Subwavelength optical field confinement and low-loss propagation are important for compact photonic integration. However, the field confinement capability of noble metal-based plasmonic devices is always accompanied by the inherent ohmic loss. These structures perform well at the near-infrared and visible frequencies. However, in the mid- and far-infrared regions, the lack of tunability in their electromagnetic response and poor modal field confinement ability hinder their applications at the nanoscale. Recently, both theoretical and experimental reports have demonstrated that graphene can support surface plasmons in the infrared range with unique tunability, extremely strong modal field confinement, and a large field enhancement. Owing to the tradeoff between modal loss and confinement in plasmonic structures, it is difficult to obtain a deep-subwavelength modal field and long-range propagation simultaneously. Although a graphene-based hybrid waveguide integrated with a triangle wedge substrate has been proposed to achieve an ultra-small normalized mode size, the modal loss remains relatively large (with a propagation length less than 10 μm). Thus, obtaining a better balance between the modal loss and field confinement remains a significant challenge. In this study, a hybrid plasmon waveguide consisting of a cylindrical silicon nanowire and graphene-coated triangular nanowire is designed. The designed waveguide exhibits strong optical field confinement capability, low loss propagation, and a high figure of merit; thus, it is suitable as a building block for subwavelength photonic devices, such as modulators and nanolasers.

The proposed plasmon waveguide consists of a graphene-coated triangular nanowire separated from a cylindrical silicon nanowire by a nanoscale dielectric gap with width (h_gap)(Fig. 1). For the modeling and simulation, the wave optics module of COMSOL software is employed. The eigenvalue solver is used to obtain the complex effective mode index (N_eff) and effective mode area (A_eff). In the simulation, the graphene layer is modeled as an electric field-induced surface current J=σ_gE without thickness, where J is the surface current and E is the electric field. The calculation domain is 2λ₀×2λ₀, and a perfectly matched layer (PML) is applied to the surroundings of the geometry to avoid the influence of reflection. A convergence analysis is performed to ensure that the numerical boundaries and meshing do not interfere with the solutions.

The proposed waveguide exhibits well-confined modal fields with the focal spot area of 13.5×10 nm² (even less), corresponding to size of λ02/(7.4×10⁵). The modal transmission properties are highly dependent on angle θ. When θ decreases, both the propagation loss and modal area decrease, indicating that the tradeoff between modal loss and confinement is broken to some extent. Additionally, a smaller gap distance results in a larger figure of merit (Fig. 3). Interestingly, when the radius of the Si nanowire increases, the propagation length decreases slightly at first and then appears to be invariable, while the modal area decreases monotonically. This is because the loss is mainly from the graphene layer, and when R increases, the graphene-light interaction region is nearly invariable. When R ranges from 40 nm to 100 nm, the loss is invariable, while the modal area reduces (Fig. 4). When the triangular nanowire permittivity ranges from 2 to 12, the figure of merit decreases from 1190 to 245, implying that a smaller permittivity can improve waveguide performance (Fig. 5). The tunability of the Fermi energy of graphene allows the active tuning of the modal properties. When the Fermi energy ranges from 0.4 eV to 1.4 eV, the normalized modal areas of the fundamental modes are on the order of ~10^-6. In particular, when the Fermi energy is above 1 eV, a propagation length of several tens of micrometers and a figure of merit of over 2500 can be achieved (Fig. 6). Finally, the proposed waveguide is demonstrated to have better comprehensive performance than a similar waveguide structure (Fig. 8) and a two-wire system (Fig. 9).

In this work, the subwavelength transmission characteristics of a graphene-dielectric nanowire hybrid waveguide are proposed and examined. The proposed waveguide exhibits extremely strong optical field confinement capability. In particular, when θ decreases, both the modal loss and modal area decrease. This is due to the tip focusing effect of the triangular-shaped GNW, and the decrease in θ results in a reduction in the graphene-light interaction region, which in turn reduces the loss. Thus, the tradeoff between modal loss and confinement is broken to some extent. When the frequency ranges from 20 THz to 40 THz, the normalized modal areas of the fundamental modes in the designed waveguide are on the order of ~10^-6 with a propagation length of approximately several tens of micrometers, as well as a high figure of merit of over 2500. Compared with a similar waveguide structure, the mode field area is reduced by one order of magnitude while maintaining a comparable propagation distance. These findings are expected to have potential applications in waveguide-integrated plasmonic devices and greatly reduce the device size, such as modulators and nano-lasers.