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Broadband omnidirectional antireflection (AR) coatings for solar cells optimized using simulated annealing (SA) algorithm incorporated with the solar (irradiance) spectrum at Earth’s surface (AM1.57 radiation) are described. Material dispersions and reflections from the planar backside metal are considered in the rigorous electromagnetic calculations. Optimized AR coatings for bulk crystalline Si and thin-film CuIn1–xGaxSe2 (CIGS) solar cells as two representative cases are presented and the effect of solar spectrum in the AR coating designs is investigated. In general, the angle-averaged reflectance of a solar-spectrum-incorporated AR design is shown to be smaller and more uniform in the spectral range with relatively stronger solar irradiance. By incorporating the transparent conductive and buffer layers as part of the AR coating in CIGS solar cells (2μm-thick CIGS layer), a single MgF2 layer could provide an average reflectance of 8.46% for wavelengths ranging from 350 nm to 1200 nm and incident angles from 0° to 80°.
©2011 Optical Society of America
1. Introduction
Designed to minimize light reflection at the refractive index discontinuity, antireflection (AR) coating has long been a fundamental yet vital subject in various applications in optics. The most well-known AR coating may be the quarter-wavelength film having a refractive index equal to the geometric mean of the indices of two different materials between which the quarter-wavelength film is applied. With the increasing demands in reducing green house gases, solar cells have been gaining much renewed attention worldwide in recent years for one purpose only: to cost-effectively enhance their efficiency. Depending on the materials used for the absorbing layer(s) and the transparency regime of the conductive/buffer layers, the typical spectral range of modern solar cells is in general from 350 nm to 1200 nm. This in turn demands some super broadband AR coatings with the upper-to-lower wavelength limit larger than three. Moreover, in the absence of the sunlight tracking system, low angle-averaged reflectance at the top surface of solar cells over a wide angle of incidence up to 80° or larger is also essential to maintain a high collection efficiency in the daytime. As a result, an AR coating that is of super broadband and effective over a wide range of incident angles is highly desirable and may still remain a challenge in solar cell applications.
As far as the broadband characteristic is concerned, various approaches have been reported in the literature and many of them rely heavily on the surface texturing techniques. They include the use of subwavelength structures [1–3], an SiOxNy-porous silicon double layer AR coating [4], and a multilayer graded porous silicon [5]. The results reported in these work appear to be obtained at normal incidence. More recently, a broadband omnidirectional AR coating using a solution-processed monolayer of silica microspheres has been demonstrated on commercial amorphous Si solar cells [6]. The most comprehensive studies on AR coatings with broadband and omnidirectional characteristics may be found in references [7–10]. A step-graded quantic profile [9] and a graded-index profile designed by the genetic algorithm (GA) [7, 8, 10] have been theoretically and/or experimentally demonstrated. Using a nano-structured SiO2/dense SiO2/dense TiO2 three-layer AR coating on polished crystalline silicon (Si), the measured angle- and wavelength-averaged reflectance for angle of incidence θ = [40°, 80°] and wavelength λ = [400, 750] nm can be as low as 5.9% [8].
Over the years, research on AR coatings for solar cell applications may have long been conducted with minimum consideration of the cell’s back reflector now commonly used in crystalline Si and thin-film solar cells. In the presence of the backside metal, the reflectance at the top surface can be stronger than that without the metal if the incident light is not completely absorbed. Hence AR coatings have to be more judiciously optimized in cases where the absorption region is no longer electromagnetically infinite in extent.
In view of the demands for super broadband and omnidirectional AR coatings for low-cost solar cell applications, simulated annealing (SA) algorithm incorporated with the solar (irradiance) spectrum at Earth’s surface are used in this work to optimize the AR coatings for metal-backed solar cells. Bulk crystalline Si and thin-film CuIn1–xGaxSe2 (CIGS) solar cells are considered as two representative cases. The reflectance is calculated rigorously electromagnetically, taking into account the material dispersions and the reflections from the back reflector. Comparisons are made between the angle-averaged reflectance spectra (averaged over two different polarizations) with and without the solar spectrum consideration to illustrate its effect on the AR performance. Furthermore, the use of the transparent conductive and buffer layers as a constituent part of the AR coating for CIGS solar cells is conceived and theoretically demonstrated.
2. Formulation
The iterative method utilized in this work for the optimization of AR coatings is the SA algorithm in conjunction with the rigorous two-dimensional transmission line network approach. The use of SA has the following important reasons. First of all, the global optimum (or near-optimum) found by SA is largely insensitive to the initial values that are often critical in conventional optimization algorithms [11]. In addition, the cost function that quantitatively describes the problem of interest can be defined in a discrete or continuous domain [12, 13]. Moreover, while the genetic algorithm shows difficulty with a flat parameter space [14], SA can deal more easily with functions having ridges and plateaus [15] and is thus in particular suitable for a problem whose cost function is multimodal. Finally, SA also allows restricted parameter searching range(s) and the optimization to a particular subset of the parameter space, making the optimized results more practically applicable. A succinct description of SA applied to the AR coating optimization is given below.
2.1. Simulated annealing optimization
SA is a global optimization algorithm that finds the global minimum of the cost function associated with the problem of interest through a number of function evaluations. It explores the image of the cost function by performing random walks in m space, where m denotes the number of parameters to be optimized. In the present work, the m variables are the physical layer parameters, including the individual layer thickness t and the corresponding wavelength-dependent permittivity ɛ(λ). A parameter vector X̲ = [x 1, x 2, … , xm]T (the superscript T denotes transpose) may thus be defined with its elements being the physical parameters ti and ɛi(λ) associated with the i-th layer. The number m is then two times the number of layers used in the AR coating. The lower and upper limits of xu (i.e. au ≤ xu ≤ bu, u = 1, 2, … , m) correspond to the smallest and largest physically realizable values of the layer thickness or relative permittivity. Let C(X̲) be the cost function evaluating the average reflectance over certain ranges of incident angles and wavelengths. The algorithm attempts to minimize C(X̲) that can be analytically obtained through the formalism of a general scattering problem associated with a multi-layer structure.
Starting from an arbitrary initial setting of X̲ (i.e., the initial configuration of an AR coating) which in turn initializes the performance parameter (i.e., the reflectance) by calculating C(X̲), SA randomly generates new physical parameter vector X̲ k +1 by applying random moves along each coordinate direction u
where k is the iteration index, r a random number in the range of [−1, 1], V͇ the step vector in the form of diagonal matrix, and e̲ u a row vector so that {e̲ u, u = 1, 2, … , m} forms the standard ordered basis for ℝm [16] [for instance, if m = 4, then e̲ 3 = (0, 0, 1, 0)]. The change in xu is valid only if it falls within the domain of C(X̲), namely au < xu < bu. The acceptance or rejection of the newly-generated parameter vector X̲ k +1 is determined by the Metropolis criterion [17] in which downhill moves [ΔC k,k+1 < 0, where ΔC k,k+1 = C(X̲ k +1) −C(X̲ k)] are always accepted, whereas uphill moves are allowed if a randomly-generated number p satisfies the expression p < exp (−ΔC k,k+1/T), where 0 ≤ p ≤ 1 and T is the temperature targeted for each individual phase during the cooling process. Note that the exponential term is the Boltzmann distribution. Performing the above steps for every element in the vector X̲ (i.e. u from 1 to m) completes one cycle of random moves and the optimum AR coating configuration is updated every time when ΔC k,k+1 < 0 occurs.To have the algorithm closely follow the “function behavior”, a 1 : 1 rate between accepted and rejected random moves in each coordinate direction is preferred. To this end, a step vector adjustment after every NS cycles of random moves is required and is suggested by [13]
where , , and cu are the new step vector component in V͇, the number of accepted AR coating configurations, and the multiplication factor in the u-th coordinate direction, respectively. In this work, cu = 2 for all u [13]. Thus the step length is increased (decreased) every NS moves if the ratio is larger than 0.6 (smaller than 0.4), which occurs when too small (large) steps are used. It remains unchanged when the ratio stays between 0.4 and 0.6.As the function value for a sequence of AR coating configurations converges (i.e., the “system” reaches the thermal equilibrium at some specific temperature), the temperature T is decreased by a reduction coefficient rT, and a new iteration of generating trial vector X̲ based on the previously-optimized configuration and step vector continues until the thermal equilibrium is reached again. The timing as to when to lower the temperature can also be specified by a parameter NT, the number of step adjustments that needs to be implemented at a temperature T. The process progresses until the stopping criterion described in Ref. [13] is satisfied.
It is worth mentioning that, for the efficient use of limited computational resources while maintaining the initial temperature high enough for rendering a rough global view of the image of C(X̲), the initial value of the control parameter (i.e., the temperature in the cooling process) suggested in [18] is adopted,
where m 1 and m 2 are the respective numbers for which ΔC k,k+1 ≤ 0 and ΔC k,k+1 > 0, χ denotes the minimum acceptance rate at T 0, and represents the average increase in cost (i.e. the average value of ΔC k,k+1 for which ΔC k,k+1 > 0). The minimum acceptance rate χ is assumed 0.8 at T 0 throughout this work.2.2. Transmission-line network formalism for a general multilayer scattering problem
Figure 1 depicts a general multilayer structure and its associated rigorous transmission-line network used in the investigation of the scattering problem associated with solar cells. Each layer is assumed homogeneous and is characterized by its thickness ti and complex relative permittivity ɛi, i = {1, 2, … , n}. A plane wave is incident from the semi-infinite air region (ɛa). For simplicity, the entire absorption layer is described by the bottommost layer (ɛn) that is terminated on a planar back reflector. The transmission line associated with the absorption layer is therefore short-circuited, corresponding to a perfect-electric-conductor (PEC) termination condition.
Fig. 1 Two-dimensional structure and the associated transmission-line network for the reflectance calculation of a general solar cell with a backside metal.
The characteristic impedance Z 0,i of the transmission line associated with the i-th layer for transverse electric (TE) and transverse magnetic (TM) polarization is given by [19]
in which ω is the angular frequency, μ 0 the free-space permeability, ɛ 0 the free-space permittivity (ɛa = 1.0 in Fig. 1), and κi the complex propagation constant along the line, where λ is the free-space wavelength and θ the angle of incidence measured from the surface normal. Note that κi is in the direction transverse to the layer interface.The reflection coefficient at the air-AR coating interface can be conveniently obtained using the following expression
in which Z L,0+ is the load impedance seen looking downward from the position z = 0+ (see Fig. 1) and Z 0, a the characteristic impedance of the line associated with the input air region. The load impedance Z L,0+ may be calculated successively, starting from the bottommost layer with the following general equation [20] where Z in, i designates the input impedance seen looking toward the end of the i-th layer from right below the (i − 1)−i interface and Γi, i+1 the reflection coefficient at the i−(i + 1) interface The expression for Z in, i +1 is the same as that of Z in, i, except that i is now replaced by i + 1. Since the characteristic impedance Z 0, i is both wavelength- and angle-of-incidence-dependent, so is the reflection coefficient.The cost function C (X̲) for the SA optimization is then defined as the normalized solar spectral irradiance w(λ) times the arithmetic average of TE and TM reflectance averaged over the angular and wavelength spectra,
with where Iλ represents the solar irradiance spectrum and ΓTE and ΓTM are the reflection coefficients [obtained using Eq. (6)] for TE and TM polarization, respectively. It is worth mentioning that the unit of C (X̲) is irrelevant to the final optimization result since the SA algorithm finds the global minimum in the image of C (X̲). As a result, the solar irradiance spectrum itself can also be directly used as the weighting function in Eq. (9) without losing the generality.3. Results and discussions
The results presented in this work, except the comparisons between simulated annealing and genetic algorithms, correspond to structurally-uniform solar cells with back reflectors. The reflectance calculations thus include the reflections from the back reflector beneath the absorption layer. The material dispersions of all the materials were considered through the expression ɛ(λ) = ɛ′ (λ) – j ɛ″ (λ), in which ɛ′ = n 2 − k 2 and ɛ″ = 2nk with n and k being the respective real and imaginary parts of the complex refractive index. To optimize and better evaluate the AR coating performance, the ɛ″ term is ignored for layers other than the absorption region where the full description of ɛ was adopted. This would exclude the attenuation in layers other than the absorption region, thus intentionally revealing the highest estimated reflectance.
3.1. Comparisons between the genetic and simulated annealing algorithms
Before the extensive optimizations of AR coatings for representative crystalline Si and CIGS solar cells, comparisons were made with published theoretical and experimental results. The structure under consideration is described in Ref. [8] where the absolute reflectance of a GA-optimized triple-layer AR coating on top of a polished crystalline Si substrate has been theoretically calculated and experimentally demonstrated. The AR coating consists of a stack of 80% porous SiO2 (n = 1.07 at 550 nm), bulk SiO2 (n = 1.47 at 550 nm), and bulk TiO2 (n = 2.66 at 550 nm). The GA-based optimization was conducted in the wavelength range of 400 – 1100 nm and the angle of incidence from 0° to 90°.
To render a fair comparison between the genetic algorithm and SA in AR coating applications, the layer compositions and their corresponding refractive indices were kept unaltered as in Ref [8] whereas the layer thicknesses were optimized based on SA. The dielectric function of a bulk crystalline Si in Ref. [21] was taken into account here and throughout this work. Using Ns = 10 and NT = 5 in the SA optimization, the optimized average reflectance R ave is 3.54% for λ = [400, 750] nm and θ = [40°, 80°], as opposed to 4.90% for the same wavelength and angle-of-incidence ranges reported in Ref. [8]. The detailed layer thickness and performance comparisons are given in Table 1.
Table 1. Comparisons Between Layer Thickness (in nm) and Average Reflectance R ave (Averaged over Incident Angles, Wavelengths, and Polarization) for a Three-Layer AR Coating on Top of a Polished Crystalline Si Substrate Obtained Using a Genetic Algorithm [8] and the Present SA Algorithm
Figure 2 illustrates the reflectance R(λ, θ) (averaged over TE and TM polarization) as a function of the incident angles and wavelengths. It is interesting to observe that the SA algorithm minimizes the reflectance for most of the wavelengths and incident angles at the cost of relatively high reflectance for λ < 435 nm over the span of θ. The SA-optimized average reflectance R ave is merely 3.40% for λ = [400, 1100] nm and θ = [0°, 80°].
Fig. 2 Calculated reflectance R(λ, θ) (averaged over TE and TM polarization) of a polished crystalline Si substrate with an SA-optimized three-layer AR coating given in Table 1 for (a) λ = [400, 750] nm, θ = [40°, 80°] and (b) λ = [400, 1100] nm, θ = [0°, 80°].
3.2. Bulk crystalline silicon solar cells
Because of its low cost character, Si solar cells are the most widely used photovoltaic devices in today’s renewable-energy market. While one- and two-layer AR coatings have long been commonly used for reducing the reflection, often they are seldom optimized for broadband and wide angle-of-incidence operations.
Using the SA algorithm incorporated with the solar spectrum at Earth’s surface (AM1.57 radiation) [22], AR coatings for bulk crystalline Si solar cells having back reflectors with up to six layers were optimized. The Si layer is assumed to have a thickness of 300 μm. The spectral range of interest is from 400 nm to 1100 nm. The index domain of the cost function C(X̲) is defined by some practically realizable refractive indices ranging from 1.05 to 2.66 [8, 23], whereas the range of thickness needs to be judiciously chosen in order not to tightly restrict the random moves during the SA optimization. Table 2 lists the SA-synthesized AR coatings for angle of incidence from 0° to 80°. Note that the refractive index of each layer is a weighted average averaged over the polarization and the spectral range of interest. It is apparent that the optimized index vs. cumulative layer thickness is not necessarily of graded index. For AR coatings with layers less than five, the SA algorithm does produce a graded-index profile; however, for the five- and six-layer coatings, alternating low-index and high-index layers with differing thicknesses were generated by SA. Also, the average reflectance R ave converges as the smallest and largest indices are close to the limits of index domain (i.e. 1.05 and 2.66). When this occurs, adding more layers with indices between the upper and lower limits can hardly improve the average reflectance.
Table 2. Refractive Index n and Thickness t (in nm) of Individual Layers in SA-Optimized AR Coatings for Bulk Crystalline Si Solar Cells with Back Reflectors*
To better visualize the AR coating performance reported in Table 2, the angle-averaged reflectance Rθ −ave(λ) defined by
for each coating configuration is depicted in Fig. 3. Because of the small absorption coefficient of the crystalline Si at longer wavelengths, the angle-averaged reflectance increases precipitously beyond approximately λ = 1000 nm, regardless of the number of layers used in the AR coating. The average reflectance R ave decreases drastically if the contributions from λ ≥ 1000 nm is excluded, as listed in Table 2. In addition, the number of local Rθ –ave minima is in general not equal to the number of AR coating layers. This is very different from the GA-based results where the number of local minima in reflectance is equal to the number of layers used [7].Fig. 3 Calculated angle-averaged reflectance spectrum of an SA-optimized AR coating for metal-backed Si solar cells with (a) and without (b) the solar spectrum consideration. The Si layer is assumed 300 μm in thickness and the incident angle θ ranges from 0° to 80°. The electromagnetic model is shown as an inset; ARC: antireflection coating, PEC: perfect electric conductor.
The effect of incorporating solar spectrum as a weighting function in the AR coating optimization may be understood through the comparisons between Fig. 3(a) and (b). For those with the layer number less than four, the local minima obtained in the presence of solar spectrum tend to shift toward the shorter wavelength range when compared to those obtained without the solar spectrum. Since the solar irradiance peaks at approximately 500 nm, introducing the solar spectrum into the cost function C(X̲) can effectively synthesize optimized AR coating designs that are in favor of the spectral range with strong solar irradiance. For AR coatings with more than three layers, the solar-spectrum-incorporated AR coatings are shown to produce smaller and more uniform Rθ –ave curves for λ = [400, 950] nm when compared to those shown in Fig. 3(b).
As the cost-effective fabrication of Si solar cells is paramount, an AR coating that is easy to fabricate yet offers good performance is highly desirable. Accordingly, it may be valuable to investigate in more detail the SA-optimized double-layer AR coating composed of SiO2 and TiO2 only. Again, a crystalline Si solar cell with a 300-μm-thick Si layer and a backside metal is of interest. The dielectric functions of SiO2 [21], TiO2 [24], and crystalline Si [21] were all considered in order to have the SA-optimized results more practically applicable. The individual layer thicknesses thus obtained for differing angle-of-incidence intervals are listed in Table 3.
Table 3. Layer Thickness (in nm) and Average Reflectance R ave of an SiO2/TiO2 Double-Layer AR Coating for Metal-Backed 300-μm-Thick Crystalline Si Solar Cells
3.3. CIGS solar cells
A typical CIGS solar cell is composed of a transparent conductive ZnO layer followed by a thin n-type CdS buffer layer for lattice matching to the molybdenum-backed CIGS absorber [25]. The structure considered in the rigorous electromagnetic model then consists of a ZnO/CdS/CIGS stack terminated by a PEC boundary condition. The dielectric functions associated with ZnO [26], CdS [27], and CuIn1– xGaxSe2 with x = 0.31 [28], as well as the AM1.57 solar spectrum, were all taken into account. The gallium composition ratio of 0.31 was chosen since it is typical of most high-efficiency CIGS solar cells [29, 30]. In the SA optimization, the CIGS layer was assumed 2 μm in thickness and the refractive index interval for all AR coating layers was limited from 1.05 to 2.00. Since, in general, both the external quantum efficiency and solar irradiance spectrum decrease precipitously for λ < 350 nm [31, 32], the contributions from λ = [300, 350] nm may be insignificant and having the AR coating be optimized in favor of the spectrum covering stronger external quantum efficiency and solar irradiance may be of more practical importance.
As indicated in Eq. (6) and in Fig. 1, the reflectance at the top surface of solar cells is determined by the input impedance seen looking toward the bottom layer from the surface. Hence every single layer above the backside molybdenum contributes to the reflection coefficient. In other words, while the CIGS layer thickness is fixed, ZnO and CdS layers could serve as a constituent part of the AR coating so long as the solar cell efficiency and the low-cost fabrication requirement are not compromised. The SA-optimized AR coatings for CIGS solar cells with up to five layers are given in Table 4. The optimized thicknesses of ZnO and CdS layers are well within the range suggested previously [25, 29]. It is interesting to note that the optimized thickness of CdS layer is nearly unchanged while that of ZnO decreases monotonically as the number of AR coating layers increases. This suggests that with only a single- or two-layer AR coating presents, the low-index layer has to be made thicker in order to reduce the average effective index for minimizing the average reflectance.
Table 4. Refractive Index n and Layer Thickness t (in nm) of Individual Layers in SA-Optimized AR Coatings for CuIn1−xGaxSe2|x=0.31 Solar Cells*
The Rθ –ave spectra for AR coatings optimized with and without the solar spectrum are shown in Fig. 4. The rapidly-varying behavior of Rθ –ave in the wavelength range from about 370 nm to 415 nm is attributed to the 3.37 eV direct bandgap energy of ZnO [26], leading to an abrupt change in its dielectric functions associated with both the ordinary and extraordinary waves. Note that the ordinary wave (E || c) and extraordinary wave (E ⊥ c) correspond to the TE-and TM-polarized waves in our multilayer model since the c-axis is normal to the ZnO layer interface [26].
Fig. 4 Calculated angle-averaged (θ = [0°, 80°]) reflectance spectrum for CIGS solar cells with (a) and without (b) the solar spectrum consideration. The CIGS layer has a thickness of 2 μm. The electromagnetic model is shown as an inset; ARC: antireflection coating, PEC: perfect electric conductor.
Similar to those in crystalline Si solar cells, the solar spectrum minimizes Rθ –ave associated with the single- and two-layer AR coatings in the range λ = [350, 586] nm and λ = [350, 664] nm, respectively, corresponding to the spectral range with stronger solar irradiance. While the reflectance performance is in general quite similar in the range λ = [650, 1000], appreciable differences in Rθ –ave between Fig. 4(a) and 4(b) in shorter (λ < 650 nm) and longer (λ > 1000 nm) wavelengths are readily observed.
In addition to the AR coatings listed in Table 4, it may be equally important to provide the data of an SA-optimized single-layer MgF2 coating commonly used in the CIGS solar cell. The birefringence of MgF2 was temporarily ignored for simplicity and its refractive index was limited between 1.3724 and 1.3996 [33] during the SA optimization. The optimized thicknesses of planar MgF2, ZnO, and CdS layers are 124.97 nm, 354.74 nm, and 45.09 nm, respectively, giving rise to an average reflectance R ave = 8.86% for λ = [350, 1200] nm and θ = [0°, 80°] without the introduction of any surface texturing. The reflectance R(λ, θ) plots associated with an SA-optimized single MgF2 layer and a two-layer coating are shown in Fig. 5.
Fig. 5 The reflectance R(λ, θ) averaged over TE and TM polarization of (a) a single MgF2 layer and (b) a two-layer AR coating for CIGS solar cells. The CIGS structure is shown in the inset of Fig. 4.
4. Summary
Broadband omnidirectional AR coatings for metal-backed solar cells optimized using simulated annealing algorithm incorporated with the solar spectrum have been theoretically demonstrated. The reflectance calculations are based upon the rigorous transmission-line network approach in which material dispersions and reflections from the planar backside metal are all taken into consideration. AR coatings for representative bulk crystalline Si and thin-film CuIn1−xGaxSe2|x=0.31 solar cells are synthesized and analyzed to illustrate the advantages of the present methodology. The results show that solar-spectrum-incorporated AR coatings could in general minimize and flatten the angle-averaged reflectance over the spectral range with stronger solar irradiance. For low-cost fabrication purposes, a SiO2/TiO2 two-layer AR coating on a metal-backed crystalline Si of thickness 300 μm is shown to reduce the average reflectance to 11.30% over λ = [400, 1100] nm and θ = [0°, 80°]. The angle-averaged reflectance spectra show that the decreasing absorption coefficient of the Si produces high angle-averaged reflectance of > 20% for λ > 1050 nm, regardless of the number of layers used in the AR coating. On the other hand, by incorporating the transparent conductive ZnO layer and CdS buffer layer as part of the AR coating in CIGS solar cells (2μm-thick CIGS layer with a back reflector), one single MgF2 layer is shown to possibly provide an average reflectance of 8.46% for wavelengths ranging from 350 nm to 1200 nm and incident angles from 0° to 80°.
Acknowledgments
This research was supported in part by Grant NSC-99-2221-E-008-057 from the National Science Council, Republic of China (Taiwan).
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