Two important aspects must be considered when optimizing antireflection coatings (ARCs) for multijunction solar cells to be used in concentrators: the angular light distribution over the cell created by the particular concentration system and the wide spectral bandwidth the solar cell is sensitive to. In this article, a numerical optimization procedure and its results are presented. The potential efficiency enhancement by means of ARC optimization is calculated for several concentrating PV systems. In addition, two methods for ARCs direct characterization are presented. The results of these show that real ARCs slightly underperform theoretical predictions.
© 2012 OSA
Antireflection coatings (ARCs) have been widely investigated since the beginning of photovoltaics . The high refractive indexes of semiconductors (n~4) will lead to unacceptable reflection losses if no ARC is used. Multijunction (MJ) solar cells have shown the highest efficiency for a photovoltaic device ever measured at 43.5% . ARC design for III-V semiconductor MJ solar cells is especially challenging as these cells are typically sensitive to a wider spectral bandwidth than silicon. In addition, their subcells are connected in series. Hence, if the ARC performs worse in a particular region of the spectrum, photogenerated current for the corresponding subcell will be limited and, as a consequence, total cell efficiency will significantly decrease. In terrestrial applications, MJ solar cells are only economically viable when used in concentrator systems. Concentration implies that the bundle of rays impinging the cell consists of a cone with large interior angle, in some systems as wide as ± 70°. Hence, besides considering the wide spectral sensitivity and the series connected structure, the angular distribution of the incident light over the cell must be taken into account to properly design an ARC for a concentrator cell.
When using optics to concentrate light, the product of concentration and acceptance angle, known as CAP, is the most significant figure of merit as it indicates how close a real concentrator is to ideality . Attainable CAP is limited by the étendue conservation theorem. In rotationally symmetric optical systems in air it can be stated as:
Previous work has only considered one of these two aspects at time when approaching ARC optimization for MJ concentrator cells. On the one hand, Aiken  described the increased complexity when designing ARC for MJ solar cells considering normal incidence and he proposed the ARC design as a subcells current matching technique. On the other hand, Algora et al. [5-6] optimized the thicknesses of a single and double layer ARC for a cone of light with semi-angle between 0° and 90° impinging a single junction GaAs solar cell. Their approach consisted on performing a simultaneous optimization of all the materials thicknesses (not only those of the ARC but also those of the semiconductor structure). Their simulated results showed that only the ARC and the window layers optimum values depends on the semi-angle of the incident cone of light. Additionally, the resulting optimum ARC thickness does not change appreciable for cones narrower than ± 60°. Valdivia et al.  presented a numerical optimization of ARCs composed of 1 to 4 layers when a MJ solar cell is illuminated by the angular distribution created by a CPV system composed of a Fresnel lens and a refractive secondary optical element (SOE). However, for this particular CPV system most of the rays are contained in a narrow cone of light (semi-angle<40°). This condition leads to a set of optimum layer thicknesses very close to the ones obtained for normal incidence.
This paper aims to apply the optimization of ARC to real CPV systems taking into account simultaneously the restriction imposed by MJ solar cells and the angular light distribution created by concentrators which illuminate the solar cell with wide-angle ray bundles. Although specific aspects of the problem have been previously considered by other authors, as summarized above, we believe that the current combination of new CPV system with ever-increasing concentration levels together with new MJ solar cell architectures with more restrictive current matching conditions  requires a global ARC optimization procedure.
The paper is organized as follows: Section 2 discusses basic ARC theory and also summarizes the available materials to manufacture ARC. Section 3 includes a brief description of ARC transmittance modeling. The definition of the figure of merit to evaluate each ARC is presented and the numerical optimization procedure is described. The optimization procedure is applied to several real CPV systems and the transmission enhancement when the actual angular distribution is considered compared with ARC optimized for normal incidence is calculated for each optical system. As it will be shown later, ARC performance sensibility to manufacturing errors is non-negligible and, as a consequence, a method to experimentally analyze the ARC once it has been manufactured is desirable. Section 4 presents two such methods to spectrally and angularly characterize ARC when the solar cell is illuminated by the particular CPV system the ARC has been designed for.
Creating an antireflection coating consists of depositing layers of materials with specific indices of refraction. As a general guide, wider incidence angles require thicker layers. Consider, for example, the well-known quarter-wave ARC design based on building a layer which refractive index is obtained as the square root of the product of the refractive indexes of the two medium adapted (i.e., solar cell and air). The layer thickness must fulfill the condition of creating destructive interference for the design wavelength, λ. In the case of oblique incidence (θ≠0) that condition leads to a thickness (d*) greater than when incidence is normal (d).9].
From the theoretical point of view, an ideal ARC would be one in which refractive index will change smoothly from that of the solar cell to that of the surrounding medium [10-11]. Different experimental processes have been proposed to obtain ideal or quasi-ideal ARC, including depositing very thin alternating layers of high-index and low-index materials to create an intermediate refractive index [12-13], depositing materials with oblique angle which causes self-shadowing and consequently porous materials whose refractive index will be between that of the air and that of the bulk material [14-15] or co-sputtering of different materials . Those processes, which aim to accurately control the refractive indexes of the deposited layer, are usually expensive and time-consuming, and furthermore are currently not used on an industrial scale. In this work we seek not to improve ARC coating technology, but to optimize the ARC for concentration applications using current deposition procedures and materials, and therefore obtain efficiency increases at no extra cost.
In industry, step-down coating structures with several layers are used. In them, refractive indexes for each layer decrease progressively from that of the cell to that of the air. A rule of thumb  can be expressed as: for an N-layers ARC, the layer-i ideal refractive index ni can be obtained as:
Table 1 summarizes the ideal refractive indexes for an ARC composed of one, two or three layers when a MJ solar cell is surrounded by air and by a medium with n = 1.5 (for example, a BK7 glass secondary or encapsulating silicone). It also shows materials whose refractive index is close to the theoretical values and are currently used by the industry .
3. Antireflection coating numerical optimization
The general process of the ARC optimization for a given concentrator can be described as follows. First, the number of layers to be used is chosen, then available materials whose refractive indexes are closest to the ideal values are selected and finally a numerical optimization process is followed to obtain the optimum thickness for each layer.
The optimization figure of merit, Tweighted, is the ARC transmittance weighted by the actual distribution of angles and wavelengths found at the entrance of the solar cell, compared by the ideal. First, the ARC transmittance T (λ, θ, d), for each wavelength and incidence angle is calculated by multiplying the transmission matrix for each layer . The transmission matrix is mainly dependent on the layer thicknesses and material refractive indexes [16-17] (not only the real part but also the imaginary part is considered to account for absorption). Secondly, using ray-tracing simulations the optical transmittance Topt (λ, θ), of the concentrator optics under study is obtained. The angular distribution of the light impinging the solar cell L(θ), the reference spectral distribution of the light B(λ), and the MJ solar cell spectral response for each subcell, SRj (λ), are also used to weight the integrand. As the three subcells are series–connected, the minimum of the three photogenerated currents is selected. To summarize, Tweighted is a measure of how close is the actual short-circuit current in the cell compared to what it will be if the transmittance at the cell entrance were perfect, similar to the solar weighted transmittance SWR, that has been proposed previously [4,7,18].
The evaluation of the figure of merit at each point is time consuming and the presence of local minima makes convergence difficult. A two-steps convergence algorithm has been implemented using brute force approach in a first step to determine the starting point for a simple nonlinear optimization method to determine the combination of layer thicknesses that maximize weighted transmittance.
3.1 Case Example: Optimized ARC for the FluidReflex concentrator
The optimization procedure described above has been used to optimize the ARC of a cell to be used in FluidReflex concentrator. A detailed description of this concentrator can be found in . It consists of a reflective paraboloid that illuminates the cell creating an incident cone of light with a semi-angle of 65°. All the volume between the mirror and the cell is surrounded by a fluid dielectric whose refractive index is 1.47 (at 550 nm). Taking into account the ideal materials to be used and their availability, the selected materials are TiO2 for the monolayer, Al2O3/TiO2 for the bilayer and SiO2/Al2O3/TiO2 for the trilayer. Optimized thicknesses are summarized in Table 2 . As it was expected, the optimum thickness (75 nm) of the TiO2 monolayer is greater than the optimum when incidence is normal (64 nm). The same general conclusion applies for to the bilayer and trilayer coatings.
The sixth column of Table 2 (Δ1) shows the efficiency benefit obtained for each extra layer, for example, adding a monolayer coating compared to no ARC increases Tweighted by 16.9 absolute percentage points, going from one to two layers increases transmittance by 2.9 absolute percentage points, and only 0.3 is gained by adding the third layer. As a general conclusion, we can state that the cost of adding up to two layers will be compensated by the efficiency improvement, while adding the third layer may or may not be justified depending on the additional cost incurred.
Figure 1 shows transmittance as a function of wavelength and incidence angle for different ARC structures, it can be observed how the number of layers the ARC is composed of is equal to the number of transmittance maximums that can be observed in the figures. As a general rule, an ARC with a number of layers equals to the number of junctions the cell is composed may be justified in terms of efficiency improvement. Although customary Ge-based MJ solar cells are composed of three subcells, only two of them are matched (top and middle) while the other (bottom) has an excess of photogenerated current under the standard spectrum. Hence, a two-layers ARC is clearly justified for these cells, but a trilayer is most likely is not. This is in agreement with previously reported results .
Table 2 seventh column (Δ2) shows efficiency enhancement when the ARC has been optimized for the particular concentrator light distribution, compared to the case when only normal incidence has been considered in the optimization. Both bilayer and trilayer ARCs show an efficiency increase higher than 2 percentage points if the particular FluidReflex light distribution is taken into account.
An additional remark need to be made regarding ARCs: values shown in Table 2 will represent average transmissions when the layers thicknesses as manufactured perfectly match the optimized ones but in reality manufacturing processes have certain tolerances that will influence the ARC performances. Figure 2 shows the losses in the average transmission values for the trilayer ARC when two of the layers have been evaporated correctly but an error was committed in the third layer optical path. That is, either the evaporated thickness or the material refractive index resulted different to what it was intended. From the graph we conclude that performance sensitivity to manufacturing errors is not the same for all layers. In fact, for this particular configuration, the average transmission is far more sensitive to errors in the layer closer to the cell. This is in agreement with a similar two-layer ARC tolerance analysis reported in . An accurate control of the evaporated thickness must be guaranteed since, if for example, an error of −50 nm happens in the optical thickness (physical thickness multiplied by refractive index) deposited for the TiO2 layer, average transmission will decrease by 2 relative points (which is similar to the advantage that it has been obtained by designing the ARC for the specific angular light distribution created by FluidReflex concentrator).
3.2 Case example: Optimized ARC for a concentrator composed of a Fresnel lens and a secondary optical element
Fresnel lenses are widely used to concentrate light on MJ solar cells. Reflective or refractive secondary optical elements (SOEs) are added to increase attainable concentration, improve acceptance angle and/or homogenize light distribution over the cell. Each particular SOE will create a different angular distribution of incident light over the cell that depends mainly on how rays bounce in the SOE walls before reaching the cell. To analyze the potential enhancement of optimizing the ARC design for a particular SOE, ray-tracing simulations were used to obtained the incidence angle light distribution over the cell (Fig. 3 ) created by several SOEs. More information on the characteristics and performances of the SOE under study can be found in .
Table 3 presents a summary of the ARC optimization results obtained assuming an Al2O3/TiO2 bilayer, the non-optimized thicknesses account for the optimization results if only normal incidence is considered while for the optimized thicknesses each particular light angular distribution has been taken into account. For the majority of SOEs the maximum incidence angle reaching the cell is lower than 45°, and as a consequence optimized and non-optimized results are very similar and the potential efficiency enhancement by a SOE-specific ARC design is negligible. However, for the dielectric CPC (Compound Parabolic Concentrator), which is a more ideal concentrator that creates a wider cone of light impinging the cell, the optimized thicknesses differ from the non-optimized values and an efficiency increase of 0.7 percentage points can be obtained by an optimized ARC. For an ideal concentrator that illuminates the cell isotropically an efficiency increase of 3.1 percentage points will be obtained by optimizing a bilayer ARC.
In addition to a different angular light distribution over the cell, every SOE creates an irradiance distribution that is not uniform either spatially nor spectrally. This may translate into significant efficiency losses due to increment of effective series resistance and a subcells currents mismatch. The real effect of a particular non-uniform profile strongly depends on the sheet resistance between subcells. This aspect has been an active area of research recently [22–26], but is outside the scope of this article.
4. Antireflection coating characterization
To analyze the performance of an ARC once it has been manufactured different techniques can be used. It is of interest to characterize ARC performance not only spectrally but also angularly. A direct way of characterizing ARC behavior at different wavelengths could be to measure reflectance by using a spectrometer. For measuring the ARC performance at different angles of incidence, a simple experiment consists on measuring short-circuit current of the solar cell when light strikes the cell with different angles can be carried out by using a goniometer and a cavity that only admits collimating light [27-28].
In this section, we propose a method to analyze the spectral and angular aspects of ARC performance when the cell is illuminated by the actual light field created by the concentrator system with which it is to be used, which provides results that accurately predict the efficiency gain produced by the ARC in the intended application. In our experiments, we illuminate the concentrator with collimated light produced by the Helios 3198 Solar simulator , although measurements could also be done under real sun if adequate tracking and temperature and spectrum controls are used.
In our method, the spectral and angular characterization is carried out separately. First, a set of band-pass filters is used to measure the concentrator optical efficiency at different wavelengths, including the ARC. Optical efficiency ηopt, is defined as the ratio between the effective concentration Xeff, and the geometrical concentration Xgeom, which in turn is obtained dividing the concentrator aperture area Aentrance by its exit area Aexit, this is, the area of the solar cell. The measurement of the short-circuit current with the set of filters gives an estimate of the system spectral response. This response is used to obtain a discrete measurement of the spectral optical efficiency, this is, the optical efficiency at a set of particular wavelengths, ηopt,λi. Every sample can be calculated using the following expression:Figure 4 shows the result for two cells with different ARC structures (monolayer and bilayer) under the FluidReflex concentrator system. The available materials instead of the ideal ones where used to manufacture the ARC. ARCs to be analyzed were evaporated over a single junction GaAs since subcell current limitation when using multijunction solar cell will prevent the measurement. Solid lines represent simulated transmission for each particular wavelength. Dots represent the measured spectral total optical efficiency ηopt,λi. As 80 nm band-pass filters were used in the measurements, an 80 nm-wide moving-average filter has been applied to the spectral transmittances predicted by simulation in order to equal data resolutions. For each band pass filter, the total optical efficiency depends on the ARC transmission for that bandwidth and on the spectral behavior of the rest of the concentrator (fluid transmittance, mirror reflectance, etc.). Then, although ARC simulated averaged transmission and measured total optical efficiency do not represent exactly the same magnitude, the fact that display a very similar tendency (Fig. 4) proves the effectiveness of the proposed method results spectrally characterize ARC performance within a concentrator. For the monolayer ARC, transmission is lower in the wavelengths region useful for the top cell than in those useful for the middle cell. If the photogenerated current for each subcell is calculated using the expression shown in Eq. (6) and assuming a matched MJ solar cell under the reference spectrum, the top subcell will generate 6% less current than the middle one, reducing significantly the total cell efficiency.
The second proposed method consists on masking the aperture of the CPV system to analyze the optical efficiency when the angular distribution reaching the cell varies. For the FluidReflex system when circular masks with different radii are used, it directly translates into the cell being illuminated by cones of light with different semi-angles. For other more complex systems, ray-tracing simulation must be used to study the angular light distribution over the cell that will create each particular mask. Figure 5 shows optical efficiency measurements for the FluidReflex system under different semi-angles of incidence of the light cone. As was expected, for almost normal incidence (cone semi-angle < 30°), trilayer and bilayer ARCs transmittances increase significantly with respect to bare cell. Transmittance for the bare cell decreases for semi-angle cones wider than 52°, while for the bilayer ARC transmittance remains almost constant up to 58 ° semi-angle cones. Trilayer ARC not only performs better than the others but its transmittance peaks for a cone with 58° semi-angle, close to the 65° design semi-angle. Note that even coatings optimized for large semi-angles, may show higher transmittances at lower angles because lower incident rays always exhibit lower reflection.
The greatest discrepancy between our experimental and simulated results is the fact that the trilayer ARC transmittes 6 absolute points more than bilayer (Fig. 5), whereas the expected increment when going from a bilayer to a trilayer predicted by numerical simulation (that is Δ1 in Table 2 but calculated for the materials and thicknesses of Fig. 5) was only 2 absolute points. Figure 5 suggests that this is due to underperformance of the bilayer ARC, which is possibly due to manufacturing tolerances. As Fig. 2 showed if an error is committed in one or a combination of several of the layers optical thicknesses ARC performance will considerably decrease. As the bilayer optical thicknesses are lower, manufacturing tolerances of a given magnitude will represent higher relative layer optical thickness changes, and cause greater performance variations in the coating as a whole. Also, the silicon oxide deposition process has been shown to have higher variability than other materials , and since the bilayer depends more on the SiO2 layer, this further increases the susceptibility of this coating system to manufacturing errors.
A numerical model to optimize MJ-cell antireflection coatings (ARCs) has been presented. It takes into account the characteristics of MJ solar cells (broad spectral response and increased spectral sensitivity due to series connection between subcells) and the angular light distribution created by the particular concentration system that will be used to illuminate the cell.
If the angle of incidence of the light reaching the cell under a concentration system is lower than 45°, ARC optimized for normal incidence can be used without expecting significant reflection losses at the cell entrance. However, for systems with wider angles of incidence over the cell an accurate ARC optimization can lead to significant efficiency improvement. A concentrator system composed of a Fresnel lens and a dielectric CPC (Compound Parabolic Concentrator), with an f-number equals to 1.1, may benefit of additional 0.7 percentage points of efficiency increase due to ARC optimization. In the case of reflective systems such as the FluidReflex, where light reaches the cell contained in a 65° semi-angle cone the efficiency increase can be up to 2.9 absolute points.
Furthermore, two experimental methods for characterizing ARC performance by means of illuminating the concentrator have been presented, allowing spectral and angular characterization of the ARC performance to be obtained directly.
This work has been partially supported by European commission within the project NACIR (226409-2) under the VII Framework Program. M. Victoria work is directly supported by Spanish Ministry of Science and Innovation under an FPI grant. Authors would like to acknowledge comments and suggestions by S. Askins that have significantly contributed to clarify the objectives and approaches followed in this work.
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