Fundamentals of the Physics of Solids: Volume 2: Electronic Properties

659 Pages·2009·9.92 MB·English

by  Jenő Sólyom (auth.)

18 VIIIA 2He21sHelium 10Ne262s2pNeon 18Ar263s3pArgon 36Kr10263d4s4pKrypton 54Xe10264d5s5pXenon 86Rn10265d6s6pRadon
of elements 13 IIIA 14 IVA 15 VA 16 VIA 17 VIIA 5 6 7 8 9BCNOF21222324252s2p2s2p2s2p2s2p2s2pBoronCarbonNitrogenOxygenFluorine 1314151617AlSiPSCl 8 9 1021222324253s3p3s3p3s3p3s3p3s3p 6 VIB 7 VIIBVIIIB 11 IB 12 IIBAluminumSiliconPhosphorusSulfurChlorine 282930313233343524252627NiCuZnGaGeAsSeBrCrMnFeCo8210110210211022102310241025515262723d4s3d4s3d4s3d4s4p3d4s4p3d4s4p3d4s4p3d4s4p3d4s3d4s3d4s3d4sNickelCopperZincGalliumGermaniumArsenicSeleniumBromineChromiumManganeseIronCobalt 464748495051525342434445PdAgCdInSnSbTeIMoTcRuRh1010110210211022102310241025515271814d4d5s4d5s4d5s5p4d5s5p4d5s5p4d5s5p4d5s5p4d4s4d5s4d5s4d5sPalladiumSilverCadmiumIndiumTinAntimonyTelluriumIodineMolybdenumTechnetiumRutheniumRhodium 787980818283848574757677PtAuHgTlPbBiPoAtWReOsIr9110110210211022102310241025425262725d6s5d6s5d6s5d6s6p5d6s6p5d6s6p5d6s6p5d6s6p5d6s5d6s5d6s5d6sPlatinumGoldMercuryThalliumLeadBismuthPoloniumAstatineTungstenRheniumOsmiumIridium 110111106107108109DsRgSgBhHsMt91101425262726d7s6d7s6d7s6d7s6d7s6d7sDarmstadtiumRoentgeniumSeaborgiumBohriumHassiumMeitnerium 646566676869707160616263GdTbDyHoErTmYbLuNdPmSmEu712921021121221321421412425262724f5d6s4f6s4f6s4f6s4f6s4f6s4f6s4f5d6s4f6s4f6s4f6s4f6sGadoliniumTerbiumDysprosiumHolmiumErbiumThuliumYtterbiumLutetiumNeodymiumPromethiumSamariumEuropium 9293949596979899100101102103CmBkCfEsFmMdNoLrUNpPuAm71281210211212213214214123125262725f6d7s5f6d7s5f7s5f7s5f7s5f7s5f7s5f6d7s5f6d7s5f7s5f7s5f7sCuriumBerkeliumCaliforniumEinsteiniumFermiumNobeliumLawrenciumUraniumNeptuniumPlutoniumAmericiumMendelevium
1 IA 11HPeriodic table 11s 2 IIAHydrogen 4 32LiBe12[He] 2s1sLithiumBeryllium 12113NaMg12[Ne] 3s3s 3 IIIB 4 IVB 5 VBSodiumMagnesium 20212223194KCaScTiV12122232[Ar] 4s4s3d4s3d4s3d4sPotassiumCalciumScandiumTitaniumVanadium 38394041375RbSrYZrNb12122241[Kr] 5s5s4d5s4d5s4d5sRubidiumStrontiumYttriumZirconiumNiobium 55565772736CsBaLaHfTa12122232[Xe] 6s6s5d6s5d6s5d6sCesiumBariumLanthanumHafniumTantalum 8788891041057FrRaAcRfDb12122232[Rn] 7s7s6d7s6d7s6d7sFranciumRadiumActiniumRutherfordiumDubnium Lanthanoids5859CePr22324f6s4f6sCeriumPraseodymium 9091 ActinoidsThPa222126d7s5f6d7sThoriumProtactinium
FundamentalsofthePhysicsofSolids
Jeno˝ So´lyom
Fundamentals
of the Physics
of Solids
Volume II
Electronic Properties
Translated by Attila Piro´th
With238Figuresand40Tables
123
Author
ProfessorDr.Jeno˝ So´lyom
HungarianAcademyofSciences
ResearchInstitutefor
SolidStatePhysics&Optics
P.O.Box49,1525Budapest
Hungary
and
DepartmentofPhysics
Eo¨tvo¨sLora´ndUniversity
1171Budapest
Pa´zma´nyse´ta´ny1/A
Hungary
[email protected]
Translator
AttilaPiro´th
www.pirothattila.com
TitleoftheHungarianedition:AModernSzila´rdtestfizikaAlapjaiII,Elektronoka
szila´rdtestekben,ISBN963463589X(cid:2)c ELTEEo¨tvo¨sKiado´ 2003
ISBN:978-3-540-85315-2 e-ISBN:978-3-540-85316-9
LibraryofCongressControlNumber:2007929725
(cid:2)c Springer-VerlagBerlinHeidelberg2009
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orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
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To Márta, Gyöngyvér, Tünde, and Iringó
Preface
Thereader isholding thesecondvolume ofathree-volume textbook onsolid-
state physics. This book is the outgrowth of the courses I have taught for
many years at Eötvös University, Budapest, for undergraduate and graduate
studentsunderthetitlesSolid-State Physics andModern Solid-State Physics.
The main motivation for the publication of my lecture notes as a book
was that none of the truly numerous textbooks covered all those areas that
I felt should be included in a multi-semester course. Especially, if the course
strives to present solid-state physics in a unified structure, and aims at dis-
cussing not only classic chapters of the subject matter but also (in more or
less detail) problems that are of great interest for today’s researcher as well.
Besides, the book presents a much larger material than what can be covered
in a two- or three-semester course. In the first part of the first volume the
analysis of crystal symmetries and structure goes into details that certainly
cannotbeincludedinausualcourseonsolid-statephysics.Thesameapplies,
among others, to the discussion of the methods used in the determination of
bandstructure,thepropertiesofFermiliquidsandnon-Fermiliquids,andthe
theory of unconventional superconductors in the present and third volumes.
Thesepartscanbeassignedassupplementaryreadingforinterestedstudents,
or can be discussed in advanced courses.
The line of development and the order of the chapters are based on the
prerequisites for understanding each part. Therefore a gradual shift can be
observedinthestyleofthebook.Whiletheintermediatestepsofcalculations
arepresentedinconsiderabledetailandexplanationsarealsomorelengthyin
the first and second volumes, they are much sparser and more concise in the
thirdone,thusthatvolumereliesmoreontheindividualworkofthestudents.
Onaccountoftheprerequisites,certaintopicshavetoberevisited.Thisiswhy
magnetic properties are treated in three, and superconductivity in two parts.
Themagnetismofindividualatomsispresentedinanintroductorychapterof
thefirstvolume.Thestructureanddynamicsofmagneticallyorderedsystems
builtupoflocalizedmomentsarebestdiscussedafterlatticevibrations,along
the same lines. Magnetism is then revisited in the third volume, where the
VIII Preface
roleofelectron–electroninteractionsisdiscussedinmoredetail.Similarly,the
phenomenologicaldescriptionofsuperconductivityispresentedinthisvolume
after the analysis of the transport properties of normal metals, in contrast to
them,whilethemicroscopictheoryisoutlinedlater,inthethirdvolume,when
the effects of interactions are discussed.
Separating the material into three similar-sized volumes is a necessity in
view of the size of the material – but it also reflects the internal logical struc-
ture of the subject matter. At those universities where the basic course in
solid-state physics runs for three semesters working through one volume per
semester is a natural schedule. In this case the discussion of the electron
gas – which is traditionally part of the introduction – is left for the second
semester. This choice is particularly suited to curricula in which the course
on solid-state physics is held parallel with quantum mechanics or statistical
physics. If the lecturer feels more comfortable with the traditional approach,
the discussion of the Drude model presented in this volume can be moved to
thebeginningofthewholecourse.NeverthelessthediscussionoftheSommer-
feld model should be postponed until students have familiarized themselves
with the fundamentals of statistical physics. For the same reason the lecturer
may prefer to change the order of other chapters as well. This is, to a large
extent, up to the personal preferences of the lecturer.
In presenting the field of solid-state physics, special emphasis has been
laid on discussing the physical phenomena that can be observed in solids.
Nevertheless I have tried to give – or at least outline – the theoretical inter-
pretation for each phenomenon, too. As is common practice for textbooks, I
have omitted precise references that would give the publication data of the
discussed results. I have made exceptions only for figures taken directly from
published articles. At the end of each chapter I have listed textbooks and
review articles only that present further details and references pertaining to
the subject matter of the chapter in question. The first chapter of the first
volume contains a longer list of textbooks and series on solid-state physics.
Bulky as it might be, this three-volume treatise presents only the funda-
mentals of solid-state physics. Today, when articles about condensed matter
physics fill tens of thousands of pages every year in Physical Review alone, it
would be obviously overambitious to aim at more. Therefore, building on the
foundationspresentedinthisseries,studentswillhavetoacquireasubstantial
amount of extra knowledge before they can understand the subtleties of the
topicsintheforefrontoftoday’sresearch.Neverthelessattheendofthethird
volume students will also appreciate the number of open questions and the
necessity of further research.
A certain knowledge of quantum mechanics is a prerequisite for study-
ing solid-state physics. Various techniques of quantum mechanics – above
all field-theoretical methods and methods employed in solving many-body
problems – play an important role in present-day solid-state physics. Some
essential details are listed in one of the appendices of the third volume, how-
ever, I have omitted more complicated calculations that would have required
Preface IX
the application of the modern apparatus of many-body problems. This is
especially true for the third volume, where central research topics of present-
day solid-state physics are discussed, in which the theoretical interpretation
of experimental results is often impossible without some extremely complex
mathematical formulation.
The selection of topics obviously bears the stamp of the author’s own
research interest, too. This explains why the discussion of certain important
fields – such as the mechanical properties of solids, surface phenomena, or
amorphous systems, to name but a few – have been omitted.
I have used the International System of Units (SI), and have given the
equations of electromagnetism in rationalized form. Since nonrationalized
equations as well as gaussian CGS (and other) units are still widely used
in the solid-state physics literature, the corresponding formulas and units are
indicated at the appropriate places. In addition to the fundamental physical
constants used in solid-state physics, the commonest conversion factors are
also listed in Appendix A of the first volume. I deviated from the recom-
mended notation in the case of the Boltzmann constant using k instead of
B
k – reserving the latter for the wave number, which plays a central role in
solid-state physics.
To give an impression of the usual values of the quantities occurring in
solid-statephysics,typicalcalculatedvaluesormeasureddataareoftentabu-
lated.Toprovidethemostprecisedataavailable,IhavereliedontheLandolt–
Börnstein series, the CRC Handbook of Chemistry and Physics, and other
renowned sources. Since these data are for information only, I have not indi-
cated either their error or in many cases the measurement temperature, and
I have not mentioned when different measurement methods lead to slightly
disparate results. As a rule of thumb, the error is usually smaller than or on
the order of the last digit.
I would like to thank all my colleagues who read certain chapters and
improved the text through their suggestions and criticism. Particular thanks
go to professors György Mihály and Attila Virosztek for reading the whole
manuscript. In spite of all efforts, some mistakes have certainly remained in
the book. Obviously, the author alone bears the responsibility for them.
Special thanks are due to Károly Härtlein for his careful work in drawing
the majority of the figures. The figures presenting experimental results are
reproducedwiththepermissionoftheauthorsorthepublishers.Thechallenge
of translating the first and second volumes of the book from the Hungarian
original was taken up by Attila Piróth. I acknowledge his work.
Finally, I am indebted to my family, to my wife and children, for their
patience during all those years when I spent evenings and weekends with
writing this book.
Budapest, August 2008 Jenő Sólyom
Contents
16 Free-Electron Model of Metals............................. 1
16.1 Classical Drude Model ................................... 2
16.1.1 Basic Assumptions of the Model..................... 2
16.1.2 Electrical Conductivity............................. 5
16.1.3 Heat Conduction .................................. 9
16.1.4 Hall Resistance ................................... 11
16.1.5 AC Conductivity .................................. 14
16.1.6 High-Frequency Behavior of a Classical Electron Gas .. 16
16.1.7 Magnetic Properties ............................... 20
16.1.8 Failures of the Drude Model ........................ 22
16.2 Quantum Mechanical Sommerfeld Model ................... 24
16.2.1 Quantum Mechanical States of Free Electrons......... 24
16.2.2 Ground State of the Electron Gas ................... 28
16.2.3 Excited Electron and Hole States.................... 31
16.2.4 Density of States of the Electron Gas ................ 32
16.2.5 Ideal Electron Gas at Finite Temperatures............ 34
16.2.6 Sommerfeld Expansion............................. 37
16.2.7 Specific Heat of the Electron Gas.................... 40
16.2.8 Equation of State for the Ideal Electron Gas .......... 42
16.2.9 Susceptibility of the Electron Gas ................... 44
16.3 Electric and Heat Currents in an Electron Gas .............. 47
16.3.1 Noninteracting Electrons in a Uniform Electric Field... 47
16.3.2 Stationary Distribution Function .................... 49
16.3.3 Electric and Heat Currents ......................... 53
16.3.4 Thermoelectric Phenomena ......................... 56
16.3.5 Galvanomagnetic and Thermomagnetic Phenomena.... 61
16.4 Scattering of Free Electrons by Impurities .................. 64
16.4.1 Formal Solution of the Schrödinger Equation ......... 64
16.4.2 Approach Based on Scattering Theory ............... 65
16.4.3 Friedel Oscillations Around Impurities ............... 69
16.4.4 Bound States Around Impurities .................... 73
16.5 Inadequacies of the Free-Electron Model.................... 74
XII Contents
17 Electrons in the Periodic Potential of a Crystal............ 77
17.1 Band Structure of Electronic States........................ 78
17.1.1 Bloch States ...................................... 78
17.1.2 Energy Levels of Bloch States....................... 80
17.1.3 Eigenvalue Problem for Equivalent k Vectors ......... 82
17.1.4 Role of the Spin–Orbit Interaction................... 83
17.2 Representation of the Band Structure ...................... 84
17.2.1 Reduced-, Repeated-, and Extended-Zone Schemes .... 85
17.2.2 Constant-Energy Surfaces and the Fermi Surface ...... 88
17.3 Metals, Insulators, Semiconductors ........................ 89
17.4 Bloch Electrons as Quasiparticles.......................... 92
17.4.1 Creation and Annihilation Operators of Bloch States... 92
17.4.2 Effective Mass of Bloch Electrons.................... 93
17.4.3 Bloch Electrons and Holes.......................... 95
17.4.4 Density of States for Bloch Electrons ................ 96
17.4.5 Specific Heat and Susceptibility of Bloch Electrons .... 98
17.5 Wannier States..........................................100
17.5.1 Wannier Functions ................................100
17.5.2 Creation and Annihilation Operators of Wannier States 103
17.6 Electron States Around Impurities.........................104
18 Simple Models of the Band Structure......................109
18.1 Nearly-Free-Electron Approximation .......................109
18.1.1 Band Structure in the Empty Lattice ................110
18.1.2 Fermi Surface in the Empty Lattice..................115
18.1.3 Effects of a Weak Periodic Potential .................120
18.1.4 Lifting of Accidental Degeneracies ...................126
18.1.5 Fermi Surface for Nearly Free Electrons ..............136
18.2 Tight-Binding Approximation.............................139
18.2.1 Broadening of Atomic Levels into Bands .............140
18.2.2 Band of s-Electrons................................143
18.2.3 Band of p-Electrons ...............................145
19 Methods for Calculating and Measuring the Band
Structure ..................................................151
19.1 Matrix Methods.........................................152
19.1.1 General Formulation of the Problem .................152
19.1.2 LCAO Method....................................154
19.1.3 Plane-Wave Method ...............................156
19.1.4 Orthogonalized-Plane-Wave Method .................157
19.1.5 Pseudopotential Method ...........................160
19.2 Variational Methods and Methods Based on Scattering Theory164
19.2.1 Augmented-Plane-Wave Method ....................164
19.2.2 Green Function or KKR Method ....................168
19.2.3 Physical Interpretation of the KKR Method ..........173