Scanning Transmission Electron Microscopy

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2

ScanningTransmission

ElectronMicroscopy

P.D.Nellist

1. Introduction

Thescanningtransmissionelectronmicroscope (STEM)isaverypowerfulandhighlyversatile instrumentcapableofatomicresolutionimaging and nanoscale analysis. The purpose of this chapter is to describe what STEM is, to high-lightsomeofthetypesofexperimentsthatcan be performed using a STEM, to explain the principlesbehindthecommonmodesofopera-tion,toillustratethefeaturesoftypicalSTEM instrumentation,andtodiscusssomeofthelim-itingfactorsinitsperformance.

1.1 ThePrincipleofOperation

ofaSTEM

Figure 2–1 shows a schematic of the essential elementsofanSTEM.MostdedicatedSTEM instruments have their electron gun at the bottomofthecolumnwiththeelectronstravel-ingupward,whichishowFigure2–1hasbeen drawn. Figure 2–2 shows a photograph of a dedicatedSTEMinstrument.

More commonly available at the time of writing are combined conventional transmis-sion electron microscope (CTEM)/STEM instruments.Thesecanbeoperatedinboththe CTEMmode,wheretheimagingandmagnifi-cation optics are placed after the sample to provide a highly magnified image of the exit wavefromthesample,ortheSTEMmodeas described in Section 8. Combined CTEM/ STEM instruments are derived from conven-tionaltransmissionelectronmicroscopy(TEM)

columns and have their gun at the top of the column. The pertinent optical elements are identical, and for a TEM/STEM Figure 2–1 shouldberegardedasbeinginverted.

In many ways, the STEM is similar to the more widely known scanning electron micro-scope (SEM). An electron gun generates a beamofelectronsthatisfocusedbyaseriesof lensestoformanimageoftheelectronsource ataspecimen.Theelectronspot,orprobe,can bescannedoverthesampleinarasterpattern byexcitingscanningdeflectioncoils,andscat-teredelectronsaredetectedandtheirintensity plottedasafunctionofprobepositiontoform animage.IncontrasttoanSEM,whereabulk sampleistypicallyused,theSTEMrequiresa thinned, electron transparent specimen. The most commonly used STEM detectors are therefore placed after the sample, and detect transmittedelectrons.

Since a thin sample is used (typically less than 50nm thick), the probe spreading within the sample is relatively small, and the spatial resolution of the STEM is predominantly controlledbythesizeoftheprobe.Thecrucial image forming optics are therefore those beforethesamplethatareformingtheprobe. Indeed the short-focal-length lens that finally focusesthebeamtoformtheprobeisreferred toastheobjectivelens.Othercondenserlenses are usually placed before the objective to controlthedegreetowhichtheelectronsource isdemagnifiedtoformtheprobe.Theelectron lensesusedarecomparabletothoseinacon-ventionalTEM,asaretheelectronaccelerating

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voltages used (typically 100–300kV). Probe sizes below the interatomic spacings in many materials are often possible, which is the great strength of STEM. Atomic resolution images can be readily formed, and the probe can then be stopped over a region of interest for spectroscopic analysis at or near atomic resolution.

To form a small, intense probe we clearly needacorrespondinglysmall,intenseelectron source. Indeed, the development of the cold field emission gun by Albert Crewe and co-workers nearly 40 years ago (Crewe et al., 1968a)wasanecessarystepintheirsubsequent construction of a complete STEM instrument (Crewe et al., 1968b).The quantity of interest foranelectrongunisactuallythesourcebright- ness,whichwillbediscussedinSection9.Field-emissiongunsarealmostalwaysusedforSTEM, either a cold field emission gun (CFEG) or a Schottkythermallyassistedfieldemissiongun. InthecaseofaCFEG,thesourcesizeistypi-callyaround5nm,sotheprobe-formingoptics must be capable of demagnifying its image of

Figure2–1. Aschematicoftheessentialelementsof a dedicated STEM instrument showing the most commondetectors.

Figure 2–2. A photograph of a dedicated STEM instrument (VG Microscopes HB501). The gun is belowthetablelevel,withmostoftheelectronoptics abovethetable.Atthetopofthecolumncanbeseen amagneticprismspectrometerforelectronenergy-lossspectroscopy.

L1 theorderof100timesifanatomicsizedprobe istobeachieved.InaSchottkygunthedemag-nificationmustbeevengreater. Thesizeoftheimageofthesourceisnotthe onlyprobesizedefiningfactor.Electronlenses suffer from inherent aberrations, in particular sphericalandchromaticaberrations.Theaber-rations of the objective lens generally have greatesteffect,andlimitthewidthofthebeam that may pass through the objective lens and still contribute to a small probe. Aberrated beamswillnotbefocusedatthecorrectprobe position,andwillleadtolargediffuseillumina-tion thereby destroying the spatial resoluposition,andwillleadtolargediffuseillumina-tion. To prevent the higher angle aberrated beams fromilluminatingthesample,anobjectiveaper-tureisused,andistypicallyafewtensofmicrons indiameter.Theexistenceofanobjectiveaper-tureinthecolumnhastwomajorimplications: (1)Aswithanyaperturedopticalsystem,there willbeadiffractionlimittothesmallestprobe that can be formed, and this diffraction limit may well be larger than the source image. (2) Thecurrentintheprobewillbelimitedbythe amount of current that can pass through the aperture,andmuchcurrentwillbelostasitis blockedbytheaperture.

Because the STEM resembles the more commonly found SEM in many ways, several ofthedetectorsthatcanbeusedarecommon to both instruments, such as the secondary electron (SE) detector and the energy-dispersive X-ray (EDX) spectrometer. The highestspatialresolutioninSTEMisobtained by using the transmitted electrons, however. Typicalimagingdetectorsusedarethebright-field (BF) detector and the annular dark-Typicalimagingdetectorsusedarethebright-field (ADF)detector.Boththesedetectorssumthe electron intensity over some region of the far field beyond the sample, and the result is dis- playedasafunctionofprobepositiontogener-ateanimage.TheBFdetectorusuallycollects overadiscofscatteringanglescenteredonthe opticaxisofthemicroscope,whereastheADF detector collects over an annulus at higher angle where only scattered electrons are detected.TheADFimagingmodeisimportant and unique to STEM in that it provides inco-herent images of materials and has a strong sensitivitytoatomicnumberallowingdifferent

elements to show up with different intensities intheimage.

Twofurtherdetectorsareoftenusedwiththe STEMprobestationaryoveraparticularspot: (1)ARonchigramcameracandetecttheinten-sityisafunctionofpositioninthefarfield,and shows a mixture of real-space and reciprocal- spaceinformation.Itismainlyusedformicro-scopediagnosticsandalignmentratherthanfor investigationofthesample.(2)Aspectrometer canbeusedtodispersethetransmittedelectrons as a function of energy to form an electron energy-loss(EEL)spectrum.TheEELspectrum carriesinformationaboutthecompositionofthe material being illuminated by the probe, and evencanshowchangesinlocalelectronstruc-turethrough,forexample,bondingchanges.

1.2 OutlineofChapter

The crucial aspect of STEM is the ability to focusasmallprobeatathinsample,sowestart bydescribingtheformoftheSTEMprobeand howitiscomputed.Tounderstandhowimages areformedbytheBFandADFdetectors,we needtoknowtheelectronintensitydistribution in the far field after the probe has been scat-teredbythesample,whichistheintensitythat would be observed by a Ronchigram camera. This allows us to go on and consider BF and ADFimaging.

Movingontotheanalyticaldetectors,there isasectionontheEELspectrumthatempha-sizessomeaspectsofthespatiallocalizationof theEELspectrumsignal.Otherdetectors,such as EDX and SE, that are also found on SEM instrumentsarebrieflydiscussed.

HavingdescribedSTEMimagingandanaly-sis we return to some instrumental aspects of STEM.We discuss typical column design, and thengoontoanalyzetherequirementsforthe electron gun in STEM. Consideration of the effectofthefinitegunbrightnessbringsustoa discussion of the resolution limiting factors in STEM where we also consider spherical and chromatic aberrations. We finish that section withadiscussionofsphericalaberrationcorrec-tion in STEM, which is arguably having the greatestcontributioninthefieldofSTEMand isproducingarevolutioninperformance.

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Therehavebeenseveralreviewarticlespre-viously published on STEM (for example, Cowley,1976;Crewe,1980;Brown,1981).More recently, instrumental improvements have increased the emphasis onatomic resolution imagingandanalysis.Inthischapterwetendto focus on the principles and interpretation of STEM data when it is operating close to the limitofitsspatialresolution.

2. TheSTEMProbe

ThecrucialaspectofSTEMperformanceisthe abilitytofocusasubnanometer-sizedprobeat thesample,sowestartbyexaminingtheform ofthatprobe.Wewillinitiallyassumethatthe electron source is infinitesimal, and that the beam is perfectly monochromatic.The effects oftheseassumptionsnotholdingareexplored inmoredetailinSection10.

Theprobeisformedbyastrongimaginglens, known as theobjective lens, that focuses the electronbeamdowntoformthecrossoverthat istheprobe.Typicalelectronwavelengthsinthe STEM range from 3.7pm (for 100-keV elec-trons)to1.9pm(for300-keVelectrons),sowe mightexpecttheprobesizetobeclosetothese values. Unfortunately, all circularly symmetric electron lenses suffer from inherent spherical aberration, as first shown by Scherzer (1936), andformostTEMsthishastypicallylimitedthe resolution to about 100 times worse that the wavelengthlimit.

Theeffectofsphericalaberrationfromageo-metricopticsstandpointisshowninFigure2–3.

Sphericalaberrationcausesanoverfocusingof thehigherangleraysoftheconvergentsothat they are brought to a premature focus. The Gaussianfocusplaneisdefinedastheplaneat whichthebeamswouldhavebeenfocusedhad theybeenunaberrated.AttheGaussianplane, spherical aberration causes the beams to miss theircorrectpointbyadistanceproportionalto thecubeoftheangleofray.Sphericalaberra-tionisthereforedescribedasbeingathird-order aberration,andtheconstantofproportionality isgiventhesymbol,CS,suchthat Dx=CSq3 (2.1) Iftheconvergenceangleoftheelectronbeam islimited,thenitcanbeseeninFigure2–3that theminimumbeamwaist,ordisc of least confu-sion, islocatedclosertothelensthantheGauss-ian plane, and that the best resolution in a STEM is therefore achieved by weakening or underfocusing the lens relative to its nominal setting.Underfocusingthelenscompensatesto some degree for the overfocusing effects of sphericalaberration. Theaboveanalysisisbasedupongeometric optics,andignoresthewavenatureoftheelec-tron.Amorequantitativeapproachisthrough waveoptics.Becausethelensaberrationsaffect theraysconvergingtoformtheprobeasafunc-tion of angle, they can be incorporated as a phaseshiftinthefront-focalplane(FFP)ofthe objectivelens.TheFFPandthespecimenplane are related by a Fourier transform, as per the Abbetheoryofimaging(BornandWolf,1980). Apointinthefront-focalplanecorrespondsto onepartial-planewavewithintheensembleof

Figure2–3. Ageometricopticsviewoftheeffectofsphericalaberration.AttheGaussianfocusplanethe aberratedraysaredisplacedbyadistanceproportionaltothecubeoftherayangle,q.Theminimumbeam

L1 planewavesconvergingtoformtheprobe.The

deflection of the ray by a certain distance at thesamplecorrespondstoaphasegradientin theFFPaberrationfunction,andthephaseshift duetoaberrationintheFFPisgivenby c(K)=(pzl|K|2₊ 1_2pCSl3|K|4) (2.2) wherewehavealsoincludedthedefocusofthe lens,z,andKisareciprocalspacewavevector that is related to the angle of convergence at thesampleby

K=qλ (2.3)

ThusthepointKinthefront-focalplaneofthe objective lens corresponds to a partial plane wave converging at an angleq at the sample. Once the peak-to-peak phase change of the rays converging to form the probe is greater thanp/2,therewillbeanelementofdestructive interference,whichwewishtoavoidtoforma sharp probe. Equation (2.3) is a quartic func- tion,butwecanusenegativedefocus(underfo-cus) to minimize the excursion ofc beyond a

peak-to-peakchangeofp/2overaswidearange of angles as possible (Figure 2–4). Beyond a critical angle,a, we use a beam-limiting aper-ture,knownastheobjectiveaperture,toprevent the more aberrated rays contributing to the probe.Thisaperturecanberepresentedinthe FFP by a two-dimensional top-hat function, H_a(K).Nowwecandefineaso-calledaperture function,A(K), that represents the complex wavefunctionintheFFP,

A(K)=Ha(K)exp[ic(K)] (2.4) Finally we can compute the wave function of theprobeatthesample,orprobe function,by takingtheinverseFouriertransformof(2.4)to give

P

(

R

)

=∫A

(

K

)

exp

(

−i2πK R K⋅

)

d (2.5) ToexpresstheabilityoftheSTEMtomovethe probe over the sample, we can include a shift termin(2.5)togive P A i i d R R K K R K R K −

(

)

=∫

(

)

(

−

)

(

)

0 exp exp 2 2 0 π π ⋅ ⋅ (2.6)

Figure2–4. Theaberrationphaseshift,c ,inthefront-focal,oraperture,planeplottedasafunctionofcon-vergenceangle,q,foranacceleratingvoltageof200kV,CS=1mmanddefocusz =-35.5nm.Thedottedlines

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Moving the probe is therefore equivalent to adding a linear ramp to the phase variation acrosstheFFP.

Theintensityoftheprobefunctionisfound by taking the modulus squared ofP(R), as is plotted for some typical values in Figure 2–5 Notethatthisso-calleddiffraction limited probe has subsidiary maxima sometimes known as Airyrings,aswouldbeexpectedfromtheuse ofanaperturewithasharpcut-off.Thesesub-sidiary maxima can result in weak features observed in images (see Section 5.3) that are imageartifactsandnotrelatedtothespecimen structure.

Letusexaminethedefocusandaperturesize that should be used to provide an optimally smallprobe.Differentwaysofmeasuringprobe sizeleadtovariouscriteriafordeterminingthe optimaldefocus(see,forexample,Moryetal., 1987),buttheyallleadtosimilarresults.Wecan againusethecriterionofconstrainingtheexcur-sionsofcsothattheyarenomorethanp/4away

fromzero.Foragivenobjectivelensspherical aberration,theoptimaldefocusisthengivenby z=-0.71l1/2_C S1/2 (2.7) allowinganobjectiveaperturewithradius a=1.3l1/4_C S-1/4 (2.8) tobeused.AusefulmeasureofSTEMresolu-tionisthefull-widthathalf-maximum(FWHM) of the probe intensity profile. At optimum defocusandwiththecorrectaperturesize,the probeFWHMisgivenby

d=0.4l3/4_C

S1/4 (2.9) Note that the use of increased underfocusing canleadtoareductionintheprobeFWHMat theexpenseofincreasedintensityinthesubsid-iarymaxima,therebyreducingtheusefulcurrent inthecentralmaximumandleadingtoimage artifacts.Alongwithotherwaysofquotingreso-lution, the FWHM must be interpreted care-fullyintermsoftheimageresolution.

Figure2–5. Theintensityofadiffraction-limitedSTEMprobefortheilluminationconditionsgiveninFigure 2–4.Anobjectiveapertureofradius9.3mradhasbeenused.

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3. CoherentCBED

andRonchigrams

MostSTEMdetectorsarelocatedbeyondthe specimen and detect the electron intensity in the far field. To interpret STEM images, it is therefore first necessary to understand the intensityfoundinthefarfield.Incombination CTEM/STEM instruments, the far-field inten-sitycanbeobservedonthefluorescentscreen at the bottom of the column when the instru-mentisoperatedinSTEMmodewiththelower column set to diffraction mode. In dedicated STEMinstrumentsitisusualtohaveacamera consisting of a scintillator coupled to a CCD arrayinordertoobservethisintensity.

Inconventionalelectrondiffraction,asample is illuminated with a highly parallelized plane wave illumination. Electron scattering occurs, and the intensity observed in the far field is given by the modulus squared of the Fourier transformofthewavefunction,ψ(R),attheexit surfaceofthesample, I i d K K R K R R

(

)

=

(

)

= ∫

(

)

[

]

Ψ 2 2 2 y exp π ⋅ (3.1) Thescatteringwavevectorinthedetectorplane, K,isrelatedtothescatteringangle,q,by K=qλ (3.2) Adetaileddiscussionofelectrondiffractionis ingeneralbeyondthescopeofthistext,butthe reader is referred to the many excellent text-books on this subject (Hirsch et al., 1977; Cowley, 1990, 1992). In STEM, the sample is illuminated by a probe that is formed from a collapsingconvergentsphericalwavefront.The electrondiffractionpatternisthereforebroad-enedbytherangeofilluminationanglesinthe convergent beam. In the case of a crystalline samplewhereonemightexpecttoobservedif-fractedBraggspots,intheSTEMthespotsare broadened intodiscs that may even overlap with their neighbors. Such a pattern is known as a convergent beam electron diffraction (CBED) or microdiffraction pattern because theconvergentbeamleadstoasmallillumina-tion spot. See Spence and Zuo (1992) for a

textbook covering aspects of microdiffraction andCBEDandCowley(1978)forareviewof microdiffraction.

3.1 Ronchigramsof

CrystallineMaterials

If the electron source image at the sample is muchsmallerthanthediffractionlimitedprobe, then the convergent beam forming the probe canberegardedasbeingcoherent.Acrystalline sample diffracts electrons into discrete Bragg beams,andinaSTEMthesearebroadenedto give discs. The high coherence of the beam means that if the discs overlap then interfer-encefeaturescanbeseen,suchasthefringesin Figure2–6.SuchcoherentCBEDpatternsare also known as coherent microdiffraction pat-terns or even nanodiffraction patpat-terns. Their observation in the STEM has been described extensivelybyCowley(1979,1981)andCowley and Disko (1980) and reviewed by Spence (1992).

Tounderstandtheformoftheseinterference fringes, let us first consider a thin crystalline samplethatcanbedescribedbyasimpletrans-mittancefunction,f (R).Theexit-surfacewave-functionwillbegivenby,

Figure2–6. AcoherentCBEDpatternofSi<110>. Note the interference fringes in the overlap region that show that the probe is defocused from the sample.

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y=P(R-R0)f(R) (3.3) BecauseEq.3.3isaproductoftwofunctions, taking its Fourier transform [inserting into Eq. (3.1)] results in a convolution between the Fourier transform of P(R) and the Fouriertransformoff(R).TakingtheFourier transformofP(R),fromEq.(2.5)simplygives A(K). For a crystalline sample, the Fourier transformoff(R)willconsistofdiscreteDirac d-functions, which correspond to the Bragg spots, at values ofK corresponding to the reciprocallatticepoints.Wecanthereforewrite the far field wavefunction,Y(K), as a sum of multiple aperture functions centered on the Braggspots, Ψ K K g K g R

(

)

=∑

(

−

)

−

(

)

  fg g A i exp 2π i 0 (3.4) wherefgisacomplexquantityexpressingthe amplitudeandphaseofthegdiffractedbeam. Equation3.4issimplyexpressingthearrayof discsseeninFigure2–6. Toexaminejusttheoverlapregionbetween thegandhdiffractedbeam,letusexpand(3.4) using (2.4). Since we are just interested in the overlap region we will neglect to include thetop-hatfunction,H(K),whichdenotesthe physicalobjectiveaperture,leaving Y(K)=fgexp[ic(K-g)+i2p(K-g)·R0 +fhexp[ic(K-h) +i2p(K-h)·R0] (3.5) andwefindtheintensitybytakingthemodulus squaredofEq.(3.5), I(K)=|fg|2+|fh|2+2|fg||fh| cos[c(K-g)-c(K-h)+ 2p(h-g)·R0+–fg -–fh] (3.6) where–fgdenotesthephaseofthegdiffracted

beam. The cosine term shows that the disc overlap region contains interference features, andthatthesefeaturesdependonthelensaber-rations,thepositionoftheprobe,andthephase differencebetweenthetwodiffractedbeams.

Ifweassumethattheonlyaberrationpresent is defocus, then the terms includingc in (3.6) become c(K-g)-c(K-h)= pzlÎ(K-g)2_-(K-h)2_˚= pzlÎ2K·(h-g)+|g|2_+|h|2_{˚ (3.7)} BecauseEq.(3.7)islinearinK,auniformset offringeswillbeobservedalignedperpendicu- lartothelinejoiningthecentersofthecorre-sponding discs, as seen in Figure 2–6. For interference involving the central, or bright-field,discwecansetg =0.Thespacingoffringes in the microdiffraction pattern from interfer-encebetweentheBFdiscandthehdiffracted beam is (zl|h|)-1_{, which is exactly what would} be expected if the interference fringes were a shadow of the lattice planes corresponding to theh diffracted beam projected using a point sourceadistancezfromthesample(Figure2– 7).Whentheobjectiveapertureisremoved,or ifaverylargeapertureisused,thentheinten-sity in the detector plane is referred to as a shadowimage.Ifthesampleiscrystalline,then theshadowimageconsistsofmanycrossedsets of fringes distorted by the lens aberrations. These crystalline shadow images are often referredtoasRonchigrams,derivingfromthe useofsimilarimagesinlightopticsforthemea-surementoflensaberrations(Ronchi,1964).It iscommoninSTEMforshadowimagesofboth crystalline and nonperiodic samples to be referredtoasRonchigrams,however.

The term containingR0 in the cosine argu-mentinEq.(3.6)showsthatthesefringesmove astheprobeismoved.Justaswemightexpect forashadow,weneedtomovetheprobeone lattice spacing for the fringes all to move one fringespacingintheRonchigram.Theideaof theRonchigramasashadowimageisparticu-larly useful when considering Ronchigrams of amorphous samples (see Section 3.2). Other aberrations, such as astigmatism or spherical aberration, will distort the fringes so that they are no longer uniform.These distortions may be a useful method of measuring lens aberrations, though the analysis of shadow images for determining lens aberrations is morestraightforwardwithnonperiodicsamples (Dellbyetal.,2001).

TheargumentofthecosineinEq.(3.6)also contains the phase difference between theg

andhdiffractedbeams.Bymeasuringtheposi-L1 tion of the fringes in all the available disc

overlap regions, the phase difference between pairsofadjacentdiffractedbeamscanbedeter-mined.Itisthenstraightforwardtosolveforthe phase of all the diffracted beams, thereby solving the phase problem in electron diffrac-tion.Knowledgeofthephaseofthediffracted beamsallowsimmediateinversiontothereal-space exit-surface wavefunction. The spatial resolutionofsuchaninversionislimitedonly by the largest angle diffracted beam that can giverisetoobservablefringesinthemicrodif-fraction pattern, which will typically be much largerthanthelargestanglethatcanbepassed throughtheobjectivelens(i.e.,theradiusofthe BF disc in the microdiffraction pattern). The methodwasfirstsuggestedbyHoppe(1969a,b, 1982) who gave it the name ptychography. Usingthisapproach,Nellistetal.(1995;Nellist and Rodenburg, 1998) were able to form an imageoftheatomiccolumnsinSi<110>inan STEMthatconventionallywouldbeunableto image them. Ptychography has not become a

commonmethodinSTEM,mainlybecausethe phasing method described above works only forthinsamples.Inthickersamples,forwhich dynamic diffraction theory is applicable, the phase of the diffracted beams can depend on the angle of the incident beam. The inherent phaseofadiffractedbeammaythereforevary across its disc in a microdiffraction pattern, makingthesimplephasingapproachdiscussed above fail. Spence (1998a,b) has discussed in principle how a crystalline microdiffraction patterndatasetcanbeinvertedtothescatter-ingpotentialfordynamicallyscatteringsamples, thoughasyettherehasnotbeenanexperimen-taldemonstration.

3.2 Ronchigramsof

NoncrystallineMaterials

When observing a noncrystalline sample in a Ronchigram,itisgenerallysufficienttoassume thatmostofthescatteringinthesampleisat

anglesmuchsmallerthantheilluminationcon-Figure2–7. Iftheprobeisdefocusedfromthesampleplane,theprobecrossovercanbethoughtofasa point source located distant from the sample. In the geometric optics approximation, the STEM detec- torplaneisashadowimageofthesample,withtheshadowmagnificationgivenbytheratiooftheprobe-detectorandprobe-sampledistances.Ifthesampleiscrystalline,thentheshadowimageisreferredtoasa Ronchigram.

L1 vergenceangles,andthatwecanbroadlyignore theeffectsofdiffraction.Inthiscaseonlythe BFdiscisobservabletoanysignificance,butit containsanimageofthesamplethatresembles aconventionalbright-fieldimagethatwouldbe observedinaconventionalTEMatthedefocus usedtorecordtheRonchigram(Cowley,1979b). The magnification of the image is again given byassumingthatitisashadowprojectedbya point source a distancez (the lens defocus) fromthesample.Asthedefocusisreduced,the magnification increases (Figure 2–8) until it passesthroughaninfinitemagnificationcondi-tion when the probe is focused exactly at the sample. For a quantitative discussion of how Eq.(3.6)reducestoasimpleshadowimagein thecaseofpredominantlylowanglescattering, seeCowley(1979b)andLupini(2001).

Aberrations of the objective lens will cause thedistancefromthesampletothecrossover pointoftheilluminatingbeamtovaryasafunc-tionofanglewithinthebeam(Figure2–3),and thereforetheapparentmagnificationwillvary withintheRonchigram.Wherecrossoversoccur at the sample plane, infinite magnification regionswillbeseen.Forexample,positivespher-icalaberrationcombinedwithnegativedefocus can give rise to rings of infinite magnification (Figure 2–8). Two infinite magnification rings occur,onecorrespondingtoinfinitemagnifica- tionintheradialdirectionandoneintheazi-muthaldirection(Cowley,1986;Lupini,2001).

Measuring the local magnification within a noncrystallineRonchigramcanreadilybedone by moving the probe a known distance and measuring the distance features move in the Ronchigram.Thelocalmagnificationsfromdif-ferent places in the Ronchigram can then be inverted to values for aberration coefficients. ThisisthemethodinventedbyKrivaneketal. (Dellbyetal.,2001)forautotuningofanSTEM aberrationcorrector.Evenforanonaberration-corrected machine, the Ronchigram of a non-periodic sample is typically used to align the instrument(Cowley,1979a).Thecomafreeaxis is immediately obvious in a Ronchigram, and astigmatismandfocuscanbecarefullyadjusted by observation of the magnification of the speckle contrast. Thicker crystalline samples also show Kikuchi lines in the shadow image,

which allows the crystal to be carefully tilted andalignedwiththemicroscopecoma-freeaxis simplybyobservationoftheRonchigram.

Finally it is worth noting that an electron shadowimageforaweaklyscatteringsampleis actuallyanin-linehologram(LinandCowley, 1986)asfirstproposedbyGabor(1948)forthe correctionoflensaberrations.Theextensionof resolutionthroughtheptychographicalrecon-struction described in Section (3.1) can be extended to nonperiodic samples (Rodenburg and Bates, 1992), and has been demonstrated experimentally(Rodenburgetal.,1993).

4. Bright-FieldImaging

andReciprocity

InSection3weexaminedtheformoftheelec-tron intensity that would be observed in the detectorplaneoftheinstrumentusinganarea detector,suchasaCCD.InSTEMimagingwe detect only a single signal, not a two-dimen-sional array, and plot it as a function of the probeposition.Anexampleofsuchanimageis anSTEMBFimage,forwhichwedetectsome oralloftheBFdiscintheRonchigram.Typi- callythedetectorwillconsistofasmallscintil-lator,fromwhichthelightgeneratedisdirected intoaphotomultipliertube.SincetheBFdetec-tor will just be summing the intensity over a regionoftheRonchigram,wecanusetheRon-chigramformulationinSection3toanalyzethe contrastinaBFimage.

4.1 LatticeImaginginBFSTEM

InSection3.1wesawthatifthediffracteddiscs intheRonchigramoverlapthencoherentinter-ferencecanoccur,andthattheintensityinthe discoverlapregionswilldependontheprobe position,R0. If the discs do not overlap, then there will be no interference and no depen-denceonprobeposition.Inthislattercase,no matterwhereweplaceadetectorintheRon-chigram,therewillbenochangeinintensityas theprobeismovedandthereforenocontrast inanimage. ThetheoryofSTEMlatticeimaginghasbeen described (Spence and Cowley, 1978). Let us

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a

b

Figure2–8. RonchigramsofAunanoparticlesonathinCfilmrecordedatdifferentdefocusvalues(aand b).Noticethechangeinimagemagnification,andtheradialandazimuthalringsofinfinitemagnification.

L1 firstconsiderthecaseofaninfinitesimaldetec-torrightontheaxis,whichcorrespondstothe centeroftheRonchigram.FromFigure2–9it isclearthatwewillseecontrastonlyifthedif- fractedbeamsarelessthananobjectiveaper-tureradiusfromtheopticaxis.Thediscsfrom three beams now interfere in the region detected. From (3.5), the wavefunction at the pointdetectedwillbe Y(K=0,R0)=1+fgexp[ic(-g) -i2pg·R0]+f-g exp[ic(g)+ i2pg·R0] (4.1) whichcanalsobewrittenastheFouriertrans-formoftheproductofthediffractionspotsof thesampleandthephaseshiftduetothelens aberrations, Ψ K 0 R K K g K g =

(

)

=∫

[

(

′

)

+

(

′ +

)

+ −

(

′ −

)

]

, exp 0 δ φ δ φ δ g g iχχ π ′

(

)

[

]

′

(

)

′ K K R K exp i2 ⋅ 0 d (4.2) Equations(4.1)and(4.2)areidenticaltothose for the wavefunction in the image plane of a

CTEMwhenforminganimageofacrystalline sample. In the simplest model of a CTEM (Spence, 1988), the sample is illuminated with planewaveillumination.Inthebackfocalplane oftheobjectivelenswecouldobserveadiffrac-tionpattern,andthewavefunctionforthisplane correspondstothefirstbracketintheintegrand of (4.2). The effect of the aberrations of the objective lens can then be accommodated in the model by multiplying the wavefunction in the back focal plane by the usual aberration phase shift term, and this can also be seen in (4.2). The image plane wavefunction is then obtainedbytakingtheFouriertransformofthis product.ImageformationinanSTEMcanbe thoughtofasbeingequivalenttoaCTEMwith thebeamtrajectoriesreversedindirection.

What we have shown here, for the specific caseofBFimagingofacrystallinesample,isthe princple of reciprocityinaction.Whentheelec-tronsarepurelyelasticallyscattered,andthere isnoenergyloss,thepropagationoftheelec-trons is time reversible. The implication for STEMisthatthesourceplaneofanSTEMis equivalenttothedetectorplaneofaCTEMand viceversa(Cowley,1969;ZeitlerandThomson, 1970).CondenserlensesareusedinanSTEM todemagnifythesource,whichcorrespondsto projectorlensesbeingusedinaCTEMformag-nifying the image. The objective lens of an STEM(oftenusedwithanobjectiveaperture) focusesthebeamdowntoformtheprobe.Ina CTEM,theobjectivelenscollectsthescattered electronsandfocusesthemtoformamagnified image. Confusion can arise with combined CTEM/STEMinstruments,inwhichtheprobe-forming optics are distinct from the image-formingoptics.Forexample,thetermobjective apertureisusuallyusedtorefertotheaperture after the objective lens used in CTEM image formation. In STEM mode, the beam conver-genceiscontrolledbyanaperturethatisusually referredtoasthecondenser aperture,although byreciprocitythisapertureisactingopticallyas an objective aperture.The correspondence by reciprocitybetweenCTEMandSTEMcanbe extendedtoincludetheeffectsofpartialcoher-ence. Finite energy spread of the illumination beam in CTEM has an effect on the image similar to that in STEM for the equivalent

Figure2–9. Aschematicdiagramshowingthatfora crystalline sample, a small, axial bright-field (BF) STEMdetectorwillrecordchangesinintensitydue to interference between three beams: the0 unscat-teredbeamandthe+gand-gBraggreflections.

L1 imagingmode.ThefinitesizeoftheBFdetector

inanSTEMgivesrisetolimitedspatialcoher-ence in the image (Nellist and Rodenburg, 1994),andcorrespondstohavingafinitediver-genceoftheilluminatingbeaminanSTEM.In STEM, the loss of the spatial coherence can easily be understood as the averaging out of interferenceeffectsintheRonchigramoverthe areaoftheBFdetector.Attheotherendofthe columnthereisalsoacorrespondencebetween thesourcesizeinSTEMandthedetectorpixel sizeinaCTEM.MovingthepositionoftheBF STEMdetectorisequivalenttotiltingtheillu-mination in CTEM. In this way dark-field images can be recorded. A carefully chosen positionforaBFdetectorcouldalsobeusedto detect the interference between just two dif-fracted discs in the microdiffraction pattern, allowinginterferencebetweenthe0beamand abeamscatteredbyuptotheaperturediameter tobedetected.Inthiswayhigher-spatialresolu- tioninformationcanberecorded,inanequiva-lent way to using a tilt sequence in CTEM (Kirklandetal.,1995).

Althoughreciprocityensuresthatthereisan equivalence in the image contrast between CTEM and STEM, it does not imply that the efficiency of image formation is identical. Bright-fieldimaginginaCTEMisefficientwith electrons because most of the scattered elec-trons are collected by the objective lens and used in image formation. In STEM, a large range of angles illuminates the sample and thesearescatteredfurthertogiveanextensive Ronchigram.ABFdetectordetectsonlyasmall fraction of the electrons in the Ronchigram, andisthereforeinefficient.Notethatthiscom-parisonappliesonlyforBFimaging.Thereare otherimagingmodes,suchasannulardark-field (Section5),forwhichSTEMismoreefficient.

4.2 PhaseContrastImaging

inBFSTEM

Thin weakly scattering samples are often approximatedasbeingweakphaseobjects(see, forexample,Cowley,1992).Weakphaseobjects simplyshiftthephaseofthetransmittedwave suchthatthespecimentransmittancefunction canbewritten f(R0)=1+isV(R0) (4.3) wheres is known as the interaction constant andhasavaluegivenby

s=2pmel/h2 _(4.4) wheretheelectronmass,m,andthewavelength,

l ,arerelativisticallycorrected,andVisthepro-jectedpotentialofthesample.Equation(4.3)is simply the expansion of exp[isV(R0)] to first order,andthereforerequiresthattheproduct

sV(R0)ismuchsmallerthanunity.TheFourier transformof(4.3)is

F(K¢)=d(K¢)+isV˜(K¢) (4.5) and can be substituted for the first bracket in theintegrandof(4.2) Ψ K 0 R K K K =

(

)

=∫

[

(

′

)

+

(

′

)

]

′

(

)

[

]

, exp exp 0 2 δ σ χ i V i i  ππ ′

(

K R.

)

K′ 0 d (4.6) Noticingthat(4.6)istheFouriertransformof a product of functions, it can be written as a convolutioninR0. Y(K=0,R0)=1+isV(R0) FT{cos[c(K¢)]+isin[c(K¢]} (4.7) Takingtheintensityof(4.7)givestheBFimage I(R0)=1-2sV(R0) FT{sin[c(R0]} (4.8)

where we have neglected terms greater than firstorderinthepotential,andmadeuseofthe factthatthesineandcosineofcareevenand thereforetheirFouriertransformsarereal.

Notsurprisingly,wehavefoundthatimaging aweak-phaseobjectusinganaxialBFdetector results in a phase contrast transfer function (PCTF) (Spence, 1988) identical to that in CTEM, as expected from reciprocity. Lens aberrationsareactingasaphaseplatetogener- atephasecontrast.Intheabsenceoflensaber-rations, there will be no contrast.We can also interpretthisresultintermsoftheRonchigram in an STEM, remembering that axial BF imaging requires an area of triple overlap of discs(Figure2–9).Intheabsenceoflensaber-rations, the interference between the BF disc

L1

andascattereddiscwillbeinantiphasetothat between the BF disc and the opposite, conju- gatediffracteddisc,andtherewillbenointen-sity changes as the probe is moved. Lens aberrationswillshiftthephaseoftheinterfer-ence fringes to give rise to image contrast. In regions of two disc overlap, the intensity will alwaysvaryastheprobeismoved.Movingthe detectortosuchtwobeamconditionswillthen givecontrast,justastwo-beamtiltedillumina-tioninCTEMwillgivefringesintheimage.In such conditions, the diffracted beams may be separatedbyuptotheobjectiveaperturediam-eter,andstillthefringesresolved.

4.3 LargeDetectorIncoherent

BFSTEM

IncreasingthesizeoftheBFdetectorreduces thedegreeofspatialcoherenceintheimage,as alreadydiscussedinSection4.1.Oneexplana-tion for this is the increasing degree to which interference features in the Ronchigram are beingaveragedout.EventuallytheBFdetector can be large enough that the image can be described as being incoherent. Such a large detectorwillbethecomplementofanannular dark-fielddetector:theBFdetectorcorrespond-ingtotheholeintheADFdetector.Electron absorption in samples of thicknesses usually used for high-resolution microscopy is small compared to the transmittance, which means thatthelargedetectorBFintensitywillbe

IBF(R0)=1-IADF(R0) (4.9) Wewilldeferdiscussionofincoherentimaging to Section 5. It is, however, worth noting that becauseIADFisasmallfractionoftheincident intensity(typicallyjustafewpercent),thecon-trastinIBFwillbesmallcomparedtothetotal intensity. The image noise will scale with the total intensity, and therefore it is likely that a largedetectorBFimagewillhaveworsesignal tonoisethanthecomplimentaryADFimage.

5. AnnularDark-FieldImaging

Annulardark-field(ADF)imagingisbyfarthe most ubiquitous STEM imaging mode [see Nellist and Pennycook (2000) for a review of

ADFSTEM].Itprovidesimagesthatarerela-tively insensitive to focusing errors, in which compositionalchangesareobviousinthecon-trast, and atomic resolution images that are much easier to interpret in terms of atomic structure than their high-resolution TEM (HRTEM) counterparts. Indeed, the ability of an STEM to performADF imaging is one of the major strengths of STEM and is partly responsibleforthegrowthofinterestinSTEM overthepasttwodecades. TheADFdetectorisanannulusofscintilla-tormaterialcoupledtoaphotomultipliertube inawaysimilartotheBFdetector.Ittherefore measuresthetotalelectronsignalscatteredin angle between an inner and an outer radius. These radii can both vary over a large range, but typically the inner radius would be in the rangeof30–100mradandtheouterradius100– 200mrad.Oftenthecenterofthedetectorisa hole,andelectronsbelowtheinnerradiuscan passthroughthedetectorforuseeithertoform a BF image, or more commonly to be energy analyzedtoformanelectronenergy-lossspec-trum.Bycombiningmorethanonemodeinthis way,theSTEMmakeshighlyefficientuseofthe transmittedelectrons.

Annular dark-field imaging was introduced inthefirstSTEMsbuiltinCrewe’slaboratory (Crewe,1980).Initiallytheirideawasthatthe high angle elastic scattering from an atom would be proportional to the product of the numberofatomsilluminatedandZ3/2_,whereZ is the atomic number of the atoms, and this scattering would be detected using the ADF detector. Using an energy analyzer on the lower-anglescatteringtheycouldalsoseparate theinelasticscattering,whichwasexpectedto varyastheproductofthenumberofatomsand Z1/2_{.Byformingtheratioofthetwosignals,it} washopedthatchangesinspecimenthickness wouldcancel,leavingasignalpurelydependent oncomposition,andgiventhenameZcontrast. Such an approach ignores diffraction effects within the sample, which we will see later is crucial for quantitative analysis. Nonetheless, thehigh-angleelasticscatteringincidentonan ADF detector is highly sensitive to atomic number.As the scattering angle increases, the scattered intensity from an atom approaches

L1 theZ2_{dependencethatwouldbeexpectedfor}

Rutherford scattering from an unscreened Coulombpotential.Inpracticethislimitisnot reached, and the Z exponent falls to values typicallyaround1.7(see,forexample,Hartelet al.,1996)duetothescreeningeffectoftheatom coreelectrons.Thissensitivitytoatomicnumber resultsinimagesinwhichcompositionchanges aremorestronglyvisibleintheimagecontrast than would be the case for high-resolution phase-contrastimaging.Itisforthisreasonthat usingthefirstSTEMoperatingat30kV(Crewe et al., 1970), it was possible to image single atomsofThonacarbonsupport.

Once STEM instruments became commer-ciallyavailableinthe1970s,attentionturnedto using ADF imaging to study heterogeneous catalystmaterials(Treacyetal.,1978).Oftena heterogeneous catalyst consists of highly dis-persedpreciousmetalclustersdistributedona lighterinorganicsupportsuchasalumina,silica, or graphite. A system consisting of light and heavy atomic species such as this is an ideal subjectforstudyusingADFSTEM.Attempts weremadetoquantifythenumberofatomsin themetalclustersusingADFintensities.Howie (1979)pointedoutthatiftheinnerradiuswas high enough, the thermal diffuse scattering (TDS)oftheelectronswoulddominate.Because TDSisanincoherentscatteringprocess,itwas assumedthatensemblesofatomswouldscatter inproportiontothenumberofatomspresent. Itwasshown,however,thatdiffractioneffects can still have a large impact on the intensity (Donald and Craven, 1979). Specifically, when aclusterisalignedsothatoneoftheloworder crystallographic directions is aligned with the beam,aclusterisobservedtobeconsiderably brighterintheADFimage.

An alternative approach to understanding the incoherence ofADF imaging invokes the principleofreciprocity.Phasecontrastimaging inanHREMisanimagingmodethatrelieson a high degree of coherence in order to form contrast.Thespecimenilluminationisarranged tobeasplanewaveaspossibletomaximizethe coherence.Byreciprocity,anADFdetectorin anSTEMcorrespondshypotheticallytoalarge, annular, incoherent illumination source in a CTEM.Thistypeofsourceisnotreallyviable

for a CTEM, but illumination of this sort is extremely incoherent, and renders the speci-meneffectivelyself-luminousasthescattering fromspatiallyseparatedpartsofthespecimen are unable to interfere coherently. Images formedfromsuchasamplearesimplertointer-pretastheylackthecomplicatinginterference featuresobservedincoherentimages.Alight-opticalanalogueistoconsiderviewinganobject withilluminationfromeitheralaseroranincan-descent light bulb. Laser beam illumination wouldresultinstronginterferencefeaturessuch asfringesandspeckle.Illuminationwithalight bulbgivesaviewmucheasiertointerpret. AlthoughADFSTEMimagingisverywidely used,therearestillmanydiscrepanciesbetween thetheoreticalapproachestaken,whichcanbe veryconfusingwhenreviewingtheliterature.A pictureoftheimagingprocessthatbridgesthe gap between thinking of the incoherence as arisingfromintegrationoveralargedetectorto thinkingofitasarisingfromdetectingpredomi-nantlyincoherentTDShasyettoemerge.Here wewillpresentbothapproaches,andattemptto discussthelimitationsandadvantagesofeach.

5.1 IncoherentImaging

To highlight the difference between coherent andincoherentimaging,westartbyreexamin-ing coherent imagandincoherentimaging,westartbyreexamin-ing in a CTEM for a thin sample.Considerplanewaveilluminationofa thin sample with a transmittance function,

f(R0).Thewavefunctioninthebackfocalplane isgivenbytheFouriertransformofthetrans-mittancefunction,andwecanincorporatethe effectoftheobjectiveapertureandlensaber-rationsbymultiplyingthebackfocalplaneby theaperturefunctiontogive F(K¢)A(K¢) (5.1) whichcanbeFouriertransformedtotheimage wavefunction, which is then a convolution betweenf(R0) and the Fourier transform of A(K¢),whichfromSection2isP(R0).Theimage intensityisthen

I(R0)=|f(R0)P(R0)|2 _(5.2) Althoughforsimplicitywehavederived(5.2) fromtheCTEMstandpoint,byreciprocity(5.2)

L1

applies equally well to BF imaging in STEM withasmallaxialdetector.

For theADF case we follow the argument first presented by Loane et al. (1992). Similar analyses have been performed by Jesson and Pennycook (1993), Nellist and Pennycook (1998a),andHarteletal.(1996).Followingthe STEM configuration, the exit-surface wave-functionisgivenbytheproductofthesample transmittanceandtheprobefunction,

f(R)P(R-R0) (5.3) We can find the wavefunction in the Ronchi-gramplanebyFouriertransforming(5.3),which results in a convolution between the Fourier transformoffandtheFouriertransformofP [giveninEq.(2.6)].Takingtheintensityinthe Ronchigram and integrating over an annular detectorfunctiongivestheimageintensity I D A i ADF R ADF K K K K K R 0 0 2

(

)

=∫

(

)

− ′

(

)

∫

(

′

)

′

(

)

Φ exp π ⋅ ddK2dK (5.4) TakingtheFouriertransformoftheimageallows simplification after expanding the modulus squaredtogivetwoconvolutionintegrals  I i D A ADF Q Q R ADF K K K K

(

)

=∫

(

)

∫

(

)

− ′

(

)

(

′

)

∫ exp 2π ⋅ 0 Φ

{{

′

(

)

′

}

×

{

∫

(

− ′′

)

(

′′

)

− exp exp * * i d A i 2πK R0 K K K K ⋅ Φ 22 0 0 π ′′

(

)

′′

}

K R K K R ⋅ d d d (5.5)

Performing theR0 integral first results in a Diracd-function,  I D A A ADF Q ADF K K K K K K

(

)

=∫∫∫

(

)

(

− ′

)

′

(

)

(

− ′′

)

′ Φ Φ* * ′′

(

)

(

+ ′ − ′′

)

′ ′′ K Q K K K K K δ d d d (5.6)

which allows simplification by performing the K≤integral,  I_ADF Q D_ADF K A K A K Q K K K

(

)

=∫∫

(

)

(

′

)

′ +

(

)

(

− ′

)

− ′ * * Φ Φ K K −Q K K

(

)

d d ′ (5.7) Equation(5.7)isstraightforwardtointerpretin terms of interference between diffracted discs intheRonchigram(Figure2–10).Theintegral overK¢isaconvolution,sothat(5.7)couldbe written,  I_ADF Q D_ADF K AK A K Q K K Q K

(

)

=∫

(

)

[{

(

)

+

(

)

]

[

(

)

− * * ⊗ Φ Φ

((

)

]

}

dK (5.8) The first bracket of the convolution is the overlap product of two apertures, and this is then convolved with a term that encodes the interference between scattered waves sepa-ratedbytheimagespatialfrequencyQ.Fora crystallinesample,F(K)willhavevaluesonly fordiscreteKvaluescorrespondingtothedif-fracted spots. In this case (5.8) is easily inter-pretable as the sum over many different disc overlap features that are within the detector function.An alternative, but equivalent, inter-pretationof(5.8)isthatforaspatialfrequency, Q,toshowupintheimage,twobeamsincident onthesampleseparatedbyQmustbescattered bythesamplesothattheyendupinthesame final wavevectorK where they can interfere (Figure2–10).ThismodelofSTEMimagingis applicable to any imaging mode, even when TDS or inelastic scattering is included. It was immediatelyconcludedthatSTEMisunableto resolve any spacing smaller than that allowed by the diameter of the objective aperture, no matterwhichimagingmodeisused.

Figure2–10showsthatwecanexpectthatthe apertureoverlapregionissmallcomparedwith thephysicalsizeoftheADFdetector.Interms of Eq. (5.7) we can say the domain of theK¢ integral(limitedtothediscoverlapregion)is smallcomparedwiththedomainoftheKinte-gral,andwecanmaketheapproximation,  I A A d D ADF ADF Q K K Q K K K K

(

)

=∫

(

′

)

(

′ +

)

′×

(

)

(

− ′

)

∫ * Φ Φ** K K− ′ −Q K

(

)

d (5.9) Inmakingthisapproximationwehaveassumed that the contribution of any overlap regions thatarepartiallydetectedbytheADFdetector issmallcomparedwiththetotalsignaldetected. Theintegralcontainingtheaperturefunctions

L1 is actually the autocorrelation of the aperture

function. The Fourier transform of the probe intensity is the autocorrelation of A, thus Fourier transforming (5.9) to give the image resultsin

I(R0)=|P(R0)|O(R0) (5.10) whereO(R0) is the inverse Fourier transform oftheintegraloverKin(5.9).

Equation (5.10) is essentially the definition of incoherent imaging. An incoherent image can be written as the convolution between theintensityofthepoint-spreadfunctionofthe image (which in STEM is the intensity of the probe) and an object function. Compare this with the equivalent expression for coherent imaging,(5.2),whichistheintensityofaconvo-lutionbetweenthecomplexprobefunctionand the specimen function. We will see later that O(R0)isafunctionthatissharplypeakedatthe atomsites.TheADFimageisthereforeasharply peakedobjectfunctionconvolved(orblurred) withasimple,realpoint-spreadfunctionthatis simplytheintensityoftheSTEMprobe.Such an image is much simpler to interpret than a

coherentimage,inwhichbothphaseandampli-tudecontrasteffectscanappear.Thedifference betweencoherentandincoherentimagingwas discussed at length by Lord Rayleigh in his classicpaperdiscussingtheresolutionlimitof themicroscope(Rayleigh,1896).

A simple picture of the origins of the inco- herencecanbeseenschematicallybyconsider-ingtheimagingoftwoatoms(Figure2–11).The scatteringfromtheatomswillgiverisetointer-ference features in the detector plane. If the detector is small compared with these fringes, then the image contrast will depend critically onthepositionofthefringes,andthereforeon the relative phases of the scattering from the two atoms, which means that complex phase effectswillbeseen.Alargedetectorwillaverage over the fringes, destroying any sensitivity to coherenceeffectsandtherelativephasesofthe scattering.Byreciprocity,useoftheADFdetec-torcanbecomparedtoilluminatingthesample with large angle incoherent illumination. In optics,theVanCittert–Zernicketheorem(Born and Wolf, 1980) describes how an extended sourcegivesrisetoacoherentenvelopethatis the Fourier transform of the source intensity function. An equivalent coherence envelope

Figure2–10. AschematicdiagramshowingthedetectionofinterferenceindiscoverlapregionsbytheADF detector.Imagingofaglatticespacinginvolvestheinterferenceofpairsofbeamsintheconvergentbeam thatareseparatedbyg.TheADFdetectorthensumsovermanyoverlapinterferenceregions.

L1

exists for ADF imaging, and is the Fourier transform of the detector function,D(K). As longasthiscoherenceenvelopeissignificantly smallerthantheprobefunction,theimagecan bewrittenintheformof(5.10)asbeinginco- herent.Thisconditionisthereal-spaceequiva-lentoftheapproximationthatallowedustogo from(5.7)to(5.9).

The strength at which a particular spatial frequency in the object is transferred to the imageisknown,forincoherentimaging,asthe optical transfer function (OTF).The OTF for incoherentimaging,T(Q),issimplytheFourier transform of the probe intensity function. In generalitisapositive,monatonicallydecaying function (see Black and Linfoot (1957) for examples under various conditions), which compares favorably with the phase contrast transferfunctionforthesamelensparameters (Figure2–12).

It can also be seen in Figure 2–12 that the interpretableresolutionofincoherentimaging extendstoalmosttwicethatofphase-contrast imaging.ThiswasalsonotedbyRayleigh(1896) forlightoptics.Theexplanationcanbeseenby

comparingthediscoverlapdetectioninFigure 2–9 and Figure 2–10. ForADF imaging single overlapregionscanbedetected,sothetransfer continuestotwicetheapertureradius.TheBF detectorwilldetectspatialfrequenciesonlyto theapertureradius.

An important consequence of (5.10) is that the phase problem has disappeared. Because the resolution of the electron microscope has always been limited by instrumental factors, primarilythesphericalaberrationoftheobjec-tive lens, it has been desirable to be able to deconvolve the transfer function of the microscope. A prerequisite to doing this for coherentimagingistheneedtofindthephase of the image plane. The modulus-squared in (5.2)losesthephaseinformation,andthismust be restored before any deconvolution can be performed. Finding the phase of the image planeintheelectronmicroscopewasthemoti-vation behind the invention of holography (Gabor, 1948).There is no phase problem for incoherent imaging, and the intensity of the probe may be immediately deconvolved. Various methods have been applied to this

Figure2–11. Thescatteringfromapairofatomswillresultininterferencefeaturessuchasthefringesshown here.Asmalldetector,suchasaBF,willbesensitivetothepositionofthefringes,andthereforesensitive totherelativephaseofthescatteredwavesandphasechangesacrosstheilluminatingwave.Alargerdetec-tor,suchasanADF,willaverageovermanyfringesandwillthereforebesensitiveonlytotheintensityof thescatteringandnotthephaseofthewaves.

L1 deconvolution problem (Nellist and

Penny-cook,1998a,2000)includingBayesianmethods (McGibbonetal.,1994,1995).Asalwayswith deconvolution,caremustbetakennottointro-duceartifactsthroughnoiseamplification.The ultimategoalofsuchmethods,though,mustbe thefullquantitativeanalysisofanADFimage, alongwithameasureofcertainty;forexample, the positions of atomic columns in an image alongwithameasureofconfidenceinthedata. Suchagoalisyettobeachieved,andtheinter-pretation of most images is still very much qualitative.

Theobjectfunction,O(R0),canalsobeexam-inedinrealspace.Byassumingthatthemaximum Qvectorissmallcomparedtothegeometryof thedetector,andnotingthatthedetectorfunc-tion is either unity or zero, we can write the Fouriertransformoftheobjectfunctionas  O D D d Q K K K Q K Q K

(

)

=∫

(

) ( )

−

(

)

(

−

)

ADF Φ Φ* _(5.11)

This equation is just the autocorrelation of D(K)f(K),andsotheobjectfunctionis

O(R0)=|D˜(R0)F(R0)|2 _(5.12) Neglecting the outer radius of the detector, wherewecanassumethestrengthofthescat-tering has become negligible, D(K) can be thoughtofasasharphigh-passfilter.Theobject function is therefore the modulus-squared of the high-pass filtered specimen transmission function. Nellist and Pennycook (2000) have takenthisanalysisfurtherbymakingtheweak- phaseobjectapproximation,underwhichcon-ditiontheobjectfunctionbecomes O J k V R R R R R 0 1 0 2 2 2

(

)

= ∫

(

)

+

(

)

[

π π σ inner half plane / −− −

(

)

]

σVR0 R 2 dR 2 / (5.13) wherek inneristhespatialfrequencycorrespond-ingtotheinnerradiusoftheADFdetector,and J1isafirst-orderBesselfunctionofthefirstkind. ThisisessentiallytheresultderivedbyJesson andPennycook(1993).Thecoherenceenvelope expectedfromtheVanCittert–Zernicketheorem isnowseenin(5.13)astheAiryfunctioninvolv-Figure2–12. Acomparisonoftheincoherentobjecttransferfunction(OTF)andthecoherentphase-con-trasttransferfunction(PCTF)foridenticalimagingconditions(V =300kV,CS=1mm,z =-40nm).

L1

ingtheBesselfunction.Ifthepotentialisslowly varying within this coherence envelope, the valueofO(R0)issmall.ForO(R0)tohavesig-nificant value, the potential must vary quickly within the coherence envelope. A coherence envelopethatisbroadenoughtoincludemore than one atom in the sample (arising from a small hole in the ADF), however, will show unwanted interference effects between the atoms. Making the coherence envelope too narrow by increasing the inner radius, on the otherhand,willleadtotoosmallavariationin thepotentialwithintheenvelope,andtherefore nosignal.IfthereisnoholeintheADFdetector, thenD(K)=1everywhere,anditsFouriertrans-form will be a delta-function. Eq. (5.12) then becomesthemodulus-squaredofF,andthere will be no contrast. To get signal in an ADF image,werequireaholeinthedetectorleading toacoherenceenvelopethatisnarrowenough todestroycoherencefromneighboringatoms, butbroadenoughtoallowenoughinterference inthescatteringfromasingleatom.Inpractice, therearefurtherfactorsthatcaninfluencethe choiceofinnerradius,asdiscussedinlatersec-tions.Atypicalchoiceforincoherentimagingis thattheADFinnerradiusshouldbeaboutthree timestheobjectiveapertureradius.

5.2 ADFImagesofThickerSamples

Oneofthegreatstrengthsofatomicresolution ADF images is that they appear to faithfully representthetrueatomicstructureofthesample evenwhenthethicknessischangingoverranges oftensofnanometers.Phasecontrastimagingin a CTEM is comparatively very sensitive to changes in thickness, and displays the well-known contrast reversals (Spence, 1988). An importantfactorinthesimplicityoftheimages istheincoherentnatureofADFimages,aswe haveseeninSection5.1.Thethinobjectapproxi-mation made in Section 5.1, however, is not applicabletothethicknessofsamplesthatare typicallyused,andweneedtoincludetheeffects ofthemultiplescatteringandpropagationofthe electronswithinthesample.Thereareseveral suchdynamicmodelsofelectrondiffraction(see Cowley,1992).Thetwomostcommonarethe Bloch wave approach and the multislice

approach.Attheanglesofscattertypicallycol-lectedbyanADFdetector,themajorityofthe electronsarelikelytobethermaldiffusescatter-ing,havingalsoundergoneaphononscattering event.AcomprehensivemodelofADFimaging thereforerequiresboththemultiplescattering and the thermal scattering to be included.As discussedearlier,someapproachesassumethat theADFsignalisdominatedbytheTDS,and this is assumed to be incoherent with respect to the scattering between different atoms. The demonstration of transverse incoherence through the detector geometry and the Van Cittert–Zernicketheoremisthereforeignored by this approach. For lower inner radii, or increasedconvergenceangle(arisingfromaber-rationcorrection,forexample)agreateramount ofcoherentscatterislikelytoreachthedetector, and the destruction of coherence through the detector geometry will be important for the coherentscatter.Asyet,aunifyingpicturehas yet to emerge, and the literature is somewhat confusing.Herewewillpresentthemostimpor-tantapproachescurrentlyused.

Initiallyletusneglectthephononscattering. Byassumingacompletelystationarylatticewith no absorption, Nellist and Pennycook (1999) were able to use Bloch waves to extend the approachtakeninSection5.1toincludedynamic scattering. It could be seen that the narrow detectorcoherencefunctionactedtofilterthe statesthatcouldcontributetotheimagesothat the highly bound 1s-type states dominated. Because these states are highly nondispersive, spreadingoftheprobewavefunctionintoneigh-boringcolumn1sstatesisunlikely(Raffertyet al., 2001), although spreading into less bound states on neighboring columns is possible. Althoughthisanalysisisusefulinunderstanding how an incoherent image can arise under dynamic scattering conditions, its neglect of absorptionandphononscatteringeffectsmeans thatitisnoteffectiveasaquantitativemethod ofsimulatingADFimages.

Early analyses of ADF imaging took the approachthatathighenoughscatteringangles, theTDSarisingfromphononswoulddominate theimagecontrast.IntheEinsteinapproxima-tion,thisscatteringiscompletelyuncorrelated betweenatoms,andthereforetherecouldbeno

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coherentinterferenceeffectsbetweenthescat-tering from different atoms. In this approach the intensity of the wavefunction at each site needstobecomputedusingadynamicelastic scatteringmodelandthentheTDSfromeach atom summed (Pennycook and Jesson, 1990). When the probe is located over an atomic column in the crystal, the most bound, least dispersivestates(usually1s-or2s-like)arepre-dominantly excited and the electron intensity “channels”downthecolumn.Whentheprobe is not located over a column, it excites more dispersive,lessboundstatesandspreadsleading to reduced intensity at the atom sites and a lower ADF signal. Both the Bloch wave (for example, Pennycook, 1989; Amali and Rez, 1997;Mitsuishietal.,2001;Findlayetal.,2003) andmultislice(forexample,Dingesetal.,1995; Allenetal.,2003)methodshavebeenusedfor simulating the TDS scattering to the ADF detector.Typically,adynamiccalculationusing the standard phenomenological approach to absorption is used to compute the electron wavefunction in the crystal.The absorption is incorporated through an absorptive complex potentialthatcanbeincludedinthecalculation simultaneously with the real potential. This method makes the approximation that the absorptionatagivenpointinthecrystalispro-portional to the product of the absorptive potentialandtheintensityoftheelectronwave-functionatthatpoint.Ofcourse,muchofthe absorptionisTDS,whichislikelytobedetected bytheADFdetector.Itisthereforenecessary toestimatethefractionofthescatteringthatis likelytoarriveatthedetector,andthisestima-tion can cause difficulties. Many estimates of the scattering to the detector, however, make the approximation that the TDS absorption computed for electron scattering in the kine-maticapproximationtoagivenanglewillend upbeingatthesameangleafterphononscat-tering.Thecrosssectionforthesignalarriving attheADFdetectorcanthenbeapproximated byintegratingthisabsorptionoverthedetector (Pennycook,1989;Mitsuishietal.,2001), σADF π π λ ADF =

(

)(

)

∫

(

)

−

(

−

)

[

]

4 2 1 0 2 2 2 m m f s Ms d / / exp ss (5.14) 11 wheres =q/2l andthef(s)istheelectronscat-tering factor for the atom in question. Other estimateshavealsobeenmade,someincluding TDSinamoresophisticatedway(Allenetal., 2003). Caution must be exercised, though. Because this approach is two step—first elec-trons are absorbed, then a fraction is reintro-duced to compute the ADF signal—a wrong estimation in the nature of the scattering can leadtomoreelectronsbeingreintroducedthan were absorbed, thus violating conservation laws.

Makingtheapproximationthatalltheelec-tronsincidentonthedetectorareTDSneglects anyelasticscatteringthatmightbepresentat thedetectionangles,whichmightbecomesig-nificant for lower inner radii. In most cases, including the elastic component is straightfor-wardbecauseitisalwayscomputedinorderto findtheelectronintensitywithinthecrystal,but thisisnotalwaysdoneintheliterature.

Note that the approach outlined above for incoherent TDS scatterers is a fundamentally different approach to understanding ADF imaging,anddoesnotinvoketheprinciplesof reciprocityortheVanZittert–Zernicketheorem. It does not rely on the large geometry of the detector,butjustonthefactthatitdetectsonly athighanglesatwhichtheTDSdominates.

The use of TDS cross sections as outlined above also neglects the further elastic scatter- ingoftheelectronsaftertheyhavebeenscat-teredbyaphonon.ThefamiliarKikuchilines visible in the TDS are manifestations of this elastic scattering. Such scattering occurs only for electrons traveling near Bragg angles, and themajoreffectistoredistributetheTDSinan angle. It may be reasonably assumed that an ADF detector is so large that theTDS is not redistributedoffthedetector,andthattheelec-trons are still detected. In general, therefore, the effect of elastic scattering after phonon scatteringisusuallyneglected.

A type of multislice formulation that does include phonon scattering and postphonon elastic scattering has been developed specifi-callyforthesimulationofADFimages,andis knownasthefrozenphononmethod(Kirkland etal.,1987;Loaneetal.,1991,1992).Anelectron accelerated to a typical energy of 100keV is

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traveling at about half the speed of light. It therefore transits a sample of thickness, say, 10nmin3¥10-17_{s,whichismuchsmallerthan} thetypicalperiodofalatticevibration(~10-13_s). Eachelectronthattransitsthesamplewillseea latticeinwhichthethermalvibrationsarefrozen insomeconfiguration,witheachelectronseeing a different configuration. Multiple multislice calculations can be performed for different thermal displacements of the atoms, and the resultant intensity in the detector plane is summedoverthedifferentconfigurations.The frozen phonon multislice method is therefore notlimitedtocalculationsforSTEM;itcanbe used for many different electron scattering experiments.InSTEM,itwillgivetheintensity at any point in the detector plane for a given illuminating probe position. The calculations faithfullyreproducetheTDS,Kikuchilines,and higher-order Laue zone (HOLZ) reflections (Loaneetal.,1991).TocomputetheADFimage, the intensity in the detector plane must be summed over the detector geometry, and this calculationrepeatedforalltheprobepositions intheimage.Thefrozenphononmethodcanbe arguedtobethemostcompletemethodforthe computationofADFimagesandhasbeenused to compute contrast changes due to composi-tionandthicknesschanges(Hillyardetal.,1993; HillyardandSilcox,1993).Itsmajordisadvan-tage is that it is computational expensive. For mostmultislicesimulationsofSTEM,onecal-culationisperformedforeachprobeposition. In a frozen phonon calculation, several mul-tislicecalculationsarerequiredforeachprobe positioninordertoaverageeffectivelyoverthe thermallatticedisplacements. Mostoftheapproachesdiscussedsofarhave assumedanEinsteinphonondispersioninwhich thevibrationsofneighboringatomsareassumed tobeuncorrelated,andthustheTDSscattering fromneighboringatomsincoherent.Jessonand Pennycook(1995)haveconsideredthecasefor amorerealisticphonondispersion,andshowed thatacoherenceenvelopeparalleltothebeam direction can be defined. The intensity of a columncanthereforebehighlydependenton thedestructionofthelongitudinalcoherenceby thephononlatticedisplacements.Considertwo atoms,AandB,alignedwiththebeamdirection, andletusassumethatthescatteringintensityto

the ADF detector goes as the square of the atomic number (as for Rutherford scattering fromanunscreenedCoulombpotential).Ifthe longitudinal coherence has been completely destroyed,theintensityfromeachatomwillbe independentandtheimageintensitywillbeZA2 +ZB2_{.Conversely,ifthereisperfectlongitudinal} coherencetheimageintensitywillbe(ZA+ZB)2_. Apartialdegreeofcoherencewithafinitecoher-ence envelope will result in scattering some-where between these two extremes. Frozen phonon calculations by Muller et al. (2001) suggest that for a real phonon dispersion, the ADFimageisnotsignificantlychangedfromthe Einsteinapproximation. Latticedisplacementsduetostraininacrystal canberegardedasanensembleofstaticphonons, andthereforestraincanhavealargeeffecton anADFimage(Perovicetal.,1993),givingrise toso-calledstraincontrast.Thedegreeofstrain contrastthatshowsupinanimageisdependent ontheinnerradiusoftheADFdetector.Asthe innerradiusisincreased,theeffectofstrainis reduced and the contrast from compositional changesincreases.Changingtheinnerradiusof thedetectorandcomparingthetwoimagescan oftenbeusedtodistinguishbetweenstrainand compositionchanges.Afurthersimilarapplica-tionistheobservationofthermalanomaliesin quasicrystallattices(Abeetal.,2003).

It is often found in the literature that the veracity of a particular method is justified by comparing a calculation with an experimental imageofaperfectcrystallattice.Animageofa crystal contains little information: it can be expressedbyahandfulofFouriercomponents andisnotagoodtestofamodel.Muchmore interestingistheinterpretationofdefects,such as impurity or dopant atoms in a lattice, and particularly their contribution to image when they are at different depths in the sample. Of particularinterestistheeffectofprobedechan- neling.IntheBlochwaveformulation,theexci-tation of the various Bloch states is given by matching the wavefunctions at the entrance surfaceofacrystal.Whenasmallprobeislocated overanatomiccolumn,itislikelythatthemost excitedstatewillbethetightlybound1s-type state.Thisstatehashightransversemomentum, andispeakedattheatomsiteleadingtostrong absorption. Whichever model of ADF image

L1 formationisused,itmaybeexpectedthatthis

willleadtohighintensityontheADFdetector andthattherewillbeapeakintheimageatthe columnsite.The1sstatesarehighlynondisper-sive, which means that the electrons will be trappedinthepotentialwellandwillpropagate mostlyalongthecolumn.Thischannelingeffect is well known from many particle scattering experiments,andisimportantinreducingthick-nesseffectsinADFimaging.The1sstatewillnot betheonlystateexcited,however,andtheother stateswillbemoredispersive,leadingtointen-sity spreading in the crystal (Fertig and Rose, 1981; Rossouw et al., 2003). Spreading of the probe in the crystal is similar to what would happeninavacuum.Therelativelyhighprobe convergenceanglemeansthatthefocusdepthof field is low, and beyond that the probe will spread. Calculations suggest that this dechan-nelingcanleadtoartifactsintheimagewhereby the effect of a heavy impurity atom substitu-tionalinacolumncanbeseenintheintensityof neighboringcolumns.Thedegreetowhichthis occurs,however,isdependentonthemodelof ADFimagingused,andtheliteratureisstillfar fromagreementonthisissue.

5.3 ExamplesofStructure

DeterminationUsingADFImages

Despite the complications in understanding ADF image formation, it is clear that atomic resolution ADF images do provide direct

images of structures. An atomic resolution imagethatiscorrectlyfocusedwillhavepeaks inintensitylocatedattheatomiccolumnsinthe crystalfromwhichtheatomicstructurecanbe simplydetermined.TheuseofADFimagingfor structure determination is now widespread (Pennycook,2002).

Thesubsidiarymaximaoftheprobeintensity (seeSection2)willgiverisetoaweakartifac-tual maxima in the image (Figure 2–13) [see alsoYamazaki et al. (2001)], but these will be small compared with the primary peaks, and oftenbelowthenoiselevel.TheADFimageis somewhat“fail-safe”inthatincorrectfocusing leadstoverylowcontrast,anditisobviousto anoperatorwhentheimageiscorrectlyfocused, unlike phase contrast CTEM for which focus changesdonotreducethecontrastsoquickly, andjustleadtocontrastreversals.

Therearenowmanyexamplesinthelitera- tureofstructuredeterminationbyatomicreso-lutionADFSTEM.Anexcellentrecentexample is the three-dimensional structural determina-tion of a NiS2/Si(001) interface (Falke et al., 2004)(Figure2–14).Theabilitytoimmediately interpretintensitypeaksintheimageasatomic columns allowed this structure to be deter-mined, and to correct an earlier erroneous structuredeterminationfromHREMdata.

Adisadvantageofscannedimagessuchasan ADFimagecomparedtoaconventionalTEM imagethatcanberecordedinoneshotisthat instabilities such as specimen drift manifest

Figure2–13. AnADFimageofGaAs<110>takenusingaVGMicroscopesHB603Uinstrument(300kV, CS=1mm).The1.4-Åspacingbetweenthe“dumbbell”pairsofatomiccolumnsiswellresolved.Anintensity

profileshowsthepolarityofthelatticewiththeAscolumnsgivinggreaterintensity.Theweaksubsidiary maximaoftheprobecanbeseenbetweenthecolumns.

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themselvesasapparentlatticedistortions.There havebeenvariousattemptstocorrectforthis byusingtheknownstructureofthesurround-ingmatrixtocorrectfortheimagedistortions before analyzing the lattice defect of interest (see,forexample,Nakanishietal.,2002).

5.4 ExamplesofCompositionally

SensitiveImaging

The ability ofADF STEM to provide images with high composition sensitivity enabled the very first STEM, operating at 30kV, to image individual atoms of Th on a carbon support (Creweetal.,1970).Insuchasystem,theheavy supportedatomsareobviousintheimage,and littleisrequiredinthewayofimageinterpreta-tion.Ausefulapplicationofthiskindofimaging isinthestudyofultradispersedsupportedhet-erogeneous catalysts (Nellist and Pennycook, 1996).Figure2–15showsindividualPtatomson thesurfaceofagrainofapoweredg-alumina support.DimersandtrimersofPtmaybeseen, andtheirinteratomicdistancesmeasured.The simultaneously recorded BF image shows fringesfromthealuminalattice,fromwhichits orientationcanbedetermined.Byrelatingthe BFandADFimages,informationonthecon-figuration of the Pt relative to the alumina support may be determined. The exact loca-tionsofthePtatomswerelaterconfirmedfrom calculations(Sohlbergetal.,2004).

When imaging larger nanoparticles, it is foundthattheintensityoftheparticlesinthe

image increases dramatically when one of the particle’s low-order crystallographic axes is aligned with the beam. In such a situation, quantitative analysis of the image intensity becomesmoredifficult. Amorecomplexsituationoccursforatoms substitutionalinalattice,suchasdopantatoms. Modernmachineshaveshownthemselvestobe capableofdetectingbothBi(LupiniandPen-nycook,2003)andevenSbdopants(Voyleset al.,2002)inanSilattice(Figure2–16).InVoyles etal.(2004)itwasnotedthattheprobechan-nelingthendechannelingeffectscanchangethe intensity contribution of the dopant atom depending on its depth in the crystal. Indeed thereissomeoverlapintherangeofpossible intensitiesforeitheroneortwodopantatoms inasinglecolumn.Anothersimilarexampleis the observation of As segregation at a grain boundaryinSi(Chisholmetal.,1998).

Naturally, ADF STEM is powerful when appliedtomultilayerstructuresinwhichcom-position sensitivity is desirable. There have

Figure 2–14. An ADF image of an NiS2/Si(001)

interface with the structure determined from the image overlaid. [Reprinted with permission from Falkeetal.(2004).Copyright(2004)bytheAmerican PhysicalSociety.]

Figure2–15. AnADFimageofindividualatomsof Ptonag-Al2O3

supportmaterial.TheBFimagecol-lected simultaneously showed fringes that allowed theorientationoftheg-Al2O3

tobedetermined.Sub-sequenttheorycalculations(seetext)confirmedthe likelylocationsofthePtatoms.

L1 been several examples of the application to

AlGaAs quantum well structures (see, for example, Anderson et al., 1997). Simulations havebeenusedtoenabletheimageintensityto beinterpretedintermsofthefractionalcontent ofAl,whereithasbeenassumedthattheAlis uniformlydistributedthroughoutthesample.

6. ElectronEnergy

LossSpectroscopy

Sofarwehaveconsideredtheimagingmodes of STEM, which predominantly detect elastic or quasielastic scattering of the incident elec-trons.An equally important aspect of STEM, however,isthatitisanextremelypowerfulana-lyticalinstrument.Signalsarisingfrominelastic scatteringprocesseswithinthesamplecontain much information about the chemistry and electronic structure of the sample. The small, bright illuminating probe combined with the useofathinsamplemeansthattheinteraction volumeissmallandthatanalyticalinformation canbegainedfromaspatiallyhighlylocalized regionofthesample.

Electron energy-loss spectroscopy (EELS) involves dispersing in energy the transmitted electrons through the sample and forming a spectrum of the number of electrons inelasti-callyscatteredbyagivenenergylossversusthe energy loss itself.Typically, inelastic scattering events with energy losses up to around 2keV areintenseenoughtobeusefulexperimentally.

The energy resolution of EELS spectra can bedictatedbyboththeaberrationsofthespec-trometerandtheenergyspreadoftheincident electronbeam.Byusingasmallenoughentrance aperturetothespectrometertheeffectofspec-trometeraberrationswillbeminimized,albeit withlossofsignal.Insuchacase,theincident beamspreadwilldominate,andenergyresolu-tions of 0.3eV with a CFEG source of about 1eVwithaSchottkysourcearepossible.Inelas-ticscatteringtendsbelowangledcomparedto elasticscattering,withthecharacteristicscatter-ingangleforEELSbeing(forexample,Brydson, 2001) θE = ∆E_2E 0 (6.1) For100-keVincidentelectrons,qEhasavalue of 1mrad for a 200eV energy loss ranging up

Figure2–16. AnADFimage(left)ofSi<110>withvisibleSbdopantatoms.Ontheright,thelatticeimage hasbeenremovedbyFourierfilteringleavingtheintensitychangesduetothedopantatomsvisible.[From Voylesetal.(2002),reprintedwithpermissionofNaturePublishingGroup.]