Study on Properties of Plasmonic Waveguide of Graphene-Coated Nanotube with a Dielectric Substrate

doi:10.21203/rs.3.rs-1195798/v1

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Study on Properties of Plasmonic Waveguide of Graphene-Coated Nanotube with a Dielectric Substrate

https://doi.org/10.21203/rs.3.rs-1195798/v1

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A hybrid plasmonic waveguide structure composed of graphene-coated nanotube with a dielectric substrate is proposed in this paper. Transmission properties of the fundamental mode are studied by the finite element method (FEM). The results reveal that the mode transmission properties are greatly dependent on the nanotube radius and internal-external diameter ratio, gap distance, nanotube permittivity, as well as chemical potential of graphene. Meanwhile, it is compared with the graphene-coated nanowires with a dielectric substrate [40]. By optimizing the parameters, the structure could achieve long-range propagation with the propagation length about 12.56um. Moreover, it could realize deep-subwavelength confinement with the normalized mode area only ~10⁻⁷. This structure may offer certain theoretical basis for integrated nanophotonic devices to achieve long-distance transmission in the deep-subwavelength range.

Waveguides

Graphene

Surface plasmonic

Transmission properties

Surface plasmons (SPs) [1–3] are the surface electromagnetic wave propagating at the interface between a metal and a dielectric. Because of breaking through the diffraction limit and controlling light at subwavelength scale, SPs have attracted extensive attention of scholars worldwide and developed rapidly [4–8]. Usually, noble metals are used to stimulate and support surface plasmons (SPs). Various metal-based SPs waveguides have been proposed [9–23]. However, the metal-based SPs waveguides usually suffer from large ohmic loss and weak field confinement in the mid-infrared (mid-IR) to terahertz (THz) frequency range [24].

Recently, it has been found that graphene can exhibit “metal-like” properties in mid-infrared (mid-IR) to terahertz (THz) wavelengths [25], which can stimulate SPs. Compared to metal SPs, graphene SPs has the advantages of low propagation loss, tight field confinement and tunable electromagnetic properties [26–27]. Therefore, a large number of graphene SPs devices have been proposed, such as graphene nanoribbon waveguide [28–29], graphene nanowire waveguide [30–31], slot/wedge waveguide [32], hybrid waveguide [33–34], and modulator [35–36]. Among them, the hybrid waveguide has a better tradeoff between strong field confinement and low propagation loss, which has been widely studied in recent years. The graphene-coated nanowire waveguide [37–44] is a research hotspot in hybrid waveguides because of its simple structure, easy fabrication and no cut-off of fundamental mode. It has been shown that the graphene-coated nanowire waveguide has stronger field confinement than the metal nanowire waveguide, and the normalized mode area is about 10⁻³ [37]. In order to further improve the field confinement and reduce the mode propagation loss, researchers have proposed a graphene-based cylindrical hybrid waveguide [38–39] and the graphene-coated nanowire waveguide with dielectric substrate [40–42]. In reference [38], although the normalized mode area of the graphene-based cylindrical waveguide is reduced to 10⁻⁶, the propagation distance is less than 2um. In reference [40], the propagation loss of the graphene-coated nanowire waveguides with dielectric substrate is smaller, and the propagation distance is about 7 µm. But the mode confinement is also reduced, and the mode area is one order of magnitude larger than the former.

Considering the advantages of the hybrid plasmonic waveguide, a new type of plasmonic waveguide structure composed of the graphene-coated nanotube with the dielectric substrate is designed in this paper. The graphene-coated nanotube waveguide structure has not been reported in related studies, and it has better performance than graphene-coated nanowire waveguide structure. The fundamental mode transmission properties of the waveguide are analyzed in detail by the finite element method. By adjusting the geometric parameters and electromagnetic parameters, the transmission properties are optimized. At the same time, the transmission properties are compared with that of the graphene-coated nanowire waveguides with the dielectric substrate [40]. It is found that the proposed waveguide has long propagation length and small mode area, which has important application value in the field of integrated nanophotonic devices.

Figure 1 presents a two-dimensional structural diagram of the proposed waveguide, which consists of monolayer graphene layer and dielectric nanotube as well as rectangle dielectric substrate. The studied structure is embedded in air. The inner and outer radii of the dielectric nanotube are r_i and r_o with the ratio S₁(S₁=r_i/r_o). The inner filling media and outer filling media of the nanotubes are different, and their permittivities areand . The outer layer of the rectangular dielectric substrate is also filled with the medium(with a height of h_o and a width of w_o), and the inner layer of the rectangular dielectric substrate is filled with the medium(with a height of h_i and a width of w_i). The height/width ratio of inner and outer layers of the rectangular dielectric substrate is S₂(S₂=h_i/h_o=w_i/w_o). Considering the large loss of high index, the interiors of nanotube and rectangular dielectric substrate are filled with air (ε₁ = 1). The gap height between the graphene-coated nanotube and the rectangular dielectric substrate is g. Graphene is wrapped in the outer layer of the dielectric nanotube, which can be achieved by chemical vapor deposition of Vander Ed Ley. In the mid infrared frequency range, the permittivity of graphene [43] can be calculated by the following formula:

$${\varepsilon _g}=1+\frac{{\text{i}{\sigma}\left( \omega \right)}}{{\omega {\varepsilon _0}\Delta }}$$

Where$\Delta =0.335nm$is the thickness of single layer graphene [40],$\omega$is the radiation angle frequency of incident light, and ${\varepsilon _0}$is the vacuum dielectric constant. ${\sigma}\left( \omega \right)$is the surface conductivity of the graphene film, which can be derived from the famous Kubo’s formula, [44, 45]:

$${\sigma}\left( \omega \right)={\sigma _{\operatorname{int} ra}}+{\sigma _{\operatorname{int} er}}$$

$${\sigma}{\left( \omega \right)_{_{{\operatorname{int} ra}}}}=\frac{{2i{e^2}{k_B}T}}{{\pi {\hbar ^2}\left( {\omega +\text{i}{\tau ^{ - 1}}} \right)}}\text{l}\text{n}\left[ {2\cosh \left( {\frac{{{\mu _\text{c}}}}{{2{k_B}T}}} \right)} \right]$$

$${\sigma}{\left( \omega \right)_{_{{\operatorname{int} er}}}}=\frac{{{\text{e}^2}}}{{4\hbar }}\left[ {\frac{1}{2}+\frac{1}{\pi }\arctan \left( {\frac{{\hbar \omega - 2{\mu _\text{c}}}}{{2{k_\text{B}}T}}} \right) - \frac{i}{{2\pi }}\ln \frac{{{{\left( {\hbar \omega +2{\mu _\text{c}}} \right)}^2}}}{{{{\left( {\hbar \omega - 2{\mu _\text{c}}} \right)}^2}+4{{\left( {{k_\text{B}}T} \right)}^2}}}} \right]$$

Where${\sigma _{\operatorname{int} ra}}$ is the intraband conductivity of graphene and ${\sigma _{\operatorname{int} er}}$is the interband conductivity of graphene. Here, the temperature${\text{T}}={\text{300K}}$, the charge of the electron $\text{e}=1.6 \times {10^{ - 19}}\text{C}$, the relaxation time $\tau =0.5\text{p}\text{s}$[46], is the reduced plank constant, is the Boltzmann’s constant, and ${\mu _\text{c}}$is the chemical potential of graphene. It can be seen from the above formula that the conductivity of graphene changes with changing the frequency and chemical potential.

Because the waveguide structure proposed in this paper is complex, it is not easy to directly solve the wave equation to calculate the mode properties. Therefore, the COMSOLMultiphysics software based on the finite element method is used to calculate the mode transmission properties of the waveguide. In the process of numerical simulation, graphene is treated as an ultra-thin film with surface conductivity of ${\sigma}\left( \omega \right)$. The effective refractive index is defined as ${n_{\text{e}\text{f}\text{f}}}=\beta /{k_0}$, where is the propagation constant which can be obtained directly by COMSOL software and is the wavenumber in free space; the propagation lengthcan be defined by , where is the vacuum wavelength and is the imaginary part of the effective refractive index. The normalized mode area is defined as ${A_{\text{e}\text{f}\text{f}}}/{A_0}$, where ${A_0}={\lambda _0}^{2}/4$is the diffraction limit mode area and the effective mode area, which reflects the mode confinement ability of waveguide, is defined as:

$${A_{\text{e}\text{f}\text{f}}}=\iint {W\left( r \right)}{\text{d}^2}r/\hbox{max} \left[ {W\left( r \right)} \right]$$

whereis the electromagnetic energy density [47–48]. In order to better describe the relationship between the mode propagation length and the normalized mode area, the figure of merit is defined as $\text{F}\text{O}\text{M}=\operatorname{Re} \left( {{n_{\text{e}\text{f}\text{f}}}} \right)/\operatorname{Im} \left( {{n_{\text{e}\text{f}\text{f}}}} \right)$[49].

Figure 2 (a) - (f) show the mode profiles of the electric field |E| of the fundamental mode of the waveguide structure when the air core size in the nanotube/dielectric substrate and the gap height between the nanotube and dielectric substrate are different. Here, the parameters are f₀=30Thz, µ_c = 0.5eV, w_o=200nm, h_o=100nm, r_o=30nm, ε₁ = 1, ε₂ = 2.09. Meanwhile, when S₁=S₂=0, the waveguide structure is similar to the structure proposed in reference [40]. It can be seen from Figure 2 that most of the electromagnetic energy is confined to the gap between the nanotube and the dielectric substrate, and the mode field confinement is very strong. As the air core size increases, the electromagnetic energy gradually diffuses from the gap between the nanotube and the dielectric substrate to the periphery of the nanotube, which will lead to weak mode field confinement and low mode propagation loss. In addition, it can be seen that with the increase of gap height g, the size of electromagnetic field distributed in the gap will increase, and the mode area will also increase, but the propagation loss decreases, resulting in the increase of propagation length.

The dependences of the mode properties on the ratio S₁ are shown in figures 3 under different S₂. As shown in Fig. 3(a), when S₂ is fixed, the effective mode index Re(n_eff) decreases slowlys with increasing S₁. As S₁ increases, the propagation length L_m increases monotonically, and the normalized mode field area Aeff/A₀ also increases monotonically, but the variation range is very small. Since the propagation length increases faster than the normalized mode area, the figure of merit increases slowly. At the same time, it can be seen that when S₁ is small, the change of mode properties is slow, but when S₁ is large, the change of mode properties is fast. This is because when S₁ is large, the electromagnetic field will diffuse to the dielectric layer around the nanotube, the mode confinement becomes worse, and the mode loss decreases faster. That is to say, when S₂ = 0, 0.3, the change of mode properties is close; when S₂ = 0.6, 0.9, the propagation length and normalized mode area change more. Especially, when S₁ = S₂ = 0, the waveguide mode properties are similar to those mentioned in reference [40]. It can be seen from Fig. 3 that when both S₁ and S₂ are greater than 0, the propagation length and the figure of merit of the proposed waveguide are higher than those in reference [40], and the normalized mode field area does not increase much than that in reference [40]. Therefore, it can be said that within a certain parameter range, the mode properties of the proposed structure are better than those in reference [40].

The fundamental mode properties of the waveguide structure as a function of the gap height g are depicted in Fig. 4. When g gradually increases, the coupling between graphene nanotubes and dielectric substrate decreases, resulting in the decrease of effective mode index Re(n_eff). Increasing the gap height leads to a larger mode area and weaker confinement. At the same time, the mode propagation loss decreases with increasing g, resulting in a longer propagation length. According to Fig. 4 (d), FOM first increasess and then tends to be flat. Therefore, the increase of g can appropriately improve the waveguide performance, but too large g has little effect on the waveguide performance. Taking into account the propagation length and the mode area, g=20nm is selected for the following research.

Another geometry parameter relating to the mode properties is the outer radius r_o of the nanotube. The dependences of the mode properties on parameter r_o are shown in figures 5, where r_o varies from 30 nm to 100 nm. As can be seen from Fig. 5 (a), Re(neff) first decreases and then increases with increasing r_o. The normalized mode area increases first and then decreases with increasing r_o, and the later change range is small, indicating that when r_o is large ( r_o > 80nm), r_o has little effect on the mode field confinement. Moreover, as r_o increases, both L_m and FOM gradually decrease, and the variation range is large, as shown in Fig. 4 (b) and 4 (d). This is due to the fact that increasing the outer radius r_o of the nanotube can increase the surface area of the graphene layer, so the mode propagation loss increases and the propagation distance decreases. It can be seen from the above when r_o is the smallest, the mode properties are the best, and the figure of merit is the highest with the longest propagation length and the smallest normalized mold area. However, considering the difficulty of manufacturing, r_o should not be too small. Therefore, r_o=30nm is selected for the following study in this paper.

The permittivity ε₂ of nanotube/dielectric substrate also has a great influence on the waveguide performance. Fig. 6 shows the dependence of the mode properties on the permittivity ε₂, where ε₂ increases from 2 to 8. In Fig. 6(a), we can see that increasing ε₂ the effective mode index Re(n_eff) increases linearly and the mode propagation loss also increases, resulting in a smaller propagation length L_m, as shown in Fig. 6(b). As can be seen in Figure 6 (c), as ε₂ increases, the normalized mode area decreases gradually and the maximum variation range of normalized mode area is about 1×10⁻⁷~6×10⁻⁷, meaning that the field confinement is very tight. Figure 6(d) depicts the dependence of the figure of merit FOM on the permittivity ε₂, and it can be seen that the smaller the permittivity ε2, the higher the figure of merit FOM, and the better the mode properties. Therefore, the permittivity of nanotube/dielectric substrate should be as small as possible in application. SiO₂ with the permittivity ε₂ = 2.09 is selected as the filling medium in this paper [50].

Compared to noble metals, the greatest advantage of the graphene is that the graphene conductivity can be dynamically tuned when its geometric structure is fixed [51]. Fig. 7 shows the dependence of mode properties on frequency f₀ under different chemical potentials. It can be seen that as f₀ increases, for four different chemical potentials µ_c, Re(n_eff) increases gradually, L_m decreases gradually, and the normalized mode field area is complex, but it varied little. FOM gradually increases with increasing f₀. As can be seen from Fig. 7(a) and (c), when f₀ is fixed, as µ_c increases, Re(n_eff) decreases and the normalized mode field area increases due to the weak field confinement. At the same time, as µ_c increases, the carrier relaxation time increases, which reduces the inherent loss of graphene, so L_m increases, as shown in Fig. 7 (b). Because the propagation length increases faster than the mode field area, FOM also increases by f₀, as shown in Fig. 7 (d). For example, when µ_c = 0.5eV and f₀=20Tz, the propagation length and normalized mode area are 12.56µm and 6.6×10⁻⁷ respectively. Compared with the research in related fields, the proposed waveguide has the smaller normalized mold area(~10⁻⁷). In the case of the same propagation length, the normalized mold area of the waveguide proposed in this paper is two orders of magnitude smaller than that of the graphene-coated nanowire waveguide with the dielectric substrate(~10⁻⁵) [40]. And the normalized mold area of the waveguide proposed is one order of magnitude smaller than that of the triangular-shaped graphene-coated nanowires on substrate(~10⁻⁶) [41].

A graphene-coated nanotube plasmon waveguide structure with a dielectric substrate is designed and its transmission properties are studied in detail. The results have revealed that the structure proposed in this paper has better mode properties than that proposed in reference [40] when the appropriate parameters are selected. The smaller the outer diameter of the nanotube, the longer the propagation length, the smaller the mode area, and the better the mode properties; meanwhile, it is shown that the waveguide structure can locally break through the restrictive relationship between surface plasmon mode loss and mode area, that is, the mode loss and mode area are reduced at the same time. The larger the gap height is, the better the mode properties are, but the mode properties are basically unchanged when the gap height is too large. Reducing the permittivity of nanotube or increasing the chemical potential of graphene can improve the figure of merit. In particular, the proposed waveguide can realize the propagation distance in the order of ~ µm, and the normalized mode area is compressed to ~10⁻⁷. This structure has important applications in the field of deep sub-wavelength transmission and integrated nanophotonic devices.

Funding

This work was supported by Chongqing higher education teaching reform research project (Grant numbers [213143]). Author Miao Sun has received research support from Chongqing Normal University.

Competing Interests

Financial interests: Author Miao Sun, Zhuanling He and Libing Huang declare they have no financial interests.

The authors have no relevant financial or non-financial interests to disclose.

Author Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Miao Sun, Zhuanling He and Libing Huang. The first draft of the manuscript was written by Miao Sun and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Data Availability

The datasets generated during and analysed during the current study are available from the corresponding author on reasonable request.

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Study on Properties of Plasmonic Waveguide of Graphene-Coated Nanotube with a Dielectric Substrate

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