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Abstract
We propose a novel simple patterned monolayer graphene metamaterial structure based on tunable terahertz plasmon-induced transparency (PIT). Destructive interference in this structure causes pronounced PIT phenomenon, and the PIT response can be dynamically controlled by voltage since the existence of continuous graphene bands in the structural design. The theoretical transmission of this structure is calculated by coupled mode theory (CMT), and the results are highly consistent with the simulation curve. After that, the influence of the graphene mobility on the PIT response and absorption characteristics is researched. It is found that the absorption efficiency of our designed structure can reach up to 50%, meaning the structure is competent to prominent terahertz absorber. Moreover, the slow-light performance of this structure is discussed via analyzing the group refractive index and phase shift. It shows that the structure possesses a broad group refractive index band with ultra-high value, and the value is up to 382. This work will diversify the designs for versatile tunable terahertz devices and micro-nano slow-light devices.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Plasmon-induced transparency (PIT) is an analog of electromagnetic induction transparency (EIT) in atomic systems, and it utilizes the interaction between surface plasmon waves to obtain a phenomenon similar to atomic EIT [1,2]. It is widely known that the atomic EIT effect is not conducive to application research since its extreme requirement for experimental conditions. However, its special optical properties, such as the sudden shift on phase of electromagnetic waves at the transparent window, still attract much attention of researchers. This dispersion characteristic can effectively slow down the propagation velocity of optical pulses and has significant applications around the domain of sensing equipment and optical storage [3,4]. Therefore, people hope to find some other systems, which can also achieve this special phenomenon and keep corresponding characteristics. In recent years, scientists have successfully discovered and achieved induced transparency effects in a lot of non-atomic systems, for instance, metal gratings, MDM waveguides and metamaterials [2,5–10]. Most of the studies are based on waveguide or metal metamaterial structure which unavoidably has some defects, such as fixed spectral response and operating frequency, the finite function of dielectric constant, limited plasma lifetime and high ohmic loss. All above limit its scope of application.
Graphene [11], a new two-dimensional material, is equipped with many excellent properties such as high electron mobility, optical transparency, etc [12]. All of those make it widely explored and applied in photonics and optoelectronics as polarizers, modulators, photodetectors, etc [13,14]. Further research shows that graphene exhibits metalloid properties when the imaginary part of the graphene surface conductivity is positive, the real part of its equivalent permittivity is negative. This makes it a preeminent supporting material for propagating the surface transverse magnetic (TM) mode polarization wave. Also, the imaginary part of the equivalent dielectric constant of graphene is affected by the real part of the complex surface conductivity and indicates the propagation loss of surface plasmon waves [15]. Besides, the resonant region of these graphene-based systems is usually in the terahertz band, which is immensely attractive to researchers [16]. Recently, some tunable single-layer patternless graphene PIT structures have been proposed. However, they always weakened in photoelectric modulation, even if the multilayer structures with better photoelectric modulation performance have been introduced after that, the manufacturing processes are still much more complicated than the single-layer graphene PIT metamaterial devices. Only a few devices have the advantages of both good performance and simple manufacturing process.
In this work, a tunable PIT metamaterial structure with simply patterned monolayer graphene in the terahertz band is presented. It consists of a patterned monolayer graphene array, substrate, and electrodes. The monolayer graphene pattern is periodically deposited on the SiO2-Si substrate, which divides into two parts as the bright and the dark radiation mode. Destructive interference between two radiation modes causes a significant PIT phenomenon, and the PIT response can be dynamically controlled by the EF (Fermi energy) of continuous graphene bands instead of the geometry parameter. This is the potent advantage compared with most metal waveguide structures [17,18], because it can be applied to extend the scope of frequency operation to explore more potential applications. What’s more, compared to other discrete graphene pattern device [19–26], the continuous graphene bands in our work keep high mobility of graphene and reduce the difficulty of EF modulation. By the way, the EF of graphene is regulated by voltage in this work. For further exploring the internal mechanism, coupled mode theory (CMT) is introduced to calculate the transmission efficiency theoretically [27]. The validity of CMT is further evidenced by the high degree of agreement between theoretical data and simulation results. Moreover, the absorption efficiency of graphene from visible light to terahertz band is 2.3% per layer [14,28], which remind us of its absorbency potential. Therefore, the absorption characteristic of the system is analyzed while the effect of graphene mobility on transmission is researched. It is found that when the mobility of graphene is low, the absorption performance of the system can maintain a high level, which means the structure can achieve significant terahertz absorbers. Also, by analyzing the phase shift of the system, the group refractive index and the slow-light function of the system is calculated and discussed. The phenomena reveal that the structure possesses a broad group refractive index band with ultra-high value, and the value is up to 382. Such a high group refractive index reflects the superior slow-light performance of the structure and put other structure [18,29] into the shade. Besides, our structure put up palpable advantages in experiment and production because of its simplicity design. With the above benefits, the structure can be applied to terahertz modulators, terahertz absorbers, slow-light devices, etc. Furthermore, this work also provides theoretical guidance for the manufacture of the device and diversify the designs for multi-function tunable terahertz devices and micro-nano slow-light devices.
2. Structure design and theoretical analysis
Figure 1(a) is the three-dimensional structure sketch of the proposed tunable graphene PIT metamaterial in this paper. The designed monolayer graphene pattern is deposited on the SiO2-Si substrate, and the thickness of Si and SiO2 are both 0.1μm. Besides, the bias circuit connects graphene to the doped Si substrate. We can see the composition of structure in the vertical direction (y-direction) from the area surrounded by grey dashed line clearly. There is a thin SiO2 layer over the graphene pattern for reducing the error in theoretical calculation caused by the transformation of the dielectric. Then a polarization plane wave propagates along the negative direction of the y-axis serve as the excitation source. In this paper, FDTD (Time Domain Finite Difference Method) is used as the numerical simulation method. The boundary conditions of x-directions and z-directions are periodic absorption layers, yet y-direction is perfectly matched layers.
Fig. 1 (a)Three-dimensional structure sketch of tunable graphene-based PIT metamaterial in terahertz; (b)Top view of unit cell of the structure in Fig. 1(a) with its concrete structural parameters: L = 4μm, a = 0.85μm, b = 1.2μm, c = 2.7μm, d = 0.175μm.
The top view of a single-period of the proposed terahertz graphene PIT metamaterial structure is revealed in Fig. 1(b), and the unit cell of graphene pattern is centrosymmetric at the center point O of the square. The graphene piece is named as Piece1 (P1), Piece2 (P2), Piece3 (P3), from left to right, where P1 and P3 are discrete graphene sheets, and P2 is continuous graphene band (all of P2 connect at the electrode, and the EF of P2 could be adequately adjusted by tuning the bias voltage between electrode and substrate). The geometric parameters of the unit cell are specifically shown in the caption of Fig. 1(b). In this structure, the discrete graphene P1 and P3 arrays constitute a bright mode called element B (Bright) that can be directly excited by incident polarization plane wave, yet the continuous graphene band P2 array acts as a dark mode called element D (Dark) that only can be indirectly excited by the bright radiation mode instead of directly excited by incident wave. According to the simulation data, the EF of the element B is set to a fixed value of 1.0eV, as well as the EF of the element D is adjusted to be within the range of 0.5eV ~1.0eV with the change in the bias voltage across the electrode.
Figure 2(a) presents the simulant transmission spectrum of designed graphene PIT metamaterials. When a terahertz plane-polarized wave perpendicularly enters the surface of the metamaterial (along the negative direction of y-axis), a transmission trough emerges in the spectrum due to the strong coupling interaction of graphene in element B with the incident wave. Surface plasmon polaritons (SPPs) are tightly trapped in the interface between graphene and the dielectric layer. Thus, it can form a broad continuum mode, which is called the bright radiation mode, as depicted by red-line in Fig. 2(a). The graphene in element D as a dark radiation mode is hard to couple with the incident wave and form a narrow discrete spectrum, as shown by orange-line. When element B and element D work as integration, a significant PIT phenomenon is produced by destructive interference between the bright mode and the dark mode. It could be observed that a distinct transparency window around 5.53THz, as shown by green-line. When the SPPs reach the corresponding frequency, a large amount of energy is trapped in the interface between graphene layer and dielectric layer. To further master the law of dynamic transmission and find out the general mechanism, the theoretical model of CMT is introduced, where element B and element D is considered as two resonators to describe the coupling mechanism between bright mode and dark mode. Figure 2(b) reveals the mechanism of interaction. Among them, the cyan-region and the purple-region represent element B and element D, respectively; The incident and exit radiation waves in the mode are distinguished by the superscript “in” and “out”; The directions of radiance propagation are described via subscript “±”.
Fig. 2 (a)The transmission spectrum of the proposed graphene PIT metamaterial in this paper when the polarization plane wave incident along the negative direction of y-axis. Among them, the red-line is the transmission curve of the periodic array consist of the graphene in element B, orange-line is the transmission curve of the periodic array composed of graphene in element D, and green-line is the transmission curve of the graphene periodic array equip with both element B and element D. At this situation, EF of element B and element D is 1.0eV and 0.7eV, respectively, and the graphene mobility is 2.5m2/Vs; (b) Equivalent CMT model for the proposed structure in this work.
Therefore, the two oscillations mB and mD (the bright and the dark radiation modes are distinguished by subscripts “B” and “D”. The superscript “in” and “out” represent the incident and exit radiation waves in the mode, severally. The subscript “±” indicates the direction of radiation.) can be expressed as [27,30]:
where , ω is the angular frequency of the incident wave, express the attenuation rate caused by internal loss, express the attenuation rate of energy that escaping from mode to outer space, ωB(D) express the angular frequency of radiation mode, and the radiation mode coupling coefficient is described by κBD or κDB. QiB(D) and QeB(D) represent the radiation mode quality factor associated with the intrinsic loss and the loss of diffusion into outer space, respectively. After that, according to the definition of the effective refractive index neff = β/k0, QiB(D) could be deduced as [31]:Combined with the simulant transmission spectrum, the aggregate quality factor can be readily determined on the definition of QiB(D) and QeB(D) described above. The relationship among them is:In which, QiB(D) = f /∆f is the aggregate quality factor of the radiation mode (f is the resonant frequency, Δf expresses the full width at half maximum).Following the principle of conservation of energy, the relationship exists around two radiation modes as:
Theoretically, combined Eqs. (4) and (5) with the condition that only single incident wave enters along the negative direction of y-axis, that is, D-in = 0, the transmission coefficient of proposed system will describe as: where , thus the transmittance and absorbance can be obtained from Eqs. (6) and (7):The structure is placed in an environment at room temperature T = 300K, and the equivalent complex relative dielectric constant of graphene is obtained as follow [32]:where σ is surface conductivity of graphene, ε0 is the dielectric constant of vacuum, t is the thickness of graphene. It is observed that the imaginary part of conductivity exerts an enormous function on the propagation of SPPs. When Im(σ) > 0, only transverse magnetic (TM) waves exist. The Kubo formula is applied to investigate the conductivity of graphene. Since our simulant region is at the terahertz band, the contribution of the intra-band conductivity far exceeds the contribution of the inter-band conductance in the formula. Hence, the graphene complex conductivity σ could be defined by the metal-like Drude model in terahertz band as [33]:where the simplification of above formulas is based on condition EF >> (ħω, kBT), here, e is the elementary charge, kB is the Boltzmann constant, ћ expresses the reduced Planck constant, τ expresses the time of carrier relaxation, which satisfies the relationship τ = μEF / (evF2), and vF = 106m/s represent the Fermi velocity [11]. According to antecedent findings, graphene mobility μ is up to 4m2/Vs at room temperature [34]. In this work, considering the convenience of simulation and the feasibility of later experiment, μ varies at the range of 0.5m2/Vs~3.0m2/Vs.In view of the single-layer form of graphene in our design, the resonance quality factor of both two modes can be approximated from the dispersion relation of complete monolayer graphene. The dispersion relationship that TM SPPs propagate on the graphene is given [32]:
In the above formula, β stands for the propagation constant, εd is set to 3.9 represents the relative dielectric constant of SiO2, and the relative dielectric constant of Si is 11.9 [35], k0 and η0 stands for the wave vector of the propagating wave and the intrinsic impedance in vacuum, separately.Besides, graphene possesses excellent tuning property. Such property makes it possible to actively control the PIT response by tuning bias voltage (Vg) without changing the structural geometry. It has unparalleled advantages over other structures. The regulatory relationship between Vg and EF can be express as [36,37]:
Here, dc represents the distance from graphene to the electrode.3. Simulation and discussion
Associated with above theoretical analysis, the simulant transmission spectra and the transmission spectra of theoretical calculation are compared when the EF of element D is increased from 0.5eV to 1.0eV. The modulation of EF is achieved by the bias voltage between the graphene and substrate, as shown in Fig. 3(a). The green line in Fig. 3(a) indicates the simulation result, and the yellow circles indicate the theoretical calculation result of CMT. Compared with the above two results, the phenomena reflect that the CMT theoretical results are in good agreement with the FDTD simulation results. When EF increases from 0.5eV to 1.0eV, the transmission peak gradually deforms from asymmetry to a nearly perfect Lorentz symmetry and then becomes asymmetrically. Meanwhile, the increase in Fermi energy leads to an increase of the resonance energy. Thus, the resonance frequencies of trough1, trough2 reveal an increscent trend, that is, the PIT curves have a blue-shift trend overall, as shown in Fig. 3(b). Moreover, to further investigate the relationship among EF, the resonance frequency and the transmission efficiency of our PIT structure, the theoretical transmittance of the structure is calculated by CMT, and the results are shown in Fig. 3(c). As the EF increases, the PIT curve presents a blue-shift trend as an entirety, which coincides with the results by FDTD numerical simulation in Fig. 3(b).
Fig. 3 (a)The transmission spectra of the graphene PIT metamaterial at terahertz band when EF in element B is maintained at 1.0eV and in element D is 0.5eV, 0.6eV, 0.7eV, 0.8eV, 0.9eV, 1.0eV, from top to bottom. Here, the graphene mobility defaults to 2.5m2/Vs; (b)The function relationship between EF and the frequency corresponding to trough1, trough2 and peak according to Fig. 3(a); (c)The functional relationship between frequency and transmittance by theoretical calculation when EF varies in a certain range continuously based on the proposed structure.
For discussing the effect of mobility on transmittance, the transmission spectra are compared when the mobility of graphene is changed, as shown in Fig. 4(a). The spectrum indicates that trough1 and trough2 are linearly blue-shifted as μ decreases. This is because an increase of the electronic activity for the rise of mobility leads to a reduction of the required resonance energy namely blue-shifted. However, the blue-shifted slope of trough1 is close to 0, and the blue-shift slope of trough2 is slightly larger than that of trough1, i.e., there is no significant blue or red shift in transmission curve, but the quality factor Q of the PIT peak increase. At the same time, the absorption efficiency of the structure under the corresponding conditions is simulated. When μ stay at a low level, high loss in graphene makes the absorption dominant. As depicted in Fig. 4(b), with the graphene mobility decreases from 3.0m2/Vs to 0.5m2/Vs, the absorption efficiency increases adversely by nearly 200%. Apparently, the low graphene mobility can bring benefit for the graphene PIT metamaterial to obtain better absorption performance in the terahertz band according to Fig. 4(b). It is easy to find that when the graphene mobility μ = 0.5m2/Vs, the absorption efficiency of the structure reaches 50%, superior to some previous studies [38] on the absorption efficiency of the metamaterials system.
Fig. 4 (a)The simulant transmission spectra of the proposed structure when the graphene mobility is reduced from 3.0m2/Vs to 0.5m2/Vs. Here, EF of element B and element D is fixed at 1.0eV and 0.7eV, respectively; (b)The simulant absorption spectra when the mobility of graphene corresponds to the left transmission spectrum.
Another discussion in this paper is based on the remarkable dispersion properties of two-dimensional materials like graphene, and such property can be applied to slow-light devices to slow down the propagation velocity of light. The group index represented the slow-light performance is theoretically calculated [31]:
where c represents the light velocity in vacuum, vg represents the group velocity of light, H indicates the propagation length of wave in the structure, τg indicates the optical delay time, and θ indicates the phase shift of transmission. Then, Figs. 5(a)-5(f) give the relationship between group refractive index or phase shift and the frequency when EF increase. The phase is calculated by the theoretical transmission coefficient obtained above, namely, θ = arg(tr). The dispersion of the surface polarization wave become intensely, while the frequency approach to the transparent window. Then, the sudden shift in phase caused by the strong interference gives rise to a sharply varies in the group index. The phenomenon is consistent with the results that analysis above. From Figs. 5(a)-5(f), we can find that there are significant shifts in phases caused by PIT at two resonance frequencies.Fig. 5 (a)-(f) Group refractive index and phase shift versus frequency, when the dark mode EF at 0.5eV, 0.6eV, 0.7eV, 0.8eV, 0.9eV, 1.0eV, respectively. Here, the EF of the bright radiation mode is fixed at 1.0eV and the graphene mobility defaults to 2.5m2/Vs.
When EF = 0.5eV, the group refractive index reaches a maximum at the smaller resonant frequency, and the value is up to 382. In Fig. 3(a), when EF increases from 0.5eV to 1.0eV, the transmission curve deforms from asymmetry to symmetry and then back to the state of asymmetrical, the transmission spectrum is the most of asymmetrical when EF = 0.5eV. At the same time, observing Figs. 5(a)-5(f), it is not difficult to find that the group refractive index curve of the system follows similar law. As the EF increases from 0.5eV to 1.0eV, the group refractive index curve of the system deforms from asymmetry to symmetry, while the maximum value gradually decreases. When EF = 0.5eV, the stronger dispersion on the formant at the small frequency makes the phase shift at that point more visible, and then the sharp phase shift achieves the system to a higher group refractive index. Combined with the above study of the tuning performance in the system, it indicates that the device can provide better slow-light capability when the transmission curve linear detuning increases. Therefore, we can roughly analyze the trend of slow-light according to the transmission spectrum, which provides a new idea for designing devices with high-performance slow-light index in the future. Compared with previous studies, the proposed structure has a very high group index and a wide bandwidth [18,29]. With these advantages, our proposed structure is expected to develop a new miniaturized slow-light device.
4. Conclusion
In summary, we propose a simple novel single-layer patterned graphene metamaterial structure based on terahertz tunable PIT. The EF of graphene is adjusted by the voltage, and this makes the dynamic modulation of the induced transparency more accessible than those structure with discrete graphene. The dynamic tuning characteristics of the system are reasonably analyzed by CMT, and the fitting results are identical with the numerical simulation results. Then, the impact of graphene mobility on PIT response and absorption characteristics of the system are scientifically analyzed together. The result reflects that the structure can maintain good absorption performance under the condition that graphene mobility stays low level. Additionally, the performance of slow-light in our system is analyzed by calculating the group refractive index and shift of phase. The results present that the structure possesses a broad group refractive index band with ultra-high value, and the value is up to 382. This work will diversify the design for versatile tunable terahertz devices and lay a solid foundation for the research work on micro-nano slow-light devices.
Funding
National Natural Science Foundation of China (NSFC) (61275174); Fundamental Research Funds for the Central Universities of Central South University (2018zzts105)
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