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Abstract
By depositing the trapezoidal dielectric stripe on top of the 3D Dirac semimetal (DSM) hybrid plasmonic waveguide, the thermal tunable propagation properties have been systematically investigated in the terahertz regime, taking into account the influences of the structure of the dielectric stripe, temperature and frequency. The results manifest that as the upper side width of the trapezoidal stripe increases, the propagation length and figure of merit (FOM) both decrease. The propagation properties of hybrid modes are closely associated with temperature, in that when the temperature changes in the scope of 3-600 K, the modulation depth of propagation length is more than 96%. Additionally, at the balance point of plasmonic and dielectric modes, the propagation length and FOM manifest strong peaks and indicate an obvious blue shift with the increase of temperature. Furthermore, the propagation properties can be improved significantly with a Si-SiO2 hybrid dielectric stripe structure, e.g., on the condition that the Si layer width is 5 µm, the maximum value of the propagation length reaches more than 6.46 × 105 µm, which is tens of times larger than those pure SiO2 (4.67 × 104 µm) and Si (1.15 × 104 µm) stripe. The results are very helpful for the design of novel plasmonic devices, such as cutting-edge modulator, lasers and filters.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nowadays, due to the unique features such as low absorption by materials and the ability to penetrate biomedical samples, terahertz (THz) technology shows prospects in various applications, such as wireless communications, bioimaging, astronomical observation, and spectroscopy analysis [1–5]. For example, H. H. Ruan et al. proposed an efficient terahertz image super-resolution model, the resolution reached 31.67 dB on the peak signal-to-noise ratio index and 0.86 on the structural similarity index, which can be utilized to extract low resolution image features and acquire the mapping of high-resolution images [6]. Surface plasmons (SPs) are the highly localized surface electromagnetic wave, stimulated by the interaction between free electrons in metal and photons, which propagate along metal-dielectric interface and manipulate the electro-magnetic waves on sub-wavelength structure [7–10]. Different types of SPs waveguide structures have been widely investigated, such as metal-dielectric-metal structure, metal wire waveguides [11], metallic V-grooves structure [12] and hybrid surface plasmonic structures [13]. As a typical example of hybrid plasmonic structure, a dielectric loaded waveguide is referred to a dielectric stripe deposited on a metal layer and manifests the advantages of strong mode confinement and small loss, which can be used to design low threshold lasers, micro-ring resonator, and filters [14–16].
For the practical applications in the fields of resonators and splitters, the tunable hybrid plasmonic devices are more preferable [17–19]. With the merits of high mobility, strong mode confinement and good tunable properties, graphene is a typical example of 2D Dirac semimetals and acts as a good candidate for the tunable medium [20–22]. Over the past few years, graphene-based tunable hybrid plasmonic structures have been proposed. For example, by inserting graphene between the gold nanowire array and the dielectric spacer HfO2, J. Y. Li et al. proposed a tunable hybrid localized surface plasmon modes structure in mid-infrared wavelengths. With the assistance of graphene layer, the p-polarized emissivity of the hybrid plasmonic structure can be enhanced, resulting in a five-fold enhancement of the p-polarized emissivity for large emission angles. The peak positions and spectral emissivity can be modulated through graphene Fermi level, with nearly 90% of the variance in the absorption spectrum around 10.5 µm achieved [23]. Based on the graphene supported hybrid plasmonic structures, C. Donnelly et al. investigated the nonlinear properties in the near-IR spectral range, the results showed that the nonlinear parameters of graphene dielectric loaded surface plasmon waveguides exceeded 105 W−1/m for the gold-graphene-Si hybrid waveguide, which are several orders of magnitude larger than those of waveguides without graphene [24]. By using a dielectric ridge-graphene plasmonic structure, J. Gosciniak et al. demonstrated a tunable hybrid waveguide and it could be modulated to 3 dB with a 65 nm-long waveguide, with an energy per bit of only 0.08 fJ/bit; Furthermore, with a high refractive index Si ridge, the influence of graphene layer on the hybrid mode was enhanced, and its figure of merit increased over 17.3, which exceeded the values of the conventional modulators (the figure of merit was 3.5) [25]. By inserting a graphene micro-ribbon between a dielectric ridge and SiO2 layer, a compact dielectric-loaded plasmonic waveguide with a length in the range of millimeter range was shown and can be utilized as a biochemical sensing, which reduced Ohmic losses and enhanced the sensor sensitivity significantly. For example, the propagation length increased from 60 µm to 16 mm, and the average limit of detection was in the range of 3- 6 µRIU [26].
Nowadays, there is a high demand for the tunable plasmonic functional devices with fine performances. Although there are some researches about the graphene hybrid waveguides, due to the thin thickness of graphene membrane, the tunable properties are highly limited. Fortunately, the emergence of 3D DSM has solved this problem well. Similar to graphene, the dielectric constant of 3D DSM can also be flexibly adjusted by applying an external gate voltage. 3D DSM also has a higher mobility, reaching 9 × 106 cm 2 V−1 s−1 at 5 K, much higher than that of the graphene (2 × 105 cm 2 V−1 s−1 at same temperature) [27–29]. Compared to graphene membrane, 3D DSM adds a degree of structural freedom due to its three-dimensional properties, which provides more choices for the structural design of waveguides, and it is also more stable and convenient to handle [30–33]. These advantages indicate that the 3D DSM is expected to be a new generation of plasmonic materials and effective manipulation plasmonic devices at different frequencies [34–36]. Here, we study the trapezoidal dielectric stripe-3D Dirac semimetal hybrid plasmonic waveguide structure. The results show that the propagation length of hybrid mode is flexibly manipulated by changing the temperature in the scope of 3 K-600 K, and the modulation depth of propagation length reaches more than 96%. Additionally, the propagation length and FOM both show the strong peaks at the balance point with the plasmonic and dielectric modes, and the peak position indicates a blue shift with the increase of temperature. These results show great promise for designing state-of-the-art plasmonic devices, chip-scale integrated plasmonic circuitry.
2. Research method
Figure 1(a) shows the proposed dielectric structure deposited on the 3D DSM hybrid waveguide structure, where the thickness of the 3D DSM layer is 10 µm. A trapezoidal SiO2 stripe is deposited on the 3D DSM layer, with height h of 120 µm, and wa and wb are the lengths of the upper and bottom sides of the trapezoidal stripe, respectively. In addition, to improve the propagation properties, a modified hybrid waveguide structure with rectangular Si is inserted into the middle of SiO2 trapezoidal dielectric (SiO2-Si hybrid structure), as shown in Fig. 1 (b), in which wsi is the width of the inserted rectangle Si. The permittivity of silicon and silica are 11.70 and 3.92, respectively.
Fig. 1. (a) Section of trapezoidal dielectric waveguide made of SiO2 and (b) section of a waveguide with a rectangular Si inserted in the middle of the SiO2 trapezoidal dielectric. The wa and wb are the lengths of the upper and bottom of the trapezoidal stripe, h is the height of the trapezoidal stripe, and wSi is the width of the inserted rectangle Si.
We use the Kubo formula to calculate the longitudinal dynamic conductivity of 3D DSM in the long-wave limit q≪kF (the local response approximation). The dynamic conductivity of 3D DSM can be expressed as [37]:
In which G(E)=n(-E)-n(E) with n(E) being the Fermi distribution function, kF = EF/ℏvF, vF = 106 m/s is the Fermi velocity, EF is the Fermi level, εc = Ec/EF = 3 (Ec is the cutoff energy beyond which the Dirac spectrum is no longer linear), Ω=ℏω/EF + iℏτ-1/EF, g = 40 is the degeneracy factor, τ=4.5 × 10−13 s is the intrinsic relaxation.
The complex dielectric constant of 3D DSM can be represented by a two-band model that takes into account the electron transitions between the bands [36]:
ɛ0 is the permittivity of vacuum, εb = 1.The normalized effective mode area Am/A0 reflects the capacity of the area of the constrained mode field, A0=λ2/4 is the diffraction-limited area, Am is the mode area and defined as the ratio of the total mode energy and the peak energy density, which can be given by [13]:
The real part of effective index Re(neff) and propagation length LSP can be defined by Re(neff)=Re(β)/k0 and LSP=λ/4πIm(neff), where Im(neff) is the imaginary part of the mode effective refractive index neff, and β is the complex propagation constant, k0 is the wave vector.
The confinement factor Γ is defined as [38]:
in which Wd is the electromagnetic energies in the dielectric stripe, We use a figure of merit (FOM) to quantify the trade-off between propagation length and model area, defining as: the modulation depth (MD) can be defined as:3. Results and discussions
We used finite element analysis software COMSOL Multiphysics to study the propagation properties of the proposed 3D DSM supported dielectric deposited hybrid plasmonic waveguide structure. The real part of the effective mode index (Re(neff)) and propagation length (Lsp) of the hybrid modes versus frequency for different shapes of trapezoid dielectric stripe, as shown in Fig. 2. With the increase of frequency, the dielectric constant decreases. 3D DSM layer exhibits worse plasmonic properties, and more mode infiltrates into the 3D DSM layer, resulting into the real part of the effective mode index increasing. The mode confinement is determined by the Am/A0 (A0=λ2/4 is the diffraction-limited area, and Am is the mode area) value, whose smaller value indicates a better mode concentration. Figure 2(b) shows the normalized mode area of the dielectric stripe with different shapes. As frequency increases, the wavelength decreases, the dielectric mode plays a larger role, leading to an increase in the mode area. Figure 2(c) describes the relationships between propagation length and frequency. The contribution of low lossy dielectric mode increases and plays an important role with frequency, and loss decreases. For example, when the upper side of the trapezoid wa is 90 µm, at low frequency, the normalized effective mode area is small, but the loss is large. As frequency increases, the propagation lengths enhance drastically, reaching about 4.67 × 104 µm at 2.0 THz. The FOM represents the trade-off between propagation length and mode constraints, as shown in Fig. 2(d). With the increase of frequency, due to the fact that as the propagation length of the hybrid mode increases, the value of FOM increases significantly. We also compare the influences of the shapes of the dielectric stripe on the waveguide. As shown in Fig. 2(c) and 2(d), with the increase (decrease) of the upper (bottom) length wa (wb), the contact area between the dielectric block and the 3D DSM layer decreases, the effect of the low-loss dielectric fiber mode is weakened, and the loss increases. Consequently, at sharper dielectric stripe, the dielectric block and mode plays a more important role, resulting into the gradual increase of propagation length and FOM. In summary, from the perspective of low dissipation and FOM, the optimized value of the wa is found to be 90 µm.
Fig. 2. (a) The real part of effective mode index, (b) normalized effective mode area, (c) the propagation length and (d) FOM as a function of frequency in trapezoidal dielectric with different upper and lower bottom lengths. The cross-section area of the dielectric stripe is set to a fixed value of 14400 µm2. The Fermi level of the 3D DSM layer is 0.01 eV.
In order to better study the effects of trapezoidal stripe shapes on waveguide performances, we simulated the field distribution diagrams of the proposed 3D DSM hybrid waveguide, as shown in Fig. 3. To have a fair comparison, we set the cross-sectional area of the dielectric stripe to a fixed value of 14400 µm2. It can be found that with the increase of wa and the decrease of wb, the contact area between trapezoidal stripe and 3D DSM layer decreases, and more mode leaks into the air, resulting in the decrease of the Re (neff) and the propagation length. For example, Fig. 3(a), 3(d) and 3(f) indicate the field distribution of wa = 80 µm, 120 µm and 150 µm, the real parts of effective mode index (propagation length) are 1.724 (5.8 × 104 µm), 1.711 (2.8 × 104 µm) and 1.682 (1.7 × 104 µm), respectively. The confinement factor Γ is defined by Eq. (6), which means that the fraction of energy in the dielectric stripe. When the width of upper (bottom) side of trapezoidal stripe is small (large), most of the mode is confined in the dielectric stripe. For example, if wa = 80 µm, the confinement factor is 0.931. However, as the width of upper (bottom) side of trapezoidal stripe increases (decrease), more modes permeate into the air and the 3D DSM layer, reducing the confinement factor, e.g., as given in Fig. 3(f) on the condition that wa = 150 µm, the confinement factor is 0.899.
Fig. 3. Field distributions for different stripe shapes, wa and wb are (a) [wa, wb] = [80,160] µm, (b) [wa, wb] = [90,150] µm, (c) [wa, wb] = [110,130] µm, (d) [wa, wb] = [120,120] µm, (e) [wa, wb] = [130,110] µm, (f) [wa, wb] = [150,90] µm. The cross-sectional area and height of the dielectric stripe were set to a fixed value of 14400 µm2 and 120 µm, respectively. The 3D DSM Fermi level is 0.01 eV. The work frequency is 1.0 THz.
The effects of temperatures on the dielectric constants of 3D Dirac semimetal at different Fermi levels are illustrated in detail in Fig. 4. It can be seen that if the Fermi level is smaller, i.e., 0.01 eV, the carrier concentrations are relatively low, thus the differences of dielectric constants between temperatures are larger. At the frequency of 0.5 THz, the dielectric constants of 3D DSM are -2.789 × 102 + 2.125 × 102i, -6.643 × 103+ 4.706 × 103i, and -2.572 × 104 + 1.820 × 104i on the condition that the temperatures are 3 K, 300 K, and 600 K, respectively. However, if the Fermi level is large, the carrier concentrations increase significantly, the effects of temperatures are relatively small. For example, at the frequency of 0.5 THz, if the Fermi level is 0.10 eV, the dielectric constants of the 3D DSM are -2.894 × 104 + 2.049 × 104i, -3.529 × 104 + 2.498 × 104i, and -5.437 × 104+ 3.848 × 104i at the temperatures of 3 K, 300 K, and 600 K, respectively. Thus, at the Fermi level of 0.01 eV (0.10 eV), the variance range of real and imaginary parts of DSM permittivity are 98.91% (46.78%) and 98.84% (46.88%). Therefore, at smaller Fermi level, the influences of temperatures on the DSM layer and propagation properties of hybrid modes are more obvious. Thus, the Fermi level of 0.01 eV is utilized.
Fig. 4. (a) The real and (b) imaginary parts of the 3D DSM permittivity versus frequency at different Fermi levels. The solid, dash, and dot lines are the results of calculations at the 3 K, 300 K, and 600 K, respectively.
Figure 5 shows the effects of temperatures on the propagation properties of the hybrid mode of the trapezoidal stripe- 3D DSM hybrid waveguide. With the increase of temperature, the value of Re(neff) decreases and the effective mode area Am/A0 increases. The reasons are as follows. As the temperatures increase, the dielectric constants of DSM increase, 3D DSM layer exhibits better plasmonic properties, and fewer mode infiltrates into the 3D DSM layer, the skin depths decrease, which are defined by δ= (2/ (wµσ))1/2 (σ is the dielectric constant of the 3D DSM, ω is angular frequency, µ is the permeability [4π×10−7 (H/m)]) [39]. For example, the dielectric constants (skin depths) of 3D DSM at 1 THz at 3 K, 300 K, and 600 K are -86.04 + 37.77i (10.96 µm), -2211 + 785.9i (2.41 µm) and -8573 + 3034i (1.23 µm), respectively. Therefore, at higher temperature, the larger dielectric constant of the 3D DSM layer leads into smaller skin depth, and the contribution of plasmonic mode reduces, and the low lossy fiber mode plays a more important role, resulting in a decrease in the value of the effective index of the hybrid mode. Figure 5(c) and 5(d) depict the influences of temperatures on the propagation lengths and FOM of the trapezoidal stripe-DSM hybrid mode structures. At low temperature of 3 K, the dielectric constant of DSM is small, and the skin depth is relatively large. Furthermore, as frequency increases, the dielectric constant decreases, the skin depth increases. For example, at the frequency of 0.5 THz, 1.0 THz, and 2.0 THz, the according skin depths are 9.26 µm, 10.96 µm, and 13.12 µm, respectively. The larger skin depth at high frequency means more modes penetrate into DSM layer, resulting into more loss and small propagation length, as the black line shown in Fig. 5(c). On the contrary, if the temperature is large, e.g., > 300 K, due to the large dielectric constant, the 3D DSM layer manifests better metal and plasmonic properties. In this case, the according skin depth is smaller, the contribution of plasmonic mode decreases. Furthermore, as frequency increases, the wavelength decreases, the effect of the low lossy fiber mode increases. Thus, at high temperature, with the increase of frequency, the loss decreases and the propagation length increases, as the cyan and violet lines shown in Fig. 5. Interestingly, in the medium temperature range, the propagation length versus frequency shows a peak, as the red and green lines given. We take the results of 200 K as an example. The reasons are given in the following. At low frequency, the dielectric constant of 3D DSM is large, the skin depth is small, the effect of the plasmonic mode is relatively weak. Furthermore, as frequency increases, the wavelength decreases, the low lossy dielectric mode play more important role. Thus, the propagation length increases with frequency, as the region I given in Fig. 5(c). However, as frequency increases further, the dielectric constant of DSM decreases, the skin depth increases, the contribution of high lossy plasmonic mode increases, resulting into the propagation length decreasing, as the region II given. Consequently, about the frequency of 1.2 THz, the propagation length and FOM shows a peak, with the values of about 1.01 × 106 µm and 9.79 × 103, respectively. It should be noted that with the increase of temperatures, the peak values of the propagation lengths and FOM indicate the blue shift. The dielectric constant of 3D DSM is smaller at lower temperature, e.g., 3 K, the skin depth and loss are larger, the peak value is smaller than 0.5 THz. As the temperatures increase, the skin depths and the losses decrease, the peak positions of propagation lengths also move to high frequency. For instance, at the temperatures of 77 K and 200 K, the peak positions of the propagation lengths are about 0.57 THz and 1.20THz. Furthermore, if the temperature increases further, e.g., > 300 K, the skin depth becomes smaller, the peak position of propagation length obviously manifests a blue shift. For example, if the temperature is 300 K, the peak position of propagation length is about 2.25 THz. Additionally, the propagation lengths of the hybrid plasmonic modes can be modulated by temperatures over a wide range. For example, at 1.0 THz, the Lsp of 3 K, 77 K and 200 K are 1.3 × 103 µm, 3.4 × 103 µm and 3.9 × 104 µm, respectively. Accordingly, the modulation depth (MD) is 96.7%. Correspondingly, the value of FOM can also be manipulated with thermal control method. For example, at 1.0 THz, for temperature of 3 K, 77 K and 200 K, the FOM is 12.75, 31.04 and 343.4, respectively, and the modulation depth is 96.3%.
Fig. 5. (a) The real part of effective mode index, (b) normalized effective mode area, (c) the propagation length and (d) FOM versus frequency at different 3D DSM temperatures. The Fermi level of 3D DSM layer is 0.01 eV. The width of the upper and bottom sides of the trapezoidal strip are [wa,wb] = [90, 150] µm.
To illustrate the effects of temperatures on waveguide performances, we simulated the field distribution diagrams of the proposed 3D DSM hybrid waveguide, as shown in Fig. 6. At low temperature, the dielectric constant of 3D DSM is small, the skin depth is relatively larger with the value of about several micrometer. Thus, the real part of the effective refractive index is larger, e.g., at the temperature of 3 K, the value of Re(neff) is 1.808. As temperature increases, the dielectric constant increases significantly, 3D DSM layer shows better plasmonic properties, resulting into the decreasing of the Re (neff). For instance, if the temperature is 600 K, the Re(neff) is 1.714. Next, we discuss the influences of temperatures on the dissipations. At low temperature, the dielectric constant of DSM is small, the skin depth is larger, e.g., the skin depth at 3 K is 10.96 µm, so more modes infiltrate into the DSM layer, leading to a larger loss, and the imaginary part of the effective refractive index is 1.871 × 10−2. As temperature increases, the dielectric constant of 3D DSM layer increases, the skin depth decreases, and leading into the loss decreasing. For example, the dielectric constant of DSM and skin depth at 200 K are -1.032 × 103+ 3.701 × 102i and 3.51 µm, and the according Im(neff) is 6.095 × 10−4. As the temperatures increase further, e.g., > 300 K, the carrier concentrations increase, the dissipations of hybrid modes increase. For instance, if the temperatures are 300 K, 500 K, and 600 K, the according Im(neff) are 2.287 × 10−3, 3.584 × 10−3, and 3.896 × 10−3, respectively.
Fig. 6. Field distributions for different temperatures, temperatures are (a) 3 K, (b) 77 K, (c) 200 K, (d) 300 K, (e) 500 K, (f) 600 K. The cross-sectional area and height of the dielectric stripe were set to a fixed value of 14400 µm2 and 120 µm, respectively. The width of the upper and bottom sides of the trapezoidal strip are [wa,wb] = [90, 150] µm. The 3D DSM Fermi level is 0.01 eV. The work frequency is 1.0 THz.
In order to further improve the performances of the trapezoidal stripe hybrid plasmonic modes, a modified dielectric stripe structure is utilized, i.e., a rectangular Si stripe inserted into the original trapezoidal SiO2 stripe (hybrid SiO2-Si dielectric stripe), as shown in Fig. 1 (b). Figure 7 shows the influences of the widths of different rectangular Si (wSi) layers on the propagation performances. The values of wSi are 5 µm, 10 µm, 30 µm, 50 µm, and 80 µm, respectively. The black and red lines represent pure trapezoidal SiO2 dielectric and pure Si dielectric, respectively. It can be seen from Fig. 7(a) and (b), with the increase of wSi, the portion of Si increases, which enhances the constraint of the dielectric stripe, the Re(neff) increases and Am/A0 decreases. Additionally, it can be found from Fig. 7(c) that compared with the Si and SiO2 trapezoidal structures, the modified structure has significantly higher propagation length. For example, at the frequency 1.65 THz (i.e., C point in Fig. 7(c)), for the trapezoidal SiO2, Si (wSi = 5 µm)- SiO2 hybrid dielectric stripe, and trapezoidal Si, the propagation lengths are 2.21 × 104 µm, 6.46 × 105 µm, and 3.22 × 103 µm, respectively. The analysis is given in the following passage. For the SiO2-Si hybrid layers, the THz wave is strong confined in the middle section of the dielectric stripe, the interaction area with the lossy DSM layer decreases, reducing the loss. However, as the width of Si layer increases further, the refractive index of the whole dielectric stripe increases, the hybrid mode is stronger confined near the lossy DSM bottom layer, which leads into the propagation length decreasing. For instance, at the frequency of 1 THz, if the Si layer widths are 10 µm, 30 µm and 50 µm, the propagation lengths are 3.5 × 104 µm, 1.2 × 104 µm, 8.1 × 103 µm, respectively. Additionally, for the hybrid Si-SiO2 dielectric stripe, the propagation length shows a peak with the increase of frequency, and the peak position is also closely associated with the width of the inserted Si layer. As the width of Si layer increases, the mode is strongly confined in the dielectric stripe, the peak position indicates a red shift. The reasons are given in the following. Since the peak position of propagation length corresponds with the balance point of plasmonic and dielectric mode, at larger width of Si layer, the contribution of dielectric mode increases, and thus a stronger plasmonic mode and lower frequency is required, as shown in Fig. 7(c). For example, if the widths of Si layer are 5 µm, 10 µm and 30 µm, the resonant peak positions are 1.65 THz, 1.19 THz and 0.64 THz. If the width of Si layer increases further, > 50 µm, the mode is well confined in the dielectric stripe. In this case, as frequency increases, the dielectric constant of DSM layer decreases, the loss increases, and the propagation length decreases, as the wSi = 80 µm line in Fig. 7(c). Figure 7(d) plots the figure of merits of hybrid modes at different widths of Si layers. Similar to the trends of propagation lengths, the FOM show the strong peak at suitable width of Si layer, as given in Fig. 7(d). On the condition that the widths of Si layers are 5 µm, 10 µm and 30 µm, the values of FOM reach 9.3 × 103, 6.5 × 103, and 6.8 × 103, respectively.
Fig. 7. (a) The real part of effective mode index, (b) normalized effective mode area, (c) the propagation length and (d) FOM as a function of frequency for different inserted rectangular Si widths. The cross-sectional area of the trapezoidal dielectric stripe was set to a fixed value of 14400 µm2. The Fermi level is 0.01 eV.
Figure 8 shows the field distribution diagrams at different Si layer widths wSi for the 3D DSM hybrid plasmonic waveguides. As the widths of Si layer increase, the hybrid modes are better confined in the dielectric stripe, and the values of Re(neff) gradually increase close to the Si trapezoidal structure. For example, Fig. 8(a), Fig. 8(b), Fig. 8(e), and Fig. 8(f) show the field distribution of pure SiO2 dielectric, wSi = 5 µm, wSi = 80 µm, and pure Si dielectric, and the according Re(neff) are 1.723, 1.857, 3.132, and 3.260, respectively. Accordingly, with the increase of the widths of Si layer, the confinements increase, with the confinement factors being 0.928 (SiO2), 0.954 (5 µm), 0.995 (80 µm) and 0.986 (Si), respectively. For the case of dissipation, the situation is a little complex. Firstly, if the widths of the inserted Si layer are small, more modes are confined in the middle of the dielectric stripe, the contact areas with the high lossy DSM layer decrease, reducing the losses. For example, if wsi are 5 µm and 10 µm, the imaginary parts of the effective refractive index are 1.552 × 10−3 and 6.766 × 10−4, respectively. However, if the widths of Si layer increase further (>50 µm), the hybrid modes are strongly confined in the dielectric stripe, more modes push into the high lossy DSM layer, leading into the loss increasing. For instance, if wsi are 50 µm and 80 µm, the imaginary parts of the effective refractive index are 2.933 × 10−3 and 3.337 × 10−3, respectively.
Fig. 8. The field distributions of the trapezoidal dielectric stripe- DSM hybrid plasmonic modes. The trapezoidal dielectric material is (a) SiO2 and the trapezoidal dielectric material is (f) Si. The widths of rectangular Si inserted into the trapezoidal dielectric SiO2 are (b) wSi = 5 µm, (c) wSi = 10 µm, (d) wSi = 50 µm and (e) wSi = 80 µm. The substrate is 3D DSM with Fermi level of 0.01 eV. The operating frequency is 1.0 THz.
The effects of temperatures on the propagation properties of the SiO2-Si hybrid waveguides are shown in Fig. 9. With the increase of temperature, the dielectric constant of 3D DSM increases, DSM layer indicates better plasmonic properties. Thus, as shown in Fig. 9(a) and 9(b), the value of Re(neff) gradually decreases, and the effective mode area Am/A0 increases. Figures 9(c) and 9(d) show the influences of temperatures on propagation lengths and FOM, respectively. When the temperature is small, the dielectric constant of 3D DSM is small, the loss is large, resulting into the smaller propagation length and FOM. In this case, the peak of propagation constant locates at low frequency, e.g., if the temperature is 3 K, the peak position is less than 0.5 THz. As temperature increases, the dielectric constant of DSM increases, the loss decreases, leading to the propagation length and FOM increasing gradually. Meanwhile, as temperatures increase, the peak positions of propagation lengths gradually show the blue shift. For instance, at the temperatures of 200 K and 300 K, the peak positions are about 0.75 THz and 1.19 THz, respectively. If the temperature is large enough, e.g., > 500 K, because of the large dielectric constant, DSM layer shows better plasmonic properties. Thus, to compensate the contribution of the dielectric fiber mode, the peak of the propagation length shifts a large frequency, in which the contribution of the low lossy dielectric fiber mode also increases as well. For example, if the temperature is 500 K, the peak value locates about 4.5 THz. Thus, only the increment trend is observed, as the cyan lines given in Fig. 9. Consequently, for the high temperatures of 500 K, as frequency increases, the loss decreases, and the propagation length and FOM increase.
Fig. 9. (a) The real part of effective mode index, (b) normalized effective mode area, (c) the propagation length and (d) FOM versus frequency at different temperatures. The Fermi level of 3D DSM layer is 0.01 eV. The width of Si layer is 10 µm.
4. Conclusion
The tunable propagation characteristics of the trapezoidal dielectric stripe-DSM hybrid plasmonic waveguide at different temperatures are systematically studied, including the influences of the dielectric stripe structure, frequency, and temperature. The results show that as the upper (bottom) side width wa of the trapezoidal stripe increases (decrease), the propagation length and FOM of the hybrid modes both decrease. The temperature affects the propagation properties significantly, which indicates a peak in the medium temperature region. For example, at temperature of 200 K, the peak value of propagation length and FOM reach 1.01 × 106 µm and 9.79 × 103, respectively. The propagation characteristic of the hybrid mode can also be modulated over a wide range by the temperature, the modulation depth of the propagation length reaches more than 96% if the temperature varies in the range of 3 K-600 K. In addition, the propagation length and FOM show a strong peak at the balance point of plasmonic and dielectric mode, the peak value also shows obvious blue shift if temperature varies in the range of 3 K-600 K. Furthermore, to improve waveguide properties, a hybrid SiO2-Si dielectric stripe is utilized. If Si layer width is 5 µm, the propagation length can reach 6.46 × 105 µm, which is much larger than those pure SiO2 (4.67 × 104 µm) and Si (1.15 × 104 µm) dielectric stripe. The results are very useful for understanding the tunable 3D DSM hybrid plasmonic waveguides and designing novel low threshold lasers, resonators and integrated circuitry.
Funding
National Natural Science Foundation of China (62205204, 12073018, 12141303, 61674106); Natural Science Foundation of Shanghai (21ZR1446500); Shanghai Local College Capacity Building Project (21010503200, 22010503300); Program of Shanghai Academic Research Leader (22XD1422100); Funding of Shanghai Normal University (SK202240).
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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