1. Introduction

Graphene is a flat monolayer of graphite with carbon atoms closely packed in a two-dimensional honeycomb lattice [1, 2]. It can be formed from the bulk pyrolytic graphite by a sticky tape [3] or fabricated by cutting the carbon nanotubes based on chemical approaches [4, 5]. The promising method may be the chemical vapor deposition (CVD) technique [6, 7]. One of the most important features of graphene is that its conductivity or dielectric function could be tuned by varying the chemical potential of the graphene sheets via electrostatic biasing. This feature provides a robust and flexible method to tune the optical properties of the photonic crystals that incorporate graphene. Transmission properties of one-dimensional graphene photonic crystal were investigated by using transfer matrix method, and it was found that the structure has a new type of the photonic band gap in the THz region which is almost omnidirectional and insensitive to the polarization [8–10]. Transmission properties of one-dimensional (1D) graphene photonic crystals with Fibonacci quasi-periodic structure were also investigated, and omnidirectional photonic band gaps appear due to the existence of the graphene sheets, and the gaps can be controlled via a gate voltage [11]. By coating the coupling region with a graphene nanoribbon, the tunable directional coupler was obtained by using an electric gate to modify the graphene chemical potential [12]. The transverse magnetic surface plasmon dispersion relations for a monolayer graphene and a graphene parallel-plate waveguide in the presence of a one-dimensional photonic crystal were also obtained [13].

To provide advanced functionalities of photonic crystal, one has to introduce the controlled defect into the graphene photonic crystal otherwise perfect periodic structure. As graphene layers have a significant effect on the absorption [14], the dual-band absorption of 1D graphene photonic crystals were theoretically analyzed by introducing single-layer graphene defect [15]. Narrow-band absorption is achieved in an asymmetric 1D dielectric photonic crystal with a monolayer graphene defect, and the perfect absorber may be changed to be a mirror by changing voltage of the graphene [16]. By placing defect layers in the middle of the graphene photonic crystal, the tunable defect mode can be utilized as an active radome or a force generator for the design of high-Q antenna [17]. Similar to the traditional photonic crystal, sharp transmission peak appears within the bandgap when a defect dielectric layer is properly selected in the periodic graphene photonic crystal structure, and only normal incidence was considered [18].

For the graphene photonic crystal with defects, most works focused on their absorption properties [14–16] or the transmission of defect mode with single-layer defect structure [17, 18]. In this study, dual-layer defects are introduced in a 1D graphene photonic crystal, and it is found that two defect modes appear among the original photonic band gaps, and the defect modes can be tuned by chemical potential of graphene. More interesting, both the magnitudes and number of the defect modes can be changed by locations of the two defect layers and incident angles. These properties of the 1D graphene photonic crystals have potential applications in tunable terahertz multiband narrow filters.

2. Theoretical model and mathematical method

Figure 1 (a) shows the 1D graphene photonic crystals structure containing alternating N layers of the graphene and dielectric A along the z axis. The dimensions in the x and y directions are assumed to be infinite for the 1D structure. Figure 1 (b) and (c) show the single-layer defect and dual-layer defect structures formed by inserting dielectric B in the center or beside the middle layers of the period structure, respectively. N₁ and N₃ are period numbers in the front and back of the defect layer, N₂ is the period numbers between the two defect layers. Here N₁ = N₃ = 4, N₂ = 1, N = N₁ + N₂ + N₃ = 9. Thickness of the graphene, dielectric A and dielectric B are assumed to be d_G, d_A and d_B with the relative dielectric constant ε_G, ε_A and ε_B, respectively.

 

Fig. 1 One-dimensional graphene photonic crystal with (a) Period, (b) Single-layer defect and (c) Dual-layer structure.

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Graphene is an optically uni-axial anisotropic materials because of its 2D nature, whose permittivity tensor can be given by (when graphene lies in the x-y plane) [19]

The normal component of graphene permittivity is ε_G,⊥ = 1 as the normal electric field cannot excite any current in the graphene sheet [20]. The tangential component of graphene permittivity ε_G,t is express as [21–23]

Here ω is the angular frequency, d_G is thickness of the single-layer graphene, ε₀ is the permittivity in vacuum. σ(ω) = σ_intra + σ_inter is the surface conductivity of graphene, σ_intra and σ_inter are the intraband conductivity and the interband conductivity and defined in the following

The transmission characteristics of the graphene photonic crystal can be described by Transfer Matrix Method (TMM). In the calculation, incident wave can be divided into the TE polarization with E = (0, E_y, 0) and H = (H_x, 0, H_z), and the TM polarization with E = (E_x, 0, E_z) and H = (0, H_y, 0).

The dispersion relations of the TE and TM polarizations in the anisotropic graphene are determined by [19, 24]

If we write the transfer matrixes of the graphene, dielectric A and dielectric B layers as M_G, M_A, and M_B, respectively. Transfer matrix M_i (i = G, A, and B) for the three different layers can be given by [17, 25–28] (Supplementary information for details of the M_G)

For the TE polarization, $k_{iz}=\sqrt{k_0^2{\epsilon }_i-k_{ix}^2}$, $k_{ix}^{}=k_x^{}=k_0^{}\sin {\theta }_0$ is the tangential component of the wave vector in each layer, which is equal to that of the incident wave [24]. ε_i denote ε_G,⊥, ε_A and ε_B for the graphene, dielectric A and dielectric B layers, respectively. For the TM polarization, $k_{Gz}=\sqrt{k_0^2{\epsilon }_{G,t}-k_x^2({\epsilon }_{G,t}/{\epsilon }_{G,\perp })}$, $k_{Az}=\sqrt{k_0^2{\epsilon }_A-k_x^2}$and $k_{Bz}=\sqrt{k_0^2{\epsilon }_B-k_x^2}$ for the graphene, dielectric A and dielectric B layers, respectively. At normal incidence for $k_x^{}=0$, $k_{Gz}=k_0^{}\sqrt{{\epsilon }_{G,t}}$both for the TE and TM polarizations, then, graphene permittivity is only determined by the tangential component ${\epsilon }_{G,t}$.

Transfer matrix for N periods in Fig. 1 (a) is

For the single-layer defect structure in Fig. 1 (b), transfer matrix M is written as

For the dual-layer defect structure in Fig. 1 (c), transfer matrix M is written as

The reflection R, transmission T and absorption A can be expressed by

3. Typographical style

For the 1D graphene photonic crystals in Fig. 1(a), we define period numbers N = 9. SiO₂ is selected as the dielectric A with permittivity ε_A = 2.2 and thickness d_A = 100 μm [20], multilayers of SiO₂ and the graphene can be formed by a plasma enhanced chemical vapor deposition technique. Polyimide is the defect layer with permittivity ε_B = 3.2 and thickness d_B = 150 μm which can be fabricated by spin coating technique [29, 30]. The graphene has parameters of the chemical potential of${\mu }_c$ = 0.9 eV, thickness d_G = 0.34 nm, temperature T = 300 K and scattering rate Γ = 1 THz. For the single-layer defect structure in Fig. 1(b), N₁ = N₃ = 4. For the dual-layer defect structure in Fig. 1(c), N₁ = N₃ = 4, N₂ = 1.

Figure 2 shows the transmission and absorption curves of the period, single-defect and dual-defect structures in Fig. 1 at normal incidence θ₀ = 0. In Fig. 2(a), the dot, dash and solid lines represent transmission curves of the period, single-defect and dual-defect structures, respectively. For the period structure, it is seen that there are two photonic band gaps in the frequency regions of 5.5-7.5 THz. The first gap locates from 5.8 THz and 6.1 THz, and the second gap locates from 6.8 THz to 7.1 THz. When the single-layer defect is introduced in the periodic structure, two transmission peaks with 5.939 THz and 7 THz appear in the first and the second gap, respectively. For the dual-layer defect structure, two transmission peaks with f₁ = 5.869 THz and f₂ = 5.980 THz appear in the first gap, and another two transmission peaks with f₃ = 6.940 THz and f₄ = 7.05 THz appear in the second gap. Compared with the single-layer defect structure, the dual-layer defect structure can be used as multiband narrow filters and has the higher transmission peaks. On the other hand, as the single-layer defect in graphene photonic crystals have been discussed in [17, 18], only properties of dual-layer defect modes are given in the following. For the four resonant peaks in the dual-layer defect structure, very high quality factor Q = 309, 374, 289 and 307 are obtained for f₁, f₂, f₃ and f₄, respectively, where Q = f_i /Δf (f_i is the resonance frequency, Δf is the full-width at half maximum bandwidth).

 

Fig. 2 Transmission and absorption curves of the period, single-defect and dual-defect structures in Fig. 1 with ε_A = 2.2, ε_B = 3.2, d_A = 100 μm, d_B = 150 μm, d_G = 0.34 nm, T = 300 K, Γ = 1 THz, ${\mu }_c$= 0.9 eV.

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Figure 2(b) shows the absorption curves of the three structures, the dot, dash and solid lines represent results of the period, single-defect and dual-defect structures, respectively. It is seen that the corresponding numbers of absorption peaks at the same frequencies appear in the two gaps for defect structures. For the single-layer defect structure, magnitudes of transmission and absorption peak are 0.374 and 0.237 in the first band, and they are 0.644 and 0.173 in the second band. Then, most losses come from the refection. Table 1 shows the transmission, absorption and reflection of the four peaks of the dual-layer defect structure. Absorption is the main loss for the first, second and forth peaks while the reflection is little higher than the absorption for the third peak.

Table 1. Transmission (T), absorption (A) and refection (R) magnitude of the four peaks

View Table

As permittivity of the graphene can be modified by the chemical potential [31, 32], the transmission of the dual-layer defect structures will be modified accordingly. Figure 3 shows the transmission and absorption curves under different chemical potential for the dual-layer defect structure with ε_A = 2.2, ε_B = 3.2, d_A = 100 μm, d_B = 150 μm, d_G = 0.34 nm, T = 300 K, Γ = 1 THz. The dash and solid lines denote the results of ${\mu }_c$ = 0.5 eV and ${\mu }_c$ = 0.9 eV, respectively. For the transmission curves in Fig. 3(a), it is seen that the four transmission peaks shift to the lower frequencies with magnitude decreasing when the chemical potential${\mu }_c$increase from 0.5 eV to 0.9 eV, and width of the two photonic band gaps increases correspondingly. For absorption curves in Fig. 3(b), the four absorption peaks shift to the lower frequencies and the absorption magnitudes increase with ${\mu }_c$ increasing. To generally study the dependency of defect modes on different chemical potentials at normal incidence. Figure 4 shows the color maps of transmission and absorption versus the frequency and the chemical potential for the dual-layer defect structure. The frequency range in the horizontal axis is from 5.5 THz to 7.5 THz. The chemical potential in the vertical axis varies from 0.1 eV to 1.0 eV. The four defect modes are denoted by 1, 2 in the first gap, and 3, 4 in the second gap. It is observed that defect modes in Fig. 4(a) shift to the lower frequencies with their transmission magnitude decreasing when the potential chemical increases from 0.1 eV to 1.0 eV. At the same time, the two gaps where defect modes appear turn to enlarge. In Fig. 4(b), the absorptions of the four peaks shift to the lower frequencies while their magnitudes increase with potential chemical increasing.

 

Fig. 3 Transmission and absorption curves under different chemical potential ${\mu }_c$ = 0.5 and 0.9 eV for the dual-layer defect structure with ε_A = 2.2, ε_B = 3.2, d_A = 100 μm, d_B = 150 μm, d_G = 0.34 nm, T = 300 K, Γ = 1 THz.

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Fig. 4 Color map of transmission and absorption under varying from 0.1 and 1.0 eV for the dual-layer defect structure with ε_A = 2.2, ε_B = 3.2, d_A = 100 μm, d_B = 150 μm, d_G = 0.34 nm, T = 300 K, Γ = 1 THz.

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To explain the tunable properties of the these defect modes, the real part Real(ε_G,t) and imaginary part Imag(ε_G,t) of tangential component of graphene permittivity versus chemical potential ${\mu }_c$ are shown in Fig. 5 (a). For normal incidence, the graphene permittivity is only determined by its tangential component ε_G,t. In the frequency ranges from 1 THz to 10 THz, the real part is negative while the imaginary part is positive, they strongly depend on the frequency in the low frequencies and varies slowly in the high frequencies. The absolute value of ε_G,t increases as chemical potential increases, which makes the effective dielectric constant of the graphene photonic crystal increases. According to the variational principle [33, 34], increasing of dielectric constant leads to decreasing of frequency modes, then the band gap shifts toward lower frequencies as well as the defect modes. On the other hand, the difference of dielectric function between graphene and dielectric becomes larger as dielectric constant of dielectric increases, which makes the gap enlarge. Therefore, the two defect modes in the gaps shift to the lower frequencies and the two gaps turns to enlarge with chemical potential increasing.

 

Fig. 5 (a) The real part (solid line) and the imaginary part (dot-dash line) of the tangential permittivity ε_G,t of single-layer graphene for different chemical potential ${\mu }_c$ = 0.1, 0.3, 0.5, 0.7 and 0.9 eV with d_G = 0.34 nm, T = 300 K and Γ = 1 THz. (b) The enlarged imaginary part valued from 0 to 500.

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Figure 5(b) shows the enlarged imaginary part Imag(ε _{G , t}) of the graphene valued from 0 to 500. It is seen that the imaginary part increases with chemical potential ${\mu }_c$ increasing, which would give rise to the larger absorption. Therefore, the absorption of the dual-layer defect structure increases with chemical potential ${\mu }_c$ increasing as seen in Fig. 4(b). In Fig. 5(a), the imaginary part of ε_G,t is much larger than 500 in the lower frequencies. Actually, similar defect modes can appear below 5.5 THz for the dual-layer defect structure. However, their transmission peaks are very small and not discussed due to the large absorption in the lower frequencies.

Figure 6 shows dependency of the defect modes on locations of the dual-defect layers. Figure 6 (a) shows the results for different numbers of the periods N₁ and N₃, where (N₁ = N₃), and Fig. 6 (b) shows the results for different numbers of the periods N₂. It is found that defect modes in the first and second gaps show similar variation trend. As number of the N₁ = N₃ increases in Fig. 6(a), magnitudes of the defect modes decrease and tend to disappear at N₁ = N₃ = 8. As number of the N₂ increases in Fig. 6(b), the two defect modes in the two gaps firstly tend to combine together with the larger magnitudes at N₂ = 4, and then disappear as N₂ increases to 12 and 15 in the first gap and the second gap, respectively. These phenomena means defect mode will not appear for graphene photonic crystal with the dual-layer defects as the defect layers can be neglected for large period number. Therefore, locations of the dual-defect layers will affect the frequency, magnitude and number of defect modes.

 

Fig. 6 Transmission versus location for the dual-defect layer structure with ε_A = 2.2, ε_B = 3.2, d_A = 100 μm, d_B = 150 μm, d_G = 0.34 nm, T = 300 K, Γ = 1 THz and. ${\mu }_c$ = 0.9 eV.

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In the former analysis, only the normal incidence is considered. Figure 7 shows the transmission of the TM and TE polarization with oblique incidence θ₀ varying from 0° to 60°. Other parameters are ε_A = 2.2, ε_B = 3.2, d_A = 100 μm, d_B = 150 μm, d_G = 0.34 nm, T = 300 K, Γ = 1 THz, and${\mu }_c$ = 0.9 eV. For the TM polarization, it is seen that defect modes shift to the higher frequencies with incident angle increasing. For the first gap, mode 1 disappears while another defect mode appears at around 30°. Mode 2 disappears at around 38°. In the second gap, mode 3 also disappears at around 30°. In the lower frequencies below the first gap, another two defect modes appear at around 35° and shift to the higher frequencies with incident angle increasing. For the TE polarization, the defect modes have the same trend with that of the TM polarization, but the defect modes disappear at the larger incident angles and new modes appear at the smaller incident angles. Actually, gaps of the period structure of graphene photonic crystals will shift to the higher frequency with incident angle increasing. As defect modes always appear among the gaps, then the shifting phenomena of the defect modes is due to the varying of the gaps of the periodic structure with incident angles.

 

Fig. 7 Color map of the transmission versus the frequency and the incidence angle for (a) TM and (b) TE polarization, where ε_A = 2.2, ε_B = 3.2, d_A = 100 μm, d_B = 150 μm, d_G = 0.34 nm, T = 300 K, Γ = 1 THz, and. ${\mu }_c$ = 0.9 eV.

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4. Conclusion

1D graphene photonic crystal with symmetrical dual-layer defects are proposed, and the defect modes versus the chemical potential, location of the defect layers and the incident angles are discussed, respectively. Difference with the 1D single-layer defect graphene photonic crystal [17, 18], two narrow defect modes instead of one defect mode appear among the gaps for the dual-layer defect structure, and the defect modes shift to the lower frequencies with magnitude reducing as chemical potential increases. Frequency, magnitude and numbers of defect modes can also be changed by varying locations of the defect layers. For oblique incidence, defect modes of TE and TM polarization have similar tendency, they first shift to the higher frequencies with angle increasing and then disappear at certain angle, however, new defect modes will disappear at the larger incident angles. These properties of the dual-layer defects graphene photonic crystal would provide potential applications in tunable narrow multi-band filters.

Appendix Transfer matrix M_G for the graphene layer

Figure 8 shows the electromagnetic field distribution of TM polarization in 1D graphene and dielectric A. Graphene permittivity tensor can be given by (when graphene lies in the x-y plane)

 

Fig. 8 Electromagnetic field distribution of TM polarization in 1D graphene and dielectric A.

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(15)$${\epsilon }_G=\left[\begin{array}{l}{\epsilon }_{G,t}\text{0}\text{0}\\ \text{0}{\epsilon }_{G,t}\text{0}\\ \text{0}\text{0}{\epsilon }_{G,\perp }\end{array}\right].$$

The normal component of graphene permittivity is given by ${\epsilon }_{G,\perp }$ and the tangential component of graphene permittivity is ${\epsilon }_{G,t}$

Maxwell equations for the graphene layer are given by

$$\nabla \times E=-{\mu }_0\frac{\partial H}{\partial t}.$$ (16.a)

$$\nabla \times H={\epsilon }_0{\epsilon }_G\frac{\partial E}{\partial t}\text{.}$$ (16.b)

In the case of TM polarization, solutions of the Maxwell equations have the form E=(E_x, 0, E_z)e^-jωt and H=(0, H_y, 0)e^-jωt, then the Maxwell curl equations in this case are

$$\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}=-j\omega {\mu }_0{H}_y.$$ (17.a)

$$\frac{\partial H_y}{\partial z}=j\omega {\epsilon }_0{\epsilon }_{G,t}{E}_x.$$ (17.b)

$$\frac{\partial H_y}{\partial x}=-j\omega {\epsilon }_0{\epsilon }_{G,\perp }{E}_z.$$ (17.c)

Eliminating E_x and E_z from these equations, we can obtain H_y satisfying the equation

(18)$$\frac{1}{{\epsilon }_{G,t}}\frac{{\partial }^2{H}_y}{\partial z^2}+\frac{1}{{\epsilon }_{G,\perp }}\frac{{\partial }^2{H}_y}{\partial x^2}+k_0{}^2{H}_y=0.$$

The dispersion relation of TM polarization in the anisotropic graphene is calculated

(19)$$\frac{k_{Gx}^2}{{\epsilon }_{G,\perp }}+\frac{k_{Gz}^2}{{\epsilon }_{G,t}}=k_0^2.$$

For boundary I, solution of magnetic field has the form

(20)$$H_y{}^I=(A_{+}{e}^{j{k}_{Gz}z}+A_{-}{e}^{-j{k}_{Gz}z})e^{-j(\omega t-k_{Gx}x)}=H_{yt}{}^I+H_{yr}{}^{II\text{'}}.$$

Where $H_{yt}{}^I=A_{+}{e}^{j{k}_{Gz}z}{e}^{-j(\omega t-k_{Gx}x)}$, $H_{yr}{}^{II\text{'}}=A_{-}{e}^{-j{k}_G{}_{z}z}{e}^{-j(\omega t-k_{Gx}x)}$

H_y^I and H_yt^I is the total and transmitted magnetic field in y direction for boundary I, H_yr^II’ is the reflected magnetic field reflecting from boundary II in y direction for boundary I. For boundary I and II, E_x and H_y satisfy the continuity condition

(21)$$\begin{array}{l}{E}_x{}^I=E_{xt}{}^I+E_{xr}{}^{II\text{'}}\\ H_y{}^I=H_{yt}{}^I+H_{yr}{}^{II\text{'}}.\end{array}$$

(22)$$\begin{array}{l}{E}_x{}^{II}=E_{xi}{}^{II}+E_{xr}{}^{II}\\ H_y{}^{II}=H_{yi}{}^{II}+H_{yr}{}^{II}.\end{array}$$

(23)$$\left(\begin{array}{l}{H}_{yi}{}^{II}\\ H_{yr}{}^{II}\end{array}\right)=\left(\begin{array}{l}{e}^{j{k}_{Gz}{d}_G}\text{0}\\ 0e^{-j{k}_{Gz}{d}_G}\end{array}\right)\left(\begin{array}{l}{H}_{yt}{}^I\\ H_{yr}{}^{\text{II'}}\end{array}\right).$$

where H_y^II, H_yi^II and H_yt^II are the total, incident and reflected wave of H_y for boundary II, respectively. E_x^I and E_xt^I are the total and transmitted electric field in boundary I, E_xr^II’ is the reflected E_x reflecting from boundary II for boundary I. E_x^II, E_xi^II and E_xr^II are the total, incident and reflected E_x for boundary II, respectively. E_x can be expressed by H_y from Eq. 17.b.

Doing simultaneous Eq. 21, Eq. 22 and Eq. 23, E_x and H_y can be connected by a transfer matrix M between boundary I and II.

(24)$$\left(\begin{array}{l}{E}_x{}^I\\ H_y{}^I\end{array}\right)=M\left(\begin{array}{l}{E}_x{}^{II}\\ H_y{}^{II}\end{array}\right).$$

where $M=\left(\begin{array}{l}\cos (k_G{}_z{d}_G)-\frac{j}{p_G}\text{sin(}{k}_G{}_z{d}_G\text{)}\\ -j{p}_G\text{sin(}{k}_G{}_z{d}_G\text{)}\cos (k_G{}_z{d}_G)\end{array}\right)$, $k_{Gz}=\sqrt{k_0^2{\epsilon }_{G,t}-k_x^2({\epsilon }_{G,t}/{\epsilon }_{G,\perp })}$ $p_G=\frac{\omega {\epsilon }_0{\epsilon }_{G,t}}{k_{Gz}}.$

Funding

This work is supported by the National Natural Science Foundation of China (NSFC) (61107030), Fundamental Research Funds for the Central Universities of China; and Opening Foundation of the State Key Laboratory of Millimeter Waves (K201703).

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