1. Introduction

During the last decade, the study of the interaction between light and nanometer-sized metallic particles has turned into a central and multidisciplinary area of research in nanoscience [1–3]. This emerging field, called plasmonics, has been mainly boosted by the convergence of excellent computational tools and improved fabrication and optical characterization methods, which have led to extremely interesting potential applications in biosensing, surface-enhanced spectroscopies, cancer therapies, renewable energies and active devices [4–8]. A key property of metallic nanoparticles is the dependence of the energies of their Localised Surface Plasmon Resonances (LSPRs) on the geometry of the structure, as well as the sensitivity of the energy of these resonances to the dielectric environment. In order to tailor the resulting optical properties, a huge variety of nanostructures has been engineered showing different shapes, materials or geometric distributions [9–11].

Among different designs, dimers have emerged as a useful canonical nanostructure to understand many basic processes in plasmonics [12–16]. In a dimer, two nanoparticles are placed close to each other giving rise to new resonant modes which are a result of the coupling between particles [12]. For nearly touching particles, the optical response is mainly governed by a Bonding Dimer Plasmon (BDP) resonance, which arises from the hybridization of the dipolar modes of the individual nanoparticles [13, 14]. This BDP mode presents strongly localised charge densities of opposite sign and enormously enhanced local electromagnetic fields at the cavity. In contrast, when a conductive path is established between both particles of the dimer, a Charge Transfer Plasmon (CTP) mode is allowed, with current density crossing through the cavity, involving an oscillating distribution of net charge at every individual nanoparticle [13, 14, 17, 18]. In most of the studies dealing with these effects, classical electrodynamical approaches have been applied. However, recent works incorporating quantum effects have shown that special care must be taken in sub-nanometer wide gaps, because electron tunneling through the junction gives rise to redistribution and modification of optical modes [19, 20]. This regime has been recently revealed experimentally, opening up new prospects in the field of quantum plasmonics [21, 22].

On the other hand, focusing on the field of molecular electronics, single molecules with well-defined molecular conductance [23] are commonly used as interconnects in nanoelectronics and optoelectronics. Thus, we explore here the possibility of using different molecular linkers to establish conductive paths in plasmonic dimers and to understand their role in determining the optical properties of the entire molecule-plasmonic electrodes system. The interaction between plasmonic resonances and the excitonic states supported by molecules has become a problem of current interest [8, 24–28]. Excitonic resonances in molecules or quantum dots, which consist of electron-hole pairs which can be created by the absorption of photons, couple to surface plasmon resonances creating plasmon-exciton states. These mixed states, so called plexcitons, are attractive due to their potential applications for optical devices but also because of the inherent interest of their physical properties. This problem has also been studied within a quantum mechanical approach to address the coupling between a metallic nanoparticle dimer and an excitonic quantum emitter placed in the gap [29], showing that the combined emitter-dimer structure undergoes dramatic changes when the excitation level of the emitter is tuned across the gap-plasmon resonance.

In this paper we present a theoretical study of the optical properties of a plasmonic cavity which consists of a strongly coupled nanoparticle dimer linked by a molecular junction characterized by the presence of an excitonic transition in its constituent molecules. We model the junction with a Drude-Lorentz dielectric function, which allows to tune in a simple way both the energy of the molecules and the corresponding dynamical conductance across the junction, and also to observe the consequences in the optical behaviour of the system.

One of the main interests of plasmonic nanostructures is their potential applications as sensors. Thus, structures with increasing asymmetry producing narrow spectral lines and Fano-resonances have been exploited during the last years to take advantage of the sensitivity of their plasmonic resonances to the dielectric environment [30–34]. The potential use of these systems for sensing is based on the shifts of their plasmonic resonances as the dielectric environment is changed. In this work, in addition to a study of the usual shift-based LSPR sensing, we propose an alternative concept for LSPR sensing based on the analysis of the relative intensities of the plexcitonic resonances as the dielectric function of the environment is tuned.

In order to analyse the optical response of the system, we solve Maxwells equations in the presence of inhomogeneous media using the Boundary Element Method (BEM) to obtain the electromagnetic fields and the optical extinction spectra [35]. Local dielectric functions, described in next section, are used as an input for these calculations.

2. Dimer with a molecular linker

Our system consists of a dimer formed by two gold nanoparticles of radii R = 50 nm separated by a gap of d = 1 nm between them (see Fig. 1(a)), filled with molecules modelled as a cylindrical junction of radius a with curved endings, which match perfectly the spherical surfaces of the nanoparticles. The interparticle distance d chosen for this study is longer than the critical value for quantum tunneling effects to emerge [19], but it allows for strong plasmonic resonances in the cavity, so that the effects of loading the gap are easy to trace. Nonetheless, when a specific molecule is considered, this distance is fixed by the geometry of the molecule itself. The separation of 1 nm, chosen here, is a general separation between particles that reveals the main ingredients of the Physics of our problem. We have checked that our results are also present for other separation distances, establishing very robust spectral trends that we will detail below.

 

Fig. 1 (a) Schematic representation of a gold nanoparticle dimer connected by a molecular linker modelled as a cylinder of radius a and length d. The radius of the gold nanoparticles is R = 50 nm and the minimum separation distance between them is d = 1 nm (proportionality is not respected in this sketch). k is the wave vector of the incident electromagnetic plane wave with polarization of the electric field E along the vertical symmetry axis of the system. (b) Resonant behaviour of the conductance G for molecular linkers of radii a = 1 nm, 5 nm and 10 nm represented by the dielectric function of Eq. (1) with parameters E_ex = ħω_ex = 1.51 eV, f = 1.5 and γ_ex = 0.1 eV. (c) Calculated normalised optical extinction cross-section of a gold nanoparticle dimer bridged by a gold linker as a function of its radius a, and therefore as a function of conductance G as well. G_CTP is the conductance threshold of the cavity load for the emergence of the CTP mode, as given by Eq. (3). (d) Analogous calculations as in (c), but with a molecular load described by a Drude-Lorentz dielectric function using the same parameters as in (b).

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The permittivity of gold nanoparticles is taken from the literature [36]. All the molecules in the linker are assumed to be identical, characterized by a single molecular transition of energy E_ex = ħω_ex, so that their dielectric response as a function of frequency ε_l(ω) can be described by a Drude-Lorentz function as [37, 38]:

It is possible to connect the optical response with the conductance through the junction, similarly to our previous works for pure metallic linkers [17, 18]. The conductivity of the molecules κ(ω) is calculated using the relationship κ(ω) = −iω[ε_l(ω) − 1]/4π[37]. Then, considering the particular geometry of the linker, a spherically indented cylinder (see Fig. 1(a)), the dynamical conductance G(ω) can be obtained, by means of a geometrical factor, from the real part of the conductivity κ₁(ω) as follows [18, 39]:

The resonant behavior of the dynamical conductance provided by the present model is shown in Fig. 1(b), where the conductance is expressed in units of the quantum of conductance G₀ = 2e²/h = 7.7 × 10⁻⁵ S (e is the electron charge and h the Planck constant). We notice that the conductance through the molecule shows a lorentzian profile with a peak centred at energy E = E_ex, and its maximum value increases when wider junctions are considered.

To analyse the optical properties of the system as a function of the radius of the linker, i.e., when increasing the number of molecules connecting the nanoparticle dimer, and thus the corresponding conductance, we calculate the optical extinction cross-section when the dimer is illuminated by light with the electric field polarized along the longitudinal axis of the linker, as sketched in Fig. 1(a). As a reference, we first review the main trends in the optical response of dimers linked by pure metallic materials in Fig. 1(c), where the results for two gold nanoparticles connected by a gold load at the gap are displayed. It is well known that the BDP and the CTP are the main modes in this case, and that there is also a transfer of spectral weight from the BDP to the CTP mode as the radius a (and so the conductance G) are increased [13, 14, 16–18]. The BDP mode, arising from the hybridization of the dipolar modes (l = 1) of the individual nanoparticles, is the dominant mode when the particles are disconnected, and also when narrow linkers are considered. The BDP blue-shifts towards higher energies as the linker becomes wider (the conductance of the linker is thus increased). In contrast, the CTP mode, not allowed for the empty cavity and arising from the hybridization of the monopolar modes (l = 0) of the nanoparticles, emerges for a given width of the molecular load, blue-shifts and eventually saturates at a given energy E_CTP = ħω_CTP as the radius of the linker is increased (implying an increase of the conductance).

The conductance threshold value, which estimates the necessary conductance for the emergence of the CTP mode in conductive linkers, is found to be [17]:

In addition to the BDP and CTP modes, a Bonding Quadrupolar Plasmon (BQP) mode is also observed as a small spectral feature around E_BQP = 2.30 eV (λ_BQP = 540 nm). This BQP mode, arising from the hybridization of the quadrupolar modes (l = 2) of the individual nanoparticles, loses intensity as the conductance of the plasmonic cavity increases [18].

We describe now the results shown in Fig. 1(d), where a molecular linker with an excitonc excitation is considered. In this case the plasmonic resonances of the cavity show a more complex optical response than in the case of the metallic load, since the plasmon-exciton coupling produces a splitting of the BDP mode, which also affects the CTP. We have checked the nature of each plasmon resonance and assigned their character by analysing the symmetry of the near-field modes calculated for each branch [40]. For the molecular linker, the BDP and BQP modes, initially found at E_BDP = 1.85 eV (λ_BDP = 670 nm) and E_BQP = 2.30 eV (λ_BQP = 540 nm), corresponding to the situation where there is no molecule in the gap, also blue-shift and lose intensity progressively as a, and thus conductance G, are increased. However, two resonant modes appear in the Near-InfraRed (NIR) range of the spectrum, in contrast to the only presence of the CTP resonance in the case of the metallic linker at long wavelength region in Fig. 1(c). From the near-field maps [40], we have checked that the most red-shifted resonance, around E ≈ 1.24 eV (λ ≈ 1000 nm), also has a BDP character, exactly as the blue-shifted BDP. In contrast, the resonance around E ≈ 1.55 eV (λ ≈ 800 nm) presents a clear CTP profile. Another remarkable difference between the response of metallic (Fig. 1(c)) and molecular (Fig. 1(d)) linkers is that, for the molecular case, the CTP mode does not emerge blue-shifting from the NIR range of the spectrum, as for the metallic linker, but it appears approximately at the transition energy E_ex of the exciton. Consequently, these results show that a more complex description to characterize the optical response of the load strongly affects the behaviour of the plasmonic cavity modes. We have also performed additional calculations for larger values of the interparticle distance d, observing a completely analogous splitting of the modes and the same physical trends, thus establishing a robust pattern of spectral splittings in all cases.

In order to distinguish between the coupled modes of higher and lower energy, we have labelled them as BDP⁺ and BDP⁻ (upper and lower branches of the splitting), respectively. In general, the optical response of the nanostructure is mainly governed by three resonant modes, BDP⁺, BDP⁻ and CTP. Nevertheless, the latter mode is not always present. When a smaller oscillator strength (f = 0.2) is considered in the dielectric response of the linker (Eq. (1)), it is also possible to observe the splitting of the BDP mode into two modes, but the conductance through the linker is not enough to allow the emergence of the CTP mode. In the case of the molecular linker under consideration in Fig. 1(d), the CTP mode saturates around E_CTP = 1.52 eV (λ_CTP = 815 nm), leading to a threshold of conductance G_CTP ≈ 672G₀, as calculated from Eq. (3). This threshold value is in good agreement with the emergence of the calculated CTP mode observed in Fig. 1(d), indicating that the concept of threshold of conductance introduced for pure metallic linkers is still valid when considering linkers described by a more sophisticated response. The prediction of the emergence of the CTP mode can be very useful when different molecular linkers are present in the cavity, as we consider in next section, where the energy of the molecules is tuned across the cavity plasmon resonances.

3. Plexcitonic splitting

It is well known in the context of atomic physics that the presence of an atom or molecule produces the splitting of the resonances in an optical cavity [41–43]. Similarly, in our system, the cavity BDP and BQP plasmon resonances can couple to the excitons of the molecular linker, giving rise to mixed plasmon-exciton modes so called plexcitons [24–26]. This coupling of the plasmon modes to the excitons can be interpreted in terms of a simple model of two coupled oscillators [27, 44]. The resulting mixed BDP-exciton modes are called BDP⁺ and BDP⁻ modes, while the mixing of the BQP to the exciton produces the BQP⁺ and BQP⁻ modes, where + and − refer to higher and lower energies, respectively. According to this coupled-oscillator interpretation of plexcitons, their energies can be expressed as follows [27, 39, 43]:

In order to explore the mixed modes in dimer cavities, we have tuned the energy of the excitonic resonance of the ensemble of molecules connecting both nanoparticles. We have changed the energy of the exciton E_ex so that the product $f{\omega }_{ex}^2$ in Eq. (1) is kept constant. We are thus varying at the same time the frequency ω_ex of the excitonic transition of the molecules and the reduced oscillator strength f describing them, while we keep constant the damping factor γ_ex = 0.1eV in Eq. (1). In this manner, the conductivity and the maximum conductance for a given volume of the molecular load is fixed, i.e, the centre of the lorentzian function describing the conductance in the cavity G (see Fig. 1(b)) shifts while the intensity of its maximum does not vary.

Figure 2 shows the calculated normalised optical extinction cross-section of a gold nanoparticle dimer bridged by a molecular linker as the energy is tuned as described above, for a given material load volume (a fixed). In this case, four different radii have been considered, a = 1 nm, 5 nm, 10 nm and 15 nm, describing different volumes of molecular loads in the gap. Even for the thinnest linker, we observe that both the BDP and the BQP resonances split into two branches, showing an anti-crossing behaviour centred at the point of intersection of the exciton energy and the BDP and BQP energy lines, indicated by white, solid lines in Fig. 2. The situation is much more evident in the case of the BDP than in the BQP due to the difference in spectral weight between both resonances (BQP is a minor spectral feature in comparison to BDP). In the splitting of the BDP mode into the BDP⁻ and the BDP⁺ modes in Fig. 2, the energy of one of the branches, $E_{BDP}^{-}$, is below the energy of the plasmon cavity mode, while the energy of the other one, $E_{BDP}^{+}$, is above the corresponding plasmon energy. An analogous behaviour is observed in the splitting of the BQP mode into the BQP⁻ and the BQP⁺ modes.

 

Fig. 2 Calculated normalised optical extinction cross-section of a gold nanoparticle dimer bridged by a molecular linker filling the interparticle separation of d = 1 nm with fixed size characterized by a molecular load radius a, as the energy and the oscillator strength of the excitonic transition in the cavity is varied. (a) a = 1 nm, (b) a = 5 nm, (c) a = 10 nm and (d) a = 15 nm. The white, solid lines included indicate the following: Ex the exciton energy line, BDP and BQP the energy lines of the dipolar and quadrupolar bonding plasmon modes when there is no linker. Finally, the white, dashed lines $E_{BDP}^{+}$, $E_{BDP}^{-}$, $E_{BQP}^{+}$ and $E_{BQP}^{-}$ indicate the energies of the coupled modes derived from Eq. (4).

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Figure 2 also shows that, as the linker becomes wider (from (a) to (d), a = 1 nm – a = 15 nm), the energy splitting between the coupled modes $E_{BDP}^{-}$ and $E_{BDP}^{+}$ increases. As already mentioned, the magnitude of the Rabi splitting ħΩ_R in Eq. (4) is estimated from the full electrodynamic simulations when E_P = E_ex and, for the cases under consideration in this study, it is of the order of hundreds of meV. From our calculations, we have also observed that the Rabi splitting increases rapidly as the radius of the molecular load grows, but presents a saturation trend for a > 10 nm.

The results of the approximation derived from Eq. (4) to obtain the energy lines of the coupled modes are shown in Fig. 2 as white dashed lines superimposed to the full electrodynamical calculation ( $E_{BDP}^{-}$, $E_{BDP}^{+}$, $E_{BQP}^{-}$ and $E_{BQP}^{+}$), showing a good agreement with the simulations. This agreement is slightly affected by the intense interaction between the coupled modes of the BDP and the BQP resonances, which increases as the linker becomes wider, and by the presence of the CTP mode as well. We would like to notice that this coupled oscillator model considers the coupling of each plasmon mode with the exciton individually, as if each plasmon mode was the only resonance in the cavity, whereas in practise both plasmon modes are present, and we therefore observe the interaction between the BDP⁺, BDP⁻, BQP⁺, BQP⁻ and CTP modes.

It is also remarkable that in Figs. 2(c) and 2(d) (a = 10 nm and 15 nm), where big cavity loads of molecules are considered, we observe the emergence of the CTP mode, in contrast to the cases with small loads in Figs. 2(a) and 2(b) (a = 1 nm and 5 nm), where the CTP does not appear. This effect is again interpreted with the help of the concept of threshold of conductance G_CTP expressed by Eq. (3) [17, 18]. We have described how for very large conductance, the CTP mode saturates at energy E_CTP = 1.52 eV, leading to a threshold value of the conductance G_CTP ≈ 672 G₀. For the small molecular loads in Figs. 2(a) and 2(b) the conductance of the linker is below the threshold value, thus, the CTP does not emerge. In contrast, for the big loads in Figs. 2(c) and 2(d), the conductance of the linker at the exciton energy is above the threshold value, G(a = 10 nm) ≈ 1000 G₀ > G_CTP and G(a = 15 nm) ≈ 1500 G₀ > G_CTP. Consequently, in these cases the CTP mode is active.

We have performed additional calculations keeping fixed f = 0.5 and changing only the excitonic energy E_ex[39]. The plexciton splitting obtained in this way is similar to that of Fig. 2, but the CTP mode is not activated in such a case, because the conductance is smaller for the same load volume and the conductance threshold value is not reached at the range of energies considered. The concept of conductance threshold is thus a very useful tool to identify and predict the emergence of the CTP mode and, therefore, to analyse high-frequency transport in coupled cavities filled with optically active molecules.

4. Plexcitonic sensing

Let us explore now the sensing capabilities of the plexcitonic resonances studied above. Fig. 3 shows the calculated normalised optical extinction cross-section of a gold nanoparticle dimer with a gap between the particles of distance d = 1 nm, as the dielectric function of the embedding medium ε_d is modified. In Fig. 3(a) we consider, as a reference, the dimer with no load in the gap, while in Fig. 3(b) the particles are bridged by a molecular linker of radius a = 3 nm, characterized by an excitonic transition with energy E_ex = 1.24 eV, corresponding to λ_ex = 1000 nm. In the case of disconnected nanoparticles, the BDP mode, which is the plasmonic resonance governing the optical spectrum, red-shifts as the dielectric constant of the embedding medium is increased. The behaviour shown in Fig. 3(a) is similar to that of other nanostructures under similar conditions, and is the basis for conventional LSPR sensing [18, 30]. The BDP⁺ plexciton in Fig. 3(b) also presents a similar behaviour, red-shifting as the surrounding dielectric constant ε_d is increased. However, the BDP⁻ plexciton hardly red-shifts as ε_d is increased, indicating that this is not an adequate resonant mode for this type of sensing. This is also observed in Fig. 3(c), where we summarize the evolution of the shifts of the BDP (black), BDP⁺ (blue) and BDP⁻ (red) modes as a function of the dielectric constant of the embedding medium. The efficiency of plasmonic systems as sensors in shift-based LSPR sensing is usually estimated by its figure of merit (FOM), which is defined as [30]:

 

Fig. 3 (a) Calculated normalised optical extinction cross-section of a gold nanoparticle dimer with a minimum separation distance between the particles d = 1 nm, as the dielectric embedding constant ε_d is varied. (b) Calculated normalised optical extinction cross-section of a gold nanoparticle dimer bridged by a molecular linker with length d = 1 nm, load radius a = 3 nm and excitation energy E_ex = 1.24 eV (λ_ex = 1000 nm), as the dielectric embedding constant ε_d is varied. (c) Shift of the BDP, BDP⁺ and BDP⁻ modes in Figs. 3 (a) and (b) as a function of the dielectric constant of the embedding medium. (d) Variation of the relative intensity of the BDP⁺ and BDP⁻ modes in (b) as a function of the dielectric constant of the embedding medium.

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In the following, we focus our attention on the behaviour of the BDP⁻ mode, since this resonance shows a nonconventional behaviour. This plexciton, which is a minor spectral feature when the embedding medium is vacuum, hardly red-shifts as the surrounding dielectric constant ε_d is increased. However, by increasing ε_d the BDP⁻ mode becomes a spectral feature as strong as the the BDP⁺ mode in terms of intensity, as shown in Fig. 3(d), where the relative intensities of the BDP⁺ and BDP⁻ modes change as the embedding dielectric constant ε_d is varied. This figure shows that, for large values of the dielectric constant, the BDP⁻ mode gains enough spectral weight to become more intense than the BDP⁺, indicating that the balance of spectral weight of both plexcitons can be inverted by means of a change in the dielectric embedding medium. We believe that this dramatic variation of the relative intensity of the lower energy plexciton mode might be the key for a new strategy for LSPR sensing based on the variation of the intensity of the peaks rather than on their shift.

This different behaviour of the BDP⁺ and BDP⁻ plexciton modes can be understood with help of Fig. 2. The BDP⁺ mode at the considered excitation energy E_ex = 1.24 eV presents a plasmon-like behaviour, thus red-shifting as the dielectric constant of the embedding medium is increased, as plasmon resonances do in standard localised plasmon sensing. In contrast, the BDP⁻ plexciton presents an excitonic-like character, thus keeping its energy constant as the medium is varied. This suggests that, depending on the desired purpose, plexciton modes can be tuned by means of the excitation energy E_ex to exhibit a more or less pronounced exciton/plasmon-like behaviour.

Finally we explore the sensing capabilities of the other important spectral feature of the plexcitonic system, the CTP mode, which is present when the conductance through the junction is above the conductance threshold G_CTP, as explained in the previous section. Fig. 4(a) shows the calculated normalised optical extinction cross-section of a gold nanoparticle dimer bridged by a molecular linker, of length d = 1 nm, load radius a = 10 nm and excitation energy E_ex = 0.5 eV (λ_ex = 2480 nm), as ε_d is varied. In Fig. 4(b) we observe clearly that the CTP mode in Fig. 4(a) is more red-shifted, as ε_d is increased, than the BDP mode. Figure 4(c) shows the shift of the CTP in Fig. 4 (a) versus the index of refraction of the embedding medium, n. The slope given by the linear regression is m = 0.393 eV/RIU and the f whm = 0.032 eV. Thus the FOM parameter for the CTP of gold dimers connected by molecular linkers with a = 10 nm and excitation energy E_ex = 0.5 eV (λ_ex = 2480 nm) is FOM= 12.4. This large FOM for the CTP mode is mainly due to the remarkable shift which occurs when changing the dielectric constant of the embedding medium and also due to the sharpness of the peak. The largest FOM values in the literature, of the order of 10 − 20, have been achieved using Fano resonances [3]. Therefore, we consider that the prominent CTP mode is a good candidate for standard LSPR shift-based sensing.

 

Fig. 4 (a) Calculated normalised optical extinction cross-section of a gold nanoparticle dimer bridged by a molecular linker, with length d = 1 nm, load radius a = 10 nm and excitation energy E_ex = 0.5 eV (λ_ex = 2480 nm), as the dielectric embedding constant ε_d is varied. (b) Shift of the BDP and CTP modes in Fig. 3 (a) as a function of the dielectric constant of the embedding medium. (c) Linear plot of the CTP shifts vs. refractive index of the embedding medium.

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5. Concluding remarks

In conclusion, we have studied theoretically the optical properties of a plasmonic cavity consisting of a strongly coupled metallic dimer when an ensemble of molecules, characterized by an excitonic transition, is placed in the cavity. The Bonding Dimer Plasmon (BDP) and the Bonding Quadrupolar Plasmon (BQP) resonances, arising respectively from the hybridization of the dipolar and quadrupolar modes of the individual nanoparticles, couple to the excitonic transition. As a consequence, these modes split into two coupled plasmon-exciton modes, called plexcitons. The Charge Transfer Plasmon (CTP) resonance, arising from the hydridization of the monopolar modes of the individual nanoparticles, is highly blue-shifted from its original spectral position and it depends strongly on the conductance of the molecular linker. The concept of conductance threshold for the emergence of the CTP mode previously introduced for metallic linkers is still valid, in spite of the complexity introduced in the linker with respect to a pure conductor.

We have explored the efficiency of the new mixed states for LSPR sensing, showing that the CTP mode is a good candidate for shift-based sensing, with a FOM value around 12. Furthermore, for the BDP-exciton mixed states, we have observed an interesting behavior for sensing based on the change of the relative intensity of the resonances, thus introducing a new framework for sensing based on the evolution of plexcitonic intensities rather than on spectral shifts.

We believe that the study of this kind of structures may contribute to improve the knowledge of plasmonic nanostructures for their use as active devices.