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 *f**0*=30Thz, *µ*c = 0.5eV, *w*o=200nm, *h*o=100nm, *r*o=30nm, *ε*1 = 1, *ε*2 = 2.09. Meanwhile, when *S*1=*S*2=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 *S*1 are shown in figures 3 under different *S*2. 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 Lm 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 *ε*2 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 *ε*2, where *ε*2 increases from 2 to 8. In Fig. 6(a), we can see that increasing *ε*2 the effective mode index Re(neff) increases linearly and the mode propagation loss also increases, resulting in a smaller propagation length Lm, as shown in Fig. 6(b). As can be seen in Figure 6 (c), as *ε*2 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 *ε*2, 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 *f*0 under different chemical potentials. It can be seen that as *f*0 increases, for four different chemical potentials *µ*c, Re(*n*eff) increases gradually, Lm decreases gradually, and the normalized mode field area is complex, but it varied little. FOM gradually increases with increasing *f*0. As can be seen from Fig. 7(a) and (c), when *f*0 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 *f*0, as shown in Fig. 7 (d). For example, when *µ*c = 0.5eV and *f*0=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].