Computational approach to finite size and shape ef

arXiv:cond-mat/0701225v1[cond-mat.mtrl-sci]10Jan2007ComputationalapproachtofinitesizeandshapeeffectsinironnanomagnetsMichaelMcGuiganJamesDavenportComputationalScienceCenterBrookhavenNati[e-mail]@[e-mail]alidateacomputationalapproachtonanomagnets.ItisbuiltonthespinwaveapproximationtoaHeisenbergferromagnetwhosepa-rameterscanbecalculatedfromafirstprinciplestheory(e.g.densityfunctionaltheory).Themethodcanbeusedforhighthroughputanalysisofavarietyofnanomagneticmaterials.Wecomputethedependenceofthemagnetizationofanironnanomagnetontemperature,sizeandshape.Theapproachisappliedtonanomagnetsintherangeof432atomsto59millionatoms,asizewhichisseveralordersofmagnitudebeyondthescalabilityofdensityfunctionaltheory.1IntroductionNanomagnetsarepromisingmaterialsforhighdensitydatastoragedevices,astheyrepresentabitbyasingledotormagneticdomain.Insuchsystemsdatastoragedensitiesgreaterthan1012bits/in2maybepossiblewhilebeingthermallystableatroomtemperature[1].Importantdesignchoicesforsuchsystemsarethematerialcomposition,thenumberofatoms,andtheshapeofthedot.Allofthesechoicescanstronglyaffectthemagneticpropertiesofthesystem.However,anexperimen-talsearchofdifferentcombinationsistimeconsuming.Thus,itisimportanttoinvestigateefficientcomputationalapproachestocalculatingmagneticproperties.Onemightexpectsuchcomputationalapproachestoreducethecostofbringingnanomagneticstoragedevicestomarket.1Asignificanteffortisdevotedtothecalculationofmagneticpropertiesofmateri-alsfromfirstprinciples.Thesestudiestypicallydescribezerotemperaturepropertiesofasmallnumberofatoms(lessthanathousand).Aeffectivespinmodelwithpa-rameterstakenfromsuchafirstprinciplemodelcanbesimulatedtopredictthebehavioroflargersystemsathighertemperatures.Thusfrombothfundamentalandpracticalpointsofview,anefficientcompu-tationalapproachtomagnetsisimportant.OnesuchapproachisgivenbytheHeisenbergmodelwiththeHamiltonian:H=−Xi,jJ(i−j)~Si·~Sj+gμB~Bext·Xi~Si+Hcrystal−aniso,(1.1)where~Siisathreecomponentspinatlatticesitei=(i1,i2,i3)satisfyingtheO(3)invariantconstraintSx2i+Sy2i+Sz2i=S(S+1).Intheabove,gistheLandeg-factor,μBtheBohrmagneton,~BextanexternalmagneticfieldandSisthespinoftheHeisenbergmodel.ThefirsttermiscalledtheexchangeHamiltonian,thesecondtheZeemanHamiltonian,andthethirdthecrystalanisotropy.Inthispaperweshallrestrictourselvestothefirsttermonly.Itprovidesagooddescriptionofthemagnetizationcurveinzeromagneticfield.Weshallreturntothephysicaleffectsoftheremainingtermsinafuturestudy.ThenumberoflatticesitesorspinswillbedenotedbyN=2NxNyNzwhereNx,Ny,Nzarethenumberofatomsalongabcclatticedimensionforarectangularparal-lelepiped.ForthequantumHeisenbergmodelthespinsdonotcommuteandobeyanalgebraforwhich[Sx,Sy]=iSz.TheparametersJareexchangeconstantswhichcanbecalculatedfromafirstprinciplesmodel.AlthoughtheHeisenbergmodelcanbeformulatedforanymaterialwheretheJvaluesareknown,inthispaperwestudythequantumHeisenbergmodelwithS=1usedtodescribebcciron,whichisanimportantmaterialfromapractical[2]aswellasafundamentalpointofview.TheexperimentaldatafortheratioofthemagnetizationofbccirontoitsvalueatT=0isfoundin[3]tofourdigitaccuracy.Wewillextendouranalysistoothermaterialsinfuturework.ForbccirontheJvalueshavebeencalculated[4]andarelistedinTable1.HereonemRycorrespondsto13.6056923meVand157.887324Kelvin.WeplottheseJvaluesasafunctionofr=|i−j|inFig.1.Wecanseethatthecouplingsareferromagneticouttosecondnearestneighbor.UsingsomewhatdifferentvaluesforJ,theCurietemperaturewasestimatedtobe1414Kinameanfieldapproximationand950KusingtheRandomPhaseapproximation[5].Thelatteriswithin10percentoftheobservedvalue1043K.Anevenbetterestimate,obtainedusingalocalspindensityapproximation[6]gives2i−jJ[mRy](12,12,12)1.97(1,0,0).623(1,1,0)-.132(32,12,12)-.166(1,1,1)-.271(2,0,0).056(32,32,12)-.028(2,1,0).051(2,1,1)-.033(32,32,32).096Table1:Jvaluesfromreference[4]1070K.TheaboveresultsindicatethattheHeisenbergmodelofferromagnetismcanbederivedfromfirstprinciplesandgivesagooddescriptionofferromagnetisminbulkbcciron.Inthenexttwosectionsweapplythismodeltothedescriptionofbccironnanomagnets,whichareofgreatexperimentalandtheoreticalinterest.Thispaperisorganizedasfollows.InSection1wesummarizeourcomputationalmethod.InSection2wecomputethefinitevolumeeffectsinironnanomagnets.InSection3wecomputethefiniteshapeeffectsinironnanomagnetsatfixedvolume.InSection4wepresentourmainconclusions.Ourcomputationalmethodcanbesummarizedasfollows:(1)DeterminetheparametersJofaneffectiveHeisenbergmodelfromafirstprincipleszerotemperaturecalculationand/orlowtemperaturemeasurement.(2)Choosealatticegeometrydescribingthenanomagnet.Thiswilldetermineitssizeandshape.Inthispaperwewillchoosebccironandaparallelepipedlatticegeometry,butotherchoicesarepossible.(3)ApplytheHolstein-PrimakofftransformationtomapthespinvariablestoasetofharmonicoscillatorswhosenearestneighborinteractionsdeterminealatticeLaplacian.(4)SolvefortheeigenstatesandeigenvaluesofthislatticeLaplacian.(5)Usetheseeigenvaluestodeterminethemagnondispers