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W iley Name Actinium Aluminum Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium Chlorine Chromium Cobalt Copper Curium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Cadolinium Gall hi in Germanium Gold Symbol Ac Al Am Sb Ar As At Ba Bk Be Bi B Br Cd Ca Cf C Ce Cs Cl Cr Co Cu Cm Dy Es Er Eu Fm F Fr Cd Ga Ge Au Name Symbol Name Hafnium Helium Hoi mill m Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Mendelevium Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium Hf He Ho H In I Ir Fe Kr La Lr Pb Li Lu Mg Mn Md Hg Mo Nd Ne Np Ni Nb N No Os O Pd P Pt Pu Po K Symbol Pr Pm Pa Ra Rn Re Rh Rb Ru Sm Sc Se Si Ag Na Sr S Ta Tc Te Tb TI Th Tm Sn Ti W U V Xe Yb Y Zn Zr I Introduction to Solid State Physics GLOBAL EDITIO N Charles Kittel Professa r Emeri ins Univcrsitỉ / o f California, Berkeley Chapter 18, Nanostructures, was written by Professor Paul McEuen of Cornell University nrRƯỜMG 0ẬÍ ^ ' hÇc Cp- n i ý' i/íậiv 5J / A J l ï W iley Copyright © 2018 The content provided in this textbook is based on K ittels Introduction to Solid State Physics, 8th edition [2005] John Wiley & Sons Singapore Pte Ltd Cover image: © Libya Linnik/Shutterstock Contributing Subject M atter Experts: Dr Mamta Dahiya and D r Kuldeep K Kapil, Delhi University Founded in 1807, John Wiley & Sons, Inc has been a valued source of knowledge and understanding lor more than 200 years, helping people around the world m eet their needs and fulfill their 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and costs, including but not limited to lost prolits and attorney’s fees, in the event legal action is required No part of this publication may be reproduced, stored in a retrieval svstem, or transm itted in any form or by am means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as perm itted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Darners, MA 01923, website wAvw.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Departm ent John Wilev & Sons, Inc., 11 River Street, Hoboken, NJ 0/030, (201) /48-6(1 1, tax (201) 748-6008 website http://wAvw.wilev.com/go/permissions ISBN: 978-1-119-45416-8 Printed at Markono Print Media Pte Ltd 109 About the Author Charles Kittel did his undergraduate work in physics at M.I.T and at the Cavendish Laboratory of Cambridge University He received his Ph.D from the University of Wisconsin He worked in the solid state group at Bell Laboratories, along with Bardeen and Shockley, leaving to start the theoretical solid state physics group at Berkeley in 1951 His research has been largely in magnetism and in semiconductors In magnetism he developed the theories of ferromagnetic and antiferromagnetic resonance and the theory of single ferromagnetic domains, and extended the Bloch theory of magnons In semi conductor physics he participated in the first cyclotron and plasma resonance experiments and extended the results to the theory of impurity states and to electron-hole drops He has been awarded three Guggenheim fellowships, the Oliver Buckley Prize lor Solid State Phvsics, and, lor contributions to teaching, the Oersted Medal of the American Association of Physics Teachers He is a member of the National Academy of Science and of the American Academy of Arts and Sciences Preface This book is the Global edition of an elementary text on solid state/ condensed matter physics for seniors and beginning graduate students of the physical sciences, chemistry, and engineering In the years since the first edi tion was published the field has developed vigorously, and there are notable applications The challenge to the author has been to treat significant new areas while maintaining the introductory level of the text It would be a pity to present such a physical, tactile field as an exercise in formalism At the first edition in 1953 superconductivity was not understood; Fermi surfaces in metals were beginning to be explored and cyclotron resonance in semiconductors had just been observed; ferrites and permanent magnets were beginning to be understood; only a few physicists then believed in the reality of spin waves Nanophysics was forty years off In other fields, the structure of DNA was determined and the drift of continents on the Earth was demon strated It was a great time to be in Science, as it is now I have tried with the successive editions of 7SSP to introduce new generation to the same excitement In the past editions, several changes as well as much clarification have been bought into the text: • An important chapter has been added on nanophysics, contributed by an active worker in the field, Professor Paul L McEuen of Cornell University Nanophvsics is the science of materials with one, two, or three small dimen sions, where small means nanometer 10-9 m) This field is the most excit ing and vigorous addition to solid state science in the last ten years • The text makes use of the simplifications made possible by the universal availability of computers Bibliographies and references have been nearly eliminated because simple computer searches using keywords on a search engine such as Google will quickly generate many useful and more recent references As an example of what can be done on the Web, explore the entry http://www.phy■sicsweb.org/bestof/cond-niat No lack of honor is in tended by the omissions oi early or traditional references to the workers who First worked on the problems oFthe solid state • The oulei of the chapters has been changed: superconductivity and magnetism appeal earlier, thereby making it easier to arrange an interesting one-semester course In this Global edition, Further changes made are: • Chaptei on dielectrics and (erroeleetrics has been moved beFore plasmons, polaiitons, and polarons as the knowledge of former is required For dis cussing optical properties and processes Preface • A number of new problems have been added to most of the chapters, while many of the problems are those of the earlier editions The crystallographic notation conforms with current usage in physics Im portant equations in the body of the text are repeated in SI and CGS-Gaussian units, where these differ, except where a single indicated substitution will translate from CGS to SI The dual usage in this book has been found helpful and acceptable Tables are in conventional units The symbol e denotes the charge on the proton and is positive The notation (IS) refers to Equation IS of the current chapter, but (3 IS) refers to Equation 18 of Chapter A caret ( ) over a vector denotes a unit vector Few of the problems are exactly easy: Most were devised to carry forward the subject of the chapter The notation QTS refers to my Quantum Theory of Solids, with solutions by C V Fong; TP refers to Thermal Physics, with IT Kroemer This edition owes much to detailed reviews of the entire text by Professor Paul L McEuen of Cornell Universitv and Professor Roger Lewis of Wollongong University in Australia They helped make the book much easier to read and un derstand However, I must assume responsibility for the close relation of the text to the earlier editions, Many credits for suggestions, reviews, and photographs are given in the prefaces to earlier editions I have a great debt to Stuart Johnson, my publisher at Wiley; Suzanne Ingrao, my editor; and Barbara Bell, my per sonal assistant Corrections and suggestions will be gratefully received and may be addiessed to the author bv email to kittel@berkeley.edu The Instructors Manual is available for download at \vww.wiley.com/ college/kittel Charles Kittel Contents C hapter 1: crystal structure Periodic Arrays of Atoms Lattice Translation Vectors Basis and the Crystal Structure Primitive Lattice Cell ^ Fundamental Types of Lattices Two-Dimensional Lattice Types Three-Dimensional Lattice Types Index Systems for Crystal Planes 11 Simple Crystal Structures Sodium Chloride Structure Cesium Chloride Structure Hexagonal Close-Packed Structure (hep) Diamond Structure Cubic Zinc Sulfide Structure 13 13 I“! 13 13 1< Direct Imaging of Atomic Structure 1$ Nonideal Crystal Structures Random Stacking and Polvtvpism 18 19 Crystal Structure Data Summary 19 22 Problems 22 C hapter 2: wave diffraction and the reciprocal LATTICE 25 Dill ruction ol Waves by Crystals The Bragg Law -' Scattered Wave Amplitude -8 Fourier Analysis Reciprocal 1dittice Vectors Dillraction Conditions Lane Equations B rillouin Z ones Reciprocal Lattice' to sc Lattice Reciprocal Lattice to bee Lattice' Reciprocal Lattice' to fee Lattice 31 33 39 viii Contents Fourier Analysis of the Basis Structure Factor of the bcc Lattice Structure Factor of the fee Lattice Atomic Form Factor 41 42 42 43 Summary 45 Problems 45 C hapter 3: crystal binding and elastic constants 49 Crystals of Inert Gases Van der Waals-London Interaction Repulsive Interaction Equilibrium Lattice Constants Cohesive Energy 51 55 58 60 61 Ionic Crystals Electrostatic or Madelung Energy Evaluation of the Madelung Constant 62 62 66 Covalent Crystals 69 Metals 71 Hydrogen Bonds 72 Atomic Radii Ionic Crystal Radii 72 74 Analysis of Elastic Strains Dilation Stress Components 75 77 77 Elastic Compliance and Stiffness Constants Elastic Energy Density Elastic Stillness Constants of Cubic Crystals Bulk Modulus and Compressibility Elastic Waves in Cubic Crystals Waves in the f 100] Direction Waxes in the [1 10] Direction 79 79 60 82 82 83 84 Summary 87 Problems 87 C hapter 4: phonons i crystal vibrations Vibrations of Crystals with Monatomic Basis 91 93 F irst B rillouin Z o n e 95 G ro u p Y elocitv 96 Contents Long Wavelength Limit Derivation of Force Constants from Experiment Two Atoms per Primitive Basis ^ 96 ^ Quantization of Elastic Waves Phonon Momentum 10? Inelastic Scattering by Phonons 10^ Summary Problems C hapter 5: phonons 11 thermal properties Phonon Heat Capacity Planck Distribution Normal Mode Enumeration Density of Slates in One Dimension Density of States in Three Dimensions Debye Model for Density of States Debye T' Law Einstein Model of the Density of States General Result for Df to) ^ 107 1" H9 110 f13 Hh 1I9 Anharmonic Ciystal Interactions Thermal Expansion 1-1 Thermal Conductivity Thermal Resistivity of Phonon Gas Umklapp Processes Imperfections Problems 1-3 1-3 1-/ l - lS 130 C hapter 6: free electron fermi gas Energy Levels in One Dimension 133 Effect of Temperature on the FermiDirac Distribution 133 Free Electron Gas in Three Dimensions E39 Heat Capacity of the Electron Gas Experimental Meat Capacity of Metals IIoavy Fermions Electrical Conductivity and Ohm’s Law l ‘E 1“** Experimental Electrical Resistivity ol Metals 130 Umklapp Scattering 153 ix A p p e n d ix From (12) we have g(£ ) c E —€ (13) and clE , N(E') E' — e ' (14) With N (E') approximately constant and equal to NF over the small energy range between 2em and 2er, we take it out of the integral to obtain 1= NfVJ T dE'Ë^~e = NpVlog “ ^ • ( 15) Let the eigenvalue e of (15) be written as e = 2eF — A , (Ifi) which defines the binding energ}' A of the electron pair, relative to two free electrons at the Fermi surface Then (15) becomes 2e = Nf V log — —26;.- + A JVFVlog ~ A 2/¡cop + A A ( 17) or 1/NFV = log( + 2fi(oD/A) (IS) This result for the binding energy of a Cooper pair mav he written as 2ft (Op A = - exp(l/AVV) - ' a (19) For V positive (attractive interaction) the energy of the system is lowtnt'd hy excitation of a pair of electrons above the Fermi level Therefore the Fermi gas is unstable in an important way The binding energy (19) is closely related to the superconducting energy gap Zy, The BCS calculations show that a high density of Cooper pairs may form in a metal APPENDIX I: GINZBURG-LANDAU EQUATION We owe to Ginzburg and Landau an elegant theory of the' phenomenology oi the superconducting state and of the spatial variation of the order parameter in that state An extension of the theory bv Abrikosov describes the structure oi the vortex state which is exploited technologically in superconducting mag nets The attractions of the CL theory are the natural introduction of the' 679 680 Appendix coherence length and of the wavefunction used in the theory of the Josephson effects in Chapter 12 We introduce the o r d e r p a r a m e t e r i/r(r) with the property that ^ ( # ) = n s (r) , (1 ) the local concentration of superconducting electrons The mathematical for mulation of the definition of the function will come out of the BCS the ory We first set up a form for the free energy density Fs(r ) in a superconduc tor as a function of the order parameter We assume that in the general vicinity of the transition temperature Fs(r) = Fv —a\i{/\2 + |/3|i/'|4 + (l/2m)\(—ihV—qA/c)ij/\2 — p M -r /B ,, (2) with the phenomenological positive constants a , /3, and m, of which more will be said Here: Fv is the free energy density of the normal state —a\if/\2 + } is a typical Landau form for the expansion of the free energy in terms of an order parameter that vanishes at a second-order phase transition This term may be viewed as —ans + oPns and by itself is a mini mum with respect to ns when ns(T) = a/(3 The term in |grad ip\2 represents an increase in energy caused by a spa tial variation of the order parameter It has the form of the kinetic energy in quantum mechanics.1 The kinetic momentum —ihV is accompanied by the field momentum —qAJc to ensure the gauge invariance of the free energy, as in Appendix G Here q = —2c for an electron pair The term —JM • (IB,;, with the fictitious magnetization M = (B —B„)/477\ represents the increase in the superconducting free energy caused bv the ex pulsion of magnetic (lux from the superconductor The separate terms in (2) will be illustrated by examples as we progress further First let us derive the GL equation (6) We minimize the total free energv JdV Fs(r) with respect to variations in the function i/r(r) We have 8FS(r) = [-aifj + /3|i//|2i/' + (1/2m)( —ifiV ~ qA/c)^ • (ihV - r/A/c)5i//* + c.c.] (3) We integrate bv parts to obtain JY/V(Vgraphv 494 surface electronic structure 498 surface nets, 494 surface plasmons 460 surface resistance 161 surface states 499 surface transport, 501 Surface plasmon resonance, 553 Susceptibility dielectric 401 TA modes 97 Temperature, Delm* 114 Temporal lire dependence reflection lines 653 Tetrahedral angles 22 Thermal conductivity 123 158 glasses 587 isotope ef fect 129 metals 158 one-dimension 567 table 118 N Thermal dilation 130 Thermal effective mass, I 17 Thermal excitation, magnons 336 Thermal expansion 122 Thermal ioni/ation 215 Thermal resistivity phonon gas 125 Thermionic emission 499 Thermodvuamics superconducting transition 272 Thermoelectric effects 16 ThomasTYrmi approximation 440 Thomas-f'crmi dielectric function 44 Three-level maser 390 'fight-binding method 234 T ( ) modes 97 Tomograph y magnetic resonance 365 Transistor MOS 501 Transition, displacive 413 first-order 19 metal insulator 109 order-disorder 41-4 second-order I In d ex 692 Transition metal allovs, 646 Tran siti on tenipera111re,gl as s, 581 Translation operation, Translation vector, Transmission electron microscopy, 526 Transmission probability, 540 Transparency, alkali metals, 434 Transport, surface, 501 Transverse optical modes, plasma, 434 Transverse relaxation time, 370 Triplet excited states, 320 Tunneling, 289 Josephson, 291 probability, 530 Twinning, 611 Two-fluid model, 297 Two-level svstem, 322 Type I superconductors, 266 Tvpe II superconductors 266, 2S5 Ultraviolet transmission limits, 435 Umklapp processes, 127 Unit cell, Upper critical field, 682 UPS, 485 Valence band edge, 193 van der Waals interaction, 55 Van Hove singularities, 121, 534 Van Vleck paramagnetism, 314 Vector potential, 673 Vickers hardness number, 628 Viscosity, 582 Vitreous Silica, 578 Vortex state, 266, 286, 298 Wannier functions, 256 Wave equation, continuum, 105 periodic lattice, 174 Weiss field, 325 Whiskers, 626 Wiedemann-Franz law, 158 Wigner-Seitz boundary’condition, 239 Wigner-Seitz cell, 6, 8, 36, 240 Wigner-Seitz method, 238 Work function, 498 Work-hardening, 624 XPS, 485 Youngs modulus, 89 Yttrium iron garnet, 383 Zener breakdown, 219 Zener tunneling, 219 Zero-field splitting, 388 Zero-point motion, 56, 85 Table of Values Q uantity Symbol Value V elocity o f light P roton charge c P lan ck ’s constant h 2.997925 1.60219 4.80325 6.62620 1.05459 A vogadro s num b er A tom ic m ass unit E lectron rest mass P roton rest mass P roton m ass/electron mass N amu M p /m 6.02217 X 1023 m o l'1 1.66053 9.10956 1.67261 1836.1 R eciprocal fine structure constant f i c / e E lectron radius c ' / m c E lectron C om pton w avelen gth f i / m c B ohr radius h 2/ m e Bohr m agneton e h / m c R ydberg constan t m e 4/ h 1/ a 137.036 e oq II m M p 2.81794 3.86159 1r K CGS SI 1010 cm s-1 10s m s 10" 19 C - 10“ 10 esu 10~2‘ erg s 10_2‘ erg s - 10“■34Js 10" 34 Js — '24g 10“ 27 kg 10" 31 kg 10“27 kg -“ R '24g — 10“ 15 m 10"13 cm lo 13 m " " cm „ 5.29177 9.27410 2.17991 13.6058 eV ro Mb R* or Ry i n - 11 m lu 111 10 cm -i i _i 04 Trn—1 10" JT 10 " erg G IS T 10" J '" erg V electro n volt B oltzm ann constant P erm ittivity o f free space P erm eability o f free space eY eV//i eV//ic eV/*„ 1.60219 2.41797 X 1014 Hz 8.06546 1.16048 X 104 K ^B *0 - 1.38062 Mo 10-12 erg 10" 19I - - lO ’ e m -1 105 m"1 - - " erg K"1 10“23 I K ' 1 10747TỴ2 47T X 10"7 " S o u rce : B N Taylor, W H Parker, and D N Langenberg, Rev Mod Pins 41, 375 (1969) See also E R Cohen and B N Taylor, Journal of Physical and Chemical Reference Data 2(4) 663 (1973)