1 Introduction

Solar absorbers are absorbing solar energy and convert it to thermal energy which can be used for different applications of heating, etc. (Wan et al. 2015; Muhammad et al. 2019). The efficiency of these solar absorbers is very important as the rise in efficiency will make them more affordable and efficient in renewable and sustainable energy generation (Patel et al. 2022a). The material of the solar absorber plays a vital role in its absorption. Many researchers all over the world are working on different materials to improve the absorption of the solar absorber (Sekhi et al. 2020; Baqir et al. 2021; Te et al. 2020). The solar absorber efficiency can be improved by incorporating graphene and loading metasurface in them (Cen et al. 2019; Mahmud et al. 2021; Surve et al. 2021). In the following, we are reviewing the literature based on graphene solar absorbers and metasurface solar absorbers.

Metamaterials is introduced by vaselago theoretically in 1968 (Veselago 1968) and practically proved by smith and his team later (Smith et al. 2000). Metamaterials are artificial materials that can be introduced into natural materials to improve their capacity and properties (Patel et al. 2020a). The split-ring resonator and thin wires are the two components that introduced the negative behavior in metamaterials (Patel and Parmar 2021). Metamaterials are used in several applications like sensing, absorbing, antennas, etc. (Shen and Xiong 2022; Vafapour et al. 2021; Nguyen et al. 2022; Ahmadivand et al. 2021). One of the effective applications is in designing solar absorbers. These metamaterial components are introduced in solar absorbers to improve their efficiency and reduce their cost. Broadband behavior can be achieved using a metamaterial absorber made with germanium-based resonating structures (Guo et al. 2022). Solar thermal absorbers are created using metamaterial loading which is applicable in heating applications (Li et al. 2021). Broadband metamaterial-based solar absorber design with absorptivity of about 80% is achieved in triple-layer thin-film design (Hoque and Islam 2020). A metamaterial absorber with swastika-shaped is achieved. The absorption level is increased by the swastika-shaped design and broadband results are achieved. The polarization independent ness is also achieved (Patel et al. 2022b). The metamaterial-based solar absorber is designed with different material disks. The broadband and polarization-independent behavior are achieved using these structures (Liang et al. 2019) Gold material-based metamaterial resonators are used to achieve broadband infrared absorbers. The array of gold disks is used to obtain this response (Patel et al. 2020b).

Graphene is a single-atom-thick graphite material that can be used in solar absorbers because of its extensive electrical and optical properties. Its usage can increase solar absorber’s absorption. The graphene layer can be used as a spacer for this improvement. Graphene-based solar absorbers can be used to achieve broadband, polarization-independent, and wide-angle solar absorbers. Graphene oxide is a material that can be incorporated with metamaterials which improve broadband absorption (Saurabh et al. 2022). Plus-shaped metasurface based on graphene material creates a resonating structure that absorbs broadband absorption which results in a broadband absorber (Patel et al. 2019). The design is giving 70% absorption in the solar region (Rufangura and Sabah 2017). The pyramid-shaped metamaterial absorber based on graphene can get an absorption of around 89% in the visible region (Charola et al. 2020). About 90% efficient solar to thermal absorber is designed with the use of graphene metamaterial array structure placed over the substrate (Patel et al. 2021a; Jadeja et al. 2020; Lin et al. 2019). The solar absorber can be used for many applications and some of the applications are solar thermal heating, solar cells, heat transfer systems, and optoelectronic devices (Klug et al. 2020; Bilal et al. 2020; Ogawa and Kimata 2018; Zhang et al. 2020).

The solar absorber design needs to be efficient to be applicable in solar thermal energy applications. The solar absorber design needs to be absorbing visible, infrared, and ultraviolet energies to be efficiently applicable in major solar applications. We proposed a graphene-based metamaterial design that gives a broadband absorption region which covers the visible and infrared regions with nearly more than 90% average absorption. The design results are also investigated for electric field and variation in different parameters. The following sections present the solar absorber design and its results.

2 Metasurface designs

The metasurface design is created by placing a line between three parallel lines as presented in Fig. 1. The metasurface design is created with the gap between the lines producing capacitance and the lines producing inductance and thus the resonance is affected by the metasurface design. Two different metasurface designs are created by reducing the size of resonator as shown in Fig. 1. The metasurface design is made up of Titanium material and it is placed over SiO2 material. The titanium material is used as resonator as it is a low-cost material compared to gold and silver metals. The graphene layer is sandwiched between the titanium and SiO2 layers. A single layer of graphene material is used having a thickness 0.34 nm. The two metasurface design and their shapes are presented in Fig. 1. The Metasurface 1 design is having a length (L2) of 3000 nm and width (W1) of 600 nm. The Metasurface 2 design is having a length (L3) of 2000 nm and width (W2) of 400 nm. The results of metasurface designs discussed later.

Fig. 1
figure 1

Metasurface design with two different sizes for solar absorber a metasurface Resonator 1 design with W1 and L2, b 3D view of Resonator 1 design (2) Front view of Resonator 1 design b Metasurface Resonator two design with W2 and L3. The different parameters are L1 = 6000 nm, L2 = 3000 nm, W1 = 600 nm, L3 = 2000 nm, W2 = 400 nm, S = 2500 nm. The graphene layer is 0.34 nm thick. The figure is not upto the scale

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Absorption analysis is presented in Patel et al. 2021b and equations are presented below.

$$r\left( {\omega ,\theta_{i} } \right) = \frac{{\omega \cos \theta_{i} \mathop \prod \nolimits_{00} \left( {\omega ,\theta_{i} } \right)}}{{2i\hbar ck^{2} + \omega \cos \theta_{i} \mathop \prod \nolimits_{00} \left( {\omega ,\theta_{i} } \right)}}$$
(1)
$$\sigma_{||} \left( {\omega ,k} \right) = - i \frac{\omega }{{4\pi \hbar k^{2} }}\mathop \prod \limits_{00} \left( {\omega ,k} \right)$$
(2)
$$r\left( {\omega ,\theta_{i} } \right) = \frac{{2\pi \cos \theta_{i} \sigma_{||} \left( {\omega ,k} \right)}}{{c + 2\pi \cos \theta_{i} \sigma_{||} \left( {\omega ,k} \right)}}$$
(3)
$${\mathcal{R}}\left( {\omega ,\theta_{i} } \right) = \left| {r\left( {\omega ,\theta_{i} } \right)} \right|^{2}$$
(4)
$${\mathcal{R}}\left( {\omega ,\theta_{i} } \right) = \frac{{4\pi^{2} \cos^{2} \theta_{i} \left[ {{\text{Re}}^{2} \sigma_{||} \left( {\omega ,k} \right) + {\text{Im}}^{2} \sigma_{||} \left( {\omega ,k} \right)} \right]}}{{\left[ {c + 2\pi \cos \theta_{i} {\text{Re}}\sigma_{||} \left( {\omega ,k} \right)} \right]^{2} + 4\pi^{2} \cos^{2} \theta_{i} {\text{Im}}^{2} \sigma_{||} \left( {\omega ,k} \right)}}$$
(5)
$${\mathcal{R}}\left( \omega \right) = {\mathcal{R}}\left( {\omega ,0} \right) = \frac{{4\pi^{2} \left[ {{\text{Re}}^{2} \sigma \left( \omega \right) + {\text{Im}}^{2} \sigma \left( \omega \right)} \right]}}{{\left[ {c + 2\pi {\text{Re}}\sigma \left( \omega \right)} \right]^{2} + 4\pi^{2} {\text{Im}}^{2} \sigma \left( \omega \right)}}$$
(6)
$$A\left( \omega \right) = 1 - {\mathcal{R}}\left( {\omega ) - T(\omega } \right)$$
(7)

where k refers to the wave vector.

The graphene chemical potential and its effect are presented in Patel et al. 2022c and equations are presented below.

$$\varepsilon \left( \omega \right) = 1 + \frac{{\sigma_{s} }}{{\varepsilon_{0} \omega \Delta }}$$
(8)
$$\sigma_{intra} = \frac{{ - je^{2} k_{B} T}}{{\pi \hbar^{2} \left( {\omega - j2\Gamma } \right)}}\left( {\frac{{\mu_{c} }}{{k_{B} T}} + 2\ln \left( {e^{{ - \frac{{\mu_{c} }}{{k_{B} T}}}} + 1} \right)} \right)$$
(9)
$$\sigma_{inter} = \frac{{ - je^{2} }}{4\pi \hbar }\ln \left( {\frac{{2\left| {\mu_{c} } \right| - \left( {\omega - j2\Gamma } \right)\hbar }}{{2\left| {\mu_{c} } \right| + \left( {\omega - j2\Gamma } \right)\hbar }}} \right)$$
(10)
$$\sigma_{s} = \sigma_{inter} + \sigma_{intra}$$
(11)

3 Results and discussions

The two metasurface designs made up of changing their length and width are simulated using COMSOL Multiphysics. The simulator is used effectively to achieve the broadband absorption and results are presented. The results for the two designs are compared in Fig. 2. The metasurface 1 design (with higher length and width)’s result is presented in Fig. 2a which shows that the absorption is high for the whole wavelength range while the metasurface 2 design (with lower length and width)’s absorption is reducing as it moves ahead from 0.67 µm and reduces to almost zero. The average absorption for both designs is compared in Table 1 for visible and infrared regions. The average absorption of design 1 is 90% in the visible region compared to 63% of design 2. Design 1 has an 88% average absorption in the infrared range of 0.7–1.5 µm. The highest absorption peaks of 97% and 97.6% are achieved for design-1 and design-2 respectively.

Fig. 2
figure 2

Absorption result for visible and infrared regions a metasurface design 1 with high length and width b metasurface design 2 with low length and width. The metasurface 1 design having higher length and width is giving better performance. The 90% for metasurface design 1 with the highest peak absorption of 97% for 0.75 µm. The metasurface design 2 with 63% with a peak absorption of 97.6% at 0.44 µm

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Table 1 Two metasurface design comparison
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3.1 Optimization

Parametric optimization is very important in designing any solar absorber so that the best and most efficient solar absorber can be designed (Patel et al. 2020c). The parametric optimization can be performed for different substrate and metasurface parameters. The optimization can also be performed for an angle of incidence and graphene chemical potential. We have used all these optimizations to get the optimized design. In the series of these optimizations, the first is metasurface thickness optimization. Optimized values are obtained by varying the metasurface thickness from 1000 to 2000 nm. The wavelength is varied between 0.4 and 1.5 µm. The initial value of metasurface thickness gives the absorption of the solar absorber having good values throughout the spectrum observed. As soon as the metasurface thickness is increased the inductance of the overall structure is increasing which is affecting the absorption of the design. The increase in metasurface thickness gives good absorption in the initially visible region but in an infrared region, the absorption is decreasing. The decrease in absorption for the infrared region is visible for the highest metasurface thickness of 2000 nm. The green color dot marker curve clearly shows this observation. The response of 1750 nm thickness is also very similar to 2000 nm and for the infrared region, the absorption is very less. So from the observation, it is clear that the overall absorption is good for 1000 nm metasurface thickness. For all the other variations the absorption is changing in different regions, but the overall absorption is similar to or less than the 1000 nm metasurface thickness.

The second optimization observed is for Substrate thickness. Optimized values are obtained by varying the substrate thickness from 2500 to 3500 nm. The wavelength is varied between 0.4 and 1.5 µm. The initial value of substrate thickness gives the absorption of the solar absorber having good values throughout the spectrum observed. As soon as the substrate thickness is increased, the absorption of the visible region is not affected much but for the infrared region absorption drastically decreases. The decrease in absorption for the infrared region is visible for the highest substrate thickness 3500 nm. The middle thickness of 3000 nm has good absorption in visible and good absorption in half infrared range till 1 µm but above this wavelength, the absorption is decreasing drastically. So from the observation, it is clear that the overall absorption is good for 2500 nm substrate thickness. For all the other variations the absorption is changing in different regions, but the overall absorption is similar to or less than the 2500 nm substrate thickness (Figs. 3, 4, 5, and 6).

Fig. 3
figure 3

Optimization of resonator thickness. a Line plot, b color plot. The thickness varied from 1000 to 2000 nm. The absorption for the initial thickness of 1000 nm has good absorption for the whole range and or higher thickness of 2000 nm has good absorption in visible while less absorption in the infrared range. The optimized value of resonator thickness is 1000 nm

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Fig. 4
figure 4

Optimization of substrate thickness. a Line plot, b color plot. The thickness varied from 1000 to 2000 nm. The absorption for the initial thickness of 2500 nm has good absorption for the whole range and or higher thickness of 3500 nm has average absorption in visible while less absorption in the infrared range. The optimized value of resonator thickness is 1000 nm

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Fig. 5
figure 5

Optimization of substrate length. a Line plot, b color plot. The length varied from 6000 to 9000 nm. The absorption for the initial length of 6000 nm and 7000 nm has good absorption for the whole visible range but it is decreasing in the infrared region for 7000 nm. For the 8000 nm and 9000 nm, the tuning in wavelength is achieved as absorption is shifted in the visible region. The optimized value of substrate length is 6000 nm

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Fig. 6
figure 6

Optimization of substrate width. a Line plot, b color plot. The width varied from 6000 to 9000 nm. The absorption for the initial width of 6000 nm is less initially in the visible region compared to the 7000 nm width. Overall, the absorption is good for 6000 nm width. The wavelength tuning is visible for 8000 nm and 9000 nm widths

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The third optimization observed is for Substrate length. Optimized values are obtained by varying the substrate length from 6000 to 9000 nm. The wavelength is varied between 0.4 and 1.5 µm. The initial value of substrate length of 6000 nm and 7000 nm gives the absorption for the visible region but it is decreasing in the infrared region for 7000 nm length. The length is further increased to 8000 nm which shifts the absorption spectrum to 0.1 µm and the absorption is decreasing in the infrared region. The 9000 nm length further shifts the spectrum to 0.1 µm. The tuning in the absorption spectrum is visible for the higher length of the substrate. So from the observation, it is clear that the overall absorption is good for 6000 nm substrate length. For all the other variations the absorption is changing in different regions, but the overall absorption is higher for 6000 nm substrate length.

The fourth optimization observed is for Substrate width. Optimized values are obtained by varying the substrate width from 6000 to 9000 nm. The wavelength is varied between 0.4 and 1.5 µm. The initial value of substrate width of 6000 nm is less initially in the visible region compared to 7000 nm width. For the rest of the wavelength range, the 6000 nm has higher absorption compared to the 7000 nm range. The length is further increased to 8000 nm which shifts the absorption spectrum to 0.1 µm and the absorption is decreasing in the infrared region. The 9000 nm length further shifts the spectrum to 0.1 µm. The tuning in the absorption spectrum is visible for the higher length of the substrate. So from the observation, it is clear that the overall absorption is good for 6000 nm substrate length. For all the other variations the absorption is changing in different regions, but the overall absorption is higher for 6000 nm substrate length.

3.2 Angle variation and graphene variation

Angle vairiation changes the light variation and the light intensity changes during the day so it is very importabnt. The variation in angle is observed in Fig. 7a. The angle variation clearly shows that for angles 30° and 80°. The absorption is minimum which is visible with blue color in the plot. All other angles on incidences show good absorption for the whole wavelength range. The initial wavelength of 0.4 µm is having minimum absorption but it increases immediately as it is increased. For the angle, 60° and 70° has good values. The variation of graphene chemical potential and its effect on is presented in Fig. 7b. The variation results clearly show that there is no variation in absorption results for different variations in graphene chemical potential. The graphene potential variation is not affecting the absorption spectrum because of its observed wavelength range (Fig. 8).

Fig. 7
figure 7

Absorption results of metasurface 1 design for different values of a angle of incidence b graphene chemical potential. There is no variation in absorption results for the variation of graphene chemical potential for 0.1–0.9 eV. The angle of incidence is showing good absorption for 0–20° and 40–70°

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Fig. 8
figure 8

Electric field intensity in V/m. a 0.4 µm, b 0.5 µm, c 0.6 µm, and d 0.7 µm. The minimum absorption is at 0.4

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3.3 Electric field Intensity (EFI)

The EFI of the metasurface absorber is investigated with four different wavelengths of the visible region. To observe the effect of the electric field in the major part of the visible region, four wavelengths are selected. The wavelengths 0.4 to 0.7 µm are investigated with a 0.1 µm step size. The absorption is minimum for 0.4 µm and maximum for 0.6 µm which is also similar to the absorption response achieved. The comparison of the proposed design is done with other designs in Table 2.

Table 2 Comparison of the proposed designs with similar published designs from reference
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4 Conclusion

We have designed two metasurface-based solar absorbers using graphene material. The two designs are having different sizes with a similar shapes. Both designs are investigated to observe the solar absorber’s absorption response in the wavelength range of 0.4–1.5 µm. The visible region and infrared regions are investigated in these proposed design results. The metasurface design 1 is giving 90% average absorption and 97% highest absorption in the visible region and 88% average absorption in the infrared range of 0.7–1.5 µm. The design results of metasurface design-1 are optimized for better absorption. The optimized parameter obtained for different parameters is substrate thickness of 2500 nm, substrate length and width of 6000 nm each, and resonator thickness of 1000 nm. The absorber is also showing the wide-angle of incidence covering the whole range except 30° and 80°. The graphene's potential effect is not visible in observed wavelength region. The proposed solar absorber has the potential to be used in solar thermal energy conversion devices.