Tunable properties of the absorption in a binary photonic crystal having a metamaterial as a defect layer and two graphene sheets in the range of GHz

doi:10.21203/rs.3.rs-1600410/v1

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Tunable properties of the absorption in a binary photonic crystal having a metamaterial as a defect layer and two graphene sheets in the range of GHz

https://doi.org/10.21203/rs.3.rs-1600410/v1

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Light absorption is essential in the construction of photodetectors and photovoltaic applications. In this work, the absorption properties of a defective photonic crystal are analyzed in the GHz region using the transfer matrix method. The defect layer is assumed to be a metamaterial of negative optical parameters sandwiched between two sheets of graphene. The tunable properties of absorption are investigated with the graphene chemical potential and phenomenological scattering rate and the thickness of the metamaterial layer. The results reveal that the number, position and height of the absorption resonant peaks are tuned by graphene optical properties. The spectra are studied with and without graphene sheets and an extra peak is created when two sheets of graphene are inserted on both sides of the metamaterial. The proposed photonic device can be useful in designing graphene and metamaterial-based optical devices such as absorbers, filters and sensors in the GHz region.

Photonic crystal

Transfer matrix method

Metamaterial

Defect layer

Graphene

Absorption.

Due to its physical properties and prospective uses in electrical, optoelectronic and energy storage devices, graphene-based structures have gotten a lot of interest in recent years [1–5]. The optical characteristics of graphene can be modified by altering its plasmon frequency using an externally supplied electric field to manipulate the Fermi level [2]. Light absorption, on the other hand, is essential in the construction of photodetectors and photovoltaic applications. Different approaches, such as graphene-based structures [5] and ordered periodic structures [4], have been used to achieve complete absorption. Several scientists have looked at the optical absorption of the structures based on graphene so far. Thongrattanasiri et al., for example, demonstrated that a single graphene layer structured into an array of nano-disks has shown 100% light absorption [3]. Ghasempour [6] examined the temperature dependency of absorbance in the graphene layer. The dielectric–graphene–metal groove-grating absorber was studied by Zhu et al. They found that adjusting the applied potential difference on graphene can result in a high absorption efficiency [7]. Deng et al. studied the absorption in the spectral range THz in graphene-based devices and found that absorption may be regulated by altering chemical potentials [8]. Absorbers and sensors have been the topic of numerous research papers in recent years [9–11]. Graphene-metamaterials based structures are also promising prospects for producing new IR and THz absorbers [12]. Metamaterials are man-made materials with sub-wavelength electric circuits. These materials have optical properties that differ from those of natural materials [13]. Watts et al. [14] have investigated the theory, characterization, and application of metamaterials as absorbers. Nefedov et al. demonstrated that a graphene-based hyperbolic metamaterial can achieve complete light absorption [15]. In the far-infrared wavelength band, ideal absorbers based on graphene microribbon metamaterial were proposed by Alaee et al. [16]. Andryieuski et al. used the effective surface conductivity to construct a graphene-based metamaterial absorber in the THz range [17]. Linder et al. investigated the absorption of structures based on graphene-anisotropic metamaterial and found that they could achieve near-perfect absorption over a wide range of angles of incidence [18]. They attributed this to the coupling of the incident light and a fast-wave propagating along with the structure. Some research groups have recently investigated one-dimensional graphene photonic crystals (PCs) [19–21].

We propose a novel PC to achieve tunable properties of absorption in the GHz band in this work. A metamaterial sandwiched between two graphene layers is used as a defect layer to enhance absorption. The presence of graphene and metamaterials have a notable effect on the optical properties of the PC and hence on the absorbance, reflectance and transmittance spectra. The proposed photonic device is useful to design graphene metamaterial-based absorbers, filters and sensors.

The remaining parts of the paper are organized as follows: the theoretical model using the transfer matrix approach is presented in Section 2, and the numerical results and discussions are analyzed and reported in Section 3. At the last, the conclusions are explained and discussed in Section 4.

A defective binary photonic crystal (BPC) of the structure (Si/SiO₂)^N/ defect / (Si/SiO₂)^N is proposed for the investigation of the absorption properties as shown in Fig. 1. This structure is surrounded by air from both sides. The defect is assumed as a metamaterial layer of simultaneously negative permittivity and permeability surrounded by two graphene sheets.

The permittivity (${\epsilon }_{L}$) and permeability (${\mu }_{L}$) of the metamaterial are assumed to obey Lorentz medium model as [22]

$${\epsilon }_{L}\left(\omega \right)={\epsilon }_{0}(1-\frac{{f}_{ep}^{2}-{f}_{eo}^{2}}{{f}^{2}-{f}_{eo}^{2}+i {\varGamma }_{e}\omega })$$

$${\mu }_{L}\left(\omega \right)={\mu }_{0}(1-\frac{{f}_{mp}^{2}-{f}_{mo}^{2}}{{f}^{2}-{f}_{mo}^{2}+i {\varGamma }_{m}\omega })$$

where $\omega$ is the angular frequency and ${f}_{eo}$, ${f}_{ep}$ and ${\varGamma }_{e}$ are the electric resonance, plasma and damping frequencies. ${f}_{mo}$, ${f}_{mp}$ and ${\varGamma }_{m}$ are the same but magnetic instead of electric.

The graphene permittivity can be written as [23]

$${\epsilon }_{G}=1+\frac{i {\sigma }_{G}}{\omega {\epsilon }_{0}{d}_{G}}$$

where ${d}_{G}$ and ${\epsilon }_{0}$ are the thickness of the graphene sheet and permittivity of vacuum. The surface conductivity ${\sigma }_{G}$ of a graphene sheet is given by

$${\sigma }_{G}=\frac{{e}^{2}}{\pi {h}^{2}}\frac{{k}_{B}T}{{\varGamma }_{G}-i\omega }[\frac{{\mu }_{C}}{{k}_{B}T}+2\text{l}\text{n}({e}^{\frac{-{\mu }_{C}}{{k}_{B}T}}+1\left)\right]$$

where${k}_{B}$ is the Boltzmann constant,$T$ is the temperature, ${\varGamma }_{G}$ is the phenomenological scattering rate of graphene, and${\mu }_{C}$ is the chemical potential.

The PC optical properties are designed and simulated using a variety of approaches. The transfer matrix method is a simple and flexible tool for analyzing the optical properties of PCs. In this work, the structure of air/multilayer/substrate is considered. The parameters needed for the air and substrate materials are just their refractive indexes, n₀ and n_s. In terms of E_s and H_s (fields in the substrate layer), E₀ and H₀ (incident fields) can be expressed in a matrix form as

$$\left[\begin{array}{c}{\text{E}}_{0} \\ {\text{H}}_{0}\end{array}\right]=\prod _{j=1}^{N}{D}_{j}\left[\begin{array}{c}\text{E}\text{s} \\ \text{H}\text{s}\end{array}\right]=\left[\begin{array}{cc}{b}_{11}& {b}_{12}\\ {b}_{21}& {b}_{22}\end{array}\right]\left[\begin{array}{c}\text{E}\text{s} \\ \text{H}\text{s}\end{array}\right]$$

where ${D}_{j}$ is the characteristic matrix of one layer and b_ij are the ${D}_{j}$ elements. The matrix D_j can be expressed as

$${D}_{j}=\left[\begin{array}{cc}\text{cos}\left({\delta }_{j}\right)& -\frac{i \text{sin}\left({\delta }_{j}\right)}{{\gamma }_{j}}\\ -i{\gamma }_{j}\text{sin}\left({\delta }_{j}\right)& \text{cos}\left({\delta }_{j}\right)\end{array}\right]$$

where ${\delta }_{j}$ is the phase change due to the propagation through the j^th layer.

$${\delta }_{j}=\frac{2\pi }{\lambda }{n}_{j}{h}_{j}{\text{cos}\theta }_{j}$$

where ${n}_{j}$ and h_j are the refractive index and thickness of the layer. ${\theta }_{j}$ is the angle of incidence into the layer. In terms of the incidence angle ${\theta }_{0}$, ${\theta }_{j}$ is given as

$${\text{cos}\theta }_{j}=\sqrt{1-{\left(\frac{{n}_{0}\text{ sin}\left({\theta }_{0}\right)}{{n}_{j}}\right)}^{2}}$$

${\gamma }_{j}={n}_{j}\text{cos}\left({\theta }_{j}\right)$ for transverse electric (TE) and ${\gamma }_{j}=\text{cos}\left({\theta }_{j}\right)/{n}_{j}$ for transverse magnetic (TM) waves. ${n}_{0}$ is the refractive index of the incidence medium. The transfer matrix D₀ for one period is written as D₀ = D_Si D_SiO2. The transfer matrix D of the whole BPC with a metamaterial layer between two graphene sheets can be written as

$$D={\left({D}_{0}\right)}^{N}{D}_{G}{D}_{M}{D}_{G}{\left({D}_{0}\right)}^{N}=\left[\begin{array}{cc}{D}_{11}& {D}_{12}\\ {D}_{21}& {D}_{22}\end{array}\right]$$

where D_M and D_G are the transfer matrices of the metamaterial and graphene layers and D_ij are the elements of the D matrix. In terms of D_ij, the transmittivity can be written as

$$T=\frac{{\gamma }_{out}}{{\gamma }_{in}}{\left|\frac{2{\gamma }_{in}}{\left({D}_{11}+{D}_{12}{ \gamma }_{out}\right){\gamma }_{in}+({D}_{21}+{D}_{22 }{\gamma }_{out})}\right|}^{2}$$

and the reflectivity can have the form

$$R={\left|\frac{\left({D}_{11}+{D}_{12} {\gamma }_{out}\right){\gamma }_{in}-({D}_{21}+{D}_{22} {\gamma }_{out})}{\left({D}_{11}+{D}_{12} {\gamma }_{out}\right){\gamma }_{in}+({D}_{21}+{D}_{22} {\gamma }_{out})}\right|}^{2}$$

Finally, the absorption (A) is obtained as

$$A=1-T-R$$

For TE and TM modes, ${\gamma }_{in}={\gamma }_{out}= \text{cos}\left({\theta }_{0}\right)$ since the BPC is assumed to be surrounded by air from both sides.

The absorption properties of a BPC having the structure (Si/SiO₂)^N/ Graphene/ metamaterial/ Graphene/ (Si/SiO₂)^N are investigated using Mathematica software. We are interested in the region in which the real parts of 𝜀_L and µ_L of the metamaterial layer are negative. The following parameters are used in the calculations as in Ref. [22]: ${f}_{eo}$ = 10.30 GHz, ${f}_{ep}$ = 12.80 GHz, ${f}_{mo}$ = 8.0 GHz and ${f}_{mp}$ = 13.0 GHz. The electric and magnetic damping frequencies are taken to be equal ${\varGamma }_{e}$ = ${\varGamma }_{m}$ = ${\varGamma }_{L}$ = 1 GHz [22]. Figure 2 shows the real parts of the permittivity (a) and permeability (b) of the metamaterial as a function of the frequency in the spectral range 0.1–25 GHz according to the Lorentz model. The figure reveals that the real parts of 𝜀_L and µ_L of the metamaterial layer are negative in the spectral range 10.4–12.7 GHz. In the next calculations, we will adopt this frequency range. We consider a metamaterial layer of 1 µm thickness sandwiched between two graphene sheets each of thickness of 0.34 nm. The thicknesses of Si and SiO₂ layers are fixed as h_Si = 10 mm and h_SiO2 = 25 mm. The number of periods of the PC is considered N = 5. The chemical potential and the phenomenological scattering rate of graphene are taken as ${\mu }_{C}$ = 0.378 eV and ${\varGamma }_{G}$ = 250 GHz, respectively.

Figure 3 shows the transmission, reflection and absorption spectra of the proposed photonic structure at the prescribed parameters. As observed from the figure, the transmittance is nearly zero. There exist five dips in the reflectance profile and five peaks in the absorbance profile. The reflectance dips and the absorbance peaks occur at the same frequencies. The number of dips and peaks can be attributed to the number of periods of the photonic device. That means the dips and peaks in the reflectance and absorbance profiles are proportional to the number of periods. In the next, we will investigate the number of dips and peaks with the number of periods. Besides, we will investigate the effects of some factors such as the chemical potential and phenomenological scattering rate of graphene and the thickness of the metamaterial layer on the absorption properties of the proposed structure. Figure 4 shows the absorption spectra of the proposed BPC for different periods. We have taken N = 3, 5, 7 and 9. It is clear from these figures that the number of absorption peaks increases proportionally to the number of periods. In addition, as the number of periods increases, the absorption spectra occur in a wider frequency range. The frequency separations between the first and the last peaks are found as 0.97, 1.16, 1.24 and 1.28 GHz for N = 3, 5, 7 and 9. The absorption spectra are shown in Fig. 5 for different chemical potentials of the graphene layers. We have assumed the values of ${\mu }_{c}$ as 0.378, 0.6, 0.8 and 1 eV. Other parameters are kept the same as in Fig. 4. The number of peaks in the absorption profile and the frequency range in which these peaks occur do not change with the variation of graphene chemical potential. The peak values show a dramatic decay when the chemical potential is increased. For more investigation of the peak decay with the increase of the chemical potential, we consider the fifth peak of each panel in Fig. 5. The maximum intensities of absorption of these peaks are presented in Table 1. More values of the chemical potential have been investigated and added to the Table. In Fig. 6, the variation of the maximum absorption of the fifth peak is plotted as a function of graphene chemical potential. The decay of the maximum absorption for ${\mu }_{c}$ = 0.378–0.7 eV is faster than that for ${\mu }_{c}$ > 0.7 eV. The main reason behind the fast decay of the absorption when ${\mu }_{c}$ is in the range 0.378–0.7 eV is the greater effect of the chemical potential on the dielectric constant of graphene during this range than in other ranges. The chemical potential of graphene can be varied by using the applied gate voltage. Hence the absorbance of the proposed structure is effectively tunable with the chemical potential of graphene and the applied gate voltage.

Table 1

Maximum intensity of the absorption as a function of chemical potential. (h_L = 1 µm, N = 5, ${\varGamma }_{G}$ = 250 GHz$,$ ${\varGamma }_{L}$ = 1GHz).
${\mu }_{c}$ (eV)	0.378	0.5	0.6	0.7	0.8	0.9	1
Maximum absorbance	0.48	0.41	0.35	0.3	0.26	0.23	0.21

The absorption spectra of the proposed BPC are investigated for different graphene phenomenological scattering rates in Fig. 7. We have assumed the values of ${\varGamma }_{G}$ as 250, 450, 650 and 850 GHz. The number of absorption peaks does not depend on the phenomenological scattering rate of graphene. Moreover, the frequency range in which these peaks occur does not show a significant dependence on the graphene phenomenological scattering rate. The absorption peak values show an essential increase when the phenomenological scattering rate increases. For more investigation of the peak enhancement with the increase of the phenomenological scattering rate, we consider the fifth peak of each panel in Fig. 7. The maximum intensities of absorption of these peaks as a function of the phenomenological scattering rate are presented in Table 2. More values of the phenomenological scattering rate have been investigated and added to the Table. These values are plotted in Fig. 8. The absorption enhancement for ${\varGamma }_{G}$ = 250–450 GHz is much greater than that for ${\varGamma }_{G}$ > 450 GHz. The absorbance of the proposed structure is effectively tunable with graphene phenomenological scattering rate.

Table 2

Maximum intensity of absorption as a function of ${\varGamma }_{G}$. (h_L = 1 µm, N = 5, ${\mu }_{c}$ = 0.378 eV and ${\varGamma }_{L}$ = 1 GHz).
${\varGamma }_{G}$ (GHz)	250	350	450	550	650	750	850
Maximum absorbance	0.48	0.59	0.70	0.75	0.82	0.86	0.90

Figure 9 shows the absorption spectra of the proposed PC for different values of the metamaterial layer thickness. We have considered the values of h_L as 100, 200, 300 and 400 µm. The maximum absorption can be enhanced with the metamaterial thickness increase. The maximum absorption of the fifth peak is found as as 57%, 64%, 67% and 70% for metamaterial layer thickness of 100, 200, 300 and 400 µm, respectively. This enhancement can be attributed to the longer optical path as the metamaterial layer thickness increases.

To show the role of the graphene sheets in the spectra of the proposed structure, the transmittance, reflectance and absorption spectra are shown without (Fig. 10a) and with (Fig. 10a) graphene. As observed (Fig. 10a), no absorption in the spectra and only four peaks appear in the transmittance and reflectance spectra in the absence of graphene sheets. From Fig. 10b, two important effects are noted in the presence of two graphene sheets. The first effect is the reduction of transmittance to very low values. The second effect is the appearance of an additional peak in the reflectance spectra. This effect is due to the large alteration in the optical path and the effective refractive index of the proposed device with the presence of two graphene sheets. The surface conductivity of graphene is responsible for the creation of additional peaks in the reflectance and absorbance profiles. Here, two graphene sheets behave like a single repetitive layer and it reflects in the result as an extra absorption peak with a high magnitude.

The absorption properties of a defective one-dimensional BPC are analyzed in the GHz region. The defect layer is assumed a metamaterial of simultaneously negative epsilon and mu sandwiched between two sheets of graphene. Tuning properties of the absorption spectra are investigated with the variation of chemical potential and phenomenological scattering rate of graphene and thickness of the metamaterial layer. The results reveal that the number of absorption peaks in the absorbance spectrum increases proportionally to the number of periods. The absorbance of the proposed structure is effectively tunable with the chemical potential of graphene. The chemical potential of graphene can be varied by using the applied gate voltage. Hence, the absorbance can be tuned by changing the applied gate voltage. The absorption peak has shown an essential enhancement when the phenomenological scattering rate of graphene increases. The absorption can be also enhanced with the metamaterial thickness increase. The reflection and absorption spectra have been investigated with and without graphene and an extra peak is created when two sheets of graphene are inserted on both sides of the metamaterial.

Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding Info: The authors are thankful to the Deanship of Scientific Research at Najran University for funding this work under the Research Groups Funding program grant code (NU/RG/SERC/11/16).

Author contribution: All authors contributed to the study conception and design. Mathematical derivation and part of the editing were performed by [Malek G. Daher]. The code and part of the editing were performed by [Sofyan A. Taya] and [Ilhami Colak]. The first draft of the manuscript was written by [Shobhit K. Patel] and [Arvind Sharma]. Discussion of the results was performed by [Sofyan A Taya] and [Abdulkarem H. M. Almawgani]. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Consent to Participate (Ethics): No consent to participate is required for this study.

Consent for Publish (Ethics): No consent for publication is required for this study.

Acknowledgement: The authors are thankful to the Deanship of Scientific Research at Najran University for funding this work under the Research Groups Funding program grant code (NU/RG/SERC/11/16).

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Tunable properties of the absorption in a binary photonic crystal having a metamaterial as a defect layer and two graphene sheets in the range of GHz

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Abstract

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1. Introduction

2. Design And Theoretical Model

3. Results And Discussions

4. Conclusions

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References

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