Resonance coupling in plasmonic nanomatryoshka homo- and heterodimers

The optical characteristics of noble metallic structures in nanoscale dimensions have been described by localized surface plasmon resonances (LSPRs), which correspond to free electron coherent oscillations in the conduction band when illuminated by an incident light at the optical frequencies.¹ Metallic nanostructures with subwavelength dimensions have been extensively employed in the design and fabrication of integrated nanoplasmonic structures and devices for numerous applications such as biological and chemical sensing,² and surface-enhanced Raman spectroscopy (SERS).³ Studies have strongly proved that various parameters such as shape, geometrical dimensions, chemical composition, and optical characteristics of the nanostructures have a significant impact on intensity, position, and frequency of the plasmon resonant modes.^4,5 In addition, the behavior of plasmon resonances strongly depends on the existence of an adjacent individual nanoparticle. Subtle variations in the elucidated chemical, physical, and optical properties influence the plasmon resonance condition due to the near-field coupling of dipoles, quadrupoles, and higher-order poles under the excitation of an incident beam.⁶ Coupling of plasmon resonances to proximal nanoparticles facilitates an opportunity to adjust the intensity of the LSPR at the desired spectrum via red- or blue-shifts. Moreover, by changing the distance between nanoparticles, scattering cross-section spectra can be tuned, which can be exploited for imaging and sensing purposes.⁷ Individual modifications in the structural and environmental parameters of a given nanostructure influence the precision of LSPR sensing significantly.⁸ To improve the quality of designed plasmonic devices, all alterations must be performed correctly and accurately. For instance, it is well accepted that increments in the refractive index of the local environment cause red-shift of the plasmon resonance frequency to the longer wavelength spectra.⁹ More studies have also shown that this sensitivity of the plasmon resonance frequency can be improved by changing the shape and size of the nanoparticles simultaneously in the correct manner.^8,9 Considering the dipolar coupling,^1,10 and Drude model,¹ the nanoparticle geometry-dependent factor (shape factor) as κ, and the refractive index of host substance by n_h, we can describe the precise surface plasmon wavelength as:

where, N is the electron density of the employed metal, ε₀ and ε_∞ are the permittivity of free space and high-frequency response respectively, and m_e is the effective electron mass. The equation above indicates that increasing the refractive index of the surrounding medium and the shape factor directly increases the size of the LSPR movement (red-shift to the longer wavelengths), and Parshant et al.⁹ have proved that this modification raises the sensitivity and accuracy of plasmon resonance coupling and sensing. Earlier computational results verified that nanoparticles with complex geometrical configurations show larger shape factors due to the number of tunable parameters in their structural geometry. For example, a simple sphere with a radius of r<50nm has a shape factor of approximately κ∼2,⁹ and for a simple nanoshell with the overall size of <50nm, the shape factor is κ ∼3.1.¹⁰ Hence, for a given structure and combination of nanoparticles, increasing complexity could help achieve larger red-shifts of LSPR to the near infrared spectrum and also more accurate sensing ability.

Two proximal nanoparticles (Ψ₁,Ψ₂) that are located in close proximity to each other are recognized as a two-member dimer configuration, which provides a significant EM field enhancement and localization of plasmon resonances at the coupling region (the offset gap distance between nanoparticles). Recently, dimers have attracted great attention for use in ultra-sensitive detectors and nanostructures, as well as SERS applications.^10–13 Several nanoparticles with different shapes and sizes in a symmetric configuration (the center of both particles is in the same axis and completely similar in structural and chemical properties leading to in-phase plasmon modes (Ψ₁ + Ψ₂)), have been discussed and studied previously.^9–13 Homodimer nanoassembly reflect only a bonding plasmon resonance mode under longitudinal-polarized excitation, and the peak of the dipolar mode appears in the gap between nanoparticles. Likewise, asymmetric dimer nanostructures (completely different particles with respect to the structural and chemical properties leading to out-of-phase plasmon modes (Ψ₁-Ψ₂)), of simple particles such as spheres and rods have been studied experimentally and theoretically for their energy levels and extinction cross-section properties.¹⁴ 

Illumination of two adjacent nanoparticles in a homodimer structure by an incident longitudinal-polarized light results in excitation of a bonding plasmon mode in the extinction diagram, which leads to a significant enhancement in the electric field at the coupling area, and consequently, a large red-shift of the plasmon resonance denoted by σ.^14,15 Additionally, reducing the offset gap between nanoparticles to a touching regime gives rise to a remarkable blue-shift of the LSPR to the shorter wavelengths, which is not considered for the proposed heterodimer structure in this study.¹¹ In contrast, in the heterodimer regime (out-of-phase mode), the adjacent heterodimer shows the antibonding modes as well as bonding modes resulting localized electric field enhancement at the outer sides of the dimer configurations and blue-shift of the plasmon resonance frequency, which is denoted by σ^∗.¹⁶ Both bonding and antibonding plasmon modes and their hybridized models are explained in the following sections. The plasmon response of the proposed nanostructures with expected red- and blue-shifts are also examined theoretically and numerically. Moreover, the effect of changes in the polarization direction and plasmon resonance reflection in both in-phase and out-of-phase plasmon modes are of interest for this study. Earlier works have shown that modifications in the polarization direction of an incident wave to the transverse polarized mode (perpendicular to the dimer axis) leads to an exactly opposite results in the shift of the plasmon resonant modes and it is important to note that corresponding shifts of the LSPR are noticeably weaker than the former case.¹⁶ In this limit, an antibonding mode is reported for the homodimer structure, while a bonding mode is observed for the heterodimer structure which are denoted by π and π^∗, respectively.

In this work, we investigate the plasmon response and optical properties of plasmonic nanostructures composed of two concentric nanoshells that are placed in close proximity to each other and are known as NM dimers. The in-phase regime of NM dimer have already been studied by Prodan et al.¹⁵ and the behavior of bonding modes are examined previously. In the antisymmetric regime, we show that the proposed dimer is able to support both bonding and antibonding plasmon modes, concurrently. Most of the previous investigations were limited to dimers with ordinary shapes or materials with similar compositions.^13–18 In this report, we extend our study to the NM structure with different metal compositions and sizes. This method allows us investigating the plasmon resonance behavior in symmetry-lacking structures. Moreover, we examine the quality of the plasmon resonance coupling in the NM structures with various geometrical dimensions to achieve enhanced and ultra-sensitive nanoplasmonic configurations.

To investigate the behavior of the presented subwavelength dimer composed of NM structures, we used a three dimensional Finite-Difference Time-Domain (FDTD) method (Lumerical package). To this end, we used a model with spatial cell size dx=dy=dz=0.1 nm, with the number of cell units 12000, and perfectly matched layers (PMLs) for the boundary condition with 16 compacted layers with a high ratio of absorption of scattered fields. Moreover, by employing a linear plane wave as an incident light source, we excited surface plasmon resonances in the studied nanocomposition. For the analytical analysis, plasmon hybridization theory was used to calculate the extinction profiles of the dimer structure in symmetric and antisymmetric compositions.¹⁵ Plasmonic nanoantennas that are located in closely-spaced regime support strong resonant modes with different energies and properties in the offset gap spot. Adjusting their size, material, and geometry, these plasmonic nanostructures can be tailored to sustain unique spectral lineshapes in the cross-sectional spectra such as Fano resonances characterized by a deep and narrow spectral windows.^19,20 Recently, methods such as multiparticle Mie theory,²¹ and ab initio method,²² have been used to analyze dipolar and multipolar electromagnetic field interference and spectral response of these structures. However, comparing various analytical methods, plasmon hybridization theory have been considered as a major and promising solution to characterize the plasmonic response of multiparticle structures.^15,23 The symmetry and antisymmetry of the entire structure play fundamental role in corresponding spectral response. It is well accepted that breaking the symmetry of nanoscale assemblies gives rise to formation of new antibonding modes across the plasmonic response.^24,25 To underlay the physical mechanism behind the formation of antibonding resonant modes, the dipolar momentum of bonding and antibonding modes can be compared. In complex nanoparticle assemblies, and in the nonretarded limit, the plasmonic antibonding mode does not have any net dipole momentum and cannot be coupled to the incident beam efficiently. On the other hand, in the retarded limit, antibonding mode becomes subradiant and bonding mode remains superradiant, therefore, any destructive interference between these modes gives rise to antisymmetric lineshapes on the extinction curve.^19–25 

In this section, first, we examine the quality of bonding and antibonding plasmon resonance modes for a NM homodimer (symmetric (Ψ₁ + Ψ₂)) with the same geometrical parameters and metal compositions (gold) for both sides, while the separation or gap distance between NM units is approximately D_in∼15 nm, which resembles the strong coupling regime and also the thickness of the rings is kept fix as 25 nm throughout the study. This examination helps us observe bonding and antibonding plasmon modes in heterostructures under illumination of incident polarized light. Figure 1(a) shows a three-dimensional schematic picture of a symmetric homodimer, along with the geometrical parameters in the top-view picture (not to scale). It is already proved that the interaction of surface plasmon polaritons (SPPs) in the interior and exterior surfaces of a nanoshell can be described by the plasmon hybridization theory,^15,23 and the corresponding frequency for antisymmetric and symmetric modes can be specified by $ω+$ and $ω−$⁠, respectively. Figure 1(b) illustrates the extinction cross-section profile for the observed plasmon resonant modes under excitation of longitudinal and transverse electric field polarization modes in strong coupling regime. Obviously, the two bonding plasmon modes result from the longitudinal (σ) (larger shift) and transverse (π^∗) polarization modes with the extinction peaks at λ∼1390nm and 780nm, respectively (see Fig. 1(c)). Figure 1(d) shows the plasmon resonance excitation as E-field maps for the homodimer antenna. One should note that these results have been predicted from the plasmon hybridization model for most of the comparable and simple homodimers with short red-shifts under the excitation by an incident longitudinally polarized light.^13,14 Considering the geometrical dimensions (r₁, r₂, r₃, r₄) = (60,80,105,135) nm for gold NM, we calculated the extinction spectra for the homodimer system in the strong interaction of the excited dipolar and multipolar fields (bonding modes only). Further, we evaluated the plasmon resonance behavior (coupling and shifting of the LSPR) during symmetry breaking for the NM dimer based on modifications in the geometrical sizes, while the chemical properties of the employed materials are the same in both sides. Accordingly, we modified the radii of one of the gold NM units (Ψ₁) to (50,65,95,125) nm, while keeping the ones for the other (Ψ₂) at (60,80,105,135)nm (Heterodimer antenna, see Fig. 2(a)). The corresponding extinction cross-sectional profile is illustrated in Fig. 2(b), where two distinct extremes induced for each one of the incident polarized modes in the symmetry breaking regime. The incident longitudinal polarization excites the peaks at λ∼1360nm (σ), 510nm (σ^∗) while the incident transverse polarization results the extremes at λ∼1080nm (π), 470nm (π^∗) for bonding and antibonding plasmon modes in the antisymmetric regime (Ψ₁-Ψ₂), respectively. Utilizing the coupled dipole-dipole model and discrete approximation (DDA) method,^26,27 for the multipolar modes, we calculated the energy diagram for different regimes as shown in Fig. 2(c). It is shown that DDA method is a highly flexible analytical method for computing the spectral responses and corresponding extinction profiles of nanoscale structures.^28,29 Considering the proposed method in Ref. 27, therefore, the extinction cross-section for a NM unit can be written as:²⁸ 

where k is the wave vector, E_inc is the incident electric field, p is the dipole moment depending on the polarizability of NM unit and dipolar field. The polarizability and spectral response of a plasmonic NM is calculated and discussed in previous studies comprehensively.^30–32 Using previously studied analysis, we extracted the plasmonic responses for both homo and heterodimers accurately. Considering the same chemical properties of plasmonic NM units, it is obvious that the antibonding mode as an abnormal mode repeals the excited dipoles of the small and large gold concentric nanoshells. These results encouraged us to develop a modified version of computations for analyzing the behavior of a dimer structure composed of two different NM units that will be discussed further. We plotted the E-field maps for the plasmon resonance hybridization in the gap spots between nanoparticle units for bonding modes (σ and π) positions in Fig. 2(d).

In the extinction cross-sectional diagrams, the compounds of the in-phase modes (σ and π^∗) are significantly severe in comparison to the out-of-phase modes (σ^∗ and π) combinations. Therefore, the presented results based on the adapted simulation model for the NM heterodimer (asymmetric regime) are in the good agreement with the hybridization model.¹⁵ Lassiter et al.¹¹ and Prodan et al.¹⁵ have demonstrated that the extent of shifts in longitudinal polarization mode coupling is dramatically longer than the transverse ones (blue-shift), which is in agreement with our results and the plasmon hybridization model as well. The noteworthy feature of the studied plasmonic heterodimer is the noticeable red-shift of the LSPR to longer wavelengths, which is a significant shift in comparison with the behavior of analogous dimers composed of spherical compositions in the same simulation workplace (e.g. spheres, rods, and rings). This feature helps us to understand the sense of the plasmon energy shifts driven from the intense coupling of a gold NM dimer for bonding and antibonding plasmon resonant modes.

In continue, we present the influence of symmetry breaking on the optical response of the heterodimer structures with modifications in both chemical and physical characteristics. First, we used the previously used geometrical sizes of (60,80,105,135) nm for Ψ₁ and (50,65,95,125) nm for Ψ₂, but changed the metal of Ψ₂ to silver while keeping the Ψ₁ gold (see Fig. 3(a)). It is well accepted that individual silver nanoparticles show larger scattering spectra for an interacted light in comparison with the gold nanoparticles in the same condition,^33–35 and as a result, the corresponding sizes of gold unit must be larger than those for the silver unit in the NM heterodimer. This adjustment helps us provide an equal spectra of scattering for both sides to obtain corresponding extinction spectra. Illuminating the proposed silver-gold heterodimer by longitudinal and transverse polarization modes, we calculated the extinction cross-sectional diagram based on the DDA method with experimentally determined Palik constants (see Fig. 3(b)).³³ Considering the extinction profile in Fig. 3(b), we realized that bonding and antibonding extremes appear and noticeable red-shifts are observed for all of the observed modes (lower and higher energies). For the longitudinal polarization excitation, the maxima of bonding and antibonding modes (σ and σ^∗) have been recorded around λ∼1390nm and 700nm, respectively. In contrast, for the transverse modes, π and π^∗-modes detected at λ∼1090nm and 380nm, respectively. In this regime, plasmon resonance peak experienced a significant red-shift to the longer spectra under longitudinal polarization mode excitation. Figure 3(c) demonstrates how the bonding and antibonding extremes of plasmon hybridization model contributed to the examined heterostructure. In the position of the high-energy modes (σ^∗ and π^∗), the size of the red-shift in the longitudinal polarization mode (σ^∗) is larger than that in the π^∗-mode. This condition verifies that the coupling strength in silver-gold NM heterodimer gives rise to a shift of the plasmon resonances to longer wavelengths in comparison with the isolated nanocomposition of silver shells. We also plotted corresponding E-field maps for the plasmon resonance confinement in the compositional dimer nanoantenna at the bonding mode wavelengths, as shown in Fig. 3(d). Herein, we observed a red-shift in the high energy modes of the incident longitudinal and transverse polarization modes, which caused by the presence of the silver shell unit on the other side of the dimer. Bachelier et al.³⁶ have verified that plasmon resonance shift in silver nanoparticles is in the direction of shorter wavelengths (blue-shift), however, this condition must be suppressed by coupling of the interband absorption of plasmon resonant modes in the gold nanoshells to the LSPR modes in the silver nanoshells. In addition, the interband transitions in the gold NM unit are prominent in the adjacent and above sections of the silver NM unit, and this situation leads to a strong coupling of interband transitions of gold particle to the silver plasmon resonances. A comparison of the coupling strength of the latest heterodimer and the homodimer structure with identical material and sizes shows the dominant behavior of the final structure.

Further, we attempted to explain the mentioned deviation of the plasmon hybridization in the presented results, by employing full dielectric function in our calculations. To this end, we studied the feasible solutions to explain the observed abnormal red-shift in silver-gold nanoshells by the plasmon resonances due to the interband transitions in gold nanoshells. Considering the real and imaginary parts of refractive index of the interband contributions and the Drude model, one can estimate the corresponding spectra by using dielectric function calculations. Sheikholeslami et al.¹⁴ have provided a method to explain the red-shift of LSPR in compositional nanosphere heterodimers based on a revision of the plasmon hybridization model in nanosphere investigations. Here, we further modified this method for a NM heterodimer in a quasi-approximation technique. In this method, plasmon resonance energy levels appear orderly for both transverse and longitudinal polarization modes, which includes the interband transitions in the gold NM and scattering of silver unit. Figure 3(c) exhibits the modified hybridization model for the silver-gold heterodimer. Due to the interband absorption of the optical power in the gold NM unit, we modified this model based on the material and orientation of the NM structures.

Ultimately, in order to calculate the coupling between NM units, we employed the modified hybridization model for plasmon resonance coupling, which contains the interband absorption of gold unit plus the scattering spectra (Figs. 3(c) and 4(a)). As we explained already, the interband absorption moved the plasmon resonance of the silver NM to the longer wavelengths (red-shift). Also, a comparison between the longitudinal plasmon antibonding mode (σ^∗) for the heterodimer nanostructure and the transverse plasmon antibonding mode (π^∗) indicates that the corresponding coupling is purely of bonding form. Figure 4(b) demonstrates formation of the new bonding and the antibonding plasmon resonant modes in the revised hybridization model in which the upper levels designate the interband absorption spectra. The figure contains the anticipated interband absorption of gold and plasmon resonance of silver shells as well.

In this study, we have examined the plasmon resonance coupling quality between NM units in an individual dimer structure using numerical analysis. Illuminating a symmetric dimer composed of gold NM with longitudinal- and transverse-polarized beams, we demonstrated the spectral response of the structure numerically. Then, breaking the symmetry of the dimer structure by changing the structural dimensions and choosing the same metal for both units, we observed bonding (σ and π) and antibonding (σ^∗ and π^∗) plasmon modes for both of the incident polarized spectra. In this regime, we also observed a minor blue-shift in the bonding mode due to the symmetry breaking which is in complete agreement with the plasmon hybridization model. Finally, making changes in the physical dimensions and chemical compositions (silver-gold) of the utilized NM units showed a plasmon coupling and an interband absorption corresponding to the gold nanoshells. This led to an abnormal red-shift of higher energy modes (σ^∗ and π^∗), which caused by coupling between interband absorption of gold shells and plasmon resonance of silver shells. Such a shift of LSPR is not expected by the plasmon hybridization model. Utilizing a modified hybridization model based on full dielectric function computations and the Drude model which have previously used for nanospheres and extended to multilayer shells, we obtained an expected blue-shift of plasmon resonance for higher energies. The variation of the optical response and plasmon resonance which caused by subtle structural and compositional alterations could transmute the modified nanostructures into a multipurpose scheme for LSPR sensing applications.

This work is supported by NSF CAREER program with the Award number: 0955013, and by Army Research Laboratory (ARL) Multiscale Multidisciplinary Modeling of Electronic Materials (MSME) Collaborative Research Alliance (CRA) (Grant No. W911NF-12-2-0023, Program Manager: Dr. Meredith L. Reed). Raju Sinha gratefully acknowledges the financial support provided through presidential fellowship by the University Graduate School (UGS) at Florida International University.