Abstract

We investigate THz absorption properties of graphene-based heterostructures by using characteristics matrix method based on conductivity. We demonstrate that the proposed structure can lead to perfect THz absorption because of strong photon localization in the defect layer of the heterostructure. The THz absorption may be tuned continuously from 0 to 100% by controlling the chemical potential through a gate voltage. By adjusting the incident angle or the period number of the two PCs with respect to the graphene layer, one can tailor the maximum THz absorption value. The position of the THz absorption peaks can be tuned by changing either the center wavelength or the thicknesses ratio of the layers constituting the heterostructure. Our proposal may have potentially important applications in optoelectronic devices.

© 2014 Optical Society of America

1. Introduction

As a two-dimension substance composed of a single layer carbon atom arranged in a honey comb lattice, graphene has attracted significant attention in recent years due to its unique optical, electronic, and mechanical properties [13]. It has had a notable impact on the fields of photonics and optoelectronics [35]. In the photonics context, graphene is a two-dimensional conductor that graphene has strong plasmonic effects which can be modified by gating, doping, and so on [4, 5].

Optical absorption plays an important role in a variety of applications such as photodetectors, saturable absorbers and photovoltaics. The concept of perfect absorbers has initiated a new research area and has important applications in optoelectronics [68]. Various microstructures have been proposed to get complete absorption in graphene, e.g., the periodically patterned graphene, the microcavity, the graphene negative permittivity metamaterials, and the attenuated total reflectance, graphene-based hyperbolic metamaterials, etc [913]. In recent years, various types of THz absorbers have attracted an explosion of research interest in many scopes, like spectroscopy, medical imaging, communications and so on [14]. By changing the various bias voltages, the absorption is controlled in the graphene fishnet metamaterials [15] and the biperiodic graphene metasurfaces [16].

In this paper, we theoretically investigate THz absorption properties of graphene-based one-dimensional photonic crystals (1DPC) heterostructure as shown in Fig. 1. We demonstrate that the proposed structure can lead to perfect THz absorption because of strong photon localization in the defect layer of the heterostructure. The THz absorption may be tuned continuously from 0 to 100% by controlling the chemical potentialsμc. By adjusting the incident angle or the period number of the two PCs with respect to the graphene layer, one can tailor the maximum THz absorption value. The position of the THz absorption peaks can be tuned by changing either the center wavelengthλ0or the thicknesses ratio of the layers constituting the heterostructure. Our proposal is very easy to implement using the existing technology and may have potentialsly important applications in optoelectronic devices.

 figure: Fig. 1

Fig. 1 Sketch of the heterostructure (ab)MGr(ba)Nwith a monolayer graphene defect layer.

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2. Theoretical model and numerical method

Previous research has shown that the conductivity of graphene came from the contribution of intraband and interband [1719]. The interband conductivity tends to be ignorable for the THz frequencies because of the photon energyωμc, where μcis chemical potential [20]. Therefore, in the THz range graphene is well described by the Drude-like conductivity [21]

σ=e2π2kBTΓiω[μckBT+2ln(eμc/kBT+1)].
where Γ is the relaxation rate, kBis the Boltzmann constant, Tis the absolute temperature,ω is the frequency of the incoming radiation. It should be noted that the conductivity σ is responsible for the optical behaviour of graphene in the THz spectral range.

To determine the THz absorption we have to compute the amount of light reflected and transmitted by and through the heterostructure, respectively. A light beam is incident at theangle of θ onto the heterostructure (ab)MGr(ba)N from the left along the z-direction in Fig. 1. Based on Maxwell’s equations, the electric field and the magnetic field of the light beam in the jth layer is given by

Ej=(Aj+eikj,zz+Ajeikj,zz)e^x,
Hj=1iωμ0μj×Ej=kj,zμ0μjω(Aj+eikj,zzAjeikj,zz)e^y,
where kj,z=(ω/c)2εjkx2 is the z component of the wave vector in the jth layer, εl is the dielectric constant of the jth layer. The transverse wave vector component, kx=k0sinθ, is preserved across all interfaces, k0is the wave vector in incident medium. The dynamic matrix referring to the transfer through the boundary of two dielectrics i and j and the propagation matrix characterizing the free propagation through dielectric j are given respectively by
Di,j=12(1+κj,z/κi,z1κj,z/κi,z1κj,z/κi,z1+κj,z/κi,z),
Pj=(exp(-ikj,zdj)00exp(ikj,zdj)),
where κl,z=kl,z with l=i,j for TE case and kl,z/εl for TM case. At the graphene interface, we make use of the boundary conditions of the electric field and the magnetic field:
n^×(ELER)=0,
n^×(HLHR)=σER,
where the subscript L(R) stands for the fields to the left (right) of graphene. From Eqs. (2), (3), (6) and (7), we can get
(AL+AL)=Tg(AR+AR),
where, for TE case Tg=DL,R+(μ0μLωσ/2kL,z)JTEwith JTE=(1111); for TM case Tg=DL,R(kR,zσ/2ε0εRω)JTM withJTM=(1111). We can relate the fields on the left of the heterostructure to the fields on the right as
(A0+A0)=TLTgTR(Asub+0)=(T11T12T21T22)(Asub+0),
where, TL(TR) stands for the transfer matrix to the left (right) structure of graphene, for the heterostructure(ab)MGr(ba)Nin the air in Fig. 1, we can get TL=Dair,b(Db,aPaDa,bPb)Mand TR=(PbDb,aPaDa,b)NDb,air.Then the reflectanceand the transmittanceTof the heterostructure can be obtained by=(A0/A0+)2=(T21/T11)2andT=(Asub+/A0+)2=(1/T11)2, respectively, the absorption can also be obtained byA=1T.

3. Results and discussion

In the following calculations, we choose the center wavelength λ0 = 80μm, the absolute temperature T = 300 K, and the relaxation rate Γ = 2.5 meV/. In the THz range we approximate the dielectric constants of Si and SiO2 by their static values, εSi = 11.9 and εSiO2 = 3.9. The thicknesses of the Si and SiO2 layers in the periodic structure are given bydSiO2=λ0/4(εSiO2)1/2 and dSi=λ0/4(εSi)1/2, respectively.In what follows, the THz absorption of graphene on a system such as the represented in Fig. 1 is computed. In Fig. 2(a) we represent the THz absorption of graphene-based heterostructures as a function of the light frequencyω. We compare the THz absorption at normal incidence for bare graphene [red dashed curve in Fig. 2(a)], and graphene within the defect layer of the1DPC heterostructure (Si/SiO2)MGr(SiO2/Si)N in Fig. 1 [black solid curve for M = 2 and N = 5, olive dash dotted for M = 3 and N = 5, and blue dotted curve for M = 4 and N = 4]. Since the dielectric constants we have used for SiO2 and Si are real, all the THz absorption comes of graphene alone. Based on the presented results, one observes that the THz absorption of the graphene monolayer without the 1DPC decreases form 16.7% to 5% when the light frequency ωincreases from 10 meV to 22 meV. In contrast, the graphene is introduced in the defect state of the PC heterostructure, a more dramatic enhancement is obtained, THz absorptions near the center of the photonic band gap are nearly 100% for M = 2 and N = 5, 73.8% for M = 3 and N = 5, and 32% for M = 4 and N = 4, respectively. To get a perfect matching of M and N to achieve the maximum absorption value, we consider more {M, N} case, in Fig. 2(b), we represent the absorption peak value with respect to M and N, we can observe the presence of an optimal number of periods M and N that maximize absorption. As is shown in Fig. 2(b) M = 2, N has little or no influence when N5. Therefore, by varying the period number M and N, one can tailor the maximum THz absorption value in the heterostructure.

 figure: Fig. 2

Fig. 2 (a) Absorption as a function of frequencies ω of light for bare graphene and graphene within the defect layer of the heterostructure (Si/SiO2)MGr(SiO2/Si)N in Fig. 1 with μc = 0.378 eV andεSiO2=3.9, (inset) electric field distribution in the 1DPC with εSiO2=3.9. (b) Map of the absorption peak value as a function of the period number M and N. (c) THz absorption and electric field distribution for the case of M = 2 and N = 5 with a finite absorption loss in SiO2. A typical value, for reference, 4.14 meV corresponds to 1 THz.

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To understand the almost 100% absorption of incident energy by the heterostructure, we show the electric field distribution in the heterostructure (Si/SiO2)MGr(SiO2/Si)N in the inset of Fig. 2(a), the spectra reveal the electric field dramatic enhanced in the defect layer, a peak of the electric field appears at the position of graphene in the 1DPC heterostructure. The PC heterostructure [(Si/SiO2)MGr(SiO2/Si)N] formed a λ/2SiO2 defect layer, thus the graphene layer and the SiO2 defect act as the defect layer of the heterostructure, the presence of a localized state in the defect layer improves field localization at the graphene position, thanks to strong photon localization in the defect layer, there is a dramatic enhancement of THz absorption near the center of the gap of the PC heterostructure. Why {M = 2, N5} case is the best in terms of absorption, it can be understood from the following explanation. PC1 and PC2 form a Bragg cavity, PC1 acts as a Bragg reflector, and also as the medium which introduces incident light into the Bragg cavity, while PC2 simply act as a Bragg mirror, when M and N are relatively large, the light inside the cavity can be sufficiently reflected and localized in the cavity, however, when M is relatively small, the incident light outside the cavity can be sufficiently introduced into the cavity, so when M = 2 and N5, light is fully localized and absorbed in the cavity shown in Fig. 2(b).

To examine how a finite THz absorption in SiO2 will affect the performance of the proposed structure, in Fig. 2(c), we represent the THz absorption and the electric field distribution of the proposed structure for εSiO2=3.9+0.025i [22], we can see that, when taking into account the absorption loss of SiO2, the near perfect THz absorption and the electric field distribution are almost unchanged, while for higher frequencies (ω>18 THz), the THz absorption slightly fluctuates, strong photon localization in SiO2-filled region of the heterostructure causes broadening and suppression of the resonance.

In Fig. 3, we plot the THz absorption maps of a graphene-based heterostructure as a function of incident frequency and angle of incidence for TE and TM polarized incident lights. It is shown that the THz absorption peak moves toward higher frequencies with increasing incident angle θ both for the TE and TM polarizations. It can be understood from the relationship between the propagation angleθ'and the resonant frequencyωin a microcavityω1/cosθ' [23], when the incident angle increases, cosθ'can be decreased, the resonant peak moves toward higher frequencies. Consequently, the THz absorption of graphene can be tuned by varying incident angle θ.

 figure: Fig. 3

Fig. 3 THz absorption maps as a function of the light frequencies and the incident angles for TE and TM incident lights for the heterostructure (Si/SiO2)2Gr(SiO2/Si)5 withμc = 0.25 eV.

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In order to better control THz absorption, in Fig. 4(a) we represent the absorption as a function of chemical potentials μc for graphene within the defect layer of the heterostructure (Si/SiO2)2Gr(SiO2/Si)5 with ω = 16 meV, it is shown that the absorption increases from 0 to 100% when chemical potentials μc increases from 0 to 0.5 eV, while the absorption decreases from 100% to 12% when chemical potentials μc increases from 0.5 eV to 1 eV. In Fig. 4(b) we plot intensity of the absorption as a function of the chemical potentials μc and the light frequenciesω. It is shown that the THz absorption peak moves toward higher frequencies with increasing chemical potentialsμc. We note that, since chemical potentials μc can be tuned by varying the density of charge carriers through a gate voltage, the conductivity σ of graphene can also be changed. So we can control conveniently the optical absorption of the proposed structure in the THz spectral range by varying chemical potentials μc.

 figure: Fig. 4

Fig. 4 (a) THz absorption as a function of the chemical potentials μc for graphene within the defect layer of the heterostructure (Si/SiO2)2Gr(SiO2/Si)5 with ω = 16 meV. (b) Intensity plot of the absorption as a function of the chemical potentials μc and the light frequenciesω.

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In Fig. 5(a) we represent the absorption as function of the parameter dSiO2/dSi. Recall that the parameter dSiO2/dSi controls the thickness of the SiO2 layer,when changing dSiO2/dSi, the optical path of the SiO2 layer can be changed, leading to a resonant wavelength altered, and absorption peaks moved. We see that the absorption arrive a maximum value almost 100% when dSiO2/dSi = 1.8, 5.2, and 8.5, respectively. In Fig. 5(b) we represent the absorption as a function of the parameter λ0 and ω. Sinceλ0influences the thicknesses of the layers constituting the periodic structure, when varyingλ0, leading to resonant frequencies ω altered, and the position of absorption peaks can be changed. As seen in Fig. 5(b), the position of the absorption peaks is sensitive to the parameterλ0,the absorption peaks move to higher frequencies with decreasingλ0.These type of plots allow the optimization of the heterostructure in order to maximize the absorption.

 figure: Fig. 5

Fig. 5 (a) THz absorption as a function of dSiO2/dSi for graphene within the defect layer of the 1DPC heterostructure (Si/SiO2)2Gr(SiO2/Si)5 with ω = 16 meV and μc = 0.25 eV in Fig. 1. (b) Intensity plot of the absorption as a function of λ0 and ω withμc = 0.25 eV.

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4. Conclusion

In conclusion, we investigated THz absorption properties of graphene-based heterostructures by using characteristics matrix method based on conductivity. We demonstrated that the proposed structure can lead to perfect THz absorption because of strong photon localization in the defect layer of the heterostructure. The THz absorption may be tuned continuously from 0 to 100%. Our proposal is very easy to implement using the existing technology and may have potentially important applications in optoelectronic devices.

Acknowledgments

This work was supported by the NSFC Grant Nos. 61464007, 11364033, 11264030 and 11264029, the Open Research Fund of State Key Laboratory of Millimeter Waves (NO. K201216), and the Postdoctoral Science Foundation of Jiangxi Province No. 2014KY32.

References and links

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

2. N. M. R. Peres, “Transport properties of grapheme,” Rev. Mod. Phys. 82, 2673–2700 (2010). [CrossRef]  

3. F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]  

4. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]  

5. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011). [CrossRef]   [PubMed]  

6. 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]  

7. 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]  

8. C. Hägglund and S. P. Apell, “Resource efficient plasmon-based 2D-photovoltaics with reflective support,” Opt. Express 18(S3Suppl 3), A343–A356 (2010). [CrossRef]   [PubMed]  

9. 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]  

10. A. Ferreira and N. M. R. Peres, “Complete light absorption in graphene metamaterial corrugated structures,” Phys. Rev. B 86(20), 205401 (2012). [CrossRef]  

11. G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013). [CrossRef]   [PubMed]  

12. Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013). [CrossRef]  

13. Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014). [CrossRef]   [PubMed]  

14. A. Fallahi and J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86(19), 195408 (2012). [CrossRef]  

15. A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013). [CrossRef]   [PubMed]  

16. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007). [CrossRef]  

17. A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]  

18. K. Ziegler, “Robust transport properties in graphene,” Phys. Rev. Lett. 97(26), 266802 (2006). [CrossRef]   [PubMed]  

19. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006). [CrossRef]   [PubMed]  

20. S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007). [CrossRef]   [PubMed]  

21. G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008). [CrossRef]  

22. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998).

23. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

References

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  1. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [Crossref] [PubMed]
  2. N. M. R. Peres, “Transport properties of grapheme,” Rev. Mod. Phys. 82, 2673–2700 (2010).
    [Crossref]
  3. F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
    [Crossref]
  4. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
    [Crossref]
  5. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
    [Crossref] [PubMed]
  6. 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]
  7. 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]
  8. C. Hägglund and S. P. Apell, “Resource efficient plasmon-based 2D-photovoltaics with reflective support,” Opt. Express 18(S3Suppl 3), A343–A356 (2010).
    [Crossref] [PubMed]
  9. 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]
  10. A. Ferreira and N. M. R. Peres, “Complete light absorption in graphene metamaterial corrugated structures,” Phys. Rev. B 86(20), 205401 (2012).
    [Crossref]
  11. G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013).
    [Crossref] [PubMed]
  12. Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
    [Crossref]
  13. Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
    [Crossref] [PubMed]
  14. A. Fallahi and J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86(19), 195408 (2012).
    [Crossref]
  15. A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013).
    [Crossref] [PubMed]
  16. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
    [Crossref]
  17. A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
    [Crossref]
  18. K. Ziegler, “Robust transport properties in graphene,” Phys. Rev. Lett. 97(26), 266802 (2006).
    [Crossref] [PubMed]
  19. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006).
    [Crossref] [PubMed]
  20. S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
    [Crossref] [PubMed]
  21. G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
    [Crossref]
  22. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998).
  23. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

2014 (1)

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

2013 (3)

G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013).
[Crossref] [PubMed]

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013).
[Crossref] [PubMed]

2012 (4)

A. Fallahi and J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86(19), 195408 (2012).
[Crossref]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

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]

A. Ferreira and N. M. R. Peres, “Complete light absorption in graphene metamaterial corrugated structures,” Phys. Rev. B 86(20), 205401 (2012).
[Crossref]

2011 (1)

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

2010 (4)

N. M. R. Peres, “Transport properties of grapheme,” Rev. Mod. Phys. 82, 2673–2700 (2010).
[Crossref]

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

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]

C. Hägglund and S. P. Apell, “Resource efficient plasmon-based 2D-photovoltaics with reflective support,” Opt. Express 18(S3Suppl 3), A343–A356 (2010).
[Crossref] [PubMed]

2009 (1)

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

2008 (2)

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]

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

2007 (3)

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

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[Crossref] [PubMed]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

2006 (2)

K. Ziegler, “Robust transport properties in graphene,” Phys. Rev. Lett. 97(26), 266802 (2006).
[Crossref] [PubMed]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006).
[Crossref] [PubMed]

Andryieuski, A.

Apell, S. P.

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006).
[Crossref] [PubMed]

Chang, D. E.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Chen, X. D.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Dai, X.

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

Deng, Z. C.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Fallahi, A.

A. Fallahi and J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86(19), 195408 (2012).
[Crossref]

Ferrari, A.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Ferreira, A.

A. Ferreira and N. M. R. Peres, “Complete light absorption in graphene metamaterial corrugated structures,” Phys. Rev. B 86(20), 205401 (2012).
[Crossref]

García de Abajo, F. J.

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]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Geim, A. K.

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

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

Giessen, H.

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]

Gómez Rivas, J.

G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013).
[Crossref] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Guinea, F.

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Guo, J.

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006).
[Crossref] [PubMed]

Hägglund, C.

Hanson, G. W.

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Hentschel, M.

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]

Kong, X. T.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Koppens, F. H. L.

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]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Landy, N. I.

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]

Lavrinenko, A. V.

Liu, N.

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]

Liu, Z.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Lozano, G.

G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013).
[Crossref] [PubMed]

Martín Moreno, L.

G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013).
[Crossref] [PubMed]

Mesch, M.

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]

Mikhailov, S. A.

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[Crossref] [PubMed]

Mock, J. J.

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]

Neto, A. C.

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Novoselov, K. S.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

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

Padilla, W. J.

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]

Peres, N.

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Peres, N. M. R.

A. Ferreira and N. M. R. Peres, “Complete light absorption in graphene metamaterial corrugated structures,” Phys. Rev. B 86(20), 205401 (2012).
[Crossref]

N. M. R. Peres, “Transport properties of grapheme,” Rev. Mod. Phys. 82, 2673–2700 (2010).
[Crossref]

Perruisseau-Carrier, J.

A. Fallahi and J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86(19), 195408 (2012).
[Crossref]

Pirruccio, G.

G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013).
[Crossref] [PubMed]

Polini, M.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

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]

Sharapov, S. G.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006).
[Crossref] [PubMed]

Smith, D. R.

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]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Tang, D.

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

Thongrattanasiri, S.

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]

Tian, J. G.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Wang, J.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

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, S.

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

Xiang, Y.

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

Xing, F.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Ye, Q.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Zhang, C. P.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Zhang, H.

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

Zhou, W. Y.

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

Ziegler, K.

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[Crossref] [PubMed]

K. Ziegler, “Robust transport properties in graphene,” Phys. Rev. Lett. 97(26), 266802 (2006).
[Crossref] [PubMed]

ACS Nano (1)

G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7(6), 4810–4817 (2013).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

Q. Ye, J. Wang, Z. Liu, Z. C. Deng, X. T. Kong, F. Xing, X. D. Chen, W. Y. Zhou, C. P. Zhang, and J. G. Tian, “Polarization-dependent optical absorption of graphene under total internal reflection,” Appl. Phys. Lett. 102(2), 021912 (2013).
[Crossref]

J. Appl. Phys. (1)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

Nano Lett. (2)

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

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]

Nat. Mater. (1)

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

Nat. Photonics (3)

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Opt. Express (2)

Phys. Rev. B (2)

A. Ferreira and N. M. R. Peres, “Complete light absorption in graphene metamaterial corrugated structures,” Phys. Rev. B 86(20), 205401 (2012).
[Crossref]

A. Fallahi and J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86(19), 195408 (2012).
[Crossref]

Phys. Rev. Lett. (5)

K. Ziegler, “Robust transport properties in graphene,” Phys. Rev. Lett. 97(26), 266802 (2006).
[Crossref] [PubMed]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006).
[Crossref] [PubMed]

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[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]

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]

Rev. Mod. Phys. (2)

N. M. R. Peres, “Transport properties of grapheme,” Rev. Mod. Phys. 82, 2673–2700 (2010).
[Crossref]

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Sci Rep (1)

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci Rep 4, 5483 (2014).
[Crossref] [PubMed]

Other (2)

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

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Figures (5)

Fig. 1
Fig. 1 Sketch of the heterostructure (ab) M Gr (ba) N with a monolayer graphene defect layer.
Fig. 2
Fig. 2 (a) Absorption as a function of frequencies ω of light for bare graphene and graphene within the defect layer of the heterostructure (Si/SiO2)MGr(SiO2/Si)N in Fig. 1 with μ c = 0.378 eV and ε Si O 2 = 3.9, (inset) electric field distribution in the 1DPC with ε Si O 2 = 3.9. (b) Map of the absorption peak value as a function of the period number M and N. (c) THz absorption and electric field distribution for the case of M = 2 and N = 5 with a finite absorption loss in SiO2. A typical value, for reference, 4.14 meV corresponds to 1 THz.
Fig. 3
Fig. 3 THz absorption maps as a function of the light frequencies and the incident angles for TE and TM incident lights for the heterostructure (Si/SiO2)2Gr(SiO2/Si)5 with μ c = 0.25 eV.
Fig. 4
Fig. 4 (a) THz absorption as a function of the chemical potentials μ c for graphene within the defect layer of the heterostructure (Si/SiO2)2Gr(SiO2/Si)5 with ω = 16 meV. (b) Intensity plot of the absorption as a function of the chemical potentials μ c and the light frequenciesω.
Fig. 5
Fig. 5 (a) THz absorption as a function of d Si O 2 / d Si for graphene within the defect layer of the 1DPC heterostructure (Si/SiO2)2Gr(SiO2/Si)5 with ω = 16 meV and μ c = 0.25 eV in Fig. 1. (b) Intensity plot of the absorption as a function of λ 0 and ω with μ c = 0.25 eV.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

σ= e 2 π 2 k B T Γiω [ μ c k B T +2ln(e μ c / k B T +1)].
E j =( A j + e i k j,z z + A j e i k j,z z ) e ^ x ,
H j = 1 iω μ 0 μ j × E j = k j,z μ 0 μ j ω ( A j + e i k j,z z A j e i k j,z z ) e ^ y ,
D i,j = 1 2 ( 1+ κ j,z / κ i,z 1 κ j,z / κ i,z 1 κ j,z / κ i,z 1+ κ j,z / κ i,z ),
P j =( exp(-i k j,z d j ) 0 0 exp(i k j,z d j ) ),
n ^ ×( E L E R )=0,
n ^ ×( H L H R )=σ E R ,
( A L + A L )= T g ( A R + A R ),
( A 0 + A 0 )= T L T g T R ( A sub + 0 )=( T 11 T 12 T 21 T 22 )( A sub + 0 ),

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