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Abstract
Perfect state transfer of the bus topological system enables the sharing of information or excitation between nodes. Herein we report groundbreaking research on the transfer of the graphene-bridged bus topological network structure to an electromagnetic metamaterial setting, named “bus topological network metamaterials (TNMMs).” Correspondingly, the electromagnetic response imprints onto the topological excitation. We find that the bus-TNMMs display a perfect modulation of the terahertz response. The blue-shift of resonance frequency could increase to as large as 1075 GHz. The modulation sensitivity of the bus-TNMMs reaches 1027 GHz/Fermi level unit (FLU). Meanwhile, with the enhancement of modulation, the line shape of the reflection keeps underformed. Parabola, ExpDec1, and Asymptotic models are used to estimate the modulation of the resonance frequency. Besides, the bus-TNMMs system provides a fascinating platform for dynamic cloaking. By governing the Fermi level of graphene, the bus-TNMMs can decide whether it is cloaking or not in a bandwidth of 500 GHz. Also, the bus-TNMMs exhibit the immense potential for dynamically detecting the vibrational fingerprinting of an analyte. These results give a far-reaching outlook for steering dynamically the terahertz response with the bus-TNMMs. Therefore, we believe that the discovery of bus-TNMMs will revolutionize our understanding of the modulation of the electromagnetic response.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Artificially designed arrays of micro- and nanometer-scale structures named metamaterials (MMs) possess the capability to generate exotic optical response owing to the interaction between light and free electrons or phonons, showing novel properties unavailable in natural materials [1–5]. Since Shelby et al. using the split ring block demonstrated the existence of negative refraction in MMs [6], different geometric shapes (e.g., C-shape, V-shape, H-shape, etc.) have been designed to constitute distinct functional MMs [7–12]. Then MMs open epoch-making prospects in the fields of “invisible” cloaks [13–14], perfect absorbers [15–16], and sensors [17–18]. To enhance the performance of these optical meta-devices, the realization of active modulation of optical response remains one of the most urgent tasks, which requires a broad and accurate tunability of resonance response [19,20]. The corresponding strategies are implemented depending on the alteration of the pixel of meta-atoms or resonant environment [21–24]. To date, numerous means for controlling resonance response have demonstrated the extraordinary ability to overcome the obstacles originating from the inherent physical characteristics of the conventional MMs, such as the electrical [25–27], optical [28–29], thermal [30], or mechanical methods [31–32]. However, once we accomplish the fabrication of such MMs, the meta-atoms with the fixed structure, size, and arrangement can only govern the static electromagnetic (EM) response. As a result, the modulation of the resonance response guided through these extra-means is hardly achievable. Alternative strategies of directly changing the inherent physical characteristics of meta-atoms through hybridizing a unique photo-responsive material (such as graphene, perovskite, etc.) show more advanced capability to solve this issue [8,33]. Although this scheme could tune the resonant frequency, it makes scattering spectrum imperfect, such as the deformation of the spectral shape, the reduction of Q-factor, and even the disappearance of resonance peaks (dips).
In MMs, the exotic resonant response may enable us to effectively eliminate the scatterings of EM waves, providing a precious chance to acquire the “invisible” cloak. Since the initial certification of “invisible” cloak in meticulously designed MMs, it encourages more research interests in the optical community. At present, various strategies reveal the “invisible” cloak. However, the resonant response of the EM scatterings for these methods is painful to achieve dynamic tunability. Thus, it would be more significant to obtain dynamic invisible cloaks with advanced MM structures. Meta-sensor, another important application of MM, has also attracted much-growing attention, which has emerged as a more promising photonic platform for the identification of resonance frequencies, owing to a spectrally narrow response and high local field concentration. The multi-pixel arrays of MMs are successful cases used as detectors, whose resonant response can be tuned to that of the vibrational modes of a target analyte. However, the flexibility and integration of the application for this kind of detectors are still restricted.
Topology relates directly to the geometric layout of the various nodes in a network. A generalization of the point-to-point topology to multiple nodes network is the alleged bus topology, which enables a node directly connected to the bus through the shared channel [34]. The excitation of any node can be transmitted and diffused along the bus and can be received by any node in the network, showing a keen ability to share resources. Owing to the superior ability, we can transfer the structure of the bus topology network to an electromagnetic MM setting, where the electromagnetic response of the MMs system imprints onto the topological excitation. We define this idiographic metamaterial as bus topology network metamaterials. It is found that the bus topology network metamaterials (TNMMs) may display a perfect modulation of electromagnetic response.
2. Results and discussion
As shown in Fig. 1(a), the entire architecture depicts schematically the bus topology MMs structure along with the geometrical parameters of meta-atom revealed in the inset of the Fig. 1(a). The structure of meta-atoms consists of a bottom gold film of 200 nm thicknesses with the capability to reflect THz waves, a polyimide substrate with the permittivity ε of 3.1, and the thicknesses of 10 µm, and a meta-atom structure with a bus topology. A meta-atom contains a horizontal gold bar serving as a bus, five rectangular gold blocks acting as nodes, and five vertical gold bars connecting each node directly to the bus. Besides, four graphene belts bridges form topology connections between the meta-atoms. A horizontal graphene belt attaches to the bus, and the other three vertical graphene belts link to a node with longer vertical gold bars, respectively. In the bus topology MM, the diversification of the conductivity of graphene belts enables the electromagnetic excitation of the node to convert utilizing Fermi level change. Moreover, the electromagnetic excitation sent by one node can be “listen” by other nodes in the bus topology MMs, indicating that all nodes can share the electromagnetic excitation. The relationship between the Fermi level and the conductivity of graphene ribbons is described by Kubo’s formalism [34,35]. The optical feature of graphene can be characterized by the surface conductivity the intraband carrier and the interband carrier transition [5,36]. This can be explained using Kubo’s expression as [37]
Fig. 1. Design of electrically controllable the bus-TNMMs. a) Schematic image of the bus-TNMMs. The incident THz wave is along the z-axis. The geometrical parameters are presented in the inset of the Fig. b) The reflection spectra of the bus-TNMMs without graphene. c) The corresponding electric field distribution for dipA. d) The corresponding electric field distribution for dipB. e) The reflection spectra of the bus-TNMMs with graphene of 0 eV. f) The corresponding electric field distribution for dipC. g) The corresponding electric field distribution for dipD. h)The reflection spectra of the bus-TNMMs with graphene of 0.2 eV. i) The corresponding electric field distribution for dipE
In simulation, the relax time of graphene belts is 0.4ps, the thickness of graphene is 0 nm, the temperature is 293 K. Then, the gate voltage and Fermi level (chemical potential) of graphene can be described as following [38],
where μc is the chemical potential, n is the carrier concentration and vF is the Fermi velocity of the graphene, Vg is gate voltage, ε0 and ε are the permittivities of free space and insulated substrate material, respectively, and d is the thickness of insulated substrate material.Experimentally, the metal structure can be engineered by current micro-nano technology [4,36,39]. Then the graphene layer is transferred to the pre-patterned substrates with bus topological network structure. After that, the graphene is patterned by conventional photolithography processes [8]. Finally, Ion-Gel Layer polystyrene is spread on the top of the bus-TNMMs [8].
In comparison to the previous design of meta-atom, the micro- and nanometer-scale structure of the bus topology presents a strong ability to share the dielectric resources among the electromagnetic nodes, and excellent electromagnetic modulation, which perfectly balances the influences on the electromagnetic excitation. Besides, the feature that all nodes can share dielectric resources may lead to ultra-sensitive modulation of the resonance response. To study the nature of the bus topology MMs, we applied the finite integration technique to evaluate the resonance response of the bus-TNMMs unit cell. The conductivity of lossy gold metal is 4.561 × 107 S m−1. As demonstrated in Fig. 1(b), the reflection spectra of the bus-TNMMs without graphene shows hybrid resonance mode, including a Fano resonance and a plasmonic resonance. The two rectangular gold blocks connecting short gold bars make the topology structure asymmetric in the plane of the structure. Such symmetry breaking directly couples between bright and dark modes. the electric field distribution of the dip A is shown in Fig. 1(c), it can be found that the electric field distribution for the two rectangular gold blocks connecting short gold bars and the three rectangular gold blocks connecting long gold bars are the opposite. This result proves that the asymmetric structure not only directly couples the y-polarized incident light, but also couples the dark mode to the perpendicular (x) polarization [21]. Figure 1(d) demonstrates the electric field distribution of the dip B, where it displays a dipole plasmonic resonance.
However, when the graphene belts with the fermi level of 0 eV are attached to the bus topology network, the Fano resonance mode weakens, the plasmonic resonance becomes the primary resonant mode (see Fig. 1(e)). The corresponding electric field distribution for the dip C and D are exhibited in Figs. 1(f) and (g). As the Fermi level changes to 0.2 eV, the Fano resonance mode completely disappears. The hybrid resonance mode transforms into a single plasmonic mode, which has been verified by the corresponding electric field distribution, as can be seen in Fig. 1(i). These results indicate that the bus-TNMMs have an excellent performance in actively tuning electromagnetic resonance response because all of the nodes can share the electromagnetic excitation.
To study the intrinsic physical mechanism of the electroexcitation-mediated modulation of the electromagnetic resonance response through the bus-TNMMs, we analyze the topological characteristics of nodes in the network. As shown in Fig. 2(a), the bus-TNMMs system consists of five rectangular gold nodes labeled by 0, 1, 2, 3, 4, which couple to each other through the bus [34,40]. As the Fermi level is 0 eV, the five sites are in the electromagnetic ground state. However, with the change of Fermi level, the source node is in the electromagnetic excited state. Thus the source node (0, 2, 4) may preferentially transfers the electromagnetic excitation to other nodes (1, 3).
Fig. 2. The modulation of resonance frequency for reflection at different Fermi levels. a) State transfer of electromagnetic resonance response of the bus topology system b) The reflection spectra of the bus-TNMMs with single-layer graphene at the Fermi levels ranging from 0 to 1 eV. c) The reflection spectra of the bus-TNMMs with double-layer graphene at the Fermi levels ranging from 0 to 1 eV. d) The reflection spectra of the bus-TNMMs with three-layer graphene at the Fermi levels ranging from 0 to 1 eV. e) The reflection spectra of the bus-TNMMs with a single-layer graphene at the Fermi levels ranging from 0.28 to 0.35 eV f). The resonance frequency shift (Δf) under the different Fermi levels of single-layer graphene extracted from (c).
The effective Hamiltonian, utilizing perturbation theory can describe the problem of the electromagnetic excitation transfer for electromagnetic waves [41], and a hypothesis of perturbed Hamiltonian is ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over H} _{ \uparrow / \downarrow }} = {v_D}{\hat{\tau }_0}{\hat{s}_0}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\mathrm{\sigma}} } _\parallel }\cdot \delta {\textrm{k}_\parallel }\textrm{ + }m{\hat{\tau }_3}{\hat{s}_3}{\hat{{\mathrm{\sigma}} }_3}$, where${\hat{\tau }_i}{\kern 1pt} \textrm{and}{\kern 1pt} {\kern 1pt} {\hat{s}_i}$are the inter-valley and pseudo-spin Pauli matrices, vD is the group velocity near the Dirac point. m is the mass term describing the degree of σz inversion-symmetry breaking [42]. To verify the above statements, we obtain the reflection spectra for varying Fermi levels of single- and double-layer graphene. The changes of the Fermi levels from 0 to 1 eV enable the resonance frequency of the bus-TNMMs to undergo a wide-ranging blue-shift (see Figs. 2(b)–(d)). For single-layer graphene, by applying Fermi levels from 0 to 1 eV, the reflection spectra could be modulated uncommonly. The bus-TNMMs shows a blue-shift as up to 876 GHz from the resonance frequency of 0 eV (Fig. 2(b)). For the double-layer graphene, the blue-shift could increase as large as 1075 GHz (Fig. 2(c)). For the three-layer graphene, the blue-shift could achieve to 1060 GHz (Fig. 2(d)). Besides, the observations from Figs. 2(b)–(d) show that with the enhanced modulation for resonance frequency, the line shape of the reflection keeps not deformed. These results highlight the performance of modulation of the bus-TNMMs, as compared to the modulations in any previous works on active THz MMs [8,33,37]. Therefore, we may conclude that the bus-TNMMs display a perfect modulation.
Moreover, to better understand the modulated performance of the bus-TNMMs, we also obtain the reflection spectra for tiny changes of Fermi levels, as illustrated in Fig. 2(e). The difference of just 0.01 eV Fermi levels between 0.28 and 0.29 eV can cause a significant blue-shift of 9.6 GHz in resonance frequency. While the difference of Fermi levels expands to 0.07 eV, the blue-shift of the corresponding resonance frequency reaches to 72 GHz. Thus, we employ the concept of modulation sensitivity to describethe performance of the bus-TNMMs, which can be defined by the derivative of resonance frequency shift to the change of Fermi levels. The dependence of the blue-shifts Δf = f (end) - f (begin) on the Fermi levels of the graphene are extracted and displayed in Fig. 2(f). The curve presents the linear relationship of blue-shifts (Δf) on Fermi levels in a minimal local frequency range. The modulation sensitivity of the bus-TNMMs achieves to 1027 GHz/FLU (FLU, Fermi level Unit). These results clearly demonstrate unprecedented modulation capabilities of the THz wave by bus-TNMMs.
To the best of our knowledge, there is no research reported so far on the modulation of resonance frequency by empirical models. The objectives of this section are to choose the best mathematical models to describe the modulation of resonance frequency accurately in the bus-TNMMs and to estimate the resonance frequency modulation process. The embedded tables in Figs. 3(c)–(e) show three empirical models (Parabola, ExpDec1, and Asymptotic model), utilized to fit the simulation data (resonance frequency shift versus Fermi level) of modulation in the bus-TNMMs system. To determine the coefficient of equations and statistical test parameters (the coefficient of correlation (R2)), we employ the non-linear regression analysis to fit the three empirical models. The higher R2 value is selected as the criteria for the goodness of fit. R2 parameter can be obtained from [43]
Fig. 3. The comparison between the simulated resonance frequency and theoretical fitting results with the Fermi levels from 0 to 1 eV. a) The reflection spectra of the bus-TNMMs with a single-layer graphene at the Fermi levels ranging from 0 to 1 eV. b) The Q-Factor with Fermi levels ranging from 0 to 1 eV. c) Parabola mode applied to control of resonance frequency with the Fermi levels from 0 to 1 eV d) ExpDec 1 mode applied to control of resonance frequency with the Fermi levels from 0 to 1 eV e) Asymptotic1 mode applied to control of resonance frequency with the Fermi levels from 0 to 1 eV.
Figure 4(a) schematically shows the modulated principle of the dynamic cloaking system. The THz scatterings would be freely controlled through a correct switch of the Fermi level in the bus-TNMMs system. Here, THz energy is mainly scattered in the form of reflection, whereas the modulation of resonance frequency could determine the reflection or perfect absorption of THz energy. There is a perfect absorption at the dip of the resonance frequency, which eliminates THz scatterings. Therefore, the bus-TNMMs system may provide a fascinating platform for dynamic cloaking. By governing the Fermi level of graphene, the bus-TNMMs system can decide whether it is cloaking or not. The evolution of the on-resonance absorption is depicted as a function of the Fermi level of graphene in Figs. 4(b) and (c). We can observe that the absorption peak undergo significant blue-shift, as Fermi level increases from 0 to 1 eV (see Figs. 4(b) and (c)), providing a chance for the dynamic scattering cancellation-based cloak. Initially, when the Fermi level of single-layer graphene is 0.2 eV, the absorption peak value is up to 99.8%. Meanwhile, the corresponding reflection value is 0.2% at 1.18THz. In contrast, this absorption peak experiences a dramatic decrease of 86.2% at the Fermi level of 1 eV (Fig. 4(b)). Such a phenomenon accurately demonstrates that the bus-TNMMs system could realize the switch of cloaking at 1.18THz when the Fermi level converts back and forth between 0.2 and 1 eV. Concurrently, by switching back and forth between 0.2 and 1 eV, the dynamic control of the scattering cancellation-based cloaking is running at 1.73THz. With further control of the Fermi level, the modulation of the absorption peak would experience an ultra-wideband of 500 GHz blue-shift. In other words, the switch of cloaking is tunable in an ultra-wideband of approximately 500 GHz.
Similarly, the bus-TNMMs with double-layer graphene also shows the outstanding performance of the scattering cancellation-based dramatic cloaking (see Fig. 4(c)). Wherein, when the Fermi level of double-layer graphene is 0 eV, the absorption peak value is up to 99.98%, the corresponding reflection value is 0.02% at 0.85THz. In contrast, the absorption peak undergoes a dramatic decrease of 89.5% at the Fermi level of 1 eV, indicating the predominant switch of cloaking at 0.85THz. However, compared to the single-layer graphene, the reflection of THz wave enhances as the Fermi level, which reduces the cloaking effect.
Fig. 4. Schematic of dynamic cloaking and properties of the absorption in the bus-TNMMs. a) Illustration of the bus-TNMMs-based reflection-control system for a dynamic cloak. b) Absorption spectra of the bus-TNMMs with single-layer graphene. c) Absorption spectra of the bus-TNMMs with a double-layer graphene.
To highlight the tunable application, we apply the bus-TNMMs to serving as a meta-sensor, which can conveniently identify the resonance frequencies of an analyte in the THz region. The detection of resonance frequencies of an analyte depends on the calculation of the difference of the reflectivity between the bare and functionalized meta-sensor (ΔR = Rbare - Rfunc). The optical transduction of vibrational fingerprinting of an analyte to the bus-TNMMs is carried out by calculating the normalized differential reflectivity D=ΔR/R for each Fermi level [21]. By calculating the first frequency derivative of the reflectivity spectrum G(ωD)(ω)= −(d(ΔR(ωD)/dω)/ (d(R(ωD)/dω)(ω=ωD), a local maximum G(ωD)(ω) describes that the resonance frequency of the bus-TNMMs w is matched to that of the vibrational mode of an analyte [21]. Figure 5(a) illustrates the plots of G(ωD)(ω) for different Fermi level couples to an analyte with the thicknesses of 10 nm and the non-specific refractive index of 1.55. The observation from Figs. 5(a) and (b) confirms that the maxima of G(ωD)(ω) is matched to a vibrational model of the analyte as the Fermi level of graphene is 0.6 eV, indicating that there is a strong coupling between the bus-TNMMs and an analyte with the most energetic vibrations. Compared with previous literature, for example, Wu at el. fabricated the multipixel Fano-resonant asymmetric metamaterials samples to study the resonance frequency of a protein A/G monolayer [21]. This method has a shortcoming in which the resonance frequency of samples cannot be modulated. To realize the objective of finding out the vibrational fingerprinting of a protein A/G monolayer, they must manufacture lots of samples with different pixels unit cells. Therefore, this strategy is quite clumsy. Anyway, our strategy exhibits the merits of the flexibility, accurateness, and integration for detecting the vibrational fingerprinting of an analyte. It may provide a promising platform for meta-sensor.
Fig. 5. Identification of resonance frequencies for an analyte employing the normalized first-frequency-derivative spectra from the bus-TNMMs system whose resonance frequency can be dynamically controlled by the change of Fermi level of graphene. a) The resonance frequency under different Fermi levels. b) The normalized first-frequency-derivative spectra under different Fermi levels enable the bus-TNMMs system to possess different resonance frequencies.
3. Conclusions
In summary, we have realized the concept of the bus-TNMMs, consisting of the hybrid graphene and metal topological structure. This strategy opens up new possibilities for steering dynamically electromagnetic response in the THz region. Such a platform gives a dynamic and perfect topological method towards modulating THz waves scattering. Besides, this novel platform enables the research of various Far-reaching electromagnetic applications such as perfect modulation, dynamic cloaking, meta-sensor, and optical switching. In particular, the perfect modulation of electromagnetic response proposed here exhibits the versatility of the bus-TNMMs system for enhancing the performance of these optical meta-devices. The exploration of governing conversion of cloaking and dynamic detecting the vibrational fingerprinting of analytes is also of great significance. The discovery of bus-TNMMs would be a milestone for expanding the performance of the optical meta-devices.
Funding
National Natural Science Foundation of China (NSFC) (61701434, 61735010, 61675147, 11974304); Taishan Scholar Project of Shandong Province (tsqn201909150); Project Funding for Qing Tan Scholar; Natural Science Foundation of Shandong Province (ZR2017MF005, ZR2018LF001); National Key Research and Development Program of China (2017YFB1401203, 2017YFA0700202); Shandong Province Higher Education Science and Technology Program (J17KA087); Program of Independent and Achievement Transformation plan for Zao Zhuang (2016GH19, 2016GH31); Qingchuang Science and Technology Plan of Shandong Universities (2019KJN001); Zao Zhuang Engineering Research Center of Terahertz; Natural Science Foundation of Jiangsu Province (BK20180862); China Postdoctoral Science Foundation (2019M651725); Graduate Research and Innovation Projects of Jiangsu Province (KYCX19_1583).
Disclosures
The authors declare no competing financial interest.
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