## Spontaneous emission in paired graphene plasmonic waveguide structures |

Optics Express, Vol. 21, Issue 7, pp. 7897-7907 (2013)

http://dx.doi.org/10.1364/OE.21.007897

Acrobat PDF (2329 KB)

### Abstract

The coupling between a single emitter and surface plasmons in paired graphene layers and in paired graphene ribbons are studied. For paired graphene layers, the coupling between surface plasmons in graphene layers is strong at low photon energy and small gap between layers, which results in strong enhancement of the emitter’s emission. The excitation efficiency of surface plasmons by a single emitter can be increased to nearly 1 in paired graphene layers. With the increase of the photon energy, emitter’s emission in paired layers is weakened and could be lower than that in graphene monolayer. For graphene paired ribbons, numerical simulations show similar properties of emission enhancement and high excitation efficiency of surface plasmons. The emission enhancement and the excitation efficiency of surface plasmons can be improved by narrowing the ribbon width.

© 2013 OSA

## 1. Introduction

1. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics **4**, 83–91 (2010) [CrossRef] .

2. D. E. Chang, A. S. Sorensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**, 053002 (2006) [CrossRef] [PubMed] .

3. A. K. Ekert, “Quantum cryptography based on Bell s theorem,” Phys. Rev. Lett. **67**, 661–663 (1991) [CrossRef] [PubMed] .

5. K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation,” Phys. Rev. A **72**, 022317 (2005) [CrossRef] .

## 2. Emission in paired graphene layers

*E*|) of the fundamental SPs mode [11

11. C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B **85**, 125431 (2012) [CrossRef] .

12. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS. Nano . **6**, 431–440 (2012) [CrossRef] .

*l*. To efficiently couple the emitter with the fundamental SPs mode in PGL, the dipole is set to be polarized vertically to graphene layers to match the symmetry of the mode. Similar schematic of PGR is illustrated in Fig. 1(b) and the width of ribbons is denoted as

*w*.

*et. al.*[6

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

*ε*

_{1}and

*ε*

_{2}, for a dipole |

*p*| situated in medium

*ε*

_{1}and polarized vertically to graphene, the total decay rate Γ and the decay rate Γ

*into a single SP can be express as and respectively.*

_{sp}*r*is the reflectivity for a parallel-polarized plane wave from the medium

*ε*

_{1}where

*q*is the wave vector parallel to the graphene monolayer,

*k*

_{0}is the wave vector in vacuum and

*σ*is the conductivity of graphene. The zero of the denominator in

*r*gives the wave number

*q*for the SPs and, correspondingly,

_{sp}*λ*= 2

_{sp}*π*/

*q*.

_{sp}*ε*

_{1}and the medium outside the graphene structures with

*ε*

_{2}. The reflection between two graphene layers takes the form Here

*k*

_{⊥}is the vertical component of wave vector in the gap [14

14. Y. C. Jun, R.D. Kekatpure, J. S. White, and M. L. Brongersma, “Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures,” Phys. Rev. B **78**, 153111 (2008) [CrossRef] .

15. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. **113**, 195–287 (1984) [CrossRef] .

*p*| by the electric field induced by the dipole [16

16. L. Novotny and B. Hecht, *Principles of Nano-Optics* (Cambridge University Press, 2006) [CrossRef] .

*at q*≫

_{sp}*k*

_{0}is For a well defined SP mode in PGL with wavelength

*λ*= 2

_{sp}*π*/

*q*, the integral in Γ includes a sharp pole associated with the emission into the SP mode and the pole contribution yields the decay rate into the SP With

_{sp}*Q*(

*l*,

*λ*) defined as If

_{sp}*ε*

_{1}=

*ε*

_{2}, Γ

*is simplified into*

_{sp}*σ*is given as within the random-phase approximations in the local limit [17

17. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New. 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] .

*θ*is a step function. The Fermi level is set to be

*E*= 0.5

_{f}*eV*unless specified. We use a value for relaxation time

*τ*= 5 ×10

^{−13}

*s*, which is estimated from the measured, impurity-limited DC mobility [19

19. 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**, 666–669 (2004) [CrossRef] [PubMed] .

20. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature **438**, 197–200 (2005) [CrossRef] [PubMed] .

10. A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B **84**, 161407 (2011) [CrossRef] .

*ε*

_{1}=

*ε*

_{2}= 1.

*in red. The separation between the dipole and graphene layer*

_{sp}*l*is 50nm. There exists an optimal photon energy at which Γ

*reaches its maximum. Away from the optimal photon energy, Γ*

_{sp}*falls gradually. The total decay rate Γ behaves similarly to that of Γ*

_{sp}*except for the part of low photon energy. As shown by Fig. 2(a), Γ get increased as the photon energy decreases. The large discrepancy between Γ and Γ*

_{sp}*in the low energy regime indicates that a large amount of energy decays into lossy channels [6*

_{sp}6. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interaction,” Nano Lett. **11**, 3370–3377 (2011) [CrossRef] [PubMed] .

*monotonically decrease with the photon energy. Though Γ for PGL is only one order of magnitude higher than that for GML, Γ*

_{sp}*is several order of magnitude higher at low photon energy than GML. Actually, the higher decay rates for PGL than GML originate from the fact that the fundamental SP mode in PGL has the electromagnetic field concentrated in the gap between graphene layers as shown in Fig. 1(a). The inset in Fig. 2(b) plots the excitation efficiency of SPs defined as Γ*

_{sp}*/Γ. For GML, the excitation efficiency is nearly 1 at high photon energy and approaches 0 at low photon energy. In contrast, the excitation efficiency of SPs for PGL is always around 1 regardless of the photon energy.*

_{sp}*q*. The first observation in Fig. 2(c) is that the normalized wave vector in both PGL and GML goes up with the photon energy monotonically which leads to strong confinement of electromagnetic field to graphene. The increasing confinement with the photon energy weakens the coupling between the emitter and the graphene layers which explain the decrease of the decay rates when the photon energy moves to high values. When photon energy moves to low values, the confinement in GML is weak and results in large mode volume, thus the decay rates of SPs also get decreased in low photon energy. But for PGL, the field of SPs is confined in the gap between layers and thus Γ

_{sp}*also gets increased in low photon energy. The second one is that the confinement of the electromagnetic field to graphene for PGL is always much stronger than that for GML. With the separation between graphene layers 2*

_{sp}*l*varying from 10 nm to 200 nm, the confinement of the electromagnetic field in PGL is lessened and, due to the fact that the coupling between the excitations of collective charge oscillation in the two graphene layers becomes weakened with

*l*, the dispersion relation of the fundamental SP mode in PGL approaches to the one in GML. In addition, we present the ratio between the propagation lengths

*L*and the wavelength

*λ*of the fundamental SP mode in Fig. 2(d), which shows that SPs in PGL have longer life times than those in GML.

_{sp}## 3. Anomalous emission enhancement in paired layers

*η*defined as where the subscripts

*p*and

*m*denote the quantities for PGL and GML, respectively. The factor 2 before Γ

_{sp,m}takes account of two graphere layers in PGL and the decay rates for PGL should be twice as that in GML when

*l*tends to infinity. The results from Eq. (11) are plotted in Fig. 3(a) with

*l*= 50

*nm*. Generally speaking,

*η*tends to 1 at sufficiently high photon energy since strong confinement of SPs in the graphene layers leads to the decoupling between layers. On the other hand,

*η*increases from 1 to 10

^{3}with decreasing the photon energy which suggests that PGL is much better in the excitation of SPs by an emitter than GML at low photon energy. Scrutinizing

*η*against the photon energy, we find an interesting regime of the photon energy in which

*η*is less than 1. The anomalous behavior of

*η*indicates that the coupling of SPs in different graphene layers could play a negative role in the emission enhancement. The anomalous

*η*exists for the photon energy larger than 0.17

*eV*as shown by the inset of Fig. 3(a) and

*η*may goes down to

*η*= 0.9. Furthermore, we find that the separation between graphene layers does not change qualitatively how

*η*depends on the photon energy. As shown in Fig. 3(b), when the coupling between SPs in graphene layers are positive to

*η*, the separation 2

*l*is manifested its effects by the value of

*η*, i.e., small separation leads to high

*η*due to the stronger coupling between graphene layers for narrower gap. For anomalous

*η*, it is noted that the separation 2

*l*does not change the minimum value of

*η*and narrowing the separation shifts the anomalous region of

*η*to high photon energy.

*γ*= 2

*π*|

*g*|

^{2}

*D*

_{2D}(

*ω*).

*D*

_{2D}(

*ω*) is the density of states. |

*g*| is the coupling strength between the dipole and the electromagnetic field at the emitter position

*x*and is given by in which

_{p}*ε*is the relative permittivity of the dielectric medium surrounding the emitter. For the PGL in this work, the effective mode volume

_{r}*V*is defined as [14

_{eff}14. Y. C. Jun, R.D. Kekatpure, J. S. White, and M. L. Brongersma, “Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures,” Phys. Rev. B **78**, 153111 (2008) [CrossRef] .

*V*=

_{eff}*L*

_{eff}h^{2}with

*h*an arbitrary quantization length.

*L*is the effective mode length across the gap:

_{eff}*E*(

*x*,

_{p}*ω*) is the electric field at the dipole’s position. The density of states can be expressed as

*D*

_{2D}(

*ω*) =

*h*

^{2}

*ω*/(2

*πυ*) with

_{p}υ_{g}*υ*and

_{p}*υ*the phase and group velocities of SPs mode, respectively. Consequently, the emission enhancement

_{g}*η*can be express as the product of three factors,

*η*are contributed by the effective mode volume

*V*, the phase velocity

_{eff}*υ*and the group velocity

_{p}*υ*, respectively. The phase velocities and the group velocities for PGL and GML can be extracted from Fig. 2(c). From Eq. (13) and the field distribution of the fundamental SP mode, we have The emission enhancement

_{g}*η*calculated from the Fermi golden rule is plotted against the photon energy in red in Fig. 4(a) under the same circumstance as Fig. 3(a). For a better comparison, we replot

*η*from Eq. (11) in black in Fig. 4(a). As shown in Fig. 4(a), the Fermi golden rule gives an similar anomalous

*η*but for a larger photon energy regime. To reveal what is responsible for the anomalous

*η*, the three factors on the right side of Eq. (14) are plotted against the photon energy in Fig. 4(b). As shown in Fig. 4(b), the factor denoting the contribution from the phase velocity decreases monotonically and approaches to 1 at high photon energy, which means that the phase velocity is a irrelevant factor to the anomalous

*η*. In contrast, the other two factors denoting the contributions from the group velocity and the effective mode volume become less than 1 for high photon energy. Furthermore, the ratio of the effective mode volumes between PGL and GML may go much lower than the ratio of the group velocities and, especially, the minimum of the factor from the effective mode volume seems to coincide with that in

*η*. Based on these observations, it could be concluded that the anomalous

*η*is mainly caused by the factor from the effective mode volume.

*log*(

*η*) for different

*E*in the plane of the photon energy and the distance between the emitter and the graphene layers

_{f}*l*. The parameter space is divided into two regimes by the yellow solid curves on which

*η*= 1 in the plots and, above the curves, the anomalous

*η*occurs. It can be drawn from Fig. 5 that both decreasing

*E*and increasing

_{f}*l*disfavor the emission enhancement. The results in Fig. 5 could be helpful in possible applications using the coupling between SPs in graphene layers to enhance the photon emission of a emitter.

## 4. Emission in paired graphene ribbons

*w*= 300

*nm*and the separation between the emitter and the ribbons

*l*= 50

*nm*, the normalized wave vector for the first four modes of PGR are plotted against the photon energy in Fig. 6(b). Typical profiles of the electric field

*E*

_{⊥}vertical to graphene ribbons are shown in Fig. 6(c) for these four SP modes. These modes are characterized by different symmetries and each mode has the field concentrated at the edges of graphene ribbons. Take the first mode denoted by

*M*1 in the figure for instance, the profile of

*E*

_{⊥}shows that the two ribbons have opposite charges and each ribbon has the same charge at its opposite edges, which indicates that the mode is antisymmetric in the vertical direction and symmetric in the parallel direction.

*l*= 50

*nm*. We consider the emitter situated in the center of PGR and polarized vertically to the graphene ribbons as shown in the inset in Fig. 7(b). According to the symmetry of

*E*

_{⊥}in these four modes, the emitter only interacts with the fundamental SP mode (

*M*1) in PGR. Figure 7(a) shows the simulation results about the total decay rate Γ and the decay rate into SPs Γ

*for PGR with the ribbon width*

_{sp}*w*varied from 300

*nm*to 100

*nm*. Here Γ is obtained by calculating the total energy flowing out of the emitter and Γ

*is by integrating the energy flowing in the direction the fundamental SP mode*

_{sp}*M*1 propagates. Compared with the results for PGL plotted in Fig. 7 in bold curves, the decay rates for PGR get higher enhancement at low photon energy. When the width of ribbons

*w*get reduced, the effective mode volume of PGR will be reduced either and the decay rates get increased as shown in Fig. 7(a). The excitation efficiencies of SPs by the emitter in PGR for different width of ribbons are plotted in Fig. 7(b). Regardless the width of the ribbons, the excitation efficiency of SPs in PGR is higher than that for PGL and is always close to 1.

*nm*to 15

*nm*, could lead the emission enhancement to become weakened. However, such a weakening emission enhancement for extremely narrow ribbon might be a delusion since the classical electromagnetic treatment employed here may be invalid as mentioned in reference [21

21. S. Thongrattanasiri, A. Manjavacas, and F. Javier García de Abajo, “Quantum Finite-Size Effects in Graphene Plasmons,” ACS Nano **6**, 1766–1775 (2012) [CrossRef] [PubMed] .

## 5. Emission in paired unequally doped graphene structures

22. S. Thongrattanasiri, Iván Silveiro, and F. Javier García de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. **100**, 201105 (2012) [CrossRef] .

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

*E*

_{f1}and

*E*

_{f2}are the Fermi levels for the two layers. Denote the reflections at mono graphene layer by

*r*

_{1}and

*r*

_{2}which are functions of

*E*

_{f1}and

*E*

_{f2}, respectively, we have the total emission rate

*E*

_{f2}at

*E*

_{f1}=

*E*

_{f0}= 0.5

*eV*. The dependency of the emission rate on

*E*

_{f2}is similar to that in equally doped condition but less sensitive. Furthermore for

*E*

_{f1}<

*E*

_{f2}, we find that Γ

_{Ef1Ef1}> Γ

_{Ef1Ef2}> Γ

_{Ef2Ef2}at low photon energy and, conversely, Γ

_{Ef1Ef1}< Γ

_{Ef1Ef2}< Γ

_{Ef2Ef2}at high photon energy. An example is shown in Fig. 8(d) in which three combinations of

*E*

_{f1}and

*E*

_{f2}(|

*E*

_{f0},

*E*

_{f0}|, |1.5

*E*

_{f0}, 1.5

*E*

_{f0}|, and |

*E*

_{f0}, 1.5

*E*

_{f0}|) are used. For PGR, Γ

_{Ef1Ef2}is presented in Figs. 8(e) and 8(f), which show that the results are similar to those for PGL.

## 6. Conclusion

*η*. For graphene ribbon structures, PGR has similar properties to PGL that the emission of an emitter gets strong enhancement at low photon energy and SPs efficiency can reach nearly 1. The emission enhancement can be improved by shortening the ribbon width provided that the ribbon width is above a certain value. The schematic discussed in this work may serve as a platform for investigating various phenomena in quantum optics such as vacuum Rabi splitting [6

**11**, 3370–3377 (2011) [CrossRef] [PubMed] .

23. A. Manjavacas, S. Thongrattanasiri, D. E. Chang, and F. J. García de Abajo, “Temporal quantum control with graphene,” New. J. Phys. **14**, 123020 (2012) [CrossRef] .

## Acknowledgments

## References and links

1. | D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics |

2. | D. E. Chang, A. S. Sorensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. |

3. | A. K. Ekert, “Quantum cryptography based on Bell s theorem,” Phys. Rev. Lett. |

4. | H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. |

5. | K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation,” Phys. Rev. A |

6. | F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interaction,” Nano Lett. |

7. | Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. H. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. |

8. | D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett. |

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

10. | A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B |

11. | C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B |

12. | J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS. Nano . |

13. | A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B |

14. | Y. C. Jun, R.D. Kekatpure, J. S. White, and M. L. Brongersma, “Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures,” Phys. Rev. B |

15. | G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. |

16. | L. Novotny and B. Hecht, |

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

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

19. | 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 |

20. | K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature |

21. | S. Thongrattanasiri, A. Manjavacas, and F. Javier García de Abajo, “Quantum Finite-Size Effects in Graphene Plasmons,” ACS Nano |

22. | S. Thongrattanasiri, Iván Silveiro, and F. Javier García de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. |

23. | A. Manjavacas, S. Thongrattanasiri, D. E. Chang, and F. J. García de Abajo, “Temporal quantum control with graphene,” New. J. Phys. |

**OCIS Codes**

(020.5580) Atomic and molecular physics : Quantum electrodynamics

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 24, 2013

Revised Manuscript: March 9, 2013

Manuscript Accepted: March 10, 2013

Published: March 25, 2013

**Citation**

Lei Zhang, Xiuli Fu, Mei Zhang, and Junzhong Yang, "Spontaneous emission in paired graphene plasmonic waveguide structures," Opt. Express **21**, 7897-7907 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-7897

Sort: Year | Journal | Reset

### References

- D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010). [CrossRef]
- D. E. Chang, A. S. Sorensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97, 053002 (2006). [CrossRef] [PubMed]
- A. K. Ekert, “Quantum cryptography based on Bell s theorem,” Phys. Rev. Lett.67, 661–663 (1991). [CrossRef] [PubMed]
- H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett.81, 5932–5935 (1998). [CrossRef]
- K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation,” Phys. Rev. A72, 022317 (2005). [CrossRef]
- F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interaction,” Nano Lett.11, 3370–3377 (2011). [CrossRef] [PubMed]
- Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. H. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008). [CrossRef]
- D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett.105, 256805 (2010). [CrossRef]
- M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009). [CrossRef]
- A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B84, 161407 (2011). [CrossRef]
- C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B85, 125431 (2012). [CrossRef]
- J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS. Nano. 6, 431–440 (2012). [CrossRef]
- A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B84, 195446 (2011). [CrossRef]
- Y. C. Jun, R.D. Kekatpure, J. S. White, and M. L. Brongersma, “Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures,” Phys. Rev. B78, 153111 (2008). [CrossRef]
- G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep.113, 195–287 (1984). [CrossRef]
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006). [CrossRef]
- B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New. J. Phys.8, 318 (2006). [CrossRef]
- E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B75, 205418 (2007). [CrossRef]
- 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,” Science306, 666–669 (2004). [CrossRef] [PubMed]
- K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005). [CrossRef] [PubMed]
- S. Thongrattanasiri, A. Manjavacas, and F. Javier García de Abajo, “Quantum Finite-Size Effects in Graphene Plasmons,” ACS Nano6, 1766–1775 (2012). [CrossRef] [PubMed]
- S. Thongrattanasiri, Iván Silveiro, and F. Javier García de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett.100, 201105 (2012). [CrossRef]
- A. Manjavacas, S. Thongrattanasiri, D. E. Chang, and F. J. García de Abajo, “Temporal quantum control with graphene,” New. J. Phys.14, 123020 (2012). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.