Tuning the Convergence Angle for Optimum STEM Performance

102-218 Bulletin JDC_(v9)FINAL.qxd 7/23/05 3:20 PM Page 24

Tuning the convergence angle for optimum STEM performance

M.Weyland and D.A. Muller Introduction Cornell University, Department of Applied and Engineering Physics, Ithaca, NY 14853, USA Scanning transmission electron microscopy (STEM), and in particular high angle annular dark field (HAADF) “Z-contrast” STEM, is becoming a key tool in the resolution of structural and functional problems at the atomic ABSTRACT scale. The reasons for this are varied but the most com- pelling are the combination of the high resolution, on the order of an Angstrom, and the interpretability of The achievable instrumental performance of a scanning the image contrast, which is strongly sensitive to the transmission electron microscope (STEM) is determined atomic number of the scattering atoms and the mass- by the size and shape of the incident electron probe. thickness of the specimen[1-4]. While there are a large The most important optical factor in achieving the opti- number of factors that control the ultimate perfor- mum probe profile is the radius of the probe-forming mance of a STEM instrument, including electron source aperture, which determines the convergence semi- brightness, accelerating voltage and lens aberrations, angle of the illumination. What is often overlooked many of these are set by the design and specification of however is that small deviations from this optimum can the microscope. The operator is left to optimize the gun degrade both the resolution and interpretability of settings and lens strengths and choose the probe-form- image contrast. A 30% error in aperture radius can lead ing aperture that determines the convergence semi- to a factor of 2 contrast reduction in typical lattice angle (α) of the electron probe. This aperture is known spacings, and a 5 Å error in the thickness measurement as the objective aperture in dedicated STEM instruments of thin layers (such as gate oxides). Theoretical calcula- and the condenser aperture in a combined TEM/STEM tions of the optimum convergence angles, from a olutions

S (to avoid confusion the term probe forming aperture is wave-optical consideration of the probe forming condi- used throughout this paper). With the convergence

nano tions, are explained and their consequences discussed. angle fixed, source size can be traded off against probe An experimental approach to the measurement and current. tuning of the convergence angle is then introduced. 24 The choice of probe forming aperture, and the resultant α, is often overlooked (with devastating consequences) as the choice is often compared to the choice of objective aperture in parallel beam illumination. The selec- tion of an objective (post specimen) aperture in TEM is NanoResearch usually made to improve contrast in the recorded image through exclusion of scattered electrons. High resolution (lattice images) in Bright-field TEM are achieved as long as the aperture is not too small for the collection of electrons scattered to the Bragg spot associated with the lattice spacing required. Indeed even without an objective aperture a high resolution TEM image can be obtained by allowing the entire range of frequencies to 102-218 Bulletin JDC_(v9)FINAL.qxd 7/23/05 3:20 PM Page 25

a) b) Current (pA) Contrast Transfer Function Contrast Transfer

k (Å-1)

Figure 1. a) Contrast transfer function (CTF) for a conventional, almost-parallel beam, TEM calculated for an FEI tecnai F20 SuperTWIN. The dashed arrow marks the potential position of the objective aperture “cutoff” which would exclude contrast reversals from higher frequencies in the resultant TEM image. The STEM CTF is calculated for this sized aperture, has a 5% information limit almost double the aperture size (solid arrow). b) Beam current enclosed within a given diameter for the F20-ST (200 kV, Cs=1.2 mm, 1Å source size) for the optimal 9.6 mrad and too- large 13 mrad aperture as an illustration of the effect of condenser aperture size on STEM 2D probe point spread function (PSF). The larger aperture provides almost double the beam current, but all of this extra current falls outside the central peak. Increasing the aperture size beyond the optimum only reduces the signal/background ratio.

be included, damped by the natural contrast transfer A holistic approach to optimizing the convergence function (CTF) of the instrument. Essentially the objec- semi-angle (α) is described, based on the wave optical tive aperture is a linear filter determining the frequen- theory of probe formation. The experimental method cies to be included in the TEM image, for example in a for calibrating the convergence angles of probe forming olutions

weak phase object filtering out the high frequency apertures in the STEM is described and suggestions S oscillations in the CTF which show inverted contrast, made on how to fine tune any mismatch between theo- nano see Fig.1 a). Increasing the aperture size does not affect retical optimum and instrumental values. the lower frequencies. However the effect of aperture radius on the CTF for HAADF STEM imaging is a far 25 more complex issue. The ADF image point spread func- Calculating optimum convergence angles tion is proportional to the square of the probe wavefunction and not the wavefunction itself (as in Bright There are three classical contributors to the probe shape field TEM). The resultant response to increasing the in the electron microscope; the effect of a finite source aperture size is non-linear and affects all spatial fre- size (the ‘gun’) contribution, the aberrations induced by NanoResearch quencies, not just the highest (squaring the wavefunc- the primary imaging lens (the spherical aberration (Cs) tion in real space is equivalent to self-convolving its term), and the diffraction limit. In a simple geometric Fourier transform in diffraction space, so high and low optics approach these factors can be assumed to be frequencies are mixed). Essentially phase errors at large Gaussian and hence can be added in quadrature[5, 6]. angles to the optic axis are mixed in to the lower fre- Achieving optimum performance is a balance between quencies (which would be unaffected in BF), degrading Cs and diffraction, see Fig. 2. The source size term due both contrast and image localization, even as the infor- to the finite gun brightness can also be added in mation limit is increased. quadrature and has the same angular dependence as the χ ( α )  0 2 λ π d 1 4 = 4 C = 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage26 0 26 NanoResearch nanoSolutions 1 α 0 semi-angle ( (FWHM) probesizeandtheoptimumconvergence the minimum(d Appendix A.Thisleadstoasimpleexpressionforboth forming conditions.Adetailedderivationisgivenin angle, is toleratedandsolvingfordefocusmaximum π ture. Inpractice,amaximumallowablephaseshiftof ideal lensforADFhaszerophaseshiftacrosstheaper- quencies byapplyinganegativedefocus(Fig.3).An partially compensatedforalimitedbandofspatialfre- ideal lens,leadingtoapositivephaseshift.Thiscanbe rays off-axistobedeflectedmorestronglythanforan value. Sphericalaberrationfromaroundlenscausesthe where the opticaxis, which describesthephaseshiftofawaveatangle applied basedontheScherzeraberrationfunction[9] more accuratewave-opticalformulation[7,8]canbe gence semi-angleforhigh-resolutionSTEMimaging.A probe sizeandunderestimatestheoptimumconver- be dominant,itconsistentlybothoverestimatesthe large analyticalprobeswherethesource sizetermcan diffraction limit.Whilethisapproachissuitablefora SuperTWIN (C If thesearecalculatedfora200kVFEITecnai F20 for particular formed thecontrasttransferfunction(CTF), ate thepointspreadfunction,andifFouriertrans- over theprobe-formingapertureandsquaredtogener- ration functioncanthenbeintegratednumerically cal methods(~2.8Åforan~7mradaperture).Theaber- performance thanthatsuggestedbythenonwave-opti- semi-angle of9.6mrad.Thisisasignificantlyhigher achievable is1.6Åwithanoptimumconvergence α 4 /2 (i.e.aquarterwavelength)acrosstheobjectivelens . α λ 0 is theelectronwavelengthand 43 allows theassessmentofoptimumprobe C α 3 0 α s ) [9]:   s χ(α) =1.2mm) theminimumprobesize and 0 ) attainablefull-widthhalfmaximum : = 4 ∆ s f , [7, 8]. 1 2     2    C 4 1 ∆ s f is thedefocus λ 4 λ f α α to -(1) -(2)     − Si/SrTiO ration limitedprobeareexaminedinFig7fora The imagingproblemsassociatedwithasphericalaber- ings comeintofocus. decrease, evendisappearentirely, asthesmallerspac- Effectively, thecontrastoflargerlatticespacingswill expense ofalossinresponseforlowerfrequencies. convergence athigherdefocusvalues,theycomethe While highfrequenciesareattainablewiththe13mrad anced frequencyresponsewiththelargeraperture. EELS) recorded.Afurtherproblemisthelackofabal- a largedelocalizationofanyanalyticalsignal(suchas reducing thecontrastfromanimagedlatticeandcause probe currenttobespreadoutsidethecentralmaximal increasing theFWHMofprobeandcausing large probetails,seeFig.4d).Thishastheeffectof the aperturealsocausingsubstantialdelocalizationand defocus howevertheaccumulatedphaseshiftsacross lattice spacingsatthisdefocusvalue,seeFig.6.At which isduetothestrongpeaksinCTFattypical SrTiO The largeprobetailsspreadtheintensityofhigh-Z aberration limitedexampleshowssignificantproblems. with nonoptimalconvergencesemi-angle,thespherical tails fromanoversizedaperturecanwashoutthe ( conditions inFig.4.Theoptimumdefocusfor9.6mrad (9.6 mrad)andsphericalaberrationlimited(13 Example plotsofPSFandCTFareshownforoptimum the secondary optimaldefocus( the secondary ma. Focusingbyeyewillmorelikelyleadtofocusingat ture, thereare2defocussettingswhichlocalmaxi- will lookthe“sharpest”.Howeverfor13mradaper- There isalsoonly1defocussettingatwhichtheimage low spatialfrequenciesthanforthebiggeraperture. with theoptimalapertureisroughly2-3timeslargerat 5, andminimalprobetailsinthePSF. Thecontrast a HfO high-Z layers(suchasanaccidentalSiO thin,low-Zlayerissandwichedbetweentwo If avery in bothlayers. a significantlossincontrastfromtheatomiccolumns face itselfalsoappearssignificantlybroaderandthereis intensity towardstheinterface.Thewidthofinter- ∆ f opt 3 2 ~ 500Å)generatesaprobewithsimpleCTF, Fig. layer intotheSiresultinginaslopeofincreased gate oxideanditssiliconsubstrate),theprobe 3 interface. Whilebothimagesareacquired ∆ ∆ f opt2 at ~1300Å), 2 layer between 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage27 limit ofaSTEM[1,10]. andtestingtheinformation imaging ofperfectcrystals information limitmakesitusefulforhigh-resolution atomic resolutionanalysisimpossible.Theincreased taster foranalyticalwork–thedelocalizedtailsmake Although usinganoversizedapertureisclearlyadis- Diebold etal.[4]. of thismetrologyproblemarecoveredinthearticleby suring thewidthof15Åthickgateoxide–moredetails its width.Thiscanleadtoerrorsaslarge5Åinmea- sure thethicknessoflow-Zlayerwilloverestimate the heavierlayerscanstilloverlapandattemptstomea- slightly thickerlow-Zlayers(10-20Å),tailsfromeachof will notevenbedetectableabovethebackground.For details ofthelightlayercompletely, andsometimesit Phase Shift (rad) , , α (mrad) , convergence semi-angle( Figure 2.Balancingsphericalaberrationagainstdiffraction.Atlow (d dominates, whileathigh optical treatment. size of~2.8Å,whichisamorepessimisticestimatethanthewave d (d s = t s Probe size (nm) ). Noteresultantconvergenceangleis6-7mrad,togiveaprobe ) isdominant.Thetermsarecalculatedbydd=0.61 1 / 2 C s α 3. Terms areaddedinquadraturetogeneratethetotal (1/4C cancel thesphericalaberrationcontribution defocus contribution(the1/2 low convergencesemi-angle( shift acrosstheprobeformingaperture.At against defocustoobtainauniformphase Figure 3.Balancingsphericalaberration ture size, the optimaldefocusandmaximumaper- band shownbytheshading,whichsetsboth ered tobeaquarterwavelength,i.e. shift, butatolerableerrorisusuallyconsid- χ aberration dominatesthefunction ( α Convergence semi-angle( ). Anideallenswouldhavezerophase convergence semi-angle s Optimum α α α 2 ) thediffractioncontribution(d the sphericalaberrationcontribution term), whileathigh α 0 . α ∆ α f ) anegative the spherical α 2 term) can λ α / α ) π and d /2 ) term 27 NanoResearch nanoSolutions 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage28 28 NanoResearch nanoSolutions settings ( a) PSFandc)CTFfora9.6mradconvergenceangle.b)d)13Markedareoptimaldefoc carried outfortheequivalentofatecnaiF20SuperTWIN:200kVacceleratingvoltage,C Figure 4.Contrasttransferfunctions(CTF)andpointspread(PSF)calculatedfromwaveopticalconsiderations.Allc lens aberrations,ifthebeamisfocusedbefore the sampleindiffractionplane.Inabsenceof Out offocus,theRonchigramgivesashadowimage forming apertureismuchlargerthantheBraggangles. wheretheprobe of anamorphousregionoracrystal Ronchigram istheconvergentbeamdiffractionpattern order toseeallthedetailsinRonchigram).The al[11, 12](usingthelargestprobe-formingaperturein Ronchigram formedonanareaofamorphousmateri- correct apertureistomakeuseofthecoherent The idealapproachtoselectionandalignmentofthe measuring convergencesemi-angles Ronchigrams, selectingaperturesand

Defocus (Angstroms) Defocus (Angstroms) ∆ ƒ opt ), foralargeraperturethereissecondoptimal( radius (Angstroms) k (Angstroms -1 ) ∆ ƒ opt2 ) due to the secondary maximaindefocus. ) duetothesecondary small diskofinfinitemagnification.Imagereversals leaving adistortedshadowedimagetosurroundthe At largeangles,thebeammustcrossbeforesample, only bebroughttocrossoverforsmallangles(Fig.8d). When thelenshassphericalaberrations,beamcan thinsample. smooth andfeaturelessforavery infinite magnification(defocusis0),whichshouldlook on thesample(Fig.8c),Ronchigramisanimageat Ronchigram isinverted.Whenthebeamatcrossover beam isfocusedafterthesample(Fig.8b), magnification isthecameralength/defocus).When image oftheilluminatedportionsample(and sample (Fig.8a),theRonchigramisanerect,magnified

Defocus (Angstroms) Defocus (Angstroms) s of 1.2mmandadefocusrange-1000to+2000Å. radius (Angstroms) k (Angstroms -1 ) alculations us 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage29 men scancoils. the beamthroughreciprocalspaceusingpostspeci- probecanbeformedontheBFdetectorbyrastering ary STEM instrumentthediffractionpatternfromastation- era (orplatefilmifthisisnotavailable).Inadedicated screen, whichcanbedirectlycapturedonaCCDcam- instrument thisisusuallyconjugatewiththeviewing tion) planeoftheSTEM.InacombinedTEM/STEM bration oftheCBEDpatternindiffraction(detec- vergence semi-angleisfairlysimple;relyingonthecali- The experimentalmethodformeasurementofthecon- have tobeadjusted. formance thebalanceofprobeforminglenseswill reality thisisrarelythecaseandtoachieveoptimalper- give theoptimumconvergencesemi-angle.Howeverin forming aperturesisalmostpreciselythecorrectsizeto the microscopeshouldbesuchthatoneofprobe stop/mark. Ideallythecondenserandobjectiveopticsof aperture canbecenteredwithreferencetothebeam size hidesmuchoftheRonchigramdetail),but center theapertureonRonchigram(assmall size. With thecorrectapertureitbecomesdifficultto and changingaperturesuntiloneisfoundofthecorrect (either bythebeamstopormarkonsmallscreen) achieved bymarkingthecenterofronchigram the rings.Choosing,andaligning,apertureisbest diffraction limited.Anidealaperturewillsitjustinside highest frequenciesavailable,andtheprobebecomes However toosmallanaperturewillnotincludethe sen thatissmallenoughnottoincludetheserings. at whichC defocus). Wheretheringsstartindicatesfrequency cannot bothbeatinfinitemagnificationthesame replaced byfaintrandommist,asthetopandbottom is morethanafewnmthick,thesmoothregionwillbe information increasingwiththeradius.(Ifsample nite magnification,withthefrequencyofimage center isanimageoftheamorphousspecimenatinfi- tern aconsequenceofC surrounding thecentralregionofinterferencepat- is areflectionoftheimagingoptics,with“rings” is properlystigmated).ThisfocusedRonchigram,Fig.9, tings, leadingtoringsintheRonchigram(ifprobe from invertedtoerectmustoccurforalloverfocusset- s dominates andanapertureshouldbecho- s and thesmoothregionin threefold. aperture reducesthecontrastatlowerspatialfrequenciestwoto point-spread functionforeachaperture.Noticehowthelarger 13 mradapertureatthedefocussettingtogivemost-peaked Cs=1.2 mm,1Åsource size)fortheoptimal9.6mradandtoo-large Figure 5.ContrastTransfer FunctionfortheT20-ST(200kV, maximum at1300Åwhichcontainsallspatialfrequencies. by eye(orFFT)forthesharpestimagewilllikelypicksecondary defocus settingsfromthelowerspatialfrequencies.Auserfocusing cies aretransmittedthroughthelensmostefficientlyatdifferent Due tothephaseshiftacrossaperture,higherspatialfrequen- 4 latticespacingsofsilicon(3.13,1.92,1.63,1.36Årespectively). kV, Cs=1.2mm,1Åsource size)asafunctionofdefocusforthefirst Figure 6.ContrastTransfer FunctionfortheT20-SuperTWIN(200 , , 29 NanoResearch nanoSolutions 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage30 30 NanoResearch nanoSolutions length) andtheconvergencesemi-angle( between thespacingBraggdiscs( W By measuring angle ( aperture diameter( the samplearenotadjusted).InFig.10experimental Bragg disc(a)foreachaperture(providedthelensesand sured bymerelymeasuringthediameterofcentral once astherestofconvergenceanglesmaybemea- length): assuchthefullpatternneedonlyberecorded this isdeterminedbytheBraggangleandcamera betweenthem(as between theBraggdiscswillnotvary forming aperturesinagiveninstrumentthespacing While thereareusuallyahandfulofdifferentprobe- lation issilicon(200)or(220),asdemonstratedinFig10. determined. Acommonreflectionchosenforthiscalcu- ith a known crystal andaknownorientationtheratio ith aknowncrystal b a = θ b ) foraparticularreflection(forgivenwave- θ α b

a Normalized intensity and a ) willbeproportionaltotheBragg b for aknownspacingallows Profile position(Å) α b ) oftheprobe: ) andthe α to be and adjustC2tofocus.Howevertheanglessubtended objective lensattheoptimumvalueforeucentricfocus modern TEM/STEMinstrumentsitisusualtofixthe in muchthesamewayasobjectivelens.Indeed used tofocus,intheplaneofspecimenandprobe, In STEMimagingthefinalcondenserlens(C2)canbe are easiertomodifythanthediameterofaperture. objective lenses,havesignificantinfluenceover involved: theimagingoptics,inparticularC2and the convergencesemi-angleitisnotonlyfactor While thechoiceofaperturehasalargeinfluenceon T optical calculations. determined astheoptimumsemi-anglefromwave- mrad. Noneoftheseaperturesisclosetothe9.6mrad probe formingaperturesis:5.6,8.0,11.1and16.9 measurements ofconvergencesemi-angleforthefour uning convergencesemi-angle a) C images acquiredfromthesameareawith between SiandSrTiO width gence anglesonapparentinterface Figure 7.Effectofapertureradius/conver- contrast ontheatomiccolumns. andthedropin broadening oftheinterface sity oftheSrTiO of theprobetails:a“spreading”inten- tions. Theseprofilesclearlyshowtheeffects at thepositionsmarked,fortwocondi- aged andnormalized)throughtheinterface, semi-angle respectively. c)Linetraces(aver- s and b)diffractionlimitedconvergence 3 layer intotheSi,apparent 3 . a),b)STEMHAADF α and 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage31 lens rangesuitableforformingasmallSTEMprobeis SuperTWIN, Fig.11a).Itisapparentthattheobjective excitations oftheobjectivelensforaTecnai F20 series ofsemi-angleshavebeenmeasuredforvarying the other, tostayinfocusyetchangetheeffective it ispossible,bymodifyingoneandcompensatingwith equivalent changeinconvergencesemi-angle.Assuch lent defocuswithbothlenseswillnotresultinan outanequiva- by bothlensesaredifferentandcarrying only bemaintainedatsmallangles. (d) Nearcross-overwithsphericalaberration-raysoff-axistocomeafocusbeforethesample.Theregionofnear-infinite image ismagnifiedandinverted.(c)Intheabsenceoflensaberrations,ifbeamnearcrossover, theimagemagnificatio magnified anderect,withthemagnificationbeingratioofcameralength/defocus.(b)Ifcrossoverisaftersam of sphericalaberration.(a)Iftheprobeisfocusedtoacrossoverbeforesample,shadowimageprojectedonvie Figure 8.RaydiagramsshowingtheformationofRonchigramshadowimagesfordifferentdefocusconditions.Cases(a)-(c)arein (a) (c) C s =0 C s >0 ? f α . A performance inacombinedsystemimpractical. variables thatwouldmakeachievingacceptableSTEM strength atacertainspecimenheightremovesthefree TEM/STEM, providingafixedvalueofobjectivelens importance oftheeucentricfocussettinginamodern approximately 7mrad(~4-11mrad).Thisillustratesthe to focus)coveringaconvergenceanglechangeof lens strength(andthebalancingchangeinC2tobring limited,witha3%percentvery changeinobjective C (d) (b) s =0 C s >0 ?

magnification can n isalmostinfinite. wing screenis ple, theshadow f

the absence 31 NanoResearch nanoSolutions 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage32 32 NanoResearch nanoSolutions selected outbytheprobe-formingaperture,howevertoolargeanaperturewillincludeC movements oftheincidentprobeandspecimen.Thespatialfrequency(k)STEMimagewillincreasewithradius presence oftheshadowimageinsiderings.Inrealcase“blobs”incenterronchigramflashrandomlydue mental Ronchigram. dotted lineinsidethe50 is independentofthechosenapertureotherthreeaperturescanbecalibratedbyjustrecordingwidthzeroorder of Si,orientedontotothe110axis.Theconvergencesemi-angle( Figure 10.MeasurementofSTEMconvergenceanglesinanFEIF20SuperTWIN.Thediffractionpatternis“calibrated”onthe2 probe-forming aperture(whichdefinestheconvergencesemi-angle)isjustinsideC the largestprobeformingaperture.b)Asimplifiedconceptual“map”ofronchigramshowinglocationringscaused Figure 9. Description ofthecoherentelectronronchigram(formedfromamorphousmaterial).a)Anexperimentalronchigram,formedinside µ m aperturerepresentstherelativescaleof50 α ) isproportionaltotheratioofdiscwidthspacing( µ m aperture. s rings andismarkedwithadottedlineintheexperi- s rings. Thereforetheideallocationfor

to thesmall disk ( Ronchigram

by C 00 reflection a a s ). The / , andthe b ). As b 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage33 and externalnoise. where themicroscopeismostsensitivetostrayfields gun lens,sothefinalchoiceofapertureswilldependon ness isfixed).Thisagaincanbecompensatedbythe portionally largerbeamcurrent(becausebright- tures willproducealargersource sizeblurringandpro- angle. Foragivenconvergenceangle,thelargeraper- cal aperturescanbetunedtotheoptimalconvergence W semi-angle, Fig.11b). and balancingwithC2,alsohasacleareffectonthe vergent (STEM)illumination.Changingthislensvalue, is instrumentalinswitchingbetweenparallelandcon- FEI systemsitisactuallypartoftheobjectivelens)and condenser sitsbetweentheC2andobjectivelenses(on systems), balancingoutwithC2orobjective.Themini- adjust theminicondenserlens(theTWINonFEI convergence semi-angle.Analternativeapproachisto rate balanceofobjective/C2toachieveanoptimum out here,shouldbesufficienttodetermineamoreaccu- A shortthrough-convergenceseries,ofthekindcarried indicates aconvergedbeammode(STEM)while+veisforparallel(CTEM). In allcasesthespecimenwasrefocusedusingC2.Notetwinlensvaluesare–veduetoconventioninFEIinstruments, Effect oflensstrengthsonmeasuredconvergencesemi-angles.Fora)Objectiveandb)Twin (mini-condenser)lens Figure 11. α ith appropriatelenssettingsalmostanyofthephysi- Convergence semi-angle ( ) Objective lensstrength(%)

Convergence semi-angle (α) as optimalinterpretablecontrastinthelatticeimage. measurement offeaturessuchasinterfacelayerswell performance. Thisapproachshouldleadtomorecertain probe forminglenses,theoptimizationofmicroscope in combinationwithcalibratedadjustmentstothe imental semi-anglehasbeendescribedandthisallows, semi-angle. Theapproachtomeasurementoftheexper- image contrastrequirestightcontroloverconvergence that tocombineoptimumresolutionwithinterpretable tions fromwave-opticalconsiderationsclearlyshow Calculations ofpointspreadandcontrasttransferfunc- Conclusions imaging intheprescenceofsphericalaberration[9]. estimate ofanoptimalaperturesizeforincoherent microscopy containsthenowoften-citedwaveoptical Scherzer’s 1949paperonspatialresolutioninelectron Aperture SizeforSTEMImaging Appendix A–DerivingtheOptimal Tw in lensestrength(%) a –vevalue

values. 33 NanoResearch nanoSolutions χ ( α )  2 λ π 0 1 4 d 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage34 4 C = 34 NanoResearch nanoSolutions 0 ϕ = i.e. The minimumofoccursat The problemcanbesimplifiedbywritingso prefactors inequation2(maintext)arealtered. ble phaseerrorischangedfrom i.e. forallIfthemaximumtolera- allow amaximumphaseerrorofquarterwavelength– focal planeasafunctionofscatteringsemi-angle bly smallbybalancing where thephaseshiftacrossapertureiskepttolera- We the higherordertermstokeepphaseerrorssmall). tions (whichinvolvesbalancingCsanddefocusagainst to generalizetheapproachincludefifthorderaberra- understand boththeunderlyingassumptionsandhow A fullderivationisgivenheresothereadercan it,,inkeepingwithLordRayleigh’s Our goalhereistofindthemaximumaperturesize, resolution criterion[13]. width, functionwith the PSFbecomessquareofAiry For anideallens,,andfiniteaperture and squaringthewavefunction. the probecanbecalculatedbyFouriertransforming The shape,orrather, thepointspreadfunction(PSF)of where thephaseintroducedbylensis α

start bywritingtheelectronwavefunctioninfront = () 0 α χ d . e i 61 () α   χ ( x ) 0 ∆ s = f λ against 1 2 0 π C /2 onlythenumerical 0-(A.4). =0 α s . Scherzerchoseto . 2    61 x α 0 χ(α) = as f α λ -(A.5). -(A.3). -(A.2). -(A.1). 0 2 , α α 0 , α that allthephaseshiftsatangleslessthan The basicassumptionatthestartofthisderivationwas aperture. (1.56 ),whichisalsocalledtheScherzer optimal aperturesizeforcoherentimaginginTEM Note thatthisoptimalaperturesizeissmallerthanthe defocus, andnotethattoget We ewn ( we want The optimaldefocuscannowbefoundbynotingthat For wegettheoptimal we find aperture size ed bythediffractionlimitforincoherentimagingwith required tobesmallsotheimageblur, these constraintswefindinstead, length orless,i.e.solvingforWith phase errorisusuallytakentobeatenthofwave- It isinterestingtonotethatinlightoptics,thetolerable equation (2)inthemainbodyoftext. Equations (A.9)and(A.10)arethedesiredresultfor The largestaperturesize find χ ( −

α now useequation(A.6)toeliminatetheoptimal x ) ≠ = 0 0 x , α x > min α 0 )= – 0 , in Fig.3.Setting(A.3)equalto0, π 2Substituting(A.5)into(A.3)we /2 ∆ α 0 , issetbythepointwhere x 0 as afunctionofdefocus d 0 , willbelimit- α 0 were -(A.10). -(A.9). -(A.8). -(A.7). -(A.6). 102-218 BulletinJDC_(v9)FINAL.qxd7/23/053:20PMPage35 .J.R.MichaelandD.B.Williams, "AConsistent 6. V.E. Cosslett,"ProbeSizeAndProbeCurrent 5. .A.C.Diebold,B.Foran,C.Kisielowski,D.A. 4. width athalfmaximum(large whichfallsbetween ing tomakebetweenachievingthesmallestprobefull that ischosendependsonthecompromiseonewill- of aperturesizeanddefocus[8].Theprecisevalue ered waveopticalestimatesofprobesizeasafunction havealsoconsid- our twoestimates.ColliexandMory disk[7]. Thisgives the probetails,butatpriceofincreasingcentral has calculatedthepointspreadfunctionthatminimizes .P.E. Batson,N.Dellby, andO.L.Krivanek, 3. S.J.Pennycook,"ZContrastSTEMfor 2. D.H.Shin,E.J.Kirkland,andJ.Silcox, 1. References probe tails(smallestphaseshifterror). euigtepaeerrfo togivesa beam current(whichisproportionalto 20% worsespatialresolutionanda40%reductionin Reducing thephaseerrorfrom Microscope". Resolution InTheAnalyticalElectron- Definition ofProbeSizeAndSpatial- Microscope". In ScanningTransmission Electron- 289-303 (1987) Microanal Electron Microscopy". Determination byAdvancedTransmission S. Stemmer, "ThinDielectricFilmThickness Muller, S.J.Pennycook,E.Principe,and 617-620 (2002) corrected electronoptics". "Sub-Angstrom resolutionusingaberration 30 Materials Science". 2458 (1989) 100 kV".Appl.Phys.Lett. images withbetterthan2Åresolutionat "Annular darkfieldelectronmicroscope 58-69 (1989) . 9 493–508 (2003) Journal ofMicroscopy Optik . Ultramicroscopy 36 α Microsc. and (1) 85(1972) 0 ) andthesmallest 55 Nature 2456 - α 0 . 2 . ). Kirkland . 418 147 -(A.11). 13. Lord Rayleigh, "On the theory ofoptical LordRayleigh,"Onthetheory 13. E.M.JamesandN.D.Browning,"Practical 12. 1 J.M.Cowley, "AdjustmentOfASTEM 11. P.D. NellistandS.J.Pennycook,"SubAngstrom 10. .O.Scherzer, "TheTheoreticalResolutionLimit 9. .C.Mory, C.Colliex,andJ.Cowley, "Optimum 8. .E.J.Kirkland, 7. 167-195 (1896) microscope". images, withaspecialreferencetothe 78 analysis inSTEM". aspects ofatomicresolutionimagingand Ultramicroscopy Instrument ByUseOfShadowImages". Rev. Lett. transmission electronmicroscopy". incoherent resolution byunderfocused Applied Physics of theElectronMicroscope". Ultramicroscopy defocus forSTEMimagingandmicroanalysis". Electron Microscopy (1-4) 125-139(1999) 81 4156-4159 (1998) Advanced Computingin Phil. Mag . . . 20 21 4 Ultramicroscopy . 1998,NY: Plenum. 413-418 (1979) 20-29 (1949) 171-178 (1987) . XLII-fifth series Journal Of . Phys. 35 NanoResearch nanoSolutions