We study an unusual effect of spectral-band replication in the optical spectra of dimers, consisting of spherical nanoparticles or nanodisks with a silver core and a J-aggregate shell of TDBC-dye. It consists in the emergence of a doubled number of plexcitonic spectral bands compared to the case of a plasmonic dimer and in narrow peaks associated with the resonances of the J-aggregate shell. The plexcitonic bands can be divided into two groups: the “original” bands, accurately reproducing plasmonic peaks, and their “replicas,” with a specific mutual arrangement and intensity distributions. The effect is interpreted using the multi-state effective Hamiltonian model describing a strong coupling between the quasi-degenerate Frenkel excitonic modes in the organic shells and multiple plasmonic modes in the pair of Ag-cores. We quantitatively explain some available experimental data on the optical properties of nanodisks and suggest a way for the observation of the replication effect. Our results extend the understanding of the nature of plexcitonic coupling to more complex systems compared to individual metal/J-aggregate nanoparticles.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The intensive development of a number of fundamental and applied directions of nanophotonics and optoelectronics proceeds together with the studies of the optical properties of various hybrid organic/inorganic nanostructures and materials. A promising approach toward high performance photonic devices is to utilize the excitonic supramolecular structures as an organic component and various plasmonic nanostructures as an inorganic component of composite systems [1–3]. Over the past decade, studies of hybrid systems consisting of metallic nanostructures of various shapes and sizes and J- or H-aggregates of cyanine dyes attracted significant interest. Optical properties of such hybrid systems are determined by both the metallic and molecular subsystems and strong coupling between them [4, 5]. This allows us to combine the unique properties of metallic nanostructures, associated with the excitation of localized or guided surface plasmons [6–8], with the remarkable properties of J-aggregates of cyanine dyes. The narrow and intense absorption features of J-aggregates are highly desirable for many photonic applications [9–11] and can be successfully described in terms of the Frenkel exciton model .
A number of works was devoted to the investigations of the effects associated with the interaction of Frenkel excitons with surface plasmons in metal-organic nanostructures of planar geometry (e.g., in thin metal films coated by ordered molecular J-aggregates ). In this case, Frenkel excitons are electromagnetically coupled with surface plasmons propagating along the planar metal–insulator or metal–semiconductor interface. There is a series of experimental and theoretical studies of spectral characteristics of hybrid nanoparticles of different shapes and composition consisting of a metallic core and an outer J-aggregate shell of cyanine dyes. This includes a series of studies of two-layer and three-layer spherical nanoparticles [14–22] as well as metal-organic nanorods [23–26], nanodisks [27,28], nanostars , and nanoplatelets . In general, the spectra of such nanoparticles cannot be described as a simple superposition of the spectra of individual components. They are determined by the effects of electromagnetic coupling of molecular exciton in the outer J-aggregate shell with localized plasmon in a metallic core. The most remarkable phenomena appear in the regimes of strong and ultra-strong plexcitonic coupling [31–33]. In such regimes, hybrid states of the entire nanosystem are formed with qualitatively new properties as compared to those of each individual components . We note that similar effects of strong plexcitonic coupling were studied  for three-layer spherical nanoparticles consisting of a metallic nanoshell coated with an outer layer of the dye J-aggregate. Ultrafast optical dynamics of excitons in nanoshell/J-aggregate complexes were investigated in . It was shown that such hybrid nanostructures might possess enhanced, tunable, on- and off-resonance nonlinear optical properties. The strength of plexcitonic coupling and properties of such hybrid systems strongly depend on their shape and geometrical parameters as well as on the optical constants of metallic and organic components including the oscillator strength of the J-aggregate shell layer and spectral positions of plasmonic and excitonic peaks [18,22,25,32].
There is also a series of papers devoted to the study of the optical properties and the effects of near-field electromagnetic coupling in systems of several plasmonic nanoparticles (dimers, trimers, quadrumers and more complex clusters) as well as in one-dimensional and two-dimensional arrays of plasmonic and plexcitonic nanoparticles. Until recently, these studies were most intensively carried out for pure metallic plasmonic systems [36–39]. However, some interesting results have been obtained on the plexcitonic coupling of a monomer dye molecule or a molecular J-aggregate with a metallic nanoparticle  or individual nanoparticle dimers or clusters [41–44] as well as on the plexcitonic coupling in metal-organic nanoparticle arrays [45–48]. Nevertheless, on the whole the nature of the electromagnetic coupling of closely spaced particles in metal/J-aggregate plexcitonic nanosystems and the effects resulting from this coupling in light absorption and scattering remain unexplored in contrast to the case of the individual plexcitonic nanoparticles.
All results obtained in the aforementioned works stimulate further interest in studies of nanoengineered plexcitonic systems with controllable spectral and nonlinear-optical properties that can be used for the fabrication of novel functional nanomaterials. It should be noted that available works on plexcitonic effects are not limited to their studies in various metal-organic systems. So, the effects of the plasmon-exciton interaction have been intensively studied in hybrid systems consisting of metallic nanoparticles, thin metallic films or metallic nanowires and colloidal quantum-confined structures such as quantum dots or quantum wells (see review articles [2–4]). Recent advances in this field are also related to studies of the strong interaction effects of individual quantum emitters with resonant cavities. They are of fundamental interest for understanding light–matter interactions including cavity quantum-electrodynamics research, quantum optics and quantum information applications. For example, the authors of  have investigated vacuum Rabi splitting, a manifestation of strong coupling, using silver bowtie plasmonic cavities loaded with semiconductor quantum dots. It was also recently demonstrated the light-emitting plexcitons from the coupling between the neutral excitons in monolayer of tungsten diselenide and highly-confined nanocavity plasmons in nanocube-over-mirror system . All-solid-state, room temperature, and actively controllable strong coupling in WSe2 monolayer coupled to a single plasmonic (Au or Ag) nanorod was realized and studied in [51,52]. The results obtained in these works clearly indicate a possibility to modify the properties of plexcitons that can be used in many nanophotonic and optoelectronic applications.
In this work we perform a theoretical study and a computer simulation of the optical properties of single pairs of closely spaced and partially merging two-layer plexcitonic nanoparticles. The nanoparticles under study consist of a silver core and an outer shell of the TDBC cyanine dye [5,5′,6,6′-tetrachloro-1-1′-diethyl-3,3′-di(4-sulfobutyl)-benzimidazolo-carbocyanine] in J-aggregated state. First, we consider a single pair of such hybrid nanoparticles of spherical shape located in a homogeneous surrounding medium (see Figs. 1(a)–1(c)). Our main goal here is to demonstrate a new effect in the behavior of the absorption spectra of hybrid nanoparticle dimers. It consists in an unusual “replication” of the plasmonic spectral bands as a result of strong coupling of Frenkel excitons of the organic shells with surface plasmons of electromagnetically interacting metallic cores. Particular attention in the work is paid to the discussion of two available experiments  and  dealing with isolated core-shell Ag/TDBC nanodisks and pairs of silver disks with small interparticle separation (see Figs. 1(d)–1(f)). We demonstrate that the effect of “replication” of spectral bands can be observed in hybrid silver/J-aggregate dimers, fabricated on the basis of available techniques.
The plexcitonic phenomenon under study is inherent to hybrid nanoparticle dimers consisting of metal-organic concentric spheres or to metal-organic nanodisk pairs (see Figs. 1(a)–1(f)). It is of particular importance not only for the fundamental studies of light interaction with complex composite nanostructures but also for a number of potential applications in hybrid organo-inorganic photonics and optoelectronics including novel nanosensors, light-emitting devices, optical switches and nanowaveguides. By its physical nature and manifestation in the absorption and scattering spectra, it is qualitatively different from all known plexcitonic effects discussed previously. In particular, it is radically different from the famous phenomenon of a dip in the absorption and scattering spectra of light by individual metal/dye core-shell nanoparticles appearing in the strong plexcitonic coupling regime [15,22,25,26,32] as well as from the effect of induced transparency in plasmon–exciton nanostructures discussed in [25,53].
2. Theoretical approach and simulation method
We perform numerical simulation of light absorption and scattering processes by the plexcitonic nanoparticle dimers using the finite-difference time-domain (FDTD) method implemented in a software package MIT Electromagnetic Equation Propagation (MEEP). The optical properties of the materials composing the nanostructures under study are determined by their dielectric functions. The dielectric function of silver core of hybrid nanoparticles is described here similarly to our previous works [18, 26, 32] by taking into account the contributions of free and bound electrons and the size effect caused by electron scattering at the metal-organic interface. The dielectric function of J-aggregate is expressed through the central frequency of the J-band, ωJ; its full width, γJ; the reduced oscillator strength, fJ; and the permittivity outside the J-band, :27].
Our goal is to determine three types of quantities for the plexcitonic nanoparticle dimers: total absorption (extinction) cross section, extinction coefficient and dark-field scattering spectra. To calculate the first two quantities we suppose that the incident plane wave propagates normally to the sample surface (see Fig. 1(d)). Then, we run simulations for two directions of polarization: along the axis connecting the centers of the particles (X-axis in Fig. 1(d)) and perpendicular to it (i.e. along the Y-axis in Fig. 1(d)). In the case of equally probable orientations of the dimer on the substrate (or naturally polarized light), the absorption (extinction) cross sections, σx and σy, are obtained for two polarizations, X and Y, and further are averaged as 〈σ〉 = (σx + σy) /2. To determine the dark-field scattering spectra we calculate the scattering cross sections for two incidence directions: along the XZ plane, σXZ, and along YZ plane, σYZ(see Figs. 1(e)–1(f)) by choosing the angle of α = 60° to reproduce the experimental conditions . The final result is given by 〈σDF〉 = (σXZ + σYZ) /2.
The main features in spectral behavior of plexcitonic dimers were obtained in this work by using the FDTD simulations. However, to clarify the mechanism of spectral band formation caused by the electromagnetic coupling of Frenkel excitons with the localized surface plasmons we employ the approach based on the suitable choice of the Hamiltonian of a pair of Ag/J-aggregate nanoparticles. Such an approach is equivalent to the coupled oscillator model . In the simplest case, when a single plasmonic mode interacts with a single excitonic mode, the Hamiltonian, describing the plexcitonic coupling effects in two-component system, is given by:
3. Results and discussions
3.1. Demonstration of the spectral band replication effect for plexcitonic dimers consisting of spherical Ag/J-aggregate nanoparticles
First, we discuss the optical properties of a pair of plexcitonic nanoparticles (NPs) caused by their electromagnetic coupling. We consider the particles of spherical shape and sufficiently small sizes to suppress the contribution from the multipole plasmonic resonances in the spectra of individual particles. Our calculations of the absorption spectra of the hybrid dimer (Ag/TDBC NP dimer) were performed for a pair of two concentric spheres with a silver core with a diameter of 20 nm and a J-aggregate shell of TDBC-dye with a thickness s = 3 nm. In Figs. 1(a)–1(c) we display the configurations of such dimer for several values of the distance, L, between the centers of the spheres. It is seen that at L < 20 nm the Ag-cores of the nanoparticles partially overlap, while at L < 26 nm a partial overlap of their shells occurs. We compare spectral properties of the plexcitonic Ag/TDBC NP dimer and the plasmonic Ag NP dimer with the same size as that of the silver core of hybrid dimer. In real experimental conditions, pairs of closely spaced nanoparticles are placed on the surface of a substrate. To simplify the analysis, the substrate effect is modeled using an effective medium approximation [36, 54]. In our calculations, we consider homogeneous host medium with a permittivity of water ɛh = 1.78 in the visible range.
Figures 2(a)–2(h) shows the results of calculations of the absorption cross sections by a plasmonic Ag NP dimer and by a plexcitonic Ag/TDBC NP dimer as functions of light wavelength in vacuum. The cross sections are averaged over the polarizations of the incident light as described in Sect. 2. In Figs. 3(a)–3(d) we additionally represent the absorption spectra of the Ag/TDBC NP dimer calculated separately for two mutually perpendicular polarizations of the incident light: along the axis of rotation of the system (the X axis in Figs. 1(a)) and perpendicular to it (the Y axis in Fig. 1(a)). This figure helps to reveal the contributions of the “longitudinal”, σx, and “transverse”, σy, electromagnetic modes to the resulting absorption cross sections averaged over the orientations. In contrast to Figs. 2(a)–2(h), the spectra in Figs. 3(a)–3(d) are given as functions of the photon energy. This makes it possible to more clearly compare not only the positions, but also the widths of the absorption bands. To clarify the nature of the plexcitonic coupling in a pair of Ag/TDBC nanoparticles as well as to identify the similarities and differences in the behavior of shifts of various bands we display in Figs. 4(a) and 4(b) the dependences of the peak frequencies in their absorption spectra on the distance, L, between the particle centers.
The absorption spectra of Ag/TDBC NP dimers contain three groups of plexcitonic peaks, designated as s1, s2; o1, ..., o4; and r1, ..., r4. The narrow s1 and s2 peaks lie near the wavelength of the J-band absorption of the TDBC-dye aggregate (λJ = 587.6 nm). Unlike other o1, ..., o4, and r1, ..., r4 bands, the spectral positions of s1 and s2 peaks vary little with the change of the interparticle distance, L (see Figs. 4(a)–4(b)). To clarify the nature of the s1 and s2 peaks we calculated the absorption spectra of dimers, in which silver cores of nanoparticles have been replaced by the optically passive medium with a dielectric constant equal to . The results are presented in Figs. 4(e)–4(f). We obtained two narrow bands with absorption maxima located near the peak wavelengths of s1 and s2 bands in the spectra of Ag/TDBC NP dimers of same sizes. Therefore, s1 and s2 peaks can be attributed to the excitonic resonances of the J-aggregate shell of a pair of hybrid nanoparticles. Further we call these peaks “shell resonances”.
The s1-peak is sufficiently strong in the absorption spectrum of light polarized along the X-axis of a dimer. However, it is practically suppressed for the polarization of light along the Y-axis (see Fig. 1(a)). With increasing L its position shifts toward large wavelengths approaching to the frequency of the absorption maximum, ωJ, of the TDBC J-aggregate, whereas its intensity decreases. The s2-peak dominates in the absorption spectra of light polarized along both X and Y directions. It is also present in the absorption spectrum of the isolated Ag/J-aggregate nanoparticles as well as in spectra of their single pairs for all magnitudes of L. The frequencies of the s1 and s2 peaks slightly differ from each other for different polarizations.
The peak positions of the o1, ..., o4 bands in the absorption spectra of Ag/TDBC NP dimers are close to those of spectral p1, ..., p4 bands of plasmonic Ag NP dimers (see Fig. 2(a)–2(h)). However, the plexcitonic dimers have an additional group of spectral r1, ..., r4 peaks. These peaks are always located at the opposite side of the “shell resonance” s1 and s2 peaks. The mutual arrangement, relative widths and intensities of plasmonic p1, ..., p4 bands in spectra of Ag NP dimers are accurately reproduced by the o1, ..., o4 bands and are mimicked by the r1, ..., r4 bands of Ag/TDBC NP dimers in quite distant spectral region. This is most evident when the plasmonic spectra of a pair of silver particles are particularly diverse and saturated (see Figs. 2(c)–2(d)). With the variation of the interparticle distance L, the changes in the configuration of plexcitonic o1, ..., o4 and r1, ..., r4 bands of hybrid dimer follow the corresponding changes in plasmonic p1, ..., p4 spectral bands of Ag NP dimer. This is clearly seen from Figs. 4(a)–4(c). The shapes of curves, representing dependences of the absorption peak frequencies on L, are similar to each other for pairs o1x − r1x, o2x − r2x, o3x − r3x, o4x − r4x and o4y − r4y. Thus, the near-field electromagnetic coupling of plasmons with Frenkel excitons leads to the replication of plasmonic peaks, i.e. to the duplication and frequency conversion of plasmonic spectral bands. Further we call the “original” bands those absorption o1, ..., o4 bands of the Ag/J-aggregate dimer, the spectral positions of which are close to the positions of plasmonic p1, ..., p4 peaks of bare Ag dimer. At the same time, the additional r1, ..., r4 peaks, which mimic the relative intensities and positions of the “originals” but are located on the other side of the “shell resonance” bands, are called “replicas” in the present work. Of course, such notations are conditional, and are used only for the convenience of identifying peaks in the absorption spectra. Figures 3(a)–3(d) demonstrate that the “original” bands of o1, o2, o3 and their replicas of r1, r2, r3 in the absorption spectra of hybrid dimers, belong to longitudinal modes o1x, o2x, o3x and r1x, r2x, r3x. The o4 and r4 bands appear for two different polarizations of light (X and Y, see Fig. 1(a)) as o4x, r4x and o4y, r4y, whereas their frequencies slightly differ from each other for different polarizations.
The main pair of plexcitonic o4 and r4 peaks are the “original” and “replica” bands of the dipole plasmon resonance of the Ag-core of an isolated particle. Their intensities and spectral positions depend weakly on the interparticle distance. The pairs of o1 − r1 and o2 − r2 bands are “originals” and “replicas” of two longitudinal plasmonic modes of the elongated structure, formed by partially merging Ag-cores. These peaks vanish when the electric contact between the cores disappears. Essential feature of the pair of o1 − r1 bands is that the original peak, o1, is located at the long-wavelength side from the “shell resonance” bands, s1 and s2, while its replica, r1, is at the short-wavelength side. This is an opposite configuration to the rest of the peaks in the spectra of the plexcitonic dimer. They cannot be observed in the case of fully separated particles, when the “original” peaks are located at the short-wavelength side and “replicas” are at the long-wavelength side from the excitonic peaks. As the distance between the particle centers increases, the o1 peak rapidly shifts to the IR range, vanishing when the electric contact between the cores disappears. Under such conditions, its “replica”, r1, also turns out to be shifted toward longer wavelengths, clinging to the “shell resonance” s1 peak, and rapidly decreases in intensity.
At L = 16 nm and larger, another pair o3 − r3 of longitudinal plexcitonic bands begins to segregate from the corresponding bands of o4 − r4 pair. This segregation is due to the near-field electromagnetic coupling between the Ag-cores and remains after the electric contact between them vanishes. For both small and large values of L, the “originals” o3 and o4 as well as their “replicas” r3 and r4 merge. This is because at small distances of L the nanoparticle cores overlap too match, so that the shape of the dimer is close to nanorod (see Fig. 1(a)). At large distances of L, the electromagnetic coupling between the nanoparticles is weakened and disappears. For all plexcitonic bands mentioned above, the shapes of curves, which show the dependences of the frequencies of the absorption peaks on the interparticle distance, are similar for corresponding “original” and “replica” bands (see Figs. 4(a)–4(c)).
The analysis of spectra presented above makes it possible to interpret, in terms of replication, the absorption bands and the spectra of individual metal/J-aggregate nanoparticles (see Fig. 2(h)). Actually, the band with a maximum of 396 nm is formed by the merged o3 and o4 bands, while the band with a maximum of 596 nm is formed by the merged replicas of these bands. The confluence of these bands is caused by the degeneracy of the electromagnetic modes in the system due to spherical symmetry. The deformation of the shape of the silver core or a sufficiently strong near-field electromagnetic interaction with another system can remove this degeneracy, demonstrating the described effect of the appearance of plexcitonic replicas of plasmonic bands of metallic cores. It is important to stress, that the positions and shapes of merged o3 and o4 bands almost coincide with positions and shapes of the plasmonic 3 and 4 bands of Ag NP dimers, while replicas are shifted quite far away from the originals.
In Fig. 5 we show the spatial distributions of the electromagnetic energy densities of light fields inside and outside the dimer consisting of hybrid Ag/J-aggregate nanoparticles. Each panel represents the energy distributions for the wavelengths corresponding to absorption maxima of the respective spectral bands marked in Fig. 3(a) with arrows. The spatial structures of the field distribution inside the Ag NP dimer for certain plasmonic bands are the same as for the corresponding plexcitonic modes associated with the “originals” and “replicas” of the absorption bands of Ag/TDBC NP dimer. It is important to stress that in most cases the energy distributions at the maxima of the plasmonic bands of bare silver dimer and of the “original” bands of hybrid dimer almost coincide. Figure 5 confirms that at the wavelengths of the plexcitonic bands of “originals”, the presence of the external J-aggregate shell practically does not change the spatial structures of the electromagnetic modes. At the same time, the spatial structures of the electromagnetic modes associated with the bands of “replicas” of Ag/TDBC NP dimer differ from the “originals” mainly by the appearance of additional regions of the concentration of electromagnetic energy near the outer boundary of the J-aggregate shell.
3.2. Analysis of the results obtained in terms of the effective multi-state Hamiltonian and evaluation of the plexcitonic coupling strengths
To clarify the mechanism of spectral band replication, we employ the coupled oscillator model . Within the framework of this model, the formation of pairs of plexcitonic spectral bands are treated in terms of plasmonic and excitonic modes mixing, caused by the near-field electromagnetic interaction. The particular form of the equivalent Hamiltonian should take into account three main features, extracted from the analysis given in Sect. 3.1. First, the absorption spectrum of a hybrid dimer contains 2N plexcitonic bands provided that the absorption spectrum of metallic cores of a dimer consists of N plasmonic bands. This means that in addition to N plasmonic modes the Hamiltonian should contain at least N excitonic modes, to keep the required matrix dimension defined by the number of plexcitonic spectral peaks. Second, the changes in frequency and width of each plasmonic band affect only one pair of coupled “original” and “replica” plexcitonic bands. Thus, for each plasmonic mode the corresponding row (and the column) in the Hamiltonian matrix contains only one non-zero off-diagonal element, which describes the coupling of this plasmonic mode with a single excitonic mode. Finally, the absorption spectrum of hybrid dimer additionally contains P narrow “shell resonance” bands, the parameters of which are not coupled with other bands. We attribute these modes to the excitonic modes of the organic shell. Hence, the Hamiltonian matrix contains at least P excitonic modes, which are uncoupled with any other modes in the system. As a result, the total number of excitonic modes is N + P, where N modes are coupled with the plasmonic modes and the rest P are uncoupled. We suppose that N excitonic modes, coupled with plasmonic modes, belong to the degenerate (or quasidegenerate) manifold of excitonic bands of the “shell resonances”. So, their frequencies and widths can be close (or even coincide) with the parameters of some of the P excitonic bands observed in the spectra of hybrid metal-organic dimer. The simplest equivalent form of the Hamiltonian, which satisfy all the requirements given above, is as follows:Eq. (3) actually reduces to the diagonalization of N matrices of 2 × 2 dimension. The pairs of eigenvalues of these matrices determine the frequencies and widths of the pairs of “original” and “replica” plexcitonic bands in the absorption spectra.
Using this approach, which is equivalent to the coupled oscillator model, we are able to reconstruct the spectra calculated in Sect. 3.1 by using the FDTD-method. Such reconstruction is performed separately for each of the polarizations of the incident light (along the X and Y-axes in Fig. 1(a)). To obtain the correct results for the frequencies, , and widths, , of the plasmonic modes, we evaluate the absorption spectra of pairs of two-layer nanoparticles with an optically passive organic shell, which does not have the absorption J-band but possesses a permittivity equal to the value of for the dye (see Eq. (1) at fJ = 0). Hence, the cross sections of light absorption by these specific type of two-layer dimers are fitted by a set of Lorentz shapes:Eq. (3) would be closest to the maxima of these absorption bands. For each polarization of the incident light we perform this procedure for two sets of the frequencies, and widths, , of the “shell resonances” s1 x/y, s2 x/y and take the best result. As follows from our calculations and detailed analysis, in most cases the s1 x/y “shell resonance” is predominantly coupled with the plasmonic modes under study. It should be noted that the assumption that each plasmonic mode is coupled with only one quasi-degenerate excitonic mode of the organic shell well describes even the rather complicated structure of plexcitonic spectra observed in the case of elongated nanostructures with partially merging metallic cores.
The results of such reconstruction are shown in Figs. 2(a)–2(h) and Figs. 3(a)–3(d) by red dashed curves. Since the intensities of the absorption bands cannot be obtained solely using the coupled oscillator model, they were chosen for the best agreement with the results of the FDTD simulations. The frequencies and widths of the plexcitonic absorption bands were calculated using the effective multi-state Hamiltonian approach. One should note that this allows us to obtain accurate estimations both for the frequencies and for the widths of the absorption bands. This is evident from Figs. 3(a)–3(d), where the absorption cross sections are presented as functions of the photon energy. For a single pair of Ag/TDBC nanoparticles of spherical shape up to 10 plexcitonic bands form the resulting absorption spectrum. Using this approach frequencies and widths of the absorption bands of Ag/TDBC NP dimers (i.e. 20 parameters) are well reconstructed by adjusting only five values of the Rabi frequencies, Ωn, that determine the plexcitonic coupling strength parameters. The dependences of the ħΩn magnitudes on the interparticle separation L are represented in Fig. 4(c). As is seen from this figure, the values of ħΩn vary in the range of 500 ÷ 700 meV, so that the coupling matrix elements, Vn = ħΩn/2, are around 250 ÷ 350 meV.
3.3. Calculation of the extinction coefficient and its comparison with the experimental data for a plexcitonic nanodisk
We have demonstrated above the effect of replication of the absorption bands in the hybrid metal-organic dimer made of a pair of spherical nanoparticles. It is interesting to find out whether is it possible to observe this phenomenon in dimers consisting of nanoparticles of other shapes. Our theoretical analysis shows that the replication effect is convenient for observing in dimers where the absorption band associated with Frenkel exciton in the organic shell (the s1 band in Figs. 2(a)–2(h)) is located on the long-wavelength side of the main (transverse) plasmonic resonance (the p4 band in Figs. 2(a)–2(h)) at a considerable distance on the frequency scale. Thus, it is most favorable to consider hybrid dimers, in which the transverse plasmonic resonances of the metallic core lie in the blue and violet regions of the spectrum. From this point of view, silver dimers and hybrid silver/J-aggregate dimers are of particular interest.
The experimental studies of arrays of silver nanodisks located on a glass substrate and coated with a J-aggregate of TDBC-dye were carried out in [27,28]. In these works the distance between adjacent disks was chosen sufficiently large to exclude the near-field electromagnetic interaction between neighboring nanostructures so that the resulting spectral properties of the entire ensemble are determined by the spectra of individual plexcitonic nanoparticles. We performed the computer simulations of the optical properties of bare Ag-disk and Ag-disk coated with J-aggregate of the TDBC dye and made the comparison of the results obtained with available experimental data.
Figures 6(c) and 6(d) shows the spectral dependences of light extinction coefficient by silver nanoparticle placed on a glass substrate without and with a J-aggregate coating. The diameter of the disk is D = 2R = 111 nm, its height is h = 60 nm. The thickness of the TDBC-dye J-aggregate coating is s = 12 nm. We specifically chose the diameter of the disk equal to 111 nm among the entire set of D values studied in  since at this value of D the plasmonic band is most distant from the absorption band of the J-aggregate of the TDBC-dye. In Figs. 6(b) and 6(c) we present a comparison of our calculations with experimental data. It is seen that our results are in good agreement with the experimental data. The two plexcitonic o1 and r1 bands are observed in Fig. 6(c). Similar to the case of an isolated hybrid spherical particle (see Fig. 2(h)) the spectrum of the Ag/TDBC nanodisk can also be interpreted in terms of replication of the absorption bands. In this case, the o1-band is interpreted as the “original”, and the r1-band as its “replica”. Again, the plasmonic resonance (the p1 band) in the spectrum of the bare silver disk (see Fig. 6(b)) practically coincides with the “original” plexcitonic o1-band in the spectrum of Ag/TDBC nanodisk (see Fig. 6(c)), while the “replica” lies in quite distant spectral region.
3.4. Analysis of the dark-field scattering spectra of plexcitonic and plasmonic nanodisk dimers
The authors of the work  have fabricated samples of compact silver nanodisk dimers by using the available lithography technique and have experimentally studied their dark-field scattering spectra. We have performed computer simulations of such scattering spectra as well as the dark-field scattering spectra of silver nanodisk dimers coated with the J-aggregate of TDBC-dye. This will allow us to demonstrate the possibility of observing the effect of band replication for a single pair of hybrid silver/J-aggregate nanodisks by using the dark-field scattering method. Such hybrid dimers can be made using a technique similar to that used in the works [27,28].
We consider two pairs of silver nanodisks (Ag ND dimers) placed onto the glass substrate: (i) one dimer is fabricated from the nanodisks, which are in contact because of the metallic waist, and (ii) another one is made of nanodisks, which are fully separated. To compare our theoretical results with the experimental data we chose those sizes of the plasmonic dimers, which reproduce the sizes of the dimers used in measurements  of the dark-field scattering spectra (see curves (C) and (D) in Fig. 9 of ). For each dimer we perform a series of three calculations. First, we evaluate the dark-field scattering spectra for the plasmonic Ag ND dimers. A comparison of our results with the measured spectra allows us to confirm the reliability of the simulation method and to adjust the sizes of the nanodisk dimers more accurately. Second, we calculate the dark-field scattering spectra of the same nanodisk dimers coated by the J-aggregate of TDBC dye (Ag/TDBC ND dimers). Finally, we determine the total extinction cross section of the same Ag/TDBC ND dimers for the case of the plane wave incident normally to the sample.
Figures 7(a) and 8(a) represent the schematic views of hybrid nanodisk dimers for two different cases: (i) with a contact between its Ag-cores and (ii) with a gap between the disks, respectively. The radii of the nanodisks R = 40 nm and their heights h = 40 nm are the same for both dimers studied. For the dimer shown in Fig. 7(a) the interparticle distance is L = 83 nm, while the width of the metallic waist is 2w = 60 nm. For the dimer shown in Fig. 8(a), the interparticle distance is equal to L = 110 nm. These sizes are close to those of silver nanostructures studied in . The organic shell thickness is set to be s = 12 nm for both Ag/TDBC ND dimers following . In Figs. 7(b) and 8(b) we demonstrate a comparison of the calculated dark-field scattering spectra of the Ag ND dimers with those measured in  (see curves (C) and (D) in Fig. 9 of that work, respectively). Here and further the results are calculated for the incident light polarization parallel to the axis connecting the centers of the nanodisks in the dimers (X-axis in Figs. 7(a) and 8(a)), following the experiment . It is seen that there is good agreement between our theoretical results and the experimental data.
The dark-field scattering spectra and the total extinction cross sections of the Ag/TDBC ND dimers are shown by full black curves in Figs. 7(c), 8(c) and 7(d), 8(d), respectively. Dashed red curve represents the results of spectra reconstruction, performed using the coupled oscillator model. During the spectra reconstruction we have confirmed the spectral positions of “original” and “replica” bands. They are marked with arrows following the notations used throughout this paper. “Shell resonance” spectral bands are denoted by s1 and s2. To perform the spectra reconstruction, as described in Sect. 3.2, we used the frequency, ωs1, and full width, γs1, of the “shell resonance” band, s1, as parameters and of that excitonic mode in Eq. (3) which is coupled with the plasmonic modes of the silver cores. According to the results of calculations presented in Figs. 7(c)–7(d) and Figs. 8(c) and 8(d), the values of the Rabi frequencies (see Eq. (3)) vary in the ranges: 400 ÷ 600 meV and 600 ÷ 700 meV for the hybrid dimers shown in Figs. 7(a) and 8(a), respectively. This means that the characteristic values of coupling matrix elements, Vn = ħΩn/2, lie in the ranges of 200 ÷ 300 meV and 300 ÷ 400 meV in both cases. Figures 7(c), 8(c) and 7(d), 8(d) indicate that the effect of spectral band replication can be observed in the extinction and dark-field scattering spectra.
We have demonstrated the spectral-band replication effect in the absorption and scattering spectra of plexcitonic dimers consisting of both nanospheres and nanodisks with an Ag-core and a J-aggregate shell. In the spectra of Ag-dimers there are up to five plasmonic absorption bands instead of only one band for an isolated Ag-nanoparticle. Some of these bands refer to the longitudinal plasmonic mode of a nanoparticle pair with contacting Ag-cores, so that they disappear when the electric contact between the cores vanishes. Other bands are formed by the splitting of the plasmonic bands of the isolated nanoparticles caused by the near-field electromagnetic interaction between the Ag-cores. The number of plexcitonic bands in the absorption and scattering spectra of the Ag/J-aggregate dimers is doubled number of the plasmonic bands of bare silver dimers of the same sizes. There are also several narrow spectral peaks, associated with resonances in the J-aggregate coating of hybrid dimer (“shell resonance” bands).
The plexcitonic bands in the spectrum of metal-organic dimer could be separated into two groups, which we called the “original” bands and their “replicas”. Spectral positions of the “original” bands are close to the corresponding positions of plasmonic bands of silver dimer of the same size. The “replicas” always lie at the opposite side of the “shell resonance” bands, which are located in close vicinity to the J-band absorption of the TDBC-dye aggregate. Configuration of plasmonic peaks in the spectra of silver dimer is accurately reproduced in the spectra of Ag/TDBC dimer by the “original” bands and mimicked by the “replica” bands in quite distant spectral region. With varying the interparticle distance, the changes in mutual arrangement, relative widths and intensities of “original” and “replica” plexcitonic bands in the spectra of hybrid dimer follow the changes in plasmonic bands in the spectra of a bare silver dimer. The effect results from the coupling of Frenkel excitons in the J-aggregate shells of hybrid Ag/TDBC nanoparticles with surface plasmons generated in a pair of silver cores.
We have introduced the multi-state Hamiltonian, which describes the observed replication effect in terms of plexcitonic coupling. The strengths of this coupling (or Rabi frequencies) have been determined from the differences in frequencies of the absorption maxima of the “original” and “replica” spectral bands of hybrid dimer. The proposed model accurately reproduces the shapes of these spectral bands for all interparticle distances. The Rabi frequencies are about 400 ÷ 700 meV. This indicates that formation of the pairs of “original” and “replica” spectral bands is caused by the strong plexcitonic coupling. However, by its physical nature and by the manifestation in the light absorption and scattering spectra, the new effect is drastically different from the well-known phenomenon consisting in the appearance of a dip in the spectra of individual core-shell nanoparticles in the strong plexcitonic coupling regime (see, e.g., [15,26,31–34]). The dip is formed when the plasmonic and excitonic bands overlap. On the contrary, the effect of band replication appears when the plexcitonic coupling regime is strong but the excitonic band of J-aggregate lies at some distance from the plasmonic band at the long-wavelength side.
We have confirmed the reliability of our theoretical approach by a direct comparison of the calculated scattering spectra with experimental data obtained in two works dealing with isolated core-shell Ag/TDBC nanodisks  and pairs of silver disks with small interparticle separations . We have performed a computer simulation of the spectra of the hybrid silver/J-aggregate nanodisk dimers under conditions mimicking the experimental ones and predict that the replication effect could be experimentally observed by spectroscopic studies of the metal-organic dimer samples. Such samples can be fabricated by using the available lithography technique  and then by coating Ag-nanodisks with J-aggregates of the TDBC dye using methods of [27,28].
The results of the work have demonstrated qualitatively new manifestations of the effects of a strong plasmon-exciton coupling in optics and spectroscopy of hybrid metal-organic nanostructures. Systems consisting of several plexcitonic nanoparticles and their arrays are promising for the development of some photonic and optoelectronic devices of next generation. This particularly concerns highly sensitive optical nanosensors, hybrid nanowaveguides and nanophotonic integrated circuits functioning on the basis of plexcitonic effects.
Russian Science Foundation (14-22-00273).
We acknowledge funding support listed above. The authors are grateful to A.A. Narits for the valuable discussions.
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