Superimposed multilayer randomly distributed graphene quantum dots and broadband terahertz perfect absorber

For investigating broadband perfect absorption in the Terahertz region, a multilayer structure using superimposed Graphene quantum dots with Gaussian distribution radius is proposed. At first, a three-layer metamaterial structure, which is composed of graphene quantum dots at the top layer, dielectric material at the middle layer, and finally gold layer at the bottom is considered and investigated. The radius Gaussian distribution for three radii 400, 450, and 500 nm is considered. Results for the proposed structure including 3, 5, and 7 layers are extracted and compared. Simulation results for the 7-layer case show a broadband absorption range of (4.2 to 8.1 THz), and (4.3 to 8.71 THz) absorption value above 70%. The dielectric material is Si3N4 in the proposed structure.

In this study, we propose a broadband multi-layer terahertz absorber in the range of (4.2 to 8.1 THz) and (4.3 to 8.71 THz). In both conditions, the structures consist of 7 layers. The top, 3rd and 5th layers, consists of a single layer of graphene quantum dots with Gaussian distribution. The 2nd, 4th, and 6th layers are dielectric and finally, the bottom layer is a lossless metal.

Structure and design
Typically, our proposed structure is depicted in Fig. 1. We concluded in 2 optimized structures. The top layer consists of a single layer of graphene quantum dots with Gaussian distribution in which R = 400 nm and δ (standard deviation) = 200 nm. The 2nd layer is Si 3 N 4 . The 3rd layer consists of a single layer of graphene quantum dots with Gaussian distribution in which R = 450 nm and δ (standard deviation) = 160 nm. The 4th layer is Si 3 N 4 . The 5th layer consists of a single layer of graphene quantum dots with Gaussian distribution in which R = 400 nm and δ (standard deviation) = 200 nm. The 6th layer is Si 3 N 4 and finally, the bottom layer is gold. The thickness of the 2nd, 4 th, and 6th layers (dielectric layers) respectively are 1250 nm, 1250 nm, and 2500 nm in the optimized structures. While the thickness of the 2nd, 4th, and 6th layers in the other optimized structure are respectively 625 nm, 1250 nm, and 2500 nm. The length of these layers is 29.4 μm and the width of The top, 3rd, and 5th layers, consist of graphene quantum dots with Gaussian distribution. The 2nd, 4th, and 6th layers are Si 3 N 4 and finally, the bottom layer is gold. b Graphene quantum dots with Gaussian distribution. In top layer R = 400 nm and δ = 200 nm, in 3rd layer R = 450 nm, and δ = 160 nm and in 5th layer R = 400 nm, and δ = 200 nm these layers is 4.2 μm. We considered gold at the bottom layer which its electric conductivity σ gold = 4.56 × 10 7 s/m. The thickness of this layer is 0.24 μm. The length of the gold layer is equal to 29.4 μm and the width is 4.2 μm.
To approach the actual model for construction, we considered graphene quantum dots in 27 columns and 3 rows as shown in Fig. 1. We used the formula: for acquiring the radius of graphene quantum dots. Where R is the radius that we want to extract the radius of graphene quantum dots around that radius, R i is the radius of the i th quantum dot, δ is the standard deviation, and Rand is the command used to randomly extract a number between (0,1).
The graphene surface conductivity consists of interband and intraband contributions taking into account the Kubo formula [18,29], , T is the temperature, E F is electrochemical potential (Fermi energy), is the frequency of the electromagnetic wave, is relaxation time, and e is the charge of an electron.
Because the photon energy − h ≪ E F in the THz range, so we can neglect the interband part in comparison with the intraband part. As a result, the surface conductivity of graphene in the THz range can be described by the Drude model [19]: Graphene conductivity changes by changing frequency at a specified E f , whereas the imaginary part of the graphene conductivity, determines resonance spectral shift and the real part of graphene conductivity determines resonance amplitude modulation, we can tune the absorption by controlling the Fermi level by applying an external electric field or using an optical pump. TE wave, in which the propagation direction is along the z-direction is illuminated on the proposed structure. In this study, to carry out numerical simulations, CST Microwave Studio commercial software was used. Unit cell boundary conditions were applied in x and y directions. The open space boundary conditions were used in the z-direction. Absorption can be calculated using where S 11 and S 21 are reflection and transmission coefficients respectively. These parameters can be acquired from CST simulation results. Since the thickness of the bottom metallic layer is thicker than the skin depth at frequencies that we carry out our study, approximately no electromagnetic wave penetrates the structure. So we can suppose S 21

Results and discussion
We have used the TE wave to study the proposed absorber. We have supposed the Fermi energy of graphene as E_f = 0.6 eV and relaxation time as τ = 0.5ps. The main goal of this research is to design a broadband THz detector using graphene quantum dots. To this end, using formula (1) we considered R = 400, 450, and 500 nm. We considered different δs for each R. The simulation is done on the structure as shown in Fig. 2. For R = 500 nm and δ = 20, 40, and 60 nm. The sweep results are demonstrated for SiO 2 and Si 3 N 4 in Fig. 3. According to [11], we have considered a structure with three layers. The bottom layer consists of gold with a height of 240 nm, the middle layer is dielectric with a height of 3900 nm, and the top layer is composed of a single layer of graphene quantum dots. The length of the structure is equal to 29.4 μm, and the width is 4.2 μm, as shown in Fig  and broadening happen when R = 400 nm, and = 200nm , while the dielectric material can be SiO 2 or Si 3 N 4 . To broaden the absorption bandwidth, we decided to place the structure with maximum absorption and broadening over each other and survey the results. By placing two structures over each other, the top layer is graphene quantum dots with R = 450, and δ = 160 nm, 2nd layer is dielectric, 3rd layer is graphene quantum dots with R = 400, and δ = 200 nm, 4th layer is dielectric, and, finally the last layer is gold. Considering the thickness of the 4th layer as 3900 nm and sweeping the thickness of the 2nd layer the results are depicted in Figs. 6, 7, 8, 9 and 10. Considering the thickness of the 4th layer as 3900 nm and the thickness of the 2nd layer as 20 nm the results are shown in Fig. 6.
Considering the thickness of the 4th layer as 3900 nm and the thickness of the 2nd layer as 200 nm the results are shown in Fig. 7.
Considering the thickness of the 4th layer as 3900 nm and the thickness of the 2nd layer as 500 nm the results are shown in Fig. 8. Considering the thickness of the 4th layer as 3900 nm and the thickness of the 2nd layer as 1500 nm the results are shown in Fig. 9.
Considering the thickness of the 4th layer as 3900 nm and the thickness of the 2nd layer as 3900 nm the results are shown in Fig. 10.
Comparing Figs. 6, 7, 8, 9 and 10 it can be distinguished that Fig. 8 has broadband absorption. To broaden the absorption bandwidth, we survey the structure (Fig. 1) in which the top, 3rd, and 5th layers are composed of graphene quantum dots, 2nd, 4th, and 6th layers are composed of a dielectric material and the bottom layer is gold. We change the thickness and dielectric material to observe the effect of this action in value and bandwidth of absorption. Considering different arrangements by changing the thickness and dielectric material in Fig. 1, the figures related to absorption diagrams are depicted in Fig. 11 to27. Tables 1, 2, 3 , 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16, demonstrate the values of R, and δ of graphene quantum dots in 1st, 3rd, and 5th layers, and also the materials used as dielectric material in layers 2,4, and 6, also the thickness Fig. 6 The thickness of the 4th layer = 3900 nm and the thickness of the 2nd layer = 20 nm. The left figure is for the condition that both dielectric materials in the 2nd and 4th layers are Si 3 N 4 . The right figure is for the condition that both dielectric materials in the 2nd and 4th layers are SiO 2 .    Figs. 18 and 19 respectively have broadband absorption spectrum in the range of (4.3THz to 8.71THz) and (4.2 THz to 8.1THz) while the absorption value is above 70%. The comparison between broadening of "3 layers", "5 layers", and "7 layers" are represented in      Table 1 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 11 1st

Conclusion
In this work, a multilayer structure including superimposed Graphene quantum dots as a broadband Terahertz absorber was designed and investigated. The top layer of the structure consists of a single layer of graphene quantum dots with a Gaussian distribution radius with R = 400 nm and δ (standard deviation) = 200 nm. The 2nd layer was Si 3 N 4 . Table 2 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 12 1st  Table 3 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 13 1st   Table 4 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig  The 3rd layer consists of a single layer of graphene quantum dots with Gaussian distribution radius and R = 450 nm and δ (standard deviation) = 160 nm. The 4th layer was Si 3 N 4 . The 5th layer consists of a single layer of graphene quantum dots with a Gaussian distribution radius in which R = 400 nm and δ (standard deviation) = 200 nm. The 6th layer was Si 3 N 4 and finally, the bottom layer was gold. Broadband absorption occurs in Table 5 Data applied to the structure in Fig. 1 Related absorption diagram is depicted in Fig. 15 1st layer = GQDs R = 400 nm δ = 200 nm 2nd layer = Dielectric Material = SiO 2 Thickness = 1250 nm 3rd layer = GQDs R = 450 nm δ = 160 nm 4th layer = Dielectric Material = Si 3 N 4 Thickness = 625 nm 5th layer = GQDs R = 400 nm δ = 200 nm 6th layer = Dielectric Material = Si 3 N 4 Thickness = 2500 nm 7th layer = lossless material Material = Gold Thickness = 240 nm Fig. 15 The absorption diagram is related to the data in Table 5 Table 6 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 16 1st   Table 7 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 17 1st  Table 8 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 18 1st  Table 9 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 19 1st   Table 10 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 20 1st   Table 11 Data applied to the structure in Fig. 1 Related absorption diagram is depicted in Fig. 21 1st  Table 12 Data applied to the structure in Fig. 1 Related absorption diagram is depicted in Fig. 22 1st   Table 13 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 23 1st   Table 14 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 24 1st layer = GQDs R = 400 nm δ = 200 nm 2nd layer = Dielectric Material = SiO 2 Thickness = 400 nm 3rd layer = GQDs R = 450 nm δ = 160 nm 4th layer = Dielectric Material = Si 3 N 4 Thickness = 700 nm 5th layer = GQDs R = 400 nm δ = 200 nm 6th layer = Dielectric Material = Si 3 N 4 Thickness = 3900 nm 7th layer = lossless material Material = Gold Thickness = 240 nm Fig. 24 The absorption diagram is related to the data in Table 14 Table 15 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 25 1st   Table 16 Data applied to the structure in Fig. 1 The related absorption diagram is depicted in Fig. 26 1st