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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 12 — Jun. 6, 2011
  • pp: 11236–11241
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Transverse electric plasmons in bilayer graphene

Marinko Jablan, Hrvoje Buljan, and Marin Soljačić  »View Author Affiliations


Optics Express, Vol. 19, Issue 12, pp. 11236-11241 (2011)
http://dx.doi.org/10.1364/OE.19.011236


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Abstract

We predict the existence of transverse electric (TE) plasmons in bilayer graphene. We find that their plasmonic properties are much more pronounced in bilayer than in monolayer graphene, in a sense that they can get more localized at frequencies just below h̄ω = 0.4 eV for adequate doping values. This is a consequence of the perfectly nested bands in bilayer graphene which are separated by ∼ 0.4 eV.

© 2011 OSA

Throughout this work we consider bilayer graphene as an infinitely thin sheet of material with conductivity σ(q,ω). We assume that air with ɛr = 1 is above and below bilayer graphene. Given the conductivity, by employing classical electrodynamics, one finds that self-sustained oscillations of the charge occur when (see [19

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

] and references therein)
1+iσ(q,ω)q2ω2/c22ɛ0ω=0
(1)
for TM modes, and
1μ0ωiσ(q,ω)2q2ω2/c2=0
(2)
for TE modes. The TM plasmons can considerably depart from the light line, that is, their wavelength can be considerably smaller than that of light at the same frequency. For this reason, when calculating TM plasmons it is desirable to know the conductivity as a function of both frequency ω and wavevector q. However, it turns out that the TE plasmons (both in monolayer [19

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

] and bilayer graphene, as will be shown below) are quite close to the light line q = ω/c, and therefore it is a good approximation to use σ(ω) = σ(q = 0,ω). Moreover, these plasmons are expected to show strong polariton character, i.e., creation of hybrid plasmon-photon excitations. At this point it is worthy to note that if the relative permittivity of dielectrics above and below graphene are sufficiently different, so that light lines differ substantially, then TE plasmon will not exist (perhaps they could exist as leaky modes).

The conductivity σ(ω) = ℜσ(ω) + iσ(ω) is complex, and plasmon dispersion is characterized by the imaginary part ℑσ(ω), whereas ℜσ(ω) determines plasmon losses, or more generally absorption of the sheet. From Eq. (2) it follows that the TE plasmons exist only if ℑσ(ω) < 0 [19

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

].

In order to calculate the imaginary part of the conductivity, we employ Kramers-Kronig relations and the calculation of absorption by Nicol and Carbotte [24

24. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

], where ℜσ(ω) [see Eqs. (19)–(21) in Ref. [24

24. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

]] was calculated by using the Kubo formula. The optical conductivity has rich structure due to the fact that the single-particle spectrum of graphene is organized in four bands given by [24

24. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

],
ɛ(k)γ=±14+(h¯vFkγ)2±12,
(3)
where vF = 106 m/s, the parameter γ ≈ 0.4 eV is equal to the separation between the two conduction bands (which is equal to the separation between the valence bands). The band structure [Eq. (3)] is calculated from the tight binding approach, where vF is connected to the nearestneighbour hopping terms for electrons to move in each of the two graphene planes, and the distance between Carbon atoms in one monolayer (see Ref. [24

24. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

]), whereas γ is the hopping parameter corresponding to electrons hoping from one layer to the other and vice versa [24

24. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

]. The two graphene layers are stacked one above the other according to the so-called Bernal-type stacking (e.g., see Ref. [11

11. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009). [CrossRef]

]). We emphasize that the perfect nesting of bands gives rise to the stronger plasmon like features of TE plasmons in bilayer than in monolayer graphene. The four bands are illustrated in Fig. 1 along with some of the electronic transitions which result in absorption. Absorption depends on γ and the Fermi level μ; the latter can be changed by applying external bias voltage.

Fig. 1 The band-structure of bilayer graphene. The two upper bands (as well as the two lower bands) are perfectly nested and separated by γ ∼ 0.4 eV; q 0 = γ/h̄vF. Horizontal line depicts one possible value of the Fermi level, and arrows denote some of the possible interband electronic transitions. See text for details.

The imaginary part of the conductivity can be calculated from ℜσ(ω) by using the Kramers-Kronig relations
σ(ω)=2ωπ𝒫0σ(ω)ω2ω2dω,
(4)
which yields
σ(ω)σ0=f(Ω,2μ)+g(Ω,μ,γ)+[f(Ω,2γ)+g(Ω,γ,γ)]Θ(γμ)+[f(Ω,2μ)+g(Ω,μ,γ)]Θ(μγ)+γ2Ω2[Ωπ(2μ+γ)+f(Ω,2μ+γ)]+γ2Ω2[Ωπγ+f(Ω,γ)]Θ(γμ)+γ2Ω2[Ωπ(2μγ)+f(Ω,2μγ)]Θ(μγ)+a(μ)πΩ+2Ωb(μ)π(Ω2γ2),
(5)
where
f(x,y)=12πlog|xyx+y|,g(x,y,z)=z2π(xz)log|x2y|+(x+z)log|x+2y|2xlog|2y+z|x2z2,a(μ)=4μ(μ+γ)2μ+γ+4μ(μγ)2μγΘ(μγ),b(μ)=γ2[log2μ+γγlog2μγγΘ(μγ)],
(6)
σ 0 = e 2/2, Θ(x) = 1 if x ≥ 0 and zero otherwise, and Ω = h̄ω. Here we assume zero temperature T ≈ 0, which is a good approximation for sufficiently doped bilayer graphene where μkBT. Formulae (5) and (6) are used to describe the properties of TE plasmons.

Fig. 2 The real (red dotted lines) and imaginary (blue solid lines) part of the conductivity of bilayer graphene for two values of doping: μ = 0.4γ (a), and μ = 0.9γ (b). The conductivity is in units of σ 0 = e 2/2, and the frequency is in units of ω 0 = γ/. The δ-functions in ℜσ(ω) at ω = 0 (intraband transitions) and ω = γ/ (transitions from the lower to the upper conduction band depicted as green solid arrows in Fig. 1) are not shown (see [24]).

In Fig. 3 we show the plasmon dispersion curves for μ = 0.4γ and μ = 0.9γ; in the spirit of Ref. [19

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

], we show Δq = qω/c as a function of frequency ω. Plasmons are very close to the light line and thus one can to a very good approximation write the dispersion curve as
Fig. 3 The plasmon dispersion curve Δq = qω/c vs. ω for μ = 0.4γ (a), and μ = 0.9γ (b) is shown as blue solid line. To the right of the vertical red dotted lines plasmons can be damped via excitation of electron-hole pairs, whereas to the left of this line these excitations are forbidden due to the Pauli principle. Black dashed line in (b) (which closely follows the blue line) corresponds to Eq. (8). The wave vector is in units of q 0 = γ/h̄vF, and the frequency is in units of ω 0 = γ/.
Δqω8ɛ02c3σ(ω)2.
(7)
To the left (right) of the vertical red dotted line in Fig. 3, plasmon damping via excitation of electron-hole pairs is (is not) forbidden.

For μ = 0.4γ, ℑσ(ω) is smaller than zero for ω in an interval of frequencies just below 2μ. From the leading term in ℑσ(ω) we find that departure of the dispersion curve from the light line is logarithmically slow: Δq 0< μ < γ /2 ∝ [log|h̄ω – 2μ|]2. The same type of behavior occurs in monolayer graphene [19

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

].

However, for μ = 0.9γ, one can see the advantage of bilayer over monolayer graphene in the context of TE plasmons. The conductivity ℑσ(ω) is smaller than zero in an interval of frequencies below γ. In this interval, the most dominant term to the conductivity is the last one from Eq. (5), that is,
Δqγ/2<μ<γωσ022π2ɛ02c3[h¯ωb(μ)γ2(h¯ω)2]2.
(8)
This approximation is illustrated with black dashed line in Fig. 3, and it almost perfectly matches the dispersion curve. Note that the singularity in ℑσ(ω) at h̄ω = γ is of the form 1/(γh̄ω), whereas the singularity at h̄ω = 2μ is logarithmic (as in monolayer graphene [19

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

]). As a consequence, the departure of the dispersion curve from the light line in bilayer graphene is much faster for μ > γ/2 than for μ < γ/2, and it is faster than in monolayer graphene as well [note the two orders of magnitude difference between the abscissa scales in Figs. 3(a) and 3(b)]. Thus, we conclude that more pronounced plasmonic features of TE plasmons (shrinking of wave length which is measured as departure of q from the light line) can be obtained in bilayer graphene. The term in ℑσ(ω) which is responsible for TE plasmons for μ > γ/2 corresponds (via Kramers-Kronig relations) to the absorption term b(μ)δ (h̄ωγ) [24

24. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

], which arises from the transitions from the first to the second valence band (shown as green solid arrows in Fig. 1), which are perfectly nested and separated by γ. Thus, this unique feature of bilayer graphene gives rise to TE plasmons with more pronounced plasmon like features than in monolayer graphene.

Before closing, let us discuss some properties and possible observation of TE plasmons. First, note that since the electric field oscillations are both perpendicular to the propagation vector q, and lie in the bilayer graphene plane, the electric current j = σ(ω)E is also perpendicular to q. Thus, j · q = 0, and the equation of continuity yields that the charge density is zero (i.e., one has self-sustained oscillations of the current). In order to excite plasmons of frequency ω with light of the same frequency, one has to somehow account for the conservation of the momentum which is larger for plasmons. Since the momentum mismatch is relatively small, the standard plasmon excitation schemes such as the prism or grating coupling methods (e.g., see [2

2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]

] and references therein) could be used for the excitation of these plasmons.

In conclusion, we have predicted the existence of transverse electric (TE) plasmons in bilayer graphene. Since they exist very close to the light line, these plasmons are expected to show strong polariton character, i.e., mixing with photon modes. However, due to the perfectly nested valence bands of bilayer graphene, their dispersion departs much more from the light line than in monolayer graphene.

Acknowledgments

We acknowledge most useful comments from Guy Bartal and Mordechai Segev from the Technion, Israel. This work was supported in part by the Croatian Ministry of Science (Grant No. 119-0000000-1015), the Croatian-Israeli scientific cooperation program funded by the Ministries of Science of the State of Israel and the Republic of Croatia. This work is also supported in part by the MRSEC program of National Science Foundation of the USA under Award No. DMR-0819762. M.S. was also supported in part by the S3TEC, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001299.

References and links

1.

D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, 1966).

2.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]

3.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]

4.

S. Vedentam, H. Lee, J. Tang, J. Conway, M. Staffaroni, and E. Yablonovitch, “A plasmonic dimple lens for nanoscale focusing of light,” Nano Lett. 9, 3447 (2009). [CrossRef]

5.

A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength lght in air,” Phys. Rev. Lett. 95, 063901 (2005). [CrossRef] [PubMed]

6.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509 (1968). [CrossRef]

7.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41 (2007). [CrossRef]

8.

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

9.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788 (2004). [CrossRef] [PubMed]

10.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666 (2004). [CrossRef] [PubMed]

11.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009). [CrossRef]

12.

K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, “Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene,” Nat. Phys. 2, 177 (2006). [CrossRef]

13.

X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in a graphene bilayer,” Phys. Rev. B 75, 041404 (2007). [CrossRef]

14.

G. Borghi, M. Polini, R. Asgari, and A. H. MacDonald, “Dynamical response functions and collective modes of bilayer graphene,” Phys. Rev. B 80, 241402 (2009). [CrossRef]

15.

X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in biased bilayer graphene,” Phys. Rev. B 81, 081402 (2010). [CrossRef]

16.

R. Sensarma, E. H. Hwang, and S. Das Sarma, “Dynamic screening and low-energy collective modes in bilayer graphene,” Phys. Rev. B 82, 195428 (2010).

17.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” N. J. Phys. 8, 318 (2006). [CrossRef]

18.

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

19.

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

20.

F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnology 7, 91 (2008). [CrossRef]

21.

C. Kramberger, R. Hambach, C. Giorgetti, M. H. Rümmeli, M. Knupfer, J. Fink, B. Büchner, L. Reining, E. Einarsson, S. Maruyama, F. Sottile, K. Hannewald, V. Olevano, A. G. Marinopoulos, and T. Pichler, “Linear plasmon dispersion in single-wall carbon nanotubes and the collective excitation spectrum of graphene,” Phys. Rev. Lett. 100, 196803 (2008).

22.

Y. Liu, R. F. Willis, K. V. Emtsev, and Th. Seyller, “Plasmon dispersion and damping in electrically isolated two-dimensional charge sheets,” Phys. Rev. B 78, 201403 (2008).

23.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009). [CrossRef]

24.

E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

OCIS Codes
(240.5420) Optics at surfaces : Polaritons
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: January 13, 2011
Manuscript Accepted: March 15, 2011
Published: May 25, 2011

Citation
Marinko Jablan, Hrvoje Buljan, and Marin Soljačić, "Transverse electric plasmons in bilayer graphene," Opt. Express 19, 11236-11241 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11236


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References

  1. D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, 1966).
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]
  3. S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]
  4. S. Vedentam, H. Lee, J. Tang, J. Conway, M. Staffaroni, and E. Yablonovitch, “A plasmonic dimple lens for nanoscale focusing of light,” Nano Lett. 9, 3447 (2009). [CrossRef]
  5. A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength lght in air,” Phys. Rev. Lett. 95, 063901 (2005). [CrossRef] [PubMed]
  6. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
  7. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41 (2007). [CrossRef]
  8. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
  9. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788 (2004). [CrossRef] [PubMed]
  10. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666 (2004). [CrossRef] [PubMed]
  11. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009). [CrossRef]
  12. K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, “Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene,” Nat. Phys. 2, 177 (2006). [CrossRef]
  13. X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in a graphene bilayer,” Phys. Rev. B 75, 041404 (2007). [CrossRef]
  14. G. Borghi, M. Polini, R. Asgari, and A. H. MacDonald, “Dynamical response functions and collective modes of bilayer graphene,” Phys. Rev. B 80, 241402 (2009). [CrossRef]
  15. X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in biased bilayer graphene,” Phys. Rev. B 81, 081402 (2010). [CrossRef]
  16. R. Sensarma, E. H. Hwang, and S. Das Sarma, “Dynamic screening and low-energy collective modes in bilayer graphene,” Phys. Rev. B 82, 195428 (2010).
  17. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” N. J. Phys. 8, 318 (2006). [CrossRef]
  18. E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007). [CrossRef]
  19. S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99, 016803 (2007). [CrossRef] [PubMed]
  20. F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnology 7, 91 (2008). [CrossRef]
  21. C. Kramberger, R. Hambach, C. Giorgetti, M. H. Rümmeli, M. Knupfer, J. Fink, B. Büchner, L. Reining, E. Einarsson, S. Maruyama, F. Sottile, K. Hannewald, V. Olevano, A. G. Marinopoulos, and T. Pichler, “Linear plasmon dispersion in single-wall carbon nanotubes and the collective excitation spectrum of graphene,” Phys. Rev. Lett. 100, 196803 (2008).
  22. Y. Liu, R. F. Willis, K. V. Emtsev, and Th. Seyller, “Plasmon dispersion and damping in electrically isolated two-dimensional charge sheets,” Phys. Rev. B 78, 201403 (2008).
  23. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009). [CrossRef]
  24. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B 77, 155409 (2008). [CrossRef]

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