## Transverse electric plasmons in bilayer graphene |

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.

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*σ*(

**q**,

*ω*). We assume that air with

*ɛ*= 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

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

*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]

*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).

*σ*(

*ω*) = ℜ

*σ*(

*ω*) +

*i*ℑ

*σ*(

*ω*) is complex, and plasmon dispersion is characterized by the imaginary part ℑ

*σ*(

*ω*), whereas ℜ

*σ*(

*ω*) determines plasmon losses, or more generally absorption of the sheet. From Eq. (2) it follows that the TE plasmons exist only if ℑ

*σ*(

*ω*) < 0 [19

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

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]

*σ*(

*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]

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]

*v*= 10

_{F}^{6}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

*v*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

_{F}**77**, 155409 (2008). [CrossRef]

*γ*is the hopping parameter corresponding to electrons hoping from one layer to the other and vice versa [24

**77**, 155409 (2008). [CrossRef]

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]

*γ*and the Fermi level

*μ*; the latter can be changed by applying external bias voltage.

*σ*(

*ω*) by using the Kramers-Kronig relations which yields

*σ*

_{0}=

*e*

^{2}/2

*h̄*, 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

*μ*≫

*k*. Formulae (5) and (6) are used to describe the properties of TE plasmons.

_{B}T*μ*= 0.4

*γ*and

*μ*= 0.9

*γ*(we focus on the electron doped system

*μ*> 0). Because plasmons are strongly damped by interband transitions, it is instructive at this point to discuss the kinematical requirements for the excitation of electron-hole pairs. If the doping is such that

*μ*<

*γ*/2, a quantum of energy

*h̄ω*(plasmon or photon) with in-plane momentum

*q*= 0 can excite an electron-hole pair only if

*h̄ω*> 2

*μ*(excitations from the upper valence to the lower conduction band shown as red dot-dashed line in Fig. 1). If

*μ*>

*γ*/2, the (

*q*= 0,

*ω*)-quantum can excite an electron-hole pair only for

*h̄ω*≥

*γ*(excitations from the lower to the upper conduction band shown as green solid lines in Fig. 1 occur at

*h̄ω*=

*γ*). If the plasmon/photon has in-plane momentum

*q*larger than zero, then interband transitions are possible for smaller frequencies (see blue dashed lines in Fig. 1). There is a region in the (

*q*,

*ω*)-plane where electron-hole excitations are forbidden due to the Pauli principle (e.g., see figures in Refs. [14

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]

*μ*= 0.4

*γ*and

*μ*= 0.9

*γ*; in the spirit of Ref. [19

**99**, 016803 (2007). [CrossRef] [PubMed]

*q*=

*q*−

*ω*/

*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 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.

*μ*= 0.4

*γ*, ℑ

*σ*(

*ω*) is smaller than zero for

*ω*in an interval of frequencies just below 2

*μ*. From the leading term in ℑ

*σ*(

*ω*) we find that departure of the dispersion curve from the light line is logarithmically slow: Δ

*q*

_{0<}

_{μ}_{<}

_{γ}_{/2}∝ [log|

*h̄ω*– 2

*μ*|]

^{2}. The same type of behavior occurs in monolayer graphene [19

**99**, 016803 (2007). [CrossRef] [PubMed]

*μ*= 0.9

*γ*, one can see the advantage of bilayer over monolayer graphene in the context of TE plasmons. The conductivity ℑ

*σ*(

*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, This approximation is illustrated with black dashed line in Fig. 3, and it almost perfectly matches the dispersion curve. Note that the singularity in ℑ

*σ*(

*ω*) at

*h̄ω*=

*γ*is of the form 1/(

*γ*–

*h̄ω*), whereas the singularity at

*h̄ω*= 2

*μ*is logarithmic (as in monolayer graphene [19

**99**, 016803 (2007). [CrossRef] [PubMed]

*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 ℑ

*σ*(

*ω*) which is responsible for TE plasmons for

*μ*>

*γ*/2 corresponds (via Kramers-Kronig relations) to the absorption term

*b*(

*μ*)

*δ*(

*h̄ω*−

*γ*) [24

**77**, 155409 (2008). [CrossRef]

*γ*. Thus, this unique feature of bilayer graphene gives rise to TE plasmons with more pronounced plasmon like features than in monolayer graphene.

**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]

## Acknowledgments

## References and links

1. | D. Pines and P. Nozieres, |

2. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

3. | S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

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. |

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. |

6. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

7. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

8. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

9. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

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 |

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. |

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 2 |

13. | X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in a graphene bilayer,” Phys. Rev. B |

14. | G. Borghi, M. Polini, R. Asgari, and A. H. MacDonald, “Dynamical response functions and collective modes of bilayer graphene,” Phys. Rev. B |

15. | X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in biased bilayer graphene,” Phys. Rev. B |

16. | R. Sensarma, E. H. Hwang, and S. Das Sarma, “Dynamic screening and low-energy collective modes in bilayer graphene,” Phys. Rev. B |

17. | B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” N. J. Phys. |

18. | E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B |

19. | S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. |

20. | F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnology |

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. |

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 |

23. | M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B |

24. | E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Phys. Rev. B |

**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

- D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, 1966).
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41 (2007). [CrossRef]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788 (2004). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in a graphene bilayer,” Phys. Rev. B 75, 041404 (2007). [CrossRef]
- 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]
- X. F. Wang and T. Chakraborty, “Coulomb screening and collective excitations in biased bilayer graphene,” Phys. Rev. B 81, 081402 (2010). [CrossRef]
- 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).
- B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” N. J. Phys. 8, 318 (2006). [CrossRef]
- E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007). [CrossRef]
- S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99, 016803 (2007). [CrossRef] [PubMed]
- F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnology 7, 91 (2008). [CrossRef]
- 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).
- 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).
- M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009). [CrossRef]
- 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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