## Abstract

Tunable propagation properties of graphene asymmetrical bowtie metamaterials (MMs) structures have been investigated in the terahertz regime, including the effects of graphene Fermi levels, structural parameters and operation frequencies. Because of its thin film thickness, the strong resonant curves of the proposed graphene MMs structures are dominated by the plasmonic mode instead of the Fabry-Perot mode for the metal structures. Compared with existing tunable graphene devices, the sharp Fano resonant curve manifests a large *Q*-factor of more than 40. In addition, as the width of graphene bowtie aperture increases, the resonant frequency dip shifts low frequency, and the resonant amplitude and figure of merit increase. The results are very helpful in order to understand the tunable mechanisms of graphene components and design high sensitivity functional devices, sensors, modulators, and antenna.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, terahertz (THz) technology has made great advances in the aspects of radiation sources and detectors, which proper its practical applications in the fields of image, biological sensors, high-speed wireless communication and so on [1–5]. For instance, based on the metal-metal waveguide structures, radio frequency modulation of terahertz quantum cascade lasers have been experimentally demonstrated, emitting laser at 4.3 THz, and the tunable frequency range covers 600 GHz [4]. To the further progress of THz technology, there is also a high demand for functional waveguide devices with fine performance. But usually the natural materials interacted with THz waves weakly, its manipulation is still a challenging task. With suitable unit cells (meta-molecule), the emergence of metamaterials (MMs) can help to surmount this dilemma [6–8]. Regretfully, the properties of MMs and surface plasmons (SPs) devices suffer from large dissipation losses and low quality-factor (*Q*-factor) [9,10], hampering its functions and applications. This problem can be overcome by introducing asymmetrical structure into MMs to excite Fano resonances [11–13]. Plasmonic induced transparency (PIT) is an important type of Fano resonances and comes from the destructive interference between bright and dark modes mimicked by metamaterials [14,15].

Consisting of two triangular plasmonic structures with their apex facing each other, bowtie antenna is a widely adopted kind of MMs unit cells and Fano nanostructures, displaying the merits of simple structure, strong near-field enhancement and high *Q*-factors [16,17]. Bowtie aperture processes two distinct electromagnetic effects, high electric field accumulation due to the sharp metallic tips and the strong resonant response to the driving field, which can be applied to optical trapping, sub-diffraction optical lithography [18–20]. Recently, on the base of bowtie meta-molecule MMs structure much research have been carried out in the frequency ranges of microwave, THz waves, visible and extreme-ultraviolet spectral curves. By using a bowtie antenna structure composed of alternative layers of SiO_{2} and Au, M. Morshed *et al.* displayed that the electric field enhancement improved six times, indicating potential applications in the fields of surface enhanced Raman spectroscopy [21]. Based on the semiconductive Si bowtie antennas integrated with a conical waveguide structure, M. C. Schaafsma *et al.* demonstrated that the focus of THz waves beyond diffraction limitation and the intensity improved a factor of 10, resulting from the fact that the bowtie structure supports localized SPs strongly and suppresses the background radiation [22]. For a gold bowtie antennas structure, A. Bhattacharya group investigated the extinction properties of the resonant spectrum, which manifested that there existed a large shift of resonant frequencies between near-field and far-field spectra, which can be explained better by the oscillator model with a Fano model [23].

To investigate the tunable Fano resonances with large values of *Q*-factor supported by metamaterials device is an important and active topic nowadays [5,24]. For current systems, the manipulation of Fano curves can be achieved by utilizing the thermal or electrical control of semiconductor substrates [25]. However, there are several important limitations yet to be addressed in the existed methods. The further development of tunable PIT devices requires the exploration of high performance materials beyond conventional metals. With the advantages of high tunability, strong mode confinement, and excellent responses to incident waves [26–29], graphene provides a promising choice to manipulate the resonant curves of plasmonic devices [30–33]. In addition, asymmetrical bowtie antenna is one of the promising unit cells for achieving sharp Fano resonance. Therefore, by depositing complementary graphene bowtie ribbons on the SiO_{2}/Si/polymer substrate, the tunable Fano resonances have been explored, taking into account the effects of graphene Fermi levels, structural parameters, and operation frequency. The results manifest that because the graphene layer is very thin, the strong resonant curves of the graphene asymmetrical bowtie structures are dominated by the plasmonic mode instead of Fabry-Perot mode for metal structures. The *Q*-factor of the complementary graphene MMs structure can reach more than 40.

## 2. Theoretical model and methods

Figure 1(a) depicts the side view of complementary graphene asymmetrical bowtie metamaterials structures, situating on the polyimide substrate. The graphene patterns are integrated with the SiO_{2}/Si layers, the thicknesses of SiO_{2} and doped Si layers are 10 nm and 5 μm, respectively. Figure 1(b) gives the top view of graphene ribbons metamaterials structures, respectively. The period lengths along x and y directions are *p*_{x} and *p*_{y} with the values of 140 μm. The incident waves normally transmit along *z* direction.

Graphene can be regarded as a 2D material and described by a complex surface conductivity σ_{g}, which is related to the operation frequency ω, chemical potential μ_{c}, Fermi level *E*_{f}, the environmental temperature *T*, and the relaxation time τ. Under the Kubo model, the real part grapheme conductivity can be written as [34]:

*f*

_{0}

*(ω)*is the Fermi distribution function,

*ψ*is the digamma function,

*Γ*and

*Γ´*are the phenomenological scattering rates,

*ω' = ω + Ω*,

*N*

_{f}is the number of spin components, Δ is an excitonic gap and directly related to the electron density imbalance of the bi-particle hexagonal lattice of graphene, and

*Ω*is the frequency. The carrier scattering rate of graphene layer is 1 ps. The imaginary part of the complex conductivity can be calculated by using the Kramers-Kronig relation:

_{0}is the permittivity of free space.

The value of *Q*-factor is given as:

The permittivity and permeability can be written as *ɛ* = *n*/*z*, *μ* = *nz*, and the value of refractive index n and the wave impedance *z* can be expressed as [35]:

*S*

_{11}and

*S*

_{21}are the

*S*-parameters of the MMs structure.

The group index of graphene MMs structure can be written as:

*n*

_{eff}(ω) is the effective refractive index, and it can be extracted from the numerical simulations of the transmission and reflection coefficients. The slow light effects is determined by the group delay,

## 3. Results and discussion

Figure 2 shows that the Fano spectral curves of complementary graphene asymmetrical bowtie MMs, displaying that PIT resonance strongly depends on the structural parameters, *i.e.* the asymmetry degree δ (offset, length difference) between graphene bowtie aperture. The length of bottom graphene triangular ribbon (L_{B}) is 30 μm, the top triangular lengths are 35, 40, 45, 50, 55, and 60 μm, and thus the according asymmetrical degrees are 5, 10, 15, 20, 25, and 30 μm, respectively. The period lengths along x and y directions are both 140 μm. The gap distance is 2 μm. The polarization of incident waves is along y direction. The Fermi level is 1.0 eV. The resonances are very sensitive to the top triangular length (L_{T}). If the value of L_{T} is 30 μm, *i.e.* the asymmetric degree is zero, the bowtie structure gives obvious single transmission peak. In this case, Fano resonance vanishes and confirms the requirement of offset for the excitation of Fano resonance. With the value of L_{T} increases, the bowtie meta-molecular becomes asymmetrical, and the offset δ excites quadrupolar mode. The interference between electric dipolar and quadrupole modes operates as the bright and dark modes, leading into an obvious dip near the transmission peak. This resonance can be explained by the dipole-like charges oscillation arises, resulting from the bowtie aperture edge. Different from Fabry–Pérot (FP) resonances for the metal structure, the resonant curves of graphene bowtie MMs are related to plasmonic mode resonance and insensitive to the graphene thickness. In this case, FP resonances diminish because of the thin thickness of graphene layer. If the degree of asymmetry is small, the focused electromagnetic fields near the bowtie aperture cause strong dissipation by Joule loss and results in a weak resonant strength. As the length of upper triangular graphene ribbon increases, the offset δ(x) and degree of asymmetry augments, and the interaction between complementary graphene bowtie and THz waves improves. Consequently, the resonant frequency of Fano dip shifts upward with the length of graphene ribbons. This provides us with one freedom degree to manipulate the spectral resonances. For instance, when the length of upper triangular graphene ribbons are 35 μm, 45 μm, and 60 μm, the values of Fano dip amplitude are 0.5351, 0.1437, and 0.03421, respectively, and the modulation depth of peak amplitude is 93.61%. Correspondingly, when the L_{T} length changes in the range of 35-60 μm, the resonant frequency of Fano curves can be shifted from 1.352 THz to 1.202 THz, and the modulation depth of frequency is 11.18%. The reflection and absorption resonant curves have also been shown in Fig. 2(b)-2(c), respectively. The interaction between graphene bowtie slots and THz wave increases with the length of graphene ribbon, much more energy transfers from “bright” to “dark” mode, resulting into the obvious reflection peak increasing. The *Q*-factor is a good tool to measure the resonant quality of spectral curves, as shown in Fig. 2(d). In order to quantize the trade-off between *Q*-factor and resonant strength, figure of merit (FOM) can be defined as FOM = *Q* × *A*_{max}, *A*_{max} is the amplitude of Fano resonances. As the value of L_{T} and asymmetry degree increases, the coupling strength between incident wave and quadrupole mode becomes stronger. Thus, the Fano dip resonance of transmission curves broaden, leading into the *Q*-factor reducing. For example, the length of upper triangular graphene ribbon are 35 μm, 45 μm, and 60 μm, the *Q*-factors are 36.38, 16.67, and 9.943, respectively. This value is much larger than that of original MMs structure, and is comparable to that of metal MMs structure. But, as the value of L_{T} increases, the interaction between the complementary graphene bowtie increases, the amplitude of Fano dip resonance decreases as well. Consequently, the value of FOM shows a peak near the value of L_{T} is 45 μm.

The influences of bowtie width (T_{R}) on the resonant curves are depicted in Fig. 3. The bottom and upper lengths of triangular graphene ribbons are 30 μm and 45 μm, *i.e.* the offset is 15 μm. The polarization of incident waves is along y direction. The Fermi level is 1.0 eV. The values of T_{R} of upper triangular graphene ribbons are 20, 25, 30, 35, 40, 45, 50, 55, and 60 μm, respectively. On condition that the value of T_{R} is small, the strength of Fano resonance is not very strong. As the width of triangular graphene ribbon increases, the resonant intensity enhances, the spectra broaden, and the Fano resonance becomes stronger. For example, when the bowtie widths are 35 μm, 45 μm, 60 μm, the amplitude values of Fano dips are 0.2392, 0.2284, and 0.1437, and the dip frequencies are 1.734 THz, 1.518 THz, and 1.360 THz, respectively. The resonant Fano dip shifts low frequency with increase of T_{R}, which can be explained in the following. The resonant wavelength of graphene bowtie aperture can be expressed as λ_{res} = n_{1} + n_{2}*L*/λ_{p} [36], *L* is the aperture perimeter, n_{1} and n_{2} are the coefficients determined by geometry and the properties of surrounding medium, λ_{p} is the plasma wavelength. The above equation means that the resonance wavelength depends on the perimeter of graphene bowtie structure and plasma length λ_{p}. Consequently, as the width of triangular graphene ribbon increases, the aperture perimeter enhances, leading into resonant frequencies moving to low frequency. The transmission curves of complementary graphene bowtie structures can modulate via T_{R}. If the width of graphene bowtie changes in the range of 30-60 μm, the resonant frequency of Fano curves shifts from 1.734 THz to 1.360 THz, and the value of transmission dip vary in the scope of 0.2392-0.1437. The according modulation depth of frequency is 21.59%, and the modulation depth of dip amplitude is 39.92%. The reflection and absorption resonance curves have also been shown in Fig. 3(b)-3(c), respectively. As the width of graphene bowtie aperture increases, the interaction between THz waves and graphene ribbons enhances, which leads into obvious increasing reflection peaks. The influences of bowtie width on the *Q*-factor can be found in Fig. 3(d). As the value of bowtie width increases, the resonant interaction and loss enhances, the transmission resonant curves sharpness decreases, resulting into *Q*-factor reducing. For example, when the bowtie widths are 35 μm, 45 μm, and 60 μm, the *Q*-factor are 31.42, 21.82, and 16.67, respectively. But, it should be noted that as the value of bowtie width increases, the interaction between the complementary graphene ribbon increases, the amplitude of Fano dip resonance increases. Consequently, the value of figure of merits shows a peak increases with bowtie width.

The effects of Fermi level on the resonant spectral curves of graphene bowtie apertures can be found in Figs. 4(a). With the increase of Fermi level, the Fano resonance enhances. For example, when the Fermi levels are 0.3 eV, 0.6 eV, and 1.0 eV, the values of Fano dip amplitude are 0.2479, 0.1869, and 0.1436, respectively. The resonant frequency of Fano curves shifts upward with increase of E_{f}, which can be explained in the following. As mentioned above, the plasmonic Fano resonant wavelength is closely relate to the aperture perimeter and plasma frequency of plasmonic structure λ_{p}. In the THz regime, the intraband contribution dominates, the carrier concentration increases with Fermi level, graphene layer shows better plasmonic properties, the according resonant plasmon frequency enhances at larger Fermi level. Consequently, the resonant wavelength of graphene bowtie antennas, *i.e.* λ_{res} = n_{1} + n_{2}*L*/λ_{p}, decreases with Fermi level. The interaction between THz waves and graphene bowtie patterns increase with E_{f}, and surface current density is better localized to the aperture edge, therefore, the Fano resonant frequency increases with Fermi level. Compared with conventional metallic system, the transmission curves of graphene bowtie structure can be manipulated in a wide scope at fixed structure geometry by changing E_{f}. If the Fermi level of graphene layer changes in the range of 0.3-1.0 eV, the resonant frequencies of Fano curves shift from 1.281 THz to 1.360 THz, and transmission dip changes in the range of 0.2478-0.1436. Correspondingly, the modulation depths of frequency and amplitude are 5.82% and 42.05%, respectively. The reflection and absorption resonance curves are shown in Fig. 4(b)-4(c), respectively. As the Fermi level increases, graphene layer shows better “plasmonic” properties, the coupling between graphene bowtie increases, much more energy transfer from “bright” to “dark” mode with low loss, which results into the absorption peak decreasing. The *Q*-factor of transmission curves can be found in Fig. 4(d). As E_{f} increases, the carrier concentration of graphene layer increases, the transmission resonance sharpness reduces, the Fano spectral curves becomes sharping due to the reducing losses, leading into the *Q*-factor enhancement. For example, when the Fermi levels of graphene layers are 0.3, 0.6 and 1.0 eV, the *Q*-factors are 10.46, 13.89, and 16.67, respectively.

Group index (GI) dispersion can be determined from the slope of the real part of refractive index. Figure 5(a) depicts the group indices of the complementary graphene bowtie MMs structures at different Fermi levels. Within the PIT window, the group index indicates strong dispersion. A significantly large group index is about 47.24, which means that THz waves pass through the bowtie graphene ribbons with a group velocity nearly 50 times slower. Furthermore, as Fermi level increases the maximum value of n_{g} gradually augments, this implies that Fano resonance becomes stronger at large value of Fermi level. We can achieve a dynamic control of group index by altering the graphene Fermi level. For instance, when the *E*_{f} changes in the range of 0.3-1.0 eV, the peak value of GI can be modulated in the scope of 11.44-47.24. The group delay (τ_{g}) is more suitable for describing the light slowing capability because it does not require the effective thickness of the device and substrate. The group delay is relate to PIT characters and defined as the opposite slope of transmission phase response, *i.e.* τ_{g} = -dφ/dω. Figure 5(b) manifests that the group delays can reach about 0.2 ns. The large positive group delays locate near PIT peaks. An optical pulse in the positive and negative group delays corresponds to slow and fast light, respectively. As shown in Fig. 5(b), the negative group delay corresponds to fast light effect nearby the transparency window, τ_{g} becomes negative (τ_{g} < 0), indicating that graphene bowtie structure completes the transition from slow to fast light.

To improve the tunable properties of proposed complementary graphene bowtie aperture structure, the stacked graphene-dielectrics superlattice structures have been adopted in the active region, *i.e.* consisting of graphene patterns separated by dielectric spacer layer along the propagation direction. The polarization is along y direction. The influences of stack number on the transmission curves can be found in Fig. 6(a). As period number increases, the conductive and plasmonic properties of active region increases, the resonant strength becomes evident, and the dip value of transmission curve decreases. In addition, the carrier concentration increases with periodic stack number, resulting into the kinetic inductance reducing and dip position moves to higher frequency. For example, on condition that the Fermi level is 0.5 eV, if the periodic numbers are 1, 5, and 10, the peak dips are 0.2091, 0.1715 and 0.1503, and dip resonance frequencies are 1.324 THz, 1.240 THz and 1.256 THz, respectively. If the periodic number is large, *e.g.* > 6, the effects of period number on transmission curves are not obvious. When the Fermi level varies in the range of 0.2-1.0 eV, on condition that the period numbers are 1, 5 and 10, the dip value of Fano resonance modulates in the scope of 0.2517-0.1437, 0.2625-0.1265, 0.2438-0.1171, and the dip position can be modulated in the range of 1.2304-1.36 THz, 1.3384-1.396 THz, 1.36-1.3984 THz, respectively. The corresponding modulation depths of amplitude are 42.91%, 51.81%, 51.97%, and the frequency MD are 9.53%, 4.13%, and 2.75%, respectively. Figure 6(b) and 6(c) illustrate that the reflection and absorption curves become stronger with the increase of period number. The reflection and absorption peaks roughly correspond with the Fano dips of transmission curves. As the period number increases, the graphene stack multilayer structure displays better conductive properties, the reflection curves become stronger, as shown in the Fig. 6(b). The effects of period number on the *Q*-factor and FOM have been shown in Fig. 6(d). The resonant interaction becomes stronger with the increase of period number, the value of *Q*-factor increases. For example, on condition that the Fermi level is 0.5 eV, when the period number is 1, 5, 10, the *Q*-factor are 13.15, 17.44, and 19.29, respectively. Furthermore, the resonant strength and amplitude of Fano resonance also enhance with increase of period number, thus the values of FOM increase consequently.

To have a deep understand of sharp Fano resonant curves, we detect the surface current density and magnetic fields along z direction. The polarization direction is along y direction. The graphene Fermi level is 1.0 eV. Figure 7 shows the surface current density and magnetic components for the complementary graphene bowtie structures. The resonant frequencies are 1.202 THz, 1.360 THz, and 1.463 THz, respectively. The magnetic fields show that charges accumulate at the sharp edge of bowtie aperture, and surface current density distribution induces surface current flows around the aperture edge. The surface currents accumulate at two edges of asymmetrical apertures form a “giant electric dipole”, enhancing the electric fields. This phenomenon is similar to that of nanoparticles at fundamental plasmonic resonances. At the low frequency of 1.202 THz, the electric dipolar mode is excited in the bottom triangular graphene ribbon, as shown in Fig. 7(a). While at the high frequency of 1.463 THz, the surface current density focuses on the upper triangular graphene ribbon structure. Because of the asymmetrical bowtie meta-molecular structure, the length of upper triangular of graphene bowtie is smaller than that of bottom section, which results into two resonant peaks at different frequencies. The bottom and upper arms of graphene bowtie structure are antiparallel electric dipoles in the meta-surface. While at the Fano dip frequency of 1.360 THz, the low and high frequencies of electric dipolar mode have both been excited, as shown in Fig. 7(b). For the asymmetrical bowtie MMs structure, the surface currents of bottom and upper triangular graphene ribbons are opposite, a net dipole moment arises, a quadrupole-like mode is excited, which interacts with the incident field very weakly and can be regarded as a dark mode. While the quadrupolar mode can interfere with the original electric dipolar resonance strongly, leading into Fano resonance. For the Fano resonance, the interaction between anti-parallel electric dipoles of upper and bottom triangular graphene ribbons occurs in the coupled asymmetric structure, dominating the local field distribution. These antiphase currents near Fano dip resonance result into a reduction of radiative loss and a very sharp Fano resonance with high value of *Q*-factor. This point can also be confirmed by the larger values of surface current density and magnetic fields of H_{z} near Fano resonance. For instance, at the frequencies of 1.202 THz, 1.360 THz, and 1.463 THz, the values of H_{z} of 14170, 13691, and 11160 A/m, respectively. At the Fano resonance, intensity of H_{z} concentrates in the gap, while is strong along each bar at the dipole resonance.

Figure 8 shows that the surface current density and magnetic components for the graphene bowtie structure at different Fermi levels. The polarization direction is along y direction. The graphene Fermi levels are 0.3, 0.6, and 1.0 eV, and the resonant frequencies are 1.281 THz, 1.334 THz, and 1.360 THz, respectively. The analysis reveals that H_{z} concentrates in the edge of complementary bowtie structure at Fano resonance. As Fermi level increases, the interaction of THz waves with graphene bowtie aperture improves, the surface current density and value of magnetic field of H_{z} increases. For instance, if the Fermi levels are 0.3, 0.6, and 1.0 eV, the values of H_{z} of are 5745, 10216, and 13691 A/m, respectively. Because of a large amount of H_{z} stored electromagnetic energy, the *Q*-factors of Fano resonances show large values as Fermi level increases. This means that Fano resonances become stronger, leading into larger values of *Q*-factors and FOM. For example, when the Fermi levels are 0.3, 0.6, and 1.0 eV, the values of *Q*-factors are 10.46, 13.89, and 16.67, respectively.

## 4. Conclusion

Based on the graphene asymmetrical bowtie MMs structure, the tunable PIT phenomena have been investigated in the THz regime, including the influences of Fermi levels, operation frequency, and structural parameters. The results manifest that different from the FP resonances for the metal structures, the resonant curves are dominated by the plasmonic mode because of the thin thickness of graphene layer. As Fermi level increases, the amplitude of Fano dip decreases, and the resonant frequency shifts to high frequency. If Fermi level changes in the range of 0.2-1.0 eV, the amplitude modulation depth of Fano resonance reaches more than 40%. Compared with existed graphene tunable devices, the Fano resonant curve of the graphene bowtie is sharp, indicating a large *Q*-factor with the value of more than 40. In addition, as the upper length of graphene bowtie aperture increases, the resonant dip shifts low frequency, the resonant amplitude and the values of FOM increase. The results are very helpful to understand the tunable mechanisms of graphene based Fano systems and design high sensitivity functional devices, *e.g.* sensors, modulators, and antenna.

## Funding

National Natural Science Foundation of China (61674106 and U1531109); Shanghai Pujiang Program (15PJ1406500); the Natural Science Foundation of Shanghai (16ZR1424300); Shanghai Normal University (KF201861); the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China.

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