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Abstract
Based on the Dirac semimetals (DSM) modified hybrid plasmonic waveguides, the tunable propagation properties have been systematical investigated by using the finite element method in the THz region, including the influences of structural parameters, the shape of dielectric fiber and Fermi levels of DSM layers. The results show that as the operation frequency increases, the real part of propagation constant increases, and the loss shows a peak. The shape of dielectric fiber (the elliptical structural parameter δ) affects the propagation property obviously, as the structural parameter decreases, the confinement and figure of merit increase, the loss reduces. With the increase of Fermi level of DSM layer, the imaginary part of propagation constant decreases, the modulation depth of loss is more than 95% if the Fermi level changes in the range of 0.01-0.15 eV. In addition, as the permittivity of dielectric material filling in the slit increases, the mode confinement and loss increases, FOM decreases. The results are very helpful to understand the tunable mechanisms of hybrid waveguides and design novel plasmonic devices in the future, e.g. modulators, filters, lasers and resonators.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, terahertz (THz) wave sheds potential applications in the fields of astronomy observation, homeland security identification, spectroscopy, and high-speed 6G wireless communication (in the range of several hundred Gbit s−1) [1–5]. For example, THz quantum cascade laser dual-comb sources with equally spaced and low phase noise frequency lines was demonstrated by phase locking one dual-comb line, which significantly reduced the phase noise of other dual-comb lines and produced pulsed-type waveforms, indicating great importance in the fields of high resolution spectroscopy, gas sensing, and metrology [5]. To the further substantial developments of THz technology, it is a high requirement to explore functional devices and components with high performances. Surface plasmons (SPs) are two-dimensional electromagnetic waves confined at the metal-dielectric interfaces, exhibiting sub-wavelength confinement of electric field in the direction normal to a conductor-dielectric interface [6–8]. Composed of a dielectric fiber and SPs waveguides, the hybrid plasmonic mode structure has the merits of good confinement and low loss simultaneously [9–11]. As a typical example of hybrid plasmonic waveguides, dielectric fiber-gap-metal layer structure consists of a high permittivity dielectric nanowire separated from a metal surface by a low permittivity dielectric gap [12,13].
The further development of hybrid SPs waveguides needs the flexible control of surface modes. With the help of novel materials, e.g. transition metal molybdenum disulfide, black phosphorous, and topological semimetals [14–21], the tunable manipulation of SPs modes can be conveniently achieved. Topological semimetal features the band touching points or nodes near Fermi levels, mainly including the Weyl and Dirac semimetals (DSM) [22–26]. Graphene is a typical 2D DSM and widely investigated in the design of tunable plasmonic devices [27–33]. For example, Qian et al. proposed a dynamic manipulation of hybrid modes in the visible spectral range by utilizing the monolayer and bilayers graphene [33], displaying that the charge density and electromagnetic energy in the gap region can be improved significantly and provided a possible new way to achieve electrically controlled electron-phonon coupling. Since the carriers can be described by Dirac physics, 3D DSM is also regarded to manifest similar tunable properties to graphene, such as strong light confinement, high mobility and the dynamically control of conductivity through Fermi energy [34–36]. Compared with graphene membrane, 3D DSM layer also has several advantages. For example, the Fermi velocity and mobility of DSM are higher than those of graphene (angle-resolved photoemission spectroscopy reveals its Fermi velocity is about 2×106 m/s and twice that of graphene) [37–41], resulting in better plasmonic and tunable properties simultaneously. Next, due to the bulk properties, 3D DSM provides an additional structural degree-of-freedom advantage over graphene in the design of plasmonic devices. Finally, 3D DSM is more stable and less susceptible to environmental defects, substrate bulk phonon and scattering effects.
Nowadays, the exploration of tunable THz functional device is an interesting and essential research topic. With the advantages of strong mode confinement and low dissipation, the exploration of tunable hybrid plasmonic waveguides is a burgeoning investigated nowadays. The existed researches focus on 2D materials supported hybrid SPs waveguides, which is limited to the thin layer thickness [19,24,33]. Inhibiting the good plasmonic and modulation properties, 3D DSM acts as efficiently tunable medium to design novel THz functional devices. To achieve better mode confinement and tunable performances simultaneously, the DSM layer modified hybrid plasmonic waveguides have been systematically explored, including the influences of structural parameters, DSM Fermi levels, and dielectric filling materials in the slit. The results show that the shape of dielectric fiber affects the propagation properties obviously, as the elliptical structural parameter (δ) decreases, the confinement and figure of merit (FOM) increase. The propagation properties can be manipulated in a wide range via Fermi levels, e.g. the propagation constant indicates obvious blue shift, and the modulation depth of dissipation is more than 95% if Fermi level changes in the scope of 0.01-0.15 eV.
2. Structural design and simulation methods
Figure 1 depicts the geometry configurations of the DSM modified hybrid plasmonic structure of dielectric fiber-gap-DSM layer. The thicknesses of DSM, SiO2 and doped Si layers are 10 μm, 10 nm and 5 μm, respectively. The bias voltage is applied between the DSM and doped Si layers to vary the Fermi level. The incident waves normally transmit through the hybrid plasmonic waveguides along the z direction.
Fig. 1. (a) The 3D sketch of the DSM modified hybrid plasmonic waveguides. (b) The side view of DSM hybrid plasmonic waveguide structure. The thicknesses of DSM, SiO2 and doped Si layers are 10 μm, 10 nm and 5 μm, respectively. Here, ax and ay represent the semi-major and semi-minor axes lengths of elliptical fiber, respectively. The slit width and thickness are 60 μm and 5 μm, respectively.
Under the framework of Kubo formalism in random phase approximation, the longitudinal complex dynamic conductivity of 3D DSM is expressed as [42,43]:
The permittivity of 3D Dirac semimetals can be obtained by using the following formula,
The permittivity of metal in the THz region is expressed as [44]:
The mode area is defined as the ratio of the total mode energy and peak energy density, which is given by [12]:
3. Results and discussion
The characteristics of the DSMs modified hybrid plasmonic waveguide in the THz spectral range, such as geometrical parameter, effective indices, and mode area, have been systematically investigated by using finite element method (FEM), which was performed by using COMSOL MULTIPHYSICS 4.2. Figure 2 shows the propagation properties of DSM hybrid plasmonic waveguides versus radius at different slit depths. The operation frequency is 1.0 THz. The effective index is defined as neff=β/k0, β is the propagation constant of hybrid mode, k0 is the wave-vector. As dielectric fiber radius increases, the real part of effective index increases. For example, on the condition that the slit depth is 5 μm, if the radii of dielectric fibers are 30, 50, 100, and 200 μm, the effective indices are 1.551 + 9.057×10−4i, 2.032 + 1.070×10−3i, 2.993 + 7.373×10−5i, and 3.307 + 6.477×10−6i, respectively. The reasons are shown in the following. The propagation properties of hybrid plasmonic modes are mainly determined by the contributions of low lossy fiber mode and strong confinement of plasmonic mode. As fiber radius increases, the effect of dielectric mode enhance, the interaction area between dielectric and plasmonic modes increases, the real part of effective index and mode confinement increase. However, for the case of dissipations, the situation is a little complex. Firstly, as the radius of dielectric fiber increases, the interaction area of dielectric fiber and DSM layer increases, enlarging the losses of hybrid mode. Simultaneously, the effect of dielectric fiber mode increases, the dissipation reduces. Thus, the loss of hybrid mode reaches a peak at certain value. In addition, as slit depth increases, the real part of effective index and the losses decrease, which result from the reduction of the interaction area between dielectric fiber and DSM layer. To measure the mode confinement, the figure of merit is defined as FOM = Re(neff)/Im(neff). Figure 2(c) shows FOM versus dielectric fiber radii at different slit depths. As the radius of dielectric fiber increases, the confinement of dielectric fiber increases, the loss reduces, leading to FOM increasing. Additionally, as the slit depth increases, the interaction of dielectric fiber and SPs mode decreases, the loss reduces obviously, therefore FOM increases significantly. The inverse of normalized mode area (Am/A0) of SPs mode is a good measurement of confinement, as shown in Fig. 2(d). The effective mode area Am is given in Eq. (5), and A0 is the diffraction-limited area, A0 =λ2/4. As fiber radius increases, the real part of effective index increases, the mode area decreases, indicating better mode confinement. But if the fiber radius is large enough, the dielectric mode dominates, the mode area increases, the confinement decreases. Therefore, the mode area shows a dip near certain fiber radius (about 50-60 μm). Additionally, as the slit depth increases, the effective indices decrease, leading to bad confinement, which agree with the results in Fig. 2(c). If the fiber radius is large enough, the influence of slit depth on mode area is not obvious.
Fig. 2. (a) Real and (b) imaginary parts of the effective indices for DSM modified hybrid plasmonic waveguide versus fiber radii at different slit depths. (c) FOM and (d) the normalized mode area Am/A0 versus fiber radius in the THz spectral range. The slit width is 60 μm, the slit depths are 1, 2, 5, 8, 10, and 20 μm, respectively. The operation frequency is 1.0 THz.
Figures 3(a) and 3(b) show the dispersion properties of DSM supported hybrid plasmonic mode waveguides for different types of dielectric fibers. The Fermi level of Dirac semimetal is 0.10 eV. In the THz region, the intraband contribution dominates, the real part of permittivity is negative, and the DSM behaves a metal membrane. As frequency increases, the DSM permittivity decreases, the contribution of plasmonic mode reduces and dielectric fiber mode increases. Thus, the real part of the effective index increases, the imaginary part of neff shows a peak, which can be found in Fig. 3(a) and 3(b). For example, at the frequencies of 0.5, 1.0, and 2.0 THz, the permittivity of DSM is -2.893×104+2.048×105i, -9.638×103+3.417×103i, and -2.620×103+4.672×102i, the according effective indices are 1.537 + 1.020×10−3i, 2.032 + 1.070×10−3i, and 2.979 + 2.145×10−5i, respectively. The reasons are given in the following. As frequency increases, the DSM layer permittivity decreases, the loss increases. However, if the radius of dielectric fiber is large enough, the dielectric fiber mode dominates, resulting in the loss decreasing. Therefore, the loss of hybrid mode shows a peak versus frequency. The shape of dielectric fiber also greatly affects the propagation property. To have a roughly fair comparison, we set the section areas of different dielectric fiber equally. The elliptical structural parameter δ is defined as δ=ax/ay. In the simulation, the semi-major and semi-minor axes of the dielectric wire are 40×62.5 μm (A-type), 50.0×50.0 μm (B-type), and 62.5×40 μm (C-type), and the according elliptical structural parameters δ are 0.64, 1, and 1.56, respectively. Compared with B-type and C-type, type-A dielectric fiber is relatively sharper, the length along the y direction ay is longer. Thus, the interaction area of A-type is smaller, the field enhancement near the dielectric fiber and DSM layer is improved remarkable, and high energy localization is achieved. Consequently, the dielectric fiber mode makes more contribution, resulting in the real (imaginary) part of effective increasing (decreasing), as shown in Fig. 3(a) and 3(b). For example, at the frequency of 1.0 THz, for type-A, type-B, and type-C of dielectric fibers, the effective indices are 2.196 + 5.621×10−4i, 2.032 + 1.070×10−3i, and 1.914 + 1.580×10−3i, respectively.
Fig. 3. (a) Real and (b) imaginary parts of the effective indices for the hybrid plasmonic waveguides versus frequency for different types of dielectric fibers. The inset in Fig. 3(b) depicts different shapes of dielectric fibers. (c) FOM and (d) the effective mode area of Am/A0 versus frequency. The semi-axes along x and y directions of the dielectric fiber [ax, ay] are [40 μm, 62.5 μm], [50 μm, 50 μm], and [62.5 μm, 40 μm], respectively.
Figure 3(c) shows the FOM versus frequency for different kinds of dielectric fibers. At low frequency, the wavelength is large, the dielectric fiber can’t provide good confinement, FOM is small. As frequency increases, the DSM layer permittivity decreases, the contribution of plasmonic mode reduces, the loss increases drastically, resulting in FOM decreasing. But if frequency is large enough (>1.0 THz), the wavelength reduces, the low dissipation of dielectric fiber mode plays more important roles, the loss decreases obviously, the FOM increases at large frequency. Additionally, because the length of type-A dielectric fiber along the y direction (ay) is larger, and the interaction area of dielectric fiber with DSM substrate is smaller, thus the dielectric fiber mode make more important role, resulting in the effective index reducing. Thus, the FOM of type-A fiber is larger than that of type-B or type-C. Figure 3(d) shows the normalized mode area versus frequency. At low frequency, DSM layer manifests good plasmonic property, the mode area is small. As frequency increases, the DSM permittivity and mode area decreases, much more modes penetrate into DSM layer, resulting in loss increasing. The dip region of mode area roughly agrees well with the loss peak position. If frequency is larger enough, the dielectric fiber dominates, the effective mode area increases with frequency. Additionally, Fig. 3(d) also demonstrates the effect of dielectric fiber shape on the effective mode area. Compared with type-A and type-B fibers, the interaction area of type-C is largest, resulting in more loss. Thus, the contribution of plasmonic mode dominates in a wider frequency range.
To have a deep understand of the propagation properties of DSM modified hybrid modes, we characterize the energy density distributions at different frequencies for three types of dielectric fiber shapes, as shown in Fig. 4. As frequency increases, the wavelength decreases, the permittivity of DSM decreases, e.g. at the frequencies of 0.5, 1.0, and 1.5 THz, the permittivity of DSM layers is -2.893×104+2.048×104i, -9.638×103+ 3.417×103i, and -4.560×103+1.080×103i, respectively. Thus, the contribution of plasmonic mode decreases at larger frequency, and the dielectric fiber mode makes more important role, which results in the real part of effective index increasing. For instance, for the circular type-B dielectric fiber, at the frequencies of 0.5, 1.0, and 1.5 THz, the effective indices of hybrid modes are 1.537 + 1.022×10−3i, 2.032 + 1.070×10−3i, and 2.662 + 1.245×10−4i, respectively, which can be found in Figs. 4(b), 4(e) and 4(h). At low frequency, the hybrid mode can’t provide good confinement due to large wavelength, the effective index is small. As frequency increases, the wavelength decreases, e.g. frequency is 1.0 THz, the hybrid structure provide relatively good confinement. However, the permittivity of DSM layer decreases with the increase of frequency, much more mode penetrates into the high lossy layer, resulting into the real and imaginary parts of the hybrid modes increasing. If frequency increases further, the DSM layer indicates worse metal property, the influences of plasmonic mode reduce. Simultaneously, the wavelength becomes very small, the contribution of dielectric fiber mode dominates at the high frequency, a large portion of mode penetrates into dielectric fiber, leading into the loss reducing, as shown in Figs. 4(h). Consequently, the real part of effective index increases with frequency, and loss shows a peak. Additionally, Fig. 4 depicts the influences of fiber shapes on the hybrid plasmonic modes. If the frequency is low, the wavelength is very large, e.g. λ=600 μm if frequency is 0.5 THz, the hybrid structure can’t provide good confinement. In this case, much of modes penetrate the surrounding low dissipative environment, as given in Figs. 4(a)–4(c). For the type-C fiber, the length along the y direction is smallest, the confinement is worst, resulting into small dissipation. As frequency increases, the hybrid mode can be well confined, the elliptical fiber shape affects the propagation property in a different way. The length of type-A dielectric fiber along the y direction is largest, the dielectric mode makes more contribution. Furthermore, the interaction area of type-A dielectric fiber with the DSM substrate is smallest, the effects of plasmonic mode decrease, the loss reduces. Therefore, the real part of effective index for type-A is largest, and the loss is the smallest, which agrees with the calculation results shown in Fig. 3. For example, if the frequency is 1 THz, for type-A, type-B and type-C dielectric fibers, the effective indices are 2.196 + 5.621×10−4i, 2.032 + 1.070×10−3i, and 1.914 + 1.583×10−3i, respectively. That is to say, the type-A dielectric fiber with smaller elliptical structural parameter δ manifests better confinement and smaller loss if frequency is relatively large.
Fig. 4. Energy density distributions of the DSM hybrid plasmonic waveguides for different types of dielectric fibers. The slit depth between dielectric fiber and DSM substrate is 5 μm. The slit width is 60 μm. The resonant frequencies are 0.5, 1.0, 1.5 THz, respectively. The radii of dielectric fiber [ax, ay] are [40 μm, 62.5 μm] (Fig. (a), (d), (g)), [50 μm, 50 μm] ((b), (e), (h)), and [62.5 μm, 40 μm] ((c),(f) and (i)), respectively.
One peculiar property of Dirac semimetals is the linear dispersion relationship and good tunability, i.e. its conductivity can be modulated in a wide range by changing the Fermi levels. The influences of Fermi levels on the propagation properties of DSM hybrid modes are given in Fig. 5. Since the permittivity increases with Fermi level, the DSM indicates better plasmonic modes, resulting in the values of effective indices decreasing. For example, if the frequency is 1.0 THz, at the Fermi levels of 0.01, 0.05, 0.10, 0.15 eV, the DSM permittivity is -8.529×101+3.789×101i, -2.402×103+8.569×102i, -9.639×103+3.417×103i, -2.170×104+ 7.683×103i, and the effective indices are 2.093 + 1.615×10−2i, 2.038 + 2.119×10−3i, 2.032 + 1.070×10−3i, and 2.030 + 7.072×10−4i, respectively. Accordingly, the modulation depth of real (imaginary) part of effective index is 3.01% (95.62%). The quality of plasmonic properties can be roughly measured by the ratio of real part to imaginary part, i.e. the ratio of Re(εmetal)/Im(εmetal), the larger value of ratio means better plasmonic properties. For example, for the metal cases, at the frequency of 1.0 THz, the permittivity of Au, Ag, Cu, and Fe is -1.119×105+7.216×105i, -2.387×105+1.038×106i, -5.489×105+1.205×106i, and -4.792×104+ 2.112×105i, which can be obtained from Eq. (4). Correspondingly, the ratios of Re(εMetal)/Im(εMetal) are 0.412, 0.230, 0.4555, and 0.2269, respectively. However, for the case of DSM layers, at the Fermi levels of 0.01, 0.05, 0.10, and 0.15 eV, the values of Re(εMetal)/Im(εMetal) are 2.251, 2.803, 2.821, and 2.824, respectively. Compared with the metal, the ratios of Re(εDSM)/Im(εDSM) for DSM layers are much larger, which means better plasmonic properties and lower losses. It should be noted that though the effect of Fermi level on the real part of effective index is little, but it affects the loss significantly. Figure 5(c) shows FOM versus frequency at different Fermi levels. As Fermi levels increases, DSM layer manifests good plasmonic properties, the loss decreases obviously, resulting in FOM increasing. Additionally, as Fermi level increases, the penetration depth and interaction area of hybrid modes decrease, the mode confinement decreases, the effective mode area increases, as given in Fig. 5(d).
Fig. 5. (a) Real and (b) imaginary parts of the effective indices for the DSM modified hybrid plasmonic waveguides versus frequency at different Fermi levels. (c) Figure of merit and (d) the values of Am/A0 versus frequency. The slit width and depth are 60 μm and 5 μm, respectively. The radius of circular fiber is 50 μm. The Fermi levels are 0.01, 0.03, 0.05, 0.08, 0.10, and 0.15 eV, respectively.
Figure 6 shows the propagation properties of DSM supported hybrid plasmonic mode waveguides for different kinds of dielectric materials filling in the slit. The dielectric materials filling in the slit are air, ovalbumin, Teflon, DNA, TNT and HMX with the refractive indices of 1, 1.15, 1.45, 1.65, 1.76, and 1.81, respectively [45,46]. The real and imaginary parts of effective indices versus frequency for different kinds of dielectric filling materials in the slit are shown in Fig. 6(a) and 6(b). By inserting the dielectric material into the slit, the hybrid mode is well confined, the effective index increases, the resonant absorption peak also manifests obvious red shift. For example, for the dielectric filling materials of air, ovalbumin, Teflon, DNA, TNT, and HMX, the effective indices are 1.691 + 1.190×10−3i (0.72 THz), 1.715 + 2.224×10−3i (0.70 THz), 1.775 + 0.00293i (0.68 THz), 1.780 + 3.340×10−3i (0.65 THz), 1.804 + 3.560×10−3i (0.65 THz), and 1.815 + 3.66×10−3i (0.65 THz), respectively. Figure 6(c) shows the FOM versus frequency. As the refractive index of dielectric filling materials in the slit increases, the polarizability around the interface between the dielectric fiber and DSM layer enhances, much more mode penetrates into the DSM layer. In this case the hybrid mode is better confined in the slit, but the losses increase also obviously, resulting in FOM reducing. The normalized mode areas versus frequency for different kinds of materials can be found in Fig. 6(d). With the increase of refractive index increases, the difference of permittivity of dielectric materials filling in the slit and dielectric fiber decreases, i.e. the permittivity offset becomes smaller, reducing the confinement. Thus, the mode area increases if ovalbumin or Teflon is inserted into the slit.
Fig. 6. (a) Real and (b) imaginary parts of the effective indices for the hybrid plasmonic waveguides versus frequency for different dielectric filling materials in the slit. (c) Figure of merit and (d) the effective mode area Am/A0 versus frequency. The dielectric fiber radius is 50 μm. The Fermi level of Dirac semimetal is 0.10 eV.
4. Conclusion
Based on the DSM modified hybrid plasmonic structures, the tunable propagation properties have been investigated by using FEM in the THz spectral region, including the effects of structural parameters, the shape of dielectric fiber and Fermi level of DSM layers. The results manifest that as frequency increases, the real part of effective index increases, and the imaginary part of effective index shows a peak. The shape of dielectric fiber affects the propagation properties obviously, as the elliptical structural parameter decreases, the confinement and FOM increase, the dissipation reduces. With the Fermi level of DSM layer increases, the real (imaginary) part of propagation constant increases (decreases), the modulation depth of dissipation peak is 95.02% if Fermi level changes in the range of 0.01-0.15 eV. As the permittivity of dielectric materials in the slit increases, the confinement and loss increase, the value of FOM decreases. The results are very helpful to understand the tunable mechanisms of DSM hybrid modes and design novel plasmonic devices in the future, e.g. modulators, filter, lasers and resonators.
Funding
Natural Science Foundation of Shanghai (21ZR1446500); Shanghai Local College Capacity Building Project (21010503200); Key Project of National Natural Science Foundation of China (U1931205); National Natural Science Foundation of China (12073018, 61674106); Shanghai Municipal Education Commission (2019-01-07-00-02-E00032); Funding of Shanghai Municipality Science and Technology Commission (19590746000, 20070502400, YDZX20203100002498).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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