Modern Theory of Magnetism in Metals and Alloys

342 Pages·2013·5.5 MB·English

by  Yoshiro Kakehashi (auth.)

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Yoshiro Kakehashi
Modern Theory
of Magnetism in
Metals and Alloys
YoshiroKakehashi
FacultyofScience,DepartmentofPhysics
UniversityoftheRyukyus
Nishihara,Japan
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Max-Planck-InstitutfürFestkörperforschung,Heisenbergstrasse1,70569Stuttgart,Germany
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DepartmentofPhysics,UniversityofMichigan
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ProfessorDr.HorstStörmer
Dept.Phys.andDept.Appl.Physics,ColumbiaUniversity,NewYork,NY10027,USAand
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ISSN0171-1873 SpringerSeriesinSolid-StateSciences
ISBN978-3-642-33400-9 ISBN978-3-642-33401-6(eBook)
DOI10.1007/978-3-642-33401-6
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Preface
Metallicmagnetismhasalonghistorybecausetherehavebeencontinuousdiscov-
eriesofmanyintriguingphenomenaanddifficultiesintheirtheoreticaldescription.
One of the long-standing problems has been known as the itinerant vs localized
behavior of the magnetism.The ground-state properties of Fe, Co, and Ni such as
magnetizationandthe T-linearspecificheatatlowtemperatures,forexample,are
explained by the band model, while their finite temperature properties such as the
paramagneticsusceptibilityandthelargespecificheatanomalyattheCurietemper-
atureareexplainedwellbythelocalizedmodel.
Thedualpropertyofmetallicmagnetismledtotwopathsintheoreticalinvestiga-
tions.Oneistodevelopthebandtheoryatthegroundstatetakingintoaccountcorre-
lationeffectsontheoneelectronpotentialforelectrons.Therethedensityfunctional
theory (DFT) has played an important role. Theoretical improvement of metallic
magnetism at the ground state has been achieved as a part of the developments of
theDFTintheelectronicstructurecalculations.
Anotherdirectionofthedevelopmenthasbeentotakeintoaccountthespinfluc-
tuations in order to describe local-moment behaviors of metallic magnetism at fi-
nitetemperatures.Theoreticalresultsinthisdirectionuntil1980aresummarizedin
thebookbyMoriya(SpinFluctuationsinItinerantElectronMagnetism(Springer,
Berlin,1985)).Althoughspinfluctuationtheorieshavesucceededindescribingthe
localmomentbehavioratfinitetemperaturesinmetallicmagnetism,theunderlying
electronicstructurerelatedtothemagnetismofacertainindividualmaterialseems
tobeoversimplified.Abookwhichunifiesthetwopathsonthesamefootingwould
bevaluableforreaderstounderstandthemetallicmagnetism.
Thisbookaimstodescribethetheoriesofmetallicmagnetismfrombothview-
points, namely spin fluctuations and the electronic structure. It attempts to clarify
themagnetismfrommetalstodisorderedalloystoamorphousalloys.
The book covers most of the traditional topics of metallic magnetism such as
electron correlation effects on the ferromagnetism, magnetic excitations, as well
as the stability of antiferromagnetism and spin density waves. But it also includes
topicswhichhavebeendevelopedinthepastthreedecades.Thefirstisthedevel-
opmentofthedynamicalCPA(coherentpotentialapproximation),whichdescribes
v
vi Preface
thedynamicalspinandchargefluctuationsonthebasisofthemicroscopicelectronic
structurewithinthesingle-siteapproximation.Inparticular,thefirst-principlesdy-
namicalCPAhasreducedthegapbetweenthespinfluctuationtheoryandtheband
theoreticalapproachtoalargeextent,thusallowingtheinvestigationoftherelation-
shipbetweenmetallicmagnetismandelectronicstructure.Weelucidatethistheory
inChap.3.WealsopointoutinthesamechapterthatthedynamicalCPAisequiv-
alenttothedynamicalmean-fieldtheory(DMFT)inthemetal-insulatortransition.
Thesecondtopicisthetheoryoflocalenvironmenteffects(LEE)indisordered
alloys,whichgoesbeyondthesingle-siteCPAtheoryofmagnetism.InChap.8,we
describe the theory and clarify the magnetic behavior in the vicinity of the mag-
neticinstabilityofFe–Ni,Ni–Mn,andNi–Cualloys.Thischapteralsoincludesthe
molecular dynamics approach, which automatically determines the complex mag-
netic structure in metals and alloys. The third topic is the theoretical development
ofmagnetisminamorphousmetalsandalloys.Thefinite-temperaturetheorysheds
lightontheamorphousmagnetismfromtheviewpointofspinfluctuationsandthe
LEE, and clarifies how structural disorder drastically changes the magnetic prop-
erties of metals and alloys. This development is discussed in Chap. 9. Chapter 1
presentsanintroductionforthereaderswhoarenotfamiliarwiththemagnetism.
The frustrated system with heavy effective mass (e.g., YMn and LiV O ) is
2 2 4
not described in this book, because it is still under development. Recent topics on
thespintronicsarealsoomittedforthesamereason.Non-localtheoryofdynamical
spin fluctuations which goes beyond the dynamical CPA is left as a problem of
futureconcern.
I would like to express my sincere thanks to Professor Peter Fulde, Professor
MatinC.Gutzwiller,andProfessorHiroshiMiwafortheircontinuoussupportand
encouragementover30years.IwouldliketothankProfessorTakashiUchidaand
ProfessorMingYufortheircriticalreadingofthemanuscriptaswellastheirvalu-
ablecomments;ProfessorShi-YuWuandProfessorSungG.Chungfortheirvalu-
ablesuggestionsforimprovement;andProfessorTakeoFujiwaraandProfessorMo-
jmirSobfortheirkindcomments.Dr.M.A.R.Patoarykindlypreparedusefulfigures
towhomIammostindebted.ThanksarealsogiventoMr.K.Chungforhishelpin
proofreadingthemanuscript.Finally,IamgratefultoDr.C.Ascheron,Publishing
EditorofSpringerVerlagforhissupporttowardsthepublicationofthisbook.
Okinawa Y.Kakehashi
July2012
Contents
1 IntroductiontoMagnetism . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 MagneticMoments . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 BasicHamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 FormationofAtomicMoments . . . . . . . . . . . . . . . . . . . 6
1.4 MetalandInsulatorinSolids . . . . . . . . . . . . . . . . . . . . 11
1.5 QuenchingofOrbitalMagneticMoments . . . . . . . . . . . . . . 18
1.6 HeisenbergModel . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 MagneticStructure . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 MetallicMagnetismattheGroundState . . . . . . . . . . . . . . . . 29
2.1 BandTheoryofFerromagnetism . . . . . . . . . . . . . . . . . . 29
2.2 ElectronCorrelationsonMagnetism . . . . . . . . . . . . . . . . 36
2.2.1 StonerConditionintheCorrelatedElectronSystem . . . . 36
2.2.2 StabilityofFerromagnetismintheLowDensityLimit . . . 37
2.2.3 GutzwillerTheoryofElectronCorrelations . . . . . . . . . 40
2.3 DensityFunctionalApproach . . . . . . . . . . . . . . . . . . . . 47
2.3.1 Slater’sBandTheory . . . . . . . . . . . . . . . . . . . . 47
2.3.2 DensityFunctionalTheory . . . . . . . . . . . . . . . . . 48
2.3.3 Tight-BindingLinearMuffin-TinOrbitals . . . . . . . . . 53
2.3.4 FerromagnetisminTransitionMetals . . . . . . . . . . . . 58
3 MetallicMagnetismatFiniteTemperatures . . . . . . . . . . . . . . 63
3.1 StonerTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 FunctionalIntegralMethod . . . . . . . . . . . . . . . . . . . . . 67
3.3 Single-SiteTheoryintheStaticApproximation. . . . . . . . . . . 74
3.4 DynamicalCPATheory . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 DynamicalCPAwithHarmonicApproximation . . . . . . . . . . 90
3.6 DynamicalCPAandDynamicalMean-FieldTheory . . . . . . . . 95
3.6.1 The Many-Body CPA and Its Equivalence
totheDynamicalCPA . . . . . . . . . . . . . . . . . . . . 95
3.6.2 TheDMFTandItsEquivalencetotheDynamicalCPA. . . 96
vii
viii Contents
3.6.3 The Projection Operator Method CPA and Summary
ofRelations . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.7 First-PrinciplesDynamicalCPAandMetallicMagnetism . . . . . 104
4 MagneticExcitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1 SpinWavesintheLocalMomentSystem . . . . . . . . . . . . . . 115
4.2 SpinWavesinItinerantFerromagnets . . . . . . . . . . . . . . . . 118
4.3 DynamicalSusceptibility . . . . . . . . . . . . . . . . . . . . . . 124
4.4 DynamicalSusceptibilityintheRPAandSpinWaveExcitations. . 129
5 SpinFluctuationTheoryinWeakFerromagnets. . . . . . . . . . . . 135
5.1 Free-EnergyFormulationoftheStonerTheory . . . . . . . . . . . 135
5.2 Self-ConsistentRenormalizationTheory . . . . . . . . . . . . . . 140
6 AntiferromagnetismandSpinDensityWaves . . . . . . . . . . . . . 149
6.1 AntiferromagnetisminMetals . . . . . . . . . . . . . . . . . . . . 149
6.2 GeneralizedStaticSusceptibilityandAntiferromagnetism . . . . . 157
6.3 MolecularDynamicsTheoryforComplexMagneticStructures . . 162
6.4 PhenomenologicalTheoryofMagneticStructure . . . . . . . . . . 169
6.4.1 MultipleSDWwithCommensurateWaveVectors . . . . . 172
6.4.2 MultipleSDWwithIncommensurateWaveVectors . . . . 177
7 MagnetisminDiluteAlloys . . . . . . . . . . . . . . . . . . . . . . . 181
7.1 MagneticInteractionsandSpinGlassesinDiluteAlloys . . . . . . 181
7.2 MagneticImpurityinNobleMetals . . . . . . . . . . . . . . . . . 193
8 MagnetismofDisorderedAlloys . . . . . . . . . . . . . . . . . . . . 203
8.1 Slater–PaulingCurves . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2 Single-SiteTheoryofDisorderedAlloys . . . . . . . . . . . . . . 206
8.3 TheoryofLocalEnvironmentEffectsinMagneticAlloys . . . . . 217
8.4 ComputerSimulationsforDisorderedMagneticAlloys . . . . . . 242
9 MagnetismofAmorphousMetalsandAlloys . . . . . . . . . . . . . 253
9.1 IntroductiontotheAmorphousMetallicMagnetism . . . . . . . . 253
9.2 AmorphousStructureandElectronicStructure . . . . . . . . . . . 255
9.3 TheoryofAmorphousMetallicMagnetism . . . . . . . . . . . . . 258
9.4 MagnetismofAmorphousTransitionMetals . . . . . . . . . . . . 266
9.4.1 GeneralSurvey . . . . . . . . . . . . . . . . . . . . . . . 267
9.4.2 MagnetismofAmorphousFe,Co,andNi. . . . . . . . . . 270
9.4.3 DegreeofStructuralDisorderandNonuniqueMagnetism . 276
9.5 TheoryofMagnetisminAmorphousAlloys . . . . . . . . . . . . 280
9.6 MagnetismofAmorphousTransitionMetalAlloys . . . . . . . . . 287
9.6.1 TM–TMAmorphousAlloys . . . . . . . . . . . . . . . . . 287
9.6.2 TheOtherTMAlloys . . . . . . . . . . . . . . . . . . . . 295
AppendixA EquivalenceoftheCPAEquations(3.83),(3.85),
and(3.89) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
AppendixB DynamicalCPABasedontheMultipleScatteringTheory . 305
Contents ix
AppendixC DerivationoftheSingle-SiteSpinFluctuationTheory
fromtheDynamicalCPA . . . . . . . . . . . . . . . . . . . . . . . . 309
AppendixD ExpansionofD withRespecttoDynamicalPotential . . 313
νσ
AppendixE LinearResponseTheory . . . . . . . . . . . . . . . . . . . 319
AppendixF IsothermalMolecularDynamicsandCanonical
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
AppendixG RecursionMethodforElectronicStructureCalculations . 325
AppendixH AnIntegralintheRKKYInteraction . . . . . . . . . . . . 329
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Chapter 1
Introduction to Magnetism
Magneticpropertiesoriginateinthespindegreesoffreedomofelectronsandtheir
associatedmotioninsolids.Wefirstdescribethemicroscopicmagneticmomentsof
electrons, and the formation of atomic magnetic moments due to strong Coulomb
interactionsinanatomwithunfilledshell.Atomicmagneticmomentschangetheir
naturewhentheatomsformasolid.Thekeytounderstandingthebehaviorofmag-
neticmomentsinsolidsisthedegreeofelectronlocalization.Webrieflyintroduce
theconceptofthemetalandtheinsulator(i.e.,theMottinsulator).Electronsinthe
latter are localized on each atom, so that their magnetic properties are described
bytheatomicmagneticmomentsandthemagneticinteractionsbetweenthem.We
present in this chapter only a minimal discussion on the magnetism in insulators.
Ontheotherhand,electronsmovefromsitetositeovertheentirecrystalinmetals,
so that the magnetic properties are connected to the whole degrees of freedom of
correlatedelectrons.Needlesstosay,themainthemeofthisbookisthemagnetism
ofmetalsandalloys.Inthelastsection,weelucidatevariousmagneticstructuresin
solidstoprovideabasicknowledgeonmagnetism.
1.1 MagneticMoments
MagneticmomentMintheelectromagneticsisdefinedbythetorqueN onamagnet
underthemagneticfieldH as
N ≡M×H. (1.1)
Itoriginatesinelectronsinthemagnet.Accordingtotheclassicalelectromagnetics,
thecoil-typelocalcurrenti(r) duetoelectronmotioncausesamagneticmoment.
ItisgivenintheCGS-Gaussunitas[1]
(cid:2)
1
M = r×i(r)d3x. (1.2)
L
2c
Y.Kakehashi,ModernTheoryofMagnetisminMetalsandAlloys, 1
SpringerSeriesinSolid-StateSciences175,DOI10.1007/978-3-642-33401-6_1,
©Springer-VerlagBerlinHeidelberg2012