Dual band and tunable perfect absorber based on dual gratings-coupled graphene-dielectric multilayer structures

Yi Zhao; Qiuping Huang; Honglei Cai; Xiaoxia Lin; Hongchuan He; Tian Ma; Yalin Lu

doi:10.1364/OE.27.005217

1. Introduction

Metamaterial perfect absorbers (MPAs), a kind of metamaterials, can absorb the entire incident electromagnetic wave and have promising applications in sensitive sensors [1], antenna systems [2], and thermal emitters [3], and so on. Since the first metamaterial perfect absorber was fabricated by Landy et al [4] in 2008, the MPAs have been studied deeply. MPAs with different structures have been designed at various frequencies including optical [5], infrared [6,7], terahertz [8] and microwave [9] regions. Up to now, great efforts have been dedicated to achieve polarization-insensitive absorption [10–12], wide angle absorption [13,14], and broadband or multi-band absorption [15,16]. However, the issues of un-tunable absorption-band, large thickness of device, etc., hamper the actual applications of MPAs, and need to be effectively solved.

In recent years, graphene has attracted great interest in the design of meta-devices due to its unique physical properties [17]. On one hand, graphene exhibits high conductivity in the mid-infrared and terahertz ranges, and thus supports surface plasmons [18,19]. On the other hand, compared to the conventional metal, graphene shows an advantage that its conductivity can be controlled by chemical doping, electrical gating or optical pumping [20–23]. It thus seems to be a good candidate for designing tunable meta-devices. Most recently, graphene-based MPAs have been proposed and investigated widely [24–29]. In some studies of them, there are some issues: (i) thickness of absorber is relatively thick due to that a thick dielectric layer between graphene and metal film is employed to match the free space to the impedance of absorber [27]; (ii) The absorption does not reach perfect absorption in some work because resonance of single-layer graphene is relatively weak [28,29]. Therefore, graphene-based hyperbolic metamaterials composed of multilayer of graphene and dielectric layers should be a good candidate [30].

Hyperbolic metamaterial (HMM) is an anisotropic medium exhibiting the hyperbolic shape of the dispersion relation [31]. The hyperbolic metamaterial is usually realized by stacking metallic layers and dielectric layers in period, and surface plasmons is supported on the interface of metallic layers and dielectric layers. In the HMM, surface plasmons in different interfaces couple together, so high k wave can transport in the HMM. Based on this properties, HMM can be applied in high-resolution imaging [32], spontaneous emission engineering [33], and thermal emission engineering [34]. In the graphene-based hyperbolic metamaterials (GHMM) [35,36], with metal layers replaced by the graphene layers, the permittivity tensor of GHMM can be controlled actively by varying the Fermi level of graphene, which makes it more attractive than the metal HMM. It has been reported that such GHMM can be designed for the emission enhancement [37], tunable broadband hyper-lens for imaging [38], realizing negative refraction [39], tunable infrared waveguide [40], and so on. In cases of absorbers, only the absorber based on graphene/MgF₂ multilayer stacking unit cells arrayed on an Au film plane was reported for the terahertz wave [30]. Taking into account the fact that GHMM is able to confine the electromagnetic field in a very small volume and has a high k value, one should design more structures to make use of them to achieve ultrathin and tunable perfect absorbers. Moreover, the structure of most perfect absorbers is complex and it is difficult to theoretically calculate absorption spectra using structural parameters. So the quantitative relationship between the absorption spectrum and the structural parameters of MPAs need to be more accurately studied.

In this paper, we propose a mid-infrared absorber based on the dual gratings-coupled graphene-dielectric multilayer structures (DGC-GDM), in which GDM is sandwiched between two Au gratings. We explored an alternative way for perfect absorption employing DGC- GDM that offers dual-band absorption behavior, polarization independent, wide angle range absorption. Furthermore, the wavelength of the absorption peak can be effectively changed by varying the Fermi level of graphene. Most importantly, the main novelty of the present work lies in the fact that an analytic formulas describing the relationships between the parameters of the absorber and the absorption spectra have been derived. Also, the accuracy of the theoretical formulas is verified by comparing the simulation results with the theoretically calculated ones.

2. Structure design and simulation method

The schematic diagram of the structure proposed to realize dual-band metamaterial perfect absorber is shown in Fig. 1. Figure 1(a) shows the 3D structure of the designed absorber, and Fig. 1 (b) presents the sectional view of the unit structure. As shown in Fig. 1, the graphene-dielectric multilayer is sandwiched between two optical gratings made of Au. And a reflective layer of Au is placed at the bottom of the lower grating. The GDM in our simulation consists of 8 periods of graphene/Al₂O₃ bilayers, in which the thickness of Al₂O₃ is 10nm. And a layer of 40nm thick Al₂O₃ is set between the upper/lower grating and the GDM, respectively, in order to reach great wave vector match and also to act as capacitor layers for gating described in the following section. The lower Au grating is also filled with Al₂O₃ dielectric. The parameters of our structure are given in Fig. 1(b), in which the thickness h₂ is 700 nm, h₁ is 150nm and the period P is 800 nm. Both the upper and lower gratings have a duty cycle of 0.5. And the proportion of width of lower grating w₂ and width of upper grating w₁ is set to 2. Although we only presented the simulation of our structure, here we briefly describe the feasibility of device fabrication. The GDM have been demonstrated experimentally in other work [35], in which graphene layers were grown by chemical vapor deposition (CVD) and transferred by the poly (methyl methacrylate) (PMMA) method to the Al₂O₃ layers grown by atomic layer deposition (ALD). The lower and upper gratings can be fabricated by electron beam lithography technique.

Fig. 1 (a) The schematic diagram of the absorber. (b) The sectional view of the absorber, and a potential scheme to electrically gate the graphene layers.

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To demonstrate the electromagnetic absorption, we utilized the FDTD solution software to simulate the absorption behaviors of our designed structure. The propagation direction of the incident electromagnetic wave is perpendicular to the surface of the structure, while the (E, H) plane is parallel to the surface of the structure, as shown in Fig. 1(a). The incident wave is generated by the port boundary conditions, and the periodic boundary conditions are employed. Since transmission coefficient S₂₁ and reflection coefficient S₁₁ can be obtained from the simulations, the absorptivity A is extracted by A = 1−T−R, where transmissivity T = |S₂₁|² and reflectivity R = |S₁₁|², respectively. Here the first subscript denotes the receiving port and the second subscript denotes the excitation port.

In our simulation, the refractive index of Al₂O₃ is based on the data in [41]. For graphene sheet, the conductivity σ of graphene is described as follows [42]:

σ_{G} = \frac{i e^{2} k_{B} T}{π ℏ^{2} (ω + \frac{i}{τ})} (\frac{μ}{k_{B} T} + 2 \ln (e^{- \frac{E_{f}}{k_{B} T}} + 1)) + \frac{i e^{2}}{4 π ℏ} \ln | \frac{2 μ - ℏ (ω + \frac{i}{τ})}{2 μ + ℏ (ω + \frac{i}{τ})} |

Where k_B is the Boltzmann constant, ω is the radian frequency, T is the temperature, τ is the phonon relaxation time, E_f is the Fermi energy of graphene. In our work, we set the parameters as: T = 300K, τ = 5 × 10⁻¹³s. As it is presented in formula (1), the conductivity σ_G is mainly dependent on the radian frequency ω and Fermi energy E_f. The Fermi energy E_f can be electrically controlled by adding a gated voltage. In order to realize the Fermi energy control, each layer of graphene is applied with a separate gated voltage, as shown in Fig. 1(b). A thin capacitor layer made by 40nm thick Al₂O₃ is placed under the graphene layers, acting as the electric gate. The lower gratings and Au layer act as an electrode. The relationship between Fermi level and gated voltage can be calculated from [43,44]:

\frac{C_{g c}}{e} (V_{G} - V_{0}) = n_{c} = \frac{2 sgn (E_{f})}{π ℏ^{2} v_{F}^{2}} \int_{0}^{\infty} ε [f_{d} (ε - E_{f}) - f_{d} (ε + E_{f})] d ε

Where V_G is the gate voltage, and n_c stands for the carrier density and is variable. f_d is the Fermi-Dirac distribution. C_gc is the geometrical capacitance, which represents the ideal capacitance of Al₂O₃. V₀ is the gate voltage when Fermi level just reaches the Dirac point. For the pristine graphene, V₀ equals to zero. However, taking into account that V₀ varies much for different graphene samples, in this work we replace the gate voltage by Fermi level in the following sections to maintain generality.

In the THz and far-infrared range, the intraband transition contribution dominates and thus the surface conductivity in formula (1) simplifies to $σ_{G} = \frac{i e^{2} E_{f}}{π ℏ^{2} (ω + i τ^{- 1})}$ with $E_{f} ≫ k_{B} T$ , where e is the electron charge and k_B is Boltzmann constant. In our study, the wavelength range is set from 6 μm to 9.5 μm in the air and the chemical potential of graphene is assumed to be E_f = 0.3 eV.

Under the assumption that the electronic band structure of graphene sheet cannot be affected by the neighboring graphene sheets, graphene’s effective permittivity 𝜀_G can be calculated as follows:

ε_{G} = 1 + \frac{i σ_{G}}{ε_{0} ω d_{G}}

We use the Drude model to set the conductivity of Au in our simulation, which is expressed as

ε_{A u} = 1 - \frac{ω_{p}^{2}}{(ω^{2} + i γ ω)}

, where ω_P = 1.37 × 10¹⁶ Hz, and γ = 4.07 × 10¹³ Hz [46].

3. Results and discussions

The absorption spectra of the designed absorber is firstly investigated and shown in Fig. 1. The simulated result is exhibited with a red line in Fig. 2. It can be seen that the absorber based on the DGC-GDM has two absorption peaks at 7.2 μm and 8.6 μm respectively, and the corresponding absorptivity is over 95%. To investigate the mechanism of the absorption, the structure only having the top grating and the structure only having the bottom grating are simulated separately, with the results shown in Fig. 2 by grey solid lines and grey dotted lines, respectively. It can be seen that only one absorption peak emerges in the structure with only one grating, and the positions of the peaks correspond to the two peak positions of our designed structures respectively. In the following paragraphs and pictures, the two peaks are called λ₁ and λ₂ as shown in Fig. 2. The absorption mechanism is that the top and bottom gratings couple the incident light into the GDM in the form of surface plasmon polaritons (SPPs), whose resonant wavelength mainly relates to the parameters of grating. Thus two kinds of SPPs in the GDM layer can be excited by the upper and lower Au gratings, respectively, which confine the incident waves into the GDM and also make them lost in the GDM layer. This can explain why this structure consists of two different absorption peaks, and it also explains why the peak of the structure with only one grating can correspond to that of our designed structure.

Fig. 2 The absorption spectra of our DGC-GDM structure (red line), the absorption spectra of the structure with only upper grating (grey solid line) and the absorption spectra of the structure with only lower gratings (grey dotted line).

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To further explain this result, we calculated the electric field distribution and the Poynting vector distribution at 7.2 μm and 8.6 μm respectively, with the results shown in Fig. 3. It is revealed that the upper grating is contributed to the excitation of surface plasmon polaritons at 7.2 μm, while the lower grating is responsible for exciting plasmon at 8.6 μm. From Fig. 3(a) and Fig. 3(b), it can be seen that the Poynting vector distribution is consistent with the electric field distribution. Therefore, for clarity, we employed a red arrow to denote the main direction of the Poynting vector in the electric field distribution of Figs. 3(c) and 3(d), respectively. And it is found that there is an angle between the direction of Poynting vector (in GDM) and the horizontal direction.

Fig. 3 (a) The Poynting vector distribution at 7.2 μm. (b) The Poynting vector distribution at 8.6 μm. (c) The electric field distribution at 7.2 μm. (d) The electric field distribution at 8.6 μm. And θ is an angle between the direction of propagation of electromagnetic waves (in GDM) and the horizontal direction.

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We then discussed the physical mechanism of the angle between the direction of electromagnetic propagation in the GDM and the horizontal direction. Firstly, we consider the dispersion relationship of the SPPs in GDM. A diagram of GDM is presented in Fig. 4. The graphene sheets with a period of d_m are embedded in the dielectric medium with a relative permittivity denoted by ε_m. The surface conductivity of graphene is denoted by σ_G. The magnetic field of the TM polarized SPPs can be written as:

H_{z} = {\begin{cases} A^{+} e^{- κ (y + d_{m})} + A^{-} e^{κ y} - d_{m} < y < 0 \\ B^{+} e^{- κ y} + A^{-} e^{κ (y - d_{m})} 0 < y < d_{m} \end{cases}

Where

κ = \sqrt{k_{x}^{2} - ε_{m} k_{0}^{2}}

with k_x the wave vector of SPPs in the x direction and

k_{0} = \frac{2 π}{λ}

being the wave vector in air. A ^± and B ^± represent the amplitudes of SPPs modes damping toward ± y directions between adjacent graphene sheets. According to the Bloch theorem, we have:

B^{±} = A^{±} e (i k_{y} d_{m})

Where k_y is the Bloch wave vector along the y direction. According to Maxwell’s equations, the tangential electric field:

E_{y} = \frac{i η_{0}}{k_{0} ε_{m}} \frac{\partial H_{z}}{\partial x}

Where η₀ is the impedance of air. Considering the boundary conditions at y = 0:

E_{y}^{+} = E_{y}^{-}

H_{z}^{+} - H_{z}^{-} = σ_{G} E_{y}

Where E_y^± and H_z^± are the fields at the two side of graphene. From Eqs. (4)-(8) we can get the dispersion relation [36,45]:

\cos (φ) = \cosh (κ d_{m}) - \frac{κ ξ}{2} \sin h (κ d_{m})

Where

ξ = \frac{η_{0} σ_{G}}{(i ε_{m} k_{0})}

and

φ = k_{y} d_{m}

. In our simulation d_m is small, so that φ and κd_m are close to zero. So one can obtain:

\cos (φ) \approx 1 - \frac{φ^{2}}{2}

\cos h (κ d_{m}) \approx 1 + \frac{{(κ d_{m})}^{2}}{2}

\sin h (κ d_{m}) \approx κ d_{m}

The formula (9) can be rewritten as:

k_{x}^{2} - \frac{φ^{2}}{d_{m} (ξ - d_{m})} = ε_{m} k_{0}^{2}

The direction of the Poynting flux is determined by the group velocity and is given by:

θ = - arc \tan (\frac{d_{m} \partial k_{x}}{\partial φ}) |_{φ \to 0}

According to Eqs. (13) and (14), one can get:

Fig. 4 Schematic of GDM and diffraction relation of SPPs.

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θ = arc \tan (\sqrt{\frac{d_{m}}{ξ - d_{m}}}) (d_{m} < ξ)

On the other hand, in our structure, since the SPPs are excited by the grating structure, k_x in the formula (14) satisfies the equation: $k_{x} = m \cdot \frac{2 π}{P}$ , and k_y is related to d_m, so the angle θ in the formula (14) should be also able to express with P and t_GDM (t_GDM = nd_m, n is the number of layers of the dielectric). Then we can get a conclusion, in our designed absorber, when the absorptivity of the absorber reaches a peak, there is a functional relationship between the angle θ and the geometric parameters of the gratings in the structure. With the optimal design, we get the relationship from Figs. 3(c) and 3(d):

θ_{1} = arc \tan (\frac{2 t_{GDM}}{w_{1}})

θ_{2} = arc \tan (\frac{2 t_{GDM}}{w_{2}})

Meanwhile, the angle θ is always satisfied the Eq. (15). Therefore, Compare Eqs. (16) and (17) with Eq. (15), so ξ can be calculated, which means that ξ can be calculated by the geometric parameters of the structure. Therefore, we obtain the relationship between ξ and the parameters of the structure at the two peak positions of the absorption spectra.

\sqrt{\frac{d_{m}}{ξ_{1} - d_{m}}} = \frac{2 t_{G D M}}{w_{1}}

\sqrt{\frac{d_{m}}{ξ_{2} - d_{m}}} = \frac{2 t_{G D M}}{w_{2}}

ξ₁ and ξ₂ are the values of ξ at λ₁ and λ₂, and they can be calculated by Eqs. (18) and (19) with the known geometric parameters of the structure, in our structure, the duty cycle of both the upper and lower gratings is 0.5, so from Eqs. (18) and (19) we can get:

ξ_{1} = (\frac{P^{2}}{16 t_{G D M}^{2}} + 1) \cdot d_{m}

ξ_{2} = (\frac{P^{2}}{64 t_{G D M}^{2}} + 1) \cdot d_{m}

Through

ξ = \frac{η_{0} σ_{G}}{(i ε_{m} k_{0})}

, one can obtain:

λ_{P e a k} = 2 π \sqrt{\frac{π ξ ε_{m} ℏ^{2} c}{η_{0} e^{2} E_{f}}}

Bring Eqs. (20) and (21) to (22), we can get:

λ_{1} = 2 π \sqrt{\frac{π (P^{2} + 16 t_{G D M}^{2}) d_{m} ε_{m} ℏ^{2} c}{16 t_{G D M}^{2} η_{0} e^{2} E_{f}}}

λ_{2} = 2 π \sqrt{\frac{π (P^{2} + 64 t_{G D M}^{2}) d_{m} ε_{m} ℏ^{2} c}{64 t_{G D M}^{2} η_{0} e^{2} E_{f}}}

Therefore, the position of the absorption peak can be calculated using Eq. (23) and (24). Additionally, it can be seen that the wavelength is related to geometric parameters of the structure and E_f , which means that the resonance wavelengths of λ₁ and λ₂ are determined by the geometric parameters of the structure and the Fermi level of graphene. In the following chapter, the accuracy of the deduced formula is verified by comparing the simulation results with the theoretically calculated ones.

Firstly, we verify the relationship between the wavelength of absorption peak and the Fermi level of graphene. We set the geometric parameters to be the fixed values, then simulate the absorption spectra with different Fermi levels. The simulation results are presented in Fig. 5(a). It can be observed that both λ₁ and λ₂ have an obvious blue shift when E_f increases slightly. Moreover, lager E_f makes the absorptivity at the peak wavelength λ₁ and λ₂ get closer to 1, respectively. In short, our proposed perfect absorber can be tunable by changing the Fermi level of graphene which can be controlled by the external gating voltage. And a little change like tens of meV of E_f is enough to make the absorption peak have an obvious shift, which means that only a small gating voltage is needed to tune the absorption peak. On the other hand, the comparison between the simulation results and calculation results are given in Fig. 5(b). In Fig. 5(b), the calculation result is shown with the blue line, and the simulation result is presented with the red scatter. It can be seen the simulation results agree very well with the calculation ones, demonstrating the accuracy of our deduced formulas (23) and (24). The electric field distributions at λ₁ and λ₂ with E_f = 280meV are shown in Figs. 5(c) and 5(d), respectively, which is employed to illustrate that the relationship between angle θ and the geometric parameters still works under different Fermi level.

Fig. 5 (a) The relationship between absorption peak and the Fermi level. (b) The comparison of simulated results and calculated results. (c) Electric field distribution at λ₁ when E_f = 280meV. (d) electric field distribution at λ₂ when E_f = 280meV.

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Next, we demonstrate the relationship between the wavelength of the absorption peak and the number of graphene/Al₂O₃ bilayers. At this time, values of Fermi level of graphene and parameters of the gratings are fixed. The simulation results are shown in Figs. 6(a) and 6(b). From Fig. 6(a) we can see that when the number of graphene/Al₂O₃ bilayers increases, both peak wavelengthsλ₁ and λ₂ shift to the shorter wavelengths significantly, and the absorption at λ₂ becomes stronger, while absorption at λ₁ varies slightly. In Fig. 6(b) we can see that when thickness of GDM t_GDM increases, λ₁ and λ₂ also have a blue shift calculated with the deduced formulas (23) and (24) (t_GDM replacing the number of graphene/Al₂O₃ bilayers). The comparison between simulation results and calculation results are shown in Fig. 6(b), denoted by the red scatters and blue lines, respectively. Also the simulation results accord very well with the calculation results. Electric field distributions at λ₁ and λ₂ when the number of graphene/Al₂O₃ bilayers set to 7 are shown in Figs. 6(c) and 6(d), respectively. It can be seen that the relationship between angle θ and the geometric parameters still work under different number of bilayer or thickness of GDM.

Fig. 6 (a) The relationship between absorption peak and the number of graphene/Al₂O₃ bilayers. (b) The comparison of simulated results and calculated results. (c) Electric field distribution at λ₁ when number of bilayers is 9. (d) Electric field distribution at λ₂ when number of bilayers is 9.

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Finally, we verify the relationship between the wavelength of absorption peak and the value of period P. During this time, the duty cycle of both the upper and lower gratings is fixed to 0.5, hence the width of upper grating w₁ and width of lower grating w₂ will vary with the period P. The Fermi level is also kept as a fixed value 300meV. The simulation results are shown in Figs. 7(a) and 7(b). In Fig. 7(a) we can see that when P increases, both λ₁ and λ₂ have a notable red shift, and the absorptivity at λ₁ gets closer to 1, while the absorptivity at λ₂ changes slightly. The comparison between simulation results and calculation results is shown in Fig. 7(b). In Fig. 7(b), the calculation results are denoted by the blue line, and the simulation results are denoted by red scatter. Also, they agree very well with each other. The electric field distributions at λ₁ and λ₂ when P = 900nm are shown in Figs. 7(c) and 7(d), respectively, which demonstrating the relationship between angle θ and the geometric parameters still works under different period of the structure. From the above results, it can be concluded that the empirical formulas (23) and (24) for our DGC-GDM structure is credible. It offers great convenience for us to select the exact values of parameters of the structure for an absorption peak as required.

Fig. 7 (a) The relationship between absorption peak and the period of structure P. (b) The comparison of simulated results and calculated results. (c) Electric field distribution at λ₁ when P = 900nm. (d) Electric field distribution at λ₂ when P = 900nm.

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We also study the dependence of the incident angle on the absorption property. Figure 8 shows the absorption spectra when the incident angle varies from 0° to 80°. It can be seen that the dual-band absorption maintain well until the incident angle increases to 60°. Therefore, our proposed tunable dual-band absorber works well in wide incident angles. Additionally, the DGC-GDM structure is symmetrical, so it is insensitive to the polarization of the incident light.

Fig. 8 The absorption spectra as a function of incident angle.

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4. Conclusions

In conclusion, we have designed a dual-band, tunable, polarization independent, ultrathin, mid-infrared metamaterial absorber based on the dual gratings-coupled graphene-dielectric multilayer structures (DGC-GDM), in which GDM is sandwiched between two Au gratings. The dual-band absorption mechanism is that the upper and lower gratings couple the incident light into the GDM in the form of surface plasmon polaritons, respectively. We also have deduced the theoretical formula to describe the relationships between the parameters of the absorber and wavelength of the absorption peak. The simulation results and the calculated results agree very well, which demonstrates the accuracy of the formula. Therefore, we can control the position of the two absorption peaks through the structural design, also can tune the position of the peak by changing the Fermi level of graphene. The proposed structure can be applied to absorbers working at other frequencies.

Funding

National Natural Science Foundation of China (Grant Nos. 11704373 and 51627901); the National Key Research and Development Program of China (Grant No. 2016YFA0401004).

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Dual band and tunable perfect absorber based on dual gratings-coupled graphene-dielectric multilayer structures

Abstract

1. Introduction

2. Structure design and simulation method

3. Results and discussions

4. Conclusions

Funding

References

Cited By

Figures (8)

Equations (24)

Optics Express