Abstract

Electromagnetic (EM) wave absorption plays a vital role in photonics. While metasurfaces are proposed to absorb EM waves efficiently, most of them exhibit limited bandwidth and fixed functionalities. Here, we propose a broadband and tunable terahertz (THz) absorber based on a graphene-based metasurface, which is constructed by a single layer of closely patterned graphene concentric double rings and a metallic mirror separated by an ultrathin SiO2 layer. Plasmonic hybridization between two graphene rings significantly enlarges the absorption bandwidth, which can be further tuned by gating the graphene. Moreover, the specific design also makes our device insensitive to the incident angle and polarization state of impinging EM waves. Our results may inspire certain wave-modulation-related applications, such as THz imaging, smart absorber, tunable sensor, etc.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Perfect electromagnetic (EM) absorbers have drawn considerable interests in photonic research due to many potential applications, such as solar photovoltaics cells, heat radiations, photo-detections [1,2]. Recently, metamaterials, artificial materials composed by subwavelength micro-units (or meta-atoms) arranged in specific orders, have shown unprecedented abilities to manipulate EM waves. Many fascinating physical effects were realized, such as negative refraction [3,4], super resolution imaging [5], optical cloaking [6,7]. In particular, Landy et al. reported a metamaterial-based prefect absorber (MPA) in 2008, designed based on impedance-matching principle via manipulating the system’s electric and magnetic responses simultaneously [8]. Almost at the same time, Teperik et al. proposed a nanostructured metal surface to achieve omnidirectional light absorption aided by surface plasmon resonance [9]. Soon after that, a class of reflection-type metasurfaces, typically consisting of metal-insulator-metal sandwiched structures, were widely adopted to achieve perfect EM-wave absorption, utilizing the magnetic resonance between the top and bottom metallic layers [10,11]. These ideas inspire a series of works on EM wave absorptions ranging from microwave, THz to optical frequency [12–15]. However, while these MPAs have exhibited advantages in miniaturization, most of them still suffer from restrictions such as limited working bandwidth and fixed functionality, being unfavorable for certain applications.

As an important family member of MPAs, THz absorbers attracted much attention recently [13,16]. In particular, graphene-based MPAs exhibit exotic properties thanks to the exotic electronic, optical, mechanical, and gate-tunable properties of graphene [17]. Many graphene-based MPAs were proposed [18–25], which typically consist of graphene microstructures in different shapes, operating based on the plasmonic resonances in such graphene microstructures. However, most of these proposed MPAs still exhibit narrow working bandwidth. Although several approaches [26–32] were established to solve this issue, the proposed structures still possess some limitations, such as complex structure, decreased absorptivity.

In this paper, we propose a broadband and tunable THz wave absorber based on a simple graphene metasurface. Our device is a sandwich structure composed by a single-layer of graphene concentric double ring (GCDR) array and a metallic mirror separated by a thin SiO2 spacer. Lateral couplings between two graphene rings hybridize the plasmonic resonances in two individual rings, which can significantly enlarge the device’s working bandwidth under certain conditions. Meanwhile, the functionality of our device is insensitive to both incident angle and polarization of impinging THz wave, thanks to the ring geometry and the deep-subwavelength responses of graphene microstructures. Finally, the working frequency band of the proposed device can be efficiently tuned through modulating the Femi energy of the graphene. Our results may point out a new possibility to achieve broadband tunable meta-devices in THz and other frequency domains.

2. Results and discussions

2.1 Geometry and principle of the proposed graphene absorber

Figure 1 depicts the schematic of the proposed graphene-based MPA, which is constructed by a single layer of the GCDR array and an optically thick gold mirror, separated by a 28μm-thicklossless SiO2 spacer withεd=3.9. Here, the concentric inner and outer graphene rings are optimized with1.5μmseparation to induce the plasmonic hybridization effect and thus achieve the desired broadband absorption (see detailed structural parameters in the caption of Fig. 1). Here, the metallic ground plate with a conductivity σgold=4.09×107S/m will guarantee the complete reflection for the impinging THz wave like a “mirror”. In our proposed device, a layer of ion-gel was spin-coated on the graphene patterns and contacted to the Au electrodes serving as the top gate. Meanwhile, an ultrathin and transparent conductive thin layer was deposited between SiO2 layer and graphene rings serving as the bottom gate [21].

 figure: Fig. 1

Fig. 1 Schematic of the proposed broadband THz absorber consisting of the GCDRs and the gold mirror separated by a thin SiO2 spacer. The ion-gel was spin-coated on the graphene nanostructures and contacted to the Au electrodes as the top gate. And an ultrathin and transparent conduct thin layer was deposited between the SiO2 and graphenes as the bottom gate. The geometrical parameters of GCDRs are a=5.5μm,b=4μm,c=2.5μm,d=2.2μm,t=28μm, and the lattice constant isL=15μm.

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In our numerical simulations, we utilize an effective surface conductivity model to describe the graphene layer. It is known that, the surface conductivity of graphene σgra is governed by the Kubo formula that takes both the intraband and interband transitions into considerations [33–35]

σgra=σintra+σinter=2e2kBTπ2iω+i/τln[2cosh(Ef2kBT)]+e242[12+1πarctan(ω2Ef2kBT)i2πln(ω+2Ef)2(ω2Ef)2+4(kBT)2]
where ω is the angular frequency, Ef is the Femi energy level, τ is the electron-phonon relaxtion time, and T is the absolute temperature of the environment. And kB,e, are Boltzman constant, electron charge and reduced Plank constant (=h/2π), respectively. For the THz frequency domain (hω2Ef) at room temperature, the interband transition is negligibly small according to Pauli exclusion principle and therefore the surface conductivity can be safely simplified to a Drude-like model [23,36,37]
σgra=e2Efπ2i(ω+i/τ)
One of the most important features of graphene is that its Femi energy can be freely modulated within a broad range by applying an electrostatic biasing [37]. In our study, we choose an arbitrary initial value of Ef=0.5eV and then sweep its value to study the tunable functionality of the proposed graphene MPA. We choose a fixed relaxation time of τ=1ps(corresponding to the electron mobilityμ=2×104cm2V1s1) for simplicity, which is large but achievable for the high-quality graphene obtained by the mechanical expholiation method [17,38]. Rigorously speaking, the relaxation time in graphene should depend on both the frequency and Fermi energy [39]. However, such dependences are typically weak and we believe that considering them should not significantly affect the main conclusions drawn in the present paper.

The working principle of such reflective MPAs can be understood as follows. While the incident light illuminates the device, anti-parallel currents are excited on both the top graphene layer and the bottom metallic mirror, leading to a strong magnetic resonance [10,11,40,41]. Since the graphene layer and the metallic mirror are very close (i.e.,28μm or aboutλ/8separation), strong coupling effect leads to an extreme strong near fields that may be dissipated by the lossy graphenes. Guided by our previous work [42], the quality factors of the radiation loss and absorption loss of our device is optimized to equal to each other, which is a straightforward criterion to achieve the perfect absorption functionality around the resonance frequency. In another similar picture, illuminated by the incident light, localized surface plasmon modes or gap plasmon modes will be excited inside the graphene microstructures [21,43,44], resulting in considerable enhancement of light absorption.

2.2 Numerical demonstrations of the broadband absorption

We perform finite element method (FEM) numerical simulations to investigate the proposed graphene based THz meta-absorber. To obtain its EM responses, we put one unit cell inside the simulation box that is surrounded by periodic boundaries to describe equivalently an infinitely large MPA slab. A plane wave port is adopted as both the source to shine the device and the receiver to obtain its reflectance spectrum, i.e.,R(ω)=|S11(ω)|2. Since there is no transmission for our device designed in a total reflection geometry, its absorption efficiency can be described easily by the equationA(ω)=1R(ω).

First, we consider the case that x-polarized THz waves are normally illuminated on the device, as illustrated in Fig. 1. Figure 2(a) depicts the calculated absorption spectra of the proposed graphene MPA (see black line), with its top view shown in Fig. 2(d). It is noted that the proposed device exhibits a high absorptivity of above 90% from 1.18 to 1.64THz. Obviously, there exsits two absorption peaks located respectively at 1.26THz and 1.54THz within the band. To clarify their physical origins, we further calculate the absorption spectra of two simplified graphene metasurfaces with the original GCDRs at the top layer replaced by either the isolated inner ring (see Fig. 2(b)) or the isolated outer ring (see Fig. 2(c)). Obviously, while the impinging wave is incident upon the absorber shown in Fig. 2(b) (top view), a resonance peak appears at about 1.48THz (see red line in Fig. 2(a)), that should correspond to the high-frequency absorption peak of the GCDR structure (see black line in Fig. 2(a)). Analogously, the absorption peak at 1.28THz for the absorber composed by the outer graphene ring (see green line in Fig. 2(a)) should correspond to the low-frequency absorption peak of the GCDR structure (see black line in Fig. 2(a)). Quite interestingly, although both of the two graphene single ring (GSR) metasurfaces (see Figs. 2(b, c)) exhibit limited absorptivity and narrow bandwidth, the GCDRs absorber can function as a perfect THz absorber with a absorptivity of high above 0.9 and a broad working band from 1.18 to 1.64THz. Moreover, while the low-frequency peak is red-shifted from 1.28 to 1.26 THz, the high-frequency peak is blue-shifted from 1.48 to 1.54THz, compared with the two GSR structures. Therefore, the GCDRs absorber can work well within much larger frequency band. Indeed, we purposely utilize the coupling effect between the two GSRs to achieve the good performance of the proposed GCDR absorber [45–47].

 figure: Fig. 2

Fig. 2 (a) Simulated absorption spectra of the proposed GCDR absorber (black line) and two GSR absorbers with the top graphene double ring replaced by the isolated inner ring (red line) and outer ring (green line). (b-d) Top views of three graphene MPAs comprised by the single inner ring, single outer ring and double rings, respectively.

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Next, we perform full wave simulations to verify the plasmonic hybridization model utilized in the proposed GCDR metasurface. Figures 3(a, b) shows the calculated electric field patterns (arrows for the direction and color for the intensity) at the top graphene layer of two GSR absorbers at their absorption peaks, i.e., 1.28THz and 1.48THz. Obviously, illuminated by the impinging light, positive and negative induced charges are accumulated at opposite sides of the graphene ring. The electric dipolar responses depicted in Figs. 3 (a, b) give rise to the absorption peaks at 1.28 and 1.48THz, respectively (see green and red lines of Fig. 2). The electric field patterns for the GCDR absorber at its two absorption peaks are presented in Figs. 3(c, d). The directions of electric field (see arrows) clarify clearly the induced charges of the coupled plasmonic modes being “++” at 1.26THz and “++” at 1.54THz, respectively. It implies that the low (high) frequency hybridized mode of the GCDR absorber depicted in Fig. 3(c) (Fig. 3(d)) is formed by two anti-parallel (parallel) dipolar modes of the graphene single rings. Since the inner and outer graphene rings are close in the concentric configuration, their dipolar modes can interact with each other and form the hybridized plasmonic modes. Moreover, due to the coupling effect, the low (high) frequency hybridized mode is red (blue) shifted to 1.26THz (1.54THz), as compared to the resonance mode of GSR absorber shown in Fig. 3(a) (Fig. 3(b)). Such behaviors perfectly justify our idea to realize the broadband THz wave absorption based on the plasmonic hybridization model [45–47], as schematically shown in Fig. 3(e).

 figure: Fig. 3

Fig. 3 (a-d) Simulated electric field patterns (arrows for the direction and color for the intensity) at the top graphene layers of three absorbers shown in Fig. 2 at their corresponding absorption peaks (see inset for the frequency). (e) Energy-level diagram to describe the plasmonic hybridization effect in the proposed GCDR absorber due to the mutual coupling between its inner and outer graphene rings.

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2.3 Verification of incidence-angle-insensitive, polarization-independent and frequency-tunable properties

The proposed GCDR metasurface exhibits robust functionalities that are highly desired in practical applications. Figures 4(a, b) depict the simulated absorption efficiencies versus the frequency (x-axis) and incident angle (y-axis) for transverse electric (TE, i.e., E field along y direction) and transverse magnetic (TM, i.e., H field along y direction) polarization. Here, the Fermi energy of the graphene is fixed atEf=0.5eV. It is noted that, for both polarizations, our structure can sustain a rather broad absorption band while the incident angle varies from 0° to 60°. In addition, some minor differences show up here. For TE polarization case (see Fig. 4(a)), as the incident angle increases, the two resonance modes red shift slightly with an additional resonance absorption peak appearing in the high frequency domain, leading to an enlarged absorption band. As for TM polarization case (see Fig. 4(b)), with the increase of incidence angle, the two resonance modes almost do not shift with their quality factors increased a little bit, leading to a slightly shrunk absorption bandwidth. The overall reduction of the absorptivity is inevitable because the interactions between incident electric field and the patterned graphene rings become weaker as the incident angle increases [48]. In general, the functionality of our device is insensitive to the incident angle, thanks to the deep subwavelength property of the graphene based metasurfaces.

 figure: Fig. 4

Fig. 4 Calculated absorptivity (color map) as function of the frequency (x-axis) and the incident angle (y-axis), illuminated by (a) TE and (b) TM polarized waves.

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Next, we investigate the polarization-angle dependence of the proposed graphene absorber. Figure 5(a) depicts its absorption spectra with the polarization angle φof incident waves varied to 0,15,30,45, respectively, for the normal incident case. Obviously, the absorptivity of our device is invariant strictly to the polarization angleφ. It is quite understandable considering that the proposed GCDR absorber possesses the rotation-invariance symmetry. Figure 5(b) depicts the electric field pattern on the top graphene layer at 1.26THz (one of the absorption peaks) for several differentφcases. As the polarization angle is varied, the hybridized plasmonic modes can be always excited inside the structure, just with their orientation angles rotating with the incident electric field simultaneously.

 figure: Fig. 5

Fig. 5 (a) Absorption characteristics of the proposed absorber for normally incident waves with different polarization angleφ. Here,φrepresents the angle of the incident electric field relative to x-axis. (b) The simulated electric field patterns on the top graphene layer at the first absorption peak (1.26THz) forφ=0o,φ=15o,φ=30o,φ=45o.

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The absorption band of the proposed GCDR absorber can be further modulated via tuning the gating voltage applied on the graphene. As described by Eq. (2), the surface conductivity of graphene is almost linearly dependent on its Fermi energy in THz regime. Supposing that an electric gate is applied on our device, the Fermi energy of the graphene can be modulated continuously with the tuned gate voltage, thus enabling its electrically tunable property. Figure 6 shows the broadband absorption spectra as the function of the frequency (x-axis) and the Fermi energy (y-axis). Interestingly, as the Fermi Energy changes from 0.35eV to 0.65eV, the broad absorption band moving form 0.98~1.36 THz to 1.36~1.94THz with the absorptivity kept at above 80%. The increased bandwidth is due to the different blue shifting tendencies of two hybridized plasmonic modes.

 figure: Fig. 6

Fig. 6 Calculated absorptivity (color map) as function of the frequency (x-axis) and the Fermi energy (y-axis) of the proposed GCDR absorber for normal incident case.

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3. Conclusion

In summary, we proposed a simple graphene metasurface to achieve broadband, tunable, polarization-independent and incident-angle-insensitive THz perfect absorption. Plasmonic hybridization [49] helps enlarge the absorption bandwidth, while the working frequency can be further tuned through gating the graphene. Compared to the conventional metallic meta-absorbers, the graphene-based devices can be much more subwavelength. The deep-subwavelength resonances and the specific ring geometry eliminate the incidence-angle and polarization-state sensitivities of the proposed device. Compared to previous multi-layer broadband absorbers [22,30], our deign is much simpler and is thus convenient for practical applications. We believe that this idea may inspire other tunable and broadband meta-devices working in both THz and other frequency domains, including heat radiators, smart absorber, tunable sensors, etc.

Funding

National Natural Science Foundation of China (NSFC) (Nos. 11404063, 11474057, 11674068, 11734007, 61675219, 61475173); National Basic Research Program of China (2017YFA0303500); Shanghai Science and Technology Committee (Grant No. 16ZR1445200, 16JC1403100, 18ZR1403400); Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences.

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References

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  1. C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
    [PubMed]
  2. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
    [Crossref] [PubMed]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [Crossref] [PubMed]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
    [Crossref] [PubMed]
  5. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [Crossref] [PubMed]
  6. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref] [PubMed]
  7. U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006).
    [Crossref] [PubMed]
  8. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
    [Crossref] [PubMed]
  9. T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
    [Crossref]
  10. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
    [Crossref] [PubMed]
  11. J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
    [Crossref]
  12. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
    [Crossref] [PubMed]
  13. H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
    [Crossref] [PubMed]
  14. X. Shen, T. J. Cui, J. Zhao, H. F. Ma, W. X. Jiang, and H. Li, “Polarization-independent wide-angle triple-band metamaterial absorber,” Opt. Express 19(10), 9401–9407 (2011).
    [Crossref] [PubMed]
  15. M. Lei, N. Feng, Q. Wang, Y. Hao, S. Huang, and K. Bi, “Magnetically tunable metamaterial perfect absorber,” J. Appl. Phys. 119(24), 244504 (2016).
    [Crossref]
  16. N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
    [Crossref]
  17. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [Crossref] [PubMed]
  18. R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
    [Crossref] [PubMed]
  19. S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
    [Crossref] [PubMed]
  20. Y. Zhang, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “Graphene based tunable metamaterial absorber and polarization modulation in terahertz frequency,” Opt. Express 22(19), 22743–22752 (2014).
    [Crossref] [PubMed]
  21. Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
    [Crossref] [PubMed]
  22. Z. Su, J. Yin, and X. Zhao, “Terahertz dual-band metamaterial absorber based on graphene/MgF2 multilayer structures,” Opt. Express 23(2), 1679–1690 (2015).
    [Crossref] [PubMed]
  23. S. Xiao, T. Wang, Y. Liu, C. Xu, X. Han, and X. Yan, “Tunable light trapping and absorption enhancement with graphene ring arrays,” Phys. Chem. Chem. Phys. 18(38), 26661–26669 (2016).
    [Crossref] [PubMed]
  24. P. Y. Chen, M. Farhat, and H. Bağcı, “Graphene metascreen for designing compact infrared absorbers with enhanced bandwidth,” Nanotechnology 26(16), 164002 (2015).
    [Crossref] [PubMed]
  25. P. Y. Chen, C. Argyropoulos, M. Farhat, and J. S. Gomez-Diaz, “Flatland plasmonics and nanophotonics based on graphene and beyond,” Nanophotonics 6(6), 1239–1262 (2017).
    [Crossref]
  26. L. Huang, D. R. Chowdhury, S. Ramani, M. T. Reiten, S.-N. Luo, A. J. Taylor, and H.-T. Chen, “Experimental demonstration of terahertz metamaterial absorbers with a broad and flat high absorption band,” Opt. Lett. 37(2), 154–156 (2012).
    [Crossref] [PubMed]
  27. Q. Feng, M. Pu, C. Hu, and X. Luo, “Engineering the dispersion of metamaterial surface for broadband infrared absorption,” Opt. Lett. 37(11), 2133–2135 (2012).
    [Crossref] [PubMed]
  28. F. Ding, Y. Cui, X. Ge, Y. Jin, and S. He, “Ultra-broadband microwave metamaterial absorber,” Appl. Phys. Lett. 100(10), 103506 (2012).
    [Crossref]
  29. J. Zhu, Z. Ma, W. Sun, F. Ding, Q. He, L. Zhou, and Y. Ma, “Ultra-broadband terahertz metamaterial absorber,” Appl. Phys. Lett. 105(2), 021102 (2014).
    [Crossref]
  30. M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
    [Crossref] [PubMed]
  31. S. Yi, M. Zhou, X. Shi, Q. Gan, J. Zi, and Z. Yu, “A multiple-resonator approach for broadband light absorption in a single layer of nanostructured graphene,” Opt. Express 23(8), 10081–10090 (2015).
    [Crossref] [PubMed]
  32. X. Shi, L. Ge, X. Wen, D. Han, and Y. Yang, “Broadband light absorption in graphene ribbons by canceling strong coupling at subwavelength scale,” Opt. Express 24(23), 26357–26362 (2016).
    [Crossref] [PubMed]
  33. E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
    [Crossref]
  34. L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).
    [Crossref]
  35. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
    [Crossref]
  36. S. Ke, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Plasmonic absorption enhancement in periodic cross-shaped graphene arrays,” Opt. Express 23(7), 8888–8900 (2015).
    [Crossref] [PubMed]
  37. G. Yao, F. Ling, J. Yue, C. Luo, Q. Luo, and J. Yao, “Dynamically electrically tunable broadband absorber based on graphene analog of electromagnetically induced transparency,” IEEE Photonics J. 8(1), 7800808 (2016).
    [Crossref]
  38. H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
    [Crossref] [PubMed]
  39. H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
    [Crossref]
  40. J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating Electromagnetic Wave Polarizations by Anisotropic Metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007).
    [Crossref] [PubMed]
  41. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
    [Crossref] [PubMed]
  42. C. Qu, S. Ma, J. Hao, M. Qiu, X. Li, S. Xiao, Z. Miao, N. Dai, Q. He, S. Sun, and L. Zhou, “Tailor the functionalities of metasurfaces based on a complete phase diagram,” Phys. Rev. Lett. 115(23), 235503 (2015).
    [Crossref] [PubMed]
  43. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
    [Crossref] [PubMed]
  44. F. Ding, J. Dai, Y. Chen, J. Zhu, Y. Jin, and S. I. Bozhevolnyi, “Broadband near-infrared metamaterial absorbers utilizing highly lossy metals,” Sci. Rep. 6(1), 39445 (2016).
    [Crossref] [PubMed]
  45. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
    [Crossref] [PubMed]
  46. Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
    [Crossref] [PubMed]
  47. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007).
    [Crossref]
  48. H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
    [Crossref]
  49. Y. T. Zhao, B. Wu, B. J. Huang, and Q. Cheng, “Switchable broadband terahertz absorber/reflector enabled by hybrid graphene-gold metasurface,” Opt. Express 25(7), 7161–7169 (2017).
    [Crossref] [PubMed]

2017 (2)

P. Y. Chen, C. Argyropoulos, M. Farhat, and J. S. Gomez-Diaz, “Flatland plasmonics and nanophotonics based on graphene and beyond,” Nanophotonics 6(6), 1239–1262 (2017).
[Crossref]

Y. T. Zhao, B. Wu, B. J. Huang, and Q. Cheng, “Switchable broadband terahertz absorber/reflector enabled by hybrid graphene-gold metasurface,” Opt. Express 25(7), 7161–7169 (2017).
[Crossref] [PubMed]

2016 (5)

F. Ding, J. Dai, Y. Chen, J. Zhu, Y. Jin, and S. I. Bozhevolnyi, “Broadband near-infrared metamaterial absorbers utilizing highly lossy metals,” Sci. Rep. 6(1), 39445 (2016).
[Crossref] [PubMed]

G. Yao, F. Ling, J. Yue, C. Luo, Q. Luo, and J. Yao, “Dynamically electrically tunable broadband absorber based on graphene analog of electromagnetically induced transparency,” IEEE Photonics J. 8(1), 7800808 (2016).
[Crossref]

S. Xiao, T. Wang, Y. Liu, C. Xu, X. Han, and X. Yan, “Tunable light trapping and absorption enhancement with graphene ring arrays,” Phys. Chem. Chem. Phys. 18(38), 26661–26669 (2016).
[Crossref] [PubMed]

X. Shi, L. Ge, X. Wen, D. Han, and Y. Yang, “Broadband light absorption in graphene ribbons by canceling strong coupling at subwavelength scale,” Opt. Express 24(23), 26357–26362 (2016).
[Crossref] [PubMed]

M. Lei, N. Feng, Q. Wang, Y. Hao, S. Huang, and K. Bi, “Magnetically tunable metamaterial perfect absorber,” J. Appl. Phys. 119(24), 244504 (2016).
[Crossref]

2015 (5)

2014 (4)

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

Y. Zhang, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “Graphene based tunable metamaterial absorber and polarization modulation in terahertz frequency,” Opt. Express 22(19), 22743–22752 (2014).
[Crossref] [PubMed]

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref] [PubMed]

J. Zhu, Z. Ma, W. Sun, F. Ding, Q. He, L. Zhou, and Y. Ma, “Ultra-broadband terahertz metamaterial absorber,” Appl. Phys. Lett. 105(2), 021102 (2014).
[Crossref]

2013 (3)

M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
[Crossref] [PubMed]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

2012 (7)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

L. Huang, D. R. Chowdhury, S. Ramani, M. T. Reiten, S.-N. Luo, A. J. Taylor, and H.-T. Chen, “Experimental demonstration of terahertz metamaterial absorbers with a broad and flat high absorption band,” Opt. Lett. 37(2), 154–156 (2012).
[Crossref] [PubMed]

Q. Feng, M. Pu, C. Hu, and X. Luo, “Engineering the dispersion of metamaterial surface for broadband infrared absorption,” Opt. Lett. 37(11), 2133–2135 (2012).
[Crossref] [PubMed]

F. Ding, Y. Cui, X. Ge, Y. Jin, and S. He, “Ultra-broadband microwave metamaterial absorber,” Appl. Phys. Lett. 100(10), 103506 (2012).
[Crossref]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
[Crossref] [PubMed]

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
[PubMed]

2011 (3)

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

X. Shen, T. J. Cui, J. Zhao, H. F. Ma, W. X. Jiang, and H. Li, “Polarization-independent wide-angle triple-band metamaterial absorber,” Opt. Express 19(10), 9401–9407 (2011).
[Crossref] [PubMed]

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

2010 (3)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref] [PubMed]

2009 (1)

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
[Crossref]

2008 (5)

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[Crossref] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[Crossref]

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[Crossref]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

2007 (5)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007).
[Crossref]

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating Electromagnetic Wave Polarizations by Anisotropic Metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007).
[Crossref] [PubMed]

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).
[Crossref]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref] [PubMed]

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

2005 (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

2003 (1)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Abdelsalam, M.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[Crossref]

Ajayan, P. M.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref] [PubMed]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

Alaee, R.

Amin, M.

Argyropoulos, C.

P. Y. Chen, C. Argyropoulos, M. Farhat, and J. S. Gomez-Diaz, “Flatland plasmonics and nanophotonics based on graphene and beyond,” Nanophotonics 6(6), 1239–1262 (2017).
[Crossref]

Atwater, H. A.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref] [PubMed]

Averitt, R. D.

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[Crossref] [PubMed]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

Avouris, P.

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Aydin, K.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

Bagci, H.

P. Y. Chen, M. Farhat, and H. Bağcı, “Graphene metascreen for designing compact infrared absorbers with enhanced bandwidth,” Nanotechnology 26(16), 164002 (2015).
[Crossref] [PubMed]

M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
[Crossref] [PubMed]

Bartlett, P. N.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[Crossref]

Baumberg, J. J.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[Crossref]

Bechtel, H. A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

Bi, K.

M. Lei, N. Feng, Q. Wang, Y. Hao, S. Huang, and K. Bi, “Magnetically tunable metamaterial perfect absorber,” J. Appl. Phys. 119(24), 244504 (2016).
[Crossref]

Bingham, C. M.

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
[Crossref]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[Crossref] [PubMed]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

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Nordlander, P.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref] [PubMed]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Novoselov, K. S.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref] [PubMed]

Padilla, W. J.

C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
[PubMed]

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
[Crossref]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[Crossref] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
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J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
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L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).
[Crossref]

Pilon, D.

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref] [PubMed]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Pu, M.

Qiu, M.

C. Qu, S. Ma, J. Hao, M. Qiu, X. Li, S. Xiao, Z. Miao, N. Dai, Q. He, S. Sun, and L. Zhou, “Tailor the functionalities of metasurfaces based on a complete phase diagram,” Phys. Rev. Lett. 115(23), 235503 (2015).
[Crossref] [PubMed]

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

Qu, C.

C. Qu, S. Ma, J. Hao, M. Qiu, X. Li, S. Xiao, Z. Miao, N. Dai, Q. He, S. Sun, and L. Zhou, “Tailor the functionalities of metasurfaces based on a complete phase diagram,” Phys. Rev. Lett. 115(23), 235503 (2015).
[Crossref] [PubMed]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Ramani, S.

Ran, L.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating Electromagnetic Wave Polarizations by Anisotropic Metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007).
[Crossref] [PubMed]

Reiten, M. T.

Rockstuhl, C.

Sajuyigbe, S.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Schlather, A.

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

Schlather, A. E.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref] [PubMed]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

Shen, X.

Shen, Y. R.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

Shi, X.

Shrekenhamer, D.

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

Smith, D. R.

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
[Crossref]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

Strikwerda, A. C.

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

Su, Z.

Sugawara, Y.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[Crossref]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Sun, S.

C. Qu, S. Ma, J. Hao, M. Qiu, X. Li, S. Xiao, Z. Miao, N. Dai, Q. He, S. Sun, and L. Zhou, “Tailor the functionalities of metasurfaces based on a complete phase diagram,” Phys. Rev. Lett. 115(23), 235503 (2015).
[Crossref] [PubMed]

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Sun, W.

J. Zhu, Z. Ma, W. Sun, F. Ding, Q. He, L. Zhou, and Y. Ma, “Ultra-broadband terahertz metamaterial absorber,” Appl. Phys. Lett. 105(2), 021102 (2014).
[Crossref]

Tao, H.

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[Crossref] [PubMed]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

Taylor, A. J.

Teperik, T. V.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[Crossref]

Thongrattanasiri, S.

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
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S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
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Tyler, T.

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
[Crossref]

Wang, B.

Wang, F.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

Wang, J.

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

Wang, K.

Wang, Q.

M. Lei, N. Feng, Q. Wang, Y. Hao, S. Huang, and K. Bi, “Magnetically tunable metamaterial perfect absorber,” J. Appl. Phys. 119(24), 244504 (2016).
[Crossref]

Wang, T.

S. Xiao, T. Wang, Y. Liu, C. Xu, X. Han, and X. Yan, “Tunable light trapping and absorption enhancement with graphene ring arrays,” Phys. Chem. Chem. Phys. 18(38), 26661–26669 (2016).
[Crossref] [PubMed]

Wang, Y.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref] [PubMed]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

Watts, C. M.

C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
[PubMed]

Weiss, T.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

Wen, X.

Wu, B.

Wu, Y.

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Xia, F.

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Xiao, S.

S. Xiao, T. Wang, Y. Liu, C. Xu, X. Han, and X. Yan, “Tunable light trapping and absorption enhancement with graphene ring arrays,” Phys. Chem. Chem. Phys. 18(38), 26661–26669 (2016).
[Crossref] [PubMed]

C. Qu, S. Ma, J. Hao, M. Qiu, X. Li, S. Xiao, Z. Miao, N. Dai, Q. He, S. Sun, and L. Zhou, “Tailor the functionalities of metasurfaces based on a complete phase diagram,” Phys. Rev. Lett. 115(23), 235503 (2015).
[Crossref] [PubMed]

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Xu, C.

S. Xiao, T. Wang, Y. Liu, C. Xu, X. Han, and X. Yan, “Tunable light trapping and absorption enhancement with graphene ring arrays,” Phys. Chem. Chem. Phys. 18(38), 26661–26669 (2016).
[Crossref] [PubMed]

Xu, Q.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Yan, H.

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Yan, X.

S. Xiao, T. Wang, Y. Liu, C. Xu, X. Han, and X. Yan, “Tunable light trapping and absorption enhancement with graphene ring arrays,” Phys. Chem. Chem. Phys. 18(38), 26661–26669 (2016).
[Crossref] [PubMed]

Yang, Y.

Yao, G.

G. Yao, F. Ling, J. Yue, C. Luo, Q. Luo, and J. Yao, “Dynamically electrically tunable broadband absorber based on graphene analog of electromagnetically induced transparency,” IEEE Photonics J. 8(1), 7800808 (2016).
[Crossref]

Yao, J.

G. Yao, F. Ling, J. Yue, C. Luo, Q. Luo, and J. Yao, “Dynamically electrically tunable broadband absorber based on graphene analog of electromagnetically induced transparency,” IEEE Photonics J. 8(1), 7800808 (2016).
[Crossref]

Yi, S.

Yin, J.

Yu, Z.

Yuan, Y.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating Electromagnetic Wave Polarizations by Anisotropic Metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007).
[Crossref] [PubMed]

Yue, J.

G. Yao, F. Ling, J. Yue, C. Luo, Q. Luo, and J. Yao, “Dynamically electrically tunable broadband absorber based on graphene analog of electromagnetically induced transparency,” IEEE Photonics J. 8(1), 7800808 (2016).
[Crossref]

Zettl, A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

Zhang, X.

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[Crossref]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Zhang, Y.

Zhao, J.

Zhao, X.

Zhao, Y. T.

Zhou, L.

C. Qu, S. Ma, J. Hao, M. Qiu, X. Li, S. Xiao, Z. Miao, N. Dai, Q. He, S. Sun, and L. Zhou, “Tailor the functionalities of metasurfaces based on a complete phase diagram,” Phys. Rev. Lett. 115(23), 235503 (2015).
[Crossref] [PubMed]

J. Zhu, Z. Ma, W. Sun, F. Ding, Q. He, L. Zhou, and Y. Ma, “Ultra-broadband terahertz metamaterial absorber,” Appl. Phys. Lett. 105(2), 021102 (2014).
[Crossref]

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating Electromagnetic Wave Polarizations by Anisotropic Metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007).
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Zhou, M.

Zhu, B.

Zhu, J.

F. Ding, J. Dai, Y. Chen, J. Zhu, Y. Jin, and S. I. Bozhevolnyi, “Broadband near-infrared metamaterial absorbers utilizing highly lossy metals,” Sci. Rep. 6(1), 39445 (2016).
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J. Zhu, Z. Ma, W. Sun, F. Ding, Q. He, L. Zhou, and Y. Ma, “Ultra-broadband terahertz metamaterial absorber,” Appl. Phys. Lett. 105(2), 021102 (2014).
[Crossref]

Zhu, W.

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Zhu, X.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref] [PubMed]

Zi, J.

ACS Nano (1)

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

Adv. Mater. (1)

C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
[PubMed]

Appl. Phys. Lett. (3)

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
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F. Ding, Y. Cui, X. Ge, Y. Jin, and S. He, “Ultra-broadband microwave metamaterial absorber,” Appl. Phys. Lett. 100(10), 103506 (2012).
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J. Zhu, Z. Ma, W. Sun, F. Ding, Q. He, L. Zhou, and Y. Ma, “Ultra-broadband terahertz metamaterial absorber,” Appl. Phys. Lett. 105(2), 021102 (2014).
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IEEE Photonics J. (1)

G. Yao, F. Ling, J. Yue, C. Luo, Q. Luo, and J. Yao, “Dynamically electrically tunable broadband absorber based on graphene analog of electromagnetically induced transparency,” IEEE Photonics J. 8(1), 7800808 (2016).
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J. Appl. Phys. (2)

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
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M. Lei, N. Feng, Q. Wang, Y. Hao, S. Huang, and K. Bi, “Magnetically tunable metamaterial perfect absorber,” J. Appl. Phys. 119(24), 244504 (2016).
[Crossref]

Nano Lett. (3)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
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Nanophotonics (1)

P. Y. Chen, C. Argyropoulos, M. Farhat, and J. S. Gomez-Diaz, “Flatland plasmonics and nanophotonics based on graphene and beyond,” Nanophotonics 6(6), 1239–1262 (2017).
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Nanotechnology (1)

P. Y. Chen, M. Farhat, and H. Bağcı, “Graphene metascreen for designing compact infrared absorbers with enhanced bandwidth,” Nanotechnology 26(16), 164002 (2015).
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Nat. Commun. (1)

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
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Nat. Mater. (3)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
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Figures (6)

Fig. 1
Fig. 1 Schematic of the proposed broadband THz absorber consisting of the GCDRs and the gold mirror separated by a thin SiO2 spacer. The ion-gel was spin-coated on the graphene nanostructures and contacted to the Au electrodes as the top gate. And an ultrathin and transparent conduct thin layer was deposited between the SiO2 and graphenes as the bottom gate. The geometrical parameters of GCDRs are a=5 .5 μ m, b=4 μ m, c=2 .5 μ m, d=2 .2 μ m, t=28 μ m, and the lattice constant is L=15 μ m .
Fig. 2
Fig. 2 (a) Simulated absorption spectra of the proposed GCDR absorber (black line) and two GSR absorbers with the top graphene double ring replaced by the isolated inner ring (red line) and outer ring (green line). (b-d) Top views of three graphene MPAs comprised by the single inner ring, single outer ring and double rings, respectively.
Fig. 3
Fig. 3 (a-d) Simulated electric field patterns (arrows for the direction and color for the intensity) at the top graphene layers of three absorbers shown in Fig. 2 at their corresponding absorption peaks (see inset for the frequency). (e) Energy-level diagram to describe the plasmonic hybridization effect in the proposed GCDR absorber due to the mutual coupling between its inner and outer graphene rings.
Fig. 4
Fig. 4 Calculated absorptivity (color map) as function of the frequency (x-axis) and the incident angle (y-axis), illuminated by (a) TE and (b) TM polarized waves.
Fig. 5
Fig. 5 (a) Absorption characteristics of the proposed absorber for normally incident waves with different polarization angle φ . Here, φ represents the angle of the incident electric field relative to x-axis. (b) The simulated electric field patterns on the top graphene layer at the first absorption peak (1.26THz) for φ =0 o , φ =15 o , φ =30 o , φ =45 o .
Fig. 6
Fig. 6 Calculated absorptivity (color map) as function of the frequency (x-axis) and the Fermi energy (y-axis) of the proposed GCDR absorber for normal incident case.

Equations (2)

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σ gra = σ int r a + σ int e r = 2 e 2 k B T π 2 i ω + i / τ ln [ 2 cos h ( E f 2 k B T ) ] + e 2 4 2 [ 1 2 + 1 π arc tan ( ω 2 E f 2 k B T ) i 2 π ln ( ω + 2 E f ) 2 ( ω 2 E f ) 2 + 4 ( k B T ) 2 ]
σ gra = e 2 E f π 2 i ( ω + i / τ )

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