Coherent Stranski-Krastanov growth in 1+1 dimension

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2v0811000/tam-dnco:viXraCoherentStranski-Krastanovgrowthin1+1dimensionswithanharmonicinteractions:Anequilibriumstudy1ElkaKorutcheva1,∗,2andIvanMarkov3,∗∗DepartamentodeFisicaFundamental,UniversidadNacionaldeEducacionaDistancia,28040Madrid,Spain2LaboratoiredePhysiqueStatistique,EcoleNormalSuperieure,24,rueLhomond,75231ParisCedex-05,France3InstituteofPhysicalChemistry,BulgarianAcademyofSciences,1113Sofia,Bulgaria(February1,2008)Theformationofcoherentlystrainedthree-dimensionalislandsontopofthewettinglayerinStranski-Krastanovmodeofgrowthisconsideredinamodelin1+1dimensionsaccountiownthatcoherent3DislandscanbeexpectedtoformincompressedratherthaninexpandedoverlayersbeyondacriticallatticemisfiattercasetheclassicalStranski-Krastanovgrowthisexpectedtooccurbecausethemisfirmodynamicreasonforcoherent3Dislandyerheightisterexplai2D-3Dtransformationtakesplacebyconsecutiverearrangementsofmono-tobilayer,bi-totrilayerislands,etc.,rrangementsareinitiatedbynucleelismbers:,1

UCTIONThepreparationofarraysofdefectfreethree-dimen-sional(3D)nanoscaleislandsisasubjectofintensere-searchinthelterarepromis-ingforfabricationoflasersandlightemittingdiodes1–nttimetheinstabilityoftwo-dimensional(2D)growthagainsttheformationofcoherentlystrained3DislandsinhighlymismatchedhethewellknownStranski-Krastanov(SK)growthmodewherethedecrementofthestrainenergyieadhesionforcesbetweenthesubstrateandfilmmaterialsovercompensatethestrainenergystoredintheoverlayerowingtothelatticemismatch,athinpseudomorphouswettinglayerconsistingofanintegernumberofmonolayersisfindofgrowthcannotcontinueindefinitelybecauseoftheaccumulationofstrainenergyandthedisappearanceoftheenergeticinfl,inthethermodynamiclimit,unstrained3Dislandsareformedandgrowontopofthewettinglayer,thelatticemis-fitbeingaccommodatedbymisfitdislocations(MDs)atthewettinglayer-3Dislandsboundary5,ewet-tinglayerandthe3DislandsrepresentdifftheclassicalStranski-Krastanovmech-anismofgrowth8(seeFig.(1a)).However,ithasbeenfoundthatundercertainconditionscoherentlystrained(dislocationfree)3Dislandsareformedontopofthewettinglayer(Fig.(1b)).Theseislandsarestrainedtofitthewettinglayerinthemiddleoftheirbasesbutaremoreorlessstrain-freeneartheirtopandsidewalls9,herentlystrainedislandsareformedatlargepos-itivemisfitswhenthelatticeparameteroftheovelsveobservationshavebeenreportedforthegrowthofGeonSi(100)2,4,11–16,InAsonGaAs(100)17–22,InGaAsonGaAs3,23–25,asesthelatticemisfitispositiveandverylarge(4.2,7.2,and≈3.8%forGe/Si,InAs/GaAs,andInP/In0.5Ga0.5P,re-spectively)forsemiconductormateyexceptiontotheauthors’knowledgeisthesystemPbSe/PbTe(111)inwhichthemisfitisnegative(-5.5%)r,theauthorsofRef.(27)notethatwhereasthein-planelatticeparame-terofthePbSewettinglayerisstrainedtofitexactlythePbTesubstrate,theparameterofthe3Dislandsrapidlydecreases,reaching95%ldspeculatethatthelatticemisfistheclassicalSKgrowthismoreorlessclearfromboththermodynamicandkineticpointsofview,tonsiderasafirstapproximationtheformationofcoherent3DislandsinSKgrowthashomoepitaxialgrowthonuniformlystrainedcrystalsur-facebothfi,itisnotclearwhatisthethermo-dynamicdrivingforcefor3Dislandingifsonotclearwhycoherentrquestionwhichshouldbeanswerediswhytheformationofcoherent3Dislandsrequiresverylargevalueofthepositivemisfisonforthenarrowsizedistributionisstillunclearalthoughmuchefforthasbeenmadetoelucidatetheproblem28,y,orapproximationsareusuallymadewhendefirstistheuseofthelineartheoryofelasticityinordertocomputethestraincon-tributiontothetotalenergyoftheislands9,10,28,30–r,thevalidityofthelattbeshownbelowtheMDsdifferdrasticallyincom-pressedandexpandedfi,itiscommonlyac-ceptedthattheinterfacialenergybetweenthewettinglayerandthedislocationfree3Dislandsissuffiterisequivalenttotheassumptionthatthesubstrate(thewettinglayer)wetscompletelythe3Dislands28,31–33,35–thisassumptionrulesoutthe3Dislandingfromthermodynamicpointofviewas3Dislandsareonlypossibleatincompletewetting,orinotherwords,whentheinterfacialenergyisgreaterthanzero8,41–nbelowtheadhesionoftheatomstothewettinglayerisalsodistributedalongtheislandinadditiontothestraindistributionandplaysamoresig-nifihelatticemisfittheatomsaredisplacedfromtheirequilibriumpositionsinthebottomsofthepotentialtroughstheyshouldocupyatzeromisfiawaytheadhesionoftheatomstothesurageadhesionofanislandofafinitesizeisthusweakercomparedwiththatofaninfirfacialboundaryappearsandthewettingoftheislandbythesubstrate(thewettinglayer)isincompletewettingwhichdrivestheformatiresentpaperwemakeuseofamorerealisticinteratomicpotentialwhichischaracterizedbyitsan-2

harmonicity,inthesensethattherepulsivebranchissteeperthantheattractivebranch,andbyitsnoncon-vexity,whichmeansthatitpossessesaninfllyTamandLamhaveusedaMiepor,theabovementionedauthorsdidnotstudytheeffectofmisfier,thedistributionofthestressinthe3adhukar46computedtheenergyandthedistributionofstraininco-herentGeislandsonSi(001)usingamoleculardynamicscoupledwiththeStillinger-Weberpotential47butdidnotstudytheeffofsuchapotentialallowsustoanswerthequestionwhycohereingtheen-ergiesofmono-andmultilayerislandsallowstomakedefiniteconclusionsconcerningthemechanismofforma-tionandgrowthofthe3Dislands,soutthatthereisacritical2Dislandsizeabove,ashasbeenshownearlierbyStoy-anovandMarkov48,6,PriesterandLannoo28andChenandWashburn31,themonolayanothercriticalsizethebilayerislandsbecomeunstableagainsttrilayerislands,,ticalsizeforthemono-bilayertransformationsin-creasessharplywiththedecreaseofthelatticemisfitgo-ingasymptoticallytoinfinityatsomecriticalmisfiolayerislandsarethusalwaysstableagainstthemul-tilayerislandsbelowthiscriticalmisfitwhichexplainsthenecessityoflargemisfiticalmisfitinexpandedoverlayersisnearlytwicegreaterinabsolutevaluethanthatincompressedoverlayerswhichinturnexplainswhycoherent3Dis-landingisveryrarely(ifatall),the2D-3Dtransformationtakesplacebytransportofatomsfromtheedgesofthemonolayerislands,wheretheyareweaklybound,ontopoftheirsurfacestoformislandsoftheupperlayerwheretheyaremorestronglybound6,ocessisthenrepeatedinthetransfor-mationofbilayertotrilayerislands,ticalsizeforthe2D-3Dtransformationtooccuristhaseofexpandedoverlayerstheatomsinteractwitheachotherthroughtheweakerattractivebranchofthepotentieeffectisvery3weak,theaverageadhesionissufficientlystrong,andthecriticalsizesfor2D-3Dtransformationeitherdonotexistorappearunderextremeconditionsofverylargeabso-lutevalueofthemisfierentmonolayerislandsareeitherenergeticallystableagainsteconsideramodelin1+1dimensions(substrate+height)whichwetreatasacross-sectionofthereal2+icitassumptionisthatinthereal2+1casethemonolayerislandshaveacompactratherthanafractalshapeandthelatticemisfighthemodelisqualitativeitgivescorrectlyalltheessentialpropertiesofthereal2+1systemasshownbySnymanandvanderMerwe49–modelthemonolayerislandisrepresentedbyafinitediscreteFrenkel-Kontorovalinearchainofatomssubjecttoanexternalperiodicpotentialexertedbyarigidsubstrate(Fig.(4))52–iderasasubstratetheuniformlystrainedwettinrwords,firststageisaFrank-vanderMerwe(layer-by-layer)paperwerestrictourselvestotheconsir-geticinfluenceoftheinitialsubstrateisalreadylostandthebondingbetweentheatomsinthe3Dislandsisthesameasthatoftheatomsofthefirstatomicplaneofthe3msofthechainareconnectedwithbondsthatobeythegeneralizedMorsepotential55–57