Optimization of a perfect absorber multilayer structure by genetic algorithms

During last decade a huge interest has been grown in the development of nearly perfect absorber materials for applications as solar energy collectors, and, more in general, for all the thermophotovoltaic applications (TPV), so to increase the absorbed power for energy harvesting, storage and conversion as well as for the reprocessing of wasted heat in industrial processes [1, 2].

The use of carbon nanotube technology improved a lot the performance of the absorbing materials. In 2009 nanomaterials based on vertically aligned single-walled carbon nanotubes shown an absorbance of about 98% of the incoming light in a wide spectral range UV-VIS-NIR [3]. But despite recent advances in the development of carbon nanotubes (CNT) purity assessment tools, the macroscale assessment of the overall surface qualities of commercial CNT materials remains a great challenge, bringing negative impacts on the reliable and consistent nano-manufacturing of CNT products [4, 5].

A new concept of perfect absorber is now based on metamaterials where the enhancement of the absorption can be obtained thanks to the excitation of the surface plasmon resonance for example in gold and silver [6–13]. This can be achieved with structured metallic surfaces [14], microcavities [15], subwavelength hole arrays and opals [16, 17]. Alternatively structured phase change materials [18–23] and chiral metamaterials [24–27] have been recently used so to obtain an active switching of the absorption properties thanks to the metal-insulator transition in one case, or to intrinsic/extrinsic dichroism in the other case. However, the realisation of broadband absorber metamaterials requires long and expensive procedures with multiple steps of film deposition, photoresist coating, etching and photoresist removing [28].

It should be underlined that the design and realisation of perfect absorbers with multilayers can be more technically and economically convenient [29]. The multilayer structure can be easily theoretically tested by numerical simulations, and optimized so to obtain broadband absorbers for a wide range of angles of incidence by using several standard search methods (Genetic Algorithms [30], Neural Network [31–33], Singular Value Decomposition [34–36], Steepest Descent Methods [37, 38], etc.…). In addition the multilayered opaque structures are also easy to be characterized by using many different diagnostic techniques: photothermal, photoacoustic, photopyroelectric, and thermographic techniques [39–45].

Concerning the materials, an increasing interest is for the inorganic ceramic materials such as semiconductor-based oxides and transition-metal nitrides (TiN) which represent the alternative plasmonic materials in the visible frequencies [46–50], with a good thermal stability [51, 52].

Other common materials for optical electronic device applications and solar cells are the transparent conducting oxides (i.e. ITO) which can support surface plasmon polariton excitations [53, 54].

In this paper we study and optimize a multilayer structure based on a stack of ITO/TiN layers deposited onto a silver thick layer so to design the coating of a solar thermal collector. The idea of such a structure has been recently proposed in Ref. [49]. The authors designed an ITO/TiN multilayer finding an average absorbance of 90% in the visible for a wide range of incidence angle with 7 layers but without studying the infrared emissivity. Our purpose is to optimize this configuration by introducing a silica top layer in order to reach a higher average absorbance and by analysing the low thermal emissivity properties of the structure at different temperatures. We performed numerical simulations by changing the number of layers, and the layer thicknesses, and we applied the Genetic Algorithms to find the optimal thicknesses of the multilayer structure.

In this section we discuss the approach to design a ITO/TiN multilayered structure to be an efficient coating for solar thermal collectors. The structure will be designed so to exhibit the highest average absorbance in the spectral range from 400 nm to 750 nm (visible window) for a wide range of incidence angles from 0° to 50°, but also satisfying the requirement on a low thermal emissivity (< 20%) in the infrared, so to minimize the radiative losses of the solar collector.

Starting from this state of the art, we want to show that it is possible to achieve better results by replacing the metal layers with Titanium Nitride layers (TiN) [50].

In fact TiN is an inexpensive metal alloy with smaller free carrier concentrations than noble metals. It behaves as a diluted metal but with material losses comparable to that of gold, showing an optical penetration depth of about 20–30 nm, in the visible range (see Fig. 1a obtained from the data in ref. [61]). In addition TiN has also good mechanical hardness, high corrosion resistance, low frictional constant, thermodynamic stability, relatively good thermal conductivity (≈20 W/m K), low thermal emissivity (< 0.2), very high melting temperature of 2930 °C, and a thermal expansion of 9.3 × 10^− 6 K^− 1 that that is lower than for gold (14 × 10^− 6 K^− 1) and silver (18 × 10^− 6 K^− 1). These properties makes it suitable for applications such as thermal radiation engineering and thermophotovoltaics [62, 63].

In our case the TiN layers are alternated with transparent Indium tin oxide layers (ITO) so to design the ITO/TiN multilayer structure made of M layers as shown in Fig. 2. ITO is a transparent conductive oxide, widely used in electronic and optical devices, and compatible with TiN during the deposition process. ITO shows a sufficiently large optical penetration depth of about 700–1000 nm in the visible (see Fig. 1a [64]), with a high refractive index (see Fig. 1b) in the visible, high melting point (> 1500 °C), and relatively low thermal emissivity.

An earlier study concerning the combination of these two materials has already shown how the average absorbance can reach 90% in the visible and for a wide range of incidence angles by using only 7 layers [49]. However we show here that this scheme can be further improved by introducing a silica top layer which acts as an additional antireflection coating (being \( {n}_{SiO2}\cong \sqrt{n_{ITO}\cdot {n}_{air}} \)), without any relevant increase of the thermal emissivity, (see Fig. 1b), and by searching the optimal thicknesses of the whole structure with Genetic Algorithms (GAs).

GAs have been introduced in the 60s by John Holland for two purposes [65]: to explain the adaptive processes of natural systems, and to design artificial systems software capable to emulate the mechanisms of natural systems. For many years GAs have been applied to solve both optimization and inverse problems in many different scientific fields: in biology [66, 67] in computer science [68, 69], in engineering and physics for adaptive filter design [70], in the synthesis of fiber gratings, in thin film metrology [71], in image processing [72], and in metallurgy for the nondestructive testing (NDT) of materials [73–75].

Adopting the terminology of the biological sciences, GAs evaluate, process and manipulate a population of chromosomes that represent possible solutions in the research space. In our case the chromosome is a string containing the values of the layer thicknesses of the ITO/TiN structure: for example for M layers, the chromosome is a string containing M genes {d₁, d₂, d₃, d₄, …, d_M} where d_i is the thickness of the generic i-th layer (from top to bottom) (see Fig. 2). For each chromosome, a numerical simulation should be performed to calculate the absorbance of the multilayer by using the transfer-matrix method [76], and to quantify how much the absorbance is close to the ideal case of 100% through the fitness of the chromosome. Each chromosome belongs to the population of N_pop individuals. Thanks to the mutual interactions among individuals and to the natural selection mechanisms, the population can evolve and adapt to the environment (research domain), increasing the fitness of all individuals, and eventually finding the best chromosome representing the optimal ITO/TiN multilayer structure.

where R(λ,θ) is the reflectance for unpolarised light at the wavelength λ, and at the incident angle θ and is calculated by using the transfer-matrix method. Note that transmittance is neglected due to the thick silver layer. A_av well represents the figure of merit of the perfect absorber.

In this section we show how Genetic Algorithms (GAs) represent a useful tool to find an optimal ITO/TiN multilayer coating for heat solar collectors, giving some examples which demonstrate how the mechanic of the GAs is surprisingly simple and efficient.

According to the methods described in the previous section, the figure of merit of the multilayer coating has been identified in the average absorbance A_av or better in the fitness function f = 1/(1-A_av)⁴ which is more appropriate to enhance the differences among quasi perfect absorbers (when A_av ≈ 1).

As a simple example we start from a ITO/TiN multilayer coating made of M = 6 layers. We apply GAs to maximize the fitness f (and hence A_av), finding the optimal thickness of each layer. In this case the generic chromosome has M = 6 genes and is represented by the string {d₁, d₂, d₃, d₄, d₅, d₆} where d_i is the thickness of the generic i-th layer (from top to bottom). Note that according to the scheme in Fig. 2, the silica top layer is included only if M is even. Then for M = 6, d₁ is the thickness of the silica top layer, d₂, and d₃ are the thicknesses of the 1st layer of ITO and TiN respectively, d₄, and d₅ are the thicknesses of the 2nd layer of ITO and TiN respectively, while d₆ = 200 nm is the thickness of the last silver layer that is kept constant (see Table 1).

As said each chromosome identifies a specific individual who belongs to the population of N_pop individuals. As a result of mutual interactions among individuals, the population can evolve and adapt to the environment, increasing the fitness of all individuals.

Table 1 shows the initial population of N_pop = 8 chromosomes. Each chromosome is made of M = 6 genes (rows from 6 to 11). All genes are randomly chosen within the ranges described in section 2.

Both absorbances A_o, and A_av are calculated for each chromosome, and shown in rows 2, and 3 respectively. The corresponding fitness function f = 1/(1-A_av)⁴ is shown in row 4. Looking at the fitness the best chromosome of the population is found to be the N.6 (the column is highlighted in black in Table 1).

From Table 1 the best chromosomes are randomly selected for the reproduction with a probability proportional to the fitness value so to give more chances to the best chromosomes respect to the worst ones which risk the extinction. Table 2 shows the new population selected for the reproduction. The best chromosomes can be selected more than once so to transfer the best chromosomic strings to future generations: in fact in Table 2 there are 4 replica of the best chromosome of Table 1.

After the selection, N_pop/2 = 4 couples are formed for reproduction. The reproduction process consists in mixing the genes of each parent couple (X, Y) so to generate the genes of the two sons (S₁, S₂) according to the following rule: S₁ = p₁X +(1-p₁)Y, and S₂ = p₂X +(1-p₂)Y where p₁ and p₂ are weights randomly chosen from 0 to 1. For example in Table 2 the couple of parents N.5 and N.6 generates the couple of sons N.5 and N.6 in Table 3. The first gene of the sons (S₁ = 65, S₂ = 72) (cells are highlighted in black in Table 3) are obtained by mixing the first gene of their parents (X = 49, Y = 89) (cells are highlighted in black in Table 2) by using the weights p₁ = 0.6 and p₂ = 0.42. This procedure is repeated for all the genes by changing randomly p₁ and p₂. In this example the fitness of the sons (414, 358) is higher than the ones of their parents (74, 63). In general after reproduction a new population of children is generated (see Table 3), which statistically has an average fitness better than the previous one (see Table 2), thanks to the selection rules, and to the general assumption that good parents generate good children. In synthesis the new generation tends to adapt more to the environment. Sometimes it happens that after several generations, the individuals become too similar to each other. This unwanted phenomenon may produce an evolutionary stop (epistasis) which inhibits further improvement of the population. To avoid the epistasis, a random mechanism of mutation of the genes should be introduced. Accordingly each single gene may be mutated and substituted by a random value (within the appropriate range seen in section 2) but this may happen only with a small probability (probability of mutation is set to P_m = 5%. This value is given by a tradeoff as shown in a previous work [30]. In fact P_m«5% could cause the epistasis, while P_m»5% generates random mechanisms for the research of the optimal solution which become dramatically time consuming). This mutation creates a new chromosome, sometimes extremely different from its parents, but anyway useful for the renewal of the population. For example in Table 4, the four genes highlighted in black are mutated with respect to the original genes in Table 3, and by chance the fitness of the chromosome N.2 increases.

Last mechanism of GAs is the elitism, which allows to clone the best chromosome of the previous generation and keep it unchanged for the next generation so to avoid a possible regression of the evolutionary process that might statistically occur: for example chromosome N.1 in Table 4 is the clone of the best chromosome in Table 1.

In synthesis the new generation is formed from the previous one by applying the sequence of the following procedures: selection, reproduction, mutation, and elitism. By iterating these procedures it has been demonstrated that the fitness of the population increases, generation by generation, and that the best chromosome slowly tends to the solution of the optimization problem [65, 30].

One fundamental question is now how the evolutionary process can be speeded up so to reduce the computation time needed for finding the optimal structure for the coating of the heat solar collector (which optimizes both A_av and fitness f). One way is by changing the size of the population N_pop that is an adjustable parameter of the GAs. The comparison on the performance of GAs for different N_pop is shown in Fig. 3 where the average absorbance A_av of the best chromosome is plotted as a function of the number of generations. Each curve refers to a particular value of N_pop that should be obviously even for reproductive reasons (8, 10, 12, 14). Looking at Fig. 3 there is an optimal value of N_pop which conjugates the right complexity with the reasonable computational time to find the best solution. In our case the best performances are obtained for N_pop = 12, which allows to reach a high value of A_av (about 97%) already after 100 generations. The corresponding absorbance spectra at normal incidence are shown in Fig. 4 during the iterative process of the GAs, taking the results after 5, 10 and 100 generations. The best chromosome quickly tends to the optimal string {46, 34, 19, 64, 60, 200} also shown in Table 5 (for M = 6) where the best multilayer structures, calculated by GAs, are summarized for any M number of layers.

In fact GAs have been also applied to optimize the multilayer structures by changing the number of layers M (from M = 3 to M = 10, and according to the scheme in Fig. 2). Figure 5 shows the results of the GAs evolution to optimize the average absorbance A_av. The curves refer to several values of M, keeping fixed N_pop = 12. All curves saturate after several hundreds of generations tending asymptotically to the optimized multilayer structures reported in Table 5. In principle the maximum achievable absorbance A_av should increase with the number of layers M, but with some exceptions. In fact the best performances are obtained when M is even, when a top silica layer is introduced as an efficient λ/4 antireflection coating. This explains why the optimized structure for M = 8 is better than the one for M = 9, and why the one for M = 6 is almost equivalent to the one for M = 7.

The absorbance spectra at normal incidence are shown in Fig. 6 only for M = 6, 8, and 10. The corresponding optimized structures are {46, 34, 19, 64, 60, 200} (M = 6), {58, 29, 14, 67, 21, 48, 59, 200} (M = 8), and {73, 15, 10, 61, 14, 59, 26, 46, 60, 200} (M = 10), reaching a very high average absorbance A_av of 96.9%, 99.1% and 99.5% respectively.

In order to evaluate the infrared properties of these three optimal structures, we calculate the emissivity spectra from 0.5 μm to 10 μm, integrated for all angles and averaged for both polarizations (see Fig. 7). By a first inspection of Fig. 7 the emissivity drops in the infrared range for all the curves. Finally the hemispherical total emissivity has been also evaluated at two different temperatures (200 °C and 400 °C) representing the typical thermodynamic range of most solar collectors. In practice the hemispherical total emissivity at 200 °C and at 400 °C are calculated by integrating, over the whole spectrum, the emissivity spectra weighted by the black body radiation curves at 200 °C and at 400 °C that are shown in Fig. 7 in normalized scale. The values of the hemispherical total emissivity ɛ_IR at 200 °C and at 400 °C are reported in Table 5 for all the structures. The quantity ɛ_IR slightly increases with the number of layers and with temperature, always satisfying the requirement ɛ_IR < 20%.

To better understand the absorbing properties of the best optimized multilayer obtained with M = 10, it is useful to study the internal electric field E in the whole structure. Figure 8 shows the amplitude of the E field in each layer at normal incidence. For a better clarity in abscissa the scale of thickness is amplified so to have equally spaced layers. Looking at Fig. 8 an efficient mechanism of distributed absorption is seen at the two wavelengths of 600 nm and 700 nm. Light penetrates without relevant reflection through the structures, being absorbed by the TiN layers and eventually by the silver layer. Conversely the absorption mechanism at 500 nm is related to the light trapping and quasi-resonant absorption [77].

The evidence of the different absorption mechanisms is given by showing the absorbance in each layer along the thin film stack. It is calculated from the difference between the incoming and outgoing Poynting vectors in each layer. Figure 9 show the distribution of the absorbance for the optimized structure with M = 10 {73, 15, 10, 61, 14, 59, 26, 46, 60, 200}, for normal incidence, at λ = 500 nm (blue histogram), λ = 600 nm (green histogram), and λ = 700 nm (red histogram). The absorption occurs prevalently in the TiN layers, but with some differences: at 700 nm the light is absorbed in the first TiN layers, while at 500 nm, the absorption is enhanced in the internal layers due to a resonance process.

To better analyse the absorbing properties of the structures the contour plots in Fig. 10 represent a useful tool to put together both the wavelength and angular behaviour of the absorbance for two optimized multilayers: for M = 6 layers (Fig. 10a) and for M = 10 layers (Fig. 10b). By comparing the two plots the optimized structure with M = 10 layers obviously shows the best performances in the dotted rectangular area for angles in the range [0° - 50°] and for wavelengths in the visible range [400 nm - 750 nm] which guarantees a huge average absorbance of 99.5% inside the rectangle. However the results for M = 6 layers are also interesting, showing A_av ≈ 97% but by using a smaller number of layers. In other words the performance increases with the number of layers (see Table 5), and it is understandable that the best sustainable practical solution should be always given by a trade off between the required performance and the structure complexity. It is also worth noting that all structures show a low hemispherical total emissivity till 400 °C (< 20%), fulfilling the requirement for the efficient solar heat collectors.

The design of quasi perfect absorbers based on ITO/TiN multilayered structures is here discussed. The practical purpose is to find new optimized coatings for solar thermal collectors with the highest achievable absorbance in the visible range from 400 nm to 750 nm, working for a wide range of angles of incidence from 0° to 50°, for both polarizations, and with a low hemispherical total emissivity, so to minimize the radiative losses. Genetic Algorithms are introduced and adjusted for searching the optimal thicknesses for several ITO/TiN multilayered structures with 6, 8, 10 layers reaching a very high average absorbance of 97%, 99% and 99.5% respectively and a low hemispherical total emissivity (< 20%) from 200 °C till 400 °C. The proposed multilayer structures use materials with high thermal stability, and high melting temperature, can be fabricated with simple thin film deposition techniques, appearing to have very promising applications in solar thermal energy harvesting.