the distance a between the atomic cores, the lattice constant, is the position of minimum energy for the bond (II). like crystal. The energy of the atomic arrangement is lowered to a minimum when all atoms lie approximately at a separation equal to the lattice constant a (the lattice constant is the equilibrium distance that separates 3 atoms of a lattice). The s p orbitals of neighboring atoms overlap to form new shared orbitals. The electrons associated with these new states are effectively shared by the neighboring atoms, but localized in the bond. The bonding process does not alter the orbitals of the other electrons signiﬁcantly. The potential energy plot of Figure 2.17 (II) is necessarily only approximate, yet contains the necessary features for a single ionic bond. In fact, it is a plot of the Morse potential, which contains a weaker attractive and a very strong, repulsive constituent, localized at the core 52 Semiconductors for Micro and Nanosystem Technology Elastic Properties: The Stressed Uniform Lattice E ( r ) = D e ( 1 – exp [ – β ( r – r eq ) ] ) 2 (2.10) The potential energy for bond formation D e and the equilibrium bond length r eq depend on the constituent atoms, and may be obtained by experiment. The parameter β controls the width of the potential well, i.e., the range of the interparticle forces. For covalent crystals, such as silicon, this picture is too simplistic. The tetrahedral structure of the orbitals forms bonds that can also take up twisting moments, so that, in addition to atom pair interactions, also triplet and perhaps even larger sets of interacting atoms should be considered. A threeatom interaction model that accounts well for most thermodynamic quantities of Si appears to be the StillingerWeber potential [2.8], which, for atom i and the interaction with its nearest neighbors j , k , l and m is 1 E SiSi(r i, r j, r k, r l, r m) =  [ E 2(r ij) + E 2(r ik) + E 2(r il) + E 2(r im) ] 2 (2.11) + E 3(r ij, r ik, θ jik) + E 3(r im, r ik, θ mik) + E 3(r il, r im, θ lim) + E 3(r ij, r im, θ jim) + E 3(r ij, r il, θ jil) + E 3(r il, r ik, θ lik) In this model, the twoatom interaction is modelled as E 2(r ij) EG ( H e = –p – r ij ⁄ a –q –1 ) exp ( ( r ij ⁄ a – c ) ) 0 r ij ⁄ a < c (2.12) r ij ⁄ a > c and the threeatom interaction by E 3(r ij, r ik, θ jik) –1 r ik r –1 Eλ exp γ ij – c +  – c a a = 1 2 × cos ( θ jik ) +  3 0 , r ij ⁄ a < c (2.13) otherwise Semiconductors for Micro and Nanosystem Technology 53 The Crystal Lattice System For silicon, this model works well with the following parameters: G = 7.0495563 , H = 0.60222456 , p = 4 , q = 0 , λ = 21 , γ = 1.2 , the cutoff radius c = 1.8 , the “lattice” constant – 19 a = 0.20951nm and the bond energy E = 6.9447 ×10 J ⁄ ion . Note that in the above r ij = r i – r j . Linearization We assume that the atoms of the crystal always remain in the vicinity of their lattice positions, and that the distance they displace from these positions is “small” when measured against the lattice constant a , see Figure 2.18. This is a reasonable assumption for a solid crystal at typical k uk rk = Rk + uk i Rk Figure 2.18. Instantaneous snapshot of the atom positions of a regular square lattice with respect to their average lattice site positions. On the right is shown the relation between the lattice position vector R k , the atom position vector r k and the atom displacement vector u k for atom k. The shading around atom i indicates how the interatom interaction strength falls off as a function of distance. operating temperatures; the covalent and ionic bonds are found to be strong enough to make it valid. 54 Semiconductors for Micro and Nanosystem Technology Elastic Properties: The Stressed Uniform Lattice Linearized Potential Energy To illustrate the derivation of the elastic energy in terms of the strain and the elastic constants of the crystal, we will use an interatomic potential that only involves twoatom interactions. For a more general derivation, see e.g. [2.1]. Denote the relative vector position of atom i by u i , and its absolute position by r i = R i + u i , where R i localizes the regular lattice site. For a potential binding energy E between a pair of atoms as depicted in Figure 2.17 (II), we form the potential energy of the whole crystal with N 1 U = 2 N ∑∑ N 1 E ( r i – r j ) = 2 i=1j=1 N ∑ ∑ E(R – R i j + ui – u j ) i=1j=1 N 1 = 2 . N ∑ ∑ E(R ij (2.14) + u ij ) i=1j=1 where R ij = R i – R j and u ij = u i – u j . The factor 1 ⁄ 2 arises because the double sum counts each atom pair twice. We expand the energy about the lattice site R ij using the Taylor expansion for vectors 1 2 E ( R ij + u ij ) = E ( R ij ) + ( u ij ⋅ ∇ )E ( R ij ) +  ( u ij ⋅ ∇ ) E ( R ij ) + … (2.15) 2 because of the assumption that the atom displacements u i and hence n their differences u ij are small. The terms ( u ij ⋅ ∇ ) E ( R ij ) must be read n as ( u ij ⋅ ∇ ) operating n times on the positiondependent energy E evaluated at the atom position R ij . Applying (2.15) to (2.14) we obtain N 1 U = 2 N N ∑∑ 1 E ( R ij ) + 2 i=1j=1 N N 1 + 4 ∑ ∑ (u N ∑∑u ij • ∇E ( R ij ) i=1j=1 (2.16) 2 ij • ∇ ) ( E ( R ij ) ) + … i=1j=1 The ﬁrst term in (2.16) is a constant for the lattice, and is denoted by U o . The second term is identically zero, because the energy gradient is evaluated at the rest position of each atom where by deﬁnition is must be zero Semiconductors for Micro and Nanosystem Technology 55 The Crystal Lattice System N 1 2 N ∑∑u ij • ∇E ( R ij ) = 0 (2.17) i=1j=1 This leaves us with the zeroth and second and higher order terms N 1 U = U o + 4 N ∑ ∑ (u 2 ij • ∇ ) E ( R ij ) + … (2.18) i=1j=1 The ﬁrst term in (2.18) is the energy associated with the atoms at the lattice positions, i.e., at rest, and represents the datum of energy for the crystal. The second term is the harmonic potential energy, or the smalldisplacement potential energy. When it is expanded, we obtain N 1 h U = 4 N ∑∑u ij • ∇( ∇E ( R ij ) ) • u ij (2.19) i=1j=1 Elasticity Tensor Equation (2.19) illustrates two terms, the deformation felt between two sites u ij , and the second derivative of the interatomic potential at zero deformation ∇( ∇E ( R ij ) ) . The second derivative is the “spring constant” of the lattice, and the deformation is related to the “spring extension”. We can now go a step further and write the harmonic potential energy in terms of the elastic constants and the strain. Moving towards a continuum view, we will write the quantities in (2.19) in terms of the position R i alone. Consider the term ∇[ ∇E ( R ij ) ] = ∇[ ∇E ( R i – R j ) ] . We expect E ( R ij ) to fall off rapidly away from R i when R j is far removed, so that for the site R i , most of the terms in (2.19) in the sum over j are effectively zero. To tidy up the notation, we denote D ( R i ) αβ as the components of the dynamical tensor D ( R i ) = ∇[ ∇E ( R ij ) ] . The remainder of the sites R j are close to R i , making R ij small, so that we are justiﬁed in making an expansion of the components of D ( R ij ) about R i D ( R ij ) αβ = D ( R i ) αβ + ∇ { D ( R i ) αβ } • R ij + … 56 Semiconductors for Micro and Nanosystem Technology (2.20) Elastic Properties: The Stressed Uniform Lattice for which we only keep the leading term. We now apply a similar series expansion to the term u j , because we expect it to vary little in the vicinity of R i , and from now on we do not consider atom sites that are far away, to obtain u j = u i + R ij • ∇u i(R i) + … (2.21) Taking the ﬁrst righthandside term of (2.20) and the ﬁrst two terms of (2.21), and inserting these into (2.19), we obtain N 1 h U = – 4 N ∑ ∑ (R T ij • ∇u i ) • D ( R i ) • ( R ij • ∇u i ) (2.22) i=1j=1 The result of (2.22) can now be rewritten again in terms of the original quantities, to give N 1 h U = – 4 ∑ N T ( ∇u i ) : i=1 ∑ {R ij D ( R i )R ij } : ∇u i j=1 (2.23) N 1 = – 2 ∑ ( ∇u ) :F(R ): ∇u T i i i i=1 The rank four elastic material property tensor F(R i) in the vicinity of the lattice site R i is deﬁned as N 1 F(R i) = 2 ∑ (R ij D ( R i )R ij ) (2.24) j=1 in terms of the crystal constituent positions and the resulting net interatom binding energy. We expect this tensor to be translationally invariant with respect to the lattice. Therefore, we can now move from a discrete crystal description to the continuum, by considering the crystal as a collection of primitive cells of volume V with an average “density” of elastic material property in any particular unit cell to be E = F ⁄ V , so that we can replace the sum in (2.23) by an integral to obtain Semiconductors for Micro and Nanosystem Technology 57 The Crystal Lattice System 1 T h U = –  ( ∇u i ) • E • ∇u i dV 2 ∫ (2.25) Ω Strain We now wish to bring the gradient of deformation ∇u , a vectorvalued ﬁeld over the crystal, in relation to the strain, which is a rank two tensorvalued ﬁeld. We consider Figure 2.19, where we follow the behavior of x P′ dx′ Q′′ X P dx Q Figure 2.19. An arbitrary body before and after deformation. The points P and Q are “close” to each other. two points P and Q before and after deformation. The points are chosen to be close to each other, and we assume that the body has deformed elastically, i.e., without cracks forming, and without plastically yielding. To a very good approximation, the movement of each point in the body can then be written as a linear transformation P′ = α o + ( δ + α ) • P 58 Semiconductors for Micro and Nanosystem Technology (2.26) Elastic Properties: The Stressed Uniform Lattice where α o represents a pure translation and α a rotation plus stretch. Next, we consider what happens to the line element d x under the deformation ﬁeld u T T d x′ = Q′ – P′ = α o + ( δ + α ) • Q – ( α o + ( δ + α ) • P ) T (2.27) = d x + α • d x = d x + du T We can rewrite du via the chain rule as du = ( ∇u ) • d x , thereby making the association α = ∇u , to ﬁnd that we can now rewrite the expression for d x' only in terms of the original quantities that we measure T d x' = d x + ( ∇u ) • d x (2.28) We are interested in how the line element deforms (stretches), and for this we form an expression for the square of its length 2 T T T T T ( d x' ) = ( d x ) • d x + [ ( ∇u ) • d x ] • d x + ( d x ) • [ ( ∇u ) • d x ] T T T + [ ( ∇u ) • d x ] • [ ( ∇u ) • d x ] T T (2.29) T = ( d x ) • [ δ + ∇u + ( ∇u ) + ( ∇u ) • ∇u ] • d x T T Consider the argument [ δ + ∇u + ( ∇u ) + ( ∇u ) • ∇u ] . If the stretch is small, which we have assumed, then we can use 1 + s ≅ 1 + s ⁄ 2 , which deﬁnes the strain ε by 1 1 T T T ε =  [ ∇u + ( ∇u ) + ( ∇u ) • ∇u ] ≈  [ ∇u + ( ∇u ) ] 2 2 (2.30) The components of the small strain approximation may be written as 1 ∂u α ∂u β + ε αβ = 2 ∂ xβ ∂ xα (2.31) The small strain is therefore a symmetric rank two tensor. Semiconductors for Micro and Nanosystem Technology 59 The Crystal Lattice System Crystal Energy in Terms of Strain We return our attention to equation (2.25), where our goal is to rewrite h the harmonic crystal energy U in terms of the crystal’s strain. Clearly, h U should not be dependent on our choice of coordinate axes, and is therefore invariant with respect to rigidbody rotations. We now choose a deformation ﬁeld u = θ × R which simply rotates the crystal atoms from their lattice positions by a constant deﬁned angle θ . The gradient of this ﬁeld is ∇u = ∇θ × R + θ × ∇R = 0 , so that the energy associated with pure rotations is clearly zero. The gradient of u can always be written as the sum of a symmetric and an antisymmetric part T T ∇u = ∇u s + ∇u a = [ ∇u + ( ∇u ) ] ⁄ 2 + [ ∇u – ( ∇u ) ] ⁄ 2 = (ε + κ) Note that ε = ε obtain T (2.32) T and κ = – κ . We substitute ε and κ into (2.25) to 1 h T T T T U = –  ( ε :E:ε + ε :E:κ + κ :E:ε + κ :E:κ ) dV 2 ∫ (2.33) Ω for the harmonic energy of the crystal. Independent Elastic Constants The elastic tensor E has inherent symmetries that can be exploited to simplify (2.33). We ﬁrst write an expression for the components of the tensor E, centered at site i, in cartesian coordinates N E(R i) αβγµ 1 = 2V ∑ ( R j=1 ∂ ij ) α ∂x β ∂E ( R i )  ( R ij ) γ ∂x µ (2.34) Consider the argument of the sum. Clearly, E is symmetric with respect to the indices β and µ , since the order of differentiation of the energy with respect to a spatial coordinate is arbitrary. Furthermore, swapping T T α and γ also has no effect. Now look at the terms ε :E:κ and κ :E:ε in (2.33). They vanish because, due the above symmetries of E , T T ε :E:κ = – κ :E:ε . Finally, because the antisymmetric strain κ repreT sents a pure rotation, the last term κ :E:κ in (2.33) must also vanish. As 60 Semiconductors for Micro and Nanosystem Technology Elastic Properties: The Stressed Uniform Lattice a consequence, the tensor E must be symmetric with respect to its ﬁrst and second pairs of indices as well, because it now inherits the symmetry of the symmetric strain. Thus we obtain the familiar expression for the elastic energy as 1 h T U = –  [ ε :E(R):ε ] dV 2 ∫ V N 3 1 T 1 ( R – R j ) D ( R ) ( R – R j ) :ε dR = –  ε : 2V 2 V j=1 (2.35) ∑ ∫ Of the possible 81 independent components for the fourth rank tensor E, only 21 remain. We can organize these into a 6 × 6 matrix C if the following six index pair associations are made: E ( ij ) ( kl ) ≡ C ( m ) ( n ) 11 → 1 23 → 4 ij → m 22 → 2 31 → 5 kl → n 33 → 3 12 → 6 (2.36) In order to obtain analog relations using the reduced index formalism, we have to specify the transformation of the stress and strain as well. For the stress we can use the same index mapping as in (2.36) σ ij ≡ s m 11 → 1 23 → 4 ij → m 22 → 2 31 → 5 33 → 3 12 → 6 (2.37) For the strain, however, we need to combine the index mapping with a scaling e 1 = ε 11 e 2 = ε 22 e 3 = ε 33 e 4 = 2ε 23 e 5 = 2ε 31 e 6 = 2ε 12 (2.38) In this way, we retain the algebraic form for the energy terms Semiconductors for Micro and Nanosystem Technology 61 The Crystal Lattice System 1 1 T 3 T 3 h U = –  [ e • C(R) • e ] dR = –  [ e • s ] dR 2 2 ∫ ∫ V (2.39) V The further reduction of the number of independent elastic coefﬁcients now depends on the inherent symmetries of the underlying Bravais lattice. Silicon has a very high level of symmetry because of its cubic structure. The unit cell is invariant to rotations of 90° about any of its coordinate axes. Consider a rotation of the xaxis of 90° so that x → x , y → z and z → – y . Since the energy will remain the same, we must have that C 22 = C 33 , C 55 = C 66 and C 21 = C 31 . By a similar argument for rotations about the other two axes, we obtain that C 11 = C 22 = C 33 , C 21 = C 31 = C 23 and C 44 = C 55 = C 66 . Since the other matrix entries experience an odd sign change in the transformed coordinates, yet symmetry of C is required, they must all be equal to zero. To summarize, Si and other cubicsymmetry crystals have an elasticity matrix with the following structure (for coefﬁcient values, consult Table 2.1) but with only three independent values: C 11 C 12 C 12 C 12 C 11 C 12 C = C 12 C 12 C 11 (2.40) C 44 C 44 C 44 C can be inverted to produce S with exactly the same structure and the following relation between the constants: –1 –1 ( S 11 – S 12 ) = ( C 11 – C 12 ) –1 ( S 11 + 2S 12 ) = ( C 11 + 2C 12 ) S 44 = C 44 (2.41) The bulk modulus B and compressibility K of the cubic material is given by 62 Semiconductors for Micro and Nanosystem Technology Elastic Properties: The Stressed Uniform Lattice 1 1 B =  ( C 11 + 2C 12 ) = 3 K (2.42) Isotropic materials have the same structure as (2.40) but with only two independent constants. In the literature, however, no less that three notations are common. Often engineers use the Young modulus E and Poisson ratio ν , whereas we ﬁnd the Lamé constants λ and µ often used in the mechanics literature. The relations between the systems are summarized by: C 11 – C 12 C 44 =  = µ 2 Eν C 11 = λ + 2µ C 12 =  ( 1 + ν ) ( 1 – 2ν ) E E(1 – ν) C 11 =  C 44 = 2(1 + ν) ( 1 + ν ) ( 1 – 2ν ) µ ( 3λ + 2µ ) λ E = ν =  λ+µ 2(λ + µ) C 11 – 2C 44 C 44 ( 3C 11 – 4C 44 ) ν =  E = C 11 – C 44 2 ( C 11 – C 44 ) C 12 = λ (2.43) The α Quartz crystal has the following elastic coefﬁcient matrix structure (for coefﬁcient values, consult Table 2.1): C 11 C 12 C 12 C 14 C 12 C 11 C 12 – C 14 C Quartz = C 12 C 12 C 11 C 14 – C 14 (2.44) C 44 C 44 C 44 Semiconductors for Micro and Nanosystem Technology 63 The Crystal Lattice System 2.4 The Vibrating Uniform Lattice The elastic, springlike nature of the interatomic bonds, together with the massive atoms placed at regular intervals; these are the items we isolate for a model of the classical mechanical dynamics of the crystal lattice (see Box 2.3 for brief details on Lagrangian and Hamiltonian mechanics). Here we see that the regular lattice displays unique new features unseen elsewhere: acoustic dispersion is complex and anisotropic, acoustic energy is quantized, and the quanta, called phonons, act like particles carrying information and energy about the lattice. 2.4.1 Normal Modes As we saw in Section 2.3, an exact description of the forces between the atoms that make up a crystal are, in general, geometrically and mathematically very complex. Nevertheless, certain simpliﬁcations are possible here and lead both to an understanding of what we otherwise observe in experiments, and often to a fairly close approximation of reality. We assume that: • The atoms that make up the lattice are close to their equilibrium positions, so that we may use a harmonic representation of the potential binding energy about the equilibrium atom positions. The spatial gradient of this energy is then the positiondependent force acting on the atom, which is zero when each atom resides at its equilibrium position. • The lattice atoms interact with their nearest neighbors only. • The lattice is inﬁnite and perfect. This assumption allows us to limit our attention to a single WignerSeitz cell by assuming translational symmetry. • The bound inner shell electrons move so much faster than the crystal waves that they follow the movement of the more massive nucleus that they are bound to adiabatically. 64 Semiconductors for Micro and Nanosystem Technology The Vibrating Uniform Lattice Box 2.3. A Brief Note on Hamiltonian and Lagrangian Mechanics. Hamilton’s Principle. Hamilton’s principle states that the variation of the action A, also called the variational indicator, is a minimum over the path chosen by a mechanical system when proceeding from one known conﬁguration to another δA t2 t1 = 0 (B 2.3.1) The Action. The action is deﬁned in terms of the lagrangian L and the generalized energy sources Ξ j ξ j of the system A = t2 ∫t 1 n L+ ∑ Ξ jξ j dt (B 2.3.2) j=1 where Ξ j is the generalized force and ξ j the generalized displacement The Lagrangian. The mechanical Lagrangian of a system is the difference between its kinetic coenergy T ∗ ( ξ˙ j, ξ j, t ) (the kinetic energy expressed in terms of the system’s velocities) and its potential energy U ( ξ j, t ) L M ( ξ˙ j, ξ j, t ) = T ∗ ( ξ˙ j, ξ j, t ) – U ( ξ j, t ) (B 2.3.3) In general, the lagrangian for a crystal is written in terms of three contributions: the mechanical (or matter), the electromagnetic ﬁeld and the ﬁeldmatter interaction L = ∫ ( L M + LF + LI ) dV (B 2.3.4) V 1 L M =  ( ρu̇ + ε • C • ε ) 2 (B 2.3.5) 1 Ȧ L F =  ( E + B ) L I =  – qψ̇ c 2 Generalized Energy Sources. This term groups all external or nonconservative internal sources (sinks) of energy in terms of generalized forces Ξ j and the generalized displacements ξ j . Generalized Displacements and Velocities. For a system, we establish the m independent scalar m degrees of freedom d ∈ ℜ required to describe its motion in general. Then we impose the p constraint equations B • d = 0 that specify the m required kinematics (the admissible path in ℜ ). This reduces the number of degrees of freedom by p, and we obtain the n = m – p generalized scan m lar displacements ξ ∈ ℜ ⊃ ℜ of the system. The generalized velocities are simply the time rate of change of the generalized displacements, ξ̇ = dξ ⁄ dt . Lagrange’s Equations. An immediate consequence of (B 2.3.2) is that the following equations hold for the motion d ∂L ∂L = Ξj – d t ∂ ξ̇ j ∂ ξ j (B 2.3.6) These are known as the Lagrange equations of motion. For a continuum, we can rewrite (B 2.3.6) for the Lagrange density as d d ∂L ∂L ∂L   =  –  dt ∂ ẋ j ∂ x j d Xk ∂(∂ x j ⁄ ∂ Xk) (B 2.3.7) In the continuum crystal lagrangian the variables X represent the material coordinates of the undeformed crystal; x the spatial coordinates of the deformed crystal. The Lagrange equations are most convenient, because they allow us to add detail to the energy expressions, so as to derive the equations of motion thereafter in a standard way. • The valence electrons form a uniform cloud of negative space charge that interacts with the atoms. Semiconductors for Micro and Nanosystem Technology 65 The Crystal Lattice System Normal Modes Normal modes are the natural eigenshapes of the mechanical system. We already know these from musical instruments: for example from the shapes of a vibrating string (onedimensional), the shapes seen on the stretched surface of a vibrating drum (twodimensional), or a vibrating bowl of jelly (threedimensional). Normal modes are important, because they are orthogonal and span the space of the atomic movements and thereby describe all possible motions of the crystal. To obtain the normal modes of the crystal, we will assume timedependent solutions that have the same geometric periodicity of the crystal. We will now derive an expression for the equations of motion of the crysh tal from the mechanical Lagrangian L M = T ∗ – U (see Box 2.3.) From h T the previous section we have that U = 1 ⁄ 2 ( ε :E:ε ) dV . The kinetic V coenergy is added up from the contributions of the individual atoms ∫ n 1 T ∗ = 2 ∑ mṙ 2 i (2.45) i=1 As for the potential energy, this sum can also be turned into a volume integral, thereby making the transition to a continuum theory. We consider a primitive cell of volume V n 1 T ∗ = 2 ∑ 1 2 m  ṙ V = V i 2 n ∑ ρṙ V ≅ 2 ∫ ρṙ dV 2 i i=1 i=1 1 2 i (2.46) V Thus we have for the mechanical Lagrangian that LM = ∫L M dV V LM 1 2 T =  ( ρṙ i – ε :E:ε ) dV , with 2 ∫ V 1 2 T =  ( ρṙ i – ε :E:ε ) 2 (2.47) The continuum Lagrange equations read [2.7] ∂L d d ∂L ∂L   =  –  dt ∂ẋ j ∂x j d X k ∂ ( ∂x j ⁄ ∂X k ) 66 Semiconductors for Micro and Nanosystem Technology (2.48) The Vibrating Uniform Lattice where we have chosen the coordinates X to describe the undeformed conﬁguration of the crystal, and x the coordinates in the deformed crystal, see Figure 2.19. 1D Monatomic Dispersion Relation The complexity in applying the lagrangian formulation to a general 3D crystal can be avoided by considering a 1D model system that demonstrates the salient features of the more involved 3D system. A large 1D lattice of N identical bound atoms are arranged in the form of a ring by employing the Bornvon Karmann boundary condition, i.e., u ( N + 1 ) = u ( N ) . From equation (2.31), the strain in the 1D lattice is simply 1 du T ε 11 =  ( ∇u + ∇u ) = 2 dX (2.49) which gives a potential energy density of 1 du 2 U =  E  2 dX (2.50) where u is the displacement of the atom from its lattice equilibrium site and E is the linear Young modulus of the interatomic bond. Note that we only consider nearestneighbour interactions. The kinetic coenergy density is 1 2 * T =  ρu̇ 2 (2.51) where the mass density is ρ = m ⁄ a for an atomic mass m and interatomic spacing a . The lagrangian density for the chain is the difference between the kinetic coenergy density and the potential energy density, * L = T – U . Note that u = X – x and hence that ∇u = Id – ∇x and u̇ = – ẋ . We insert the lagrangian density in equation (2.48) to obtain d du ρu̇˙ = – E   dX dX Semiconductors for Micro and Nanosystem Technology (2.52) 67 The Crystal Lattice System Since the potential energy is located in the bonds and not the lattice site, and therefore depends on the positions of the neighboring lattice sites, 2 2 the term d u ⁄ d X is replaced by its lattice equivalent for the lattice site 2 i ( 2u i – u i – 1 – u i + 1 ) ⁄ a . This ﬁnite difference formula expresses the fact that the curvature of u at the lattice site i depends on the nextneighbor lattice positions. This step is necessary for a treatment of waves with a wavelength of the order of the interatomic spacing. If we use the simpler site relation, we only obtain the long wavelength limit of the dispersion relation, indicated by the slope lines in Figure 2.20. We now look for solutions, periodic in space and time, of the form u ( X, t ) X = ia = exp [ j ( kX – ωt ) ] X = ia (2.53) which we insert into equation (2.52) and cancel the common exponential E 2 – ρω = – 2 ( 2 – exp [ kia ] – exp [ – kia ] ) a (2.54) Reorganizing equation (2.54), we obtain ω = ka E 2E ( 1 – cos [ ka ] )  = 2 2 sin 2 2 ρa ρa (2.55) Equation (2.55) is plotted in Figure 2.20 on the left, and is the dispersion relation for a monatomic chain. The curve is typical for an acoustic wave in a crystalline solid, and is interpreted as follows. In the vicinity where ω is small, the dispersion relation is linear (since sin [ ka ⁄ 2 ] ≈ ka ⁄ 2 ) and the wave propagates with a speed of E ⁄ ρ as a linear acoustic wave. As the frequency increases, the dispersion relation ﬂattens off, causing the speed of the wave ∂ω ⁄ ∂k to approach zero (a standing wave resonance). 1D Diatomic Dispersion Relation Crystals with a basis, i.e., crystals with a unit cell that contains different atoms, introduce an important additional feature in the dispersion curve. We again consider a 1D chain of atoms, but now consider a unit cell containing two different atoms of masses m and M . 68 Semiconductors for Micro and Nanosystem Technology The Vibrating Uniform Lattice unit cell = a unit cell = a slope = 0 ω(k) v ω(k) e= v slope = Optical branch slop Acoustic branch Acoustic branch π a π – a X ∆ Γ ∆ π a π – a k X ∆ X Γ ∆ k X Figure 2.20. The monatomic and diatomic onedimensional chain lattices lend themselves to analytical treatment. Depicted are the two computed dispersion curves with schematic chains and reciprocal unit cell indicated as a gray background box. Γ is the symmetry point at the origin, ∆ the symmetry point at the reciprocal cell boundary. Only nearestneighbor interactions are accounted for. The kinetic coenergy is added up from the contributions of the individual atoms 1 T ∗ = 2 n, 2 ∑ m u̇ 2 α iα (2.56) iα The index α counts over the atoms in an elementary basis cell, and the index i counts over the lattice cells. Similarly, the bond potential energy is dependent on the stretching of the interatom bonds 1 U = 4 n, 2, n, 2 ∑ iα, βj E αiβj 2  ( u iα – u jβ ) a Semiconductors for Micro and Nanosystem Technology (2.57) 69 The Crystal Lattice System The restriction to nextneighbor interactions and identical interatomic force constants yields the following equations of motion when the expressions (2.56) and (2.57) are inserted into the Lagrange equations E mu̇˙im = –  ( 2u im – u iM – u ( i – 1 )M ) a (2.58a) E Mu̇˙iM = –  ( 2u iM – u ( i + 1 )m – u im ) a (2.58b) At rest the atoms occupy the cell positions (due to identical static forces) ( i – 1 ⁄ 4 )a and ( i + 1 ⁄ 4 )a . Hence we choose an harmonic atom displacement ansatz for each atom of the form 1 1 u im =  c m exp j ka i –  – ωt 4 m (2.59a) 1 1 u iM =  c M exp j ka i +  – ωt 4 M (2.59b) The second ansatz can be written in terms of u im u iM = ka m cM   exp j  u im 2 M cm (2.60) Equations (2.59b) and (2.60) are now inserted into (2.58a). Eliminating common factors and simplifying, we obtain an equation for the amplitudes c m and c M ka 2E 2E 2 – ω m –  cos  a m a M 2 cm = 0 (2.61) c ka 2E 2E 2 –  cos   – ω M M 2 a m a M This is an eigensystem equation for ω , and its nontrivial solutions are obtained by requiring the determinant of the matrix to be zero. Performing this, we obtain 70 Semiconductors for Micro and Nanosystem Technology The Vibrating Uniform Lattice 4 ka 2 E 1 1 1 1 2 2 ω =   +  − +  +  –  sin  mM 2 a m M m M (2.62) The two solutions are plotted in Figure 2.20 on the right. Discussion The case when the two masses of the unitcell atoms differ only by a small amount, i.e., M = m + µ with µ small, is instructive. The optical and acoustic branches approach then each other at the edge of the reciprocal cell, i.e., at k = − + π ⁄ a . As the mass difference µ goes to zero, the lattice becomes a monatomic lattice with lattice constant a ⁄ 2 , so that the branches touch each other at the reciprocal cell edge. The interpretation of the two branches is as follows. For the lower branch, all the atoms move in unison just as for an acoustic wave, hence the name acoustic branch. In fact, it appears as a centerofmass oscillation. For the upper or optical branch, the center of mass is stationary, and the atoms of a cell only move relative to each other. Its name refers to the fact that for ionic crystals, this mode is often excited by optical interactions. 2D Square Lattice Dispersion Relation The next construction shows the richness in structure that appears in the dispersion relation when an additional spatial dimension and one level of interaction is added to the 1D monatomic lattice. It serves as an illustration that the anisotropy of the interatomic binding energy enables more involved crystal vibrational modes and hence additional branches in the dispersion curves. At the same time it shows that most of the essential features of the expected structure is already clear from the simple 1D models. The model considers a 2D monatomic square la