Plasmonic Quasicrystalline Arrays

Optimized quasicrystalline spherical nanoparticle arrays

Hybrid Photonic-Plasmonic Resonance in Periodic Arrays of Metallic Nanoparticles

In recent years there has been immense interest in the optical properties of metallic arrays of nanoparticles (NPs). It is well known that certain noble metals such as Ag, Au, behave as plasmonic materials in the optical frequencies which can lead to highly enhanced local fields known as localized surface plasmon resonance.

These plasmonic arrays are ideal candidates for a variety of applications such as enhanced absorption with large angular tolerance for solar cells, surface-enhanced Raman scattering substrates, and broadband plasmonic enhancement.

The optical response of plasmonic arrays of NPs can be explained in terms of the plasmonic response of the NPs (plasmonic resonance) and the Bragg resonance of the array (photonic resonance). The plasmonic resonance is a function of the geometry of the particle and the dielectric function of the metal and photonic resonances can be analyzed as the Bragg grating modes, which are due to the coherent scattering which occur as the incident wavelength approaches the lattice constant.

This phenomenon is of particular interest when the critical grating constant also coincides with the localized surface plasmon wavelength of the NP. The result is the excitation of a very sharp, so-called “hybrid photonic-plasmonic resonance”. In periodic lattices, the hybrid resonance, while very sharp has a very narrow width. This is due to the inherent relatively high-Q nature of photonic resonances that occur in periodic lattices.

Au/SiO2 core-shell nanoparticles

Hybrid Resonance in Quasicrystalline Arrays of Metallic Nanoparticles

Hybrid resonances can also exist in quasicrystalline (QC) lattices since they possess photonic resonances that can be deduced from their diffraction patterns. In fact since QC lattices can possess multiple photonic resonances in close proximity, in theory when properly scaled, QC arrays of plasmonic NPs can possess multiple hybrid resonances or broadband resonances which are not feasible in periodic lattices.

A key obstacle in analyzing the optical properties of QC arrays of plasmonic NPs and QC morphologies in general is the lack of accurate is the lack of analytical tools to accurately and efficiently model them. In periodic lattices, the solution domain of an infinite structure is reduced to a single unit cell by exploiting the translational symmetry. QC by definition lack translational symmetry and hence cannot be modeled accurately using periodic boundary conditions. Currently, the primary analytical method for QCs is the so-called super-cell approach. The method essentially takes a large segment of the structure and applies periodic boundary conditions to. Analyzing plasmonic structures using traditional full-wave finite-difference and finite-element techniques requires considerable computational resources even for relatively simple configurations.

For this reason, there has been recent interest in developing more efficient analytical tools which offer solutions several orders of magnitude faster than is possible with finite-difference and finite-element methods.

Generalized Multiparticle Mie Theory (GMT)

In the case of spherical particles, generalized multiparticle Mie theory (GMT) can be applied to evaluate the scattering properties of arrays with arbitrary morphologies. This is a rigorous multiparticle approach which provides a complete solution to Maxwell's equations and takes into account all the multipolar scattering orders. GMT was developed by Yu-lin Xu in 19951. Upon publication, he also developed a very efficient open-source numerical code to implement the method. The method is essentially a T-matrix method based on expanding all incident and scattered fields in terms of vector spherical wave functions (VSWF).

In theory any physically realizable incident wave can be expanded in terms of VSWFs. However in the original paper and code that was subsequently developed an incident plane wave was assumed. Using an incident plane wave greatly simplifies the problem on two fronts. First, a plane wave has a well-known simple expansion in terms of VSWFs. Secondly it can be shown that for an incident plane wave, the expansion coefficients at the displaced systems differ from the primary expansion coefficients only by a constant phase term and thus does not require the application of addition theorems for VSWFs. However since a plane wave has an infinite beamwidth, it is not possible to define reflection and transmission coefficients in the usual sense when considering the analysis of finite-size arrays.

Previous Applications of GMT to Aperiodic Plasmonic NP Arrays

Optical properties of aperiodic Au NP using the GMT method where first studied in Ref. 2 where aperiodic arrays of cylindrical Au NPs with a height of 30 nm and a diameter of 200 nm were fabricated and dark-field scattering spectroscopy was employed to characterize the scattering properties of the aperiodic arrays. For the electrodynamic calculations, the GMT method was utilized to calculate the scattering efficiencies of finite-size arrays composed of approximately 100 Au nano-spheres with radii of 100 nm. Here two important points are noted regarding the results reported in Ref. 2:

It was argued that since the main aim of the study was to reveal the role of array morphology, simulations based on spherical particles could be used to explain the scattering features of cylindrical NP arrays.
The second issue had to do with the incident field used in the GMT simulations. The authors used the original code developed by Xu based on an assumed plane wave excitation. However since a plane wave has an infinite beamwidth, it is not possible to define reflection and transmission coefficients in the usual sense when considering the analysis of finite-size arrays.

Due to these inconsistencies simulated and measured results were never compared directly and in some cases there was very little resemblance between measured and simulated patterns. Some of the results published in Ref. 2 are shown in the next slide.

More Accurate Results Based on Accurate Fabrication and Improved Analytical Methods

As it can be seen from the previous slide, there is very little resemblance between the simulated and measured results. These are mostly due to two issues that were mentioned in slide 6. We resolved both of these issues and have obtained quantitative results which can be directly compared with experimental measurements and therefore can be effectively used to characterize the scattering properties of QC lattices. In doing so we have taken two major steps:

First, we were able to fabricate spherical NP arrays with great accuracy.
We implemented a modified GMT approach introduced in Ref. 3 for an incident wave excitation with a finite beamwidth. The incident beam was obtained by placing a circular aperture in front of a plane wave to obtain a beamwidth smaller than the array dimensions thereby avoiding diffraction by spheres at the edges of the lattice. Such a setup is very similar to realistic experimental conditions. Using the far-field expressions for scattered fields, generalized transmission and reflection coefficients were then defined based on total far-field energy fluxes.

Accurate Fabrication of Spherical NPs

The optimized quasicrystalline spherical NP arrays were fabricated by adopting a novel nanofabrication approach that employs electron beam lithography and subsequent thermal treatment. This process begins by defining QC arrays (3 mm × 3 mm) of cylindrical amorphous-Si/Au NPs on a fused silica substrate using electron-beam lithography followed by an amorphous Si/Au lift-off process. The resulting QC cylindrical NP arrays have the desired spacing and aperiodic arrangement. The resulting amorphous-Si/Au NP array is then converted into a Au/SiO2 core-shell NP array by thermally oxidizing the entire structure in dry O2. During the thermal treatment, the Au core is transformed from a cylinder into a sphere to reduce the interfacial energy between the Au core and the SiO2 shell.

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