## Photon emission rate engineering using graphene nanodisc cavities |

Optics Express, Vol. 22, Issue 6, pp. 6400-6415 (2014)

http://dx.doi.org/10.1364/OE.22.006400

Acrobat PDF (1549 KB)

### Abstract

In this work, we present a systematic study of the plasmon modes in a system of vertically stacked pair of graphene discs. Quasistatic approximation is used to model the eigenmodes of the system. Eigen-response theory is employed to explain the spatial dependence of the coupling between the plasmon modes and a quantum emitter. These results show a good match between the semi-analytical calculation and full-wave simulations. Secondly, we have shown that it is possible to engineer the decay rates of a quantum emitter placed inside and near this cavity, using Fermi level tuning, via gate voltages and variation of emitter location and polarization. We highlighted that by coupling to the bright plasmon mode, the radiative efficiency of the emitter can be enhanced compared to the single graphene disc case, whereas the dark plasmon mode suppresses the radiative efficiency.

© 2014 Optical Society of America

## 1. Introduction

2. E. Fort and S. Grésillon, “Surface enhanced fluorescence,” J. Phys. D: Appl. Phys. **41**, 013001 (2008). [CrossRef]

3. M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. **102**, 107401 (2009). [CrossRef] [PubMed]

4. S. Lee, J. H. Ryu, K. Park, A. Lee, S.-Y. Lee, I.-C. Youn, C.-H. Ahn, S. M. Yoon, S.-J. Myung, D. H. Moon, X. Chen, K. Choi, I. C. Kwon, and K. Kim, “Polymeric nanoparticle-based activatable near-infrared nanosensor for protease determination in vivo,” Nano Lett. **9**, 4412–4416 (2009). PMID: [PubMed] . [CrossRef]

5. R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, “Fluorescence and energy transfer near interfaces: The complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. **63**, 1589–1595 (1975). [CrossRef]

6. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics **1**, 449–458 (1975). [CrossRef]

3. M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. **102**, 107401 (2009). [CrossRef] [PubMed]

7. J. B. Khurgin and A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials,” MRS Bull. **37**, 768–779 (2012). [CrossRef]

8. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mat. **6**, 183–191 (2007). [CrossRef]

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

10. T. Ando and T. Nakanishi, “Impurity scattering in carbon nanotubes – absence of back scattering –,” J. Phys. Soc. Jpn **67**, 1704–1713 (1998). [CrossRef]

11. 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]

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

13. H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. **14**, 125001 (2012). [CrossRef]

## 2. Methods

### 2.1. Modeling electrodynamic response of graphene

14. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B **56**, 281–284 (2007). [CrossRef]

*H*(

*ω*,

*T*) = sinh(

*h̄ω/kT*)/[cosh(

*μ/kT*) + cosh(

*h̄ω/kT*)].

*τ*is the electron relaxation time. The relaxation time typically has contributions from 1) scattering from impurities in infinite graphene, 2) coupling to acoustic and optical phonons (

*h̄ω*= 0.2 eV) in graphene and phonon modes of polar substrates and 3) edge scattering in the case of finite nanostructures, such as the one discussed in the present paper ([15

_{OPh}15. M. Jablan, M. Soljacic, and H. Buljan, “Plasmons in graphene: Fundamental properties and potential applications,” Proceedings of the IEEE **101**, 1689–1704 (2013). [CrossRef]

16. C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronic,” Nature Nanotech. **5**, 722–726 (2010). [CrossRef]

17. K. Bolotin, K. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. **146**, 351–355 (2008). [CrossRef]

*τ*∼ 50 fs [9

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

*τ*= 50 fs and

*T*= 300 K. For the plane-wave excitation result, a larger

*τ*of 100 fs is used since for smaller values, the extinction peak for the dark mode is too broad to be separated out from the background, dominated by the bright mode. However, both these values we used for

*τ*are on the very conservative side of the range of experimentally measured values.

### 2.2. Simulation of graphene plasmon modes

18. H. Reid, scuff-em suite version 0.95, http://homerreid.com/scuff-em.

19. M. T. H. Reid and S. G. Johnson, “Efficient computation of power, force, and torque in bem scattering calculations,” (2013), http://arxiv.org/abs/1307.2966.

20. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science **332**, 1291–1294 (2011). [CrossRef] [PubMed]

### 2.3. Calculation of decay rates

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

_{0}is the decay rate of the emitter, if it were in free space.

*/Γ*

_{rad}_{0}=

*P*

_{rad}/P_{0}= 1 +

*P*

_{sca}/P_{0}, where

*P*is the power radiated to the far field,

_{rad}*P*is the total scattered power and

_{sca}*P*

_{0}is the power radiated by the emitter when placed in free-space. The non-radiative decay rate, which is the dominant contribution from the decay into the plasmon mode is given by Γ

*Γ*

_{abs}/_{0}=

*P*

_{abs}/P_{0}≈ Γ

*/Γ*

_{plasmon}_{0}, where

*P*is the power absorbed in the graphene nanostructure.

_{abs}*ω*

_{0}is resonant frequency of the quantum emitter and

*ω*is the frequency of the plasmon mode.

*σ*

^{+}(

*σ*

^{−}) are the atomic raising (lowering) operators and

*a*

^{†}(

*a*) are the creation (annihilation) operators for a cavity photon.

23. T. Hümmer, F. J. García-Vidal, L. Martín-Moreno, and D. Zueco, “Weak and strong coupling regimes in plasmonic qed,” Phys. Rev. B **87**, 115419 (2013). [CrossRef]

*κ*is the rate of decay of the plasmon mode.

*κ*contains both the radiation as well and absorption mechanisms for broadening the plasmon resonance [24

24. E. Waks and D. Sridharan, “Cavity qed treatment of interactions between a metal nanoparticle and a dipole emitter,” Phys. Rev. A **82**, 043845 (2010). [CrossRef]

_{0}, the spontaneous emission rate in free space, in accordance with Wigner-Weisskopf theory.

*g*〉 ⊗ |1〉, |

*e*〉 ⊗ |0〉, |

*g*〉 ⊗ |0〉} need to be retained. Here |

*g*〉 and |

*e*〉 are the ground and excited states of the atom and |0〉 and |1〉 denote the number of photons in the cavity mode. It can then be shown [25

25. J.-M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” in “Single Quantum Dots,”, vol. 90 of *Topics in Applied Physics* (SpringerBerlin Heidelberg, 2003), pp. 269–314. [CrossRef]

*g*/(

*κ*− Γ′)| > 1/2.

*g*is determined by the details of the cavity field mode and the atomic dipole matrix element. For our purpose, we determine

*g*classically, using the limit of a low finesse cavity. In this limit,

*g*satisfies the following equation: where Δ is the detuning between the resonant frequency of the plasmon mode and that of the quantum emitter. For the present work, the typical spontaneous emission rate of the emitter is much smaller than the cavity line-width. Thus, Eq. (6) suggests that on resonance, Γ

*= (*

_{tot}/κ*g/κ*)

^{2}. Hence the

*g*factor can be determined. This expression also points out that Rabi oscillations should exist when Γ

*/*

_{tot}*κ*> 1/4.

## 3. Results and discussion

### 3.1. Calculation of the eigen-modes in the quasistatic limit

26. A. L. Fetter, “Magnetoplasmons in a two-dimensional electron fluid: Disk geometry,” Phys. Rev. B **33**, 5221–5227 (1986). [CrossRef]

#### 3.1.1. Mathematical framework for a stacked dimer of discs

26. A. L. Fetter, “Magnetoplasmons in a two-dimensional electron fluid: Disk geometry,” Phys. Rev. B **33**, 5221–5227 (1986). [CrossRef]

27. W. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B **86**, 125450 (2012). [CrossRef]

28. J. I. Gersten, “Disk plasma oscillations,” J. Chem. Phys. **77**, 6285–6288 (1982). [CrossRef]

*R*, stacked vertically with a separation

*D*in between (see Fig. 1). The location of the discs in our chosen coordinate system is

*z*= ±

*D*/2. The approach that will be presented here can easily incorporate the case where the two discs are non-identical. However, for the sake of clarity for our specific case, we will only consider identical discs for now.

**r**) = Φ(

*r*,

*z*)

*e*, in cylindrical coordinates.

^{iLϕ}- Express the surface potential Φ(
*r*,*z*= ±*D*/2) in terms of the surface charge density*σ*, using the Laplace equation and the normal electric field boundary condition_{b} - Express surface charge density
*σ*in terms of the surface potential Φ(_{b}*r*,*z*= ±*D*/2), using the continuity equation and the current-field relation

*σ*: The Poisson equation in this case is given by: where

_{b}*σ*is the surface charge density (and not the surface conductivity, which is represented by

_{b}*σ*).

*z*≠ 0 and then use the boundary condition for the normal electric field. To be specific, these equations are given below: where

*ε*is the relative permittivity of the medium in between the discs. Note that the

_{m}*e*dependence was suppressed in the boundary condition equation. (Note that there is another boundary condition which is the continuity of the potential across

^{iLϕ}*z*= ±

*D*/2.)

*z*≠ ±

*D*/2, Eq. (8) holds for the potential Φ(

**r**) in real coordinates. Equivalently, for

*z*≠ ±

*D*/2, Eq. (12) holds for the Hankel transformed potential. We can write down the form of the solution in the three different regions as follows: There are four unknowns

*A*,

_{u}*A*. We also have four equations, two for the continuity of the potential across the discs and the other two for the normal electric field boundary condition.

_{d}*ε*=

_{u}*ε*=

_{m}*ε*=

_{d}*ε*for simplicity. Solving the above linear system of equations, we get the solution for the Hankel-transformed potential on the discs: Now we go to real space, by taking the inverse Hankel transform on each side of the above equation. For brevity, we denote the Hankel transform operator as

*σ*in terms of Φ: There are two equations that we need to express

_{b}*σ*in terms of Φ. One is the continuity equation for surface current density and the other is the relation between surface current density and the electric field. These equations are given below: Now using the relation

_{b}**E**

_{||}= −∇

_{||}(Φ(

*r*,

*z*= 0)

*e*), we arrive at the relation: Thus, combining Eq. (15) and Eq. (18), we arrive at the final eigenvalue equation for the stacked discs case:

^{ı}^{Lϕ}#### 3.1.2. Comparison with full wave simulation

*L*= 1,

*n*= 1 mode since that is the mode that we will be concerned with in the rest of the paper, when talking about photon emission rate engineering. It should also be noted that for the simulations, we use a realistic absorption in the graphene conductivity. The comparison is presented in Fig. 2. Figure 2 suggests that there is a good overall match between the resonant frequencies found from the BEM and the quasistatic result. The resonant frequencies in the simulation were obtained from the LDOS spectrum.

*E*. This is due to the fact that increasing

_{F}*E*results in an increase in the carrier density, which in turn causes an increase in the restoring force. This explanation is similar to how the plasma frequency in noble metals increases with carrier concentration. Secondly, the frequency splitting

_{F}*ω*−

_{B}*ω*increases with

_{D}*E*. This is due to the fact that at higher

_{F}*E*, the plasmon modes of individual discs are more leaky. This results in the interaction between the two discs being even stronger, resulting in a larger splitting.

_{F}### 3.2. Eigen-response theoretic framework and calculation of overlap

29. K. H. Fung, A. Kumar, and N. X. Fang, “Electron-photon scattering mediated by localized plasmons: A quantitative analysis by eigen-response theory,” Phys. Rev. B **89**, 045408 (2014). [CrossRef]

*α*and

_{A,L}*α*are the eigen-polarizabilities of the antisymmetric and symmetric modes and

_{S,L}*P*and

_{A,L}*P*are the eigenmodes.

_{S,L}*i*〉 is a quantity proportional to the surface polarization, for each mode. In the following section we present the calculation of the surface polarization, which will help us calculate these overlap terms.

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

**r**

_{0}=

*X*+

_{d}x̂*Y*+

_{d}ŷ*Z*and the location of the infinitesimal dipoles

_{d}ẑ**r**=

*xx̂*+

*yŷ*+

*zẑ*.

### 3.3. Decay rate engineering

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

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

30. C. Vandenbem, D. Brayer, L. S. Froufe-Pérez, and R. Carminati, “Controlling the quantum yield of a dipole emitter with coupled plasmonic modes,” Phys. Rev. B **81**, 085444 (2010). [CrossRef]

30. C. Vandenbem, D. Brayer, L. S. Froufe-Pérez, and R. Carminati, “Controlling the quantum yield of a dipole emitter with coupled plasmonic modes,” Phys. Rev. B **81**, 085444 (2010). [CrossRef]

31. M. K. Schmidt, S. Mackowski, and J. Aizpurua, “Control of single emitter radiation by polarization- and position-dependent activation of dark antenna modes,” Opt. Lett. **37**, 1017–1019 (2012). [CrossRef] [PubMed]

#### 3.3.1. Case A: Emitters inside the stacked disc cavity

*Bright Mode*: Since the sign of the infinitesimal dipole moment does not change, we only need to consider the sign change in the*z*coordinate of the two discs. This results in the total sum of the overlap term for the*z*= ±*d*/2 giving a zero for the*z*–polarization of the emitter. The other two terms for the*x*and*y*polarizations survive and are basically twice of the corresponding value for the single disc case.*Dark Mode*: In this case, the sign of the infinitesimal dipole moment does change, and so does the sign change in the*z*coordinate of the two discs. This results in the total sum of the overlap term for the*z*= ±*d*/2 giving a zero for the*x*– and*y*–polarizations of the emitter. The only nonzero term is the one for the*z*–polarizations and is just twice of the corresponding value for the single disc case.

#### 3.3.2. Case B: Emitters outside the stacked disc cavity

*X*= 0 nm, both the modes are excitable, with highest probability.

_{d}*X*, 0,

_{d}*Z*) couples to either of the modes at their respective resonant frequencies. An example spectrum, for

_{d}*X*= 0 nm is shown in Fig. 7(a). In this case, the total decay rate enhancement, which is close to the non-radiative part, is almost the same for the two modes. This is consistent with our analytical calculation of the overlap terms which show that at

_{d}*X*= 0 nm both the modes are equally excitable. However, there is a difference between the radiative decay rate enhancement. This situation results because of partial cancellation of the induced moments on two discs, for the dark mode.

_{d}*E*. The dark mode becomes less, and the bright mode, more radiative, as the carrier concentration increases. As mentioned before, this is because at higher carrier concentrations, the modes of the two discs can interact more strongly.

_{F}*X*of the emitter, in Fig. 8(a). Clearly, the bright mode has a higher radiative efficiency compared to the dark mode, for various locations of the emitter. We find that the radiative efficiency drops from a maximum at

_{d}*X*= 0 nm to a local minimum, as the emitter approaches a certain horizontal distance (

_{d}*X*= 40 nm in the specific case of Fig. 8(a)) and then rises again. This behaviour can be qualitatively explained by looking at the overlap term (not shown), which predicts that the mode excitability being maximum at

_{d}*X*= 0 nm, drops to zero at a certain

_{d}*X*to subsequently rise again. Similar pattern in observed for the dark mode.

_{d}12. F. H. L. Koppens, D. E. Chang, and F. J. Garcia de Abajo, “Graphene plasmonics: A platform for strong light-matter interactions,” Nano Lett. **11**, 3370–3377 (2011). [CrossRef] [PubMed]

*≈ Γ*

_{tot}*>*

_{plasmon}*κ*on resonance. This condition ensures the existence of coherent coupling between the plasmon and the emitter and the possibility of observing vacuum Rabi splitting [23

23. T. Hümmer, F. J. García-Vidal, L. Martín-Moreno, and D. Zueco, “Weak and strong coupling regimes in plasmonic qed,” Phys. Rev. B **87**, 115419 (2013). [CrossRef]

*g*factors for different Fermi-level energies, ranging from 0.2 – 0.8 eV, for different positions of the emitter both inside and outside the cavity. Our calculations for the normalized Rabi splitting are shown in Fig. 8(b), for an example case of the emitter placed outside the cavity (same geometr y as considered earlier in this section). We find that for a range of values of the doping level, we do obtain

*g/κ*> 1/2 and we predict Rabi splitting values

*g*of up to 10 meV at room temperature. Here for calculating the absolute value of

*g*, we have used Γ

_{0}≈ 5 × 10

^{7}

*s*

^{−1}[12

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

*τ*is larger. Another important trend suggested by Fig. 8(b) is that the value of

*g/κ*decreases with increasing

*E*. This can be qualitatively understood as resulting from the increasing leakiness of the plasmon mode as the carrier concentration is increased. This results in a larger mode volume

_{F}*V*, which is expected to cause a decrease in the decay rate Γ

*. Since*

_{tot}*κ*∼ 1/

*τ*is almost independent of Fermi Level

*E*, therefore

_{F}*E*.

_{F}## 4. Conclusion

## Acknowledgments

## References and links

1. | S. A. Maier, |

2. | E. Fort and S. Grésillon, “Surface enhanced fluorescence,” J. Phys. D: Appl. Phys. |

3. | M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. |

4. | S. Lee, J. H. Ryu, K. Park, A. Lee, S.-Y. Lee, I.-C. Youn, C.-H. Ahn, S. M. Yoon, S.-J. Myung, D. H. Moon, X. Chen, K. Choi, I. C. Kwon, and K. Kim, “Polymeric nanoparticle-based activatable near-infrared nanosensor for protease determination in vivo,” Nano Lett. |

5. | R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, “Fluorescence and energy transfer near interfaces: The complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. |

6. | S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics |

7. | J. B. Khurgin and A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials,” MRS Bull. |

8. | A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mat. |

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

10. | T. Ando and T. Nakanishi, “Impurity scattering in carbon nanotubes – absence of back scattering –,” J. Phys. Soc. Jpn |

11. | 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 |

12. | F. H. L. Koppens, D. E. Chang, and F. J. Garcia de Abajo, “Graphene plasmonics: A platform for strong light-matter interactions,” Nano Lett. |

13. | H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. |

14. | L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B |

15. | M. Jablan, M. Soljacic, and H. Buljan, “Plasmons in graphene: Fundamental properties and potential applications,” Proceedings of the IEEE |

16. | C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronic,” Nature Nanotech. |

17. | K. Bolotin, K. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. |

18. | H. Reid, scuff-em suite version 0.95, http://homerreid.com/scuff-em. |

19. | M. T. H. Reid and S. G. Johnson, “Efficient computation of power, force, and torque in bem scattering calculations,” (2013), http://arxiv.org/abs/1307.2966. |

20. | A. Vakil and N. Engheta, “Transformation optics using graphene,” Science |

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

22. | A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics |

23. | T. Hümmer, F. J. García-Vidal, L. Martín-Moreno, and D. Zueco, “Weak and strong coupling regimes in plasmonic qed,” Phys. Rev. B |

24. | E. Waks and D. Sridharan, “Cavity qed treatment of interactions between a metal nanoparticle and a dipole emitter,” Phys. Rev. A |

25. | J.-M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” in “Single Quantum Dots,”, vol. 90 of |

26. | A. L. Fetter, “Magnetoplasmons in a two-dimensional electron fluid: Disk geometry,” Phys. Rev. B |

27. | W. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B |

28. | J. I. Gersten, “Disk plasma oscillations,” J. Chem. Phys. |

29. | K. H. Fung, A. Kumar, and N. X. Fang, “Electron-photon scattering mediated by localized plasmons: A quantitative analysis by eigen-response theory,” Phys. Rev. B |

30. | C. Vandenbem, D. Brayer, L. S. Froufe-Pérez, and R. Carminati, “Controlling the quantum yield of a dipole emitter with coupled plasmonic modes,” Phys. Rev. B |

31. | M. K. Schmidt, S. Mackowski, and J. Aizpurua, “Control of single emitter radiation by polarization- and position-dependent activation of dark antenna modes,” Opt. Lett. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Plasmonics

**History**

Original Manuscript: February 11, 2014

Manuscript Accepted: February 28, 2014

Published: March 11, 2014

**Citation**

Anshuman Kumar, Kin Hung Fung, M. T. Homer Reid, and Nicholas X. Fang, "Photon emission rate engineering using graphene nanodisc cavities," Opt. Express **22**, 6400-6415 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6400

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### References

- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
- E. Fort, S. Grésillon, “Surface enhanced fluorescence,” J. Phys. D: Appl. Phys. 41, 013001 (2008). [CrossRef]
- M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009). [CrossRef] [PubMed]
- S. Lee, J. H. Ryu, K. Park, A. Lee, S.-Y. Lee, I.-C. Youn, C.-H. Ahn, S. M. Yoon, S.-J. Myung, D. H. Moon, X. Chen, K. Choi, I. C. Kwon, K. Kim, “Polymeric nanoparticle-based activatable near-infrared nanosensor for protease determination in vivo,” Nano Lett. 9, 4412–4416 (2009). PMID: . [CrossRef] [PubMed]
- R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: The complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975). [CrossRef]
- S. Noda, M. Fujita, T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (1975). [CrossRef]
- J. B. Khurgin, A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials,” MRS Bull. 37, 768–779 (2012). [CrossRef]
- A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nature Mat. 6, 183–191 (2007). [CrossRef]
- M. Jablan, H. Buljan, M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009). [CrossRef]
- T. Ando, T. Nakanishi, “Impurity scattering in carbon nanotubes – absence of back scattering –,” J. Phys. Soc. Jpn 67, 1704–1713 (1998). [CrossRef]
- K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). [CrossRef] [PubMed]
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