Abstract

The effective medium properties of graphene sheet stacks are calculated, and it is shown that such stacks can have very low reflectivity and high absorbance. These properties make graphene-sheet-stack-based materials darker than recently studied carbon nanotube materials. Graphene stacks thus hold promise for realizing lower reflectivity coatings and enhanced photodetectors. The bounds of the effective medium approximation and the possible benefits of using graphene sheet stacks in a regime where this approximation does not hold are discussed.

© 2011 Optical Society of America

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  1. J. H. Lehman, C. Engtrakul, T. Gennett, and A. C. Dillon, Appl. Opt. 44, 483 (2005).
    [CrossRef] [PubMed]
  2. E. Theocharous, R. Deshpande, A. C. Dillon, and J. Lehman, Appl. Opt. 45, 1093 (2006).
    [CrossRef] [PubMed]
  3. Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
    [CrossRef] [PubMed]
  4. F. J. García-Vidal, J. M. Pitarke, and J. B. Pendry, Phys. Rev. Lett. 78, 4289 (1997).
    [CrossRef]
  5. W. Lü, J. Dong, and Z. Y. Li, Phys. Rev. B 63, 033401 (2000).
    [CrossRef]
  6. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
    [CrossRef] [PubMed]
  7. T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
    [CrossRef]
  8. K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
    [CrossRef] [PubMed]
  9. A. Ludwig and K. J. Webb, Phys. Rev. B 81, 113103 (2010).
    [CrossRef]
  10. A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
    [CrossRef]
  11. K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
    [CrossRef] [PubMed]

2010 (1)

A. Ludwig and K. J. Webb, Phys. Rev. B 81, 113103 (2010).
[CrossRef]

2009 (2)

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

2008 (4)

Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
[CrossRef] [PubMed]

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef] [PubMed]

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

2000 (1)

W. Lü, J. Dong, and Z. Y. Li, Phys. Rev. B 63, 033401 (2000).
[CrossRef]

1997 (1)

F. J. García-Vidal, J. M. Pitarke, and J. B. Pendry, Phys. Rev. Lett. 78, 4289 (1997).
[CrossRef]

Ahn, J.-H.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Ajayan, P. M.

Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
[CrossRef] [PubMed]

Bulovic, V.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Bur, J. A.

Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
[CrossRef] [PubMed]

Carbone, F.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef] [PubMed]

Choi, J.-Y.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Ci, L.

Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
[CrossRef] [PubMed]

Deshpande, R.

Dillon, A. C.

Dong, J.

W. Lü, J. Dong, and Z. Y. Li, Phys. Rev. B 63, 033401 (2000).
[CrossRef]

Dresselhaus, M. S.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Engtrakul, C.

García-Vidal, F. J.

F. J. García-Vidal, J. M. Pitarke, and J. B. Pendry, Phys. Rev. Lett. 78, 4289 (1997).
[CrossRef]

Geim, A. K.

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

Gennett, T.

Heinz, T. F.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

Ho, J.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Hong, B. H.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Jang, H.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Jia, X.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Kim, J. M.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Kim, K. S.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Kim, P.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Kong, J.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Kuzmenko, A. B.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef] [PubMed]

Lee, S. Y.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Lehman, J.

Lehman, J. H.

Li, Z. Y.

W. Lü, J. Dong, and Z. Y. Li, Phys. Rev. B 63, 033401 (2000).
[CrossRef]

Lin, S. Y.

Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
[CrossRef] [PubMed]

Lü, W.

W. Lü, J. Dong, and Z. Y. Li, Phys. Rev. B 63, 033401 (2000).
[CrossRef]

Ludwig, A.

A. Ludwig and K. J. Webb, Phys. Rev. B 81, 113103 (2010).
[CrossRef]

Lui, C. H.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

Mak, K. F.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

Misewich, J. A.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

Nezich, D.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Pendry, J. B.

F. J. García-Vidal, J. M. Pitarke, and J. B. Pendry, Phys. Rev. Lett. 78, 4289 (1997).
[CrossRef]

Peres, N. M. R.

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

Pitarke, J. M.

F. J. García-Vidal, J. M. Pitarke, and J. B. Pendry, Phys. Rev. Lett. 78, 4289 (1997).
[CrossRef]

Reina, A.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Sfeir, M. Y.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

Son, H.

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Stauber, T.

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

Theocharous, E.

van der Marel, D.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef] [PubMed]

van Heumen, E.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef] [PubMed]

Webb, K. J.

A. Ludwig and K. J. Webb, Phys. Rev. B 81, 113103 (2010).
[CrossRef]

Wu, Y.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

Yang, Z. P.

Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
[CrossRef] [PubMed]

Zhao, Y.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Appl. Opt. (2)

Nano Lett. (2)

Z. P. Yang, L. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, Nano Lett. 8, 446 (2008).
[CrossRef] [PubMed]

A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Nature (1)

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, Nature 457, 706 (2009).
[CrossRef] [PubMed]

Phys. Rev. B (3)

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

W. Lü, J. Dong, and Z. Y. Li, Phys. Rev. B 63, 033401 (2000).
[CrossRef]

A. Ludwig and K. J. Webb, Phys. Rev. B 81, 113103 (2010).
[CrossRef]

Phys. Rev. Lett. (3)

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef] [PubMed]

F. J. García-Vidal, J. M. Pitarke, and J. B. Pendry, Phys. Rev. Lett. 78, 4289 (1997).
[CrossRef]

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Graphene sheet stack illuminated by a plane wave.

Fig. 2
Fig. 2

Side view of a GSS illuminated by a TM-polarized plane wave, to accompany the derivation given. The derivation for TE illumination is similar and is not dwelt upon.

Fig. 3
Fig. 3

Reflected (R) and transmitted (T) power (normalized to the incident power) for a GSS illuminated by a normally incident plane wave at a wavelength λ = 630 nm as a function of the GSS thickness, D, for different intersheet distances, d.

Fig. 4
Fig. 4

The deviation of the real part of the effective refractive index from unity Re ( n ¯ 1 ) versus the effective absorption constant α ¯ . Results are shown for GSSs of different intersheet distances, d (unit cell dimension), and at different wavelengths of operation, λ, and compared with results for two-dimensional arrays of CNTs [5] for the same unit cell dimensions and wavelengths. Results for the curves of the GSSs are obtained from Eq. (6), while results for the shaded regions below these curves are obtained directly from Eqs. (2, 3, 4, 5) for all incident angles and both TM and TE incidence.

Equations (6)

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r = α ˜ cos θ / ( ϵ + α ˜ cos θ ) ,     t = ϵ / ( ϵ + α ˜ cos θ ) ,
T = [ ϵ α ˜ cos θ ϵ e i ϵ k 0 d cos θ α ˜ cos θ ϵ α ˜ cos θ ϵ ϵ + α ˜ cos θ ϵ e i ϵ k 0 d cos θ ] ,
ϵ ¯ t = β ¯ / ( Z ¯ ϵ 0 ) , μ ¯ t ( ϵ sin 2 θ ) / ϵ ¯ n = β ¯ Z ¯ / μ 0 ,
Z ¯ = μ 0 ϵ ϵ 0 ( T 22 + T 21 ) 2 1 ( T 22 T 21 ) 2 1 cos θ
β ¯ = arccos { [ 1 + ( T 22 2 T 21 2 ) ] / 2 T 22 } / d
{ ϵ ¯ t = ϵ + i γ / π ϵ ¯ n = ϵ , where γ = π σ 0 c ϵ 0 k 0 d .

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