Elemental thin film depth profiles by ion beam analysis using simulated annealing—a new tool

30 Pages·2003·0.72 MB·English

INSTITUTEOFPHYSICSPUBLISHING JOURNALOFPHYSICSD:APPLIEDPHYSICS
J.Phys.D:Appl.Phys.36(2003)R97–R126 PII:S0022-3727(03)34952-6
TOPICAL REVIEW
Elemental thin film depth profiles by ion
beam analysis using simulated
annealing—a new tool
CJeynes1,NPBarradas2,6,PKMarriott3,GBoudreault1,MJenkin4,
EWendler5 andRPWebb1
1UniversityofSurreyIonBeamCentre,Guildford,GU27XH,UK
2InstitutoTecnolo´gicoeNuclear,E.N.10,Sacave´m,Portugal
3DepartmentofStatistics,NationalUniversityofSingapore,Singapore
4SchoolofElectronicsComputingandMathematics,UniversityofSurrey,Guildford,UK
5Friedrich-Schiller-Universita¨tJena,Institutfu¨rFestko¨rperphysik,Jena,Germany
Received18March2002
Published19March2003
Onlineatstacks.iop.org/JPhysD/36/R97
Abstract
Rutherfordbackscatteringspectrometry(RBS)andrelatedtechniqueshavelongbeen
usedtodeterminetheelementaldepthprofilesinfilmsafewnanometrestoafew
micronsthick. However,althoughobtainingspectraisveryeasy,solvingtheinverse
problemofextractingthedepthprofilesfromthespectraisnotpossibleanalytically
exceptforspecialcases. Itisbecausethesespecialcasesincludeimportantclasses
ofsamples,andbecauseskilledanalystsareadeptatextractingusefulqualitative
informationfromthedata,thationbeamanalysisisstillanimportanttechnique.
Wehaverecentlysolvedthisinverseproblemusingthesimulatedannealing
algorithm. Wehaveimplementedthesolutioninthe‘IBADataFurnace’code,which
hasbeendevelopedintoaveryversatileandgeneralnewsoftwaretoolthatanalysts
cannowusetorapidlyextractquantitativeaccuratedepthprofilesfromrealsamples
onanindustrialscale. Wereviewthefeatures,applicabilityandvalidationofthis
newcodetogetherwithotherapproachestohandlingIBA(ionbeamanalysis)data,
withparticularattentionbeinggiventodeterminingboththeabsoluteaccuracyof
thedepthprofilesandstatisticallyaccurateerrorestimates.
WeincludeexamplesofanalysesusingRBS,non-Rutherfordelasticscattering,
elasticrecoildetectionandnon-resonantnuclearreactions. Highdepthresolution
andtheuseofmultipletechniquessimultaneouslyarebothdiscussed. Thereis
usuallysystematicambiguityinIBAdataandButler’sexampleofambiguity
(1990Nucl.Instrum.MethodsB45160–5)isreanalysed. Analysesareshown:of
evaporated,sputtered,oxidized,ionimplanted,ionbeammixedandannealed
materials;ofsemiconductors,opticalandmagneticmultilayers,superconductors,
tribologicalfilmsandmetals;andofoxidesonSi,mixedmetalsilicides,boron
nitride,GaN,SiC,mixedmetaloxides,YBCOandpolymers.
1. Overview Glossaryinsection2foranexplanationoftheacronyms)inthe
contextoftheworkofthecommunity;discussingthescientific
We will review our recent software developments in thin implications of these developments. We hope to show that
film depth profiling with ion beam analysis (IBA, see the these developments improve the usability of IBA to such an
extentastoeffectivelyestablishanewdepthprofilingtool.
6 CentredeF´ısicaNucleardaUniversidadedeLisboa,AvenidaProg.Gama We start (section 2) with a simplified overview of
Pinto2,1699LisboaCodex,Portugal IBA since we believe that our new software tool (the IBA
0022-3727/03/070097+30$30.00 ©2003IOPPublishingLtd PrintedintheUK R97
TopicalReview
DataFurnace)willmakeIBAattractivetomanyusersforthe with magnetic tunnel junctions (used for magnetic random
firsttime. Theessenceofthisnewtoolisitssolutionofthe access memory applications) fabricated with sophisticated
‘inverse problem’ of automatically extracting depth profiles metallic (and semi-metallic) multilayers and analysed with
from Rutherford backscattering (RBS) data: we therefore highresolutionRBS.WithRossendorftheyhaveprofiledMBE
describethisinverseproblem(section3)andcontinuewithan GaN on Si (Alves et al 2002). The Rossendorf group has
introductiontothesimulatedannealingalgorithmonwhichthe investigatednitridingofAl(Telbizovaetal2001a,b,Fitzand
inverseproblem’ssolutionisbuilt(section4). Thisalgorithm Mo¨ller2002).
was first applied to RBS in 1997 by Barradas et al, and the ThegroupatSurreyhasalso:measuredlateralstressinim-
mathematical analogy of ‘annealing’ is why we have called plantedgoldfilmsusingathreecrystalquartzresonatormethod
thecode‘DataFurnace’. andadetailedself-consistentRBSanalysis(Wayetal1999);
After some RBS examples exemplifying the scope of investigatedsurfactantbehaviourintheformationoflatexfilms
the tool (section 5) we give details of the physics (the (Tzitzinouetal 1999); characterizedCVDdepositedSiCon
‘forward model’) and the algorithm used (section 6). We Si (Toal et al 1999) and sol–gel deposited Ta2O5 and TiO2
havealsoincludedheresomerecentdevelopmentsextending dielectricfilmsonsilicon(Cappellanietal1999);usedcom-
plementaryRBSandXPStoinvestigatetribologicalcoatings
the code to moderately rough surfaces. Another problem is
(TiN,TiAlNandMoS :Ti)onsteel(Bakeretal2000);inves-
thatweareforcedbyconsiderationofcomputationtimetouse 2
tigated low temperature growth of GaN (Young et al 2000);
asimpleforwardmodelinvolvingonlysinglescattering,and
characterized amorphous iron dicilicide (Milosavljevic´ et al
therefore multiple scattering effects cannot be fitted directly.
2001,2002);andmeasuredHandNcontentofa-CbyHi-ERD
However, we have demonstrated an effective method for
(Careyetal (2000,2001):thisworkwasdoneatRossendorf
accountingforthisandrelatedeffects(section7).
andcitedBarradasetal(2000a,b)).
We describe and discuss the intrinsic ambiguity of IBA
TheCambridgegrouphascharacterizedsiliconanodically
data and the systematic approaches to valid analysis in the
oxidized using a wave resonance plasma (WARP) source by
presenceofthisambiguity(section8). Wespecificallyrevisit
RBS (Uchikoga et al 1999), and also characterized WARP
Butler’s (1990) interesting example of ambiguity and show
deposited a-C:N:H films by He ERD/RBS (Rodil et al
howDataFurnaceisanexcellenttooltoexplorevalidinterpre-
2000). The Sussex group has profiled metal nanoparticles
tationsofthedataunderavarietyofconstraints. Wealsoshow
formed in glass by high dose implantation (Stepanov et al
anumberofexamplesusingvariousforwardrecoiltechniques.
1999, 2001a,b). Other British RBS applications include
It is one of the major benefits of this new tool that the
the characterization of gate oxides on SiGe MOSFETs
determinationoftheconfidencelimitsonthecalculateddepth
(Riley et al (1999): Liverpool/London/Southampton); and
profilescanbedoneasanaturalextensionofthealgorithm. We
the investigation of polycarbonate/polymethylmethacrylate
discusstheprecisionthatisavailableinthisway(section9).
interfacechemistry(Hutchingsetal2001).
We also discuss the use of accurate calculations of energy
The Go¨ttingen group has followed ion beam mixing
resolutionasafunctionofdepthtoenableustodoveryhigh
of Si/C multilayers (Harbsmeier et al 2000) and Ta/Si
depthresolutionanalysis(section10). Bothofthesefeatures
bilayers (Bibic et al 2000, Dhar et al 2002) with RBS
areusedintheanalysisofbothSiGemultilayersanalysedwith
and other methods. They have also characterized cubic
RBSanddeuteratedpolymermultilayersanalysedwithNRA
boron nitride (Zinin et al 2002), fluorinated amorphous
(nuclearreactionanalysis).
carbon(Ronningetal2001)andbeta-irondisilicide(Wagner
Theaccuracyofananalysis,discussedinsection11,isin et al 2002). The Darmstadt group has investigated heavy
theendthemostimportantissue. Theestablishmentofanew ion induced metal–ceramic interface diffusion with RBS
tooldependsfirstofallonbeingabletodemonstratethatthe (Nagel and Balogh 1999, Nagel et al 1999). The Chinese
answers are correct. At last a routine analysis of non-trivial group have followed the interdiffusion of superconducting
samplesisavailablewhichisbothcompleteandatstate-of-the- heterostructures(PZT/YBCO)(Xieetal1999).
artaccuracy. Accuracyofanalysisdependsultimatelyonthe
accuracyoftheforwardmodel:weendwithasubtlediscussion
2. IntroductiontoIBA
oflimitationsofthecurrentforwardmodelinthecontextofan
extraordinarilypreciseanalysisofanopticalmultilayersample
IBAisaclusteroftechniquesinvolvingmaterialsanalysisby
(section12). MeVionbeams. Whenanenergeticionstrikesatargetthere
Inconclusion,wepointoutpossiblefuturedevelopments areavarietyofenergylossmechanisms,any(orall)ofwhich
(section13). Thereviewisillustratedwithanumberofstriking can be used (together or separately) to obtain information
examplesshowingthewiderangeofsamplesthathavebeen about the target. Table 1 summarizes these techniques, and
analysed using the whole range of the IBA depth profiling indicatesthoseforwhichtheIBADataFurnacecanbeusedto
techniques. It is worth mentioning that DataFurnace is in obtain elemental depth profiles. We emphasise that the IBA
increasingusebyanalystsforavarietyofmaterialsanalyses DataFurnace software we describe applies only to thin film
andwebrieflyreviewthisworkhere. compositiondepthprofilingandnottothevariousmicrobeam
The Lisbon group have used high resolution RBS to imagingorchannellingapplications.
monitor the fabrication of MnIr spin valves with nano- IBAisnowamaturescientifictechnique,verywidelyused
oxide layers formed from plasma oxidation of CoFe inelectronicmaterialsresearch,andthroughoutthinfilmsci-
layers (Veloso et al 2000). This process is to enhance ence:themostusefulsinglevolumeoverviewsareintheHand-
the magnetoresistance for magnetic memory application. books(MayerandRimini1977,TesmerandNastasi1995). It
Zhang et al (2001) have investigated processing problems is a very large and diverse field which we do not intend to
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Table1. GlossaryofIBAtechniques.
Technique Explanation DataFurnace?
RBS Rutherfordbackscattering NuclearCoulombscattering, Yes
incidentionisdetected(Q=0)
EBS Elastic(non-Rutherford) NuclearscatteringasRBS,butwherethe Yes
backscattering Coulombbarrierisexceeded(Q=0)
ERD Elasticrecoildetection SynonymforFRS(Q=0) Yes
FRS Forwardrecoilspectrometry SynonymforERD.AsRBS(orEBS)but Yes
whererecoiledtargetatomisdetected.
Thishastobeinaforwardscattering
directionforkinematicalreasons
NRA Nuclearreactionanalysis Non-elasticinteraction(Q(cid:1)=0). Yes
Resultantreactionproductdetected
(usuallyproton,alphaordeuterium)
PIXE Particleinduced X-raysfrominelasticcollisionsofthe No
x-rayemission incidentionwithinnercoreelectrons
areeasytodetect:similarspectratoEDAX
PIGE Particleinduced SpecialcaseofNRA,whereaphotonisdetected No
gammarayemission
Channelling UsedwithanyIBAtechniqueforquantifying No
andprofilingdamageinsinglecrystalsamples
Microbeam Theionbeamcanbefocussedandscannedand Yes
usedasascanningionmicroscope. RBSand
otherspectracanbecollectedfromspecificregions
ofthesample. UsuallyusedwithPIXE
IL Ionoluminescence AsPIXE,butlowerenergyphotonsdetected No
IBIC Ionbeaminducedcurrent Microbeamtechniqueforinvestigatingsemiconductor No
devices,whichrespondtosingleionimpacts
STIM Scanningtransmission Detectsenergylossofsingleionspenetrating No
ionmicroscopy (relatively)thicktargets
review here, only making a few general comments that may An ion beam striking a solid target has many types of
be useful for readers not analysts themselves. These read- interaction,anyofwhichcanbeusedforanalysis. Weshould
ers should note that practitioners will probably quibble with pointoutthatnothingpreventstheanalystinstallingmultiple
almostallofourgeneralizationsherewhichareintendedonly detectors in the target chamber to detect any or all of these
toallownewcomerstograspIBAinaconcreteway. interactionssimultaneously.
The first of the international biennial IBA Conference
serieswasheldin1973(MayerandZiegler1973):theseare 2.1. Photonemission
nowverylargemeetings,IBA-14(Mo¨lleretal2000)had362
Ionoluminescence has the highest cross-section of all the
participantsandissupplementedbyseveralotherinternational
interactions: this is a very young field (see, e.g. Bettiol et al
conference series: both PIXE (see Malmqvist (1999)) and
(1994)andYangetal(1994)). PIXE(JohanssenandCampbell
the microbeam (see Prozesky et al (1999)) have their own
1988) typically uses a 3MeV proton beam, although helium
large conferences and there are many regional meetings.
canalsobeused. Itisdirectlycomparablewithelectronprobe
IBA is ubiquitous in the scientific thin film community: for
microanalysis(EPMA)withverysimilarspectra,exceptthat
example, IBA was used in 38% of the papers published at
there is negligible primary bremsstrahlung background due
a recent IBMM (conference on the ion beam modification
to the much higher particle mass. PIXE therefore has much
of materials: Vredenberg et al (1999)). And not only the
highersensitivitythanEPMA.Cross-sectionscanbehundreds
materials community: for example, the Louvre Museum in of barns (1barn = 10−24cm2). However, it is hard to get
Parishasitsowndedicatedaccelerator(Amseletal1990),and
depthprofilesfromPIXEspectra,andthistechniqueisnotyet
thereisextensiveinternationalapplicationofIBAtechniques
implementedinDataFurnacealthoughitisoftenaveryuseful
to cultural and archaeological artefacts (see Respaldiza and
complementarytechnique(see,e.g.Lohetal(1993)).
Go´mez-Camacho(1997)).
IBA uses incident beam energies ranging perhaps from
2.2. Elasticscattering
100keV to 200MeV. At the low energy end the detectors
become very expensive, as do the accelerators at the high RBS(Chuetal 1978)typicallyusesa2MeVHebeamwith
end: we will not attempt to summarize all the detectors and typicalcross-sectionsofabarn:butforasimilarenergyproton
accelerators in use today. IBA really started when nuclear beam the Coulomb barrier of light target nuclei (up to P)
physics groups looked for new uses for their obsolete little is exceeded and the cross-sections are no longer Rutherford.
2MVVandeGraaffacceleratorsinthe1960sand1970s,and Where the interactions remain elastic we call the method
there are still some of these machines around. The typical EBS. EBS cross-sections can exceed RBS ones by orders of
machine being installed today is a 2MV tandem machine magnitude. Of course, two body interactions between the
which can deliver multiply charged beams up to perhaps incident ion and the target nucleus will both scatter the ion
10MeV. and recoil the target nucleus into forward directions. The
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TopicalReview
recoilscarrymuchinformationaboutthesampleandcanalso 3.2. Elementaldepthprofiles
bedetected. ThisisknownvariouslyasFRSorERD(Tirara
Itturnsoutthatsimplesilicondiodesveryconvenientlystop
etal1996).
alphasofMeVenergies,convertingtheirenergyintoacloud
ofelectron–holepairswhichcanbeseparatedbytheelectric
2.3. Nuclearreactions fieldifthediodeisreversebiased. Highqualityelectronicsto
determinethenumberofelectronsarrivingattheanode,and
Where the interaction energy is sufficiently high the nuclear
hencetheenergyoftheparticle,havebeenavailableforover
structure becomes involved, the interaction may become
threedecades. Thesedetectorsareeffectively100%efficient,
inelastic, and nuclear reactions can occur: NRA is often
detecting every particle that strikes them. Thus, an alpha
very useful since it is isotope specific and there is often no
scattering experiment can determine the mass of the target
backgroundtothesignal, butcross-sectionstypicallyfallby
nucleus (from the energy of the scattered alpha, using the
ordersofmagnitude(Vizkelethy1995,Hirvonen1995).
kinematics:conservationofenergyandmomentum),thedepth
of the target nucleus (also from the energy of the scattered
2.4. Imagingandchannelling alpha, this time using the energy loss as a function of depth
in the target material), and the concentration of the target
Turning to beam and sample manipulation, microbeams as nucleus(fromtheabsolutenumberofscatteredalphas,using
smallas50nmhavebeenformed,and1µmmicrobeamsare thecross-sectionsthatRutherfordcalculated).
in widespread routine use: thus lateral maps of the major,
minorandtraceelementsofthesamplearereadilyavailable
byPIXE(seeWattandGrime(1987)). NotethatforPIXEthe 3.3. Mass-depthambiguity
excitationvolumewherethex-raysaregeneratedisessentially
RBS spectrometry can therefore, in principle, determine the
determined by the beam spot size since protons are not
completeelementaldepthprofileofasamplefromtheenergy
deflectedmuchbyelectrons. Incontrast,theexcitationvolume
spectrumofthescatteredparticles. However,thereaderwill
for x-ray analysis in the SEM is determined by the electron
havenoticedthatthescatteredparticleenergyisafunctionboth
beamenergy,andisoftenlargerthan5µmacross. Therefore,
ofthemassofthetargetnucleus,andofthedepthofthetarget
PIXEx-raymapsaretypicallyofmuchhigherlateralresolution
nucleusinthesample. Thisambiguityintheinterpretationof
thanEPMAones. Ofcourse,secondaryelectronmapsareof
thedetectedparticleenergyistheoriginoftheinverseproblem
muchlowerresolutionforPIXEthanfortheSEM.
thatisattheheartofthispaper.
Highenergyionbeamscanbeverywellcollimatedand Brice (1973) has shown that the elastic backscattering
aligned with symmetry directions in single crystal samples, spectrumhastheformofatripleintegral:
makinglatticesiteanddamageinformationavailable. Known (cid:1) (cid:1) (cid:1)
∂ ∞ t E σ(E )
as channelling, this is a very large field in its own right (see (cid:1) (E,E )=A dx(cid:3) N dx dE P P 1
Feldmanetal(1982),Go¨tzandGa¨rtner(1988)). e 3 ∂E3 0 0 e E(cid:3) 1 in outS(E1)
Other interesting emerging fields include in situ obser- (1)
vation of the electrical behaviour of working semiconductor where (cid:1) is the number of incident particles energy E
e
devicesbytheionbeaminducedcurrent(IBIC:see,e.g.Breese backscatteredwithenergyE intothedetectorfromelemente
3
et al (1998a)), and the characterization of the defect struc- in the target. N (t) is the number of scattering centres of
e
ture of thick crystalline samples using a channelled beam in element e at depth t in the target. For target atoms at the
transmission(STIM,see,e.g.Breeseetal(1998b)). surface E is given by the kinematical factor k : E = k E,
3 e 3 e
and in general E(cid:3)(E ) = E /k . E is the beam energy
3 3 e 1
before scattering for atom e at depth t in the sample. σ is
3. TheIBAinverseproblem
the differential cross-section and S = dE/dR where dR is
theaveragedistancetravelledbyanionenergyEwhilelosing
3.1. Rutherfordscattering
energydE. Thepathlengthintothetargetbeforescatteringis
In1911,ErnestRutherfordexplainedthestructureoftheatom givenbyx andthepathlengthafterscatteringbyx(cid:3). P and
in
asapositivelychargednucleussurroundedbyelectrons,where P are respectively the distributions in the projected range
out
thenucleuswastinywithrespecttothesizeoftheatom(already alongtheincomingandoutgoingdirections.
known then to be around 1Å). He used a simple Coulomb Ofcourse,themeasuredspectrumisthesumofthe(cid:1)efor
potentialandcontradictedJ.J.Thompson’s‘pudding’model. alltheconstituentelementsofthetarget. Thereisusuallyno
RutherfordbasedhismodelonGeigerandMarsden’s(1909) waytomeasureindividualpartialspectra(cid:1)e.
data on the ‘diffuse scattering’ (in backward directions) of
alpha particles by metal foils, and Geiger’s (1910) data on
3.4. Fittingbysimulation
themostprobableangleofscatteringoftransmittedparticles.
Although the most probable scattering angle is very low, Clearly,tocalculatetheinverseN((cid:1))isnoteasyanalytically
indicating a distributed charge as in Thompson’s model, and we describe a new approach to this inverse problem
backscatteringeventsimplyaconcentratedchargeatthecentre below. Themostwidespreadapproachcurrentlyisapragmatic
oftheatom(requiringtheconceptofthenucleus). Geigerand application of trial and error. The standard treatment is to
Marsden(1913)subsequentlyverifiedRutherford’scalculation assume that the target has a certain structure, to calculate
ofthescatteringprobability,orcross-section. the spectrum that would be obtained from such a target, to
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TopicalReview
comparewiththecollectedspectrumandthentoiterateuntil 4. Introductiontosimulatedannealing
a reasonable match is obtained. The skilled analyst, from
long experience and cunning, recognizes certain features of In this section, we describe the various elements of the
the spectrum and will, usually quite quickly, converge to an simulated annealing (SA) algorithm that lies at the heart of
acceptablesolution. theIBADataFurnace. Thisdiscussioniscentraltothispaper.
Thisapproachhasanumberofmajordrawbacks. First, TheessentialelementsoftheSAalgorithmwereinvented
it is entirely manual, requiring a skilled analyst to give each 50yearsagobyMetropolisetal(1953)withagroupincluding
spectrumatreatmentthatcaneasilytakehours. (Itshouldbe Edward Teller, in the context of the discussion of the use
notedthatmanytypesofspectracanbetreatedfaster,orrequire of ‘fast computing machines’ and Monte Carlo methods
only limited data to be extracted. And of course, analysts to calculate the equilibrium value of any (thermodynamic)
havebecomeadeptatthis.) Second,largenumbersofsamples quantityusingthecanonicalensembleofstatisticalmechanics.
cannotbehandledwhereafulltreatmentisneeded,sincethe They successfully calculated the equation of state for a two-
analyst simply runs out of patience. Third, because of the dimensional ensemble with these methods using the Los
difficultyofdataanalysisthetemptationfortheexperimentalist AlamosMANIACcomputer.
is to make samples especially for the analysis, instead of The fundamental SA theorem establishes that provided
analysingthesamplesthatresultfromtheexperiment. Fourth, certainformalcriteriaareobserved,thealgorithmisguaranteed
incaseswheretheexperimenterhasfailedtokeepthesamples to find the absolute minimum of any piecewise continuous
simple, the spectra can be complex so that the analyst has function. The implementation of the algorithm is always
difficulty extracting definite depth profiles from the data: in computationallyrelativelyexpensive,andarevivalofinterest
these cases the analyst may remain uncertain of the validity in it followed the explosion of computing power in the
oftheresult. Fifth,trainingtheseskilledanalystsisdifficult: 1980s. We have found the summary of Aarts and Korst
theyarealmostinvariablyatleastpostgraduatestudentlevel. (1989)useful. Ithasbeenusedtosolvepreviouslyintractable
Sixth,wehavedevelopedthe‘ruleofthumb’(roughestimate), problems,knownas‘NP-complete’,thatis‘non-deterministic
over many years of analysis, that to analyse reasonably polynomialtimecomplete’wherethecomputationtimeisnot
simple data collected over one day requires three further boundedbyanypowerofthenumberofindependentvariables
days subsequent concentrated work by the analyst. Difficult oftheproblem. Examplesoftheseare:thetravellingsalesman
data (or the inexperienced analyst) may take much longer! problem (see Press et al (1992)), and the construction of
Clearly, to fully staff an accelerator will be very expensive. automatic sentence parsers for natural language processing
Thereare,currently,fewlabsofferingafullcommercialRBS (see Wilks et al (1996)). It was actually in the context of
service. thelastapplication,serendipitously,thatwecameacrossSA.
In fact, the NP-complete class of problems has provided an
importantstimulusforthedevelopmentofSA,asKirkpatrick
3.5. Fittingbysimulatedannealing
etal(1983)pointoutinaninterestingarticlethatusesSAto
We have taken a completely different approach. At its solvethesemiconductorchiplayoutdesignproblem(aswell
heartisthesimulatedannealingalgorithm,whichisaglobal asthetravellingsalesmanproblem).
minimizationalgorithmusingcombinatorialoptimization(see
section 4). We have applied this algorithm to IBA data,
4.1. Theforwardmodel
yieldinganalgorithmicsolutionoftheIBAinverseproblem.
Notethatthemanualproceduredescribedaboveisnotatrue Wewillrepresenttheelementaldepthprofileofathinfilmas
algorithmsinceitcannotbehandledbyamachineinthegeneral a sequence of layers of specified thickness, each containing
case. Thefunctionoftheanalystisnowdifferent:insteadof certainelementsinagivenproportion. Thereareotherways
trying to guess a rough structure that may be approximately of representing profiles, but this is quite general and is the
consistentwiththedata,theanalystconcentratesonspecifying naturalrepresentationforthecalculationoftheexpectedIBA
themachinecalibrations,onspecifyinganypriorknowledgeof energy spectra. Consider an arbitrary depth profile x which
thesample,andonanappropriatemetricforthevalidityofthe is a structure {l,λ }, where the vector l is the list of layer
i ij
result. Foranygivensetofpriors(theseincludeinstrumental thicknesses,andthematrixλisthelistofr(cid:2)elativeproportions
parameters)thealgorithm,nottheanalyst,willdeterminethe oftheelementsforeachlayer(andhence λ =1). Then
j ij
depth profile. We should point out that the algorithm itself, theforwardmodelisencapsulatedinthefunctionF(x)which
aswehaveimplementedit,hasalargenumberofparameters calculatestheenergyspectrumY(E)(cfequation(1))expected
which could be chosen by the analyst: our implementation forthesamplewithdepthprofilex.
has value largely because we have shown that we have
foundvaluesofgeneralvalidityformostoftheseparameters, Y(E)=F(x)
(cid:3)
and we give the analyst clear rules for the choice of the (2a)
F(x)= f (x)
remainder. e
e
We believe that the advent of this new algorithm will
completelyrevolutionizetheapplicabilityofIBAtechniquesto wheretheforwardmodelcalculatesapartialspectrumf for
e
thinfilmanalysissinceitaddresseseachofthesixdrawbacks eachchemicalelementinthesample,andthetotalspectrumis
enumeratedaboveatafundamentallevel. Theimprovements justthesumofthepartials.
aresodramaticthatweeffectivelyhaveanewtoolforthinfilm This forward model, which is discussed in detail later,
compositionanalysis. is the essence of the calculation: without a forward model
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TopicalReview
the spectra cannot be interpreted. All existing simulation ForSAtheMarkovchainsareconstructedinaparticular
programsareessentiallysimplyimplementationsofaforward way. The state space S is the space of all possible depth
model. Weconsiderlimitationsoftheexistingforwardmodel profiles x. This is the same as the space of all conceivable
attheappropriateplaces. samples. The Markov chains are then sequences of profiles
{x ,x ,...,x }. Thegenerationfunctionforconstructingx
1 2 m i+1
4.2. Theobjectivefunction fromxi canbeanymethodofselectinganewprofilexfrom
state space. It is the acceptance criterion that gives SA its
We have to have a way of comparing some profile x with remarkableproperties.
the measured spectrum Y. There are many possibilities,
some of which we will discuss later. We could use the very
4.4. Theacceptancecriterion
simple and general χ2 function to compare the measured
n-channelspectrumY = {y ,y ,...,y }withthecalculated Aproposedstatex isaccepteddependingonhowtheobjective
1 2 n p
oneF(x)={ψ ,ψ ,...,ψ },givenby functionhaschangedfromitsvalueforthelastelementx of
1 2 n i
(cid:3) thecurrentMarkovchainunderconstruction. Weevaluatethe
χ2(Y,x)= (yi−ψi)2. (2b) change (cid:8) = O(Y,xp)−O(Y,xi) in the objective function
i impliedbymovingfromstatex tox ,andifitisimproving
i p
It is not necessary to use the χ2 function, but some such ((cid:8) < 0) we accept the state: xi+1 = xp. However, if it is
functionmustbespecifiedandthisisknownastheobjective gettingworse((cid:8)>0)wemayalsoacceptitaccordingtothe
function O(Y,x). To solve the inverse RBS problem the probabilityP givenbytheMetropoliscriterion:
(cid:4) (cid:6)
objective function has to be minimized. When O(Y,x) is (cid:8)
minimized an optimal solution x has been found: O = P ∝exp− (2e)
o min T
O(Y,x ). Note that there may be many indistinguishable
o
solutions x . Normally we are content with any particular where T is a control parameter characteristic of the Markov
o
solution x that is near an optimal solution x , but we show chain under construction. It is not hard to recognize the
o
later how to find both a most probable solution and also an similarity of this probability to the Boltzmann factor, with
estimateoftheerrorofthissolution. T analogous to temperature. Thus, for the construction of
the Markov chains used in SA the Metropolis criterion (or
somegeneralizationofit)isusedfortheacceptancecriterion:
4.3. Markovchains
this gives the Markov chains a natural thermodynamic
SA depends on the mathematics of Markov chains (see interpretation. IftheMarkovchainsarelongenoughtheywill
Gilks and Richardson (1996)). These are simply sequences reacha‘thermodynamicequilibrium’,thatis,theexpectation
M of states s selected from a state space S such that value of certain statistics will stabilize. The introduction of
s i
M ={s ,s ,...s,...s }wherethes thentryinthechain aprobabilityintothetransitionfunctionmeansthatitcanbe
s 1 2 i m i+1
dependsonlyonthesthentryandnotonanypreviousones. viewed as a Monte Carlo process, and these Markov chain
i
Moreproperly, MonteCarlo(MCMC)techniquesarecurrentlyofenormous
mathematicalinterest(see,e.g.GilksandRichardson(1996)).
P(s |s ,s ,...,s)=P(s |s) (2c)
i+1 1 2 i i+1 i
that is, the probability of si+1 given si is independent of all 4.5. Thecoolingschedule
statesbeforei.
ThefinalcomponentinSAisthecoolingschedule. Asequence
Then the Markov chain is constructed by the transition
distribution T(s) = s . This distribution is composed of of Markov chains is constructed with steadily reducing
i i+1
‘temperature’. ThentheSAtheoremstatesthattheobjective
a generation distribution from which a proposed new state
functionwillbeminimizedprovidedthateachMarkovchain
is generated, and an acceptance criterion which determines
islongenoughtohavereachedequilibrium,andprovidedthat
whetherornottoacceptthisproposedstateasthenextentry
thecoolingscheduleissufficientlyslow.
intheMarkovchain.
Barradas et al (1997) have pointed out that although S
is extremely large, it is finite in principle because there are 4.6. Summary
only 92 elements, and both sensitivity and energy resolution
SA is an algorithm for finding the global minimum of an
arefinite. Asanestimate, foranygivenbeamanddetection
objective function. The entire state space of this function is
system (cid:4)(cid:5) (cid:6) explored. Asequenceofstates(aMarkovchain)isconstructed
e 1
log|S|≈ i=1 logn (2d) in which succeeding states have an objective function that
φi is either reducing or has a Boltzmann-like probability of
where |S| is the number of states s in the space S, there are increasingaccordingtoaparameteranalogoustotemperature.
e elements in the sample, φ is the relative sensitivity for Hencetheideaof‘annealing’. AsequenceofMarkovchains
i
element i and n is the number of separable layers. Note isthenconstructedwithreducingtemperature; theendpoint
that typically no more than 50 layers can be distinguished being an optimal solution. For IBA, the objective function
(n < 50) and in each layer we can probably not determine foraproposeddepthprofileisconstructedfromthedifference
the concentration of an element at better than 1% precision between the spectrum being fitted and that calculated with a
(1/φ < 100). Therefore, theproblemmaybetreatedeither forwardmodel(astandardsimulationcode)fromtheproposed
i
asadiscreteor(sinceSissolarge)asacontinuousproblem, depthprofile. Thenthestatespaceexploredisthespaceofall
dependingwhatismathematicallymoreconvenient. possibledepthprofiles.
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4.7. Algorithmicissues
The SA theorem proves that in the general case a global
minimum in the objective function is reached in logarithmic
time,thatis,forMarkovchainkthecoolingscheduleisgiven
by a temperature sequence T = T /log(10+k), provided
k 0
thattheinitialtemperatureT issufficientlylarge(Gemanand
0
Geman 1984). Unfortunately, this implies ridiculously long
computation times. In practical cases, geometrical cooling
schedulesarealwaysused,thatisT =akT ,wheretypically
k 0
inourcasea ∼ 1. ThisimpliesthatSAisefficientforsome
2
classesofproblem,butmaybeveryinefficientforothers.
An interesting paper by Sorkin (1991) has given an
indicationofthecircumstancesinwhichSAcanbeexpected
to be efficient. He proves that for a state space with certain
(rather artificial) fractal properties, a value of the objective
functionclosetoaglobalminimumcanbefoundingeometrical
time. Butheshowsthatitislikelythatamoregeneraltheorem
forrealisticfractalstatespacesisalsotrue. Inapreliminary
analysisofthestatespaceforarepresentativeRBSspectrum
wehavefoundthat,inthevicinityoftheglobalminimumand
in certain representations (not the χ2 one described above),
the objective function has only one minimum (the global
minimum). WethereforesuspectthatIBAspectraarelikely
tohavequitewellbehavedobjectivefunctionswhichSAcan
solveeasily. Wewillreportonthisinanotherplace. Depth(1015 atoms/cm2)
To demonstrate that our SA algorithm is well behaved
Figure1. 1.5MeV3HeRBSof200keV2×1017Fecm−2
we need to show that it converges properly. Barradas et al
implantedintoSi. (a)RBSspectrumwithfit(linethroughdata
(1998e) have shown the variation of the average Si content points)andtheFe,SiandOpartialfittedspectra(otherlines).
of the last 50 elements (states x) of each Markov chain in (b)fitteddepthprofileforFe(afterBelsenetal(1999)). Probably
the cooling schedule at various depths for an annealed and theFedoesnotpenetratesodeeplyintothetarget(thereisstill
apparently‘significant’Feat24×1017atomscm−2):thisisan
oxidizedironsilicide,plottedfortheentirecoolingschedule.
ambiguityduetouncertaintyinthechargecollection. Thereis
Inthesubstrate,thesiliconcontentreachesitsfinalvalueata
noticeablesurfaceoxidationinthissample. Noteon‘thinfilmunits’
hightemperature,butnearthesurface,wheretherearemany (cfsection6,equation(3)):forsiliconofdensity2.32gcc−1or
more possible states due to the presence of lots of oxygen 5×1022atomscc−1,onethinfilmunit(1015atomscm−2)is0.2nm.
(with its small signal and consequent small influence on the
χ2),thefinalvalueisnotreacheduntiltheendofthecooling.
Marriottetal(1999)havealsodemonstratedthatourMCMC toextractfromspectralikethiseventhoughinthiscaseonly
algorithmiswellbehaved,inparticularthatitconvergesand twoelements(SiandFe)areinvolved. TheSQUEAKIEcode
‘mixes’ properly. Mixing means that the sampling process of Børgesen et al (1982) is a matrix inversion type of code
exploresallpossiblesolutionsinanefficientmanner. which reconstructs the depth profile by a closed calculation
fromtheseparatedpartialspectra,alsoshowninfigure1. In
thiscase, thepartialspectraoverlapandSQUEAKIEcanbe
5. Mixedmetalsilicides used iteratively, starting from the surface. Alkemade et al
(1990) use the same type of iterative procedure. (There are
Our first set of examples is in the silicide system. One other more general analytical approaches which we discuss
form of iron disilicide is a semiconductor, and in principle below.)
it should be possible to fabricate a heterojunction laser in Figure2isaspectrumfromamixedFeandCosilicide,
silicon. On the other hand, cobalt disilicide is a metal, and in this case with three elements we believe that no
andinterestingthree-dimensionalmetallizationschemesseem iterative procedure for SQUEAKIE exists. The kinematical
feasible. A successful fabrication method for thin films of gap between Fe and Co for these conditions is 18keV, only
eithermaterialinsiliconwaferscouldhaveadramaticimpact slightly higher than the system energy resolution of about
onsiliconprocessingformicroelectronics. Ionbeamsynthesis 15keV(equivalenttoabout30nmofSi). Therefore,onewould
ofβFeSi2 (usinghighdoseionimplantation)hasresultedin notexpecttheretobemuchdiscriminationbetweenFeandCo.
light emitting diode structures (Leong et al 1997), and the The manual method using spectrum simulation described in
examplesinthissectionaretakenfromrelatedwork. section3aboveistrickytousesinceitisnotobvioushowthe
contrast in the metal signal arises. Because of the difficulty
anduncertaintyofextractingthedepthprofilefromtheRBS
5.1. Analyticalsolutions
spectrum, cross-sectional transmission electron microscopy,
Figure1showsahighdose,highenergyFeimplantintoSisuch x-ray photoelectron spectroscopy and other techniques were
thattheFeandSisignalsoverlap. Depthprofilesarenoteasy deployedtodeterminethedepthprofile(Harryetal1996).
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TopicalReview
(a) stronglyconstrainsthepossibleaverageZ(atomicnumber)of
thesample. ThesameappliestoalltheIBAtechniquessince
theyallhavestronglyZ-dependentcross-sections.
With the manual method of spectrum simulation the
charge is often treated as a free parameter (the analyst may
simply ‘normalize’ the spectrum at an appropriate place).
However, with the algorithmic approach of SA no tacit
assumptions are allowed, and the explicit value given to the
charge·solid-angle product is a very important constraint of
thefittingprocess.
(b) 5.4. Batchanalysisbysimulatedannealing
In Barradas et al (1998b) examples are shown of data from
asetofover50samplesinvestigatingthekineticsoftheion
beammixingandannealingbehaviourofsputteredmetalfilms
onsilicon(figure3). Thesespectraarealsohardtoanalyse,
especiallysinceunwantedsevereoxidationwasexperiencedin
thefurnaceannealing. ByusingSA,reliablequantitativedata
fromthislargesetofhardspectracouldberapidlyobtained,
allowingusefulconclusionstobedrawneveninthepresence
of the oxidation (see also Milosavljevic´ et al (2001, 2002)).
With manual methods sufficiently precise results would not
havebeenobtainableandtheexperimentswouldhavehadto
Figure2. 1.5MeV4HeRBSof200keV25×1016cm−2double
berepeated.
implantsofCofollowedbyFeintoSi. Implantationtemperature
Figure 3 shows a set of spectra (left-hand side) with
wasabout350˚C.(a)RBSspectrumwithfit(——)andCoandFe
partialfittedspectra(----). (b)Fitteddepthprofile(figure3of the corresponding extracted depth profiles (right-hand side).
Barradasetal(1997).) Thesurfaceissignificantlyoxidised,butthe The spectral plots show the data and the calculated fits
Oprofileisomittedforclarity. ThedepressionintheSi (which go through almost all of the data points), and also
concentrationbetweenabout10and15×1017cm−2correspondsto the calculated partial spectra. The profiles (right-hand side)
acubicsilicide,whilebelowabout10×1017cm−2thelayerhasa
wereobtainedusingamodifiedversionofthematrixinversion
stoichiometrysimilartotheSi-richα-FeSi phase.
2
code (SQUEAKIE) of Børgesen et al (1982) which can be
applieddirectlysincethepartialspectraareavailableoncethe
5.2. Simulatedannealingsolution SAcalculationiscomplete. Thelayerstructuredeterminedby
SA is shown by the histogram lines in the top depth profile
It is because the spectrum of figure 2 is very contrasty, with
(figure3(a),right-handside).
many strong edges and peaks, that it is ideally suited to
SA: even a rapidly prototyped SA code was able to find
a solution very rapidly, taking only minutes on a 100MHz 5.5. Discontinuityofsimulatedannealingsolutions
486 PC (Barradas et al 1997). This example was the first
Figure3raisestheinterestingquestionofwhichrepresentation
demonstrationthatSAwasapowerfulalgorithmforsolving
is more fundamental, the discontinuous layer structure
theinverseRBSproblem.
determined by SA, or the continuous profile determined by
subsequently displaying the partial spectra calculated by SA
5.3. Sensitivitytocollectedcharge asconcentrationversusdepth,usingSQUEAKIE?Clearly,the
realdepthprofileiscontinuous,soexperimentalistsmayprefer
In Barradas et al (1998a) the data of Harry et al (1996) to see the inverted partial spectra rather than the raw layer
are re-analysed by SA, this time each fit taking about 2min structure. However,althoughSQUEAKIEhastheadvantage
on a 200MHz Pentium PC. The optimization of the cooling thattheextracteddepthprofileusesallthespectraldatainan
scheduleisdiscussed,togetherwithademonstrationthatinthis (nominally)internallyconsistentway,thematrixinversionstill
case,withcontrastfrombarelyseparatedmasses,theFe:Co hasanumberofproblems.
ratioisextremelysensitivetoerrorsinthechargecollection. First, the partial spectra are completely dependent on
A1%errorinthechargegivesa20%errorintheFecontent. SA. Unavoidable statistical error in the estimation of the
Doubtless a careful analytical procedure might increase the layer structure will cascade into the partial spectra and the
precisionofthistypeofmeasurement,butno-onehasyetgiven profiles calculated from them. Moreover, the energy loss
acriticaldemonstrationofchargecollectionbetterthan1%. variation with depth used by the matrix inversion code is a
We will return repeatedly to the issue of instrumental function of the composition, where the layer structure is
parameters and the determination of analytical errors, here determinedbySA.Therefore,theenergylossisanunavoidably
weemphasisethatthecollectedchargeisacriticalparameter discontinuousfunctionofdepth. Second,aniterativeprocess
in the interpretation of IBA data. This is because for RBS ofusingtheinvertedprofilesasthebasisforanewproposed
spectra the cross-section is proportional to the square of the layer structure is not simple (and may be not possible) to
atomicnumber,sotheabsolutenumberofcountscollectedvery implementsinceasthecodestandstheinvertedprofilestake
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TopicalReview
(a)
(b)
(c)
(d)
(e)
Figure3. RBSdatawithsimulatedannealingfits(left),andcorrespondingderiveddepthprofiles(right),ofsputterdeposited120nmFe
layersonSiionbeammixedwitheitherAsorXeandannealedat900˚Cfor150min. Intheleftcolumn,thesolidlinesarethefittedspectra,
andthepartialO,SiandFespectraareshownasdashedlines. Intheright-handcolumnthepartialspectraextractedwithDataFurnaceare
replottedas‘depthprofiles’usingSQUEAKIE(Børgesenetal1982). Forclarity,onlyonefitteddepthprofileisshownasahistogram(top
spectrum,right-handsidecolumn). Noticethedifferentdepthscaleintheright-handsidecolumnofe). Figure2ofBarradasetal(1998b).
noaccountofthedetectorresolutionorthevariationofenergy Tosummarize:wecanreplotthespectrumpointbypoint
resolution with depth (energy straggling). In fact, Børgesen onanelementalconcentrationversusdepthscale,butthispro-
etalspecificallypointoutthattheiralgorithmwillnothandle cedureiscarriedoutoneachpartialspectrumindependently
energy straggling (or the surface signals) correctly. Third, withoutattemptingtoforceself-consistencybetweenthevar-
theinversionnecessarilyassumestheaccuracyoftheforward iousreplottedpartialspectra. ItisdoneonlyafterSAdeter-
model,buterrorsinthestoppingpowerdatabase(whichcan minesavalidstructure(depthprofile)forthesample,andusing
beatthe10%levelasdiscussedindetailbelow)meanthatthe thedepthdependentenergylosscalculatedfromthisstructure.
depthsofinterfacesmaybeinconsistentlyestimatedbetween The plots obtained take no account of energy resolution and
thedifferentelementsinthesample. Wehaveseenverylarge straggling,buttheyareveryusefulinsomecircumstances.
artefactsdemonstrablyduetothiseffect.
We have therefore concluded that the layer structure
5.6. Occam’sRazor
determinedbySAisthemorefundamental,andfortheseand
otherreasonswenolongerusetheBørgesenmatrixinversion We have up to now invoked the principle of Occam’s Razor
code (SQUEAKIE). As we mentioned above, the apparent (non sunt multiplicanda entia praeter necessitatem, that is,
advantage of SQUEAKIE is that it nominally imposes self- entitiesarenottobemultipliedexceptofnecessity; inother
consistencyonthedepthprofilesobtained. Butsincewecan words,‘minimizeyourassumptions’)instatingthatwechoose
demonstrate that actually these profiles are not internally the layer structure with the fewest layers to represent the
consistentingeneralwearebetteroffusingasimplerinversion sample. See Garrett (1991) for an interesting application of
algorithm that does not claim self-consistency, treating each Occam’s Razor to Bayesian probability. Thus, we make the
partial spectrum independently, while using a correct depth layer structure as coarse as possible consistent with the data
scalecalculatedfromthefittedstructure. (thecalculationmethodisdescribedinBarradasetal(1999b)).
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TopicalReview
Notethatthequalityoffitobtainedisoutstandinglygoodeven Børgesen et al’s matrix inversion code (‘SQUEAKIE’,
with these very discontinuous solutions. From an analytical 1982)isamathematicalattempttosolvetheinverseproblem
point of view it serves to emphasise that we find an optimal which we have already described (section 5) and which has
solution;thesolutionisconsistentwiththedata,leavingopen severelimitations. Asimilarbutmuchmoregeneralapproach
the question of what other different structures may also be hasbeentakenbyKoganetal(1994)basedonthereduction
consistentwiththedata,andhowdifferenttheymightbe. We method for ill-posed problems developed by Pyt’ev (1983).
returntothesequestionslater,butwenoteherethatitisclear Pyt’ev’smethodisforlinearcasesbutKoganetalhavegen-
thatsomesortofcontinuityconditionoughttobeappliedto eralizedtheirtreatmentforthisnon-linearproblem. Kogan’s
thistypeofdataandwehavenotyetsatisfactorilydetermined ‘BeamExpert’doesappeartobetrulygeneralpurpose,thatis,
howsuchaconditioncouldbespecified. itiscapableofinvertingspectrawherethepartialspectraover-
laporwherethereisnouniquesolution. However, itisalso
notclearhowrobustitistoproblemsoftheforwardmodel,in
5.7. RBSbytheunskilled
particularthoseduetoerrorsintheenergylossdatabase. Nor
Finally,returningtofigure1,thisspectrumwasfromoneofa isitclearwhetherornotitisrobustinuse,orwhetherithas
setoftwelvesamplescreatedinanexperimentalset-upasan instabilitiessimilartothoseexhibitedbySerruys’code.
exerciseforagroupofschool-leavers(18-year-olds)aspartof Difficulties and shortcomings of this type of inversion
anationalschemetogivethisagegroupatasteforuniversity code have been comprehensively and elegantly described
research. Theseyoungpeoplehadapproximatelyfourhours by Cumpson (1995) for the (related) technique of angle-
traininginRBSdatacollectionanddatareductiontechniques. resolved XPS. Cumpson also compares the properties of
Thedatawerecollectedforthemandtheanalysiswasset-up, inversionandmaximumentropyalgorithms(wedescribethe
andthenthey,essentiallyunaided,usedtheDataFurnacesoft- MaxEnt code of the Garching group below, section 9), and
waretoextractdepthprofilesfromthewholedataset,writing he comments on the usefulness of the MaxEnt approach,
up the results (including a full presentation of the methods, especially when uncertainty estimates are required. Our
aimsandcomplementaryopticalresults)inasingleday. This approachismathematicallycloselyrelatedtotheMaxEntone.
expressesexactlythebenefitsofusingthisnewsoftwaretool
forsolvingRBSspectra. Thedataturnedouttobeveryinter-
6.3. TheRBSforwardmodel
estingandweresubsequentlypublished(Belsonetal1999).
The RBS forward model is straightforward to calculate
numerically. The sample is divided into thin layers and the
6. Theforwardmodel scattering of the incident beam (of energy E ) from each
0
layer is considered. Then the energy of the beam E before
q
In this section, we explicitly state the forward model that scatteringaftercrossinglayerq isgivenbyanintegrationof
we use, and also describe the code structure of the IBA theenergylossesontheinwardpathtothelayer:
DataFurnace,andanextensiontomoderatelyroughsurfaces.
Wealsobrieflyreviewothersimulationandfittingcodes. (cid:3)q−1
E =E −secθ tε(p,E) (3)
q 0 1 i i i
i=0
6.1. Simulation
where θ is the incident beam angle from normal, t is the
1 i
All IBA labs have simulation codes to calculate spectra to
thickness of the ith layer (in thin film units, that is, in
comparewithcollecteddata:theseareequivalenttowhatwe µgcm−2 or equivalent units), and E is the incident beam
0
have called the forward model and have long been available
energy. ε(p,E) is the energy loss (per unit depth, in thin
(Brice1973). Thereareanumberofcodes,withvarioususeful i i
film units) in the ith layer which(cid:2)has a composition given
userinterfacefeatures,whichhavebeenwellreviewedbyKo´tai by p = {p ,...,p } where e p = 1 and there
(1994). TheDataFurnaceisnotasimulationcode;itisafitting i 1,i e,i j=1 j,i
are e elements. The number of scattering centres C of
e,i
code,incorporatingasimulationcodeinitscore. Itreplaces element e in layer i is then given by C = tp. Note that
all existing single scattering simulation codes (we discuss this structure {t,p } has the same cheimicali eilements but
i ij
multiple scattering codes below in section 7). Because the
a different (larger) number of layers from the depth profile
forwardmodelisexecutedrepeatedlyforeachfititisoptimized x={l,e }definedpreviously. Thelayerthicknessest must
i ij
forspeed. However,someimprovementofthealgorithmmay bechosensmallenoughthatε(Ei−1) ∼= ε(Ei)otherwisethe
beobtainableusingSerruys’retrogrademethod(1991).
numericalintegrationwillcumulateerrors.
The energy loss ε is calculated from a database of the
6.2. Fitting energylossofionsinmatter(Ziegleretal1985)thatextends
acrosstheentireperiodictableforbothbeamandtarget,andfor
Ko´taidoesnotmentionSerruysetal’sinteresting‘PERM’code
allbeamvelocitiesaboveaboutthatof50keVH.Theenergy
(1993)whichisatruefittingcodebasedonanovelsimulation
lossinanytargetcanbecalculatedbylinearsuperpositionof
algorithm (Serruys 1991) and a novel and efficient spectrum
elementalenergylosses. Thus
smoothingalgorithm(Serruys1990). However,Serruys’code
isnotatruealgorithm,relyingasitdoesbothonareasonable (cid:3)e
initialguessbytheuserandalsoonsomeguidancebytheuser ε(p,E)= p ε(j,E) (4)
i i ij i
duringexecution. j=1
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