Analytical model for plasmon modes in graphene-coated nanowire

Yixiao Gao; Guobin Ren; Bofeng Zhu; Huaiqing Liu; Yudong Lian; Shuisheng Jian

doi:10.1364/OE.22.024322

1. Introduction

Surface plasmons (SPs), which are charge density waves coupled to electromagnetic (EM) waves at the interface between dielectric and metal, offer a promising solution to control EM waves at subwavelength scale. Noble metals are usually chosen to support SPs for their relatively low loss among the available plasmonic materials, the frequency range of which lies in the visible to near-infrared frequencies. Various SPs waveguiding structures were proposed such as metal film [1], metal nanowire [2], metal groove/wedge [3], and dielectric/metal nanowire hybrid waveguide [4, 5]. Recently, experiment confirms that doped graphene supports surface plasmons as well [6]. Graphene plasmons (GPs) modes on graphene sheet [7], graphene nanoribbon [8] and graphene groove/wedge [9] were investigated intensively. Plasmons on graphene had not only a lower ohmic loss than conventional plasmonic materials [7], but also an extremely subwavelength confinement of EM field. Moreover, carrier density of graphene could be electrostatically tuned which further leads to a dynamically tunable GPs performance on graphene. Due to these attracting features, graphene is considered as a promising candidate for future compact plasmon devices [8].

Classical electromagnetic theory is successfully applied to describe SPs mode characteristics both in metal [1–5, 10–13] and graphene [7–9, 14–16] numerically or analytically depending on the complexity of waveguiding structure. The mode characteristics could be analyzed by numerical methods at the expense of computing time and resources. However analytic method provides a simple and efficient way to get a deeper insight into the mode behavior, such as modal cutoff characteristic, which is usually preferred.

Several experiments [17–20] showed graphene layer can tightly coat the dielectric nanowire due to van der Waals force. Graphene-microfiber/nanowire combined system has been proposed for various applications [17–22], but GPs modes in the graphene-coated nanowire waveguide, which is an analogy of metal nanowire with azimuthal symmetry, have not been characterized yet.

In this paper, we present an analytical model for plasmon modes in graphene-coated dielectric nanowire (GNW). Eigen GPs modes of GNW are obtained by solving Maxwell equations in cylindrical coordinate. Equation for dispersion relation is derived. Mode patterns and the characteristics of GNW including dispersion relation and propagation length are illustrated, and their dependence on the nanowire radius, nanowire permittivity and chemical potential of graphene are studied as well. In the last part, we compare the performance of GNW and Au-coated nanowire in the mid-infrared frequencies.

2. Theoretical model

Graphene-coated nanowire could be regarded as rolling graphene ribbon around a dielectric nanowire with a permittivity of ε₁, as depicted in Fig. 1. The studied structure is embedded in medium with permittivity ε₂. Radius of nanowire is R. Graphene is treated as a thin layer with surface conductivity σ_g characterized by Kubo formula [23, 24].

Fig. 1 Schematic of graphene-coated nanowire.

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Due to the azimuthal symmetry of GNW, we solve Maxwell equations in cylindrical coordinate with its origin set at the center of nanowire. Electric and magnetic field could be written as $\vec{E} (ρ, φ, z) = E_{ρ} \hat{ρ} + E_{φ} \hat{φ} + E_{z} \hat{z}$ and $\vec{H} (ρ, φ, z) = H_{ρ} \hat{ρ} + H_{φ} \hat{φ} + H_{z} \hat{z}$ , respectively. GPs mode propagates in z-direction and the EM field of eigen mode has the form $\vec{F} (ρ, φ) \exp (i β z) \exp (- i ω t)$ , in which β is the propagation constant, and $\vec{F}$ stands for either electric or magnetic field. We first treat the longitudinal EM field (E_z and H_z), and using the separation of variables approach upon Maxwell equations by setting $F_{z} (ρ, φ) = R (ρ) Φ (φ)$ and considering the fact that surface plasmon mode always has a larger propagation constant β than the wavevector of light in dielectric $k_{0} \sqrt{ε_{i}}$ (i = 1, 2), i.e. electromagnetic wave exists as an evanescent wave in the away-from-graphene direction, R(ρ) is the solution of modified Bessel equation and $Φ (φ) = \exp (i m φ)$ . Thus longitudinal component of electric field E_z and magnetic field H_z could be represented as

{\begin{cases} E_{z}^{(1)} = A_{m} I_{m} (μ_{1} ρ) \exp (i m φ) \exp (i β z) \\ H_{z}^{(1)} = B_{m} I_{m} (μ_{1} ρ) \exp (i m φ) \exp (i β z) \end{cases} (ρ < R)

{\begin{cases} E_{z}^{(2)} = C_{m} K_{m} (μ_{2} ρ) \exp (i m φ) \exp (i β z) \\ H_{z}^{(2)} = D_{m} K_{m} (μ_{2} ρ) \exp (i m φ) \exp (i β z) \end{cases} (ρ > R)

where

μ_{1} = \sqrt{β^{2} - ω^{2} ε_{1} μ_{0}}

and

μ_{2} = \sqrt{β^{2} - ω^{2} ε_{2} μ_{0}}

.

I_{m} (μ_{1} ρ)

and

K_{m} (μ_{2} ρ)

are the modified Bessel functions of the first and second kind (m = 0, 1, 2…). A_m, B_m, C_m and D_m are arbitrary constants which is determined by boundary conditions in the follow. By applying the relation between the longitudinal and transverse EM field components

\begin{array}{l} {\vec{E}}_{t} = \frac{1}{- μ_{i}^{2}} (\nabla_{t} \frac{\partial E_{z}}{\partial z} + j ω μ_{0} \hat{z} \times \nabla_{t} H_{z}) \\ {\vec{H}}_{t} = \frac{1}{- μ_{i}^{2}} (\nabla_{t} \frac{\partial H_{z}}{\partial z} - j ω ε_{i} \hat{z} \times \nabla_{t} E_{z}) \end{array}

in which

{\vec{E}}_{t} = E_{ρ} \hat{ρ} + E_{φ} \hat{φ}

(H field is the same) and

\nabla_{t} = \hat{ρ} \frac{\partial}{\partial ρ} + \hat{φ} \frac{1}{ρ} \frac{\partial}{\partial φ}

. We could get field components E_φ and E_ρ .When ρ < R, we have

{\begin{cases} E_{φ}^{(1)} = \frac{j}{- μ_{1}^{2}} {\frac{j m β}{ρ} A_{m} I_{m} (μ_{1} ρ) - ω μ_{0} B_{m} μ_{1} I_{m}^{'} (μ_{1} ρ)} \exp (i m φ + i β z) \\ E_{ρ}^{(1)} = \frac{j}{- μ_{1}^{2}} {β A_{m} μ_{1} I_{m}^{'} (μ_{1} ρ) + \frac{j m ω μ_{0} B_{m}}{ρ} I_{m} (μ_{1} ρ)} \exp (i m φ + i β z) \\ H_{φ}^{(1)} = \frac{j}{- μ_{1}^{2}} {\frac{j m β}{ρ} B_{m} I_{m} (μ_{1} ρ) + ω ε_{1} A_{m} μ_{1} I_{m}^{'} (μ_{1} ρ)} \exp (i m φ + i β z) \\ H_{ρ}^{(1)} = \frac{j}{- μ_{1}^{2}} {β B_{m} μ_{1} I_{m}^{'} (μ_{1} ρ) - \frac{j m ω ε_{1} A_{m}}{ρ} I_{m} (μ_{1} ρ)} \exp (i m φ + i β z) \end{cases}

when ρ > R,

{\begin{cases} E_{φ}^{(2)} = \frac{j}{- μ_{2}^{2}} {\frac{j m β}{ρ} C_{m} K_{m} (μ_{2} ρ) - ω μ_{0} D_{m} μ_{2} K_{m}^{'} (μ_{2} ρ)} \exp (i m φ + i β z) \\ E_{ρ}^{(2)} = \frac{j}{- μ_{2}^{2}} {β C_{m} μ_{2} K_{m}^{'} (μ_{2} ρ) + \frac{j m ω μ_{0} D_{m}}{ρ} K_{m} (μ_{2} ρ)} \exp (i m φ + i β z) \\ H_{φ}^{(2)} = \frac{j}{- μ_{2}^{2}} {\frac{j m β}{ρ} D_{m} K_{m} (μ_{2} ρ) + ω ε_{2} C_{m} μ_{2} K_{m}^{'} (μ_{2} ρ)} \exp (i m φ + i β z) \\ H_{ρ}^{(2)} = \frac{j}{- μ_{2}^{2}} {β D_{m} μ_{2} K_{m}^{'} (μ_{2} ρ) - \frac{j m ω ε_{2} C_{m}}{ρ} K_{m} (μ_{2} ρ)} \exp (i m φ + i β z) \end{cases}

where

I_{m}^{'} (μ_{1} ρ) = I_{m + 1} (μ_{1} ρ) + \frac{m}{μ_{1} ρ} I_{m} (μ_{1} ρ)

and

K_{m}^{'} (μ_{2} ρ) = - K_{m + 1} (μ_{2} ρ) + \frac{m}{μ_{2} ρ} K_{m} (μ_{2} ρ)

.

Due to the existence of graphene surface conductivity, the boundary conditions at ρ = R are $E_{z}^{(1)} = E_{z}^{(2)}$ , $E_{φ}^{(1)} = E_{φ}^{(2)}$ , $H_{z}^{(2)} - H_{z}^{(1)} = - σ_{g} E_{φ}^{(1)}$ and $H_{φ}^{(2)} - H_{φ}^{(1)} = σ_{g} E_{z}^{(1)}$ to ensure the continuity of the tangential components of the EM fields at graphene layer. Substituting Eq. (4) and Eq. (5) into the boundary conditions, we have the eigen equation for the m-th order mode.

| \begin{matrix} I_{m} (μ_{1} R) & 0 & - K_{m} (μ_{2} R) & 0 \\ \frac{j m β}{- μ_{1}^{2} R} I_{m} (μ_{1} R) & \frac{ω μ_{0}}{μ_{1}} I_{m}^{'} (μ_{1} R) & \frac{j m β}{μ_{2}^{2} R} K_{m} (μ_{2} R) & - \frac{ω μ_{0}}{μ_{2}} K_{m}^{'} (μ_{2} R) \\ \frac{j ω ε_{1}}{μ_{1}} I_{m}^{'} (μ_{1} R) - σ_{g} I_{m} (μ_{1} R) & - \frac{m β}{μ_{1}^{2} R} I_{m} (μ_{1} R) & - \frac{j ω ε_{2}}{μ_{2}} K_{m}^{'} (μ_{2} R) & \frac{m β}{μ_{2}^{2} R} K_{m} (μ_{2} R) \\ σ_{g} \frac{m β}{- μ_{1}^{2} R} I_{m} (μ_{1} R) & I_{m} (μ_{1} R) - σ_{g} \frac{j ω μ_{0}}{μ_{1}} I_{m}^{'} (μ_{1} R) & 0 & - K_{m} (μ_{2} R) \end{matrix} | = 0

When m = 0, Eq. (6) could be simplified as

\frac{ε_{1}}{μ_{1}} \frac{I_{1} (μ_{1} R)}{I_{0} (μ_{1} R)} + \frac{ε_{2}}{μ_{2}} \frac{K_{1} (μ_{2} R)}{K_{0} (μ_{2} R)} = \frac{σ_{g}}{j ω}

By solving Eq. (6) or Eq. (7) and combining the boundary conditions, we could get the propagation constant β of m-th order mode and corresponding EM field. The real part of β relates to the SPs wavelength $λ_{S P s} = 2 π / Re (β)$ , and effective mode index is defined as Re(β)/k₀, where k₀ is the wavenumber in free space. The imaginary part of β relates to propagation loss of SPs, and propagation length can be defined as 1/2Im(β), which means SPs mode decays to 1/e of its original power after travelling such a distance. μ₁ and μ₂ are related to the field confinement of mode. A larger μ₁ (μ₂) leads to $I_{m} (μ_{1} ρ)$ [ $K_{m} (μ_{2} ρ)$ ] increasing (decreasing) more quickly, which means the field is better confined near graphene.

3. Results and discussion

3.1 Plasmon modes in GNW

Graphene complex surface conductivity σ_g relates to radian frequency ω, ambient temperature T, chemical potential μ_c and relaxation time τ. Impurities in graphene and electron-phonon coupling contribute to relaxation time ( $τ^{- 1} = τ_{imp}^{- 1} + τ_{e-ph}^{- 1}$ ) and additional loss caused by optical phonon can be neglected when photon energy is less than 0.2 eV [25]. In this paper, we consider GPs modes in the frequency range from 10 THz to 50 THz neglecting the impact from optical phonon at the whole frequency range and τ is assumed to be 0.5 ps based on recent studies [26]. We set ε₁ = 3ε₀, ε₂ = ε₀, R = 100 nm, T = 300 K and μ_c = 0.5 eV, unless otherwise stated. Quantum finite-size effect of graphene is also ignored for a larger graphene structures size compared with 20 nm considered in the whole paper [27].

Figure 2(a) presents mode patterns of the first 5 order modes. The plasmon modes in GNW originate from the waveguide modes in graphene ribbon [28]. For m = 0 mode, EM field distribution is independent of φ. There only exist field components E_z, E_r and H_φ, which could be regarded as TM mode. For m≠0 mode, 2m nodes could be observed in the EM field along the circular graphene layer. Mode order is characterized by the phase factor exp(imφ) in Eq. (1) and Eq. (2). Graphene also supports TE modes when Im(σ_g) is negative [29], but TE mode will not occur in the frequency and chemical potential (0.2 eV~1 eV) range considered in this paper.

Fig. 2 (a) Mode patterns of the first 5 order modes at 50THz. (b) Amplitude of Poynting vector of m = 0 mode at frequencies of 20THz, 30THz, 40THz and 50THz. (c) Dispersion of GPs modes in GNW. (d) Propagation length of GPs modes in GNW. In both (c) and (d), solid blue curves are obtained by Eq. (6) and red dashed curves are by FEM.

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Figure 2(c) demonstrates the dispersion relations of GPs modes in GNW of R = 100 nm. We noticed that m = 0 mode is cutoff-free and the effective mode indexes of all modes decrease monotonically with frequency decreasing. At high frequencies, a larger proportion of mode energy reside inside GNW near the graphene-nanowire interface, while at low frequencies, they are mainly localized outside GNW, showing a weaker EM field confinement. This is evidenced by the Poynting vector distribution in the radial direction depicted in Fig. 2(b). High order plasmon mode (m≠0) cuts off when $Re (β) / k_{0} < \sqrt{ε_{1}}$ , and waveguide-like mode may forms [10], the longitudinal electric field component E_z of which is ∝ $J_{m} (μ_{1} ρ) \exp (i m φ + i β z)$ in nanowire (J_m is m-th Bessel function). However, the waveguide-like modes are radiative due to the small nanowire radius compared to EM wavelength and we neglect this situation in our paper. Figure 2(d) shows the propagation length of GPs modes in GNW as a function of frequencies. In the high frequency range, the modes are strongly confined and result in an increasing absorption loss. At the frequencies near the modal cutoff, the group velocity decrease and the propagation loss is accumulated in a short length. High order mode shows a maximum in the propagation length, which originates from the balancing between mode confinement and low group velocities [28]. We also perform numerical simulation based on finite element method (FEM) to obtain the dispersion relations and propagation lengths (dashed lines in Fig. 2(c) and 2(d)). Here graphene layer is treated as a t = 0.5 nm-thick layer with effective permittivity $1 + i σ_{g} / (ε_{0} ω t)$ [9]. Solid lines in Fig. 2(c) and 2(d) are obtained by Eq. (6). Two methods show good agreement with each other. But at high frequency range, slight deviation is observed. We believe it results from different treatments towards graphene thickness between analytic and numerical methods.

Radius of nanowire has a strong impact on the modal behavior in GNW, such as the number of supported modes. Figure 3 shows the radius-dependent effective mode index and propagation length at the frequency of 25THz. Radius of GNW varies from 50 nm to 300 nm. Effective mode indexes decrease with radius reducing and a cross phenomenon between the effective mode indexes of m = 0 and m = 1 mode could be observed. m = 1, 2, 3, 4 order modes cut off at radii of 67 nm, 135 nm, 202 nm and 270 nm, respectively. This indicates single mode operation could be achieved by reducing nanowire radius. In single mode region (R < 67 nm), propagation length increases with nanowire radius decreasing. The number of supported GPs modes in GNW can be estimated as $2 π R n_{eff}^{2DGSP} / λ$ , in which $n_{eff}^{2DGSP} = i (ε_{1} + ε_{2}) c / σ_{g}$ is the mode index of GPs mode in an infinite graphene sheet placed at the interface between two dielectric half spaces with permittivities of ε₁ and ε₂ [28] and λ is EM wavelength in free space.

Fig. 3 Effective mode index (a) and propagation length (b) as a function of nanowire radius at the frequency of 25THz.

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For SPs mode in silver nanowire, higher permittivity value of surrounding dielectric has a tendency to drag EM field to the interface between metal and dielectric [30]. Permittivity of nanowire also has a similar impact on GPs mode in GNW. Figure 4 depicts the permittivity-dependent effective mode index and propagation length at the frequency of 30THz. Inset of Fig. 4(b) shows the influence of different nanowire permittivities on electric field distribution of m = 0 mode along ρ direction. The larger ε₁ is, the more tightly mode is localized near graphene layer. With decreasing nanowire permittivity, effective mode index almost linearly decrease, as depicted in Fig. 4(a), which indicates a weaker localization of GPs mode, and high order modes is no more sustained at an enough small value of nanowire permittivity. In SPs mode, a tradeoff between mode confinement and propagation loss is well-known. The same characteristic could also be observed by comparing Fig. 4(a) and Fig. 4(b).

Fig. 4 Effective mode index (a) and propagation length (b) as a function of nanowire permittivity at the frequency of 30 THz. Inset of (b) shows the normalized |E| distribution of m = 0 mode along ρ direction under different nanowire permittivity. R = 100 nm.

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Graphene conductivity σ_g could be tuned by changing chemical potential μ_c. In monolayer graphene $μ_{c} = ℏ v_{f} \sqrt{n_{c} π}$ [31], $v_{f} = 10^{6} m / s$ , n_c is carrier density. In recent experiments carrier density reaches as high as 10¹⁴ cm⁻² [32], equaling to μ_c = 1.17 eV. Figure 5 shows the chemical potential dependent characteristics of GPs modes in GNW. We change μ_c from 0.2 eV to 1 eV. As shown in Fig. 5(a), when chemical potential is increased, effective mode index decreases, implying the GPs modes becoming less localized and a consequent lower propagation loss, i.e. an increase in propagation length as depicted in Fig. 5(b). High order mode is no longer sustained by GNW at high doping level. m = 2, 3, 4 modes cut off at μ_c = 0.54 eV, 0.37 eV and 0.28 eV, respectively. This indicates the possibility of realizing single mode operation by simply tuning chemical potential. For instance, at the frequency of 20THz, single mode operation in GNW with a radius of 100 nm is achieved when chemical potential is larger than 0.48 eV.

Fig. 5 Effective mode index (a) and propagation length (b) as a function of chemical potential of graphene at the frequency of 30 THz. R = 100nm.

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3.2 Comparison with metal-coated nanowire

In this section, we will compare the SPs modes behavior of metal-coated and graphene-coated nanowire in the 30THz-50THz frequency range. Analytical investigation for SPs modes in metal-coated dielectric nanowire has been carried out in several papers [10, 11, 33]. For example, Ref [33]. studied SPs modes in GaAs nanowire with 30nm-thick silver cladding, the dispersion equation of which is obtained by applying the boundary conditions both at the inner and outer radii of silver cladding, and noble metal is characterized by dielectric constant obtained by either Drude model or measured data. However, due to a much larger thickness of noble metal coating compared with atomically thin graphene layer, the previous model could not readily be applied in the GNW case, moreover graphene is usually treated as a zero-thickness layer with surface conductivity in the analytical approach [7, 34].

Silver has the smallest relaxation rate among the high-conductivity metals and is the best choice for plasmonic applications at optical frequencies. Gold performs better at lower infrared frequencies and is chemically stable in various environments [35]. Here, we choose gold-coated nanowire (AuNW) for comparison. The permittivity data of gold is from Palik’s handbook [36] in the range of 5 μm to 10 μm.

Figure 6 shows the comparison of plasmonic properties between GNW and AuNW and we only consider m = 0 mode in both cases because the latter one is single-mode guiding in the studied frequency range. As shown in Fig. 6(a), the effective mode index of SPs mode in GNW (ranges from 11.1 to 25.4) is much larger than that of AuNW (around 1.03 in the whole considered frequency range), indicating plasmon mode in GNW has a much shorter SPs wavelength and better mode confinement. In order to compare mode confinement quantitatively, we introduce effective mode area A_eff, which is defined as $A_{e f f} = {(\int W (r) d s)}^{2} / \int W {(r)}^{2} d s$ and W(r) is energy density of SPs mode. The calculated mode areas of GNW and AuNW are normalized by $λ^{2}$ and depicted in Fig. 6(c). SPs mode in GNW has a much smaller mode area, roughly one order of magnitude smaller, than its counterpart in AuNW. In addition, the mode energy of GNW is mainly localized inside the nanowire, whereas the mode energy of AuNW resides outside the Au coating, as seen from inset of Fig. 6(c). Better confined mode energy is preferred to avoid crosstalk in nano-circuit.

Fig. 6 Comparison of the plasmonic properties of GNW and AuNW. (a) Effective mode index, (b) Propagation length and (c) mode area of m = 0 SPs mode as a function of frequency for GNW and AuNW. Inset of (a) shows the structure of AuNW. The outer radii of both kinds of nanowire are 100 nm, and Au layer is 30 nm. Chemical potential of graphene is fixed at 1 eV. Inset of (c) is normalized |E| of m = 0 SPs modes at 50THz.

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The propagation length of SPs mode in AuNW is larger than that in GNW. However, at lower frequencies, the difference of propagation lengths between GNW and AuNW is not significant. For example, the propagation lengths are 5.44 μm (GNW) and 6.36 μm (AuNW) at 30THz. Moreover, surface imperfections of gold coating may cause additional scattering loss, which is neglected in calculation and will decrease the propagation length, whereas graphene can be produced with defect-free lattice structure over several SPs wavelengths due to its inherent chemical characteristic [8].

5. Conclusion

In summary, we presented an analytical model to characterize the plasmon modes in graphene-coated nanowire which shows good agreement with rigorous numerical simulation. An eigen equation for dispersion relation is derived. The plasmon modes show an azimuthal symmetry and mode order could be characterized by the phase factor exp(imφ) in EM field distribution. Nanowire radius, nanowire permittivity and chemical potential of graphene layer affect the performance of GNW: an enough small nanowire radius will prohibit the existence of high order modes. Nanowire with a higher permittivity value will show better mode confinement near the graphene layer. A higher graphene doping level not only suppresses high order modes but increases the propagation length, especially for fundamental mode (m = 0). Compared with metal-coated nanowire, GNW has a much higher mode index and a better mode confinement. GNW may be exploited for many applications such as sensing and integrated plasmonic circuit in the mid-infrared to terahertz frequencies and the proposed model provides valuable references for GNW-based applications.

Acknowledgment

This work is supported in part by the Major State Basic Research Development Program of China (Grant No. 2010CB328206), the National Natural Science Foundation of China (NSFC) (Grant Nos. 61178008, 61275092), and the Fundamental Research Funds for the Central Universities, China.

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Analytical model for plasmon modes in graphene-coated nanowire

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

3.1 Plasmon modes in GNW

3.2 Comparison with metal-coated nanowire

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Equations (7)

Optics Express