- Email: [email protected]
Metal-loaded graphene surface plasmon waveguides working in the terahertz regime
Optics Communications 355 (2015) 602–606
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Metal-loaded graphene surface plasmon waveguides working in the terahertz regime Binggang Xiao, Kang Qin, Sanshui Xiao, Zhanghua Han n Centre for Terahertz Research, China Jiliang University, Hangzhou 310018, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 April 2015 Received in revised form 27 June 2015 Accepted 15 July 2015
A metal-loaded graphene surface plasmon waveguide composed of a thin silica layer sandwiched between a graphene layer and a metal stripe is proposed and the waveguiding properties in the THz regime are numerically investigated. The results show that the fundamental mode of the proposed waveguide is tightly confined in the middle silica layer with an acceptable propagation loss. Compared with most other graphene waveguides proposed in the literature, the realization of this waveguide does not need to pattern or deform the graphene layer, thus retaining the superior properties of bulk graphene material. The tight modal confinement and the ease of fabrication suggest the high potential use of this waveguide in high-density THz photonic integration. & 2015 Elsevier B.V. All rights reserved.
Keywords: Plasmonics waveguides graphene
1. Introduction The use of surface plasmons (SPs) for radiation guiding at a scale beyond the diffraction limit has been widely investigated in the optical frequencies [1,,2]. Due to the high level of free electron concentration, noble metals have the plasma frequencies normally in the ultraviolet, making them good plasmonic materials in the optical regime. However, in the lower frequency bands such as Terahertz (THz) or microwave frequencies, where the frequency is far below the plasma frequency of metals, noble metal usually cannot be used to support SPs due to the near-zero skin depth. Although the concept of spoof surface plasmons (SSPs) [3,,4] have been proposed in the THz/Microwave band to support the propagation of artificial surface wave, recently another material of graphene has emerged as a good candidate to support SPs in those frequencies. The energy band structure of single-layer graphene is linear with no band gap [5–7], making graphene possess unique properties which other substances do not have, such as zero effective carrier mass near the Dirac point and tunable carrier density by chemical potential or Fermi level [8]. As a result, the plasma frequency of graphene can be tuned to be in the THz regime and graphene can work as a good plasmonic material. Compared with noble metals, graphene has many advantages, such as extreme confinement in the THz band, and high tunability [9] by electrostatic gating. Actually graphene as a one-atom-thick plasmonic material [1] has achieved wide applications in THz n
Corresponding author. E-mail address: [email protected] (Z. Han).
http://dx.doi.org/10.1016/j.optcom.2015.07.031 0030-4018/& 2015 Elsevier B.V. All rights reserved.
metamaterials and plasmonics [2]. To date various graphene based plasmonic waveguides working in the far infrared or lower frequencies have been proposed, with the graphene in the form of ribbons [10–13], nanowire [14,,15], rings [16], wedge/groove waveguide [17] and graphene on the dielectric [18]. However in most of those proposed structures the two dimensional (2D) material of graphene layer needs to be either deformed or structured, which may not only add much complication to the fabrication process but also deteriorate the original unique properties of bulk graphene because the new boundaries in structured graphene may introduce some defect states into the energy band of graphene. The effect will be more evident at lower frequencies because the period of electromagnetic wave is long enough and the moving electrons in graphene will suffer from additional scattering from the new edges. Many waveguides proposed in the literature based on structured graphene may be too optimistic in terms of propagating loss when the theoretical estimation is based on the property of bulk graphene material. Besides the tunability, graphene also shows other interesting optical properties such as high nonlinear kerr effect [19] and saturable absorption [20], etc. In this paper a new type of waveguide referred to as the metalloaded graphene surface plasmon (MLGSPP) waveguide, is proposed and numerically investigated. Quite similar to the dielectricloaded surface plasmon polariton waveguide (DLSPPW) [21] in the optical frequencies, this waveguide does not need one to change the original 2D shape of graphene layer or pattern it, thus circumventing the problems of using graphene as the plasmonic materials mentioned above. The modal properties of the proposed waveguide, including mode effective index Re(neff), normalized
B. Xiao et al. / Optics Communications 355 (2015) 602–606
attenuation constant Im(neff) and the coupling length, are numerically calculated.
2. Calculation methods The unique property of graphene lies in its complex conductivity, which consists of both interband and intraband contributions. With random-phase approximation [22], the conductivity of graphene is calculated and investigated as [6]: σ g (ω, μ c , τ, T ) =
ie 2 (ω + i /τ ) π ℏ2 −
∫0
∞
ε(
[
1 (ω +
i /τ )2
∂fd ( − ε ) (ω + i /τ )2
−
]
∫0
∞
ε(
∂fd (ε ) 4 (ε/ℏ)2
∂fd (ε ) ∂ε
−
∂fd ( − ε )
) dε
∂ε
) dε
(1)
If the condition KBT«mc is satisfied, the above equation can be simplified as:
σinter ≈
σintra ≈
⎡ 2 μ − (ω + i/τ )ℏ ⎤ ie2 c ⎥ ln ⎢ 4π ℏ ⎢⎣ 2 μ c + (ω + i/τ )ℏ ⎥⎦
(2)
⎡ μc ⎤ + 2 ln (e−μc / K B T + 1) ⎥ ⎢ ⎦ π ℏ2 (ω + i/τ ) ⎣ KB T
(3)
ie2KB T
In our calculations, the temperature T in the above equations is set as 300 K. The τ is the relaxation time of the electrons, which equals to 0.6 ps [23] through τ ¼1/2Г (Г is scattering rate). mc is chemical potential, which is equal to Fermi level Ef for KBT«mc. The dielectric constant of the graphene [24] is derived by:
ε (ω) = 1 + iσ g /(ε 0 ωh)
(4)
where h is the thickness of graphene and Ɛ0 is the permittivity of free space. With the above equations one can calculate that in visible frequencies, graphene serves as a lossy dielectric, which does not support spp. However, from mid-infrared to THz regime, the intraband contribution dominates the over conductivity, and the real part of graphene permittivity becomes negative. As a result, the graphene behaves like an ultrathin metal film in the optical frequencies, which can support SPs. Using the above conductivity model for graphene and the finite element method (FEM) based mode solver, the eigen mode properties of the MLGSPP waveguides working in the THz regime are calculated and presented in the following sections of the paper.
3. Schematic and modal properties of MLGSPP A schematic of the proposed waveguide is shown in Fig. 1, where a single layer of graphene (thickness of graphene is assumed to be 0.5 nm) is placed on the SiO2 (Ɛ¼3.9Ɛ0) substrate. Similar to DLSPPW [21], a metal (Cu with a conductivity s ¼4 107 S/m is used in this paper) stripe with width w and height
Fig. 1. Schematic of the proposed waveguide, and surface plasmons propagates along the z direction.
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d is laid on the top of the spacer material, which is assumed to be SiO2 in this paper with a thickness t. Besides the fact that the bulk property of graphene is not affected in this waveguide, another advantage of using a full unpatterned graphene layer lies in the ease of fabrication. After the transfer of graphene flake onto the SiO2 substrate and the deposition of the spacer layer, only one step of photolithography and metal evaporation are required to realize the structure after lift-off process. The main purpose of this paper is to comprehensively look into the modal properties of the waveguide, with the emphasis on the investigation of the dependence of the waveguide's property on the geometrical parameters as well as the chemical potential of graphene. To have a tight confinement, the thickness of the spacer layer is chosen to be quite small throughout this paper. A typical distribution of the electric field amplitude |E| for the fundamental TM mode at the frequency of 7 THz is shown in Fig. 2 (a). As can be seen, the electric field is tightly confined in the region of spacer material between the metal stripe and graphene layer. Fig. 2(b) and (c) gives the normalized |E| distribution along x and y direction respectively. One can see that the mode in the y direction is like the metal–insulator–metal (MIM) mode in optical plasmonics while in x direction it has a Gaussian profile. Note that in the THz region, the amplitude of the metal's dielectric constant is extremely large, leading to a near-zero penetration depth of electromagnetic wave into the metal. In this case the metal is similar to perfect electric conductor (PEC) [23]. However, the amplitude of graphene dielectric constant is relatively smaller compared to that of metals in THz regime, making graphene a good plasmonic material in this frequency band. In this context, the mode in the y direction should be referred to as PEC-insulatorplasmonic mode, not MIM mode, as there is only one plasmonic mode at the interface between the central insulator and the surround material. The Gaussian distribution of the mode in the x direction is due to the index-guiding mechanism of the mode confinement in this direction, quite similar to the mode confinement of DLSPPWs. The mode width is defined as the full width where the electrical density decays to 1/e of its peak value. The mode width in the x direction and y direction is 2.4 mm and 0.05 mm respectively, which corresponds to the phenomenon of the effective mode area. These two dimensions correspond to 1/ 125 and 1/6000 of the operation wavelength respectively, demonstrating the high capacity of the waveguide for extreme mode confinement. We numerically investigate the modal properties of MLGSPP waveguide including the mode effective index defined by Re(neff) ¼ β/K0 and the propagation loss is connected with Im(neff) [25,,26] where β is the complex propagation constant of the waveguide mode and K0 ¼2π/λ0 is the free space wave vector. Dependence of Re(neff) and Im(neff) on width w of Cu stripe is shown in Fig. 3(a) as w increases from 0.5 mm to 4 mm when t is fixed as 200 nm. One can see that Re(neff) is quite high in this waveguide, between 8 and 18 in the investigated w range, much larger than the refractive index of any dielectric material involved in the waveguide. This is due to an ultra-small value of the spacer material thickness (200 nm) compared to the wavelength ( 42.8 mm), resulting in an extremely high effective index of the central region of the waveguide [21]. Actually from the effective index method point of view, the cross section of the MLGSPP waveguide can be divided into three regions from the left to the right. The central region composed of Cu/SiO2/graphene supporting the aforementioned PEC-insulator-plasmonic mode is found to have an effective index denoted as neff1 up to a few tens depending on the thickness of the SiO2 layer [23]. The left or right region consisting of air/SiO2/graphene configuration supports the regular SPP mode at the graphene/SiO2 interface, thus has an effective index of neff2. The mode confinement in the x direction is due to the contrast
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Fig. 2. (a) |E| picture of the fundamental TM mode. (b, c) Normalized |E| distribution (b) along x direction and (c) along y direction, where we set w¼2 mm, d ¼0.5 mm, t¼ 200 nm, f¼ 7 THz, mc ¼0.7 eV.
between neff1 and neff2, quite like that in a DLSPPW in the optical frequencies. That is why the mode profile in this direction is Gaussian as shown in Fig. 2(b). Quite similar to the modal confinement mechanism in a regular optical slab waveguide, the real part of the mode effective index of the MLGSPP waveguide, Re(neff), increases as the width w of the central region is larger and approaches neff1. Another mode with higher order will appear as w is above a certain value, as can be seen in Fig. 3(a). To ensure a single mode operation, the width w should be small enough. One can also notice in Fig. 3(a) that as w increases, the imaginary part of the mode effective index of the MLGSPP waveguide for both the 1st and the 2nd order also increases before the high modes appear (for example, before the 2nd order appears, Im(neff) of the 1st order increases), indicating that the propagation loss increases. This is because a larger proportion of the modal power is in the central region which has a higher loss. The modal property of a slab waveguide is affected not only by
the core width, but also by the index of the core material. As is investigated in the literature, the effective index of the central Cu/SiO2/graphene region, neff1, depends significantly on the thickness of the SiO2 layer [23]. As the SiO2 thickness increases, both the real part and the imaginary part of neff1 will decrease. Then the modal property of the MLGSPP waveguide will be affected. In Fig. 3(b), the relation between the Re(neff) and Im(neff) of the MLGSPP waveguide and the thickness of the buffer layer is presented. It is clear that as t increases, both Re(neff) and Im(neff) of the MLGSPP waveguide decrease. As is known, plasmons in graphene depend strongly on the chemical potential mc or Fermi level of the graphene material, so its properties can be tuned by electric gating or chemical doping. This makes THz plasmonic waveguide based on graphene superior to regular plasmonic waveguides which are based on noble metals such as Ag and Au in the optical frequencies. In the following part, the influence of the chemical potential of graphene to the
Fig. 3. (a, b) Dependence of Re(neff) and Im(neff) on (a) width w of metal Cu and (b) thickness t of middle-cladding silica, where we set d¼ 0.5 mm, f ¼7 THz, mc ¼ 0.7 eV obtained by FEM mode solver.
B. Xiao et al. / Optics Communications 355 (2015) 602–606
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Fig. 4. (a, b) Dependence of Re(neff) and Im(neff) on (a) chemical potential mc at different width of Cu stripe and (b) working frequency f, where we set w¼ 1 mm, d ¼0.5 mm, t¼ 200 nm, obtained by FEM mode solver.
performance of the MLGSPPW is numerically analyzed. In our calculations, the geometric parameters are set to be w ¼1 mm, d ¼0.5 mm and t¼200 nm. The results given in Fig. 4(a) show that with the chemical potential increasing, both Re(neff) and Im(neff) decrease, indicating longer propagation distances achieved at the cost of weaker confinement at a larger chemical potential. Unlike the change of copper stripe width, tuning chemical potential mc within the range of 0.3–1.0 eV to change the performance of the waveguide at a given frequency f¼ 7 THz does not produce higher order modes. This is because the tuning of mc will lead to a synchronous change of both neff1 and neff2, as used above in the interpretation of the modal confinement using the effective index method. As a result, the contrast between neff1 and neff2 is mitigated. This conclusion can also be found for other waveguide width values, as can be seen in Fig. 4(a) from the dependence of the real part of effective index of MLGSPPW as a function of Fermi level for waveguide widths of 0.5 mm and 1.5 mm. Note that for w¼ 1.5 mm there are two orders of modes present for all these Fermi levels and no additional mode order appears. Thus it’s beneficial for one to achieve single-mode operation by tuning the chemical potential. Next, the chemical potential of the graphene is fixed at mc ¼ 0.7 eV and the dispersion of the waveguide is studied. As shown in Fig. 4(b), from the operation frequency of 1–10 THz, the real part of the MLGSPPW effective mode index increases while the imaginary part decreases in contrast. This implies a tighter confinement and lower propagation loss at a higher frequency. From Eqs. (1)–(3), one knows that the conductivity of graphene is a function of both working frequency and the chemical potential. Since the mode properties of the MLGSPPW are affected by both the real part and imaginary part of the graphene permittivity, the spectral dependence presented in Fig. 4(b) show that as the frequency increases, both the real and imaginary part of the graphene permittivity become smaller (the absolute value of the real part of the graphene permittivity becomes larger). Besides the modal properties, another factor which is quite important in photonic integration is the cross talk between adjacent waveguides, because it will determine the device packing density. To evaluate the performance of our proposed structure in integrated THz circuit, the crosstalk is characterized by the use of coupling length Lc [26], which can be calculated by Lc ¼ λ0/|Nc Na|, where Nc and Na are the effective mode index of the symmetric and antisymmetric super modes of coupled waveguides respectively, are investigated at the frequency of 7 THz. The waveguide configuration is as shown in Fig. 5, the dependence of the coupling length Lc on the edge-to-edge spacing of the waveguide indicates that crosstalk would be weaker with the increase of waveguide distance. In order to ensure a low crosstalk, Lc must be large enough to make Lc/Lspp»1, which means energy of mode decays to 1/e of its original value before it is coupled to the adjacent waveguide.
Fig. 5. Dependence of coupling length Lc on gap between Cu, where we set same geometric parameter of two Cu, w¼ 1 mm, d¼ 0.5 mm, t¼ 200 nm, f¼ 7 THz, mc ¼0.7 eV, obtained by FEM mode solver.The inset shows the waveguide coupling structure.
In our previous calculation, propagation distance Lspp of metalloaded waveguide is found to be 8.5 mm according to L spp = 1/[2*K0* Im (neff )] [25]. So as long as the edge-to-edge gap is above 0.2 mm, crosstalk will be small enough that it can be almost ignored.
4. Conclusions In conclusion, in this paper we have proposed a novel graphene plasmonic waveguide working in the THz. The modal properties of the proposed waveguide as well as their dependence on the geometrical parameters and the chemical potential of graphene are numerically investigated using finite element method. Compared to other plasmonic waveguides with graphene working as the plasmonic material investigated in the literature, our proposed waveguide doesn’t need any patterning of the graphene layer, thus overcomes the difficulty of fabricating THz device based on a sheet of single-layer graphene while retaining the original material properties of bulk graphene. Different from other waveguides like the spoof surface plasmon waveguide in the THz, graphene based plasmonic waveguide has additional advantage of high tunability. We believe the proposed structure could become an interesting candidate for device miniaturization and photonic integration in the THz regime.
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B. Xiao et al. / Optics Communications 355 (2015) 602–606
Acknowledgments The research work was supported by 2015 Zhejiang Province Public Welfare of International Cooperation Project under Grant no. 2015C34006. The authors also acknowledge financial support from National Natural Science Foundation of China (61377108).
References [1] D.K. Gramotnev, S.I. Bozhevolnyi, Plasmonics beyond the diffraction limit, Nat. Photonics 4 (2010) 83–91. [2] Z. Han, S.I. Bozhevolnyi, Radiation guiding with surface plasmon polaritons, Rep. Prog. Phys. 76 (2013) 016402. [3] T. Jiang, L. Shen, J.-J. Wu, T.-J. Yang, Z. Ruan, L. Ran, Realization of tightly confined channel plasmon polaritons at low frequencies, Appl. Phys. Lett. 99 (2011) 261103. [4] J.B. Pendry, L. Martín-Moreno, F.J. Garcia-Vidal, Mimicking surface plasmons with structured surfaces, Science 305 (2004) 847–848. [5] A. Vakil, N. Engheta, Transformation optics using graphene, Science 332 (2011) 1291–1294. [6] X. Luo, T. Qiu, W. Lu, Z. Ni, Plasmons in graphene: recent progress and applications, Mater. Sci. Eng. R Rep. 74 (2013) 351–376. [7] M. Jablan, M. Soljacic, H. Buljan, Plasmons in graphene: fundamental properties and potential applications, Proc. IEEE 101 (2013) 1689–1704. [8] B. Zhu, G. Ren, S. Zheng, Z. Lin, S. Jian, Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices, Opt. Express 21 (2013) 17089–17096. [9] J.S. Gómez-Díaz, J. Perruisseau-Carrier, Graphene-based plasmonic switches at near infrared frequencies, Opt. Express 21 (2013) 15490–15504. [10] S. He, X. Zhang, Y. He, Graphene nano-ribbon waveguides of record-small mode area and ultra-high effective refractive indices for future VLSI, Opt. Express 21 (2013) 30664–30673. [11] Y. a, F. Nikitin, F.J. Guinea, L. García-Vidal, Martín-Moreno, Edge and waveguide terahertz surface plasmon modes in graphene microribbons, Phys. Rev. B 84
(2011) 161407. [12] X. Zhu, W. Yan, N.A. Mortensen, Bends and splitters in graphene nanoribbon waveguides abstract, Opt. Express 21 (2013) 630–634. [13] J. Christensen, A. Manjavacas, S. Thongrattanasiri, F.H.L. Koppens, F.J.G. de Abajo, Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons, ACS Nano 6 (2012) 431–440. [14] Y. Gao, G. Ren, B. Zhu, J. Wang, S. Jian, Single-mode graphene-coated nanowire plasmonic waveguide, Opt. Lett. 39 (2014) 5909–5912. [15] J. Zhao, X. Liu, W. Qiu, Y. Ma, Y. Huang, J.-X. Wang, et al., Surface-plasmonpolariton whispering-gallery mode analysis of the graphene monolayer coated InGaAs nanowire cavity, Opt. Express 22 (2014) 5754–5761. [16] J. Hu, W. Lu, J. Wang, Highly confined and tunable plasmonic waveguide ring resonator based on graphene nanoribbons, EPL (Europhys. Lett.) 106 (2014) 48002. [17] P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, J. Xu, Surface plasmon modes in graphene wedge and groove waveguides, Opt. Express 21 (2013) 32432–32440. [18] Y. Sun, Z. Zheng, J. Cheng, J. Liu, Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission, Opt. Commun. 328 (2014) 124–128. [19] A. Auditore, C. De Angelis, A. Locatelli, S. Boscolo, M. Midrio, M. Romagnoli, et al., Graphene sustained nonlinear modes in dielectric waveguides, Opt. Lett. 38 (2013) 631–633. [20] Z. Zheng, C. Zhao, S. Lu, Y. Chen, Y. Li, H. Zhang, et al., Microwave and optical saturable absorption in graphene, Opt. Express 20 (2012) 23201. [21] T. Holmgaard, S. Bozhevolnyi, Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides, Phys. Rev. B. 75 (2007) 245405. [22] G.W. Hanson, Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene, J. Appl. Phys. 0 (2007) 1–18. [23] X. Gu, I.-T. Lin, J.-M. Liu, Extremely confined terahertz surface plasmon-polaritons in graphene–metal structures, Appl. Phys. Lett. 103 (2013) 071103. [24] Z. Lu, W. Zhao, Nanoscale electro-optic modulators based on graphene-slot waveguides, J. Opt. Soc. Am. B 29 (2012) 1490. [25] J. Tao, X. Yu, B. Hu, A. Dubrovkin, Q.J. Wang, Graphene-based tunable plasmonic Bragg reflector with a broad bandwidth, Opt. Lett. 39 (2014) 271–274. [26] Y. Bian, Z. Zheng, X. Zhao, J. Zhu, T. Zhou, Symmetric hybrid surface plasmon polariton waveguides for 3D photonic integration, Opt. Express 17 (2009) 21320–21325.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Metal-loaded graphene surface plasmon waveguides working in the terahertz regime Binggang Xiao, Kang Qin, Sanshui Xiao, Zhanghua Han n Centre for Terahertz Research, China Jiliang University, Hangzhou 310018, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 April 2015 Received in revised form 27 June 2015 Accepted 15 July 2015
A metal-loaded graphene surface plasmon waveguide composed of a thin silica layer sandwiched between a graphene layer and a metal stripe is proposed and the waveguiding properties in the THz regime are numerically investigated. The results show that the fundamental mode of the proposed waveguide is tightly confined in the middle silica layer with an acceptable propagation loss. Compared with most other graphene waveguides proposed in the literature, the realization of this waveguide does not need to pattern or deform the graphene layer, thus retaining the superior properties of bulk graphene material. The tight modal confinement and the ease of fabrication suggest the high potential use of this waveguide in high-density THz photonic integration. & 2015 Elsevier B.V. All rights reserved.
Keywords: Plasmonics waveguides graphene
1. Introduction The use of surface plasmons (SPs) for radiation guiding at a scale beyond the diffraction limit has been widely investigated in the optical frequencies [1,,2]. Due to the high level of free electron concentration, noble metals have the plasma frequencies normally in the ultraviolet, making them good plasmonic materials in the optical regime. However, in the lower frequency bands such as Terahertz (THz) or microwave frequencies, where the frequency is far below the plasma frequency of metals, noble metal usually cannot be used to support SPs due to the near-zero skin depth. Although the concept of spoof surface plasmons (SSPs) [3,,4] have been proposed in the THz/Microwave band to support the propagation of artificial surface wave, recently another material of graphene has emerged as a good candidate to support SPs in those frequencies. The energy band structure of single-layer graphene is linear with no band gap [5–7], making graphene possess unique properties which other substances do not have, such as zero effective carrier mass near the Dirac point and tunable carrier density by chemical potential or Fermi level [8]. As a result, the plasma frequency of graphene can be tuned to be in the THz regime and graphene can work as a good plasmonic material. Compared with noble metals, graphene has many advantages, such as extreme confinement in the THz band, and high tunability [9] by electrostatic gating. Actually graphene as a one-atom-thick plasmonic material [1] has achieved wide applications in THz n
Corresponding author. E-mail address: [email protected] (Z. Han).
http://dx.doi.org/10.1016/j.optcom.2015.07.031 0030-4018/& 2015 Elsevier B.V. All rights reserved.
metamaterials and plasmonics [2]. To date various graphene based plasmonic waveguides working in the far infrared or lower frequencies have been proposed, with the graphene in the form of ribbons [10–13], nanowire [14,,15], rings [16], wedge/groove waveguide [17] and graphene on the dielectric [18]. However in most of those proposed structures the two dimensional (2D) material of graphene layer needs to be either deformed or structured, which may not only add much complication to the fabrication process but also deteriorate the original unique properties of bulk graphene because the new boundaries in structured graphene may introduce some defect states into the energy band of graphene. The effect will be more evident at lower frequencies because the period of electromagnetic wave is long enough and the moving electrons in graphene will suffer from additional scattering from the new edges. Many waveguides proposed in the literature based on structured graphene may be too optimistic in terms of propagating loss when the theoretical estimation is based on the property of bulk graphene material. Besides the tunability, graphene also shows other interesting optical properties such as high nonlinear kerr effect [19] and saturable absorption [20], etc. In this paper a new type of waveguide referred to as the metalloaded graphene surface plasmon (MLGSPP) waveguide, is proposed and numerically investigated. Quite similar to the dielectricloaded surface plasmon polariton waveguide (DLSPPW) [21] in the optical frequencies, this waveguide does not need one to change the original 2D shape of graphene layer or pattern it, thus circumventing the problems of using graphene as the plasmonic materials mentioned above. The modal properties of the proposed waveguide, including mode effective index Re(neff), normalized
B. Xiao et al. / Optics Communications 355 (2015) 602–606
attenuation constant Im(neff) and the coupling length, are numerically calculated.
2. Calculation methods The unique property of graphene lies in its complex conductivity, which consists of both interband and intraband contributions. With random-phase approximation [22], the conductivity of graphene is calculated and investigated as [6]: σ g (ω, μ c , τ, T ) =
ie 2 (ω + i /τ ) π ℏ2 −
∫0
∞
ε(
[
1 (ω +
i /τ )2
∂fd ( − ε ) (ω + i /τ )2
−
]
∫0
∞
ε(
∂fd (ε ) 4 (ε/ℏ)2
∂fd (ε ) ∂ε
−
∂fd ( − ε )
) dε
∂ε
) dε
(1)
If the condition KBT«mc is satisfied, the above equation can be simplified as:
σinter ≈
σintra ≈
⎡ 2 μ − (ω + i/τ )ℏ ⎤ ie2 c ⎥ ln ⎢ 4π ℏ ⎢⎣ 2 μ c + (ω + i/τ )ℏ ⎥⎦
(2)
⎡ μc ⎤ + 2 ln (e−μc / K B T + 1) ⎥ ⎢ ⎦ π ℏ2 (ω + i/τ ) ⎣ KB T
(3)
ie2KB T
In our calculations, the temperature T in the above equations is set as 300 K. The τ is the relaxation time of the electrons, which equals to 0.6 ps [23] through τ ¼1/2Г (Г is scattering rate). mc is chemical potential, which is equal to Fermi level Ef for KBT«mc. The dielectric constant of the graphene [24] is derived by:
ε (ω) = 1 + iσ g /(ε 0 ωh)
(4)
where h is the thickness of graphene and Ɛ0 is the permittivity of free space. With the above equations one can calculate that in visible frequencies, graphene serves as a lossy dielectric, which does not support spp. However, from mid-infrared to THz regime, the intraband contribution dominates the over conductivity, and the real part of graphene permittivity becomes negative. As a result, the graphene behaves like an ultrathin metal film in the optical frequencies, which can support SPs. Using the above conductivity model for graphene and the finite element method (FEM) based mode solver, the eigen mode properties of the MLGSPP waveguides working in the THz regime are calculated and presented in the following sections of the paper.
3. Schematic and modal properties of MLGSPP A schematic of the proposed waveguide is shown in Fig. 1, where a single layer of graphene (thickness of graphene is assumed to be 0.5 nm) is placed on the SiO2 (Ɛ¼3.9Ɛ0) substrate. Similar to DLSPPW [21], a metal (Cu with a conductivity s ¼4 107 S/m is used in this paper) stripe with width w and height
Fig. 1. Schematic of the proposed waveguide, and surface plasmons propagates along the z direction.
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d is laid on the top of the spacer material, which is assumed to be SiO2 in this paper with a thickness t. Besides the fact that the bulk property of graphene is not affected in this waveguide, another advantage of using a full unpatterned graphene layer lies in the ease of fabrication. After the transfer of graphene flake onto the SiO2 substrate and the deposition of the spacer layer, only one step of photolithography and metal evaporation are required to realize the structure after lift-off process. The main purpose of this paper is to comprehensively look into the modal properties of the waveguide, with the emphasis on the investigation of the dependence of the waveguide's property on the geometrical parameters as well as the chemical potential of graphene. To have a tight confinement, the thickness of the spacer layer is chosen to be quite small throughout this paper. A typical distribution of the electric field amplitude |E| for the fundamental TM mode at the frequency of 7 THz is shown in Fig. 2 (a). As can be seen, the electric field is tightly confined in the region of spacer material between the metal stripe and graphene layer. Fig. 2(b) and (c) gives the normalized |E| distribution along x and y direction respectively. One can see that the mode in the y direction is like the metal–insulator–metal (MIM) mode in optical plasmonics while in x direction it has a Gaussian profile. Note that in the THz region, the amplitude of the metal's dielectric constant is extremely large, leading to a near-zero penetration depth of electromagnetic wave into the metal. In this case the metal is similar to perfect electric conductor (PEC) [23]. However, the amplitude of graphene dielectric constant is relatively smaller compared to that of metals in THz regime, making graphene a good plasmonic material in this frequency band. In this context, the mode in the y direction should be referred to as PEC-insulatorplasmonic mode, not MIM mode, as there is only one plasmonic mode at the interface between the central insulator and the surround material. The Gaussian distribution of the mode in the x direction is due to the index-guiding mechanism of the mode confinement in this direction, quite similar to the mode confinement of DLSPPWs. The mode width is defined as the full width where the electrical density decays to 1/e of its peak value. The mode width in the x direction and y direction is 2.4 mm and 0.05 mm respectively, which corresponds to the phenomenon of the effective mode area. These two dimensions correspond to 1/ 125 and 1/6000 of the operation wavelength respectively, demonstrating the high capacity of the waveguide for extreme mode confinement. We numerically investigate the modal properties of MLGSPP waveguide including the mode effective index defined by Re(neff) ¼ β/K0 and the propagation loss is connected with Im(neff) [25,,26] where β is the complex propagation constant of the waveguide mode and K0 ¼2π/λ0 is the free space wave vector. Dependence of Re(neff) and Im(neff) on width w of Cu stripe is shown in Fig. 3(a) as w increases from 0.5 mm to 4 mm when t is fixed as 200 nm. One can see that Re(neff) is quite high in this waveguide, between 8 and 18 in the investigated w range, much larger than the refractive index of any dielectric material involved in the waveguide. This is due to an ultra-small value of the spacer material thickness (200 nm) compared to the wavelength ( 42.8 mm), resulting in an extremely high effective index of the central region of the waveguide [21]. Actually from the effective index method point of view, the cross section of the MLGSPP waveguide can be divided into three regions from the left to the right. The central region composed of Cu/SiO2/graphene supporting the aforementioned PEC-insulator-plasmonic mode is found to have an effective index denoted as neff1 up to a few tens depending on the thickness of the SiO2 layer [23]. The left or right region consisting of air/SiO2/graphene configuration supports the regular SPP mode at the graphene/SiO2 interface, thus has an effective index of neff2. The mode confinement in the x direction is due to the contrast
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Fig. 2. (a) |E| picture of the fundamental TM mode. (b, c) Normalized |E| distribution (b) along x direction and (c) along y direction, where we set w¼2 mm, d ¼0.5 mm, t¼ 200 nm, f¼ 7 THz, mc ¼0.7 eV.
between neff1 and neff2, quite like that in a DLSPPW in the optical frequencies. That is why the mode profile in this direction is Gaussian as shown in Fig. 2(b). Quite similar to the modal confinement mechanism in a regular optical slab waveguide, the real part of the mode effective index of the MLGSPP waveguide, Re(neff), increases as the width w of the central region is larger and approaches neff1. Another mode with higher order will appear as w is above a certain value, as can be seen in Fig. 3(a). To ensure a single mode operation, the width w should be small enough. One can also notice in Fig. 3(a) that as w increases, the imaginary part of the mode effective index of the MLGSPP waveguide for both the 1st and the 2nd order also increases before the high modes appear (for example, before the 2nd order appears, Im(neff) of the 1st order increases), indicating that the propagation loss increases. This is because a larger proportion of the modal power is in the central region which has a higher loss. The modal property of a slab waveguide is affected not only by
the core width, but also by the index of the core material. As is investigated in the literature, the effective index of the central Cu/SiO2/graphene region, neff1, depends significantly on the thickness of the SiO2 layer [23]. As the SiO2 thickness increases, both the real part and the imaginary part of neff1 will decrease. Then the modal property of the MLGSPP waveguide will be affected. In Fig. 3(b), the relation between the Re(neff) and Im(neff) of the MLGSPP waveguide and the thickness of the buffer layer is presented. It is clear that as t increases, both Re(neff) and Im(neff) of the MLGSPP waveguide decrease. As is known, plasmons in graphene depend strongly on the chemical potential mc or Fermi level of the graphene material, so its properties can be tuned by electric gating or chemical doping. This makes THz plasmonic waveguide based on graphene superior to regular plasmonic waveguides which are based on noble metals such as Ag and Au in the optical frequencies. In the following part, the influence of the chemical potential of graphene to the
Fig. 3. (a, b) Dependence of Re(neff) and Im(neff) on (a) width w of metal Cu and (b) thickness t of middle-cladding silica, where we set d¼ 0.5 mm, f ¼7 THz, mc ¼ 0.7 eV obtained by FEM mode solver.
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Fig. 4. (a, b) Dependence of Re(neff) and Im(neff) on (a) chemical potential mc at different width of Cu stripe and (b) working frequency f, where we set w¼ 1 mm, d ¼0.5 mm, t¼ 200 nm, obtained by FEM mode solver.
performance of the MLGSPPW is numerically analyzed. In our calculations, the geometric parameters are set to be w ¼1 mm, d ¼0.5 mm and t¼200 nm. The results given in Fig. 4(a) show that with the chemical potential increasing, both Re(neff) and Im(neff) decrease, indicating longer propagation distances achieved at the cost of weaker confinement at a larger chemical potential. Unlike the change of copper stripe width, tuning chemical potential mc within the range of 0.3–1.0 eV to change the performance of the waveguide at a given frequency f¼ 7 THz does not produce higher order modes. This is because the tuning of mc will lead to a synchronous change of both neff1 and neff2, as used above in the interpretation of the modal confinement using the effective index method. As a result, the contrast between neff1 and neff2 is mitigated. This conclusion can also be found for other waveguide width values, as can be seen in Fig. 4(a) from the dependence of the real part of effective index of MLGSPPW as a function of Fermi level for waveguide widths of 0.5 mm and 1.5 mm. Note that for w¼ 1.5 mm there are two orders of modes present for all these Fermi levels and no additional mode order appears. Thus it’s beneficial for one to achieve single-mode operation by tuning the chemical potential. Next, the chemical potential of the graphene is fixed at mc ¼ 0.7 eV and the dispersion of the waveguide is studied. As shown in Fig. 4(b), from the operation frequency of 1–10 THz, the real part of the MLGSPPW effective mode index increases while the imaginary part decreases in contrast. This implies a tighter confinement and lower propagation loss at a higher frequency. From Eqs. (1)–(3), one knows that the conductivity of graphene is a function of both working frequency and the chemical potential. Since the mode properties of the MLGSPPW are affected by both the real part and imaginary part of the graphene permittivity, the spectral dependence presented in Fig. 4(b) show that as the frequency increases, both the real and imaginary part of the graphene permittivity become smaller (the absolute value of the real part of the graphene permittivity becomes larger). Besides the modal properties, another factor which is quite important in photonic integration is the cross talk between adjacent waveguides, because it will determine the device packing density. To evaluate the performance of our proposed structure in integrated THz circuit, the crosstalk is characterized by the use of coupling length Lc [26], which can be calculated by Lc ¼ λ0/|Nc Na|, where Nc and Na are the effective mode index of the symmetric and antisymmetric super modes of coupled waveguides respectively, are investigated at the frequency of 7 THz. The waveguide configuration is as shown in Fig. 5, the dependence of the coupling length Lc on the edge-to-edge spacing of the waveguide indicates that crosstalk would be weaker with the increase of waveguide distance. In order to ensure a low crosstalk, Lc must be large enough to make Lc/Lspp»1, which means energy of mode decays to 1/e of its original value before it is coupled to the adjacent waveguide.
Fig. 5. Dependence of coupling length Lc on gap between Cu, where we set same geometric parameter of two Cu, w¼ 1 mm, d¼ 0.5 mm, t¼ 200 nm, f¼ 7 THz, mc ¼0.7 eV, obtained by FEM mode solver.The inset shows the waveguide coupling structure.
In our previous calculation, propagation distance Lspp of metalloaded waveguide is found to be 8.5 mm according to L spp = 1/[2*K0* Im (neff )] [25]. So as long as the edge-to-edge gap is above 0.2 mm, crosstalk will be small enough that it can be almost ignored.
4. Conclusions In conclusion, in this paper we have proposed a novel graphene plasmonic waveguide working in the THz. The modal properties of the proposed waveguide as well as their dependence on the geometrical parameters and the chemical potential of graphene are numerically investigated using finite element method. Compared to other plasmonic waveguides with graphene working as the plasmonic material investigated in the literature, our proposed waveguide doesn’t need any patterning of the graphene layer, thus overcomes the difficulty of fabricating THz device based on a sheet of single-layer graphene while retaining the original material properties of bulk graphene. Different from other waveguides like the spoof surface plasmon waveguide in the THz, graphene based plasmonic waveguide has additional advantage of high tunability. We believe the proposed structure could become an interesting candidate for device miniaturization and photonic integration in the THz regime.
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Acknowledgments The research work was supported by 2015 Zhejiang Province Public Welfare of International Cooperation Project under Grant no. 2015C34006. The authors also acknowledge financial support from National Natural Science Foundation of China (61377108).
References [1] D.K. Gramotnev, S.I. Bozhevolnyi, Plasmonics beyond the diffraction limit, Nat. Photonics 4 (2010) 83–91. [2] Z. Han, S.I. Bozhevolnyi, Radiation guiding with surface plasmon polaritons, Rep. Prog. Phys. 76 (2013) 016402. [3] T. Jiang, L. Shen, J.-J. Wu, T.-J. Yang, Z. Ruan, L. Ran, Realization of tightly confined channel plasmon polaritons at low frequencies, Appl. Phys. Lett. 99 (2011) 261103. [4] J.B. Pendry, L. Martín-Moreno, F.J. Garcia-Vidal, Mimicking surface plasmons with structured surfaces, Science 305 (2004) 847–848. [5] A. Vakil, N. Engheta, Transformation optics using graphene, Science 332 (2011) 1291–1294. [6] X. Luo, T. Qiu, W. Lu, Z. Ni, Plasmons in graphene: recent progress and applications, Mater. Sci. Eng. R Rep. 74 (2013) 351–376. [7] M. Jablan, M. Soljacic, H. Buljan, Plasmons in graphene: fundamental properties and potential applications, Proc. IEEE 101 (2013) 1689–1704. [8] B. Zhu, G. Ren, S. Zheng, Z. Lin, S. Jian, Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices, Opt. Express 21 (2013) 17089–17096. [9] J.S. Gómez-Díaz, J. Perruisseau-Carrier, Graphene-based plasmonic switches at near infrared frequencies, Opt. Express 21 (2013) 15490–15504. [10] S. He, X. Zhang, Y. He, Graphene nano-ribbon waveguides of record-small mode area and ultra-high effective refractive indices for future VLSI, Opt. Express 21 (2013) 30664–30673. [11] Y. a, F. Nikitin, F.J. Guinea, L. García-Vidal, Martín-Moreno, Edge and waveguide terahertz surface plasmon modes in graphene microribbons, Phys. Rev. B 84
(2011) 161407. [12] X. Zhu, W. Yan, N.A. Mortensen, Bends and splitters in graphene nanoribbon waveguides abstract, Opt. Express 21 (2013) 630–634. [13] J. Christensen, A. Manjavacas, S. Thongrattanasiri, F.H.L. Koppens, F.J.G. de Abajo, Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons, ACS Nano 6 (2012) 431–440. [14] Y. Gao, G. Ren, B. Zhu, J. Wang, S. Jian, Single-mode graphene-coated nanowire plasmonic waveguide, Opt. Lett. 39 (2014) 5909–5912. [15] J. Zhao, X. Liu, W. Qiu, Y. Ma, Y. Huang, J.-X. Wang, et al., Surface-plasmonpolariton whispering-gallery mode analysis of the graphene monolayer coated InGaAs nanowire cavity, Opt. Express 22 (2014) 5754–5761. [16] J. Hu, W. Lu, J. Wang, Highly confined and tunable plasmonic waveguide ring resonator based on graphene nanoribbons, EPL (Europhys. Lett.) 106 (2014) 48002. [17] P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, J. Xu, Surface plasmon modes in graphene wedge and groove waveguides, Opt. Express 21 (2013) 32432–32440. [18] Y. Sun, Z. Zheng, J. Cheng, J. Liu, Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission, Opt. Commun. 328 (2014) 124–128. [19] A. Auditore, C. De Angelis, A. Locatelli, S. Boscolo, M. Midrio, M. Romagnoli, et al., Graphene sustained nonlinear modes in dielectric waveguides, Opt. Lett. 38 (2013) 631–633. [20] Z. Zheng, C. Zhao, S. Lu, Y. Chen, Y. Li, H. Zhang, et al., Microwave and optical saturable absorption in graphene, Opt. Express 20 (2012) 23201. [21] T. Holmgaard, S. Bozhevolnyi, Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides, Phys. Rev. B. 75 (2007) 245405. [22] G.W. Hanson, Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene, J. Appl. Phys. 0 (2007) 1–18. [23] X. Gu, I.-T. Lin, J.-M. Liu, Extremely confined terahertz surface plasmon-polaritons in graphene–metal structures, Appl. Phys. Lett. 103 (2013) 071103. [24] Z. Lu, W. Zhao, Nanoscale electro-optic modulators based on graphene-slot waveguides, J. Opt. Soc. Am. B 29 (2012) 1490. [25] J. Tao, X. Yu, B. Hu, A. Dubrovkin, Q.J. Wang, Graphene-based tunable plasmonic Bragg reflector with a broad bandwidth, Opt. Lett. 39 (2014) 271–274. [26] Y. Bian, Z. Zheng, X. Zhao, J. Zhu, T. Zhou, Symmetric hybrid surface plasmon polariton waveguides for 3D photonic integration, Opt. Express 17 (2009) 21320–21325.