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A refractive index from negative to positive of graphene plasmonic crystal at the Dirac-like cone in mid-infrared region
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
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Photonics and Nanostructures - Fundamentals and Applications journal homepage: www.elsevier.com/locate/photonics
A refractive index from negative to positive of graphene plasmonic crystal at the Dirac-like cone in mid-infrared region
T
⁎
Zeyu Wanga, Weibin Qiua, , Junbo Rena, Zhili Lina, Pingping Qiub,c, Qiang Kanb,c a
Fujian Key Laboratory of Light Propagation and Transformation, College of Information Science and Engineering, Huaqiao University, Xiamen, 361021, China College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing, 100086, China c Institute of Semiconductors, Chinese Academy of Sciences, Beijing, 100086, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Graphene plasmon Negative refractive index Plasmonic crystals Dirac-like cone
In this work, two-dimensional plasmonic crystals composed of graphene nanodisks where the chemical potential varies in triangular symmetry are proposed. A special band structure of Dirac-like cone is obtained at the center of Brillouin zone in mid-infrared region. A refractive index of the plasmons from negative to positive is calculated by the method of eigen-fields averaging. Also, the isotropic distribution of the negative/positive effective permeability and permittivity is numerically demonstrated. In addition, the significant propagation characteristics including the reverse Snell’s law effect, deep-sub-wavelength focusing and negative Goos-Hänchen shifts are numerical simulated. The proposed structure may find broad applications in the fields of on-chip plasmonic interconnect, high density plasmonic integrated circuit techniques and nano-imaging in the future.
1. Introduction Negative refractive index(NRI) materials were originally proposed by Veselago in double-negative (DNG) materials [1], which reverse the directions of the phase and group velocities during the propagation of electromagnetic (EM) waves. Further on, NRI materials achieved series of significant applications, such as superfocusing, negative Goos-Hänchen shifts, cloaking [2–5] due to the extraordinary propagation nature. In the early stages, NRI medium based on surface plasmon polaritons (SPP) hosted by the noble metals sparked the extensive interest. The literatures were concentrated on the composite NRI media by embedding arrays of metallic split-ring resonators, bulk fishnet and doublefishnet (DF) structures [6–8]. However, such noble metal based structures have difficulty in reducing dissipative losses to prolong the propagation distance of SPP. Also, the working frequency is determined once the geometry structures of the devices are defined due to the definite dispersion function of the noble metal. All the obstacles constrain the functionality and applications of the NRI materials. Graphene, a monolayer of carbon sheet where the atoms are arranged in a two-dimensional (2-D) honeycomb lattice, has been widely studied as a promising material for its superior properties [9]. Comparing with the SPPs at a metal-dielectric interface, graphene supported SPP has superiority in field localization and lower loss [10–12]. Besides, the most appealing property is the frequency tunability, whereas the conductivity of graphene can be tuned dynamically by changing the ⁎
chemical potential from terahertz (THz) to mid-infrared. Graphene has been introduced to compose the negative refractive index materials recently. For example, a NRI media constructed by graphene-dielectric multilayered structure is proposed by K. V. Sreekanth [13]. However, the multilayered structure is inconvenient for on-chip integration, where another problem raised here is the anisotropic refractive index distribution. Hence, an in-plane, isotropic negative refractive index distribution is highly desired. It is well known that a peculiar band structure called ‘Dirac cone’ of plasmonic crystal of graphene plasmonic crystal(GPC) shows the distinct superiority in implementing the isotropic distribution of effective permittivity and permeability in plane, which offers the convenient for on-chip integration [14–16]. Thus, combining concepts of graphene hosted plasmon and the Dirac cone band structure is the practical way to get the in-plane and isotropic negative refractive index material. In this article, we design a graphene plasmonic crystal structure whose effective electromagnetic parameters evolve from negative to positive continuously, which is beneficial to achieve two-dimensional on-chip integration. In our work, the graphene nanodisks are arranged in a triangular lattice which is constructed by periodically modulating the chemical potential of grapheme [17]. Further on, we study the dispersion relation for the transverse magnetic SPP modes and the band structure of the GPC. The interaction between a monopolar mode and two dipolar modes results in a three-fold accidentally degenerate state called Dirac-like cone dispersion. The connection between the
Corresponding author. E-mail address: [email protected] (W. Qiu).
https://doi.org/10.1016/j.photonics.2019.100745 Received 18 October 2018; Received in revised form 10 June 2019; Accepted 26 September 2019 Available online 27 September 2019 1569-4410/ © 2019 Elsevier B.V. All rights reserved.
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
Z. Wang, et al.
Fig. 1. (a) Schematic diagram of the surface of array of graphene nanodisks arranged in triangular lattice. (b) The single layer of graphene sits on a silica layer on silicon substrate. The silica layer with periodic thickness d1 and d2 corresponding to periodic chemical potentials μc1 and μc2 under an external gate voltage.
Fig. 2. (a) The diagram of the unit cell of graphene plasmonic crystals. (b) The reduced Brillouin zone with the path of Γ-M-K-Γ. (c) The band structure of GPC with r = 0.2517a, P is the three-fold degeneracy point. (d), (e)Three-dimensional dispersion surfaces near the point P in k-space when r = 0.2517a and r = 0.4a, respectively. (f) The band structures when r = 0.248a. The electric field distributions profiles of the dipole (f) and monopole (g–h) interact at point P when the frequency is 70.827 THz.
isofrequency contours show an anisotropic characteristic when the three-fold degenerate splits into doubly degenerate point of dipolar modes and a single mode of monopole. Moreover, fantastic electromagnetic properties including reverse Snell’s law, negative Goos-
electromagnetic parameters and frequency is revealed by using the method of eigen-field averaging. The effective electromagnetic parameters from negative to positive continuously are demonstrated respectively in two isotropic behaviors frequency ranges. However, the 2
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
Z. Wang, et al.
Fig. 3. (a) The effective permittivity and permeability of the GPC obtained by the field averaging theory. (b–e) The isofrequency contours of the TM2 and TM4 when the r = 0.2517a and r = 0.4a, respectively.
Hänchen shifts, and superfocusing are investigated. The proposed structure has significant potential applications in the fields of on-chip integration and nano-imaging technologies.
and μc2 ) of the graphene can be realized by periodically modulating the thickness(d1 and d2 ) of the SiO2 layer under an external gate voltage [18,19]. And r is the radius of the Si rod which corresponds to the radius of graphene nanodisk. As shown in Fig. 1(b), the connection between with silica layer thicknesses of the graphene regions and the chemical potentials is μc1 / μc 2 = (d2/ d1)1/2 [17]. The dispersion relation for transverse magnetic (TM) polarized SPP modes in two dimensional GPC can be deduced by solving Maxwell’s equations with boundary conditions [20]
2. Band structure and the effective parameters of graphene In our work, a single layer of graphene sits on a silica layer on silicon substrate. The two-dimensional plasmonic crystal is composed of an array of graphene nanodisks arranged in triangular lattice which is displayed in Fig. 1(a). Specifically, the variant chemical potential( μc1 3
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
Z. Wang, et al.
Fig. 4. Electric field distribution of the propagation the GPC: (a) The schematic figure of the triangle formed by the graphene plasmonic lattice with a chemical potential μc1 = 0.4eV at the nanodisks and μc2 = 0.7eV at the large area. The whole triangle structure is surrounded by the graphene sheet with μc2 = 0.7eV , and the refraction angle α is equal to the angle θ. The transmission characteristics of the beams are plotted in (b–e) with r = 0.2517a, when the frequency is 73.2, 70.0827, 67 and 65.5 THz, respectively. (f) The electric field distribution of the beam when r = 0.4a.
εAir β2 − k 0 εAir
+
εsio2 β2 − k 0 εsio2
=
σg iϖε0
graphene layer can be calculated by the equation neff = β / k 0 . The unit cell of the graphene plasmonic crystals is depicted in Fig. 2(a), where the graphene nanodisks with μc1 = 0.4eV are surrounded by the graphene with μc2 = 0.7eV . a and r are the lattice constant, the radius of the graphene nanodisk, respectively. The band structures and the propagation properties are numerically studied by the commercial software COMSOL Multiphysics, RF module Ver.5.2a based on finite-element method (FEM). The Γ-M-K-Γ path of the reduced Brillouin zone (BZ) is displayed in Fig. 2(b). Fig. 2(c) shows the corresponding band structures of the triangular lattice when a is 45 nm and r is 0.2517a. It is distinct to find that TM2 , TM3 and TM4 bands intersect at the center of BZ (Γ point) and form a triply degenerate state marked with P. The three-dimensional dispersion surfaces of conical behavior near the P point in k-space are exhibited in Fig. 2(d). Further on, when the radius of graphene nanodisk shifts to 4a, the Dirac-like cone structure opens a gap in Fig. 2(f). On the one hand, the Dirac-like cone is a triply accidental degenerate behavior because the band structures are constructed by the definite structural parameters. On the other hand, the electric field profiles at triply degeneracy point are formed by a monopole mode (Fig. 2(g)) and dipole modes (Fig. 2(h) and (i)) interaction. Therefore, this triply degenerate state is a Dirac-like point at the frequency of 70.827 THz. The effective permittivity and permeability are obtained by the method of eigen-field averaging [22].
(1)
Here εAir and ε sio2 are the relative permittivity of air and silica, respectively. k 0 is the wave number in the free space, and ε0 is the vacuum permittivity. In particular, the surface conductivity of graphene σg is composed of the intraband electron-photon scattering σintra and the inter-band electron transitions σinter , which are determined by the Kubo’s formula [21].
σg = σintra + σinter σintra =
σinter =
β = ε0
(2)
ie 2kB T ⎡ μc + 2 ln ⎛1 + exp(− μc )⎞ ⎤ ⎜ π ℏ2 (ω + i/ τ ) ⎢ kB T ⎠ ⎥ ⎝ ⎦ ⎣ KB T
(3)
ie 2 ⎡ 2 |μc | − ℏ(ω + i/ τ ) ⎤ ⎥ 4π ℏ2 (ω + i/ τ ) ⎢ ⎣ 2 |μc | + ℏ(ω + i/ τ ) ⎦
(4)
⎟
εAir + εsio2 2iω 2 σg
(5)
where e, kB and ω are the electron charge, Boltzmann constant and the angular frequency of the plasmonic crystal, respectively. T is the temperature, ℏ is the reduced Planck constant. μc and τ represent the chemical potential and the electron momentum relaxation time, respectively. β is the propagation constant of SPPs based on graphene layer and the effective refractive index neff for the SPP modes on 4
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
Z. Wang, et al.
Fig. 5. (a) The energy flux density intensity distribution of the wave propagating through the GPC slab with a negative refraction when f =62 THz. (b) Electric field distribution of Gaussian beams transmit with a negative Goos-Hänchen shift Δx1 when f =67.5 THz. (c) Δx1 as a function of the frequency when θ = 45°.
ψi(av) =
∫ unitcellψi(k ) (x , y ) ∂xdxdy (av )
ε eff =
∂Ez 1 j ωHx(av) ∂x
μ eff =
∂Ez 1 j ωHx(av) ∂x
=
k Hx(av) ω Ez(av)
(7)
=
(av ) k Ez ω Hx(av)
(8)
(av )
ψi(av)
Ei
The GPC is designed in a triangular structure where the graphene nanodisks with μc1 = 0.4eV are surrounded by the graphene with μc2 = 0.7eV , r = 0.2517a. In our simulation, variant frequencies of TM polarized plane wave propagate from the b ottom of the triangle. When the frequency is 73.2 THz(PRI frequency range), the refraction beam and incidence beam emerge on the different side of the normal line in Fig. 4(b), which confirms the regular Snell’s law. The refraction angle θ is zero at the Dirac-like point frequency (70.827 THz) in Fig. 4(c), The phenomenon of optical beam shaping confirms the zero refractive index characteristics of GPC [23]. In particular, as shown in Fig. 4(d–f), the refraction and incidence wave emerges on the same side of the normal line when frequencies respectively are 67 THz and 65.5 THz (NRI frequency range) in Fig. 4(d–e). This especial propagation phenomenon is called Inverse Snell’s law which verifies the negative refractive index of GPC. Further on, when the frequency is close to 70.827 THz, the angle θ approaches to zero, which confirms the values of the effective permittivity and permeability in Fig. 4(a). In addition, the refraction waves experience different refractive indices in different directions when the r shifts to 0.4a and frequency is 62.6 THz. This phenomenon relates to Fig. 4(d) where the isofrequency contours is no longer a circular shape in this specific frequency ranges. Thus, the GPC exhibits an anisotropy characteristic when the structural parameters are changed. Hence, the Dirac-like cone is crucial for the isotropy of the GPC.
(6)
A
Hi
where = , , is the averaging eigen-fields in the i direction, i=x, z, and A is the area of the unit cell. ε eff and μ eff are effective permittivity and permeability of GPC, respectively. Through our calculation, the relationship between the effective permittivity (permeability) and the frequency is shown in Fig. 3(a). It should be noted that the effective permittivity and permeability increase from the negative to positive monotonously with the frequency increasing. In particular, the permittivity and permeability reach zero when the frequency is Dirac-like point frequency (70.827 THz). Thus, the GPC exhibits a zero refractive index(ZRI) at 70.827 THz. Moreover, by analyzing the isofrequency contours of TM2 and TM4 bands in Fig. 3(b) and (c), it is clear to find that the contours maintain a nearly circular shape which means the GPC can be essentially regarded as an isotropic material in two frequency ranges. These two frequency ranges are 66.49–70.827 THz and 70.827–73.31 THz which can be defined as negative refractive index frequency range and positive refractive index(PRI) frequency range, respectively. However, when the radius of graphene nanodisk pillar becomes 0.4a, the isofrequency contours of TM2 and TM4 bands are displayed in Fig. 3(e) and (f). One can find that the isofrequency contours lose the circular profiles when the nature of three-fold accidental degeneracy is broken. Hence, Dirac-like cone is crucial in constructing an isotropic GPC in these frequency ranges.
4. Negative Goos-Hänchen shift in the proposed structure Goos-Hänchen shift is an intriguing optical phenomenon in which a physical beam experiences a lateral shift from the position predicted by geometrical optics under a totally reflection condition. This deviation was originally studied by Goos and Hänchen so-called Goos-Hänchen shift and theoretically analyzed by Artmann in the late 1940s [24,25]. Since then, such shift effect has been demonstrated to be a universal phenomenon, which can take place at the interface between the NRI and PRI [26]. Especially some studies have been concentrated on the shift in an isotropic NRI flat which has extensive applications including slow light and optical sensors [27,28]. To get insight into the negative Goos-Hänchen shift, a Gaussian wave is incident to the GPC flat a slab which is constructed by the proposed GPC structure with different
3. Inverse Snell's law The most symbolic characteristic of NRI is the inverse Snell's law, where the propagation direction is inversed but the Maxwell's equations is still satisfied. Hence, to further verify the NRI transmission of graphene plasmonic crystals, the proposed structure is shown in Fig. 4(a). 5
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
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frequency and incident angle. And the energy flux density intensity distribution of the transmission of the wave is displayed in Fig. 5(a). The transmission of the Gaussian beam displays a negative refraction characteristic when the frequency is 62 THz and the angle of incidence θ is 20°. The phenomenon verifies the NRI of the GHC again. Contrarily, the electric field distribution of the totally reflection occurs at 45°, as displayed in Fig. 5(b). The negative Goos-Hänchen shift Δx1 presents at the interface when the frequency f is 67.5 THz. The length of Δx1 approachs 0.041λ which is much less than the wavelength of incident wave. Further on, the Δx1 goes up as increasing frequency in Fig. 5(c), and it flattens out when the frequency reaches ZRI frequency (70.827 THz).
effect is shown in Fig. 6(b), it is evident that the incident beams are focused into a spot by the GPC lens. Besides, the normalized intensity distribution of the image is displayed in Fig. 6(c) where FWHM is 9.18 × 10−3λ . The FWHM value far less than the wave length of the point source which breaks through the diffraction limit so-called deepsub-wavelength focusing. Although GPC flat lens shows the superiority in focusing the point source, there is limitation in plane wave imaging. So, a planoconcave lens is designed to further overcome the restriction, which is constructed by cutting the exit surface of GPC slab into a triangle as illustrated in Fig. 7(a). A spot is focused by the GPC planoconcave lens on the cutting side of the lens in Fig. 7(b) when the frequency of plane wave is 68.9 THz. Moreover, as shown in Fig. 7(c), we also plot the intensity distribution whose FWHM along the y-axis approaches 9. 79 × 10−3λ . In addition, to verify superiority of NRI on subwavelength imaging, we simulate the imaging effect and calculate the FWHM value with the frequency of the PRI range and Dirac-like frequency (70.827 THz), respectively. The size of the imaging spot, loss and FWHM is much larger than the NRI frequency range. Hence, this structure in negative refractive index frequency range has better focusing effect. Actually, when the radius of graphene nanodisk is one atom more or less, the reduced degenerate Dirac-like cones at Γ point open negligible gaps (0.28 THz for one more atom, and 0.231Thz for one atom less). There are two zero refractive index points and two discontinuity points correspond to two zero slope points of the bands and the gap of Dirac points. However, the relationship between the frequency and effective permittivity (permeability) still maintain a linear relationship in negative and positive frequency ranges for interested (66.49–70.827 THz for negative refractive index and 70.827–73.31 THz for positive refractive index (PRI) frequency range).
5. Deep-sub-wavelength focusing Current research in nanotechnology requires the images with a resolution in deep nanometer scale. To break the diffraction limit, the idea of SPP was proposed to manipulate wave [29,30]. Further on, a negative refraction flat lens of silver basing on SPP has been proposed by Pendry to achieve super-resolution [31]. But the silver flat lens also has limitations on the minimum size of metallic nanostructure. For further breakthrough the nano-image resolution, we design the perfect lens basing on GPC flat structure. The schematic diagram of the NRI lens is illustrated in Fig. 6(a). A TM polarized point source is placed above the GPC lens. D is the distance between the point source and upper of the GPC lens. The beam of point source p1 is focused into in p2 through the GPC flat lens because of the negative refraction. Moreover, we should note that the perfect lens also focuses the evanescent wave which transmits the deep-sub-wavelength details of the source. We adjust different structure parameters to obtain the optimal focusing effect of the image. On the one hand, the full width at half maximum (FWHM) of the image is very crucial to the resolution of the image. Thus, we plot the FWHM of the image spot along the x-axis as a function of D, the layer number of GPC N and the frequency f, respectively in Fig. 6(d–f). On the other hand, The energy loss during the imaging process is inevitable due to the impedance mismatch. There is a tradeoff between the resolution and the energy loss. Particularly, the FWHM essentially decreases continuously as frequency increasing. However, the energy loss sharply increases when frequency exceeds 68 THz. Combining all the factors, we find the optimal parameters as D = 0.172um, N = 3 and F = 68 THz. The energy flux density of image
6. Conclusions In conclusion, Dirac-like conical dispersion of the two-dimensional GPC structure composed of graphene nanodisks are designed to obtain the NRI and PRI medium. A three-fold accidental degeneracy conical dispersion at the center of BZ called Dirac-like cone has been got, and the band structure of the GPC has been investigated numerically. The frequency of Dirac-like point is 70.287 THz when the lattice constant and radius of nanodisk is 45 nm and 0.2517a, respectively. Moreover,
Fig. 6. (a) Structure of the GPC flat lens. (b) The energy flux density of the nano-image point is focused by the GPC lens with the optimal structure parameters. (c) The normalized intensity distribution of the nano-image along the x-axis the FWHM is 9.18 × 10−3λ . (d–e) The FWHM as a function of D, N and F. 6
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
Z. Wang, et al.
Fig. 7. (a) Schematic diagram of the GPC planoconcave lens. (b) The intensity distributions of electric field for a plane wave with 68.9 THz. (c) Normalized intensity distribution of the plane wave whose FWHM value is 9.79 × 10−3λ .
the negative permittivity and permeability of the GPC are calculated by the theory of the eigen-field averaging. The distributions of isotropic ranges of electromagnetic parameters are obtained by isofrequency contours and the NRI frequency range is 66.49–70.827 THz. However, the isotropy is fading away when the radius of the graphene nanodisk are shifted gradually. Moreover, the propagation of GPC in the isotropic range is simulated to verify Inverse Snell’s law and negative GoosHänchen shifts. In addition, perfect lens is constructed by negative refractive index GPC which is significant in deep-sub-wavelength focusing field. The proposed GPC structure has enormous potential significance in the on-chip interconnect and nanotechnology.
coated InGaAs nanowire cavity, Opt. Express 22 (2014) 5754. [11] L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H.A. Bechtel, X. Liang, A. Zettl, Y.R. Shen, F. Wang, Graphene plasmonics for tunable terahertz metamaterials, Nat. Nanotechnol. 6 (2011) 630–634. [12] W. Qiu, X. Liu, J. Zhao, S. He, Y. Ma, J.-X. Wang, J. Pan, Nanofocusing of midinfrared electromagnetic waves on graphene monolayer, Appl. Phys. Lett. 104 (2014) 041109. [13] K.V. Sreekanth, A. De Luca, G. Strangi, Negative refraction in graphene-based hyperbolic metamaterials, Appl. Phys. Lett. 103 (2013) 023107. [14] G.-H. Lee, G.-H. Park, H.-J. Lee, Observation of negative refraction of Dirac fermions in graphene, Nat. Phys. 11 (2015) 925–929. [15] Y. Li, S. Kita, P. Muñoz, O. Reshef, D.I. Vulis, M. Yin, M. Lončar, E. Mazur, On-chip zero-index metamaterials, Nat. Photonics 9 (2015) 738–742. [16] P. Qiu, W. Qiu, Z. Lin, H. Chen, J. Ren, J.-X. Wang, Q. Kan, J.-Q. Pan, Double Dirac point in a photonic graphene, J. Phys. D Appl. Phys. 50 (2017) 335101. [17] W. Qiu, P. Qiu, J. Ren, Z. Lin, J.-X. Wang, Q. Kan, J.-Q. Pan, Realization of conical dispersion and zero-refractive-index in graphene plasmonic crystal, Opt. Express 25 (2017) 33350. [18] B. Shi, W. Cai, X. Zhang, Y. Xiang, Y. Zhan, J. Geng, M. Ren, J. Xu, Tunable bandstop filters for graphene plasmons based on periodically modulated graphene, Sci. Rep. 6 (2016). [19] Z. Fei, A.S. Rodin, G.O. Andreev, W. Bao, A.S. McLeod, M. Wagner, L.M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M.M. Fogler, A.H.C. Neto, C.N. Lau, F. Keilmann, D.N. Basov, Gate-tuning of graphene plasmons revealed by infrared nano-imaging, Nature 487 (2012) 82–85. [20] H.B.M. Jablan, M. Soljačić, Plasmonics in graphene at infrared frequencies, Phys. Rev. B 80 (2009). [21] H. Lu, C. Zeng, Q. Zhang, X. Liu, M.M. Hossain, P. Reineck, M. Gu, Graphene-based active slow surface plasmon polaritons, Sci. Rep. 5 (2015) 8443. [22] Z.L. Lu, D.W. Prather, Calculation of effective permittivity, permeability, and surface impedance of negative-refraction photonic crystals, Opt. Express 15 (2007) 8340–8345. [23] J.W. Dong, M.L. Chang, X.Q. Huang, Z.H. Hang, Z.C. Zhong, W.J. Chen, Z.Y. Huang, C.T. Chan, Conical dispersion and effective zero refractive index in photonic quasicrystals, Phys. Rev. Lett. 114 (2015). [24] H.H.F. Goos, Ein neuer und fundamentaler Versuch zur Totalreflexion, Ann. Phys. 436 (1947) 333–346. [25] K. Artmann, Berechnung der Seitenversetzung des totalreflektierten Strahles, Ann. Phys. 87 (1948). [26] P.R. Berman, Goos-Hanchen shift in negatively refractive media, Phys. Rev. 66 (2002) 067603. [27] K.L. Tsakmakidis, A.D. Boardman, O. Hess, ’Trapped rainbow’ storage of light in metamaterials, Nature 450 (2007) 397–401. [28] X. Yin, L. Hesselink, Goos-Hänchen shift surface plasmon resonance sensor, Appl. Phys. Lett. 89 (2006) 261108. [29] S. Kawata, Y. Inouye, P. Verma, Plasmonics for near-field nano-imaging and superlensing, Nat. Photonics 3 (2009) 388–394. [30] V. Shalaev, Optical Negative-Index Metamaterials Nature 1 (2007). [31] J.B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85 (2000) 3966–3969.
Acknowledgements The authors are grateful to the support by the Natural Science Fund of China under grant No. 11774103, and Project for Cultivating Postgraduates’ Innovative Ability in Scientific Research of Huaqiao University (17013082023) References [1] V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ, Sov. Phys. Usp. 10 (1968) 509–515. [2] I.I. Smolyaninov, Y.-J. Hung, C.C. Davis, Imaging and focusing properties of plasmonic metamaterial devices, Phys. Rev. B 76 (2007). [3] X. Chen, X.-J. Lu, Y. Ban, C.-F. Li, Electronic analogy of the Goos–Hänchen effect: a review, J. Opt. 15 (2013) 033001. [4] S. Foteinopoulou, C.M. Soukoulis, Negative refraction and left-handed behavior in two-dimensional photonic crystals, Phys. Rev. B 67 (2003). [5] P. Kolinko, D.R. Smith, Numerical study of electromagnetic waves interacting with negative index materials, Opt. Express 11 (2003) 640–648. [6] A. Mary, S.G. Rodrigo, F.J. Garcia-Vidal, L. Martin-Moreno, Theory of negativerefractive-index response of double-fishnet structures, Phys. Rev. Lett. 101 (2008) 103902. [7] R. Marqués, J. Martel, F. Mesa, F. Medina, Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides, Phys. Rev. Lett. 89 (2002). [8] T.Z.X.Z.J. Valentine, S. Zhang, Development of bulk optical negative index fishnet metamaterials: achieving a low-loss and broadband response through coupling, Proc. IEEE 99 (2011) 1682–1690. [9] A.A. Balandin, Thermal properties of graphene and nanostructured carbon materials, Nat. Mater. 10 (2011) 569–581. [10] J. Zhao, X. Liu, W. Qiu, Y. Ma, Y. Huang, J.-X. Wang, K. Qiang, J.-Q. Pan, Surfaceplasmon-polariton whispering-gallery mode analysis of the graphene monolayer
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A refractive index from negative to positive of graphene plasmonic crystal at the Dirac-like cone in mid-infrared region
T
⁎
Zeyu Wanga, Weibin Qiua, , Junbo Rena, Zhili Lina, Pingping Qiub,c, Qiang Kanb,c a
Fujian Key Laboratory of Light Propagation and Transformation, College of Information Science and Engineering, Huaqiao University, Xiamen, 361021, China College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing, 100086, China c Institute of Semiconductors, Chinese Academy of Sciences, Beijing, 100086, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Graphene plasmon Negative refractive index Plasmonic crystals Dirac-like cone
In this work, two-dimensional plasmonic crystals composed of graphene nanodisks where the chemical potential varies in triangular symmetry are proposed. A special band structure of Dirac-like cone is obtained at the center of Brillouin zone in mid-infrared region. A refractive index of the plasmons from negative to positive is calculated by the method of eigen-fields averaging. Also, the isotropic distribution of the negative/positive effective permeability and permittivity is numerically demonstrated. In addition, the significant propagation characteristics including the reverse Snell’s law effect, deep-sub-wavelength focusing and negative Goos-Hänchen shifts are numerical simulated. The proposed structure may find broad applications in the fields of on-chip plasmonic interconnect, high density plasmonic integrated circuit techniques and nano-imaging in the future.
1. Introduction Negative refractive index(NRI) materials were originally proposed by Veselago in double-negative (DNG) materials [1], which reverse the directions of the phase and group velocities during the propagation of electromagnetic (EM) waves. Further on, NRI materials achieved series of significant applications, such as superfocusing, negative Goos-Hänchen shifts, cloaking [2–5] due to the extraordinary propagation nature. In the early stages, NRI medium based on surface plasmon polaritons (SPP) hosted by the noble metals sparked the extensive interest. The literatures were concentrated on the composite NRI media by embedding arrays of metallic split-ring resonators, bulk fishnet and doublefishnet (DF) structures [6–8]. However, such noble metal based structures have difficulty in reducing dissipative losses to prolong the propagation distance of SPP. Also, the working frequency is determined once the geometry structures of the devices are defined due to the definite dispersion function of the noble metal. All the obstacles constrain the functionality and applications of the NRI materials. Graphene, a monolayer of carbon sheet where the atoms are arranged in a two-dimensional (2-D) honeycomb lattice, has been widely studied as a promising material for its superior properties [9]. Comparing with the SPPs at a metal-dielectric interface, graphene supported SPP has superiority in field localization and lower loss [10–12]. Besides, the most appealing property is the frequency tunability, whereas the conductivity of graphene can be tuned dynamically by changing the ⁎
chemical potential from terahertz (THz) to mid-infrared. Graphene has been introduced to compose the negative refractive index materials recently. For example, a NRI media constructed by graphene-dielectric multilayered structure is proposed by K. V. Sreekanth [13]. However, the multilayered structure is inconvenient for on-chip integration, where another problem raised here is the anisotropic refractive index distribution. Hence, an in-plane, isotropic negative refractive index distribution is highly desired. It is well known that a peculiar band structure called ‘Dirac cone’ of plasmonic crystal of graphene plasmonic crystal(GPC) shows the distinct superiority in implementing the isotropic distribution of effective permittivity and permeability in plane, which offers the convenient for on-chip integration [14–16]. Thus, combining concepts of graphene hosted plasmon and the Dirac cone band structure is the practical way to get the in-plane and isotropic negative refractive index material. In this article, we design a graphene plasmonic crystal structure whose effective electromagnetic parameters evolve from negative to positive continuously, which is beneficial to achieve two-dimensional on-chip integration. In our work, the graphene nanodisks are arranged in a triangular lattice which is constructed by periodically modulating the chemical potential of grapheme [17]. Further on, we study the dispersion relation for the transverse magnetic SPP modes and the band structure of the GPC. The interaction between a monopolar mode and two dipolar modes results in a three-fold accidentally degenerate state called Dirac-like cone dispersion. The connection between the
Corresponding author. E-mail address: [email protected] (W. Qiu).
https://doi.org/10.1016/j.photonics.2019.100745 Received 18 October 2018; Received in revised form 10 June 2019; Accepted 26 September 2019 Available online 27 September 2019 1569-4410/ © 2019 Elsevier B.V. All rights reserved.
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
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Fig. 1. (a) Schematic diagram of the surface of array of graphene nanodisks arranged in triangular lattice. (b) The single layer of graphene sits on a silica layer on silicon substrate. The silica layer with periodic thickness d1 and d2 corresponding to periodic chemical potentials μc1 and μc2 under an external gate voltage.
Fig. 2. (a) The diagram of the unit cell of graphene plasmonic crystals. (b) The reduced Brillouin zone with the path of Γ-M-K-Γ. (c) The band structure of GPC with r = 0.2517a, P is the three-fold degeneracy point. (d), (e)Three-dimensional dispersion surfaces near the point P in k-space when r = 0.2517a and r = 0.4a, respectively. (f) The band structures when r = 0.248a. The electric field distributions profiles of the dipole (f) and monopole (g–h) interact at point P when the frequency is 70.827 THz.
isofrequency contours show an anisotropic characteristic when the three-fold degenerate splits into doubly degenerate point of dipolar modes and a single mode of monopole. Moreover, fantastic electromagnetic properties including reverse Snell’s law, negative Goos-
electromagnetic parameters and frequency is revealed by using the method of eigen-field averaging. The effective electromagnetic parameters from negative to positive continuously are demonstrated respectively in two isotropic behaviors frequency ranges. However, the 2
Photonics and Nanostructures - Fundamentals and Applications 37 (2019) 100745
Z. Wang, et al.
Fig. 3. (a) The effective permittivity and permeability of the GPC obtained by the field averaging theory. (b–e) The isofrequency contours of the TM2 and TM4 when the r = 0.2517a and r = 0.4a, respectively.
Hänchen shifts, and superfocusing are investigated. The proposed structure has significant potential applications in the fields of on-chip integration and nano-imaging technologies.
and μc2 ) of the graphene can be realized by periodically modulating the thickness(d1 and d2 ) of the SiO2 layer under an external gate voltage [18,19]. And r is the radius of the Si rod which corresponds to the radius of graphene nanodisk. As shown in Fig. 1(b), the connection between with silica layer thicknesses of the graphene regions and the chemical potentials is μc1 / μc 2 = (d2/ d1)1/2 [17]. The dispersion relation for transverse magnetic (TM) polarized SPP modes in two dimensional GPC can be deduced by solving Maxwell’s equations with boundary conditions [20]
2. Band structure and the effective parameters of graphene In our work, a single layer of graphene sits on a silica layer on silicon substrate. The two-dimensional plasmonic crystal is composed of an array of graphene nanodisks arranged in triangular lattice which is displayed in Fig. 1(a). Specifically, the variant chemical potential( μc1 3
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Z. Wang, et al.
Fig. 4. Electric field distribution of the propagation the GPC: (a) The schematic figure of the triangle formed by the graphene plasmonic lattice with a chemical potential μc1 = 0.4eV at the nanodisks and μc2 = 0.7eV at the large area. The whole triangle structure is surrounded by the graphene sheet with μc2 = 0.7eV , and the refraction angle α is equal to the angle θ. The transmission characteristics of the beams are plotted in (b–e) with r = 0.2517a, when the frequency is 73.2, 70.0827, 67 and 65.5 THz, respectively. (f) The electric field distribution of the beam when r = 0.4a.
εAir β2 − k 0 εAir
+
εsio2 β2 − k 0 εsio2
=
σg iϖε0
graphene layer can be calculated by the equation neff = β / k 0 . The unit cell of the graphene plasmonic crystals is depicted in Fig. 2(a), where the graphene nanodisks with μc1 = 0.4eV are surrounded by the graphene with μc2 = 0.7eV . a and r are the lattice constant, the radius of the graphene nanodisk, respectively. The band structures and the propagation properties are numerically studied by the commercial software COMSOL Multiphysics, RF module Ver.5.2a based on finite-element method (FEM). The Γ-M-K-Γ path of the reduced Brillouin zone (BZ) is displayed in Fig. 2(b). Fig. 2(c) shows the corresponding band structures of the triangular lattice when a is 45 nm and r is 0.2517a. It is distinct to find that TM2 , TM3 and TM4 bands intersect at the center of BZ (Γ point) and form a triply degenerate state marked with P. The three-dimensional dispersion surfaces of conical behavior near the P point in k-space are exhibited in Fig. 2(d). Further on, when the radius of graphene nanodisk shifts to 4a, the Dirac-like cone structure opens a gap in Fig. 2(f). On the one hand, the Dirac-like cone is a triply accidental degenerate behavior because the band structures are constructed by the definite structural parameters. On the other hand, the electric field profiles at triply degeneracy point are formed by a monopole mode (Fig. 2(g)) and dipole modes (Fig. 2(h) and (i)) interaction. Therefore, this triply degenerate state is a Dirac-like point at the frequency of 70.827 THz. The effective permittivity and permeability are obtained by the method of eigen-field averaging [22].
(1)
Here εAir and ε sio2 are the relative permittivity of air and silica, respectively. k 0 is the wave number in the free space, and ε0 is the vacuum permittivity. In particular, the surface conductivity of graphene σg is composed of the intraband electron-photon scattering σintra and the inter-band electron transitions σinter , which are determined by the Kubo’s formula [21].
σg = σintra + σinter σintra =
σinter =
β = ε0
(2)
ie 2kB T ⎡ μc + 2 ln ⎛1 + exp(− μc )⎞ ⎤ ⎜ π ℏ2 (ω + i/ τ ) ⎢ kB T ⎠ ⎥ ⎝ ⎦ ⎣ KB T
(3)
ie 2 ⎡ 2 |μc | − ℏ(ω + i/ τ ) ⎤ ⎥ 4π ℏ2 (ω + i/ τ ) ⎢ ⎣ 2 |μc | + ℏ(ω + i/ τ ) ⎦
(4)
⎟
εAir + εsio2 2iω 2 σg
(5)
where e, kB and ω are the electron charge, Boltzmann constant and the angular frequency of the plasmonic crystal, respectively. T is the temperature, ℏ is the reduced Planck constant. μc and τ represent the chemical potential and the electron momentum relaxation time, respectively. β is the propagation constant of SPPs based on graphene layer and the effective refractive index neff for the SPP modes on 4
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Fig. 5. (a) The energy flux density intensity distribution of the wave propagating through the GPC slab with a negative refraction when f =62 THz. (b) Electric field distribution of Gaussian beams transmit with a negative Goos-Hänchen shift Δx1 when f =67.5 THz. (c) Δx1 as a function of the frequency when θ = 45°.
ψi(av) =
∫ unitcellψi(k ) (x , y ) ∂xdxdy (av )
ε eff =
∂Ez 1 j ωHx(av) ∂x
μ eff =
∂Ez 1 j ωHx(av) ∂x
=
k Hx(av) ω Ez(av)
(7)
=
(av ) k Ez ω Hx(av)
(8)
(av )
ψi(av)
Ei
The GPC is designed in a triangular structure where the graphene nanodisks with μc1 = 0.4eV are surrounded by the graphene with μc2 = 0.7eV , r = 0.2517a. In our simulation, variant frequencies of TM polarized plane wave propagate from the b ottom of the triangle. When the frequency is 73.2 THz(PRI frequency range), the refraction beam and incidence beam emerge on the different side of the normal line in Fig. 4(b), which confirms the regular Snell’s law. The refraction angle θ is zero at the Dirac-like point frequency (70.827 THz) in Fig. 4(c), The phenomenon of optical beam shaping confirms the zero refractive index characteristics of GPC [23]. In particular, as shown in Fig. 4(d–f), the refraction and incidence wave emerges on the same side of the normal line when frequencies respectively are 67 THz and 65.5 THz (NRI frequency range) in Fig. 4(d–e). This especial propagation phenomenon is called Inverse Snell’s law which verifies the negative refractive index of GPC. Further on, when the frequency is close to 70.827 THz, the angle θ approaches to zero, which confirms the values of the effective permittivity and permeability in Fig. 4(a). In addition, the refraction waves experience different refractive indices in different directions when the r shifts to 0.4a and frequency is 62.6 THz. This phenomenon relates to Fig. 4(d) where the isofrequency contours is no longer a circular shape in this specific frequency ranges. Thus, the GPC exhibits an anisotropy characteristic when the structural parameters are changed. Hence, the Dirac-like cone is crucial for the isotropy of the GPC.
(6)
A
Hi
where = , , is the averaging eigen-fields in the i direction, i=x, z, and A is the area of the unit cell. ε eff and μ eff are effective permittivity and permeability of GPC, respectively. Through our calculation, the relationship between the effective permittivity (permeability) and the frequency is shown in Fig. 3(a). It should be noted that the effective permittivity and permeability increase from the negative to positive monotonously with the frequency increasing. In particular, the permittivity and permeability reach zero when the frequency is Dirac-like point frequency (70.827 THz). Thus, the GPC exhibits a zero refractive index(ZRI) at 70.827 THz. Moreover, by analyzing the isofrequency contours of TM2 and TM4 bands in Fig. 3(b) and (c), it is clear to find that the contours maintain a nearly circular shape which means the GPC can be essentially regarded as an isotropic material in two frequency ranges. These two frequency ranges are 66.49–70.827 THz and 70.827–73.31 THz which can be defined as negative refractive index frequency range and positive refractive index(PRI) frequency range, respectively. However, when the radius of graphene nanodisk pillar becomes 0.4a, the isofrequency contours of TM2 and TM4 bands are displayed in Fig. 3(e) and (f). One can find that the isofrequency contours lose the circular profiles when the nature of three-fold accidental degeneracy is broken. Hence, Dirac-like cone is crucial in constructing an isotropic GPC in these frequency ranges.
4. Negative Goos-Hänchen shift in the proposed structure Goos-Hänchen shift is an intriguing optical phenomenon in which a physical beam experiences a lateral shift from the position predicted by geometrical optics under a totally reflection condition. This deviation was originally studied by Goos and Hänchen so-called Goos-Hänchen shift and theoretically analyzed by Artmann in the late 1940s [24,25]. Since then, such shift effect has been demonstrated to be a universal phenomenon, which can take place at the interface between the NRI and PRI [26]. Especially some studies have been concentrated on the shift in an isotropic NRI flat which has extensive applications including slow light and optical sensors [27,28]. To get insight into the negative Goos-Hänchen shift, a Gaussian wave is incident to the GPC flat a slab which is constructed by the proposed GPC structure with different
3. Inverse Snell's law The most symbolic characteristic of NRI is the inverse Snell's law, where the propagation direction is inversed but the Maxwell's equations is still satisfied. Hence, to further verify the NRI transmission of graphene plasmonic crystals, the proposed structure is shown in Fig. 4(a). 5
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frequency and incident angle. And the energy flux density intensity distribution of the transmission of the wave is displayed in Fig. 5(a). The transmission of the Gaussian beam displays a negative refraction characteristic when the frequency is 62 THz and the angle of incidence θ is 20°. The phenomenon verifies the NRI of the GHC again. Contrarily, the electric field distribution of the totally reflection occurs at 45°, as displayed in Fig. 5(b). The negative Goos-Hänchen shift Δx1 presents at the interface when the frequency f is 67.5 THz. The length of Δx1 approachs 0.041λ which is much less than the wavelength of incident wave. Further on, the Δx1 goes up as increasing frequency in Fig. 5(c), and it flattens out when the frequency reaches ZRI frequency (70.827 THz).
effect is shown in Fig. 6(b), it is evident that the incident beams are focused into a spot by the GPC lens. Besides, the normalized intensity distribution of the image is displayed in Fig. 6(c) where FWHM is 9.18 × 10−3λ . The FWHM value far less than the wave length of the point source which breaks through the diffraction limit so-called deepsub-wavelength focusing. Although GPC flat lens shows the superiority in focusing the point source, there is limitation in plane wave imaging. So, a planoconcave lens is designed to further overcome the restriction, which is constructed by cutting the exit surface of GPC slab into a triangle as illustrated in Fig. 7(a). A spot is focused by the GPC planoconcave lens on the cutting side of the lens in Fig. 7(b) when the frequency of plane wave is 68.9 THz. Moreover, as shown in Fig. 7(c), we also plot the intensity distribution whose FWHM along the y-axis approaches 9. 79 × 10−3λ . In addition, to verify superiority of NRI on subwavelength imaging, we simulate the imaging effect and calculate the FWHM value with the frequency of the PRI range and Dirac-like frequency (70.827 THz), respectively. The size of the imaging spot, loss and FWHM is much larger than the NRI frequency range. Hence, this structure in negative refractive index frequency range has better focusing effect. Actually, when the radius of graphene nanodisk is one atom more or less, the reduced degenerate Dirac-like cones at Γ point open negligible gaps (0.28 THz for one more atom, and 0.231Thz for one atom less). There are two zero refractive index points and two discontinuity points correspond to two zero slope points of the bands and the gap of Dirac points. However, the relationship between the frequency and effective permittivity (permeability) still maintain a linear relationship in negative and positive frequency ranges for interested (66.49–70.827 THz for negative refractive index and 70.827–73.31 THz for positive refractive index (PRI) frequency range).
5. Deep-sub-wavelength focusing Current research in nanotechnology requires the images with a resolution in deep nanometer scale. To break the diffraction limit, the idea of SPP was proposed to manipulate wave [29,30]. Further on, a negative refraction flat lens of silver basing on SPP has been proposed by Pendry to achieve super-resolution [31]. But the silver flat lens also has limitations on the minimum size of metallic nanostructure. For further breakthrough the nano-image resolution, we design the perfect lens basing on GPC flat structure. The schematic diagram of the NRI lens is illustrated in Fig. 6(a). A TM polarized point source is placed above the GPC lens. D is the distance between the point source and upper of the GPC lens. The beam of point source p1 is focused into in p2 through the GPC flat lens because of the negative refraction. Moreover, we should note that the perfect lens also focuses the evanescent wave which transmits the deep-sub-wavelength details of the source. We adjust different structure parameters to obtain the optimal focusing effect of the image. On the one hand, the full width at half maximum (FWHM) of the image is very crucial to the resolution of the image. Thus, we plot the FWHM of the image spot along the x-axis as a function of D, the layer number of GPC N and the frequency f, respectively in Fig. 6(d–f). On the other hand, The energy loss during the imaging process is inevitable due to the impedance mismatch. There is a tradeoff between the resolution and the energy loss. Particularly, the FWHM essentially decreases continuously as frequency increasing. However, the energy loss sharply increases when frequency exceeds 68 THz. Combining all the factors, we find the optimal parameters as D = 0.172um, N = 3 and F = 68 THz. The energy flux density of image
6. Conclusions In conclusion, Dirac-like conical dispersion of the two-dimensional GPC structure composed of graphene nanodisks are designed to obtain the NRI and PRI medium. A three-fold accidental degeneracy conical dispersion at the center of BZ called Dirac-like cone has been got, and the band structure of the GPC has been investigated numerically. The frequency of Dirac-like point is 70.287 THz when the lattice constant and radius of nanodisk is 45 nm and 0.2517a, respectively. Moreover,
Fig. 6. (a) Structure of the GPC flat lens. (b) The energy flux density of the nano-image point is focused by the GPC lens with the optimal structure parameters. (c) The normalized intensity distribution of the nano-image along the x-axis the FWHM is 9.18 × 10−3λ . (d–e) The FWHM as a function of D, N and F. 6
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Fig. 7. (a) Schematic diagram of the GPC planoconcave lens. (b) The intensity distributions of electric field for a plane wave with 68.9 THz. (c) Normalized intensity distribution of the plane wave whose FWHM value is 9.79 × 10−3λ .
the negative permittivity and permeability of the GPC are calculated by the theory of the eigen-field averaging. The distributions of isotropic ranges of electromagnetic parameters are obtained by isofrequency contours and the NRI frequency range is 66.49–70.827 THz. However, the isotropy is fading away when the radius of the graphene nanodisk are shifted gradually. Moreover, the propagation of GPC in the isotropic range is simulated to verify Inverse Snell’s law and negative GoosHänchen shifts. In addition, perfect lens is constructed by negative refractive index GPC which is significant in deep-sub-wavelength focusing field. The proposed GPC structure has enormous potential significance in the on-chip interconnect and nanotechnology.
coated InGaAs nanowire cavity, Opt. Express 22 (2014) 5754. [11] L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H.A. Bechtel, X. Liang, A. Zettl, Y.R. Shen, F. Wang, Graphene plasmonics for tunable terahertz metamaterials, Nat. Nanotechnol. 6 (2011) 630–634. [12] W. Qiu, X. Liu, J. Zhao, S. He, Y. Ma, J.-X. Wang, J. Pan, Nanofocusing of midinfrared electromagnetic waves on graphene monolayer, Appl. Phys. Lett. 104 (2014) 041109. [13] K.V. Sreekanth, A. De Luca, G. Strangi, Negative refraction in graphene-based hyperbolic metamaterials, Appl. Phys. Lett. 103 (2013) 023107. [14] G.-H. Lee, G.-H. Park, H.-J. Lee, Observation of negative refraction of Dirac fermions in graphene, Nat. Phys. 11 (2015) 925–929. [15] Y. Li, S. Kita, P. Muñoz, O. Reshef, D.I. Vulis, M. Yin, M. Lončar, E. Mazur, On-chip zero-index metamaterials, Nat. Photonics 9 (2015) 738–742. [16] P. Qiu, W. Qiu, Z. Lin, H. Chen, J. Ren, J.-X. Wang, Q. Kan, J.-Q. Pan, Double Dirac point in a photonic graphene, J. Phys. D Appl. Phys. 50 (2017) 335101. [17] W. Qiu, P. Qiu, J. Ren, Z. Lin, J.-X. Wang, Q. Kan, J.-Q. Pan, Realization of conical dispersion and zero-refractive-index in graphene plasmonic crystal, Opt. Express 25 (2017) 33350. [18] B. Shi, W. Cai, X. Zhang, Y. Xiang, Y. Zhan, J. Geng, M. Ren, J. Xu, Tunable bandstop filters for graphene plasmons based on periodically modulated graphene, Sci. Rep. 6 (2016). [19] Z. Fei, A.S. Rodin, G.O. Andreev, W. Bao, A.S. McLeod, M. Wagner, L.M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M.M. Fogler, A.H.C. Neto, C.N. Lau, F. Keilmann, D.N. Basov, Gate-tuning of graphene plasmons revealed by infrared nano-imaging, Nature 487 (2012) 82–85. [20] H.B.M. Jablan, M. Soljačić, Plasmonics in graphene at infrared frequencies, Phys. Rev. B 80 (2009). [21] H. Lu, C. Zeng, Q. Zhang, X. Liu, M.M. Hossain, P. Reineck, M. Gu, Graphene-based active slow surface plasmon polaritons, Sci. Rep. 5 (2015) 8443. [22] Z.L. Lu, D.W. Prather, Calculation of effective permittivity, permeability, and surface impedance of negative-refraction photonic crystals, Opt. Express 15 (2007) 8340–8345. [23] J.W. Dong, M.L. Chang, X.Q. Huang, Z.H. Hang, Z.C. Zhong, W.J. Chen, Z.Y. Huang, C.T. Chan, Conical dispersion and effective zero refractive index in photonic quasicrystals, Phys. Rev. Lett. 114 (2015). [24] H.H.F. Goos, Ein neuer und fundamentaler Versuch zur Totalreflexion, Ann. Phys. 436 (1947) 333–346. [25] K. Artmann, Berechnung der Seitenversetzung des totalreflektierten Strahles, Ann. Phys. 87 (1948). [26] P.R. Berman, Goos-Hanchen shift in negatively refractive media, Phys. Rev. 66 (2002) 067603. [27] K.L. Tsakmakidis, A.D. Boardman, O. Hess, ’Trapped rainbow’ storage of light in metamaterials, Nature 450 (2007) 397–401. [28] X. Yin, L. Hesselink, Goos-Hänchen shift surface plasmon resonance sensor, Appl. Phys. Lett. 89 (2006) 261108. [29] S. Kawata, Y. Inouye, P. Verma, Plasmonics for near-field nano-imaging and superlensing, Nat. Photonics 3 (2009) 388–394. [30] V. Shalaev, Optical Negative-Index Metamaterials Nature 1 (2007). [31] J.B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85 (2000) 3966–3969.
Acknowledgements The authors are grateful to the support by the Natural Science Fund of China under grant No. 11774103, and Project for Cultivating Postgraduates’ Innovative Ability in Scientific Research of Huaqiao University (17013082023) References [1] V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ, Sov. Phys. Usp. 10 (1968) 509–515. [2] I.I. Smolyaninov, Y.-J. Hung, C.C. Davis, Imaging and focusing properties of plasmonic metamaterial devices, Phys. Rev. B 76 (2007). [3] X. Chen, X.-J. Lu, Y. Ban, C.-F. Li, Electronic analogy of the Goos–Hänchen effect: a review, J. Opt. 15 (2013) 033001. [4] S. Foteinopoulou, C.M. Soukoulis, Negative refraction and left-handed behavior in two-dimensional photonic crystals, Phys. Rev. B 67 (2003). [5] P. Kolinko, D.R. Smith, Numerical study of electromagnetic waves interacting with negative index materials, Opt. Express 11 (2003) 640–648. [6] A. Mary, S.G. Rodrigo, F.J. Garcia-Vidal, L. Martin-Moreno, Theory of negativerefractive-index response of double-fishnet structures, Phys. Rev. Lett. 101 (2008) 103902. [7] R. Marqués, J. Martel, F. Mesa, F. Medina, Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides, Phys. Rev. Lett. 89 (2002). [8] T.Z.X.Z.J. Valentine, S. Zhang, Development of bulk optical negative index fishnet metamaterials: achieving a low-loss and broadband response through coupling, Proc. IEEE 99 (2011) 1682–1690. [9] A.A. Balandin, Thermal properties of graphene and nanostructured carbon materials, Nat. Mater. 10 (2011) 569–581. [10] J. Zhao, X. Liu, W. Qiu, Y. Ma, Y. Huang, J.-X. Wang, K. Qiang, J.-Q. Pan, Surfaceplasmon-polariton whispering-gallery mode analysis of the graphene monolayer
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