Spherical Model for Anisotropic Ferromagnetic Films

The corrections to the Curie temperature T_c of a ferromagnetic film consisting of N layers are calculated for N \gg 1 for the model of D-component classical spin vectors in the limit D \to \infty, which is exactly soluble and close to the spherical model.

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9/tam-dnco:viXraJ.Phys.A:Math.Gen.29(1996)L257–L262.PrintedinHamburgLETTERTOTHEEDITORSphericalmodelforanisotropicferromagnetic lmsDAGaranin I.Institutf¨urTheoretischePhysik,Universit¨atHamburg,Jungiusstr.9,D-20355Hamburg,GermanyReceived26October1995Abstract.ThecorrectionstotheCurietemperatureTcofaferromagnetic lmconsistingofNlayersarecalculatedforN 1forthemodelofD-componentclassicalspinvectorsinthelimitD→∞,whichisexactlysolubleandclosetothesphericalmodel.Thepresentapproachaccounts,however,forthemagneticanisotropyplayingthecrucialroleinthecrossoverfrom3to2dimensionsinmagnetic lms.InthespatiallyinhomogeneouscasewithfreeboundaryconditionstheD=∞modelisnonequivalenttothestandardsphericaloneandalwaysleadstothediminishingofTc(N)relativetothebulk.Theapplicationofthesphericalmodeltospatially-inhomogeneousmagneticsystemssuchasferromagnetic lmswithfreeboundaryconditionsbyBarberandFisher[2]hasrevealedtheunphysicalbehaviourofthesolutionbeingtheconsequenceoftheglobalspinconstraint.ThedependenceTc(N)forad-dimensionalhypercubiclatticein niteind′=d 1dimensionsandhavingNlayersinthedthdimensionhasbeenfoundtobeford≥4anon-monotonousfunctionwithamaximum,i.e.,Tc(N)forN 1waslargerthaninthebulk.OthersingularfeaturesofthesphericalmodelwerefoundbyAbrahamandRobert[3]byconsideringtheproblemofphaseseparation(i.e.,thedomainwallformation).Besidesthenumerouspublicationsusingthesphericalmodelforinhomogeneoussystemsinitsoriginalform(see,e.g.,[4,thereisaworkbyCostache,MaziluundMihalache[6]inwhichtheglobalspinconstraintwasreplacedforaferromagnetic lmbyseparateconstraintsineachlayer.Althoughthismodelislessconvenientforanalyticalcalculations,itwasshownthatford≥4thevalueofTc(N)monotonicallyincreasestoitsbulkvalueTc(∞),asitshouldfromthephysicalgrounds.Earlier

Knops[7]hadprovedthatinageneralinhomogeneoussituationthesphericalmodelwithaspinconstraintoneachlatticesiteisequivalenttotheD-componentclassicalvectormodelbyStanley[8]inthelimitD→∞.Thelatterisnotonlymorephysicallyappealingthantheoriginalsphericalmodel,butitalsoallowsonetotakeintoaccountthespinanisotropy[9]andtoproducethe1/DexpansionsAconvenienttooltohandletheD-vectormodelistheclassicalspindiagramtechniqueThepossibilityofconsideringanisotropicsystemsmakestheanalyticallysolubleD=∞model,whichcanalsobecalledforsimplicitythesphericalmodel,ratherattractiveforapplications.InRef.[14]itwasusedtoinvestigatetheroleof uctuationsinthephasetransitionfromBlochtolineardomainwallsinbiaxialferromagnets. E-mail:garanin@physnet.uni-hamburg.de

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