Scaling properties of step bunches induced by subl

arXiv:cond-mat/0409013v1[cond-mat.mtrl-sci]1Sep2004Scalingpropertiesofstepbunchesinducedbysublimationandrelatedmechanisms:AunifiedperspectiveJ.KrugInstitutf¨urTheoretischePhysik,Universit¨atzuK¨oln,Z¨ulpicherStrasse77,50937K¨oln,GermanyV.Tonchev,S.StoyanovInstituteofPhysicalChemistry,BulgarianAcademyofSciences,1113Sofia,BulgariaA.PimpinelliLASMEA,UMR6602CNRS/Universit´eBlaise-Pascal–Clermont2,F-63177Aubi´ereCedex,France(Dated:February2,2008)ThisworkprovidesagroundforaquantitativeinterpretationofexperimentsonstepbunchingduringsublimationofcrystalswithapronouncedEhrlich-Schwoebel(ES)barrierintheregimeofweakdesorption.Astrongstepbunchinginstabilitytakesplacewhenthekineticlengthdd=Ds/Kdislargerthantheaveragedistancelbetweenthestepsonthevicinalsurface;hereDsisthesurfacediffusioncoefficientandKdisthestepkineticcoefficient.Intheoppositelimitdd≪ltheinstabilityisweakandstepbunchingcanoccuronlywhenthemagnitudeofstep-steprepulsionissmall.ThecentralresultarepowerlawrelationsoftheformL∼Hα,lmin∼H−γbetweenthewidthL,theheightH,andtheminimuminterstepdistancelminofabunch.Theserelationsareobtainedfromacontinuumevolutionequationforthesurfaceprofile,whichisderivedfromthediscretestepdynamicalequationsforthecasedd≫l.Theanalysisofthecontinuumequationrevealstheexistenceoftwotypesofstationarybunchprofileswithdifferentscalingproperties.Throughcomparisonwithnumericalsimulationsofthediscretestepequations,weestablishthevalueγ=2/(n+1)forthescalingexponentoflminintermsoftheexponentnoftherepulsivestep-stepinteraction,andprovideanexactexpressionfortheprefactorintermsoftheenergeticandkineticparametersofthesystem.ForthebunchwidthLweobservesignificantdeviationsfromtheexpectedscalingwithexponentγ=1−1/α,whichareattributedtothepronouncedasymmetrybetweentheleadingandthetrailingedgesofthebunch,andthefactthatbunchesmove.Throughamathematicalequivalenceonthelevelofthediscretestepequationsaswellasonthecontinuumlevel,ourresultscarryovertotheproblemsofstepbunchinginducedbygrowthwithastronginverseESeffect,andbyelectromigrationintheattachment/detachmentlimitedregime.Thusourworkprovidessupportfortheexistenceofuniversalityclassesofstepbunchinginstabilities[A.Pimpinellietal.,Phys.Rev.Lett.88,206103(2002)],butsomeaspectsoftheuniversalityscenarioneedtoberevised.PACSnumbers:68.35.-p,81.10.-h,05.70.Np,89.75.DaI.INTRODUCTIONTheformationofstepbunchesatavicinalsurfaceisaproblemofgreatcurrentinterest,bothfromafundamentalviewpointandwithregardtothepossibleusesofstepbunchesasnanotemplatesornanostructures1,2,3,4,5,6.Mech-anismscausingstepbunchinginstabilitiesincludestraineffects1,2,7,8,sublimationunderconditionsofasymmetricdetachmentkineticsknownastheEhrlich-Schwoebel(ES)effect9,10,11,growthwithaninverseESeffect9,12,13,14,andsurfaceelectromigration15,16,17,18,19,20,21,22,23,24,25,26,27,28.Quiterecentlyitwasrealisedthatstepbunchingisapromisingwaytostudytheinteractionsbetweenthesteps29,30,31,32,33,34.Thephysicalgroundissimple:Thestepsinthebunchkeepacertaindistancefromeachotherbecausethestep-steprepulsionbalancesthetendencytofurthercompressionofthebunch.Thefreeenergyrelatedtothestep-stepinteractionisoftheformA/ln,wherelistheinterstepdistance.Whenn=2,theamplitudeA(T)accountsforbothelasticandentropicrepulsionbetweenthesteps35.Undercrystal-vapourequilibriumonehastherelationA(T)∼g(T)whereg(T)isthesteprepulsioncoefficientintheexpressionf(ρ)=f(0)+κρ+gρ3(1)forthesurfacefreeenergy(perunitprojectedarea)ofavicinalcrystalsurfacewithadensityofstepsρ.Toinferinformationaboutthestep-stepinteractionsfromexperimentalobservationsofbunchmorphology,onemakesuseofscalingrelationsbetweenthelengthandtimescalescharacterizingthebunches.TherelevantlengthscalesarethewidthLandtheheightHofthebunch,andthespacingξbetweensubsequentbunches(Fig.1).Thelengthξisalsosometimesreferredtoastheterracewidth;thisnomenclatureissomewhatambiguous,however,2ξLHFIG.1:Schematicofabunchedvicinalsurface,illustratingthedefinitionofthebunchwidthL,thebunchheightHandthebunchspacingξ.becausetheregionbetweentwobunchesmaycontainseveralmonoatomicsteps,and,hence,severalwideterraces.ThebunchheightisrelatedtothenumberofstepsNinthebunchbyH=Nh0,whereh0istheheightofanatomicstep.Aquantitythatisdirectlyaccessibletoexperimentalobservations31istheminimalterracesizelmininsidethebunch,whichisrelatedtothemaximalslopemmaxthroughlmin=h0/mmax.FollowingthenotationofRef.36,weintroducescalingexponentsαandγcharacterizingtheshapeofindividualbunchesthroughtherelationsH∼Lα,lmin∼N−γ.(2)Assumingthattheminimalterracesizelminisofthesameorderasthemeansize¯l=L/Noftheterracesinsidethebunch,oneexpectstheexponentidentityγ=1−1/α.(3)Furthermorethecoarseningofthebunchmorphologywithtimeisdescribedbyadynamicexponentzdefinedthrough36ξ∼N∼tα/z.(4)Becausetheratioξ/Nhastobeequaltothemeanterracewidthl,whichisfixedbytheoverallvicinalityofthesurface,thebunchspacingξandthebunchsizeNgrowwithtimeinthesamemanner.Forstepbunchinginducedbysurfaceelectromigration,scalingrelationsoftheform(2)havebeenderivedbothfornon-transparentandtransparentsteps26,29,30,34.Theirapplicationtotheanalysisofexperiments31,32