Quantum Theory of Molecular Magnetism

arXiv:cond-mat/0501625v1[cond-mat.str-el]26Jan2005X5QuantumTheoryofMolecularMagnetismJ¨urgenSchnackFachbereichPhysikUniversit¨atOsnabr¨uckContents1Introduction22Substances23Theoreticaltechniquesandresults43.1Hamiltonian....................................43.2Evaluatingthespectrum..............................63.3Evaluationofthermodynamicobservables....................153.4Propertiesofspectra...............................164Dynamics244.1Tunneling.....................................244.2Relaxationdynamics...............................255Magnetocalorics27X5.2J¨urgenSchnack1IntroductionThesynthesisofmolecularmagnetshasundergonerapidprogressinrecentyears[1,2,3,4,5,6].Eachoftheidenticalmolecularunitscancontainasfewastwoanduptoseveraldozensofparamagneticions(“spins”).Oneofthelargestparamagneticmoleculessynthesizedtodate,thepolyoxometalate{Mo72Fe30}[7]contains30ironionsofspins=5/2.Althoughthesematerialsappearasmacroscopicsamples,i.e.crystalsorpowders,theintermolecularmag-neticinteractionsareutterlynegligibleascomparedtotheintramolecularinteractions.There-fore,measurementsoftheirmagneticpropertiesreflectmainlyensemblepropertiesofsinglemolecules.Theirmagneticfeaturespromiseavarietyofapplicationsinphysics,magneto-chemistry,bi-ology,biomedicineandmaterialsciences[1,3,4]aswellasinquantumcomputing[8,9,10].Themostpromisingprogresssofarisbeingmadeinthefieldofspincrossoversubstancesusingeffectslike“LightInducedExcitedSpinStateTrapping(LIESST)”[11].Itappearsthatinthemajorityofthesemoleculesthelocalizedsingle-particlemagneticmomentscoupleantiferromagneticallyandthespectrumisratherwelldescribedbytheHeisenbergmodelwithisotropicnearestneighborinteractionsometimesaugmentedbyanisotropyterms[12,13,14,15,16].Thus,theinterestintheHeisenbergmodel,whichisknownalreadyforalongtime[17],butusedmostlyforinfiniteone-,two-,andthree-dimensionalsystems,wasrenewedbythesuccessfulsynthesisofmagneticmolecules.Studyingsuchspinarraysfocusesonqualitativelynewphysicscausedbythefinitesizeofthesystem.Severalproblemscanbesolvedwithclassicalspindynamics,whichturnsouttoprovideaccu-ratequantitativeresultsforstaticproperties,suchasmagneticsusceptibility,downtothermalenergiesoftheorderoftheexchangecoupling.However,classicalspindynamicswillnotbethesubjectofthischapter,itiscoveredinmanypublicationsonMonte-Carloandthermostatedspindynamics.OneoverviewarticlewhichdiscussesclassicalspinmodelsinthecontextofspinglassesisgivenbyRef.[18].Theoreticalinorganicchemistryitselfprovidesseveralmethodstounderstandanddescribemolecularmagnetism,seeforinstanceRef.[19].Inthischapterwewouldliketofocusonthosesubjectswhichareofgeneralinterestinthecontextofthisbook.2SubstancesFromtheviewpointoftheoreticalmagnetismitisnotsoimportantwhichchemicalstructuresmagneticmoleculesactuallyhave.Nevertheless,itisveryinterestingtonotethattheyappearinalmostallbranchesofchemistry.Thereareinorganicmagneticmoleculeslikepolyoxometa-lates,metal-organicmolecules,andpurelyorganicmagneticmoleculesinwhichradicalscarrythemagneticmoments.Itisalsofascinatingthatsuchmoleculescanbesynthesizedinahugevarietyofstructuresextendingfromratherunsymmetricstructurestohighlysymmetricrings.OneofthefirstmagneticmoleculestobesynthesizedwasMn-12-acetate[20](Mn12)–[Mn12O12(CH3COO)16(H2O)4]–whichbynowservesasthe“drosophila”ofmolecularmag-netism,seee.g.[1,21,4,22,23].AsshowninFig.1themoleculescontainsfourMn(IV)ions(s=3/2)andeightMn(III)ions(s=2)whicharemagneticallycoupledtogiveanS=10groundstate.Themoleculespossessesamagneticanisotropy,whichdeterminestheobservedrelaxationofthemagnetizationandquantumtunnelingatlowtemperatures[21,24].Althoughtheinvestigationofmagneticmoleculesingeneral–andofMn-12-acetateinpartic-QuantumTheoryofMolecularMagnetismX5.3Fig.1:StructureofMn-12-acetate:Onthel.h.s.theMnionsaredepictedbylargespheres,onther.h.s.thedominantcouplingsaregiven.WithfriendlypermissionbyG.Chaboussant.ular–hasmadegreatadvancesoverthepasttwodecades,itisstillachallengetodeducetheunderlyingmicroscopicHamiltonian,eveniftheHamiltonianisofHeisenbergtype.Mn-12-acetateisknownforabout20yearsnowandinvestigatedlikenoothermagneticmolecule,butonlyrecentlyitsmodelparameterscouldbeestimatedwithsatisfyingaccuracy[25,26].Fig.2:Structureofachromium-4andachromium-8ring.TheCrionsaredepictedbylargespheres.Anotherverywellinvestigatedclassofmoleculesisgivenbyspinringsamongwhichironrings(“ferricwheels”)aremostpopular[27,28,29,30,31,32,33,34].Iron-6ringsforinstancecanhostalkaliionssuchaslithiumorsodiumwhichallowstomodifytheparametersofthespinHamiltonianwithinsomerange[16,35].Anotherrealizationofringsispossibleusingchromiumionsasparamagneticcenters.Figure2showsthestructureoftworings,onewithfourchromiumionstheotheronewitheightchromiumions.Anewroutetomolecularmagnetismisbasedonso-calledKepleratestructureswhichallowthesynthesisoftrulygiantandhighlysymmetricspinarrays.Themolecule{Mo72Fe30}[7,36]containing30ironionsofspins=5/2mayberegardedasthearchetypeofsuchstructures.Figure3showsonthel.h.s.theinnerskeletonofthismolecule–FeandO-Mo-Obridges–aswellastheclassicalground