This is an open access article published under an ACS AuthorChoice License, which permits copying and redistribution of the article or any adaptations for non-commercial purposes. Letter Cite This: Nano Lett. 2019, 19, 5790−5795 pubs.acs.org/NanoLett Energy Transfer and Interference by Collective Electromagnetic Coupling Mayte Goḿ ez-Castaño,†,‡ Andreś Redondo-Cubero,§ Lionel Buisson,† Jose Luis Pau,§ Agustin Mihi,‡ Serge Ravaine,† Renaud A. L. Valleé ,*,† Abraham Nitzan,∥,⊥ and Maxim Sukharev*,#,⊗ † CNRS, University of Bordeaux, CRPP, UMR5031, 115 Avenue Schweitzer, 33600, Pessac, France ‡ Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus de la UAB, 08193 Barcelona, Spain § Electronics and Semiconductor Group, Department of Applied Physics, Universidad Autónoma de Madrid, E-28049 Madrid, Spain ∥ Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States ⊥ School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel # Department of Physics, Arizona State University, Tempe, Arizona 85287, United States ⊗ College of Integrative Sciences and Arts, Arizona State University, Mesa, Arizona 85201, United States * S Supporting Information ABSTRACT: The physics of collective optical response of molecular assemblies, pioneered by Dicke in 1954, has long been at the center of theoretical and experimental scrutiny. The influence of the environment on such phenomena is also of great interest due to various important applications in, e.g., energy conversion devices. In this Letter, we demonstrate both experimentally and theoretically the spatial modulations of the collective decay rates of molecules placed in proximity to a metal interface. We show in a very simple framework how the cooperative optical response can be analyzed in terms of intermolecular correlations causing interference between the response of different molecules and the polarization induced on a nearby metallic boundary and predict similar collective interference phenomena in excitation energy transfer between molecular aggregates. KEYWORDS: Collective fluorescence, superradiance, energy transfer T he collective optical response of atomic and molecular emitters has been discussed throughout the evolving science of light−matter interaction. In particular, the Dicke Both this collective feature and Dicke’s superradiance physics have been discussed recently with regards to their implications for the performance of energy conversion devices.12−21 It theory of superradiance emission1 and variants such as should be noted that the theme exploited in all these studies is superfluorescence2 have been extensively discussed and the correlated behavior of a many-body system (often modeled different aspects of the phenomenon, including its quantum as a cluster of N two-level atoms) supported by their mutual and classical aspects3 as well as the effect of static disorder coupling to the radiation field, sometimes in the form of strong (inhomogeneous broadening) and dephasing (homogeneous coupling to a cavity mode. The general physics of such systems broadening), were elucidated.3−7 More recent discussions of has been extensively discussed since the middle of the previous cooperative molecular effects have focused on strong coupling century.22−31 More recent studies have placed such systems in between molecular excitons and plasmons or cavity modes, the vicinity of metal nanostructures, where interaction between mostly manifested through the N scaling of the observed molecular emitters is mediated not only by free or cavity Rabi splitting in such systems, where N is the number of photon modes but also via their mutual interactions with metal involved molecules.8−10 It should be noted that while both plasmons.11,32−38 phenomena are manifestations of a collective response, Dicke While not usually phrased in this language, cooperative superradiance is observed during the decay of a state in which all molecules in a given cluster are initially excited while the optical response can often be analyzed as interference between collective Rabi splitting usually involves the bright single the response of different molecules, where correlations exciton state where a single molecule excitation is delocalized determine whether one observes constructive or destructive over N molecules. Significantly, the radiative lifetime of such a bright exciton, which carries most of the oscillator strength of Received: June 21, 2019 the N-molecule cluster, is τR(N) = τR(1)/N, a property which Revised: July 25, 2019 expresses itself also in energy transfer out of such a cluster.11 Published: July 26, 2019 © 2019 American Chemical Society 5790 DOI: 10.1021/acs.nanolett.9b02521 Nano Lett. 2019, 19, 5790−5795 Nano Letters Letter Figure 1. Panel a shows the scheme of the fabrication process. Panel b shows the experimental setup for optical measurements. Panels c−f show the semilog decay profiles of ensembles of molecules (3 × 1022 m−3 layer density in panels c and e, 3 × 1023 m−3 in panels d and f) deposited either on a bare glass substrate (reference sample, magenta dots in panel c) or at a given distance (80 nm in panel c, 380 nm in panel d, 200 nm in panel e, and 560 nm in panel f), fixed by a silica layer, on top of the gold mirror structure (black dots). For the reference sample, the decay is monoexponential (panel c, white line) with a rate γ = 0.178 μeV. In the other cases, the decay is found to be best fitted by a biexponential with rates γL (γS) equal to 1.68 (0.224) μeV (panel c, blue line), 5.52 (0.191) μeV (panel d, red line), 1.64 (0.145) μeV (panel e, blue line), and 1.90 (0.184) μeV (panel f, red line). interference. In the simplest case of a single excited molecule absorption (1.25105 M−1 cm−1), high fluorescence quantum placed in front of a mirror, emission from the molecule and the yield (0.3), and high thermal and photostability for a red dye polarization induced in the metal (or the molecule and its emitting at 680 nm, e.g., at a wavelength where autofluor- image) interfere, leading to oscillations in fluorescence escence of the sample is drastically reduced, ensuring an lifetimes as a function of the distance from the surface.39−43 extremely clean collection of the photon counts. A layer of Obviously, this phenomenon is not limited to a single poly(methyl methacrylate) (PMMA) is finally deposited on molecule. The response of a molecular cluster, including all top of the structure in order to fix the position of the the intricacies mentioned above, is expected to interfere with fluorescent dyes and provide similar electromagnetic boundary the response of the corresponding polarization induced on a conditions on both sides of the layer. As a reference sample, nearby metallic boundary or a metal particle. Indeed, such the ensemble of fluorophores and PMMA were deposited in effects have been very recently discussed37 and observed.44 the same conditions on a bare glass substrate. Once prepared, In this Letter, we experimentally and theoretically the samples underwent structural and ellipsometry character- demonstrate such collective molecular phenomena. We note izations, ensuring that they were designed as prescribed. We that with strong dependence on intermolecular correlations, then investigated the samples by time-resolved fluorescence specific behaviors may reflect properties of specific structures. microscopy in order to determine the emission properties of By studying the optical properties of such structures, we can the system. As presented in Figure 1b, a pulsed laser operating characterize their collective nature as reflected in their steady at 654 nm is normally focused on the sample surface. The state and relaxation behaviors, their dependence on emitter emitted fluorescence is then collected through the same density (or number), system dimensionality and structure, as objective and sent to the detector for the following well as effects of structural and dynamical disorder. Our reconstruction of the decay profiles. experimental results demonstrating such collective behavior are The decay profiles of Atto655 molecules deposited on a bare presented in Experimental Results. We discuss these glass substrate from a 10−8 M concentration of dyes in ethanol phenomena within a simple theoretical model in Theoretical exhibit a clear single exponential decay profile with a decay rate Model and Computational Results and present results of of 0.178 μeV (Figure 1c: magenta dots fitted by a white line). numerical simulations based on a Maxwell−Bloch model for This sample, with a molecular number density of 3 × 1022 m−3, the radiation field−molecules system. Even on this simple is taken as reference because its measured decay rate is level, we can demonstrate dramatic effects of structure, order, characteristic of nonaggregated molecules. Larger concen- and emitter density, implying new ways to control the optical trations (typically larger than 10−6 M) lead to nonexponential properties of molecular nanodevices. decay profiles evidencing dyes’ aggregation on the bare glass Experimental Results. The proposed samples are substrate, a behavior which we want to avoid. The decay fabricated as described in Figure 1a (see Experimental methods profiles of similar molecular number densities (3 × 1022 m−3) in the Supporting Information for fabrication details). First, a deposited at a controlled distance (silica spacer) of the gold flat gold substrate is covered by a silica layer with a well- mirror structure significantly deviate from a single exponential controlled thickness. Such a film works as a spacer between the decay profile. metallic mirror and a postdeposited molecular layer of Atto A biexponential function I(t) = IS exp(−γSt) + IL exp(−γLt) 655 fluorophores, whose number density can be varied with convoluted to the instrumental response function (IRF) of the the concentration of molecules in solution. The fluorescent dye setup provides a good estimate of the small γS and large γL was chosen due to its characteristic features of strong decay rates (Figure 1c−f). Both rates depend on the molecular 5791 DOI: 10.1021/acs.nanolett.9b02521 Nano Lett. 2019, 19, 5790−5795 Nano Letters Letter layer−gold mirror distance as well as on the molecular density Nevertheless, one can clearly see that the large decay rates are of the film. We show results for molecular number densities 3 significantly higher for the 3 × 1023 m−3 layer density (Figure × 1022 m−3 (Figure 1c,e) and 3 × 1023 m−3 (Figure 1d,f) and 2b) than for the 3 × 1022 m−3 layer density (Figure 2a). For for the molecular layer to the gold mirror distances in the the largest concentrations of molecules (up to 3 × 1025 m−3), performed range extending from 20 nm up to 680 nm with the trend followed by the small decay rates is similar to those steps of 60 nm. shown in Figure 2c,d. One notices immediately the increase of the large decay rate, Theoretical Model and Computational Results. For γL, while going from lower (Figure 1c,e) to higher (Figure the simplest model of collective response from a molecular 1d,f) concentrations of molecules, a feature not observed for layer, we assume the molecules are oriented parallel to the gold the small decay rate γS. This rapid behavior, as better exhibited mirror, thus leading to a one-dimensional electromagnetic in the fitted values reported in Figure 2a,b, indicates a setup. The corresponding Maxwell’s equations governing the electrodynamics of molecule-to-molecule interactions are ∂By ∂Ex =− , ∂t ∂z ∂Ex 1 ∂By ε0 =− − Jx ∂t μ0 ∂z (1) where z corresponds to the longitudinal coordinate perpen- dicular to the mirror, and Jx is the polarization current density of the molecular layer. The molecules are assumed to be two- level emitters described by the density matrix ρ̂ and driven by a local field in accordance with jij Exμ12 zy jj (ρ22 − ρ11)zzz jj ℏ zz Exμ12 d ijj ρ11 ρ12 yz jjj zz zz (ρ12 − ρ12* ) − jj zz = jjj zz z ℏ dt jk ρ21 ρ22 z jjj zz zz { jj E μ zz − ω12ρ12 jj x 12 i (ρ12* − ρ12 ) zzz Figure 2. Panels a and b show the evolution of γL as a function of jj− Exμ12 k { molecule−metal distance for 3 × 1022 m−3 and 3 × 1023 m−3 (ρ22 − ρ11) − ω12ρ12* molecular number densities, respectively. Similarly, panels c and d ℏ ℏ show the evolution of γS. Experimental γL (dots and squares) in panels (2) a and b were fitted by an appropriate sinusoidal signal, while the γS Here μ12 and ω12 are the transition dipole and the transition (dots and squares) in panels c and d were fitted by a damped sinusoid. frequency of a single molecule, respectively. The polarization The frequency (related to the emission frequency of the molecules and the refractive index of the surrounding medium) found in panel a current is then calculated as was used as a fixed parameter for the three other fits. The amplitude ∂Px ∂ of the sinusoid in panel b was merely adjusted by hand, which makes Jx = = n0μ12 (ρ12 − ρ12* ) ∂t ∂t (3) it more a guide to the eye than a fit. where n0 is the number density of molecules. collective relaxation behavior. Unfortunately, we could not We consider the dynamics of molecules initially prepared by experimentally test the density dependence in a large enough a sudden (short time) excitation of all molecules in a coherent range to observe a potential linear density dependence since superposition state such that the population of excited for larger concentrations (up to 3 × 1025 m−3), the large decay molecules is significantly less than 1 throughout the time rate was beyond the time resolution of our apparatus. We propagation (this corresponds to the optical linear regime). could then only see a single exponential profile corresponding We set it to 0.01, although we note that for the range of to the small decay rate, γS. Thus, while the increase of the large molecular densities and material parameters considered in this decay rate with density is clearly observed, we cannot speak of manuscript, an excited state population as high as 0.5 leads to a linear scaling behavior given just two points. qualitatively similar results. Thus, the initial conditions for the Figure 2 exhibits the evolution of these decay rates as coupled Maxwell−Bloch equations are functions of number density and distance D from the gold ρ11 = 1 − ρ22 = 0.99 mirror structure. Obviously, the small decay rates exhibit damped oscillations (Figure 2c,d), as already observed in ρ12 = ρ11ρ22 exp(iφ) (4) Drexhage’s experiments over 50 years ago.39 These observa- tions were shown to result from interference between an where the initial phase φ can be taken the same (and then emitter and its image in the mirror.39,40,42,43,45 Strikingly chosen to be 0) for all molecules or with a random component enough, this behavior is not limited to individual molecules but to represent disorder, as discussed below. is also seen in the collective relaxation. Remarkably, the γL The dynamics of molecules obtained from Maxwell−Bloch oscillations are not damped over the observed distance, as seen equations with the initial conditions (eq 4) shows that the by comparing Figure 2a,c. It should be noted that the large ensemble average excited state population as a function of time decay rates exhibited in Figure 2b are on the edge of our time follows an exponential decay law, exp(−γt). The rates γ, resolution. The error bars shown in this figure are merely calculated by fitting the numerical time dependent excited state standard deviations of various measurements performed on the population, are displayed in Figure 3. same sample and do not reflect the setup’s time resolution. As Figure 3a shows the dependence of the decay rate on the such, not all decay profiles could be fitted appropriately. molecular number density for different thicknesses of the 5792 DOI: 10.1021/acs.nanolett.9b02521 Nano Lett. 2019, 19, 5790−5795 Nano Letters Letter the silica layer). Since the light emitted by the molecules is reflected by the mirror, it undergoes a π-phase change (for an ideal mirror). Thus, the resonant condition corresponding to the destructive interference of the reflected and emitted waves leading to a near-zero decay rate is 2·d nmin ·λ12 = c·troundtrip = c· ⇒ d = nmin ·λ12 /2, c nmin = 1,2,Κ (5) where c is the speed of light in the media between the mirror and molecules, d is the layer-to-mirror distance, and λ12 is the molecular transition wavelength (since molecules are embed- ded in PMMA the corresponding wavelength is shorter by the factor of the PMMA refractive index compared to its value in vacuum). Small deviations from (eq 5) are attributed to the finite thickness of the molecular layer (as clearly seen it affects the positions of minima in Figure 3b) and the phase change of the emitted radiation when it bounces off of a gold surface. Figure 3. Panels a and b show simulations for a single molecular layer in the experimental geometry (Figure 1a,b). Panel a shows the decay There is also a small mismatch of the refractive indices for the rate extracted from the ensemble average excited state population as a silica and PMMA, in which the molecules are embedded. function of the molecular density for three different thicknesses of the Nonetheless the spatial modulations of the decay rate obtained molecular layer: black, 5 nm; red, 10 nm; and blue, 15 nm. The from one-dimensional Maxwell−Bloch equations agree per- thickness of the silica spacer is 300 nm. Panel b shows the spatial fectly with experimental results. Importantly, this behavior, dependence of the decay rate on the thickness of the silica layer long known for individual molecules, is predicted and observed (distance from the gold mirror) for the same three molecular layers’ also for the collective decay of a molecular aggregate, here a thicknesses as in panel a. The molecular number density is 3 × 1023 molecular layer parallel to the mirror, as indicated by the m−3. Panels c and d explore the collective evolution in a system of two density dependence of the rate seen in Figure 3a. molecular layers separated by a silica spacer without mirrors. In panel It is also important to note that dephasing of intermolecular c, both layers start in the same coherent superposition as defined by eq 4 and the decay of the overall excited state population is shown as coherence is what causes the transition from collective (large a function of the interlayer distance (thickness of a silica spacer) for decay rate) to single molecule dynamics (small decay rate) the same three molecular layer’s thicknesses as in panel a, with the observed experimentally. In the present one-dimensional molecular number density in both layers taken as 3 × 1023 m−3. The model, such dephasing, associated with dynamic or static initial state in panel d corresponds to one of the layers prepared in the disorder (inhomogeneous broadening), can be imposed only state in eq 4 while the molecules in the other layer are initially in the partially along the direction normal to the layers, since the ground state, and the dynamics of interlayer energy transfer and molecular layers respond collectively by the nature of the 1-D overall relaxation are observed. Black lines correspond to a spacer calculation. Thus, in the simulations discussed in Figure 3, the thickness of 20 nm, red lines are for 50 nm, and blue lines correspond phase in eq 4 is taken the same for all system units, i.e., all to 80 nm. Other parameters are the molecular number density is 1026 points of our one-dimensional grid. If the phase is chosen m−3 and the layer’s thickness is 10 nm. In all simulations, the phase of the coherence in eq 4 is set to 0. randomly between 0 and 2π for each grid point, representing system disorder, the same oscillations in the decay rate vs distance are seen, albeit with a significantly reduced amplitude molecular layer according to the experimental geometry than experimentally observed. Another important difference (Figure 1a,b). It is seen that the decay rate scales linearly between the behaviors of the coherent and incoherent with the molecular number density, as expected from a molecular systems lies in the observed distance dependence. delocalized single exciton state.33 These observations nicely For the geometry studied here, a system with a fully coherent complement the experimental trend shown earlier and are molecular layer is truly one-dimensional, implying that the rate based on two points only. For a silica thickness of 300 nm and oscillates with a distance independent amplitude, as seen in the a molecular number density of 3 × 1023 m−3, our model experimental fast decay (Figure 1c−f) and captured by our calculation predicts the large (collective) decay rate of 22 μeV, one-dimensional simulations. In contrast, when the emitters which overestimates the experimental large rate of 6.6 μeV oscillate with random phases, the observed emission is a sum (Figure 2b). The overestimation is likely due to the fact that of signals generated by individual emitters so that the pertinent the calculation takes all molecules to be perfectly aligned geometry is three-dimensional, characterized by an oscillation parallel to the interface and to each other. In contrast, for an amplitude that decays with distance as already reported in refs isotropic molecular system the averaging over all orientations 39 and 40 and seen in Figure 2c,d. The presence of an of the molecular dipole leads to an effective coupling between incoherent component in the molecular response may also the macroscopic polarization and the electric field which is explain why a perfect destructive interference is not observed reduced by a factor of 1/3.9 This results in a decay rate of 7.3 experimentally. μeV, remarkably close to the observed rate. Our successful modeling of the experimental observation Next consider the results presented in Figure 3b, which implies the important prediction that collective interference shows the observed decay rate as a function of the distance phenomena should be observed for interacting molecular between the molecular layer and the gold film (as per assemblies. This is demonstrated in Figure 3c,d where the two experimental setup this corresponds to varying the thickness of assemblies are taken as two parallel molecular layers separated 5793 DOI: 10.1021/acs.nanolett.9b02521 Nano Lett. 2019, 19, 5790−5795 Nano Letters Letter by a silica spacer. One can envision various scenarios for the time dynamics depending on initial conditions. If molecules in ■ ACKNOWLEDGMENTS This work is sponsored in part by the Air Force Office of both layers are prepared in the same coherent superposition Scientific Research under Grant Nos. FA9550-15-1-0189 and (Figure 3c), we observe the familiar exponential decay as in FA9550-19-1-0009 (M.S.), the U.S. National Science Founda- Figure 3a,b depending on the spacer’s thickness, albeit with a tion, Grant Nos. CHE1665291 (A.N.), CTQ2017-84309-C2- change of phase: for a “real” molecular layer (as opposed to an 2-R (A.R.-C. and J.L.P.), P2018/NMT4349 Comunidad de “image” layer) the condition for the destructive interference Madrid (A.R.-C. and J.L.P.), RYC-2015-18047 (A.R.-C.), and between electromagnetic waves emitted by both layers is d = the US-Israel Binational Science Foundation, Grant No. (nmin + 1/2)λ12, nmin = 0,1,K, showing a phase shift compared 2014113 (M.S. and A.N.). We thank the Spanish Ministry of to eq 5, as seen in Figure 3c. Again, the resonant condition is clearly maintained throughout all the spacer’s distances with a Economy and Competitiveness (MINECO) for its support slight deviation due to the finite molecular layers’ thickness. through Grants MAT2016-79053-P and Grant SEV-2015- In a different scenario (Figure 3d), only molecules in one of 0496. This project has received funding from the European the layers are prepared in the excited coherent superposition. Research Council (Grant Agreement No. 637116). This work Here, collective relaxation and energy transfer are seen in the was also financially supported by the French National Research time evolution. The time dependence of the excited state Agency (ANR) as part of the Initiative for Excellence IdEx populations reveals complex dynamics, exhibiting a clear sign Bordeaux program (Grant ANR-10-IDEX-03-02). of interference through the fact that the decay rate is significantly influenced by the thickness of the silica spacer. While the excited layer begins to decay by emitting the ■ REFERENCES (1) Dicke, R. H. Coherence in Spontaneous Radiation Processes. radiation, the initially relaxed molecules are picking up this Phys. Rev. 1954, 93, 99−110. radiation. The time at which the initially relaxed molecular (2) Bonifacio, R.; Lugiato, L. A. Cooperative radiation processes in layer reaches its maximum excitation clearly depends on the two-level systems: Superfluorescence. Phys. Rev. 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