Plasmon modes of circular cylindrical double-layer graphene

Tao Zhao; Min Hu; Renbin Zhong; Xiaoxing Chen; Ping Zhang; Sen Gong; Chao Zhang; Shenggang Liu

doi:10.1364/OE.24.020461

1. Introduction

Graphene, a two-dimensional atomic thin carbon material with honeycomb lattice, has attracted tremendous research attention due to its unique and superior electronic and optical properties [1–6]. Both theoretical and experimental works demonstrated that graphene can support surface plasmon polaritons (SPPs) in the terahertz (THz) and mid-infrared frequency regimes [7,8]. Comparing to the SPPs in the metal films, graphene SPPs exhibit remarkable features such as strong mode confinement and low propagation loss [9–11]. Moreover, these features can be tuned by adjusting the electrostatic gating or the chemical doping [12–14]. Therefore, graphene-based plasmonics can play a significant role in photonic and optoelectronic applications, such as enhanced THz radiation sources [15–18], transformation optics [13], tunable metamaterials [12,19], nonlinear optics [20,21] and cloaks [22,23]. Recent theoretical and experimental interest focus on SPPs in double-layer graphene structures [24–28], in which SPPs act as powerful tools for probing many-body properties such as screening and drag [27,29]. The SPPs in the two graphene layers are coupled via interlayer electromagnetic interaction resulting in the formation of two branches of plasmon spectra called optical plasmon mode with a square-root dispersion and acoustic plasmon mode with a linear dispersion in the long-wavelength limit [24,26].

Circular cylindrical graphene structure in the scale of micrometer, rolling by a graphene ribbon, has attracted intense attention and has shown wide applications [30–37]. SPPs can be sustained in the circular cylindrical graphene structure as its planar counterpart [33–36], but their behaviors are quite different arising from the structure diversity. For the circular cylindrical graphene structure, due to the naturally azimuthal symmetry, SPPs modes consist of the fundamental mode and hybrid modes, whose mode orders are determined by the fields nodes in the azimuthal direction [33–35]. This azimuthal characteristic is similar to that of SPPs in the carbon nanotubes, while the plasmon frequencies of the carbon nanotubes are much higher [38, 39]. In both circular cylindrical monolayer and double-layer graphene structures with dielectric loading, SPPs can be excited by a cyclotron electron beam and transformed into enhanced coherent and tunable THz radiation because of the natural periodicity of 2π [35]. This transformation process is impossible for the planar graphene since SPPs modes always lie below the light line in dielectrics. In particularly, two-color THz radiation could be generated in the double-layer graphene structure. It has been shown that SPPs in circular cylindrical graphene structure have much higher mode confinement and longer propagation length compared with those of circular cylindrical metal-film structure in the mid-infrared frequency regime [34,37].

The SPPs modes of a cylindrical graphene plasmon waveguide, bent vertically offset graphene ribbon pairs, has been studied in a recent work by Yang et al [40]. However, Yang’s work ignores the coupling between two graphene layers. As a result the effect of SPPs modes split was completely missing in Yang’s work. Moreover, the circular cylindrical structure has a natural periodicity of 2π in the azimuthal direction, so it is not necessary to regard graphene as an anisotropic layer in [40].

In this paper, we present a theoretical study on plasmon modes in a circular cylindrical double-layer graphene (CDLG) structure. The results show distinct properties of mode pattern, effective mode index and propagation loss of optical plasmon mode and acoustic plasmon mode. We analyze the mode dependence on the distance between two graphene layers, chemical potential of graphene and permittivity of interlayer dielectric. We found the breakup of tradeoff between mode confinement and propagation loss in the distance-dependent modal behavior. In the last part, the performance of SPPs in circular cylindrical double-layer and monolayer graphene systems are compared.

2. Theoretical formulation

The CDLG structure consists of two graphene ribbons with a spatial separation rolling around dielectrics, as depicted in Fig. 1. This structure may be constructed by the methods proposed in [31, 41] The radii of the inner and outer graphene layers are r_a and r_b, respectively and the distance between two graphene layers is r_b-r_a. Graphene is considered to be infinitely thin with a surface conductivity $σ_{g}$ described by the Kubo formula [7, 42–44]. The scheme can be divided into three regions depending on the relative permittivities of dielectrics $ε_{1}$ , $ε_{2}$ and $ε_{3}$ . By solving the Maxwell equations together with boundary conditions in the cylindrical-coordinate system, electromagnetic fields in each region can be obtained as follow.

Fig. 1 Schematic of CDLG structure, the radii of the inner and outer graphene layers are r_a and r_b, respectively.

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In the region I ( $r < r_{a}$ ):

\begin{array}{l} E_{z}^{I} = A_{1 m} k_{c 1}^{2} J_{m} (k_{c 1} r) H_{z}^{I} = A_{2 m} k_{c 1}^{2} J_{m} (k_{c 1} r) \\ E_{r}^{I} = j A_{1 m} k_{z} k_{c 1} J_{m}^{'} (k_{c 1} r) - A_{2 m} ω μ_{0} \frac{m}{r} J_{m} (k_{c 1} r) \\ E_{θ}^{I} = - A_{1 m} k_{z} \frac{m}{r} J_{m} (k_{c 1} r) - j A_{2 m} ω μ_{0} k_{c 1} J_{m}^{'} (k_{c 1} r), \\ H_{r}^{I} = A_{1 m} ω ε_{0} ε_{1} \frac{m}{r} J_{m} (k_{c 1} r) + j A_{2 m} k_{z} k_{c 1} J_{m}^{'} (k_{c 1} r) \\ H_{θ}^{I} = A_{1 m} j ω ε_{0} ε_{1} k_{c 1} J_{m}^{'} (k_{c 1} r) - A_{2 m} k_{z} \frac{m}{r} J_{m} (k_{c 1} r) \end{array}

where:

k_{c 1}^{2} = ε_{1} k_{0}^{2} - k_{z}^{2}

,

k_{z}

is the wave vector of SPPs modes in z direction,

J_{m} (k_{c 1} r)

is the first kind of Bessel function, m = (0, 1, 2, 3…),

J_{m}^{'} (k_{c 1} r) = \frac{1}{2} [J_{m - 1} (k_{c 1} r) - J_{m + 1} (k_{c 1} r)]

,

J_{0}^{'} (k_{c 1} r) = - J_{1} (k_{c 1} r)

A_{1 m}

and

A_{2 m}

are fields coefficients.

In the region II ( $r_{a} \leq r < r_{b}$ ):

\begin{array}{l} E_{z}^{I I} = A_{3 m} k_{c 2}^{2} I_{m} (k_{c 2} r) + A_{4 m} k_{c 2}^{2} K_{m} (k_{c 2} r), H_{z}^{I I} = A_{5 m} k_{c 2}^{2} I_{m} (k_{c 2} r) + A_{6 m} k_{c 2}^{2} K_{m} (k_{c 2} r) \\ E_{r}^{I I} = - j k_{z} k_{c 2} [A_{3 m} {I^{'}}_{m} (k_{c 2} r) + A_{4 m} K_{m}^{'} (k_{c 2} r)] + ω μ_{0} \frac{m}{r} [A_{5 m} I_{m} (k_{c 2} r) + A_{6 m} K_{m} (k_{c 2} r)] \\ E_{θ}^{I I} = k_{z} \frac{m}{r} [A_{3 m} I_{m} (k_{c 2} r) + A_{4 m} K_{m} (k_{c 2} r)] + j ω μ_{0} k_{c 2} [A_{5 m} {I^{'}}_{m} (k_{c 2} r) + A_{6 m} {K^{'}}_{m} (k_{c 2} r)], \\ H_{r}^{I I} = - j k_{z} k_{c 2} [A_{5 m} {I^{'}}_{m} (k_{c 2} r) + A_{6 m} {K^{'}}_{m} (k_{c 2} r)] - \frac{ω ε_{0} ε_{2}}{r} m [A_{3 m} I_{m} (k_{c 2} r) + A_{4 m} K_{m} (k_{c 2} r)] \\ H_{θ}^{I I} = \frac{k_{z}}{r} m [A_{5 m} I_{m} (k_{c 2} r) + A_{6 m} K_{m} (k_{c 2} r)] - j ω ε_{0} ε_{2} k_{c 2} [A_{3 m} {I^{'}}_{m} (k_{c 2} r) + A_{4 m} {K^{'}}_{m} (k_{c 2} r)] \end{array}

where:

k_{c 2}^{2} = k_{z}^{2} - ε_{2} k_{0}^{2}

,

I_{m} (k_{c 2} r)

and

K_{m} (k_{c 2} r)

are the modified Bessel functions of the first and second kind, m = (0, 1, 2, 3…),

I_{m}^{'} (k_{c 2} r) = \frac{1}{2} [I_{m - 1} (k_{c 2} r) + I_{m + 1} (k_{c 2} r)]

,

I_{0}^{'} (k_{c 2} r) = I_{1} (k_{c 2} r)

,

K_{m}^{'} (k_{c 2} r) = - \frac{1}{2} [K_{m - 1} (k_{c 2} r) + K_{m + 1} (k_{c 2} r)]

,

K_{0}^{'} (k_{c 2} r) = - K_{1} (k_{c 2} r)

,

A_{3 m}

,

A_{4 m}

,

A_{5 m}

and

A_{6 m}

are fields coefficients.

In the region III ( $r \geq r_{b}$ ):

\begin{array}{l} E_{z}^{I I I} = A_{7 m} k_{c 3}^{2} K_{m} (k_{c 3} r), H_{z}^{I I I} = A_{8 m} k_{c 3}^{2} K_{m} (k_{c 3} r) \\ E_{r}^{I I I} = - j k_{z} k_{c 3} A_{7 m} K_{m}^{'} (k_{c 3} r) + ω μ_{0} \frac{m}{r} A_{8 m} K_{m} (k_{c 3} r) \\ E_{θ}^{I I I} = k_{z} \frac{m}{r} A_{7 m} K_{m} (k_{c 3} r) + j ω μ_{0} k_{c 3} A_{8 m} {K^{'}}_{m} (k_{c 3} r), \\ H_{r}^{I I I} = - j k_{z} k_{c 3} A_{8 m} {K^{'}}_{m} (k_{c 3} r) - \frac{ω ε_{0}}{r} m A_{7 m} K_{m} (k_{c 3} r) \\ H_{θ}^{I I I} = \frac{k_{z}}{r} m A_{8 m} K_{m} (k_{c 3} r) - j ω ε_{0} k_{c 3} A_{7 m} {K^{'}}_{m} (k_{c 3} r) \end{array}

where:

k_{c 3}^{2} = k_{z}^{2} - k_{0}^{2}

,

K_{m}^{'} (k_{c 3} r) = - \frac{1}{2} [K_{m - 1} (k_{c 3} r) + K_{m + 1} (k_{c 3} r)], K_{0}^{'} (k_{c 3} r) = - K_{1} (k_{c 3} r)

,

A_{7 m}

, and

A_{8 m}

are fields coefficients.

The phase factor $e^{j k_{z} z + j m θ - j ω t}$ is omitted in the expressions of electromagnetic fields, the real part of k_z representing phase constant is always larger than the wave vector of light, so SPPs modes are surface waves decaying away from the graphene layer. Its imaginary part is the attenuation constant representing propagation loss in the z direction. m represents the order of SPPs mode. The m = 0 mode is the fundamental TM mode which has three fields components $E_{z}$ , $E_{r}$ , $H_{θ}$ . For m≥1, to satisfy boundary conditions, TM and TE fields co-exist to form hybrid SPPs modes.

The dispersion equation of CDLG structure can be achieved by method of field matching at boundaries. The boundary conditions can be written as,

\begin{array}{l} {E_{z}^{I} |}_{r = r_{a}} = {E_{z}^{I I} |}_{r = r_{a}}, {E_{θ}^{I} |}_{r = r_{a}} = {E_{θ}^{I I} |}_{r = r_{a}}, \\ {(H_{z}^{I} - H_{z}^{I I}) |}_{r = r_{a}} = {σ_{g} E_{θ}^{I} |}_{r = r_{a}}, {(H_{θ}^{I I} - H_{θ}^{I}) |}_{r = r_{a}} = {σ_{g} E_{z}^{I} |}_{r = r_{a}} . \end{array}

\begin{array}{l} {E_{z}^{I I} |}_{r = r_{b}} = {E_{z}^{I I I} |}_{r = r_{b}}, {E_{θ}^{I I} |}_{r = r_{b}} = {E_{θ}^{I I I} |}_{r = r_{b}}, \\ {(H_{z}^{I I} - H_{z}^{I I I}) |}_{r = r_{b}} = {σ_{g} E_{θ}^{I I I} |}_{r = r_{b}}, {(H_{θ}^{I I I} - H_{θ}^{I I}) |}_{r = r_{b}} = {σ_{g} E_{z}^{I I I} |}_{r = r_{b}} . \end{array}

Submitting the electromagnetic fields into the above boundary conditions, the dispersion equation is obtained. For m = 0, the dispersion equation can be simplified as

\begin{array}{l} \frac{[σ_{g} k_{c 3} K_{0} (k_{c 3} r_{b}) - j ω ε_{0} ε_{3} K_{1} (k_{c 3} r_{b})] \frac{k_{c 2} I_{0} (k_{c 2} r_{b})}{k_{c 3} K_{0} (k_{c 3} r_{b})} - j ω ε_{0} ε_{2} I_{1} (k_{c 2} r_{b})}{[σ_{g} k_{c 1} J_{0} (k_{c 1} r_{a}) - j ω ε_{0} ε_{1} J_{1} (k_{c 1} r_{a})] \frac{k_{c 2} I_{0} (k_{c 2} r_{a})}{k_{c 1} J_{0} (k_{c 1} r_{a})} + j ω ε_{0} ε_{2} I_{1} (k_{c 2} r_{a})} \\ = \frac{[σ_{g} k_{c 3} K_{0} (k_{c 3} r_{b}) - j ω ε_{0} ε_{3} K_{1} (k_{c 3} r_{b})] \frac{k_{c 2} K_{0} (k_{c 2} r_{b})}{k_{c 3} K_{0} (k_{c 3} r_{b})} + j ω ε_{0} ε_{2} K_{1} (k_{c 2} r_{b})}{[σ_{g} k_{c 1} J_{0} (k_{c 1} r_{a}) - j ω ε_{0} ε_{1} J_{1} (k_{c 1} r_{a})] \frac{k_{c 2} K_{0} (k_{c 2} r_{a})}{k_{c 1} J_{0} (k_{c 1} r_{a})} - j ω ε_{0} ε_{2} K_{1} (k_{c 2} r_{a})} . \end{array}

By solving the dispersion equation in the complex plane of $k_{z}$ as a function of real frequency, two types of solutions can be obtained. One is the optical plasmon mode, and another is the acoustic plasmon mode. The dispersion curves of SPPs modes are characterized by $Re (k_{z})$ as a function of frequency and effective mode index is defined as $Re (k_{z}) / k_{0}$ . Because of the transverse wave vector is nearly proportional to $k_{z}$ , higher effective mode index indicates stronger mode confinement or faster decay of the fields away from graphene layer. $Im (k_{z})$ represents the propagation loss of SPPs modes in the z direction. $1 / 2 Im (k_{z})$ is defined as the propagation length, after which the SPPs intensity decreases to $1 / e$ .

3. Results and discuss

3.1 SPPs modes in CDLG structure

Quantum finite-size effect of graphene can be ignored for circular cylindrical graphene structure with a larger radius than 20nm [45]. For a circular structure of this size the conductivity is the same as that of planar graphene layer [34, 36]. In our consideration, the two graphene layers are with identical conductivity, which is determined by frequency, chemical potential $μ_{c}$ , temperature $T$ , and relaxation time $τ$ . In the calculations, the parameters are $μ_{c}$ = 0.2eV, $T$ = 300K, and $τ$ = 1.2ps [46], r_a = 100nm, r_b = 150nm, and $ε_{1}$ = $ε_{2}$ = $ε_{3}$ = 1. Unless otherwise stated, in all figures shown below, the solid curves are for the acoustic plasmon modes, and dashed curves are for the optical plasmon modes.

The mode patterns of first four SPPs modes for optical and acoustic plasmon modes are shown in Fig. 2(a). The optical modes are symmetric and the acoustic modes are asymmetric with respect to the circular cylindrical plane at middle of two graphene layers. This symmetry property can also be seen in the E_z distribution in the radial direction depicted in Fig. 2(b). Two maximum of SPPs fields exist at the inner and outer graphene layer for optical plasmon mode, while there is a maximum and a minimum for the acoustic plasmon mode. The fields distribution of fundamental mode for the optical and acoustic plasmon modes is uniform in azimuthal direction, and hybrid modes show 2m nodes.

Fig. 2 (a) Mode patterns of the first four modes at the frequency of 40THz. (b) Normalized Ez field of m = 0 mode at 10THz, 20THz, 30THz and 40THz. (c) Dispersion of SPPs modes (d) Normalized propagation length of SPPs modes.

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Figure 2(c) shows the dispersion curves for the two types of SPP modes. Due to the electromagnetic coupling between the two graphene layers, the dispersion curves of the acoustic plasmon modes shift upward, while those of optical plasmon modes move down. Therefore, for the same order modes, effective mode indexes are much larger for the acoustic plasmon mode, especially in the long-wavelength limit. It indicates that a large proportion of mode energy for the acoustic plasmon mode reside inside the structure. This is demonstrated by both mode patterns and the E_z field distribution in the radial direction. We also notice that fundamental modes are cutoff-free, while hybrid modes are not. The optical plasmon modes have higher cutoff frequencies because of the coupling effect in the double-layer system.

The propagation length of SPPs modes decreases monotonically as the frequency increases. This frequency dependence is an opposite to that of the effective mode index and it indicates a tradeoff between mode confinement and propagation loss [47–49]. Figure 2(d) shows the propagation length normalized by the free-space wavelength of SPPs. It is inversely proportional to the imaginary part of complex effective index $Im (k_{z}) / k_{0}$ which can be viewed as the figure of merit of SPPs [47, 48]. We notice that the normalized propagation length of the fundamental optical mode crosses with its counterpart at the frequency of 27.6THz, conveniently called as crossover frequency f_c. This is caused by the unique properties of a double-layer graphene structure. The electromagnetic interaction between two graphene layers results in the formation of the optical and the acoustic plasmon modes. The optical modes have a square-root dispersion ( $ω_{o} \propto k_{z}^{1 / 2}$ ) and the acoustic modes have linear dispersion ( $ω_{a} \propto k_{z}$ ) [24–26]. The propagation length is inversely proportional to $Im (k_{z})$ , which leads to inverse square and inverse linear dependence on the frequency for the optical and the acoustic plasmon modes. Although the propagation length is longer for the optical plasmon mode in the low frequency regime, faster decreasing of its propagation length with increasing frequency results in the occurrence of cross phenomenon. Therefore, a stronger mode confinement and a lower propagation loss, which are always pursued in the subject of subwavelength optics [9], can be achieved by the acoustic plasmon modes in the present system.

The distance between two graphene layers has a strong influence on the characteristics of SPPs modes due to the coupling effect. Figure 3 shows the distance-dependent the effective mode index and the normalized propagation length at the frequency of 40THz. At short distances, the two types of plasmon modes exhibit opposite variation trends. At large distances, due to vanishing interlayer electromagnetic coupling, each mode approaches the limit of the SPP modes of a circular cylindrical monolayer graphene (CMLG) of fixed radius.

Fig. 3 Effective mode index (a) and normalized propagation length (b) as a function of the distance between two graphene layers at the frequency of 40THz.

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For both the optical modes and the acoustic modes at short distances, the effective mode index and the propagation length have the opposite frequency dependence. This indicates that the tradeoff between mode confinement and propagation loss. At large distances, this tradeoff is broken. For the acoustic plasmon modes, the wave vector $k_{z}$ is inversely proportional to square-root of distance d in the long-wavelength limit under the strong coupling condition (short distance) [24, 26], thus the propagation length is dependent on the distance with a square-root relation. This leads to the propagation length increasing with the distance. If this condition is unsatisfied, both the propagation length and the mode confinement decrease with the distance, leading the disappearance of the tradeoff. The propagation length of the acoustic plasmon modes are longer than that of the optical plasmon modes unless at the very short distance. The strong interlayer coupling at small distances gives rise to a crossover frequency, as described in Fig. 2(d). At high frequencies the propagation length for the optical plasmon mode is longer.

The coupling becomes weak at large distances and leads to a lower crossover frequency. In turn the propagation length of the optical plasmon mode becomes shorter. Hence, stronger mode confinement and lower propagation loss for acoustic plasmon modes can also be obtained in the distance-dependent modal behavior.

An essential difference of SPPs between graphene and metal films is that characteristics of SPPs can be effectively tuned by adjusting the chemical potential. Figure 4 shows the dependence of the effective mode indexes and normalized propagation length on the chemical potential. The m = 1, 2, 3 optical plasmon modes and m = 3 acoustic plasmon mode cutoff at chemical potential of 0.7eV, 0.39eV, 0.29eV and 0.59eV, respectively. For the same order modes, optical plamson mode has a much lower cutoff chemical potential because of higher plasmon energy. The propagation length of SPPs modes increases with chemical potential. It can reach up to several times of SPPs wavelength at expense of low mode confinement. A crossover phenomenon between propagation length of the fundamental optical and acoustic plasmon modes occurs at 0.42eV. This energy is much higher for optical plasmon mode in the high chemical potential regime. For high chemical potential, 40THz is lower than the crossover frequency described in Fig. 2(d), thus the propagation length of the fundamental optical plasmon mode is longer. However, the high order acoustic plasmon modes with larger propagation length are observed. The high order modes show a maximum due to the balance between mode confinement and low group velocities [34].

Fig. 4 Effective mode index (a) and normalized propagation length (b) as a function of the chemical potential at the frequency of 40THz.

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The surrounding dielectrics have a direct impact on the modal behavior of SPPs in the CDLG structure. Figure 5 shows the dependence of effective mode indexes and normalized propagation length on the permittivity of interlayer dielectric. The effective mode indexes of acoustic SPPs modes are much larger than those of the optical modes. Therefore the high order acoustic modes can be sustained in the low permittivity regime. The crossover phenomenon also occurs in the permittivity-dependent propagation length. The acoustic plasmon modes have larger propagation length in the high permittivity range. The wave vectors of the optical plasmon mode and the acoustic plasmon mode are dependent on the permittivity with linear and square-root relation [24–26]. The propagation length of the fundamental optical plasmon mode decreases more rapidly than that of acoustic plasmon mode, leading to the crossover phenomenon. As a result, higher mode confinement and lower propagation loss of acoustic plasmon modes can also be obtained.

Fig. 5 Effective mode index (a) and normalized propagation length (b) as a function of the permittivity of dielectric sandwiched between two grpahene layers at the frequency of 20THz.

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3.2 Comparison with the CMLG structure

In the CDLG structure, the interlayer electromagnetic interaction leads to splitting of the plasmon mode into two plasmon modes. Comparing to the effective mode index of fundamental plasmon mode in the CMLG structure, Fig. 6(a) shows that of the optical plasmon mode in the CDLG structure shifts lower while acoustic plasmon mode shifts upward. It indicates a larger proportional of mode energy of the acoustic plasmon mode resides inside the structure. This is evidenced by mode patterns of the acoustic plasmon mode and the fundamental mode of the CMLG structure depicted in Fig. 6(c). Although the structure size is a little larger for the CDLG structure and SPPs fields are localized near the two graphene layers, the acoustic plasmon mode exhibit higher mode confinement than that in the CMLG structure. The high mode confinement is preferred to avoid crosstalk in nano-circuit.

Fig. 6 Comparison of SPPs properties between CDLG and monolayer graphene structure. (a) Effective mode index and (b) Normalized propagation length of fundamental mode as a function of frequency. Mode patterns in the (c) X-Y and (d) Y-Z planes.

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The normalized propagation length of fundamental SPPs mode of the CDLG and the CMLG structure is shown in Fig. 6(b). The m = 0 optical plasmon mode firstly increasing and then decreasing with the frequency increasing. This behavior is similar to that of the CMLG structure. The energy of the fundamental optical plasmon mode of the CDLG is much higher than that of the CMLG structure below the frequency of 30THz, indicating a lower propagation loss. This property can also be seen in the mode patterns of Ez fields at 12 THz in the Y-Z plane shown in Fig. 6(d). The propagation loss for the CDLG structure is very low in one wavelength propagating distance. The maximum of normalized propagation length of the optical plasmon mode can be further increased by improving the coupling effect, such as higher chemical potential and shorter interlayer distance. It can reach up to 0.892 which is much larger than 0.511 for the CMLG structure. The acoustic plasmon mode has better performance in propagation length above the frequency of 21.7THz than that of the CMLG structure. Therefore, SPPs modes of the CDLG structure can propagate longer in the whole frequency regime.

4. Conclusions

In summary, we presented a theoretical investigation on plasmon modes in the CDLG structure. The dependence of modal behavior of SPPs modes on the distance between two graphene layers, chemical potential of graphene and permittivity of interlayer dielectric is analyzed in detail. In the distance-dependent modal behavior, we discover the breakup of tradeoff between mode confinement and propagation loss because of the unique dispersion properties of double-layer graphene system. Both higher mode confinement and longer propagation length for acoustic plasmon modes can be achieved under certain parameters. In comparison to the CMLG structure, advances of SPPs features in both mode confinement and propagation loss have been demonstrated in the CDLG structure. Therefore, the CDLG structure may provide good opportunities for applications such as sensing systems, integrated plasmonic circuit and subwavelength optics in the THz and mid-infrared frequency regimes.

Funding

Fundamental Research Funds for the Central Universities (FRFCU) (ZYGX2016KYQD113); Program 973 (2014CB339801); Natural Science Foundation of China (NSFC) (61231005, 11305030, 612111076); National High-tech Research and Development Project (NHRDP) (2011AA010204).

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Plasmon modes of circular cylindrical double-layer graphene

Abstract

1. Introduction

2. Theoretical formulation

3. Results and discuss

3.1 SPPs modes in CDLG structure

3.2 Comparison with the CMLG structure

4. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (6)

Optics Express