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SpringerSeriesin solid-state sciences 134 SpringerSeriesin solid-state sciences SeriesEditors: M.Cardona P.Fulde K.vonKlitzing R.Merlin H.-J.Queisser H.Sto¨rmer The Springer Series in Solid-State Sciences consists of fundamental scientific books preparedbyleadingresearchersinthef0ield.Theystrivetocommunicate,inasystematic andcomprehensiveway,thebasicprinciplesaswellasnewdevelopmentsintheoreticaland experimentalsolid-statephysics. 136 NanoscalePhaseSeparation 143 X-RayMultiple-WaveDiffraction andColossalMagnetoresistance TheoryandApplication ThePhysicsofManganites ByS.-L.Chang andRelatedCompounds 144 PhysicsofTransitionMetalOxides ByE.Dagotto ByS.Maekawa,T.Tohyama, 137 QuantumTransport S.E.Barnes,S.Ishihara, inSubmicronDevices W.Koshibae,andG.Khaliullin ATheoreticalIntroduction 145 Point-ContactSpectroscopy ByW.MagnusandW.Schoenmaker ByYu.G.NaidyukandI.K.Yanson 138 PhaseSeparation 146 OpticsofSemiconductors inSoftMatterPhysics andTheirNanostructures MicellarSolutions,Microemulsions, Editors:H.KaltandM.Hetterich CriticalPhenomena 147 ElectronScattering ByP.K.Khabibullaevand inSolidMatter A.A.Saidov ATheoretical 139 OpticalResponse andComputationalTreatise ofNanostructures ByJ.Zabloudil,R.Hammerling, MicroscopicNonlocalTheory L.Szunyogh,andP.Weinberger ByK.Cho 148 PhysicalAcoustics 140 FractalConcepts intheSolidState inCondensedMatterPhysics ByB.Lu¨thi ByT.NakayamaandK.Yakubo 149 SolitaryWaves 141 ExcitonsinLow-Dimensional inComplexDispersiveMedia Semiconductors Theory·Simulation·Applications Theory,NumericalMethods, ByV.Yu.Belashovand ApplicationsByS.Glutsch S.V.Vladimirov 142 Two-DimensionalCoulomb 150 TopologyinCondensedMatter LiquidsandSolids Editor:M.I.Monastyrsky ByY.MonarkhaandK.Kono Volumes90–135arelistedattheendofthebook. Peter Mohn Magnetism in the Solid State An Introduction CorrectedSecondPrinting With72Figuresand7Tables 123 Prof.Dr.PeterMohn ViennaUniversityofTechnology CenterforComputationalMaterialsScience Getreidemarkt9/134 1060Vienna,Austria E-mail:[email protected] SeriesEditors: ProfessorDr.,Dres.h.c.ManuelCardona ∗ ProfessorDr.,Dres.h.c.PeterFulde ProfessorDr.,Dres.h.c.KlausvonKlitzing ProfessorDr.,Dres.h.c.Hans-JoachimQueisser Max-Planck-Institutfu¨rFestko¨rperforschung,Heisenbergstrasse1,70569Stuttgart,Germany ∗Max-Planck-Institutfu¨rPhysikkomplexerSysteme,No¨thnitzerStrasse38 01187Dresden,Germany ProfessorDr.RobertoMerlin DepartmentofPhysics,5000EastUniversity,UniversityofMichigan AnnArbor,MI48109-1120,USA ProfessorDr.HorstSto¨rmer Dept.Phys.andDept.Appl.Physics,ColumbiaUniversity,NewYork,NY10027and BellLabs.,LucentTechnologies,MurrayHill,NJ07974,USA ISSN0171-1873 Corrected2ndPrinting2006 ISBN-10 3-540-29384-1 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-29384-2 SpringerBerlinHeidelbergNewYork ISBN3-540-43183-7 1stEd.Springer-VerlagBerlinHeidelbergNewYork LibraryofCongressControlNumber:2005933346 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproduction onmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationorpartsthereofis permittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion, andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliabletoprosecutionunderthe GermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com ©Springer-VerlagBerlinHeidelberg2003,2006 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. Typesettingbytheauthors Coverconcept:eStudioCalamarSteinen Coverproduction:design&productionGmbH,Heidelberg Productionanddataconversion:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Printedonacid-freepaper SPIN:11568209 57/3141/YL-543210 To the memory of E. Peter Wohlfarth, a great magnetician and humanist Preface During1986and1987,whenIwasworkingattheImperialCollegeofScience andTechnologyinLondonwithE.P.Wohlfarth,Iwasaskedtopreparesome lecturenotesfollowinghislecture“Magneticpropertiesofmetalsandalloys”. Peter Wohlfarth had planned to use these lecture notes in a book about itinerant magnetism he wanted us to write together. Due to his untimely death this project was never realized. Back in Vienna I started to write up the lecture notes for a graduate course, which to some extent was based on these original lectures but in manyrespectshasbeenmodernizedbyincludingmorerecentresultsofband structurecalculationsandbandtheoreticalresults.Duringatwomonthvisit of Melbourne University in 1993, I met G. Fletcher from Monash Univer- sity (Melbourne) who kindly offered to critically read the manuscript. His comments are highly appreciated. Thepresentgreatlyenlargedversionwasmainlywrittenduringasabbati- calattheuniversityofUppsaladuring2000.IamgratefultoBo¨rjeJohansson for making this possible and to Clas Persson for reading the manuscript and providing me with lots of useful comments. Theaimofthebookistopresentalargelyphenomenologicalintroduction tothefieldofsolidstatemagnetismatarelativelyelementarylevel.Thetwo basic concepts of magnetism in solids namely the localized and the delocali- zed description are presented as the extreme approaches. The true nature of magnetism lies, as often in life, somewhere in between, sometimes showing a tendency towards the more localized side, sometimes tending to the delo- calized side. It is perhaps this mixing of concepts which makes magnetism appear complicated and difficult. Another source of confusion is the diffe- rent language used by theoreticians and experimentalists. I have tried very hard to clarify these rather more semantic problems and to use a uniform nomenclature throughout the book. It is my belief and my experience that the approach presented here provides a useful introduction not only for the physicist,butalsofortheinterestedreadercomingfromfieldslikechemistry, electrical engineering or even geo-sciences. The mathematical concepts used arekeptrathersimpleandhardlyevergobeyondanundergraduatecoursein mathematics for physicists, chemists or engineering. Since the book emerged fromalecturecourseIhavegivenatViennaUniversityofTechnologyforthe VIII Preface last15years,thechaptersinthebookarenotcompletelyself-contained.The first-time reader is thus advised to read the chapters in the sequence that they appear in the book. It is my sincere hope that after having read this book the reader will agree that for once the Encyclopedia Brittanica is in error when it states Few subjects in science are more difficult to understand than magnetism, (Encyclopedia Brittanica, 15th edition 1989). The present book does not attempt to cover the whole field of solid state magnetism, but tries to provide an overview by selecting special topics. The idea is to create an interest in this fascinating field in which quantum me- chanics, thermodynamics and computer simulations join forces to explain “Magnetism in the Solid State”. Vienna, Peter Mohn June 2002 Since this book has been so well received by the scientific community that the first printing has been sold within three years, I was asked by the publi- shers to produce an updated version for a second printing. I am grateful to all colleagues (mainly students) who reported typos, ambiguous or unclear formulations etc. I considered all of them seriously and thus made a number of changes, which I hope will improve the text. Vienna, Peter Mohn September 2005 Contents 1. A Historical Introduction................................. 1 2. Consequences of Fermi Statistics ......................... 13 2.1 Quantum Statistics of Fermions .......................... 13 2.2 Free Energy of the Fermi Gas ............................ 16 3. Paramagnetism ........................................... 25 4. Energy Bands in the Crystal.............................. 31 5. Experimental Basis of Ferromagnetism ................... 37 5.1 Nickel Alloys........................................... 42 5.2 Iron Alloys ............................................ 43 5.3 Palladium Alloys ....................................... 44 5.4 Iron–Nickel Alloys ...................................... 44 5.5 Effects of Strong Magnetic Fields......................... 44 5.6 Effects of High Pressure ................................. 46 5.7 Effects of Finite Temperature ............................ 47 5.8 Susceptibility above Tc .................................. 47 5.8.1 Susceptibility of “Classical Spins” .................. 48 5.9 Critical Exponents...................................... 50 5.10 Neutron Diffraction..................................... 51 5.11 Further Experimental Methods........................... 52 6. Weiss Molecular Field Model ............................. 53 6.1 Rhodes–Wohlfarth Plot ................................. 59 7. Heisenberg Model ........................................ 63 7.1 Magnon Operators...................................... 63 7.2 Heisenberg Hamiltonian in Magnon Variables .............. 66 7.3 Magnon Dispersion Relation ............................. 67 7.3.1 Specific Heat of Magnons.......................... 69 7.3.2 Ordering Temperature ............................ 70 7.4 Approximations for the Heisenberg Model ................. 71 7.4.1 Ising Model...................................... 71 X Contents 7.4.2 XY Model....................................... 72 7.4.3 Mean Field Solutions of the Heisenberg Model ....... 73 8. Itinerant Electrons at 0 K ................................ 75 8.1 Pauli Susceptibility of the Itinerant Electrons .............. 80 8.2 Susceptibility of the Interacting Itinerant Electrons ......... 80 8.3 Non-linear Effects ...................................... 83 8.4 Effects of High Fields at 0 K............................. 83 8.4.1 Non-magnetic Limit .............................. 84 8.4.2 Strong Ferromagnets.............................. 85 8.4.3 Weak Ferromagnets............................... 85 8.4.4 bcc Iron and hcp Cobalt .......................... 85 8.4.5 Extremely High Fields ............................ 87 8.4.6 Metamagnetism .................................. 87 8.5 Susceptibility of Paramagnetic Alloys ..................... 89 9. Band Gap Theory of Strong Ferromagnetism ............. 93 9.1 Magnetism of Alloys .................................... 97 10. Magnetism and the Crystal Structure – Covalent Magnetism...................................... 101 10.1 Crystal Structure of Mn, Fe, Co, and Ni................... 103 10.2 Covalent Magnetism .................................... 105 10.3 Covalent Polarization ................................... 111 11. Magnetic Impurities in an Electron Gas .................. 117 11.1 Impurity Potential in the Jellium......................... 117 11.2 Strong Perturbations in the Jellium....................... 119 11.3 Layer and Line Defects.................................. 119 11.4 Magnetic Impurities and Oscillations of the Magnetization... 120 12. Itinerant Electrons at T > 0: A Historical Survey......... 123 12.1 Excitations at Low Temperatures......................... 127 12.1.1 Strongly Ferromagnetic Systems.................... 127 12.1.2 Weakly Ferromagnetic Systems..................... 129 12.2 Stoner Theory for a Rectangular Band .................... 130 12.3 Weak Excitations with ζ <<1 ........................... 132 13. Hubbard Model .......................................... 135 13.1 Beyond Hartree–Fock ................................... 137 14. Landau Theory for the Stoner Model ..................... 139 14.1 General Considerations.................................. 139 14.2 Application to the Stoner Model ......................... 141 15. Coupling Between Itinerant and Localized Moments...... 147