93Flat (001) surfaces of II-VI semiconductors A la

arXiv:cond-mat/0111114v1[cond-mat.stat-mech]7Nov2001Flat(001)surfacesofII-VIsemiconductors:AlatticegasmodelMartinAhr∗,MichaelBiehlInstitutf¨urTheoretischePhysikundAstrophysikJulius-Maximilians-Universit¨atW¨urzburgAmHubland,D-97074W¨urzburg,GermanyFebruary6,2008AbstractWepresentatwo-dimensionallatticegaswithanisotropicinteractionswhichmodeltheknownpropertiesofthesurfacereconstructionsofCdTeandZnSe.Incontrasttoanearlierpublication[12]theformationofaniondimersisconsidered.Thisaltersthebehaviourofthemodelconsiderably.WedeterminethephasediagramofthismodelbymeansoftransfermatrixcalculationsandMonteCarlosimulations.Wefindqualitativeagreementwiththeresultsofvariousexperimentalinvestigations.Keywords:Equilibriumthermodynamicsandstatisticalmechanics,MonteCarlosimulations,Surfacerelaxationandreconstruction,Surfacethermodynamics(includ-ingphasetransitions),Cadmiumtelluride,Zincselenide,Lowindexsinglecrystalsurfaces.1IntroductionWithinthelastyears,potentialtechnologicalapplicationsofelectronicdevicesbasedonII-VIsemiconductors[1]haveinspiredbasicresearchconcerningsurfacesofthesematerials.Inthiscontext,variousstudieshaveaddressedthepropertiesofsurfacereconstructions.Experimentalstudieshaveinvestigatedwhichreconstructionsarepresent[2,3]andhowthereconstructionofthesurfaceisinfluencedbyparametersliketemperatureandparticlefluxinanMBEenvironment[4,5,6].ThemajorityofthisworkhasbeendevotedtoCdTeandZnSe,whereafairlycompletequalitativeoverviewoverthephasediagramhasbeengained.AnoverviewoverthepropertiesofCdTecanbefoundin[7].Ontheotherhand,therehavebeentheoreticalinvestigationsofthereconstructionsofCdTe[8]andZnSe[9,10]usingdensityfunctionaltheory.Inthesestudies,knowledgeaboutthechemicalbondingofsurfaceatomsandgroundstateenergiesofvariousreconstructionshasbeengained.Beingbasedonquantummechanics,densityfunctionaltheoryisbelievedtobeexactapartfromapproximationsmadeinthecalculation.Thecomputationalburdenofthismethodiscomparativelyhigh,whichrestrictsitspracticalapplicabilitytosystemscon-sistingofonlyafewatoms.Duetotheperiodicityofcrystalsurfacesthisisnotasevere∗Correspondingauthor.Phone:[phone]Fax:+49(0)[e-mail]1(a)(b)(c)εbεtεx(d)εdFigure1:Panels(a),(b),(c):SketchesofthereconstructionsofCdTewhicharediscussedinthiswork.Thegreyrectanglesshowthesurfaceunitcells.The[110]axisisalignedhorizontally.(a),(b):Cd-terminatedreconstructions.Panel(a)showsthec(2×2)Cdreconstruction,panel(b)the(2×1)Cdreconstruction.Panel(c)showsthe(2×1)TereconstrutionofsurfacesterminatedwithacompletemonolayerofTe.Panel(d)showstheattractivecouplingsinourmodel.restrictionifoneisinterestedingroundstatepropertiesofthesystemwhichhowever,arestrictlyrelevantonlyatzerotemperature.Athighertemperature,thepropertiesofthesurfacewillbeinfluencedbythermodynamiceffects.Theseareparticularlyimportantifphasetransitionsbetweendifferentreconstructionsoccur.Theirtheoreticalinvestigationrequiresthestudyofsystemswithalargenumberofatomswhichisbeyondthescopeoffirstprinciplesmethodsormoleculardynamicssimulationsusingrealisticempiricalpotentials.Consequently,simplifyingmodelsareneededwhichpreserveessentialfeaturesofatomicinteractionsandcanbeinvestigatedwithmoderatenumericaleffort.Inmanycases,two-dimensionallatticegaseshavebeenusedsuccessfullytomodelatomsadsorbedonasingularcrystalsurfaceortheterminatinglayerofsuchacrystal[11,12,13,14].Inspiteoftheconceptualsimplicityofsuchmodelstheinterplayofattractiveandrepulsiveshortrangeinteractionscanresultinhighlynontrivialcriticalbehaviourandcomplexphasediagrams.Inthispaper,wewillfollowthisapproachtomodelthereconstructionsof(001)surfacesofCdTeandZnSe,ourmainfocusbeingonCdTe.Theoutlineofthispaperisasfollows:Insection2,wewillgiveashortreviewoftheknownfactsaboutthereconstructionsofthe(001)surfacesofCdTeandZnSe.Insection3,weintroducealatticegasmodelwhichconsiderstheoccupationofCdsitesandthedimerizationofTeatomsanddiscussitsphasediagram.Weconcludewithacomparisonofthefeaturesofourmodelwithexperimentalresultsinsection4.2SurfacereconstructionsofCdTeandZnSeBothCdTeandZnSecrystallizeinthezinc-blendelattice.Thislatticestructureiscom-posedofalternatinglayersofcationsandanionswhichareparalleltothe(001)surface,suchthatanideal(001)surfacewouldbeterminatedbyacompletelayerofoneparticlespecies.Thepositionsoftheatomsinonelayerlieonaregularsquarelatticewithitsaxesorientedinthe[110]directionandthe[110]direction.Undervacuum,the(001)surfaceofCdTeisCdterminated.Thesurfaceischaracter-izedbyvacancystructureswherelessthanonehalfofthepotentialCdsitesinthetop2layerareoccupied[5,6].Thiscanbeunderstoodfromsimplequantum-mechanicalcon-siderationsliketheelectroncountingrule[15,16],whichstatesthatasurfaceterminatedbyacompletelayerofCdisenergeticallyunfavourable.Atlowtemperature,onefindsac(2×2)Cdreconstruction[7,2],whereCdatomsandvacanciesarrangeinacheckerboardpattern(figure1a).Frequently,acontributionofa(2×1)Cdarrangementcanbefound.Inthe(2×1)Cdstructure,theCdatomsarrangeinrowsalongthe[110]directionwhichalternatewithrowsofvacancies(figure1b).Densityfunctionalca