Geometric frustration in a hexagonal lattice of plasmonic nanoelements

We introduce the concept of geometric frustration in plasmonic arrays of nanoelements. In particular, we present the case of a hexagonal lattice of Au nanoasterisks arranged so that the gaps between neighboring elements are small and lead to a strong near-field dipolar coupling. Besides, far-field interactions yield higher-order collective modes around the visible region that follow the translational symmetry of the lattice. However, dipolar excitations of the gaps in the hexagonal array are geometrically frustrated for interactions beyond nearest neighbors, yielding the destabilization of the low energy modes in the near infrared. This in turn results in a slow dynamics of the optical response and a complex interplay between localized and collective modes, a behavior that shares features with geometrically frustrated magnetic systems. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Plasmonic nanoelement arrays have been widely used for a variety of applications including enhanced spectroscopies [1][2][3][4][5], perfect absorbers [6][7][8], plasmonic lasing [9][10][11] or energy harvesting [12].Innumerable research has been done in fully characterizing the spectral response or the field enhancement (FE) of a variety of nanoantennas [2,4,[13][14][15][16].Besides, many studies have analyzed the response of nanoparticle arrays where interparticle interactions are significant [11,[14][15][16][17][18][19][20][21][22].In these systems, when the interparticle separation is of the order of the resonance wavelength, far-field interactions between the nanoelements in the array may result in constructive interferences giving rise to collective modes known as Surface Lattice Resonances (SLR).In this case, each element in the array is excited not only by the impinging radiation but also by the light scattered by other nanoelements, yielding a collective coupling among all of them.Moreover, the optical response of the system may also be governed by the near-field coupling between nearest neighbors giving rise to Localized Surface Resonances (LSR).Whereas LSR usually exhibit broad extinction peaks due to radiative damping, SLR present sharp linewidths that could be more suitable for applications profiting from a high frequency specificity [21][22][23][24][25]. Interestingly, for incoming radiation around normal incidence and with a wavelength matching one of the diffraction edges of the Bravais lattice of the array, a SLR mode due to the lattice diffraction shows up.This mode may interfere with one of the LSR modes, resulting in a sharp asymmetric dip superimposed to the LSR peak of the extinction spectrum that corresponds to a kind of Fano resonance [14,18].
The effect that the symmetry of the array has on the plasmonic response, and in particular, on the occurrence of SLR and Fano resonances, has been studied in a number of different lattices [17][18][19].At normal incident radiation, slight differences in the spectra have been observed as a function of lattice geometries for the same diffraction edge [19], and only for oblique incident light those differences became more noticeable [17].This brings up the question whether it is possible to take advantage of the delocalized nature of the SLR to where inter the excitati features wi systems suc been studie properties that honeyc modes simi control of metamateri honeycomb related to th Moreov materials a combinatio magneto-pl Fig cel In this numerical hexagonal may occur The six re nearest-nei strong near modes) of bars formi interpenetra of the gaps are excited [32].Conse delocalized modes.A t decaying ti variety of basic mode compatible mode of th o explore exc racting electri ion of a variet ith those kind ch as spin ice ed due to its s [17,20,28].Fo comb arrays o ilar to electron the light pro ials [28].Th b lattices [17-he geometrica ver, SLR mod are combined on of near-fiel lasmonic nano

Sample fabrication
Arrays of asterisk-shaped Au nanostructures of 200 μm x 200 μm in total area have been fabricated by means of Electron Beam Lithography (EBL) in a metal-insulator-metal (MIM) configuration.MIM configurations are conventionally used in metamaterial absorbers or Surface Enhanced Infrared Absorption (SEIRA) applications to maximize light absorption [4,16,18].There are two different approaches for the MIM configuration: (i) the near-field coupling scheme, where the thickness of the dielectric spacer layer is much smaller than the incident light wavelength λ; and (ii) the far-field coupling scheme, based on the Salisbury screen [33], where the thickness of the spacer layer is about nλ/4, being n the refractive index of the spacer material.In this case, destructive interference is obtained so that there is a total reflection of the incoming light [7].The latter configuration has been adopted in our samples.
In order to form the MIM stack [Fig.1(a)], a 60 nm thick Au layer was deposited by electron beam evaporation on a conventional Si wafer, using a 5 nm Ti layer to promote adhesion, followed by the deposition of SiO 2 by Plasma Enhanced Chemical Vapor Deposition (PECVD).The thickness of the spacer layer has been chosen so that total reflection takes place at a wavelength around the SLR peak to enhance collective excitations.The SiO 2 deposition took place in two steps.First, a thin layer was deposited at 50 W of radio frequency (RF) power at 350 °C for 20 seconds, to obtain a thickness of around 20 nm.This layer, which typically accounts for 10-20% of the final total thickness of the SiO 2 layer, serves as a protection for the underlying Au layer.Then, a second deposition took place at a power of 200 W and at a temperature of 380 °C.
Then, poly(methyl methacrylate) (950 PMMA A2, MicroChem) was spin-coated at 1200 rpm for 1 minute onto the substrate and cured at 180 °C for 1 minute.The EBL exposure was done at 20 kV with a 20 μm gun aperture and a step size of 10 nm.The dose was fixed to 180 μA/cm 2 and the dose factor was changed from area to area to finely tune the size.After the exposure, the sample was developed by dipping it in a 1:3 solution of methyl isobutyl ketone (MIBK) in isopropanol (IPA) for 30 seconds and then rinsed in IPA for 30 seconds to stop the reaction.Then, samples were metallized by electron beam evaporation (ATC Orion, AJA International, Inc).For all samples, a thin (0.5-2 nm) layer of either Ti or Cr was deposited to promote adhesion before growing the Au layer (20-30 nm).Finally, the sample was immersed in acetone at 40 °C for 5 minutes to soften the resist and then ultrasounds were applied to perform the lift-off.
A schematic plot of the samples is shown in Fig. 1(a).Several geometries ranging from asterisk-[Fig.2(a)] to star-shaped [Fig.2(d)] nanoelements were prepared.Besides, the pitch of the array and the gap size were varied from 400 to 500 nm and from 20 to 100 nm, respectively.

FTIR
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FDTD simulations
For a better understanding of the plasmonic response of the arrays, Finite-Difference Time Domain (FDTD) simulations using the Lumerical FDTD Solutions package [34] were carried out to calculate the absorption spectrum, the near-field intensity, the charge distribution, and the time evolution of the optical response.The refractive indices were taken from the data set included in Lumerical's database and correspond to the data from Palik [35] for the case of Cr, Ti, Si and SiO 2 and Johnson's and Christy [36] data for the case of Au.The background dielectric constant was taken to be 1.The array was excited by an incident pulse normal to the plane containing the system, and the electric field was parallel to the x-axis.Symmetric and antisymmetric conditions were used in the x-and y-axes to simulate only one quarter of unit cell and minimize the computing time.Perfectly matched layer (PML) boundaries were used in the z-axis.Simulations of a single element were also performed using a Total-Field Scattered-Field source and two detector boxes placed outside and inside the source to measure the scattering and absorption cross-sections, respectively.
It has been shown that, in most cases, the Ti or Cr layers used to promote adhesion of Au may yield critical effects on the optical response producing a strong weakening of it [4,37,38].Therefore, all the simulations include a 0.5 nm-thick Cr layer under the Au nanostructures to make them comparable to the fabricated samples.Figure 4 shows the response of a single asterisk and an array of 500 nm / 50 nm / 45 nm / 30 nm (pitch / gap / width / thickness) asterisk-shaped nanoelements.
Although the fabricated samples were metallized with a 20 nm-thick Au layer, the simulations were performed for 30 nm-thick Au which optimizes the absorption (see Appendix for further information).We did not succeed in manufacturing samples with 30 nm-thick Au layer due to some issues in the lift-off process.The absorption spectra, calculated with and without the MIM configuration, are shown in Fig. 4(b), which also shows that transmitted light in the MIM configuration is negligible.In the former, a 100 nm-thick SiO 2 layer was used as a spacer [Fig.1(a)].The absorption of the system is clearly higher in MIM configuration [Fig.4(b)], whose spectrum exhibits two intense peaks, one wide NIR peak (2277 nm) and a narrow one slightly above the visible region (764 nm).The latter displays a quality factor of 19 and reaches 98.4% of absorbance, making these structures nearly-perfect absorbers.Besides, a less intense but also sharp peak is located at 624 nm, which is only present in MIM configuration.
Simulations show a slight change in the location of the high-order energy peak as a function of pitch, gap width or length of the bars, while the wavelength of the NIR peak is strongly dependent on those parameters (see Appendix), in good agreement with the experimental data shown in Fig. 3.However, the VIS peak in the FTIR measurements [Fig.3] is much broader than in the simulations.This difference may be attributed to several factors, such as the imperfections at the interfaces between both the substrate and the Cr layer, and the Cr and Au layers, or the variability in the shape of the nanoelements due to imperfections in the fabrication process.The latter gives rise to a width distribution of the gaps, roughness and rounded edges of the fabricated nanoelements [Fig.2], as compared to the ideal asterisk-shaped nanostructures used in the simulations (see Appendix for further information).Moreover, the secondary VIS peak at 624 nm observed in the simulation in MIM configuration is likely merged into the high-order broad peak of the experimental data.The sp logarithmic the simulat Fig. 4(c is R collective hin the gaps and being compatible with the translational symmetry of the hexagonal lattice of nanoelements.

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