1. Introduction

Interaction of plasmons with excitons in molecules or quantum dots attracts considerable interest and research efforts in hopes of realizing high-end emerging applications like single-atom lasers [1], single-photon switches [2], and quantum cryptography [3] to name a few. Use of plasmonic nanostructures, rather than photonic crystals [4] and microring resonator cavities [5], is dictated by the ability of the former to capture and guide light in deep subwavelength volumes [6]. Experimental and theoretical studies so far have demonstrated various degrees of interaction strengths between metal nanoparticles and excitons [7] in both large structures [8–13] as well single nanoparticles [14,15] and dimer nanostructures [16]. These structures are in an overwhelming majority made either of silver or gold [17] – the workhorses of plasmonics due to low losses, though in principle in the UV aluminum is viable due to its optical properties [18].

Using Ag and Au means that the field enhancement near the surface of the metal nanoparticles easily exceeds one order of magnitude in the case of lone objects and for closely spaced dimers or rough surfaces can reach in excess of several hundreds. Interactions between plasmons and objects placed in these hot spots are thus greatly amplified leading to e.g. single molecule detection through Raman spectroscopy [19–21], enhanced absorption [22] and efficient solar harvesting [23].

It is usually accepted that in the case of plasmon-exciton nanostructures a large field enhancement is an important factor enabling strong coupling and Rabi splitting. This is true, however, simultaneously we stress that large field enhancement is not important for the values of Rabi splitting as such, as here we demonstrate virtually unchanged peak spitting for field enhancements spanning close to one decade. Although this seems surprising at first glance, in fact this just reflects that Rabi splitting alone is not an appropriate figure-of-merit for strong coupling. Rather the figure-of-merit should adequately consider losses. There are two alternative ways of doing this: (i) by taking a ratio between Rabi splitting and loss, 2g/γ_pl, and (ii) by considering cooperativity, $2g/\sqrt{{\gamma }_{\text{pl}}{\gamma }_0}$, where g is the coupling strength and γ_pl and γ₀ are the plasmon and exciton dissipation rates, respectively. Importantly, both of these parameters scale with the field enhancement.

The coupling strength of plasmon-exciton hybrids is given by

In this contribution we perform an in-depth study of various material-related and geometrical parameters on strong coupling between plasmons and excitons in the classical regime. This treatment is valid whenever the number of excitations in the system does not exceed the number of excitons [7]. For small structures considered here Ohmic losses dominate over radiative ones, which implies that material losses essentially determine Q of the plasmon resonance. We monitor the coupling strength as a function of the mode volume, that is, the distribution of electromagnetic energy density, ρ = ∊|E|², around the nanoantenna. In the first approximation the mode volume does not depend on the material loss, but instead it is determined by the arrangement of metal nanoparticles comprising the nanocavity. We study the effect of mode distribution and mode extraction on strong coupling. Finally, we show that upon sufficient compression of plasmonic modes, a single quantum dot may reach strong coupling with plasmon mode under realistic conditions. This latter observation although has been predicted theoretically [26–28], was reported in experiments only very recently [29–31] and should be further verified.

2. Model hybrid systems

In this study, we exploit three specific hybrid nanostructures as shown in Fig. 1, which we investigate using the finite-difference time-domain method (FDTD Solutions, Lumerical) and the T-Matrix method in one case as noted below. First, we consider a metal sphere of 30 nm diameter in whose vicinity we place a small dielectric sphere with dispersive permittivity mimicking that of a dye molecule. Maximum field enhancement near the metal nanosphere is about 10. Placing two spheres in a dimer arrangement [Fig. 1(b)] 10 nm apart pushes the field enhancement up to 30 at the center of the gap. Finally, we maximize the field enhancement by placing the dye nanoparticle between two triangular nanoprisms arranged in a bow tie fashion as in Fig. 1(c). In all these cases metal nanoparticle spectra are dominated by absorption, as illustrated by cross section plots shown in the top row of Fig. 1(d). Due to varying geometrical parameters, the resonance frequency of each nanostructure (sphere or both dimers) is different; hence, the parameters of the Lorentz permittivity of the dye change so that its absorption resonance coincides with that of the structure [note the dashed line in the top row of Fig. 1(d)]. However, to assure that conditions are equal despite varying resonance positions, the parameters of the Lorentz material are chosen such, that the transition dipole moment of the dye sphere is the same in all three cases and is equal to ∼ 60 Debye. We intentionally use the transition dipole moment that is 2–4 times higher than realistic values of colloidal quantum dots to make the conclusions more instructive. Further, we consider more realistic transition dipole moment values. The parameters of the Lorentz spheres are summarized in Table 1 and the functions themselves are plotted in the bottom row of Fig. 1(d). The permittivity of the metal is described by a Drude function ${∊}_{\text{metal}}(\omega )=1-{\omega }_p^2/\left({\omega }^2+i{\gamma }_{\text{p}}\omega \right)$, where ω is the (angular) frequency, ħω_p = 6.25 eV is the plasma energy, and γ_p is the collision frequency which varies from 0.2 to 2 eV.

 

Fig. 1 Investigated metal-dye nanostructures. (a) Metal sphere d_m = 30 nm in diameter with a spherical dye particle d_d = 5 nm in size, separation of 2.5 nm. (b) Dimer of identical spheres as in (a) separated by 10 nm with a centrally placed dye particle. (c) Two thin nanoprisms with an edge of 70 nm, corner rounding 7 nm radius, and thickness of 10 nm. Prism separation is 10 nm and the dye particle is as previously. (d) Top row: cross sections of the metal nanostructure (solid lines) and absorption cross section of the Lorentz sphere (dashed line, multiplied by indicated factor for readability). Bottom row: real and imaginary parts of the Lorentz permittivity.

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Table 1. Lorentz permittivity ${∊}_{\text{j}}(\omega )=1-f{\omega }_{\text{j}}^2/\left({\omega }_{\text{j}}^2-{\omega }^2-2i{\gamma }_{\text{j}}\omega \right)$ parameters used to model the optical properties of the dye: resonance frequency, line width, Lorentz permittivity, and transition dipole moment; λ₀ is the resonance wavelength of the nanostructure. These parameters give in all cases a transition dipole moment of μ_e = 60 D.

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3. Results

We begin by considering a simple case of a metal sphere placed adjacent to a Lorentz sphere. The extinction spectra [dominated by absorption, see Fig. 1(d)] are shown in Fig. 2 for five Drude damping values (with all other parameters constant). In all cases, interaction between the plasmon and exciton resonance results in a clear dip. The peak splitting, Ω, is constant (within calculation error) and equal to ∼ 130 meV, as illustrated by the vertical dashed lines in the top row of Fig. 2. In order to appreciate the significance of a constant Ω, we analyze the optical properties of the metal resonator. Figure 2(d) displays the electric field enhancement generated by the metal sphere as measured at the position of the center of the Lorentz sphere (5 nm from the metal surface). Due to increasing damping, the enhancement decreases by a factor of more than 5 when increasing γ_p from 0.2 to 2 eV. However, this decrease of the field enhancement is not accompanied by a concomitant reduction of the peak splitting. Instead we observe a nearly constant splitting for all ranges of losses up to very high values of γ_p = 2 eV, which correspond to extremely bad plasmonic metals such as chromium. What does change with variation in material loss is the depth of the dip. We argue that despite observation of Rabi spitting in such lossy systems, one is hardly able of claiming strong coupling in this case, as the losses dominate over the coupling rate by more than one order of magnitude. Thus, it is important to realize that even if splitting is visible in the spectra, one cannot immediately conclude this is due to strong coupling [7].

 

Fig. 2 (a–c) Extinction cross sections of a coupled metal sphere – dye nanoparticle structure. The damping of the metal increases by one order of magnitude and the peak splitting Ω (see the vertical dotted lines and the table) remains practically constant, only its visibility decreases up to a point when it almost disappears for the highest losses γ_p = 2 eV. (d–f) Field enhancement at the center of the Lorentz nanosphere (left y-axis, circles) and mode volume of the Drude metal sphere (right y-axis, squares) as function of the loss parameter γ_p of the metal sphere. Note the considerable decrease of the field enhancement accompanied by an almost constant mode volume. Extinction cross sections of dimers composed of (a) two spheres and (c) two triangles coupled to a dye nanoparticle. (b,d) Field enhancements (left y-axis) and mode volumes (right y-axis) of the sphere dimer and triangle dimer structures, respectively. The field enhancement values for the dimers are considerably larger than for the single sphere, yet the peak splitting, when normalized to the resonance energy of the plasmonic structure, is only slightly larger for the dimers. The mode volumes are, on the other hand, slightly smaller for the dimers than the single sphere, in agreement with the peak splitting dependence. Note, that while the field enhancement is strongly dependent on the damping, the mode volume and the peak splitting are not.

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Having observed that Ω does not vary, we calculate the mode volume of the metal sphere for each Drude damping parameter. The results are plotted with squares in Fig. 2(d) (right y-axis). There exist several approaches to calculating the mode volume of plasmonic nanostructures [32–34]. In particular we follow Koenderink [32] and define the mode volume as

To verify if such a picture holds in other types of plasmonic resonators, we now turn to structures with much larger field enhancements – nanoparticle dimers. The optical spectra of sphere- and triangle-dimers coupled to the Lorentz spheres are shown in Fig. 2(b,e) and Fig. 2(c,f) respectively. Measured splitting is 160–190 meV and 120–140 meV for dimers made of spheres and triangles, respectively. Comparing these values to the field enhancements for various loss parameters calculated at the positions of the Lorentz spheres [Fig. 2(e,f)] we note again the lack of any relationship. Only when we analyze the mode volume, which in both cases is lower than for the sphere, though not by a lot, we see again the basis for stable splitting. Estimating the coupling rates g from the calculated mode volumes V = 1.0 × 10⁴ nm³ and taking into account red shift in the plasmon resonance frequency for the dimer structures we get ∼66 meV and 47 meV, respectively. These values compare rather well with the splitting that is read off the cross section plots, however, we observe that the equation for the coupling rate g = μ_e|E_vac| does underestimate the splitting in case of dimer structures by about 20%. We note that a small reduction in the mode volume (from 1.3 × 10⁴ nm³ to 1.0 × 10⁴ nm³) cannot account for larger splitting in case of dimers even after taking the plasmon resonance shifts into account. This extra splitting arises from (i) better mode overlap between excitons and plasmons in case of dimer structures, as the maximum energy density ρ_max is efficiently pulled out of the metal due to nanoparticle interaction and (ii) additional mode concentration in the gap region due to presence of high refractive index dielectric [35]. Indeed, the energy density plots show that the maximal energy density is located inside metal for the single sphere [inset Fig. 2(d)], while it is mostly concentrated in the gap region for both dimer nanostructures [insets Fig. 2(e,f)]. Based on these observations, in the remaining part of the manuscript, we focus on properties of the mode in terms of its localization and distribution. This knowledge is essential for optimizing plasmon exciton interactions.

We start by studying the effect of mode overlap between excitons and plasmons (see Fig. 3, here we used the T-Matrix method). The spatial extent of the mode, η(r) = ρ(r)/ρ_max ≤ 1, not only determines V but also allows to calculate the field enhancement and the Rabi splitting at arbitrary position r near the nanostructure. Both M(r) and Ω(r) scale as $~\sqrt{\eta (\mathbf{r})}$, which describes the distribution of local density of states around the nanoantenna. By displacing the Lorentz sphere up to 16 nm away from the center of the dimer, different local density of states is probed [Fig. 3(a)]. In agreement with the above, the splitting becomes smaller and the dip shallower. This can be compared to

 

Fig. 3 (a) Absorption cross sections for sphere dimers with the Lorentz nanoparticle moving out of the hot spot as indicated in the inset. The electric field is polarized along the axis joining the two metal spheres. As the exciton leaves the hot spot and experiences a deceasing field enhancement the peak splitting decreases (mainly due to smaller overlap between the molecules and the mode) and the dip becomes shallower (mainly due to a weaker field) in line with $\mathrm{\Omega }(\mathbf{r})=2\sqrt{\eta (\mathbf{r})}{\mu }_{\text{e}}\left|E_{\text{vac}}\right|$. (b) Comparison of measured (squares) and calculated (circles) splitting. The calculated splitting is multiplied by ca. 1.6 to match the measured values (see Fig. 5).

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Fig. 4 Compression of the mode volume of a silver bow-tie dimer loaded with a 4 nm dielectric sphere in the gap as a function of the sphere’s permittivity. Increase of ∊ from 1 to 25 deceases the mode volume to ca. 0.2 of the initial value. Although 25 is unrealistic at optical frequencies, ∊_∞ = 6 is typical for quantum dots and this decreases V about 2.5 times, which increases the coupling strength 1.6-fold. This is consistent with the multiplicative factor used in Fig. 3.

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We now turn to the role of the mode volume itself. By appropriately designing the nanostructure the mode volume can be made very small, however, there are limits to what can be achieved [25, 32–34]. For single metal nanoparticles in the 50 nm size range V is smaller than the physical volume; however, as the diameter decreases the mode volume approaches the physical one [32]. In Fig. 2(a) we see that the mode volume is well approximated by the geometrical volume of the spherical nanoparticle 4πR³/3 for R = 15 nm, which is V_geom = 1.4 × 10⁴ nm³. Thus, an appreciable increase of the coupling rate can be achieved by shrinking the particle diameter. An alternative way of compressing the mode is to utilize dimers to localize the electric field in the gap. In Fig. 5(a) we show field enhancement and mode volume for the bowtie geometry calculated as a function of interparticle gap within the range of 1–10 nm. We show that for 1 nm gap the mode can be compressed as much as V = 300 nm³, which is about 10² times more compact than for an isolated particle and about 10⁶ than the diffraction limit λ³.

 

Fig. 5 (a) Dashed line shows $M~1/\sqrt{V}$. Blue circles is an actual calculation of the field enhancement. Orange squares is calculated V. For all points loss is 0.59 eV. (b) Energy density distribution at loss equal 0.59 eV. (c) Scattering cross-section of a silver bow-tie nanoantenna loaded with (red) and without (blue) a CdSe QD having μ = 13 D (∊_∞ = 6, f = 0.1, line position 1.77 eV, width 5 meV).

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Strong compression of the mode volume, among other things, implies that a single quantum emitter with a realistic value of transition dipole moment may reach the strong coupling regime. To demonstrate this, in Fig. 5(c) we show that a 7 nm gap bowtie can produce splitting with an individual quantum emitter placed inside the gap and for realistic parameters (μ_e ≈ 13 D) of colloidal CdSe quantum dots and exciton width corresponding to low temperature studies [26,36]. Thus a single nanoantenna – single quantum emitter geometry is indeed feasible and promising for quantum optics using plasmonic nanocavities, but at the same time that no unrealistically low losses on the metal side is necessary. We note though that as the gap is reduced, the resonance shifts to the red rather considerably (note that mode volumes were always calculated at the resonance wavelength).

Finally, we would like to mention that analysis of strong coupling in terms of the Q and V allows to draw a connection between them and the field enhancement. In particular from Fig. 5(a) we learn that the field enhancement depends on the mode volume in accordance with $M~1/\sqrt{V}$ [dashed line in Fig. 5(a)], provided Q is constant. From Fig. 2, on the other hand, we observe that M ∼ Q (see dashed lines in panels d–f), provided V is constant. By merging these two observations together we obtain $M~Q/\sqrt{V}$ in agreement with previous studies [25,37]. This is precisely the figure-of-merit for strong coupling, which implies that one should maximize M but at the same time this is not crucial for Ω.

4. Conclusions

In this work we have made a surprising observation that the mode volume and thus the splitting does not depend on material loss. In particular, the three nanostructures we have investigated clearly demonstrate that to observe plasmon-exciton peak splitting it is not critical to have low loss materials. The smallest loss we considered, 200 meV, is approximately one order of magnitude larger than in the lowest accepted Drude fits of experimental data of Johnson and Christy for silver [38], however, other experiments report larger losses and hence a greater imaginary part [39]. Yet for all γ_p ∈ (0.2, 2) eV we observe a constant peak splitting. We argued that judging on whether all the spectra shown in Fig. 2 can be attributed to strong coupling based solely on observation of splitting may be misleading. Instead, more thoroughly one should observe anti-crossing in absorption of a strongly coupled system [40]. This is hard to achieve in experiments, therefore one should optimize strong coupling in terms of e.g. cooperativity.

We have also analyzed the mode volume and its overlap with excitons. Our findings indicate that while dimer structures are not able to compress the mode much in comparison to individual nanoparticles (except for very narrow gaps), instead extract the mode out of the metal and thus make it accessible for interaction with excitons. Finally, we note that plasmonic modes can be compressed in sharp gaps and placing high dielectrics in the gap region, though to a limit of about ∼100 nm³, which opens up possibilities for strong coupling with individual quantum emitters and thus possible quantum optics applications.