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OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
Molecular decay rate near nonlocal plasmonic particles Christian Girard,* Aurélien Cuche, Erik Dujardin, Arnaud Arbouet, and Adnen Mlayah CEMES, CNRS, Université Paul Sabatier, 29 rue Jeanne Marvig, 31055 Toulouse, France *Corresponding author: [email protected] Received January 28, 2015; revised April 10, 2015; accepted April 10, 2015; posted April 10, 2015 (Doc. ID 233146); published May 1, 2015 When the size of metal nanoparticles is smaller than typically 10 nm, their optical response becomes sensitive to both spatial dispersion and quantum size effects associated with the confinement of the conduction electrons inside the particle. In this Letter, we propose a nonlocal scheme to compute molecular decay rates near spherical nanoparticles which includes the electron-electron interactions through a simple model of electronic polarizabilities. The plasmonic particle is schematized by a dynamic dipolar polarizability αNL ω, and the quantum system is characterized by a two-level system. In this scheme, the light matter interaction is described in terms of classical field susceptibilities. This theoretical framework could be extended to address the influence of nonlocality on the dynamics of quantum systems placed in the vicinity of nano-objects of arbitrary morphologies. © 2015 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (260.2510) Fluorescence; (050.6624) Subwavelength structures; (300.3700) Linewidth. http://dx.doi.org/10.1364/OL.40.002116
The new optical concepts currently developed in the research field of plasmonics are expected to yield significant practical applications in the miniaturization and integration of optical devices [1], as well as in molecular sensing [2]. Particularly, these new devices offer interesting opportunities for optical addressing of quantum systems (QS) [3–6]. Although electromagnetic field enhancement and confinement have been extensively studied on lithographically designed nanostructures, the investigations of spatial distribution of near-field local density of electromagnetic states (LDOS) and the role of surface plasmon modes in assemblies made of very small colloidal particles remain sparse [7–10]. Recently, a considerable breakthrough in terms of experimental resolution has been achieved with electron energy loss spectroscopy (EELS) [11–16] and cathodoluminescence spectrocopy (CL) [17] so that it becomes possible to probe smaller and smaller plasmonic entities and access the characteristic length scale at which nonlocality effects are expected to play a significant role. Several recent investigations have shown that nonlocal descriptions yield weaker near-field intensities and blueshifted surface plasmon resonances as compared to local ones [9,18]. These spatial and spectral characteristics of the surface plasmon resonances supported by metallic nanosystems have a dramatic influence on the dynamics of neighboring QS. Depending on the wavelength and arrangement of the metallic system and light emitter, either a quenching or an enhancement of the fluorescence yield can be obtained. These phenomena have been extensively studied and well accounted for by local descriptions, but the question of the validity of these descriptions for very small particles then arises or, equivalently, of the influence of nonlocality on the decay rate of QS coupled to a very small particle. Indeed, when a quantum system is placed in the vicinity of very small particles, the photon emission rate will be modified and becomes sensitive to electron-electron interactions in the metal. In fact, the effect of nonlocality of the electron gas can be included in the multipolar responses of the metallic particles [7]. These interactions alter the 0146-9592/15/092116-04$15.00/0
field susceptibility experienced around the particles and, thus, modify the photonic LDOS. In this Letter, we propose a new analytical derivation of the photon field susceptibility generated around nonlocal spherical particles. The approach we present here contrasts markedly with the earlier works devoted to the nonlocal response of individual metallic particles. In fact, the main ingredients of our method are (1) the formalism of field propagators [19,20] also known as susceptibilities of the field, and (2) the well-established theory of the nonlocal response of small metallic particles [7,8]. The combination of these two formalisms by a Dyson type equation allows us to establish a new propagator which takes into account the electron-electron interactions inside the metal. In this way, the self-consistent coupling of various field susceptibilities improves the understanding of light-matter interactions in the vicinity of metal nanoparticles, opening the way to advance in the precision and accuracy of molecular decay rate computation near sub-10 nm noble metal particles. The total decay rate of a quantum system placed at nanometric distances from the metallic nano-object is then derived by taking the imaginary part of this dyadic tensor projected along the molecular axis. Within this scheme, in which the dynamic response of the metal is described by nonlocal polarizabilities, it is a simple matter to include spatial dispersion effects in very small metallic particles [7,8,10]. In this Letter, although all spatial configurations can easily be explored, we have chosen a particular geometry in which the molecule is perpendicular to the particle surface. The field susceptibility Sr; r0 ; ω between two arbitrary points r and r0 enables us to know how the electric field at location r0 is modified near a solid body when a punctual dipolar source is placed in r. The computation for a particle of complex shape needs a specific procedure that can be based on solving electrodynamic Dyson equation outside the source region [21]. However, in the quasistatic approximation valid for small spherical particles, that second rank tensor can be written with simple analytical formula [22]. For example, according to Ref. [22], for a single particle located at the origin of the framework © 2015 Optical Society of America
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS
Fig. 1. Schematic view illustrating the field susceptibility between two arbitrary points, r and r0 , near a spherical metal particle.
[see Fig. (1)], the dyad Sr; r0 ; ω can be deduced from the field susceptibility in vacuum S0 r; r0 ; ω and the dipolar polarizability αω of the nanoparticle: Sr;r0 ;ω S
0
r;r0 ;ω S
0 r;0;ω · αω · S0
0;r0 ;ω;
3r β r γ −r 2 δβ;γ ; (2) S0β;γ r;0;ωS0β;γ 0;r;ωexpik0 r× r5 with k0 ω∕c, (r 1 x; r 2 y; r 3 z) and r p x2 y2 z2 . This simple relation is only appropriate to describe the near-field zone. For a longer approach distance, the complete relation presented in [22] must be applied. In the absence of spatial dispersion effects, the dipolar polarizability of a single metallic sphere of radius a ≪ λ0 2πc∕ω placed in a vacuum reads, in CGS units (see Eq. (7) of Ref. [22]), αL ω a
3
εm ω − 1 ; εm ω 2
from the seminal work of Newns [26], was successfully applied to the calculation of absorption cross section of small colloidal particles [7]. It is based on the random phase approximation (RPA) which provides a quite successful model of the linear bulk dielectric response of simple metals. Within this scheme, a jellium model with infinite barriers is used to describe the electronic properties of the metal in which scattering by the ion cores may be neglected. From this model, the multipolar polarizabilities of noble metal particles can be described by supplementing the contribution of the interband electronic transition obtained from standard experimental data with a nonlocal description of the conduction electrons, which includes electron-electron interaction through the RPA frame described in [7]. For the dipolar polarizability, such a procedure leads to αNL ω a
(4)
with −1 Z 6a ∞ j21 ka F 1 a; ω dk ; π 0 εm ω; k
(5)
where j1 is the first-order spherical Bessel function, and εm ω; k represents the nonlocal bulk dielectric constant of the metal. To describe the spatial dispersion effects, we use the hydrodynamical model [7,27]: εm ω; k ϵ0 ω −
ω2p ; ωω iγ − δ2 k2
(6)
where the interband contribution ϵ0 ω can be extracted from experimental data [28], ωp labels the plasma frequency, and δ is related to the Fermi velocity vF of the metal:
(3)
where εm ω is the permittivity of the metal. When the diameter of a metallic sphere becomes smaller than the electron mean free path, the confinement of the electron motion in the particle leads to a strong damping of the surface plasmon resonances that can be described phenomenologically by an ad hoc modification of the imaginary part of the dielectric constant of the metal [23]. In the context of optical properties of metallic colloids, the impact of the spatial dispersion of the dielectric constant has been theoretically investigated in the case of spherical nanoparticles [7,8,24], nanowires, dimers [9], or two-dimensional arrays of metallic nanoparticles, and it has been shown that nonlocal effects can induce spectral shifts of the surface plasmon resonances, as well as a modification of their intensity. Such spatial dispersion effects can weaken the interactions and lead to significant blue shifts of the plasmon resonance and, consequently, modify absorption or EELS spectra [25]. In our work, the spatial dispersion is introduced by applying the method of Fuchs and Claro [7] to an electron gas confined in a spherical well. This method, inspired
F 1 a; ω − 1 ; F 1 a; ω 2
3
(1)
where r x; y; z and r0 x0 ; y0 ; z0 . In a cartesian frame, the second rank tensor S0 has nine components (β, γ), where both β and γ take the three values x, y, or z (see Section (1.3) of Ref. [20]). At a short distance, it may be deduced from Eq. (4) of Ref. [20]:
2117
δ2
3v2F : 5
(7)
When this factor vanishes, the local dielectric constant is recovered. As described in many previous papers, the density of optical modes, characterized by the photonic LDOS function, changes dramatically when approaching the surface of materials [29,30]. In particular, the increasing contribution of the density of evanescent modes tends to enhance this quantity near the particle surfaces. The first consequence of this effect is the modification of the lifetime of molecular fluorescence levels [20,29]. For a given orientation u of the molecule—or more generally of the transition dipole moment meg of the QS—we can write the total decay rate as follows [20]: Γtot Rm
2jmeg j2 IfSRm ; Rm ; ω:uug; ℏ
(8)
where Rm labels the molecule location. This general equation can also be expressed with Γ0 4k30 jmeg j2 ∕ 3ℏ, which is the common expression for the decay of a quantum state in vacuum [20]:
OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015 450 400
(9)
If we adopt a simple nanoparticle/molecule geometry where Rm 0; 0; Rm , only the diagonal terms of S0 survive. That leads to 3 Γtot Rm Γ0 1 3 Ifαω 2k0 X β;β β;β 2 S 0 Rm ; 0; ωS 0 0; Rm ; ωuβ g : (10) × βx;y;z
In addition, in the near-field zone around the particle, the field propagator can be reasonably described by the following asymptotic form [see Eq. (4) of Ref. [20]]: 0 1 −1 0 0 eik0 Rm @ 0 −1 0 A: (11) S0 Rm ; 0; ω 3 Rm 0 0 2 After substitution of this expression (11) into (10), we find a simple and analytical expression of the total decay rate: 3 Γtot Rm Γ0 1 3 6 u2x u2y 4u2z 2k0 Rm 0 00 × α ωsin2k0 Rm α ωcos2k0 Rm ; (12) that depends on both real and imaginary parts, α0 , and α00 , of the polarizability. From this relation, it is a simple matter to account for the dispersion spatial effect by using the nonlocal polarizability αNL ω defined by relations (4) and (5). Using relations (4)–(7) and (12), we have performed various simulations of the normalized decay rate Γtot Rm ∕Γ0 in the vicinity of small gold particles. Thus, in Fig. (2), we present the spectral variation when reducing the sphere diameter 2a: (red) a 6 nm; (magenta) a 5 nm; (green) a 4 nm; (blue) a 3 nm; (yellow) a 2 nm; and (black) a 1.5 nm. In the resonance region, the dipolar plasmon peak (located around 2.43 eV) is gradually damped as the particle size decreases. This phenomena, originating from the electron– electron interactions in the confined sphere volume, is always accompanied by a blue shift of the plasmon resonance. This blueshift is clearly illustrated in Fig. (3), which maps the normalized total decay rate as a function of particle size and energy. In the first map, we observe a displacement of about 45 meV when passing from a 6 to a 2 nm. Similar blueshift of the plasmon resonance has been recently observed with EELS measurement near small plasmonic particles [25,31]. In this Letter, we have discussed the elementary mechanisms that modify the total decay rate of a two-level quantum system coupled to a nonlocal plasmonic particle. This theoretical scheme based on a pure Fermi golden rule approach [20] shows that the physics related to the photon–plasmon coupling can be described by
2a
350
Γo
3 Γtot Rm Γ0 1 3 IfS0 Rm ; 0; ω· 2k0 × αω · S0 0; Rm ; ω:uug .
Rm
300 250
Γtot
2118
200 150 100 50 0 1.6
1.8
2
2.2 2.4 h ω (eV)
2.6
2.8
3
Fig. 2. Normalized decay rate spectra computed in the vicinity of a gold sphere. This set of curves display the nonlocal Γtot Rm ∕Γ0 spectrum evolution obtained when reducing the sphere diameter 2a: (red) a 6 nm; (magenta) a 5 nm; (green) a 4 nm; (blue) a 3 nm; (yellow) a 2 nm; and (black) a 1.5 nm. These six spectra have been computed by keeping the same distance d Rm − a 5 nm between the sphere surface and the molecule (the orientation vector u is perpendicular to the surface).
using generalized forms of field propagators, which include nonlocal descriptions of the dielectric constant of the metal. Our formalism could be extended to meet more complex physical situations [16]. In fact, all the information
Fig. 3. Illustration of the resonance blue-shift versus the particle radius. (a) Nonlocal and (b) local color maps of the normalized decay rate Γtot Rm ∕Γ0 as a function of the particle radius a and the two-level system energy ℏω. The two magenta lines are introduced to vizualize, in both nonlocal and local cases, the plasmon resonance trajectory in the radius–energy plane.
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS
on the structure and the physical properties probed by the QS is carried by the field propagator SRm ; Rm ; ω associated with the sample. Consequently, the present treatment can be generalized to large assemblies of metallic particles of arbitrary nature deposited on the surface, provided we access the field propagator of the composite system. Recently, we have developed a self-consistent scheme well suited to compute the field susceptibility near complex plasmonic nanoparticles networks that can be merged with the present formalism. In particular, by applying the numerical technique described in Ref. [22], we will have the opportunity to represent maps Γtot Rm near complex patterns formed of interacting nonlocal metallic particles. The authors thank Alexandre Teulle for fruitful discussions on the physics of small plasmonic particles. This work was supported by the Agence Nationale de la Recherche (ANR) (Grant ANR-13-BS10-0007-PlaCoRe) and the computing center CALMIP in Toulouse. References 1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). 2. M. D. Sonntag, J. M. Klingsporn, A. B. Zrimsek, B. Sharma, L. K. Ruvuna, and R. P. Van Duyne, Chem. Soc. Rev. 43, 1230 (2014). 3. A. W. Schell, P. Engel, J. F. M. Werra, C. Wolf, K. Busch, and O. Benson, Nano Lett. 14, 2623 (2014). 4. G. Colas des Francs, C. Girard, T. Laroche, G. Lévèque, and O. J. F. Martin, J. Chem. Phys. 127, 034701 (2007). 5. A. Cuche, O. Mollet, A. Drezet, and S. Huant, Nano Lett. 10, 4566 (2010). 6. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Memmer, H. Park, and M. D. Lukin, Nature 450, 402 (2007). 7. R. Fuchs and F. Claro, Phys. Rev. B 35, 3722 (1987). 8. C. Girard, S. Maghezzi, and F. Hache, J. Chem. Phys. 91, 5509 (1989). 9. F. J. Garcia de Abajo, J. Phys. Chem. C 112, 17983 (2008).
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10. V. V. Datsyuk and O. M. Tovkach, J. Opt. Soc. Am. B 28, 1224 (2011). 11. M. Bosman, V. J. Keast, M. Watanabe, A. I. Maaroof, and M. B. Corties, Nanotechnol. 18, 165505 (2007). 12. J. Nelayah, M. Kociak, O. Stéphan, F. J. Garcia de Abajo, M. Tencé, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, Nat. Phys. 3, 348 (2007). 13. L. Gu, W. Sigle, C. T. Koch, B. Ogut, P. A. van Aken, N. Talebi, R. Vogelgesang, J. Mu, X. Wen, and J. Mao, Phys. Rev. B 83, 195433 (2011). 14. D. Rossouw and G. A. Botton, Phys. Rev. Lett. 110, 066801 (2013). 15. M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, and M.-Y. Han, Sci. Rep. 3, 1312 (2013). 16. A. Teulle, M. Bosman, C. Girard, K. L. Gurunatha, M. Li, S. Mann, and E. Dujardin, Nat. Mater. 14, 87 (2015). 17. E. J. R. Vesseur, R. de Waele, M. Kuttge, and A. Polman, Nano Lett. 7, 2843 (2007). 18. J. M. McMahon, S. K. Gray, and G. C. Schatz, Nano Lett. 10, 3473 (2010). 19. G. S. Agarwal, Phys. Rev. A 11, 230 (1975). 20. C. Girard, Rep. Prog. Phys. 68, 1883 (2005). 21. J. C. Weeber, C. Girard, J. R. Krenn, A. Dereux, and J. P. Goudonnet, J. Appl. Phys. 86, 2576 (1999). 22. C. Girard, E. Dujardin, R. Marty, A. Arbouet, and G. Colas des Francs, Phys. Rev. B 81, 153412 (2010). 23. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters, Springer Series in Materials Science (Springer, 1995), Vol. 25. 24. R. Ruppin, Phys. Rev. B 45, 11209 (1992). 25. S. Raza, W. Yan, N. Stenger, M. Wubs, and N. A. Mortensen, Opt. Express 21, 27344 (2013). 26. D. M. Newns, Phys. Rev. B 1, 3304 (1970). 27. P. J. Feibelman, Prog. Surf. Sci. 12, 287 (1982). 28. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972). 29. R. R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 60, 2744 (1974). 30. R. Ruppin, J. Chem. Phys. 76, 1681 (1982). 31. J. A. Scholl, A. L. Koh, and J. A. Dionne, Nature 483, 421 (2012).
OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
Molecular decay rate near nonlocal plasmonic particles Christian Girard,* Aurélien Cuche, Erik Dujardin, Arnaud Arbouet, and Adnen Mlayah CEMES, CNRS, Université Paul Sabatier, 29 rue Jeanne Marvig, 31055 Toulouse, France *Corresponding author: [email protected] Received January 28, 2015; revised April 10, 2015; accepted April 10, 2015; posted April 10, 2015 (Doc. ID 233146); published May 1, 2015 When the size of metal nanoparticles is smaller than typically 10 nm, their optical response becomes sensitive to both spatial dispersion and quantum size effects associated with the confinement of the conduction electrons inside the particle. In this Letter, we propose a nonlocal scheme to compute molecular decay rates near spherical nanoparticles which includes the electron-electron interactions through a simple model of electronic polarizabilities. The plasmonic particle is schematized by a dynamic dipolar polarizability αNL ω, and the quantum system is characterized by a two-level system. In this scheme, the light matter interaction is described in terms of classical field susceptibilities. This theoretical framework could be extended to address the influence of nonlocality on the dynamics of quantum systems placed in the vicinity of nano-objects of arbitrary morphologies. © 2015 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (260.2510) Fluorescence; (050.6624) Subwavelength structures; (300.3700) Linewidth. http://dx.doi.org/10.1364/OL.40.002116
The new optical concepts currently developed in the research field of plasmonics are expected to yield significant practical applications in the miniaturization and integration of optical devices [1], as well as in molecular sensing [2]. Particularly, these new devices offer interesting opportunities for optical addressing of quantum systems (QS) [3–6]. Although electromagnetic field enhancement and confinement have been extensively studied on lithographically designed nanostructures, the investigations of spatial distribution of near-field local density of electromagnetic states (LDOS) and the role of surface plasmon modes in assemblies made of very small colloidal particles remain sparse [7–10]. Recently, a considerable breakthrough in terms of experimental resolution has been achieved with electron energy loss spectroscopy (EELS) [11–16] and cathodoluminescence spectrocopy (CL) [17] so that it becomes possible to probe smaller and smaller plasmonic entities and access the characteristic length scale at which nonlocality effects are expected to play a significant role. Several recent investigations have shown that nonlocal descriptions yield weaker near-field intensities and blueshifted surface plasmon resonances as compared to local ones [9,18]. These spatial and spectral characteristics of the surface plasmon resonances supported by metallic nanosystems have a dramatic influence on the dynamics of neighboring QS. Depending on the wavelength and arrangement of the metallic system and light emitter, either a quenching or an enhancement of the fluorescence yield can be obtained. These phenomena have been extensively studied and well accounted for by local descriptions, but the question of the validity of these descriptions for very small particles then arises or, equivalently, of the influence of nonlocality on the decay rate of QS coupled to a very small particle. Indeed, when a quantum system is placed in the vicinity of very small particles, the photon emission rate will be modified and becomes sensitive to electron-electron interactions in the metal. In fact, the effect of nonlocality of the electron gas can be included in the multipolar responses of the metallic particles [7]. These interactions alter the 0146-9592/15/092116-04$15.00/0
field susceptibility experienced around the particles and, thus, modify the photonic LDOS. In this Letter, we propose a new analytical derivation of the photon field susceptibility generated around nonlocal spherical particles. The approach we present here contrasts markedly with the earlier works devoted to the nonlocal response of individual metallic particles. In fact, the main ingredients of our method are (1) the formalism of field propagators [19,20] also known as susceptibilities of the field, and (2) the well-established theory of the nonlocal response of small metallic particles [7,8]. The combination of these two formalisms by a Dyson type equation allows us to establish a new propagator which takes into account the electron-electron interactions inside the metal. In this way, the self-consistent coupling of various field susceptibilities improves the understanding of light-matter interactions in the vicinity of metal nanoparticles, opening the way to advance in the precision and accuracy of molecular decay rate computation near sub-10 nm noble metal particles. The total decay rate of a quantum system placed at nanometric distances from the metallic nano-object is then derived by taking the imaginary part of this dyadic tensor projected along the molecular axis. Within this scheme, in which the dynamic response of the metal is described by nonlocal polarizabilities, it is a simple matter to include spatial dispersion effects in very small metallic particles [7,8,10]. In this Letter, although all spatial configurations can easily be explored, we have chosen a particular geometry in which the molecule is perpendicular to the particle surface. The field susceptibility Sr; r0 ; ω between two arbitrary points r and r0 enables us to know how the electric field at location r0 is modified near a solid body when a punctual dipolar source is placed in r. The computation for a particle of complex shape needs a specific procedure that can be based on solving electrodynamic Dyson equation outside the source region [21]. However, in the quasistatic approximation valid for small spherical particles, that second rank tensor can be written with simple analytical formula [22]. For example, according to Ref. [22], for a single particle located at the origin of the framework © 2015 Optical Society of America
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS
Fig. 1. Schematic view illustrating the field susceptibility between two arbitrary points, r and r0 , near a spherical metal particle.
[see Fig. (1)], the dyad Sr; r0 ; ω can be deduced from the field susceptibility in vacuum S0 r; r0 ; ω and the dipolar polarizability αω of the nanoparticle: Sr;r0 ;ω S
0
r;r0 ;ω S
0 r;0;ω · αω · S0
0;r0 ;ω;
3r β r γ −r 2 δβ;γ ; (2) S0β;γ r;0;ωS0β;γ 0;r;ωexpik0 r× r5 with k0 ω∕c, (r 1 x; r 2 y; r 3 z) and r p x2 y2 z2 . This simple relation is only appropriate to describe the near-field zone. For a longer approach distance, the complete relation presented in [22] must be applied. In the absence of spatial dispersion effects, the dipolar polarizability of a single metallic sphere of radius a ≪ λ0 2πc∕ω placed in a vacuum reads, in CGS units (see Eq. (7) of Ref. [22]), αL ω a
3
εm ω − 1 ; εm ω 2
from the seminal work of Newns [26], was successfully applied to the calculation of absorption cross section of small colloidal particles [7]. It is based on the random phase approximation (RPA) which provides a quite successful model of the linear bulk dielectric response of simple metals. Within this scheme, a jellium model with infinite barriers is used to describe the electronic properties of the metal in which scattering by the ion cores may be neglected. From this model, the multipolar polarizabilities of noble metal particles can be described by supplementing the contribution of the interband electronic transition obtained from standard experimental data with a nonlocal description of the conduction electrons, which includes electron-electron interaction through the RPA frame described in [7]. For the dipolar polarizability, such a procedure leads to αNL ω a
(4)
with −1 Z 6a ∞ j21 ka F 1 a; ω dk ; π 0 εm ω; k
(5)
where j1 is the first-order spherical Bessel function, and εm ω; k represents the nonlocal bulk dielectric constant of the metal. To describe the spatial dispersion effects, we use the hydrodynamical model [7,27]: εm ω; k ϵ0 ω −
ω2p ; ωω iγ − δ2 k2
(6)
where the interband contribution ϵ0 ω can be extracted from experimental data [28], ωp labels the plasma frequency, and δ is related to the Fermi velocity vF of the metal:
(3)
where εm ω is the permittivity of the metal. When the diameter of a metallic sphere becomes smaller than the electron mean free path, the confinement of the electron motion in the particle leads to a strong damping of the surface plasmon resonances that can be described phenomenologically by an ad hoc modification of the imaginary part of the dielectric constant of the metal [23]. In the context of optical properties of metallic colloids, the impact of the spatial dispersion of the dielectric constant has been theoretically investigated in the case of spherical nanoparticles [7,8,24], nanowires, dimers [9], or two-dimensional arrays of metallic nanoparticles, and it has been shown that nonlocal effects can induce spectral shifts of the surface plasmon resonances, as well as a modification of their intensity. Such spatial dispersion effects can weaken the interactions and lead to significant blue shifts of the plasmon resonance and, consequently, modify absorption or EELS spectra [25]. In our work, the spatial dispersion is introduced by applying the method of Fuchs and Claro [7] to an electron gas confined in a spherical well. This method, inspired
F 1 a; ω − 1 ; F 1 a; ω 2
3
(1)
where r x; y; z and r0 x0 ; y0 ; z0 . In a cartesian frame, the second rank tensor S0 has nine components (β, γ), where both β and γ take the three values x, y, or z (see Section (1.3) of Ref. [20]). At a short distance, it may be deduced from Eq. (4) of Ref. [20]:
2117
δ2
3v2F : 5
(7)
When this factor vanishes, the local dielectric constant is recovered. As described in many previous papers, the density of optical modes, characterized by the photonic LDOS function, changes dramatically when approaching the surface of materials [29,30]. In particular, the increasing contribution of the density of evanescent modes tends to enhance this quantity near the particle surfaces. The first consequence of this effect is the modification of the lifetime of molecular fluorescence levels [20,29]. For a given orientation u of the molecule—or more generally of the transition dipole moment meg of the QS—we can write the total decay rate as follows [20]: Γtot Rm
2jmeg j2 IfSRm ; Rm ; ω:uug; ℏ
(8)
where Rm labels the molecule location. This general equation can also be expressed with Γ0 4k30 jmeg j2 ∕ 3ℏ, which is the common expression for the decay of a quantum state in vacuum [20]:
OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015 450 400
(9)
If we adopt a simple nanoparticle/molecule geometry where Rm 0; 0; Rm , only the diagonal terms of S0 survive. That leads to 3 Γtot Rm Γ0 1 3 Ifαω 2k0 X β;β β;β 2 S 0 Rm ; 0; ωS 0 0; Rm ; ωuβ g : (10) × βx;y;z
In addition, in the near-field zone around the particle, the field propagator can be reasonably described by the following asymptotic form [see Eq. (4) of Ref. [20]]: 0 1 −1 0 0 eik0 Rm @ 0 −1 0 A: (11) S0 Rm ; 0; ω 3 Rm 0 0 2 After substitution of this expression (11) into (10), we find a simple and analytical expression of the total decay rate: 3 Γtot Rm Γ0 1 3 6 u2x u2y 4u2z 2k0 Rm 0 00 × α ωsin2k0 Rm α ωcos2k0 Rm ; (12) that depends on both real and imaginary parts, α0 , and α00 , of the polarizability. From this relation, it is a simple matter to account for the dispersion spatial effect by using the nonlocal polarizability αNL ω defined by relations (4) and (5). Using relations (4)–(7) and (12), we have performed various simulations of the normalized decay rate Γtot Rm ∕Γ0 in the vicinity of small gold particles. Thus, in Fig. (2), we present the spectral variation when reducing the sphere diameter 2a: (red) a 6 nm; (magenta) a 5 nm; (green) a 4 nm; (blue) a 3 nm; (yellow) a 2 nm; and (black) a 1.5 nm. In the resonance region, the dipolar plasmon peak (located around 2.43 eV) is gradually damped as the particle size decreases. This phenomena, originating from the electron– electron interactions in the confined sphere volume, is always accompanied by a blue shift of the plasmon resonance. This blueshift is clearly illustrated in Fig. (3), which maps the normalized total decay rate as a function of particle size and energy. In the first map, we observe a displacement of about 45 meV when passing from a 6 to a 2 nm. Similar blueshift of the plasmon resonance has been recently observed with EELS measurement near small plasmonic particles [25,31]. In this Letter, we have discussed the elementary mechanisms that modify the total decay rate of a two-level quantum system coupled to a nonlocal plasmonic particle. This theoretical scheme based on a pure Fermi golden rule approach [20] shows that the physics related to the photon–plasmon coupling can be described by
2a
350
Γo
3 Γtot Rm Γ0 1 3 IfS0 Rm ; 0; ω· 2k0 × αω · S0 0; Rm ; ω:uug .
Rm
300 250
Γtot
2118
200 150 100 50 0 1.6
1.8
2
2.2 2.4 h ω (eV)
2.6
2.8
3
Fig. 2. Normalized decay rate spectra computed in the vicinity of a gold sphere. This set of curves display the nonlocal Γtot Rm ∕Γ0 spectrum evolution obtained when reducing the sphere diameter 2a: (red) a 6 nm; (magenta) a 5 nm; (green) a 4 nm; (blue) a 3 nm; (yellow) a 2 nm; and (black) a 1.5 nm. These six spectra have been computed by keeping the same distance d Rm − a 5 nm between the sphere surface and the molecule (the orientation vector u is perpendicular to the surface).
using generalized forms of field propagators, which include nonlocal descriptions of the dielectric constant of the metal. Our formalism could be extended to meet more complex physical situations [16]. In fact, all the information
Fig. 3. Illustration of the resonance blue-shift versus the particle radius. (a) Nonlocal and (b) local color maps of the normalized decay rate Γtot Rm ∕Γ0 as a function of the particle radius a and the two-level system energy ℏω. The two magenta lines are introduced to vizualize, in both nonlocal and local cases, the plasmon resonance trajectory in the radius–energy plane.
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS
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