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]

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

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

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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]

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

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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

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

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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 ns. In terms of Es and Hs (fields in the substrate layer), E0 and H0 (incident fields) can be expressed in a matrix form as

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

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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

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

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where \({\delta }_{j}\) is the phase change due to the propagation through the jth layer.

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

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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

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

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\({\gamma }_{j}={n}_{j}\text{cos}\left({\theta }_{j}\right)\) 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 *D*0 for one period is written as *D*0 **=** *D*Si *D*SiO2. The transfer matrix *D* of the whole BPC with a metamaterial layer between two graphene sheets can be written as

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

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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

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

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and the reflectivity can have the form

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

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Finally, the absorption (*A*) is obtained as

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.