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Abstract
In this paper, we theoretically propose an effective broadband absorption architecture in mid-infrared region based on strong coupling between the plasmonic resonance of graphene nanoribbons and the waveguide mode of a metal tapered groove. The special architecture facilitates two new hybrid modes splitting with very strong energy distribution on graphene ribbon, which results in the broadband absorption effect. To well explain these numerical results, an analytical dispersion relation of waveguide mode is obtained based on the classical LC circuit model. The fluctuating range of absorption passband is investigated by adjusting the filled medium inside of the grooves. Leveraging the concept and method, a broadband flat-top (bandwidth ≈2.5 µm) absorption with absorption rate over 60% is demonstrated. Such a design not only enhances the intrinsic weak plasmons resonance in mid-infrared spectral region, but also reduces the absorption fluctuations caused by coupling, which are the key features for developing next-generation mid-infrared broadband optical devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light absorption is one of the intrinsic properties that exists in various materials, and it plays a pivotal role in the applications of photodetectors, modulators, photovoltaics and thermal emitters [1–3]. For conventional materials, the low absorption and narrow parameter spaces hamper the development of miniaturized mid-infrared devices. In contrast, graphene, as a monatomic layer of thin film materials, has been reported to possess ultra-high carrier mobility, excellent tunability and nonlinear effects, which can be used to fabricate various functional devices for optical and electronic applications [4,5]. More important, interactions between mid-infrared photons and massless electrons in graphene produce the collective oscillations of charge carriers, forming graphene surface plasmons (GSPs). Compared to conventional metal plasmons, the excited GSPs mode exhibits ultra-high field confinement, low power dissipation, as well as excellent tunability, which is very beneficial in developing nanoscale mid-infrared optical device [6–12]. In spite of these advantages, applications of graphene have severely limited due to its low mid-infrared absorption in graphene (absorption close to zero). While a diverse range of ways including plasmons resonance enhancing, critical coupling theory, and interference [13–16] have been suggested to improve the absorption efficiency of graphene, nearly all efforts are aimed to raise the narrow-band absorption efficiency of graphene, leaving the potential of the broadband research unexplored. Recently, wide-band absorbers based on graphene have been reported [17,18], but most of them focus on enhancing absorption of the whole configuration, with the absorption in monolayer graphene unknown. Utilizing the multi-resonators structure is an alternative method to broaden the absorption bandwidth of graphene [19–22]. But the plasmonic response of graphene nanostructures in mid-infrared are usually weak (only produce 5%-7% absorption efficiency) [22], and the mode splitting due to the coupling among multiple localized resonant modes will cause the unavoidable absorption fluctuations. Therefore, a broadband, non-fluctuated and high intensity mid-infrared absorption strategy is extremely desirable for future researches and applications.
In this work, we achieve the mid-infrared broadband light absorption in graphene nanoribbon by constructing a strong coupling system [23–25], which is composed of a tapered metal groove (cavity) covered with graphene nanoribbon. Different from the previous reported, the splitting hybrid modes in our proposed microstructure possess very strong energy distribution on graphene ribbon, which lead to the enhanced broadband absorption effect. Through a classical LC circuit theory, the analysis dispersion relation of waveguide mode favored in the groove is obtained, and the possible dissipation mechanism of graphene plasmons resonance in the groove is well explained by the reason of the strongly edge loss contacted with metal. Besides, we show that the absorption fluctuations caused by coupling can be eliminated by searching a suitable filled medium inside of the grooves. The giant non-fluctuated absorption bandwidth (≈2.5 μm) design with a high average absorption efficiency (near to 60%) in the graphene nanoribbon provides a promising platform for designing next-generation broadband integrated optical device.
2. Model design and simulation
Three-dimensional (3D) schematic diagram of designed microstructure is shown in Fig. 1, with detailed geometrical parameters illustrated in the inset. The graphene nanoribbons array with width d = 67 nm is placed on one side of the tapered metal groove array filled with silica. And the tapered groove is made by metal silver with bottom width w1 = 700 nm, top width w2 = 300 nm, height h0 = 1.4 μm and period p = 4 μm, respectively. The whole structure length is assumed infinitely in z direction and the ground plane is thick enough to eliminate the transmission. A transverse magnetic (TM) polarized wave (whose magnetic field is perpendicular to the x-y plane) normally illuminates on the sample plane, the optical properties are investigated using the Finite Element Method (FEM).
Fig. 1 3D schematic of the graphene-based tapered metal groove structure. The inset shows the cross section of the proposed structures.
In simulation, the periodic boundary condition and ports are used. The optical constants of silver are modeled by the Drude model with a plasma frequency of 1.374 × 1016 rad/s and collision frequency of 2.02 × 1014 rad/s [26]. For the anisotropic graphene, its out-of-plane permittivity is set to 2.5, and the in-plane conductivity can be obtained from random-phase approximation [8], including the effect of finite temperature (T = 300 K):
Here σg and kB are the conductivity of graphene and the Boltzmann’s constant. Ef and ɷ are the Fermi energy level and radian frequency, respectively. The carrier relaxation time τ = μEf/evf2, where the Fermi velocity vf = 106 m/s and carrier mobility μ = 10000 cm2/ (V·s). The effective permittivity of graphene ɛg can be described by means of the following expression [27]:where ɛ0 and tg = 0.33 nm represent the vacuum permittivity and thickness of graphene. In addition, the permittivity of silica is set as 2.25.3. Theory analysis and result discussions
The oscillating magnetic field perpendicular to x-y plane creates a closed current loop around the trench, and the corresponding resonant behavior excited in tapered metal groove is firstly investigated in Fig. 2. An obvious absorption peak occurs at the wavelength of 11.5 µm plotted in Fig. 2(b) is associated with a fundamental waveguide mode, which is a typical existence in waveguide structure. The Ey electric field distribution of peak A in inset indicates the stronger electric fields locate on the edge of groove, identifying the extreme coupling on the narrow gap of top groove. As a result of the coupling, the confined vectorial field inside groove is mainly along the paralleled direction, which guarantees the strong interaction with graphene [28]. Figures 2(c) and 2(d) show the absorption contours with respect to the structure parameter w2 and h0, respectively. By observing these contours, it can be seen that the resonant wavelength depends on the parameter of top groove width (w2) and height (h0), while the resonant intensity is almost unchanged with the groove height. Both of these further implying the confined fields inside of the groove are mostly concentrated on the top of trench caused by the extreme coupling between two narrow walls.
Fig. 2 (a) The schematic of tapered metal groove, with the LC circuit model in inset. (b) The calculated absorption spectrum of the proposed tapered metal groove structure with Ey field distribution and vectorial field distribution in inset. (c) and (d) show the absorption contours with respect to wavelength by changing the top groove width and height, respectively. The red solid lines on these diagrams are the fitting dispersion curve through LC circuit model.
Since such a waveguide mode supported on groove is actually a magnetic resonance, the dispersion relation can be described using the equivalent circuit method seen inset in Fig. 2(a) [29–32]. We regard the tapered groove as the composition of infinitesimally thick parallel-gratings with different intervals, the mutual inductance Lm can be easily deduced by integrating the variables dLp = µ0·w·dh/l along the side-wall of groove according to the parallel-plate inductance formula [31,32] (here, l is the side-length along the z-direction, µ0 is the vacuum permeability, w is the interval between two walls at arbitrarily height h):
Where l is set to unity for the periodic groove structures, the interval can be expressed as a function of h. w = w1-2·h·tanθ, θ is the angle between the vertical direction and side wall. Thanks to the kinetic energy of mobile charge carriers, the kinetic inductance Lk should be introduced and it can be expressed as:Where b is the wall length of tapered groove, ε′ and ε″ are the real and imaginary parts of the metal silver, δ and ω are the penetration depth of electric field and radian frequency, respectively. Because the metal in calculated wavelength range is approximate to a perfect conductor, we choose δ = 0.8 nm in following fitted process and the total inductance L = Lm + Lk in this system. The dielectric trench between two walls is considered as the capacitance C. For the capacitance, it has similarly inferred process combined with parallel-plate capacitance formula dC = α·ε0·dh /w [31,32] (α = 0.4 is a correction coefficient due to the inhomogeneous potential distribution and it is determined by analyzing the obtained numerical results):Here, it should to be noted when the bottom width of groove is equal to the top width, the above expression can be reduced to the parallel-plate capacitance formula. The resonant wavelength λ is given by [13,31,32]:Where c0 is the light speed. Solving the Eq. (6), the fitting dispersion curves based on the LC model (red solid line) are acquired in Figs. 2(c) and 2(d). There is good agreement between theory values and simulation results, confirming the correctness of the theoretical formula.Next, we add the graphene nanoribbon on top of the tapered metal groove with groove height h0 = 2.1 µm. Noted that the waveguide mode with very strong electromagnetic field localized on the grooved edge, thus it is reasonable to place the ribbon adjacent to one edge of the groove for enhancing the light-graphene interaction. The near-field characteristic of excited modes and the absorption spectra for various ribbon widths at Fermi energy of 0.4 eV are respectively investigated in Fig. 3. When the ribbon width d is less than the groove width w2 [Fig. 3(a)], there are two apparent absorption peaks B and C at wavelengths 9.5 µm and 16.6 µm. Through checking the field distribution zones in Fig. 3(c), we can infer that peak B is resulted from a GSPs mode, while peak C is a waveguide mode, which is in accordance with prediction in Eq. (6). For the excited GSPs mode and waveguide mode, both absorption intensity and resonant wavelength are increase with the ribbon widths. The red-shift of GSPs mode can be analyzed by the Fabry-Pérot model [8,33,34], while the frequency-shift of peak C may be attributed to the unavoidable far-field coupling between GSPs mode and waveguide mode. The dominant enhancing resonance mechanism of peak B involves the edge scattering on graphene ribbon, which is a more stronger resonance with a wider graphene ribbon [35,36]. However, when the ribbon width w2 = 300 nm, there is a very weak absorption peak D on the left side of peak E shown in Fig. 3(b). Through analyzing the electric field distribution, we can distinguish the peak D as a GSPs mode and the peak E as the waveguide mode. The extremely weak resonance means a huge dissipation in graphene ribbons and the obviously changing of the resonant wavelength and strength indicated the important influence of boundary condition on ribbon.
Fig. 3 (a) The calculated absorption spectra of the proposed tapered metal groove with different ribbon width at Fermi energy of 0.4 eV. (b) The absorption spectrum when ribbon width d = 300 nm. (c) The corresponding Ey electric field distribution of peaks (B, C, D, E).
By comparing the electric field distributions of peak B with D in Fig. 3(c), we will get the possible loss mechanism of graphene plasmons resonance in the groove. Here, we find the mode B exhibits a very strong field near the edge of ribbon that adjacent to dielectric, while the excited field on the edge that contacted with metal is negligible. This can be attributed to the high conductivity of metal Ag result in the disappearance of tangential electric field. Similarly, for mode D, the existed Ey field is relatively weak and it mainly confined on the center of ribbon with no field located near the ribbon edges because the both edges are contacted with metal. All of the above results implying the huge intrinsic loss in metal causes the more higher edge dissipation of graphene nanoribbon, which further dominate the quality factor and damping of graphene plasmons resonance in the groove. Besides, we also study the case that graphene ribbon on the central of metal groove, in order to better present the excited GSPs mode in ribbon, the absorption spectra of graphene nanoribbon laying on the silica is given in the Appendix of Fig. 6(a), and the strong GSPs resonance without contact with metal in the cavity will lead to a deep absorption dip, which is hard to eliminate in the broadband design [See Figs. 6(b) and 6(c) in the Appendix].
According to the above analysis, if we choose the single-edge contacted graphene nanoribbon inside of groove with proper structure and parameters, the strongly near-field interaction will lead to the GSPs mode resonant with waveguide mode in a strong coupling regime. The modification of spatial field distribution after mode splitting may provide possibility to broaden the absorption bandwidth of graphene. Figure 4(a) shows the calculated graphene absorption spectrum of the tapered groove structure, as is expected, a wide bandwidth ≈2.7 µm with a high average absorption ≈78% is achieved in this graphene nanoribbon (here, the absorption bandwidth defined as the bandwidth of 68% absorption that is dip magnitude). To demonstrate the splitting spectrum is induced by the strong coupling, we calculate the absorption contour with groove height in Fig. 4(b), the anti-crossing phenomenon in the figure can be analyzed by a coupled harmonic oscillator model [37,38]:
Where ωGSPs and ωgroove are the resonant frequencies of graphene plasmons and waveguide mode, ∆ωR = 2π(c/λF-c/λG) stands for the separation strength, λF and λG are the resonant wavelength of mode F and G, respectively. Combined the dispersion relation of waveguide mode given by Eq. (6) with the GSPs resonance fixed at 11.2 µm, the positions of splitting peaks for different groove height can be predicted by Eq. (7). Clearly, a good agreement between electromagnetic theory using the FEM method and coupled harmonic oscillator model (green dots) is achieved in Fig. 4(b), which well verifying the strong coupling effect in our designed structure. Then, in order to explain the origin of the broadband absorption effect, we calculate the electric field distribution of the splitting modes (F and G) in inset 4a. It is obvious that mode F has a similar electric field distribution as mode B excepting a stronger field located on the left edge of groove, while mode G possesses a uniform field distribution on both edge of groove and ribbon. The spatial field modulation of strong coupling makes the splitting modes with majority energy distribution on graphene, which guarantee the strong broadband absorption effect. In the design structure, the absorption fluctuations caused by coupling can be adjusted by changing the filled medium inside of groove. As shown in Fig. 5, the absorption of graphene nanoribbon with a refractive n = 3 filled inside of groove exhibits a flat-top absorption spectrum in a broad wavelength range. This result may be interpreted by the point of initial coupling by tailoring the dissipation of groove of filled medium. Table 1 gives the detail variations of absorption fluctuation and bandwidth with the filled medium, we can see that when the refractive index of filled medium n = 3.4, it can reach an initial coupling point, where the waveguide mode and GSPs mode just starting coupling. Besides, the absorption strength and bandwidth of graphene nanoribbon in the whole process keep in good performance, which is benefit to various broadband graphene device design.Fig. 4 (a) The calculated graphene absorption spectrum of single-edge contacted nanoribbon with the same geometry parameters in section 2. The filled medium inside of groove is silica and the Fermi energy of graphene is 0.4 eV. Besides, insets show the field distribution of peak F and G at wavelength 10.8 µm and 11µm, respectively. (b) The absorption contour of proposed structure with the change of groove height. Here, the red line represents the dispersion curve of tapered groove, red dots are the GSPs resonance at the ribbon width d = 67 nm, and green dots are the fitted results of harmonic oscillator model.
Fig. 5 The flat-top absorption spectrum of graphene nanoribbon with the refractive index n = 3 medium filled in groove, the Fermi energy is 0.4 eV.
Table 1. Performance comparison of different filled medium
4. Conclusions
In summary, we investigate the broadband light absorption in graphene nanoribbon by constructing a metal tapered groove filled with dielectric. A 2.5 µm bandwidth with absorption strength over 60% broadband flat-top absorption is demonstrated in monolayer graphene. The calculated electric distributions indicate the broadband absorption effect comes from the strong coupling due to the splitting of two hybrid modes with very strong energy distribution on graphene. Utilizing the classical LC circuit theory, these numerical results can be well explained. Besides, we investigate the corresponding loss mechanism of graphene plasmons resonance in the groove, and find by varying the filled medium in groove, the problem of absorption fluctuations induced by coupling can be also solved. With the development of modern nanofabrication technology, it is possible to fabricate the proposed sample structure with various techniques. For example: silicon film can be formed on the Ag wafer by Plasma Enhanced Chemical Vapor Deposition (PECVD) method at first. An E-beam lithography process with photoresist can be carried out to make the pattern, which is then transferred to the silicon layer with an inductively coupled plasma (ICP) etching process. Depending on this Si patterned substrate, the Ag tapered groove structure can be deposited by magnetron sputtering process. The surplus silver layer can be removed by etching or polishing. Finally, we can directly transfer the chemical vapor deposition (CVD) grown graphene sheet onto the groove substrate, the nanoribbon can be fabricated by electron beam lithography (EBL) method. Our work takes full use of the advantages of strong coupling, which may provide a promise route for designing next-generation mid-infrared broadband optical device.
Appendix
Here, we give the absorption spectra of graphene nanoribbon in groove without contact with the metal edge, as shown in Figure 6.
Fig. 6 (a) The calculated absorption spectrum of the graphene nanoribbon laying on the silica substrate. The graphene is deposited on the silica substrate with a GSPs mode excited in inset. (b) The calculated absorption spectra of the tapered metal grooves with central-placed graphene ribbons. The case that groove height h0 = 2.1 µm, the excited GSPs mode almost completely decouple with the waveguide mode, and obviously, there is a higher quality factor of GSPs mode with the change of ribbon widths. (c) The calculated absorption spectrum of the tapered metal grooves with central-placed graphene ribbons. The case that groove height h0 = 1.4 µm. Here, the strong coupling is excited and as it expected, a deep absorption fluctuation is produced in this system, which is hard to eliminate it due to the huge refractive material that do not exist in nature.
Funding
State Key Program of National Natural Science Foundation of China (61535012); Natural Science Foundation of Jiangsu Province Grant (number BK. 20161429), and National Natural Science Foundation of China (61601118).
Acknowledgments
The authors thank Prof. Shan Wu (Fuyang Normal University, China) for beneficial discussion.
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