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Abstract
A novel design of a dynamic electromagnetically induced transparency (EIT) metamaterial is proposed. The metamaterial consists of two kinds of cut-wire metal resonators with a graphene strip placed between them. The destructive interference between the two resonators gives rise to a transparency window. By varying the Fermi energy of the graphene through external gating, the EIT effect can be manipulated dynamically and the maximum modulation depth can reach up to 81%. The efficient modulation is controlled only in the EIT window with slight changes in transmission dips, which may avert the additional noises at adjacent frequencies in the modulation process. Moreover, the actively controlled slow-light effect and sensing performances can also be realized as the corresponding EIT window is modulated. This work provides a strategy to achieve a tunable EIT effect in a metal-graphene hybrid structure and exhibits potential applications in designing terahertz modulators, environmental sensors and slow-light devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is a quantum coherent effect which was first observed in a three-level atomic system [1]. This phenomenon leads to a narrow transparent spectrum through an opaque medium accompanied with strong dispersion, and hence produces ultraslow group velocity and enhanced light-matter interactions [2,3]. Recently, mimicking EIT effect using metamaterials has attracted enhanced attention since it avoids the extreme experimental requirements in atomic systems, such as low temperature and stable gas lasers [4]. The EIT-like behaviour in metamaterials is generally obtained through two main approaches of near field coupling [5]: bright-bright mode coupling and bright-dark mode coupling. Theoretical predictions of EIT analogues in metamaterials have been reported and experimentally demonstrated in microwave, terahertz and near-infrared regions by different groups [6–9]. The EIT metamaterials exhibit potential applications including slow light [10], sensing [11], and optical modulators [12]. However, most of these EIT metamaterials lack active manipulation once fabricated, which severely hinders their further applications.
Up to now, some approaches have been reported to achieve a dynamical control of the EIT effect based on metamaterial. Thermal control of the EIT window was achieved by employing superconducting materials as a part of the EIT system [13]. Optical tuning was demonstrated using photoconductive material as the dielectric [14]. Micro-Electro-Mechanical Systems (MEMS) based metamaterials have recently also been reported to actively manipulate the transparency window [15,16]. Nevertheless, the optical and thermal excitations result in low modulation depth. And the MEMS technologies are limited by complex structures for massive fabrication. In the past decade, graphene, a two-dimensional material has attracted tremendous interest due to its peculiar electric, mechanical, and thermal properties [17–19]. Its conductivity can be dynamically controlled by tuning the Fermi energy through electrostatic gating or chemical doping which can help realize large-scale modulations in metamaterials [20–23]. The monolayer graphene sheet is flexible and easy to be integrated. Therefore, graphene is a good candidate for designing and fabricating tunable metamaterial to actively control the EIT effect.
So far, various graphene-based metamaterials have been proposed to realize selective control of the coupled resonators in EIT systems. For example, Izadshenas et al. investigated a graphene-based metamaterial with tunable EIT for sensing and slow light purpose [24]. Ding et al. proposed a terahertz modulator using coupled graphene cut-wire pairs based on bright-bright coupling [25]. However, the special patterned design of graphene resonators make fabrication more difficult and the Fermi energy of the isolated graphene cannot be easily tuned since bias gating is a challenging task in practice. Additionally, most recent dynamic modulations of EIT metamaterials through employing graphene as tunable bright or dark resonators have been reported [26–30]. In this circumstance, not only the EIT peak but also the adjacent frequency spectra especially the transmission dips were changed, which may introduce additional noises in adjacent frequency spectra in the modulation process [31].
In this paper, we proposed an EIT metamaterial based on a metal-graphene hybrid structure which can be electrically controlled without shifting adjacent resonance frequency. The unit cell consists of a monolayer graphene sheet with two kinds of cut-wire resonators closely placed on two sides. The transparency window can be induced through the destructive interference of the detuned resonators in the investigated frequency range. Most importantly, the transparency window can be actively manipulated by tuning the Fermi energy of graphene strip which is right on the coupling path of the system. Moreover, dynamical control of the group delay and sensitivity has also been realized via the tunable EIT effect. Compared with the previous structures, this metamaterial is relatively simple and efficient modulation of transmission amplitude is controlled only in the transparency window without affecting the adjacent resonant frequency. This may avert the unnecessary noise in the modulation process. The proposed scheme may inspire design in terahertz modulators, switches, sensors and slow light devices which are highly desired in the THz wireless communications.
2. Structure design and graphene model
The schematic structure and the unit cell of the metal-graphene metamaterial are depicted in Fig. 1(a) and (b). The periodic length p is 80 μm in both x and y directions. The unit cell consists of two resonators: a pair of vertical cut-wires (left) and a single warped cut-wire (right) with different resonant frequencies. The indirect near coupling can be induced between the cut-wires due to the similar radiation mode. The geometric parameters of the cut-wires are w1 = 10 μm, d = 16 μm, l1 = 38 μm, g = 12 μm and l2 = 75 μm, respectively. A monolayer graphene strip with a width of w2 = 7 μm is placed between two cut-wire resonators along the y direction. The cut-wire pair and the monolayer graphene are placed on the substrate Si covered with a thin SiO2 layer. The thickness of SiO2 layer is 100 nm and the thickness of Si substrate is 1 μm.
Fig. 1 (a) Schematic of the metal-graphene hybrid metamaterial, the unit cells are arranged in a periodic array with periodic lengths px = py = 80 μm, respectively, (b) the unit cell of the structure.
In our calculations, the graphene strip is modeled as an ultrathin conductive sheet. The surface conductivity can be expressed as , the and denote the interband and intraband conductivity, respectively. In the lower terahertz range, since is effective, the interband transitions in graphene are forbidden by the Pauli exclusion principle [32]. Thus, the intraband conductivitydominates and can be obtained from the well-known Kubo formula [33]:
where e is the electron charge, T is the temperature of the environment, ω is the circular frequency, kB is the Boltzmann’s constant, ћ represents the reduced Planck’s constant, EF is the Fermi energy of graphene, τ is the relaxation time which is related to the electronic impurities, flaws, electron–phonon scattering, and the intrinsic losses. The relaxation time τ determines the carrier mobility μ in graphene as μ = EFvF2τ/e, vF = 106 m s−1 denotes the Fermi velocity in graphene. Here, we assumed μ = 10,000 cm2/V, which is a compromise value lower than the experimental results in ref [34] and [35].By applying a bias voltage Vg to the structure as shown in Fig. 1(a), the Fermi energy and thus the surface conductivity of graphene can be actively modified. The tuning conductivity of graphene leads to the dynamic control of electromagnetic wave propagation. An approximate theoretical relation between EF and Vg is given by [17]
where ε0 is the permittivity of vacuum, εr is the permittivity of silicon dioxide, and ds is the thickness of the silicon dioxide layer.The proposed structure is predicted to display a tunable EIT effect as the conductivity of graphene is changed by tuning Fermi energy through external voltages. The procedure of structure fabrication and modulation process could be performed as follows [36, 37]: Firstly, a monolayer graphene synthesized via chemical vapor deposition (CVD) is transferred onto SiO2 covered Si substrate. Secondly, the monolayer graphene are patterned into strips using electron beam lithography (EBL) and oxygen plasma etching. Thirdly, the gold cut-wire pairs are fabricated on both sides of the graphene strips using another EBL. Finally, a pair of gold electrodes, respectively connected with the graphene strips and the silicon layer, is deposited at the opposite sides of the structure. Consequently, by applying the external voltage between the electrodes as shown in Fig. 1 (a), the Fermi energy of graphene can be actively tuned and the dynamic surface conductivity can be realized.
3. Results and discussions
To investigate the EIT behaviour of the designed structure, numerical calculations based on the finite integration method are performed by the commercial software CST Microwave Studio package. The unit cell boundary condition is used in x- and y-directions. The absorbing boundary conditions are applied in z-direction. The structure is illuminated by a normally incident wave polarizing along the y-direction as shown in Fig. 1(a). The investigated frequency ranges from 0.3 THz to 1.8 THz.
The optical properties of the designed hybrid metamaterial can be understood by first numerically calculating the transmission spectra of four different structures, where the Fermi energy of graphene is set as 0.05eV. As shown in Fig. 2(a), the vertical cut-wire (CW) pairs display a single broad dip at 1.25 THz visible in the transmission spectra caused by the electric dipole resonance. The sole warped CWs exhibit the similar behaviour with a transmission dip at 0.90 THz. Then, as the two kinds of CWs are combined, a transparency peak located at 0.99 THz within two resonance dips in the transmission spectra is activated through the coupling in the asymmetric structure, as can be seen from Fig. 2(b). The two new resonance frequencies (0.86THz and 1.28THz) are close to the resonance frequencies of the corresponding CW alone. Here, the EIT effect phenomenon can be attributed to the coupling between two radiative modes and its response strength mainly depends on the difference of two resonance lengths and the coupling distance. Furthermore, as the graphene strip with EF = 0.05eV is integrated between two CWs, the EIT effect still exists despite a slight decrease in the transparency peak which indicates unnoticeable attenuation of the coupling of the radiative modes.
Fig. 2 Transmission spectra of four structures: (a) the sole vertical cut-wire pairs, the sole warped cut-wires, (b) the combined EIT structure and the hybrid structure with graphene strips.
To get a closer look into the origin of the EIT window in the metamaterial, the surface current distributions in the case of combined cut-wire structure at different frequencies are depicted in Fig. 3. It can be found that the surface currents mainly center on the right CWs at resonant dip 0.86 THz and on the left CWs at resonant dip 1.28 THz, which indicates strong interaction with the electric field of incident waves at corresponding resonance. Therefore, both two kinds of CWs are generally called bright elements. It is worthy to be noted that only one kind of CWs is excited strongly at either resonant dip while the other is weakly coupled with the incident light. By contrast, at the transparency peak, both two kinds of CWs are excited simultaneously due to detuned bright modes. Meanwhile, the induced currents around two CWs oscillate in antiphase and the destructive interference occurs. Therefore, the scattered electromagnetic waves induced by the surface currents interfere destructively. As a result, the radiative losses of the transmitted waves are suppressed, giving rise to a transparency peak at 0.99 THz.
Fig. 3 (a–c) The surface current distributions of the combined cut-wire structure at the resonant frequencies corresponding to the transmission dips (0.86 and 1.28THz) and the transparency peak (0.99THz).
In the case when the graphene strips are integrated right between the two kinds of CWs in the metamaterial, the EIT window remains noticeable despite a slight decrease of the peak as mentioned above. Compared with the traditional metal-based EIT metamaterial, the important property of the hybrid structure is the tunability of the graphene. By applying a gate voltage, the Fermi energy of the graphene strips can be tuned and the transmission of the hybrid metamaterial can be dynamically modulated without refabricating the structure, which is highly desirable for many practical occasions. Figure 4(a) shows the response of the designed hybrid EIT structure with different Fermi energy. As observed, the amplitude manifests an evident decrease at the transmission peak as Fermi energy increases from 0.05 eV to 0.35 eV, while the adjacent transmission dips keep almost unchanged. To illustrate more details on the modulating performance of the metamaterial, we plot the transmission amplitude at three characteristic frequencies as a function of Fermi energy in Fig. 4(b). The transmission amplitude at 0.99 THz displays rapid decrease from 0.62 to 0.12 as Fermi energy increases from 0.05 eV to 0.40 eV, realizing a maximum modulation depth (Tmod, Tmod = ΔT/Tmax) up to 81%. By contrast, the amplitudes of the transmission dips at 0.86 THz and 1.28 THz keep constantly at low values with slight fluctuations. It is worth noting that the amplitude of the transmission dips rises drastically in modulating process in most previous graphene-based metamaterials [25–30]. And this unnecessary modulation may cause additional noise in practice. For our proposed design, the problem can be solved since only the transparency peak is modulated and the modulator may effectively avert the noises brought by unnecessary changes of adjacent frequencies. Furthermore, the adjustment to the Fermi energy required for modulation is much smaller than that described in Ref [21,23]. where an increase of Fermi energy from 0.1 eV to 1.0 eV is demanded.
Fig. 4 (a) Transmission spectra with different Fermi energy, (b) transmission versus Fermi energy at three resonant frequencies of the transmission peak (0.99 THz) and dips (0.86 THz and 1.28 THz).
To further investigate the dynamical control of EIT window, we calculate the surface current distributions at transmission peak of the hybrid metamaterial with EF = 0.4 eV and EF = 0.05 eV as depicted in Fig. 5. It can be found that as EF = 0.4eV, the graphene strip is evoked strongly by the incident waves and the surface currents in the strip are intensive. Worth noting is that since the graphene strip is right between the two CWs, the surface currents in the strip are excited via the following three pathways: one is the direct response to the electric filed of the incident waves, one is the indirect coupling from the left vertical cut-wires, another is the indirect coupling from the right warped cut-wire. As a consequence, these various pathways lead to complicated current distributions along the graphene strip and the surface currents which are parallel and anti-parallel with those in the CWs on both sides exist at the same time. Then, the scattered electromagnetic waves induced by the surface currents in turn couple with the CWs which causes not only destructive interference but also constructive interference at various phase difference. Therefore, the radiative loss is enhanced by the interactions between CWs and graphene, leading to the amplitude decrease of the transparency peak. However, as EF = 0.05eV, the surface currents are slightly evoked in the graphene strip due to the weak interaction with the incident waves and the CWs since the graphene sheet is much less metallic. Thus, the transparency peak of the metamaterial can be actively controlled by tuning Fermi energy through an external gate.
Fig. 5 The surface current distributions at transmission peak of the hybrid metamaterial with (a) EF = 0.4 eV and (b) EF = 0.05 eV
As mentioned above, in most previous studies on tunable EIT metamaterials, the graphene sheet is employed as tunable bright element or dark element. In such case, both the EIT peak and the adjacent transmission dips went through drastic changes with different Fermi energy, which may introduce additional noise in the modulating process for undesirable modulation. By contrast, for our designed structure, this unnecessary modulation can be avoided since the amplitudes of the transmission dips (0.86 THz and 1.28THz) keep constantly at low values with slight fluctuations as can be seen from Fig. 4(b). This can be attributed to that the resonances of gold cut-wires acting as bright element are not influenced by the graphene strip. Figure 6(a) and (b) display the transmission spectra of the sole warped cut-wires and the sole vertical cut-wire pairs with the graphene strip of different Fermi energy. An obvious change in the transmission spectrum is not observed in both two cases, which coincides with the results obtained from Fig. 4(b). The reason for this independence is that, for the sole CW array, the electric dipole resonance is not considerably affected by the varying conductivity of graphene strips with an alteration of Fermi energy. Therefore, for the proposed hybrid structure, the efficient modulation of transmission amplitude is controlled only in the transparency window with slight influence on the transmission dips, which may avert the unnecessary noise in the modulation process.
Fig. 6 Transmission spectra of (a) the sole warped cut-wires and (b) the sole vertical cut-wire pairs with the graphene strip of different Fermi energy.
Slow light is an important concomitant phenomenon of the EIT effect, which results from the strong dispersion properties of the EIT window. Generally, the group delay defined as tg = -dφ/dω is introduced to measure the slow light capability, where φ is the transmission phase shift [38]. The positive and negative value of the group delay is related to the slowness of slow light devices. Due to the tunable characteristics of the transparency window, the group delay can also be flexibly manipulated by actively tuning the Fermi energy in the proposed metamaterial. Figure 7 shows the transmission phase shift and group delay of the metamaterial with various Fermi energy of graphene. The positive and negative group delays can be observed corresponding to slow and fast light. Moreover, the steep phase slope can be seen in the vicinity of the transparency window as EF = 0.05 eV, which results in the maximum group delay of 3.6 ps corresponding to a 1.08 mm-distance delay propagating in free space. As the Fermi energy increases, the transmission phase slope turns more gentle and the hybrid structure gradually lose the slow light capability. For example, the calculated group delay has a relatively large value of 3.6 ps at 0.99 THz for EF = 0.05 eV. Whereas as EF = 0.35 eV, a group delay value of only 1.4 ps is obtained at 1.1 THz. Therefore, the tunable control of slow light effect can be realized in our proposed metamaterial.
Fig. 7 (a) The transmission phase shift and (b) group delay of the EIT metamaterial as the Fermi energy of graphene increases.
Moreover, considering the practical application, the bandwidth is also an important feature that should be taken into consideration. Here, the delay bandwidth product (DBP) which is defined as the product of maximum group delay and the band width (DBP = tg × Δf) is introduced. The DBP is a critical index which indicates the buffering capacity of slow-light devices [39]. Table 1 displays the group delay, DBP, Q-factor, and EIT peak amplitude for our designed structure with various Fermi energy. It can be found that the group delay, DBP, Q-factor and the corresponding transparency peak all decreases gradually with the increasing Fermi energy. This manifests that the slow light effect can be completely controlled with the same trend by changing the coupling between bright modes through adjusting the gate voltages. Therefore, this manipulation enables more efficient application of the controllable slow light devices by overcoming the common obstacle that the group delay is fixed once the structures are prepared.
Table 1. The group delay, DBP, Q-factor, and transparency peak amplitude of hybrid metamaterial with various Fermi energy
As mentioned above, a remarkable potential application of the EIT phenomenon is the sensing performance. Up to now, among researches studying the sensitivity of the EIT structures as sensors, most of the current sensors are reported to be untunable [40,41]. This restricts the range of their practical applications. Here, to solve this problem, we investigate the tunable sensing performance based on our proposed metamaterial with different Fermi energy of graphene. Figure 8(a) shows the calculated transmission spectra of the hybrid metamaterial with EF = 0.05eV, as environmental refractive index n increases from 1.0 to 1.8. It can be observed that noticeable red shifts in the EIT window are realized despite the small index variations. To evaluate the sensing performance, the corresponding shift of the EIT window relative to the refractive index of the surrounding media is plotted, as shown in Fig. 8(b). The frequency shifts of the EIT windows manifest nearly linear relation with the increasing refractive index as Fermi energy changes. We define the refractive index sensitivity S as the frequency shift of the EIT window over per refractive index change unit (RIU) which can be expressed as . The sensitivity of 339 GHz/RIU can be obtained from the figure as EF = 0.05eV. Moreover, by varying the Fermi energy through manipulating the gate voltages, the sensitivity of the structure can be actively tuned, as displayed in Fig. 8(c). For example, the sensitivity increasing of 98 GHz/RIU (from 339GHz/RIU to 437GHz/RIU) can be obtained as Fermi energy rises from 0.05 eV to 0.30 eV. These results indicate that the sensitivity of the hybrid metamaterial can be dynamically controlled, which is desirable in designing tunable sensors.
Fig. 8 (a) Transparency spectra and (b) peak frequency shift with different refractive index as EF = 0.05eV, (c) the sensitivities for different Fermi energy.
4. Conclusions
In summary, we have numerically and theoretically investigated tunable EIT effect based on a metal-graphene hybrid metamaterial consisting of two coupled cut-wire resonators with graphene strips placed between them. The surface current distributions on the structure indicate that the destructive interference between the two kinds of cut-wire resonators leads to the transparency window. By adjusting the Fermi energy of the graphene strips on the coupling path, the amplitude modulation of 81% at transmission peak can be realized. The modulation is controlled only in the EIT window with slight changes on the transmission dips. This may help avert the noises brought by unnecessary modulation in adjacent frequency range of the EIT window. Moreover, the actively controlled slow-light effect and sensing performances are achieved, as the corresponding EIT window is modulated. We believe that this research is helpful in potential applications in modulators, tunable slow-light devices and environmental sensors. It may also inspire interest in developing metal-graphene hybrid terahertz devices that could realize electrically tunable EIT effect.
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