Published in Journal of Applied Physics 88, issue 1, 148-160, 2000which should be used for any reference to this work

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Optical absorption and light scattering in microcrystalline silicon thin filmsand solar cells

A. Poruba, A. Fejfar,a) Z. Remes, J. Springer, M. Vanecek, and J. KockaInstituteof Physics,Academyof Sciencesof the CzechRepublic,Cukrovarnicka´ 10, CZ-16253Praha 6, CzechRepublic

J. Meier, P. Torres, and A. ShahInstitut de Microtechnique,Universitede Neuchaˆtel, rue Breguet2, CH-2000Neuchaˆtel, Switzerland

Optical characterizationmethodswereappliedto a seriesof microcrystallinesilicon thin films andsolarcells depositedby the very high frequencyglow dischargetechnique.Bulk andsurfacelightscatteringeffects were analyzed.A detailed theory for evaluation of the optical absorptioncoefficienta from transmittance,reflectanceandabsorptance~with thehelpof constantphotocurrentmethod! measurementsin a broadspectralregionis presentedfor thecaseof surfaceandbulk lightscattering.The spectral dependenceof a is interpretedin terms of defect density, disorder,crystalline/amorphousfraction and material morphology. The enhancedlight absorption inmicrocrystallinesilicon films andsolarcells is mainly dueto a longeroptical pathasthe resultofan efficient diffuse light scatteringat the texturedfilm surface.This light scatteringeffect is a keycharacteristicof efficient thin-film-silicon solarcells.

t

I. INTRODUCTION

Hydrogenatedmicrocrystallinesilicon (mc-Si:H) depos-ited by very high frequency glow discharge ~VHF-GD!methodshasbeenintroducedasa newthin film photovoltaic~PV! material1 and leadsto stablesingle-junctioncells withover8% efficiencyandover25 mA/cm2 short-circuitcurrentdensity.2 The stable efficiency of the tandem micro-crystalline–amorphoussilicon ~‘‘micromorph’’ ! cells, pro-ducedby VHF-plasmadeposition,reached12%.3 Similar re-sults were reportedrecently by other groups.4,5 High effi-cienciesarepossiblethanksto a goodpassivationof defectsby hydrogenin themc-Si:H plasmagrowthprocessanddueto an enhancedoptical absorptionleading to efficient sun-light absorptionin just 1–4 mm thick films andsolarcells.6,7

The importantkey for the successof the mc-Si:H as aPV absorbentmaterial is its enhancedabsorptioncomparedto themonocrystallinesilicon.Therehasbeena discussioninthe literaturewhetherthe observedabsorptionenhancemenshouldbe attributedto high internal surfacesof small crys-tallites, internalstress,amorphousfraction in the films or tothelight scatteringeffects.6,8–10 Recentlywehaveshownthatthe absorption enhancementis mainly due to the lightscattering7,11 andwe havesuggesteda procedureto identifythe influenceof scatteringand to extractthe ‘‘true’’ opticalabsorptioncoefficienta(E) as a functionof photonenergyEfrom the experimentallyobservedspectra.

This procedureis important,especiallyfor understand-ing the subgappart of the optical absorptionspectrarelatedto thedefectstateswithin theenergygap.A low defectden-sity is anotherprerequisitefor an efficient microcrystalline

a!Author to whom correspondenceshould be addressed;electronic mail:fejfar@fzu.cz

solarcell. In mc-Si:H thesubgapabsorptionis howevertypi-cally very low andsowe needa methodwhich measurestheabsorptancedown to valuesof 1026. Both the photothermaldeflectionspectroscopy~PDS! andtheconstantphotocurrentmethod ~CPM!, well known from the field of amorphoussilicon, can be used.However, both methodsresult in an‘‘apparent’’ optical absorptioncoefficient aapp affectedbyscatteringeffects.

In this studywe will discussthe influenceof light scat-tering on CPM absorptionspectra.CPM detectsthe lightabsorbed~either directly or after one or more scatteringevents! inbetweentwo metal electrodesusedfor the photo-current measurement.Dimensionsof the gap influencethecontributionof the scatteredlight to the measuredphotocur-rentand,thus,to theaapp. We will presentherea procedureon how to extractfrom the measuredaapp both the true op-tical absorptioncoefficienta true(E) and the contributionoflight scattering.

In the caseof a very weak volume scatteringwe havepresenteda theory for the evaluationof the true optical ab-sorption coefficient a true(E).12 In this study we presentadetailedtheory for the evaluationof a true(E) in scatteringmc-Si:H samplesin the subgapandabove-gapregionsfromspectrophotometricmeasurementsof transmittanceT(l) andreflectanceR(l) and from the absorptanceA(l) measuredby CPM. Both scatteringeffects, i.e., at the surfaceand inthe bulk of the film, are includedin our analysis.We verifyour theory with the experimental results for differentmc-Si:H films. We show how the spectraldependenceofa true(E) canbe interpretedin termsof defectdensity,disor-der, crystalline/amorphousfraction and material morphol-ogy. Finally, we will discussthe correlation of mc-Si:H

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propertiesandlight trappingeffectswith a boostof thesolarcell performance.

II. THEORY

A. Parameters used to describe light scattering in thevolume and at the surface of mc -Si:H

In the following let usconsidera light beamincidentingperpendicularlyon a thin mc-Si:H film depositedon a thicknonabsorbingsubstratewith a correspondingrefractiveindexns andsurroundedby anambientwith a real refractiveindexna . The parametersof the film are denotedby the index f:thus,thefilm thicknessis df andits complexrefractiveindexis Nf5nf1 ik f . The imaginarypartof the refractiveindexkis relatedto the optical absorptioncoefficienta by

k5l

4pa. ~1!

For thedispersionof themc-Si:H refractiveindexnf we usea modification of the Sellmeier formula suggestedbyHerzberger13

nf5nA1nB

l220.028, ~2!

where nA and nB are material constantswhich have to bedeterminedexperimentally, the vacuum wavelength l isgiven in micrometersandthechoiceof theconstant0.028isarbitraryand independentof the material.The advantageofHerzberger’sformula is the linear relation betweenthe re-fractive index and its parameters,which will be useful forfitting the interferencefringesin theR(l) andT(l) spectra.

Light scatteringoccursboth at inhomogeneitiesin thevolume ~mostly voids! and at the surfaceof the mc-Si:Hfilm. Both effectslead to the attenuationof the specularre-flectedandtransmittedbeamsandto an enhancementof theabsorption.The typical rms surfaceroughnesss, observedby atomic force microscopy~AFM!, is 10–40 nm for 2 mmthin films.11 Fourier transformanalysisof the AFM datare-vealeda small correlationdistance.The grainsin the layer,typically prolongedin the directionof growth,havea diam-eteron theorderof tensof nanometers.14 Hence,we canusethescalarscatteringtheoryfor randomroughsurfaceswith asmall correlationdistance.15

The lossesfrom the specularlight beamby scatteringatthe rough interfacebetweenmedia 1 and 2 are describedwithin thescalartheoryby thescatteringfactorss.15 Thus,incomparisonwith a smoothinterface,the Fresnelcoefficientr 12 for the amplitudeof the beamreflectedfrom the roughsurfacebackto the medium1 is reducedby a factor

s12r 5expF2

1

2 S 4pn1s

l D 2G ~3!

andthe Fresnelcoefficientt12 for the amplitudeof the beamtransmittedthroughthe rough interfaceto medium2 is re-ducedby a factor

s12t 5expF2

1

2 S 2p~n12n2!s

l D 2G , ~4!

wheres is the rms surfaceroughnessof the boundaryinter-face,n1 andn2 arethe refractiveindicesof the correspond-ing media,andl is the light wavelengthin vacuum~seealsoAppendix A!. Formulas~3! and ~4! assume:~i! s!l, ~ii !small correlation length of the surfaceroughnessand ~iii !normalincidenceof the light. In the caseof an obliqueinci-dencethe surfaceroughnesss hasto be reducedby the fac-tor sin~g!, whereg is the anglebetweenthe surfaceandthedirectionof the incominglight beam.

On the other hand,the attenuationof the specularlightbeamby an isotropic volume scatteringmay be simply de-scribedby the volume scatteringcoefficient asc.

16 There-fore, theattenuationof thespecularbeamin thefilm is givenby the coefficienta f

a f5a true1asc, ~5!

wherea true is the true optical absorptioncoefficient in thefilm.

The descriptionof the scatteringby s factorsandby asc

is the simplestdescriptionof the otherwisecomplex phe-nomenonandexpressesthe level of approximationswe use.

B. Specular reflectance and transmittance ofscattering samples and determination of the opticalconstants

Thedescriptionof thespecularreflectanceandtransmit-tanceof a mc-Si:H layer on a thick glasssubstrateneedstotake into accountboth mc-Si:H interfaces,multiple reflec-tions within the layer as well as the influenceof the backsubstrate–ambientinterface.The resulting formulas for thespectraof reflectancefrom the film side Rf(l) or from thesideof thesubstrate~typically glass! Rg(l) andthetransmit-tanceT(l) as a function of the sampleparametersnf , a f ,df , s, na , ns arecomplicatedfunctionsgivenin full detail inAppendix A. As an example,the reflectanceand transmit-tancecalculatedfor a 1 mm thick mc-Si:H layer on a thickglasssubstrateareshownin Fig. 1. Theresultsplottedby fulllinescorrespondto thesamplewith a smoothtop surfaceandthoseplottedby the dashedlines for a surfacewith the rmsroughnesss530 nm.Onecanobservethat theroughsurfaceleadsto a decreaseof both reflectanceandtransmittanceandalsoto a partial smoothingof the interferencefringes.

Figure 1 shows the T and R results of modeling forknown film parameters,but in the experiment,we needtosolve the inverse problem, i.e., the determinationof thesampleparametersnf , a f , df and s from the measuredRf(l), Rg(l) and T(l). For that purposewe divide thespectrainto threeregions:

~i! a region of strong ~complete! absorption (Tf5Tg

;0) wherethesurfaceroughnessandrefractiveindexnf(l) of the film canbe estimated;

~ii ! a regionwith negligibleabsorption(a f;0) wherethefilm thicknessandrefractiveindex nf(l) canbe esti-mated;and

~iii ! a regionof mediumandhigh absorption~Tf5Tg.0wherethea fdf productis in therange0.05–5! wherethe absorptioncoefficienta f(l) canbe determined.

.

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Thesespectralregionsaremarkedin Fig. 1 togetherwith theparametersthat canbe obtainedfrom them.Let us considertheseregionsin moredetailasmethodicalsteps~1–3! for theevaluation.

1. Determination of the surface roughness and realpart of the refractive index in the highabsorption region

The real part nf of the film refractive index and thesurfaceroughnesss canbe found from themeasurementsofthe samplereflectanceRf andRg in the regionof completeabsorptionwhere the interferenceeffects are suppressedThus,we mayneglectthemultiple reflectionsin thefilm andconsideronly the first-ordertermsfor reflectedlight in Eq.~A5! of AppendixA. The formula ~A11! for the reflectanceof the sampleRsfa from the substrateside is simplified to

Rsfa5Rsf5UNf2ns

Nf1nsU2

. ~6!

FIG. 1. Calculatedspectraof reflectanceRg , Rf andtransmittanceTf of amc-Si:H layer on a glasssubstrateaccordingto the formulasgiven in Ap-pendix A for a smooth surface~solid lines! and for a layer with a rmsroughnessof s530 nm ~dashedlines!. The subscriptsf andg indicatethatwe considerlight incidenceon thesamplefrom thesideof thefilm, respec-tively from the glasssubstrate.Note that Tf5Tg . The refractiveindex ofthe mc-Si:H layer wasapproximatedby nf53.310.2/(l220.028) andtheabsorptioncoefficient by af515000* (E21.1)4(cm21). The boxesmarkthe spectralregionsfrom which the optical parametersof the sampleareobtained:~1! refractiveindexof thefilm nf is obtainedfrom thevalueof Rg

in theregionof strongabsorption;~2! rmsroughnesss is obtainedfrom thedifferenceof the Rf expectedfor the smoothfilm andactuallyobservedintheregionof thestrongfilm absorption;~3! thefilm thicknessdf andrefrac-tive indexdependencenf(l) arefoundfrom theregionof negligibleabsorp-tion; ~4! the absorptioncoefficientof the film af(l) is obtainedfrom theregionof mediumabsorption.

Neglectingthe imaginarypart kf of the film refractiveindexNf5nf1 ik f ~in this spectralrangekf<0.1!nf;3.8! we canexpressthe film refractiveindex nf as

nf5ns

11Rsf

12Rsf, ~7!

where Rsf can be calculatedfrom the experimentallyob-servedreflectanceRg usingEq. ~A14!

Rsf8Rsfa5Rg2Ras

Tas2 1Ras•~Rg2Ras!

. ~8!

Here Ras is the reflectanceand Tas the transmittanceof thesubstrate–ambient interface which can be calculatedfromthe known refractiveindexesof the substrateand the ambi-ent.

While the reflectanceRg measuredfrom the substrateside is not influencedby the rough free film surface,thereflectancemeasuredfrom the side of the film Rf will bereducedby scattering.With the value of nf determinedbyEq. ~7! we cannow calculatethesurfaceroughnessfrom themeasuredRf at the samewavelength:

s5l

4pnalnARf•S nf1na

nf2naD 2

. ~9!

An estimatedrms surfaceroughnesss by Eq. ~9! simplifiesfurther calculations.

2. Determination of the sample thickness andrefractive index from the interference fringes in thelow absorption spectral region

Thethicknessof thefilm df andthespectraldependenceof nf ~i.e., the parametersnA andnB @Eq. ~2!# of Herzberg-er’s formula! canbe calculatedfrom the interferencefringesin the reflectanceandtransmittancespectrausinga multipa-rameterfitting procedure.17 We needto fit threeparametersandin orderto avoid any divergenceproblemwe divide thefitting procedureinto two parts.

First we fit the spectraldependenceof the refractivein-dex.Theformulasfor reflectanceRg or Rf of thesample,Eq.~A14!, canbe rewritten into the form

y~l!51

12R5a~l!1b~l!•cos~w!, ~10!

wherea(l) and b(l) are functionsof the index of refrac-tion: nf , ns , andna andw54pnfdf /l. A usefulexpressioncan be obtainedby consideringthe envelopesaround themaxima ymax(l)5a(l)1b(l) and minima ymin(l)5a(l)2b(l) of the y(l) spectrum.Theseenvelopesareconsideredto be continuousfunctions of l. We obtain the functionsa(l) and b(l) by a spline interpolationof the experimen-tally observedymax and ymin . Once a(l) and b(l) areknown, we can calculatecos(w)5(y(l))2a(l))/b(l). NowusingHerzberger’sdispersionformula, i.e., Eq. ~2!, we canexpresscos~w! as

4

cos~w!5cosS 4pnfdf

l D5cosF 4pE

1.2389S pA1pB

~1.2389/E!220.028D G , ~11!

whereE51.2389/l is the photonenergyin eV and pA andpB are parametersrelatedto nA and nB by the expressionspA5nAdf and pB5nBdf ~wheredf and l are expressedinmicrometers!. ParameterspA andpB are found by a nonlin-earleast-squaresfitting of the spectraldependenceof cos~w!obtainedfrom measuredT/R usingEq. ~11!.

UsingtheobtainedpA andpB we cancalculatecos~w! ateachl in the formulasfor the reflectanceandtransmittanceof the film @Eqs. ~A10!–~A11!#. A nonlinear least-squaresfitting routineusedto fit thetransmittanceor reflectancedatain the transparentregion then requiresonly the free param-eterdf , which is easyto obtain.

3. Imaginary part of the index of refraction from theregion of medium absorption

When the spectraldependenceof nf(l) and the thick-nessof the film df are known we can find the spectrumofaf(E). In the spectralregion wherethe film absorptionco-efficient is high enoughandthe interferencesin the film areabsentthe absorptioncoefficient can be calculateddirectlyfrom the transmittanceusingEq. ~A12! reducedto the form

T85TafTfsTsaexp~2afdf !

12RsfRsa, ~12!

whereTaf , Tfs , Tsacanbecalculatedusingtheknownvaluesfor s, df , na , ns and nf ~nf is extrapolatedfrom theHerzberger’sformula using nA and nB determinedin thetransparentregion!.

Theapproachbasedonly on thetransmittancefails in theregion of week absorptance,becausethe influence of theinterferenceeffectson the transmittancespectradrownstheinfluenceof the absorption.The inspectionof Eqs. ~A10!–~A12! showsthat the function (12Rg)/T is almost free ofinterferencefringes18 and is, thus, suitableto solve a f(E).This approachmakesit possibleto calculatevaluesof a f inthe spectralregion,wherea fdf>0.05.

C. Measurement of the absorptance in samples withthe bulk or the surface scattering by CPM

CPM usesthe photocurrentproportionalto the amountof light absorbedin the film. For nonscatteringsamplesonnonabsorbingsubstratesthe photocurrentis simply propor-tional to the absorptancedefinedas

A512R2T, ~13!

where R and T are the reflectanceand the transmittance.More careis requiredfor the analysisof scatteringsamples.Here, a large part of the light scatteredin the silicon film~with relativelyhigh refractiveindex! will remaintrappedinthefilm by total internalreflectionsand,thus,scatteringwillactuallyleadto a muchmoreefficientphotogenerationof thecarriers mainly in the low absorptionrange.Finally, it isimportantto realizethatdueto theshortdiffusion lengthsof

the carriersonly the light absorbedbetweenthe electrodeswill contributeto the photocurrent.The geometryof the co-planarelectrodeswith width W anddistanceD usedfor CPMmeasurementsis shownin Fig. 2.

The detailsof the CPM methodand evaluationare de-scribedelsewhere.19 Themostsimplestandardinterpretationof the measuredCPM spectra20 usesthe Beer’s law expres-sion for the absorptanceof the film

A5~12R!@12exp~2ad!#. ~14!

For small valuesof a this leadsto A8(12R)ad. Further-more, changesin the reflectanceR as a function of photonenergyE canbe neglected,whencomparedwith changesina over severalordersof magnitude.A similar assumptionhasbeenmadein PDS.21 Keepingthe photocurrentconstantby changing the intensity of the illumination leads to asimple evaluationof the a(E) spectrum,which is propor-tional to the inverseof the photonflux. In the following weshall denotethe resultof this procedureas the apparentab-sorptioncoefficientaapp. More preciseexpressionsfor theabsorptance,which would include the effect of the multiplereflections,canbeused~seeAppendixA! but theargumentsof Sec.II C1 remainvalid.

1. Model of multiple bulk scattering

If we usethesimpleapproach~Beer’slaw! for bulk scat-tering in the films, the attenuationof the specularbeamwillbe given by Eq. ~14! with absorption coefficient a f

5a true1asc @Eq. ~5!#. Hence,neglectingthe spectraldepen-denceof reflection,thepartof light directly absorbedwill be

Adir'atrue

a f•@12exp~2a fdf !# ~15!

andthe part of light scatteredin the volumeof the film willbe

Asc'asc

a f•@12exp~2a fdf !#. ~16!

Correspondinglythe photocurrentwill be composedof twocomponents

I ph5I dir1I sc, ~17!

whereI dir is proportionalto Adir andrepresentsthephotocur-rent excited by photonsabsorbedfrom the specularbeam,i.e., without any scattering

FIG. 2. Electrodeconfigurationfor the measurementof the photocurrentinthe constantphotocurrentmethod~CPM!.

5

I dir'NphotonsAdir'Nphotons•atrue

a f•@12exp~2a fdf !#,

~18!

whereNphotonsis the incidentphotonflux.The componentI sc representsthe contribution of pho-

tonsabsorbedafteroneor morescatteringevents.This com-ponentis importantmainly at the spectralregion with lowopticalabsorptioncoefficienta true. It will beproportionaltothe numberof photonsscatteredfrom the specularbeamAsc

definedby Eq. ~16!, howeverwe needalso to considertheprobability that the scatteredphotonwill be absorbedin thespacebetweenthe electrodesandthuswill contributeto theobservedphotocurrent.

First, let us define a very important parameterfor thelight trapping scheme,the critical angle u for total reflec-tions. u definesthe escapeconeandcanbe calculatedfromthetotal internalreflectionconditionsin(u)5n/nf , wheren istheindexof theoutermedium.For therefractiveindexof 3.5~for mc-Si:H films in thenearIR region! this givesu816.6°for the film–air interface and u825.4° for the film–glassinterface.If thephotonfrom thespecularbeamis scatteredinthe direction outsideof the escapeconeit will be confinedwithin the film by total internal reflectionsand will remaintrappeduntil it is either absorbedor scatteredagain.If thephoton is scatteredwithin the escapecone it will probablyescapefrom the film after a few internal reflections.A nu-mericalsimulationshowsthat the contributionof suchpho-tons to the photocurrentis negligible. Assuming isotropicscatteringwe canwrite the probability Ptr that the photonisscatteredoutsidethe escapeconeas

Ptr5cosu. ~19!

Whenthe scatteredphotonis trappedby the waveguid-ing it mustfinally be absorbedor scatteredagain.The prob-ability that this photonwill bescatteredis determinedby thevalue of the meanfree path betweentwo scatteringevents.The mean free path is given as a reciprocal value of thescatteringcoefficientasc ~typical valuesin microcrystallinesilicon for asc are &503E4 cm21, where E is the photonenergy in eV7!. The probability that the waveguidedphoton is absorbed within its mean free path is12exp(2atrue/asc). If thephotonis not absorbedduring themeanfree path1/asc it is scatteredagainandthe samekindof considerationsarevalid. Thephotocurrentexcitedby scat-tered photonscan then be calculatedby the sum of thesecontributions,leadingto a geometricalseries~seeAppendixB for details!.

However, the absorbedphoton will contribute to thephotocurrentwith probability PWD only if it is absorbedwithin the areabetweenelectrodes~seeFig. 2!. Finally thecontribution of the scatteredlight to the photocurrentob-servedby CPM will beproportionalto theproduct~with thesameconstantasthe I dir component!

I sc;NphotonsAscPtrPWD , ~20!

which leadsto the expression~for detailsseeAppendixC!

I sc'Nphotons

asc

a fexp~2a fdf !•cosu

•F12expS 2a true

ascD G Fcosu•expS 2

a true

ascD GNbulk

21

cosu•expS 2a true

ascD21

,

~21!

whereNbulk is the numberof the scatteringeventsbetweenthe electrodes.For the determinationof this parameterweusethe solutionof the classicalprobability task—arandomwalk:

Nbulk5@asc•X~W,D !] 2, ~22!

where X(W,D) is the function of electrode dimensionswhich givesa valueapproximatelyequalto the distancebe-tween the electrodeswhen the ratio of the length of elec-trodesandthe interelectrodespacingis closeto the valueof6.6—for detailsseeAppendixC.

If the standardCPM evaluationprocedurewill be ap-plied to the measuredspectrum,we will obtainan apparentoptical absorptioncoefficient aapp for which the followingequationis valid:

12exp~2aapp•df !

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a f•@12exp~2a f•df !#•F a true1asc•cosu

•F12expS 2a true

ascD G

•

Fcosu•expS 2a true

ascD GNbulk

21

cosu•expS 2a true

ascD21

G . ~23!

This equationshowshow the CPM measuredaapp dependson the electrodegeometry,i.e., the interelectrodespacingDand the electrodelength W ~seeFig. 2!, and on the param-etersa true, asc andnf ~nf influencesthevalueof thecriticalanglefor the total reflection!.

The combinationof Eqs.~23! and ~5! enablesthe mod-eling of the apparentoptical absorptioncoefficient aapp ofthe film with the known parametersa true, asc, u, W andD.Note that although the film thicknessappearsalso in Eq.~23!, this parameterdoesnot influencethe result of model-ing. The inverseproblem,i.e., the evaluationof the spectraldependenceof the absorptioncoefficienta true and the scat-tering coefficientasc cannotbe solvedanalytically and nu-mericalmethodshaveto be used.First we roughly estimatethe scatteringcoefficientasc from a comparisonof severalCPM measurements~for different interelectrodespacing!with themodeloutputdataof thesamecoplanargeometries.In agreementwith Ref. 22 we supposethe spectraldepen-denceof the scatteringcoefficient.For the valueof the scat-

6

tering coefficientasc ~at photonenergyE51 eV! up to 50cm21 we supposethe Rayleightype of scattering,i.e., a de-pendence'E4. The estimatedscatteringcoefficient~anditsspectraldependence! togetherwith the smoothedCPM dataenterthe iterativeprocedureof evaluationof thetrueabsorp-tion coefficient.

2. Model of surface scattering

Most of thehigh quality microcrystallinesilicon films donot exhibit a mirrorlike free surface.Theselayers with a‘‘milky’’ appearancehavea typical rmssurfaceroughnessof15–35 nm, which leadsto thesurfacescattering.We presentherethetheoryfor surfacescatteringderivedby thehelpof sfactorsfor the caseof light enteringfrom the film side.Forthesecalculationswe introducenew factorsSt , Sr and Sr0

which decreasethe intensityof the specularlytransmittedorreflectedwave.Thesefactorsaredefinedas15

St5saf2 5sfa

2 5expF2S 2p~nf2na!s

l D 2G , ~24!

Sr05sff25expF2S 4pnfs

l D 2G , ~25!

Sr5expF2S 4pnfs cosw

l D 2G , ~26!

whereSr0 correspondsto the perpendicularinternal reflec-tion and Sr to the oblique incidencewith the angleof inci-dencew ~with respectto the surfacenormal!. Sincewe cal-culateanaverageamountof thescatteredlight for all angles,we usethe following empirical formula ~taking into accounta changeof thecritical angleof total reflectionwith a changeof the index of refraction!:

Sr5expF2S 4pnfs cos~p/nf !

l D 2G . ~27!

For CPM modeling in the caseof rough samples,westartagainwith Eq. ~17! for thephotocurrentpassingthroughthesample~consistingof I dir andI sc!. I dir ~photocurrentafterphotonabsorptionwithout anyscatteringeventin thin film ofthicknessdf! is given by

I dir'Nphotons•St•@12exp~2a true•df !#. ~28!

The componentI sc representsthe contribution of pho-tonsabsorbedafteroneor morescatteringevents.This com-ponentis importantnot only in the low absorptionrange~asin the caseof bulk scattering! but alsoat the spectralregionwith medium~andhigh! optical absorptioncoefficienta true.It is proportionalto thenumberof photonsscatteredfrom thespecularbeamby the rough interfaceand absorbedin theregionof electricfield.

The number of photons scatteredduring the passagethroughthe rough ambient–film interfaceis proportionalto(12St). Nevertheless,we also considerhere the contribu-tion of photonsscatteredby more complicatedways afterspeculartransmissionthroughthefirst interface,thepenetra-tion through the film, then the reflection at the second~smooth! film–substrateinterface, followed again by the

penetrationthroughthe film, andfinally the diffusive reflec-tion at the first interface.The total amountof the scatteredlight is thenin relationto the probability P0

P05~12St!1St•Rfs•exp~22a truedf !Rfa~12Sr0!.~29!

The parametersRfs andRfa are the reflectioncoefficientsatthe interfacefilm–substrateand film–air, respectively.Thesecondpart of the right hand side of Eq. ~29! is usuallysmallerbut not negligible in comparisonwith the first one,because(12Sr0) canbe an orderof magnitudehigher than(12St).

Waveguidingpropertiesof the materialare determinedagain by the critical angle u for the total reflection whichdefinesthe escapecone.The probability Pesc that a photonwill bescatteredwithin theescapeconeis givenfor theidealLambertian~cosine! distributionas

Pesc5S ns

nfD 2

. ~30!

For surfacescatteringthesmallestdistancebetweentwoscatteringeventsis given by twofold thicknessof the film~for the photonsscatteredin the speculardirection!. On theotherhand,the longestdistanceis infinite for the light scat-teredparallel to the film plane.The total Lambertianangledistribution~w from 0 to p/2! is givenby cos~w!sin~w! prod-uct with a maximumat theanglew5p/4. Themultiplicationfactor Y for the averageincreaseof the optical path of thescatteredlight, comparedto the samplethickness,can becalculatedfrom the angulardistributionas

Y52•*u

p/2 sinwdw

*up/2 sinw coswdw

, ~31!

wherethe factor 2 meanstwofold path becauseof the totalinternal reflection at the film–substratesmooth interface.@The parameterY varies for mc-Si:H with the index of re-fraction ~wavelength! from about4.2 to 4.6.# Hence,for veryroughsamples,the Y•df productdenotessomethinglike themeanfree path.For microcrystallinesilicon films with rmssurfaceroughnesstypically varying from 15 to 35 nm, theprobability of a photonbeing absorbedduring this distance(Y•df) is equalto @12exp(2Yatruedf)# and the probabilityof being scatteredat this rough interface is given by (12Sr). If thephotonis neitherabsorbednorscatteredor if thephoton is not absorbedbut is scatteredoutsidethe escapecone,thenthesamekind of considerationshaveto beappliedagain.The calculationof the photocurrentI sc leadsto thesolutionof a geometricalseriesandis given by

I sc'P0A11P0P1A2•P2

Nsurf21

P221, ~32!

where

A15~12Pesc!•@12exp~2Ya truedf !#

1Pesc•@12exp~21.05a truedf !#, ~33!

where A1 representsthe relative amountof scatteredlightabsorbedbetweenthe first and the secondscatteringevent

7

andthefactor1.05is theaverageincreaseof theopticalpathof thescatteredlight within theescapecone~for themc-Si:Hfilm/glassinterface!.

P15~12Pesc!•exp~2Ya truedf ! ~34!

is the relative reductionof the light intensity betweenthefirst andthe secondscatteringevent

A25@~12Pesc!•~12Sr !1Sr #•@12exp~2Ya truedf !#

1Pesc•@12exp~21.05a truedf !#

•@11Rfs•exp~21.05a truedf !# ~35!

is therelativeamountof thescatteredlight absorbedbetweenthe next two scatteringevents~secondand third, third andfourth, ...! and

P25@~12Pesc!•~12Sr !1Sr #•exp~2Ya truedf !] ~36!

is again the relative light intensity reduction betweenthenext two scatteringevents.

The total numberof scatteringeventsinside the regionof theelectricfield Nsurf in thecaseof surfacescatteringcanbe calculatedby weighting the probability (12Sr) of pho-tons to be scatteredat a distanceYdf and probability Sr ofphotonsto be specularlyreflected

Nsurf5S X~W,D !

YdfD 2

•~12Sr !1S X~W,D !

YdfD •Sr , ~37!

where X(W,D) is the interelectrodespacingfunction ~seeAppendixC!.

For the modelingwe takeinto accountthat a part of thelight escapingto the ~nonabsorbing! glass substrate~withindex of refractionns'1.5 and substratethicknessds! canreturnto theareaof theelectricfield betweentheelectrodes.This considerationleadsto the modificationof the relativeamountof the photonswithin the escapeconeand,thus,theprobability Pesc, which will vary not only with the index ofrefractionof thefilm andsubstrate~nf andns!, but alsowiththe substratethicknessds and the interelectrodefunctionX(W,D). Pesc is given by

Pesc5S nsef

nfD 2

, ~38!

wherensef ~varying from na to ns! representsthe effectiveindex of refractionof the substratewhich canbe calculated~for the cosinedistributionof the scatteredlight! as

nsef5S 11ns22H ns•sinFarctanS X~W,D !

2dsD G J 2D 0.5

. ~39!

Now, we can write for the apparentoptical absorptioncoefficientasmeasuredby CPM

12exp~2aapp•df !5St@12exp~2a true•df !#

1P0S A11P1A2

P2Nsurf21

P221D . ~40!

Similarly as in the caseof the multiple bulk scattering,this equationcanbe usedfor modelingof the apparentopti-cal absorptioncoefficientaapp with the help of the knownparametersa true(E), s, d, ds , n(E), W andD. The evalua-

tion of a true(E) from theCPM measurementis donenumeri-cally on the basisof the known rms surfaceroughnesss,determinedoptically.

III. EXPERIMENT

Microcrystallinesilicon layersand solar cells were de-positedby theVHF-GD methodusinghigh dilution of silanein hydrogen,with and without a purifier.2,3,23 Layers weredepositedon AF45 glass,underconditionssimilar to thecor-respondingcells. The typical film thicknesswas around2mm. With respectto the depositionconditionsthe layersre-sult in either a smooth,mirrorlike or a rough,hazy appear-ance.Somerough ~textured! layerswereafterwardschemo-mechanicallypolishedto a mirrorlike surfaceto excludetheinfluenceof theroughsurfaceon themeasurementof opticalproperties.This is a delicateprocedure,frequently endingwith the layer peelingoff from the substrate.

The typical rms surfaceroughnessof ~220! texturedmi-crocrystallineSi layersis below40 nm for about2 mm thickfilms, as directly observedby the AFM. The grains in thelayer,prolongedin thedirectionof growth,haveaneffectivediameterof 10–30 nm, asseenby transmissionelectronmi-croscopy~TEM!.14

A computer-controlledsingle-beamspectrometerwasusedfor the transmittance/reflectancemeasurementsin the0.6–3 eV spectralregion. The diameterof the light beamwas limited to 1 mm in order to suppressthe influenceofpossiblevariationsof the layer thicknesson the modulationdepthof the interferencefringes. The detector~or the inte-grated sphere with a detector! is placed far behind thesample;hence,just thespecularreflectanceandtransmittancearemeasured.

TheCPM setupdescribedelsewhere19 wasusedto mea-sure the absorptanceof the films both in the standardandabsoluteCPM mode. On top of the layers coplanarAl orCr/Ag electrodeswere evaporatedwith the interelectrodespacingD varying from 30 mm to 3 mm andelectrodewidthW between2 and 8 mm ~Fig. 2!. The contribution of thescatteredlight to the CPM signal is changedby this geo-metricalsetupof the electrodes.7

IV. RESULTS

The spectraldependenceof the speculartransmittanceandreflectanceof a typical layerbeforeandafter thepolish-ing is shownin Fig. 3. Justa part of the reflectancedataisdisplayedhere.In the caseof texturedsurfaces,the modula-tion depthof the interferencefringesis reduceddueto lightscattering.The ‘‘scalar scatteringtheory,’’ which considersjust the phasemodulationof the incidentandoutgoinglightby the heightvariationsalong the surface,hasbeenusedtointerpretthedataandto evaluatethe root meansquare~rms!surface roughnesss by the proceduredescribedin Sec.II A.24

Calculatedvaluesof the optical absorptioncoefficientafor thesamplefrom Fig. 3 areshownin Fig. 4 andcomparedwith the absorptioncoefficientof crystallinesilicon.25 Per-fect agreementof the a true(E) evaluatedfrom the T and Rmeasurementson thepolished~s50! andon the‘‘as grown’’

8

texturedlayers~s524 nm! can be seen.Note, on the otherhand,thatusingthestandardtheoryfor theevaluationof theT/R data~it meansneglectingtheeffectof thesurfacerough-ness! resultsin a false, ‘‘apparent’’ optical absorptioncoef-ficient for texturedlayers.

A. Study of textured and smooth layers by constantphotocurrent method „CPM…

CPM detectsthe light absorbed~either directly or afteroneor morescatteringevents! inbetweentheelectrodesusedfor the photocurrentmeasurement.The gap ~spacing! be-tweenthe electrodesinfluencesthe contributionof the scat-tered light to the measuredphotocurrentand thus, to the

FIG. 3. Transmittanceandreflectancespectraof a mc-Si:H sample,depos-ited with 5% dilution of silane in the total gasflow at a VHF depositionpower of 19 W. Dashedlines show the resultsbeforepolishing ~texturedsurfacewith rmsroughnesss524 nm! andfull linesaftera chemomechani-cal polishing.Thedifferent spacingof theinterferencefringesindicateshowthepolishingdecreasedthethicknessof thesample.Both spectraweremea-suredwith light incidentfrom thesideof thefilm. Only a partof the reflec-tancespectrais shown.

FIG. 4. The apparentabsorptioncoefficient aapp ~calculatedwith the as-sumptions50, i.e., by a procedurewhich doesnot take into accountanysurfaceroughness! and true absorptioncoefficienta of the film evaluatedfrom the reflectanceandtransmittancedatain Fig. 3. The error barscorre-spondto the estimatedprecisionof the a evaluationbasedon the relativeerror of transmittancemeasurement~;1%!. The absorptioncoefficientofmonocrystallinesilicon is shownfor comparison.

deducedCPM apparentoptical absorptioncoefficient aapp

~proportionalto the inverseof photonflux necessaryto keepthe photocurrentconstant!. CPM resultsfor different inter-electrodespacingtogetherwith true a(E) determinedfromT/R measurementsandcomputedfrom CPM dataareshownin Figs. 5 and6 for the sampleswith preferential~220! tex-ture. To evaluateour datawe haveusedthe theoreticalap-proachdescribedin Sec. II. Figure 5~a! showsthe typicalsampleA with an as-grown~textured! surfaceand in Fig.5~b! the samesamplewith a polished~mirrorlike! surface.We can seethat the effect of light scatteringon the CPMspectradisappearedwith the polishing procedureand CPMmeasuresin this casedirectly the true a(E).

Figure6 showsa different classof samples~sampleB!againwith the as-grown~textured! @Fig. 6~a!# and polished@Fig. 6~b!# surface.For thesampleB thereis a strongdiffer-encein CPM spectrameasuredwith different interelectrodespacingalso for the polishedsurface.This samplewaspre-paredat a high depositionrate~over1 nm/s! andhasa lowermassdensity,presumablydueto voids.

The increasein the depositionrate and the decreaseinthe defectdensity~subgapabsorption! is an importantissuefor microcrystallinesilicon solar cells. ResultspresentedinFig. 7 showthat it is possibleto depositmaterialwith a verylow subgapabsorptionat a depositionrate of around5 Å/s

FIG. 5. Apparentoptical absorptioncoefficientsof sampleA measuredbyCPM with different interelectrodespacing~gap! and calculatedfrom T/Rmeasurements:~a! in the as-grownstate~texturedsample! and ~b! after achemomechanicalpolishing.The evaluatedspectraldependenceof the trueabsorptioncoefficienta(E) is shownas the main result by full diamonds;a(E) of crystallinesilicon is shownfor comparison.The valueof the rmssurfaceroughnessof s521 nm of theas-grownsamplewasevaluatedfromthe T/R data.

9

~for depositionconditionsseeRef. 26!. Thesesamples@pre-paredat very high frequency~VHF! power 11 and 13 W#representthelowestsubgapabsorptiondatafor microcrystal-line silicon presentedin the literatureso far.

V. DISCUSSION

A. Surface and bulk scattering

The resultsin Figs.5~a! and5~b! for the typical sampleA ~preparedat a mediumgrowth rateof 6 Å/s! demonstratethe dominant surface scattering to be present in suchsamples.After a standardchemomechanicalpolishingof therough samplesurfacethe scatteringdisappearsand the trueoptical absorptioncoefficient a true is directly measuredbyCPM. On thecontrary,sampleB ~preparedat a higherdepo-sition rate of over 10 Å/s! also exhibits a bulk scatteringcontribution,7,12 which cannotberemovedby polishing,asitis seenin Fig. 6~b!. Theevaluatedbulk scatteringcoefficientis asc528•E4, where E is the photon energy ~eV!. Thiswould point to a hypothesisthat small voids are presentinmicrocrystallinesilicon preparedat high deposition rates,similarly to the caseof amorphoussilicon.12,27 This is fur-thermoresupportedby a lower massdensityof suchfilms.

It canbeusefulto unify the theoriesof bulk andsurfacescattering,asdescribedin Secs.II C1 andII C2, in termsof

FIG. 6. Apparentoptical absorptioncoefficientsof sampleB measuredbyCPM with different interelectrodespacing~gap! and calculatedfrom T/Rmeasurements:~a! in the as-grownstate~texturedsample! and ~b! after achemomechanicalpolishingandH2 plasmatreatment.The evaluatedspec-tral dependenceof thetrueabsorptioncoefficienta(E) is shownasthemainresultby solid or emptydiamonds;a(E) of crystallinesilicon is shownforcomparison.The rms surfaceroughnesss526 nm of the as-grownsamplewasevaluatedfrom the T/R data.

thescatteringcoefficientasc.28 Thecaseof light scatteringat

a rough interface,given by the scatteringfactorsS, can beexpressedon thebasisof thermssurfaceroughnesss by thescatteringcoefficientasc

asc51

dfS 2p~nf2na!s

l D 2

, ~41!

whereall termshavetheir usualmeaning.Now we canusefor thecalculationof surfacescatteringthetheoryof multiplebulk scatteringdescribedin Sec.II C1.

Surfacescatteringis of a vital importancefor thin filmsilicon solar cells. Becauseof indirect optical transitionsincrystallinesilicon, optical absorptionin thin silicon films isratherlow andhasto beincreasedin orderto absorbmostofthe solar spectrumin film a few micrometersthick. There-fore, complicated light trapping schemes have beensuggested.29

An alternativeis an ideal ‘‘Lambertian’’ light scatteringdueto a randomroughsurface.30 As a specialcase,we dis-cuss here a random rough surface with rms roughnesssmallerthanthe wavelengthof the light ~nanotexturedfilm!.This is exactlythecasewhich hasalreadybeenrealizedin areal thin film mc-Si:H cell ~2 mm! dueto the naturalgrownsurfacetexture.3,4,31The reasonfor this optical enhancementis presentedin a theoryhere,wherebythes factorsaregivenin Eqs. ~3! and ~4!. In Fig. 8, the squareof the scatteringfactorss is plottedbecausethe reflectedor transmittedlightintensity is proportionalto the squareof amplitude.In thecaseof internal reflection in silicon at the rough silicon/airinterface, the rms surfaceroughnessof s540–50 nm re-movesthe specularpart of the reflectionbecause(sint)

2 ap-proacheszero and one observesjust the diffuse reflection~for the wavelengthof 900 nm!. This is not the casefor thetransmissionthrougha surfaceof the sameroughness,as itcanbe seenfrom Fig. 8. The analysisof this figure ~dashedcurve for sext

2 ! also explainsthe appearanceof the nanotex-

FIG. 7. Comparisonof the true optical absorptioncoefficient atrue for aseriesof mc-Si:H films preparedat different dischargepowersandat dilu-tion of silanein hydrogenof 5%.

10

tured silicon films for the nakedeye. A root meansquareroughnesssmaller than 10 nm is difficult to detectby theeyes,a typical roughness20–30 nm can be easily detectedandfilms with a rms roughnessof 200 nm haveno specularreflectance~for random,noncorrelatedroughness!.

All this has a crucial importancefor thin film siliconsolarcells.Figure9 showstheMonteCarlomodelresultsoftheexternalquantumefficiencyof a thin texturedmicrocrys-talline silicon p- i -n solarcell. Detailsof this modelingwillbe publishedelsewhere.32 The cell consistsof a glasssuper-strate,a front transparentconductingoxide ~TCO! layer, 2mm thin p- i -n mc-Si:H, TCO backcontactanda metalbackreflector ~Ag!. All optical material constantsand all thick-nessesenter in the model, togetherwith the roughnessofeachinterface.In Fig. 9 we assumethat all interfacescopytheroughnessof thefront TCO andthat thermsroughnesssvaries from 0 to 100 nm. One can observea pronouncedenhancementof the responsein the infraredregiondueto adiffusive light scatteringat roughinterfaces.

FIG. 8. Reductionof the intensitiesof light specullaryreflectedor transmit-ted at the silicon film–air interface due to light scatteringat the roughsurface,as a function of the rms surfaceroughnesss, for wavelengthsof900nm ~full lines! and550nm ~dashedlines!. Thereductionis proportionalto thesquareof thescatteringfactorsdefinedby Eqs.~3! and~4!. The index~t! standsfor transmitted,theindex~ext! for externallyreflectedand~int! forinternally reflectedlight in Si films.

FIG. 9. Calculatedspectraldependenceof the quantumefficiency for a 2mm thick microcrystallinesolarcell asa functionof theinterfaceroughness.The cell consist of the structure: thick glass substrate/ZnO~1 mm)/p1(10 nm)/mc-Si:H~2 mm)/n1(20 nm!/ZnO~1mm!/Ag back reflector. Thenumbersin bracketsindicatethe thicknessesof the correspondinglayers.

B. Material parameters obtained from the true a„E…:Differences between microcrystalline and singlecrystalline silicon

The spectraldependenceof the optical absorptioncoef-ficient a(E) of microcrystallineSi givesus the informationon thematerialstructure,whereasthesubgappartreflectsthedefectstatesin the material.In order to comparea standardcrystallinesilicon materialwith a typical ‘‘device-quality’’microcrystalline silicon with hydrogen content about 5%~from IR spectra! and10%amorphousfraction ~from Ramanscattering! we divide the spectralregioninto threeparts:

~a! the regionbetween1.2 and1.4 eV describingthepara-bolic behaviorcloseto the bandedgesfrom which theindirect bandgapof silicon may be determined:

~b! the intrinsic absorptionregionabove1.5 eV; and~c! the region below 1.1 eV describing the defect-

connectedabsorption.

Thedetailedstudyof region~a! on a largesetof samplespointsto theconclusion,thataa(E) reachessimilar valuesasin the caseof crystallinesilicon ~seeFigs. 4–7!, hence,theindirect gapof typical device-qualitymicrocrystallineSi hasa valueof approximately1.1 eV at room temperature.Smallvariancebetweendifferent samplescanbe relatedto the in-ternal strain in the layers,as also observedfor the caseofmonocrystallineSi.33

The true optical absorptioncoefficientaa(E), measuredeitheron nonscatteringsamplesor determinedwith the helpof proceduresdescribedin this paperfor texturedlayers,isalwayshigher than the absorptioncoefficientof crystallinesilicon in the above-gapregion ~b!. This enhancedopticalabsorptionbrings an important advantagefor PV applica-tions.The highera(E) coefficientcanbe understoodby theeffective media approximation13 taking the combinationofthreedifferentmaterialcomponentsinto account:thecrystal-line silicon grains,thesurfaceregionof grainsandtheamor-phoussilicon tissue.It is temptingto quantitativelycorrelateeachcomponentwith the correspondingpart ~peak! in Ra-manspectasituatedalternativelyat 520,500 and480 cm21.

In a singlecrystallinesilicon, the absorptioncoefficientaround 1.1 eV is related to phonon assistedtransitions.25

This mechanismis maskedin microcrystallinesilicon by anexponentialdecay~Urbach tail! with a typical slope of 50meV. The appearanceof an exponentialdecayin mc-Si:Hmay be interpretedas a result of the loss of translationalsymmetryat the grain boundaries.

The defect-connectedabsorption@a(E) below 1 eV# isattributed to silicon dangling bonds mainly at the grainboundariesand in the amorphoustissue.6,34 This absorptionpart generallyincreaseswith hydrogenevolutionat high an-nealing temperature and decreases with posthydro-genation.6,34,35At photonenergyof 0.8 eV device-grademi-crocrystallinesilicon showsa typical valueof thetrueopticalabsorptioncoefficient below 0.1 cm21.7 One can even ob-servea true valuesat the photonenergyof 0.8 eV as low as0.01cm21 ~seeFig. 7!. A low defect-relatedabsorptionis anindication of the high quality of this material.One can ob-servethat at higherappliedVHF power, the quality of ma-

11

terial decreases~increasedsubgapabsorption,reducedsolarcell efficiency!.26

Finally, we want to commentthat the evaluationof theoptical absorptioncoefficientof nanotexturedmicrocrystal-line silicon films in somepreviouspapers6,10,34,36shouldbereconsideredwith the help of theorypresentedhere.

VI. CONCLUSIONS

We have experimentallydemonstratedthat the absorp-tion enhancementin the infraredregionof nanotexturedmi-crocrystallinesilicon thin films andsolarcellscomesmainlyfrom thesurfacescattering.Scatteringin thebulk of materialcansignificantlycontribute~to the CPM results! in the sub-gap spectralregion.A detailedtheory for the evaluationofthe true optical absorptioncoefficienta from transmittance,reflectanceand absorptance~CPM! measurementshasbeenpresented.Both the scatteringat the surfaceand in the bulkof the film havebeentakeninto accountin our theory.Thespectraldependenceof the absorptioncoefficienta(E) hasbeeninterpretedin termsof defectdensity,disorder,crystal-line andamorphousfractionsandmaterialmorphology.

ACKNOWLEDGMENTS

This work was supported~Contract No. JOR3-CT97-0145! by theEuropeanCommission,projectNEST,by GrantNos. GAAV A1010809,GACR 202/98/0669,andpartly byNEDO, JapanandGrantNo. GACR 202/99/0403.

APPENDIX A: SCALAR SCATTERING THEORY OFTHE REFLECTANCE AND TRANSMITTANCEOF THE LAYER WITH A ROUGH INTERFACE ON ATHICK SUBSTRATE

Supposinga light wave is incidentperpendicularlyto asmoothinterfacebetweenthe media i and j and with com-plex refractive indexesNi and Nj . Then the reflectionandtransmissioncanbe describedby the Fresnelcoefficients

r i j0 5

Ni2Nj

Ni1Nj, ~A1!

t i j0 5

2Ni

Ni1Nj, ~A2!

wherethe superscript0 correspondsto roughnesss valueofzero.

If the interfacebetweenmediai andj is roughthenscat-teringwill leadto a lossof light from thespecularlyreflectedand transmittedbeams.The scalarscatteringtheory15,28 de-scribestheselossesby reducingthe Fresnelcoefficientsbythesocalledscatteringfactors.Thus,thecoefficientsfor am-plitudes of reflectedand transmittedlightwave at a roughinterfacearegiven by

r i j 5r i j0 si j

r 5Ni2Nj

Ni1Nj•expF2

1

2 S 4pnis i j

l D 2G , ~A3!

t i j 5t i j0 si j

t 52Ni

Ni1Nj•expF2

1

2 S 2p~ni2nj !s i j

l D 2G , ~A4!

wheres i j is the rms roughnessof the interfaceandl is thelight wavelengthin vacuum.Thescatteringfactorssi j

r andsi jt

areequalto unity for smoothsurface~s50!.Thereflectionandtransmissioncoefficientsof a thin film

on a semi-infinitenonabsorbingsubstratecan be calculatedby the sumof the amplitudesof multiple beamsoriginatingfrom the split incidentbeam

r 1235t12e2 idr 23e2 idt21

1t12e2 idr 23e2 idr 21e2 idr 23e2 idt211...

5r 121t12t21r 23e22id

12r 21r 23e22id , ~A5!

t1235t21e2 idt231t12e2 idr 23r 21e22idt23

1t12e2 idr 232 r 21

2 e24idt231...

5t12t23e

2 id

12r 21r 23e2 i2d . ~A6!

The subscript123 indicatesthat the light is incident frommedium 1 on a thin film with index N2 on the substrate~medium3! andthe phasefactor d canbe written as

d52pdN2

l. ~A7!

The phasefactor is the sum of two terms d5w/21 iad/2,where w54pn2d/l describesthe interferences~n2 is theindex of refraction! and a54pk2 /l is the extinction coef-ficient.

We needto convertthe amplitudequantitiesinto inten-sities observedin the experiment.The intensitiesof the re-flectedandtransmittedbeamsat an interfacebetweenmedia1 and2 ~with or without any interfacial layers! aregiven by

R125r 12r 12* , ~A8!

T125n2

n1t12t12* . ~A9!

Using the formulas aboveand substitutingthe param-etersfor mc-Si:H (nf ,a,d,s) andtheindicesof ~nonabsorb-ing! substratens andambientna onecancalculatethereflec-tance and transmittanceof the film on a semi-infinitesubstrate.Thusoneobtainsfor incidenceeitherfrom thesideof ambient~afs! or substrate~sfa!

Rafs5Raf1Rfs e22ad12ARafRfs e2ad cos~w!

11RafRfs e22ad12ARafRfs e2ad cos~w!, ~A10!

Rsfa5Rsf1Rfa e22ad12ARsfRfa e2ad cos~w!

11RsfRfa e22ad12ARsfRfa e2ad cos~w!, ~A11!

Tafs5Tsfa5TafTfs e2ad

11RafRfs e22ad12ARafRfs e2ad cos~w!.

~A12!

We still needto include the influenceof the substratebackinterfacein our results.If the substrateis thick enough

12

~thicker than the coherencelength of the monochromaticlight used! then the interferenceeffectsin the substratewillbe suppressedandonehasto takethe sumof the intensitiesof the multiple split beams.Thus, in caseof incident lightfrom the film sideonegets:

Rf5Rafs1TafsRsaTsfa1TafsRsaRsfaRsaTsfa1...

5Rafs1TafsTsfaRsa

12RsfaRsa. ~A13!

In the caseof incident light from the glassside

Rg5Ras1TasRsfaTsa1TasRsfaRsaRsfaTsa1...

5Ras1TasTsaRsfa

12RsfaRsa. ~A14!

The transmittancedoesnot dependon the direction of thelight beamandis the samefor light incidentfrom eitherthefilm or glasssideof the sample

Tf5Tg5TafsTsa1TafsRsaRsfaTsa1...5TafsTsa

12RsfaRsa.

~A15!

APPENDIX B: EFFECT OF MULTIPLE BULKSCATTERING ON THE PHOTOCURRENTMEASUREMENT IN A COPLANAR ELECTRODECONFIGURATION

The photocurrentI sc arisesfrom the photonsabsorbedafter oneor morescatteringevents.In the caseof bulk scat-tering in the thin film, onecanwrite

~B1!

whereNphotons representsthe numberof photonsimpingingperpendicularlyon the samplesurface,and u is the criticalanglefor total reflection.In this caseonecanassumethatthescatteringmeanfree path ~optical length betweentwo scat-tering events! is 1/asc. Thereforeeachrow of Eq. ~B1! rep-resentsa contributionto thephotocurrentafter thenextscat-tering event.The term ‘‘cos u’’ representsthe optical lossesof the light within two escapecones~givenby the refractionindexesof Si and surroundingmedium!. Here we consider~insteadof two anglesu1 andu2 for interfacesair–film andsubstrate–film! either the higher angle enabling thewaveguiding of the scatteredlight ~for the interelectrodespacingbelow1 mm! or theeffectiveanglecorrespondingtothe effective index of refraction.The valueof this effectiverefractiveindex is given not only by the surroundingmediabut also by the geometryof the gap ~areaW•D! becausesomepart of the light escapingfrom thin film to glasssub-stratecanreturnbackto the areaof electricfield

cosu5cosu11cosu22cosu3 , ~B2!

whereu3 is calculatedon the basisof index of refractionofthefilm nf , of thesubstratens , of thesubstratethicknessds

andof the interelectrodefunction X(W,D)

u35arcsinFns

nf•sinS arctan

X

2dsD G . ~B3!

Equation~B1! canbe simply rewrittenas

I sc'Nphotons@12exp~2asc•df !#cosu

3@12exp~2a true/asc!#•F11cosu expS 2a true

ascD

1cos2 u expS 22a true

ascD1cos3 u expS 23

a true

ascD1...G .

~B4!

The terms of Eq. ~B4! in the bracketsform a geometricalseries,thereforeonecanwrite

I sc'Nphotons@12exp~2asc•df !#cosu

3F12expS 2a true

ascD G Fcosu•expS 2

a true

ascD GNbulk

21

cosu•expS 2a true

ascD21

,

~B5!

whereNbulk is the numberof the scatteringeventsinsidetheregion of an electric field between the electrodes~areaW•D!.

APPENDIX C: CALCULATION OF THEINTERELECTRODE SPACING FUNCTION X„W,D…

The interelectrodespacingfunction X(W,D) representsthe averagedistancetravelledby the photonsafter they arescatteredin a waveguideddirection of propagationbeforethey leave the active areaof the electric field betweentheelectrodes.Sincethefilm thicknessis muchsmallerthanthedimensionsof electrodes~W andD!, thedeterminationof theX(W,D) may be solvedasa two dimensionalproblem.

13

For photonsoriginating in a point (w,d) betweentheelectrodesthe averagedistancebefore loss x(W,D,w,d) iscalculatedas the averageof the distanceto the edgeof theinterelectrodeareaover the possibledirectionsof the light,i.e., for wP,0,2p!. The spacingfunction X(W,D) is thentheaverageof thex(W,D,w,d) overthewholeareabetweenthe electrodes,i.e., for w rangingfrom 0 to W andd from 0to D

X~W,D !5E0

WE0

D x~W,D,w,d!

WDdwdd. ~C1!

Unfortunately,this integralcannotbesolvedanalyticallyandthereforeonehasto usea numericalmethodfor the calcula-tion. Examplesof theX(W,D) resultsof typically usedgapsW•D are listed in Table1.

1J.Meier,R. Fluckiger,H. Keppner,andA. Shah,Appl. Phys.Lett. 65, 860~1994!.

2J. Meier et al., Proceedingsof the 13th EuropeanPV Conference,Nice1995,p. 1445.

3J. Meier et al., J. Non-Cryst.Solids227–230, 1250 ~1998!.4K. Yamamoto,M. Yoshimi, T. Suzuki,Y. Tawada,Y. Okamoto,andA.Nakajima,Proceedingsof the2ndWorld Conferenceon PhotovoltaicEn-ergy Conversion,Vienna,July 1998,p. 1284.

5K. Saito,M. Sano,K. Matuda,T. Kondo,T. Nishimoto,K. Ogawa,andI.Kajita, Proceedingsof the2ndWorld Conferenceon PhotovoltaicEnergyConversion,Vienna,July 1998,p. 351.

6N. Beck,J. Meier, J. Fric, Z. Remesˇ, A. Poruba,R. Fluckiger,J. Pohl,A.Shah,andM. Vanecek, J. Non-Cryst.Solids198–200, 903 ~1996!.

7M. Vanecek, A. Poruba,Z. Remesˇ, N. Beck, and M. Nesladek, J. Non-Cryst.Solids227–230, 967 ~1998!.

8A. Masuda,R. Iiduka, A. Heya,C. Niikura, and H. Matsumura,J. Non-Cryst.Solids227–230, 987 ~1998!.

TABLE I. Resultsof the interelectrodespacingfunction X(W,D) for thetypically usedinterelectrodedistanceD andelectrodewidth W ~seeFig. 2!.

W (mm) 8 8 2.2 2.2 2.2D (mm) 2 0.5 0.2 0.08 0.03X(W,D) (mm) 1.695 0.635 0.231 0.115 0.052

9F. Diehl, M. Scheib,B. Schroder, andH. Oechsner,J. Non-Cryst.Solids227–230, 973 ~1998!.

10S. Veprek, F.-A. Saarott,andZ. Iqbal, Phys.Rev.B 36, 3344 ~1987!.11A. Poruba,Z. Remesˇ, J. Springer, M. Vanecek, A. Fejfar, J. Kocka, J.

Meier, P. Torres,andA. Shah,Proceedingsof the2ndWorld Conferenceon PhotovoltaicEnergyConversion,Vienna,July 1998,p. 781.

12M. Favre, H. Curtins, and M. Vanecek, J. Non-Cryst. Solids 114, 405~1989!.

13Handbookof Optics, 2nd ed., edited by M. Bass ~McGraw-Hill, NewYork, 1995!.

14A. Shahet al., Mater.Sci. Eng.,B 69–70, 219 ~2000!.15P. Beckmannand A. Spizzichino, The Scattering of Electromagnetic

Wavesfrom RoughSurfaces~Pergamon,Oxford, 1963!, Chap.5.16H. C. Van de Hulst, Light Scatteringby Small Particles ~Wiley, New

York, 1957!.17W. H. Press,S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,

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