AbstractInthisthesisIpresenttheprogressImadeinunderstandingimprovingandapplyingadigitalmethodforlatticefringespacingmeasurementsinHRTEMimagesThemethoduses lteringinFourierspacetoformcomplexvaluedda ID: 282263
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SpacingmeasurementsoflatticefringesinHRTEMimagesusingdigitaldark elddecompositionMartinRoseAThesisSubmittedtoTheGraduateSchoolattheUniversityofMissouri-St.Louisinpartialful llmentoftherequirementsforthedegreeMasterofScienceinPhysicsJuly2006AdvisoryCommittee:P.Fraundorf,Ph.D.ChairpersonB.Feldman,Ph.D.D.Leopold,Ph.D. AbstractInthisthesisIpresenttheprogressImadeinunderstanding,improv-ingandapplyingadigitalmethodforlatticefringespacingmeasurementsinHRTEMimages.Themethoduses lteringinFourierspacetoformcomplexvalueddark eldimages.Theseimagesarebrightwheretheim-agecontainstheselectedperiodicityanddarkotherwise.Theamplitudeandphaseinformationinthesedark eldimagescanbeusedtoanal-ysethedistributionofperiodicitiesandthechangeinlatticespacings.Itturnsoutthatthephasegradientcanbeinterpretedasapositiondepen-dentwavevector.Themethodhasbeenanalyzedindetailinordertoquantifythesystematicerrorandtheerrorintroducedbynoise.Win-dowingandWiener lteringtechniqueshavebeenappliedtoimprovethemeasurement.Anewapproachtoreducethee ectofnoiseusesmultipleperiodicitiestodeterminethebasevectorsineveryunitcellwhichleadstoareductionofnoise.Themethodhasbeenappliedtosimulatedimageswithknownnoiselevelstomeasuretheerror.Finallyithasbeenappliedtoexperimentalimagesofheterostructures.1 Contents1Introduction31.1Motivation..............................31.2Crystalsandtransmissionelectronmicroscopy...........41.3Thebasicassumptionforimageprocessing............61.4Di erentmethodsforspacingmeasurements............71.5TheDFmethod...........................92MathematicalfoundationoftheDFmethod112.1The2DDFT.............................112.2Formationofadark eldimage...................122.3MeasurementofspacingsfromtheDFimage...........122.4SimulationofTEMimages.....................142.5Datawindowing...........................212.6Errorbychangingwavevector...................243Thenatureofnoise293.1AmodelfornoiseinaHRTEMimage...............303.2EstimationoftheerrorinthephaseoftheDFimage.......343.3Measurementoftheerrorinspacingmeasurements........363.4Removalofnoise...........................403.5Multiplespotmeasurement.....................424ApplicationoftheDFmethod444.1computergeneratedimages.....................444.2experimentalimages.........................465Summary,conclusionsandoutlook606Acknowledgment62AImplementationoftheDFmethod63ListofFigures64References652 1IntroductionThisthesispresentstheresultsoftheworkIhavedoneattheCenterforMolec-ularElectronicsattheUniversityofMissouri-St.LouisbetweenNovember2005andJune2006inthegroupofProf.Dr.P.Fraundorf.Iworkedonamethodcalleddigitaldark eldimagingorsimplyDFmethodwhichisusedtoanalysedigitizedhighresolutiontransmissionelectronmicroscopy(HRTEM)images.ThemethodusesFourierspace lteringtoformsocalleddark eldimagesindirectspace.Thebrightnessandthephaseofthesecomplexvaluedimagescanbeusedtoanalyseperiodicitiesandchangestherein.Theselectionoffrequen-ciesinFourierspaceusedinthisdigitalmethodisanalogoustotheselectionofabeamofscatteredelectronsinaTEMinordertoformdark eldimages-thereforethename.1.1MotivationTransmissionelectronmicroscopyprovidesuswithhighresolutionTEM(HRTEM)imagesthatshowstructuresattheatomicscale.Thatmeansweareabletoidentifyfeaturesonthesizeof2A(210 10m).Wecanusethistooltotakealookatthearrangementofatomsinacrystal.Asthestructureofcrystalsisknownthisprovidesnonewinformationat rstglance,butitgetsveryinter-estingwhenweexamineinterfacesbetweencrystalsordefectsinacrystal.Aninterfacecanbecreatedforexamplebyepitaxialgrowthwhereonesub-stanceisdepositedontopofanother,layerbylayer.Thetwosubstancesmustbeofthesamelatticetypeandhavesimilarlatticespacingsinorderto t.Thislayerwisedepositionwillresultinacrystallinestructure.Farawayfromtheinterfacebothsubstanceswillhavethesameproperties,i.e.latticeparameters,astheyhaveinaperfectcrystal.Butattheinterfacetheseparametershavetochange,asitconnectstwosimilarbutnotidenticallattices.Theresultisadeformationofthelatticecalledstrain.Formsofstrainareforexamplecom-pression,expansionandshear.WealsoexpectthesechangestobeverysmallsothattheycannotbeseeninadigitizedHRTEMimagewiththenakedeye.Let'ssaywewanttoexaminechangesofabouttwopercentina2.5Aspacing,whichcorrespondsto5picometers(pm).Thisiswaybelowtheresolutionofanymicroscopeavailable!TheheterostructuresImentionedbeforearethebasicmaterialindevicemanufacturingfromLEDsandtransistorstoverylargescaleintegratedcircuits.Asthepropertiesofasemiconductordependstronglyonlatticeparametersitiscrucialtoknowthemexactly.Anotherimportanttopicinmaterialsciencearedefects.Theyoftenoccuratinterfacesandchangetheelectricpropertieslocally.Insomeapplicationstheyareveryusefull,forexampletheyscatterelectronsinacrystalandallowanelectricalcurrentto\rowbutinotherstheyarejustadisturbingsidee ectofthemanufacturingprocess.Insolidstatephysicsdefectscanbetreatedasperturbations,soagaintheparameterstodescribeadefectareveryimportant.Sohowdoweobtainthespacingsatanatomicscaleandmoreimportanthowcanwemeasuretinychangesofspacingsifwearenotworkingwithaperfect3 singlecrystal?Onetheoreticalapproachiselastictheorywhichcalculatesatompositionfrombalancingforces.Thismethodisnotfurtherconsideredinthisthesis.AnexperimentalapproachistoextractthisinformationfromdigitizedHRTEMimages.Thecharacterization,improvementandapplicationoftheDFmethodisthemaintopicofthisthesis.Iwill rstdescribetheDFmethodmathematically,afterthatthee ectofnoiseandotherlimitingfactorswillbeconsidered.FinallythemethodwillbeappliedtosimulatedandexperimentalHRTEMimagesofdefectfreestructures,astheDFmethodturnsouttobemostpreciseinthiscase.Iwillfocusonmeasuringchangesinspacingsofabout15%inheterostructures.1.2CrystalsandtransmissionelectronmicroscopyAtthispointIwanttoreviewthebasicsofcrystallography,transmissionelectronmicroscopyandtheprocessofimageformationinaTEM.Anintroductiontocrystallographyanddi ractioncanbefoundin[2].Ifound[9]togivethebestintroductiontotransmissionelectronmicroscopy.crystallographyThemainpropertyofcrystalsistheperiodic,threedimensionalarrangementoftheatomstheyaremadeof.Crystalsarehighlyorderedcomparedtootherstatesofmatterlikegasesor\ruidswhichcontainonlylocalorderornoneatall.AtomsinacrystalarelocatedatpointsthatcanbedescribedasathrdimensionalgridcalledtheBravaislattice.Thisgridinturncanbeconstructfromthreebasisvectors.Duetoperiodicarrangement,somephysicalpropertiesarenotisotropicanymore,i.e.theydependontheorientationofthecrystal.Wethereforeneedawaytoidentifydirectionsandplanesinacrystal.Agroupofthreeatomsthatarenotononelinede neaplane.Allparallelplanesformasetofplanes,de nedbythreenumbers,theMillerindices.Theseindicesaredeterminedinthefollowingway:Theoriginofthelatticemustbeputinaneighboringplane.Thepointswheretheplaneintersectsthecoordinateaxesmustbedetermined.Theinverseofthesepointsmustbemultipliedinordertoobtainintegernumbers.IfaplanedoesnotintersectanaxisthecorrespondingMillerindexiszero.Directionsinacrystalarede nedsimplybythecomponentsofavectorpointinginthespeci cdirection.Thebasisofthisvectoristhebasisofthecrystal.di ractionofelectronsBraggdiscoveredin1913thatx-raysscatteredbycrystalscreatecharacteristicpatternscalleddi ractionpatterns.Thiscanbeexplainedbyinterferenceofraysscatteredbyneighboringplanes.ConstructiveinterferenceoccurswhentheBraggconditionisfull lled:n=2dsin()4 whereisthewavelengthofthex-rays,dthedistancebetweentheplanesandtheangleofincidence.nisanintegerdescribingtheorder.Thesameexperimentcanbeperformedwithelectronsduetotheirwavenature.Usually,whenacrystalisexaminedinaTEMthesampleisrotatedsothattheBraggconditionisfull lledsothatbrightanddark eldtechniquescanbeused.harmonicanalysisTheanalysisofperiodicstructurescanbesummarizedasharmonicanalysis.Bragg'slawconnectsscatteringanglesandspacingsoflatticeplanesinacrystal.Thereforeadi ractionpatterncanbeusedtodeterminelatticeplanespacings.Thisisanoptical,analogmethodwhichusesfrequencyspacedata.PeriodicitiescanalsobeanalyzedstartingfromdirectspacedataforexamplefromaHRTEMimagewhichisaprojectionofthecrystal.AFouriertransformcanbeusedtocalculatethePS.ThisPSisverysimilartothedi ractionpatternasitcontainsspotsinthesamedirectionforallperiodicitiesthatarevisibleasfringes.Wecanalsoanalyseperiodicitieslocallyusingselectedareaelectrondi raction(SAED).Afuturemethodintermediatebetweendirectandfrequencyspacemighttakeadvantageofwavelets.ThewindowedFouriertransformusedintheDFmethodcanberegardedasasimplewavelet.Usingmoreadvancedwindowfunctionsitshouldbepossibletoanalyseandlocateperiodicitiesmoreaccurately.[8]focusesonthistopicwhile[5]showsapplicationsofwaveletsinsignalprocessing.TEMandhighresolutionimagingInaTEMaverythinspecimenisirradiatedbyanelectronbeam.Magneticlenses,solenoids,areusedtomanipulatetheelectronbeam.Theelectronstravelalongthesymmetryaxisofthelenseswhereanaxialandtransversalmagnetic eldispresent.Theaxialcomponentmakestheelectronscirclearoundtheaxis.Thiscircularmotionandthetransversalcomponentofthemagnetic eldareresponsibleforaforcepointingtowardsthecenterofthelens.Thepointoffocuscanbecontrolledbythe eldstrength,i.ebytheelectriccurrent.Theabilitytofocuselectronsinatinyareaisthekeytoelectronmicroscopy.Tounderstandthedi erentimagingmodesitisnecessarytounderstandtheimagingsystemofaTEM.Itisassumedthatthespecimenisirradiatedbyanelectronbeam.The rstlensbelowthespecimeniscalledobjectivelens.Thislensformsanintermediateimagewithamagni cationof20to50timesinit'simageplane.Raysthatleavethespecimenatthesameanglearecollectedinthesamespotinthebackfocalplaneoftheobjectivelens.Eitheroftheseplanescanbeprojectedontheviewingscreenbyasecondlens.Toformarealimagetheimageplaneisprojected,toexaminethedi ractionpatternthebackfocalplaneisprojected.Thebackfocalplaneiswheretheobjectiveaperture,asmallholeinametallicplate,islocated.Thisapertureallowssinglespotsinthebackfocalplanetobeselectedforimageformation.Whentheunscatteredbeamisselectedtheresultingimageiscalledabright eldimage,whileadark eldimageisformedbyscatteredelectrons.Whennoobject5 apertureisusedallelectrons,scatteredandunscattered,areusedtoformtheimage.Theseimagesusuallyhavelittlecontrastcomparedtodarkorbright eldimagesifnoncrystallinespecimensareexamined.Inthiscasethecontrastoriginatesfrominelasticscatteringoftheelectronsbyatomsinthespecimen.Thiscontrastmechanismisreferredtoasmass-thicknessorsimplyZcontrast.WhencrystallinespecimensareimagedtheDFandBFimagesdescribedabovecanbeusedbecausecrystalsscatterelectronsasdescribedbyBragg'slaw.ThebackfocalplanerepresentstheFouriertransformofthedistributionofatomsinthespecimen,multipliedbythecontrasttransferfunction.Ingen-eralsmoothfunctionshaveonlyfewFouriercoecientsinasmallrange,whilerapidlyvaryingfunctionshaveFouriercomponentsinamuchwiderrange.Thedistributionofatomsisde nitelyarapidlyvaryingfunction.ThesmallaperturusedinDForBFmodetruncatestheFouriertransformsothatonlycoecientsinasmallrangecancontributetotheimage.Thereforetheseconventionalimag-ingmodescannotproducehighresolutionimages.EvenifnoaperturewouldbeusedinordertoincludeallFouriercoecientsthespacialresolutionislim-itedbythephasedistortionW(k)whichincreaseswithk.Thecontrasttransferfunctionmentionedbeforeisde nedbyexp(iW(k))AsW(k)increasesrapidlywithkhighfrequenciescannotbeusedforimageformation.ThislimitstheresolutionofastandardTEMtoabout2A.Toobtainhighresolutionimagescontaininglatticefringes,theinterferenceoftheunscatteredandthescatteredbeamisused.Theelectronscanbedescribedasawavefront.Thephaseofthiswavefrontisshiftedwhentheelectronspassthespecimen.Thephaseshiftdependsonthepotentialinthespecimen.Aquantummechanicalderivationoftheimageformationinthehighresolutiomodecanbefoundinchapter10of[9].1.3ThebasicassumptionforimageprocessingBeforeIwillexplaindi erentmeth-odstomeasurespacingsbetweenlatticefringes,IhavetoexplainwhyspacingsonanatomicscalecanbededucedfromHRTEMimages.Theimagetotherightisanexampleoftheimageswewanttoanalyse.Thebrightstripesarecalledlatticefringes,theycanappearinoneormoredirections.Ifthespecimenisverythin,thesefringescanbeinter-pretedastheprojectionoftunnelsbe-tweencolumnsofatoms,whilethedarklinesaretheatomsthemselves. 6 Thisinterpretationistruewhenthe rstzerointhecontrasttransferfunctionisathigherfrequenciesthantheperiodicityexamined.Whenthespecimengetsthickeraphaseshiftmightoccurandthemeaningofdarkandbrightfringesisinterchanged.Thisphaseshiftoccurswhenthe rstzerointhetransferfunctionisatlowerfrequenciesthantheexaminedfrequency.Article[11]analysestheextenttowhichthespacingsbetweenfringesinaHRTEMimagecorrespondtoatomicspacings.Lenstransfertheoryisusedtogivesomepracticalrulesfortapplicationofthegeometricphasemethod,whichissimilartotheDFmethod:Thethicknessofthespecimenandthedefocusshouldbechosensothatthelatticefringecontrastismaximal.Thisleadstolowererrorlevelswthespacingsaremeasured.Regionswherefringecontrastchangesrapidlyshouldnotbeused.Theanalysisofspacingsordisplacementsshouldbecon rmedusingdif-ferentre\rections,seemultispotmethodinchapter3.5.Theanalysisshouldbecarriedoutatdi erentdefocusvalues.Fromthispointonitisassumedthatthefringesrepresentcolumnsbetweenatomssothatthespacingsbetweenlatticefringescorrespondtoatomicspacings.Theunitforspacingsitthepixelfromnowon.Thescalefactorbetweenpixelspacingsandspacingsinnanometershastobedeterminedbytheuser.Thisisnotthateasybecausethemagni cationisnotknownwithhighaccuracy,buttherearevariousmethodstodothat:The rstpossibilityistousespacingmeasurementsinanundisturbedregionoftheimageasreference,assumingthematerialandthereforethelatticespacingsareknownatthisposition.Anotherpossibilityistodepositknownstructuresontopofthespecimenthatcanserveasareference.Forexampleathinlayerofpolycrystalinealuminumcanbeused.Becauseoftherandomorientationofthesinglecrystalsthislayerwillcreateringsinthepowerspectrumcorrespondingtothespacingsinthealuminumcrystals.Theseringsde neascalefromwhichspacingsofotherspotscanbededuced.Ifthemagni cationisknownwithincalibratedlimitsthescalefactorcanbecalculatedfromtheresolutionofthescannerandthemagni cation.Thismethodoftenyieldslargerabsolutespacinguncertainties.1.4Di erentmethodsforspacingmeasurementsNowthattheconnectionbetweentheimagesandtherealcrystalhasbeenmadewecanfocusonmethodstomeasurethespacingsbetweenlatticefringes.Anearlyreference-ausermanualofanimageprocessingunit-thatdescribesamethodinvolvingFouriertransformsgoesbackto1990[17].Ingeneralall7 methodsforspacingmeasurementscanbedividedintwogroups:thoseworkingindirectspaceonlyandthoseinterpretingFourierspacedata.The rstdirectspacemethodtomeasurethespacingsbetweenlatticefringesthatcomestomindissimplyusingsomekindofrulertomeasurespacingsbetweenadjacentfringes.Intheimageabovethespacingsareabout6pixels.Withthissimplemethodthespacingscanbedeterminedwithanaccuracyofabout0.5pixel.Thiscanbeimprovedwhenthemeasurementisaveragedovermorefringes,butatthesametimethisaveragingprocesslimitsthespacialresolutionsothatspacingchangesclosetoasharpinterfacecannotberesolved.Amoresophisticateddirectspacemethodhasbeendescribedin[4].ThismethodusestheoccurrenceofMoirestructureswhentwopatternsaresuper-imposed.Firstofallareferencelatticeiscalculatedfromanundisturbedareaoftheimage.Thispatterndescribesthepositionofbrightspotsinthisarea.Thenthisreferencepatternisextrapolatedoverthewholeimage.Nowthesep-arationbetweenthereferencelatticepointsandtheclosestexperimentalpointismeasured.ThismethodhasbeenimplementedinasoftwarepackagecalledLADIA,see[7].Thesedisplacementmeasurementsarethebasisforchemicalmappingandstrainanalysis.FourierspacemethodsanalysethepowerspectrumandtheFouriercoef- cientsofaHRTEMimage.AperfectunstrainedlatticegivesrisetoverysharplypeakedspotsinthePS.Thespacingandangleoflatticefringescanbecalculatedfromthepositionofthecorrespondingspotinthepowerspectrum,asshownin[6].InthisarticlethespacingsbetweenfringesinTiO2andTiNparticlesinacatalystaremeasured.AsthespacingisconstantwithinaparticlthespotinthePShasverylittlestructure,it'snearlyapoint.Byinterpolationtheexactcenterofaspotcanbefound.Thismethodcanbeusedtomeasureconstantspacingswithanaccuracyof0.001-0.05Adependingonthespecimenandimagingconditions.Howeverthismethodcannotspatiallyresolvetinychangesinspacings.Changesinspacingsandtheshapeoftheregioncontain-ingperiodicitieschangethepro leofthecorrespondingpeakinthePS,itgetswiderandsmoother.AnotherwellestablishedFourierspacemethodisthegeometricphasetech-nique,describedin[12]andappliedin[15]and[10].Ituses lteringinFourierspacetocalculateaDFimage.ThephasePg(~r)oftheDFimageisrelatedtothedisplacement eld~uby:Pg(~r)=2~g~uThephasesPg1(~r)andPg2(~r)ofDFimagescorrespondingtotwodi erentperiodicitiesatposition~g1and~g2areusedtocalculatethedisplacement eld:~u1 2(Pg1(~r)~a1Pg2(~r)~a2)where~a1and~a2arethecorrespondingdirectspacevectorsto~g1and~g2.Thismethodcanbeusedtomeasurethefringepositionwithanaccuracyof1%ofthespacing.8 1.5TheDFmethodTheDFmethodissimilartothegeometricphasetechniqueasitinterpretsthephaseofaDFimage.TheDFimageisintermediatebetweendirectandreciprocalspaceasitcontainslocalizedinformationaboutperiodicities,see[8].Usingthephasegradientitmeasuresthelocalspacingbetweenlatticefringes.TheamplitudeoftheDFimagecanbeusedtolocateareascontainingperiodicities.Fromthesefundamentalmeasurementsotherquantitieslikethedisplacement eldandthestraintensorcanbederived.Todothat,morethanonesetoffringesisrequiredofcourse.Figure1showsthestepsoftheDFmethodindetail.Firstofalltheinputimageisloaded,shiftedandscaled.TheshiftingisdonetodecreasethemagnitudeoftheDCpeakwhichleedstoabettervisibilityoftheperiodicitiesinthePS.Scalingisnotnecessarybutitincreasesthecontrastwhentheinputissavedagain.Theseoperationshavenoin\ruenceonthemeasurements,seechapter3.Awindowfunctionisappliedtotheinputinordertodecreasethee ectofleaking,seechapter2.5.NowtheFouriertransformiscalculatedusingtheFFTalgorithm.ThepowerspectrumcanbeexaminedtodeterminetheSNRwhichisameasureforthequalityoftheimage.ForadetaileddescriptionofthenoiseinaTEMimageandthee ectithasonthespacingmeasurementsseechapter3.Oncethenoisemodelhasbeenestablished,aWiener ltercanbeappliedinordertoreducethenoiselevel.Noisecanalsobeaddedwhensimulatedimagesareusedinordertounderstandthee ectofnoise.Acircularapertureisthencenteredaroundthebrightestpointofaspotinordertoselectaperiodicityi.e.asetoflatticefringes.Thee ectofthechoiceofcenterandradiusisdiscussedinchapter4.Anotherwindowisappliedtotheaperture.InamethodpreviouslyusedtheFouriercoecientswereshiftedsothatthecenteroftheaperturebecometheDCpeak.Thisstepisnotusedhere.NowtheDFimageiscalculatedusingtheinverseFouriertransform.FromtheDFimagethespacingsbetweenfringescanbemeasuredbythephasegradient,seechapter2.3.TheamplitudeoftheDFimagecanbeusedtolocateareascontainingtheperiodicityselectedbytheaperture.FinallymeasurementsfrommultipleDFimagescanbecombinedinordertoincreasetheaccuracyofthemeasurementswithoutloosingspacialresolution.Thismultispotmethodisdescribedinchapter3.5.9 Figure1:singlestepsoftheDFmethod10 2MathematicalfoundationoftheDFmethodThischapteroutlinesthemathematicsbehindtheDFmethodandthesimula-tionofHRTEMimages.Theunitforspacialcoordinatesisonepixel.Allpixelvaluesmustbeinterpretedtoconnectthemtophysicaldistanceswhichisdonewhenthemethodisappliedinchapter4.Therearesomeparametersusedtode netheinputandtocontrolspecialfunctions.Someoftheseparametershavealreadybeenmentionedinthedescriptionofthemethodandwillbede nedinthismathematicalcontext.AppendixAlistsallavailableparametersandthefunctionstheycontrol.FirstofallanimageofsizeNNcontaininglatticefringesisrequiredasinput.The rstparameteristhereforeN,apowerof2,usually512,1024or2048inordertotakeadvantageoftheFFTalgorithm.2.1The2DDFTThetwodimensionaldiscreteFouriertransform^(h;hy)=N 1X=0;y=0exp2i(hxhyy) N(x;y)anditsinverse(x;y)=1 N2N 1Xhx=0;hy=0exp2i(hxhyy) N^(h;hy)asde nedin[16]isusedtotranslatebetweendirectandreciprocalspace.TheunitoftheFouriercoecients^F(h;hy)ispixel 1.Withthehelpofthesetwoquantities~k2hhyand~r1 Nxytheinversetransformcanberewritten:(~r)=1 N2X~kexpi~k~r^(~k)where~kisthediscretewavevectorandeach^(~k)correspondstoone^(h;hy).Thecomponentsof~r(randry)varybetween0and1.Thisformoftheinversetransformmakesitobviousthattheimageindirectspacecanberepresentedasasumofcomplexvaluedplanewaves.AstheinputisrealtheFouriercoecientsmusthaveconjugatesymmetrywhentheFouriertransform(FT)oftheinputiscalculated:^(~k)=^(~k).Thissymmetryisthereasonforthesymmetryofthepowerspectrum(PS)withrespecttotheDCpeak.ThevalueofthePSatagivenpointisde nedinanotsocommonwayastheabsolutevalueoftheFouriercoecientatthispoint.11 2.2Formationofadark eldimageIntransmissionelectronmicroscopyadark eldimage(DF)isformedbyal-lowingonlyelectronsthatwerescatteredatacertainangletoformtheimageaccordingtochapter2.3in[9].Acircularmechanicalaperturemadeoutofmetalcalledtheobjectiveapertureispositionedinthebackfocalplanetose-lectelectronsthatwerescatteredintothesameangle.Whenscatteredelectronsareallowedtopasstheaperturewecalltheresultingimageadark- eldimage.TheanalogousprocessintheDFmethodis lteringinFourierspacewheretheDCpeakcorrespondstotheunscatteredelectronsandthespotscorrespondtodi ractionspotsi.e.thescatteredelectrons.Thisisjustananalogy,thePSofaHRTEMimageshouldnotbeconfusedwiththedi ractionpatternwecanobserveintheTEM!ToapplyanapertureonepointinFourierspaceischosenasthecenteroftheaperture.AllFouriercoecientsoutsideacirclewithra-diusrAParesettozero.InsidetheapertureaHammwindowisapplied.Thechoiceofthecenterpointandthee ectoftheradiuswillbediscussedin4.1.This lteringprocessinFourierspacedestroystheconjugatesymmetryoftheFouriercoecients.Theresultoftheinversetransformoftheseasymmetriccoecientsiscalledthedark eldincloseanalogytoopticaldark eldimages.RegionsofthespecimenthatcontainnoperiodicitieswilllookdarkiftheDCpeakisoutsidetheaperturewhileregionscontainingperiodicitieslightup.ThevalueoftheDFimageDF(~r)ateachpointisgivenbythesuperpositionoftheplanewavescorrespondingtotheFouriercoecientsinsidetheapertureasdescribedintheprevioussection.DF(~r)=1 N2X~kexpi~k~r^(~k)=ARes(~r)exp(iRes(~r))whereARes(~r)istheresultingamplitudeandRes(~r)istheresultingphaseateachpoint.2.3MeasurementofspacingsfromtheDFimageLet'sassumeforamomentthatwewouldalsousetheFouriercoecientsat~kforeach~kinsidetheaperturetoformtheDFimage.Inthiscaseitwouldbearealimageshowingtheperiodicitieschosenbytheaperture:DFR(~r)=ARes(~r)cos(Res(~r))Res(~r)canbewritteninthefollowingway:Res(~r)=~kRes(~r)~r12 SotheDFimagecanbeinterpretedasaplanewavewithpositiondependentwavevector~kRes(~r).The guretotherightshowsthefunctioncos(k(x)x).Theuppergraphisk(x),thewavevec-torwhichincreaseslogarithmicallywithx.Thee ectofthepositiondependentwavevectorcanbeseenbestattheleftsideoftheplot. k(x)cos(k(x))Nowitisassumedthatthewavevectorvariesonlyslightlywithinoneperiodsothatthespacingdbetweentwomaximaisgivenbyd(~r)=2 ~kRes(~r)Thedistancedisaboutsixpixelsinthesimulatedimagessoameasurementof~kwithinthisdistanceisassumedtogivethesamevalueford.HoweverthephaseRes(~r)cannotbemeasuredintherealDFimagewithhighaccuracy,becauseit'sdiculttodeterminetheintensitymaximawithhighaccuracy.Thesameproblemsetslimitsforthesimplerulermethoddescribedinchapter1.4.ThereforethecomplexDFimagedescribedinthesectionbeforeisused:DFC(~r)=ARes(~r)fcos(Res(~r))+isin(Res(~r))gNowthephaseisgivenateachpointbyRes(~r)=~kRes(~r)~rarctan=(DFC(~r)) (DFC(~r))FromthisthegradientofRes(~r)canbecalculated~r~k~r~k~r~r~r~r~kwhere~k~kRes(~r)=(k@ky@y)~r+(r@ry@y)~kk^xky^y+(r@ry@y)(k^xky^y)~k+^x(r@kry@yk)+^y(r@kyry@yky)Makingthesameassumptionasbefore,namelythat~kRes(~r)variesonlyslightlywithpositionthegradientofthephaseisapproximately~kRes(~r).There-forethelocalspacingdcanbecalculatedasa rstapproximationfromthephasegradientbyd(~r)=2 ~rRes(~r)Thisassumptionwillbecheckedandtheerrorinthespacingmeasurementsduetothechangeof~kwillbeexaminedinsection2.6.13 ThemagnitudeofavalueintheDFimage,jDFC(~r)j,canbeinterpretedaswellincloseanalogytotheDFimageintransmissionelectronmicroscopy.InaTEMDFimageareasofthespecimenwhichscatterelectronsappearbrightwhilethoseareaswhichdonot,appeardark.Forexampleinapolycrystalinesamplethiscontrastmechanismwouldrevealcrystalsinthesameorientation.IntheDFmethodthisvalueisusedasabinarymask,becausethevalueishighinareaswithlownoiselevelsandclearperiodicitieswhileittendstozeroinareaswherethespecimenisamorphousandnoperiodicitiesarepresent.TheDFmethodisatoolformeasuringspacingsbetweenlatticefringes.Themeasurementofdistancesisthemostbasicoperationingeometry.Fromdistancesbetweenatomsmorecomplicatedquantitieslikethedisplacement eld~uandthestraintensor,whichiscomposedofthepartialderivativesof~u,canbecalculated.Thestresstensorisconnectedtothestraintensorbymaterialrelations.Iwillthereforedescribeawaytoobtainthedisplacement eldfromthespacingmeasurements,asitmakestheconnectiontoothermethodsliketheoneusedbyHytchetal.(seesection1.4).Letsassumewewanttodeterminethedisplacement eldforastructureasshownin gure21wherethedarkdotscorrespondtocolumnsbetweenatoms.TheDFmethodsprovidesthewavevectorsatsomespotthatcanbechoosenarbitrarily,forexampleinanareawithoutdistortions.Fromthatwecancom-putethepositionsoftheneigbouringatomsortunnelsrespectively.Thenextstepistomeasurethewavevectorsatthenewpositionsinordertorepeatthepreviousstep.Thisprocedureyieldsthepositions~aiofallthespotsintheHRTEMimage.Asthedisplacement elddescribesthedistancefromsomeref-erencepositionweneedtointroduceareferencelattice.Thisreferencelatticecanbecalculatedforexamplefromthepositionsofspotsinanundistortedarea.2.4SimulationofTEMimagesAbasicmethodincrystallographyandmaterialscienceistocompareexper-imentalHRTEMimagestocomputersimulatedimagesinordertochecktheatomicmodels.AmajorpartofmyworkwastoapplytheDFmethodtosimu-latedimagesinordertoseehowaccurateitworksunderperfectconditionsandtodeterminetheerrorinthemeasurementsobtainedbytheDFmethod.Chap-ter10.4in[9]givesanintroductiontoasimulationmethodcalledmultislicemethod.Theinputtoanysimulationisthepositionofatomsinsidetheregioofinterest.Fromthistheelectronwavefunctionatthespecimenexitsurface,atthebackfocalplaneandattheimageplanecanbecalculated.Thesecondstepistomodifythecalculatedwavefunctionatthevariouspositionsinordertosimulatethecharacteristicsandde cienciesofthemicroscopee.g.sphericalaberration,astigmatismanddefocus.Iusedadi erentapproachtogeneratetestimagessimilartohighresolutionTEMimages.Intheintroductionof[18]itismentionedthatthetunnelsbetweencolumnsofatomsinaperiodicstructureappearasnearlygaussianshapeddotsinaHRTEMimage.Thedotsareeitherblackorwhitedependingonthe14 thicknessofthespecimenandthefocus.Thedistancebetweenthecentersofthesedotscorrespondstothelatticespacingasdescribedinchapter1.3.Thepro leofalatticefringecanbeexaminedusingthe3DvisualisationpluginiImageJandtheshapeofagaussianisrevealed.ThereforeGausspro lesatpositionsgivenbya2DBravaislatticeareusedtosimulateHRTEMimages.ImageslikethesecanbeobtainedfromamicroscopewithGaussianpointspreadbutnosphericalaberration.Anewlyimplementedtechniqueofaberrationcorrectionmightmakesuchmicroscopesavailable,see[3].Thissimulationdoesnottakede cienciesofthemicroscopeintoaccount.Thebrightnessatapositionx/yinthemicrographduetoatunnelatpositionp/qisthereforegivenbyexp ((xp)2+(yq)2)TheimagetotherightshowsaGausspro le.Theconstantischosentobe0:5.Astheimageismadeoutofpixelsatdiscreteintegerpositionsitisneces-sarytointegratetheintensitythatcon-tributestoonepixelovertheareaofthepixel.Thepixelsarealignedtothemi-crographinsuchawaythatthecenterofapixelcorrespondstoanintegercoordi-nateinthemicrograph.ThebrightnessBxyofapixelisthenobtainedbyinte-gratingtheGausspro leovertheareaofonepixel. -5 -5 -4 -4 -3y x-3 -2 -2 -1 -1 00 0.0 0.1 1 1 0.2 2 0.3 2 0.4 3 3 0.5 4 0.6 4 0.7 5 0.8 5 0.9 1.0 Bxy+0:5 0:5y+0:5y 0:5exp ((xp)2+(yq)2) 4 p (x0:5p) p (x+0:5p) p (y0:5q) p (y+0:5q) where =2 p 0expt2 2dtComparedtoamethodpreviouslyusedtosimulateimagesthisapproachallowsustoputtunnelsatarbitrarypositions.Thegrainsizeoftheparticlesonthe lmwhichdetecttheelectronsisverysmallandtheyarerandomlydistributed.Thereforeitisassumedthatthe lmcandetectelectronscontinouslyinspacewithoutintroducinganygrid.Thebrightnessisproportionaltothenumberofelectronshittingthe lmatacertainposition.Theintegrationovertheareaofapixelinthesimulatedmicrographcorrespondstotheprocessofdigitalizatiointhescanner.Asthesamepro leisusedforalltunnelstheintegrationensuresthatthebrightnessofallspotsisequal.Inthemicroscopethismeansthatthesamenumberofelectronswasdetectedinonespot.Figure2showsthebasicgeometryofthesimulatedimages.TheBravaislatticewiththebasisvectors~aand~bde nesthepositionwheretheGausspro les15 willbecentered.Thislatticecontainsanumberofperiodicitiesrunningindi erentdirectionsasindicatedbythevariouspairsofparallellines.ThespacingmarkedbydwillbeexaminedwhentheDFmethodisappliedtosimulatedimages.Thefollowingbasisisused:~a6 109103and~b7 3716 \rd~a~b90\rFigure2:BravaislatticeandtheexaminedspacingsThespacingdiscalculatedasfollows:dacos(90\r)=5:385326and\rarcos0@~a~b j~aj~b1ATointroduceachangeinthisspacingthebasevector~aismultipliedbyascalefactor,seeforexampled.ThesimulatedimagesdonotobeytheperiodicboundaryconditionassumedbytheDFT.Theimagecanberegardedasasquarewindowfunctionmultipliedbyanin niteimage.ThePSisthereforeaconvolututionoftheFTsofthe2Dsquarewindowandtheimage.LinesrunningperpendiculartotheboundaryoftheimageappearinthePS.BecauseofthisnoiseinthePStheexaminedperiodicityshouldnotrunperpendiculartoaboundary.TherearetwostructureswhichwillbeusedtotesttheDFmethod.Bothstructuresstartfromthelatticedescribedabove.Theydi erinthewaythespacingsarechanged.Figure3and4showthespacingbetweenacertainsetofparallelrowsofGaussianpeaks.Thespacingisde nedasthedistancebetweentwoneighboringrows.Thequestionthatarisesiswheretomeasureaspacing,orwhichpoint16 isthespacingassignedto.Byde nitionthespacingdoesnotchangebetweentwopeaks.Thereforethespacingisassignedtothecenterofthepeakasitislocatedinthemiddleofcolumnsofatoms.Thespacingsareanoutputoftheprogramwhichsimulatestheimages,andtheywillbeusedinchapter4.1asareferenceforthespacingmeasurements.SharpstepAlthoughtheabruptchangeofthespacinginthisgeometryissomehowunphysical,itisveryusefultounderstandthee ectofnoisewhichisoftenabundantinexperimentalimages. 02004006008001000100080060040020005,385,405,425,445,465,48 y [pixel]spacing [pixel]x [pixel] Figure3:localspacingsforthesharpstepgeometryThegeometryisdividedintothreeregionsofequalspacings:inthelowerleftthespacingisthatofthestartinggeometryd0=5:385326.Atthe rststepthespacingismultipliedby1:012leadingtod1=5:44995.Atthesecondstepthespacingismultipliedby1:006resultinginaspacingofd3=5:48265.Thelineswhichdivideareasofconstantspacingintersectthex-axisatx1=259and17 x2=535.Widestep-interfaceThisgeometryisamodelforahomogeneousinterface-achangeinspacingsoverawiderrange. 020040060080010005,405,455,505,555,605,65 10008006004002000 spacing [pixel]y [pixel]x [pixel] Figure4:localspacingsforthewidestepgeometryThescalefactorisasteadyfunctionsimilartotheFermiDiracdistribution:(x)=1+a 1+exp( 0 w)whereade nestheheightofthestepandwdeterminesthecharacteristiclengthonwhichthespacingchanges.Thepointwherethespacinghaschangedbya=2issetbyx0.Thewidthwsonwhich90%ofthespacingchangehappenscanbecalculatedtobews=2ln(19)w18 Thisenablesustoconnecttheparameterwtoawidthwsinpixels.Themaximumslopeisa=4watxx0.Withthisgeometryitwillbepossibletomeasuretheerrorintroducedbyachangingwavevector,asdescribedinchapter2.6.Wewillbeabletosetlimitsfortheaccuracyofthemethoddependingonthegeometry,inotherwordswecantellhowabruptandhowbigthechangeinaspacingcanbe,inordertobemeasuredwithacertainaccuracy.Thiswaythetwomajorsourcesoferror-noiseandachangingwavevector-canbecompared.19 ThepowerspectrumAsbothgeometriesareverysimilartheyhavenearlyequalpowerspectra.Theleftof gure5showsthepowerspectrumofthestepgeometryinlinearscale,wherethecolorhasbeeninvertedforbettervisualisa-tion.TherightshowsthesamePSinlogarithmicscale.ThePSissymmetrictotheDCpeak,thereforethepartofthePSthatisshowncontainsalltheinformation. DCDC111101010202101020201111222221211212~a~a~b~bFigure5:PSofthestepgeometry,rightlogscale,leftlinscaleThespotsinthePSappearveryfaintinthelinearlyscaledimage,itisusefulltoidentifythestrongestperiodicities.Inthelogarithmicallyscaledimagetheweakerperiodicitiesbutalsotheartifactsrunningperpendiculartotheedgesoftheimagebecomevisible.Thespotshavebeenindexedusingthearbitrarilychosenbasevectors~aand~b.Thespotsthatarehardlyvisibleintheleftimagecanbefoundeasilyintherightimage.20 2.5DatawindowingDatawindowingisamathematicalmethodusedtoimprovetheaccuracyoftheDFT.Anysignalisallwayssampledonalimitedintervalforexampleintimeorspace.Theprocessofmeasuringcanbethoughtofastakingalookatthesignalthroughawindowwhichisopenedwhenwestarttomeasureandclosedwhenthemeasurementis nished.Inmathematicaltermsthismeansthatthemeasureddataisequaltothesignalmultipliedwithawindowfunctionwhichvariesbetweenzeroandonewhilethemeasurementisinprogress.WindowingisallwayspresenteveniftheuseroftheDFTisnotawareofit:Thesquarewindowsmultiplieseverydatapointbyone,it'sthesimplestwindowandgivesthepoorestaccuracyofthePS.TheaccuracyofthePSislimitedbyaprocesscalledleakagewhichisdescribedindetailinchapter13.4in[16].Leakageisthein\ruenceonaFouriercoecientbynearbycoecients.Highleakagemeanscoecientsfarawayinthespectrumcontributetoacoecient,whilelowleakagemeansthatonlycoecientsclosebyhaveanin\ruence.Leakagecanbeminimizedbydatawindowingi.e.bymultiplyingtheinputwithawindowfunctionasdescribedin[16]wheresome1Dwindowfunctionsarede ned.The2Dwindowfunctionatposition(x,y)istheproductoftwo1Dfunctions,oneevaluatedatxtheotheroneaty.Thee ectofthefollowingwindowfunctionsonthexcomponentofthephasegradienthasbeencompared:rectanglewxywwy=1Bartlettwxy1xN 22 N1yN 22 NWelchwxy 1xN 22 N2! 1yN 22 N2!Hannwxy0:50:5cos2x N0:50:5cos2y NHammingwxy0:540:46cos2x N0:540:46cos2y NTheindicesxandyrunfrom0toN1.Figure6showsthedi erentwindowfunctions.Figure7andFigure8showastepinthex-componentofthephasegradientwithdi erentwindowsappliedtotheinputandtotheaperture.Thenamingconventioninthelegendisthefollowing:thewindowappliedtotheinputindirectspaceisontheleftwhilethewindowappliedtotheapertureinFourierspaceisontheright.Figure7clearlyshowstheimpactofwindowingonthephasegradient.Usingrectangularwindowsthephasegradientisnotverysmooth.Thewindowsthatareshowninthis gurehaveaverysimilare ectonthephasegradient.Themaindi erenceisattheborderoftheimagewheredi erentwindowsproducedi erentoscillations.Thisregionisnotshownhereasitisnotimportantforthe21 020040060080010000,00,51,0 wxX Axis Title rect Hamming Hann Bartlett Welch Figure6:thevarious1Dwindowfunctions.(N=1024)measurements.Withtheinformationinthis gureIdecidetouseaHammingwindowasthestandardwindowfortheinput.Figure8illustratesthechangeswhenawindowisappliedtotheaperture.AsacombinationofHammingwin-dowfortheinputandaHannwindowfortheapertureresultsinthesmoothestcurveandthelowestoszillationattheborder,thiscombinationofwindowsisusedbydefault.Alsonotehowtheslopeofthegraphgetssmallerwhenusingwindows.Thise ectcanbecompensatedbyabiggeraperture.22 2003004001,131,141,151,16 x component of the phase gradientPosition rect - rect Welch - rect Hann - rect Figure7:di erentwindowfunctionsfortheinput,nowindowonaperture.(N=1024) 2003004001,131,141,151,16 x component of the phase gradientPosition Hamming - rect Hamming - Hann Figure8:di erentwindowsappliedtotheaperture,Hammingwindowfortheinput.(N=1024)23 2.6ErrorbychangingwavevectorNowthatallthepartsoftheDFmethodareunderstoodtheerrorinherentinthemethodwillbeexamined.Insection2.3weassumedthatthephasegradientisthewavevectorifthewavevectorchangesonlyslightlywithposition.WewillcheckthisassumptionandquantifytheintroducederrorbyapplyingtheDFmethodtocomputergeneratedimagesandcomparingtheobtainedspacingstothecorrectvalues.Thewidestepgeometrydescribedin2.4isusedbecauseitallowsustogeneratesmoothlychangingspacings.Thegeometryisdescribedbytwoparameters,theheightofthestepanditswidth.Figure9showstwogeometrieswithextremeparameters,bothatthesamescale:(a)isanarrow(w=3)5%stepwhile(b)isawide(w=100)1%step.Themaximumoftherelativeerrorwillbemeasuredfordi erentcombina-tionsoftheparameters.Theheightofthestepvariesbetween1and5percentwhilewvariesbetween60and1.Figure10and11showexamplesofsuchmea-surements,themaximumerroroccurswherethespacingchangesmost.The guresinthissectionshowtheprojectionofthedatapointsontothex-zplanetomakeiteasierforthereadertoseethe3dimensionaldistribution.Table1showstheresultsoftheerrormeasurements.Eachrowcontainstherelativeerrormeasuredinforacertainstepheight.Di erentcolumnsrepresentdi erentvaluesofw,theparameterforthewidth. 60 40 20 10 5 3 1 1% 0.16 0.16 0.42 1.06 2.09 3.01 4.082% 0.24 0.37 0.87 2.17 4.42 5.67 8.553% 0.37 0.57 1.33 3.36 6.53 9.24 12.084% 0.51 0.78 1.81 4.40 8.82 11.82 16.565% 0.64 0.98 2.27 5.53 11.23 15.27 22.25Table1:relativeerror[]inspacingmeasurementsinsimulatedimages24 020040060080010005,405,455,505,555,605,65 10008006004002000 Spacing [pixel]Y AxisX Axis(a) 020040060080010005,405,455,505,555,605,65 10008006004002000 (b)Spacing [pixel]y [pixel]x [pixel] Figure9:(a)5%sharp(w=3)step(b)1%wide(w=100)step25 020040060080010000,00,20,40,60,81,0 10008006004002000 relative error [ ]y [pixel]x [pixel](a) 020040060080010000,00,20,40,60,81,0 10008006004002000 relative error [ ]y [pixel]x [pixel](b) Figure10:width:40(a)1%step(b)5%step26 0200400600800100005101520 10008006004002000 relative error [ ]y [pixel]x [pixel](a) 0200400600800100005101520 10008006004002000 relative error [ ]y [pixel]x [pixel](b) Figure11:width:1(a)1%step(b)5%step27 Fromtheerrorlevelswefoundwecandrawthefollowingconclusions:Besidesthenoiselevelthegeometryitselfmustbeconsideredasareason-ablesourceoferrorwhenthemethodisappliedtoexperimentalimages.Asharpstepthatcanbedescribedasa5%changeinspacingswithinabout3fringescanbedetectedwithanaccuracyofabout2:2%.Thiscorrespondstoanerrorofabout6pmina2.5Aspacing.Evensharperorhigherstepscanbemeasuredwithlessaccuracy,soothermethodsforspacingmeasurementsshouldbeconsideredforthesegeome-tries.28 3ThenatureofnoiseNoiseisafundamentalphenomenonpresentineveryphysicalmeasurement.Itisde nedinchapter13.3of[16]asarandomsignaln(t)thatisaddedtotheoriginalsignals(t).Itismostconvenienttodiscussthenatureofnoiseanditse ectsinreciprocal(frequency)space.Anysignals(t)canberepresentedbyitscomplexvaluedFouriercomponents^s()whichareobtainedbytheFTofs(t).Chapter12of[16]givesanoverviewofthepropertiesoftheFT.AveryimportantpropertyoftheFTwhichfollowsdirectlyfromthede nitionistadditivity:dsn=^s+^nWhiletheamplitudesofthesignals(t)andthenoisen(t)areaddedateveryinstantindirectspace,thecorrespondingFouriercomponents^s()and^n()areaddedinreciprocalspace.Thiswillallowuslaterto ndanoisemodelinordertoreducethee ectofnoise!Amoreprecisede nitionofnoisecanbegivenbythepropertiesofitsFouriercomponents^n():theyhaverandomphaseswithoutanycorrelationamongthemtheirabsolutevaluechangesrandomlybutitcanusuallybeapproximatedbyasteadyfunctionn()Theimagetotherightillustratestheadditiv-ityoftheFouriercoecients:Thearrowpoint-ingtothecenterofthecirclerepresentsthesignal^swhilethecirclerepresentstherandomFouriercomponents^nwithaconstantabsolutevaluethatareaddedto^s.^s(u;v)issimply^n,theamplitudeofthenoisecoecient,while'^s(u;v)=arcsin ^n ^s. Re Im ^s^n^s'^s+^nAsthecoecientsareaddedthearrowspointingtotheedgeofthecirclerep-resenttheFouriercoecientsofthemeasuredsignal^s+^n.Notehoweverthattheerrorinthephaseofthecoecientisabout 10whiletheerrorintheabsolutevalueisabout24%inthisexample.Theerrorinthephaseissmalliftheamplitudeofthenoiseissmallerthantheamplitudeoftheoriginalsignalbuttheerrorincreaseswithincreasingnoiselevel.Theratioofthesignalandthenoiseamplitudesdeterminesthequalityofmeasureddata.Acommonwaytoquantifythequalityisthesignaltonoiseratio(SNR)de nedinthiscontextas:SNR=20logj^sj j^nj29 wherej^sjandj^njaretheabsolutevaluesoftheFouriercomponents.UsuallytheSNRisusedindirectspacebutitisasusefulinreciprocalspace.DoesimageprocessingchangetheSNR?No,itcanbeshownthatshiftingandscalingtheinputimageindirectspacedoesnotchangetheSNRinreciprocalspace.Acommonmethodiscontrastenhancementwhichisacombinationofshiftingandscaling.Shiftingindirectspaceaddsaconstantatotheinputimage.Thisconstanthasane ectontheDCcoecientonly,thustheSNRinreciprocalspaceisnota ected.Scalingtheinputindirectspaceisamultiplicationwithaconstant.TheFouriercoecientsaremultipliedbythisconstant,sotheycanceloutintheSNR.Neverthelessattentionmustbepaidwhentheinputimageisconvertedtoorfromanotherimageformat.ThecurrentimplementationoftheDFmethodrequires16bitTIFFimagesasinput.TheTIFFformato erslosslessLZWcompression.JPEGimages,forexample,cannotbeusedbecausetheJPEGalgorithmachievescompressionbynotsavinghighfrequencyFouriercompo-nents!TheSNRwillbeusedlatertomeasurethelevelofsimulatednoiseandasaquantityfromwhichtheerrorofspacingmeasurementscanbedeterminedinexperimentalimages.3.1AmodelfornoiseinaHRTEMimageInthepreviouschaptertheSNRwasde ned.NowthepowerspectrumofaregioninanexperimentalHRTEMimagewillbeexaminedinordertoobtainvaluesforj^sjandj^nj.Figure12showsthePSofanexperimentalimagecontain-inglatticefringes.Incomparison gure13showsthePSofasimulatedimage.The3DvisualisationpluginofImageJwasusedtocreatethesegraphics.In gure12thenoisebackgroundandthepeakwhichcorrespondstotheex-aminedperiodicitycanbeseenclearly.Thepluginallowstorotatethegraphic.Doingso,onenoticesthatthePSisazimuthallysymmetricexceptforthesig-nalofcourse.Thisistrueforimagesiftheastigmatismhasbeencorrected.Ifastigmatismwerepresentthecontourlineswouldbeellipsesinsteadofcircles.BecauseofthesymmetrytheazimuthalaverageofthePSwiththeDCpeakascenterwillbecalculated.Thishastheadvantagethataonedimensionalnoisemodelcanbemadewhichismuchsimplerthanatwodimensionalmodel.Figure14and15showexamplesofazimuthallyaveragedpowerspectra.Nowthesignalandthenoiselevelscanbemeasuredfromthesegraphs.Thesignallevelj^sjisde nedasthemaximumvalueinsidetheaperture.Thisvalueandit'sposition0ismeasuredbytheprogram.Tomeasurej^njthefollowingnoisemodelismade:Weassumethattheintensityofthenoisebackgrounddecreaseslinearlytotheleftofthepeakinthislogarithmicgraph.Figure14isaniceexampleforthatlineardependence.Ifasignalispresent-asin gure1530 Figure12:experimentalPS Figure13:theoreticalPS-thislinearrelationisnotvalidanymore.Thereforeweapproximatetheregiontotheleftofthesignalwhereit'se ectisnegligiblebylinearinterpolation.Whentheparametersforthislinehavebeenfoundwede neit'svalueat0asthenoiselevelj^nj.FromheretheSNRcanbecalculated.Table2showstheparametersfortheinterpolationofthepowerspectrashownin gure15andtheresultingsignaltonoiseratios.j^njisapproximatedbyj^n()jm.negative m[int/pix] c[int] j^sj[int]@0 0[1/pix] SNR[dB]@0 5001 -5.26 7.069 8.81 0.1346 495001 -3.35 6.576 8.65 0.1338 505209 -3.93 6.795 8.1938 0.1641 415209 -3.68 7.07 8.3212 0.1638 37Table2:SNR'sofexperimentalimagesNowthatweknowthequalityofourexperimentalimagesweneedtoexaminethee ectthenoisehasonthespacingmeasurement.31 0,00,10,20,30,40,556789 magnitude - log10 scalespatial frequency500152095374 Figure14:azimuthallyaveragedPSofimagesshowingaholeinthespecimen32 0,00,10,20,30,40,556789 0,060,080,100,120,140,166,06,57,0 (a)intensity - log10 scalespatial frequency 0,00,10,20,30,40,556789 0,100,120,140,160,180,206,06,57,0 intensity - log10 scalespatial frequency(b) Figure15:(a)negative5001,fringes(b)negative5029,fringesandamorphousmaterial33 3.2EstimationoftheerrorinthephaseoftheDFimageTheerrorinthephasegradientwillbeapproximatedusingGauss'serrorprop-agation.Eachpoints(x;y)S(x;y)exp(i's(u;v))intheDFimageisgivenbytheinverseFToftheFouriercoecients^s(u;v)^S(u;v)exp(i'^S(u;v))bys(x;y)=1 N2u;v=N 1Xu;v=0expi2uxvy N^s(u;v)1 N2u;v=N 1Xu;v=0^S(u;v)expi2uxvy Ni'^S(u;v)Capitallettersdenotetheabsolutevalueofacomplexnumberwhilelowercaselettersdenotecomplexnumbers,thehatdenotesquantitiesinFourierspace.WithEuler'sformulawegets(x;y)=1 N2Xu;v^S(u;v)cos2uxvy N'^S(u;v)isin2uxvy N'^S(u;v)1 N2Xu;v^Scos2uxvy N'^SiXu;v^Ssin2uxvy N'^S1 N2(p(x;y)+iq(x;y))wherep(x;y)isthe rstsumintheequationaboveandq(x;y)isthesecond.Nowthephase'scanbecalculated's(x;y)=arg((x;y))=arctanq(x;y) p(x;y)Gauss'serrorestimationisusedtoquantifytheerrorin's.The(x,y)depen-dencyisnotwritten.'2s@'s @p2(p)2@'s @q2(q)2Withthede nitionof'sfollows'2sq2 (p2q2)2(p)2p2 (p2q2)2(q)2q2(p)2p2(q)2 (p2q2)2Thenextstepisto ndpandq.AgainGauss'sestimationisappliedtopandq.p2Xu;v @p @^S(u;v)!2(^S(u;v))2@p @'^S(u;v)2('^S(u;v))2q2Xu;v @q @^S(u;v)!2(^S(u;v))2@q @'^S(u;v)2('^S(u;v))2Fromthispointonthefollowingsimpli cationsandassumptionsaremade:34 ^S(u;v)^Nwhere^Ndoesnotdependonuandvwhichmeansthatwhitenoiseisassumedhere.Thisisareasonableassumptionastheradiusoftheapertureissmallcomparedtothefrequencyrangeonwhichthenoiselevelchanges.Thisassumptionisusedonlyhere,itdoesnotapplytothenoisemodelusedforWiener ltering.^Nhastobemeasuredintheimagebythemethoddescribedintheprevioussection.^N=0outsidetheaperturesothenumberofvaluescontributingtothesumisthenumberofpixelsinsidetheaperture:#APTheTaylorexpansionofarcsin ^n ^sisusedtoapproximate'^S^N ^SO^N ^S2.Thisisaverygoodapproximationforanexperimentalimage,becausethetypicalSNRisabout40dBwhichcorrespondsto^N ^S=10 2.Nowp2andq2takethefollowingformp2Xu;v(cos22uxvy N'^S^N2^S2sin22uxvy N'^S ^N2 ^S2!)^N2Xu;v1=^N2#AP=q2Sop2=q2^N2#AP2andp2q2N4S2leadsto'2s22 p2q2=2#AP^N2 N4S2'sdependsonN,thenumberofpointsoftheFFTandonthenumberofpointsintheaperture.HoweverSismeasuredindirectspacewhile^NismeasuredinFourierspace!SoaconnectionbetweenSand^Smustbemade.ThemaximumsignalSisobtainedwhenallwavesinsidetheapertureareaddedinphase.TheupperlimitforSisthesumofallFouriercoecientsinsidetheaperture.Thissumiswrittenasamultipleof^S:S^Sm N2wheremisaunknownconstant.The nalexpressionfortheerrorinthephaseistherefore:'sp 2#AP m^N ^Swhere#APr2APThespacingdisthequantityweareinterestedin.Itiscalculatedfromthephasegradientwhichisobtainedbysubtractingnearbyphases,dependingonwhichnumericalexpressionisused.Theerrorinthephasegradientis'swhiledependsonthenumericalexpression.p 2fortheforwardorbackwardderivativeand 1=2forthecentralderivativewhichismoreaccurate.Usingthede nitionofthespacingandGauss'serrorpropagationtherelativeerrorinaspacingmeasurementisfoundtobed drAP^N ^Sp 2 mk35 ItincreaseslinearlywiththeradiusoftheaperturerAPandthenoisetosignalratio~N=~S.Asd=2=ktheerrorinkhasabiggerimpactondifkissmall.Thereforetherelativeerrorincreaseswiththespacingd.Theconstantmisknownapproximatelyonly.3.3MeasurementoftheerrorinspacingmeasurementsInthissectiontheerrorinspacingmeasurementswillbemeasuredinordertocheckthedependenciesonapertureradiusandnoiselevel.Todosothespacingsmustbeknownofcourse,sothespacingsincomputergeneratedimageswillbemeasured.Evenifwehadaperfectsampleandaperfectmicrographwewouldstillnotknowthespacingsbetweenlatticefringesinthemicrographwithhighspacialresolutionbecausethemagni cationofthemicroscopeisnotknowwithhighaccuracy.Asindicatedbeforethesemeasurementswillbeusedtoestimatetheerrorinthemeasurementofaspacinginanexperimentalimage.Arelationbetweentheerrorandthequalityoftheimageisrequiredtodothat.Thisrelationhasbeenderivedintheprevioussectionbuttheconstantmisjustapproximatelyknown.Tomeasurethisrelationacertainamountofwhitenoiseisaddedtotheimageandthespacingmeasurementisperformed.Atthispointitisassumedthatthesimulatedimagescontainnonoise.Thisisareasonableassumptionastheperiodicityisnotrunningparalleltoanedgeoftheimage,sotheartifactsduetononperiodicboundariesdonotoverlapwiththepeakwhichcorrespondstotheperiodicity.TheonlynoisepresentinthewholePSisduetoroundo errorsthatoccurduringtheFFT,butweexpectthemtobenegligible.ThenoisewillbeaddeddirectlyinFourierspacesowedonotneedtocalculatetheimageofthenoiseindirectspace.Anoiselevel^NissetandeachFouriercoecientinsidetheapertureismodi edinthefollowingway:^(u;v)=^(u;v)+^N(1+r)exp(i')whererisarandomnumberbetween0:1and0:1and'isarandomphasebetween0and2.rcreatesamplitudenoie,while'createsphasenoise.Whenthenoisehasbeenaddedthemeasurementisperformedasusually.500measurementswiththesamenoiselevelaremadeatacertainlocationinordertohaveenoughvaluesforastatisticalanalysis.Asthenoisecoecientsthemselvesaredi erenteachtimetheresultofthemeasurementisdi erenteverytime.Toexaminethedistributionofthemeasurementsthenumberofmeasure-mentsthatlieinacertainintervaliscounted.ThefrequencycountfeatureofOriginPro[1]isusedtodothat.Theobtaineddistributioncanbeinterpolatedbyagaussianpro le.TheFWHMofthisgaussianpro leisde nedastheerrorinthemeasurement.Figure16(a)showsanexampleofsuchdistributions.Thespacinghasbeenmeasuredat500/500withanapertureof20inthesharpstepgeometrydescribedin2.4.Theregionswherethespacinghasbeenmeasuredhaveaconstantspacing.Thereforetheerrorinthemeasurementsisexclusivelyduetothenoise.Thee ectofthechangein~kwillbeexaminedinchapter4.1.36 Figure16(b)showsthelinearincreaseoftheerrorwithincreasingN=S.Eachpointinthis gurehasbeenobtainedfromadistributionof500measure-ments.Theyaxisshowstherelativeerrorinthemeasuredspacing.37 5,4445,4465,4485,4505,4525,4540,00,20,40,60,81,0 number of measurements in intervalspacing [pixels](a) 0,0000,0010,0020,0030,0040,0050,0060,007012345 @500/500 aperture 30 @500/500 aperture 15 @200/600 aperture 30 @200/600 aperture 15Error [ ] N/S(b) Figure16:(a)statisticaldistributionofmeasurements(b)linearincreaseoferrorwithN/S38 Nowthein\ruenceoftheapertureontheerrorwillbeexamined.Figure17showstherelativeerrorinspacingmeasurementsandthecenterofthedistribu-tionofmeasurementsfordi erentradiioftheaperture.Thenoiselevelisthesameforallmeasurements.Again500measurementsweremadeateachpoint.Thesameanalysisofthedistributionasdescribedbeforeisused.ThelinearincreaseoftheerrorwithincreasingapertureradiusrAPcanbeseenclearly.Wecanalsoseethatthedistributionofmeasurementsisnotcenteredaroundthecorrectvalue(horizontalline)forsmallradii. 510152025300,20,40,60,81,01,21,41,6 5,4465,4485,4505,4525,454 relative error - width of distribution center of distributionrelative Error [ ]radius of aperture [pixel]center of distribution [pixel] Figure17:linearincreaseoferrorwithaperture,N=S0:006SNR=44bBWecanconcludethattheremustbeanidealradiusfortheaperture.Theerrordecreasesaslongasdecreasingtheapertureexcludespointscontainingmostlynoise.Iftheaperturegetstoosmallpointscontainingthesignalwillbeexcluded.Theerrorresultingfromthatishardtopredict.Astheaperturegetssmallerthepositionofthecenteroftheaperturebecomesmoreimportant.Thesee ectswillbestudiedmoreaccuratelyinchapter4.1.Fromthelinearinterpolationsin gure16band17valuesformcanbedetermined.Todothis,theerrorislinearlyinterpolated:d dB^N ^Sord dBrAPdependingonwhichquantityisvaried.Bistheslopeofthelinearinterpolation.39 missimplygivenbymrApp 2 kBorm^N ^Sp 2 kBwhererAP( gure16(b))and^N=^S( gure17)isconstantrespectively.InthecasewhererAPiskept xedand^N=^Sisvariedthefollowingvaluesformarefound:B rAp Position k m 0.681 30 200/600 1,1529 660.361 15 200/600 1,1529 630.228 30 500/500 1,1668 2010.100 15 500/500 1,1668 228Table3:mobtainedfrom gure16(b)InthecasewhererAPisvariedwe ndaslopeofB=0:05whichleadstom=185wherethe^N=^Sratiois0:006.Thevaluesthathavebeendeterminedformaremuchbiggerthanexpected.Thereforetheerrorneedstobemeasuredwhenbothsourcesoferrorarepresent.Thiswillbedoneinsection4.1.3.4RemovalofnoiseNowthatanoisemodelforexperimentalandsimulatedimageshasbeenes-tablishedandthee ectofnoiseontheFouriercoecientsisunderstood,itsuggestsitselftoremovethenoiseoratleasttoreduceitse ect.ThenoisemodelsdescribedabovemodeltheamplitudeoftheFouriercoecientsofthenoise,notthephase!Amodelforthephaseiswaymorecomplicatedorevenimpossibletocomeupwith,becausetheprocessescreatingthenoisearenotknownindetail.Forthesimulatedwhitenoiseit'sde nitelyimpossiblebecausthephaseisrandom.Thesimplestwaytoreducethee ectofnoiseistosubstracttheabsolutevalueofthenoisecoecientfromtheabsolutevalueofthesignalcoecient.Thiswouldcorrecttheerrorifbothcoecientshadthesamephase,whichisingeneralnotthecase.AmoresophisticatedmethodisWiener lteringwhichisderivedinchap-ter13.3of[16].Thistechniquerequiresthenoisysignalandanoisemodeltocalculatea lterfactor.EachnoisyFouriercoecientismultipliedbythecorrespondingfactor()whichyieldsacoecient^U0()thatisascloseaspos-sibletotheuncorruptedcoecient^U().Thatmeanstheobtainedcoecientsminimizethesumofthesquareddeviations:1 1^U0()^U()2min40 isfoundtobe:()=^S()2 ^S()2^N()2j^S()jandj^N()jaretheFouriercoecientsofthesignalandthenoiseatfrequency,bothcanbedeterminedfromthePS.j^S()jissimplygivenbythenoisyFouriercoecientminusj^N()j,see gure3.4(a)foranexampleofanoisemodeland(b)fortheextractedsignal.()iscloseto1wherethenoiseisnegligibleand0wherethenoisedominates.Figure3.4(c)showsthefactorand(d)showsthe lteredsignal.Becausethefactor()resultsfromaminimizationproblemtheobtainedresult^U0()di ersfrom^U()byanamountthatissecondorderintheprecisiontowhich()hasbeendetermined.ThismeansthatthenoisemodelweestablishedforexperimentalimagescanbeusedforWiener ltering,althoughitisjusta rstorderapproximation.Inchapter4this lteringtechniquewillbeappliedtosimulatedimagesafterwhitenoisehasbeenaddedandtoexperimentalimagesusingthe(inlog10scale)linearnoisemodel. 0,050,100,150,206,006,256,506,757,00 intensity - log10 scalefrequency power spectrum noise model 0,050,100,150,200,00,20,40,60,81,0 frequency Wiener filter 0,050,100,150,201,01,52,02,53,03,54,04,55,05,56,06,57,0 intensity - log10 scalefrequency signal 0,050,100,150,201,01,52,02,53,03,54,04,55,05,56,06,57,0 intensity - log10 scalefrequency filtered signal Figure18:(a)azimuthallyaveragedPSandnoisemodel(b)extractedsignal(c) lterfactor()(d) lteredsignal41 3.5MultiplespotmeasurementInsection3.3thestatisticaldistributionofspacingmeasurementshasbeenexamined.Ifameasurementofanoisyquantityisperformeditisnaturaltorepeatthemeasurementinordertoobtainmoredataandto ndthe nalanswerbysomekindofaveragingoveralldatapoints.Figure16(a)showsadistributionofmeasurements.Themaximumofthedistributionisattheaveragevalueifagaussiandistributionisassumed.Theaveragevalueisveryclosetotheexactvalue,asweexpected!HowcanwemakeuseoftheideatoaveragemultiplemeasurementsintheDFmethod?Well,therearebasicallytwodi erentpossibilities:The rstoneistocollectmeasurementsfromdi erentimagestakeninsequence.Thiswouldbeagoodsolutioniftheimagescouldbeobtainedeasilyfromadigitalcameraconnectedtothemicroscopeaswecouldassumethen,thatthenoiselevelandthecorrespondingnoisemodelisidenticalineveryimage.Weuseananalogcameratorecordtheimageonamicrographwhichisdevelopedand nallydigitized.Wedonotexpecttohavethesamenoisemodelintwodi erentmicrographsbecausetherearetoomanyprocessesinvolvedinobtainingtheimage.Thereforeweuseamethodthatcollectsdi erentmeasurementsfromthesamemicrograph.IntheorywecanmeasureaspacingatagivenpositionintheimageforeveryspotthatappearsinthePS.In gure5abasiswasintroducedandthespotswereindexedusingthisbasis,sothatitispossibletorelatethemeasurede ectivewavevector~keff~kitothebasisvectors.Foreachspotweobtainavector~kiandalinearequation:~kii~adi~bwhereianddiaretheindicesassignedtothespoti.WeendupwithasetofNequationscontainingwavevectorsandindicesfromNdi erentspots.Thislinearsystemneedstobesolvedforthebasisvectorsfromwhichallspacingsatacertainpositioncanbecalculated.Ncanbelargerthan2sotherecanbemoreequationsthanvariableswhichisassumedtobethecase.Forexamplethesetofequationsforthexcomponentsofthebasisvectorscanbewrittenlikethis:0BB@1d12d2::::::::::::1CCAab0BB@k1k2::::::1CCAA~x~bWearelookingforthesolution~xwhichminimizesthenormA~x~bWiththehelpofthesingularvaluedecompositionthematrixAcanbewrittenasAUSVT,asdescribedinchapter2.6of[16].UandVareorthogonalandSisdiagonalcontainingthesingularvaluesofA.Thesolutionisgivenby~xA 1~bwhereA 1VS 1UT42 iscalledthepseudoinverseofA.Thesolution~xisdeterminedfrommultiplepoints.Thisminimizationpro-cessisnottobeconfusedwithspatialaveraging.ThespatialresolutionoftDFmethodisnotreducedbythemultispotmethodbecauseeachspotprovidesanadditionaldatapoint.Itisexpectedthatthismethodreducesthee ectofnoisebecausenoisetendstocancelinanaveragingprocess,howeveritcannotreducetheerrorintroducedbythechangein~k.Thenoisemodelintroducedinsection2ofthischapterassumesalinear(inlogscale)decreaseofthenoiseamplitudewithfrequency,sousingspotsathigherfrequenciesautomaticallyhastheadvantageoflowernoiselevels!Howeverthesignalamplitudeisweakerforhigherfrequenciesaswell.TheSNRofeachspotcanthereforebeusedtoweigththevaluesobtainedfromdi erentspots.MeasurementsobtainedfromaspotwithahigherSNRaretobetrustedmorethanmeasurementsobtainedfromnoiserspots.Iftheimagesweretakenbyadigitalcamera,bothaveragingtechniquesdescribedabovecouldbeusedinparallel.43 4ApplicationoftheDFmethodInthischaptertheDFmethodwillbeappliedtosimulatedandexperimentalimages.Firstitwillbeappliedtosimulatedimagesinordertolearnmoreabouttheerrorwhenbothsourcesoferror,namelynoiseandchangingspacingsarepresent.Wealsowanttounderstandthein\ruenceofthecenterandtheradiusoftheaperture.Examplesfornoiseremovalandmultispotmeasurementsinsimulatedandexperimentalimageswillbeshown.4.1computergeneratedimagesunstrainedgeometrywithnoiseThe rstimagethatwillbeexaminedhereisaunstrainedgeometrywithanoiselevelof40bB.Weusethebasicgeometrydescribedin2.4andaddnoise.Themaximumintheapertureis109:148whichisthesignalstrength.Weaddnoisewithanamplitudeof107whichcorrespondstoaSNRof43bB.Thetablebelowshowstheresultsforvarioussettings.Aperturecenter rAp Error[] 697/481 25 10697/481 20 6697/481 15 2699/480 10 1.2Weseethatevenwithoutnoiseremovalthespacingscanbemeasuredeasilywithanaccuracyof1intheunstrainedgeometry.ThisisnottoosuprisingthoughbecausethespotinthePShaslittlestructure,sotheaperturecanbeverysmall.widestepgeometrywithnoiseThemostinterestingsimulatedimageisprobablythewidestepgeometrywithextremeparametersasshownin gure9(a)withnoise.Againweaddnoisewithanamplitudeof107,thesignallevelisthesameasbefore.Theparametersforthegeometryarew=1anda=5%.Thetablebelowshowstheresultsofthemeasurements.Aperturecenter rAp Error[] comment 690/482 25 25 690/482 20 22 690/482 20 20 nonoise690/482 10 - stepisdarkinDFimage695/482 10 22 695/482 10 32 SNR23dB44 Thethirdmeasurementshowsaninterestinge ect:Theradiusoftheaper-turehasbeendecreasedandonesideofthestepappearsdarkintheDFimage,see gure19middle.ThismeanssomeimportantFouriercoecientswerenotincluded.Whentheapertureisshiftedtotherightby5pixels,bothsidesofthestepappearbrightagain, gure19left.Thise ecttellsustopositiontheaperturesothatthewholespotisinsidetheaperture.Inanexperimentalimagethisishardertodobecausethespotisspreadoutmore,buttheDFimagecanbeusedtoseewhetherenoughdatawasincludedintheaperturefortheregionofinterest.Therightimagein gure19showsaDFimagewithhighnoiselevels. Figure19:DFimagesfordi erentsettingsandnoiselevelsTheimagetotherightshowstheposition,thesizeandthecontentoftheapertureforthedi erentmeasurements.Thebrightnesschangesbecauseofdi erentscalefactors,thisandthewhitedotsareduetovisualization.Intheimageonthetoptheradiusis25pixels,thewholespotisinsidetheaperture.Wecanrecognizetwobrightareascon-nectedbyadimmerarea.Inthesecondimagetheradiusisde-creasedto20pixelsandtheaperturestillcontainsbothbrightareas.Thenumberofpointscontainingnoiseonlyhasbeenre-duced.ThethirdimageexplainswhythesideofthestepwhichcontainsthesmallerspacingsappearsdarkintheDFimage:TheDCpeakistothelowerleftintheseimages,sothebrightareafurtherawayfromtheDCpeak(tothetopright)corre-spondstohigherfrequenciesorsmallerspacingsrespectively.AsthisareaisnotincludedintheaperturetheDFimageisdarkinregionscontainingthesmallerspacing.Thefourthimageshowstheapertureshiftedalittletotheright.Thewholespotisinsidetheapertureagain,theDFimageisbrightinbothareas. Werealizethatresultofthemeasurementisnotverysensitivetothechoiceofthecenteroftheaperture.Whenplacingtheaperturecareshouldbetakensothatthewholespotisinsidetheaperture.Thentheradiusshouldbeminimizedinordertoreducethenumberofpointscontainingnoiseonly.45 4.2experimentalimages3buriedquantumdotsInthissectiontheDFmethodwillbeappliedtoaGaInP/InPheterostructuregrownonGaAs(001)substrate.Asdescribedin[14]thesamplewasgrownbymolecularbeamepitaxywhichallowstodepositsinglelayersofatomsoasubstrate.Firstofalla200nmbu erGaAslayerwasgrown,followedbya45nmGaInPlayer.ThelatticeconstantofthislayerwasmatchedtoGaAs.ThismatchingisdonebychangingtheGa/Inratio,hereGa0:52In0:48Pisused.NowthreemonolayersofInParedepositedfollowedbyaGaInPspacinglayer.ThisInP/GaInPsequenceisrepeatedthreetimeswithspacinglayerwidthsbetween2and16nm.Finallythestructurewascappedbya45nmGaInPlayer.ThelatticemismatchbetweenGaInPandInPis3:7%whichmakesitanidealteststructurebecausetheerrorinherentinthemethodisexpectedtobesmallerthan2%.DuetothislatticemismatchtheatomshavetorearrangeclosetoeveryInP/GaInPinterfacewhichleadstosocalledburiedquantumdots.Theelectricalandopticalpropertiesofthesequantumdotsdependontheirsizeandtheirshapewhichinturncanbederivedfromspacingmeasurements. ~a~b001011121320212223010211 0,00,10,20,30,40,56789 0,030,040,050,060,070,080,090,100,110,126,46,66,87,07,27,4 intensity - log10 scalefrequency Figure20:PSandazimuthalaverageofPS46 Figure21showsaHRTEMimageoftheregioncontainingthestructuredescribedabove,thisimageandimage25waskindlyprovidedbyDr.N.Y.Jin-Phillipp,Max-Planck-InstitutfurMetallforschungatStuttgart.Figure20showsthepowerspectrumanditsazimuthalaverage.ThePSshowsmanyhighfrequencyspotswhichindicatesthatonlyslightchangesinspacingsarepresent.Thespotshavebeenindexedusingthebasisvectors~aand~b. Figure21:HRTEMimageof3buriedInPquantumdots47