Applic ationofStereo Im agingto Atom ic Forc e
M ic roscopy
B ernard o D.Aum ond ,and K am alY ouc ef
-T oum i
Abstrac t| M etrologic ald ata from sam ple surfac esc anb e ob tained b yusinga varietyofpro¯lom etrym ethod s.Atom ic Forc e M ic rosc opy (AF M ), w hic h relies on c ontac t in ter-atom ic forc esto extrac t top ographic alim agesofa sam ple, isone suc h m ethod that c anb e used ona w id e range ofsur-fac e types,w ith possib le nanom eter range resolution.How -ever, AF M im agesare c om m only d istorted b y c onvolution, w hic h red uc esm etrologic alac c urac y. T histyp e ofd istor-tionis m ore signi¯c ant w henthe sam ple surfac e c ontains high aspec t ratio featuressuc h aslines,stepsor sharp ed ges - struc turesc om m only found insem ic ond uc tor d evic esand applic ations.Aim ingat m itigatingthese d istortionsand re-c overing m etrology sound ness, w e introd uc e a novelim age d ec onvolutionschem e b ased onthe princ iple ofstereo im ag-ing.M ultiple im agesofa sam ple, takenat d i®erent angles, allow for separationofc onvolutionartifac tsfrom true to-p ograto-phic d ata.Asa result, perfec t sam ple rec onstruc tion and prob e shap e estim ationc anb e ac hieved inc ertainc ases. Ad d itionally, shad ow z ones, w hic h are areasofthe sam ple that c annot b e prob ed b y the AF M , are greatly red uc ed . M ost im portantly, thistec hnique d oesnot require a priori prob e c harac teriz ation.It also red uc esthe need for slend er or sharper prob es, w hic h, onone hand , ind uc e lessc on vo-lutiond istortionb ut,onthe other hand ,are m ore prone to w ear and d am age,thusd ec reasingoverallsystem reliab ility.
K eyw ord s| M etrology,P ro¯lom eter,Atom ic Forc e M ic ro-scop e, Dec onvolution, Stereo Im aging
I. Introd uction
S
U R FA CE characteristics such as topography and criti-caldimensions, roughness and the area density,shape and location ofdefects often serve as importantindicators ofproductqualityandmanufacturingprocessperformance. Forsuchreasons,surfacecharacterizationproceduresareof primary importancein awide rangeoftechnological¯elds and across industries. In addition, high precision char-acterization has played an increasingly importantrole as therequireddimensions ofsemiconductorandothermicro-fabricated devices continue to shrink into the nanometer and micrometerdomains [1 ].V arious technologies existthatcanbeusedforobtaining high precision images ofsurfaces. T he A tomic Force M i-croscope(A FM )is onesuchtoolthatcanbeusedtopro¯le samples,with possible nanometerlevelresolution [2].T he A FM generates topographical images via van der W aals forces thatarisefrom directcontactbetweenasharp probe and a surface. T herefore, this imaging toolcan be used tomeasure severaldi®erenttypes ofsurfaces regardless of otherphysicalattributes such as re° ectivity, conductivity ormagnetism.Itobviously circumvents resolution limita-tionsintroducedbydi®ractionphenomena,associatedwith opticaltools,orby¯niteelectron escapedepth,associated with SEM imaging.In addition tothat,A FM images con-sistofthree dimensionaltopographicmaps ofthe surface and are, forthis reason, idealforcross
sectionalmetrol-F ig.1. T he prob e apex isnot nec essarily the sole point ofc ontac t for high aspec t ratio sam ples. Variouspositionsofthe prob e along the scanare portrayed .
ogy applications.H owever, despite the A FM 's versatility and high resolution, its metrology accuracy is limited by the sizeand shapeoftheemployed probe[3]duetoimage convolution.
Image convolution expresses itselfin the form ofloss of surfacedetailanddullingofhighaspectratiofeatures.T his type ofdistortion occurs when, during the scanning pro-cess,the contactpointbetween the probe and the sample is notthe apex ofthe probe butits sideinstead,as shown in Figure 1 . In otherwords, during the imaging process, the probe is always externally tangenttothe surface,and as aresult,theimageis adilatedversion oftheunderlying sample.D ilationdistortions introduceerrorsinthemetrol-ogydataobtainedfrom thesurface[4].A s aresult,critical dimensions such as linewidth ofsteps orradius ofcurva-ture ofhigh aspectratiostrucofcurva-tures (such as ¯eld emission probes and high precision cutting edges)become inaccu-rate [5].
For the sake of semantic accuracy, it should be noted that the term convolution does not strictly apply to the mechanism ofimage formation in A FM .Ithas been com-monlyemployedeventhoughnoconvolutionoccurs during imaging;dilation instead,is thecorrectinteraction model.
In order to circumvent these limitations, deconvolu-tion algorithms must be developed. T hese restoradeconvolu-tion procedures should eliminate metrology distortions in the nanometerleveland should also be compatible with high volumeinspection tasks.P roposed methodologies mustbe robust,reliable and easy toimplementwith commercially availableA FM units.
In section II, we introduce the mechanisms of image formation and how they give rise to convolution. In sec-tion III,webrie° y explain stateofthearttechniques that can beused forimagerestoration.In section IV ,weintro-duceanovelstereoimagingprocedurethatcanbeusedfor restoringsampletopographiesandthatcanalsobeusedfor probe shape estimation and ¯nally in section V , we show introductoryresults thatillustratethecapabilitiesandver-satility ofthis newtechnique.
II. IM AG E C O NV O L U T IO N
T here exist two widely accepted mechanisms of image formationforA FM .O nemechanism modelis basedonthe conceptofL egendre T ransforms [6], [7]and relies on the assumption thatatcontactpoints duringscan, the probe andthesamplesurfacesharethesamegradientortangent.
In Figure 2, let S (m)be the intercept of the tangent linethrough thetruecontactpointwith theverticalimage axis;atthatcontactpointthe sample slope is denoted by m and S (m)is denoted the \ L egendre T ransform ofthe probe shape atthe derivative m".L etI(m)be the inter-cept of the parallel line through the image point (probe apex)with theverticalimageaxis.Ithas been shown else-where[7]thattheimagewillalsohavederivativem atthat point.Finally,letP (m)betheinterceptofthetangentline throughthetruecontactpointwiththeprobeverticalaxis. T he followingrelationship holds:
S (m)= I(m)+ P (m) (1 ) G ivenasurfacedescriptiony = y(x),itsL egendreT rans-form atany derivativem =dxdy is given by:
Y (m)= y(x(m))¡m:x (2) Conversely, if the L egendre T ransform of a curve is known, its Cartesian description can be obtained by ap-plyingthe followinginversetransform: x = ¡d[Y (m)] dm y = Y (m)¡m(d[Y (m)] dm ) y = y(x) (3)
T herefore, if the shape of the probe and the shape of the sample are known, theirL egendre T ransforms can be obtained according to Equation (2). T hen, the L egendre T ransform oftheresultingimagecan becomputedaccord-ing to Equation (1 ). A nd ¯nally, the shape of the re-sulting image can be calculated using Equation set (3). H owever,even though the linearrelationship expressed in Equation (1 )is simple and straightforward, some implicit assumptions exist. Firstly, the sample and probe geome-tries must be continuous and, secondly, the sample and probe geometries mustnothave repeated slopes, thatis, theymustbeconvex.T his reduces theapplicabilityofthis mechanism ofimageformation tosimplerand wellde¯ned problems where the implicitassumptions hold [5],[7].
F ig.2 . M ec hanism ofim age form ationb ased onthe Legend re T rans-form m od el.
F ig.3. M ec hanism ofim age form ationb ased onthe d ilationm od el. (a) T he prob e isexternallytangent to the sam ple d uringthe sc anning, generatingthe im age.(b ) A d ualand equivalent interpretationisthat the im age isthe c omb ined volum e ofthe translatesofthe (re°ec ted ) prob e.
A second mechanism ofimage formation thatis general and makes nogeometricassumptions is based on the con-cept ofmorphologicaldilation [4], [8], [9], [1 0 ]. T he im-age ofa sample is a dilated version ofthe sample and the structuringelementis considered tobethere° ected probe shape.T hatis:
I= S © ·P (4)
where ·P is there° ected version oftheprobeshapeP ,and S is the sample shapeand I,the resultingimage.
Inotherwords,ifoneplacesacopyofthere° ectedprobe on every single pointofthe sample surface, with the re-° ectedprobeapexcoincidingwith thatsurfacepoint,then the surface ofthe resultingcombined volume ofalltrans-lates ofthese re° ected probes willconstitute the image of the sampletaken with thatspeci¯cprobe (Figure3).
A quickanalysis ofEquation (4)(andofEquation(1 )as well)reveals somestraightforward conclusions thatcan be summarized in the followingcases:
²Case 1: Ifthe shape ofthe prob e isexac tly know nand the im -age isalso know n, thenb y applying a reverse m ec hanism ofim age form ation, the sam ple shape c ould b e c om puted , and c onvolution d istortions,elim inated .
²Case 2 : Conversely, ifthe sam ple shape isexac tly know nand the im age isalso know n,thenb y applyinga reverse m ec hanism ofim age
form ation,the prob e shape c ould b e c om puted .
²Case 3: Ifthe prob e isvery sharp c om pared to the sam ple d im en-sions,thenthe im age shape w illb e very sim ilar to the sam ple shape, the prob e c ontrib utionb eingnegligib le to im age form ation.
²Case 4 : Conversely, ifthe sam ple isvery sharp c om pared to the prob e d im ensions, thenthe im age shape w illb e very sim ilar to the re°ec ted prob e shape, the sam ple c ontrib utionb eing negligib le to im age form ation.
T heseconclusions arethemotivation behindcurrentde-convolution orbehindcurrentde-convolution minimization strategies as we discuss is section III.
III. STAT E O F T H E AR T IN IM AG E D EC O NV O L U T IO N
In ordertorecovermetrology accuracy,convolution dis-tortions mustbe eliminated from A FM images. Current deconvolution strategies are based on the realization that Equation(4)(orEquation(1 ))establishasystem withtwo unknowns, the probe and the sample geometries (ortheir L egendretransforms),andoneknown,theimagegeometry (oritsL egendretransform).Inordertosolvesuchanunder constrainedsystem,onevariablemustbedetermined;that can be accomplished by probe characterization. A nother approach is tomake the probe sosharp thatits contribu-tion to image formacontribu-tion becomes negligible. T hatwould be equivalent to making ·P in Equation (4) or P (m) in Equation (1 )vanish. T hat is, the system collapses into oneknown andoneunknown andcan besolved.T hemost popularapproach though, called blind deconvolution, es-tablishes an estimate forthe probe shape,and as aresult, asample estimatecan bealsoobtained.
A .H igh A spectR atioP robes
Convolution distortions are proportional to the probe sizerelativetothesamplesize.T herefore,sharp andsmall probes can be used to minimize convolution distortions, accordingtoCase 3 in section II.
Sharp probes can be obtained by FIB milling conven-tionalA FM probes.Sharpened probes existin themarket with typicalradius ofcurvature of5 to 20 nm. H owever, forstatisticalreasons orquality assurance considerations, high volumeinspectionhas becomearequirementin many applications.T herefore, long lasting probes mustbe em-ployedinordertominimizeprobereplacementduetowear orfailure.Sharpenedandslenderprobes aremoreproneto failurethan theirlargercounterparts and therefore reduce theoverallreliability and speed oftheinspection system. R ecently,carbonnanotubeshavebeenemployedinA FM imaging.N anotubes areused as probes duetotheirsharp geometryandmechanicalresilience.T hecarbonnanotubes consistofperfectand seamless graphiteshells with dimen-sions of typically 1 nm in diameter and several microns in length.T he slenderness ofthese nanotubes may allow forimagingofhigh aspectratiosurface features with very smallconvolution distortions. H owever, fabrication tech-niques haveyettobere¯ned [1 1 ]and,additionally,lateral ° exing of the tube is still a problem when imaging tall structures.A s a resultofthese limitations, imaging with ultrahigh aspectratioprobes has been mostlycon¯ned to
F ig.4 . B lind prob e rec onstruc tion.(a) O riginalSc an.(b ) P eaksin im age are °agged .(c ) P rob e estim atesare ob tained from the ind ivid -ualpeaksand the m inimum envelop takenasthe ¯nalprob e estim ate. (d ) Com parisonb etw eenac tualand rec onstruc ted prob e.
o®lineinspectiontasks wherescanspeeds areslowerinor-dertoavoidtip crashes,° exion orexcessivewear.U sually, intermittentcontactscanningis used in thesecases. B .P robe Characterization
D econvolution can alsobe accomplished by establishing theshapeoftheprobeandthensolvingtheinverseproblem establishedbyEquation (4)orbyEquation(1 ),depending on which mechanism ofimage formation is used toformu-late the problem.A ccording to Case 2 in section II, one could useasampleofknown shape,thatis,astandard,in ordertoestablishtheprobeshape;thisiscommonlyknown as probe characterization. Commonly used characteriza-tion standards include high precision silicon gratings [3], [7],blades,otherA FM tips,colloidalgold articles [1 2] ,la-tex particles [1 3],etc...
T heproblems associatedwithprobecharacterizationare the following:(1 )T he shape ofthe characterization stan-dard in never perfectly known and manufacturing toler-ances for commercially available standards range from 5 to 20 nm [1 4], which is on the same order of magnitude ofthe size ofthe probes themselves [1 5]. T herefore, the shapeoftheprobecannotberecovered with thenecessary precision.(2)Since probes wearwith use, frequentchar-acterizations arenecessarytore-establishtheprobeshape, decreasinginspectionthroughputand,¯nally,(3)thestan-darditselfmustbekeptintactandcleanthroughoutits life span,which adds toprocess complexity.T herefore,probe characterization has only been used on aqualitativebasis, as ameans toassess probe wearordetectprobefailure. C.B lind R econstruction
A nothercurrentlyusedmethodologyforimagedeconvo-lution is the so called blind reconstruction [1 6], [1 7], [1 8] procedure and its variations.T his technique allows fores-timation of the probe shape without a priori knowledge ofthe surface ofthe characterization standard;it, there-fore, removes the requirement for any calibration of the standard. H owever, this estimate consists ofonly an up-perbound forthe probe shape and its quality is a strong function ofthe surfacefeatures found on the standard.
B lind reconstruction has its basis on the dilation inter-pretation forimage formation.Itassumes,therefore,that theimageisessentiallythesurfaceofthecombinedvolumes ofalltranslates ofthe re° ected probe tip as the re° ected probe apex is lined up with each sample surface point. T hus, the (re° ected)probe pro¯le is always bounded by the image.Crude estimates ofthe probe pro¯le can then bemadesimply by takingthesharpestfeatures presentin the image.T hatshould serve as an upperbound forthe A FM probe shape.In fact, ifthe surface portrays an in-¯nitely sharp protrusion,the image ofthatsurface should beidenticaltothere° ectedprobeshape,accordingtoCase 4 in section II.Foran A FM probe with a single tip (dis-carding pathologicalcases ofprobes with split tips), the blind reconstruction procedure can be reduced to the fol-lowingrecipe:(1 )identify peaks in the image,(2)use the peakregions toestimatethere° ected probeshapeand (3) overlaytheseestimates,liningup atthepeaks andtakethe minimum envelopeofthecombinedpro¯les as there° ected probeestimate.Figure4 illustrates theprocess.
A s mentioned before, since probe estimation by blind reconstruction is implemented usingonly the image data, the quality ofthe probe estimate is highly dependenton theimageitself,as demonstrated in R ef.[9].Forinstance, a smooth surface would yield fewdistinctpeaks in its im-age and thus have limited utility in this mode of probe estimation.O n theotherhand,asurfacewith high aspect ratiofeatures would producean imagewith sharp features andthereforeamoreaccurateprobeestimate.InFigure5, we see the resulting probe estimate made using two sur-faces with the same peak topeak R M S butwith di®ering correlation lengths (which is theaveragewavelength ofthe surface).Clearly,thequalityoftheprobeestimationis sig-ni¯cantlybetterforthesurfacewiththesmallercorrelation length,thatis,with sharperfeatures.
Since it is hard to guess the sharpness of features on the characterization standard, the quality ofthe estimate cannotbe precisely determined either.T herefore, the de-convolvedA FM images thatusesuch probeestimates may lack accuracy due to poorprobe estimation. T his ambi-guity in probe estimation can beillustrated as depicted in Figure 6. In short, a certain image can be generated by an in¯nitenumberofappropriatesample/probepairs that satisfy Equation (4).
A s a conclusion, even though estimation schemes ex-ist,theystillnecessitatespecialcharacterizationstandards thathavefeaturesmuchsmallerthantheprobe.Suchstan-dards maybe di±cultto obtain, maintain orcharacterize. A dditionally, forthe same reasons laid outin section III-B , frequentestimation may lead to lowerthroughput. It would be desirable, then, to develop a methodology that does not require special characterization standards, that can deliverhigh quality probe estimates forimage decon-volution,and thatdoes notreduce throughputorincrease complexity signi¯cantly.
F ig.5. P rob e estim ates(ind ivid ual).(a) Sam ple surfac e w ith large c orrelationlength.(b ) R esultingprob e estim ate ispoor.(c ) Sam ple w ith sm allc orrelationlength.(d ) R esultingprob e estim ate isgood .
F ig.6. Inb oth (a) and (b ), the sam e im age c ould b e ob tained w ith a sw apped set ofprob e and sam ple. T hat is, inb oth c ases, E quation(4 ) issatis¯ed .T herefore,the b est possib le prob e estim ate (upper b ound ) isP robe 1 asshow nin(a) eventhough the realprob e geom etry m ay b e P robe 2 asshow nin(b ).
IV . ST ER EO IM AG ING
W e propose a new approach to image deconvolution called Stereo Imaging. In this approach, two images of the same sample are obtained atdi®erentvantage points. T hat is, the sample is mechanically rotated relatively to the probe priortothe second scan.Since the rotation an-gle can be speci¯ed, one obtains the following system of equations:
I1 = S © ·P
I2 = S¤© ·P
S¤ = R ot(S ;µ) (5)
IfS isthesetofallpointsfx;yg pertainingtothesurfaceof thesample,then S¤willbecomposed ofpointsfx
r;yrg as
F ig.7. T w o im agesfrom d i®erent vantage points.(a) Sam ple tilted b y¡¼=10 rad iansand resulting im age ob tained w ith the portrayed prob e.(b ) Sec ond im age ob tained b y rotatingthe sam ple b y ¼ =10 ra-d iansfrom the vertic ald irec tion.B oth im agesare b lunt and d istorted versionsofthe high aspec t ratio sam ple d ue to c onvolution.
½ xr yr ¾ = · cos(µ) ¡sin(µ) sin(µ) cos(µ) ¸½ x y ¾ (6) Equation (5)de¯nes a system with two equations and twounknowns and therefore can be solved forboth probe and sample geometry without the need for priorcharac-terization. T he rotation ofthe sample provides an extra constraintforestimation;as aresult,estimationambiguity is greatly reduced.
T he steps involved in stereo imaging include: (1 )ob-tainingtwoimages ofthesample;(2)estimatingtheprobe shape by blind deconvolution;(3)combining probe esti-mates byoverlay;(4)generatingsampleestiesti-mates by Ero-sion;(5)combiningsampleestimates byoverlay;(6)sharp-ening the probe estimates. Steps (3)though (6)are re-peated untilconvergence is reached.Furtherexplanations follow.
T he ¯rststep ofthe methodology consists ofobtaining twoimages ofthe same sample atdi®erentangles as seen in Figure 7.T he sample rotation is obtained by means of ahighprecisiontiltactuationsystem.Inthis example,the sample is chosen to have a high aspectratio.N otice how theimagesareblunterandwiderversionsoftheunderlying sample due to convolution. A lso, in this example, in the ¯rstimage the sample is tilted by¡¼=1 0 radians and in thesecond image,the sampleis tilted by ¼=1 0 radians.
N ext,blind deconvolution is used toestimate the probe geometry foreach image.T he estimates are combined by simplyoverlayingthem.A sharperestimatefortheprobeis obtained.T his probe estimate is used togenerate sample estimates based on each image. T he process of sample estimation given a probe estimate is called Erosion and corresponds totheinversemechanism ofimage formation. Itcan be done by simply scanning the underside ofthe images withthere° ectedprobeestimateandrecordingthe re° ected probe apex position ateach translate.Complete discussions on Erosion can befound in R efs.[1 0 ],[1 6],[1 7]
F ig.8. P rob e and sam ple initialestim ation.(a) and (b ) portray the resultsofprob e estim ationb y b lind d ec onvolutionfor eac h im age. T he estim atesare signi¯c ant d i®erent thanthe originalprob e shape. In(c ), estim ate c omb inationyield sa slightly tighter estim ate. (d ) and (e) are sam ple estim ationsfor eac h vantage point ob tained b y erosionw ith the new c omb ined prob e estim ate. (f) show sthe new sam ple estim ate b y overlay ofthe previousestim ates.T hisestim ate issom ehow c loser to the realsam ple shape b ut far from prec ise.
and [1 8].
T he sample estimates are combined to generate a new one by simply overlayingthem.P riortooverlay,the sam-ple estimates are broughtto an uprightposition.In Fig-ure8(f),thenotch atthetop ofnewestimateis duetothe overlay ofthe sample estimates.Itshould be clearatthis stage thatthe overlaying operation relies on an accurate knowledge of the rotation angle µ. A lso, if there is any spurious translation duringthetiltingprocess,images will be displaced and so willbe the initial sample estimates. A s a result, during the overlay, sample estimates willfall outofplace generatingan inaccurate newestimate.T his creates theneedforhighprecisiontiltactuators (lowradial runout).In addition,itrequires thatthe relative position of the rotation axis with respect to the A FM frame be known atalltimes,which can be achieved by calibration. A n alternativetothatis thescanningofreferencefeatures before and afterrotation toallowforidenti¯cation ofany spurious translation thatmighthave occured duringtilt.
N ow,thenewprobeestimatemustbesharpenedinorder tosatisfy Equation set(5).T he process ofsharpeningthe probeestimateincludesthefollowingsteps:(1 )P laceprobe estimate in acertain pointP1 alongthe image.(2)V erify
ifthe probe estimate interferes with the sample estimate. (3)T rim orsharpen interfering regions ofthe probe. (4) R epeatprocedure forallpoints Pi along one image. (5)
R epeatprocedureforthesecond image.
T hesharpeningproceduremakes surethateachimageis adilationoftheestimatedsamplebyastructuringelement with the shape ofthe re° ected probe estimate.T hatis,it forces the estimates tosatisfy Equation set(5)
T he sharpening operation reduces ambiguity around sampleand probeestimation.T his is sobecausethespace ofsolutions forprobe and sample geometry satisfyingone constraint as established in Equation (4) is necessarily
F ig.9 . (a) P rob e E stim ate must b e alw aysexternally tangent to sam ple estim ate inord er to satisfy E quation(4 ).T herefore, the in-terfering regionmust b e trim m ed from the prob e estim ate. As a result,the prob e estim ate issharpened asshow nin(b ).
F ig.10 . E volutionofestim ates.Sketc hes(a) and (d ) show prob e and sam ple estim atesb ased ona c omb inationofb lind estim ates. Since the sam ple isnot very sharp,estim atesare very inac c urate.Sketc hes (b ) and (e) are the resultsofthe ¯rst iterationofthe Stereo Im aging proc ed ure.E stim atesare greatly enhanced .Sketc hes(c ) and (f) are the results ofthe sec ond iteration. E stim ates and realgeom etries are id entic al; c onvergence isreac hed and no signi¯c ant geom etric al c hangeshappeninfurther iterations.
larger than the space of solutions that satisfy two con-straints,simultaneously,as establishedinEquationset(5). Consequently,increasingthenumberofconstraints (orim-ages at di®erentangles)willeventually reduce the space toasinglesolution and zeroambiguity.H owever,multiple imagingistootimeconsumingandtwoimagesseem enough toaccomplish high precision estimates in mostcases.
T he sharpened probe estimates are then combined by overlayagain.N ewsampleestimates aregeneratedbyero-sion and the whole process is repeated untilnonoticeable change in the estimates is detectable. T hatis, untilthe R M S ofthe di®erence between probe estimates from one iteration toanotheris su±ciently small.T he evolution of probeand sample estimates is shown in theFigure1 0 .
F ig.11. (a) ConventionalAF M im age w ith sam ple inthe upright position(µ = 0 ).(b ) T w o im agesob tained b y tilting the sam ple b y µ = §¼=10 rad ians.
V . PR EL IM INAR Y R ESUL T S
StereoImagingallows forpreciseestimationofprobege-ometries withouta priori probe characterization.T hatis, probe and sample are estimated simultaneously, and esti-mationqualityisindependentofthesamplecharacteristics. In addition tothat,tiltingofthe sample allows the probe to reach otherwise inaccessible regions. Such regions are usually called shadowzones.
Figure1 1 (a)shows thecross section ofastep feature.It alsodepicts aprobe,in this case with twoapices,and the correspondingsimulatedimage.T heimageisadilatedver-sion ofthe sample and metrology data obtained from this image would renderinaccurate results (image linewidth¼ 225 nm,samplelinewidth¼1 50 nm).In addition tothat, the side walls ofthe step are neverreally touched by the probe;therefore,noinformation aboutthis shadowzoneis stored in the image. In this simulation, the step has di-mensions compatiblewith microfeatures regularlypresent in semiconductordevices.T heprobewas chosen tohavea complicated geometry with a primary radius ofcurvature ofaround1 0 nm andasecondaryapex.Figure1 1 (b)shows twoimages obtained by tiltingthe sample.T he side walls are nowexposed toprobe.
T he results ofboth blind deconvolution estimation (ob-tainedfrom theimagedepictedinFigure1 1 (a))andstereo imagingestimation (obtained from the images depicted in Figure 1 1 (b))are shown in Figure 1 2. Since the sample has nosharp features,blind estimation provides very poor results.Infact,theprobecouldnotbeestimatedatallbe-causetherewerenosharp peaks in theimageand,as are-sult,sampleestimation is poor.O n theotherhand,stereo imaging estimation results are virtually indistinguishable from the originalgeometries (estimated linewidth¼ 1 60 nm ).T he areas ofthe estimation close to the base that signi¯cantly di®er from the original sample are actually uncertain reconstruction zones and are ° agged by the al-gorithm, and should be ignored.U ncertain zones happen when the probe estimate is in contactwith the sample
es-F ig.12 . (a)P rob e and sam ple estim ationb ased onb lind d ec onvol u-tion(State ofthe Art). (b ) P rob e and sam ple estim ationb ased on stereo im aging.R esultsafter 3 iterations.
F ig.13. P rob e and sam ple estim ationb y stereo im aging. Sam ple rotated µ = §¼=5 rad ians.P rob e w ith tip rad iusof¼ 2 0 nm .R esults after 3 iterations.
timate in more than one point simultaneously [8], for a certain position along the image. D uring scanning, such regions arenottouchedbytheA FM probe,and maybeof any depth.
Similarresults can be obtained ifaprobe ofcompletely di®erent geometry is used, as shown in Figure 1 3. A lso notice thatthe probe geometry used in the simulations is not sharp or slender, compared to the sample geometry. D econvolution is achieved in spite ofprobe size orshape. T his insensitivity to probe shape and size opens the pos-sibility forusinglarge probes.Such probes maybe coated withD iamondforextendedlifeandmaybeutilizedforsur-facestrengthmeasurements bymicro-indentation,simulta-neously with topography measurements.
O nelimitation ofthestereomethodology is thatitdoes not solve the problem of narrow trenches that are inac-cessible to the probe. O nly high aspect ratio probes at low scan rates, operated in intermittentcontact, can ac-cess areas deep into the negative steps.H owever, as seen in Figure 1 4, the reconstruction obtained with the stereo
F ig.14 . (a) P rob e estim ationb y stereo im aging.(b ) Sam ple estim a-tionb y stereo im aging. Sam ple rotated µ = §¼=2 0 rad ians. P rob e w ith tip rad iusof¼ 10 nm .R esultsafter 5 iterations.(c ) Im age of the sam ple w ith the sam e prob e inupright position.
approach renders good pitch measurementand is a better depiction ofthe underlying sample than an image ofthe samesampletaken in theusualuprightposition.T hebot-tom ofthetrenchestimation(solidgrayline,Figure1 4(b)) is an uncertain reconstruction zone thatis automatically tagged by the algorithm.In thatzone,the reconstruction sets un upperbound forthesampletopography,thatis,in thatregion therealsamplemayhaveanydepth as longas itis deeperthan the boundary setby theestimation.
V I. C O NC L USIO NS
Convolution e®ects may severely distortmetrology data obtained with A tomicForce M icroscopes.A lthough some techniques existthatallowforimage deconvolution, they mostly fail to deliver high ¯delity topography estimates and are cumbersome and time consuming, thus reducing inspection throughput.
B y obtaining two ormore images ofthe sample atdif-ferentvantage points, stereo imagingcan be used to gen-erate high precision estimates ofboth probe and sample simultaneously.T herefore,metrology accuracy is ensured, regardless ofprobe shape orsize.Since the probe geome-try is estimated atevery imagingevent,an e®ective probe monitoringscheme can be implemented.
Since allsteps involved in the StereoImagingapproach arecarriedoutbysetormorphologicaloperations,its gen-eralizationto3-D andvolumeanalysis(insteadofthecross-sectionalanalysis discussed in this paper)is simple.
Finally,thenextstep ofthisresearchprojectisunderway and includes experimental tests with calibrated samples aimingatassessingtherepeatabilityofthemethod as well as the in° uence ofnoise and scan dynamics e®ects on the ¯nalreconstruction results.
R e fe re nce s
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