Ferromagnetism

1. Ferromagnetism Characteristicsexperimentsandits „origin“ (by Florian Lüttner)

2. Contents • General expressionsfordescribingmagnetismandferromagnetism • Different kindsofmagnetism - shortoverview • Whatisferromagnetism? - thephenomenonand ist characteristics • Ferromagnetismferrimagnetismandantiferromagnetism • Observation ofmagneticstructures • „Origin offerromagnetism“ - Singletandtrippletstatesfor a twoelectronsystem - Spin Hamiltonian - Heisenbergmodel - Meanfieldapproximation - curie-temperature

3. General expressions • Magnetization: • Vectorfieldthatexpressesthedensityof permanent orinducedmagneticmoments

4. Magneticsusceptibility: Dimensionless proportionalityconstantbetweenthedegreeofmagnetizationandtheappliedmagneticfield

5. Magneticmoment: Determines the angular momentum (ortorque) a magnet will experience in an externalappliedmagneticfield isthevectorthatrelatesthealigningtorque on an objectfrom an externallyappliedmagneticfieldtothefieldvectoritself. With an unknown sample orobjectyoucanmeasurethetorqueby an appliedknownexternalmagneticfieldandgetthemagneticmoment.

6. Gyromagneticratio: Ratio ofitsmagneticmomenttoits angular momentum (classicalbody) The gyromagneticratiocanbewrittenas Thereforethemagneticmomentis For an isolatedelectron (quantummechanical) An electronhas a spin noclassicalrotation (quantummechanicalphenomenon) so

7. But with a correctionfactororelectron g-factor Forthis is Andthereforeiscalculatedas

8. Electron spin: • Followsthe same mathematicallawsasquantized angular momenta, but • Noclassicalrotation quantummechanicalphenomenon • Spin as an intrinsic form of angular momentumforelementaryparticles, hadronsandnucleiwith a definite magnitudeand a direction (upand down) • Spin quantumnumberonlytakes half-integer values (0, ½, 1, 3/2, 2, …) • Onlyspindirectioncanbechanged in spinuporspin down but not ist value • The calculationofthemagneticmomentof an electronneeds an electron g factor different from 1 (like in classicalcases) • Spin angular momentum

9. Spin angular momentum So S also canbecalculatedwith Forthistheelectron (fermion) spinquantumnumberandthe angular momentumcanbecalculatedwith forthis and in thiscase

10. Short overview • Paramagnetism • - < 0 • - magnetizationis different from 0 aslongas an externalappliedmagneticfield • exists • - atoms, moleculesandlatticedefectswith an oddnumberofelectrons • (the total spinof a system must not bezero) • - freeatomsandionswith a partlyfilledinnershell • - metals • - fewcompoundswith an evennumberofelectrons (molecularoxygen)

11. Diamagnetism • - > 0 • - magnetizationis different from 0 aslongas an externalappliedmagneticfield • exists • - theinnermagnetizationis in theoppositedirectiontotheappliedexternalfield • (wanttovanishthefield) • - Bismut • - Carbon • - superconducter • (crowdsthemagneticfieldlinesoftheappliedfield out ofthesuperconducter) • Ferromagnetism • - spontaniousmagneticmoment (saturationmoment) • - electronspinsandmagneticmomenthavetobearranged in a regular • manner

12. Whatisferromagnetism? Ferromagneticorders • Spontanious magnetization saturationmagnetization • Vanishes not  temperatureunderthe Curie-temperature () • theorderofelectronspinsvanishes after theexternalmagneticfieldisturned off (Paramagnetism) • Frominternalinteractionofthemagneticmomentsofthespins (exchangefield)

13. Characteristics(hysteresis)

14. Materials with different formsofmagnetism

15. Ferromagnetism, ferrimagnetismandantiferromagnetism • Usethelanguageappropriateofsolids (magneticionslocalizedatlatticepoints) • ferromagnetism • Herewehave in general non vanishingvectormomentsofthemagneticionsbelow a temperature (orderedspinorientation) • Same directionofmagneticmoments (spins)  addupto a netmagnetizationdensity •  ferromagnetism • Microscopicmagneticorderingisrevealedbytheexistanceof a macroscopicbulkmagnetizationdensity • Not all magneticmomentshavetobethe same • Antiferromagnetism • Magneticorderingofthe individual localmomentsvavanishestozero • Nospontaniousmagnetization • Microscopicmagneticorderingcan not berevealedby a macroscopicbulkmagnetizationdensity • Localmomentsorientatedliketwointerpenetratingsublatticeswiththe same structure

16. ferrimagnetism • Herewehaveeven a non vanishingvectormomentsofthemagneticions • Not the same directionofmagneticmoments (spins) at all • Exchange couplingbetweennearestneighboursmayfavor antiparallel orientation • Neighbourhas not the same valueofmagneticmoment • Leaving a netmagneticmomentforthe solid as a whole • Somemorecomplexarrangementsarepossible • Describtion not withtheabovethreetypes • Specifiingtheordering in termsofspindensity

17. At anypointalonganydirectionthedensityisdescribedby therefore forferrimagnetsandferromagnetsand forantiferromagnets

19. Observation ofmagneticstructure • Can beobservedbythescatteringpatternofneutronscattering • Neutrons have a magneticmoment couplestotheelctronicspin • also nonmagnetic Bragg reflectionofneutronsbyionicnuclei •  additional peaks in theelasticneutronscatteringcrosssection • Over themagneticscatteringpeaksarevanished

20. Crosssectionforneutronelectronscatteringofthe same orderofmagnitudeasforneutronnucleiinteraction • Determination ofthedistributiondirectionandtheorderofthemagneticmoments • Deteminethemagneticstructureofantiferromagnets

21. Origin offerromagnetism • Magnetic effects in a material bythepaulipriciple (evenwithnospindependentterms in theHamiltonian) • considerTwo-electronsystem(withspinindependetHamiltonian) • stationarystateistheproductof a purely orbital stationarystate • satisfies orbital Schrödingerequation • Fourspinstateswithbothelectrons in levelsof definite • Chooselinearcombinationsofthesestatestoget definite valuesofthe total spin (anditscomponent)

22. Singletstate Triplet states Combination oftwoparticlescanonly carry a total spinof 1 or 0 (occupytripletorsingletstate) Anycombination (like) canbecalculatedby Span a 4-dimensional room ( for particles) CalculatedwithClebsch-Gordon coefficients

23. Substituting in thefourbasisstates for for for for Thereforeyougetthreestateswith total spin angular momentum 1 andonestatewith total spin angular momentum 0

24. Total wavefunctionchangesignundersimultaniousninterchangeofthespin () and orbital parts () • istheproductofitsspinand orbital parts • Thereforeif do not changesignunderinterchangeof •  solutionissymmetric •  must describestateswith • changesignunderinterchangeof •  antisymmetricsolution •  must describestateswith • If andarethelowesteigenvaluesofthespin-independent orbital schrödingerequationassociatedtothesymmetricandantisymmetricsolutionofitthenthegroundstate will onlyhavespin 0 orspin 1 • This onecanrequireonlyby an examinationofthespin-independent Schrödingerequation • Fortwoelectronproblemtheground-statewavefunctionhavetobesymmetric • Onlyfortwoelectronsystem • Havetoestimatethatcanbegeneralizedtotheanologousproblemof a N-atom solid

25. Spin Hamiltonian • A wayto express dependenceof a two-electronconfigurationspin on thesinglet-triplet energysplitting • Importantfortheanalyzationofenergiesofthespinconfigurationof real insulatingsolids • protonswithgreatdistances twoindependent hydrogen atomsdescribedbythegroundstate • (fourfolddegeneratedwithtwopossibleorientationsofeachelectronspin) • Bring theprotonsclosertoeachother (molecule)  splittingofthefourfolddegeneracy (due toatominteractions) • This interactionissmallcomparedwithotherexcitationenergiesof a two-electronsystem • Simplifyingbyignoringhigherstatesoverfourfolddegenerated • Moleculeas a simple four-statesystem • Representgeneralstateofthemoleculeby linear combinationoftheforurloweststates • Convenienttohave an operator • Spin Hamiltonian

26. Same eigenvaluesas in the original Hamiltonianwithinthefourstatemanifold • Eigenfunctionsgivethespinofeachstate • Each individual spinoperatorsatisfies • Thereforethe total spinsatisfies • hastheeigenvaluesandwiththeequationaboveforwegettheeigenvaluesoftheoperatorasforthesingletstateandforeachtripletstate • Therefore is • Redefiningthezeroofenergyandomittheconstantwhichiscommonto all fourstatesweget

27. Heisenberg model • isthescalarproductofthespinoperators parallel spinsfor (leadstoferromagnets) and antiparallel spinsfor (leadstoferrimagnetsandantiferromagnets) • Coupling in thespinHamiltonian do not depend on thespatialdirectionwithrespectto but on theralativeorientationofthetwospins • isIsotropic but weneedunisotropictermstodescribeferromagnetism • Includetermsthatbraeakrotationalsymmetrie in spinspace • Spin Hamiltonianonlyforinsulatingmaterials (Hubbard modelformetals) • Therefore N widelyseperatedionswith a smalloverlapoftheelectronwavefunction • Onlyinteractionbetweennearestneighbours • Groundstate will be gegenerated • Spin Hamiltoniandescribesthesplittingofthisgroundstatewhenionsaresomeclosertogetherthatthesplittingsaresmallcomparedwithanyotherexcitationenergies • The eigenvaluesofthegivesthesplitlevels • Formanycasesofinterestis in the form ofthetwo-spin casesummedover all pairsofions

28. istheexchangecouplingconstant (exchangeenergy) • Exchange interactionbetweenlocalizedspinsonlyfromcoulombrepulsionandfrompauliprinciple • For angular momentumsdepending on orbital aswellas on spinpartstheHamiltoniandepends on the absolute spinoriantationsaswellas on the relative ones

29. Meanfieldapproximation • Early analysisoftheferromagneticeffectsby P. Weisswiththemeanfieldtheory • This theoryfailstopredictspinwavesatlowand high temperatures but a goodapproximationat • Supposewefocusourattention on a particularBravaislatticesiteR in the Heisenberg Hamiltonian • With (ferromagnet) • Representsforeachionthe total angular momentumwithspinand orbital partusualtotakethesefictitiousspinstobe parallel tothemagneticmomentoftheion

30. And isolatingfromthosetermscontaining • Hasthe form of an energyof a spin in an effectiveexternalfield • But thisis an operatordepending on a complicatedway on thedetailedspinconfigurationof all theotherspinatdifferent sitesfromR • Meanfieldapproximation • Replacewithitstermalequilibriummeanvalue • (replace all spinvaluesbyitstermalequilibrium) • In ferromagnets all spinshavethe same meanvalue • In termsofthe total magnetizationdensityitis

31. Replace eachspin in byitsmeanvalue Wearriveattheeffectivefield with and Every magneticatomexperiences not onlythemagnetizationofitsnearestneighbour but an averagemagnetizationof all theothermagneticatoms With a spontaniousmagnetizationatitisusual in ferromagneticswithnoappliedfieldwecanassume as

32. Curie temperature • It isknownthat • Sinceisgivenby • without an appliedfieldthemagnetizationisgivenby • Withthemagnetizationdensitywithnoappliedfield • Seperate in a pair ofequationslike • Putbothequationsasgraphs in a plot

33. Important aretheintersectionsof and • Ifandonlyiftheslopeofthestraightlineislessthanthatoneofattheorigin, at a nonzerovalueof x • canbecalculated in termsofzerofieldsusceptibility, calculated in theabsenceofinteractions, for

34. From theanalysisof a setofidenticalionsof angular momentum J onecangetcurieslaw • with • Comparecurieslawwith (canread off thevalueof ) • Critical temperatureisgivenby

35. Not exactly • More precisewiththeIsingmodel • Simplification oftheHeisenbergmodel onlywithspins on oneaxes (z-axes) andonlyspinswiththevalue • onlyforneighboringions

36. Sources • Books • Ashcroft/Mermin, Solid State Physics Ch. 31, 32, 33 • Charles Kittel, Introductions to Solid State Physics (eight edition) Ch.12, 13 • Weblinks • http://de.wikipedia.org/wiki/Ferromagnetismus • http://en.wikipedia.org/wiki/Gyromagnetic_ratio • http://de.wikipedia.org/wiki/Diamagnetismus • http://de.wikipedia.org/wiki/Paramagnetismus • http://en.wikipedia.org/wiki/Magnetic_susceptibility • http://en.wikipedia.org/wiki/Magnetization • http://de.wikipedia.org/wiki/Elektronenspin

37. http://www.google.de/imgres?imgurl=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fthumb%2F0%2F04%2FPermeability_by_Zureks.svg%2F220px-Permeability_by_Zureks.svg.png&imgrefurl=http%3A%2F%2Fde.wikipedia.org%2Fwiki%2FParamagnetismus&h=170&w=220&tbnid=4qdQ9MfddgwCgM%3A&zoom=1&docid=19Wq_TNMEP5fmM&ei=FSc4U4OmGsfOtQbikIHoCQ&tbm=isch&iact=rc&dur=1325&page=1&start=0&ndsp=15&ved=0CFsQrQMwAQhttp://www.google.de/imgres?imgurl=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fthumb%2F0%2F04%2FPermeability_by_Zureks.svg%2F220px-Permeability_by_Zureks.svg.png&imgrefurl=http%3A%2F%2Fde.wikipedia.org%2Fwiki%2FParamagnetismus&h=170&w=220&tbnid=4qdQ9MfddgwCgM%3A&zoom=1&docid=19Wq_TNMEP5fmM&ei=FSc4U4OmGsfOtQbikIHoCQ&tbm=isch&iact=rc&dur=1325&page=1&start=0&ndsp=15&ved=0CFsQrQMwAQ • http://hyperphysics.phy-astr.gsu.edu/hbase/solids/imgsol/hystcurves.gif • http://hyperphysics.phy-astr.gsu.edu/hbase/solids/imgsol/hyloop.gif • http://de.wikipedia.org/wiki/Ising-Modell • http://de.wikipedia.org/wiki/Heisenberg-Modell_%28Quantenmechanik%29

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