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modeling of the optical properties of hydrogenatedamorphous-sillicon-based p-i-n solar cells deposited on roughtransparent conducting oxide substratesF. Leblanc) and J. PerrinLaboratoire de Physique des Interfaces et des Couches Minces, CNRS UPR A0 2% Ecole Polytechnique,91128 Palaiseau Cedex, FranceJ. Schmittb)Solems S.A., 3 rue L&on Blum, ZL Les Glaises, 91124 Palaiseau Cedex, France(Received 19 April 1993; accepted for publication 4 October 1994)Hydrogenated amorphous-silicon (a-Si:H) -based solar cells consist of two electrodes and ap-i-nstructure, deposited on glass substrates. Depositing the p-i-n layers and the back metallicelectrode on an optically rough transparent conducting oxide (TCO) electrode enhances theabsorption of the incident light in the active i layer: Light is scattered at the rough front interfaceand is partially trapped in the high refraction index layer, as in a waveguide. In addition TCOroughness increases the front transmission coefficient, increasing the amount of light in theactive layer. TCO texture yields a relative increase of the conversion efficiency up to 30%. Asemiempirical model of thin-film solar-cell optics is presented, taking into account the interfaceroughness by introducing experimentally derived scattering coefficients and treating thepropagation of specular light in a rigorous way. Numerically simulated spectral response andtotal reflectance of standard solar cells deposited on different TCO textures are compared toexperimental data. The results show a better fit to measured characteristics than simulationsobtained by previous semiempirical modeling. Improvements mainly come from the lightpropagation calculation. According to the model, the number of passes ncident light may makethrough the active i layer reaches six for the most efficient cell. As an example of the modelsmain application, the enhancement of the conversion efficiency that would be expected from anoptimized TCO layer is calculated for each texture studied and for different back metallizations.

I. INTRODUCTIONThe main interest in a-Si:H solar cells comes fromthin-film deposition technology. This technology allows

low material costs and devices to be deposited over largeareas. We present in Fig. 1 a schematic description of atypical a-Si:H-based p-i-n solar cell. In order to enhancethe conversion efficiency of the cells, both electronic andoptical properties have to be investigated. The aim is tomaximize incident light absorption in the active layer andto collect all the photogenerated carriers. In the past fewyears, improvement of the optical properties has been adominating factor in device conversion enhancement. As amatter of fact, transparent conducting oxide (TCO) tex-ture is a key issue: It reduces optical reflection loss at theTCO/silicon interface and greatly increases light diffusion.Weakly absorbed radiation, when scattered, can be partlytrapped in the high refraction index layer because of totalinternal reflection at the interfaces. Due to the high refrac-tion index of a-Si:H (n z 3.7), optical confinement occursin the active layer, allowing light to experience severalpasses hrough this layer. When deposited on appropriatelyrough TCO substrates, large area conversion efficienciesintegrated over the conventional AM1.5 solar spectruma)Also with: SOLEMS S.A., 3 rue Leon Blum, 2.1. Lea Glaises, 91124PalaiseauCedex, France.Present address: Bakers Display Technology S.A., 5 rue L&m Blum,2.1. Lea Glaises, 91124 PalaiseauCedex, France.

reach 8.5% for a p-i-n/p-i-n tandem structure, as reportedby Yoshida et al.; so far device optical optimization wasbased on empirical approaches.The TCO texture consists of microcrystallites whosesize is comparable to visible light wavelengths. Because ofthe broad distribution of grain size and shape, the interac-tion between incident light and the multilayer structure iscomplex. A rigourous three-dimensional electromagnetictreatment would require enormous computation facilities,so a semiempirical approach is usually chosen. Differentgroups, namely Deckman et aL? Shade and Smith,3 andMorris et al.,4 have already addressed this problem bymore or less sophisticated semiempirical modeling.An original semiempirical model is presented here.Diffuse reflectances Rd and transmittances T, due to inter-face roughness are derived from angular-resolved photo-metric measurements, and are used as input parameters tothe numerical program. In our model, the electromagneticfields specular reflection and transmission coefficients areassumed to be proportionnal to the classical Fresnel coef-ficients, the proportionality factor depending on theamount of total diffused light. Consequently specular lightcoherence is kept and specular interferential effects, whichmay be observed experimentally with a moderately roughTCO substrate, are taken into account. With the support ofexperimental results and as a simplifying assumption,phase coherence between diffused light and incident light isassumed to be lost at the interface. Thus, at a given rough

1074 J. Appl. Phys. 75 (2), 15 January 1994 0021-8979/94/75(2)/l 074/l 4/$6.00 0 1994 American Institute of PhysicsDownloaded 13 Sep 2007 to 131.183.161.189. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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+ incident light

GlassTCOa-Si:H p-i-nMetallic backcontact

FICr. 1. Schematic description of a-Si:H-based -i-n solar cells depositedon rough TCO substrates.interface, diffused light is similar to a new source emittinginside the stack. After sampling the space in several direc-tions, the light propagating along each of these directions isassumed to be a plane wave. This diffuse light is once againpartly diffised and partly specularly transmitted or re-flected at the rough interfaces. Each of the componentscoming from this second diffusion is then treated as in thetist step. At each step, electromagnetic-field amplitudes ateach side of every interface are calculated, allowing thederivation of the Poynting vector flux and, thus, theamount of absorbed light in each layer. In this way, mul-tiple diffusion is taken into account. Instead of adding theabsorbed flux at each successive step, we have developed amatrix method procedure, based on the formal similaritybetween the calculation process for the n-times-diffusedlight and the once-diffused light propagation. This ap-proach is also applicable to CdTe or CuInSq thin-filmsolar cells.First of all, the consistency and reliability of the nu-merical code are verified by calculating the thermodynamiclight trapping limit analyzed by Yablonovitch5 and Yab-lonovitch and Cody.6 Angular-resolved-refectance( ARR) and transmittance (ART) measurements are thenpresented at four wavelengths covering the visible spec-trum for bare TCO tilms, metallized TCO, and a-Si:H-covered TCO films. Absorptances of standard solar cells(absorptance in a given layer being defined as the fractionof incident photons that are absorbed in the layer) arecalculated, using the derived diffuse reflectance and trans-mittance at the rough interfaces. Calculated cell total re-flectance and calculated absorptance in the i layer are com-pared to measured total reflectance R(A) and quantumspectral response Q(1) under negati ve bias of the solarcell. Under negative bias, the a-Si:H collection efficiency ofphotogenerated carriers is close to one, and therefore theabsorptance in the i layer and Q(L) are equal. Comparisonbetween the present numerical modeling and another ana-lytical model is discussed.II. STATE OF THE ARTA. Classical thin-film theory

The propagation of an electromagnetic plane wave in-cident upon a stack of thin layers separated by ideally flat

interfaces can be easily solved from the classical Maxwellequations. For non-normal incidence, the incident l ight isdivided into two polarization modes, the transverse electric(TE) and transverse magnetic (TM) modes, transversemeaning perpendicul ar to the plan of incidence. It is com-mon practice to associate the transverse electric field E,with the TE mode, and the parallel electric field E, withthe TM mode. In absorbing media, Ep is no longer linearlypolarized (the end point of Ep traces an ellipse in the planof incidence) . Thus, in this case, the direction of Ep is notconstant, and the scalar expression of the Poynting vectorB [see Eq. (6)] becomes complicated. For the TM mode, itis convenient to choose the transverse magnetic field H,,since H, remains transverse as it propagates through thestack (as does E, for the TE mode). For the TE (TM)polarization mode, the fields amplitude reflection andtransmission Coefficients rTnn and tTE,2 ( rrMn and tWn) atthe interface between two layers 1 and 2 depend on theelectromagnetic wave angle of incidence. It is convenient tointroduce the wave vector k and to choose as the angularvariable kxy , the component of k lying in the interfaceplane xy. According to Descartes-Snell s law, kx,, remainsconstant at the reflection and refraction at each interface,and therefore is an invariant variable when specular lightpropagates through the entire stack. Thus, ATE,*, tTE12,ITM,,, and tTMlz are giVen as fUnCtiOnS of kx,, by

+-k, %--rmn-kl~+~2~y --tTEn-klz+;2 IEl E2 - El--T%-kl k2 --

-212tT%,-k, k2

-..zl_zEl E2 El E2

wherekm== [ ~)2&-k$y]12 (m= 1,2),

due to

(1)

(2)

(3)where n; and E, are the complex refractive index anddielectric permittivity of the m medium, and L the wave-length in vacuum.

The total electromagnetic field at each interface of theentire stack is the sum of the components coming frommultiple reflection and refraction. Instead of calculatingand successively adding each of these components, the to-tal electromagnetic field is directly derived by using thepowerful matrix method of Abeles.l* A t a given interfacebetween two media m and m+ 1, in the case of the TEmode (for the TM mode, the magnetic-field amplitude Hreplaces the electric field amplitude E), fields propagatingin both directions Em,+ and E,,,,- in the medium m, arecorrelated to the fields Em, i,+ and E,,+ r,- propagating inthe medium m + 1 by the interface matrix Imlm+ I given byJ. Appl. Phys., Vol. 75, No. 2, 15 January 1994 Leblanc, Perrin, and Schmitt 1075Downloaded 13 Sep 2007 to 131.183.161.189. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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I (4)Field propagation through layer m is described by a diag-onal propagation matrix Pm,

exp ( - ikJ,J 0Pm 0 exp( +ikJ,,J (5)where I, is the layer thickness and ?= - 1. The total fieldis related to the field in the surrounding media by multi-plying the Imlm+t and Pm matrices. In the case of anN-layer stack surrounded by two semi-infinite media 0 andN+ 1, the boundary conditions (EN+ t,- =0 and%cident - E 0, + ) allow the calculation of the field in eachlayer, as a function Of Eincident.The Poynting vector I; correlates the total electromag-netic field with the electromagnetic energy transportthrough the stack. Its flux through a unit area of interfacesurface unit i s the power flux passing through this surface.B is given by

I;=$ Re(EXH*), (6)where Re (Im) refers to the real (imaginary) part of acomplex vector, and * to the conjugation operator. Whenthe total el ectromagnetic field is known, the flux Z, of Zthrough any xy section can easily be calculated. For the TEmode,

+2 Im(kJIm(P++E-1, (7)where the permeability p equals the permeability of vac-uum ,uo, for nonmagnetic media. For the TM mode

xz=& Re 2 (lH+l-- IH-1)IO+2Im 2 Im(p+H-) .0 1

The absorptance in each layer A,,, is obtained by takingthe difference of the power flux through the top interfaceand the bottom interface of the layer,B - %nbottomA,= zmtOp zinc .

z (9)This classical thin-film method takes into account lightcoherence and applies to flat interfaces. When using thismethod for a-Si:H p-i-n solar cells, interference peaks arepredicted in the i layer absorptance and in the R(L)curves. Experimentally the i-layer absorptance is obtainedby measuring Q(n), and R(A) is derived by measuring thesum of the specular and diffuse components ,of the reflec-tance. Although the predicted interference peaks areclearly observed in the experimental Q(a) and R(A)curves when the p-i-n layers are deposited on smooth ormoderately rough TCO, they completely disappear onhighly textured TCO. Moreover, the Q(n) curve inte-

grated over the AM1.5 solar spectrum usually underesti-mates the measured conversion efficiency of a solar celldeposited on textured TCO whereas it gives an upper limitof the conversion efficiency of a solar cell deposited onsmooth TCO. These findings reveal that the interfaceroughness has to be taken into account for a correct mod-eling of the optical absorptances and power balance in asolar cell.TCO texture features reach a few tenths of micronswhich is comparable to visible wavelengths. Therefore, theclassical dipole perturbative theory,g based on the intro-duction of current densities at the rough interface and onthe assumption that the mean spatial frequencies arehigher than the incident radiation frequency, is not useful.One must directly solve the Maxwell equations by takinginto account the exact surface topology and derive the elec-tromagnetic field in the entire space. Exact methodshave been developed by Maystre, by Saillard andMaystre, and by DeSanto.12 Their extension to three-dimensional randomized surfaces is still in progress. More-over, they require the most powerful calculation tools,which is obviously an obstacle for a routine application inan industrial research and development laboratory.B. Mie scattering theory

The scattering properties of small conducting particlesembedded in a dielectric medium have been analyticallyderived from the Maxwell equations by Mie. Mie scatteringtheory has already been introduced to analyze TCO sur-face scattering properties. 13*14he key factors are the ratioof particle size to wavelength and the relative index ofrefraction of the particles. The correlation between Mietheory and experimental diffuse transmittance of bare TCOfilms has been reported by O Dowd.13 In order to modelthe optical properties of solar cells, Shade and Smith14compared the diffuse reflectance and transmittance ofTCO/air and TCO/metal interfaces by using Mie theory,but the model they have proposed still belongs to the semi-empirical model class (see below). This theory is difficultto apply in the case of a-Si:H solar cells because the grainsize distribution has to be taken into account and becausemultiple scattering may occur in the stack. As far as weknow, a complete modeling based on Mie scattering theorystill has to be proposed.C. Light-trapping limit from statistical ray optics

The maximum increase of the absorptance in the ilayer due to TCO roughness has been studied by Yablono-vitch and Cody.6 Consider a layer with an index of refrac-tion n=n+k, covered at the back interface by an idealreflector [see Fig. 2), and such that complete randomiza-tion occurs in the layer (i.e., light i s totally uniformly dis-tributed in the layer or, in other words, the luminance isconstant). The mean transmittance coefficient from the airto the layer is written as Ti, and the mean transmittancecoefficient from the layer to the air as T,,,. According tostatistical ray optics, when the layer absorption coefficienta: tends to zero ((II =h~/d ) , the absorptance enhancement1076 J. Appl. Phys., Vol. 75, No. 2, 15 January 1994 Leblanc, Perrin, and Schmitt

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I ncident light1,int Tescno = 1 4critical angleperfect diffusorR= Rd=l

FIG. 2. Du e to the interface roughness, ncident light is randomly dis-tributed in the layer. For low absorption coefficients, ight is trapped inthe layer because f its higher index of refraction, and hence he absorp-tance is enhanced.

factor G=A/AI (A being the absorptance in the layer andAl the absorptance along one orthogonal pass) reaches amaximum value G,, given byGmax=4n2 2.esc (10)If the incident light is assumed to have an angularly

uniform distribution, then Tint= T,, . Therefore, in the redpart of the spectrum where a&i:. tends to zero and where%-Si:H z 3.7, the maximum theoretical limit of the absorp-tance enhancement due to light trapping in a-Si:H solarcell is G,,, =4x3.72do.D. Semiempirical modeling

Because of the considerable complexity of the previ-ously mentioned electromagnetic approaches, the mostcommon models encountered in the literature24 are basedon a semiempirica l treatment of light diffusion due to theTCO texture.These models share the following common assump-tions: (i) light coherence is not taken into account, p re-venting the prediction of interference pa tterns; (ii) therough inter face diffuse reflectance Rd and transmittance Tdare either experimentally determined or derived from theMie theory, and scattering angular laws are Lambertian(cosine law); (iii) only two interfaces are assumed to berough, the TCO/p interface and the n/back reflector inter-face; (iv) at a given interface, transmittance is either 1inside the optical admittance cone, or 0 outside (i.e. totalinternal reflection occurs).Because of these simplifying assumptions, light propa-gation through the different layers can be represented by amean beam. If we assume a total Lambertian diffusion,for example, this mean beam path through a layer is twicethe layer thickness. Absorptances are calculated by sum-ming the absorptances along this mean beam coming fromeach successive diffusion. Thus, the absorptance, written asthe sum of a geometrical series, has an analytical expres-sion. In Shade and Smith mode13P4 iffuse reflectance atthe back interface (n/metal) is correlated to diffuse reflec-tance and transmittance at the front interface (TCO/p)according to Mie theory. Thus, the p and n layers are takeninto account. In the model proposed by Morris et aI., dif-

5 C0 ) \ Tspecular

FIG. 3. Schematic llustration of the assumptions of the model for thespecular and diiuse transmission and reflection of the TE mode. Thesame assumptionsare made for the TM mode, where H replacesE.

fuse transmitted light is initially neglected, and the as-sumption of partial scattering at the back rough interface isthen proposed in order to obtain a better simulation of theQ(A) curve.When the back reflector exhibits a high reflectivity,these models may poorly simulate the measured Q(n) andR(A) in the red part of the visible spectrum. As discussedlater, some discrepancies4 become appearent, revealing thelimits of the above-mentioned simplifying assumptions.The model we propose is also based on a semiempiricaltreatment of interface diffusion: Diffuse reflectance andtransmittance are used as input parameters. However,specular light propagation is treated according to therigourous plane waves equations as previously presented inthe matrix method. Thus, interferential effects are takeninto account, and fewer assumptions are made. Netherthe-less, contrary to the models proposed by Deckman et aI.,Shade and Smith,3 and Morris et aZ.,4 there is no possibilityof obtaining an analytical solution.

III. DESCRIPTION OF THE MODELA. Reflection and refraction at a rough interface

At a given rough interface, incident light is dividedinto two specular and two diffuse components, as shown inFig. 3. The transverse electric (magnetic) specularly re-flected and transmitted fields amplitude coefficients r& andt+E (r& and t&) are assumed to be proportional to theFresnel coefficients rTE and tTE (rTM and t-& with a pro-portional factor yTE (j+M),

rh= KErTE %FYTMrTM), (11)

The factor 7/TE (YTM), which depends on the diffusereflectance R, and transmittance Td at the interface, i sderived by equalizing the power flux, before and after theinterface,J. Appl. Phys., Vol. 75, No. 2, 15 January 1994 Leblanc, Perrin, and Schmitt 1077Downloaded 13 Sep 2007 to 131.183.161.189. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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rough interface rFJIG.4. Normal componentsof the Poynting vectors on both sides of arough interface.

TE: 1+2 WWmhdWk,) rE-2 WWm(rdRe(kJ y~~=l-Rd-Td,

TM: 1+2 WkJWrTdRe(k) AM-2 Im(k)Im(r,,)Rdk) YTM= I--%- Tti,

(12)

(13)

where k, is the normal component of the incident wavevector k. Thus, the incident light specular propagation isdetermined by using the matrix method with the corre-sponding amplitude coefficients. From the specular fields,absorptances At in each m layer are derived.B. Propagation of light through the multilayer stack

After determining the absorptances A: along the spec-ular incident direction, the diffused flux must be analyzed.Each half-space surrounding the interface is sampled inseveral directions corresponding to an increment&ion ofthe in-plane component kxy of the wave vector k.At a given rough interface r, according to the previouscalculation [see Eqs. (7) and (S)], the z component%rLght and %left of the specular Poynting vectors on boththe right- and left-hand sides of the interface (see Fig. 4)corresponding to the specular fields are known (when theinterface is smooth, due to the power conservation law,ZLight and XLfi are equal). For the left-hand side of the rinterface (the s ame assumption is made for the right-handside), in order to separate the incident flux 2:,,+,,, on theinterface from the emergent flux Z$-teft, we assume that~~~eft=~~~,+left-2~~,--IeftY (14)where, in the case of the TE mode,%,+I&=&-Re(k,) IE, ~2+Im(kz)Im(~+~~)l (15)and

x0,,,=& [Re(k,) jE_ 12--Im(kZ)Im(E*~-)]. (16)For a nonabsorbing med ium, the interferential terms inEqs. (15) and (16) vanish, becauseIm(k,) =O. (17)We have numerically veritled that in the case of a typ-ical p-i-n sol ar-cell multil ayer stack, the interferentialterms do not exceed 10% of the quadratic terms, allowingus to artificially separate the incident flux and the emergent

flux. Thus, the st ecular calculations allow us to derive theincident flux ZZZr,+left nd Zir,-rizht on each side of therough interface r. In order to simplify our notation,$+,a and %.,-right will be replaced by X&+ and 2&-,respectively.For a given km value, we introduce the classical angu-lar direction 8,

(18)The angular diffuse reflectance Rd( 0) is defined as thereflected power around the direction 8 per unit solid angle.

It is related to the diffuse reflectance Rd according tos

rr/2Rd=h- Rd( e) Sh 8 de. (19)0Thus, the diffused reflected flux into the left-hand sidein the direction 6 corresponding to kx,, is given by2?rRd(B)sin 8 de 2zr,+. (20)The electromagnetic field due to this new source iscalculated in all theupper and lower layers. In each layerm, the absorptance Ak(k,,) coming from this first ditfu-sion and calculated along the kxy direction is derived, as are

the new incident fluxes &, + (k,) and 2& _ ( kxy at roughinterfaces.Due to the phase coherence loss inherent in diffusion,absorptances A;( kJ and incident flux on rough interfaces2ir,+ (k,,,) and X&,.- (k,) in every kx,, direction are addedin order to evaluate the total absorptance Af, and the totalincident flux on rough interfaces 8&, and X&- due to thefirst diffusion:

+%r,+(&nax), (22)and

+%,-&m,). (23)The new incident flux on the rough interfaces is onceagain diffused, and it is possible to similarly derive theabsorptances A& and the new incident flux 8ir,+ and X&-.The total absorptance A, is thus given by the successiveaddition of each diffusion step contribution.

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It is important to notice that the computation proce-dure is the same at thejth diffusion step to derive AA, X6+,and X&- from the incident flux Z&+r and Z&l. In orderto take advantage of this linearity, we have developed amatrix method that greatly decreases he computation timeand the effects of numerical errors. Let us tirst define thecolumn vectors {AA) and {Zr,+/XF} (fluxes are normal-ized to the incident flux Zr in order to have homogeneousvariables A, and Zr,*/Xy):

\(24)

The superscript j indicates that the parameters are re-lated to thejth diffusion step. {Af,} has Melements (M-2is the number of layers in the stack, and the index 0 refersto the incident medium, whereas index M- 1 refers to theemergent medium), {Z,-& /Xp} has 2 x R elements (R isthe number of rough interfaces and both + and - direc-tions are taken into account). In order to use the linearityin the computation process, we introduce the following [A]and [F] matrices:{A/m = [A]{Bj-/X)r,* a 2 (25)pj /yy= pq{I;j-/&C}s?r,* z zr,* z * (26)Each successive contribution can be expressed as afunction of the specular normalized flux incident on therough interfaces {Xk,,/X.), i

{A/,>= [A] [F]{X,;;2/X~)= [A] [F]il{Z;,.,/B~}. (27)The total diffuse absorptance {A?}, meaning the ab-sorptance due to scattering, is given byCA~ff~=CA~)+{A2,)+...+{Ai,}+...

=[A]([F]+[F]+...+[F]j-+...)x cgr,*/z$% (28)

Thus, [Id] being the identity matrix, it is easy to derive@$flh

CA:?= [Al ([Id] - [Fl) -%$.JX~), (29)and the total absorptance {A,) is the sum of the specularand diffuse components,

CA,)=#J+-L4:ffl. (30)J. Appl. Phys., Vol. 75, No. 2, 15 January 1994

no=1plane interface

incident&llimatedbeam

FIG. 5. Ideal test structure to verify the light trapping thermodynamiclimit.

In practice, {A:} and {I;~~,JX~} are tlrst calculated,and the [A] and [r;l matrices are determined when calcu-lating (AL) and {Z&,/E,) following the procedure de-scribed at the begin&g of this subsection.C. Input parameters

Consider ing Fig. 1, showing the multilayer structure ofa typical p-i-n solar cell, w e see that there are four roughinterfaces. However, due to the low p- and n-layer thick-nesses n comparison to the texture feature si ze, it is justi-fied to consider that the p/i and i/n interfaces are opticallyflat. Thus, we have assumed that in p-i-n solar cells, onlytwo interfaces are rough, namely the TCO/p (front) andn/back reflector (back) interfaces. Another simplifying as-sumption is that diffusion is symmetrical between two ad-jacent m and m+ 1 layers,R d,m+ma=Rd,m/m+ 1,Td,m+wn=Td,m/m+l.

(31)Thus, at a given incident wavelength A., the opticaleffects of the TCO texture in the stack are modeled by fourparameters: the diffuse reflectances Rd rront, Rd back nd thediffuse transmittances Td rront Td back. These coefficientshave been experimentally derived at selected wavelengths.Due to the high index of refraction of the a-Si:H layer,the sampling interval for the space variable k,,(d/2r) hasbeen fixed to [0,4] with an increment step equal to 0.1.Therefore, the number of directions in the air is 10 butreaches 40 in the a-Si:H layer.

IV.. NUMERICAL VERlFlCATlON OF THETHERMODYNAMIC LIGHT-TRAPPING LMTAfter the usual testing on plane interface structuresand in order to evaluate the effects of numerical errors onthe final accuracy, we have numericall y verified the ther-modynamic light trapping limit. Let us consider a layer ofan absorbing medium (see Fig. 5 ) having a complex indexof refraction E=n +k and with a thickness I. The frontinterface is assumed to be flat, whereas the back interface isassumed to be an ideal reflector and di&sor. The incidentbeam is collimated and normal to the layer.

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~ 0.8zg 0.622 0.44

0.20.0 0 1 2 3

(4 al

c9Bz829-weeg%apx42%

20

1510

5

0

18 16 I b II II I

,

0 1 2 3(b) al

FIG. 6. (a) Calculatedabsorptance n the medium of index n=2 (cf. Fig.5) as a function of al for a perfectly absorbing (square dots, Z+O) anda perfectly reflecting (solid line curve, Rd= 1) di ffusive back contact. (b)Absorptance enhancement actor G [ratio of absorptances f (a)] as afunction of al.

We have studied the variations of the absorptance A asa function of al, for al ( 1. n was taken to be 2 for thenumerical test. The mean power transmission coefficientsThe and T,, at the front interface have been calculated5.7according to

Tim=; INTEL'=: [~TMI' (at k,=O>, (32)

xO0seshede. (33)When the medi um becomes less and l ess absorbing,

according to the light-trapping limit, the enhancement fac-tor G reaches the maximum valueGmax=4n2 g= 19.35. (34)escWe present in Fig. 6(a) the calculated absorptancesfor different al values. When al is low, absorptance A, forone normal pass is close to al. As we can see n Fig. 6 (b) ,the ratio A/A, tends to 19.2 when al tends to zero, whichis in very good agreement with the theoretical GmaX alue.Thus, the numerical algorithm permits us to decrease thecomputation time while keeping numerical accuracy.

laser beam

Transmission : ART-I Reflection : ARR180 2 0 2 90 I 9o"reroSampleplane

9=90"

FIG. 7. Chosenconventions or the ARR and ART measurements.

V. EXPERIMENTAL DETEFMNATION OF THESCATTERING COEFFICIENTSA. Experimental setup

The experimental setup consists of a crystalline silicondetector placed on a rotating arm. The light source is achopped krypton laser beam, alternatively tuned at thefour following emitting wavelengths: 472, 568, 647, and752 nm. A lock-in amplifier is used to extract the sensorsignal. By measuring the signal corresponding to the inci-dent beam flux (provided that the sensor active area islarger than the incident beam diameter), it is possibleto deriveI the normalized diffuse angnlar-resolved-reflectance (ARR) or -transmittance (ART). The schemepresented on Fig. 7 shows the notation we have chosen.

B. SamplesTwo types of TCO substrates have been studied. Themain characteristics of the substrates are summarized inTable I. Absorptances of the bare TCO substrates placed inethanol are measured by the use of photothermal deflexionspectroscopy (PDS), the PDS spectra being calibrated bymaking use of the joule heating i nside the film~.~ TCOtexture is analyzed by the scanning electron microscopy(SEM) technique.The TCO substrates are covered either with aluminumor with a-Si:H. Thus, optical diffusion properties of theTCO/air, TCO/a-Si:H, and TCO/alumi num interfaces

have been investigated and compared.TABLE I. Main characteristicsof the TCO substratesnos. 1 and 2.

TCO substrate 1 2Glass hickness 1.1 mm 2 mmTCO thickness 700 nm 1200 nmMean roughness eature size 170 nm 300 nmTCO square sheet esistance 6.5 i-k/n 9 Q/OTCO total absorptance AM1.5) 6.9% 5.3%Haze ratio H. at 633 nm 7% 13%



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TABLE II. Summary of angular-resolved-reflectance ARR) and-transmittance (ART) measurements achieved on different polishedstructures. + ( -) refers to an incident light coming from the glass sub-strate (polished surface) side. Structures are polished n order to inde-pendently analyze he diffusion propertiesof the front and the back roughinterfaces.Structurea 472nm 568 nm 647 nm 752 nm

TCOl/a-Si:H (18 pm) ARR+ Am+ ARR+ ARR+,-polished ART+,-TC02/&i:H (7 pm) A=+ Am+ AI=-!- -+,-

polished ART+, LTCOI/Al (150 nm)/ ARR- ARR-a-Si:H (7 pm) polishedTCO2/Al (150 nm)/n-Si:H (7 pm) polished ARR- ARR-

In the case of the a-Si:H-covered samples, in order toderive the diffusion proper ties of the front and back roughinterfaces separately, thick a-Si:H layers are deposited ei-ther on the bare TCO films (front interface) or on a 2000-A-thick aluminum layer previously deposited on the TCOfilms (back interface). Free a-Si:H surfaces are then pol-ished, as summarized in Table II. This polishing isachieved by using a granular diamond paste. Thus, theresulting stacks have only one rough interface, whichmakes the analysis easier. In the near-infrared region,TCO/a-Si:H samples have been irradiated from both the+ and - sides. Moreover, assuming that the 2000~A-thickaluminum layer conformally covers the TCO texture, andthat the aluminum/a-Si :H interface scattering propertiesdo not depend on the deposition order, the back interfacediffuse reflectance is also investigated.C. Results1. Angular-resolved reflectance and transmittance

As a general result, A RR measurements for both T COroughness and for each interface present a mostly linear

s O-O2

0 20 40 60 80e (degrees)FIG. 8. Measured (square dots) and simulated (solid line curves) ARRfor the glass/rC!Ol/p (ZP=8 nm)/i (Z;= 18pm) structure, at A=472 nm.Curve 1: Rd(B) is linear with a total reflectance Rd=13%. Curve 2:RJ 0) is proportional to cos 9 with Rd= 10%. Curve 3: Rd( 8) is Lam-bertian (cosine aw) with R,= 14%.

0 20 40 60 800 (degrees)

FIG. 9. Sameas n Fig. 8, with two linear diffuse retkectanceaws Rd( 0)and different total refl ectanceRd. Curve 1: R,= 13%; curve 2: Rd= 10%.

angular dependence. The experimental ARR and ART an-gular dependence does not demonstrate any structure orany resonance. Similar results have been reported by Shadeand Smith.14In Figs. 8 and 9, the solid line curves correspond tosimulated A RR curves for the entire stack, obtained fordifferent angular diffusion laws Rd ,,,,(0> at the front in-terface, and square dots correspond to the measured ARRat the incident wavelength 472 nm. The experimental an-gular dependencies can be approximated by using a linearlaw, but cosine laws to the power 1 or 2 could also be used,as shown in Fig. 8. In order to show the effect of the totaldrffuse reflectance Rd front, we present in Fig. 9 results for.two different values of Rd front.At the near-infrared wavelength 752 nm, ARR andART measurements obtained when irradiating from bothsides are shown in Fig. 10, where solid line curves corre-spond to simulated values.

0 30 60 90 120 150 1808 (degrees)FIG. 10. Measured (points) and simulated (solid line curves) ARR andART measurements t A=752 nm for the glass/TCOl/p/i (Zi=7 pm)stack, after polishing of the i layer surface. For the simulations, bothRd(0) and Td(0) at the rough TCOl/p interface are assumed o besymmetrical, and therefore are the only free parameters.

J. Appl. Phys., Vol. 75, No. 2, 15 Januar y 1994 Leblanc, Perrin, and Schmitt 1081Downloaded 13 Sep 2007 to 131.183.161.189. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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2i c 0.6218292.5 0.4

(4 Wavelength (.nm)

o.o~,.,.,,,.,,.,..,.,,,,,,,,,,~,,,,,*,-450 550 650 750Wavelength (nm)FIG. 11. (a) Spectral dependenc e f the diffuse reflectanceRd and trans-mittance T, for the TCOl/air (-0-: Rd; -8-: Td); the TCOl/u-Si:H (-U-z Rd; I-: TJ, and the TCOl/Al (--A--: Rd) interfaces.(b) Spectral dependenc e f the diffuse reflectanceRd and transmittanceTd for the TCO2/air (-0-: Rd; -O--z TJ, the TC02/&Si:H(-C-z R,; -W---: Td), and the TCOZ/Al (-A-: Rd) interfaces.

2. Spectral dependence of diffuse .reflectance andtransmittanceWhen simulating as closely as possible each of the ex-perimental curves by using different angular diffusion laws

&,,,(e), Td ,,,(e>, Rd bade), and Td b&(e), the to-tal diffise reflectance Rd and transmittance Td for eachinterface and for the four wavelengths are estimated [seeEq. (19)l.We present in Fig. 11 the spectral variation of~the Rdcoefficients for the TCO/air, TCO/a-Si:H, and TCO/aluminum interfaces, for two different TCO roughnesses,TCOl and TC02.For both TCO roughnesses the diffuse reflectance Rd isthe lowest for the bare TCO layers, and the highest for theTCO/aluminum interfaces. Rd in the air is higher for themost textured (TC02) substrate, and it increases with de-creasing incident wavelength. This wavelength depen-dence, still valid for the moderately textured TCOl/a-Si:Hinterface, is not found anymore for the TC02/a-Si:H in-terface, where Rd reaches a maximum value at 568 nm.The former spectral dependence can be related to Raylei ghscattering, whereas the latter may signify Mie scattering, asreported by ODowd.t3 For the TCO/aluminum interface

Rd is about the same for both substrates, and it seems tosaturate at a value of 65%. Such a saturation effect hasbeen reported by Shade and Smithi when calculating, ac-cording to Mie theory, the scattering properties of silverparticles in air.ART measurements show that Td is higher than Rd forthe TCO/air and TCO/a-Si:H interfaces. For the TCO/airinterface, the increase of Td with decreasing incident wave-length is also higher than that of Rd.For the back n/Al rough interface, the experimentallyderived Rd frontdiffuse reflectance is about 60% for TCOland 40% for TC02 at an incident wavelength of 752 nm.The ARR measurements are not very accurate, since aremaining parasitic reflectance at the polished air/a-Si:Hinterface can make a non-negligible contribution to thetotal reflectance.

3. ConclusionWe have experimentally observed that the diffusionproperties of the TCOAayer m interface strongly dependson the refraction index of layer m. The TCOl/air and

TC02/air interfaces are respectively moderately and veryrough, whereas the TCOl/Al and TC02/Al interfacesshow the same diffuse reflectance. Moreover, according tothe experimentally determined diffuse reflectance of theTCO/a-Si:H interfaces, it seems that the TCO roughnesseffects on actual solar cells optical properties are not di-rectly linked to spectrophotometric measurementsachieved on bare substrates. This might be related to thefact that several studies (see, for example, Go rdon et al. 16)have reported a saturation in the solar-cell conversion ef-ficiency improvement when increasing~ the TCO diffusetransmittance. According to Mie theory, the dominatingfactor is the ratio of the texture feature size to the wave-length in the material, as reported by Shade and Smith14 bycomparing the diffusion coefficients of silver sphe res andthe same TCO spheres in air.

VI. NUMERICAL SIMULATIQN OF THE OPTICALABSORPTANCES IN SOLAR CELLSA. Quantum spectral response of devices depositedon a moderately rough TCO substrate

We have deposited two p-i-n structures on the TCOlsubstrate, covered with two different back metallizations.The first stack consists of TCOl/Z,=8 nm/lj=320nm/I,=40 n&Al. We present in Fig. 12 the experimentalQ(1) and simulated absorptances in the i layer, obtainedfor different diffuse reflectances and transmittances.4t wavelengths lower than 600 nm, due to the higha-Si:H absorption coefficient, only the front interface needbe taken into account. If we use the derived spectralRd front diffuse reflectance and a low Td front diffise trans-mittance, the simulated Q(n) is lower than the experimen-tal value, and it shows a larger interferential effect. Sincethe Td ti,,nt value can not be experimentally derived, wehave assumed that Td front is much larger than the



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w 1.0gg 0.825 0.6-ij *tjj 0.4E3 0.22a 0.0

t- TCO#l 4

400 500 600 700Wavelength (nm) 800

3 1-os,a 0.82g 0.6zg 0.4E3 0.2

cJ 0.0 400 500 600 700Wavelength (nm) 800

FIG. 12. Measured (square dots) and simulated (solid and dashed-linecurves) Q(n) for a glass/TCOl/pi-n/Al solar cell. Rd frontvaluesare theexperimentally derived values [see Fig. 11 a)]. Dashed line: T, frontas-sumed to have the same spectral variation as Rd nont solid line curves:T d rmnt= 80% at 400 mn, and T,+ rront= 50% at 800 nm, with a linearT d t&1) spectral aw. Td n&19) is assumed o be proportional to COS?0. At the back rough interface, Rd t&(e) and Td ,,,(e) are assumed obe Lambertian, with different Rd backand Td ,,a& values. Curve A:Rd back 60%, Td back= 40%; curve B: Rd brek= 40 %, T,, back= 40%;curve(=: Rd back 60%, Td b3.& 60%.

FIG. 13. Q(1) for the samesolar cell as n Fig. 12, with a n IT0 (Ino=nm)/Ag back contact. Rd front nd Td R,,ntare the same as n Fig. 12, oncurves A, B, and C. Curve A: Rd back= 40%, Td back= 10%; curve B:Rd back m%, 7-dback 20%;CUNe c: R,, back 40%, Td back= 40%;curve D: R, baek O%, Td back 100% (no light reflectedat the ITO/Aginterface).

Rd front value. For all the following simulations on TCOl,Rd front is the same as previously derived, and Td front is80% at 400 nm and 50% at 800 nm.These high values of the diffuse transmittance are re-lated to the texture effect at the TCO/p interface: Due tothe large grain size and by analogy with effective mediatheory, the transmittance is higher since the interface is nolonger optically abrupt. In order to take into account thisphenomenon (previously reported by Walker et al. l7 andother groups), we have chosen an angular dependence lawvarying as a cos3 function, which is more centered aroundthe normal direction.

with an interferential effect in the i layer, which is en-hanced by the low index of the IT0 layer. The best fit isobtained when taking Rd sack= 40% and Td back= 20%. Inorder to show the benefit of the back ITO/Ag reflection,we report the calculated absorptance corresponding toR d back = 0% and Td back = 100%. In this case, no light,either specular or diffuse, is back reflected at the ITO/Aginterface.We have also calculated the i layer absorptance for athicker IT0 back layer ( Zrro= 60 nm), keeping the otherparameters constant. The short-circuit current integra tedover the AN.5 spectrum increases from 13.1 to 13.5mA/cm2.

The rough front interface Rd and Td being determined,we present in Fig. 12 three simulated Q(L) curves obtainedfor different Rd back and Td back values. For Rd back= 40% (60% ) and Td back= 60% (40%), light diffusion iscomplete at the back interface, whereas for Rd back= T d back= 40%, 20% of the light remains specularly re-flected and refracted at the back rough interface. Thesevalues of Rd back are in good agreement with the valueexperimentall y determined (i.e., Rd back=: 4O%)..The ex-perimental and the calculated Q(n) curves present a van-ishing bump around 650 nm, which corresponds to an in-terferential effect occuring in the TCO layer. It does notoccur in the i layer since it is observable even when diffu-sion is complete at the back interface.

B. Total reflectance and quantum spectral responseof devices deposited on very rough TCOsubstratesWith the use of our numeridal program, we have ana-lyzed the experimental R(n) and Q(a) curves publishedby Morris et aL4 Different p-i-n structures ( P= 12n&Z,=525 nm/Z,=30 nm) are deposited on a TCO sub-strate (TCO M) with ZTm M=650 nm, and four backreflectors are used.

The second stack consists of TCO1/1,=8 m/Z,=460nm/Z,=30 mn/ITO(Zrro= 5 nm/Ag. We present in Fig.13 the experimental Q(d) and simulated absorptances inthe i layer obtained for different Rd back nd T, back alues.The back rough interface is assumed to be the XTO/Aginterface, the n/IT0 interface being flat. A bump clearlybecomes appearent around 640 nm, narrower than the pre-vious one (see Fig. 12). This second bump is associated

Due to the absence of interferential pattern in the ex-perimental spectra, we infer that the TCO M roughness isvery high. In order to analyze the diffuse reflectance andtransmittance of the front rough interface and to verify therefraction index values, we first study a stack with a verypoor back reflector (molybdenum reflector). Between 400and 600 nm, the reported experimental Q(A) exhibits avery high value, as shown in Fig. 14. In order to accountfor this h igh value, it i s necessary to increase Td front. Forthe following computations, we assume that Rd front s thesame as previously measured with the very rough TC02,and that Td front s such thatJ. Appl. Phys., Vol. 75, No. 2, 15 January 1994 Leblanc, Perrin, and Schmitt

IT0 / Ag contact

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aE8 "u 0.8MEi3%b$ 0.6zwIaH 0.4*.dEftag 0.2g 00

t Q(k) - TCO Mw MO contact

500 600 700Wavelength (nm)

FIG. 14. Measur ed squa re dots: from Morris et al. (Ref. 4)] and simu-lated (solid line curves) Q(L) and R(1) for a glass/E0 M/p-i-~/MOsolar cell. For the front rough interface, Rd fmnthas he samevalue as forTC02 (very rough TCO substrate), and Td fmnt = 0.95 - Rd front. Solidline curve: cTco sed n the simulation is derived from PD S measurementperformed on TCO2 Dashed ine: rcrW s given by Morris et al. (Ref. 4).

T d front=O.g5-- Rd front * (35)For the computed simulations, we use the same indicesof refraction as the values given by the authors of Ref. 4.However, when using the TCO M extinction coefficient Kvalues given by the authors, it becomes appearent that theQ(k) calculated values underestimate the experimentaldata, as shown in Fig. 14. TCO thin films show low ab-sorption levels over the visible spectrum, and it is usuallydifficult to measure K. According to the photothermal de-flection technique we have developed,15 lower extinctioncoefficient values are derived compared to the above-mentioned values. Due to the fact that a better fit is ob-

tained, we choose to use the K values derived by PDS onTC02. Eventually, we assume that TCO M and TC02have the same diffuse reflectance and the same index ofrefraction, but different thickness.

%gz-w2;%m;22%aagcn72EgBcl23

Pl.OC ,1111,1111,,,11TCO M.8 Q(h) Al contact

0.2 k-0.0 600 650 700 750 80(Wavelength (nm)

PIG. 15. Measured [square dots: from Morris et al. (Ref. 4)] and simu-lated Q(L) and R(L) for the samesolar cell as n Fig. 14,with an Al backamtact. Rci rontand Td rrontare the same as in Fig. 14. Curve A:Rd back = 25%, Td back 65%;CUNeB:& bact = 40%, Td back= 60%.

s 1.0 I I I I ,I 1 I I, 1 I I I, 1 I I I5 TCO MQw 0.8a;-2';siz 0.6gw!a& 0.4EZ$ 0.2i!cy 0.0 6

IT0 / MO contact

650 700 750 800Wavelength (nm)

FIG. 16. Measured squaredots: from Morris et al. (Ref. 4)] and simu-lated Q(L) for the same solar cell as in Fig. 14, with an ITO/Mo backCOntaCt. Rd back 20%, Td ,,ack 70%.

In the case of the MO metallization, both simulatedQ(d) and R(d) curves are in good agreement with theexperimental curves when using Rd back = 6% andT d back= 92%. For the other back metallizations, we onlychange the VEtlUeSOf Rd back and i-d back, keeping all theother parameters constant.For the three other back metallizations, experimentalQ(d) and R(d) curves are compared to calculated curvesfor different Rd &,& and Td back alues, as shown in Figs.15-17. For the aluminum (ITO/Mo) back contact, a goodagreement is obtained when Rd back = 25% and Td back= 65% (Rd back 20% and Td back 70%). As mentionedby Morris et al., the ITO/Ag back contact i s the mostinteresting case since it strongly reflects the incident light.A fairly good agreement is obtained when using Rd back= 70% and Tiback = 25%. We report in Fig. 17 the simu-lated Q(1) and R(d) curves in the case of a perfect diffu-SOT (i.e., & back 100% and Td back 0%).

% 1.0E2-a2;

0.8zz3 0.63iw

cn% 0.4E+$ 0.22a 0.0 600 650 700 750 800Wavelength (nm)

FIG. 17. Measured squaredots: from Morris et al. (Ref. 4)] and simu-lated Q(n) and R(1) for the same solar cell as in Fig. 14, with anITO/Ag back contact. Solid line curve: Rd back 7O%, T,j back 25%;long-dashed ine: Rd back= 40%, Td &k = 20%; short-dashed line:Rd back lOO%,Td back 0% (perfectdiffusor).1084 J. Appl. Phys., Vol. 75, No. 2, 15 January 1994 Leblanc, Perrin, and Schmitt


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zz32 0.6it""a 0.4zE-g 0.22cz

0.0 600 650 700 750 so0Wavelength (nm)

FIG. 18. Same as in Fig. 17. Solid line curve: Rd back= 70%, Td brck= 25%; dashed ine: from the model of Morris et al. (Ref. 4).

VII. DISCUSSIONA. Comparison between previous semiempiricalmodel and present model

As mentioned by Morris et LzL,~ oth Q(;l> and R(A)curves must be taken into account in order to model theoptical properties of a-Si:H solar cells. The back diffusereflectance and transmittance experimentally determinedby the authors and used as input parameters of their ana-lytical modeling are very close to the values we used inorder to fit the experimental curves. Thus, apart from thelight propagation treatment and the TCO extinction coef-ficient K, the input parameters of the two models are ap-proximately similar.When using the model of Morris et uZ.,~ he simulatedQ(1) and R(A) curves for the ITO/Ag back metallizationrespectively overestimate and underestimate the experi-mental curves (see Fig. 18). As mentioned by the authors,any attempt to reduce the computed Q(d) values leads toa decrease in the corresponding R (2) values which cannotreconcile with experimental observations and demonstratesthe limitation of their modeling. Figure 18 shows that abetter fit is obtained between the experimental and simu-lated optical characteristics when using our modeling. Thisimprovement mainly comes from a better treatment of thespecular light propagation through the multilayer stack.B. Optical loss in the TCO layer

One of the main aims of the a-Si:H solar-cell opticalanalysis is to evaluate the optical loss in the front trans-parent electrode. The usual TCO thin-film qualificationconsists in measuring the optical specular and diffuse trans-mittances of bare TCO substrates and tentatively relatingthese characteristics to the optical properties of the cell.The model we have proposed allows an accurate estimationof the potential enhancement of the cell short-circuit cur-rent density .Jq, hat may arise from an ideal optimizationof the TCO optical transparency. In order to calculate thisenhancement, we introduce for two TCO substrates(TCOl and TCO M) a corresponding ideal substrate hav-ing the same film thickness, the same topology and hence

-ii-i: 0.68k 0.42;: 0.2zcJ 0.0 400 500 600 700 so0(4 Wavelength (nm)

8 1.0.0gE: 0.8.8

.2-iIb 0.6.6zma 0.4.4E2 0.2.2sa 0.0.0

40000 50000 60000 70000 so0o0M Wavelength (nm)avelength (nm)

FIG. 19. Effect of TCO absorption losseson the quantum spectral re-sponseof a solar cell for two different cell structures. (a) glass/TCOl/p-i-n/Al. Solid line curve: sameQ(n) as n Fig. 12 (curve B); dashed ine:~~~~~ =0 (ideal TCOl transparency) . (b) Gla ssMCO M/p-i-n/ITO/Ag.Solid line curve: same as in Fig. 17 (solid line curve); da shed line:~~~ M=O (ideal TCO M transparency).

the same interface diffuse reflectances and transmittances,and the same real index of refraction. The extinction coef-ficient KTCO is artificially put to zero, and the same com-putations as before (i.e., with the samep-i -n structures andback metallizations) are performed in order to comparethe Q(n) curves (see Fig. 19 . For four of the previouslystudied structures, we also present in Table III the calcu-lated integrated short-circuit current density J,, and theincrease AJ,, that would be expected from an optimizedTCO transparency. AJ,, increases with the TCO roughnessand with the back contact reflectivity. For the roughestTCO and for the ITO/Ag back contact, AJ,, reaches themaximu m value 1.16 mA/cm2. For the aluminum backTABLE III. Short-circuit integrated current density J,, (mA/cm) andincrease AJ,, calculated by comparing the simulated quantum spectralresponseQ(1) of realistic cells and of identical cells depositedon a per-fectly nonabsorb ing (ideal) TCO substrate.

Moderately rough TCO (TCOl) Very rough TCO (TCO M)JSC AJx AJJJsc 4, AJsc AJSJJX

Al 11.98 0.72 6% 12.82 0.95 7.4%ITO/Ag 13.10 1.06 8.1% 13.89 1.16 8.4%

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s .m Ch; 6: TC Oz- _ IT0 I AgSC, 5:E=g 4-.ccmg, 3:

Mcontact

.1i ..

p$ 2 : , .BE 1 .ym=m, I t I t * t I I I t2 ,z 600 650 700 750 800Wavelength (nm)

FIG. 20. i layer absorptance nhancemen t actor G calculated or themost efficientsolar cells (TCO M and TO/Ag back contact). For low alvalues, G represents he equivalent number of orthogonal ight passesthroug h the i layer.

electrode, it is noticeable that AJ,, is lower in the red partof the visible spectrum, and thus that the main benefitoccurs at short wavelengt hs. Due to the a-Si:H/Al inter-face absorpti on, light trapping for red light is limited bythe back electrode absorption, and a poor en hancemen t isexpected rom the TCO transparency optimization. For theITO/Ag back contact, AJ,, is the same at wavelengths~600 n m beca use of the high absorption of a-Si:H in thisregion; however, AJ,, is higher in the red part of the spec-trum, where light trapping is more efficient.Our computations demonstrate that the highest benefitis obtained when usi ng rough TCO substrates and highlyreflecting back contacts. For a moderately rough TCO sub-strate and aluminum back electrode, the light trapping ef-ficiency and hence AJ,, are limited by the back contactspoor reflectivity.

C. Maximum number of equivalent passes throughthe active layerComparing the calculated i layer absorptance to theabsorptance Al for one orthogonal pass allows us to derivethe absorptance enhan cement factor G, which correspondsfor low al values to the number of equivalent passesthrough the active layer. In o rder to calculate Al , we

assume hat the back layer is a perfect absorber. Figure 20shows the calculated G in the case of the most efficient cell(i.e., very rough TCO substrate and ITO/Ag contact). Greaches he maximum value of 6 for SO0nm. This value ismuc h lower than the light-trapping limit (G=50) previ-ously presented. This important difference comes from re-sidual absorptances occuring in the electrodes and in the pand n layers. For the less efficient structure, G saturates at3. Our calculated maxi mum values of G for these solar cellsare in g ood agreement with values propose d by Sha de andSmith3 (G=3) and also by Walker, Hollingsworth, andMadan (G=3-5).

VIII. CONCLUSIONWe h ave reported a new semiempirical approach tomodel the optical properties of a-Si:H p-i-n solar cells. Inorder to describe the optical properties indu ced by theTCO roughness, we assume that the thin-film stacks havetwo rough interfaces: the TCO/p and the n/metal inter-faces. The specular light propagation is analyzed by theusual matrix method, where the amplitude reflection andrefraction specular coefficients are assumed to be propor-tional to the Fresnel coefficients. The light coherence andthe angular depe ndenc e of the specul ar coefficients aretaken into account. The diffuse reflectance and transmit-tance for the front and back rough interfaces, considered asinput parameters, are either experimentally measure d atfour visible wavelengths or adjusted in order to fit the ex-perimental characteristics. A second matrix algorithm pro-cedure allows the calculation of the absorptan ces due tolight scattering in reasonabl e computation time. T he dif-fuse reflectance and transmittance are comparable to val-ues found in the literature, but the modeling demonstratesbetter results than the usual analytical but more restrictiveapproaches. Improvement mainly comes from the specular

light propagation treatment. The numerical simulation al-lows the estimation of the power balance in realistic solar-cell structures. Optical losses in the electrodes, in thedope d layers and due to reflection, can be accurately cal-culated. The absorptance enhanc ement factor due to lighttrapping reaches the value G=6 for the most efficientstructure studied.

ACKNOWLEDGMENTSWe gratefully thank H. J. Drouhin and C. Hermann

from the Laboratoire d e Physique de la Mat&e Conden s&eof the Cole Polytechnique, for lending us the Kryptonlaser, R. Moran0 from the Laboratoire de Physique Nu-clbaire des Hautes Energies of Ecole Polytechnique, forassistance in the computations, and 13. Cornil fromSOLEMS S.A., for providing samples. We thank S. Vallonfrom the Laboratoire de Physique des Interfaces et desCouche s Minces for helpful discussions.

T. Yoshida, S. Jujikake, H. Fujisawa, S. Saito, T. Sasaki,Y. Ichikawa,and H. Sakai, n Proceedingsof the IOth European Photovoltaic SolarEnergy Conference,Lisbon, Portugal, April, 1991 ,edited by A. Luque,G. Sala, W. Palz, G. DOSSantos,and P. Helm (JSluwer Academic,Dordrecht, 1991), p. 1193.H. W. Dec kman, C. R. Wronski, H. W itzke, and E. Yablonovitch,Appl. Phys. Lett. 42, 968 (1983).3H. Shadeand Z. Smith, J. Appl. Phys . 57, 568 (1985).4J. Morris, R. R. A.rya, J. G. ODowd, an d S. Wie deman, . Appl. Phys.67, 1079 1990).5E. Yablonovitch, J. Opt. Sot. Am. 72, 899 (1982).E. Yablonovi tch and G. D. Cody, IEEE Trans. Electron. DevicesED-29, 300 (1982).H. A. Macleod, Thin Film optical Filters (Hilger, Bristol, 1986).*F. Abeles, Ann. Phys. Paris 5, 59 6 (1950); 5, 7 06 (195 0).P. Bousquet,F. Flory, an d P. Roche,J. Opt. Sot. Am. 7 1,11 15 (1981).D. Maystre, J. Optics (Paris) 15, 43 ( 1984).M. Saillard and D. Maystre, J. Opt. Sot. Am. A 7, 982 (1990).

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