Plasmon-Coupled Resonance Energy Transfer - The Journal of

Perspective pubs.acs.org/JPCL

Plasmon-Coupled Resonance Energy Transfer Liang-Yan Hsu,† Wendu Ding,† and George C. Schatz* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States S Supporting Information *

ABSTRACT: In this study, we overview resonance energy transfer between molecules in the presence of plasmonic structures and derive an explicit Förstertype expression for the rate of plasmon-coupled resonance energy transfer (PCRET). The proposed theory is general for energy transfer in the presence of materials with any space-dependent, frequency-dependent, or complex dielectric functions. Furthermore, the theory allows us to develop the concept of a generalized spectral overlap (GSO) J ̃ (the integral of the molecular absorption coefficient, normalized emission spectrum, and the plasmon coupling factor) for understanding the wavelength dependence of PC-RET and to estimate the rate of ̃ PC-RET WET. Indeed, WET = (8.785 × 10−25 mol) ϕDτ−1 D J, where ϕD is donor fluorescence quantum yield and τD is the emission lifetime. Simulations of the GSO for PC-RET show that the most important spectral region for PC-RET is not necessarily near the maximum overlap of donor emission and acceptor absorption. Instead a significant plasmonic contribution can involve a different spectral region from the extinction maximum of the plasmonic structure. This study opens a promising direction for exploring exciton transport in plasmonic nanostructures, with possible applications in spectroscopy, photonics, biosensing, and energy devices. wavelength (R ≪ λ, R is around 2−10 nm), the RET rate described by Förster theory is expressed as62,63

lasmonics is a multidisciplinary field that examines the interaction between electromagnetic fields and conduction electrons in metals and semiconductors. In general, plasmons can be regarded as involving collective oscillations of the conduction electrons, and it is natural to model plasmons using a dispersive and complex dielectric function embedded into the macroscopic Maxwell equations. During the past few decades, due to great advances in experimental techniques and computational modeling, plasmonic effects associated with surface plasmon polaritons (SPPs), localized surface plasmons (LSPs), and delocalized lattice plasmons (DLPs), have stimulated broad applications in spectroscopy1 (e.g., surfaceenhanced Raman spectroscopy2−5), photonics6 (e.g., nanoscale lasers,7−9 metamaterials10,11), energy devices12 (e.g., plasmonic solar cells13−15), and sensing16,17 (e.g., plasmon-enhanced fluorescence,18−20 SERS, and other techniques). These studies involve a variety of photophysical processes, including absorption, extinction, Rayleigh and Raman scattering, electron transfer, and nonlinear processes. One promising direction that we examine in this Perspective is resonance energy transfer between molecules in the presence of plasmonic materials.21−45 Resonance energy transfer (RET), including radiative and radiativeless mechanisms, is a ubiquitous photophysical process. RET has attracted considerable attention due to its promising applications in photosynthesis46−49 (e.g., light harvesting), photovoltaics50−52 (e.g., exciton diffusion in organic or inorganic photovoltaic cells, energy conversion), biomolecular structure and dynamics,53−55 single-molecule experiments,56−58 biosensing,59,60 and chemical sensing.61 Typically, for an intermolecular distance much smaller than the excitation

P

© 2017 American Chemical Society

WET =

ΦD 9000 ln 10κ 2 τD 128π 5NAnr4R6

∫

dν ̅

ϵ(ν ̅ )I(ν ̅ ) ν̅ 4

(1)

where ν̅ is wavenumber, ΦD is donor fluorescence quantum yield, τD is the emission lifetime of the donor, nr is the index of refraction of the host medium, NA is Avogadro’s number, κ is an orientation factor, and the spectral overlap J = ∫ dν̅ ν̅ −4 ϵ(ν̅) I(ν)̅ is associated with the molecular absorption coefficient of the acceptor ϵ(ν)̅ and the normalized emission spectrum of the donor I(ν̅) . Equation 1 provides a simple but useful physical picture for many complicated donor−acceptor RET systems. Nevertheless, Förster theory has several critical limitations. (I) When the donor and acceptor molecules are very close within several angstroms, electron transfer may compete with energy transfer. Also, for energy transfer, the point-dipole approximation becomes invalid. These two issues have been addressed by Dexter theory 47,64−68 and by using the transition density69−71 to calculate Coulomb matrix elements, respectively. (II) When the donor and acceptor molecules are far away from each other (R ≈ λ or R > λ), the radiative mechanism is not included in Förster theory. This issue can be addressed by a formulation based on quantum electrodynamics.72−74 (III) Förster theory is based on the weak electronic coupling limit (Fermi’s golden rule). In this limit, the Received: March 3, 2017 Accepted: May 3, 2017 Published: May 3, 2017 2357

DOI: 10.1021/acs.jpclett.7b00526 J. Phys. Chem. Lett. 2017, 8, 2357−2367

The Journal of Physical Chemistry Letters

Perspective

RET is incoherent (Markovian) and irreversible.63 Studies beyond the weak coupling limit75−77 or including coherent mechanisms78−84 have been extensively investigated especially recently. (IV) Förster theory only considers RET between a pair of donor and acceptor molecules; however, several studies have extended the theory to include RET among multiple donors and acceptors such as multichromophoric Förster resonance energy transfer.82,85−88 (V) When the index of refraction nr is a space-dependent (inhomogeneous medium), frequency-dependent (dispersive medium), or complex function (absorbing medium), eq 1 has to be modified. Indeed, concerning this last point, we note that since nr is the square root of the dielectric function, it is immediately apparent that eq 1 cannot be used to describe RET between donor and acceptor molecules in the presence of plasmonic structures. In fact, RET near surfaces and small particles has been studied since the 1980s89−92 and has motivated several theoretical studies.30,32−35 Recently RET involving donor−acceptor pairs (separated by double-stranded DNA linkers, streptavidin, or polyelectrolyte) in the presence of plasmonic structures including nanospheres, dimer nanoantennas, and nanoapertures has attracted extensive attention.37−42 In this study, we will focus on points (II) and (V) above, with the goal of providing a simple and useful picture of plasmon-coupled resonance energy transfer (PC-RET). Through the years, by using computational chemistry techniques or quantum electrodynamics, several theories have been developed to describe RET in inhomogeneous media93,94 or in dispersive media,95 but it is nontrivial to formulate a theory of RET in media that have inhomogeneous, dispersive, and absorbing properties together. To take all these properties into account, Juzeliunas and Andrews started from quantum ̅ electrodynamics and developed a theory that successfully describes both the averaged and local fields in RET.96−99 However, this theory is limited to periodic media, and the lattice constant in the periodic media has to be much smaller than the distance between donor and acceptor molecules. As a result, RET in the presence of plasmonic structures cannot be captured by this method. After this study, several pioneering works successfully treated quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,100−103 and Dung et al. derived a transition amplitude for RET in terms of dyadic Green’s functions.92 Nevertheless, for arbitrary inhomogeneous, dispersive, and absorbing environments, it is challenging to determine dyadic Green’s functions either analytically or numerically. To circumvent these issues, we here present a practical theory for PC-RET and show that (i) the complete expression for the rate of PC-RET can be reduced to the Förster expression (eq 1) as extensively used in interpreting experiments, (ii) the rate of PC-RET is dominated by a generalized spectral overlap (GSO), which describes the contribution of the molecular absorption coefficient, the normalized emission spectrum, and an electromagnetic coupling factor (the coupling between an acceptor molecule and the electric field generated by a donor molecule) that contains plasmon enhancement, (iii) the coupling factor in the frequency domain is a type of two-point near-field enhancement for plasmonic systems that is very different from the absorption or extinction spectrum of the metal nanoparticles, and (iv) the PC-RET method provides a framework for simulation that has important computational advantages compared to calculating the coupling factor using dyadic Green’s functions.

The proposed theory provides a framework for simulation that has important computational advantages compared to calculating the coupling factor using dyadic Green’s functions. We focus on PC-RET between a donor molecule at position rD and an acceptor molecule at rA in the presence of timeinvariant medium, i.e., displacement field D(r, ω) = ϵr(r, ω)ϵ0 E(r, ω), with a complex dielectric function ϵr(r, ω) and relative permeability μr = 1. The total Hamiltonian can be separated into three parts, i.e., Htot = Hmol + HEM + Hint, where Hmol, HEM, and Hint stand for molecular Hamiltonian, electromagnetic field Hamiltonian, and their interactions, respectively. We consider Coulomb gauge and minimal coupling,92 and then study PCRET in the framework of quantum electrodynamics because quantum electrodynamics can adequately describe the shortand long-range asymptotes in a unified theory.72,74,104 In this framework, HEM, the energy of the medium-assisted electromagnetic field, is expressed in terms of bosonic vector fields.92 Furthermore, one can make the electric-dipole approximation for Hint, use Fermi’s golden rule, adopt the Condon approximation for Hmol, and finally derive the rate of molecular RET as WET =

2π ℏ

∑ ∑ ∫ dEPd ′Pa |⟨ϕa ′|ϕa⟩|2 |⟨ϕd|ϕd ′⟩|2 d ,d′ a,a′

|M(E , rD, rA)|2 δ(Ed ′ − Ed − E)δ(Ea ′ − Ea − E),

(2)

where d′ and a′ (d and a) denote the vibrational states (a collection of normal modes) associated with the electronic excited (ground) state of donor and acceptor molecules, respectively; |⟨ϕd|ϕd′⟩|2 (|⟨ϕa′|ϕa⟩|2) stands for the Franck− Condon factor of a donor (acceptor) molecule; Pd′ (Pa) corresponds to the probability distribution of donor (acceptor) molecules at thermal equilibrium; M(E, rD, rA) is the transition amplitude, which depends on the donor and acceptor positions, and the photon energy E. Strictly speaking, photons here are dressed photons (or plasmon polaritons) because they come from quantization of macroscopic Maxwell’s equations, which include ϵr(r, ω).92 For simplicity, we use the term photon instead of dressed photon. In a previous study,105 we derived a relationship between electric fields generated by a donor and transition amplitudes from quantum electrodynamics, and showed that the transition amplitude can be written as M(E , rD, rA) = −μD μ A

eA ·ED(rA , E) pex (E)

(3)

= −μ A · E͠ D(rA , E)

(4)

which expresses the transition amplitude as the inner product of the transition dipole of an acceptor molecule μA (μA can be decomposed into its magnitude μA and direction eA), and the field at the acceptor location induced by the donor molecule E͠ D(r , E) = μD ED(r , E)/p (E). The scaling by p (E) A

A

ex

ex

originates from the electric field ED(rA, E) generated by a classical source dipole pex(E) (not transition dipole) from 2358



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associated with the emission spectrum η(ω) (WDr = ∫ η(ω) dω, where η(ω) = (4αeω3/3ℏc3)∑d,d′Pd′|μD|2|⟨ϕd|ϕd′⟩|2δ(ωd′ − ωd − ω) is the emission spectrum of the donor molecule without plasmonic media). Equation 6 accounts for plasmonic effects caused by media via the coupling factor, so WDr and η(ω) are not related to the fluorescence rate in the vicinity of plasmonic structures.110 From this we see that an advantage of our theory is that one can compute the rate of PC-RET based on emission spectra and absorption spectra from a spectral database of chemical compounds in media with a constant ϵr. Note that the absorption cross section and the normalized emission spectrum in the absence of plasmonic media (in Gaussian units) correspond to65,67

classical electrodynamics. Equation 4 clearly shows that the transition amplitude of RET is associated with the potential energy of an electric dipole, but the difference is that μA is the acceptor transition dipole and E͠ D(r , E) is the electric field A

induced by the donor transition dipole. Note that the transition dipoles arise from the quantum mechanical theory, which underlies the approach. Here we remark that, in practice, eq 3 is more convenient to use in our development, although eq 4 provides a clearer physical picture. From eq 3, the transition amplitude reaches a maximum (a minimum) when eA and ED(rA, E) are parallel (perpendicular). Equation 3 provides a significant computational advantage because one can compute the transition amplitude using time-domain electrodynamics methods105 instead of directly solving for the dyadic Green’s functions of a whole system. Substituting eq 3 into eq 2, one can derive the rate of PC-RET in terms of the absorption line shape of the acceptor molecule Wabs(E) and the emission line shape of the donor molecule Wem(E), WET =

where

ℏ 2π

∫

D

dEWem(E)Wabs(E)

eA ·E (rA , E) pex (E)

σ(ω) =

2

(5)

=

Wem(E) = (2π /ℏ) ∑ Pd ′|μD |2 |⟨ϕd|ϕd ′⟩|2 δ(Ed ′ − Ed − E) d ,d′

Equation 5 expresses that the rate of PC-RET is associated with the overlap of the absorption line shape of the acceptor, the emission line shape of the donor, and a coupling factor |eA·ED(rA, E)/pex(E)|2. We will refer to this overlap as the generalized spectral overlap (GSO), as it is a unique expression compared to the overlap of absorption and emission lineshapes that appears in Förster theory. The coupling factor can be also regarded as an orientation factor because the coupling depends on the relative orientation of eA and ED(rA, E) . However, it is the dependence of the terms in the GSO on frequency (or wavelength) that plays a unique role from our perspective, as the electromagnetic coupling factor can have strong dependence on frequency for applications to plasmonic systems that can significantly modify the important frequencies contributing to energy transfer compared to the conventional Förster donor/acceptor overlap. Note that the transition dipoles and vibrational frequency of the line shape functions (Wabs(E) and Wem(E)) can be theoretically computed from computational chemistry packages, Franck−Condon factors can be derived from previous literature,107−109 while the coupling factor (including eA and ED(rA, E)) can be derived from computational electrodynamics packages. By using E = ℏω and δ(ℏω) = δ(ω)/ℏ, eq 5 can be written as an expression consisting of the absorption cross section σ(ω) and the normalized emission spectra I(ω) in Gaussian units, WET =

9c 8π

∫

D

dω

a,a′

(sec)

4αeω3 3ℏc 3WrD

∑ Pd ′|μD|2 |⟨ϕd|ϕd ′⟩|2 δ(ωd ′ − ωd − ω) d ,d′

where αa and αe are the real-valued refractive indices65,67 (for the averaged fields caused by solvent) introduced to correct the absorption cross section and the normalized emission spectrum, respectively. In order to compare with the rate of RET shown by previous studies,62,65,67 we do not include local field effects due to the solvent in σ(ω) and I(ω) . The discussion of local field effects on the absorption and emission processes can be found in the literature.111 To suppress the influence of solvent on the spectra of molecules, typically the solvent is chosen to be a medium with a real dielectric constant in the frequency region which we focus on. Also αa and αe are canceled in eq 6 because αa ≈ αe when σ(ω) and I(ω) are measured in the same type of solvent. The coupling factor in eq 6 has a relationship with the dyadic Green’s function g (r, r′, ω) via105

a,a′

WrD

(cm 2),

2

Wabs(E) = (2π /ℏ) ∑ Pa|μ | |⟨ϕa ′|ϕa⟩| δ(Ea ′ − Ea − E),

4

∑ Pa|μ A |2 |⟨ϕa ′|ϕa⟩|2 δ(ωa ′ − ωa − ω)

η(ω) I(ω) = WrD

106

A2

4π 2ω 3ℏαac

σ(ω)I(ω) eA ·E (rA , ω) pex (ω) ω4

eA ·ED(rA, ω) pex (ω)

=−

ω2 e ·g (rA , rD, c 2 ϵ0 A

=−

4πω2 e ·g (rA , rD, c2 A

ω) ·eD

(SI Units),

ω) ·eD (CGS Units)

(7)

where g (r, r′, ω) satisfies ⎞ ⎛ ϵ (r, ω)ω 2 ⎜ r 2 − ∇ × ∇ × ⎟g (r, r′, ω) = δ(r − r′) c ⎠ ⎝

Equation 7 indicates that computing ED(rA, ω) and pex(ω) is equivalent to solving the dyadic Green’s function g (r, r′, ω). Experimentally, the spectral overlap given by Förster theory is usually expressed in terms of the molecular absorption coefficient and the normalized emission spectrum in wavenumber units ν̅ = (2πc)−1ω (recall eq 1). By using the following relations,

2

(6)

where c is the speed of light in vacuum, ω is the (angular) frequency of light, and WDr is the total emission rate of the donor molecule in the absence of plasmonic media; it can be regarded as a frequency-independent normalization constant 2359


The Journal of Physical Chemistry Letters I(ω) = WrD =

Perspective

I (ν ̅ ) , 2πc ϕD τD

The rate of PC-RET is determined by the overlap of three factors: molecular absorption coefficient, normalized emission spectrum, and coupling factor.

,

NA σ(ν ̅ ) (mol−1cm 2) ln 10 N = 3 A σ(ν ̅ ) (M−1cm−1) 10 ln 10

ϵ(ν ̅ ) =

noteworthy: (i) eq 6 and eq 8 can be applied to RET in any kind of media with complex, space-dependent, and frequencydependent dielectric functions, not limited to plasmonic materials; (ii) eq 6 and eq 8 are in Gaussian units; (iii) I(ν̅) is a normalized function so it has to satisfy ∫ I(ω) dω = ∫ I(ν)̅ dν̅ = 1 (i.e., I(ν)̅ = 2πcI(ω)), and the unit of I(ν)̅ is cm; (iv) the absorption cross section σ(ν)̅ (in cm2) and molar absorption coefficient ϵ(ν̅) (in cm−1 M−1) can be experimentally derived by the Beer−Lambert law; and (v) units for the coupling factor and GSO are cm−6 and mol−1, respectively. Equation 8 can be reduced to the famous Förster result (recall eq 1). Consider the Hertzian (ideal) dipole with the wavenumber νd̅ ′d = ν̅d′ − ν̅d as a source, i.e., pex(ν)̅ = (2πc)−1p0δ(ν̅d′d − ν), ̅ in dielectrics with permittivity ϵr, where p0 is the amplitude of the Hertzian (ideal) dipole and ϵr is a real number, it is a well-known result that ED(rA,ν̅) = (2πc)−1ϵ−1 r R−3(3eR(eR·eD)−eD)p0δ(ν̅d′d − ν)̅ in the near-field regime.112 Substituting pex(ν)̅ and ED(rA,ν)̅ into the coupling factor, one can derive the coupling factor as follows:

one can derive WET = =

ϕD 9000 ln 10 τD 128π 5NA ϕD τD

∫ dν ̅ ϵ(νν̅)I4(ν ̅) ̅

eA ·ED(rA, ν ̅ ) pex (ν ̅ )

(8.785 × 10−25mol × J ̃)

J ̃ ≡ ∫ dν ̅

ϵ(ν ̅ )I(ν ̅ ) ν̅ 4

eA ·ED(rA, ν ̅ ) pex (ν ̅ )

2

(mol−1)

2

(8) (9)

Equation 8 clearly shows that the rate of PC-RET is determined by the overlap of three factors: molecular absorption coefficient, normalized emission spectrum, and coupling factor (the relative orientation of an acceptor and the electric field generated by a donor). The integral of the three factors, i.e., eq 9, is the “generalized spectral overlap (GSO)” mentioned earlier. Evidently, one can optimize the rate of PC-RET by designing a system (including molecules and plasmonic structures) where the three factors have a maximal overlap. On the other hand, when one of the three factors does not overlap with the others, the rate of PC-RET will be negligibly small. In practice, the first two terms in the overlap can be directly obtained from experiment, while the third term can be computed from computational electrodynamics packages. In other words, one can predict the magnitude of PC-RET via the experimental spectrum in the absence of plasmonic structures together with electrodynamics calculations of the coupling factor. Incidentally, a previous study based on classical electrodynamics31 indicates that the energy transfer rate is proportional to the square amplitude of the donor’s electric field at the position of the acceptor (i.e., coupling factor). While this is similar to our eq 9, our theory based on quantum electrodynamics together with electronic and vibrational states of the donor and acceptor is such that the energy transfer rate is proportional to the GSO. Several additional remarks concerning eq 6 and eq 8 are

eA ·ED(rA , ν ̅ ) pex (ν ̅ )

2

=

(eA ·eD − 3(eA ·eR )(eR ·eD))2 ϵ2r R6

≡

κ2 ϵ2r R6 (10)

Obviously, the coupling factor in Förster theory is independent of frequency, so it can be taken out of the integral in eq 9. Furthermore, substituting eq 10 into eq 8, one can derive exactly the same result as in Förster theory. Alternatively, substituting eq 10 into eq 6, one can obtain the rate of RET in terms of the absorption cross section, which is also exactly the same as that in the previous literature.65,67,113 The comparison between our theory and Förster resonance energy transfer (FRET) is summarized in Table 1. The GSO is a useful concept for understanding PC-RET. To demonstrate this, we study RET between two molecules, specifically from 7-methoxycoumarin-4-acetic acid (MCA) to Coumarin 6 (C6), in the presence of a solid silver spherical nanoparticle with 120 nm diameter. The structures and

Table 1. Comparison between Theories of PC-RET and FRET

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Figure 1. Left: Structures of 7-methoxycoumarin-4-acetic acid (MCA) and Coumarin 6 (C6). Right: Absorption spectra (red lines) and normalized emission spectra (blue lines) of MCA114−116 (solid lines) and C6115−118 (dashed lines). The shaded area marks the spectral overlap between the two molecules that is important in the framework of conventional Förster theory.

Figure 2. Geometry setup for modeling energy transfer between donor MCA and acceptor C6 in the presence of a nanoparticle (NP), forming the MCA-NP-C6 system. The transition dipoles of the donor and the acceptor (red arrows) are parallel to each other in each case. They are oriented either in the z direction as System Z (a) or in the x direction as System X (b).

Figure 3. Coupling factors between MCA and C6 with dipole orientation the same as System Z (a) and System X (b) in Figure 2 determined from three different methods: Förster theory for MCA-C6 in vacuum (black), QED for MCA-C6 in vacuum (red), and TED method for MCA-NP-C6 (blue).

experimental optical spectra114−118 of the two molecules are shown in Figure 1. Clearly, there is significant spectral overlap (the shaded area in Figure 1) near 400−450 nm between the normalized emission spectrum of the donor MCA114−116,119 (blue solid line) and the absorption spectrum of the acceptor C6115−118,120 (red dashed line). As suggested by Förster theory, this indicates possible energy transfer between the two

molecules in a homogeneous environment if the alignment between them is suitable and the molecules are close enough together. Now consider the donor−nanoparticle−acceptor (MCA-NPC6) system, in which the two molecules are placed on opposite sides of a solid silver sphere with 120 nm diameter, as shown in Figure 2. A separation of 5 nm is inserted between each 2361



Perspective

Figure 4. (a) Terms in the GSO, including C6 absorption (black), MCA emission (red), and coupling factor spectra for System Z (blue) and System X (orange) calculated with TED. (b) Spectral overlap of C6 absorption (black), MCA emission (red), and Ag 120 nm sphere extinction spectra (green).

Figure 5. GSO integrand of System Z (a) and System X (b) for MCA-NP-C6 (blue curves) and for MCA-C6 in vacuum (red curves). In panel b, the GSO integrands of MCA-C6 in vacuum are multiplied by 50 to be visible. PEF for System Z (c) and System X (d) are calculated by taking the ratio of MCA-NP-C6 to MCA-C6 in vacuum in panels a and b (i.e., blue curve divided by red curve), respectively.

Johnson and Christy.121 We note that the dielectric function based on Johnson and Christy has been demonstrated to provide a good description of correlated LSPR-HRTEM measurements for silver nanocubes,122 so it should be adequate in the present case. The real part of this dielectric function is negative for wavelengths longer than 325 nm, leading to plasmon excitation especially in the range 350−500 nm, as will be discussed. The coupling factors between MCA and C6, calculated by three different methods under different conditions but all with

molecule and the surface of the sphere, to avoid any possible electron transfer between them. The transition dipoles of the donor and acceptor are taken to be parallel to each other (the red arrows in Figure 2). Two orientations are chosen as examples: one with the transition dipoles being tangential to the surface of the nanoparticle (Figure 2a, namely System Z), and the other with the dipoles being perpendicular to the surface (Figure 2b, namely System X). The 120 nm diameter of the sphere is large enough for one to use the bulk silver dielectric function, which in this case is taken from the work of 2362



Perspective D 2 −1 i.e., ν̅ −4ϵ(ν)̅ I(ν)|e ̅ A·E (rA,ν)p ̅ ex (ν)| ̅ , as shown in Figure 5a,b. The rate of RET is proportional to the area under the blue (red) curves for MCA-NP-C6 (MCA-C6). Roughness of these curves originates from the experimental MCA emission and C6 absorption spectra. For both System Z and System X, the GSO integrands for MCA-NP-C6 are mostly larger than those of MCA-C6. A plasmon enhancement factor (PEF) (see Figure 5c,d) can be defined as the ratio of the GSO integrands for MCA-NP-C6 to those of MCA-C6. Moreover, according to eq 8, one can easily show that the PEF is exactly the same as the ratio of the coupling factor for MCA-NP-C6 to that for MCAC6. Thus, we see that Systems Z and X have PEF maxima at 350 and 355 nm, respectively. In addition, the GSO integrand spectrum of MCA-NP-C6 in System Z and the corresponding PEF have a similar maxima. On the other hand, for System X, the positions of the GSO integrand and PEF maximima are distinct due to the significant tail in the coupling factor (recall the blue line in Figure 3b). Also for System X, the small peak at 365 nm in the blue curve in Figure 5b results from the maximum PEF and the small emission-absorption overlap. Figure 5 clearly shows the importance of the GSO because neither the GSO integrand spectrum nor the PEF spectrum agrees with the extinction spectrum of the Ag sphere. Traditional Förster theory has several critical limitations, and the treatment of RET mediated by plasmonic structures is especially challenging. In this study, we proposed a rigorous fully retarded quantum electrodynamic method that does not require calculating dyadic Green’s functions, and which generalizes Fö rster theory to the inclusion of arbitrary nanostructures whose dielectric functions can be both complex and dispersive. Table 1 presents generalizations of standard Förster theory rate expressions to the new approach. A key new concept in this paper is the introduction of the generalized spectral overlap (GSO) for understanding the rate of PC-RET in terms of the donor emission spectrum, the acceptor absorption spectrum, and an electromagnetic coupling factor that involves the emitted dipole field evaluated at the acceptor position. This electromagnetic coupling factor involves near-field properties evaluated at two positions, so in applications to plasmon enhanced energy transfer, the enhancement has a very different dependence on wavelength than are familiar for far-field properties such as the extinction spectrum. These expressions for transition amplitude and generalized spectral overlaps are important conceptual advances for understanding energy transfer involving heterogeneous nanostructures. While the donor−acceptor distance considered in Figure 2 is much larger than is normally considered in energy transfer studies without plasmonic particles, energy transfer is greatly enhanced by plasmon excitation, such that for some applications, PC-RET will be an important process. Moreover, our theory is not limited to PC-RET; it can be applied to any type of material with space-dependent, frequency-dependent, or complex dielectric functions.

the same donor−acceptor distance, are shown in Figure 3. A significant difference is found in the comparison between coupling factors derived from Förster theory (eq 10) and quantum electrodynamics (QED)72,74 for MCA-C6 in vacuum with dipole orientation taken as either System Z or X. This is not surprising, given that the donor−acceptor distance is 130 nm (120 + 5 + 5 = 130 nm), which is much longer than the valid range of Förster theory. Indeed, at this donor−acceptor separation, the decay rate of the distance-dependent coupling factor is no longer dominated by the R−6 term, but rather includes the influence of R−4 and R−2.72,74 As a result, the coupling factors derived from QED for MCA-C6 in vacuum (the red lines) are larger than those from Förster theory (the black lines). In addition, unlike Förster theory, the coupling factors from QED are frequency dependent, leading to a nonzero slope in the plots. Next, we focus on a comparison between the coupling factors derived from the time-domain electrodynamics (TED) method105 for the MCA-NP-C6 system and from QED for MCA-C6 in vacuum. How to calculate the coupling factor (pex(ω) and ED (rA, ω)) using the TED method can be found in a previous study,105 and the numerical details of parameters and settings are listed in the Supporting Information. The computational results show that plasmon enhancement in the systems with the Ag sphere is significant but not at all wavelengths where plasmon excitation in the nanoparticle can occur. In fact, for System Z (Figure 3a), only a small portion of the wavelength range from 300 to 600 nm has a coupling factor for MCA-NP-C6 larger than that of MCA-C6. The largest enhancement occurs around 350 nm wavelength (or 28,571 cm−1), which is close to the quadrupole peak (358 nm) in the extinction spectrum of the 120 nm silver sphere (see green line in Figure 4b). However, the coupling factor shows a small enhancement or even suppression in the range of the dipole peak (413 nm) in the extinction spectrum of the Ag nanosphere. For System X (Figure 3b), plasmon enhancement is significant over a wide range of wavelengths, yet there is suppression below 340 nm where plasmon excitation is weak. Similar to System Z, the largest enhancement is close to the Ag sphere quadrupole extinction peak. Unlike the previous case, large enhancement also exists close to the dipole extinction peak, and the coupling is 3 orders of magnitude larger in System X at 450 nm than in System Z, as discussed later in Figure 5. Recently, in order to explore plamon-enhanced effects, the concept of the overlap of the extinction spectrum of nanoparticles and molecular optical spectra has been employed to investigate plasmon enhanced fluorescence123,124 and resonance energy transfer.23−25,125,126 A remarkable aspect of the results in Figure 4a is that the electromagnetic coupling factor is fundamentally different from extinction of the plasmonic nanoparticle, which we show in Figure 4b. Indeed, although there is a large overlap between C6 absorption and MCA emission spectra in the range of 380−450 nm, the coupling factor for System Z (orange line) is small in this region. In other words, the coverage of the coupling factor spectrum misses the spectral overlap of C6 and MCA. This also contrasts with the behavior of System X (blue line), where there is a noticeable tail inside the overlap region of C6 absorption and MCA emission. To further explore the consequences of plasmon enhancement on the GSO, we calculate the GSO integrand (the wavenumber-dependent function inside the integral of eq 9),

Our theory is not limited to PCRET; it can be applied to any type of material with space-dependent, frequency-dependent, or complex dielectric functions.

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PC-RET is an exciting and promising field. Several studies have shown the influence of photonic mode density and plasmon effects on resonance energy transfer36−45 and explored the potential applications of PC-RET in plasmonic nanowaveguides,22,29 molecular imaging,23 nanospectroscopy,24 and solar energy conversion.25,26 Indeed our approach is promising for applications to plasmon-coupled exciton transport in metal organic framework materials (MOFs),127 chromophore aggregates, and DNA-linked nanoparticle superlattices.128−130 The theory has been developed in the framework of Fermi’s golden rule (weak electronic coupling limit and incoherent energy transfer) and does not consider the exchange-induced (Dextertype) RET or generalizations beyond the point dipole approximation. We hope that this paper will motivate further experimental and theoretical investigations into plasmoncoupled resonance energy transfer to explore novel device applications in nanotechnology.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b00526. The details of calculation of coupling factors using the time-domain electrodynamics method. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

George C. Schatz: 0000-0001-5837-4740 Author Contributions †

Contributed equally to this work

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. This work was supported by the U.S. National Science Foundation under Grant Number CHE-1465045.

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