Modelling plexcitons of periodic gold nanorod arrays with molecular components

Plasmonic or exciton/plasmon (plexcitonic) systems are presently described based on electromagnetic models, ignoring the need for an improved microscopic understanding. This is based on the fact that a full quantum mechanical approach on a micrometer scale still represents a considerable challenge. In this paper we report on the experimental observation of plexcitons in 2D gold nanorod array systems coupled to dye molecules and we provide a description of the experimental data using a quantum model. We show that treating the collective behavior in the array as being represented by a single quasiparticle is a suitable approximation that offers the opportunity to avoid the complicated calculation of long-distance interactions between the individual nanoparticles of the plexcitonic, periodic system. This enables us to model the optical response of plasmons in nanostructured arrays in contact with quantum emitters and to derive microscopic informations. Our work provides a potential tool for the design of plexcitonic devices, which rely on periodic metallic nanostructures.


Introduction
In the last decade, increasing attention has been addressed towards surface plasmons of metallic nanostructures and the interaction of surface plasmons with quantum emitters, for instance quantum dots or dye molecules [1][2][3][4]. Such nanostructures offer the opportunity to control light-matter interaction on subwavelength length scales. Possible applications are related to sensing, photonic information processing [5,6].
The theoretical modelling of light-matter interaction usually relies on classical electromagnetic models [7,8]. It is based on a frequency-dependent, complex dielectric function ò(ω) that is used to represent each individual part of the nanostructured system. If the size is reduced to the nanometer scale, however, the bulk dielectric function ò(ω) has to be corrected in order to extend the validity of the classical approach [9,10]. Besides this, numerical simulations of Maxwell's equations can not completely represent the interactions between quasi-particles, for example, plasmons and excitons [11]. Therefore, a more microscopic, quantum mechanical point of view is required. Such an approach would also provide a more complete understanding of plasmonexciton interactions in nanostructured, plexcitonic systems [12]. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Several quantum mechanical treatments are presently evaluated [11,[13][14][15][16][17]. In particular, time dependent density functional theory (TDDFT) relying on quantum modeling or quantum-corrected modeling, seems to be promising but it suffers from a limited calculation volume [16,17]. The Dicke model can be applied to a many emitters case within the low excitation limit [14]. The Zubarev Green's functions approach introduced by Manjavacas [11] can be applied to describe the optical absorption of systems, which consist of a single exciton-supporting quantum emitter and a single nanoparticle or a nanoparticle dimer. It is still challenging to deal with extended plexcitonic systems having a larger number of internally interacting nanostructures by exploiting a quantum treatment.
In this paper, we demonstrate the modelling of gold nanorod array-molecule plexcitonic systems that have collective plasmon modes and molecular excitons [18]. Collective phenomena within the internal interacting plasmonic structures are treated approximately as single quasiparticles which leads to an analytic and microscopic description of the optical response of the hybrid plexciton. This approach is helpful for a fundamental understanding of the underlying phenomena. It also guides the design of devices based on strong coupling between molecular excitons and plasmons, for which dielectric function simulation is only of limited suitability. Particularly in those cases a nanoparticle array is preferred rather than a group of randomly distributed nanoparticles. This is due to the fact that the former offers a better reproducibility and smaller variation of properties on a microscopic scale.

Experimental
Periodic Au nanorod arrays with a quasi-hexagonal superlattice are prepared based on electro-deposition of Au wires into anodic aluminum oxide (AAO) porous templates [19][20][21]. The templates establish the overall geometry by defining the diameter and distance of the Au nanorods, and they also contribute to the effective dielectric constant of the array. A respective preparation scheme is given in figure 1(a).
In short, AAO nanoporous templates are fabricated by a standard two-step anodization process [21,22]. The aspect ratio (AR) of the Au nanorods in the arrays is controlled by varying the duration of electrochemical deposition to be in the range of 7 to 15. A typical SEM image of the top-plane is given in figure 1(b). Here, the white dots correspond to the Au nanorods, which are surrounded by the AAO template in the plane. The Au nanorods have a diameter of D w ≈23 nm and form a hexagonal superlattice with an inter-rod distance of D int ≈63 nm. In figure 1(c) absorption spectra are shown taken at room temperature using a UV-VIS-NIR spectrometer (PE Lambda 900) of an Au nanorod array with AR=15.
A series of spectra was obtained while changing the incident angle from 0°to 40°between the incident direction and the normal of the array plane. Two maxima are observed if the incident light polarization has an electric field component both along and perpendicular to the long axis of nanorods. The two peaks at l = are attributed to the transverse (T) and longitudinal (L) resonance of the Au nanorods, respectively [18]. This assignment agrees well with the observation of an extreme low intensity of the L-mode when no electric field component parallel to the long axis of nanorods exists, as shown by the θ=0°incidence (black curve) in figure 1(c). Meanwhile, the intensity or the resonance wavelength of the T-mode should not depend on the incident angle of a linear polarized light, because of its localized surface plasmon-like behavior and the invariant extinction cross-section with different angles.
Comparing these data with observations on isolated Au nanorods, two peculiarities become apparent: (a) the energy of the L-mode is much higher compared to that of an isolated Au nanorod with the same aspect ratio. In the latter case it should be observed as a maximum in the infrared region with a wavelength larger than 1700 nm [23,24]. The pronounced blue shift from 1700 nm to 750 nm corresponds to an effective interaction between the nanorods in the array [25]. As reported earlier, such an interaction between two parallel Au nanorods depends on D int and also on the AR of each nanorod. When D int is relative small and the AR is relatively large, the blue shift of the absorption maximum is larger [26]. For our case with AR=15 and D int =63 nm it is only reasonable that not only two neighbors interact with each other. In contrast, the coherence length of the surface plasmon is estimated as l coh =t plasmon ×v plasmon =320 nm, which is a factor of 5 larger than the superlattice constant D int . The low energy maximum therefore corresponds to a collective longitudinal mode of the array.(b) The L-mode shifts with the angle of incidence [27], i.e. by 4% to higher energy while the angle of incidence increases from 10°to 40°. Figure 1(d) shows the resulting dispersion diagram of the L-mode energy as function of in-plane momentum. The resulting quasi-linear dispersion relation matches quite well to simulations [28]. Using templates with Au wire arrays is an advantage as the resonances can be tuned by varying the aspect ratio, distance and the angle of incidence on the template. Thereby, both a rough adaption and a fine-tuning of the working wavelength/energy can be achieved allowing to use emitters with excitonic energy that cannot be adapted easily. In our studies we use the highly efficient and chemically stable Aza-BODIPY (boron-dipyrromethene) [29]. For the preparation, 6 μl Aza-BODIPY solution in CHCl 3 with a concentration of 10 −5 mol L −1 is dropped on the surface of the Au nanorod array and the solvent is allowed to evaporate.
As shown in figure 2(a), Aza-BODIPY has an absorbance maximum at around 648 nm (thin line), being very close in energy to the collective L-mode of Au nanorod arrays. By tuning the aspect ratio and angle of incidence two limiting cases are achieved: (a) strong coupling with coinciding energies of the collective L-mode and the molecular excitons and (b) weak coupling with disagreeing energies. Please note that a weak coupling in our context is associated with a vanishing splitting. In figure 2(a), an equidistant splitting by ±153 meV of the previously observed absorption maximum into two broad but well separated maxima is observed. The respective maxima are hybrid plasmon-exciton branches. Such effects are characteristic for strong coupling and the two maxima can also be described within the scheme of Rabi oscillations of a two-level system [30,31]. The absorption maxima of two hybrid branches in our observation are rather broad due to (1) variation of the coupling strength between the L-mode and exciton induced by sample inhomogeneities, and (2) lifetime effects of each hybrid branches due to strong coupling between broad L-modes and excitons.
In figure 2(b), corresponding data of the same porous AAO structure but deposited Au nanorods having a larger AR are shown. The L-mode energy is now observed as a maximum at 685 nm (thick red line). Applying the Aza-BODIPY solution under the same conditions there is only a small shift of the maximum by 16 nm. This effect can be attributed to the variation of the dielectric environment around the Au nanorods by the dye molecules. Hence, there is only a weak coupling of the exciton and the plasmon. Please note that, compared with the Aza-BODIPY molecules, Au nanorods absorb the incident photons much more efficiently due to the more pronounced light-matter coupling of the plasmon. Furthermore, in both strong and weak coupling cases, the total number of Aza-BODIPY molecules on the array is so small, that no apparent peak of uncoupled Aza-BODIPY is observed. This can be derived from figure 2(b).

Modelling
In order to model the absorption of plexcitonic systems, a Zubarev's Green function method is evaluated following the work of Manjavacas et al [11]. In the latter work the interaction between one quantum emitter and one single or one pair of metallic nanoparticles has been considered. The Hamiltonian of the hybrids is described with decay in Manjavacas's model, where H 0 describes the noninteracting evolution of the emitter and plasmons; H int describes the elastic plasmon-emitter interaction; H decay describes inelastic interactions. Each term, for instance, O N 2 creation-annihilation operator pairs and corresponding coupling strengths D d d i j and D d c i , while N specifies the number of metallic particles and d † and c † (d and c) are the creation(annihilation) operators for the nanoparticle plasmon and emitter.
In our case, the hexagonal array consists of a certain density of nanorods: Within the coherence length (≈320 nm) of collective L-modes, approximately two shells of nearest neighbor Au nanorods have to be considered. Even if we consider only the first shell of one rod, seven nanorods are involved in the Hamiltonian. So, 28 distance dependent coupling strengths (D d d i j and D d c i ) are for instance involved in the H int term in the Hamiltonian. Furthermore, considering itinerant surface plasmons and long-distance interactions, the Hamiltonian becomes even more complicated and an exact diagonalization of this many-body system is not possible up to now. Therefore, an approximation should be applied.
Here, we introduce a quasi-particle approach, treating the collective L-mode as emergent from a single quasi-particle instead of calculating a representational N-particle model. In other words, a collective mode will be quantized as a boson, then this boson is applied in Manjavacas's one emitter/one particle model [11]. The basic idea of this approximation is sketched in figure 3. Thereby, our model simplifies from a system with a large number of plasmons (corresponding to a large number of metallic nanoparticles) and one molecular exciton to a system which consists of one single boson (N=1) and one exciton. The N-particle problem, therefore, can be transfered to a problem of interaction between two single particles.
Based on the one emitter/one particle model from [11] and the quasi-particle approximation, the optical absorption spectrum of the plexciton system in the array, is then given by Here, w is the energy of one incident photon;  p and  e represent the energies of surface plasmons in the Au nanowire array and the molecular exciton, respectively; δω p and δω e denote energy shifts; Γ p and Γ e are the inelastic decay rates of the plasmon and exciton, respectively; Δ pe is the plasmonexciton coupling strength; n e is the expectation value of initial occupation of the excited state of the exciton.
We note that  dw p p in equation (1) (as well as  dw e e , when n e is much smaller than 1), can be treated as one term and represents a shifted energy level.
In figure 4(a) the experimental collective L-mode data as a function of energy are shown as black dots. Normalized data are converted from the blue curve in figure 2(a), and the T-mode contribution is removed. The data were analyzed using The modeled spectrum exhibits a distinct mode splitting in accordance with the observation with the coupling constant (half of the peak splitting) being Δ pe ≈147.5 meV. This value is reasonable, if compared with the reported value of the coupling constant of ∼140 meV between plasmons of single Ag nanoparticles and molecular excitons in DTBC [32]. Besides this, the values of inelastic decay rate of the hybrid plasmon branch and exciton branch ω p , 0.246 eV and ω e , 0.3805 eV, are also in accordance with the measurement results shown in figure 2(a).
In figure 4(b), we show the pronounced effect of the coupling constant Δ pe on the line shape of the absorption. For this a series of normalized data with varying Δ pe are calculated. As shown, for Δ pe =260 meV, the dip is deep and the separation between the two resulting maxima is obvious. For decreasing Δ pe , the dip becomes more shallow and at Δ pe =100 meV it almost disappears. Comparing this with the upper experimental spectrum, the quality of the data representation using the quasiparticle model is very satisfying.

Conclusion
In summary, a quasi linear dispersion behavior of collective L-mode in Au nanorod arrays has been observed. This is a finger print of the collective nature of the plasmon within the array. The coupling between molecular excitons and this L-mode is fine-tuned via their energy separation that depends on the aspect ratio and the angle of incidence. In order to theoretically describe the interaction between collective plasmons and excitons, a quasiparticle model is used. This model extends the applicability of the Zubarev's Green function method [11] to a many-particle system. The validity of this quasiparticle model has been demonstrated based on an excellent agreement of the calculated spectra with the experimental data.