1. Introduction

Compared with microelectronics, today’s photonic waveguides and circuit elements are still too large in size. To make photonics have a microelectronics-like development, we need a low-loss nano-waveguide with a drastically reduced mode area (for small size in the transverse direction) and an ultra-high effective refractive index (so that the effective wavelength is small and consequently the device length can be short along the propagation direction). Graphene as a one-atom-thick platform for infrared and THz metamaterials [1] can play an important role in optical science and engineering. Since graphene was experimentally made for the first time from graphite in 2004, this newly available material, which is a single layer of carbon atoms, has attracted much attention due to its unique properties [2, 3]. It can serve as a platform for metamaterials and can support surface-plasmon polaritons (SPPs) at infrared or THz frequencies [4–9]. Some noble metals, such as silver and gold, are widely regarded as the best available plasmonic materials [10]. However, devices fabricated with these metals suffer from large Ohmic losses and cannot be tuned once the geometry of the structure is fixed. Compared with noble metals, graphene can be tuned flexibly via electrical gating or chemical doping [4]. It can support both TM and TE modes [4] and has strongly enhanced light–matter interactions [7]. Meanwhile, graphene exhibits relatively low Ohmic losses. The graphene layer has a single-atom thickness, which is much thinner than any metallic thin film that has ever been fabricated. Due to the above properties, graphene has been studied in many areas [11–15]. The SPP mode supported by graphene ribbons has already been studied, but the ribbon width is still large (on the order of micrometer) [9]. In this paper, the waveguide mode properties of graphene nano-ribbon (the width is smaller than 100 nm) is investigated, and low-loss plasmonic waveguides of ultra-small mode area are achieved.

2. The properties of graphene ribbons

As a unique property of graphene, the complex conductivity of a graphene layer can be tuned flexibly by controlling the chemical potential (via the applied electric field), chemical doping or ground plane evenness [3]. In this way graphene can behave as either a metal or a dielectric and support both TM and TE waveguide modes. The graphene considered throughout the present paper has a chemical potential μ_c = 0.15 eV, T = 3 K and scattering rate Г = 0.43 eV to achieve a conductivity value σ_g = 0.0009 + i0.0765 mS, which is capable of supporting TM SPP surface mode [5, 6, 8, 13] at 30 THz. The conductivity value σ_g is derived using the Kubo formula model [4, 14]. The dispersion relations of TM SPP surface mode is expressed as β² = k₀²[1-(2/η₀σ_g)²], where β and k₀ are the wave numbers in the wave guide and the free space, respectively, and η₀ is the intrinsic impedance of free space. Thus, we get β = (69.34 + i0.71) k₀ for an infinitely extended graphene sheet.

As the temperature increases, from the Kubo formula one sees that Re(σ_g) will increases. For example, Re(σ_g) at 200 K goes up by 30% compared to that at 3 K. A larger Re(σ_g) implies a smaller propagation length of graphene waveguide mode. At T = 200 K, the propagation constant β = (75.28 + i2.46) k₀, (indicating that Im(β) is three times large as compared to that at 3 K).

As a 2D material, graphene with infinitely-small thickness cannot be directly incorporated into the numerical simulation using conventional electromagnetic software such as COMSOL or CST. This difficulty can be overcome by treating the graphene layer as an ultra-thin material with an effective thickness Δ and effective bulk conductivity [4] ε_g,eq = -σ_g,i/ω△ + ε₀ + iσ_g,r/ω△, where σ_g,r and σ_g,i are the real part and imaginary part of σ_g, respectively. Because the surface conductivity σ_g remains constant when the effective thickness Δ varies, the mode properties, e.g., n_eff, are actually insensitive to the specific value of Δ, as long as Δ is small enough (Δ < 1 nm). In all our simulations, the thickness of graphene Δ is assumed to be 0.4nm (very close to the typical value of 0.34 nm measured using the interlayer space of graphite [15]) to save the simulation time.

The waveguide mode of a single freestanding graphene ribbon is studied first. The 3D propagation of a guided wave on a typical graphene ribbon with width w = 40 nm is first shown in Fig. 1(a) to give the readers an intuitive idea on the mode confinement in both the lateral direction and the propagation direction. A discrete port is placed in front of the graphene ribbon to excite the guided mode. Although the length along the propagation direction is only 350 nm, more than 4 harmonic oscillations are supported, indicating the existence of waveguide mode with an ultra-large wave number. Meanwhile, the waveguide mode is also tightly confined in the lateral direction, as can be seen from the rapid decay of the optical field in the surrounding air. Since an ultra-small waveguide with some tightly confined energy is highly desirable for nanoscale photonic integration, it is worthwhile to study the energy profile of the waveguide mode. The electromagnetic energy density W(x, y) of the corresponding guided mode in Fig. 1(a) is shown in Fig. 1(b). Here W(x, y) is calculated by W = 0.5(ε_effε₀|E|² + μ₀|H|²), where ε_eff is the effective permittivity and ε_eff = ∂ (ωε)⁄∂ω. Based on the dispersion model, our calculation gives ε_{eff, g} = 169.45 when the thickness Δ of the graphene layer is 0.4nm and ε_eff,air = 1. In Fig. 1(b), one sees that the electromagnetic energy is tightly confined inside the graphene due to the large magnitude of ε_eff,g. It is also noted that a significant amount of energy is concentrated on the edges of the graphene ribbon. In our study, the microscopic details at the edges of the graphene ribbons are not taken into consideration, similar to the treatment in some other group’s work [9].

 

Fig. 1 (a) Plasmonic mode supported by a single freestanding graphene ribbon. Here the E_y distribution of the guided wave is calculated with software CST 2009. The boundary of the graphene ribbon is indicated by the red line. The width of the graphene ribbon is w = 40 nm. The free-space wavelength used in the simulation is 10 μm. A discrete port (a dipole source) is used for the excitation of the waveguide mode. q is the propagation vector of the waveguide mode. (b) The distribution of the energy density of the waveguide mode. This figure, as well as all the mode properties of graphene ribbons throughout the paper, is calculated using the software COMSOL 3.5. One clearly sees that the energy is tightly confined inside the graphene ribbon.

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The evolution of the waveguide mode in a graphene ribbon of finite width (micron dimension) has already been studied in [9]. It is found that an edge mode will appear due to the presence of the graphene edge. Here we want to further investigate the properties of graphene ribbon waveguides when the ribbon width is reduced down to nanoscale. It is expected that the ribbon width will greatly influence the behavior of the edge mode due to the strong interaction of ribbon edges.

The three waveguide modes supported by a freestanding graphene ribbon with width w< 270 nm is plotted in blue curves in Fig. 2(a). To find out the physical origin of the three modes, the edge plasmon mode supported by a semi-infinite graphene ribbon (EGSP) and the surface plasmon mode supported by an infinitely extended two-dimensional graphene (2DGSP) are also plotted in Fig. 2(a) in aquamarine blue and in green, respectively. Now it is clear that mode #1 and mode #2 originate from the coupling of EGSP and mode #3 originate from the evolution of 2DGSP. The effective index of mode #3 would decrease as width w gets smaller and finally becomes cutoff, leaving only EGSP modes (mode #1 and mode #2). The mode #1 and mode #2 arise from the symmetric and anti-symmetric hybridization of EGSP mode, as can be clearly seen from Fig. 2(b). The E_y component of mode #1 is symmetric with respect to the y axis, and E_y component of mode #2 is anti-symmetric with respect to the y axis. The symmetric hybridized mode #1 has a higher refractive index than the mode #2. As the width further shrinks, these two modes will continue splitting: the effective index of mode #1 will increase and the effective index of mode #2 will decrease. Finally, the mode #2 is cutoff when w < 50 nm. Eventually, the graphene ribbon can operate at single-mode region (red shaded region in Fig. 2(a)) if its width < 50 nm, leaving only the symmetric hybridized EGSP mode with very large refractive index. In contrast to the mode #2 and mode #3, the confinement of this remaining mode #1 will become tighter and tighter when the ribbon width further shrinks. The effective index of a graphene nano-ribbon is 117.5 when the ribbons width is 20 nm. It goes up by 30% compared to the n_eff of EGSP, and goes up by 70% compared to the n_eff of 2DGSP. The increase of effective index of mode #1 is caused by the coupling of the edge modes, which is similar to the phenomenon occurring in metal nanowires [16].

 

Fig. 2 (a) The dependence of the effective refractive index on the width of the graphene ribbon. The dark blue lines show the variation of three SPP waveguide modes when the width of the ribbon varies. The green line marked as “2DGSP” is for the SPP waveguide mode in an infinitely-extended graphene sheet. The light blue line marked as “EGSP” is for the SPP waveguide mode in a semi-infinite graphene sheet [9]. We find that modes 1 and 2 originate from the hybridization of EGSPs. Mode 1 has an even parity of E_y with respect to the ribbon axis, and its effective index n_eff > n_eff^EGSP. In contrast, mode 2 has an odd parity of E_y with respect to the ribbon axis, and its effective index n_eff < n_eff^EGSP. Mode 2 will be cut off if the width < 50 nm. Thus, there is a single-mode region (red shaded region in (a)) for small width w. The effective index n_eff of mode 3 will increase as w increases and finally approach to n_eff^2DGSP [which is not shown in (a)]. (b) The electric fields of the three modes. (c) The mode area (A_eff /λ₀²) of the SPP waveguide in a freestanding graphene ribbon when width w is in the single-mode region.

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The waveguide mode area is defined as A_eff = ∫∫W(x,y)dxdy/W_max, where W_max is the maximum of the energy density of the whole waveguide cross section. The calculated mode area is displayed in Fig. 2(c) as a function of ribbon width. It decreases considerably as the width gets small. That is because the optical energy with the graphene ribbon dominates the whole energy due to the tight confinement, so a waveguide with smaller width can lead to a smaller mode area. In our calculation, an extremely small mode area of 1.3 × 10⁻⁷ λ₀² [Fig. 3] is obtained at w = 20 nm, which is the smallest among all the reported subwavelength waveguide (to the best of our knowledge). This mode area is much smaller than the smallest mode area that can be achieved using a thin metallic film. For example, a thin gold film with thickness t = 4.6 nm and width w = 20 nm can support a well confined surface plasmon mode. However, the confinement is quite poor compared with a graphene nano-ribbon. At λ₀ = 1 μm, the effective mode index is Re(n_eff) = 4.487, and the mode area is calculated to be 1.5 × 10⁻⁴λ₀², which is three orders larger than its graphene counterpart. Of course, the mode area supported by the gold film can further shrink by reducing further the film thickness. However, it is very difficult to fabricate a smooth metallic film with thickness t < 3 nm experimentally.

 

Fig. 3 (a) A low-low waveguide structure: a graphene buffered by a silica layer on silicon. (b) The distribution of the energy density in the waveguide structure. Most of the energy is in the graphene and silica layer. Inset of (b) shows the energy field at the edge of the graphene. It shows the energy is tightly confined in the graphene and W_max is located at the edge of the graphene. (c) and (d) are the distributions of the electric field and the magnetic field in the waveguide structure. In (c), it is found that the electric field is strong in the graphene layer and the silica (the electric field in silicon is very weak), especially at the corners of the graphene ribbon. In (d), the magnetic field is strong in all the three layers. (e) shows the effective index n_eff of the low-loss waveguide structure when h_sio2 varies. The blue solid line is Re(n_eff) and the green dashed line is Im(n_eff). (f) The solid green line is the FOM of the SPP waveguide for this structure with different h_SiO2.The dashed green line is the FOM for a freestanding graphene waveguide with the same width (20 nm). As w gets smaller, the FOM will first increase for h_SiO2 > 3.0 nm and then decreases quickly for h_SiO2 < 3.0 nm, leading to a maximum FOM = 145 and it is bigger than the FOM for a freestanding graphene waveguide with the same width. The solid blue line is the propagation length for this structure with different h_SiO2, and the dashed blue line is the propagation for a freestanding graphene waveguide with the same width. The propagation length increases as h_SiO2 increases.

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By placing the graphene nano-ribbon on a silicon substrate, the effective index of the waveguide mode can further increase (and the guided wavelength can be reduced further). However, the propagation loss also increases at the same time. A tight mode confinement is usually accompanied by a large propagation loss [17]. Here we propose to reduce the propagation loss by buffering the graphene ribbon with a silica layer, as shown in Fig. 3(a). h_SiO2 is the height of the SiO₂ layer, and h_Si is the height of the Si layer. For this structure, we choose w = 20 nm, h_Si = 20 nm, ε_si = 11.9, and ε_SiO2 = 2.09. If there is no SiO₂, the effective index is n_eff = 466.3 + 4.7i. Accordingly, the guided wavelength λ_spp = λ₀/Re(n_eff) = 21.4 nm. The propagation length L_m defined as 1/Im(β), where Im(β) = Im(n_eff)k₀. Thus, the propagation length is 338 nm, and the figure of merit (FOM) [which is defined as the ratio of Re(n_eff) to Im(n_eff) [18]] of this waveguide mode is 99.2. When a thin layer of SiO₂ exists, a strong electric field is induced within the SiO₂ layer due to the high index contrast between Si and SiO₂ (as required by the continuity of the normal displacement field components). Consequently, the percentage of the optical energy inside the graphene layer decreases, which leads to a reduced propagation loss since the optical loss is entirely due to the damping inside the graphene. Taking h_SiO2 = 5 nm as an example, we have n_eff = 180.9 + 1.3i, L_m = 1256 nm, and λ_spp = 55.3 nm. Compared with the case when the SiO₂ layer is absent, significant loss reduction is achieved. Although the Im(n_eff) of our slot structure is larger than that of the freestanding graphene ribbon (indicating short propagation length), Re(n_eff) is also greatly enhanced due to the presence of high-index substrate. It turns out that our slot waveguide has a larger FOM compared to that of the freestanding case. Si has a high index and thus will pull the light away from the graphene to the SiO₂ buffer layer. Since more energy is located in the region of the low refractive index SiO₂ layer, the effective index of the waveguide mode Re(n_eff) decreases at the same time. A good effect is that the FOM of such a graphene ribbon waveguide increases to FOM = 142.9 (much larger than that for the structure without SiO₂,). By changing h_SiO2, the effective index Re(n_eff) varies from 160 to 400 [Fig. 3(e)]. It is worth noticing that there is a maximum FOM when the height of SiO₂ varies. By comparing with the structure without SiO₂, the maximum figure of merit Re(n_eff)/Im(n_eff) ≈145 is obtained when h_SiO2 ≈3.0 nm, which is 50% larger than that for the structure without SiO₂. For the structure with a maximum FOM, we have n_eff = 204.4 + 1.4i (i.e., the effective wavelength of the guided wave is λ_spp = 48.9nm), and L_m = 1126 nm. In comparison, for a freestanding graphene waveguide with the same width, we have n_eff = 117.5 + 0.9i, λ_spp = 89.1nm and L_m = 1785 nm. Since the effective wavelength in the waveguide on Si substrate is only about half of that in the freestanding graphene waveguide, more propagation cycles can be supported (before the light is very much attenuated) in our current design. Typically the size of an optical device is on the order of λ_spp. The graphene ribbon (with a silica buffer layer) on a silicon wafer provides a much smaller λ_spp than the freestanding graphene with the same width. Here, high-index Si pulls the light away from the lossy graphene to the lossless SiO₂ layer and helps to improve FOM. It is noted that the buffering of our graphene ribbon increases slightly the device volume along the vertical direction, but the loss reduction is significant. This is very suitable for planar integration.

Finally, we would like to study the mode properties of a graphene waveguide formed by two graphene ribbons. Similar to the case of mode coupling in a conventional MIM (metal-insulator-metal) or IMI (insulator-metal-insulator) waveguide, it is expected that the coupling of the nano-ribbon waveguide can lead to further mode splitting. Thus, the waveguide constructed by two identical graphene ribbons with a nanometer gap is investigated in this work. Two kinds of coupling configurations, namely, the side-side configuration [Fig. 4(a)] and top-bottom configuration [Fig. 4(b)], are studied.

 

Fig. 4 (a) is the side-side configuration and (b) is the top-bottom configuration. d is the gap size between the two ribbons. The wave propagates in the direction of the vector q.

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For the side-side configuration, d is the distance between the two graphene ribbons, and the width of graphene ribbons is chosen to be 20 nm. The hybridization of the waveguide modes will leads to the presence of both a symmetric mode [Fig. 5(a)] and an anti-symmetric mode [Fig. 5(b)], It is found that the symmetric mode can squeeze the optical energy effectively into the gap between the two ribbons, but the anti-symmetric mode only slightly modifies the profile of the optical energy density. For the symmetric mode, n_eff and the figure of merit increase when d is reduced [Fig. 5(c)]. The waveguide mode area depends critically on the gap size [Fig. 5(d)]. A smaller gap will lead to a smaller mode area. The mode area is extremely small, only about 10⁻⁷λ₀², which is an order smaller than the smallest mode area of any waveguide that has ever been reported in the literature [19].

 

Fig. 5 (a) is for the symmetric mode and (b) is for the anti-symmetric mode. In (a), the energy is mainly in the gap between the two ribbons, and in (b) the highest energy density is at the left and right corners. The insets in (a) and (b) show the distribution of the electric field of the modes. (c), the dashed blue solid line is for Re(n_eff) of the symmetric mode, the green solid line is for Re(n_eff) of the anti-symmetric mode, the dashed red line is the figure of merit of the symmetric mode, and the light blue dashed line is the figure of merit of the anti-symmetric mode. (d) The mode area of the symmetric mode. The mode area decreases rapidly as the gap d decreases.

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For the top-bottom configuration, d is the vertical distance between the two ribbons, and the width of the graphene ribbons we use is also 20 nm. The mode with top-bottom configuration [20] is called “gap mode” in a very recent publication [21]. Here the name of “top-bottom configuration” is used for consistency and comparison with our side-side configuration. Again, there are both a symmetric mode and an anti-symmetric mode due to the presence of mode coupling. Figure 6(a)–6(f) shows the distributions of energy, electric field and magnetic field of the two modes. For the symmetric mode, more energy is confined in the area between the two ribbons. For the anti-symmetric mode, the field is mainly located at the top and bottom regions (the energy density, however, is still concentrated within the graphene ribbon due to the large effective permittivity ε_eff). Figure 6(g) displays Re(n_eff) of the two modes. The effective refractive index of the waveguide mode depends critically on distance d. If the quantum effect is not taken into account, Re(n_eff) can be extremely large, reaching a value of up to 1000 when d is smaller than 1 nm. The figure of merit is also a function of distance d, and can be more than 180. However, the mode area remains more or less the same when d changes due to the large amount of energy in graphene.

 

Fig. 6 (a)- (f) is the distribution of the energy intensity, electric field, and magnetic field for the two modes in the top-bottom configuration. In (g), the blue solid line is for Re(n_eff) of the symmetric mode, the green solid line is for Re(n_eff) of the anti-symmetric mode, the dashed red line is the figure of merit of the symmetric mode, and the light blue dashed line is the figure of merit of the anti-symmetric mode.

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For our double-ribbon configurations, the propagation length is not plotted due to the limitation of paper length. In fact, the propagation length L_m can be related to FOM as L_m = FOM*λ₀/4πRe(n_eff). For example, in the top-bottom configuration, the propagation length is Lm = 1477.3 nm [while Re(neff) = 187.21, Im(n_eff) = 1.08 and FOM = 173] when d = 5 nm.

Due to the extremely large wavenumber supported by this configuration, additional modes with high order oscillations are found in our calculation. If d is not too small, e.g. 20 nm, only two modes are supported. However, if the two ribbons get closer to each other, wavenumber k will grow significantly. Then more oscillations along the lateral direction will occur, resulting in more waveguide modes. For example, when d is 2 nm, one additional mode appears [Fig. 7]. This characteristic is potentially useful for the design of e.g. ultra-compact MMI (multimode interferometer) devices.

 

Fig. 7 (a) and (c) are for the mode (which originally is the symmetric mode) that is formed through the coupling/hybridization of the two ribbons. (b) is a high order mode in the x direction.

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Since a single freestanding graphene ribbon can support a plasmonic waveguide mode with an extremely large wave number, this configuration has high potential for design of some ultra-compact optical structure. Here as an example we give a nano-ring cavity based on a single freestanding graphene ribbon. In our design, the width of the graphene ribbon is chosen to be 20 nm. It is well known that a round-trip phase accumulation of integral multiple of 2π can result in a resonant mode. With an inner radius r = 38 nm, we see a resonant peak at 30 THz (corresponding to a vacuum wavelength of about 10 μm) in our numerical simulation with CST software [Fig. 8]. The mode profile shows that it is a 4th-order ring cavity mode. The Q value (Q = f/Δf, f is the frequency, Δf is the linewidth) is 42.3. Considering the ring size is on the order of nanometer, the cavity size (compared to the vacuum wavelength) is surprisingly small. Since the response of graphene can be tuned easily by changing its chemical potential such as doping, the resonant frequency can be controlled in a flexible way.

 

Fig. 8 (a) The proposed ring cavity with graphene ribbon width w = 20 nm and inner radius r = 38 nm. (b) The electric field distribution of the cavity mode at f = 30 THz, which is excited by a discrete port in CST.

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3. Conclusion

In summary, the unique properties of plasmonic waveguides based on graphene nano-ribbons have been investigated. Compared with traditional plasmonic materials (noble metals), graphene has several advantages: 1) The properties of graphene can be tuned via electrical gating or chemical doping. 2) It has low Ohmic losses, and the FOM can be 145, which is much larger than that for a noble metal (for the noble metal, the FOM is usually smaller than 80 [22]). 3) The effective index of graphene can be ultra-large, which can make the effective wavelength of the guided wave ultra-small. 4) It is very thin (one-atom layer, which is far thinner than any fabricated metal layer) and can be very narrow (20 nm in this paper) for waveguide. Due to these superior properties, graphene nanoribbons waveguides have the potential for future optical devices.

The guided modes are tightly confined in both the lateral direction and the propagation direction. A single mode operation region has been identified if the ribbon width is smaller than 50 nm. Due to the tight confinement, a record small mode area (an order smaller than the smallest mode area of any waveguide that has ever been reported in the literature) and extremely high effective refractive index can be achieved. The low-loss waveguide structure with an embedded low index silica layer between the graphene layer and the silicon substrate has been proposed to reduce the propagation loss and increase the FOM of the plasmonic waveguide. It is very attractive for designing optical devices with high performance. The coupled configurations with two identical graphene ribbons also exhibit interesting properties. In particular, the side-side-coupling can further reduce the waveguide mode area, while the top-bottom-coupling can result in much larger effective indices than an isolated graphene ribbon. A nano-ring cavity of extremely small size based on a graphene ribbon waveguide has been shown to support a cavity resonance at far infrared range.