材料物理性能6

8.2MAGNETICFIELDCLASSIFICATIONSIngeneral,magneticmaterialsareclassifiedintofivedistinctgroups;diamagnetic,paramagnetic,ferromagnetic,ferrimagnetic.Table8.2providesasummaryofthemagneticpropertiesoftheseclassesofmaterials.1Typicaldiamagneticmaterialshaveamagneticsusceptibilitythatisnegativeandsmall.forexample,thesiliconcrystalisdiamagneticwithWhenadiamagneticsubstancesuchasasiliconcrystalisplacedinamagneticfieldmagnetizationvectorMinthematerialisintheoppositedirectiontotheappliedfieldandtheresultingfieldBwithinthematerialislessthan.Asubstanceexhibitsdiamagnetismwhenevertheconstituentatomsinthematerialhaveclosedsubshellsandshells..

28.2.2PARAMAGNETSMParamagneticmaterialshaveasmallpositivemagneticsusceptibility.Forexample,oxygengasisparamagneticwithΧm=2.1×10-6atatmosphericpressureandroomtemperature.Eachoxygenmoleculehasanetmagneticdipolemomentμmol.Intheabsenceofanappliedfield,thesemolecularmomentsarerandomlyorientedduetotherandomcollisionsofthemolecules,asdepictedinFigure8.13a.Themagnetizationofthegasiszero.Inthepresenceofanappliedfield,themolecularmagneticmomentstakevariousalignmentswiththefield,asillustratedinFigure8.13b.ThedegreeofalignmentofμmolwiththeappliedfieldandhencemagnetizationMincreaseswiththestrengthoftheappliedfieldμ0H.Whenaparamagneticsubstanceisplacedinanonuniformmagneticfield,theinducedmagnetizationMisalongBandthereisanetforcetowardgreaterfields.Forexample,whenliquidoxygenispouredclosetoastrongmagnet,asdepictedinFigure8.14,theliquidbecomesattractedtothemagnet.38.2.3FERROMAGNETISMFerromagneticmaterialsuchasironcanpossesslargepermanentmagnetizationsevenintheabsenceofanappliedmagneticfieldThemagneticsusceptibilityΧmistypicallypositiveandverylarge(eveninfinite)and,further,dependsontheappliedfieldintensity.TherelationshipbetweenthemagnetizationMandtheappliedmagneticfieldμ0Hishighlynonlinear.4Theoriginofferromagnetismisthequantummechanicalexchangeinteraction(discussedlater)betweentheconstituentatomsthatresultsinregionsofthematerialpossessingpermanentmagnetization.Figure8.15depictsaregionoftheFecrystal,calledamagneticdomainthathasanetmagnetizationvectorMduetothealignmentofthemagneticmomentsofallFeatomsinthisregion.Thiscrystaldomainhasmagneticorderingasalltheatomicmagneticmomentshavebeenalignedparalleltoeachother.FerromagnetismoccursbelowacriticaltemperaturecalledtheCurietemperatureTC.AttemperaturesaboveTC,ferromagnetismislostandmaterialbecomesparamagnetic.58.2.5FERRIMAGNETISM

Ferrimagneticmaterialssuchasferrets(e.g.,Fe­3O4)exhibitmagneticbehaviorsimilartoferromagnetismbelowacriticaltemperaturecalledtheCurietemperatureTc.AboveTctheybecomeparamagnetic.Theoriginofferrimagnetismisbasedonmagneticordering,asschematicallyillustratedinFigure8.17.AllAatomshavetheirspinsalignedinonedirectionandallBatomshavetheirspinsalignedintheoppositedirection.AsthemagneticmomentofanAatomisgreaterthanthatofaBatomic,thereisnetmagnetizationbyelectriccurrentMinthecrystal.Sinceferrimagneticmaterialsaretypicallynonconductingandthereforedonotsufferfromeddycurrentlosses,theyarewidelyusedinhigh-frequencyelectronicsapplications.6Allusefulmagneticmaterialsinelectricalengineeringareinvariablyferromagneticorferromagnetic.78.3FERROMAGNETISMORIGINANDTHEEXCHANGEINTERACTIONThetransitionmetalsiron,cobalt,andnickelareallferromagneticatroomtemperature.Therareearthmetalsgadoliniumanddysprosiumareferromagneticbelowroomtemperature.Ferromagneticmaterialscanexhibitpermanentmagnetizationevenintheabsenceofanappliedfield;thatis,theypossessasusceptibilitythatisinfinite.Inamagnetizedironcrystal,alltheatomicmagneticmomentsarealignedinthesamedirectional,asillustratedinFigure8.15,wherethemomentsinthiscasehaveallbeenalignedalongthe[100]direction,whichgivesnetmagnetizationalongthisdirection.8Itmaybethoughtthatthereasonforthealignmentofthemomentsisthemagneticforcesbetweenthemoments,justasbarmagnetswilltendtoalignheadtotailinanSNSN…fashion.Thisisnothingto,however,thecause,asthemagneticpotentialenergyofinteractionissmall,indeedsmallerthanthethermalenergy.TwoelectronsparalleltheirspinsnotbecauseofthedirectmagneticinteractionbetweenthespinmagneticmomentsbutbecauseofthePauliExclusionPrincipleandtheelectrostaticinteractionenergy.Togethertheyconstitutewhatisknownasanexchangeinteraction,whichforcestwoelectronstotakemsandmlvaluesthatresultintheminimumofelectrostaticenergy.9Fromquantummechanics,theexchangeinteractioncanberepresentedintermsofanexchangeenergyas[8.21]whereS1andS2arethespinangularmomentaofthetwoelectronsandJeisanumericalquantitycalledtheexchangeintegralthatinvolvesintegratingthewavefunctionswithvariouspotentialenergyinteractiondistance.Forthemajorityofsolids,Jeisnegative,sotheexchangeenergyisnegativeifS1andS2areintheoppositedirections,thatis,thespinsareantiparallel(aswefoundincovalentbonding).Thisistheantiferromagneticstate.ForFe,Co,andNi,however,Jeispositive.EexisthennegativeifS1andS2areparallel.Spinsofthe3delectronsontheFeatomsthereforespontaneouslyaligninthesamedirectiontoreducetheexchangeenergy.Thisspontaneousmagnetizationisthephenomenonofferromagnetism.108.4SATURATIONMAGNETIZATIONANDCURIETEMPERATURE

ThemaximummagnetizationinaferromagnetwhenalltheatomicmagneticmomentshavebeenalignedasmuchaspossibleiscalledthesaturationmagnetizationMsat.Intheironcrystal,forexample,thiscorrespondstoeachFeatomwithaneffectivespinmagneticmomentof2.2T.Asweincreasethetemperature,latticevibrationsbecomemoreenergetic,whichleadtoafrequentdisruptionofthealignmentsofthespins.Thespinscannotalignperfectlywitheachotherasthetemperatureincreasesduetolatticevibrationsrandomlyagitatingtheindividualspins.Whenanenergeticlatticevibrationpassesthroughaspinsite,theenergyinthevibrationmaybesufficienttodisorientatethespinoftheatom.TheferromagneticbehaviordisappearsatacriticaltemperaturecalledtheCurietemperature,denotedbyTc,whenthethermalenergyoflatticevibrationsinthecrystalcanovercomethepotentialenergyoftheexchangeinteractionandhencedestroythespinalignments.AbovetheCurietemperature,thecrystalbehavesasifitwereparamagnetic.ThesaturationmagnetizationMsat,therefore,decreasesfromitsmaximumvalueMsat(0)atabsolutezerooftemperaturetozeroattheCurietemperature..11Figure8.21showsthatdependenceofMsatonthetemperaturewhenMsathasbeennormalizedtoMsat(0)andthetemperatureisthereducedtemperature,thatis,T/Tc.AtT/Tc=1,Msat=0.Whenplottedinthisway,theferromagnetscobaltandnickelfollowcloselytheobservedbehaviorforiron.WeshouldnotethatsinceforironTc=1043K,atroomtemperature,T/Tc=0.29andMsatisveryclosetoitsvalueatMsat(0).SinceattheCurietemperature,thethermalenergy,oftheorderofkTc,issufficienttoovercometheenergyoftheexchangeinteractionEexthatalignsthespins,wecantakekTcasanorderofmagnitudeestimateofEex.Foriron,Eexis～0.09eVandforcobaltthisis～0.1eV.128.4MAGNETICDOMAINS:FERROMAGNETICMATERIALS

8.4.1MagneticDomains

Asinglecrystalofirondoesnotnecessarilypossessanetpermanentmagnetizationintheabsenceofanappliedfield.IfamagnetizedpieceofironisheatedtoatemperatureaboveitsCurietemperatureandthenallowedtocoolintheabsenceofamagneticfield,itwillpossessnonetmagnetization.Thereasonfortheabsenceofnetmagnetizationisduetotheformationofmagneticdomainsthateffectivelycanceleachother,asdiscussedbelow.Amagneticdomainisaregionofthecrystalinwhichallthespinmagneticmomentsarealignedtoproduceamagneticmomentinonedirectiononly.13Figure8.22ashowsasinglecrystalofironthathasapermanentmagnetizationasaresultofferromagnetism(aligningofallatomicspins).Thecrystalislikeabarmagnetwithmagneticfieldlinesaroundit.Asweknow,thereispotentialenergy(PE),calledmagnetostaticenergy,storedinamagneticfield,andwecanreducethisenergyintheexternalfieldbydividingthecrystalintotwodomainswherethemagnetizationsareintheoppositedirections,asshowninFigure8.22b.14Theexternalmagneticfieldlinesarereducedandthereisnowlesspotentialenergystoredinthemagneticfield.Thereareonlyfieldlinesattheends.Thisarrangementisenergeticallyfavorablebecausethemagnetostaticenergyhasbeenreducedbydecreasingtheexternalfield15ThewallinFigure8.22bisa180°wallinasmuchasthemagnetizationthroughthewallisrotatedby180°.Itisapparentthatthewallregionwheretheneighbouringatomicspinschangetheirrelativedirection(ororientation)fromonedomaintotheneighboringonehashigherPEthanthebulkofthedomain,wherealltheatomicspinsarealigned.16Aswewillshowbelow,thedomainwallisnotsimplyoneatomicspacingbuthasafinitethickness,whichforironistypicallyoftheorderof0.1μm,orseveralhundredatomicspacings.Theexcessenergyinthewallincreaseswiththeareaofthewall.178.5.2MAGNETOCRYSTALLINEANISOTROPY

Ferromagneticcrystalscharacteristically­exhibitmagneticanisotropy,whichmeansthatthemagneticpropertiesaredifferentalongdifferentcrystaldirections.Inthecaseofiron(BCC),thespinsinadomainarealmosteasilyalignedinanyofthesix[100]typedirections,collectivelylabelledas‹100›,andcorrespondtothesixedgesofthecubicunitcell.Theexchangeinteractionsaresuchthatspinmagneticmomentsaremosteasilyalignedwitheachotheriftheyallpointinoneofthesix‹100›directions.Thus‹100›directionsintheironcrystalconstitutetheeasydirectionsformagnetization.18WhenamagnetizingfieldHalonga[100]directionisapplied,asillustratedinFigure8.23aandb,domainwallsmigratetoallowthosedomains(e.g.,A)withmagnetizationsalongHtogrowattheexpenseofthosedomains(e.g.,B)withmagnetizationsopposingH.TheobservedMversusHbehaviourisshowninFigure8.24.Magnetizationrapidlyincreasesandsaturateswithanappliedfieldoflessthan0.01T.

19Ontheotherhand,ifwewanttomagnetizethecrystalalongthe[111]directionbyapplyingafieldalongthisdirection,thenwehavetoapplyastrongerfieldthanthatalong[100].ThisisclearlyshowninFigure8.24,wheretheresultingmagnetizationalong[111]issmallerthanthatalong[100]forthesamemagnitudeofappliedfield.Indeed,saturationisreachedatanappliedfieldthatisaboutafactorof4greaterthanthatalong[100].The[111]directionintheironcrystalisconsequentlyknownastheharddirection.TheMversusHbehaviouralong[100],[110],and[111]directionsinanironcrystalandtheassociatedanisotropyareshowninFigure8.24.20Itisapparentthatthemagnetizationofthecrystalalong[100]needstheleastenergy,whereasthatalong[111]consumesthegreatestenergy.TheexcessenergyrequiredtomagnetizeaunitvolumeofacrystalinaparticulardirectionwithrespecttothatintheeasydirectioniscalledthemagnetocrystallineanisotropyenergyandisdenotedbyK.Foriron,theanisotropyenergyiszerofor[100]andlargestforthe[111]direction,about48kJ.m-3or3.5×10-6evperatom.Forcobalt,whichhastheHCPcrystalstructure,theanisotropyenergyisatleastanorderofmagnitudegreater.Table8.4summarizestheeasyandharddirections,andtheanisotropyenergyKfortheharddirection21Werecallthatthespinmagneticmomentsrotateacrossadomainwall.Aschematicillustrationofthestructureofatypical180°Blochwall,betweentwodomainsAandB,isdepictedinFigure8.25.Itcanbeseenthattheneighbouringspinmagneticmomentsarerotatedgradually,andoverseveralhundredatomicspacingsthemagneticmomentreachesarotationof180°8.5.3DOMAINWALLS22Example8.4Figure8.26showsthecontributionsoftheexchangeandanisotropyenergies,and,tothetotalBlochwallenergyasafunctionofwallthicknessδ.Itisclearthatexchangeandanisotropyenergieshaveopposite(orconflicting)requirementsonthewallthickness.Thereis,however,anoptimumthicknessthatminimizestheBlochwallenergy,thatis,athicknessthatbalancestherequirementsofexchangeandanisotropyforces.Iftheinteratomicspacingisa,thentherewouldbeatomiclayersinthewall.Sincethespinmomentanglechangesby180°acrossδ,wecancalculatetherelativespinorientationsofadjacentatomiclayers,andhencewecanfindtheexactcontributionsofexchangeandanisotropyenergies.Wedonotneedtheexactmathematics,butthefinalresultisthatthepotentialenergyperunitareaofthewallisapproximately:23Thefirsttermontherightsistheexchangeenergycontributions(proportionalto/),andthesecondistheanisotropyenergycontribution(proportionalto);bothhavethefeatureswediscussed.Showthattheminimumenergyoccurswhenthewallhasthethickness24TakingwhereistheCurietemperature,andforiron,K=50KJm-3,anda=0.3nm,estimatethethicknessofaBlochwallanditsenergyperunitarea.25Thedomainstructureineachgrainwilldependonthesizeandshapeofthegrainand,tosomeextent,onthemagnetizationsinneighboringgrains.Althoughverysmallgrainsperhapssmallerthan0.1µm,maybesingledomains,inmostcasesthemajorityofthegrainswillhavemanydomains.Overall,thestructurewillpossessnonetmagnetization,providedthatitwasnotpreviouslysubjectedtoanappliedmagneticfield.WecanassumethatthecomponentwasheatedtoatemperatureabovetheCuriepointandthenallowedtocooltoroomtemperaturewithoutanappliedfield.26Supposethatwestartapplyingaverysmallexternalmagneticfield(µ0H)alongsomedirection,whichwecanarbitrarilylabelas+x.Thedomainwallswithinvariousgrainsbegintomovesmalldistances,andfavorablyorienteddomains(thosewithacomponentofMalong+x)growalittlelargerattheexpenseof