Position-guided Fano resonance and amended GaussAmp model for the control of slow light in hybrid graphene&#x2013;silicon metamaterials

Maosheng Yang; Maosheng Yang; Xiaoxian Song; Xiaoxian Song; Xiaoxian Song; Haiting Zhang; Haiting Zhang; Haiting Zhang; Yunxia Ye; Yunxia Ye; Yunpeng Ren; Yunpeng Ren; Xudong Ren; Xudong Ren; Jianquan Yao; Jianquan Yao; Jianquan Yao

doi:10.1364/OE.388298

1. Introduction

Metamaterials (MMs) comprise artificial subwavelength geometric arrays and allow electromagnetic resonance to be tailored in light–matter interactions [1–6]. They offer the prospect of exciting exotic electromagnetic responses that are not easily excited in traditional media [2,7–10]. Fano resonance is one of the most bizarre electromagnetic responses in MMs, which is generated by the asymmetry of a structure with respect to the axis of incident field polarization [11]. Fano resonance originates from the interference of broad and narrow excitation modes, respectively corresponding to bright and dark modes [2,12–14]. Owing to strong enhancement of the local field, Fano resonances with an asymmetric line shape have far-reaching application in slow-light devices, ultrasensitive label-free biosensors, and nonlinear optics [15–17]. Conventional Fano resonance in metallic and all-dielectric MMs has been widely studied in recent years. In general, owing to high radiation losses, the use of metal MMs inevitably leads to an increase in light absorption, which hinders many practical device applications based on Fano resonance effects [18]. Although all-dielectric Fano-resonance MMs have high-quality factors and largely avoid radiation losses, it is not easy to meet the requirements for the dynamic control of metamaterial resonances, which is a challenge in developing real-time control photonics devices [7].

Graphene has recently been considered a desirable candidate for application as a light–matter response material and for tuning the resonance response [19,20]. Compared with traditional materials, graphene has many unique properties [21]. As an example, the electric conductivity of graphene can be flexibly tuned in a full frequency region via external electrostatic biasing [22,23]. Furthermore, graphene is more suitable in engineering a miniaturized photonic device because it is a two-dimensional material, comprising one monolayer of carbon atoms [19,22–25]. These unique properties mean that graphene-based photonic devices are dynamically tunable in a broad frequency range [19,20,26]. The use of graphene as a photoactive material for the active tuning of resonance effects has been widely demonstrated. Promisingly, graphene offers prospects for the manipulation of resonance effects and has been shown to have a novel photon response. In particular, hybrid graphene–dielectric MMs have superior optical properties for the application of the dynamic modulation of photonic devices. Meanwhile, establishing a suitable mathematical model is essential for the dynamic modulation of the Fano resonance effect in practical applications of photonic devices. Mathematical modeling is considered a core technology in the fields of communication, microelectronics, and automation control, which refers to the fact that mathematical concepts, methods, and theories are used in describing practical problems and provide accurate data or reliable guidance for control. Similarly, the development of photonic devices requires the support of mathematical models.

The present paper reports a novel observation of position-guided Fano resonance generated by hybrid graphene–silicon MMs in the terahertz (THz) region. We here propose a hybrid graphene–silicon structure for generating position-guided Fano resonance. The structure of the unit cell comprises two silicon-based rectangular parallelepiped bars on the front of a silicon substrate, and graphene ribbons are laid at asymmetric positions on the back side. In this case, the asymmetric positions of the graphene ribbons make the metal molecule asymmetric by breaking all spatial transmission symmetries in the plane of the structure [13,27–29]. Such symmetry breaking allows the hybrid graphene–silicon structure to strongly couple electric and quadrupole dipole resonances. The coupling thus gives rise to an asymmetric Fano transmission spectrum. This position-guided Fano resonance can be attributed to the position and Fermi level of the graphene ribbon. Additionally, the coupled-oscillators model is applied to illustrate that the transmission intensity at the Fano peak enhances dramatically with an increase in the Fermi level. Finally, the slow-light effect based on position-guided Fano resonance is studied. The most notable group delay is 9.73 ps at F_E = 0.6 eV, which corresponds to the group delay in the free-space propagation of 2.92 mm. Furthermore, the amending GaussAmp model is selected as a suitable tuning equation for the group delay based on the position-guided Fano resonance. Our results demonstrate that position-guided Fano resonance is promising for applications in dynamically adjustable optical switches and slow-light devices.

2. Results and discussion

The structure of hybrid graphene–silicon MMs used in this work, shown in Fig. 1, is designed for spectroscopic investigation. The front architecture of the unit cell comprises two silicon-based rectangular parallelepiped bars on a silicon substrate. The two parallel silicon-based bars are identical, and their positions on the silicon substrate are axisymmetric. On the backside, graphene ribbons are laid at asymmetric positions. The front of the hybrid graphene–silicon MMs is schematically illustrated in Fig. 1(a). The geometrical parameters are c = 20 µm and d = 20 µm. In simulations, the incident THz waves propagate along the z-direction, and electric and magnetic boundary conditions are respectively set along the x- and y-directions. The simulations are performed using commercial software, CST Microwave Studio, on the basis of the finite element method. Figure 1(b) shows a unit cell of the hybrid graphene–silicon MMs. The geometrical parameters are a = 50 µm, b = 15 µm, m = 50 µm, n = 50 µm, and p = 70 µm. The back of the hybrid graphene–silicon MMs is schematically illustrated in Figs. 1(c) and (d). The gap between the two graphene ribbons is l, whose value changes with the positions of the graphene ribbons. The distance e between a graphene ribbon and the edge of the silicon substrate also changes with the positions of the graphene ribbons. The width of each graphene ribbon is w = 15 µm. The Fermi level (F_E) of a graphene ribbon can be tuned by external electrostatic biasing on lon gel, which covers the graphene ribbons. The hybrid graphene–silicon MMs can be experimentally fabricated. First, adopting chemical vapor deposition, graphene is directly grown on the surface of a silicon substrate with a thickness of 100 µm [30]. Graphene–ribbon arrays are fabricated using standard optical lithography followed by oxygen plasma etching. An Au/Cr electrode with a thickness of 50 nm is then deposited onto the graphene ribbon in a vacuum using stencil masks [20]. Next, the graphene ribbons are covered with lon gel. Subsequently, other Au/Cr electrodes are deposited onto the surface of the lon gel, as shown in Fig. 1(c). Finally, the rectangular silicon array is fabricated adopting reactive ion etching technology based on a designed hard mask. Kubo’s formalism is adopted to describe the relationship between the Fermi level and the conductivity of graphene ribbons. The intraband carrier transitions affect the optical conductivity of graphene in the THz region [31]. The intraband and interband transitions determine the optical conductivity of the graphene ribbon. This can be explained using Kubo’s formalism expressed by [5,32–35]

(1)$${{\mathrm{\sigma}} ({\mathrm{\omega}} , \Gamma , }{{\mathrm{\mu} }_c}{, {\rm T}) = }{{{\mathrm{\sigma}} }_{\textrm{inter}}}{({\mathrm{\omega}} , \Gamma , }{{\mathrm{\mu} }_c}{, {\rm T}) + }{{{\mathrm{\sigma}} }_{\textrm{intra}}}{({\mathrm{\omega}} , \Gamma , }{{\mathrm{\mu} }_c}{, {\rm T}) }$$

(2)$${{{\mathrm{\sigma}} }_{\textrm{inter}}}{({\mathrm{\omega}} , \Gamma , }{{\mathrm{\mu} }_c}{, {\rm T}) = }\frac{{i{e^2}({{\mathrm{\omega}} + }i2{\Gamma })}}{{{{\mathrm{\pi}} }{\hbar ^2}}}\int\limits_0^\infty {\frac{{{f_d}( - \xi ) - {f_d}(\xi )}}{{{{({{\mathrm{\omega}} + }i2{\Gamma })}^2} - 4{{({\raise0.7ex\hbox{$\xi $} \!\mathord{\left/ {\vphantom {\xi h}} \right.}\!\lower0.7ex\hbox{$h$}})}^2}}}} d\xi$$

(3)$${{{\mathrm{\sigma}} }_{\textrm{intra}}}{({\mathrm{\omega}} , \Gamma , }{{\mathrm{\mu} }_c}{, {\rm T}) = }\frac{{i{e^2}}}{{{{\mathrm{\pi}} }{\hbar ^2}({{\mathrm{\omega}} + }i2{\Gamma })}}\int\limits_0^\infty {\xi (\frac{{{\mathrm{\partial}} {f_d}(\xi )}}{{{\mathrm{\partial}} \xi }} - \frac{{{\mathrm{\partial}} {f_d}( - \xi )}}{{{\mathrm{\partial}} \xi }})} d\xi$$

(4)$${f_d}(\xi ) = \frac{1}{{\exp ({{(\xi - {\mu _c})} \mathord{\left/ {\vphantom {{(\xi - {\mu_c})} {({k_B}}}} \right.} {({k_B}}}T)) + 1}}\textrm{ }$$

where μ_c is the chemical potential, ω is the angular frequency, Γ is the scattering rate, T is the environmental temperature, f_d(ξ) is the Fermi–Dirac distribution, e is the electron charge, ħ is the reduced Planck constant, and ξ is the photon energy. The chemical potential and carrier concentration of graphene are described by [20,36]

(5)$${\mu _c} = \hbar {v_F}{({\pi} n)^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}\textrm{ }$$

(6)$$n = {\varepsilon _0}\varepsilon {V_g}/ed\textrm{ }$$

where μ_c is the chemical potential, n is the carrier concentration, v_F is the Fermi velocity of graphene, V_g is the gate voltage, ɛ₀ and ɛ are respectively the permittivities of free space and the insulated substrate material, and d is the thickness of the insulating substrate material.

Fig. 1. Structure of the hybrid silicon–graphene MMs. (a) Schematic illustration of the front of the hybrid graphene–silicon MMs with a normally incident THz wave and geometrical parameters of c = 20 µm and d = 20 µm. (b) Unit cell of the hybrid graphene–silicon MMs. The geometrical parameters are a = 50 µm, b = 15 µm, m = 50 µm, n = 50 µm, and p = 70 µm. (c) Schematic illustration of the back of the hybrid graphene–silicon MMs. (d) Schematic illustration of the back of the hybrid silicon–graphene MMs with geometrical parameters; the values of e and l change with the positions of the graphene ribbons while w = 15 µm.

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As shown in Fig. 2(a) when the graphene ribbon with a Fermi level of 0.7 eV is located near the edge of the silicon unit cell, the transmission spectrum of the hybrid graphene–silicon structure under a normally incident THz wave has an asymmetric Fano line shape, indicating the Fano resonance mode is excited efficiently. To explore the formation mechanism of this type of Fano resonance, referred to as position-guided Fano resonance, the transmission spectra of three other different structures, namely a planar structure with the graphene ribbon located on the axis of symmetry on the silicon unit cell, a planar structure with the graphene fully covering the silicon unit cell, and a planar structure without graphene, are also investigated as shown in Figs. 2(b) to 2(d). No Fano resonance mode is excited for the three planar structures, as shown by the black lines in the figure. It is inferred that the position-guided Fano resonance is excited by the asymmetry of the positions where the graphene is located on the silicon plane. At present, various resonance excitations have been reported for flat graphene structures. In an experiment, Ju et al. achieved tunable plasmon resonance excitations in graphene micro-ribbon arrays. Furthermore, they employed finite element methods to simulate the graphene structure and well reproduced the experimental spectrum [20]. In simulations, Chen et al. realized the efficient excitations of multiple plasmonic resonance modes in a three-dimensional graphene metamaterial structure [30]. Most importantly, the efficient excitation of the Fano resonance mode has been observed in a hybrid graphene–silicon structure. As an example, Argyropoulos presented a hybrid graphene–silicon metasurface design that excites Fano resonance at near-infrared frequencies [37]. Additionally, Liu et al. showed the excitation of electromagnetically induced transparency based on a graphene–dielectric structure [38]. In theoretical work, Gumbs et al. observed undamped plasmon excitations in a region of the frequency–wave vector space based on free-standing gapped graphene [39]. The excitation of position-guided Fano resonance can be considered reasonable on the above basis.

Fig. 2. Transmission spectra for different hybrid graphene–silicon structures under normal incident THz waves.

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Following the characterization of structure, spectral properties of the different positions of graphene ribbons are studied. Figure 3 shows that the positions of graphene ribbons can be represented by the axis-to-axis separation g of the graphene ribbon and silicon substrate. The Fermi level of the graphene ribbons is fixed at 0.7 eV in all cases. The parameters are thus g = 0 µm, F_E = 0.7 eV; g = 2.5 µm, F_E = 0.7 eV; g = 5 µm, F_E = 0.7 eV; g = 7.5 µm, F_E = 0.7 eV; g = 10 µm, F_E = 0.7 eV; and g = 15 µm, F_E = 0.7 eV. Under the excitation of a normally incident THz pulse, the transmission spectrum for a hybrid graphene–silicon structure having a separation g = 0 µm and F_E = 0.7 eV is shown in Fig. 3(a). It is seen that the transmission spectrum does not have an asymmetric Fano resonance line shape within the frequency range of 0.45 to 0.85 THz. That is to say, the Fano resonance response is not excited when a graphene ribbon is laid at the asymmetric position on the silicon substrate. In contrast, when the axis-to-axis separation g increases from 2.5 to 15 µm, the transmission spectrum has a typical Fano resonance shape [Figs. 3(b) to 3(f)] because, on the front side of the unit cell, the two parallel silicon-based bars are identical and their positions on the silicon substrate are axisymmetric. Therefore, the first structure of hybrid silicon–graphene MMs is not coupled and does not generate a Fano resonance response. On the backside of the unit cell, if a graphene ribbon is laid at a symmetric position along the axis of the silicon substrate (g = 0 µm, F_E = 0.7 eV), no Fano resonance response is excited by the normally incident THz pulse [Fig. 3(a)]. However, the adoption of an asymmetric graphene ribbon position makes the metal molecule asymmetric by breaking all spatial transmission symmetries in the plane of the structure [13]. Such symmetry breaking results in Fano resonance, as shown in Figs. 3(b) to 3(f).

Fig. 3. Simulated transmission spectra for different positions of graphene ribbons on the silicon substrate. (Insets) Schematics of the axis-to-axis separation of the graphene ribbons and silicon substrate, where the Fermi level of the graphene ribbons is fixed at 0.7 eV. The graphene ribbons have the parameters a) g = 0 µm, F_E = 0.7 eV, b) g = 2.5 µm, F_E = 0.7 eV, c) g = 5 µm, F_E = 0.7 eV, d) g = 7.5 µm, F_E = 0.7 eV, e) g = 10 µm, F_E = 0.7 eV, and f) g = 15 µm, F_E = 0.7 eV.

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We next study the conductivity characteristics of the position-guided Fano resonance. Interestingly, the conductivity of graphene ribbons at the asymmetric positions on the silicon substrate is sensitive to the excitation of the Fano resonance response. Figure 4(a) presents the transmission spectrum for the hybrid graphene–silicon structure when g = 15 µm and F_E = 0 eV. It is observed that although the graphene ribbon is at the asymmetric position of the silicon substrate, while the Fermi level of the graphene ribbons is 0 eV, no Fano resonance is excited. As in Fig. 3(a), the transmission spectrum does not have an asymmetric Fano resonance line shape within the frequency range of 0.45 to 0.85 THz. In contrast, when the Fermi level of the graphene ribbons increases from 0.1 to 0.9 eV, Fano resonance is observed under the excitation of the normally incident THz pulse, as shown in Figs. 4(b) to 4(f). The conductivity of the graphene ribbons is therefore an indispensable factor in the generation of position-guided Fano resonance. Meanwhile, the observation shows that the Fano dip undergoes a blueshift with an increase in the Fermi level [Figs. 4(b) to 4(f)]. Additionally, the transmission intensity at the Fano dip declines dramatically with an increase in the Fermi level. Owing to slight Ohmic losses, the transmission intensity at the Fano peak enhances dramatically with an increase in the Fermi level [37,40]. The properties of the graphene–silicon structure Fano resonance, i.e., Q factor of the resonance is determined by the position of graphene ribbon as well as Fermi level. Smaller g and lower Fermi level causes higher Q factor due to coupling of the mode to the free space [28,29].

Fig. 4. Simulated transmission spectra for different positions of graphene ribbons on the silicon substrate. (Insets) Schematics of the axis-to-axis separation of the graphene ribbons and silicon substrate, where the Fermi level of graphene ribbons is fixed at 0.7 eV. The parameters of graphene ribbons are a) g = 0 µm, F_E = 0.7 eV, b) g = 2.5 µm, F_E = 0.7 eV, c) g = 5 µm, F_E = 0.7 eV, d) g = 7.5 µm, F_E = 0.7 eV, e) g = 10 µm, F_E = 0.7 eV, and f) g = 15 µm, F_E = 0.7 eV.

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To determine the intrinsic physical mechanism of position-guided Fano resonance, the near-field interaction of the hybrid graphene–silicon MMs, accompanied by electric and quadrupole dipole excitations, is presented in Fig. 5. In the hybrid structure, because the two parallel silicon-based bars are identical and their positions on the silicon substrate are axisymmetric, if the graphene ribbon is laid at a symmetric position along the axis of the silicon substrate, the dark mode (quadrupole dipole) is not excited and decoupled from the normally incident THz waves [13]. The asymmetric position of the graphene ribbon in the plane of the structure makes the metal molecule asymmetric. Such symmetry breaking results in excitation for the dark mode and directly couples the bright and dark modes. Fano interference then arises under the x-polarized incidence of the THz pulse excitation. The near-field distributions of the hybrid silicon–graphene MMs at the resonance frequencies of the bright mode (W⁰_(D), W¹_(D), and W²_(D)) and dark mode (W_(Q)) are respectively shown in Figs. 5(a), 5(b), 5(c), and 5d. Meanwhile, their manifestations in the transmission spectrum are depicted in Figs. 5(e), 5(f), and 5(h). It is seen that when the separation g is 15 µm and F_E is set at 0.7 eV [as shown by the inset in Fig. 5(e)], the transmission spectrum has a typical Fano resonance line shape that peaks at the frequency W⁰_(D) ≈ 0.738 THz of the dipole resonance. According to the literature [5], the quadrupole dipole (dark mode) resonance is within the frequency range Wpeak (W⁰_(D)) < W_(Q) < W_dip, which is around 0.745 THz. When the separation g is 15 µm and F_E is set at 0 eV [as shown by the inset in Fig. 5(f)], the transmission spectrum has a Lorentzian resonance line shape. Such results indicate that although the graphene ribbon is at an asymmetric position on the silicon substrate, the Fermi level of the graphene ribbon is fixed at 0 eV and Fano interference does not arise. Similarly, when the separation g is zero and F_E is set at 0.7 eV [as shown by the inset in Fig. 5(h)] (i.e., the graphene ribbon is laid at a symmetric position along the axis of the silicon substrate), the transmission spectrum has a Lorentzian resonance line shape. In summary, as shown in Fig. 5(i), both the asymmetric position of the graphene ribbon and the higher conductivity of the graphene ribbon lead to the coupling of the bright (electric dipole) and dark (quadrupole dipole) modes.

Fig. 5. Near fields of the hybrid graphene–silicon MMs for different frequencies and their manifestations in the transmission spectrum. (a)–(d) Field distributions for different modes. (e)–(h) Manifestations for different modes in the transmission spectrum. (i) Coupling of different modes.

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In further clarifying the effect of the conductivity of the graphene ribbon on position-guided Fano resonance, the coupled harmonic oscillator model is used to explain the near-field interference between bright- and dark-mode resonators. Coupled differential equations are adopted to analyze the near-field interference in the hybrid graphene–silicon MMs [41]:

(7)$$\ddot{x}_{1} + \gamma _{1}\dot{x}_{1} + {\mathrm{\omega}} _0^2x_{1} + \kappa x_{2} = E,$$

(8)$$\ddot{x}_{2} + \gamma _{2}\dot{x}_{2} + {({{{\omega}_0} + \delta } )^2}x_{2} + \kappa x_{1} = 0\textrm{ }$$

where x₁ and x₂ are the resonant amplitudes, κ is the coupling coefficient of the two oscillators, γ₁ and γ₂ are respectively the losses of bright and dark modes, and δ is the detuning of the resonant frequency of the dark-mode oscillator from that of the bright-mode oscillator. Using Eqs. (7) and (8) and considering ω − ω₀ « ω₀, the susceptibility χ is approximated as

(9)$$\chi = {\chi _r} + i{\chi _i} \propto \frac{{({{\omega} - {{\omega}_0} - \delta } )+ i\frac{{{\gamma _2}}}{2}}}{{\left( {{\omega} - {{\omega}_0} + i\frac{{{\gamma_1}}}{2}} \right)\left( {{\omega} - {{\omega}_0} - \delta + i\frac{{{\gamma_2}}}{2}} \right) - \frac{{{\kappa ^2}}}{4}}}$$

The energy loss is proportional to the imaginary part of χ, and the transmission is thus expressed as

(10)$$T \propto 1 - {\chi _i} = 1 - g{\chi _i}$$

Generally, the imaginary part of susceptibility χi is proportional to the energy loss. The transmission T is therefore expressed as T = 1 − gχ_i, in which g is a geometric parameter indicating the strength of the coupling of the bright mode with the incident electric field E. By solving Eqs. (9) and (10), the transmission T of the hybrid silicon–graphene MMs with different Fermi levels of the graphene ribbon is analytically fitted. Figure 6 shows that the theoretical fitting results are in good agreement with the results of simulation. The corresponding fitting parameters as a function of the Fermi level of the graphene ribbon areshown in Table 1. It is seen that the radiative damping item γ₁ decreases from5.306 to 5.170 THz as the Fermi level of the graphene ribbon increases from 0.3 to 0.9 eV. Additionally, the non-radiative damping item γ2 increases from 0.0544 to 0.065 THz. However, these results are far lower than those reported in the literature for metal- based Fano-resonance MMs [42]. Owing to the decrease in radiative damping γ1 and low non-radiative damping γ2, the transmission intensity at the Fano peak enhances gradually with an increase in the Fermi level [Figs. 4(b)–4(f)]. Meanwhile, the coupling coefficient κ reduces from 3.234 to 2.307 THz with an increase in the Fermi level from 0.3 to 0.9 eV. A smaller coupling coefficient, κ, indicates the weaker coupling of bright and dark modes, resulting in less suppression of radiative losses. Therefore, radiative losses gradually increase with the Fermi level of the graphene ribbon. Furthermore, the transmission intensity at the Fano dip declines dramatically with an increase in the Fermi level [Figs. 4(b)–4(f)].

Fig. 6. Transmission spectrum of hybrid silicon–graphene MMs for different Fermi level of the graphene ribbon (g = 15 µm) and the corresponding theoretical fitted transmission spectrum.

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Table 1. Fitting parameters γ₁, γ₂, δ, and κ.

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The active control of slow light is a frontier of optics research because the calculation of the group delay (Δtg) does not consider the adequate thickness of hybrid graphene–silicon MMs [42]. Furthermore, the control is directly related to the practical applications of slow-light devices. The group delay (Δtg) is therefore applied instead of the commonly used group refractive index [43] to represent the slow-light effect. Figure 7 shows the group delay (Δtg) of the hybrid graphene–silicon MMs. The largest group delay is 9.73 ps at an F_E of 0.6 eV, which corresponds to a group delay in free-space propagation of 2.92 mm. Additionally, as shown in Fig. 7, the bandwidth in which the group delay is better than 5 ps is about 13.75 GHz, ranging from 0.73878 to 0.75253 THz. To control the slow-light effect in practical applications flexibly, we amend the GaussAmp model that describes group delay curves above 5 ps in the hybrid graphene–silicon MMs. The amended GaussAmp model is expressed as

(11)$$y = b + \textrm{A}\exp ( - 0.5 \cdot {((x - c)/w)^2}) + \textrm{A}\exp ( - 0.5 \cdot {((x - c)/w)^3}) + \textrm{A}\exp ( - 0.5 \cdot {((x - c)/w)^4}\textrm{)}$$

where y is the group delay (ps), x is the frequency (THz), and A, b, c, and w are constants of the model.

Fig. 7. Group delay curves of hybrid silicon–graphene MMs with different Fermi levels of the graphene ribbon.

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We calculate the constants and parameters of the amended GaussAmp model though nonlinear regression analysis. The coefficient of correlation (R2), root-mean-square error (ERMS), and sum of squared errors (SSE) are adopted as primary metrics in determining whether the amended GaussAmp model can describe the variation in the group delay above 5 ps. Table 2 gives the coefficients of the amended GaussAmp model and the metrics applied used to evaluate the goodness of fit of the group delay curves at different Fermi levels. The table shows that RMSE and SSE have lower values (from 0.00421 to 0.02543 and from 0.00156 to 0.18694 respectively) for the amended GaussAmp model. All values of R² for the amended GaussAmp model exceed 0.999, indicating functional fitness. The validity of the amended GaussAmp model is confirmed by comparing the predicted group delay with the simulation results at different Fermi levels of the graphene ribbon. Figures 8(a)–8(h) shows that the data obtained using the amended GaussAmp model are in good agreement with the results of simulation. This agreement strengthens the view that the amended GaussAmp model represents the characteristics of the group delay above 5 ps.

Fig. 8. Simulated group delay curves for different Fermi levels of the graphene ribbon and the corresponding analytical fitted group delay curve.

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Table 2. Statistical results for the amended GaussAmp model.

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3. Conclusion

We observed position-guided Fano resonance in hybrid graphene–silicon MMs when graphene ribbons were laid at asymmetric positions on a silicon substrate. Through control of the position and Fermi level, the hybrid graphene–silicon structure excites strong coupling between electric and quadrupole dipole resonances. This coupling gives rise to an asymmetric Fano transmission spectrum. Additionally, the transmission intensity at the Fano peak is primarily enhanced with an increase in the Fermi level owing to low Ohmic losses. Furthermore, the amended GaussAmp model was selected as a suitable tuning equation for the group delay based on position-guided Fano resonance in the hybrid graphene–silicon structure. Our results demonstrate that position-guided Fano resonance in hybrid graphene–silicon MMs has promising application in dynamically adjustable optical switches and THz communication networks.

Funding

Natural Science Foundation of Jiangsu Province (BK20180862, BK20190839); Graduate Research and Innovation Projects of Jiangsu Province (KYCX19_1583); National Natural Science Foundation of China (61675147, 61701434, 61735010); National Key Research and Development Program of China (2017YFA0700202, 2017YFB1401203); China Postdoctoral Science Foundation (2019M651725).

Disclosures

The authors declare no conflicts of interest.

References

1. L. Cong, P. Pitchappa, C. Lee, and R. Singh, “Active Phase Transition via Loss Engineering in a Terahertz MEMS Metamaterial,” Adv. Mater. 29(26), 1700733 (2017). [CrossRef]

2. M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017). [CrossRef]

3. D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef]

4. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef]

5. C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11(1), 69–75 (2012). [CrossRef]

6. K. Bi, D. Yang, J. Chen, Q. Wang, H. Wu, C. Lan, and Y. Yang, “Experimental demonstration of ultra-large-scale terahertz all-dielectric metamaterials,” Photonics Res. 7(4), 457 (2019). [CrossRef]

7. M. Manjappa, Y. K. Srivastava, A. Solanki, A. Kumar, T. C. Sum, and R. Singh, “Hybrid Lead Halide Perovskites for Ultrasensitive Photoactive Switching in Terahertz Metamaterial Devices,” Adv. Mater. 29(32), 1605881 (2017). [CrossRef]

8. P. L. Jiahao Yan, Zhaoyong Lin, Hao Wang, Huanjun Chen, Chengxin Wang, and Guowei Yang, “Directional Fano Resonance in a Silicon Nanosphere Dimer,” ACS Nano 9(3), 2968–2980 (2015). [CrossRef]

9. H. Liu, G. X. Li, K. F. Li, S. M. Chen, S. N. Zhu, C. T. Chan, and K. W. Cheah, “Linear and nonlinear Fano resonance on two-dimensional magnetic metamaterials,” Phys. Rev. B 84(23), 235437 (2011). [CrossRef]

10. S. Zhang, K. Bao, N. J. Halas, H. Xu, and P. Nordlander, “Substrate-Induced Fano Resonances of a Plasmonic Nanocube: A Route to Increased-Sensitivity Localized Surface Plasmon Resonance Sensors Revealed,” Nano Lett. 11(4), 1657–1663 (2011). [CrossRef]

11. M. Manjappa, Y. K. Srivastava, L. Cong, I. Al-Naib, and R. Singh, “Active Photoswitching of Sharp Fano Resonances in THz Metadevices,” Adv. Mater. 29(3), 1603355 (2017). [CrossRef]

12. Y. H. Fu, J. B. Zhang, Y. F. Yu, and B. Luk’yanchuk, “Generating and manipulating higher order Fano resonances in dual-disk ring plasmonic nanostructures,” ACS Nano 6(6), 5130–5137 (2012). [CrossRef]

13. X. Yan, Z. Zhang, L. Liang, M. Yang, D. Wei, X. Song, H. Zhang, Y. Lu, L. Liu, and M. Zhang, “A multiple mode integrated biosensor based on higher order Fano metamaterials,” Nanoscale 12(3), 1719–1727 (2020). [CrossRef]

14. S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of Fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013). [CrossRef]

15. S. H. Wu, K. L. Lee, A. Chiou, X. Cheng, and P. K. Wei, “Optofluidic platform for real-time monitoring of live cell secretory activities using Fano resonance in gold nanoslits,” Small 9(20), 3532–3540 (2013). [CrossRef]

16. X. Guo, H. Hu, X. Zhu, X. Yang, and Q. Dai, “Higher order Fano graphene metamaterials for nanoscale optical sensing,” Nanoscale 9(39), 14998–15004 (2017). [CrossRef]

17. F. B. Zarrabi, M. Bazgir, M. Naser-Moghadasi, and A. S. Arezoomand, “Symmetrical metamaterial nano particle for improving the Fano mode for biological application at mid infrared,” Optik 130, 1191–1196 (2017). [CrossRef]

18. S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016). [CrossRef]

19. Q. Li, Z. Tian, X. Zhang, N. Xu, R. Singh, J. Gu, P. Lv, L.-B. Luo, S. Zhang, J. Han, and W. Zhang, “Dual control of active graphene–silicon hybrid metamaterial devices,” Carbon 90, 146–153 (2015). [CrossRef]

20. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). [CrossRef]

21. M. Losurdo, I. Bergmair, B. Dastmalchi, T.-H. Kim, M. M. Giangregroio, W. Jiao, G. V. Bianco, A. S. Brown, K. Hingerl, and G. Bruno, “Graphene as an Electron Shuttle for Silver Deoxidation: Removing a Key Barrier to Plasmonics and Metamaterials for SERS in the Visible,” Adv. Funct. Mater. 24(13), 1864–1878 (2014). [CrossRef]

22. H. Huang, H. Xia, W. Xie, Z. Guo, H. Li, and D. Xie, “Design of broadband graphene-metamaterial absorbers for permittivity sensing at mid-infrared regions,” Sci. Rep. 8(1), 4183 (2018). [CrossRef]

23. Y. Zhang, Y. Feng, T. Jiang, J. Cao, J. Zhao, and B. Zhu, “Tunable broadband polarization rotator in terahertz frequency based on graphene metamaterial,” Carbon 133, 170–175 (2018). [CrossRef]

24. T. Liu, H. Wang, Y. Liu, L. Xiao, C. Zhou, Y. Liu, C. Xu, and S. Xiao, “Independently tunable dual-spectral electromagnetically induced transparency in a terahertz metal–graphene metamaterial,” J. Phys. D: Appl. Phys. 51(41), 415105 (2018). [CrossRef]

25. X. He, Y. Yao, X. Yang, G. Lu, W. Yang, Y. Yang, F. Wu, Z. Yu, and J. Jiang, “Dynamically controlled electromagnetically induced transparency in terahertz graphene metamaterial for modulation and slow light applications,” Opt. Commun. 410, 206–210 (2018). [CrossRef]

26. Z. Zhang, X. Yan, L. Liang, D. Wei, M. Wang, Y. Wang, and J. Yao, “The novel hybrid metal-graphene metasurfaces for broadband focusing and beam-steering in farfield at the terahertz frequencies,” Carbon 132, 529–538 (2018). [CrossRef]

27. Y. K. Srivastava, L. Cong, and R. Singh, “Dual-surface flexible THz Fano metasensor,” Appl. Phys. Lett. 111(20), 201101 (2017). [CrossRef]

28. M. Manjappa, P. Pitchappa, N. Singh, N. Wang, N. I. Zheludev, C. Lee, and R. Singh, “Reconfigurable MEMS Fano metasurfaces with multiple-input-output states for logic operations at terahertz frequencies,” Nat. Commun. 9(1), 4056 (2018). [CrossRef]

29. Y. K. Srivastava, R. T. Ako, M. Gupta, M. Bhaskaran, S. Sriram, and R. Singh, “Terahertz sensing of 7 nm dielectric film with bound states in the continuum metasurfaces,” Appl. Phys. Lett. 115(15), 151105 (2019). [CrossRef]

30. X. Chen, W. Fan, and C. Song, “Multiple plasmonic resonance excitations on graphene metamaterials for ultrasensitive terahertz sensing,” Carbon 133, 416–422 (2018). [CrossRef]

31. B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat Commun3780 (2012).

32. G.-D. J. S. Correas-Serrano D, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Graphene-based plasmonic tunable low-pass filters in the terahertz band[J],” IEEE Trans. Nanotechnol. 13(6), 1145–1153 (2014). [CrossRef]

33. L.-H. Jiang, F. Wang, R. Liang, Z. Wei, H. Meng, H. Dong, H. Cen, L. Wang, and S. Qin, “Tunable Terahertz Filters Based on Graphene Plasmonic All-Dielectric Metasurfaces,” Plasmonics 13(2), 525–530 (2018). [CrossRef]

34. G. W. Hanson, “Dyadic Green’s Functions and Guided Surface Waves for a Surface Conductivity Model of Graphene,” J. Appl. Phys. 103(6), 064302 (2008). [CrossRef]

35. M. S. Yang, L. J. Liang, D. Q. Wei, Z. Zhang, X. Yan, M. Wang, J. Q. Yao, and T. University and T. University, “Dynamically tunable terahertz passband filter based on metamaterials integrated with a graphene middle layer,” Chin. Phys. B 27(9), 098101 (2018). [CrossRef]

36. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). [CrossRef]

37. C. Argyropoulos, “Enhanced transmission modulation based on dielectric metasurfaces loaded with graphene,” Opt. Express 23(18), 23787–23797 (2015). [CrossRef]

38. C. Liu, P. Liu, C. Yang, Y. Lin, and H. Liu, “Analogue of dual-controlled electromagnetically induced transparency based on a graphene metamaterial,” Carbon 142, 354–362 (2019). [CrossRef]

39. G. Gumbs, A. Iurov, and N. J. M. Horing, “Nonlocal plasma spectrum of graphene interacting with a thick conductor,” Phys. Rev. B 91(23), 235416 (2015). [CrossRef]

40. J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Near-infrared trapped mode magnetic resonance in an all-dielectric metamaterial,” Opt. Express 21(22), 26721–26728 (2013). [CrossRef]

41. X. Yan, M. Yang, Z. Zhang, L. Liang, D. Wei, M. Wang, M. Zhang, T. Wang, L. Liu, J. Xie, and J. Yao, “The terahertz electromagnetically induced transparency-like metamaterials for sensitive biosensors in the detection of cancer cells,” Biosens. Bioelectron. 126, 485–492 (2019). [CrossRef]

42. S. Xiao, T. Wang, T. Liu, X. Yan, Z. Li, and C. Xu, “Active modulation of electromagnetically induced transparency analogue in terahertz hybrid metal-graphene metamaterials,” Carbon 126, 271–278 (2018). [CrossRef]

43. J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H. T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3(1), 1151 (2012). [CrossRef]

E_F	γ₁	γ₂	δ	κ
0 eV	4.374	0.544	1.365	6.289
0.3 eV	5.306	0.054	0.105	3.234
0.5 eV	5.264	0.051	0.133	3.002
0.7 eV	5.264	0.051	0.133	3.002
0.9 eV	5.170	0.065	0.160	2.307

E_F	A	b	c	w	RMSE	$R^{2}$	SSE
0.2 eV	−23.57	74.26	0.7367	−0.0052	0.00421	0.99998	0.00156
0.3 eV	−28.74	90.12	0.7368	−0.0070	0.00794	0.99996	0.00870
0.4 eV	−31.32	97.99	0.7367	−0.0087	0.01073	0.99995	0.02062
0.5 eV	−33.02	103.02	0.7364	−0.0105	0.01360	0.99992	0.03997
0.6 eV	−34.26	106.56	0.7358	−0.0124	0.01731	0.99987	0.07460
0.7 eV	−34.71	107.84	0.7353	−0.0141	0.02082	0.99982	0.12091
0.8 eV	−35.48	109.87	0.7344	−0.0161	0.02291	0.99988	0.15952
0.9 eV	−35.68	110.32	0.7336	−0.0178	0.02441	0.99971	0.19490

E_F	γ₁	γ₂	δ	κ
0 eV	4.374	0.544	1.365	6.289
0.3 eV	5.306	0.054	0.105	3.234
0.5 eV	5.264	0.051	0.133	3.002
0.7 eV	5.264	0.051	0.133	3.002
0.9 eV	5.170	0.065	0.160	2.307

E_F	A	b	c	w	RMSE	$R^{2}$	SSE
0.2 eV	−23.57	74.26	0.7367	−0.0052	0.00421	0.99998	0.00156
0.3 eV	−28.74	90.12	0.7368	−0.0070	0.00794	0.99996	0.00870
0.4 eV	−31.32	97.99	0.7367	−0.0087	0.01073	0.99995	0.02062
0.5 eV	−33.02	103.02	0.7364	−0.0105	0.01360	0.99992	0.03997
0.6 eV	−34.26	106.56	0.7358	−0.0124	0.01731	0.99987	0.07460
0.7 eV	−34.71	107.84	0.7353	−0.0141	0.02082	0.99982	0.12091
0.8 eV	−35.48	109.87	0.7344	−0.0161	0.02291	0.99988	0.15952
0.9 eV	−35.68	110.32	0.7336	−0.0178	0.02441	0.99971	0.19490

Position-guided Fano resonance and amended GaussAmp model for the control of slow light in hybrid graphene–silicon metamaterials

Abstract

1. Introduction

2. Results and discussion

3. Conclusion

Funding

Disclosures

References

Cited By

Figures (8)

Tables (2)

Equations (11)

Optics Express