Strong and weak polarization-dependent interactions in connected and disconnected plasmonic nanostructures

We explore numerically and experimentally the formation of hybridized modes between a bright mode displayed by a gold nanodisc and either dark or bright modes of a nanorod – both elements being either separated by a nanometer-size gap (disconnected system) or relied on a metal junction (connected system). In terms of modeling, we compare the scattering or absorption spectra and field distributions obtained under oblique-incidence plane wave illumination with quasi-normal mode computation and an analytical model based on a coupled oscillator model. Both connected and disconnected systems have very different plasmon properties in longitudinal polarization. The disconnected system can be consistently understood in terms of the nature of hybridized modes and coupling strength using either QNMs or coupled oscillator model; however the connected configuration presents intriguing peculiarities based on the strong redistribution of charges implied by the presence of the metal connection. In practice, the fabrication of disconnected or connected configurations depends on the mitigation of lithographic proximity effects inherent to top-down lithography methods, which can lead to the formation of small metal junctions, while careful lithographic dosing allows one to fabricate disconnected systems with a gap as low as 20 nm. We obtained a very good agreement between experimentally measured scattering spectra and numerical predictions. The methods and analyses presented in this work can be applied to a wide range of systems, for potential applications in light–matter interactions, biosensing or strain monitoring.


Finite element simulations
Numerical simulations have been performed using two open-source softwares, namely FreeFem++ for the numerical resolution of Maxwell equations using FEM, [1][2][3] and Gmsh to create geometries and generate tetrahedral meshes. 4 For reasons of computer resources savings, symmetry (resp. anti-symmetry) conditions have been applied on the (Oxz) plane for p (resp. s) polarization. In order to prevent reflection of the scattered field on the boundaries of the simulation domain, an additional spherical PML (Perfectly Matched Layer) layer has been added around the simulation domain, delimited by the red dashed line. An example of geometry and mesh is shown on Fig. S1. Figure S1: Geometry and mesh realized with Gmsh.
We have performed both plane-wave illumination and quasi-normal modes simulations. For that purpose, the dielectric constant for gold must be written under an analytical form in order to be able to extend it to complex frequencies. We have then fitted experimental data from Johnson and Christy 5 in the wavelength range between 500 nm and 900 nm with a two-oscillators Drude-Lorentz model, following : where k 0 = ω/c = 2π/λ is the wavenumber, and: The obtained numerical values for the model parameters are listed in Table 1.

Planewave illumination
Simulations of the electromagnetic response of the system under monochromatic planewave illumination rely on the numerical resolution of the following equation : where the total electric field reads E = E b + E s , with E b the background field resulting from the reflection of the incident planewave on the glass interface (without the metal nanostructures), E s the field scattered by the nanostructures, and ϵ b (r) the position-dependent dielectric constant of the glass-air environment. This equation is then translated into its weak-form in order to be solvable with FEM. Details about the procedure to follow, including explanations about boundary conditions and finite-elements type can be found in reference. 6 Extinction, absorption and scattering cross-sections are respectively given by: where Im stands for the imaginary part and the polarization vector is defined as:

Quasi-normal modes
Quasi-normal modes are complex-frequency (or complex wavenumber) solutions of Eq. 3 without source term (E b = 0). This equation must then be rewritten in a shape allowing eigenvectors computation. Using 1, we obtain: wherek 0 is the complex eigenvalue of the QNM, and: and:ε where V is the volume of the metal nanostructure. The real part ofk 0 gives access to the resonance wavelength, λ 0 = 2π/ Re(k 0 ), and the imaginary part to its width ∆λ 0 = λ 0 Im(k 0 )/ Re(k 0 ). It is clear that any solution E s of equation 4 for a particulark 0 is an eigenvector of the operator 1/ε ∞ (r)∇ × ∇ × () + δε(r,k 0 ) for the complex eigenvaluek 2 0 . However, as this operator explicitly and non-linearly depends onk 0 , this equation is self-consistent and cannot be solved directly with the finite elements method. We choose instead to solve: where k e is a complex, close-enough estimate ofk 0 , which can be extracted for example from extinction spectra. Equation 5 is then a standard eigenvalue problem and can be solved with FreeFem++ using a generalized eigenvalue solver. The procedure to calculate the eigenmodes and eigenvalues is the following: • compute extinction or absorption spectra under planewave illumination; • extract estimates k e of eigenvalues (real and imaginary parts) using wavelengths λ e and widths ∆λ e for maxima shown in absorption or extinction spectra: k e = 2π/λ e (1 + i∆λ e /λ e ); • solve Eq. 5 to obtain a new value ofk 0 ; • iterate k e based on the obtainedk 0 and repeat until convergence is reached. The main difficulty is to extract the physically meaningfulk 0 among the very large number of eigenvalues of the operator projected on the basis of finite elements (see figure 2(a)), as most of the returned eigenvectors are modes either due to the meshing (figure S2(c)) or associated to the PMLs ( figure S2(d)). Fortunately, efficient sorting can be performed on the imaginary part ofk 0 , which is generally comprised between low-losses mesh eigenmodes and large-losses PMLs eigenmodes. In practice, generally only two or three iterations are necessary to reach a converged eigenvalue with satisfactory precision, as shown on Fig. 2(b).

Uncoupled particles
The response of isolated ND and the NR is plotted on Fig. S3. The incidence plane contains the NR long axis. Both scattering and absorption spectra of the isolated gold nanodisk are dominated by a broad resonance centered at 682 nm, and corresponds to the excitation of the in-plane dipolar mode, (i). The width of that resonance, of about 100 nm, is related to the contribution of both internal dissipation and radiation losses. The NR is characterized by longitudinal bright and dark modes, with alternating positive and negative surface charges, bright modes being antisymmetric along the NR's long axis, while dark modes are symmetric. The 84-nm-long NR shows one resonance, Figure S3: Scattering (thick lines) and absorption (thin lines) spectra of the ND and NRs with two different lengths L. The incidence plane contains the NR axis, and the polarization is p. The real part of the surface charges distribution is plotted for each associated QNM, whose wavelength is indicated by dashed lines.
(ii), close to 645 nm, corresponding to the lowest order bright mode, as evidenced by its surface charges distribution. The 244-nm-long rod shows a maximum close to 715 nm, and corresponds to the excitation of the lowest order dark mode, (iii). Note that the NR's dark modes can only be excited because the illumination direction breaks the longitudinal symmetry of the nanoparticle. The maxima are close to the wavelengths obtained from QNM modes calculations, indicated by a vertical dashed line. The discrepancy is the largest for the ND dipole mode, with a value of about 6 nm, as QNM simulations leads to a dipolar mode wavelength of about 687.7 nm and a width ∆λ 0 = 114 nm or (∆λ 0 )/λ 2 0 = 0.24 µm −1 , consistent with the spectra. For nanorods, the QNM wavelengths are 645.8 nm for the short NR (∆λ 0 = 48 nm or (∆λ 0 )/λ 2 0 = 0.12 µm −1 ) and 716.0 nm for the long NR (∆λ 0 = 37 nm or (∆λ 0 )/λ 2 0 = 0.072 µm −1 ). The widths are again consistent with the spectra, and we verify that it is larger for the bright mode than for the dark mode due to additional radiation losses.