Optimization of a perfect absorber multilayer structure by genetic algorithms
Abstract
Background
An increasing interest has been recently grown in the development of nearly perfect absorber materials for solar energy collectors and more in general for all the thermophotovoltaic applications.
Methods
Wide angle and broadband perfect absorbers with compact multilayer structures made of a sequence of ITO and TiN layers are here studied to develop new devices for solar thermal energy harvesting. Genetic Algorithms are introduced for searching the optimal thicknesses of the layers so to design a perfect broadband absorber in the visible range, for a wide range of angles of incidence from 0° to 50°, and for both polarizations.
Results
Genetic Algorithms allow to design several optimized structures with 6, 8, and 10 layers reaching a very high average absorbance of 97%, 99% and 99.5% respectively together with a low hemispherical total emissivity (<20%) from 200 °C till 400 °C.
Conclusions
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.
Keywords
Optical materials and properties Perfect absorber Multilayer structure Thermophotovoltaic Solar energy collectorsAbbreviations
 A_{av}
Absorbance averaged in the wavelength range 400 nm – 750 nm, and in the angular range from 0° to 50°
 A_{o}
Absorbance for normal incidence averaged in the wavelength range 400 nm – 750 nm
 ɛ_{IR}
hemispherical total emissivity
 f
Fitness function of the Genetic Algorithms
 GAs
Genetic Algorithms
 ITO
Indium Tin Oxide
 M
Number of layers of the structure
 N_{pop}
Size of the population of the Genetic Algorithms
 P_{m}
Probability of mutation
 SiO_{2}
silica
 TiN
Titanium Nitride
Background
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 singlewalled carbon nanotubes shown an absorbance of about 98% of the incoming light in a wide spectral range UVVISNIR [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 nanomanufacturing 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, 7, 8, 9, 10, 11, 12, 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, 19, 20, 21, 22, 23] and chiral metamaterials [24, 25, 26, 27] have been recently used so to obtain an active switching of the absorption properties thanks to the metalinsulator 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, 32, 33], Singular Value Decomposition [34, 35, 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, 40, 41, 42, 43, 44, 45].
Concerning the materials, an increasing interest is for the inorganic ceramic materials such as semiconductorbased oxides and transitionmetal nitrides (TiN) which represent the alternative plasmonic materials in the visible frequencies [46, 47, 48, 49, 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.
Methods
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].
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, 74, 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 {d_{1}, d_{2}, d_{3}, d_{4}, …, d_{M}} where d_{i} is the thickness of the generic ith 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 transfermatrix 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.
 a)
The structure is made of a sequence of ITO and TiN layers. The thickness for each ITO layer should be searched by GAs in the range [5 nm – 100 nm], thinner than λ/2 so to avoid internal interference effects. Each TiN layer should be searched by GAs in the interval [5 nm – 60 nm] so to play with its transparency/opaqueness.
 b)
A silica top layer can be optionally inserted as additional antireflection coating. Its thickness should be searched by GAs in the range [5 nm – 150 nm], without causing a relevant change of the IR emissivity.
 c)
The last 200 nm thick silver layer absorbs and stops the residual radiation. Therefore the results of the numerical simulations and optimizations are general and independent on the choice of the substrate.
 d)
The total number of layers M ranges from 3 to 10. The silica top layer is inserted only when M is an even number (see Fig. 2). The optimized multilayers found by GAs for different values of M, will be discussed and compared.
 e)
The objective to be maximized is the absorbance A_{av} averaged in the visible range from λ_{min} = 400 nm to λ_{max} = 750 nm, and averaged for a wide range of angles of incidence from 0° to λ_{max} = 50°, for unpolarised light. It is calculated as follows
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 transfermatrix method. Note that transmittance is neglected due to the thick silver layer. A_{av} well represents the figure of merit of the perfect absorber.
 f)
The hemispherical total emissivity of the whole structure is also calculated as ɛ_{IR} = 1R_{IR} in the infrared range from 1 μm to 10 μm, averaged over the solid angle and over the Planck blackbody radiation spectrum for 200 °C and for 400 °C, which is the typical temperature range for most solar collectors. The hemispherical total emissivity should be kept as small as possible (< 0.2) to minimize the radiation losses. The calculation is done by using the literature values for the infrared properties of TiN, ITO, SiO_{2}, and silver [61, 62, 63, 64].
 g)
Many parameters of the GAs has to be set, controlled or adjusted (as will be clear in the next section):

M is the number of genes (coincident with the number of layers). It will be selected in the range from 3 to 10 so to design a realistic and sustainable structure;

N_{pop} is the size of the population used for searching the maximum of A_{av}. It is an even number to be adjusted in the range from 8 to 14;

f is the fitness function of each individual and is the quantity to be maximized. It also rules the selection for the reproduction process. In order to enhance the sensitivity we implemented “ad hoc” GAs by using the fitness function f = 1/(1A_{av})^{4} which is more appropriate to distinguish the difference among highly absorbing structures (with A_{av} around unity). This choice allows to reach an optimal solution already after 1000 generations with a run of a few of minutes on a standard PC, giving in general better results with respect to commercial software;

P_{m} is the probability of mutation of each gene in the GAs. It is kept constant to 5%; This choice is driven by a previous study [30].
Results and discussions
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/(1A_{av})^{4} which is more appropriate to enhance the differences among quasi perfect absorbers (when A_{av} ≈ 1).
Initial population processed by the GAs. Number of layer M = 6. Size of the population N_{pop} = 8
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/(1A_{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).
Results after the selection procedure applied to the initial population in Table 1
Results after the crossover procedure applied to the population in Table 2
Results after the mutation procedure applied to the population in Table 3
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].
Optimized structures found by GAs for different number of layers M
Conclusions
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.
Notes
Acknowledgements
This author is indebted with Mario Bertolotti and Joseph Haus for useful discussions.
Funding
This work has been done in the framework of the project “optical metamaterial” funded by Sapienza University of Rome, and “Scherma” cofinanced by Italian Ministry of Defence.
Availability of data and materials
The numerical results and data can be reproduced by applying the optical method in Ref. [71], by using the Genetic Algorithms described in Ref. [60], and by taking the optical properties of the materials found in Refs. [47, 61, 62, 63, 64]. No additional supporting information or data are necessary.
Author's contributions
The author read and approved the final manuscript.
Author’s information
Roberto Li Voti is associate professor in Applied Physics at Sapienza University of Rome, Italy. He is author of more than 150 publications in the field of optics, photohermal and photoacoustic techniques for nondestructive testing of materials.
Competing interests
The author declares that he has no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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