SPIE Proceedings [SPIE Solar Energy + Applications - San Diego, CA (Sunday 10 August 2008)] Optical Modeling and Measurements for Solar Energy Systems II - Monte Carlo simulations

Monte Carlo simulations of quantum dot solar concentrators: ray tracing based on fluorescence mapping

A. Schüler, A. Kostro, B. Huriet, C. Galande, J.-L. Scartezzini

Laboratory of Solar Energy LESO, Ecole Polytechnique Fédérale de Lausanne EPFL, Bâtiment LE, CH-1015 Lausanne, Switzerland

e-mail: [email protected]

ABSTRACT

One promising application of semiconductor nanostructures in the field of photovoltaics might be quantum dot solar concentrators. Quantum dot containing nanocomposite thin films are synthesized at EPFL-LESO by a low cost sol-gel process. In order to study the potential of the novel planar photoluminescent concentrators, reliable computer simulations are needed. A computer code for ray tracing simulations of quantum dot solar concentrators has been developed at EPFL-LESO on the basis of Monte Carlo methods that are applied to polarization-dependent reflection/transmission at interfaces, photon absorption by the semiconductor nanocrystals and photoluminescent reemission. The software allows importing measured or theoretical absorption/reemission spectra describing the photoluminescent properties of the quantum dots. Hereby the properties of photoluminescent reemission are described by a set of emission spectra depending on the energy of the incoming photon, allowing to simulate the photoluminescent emission using the inverse function method. By our simulations, the importance of two main factors is revealed, an emission spectrum matched to the spectral efficiency curve of the photovoltaic cell, and a large Stokes shift, which is advantageous for the lateral energy transport. No significant energy losses are implied when the quantum dots are contained within a nanocomposite coating instead of being dispersed in the entire volume of the pane. Together with the knowledge on the optoelectronical properties of suitable photovoltaic cells, the simulations allow to predict the total efficiency of the envisaged concentrating PV systems, and to optimize photoluminescent emission frequencies, optical densities, and pane dimensions. Keywords: quantum dots, photoluminescence, fluorescent planar solar concentrators, photovoltaics, solar cells, Monte Carlo simulations, ray tracing, polarization

1. INTRODUCTION

Due to their fascinating optical and electronic properties, nanometer-scaled structures play an important role in solar energy conversion [1]. The three-dimensional quantum confinement in semiconductor nanocrystals (so-called quantum dots) induces a discretization of electronic states, very much like in atoms or molecules. The resulting optical properties depend on the crystal size and can thus be tuned by varying the dimensions of the nanocrystals. Quantum dots are considered as promising candidates for optoelectronic applications including light-emitting diodes [2] and quantum dot lasers [3]. Various concepts have been proposed for the use of semiconductor nanocrystals in solar energy conversion. Quantum dots might be used as frequency converters to better match the spectrum of the incoming radiation to the spectral efficiency of the solar cell [4]. Semiconductor nanostructures open the way to intermediate band solar cells with high efficiencies [5,6]. By introducing zero dimensional structures into photovoltaic cells, the creation of multiple electron-hole pairs per photon through impact ionization can be achieved, suggesting a potential capability of quantum yields >1 [7]. In quantum dot sensitized nanocrystalline TiO2 solar cells, semiconductor nanocrystals substitute for organic molecules [8,9]. Hereby the possible advantages of quantum dots over dye molecules include the tunability of optical properties with size and better heterojunction formation with solid hole conductors. Photovoltaic effects have been reported for quantum dots immersed in hole-conducting MEH-PPV [10], and it has been proposed to use a blend of electron and hole conducting polymers as host matrix [7].

Optical Modeling and Measurements for Solar Energy Systems II, edited by Benjamin K. Tsai,Proc. of SPIE Vol. 7046, 704609, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.794739

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One promising application of quantum dot containing nanocomposite materials might be planar photoluminescent concentrators where high concentration factors might be achieved for direct and even for diffuse solar radiation. Such devices have originally been designed on the basis of organic dyes [11] and might benefit from a considerably improved lifetime when replacing the organic fluorescent substances by inorganic quantum dots [12,13]. Additionally, the tunability of the optical properties by the size of the nanocrystals provide a large amount freedom for the design and optimization of such devices.

The fabrication of nanometer-sized crystallites can be achieved by growth in colloidal solutions [14,15], on surfaces in vacuum deposition processes [16-18], by electron beam lithography [19], or by sol-gel techniques [20,21]. Often core-shell structures are used to passivate the surfaces of the nanocrystals (e.g. CdSe core in ZnS shell). For colloidal quantum dots, strong and tuneable visible photoluminescence has been observed with quantum yields in the range from 50 % to 85 % [22,23]. In many cases, the influence of the process parameters on the photoluminescence spectrum can be interpreted in terms of the size dependence of the optical properties of the nanocrystals.

In a previous publication [24] we proposed to fabricate fluorescent solar concentrators by coating transparent substrates with quantum dot containing nanocomposite thin films. We used the sol-gel dip-coating process for the deposition of photoluminescent CdS nanocrystal containing silica films. The absorptance and the photoluminescence of the prepared samples were subsequently characterized by optical spectroscopies. In order to study the potential of the use of quantum dot solar concentrators in photovoltaic solar energy conversion, reliable computer simulations are needed. Together with the knowledge on the optoelectronical properties of suitable photovoltaic cells, such simulations shall allow to predict the total efficiency of the envisaged concentrating PV systems, and to optimize pane dimensions, photoluminescent emission frequencies, and choice of PV cell types. In this paper we describe the principles underlying Monte-Carlo ray tracing simulations of quantum dot solar concentrators, and we report on the development of a suitable software tool. Finally we present results of preliminary simulations of quantum dot containing systems.

2. PRINCIPLES OF POLARIZATION RAY TRACING

In a planar photoluminescent concentrator, incident solar radiation is absorbed by a photoluminescent material embedded within a pane or applied as a coating onto it. A large part of the emitted fluorescent radiation will be captured by total internal reflection. The likewise concentrated radiation can be harvested at the edge of the pane. An incident photon can be e.g. refracted at the surface, then absorbed by the photoluminescent material, which in turn might emit another photon, the latter being e.g. reflected at the interface, reabsorbed, etc. In order to stochastically model the various possible processes, they are represented by the state machine of the evolution of a ray as illustrated by Fig. 1.

After its birth, a photon has a determined direction and travels a certain distance, which has been drawn randomly using the probability distribution corresponding to the absorption coefficient of the material. If the photon does not strike an interface along its path, it will be absorbed after the drawn distance. If the absorbing material is photoluminescent, a second photon might be emitted.

At interfaces photons will be either reflected or refracted, with probabilities according to the reflection and transmission coefficients given by the Fresnel formulae [25]. Hereby the polarization state of the beam has to be taken into account. The initial polarization is characterized by the shape and orientation of the vibrational ellipsis of the electrical field vector. Let a and b be the amplitudes of the electrical field in the coordinate system of the principal axis of the ellipsis, and ψ the tilt angle of the ellipsis with respect to the interface. The amplitudes a1 and a2 in the coordinate system aligned with the interface, and the phase shift δ between the two electrical field components, can be found from the following expressions [26]:

a2 + b2 = a1

2 + a22 (1)

tan2ψ = (tan2α) cosδ (2)

sin2χ = (sin2α) sinδ (3)



Refract Yea No

ccIectio Intersection Absorbed <Cio>

Reflect

rii" "ed

with the auxiliary angles

α : tan α = a1 / a2 (0 < α ≤ π/2) (4)

and

χ : tan χ = −/+ b / a (−π/4 < χ ≤ π/4) (5)

Care has to be taken here to respect the sign conventions in all possible cases. The new shape and orientation of the ellipsis after reflection or transmission is found by using the complex Fresnel coefficients for the parallel and perpendicular field components respectively [25]. The parameters describing the new shape and orientation of the ellipsis are stored and used for the reflection/transmission at the next interface. At the borders of the system, an interface can also mean that the photon quits the system or is detected (e.g. by a photovoltaic cell). The transmission and reflection events at the interface are generated from the Fresnel formulae by a Monte Carlo algorithm based on the inverse function method. A suitable computer code has been developed from scratch in C++.

Fig. 1. State machine of the evolution of a ray.

The spectra of photoluminescent emission depend in general on the excitation energy. For a complete experimental characterization of fluorescent quantum dots, a monochromator (ORIEL 77250) is used for the selection of the excitation energy, and a spectrophotometer to acquire the emission spectra (ORIEL MS 125TM 1/8m Spectrograph, with InstaspecTM II Photodiode Array Detector and sighting optics). An argon/mercury discharge lamp serves as wavelength standard for the calibration of the spectrophotometer. An example is shown in Fig. 2. The sample consists of red light emitting colloidal CdSe/ZnO core shell quantum dots with a crystal size of approx. 5 nm (Evident Technologies). The displayed spectra have been measured with a spectral resolution of 5 nm FWHM, the bandwidth of excitation was 7 nm FWHM. A clear dependence of the emission spectra on the excitation energy is revealed.

The likewise measured fluorescence maps are imported into the simulation program, which uses the inverse function method to generate random emission events according to the corresponding emission probabilities. Hereby the internal quantum yield needs to be specified, giving the probability of photoluminescent emission per absorbed photon.



Inte

nsity

(a.u

.)

LA L

Fig. 2. Photoluminescence spectra as a function of excitation energy for red light emitting CdSe/ZnO core shell quantum

dots, crystal size 5 nm.

The illumination assumed for the simulation can be set to direct or diffuse light or a combination of both. For example, standards of the CIE for distribution of intensity along the directions of the sky such as the Moon and Spencer’s model for a covered sky [27] can be used with standard incoming spectra such as D65 or AM1.5global [28]. The energies and the angular distribution of the photons impinging on the detectors are stored and can be used e.g. in combination with spectral efficiency curves of PV cells to predict overall system efficiencies. Some output options of the program are summarized in Fig. 3.

Fig. 3. The various output options of the software. The angular distribution of edge emission can be displayed as one or two dimensional polar plots (top left and top middle), or using Tregenza's discretization of the hemisphere (top right). Bottom left: spectrum of concentrated radiation. Bottom middle: spectra at various emission angles for the back side of the last plate. At the angle close to normal it is the direct transmitted spectrum as the excitation was normal to the plate, at other angles the spectrum is dominated by the fluorescent emission. Bottom right: the distribution of the number of absorption-reemissions cycles per trajectory. A significant contribution is due to trajectories including more than one absorption-emission event.



- - white littupectrophotometer

"V collimated beam

I I

£

3. VALIDATION The generation of transmission and reflection events at the interfaces has been verified by comparison to the Fresnel coefficients for polarized light (see Fig. 4). The solid lines correspond to the theoretical transmittance/reflectance values for elliptically polarized rays with ψ = 35° and α = 30° incident at the interface of two media with the refractive indices 1.5 and 1. The simulated data (open circles in Fig.2) match perfectly the theoretical curves. Above the critical angle, total reflection occurs.

100

80

60

40

20

0

refle

ctan

ce /

trans

mitt

ance

[%

]

806040200incident angle [°]

polarization parametersof incident ray : psi = 35°alpha = 30° refractive indices:n1 = 1.5n2 = 1

theoretical reflectance simulated reflectance theoretical transmittance simulated transmittance

Fig. 4. Validation of transmission/reflection probabilities for elliptically polarized rays. The simulated data match the

theoretical curves perfectly.

In order to validate the complete model, simulation results for known reference systems were compared to experimental optical data. As well-defined reference samples, Polymethyl methacrylate (PMMA) plates (50x28x1mm) containing fluorescent dyes of different colours (orange, red and yellow) were chosen. The complex refractive index ε of these samples was determined from reflectance and transmittance spectra. Fluorescence maps were acquired by measuring photoluminescence spectra at different excitation wavelengths. Measurements of the edge emission spectra were then performed using a focused excitation spot at various distances from the edge of the plate. The configuration of the experiment is illustrated by Fig. 5.

Fig. 5. Experimental setup for measurements of concentrated light at the border of a fluorescent PMMA plate, l is the lateral

path length and θ the angle at which the spectra were recorded.



Simulations were then run based on the likewise characterized optical properties for the considered materials using the same geometry. The simulated transmitted spectrum corresponds exactly to the measured spectrum [see Fig. 5(a)]. Also the simulated spectra of the concentrated radiation being emitted from the sample edges match closely the experimental data. In Fig. 6, simulated and measured data are displayed for lateral optical path lengths of 2.5 and 40 mm. As expected, in both simulation and measurement, the redshift is larger in the case where the optical path is longer (40mm).

(a)

80

60

40

20

trans

mitt

ance

[%

]

700650600550500450400wavelength [nm]

spectral transmittance of fluorescentdye colored PMMA sample

measured simulated

(b)

1.0

0.8

0.6

0.4

0.2

0.0

inte

nsity

[a.

u.]

640620600580560540520500480wavelength [nm]

simulated, lateral path 2.5mm measured, lateral path 2.5mm simulated, lateral path 40 mm measured, lateral path 40 mm

concentrated emission at the edge offluorescent dye colored PMMA sample

Fig. 6. Comparing simulated and measured spectra. a) Transmission rate spectrum. b) Concentrated spectra at different lateral optical path lengths l = 2.5mm and 40mm.

4. RESULTS The geometry of the planar concentrators (individual panes or stacks of concentrators as tandem/triple devices etc.) is defined by the user of the software. In the geometrical configuration of single or triple planar concentrators, mirrors or virtual detectors can be placed on the edges of the panes. In the initial stage of the simulation, the three-dimensional photon trajectories are displayed in real time, thus allowing a direct visual control. After the simulation of some hundreds of photons, the real time visualization is stopped in order to speed up the simulation process. An example for the graphical representation of the photon trajectories is shown in Fig. 7. A bundle of incoming photons (spot diameter 3 mm, opening angle ± 10°) hits the first surface of a triple device. Refraction, absorption, emission, and reflection events can clearly be distinguished in Fig. 7(a). A three-dimensional representation of the same simulation is given in Fig. 7(b).

For devices consisting of a single concentrating pane (quantum dots contained in bulk or applied as coating) in combination with crystalline silicon PV cells, a systematic study of parameter variation was performed. The system efficiencies obtained in the following do thus not yet represent an upper limit, higher efficiencies can be obtained for stacked devices (e.g. tandem or triple stacks) in combination with PV cells made of different semiconductors materials with matched band gaps.

For colloidal CdSe/ZnS quantum dots, transmittance spectra have been simulated and compared to the experimental data. For the transmittance measurement, the core shell quantum dots were immersed in toluene and contained in a standard glass cuvette with an optical path length of 10 mm. The results for blue emitting CdSe/ZnS quantum dots with a crystal size of 2 nm are illustrated by Fig. 8(a). A close match between simulated and measured transmittance data has been achieved.



(a) (b)

Fig. 7. Visualization of three-dimensional photon trajectories in a triple stack concentrator.

(a)

90

80

70

60

50

40

30

trans

mitt

ance

[%

]

550500450400wavelength [nm]

measured simulated

(b)

inte

nsity

[a.

u.]

800750700650600550500450wavelength [nm]

concentrator dimensions[mm x mm x mm]

10 x 30 x 30 10 x 100 x 100 10 x 300 x 300 10 x 1000 x 1000

(c)

inte

nsity

[a.

u.]

800750700650600550500450wavelength [nm]

concentrator dimensions[mm x mm x mm]

10 x 30 x 30 10 x 100 x 100 10 x 300 x 300 10 x 1000 x 1000

Fig. 8. Spectra obtained by Monte-Carlo ray-tracing. (a) Measured and simulated transmittance, blue emitting CdSe/ZnS quantum dots, crystal size 2 nm. (b) Simulated spectra of concentrated radiation, blue emitting CdSe/ZnS quantum dots, crystal size 2 nm. (c) Simulated spectra of concentrated radiation, red emitting CdSe/ZnS quantum dots, crystal size 5 nm.



The spectra of concentrated simulation have been simulated for concentrator dimensions of 30 x 30 mm2, 100 x 100 mm2, and 1000 x 1000 mm2 (pane thickness 10 mm). As incident radiation a solar spectrum AM1.5global was chosen [28], together with an angular distribution according to the Moon-Spencer expression [26]. The results for blue emitting CdSe/ZnS quantum dots (crystal size 2 nm) and red emitting CdSe/ZnS quantum dots (crystal size 5 nm) are plotted in Fig. 8(b) and Fig. 8(c) respectively. Spectra are normalized with respect to equal flux density of incoming photons at the concentrator surface and represent the flux density of photons received by the virtual detector. With increasing concentrator area, the detector receives more photons, and in both cases the peak observed in the spectra shifts to higher wavelengths. This redshift of the emission line is typical for planar photoluminescent concentrators, and confirms the physical meaningfulness of the simulated data.

One important question is whether more or less advantageous to apply the quantum dots embedded in a coating on a transparent substrate, or if the quantum dots are dispersed in the entire volume of the pane. In corresponding simulations we compared the case of quantum dots contained in a 0.1 mm thick coating on one or two sides of the substrate to the configuration of quantum dots being immersed in the bulk material. Basically the same results are obtained: No significant energy losses are implied when the quantum dots are contained in a coating and not in the bulk material.

For crystalline silicon solar cells, the best spectral match would be a photoluminescent emission in the near infrared (NIR), with the emission line spectrally located close just next to the absorption edge of silicon. Data on the spectral response of a typical crystalline silicon solar cell were kindly provided by the group of Prof. C. Baillif (Institute of Microtechnology IMT, University of Neuchâtel). A cell efficiency of 20% was assumed.

Since NIR-emitting quantum dots were not easily available without special agreements, the optical data had to be extrapolated. The extrapolation procedure is illustrated by Fig. 9. In the simulations, the Stokes shift has been varied from 25nm to 85nm (for comparison: the Stokes shift of CdS quantum dots immersed in SiO2 coatings produced by our laboratory exhibit Stokes shifts in the order of 60nm). A quantum yield of 85% was assumed for the NIR emitting quantum dots.

1.2

1.0

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0.6

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Nor

mal

ized

inte

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[a.u

.]

1000900800700600500400Wavelength [nm]

34nm 37.5nm

20.4nm 22.9nm 25.4nm

47nm

Placid Blue Maple Red NIR QDs Stokes 25nm

Stokes Shift

FWHM

NIR QDs Stokes 45nm

Absorption and Fluorescence Emission

Fig. 9. Spectra of the absorption coefficient and the photoluminescent reemission as used in the simulations.



In general, the performance of the devices is strongly dependent on the concentration of the quantum dots (number of quantum dots per unit volume). The simulation is a powerful tool to optimize this important parameter. The effective concentration factor of the quantum dot solar concentrator can be defined as the cell area directly exposed to sunlight, orientation in plane of device, divided by the cell area on edges of device yielding the same amount of energy. In Fig. 10 the effective concentration factor is plotted as a function of the relative quantum dot concentration. In the optimized case, effective concentration factor of 9.4 is achieved, while too high or too low quantum dot concentrations yield inferior performance.

An overview of simulation results for single-pane devices in combination with crystalline silicon solar cells is given in Table 1. A surface of 1m2 was considered, with 1000W of incident solar radiation. The geometrical configuration included a back mirror. The optical density of the photoluminescent material was optimized for each type of material. For commercial CdSe/ZnS core shell quantum dots, the simulation yields only an efficiency of 0.8%. For fluorescent dye colored PMMA, an efficiency of 1.5% is obtained. Higher efficiency values can be obtained for NIR-emitting quantum dots. For a Stokes shift of e.g. 65 nm, an efficiency of 5.3% is achieved, yielding under the given conditions an electricity production of 53W. This can be easily explained by the fact that the emission by the infrared emitting QDs is spectrally much better matched to the to the band gap of the crystalline silicon PV cell. In practice, the assumed Stokes shift of 65nm might not be not out of reach: CdS nanocrystal containg SiO2 coatings produced by our laboratory exhibit a Stokes shift in the order of 60nm.

9

8

7

6

5

4

eff.

conc

. fac

tor

0.12 3 4 5 6 7 8 9

12 3 4 5 6 7 8

relative concentration [a.u.]

NIR emitting quantum dotsvariation of QD concentration

Fig. 10. Dependence of the effective concentration factor of the device as a function of the relative concentration of near infrared emitting quantum dots.

Table 1. Overview on simulation results for single-pane devices in combination with crystalline silicon solar cells. Indicated are the effective concentrations and the projected energy conversion efficiencies.

VIS QDs NIR QDs NIR QDs NIR QDs NIR QDs

Cd/ZnS Stokes sh. 25nm Stokes sh. 45nm Stokes sh. 65nm Stokes sh. 85nm

Effective Concentr. 2.0 5.1 9.4 13 15

Projected Efficiency 0.8% 2.1% 3.8% 5.3% 6.0%



5. DISCUSSION

By our simulations, the importance of two main factors is illustrated, an emission spectrum matched to the spectral efficiency curve of the photovoltaic cell, and a large Stokes shift, which is advantageous for the lateral energy transport. CdS nanocrystal containg SiO2 coatings produced by our laboratory exhibit a Stokes shift in the order of 60nm. It should be feasible to replace the CdS in the nanocomposite coatings by another semiconductor, thus aiming at a combination of the two beneficial factors. A quantum yield of 85% was assumed. Quantum yields of 50%-85% have been reported in literature for VIS emitting CdSe/ZnS core shell quantum dots [22,23]. Further investigations are necessary in order to clarify whether a comparable quantum yield can be achieved also for NIR-emitting quantum dots.

For single pane devices based on NIR emitting QDs in combination with crystalline silicon PV cells, system efficiencies up to 5% - 6% were projected. This is approximately in the same order of the efficiencies of amorphous silicon PV cells, but the sol-gel coatings could be produced at very low cost on the large scale, without the need for expensive vacuum eqiupment. System efficiencies will be higher when tandem or triple stack devices are built, in combination with high efficiency cells of different semiconductor materials with suitable energy gaps.

Nanocomposite materials can be very durable. In selective solar absorber coatings, metal nanocrystal containing films have proven excellent stability in accelerated aging tests designed for a service lifetime over 25 years in harsh conditions (elevated temperatures and humidity). Silicon dioxide is an effective oxidation barrier: In food packaging, already a 2 nm thin SiO2 layer is sufficient to create an oxygen-tight protective layer. Therefore we believe that nanocomposite coatings consisting of inorganic semiconductor nanocrystals embedded in a silicon dioxide host matrix will show superior aging stability.

By sol-gel processing, large surfaces can be coated at low price. As an example, the German CENTROSOLARGLAS company can be mentioned: for sol-gel anti-reflection coatings on solar thermal collector glazing, the market prize amounts to only 8 EUR/m2. Here lies one of the strengths of the concept: large surfaces could be coated in a low-cost process, resulting finally in low cost per kWh electricity.

6. CONCLUSIONS A software tool for Monte-Carlo ray-tracing simulations of quantum dot solar concentrators has been developed and tested. Such simulations are useful to predict the total efficiency of the envisaged concentrating PV systems, and to optimize pane dimensions, quantum dot concentrations, and the photoluminescent emission frequencies.

A large Stokes shift is advantageous in order to avoid energy losses during the lateral energy transfer. The photoluminescent emission of the quantum dots should spectrally match the used PV cells: e.g. for crystalline silicon PV cells, near infrared emitting QDs should be used. Quantum dots can be contained in a coating instead of being dispersed in the entire volume of the pane. No losses are implied due to the fact that the quantum dots are applied as a (nanocomposite) coating. Quantum dot containg sol-gel coatings can be applied at low price, quantum dots do not need to be purchased, crystals can be grown by self-organization during the thermal annealing step. Nanocomposite coatings consisting of inorganic semiconductor nanocrystals embedded in a silicon dioxide host matrix will be very durable.

For single pane devices based on NIR emitting QDs in combination with crystalline silicon PV cells, system efficiencies above 5% - 6% might be possible, depending on the optical properties which can be achieved for NIR emitting quantum dots. For stacked devices, system efficiencies will be higher than for single pane devices. Combining cost-effective sol-gel deposition, high durability and a potential for system efficiencies > 6%, the novel concept opens interesting perspectives and definitely merits future development.

The next step should be the development of coatings containing NIR emitting quantum dots made of a semiconductor material with suitable band gap.

Proc. of SPIE Vol. 7046 704609-10


7. ACKNOWLEDGMENTS The authors would like to thank P. Loesch and L. Deschamps for technical support, and the group of Prof. C. Baillif (Institute of Microtechnology IMT, University of Neuchâtel) for providing data on the spectral response of crystalline silicon solar cells. The financial support of the Swiss Federal Office of Energy SFOE is gratefully acknowledged.

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SPIE Proceedings [SPIE Solar Energy + Applications - San Diego, CA (Sunday 10 August 2008)] Optical Modeling and Measurements for Solar Energy Systems II - Monte Carlo simulations