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Abstract
Tunable polarization-nonsensitive electromagnetically induced transparency (EIT) based on Dirac semimetal films (DSF) at terahertz frequencies is numerically studied in this paper. We first numerically investigate the EIT effect in a DSF-based terahertz metamaterial structure comprising a DSF strip and two DSF L-shaped resonators. To achieve the polarization-nonsensitive EIT effect, we introduce one vertical strip and two more L-shaped resonators to the above structure. The transmission spectra of the new structure show strong polarization-nonsensitive characteristic because of the fourfold symmetry. By analyzing the surface currents and electric field distribution at the resonant frequencies, we demonstrate that an EIT effect arises from the bright–bright mode coupling. Moreover, the polarization-nonsensitive EIT structure exhibits a blue shift with the increase in the Fermi energy of the DSF without having to re-optimize its geometrical parameters. Our study opens a new path for developing terahertz devices such as terahertz switches.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is a quantum interference effect in atomic physics and can lead to a transparency window in a wide absorption spectrum [1,2]. Recently, researches have focused on the EIT effect exhibited by metamaterials owing to its significant advantages and potential applications [3–19]. Generally, the EIT effect arises from two modes: bright–dark mode coupling [20–27] and bright–bright mode coupling [28–30]. For example, Pan et al. proposed a terahertz metamaterial structure made of a coupled “bright” circular split-ring resonator and a “dark” square split-ring resonator and realized EIT-like effect [31]. Zhu et al. introduced multiple dark resonators to be coupled with a bright resonator and realized plasmon induced transparency (PIT) in a broad frequency range [32]. Zhu et al. achieved a broadband EIT-like effect by the coupling interactions between bright and dark resonators [33]. Unfortunately, most of these metamaterial structures are made of metallic materials that can only work at a fixed frequency, thus limiting the application and development of the EIT effect.
Currently, the tunable EIT effect has attracted considerable attention, and graphene has been widely used to dynamically control the EIT effect [34–38]. He et al. proposed a selectively controllable terahertz EIT metamaterial structure, realized by the coupling between bright and dark modes [39]. Zhao et al. presented a terahertz graphene-based metamaterial structure to dynamically control the PIT window [40]. Ding et al. designed a wavelength-tunable EIT metamaterial structure comprising a graphene grating and a square closed-ring gold resonator [41]. Most recently, a novel state of quantum matter called Dirac semimetal films (DSF), widely known as three-dimensional (3D) graphene [42], has drawn significant attention. Compared to graphene, 3D DSF not only have the advantages of graphene as a photosensitive material but also are more robust against environmental defects or excess conductive bulk states [43]. Moreover, the mobility in DSF is higher than that of the best graphene [44] owing to their crystalline symmetry protection against gap formation [45–47]. Most importantly, the surface conductivity of DSF can be dynamically controlled by adjusting its Fermi energy (EF) through alkaline surface doping [48]. Shen et al. demonstrated an extended concept of EIT effect (tunable electromagnetically induced reflection) with an ultrahigh Q factor based on a complementary Dirac semimetal metamaterial structure [49]. Chen et al. investigated the tunable plasmon-induced transparency effect based on DSF by bright–bright mode coupling [50]. However, these EIT metamaterial structures are sensitive to the fixed polarization of the incident light. Therefore, the EIT effect is attenuated and even disappears when changing the polarization.
In this paper, we present a numerical and theoretical study on the tunable polarization-nonsensitive EIT effect at terahertz frequencies in a Dirac semimetal. A DSF-based structure comprising a DSF strip and two DSF L-shaped (2L) resonators in the terahertz region is investigated. The proposed structure exhibits the EIT effect because of the destructive interference between the DSF strip (bright mode) and the 2L (bright mode) resonators. To achieve the polarization-nonsensitive EIT effect, we introduced a vertical strip and two more L-shaped resonators, i.e., a DSF cross and four L-shaped (4L) structure. Owing to the fourfold symmetry, the transmission spectra of the new structure show strong polarization-nonsensitive characteristic. To our knowledge, this is the first time a DSF has been introduced in a polarization-nonsensitive EIT structure. By analyzing the surface currents and electric field distribution at the resonant frequencies, we find that the EIT effect is produced by the coupling between the bright–bright modes. Furthermore, the proposed polarization-nonsensitive EIT structure can be made to work at different frequencies by changing the Fermi energy of the DSF without having to re-optimize its geometrical parameters.
2. Electromagnetically induced transparency: Design and analysis
Figure 1 shows the schematic of the DSF-based metamaterial geometry. The unit cell comprises a DSF strip and two DSF L-shaped resonators patterned on a lightly doped silicon substrate with a thin SiO2 layer. In the proposed geometry, P represents the periodicity of the structure and is taken to be 160 µm in our simulations. We set the width of DSF strip and two DSF L-shaped resonators are 4 µm. The dimension of each DSF L-shaped resonator is 33 µm. The length of DSF strip is 95 µm. The distance between the strip and L-shaped is denoted by d. The thicknesses of the SiO2 layer and silicon substrate are 0.1 and 1 µm. The permittivities of the SiO2 and Si substrates are taken as 3.9 and 11.7, respectively. The thickness and Fermi energy of the DSF layer are set as 0.5 μm and 30 meV, respectively. The designed DSF-based tunable metamaterial structures were numerically studied using the commercial simulation software CST MICROWAVE STUDIO with the method of finite difference time domain (FDTD). In the simulation, the unit cell of the structure was simulated with periodic boundary conditions in the x–y plane and an open boundary condition in the z directions. The incident waves with their electric field polarized along the x-direction were irradiated normally to the surface.
Fig. 1 Schematic of a DSF-based metamaterial geometry comprising a strip and two L-shaped resonators: (a) Side view, and (b) Top view of the unit cell with a period P = 160 µm, a = 33 µm, L = 95 µm, d = 20 µm, and w = 4 µm.
The conductivity of the DSF can be calculated using the Kubo formalism of the random-phase approximation theory (RPA) at the long-wavelength limit, including both the intra-band and inter-band processes [51]. In the simulation, at the low-temperature limit T≪EF, the real and imaginary parts of the dynamic conductivity of the Dirac semimetal can be written as follows [52]:
whereis the Fermi momentum, ћ is the reduced Planck constant, EF is the Fermi energy, and the Fermi velocity . g is the degeneracy factor,, and (Ec = 3 is the cutoff energy).Considering the interband electronic transitions, the permittivity of 3D Dirac semimetals can be obtained using the two-band model [50]:
whereis the effective background dielectric (for g = 40 (AlCuFe quasi-crystals [53])), andis the permittivity of vacuum. Using the above permittivity equation for Dirac semimetal, we can calculate the values of the permittivity under different frequencies in MATLAB. And then import these dispersion values to the characteristics of the new material in the CST, thus accomplishing the setting of DSF.To demonstrate the EIT effect, we obtained the transmission spectra of the unit cell with the standalone DSF strip and 2L structures, as shown in Figs. 2(a) and (d), respectively. The standalone DSF strip and 2L structures exhibit a transmission dip for the x-polarized incident terahertz radiation at 0.427 and 0.579 THz respectively. Both the standalone DSF strip and the 2L structure can be directly excited by the incident wave as a bright mode owing to the coupling with the incident wave. To further confirm the above analysis, we analyzed the surface currents and electric field of the standalone DSF strip and 2L structures at the transmission dip. As shown in Figs. 2(b) and (c), the directions of the induced surface currents on the standalone DSF strip structure are parallel to the excitation field, while strong electric fields are localized at the end of the strip. Thus, both the current and electric field distribution patterns confirm a typical electric dipole mode. As shown in Figs. 2(e) and (f), the same surface currents on the arm of the DSF 2L structure and the electric field are generated along the arms of the 2L structure. Therefore, when the two standalone structures are combined into a composite structure, they can be simultaneously excited as a bright mode. As a result, an EIT window can be achieved by combining the two bright resonators with similar resonances.
Fig. 2 Calculated transmission spectra and field distributions of the standalone DSF strip and 2L structure: (a) transmission, (b) surface currents and (c) electric field of the standalone DSF strip structure, and (d) transmission spectra, (e) surface currents and (f) electric field of standalone DSF 2L structure.
As expected, an EIT-like transparency window appears with a transmission peak located between the resonance frequencies of the two isolated resonators, as shown in Fig. 3(a). The frequency of the transmission dip A (0.424 THz) and transmission dip C (0.586 THz) are the same as the resonance frequencies of the standalone DSF strip and 2L structures, respectively. To better understand the physical mechanism of the EIT-like transparency phenomenon, surface currents and electric field distributions at the transmission dips A and C as well as at the transmission peak B are calculated, and shown in Figs. 3(b)–(d). At the transmission dip A, only the strip is excited strongly by the incident light, whereas the 2L structure is weakly excited, as shown in Fig. 3(b). However, at the transmission dip C, only the 2L structure is excited by the incident light, whereas the strip is weakly excited, as shown in Fig. 3(d). Thus, both the resonances of the transmission dips A and C serve as bright modes with electric dipole oscillation excited directly by the incident wave. As shown in Fig. 3(c), for the transmission peak B, we observe that the strip and 2L structures are excited simultaneously, and the surface currents are anti-parallel to each other. Therefore, a quadrupole oscillation is excited in the EIT structure because of the hybridized coupling between the two resonators. Thus, the destructive interference resulting from the near-field coupling of the two bright modes leads to a distinct transparency window.
Fig. 3 Calculated transmission spectra, surface currents, and electric field distributions of the EIT structure: (a) transmission, (b) surface currents and electric field distributions at transmission dip A, (c) surface currents and electric field distributions transmission peak B, and (d) surface currents and electric field distributions at transmission dip C.
3. Polarization-nonsensitive electromagnetically induced transparency
In the proposed EIT structure, the EIT effect occurs only when the incident polarization is parallel to the DSF strip. When the polarization of the incident wave is changed, the EIT effect will attenuate and even disappear. Therefore, we introduced a vertical strip structure and two more L structures to the above design and achieved polarization-nonsensitive EIT effect. As shown in Fig. 4, the polarization-nonsensitive EIT structure comprises DSF cross and four L-shaped resonators. Because of the fourfold symmetry, the polarization-nonsensitive EIT structure has an identical response to both x-polarized and y-polarized incident waves. Therefore, it exhibits polarization-nonsensitive for x and y polarizations. Thus, the transmission spectra of the new structure are the same for the x-polarized and y-polarized waves.
Fig. 4 Schematic of the DSF-based polarization-nonsensitive EIT structure comprising DSF cross and four L-shaped resonators, with the same material and size as those shown in Fig. 1.
To illustrate the EIT effect, we investigated the transmission property of the isolated cross structure, the isolated 4L structure, and the polarization-nonsensitive EIT structure, as shown in Fig. 5. The transmission spectra of the isolated cross and 4L structures are indicated by red and pink lines, respectively. Each of the two structures has a resonant mode. An EIT-like transparency window is observed at 0.54 THz between the two dips at 0.43 and 0.586 THz, as indicated by the blue line in Fig. 5. The two resonant modes (0.436 and 0.58 THz) are close to the two transmission dips of the EIT transmission spectra (0.43 and 0.586 THz) because the two excited modes are weakly hybridized. Therefore, the cross and 4L structures can be directly excited by the incident light as bright modes, and the transparency window appears through the coupling of the bright–bright mode.
Fig. 5 Calculated transmission spectra for cross, 4L, and EIT structurers with Fermi energy of 30 meV.
To get further insight into the physical mechanism behind the EIT effect in the proposed polarization-nonsensitive EIT structure, we investigated the surface currents and electric field distributions of the standalone DSF cross structure at 0.436 THz and of the standalone DSF 4L structure at 0.58 THz, as shown in Fig. 6. Owing to the polarization-nonsensitive characteristic of these structures, we only analyzed the results under x-polarization. As shown in Fig. 6(a), the surface currents on the horizontal strip of the isolated cross structure are parallel to the excitation field, whereas the electric fields are localized at the edge of the horizontal strip of the standalone DSF cross structure. Thus, the horizontal strip of the standalone DSF cross structure that is parallel to the incident light is excited. In addition, the surface currents and electric fields are localized at the four L-shaped structures of the isolated 4L structure, as shown in Fig. 6(b). Thus, the four L structures can be directly excited by the incident light.
Fig. 6 Calculated surface currents and electric field distributions of the standalone DSF cross and 4L structure (a) surface currents and electric field distributions of standalone DSF cross structure at 0.436THz, (b) surface currents and electric field distributions of standalone DSF 4L structure at 0.58THz.
Figure 7 shows the surface currents and electric fields of the polarization-nonsensitive EIT structure corresponding to the two transmission dips (0.43 and 0.586 THz) and transmission peak (0.54 THz). At the low-frequency transmission dip, only the DSF cross structure is excited by the incident light, whereas the DSF 4L structure is excited weakly, as shown in Fig. 7(a). However, when the frequency is 0.586 THz, only the DSF 4L structure is excited by the incident light, and the DSF cross structure is weakly excited, as shown in Fig. 7(c). Therefore, the DSF cross and 4L structures serve as bright modes exhibiting electric dipole oscillation excited directly by the incident light. At 0.54 THz, the surface currents and electric field are simultaneously excited on the DSF cross and 4L structures, as shown in Fig. 7(b). The two excited bright modes undergo hybridized coupling. The destructive interference produced by the coupling of the two bright modes completely suppressed the radiative losses and allowed the incident wave to be transmitted, therefore leading to a transparency window.
Fig. 7 Calculated surface currents and electric field distributions of the polarization-nonsensitive EIT structure: (a) surface currents and electric field distributions at 0.43 THz, (b) surface currents and electric field distributions at 0.54 THz, and (c) surface currents and electric field distributions at 0.586 THz.
To elucidate the coupling mechanism of the EIT effect in metamaterial systems, we employed a two-particle model for our structure [54,55], where both the DSF cross and 4L elements can be considered particles interacting with the incoming electric field. The coupled differential equations can be written as follows [29]:
where,,,, andare the effective mass, displacements, resonance frequencies, loss factors of the two particles (the DSF cross and 4L elements), and the coupling strength of the two particles with the incident wave, respectively. is the coupling coefficient between the two particles. The following equations can be derived by solving Eqs. (4) and (5): The effective polarization of the metamaterial can be derived as follows.By substitutingand, we can express the effective electric susceptibility χ of the DSF-based EIT structure as follows.whereis the proportionality factor. The real and imaginary parts of the effective electric susceptibility represent the dispersion and absorption within the metamaterial, respectively. Figure 8 and Table 1 present the fitted curve based on the analytical models and the corresponding fitting parameters with DSF Fermi energy of 30 meV, respectively. The result from the fitted curve is in good agreement with the numerically simulated curve observations (blue line in Fig. 5). Thus, this result demonstrates the validity of the analytical model.
Table 1. Fitting parameters of the analytical models
Next, the effect of the geometrical dimension on the response of the EIT is analyzed. Figure 9 shows the simulation transmission spectra of the standalone DSF cross structure and the polarization-nonsensitive EIT structure with various DSF cross lengths l. With the increase in l, the resonant frequencies of the standalone DSF cross structure exhibit a red shift, as shown in Fig. 9(a). The high frequency of the transmission dips is concentrated at approximately 0.586 THz, whereas the low frequency of the transmission dips exhibits a red shift with the increase in the DSF cross length l, as shown in Fig. 9(b). This is because the low-frequency resonance is generated by the DSF cross structure.
Fig. 9 Calculated transmission spectra with respect to DSF cross length l for (a) standalone DSF cross structure, and (b) polarization-nonsensitive EIT structure when the Fermi energy is 30 meV.
Figure 10 shows the transmission spectra of the standalone DSF 4L structure and polarization-nonsensitive EIT structure with respect to the DSF 4L length a. The resonant frequencies of the standalone DSF 4L structure exhibits a red shift with the increase in the DSF 4L length, as shown in Fig. 10(a). Similar to the various cross lengths l discussed above, the high frequency of the transmission dip resonances is generated by the DSF 4L structure. Thus, when the DSF 4L length l is fixed, only the high frequency of the transmission dips exhibits a red shift with the increase in the DSF 4L length a, as shown in Fig. 10(b).
Fig. 10 Calculated transmission spectra with respect to the DSF 4L length a for (a) standalone DSF 4L structure, and (b) polarization-nonsensitive EIT structure when the Fermi energy is 30 meV.
The proposed structure is found to exhibit polarization-nonsensitive EIT owing to its fourfold rotational structural symmetry. The four different polarization states are plotted to confirm the polarization-nonsensitive property of the structure. When the polarization of the incident wave is altered with θ varying from 0° to 45°, the transmission spectra remain largely unchanged, as shown in Fig. 11. For further explanation, we simulated the electric field distributions of the polarization-nonsensitive EIT structure at transmission peak with different angles when the Fermi energy is 30 meV as shown in Fig. 12. It is obvious that the electric field distributions are distributed on both the cross and 4L structures at different θ. Therefore, in the case of different θ, the two excited bright modes always have hybridized coupling at the transmission peak. It is consistent with the transmission spectra in Fig. 11. Therefore, the transmission spectra clearly demonstrate the strong polarization-nonsensitive characteristic of the structure.
Fig. 11 Calculated transmission spectra of the polarization-nonsensitive EIT structure with respect to θ when the Fermi energy is 30 meV, where θ is the angle between the polarized electric field and the x-axis of the structure.
Fig. 12 Calculated electric field distributions of the polarization-nonsensitive EIT structure at 0.54 THz with respect to θ when the Fermi energy is 30 meV, where θ is the angle between the polarized electric field and the x-axis of the structure.
Furthermore, as mentioned before, the conductivity of the DSF can be dynamically modified by varying their Fermi energy. The Fermi energy of DSF can be changed through the in situ electron doping [48] which realized by evaporating potassium onto the sample surface. This is advantageous to dynamically control the frequency of the transparent window in our polarization-nonsensitive EIT structure, i.e., by changing the Fermi energy without re-optimizing and re-fabricating the structure. To demonstrate the frequency tunability of the proposed polarization-nonsensitive EIT structure, the transmission spectra with respect to different Fermi energies are plotted, as shown in Fig. 13. With the increase in the Fermi energy, the two transmission dips and the transparent window exhibit blue shifts because of the Fermi energy-induced change in the plasma frequencies. All the transparent windows intensities unchanged under different Fermi energies within a certain range which still maintain high transmission intensities. As the tunability using the change of Fermi energy of the DSF is mentioned before, we can control the structure to function as a switch in the terahertz frequency region. For example, the transmission is about unity and tuned to nearly 0 when the Fermi energy increases from 30 meV to 40 meV at 0.75 THz. Therefore, the transmission could be in the “on” or “off” state by changing the Fermi energy of DSF.
4. Conclusions
In conclusion, we first numerically investigated the EIT effect in a DSF-based terahertz metamaterial comprising a DSF strip and two L-shaped resonators. The DSF strip and 2L resonators served as bright modes, and the transmission spectra showed a distinct transparency window at 0.54 THz, because of the destructive interference resulting from the coupling of the two bright modes. We then introduced a vertical strip and two more L-shaped resonator structures to the proposed design to achieve polarization-nonsensitive EIT effect. The transmission spectra show strong polarization-nonsensitive property because of the fourfold symmetry of the structure. Owing to the tunability of the DSF, the EIT effect can be tuned in the terahertz frequency regime by a small change in the Fermi energy. Therefore, the proposed polarization-nonsensitive EIT structure can work at different frequencies without having to re-optimize or re-fabricate the structure. These properties can be utilized in many important applications such as in sensing and terahertz switching devices.
Funding
National Natural Science Foundation of China (61875106, 61775123); Natural Science Foundation of Shandong Province, China (ZR2016FM09, ZR2016FM32); the SDUST top-notch talent project for young teachers (BJRC20160505); the Shandong graduate student tutor guidance ability promotion program project (SDYY17030); the National key research and development program of China (2017YFA0701000).
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