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Abstract
We theoretically investigate the dispersion and coupling characteristics of terahertz surface plasmons polaritons (SPPs) in bulk Dirac semimetals (BDS) sheets, which indicate that symmetric and anti-symmetric modes are attributed to the odd and even superpositions of the anti-symmetric eigenmode supported by the single-layer BDS sheet. Interestingly, the symmetric mode has better confinement than the eigenmode and anti-symmetric mode. By introducing two silicon bars, the highly-confined symmetric mode is modulated in resonance frequency and transmission intensity by the designed novel band-pass filter. Numerical results show good agreement with the theoretical analysis based on the coupled mode theory and the Fabry-Perot resonance theory. The developed Dirac semimetal plasmonic structures pave the way to the development of novel THz active devices for light modulation platform.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) are the surface electromagnetic modes resulting from the interactions between free electrons in metallic materials and photons, which propagate along the metal-dielectric interface and makes it possible for manipulating light on sub-wavelength structure beyond the diffraction limit [1]. Because of its high confinement and proper propagation distance in nanoscale, the controlling of SPPs undoubtedly play an essential role in ultra-compact device for optical communications. A large number of plasmonic designs based on noble metal, such as metal-insulator-metal waveguide [2], optical absorbers [3], rulers [4] and sensors [5], have been theoretically proposed and experimentally investigated in order to effectively tailor SPPs with desired transmission properties and modulation features. Despite of these extensive applications in the visible and near-infrared regions, the difficulties in controlling dielectric functions of metals and the existence of enormous intrinsic loss have hampered the performance of SPPs over past decades. As an alternative material, graphene, the two dimensional carbon material, has drawn growing attention because of its remarkable electro-optical properties including strong field confinement, low intrinsic loss, and especially the dynamically tunable permittivity by changing the Fermi energy via external fields [6]. Thus, the manipulation of optical features for graphene SPPs has been concentratively investigated [7–13]. Xu et al. proposed that highly confined plasmonic waves in monolayer graphene can be efficiently excited using an etched diffractive grating on silicon, which induced the guided-wave resonance of the combined structure suggesting a sharp notch on the transmission spectra [7]. Yang et al. present a compact 1 × 2 plasmonic splitter of dielectric-loaded graphene waveguide based on multimode interference [8]. Lu demonstrates the plasmonic characteristics of a nanoscale graphene resonator-coupled waveguide based on the high confinement of graphene SPPs in monolayer graphene sheet [9]. From these reports, we can find that the high confinement of graphene plasmons plays an essential role in designing integrated plasmonic devices at terahertz and infrared frequencies. More importantly, that makes it possible to manipulate light at deep subwavelength scale.
Graphene and topological insulators are typical Dirac fermion systems. Recently, the research accent of Dirac systems has shifted to the investigation of a novel state of quantum matter that can be regarded as “3-D graphene”—3D Dirac semimetals, also called bulk Dirac semimetals (BDS) [14], such as, Na3Bi [15], Cd3As2 [16] and AlCuFe quasicrystals [17]. Comparing to graphene, BDS are easier to process and more stable [18, 19]. The good news is that the crystalline symmetry protection in some samples results in ultrahigh carrier mobility up to 9 × 106 cm2V−1s−1 at 5 K, which is much higher than the best graphene of 2 × 105 cm2V−1s−1, which is essential for reducing intrinsic loss of SPPs at low temperature [20, 21]. Furthermore, the dielectric functions can also be dynamically adjusted by changing the Fermi energy of BDS through alkaline surface doping [15, 16]. People may wonder that BDS can be considered as a new class of plasmonic platform with effective tunability at different frequencies. However, the confinement of SPPs in BDS is strongly limited by the radiative loss and most of the SPP energy cannot be confined in the dielectric-BDS interface, as shown in Fig. 2(c), which will hamper its further application in ultra-compact plasmonic devices.
Herein, we propose a double-layer BDS structure for its novel transmission characteristics. By using waveguide propagation theory, we theoretically investigate the dispersion and coupling features of SPPs in BDS sheets, which demonstrate that the symmetric and anti-symmetric modes are attributed to the odd and even superpositions of anti-symmetric eigenmode supported by the single-layer BDS sheet. The symmetric mode has better confinement and transmission efficiency than the eigenmode and anti-symmetric mode. As an application of symmetric mode, we propose a highly tunable band-pass filter based on double-layer BDS sheets sandwiched by two silicon bars. It is found that the plasmonic resonance and transmission features possess the strong dependence on the bar-bar distance, width of each bar, coupling distance of double-layer BDS sheets as well as their Fermi energy. These results may find potential applications in integrated active plasmonic devices, especially frequency-selective components.
2. Theory and simulation
The schematics of the BDS systems are illustrated in Figs. 1(a)-(b), where the single layer and double-layer BDS sheets are embedded in a dielectric medium with a permittivity εr. The interlayer space between the BDS sheets is denoted by g = 2a. In our configuration, the monolayer BDS is treated as a thin film with thickness Δ = b-a. The incident light is only impinged from x-direction and the transverse magnetic (TM)-polarized SPP will be supported and propagate in the BDS sheet, which acts as a waveguide. The complex-valued conductivity of BDS is governed by Kubo formula in RPA [14,21]
where G(E) = n(-E) - n(E) with n(E) being the Fermi distribution function, EF is the Fermi energy level, kF = EF/ћvF is the Fermi momentum, vF = 106 m/s is the Fermi velocity, ε = E/EF, Ω = ћω/EF + iћτ−1/EF, where ћτ−1 = vF/(kFμ) is the scattering rate determined by the carrier mobility μ. εc = Ec/EF (Ec is the cutoff energy beyond which the Dirac spectrum is no longer linear), and t is the degeneracy factor. Correspondingly, the equivalent permittivity of the BDS based on two-band model is given by [14, 21]where ε0 is the permittivity of vacuum. For AlCuFe quasicrystal, the fitting parameters are εb = 1, t = 40. In our analysis, the thickness of monolayer BDS is reasonably set as Δ = 0.2 μm, εc = 3, EF = 70 meV and μ = 3 × 104 cm2V−1s−1 at 5 K (the corresponding intrinsic time τ = 4.5 × 10−13 s) which is rather conservative to feature the intrinsic loss of BDS at low temperature [20-22].
Fig. 1 Schematic diagrams of the BDS structure consisting of single-layer Dirac semimetal sheet (a) and double-layer Dirac semimetal sheets (b) with an interlayer space of g = 2a. The transverse magnetic (TM)-polarized SPP wave propagates along the x-direction.
The electromagnetic field of the eigenmode has the form of E(y)expi(βx-ωt) or H(y)expi(βx-ωt). Based on waveguide theory, we can obtain the TM-polarized SPP mode of the double-layer BDS system [23]
where kr2 = β2-εrk02 and kb2 = β2-εk02 with β being the propagation constant of the SPP wave. By matching the following boundary conditions and taking the limit of the BDS thickness Δ approaches zero (g >> Δ), the dispersion relation of TM-polarized SPP eigenmode can be obtainedThere are two solutions for Eq. (8) β ± 2 = εrk02 + kr ± 2, corresponding to the two propagation constants of symmetric (β+) and anti-symmetric (β-) modes, where kr ± denote the two solutions of kr. Taking the limit of the BDS sheets gap g approaches + ∞ in Eq. (8), the TM-polarized SPP wave propagation constant kSPP of single BDS sheet can be obtainedwhere η0 = 376.6 Ω is the impedance of vacuum. For small coupling distance g, β ± = kSPP + Δβ ± , i.e.,The propagation constants of the symmetric and anti-symmetric modes as a function of gap g are plotted in Fig. 2(a), where the permittivity of dielectric medium is assumed to be εr = 1 for simplicity, the Fermi energy EF = 70 meV, and incident frequency f = 1.56 THz. The real part of propagation constant of symmetric mode Re(β+) decreases as the coupling distance g increases, while that of anti-symmetric mode Re(β-) keeps nearly constant in the considered scale, which is smaller than that of single layer BDS sheet Re(kSPP) = 33.20 mm−1. In order to examine the theoretical analysis, we perform numerical calculations on the dispersion of TM mode by using 2D FDTD method with a perfectly matched layer absorbing boundary condition. The mesh size of BDS sheet in the x- and y-directions is chosen to be 0.1 × 0.02 μm. The calculated results of propagation constants are also plotted in Fig. 2(a), which show good agreement with theoretical analysis. To demonstrate the transmission efficiency of the BDS structure, we numerically calculate the transmission intensities of the single and double-layer BDS system with symmetric and anti-symmetric modes for g = 50 μm, as shown in Fig. 2(b). The transmission intensities are 0.91, 0.99 and 0.83 at the incident frequency f = 1.56 THz, which indicates that the symmetric mode suffers less propagation loss than the anti-symmetric mode. At g = 50 μm, Re(β+) = 34.33 mm−1 and Re(β-) = 32.40 mm−1. Thus, the symmetric mode has a shorter spatial oscillation period than the anti-symmetric mode. To visualize this physical picture, we plot the magnetic field (Hz) distributions of those modes in Figs. 2(c)-(e). The eigenmode of single layer BDS sheet shows anti-symmetric distribution, which propagate over quite a few SPP wavelength in single layer BDS sheet. For the double-layer BDS structure, the symmetric and anti-symmetric modes originate the odd and even superpositions of eigenmodes in separate BDS sheet. Thus, we define the coupling strength between the two BDS sheet as κg = (β--β+)/2 [22], which is also plotted in Fig. 2(a). The coupling strength decreases exponentially with the increase of coupling distance g, which also shows good agreement between the theoretical calculations and numerical results. Therefore, we get the dispersion relations of double-layer BDS structure, which may find potential applications in dynamical optical splitter, switch and Mach-Zehnder interferometer.Fig. 2 (a) Theoretical (lines) and numerical (dots) results of propagation constants of single-layer BDS sheet, symmetric and anti-symmetric modes together with coupling strength of double-layer BDS structure as functions of coupling distance g. Transmission intensities (b) and magnetic field (Hz) distributions (c)-(e) of single-layer BDS sheet, symmetric and anti-symmetric modes of double-layer BDS structure. Here the parameters are set as f = 1.56 THz, EF = 70 meV, and g = 50 μm.
3. Results and discussions
Here, we propose a novel optical filter based on symmetric coupling of terahertz SPP wave between BDS sheets, as presented in Fig. 3(a). Two Si bars are introduced into the double-layer BDS sheets separated by the coupling distance g. The permittivity of silicon is from Palik [24]. The width of each bar is d, and the distance between the bars is L. Here the induced Si bars act as two partial reflection mirrors, where the propagating SPP wave should be reflected at Si-air interface. For the double-layer BDS region between the two bars, the SPP wave would be reflected back and forth and the localized standing wave resonance will be supported. Thus, the double-layer BDS region sandwiched by the two Si bars forms a novel resonance cavity. When the incident frequency satisfies the resonance condition, the SPP waves can be coupled into the designed cavity and then transmit the BDS sheets. Accordingly, the obvious band-pass filtering effect could be realized in the entire device. To examine above analysis, we numerically calculate the transmission spectrum of the double-layer BDS sheets with two Si bars, where g = 50 μm, d = 10 μm, and L = 190 μm. Two transmission peaks can be clearly observed at the incident frequency of 1.56 and 2.22 THz. The FWHM values (full width at half maximum) of the transmission peaks are 0.10 and 0.13 THz, which suggests a novel band-pass filtering effect at terahertz region. The magnetic field distributions (|Hz|2) of the resonance peaks are plotted in Figs. 3(c) and 3(e), which indicates that the double-layer BDS region separated by two Si bars behaves as a Fabry-Perot (FP) cavity. In Fig. 3(c), the incident frequency f = 1.56 THz satisfying the first-order resonance can be coupled into the FP cavity, and then pass though the BDS sheet, which generates the resonance peak c in the transmission spectrum. Similarly, the second-order resonance is formed in the FP cavity which corresponds to the transmission peak e at the incident frequency f = 2.22 THz. Therefore, only the incident frequency satisfying the resonance condition, the SPP waves can be coupled into the designed FP cavity and finally pass through the BDS sheets. In contrast, other frequencies are prohibited. For example, the incident frequency of f = 1.83 THz, which does not satisfy the resonance condition of the FP cavity, is reflected at the first Si-air interface. The standing wave resonance cannot be formed and therefore prohibited in the left side of the FP cavity, as shown in Fig. 3 (d). Hence, with corporation of the two Si bars, the terahertz SPPs can be controlled in the double-layer BDS structure and an effective band-pass filter can be realized.
Fig. 3 (a) Schematic of the double-layer BDS structure with introduced Si bars. The coupling distance of the double-layer BDS sheets is g. The width of each Si bar is d, and the distance between the bars is L. The excited symmetric TM-polarized SPPs propagate along the x-direction. (b) Transmission spectrum of the proposed device where g = 50 μm, d = 10 μm, and L = 190 μm. (c)-(e) Magnetic field distributions (|Hz|2) of the two-dimensional cross-section (x-y plane) of the structure at the incident frequencies of f = 1.56 (c), 1.83 (d), and 2.22 THz (e).
According to the FP resonance theory, the length L of the designed FP cavity is an important factor to determine the resonance frequency. The numerical results in Fig. 4 (a) show that the resonance peak have a red shift with the increase of L. The relationship between the resonance frequency and the length L of the cavity can be presented by the standing wave equation: 2β+(ωr)L + θ = 2mπ (m = 1, 2, 3, ...), where θ is the additional phase shift from the Si-air interface. Here β+(ωr) is the propagation constant of the symmetric mode of double-layer BDS structure at the resonance frequency. The first and second-order modes indeed presents a red shift with the increasing of L, which agrees well with the numerical results, as shown in Fig. 4(b). For the cavity-coupled waveguide system, the waveguide coupling strength and cavity intrinsic loss play crucial roles in tailoring the resonance feature. In the proposed BDS filter, the waveguide coupling strength κw is controlled by the width of each Si bars, while the decay rate of SPP field κi is related to the material dispersion and loss of BDS. The transmission characteristic can be described by coupled mode theory (CMT) T = (κw)2/[(ω-ω0)2-(κw + κi)2], where ω and ω0 are the incident frequency and resonance frequency of the FP cavity, respectively. The analytical fitting based on CMT shows good agreement with the numerical calculation, as shown in Fig. 4(c), where κw = 0.063 THz and κi = 0.005 THz. Figure 4(d) shows the transmission spectra for different width d of each Si bar with L = 200 μm. A larger width d, i.e., a smaller κw indicates a narrower linewidth (FWHM), while a lower transmission peak. For the resonant frequency ω0, where the decay rate κi would be a constant, the transmission intensity T = [κw/(κw + κi)]2 should be reduced with the decrease of κw. Meanwhile, the linewidth of the transmission peak FWHM = 4πc(κw + κi)/ω02 will also decrease with κw. Thus, the resonance frequency and quality factor of transmission peak can be optimized by varying the distance between the two introduced Si bars and their widths, respectively.
Fig. 4 (a) Transmission spectra for different distance L between the two Si bars. (b) Numerical (dots) and theoretical (lines) resonant frequencies of mode 1, 2 as functions of the distance L. (c) The numerical result and CMT fitting of the transmission spectra, where d = 10 μm and L = 200 μm. (d) Transmission spectra for different width d of each Si bar with L = 200 μm.
As indicated in Eq. (10), the propagation constant of the coupled mode depends on the coupling distance g between the BDS layers. As shown in Fig. 5, the real part of propagation constant of symmetric mode decreases with coupling distance. According to the FP resonance condition, the resonance frequency should be sensitive to the coupling distance, which provides a new degree of freedom to tune the transmission feature. Figure 6(a) shows the transmission spectra with different coupling distance g, where L = 200 μm. As coupling distance increases, the transmission peak with the same order indeed tends to present a blue shift. This can be attributed to the pronounced decrease of the effective index (neff = β+/k0) of the symmetric mode. Furthermore, according to the Eqs. (1)-(2), the conductivity of BDS shows strong dependence on its Fermi energy EF. The tuning of Fermi energy of BDS can be realized by alkaline surface doping in experiment [15, 16]. The calculated transmission spectra of first-order resonance for different Fermi energy are shown in Fig. 6(b), which suggests that a small change in Fermi energy can indeed tune the resonance frequency of the filter and the transmission peak indeed exhibit a blue shift as the Fermi energy increases. According to the well-known standing wave condition in plasmonic cavity, λSPP is near a constant for the defined cavity length L. Besides, λSPP = λ0/Re(neff), where λ0 is the incident wavelength. When the Fermi energy EF is increased, Re(neff) is decreased as shown in Fig. 5(a). Thus, the incident wavelength λ0 should be decreased as well and that means blue shift in resonance frequency. On the other hand, the imaginary part of the effective index Im(neff) shown in Fig. 5(b) also tend to be decreased with the increase of Fermi energy, which suggests the decrease in dispersion loss, i.e. the decay rate of SPP field κi decreases correspondingly. Based on the CMT, the transmission intensity T = [κw/(κw + κi)]2 at the resonance frequency will increase as well, which is in accordance with the numerical results shown in Fig. 6(b). Therefore, similar with graphene plasmonic device, the dynamically tunable band-pass filter is achieved by simply tune the Fermi energy of BDS, rather than re-optimize the structural parameters of the device.
Fig. 5 Dependence of the real (a) and imaginary (b) parts of the effective index neff of symmetric mode on the Fermi energy EF and coupling distance g of the double-layer BDS structure. Here the other parameters are same as Fig. 2(a).
Fig. 6 Transmission spectra with different coupling distance g (a) and Fermi energy EF (b), where the other parameters are same as Fig. 3(b).
4. Conclusion
In conclusion, we have analytically and numerically investigated the optical coupling of terahertz surface plasmons in BDS sheets. The dispersion relations of symmetric and anti-symmetric modes have been obtained. By controlling the symmetric coupling mode, a novel band-pass filter has been constructed with double-layer BDS sheet sandwiched by two silicon bars. The resonance frequency of this filter regularly shows dependence on the cavity length and coupling distance between the two BDS sheets. Moreover, a small change in Fermi energy in BDS can dynamically tune the transmission spectra. Finally, the relationship between the transmission intensity and bars width has also been discussed. Numerical results hold good agreement with the theoretical analysis, this may provide guidelines to the design of ultra-compact BDS device in the terahertz region.
Funding
National Natural Science Foundation of China (Grant Nos, 61505052, 61775055, 61176116, 11074069).
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