Proposed El Lip Some Try Experiment Vander Slice

7/29/2019 Proposed El Lip Some Try Experiment Vander Slice

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NullEllipsometry:ABasicExperimentalProcedure

Submittedto:

Dr.StephenDucharme

ProfessorofPhysics

UniversityofNebraska-Lincoln

Submittedon:

May5,2010

Submittedby:

JeremyVanDerslice


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Introduction to Lasers and Modern Optics

University of Nebraska Lincoln

J e r e m y V a n D e r s l i c e

Purpose

Tolearnthefundamentalsofnullellipsometry

Introduction

Ellipsometryisanextremelysensitivetechniqueusedtoanalyzethepropertiesofsurfacesorthinfilms.

Ellipsometrycanbeusedtofindtheindexofrefraction,extinctioncoefficient,reflectivity,orthickness

ofasurfaceorthinfilm,amongotherthings.Inbasicterms,ellipsometrymeasuresthechangein

polarizationthatoccursaslightisreflectedoffofatargetsurface.Withthisinmind,ellipsometrycould

verywellbecalledpolarimetryexceptthatadifferingtechniquenotrelatedtoellipsometrywasalready

coinedpolarimetry.Althoughitcanbedisputedwhoactuallyinventedellipsometry,PaulDrudederived

thefundamentalequationsofellipsometryintheearly1900sandbasedonhisworkAlexanderRother

thencoinedthetermellipsometryinajournalarticlein1945.Thenamestemsfromthefactthatthe

mostbasicformofpolarizedlightiselliptic.[1]

GeneralEllipsometryTheory

SnellsLaw

Todefinethegeneralequationofellipsometry,manykeydefinitionsmustfirstbediscussed.First,

incidentlighttransmittedandreflectedatamediumisdefinedtoactinaplaneofincidence.Theplane

ofincidenceisshowninFigure1.

Figure1.Incident,reflected,andtransmittedlightintheplaneofincidence.[1]


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Snellslawdefineshowlightinteractswithamediumasitpassesthroughit.Snellsrelatesthecomplex

indexofrefractiontotheanglesofincidenceofthemedium.Theindexofrefractionisacomplex

argumentdefinedby:

=+

Wherenrelatesthespeedoflightinthemediumtothespeedoflightinavacuum, =.Theextinction

coefficient,k,isdefinedthroughtheabsorptioncoefficient()whichdescribeshowquicklythe

amplitudeofthelightwavedecreasesasittravelsthroughamedium,

=,,=4

Thus,Snellslawcanbeproperlydescribedbythefollowingequation,

11=22

Where1istheangleofincidencemeasuredfromtheboundaryunitnormaland 2istheangleof

transmissionalsomeasuredfromtheunitnormal.Foradielectricmaterialinwhichnolightisabsorbed

thecomplexportionoftheargumentiszero,thus,forthissimplifiedcase,Snellslawcanbeexpressed

as,

11=22

However,forthesakeofthisderivingthegeneralellipsometryequationsonlythemostgeneralcasewill

beconsidered.

FresnelsEquations

Ellipsometryinvestigatesthephasechangeofthesecomponentswhenreflectedoffasurface.Ingeneral

thephasechangeisnotthesameforbothcomponentsandtheresultisellipticallypolarizedlight.The

ellipticallypolarizedlightcanbedefinedbytwocomponents,p-waveands-wave.Thep-wavesis

definedasthepolarizedwaveparalleltotheplaneofincidencewhiles-waveisdefinedasplane

polarizedwaveperpendiculartotheplaneofincidence.

Fresnelreflectioncoefficientsdefinetheamplitudechangebetweentheincidentandreflectedwavesat

aninterface.Theyaredefinedbythefollowingexpression,

12=211221+1212=112211+22

Where1and2areshowninFigure2.

AzzamfurtheredtheunderstandingoftheFresnelreflectioncoefficientsbydefininghowtheamplitude

betweentheincidentandreflectedwaveschangewhenthinfilmsarepresent.Hedefinedanoptical

thicknessasthedistancethewavestravelthroughthethinfilmandthenintegratedthesefindingsto

withtheFresnelreflectioncoefficientsasfollows,


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=12+2321+12232=12+2321+12232,

=222

Whereistheopticalthickness,disthethicknessofthethinfilm,andisthewavelengthofthelight

source.

Figure2.Reflectionandtransmissionoflightatthinfilmboundary.[2]

GeneralEllipsometryEquation

Incidentlightreflectingoffasurfaceundergoesachangeinphase.Let 1describethephasedifference

betweenthes-andp-wavesfortheincidentlightand 2describethephasedifferencebetweenthes-

andp-wavesforthereflectedlight.Now,letdescribethechangeinphasebetweentheparalleland

perpendicularcomponentsoftheincidentlightthatoccurswhenitisreflectedfromasurface,

=12

Aspreviouslydiscussed,theFresnelequationsdefineamplituderatiosandcanbeuseddefinethe

changeofamplitudeofthelightwavesastheyarereflected,

=

Anewterm,,hasbeenintroducedtoquantifytheamplitudechangethatoccursuponreflection.Usingthepreviousdefinitions,thefundamentalequationofellipsometrycannowdefinedas,

=

Thus,thethicknessandcomplexrefractionofathinfilmcanbefoundifthecomplexrefractionofthe

substrateandtheoutsideenvironment(usuallyair)areknown.[2]


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NullEllipsometry

Thepreviousderivationofthefundamentalequationsofellipsometryisvalidforalltypesof

ellipsometry.However,inthisexperimentonlynullellipsometrywillbeinvestigated.Nullellipsometryis

oneoftheoldestformsofellipsometryanditisstilloneofthemostaccurate,evenwhendone

manually.However,itisoftenveryslowandthecalculationsrequiredtoextractandcanbeverytediouswhendonebyhand.

ThetypicalsetupofanullellipsometerisshowninFigure3.ItisknownasthePCSAconfigurationor

polarizer-compensator-sample-analyzerconfiguration.Therearemanydifferentwaystosetupanull

ellipsometer,includingPSCAandPSA,butPCSAisthemostcommon.ThePCSAconfigurationcanbe

usedfairlysimplytoobtainthetwoellipsometricparametersforthesample.Thebasicprocedure

consistsoffindingcomponentsettingsthatextinguishthelightatthedetector.Theazimuthsofthe

polarizerandcompensator(Po&Co)areadjustedtomaketheellipticallypolarizedlightreflectoffthe

sampleaslinearlypolarizedlight.Thereflectedlightisthenextinguishedbyasuitableanalyzerangle,A o.

Assumingthelightiscompletelyextinguished,thecomplexreflectanceratiocanbedefinedas

=ctanPoCo+tanCotanPoCotanCo1tanAo=

Wherethetransmittanceratioofthecompensatorisdenotedby c.Itisevidentthatthisisafairly

complexequationbutinthisinvestigationasimplificationwillbeusedinwhichthecompensatorwillbe

heldconstantataquarter-wave,Co=/4.Withthissimplificationitisknownthatthereflectedlightcan

beextinguishedwithtwodifferentsettingsP1andA1,andP2andA2.Thesesettingsareknowntobe

relatedtoeachotherbyP2=P1andA2=-A1.Inaddition,thesemeasurementsarerelatedtothe

ellipsometricanglesby,

=1=21+3210(1)

=1=21+21


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Figure3.Polarizer-Compensator-Sample-Analyzerconfigurationofanullellipsometer.[3]

Oncetheellipsometricanglesareknown,theextinctioncoefficient,indexofrefractionandreflectivityof

thesamplecanbecalculatedfromDrudesapproximationequations[4]

=tan2(3)

=2212(4)

=(1)2+22(+1)2+22(5)

ExperimentalProcedures

Equipment

Thefollowingequipmentwillbeusedforthisexperiment:Highpressuremercurylamp,5461green

linefilter,Gaertnerspectrometer,rails,adjustablesamplemount,polarizer,analyzer,Soliel-Babinet

compensator,Gaussianeyepiece,photomultipliertube,picoammeter,acetone,methanol,anda

crystallinesiliconwafer.

SetupandCalibration:

FastensampletoGaertnerSpectrometertableusingopticalplatemount,rails,andadjustablesamplemount.Placehighpressuremercurylampwithfilteratthecollimatorslit.UsingtheGaussianeyepiece,

adjustthetelescopeontheGaertnerSpectrometeruntilthecross-hairsarecenteredhorizontallyand

verticallyontheilluminatedgreenrectangleproducedbytheslitonthecollimator.Adjustthevertical

tiltonthetelescopeifneeded.Turnthelowerrotatabletable,showninFigure4,untilthe0 oline

matchesupwiththe180olineofthetelescope.Placethesampleontheadjustablesamplemountthat


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wasconstructed.Carefullycleanthesamplesurfacewithacetoneandthenmethanoltoremoveorganic

buildup.Slowlyslidethesamplealongtherailuntilitiscenteredbetweenthecollimatorandtelescope.

Figure4.EssentialcomponentsofPCSAnullellipsometrywithGaertnerSpectrometer.

Placethepolarizerontheendofthecollimatorandsetitto0 o.AdjustthetelescopetosiliconsBrewster

Angle(angleofincidenceof152.6o).Rotatetheentirepolarizer(whilekeepingthepolarizersetto0 o)

untilthephotomultipliertubedetectsaminimumamountoflight.Thepolarizerisnowknowntobe

orientedsuchthatthe0osettingproduceslightpolarizedvertically.Now,settheangleofincidenceto0 oandplacetheanalyzerontheendofthetelescopeandsetitorthogonaltothepolarizer.Rotatethe

entireanalyzer(whilekeepingitorthogonaltothepolarizer)untilmaximumextinctionisfoundand

tightenitdown.Attachthecompensatortothespectrometer.Withthepolarizerat+45 oandthe

analyzerat-45o,rotatethecompensatoruntilmaximumextinctionisfound.Adjusttheretardationon

thecompensatorbyturningtheknobtocorrespondwithaquarterwavephaseshift(1.33turns)as

givenbyTable1.

PhotomultiplierTube

Telescope

LowerRotatableTableUpperRotatableTable

Analyzer

Compensator Polarizer

Collimator

AdjustableSlit


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Table1.Soliel-Babinetcalibrationguidetosettheretardanceofthecompensator.Notek=0.987

[AppendixB]

Procedure

Settheangleofincidenceto170o

byrotatingthetelescope.Next,rotatethepolarizeruntilmaximumextinctionisfound.Then,adjusttheanalyzertofurtherreducethesignalproducedbythe

photomultipliertube.Continuetoadjusttheanalyzerandpolarizeruntilthemaximumextinctionis

found.Recordthepolarizerandanalyzerazimuthsettings.Next,adjustthepolarizerandanalyzerto

allowformaximumtransmission.Now,adjustanalyzerformaximumextinctionandthensequentially

adjustthepolarizerandanalyzerformaximumextinction.Iftheazimuthsoftheanalyzerandpolarizer

aredifferentthanthefirstmeasurement,recordthevalues.Iftheyarethesame,adjustthepolarizerto

anewlocationandbeginagainbyadjustingtheanalyzerformaximumextinction.Oncebothnull

positionsarefound,reducetheangleofincidenceby10 oandrepeatmeasurementsforanglesof

incidencebetween170oand60oat10oincrements.Sortthedataintotwoconsistentdatasetsandplot

thepolarizerandanalyzeranglesasafunctionofangleofincidence.

ReferencesandAdditionalReading

[1]Santer,S.(2004,July20).Ellipsometry.Retrievedfromhttp://www.imtek.de/cpi/images/images/vl-

ofa/OFA-elli.pdf

[2]Tompkins,Harland,&McGahan,William.(1999).Spectroscopicellipsometryandreflectometry.New

York:Wiley-Interscience

[3]Tompkins,Harland,&Irene,Eugene.(2005).Handbookofellipsometry.WilliamAndrew.

[4]Monk,G.S..(1963).Light:principlesandexperiments.NewYork:DoverPublications.

[5]Azzam,R.,&Bashara,N.(1987).Ellipsometryandpolarizedlight.NorthHolland.

*SeeAppendixforcriticalinformationaboutGaertnerNullEllipsometerEquipment


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CommentsforDr.Ducharme

Followingtheprocedurethatwaspreviouslyoutlined,rawdatawascollectedasshowninTable2.The

datacanbecomparedtotheexpectedresultsrememberingthatthetwonullpositionsforeachangleof

incidenceshouldhavethefollowingrelationship,P2=P1andA2=-A1.Itisquicklyevidentthattheraw

datacollecteddoesntdirectlyfollowtheguideline.

Null1 Null2

2 Analyzer Polarizer Analyzer Polarizer

170 85 62 41 66 51

165 82.5 75 54

160 80 74 31 82 60155 77.5 80 22 88 79

152 76 79 3 88 90

150 75 78 7 88 79

145 72.5 74 24 88 73

140 70 72 32 82 61

130 65 67 37 74 51

Table2.Rawdatacollectedfromanalyzerandpolarizerazimuthmeasurements

Therawdatashowsthat1!2and2!901.Theanalyzerazimuthmeasurementisthe

complementaryangletotheonethatisdesiredthus,1=901.

Aftermakingtheadjustment,2=901and2=901sotherelationshipbetweenthetwonull

setsofthemeasureddataisslightlydifferentthanthatgivenbyH.Tompkins.Thisdiscrepancyisstill

underconsideration,butitisthoughtthatthedifferencehassomethingtodowiththescaleonthe

polarizers.Althoughtheadjustmentsaresimple,thechangesaremadeandshowninTable3forclarity.

Null1 Null2

2 Analyzer Polarizer Analyzer Polarizer


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170 85 28 41 -66 -51

165 82.5 -75 -54

160 80 16 31 -82 -60

155 77.5 10 22 -88 -79

152 76 11 3 -88 -90150 75 12 7 -88 -79

145 72.5 16 24 -88 -73

140 70 18 32 -82 -61

130 65 23 37 -74 -51

Table3.Dataaftermodificationsaremadetopreparethemforuseinellipsometricequations

Theanalyzerandpolarizerazimuthmeasurementsareplottedasafunctionofangleofincidencein

Figure5.Noticethattheprincipalangleofincidenceislocatedatthedipinthecurves.Theazimuth

measurementoftheanalyzeratthislocationistheprincipalangleofazimuth.Thisisthelocationand

measurementthatwillbeusedlaterincalculations.

Figure5.Analyzerandpolarizeradjustedazimuthmeasurementsasafunctionoftheangleofincidence.

Now,theextinctioncoefficient,indexofrefraction,andreflectivitycanbeextractedfromthedatausing

equations(3),(4)&(5).Althoughitseemsthatanyangleofincidencecouldbeusedtoextractthese

properties,theangleofincidencenearesttoBrewstersangleischosenbecauseitprovidesthegreatest

changeinpolarizationsothemeasurementcanbefoundwithlesserror.Thecalculatedextinctioncoefficient,indexofrefraction,andReflectivityarecomparedtovaluesgivenbytheSOPRAN&K

DatabaseinTable4.

-90

-70

-50

-30

-10

10

30

50

65 70 75 80 85

Pola

rizer,

Analyzer

[deg]

AngleofIncidence[deg]

Null1-A

Null1-P

Null2-A

Null2-P


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SOPRAN&K

Database

Measured:Null

location1

Rel.

Error

Measured:Null

LocationII

Rel.

Error

k 0.0415 0.404 10x 0.067 68%

n 4.099 4.254 3.8% 3.901 4.8%

R .392 0.452 15.3% .400 1.9%

Table4.Extinctioncoefficient(k),IndexofRefraction(n)andReflectivity(R)asgivenbytheSOPRAN&K

databaseincomparisontovaluescalculatedfrommeasurements.

Therelativeerrorinthecalculatedextinctioncoefficientsimmediatelydrawsconcern.However,itwas

foundthattheextinctioncoefficientisextremelysensitivetotheazimuthoftheanalyzer.Forinstance,

theNull2analyzerazimuthreadingattheprincipalangleofincidenceis-88 o,whichleadstoanerrorof

68%.Ifthereadingwouldhavebeenonly1odifferent(-89o)theerrorwouldhaveonlybeen15%.Even

thoughtheindexofrefractionandreflectivityarefunctionoftheextinctioncoefficient,thedependence

isweak.Thehugereductionoferror(concerningthehypotheticalcasethat=-89 oratherthan-88o)in

theextinctioncoefficientonlyimprovestheerrorintheindexofrefractionby0.2%andtheerrorinthe

reflectivityby1.4%.ThisexplainshowtheindexofrefractionandreflectivityfortheNull1datasetare

reasonablyclosetoacceptedvalueseventhoughtheextinctioncoefficientisoffby1000%.

Throughouttheexperimentitwasfoundthatobtainingprecisemeasurementswiththepolarizerand

analyzerisfairlydifficultleadingtomuchlessprecisioninthenullellipsometryexperimentthan

advertisedinliterature.Whenreadingthroughsomeliteratureonnullellipsometry,itwasfoundthatamanualnullellipsometertypicallyusepolarizersthatareaccuratewithin0.01o.Thepolarizersusedin

thisexperimentwereaccurateto0.5owhileactuallyachievingthisaccuracywasoftendifficult.With

patienceitwasfoundthatthetoleranceontheazimuthmeasurementswasnolessthan2.5 o.Even

withthislackofprecision,valueswereobtainedthatcomparedreasonablywelltoacceptedvalues.


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Conclusion(PersonalNotes):

Mymethodforfindingbothnullpositionsforeachangleofincidenceisnotanexactscience;however,it

seemedthattheorderinwhichthepolarizerandanalyzerwereadjustedcouldinfluencethenullposition

thatwasfound.ThisistheprocedureIoutlined,butIadmitthatitdidntalwaysworkwellforme.Im

slightlyperplexedbyafewthingsintherawdata,namely,therelationshipbetweenthetwonullsets.I

foundthetheoreticalrelationshipgivenbyTompkinsdoesntdirectlymatchwhatIhave.Ioftensawin

theliteraturethatpeopleuseananalyzerthathasarangeof-180oto180

oandapolarizerthathasa

rangefrom0oto360

o.ThepolarizerandanalyzerthatIusedrangedfrom0

oto90

o.Ithinkthisdifference

inscalecamouflagesmydatatolookdifferentthanwhatTompkinspredictswhileitisactuallythesame.

Overall,thiswasafairlycomprehensiveexperimentcoveringarangeofthingsthatwerelearnedduring

thelabbutitreallyhelpedtosolidifytheconceptswelearnedduringthepolarizationexperiment.

AssumingthatIveexplainedmyprocedurewellenoughtoreproduce,Ithinkthisentireexperimentcould

beeasilyduplicatedina2-weeklab,thefirstweekprobablybeingsetupandcalibrationandthesecond

weekdatacollection.

CommentsontheClass

Theclasswasbothinterestingandchallenging.Ireallyappreciatedthefactthatitistaughtbya

professorratherthanaTA,whichissooftenseenintheengineeringcollege.Youbroughtalotof

enthusiasmaboutthesubjecttotheclassandthatmadeworkingalittlehardertolearnthingsalot

easier.Thanksfortheextratimeyouspenthelpingmegetthisextraexperimentgoingaswellasallthe

rest.Haveagoodsummer.

Regards,

Jeremy


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Appendix

Proposed El Lip Some Try Experiment Vander Slice

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