- Email: [email protected]
Fano resonance of hybrid graphene-metal gratings
Journal Pre-proof Fano resonance of hybrid graphene-metal gratings ZiZheng Guo
PII:
S0030-4026(19)31399-3
DOI:
https://doi.org/10.1016/j.ijleo.2019.163501
Reference:
IJLEO 163501
To appear in:
Optik
Received Date:
31 May 2019
Accepted Date:
30 September 2019
Please cite this article as: Guo Z, Fano resonance of hybrid graphene-metal gratings, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163501
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Fano resonance of hybrid graphene-metal gratings ZiZheng Guo* College of Electronic Engineering, South China Agricultural University, Guangzhou 510642, China
ABSTRACT:
ro of
In this paper, the phenomena of extraordinary optical transmission and Fano resonance in the hybrid graphene-metal gratings are studied. The results show that the Fano resonance in this hybrid graphene-metal grating can be enhanced by parameter adjustment and optimization. The two most effective control parameters are the proportion of metal width in the hybrid graphene-metal grating and the Fermi level of graphene. The above calculations do not involve the effects of the surface plasmon excitation, so the results should be independent of plasmon excitation, and the transmission enhancement effect is the result of structural Fabry-Pérot-type interference.
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KEYWORDS : graphene; plasmonics; Fano resonance; hybrid graphene-metal grating
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1. Introduction In recent years, the study of extraordinary optical transmission （EOT）phenomena in metal grating structures and metal nano-array structures has attracted attention [1]. The physical mechanism of this extraordinary transmission phenomenon has also led to a major international discussion. Some people think that this extraordinary transmission phenomenon is the resonance effect caused by electromagnetic wave coupling with metal surface plasmon[2]. Since the surface plasmon oscillation in the metal nano-array structure appears as a TM mode, the surface plasmon transmission enhancement of the TM mode in this structure was first examined, and a positive result was obtained. However, for metallic chromium, perfect metals, and non-metallic materials in the visible region of the spectrum, there is no surface plasmon, but the EOT behavior can also be observed. There is also transmission enhancement for the TE mode, when there is no surface plasmon [3,4]. In addition, the extraordinary transmission behavior of sound waves through sub-wavelength slit arrays has also been reported, and apparently there is no surface plasmon [5]. The above examples show that the EOT phenomenon in the metal nano-array structure is independent of surface plasmon excitation and is mainly due to the Fabry-Pérot-type structural resonance [6]. Graphene has metalloid properties. Therefore, the graphene nano-array structure, or the graphene grating structure, can still observe the transmission enhancement phenomenon [7-9]. The transmission enhancement phenomenon in the metal grating is most prominent in the mid-infrared band; while the graphene grating exhibits strong asymmetric transmission absorption characteristics in the terahertz section. The optical stability of the metal makes it difficult to externally regulate by voltage or the like. However, the properties of graphene are externally adjustable. Therefore, many authors have done research on replacing metals with graphene, such as graphene instead of metal as a reflective *
Corresponding Author , Email: [email protected].
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layer[10], or simply forming a graphene grating[11]. Further, in order to adjust the response of the grating to the incident frequency, the authors propose a graphene- metal hybrid grating structure [12-15]. This structure can be used for the study of nano-optical phenomena such as Fano resonance [12,13], negative reflection [14], resonance absorption or transmission [15] in plasmonics. Fano resonance has always been a hot spot in physics and applied research. In recent years, people have paid particular attention to the Fano resonance phenomenon in plasmonics. In optics, Fano resonance describes the resonant response of incident (excited) light as reflected by two modes of coupling and interference. In general, the Fano resonance requires a wide formant, also called the bright mode; another narrow formant, also called the dark mode. The Fano resonance occurs as a result of the coupling of the two modes. The metal in the metal-graphene hybrid grating provides a bright mode, and the graphene provides a dark mode, and their interaction is likely to induce Fano resonance. Experimentally, Chen et al. used a method of mixing graphene-metal gratings to achieve an adjustable Fano resonance [12]. However, how to regulate the bright and the dark mode, which parameters determine the role is not very clear. In the Fano resonance structures based on graphene sheets, many important results have been reported [16-18] while many problems have not been completely solved. For example, in addition to the above graphene-metal hybrid grating, what other methods can achieve Fano resonance? In addition, which metal can be combined with graphene to achieve the best Fano resonance effect? These require further research. In this paper, the Fano resonance effect of graphene-metal hybrid grating structure is studied. Materials used as transmission gratings require high transparency and low absorption, and graphene has such a property. ITO (Indium Tin Oxides, composed of indium oxide and tin oxide, also known as indium tin oxide) [19] and silver are good choices in metal materials. Silver was chosen for the calculation in this paper. In this paper, the absorption spectrum of the graphene-metal combined grating structure is calculated by the improved Fourier model method（FMM）[20,21]. A significant Fano resonance effect was observed by optimizing the grating structure parameters and changing the Fermi level of graphene. The results of this paper show that optimizing the grating structure parameters and regulating the properties of graphene are effective ways to achieve significant Fano resonance.
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2. Computational models and methods The hybrid graphene-metal grating discussed is shown in Fig.1. This is a graphene-metal nanoribbon periodic array, with period L . As depicted, the nanoribbons are infinitely long in y direction. The structure is illuminated by a TM polarized (magnetic field along the y direction)
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plane wave (having vacuum wavelength upper and lower mediums are 1 and
) whose incident angle is . The permittivities of the
2 , respectively.
This structure is actually a diffraction grating, and there are many existing methods to analyze it. It is also a kind of stripe grating. After extensive research on gratings by people for half a century, several analysis methods have been developed. Among these methods, the FMM may be the most popular one because it is very simple but efficient[20,21]. However, this method requires strict Fourier decomposition, otherwise it has poor convergence[20,21]. In the following, we only consider TM polarized waves. As we all know, the electromagnetic field in the uniform region can be expressed by the following Rayleigh expansion:
H jy (a jn e
jk jzn z
b jne
ik jzn z
)e ikxn x
(1)
n
E jx n
where
k jzn
j
(a jn e
jk jzn z
b jn e
ik jzn z
)e ik xn x
(2)
j 1, 2 ， representing regions 1 or 2. In these expressions a jn and b jn are the
amplitudes of downward and upward modes, respectively, and k xn k0n1 sin
2 n， L
k jzn k02n j 2 k xn 2 ，where n j is the refractive index of the region j and k0 2 / is the vacuum wave number. The z component of the wave vector k jzn is either negative real
ro of
(propagating wave) or positive imaginary (evanescent wave). Eqs. (1) and (2) must satisfy certain boundary conditions. Here we use the improved boundary conditions given in references [20] and [21]:
E1x x, z 0 E2 x x, z 0 0
(3)
H1 y x, z h / 2 H 2 y x, z h / 2 seff ( x) Ex ( x, z 0)
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seff x s x i h is effective surface conductivity and (1 2 ) / 2 ,
re
where
(4)
e2 EgF
i
i 2 g1
na
gs
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gs ,0 x Wg 0,Wg x Wg W1 s x m ,Wg W1 x Wg W1 Wm 0,Wg W1 Wm x L 2
(5)
2 Eg e2 i [ H 2 EgF ln ] 4 2 EgF F
(6)
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The conductivity of the metal is given according to the Drude model
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In the above formula,
m
m0 1 i m
(7)
gs ( x ) is the surface conductivity of graphene, which exists only in
the graphene band, equal to the graphene surface conductivity, otherwise zero, and is a periodic function. Since [ s ( x )]
1
will be infinite when
s ( x ) is zero, its Fourier expansion does not
satisfy the inverse rule. In addition, this reason may also lead to slow convergence. Therefore, direct Fourier decomposition according to the conventional electromagnetic wave boundary conditions is problematic. This is why the improved approximate boundary conditions are given in Refs.[20,21]. In addition , e is the electron charge,
is the Planck constant, is the frequency, E g and F
g are the Fermi energy and relaxation time of graphene and H is a step function. m , m and mm are the conductivity, relaxation time, and effective mass of the metal. h is the parameter introduced for applying the boundary conditions. We have assumed that the time dependency is
eit . For computational convenience, we substitute (1) and (2) into the above boundary conditions (3) and (4) and organize them into the following S-matrix forms:
2 1 a2 1 2 a1 a 1 I / eff 1 a 1 b a a I / eff 1 b 2 1 1 2 1 2 2 2
j is a diagonal matrix whose (n,n) element is k jzn / j ,
ro of
where I is the identity matrix and
(8)
a j and b j are column vectors whose elements are a jn and b jn , respectively.
In the grating problem shown in Fig. 1 there is no upward wave in region 2, so b2 0 and
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the only nonzero element of a1 is a10 1 . Therefore, a2 and b1 , which are representing transmission and reflection coefficients of diffracted orders, respectively, are directly calculated from Eq.(8). Finally, reflection and transmission coefficients are given by (9)
k 2 T Re 1 2 zn a2 n n 2k1zn
(10)
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and absorption coefficient is
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k 2 R Re 1zm b1n n k1z 0
A 1 T R
(11)
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3. Numerical results and discussion To analyze how the Fano resonance (wavelength) is controlled, we calculated the transmission, reflection, and absorption curves for various parameter changes, as shown in Figs. 2-6. First, we explain the parameters used in the calculation as follows. is the dielectric constant of the medium and its relationship with the refractive index n is / 0 n . The medium with 2
1 / 0 =1 can be approximated as air. For example, when
1 / 0 2 / 0 1 , graphene and metal are suspended in the air, when 1 / 0 is suspended at
1 ( 1 / 0 1 ), one side of graphene and metal is exposed to the air. The medium of
/ 0 =4
can be used as an approximation of the actual medium such as SiO2 ( / 0 = 3.9), and the medium of
/ 0 = 9 is Al2O3.
If the upper and lower media of the graphene-metal grating are the same, it is called a symmetric environment, otherwise it is an asymmetric environment. Generally, when the upper and lower media of graphene-metal are asymmetrical, the graphene-metal can still be considered to be in a homogeneous medium environment, and the dielectric constant is taken as the average value of the upper and lower media. This is the boundary conditions given in the literature [20] and [21] . L is the period of the graphene-metal nanobelt, Wg and Wm are the width of the graphene and the metal strip, respectively, W1 and W2 the gap between the graphene-metal and the
g and EgF are the relaxation time and the Fermi level of
g 0.25ps . m and m 0 are the
the graphene. In the calculations below, we fixed
ro of
metal-graphene strip, respectively.
relaxation time and DC conductivity of the metal. In the calculation below, we take the metal as
silver and fix m 3.8 1014 s and m0 6.2 10 m . h is the boundary depth of 7
1
1
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graphene set for the application of the boundary conditions. We take h 0.1nm in the following calculations. Fig.2 shows the relationship of the transmission, reflection and absorption curves of the hybrid graphene-metal grating with wavelength. The surface plasmon frequency of graphene is in the far infrared and terahertz regions, so the graphene grating resonance in Fig.2 occurs around 1×105 nm. No resonance occurs in the visible and near-infrared regions. The resonance of the metal grating occurs near 1×104 nm. Therefore, in order to interact with the scattered light of the metal grating and the graphene grating, that is, to make the Fano resonance occur, the resonance wavelength of the metal grating must be shifted to the long wave, and the resonance wavelength of the graphene grating is made to shift to the short wave. Compared to Fig.2, we made two major changes in the calculation of Fig.3. First, the period of the graphene-metal nanoribbon array was increased from 8 μm to 15 μm . Second, the
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asymmetric environment was changed to a symmetric environment. Fig.3 illustrates the transmission, reflection and absorption versus wavelength for such a hybrid graphene-metal grating. The curves of Fig. 3 are basically similar to that of Fig. 2, and the resonance positions of the metal and graphene grating are not effectively regulated, and the Fano resonance line type is not obvious. Next, we continue to adjust and optimize the parameters based on Fig.3. The period of the graphene-metal nanobelts remains at 15 μm . We have increased the width of the metal strip. The result is shown in Fig.4. It can be seen from Fig. 4 that the asymmetrical line shape of the curve basically appears, which is a typical feature of the Fano resonance, indicating that the parameter regulation has achieved initial success. Fig.4 illustrates that increasing the proportion of metal width in the hybrid graphene-metal grating, i.e., increasing Wm / Wg , favors the Fano resonance. However, the above calculations did not examine the effects of changes in graphene material parameters. In the calculation of Fig. 5, we set the grating structure parameters to be the same as in Fig. 3, but increase the Fermi level of
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graphene from 0.6 eV to 1.2 eV . The calculation results are shown in Fig. 5. It can be seen that the effect of graphene Fermi level is obvious. The curve shows the asymmetrical line shape unique to the Fano resonance. Fig.4 illustrates that increasing the proportion of metal width in the hybrid graphene-metal grating facilitates the Fano resonance. Fig.5 shows that increasing the Fermi level of graphene also favors the Fano resonance. In the calculation of Fig.6, we simultaneously increase the proportion of metal width in the mixed graphene-metal grating and the Fermi level of graphene. The calculation results are shown in Fig. 6. Fig.6 shows that the line shape of the Fano resonance is very obvious. This shows that by optimizing the grating structure and simultaneously implementing the external regulation of graphene, the Fano resonance phenomenon of the mixed graphene-metal grating can be clearly observed. In Ref.[22], the optical absorption in the graphene-insulator-metal hybrid plasmonic device was studied, and the resonance absorption obtained was around from 1440nm to 1450 nm. The Terahertz tunable graphene Fano resonance was studied in Ref.[23], and the resonance frequency was about 1.75 THz. These are consistent with the results of this paper.
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4. Summary In this paper, the transmission properties of the mixed graphene-metal gratings are studied by the FMM method and the transmission enhancement effect is observed. The above calculations do not involve the effects of the surface plasmon excitation, so the results should be independent of plasmon excitation, and the transmission enhancement effect is the result of structural Fabry-Pérot-type interference. In addition, by optimizing the grating structure and regulating the properties of the graphene material, we observed a typical Fano resonance effect. Obviously, this resonance is the result of the coupling of the two modes of metal grating and graphene grating. Our results show that the basic method for regulating this Fano resonance is to increase the proportion of metal width in the mixed graphene-metal grating and to increase the Fermi level of graphene.
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Acknowledgment It is hereby acknowledged that we used ee.sharif.ir/~khavasi/index_files/Page1077.htm.
the
code
by
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Khavasi
et
al.
at
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References [1] Andre-Pierre Blanchard-Dionne and Michel Meunier, Optical transmission theory for metal-insulatormetal periodic nanostructures ,Nanophotonics 6(1) (2017)349–355. [2] U. Schrte, D. Heitmann, Surface-plasmon-enhanced transmission through metallic gratings ,Phys. Rev. B 58(23) (1998)15419~15421. [3] A. G. Borisov, F. J. G. Abajo, S. V. Shabanov, Role of electromagnetic trapped modes in extraordinary transmission in nanostructured materials, Phys. Rev. B 71 (7) (2005) 075408. [4] D. Crouse, P. Keshavareddy, Polarization independent enhanced optical transmission in one-dimensional gratings and device applications, Opt. Exp. 15(4) (2007)1415~1427. [5] M. H. Lu, X. K. Liu, L. Feng ,J. Li , C. P. Huang , Y. F. Chen, Y. Y. Zhu, S.N.Zhu, N.B. Ming, Extraordinary acoustic transmission through a 1D grating with very narrow apertures, Phys. Rev. Lett. 99(17) (2007)174301.
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ro of
[6] Q. Cao, Ph. Lalanne, Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits, Phys. Rev. Lett. 88 (5) (2002)057403. [7]Z. Guo, Effect of dielectric environment on plasmonic resonance absorption of graphene nanoribbon arrays, Int. J. Mod. Phys. B 32(26)(2018)1850284. [8]Sukosin Thongrattanasiri, Frank H.L. Koppens, and F. Javier Garcıa de Abajo, Complete optical absorption in periodically patterned graphene, Phys. Rev. Lett. 108 (2012)047401. [9] S. Ke, B. Wang, H. Huang, H. Long, K. Wang, P. Lu, Plasmonic absorption enhancement in periodic cross-shaped graphene arrays ,Opt. Exp. 23(7) (2015)8888-8900. [10] A.Yu, Tunable strong THz absorption assisted by graphene-dielectric stacking structure, Superlattices Microstruct. 122 (2018) 461-470. [11] L. Yang ,J. Wang, C. Lu, Sensitive perfect absorber with monolayer graphene-based multi-layer dielectric grating structure, Optik 158(2018)508-513. [12]Z. Chen, J. Chen, Z. Wu, W. Hu, X. Zhang and Y. Lu，Tunable Fano resonance in hybrid graphene-metal gratings ,Appl. Phys. Lett. 104(2014)161114. [13] Bo Zhao and Zhuomin M. Zhang, Strong plasmonic coupling between graphene ribbon array and metal gratings ACS Photonics 2(2015)1611−1618. [14]X. Su, Z.Wei, C. Wu, Y. Long and H. Li, Negative reflection from metal/graphene plasmonic gratings, Opt. Lett. 41(2)(2016) 348-351. [15]Mohammad M. Jadidi, Andrei B. Sushkov, Rachael L. Myers-Ward, Anthony K. Boyd, Kevin M. Daniels, D. Kurt Gaskill, Michael S. Fuhrer, H. Dennis Drew and Thomas E. Murphy, Tunable terahertz hybrid metal−graphene plasmons, Nano Lett. 15(2015)7099−7104. [16]A. Ahmadivand, R. Sinha and N. Pala, Graphene plasmonics: multiple sharp Fano resonances in silver split concentric nanoring/disk resonator dimers on a metasurface, Plasmonics: Metallic Nanostructures and Their Optical Properties XIII. Vol. 9547. International Society for Optics and Photonics, 2015. [17] N. Dabidian, S. H. Mousavi, I. Kholmanov, K. Alici, D. Purtseladze, N. Arju, K. Tatar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, Inductive tuning of Fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared ,Nano Lett. 13(3) (2013) 1111-1117. [18] D. H.Chae, T. Utikal, S.Weisenburger, H.Giessen, K. V. Klitzing, M. Lippitz and J.Smet, Excitonic Fano resonance in free-standing graphene, Nano Lett. 11(3) (2011)1379-1382. [19] A. Melikyan and N. Lindenmann, A surface plasmon polariton absorption modulator, Opt. Exp. 19 (9) (2011) 8855-8869. [20]A. Khavasi, Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons, Opt. Lett. 38(16)(2013)3009-3013. [21]A. Khavasi, Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons, J. Opt. 14 (2012) 125502. [22]N. Matthaiakakis, X. Yan, H. Mizuta and M. D. B. Charlton, Tuneable strong optical absorption in a graphene-insulator-metal hybrid plasmonic device, Scientific Reports 7.7303(2017). [23] X. He, F. Lin, F. Liu and W. Shi, Terahertz tunable graphene Fano resonance, Nanotechnology 27 (2016) 485202.
Figure captions
Fig.1. Schematic diagram of the hybrid graphene-metal grating
ro of
Fig.2. Transmission, reflection and absorption of graphene-metal nanoribbon periodic arrays versus wavelength. The transmission, reflection and absorption are denoted as T,R and A and in red, green and blue (The same marks are used in the following figures). The calculated parameters used are as follows: L 8μm ， Wg 0.6 L ， Wm 0.2 L ， W1 W2 0.1L ， 1 3 0 ，
-p
2 7 0 ， EgF 0.6eV .
Fig.3. Transmission, reflection and absorption curves of the hybrid graphene-metal grating as a
L 15μm ，
re
function of wavelength with parameters
Wg 8μm , Wm 3μm ,
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F W1 W2 2μm , 1 2 3.5 0 ，and Eg 0.6eV .
Fig.4. Transmission, reflection and absorption of graphene-metal nanoribbon periodic arrays
na
versus wavelength. The calculated parameters used are as follows: L 15μm ， Wg Wm 0.4L ,
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F W1 W2 0.1L , 1 2 3.5 0 ， Eg 0.6eV .
Fig.5. Transmission, reflection and absorption curves of the hybrid graphene-metal grating with
L 15μm ， Wg 8μm , Wm 3μm , W1 W2 2μm ,
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wavelength at the case
1 2 3.5 0 ， EgF 1.2eV .
Fig.6. Transmission, reflection and absorption of the hybrid graphene-metal gratings as the function of wavelength. The calculated parameters used are as follows: L 15μm ， F Wg Wm 0.4L , W1 W2 0.1L , 1 2 3.5 0 ， Eg 1.2eV .
ro of -p lP
re
Fig.1
0.6
na
0.8
T
=3 =7
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Transmission, reflection and absorption
1.0
Jo
0.4
A
0.2
R
0.0
4
5.0x10
5
1.0x10
Wavelength (nm)
Fig.2
5
1.5x10
5
2.0x10
0.8 T
0.6
ro of
=3.5 =3.5
0.4 A
0.2 0.0
R 4
5.0x10
5
-p
Transmission, reflection and absorption
1.0
5
1.0x10
1.5x10
re
Wavelength (nm)
na
0.8
T
0.6
ur
Transmission, reflection and absorption
1.0
lP
Fig.3
=3.5 =3.5
Jo
0.4
A
0.2 0.0
R 4
5.0x10
5
1.0x10
Wavelength (nm)
Fig.4
5
2.0x10
5
1.5x10
5
2.0x10
0.8 T
0.6
=3.5 =3.5
A
0.2 0.0
R 4
5
5.0x10
5
1.0x10
1.5x10
5
2.0x10
re
Wavelength (nm)
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Fig.5
na
0.8
T
0.6
ur
Transmission, reflection and absorption
1.0
=3.5 =3.5
0.4
Jo
A
0.2
R
0.0
4
5.0x10
5
1.0x10
Wavelength (nm)
Fig.6
ro of
0.4
-p
Transmission, reflection and absorption
1.0
5
1.5x10
5
2.0x10
PII:
S0030-4026(19)31399-3
DOI:
https://doi.org/10.1016/j.ijleo.2019.163501
Reference:
IJLEO 163501
To appear in:
Optik
Received Date:
31 May 2019
Accepted Date:
30 September 2019
Please cite this article as: Guo Z, Fano resonance of hybrid graphene-metal gratings, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163501
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Fano resonance of hybrid graphene-metal gratings ZiZheng Guo* College of Electronic Engineering, South China Agricultural University, Guangzhou 510642, China
ABSTRACT:
ro of
In this paper, the phenomena of extraordinary optical transmission and Fano resonance in the hybrid graphene-metal gratings are studied. The results show that the Fano resonance in this hybrid graphene-metal grating can be enhanced by parameter adjustment and optimization. The two most effective control parameters are the proportion of metal width in the hybrid graphene-metal grating and the Fermi level of graphene. The above calculations do not involve the effects of the surface plasmon excitation, so the results should be independent of plasmon excitation, and the transmission enhancement effect is the result of structural Fabry-Pérot-type interference.
-p
KEYWORDS : graphene; plasmonics; Fano resonance; hybrid graphene-metal grating
Jo
ur
na
lP
re
1. Introduction In recent years, the study of extraordinary optical transmission （EOT）phenomena in metal grating structures and metal nano-array structures has attracted attention [1]. The physical mechanism of this extraordinary transmission phenomenon has also led to a major international discussion. Some people think that this extraordinary transmission phenomenon is the resonance effect caused by electromagnetic wave coupling with metal surface plasmon[2]. Since the surface plasmon oscillation in the metal nano-array structure appears as a TM mode, the surface plasmon transmission enhancement of the TM mode in this structure was first examined, and a positive result was obtained. However, for metallic chromium, perfect metals, and non-metallic materials in the visible region of the spectrum, there is no surface plasmon, but the EOT behavior can also be observed. There is also transmission enhancement for the TE mode, when there is no surface plasmon [3,4]. In addition, the extraordinary transmission behavior of sound waves through sub-wavelength slit arrays has also been reported, and apparently there is no surface plasmon [5]. The above examples show that the EOT phenomenon in the metal nano-array structure is independent of surface plasmon excitation and is mainly due to the Fabry-Pérot-type structural resonance [6]. Graphene has metalloid properties. Therefore, the graphene nano-array structure, or the graphene grating structure, can still observe the transmission enhancement phenomenon [7-9]. The transmission enhancement phenomenon in the metal grating is most prominent in the mid-infrared band; while the graphene grating exhibits strong asymmetric transmission absorption characteristics in the terahertz section. The optical stability of the metal makes it difficult to externally regulate by voltage or the like. However, the properties of graphene are externally adjustable. Therefore, many authors have done research on replacing metals with graphene, such as graphene instead of metal as a reflective *
Corresponding Author , Email: [email protected].
na
lP
re
-p
ro of
layer[10], or simply forming a graphene grating[11]. Further, in order to adjust the response of the grating to the incident frequency, the authors propose a graphene- metal hybrid grating structure [12-15]. This structure can be used for the study of nano-optical phenomena such as Fano resonance [12,13], negative reflection [14], resonance absorption or transmission [15] in plasmonics. Fano resonance has always been a hot spot in physics and applied research. In recent years, people have paid particular attention to the Fano resonance phenomenon in plasmonics. In optics, Fano resonance describes the resonant response of incident (excited) light as reflected by two modes of coupling and interference. In general, the Fano resonance requires a wide formant, also called the bright mode; another narrow formant, also called the dark mode. The Fano resonance occurs as a result of the coupling of the two modes. The metal in the metal-graphene hybrid grating provides a bright mode, and the graphene provides a dark mode, and their interaction is likely to induce Fano resonance. Experimentally, Chen et al. used a method of mixing graphene-metal gratings to achieve an adjustable Fano resonance [12]. However, how to regulate the bright and the dark mode, which parameters determine the role is not very clear. In the Fano resonance structures based on graphene sheets, many important results have been reported [16-18] while many problems have not been completely solved. For example, in addition to the above graphene-metal hybrid grating, what other methods can achieve Fano resonance? In addition, which metal can be combined with graphene to achieve the best Fano resonance effect? These require further research. In this paper, the Fano resonance effect of graphene-metal hybrid grating structure is studied. Materials used as transmission gratings require high transparency and low absorption, and graphene has such a property. ITO (Indium Tin Oxides, composed of indium oxide and tin oxide, also known as indium tin oxide) [19] and silver are good choices in metal materials. Silver was chosen for the calculation in this paper. In this paper, the absorption spectrum of the graphene-metal combined grating structure is calculated by the improved Fourier model method（FMM）[20,21]. A significant Fano resonance effect was observed by optimizing the grating structure parameters and changing the Fermi level of graphene. The results of this paper show that optimizing the grating structure parameters and regulating the properties of graphene are effective ways to achieve significant Fano resonance.
ur
2. Computational models and methods The hybrid graphene-metal grating discussed is shown in Fig.1. This is a graphene-metal nanoribbon periodic array, with period L . As depicted, the nanoribbons are infinitely long in y direction. The structure is illuminated by a TM polarized (magnetic field along the y direction)
Jo
plane wave (having vacuum wavelength upper and lower mediums are 1 and
) whose incident angle is . The permittivities of the
2 , respectively.
This structure is actually a diffraction grating, and there are many existing methods to analyze it. It is also a kind of stripe grating. After extensive research on gratings by people for half a century, several analysis methods have been developed. Among these methods, the FMM may be the most popular one because it is very simple but efficient[20,21]. However, this method requires strict Fourier decomposition, otherwise it has poor convergence[20,21]. In the following, we only consider TM polarized waves. As we all know, the electromagnetic field in the uniform region can be expressed by the following Rayleigh expansion:
H jy (a jn e
jk jzn z
b jne
ik jzn z
)e ikxn x
(1)
n
E jx n
where
k jzn
j
(a jn e
jk jzn z
b jn e
ik jzn z
)e ik xn x
(2)
j 1, 2 ， representing regions 1 or 2. In these expressions a jn and b jn are the
amplitudes of downward and upward modes, respectively, and k xn k0n1 sin
2 n， L
k jzn k02n j 2 k xn 2 ，where n j is the refractive index of the region j and k0 2 / is the vacuum wave number. The z component of the wave vector k jzn is either negative real
ro of
(propagating wave) or positive imaginary (evanescent wave). Eqs. (1) and (2) must satisfy certain boundary conditions. Here we use the improved boundary conditions given in references [20] and [21]:
E1x x, z 0 E2 x x, z 0 0
(3)
H1 y x, z h / 2 H 2 y x, z h / 2 seff ( x) Ex ( x, z 0)
-p
seff x s x i h is effective surface conductivity and (1 2 ) / 2 ,
re
where
(4)
e2 EgF
i
i 2 g1
na
gs
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gs ,0 x Wg 0,Wg x Wg W1 s x m ,Wg W1 x Wg W1 Wm 0,Wg W1 Wm x L 2
(5)
2 Eg e2 i [ H 2 EgF ln ] 4 2 EgF F
(6)
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The conductivity of the metal is given according to the Drude model
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In the above formula,
m
m0 1 i m
(7)
gs ( x ) is the surface conductivity of graphene, which exists only in
the graphene band, equal to the graphene surface conductivity, otherwise zero, and is a periodic function. Since [ s ( x )]
1
will be infinite when
s ( x ) is zero, its Fourier expansion does not
satisfy the inverse rule. In addition, this reason may also lead to slow convergence. Therefore, direct Fourier decomposition according to the conventional electromagnetic wave boundary conditions is problematic. This is why the improved approximate boundary conditions are given in Refs.[20,21]. In addition , e is the electron charge,
is the Planck constant, is the frequency, E g and F
g are the Fermi energy and relaxation time of graphene and H is a step function. m , m and mm are the conductivity, relaxation time, and effective mass of the metal. h is the parameter introduced for applying the boundary conditions. We have assumed that the time dependency is
eit . For computational convenience, we substitute (1) and (2) into the above boundary conditions (3) and (4) and organize them into the following S-matrix forms:
2 1 a2 1 2 a1 a 1 I / eff 1 a 1 b a a I / eff 1 b 2 1 1 2 1 2 2 2
j is a diagonal matrix whose (n,n) element is k jzn / j ,
ro of
where I is the identity matrix and
(8)
a j and b j are column vectors whose elements are a jn and b jn , respectively.
In the grating problem shown in Fig. 1 there is no upward wave in region 2, so b2 0 and
-p
the only nonzero element of a1 is a10 1 . Therefore, a2 and b1 , which are representing transmission and reflection coefficients of diffracted orders, respectively, are directly calculated from Eq.(8). Finally, reflection and transmission coefficients are given by (9)
k 2 T Re 1 2 zn a2 n n 2k1zn
(10)
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and absorption coefficient is
lP
re
k 2 R Re 1zm b1n n k1z 0
A 1 T R
(11)
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3. Numerical results and discussion To analyze how the Fano resonance (wavelength) is controlled, we calculated the transmission, reflection, and absorption curves for various parameter changes, as shown in Figs. 2-6. First, we explain the parameters used in the calculation as follows. is the dielectric constant of the medium and its relationship with the refractive index n is / 0 n . The medium with 2
1 / 0 =1 can be approximated as air. For example, when
1 / 0 2 / 0 1 , graphene and metal are suspended in the air, when 1 / 0 is suspended at
1 ( 1 / 0 1 ), one side of graphene and metal is exposed to the air. The medium of
/ 0 =4
can be used as an approximation of the actual medium such as SiO2 ( / 0 = 3.9), and the medium of
/ 0 = 9 is Al2O3.
If the upper and lower media of the graphene-metal grating are the same, it is called a symmetric environment, otherwise it is an asymmetric environment. Generally, when the upper and lower media of graphene-metal are asymmetrical, the graphene-metal can still be considered to be in a homogeneous medium environment, and the dielectric constant is taken as the average value of the upper and lower media. This is the boundary conditions given in the literature [20] and [21] . L is the period of the graphene-metal nanobelt, Wg and Wm are the width of the graphene and the metal strip, respectively, W1 and W2 the gap between the graphene-metal and the
g and EgF are the relaxation time and the Fermi level of
g 0.25ps . m and m 0 are the
the graphene. In the calculations below, we fixed
ro of
metal-graphene strip, respectively.
relaxation time and DC conductivity of the metal. In the calculation below, we take the metal as
silver and fix m 3.8 1014 s and m0 6.2 10 m . h is the boundary depth of 7
1
1
na
lP
re
-p
graphene set for the application of the boundary conditions. We take h 0.1nm in the following calculations. Fig.2 shows the relationship of the transmission, reflection and absorption curves of the hybrid graphene-metal grating with wavelength. The surface plasmon frequency of graphene is in the far infrared and terahertz regions, so the graphene grating resonance in Fig.2 occurs around 1×105 nm. No resonance occurs in the visible and near-infrared regions. The resonance of the metal grating occurs near 1×104 nm. Therefore, in order to interact with the scattered light of the metal grating and the graphene grating, that is, to make the Fano resonance occur, the resonance wavelength of the metal grating must be shifted to the long wave, and the resonance wavelength of the graphene grating is made to shift to the short wave. Compared to Fig.2, we made two major changes in the calculation of Fig.3. First, the period of the graphene-metal nanoribbon array was increased from 8 μm to 15 μm . Second, the
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asymmetric environment was changed to a symmetric environment. Fig.3 illustrates the transmission, reflection and absorption versus wavelength for such a hybrid graphene-metal grating. The curves of Fig. 3 are basically similar to that of Fig. 2, and the resonance positions of the metal and graphene grating are not effectively regulated, and the Fano resonance line type is not obvious. Next, we continue to adjust and optimize the parameters based on Fig.3. The period of the graphene-metal nanobelts remains at 15 μm . We have increased the width of the metal strip. The result is shown in Fig.4. It can be seen from Fig. 4 that the asymmetrical line shape of the curve basically appears, which is a typical feature of the Fano resonance, indicating that the parameter regulation has achieved initial success. Fig.4 illustrates that increasing the proportion of metal width in the hybrid graphene-metal grating, i.e., increasing Wm / Wg , favors the Fano resonance. However, the above calculations did not examine the effects of changes in graphene material parameters. In the calculation of Fig. 5, we set the grating structure parameters to be the same as in Fig. 3, but increase the Fermi level of
ro of
graphene from 0.6 eV to 1.2 eV . The calculation results are shown in Fig. 5. It can be seen that the effect of graphene Fermi level is obvious. The curve shows the asymmetrical line shape unique to the Fano resonance. Fig.4 illustrates that increasing the proportion of metal width in the hybrid graphene-metal grating facilitates the Fano resonance. Fig.5 shows that increasing the Fermi level of graphene also favors the Fano resonance. In the calculation of Fig.6, we simultaneously increase the proportion of metal width in the mixed graphene-metal grating and the Fermi level of graphene. The calculation results are shown in Fig. 6. Fig.6 shows that the line shape of the Fano resonance is very obvious. This shows that by optimizing the grating structure and simultaneously implementing the external regulation of graphene, the Fano resonance phenomenon of the mixed graphene-metal grating can be clearly observed. In Ref.[22], the optical absorption in the graphene-insulator-metal hybrid plasmonic device was studied, and the resonance absorption obtained was around from 1440nm to 1450 nm. The Terahertz tunable graphene Fano resonance was studied in Ref.[23], and the resonance frequency was about 1.75 THz. These are consistent with the results of this paper.
lP
re
-p
4. Summary In this paper, the transmission properties of the mixed graphene-metal gratings are studied by the FMM method and the transmission enhancement effect is observed. The above calculations do not involve the effects of the surface plasmon excitation, so the results should be independent of plasmon excitation, and the transmission enhancement effect is the result of structural Fabry-Pérot-type interference. In addition, by optimizing the grating structure and regulating the properties of the graphene material, we observed a typical Fano resonance effect. Obviously, this resonance is the result of the coupling of the two modes of metal grating and graphene grating. Our results show that the basic method for regulating this Fano resonance is to increase the proportion of metal width in the mixed graphene-metal grating and to increase the Fermi level of graphene.
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Acknowledgment It is hereby acknowledged that we used ee.sharif.ir/~khavasi/index_files/Page1077.htm.
the
code
by
A.
Khavasi
et
al.
at
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References [1] Andre-Pierre Blanchard-Dionne and Michel Meunier, Optical transmission theory for metal-insulatormetal periodic nanostructures ,Nanophotonics 6(1) (2017)349–355. [2] U. Schrte, D. Heitmann, Surface-plasmon-enhanced transmission through metallic gratings ,Phys. Rev. B 58(23) (1998)15419~15421. [3] A. G. Borisov, F. J. G. Abajo, S. V. Shabanov, Role of electromagnetic trapped modes in extraordinary transmission in nanostructured materials, Phys. Rev. B 71 (7) (2005) 075408. [4] D. Crouse, P. Keshavareddy, Polarization independent enhanced optical transmission in one-dimensional gratings and device applications, Opt. Exp. 15(4) (2007)1415~1427. [5] M. H. Lu, X. K. Liu, L. Feng ,J. Li , C. P. Huang , Y. F. Chen, Y. Y. Zhu, S.N.Zhu, N.B. Ming, Extraordinary acoustic transmission through a 1D grating with very narrow apertures, Phys. Rev. Lett. 99(17) (2007)174301.
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ro of
[6] Q. Cao, Ph. Lalanne, Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits, Phys. Rev. Lett. 88 (5) (2002)057403. [7]Z. Guo, Effect of dielectric environment on plasmonic resonance absorption of graphene nanoribbon arrays, Int. J. Mod. Phys. B 32(26)(2018)1850284. [8]Sukosin Thongrattanasiri, Frank H.L. Koppens, and F. Javier Garcıa de Abajo, Complete optical absorption in periodically patterned graphene, Phys. Rev. Lett. 108 (2012)047401. [9] S. Ke, B. Wang, H. Huang, H. Long, K. Wang, P. Lu, Plasmonic absorption enhancement in periodic cross-shaped graphene arrays ,Opt. Exp. 23(7) (2015)8888-8900. [10] A.Yu, Tunable strong THz absorption assisted by graphene-dielectric stacking structure, Superlattices Microstruct. 122 (2018) 461-470. [11] L. Yang ,J. Wang, C. Lu, Sensitive perfect absorber with monolayer graphene-based multi-layer dielectric grating structure, Optik 158(2018)508-513. [12]Z. Chen, J. Chen, Z. Wu, W. Hu, X. Zhang and Y. Lu，Tunable Fano resonance in hybrid graphene-metal gratings ,Appl. Phys. Lett. 104(2014)161114. [13] Bo Zhao and Zhuomin M. Zhang, Strong plasmonic coupling between graphene ribbon array and metal gratings ACS Photonics 2(2015)1611−1618. [14]X. Su, Z.Wei, C. Wu, Y. Long and H. Li, Negative reflection from metal/graphene plasmonic gratings, Opt. Lett. 41(2)(2016) 348-351. [15]Mohammad M. Jadidi, Andrei B. Sushkov, Rachael L. Myers-Ward, Anthony K. Boyd, Kevin M. Daniels, D. Kurt Gaskill, Michael S. Fuhrer, H. Dennis Drew and Thomas E. Murphy, Tunable terahertz hybrid metal−graphene plasmons, Nano Lett. 15(2015)7099−7104. [16]A. Ahmadivand, R. Sinha and N. Pala, Graphene plasmonics: multiple sharp Fano resonances in silver split concentric nanoring/disk resonator dimers on a metasurface, Plasmonics: Metallic Nanostructures and Their Optical Properties XIII. Vol. 9547. International Society for Optics and Photonics, 2015. [17] N. Dabidian, S. H. Mousavi, I. Kholmanov, K. Alici, D. Purtseladze, N. Arju, K. Tatar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, Inductive tuning of Fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared ,Nano Lett. 13(3) (2013) 1111-1117. [18] D. H.Chae, T. Utikal, S.Weisenburger, H.Giessen, K. V. Klitzing, M. Lippitz and J.Smet, Excitonic Fano resonance in free-standing graphene, Nano Lett. 11(3) (2011)1379-1382. [19] A. Melikyan and N. Lindenmann, A surface plasmon polariton absorption modulator, Opt. Exp. 19 (9) (2011) 8855-8869. [20]A. Khavasi, Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons, Opt. Lett. 38(16)(2013)3009-3013. [21]A. Khavasi, Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons, J. Opt. 14 (2012) 125502. [22]N. Matthaiakakis, X. Yan, H. Mizuta and M. D. B. Charlton, Tuneable strong optical absorption in a graphene-insulator-metal hybrid plasmonic device, Scientific Reports 7.7303(2017). [23] X. He, F. Lin, F. Liu and W. Shi, Terahertz tunable graphene Fano resonance, Nanotechnology 27 (2016) 485202.
Figure captions
Fig.1. Schematic diagram of the hybrid graphene-metal grating
ro of
Fig.2. Transmission, reflection and absorption of graphene-metal nanoribbon periodic arrays versus wavelength. The transmission, reflection and absorption are denoted as T,R and A and in red, green and blue (The same marks are used in the following figures). The calculated parameters used are as follows: L 8μm ， Wg 0.6 L ， Wm 0.2 L ， W1 W2 0.1L ， 1 3 0 ，
-p
2 7 0 ， EgF 0.6eV .
Fig.3. Transmission, reflection and absorption curves of the hybrid graphene-metal grating as a
L 15μm ，
re
function of wavelength with parameters
Wg 8μm , Wm 3μm ,
lP
F W1 W2 2μm , 1 2 3.5 0 ，and Eg 0.6eV .
Fig.4. Transmission, reflection and absorption of graphene-metal nanoribbon periodic arrays
na
versus wavelength. The calculated parameters used are as follows: L 15μm ， Wg Wm 0.4L ,
ur
F W1 W2 0.1L , 1 2 3.5 0 ， Eg 0.6eV .
Fig.5. Transmission, reflection and absorption curves of the hybrid graphene-metal grating with
L 15μm ， Wg 8μm , Wm 3μm , W1 W2 2μm ,
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wavelength at the case
1 2 3.5 0 ， EgF 1.2eV .
Fig.6. Transmission, reflection and absorption of the hybrid graphene-metal gratings as the function of wavelength. The calculated parameters used are as follows: L 15μm ， F Wg Wm 0.4L , W1 W2 0.1L , 1 2 3.5 0 ， Eg 1.2eV .
ro of -p lP
re
Fig.1
0.6
na
0.8
T
=3 =7
ur
Transmission, reflection and absorption
1.0
Jo
0.4
A
0.2
R
0.0
4
5.0x10
5
1.0x10
Wavelength (nm)
Fig.2
5
1.5x10
5
2.0x10
0.8 T
0.6
ro of
=3.5 =3.5
0.4 A
0.2 0.0
R 4
5.0x10
5
-p
Transmission, reflection and absorption
1.0
5
1.0x10
1.5x10
re
Wavelength (nm)
na
0.8
T
0.6
ur
Transmission, reflection and absorption
1.0
lP
Fig.3
=3.5 =3.5
Jo
0.4
A
0.2 0.0
R 4
5.0x10
5
1.0x10
Wavelength (nm)
Fig.4
5
2.0x10
5
1.5x10
5
2.0x10
0.8 T
0.6
=3.5 =3.5
A
0.2 0.0
R 4
5
5.0x10
5
1.0x10
1.5x10
5
2.0x10
re
Wavelength (nm)
lP
Fig.5
na
0.8
T
0.6
ur
Transmission, reflection and absorption
1.0
=3.5 =3.5
0.4
Jo
A
0.2
R
0.0
4
5.0x10
5
1.0x10
Wavelength (nm)
Fig.6
ro of
0.4
-p
Transmission, reflection and absorption
1.0
5
1.5x10
5
2.0x10