Characteristic Distance of Resonance Energy Transfer Coupled with

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Energy Conversion and Storage; Plasmonics and Optoelectronics

Characteristic Distance of Resonance Energy Transfer Coupled with Surface Plasmon Polaritons Jhih-Sheng Wu, Yen-Cheng Lin, Yae-Lin Sheu, and Liang-Yan Hsu J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b03429 • Publication Date (Web): 29 Nov 2018 Downloaded from http://pubs.acs.org on December 4, 2018

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The Journal of Physical Chemistry Letters

Characteristic Distance of Resonance Energy Transfer Coupled with Surface Plasmon Polaritons Jhih-Sheng Wu,† Yen-Cheng Lin,‡ Yae-Lin Sheu,‡ and Liang-Yan Hsu∗,‡ †Center for Nano-Optics (CeNO) and Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA ‡Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan E-mail: [email protected]

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Abstract We investigate resonance energy transfer (RET) between a donor-acceptor pair above a gold surface (including bulk and thin-film systems) and explore the distance/frequency dependence of RET enhancements using the theory we developed previously. The mechanism of RET above a gold surface can be attributed to the effects of mirror dipoles, surface plasmon polaritons (SPPs), and retardation. To clarify these effects on RET, we analyze the enhancements of RET by the mirror method, the decomposition of s- and p-polarization, and the SPP dispersion of charge-symmetric and charge-antisymmetric modes. We find a characteristic distance (approximately one tenth of the wavelength) which can be used to classify the dominant effect on RET. Moreover, the characteristic distance can be shortened by narrowing the thickness of the thin-film systems, indicating that SPPs can enhance the rate of RET at a short range. The charge-symmetric and charge-antisymmetric modes of the thin films also allow us to engineer the maximum RET enhancement. We hope that our analysis inspires further investigation into the mechanism of RET coupled with SPPs and its applications. Graphical TOC Entry

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Keywords Exciton, Spectroscopy, Energy Conversion, Thin Film, Quantum Electrodynamics, Plasmonics

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Resonance energy transfer (RET), also known as excitation energy transfer, plays a critical role in organic photovoltaics, 1,2 photosynthesis, 3,4 and imaging applications. 5,6 Since 1970s, the study of energy transfer assisted by an electromagnetic environment has attracted wide attention in the area of physical chemistry. Drexhage, Kuhn, Chance, and Silbey et al. considered energy transfer from an excited molecule to a metal and demonstrated that the fluorescence lifetime oscillated as a function of the molecule-metal distance. 7–11 Gersten and Nitzan proposed a pioneering theory of energy transfer between molecules near a solid particle. 12–14 Recently, due to advances in nanotechnology, RET between two chromophores in nanostructures as well as cavities attracts considerable interest. 15–31 Several experimental studies have observed RET enhanced by nanostructures, 21,27 the competition between RET and spontaneous emission, 28 and long-range RET on a metal surface. 32 In spite of the fact that these phenomena are thought to be related to surface plasmons or surface plasmon polaritons (SPPs), the analysis of RET coupled with SPPs is still quite lacking. To better understand the effect of SPPs on RET, we investigate RET between two chromophores above a gold surface (the most representative systems in plasmonics), study the distance and the frequency dependence of RET, and explore the origin of plasmonic enhancement. RET and its theory have been studied for more than 70 years. 33–44 It is well-known that RET is dominated by a non-radiative process at a short range (the distance between two chromophores is shorter than 10 nm) while by a radiative process at a long range. 39 The former has been extensively investigated by using Förster theory, 33 and the latter can be modeled as radiative dipoles in classical electrodynamics. 45 Several pioneering studies have shown that the two processes can be unified as a single theory in the framework of quantum electrodynamics. 46–49 However, this single theory cannot be used to investigate RET in complicated dielectric environments, i.e., inhomogeneous, dispersive, and lossy dielectrics. To address this issue, in the spirit of the form of Förster theory combined with macroscopic quantum electrodynamics, a general theory of RET is established. 30,31 This theory includes the retardation effect, enables the description of SPPs, and allows to develop the concept of a

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generalized spectral overlap as well as a coupling factor for understanding the wavelength dependence of RET in complicated dielectric environments. Furthermore, the coupling factor can be expressed as either the form of electric fields or the form of dyadic Green’s functions, which facilitates the analysis of the mechanisms of RET coupled to inhomogeneous, dispersive, and lossy dielectrics.

Figure 1: Resonance energy transfer between two chromophores (donor and acceptor) above a gold surface with SiO2 substrates. Two chromophores are separated by the distance R at the height h from the gold film with thickness d (for the bulk gold system, d = ∞). The symbols 0 , 1 , and 2 represent the dielectric functions of vacuum, gold, and SiO2 , respectively. (a) and (b) correspond to System ZZ and System XX, respectively. In System ZZ (XX), the transition dipoles of two chromophores are normal (parallel) to the gold surface. In the present Letter, we focus on RET between two chromophores above gold surfaces, including bulk gold and gold thin films, as shown in Figure 1. In System ZZ, both the transition dipoles of the chromophores are oriented in the z direction; in System XX, both the transition dipoles are oriented in the x direction. To consider the effects of retardation and metal surfaces on RET, we adopt the general theory of RET in the previous studies. 29–31 This theory states that the rate of RET between two chromophores, a donor at rD = (xD , yD , zD ) and an acceptor at rA = (xA , yA , zA ), in a dielectric environment can be expressed as 30

WET

9c4 WrD = 8π

Z

∞

dω 0

σ(ω)I(ω) F (rD , rA , ω), ω4

5


(1)


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where c is the speed of light in vacuum, WrD is the total emission rate of energy in the absence of the dielectric environment, and ω is the angular frequency of light. σ(ω) is the absorption cross section of the acceptor. I(ω) denotes normalized emission spectra, with R R∞ dωI(ω) = 1. The integral 0 dωω −4 σ(ω)I(ω)F (rD , rA , ω) is so-called generalized spectral overlap. 30 The coupling of the two chromophores, including non-radiative and radiative processes, is described by the coupling factor F (rD , rA , ω) 30 2 n ˆ · E (r , r , ω) A D A D , F (rD , rA , ω) = pex

(2)

where ED (rA , rD , ω) is the electric field at rA generated by the transition dipole of the donor pex n ˆD, n ˆ D(A) is the unit vector of the transition dipole of the donor (acceptor), and pex is the amplitude of the transition dipole of the donor. In the environment with the dielectric function r (r, ω), one can obtain ED (rA , rD , ω) from the dyadic Green’s function g¯(rA , rD , ω) in Gaussian units,

ED (rA , rD , ω) = 4πk02 g¯(rA , rD , ω)pex n ˆD,

(3)

where k0 = ω/c and g¯(rA , rD , ω) satisfy r (r, ω)k02 − ∇ × ∇× g¯(rA , rD , ω) = −δ(rA − rD ).

(4)

We consider RET above gold surfaces (two systems as shown in Figure 1). The dielectric functions r (r, ω) of the media are modeled as

r (r, ω) =

    0 ,    

0 < z,

1 (ω),       2 (ω),

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− d < z < 0, z < −d,


(5)

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where 0 , 1 (ω) and 2 (ω) correspond to the dielectric functions of vacuum, gold, and the substrate (SiO2 ), respectively. The dielectric function of gold is adopted from the work of Johnson and Christy; 50 the dielectric function of SiO2 is adopted from the work of Ghosh et al. 51 To ensure that our main results are robust against the choice of gold dielectric functions, we compare the coupling factors using different experimental gold dielectric functions and demonstrate their similarity in Figure S1 in the Supporting Information (SI). The dyadic Green’s function g¯(rA , rD , ω) for two chromophores above the gold thin film is obtained by the integration in the reciprocal space, and it can be written in terms of the free-space dyadic Green’s function g¯0 (rA , rD , ω), the s-polarized dyadic Green’s function g¯s (rA , rD , ω) and the p-polarized dyadic Green’s function g¯p (rA , rD , ω) as follows, 52,53

g¯(rA , rD , ω) = g¯0 (rA , rD , ω) + g¯p (rA , rD , ω) + g¯s (rA , rD , ω).

(6)

g¯0 (rA , rD , ω) is given by 45

g¯0 (rA , rD , ω) =

o eik0 R n 2 −2 −1 ˆ ˆ ˆ ˆ I − R ⊗ R k + 3 R ⊗ R − I R − ik R , 3 0 3 0 4πk02 R

(7)

ˆ = R/R, I3 is a 3 × 3 identity matrix, and ⊗ denotes the outer where R = rA − rD , R product; g¯s (rA , rD , ω) and g¯p (rA , rD , ω) are given by

g¯p (rA , rD , ω) =

Z

∞

0

g¯s (rA , rD , ω) =

Z 0

∞

idq ¯ (q, ω, ρ)eiKz,0 (zD +zA ) , Rp (q, ω)M p 4π idq ¯ (q, ω, ρ)eiKz,0 (zD +zA ) , Rs (q, ω)M s 4π

where ρ = (xA − xD , yA − yD , 0), Kz,i =

(8) (9)

p

i k02 − q 2 correspond to the z-component wavevec-

tors in the media with the dielectric functions i (i = 0, 1, and 2), and q is the in-plane ¯ (q, ω, ρ) and M ¯ (q, ω, ρ) are defined by component of the wavevector. The 3 × 3 matrices M p s Eqs. (S4) and (S5) in the SI. Rs (q, ω) and Rp (q, ω) are the reflection coefficients of s-polarized

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and p-polarized components, respectively. They can be expressed as 53     rα,01 + rα,12 exp 2iφ , gold film system Rα (q, ω) = 1 + rα,01 rα,12 exp 2iφ   rα,01 , bulk gold system Kz,i − Kz,j , Kz,i + Kz,j j Kz,i − i Kz,j , = j Kz,i + i Kz,j

(10)

rs,ij =

(11)

rp,ij

(12)

where α = s or p, φ = Kz,1 d, and d is the thickness of the gold layer. To clearly demonstrate whether RET is enhanced or suppressed by metal surfaces, we study the enhancement of RET instead of the rate of RET. The enhancement of RET γ is defined as WET (the rate of energy transfer in media) divided by WET,0 (the rate of energy transfer in vacuum),

γ=

WET . WET,0

(13)

Furthermore, in order to focus on the effect of metal surfaces on RET, we consider the overlap of the absorption cross section and the normalized emission spectra in a narrow frequency regime, i.e., σ(ω)I(ω) = S(ω)δ(ω − ω 0 ), where S(ω)δ(ω − ω 0 ) correspond to the overlap of σ(ω) and I(ω). By using the approximation above and Eqs. (1)-(3), and (13), one can obtain the enhancement in terms of coupling factors, 2 n ¯(rD , rA , ω)ˆ F (rD , rA , ω) ˆ g n A D , γ= = F0 (rD , rA , ω) n ˆ A g¯0 (rD , rA , ω)ˆ nD

(14)

where F (rD , rA , ω) and F0 (rD , rA , ω) correspond to the coupling factors of the chromophores above a metal surface and in vacuum, respectively. Eqs. (2), (3), and (6)-(12) enable us to numerically compute RET above a surface or layer structures. The main purpose of this Letter is to clarify the effects of mirror dipoles, SPPs, and

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retardation on RET on metal surfaces. To analyze these effects, we investigate RET from two aspects: the distance dependence and the frequency (wavenumber) dependence of the enhancements. In all simulations, we focus on the short range and the intermediate range of RET, i.e., R ≤ λ (the wavelength of a photon or a virtual photon emitted by a donor chromophore), and we would like to explore at what distance the plasmon effect becomes significant in RET. First, we investigate the distance dependence of the enhancement of RET, as shown in Figure 2. Here we choose the wavenumber ω = 18000 cm−1 and the height h = 5 nm because F (rD , rA , ω) reaches the maximum plasmonic responses in the proximity of ω = 18000 cm−1 (see Figures 3d and 4d) and the small height (h < 1 nm) may lead to electron transfer, which is not considered in our model. γzz and γxx are the enhancements of System ZZ and System XX, respectively. The effect of mirror dipoles dominates the mechanism of RET at a short range and it can be analyzed by using the mirror method. 11 The details of the mirror method can be found in Section S8 in the SI. The mirror method assumes that metal is a perfect mirror, i.e., Rp = −Rs = 1. Note that the signs of Rs (q, ω) and Rp (q, ω) depend on the ¯ (q, ω, ρ) and M ¯ (q, ω, ρ). 11,53 γ definitions of M s p zz,mirror and γxx,mirror are the enhancements of System ZZ and System XX contributed by the mirror dipoles, respectively; they are defined as

γmirror

2 n ˆ A [g¯0 (rD , rA , ω) + g¯mirror (rD , rA , ω)]ˆ nD = , n ˆ A g¯0 (rD , rA , ω)ˆ nD

(15)

where g¯mirror (rD , rA , ω) is the dyadic Green’s function obtained from the mirror method. To classify the dominant effect on RET, we use γzz(xx) − γzz(xx),mirror = γzz(xx),mirror to define the characteristic distance Rc , where γzz(xx) −γzz(xx),mirror means that the enhancement which is not contributed by the effect of mirror dipoles. According to Rc , the distance dependence of RET enhancement can be divided into two regions: (i)R . Rc (mirror dipoles) and (ii) Rc . R . λ (SPPs and retardation). Rc is roughly 0.1λ for both System ZZ and

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Figure 2: Distance dependence of RET enhancement for a donor-acceptor pair on (a) bulk gold and (b) a gold film with d = 25 nm. We use the height h = 5 nm and the wavenumber ω = 18000 cm−1 . γzz and γxx stand for the RET enhancements of System ZZ and System XX, respectively. γzz,mirror and γxx,mirror are the RET enhancements which include only the effect of mirror dipoles. According to the characteristic distance Rc , the mechanism of RET can be roughly divided into two regions: (i) the region of mirror dipoles (mirror) and (ii) the region of surface plasmon polaritons (SPP). Panels (c) and (d) are the schematic plots of the fields due to the image dipoles (mirror dipoles in the electrostatic limit) of the donors in System ZZ and System XX, respectively. When R ≈ 3 × 10−2 λ, γzz > 1 and γxx < 1 are due to the orientations of the image dipoles and they can be understood by the superposition of the field lines at Point A in (c) and (d).

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System XX in the case of bulk gold (see Figure 2a), while Rc is smaller than 0.1λ in the case of a gold thin film (see Figure 2b). Note that Rc for System ZZ and System XX are slightly different. To avoid confusion (two different Rc ), we use Rc for System ZZ to separate the two regions in Figure 2. The coincidence of γzz(xx) and γzz(xx),mirror in Region (i) indicates that mirror dipoles dominate the main mechanism of RET in Region (i). Furthermore, when R Rc , RET can be quantitatively analyzed by the image-dipole method (See SI). Note that when we refer to “image dipole”, it means “mirror dipole in the electrostatic limit (no retardation)”. In Region (i), the field ED (rA , rD , ω) is approximately the sum of the fields contributed by the donor dipole and its image dipole. The enhancement γimage can be written as γimage = |1 + β|2 , β=

n ˆ A g¯image (rD , rA , ω)ˆ nD , n ˆ A g¯0 (rD , rA , ω)ˆ nD

(16) (17)

where g¯image (rD , rA , ω) = lim g¯mirror (rD , rA , ω) is the dyadic Green’s function obtained from k0 →0

an image dipole. Note that β ≈ 0 and γ ≈ γimage ≈ 1 when R . 0.01λ and R h. It is because the effect of an image dipole can be neglected when the acceptor is much closer to the donor than to the image dipole of the donor. One can show that |β| < 1 in Systems ZZ and XX (see Eqs. (S27) and (S41) in the SI for the rigorous proof). Thus, the enhancements are between 0 and 4 in Region (i) in Figures 2a and 2b. This also indicates that the enhancement γzz ≈ 4 can be used to distinguish the regions of mirror dipoles and SPPs in experiments. It is worth noting that metal does not always enhance RET and in fact it can strongly suppress RET. For example, the RET in System XX can be nearly totally suppressed (γxx ≈ 0), as shown in Figures 2a and 2b. For R ≈ 3 × 10−2 λ, the enhancement γzz is greater than one, while the enhancement γxx is smaller than one. These behaviors can be understood by the concept of an image dipole. For System ZZ, as shown in Figure 2c, at Point A, the direction of the z-component electric field generated by the donor is the same as that of

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its image dipole. For System XX, as shown in Figure 2d, at Point A, the direction of the x-component electric field generated by the donor is opposite to that of its image dipole. The effects of SPPs and retardation on RET become significant when R & Rc (Region (ii)). To further analyze the effects of SPPs and retardation, we separate the contributions from the p-polarized component and the s-polarized component by defining the p-polarized and s-polarized enhancements, 2 n ¯0 (rD , rA , ω) + g¯α (rD , rA , ω)] n ˆ [ g ˆ A D , γα = n ˆ A g¯0 (rD , rA , ω)ˆ nD

(18)

with α = s or p. Note that γα = 1 means g¯α (rD , rA , ω) = 0, i.e., the α-polarized component has no contribution. Since the SPPs correspond to the poles of Rp (q, ω) in the complex q plane and there is no pole in Rs (q, ω), 53 in Region (ii) γs is dominated by the effect of retardation while γp is dominated by the effect of SPPs (Retardation is secondary.). The proof of no pole in Rs (q, ω) can be found in Section S4 in the SI. Assuming that Rp (q, ω) has poles at q = qp , where p = 1, 2, ..., N (SPP wavevectors), the p-polarized dyadic Green’s function can be decomposed into a non-residue part I¯non−res and a residue part I¯res , g¯p (rA , rD , ω) = I¯non−res + I¯res , √ 1X ¯ 0 (q , ω, ρ)e− qp2 −0 k02 (zD +zA ) , Res[Rp (qp , ω)]M I¯res = − p p 4 q

(19) (20)

p

where I¯res is related to the contribution of SPPs to RET while I¯non−res is not. Equation (20) ¯ p0 (q, ω, ρ) is given by replacing the Bessel function is derived in Section S3 in the SI. Here M ¯ p (q, ω, ρ) with the Hankel function of the first kind Hn(1) (x). The of the first kind Jn (x) in M residue term Res[Rp (qp , ω)] is determined by the dielectrics and it is related to SPPs. The √ ¯ 0 (q , ω, ρ)e−2 qp2 −0 k02 h , dependent on ρ and h (h = z = z ), refers to “geometric term M p D A p √2 ¯ 0 (q , ω, ρ) and e−2 qp −0 k02 h correspond to coupling” between a donor dipole and SPPs. M p p the expansion of the field of a donor dipole in the reciprocal space and the confinement of the

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¯ 0 (q , ω, ρ) monotonically increases SPP field, respectively. Furthermore, as qp increases, M p p √ √2 2 2 2 and e−2 qp −0 k0 h decreases. 52,53 Actually, e−2 qp −0 k0 h can be ignored in our discussion since q h 1/ qp2 − 0 k02 (Recall that h = 5 nm.). As a result, the geometric coupling increases with qp .

Figure 3: Distance dependence of RET enhancement in the case of bulk gold at (a) ω = 12000 cm−1 , (b) ω = 18000 cm−1 , and (c) ω = 24000 cm−1 . (d) Imaginary part of Rp (q, ω). The yellow region indicates the dispersion of SPPs. Figure 3 shows the enhancements decomposed into s- and p-polarized components in the case of bulk gold for three frequencies. Comparing γzz,p in Figures 3a, 3b and 3c, we find that the enhancements in Region (ii) are strongest at ω ' 18000 cm−1 . The effect of SPPs is associated with the residue term Res[Rp (qp , ω)], and it can be understood by the imaginary part of Rp (qp , ω), as shown in Figure 3d. 54 The yellow region in Figure 3d represents the dispersion of the SPPs. The brightest yellow region indicates the greatest contribution of SPPs to RET at ω ' 18000 cm−1 . The enhancements γzz and γzz,p coincide because there 13



is no s-polarized component from the z-oriented transition dipole of the donor (azimuthal symmetry). In Region(ii), the deviation of γxx,s and γzz,s indicates that the donor in System XX has both s-polarized and p-polarized components (Recall that γzz,s = 1 means no spolarized component.). Furthermore, in general, we find that γzz is greater than γxx in Region (ii). The origin of γzz > γxx can be attributed to System ZZ with more p-polarized components, leading to the stronger SPP effect and the RET enhancement (Recall that the mechanism of the p-polarized component is dominated by SPPs, and the degree of RET enhancement: SPPs > retardation.).

Figure 4: Distance dependence of RET enhancement in the case of a gold film with d = 25 nm at (a) ω = 12000 cm−1 , (b) ω = 18000 cm−1 , and (c) ω = 24000 cm−1 . (d) Imaginary part of Rp (q, ω). Two yellow regions indicate the dispersions of the charge-symmetric mode and the charge-antisymmetric mode. Figure 4 shows the distance dependence of RET enhancements in the case of a gold film with d = 25 nm at three frequencies (wavenumbers). Figures 4a, 4b and 4c show similar 14


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features, i.e., γzz < 4 when R . 0.1λ, indicating that the characteristic distances for the thin films are slightly smaller than that in the case of bulk gold. Compared with bulk gold, it is intriguing that γzz,p for the 25-nm gold film has two peaks (see Figure 4b and Figure 3b). The two peaks result from the charge-symmetric mode and the charge-antisymmetric mode of SPPs, which are classified according to the relative phase of surface charges. 55,56 The two modes can be visualized by the yellow regions in Figure 4d (the SPP dispersion of the thin film). The two modes depend on the thickness d and they are expressed as follows 57      1 k02 + qp =     1 k02 +

1 d2 1 d2

z,1 tanh−1 01 K Kz,0

coth−1

0 Kz,1 1 Kz,0

+

z,1 tanh−1 21 K Kz,2

+ coth−1

2 Kz,1 1 Kz,2

2 1/2

2 1/2

, charge-antisymmetric mode,

, charge-symmetric mode. (21)

The charge-antisymmetric mode has greater qp than the charge-symmetric mode does. 57 Due to the shorter wavelength (larger qp ) of the charge-antisymmetric mode, the mode begins to dominate the RET enhancement at a short distance and results in the left peak of γzz,p in Figure 4b. At a longer distance, the charge-symmetric mode emerges. The right dip of γzz,p around 3 × 10−1 R/λ in Figure 4b may originate from the destructive interference of the two modes and it deserves further exploration. Besides explaining the existence of the two peaks, Eq. (21) enables the analysis of the relationship between the characteristic distance and the thickness. Comparing γzz in Figure 3b and 4b (or γzz in Figure 2a and 2b), the characteristic distance is less than 0.1λ due to the smaller thickness. When d decreases, the qp of the charge-antisymmetric modes increases, 58 resulting in a stronger geometric coupling and thereby a shorter Rc . In order to better understand the effect of the thickness on RET, we investigate the frequency (wavenumber) dependence of RET enhancement, as shown in Figure 5. When R Rc , by adopting the image-dipole method (k0 → 0), we can obtain a closed form

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Figure 5: Frequency (wavenumber) dependence of RET enhancement for System ZZ. The donor is at rD = (0, 0, 5 nm) and the acceptor is at rA = (R, 0, 5 nm). The colors of lines correspond to the thin films with four different thickness values: 11.7 nm, 25 nm, 53 nm, and ∞ (bulk).

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expression of RET enhancement of System ZZ (see Eq. (S27) in the SI) as ρ3 (8h2 − ρ2 ) = −1 + 2 (ρ + 4h2 )5/2

γzz,image

2 ,

(22)

where ρ = |ρ| = R, because rD = (0, 0, 5 nm) and rA = (R, 0, 5 nm). For R = 5 nm (Figure 5a), Eq. (22) gives γzz,image ≈ 0.77, which is nearly the same as γzz ≈ 0.76 (green, blue, and cyan lines in the region between 21000 cm−1 and 28000 cm−1 ). The minute difference between γzz and γzz,image is due to the fact that Eq. (22) is derived from the assumption of a perfect mirror. For R = 20 nm (Figure 5b), γzz,image ≈ 1.66 is less close to γzz because the effect of retardation emerges. The effect of SPPs on RET is more substantial for a smaller thickness d and a longer distance R. Figure 5 shows that the significant variations of all the curves are around 15000 cm−1 − 20000 cm−1 , and the greatest variations around 17000 cm−1 − 18000 cm−1 can be associated with the effect of SPPs (Recall large ImRp near 18000 cm−1 in Figure 3d and Figure 4d.). In Figure 5b (R = 20 nm), in addition to retardation, the effect of SPPs also emerges, especially for the film with d = 11.7 nm. At a longer distance (Figure 5d), the effect of SPPs becomes prominent and γ reaches 95 around 17000 cm−1 (a redshift of plasmon resonance). Note that the behaviors of γ of the film with d = 11.7 nm are rather different from those of other films. The reason is that the external field can penetrate the film. Note that d = 11.7 nm is smaller than the skin depth of gold (∼ 20 − 45 nm 59 ) in the frequency range which we consider. The peak (or dip) positions in Figure 5 are gradually redshifted as the thickness d decreases. The redshifts are associated with the dispersion of the chargeantisymmetric mode in Eq. (21) (see Figure S6 in the SI). The peak (dip) strength gradually increases as the thickness d decreases because the strength is associated with the geometric coupling. In another word, when d decreases, qp of the charge-antisymmetric mode increases, ¯ p (q, ω, ρ), and thereby γ is increased. which in turn increases M Resonance energy transfer coupled with surface plasmon polaritons is a fundamental topic

17



in physical chemistry and energy sciences, but the mechanism of RET on a metal surface have not been elucidated in detail in previous studies. In this Letter, we have analyzed the effects of mirror dipoles, surface plasmon polaritons, and retardation on RET. In the case of bulk gold, we find that the characteristic distance on RET is approximately one tenth of the wavelength, i.e., Rc ' λ/10. Our analysis indicates that the enhancement γzz ' 4 can be used to separate the mirror region and the SPP region in experiments (When we adopt γzz − γzz,mirror = γzz,mirror , Rc ' 0.13λ; when we adopt γzz = 4, Rc ' 0.09λ. Both match Rc ' λ/10.). Moreover, in the case of gold thin films, the characteristic distance decreases as the thickness of thin films decreases, indicating that thinner films can be employed to enhance the rate of RET at a shorter range. The decomposition of s- and p-polarization enables the separation of the effects of surface plasmons and retardation, and it further explains the origin of γzz > γxx in Region (ii). The SPP dispersion of the charge-symmetric and chargeantisymmetric modes elucidates the origin of the two peaks in the distance dependence of RET enhancement. We would like to emphasize that the characteristic distance Rc ' λ/10 is general for RET on bulk gold and on bulk silver. The simulations of silver systems can be found in Figures S5 in the SI. Furthermore, compared with the gold systems, the RET enhancements in the silver systems are stronger at the resonance of SPPs (γzz > 100). In addition, our simulations are supported by the results from the finite-difference time domain method (see Figure S2 in the SI). The characteristic distance is also less sensitive to the height h . 50 nm (see Figures S3 and S4 in the SI). The dielectric function for gold has the shape (e.g., thickness) dependence at the nanoscale, and it has to be modified when the thickness of the thin film is smaller than 30 nm. 60 However, the conclusions in this Letter are not affected by the modification and they are robust for a variety of experimental dielectric functions for gold (see Figure S1 in the SI). We do not consider the non-local effect in our simulations because the distance between the donor and the metal surface does not satisfy the condition of the non-local effect. 61 The next step of our study is to explore a variety of plasmonic systems and extend

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our approach to investigate exciton-polariton interactions in the strong-coupling regime. 62 Recently, plasmons for energy conversion have become an active field due to their promising applications and intriguing phenomena. 63 We hope that our analysis provides important new insights into the mechanism of resonance energy transfer and inspires further investigations in the field of plasmons for energy conversion.

Supporting Information Available Comparison of coupling factors using different dielectric functions for gold; detailed forms of the dyadic Green’s functions on the metal plane; derivation of the contributions of the poles to the dyadic Green’s functions; simulation results given by the finite-difference time domain method; distance dependence of RET enhancements (gold) at a different height; distance dependence of RET enhancements in the silver systems; condition to have poles; mirror method. This material is available free of charge via the Internet at http://pubs. acs.org/.

Acknowledgement This research was supported by Academia Sinica and the Ministry of Science and Technology of Taiwan (MOST 106-2113-M-001-036-MY3).

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Characteristic Distance of Resonance Energy Transfer Coupled with

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