1 Introduction

In the past decades, significant and rapid advances have been obtained in many scientific fields by nanoparticles due to their unique properties. A lot of research has been done on the specific features of nanoparticles, such as optical properties, local surface plasmon resonance, and also absorption and scattering cross-section of nanoparticles (Bohren and Huffman 2008); Piliarik et al. 2011; Merkus 2009). In this regard, Gustav Mie developed Maxwell’s electromagnetic wave equations and improved the scattering of electromagnetic waves of spherical particles (Bohren and Huffman 2008). Furthermore, various methods such as coupled dipole approximation (Purcell and Pennypacker 1973), discrete dipole approximation (Yurkin et al. 2007; Loke et al. 2011), multiple multipole (Ballisti and Hafner 1983), the generalized Mie theory (Lechner 2005), the finite difference time domain method (Yurkin et al. 2007; Schneider et al. 1998) and T-matrix (Mishchenko et al. 1996) have been presented to obtain electromagnetic waves to describe the structure of nanoparticle array. These methods have been used to describe periodic array and deterministic arrays of nanoparticles. Additionally, plasmonic properties and nanoparticles, especially Silica and Silicon, have been used in many applications. Moreover, semiconductors especially silicons depend on their electronic and scattering properties in the RGB range. Therefore, researchers have been used the influence of factors such as electronic properties of nanoparticles, the environment of nanoparticles, and nanoparticles in various shapes, sizes, materials, array dimensions, and structures to develop devices with different applications (Kondorskiy et al. 2018; Jain et al. 2006; Gatoo et al. 2014; Zou and Schatz 2004; Forestiere et al. 2009; Ramaccia et al. 2016; Schebarchov et al. 2013). One of the recently developed applications of nanoparticles is using those nanoparticles as composite materials to realize transparent display (Hsu et al. 2014; Dolatyari et al. 2019).

Transparent displays are a new technology that has recently attracted researchers to improve the quality of that and introduce new applications. These types of displays can simultaneously demonstrate the information and the surroundings of the screen (Azuma et al. 2001; Honcapie-Ramos et al. 2014). Many techniques have been used to make transparent displays (Tsujimura 2017; Li et al. 2017; Kim et al. 2017; Yang et al. 2019; Chen et al. 2018; Numata et al. 2017; Kateb et al. 2015). One of the most uncomplicated displays is the Head-up display, which researchers have been done a lot of research to reduce the disadvantage of it including, less brightness, glare, specular reflection, and diffuse reflection (Karar and Ghost 2014; Lee et al. 2013; Mahajan et al. 2015; Okumura and Shinohara 2016; Kim and Park 2016; Wiesner et al. 2016). These types of screens are used in the windshield of cars and aircraft, and light is brought into the display by a projector. Moreover, the researchers made a HOE lens array with a planar glass as a waveguide for a transparent color display that increased the transparency of the image, reducing its image quality (Liu et al. 2017). Furthermore, a display was made to convert infrared light into visible light (Downing et al. 1996), and also in another work, Sun and Lie made a display that turns ultraviolet and infrared lights to visible light (Sun and Liu 2006). In this regard, researchers realized that to overcome the problem of contrast reduction to increase the transparency, they could use certain materials in the display construction to avoid scanning all visible wavelengths and have scattering at specific wavelengths. Therefore, in addition to high transparency, this method also provides image quality and contrast (Hsu et al. 2014; Qin et al. 2017; Ekroll and Faul 2013). Hsu et al. by Ag–SiO2 nanoparticles was able to make a display that selectively scattered light at specific wavelengths and passed through other wavelengths (Hsu et al. 2014). Moreover, Jafari et al. have shown that the scattering cross-section at their desired wavelengths can be enhanced by using Si–SiO2 instead of Ag–SiO2 and also transparency of the monitor, in this case, was enhanced (Dolatyari et al. 2019).

In this paper, we investigate the effect of different morphologies of Si–SiO2 nanoparticles on the scattering and absorption cross-section, and also we use different structures of nanoparticles such as the Fibonacci, the Thue-Morse and, the Pseudo-Brillouin zone by 12-fold quasicrystal (hex). Different shapes such as the sphere, the disk, the pyramid, the cube, the prolate, and the oblate ellipse of coated Si–SiO2 nanoparticles have been used to achieve sharper peaks and narrower bandwidths for scattering cross-section. Therefore, the goal is to find the best morphology and the most appropriate array of nanoparticles.

2 Computational method and mathematical modeling

The schematic of the transparent display is shown in Fig. 1. We use a transparent display with different morphologies of nanoparticles in periodic and aperiodic arrays to describe the absorption and scattering cross-sections. In this work, we use Si–SiO2 core–shell nanoparticles, which in detail was discussed in (Dolatyari et al. 2019). We use Palik data for optical constant, material’s relative permittivity, and complex refractive index of Si–SiO2 (Palik 1998). Since the optical properties of the nanoparticles depend on the size, the shape, the core–shell structure, the array, and the number of nanoparticles, we investigate the effect of different forms of nanoparticles on four different arrays. The optical properties of the Si–SiO2 nanosphere, nanocube, nano oblate ellipse, nano prolate ellipse, nano pyramid, and nanodisk were qualified in term of their calculated scattering and absorption cross-sections (\({\sigma }_{sca}\) and \({\sigma }_{abs}\)). Scattering cross-section is obtained by the extinction cross-section minus absorption cross-section (\({\sigma }_{sca}={\sigma }_{ext}-{\sigma }_{abs}\)). Incident light is irradiated by a plane wave field onto an array of nanoparticles to calculate the scattering and absorption cross-section.

Fig. 1
figure 1

Schematics of Color Transparent display. (Color figure online)

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The FDTD solution is one of the powerful and flexible numerical methods for computing the scattering and absorption cross-section of nanoparticles with different shapes. Therefore, we use the FDTD solution for numerical investigation of different structures of nanoparticles, and also we use Lumerical and MATLAB software for simulation. According to studies at the Lumerical site, this method has a perfect approximation to Mie theory.

2.1 Information about arrangement of nanoparticles

As it is pointed out, we use different arrays of nanoparticles, such as the periodic, the Fibonacci, the Thue-Morse, and the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) to investigate the role of deterministic aperiodic structures on core–shell nanoparticles. Unlike random structures, the position of nanoparticles in the deterministic aperiodic array follows unique rules. We use two-dimensional Fibonacci and Thue-Morse arrays in this work. We have also used a type of quasicrystal structure called the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) in this article. Although the Pseudo-Brillouin zone by 12-fold quasicrystal structure is not periodic, it has a repetitive, orderly, and symmetrical arrangement. These arrays are illustrated in Fig. 2, and their relationships have been examined in various articles (Gopinah et al. 2009; Negro and Boriskina , 2012; Vardeny et al. 2012; Raghuwanshi and Kumar 2013; Dallapiccola et al. 2008; Rostami and Matloub 2010; Rostami et al. 2010; Negro et al. 2008; Macia 2006).

Fig. 2
figure 2

Arrangement of nanoparticles in a The Fibonacci, b The Thue-Morse, c The Pseudo-Brillouin zone by 12-fold quasicrystal, and d The periodic structures

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2.2 Information about parameters and morphologies of nanoparticles

The shape of the nanoparticles due to their optical properties is one of the important factors to obtain sufficient contrast and transparency for the transparent display. Since the position of the plasmon resonance peaks and the spectral distribution of the intensities are different for the nanoparticles in different forms, even if the particle size characteristics are equal, therefore, we examine the different configurations of nanoparticles. The following shapes have been used to investigate the effect of nanoparticle shape on the scattering and absorption cross-sections.

  • Nanosphere The simplest form often used in nanoparticle analysis is Nanospheres. Two concentric spheres with different radius are used to create a core–shell spherical nanoparticle. By changing the radius of the core and shell of the nanosphere, we can adjust the scattering and absorption cross-section at RGB wavelengths. Table 1 shows the desired nanosphere and selected radius for blue, green and red wavelengths.

    Table 1 Obtained parameters for the blue, green, and red wavelengths for different shapes
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  • Nano oblate ellipse The oblate ellipse is a sphere whose equatorial radius is greater than its polar radius, a > c. Table 1 shows the coated nano oblate ellipse and its required radius to obtain the scattering and absorption cross-sections. The distance between core and clad represents h coating.

  • Nano prolate ellipse In the prolate ellipse, the equatorial radius is less than its polar radius, a < c. Coated nano prolate ellipse and its required radius to obtain the scattering and absorption cross-sections are shown in Table 1.

  • Nanocube A cube is a three-dimensional object with six squares on each side and it has six faces, twelve edges, and eight vertices. All six faces have equal lengths. Its shape and parameters are shown in Table 1.

  • Nano pyramid The pyramid is a structure whose bottom surface is a square and the outer surface of triangles that converge to a point. The length of the square and the height of the pyramid in Table 1 are specified by b and h, respectively.

  • Nanodisk The disk is a cylinder with a low height. It consists of two concentric circles that are slightly apart. The height of the inner and outer disk is denoted by h1 and h2. Information about the disk and its shape is shown in Table 1.

According to Table 1 and the sizes mentioned in it, it is important to consider some tips to get the right size of particles. The thickness between the inner and outer shapes is important because it is the principal parameter in shifting the scattering and absorption cross-section to higher or lower wavelengths. Therefore, we select the appropriate thickness between the inner and outer particles. Besides, the size of the outer shape is a significant factor because the peak value of scattering cross-section decreases if we assume the smaller size for outer shape. On the other hand, increasing the outer shape size due to the choice of the appropriate thickness between inner and outer shapes causes an increase in the size of the inner shape and the particle shifts toward the red wavelength. Consequently, there must be a compromise between the thickness of inner and outer shapes and the size of the outer shape. This compromise is very important because if we do not choose the right sizes, the scattering cross-section will be broadened in each area, while one of the goals is to find sharper peaks for scattering cross-section. Therefore, we obtain suitable sizes for each shape with the mentioned tips.

The peaks of each shape will be at different wavelengths due to the different structures of each shape, but we try to keep the peak of the scattering cross-section in the desired wavelength range. For this purpose, we repeat the simulations several times to find the size of each nanoparticle to obtain the appropriate peak of scattering cross-section in the mentioned wavelength range. We named the wavelength ranges blue, green, and red for ease. As shown in Table 1, for both oblate and prolate ellipse, when the size of nanoparticle changes 10 nm, the peak value of the scattering cross-section appears at 453, 513, and 587 nm. In both cases, we obtain the distance between the inner and outer ellipse of 10 or 20 nm. In disk, the internal disk change 10 or 20 nm, while the external disk change 5 or 10 nm, therefore, it is clear, to find the right scattering cross-section peak, the distance between the inner and outer disk in each range is different. By simulating a coated disk with the specified sizes, the peak values obtained at the wavelengths of 460, 510, and 629 nm. Getting the right size for the pyramid differs from other shapes and requires more precision. We found the appropriate distance between inner and outer pyramids to be 30 nm, and by placing the mentioned sizes, the scattering cross-section peaks obtain at 416, 501, and 649 wavelengths. In the cube, by varying the length of cube, scattering cross-section peaks appear at the wavelengths 460, 524, and 599 nm. In spherical nanoparticles, due to its highly symmetrical structure, it is possible to find different sizes with appropriate scattering cross-sections in each wavelength range, but it is important to choose the size of the outer sphere more carefully because we need to have a larger and sharper scattering cross-section. Thus, although the outer spheres in blue and green spheres are 90 nm, we find the scattering peak with more amount in the desired range by varying the distance between inner and outer spheres. We found suitable peaks at wavelengths of 467, 522, and 613 nm.

All information is about a single nanoparticle. In the simulation of more than one nanoparticle, another factor must be considered, the distance between nanoparticles. In this paper, we simulate for 360 particles with a distance of 40 nm from each other. The reason why we use 40 nm for the distance between nanoparticles is discussed in another article in the submit stage.

3 Simulation results

As mentioned in the introduction section, Hsu et al., for the first time, used nanoparticles in transparent displays for high transparency, better image quality, and better contrast. They used Ag/SiO2 coated nanoparticles and they were able to make a transparent display that scattered blue wavelength and absorbed the rest of the wavelengths (Hsu et al. 2014). Then, by examining Ag/SiO2 and Si-SiO2 coated nanoparticles, Jafari et al. found that the scattering cross-section of Si-SiO2 nanoparticles is higher than Ag/SiO2 nanoparticles. They also were able to create a transparent display by Si-SiO2 nanoparticles that scattered blue wavelengths and absorbed the rest of the wavelengths. In this method, they used 400 coated nanoparticles in the periodic array (Dolatyari et al. 2019).

The calculations carried out thus demonstrate that, along with the size and material of the nanoparticle, its shape and structure are other factors that essentially determine their optical features. By controlling the size and shape of nanoparticles, scattering and absorption cross-section can be controlled and managed at specific wavelengths. Accordingly, we investigate different morphologies of Si-SiO2 coated nanoparticles and different arrays to obtain higher scattering cross-section and lower absorption cross-section. In this regard, at the first part, as shown in Fig. 3, we obtain the scattering and absorption cross-section for a single nanoparticle for different shapes including sphere, cube, disk, pyramid, oblate and prolate ellipse at the three-wavelength region, which we call them, blue, green and red wavelengths. At the second part, in Figs. 4, 5, 6, 7, 8, 9, we simulate the scattering and absorption cross-section for a different arrangement of nanoparticles such as the periodic, the Fibonacci, the Thue-Morse and the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) separately for different nanoparticles morphologies to find out which array of nanoparticles shows the highest scattering cross-section and less absorption cross-section per different shapes of nanoparticles in the specific wavelength range. At the third part, in Figs. 10, 11, 12, 13, we repeat the simulations for each of the four configurations for different morphologies to know which morphology of nanoparticles has a better answer in specified wavelength regions. We can achieve higher scattering cross-section and less absorption cross-section than previous works by selecting the appropriate morphology and array for nanoparticles in all three wavelength regions, and as a result, we will have a transparent display with high transparency, image quality, and contrast.

Fig. 3
figure 3

Scattering (left side figures) and absorption (right side figures) cross-section for a single nanoparticle with different shapes of nanoparticles such as the cube, sphere, oblate ellipse, prolate ellipse, pyramid and disk at a blue, b green and c red wavelengths. (Color figure online)

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Fig. 4
figure 4

The effect of different arrays such as Fibonacci, Thue-Morse, periodic and hex on coated nanosphere at a blue, b green and c red wavelengths to find Scattering (left side figures) and absorption (right side figures) cross-section. (Color figure online)

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Fig. 5
figure 5

The effect of different arrays such as Fibonacci, Thue-Morse, periodic and hex on coated Si-SiO2 nano oblate ellipse at a blue, b green and c red wavelengths to find Scattering (left side figures) and absorption (right side figures) cross-section. (Color figure online)

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Fig. 6
figure 6

The effect of different arrays such as Fibonacci, Thue-Morse, periodic and hex on coated Si-SiO2 nano prolate ellipse at a blue, b green and c red wavelengths to find Scattering (left side figures) and absorption (right side figures) cross-section. (Color figure online)

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Fig. 7
figure 7

The effect of different arrays such as Fibonacci, Thue-Morse, periodic and hex on coated Si-SiO2 nanocube at a blue, b green and c red wavelengths to find Scattering (left side figures) and absorption (right side figures) cross-section. (Color figure online)

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Fig. 8
figure 8

The effect of different arrays such as Fibonacci, Thue-Morse, periodic and hex on coated Si-SiO2 nano pyramid at a blue, b green and c red wavelengths to find Scattering (left side figures) and absorption (right side figures) cross-section. (Color figure online)

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Fig. 9
figure 9

The effect of different arrays such as Fibonacci, Thue-Morse, periodic and hex on coated Si-SiO2 nanodisk at a blue, b green and c red wavelengths to find Scattering (left side figures) and absorption (right side figures) cross-section. (Color figure online)

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Fig. 10
figure 10

Scattering (left side figures) and absorption (right side figures) cross-section for different shapes of nanoparticles for the Fibonacci array at a blue, b green and c red wavelengths. (Color figure online)

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Fig. 11
figure 11

Scattering (left side figures) and absorption (right side figures) cross-section for different shapes of nanoparticles for the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) array at a blue, b green and c red wavelengths. (Color figure online)

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Fig. 12
figure 12

Scattering (left side figures) and absorption (right side figures) cross-section for different shapes of nanoparticles for a periodic array for RGB wavelength at a blue, b green and c red wavelengths. (Color figure online)

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Fig. 13
figure 13

Scattering (left side figures) and absorption (right side figures) cross-section for different shapes of nanoparticles for Thue-Morse array at a blue, b green and c red wavelengths. (Color figure online)

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As mentioned, we use the FDTD solution and we use MATLAB and Lumerial software for simulation. For this purpose, a computer with 64 GB of RAM is required. In this paper, we use CPU E5-2650 with a 128 GB RAM + 24 C computer. The average simulation time for each shape lasted between 4 and 9 hours, and the required amount of RAM for 360 nanoparticles is between 38 and 60 GB.

3.1 A single nanoparticle with different morphology

In this section, we investigate the effect of different shapes of nanoparticle on a single nanoparticle. Totally, in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, a, b and c represents the blue, green and red wavelengths, respectively. All figures on the left side drawn by solid lines are for scattering cross-section, and all figures on the right side drown by dot-dashed lines are for absorption cross-section. The X-axis represents the wavelength in nanometers, and Y-axis represents the cross-section in µm2. In Fig. 3, we simulate a single nanoparticle with different forms for the blue, green and red wavelengths to obtain the appropriate parameters for scattering and absorption cross-sections. We show the oblate ellipse in pink, the prolate ellipse in steel blue, the cube in sky blue, the disk in light green, the sphere in goldenrod yellow and the pyramid in red. We consider three important factors to determine optimal scattering cross-section: sharper and narrower scattering cross-section, lack of unwanted peaks, and higher scattering to absorption ratio (SAR). At first glance, it looks like, in the single nanoparticle, the nano pyramid has a maximum scattering cross-section for the green and red wavelengths and nano oblate ellipse has the highest scattering cross-sections for blue wavelength. Besides, the unwanted peak is not seen next to the principal peak in Fig. 3. By looking more closely at Fig. 3, we obtain different results by examining the scattering to absorption ratio (SAR) and the amount of scattering cross-section for all shapes of a single nanoparticle shown in Table 2.

Table 2 Scattering (SCATT), unwanted or broad peaks (UBP) and the scattering to absorption ratio (SAR) for all shapes of a single nanoparticle in RGB wavelengths
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According to comparisons, we observed that the nano oblate ellipse for blue and green wavelengths and the nano pyramid for the red wavelength provide the highest scattering cross-section. The scattering cross-section of Si-SiO2 nano oblate ellipse for blue and green wavelengths and coated Si–SiO2 nano pyramid for red wavelength are 1.25, 1.14 and 1.44 orders higher than ordinarily coated nanosphere, while the magnitude of absorption cross-section for blue, green and red are 1.03, 1.04 and 1.19 order less than usual core–shell nanosphere, respectively.

3.2 Different types of morphologies of nanoparticles

In this section, for a closer investigation, we simulated the periodic, the Fibonacci, the Thue-Morse and the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) arrays of nanoparticles for six different classes of nanoparticles viz. cube, sphere, oblate ellipse, prolate ellipse, pyramid and disk at RGB wavelengths. In Figs. 4, 5, 6, 7, 8, 9, the aim is to find the proper arrays of nanoparticles for different morphologies. We demonstrate the Fibonacci in light green, the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) in red, the Thue-Morse in steel blue, and the periodic in goldenrod yellow. The results are as follows:

  1. 1

    Nanosphere As shown in Fig. 4, we have an unwanted peak in all structures near the main peak, but the amount of unwanted peak in the Fibonacci and the Thue-Morse arrays are less than others. The Thue-Morse is narrower than the Fibonacci, and the Fibonacci is more than the Thue-Morse. Also, the amount of SAR for Fibonacci for blue, green, and red wavelengths is 2.31, 3.43, and 6.93, respectively. This value for the Thue-Morse array is 2.09, 3.68, and 7.17, respectively. Therefore, the Thue-Morse for red and green wavelengths and the Fibonacci array for blue one have the highest scattering cross-section for 360 nanospheres.

  2. 2

    Nano oblate ellipse As shown in Fig. 5, the Fibonacci array has the highest scattering cross-section at blue and green wavelengths, while periodic array shows the highest cross-section at a red wavelength for the core–shell nano oblate ellipse. On the other hand, periodic and the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) arrays show an unwanted peak near the principal peak. Moreover, the amount of SAR for Fibonacci at blue, green and red wavelengths is 3.05, 4.37 and 7.55, respectively. This value for the Thue-Morse array is 2.55, 3.94 and 8.29, respectively. Therefore, the Fibonacci for blue and green wavelengths and the Thue-Morse array for red one have the highest scattering cross-section for 360 nano oblate ellipse.

  3. 3

    Nano prolate ellipse For core–shell nano prolate ellipse, the Fibonacci and Thue-Morse show the highest and narrowest scattering cross-section for RGB wavelength, especially green and red wavelengths. As shown in Fig. 6, Thue-morse array is better than the Fibonacci array due to the amount of absorption cross-section, and also there is an unwanted peak in the Fibonacci array at Nano prolate ellipse structure. The amount of SAR for the Thue-Morse array is 2.05, 3.9 and 7.53 for blue, green and red wavelengths, respectively. This parameter for the Fibonacci array at blue, green and red is 1.74, 3.2 and 6.1, respectively. The Thue-Morse structure has a suitable scattering cross-section in this case.

  4. 4

    Nanocube It seems that, as shown in Fig. 7, the Fibonacci array has a maximum scattering cross-section for 360 nanocubes. The Fibonacci structure has a sharper and narrower scattering cross-section, while periodic and the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) show unwanted peaks near the main peak. The SAR calculation for the Fibonacci structure at blue, green and red is 2.16, 3.87 and 6.89, and also there is 2.17, 3.99 and 7.1 for Thue-Morse array, respectively. Therefore, the scattering cross-section for Thue-Morse array is a little better than the Fibonacci structure.

  5. 5

    Nano pyramid As shown in Fig. 8a, the Fibonacci, the Thue-Morse, the periodic, and the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) show the highest scattering cross-section for 360 nanoparticles, with SAR value of 1.57, 1.54, 1.52 and 1.33, respectively. It is also crystal clear; the results of the scattering cross-section at blue wavelength have not narrow bandwidth. For the green and red wavelengths mentioned in Fig. 8b and c, the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) and periodic arrays have an unwanted peak near the main peak. The Fibonacci and the Thue-Morse have equal SAR about 4.15 at the green wavelength. This value for Fibonacci and Thue-Morse arrays is 12.41 and 13.41 at red wavelength, respectively.

  6. 6

    Nano disk For nanodisks, the Thue-Morse array for blue wavelength and periodic array for green and red wavelength has maximum scattering cross-section, and also, as shown in Fig. 9 the absorption cross-section for these arrays have the highest amount, therefore, the scattering to absorption rate (SAR) for the Fibonacci array is more than periodic and Thue-Morse array. Furthermore, all of the structures have an unwanted peak near their main peak. In general, it seems that nanodisk does not show the right scattering cross-section.

3.3 Different arrays of nanoprticles

One of the cases discussed in this paper is the different shapes of nanoparticles. Therefore, the comparison of scattering and absorption cross-section of nanoparticles based on their shapes is necessary to find the most suitable morphology with maximum scattering cross-section and minimum absorption cross-section. For a better conclusion, we investigate the scattering and absorption cross-section for different shapes of 360 core–shell Si-SiO2 nanoparticles to find the best morphology of nanoparticles in different arrays such as the Fibonacci, the Thue-Morse, the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) and periodic separately at visible wavelengths.

In Figs. 10, 11, 12, 13, we illustrated the cube in sky blue, the disk in light green, the oblate ellipse in pink, the prolate ellipse in steel blue, the pyramid in red and the sphere in goldenrod yellow. As shown in Fig. 10a, scattering cross-section for nano oblate ellipse, nanocube, and nano prolate ellipse are narrower than nanosphere, nano pyramid, and a nanodisk. It seems the oblate ellipse has the best SAR and highest scattering cross-section at blue and green wavelengths and the pyramid has maximum SAR and scattering cross-section at red wavelength.

Figure 11 shows the result of the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) array for different shapes of nanoparticles for RGB wavelengths. As we expected, almost all shapes at RGB wavelengths show an unwanted peak along their main peak, indicating that the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) array is not suitable for simulating transparent displays. According to Fig. 12, there are similar results as the Pseudo-Brillouin zone by 12-fold quasicrystal (hex) array for the periodic array. The periodic array shows an unwanted peak for all shapes at RGB wavelength and scattering cross-section for a periodic array is not suitable. While previous work was based on periodic arrays, it seems that we can improve the scattering cross-section by using a different kinds of arrays.

As shown in Fig. 13, for the Thue-Morse array, coated nano oblate ellipse with 2.55 and 3.94 SAR has the steepest scattering cross-section in blue and green wavelengths. Moreover, the coated nano pyramid has the highest SAR and the highest scattering cross-section for red wavelength.

3.4 The results and the most optimal morphology and array of nanoparticles

In general, as shown in the italic and bold section of the Table 3, the Fibonacci arrays for nano oblate ellipse at blue and green wavelengths provide the highest scattering cross-section and the scattering to absorption ratio (SAR), and also without any unwanted or broadening peaks near the main peak. Furthermore, the Fibonacci array for nano pyramid has the highest scattering cross-section and the Thue-Morse array for nano pyramid has the Highest SAR at red wavelength. On the other hand, another critical point is that the input plane wave sources in all shapes except the nano pyramid are located in the center of them. In the coated nano pyramid, the source location is very sensitive. We must place the source in such a way that includes the core and shell of the nano pyramid. In addition, when the source approaches the bottom or tops of the pyramid, the scattering cross-section is oscillating, and it does not provide an appropriate answer. Therefore, coated nano oblate ellipse of the Fibonacci array provides the most optimal and appropriate scattering cross-section.

Table 3 Comparison of scattering and absorption cross-section for different morphologies and different arrays of nanoparticles
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4 Conclusion

In this paper, we performed a theoretical analysis of the effects of six different classes of Si–SiO2 nanoparticles viz. nanocube, nanosphere, nano oblate ellipse, nano prolate ellipse, nanodisk and nano pyramid for different arrays such as the Fibonacci, the Thue-Morse, periodic and the P-Bz by 12-fold QC (hex) arrays on the scattering and absorption cross-section. We carried out calculations based on FDTD solutions for coated nanoparticles in various shapes. The size of the structure primarily influences the specific amounts of the wavelength corresponding to the positions of the spectral feature. We obtained the appropriate size of nanoparticles in different shapes for maximum scattering cross-section and minimum absorption cross-section, by repeating the simulations. We then examined the different arrays and shapes of the nanoparticles and found the appropriate array and morphology to have the maximum scattering cross-section, highest the scattering to absorption ratio (SAR) and the least amount of unwanted or broadened peaks around the main peak. We show that the coated nano oblate ellipse with the Fibonacci structure has sharper and narrower scattering cross-section. For high resolution and brightness, the amount of scattering cross-section should be sharper and narrower. By changing the size of the core and the shell of nanoparticles in each shape, the scattering and absorption cross-section can be adjusted to the desired wavelengths.