Nonlocal Plasmonic Response and Fano Resonances at Visible

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Non-local plasmonic response and Fano resonances at visible frequencies in sub-nm gap coupling regime Simone Panaro, and Cristian Ciraci ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00716 • Publication Date (Web): 21 Nov 2016 Downloaded from http://pubs.acs.org on November 23, 2016

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Non-local plasmonic response and Fano resonances at visible frequencies in subnm gap coupling regime

Simone Panaro and Cristian Ciracì* Center for Biomolecular Nanotechnologies (CBN), Istituto Italiano di Tecnologia (IIT), Via Barsanti 14, 73010 Arnesano (LE), Italy. *corresponding author: [email protected]

ABSTRACT Plasmonic nano-assemblies endowed with ultra-small inter-particle gaps display a rich and unusual electromagnetic response both in near- and far-field read-out. The properties of such systems can be mainly ascribed to the hybridization of localized surface plasmon modes, which in turn enables a higher degree of field confinement with respect to single particles. Pushing these plasmonic resonances towards stronger coupling regimes (i.e. smaller gaps) requires particular attention to the microscopic description of the system. In this article, we theoretically show that the appearing of non-local effects in the electromagnetic response of a plasmonic dimer in strong coupling regime unveils a natural extension of its hybridization scheme. We demonstrate the arising of a Fano resonant interference in the visible range, resulting from the interaction of a hybrid quadrupolar mode and a highly localized gap mode.

KEYWORDS: Strong coupling, Sub-nm gap, Non-local response, Hydrodynamic model, Fano resonance

Plasmonic systems provide suitable strategies for manipulating light in sub-diffractive regime. The capability of metal-dielectric interfaces to sustain propagating polaritonic waves together with the possibility to engineer sophisticated architectures has allowed to access extremely challenging physical conditions, where a plethora of intriguing phenomena arise and can be experimentally observed. Just to mention some breakthrough achievements, we report negative refractive index response,1,2 superlensing,3,4 cloaking5,6 and single/few-molecule detection.7,8 One of the fundamental effects that determine the electromagnetic (EM) response of such complex systems consists in the so called plasmonic strong coupling.9,10 When the distance ACS Paragon Plus Environment

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between two particles supporting a localized surface plasmon (LSP) becomes comparable to the LSP intrinsic decaying length, the single nanoparticle modes undergo hybridization process,11,12 with the consequent generation of high energy (antibonding) and low energy (bonding) plasmonic modes. A noteworthy case of hybridization that recently drew particular attention in nanophotonics community is commonly known as Fano-like plasmonic interference.13-15 This resonant phenomenon involves a spectrally broad bright mode (super-radiant) and a spectrally narrow dark mode (sub-radiant), mutually exchanging their energy in near-field coupling condition.15 As a result, systems sustaining Fano resonance manifest a typical asymmetric line-shape in their scattering spectra in correspondence of which, remarkable local field enhancement occurs.16 One of the advantages of using dark modes is the possibility to control low-loss plasmonic modes endowed with longer lifetimes, able to potentially store a higher amount of EM energy in sub-diffractive regime.17 Typically, this kind of resonances requires the combination of several sub-systems in complex nanoassemblies, with the aim of providing the sufficient degrees of freedom to energetically promote the nearfield induced de-phasing between LSPs.13 However, this approach is rather detrimental from the applicative point of view. In fact, the necessary conditions for the arising of Fano resonances become very stringent, considering that one must simultaneously have: i) small gaps, required for the efficient LSP coupling, ii) spectral overlap of super- and sub-radiant modes and iii) sufficient number of elements for inducing the mutual LSPs de-phasing. In this perspective, a feasible way for reducing the complexity of Fano resonant plasmonic systems consists in employing ad-hoc symmetry breaking approaches,18,19 with the aim of forcing the out-of-phase oscillation between LSPs and promoting the excitation of almost-zero dipolar moments15. Within this context, we can mention several examples in which Fano resonances have been achieved in the near- and mid-infrared range, by recurring to asymmetric plasmonic trimers20-22 and quadrumers.23,24 Significantly more delicate is the transposition of these concepts into the visible spectral range, where the plasmonic behavior is greatly suppressed by the damping losses associated with interband transitions of noble metals.25 The intrinsic spectral broadening introduced by single particle excitations tends to attenuate and limit the local field enhancement associated to plasmonic modes. Moreover, it becomes very difficult to properly tune multiple plasmon resonances at once, because for small enough particles, the LSP spectral position only depends on the material properties.

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Considering the role that near-field coupling plays into plasmonic hybridization, an additional degree of freedom would be offered by pushing the distance between plasmonic elements to the order of just few nanometers, which are critical values especially for reaching the strong coupling condition in the visible range. This is however particularly challenging not only due to the complexity of the structures involved, but also because as the inter-particle distance is reduced, the harder it becomes to achieve spectral overlap between bright and dark modes required to obtain a Fano resonance. In order to overcome these issues, it becomes essential a further simplification in the geometry of these plasmonic nano-assemblies, as well as a new family of resonances that better coexist in the same spectral range. A single plasmonic element is in fact able to sustain a variety of different order modes, part of them of super-radiant and the other of sub-radiant character (e.g. the different dipolar orders of a rod-like nanoantenna10). However, these modes are defined at well separated energies, fact that hinders their spectral overlap. Moreover, the variation of the geometrical parameters in plasmonic systems typically implies the contemporary spectral shift of their intrinsic modes, rendering their mutual coupling highly inefficient. The solution to this impasse could however provide a keyapproach for simplifying the architectures of Fano resonant plasmonic systems. We anticipate that this can be achieved with plasmonic dimers that are characterized by nanometer gaps in which a hybrid quadrupolar mode and a dark gap-localized mode are strongly coupled. Although analogous systems have been investigated,26-29 in our specific geometry we find a possible strategy to excite plasmonic resonances at relatively long wavelengths, maintaining fixed the curvature radii in the profile of the gap region. When dealing with systems that are characterized by nano- or even sub-nanometer distance between inclusions, particular attention must be paid in the description of their optical response. It has been shown30,31 that at such distances the local approximation which is conventionally used to describe the macroscopic response of the metal’s electrons breaks down and one must take into account the non-local character of the electron gas response.32 Although a full quantum treatment33 is beyond the reach of systems that extends several hundreds of nanometers, the hydrodynamic approach, in which the non-local character is obtained via the introduction of an electron pressure term32, was shown to reproduce experimental results fairly well.30,31 In its simplest form (Thomas-Fermi approximation) the hydrodynamic model does not take into account spill-out and tunneling effects, but recent investigations34 have shown that this limitation can be easily overcome by including a ∇n-dependence (n is the electron density in the metal) in the internal energy


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description, although particular attention must be placed in the choice of the equilibrium density.35 However, our study will be limited to a minimum gap value of 0.5 nm, in a regime where quantum tunneling is negligible. Both theoretical and experimental results show in fact that quantum effects occur for gaps below 0.4 nm.36,37 This lower limit however can be modified for example in presence of highly intense electric fields (i.e. in non-linear regime)38 or by filling the plasmonic gap with conductive molecules.39 Nevertheless, for dielectric cavities in linear response approximation, as in our case, no appreciable tunneling effects can be observed for gaps larger than 0.4 nm. One could be however concerned by the broadening of the plasmonic resonances due to single particle excitations (Landau damping). Although it is hard to accurately estimate such effects in systems extending several hundreds of nanometers, one can safely assume that at the inter-wire distances considered, although still present the Landau damping would not sensibly affect the spectral feature of the system.31,33,37,40 Due to its onerous computational cost (with respect to the local approximation approach) and complexity, the hydrodynamic model has been up to now used only on relatively simple systems. A significant advancement would consist in the application of the hydrodynamic equation to more complex plasmonic hybridization schemes, although we will still limit our analysis to two-dimensional systems, since full threedimensional calculations are unfortunately yet computationally prohibitive. In this work, we theoretically investigate an elemental plasmonic dimer system displaying mutual coupling between a hybrid quadrupole-like mode and a dark gap-localized mode triggered in ultra-strong plasmonic coupling regime. We analyze the evolution in the optical response of the system from the strong to the ultra-strong coupling regime (inter-wire gaps below 1 nm), approaching the problem via the non-local hydrodynamic model (in the limit of a homogeneous electron density and the Thomas-Fermi approximation). In particular, we observe for sub-nm inter-particle gaps, the occurrence of a sharp Fano-like interferential feature in the visible range. In correspondence of this dip in the scattering spectrum of the system, a resonant near-field response is manifested in the inter-wire cavity, confirming the near-field activation of a plasmonic resonance. This intense field resonance is obtained in a spectral range of great interest for a wide variety of nanophotonics applications, such as single-/few-molecule detection7,41 or logic gating42,43 mediated by optical bi-stabilities.44,45


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Because of the complexity of our system, in the following, we will proceed gradually by focusing on one element at a time. Initially, we will introduce the system and its EM response in strong coupling condition. We will then investigate the critical role of non-locality for the correct description of the system response in ultra-small gap regime, performing a study on gap-localized modes. Finally, we will push the system from the strong to the ultra-strong coupling condition (sub-nm gaps), highlighting the occurrence of the Fano-like interference.

RESULTS AND DISCUSSION The system we are investigating is constituted by two infinite gold46 wires separated by a short distance and embedded in a dielectric medium of refractive index n = 1.43 (the minimum value associated to the most common oils, found in literature), whose cross-section is depicted in Figure 1(a). The linear EM response of such two-dimensional system can be obtained by solving the wave equation (Eq. 1) in combination to both the hydrodynamic equation for the free electrons (Eq. 2) and the phenomenological (local) Lorentz relation for the bound electrons (Eq. 3):

r r r r r ∇ 2 E + k 02 E = − µ 0ω 2 Pfree + Pbound

(

r r r

(

)

r

)

(1)

r

β 2 ∇ ∇ ⋅ Pfree + (ω 2 − iγω )Pfree = −ε 0ω 2p E

(2)

r r Pbound = ε 0 χ Lorentz (ω )E

(3)

r

r

where k0 is the free space wave number, Pfree and Pbound are respectively the free and bound electrons contributions to the local polarization, β ≈ 106 m/s is the speed of sound in the Fermi-degenerate plasma of conduction electrons,32,47 γ and ωp are respectively the damping coefficient and the plasma frequency of the free electrons and finally χLorentz(ω) is the part of the electric susceptibility that takes into account the phenomenological Lorentz oscillators related to the intrinsic resonances of bound electrons.46 The system of Eqs. (1)-(3) has been solved by means of a commercial software48 based on finite element method. For what concerns Eq. (2), we have employed Dirichlet boundary conditions, setting ݊ො ∙ ܲሬԦ௙௥௘௘ = 0 (with ݊ො the unity vector normal to the metal surface) around the profile of the antennas. Finally, the Eq. (3) that takes into ACS Paragon Plus Environment

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account the bound electrons contribution has been considered by defining an artificial material presenting the dispersion of gold without the Drude term. In order to perform the scattering analysis of the system, we introduced perfectly matched layer elements around the simulation domain. The system is excited by a TM-polarized plane wave impinging perpendicularly to the wires’ axis. Due to the translational symmetry of the system, we can focus exclusively on the cross-sections plane, as shown in Figure 1(a). We can recognize two identical elongated sections, in which we unambiguously identify a long axis size L and a short axis size W. The apexes of these two sections are semicircular, therefore the apical curvature radius is equal to W/2. For our purposes we have set L = 400 nm and W = 260 nm. The interstructures distance g is, in our case, a free parameter that we sweep in order to analyze the role of the LSPs coupling in the EM response of the system. Finally, we also introduce an angle θ that quantifies the tilt of the left antenna with respect to the right one (see Figure 1(a)). For θ = 0°, we are considering an aligned wires dimer, while, for non-zero θ values, we simply denote the system as a tilted wires dimer.

Figure 1. (a). Geometry of the investigated system with the main illumination configuration selected. (b). Diagram showing the hybridization scheme that involves the modes of a nanoantenna dimer. (c). Spectra of the power scattered by the system per unit of length, for tilt angles θ varying between 0° and 90° (a rigid offset has been assigned to all the spectra in order to simplify the readability of the plot) and inter-structures gap g = 30 nm. (d,e). Field enhancement plots evaluated on a plane that cuts the system perpendicularly to the axes of the wires, for θ = 60° condition. They are referred to the bonding and the antibonding mode respectively. (Superimposed to the plots, the surface charge current distributions have been reported, in addition to representative arrows guiding the interpretation of the main charge oscillation).


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The wire dimer, or more generally the rod dimer, is one of the simplest systems that guarantee a straightforward hybridization scheme via the strong coupling between the LSPs of the two rods. A dimer of rods, considered at the lower order, supports both mutual in phase (bonding) and out-of-phase (antibonding) long axis current oscillations (see the lower part of the diagram in Figure 1(b)). Immediately at higher energies, sub-radiant quadrupolar modes can be supported by each individual antenna (see the upper part of the diagram in Figure 1(b)). Therefore, considering a dimer in strong coupling condition, these modes can mutually hybridize, generating both bonding and antibonding quadrupolar modes and more complex hybridization schemes. In our study we will impose a symmetry breaking on the system, by the introduction of the tilt angle θ between the long axes of the two wires (see Figure 1(a) for the geometric sketch). This choose is intended to activate in far-field the lower order hybridized modes of the system49. In fact, in this particular low-symmetry condition, the hybridized modes under study can be associated to an instantaneous non zero dipolar moment49, producing a distinguishable resonance peak in the far-field. This behavior can be clearly observed in Figure 1(c), where we report the evolution of the dimer scattering spectrum per unit of wire length, for increasing angles θ (see Figure 1(a)). For our calculation, we employed a plane wave with polarization parallel to the long axis of the right antenna and k-vector parallel to its short axis (see Figure 1(a)). At first, we moved from an aligned dimer configuration to a gradually tilted antenna dimer configuration, keeping constant both the light incidence direction and the inter-structures gap g = 30 nm. If we focus on the spectra evolution, we can see how, for θ = 0° configuration (black curve), the scattering spectrum shows a bimodal dispersion with a main sharp peak around 600 nm and a broader peak at around 1200 nm. We identify the larger wavelength peak to the lowest order dipolar mode, while we ascribe the shorter wavelength peak to a higher order dipolar mode (see Figure S1 in the Supporting Information). As we increase θ, the spectral response of the system evolves quite significantly, with a gradual weakening of the peak at 1200 nm and the simultaneous arising of a bimodal response in the spectral window between 400 nm and 900 nm. In particular, around θ = 40° (green curve), a peak seems to arise at 600 nm. Finally, for higher tilt angles, we can see a broad convoluted peak at 1500 nm, and two well distinguished sharp peaks respectively at 450 nm and 650 nm (see from the purple curve on). Considering the evolution of the peaks, we can attribute the 1500 nm peak to the first order dipolar bonding mode, which results from the hybridization of the long axis LSPs of the two wires. On the other hand, we can ascribe the sharp peak at 650


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nm to an initially dark mode which has been activated in far-field through the symmetry breaking induced by the θ tilt angle. For a further confirmation of our hypotheses, we considered the θ = 60° configuration and plotted in Figures 1(d,e) the electric field distribution in correspondence of respectively the 1500 nm and the 650 nm scattering peaks. For a clearer interpretation of the near-field response exhibited by the system, we superimposed to the electric field maps also the surface charge current distribution in a fixed phase of the optical cycle. If we observe Figure 1(d), we can appreciate how the electric field is mainly localized in the gap region. At the same time, we notice how the charge current distribution manifests a mutually in phase oscillation of the charges in the majority of the geometry, as we can typically expect from a dipolar bonding mode9,50. If we now observe the near-field response of the system for λ = 650 nm (Fig. 1(e)), we can notice a a more complex distribution. In the peripheral regions of the dimer charges tend to oscillate mutually out-ofphase (see the white arrows at the external apexes of the antennas). However, the near-field distribution around the two antennas presents 4 lobes, fact which suggests the arising of a quadrupole-like plasmonic mode. In order to have a simpler interpretation of this plasmonic resonance, we analyzed again the local field distribution of the aligned dimer around 650 nm (see Figure S2 of the Supporting Information). As a result of the comparison between aligned and tilted configuration, the mode under analysis can be considered quadrupolar but, for θ = 60°, it presents a super-radiant character, as confirmed by the single hot spot in the gap. During the symmetry breaking process, a sharp extra peak has emerged around 450 nm, not expected from the low order hybridization scheme (Fig. 1(b)). This fact suggests that the plasmonic dimer cannot be considered as a simple two-antenna system, especially if we want to investigate its properties in ultra-strong coupling conditions.

Gap reduction and the role of non-locality In order to estimate the role of the highest energy resonance in the reported EM response of the dimer, we evaluated the scattering spectra of the dimer as a function of the inter-wire gap (Figs. 2(a,b)), keeping fixed the angle θ = 60°. We focused on the gap range between 30 nm and 5 nm, moving towards a coupling regime in which the non-local response of the electron gas inside the metallic antennas becomes crucial.30 If we


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observe the 2D map in Figure 2(a), we can appreciate how the bonding mode significantly red-shifts from 1500 nm to around 1600 nm, as a function of the gap reduction. Contemporarily, the quadrupolar peak shows a much less appreciable dynamics with the gap change, remaining centered around 650 nm, even for 5 nm gap (see the zoomed map in Figure 2(b)). On the contrary, the sharp peak observed around 450 nm for g = 30 nm, manifests a quite remarkable evolution with the gap, showing an important red-shift when g < 10 nm. The evolution of this peak as a function of g, suggests that the local fields of the associated mode are strongly confined inside the gap region. For g = 5 nm inter-wire separations, this high energy mode strongly overlaps to the quadrupolar peak. The condition we achieved is particularly critical for two reasons: i) in the gap regime we are approaching, non-local effects are not anymore negligible and ii) the spectral overlap of two different plasmonic modes implies new hybridization schemes which require a deep investigation of the involved modes.

Figure 2. (a,b) Respectively large scale and zoomed 2D maps reporting the evolution of the scattering spectra (non-local approach) for the dimer system as a function of the inter-wire gap g (the zoomed area is indicated by the squared dotted region in the lower left part of Figure 2(a)). (c,d) Respectively large scale and zoomed 2D maps reporting the evolution of the scattering spectra (local approximation) for the dimer system as a function of the inter-wire gap g (the zoomed area is indicated by the squared dotted region in the lower left part of Figure 2(c)).

In order to appreciate and quantify the non-local effect, we simulated the system employing the traditional local approximation (setting β = 0 m/s in Eq. (2)). In Figures 2(c,d), we reported the resulting maps of the scattering spectra as a function of the inter-structures gap. As we can appreciate comparing Figure 2(a) and ACS Paragon Plus Environment

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Figure 2(c), the local approximation still well describes the spectral response of the system down to around 20 nm gaps. However, if we further reduce the gap, the effects of non-localities in the gap region gradually arise. This can be noticed, if we compare the scattering spectra evaluated respectively via non-local (Fig. 2(b)) and local approach (Fig. 2(d)), for g = 10 nm. In this gap configuration, we can notice an almost 70 nm blue-shift of the highest energy feature in the former case with respect to the latter one. The introduction of a pressure term (whose intensity is determined by the β parameter) in Eq. (2), takes into account the presence of a thin region immediately below the dielectric-metal interface (Thomas-Fermi skin depth), by which free charges are distributed. This reflects in the reduction of the effective coupling between gap plasmons, in comparison to what expected from local theory that confines the entire charge distribution on the metallic surface. As a consequence, for sufficiently small gaps, this implies an appreciable blue-shift in the spectra with respect to the corresponding local counterparts. In such spectral range, we will see how a shift of this entity can determine a significant difference in the EM response of the plasmonic system. In fact, due to the pronounced spectral overlap observed in Figure 2(b) around 600 nm, the optical response of the investigated system cannot be anymore described by the simple hybridization scheme shown in Figure 1(b).

From strong to ultra-strong coupling: the arising of gap modes In order to focus on the spectral overlap around 600 nm, we concentrated our study on the wavelength window between 400 nm and 900 nm, solving Eq. (2) for β = 106 m/s (non-local approach). As we reported in dotted black curve of Figure 3(a), the polarization configuration that we chose (parallel to the long axis of the right wire) highlights how a mode presumably localized in the gap region is spectrally overlapped to the quadrupolar mode around 600 nm (to be noticed that g = 5 nm). With the aim of unbinding the convolution of spectra in the wavelength region of interest, we selected a polarization configuration that could suppress the quadrupolar mode (see Figure S3 in the Supporting Information). For this reason, we anti-clockwise rotated the polarization of 60° (see the upper inset of Figure 3(a) for the former illumination configuration and the lower inset of the same figure for the latter optical configuration).


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Figure 3. (a). Scattering spectra of the tilted dimer configuration respectively for longitudinal polarization and g = 5 nm (dotted black curve), rotated polarization and g = 5 nm (solid black curve), rotated polarization and 0.5 nm < g < 5 nm (solid grey curves) and finally rotated polarization and g = 0.5 nm (solid red curve). (b,c,d). 2D plots of the charge current density distribution on the surface of the wires in the g = 0.5 nm gap region: these plots are in correspondence to the scattering peaks of the red spectrum in Figure 3(a), respectively the λ = 760 nm, the λ = 560 nm and the λ = 440 nm. All the results have been obtained in non-local approach.

As a result of this operation, we observe i) a general lowering of the overall spectrum, due to the suppression of the quadrupolar mode (see the solid black curve in Figure 3(a)) and ii) the appearance of two features around 410 nm and 660 nm, sharper with respect to the super-radiant peak. The peak at 660 nm is presumably the same that we observed red-shifting in Figure 2, while the peak at 410 nm seems to be promoted by the rotated polarization. In order to investigate the evolution of these modes as a function of the coupling strength, we fixed the polarization and gradually decreased the inter-wire gap, monitoring the scattering spectrum. In Figure 3(a), we reported, in grey color, the scattering spectra of the dimer with g parameter spanning from 5 nm to 0.5 nm at steps of 0.5 nm (to be noticed that the polarization has been kept 60°-rotated with respect to the initial direction). Finally, the g = 0.5 nm spectrum has been traced in red color. As we can appreciate, the reduction of the inter-wire gap induces a red-shift of about 130 nm in the scattering spectra and the appearance, for g = 1 nm, of a third peak around 440 nm (see red curve in Figure 3(a)). The arising of these adjacent scattering peaks, so dependent upon the g parameter, suggests the attribution of these features to cavity-like modes that originate in the close vicinity of the inter-structures gap. Similar plasmonic modes have already been


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investigated on spheres placed in close vicinity to metallic films51 or on stacked optical antennas,52 and, from the theoretical point of view, on a system of two wires with circular section.53 These modes have demonstrated a strong dependence both on the inter-particle gap and on the curvature radii in the nearby of the cavity. Furthermore, their spectral response is similar to the one of a Fabry-Pèrot resonator, with increasingly higher order modes. As it happens in a linear cavity where the intrinsic resonance wavelengths increase with the dimension of the cavity, in the same way, the wavelengths associated to the gap-localized modes increase with the curvature radius of the gap profile. From the near-field point of view, these modes, which we identify as gap modes, are characterized by an increasing number of nodes in the local fields, in analogy to what can be observed in standing waves. The typical signature of these modes has been captured in Figures 3(b-d), where we report the charge current density distribution in the cavity region, corresponding to the 3 peaks observed in the red curve of Figure 3(a). The peak at 760 nm (Fig. 3(b)) is characterized by a strong current distribution mainly directed from right to left. If we observe the cavity response around 560 nm (Fig. 3(c)), we can notice a node in the current density distribution and two opposite current subdistributions. Finally, for impinging light at 440 nm (Fig. 3(d)), the system shows two main nodes in the current density inside the cavity, denoting a higher order gap mode. In Figures S4(a-c) of the Supporting Information, we analyzed the distribution of the out-of-plane component of the H field associated to each gap mode, in order to better identify these cavity resonances. As a result, we labeled them respectively as m = 0, 1 and 2 gap modes, considering from the larger to the smaller wavelength one.

Interaction between quadrupolar mode and dark gap mode As we could notice, in ultra-strong coupling condition, these gap modes resonate in the same spectral window of the hybrid quadrupolar mode. This condition is quite peculiar since higher order modes are, by definition, in a well separated spectral position with respect to lower order modes. In our case, this superposition is possible because the quadrupolar peak resonance remains practically fixed during the gap reduction. This is the reason why, in particularly high coupling conditions (g < 5 nm), the super-radiant mode can potentially interact with modes of different nature, widening the hybridization scheme (see the extended hybridization diagram in Figure 4(a)). In order to verify this assumption, we set the polarization in the initial condition (parallel to the long axis of the right antenna), so that the quadrupolar mode can be again


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activated in far-field (see the inset in Figure 4(a)). We then focused on the ultra-strong coupling regime, considering inter-wire gaps between 5 nm and 0.5 nm (employing the non-local approach). We plotted in Figure 4(b) the scattering spectra of the tilted dimer for gradually decreasing gap values.

Figure 4. (a). Extended diagram showing the complex hybridization scheme that involves the lower order modes with the gap modes in ultra-strong coupling regime (g < 5 nm). (b). Scattering spectra of the tilted dimer system for decreasing inter-wire gap (from 5 nm to 0.5 nm at steps of 0.5 nm). The polarization configuration is longitudinal to the non tilted antenna. Inset: charge current distribution in the cavity region associated to the Fano dip in g = 0.5 nm spectrum (magenta curve). (c). 2D distribution of the out-ofplane component of the normalized H field in Fano resonance condition for g = 0.5 nm dimer. The poynting vector distribution has been superimposed to the plot. Inset: zoom of the plot in the cavity region. (d). Spectra of the electric field enhancement in the gap region of tilted dimer for g parameter spanning from 5 nm to 0.5 nm at steps of 0.5 nm. All the results have been obtained in nonlocal approach.

As we can appreciate, for g = 5 nm (dark yellow curve in Figure 4(b)), the spectrum shows a highly convoluted spectrum, resulting from the simple summation of the quadrupolar peak and the gap mode reported in Figure 3(b). At this point, as we reduce the gap, we observe the deepening of the spectrum around 450 nm (see the green curve in Figure 4(b)) and, if we further diminish the inter-wire cavity, this dip red-shifts producing an interferential feature in the broad line-shape of the quadrupolar mode. Finally, if we consider the g = 0.5 nm spectrum (magenta curve in Figure 4(b)), we can clearly see how the broad lineshape of the quadrupole-like mode presents a sharp dip around 560 nm. We stopped the parametric sweep at this minimum gap value, since the investigation of smaller cavities is beyond the scope of our work. In fact, ACS Paragon Plus Environment

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it is well acknowledged in literature36,37 how single particle excitations start playing a critical role in the EM response of plasmonic cavities below the 0.4 nm regime, introducing additional damping phenomena (i.e. Landau damping). In the inset of Figure 4(b), we reported the charge current distribution inside the cavity region, in correspondence of the scattering dip. As we can appreciate, such current distribution presents similarities with the m = 1 gap mode analyzed in Figure 3(c). This aspect is quite interesting since, for symmetry reasons, the polarization configuration we chose is not expected to promote a split current distribution in the gap region, characterized by two opposite current directions. According to these observations, the aforementioned gap mode is strongly hindered by this polarization of the incoming light. In order to show the effect of the dimer response to the propagating EM field, we reported in Figure 4(c) the distribution of the out-of-plane component of H field at 560 nm, together with the poynting vector distribution. From the plot, the incoming light seems to surround the dimer almost without interacting with it, apart from a remarkably high activity in the inter-wire cavity region. We therefore zoomed in the gap region (see the inset in Figure 4(c)), appreciating a high EM funneling through the dimer, highlighted by the localized and intense poynting vector distribution across the cavity. Reasonably, the system does not manifest an efficient scattering condition, as confirmed by the scattering dip, even showing a strong plasmonic activity in the very gap region. We can hypothesize that, in ultra-strong coupling condition, the dimer system responds to EM radiation around 600 nm by producing a Fano-like interference between the super-radiant quadrupolar mode and a dark gap mode. The trace of the quadrupolar peak can be recovered from the spectral tails between 400 nm and 500 nm and between 700 nm and 900 nm in g = 0.5 nm lineshape of Figure 4(b) (magenta curve), which well fit the corresponding portions of spectrum in g = 5 nm lineshape (dark yellow curve). In order to confirm the resonant character of the Fano-like feature observed, we reported in Figure 4(d) the spectral response of the field enhancement in the gap region, for g parameter spanning between 5 nm and 0.5 nm at steps of 0.5 nm. If we observe the evolution of the spectra with the gap (the colors of the spectra are the same of the corresponding scattering spectra in Figure 4(b)), we can see how for g = 5 nm, where the dark mode is not triggered yet, the enhancement in the gap is quite moderate. However, as the dark gap mode gradually red-shifts and overlaps more efficiently to the quadrupolar peak, a corresponding resonance in the local field appears, following the spectral trend of the Fano dip in Figure 4(b). Finally, for g = 0.5 nm a very


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intense field peak can be observed around 600 nm, in fair spectral superposition with the corresponding Fano dip (see the magenta curve in Figure 4(d)). Apart from the high enhancement factor observed, obviously motivated by the ultra-small inter-wire gap, we want to stress how an electric field resonance occurs in correspondence of a minimum in the scattering spectrum. Finally, despite the variety of gap modes overlapping to the quadrupolar peak in ultra-strong coupling condition, only the one reported in Figure 3(c) seems to appreciably interact with it. Presumably this can be ascribed to the fact that the aforementioned gap mode is the only one, among the three analyzed in Figure 3, that presents the same affinity and symmetry54 with the quadrupolar mode, in terms of local EM response. In order to confirm this, in Figure S4(d), we reported the H field distribution of the quadurpolar mode for the rotated polarization (the same optical condition which allows to directly excite all the gap modes). In fact, in the same way in which the dark quadrupolar mode is characterized by a single magnetic hot-spot (see Figure S4(d)), also the m = 1 gap mode in Figure S4(b) shows a strong concentration of H field distribution in the plasmonic cavity.

CONCLUSIONS We presented an elemental ad-hoc plasmonic dimer sustaining the interaction of a bright quadrupolar mode and a dark gap mode. For justifying this particular EM response of the system, we gradually increased the coupling between the LSPs of the two antennas from a regime of weak/strong coupling to a condition of ultra-strong coupling. The inter-wire gap regimes explored during this analysis rendered necessary the employment of the non-local hydrodynamic approach for an accurate description of free carriers contributing to the plasmonic resonances observed. The interaction of the two plasmonic modes in sub-nm gap regime has been demonstrated by the clear appearance of a Fano-like line-shape in correspondence of the gap mode. Although we presented a purely numerical analysis, the system we introduced could be potentially realized via top-down fabrication of plasmonic wires on metal-dielectric thin films. We believe our work shows clearly the necessity of moving toward ultra-strong plasmonic coupling regimes, where sub-nm gap systems enable intense field resonances in a spectral range of great interest for a wide variety of nanophotonics applications. In fact, the plasmonic resonator we are proposing is an ideal candidate for the study of nonlinear photonic processes, triggered by high values of local field in the gap region, in concomitance with a


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remarkably sharp (or steep) spectral line-shape in far-field read-out. These two conditions constitute an optimal route towards the achievement of optical bi-stabilities at low energy, in order to realize ultrafast alloptical signal switch and modulation.

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