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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 7897–7907
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Spontaneous emission in paired graphene plasmonic waveguide structures

Lei Zhang, Xiuli Fu, Mei Zhang, and Junzhong Yang  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 7897-7907 (2013)
http://dx.doi.org/10.1364/OE.21.007897


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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

Surface Plasmons (SPs) in noble metals have been widely studied due to their abilities to enable confinement and control of electromagnetic energy at subwavelength scales [1

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

]. The small mode volume for the photons induced by the strong confinement of electromagnetic energy enables strong coupling between emitters and SPs in noble metals [2

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] .

] and such a mechanism is essential for quantum communications and the scalability of quantum computers [3

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

5

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

].

Recently, SPs in graphene have attracted intense attentions. Compared with SPs in noble metals, SPs in doped graphene have been proved with many advantages such as higher confinement in volume [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] .

], electric tunability [7

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. 4, 532–535 (2008) [CrossRef] .

, 8

8. D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105, 256805 (2010) [CrossRef] .

] and longer propagation length [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] .

, 9

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

]. Monolayer graphene has been proposed to work as an effective one dimensional micro-waveguide [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] .

,9

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

,10

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] .

] and double-layer graphene supports SPs with stronger confinement and longer life time [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

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] .

]. The coupling between emitters and SPs in graphene has been studied by several authors [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] .

, 13

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 84, 195446 (2011) [CrossRef] .

] in which strong excitation of SPs by emitters has been proved in infinite monolayer graphene and graphene nanodisks.

In this work, we focus on the coupling between a single emitter, modelling an excited molecule or a quantum dot, and SPs in paired graphene waveguide structures in the infrared regime. The graphene waveguide structures to be studied are paired infinite graphene layers (PGL) and paired graphene ribbons (PGR). For both paired structures, the emission of the emitter gets strong enhancement at low photon energies or at small separation between structures while increasing the photon energy results in less enhancement of the emission. We find that the excitation efficiency of SPs by an emitter can reach nearly 1 in PGL at low photon energy while, correspondingly, the efficiency in monolayer graphene (GML) is just around zero. We find that the coupling of SPs in graphene structures could play a negative role in the emission enhancement in which the emission of the emitter in PGL may be weaker than that in GML. We also investigate the effects of the width of ribbons in PGR on the coupling between the emitter and SPs in graphene structures.

2. Emission in paired graphene layers

The schematic of the coupling between a single emitter and SPs in PGL is illustrated in Fig. 1(a). The electric field intensity (|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

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] .

] in PGL is plotted in the inset of Fig. 1(a). As shown, the electromagnetic field in the fundamental SP mode is mainly concentrated in the gap between graphene layers and the electric field inside the gap is vertical to the layers (indicated by blue arrows). The emitter is treated as a sizeless dipole positioned at the center of the gap and the separation between the graphene layers is 2l. 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.

Fig. 1 Schematic of (a) paired graphene layers and (b) paired grapehene ribbons. Insets are the profiles of the electric field intensity (|E|) in their fundamental SPs modes. In the insets, the field directions in gaps are indicated by blue arrows. The optical dipole is placed in the center of structures and is polarized vertically to graphene. Distance between the dipole and graphene layers (or ribbons) is l, and the graphene structures are assumed to be surrounded by air.

The emission of an optical dipole in GML has been studied in an early work by Koppens 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] .

]. Considering GML surrounded by two dielectric media with the dielectric constant ε1 and ε2, for a dipole |p| situated in medium ε1 and polarized vertically to graphene, the total decay rate Γ and the decay rate Γsp into a single SP can be express as
ΓΓ0+2h¯|p|20Im(r)q2e2qldq,
(1)
and
ΓspΓ03πε1+ε2(λ0λsp)3e4πlλsp.
(2)
respectively. Γ0=4k03|p|2/3h¯ is the free space decay rate. r is the reflectivity for a parallel-polarized plane wave from the medium ε1
r=ε2k1ε1k2+k1k2σ/ωε0ε2k1+ε1k2+k1k2σ/ωε0
(3)
where k1=ε1k02q2, k2=ε2k02q2, q is the wave vector parallel to the graphene monolayer, k0 is the wave vector in vacuum and σ is the conductivity of graphene. The zero of the denominator in r gives the wave number qsp for the SPs and, correspondingly, λsp = 2π/qsp.

For PGL, we assume the medium in the gap to be the one with ε1 and the medium outside the graphene structures with ε2. The reflection between two graphene layers takes the form
rpair=re2ikl1re2ikl.
(4)
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

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

]. Letting the dominator to be zero, we get dispersion relations for the fundamental mode of SPs in PGL
re2qsp2ε1k02l=1.
(5)
By considering the work done on a dipole |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] .

] (i.e., the field reflected by the graphene layers), the total decay rate for the dipole in PGL at qspk0 is
ΓΓ01+3k030Im(re2ql1re2ql)q2dq.
(6)
For a well defined SP mode in PGL with wavelength λsp = 2π/qsp, 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
ΓspΓ0=3π(λ0λsp)32ε1e4πlλsp(ε2ε1)4πlλspe4πlλsp(1e4πlλsp)+[(ε1+ε2)(ε2ε1)e4πlλsp]Q(l,λsp).
(7)
With Q(l, λsp) defined as
Q(l,λsp)=1e4πlλsp+4πlλspe4πlλsp
(8)
If ε1 = ε2, Γsp is simplified into
ΓspΓ0=3π(λ0λsp)3e4πlλspQ(l,λsp).
(9)

The conductivity of graphene σ is given as
σ(ω)=e2Efπh¯2iω+iτ1+e24h¯[θ(h¯ω2Ef)+iπlog|h¯ω2Efh¯ω+2Ef|]
(10)
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

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 Ef = 0.5eV unless specified. We use a value for relaxation time τ = 5 ×10−13s, 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

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] .

]. The photon energy to be investigated is in the infrared regime. To be mentioned, the above analysis is insensitive to the micro structure of graphene edges, armchair edges or zigzag edges, since SPs are collective excitations of the total electronic charges [10

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] .

]. Throughout this work, we let ε1 = ε2 = 1.

We first show the decay rates versus the photon energy for GML in Fig. 2(a) with the total decay rate Γ plotted in black and the SP decay rate Γsp in red. The separation between the dipole and graphene layer l is 50nm. There exists an optimal photon energy at which Γsp reaches its maximum. Away from the optimal photon energy, Γsp falls gradually. The total decay rate Γ behaves similarly to that of Γsp 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 Γsp in the low energy regime indicates that a large amount of energy decays into lossy channels [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] .

]. The decay rates for PGL are shown in Fig. 2(b) with the same parameters as those in Fig. 2(a). Differently from GML, both Γ and Γsp monotonically decrease with the photon energy. Though Γ for PGL is only one order of magnitude higher than that for GML, Γsp 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 Γsp/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.

Fig. 2 (a) For graphene monolayer, the total decay rate Γ is plotted in black solid line and the SPs decay rate Γsp in red dashed line. (b) For paired graphene layers, the total decay rate Γ is plotted in black solid line and the SPs decay rate Γsp in red solid line. Inset: the excitation efficiency of SPs by an emitter in graphene monolayer is plotted in green and the efficiency of SPs in paired graphene layers in blue line. In (a) and (b), l = 50nm. The solid lines show (c) the real part of the propagation constant and (d) the propagation distance of SPs in paired graphene layers versus the photon energy, and l is taken as 10 nm, 20 nm, 50 nm, 100 nm and 200 nm from the top line to the bottom line. For SPs in graphene monolayer, the data are plotted in black dashed line.

To better understand the coupling between the emitter and the fundamental SP mode in graphene structures, we present the dispersion relations of SPs in PGL and of SPs in GML in Figs. 2(c) and 2(d). Figure 2(c) shows the normalized wave vector, the real part of qsp. 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 Γsp 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 2l 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 λsp of the fundamental SP mode in Fig. 2(d), which shows that SPs in PGL have longer life times than those in GML.

3. Anomalous emission enhancement in paired layers

To show how the coupling between graphene layers in PGL affects the dipole’s emission, we consider the emission enhancement η defined as
η=Γsp,p2Γsp,m=(λsp,mλsp,p)3e4πl(1λsp,p1λsp,m)Q(l,λsp,p).
(11)
where the subscripts p and m denote the quantities for PGL and GML, respectively. The factor 2 before Γsp,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 = 50nm. 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 103 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 0.17eV 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 2l 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 2l does not change the minimum value of η and narrowing the separation shifts the anomalous region of η to high photon energy.

Fig. 3 (a)η versus photon energy with η’s range from 1 to 103. Inset: η versus photon energy with η’s range from 0.8 to 1.5 and dashed red line indicates the line that η=1. Emitter distance l is 50 nm and Fermi level Ef is 0.5eV. (b)η versus photon energy with l varied between 10 nm, 20 nm, 50 nm, 100 nm and 200 nm from the right curve to the left curve.

To figure out the anomalous emission enhancement in PGL, we retrieve the results from the Fermi’s golden rule. According to the Fermi’s rule, the decay rate of an emitter takes the form γ = 2π|g|2D2D(ω). D2D(ω) is the density of states. |g| is the coupling strength between the dipole and the electromagnetic field at the emitter position xp and is given by
|g|2=ω|p|2/(2h¯εrε0Veff)
(12)
in which εr is the relative permittivity of the dielectric medium surrounding the emitter. For the PGL in this work, the effective mode volume Veff is defined as [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] .

]Veff = Leffh2 with h an arbitrary quantization length. Leff is the effective mode length across the gap:
Leff=12{ε0d(εrω)dω|E(x,ω)|2+μ0|H(x,ω)|2}dx/(εrε0|E(xp,ω)|2)
(13)
E(xp, ω) is the electric field at the dipole’s position. The density of states can be expressed as D2D(ω) = h2ω/(2πυpυg) with υp and υg the phase and group velocities of SPs mode, respectively. Consequently, the emission enhancement η can be express as the product of three factors,
η=Leff,m2Leff,pυp,mυp,pυg,mυg,p
(14)

The three factors in η are contributed by the effective mode volume Veff, the phase velocity υp and the group velocity υg, 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
Leff,m2Leff,p=λsp,mλsp,pe4πl(1λsp,p1λsp,m)1+(1+4πlλsp,p)e4πlλsp,p.
(15)
The emission enhancement 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.

Fig. 4 (a) Exact value of η in black and η by Fermi golden rule in red. (b)According to Fermi golden rule, η can be divided into three factors, effective mode volume (Leff,m/2Leff,p in red), phase velocity (υp,m/υp,p in blue) and group velocity (υg,m/υg,p in green).

To give a detail view on the dependence of the emission enhancement on the system parameters, Fig. 5 shows the contour plot of log(η) for different Ef in the plane of the photon energy and the distance between the emitter and the graphene layers 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 Ef and increasing 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.

Fig. 5 log(η) at different photon energies and different emitter distances with (a) Ef =0.3 eV, (b) Ef =0.5 eV, (c) Ef =0.7 eV and (d) Ef =1.0 eV. Yellow solid lines in theses figures indicate positions that η=1.

4. Emission in paired graphene ribbons

To investigate the emission of a single emitter in PGR, we employ the finite element method by commercial software COMSOL. Both the SP modes analysis in PGR and the three dimensional simulations of the emission by a emitter in PGR are calculated by COMSOL. In the simulations, the graphene layers are treated as conductors with thickness of 0.3 nm. According to the above knowledge from graphene layers, we expect the emission enhancement in comparison with a single graphene ribbon to be in the regime of low photon energy. Consequently, we only focus on low photon energy in simulations. The validity of the simulation method can be checked by comparing the numerical results by using COMSOL and the above theoretical results on the emission of a single emitter in PGL. Figure 6(a) shows the good agreement between the results obtained by these two methods. For the ribbon width w = 300nm and the separation between the emitter and the ribbons l = 50nm, 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 M1 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.

Fig. 6 (a) Theory results for Γ in PGL are plotted in black-square line and simulation results for Γ are in red-circle line. (b) Real part of wave vectors for the first four SPs’ modes in GPR. Ribbon width w is set to be 300 nm and emitter distance l is taken as 50 nm (c) Profiles for electric field vertical to graphene E for first four modes. Photon energies of these modes are marked by black dots in (b).

The separation between the emitter and the graphene ribbons is fixed at l = 50nm. 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 (M1) in PGR. Figure 7(a) shows the simulation results about the total decay rate Γ and the decay rate into SPs Γsp for PGR with the ribbon width w varied from 300nm to 100nm. Here Γ is obtained by calculating the total energy flowing out of the emitter and Γsp is by integrating the energy flowing in the direction the fundamental SP mode M1 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.

Fig. 7 (a) Simulation results of PGR and theory results of PGL. Γ are plotted in black-square line and Γsp are plotted in red-square line. Ribbons width w is varied between 100 nm, 200 nm and 300 nm. (b) Efficiencies for emitters in PGR at different widths, and efficiency for PGL in black solid line. Inset: schematic for interactions between PGR and single dipole.(c) Simulation results of GMR and theory results of GML. Γ are plotted in black-square line and Γsp are plotted in red-square line. Ribbons width w is varied between 100 nm, 200 nm and 300 nm. (d) Efficiencies for emitters in graphene mono ribbon at different widths, and efficiency for GML in black solid line. In all these figures, distance between emitter and graphene d is 50 nm, and Fermi level Ef is 0.5 eV. Inset: schematic for interactions between GMR and single dipole.

As a comparison, we show the simulation results on the decay rates of an emitter above graphene mono-ribbon (GMR) in Figs. 7(c) and 7(d). Figure 7(c) shows that the decay rates of the emitter for GMR at low photon energy is much higher than those for GML. Figure 7(c) also shows that the enhancement can be further improved by narrowing the ribbon width in the range we investigated. To be mentioned, in additional simulations not presented here, we find that furthering decreasing the ribbon width to a much low value, for example from 50nm to 15nm, 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] .

]. Nevertheless, the decay rates for GMR are always one order of magnitude lower than those for PGR. The excitation efficiency presented in Fig. 7(d) shows that GMR has similar behaviors to GML, that is, the excitation efficiency is low at low photon energy and then monotonically increases with the photon energy.

5. Emission in paired unequally doped graphene structures

For graphene, one can tune its Fermi energy simply by gating or doping. Thongrattanasiri et. al. investigated graphene plasmons in some realistic configurations with inhomogeneous doping and found that either dispersion or spatial profile could be considerably differe from uniformly doped ribbons [22

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

]. Then it will be interesting to investigate the coupling between an emitter and SPs in paired graphene layers (or ribbons) when the two graphene layers (or ribbons) do not have equal Fermi energies.

Firstly, we consider the emission rates against the Fermi level when the two graphene layers are the same. Figures 8(a) and 8(b) show the analytical results for GML and PGL at different Fermi levels, respectively. We get similar conclusions for both structures that increasing Fermi level results in the rise of the emission rates in the regime of low photon energy and the fall of the emission rates in the regime of high photon energy. It should be mentioned that dependency of GML on Fermi level has been studied in the reference [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] .

] and similar results have been found.

Fig. 8 Emitter’s total emission rates at different Fermi levels and different graphene structures, with Ef0 = 0.5eV. (a) Analytical results for GML at different Fermi levels and (b) analytical results for equally doped PGL at different Fermi levels. (c) Analytical results for non-equally doped PGL with Ef1 = Ef0 at different Ef2. (d) Analytical results for PGL at three combinations of Ef1 and Ef2 (|Ef0, Ef0|, |1.5Ef0, 1.5Ef0|, and |Ef0, 1.5Ef0|). (e) Simulation results for non-equally doped PGR with Ef1 = Ef0 at different Ef2. (f) Simulation results for PGR at three combinations of Ef1 and Ef2 (|Ef0, Ef0|, |1.5Ef0, 1.5Ef0|, and |Ef0, 1.5Ef0|).

For non-equally doped graphene layers, suppose that Ef1 and Ef2 are the Fermi levels for the two layers. Denote the reflections at mono graphene layer by r1 and r2 which are functions of Ef1 and Ef2, respectively, we have the total emission rate
ΓEf1,Ef2Γ01+32k030Im((1+r1e2ql)(1+r2e2ql)1r1r2e4ql1)q2dq.
(16)

Figure 8(c) shows the the emission rates for different Ef2 at Ef1 = Ef0 = 0.5eV. The dependency of the emission rate on Ef2 is similar to that in equally doped condition but less sensitive. Furthermore for Ef1 < Ef2, we find that ΓEf1Ef1 > ΓEf1Ef2 > ΓEf2Ef2 at low photon energy and, conversely, ΓEf1Ef1 < ΓEf1Ef2 < ΓEf2Ef2 at high photon energy. An example is shown in Fig. 8(d) in which three combinations of Ef1 and Ef2 (|Ef0, Ef0|, |1.5Ef0, 1.5Ef0|, and |Ef0, 1.5Ef0|) are used. For PGR, ΓEf1Ef2 is presented in Figs. 8(e) and 8(f), which show that the results are similar to those for PGL.

6. Conclusion

In conclusion, we theoretically and numerically investigate the coupling between an emitter and SPs in graphene paired structures. At low photon energy and small separation between graphene layers where the coupling between SPs in graphene layers is strong, the emission of the emitter gets enhanced and the excitation efficiency of SPs can be increased from nearly 0 to nearly 1. On the other hand, increasing either the photon energy or the separation between graphene layers weakens the coupling between SPs in graphene layers and, as a result, the emission enhancement is lowered. We also find an anomalous behavior of the emission enhancement in which the coupling between SPs in graphene layers plays a negative role on emission enhancement. We find that the larger effective mode volume of SPs in PGL is responsible for the anomalous 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

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] .

] and full temporal quantum control [23

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] .

].

To be noted, the emitter in this work is positioned with the same distance away from the graphene layers (or ribbons) and with the same distance away from the edges of graphene ribbons. When the condition of the same distance away from the graphene layers (or ribbons) is not holden, the asymmetry just has the emission rate increased, especially at large photon energy. On the other hand, when the condition of the same distance away from the edges of graphene ribbons is broken, the asymmetry brings the emission rate down provided that the emitter is not too close to the edges of the ribbons. Nevertheless, the qualitative properties of the interaction between emitter and graphene layers (or ribbons) are independent of the asymmetry of the emitter’s position.

Acknowledgments

The authors acknowledge the financial supports of this work from the project of National Natural Science Foundation of China (grants no. 51172030, 11274052, 90921015 and 11174040).

References and links

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] .

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. 81, 5932–5935 (1998) [CrossRef] .

5.

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

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. 4, 532–535 (2008) [CrossRef] .

8.

D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105, 256805 (2010) [CrossRef] .

9.

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

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] .

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] .

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 84, 195446 (2011) [CrossRef] .

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] .

16.

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

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] .

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] .

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] .

22.

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

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] .

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


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References

  1. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 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]
  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.81, 5932–5935 (1998). [CrossRef]
  5. K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation,” Phys. Rev. A72, 022317 (2005). [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]
  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.4, 532–535 (2008). [CrossRef]
  8. D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett.105, 256805 (2010). [CrossRef]
  9. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009). [CrossRef]
  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. B84, 161407 (2011). [CrossRef]
  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. B85, 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]
  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. B84, 195446 (2011). [CrossRef]
  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. B78, 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]
  16. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006). [CrossRef]
  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. B75, 205418 (2007). [CrossRef]
  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,” Science306, 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,” Nature438, 197–200 (2005). [CrossRef] [PubMed]
  21. S. Thongrattanasiri, A. Manjavacas, and F. Javier García de Abajo, “Quantum Finite-Size Effects in Graphene Plasmons,” ACS Nano6, 1766–1775 (2012). [CrossRef] [PubMed]
  22. S. Thongrattanasiri, Iván Silveiro, and F. Javier García de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett.100, 201105 (2012). [CrossRef]
  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]

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