Optical properties of nano-structured dye-sensitized solar cells

Solar Energy Materials & Solar Cells 69 (2001) 147}163

Optical properties of nano-structured dye-sensitized solar cells William E. Vargas *, Gunnar A. Niklasson
Centro de Investigacio& n en Ciencia e Ingeniern& a de Materiales and Escuela de Fn& sica, Universidad de Costa Rica, 2060 San Jose& , Costa Rica
The As ngstro( m Laboratory, Department of Materials Science, Uppsala University, Box 534, S-751 21 Uppsala, Sweden Received 25 February 2000; received in revised form 25 September 2000

Abstract Radiative transfer computations are carried out to describe the intrinsic and e!ective optical properties of light di!using and absorbing materials consisting of anatase titania pigments hosted in an electrolyte medium. The intrinsic visible absorption of some of the pigments has been increased by coating them with an absorbing dye monolayer. A multiple scattering approach is applied to compute average path-length parameters and forward-scattering ratios used in four-#ux radiative transfer calculations. It is shown that the e!ective absorption coe$cient of the inhomogeneous medium is maximized when the size of the pigments is around 12 nm in diameter, and the e!ective scattering coe$cient is optimized for diameters of the pigments around 250 nm. The intrinsic solar absorptance of the medium is optimized when the diameter of the pigments is around 60 nm.  2001 Published by Elsevier Science B.V. Keywords: Radiative transfer; Light scattering; Dye-sensitized solar cells

1. Introduction The processes involved in producing conventional solar cells are characterized by the use of high-vacuum technologies and deposition techniques, which increase the production costs of the cells being consequently limited their use in large-scale applications [1,2]. During the last years a growing e!ort is being focused in considering a new alternative to produce solar cells at lower costs by means of the use of

* Corresponding author. 0927-0248/01/$ - see front matter  2001 Published by Elsevier Science B.V. PII: S 0 9 2 7 - 0 2 4 8 ( 0 0 ) 0 0 3 8 8 - 3

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nano-sized dye-sensitized titania pigments hosted in an electrolyte [3}5]. The pigments become absorbing by coating them with a thin monolayer of dye molecules which are responsible of absorbing solar radiation mainly in the visible wavelength range [6,7]. The excited dye molecules inject electrons into the conduction band of the TiO semiconductor particles, which are subsequently responsible for the charge  transport. The dye molecules are later reduced by means of redox reactions which involve the ions in the electrolyte. The transferred electrons di!use toward the front electrode of the cell, where they are extracted as an external current by means of a transparent conductive oxide thin layer. The pigments usually are in high concentrations, with volume fractions around 50%, increasing signi"cantly the e!ective surface to be coated with a monolayer of dye molecules. The typical thicknesses of the inhomogeneous "lms are around 10
m, and we will assume perpendicular illumination with unpolarized radiation of free space wavelength
. The inhomogeneous  di!using slab is placed between transparent conductive-oxide layers deposited on glass substrates. The basic structure of this "ve-layer system is depicted in Fig. 1. In this article we consider the intrinsic and e!ective optical properties of inhomogeneous layers consisting of uncoated and dye-coated anatase titania pigments immersed in an electrolyte. The multiple scattering approach that we are using takes into account interactions between pigments within the so-called far-"eld approximation, as well as partial re#ections of the di!use radiation propagating through the particulate medium at its interfaces [8}10].

Fig. 1. Structure of a GraK tzel cell based on a light di!using layer containing anatase titania particles, some of them coated with a dye monolayer, perpendicularly illuminated with unpolarized light.

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The scattering and absorption cross sections of the pigments (C and C respec
 tively), used in the four-#ux radiative transfer model [11,12] from which we are evaluating optical properties (re#ectances, transmittances, and absorptances), are computed from the Lorenz}Mie theory for homogeneous particles [13] or as extended by Aden and Kerker to consider layered spherical ones [14], according to the algorithms devised by Wiscombe [15], and Toon and Ackerman [16] respectively, and by assuming independent scattering. We expect to consider in the future near-"eld interactions between pigments by using an improved version of the extended boundary condition method, the T-matrix approach, versatile enough to describe clustering e!ects [17]. Intrinsic and e!ective scattering and absorption coe$cients per unit length of the particulate medium are considered in terms of the size of the pigments, and those particle sizes which optimize the scattering or absorption by the inhomogeneous slab are shown. Re#ectance, transmittance, and absorptance spectra of particulate slabs containing highly scattering or absorbing pigments are depicted in the solar spectral range, with wavelengths between 0.3 and 2.5
m, and the degree of isotropy of the propagating di!use radiation intensity is characterized by means of the corresponding spectral dependence of average path-length parameters and forward scattering ratios. Polar plots of di!use intensity patterns are displayed for light di!using layers containing monodispersions of particles with di!erent sizes. Finally we establish from extensive radiative transfer computations the size that the pigments should have to optimize the solar absorptance of the material, which will be very useful for people interested in improving e$ciency of the so-called GraK tzel cells. The e!ect of the transparent conductive oxide and glass layers, containing the inhomogeneous active layer, is also considered. Recently the optical properties of dye-sensitized nanostructured solar cells, normally illuminated with unpolarized radiation, have been modelled by using four-#ux radiative transfer models. An inversion technique, based on a non-linear least-squares algorithm, has been applied to obtain the spectral dependence of the average intrinsic scattering and absorption coe$cients per unit length, as well as the forward scattering ratio and the average path-length parameter [18], which are parameters used in the standard four-#ux model of Maheu et al. [19].

2. Theoretical framework In this section we will give a brief description of the theoretical aspects involved when considering the propagation of di!use radiation through scattering and absorbing media. More detailed derivations have been published elsewhere [8}12]. We assume perpendicular illumination with unpolarized radiation of an inhomogeneous layer consisting of anatase titania particles, some of them coated with a monolayer of dye molecules, homogeneous- and randomly distributed through an electrolyte matrix. At a given optical depth  through the inhomogeneous layer we assume expansions of the di!use intensity in Legendre polynomials, being c>() and c\() L L

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(with n"0, 1, 2,...) the coe$cients used to specify the expansions of forward and backward di!use intensities in terms of Legendre polynomials  I(, )"  c>()P (), 0()1, (1a) L L L  J(, )"  c\()P (), !1)(0. (1b) L L L The coe$cients c!() are related to those de"ning the corresponding expansions of L the di!use intensity in terms of scattering-order contributions, Q>() and Q\() I I (with k"1, 2, 3,...). It is in terms of these coe$cients that the expansions of forward and backward di!use intensities can also be speci"ed taking into account the angular dependence of each scattering order by means of the generalized phase functions introduced by Hartel [20]. By considering perpendicular collimated illumination, both for semi-in"nite media [8] and slabs [10], explicit forms of the scattering-order coe$cients have been obtained by solving the radiative transfer equation for each scattering order under homogeneous boundary conditions: Q>("0)"0 and I Q\(")"0, where  is the optical thickness of the layer, and where the coe$I cients Q>() and Q\() specify the relative amounts of radiation emerging from I I k scattering events at an optical depth , propagating in forward and backward directions, respectively. In the case of particulate "lms or slabs, partial re#ection of backward di!use radiation at the front interface will give additional contributions to the total forward di!use radiation intensity, and partial re#ection of forward di!use radiation at the back interface will contribute to the total backward di!use radiation intensity as well. These internal boundary re#ections of the di!use radiation have been taken into account by means of scattering-order re#ectivities corresponding to each scattering order [10]. Once the total forward and backward di!use intensities are obtained various parameters used in four-#ux radiative transfer computations can be evaluated. They are the forward and backward average path-length parameters (FAPP"> and BAPP"\ respectively), and the forward scattering ratios for forward and backward di!use intensity impinging on any of the particles randomly distribuited through the inhomogeneous medium (FSRF"> and FSRB"\ respectively). They are given by  I(, ) d  d I(, )p(, ) d  , >"  , (2a) >"   d I(, )p(, ) d  I(, ) d  \   J(, ) d  d J(, )p(, ) d \ , >" \ , (2b) \"! \  d J(, )p(, ) d  J(, ) d \ \ \ where p("cos , "cos ) is the single-particle phase function specifying the relative intensity of the radiation scattered in the direction  for a given incidence . This function is also expanded in Legendre polynomials according to  p(, )"  P ()P (), L L L L

(3)

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where the coe$cients are evaluated from the Lorenz}Mie theory. The APPs L depend on the angular distribution of the di!use intensity patterns. Values close to unity indicate the propagation of very anisotropic di!use intensity patterns, while values around two correspond to quite isotropic di!use radiation. The FSRs give information about the fraction of energy that any of the particles scatter into the forward hemisphere, with respect to the incidence direction of the di!use radiation impinging on it. In the Rayleigh limit both > and \ approach  while in the  large-particle limit they tend to unity owing to the peaked forward distribution of the scattered radiation [9]. Other parameters obtained from the multiple scattering approach brie#y described above are the e!ective scattering and absorption coe$cients per unit length of the slab, S and K respectively. These parameters are given by S" (1! ) and K"  where "fC /< and "fC /< are the intrinsic    
 scattering and absorption coe$cients per unit length of the particulate medium, respectively, with f as the particle volume fraction, and the particle volume speci"ed by V. Moreover,  "[>\] and  "[>#\]/2 are mean values of the   APPs and FSRs respectively, [11,12]. APPs and FSRs are used in a four-#ux radiative transfer model to compute optical properties of particulate slabs or "lms. Total re#ectance and transmittance have contributions from collimated}collimated (R and T respectively) and col  limated}di!use (R and T , respectively) components according to the formulas   derived previously [11]. The e!ective refractive index of the inhomogeneous layer, and re#ection coe$cients for di!use radiation, are obtained from the extended Maxwell}Garnett model published elsewhere [21,22]. Film absorptance is obtained from energy conservation A"1![R #T ] where R "R #R and      T "T #T . At each wavelength, the expansions of the forward and backward    di!use intensities in terms of scattering order contributions are carried out until obtaining an error of 0.1% in the values of the total re#ectance and the total transmittance. By weighting R (
) and T (
) with the AM1.5 solar spectrum [23,24]   the solar re#ectance and transmittance (R and T respectively) have been computed   and used to evaluate the solar absorptance: A "1![R #T ].    3. Computed intrinsic optical properties As explained above our multiple scattering approach does not take into account particle interactions in the near-"eld regime, which is expected to have some in#uence in the intrinsic optical properties of the particulate layer owing to the high concentration of the titania particles typically found in GraK tzel cells. Even though the calculations that we are reporting in this article give signi"cant information about the general trends in the optical properties of the active layer in GraK tzel cells, in terms of the size of the particles. This information would be a desirable background when considering dependent scattering e!ects [25]. In real samples, for a given particle volume fraction f, particle aggregation will decrease the e!ective absorbing surface of the particles becoming ( f!f )/f the fraction of homogeneous anatase titania particles participating in scattering processes with

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Fig. 2. Particle size dependence of the internal roughness factor [ ], and the particle volume fraction [f ], of dye-sensitized anatase particles immersed in an electrolyte medium 10
m thick. The particle volume fraction and coordination number were set to 50% and 6 respectively [26].

low intrinsic absorption from the anatase titania particles, and f /f the fraction of dye coated particles participating in scattering processes with signi"cant absorption mainly from the dye. f " r/6 h where r is the particle radius, h is the "lm thickness, and is the so-called internal roughness factor. This factor depends on the particle volume fraction, the coordination number, and "lm thickness. Throughout our computations we are using the internal roughness factor reported by Ferber and Luther for a particle volume fraction of 50%, "lm thickness of 10
m, and six as the coordination number [26]. In the formula above given to specify f , a factor of  has  been taken into account owing to the need of each dye molecule of having two chemical bonds connected to corresponding Ti atoms, decreasing consequently the available surface of anatase particles to be coated with a monolayer of dye molecules. It can be seen from Fig. 2 that the largest internal roughness factor is found for particles with around 5 nm in radius. In this case the e!ective surface area to be covered with a dye monolayer is around 1500 times larger than a corresponding #at surface of the cell. Average values of the scattering and the extinction cross sections are calculated from C "[( f!f )/f]C #[f /f]C , (4a)    C "[( f!f )/f]C #[f /f]C , (4b)    where superscript 1 stands for cross sections of uncoated anatase titania particles, and the superscript 2 denotes scattering and absorption cross sections of dye-coated

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titania particles. Owing to the small value of the dye-coating layer thickness, around 1 nm, both uncoated and dye-coated particles are approximately characterized by the same size parameter: x"2rN/
and x x, where N is the refractive index of  the electrolyte. The average absorption cross section is obtained from C "C !
  C , and the average albedo is given by "C /C . Two set of coe$cients,      and  (with n"1, 2, 3,...), involved in the expansions of the single particle phase L L functions, are computed from the corresponding x and Q , and x and Q para  meters, and where CG "rQG for i"1 or 2. This is carried out according to the   procedure published elsewhere [27}29], and required the previous computation of the scattering order coe$cients, a and b for the uncoated particles, a and b for the L L L L dye-sensitized ones (with n"1, 2,...). According to the procedure devised by Mishchenko et al. [30] the average values of the coe$cients used to compute the average single-particle phase function are given by [(f!f )/f] C #[f /f] C ? L  ? L  , " L C 

(5)

with n"1, 2, 3, ... From these coe$cients the average path-length parameters and the forward scattering ratios are evaluated, as well as the forward scattering ratio for collimated radiation,  . The optical constants of anatase titania have been taken  from literature [31], and the imaginary part of the refractive index of the dye has been estimated from the absorptance curve reported by Ferber and Luther [26] (see Fig. 3) with a real part assumed wavelength-independent and equal to 1.50. The electrolyte has been considered non absorbing with a refractive index N"1.50. Fig. 4 displays the dependence of the intrinsic and e!ective scattering and absorption coe$cients per unit length of an unsupported inhomogeneous slab 10
m thick,

Fig. 3. Spectral dependence of the optical constants of anatase particles [31] whose real part of the refractive index is denoted by n(TiO ) and with the imaginary part given by k(TiO ). The imaginary part of   the refractive index of the dye [26], k(dye), is also displayed.

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Fig. 4. Intrinsic ( and ) and e!ective (S and K) scattering (a) and absorption (b) coe!icients (in units of
m\) of a slab containing both composite dye-coated anatase titania pigments and homogeneous anatase titania particles. The thickness of the dye-layer and the free space wavelength of the incident radiation were taken as 1.0 nm and 0.55
m, respectively. The particle volume fraction and the thickness of the inhomogeneous slab were set at 50% and 10.0
m respectively.

on the size of the anatase titania core. The maximum values of the e!ective scattering coe$cient S correspond to particles sizes around 0.125
m in radius being the fraction of dye-sensitized particles f "25%, while the more absorbing particles, with the largest K values, have sizes about 6 nm in radius with a fraction of dye-sensitized particles f "14%. The thickness of the dye coating and the free space wavelength of the incident radiation were taken as 1.0 nm and 0.55
m respectively. The particle volume fraction was set at 50%. Films containing nano-sized dye-sensitized titania particles will be characterized by signi"cant absorption of visible radiation with low di!use re#ectance, while "lms containing submicron-sized dye-sensitized titania particles will display less absorption and signi"cant di!use re#ectance values, as depicted in Fig. 5, where the optical properties have been evaluated from four-#ux radiative transfer computations. For the low-re#ecting case R "0.11, T "0.61, and   A "0.28 and for the high-re#ecting one R "0.32, T "0.50 , and A "0.18.     The di!erence between these two solar absorptance values arises mainly from the absorption peak of the dye-sensitized titania particles around 0.50
m of wavelength (see Fig. 3). Our computed optical properties corresponding to the 10
m thick "lm, containing spherical particles with 12 nm of diameter, are quite similar to the re#ectance and transmittance spectra measured by Rothenberger et al. [18], when considering a 4.5
m thick coating with particle diameters around 15 nm. They have also measured the optical properties of a high re#ecting sample 6.0
m thick, with particle sizes up to 150 nm. In this case a comparison with our computations is more di$cult because the measurements were carried out from a "lm containing more irregularly shaped particles, with a large range of sizes.

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Fig. 5. Re#ectance, transmittance, and absorptance spectra of an inhomogenous layer containing composite dye-coated anatase pigments and homogeneous anatase titania particles. The thickness of the dye layer and the radius of the titania core were put to 1.0 nm, and (a) 6.0 nm and (b) 125.0 nm respectively. The particle volume fraction and the thickness of the slab were set at 50% and 10.0
m respectively.

Fig. 6. Spectral dependence of average path-length parameters and forward scattering ratios corresponding to a "lm containing composite pigments with a titania core radius of 125 nm. The dye coating thickness is 1 nm, the particle volume fraction is 50%. The black dots correspond to a "lm containing titania particles with a radius of 6.0 nm.

When computing the APPs and FSRs corresponding to the slab whose optical properties where depicted in Fig. 5(a) any signi"cant spectral dependence was displayed, with values of around 1.77 and 0.50, respectively, over the solar spectral range (see the black dots in Fig. 6). In the case of the slab containing submicron-sized particles there is an increasing degree of isotropy towards the near infrared where the single-scattering phase function is approaching the Rayleigh limit, and consequently the FSRs are close to 0.5 as depicted in Fig. 6. In the visible wavelength range there is a signi"cant degree of anisotropy with peaked forward di!use intensity patterns.

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Fig. 7. (a) Particle size dependence of the solar absorptance (solid line) of an inhomogeneous layer containing both uncoated anatase particles as well as dye-coated ones. The long-dash line corresponds to the solar absorptance of a similar inhomogeneous layer containing only uncoated anatase particles, and the short-dash line displays the solar absorptance when the refractive index of the anatase particle is considered wavelength independent with no absorption at all. (b) Spectral dependence of the mean average path-length parameter corresponding to "lms containing particle with di!erent sizes as indicated in "gure. (c) Spectral dependence of the absorptance of light di!using "lms containing particles with di!erent sizes as indicated in the "gure. The layer thickness and pigment volume fraction were set to 10
m and 50%.

Rothenberger et al. have obtained, from inversion of the measured optical properties of the "lms mentioned in the previous paragraph, optimal values of the APP and the FSR [18]. Their analysis gives "1.90 and  "0.79 for the "lm containing nano sized particles, "1.77 and  "0.90 for the "lm containing submicron-sized par ticles. By comparing with the spectral dependence of the APP and the FSR depicted in Fig. 6, the largest discrepancy with Rothenberger et al. results corresponds to the FSR of the low-re#ecting "lm. In this case they have attributed the large value of the FSR, obtained from their non-linear least-squares "tting, to a more peaked forward distribution of the di!use radiation owing to particle aggregation.

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We have carried out extensive computations in order to optimize the size of the composite particles giving the largest absorptance values in the visible wavelength range. The results have been displayed in Fig. 7a, where we have depicted the particle size dependence of the solar absorptance (solid line). The dashed line with values around 6% corresponds to the solar absorptance of a "lm with no dye-sensitized particles. In this case the absorption comes from the anatase particles. The other dashed line is obtained when the refractive index of the anatase is considered wavelength independent and equal to 2.50, and without any absorption from the anatase particles. In this case the absorption comes from the dye molecules. The solar absorptance depicts two maxima: the "rst maximum, which corresponds to the highest intrinsic and e!ective absorption coe$cients of the di!using layer (see Fig. 4b), is displayed for "lms containing particles with around 6 nm in radius, and the second one is depicted when the particles have around 30 nm in radius. This additional size-dependent-absorption-band arises from the fact of having an enhanced degree of isotropy of the propagating visible di!use radiation, as seen in Fig. 7b where we have depicted the spectral dependence of the mean APP for "lms containing particles with di!erent sizes. The absorptance spectra of "lms containing particles with these optimized sizes are depicted in Fig. 7c. The main di!erence between these two spectra is depicted for wavelengths between 0.4 and 0.55
m, where the propagating di!use intensity pattern displays an enhanced degree of isotropy. Fig. 8 displays polar plots of the di!use intensity patterns at a given optical depth in the inhomogeneous medium ("0.10), for "lms containing particles with di!erent sizes as indicated in the "gures, illuminated with visible radiation (
"0.55
m). In  the case of the "lm containing highly absorbing particles (r"6 nm in Fig. 8a) there is a symmetric distribution of forward and backward multiple scattered radiation with very low values of the di!use intensity. In this case R "0.08, R "0.00, T "0.24,    and T "0.00, and the absorption is A(
)"0.68 whose numerical value comes from   absorption of the collimated radiation propagating through the inhomogeneous layer being neglegible the absorption of di!use radiation. A peaked forward distribution of the di!use radiation intensity is displayed when considering larger particles (r"32 nm in Fig. 8b and r"125 nm in Fig. 8c) with very signi"cant values of the di!use intensity. For the "lm with medium-sized particles (r"32 nm) R "0.08,  R "0.10, T "0.09, T "0.06, and A(
)"0.67 with signi"cant contributions     from absorption of collimated and di!use radiation. For the "lm containing large particles (r"125 nm) R "0.04, R "0.62, T "0.0, T "0.04, and A(
)"0.30.      4. E4ective optical properties In this section we will consider the fact of having the inhomogeneous slab between transparent conductive oxide layers and this three layer system placed between two glass layers. Firstly, we will consider the e!ect of the back transparent conductive oxide (TCO) layer assuming this as a substrate of the inhomogeneous light di!using layer (LDL). A doped ZnO "lm substrate is considered as the TCO layer whose optical constants have been taken from the literature [32] (see Fig. 9). Fig. 10 displays

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Fig. 8. Polar plots of the intensity patterns of di!use radiation propagating through inhomogeneous "lms 10
m thick illuminated with visible radiation (
"0.55
m), and containing dye-sensitized anatase titania  particles whose core radii are (a) 6 nm, (b) 32 nm, and (c) 125 nm. The particle volume fraction and thickness of the dye monolayer coating a fraction of the anatase particles were considered as 50% and 1 nm, respectively.

the e!ect of the substrate thickness in the solar absorptance A of the system. The  di!erent particles sizes displaying the highest solar absorptance values have been considered. The variation of the re#ection coe$cients for collimated [r ] and di!use  radiation [R ] at the interface between the inhomogeneous layer and the doped-ZnO  substrate, in terms of the substrate thickness is also shown. The re#ection coe$cients r and R are respectively evaluated according to the formulas [33]   (1!r )(1!r )r e\? F Q   r "r # , (6a)   1!r r e\? F   (1!R )(1!R )R e\? F Q   R "R # , (6b)   1!R R e\? F  

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Fig. 9. Spectral dependence of the optical constants (real and imaginary parts of the refractive index) of doped-ZnO assumed as transparent conductive oxide layer in GraK tzel cells [32].

Fig. 10. Variation of the solar absorptance [A ], and the collimated [r ] and di!use [R ] re#ection    coe$cients, with the thickness of a doped-ZnO substrate supporting an inhomogeneous layer of dyesensitized anatase particles immersed in an electrolyte. The "lm thickness and pigment volume fraction were set to 10
m and 50%.

where r (r /r ) is the re#ection coe$cient for collimated radiation going from the    inhomogeneous xlm to the substrate (substrate to the inhomogeneous xlm/substrate to the air). The corresponding re#ectivities for di!use radiation are denoted with capital letters. These formulas account for boundary re#ections of collimated and di!use

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Fig. 11. Particle size dependence of the integrated optical properties of (a) an unsupported di!using layer 10
m thick containing uncoated and dye-sensitized anatase particles, and (b) an di!using layer between transparent conductive oxide (TCO) and glass layers as indicated in the "gure. The thicknesses of the glass and the TCO layers have been set to 40
m and 0.7
m, respectively.

radiation at the interfaces substrate}air and substrate}"lm. The absorption coe$cients of the doped-ZnO substrate for collimated and di!use radiation are given by

"4k /
and "> respectively, and where k is the imaginary part of the  Q     refractive index of the ZnO and h is the thickness of the TCO layer. The re#ection  coe$cients have been evaluated at a wavelength
"0.55
m. The absorption of the  system is slightly increased owing to the presence of the conductive oxide substrate but the amount of collimated and di!use radiation re#ected by the substrate into the active inhomogeneous layer decreases slightly with the substrate thickness. The decrease is rather more signi"cant for the di!use radiation which means that the e!ective solar absorption in the active layer decreases slightly with the thickness of the conductive oxide substrate. Of course, it is desirable to use a transparent conductive oxide layer as thin as possible according to the required conduction properties. Fig. 11 compares the particle size dependence of the intrinsic optical properties integrated over the solar spectrum AM1.5 with the corresponding ones when considering the inhomogeneous active or di!using layer to be placed between two transparent conductive oxide layers and two supporting glass slabs. For di!using layers containing very small particles, radius lower than 10 nm, the intrinsic solar re#ectance (around 12%) is larger than the e!ective one (around 6%) with an opposite behavior for the solar transmittance. This is due to the larger di!erence between the e!ective refractive index of the di!using layer and air in the illuminated interface of the unsupported layer, as compared to the di!erence between the e!ective refractive index of the di!using layer and the doped-ZnO transparent conductive oxide "lm. This fact also explains the more signi"cant feature displayed when comparing Fig. 11a with Fig. 11b: the suppression of the absorption band for particles sizes around 30 nm in radius, while the absorption band centered at particles sizes of about 6 nm is still displayed.

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Fig. 12. (a) Spectral dependence of the relative refractive index of a light di!using layer (LDL) in air [ m ],  or between transparent conductive doped-ZnO (TCO) layers [ m ], and the corresponding di!use re#ec tivities R and R. (b) Angular dependence of the backward di!use radiation intensity emerging from the   front TCO}LDL interface, for the unsupported inhmogeneous slab or the layered system as indicated in the "gure.

Fig. 12a depicts the spectral dependence of the magnitude of the ratio between the e!ective refractive index of the light di!using slab and the refractive index of the neighboring media, which corresponds to air in the case of the unsupported inhomogeneous slab (see the m spectrum in Fig. 12a), and the transparent conductive  oxide "lms in the case of the layered system (see the m spectrum in Fig. 12a). In the  case of the unsupported light di!using layer the relative refractive index is in magnitude around 1.8 in the mid visible while for the layered system it is about 0.98. Consequently there is more di!use radiation, propagating initially in the backward direction, re#ected in the front interface TCO}LDL into the active inhomogeneous slab. Fig. 12a also depicts the spectral dependence of the di!use re#ectivities at this TCO}LDL interface. Small values are displayed for the layered system (R 0.02),  and large ones in the case of the unsupported light di!using layer [R 0.72]. The  larger the amount of backward di!use radiation partially re#ected in the front TCO}LDL interface, the larger energy to be absorbed in the active layer. Fig. 12b displays the relative di!use intensity emerging from the active layer into the front TCO "lm. As seen, in the case of the unsupported LDL the intensity of the emerging light di!use radiation is signi"cantly lower than the one corresponding to the layered system. One can conclude that the suppression of the absorption band centered at 30 nm in radius is due to the decreased amount of di!use radiation propagating through the active light di!using layer.

5. Conclusions It has been shown, from extensive radiative transfer computations based on a four#ux model application, that the intrinsic solar absorptance of an inhomogeneous

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layer containing anatase titania particles with a fraction of them coated with an absorbing dye monolayer, can be optimized in terms of the mean particle size. For particles sizes around 12 or 60 nm in diameter the largest intrinsic solar absorptance values are displayed and they are around 28%. The "rst maximum, displayed for particles diameters around 12 nm, is directly correlated to the maximum value of the intrinsic and e!ective absorption coe$cients per unit length of the di!using layer, and it corresponds to the absorption of collimated radiation propagating through the light di!using layer (LDL). The second maximum has been attributed to the higher degree of isotropy of the di!use radiation propagating for di!using layers containing particles whose diameters are around 60 nm. It has been shown that the presence of transparent conductive oxide (TCO) layers, in between the light di!using layer is located, suppresses the display of the second maximum in the solar absorptance. This fact has been explained in terms of the less backward di!use radiation partially re#ected at the front TCO}LDL interface, decreasing in this way the amount of di!use radiation available to be absorbed in the light di!using layer. The general trends observed in some of the parameters reported through this article, are in agreement with previous analyses including optical measurements of nano-crystalline solar cells.

Acknowledgements The authors thank the University of Costa Rica, and the Swedish Foundation for Strategic Environmental Research (MISTRA) through the As ngstroK m Solar Center at Uppsala University for the support given for carrying out this research work. We are specially indepted to Sten-Eric Lindquist, at the Department of Physical Chemistry in Uppsala University, for valuable discussions and suggestions.

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