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Abstract
Many plasmon-induced transparency (PIT) metamaterials previously reported had limited functions. Their tunabilities were realized by complex discrete structures, which greatly increased the difficulty and cost of device fabrication and adversely affected their resonance characteristics. It is an open question to adjust the Fermi levels of many graphene patterns with only a few in-plane electrodes. We propose and numerically study a novel electrically tunable and multifunctional trigate graphene metamaterial (TGGM) based on the concept of “Lakes of Wada”. Benefiting from the trigate regulation, our proposed TGGM turns out to exhibit excellent characteristics, that can not only be used for terahertz band-stop filter, terahertz refractive index sensor, near-field optical switch, slow-light device, but also for double PIT window metamaterial with broad transparency windows and large tunable frequency range.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Flexible and efficient control of electromagnetic waves has become a research focus, particularly in view of the rapid development of information technology. Since conventional three-dimensional materials require large volumes, it is not easy to use them to manufacture ultra-compact electromagnetic devices. With the development and progress of the burgeoning micro-nano processing technology, artificial metamaterials have been proposed that could support a series of novel electromagnetic properties such as negative magnetic permeability [1] and negative refraction [2]. By orderly and reasonably design of its unit structure and arrangement, metamaterial-based electromagnetic devices are becoming more compact, more diverse, and more integrated. Phase, polarization, and propagation of electromagnetic waves can be flexibly and effectively controlled by ultra-thin two-dimensional array planes that can easily be processed and integrated. With its rich and unique nature, metamaterial has been used advantageously in many fields such as antenna miniaturization, micro-nano photonics, terahertz sensing and imaging [3–5]. However, with the development of science and technology, many traditional metamaterial-based devices cannot fulfill the requirements for dynamic beam steering, wide-band polarization response, and tunability. Since most of the metal or dielectric metamaterial structures are difficult to change after fabrication, they have low regulation efficiency, and narrow frequency tuning range, and can only work within pre-set bands or resonance frequencies.
Therefore, there is an increasing demand for new controllable and highly practical metamaterials. The essence of controllable metamaterials is to change the spatial distribution of the equivalent dielectric constant or permeability of the cell structure, thereby changing the transmission characteristics of electromagnetic waves, and then realizing dynamic real-time control of the resonance frequency. This greatly expands the operating frequency bandwidth and selectivity of the device and greatly enriches the response of the metamaterial to electromagnetic waves. Graphene, as a kind of two-dimensional material, shows extraordinary ability in terms of adjustability. It has high carrier mobility [6–9], and exhibits flexibly adjustable conductivity by applying a magnetic field or a bias voltage. Graphene structures have also excellent plasmon characteristics in the terahertz and far infrared spectral range. These prominent properties of graphene have drawn strong attention from scientific researchers and industries, and there is a large number of graphene-based metamaterial devices proposed to work well in the terahertz range [10–17]. Recently, graphene metamaterials have been proposed that combine graphene and metamaterial perfectly, with flexible adjustability. Its emergence has provided a powerful tool for the design and implementation of various novel electromagnetic devices, and it is hoped to ignite a new revolution in the field of optronics.
Electromagnetically induced transparency (EIT) has great application prospects for slow light applications, Kerr effect, super Raman scattering, optical storage, quantum switch and sensing technology [18–22]. However, in order to achieve the quantum EIT effect, the selection of materials is very limited, and the experimental conditions are rigorous. In recent years, plasmon induced transparency (PIT) devices based on metamaterials have attracted wide interest due to their simple implementation conditions and flexible design. Benefiting from destructive interference between bright and dark modes, a variety of optical structures have been proposed to realize optical analogies of EIT [23–33] and PIT [25,26]. However, many of the PIT structures are made of metallic materials, that can only be adjusted by changing the structure, the size or by embedding other tunable materials. Therefore, it is difficult to achieve dynamic tuning of the PIT window in practical applications. Fortunately, Double-window PIT and dynamically tunable PIT have been reported in graphene-based metamaterials [34–36]. However, many of the PIT structures are made of metallic materials, that can only be adjusted by changing the structure, the size or by embedding other tunable materials. Therefore, it is difficult to achieve dynamic tuning of the PIT window in practical applications. Furthermore, most of the previous graphene PIT structures are composed of discrete cells. In order to electrically regulate the separated graphene structures, it is necessary to add a complicated metal connection network, which greatly increases the cost and fabrication difficulty and adversely affects the resonance characteristics of the device. In addition, many PIT metamaterial structures exhibit the transmission characteristics of a single band with only one induced transparency window, the maximum transmittance of the induced transparency window being low and the bandwidth narrow, thus limiting the practical application of the PIT devices.
In order to solve the above problems, we adopt the topological concept of “Lakes of Wada” [37] to design an electrically tunable multifunctional trigate graphene metamaterial (TGGM) with broadband terahertz PIT windows. The designed structure exhibits either a distinct single PIT window or double transparency windows in the transmission spectra. Their resonance mechanism is further discussed based on the distributions of the near-electric fields. The corresponding transmission characteristics can be electrically tuned by the Fermi levels of the two dark modes. Moreover, we achieve active regulation of the group refractive index and the bandwidth. By regulating the Fermi levels of different graphene regions, the TGGM can also be used for near-field optical switching. Compared to other tunable PIT metamaterials consisting of many discrete graphene patterns, our design has the major advantage, that it consists of only three continuous graphene patterns. Therefore, it is very simple and convenient to implement electrical tuning on a plane by simply adding a metal electrode to the end of each continuous graphene region, which greatly simplifies the sample preparation. The proposed TGGMs are multifunctional and applicable for tunable THz optical devices, band-stop filters, slow-light devices, Kerr effects, super-Raman scattering, optical storage, quantum switches, sensing, near-field optical switches, and terahertz wireless communications.
2. Structural design and research method
The structure of the metamaterial unit designed according to the concept of “Lakes of Wada” is shown in Fig. 1(a). The possible fabrication method can be e-beam lithography and oxygen plasma etching [38], or femtosecond laser direct writing [39]. The unit cell can be divided into three constituent elements: a rectangular graphene dipole antenna (bright mode) with Fermi level Efm, and two horizontal graphene monopole antenna pairs (two dark modes) connected by respective rectangular arm of continuous graphene. The monopole antenna pair on the left (right) is referred to as MAPL (MAPR), and its Fermi level is Efl (Efr). They are all made of single-layer graphene on the same plane. Figure 1(b) is a schematic view of the TGGM structure. Periodically patterned graphene is deposited on top of a SiO2/ highly doped-Si substrate. The highly doped silicon acts as an electrode with an insulating layer of silicon dioxide. Three continuous graphene regions are respectively connected to three gold electrodes by narrow graphene ribbons. From the viewpoint of graph theory, the overall structure is a forest consisting of three trees. Three gate voltages are applied between the gold electrodes and the highly doped silicon substrate to regulate the Fermi levels of different graphene regions, respectively. The graphene patterns along each tree have the same bias voltage provided by its corresponding electrode. The excitation source is a linearly polarized plane wave propagating along the negative z direction.
Fig. 1. (a) Schematic diagram of a unit cell with the geometric parameters Px = 6 µm, Py = 5 µm, a = 1.6 µm, b = 3.6 µm, c = 0.9 µm, l = 0.6 µm, L = 2.6 µm, w = 0.6 µm, g = 0.1 µm, d = 0.1 µm, dsilica = 200 nm, ddoped_Si = 500 nm; (b) Schematic diagram of the overall structure of the TGGM device.
In graph theory, a graph is a pictorial representation of a set of objects with some pairs of objects connected by links. A tree is a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, equivalently a disjoint union of trees. In a graphene-based device, individual graphene patterns are the objects termed as vertices, and the conductive connections between vertices are the links termed as edges. Most of previous graphene metamaterials or metamaterials are null graphs, and few electrodes are individually connected to each graphene pattern. It is possible, but it would be quite complex and difficult to regulate the Fermi level of every graphene pattern by out-of-plane circuits for a large number of graphene patterns. It is an open question to adjust the Fermi levels of many graphene patterns with a few in-plane electrodes. Solving this question will increase the versatility and reduce the production cost of graphene-based devices.
Taking the example shown in Fig. 2, we list some common graphene structures. There are four genus 0 homeomorphic graphs of points in Fig. 2(a), each of which can be simplified to a single vertex. Figure 2(b) shows two genus 1 homeomorphic graphs and one genus 2 graph. From the viewpoint of a single bias voltage, a graph with a genus more than 0 can also be simplified to a single vertex. Figure 2(c) shows three nested structures. Whenever it is not necessary to regulate the Fermi level of graphene inside them, they can be simplified to a single vertex from the viewpoint of a single bias voltage. However, if a different voltage has to be applied also to the inner graphene pattern, then this type of structure is beyond the scope of this article, because bias voltage cannot be applied only through in-plane circuits. Figure 2(d) shows that a graphene unit contains multiple elements that need to be regulated independently, which means that the graphene unit contains multiple vertices. Graphene patterns with the same bias voltage can be connected by a conductive circuit. By following the principles from Fig. 2(a) to Fig. 2(d), the TGGM in Fig. 1(b) can be simplified into a forest composing of three trees as shown in Fig. 2(e), whose Euler characteristic is 3.
Fig. 2. (a) homeomorphic graphs of points; (b) homeomorphic graphs of rings and multiple rings; (c) nested structures; (d) graphene unit containing multiple elements; (e) simplified schematic diagram of the TGGM.
The “Lakes of Wada” topology was proposed by the Japanese mathematician Kunizō Yoneyama. Consider a land surrounded by an ocean of salty water and two lakes, a warm one and a cold one, within the land. In order to furnish the dry land with water from the lakes and the sea, imagine three canals being digged into the dry land. The Lakes of Wada are formed by starting with a closed unit square of dry land, and then digging three canals according to a certain rule [37]. The “Lakes of Wada” are three disjoint connected open sets of the plane that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes’ boundaries also contain that point.
We can consider the blue area in Fig. 1(b) as the dry land of the “Lakes of Wada” topology, and the three metal electrodes as warm, cold and salty water sources. Since it is known from the “Lakes of Wada” structure that there exists a common boundary of these three regions on one plane, any piece of dry land adjacent to any one of the three canals, in other words, an isolated graphene pattern can be connected to any of the three in-plane electrodes. Then if only three bias voltages are required to apply on a graphene-based device, it is definitely able to design the device with three in-plane electrical connections no matter how complex the device is.
The surface conductivity σs of a single-layer graphene follows the Kubo formula [40], including interband and intraband transition contributions. At terahertz frequency (ħω≪2Ef), the Pauli exclusion principle prohibits interband transitions. Since the approximate condition kBT≪Ef is satisfied at room temperature (here assumed to mean T = 300 K), the surface conductivity can be simplified to the following Drude-like expression [7,41]:
We set the carrier mobility of single-layer graphene on a SiO2 substrate to be 40000 cm2/Vs because this value has been reported for room temperature [42]. The complex permittivity of graphene is obtained from ε = 1 + iσs/(ε0ωt), where t is the thickness of the graphene layer and ε0 is the dielectric constant of the free space. With these values we performed numerical simulations using COMSOL Multiphysics based on the finite element method (FEM).
3. Results and discussions
In the case of y-polarized normal incidence of terahertz waves, as shown by the black curve in Fig. 3(a), the graphene dipole antenna array produces a low Q factor surface plasmon resonance with a resonance frequency of 3.91 THz and a Q factor about 14. However, the graphene monopole antenna pair under this incident condition is not activated in this frequency band due to the lack of an electric field component along the axis of the monopole antenna. As shown by the pink curve in the Fig. 3(a), the transmittance is always above 88% if there are only two monopole antenna pairs. As shown by the red curve in Fig. 3(a), a significant PIT window appears around the same resonance frequency of 3.91 THz as the dipole antenna, if the dipole antenna and the two monopole antenna pairs are combined into one unit cell (Efl = 0.53 eV, Efr = 0.53 eV). In this design, the dipole antenna is directly excited by the incident terahertz wave and thus acts as a bright mode, while the monopole antenna pairs cannot be directly excited as a dark mode. As shown by the blue curve in the Fig. 3(a), when Efl = 0.80 eV and Efr = 0.45 eV, two distinct transparent windows appear at 3.29 THz and 4.35 THz, respectively.
Fig. 3. (a) Transmission spectra of graphene dipole antenna, two graphene monopole antenna pairs and TGGM in the case of y-polarized normal incidence. Black curve: dipole antenna, Efm = 0.50 eV. Pink curve: two monopole antenna pairs, Efl = 0.53 eV, Efr = 0.53 eV. Red curve: TGGM for Efm = 0.50 eV, Efl = 0.53 eV, Efr = 0.53 eV. Blue curve: TGGM for Efm = 0.50 eV, Efl = 0.80 eV, Efr = 0.45 eV; (b) Simulated transmission spectrum (black solid line) and theoretical curve (red dotted curve) of the TGGM for Efm = 0.50 eV, Efl = 0.53 eV, Efr = 0.53 eV; (c) Simulated transmission spectrum (black solid line) and theoretical curve (red dotted line) of the TGGM for Efm = 0.50 eV, Efl = 0.80 eV, Efr = 0.45 eV.
When an incident wave (${\tilde{S}_ + } = {S_{1 + }}{e^{i\omega t}}$) is incident on the metamaterial, the bright mode and the dark modes will both be excited. We adopt a widely used coupled harmonic oscillator model to analyze the coupling effect between the bright mode and the dark modes [43,44]. The equations describing the resonance process are:
To further explore the intrinsic mechanism of the PIT peak, we present the electric field distributions of the dipole antenna and the TGGM at different frequencies. Figures 4(a)–4(c) are distributions of the electric field Ex component, Ey component, and the field strength |E| of the dipole antenna at 3.91 THz respectively. It can be seen from the figure that there is a strong electric field distribution at the edges and corners of the bright mode dipole antenna when there is no coupling with the dark mode, and the electric field distribution can be determined as a typical electric dipole resonance mode. Figures 4(d)–4(l) show the electric field distributions of the TGGM at different frequencies. From the figure we can clearly observe the transfer of electric field energy from the dipole antenna to the monopole antennas. The electric field distribution around the monopole antennas can be determined as the electric monopole resonance mode. Since one end of each monopole antenna is connected to one continuous graphene arm, the electric field is mainly distributed at the other end of the monopole antennas. The excitation of the monopole resonance mode proves the near-field coupling effect between the dipole antenna and the monopole antennas. The redistribution of the electric field around the dipole antenna causes the electric field to produce an x-direction component and excites the monopole antennas. The indirectly excited monopole resonance is further coupled back to the dipole antenna. Due to the π phase difference between direct excitation and indirect excitation, destructive interference between the dipole and the monopoles causes the electric field around the bright mode to be suppressed, which leads to the appearance of induced transparency windows. It can be seen from Fig. 4 that the electric field around the dipole antenna is almost completely suppressed, which strongly proves the destructive interference effect in the induced transparency window. In Fig. 4(i), the electric field energy is mainly concentrated between the dipole antenna and the MAPR when the incident wave frequency is 3.29 THz, which is mainly due to the fact that the increase of Efr causes a redshift of the MAPR resonance frequency. It can be seen from Fig. 4(1) that when the incident wave frequency is 4.35 THz, the electric field energy is mainly concentrated between the dipole antenna and MAPL, which is mainly due to a decrease of Efl resulting in a blueshift of the MAPL resonant frequency.
Fig. 4. Dipole antenna, Efm = 0.50 eV, f = 3.91 THz, (a) distribution of the Ex component of the electric field, (b) distribution of the Ey component, and (c) distribution of the electric field strength |E|. TGGM, Efm = 0.50 eV, Efl = 0.53 eV, Efr = 0.53 eV, f = 3.91 THz, (d) distribution of the Ex component, (e) distribution of the Ey component, and (f) distribution of the electric field strength |E|. TGGM, Efm = 0.50 eV, Efl = 0.80 eV, Efr = 0.45 eV, f = 3.29 THz, (g) distribution of the Ex component, (h) distribution of the Ey component, and (i) distribution of the electric field strength |E|. TGGM, Efm = 0.50 eV, Efl = 0.80 eV, Efr = 0.45 eV, f = 4.35 THz (j) distribution of the Ex component, (k) distribution of the Ey component, and (l) distribution of the electric field strength |E|.
The impedance of the antenna has an important influence on the performance of the actual device. Therefore, we calculate the impedance of the dipole antenna and the TGGM according to the effective parameter theory. It can be seen from Fig. 5 that in the non-resonant region, the real part (resistance) of the impedance approaches 1, while the imaginary part (reactance) of the impedance approaches 0, which matches the free space impedance. At the resonance frequency, the resistance increases sharply due to the absorption and scattering of the terahertz wave by the device. When the frequency is smaller than the resonance frequency, the resonance exhibits a capacitive characteristic. When it is larger than the resonance frequency, the resonance corresponds to an inductive characteristic. The impedance near the induced transparent window matches the free space impedance, which helps to reduce scattering and absorption losses.
Fig. 5. Curves of the real part of the impedance (blue curve: resistance) and the imaginary parts (red curve: reactance) as a function of frequency for different structures: (a) dipole antenna; (b) TGGM, Efm = 0.50 eV, Efl = 0.53 eV, Efr = 0.53 eV; (c) TGGM, Efm = 0.50 eV, Efl = 0.80 eV, Efr = 0.45 eV.
The intrinsic loss (electron mobility) of graphene has an important influence on the performance of graphene-based devices, too. We calculate the effect of different carrier mobilities on the transmission spectra of the TGGM with carrier mobility ranging from 5000 cm2/Vs to 40000 cm2/Vs. As shown in Fig. 6, being consistent with the prediction, the carrier mobility does not affect the frequencies of the induced transparency windows. When the carrier mobility is low, the absorption occupies a dominant position, and thus the transmission efficiency in the PIT windows drops a lot due to high loss. The increase in carrier mobility results in an increase in the highest transmittance of the induced transparency window, in deeper transmission valleys, and in a significant increase of the induced transparency effect.
Fig. 6. Transmission spectra of the TGGM for different graphene carrier mobilities. Pink curve: µ = 5000 cm2/Vs; Blue curve: µ = 10000 cm2/Vs; Red curve: µ = 20000 cm2/Vs; Black curve: µ = 40000 cm2/Vs.
The dipole and the monopole antennae are respectively connected by continuous narrow graphene ribbons. Therefore, it is only necessary to add a metal electrode at the end of each continuous graphene structure, and the Fermi levels of the bright and dark modes can be controlled separately and conveniently. The conductive connection of the narrow graphene ribbons does not affect the characteristics of the graphene microstructures at the corresponding resonance frequencies. The manufacturing process is relatively simple, which is a great advantage for practical applications. By applying three individual voltages between the electrodes and the substrate, we can adjust the performance of the device by regulating the Fermi levels of the bright and dark modes. As shown in Fig. 7(a), when Efr and Efl both fall to 0.01 eV, the detuning of the transmission spectra between the two dark modes and the bright mode becomes large. As a result, the transparency window disappears, and the transmission spectrum becomes a broadband dipole resonance valley. As shown in Fig. 7(c), when Efl = 0.53 eV and Efr = 0.01 eV, the detuning of the transmission spectra between the dark mode MAPR and the bright mode is large. Therefore, the bright mode couples only with the dark mode MAPL, resulting in appearance of the transparency window caused by the MAPL. As shown in Fig. 7(e), when Efl = Efr = 0.53 eV, the detuning of the resonance frequencies between the bright and dark modes is very small, and there is only a PIT window at 3.91 THz. The induced transparent window becomes wide, the two neighboring valleys become deep, and the group refractive index increases. Because the induced transparency window is blue-shifted with the increase of the Fermi level of the dark mode, we can also achieve double transparent windows with a high group refractive index by adjusting Efr and Efl to suitable values. For example, when Efl = 0.80 eV and Efr = 0.45 eV, there are two induced transparency window caused by the MAPL and MAPR, as shown in Fig. 7(g).
Fig. 7. (a,c,e,g): Transmission spectra of the TGGM for different Fermi levels of the dark mode (Efm = 0.50 eV); (b,d,f,h): The black curves are the group refractive indices ng, and the red dotted lines are the imaginary parts Im(ne) of the effective refractive indices.
Within the tuning range, we can achieve a maximum transmission difference from 0.97 to 0.23, and also achieve dynamic tuning of dual transparent windows and single transparent windows, which are of great practical value. As illustrated in Fig. 7(b, d, f, h), we can realize real-time regulation of the group refractive index. In the PIT window, the group refractive index is high, the imaginary part of the effective refractive index is very small, and the maximum group refractive index is as high as 819, much larger than the previous similar work [33], which means that the TGGM has great potential for slow light applications.
As shown in Fig. 8, in addition to regulating the Fermi levels of the dark modes, we can also adjust the overall operating band of the device by controlling the Fermi level of the bright mode. When Efr = Efl = 0.01 eV, the minimum transmission decreases obviously with the increase of Efm. The center frequency has a blueshift from 3.03 THz to 5.25 THz, and the half-width of the transmission valley increases. In this case, the TGGM can be used as a tunable terahertz band-stop filter.
As shown in Fig. 9(a), when Efm = 0.53 eV and Efr = 0.01 eV, the TGGM is equivalent to previously reported metamaterials that can only adjust one of the dark modes [45]. In this case, the device has a very small tuning frequency range, and the transmission spectrum gradually becomes a wide-band dipole resonance valley with a decrease of Efl. As shown in Fig. 9(b), when Efl = 0.53 eV, Efr = 0.01 eV, and Efm is increased, the center frequency of the transparent window has a blueshift from 2.83 THz to 4.77 THz, the minimum transmittance obviously decreases, and the bandwidth of transparent window is increased from 0.92 THz to 1.60 THz. In practical applications, it is quite necessary to increase the bandwidth and group refractive index of the transparent window. Here our design can achieve this aim by adjusting Efr to get the MAPR involved. As shown in Fig. 9(c), when Efl = 0.53 eV and Efr = 0.53 eV, Efm is increased, the center frequency of the transparency window has a blueshift from 3.01 THz to 5.23 THz, the minimum transmittance obviously decreases, and the bandwidth of the transparency window is increased from 1.22 THz to 2.04 THz. As shown in Fig. 9(d), when we adjust Fermi levels of the three graphene trees simultaneously from Efm = 0.30 eV, Efl = 0.50 eV, Efr = 0.27 eV up to Efm = 0.90 eV, Efl = 1.30 eV, Efr = 0.80 eV, the center frequencies of the double transparency windows shift from 2.53 THz and 3.37 THz to 4.43 THz and 5.87 THz, respectively. The valley transmittances significantly decrease, and the bandwidths of the double transparency windows increase from 0.54 THz and 0.96 THz to 0.88 THz and 1.60 THz, respectively. That is to say that we have achieved a wide bandwidth of the device by controlling the Fermi levels. Comparing with previous PIT structures composed of discrete graphene patterns on a plane, we can tune the overall operating frequency band of the device by controlling the Fermi levels, which has great advantages in practical applications.
Fig. 9. (a) Efm = 0.50 eV, Efr = 0.01 eV, transmission spectra of the TGGM for different Efl; (b) Efl = 0.53 eV, Efr = 0.01 eV, the transmission spectra of the TGGM for different Efm; (c) Efl = 0.53 eV, Efr = 0.53 eV, the transmission spectra of the TGGM for different Efm; (d) the transmission spectra of the TGGM when Efm, Efl and Efr change simultaneously.
Then, we consider the influence of ambient refractive index on the resonance frequency when Efm = 0.50 eV, Efl = 0.80 eV, and Efr = 0.45 eV. In the simulation, the thickness of an analyte is set to be 0.7 µm, and the refractive index changes from 1.0 to 2.5. As shown in Fig. 10(a), with the increase of the ambient refractive index, the transparency peak has a significant redshift. From Fig. 10(b), we can see that the shift of the dip frequency has a linear relationship with the ambient refractive index. The refractive index sensitivity for the right dip is as high as 21.92 µm/RIU (1.224 THz/RIU), which is lower than 1.9082 THz/RIU [46]. But it is much higher than the previous results in the similar field, such as 5100 nm/RIU in graphene metamaterial [47], 1570 nm/RIU in photonic crystal [48], and 1500 nm/RIU in silicon-based resonator [49]. The super-high sensitivity of the TGGM demonstrates that the TGGM can be used for highly sensitive refractive index sensing at terahertz range.
Fig. 10. (a) Transmission spectra of the TGGM induced by different ambient refractive index (Efm = 0.50 eV, Efl = 0.80 eV, Efr = 0.45 eV); (b) Relationships between the dip frequency shifts and the ambient refractive index. The black squares, the blue triangles and the red inverted triangles are the simulated results of the left dip, the mid dip and the right dip, respectively. The solid lines are linear fits to the results.
As shown in Fig. 11, the TGGM can also be used for near-field optical switching. For example, it can be used to place quantum dot materials or optical fibers between a bright mode and a dark mode. To quantitatively describe the spatial difference of the near-fields, a figure of merit (FOM), which can describe the ability of switching, is defined as [50]
where Er is the electric field between the right dark mode and the bright mode, and El denotes the electric field between the left dark mode and the bright mode. The length and width of the integration region are selected as 0.60 µm and 0.10 µm, respectively. When Efm = 0.50 eV, Efl = 0.30 eV, Efr = 0.01 eV, f = 2.57 THz, the FOM can reach up to 0.0019. When Efm = 0.50 eV, Efl = 0.01 eV, Efr = 0.30 eV, f = 2.57 THz, the FOM can reach up to 485.7. When Efm = 0.50 eV, Efl = 0.90 eV, Efr = 0.01 eV, f = 4.19 THz, FOM can reach to 0.0005. When Efm = 0.50 eV, Efl = 0.01 eV, Efr = 0.90 eV, f = 4.19 THz, FOM can reach to 1861. We can control the Fermi levels of the bright and dark modes with a tuning range of 1.62 THz for selective excitation of the quantum dots or for control of fiber coupling. These indicate that TGGM has great potential in the field of near-field optical switches.Fig. 11. Electric field strength |E| of the TGGM (a) for Efm = 0.50 eV, Efl = 0.30 eV, Efr = 0.01 eV, f = 2.57 THz; (b) for Efm = 0.50 eV, Efl = 0.01 eV, Efr = 0.30 eV, and f = 2.57 THz; (c) for Efm = 0.50 eV, Efl = 0.90 eV, Efr = 0.01 eV, and f = 4.19 THz; (d) for Efm = 0.50 eV, Efl = 0.01 eV, Efr = 0.90 eV, and f = 4.19 THz.
The operating speed of the TGGM is limited by the RC time constant of the device. The capacitance C mainly comes from parasitic capacitance. It can be calculated by the following formula: C = ε0εdAg /dsilica, where ε0 is the vacuum dielectric constant, εd is the relative dielectric constant of silicon dioxide, dsilica = 200 nm, and Ag is the effective area of graphene. If we assume that the TGGM is a 100 × 100 array, then we can estimate that Ag is approximately equal to 0.13 mm2, and we can calculate the device capacitance to be approximately 22.4 pF. If we assume that the substrate of the device is a highly doped silicon with very low resistance, then the resistance will be mainly from graphene. According to previous reports, highly doped graphene can greatly reduce the resistance of the modulator, and the resistance can be as low as ∼125 Ωsq−1 on the order of centimeters [51,52]. Based on the above discussion, if the resistance is assumed to be 125 Ω, the relationship 1/2πRC can be used to estimate the operation rate of the device as high as ∼56.8 MHz. The operation rate can be further increased by increasing the thickness of the silica layer or by decreasing the resistance of graphene.
4. Conclusions
In summary, we propose a novel electrically tunable multifunctional trigate graphene metamaterial based on the concept of “Lakes of Wada”. We discuss the advantages of this continuous graphene structure for sample fabrication and regulation. We utilize a widely used coupled harmonic oscillator model to analyze the coupling effect between the bright mode and the dark modes. We further study the distributions of the electric field and explain the excitation mechanism of the transparency peak. We discuss how the transmission spectrum changes with the Fermi levels. By controlling the Fermi levels, we achieve the versatility of the device, high bandwidth and overall tuning of the operating frequency band, which has great advantages in practical applications. We achieve active regulation of the group refractive index with a maximum of 819. When the TGGM is used to fabricate a terahertz refractive index sensor, its refractive index sensitivity is as high as 1.224 THz/RIU. We demonstrate the potential applications of our metamaterial design in terahertz band-stop filter, terahertz refractive index sensor, double PIT windows, near-field optical switch, slow-light device, etc. Our structure based on the concept of “Lakes of Wada” provides a very meaningful guide for the design of compact and tunable graphene devices.
Funding
Natural Science Foundation of Tianjin City (17JCYBJC16700, 18JCQNJC02100); Hundred Young Academic Leaders Program of Nankai University; PCSIRT (IRT_13R29); Higher Education Discipline Innovation Project (B07013); National Natural Science Foundation of China (11674182, 11774185, 91750204); National Key Research and Development Program of China (2017YFA0303800, 2017YFA0305100).
Disclosures
The authors declare no conflicts of interest.
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