Effect of Localized Surface-Plasmon Mode on Exciton Transport and

Article pubs.acs.org/JPCB

Effect of Localized Surface-Plasmon Mode on Exciton Transport and Radiation Emission in Carbon Nanotubes Oleksiy Roslyak,†,‡ Charles Cherqui,‡,§ David H. Dunlap,*,§ and Andrei Piryatinski*,‡ †

Center for Integrated Nanotechnologies (CINT) Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States ‡ Center for Nonlinear Studies (CNLS), Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States § Department of Physics, University of New Mexico, Albuquerque, New Mexico 87131, United States ABSTRACT: We report on a general theoretical approach to study exciton transport and emission in a single-walled carbon nanotube (SWNT) in the presence of a localized surface-plasmon (SP) mode within a metal nanoparticle interacting via near-field coupling. We derive a set of quantum mechanical equations of motion and approximate rate equations that account for the exciton, SP, and the environmental degrees of freedom. The material equations are complemented by an expression for the radiated power that depends on the exciton and SP populations and coherences, allowing for an examination of the angular distribution of the emitted radiation that would be measured in experiment. Numerical simulations for a (6,5) SWNT and cone-shaped Ag metal tip (MT) have been performed using this methodology. Comparison with physical parameters shows that the near-field interaction between the exciton− SP occurs in a weak coupling regime, with the diffusion processes being much faster than the exciton−SP population exchange. In such a case, the effect of the exciton population transfer to the MT with its subsequent dissipation (i.e., the Förster energy transfer) is to modify the exciton steady state distribution while reducing the equilibration time for excitons to reach a steady sate distribution. We find that the radiation distribution is dominated by SP emission for a SWNT-MT separation of a few tens of nanometers due to the fast SP emission rate, whereas the exciton−SP coherences can cause its rotation.

1. INTRODUCTION The excitation of surface-plasmon (SP) resonances at a metal− dielectric interface results in a strong local electric field in close (i.e., nanoscale) proximity to the interface that can affect the optical properties of quantum emitters such as fluorescent molecules or semiconductor nanostructures.1,2 Basic quantities, such as emission rates, and Raman scattering and absorption cross sections, can be enhanced.1,3,4 On the other hand, energy transfer from a quantum emitter to a metal nanoparticle, and subsequent Ohmic losses, occur as a result of the same nearfield coupling. The balance between the enhancement of the optical properties and the dissipation of energy has a complex dependence on the distance between the quantum emitter and metal interface and depends as well as on the nanoparticle geometry.1,3,4 Recent advances in nanoscale fabrication allow for precise control over the geometry of metal nanoparticles, allowing one to tune their SP response across the visible and near-IR spectrum, and to control the associated SP line shapes.5−8 This allows for modification of desired properties, e.g., enhancing Förster energy transfer between molecules and/or semiconductor quantum dots.9−12 Single- and multiexciton emission rates and emitted photon statistics can be manipulated in composite semiconductor quantum dot and metal nanostruc© 2014 American Chemical Society

tures for applications such as single-photon or entangled photon-pair sources.13,14 Single-photon emission can also be controlled via the coupling of localized quantum emitters to SPpolariton modes in metal nanowires.15 Interactions within assemblies of quantum emitters, as mediated by SP modes, can lead to cooperative optical behavior such as the plasmonic Dicke effect.16,17 The strong exciton−SP interaction regime, as characterized by the interaction frequency exceeding the dissipation rate, can be achieved in specially chosen geometries of metal nanostructures and quantum emitters. In these systems, strong coupling leads to the formation of mixed exciton−SP states referred to as plexitons, and the appearance of a significant exciton energy Rabi shift.18−21 Furthermore, the interference between exciton and SP energy exchange pathways results in Fano resonances.22−24 Strong coupling between a single quantum emitter and a SP-polariton confined to a narrow metal nanowire results in the excitation of quantized SP Special Issue: James L. Skinner Festschrift Received: January 31, 2014 Revised: March 13, 2014 Published: March 25, 2014 8070

dx.doi.org/10.1021/jp501144s | J. Phys. Chem. B 2014, 118, 8070−8080

The Journal of Physical Chemistry B

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modes25−27 and allows for control of SP transmission, suggesting a potential application in single-photon transistors.28 A nanoscale plasmon resonator has been realized by combining a metal nanowire with a distributed Bragg grating.29 These studies, primarily focused on the interactions of localized quantum emitters with SP modes, constitute a new field of quantum plasmonics.30 On the other hand, localized SP modes can interact with delocalized excitons in one-dimensional semiconductor nanostractures, particularly affecting their long-range transport and emission properties. In one example, the effect of exciton−SP energy transfer on the exciton emission wavelengths has been investigated in a semiconductor nanowire decorated with metal nanoparticles.31 It has been shown that energy transfer to the metal prevents excitons from finding local minima and enhances the resulting blue-shifted emission. Another class of one-dimensional nano structures exhibiting long-range exciton transport are single-walled carbon nanotubes (SWNTs). Their optical and transport properties look very promising for potential applications in optoelectronic and photovoltaic devices.32 Exciton motion in SWNTs has diffusion-limited transport features that are likely caused by coupling to extensive environmental fluctuations.33 Measurements of exciton transport in SWNTs have been carried out using optical microscopy techniques1,33 including near-field enhancement caused by the interactions of excitons with SP modes in a metal tip (MT).34 These techniques are sensitive to the emitting dipole orientation via the angular distribution of emitted radiation in the far-field.1,35 Further application of optical microscopy techniques to study MT effects on SWNT excitons has revealed a nanoantenna effect in which there is fast near-field energy transfer to the tip that is followed by efficient SP radiative emission with a specific directional redistribution.36 To this point, these observations have been described in the context of a simple exciton random walk model with enhanced excitation and radiation rates. However, a complete understanding of the effect of the localized SP resonances on transport properties of excitons in SWNTs requires a theoretical approach capable of treating exciton transport, near-field exciton−SP interactions, dissipation processes, and radiation properties, on the same footing. In this paper we report on a systematic investigation of a localized SP mode on exciton transport and emission properties in a SWNT. The exciton, SP, and photon degrees of freedom are accounted for using a unified quantum mechanical approach. Parameters associated with the exciton−SP interactions are determined from independent numerical calculations of the quasi-static SP response. These parameters are subsequently used in simulations of coupled exciton transport that are used to calculate a combined exciton/SP radiation distribution pattern. The paper is organized as follows. Our theoretical model is introduced in section 2. In section 3 we apply the model to a (6,5) SWNT in proximity to a MT, and present the results of numerical simulations. Concluding remarks are given in section 4.

the rate equations for exciton and SP populations. We express the exciton−SP population transfer rates in terms of the SP dyadic Green function associated with the near-field interactions of the SP mode and excitons. This quantity can be determined from numerical simulations of the MT-SP response. Ultimately, the overall emission cross-section can be calculated in terms of the exciton and SP populations and coherences. 2.1. Transport Model. High energy photoexcitation of a semiconductor SWNT leads to the formation of exciton states that rapidly relax to the bottom of the lowest exciton band and further participate in the diffusion processes.32,33 Therefore, we consider an exciton wavepacket formed near the bottom of the lowest exciton band migrating in a SWNT that is aligned with the x-axis. The exciton excites SP modes in a MT a distance, d, below the center of the SWNT, as shown in Figure 1. For

Figure 1. Partitioning of the exciton delocalization length along SWNT into N sites separated by distance a. The transition dipole, μn, is assigned to each site and oriented along the SWNT axis, i.e., the xaxis. The MT is placed on the distance d below the SWNT with the symmetry axis coinciding with the z-axis. We consider the SP mode associated with the transition dipole, μsp, oriented along the MT symmetry axis.

computational purposes, the SWNT was partitioned into N + 1 sites, each labeled by a site-index n, with site n = 0 locating the center. The lattice spacing was chosen to be small enough to adequately sample variations in coupling strength in the vicinity of the MT. To model the exciton−SP interaction, we assign a transition dipole, μn = exμn, to each site, with direction defined by the unit vector ex. The SP dipole, μsp = ezμsp, has a perpendicular orientation in a direction defined by the unit vector ez. The vector connecting the centers of the μsp and μn is denoted by rn. We will assume that the SP energy ℏωsp associated with the chosen polarization is tuned to be close to the on-site exciton energies ℏωn, and yet well-separated from the energies of other SP modes. The quasi-electrostatic exciton−SP interaction will be characterized by the energy ℏgn. As discussed in Appendix A, the coupling constant gn provides frequency of a single-quantum exchange between site-n and the SP mode due to transitions characterized by the associated dipole matrix elements μn and μsp. When a rotating-wave approximation is adopted for the

2. THEORETICAL MODEL In this section we derive a set of coupled equations describing the evolution of the exciton and MT-SP populations and their coherences following a reduced density matrix approach37−39 to account for the coherent exciton−SP coupling and the effect of the environment. Integration of coherences results in a set of 8071



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Ẋ nm(t ) = iωnm ̃ X nm − i(gmYn ,sp − gn*Y m*,sp)

coupling term, the system Hamiltonian maps on a spinless Anderson−Fano Hamiltonian40 ̂ = HAF

Yṅ ,sp(t ) = iωñ ,spYn ,sp − i

†

∑ ℏωnbn̂ bn̂ + ℏωspasp̂ † asp̂ †

∑ ℏ(gnbn̂ asp̂ + gn*asp̂ † bn̂ )

where b†n̂ and bn̂ (â†sp and âsp) are creation and annihilation operators of the exciton on site n (localized SP state), respectively. Considering low exciton population, both exciton and SP operators obey Bose commutation relations. In the vicinity of the MT, exciton intersite matrix elements are weak in comparison to the exciton−plasmon interaction, justifying the neglect of exciton hopping terms in eq 1. Under the assumption of a small mean free-path, diffusive transport along the SWNT is included phenomenologically, in the Liouville equation for the reduced density operator, 1 ̂ ̂ + ρex̂ ̇ ̂ + 9̂ 2ρAF ̂ ] + 9̂ 1ρAF [HAF ,ρAF iℏ

(2)

The first term on the right-hand side describes coherent dynamics corresponding to the Anderson−Fano Hamiltonian (eq 1), and the last term described the exciton population increase due to optical pumping. The second and the third terms are relaxation terms accounted for in a general Lindblad form γ † † † ̂ bn̂ − bn̂ bn̂ ρAF ̂ − ρAF ̂ bn̂ bn̂ ) ̂ = ∑ n (2bn̂ ρAF 9̂ 1ρAF 2 n γsp ̂ asp ̂ − ρAF ̂ asp + (2asp ̂ ρAF ̂ † − asp ̂ † asp ̂ ρAF ̂ † asp ̂ ) (3) 2 ̂ = 9̂ 2ρAF

∑ nm

† † ̂ bm̂ bn̂ bn̂ bm̂ ) − ρAF

− (μm ·Gmn(ωñ ) ·μn )*(Nm − Nsp)} (4)

Yn ,sp =

Equation 3 accounts for the exciton and SP population decay processes with rates γn and γsp, respectively. These include both nonradiative and radiative channels. Equation 4 describes exciton scattering processes with the rates wnm giving rise to dephasing and diffusion. The complete set of reduced density operator matrix elements can be introduced as the following set of operator averages: the on-site exciton population is Nn = tr{b†n̂ bn̂ ρ̂AF}, the SP population is Nsp = tr{â†spâspρ̂AF}, the exciton spatial coherence between sites n ≠ m is Xnm = tr{b†n̂ bm̂ ρ̂AF}, and the exciton−SP coherence is Yn,sp = tr{b†n̂ âspρ̂AF}. We consider an experimentally realized situation in which only the excitons are optically excited but not the SP mode (e.g., the laser is tuned to an energy lying well above the SP resonance, and exciton population growth follows inter and/or intraband relaxation). These processes are accounted for by introducing an on-site exciton population gain with the rate tr{b†n̂ bn̂ ρ̂eẋ } = In. For the density matrix elements introduced above, we obtain the following coupled equations of motion from eq 2 ∂n2

− γnNn + In

̇ (t ) = i ∑ (g Yn ,sp − g *Y n*,sp) − γ Nsp Nsp sp n n n

|μm ·Gmn(ωñ ) ·μn |2

∑

gnωnm ̃

m

+

(9)

(Nn − Nm)

1 (μ ·Gnn(ωñ ) ·μn )(Nn − Nsp) gn n

(10)

respectively. Substitution of the coherences into eqs 5 and 6 leads to coupled rate equations for the exciton and SP populations: Nṅ (t ) = −Wn ,sp(Nn − Nsp) −

∑ Wnm(Nn − Nm) m≠n

+D ̇ (t ) = Nsp

∂ 2Nn ∂n2

− (γnnr + γnrd)Nn + In

(11)

∑ Wn,sp(Nn − Nsp) + ∑ ∑ Wnm(Nn − Nm) n

−

n

(γspnr

+

γsprd)Nsp

m≠n

(12)

valid in the weak coupling regime and accounting for both energy transfer to SP and SP-assisted exciton site energy transfer on the same footing. Specifically, the exciton−SP population transfer rate is

2

∂ Nn

1 {(μm ·Gmn(ωñ )·μn )(Nn − Nsp) ωnm ̃

X nm =

wnm † † † † ̂ bm̂ bn̂ − bm̂ bn̂ bn̂ bm̂ ρAF ̂ (2bn̂ bm̂ ρAF 2

Nṅ (t ) = −i(gnYn ,sp − gn*Y n*,sp) + D

(8)

Equation 5 contains a diffusion term in which the second derivative ∂2Nn/∂n2 = (Nn+1 − 2Nn + Nn−1) is defined as the finite difference of the nearest neighbor populations. The diffusion term is the long-wavelength approximation to the master equation associated with the relaxation term (eq 4) matrix element tr{b̂n†b̂sp9̂ 2 ρ̂ AF} = ∑m(wnmNm − wmnNn). Assuming inversion and translational symmetry, ωnm = ωmn → ω|m−n|, the diffusion rate D = ∑nωnn2 is given by the weighted sum of exciton site-scattering rates from eq 4. The coherences (eqs 7 and 8) Xnm and Yn,sp depend on the complex frequencies ω̃ nm = ωn − ωm + i(γn/2 + γm/2 + γ̃mn) and ω̃ n,sp = ωn − ωsp − i(γn/2 + γsp/2 + γ̃n), respectively. Here, γn and γsp are the exciton and SP population decay rates defined in eq 3, and the dephasing rates are expressed in terms of the exciton scattering rates as follows γ̃n = ∑mwmn/2 and γ̃mn = wmn + wmm/2 + wnn/2. Note that set of eqs 5−8 is exact; i.e., it does not contain higher than second order operator averages that would require factorization.a Typically, coherences approach their steady state values on the time scale faster than the populations and can be eliminated from the equations for the populations. The steady state solution for coupled eqs 7 and 8 can be found from a perturbation expansion in the small parameter gn/(γn/2 + γsp/2 + γ̃n), resulting in the following second and third order expressions

(1)

n

̂̇ = ρAF

∑ gm*Xnm − ign*(Nn − Nsp) m≠n

n

+

(7)

(5)

Wn ,sp =

(6) 8072

2 Im(μn ·Gnn(ωñ ) ·μn ) ℏ

(13)



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with ω̃ n = ωn + i(γn/2 +γ̃n), and the SP-assisted exciton site transfer rate is Wnm =

2γnm

|μn ·Gnm(ωñ ) ·μm |

(14)

with ωnm = ωn − ωm and γnm = γn/2 + γm/2 + γ̃mn. These expressions, along with eqs 9 and 10 are written in terms of the dyadic Green function Gnm(ω) for the SP mode, as described in Appendix. A. Although the connection between Gnm(ω) and gn is straightforward, ℏgngm* ω − ωsp ̃

(15)

+ − + μsp̂ ·(Ê (rsp) + Ê (rsp))

.(0) n (ω) =

q ,λ = 1,2

ℏv|q| eqλaq̂ λe iq·r 2ε0εV

+ Ê (r,t ) =

4πε0εv 2r

Σsp(ω) =

+

∑ nm

2

4πε0εv r

ωsp2 χsp a(̂ t − r /v)

1 ω − ωn + iϵ

∑ μn ·Gnn(ω − ωn,sp)·μn n

(23)

(24)

(25)

The SP Green function can be calculated in the same manner and has the form40 1 ω − ωsp − Σsp(ω) + iϵ

.sp(ω) =

(26)

Substitution eq 19 along with the Hermitian conjugate into eq 18 gives the final expression for the radiated power (17)

dP = dΩ

∑ ℏωnγnrdfnn (Ω)χnm2 Nm(t − r /v) nm

+ ℏωspγsprdfsp (Ω)χsp2 Nsp(t − r /v) +

∑∑ℏ

ωnωmγnrdγmrd χnn ′ χmm ′ X n ′ m ′(t − r /v)

nm n ′ m ′

+

∑ℏ

(18)

ωspωnγsprdγnrd fn ,sp (Ω)χnm χsp

nm

× (Ym ,sp(t − r /c) + Y m*,sp(t − r /v))

(27)

We observe that the angular distribution is described as a superposition of following ellipsoidal patterns

ωnωmχnm bm̂ (t − r /v)

μsp − e(e ·μsp )

(22)

may be written in terms of the SP dyadic Green function using eq 15, whose argument depends on the self-energy

(16)

To calculate the trace, we express the electric field operators in term of the exciton and SP operators, adopting the so-called source quantity representation.39,41 For this purpose, we solve + Heisenberg equation of motion iℏE ̂ ̇ = [Ê +, Ĥ AF + Ĥ RD] in the rotating-wave approximation to find μn − e(e ·μn )

(21)

;nm(ω) = μn ·Gnm(ω − Σsp(ω)) ·μm

and Ê − = (Ê +)† where V is a photon mode cavity volume and ε0 is the vacuum permittivity. At this point, we are ready to calculate the distribution of emitted radiation. The power radiated per solid angle may be expressed as the averaged radiation intensity operator,41 dP − + ̂ ) = 2ε0εvr 2 tr(Ê (r,t ) Ê (r,t )ρAF dΩ

dω ω 2 Im{.sp(ω)} π ωsp2

is the zero-order exciton Green function, and the exciton scattering matrix,

The free photon term explicitly depends on the photon energy ℏv|q|, where v = c/√ε is the speed of light in the surrounding environment characterized by the dielectric constant ε. The operators â†qλ and âqλ respectively create and destroy modes with the wave vector q and polarization λ. The interaction terms contain exciton and SP transition dipole operators μ̂n = μnb†n̂ + μn*bn̂ and μ̂sp = μspâ†sp + μsp * âsp, respectively, and the electric field operators are

∑

∞

where

q ,λ = 1,2

+ Ê (r) = i

∫0

(20)

(0) (0) .nm(ω) = δnm .(0) n (ω) + . n (ω) ;nm(ω) . m (ω)

+ − ℏv|q|aq̂ †λaq̂ λ + μn̂ ·(Ê (rn) + Ê (rn))

∑

χsp =

dω ω 2 Im{.nm(ω)} π ωnωm

∞

account for the photon emission line shape, and are given by integrals over the imaginary parts of exciton and SP Green functions, .nm and .sp, associated with the Anderson−Fano model. The exciton Green function is associated with the equation of motion for the operator bm̂ as determined by Ĥ AH (eq 1). Its explicit form can be written in terms of the scattering matrix,42

the preference in using the dyadic Green function representation is that Gnm(ω) can be directly determined via classical electrodynamic simulations. 2.2. Radiation Emission Pattern. To include spontaneous emission processes leading to PL, we extend our system Hamiltonian (eq 1) by adding a term describing a quantized radiation mode interacting with material dipoles39,41 ĤRD =

∫0

2

ℏ2(ωnm 2 + γnm 2)

μn ·Gnm(ω) ·μm =

χnm =

fnm (Ω) =

3 (1 − (e·en)(e·em)) 8π

(28)

fsp (Ω) =

3 (1 − |e·esp|2 ) 8π

(29)

fn ,sp (Ω) = −

(19)

Here, e = r/r, where r is the vector from the center of the SWNT to the photon detector. The quantities

3 (e·en)(e·esp) 8π

(30)

appropriately weighted by exciton and SP radiative decay (rd) rates 8073



γnrd

γsprd

=

=

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coordinates, ρ = (x2 + y2)1/2 and z, of the MT rounded cone (Figure 1) obey the equation z = t0z0 − d −t0(z02 + ρ2)1/2, where we set t0 = 1/8, z0 = 10 nm, and vary 0 ≤ ρ ≤ 400 nm. For this MT, numerical simulations of the SP response, carried out as outlined in Appendix A, result in the SP resonance progression shown as a blue curve in Figure 2. We identify the

ωn3 |μn |2 3π ℏϵ0ϵv 3

(31)

ωsp3 |μsp |2 3π ℏϵ0ϵv 3

(32)

respectively. These rates comprise part of the total exciton and rd nr rd SP decay rates, γsp = γnr sp + γsp and γn = γn + γn , appearing in the relaxation operator (eq 3) in our equations of motion. The nr contribution to these rates from nonradiative decay, γnr sp and γn , are to be determined empirically from experiment. Finally, for ωn ∼ ωsp, the ratio of photon fluxes R=

γsprdχsp2 Nsp ∑nm γnrdχnm 2 Nm

(33)

produced by the SP and the exciton states can be introduced. Provided the contribution of the coherences in eq 27 is negligible, this quantity reflects variation of the radiation pattern as a function of the exciton−SP coupling strength. If R ≫ 1 (R ≪ 1), then the radiation is dominated by the SP (exciton) dipole emission and is referred as the SP (exciton) emission. For R ∼ 1, the emission pattern represents a superposition of both emitting dipoles. In summary, the populations and coherences entering eqs 27 are solutions of eqs 5−8. In the weak coupling regime the coherences are given by eqs 9 and 10 and the populations can be determined by solving eqs 11 and 12. These sets of equations along with the expression for radiative power represent main results of our theory. They will now be used in the numerical calculations described below.

Figure 2. Comparison of experimentally measured32 (6,5) SWNT absorption (black, 1) photoluminescence (red, 2) spectra with the simulated SP resonances polarized along the z-direction (blue, 3) for the adopted Ag MT geometry.

lowest energy SP mode ℏωsp = 1.25 eV (λ = 985 nm) as asymptotically corresponding to the SP dipole response. For the adopted parameters this SP mode is in resonance with the lowest energy S1 exciton band and therefore interacts with the exciton states participating in the diffusion process and photon emission. Fitting the SP response with Lorentzian profile (eq −1 39), we find a SP nonradiative decay rate γnr (τnr sp = 33.4 ps sp = 30 fs) and a transition dipole moment μsp = 12128 D. According to eq 32, the SP radiative decay rate becomes γrd sp = 56.8 ps −1 (τrd sp = 18 fs). Photoluminescence is typically excited via high energy S2 absorption at λ = 570 nm followed by rapid internal conversion to the S1 band.32 According to Figure 2, the S2 band does not overlap with any SP resonances of the MT. As a result, this model consistently accounts for the SP effect on the exciton diffusion and emission within the S1 band only, and does not require consideration of SP enhancement of the exciton photogeneration processes. Assuming that SWNT is photoexcited homogeneously, we set the exciton population gain In = γnr n , for all n. With the parameters as discussed above, we solve numerically eqs 11−14. The solutions were benchmarked against solutions of eqs 5−8 and found to be indistinguishable, indicating that the weak exciton−SP coupling approximation is valid. Numerical evaluation of the integral quantities χnm and χsp (eqs 20 and 21) shows that χnm ≈ δnm and χsp ≈ 1 for the parameter range of interest. 3.2. Simulation Results. Key quantities defining the effect of the SP on exciton diffusion, such as exciton−SP and the SPassisted exciton site transfer rates, are plotted in (a) and (b) of Figure 3. The plot shows that the effective exciton−SP interaction range is constrained to an interval of ∼100 nm around the origin. Increasing d leads to a rapid decrease of the rates and the effective interaction length. The diffusion rate (D = 750 ps−1) significantly exceeds the population transfer rate which in turn has significant effect on the exciton populations profile, as we show below. On the other hand, the SP-assisted

3. NUMERICAL CALCULATIONS In this section we discuss the results of numerical calculations of the exciton population distribution and the associated radiation patterns for three (6,5) SWNTs of different sizes in the presence of SP mode excited in Ag MT. We consider first a nanotube of length L = 300 nm, which is on the order of the measured value lD = 202 nm of the exciton diffusion length.32,33 We also consider nanotubes of lengths L = 2600 nm and L = 5200 nm. These three will be referred below as SWNT-I, SWNT-II, and SWNT-III, respectively. 3.1. Model Parametrization. For computational purposes, each nanotube is divided into N = L/a − 1 sites with lattice constant a = 1.0 nm. This is sufficiently small to resolve the spatial dependencies in the Hamiltonian, but large enough to keep the calculation from becoming unwieldy. We assign a dipole moment μn = 10 D to each of site (Figure 1). Furthermore, we assume that the system is imbedded into a solvent with the dielectric constant set to ε = 1.36. According to eq 31 the adopted dipole value corresponds to −5 a radiative decay rate γrd ps −1 and associated n = 4.3 × 10 rd decay time τn = 23 ns. The value of the nonradiative decay rate −2 is set to γnr ps−1 to reproduce the experimentally n = 1.8 × 10 measured32,33 photoluminescence decay time τPL = 55 ps. Dephasing effects due to exciton scattering are neglected in our calculations, i.e., γ̃n = 0. Finally, the diffusion rate D = lD2/a2τPL = 750 ps −1 is determined from the measured diffusion length lD = 220 nm. To model the SP response, we consider Ag MT with the Ag bulk dielectric function represented by the Drude formula in which the parameters are fit to experiment.43 The surface 8074



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Figure 3. Calculated (a) exciton−SP population transfer rate (eq 13) vs nanotube coordinate x = na and (b) the SP-assisted exciton site transfer rate (eq 14) vs Δx = a(m − n) for various SWNT-MT distances, d. Here, we mark d = 6 nm and d = 8.5 nm curves only. All subsequent curves are obtained for d increased by Δd = 2.5 nm up to dmax = 100 nm. The inset shows the dimensionless parameter |gn|/(γnr n + γnr sp) determining the exciton−SP coupling regime. The variation in d is the same.

Figure 4. Calculated steady state exciton distribution along (a) SWNT-I, (b) SWNT-II, and (c) SWNT-III as a function of SWNT coordinate x = na. Here, we mark d = 6 nm and d = 8.5 curves. All subsequent curves are obtained with the distance increased by Δd = 2.5 nm up to dmax = 100 nm. The red dashed line shows the exciton population in the absence of the SP-assisted exciton site population transfer with the rate Wnm for d = 6 nm.

exciton site transfer rate (panel b) is small compared to the exciton−SP transfer rate (panel a), and has weak effect. Finally, nr we find that the dimensionless parameter, |gn|/(γnr n + γsp) is below 0.15 for all n, which is consistent with the weak exciton− SP coupling assumption made in deriving of the approximate rate eqs 11−14. Let us first examine how the exciton−SP interaction influences the steady state exciton distribution, Nn, obtained from solution of eqs 11−14. This quantity is plotted in (a)−(c) of Figure 4 for SWNT-I, -II, and -III, respectively. The diffusion boundary conditions33 require vanishing populations at the SWNT ends. As a result, the SP-uncoupled exciton population distributions (d ∼ 100 nm) deviate from a rectangular profile, showing a smooth drop of the end populations, the effect being strong in shorter nanotubes. Interaction with the SP modes causes a population dip around x = 0, reaching maximum depth at small d. We find that the main contribution to this dip is from the exciton−SP population transfer at the rate Wn,sp. The SP-asisted exciton site transfer at the rate Wnm slightly changes the dip. This is seen from the behavior of the red dashed curve representing the solution of eqs 11 and 12 with Wnm set to zero. We find a tiny ratio of Nsp to the exciton population within the SP-dip, showing that most of the exciton energy transferred to the SP is dissipated to heat (Ohmic loss).

Next, we examine the transient population dynamics characterized by the exciton equilibration time, τeq, required to reach the steady state solutions (Figure 4). We present τeq vs d in (a)−(c) of Figure 5 for SWNT-I, -II, and -III, respectively. At large MT-SWNT separation (d ∼ 100 nm), this quantity approaches the equilibration time for uncoupled excitons. The SWNT size (boundary) imposes an upper limit on the equilibration time which is faster for short SMNTs (compare (a) with (b) and (c)). The SP induced reduction in τeq is noticeable for d ≲ 50 nm regardless of the tube length. We also examined the effect of SP-assisted exciton transfer (not shown in the plot) and found it to be negligible. We thus conclude that the SP modifies transient dynamics primarily from the exciton− SP population transfer, accompanied by Ohmic loss. At this point we turn our attention to the radiation emission properties of the SWNT-MT system by examining the photon flux ratio, R, (eq 33) and the associated radiation angular distribution (eqs 27−32) due to the steady state exciton and SP populations (Figure 4). As pointed out in section 2.2, R distinguishes between the SP (R ≫ 1) and exciton (R ≪ 1) emissions if the contribution of the coherences Yn,sp and Xnm to eq 27 is negligible. This is exactly the case if ωn = ωsp, but it is also true if ωn ≠ ωsp provided that ωn is a random variable (reflecting diagonal energy disorder) with a mean value equal to ωsp. If there is a fixed frequency detuning Δω = ωn − ωsp for 8075



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Figure 6. Ratio of the SP and the exciton radiation fluxes, R, and associated ratio of the state populations, Ns/Nex, where Nex = ∑nNn, for (a) SWNT-I, (b) SWNT-II, and (c) SWNT-III as a function of the SWNT-MT separation d. The insets in (a) shows the color code for different segments of the curve distinguishing SP (red) and exciton (blue) emissions. The green segment denotes the crossover regime. This color code applies to all panels including the inset in (b), which shows the associated radiation angular distributions.

Figure 5. Variation of the equilibration time, τeq, required to reach the steady state exciton populations as a function of SWNT-MT separation, d, for (a) SWNT-I, (b) SWNT-II, and (c) SWNT-III.

all n, one can apply symmetry arguments, including the fact that the coupling frequency gn is antisymmetric function, i.e., gn = −g−n,b to make the case that for the MT located at the center (or more specifically away from the ends) of the nanotube (Figure 1), the coherences vanish identically. This is supported by direct numerical calculations. Panels a−c in Figure 6 present R plotted vs d for SWNT-I, SWNT-II, and SWNT-III, respectively. The radiation emission is dominated by the SP dipole (red segments) with the angular distribution shown by red curves in the inset to (b) for d in a range of a few tens of nanometers. For the short SWNT-I, SP -dominated emission occurs for d ≲ 40 nm, whereas, for the long SWNT-II and SWNT-III, it occurs for d ≲ 20 nm and d ≲ 10 nm, respectively. This observation can be rationalized by examining the exciton population distribution shown in Figure 4. For SWNT-I, the exciton−SP interaction range (Figure 3a) is about the size of the nanotube. According to Figure 4a, this results in a significant reduction of the exciton population across the SWNT-I. Therefore, the total exciton contribution to the radiated power decreases. In contrast, for long SWNT-II and -III (Figure 4b,c) only the excitons near the center of the nanotube suffer drastic population reduction, and there is significant photoexcited exciton population outside the interaction region that enhances the exciton emission component. This shows that a stronger coupling (i.e., smaller d) is needed to reach the SP-dominated emission regime. According to eq 33, there are two main quantities determining the angular distribution of emitted radiation.

One is the ratio of the steady state SP and exciton populations, Nsp/Nex with Nex = ∑nNn, plotted in Figure 6 (right scale). The rd other is the ratio of their radiative decay rates, γrd sp/γn = 1.3 × 6 10 . Figure 6 shows that the total population transferred to the SP state from the coupled exciton states is 5−7 orders of magnitude less than the exciton population, a consequence of weak coupling. However, a large ratio of the radiative decay rates (associated with the large values of the SP transition dipoles) compensates for the low population, allowing the SP state to dominate the angular distribution at small d. In other words, it is primarily the fast SP emission rate that causes the angular distribution to be SP-dominated. Increasing d leads to a rapid SP population drop, causing a shift to the excitondominated emission regime (blue segments in Figure 6). The evolution of the radiation angular distribution as a function of d can be seen in the inset to Figure 6b. Finally, we argue that the coherences can have a strong influence on the orientation of the radiation diagram when the inversion symmetry is broken, especially in small nanotubes in which the emission from excitons outside the interactin region is minimized. For example, let us break the symmetry of the gn in SWNT-I by placing the MT at the end of the tube, at x = 150 nm, and let us also detune the exciton energy from the SP by ℏΔω = 10 meV. Our results are shown in Figure 7, where we compare the radiation angular distributions associated with the 8076



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intensities (inset to Figure 6b). In contrast, a sustained oscillation of the coherences introduces a phase shift in the exciton and SP dipoles, and they behave as a a single superposition dipole whose orientation depends on the phase shift. Such a phase shift corresponds to the radiation diagram of a rotated effective dipole as observed in Figure 7. Quantum mechanically, this effect can be explained as the emission associated with spatially reoriented superposition dipole whose matrix elements are calculated in the dressed exciton−SP eigenstates. The effect is most noticeable in the crossover regime because the net contribution of the excitons and SP to the emitted power are about the same.

4. DISCUSSION AND CONCLUSIONS We have developed a general theoretical approach to study exciton transport and emission in a SWNT in the presence of a localized SP mode within a MT interacting via near-field coupling. Our primary result is a set of quantum mechanical equations of motion (eqs 5−8) that account for the exciton, SP, and the environment (including radiation mode) on the same footing. These material equations are complemented by the expressions for the radiated power (eqs 27−32), which depends on the exciton and SP populations and coherences, allowing for an examination of the angular distribution (and spectrum) of the emitted radiation. The model is capable of describing both strong and weak exciton−SP coupling regimes, as well as the transition from ballistic to diffusive transport, thus spanning a broad range of behaviors in low-dimensional semiconductor materials. In this paper, the weak exciton−SP limit has been examined in detail. Using a perturbation approach, with the exciton−SP coupling frequency normalized per dissipation rate as the expansion parameter, we have derived a minimal set of rate equations (eqs 11 and 12) for the exciton and SP populations. These equations account for the exciton−SP population transfer and SP-assisted exciton site transfer processes with their associated rates given by eqs 13 and 14, respectively. The expressions for the rates are represented in terms of the dyadic Green function, which is determined via numerical calculations of the SP response, allowing for consideration of a Ag tip in the shape of a rounded cone. The exciton−SP and site exciton transfer matrix elements required in this model have been extensively considered in the literature in the context of the Förster energy transfer to metal and between donor and acceptor sites. In fact, master equations with rates determined by the dyadic Green function have been introduced in the same manner.1,9,10,12 The difference (and advantage) of the present treatment is that the steady state coherences (eqs 9 and 10) have been retained in expressions for the radiated power. They play an important role in its analysis, as demonstrated in section 3.2. Furthermore, we have established a simple relationship (eq 15) between the dyadic Green function and the exciton−SP coupling frequency that allows one to obtain the latter quantity from simulation. Numerical simulations for various sizes of (6,5) SWNT and Ag MT have been performed using our proposed methodology. We have found that the near-field interaction between the exciton−SP is efficient on 100 nm length scales or less and occurs in the weak coupling regime with diffusion processes being much faster than the exciton−SP population exchange. Exciton−SP population transfer dominates over SP-assisted site exciton transer, resulting in fast energy transfer to the MT with subsequent dissipation (i.e., Förster energy transfer). The

Figure 7. Radiation angular distribution of SWNT-I with the energy detuned to ℏΔω = 10 meV plotted for various SWNT-MT distances ranging between d = 100 nm and d = 15 nm as indicated in parentheses. The angle is measured with respect to the x-axis in Figure 1. Solid lines mark the distributions associated with the MT placed at the end of the nanotube, i.e., x = 150 nm. Dashed lines are for the MT located at the center, i.e., x = 0.

MT placed at the end (solid lines), and at the center (dash lines), of the nanotube, parametric in d. The general trend is that the nonvanishing coherences induce rotation of the radiation diagram. The effect is strongest for the distance range corresponding to R ∼ 1, which is the crossover regime (green segment) in Figure 6. Our data analysis also indicates that the dominant contribution to the diagram rotation originates from Yn,sp.c To rationalize this effect, we should recall that an exciton dipole behaving incoherently, i.e., without any information about its oscillation phase, merely transfers energy to the SP dipole. As a result, the exciton and SP dipoles become excited together, but the overall radiation pattern is a superposition of their 8077



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Hamiltonian Ĥ AF (eq 1) with the exciton−SP interaction energy

exciton−SP energy transfer also causes a reduction in equilibration time for excitons to reach their steady sate distributions. Analysis of radiation emission properties shows that the radiation diagram is dominated by SP emission for a SWNT-MT separation of a few tens of nanometers. In small SWNTs, whose length is on the order of the effective exciton− SP interaction range, this effect requires weaker coupling to the plasmon than in long SWNTs for which delocalized excitons provide an additional contribution to the radiated power. We have found that the SP population in the steady state regime is orders of magnitude below the exciton population, and yet the fast SP radiative decay rate still makes the SP emission more efficient. Finally, we found that breaking the inversionsymmetry of the exciton−SP interaction by moving the MT away from the center of the SWNT causes a rotation in the radiation pattern that depends on the strength of the coherences.The rotation can be understood as the emission pattern of a tilted dipole composed of hybridized exciton−SP eigenstates. In conclusion, exciton diffusion in SWNT affected by a localized SP mode has been systematically modeled using a unified approach. We have calculated the influence of exciton− SP energy transfer and dissipation on radiative power and angular distribution. We find that the shape of the radiation pattern depends on the interplay between the exciton−SP state populations and radiative decay rates, and that a high ratio of the SP to exciton radiative rates is what causes the emission to be SP-dominated. We demonstrate that the coherences can play significant role in the pattern orientation.

ℏgn = μsp ·Tn·μn

To find the relationship between the dyadic Green function and the exciton−SP interaction rate, gn, we solve the frequencydomain equations of motion for the Hamiltonian system given by eqs 34 and 35, adding the SP nonradiative relaxation rate γnr sp/2 via the second term of eq 3. The outcome is an expression for the local contribution to the SP dipole induced by the exciton dipole at site m, Psp(ωm) = α(ωm)ez(ez ·Tm ·μm )

αsp(ω) =

†

n

+ bn̂ )

3nn − I 4πε0εrn3

(39)

(40)

Alternatively, one could use the dyadic Green function to express the same field, En(ωm) = Gmn(ωm) ·μm (ωm)

(41)

Comparison of the two expressions above shows that the dyadic Green function may be expressed in the form Gmn(ω) = αsp(ω)(em·Tm·ez)(ez ·Tn·en)

(42)

The substitution of eqs 37 and 39 into eq 42 results in eq 15, extensively used in section 2. Typically, the dyadic Green function is introduced as the solution to Maxwell’s equations.1,39 In our case, however, it is only the near-field interactions defined by the Poisson equation that is responsible for the exciton−SP coupling and contributes to the Green function given by eq 42. We determine Gmn(ω) numerically using the boundary element method implemented in the MNPBEM13 code.45,46 Values of the SP frequency ωsp and the nonradiative decay rate γsp are extracted from the calculated SP absorption spectrum (excited by a z-polarized plane wave) as this is proportional to the imaginary part of Gmn(ω). The SP transition dipole μsp can be determined analytically from the dyadic Green functions found in eqs 36, 39, and 42. In this calculation, a SWNT axis is separated from the MT by distance d > 100 nm to ensure that the asymptotic eq 36 holds. For distances d < 100 nm, eq 36 breaks down and the absolute values of the coupling rates gn must be determined from direct numerical simulations of Im(μn·Gnn(ωn)·μn) with the help of eq 15. The sign of gn is recovered by examining the derivative of the coupling rate, dgn/dx.

(34)

(35)

containing a tensor Tn which asymptotically corresponds to the longitudinal electric field component of a dipole, Tn ∼

ℏ(ω − ωsp + iγspnr /2)

En = α(ωm)(Tn·ez)(ez ·Tm·μm )

Here, the local field is induced by the exciton transition dipole leading to the relation

∑ Tn·μn (bn̂

μsp2

where μsp is the SP transition dipole matrix element (introduced in section 2.1). The electric field at exciton site n due to the induced SP dipole is En = Tn·Psp, so that formally

APPENDIX A: RELATION BETWEEN EXCITON−SP INTERACTION FREQUENCY AND DYADIC GREEN FUNCTION In this Appendix we introduce a quantized model for the exciton−SP interaction. A rigorous second quantization procedure for the localized SP mode in metal nanoparticles uses cavity QED.44 Here, we do not specify the geometry of the nanoparticle “cavity” but instead declare the SP to be a quantum oscillator with an effective transition dipole moment. To identify the parameters of the SP quantum oscillator, we exploit the connection between parameters of the Hamiltonian (eq 1) and the classical dyadic Green function. Consider quantum mechanical oscillator associated with a SP mode characterized by a transition dipole operator μ̂sp = μsp(âsp + â†sp) coupled to a local electric field Ê loc. The associated Hamiltonian is

Ê loc =

(38)

In eq 38, ez is the polarization direction of the SP mode (Figure 1) with polarizability

■

Ĥ sp = ℏωspasp ̂ † asp ̂ − μsp̂ ·Ê loc

(37)

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AUTHOR INFORMATION

Corresponding Authors

(36)

*D. H. Dunlap: e-mail, [email protected]. *A. Piryatinski: e-mail, [email protected].

In eq 36, n = rn/rn (Figure 1) and I is the 3 × 3 identity matrix. Substitution of eq 35 into eq 34 in the rotating wave approximation gives second and the third terms in the

Notes

The authors declare no competing financial interest. 8078



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(12) Pustovit, V. N.; Shahbazyan, T. V. Resonance Energy Transfer near Metal Nanostructures Mediated by Surface Plasmons. Phys. Rev. B 2011, 83, 085427. (13) Masuo, S.; Naiki, H.; Machida, S.; Itaya, A. Photon Statistics in Enhanced Fluorescence from a Single Cdse/Zns Quantum Dot in the Vicinity of Silver Nanoparticles. Appl. Phys. Lett. 2009, 95, 193106− 193106. (14) Park, Y.-S.; Ghosh, Y.; Chen, Y.; Piryatinski, A.; Xu, P.; Mack, N. H.; Wang, H.-L.; Klimov, V. I.; Hollingsworth, J. A.; Htoon, H. SuperPoissonian Statistics of Photon Emission from Single CdSe-CdS CoreShell Nanocrystals Coupled to Metal Nanostructures. Phys. Rev. Lett. 2013, 110, 117401. (15) Kolesov, R.; Grotz, B.; Balasubramanian, G.; Stöhr, R. J.; Nicolet, A. A.; Hemmer, P. R.; Jelezko, F.; Wrachtrup, J. Wave-Particle Duality of Single Surface Plasmon Polaritons. Nat. Phys. 2009, 5, 470− 474. (16) Pustovit, V. N.; Shahbazyan, T. V. Cooperative Emission of Light by an Ensemble of Dipoles near a Metal Nanoparticle: The Plasmonic Dicke Effect. Phys. Rev. Lett. 2009, 102, 077401. (17) Pustovit, V. N.; Shahbazyan, T. V. Plasmon-Mediated Superradiance near Metal Nanostructures. Phys. Rev. B 2010, 82. (18) Sugawara, Y.; Kelf, T.; Baumberg, J.; Abdelsalam, M.; Bartlett, P. Strong Coupling between Localized Plasmons and Organic Excitons in Metal Nanovoids. Phys. Rev. Lett. 2006, 97, 266808. (19) Gomez, D.; Vernon, K.; Mulvaney, P.; Davis, T. Surface Plasmon Mediated Strong Exciton- Photon Coupling in Semiconductor Nanocrystals. Nano Lett. 2009, 10, 274−278. (20) Gomez, D.; Vernon, K.; Mulvaney, P.; Davis, T. Coherent Superposition of Exciton States in Quantum Dots Induced by Surface Plasmons. Appl. Phys. Lett. 2010, 96, 073108−073108. (21) Manjavacas, A.; Garca de Abajo, F.; Nordlander, P. Quantum Plexcitonics: Strongly Interacting Plasmons and Excitons. Nano Lett. 2011, 11, 2318−2323. (22) Zhang, W.; Govorov, A. O.; Bryant, G. W. Semiconductor-Metal Nanoparticle Molecules: Hybrid Excitons and the Nonlinear Fano Effect. Phys. Rev. Lett. 2006, 97, 146804. (23) Artuso, R. D.; Bryant, G. W. Optical Response of Strongly Coupled Quantum Dot- Metal Nanoparticle Systems: Double Peaked Fano Structure and Bistability. Nano Lett. 2008, 8, 2106−2111. (24) Ridolfo, A.; Di Stefano, O.; Fina, N.; Saija, R.; Savasta, S. Quantum Plasmonics with Quantum Dot-Metal Nanoparticle Molecules: Influence of the Fano Effect on Photon Statistics. Phys. Rev. Lett. 2010, 105, 263601. (25) Chang, D.; Sørensen, A. S.; Hemmer, P.; Lukin, M. Quantum Optics with Surface Plasmons. Phys. Rev. Lett. 2006, 97, 053002. (26) Akimov, A.; Mukherjee, A.; Yu, C.; Chang, D.; Zibrov, A.; Hemmer, P.; Park, H.; Lukin, M. Generation of Single Optical Plasmons in Metallic Nanowires Coupled to Quantum Dots. Nature 2007, 450, 402−406. (27) Fedutik, Y.; Temnov, V.; Schöps, O.; Woggon, U.; Artemyev, M. Exciton-Plasmon-Photon Conversion in Plasmonic Nanostructures. Phys. Rev. Lett. 2007, 99, 136802. (28) Chang, D. E.; Sørensen, A. S.; Demler, E. A.; Lukin, M. D. A Single-Photon Transistor Using Nanoscale Surface Plasmons. Nat. Phys. 2007, 3, 807−812. (29) de Leon, N. P.; Shields, B. J.; Yu, C. L.; Englund, D. E.; Akimov, A. V.; Lukin, M. D.; Park, H. Tailoring Light-Matter Interaction with a Nanoscale Plasmon Resonator. Phys. Rev. Lett. 2012, 108, 226803. (30) Tame, M.; McEnery, K.; Ö zdemir, S.; Lee, J.; Maier, S.; Kim, M. Quantum Plasmonics. Nat. Phys. 2013, 9, 329−340. (31) Lee, J.; Hernandez, P.; Lee, J.; Govorov, A. O.; Kotov, N. A. Exciton-Plasmon Interactions in Molecular Spring Assemblies of Nanowires and Wavelength-Based Protein Detection. Nat. Mater. 2007, 6, 291−295. (32) Arnold, M. S.; Blackburn, J. L.; Crochet, J. J.; Doorn, S. K.; Duque, J. G.; Mohite, A.; Telg, H. Recent Developments in the Photophysics of Single-Walled Carbon Nanotubes for Their Use As Active and Passive Material Elements in Thin Film Photovoltaics. Phys. Chem. Chem. Phys. 2013, 15, 14896−14918.

ACKNOWLEDGMENTS This work was supported by Los Alamos National Laboratory Directed Research and Development (LDRD) Funds. O.R. acknowledges the support provided by the Center for Integrated Nanotechnologies (CINT), a U.S. Department of Energy, Office of Basic Energy Sciences (OBES) user facility. C.C. also acknowledges the support provided by the Center for Nonlinear Studies (CNLS). We thank Jared Crochet, Han Htoon, and Stephen Doorn for stimulating discussions.

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ADDITIONAL NOTES Quadratic Andreson−Fano Hamiltonian (eq 1) and quadratic Lindblad term (eq 3) identically generate quadratic terms in the equations of motion for Bose operators. In general, the biquadratic relaxation term (eq 4) gives rise to the third- and fourth-order operator averages in the equations of motion for Yn,sp and Xnm, respectively. However, these terms vanish identically for the adopted symmetric form of scattering rate ω|n−m|. b This can be easily demonstrated by looking at the dipole− dipole interaction form for gn and has been numerically validated for small d when the dipole−dipole approximation beaks down. c The Xnm contribution to the radiated power is weak because the system is in the regime of weak coupling. In this case Xnm is a perturbative correction to Yn,sp. This effect is similar to what we observed comparing contributions of Wn,sp associated with Yn,sp and Wnm associated with Xnm. a

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REFERENCES

(1) Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge University Press: New York, 2008. (2) Stockman, M. I. Nanoplasmonics: Past, Present, And Glimpse into Future. Opt. Express 2011, 19, 22029−22106. (3) Chance, R. R.; Prock, A.; Silbey, R. In Advances in Chemistry and Physics; Prigogin, I., Rice, S. A., Eds.; Wiley: New York, 1978; Vol. 37, pp 1−65. (4) Gersten, J.; Nitzan, A. Spectroscopic Properties of Molecules Interacting with Small Dielectric Particles. J. Chem. Phys. 1981, 75, 1139−1152. (5) Prodan, E.; Radloff, C.; Halas, N.; Nordlander, P. A Hybridization Model for the Plasmon Response of Complex Nanostructures. Science 2003, 302, 419−422. (6) Liu, N.; Prall, B. S.; Klimov, V. I. Hybrid Gold/Silica/ Nanocrystal-Quantum-Dot Superstructures: Synthesis and Analysis of Semiconductor-Metal Interactions. J. Am. Chem. Soc. 2006, 128, 15362−15363. (7) Le, F.; Brandl, D. W.; Urzhumov, Y. A.; Wang, H.; Kundu, J.; Halas, N. J.; Aizpurua, J.; Nordlander, P. Metallic Nanoparticle Arrays: A Common Substrate for Both Surface-Enhanced Raman Scattering and Surface-Enhanced Infrared Absorption. ACS Nano 2008, 2, 707− 718. (8) Achermann, M. Exciton- Plasmon Interactions in MetalSemiconductor Nanostructures. J. Phys. Chem. Lett. 2010, 1, 2837− 2843. (9) Govorov, A. O.; Lee, J.; Kotov, N. A. Theory of PlasmonEnhanced Förster Energy Transfer in Optically Excited Semiconductor and Metal Nanoparticles. Phys. Rev. B 2007, 76, 125308. (10) Durach, M.; Rusina, A.; Klimov, V.; Stockman, M. Nanoplasmonic Renormalization and Enhancement of Coulomb Interactions. New J. Phys. 2008, 10, 105011. (11) Miao, X.; Brener, I.; Luk, T. S. Nanocomposite Plasmonic Fluorescence Emitters with Core/Shell Configurations. J. Opt. Soc. Am. B 2010, 27, 1561−1570. 8079



Article

(33) Crochet, J. J.; Duque, J. G.; Werner, J. H.; Lounis, B.; Cognet, L.; Doorn, S. K. Disorder Limited Exciton Transport in Colloidal Single-Wall Carbon Nanotubes. Nano Lett. 2012, 12, 5091−5096. (34) Georgi, C.; Böhmler, M.; Qian, H.; Novotny, L.; Hartschuh, A. Probing exciton Propagation and Quenching in Carbon Nanotubes with near-Field Optical Microscopy. Phys Status Solidi B 2009, 246, 2683−2688. (35) Lieb, M. A.; Zavislan, J. M.; Novotny, L. Single-Molecule Orientations Determined by Direct Emission Pattern Imaging. J. Opt. Soc. Am. B 2004, 21, 1210−1215. (36) Böhmler, M.; Hartmann, N.; Georgi, C.; Hennrich, F.; Green, A. A.; Hersam, M. C.; Hartschuh, A. Enhancing and Redirecting Carbon Nanotube Photoluminescence by an Optical Antenna. Optics Express 2010, 16443−16451. (37) Kubo, R.; Toda, M.; Hashitsume, N. Statistical Physics II: Nonequilibrium Statistical Mechanics; Springer-Verlag: New York, 1985. (38) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford: New York, 1995. (39) Vogel, W.; Welsch, D.-G. Quantum Optics; Wiley-VCH: New York, 2006. (40) Mahan, G. E. Many-Body Physics; Kluwer Academic/Plenum: New York, 2000. (41) Loudon, R. The quantum theory of light, 2nd ed.; Oxford University Press: New York, 1983. (42) Economou, E. N. Green’s functions in Quantum Physics; SpringerVerlag: New York, 1983. (43) Johnson, P. B.; Christy, R.-W. Optical Constants of the Noble Metals. Phys. Rev. B 1972, 6, 4370. (44) Waks, E.; Sridharan, D. Cavity QED Treatment of Interactions between a Metal Nanoparticle and a Dipole Emitter. Phys. Rev. A 2010, 82, 043845. (45) Garcia de Abajo, F. J.; Howie, A. Retarded Field Calculation of Electron Energy Loss in Inhomogeneous Dielectrics. Phys. Rev. B 2002, 65, 115418. (46) Hohenester, U.; Trügler, A. MNPBEM−A Matlab Toolbox for the Simulation of Plasmonic Nanoparticles. Comput. Phys. Commun. 2012, 183, 370−381.

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Effect of Localized Surface-Plasmon Mode on Exciton Transport and

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