ONE-DIMENSIONAL SEMI-COHERENT OPTICAL MODEL FOR THIN FILM SOLAR CELLS WITH ROUGH INTERFACES Janez Krč, Franc Smole and Marko Topic Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia Key words; semiconductors, a-Si:H, Hydrogenated amorphous Silicon, thin-film solar cells, optical modelling, one-dimensional semi-coherent optical models, rough interfaces, smooth interfaces, light scattering, coherent light, scattered light, light absorption, QE, Quantum Efficiency, light trapping, TCO, Transparent Conductive Oxides, experimental results Abstract: Optical model for thin film solar cells with rough interfaces is presented. In contrast to previous optical models, the presented model considers the coherent nature of direct (non-scattered) light, which is treated as electromagnetic waves, all over the structure. Therefore, the model accounts also for the interference pattern, which appears in the measured wavelength-dependent oharaoteristics of the solar cells with rough interfaces. The model includes scattering of direct and already scattered incident light at all rough interfaces. In the paper, optical circumstances at flat and rough interfaces are described in detail. Coherent propagation of non-scattered and incoherent spreading of scattered light is defined and the calculation procedure in the model is addressed. The optical model is verified for a single junction hydrogenated amorphous silicon p-i-n solar cell deposited on Asahi U - type substrate. A good agreement, also in the interference pattern, verifies the applicability of the developed optical model. Eno-dimenzionaIni delno-koherentni optični model tankoplastnih sončnih celic s hrapavimi spoji Ključne besede: polprevodniki, a-Si:H silicij amorfni hidrogeniziran, celice sončne tankoplastne, modeliranje optično, modeli optični delno-koherentni eno-dimenzionalni, ploskve mejne hrapave, ploskve mejne gladke, razprševanje svetlobe, svetloba koherentna, svetloba razpršena, vpijanje svetlobe, QE učinkovitost kvantna, ujetje svetlobe, TCO oksidi transparentni prevodni, rezultati eksperimentalni Izvleček: Predstavljen je nov optični model tankoplastnih sončnih celic s hrapavimi spoji. Poglavitna prednost modela je, da upošteva koherentni značaj usmerjenega dela svetlobe, ki je obravnavan kot elektromagnetno valovanje preko celotne strukture. Zaradi tega model zajema tudi interferenčne pojave svetlobe v strukturi, katerih posledica so interferenčne oscilacije v valovno odvisnem kvantnem izkoristku sončne celice. Model vključuje tako razprševanje usmerjene kot tudi že razpršene vpadne svetlobe na hrapavih spojih. V članku so podrobno opisane optične razmere na gladkih in hrapavih spojih. Predstavljen je opis širjenja koherentne usmerjene in nekoherentne razpršene svetlobe. Naveden je potek izračuna svetlobne intenzitete v modelu po celotni strukturi sončne celice. Model je verificiran na podlagi izmerjenega poteka kvantnega izkoristka amorfnosilicijeve sončne celice, ki je bila izdelana na standardnemu 'Asahi U" hrapavem substratu. Dobro ujemanje simulacije z meritvijo, tudi v poteku interferenčnih oscilacij, potrjuje uporabnost razvitega optičnega modela. I. Introduction To enhance light absorption in thin film solar cells, light trapping should be implemented in the multi-layer structure of the solar cell. One of the most commonly used light trapping technique is scattering of incident light at rough internal interfaces, which is usually introduced in the solar cell structure by using rough substrates. Multi-directional spreading of scattered light increases the effective path of light propagation in the thin layers resulting in higher absorption. Additionally, higher reflection of scattered light at internal interfaces, originating from non-perpendicular incidence of scattered light beams at the interfaces, assures more efficient light trapping in the solar cell. The description of light scattering at rough interfaces in thin film solar cells presents a comprehensive problem, since the morphology of the interfaces is usually random and the root-mean-square roughness, arms, of the interfaces is in the same order as the wavelength of incident light, A. This complicates analytical description of the problem /1/. Additionally, the interference pattern, which can be observed in the wavelength-dependent characteristics of the solar cells (e.g. quantum efficiency, QE) indicates that besides the scattered light there still exists a significant amount of direct non-scattered light that propagates coherently throughout the structure. To analyse and optimise the complex optical process an optical modelling is required. In the simulations an optical model that incorporates both, light scattering and coherent propagation of direct light, should be used. In orderte describe the scattering process in the model, it has to be determined the amount of scattered light at each rough interface and how the light is scattered In reflection and in transmission at the interface. Previous publications /2-7/ indicated that the amount of scattered light at a rough interface can be approximately determined by means of scalar scattering theory /7-9/ which defines the ratio between diffused (scattered) and total (diffused + specular) amount of light. To describe how the light is scattered, an angular distribution function of scattered light, f((p), has to be determined. Some one-dimensional optical models for thin-film solar cells with rough interfaces have already been developed /2-6/. In most of the models the direct light is not taljen coherently, thus the simulated QEs can not reproduce the interference pattern observed in measurements. In this paper we present a new one-dimensional semi-coherent optical model that includes light scattering at all rough interfaces in the solar cell and considers the coherent propagation of direct light all over the structure. The optical situation at flat and rough interface is described in detail for direct and scattered light. In case of scattered light falling at a rough interface additional scattering in reflection and in transmission is taken into account. The analysis of coherent propagation of direct and incoherent spreading of scattered light is described. The model is verified with the measured Q£ of a thin film hydrogenated amorphous silicon (a-Si:H) p-i-n solar cell. The analysis of direct coherent light is based on the theory of electromagnetic waves. In general, the electromagnetic waves are presented by complex values of electric and magnetic field strengths distributed in the space. In the model all the waves of direct light are assumed to be planar and transversal (TEM)/10/, therefore they can be presented by electric field strengths E*{X,x) and F(A,x) for for-ward-going and backward-going waves only. From B*{X,x) and E'(X,x) the individual components of direct light, /dir"^(A,/», /dir"(A,/,x) and K(X,x), and the total intensity, ldir(X,x) = /dir"'(A,/,x) - /dir"(A,/,x) - K{X,x), Can be determined by Eq. 1. The symbols n(X) and k(X) present the real and the imaginary part of layer's complex refractive index A/fAj = n(X) -jk(X). Yo is the optical admittance of free space (Vo = 2.6544-10'^ S). The superscript '*' denotes a conjugated complex number whereas "Im" stands for imaginary part of the complex number. II. Optical model In the developed one-dimensional semi-coherent optical model we assume perpendicular input illumination with direct coherent light (Fig. 1). All the interfaces in the multilayer structure are assumed to be parallel. This situation can be considered as a good approximation of usual circumstances in real thin film solar cells. Because of multiple reflections and transmissions at the interfaces, the direct light (dir) and scattered light (scatt), which is formed at rough interfaces (vertical arrows in the figure), is propagating in forward (+) and backward (-) direction. In the figure the direct light is presented by wavy arrows, indicating its coherent nature, whereas the dispersed nature of scattered light is shown by a sheaf of straight arrows. In case of direct light where forward-going and backward-going waves are represented by /dir^(/,x) and /dir"(/,x), there exists also an interference component K{l,x), which has either forward or backward direction. The symbol / and x stands for wavelength- and position- dependency of the light intensities, respectively. The components of scattered light, /scatt^(/,/,x) and /scatt'(/,y,x), are also angular (j) dependent in order to indicate their directions of spreading. Since in numerical analysis the angle / is discretised, scattered light can be represented by a sheaf of light beams spreading in discrete directions //. dircct coherent iight dispersed iiieoheient ^^ lighl -V» IM") ^ 4 4 ^ ■V* -v»! -v* ^v^l-v» -v»! A^ -v» I I i 4 4 ¥ 4 #14 ^14 n{Ä)E^{Ä,x) -n(Ä)E^(Ä,x) -2k(Ä)Im = I„r(Ä,x)-K(Ä,X) (1) The scattered light is analysed incoherently in the model, therefore it can be presented by the intensities /scatt""fA, cp, x) and /scattYA, (p, x) directly and not by electric field strengths of the electromagnetic waves. In the model the optical circumstances at flat and rough interfaces are specified for direct coherent and scattered incoherent light. In sections II.A-II.D all four combinations (flat interface - direct light, flat interface - scattered light, rough interface - direct light and rough interface - scattered light) are described in detail. In section II.E the propagation of direct light and spreading of scattered light across the structure are determined. To simplify the indexing of light intensities, the subscripts "dir" and "scatt" are left out in further analysis. At the interfaces also positional dependency (x) is left out from the denotations of light intensities. II.A Incidence of direct coherent light at a flat interface Since the electromagnetic waves of direct light are transversal and since they propagate perpendicularly to the interfaces, all the components of electric field strengths are parallel to the interfaces, as depicted in Fig, 2. Fig. 1: Propagation of direct coherent light and spreading of scattered incoherent iight in a thin film solar cell structure with smooth and rough interfaces. E;a) incident reflected i & transmitted. EJÄ) Fig. 2. Electromagnetic waves of direct coherent light at a flat interface. Therefore, in the model the electric field strengths on the left (L) side, Eif/i) and Ei(X), can be calculated from the known values of electric field strengths on the right (R) side of the interface, Er(X) and ErYA;, by Eq. 2 and Eq, 3, respectively. The symbols Nt(X) and correspond to complex refractive indexes of the left and right layer forming the interface. N,iÄ) + N,{Ä) e;{Ä) + EiiÄ) (2) NM) 4- (3) In the figure El(X) and ErYAJ are the electric field strengths of the incident light which falls to the interface from left and right side, respectively, whereas Ei(X) and En(X) present the combined components of reflected and transmitted direct light. The corresponding light intensities at left side, Il(X), Il(X) and Ki(X), and right side, Ir^X). /r" (X) and Kfi(X), of the interface can be calculated by means of Eq. 1. II.B Incidence of scattered light at a flat interface In Fig. 3, the situation at a flat interface illuminated with scattered light from left side, Il(X, cpj, and right side, lpi(X, (pn), is symbolically illustrated. In this case the angular dependent intensities of reflected (r) light, krlX, (pir) and lRr*(X, (pRr), and transmitted (t) ligh Ili(X, (ftO and iRi*(X, (PftO, are shown separately in the figure. At the interface, for each discrete beam of incident light the intensity and direction (angle) of reflected and transmitted components has to be determined. Due to superposition of the intensities, which can be assumed for incoherent scattered light, the situation can be analysed for left and right side illumi- nation independently. Next we focus on left side illumination only. \ L r/ ^ luUMr)^ Fig. 3: Intensities of scattered light at a flat interface. If the intensity of the leftside illumination, li(X,(pi), is known the reflected, lLr'(X,(pLr), and transmitted, Iri*(X,(PrO, components can be calculated by Eq. 4 and Eq. 5, respectively. The angles (pi_, (pirand (pRt correspond to the directions of incident, reflected and transmitted light beams, respectively. The RHyL(X,(pL, (pntjand Twi,(X,(pi, (p^i) are reflectance and transmittance at a flat interface and will be explained in following. h,- (Plr ) = RHV L (PL ^(PR,)- h' (PL ) (4) IR' (PR, ) = T^V Lr (PR, ) ■ IL V L ) (5) Since the light beams of scattered light do not fall perpendicularly on the interface, in general the reflectance and transmittance should be defined for horizontally, (H) and vertically, (V) polarized light /10/. The reflectance of horizontal polarisation, RndKcpi, (pp.0, and vertical polarisation, RvUXqK., (pR\), are defined by Eq. 6 and Eq. 7, respectively, (6) A'';(/i)"COS (7) Since the light polarisation in the multi-layer structures with rough interfaces can not be exactly determined, we use in the model an average reflectance RHyi(X,(pi., iprtj, which was chosen as a mean value of Rh tfX, (fi, (pnO and ffvifA, (pL, (PrO- The corresponding average transmittance 7hvlr('A,(Pl, (PrO for the scattered light beams is simply calculated from the RmL(X,(pL, (pm) using Eq. 8. T„v RHV (pRt ) (8) BymeansofftHVL('A,(pL, (pROand Thvlr(^,(Pl, (jfRtJ the amount of reflected and transmitted light can be determined. To define the directions (angles) of the reflected and transmitted beams the Fresnell refractive law is used /10/. Thus, the angle in reflection, cpir, is determined as a negative value of the corresponding incident angle, (pi, whereas the angle in transmission, i^Rt, is calculated using Eq. 9 where nifAj and npfAj are the real parts of refractive indexes of the layers forming the interface. I.C =arcsin(^—sin^J Incidence of direct light at a rough interface (9) At a rough interface the amount of scattered light and its angular distribution function have to be determined. The scalar scattering theory can be used as an approximation to define the amount of scattered (diffused) and non-scattered (specular) light in reflection and in transmission. If the direct incident light were taken incoherently and the illumination were applied from left side of the interface only, the specular and diffused parts of light could be determined by specular and diffuse reflectances, flispecCAj (Eq. 10) and Ridiff'Aj (Eq. 11), and specular and diffused transmit-tances, TiRspecfAj (Eq. 12) and TiRdiffAj (Eq. 13). (10) (11) (12) (13) The parameters c, (Eq. 10) and Ct (Eq. 12) are the correction factors for specular reflectance and transmittance which can vary between 0 and 1 /7/. Ro(X) and To(X) are total (specular + diffused) reflectance and transmitance which are assumed to be the same as in the case of a flat interface and can be defined by Eq. 14 and Eq. 15, respectively (14 (15) In the model, the direct light is treated coherently, but the Eqs. 10-15 are still assumed to be valid for the light intensities. In the case it is necessary to start the analysis on electric field strengths of the specular components. Further explanation will be based on Fig. 4, where a rough interface is illuminated with direct coherent light from both sides, as in the case of flat interface in Sec. ILA. incident reflected & transmitted specular reflected diffused transmitted diffused Fig. 4: Incidence of direct coherent iight at a rough interface. At the rough interface, the electric field strengths of coherent light on the right side, £r"^CAj and ErYAJ, are firstly transformed to the corresponding intensities /pspec'^fAj, Ir (X) and Kf{(X) using Eq. 1. Then, the value of Kn(X) is added either to /pspec^fAj or to Ir(X), according to its direction (sign), resulting in two components, /r and /r k(X) defined by Eq. 16 and Eq. 17, respectively. R spec ; KJX)>o (16) ^RK y-^) ~ 1 (17) From /rand /rk(X) the components /lk(X) and /lk" (X) of specular light on the left side of the interface can be determined, using Eq. 18 and Eq, 19, respectively. 4/ a)= 1 Tu^s^M) ■4/a) - TlRspcM) ■i (18) Z. M) L spec LR spec LR spec M) ■I UK U) (19) In the equations the reflectance Rr spec(X) and transmittance TRLspecfAj correspond to the case of right side illumination. They can be determined by Eq. 10 and Eq. 12 where all indexes "L' should be substituted with "R" and vice versa. From the calculated intensities Ilk(^) and the absolute values of electric field strengths of coherent light, \El(X)\ and |£lYA;|, on the left side can be obtained if the direction (sign) of KlCAJ is known. The sign of Ki(X) can be determined if the phases, Jl(X) and Jl(X), of the electric field strengths and EC (Xj^lEClXJlei^^''^') are known. Simulations indicated that these phases have significant influence on the position of the interference pattern in the wavelength dependent QE of the solar cells. Moreover, from measured QEs of the solar cells can be observed that the position of the interference pattern is in general not affected by the (Trms. The value of the (Jrms influences only the intensity of the interference pattern. These findings lead us to the assumption that in case of a rough interface the phases of electric field strengths can be deduced from the case of flat interface. Thus, to determine ßt(X) and {}C(Ä), first the electric field strengths. El' ''(X) and El ' '(X), of the corresponding flat interface are calculated by means of Eq. 2 and Eq. 3. Then, the phases ^l(X)- z9l' '(X) and A'(XhßL '(X) can be determined by Eq. 20 where "+" and "-" signs correspond to the electric field strength of forward-going and backward-going waves, respectively. arctg arctg (20) From the relation Kl(X) = YQk(X)\m[E(X) E" '(X)], which originates from Eq. 1, it can be found out that the sign of Ki(X) can be indicated by Eq. 21. e;{ä) (22) (Ä) ■ El (A)' - (A) ■ E; (A) ■ sinCt?" (i) - &1{A))-EI{A)-I,;{A) = 0 (23) If K{_(X) is positive, the role of | ElYAJ | and | El(X) \ in Eq. 22-23 exchanges. Knowing | ElYA; |, | ElUX) \, Jl(X) and JClX), the complex electric field strengths of the specular coherent light on the left side of the rough interface are determined and the corresponding specular intensities iLspec(X), /LspecYAj and K\_(X) can be calculated. The diffused components in reflection, lumlX^cp), Int d/(X,(p), and in transmission, ludi{(X,(p), lmd/(X,(p), can be calculated from Ilk^(X) and Irk(X) which presents the illumination from left and right side of the interface, respectively. By determining the incoherent diffused parts of light, the superposition of left and right side illumination is assumed. Therefore, we will determine only the components lLrdi{(X,(p), and lmd/(X,(p) which correspond to the leftside illumination. The diffused component in reflection, /Lrdif" (X,(p), and in transmission, lmd/(X,(p), can be calculated by Eqs. 24-25. RLdiffAj and TLRdiffAj were defined by Eq. 11 and Eq. 13, respectively. The angular distribution function for the case of direct incident light, f\((p), was chosen to be a normalised cos^(q)) function /7/ in our simulations. However, f((p) is an input parameter of our optical model, thus an arbitrary selection of f((p) can be made. (P) = fi(P)- RLä, (^) • / (^) (24) a) - (A)) >0=> K,(A) >0 else K,(A)<0 (21) In further analysis we use Eqs. 16-17 and substituting index "R" with index "L', since the leftside of the interface is under scope. According to the sign of Ki(X) appropriate relations for Ilk(X) and Ilk(X) can be chosen from Eqs. 16-17. Based on Eq. 1, iLspec^(X), iLspec'(X) and Kl(X) can be written in terms of El'YAJ and Ei(X), where El(X) and Ei(X) can be represented by their absolute values and phases. Since the phases are already known the two absolute values of electric field strengths, \Ei(X) \ and |£l" {X) I, can be easily obtained. In the case of negative KiiX) the \El(X)\ is determined byEq. 22 whereas the \Ei(X)l is implicitly given by Eq. 23. (25) II.D Incidence of scattered light at a rough interface For scattered incident light it is assumed that a part of it is additionally dispersed at a rough interface (diffused part) while the rest is spread as in the case of flat interface (specular part). A general situation, where the scattered light falls on a rough interface from left and right side is symbolically illustrated in Fig. 5. incident reflected specular L ^ k (■illM transmitted / specular / reflected . r v ' diffused \ transmitted/ j : ' diffused/ Fig. 5: Incidence of scattered incotierent light at a rough interface. Due to the superposition we describe the analysis for the case of left side illumination only. To determine specular reflectance and transmittance for the case of scattered incident light, we have to introduce the angular dependency on the angle of incident light,