Introduction to solid state physics kittel pdf free download

EIGHTH EDITION

Introduction to Solid State Physics CHARLES KITTEL

Name Actinium Aluminum Americium Antimony Argon

Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium Chlorine Chromium Cobalt Copper Cur]um Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold

Symbol

~f 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 Gd Ga Ge Au

Name

Symbol

Name

Symbol

Hafnium Helium Holmium Hydrogen Ind1um Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Mendelevium Mercury Molybdenum Neodymium I\' eon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium

Hf

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

Pr Pm Pa

He Ho H

In I Ir Fe

Kr La

Lr Pb Li Lu Mg \t1n Md Hg Mo Nd r\e Np Ni Nb !\

No Os 0 Pd p Pt

Pu Po K

Ra Rn Re Rh Rb Ru Sm Sc Se Si Ag Na Sr

s Ta Tc Te Tb Tl Th Tm Sn Ti

w L'

v Xe Yb y

Zn Zr

H' Periodic Table, with the Outer Electron Configurations of Neutral Atoms in Their Ground States

Is

LP

Neto

0~'~

The notation used to describe the electronic configuration of atoms and ions is discussed in all texthooks of introductory atomic physic.:s. The letters s, p, d, signify electrons having orbital angular momentum 0, I, 2, . . . in units n; the numher to the left of the letter denotes the principal quantum number of one orhit, and the superscript to the right denotes the number of electrons in the orhit.

Si'~

AJ' 3

su•

pt5

3s

3d 2 4s 2

4s

3d!) 4s

:J({I 4s:.~

3d 5

3d" 4s 2

4d 1 5s

5s

Hf72

4d"

3d" 4s 2

3d 10

3d 10

4.~

4s 2

4d' 0

4d 5.~

5s

Qs71i

Ta 7-'1

4s2 4p 4s 2 4p 2 4s 2 4p 3 4.~2 4p 4 4s2 4p 5 4s 24p 6

Au 79

4f' .. 5d 2 6s2

6s Fr



RaMs

Ac~'~!+ 1'~ rrrr Gd•H Tb Oy Ho Er' Tm Yb Lu 6d 4J' 4fs 4f'o 4fn 4f12 4f'3 4ft4 4/u 65

7s

7.·P·

7s2

6

2

6

6sz

5d 6z

66

67

69

8

71

70

5d

5d 2

2

~~-s--~6-s-·--+--sr--~-s-·-+-6_s_-+_6_s_~~6·s--~s-~6-s-~-6s-·-~6·s-·-~6_s_-+-6_s_~ 2

2

7s2

2

2

2

2

2

2

2

I ntroductioti to Solid State Physics EIGHTH EDITION

Charles Kittel Professor Emeritus University of California, Berkeley

Chapter 18, Nanostructures, was written by Professor Paul McEuen of Cornell University.

John Wiley & Sons, Inc

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Copyright© John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval or transmitted in any scanning or otherwise, form or by any means, electronic, mechanical, photocopying, execpt as permitted under Sections or of the United States Copyright Act, without either the prior written of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. Rosewood Drive, Danvers, MA , (), fax () Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., River Street, Hoboken, NJ , (), fax () To order books or for customer service please, calllCALL WILEY ().

Library of Congress Cataloging in Publication Data: Kittel, Charles. Introduction to solid state physics I Charles Kittelth ed. www.cronistalascolonias.com.ar ISBN X 1. Solid state physics. I. Title. QCK5

~dc22

ISBN'frX \VIE ISBN Printed in the United States of America 10 9 8 7 6 5 4



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 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 L His research has been largely in magnetism and in semiconductors. In he developed the theories of ferromagnetic and antiferromagnetic resonance and the theory of single ferromagnetic domains, and extended the Bloch theory of magnons. In semiconductor physics he participated in the first cyclotron and plasma resonance experiments and extended the results to theory of impurity states and to electron-hole drops. He has been awarded three fellowships, the Oliver Buckley Prize for Solid State Physics, and, for 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 eighth 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 edition 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 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 demonstrated. It was a great time to be in Science, as it is now. I have tried with the successive editions of ISSP to introduce new generations to the same excitement. There are several changes from the seventh edition, as well as much clariflcation: • An important chapter has been added on nanophysics, contributed by an active worker in the field, Professor Paul L. McEuen of Cornell University. N anophysics is the science of materials with one, two, or three small dimensions, where "small" means (nanometer 9 m). This field is the most exciting 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 Coogle will quickly generate many useful and more recent references. As an example of what can be done on the Web, explore the entry http://w\www.cronistalascolonias.com.arstof/cond-mat. No lack of honor is intended by the omissions of early or traditional references to the workers who first worked on the problems of the solid state. • The order of the chapters has been changed: superconducth,ity and magnetism appear earlier, thereby making it easier to arrange an interesting one-semester course. The crystallographic notation conforms with current usage in physics. Important 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 arc in conventional units. The symbol e denotes the

Pre~ace

charge on the proton and is positive. The notation (18) refers to Equation 18 of the current chapter, but () refers to Equation 18 of Chapter 3. A caret r) over a vector denotes a unit vector. Few of problems are exactly easy: Most were devised to carry forward the subject of the chapter. \Vith few exceptions, the problems are those of the original sixth and seventh editions. The notation QTS refers to my Quantum Theory of Solids, with solutions by C. Y. TP refers to Thermal Physics, with H. Kroemer. This edition owes much to detailed reviews of the entire text by Professor PaulL. McEuen of Cornell University and Professor Roger Lewis of\Vollongong University in Australia. They helped make the book much easier to read and understand. 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 \Viley; Suzanne Ingrao, my editor; and Barbara Bell, my personal assistant. Corrections and suggestions will be gratefully received and may be addressed to the author by email to [email&#;protected] The Instructor's Manual is available for download at: www.cronistalascolonias.com.ar co1lege/kittel. Charles Kittel

v

Contents CHAPTER

1:

CRYSTAL STRUCTURE

1

Periodic Array of Atoms

3

Lattice Translation Vectors Basis and the Crystal Structure Primitive Lattice Cell

Fundamental Types of Lattices Two-Dimensional Lattice Types Three-Dimensional Lattice Types

6 8 9

11

Simple Crystal Structures

13 13 14 15

16 17

Direct Imaging of Atomic Structure

18

Nonideal Crystal Structures

18 19

Random Stacking and Polytypism

2:

6

Index Systems for Crystal Planes Sodium Chloride Structure Cesium Chloride Structure Hexagonal Close-Packed Structure (hcp) Diamond Structure Cubic Zinc Sulfide Structure

CHAPTER

4

5

Crystal Structure Data

19

Summary

22

Problems

22

WAVE DIFFRACTION AND THE RECIPROCAL LATTICE

Diffraction of Waves by Crystals Bragg Law

Scattered Wave Amplitude Fourier Analysis Reciprocal Lattice Vectors Diffraction Conditions Laue Equations

Brillouin Zones Reciprocal Lattice to sc Lattice Reciprocal Lattice to bee Lattice Reciprocal Lattice to fcc Lattice

23 25 25

26 27 29

30 32

33 34

36

37

viii

Fourier Analysis of the Basis Structure Factor of the bee Lattice Structure factor of the fcc Lattice Atomic Form Factor

CHAPTER

3:

40 41

43

Problems

43

CRYSTAL BINDING AND ELASTIC CONSTANTS

47

Crystals of Inert Gases

49 53 56 58

Ionic Crystals Electrostatic or Madelung Energy Evaluation of the Madelung Constant

59 60 60

64

Covalent Crystals

67

Metals

69

Hydrogen Bonds

70

Atomic Radii

70

Ionic Crystal Radii

Analysis of Elastic Strains Dilation Stress Components

Elastic Compliance and Stiffness Constants Elastic Energy Density Elastic Stiffness Constants of Cubic Crystals Bulk Modulus and Compressibility

4:

40

Summary

Van der Waals-London Interaction Repulsive Interaction Equilibrium Lattice Constants Cohesive Energy

www.cronistalascolonias.com.ar



72

73 75

75 77 77 78

80

Elastic Waves in Cubic Crstals

80

Waves in the [] Direction Waves in the [] Direction

81 82

Summary

85

Problems

85

PHONONS I. CRYSTAL VIBRATIONS

89

Vibrations of Crystals with Monatomic Basis

91 93 94

First Brillouin Zone Group Velocity

Contents

Long Wavelength limit Derivation of Force Constants from Experiment

Two Atoms per Primitive Basis Quantization of Elastic Waves Phonon Momentum Inelastic Scattering by Phonons Summary Problems

CHAPTER

5: PHONONS THERMAL PROPERTIES Phonon Heat Capacity Planck Distribution Normal Mode Enumeration Density of States in One Dimension Density of States in Three Dimensions Debye Model for Density of States Debye 'f3 Law Einstein Model of the Density of States General Result forD( w)

Anharmonic Crystal Interactions

lll







Thermal Conductivity



Problems

6:

95 99

Thermal Expansion Thermal Resistivity of Phonon Gas Umklapp Processes Imperfecions

CHAPTER

94 94

FREE ELECTRON FERMI GAS

Energy Levels in One Dimension Effect of Temperature on the FermiDirac Distribution Free Electron Gas in Three Dimensions Heat Capacity of the Electron Gas







Experimental Heat Capacity of Metals Heavy Fermions



Electrical Conductivity and Ohm's Law



Experimental Electrical Resistivity of Metals Umklapp Scattering



ix

Motion in Magnetic Fields Hall Effect

Thermal Conductivity of Metals Ratio of Thermal to Electrical Conductivity

Problems CHAPTER

7:





Nearly Free Electron Model Origin of the Energy Gap Magnitude of the Energy Gap





Bloch Functions



Kronig-Penney Model



Wave Equation of Electron in a Periodic Potential



Number of Orbitals in a Band Metals and Insulators

8:





ENERGY BANDS

Restatement of the Bloch Theorem Crystal Momentum of an Electron Solution of the Central Equation Kronig-Penney Model in Reciprocal Space Empty Lattice Approximation Approximate Solution Near a Zone Boundary

CHAPTER







Summary



Problems



SEMICONDUCTOR CRYSTALS

Band Gap Equations of Motion Physical Derivation of lik F Holes Effective Mass Physical Interpretation of the Effective Mass Effective Masses in Semiconductors Silicon and Germanium

Intrinsic Carrier Concentration Intrinsic Mobility

Impurity Conductivity Donor States Acceptor States Thermal Ionization of Donors and Acceptors













Contents

Thermoelectric Effects Semimetals Superlattices Bloch Oscillator Zener Tunneling

Summary Problems CHAPTER

9:

FERMI SURFACES AND METALS

Reduced Zone Scheme Periodic Zone Scheme

Construction ofF ermi Surfaces Nearly Free Electrons

Electron Orbits, Hole Orbits, and Open Orbits Calculation of Energy Bands Tight Binding Method of Energy Bands Wigner-Seitz Method Cohesive Energy Pseudopotential Methods

Experimental Methods in Fermi Surface Studies Quantization of Orbits in a Magnetic Field De Haas-van Alphen Effect Extremal Orbits Fermi Surface of Copper Magnetic Breakdown

Summary Problems CHAPTER

SUPERCONDUCTIVITY Experimental Survey Occurrence of Superconductivity Destruction of Superconductivity of Magnetic Fields Meissner Effect Heat Capacity Energy Gap Microwave and Infrared Properties Isotope Effect

Theoretical Survey Thermodynamics of the Superconducting Transition London Equation



















xi

Coherence Length BCS Theory of Superconductivity BCS Ground State Flux Quantization in a Superconducting Ring Duration of Persistent Currents Type II Superconductors Vortex State Estimation of Hc1 and Hc 2 Single Particle Tunneling Josephson Superconductor Tunneling De Josephson Effect Ac Josephson Effect Macroscopic Quantum Interference

High-Temperature Superconductors Summary Problems Reference CHAPTER

DIAMAGNETISM AND PARAMAGNETISM Langevin Diamagnetism Equation Quantum Theory of Diamagnetism of Mononuclear Systems Paramagnetism Quantum Theory of Paramagnetism Rare Earth Ions Hund Rules Iron Group Ions Crystal Field Splitting Quenching of the Orbital Angular Momentum Spectroscopic Splitting Factor Van Vleck Temperature-Independent Paramagnetism

Cooling by Isentropic Demagnetization Nuclear Demagnetization

CHAPTER













Paramagnetic Susceptibility of Conduction Electrons



Summary



Problems

'

FERROMAGNETISM AND ANTIFERROMAGNETISM

Ferromagnetic Order Curie Point and the Exchange Integral



Temperature Dependence of the Saturation Magnetization Saturation Magnetization at Absolute Zero

Magnons Quantization of Spin Waves Thermal Excitation of Magnons





F errimagnetic Order



Antiferromagnetic Order Susceptibility Below the N eel Temperature Antiferromagnetic Magnons

Ferromagnetic Domains Anisotropy Energy Transition Region between Domains Origin of Domains Coercivity and Hysteresis

Single Domain Particles Geomagnetism and Biomagnetism Magnetic Force Microscopy





Neutron Magnetic Scattering Curie Temperature and Susceptibility ofF errimagnets Iron Garnets

CHAPTER













Summary



Problems



MAGNETIC RESONANCE

Nuclear Magnetic Resonance Equations of Motion

Line Width Motional Narrowing

Hyperfine Splitting Examples: Paramagnetic Point Defects F Centers in Alkali Halides Donor Atoms in Silicon Knight Shift









Nuclear Quadrupole Resonance



Ferromagnetic Resonance



Shape Effects in FMR Spin Wave Resonance

Antiferromagnetic Resonance





liv

Electron Paramagnetic Resonance Exchange Narrowing Zero-field Splitting

Principle of Maser Action Three-Level Maser Lasers

CHAPTER







Problems



PLASMONS, POLARITONS, AND POLARONS

Definitions of the Dielectric Function Plasma Optics Dispersion Relation for Electromagnetic Waves Transverse Optical Modes in a Plasma Transparency of Metals in the Ultraviolet Longitudinal Plasma Oscillations



Plasmons



Electrostatic Screening



Screened Coulomb Potential Pseudopotential Component U(O) Mott Metal-Insulator Transition Screening and Phonons in Metals

Polaritons LST Relation





Summary

Dielectric Function of the Electron Gas

CHAPTER







Electron-Electron Interaction



Fermi Liquid Electron-Electron Collisions



Electron-Phonon Interaction: Polarons



Peierls Instability of Linear Metals



Summary



Problems



OPTICAL PROCESSES AND EXCITONS

Optical Reflectance Kramers-Kronig Relations Mathematical Note



Contents

Example: Conductivity of collisionless Electron Gas Electronic Interband Transitions

Excitons Frenkel Excitons Alkali Halides Molecular Crystals Weakly Bound (Mott-Wannier) Excitons Exciton Condensation into Electron-Hole Drops (EHD)

Raman Effects in Crystals Electron Spectroscopy with X-Rays

Energy Loss of Fast Particles in a Solid Summary Problems CHAPTER 16~ DIELECTRICS AND FERROELECTRICS

Maxwell Equations Polarization

Macroscopic Electric Field Depolarization Field, E 1

Local Electric Field at an Atom Lorentz Field, E 2 Field of Dipoles Inside Cavity, E 3

Dielectric Constant and Polarizability Electronic Polarizability Classical Theory of Electronic Polarizability

Structural Phase Transitions Ferroelectric Crystals Classification of Ferroelectric Crystals

Displacive Transitions Soft Optical Phonons Landau Theory of the Phase Transition Second-Order Transition First -Order Transition Antiferroelectricity Ferroelectric Domains Piezoelectricity

Summary Problems





























xvi

CHAPTER

SURFACE AND

INTERFACE PHYSICS

Reconstruction and Relaxation

Surface Crystallography Reflection High-Energy Electron Diffraction



Work Function Thermionic Emission Surface States Tangential Surface Transport



Integral Quantized Hall Effect (IQHE) IQHE in Real Systems Fractional Quantized Hall Effect (FQHE)

p-n Junctions Rectification Solar Cells and Photovoltaic Detectors Schottky Barrier

lleterostructures





n-N Heterojunction



Semiconductor Lasers



Light-Emitting Diodes Problems





Surface Electronic Structure

Magnetoresistance in a Two-Dimensional Channel

CHAPTER



NANOSTRUCTURES

Imaging Techniques for N anostructures Electron Microscopy Optical Microscopy Scanning Tunneling Microscopy Atomic Force Microscopy

Electronic Structure of lD Systems One-Dimensional Subbands Spectroscopy of Van Hove Singularities lD Metals - Coluomb Interactions and Lattice Copulings

Electrical Transport in lD Conductance Quantization and the Landauer Formula Two Barriers in Series-resonant Tunneling Incoherent Addition and Ohm's Law







C.:ontents

Localization Voltage Probes and the Buttiker-Landauer Formalism

Electronic Structure of OD Systems Quantized Energy Levels Semiconductor N anocrystals Metallic Dots Discrete Charge States

Electrical Transport in OD Coulomb Oscillations Spin, Mott Insulators, and the Kondo Effect Cooper Pairing in Superconducting Dots





Quantized Vibrational Modes Transverse Vibrations Heat Capacity and Thermal Transport



NONCRYSTALLINE SOLIDS Diffraction Pattern Monatomic Amorphous Materials Radial Distribution Function Structure ofVitreous Silica, Si0 2

Glasses Viscosity and the Hopping Rate

Amorphous Ferromagnets Amorphous Semiconductors Low Energy Excitations in Amorphous Solids Heat Capacity Calculation Thermal Conductivity

Fiber Optics Rayleigh Attenuation

Problems







Problems

CHAPTER



Vibrational and Thermal Properties of Nanostructures

Summary

CHAPTER















POINT DEFECTS



Lattice Vacancies Diffusion



Metals



XVII

x:viii

Color Centers



F Centers



Other Centers in Alkali Halides Problems

CHAPTER



DISLOCATIONS

Shear Strength of Single Crystals Slip

Dislocations Burgers Vectors Stress Fields of Dislocations Low-angle Grain Boundaries Dislocation Densities Dislocation Multiplication and Slip







Dislocations and Crystal Growth



Hardness of Materials Problems





Strength of Alloys Whiskers

CHAPTER



ALLOYS

General Considerations







Substitutional Solid SolutionsH ume-Rothery Rules



Order-Disorder Transformation



Elementary Theory of Order

Phase Diagrams Eutectics





Transition Metal Alloys



Electrical Conductivity



Problems



APPENDIX A:

TEMPERATURE DEPENDENCE OF THE REFLECTION LINES



APPENDIX B:

EWALD CALCULATION OF LATTICE SUMS



Ewald-Kornfeld Method for Lattice Sums for Dipole Arrays



Kondo Effect

Lontents

APPENDIXC:

QUANTIZATION OF ELASTIC WAVES: PHONONS Phonon Coordinates Creation and Annihilation Operators

APPENDIX D:

FERMI-DIRAC DISTRIBUTION FUNCTION

APPENDIX E:

DERIVATION OF THE

APPENDIX F:

BOLTZMANN TRANSPORT EQUATION

dkJdt EQUATION

Particle Diffusion Classical Distribution Fermi-Dirac Distribution Electrical Conductivity APPENDIX G:

VECTOR POTENTIAL, FIELD MOMENTUM, AND GAUGE TRANSFORMATIONS Lagrangian Equations of Motion Derivation of the Hamiltonian Field Momentum Gauge Transformation Gauge in the London Equation









APPENDIX H:

COOPER PAIRS



APPENDIX I:

GINZBURG-LANDAU EQUATION



APPENDIXJ:

ELECTRON-PHONON COLLISIONS



INDEX



1 Crystal Structure PERIODIC ARRAYS OF ATOMS Lattice translation vectors Basis and the crystal structure Primitive lattice cell

3 4 5

FUNDAMENTAL TYPES OF LATTICES Two-dimensional lattice types Three-dimensional lattice types

6 8 9

6

INDEX SYSTEM FOR CRYSTAL PLANES

11

SIMPLE CRYSTAL STRUCTURES Sodium chloride structure Cesium chloride structure Hexagonal close-packed structure Diamond structure Cubic zinc suHide structure

13 13 14 15 16 17

DIRECT IMAGING OF ATOMIC STRUCTURE

18

NONIDEAL CRYSTAL STRUCTURES Random stacking and polytypism

18 19

CRYSTAL STRUCTURE DATA

19

SUMMARY

22

PROBLEMS

22

I. 2. 3.

UNITS:

22 22

Tetrahedral angles Indices of planes Hcp structure

1A

1 angstrom = cm

22

= www.cronistalascolonias.com.ar

I0- 10 m.

(b)

(a)

(c)

Figure 1 Relation of the exter nal form of crystals to the form of the elementary huilding blocks . The building blocks are identical in (a) and (b). hut dilTerent crystal faces are developed. (c) Cleaving a crystal 0f rocks:alt.

CHAPTER

1: CRYSTAL

STRUCTURE

PERIODIC ARRAYS OF ATOMS

The serious study of solid state physics began "With the discovery of x-ray diffraction by crystals and the publication of a series of simple calculations of the properties of crystals and of electrons in crystals. Why crystalline solids rather than noncrystalline solids? The important electronic properties of solids are best expressed in crystals. Thus the properties of the most important semiconductors depend on the crystalline structure of the host) essentially because electrons have short wavelength components that respond dramatically to the regular periodic atomic order of the specimen. Noncrystalline materials, notably glasses, are important for optical propagation because light waves have a longer wavelength than electrons and see an average over the order, and not the less regular local order itself. We start the book with crystals. A crystal is formed by adding atoms in a constant environment, usually in a solution. Possibly the first crystal you ever saw was a natural quartz crystal grown in a slow geological process from a silicate solution in hot water under pressure. The crystal form develops as identical building blocks are added continuously. Figure 1 shows an idealized picture of the growth process, as imagined two centuries ago. The building blocks here are atoms or groups of atoms. The crystal thus formed is a three-dimensional periodic array of identical building blocks, apart from any imperlections and impurities that may accidentally be included or built into the structure. The original experimental evidence for the periodicity of the structure rests on the discovery by mineralogists that the index numbers that define the orientations of the faces of a crystal are exact integers. This evidence was supported by the discovery in of x-ray diffraction by crystals, when Laue developed the theory of x-ray diffraction by a periodic array, and his coworkers reported the first experimental observation of x-ray diffraction by crystals. The importance of x-rays for this task is that they are waves and have a wavelength comparable with the length of a building block of the structure. Such analysis can also be done "With neutron diffraction and with electron diffraction, but x-rays are usually the tool of choice. The diffraction work proved decisively that crystals are built of a periodic array of atoms or groups of atoms. With an established atomic model of a crystal, physicists could think much further, and the development of quantum theory was of great importance to the birth of solid state physics. Related studies have been extended to noncrystalline solids and to quantum fluids. The wider field is known as condensed matter physics and is one of the largest and most vigorous areas of physics. 3

4

Lattice Translation Ve.c tors An ideal crystal is constructed by the infinite repetiticm of identical groups of atoms (Fig. 2). A group is called the basis. The set of mathematic-al points to which tl11e basis is attached is called the lattice. The lattice in three dimensions may be defined by three translation vectors a" a 2 , a 3 , such that the arrangement of atoms in the crystal looks the same when viewed from the point r as when viewed from every point r 1 translated by an integral multiple of the a's:

(1) Here Ut, u2, u3 are arbitrary integers. The set of points e defined by (1) for all u 1 , u 2 ; u 3 defines the lattice. The lattice is said to be-primitive if any two points from which the atomic arrangement looks the same www.cronistalascolonias.com.ar satisfy (1) with a suitable choice of the integers u 1• This statement defines the primitive translation vectors ai. There is no cell of smaller volume than a 1 • a 2 X a 3 that can serve as a building block for the crystal structure. '\Ve often use the primitive translation vecto(s to define the crystal axes, which form three adjacent edges of the primitive parallelepiped. N onprimitive ax_e s are often used as crystal axes when they have a simple relation to the symmetry of the struchtre.

www.cronistalascolonias.com.ar 2 The www.cronistalascolonias.com.arure is formed by the addition of the basis (b) to every lattice point of the space lattice (a). By looking at (c), one can recognize the basis and then one can abstract the space lattice. It does oot matter whertl fhe basis is put _in relation to a lattice point.

1 Grystal Structure

Basis and ·th~ Crystal Structure The basis of the crystal structure can be identified once the crystal axes have been chosen. Figure 2 shows how a c1ystal is made by adding a basis to every lattice point-of course the lattice points ar.e just mathematical constructions. Every www.cronistalascolonias.com.ar in a given ctystal is· identical to every other in composition, arrangement, and orientation. The number of atoms in the basis may be one, or it may be more than one. The position of the center of an atom j of the basis relative to the associated lattice point is (2)'

We may arrange the origin, which we .have called the associated lattice point, so that 0 ; xj, yj , z1 ~ 1.

•

•

•

•

•

•

•

•

•

•

•

•

• •

•

•

•

•

•

(a)

(h)

(c)

Figw·c 3a Lattice points ofa space latti u 2 , UJ are integers and ai> a 2, a 3 are the crystal axes. • To form a crystal we attach to every lattice point an identical basis composed of s atoms at the positions~ xia1 + yia 2 + zia3 , withj = 1, 2, , s. Here x, y, z may be selected to have values between 0 and 1.

• The axes ab , a 3 are primitive for the minimum cell volume la1• a 2 X a 3 \ for which the crystal can be constructed from a lattice translation operator T and a basis at evecy lattice point.

Problems 1. Tetrahedral angles. The angles between the tetrahedral bonds of diamond are the same as the angles between the body diagonals of a cube, as in Fig. Use elementary vector analysis to find the value of the angle.

2. Indices of planes. Consider the planes with indices () and (); the lattice is fees and the indices refer to the conventional cubic cell. What are the indices of these planes when referred to the primitive axes of Fig. 11? 3. Bcp 8tructu~. Show that the cia ratio for an ideal hexagonal dose-packed structure is (~) = If cia is significantly larger than this value, the crystal structure may he thought of as composed of planes of closely packed atoms, the planes being

loosely stacked.

2 Wave Diffraction and the Reciprocal Lattice DIFFRACTION OF WAVES BY CRYSTALS The Bragg law

25 25

SCATTERED WAVE AMPLITUDE Fourier analysis Reciprocal lattice vectors Diffraction conditions Laue equations

26 27 29 30

33

BRILLOUIN ZONES Reciprocal lattice to sc lattice Reciprocal lattice to bee lattice Reciprocal lattice to fcc lattice

32 34 36 37

FOURIER ANALYSIS OF THE BASIS Structure factor of the hcc lattice Structure factor of the fcc lattice Atomic form factor

39 40 40 41

SUMMARY

43

PROBLEMS

43

1. 2. 3. 4. 5. 6.

7.

Interplanar separation Hexagonal space lattice Volume of Brillouin zone Width of diffraction maximum Structure factor of diamond Form factor of atomic hydrogen Diatomic Jine

43 44 44 44 45 45 45

10 5

'

'

"

~ r

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I

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1 ·r ""-X-ray photon



r"

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~ !Neutrons

-"'"\

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I'

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5

Figure l Wavelength versus particle energy, for photons, neutrons, and electrons.

10 Photon energy, keV Neutron energy, eV Electron energy, eV





Figure 2 Derivation of the Bragg equation 2d sin 8 = nA; here d is the spacing ofparallel atomic planes and 27m is the difference in phase between reflections from successive planes. The reflecting planes hav~ nothing to do with the surface planes b:otll1ding the particular ~pecimen . 24

CHAPTER 2: WAVE DIFFRACTION AND THE RECIPROCAL LATTICE

DIFFRACTION OF WAVES BY CRYSTALS

The Bragg law We study crystal structure through the diffraction of photons, neutrons, and electrons (Fig. 1). The diffraction depends on the crystal structure and on the wavelength. At optical wavelengths such as A, the superposition of the waves scattered elastically by the individual atoms of a crystal results in ordinary optical refraction. \Vhen the wavelength of the radiation is comparable with or smaller than the lattice constant, we may find diffracted beams in directions quite different from the incident direction. W. L. Bragg presented a simple explanation of the diffracted beams from a crystal. The Bragg derivation is simple but is convincing only because it reproduces the correct result. Suppose that the incident waves are reflected specularly from parallel planes of atoms in the crystal, with each plane reflecting only a very small fraction of the radiation, like a lightly silvered mirror. In specular (milTorHke) reflection the angle of incidence is equal to the angle of reflection. The diffracted beams are found when the reflections from parallel planes of atoms interfere constmctively, as in Fig. 2. We treat elastic scattering, in which the energy of the x-ray .is not changed on reflection. Consider parallel lattice planes spaced d apart. The radiation is incident in the plane of the paper. The path difference for rays reflected from adjacent planes is 2d sin 8, where 8 is measured from the plane. Constructive interference of the radiation from successive planes occurs when the path difference is an integral number n of wavelengths A, so that (1)

This is the Bragg law, which can be satisfied only for wavelength A :s; 2d. Although the reflection from each plane is specular, for only certain values of 8 will the reflections from all periodic parallel planes add up in phase to give a strong reflected beam. If each plane were perfectly reflecting, only the first plane of a parallel set would see the radiation, and any wavelength would be re-flected. But each plane reflects 3 to of the incident radiation, so that to planes may contribute to the formation of the Bragg-reflected beam in a perfect crystaL Reflection by a single plane of atoms is treated in Chapter 17 on surface physics. The Bragg law is a consequence of the periodicity of the lattice. Notice that the law does not refer to the composition of the basis of atoms associated 25

26

I c

·.g

Incident beam / from JH

or reactor

c::

"'-Main beam peak inteusity , c..p.m.

() reflection A =d.l6A

() A= A



::I 0

u

() reflection A= A.

I oo

wo

30° 20" Bragg angle f)



To crystal :;pecimen on rotating table

U ndeviated

cnmponf'nts of main beam Figure 3 Sketch of a monochrom 0, Cf, - C t > 0, and C,, + ZC,, > 0. For an example of the instability which results when C,, C,,, see L. R. Testardi et al., Phys. Rev. Letters 15, ().

-

Phonons I . Crystal Vibrations VIBRATIONS O F CRYSTALS WITH MONATOMIC BASIS 91 First Brillonin zone 93 Group velocity 94 Long wavelength limit 94 Derivation of force constants from experiment 94 TWO ATOMS PER PRIMITIVE BASIS QUANTIZATION O F ELASTIC WAVES PHONON MOMENTUM INELASTIC SCATTERING BY PHONONS SUMMARY PROBLEMS k

t

1. Monatomic linear lattice 2. Continuum wave equation 3. Basis of two unlike atoms 4. Kohn anomaly 5. Diatomic chain 6. Atomic vibrations in a metal 7. Soft phonon modes

Chapter 5 treats the thermal properties of phonons.

-+I+

-

Name

Field

Electron

-

Photon

Elechomagnetic -wave

Phonon

Elastic wave

Plasmon

Collective electron wave

Magnon

M ~ e t i z a t i o nwave

Puhn

Elechon + elastic deformation

Exciton

Polarization wave

Figure 1 Important elementary excitations in solids.

Figure 2 (Dashed lines) Planes of atoms when in equilibrium. (Solid lines) Planes of atoms when displaced as for a longitudinal wave. The coordinate u measures the displacement of the planes.

Figure 3 Planes of atoms as displaced during passage of a transverse wave.

CHAPTER

4: PHONONS I.

CRYSTAL VIBRATIONS

VIBRATIONS OF CRYSTALS WITH MONATOMIC BASIS

b

i i

k

/

I

1i !

1i 1

i

Consider the elastic vibrations of a crystal with one atom in the primitive cell. We want to find the frequency of an elastic wave in terms of the wavevector that describes the wave and in terms of the elastic constants. The mathematical solution is simplest in the [loo], [], and [ I l l ] propagation directions in cubic crystals. These are the directions of the cube edge, face diagonal, and body diagonal. When a wave propagates along one of these directions, entire planes of atoms move in phase with displacements either parallel or perpendicular to the direction of the wavevector. We can describe with a single coordinate u, the displacement of the planes from its equilibrium position. The problem is now one dimensional. For each wavevector there are three modes as solutions for us, one of longitudinal polarization (Fig. 2) and two of transverse polarization (Fig. 3). We assume that the elastic response of the crystal is a linear function of the forces. That is equivalent to the assumption that the elastic energy is a quadratic function of the relative displacement of any two points in the crystal. Terms in the energy that are linear in the displacements will vanish in equilibrium-see the minimum in Fig. Cubic and higher-order terms may be neglected for sufficiently small elastic deformations. We assume that the force on the planes caused by the displacement of the plane s + p is proportional to the difference us+,-us of their displacements. For brevity we consider only nearest-neighbor interactions, with p = The total force on s from planes s + 1:

F, = C(u,+, - us) + C(u,-,

- u,)

.

(1)

This expression is linear in the displacements and is of the form of Hooke's law. The constant C is the force constant between nearest-neighbor planes and will differ for longitudinal and transverse waves. It is convenient hereafter to regard C as defined for one atom of the plane, so that F, is the force on one atom in the planes. The equation of motion of an atom in the planes is

where M is the mass of an atom. We look for solutions with all displacements having the time dependence exp(-iot). Then dZu,ldt2= -ozua, and (2) becomes

u,,, = u

exp(isKa) exp(+ iKa) ,

(4)

where a is the spacing between planes and K is the wavevector. The value to use for a will depend on the direction of K. With (4), we have from (3): Mu exp(*iKa) = Cu{exp[i(s + l)Ka]+ exp[i(s - I)&] - 2 exp(isKa)J .

(5)

We cancel u exp(isKa) from both sides, to leave

With the identity 2 cos Ka relation w(K).

=

exp(iKa)

+ exp(-i&),

we have the dispersion

oz= (ZC/M)(l- cos Ka) .

(7)

The boundary of the first Brillouiu zone lies at K = + d a . We show from (7) that the slope of o versus K is zero at the zone boundary: do2/dK = (2CaIM) sin Ka = 0

(8)

at K = ?&a, for here sin Ka = sin (km) = 0. The special significance of phonon wavevectors that lie on the zone boundary is developed in (12) below. By a trigonometric identity, (7) may be written as

A plot of o versus K is given in Fig. 4.

Figure 4 Plot of o versus K . The region of K

*

l / n or A B o corresponds to the continuum approximation; here o is directly proportional to K.

4 Phonons I . Crystal Vibratim

First Brillouin Zone

What range of K is physically significant for elastic waves? Only those in the first Brillouin zone. From (4) the ratio of the displacements of two successive planes is given by

The range www.cronistalascolonias.com.ar to +.rr for the phase Ka covers all independent values of the exponential. The range of independent values of K is specified by

This range is the first Brillouin zone of the linear lattice, as defined in Chapter 2. The extreme values are G, = ? d a . Values of K outside of the first Brillouin zone (Fig. 5) merely reproduce lattice motions described by values within the limits ?ria. We may treat a value of K outside these limits by subtracting the integral multiple of www.cronistalascolonias.com.ar that will give a wavevector inside these limits. Suppose K lies outside the first zone, but a related wavevector K' defined K' = K - 2 m l a lies within the first zone, where n is an integer. Then the displacement ratio (10) becomes

because exp(i2m) = 1. Thus the displacement can always be described by a wavevector within the fust zone. We note that 2 m l a is a reciprocal lattice vector because 2 d a is a reciprocal lattice vector. Thus by subtraction of an appropriate reciprocal lattice vector from K, we always obtain an equivalent wavevector in the first zone. At the boundaries K,, = www.cronistalascolonias.com.ar of the Brillouin zone the solution u, = u exp(isKa) does not represent a traveling wave, but a standing wave. At the zone boundaries sK,,ua = ?ST, whence

i

Figure 5 The wave represented by the solid curve conveys no information not given by the dashed curve. Only wavelengths longer than 2n are needed to represent the ,notion.

93

neither to the right nor to the left. This situation is equivalent to Bragg reflection of x-rays: when the B condition is satisfied a traveling wave cannot propagate in a lattice, through successive reflections back and forth, a standing wave is set up. The critical value K,, = +m/a found here satisfies the Bragg cond 2d sin 0 = nA: we have 0 = $m, d = a , K = 2m/A, n = 1, so that A = 2a. W x-rays it is possible to haven equal to other integers besides unity because amplitude of the electromagnetic wave has a meaning in the space betw atoms, hut the displacement amplitude of an elastic wave usually has a m ing only at the atoms themselves. Group Velocity

The transmission velocity of a wave packet is the group velocity, give va = do/dK ,

or

the gradient of the frequency with respect to K. This is the velocity of en propagation in the medium. With the particular dispersion relation (9), the group velocity (Fig. 6) i vg = ( c ~ ~ / cos M )$ ~Ka

.

This is zero at the edge of the zone where K = r/a. Here the wave is a stan wave, as in (12),and we expect zero net transmission velocity for a standing w Long Wavelength Limit When Ka < 1 we expand cos Ka tion (7) becomes

-

I - ;(Ka)', so that the dispersion r

w2 = (C/M)@a2

.

The result that the frequency is directly proportional to the wavevector in long wavelength limit is equivalent to the statement that the velocity of so is independent of frequency in this limit. Thus v = o l K , exactly as in the tinuum theory of elastic waves-in the continuum limit Ka < 1. Derivation of Force Constants from Experiment

In metals the effective forces may be of quite long range and are car from ion to ion through the conduction electron sea. Interactions have b found between planes of atoms separated by as many as 20 planes. We can m a statement about the range of the forces from the observed experime

4 Phomm I. Crystal Vibration8

95

Figure 6 Group velocity u, versus K for model of Fig. 4. At the zone boundary K = wla the group velocity is zero.

dispersion relation for w . The generalization of the dispersion relation (7) t o p nearest planes is easily found to be w2 = (21M)

c. Cp(l

- cos pKa)

.

(16a)

p>n

We solve for the interplanar force constants C, by multiplying both sides by cos rKa, where r is an integer, and integrating over the range of independent values of K : via

MIIT'" dK w: cos rKa

= 2~

C p L dK (1 - cos pKa) ms rKa

P>O

WIO

The integral vanishes except for p

=

li/o

r. Thus ,./a

dK w$ cos

CP = ,./a

gives the force constant at range pa, for a structure with a monatomic basis TWO ATOMS PER PRIMITIVE BASIS

The phonon dispersion relation shows new features in crystals with two or more atoms per primitive basis. Consider, for example, the NaCl or diamond structures, with two atoms in the primitive cell. For each polarization mode in a given propagation direction the dispersion relation w versus K develops two branches, known as the acoustical and optical branches, as in Fig. 7. We have longitudinal LA and transverse acoustical TA modes, and longitudinal LO and transverse optical TO modes. If there are p atoms in the primitive cell, there are 3p branches to the dispersion relation: 3 acoustical branches and 3p - 3 optical branches. Thus germanium (Fig. 8a) and KBr (Fig. Sh), each with two atoms in a primitive cell, have six branches: one LA, one LO, two TA, and two TO.

pbonon branch

Figure 7 Optical and acoustical branches of the dispersion relation far a diatomic linear lattice, showing the limiting frequencies at K = 0 and K = K., = v t a . The lattice constant is a.

?r a

K

0 Kt&,,

in [ I l l ] direction

Figure 8a Pbonon dispersion relations in the I direction in germanium at 80 K. The huo TA phonon branches are horizontal at the zone boundary position, &, = (2/a)(+$+).The LO and TO branches coincide at K = 0;this also is a consequence of the crystal symmetry of Ge. The results were obtained with neutron inelastic scattering by G. Nilsson and G. Nelin.

Kt&,,

in I d i d o n

Figure 8b Dispersion curves in the [ I l l ] direction in KBr at 90 K, after A. D. B. Woods, B. N. Bmckhouse, R. A. Cowley, and W. Cochran. The extrapolation to K = 0 of the TO, LO branches are called mr,mL.

The numerology of the branches follows from the number of degrees of freedom of the atoms. With p atoms in the primitive cell and N primitive cells, there are pN atoms. Each atom has three degrees of freedom, one for each of the x , y, z directions, mahng a total of 3pN degrees of freedom for the crystal. The number of allowed K values in a single branch is just N for one Brillouin zone.' Thus the 'We show in Chapter 5 by application of periodic b o u n d q conditions to the modes of the crystal of volume V that there is one K value in the volume (2w)VVin Fourier space. The volume of a Brillouin zone is (Zn)'N, where V. is the volume of a crystal primitive cell. Thus the number of allowed Kvalues in a Brillouin zone is VN., which is just N, the number ofprimitive cells in the crystal.

4 Phonona I. Crystal Vibrations

Figure 9 A diatomic ctystal structure with masses M,, Mz connected by force constant C between adjacent planes. The displacements of atoms M Iare denoted by u,-,, u,, u,,,, . . . , and of atoms M, by 0,-,, v., v,,,. The repeat &stance is a in the direction o f the wavevector K . The atoms are shown in their undisplaced positions.

LA and the two TA branches have a total of 3N modes, thereby accounting for 3N of the total degrees of freedom. The remaining (3p - 3)N degrees of freedom are accommodated by the optical branches. We consider a cubic clystal where atoms of mass MI lie on one set of planes and atoms of mass M, lie on planes interleaved between those of the first set (Fig. 9). It is not essential that the masses be different, but either the force constants or the masses will be different if the two atoms of the basis are in nonequivalent sites. Let a denote the repeat distance of the lattice in the direction normal to the lattice planes considered. We treat waves that propagate in a symmetry direction such that a single plane contains only a single type of ion; such directions are [ I l l ]in the NaCl structure and [loo] in the CsCl structure. We write the equations of motion under the assumption that each plane interacts only with its nearest-neighbor planes and that the force constants are identical between all pairs of nearest-neighbor planes. We refer to Fig. 9 to obtain

We look for a solution in the form of a traveling wave, now with different amplitudes u, u on alternate planes:

We define n in Fig. 9 as the distance between nearest identical planes, not nearest-neighbor planes. On substitution of ( 1 9 )in (18) we have

97

the coefficients of the unknowns u, o vanishes:

or M,M 2C(M1 + M2)02+ 2C2(1- cos Ka) = 0

.

(22)

We can solve this equation exactly for w2, but it is simpler to examine the limiting cases Xn < 1 and Ka = +TI at the zone boundary. For small Ka we have cos Ka E 1- K2a2 . . . , and the two roots are

+

(optical branch)

;c K2a2 -

02=

MI + MZ

;

(acoustical branch)

The extent of the first Brillouin zone is -v/a 5 K 5 d a , where a is the repeat distance of the lattice. At K, = ?r/a the roots are

The dependence of o on K is shown in Fig. 7 for M, > M2. The particle displacements in the transverse acoustical (TA) and transverse optical (TO) branches are shown in Fig. For the optical branch at K = 0 we find, on substitution of (23) in (,

The atoms vibrate against each other, hut their center of Inass is fured. If the two atoms cany opposite charges, as in Fig. 10, we may excite a motion of this

Figure 10 Transverse optical and transverse amustical waves in a diatomic linear lattice, illustrated by the particle dqlacements far the two modes at the same wavelength.

Acoustical mode

4 Phonons I. Crystal Vibrations

type with the electric field of a light wave, so that the branch is called the optical branch. At a general K the ratio ulu will be complex, as follows from either of the equations (20). Another solution for the amplitude ratio at small K is u = u, obtained as the K = 0 limit of (24). The atoms (and their center of mass) move together, as in long wavelength acoustical vibrations, whence the term acoustical branch. Wavelike solutions do not exist for certain frequencies, here between (2C/M,)'" and (2C/M,)'". This is a characteristic feature of elastic waves in polyatomic lattices. There is a frequency gap at the boundary K,, = ? ~ / aof the first Brillouin zone. QUANTIZATON OF ELASTIC WAVES

ij ;

L

The energy of a lattice vibration is quantized. The quantum of energy is called a phonon in analogy with the photon of the electromagnetic wave. The energy of an elastic mode of angular frequency o is

when the mode is excited to quantum number n; that is, when the mode is occupied by n phonons. The term $ fiw is the zero point energy of the mode. It occurs for both phonons and photons as a consequence of their equivalence to a quantum harmonic oscillator of frequency w, for which the energy eigenvalues are The quantum theory of phonons is developed in Appendix C. also (n + i)fi~. We can quantize the mean square phonon amplitude. Consider the standing wave mode of amplitude

Here u is the displacement of a volume element from its equilibrium position at x in the crystal. The energy in the mode, as in any harmonic oscillator, is half kinetic energy and half potential energy, when averaged over time. The kinetic energy density is 2 p(&lat)2, where p is the mass density. In a crystal of volume V, the volume integral of the kinetic energy is ipVo2u; sin2&. The time average kinetic energy is

because = i. The square of the amplitude of the mode is

This relates the displacement in a given mode to the phonon occupancy n of the mode. What is the sign of o ? The equations of motion such as (2) are equations for oZ,and if this is positive then w can have either sign, + or -. But the

99

energy of a phonon must be positive, so it is conventional and suitable to vie o as positive. If the crystal structure is unstable, then o2will be negative and will be imaginary. PHONON MOMENTUM

A phonon of wavevector K will interact with particles such as photon neutrons, and electrons as if it had a momentum hK. However, a phonon do not carry physical momentum. The reason that phonons on a lattice do not carry momentum is that phonon coordinate (except for K = 0) involves relative coordinates of th atoms. Thus in an Hz molecule the internuclear vibrational coordinate rl is a relative coordinate and does not carry linear momentum; the center mass coordinate $(rl+ r2)corresponds to the uniform mode K = 0 and c carry linear momentum. In crystals there exist wavevector selection rules for allowed transitio between quantum states. We saw in Chapter 2 that the elastic scattering of x-ray photon by a crystal is governed by the wavevector selection rule

where G is a vector in the reciprocal lattice, k is the wavevector of the incide photon, and k' is the wavevector of the scattered photon. In the reflectio process the crystal as a whole will recoil with momentum -hG, but this un form mode momentum is rarely considered explicitly. Equation (30) is an example of the rule that the total wavevector of inte acting waves is conserved in a periodic lattice, with the possible addition of reciprocal lattice vector G . The true momentum of the whole system always rigorously conserved. If the scattering of the photon is inelastic, with th creation of a phonon of wavevector K, then the wavevector selection ru becomes

If a phonon K is absorbed in the process, we have instead the relation

Relations (31) and (32) are the natural extensions of (30). INELASTIC SCAWERING BY PHONONS

Phonon dispersion relations o(K) are most often determined experime tally by the inelastic scattering of neutrons with the emission or absorption of phonon. A neutron sees the crystal lattice chiefly by interaction with the nucl

4 Phonons I . Crystal Vibrations

of the atoms. The kinematics lattice are described by the gc

tttering of a neutron beam by a crystal evector selection m1

; the wavevector of and by the requirement of conservation of energy the phonon created (+) or absorbed ( - ) in the process, and G is (onwe choose G such that K lies in the any reciprocal lattice vector. 1 first Brillouin zone.

Wavevector, in units Snla

Figure 1 1 The dispersion curves of sodium far ~ h o n o n spropagating in the [], [], and [ I l l ] directions at 90 K, as determined hy inelastic scattering of neutrons, by Woods, Brockhouse, March and Bowers.

i

Figure 12 .4 triple ads neutron spectrometer at Bruoklravm. (Coxirtesy oCB. If. Grier.)



of the neutron. The momentum p is given by hk, where k is the wavevecto the neutron. Thus h2k2/2M,is the kinetic energy of the incident neutron. I is the wavevector of the scattered neutron, the energy of the scattered neut is fi2k'2/2M,.The statement of conservation of energy is

where h o is the energy of the phonon created (+) or absorbed (-) in process. To determine the dispersion relation using (33) and (34) it is necessar the experiment to find the energy gain or loss of the scattered neutrons function of the scattering direction k - k'. Results for germanium and KBr given in Fig. 8; results for sodium are given in Fig. A spectrometer used phonon studies is shown in Fig. SUMMARY

The quantum unit of a crystal vibration is a phonon. If the angular quency is o, the energy of the phonon is fio.

When a phonon of wavevector K is created by the inelastic scattering o photon or neutron from wavevector k to k', the wavevector selection rule t governs the process is k=kl+K+G, where G is a reciprocal lattice vector.

All elastic waves can be described by wavevectors that lie within the f Brillouin zone in reciprocal space.

If there are p atoms in the primitive cell, the phonon dispersion relation have 3 acoustical phonon branches and 3p - 3 optical phonon branches.

Problems 1. Monatomic linear lattice.

Consider a longitudinal wave u, = u cos(mt - sKa)

which propagates in a monatomic linear lattice of atoms of mass M, spacing a, nearest-neighborinteraction C. (a) Show that the total energy of the wave is

where s runs over all atoms

(h) By substitution of u, in this expression, show that the time-average total energy per atom is

where in the last step we have used the dispersion relation (9) for this problem 2. Continuum wave equation. Show that for long wavelengths the equation of motion (2) reduces to the continuum elastic wave equation

where o is the velocity of sound 3. Basis oftwo unlike a t o m . For the problem treated by (18) to (26), find the am~ l i t u d eratios ulv for the two branches at &, = ria. Show that at this value of K the two lattices act as if decoupled: one lattice remains at rest while the other lattice moves. 4. Kohn anomaly. We suppose that the interplanar force constant C, between planes s and s + p is of the form C, =A-

sin pk,a

Pa

where A and k, are constants and p runs over all integers. Such a form is expected in metals. Use this and Eq. (16a) to find an expression for 0% and also for do2/JK. Prove that JwZ/aKis infinite when K = k,. Thus a plot of wZversus K or of o versus K has a vertical tangent at k,: there is a kink at k, in the phonon dispersion relation o(K).

5. Diatomic chain. Consider the normal modes of a linear chain in which the force constants between nearest-neighbor atoms are alternately C and 10C. Let the masses he equal, and let the nearest-neighbor separation be aI2. Find o(K) at K = 0 and K = &a. Sketch in the dispersion relation by eye. This problem simulates a crystal of diatomic molecriles such as H,. 6 . Atomic vibrations in a metal. Consider point ions of mass M and charge e im-

mersed in a uniform sea of conduction electrons. The ions are imagined to be in stable equilibrium when at regular lattice points. If one ion is displaced a small distance r from its equilibrium position, the restoring force is largely due to the electnc charge within the sphere of radius r centered at the equilibrium position. Take the number density of ions (or of conduction electrons) as 3/4?rR3, which defines R. (a) Show that the frequency of a single ion set into oscillation is o = (e2/MR3)1'e. (b) Estimate the value of this frequency for sodium, roughly. (c) From (a), (b), and some common sense, estimate the order of magnitude of the velocity of sound in the metal.

'7. Soft phonon modes. Consider a Line of ions of equal mass but alternating in charge, with e,

=

e(-

1)P

'This problem is rather difficult.

as the charge on the pth ion. The interatomic potential is

Phonons II. Thermal Properties PHONON HEAT CAPACITY Planck distribution Normal mode enumeration Density of states in one dimension Density of states in three dimensions Debye model for density of states Debye T3 law Einstein model of the density of states General result for D ( o ) ANHARMONIC CRYSTAL INTERACTIONS Thermal expansion THERMAL CONDUCTMTY Thermal resistivity of phonon gas Umklapp processes Imperfections PROBLEMS 1. Singularity in density of states 2. Rms thermal dilation of crystal cell 3. Zero point lattice displacement and strain 4. Heat capacity of layer lattice

5. Griineisen constant

Figure 1 Plot of Planck distribution function. At high temperatures the occupancy of a state

approximately linear in the temperature. The function (n) + b, which is not plotted, approach the dashed line as asymptote at high temperatures.



We discuss the heat capacity of a phonon gas and then the effects of anharmonic lattice interactions on the phonons and on the crystal.

PHONON HEAT CAPACITY

By heat capacity we shall usually mean the heat capacity at constant volume, which is more fundamental than the heat capacity at constant pressure, which is what the experiments determine.' The heat capacity at constant volume is defined as Cv = (dU/dT), where U is the energy and T the temperature. The contribution of the phonons to the heat capacity of a crystal is called the lattice heat capacity and is denoted by C,.,. The total energy of the phonons at a temperature T(= k,T) in a crystal may he written as the sum of the energies over all phonon modes, here indexed by the wavevector K and polarization index p: Ui, =

2K 2v U,,

=

z zcn,NJJio,

2

K P

1 (n) = errp(ho/.r) - l '

(1)

(2)

where the () denotes the average in thermal equilibrium. A graph of (n) is

Planck Distribution Consider a set of identical harmonic oscillators in thermal equilibrium. The ratio of the number of oscillators in their (n + 1)th quantum state of excitation to the number in the nth quantum state is

N,,+,IN.

= exp(-fio/~) ,

7=

kBT ,

(3)

'A thermodynamic relation gives Cp - C, = BVT, where a is the temperature coefficient of linear expansion, V the volume, and B the bulk modulus. The fractional difference between C, and C, is usually small in solids and often may be neglected. As T- 0 we see that C,+Cv, provided a and B are constant.

We see that the average excitation quantum number of an oscillator i z s exp-shw/~) (n) =

'

z eT(-sfiolr)

The summations in (5)are

with x = exp(-ftwl~). Thus we may rewrite (5) as the Plauck distribution: 1 (n) = -x-1 - x exp(fw/7) - 1 Nomal Mode Enumeration

The energy of a collection of oscillators of frequencies on;, in th equilibrium is found from (1) and (2):

It is usually convenient to replace the summation over K by an integral pose that the crystal has DP(o)domodes of a given polarization p in th quency range o to o + d o . Then the energy is

The lattice heat capacity is found by differentiation with respect to tem ture. Let x = h o / ~= ho/kBT:then 8U/aT gives x2 exp x ~ ~ = k p, ~ I d o ~ (expx , , ( o-)

'

The central prohlem is to find D(w), the number of modes per un quency range. This function is called the density of modes or, more often sity of states. Density of States in One Dimension

Consider the boundary value prohlem for vibrations of a one-dimen line (Fig. 2) of length L carrying N 1 particles at separation a. We su

+

5 Phonons Thermal Properties

Figure 2 Elastic line of N + I atoms, with N = 10, fur boundary conditions that the end atoms s = 0 and s = 10 are k c d . The particle displacements in the normal modes for eitl~crlongitu&~d or transverse displacrme~~ts are of the form u, sin sKa. This form is antomatically zero at the atom at the ends = 0 , and we choose K to make the displacement zero at the e n d s = 10

Figure 3 Thc boundary condition sin sKa = O for s = 10 can be satisfied by choosing K = .rr/lOa, ZdIOa, . . ., www.cronistalascolonias.com.ar,where 10a is the length L of the line. The present figure is in K space. The dots are not atoms but are the allowed valucs of K. Of the N + 1 particles on the line, only N - 1 are allowcd to move, and their most general motion car1 be expressed in terms of the N - 1 allowed vali~esof K . This quantization of K has nothing to do with qnantnm mechanics but follows classically from the boondaryconditions that tlre cnd atoms be fixed.

s = 0 and s = N at the ends of the line are held fixed. Each that the norrrlal vibrational modc of polarization p has the form of a standing wave, where u , is the displacement of the particle s: v, = 4 0 ) exp-io,,+,t)

sin sKtl ,

(11)

wtiere wKl, is related to K by the appropriate dispersion relation. As in Fig. 3 , thc wavevector K is restricted by the fixed-end boundary conditions to the values

The solution for K = n/L has

u, a sin (www.cronistalascolonias.com.ar)

(13)

and vanishes for s = 0 and s = N as required. The solution for K = NT/L = d a = K,,,, has u, sin ST; this permits no otioll of any atom, because sinsz- vanishes at each atom. Thus there are N - 1 allowed independent values of K in (12). This number is equal to the number of particles allowed to Inove. Each allowed value of K is associated with a sta~ldi~ig wave. For the one-dimensional line there is one mode for each iriterval AK = T/L, so that the number of modes per unit range of K is LIT for K 5 d a , and 0 for K > rrla. Therc are three polarizations p for each value of K: in one dimension two of these are transverse and one longitudinal. In three dimensions the polarizations are this simple only for wavevectors in ccrtain special crystal directions. Another device for enumerating modes is equally valid. We consider the medium as unbounded, hut require that the solutions be periodic over a large



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=

5 Phonons Themol Properties

Figure 6 Allowed values in Fourier space of the phonon wavevector K for a square lattice of lattice constant a, with periodic boundary conditions applied over a square of side L = 10o. The uniform mode is marked with a cross. There is one allowrd value of K per area (/l&~)~ = (ZwIL)', so that within the circle of area 7iKi the smoothed number of allowed paints is ITK'(L/ZW)'.

We can obtain the group velocity doldK from the dispersion relation o versus K. There is a singularity in Dl(o)whenever the dispersion relation w(K) is horizontal; that is, whenever the group velocity is zero. Density of States in Three Dimensions We apply periodic boundary conditions over N3 primitive cells within a cube of side L, so that K is determined by the condition

whence

in K space, or Therefore, there is one allowed value of K per volume (25~lL)~

allowed values of K per unlt volume of K space, for each polarization and for each branch. The volume of the specimen is V = L3. The total number of modes with wavevector less than K is found from (18) to he (L~)~ times the volume of a sphere of radius K. Thus N = (L/25~)~(4?ik"/3)

(19)



D ( W ) = d ~ / =d( v~I C / 2 d ) ( d ~ l d w. )

(2

Debye Model for Density of States

In the Debye approximation the velocity of sound is taken as constant fo each polarization type, as it would be for a classical elastic continuum. The dis persion relation is written as w=uK ,

(2

with v the constant velocity of sound. The density of states ( 2 0 )becomes

If there are N primitive cells in the specimen, the total number of acousti phonon modes is N. A cutoff frequency oDis determined by (19)as

To this frequency there corresponds a cutoff wavevector in K space:

On the Debye model we do not allow modes of wavevector larger than K,. Th number of modes with K 5 K, exhausts the number of degrees of freedom of monatomic lattice. The thermal energy ( 9 )is given by

for each polarization type. For brevity we assume that the phonon velocity independent of the polarization, so that we multiply by the factor 3 to obtain

where x = h o / r --= www.cronistalascolonias.com.ar,T and xD = hwulk,T = BIT .

(2

This defines the Debye temperature 0 in terms of w, defined by (23 We may express 0 as

5 Phomn8 Thermal Properties



Figure 7 Heat capacity C, of a solid, according to the Debye www.cronistalascolonias.com.ar vertical scale is in J mol-' K-I. The holizuntal scale is the temperature normalized to the Debye temperature 0 . The region of the T3 law is below The asymptotic value at high values of TI0 is J mol-' deg-'.

Temperature, K

Figure 8 Heat capacity of silicon and germanium. Note the decrcase at low temperatures. To convert a value in caVmol-K to Jlmol-K, multiply by

so that the total phonon energy is

where N is the number of atoms in the specimen and XD = BIT. The heat capacity is found most easily by differentiating the middle expression of (26) with respect to temperature. Then

The Debye heat capacity is plotted in Fig. 7. At T P 0 the heat capacity approaches the classical value of 3Nkn. Measured values for silicon and germanium are plotted in Fig. 8.

limit go to infinity. We have

where the sum over s-4 is found in standard tables. Thus U T G 8, and

-

37r4Nk,P/ f

which is the Dehye T3 approximation. Experimental results for argon are plo ted in Fig. 9. At sufficiently low temperature the T3 approximation is quite good; that when only long wavelength acoustic modes are thermally excited. These are ju the modes that may be treated as an elastic continuum with macroscopic elas constants. The energy of the short wavelength modes (for which this approxim tion fails) is too high for them to he populated significantly at low temperature We understand the T3 result by a simple argument (Fig. 10). Only tho lattice modes having h o < kBTwill be excited to any appreciable extent a low temperature T. The excitation of these modes will he approximately clas cal, each with an energy close to k,T, according to Fig. 1. Of the allowed volume in K space, the fraction occupied by the excit where KTis a "thermal" waveve modes is of the order of do^)^ or (KT/KD)3, tor defined such that hvK, = k,T and K , is the Debye cutoff wavevector. Th the fraction occupied is (T/O)3of the total volume in K space. There are of t order of 3N(T/8)3excited modes, each having energy kBT. The energy -3Nk,T(T/O)3, and the heat capacity is NkB(T/O)3. For actual crystals the temperatures at which the T3 approximation hol are quite low. It may be necessary to be below T = 8/50 to get reasonably pu T3 behavior. Selected values of 8 are given in Table 1. Note, for example, in the alk metals that the heavier atoms have the lowest 8>, because the velocity sound decreases as the density increases. Einstein Model of the Density of States

Consider N oscillators of the same frequency o, and in one dimensio The Einstein density of states is D(o) = N6(o - w,), where the delta functi is centered at owThe thermal energy of the system is Nho U = N(n)ho = e""/' , with o now written in place of o,, for convenience

Figure 9 Low temperature heat capacity of solid argon, plotted against T3. In this temperature region the experimental results are in excellent agreement with the Debye T3law with B = K. (Conrtesy of L. Finegold and N. E. Phillips.)

Figure 10 To obtain a qualitative explanation of the Debye T3law, we suppose that all phonon modes of wavevector less than K , have the classical thermal energy k,T and that modes between K, and the Debye cutoff K, are not excited at all. Of the 3N possible modes, the fraction excited is (KdKDJ1 = (T/O)3,because this is the ratio of the volume of the inner sphere to the outer sphere. Tne e n e r a i s U k,T . 3N([email&#;protected], and the heat capacity is C, = JU/aT= 12NkB(T/B)3.

-

5 Phonons T h e m 1 Properties

0

no,

values of the heat capacity of diamond with values calcuFigure 11 Comparison of hted on the earliest quantum (Einstein) model, using the characteristic temperature & = W k , = K. To convert to Jlmol-deg, multiply by

The heat capacity of the oscillators is

(;gv

Cv - - =Nk,

f:y(e6iy -

,

(34)

as plotted in Fig. This expresses the Einstein () result for the contribution of N identical oscillators to the heat capacity of a solid. In three dimensions N is replaced by 3N, there being three modes per oscillator. The high temperature limit of Cv becomes 3Nk8, which is known as the Dnlong and Petit value. At low temperatures (34) decreases as exp(-fiw/~), whereas the experimental form of the phonon contribution is known to he T3as accounted for by the Debye model treated above. The Einstein model, however, is often used to approximate the optical phonon part of the phonon spectrum. General Result for D(m) We want to find a general expression for D(w), the number of states per unit frequency range, given the phonon dispersion relation o(K). The number of dlowed values of K for which the phonon frequency is between o and w + dw is Mw) dw =

($

Ishe". BK

(35)

where the integral is extended over the volume of the shell in K space hounded by the two surfaces on which the phonon frequency is constant, one surface on which the frequency is w and the other on which the frequency is o + dw. The real problem is to evaluate the volume of this shell. We let dS, denote an element of area (Fig. 12) on the surface in K space of the selected constant

Figure 12 Element of area d S , on a constant frequency surface in K space. The volume between -two surfaces of constant frequency at wand w + dw is equal to J dS,do/lV,wl.

frequency w. The element of volume between the constant frequency surfaces w and w + dw is a right cylinder of base dS, and altitude dK,, SO that

J

shell

=

J ~ S J K. ~

Here dKL is the perpendicular distance (Fig. 13) between the surface w constant and the surface w + dw constant. The value of dK, will vary from one point to another on the surface. The gradient of w, which is VKw,is also normal to the surface w constant, and the quantity

is the difference in frequency between the two surfaces connected by dKk Thus the element of the volume is

where vg = lVKwl is the magnitude of the group velocity of a phonon. For (35) we have

We divide both sides by dw and write V = L3 for the volume of the crystal: the result for the density of states is

5 Phonons Thelma1 Properties Surface o + dw = constant

Figure 13 Tlre quantity dK, is the perpendicular distance between two constant frequency surfaccs in K space, one at frequency o and the other at frequency o + dw.

(a)

(b)

Figure 14 Density of states as a function of frequency for (a) the Debye solid and (b) an actual crystal structure. The specbum for the crystal starts as o2for small o,but discontinuities develop at singular points.

Thc integral is taken over the area of the surface o constant, in K space. The result refers to a single branch of the dispersion relation. We can use this result also in electro~lband theory. There is a special interest in the contribution to D(w) frorn points at which the group velocity is zero. Such critical points produce singularitics (known as Van Hove singnlarities) in the distribution function (Fig. 14). ANHARMONIC CRYSTAL INTERACTIONS

The theory of lattice vibrations disciissed thus far has been limited in the potential energy to terms quadratic in the interatomic displacements. This is the harmonic theory; among its consequences are: Two lattice waves do not interact; a single wave docs not decay or change form with time. There is no thermal expansion. Adiabatic and isothermal elastic constants are equal. The elastic constants are independent of pressure and temperature. The heat capacity becomes constant at high temperatures T > 8.

tions may bc attributed to the neglect of anharmonic (higher than quadratic terms in the interatomic displacements. We discuss some of the simpler as pects of anharnionic effects. Beautiful demonstrations of anharmonic effects are the experiments o thc interaction of two pllonons to poduce a third phonon at a frequenc w3 = wl + 0~ Three-phonon processes are caused by third-order terms in th lattice potential energy. The physics of the phonon interaction can be state simply: the presence of one phonon canses a periodic elastic strain whid (through the anharmonic interaction) modulates in space and time the elasti constant of the crystal. A second phonon perceives the modulation of the elas tic constant and thereupon is scattered to produce a third phonon, just as from a moving three-dimensional grating.

Thermal Expansion

We may understand thermal expansion by considering for a classical osci lator the ellect of anharmonic terms in the potential energy on the mean scpa ration of a pair of atoms at a temperature T . We take the potential energy of th atoms at a displacement x from their equilibrium separation at absolute zero as

with c, g, andf all positive. The term in x3 represents the asymmetry of th mutual repulsion of the atoms and the term in x4 represents the softening of th vibration at large amplitudes. The ~ninimumat x = 0 is not an absolute mini mum, hut for small oscillations the form is an adequate representation of an in teratomic potential. We calculate the average displacement by using the Boltzmann distribu tion function, which weights the possible values of x according to thei thermodynamic probability

with p = l/k,T. For displacements such that the anharmonic terms in th energy are small in comparison with k,T, we may expand the integrands as

whence the thermal expansion is

3tz (x) = -kRT 4cZ

5 Phonons ZI. T h s m l Properties



Figure 15 Lattice constant of solid argon as a

Temperature, in K

funaion of temperature.

in the classical region. Note that in (38) we have left a2in the exponential, but we have expanded exp(pgx3+ pfi4)s 1 pgx3 pfi4 . . .. Measurements of the lattice constant of solid argon are shown in Fig. The slope of the curve is proportional to the thermal expansion coefficient. The expansion coefficient vanishes as T+ 0, as we expect from Problem 5. In lowest order the thermal expansion does not involve the symmetric termfi4 in U ( x ) ,but only the antisymmetric term gx3.

+

+

+

THERMAL CONDUCTMTY

The thermal conductivity coefficient K of a solid is defined with respect to the steady-state flow of heat down a long rod with a temperature gradient dT/&:

where jL, is the flux of thermal energy, or the energy transmitted across unit area per unit time. This form implies that the process of thermal energy transfer is a random process. The energy does not simply enter one end of the specimen and proceed directly (hallistically) in a straight path to the other end, but diffuses through the specimen, suffering frequent collisions. If the energy were propagated directly through the specimen without deflection, then the expression for the thermal flux would not depend on the temperature gradient, but only on the difference in temperaturc AT between the ends of the specimen, regardless of the Tength of the specimen. The random nature of the conductivity process brings the temperature qadient and, as we shall see, a mean free path into the expression for the thermal flux.

The e's obtained in this way refer to umklapp processes.]

'Parallel to optic axis.

From the kinetic theory of gases we find below thc tbllowing expression for the thermal conductivity: K

=

;cue ,

(42)

where C is the heat capacity per unit volu~ne,v is the average particle velocity, and Z is the mean free path of a prticle between collisions. This result was applied first by Debye to describe thermal conductivity in dielectric solids, with C as the heat capacity of the phonons, o the phonon velocity, and e the phonon mean free path. Several representative values of the mean free path are given in Table 2. We give the elementary kinetic theory which leads to (42). The flux of particles in the x direction is in(lozl),where n is the concentration of molec~iles in equilibrium there is a flux of equal magnitnde in the opposite direction. The (. . .) denote average valuc. If c is thc heat capacity of a particle, then in moving frurn a region at local temperatine T + AT to a region at local temperature 2' a particle will give up energy c AT. Now AT between the ends of a free path of the particle is given hy

where T is the average time between collisions. The net flnx of energy (from both senses of the particle flux) is therefore j,,

=

dT

-n(d)cr

dx

1

dT

= -zn(v2)c~-

dx

If, as for phonons, u is constant, we may write (43) as u --

with e = OT and C = nc. Thns K

-k~ dle-;

= $cut.

dx

.

5 Phononn ZI. Thermal Properties

Thermal Resistivity of Phonon Gas The phonon mean free path t! is determined principally by two processes, geometrical scattering and scattering by other phonons. If the forces between atoms were purely harmonic, there would be no mechanism for collisions between different phonons, and the mean free path wolild be limited solely by collisions of a phonon with the crystal boundary, and by lattice imperfections. There are situations where these effects are dominant. With anharmonic lattice interactions, there is a coupling between different phonons which limits the value of the mean free path. The exact states of the anharmonic system are no longer like pure phonons. The theory of the effect of anharmonic coupling on thermal resistivity predicts that C is proportional to l/T at high temperatures, in agreement with many experiments. We can understand this dependence in terms of the nnmber of phonons with which a given phonon can interact: at high temperature the total number of excited phonons is proportional to T. The collision frequency of a given phonon should be proportional to the number of phonons with which it can collide, whence e 1/T. To define a thermal conductivity there must exist mechanisms in the crystal whereby the distribution of phonons may be brought locally into thermal equilibrium. Without such mechanisms we may not speak of the phonons at one end of the crystal as being in thermal equilibrium at a temperature T, and those at the other end in equilibrium at T , . It is not sufficient to have only a way of limiting the mean free path, but there must also be a way of establishing a local thermal equilibrium distribution of phonons. Phonon collisions with a static imperfection or a crystal boundary will not by themselves establish thermal equilibrium, because such collisions do not change the energy of individual phonons: the frequency o2of the scattered phonon is equal to the frequency o,of the incident phonon. It is rather remarkable also that a three-phonon collision process

will not establish equilibrium, but for a subtle reason: the total momentum of the phonon gas is not changed by such a collision. An equilibrium distribution of phonons at a temperature T can move down the crystal with a drift velocity which is not disturbed by three-phonon collisions of the form (45). For such collisions the phonon momentum

is conserved, because on collision the change in J is K3 - K2 - K1 = 0. Here nK is the number of phonons having wavevector K. For a distribution with J 0, collisions such as (45) are incapable of establishing complete thermal cquilihrium because they leave J unchanged. If

+

Figure 16a Flow of gas molecules in a state of drifting equilibrium down a long open tube with frictionless walls. Elastic collision processes among the gas molecules do not change the momen tum or energy flux of the gas because in each collision the velocity of the center of mass of the col liding particles and their energy remain unchanged. Thus energy is transported from left to righ without being driven by a temperature gradient. Therefore the thermal resistivity is zero and th thermal conductivity is infinite.

Figure 16b The usual definition of thermal conductivity in a gas refers to a situation where n