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Plasmonic coupling of multipolar edge modes and the formation of gap modes Edson P. Bellido, Yue Zhang, Alejandro Manjavacas, Peter Nordlander, and Gianluigi A. Botton ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 18 May 2017 Downloaded from http://pubs.acs.org on May 19, 2017

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Plasmonic coupling of multipolar edge modes and the formation of gap modes Edson P. Bellido,† Yue Zhang,‡ Alejandro Manjavacas,¶ Peter Nordlander,∗,‡ and Gianluigi A. Botton∗,† †Department of Materials Science and Engineering, McMaster University, 1280 Main Street W. Hamilton, ON L8S 4L7, Canada ‡Department of Physics and Astronomy and Laboratory for Nanophotonics, Rice University, Houston, Texas 77005, United States ¶Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, United States E-mail: [email protected]; [email protected]

Abstract The coupling of plasmonic resonances is an effective tool to tailor the optical properties of nanostructures. However, the coupling of higher order plasmonic resonances has not received much attention, with most studies focusing on the interaction of dipolar modes. Taking advantage of the high spatial and energy resolution of modern scanning transmission electron microscopes equipped with electron energy loss spectroscopy, we analyze the coupling of edge modes in planar nanostructures with emphasis on the interaction of high order modes and the formation of gap modes. We show that coupling of edge modes can be understood by a simple and intuitive scheme, with three regimes: First, a strong coupling through the edge of the structure resulting in bonding and antibonding gap edge modes; second, coupling through the corners of the structures

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resulting in bonding and antibonding corner edge modes; and a third behaviour where the edge modes do not couple and behave independently of the rest of the structure. The formation of gap modes through the coupling of edge modes is analyzed and compared to the modes found in planar slot waveguides, finding that the properties of the symmetric and asymmetric modes on slot waveguides are equivalent to the antibonding and bonding gap edge modes respectively. Our experimental and numerical analysis of the plasmon resonances in nano-squares and waveguides shows that our scheme of plasmonic coupling of edge modes can be generalized to other planar structures with straight edges, and might inspire the design of more complex planar plasmonic devices based on the coupling of edge modes. Keywords: Coupling, Edge modes, EELS, Square dimers, Hybridization

The study of the optical properties of metallic nanostructures, and in particular the characterization of the localized surface plasmon resonances (LSPR) supported by these systems, has experienced significant growth in recent years, due to the potential applications of the strong field confinement and large cross-section provided by LSPR. Applications of LSPR span over a large range of areas, including sensing, 1 metamaterials, 2–4 photovoltaics, 5 and even cancer treatment. 6,7 For these reasons, a variety of metallic structures have been investigated and characterized by employing a variety of tools. Among them, electron energy loss spectroscopy (EELS) performed in a scanning transmission electron microscope (STEM) has become a valuable characterization tool, revealing the full spectrum of resonances present in nanostructures due to the combination of high spatial resolution and its ability to map dark modes. Since the seminal work of Nelayah et. at. 8 and Bosman et. at. 9 many geometries have been studied, including rods, 10,11 cubes, 12–14 wires 15 , 16 holes, 17 polygonal prisms, 18–20 crosses, 21 shurikens, 22 and planar structures. 23,24 These studies have shown the existence of multiple modes present on nanostructures with unprecedented spatial resolution, and have also demonstrated the ability to tailor the optical properties of a nanostructure by modifying the shape, size, and dielectric environment. An additional method for tailoring the LSPR that provides extra flexibility relies on ex2


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ploiting the coupling of plasmonic nanostructures. Due to the interaction between the LSPR of the individual structures, the optical response of the whole system is strongly modified when the distance between nanostructures becomes comparable or smaller than their lateral size and a splitting of the individual structure modes appears, as was explained by Prodan et al. 25 in their hybridization model. The interaction of closely spaced plasmonic nanostructures also induces other unique effects, such as tunneling charge transfer plasmons, 26 and Fano resonances, 27 thus expanding the toolset for modifying the optical properties of these systems. Even in the simplest case of nanoparticle dimers, a hybridization of the dipolar mode that is dependent on the inter-particle distance is observed. 28,29 This phenomenon has been used to develop the concept of plasmon rulers. 30,31 In structures where the excitation of higher order resonances is efficient, a richer modal spectrum can be obtained from the plasmonic coupling of higher order modes, and phenomena such as intermodal coupling 32 can be observed. However, most of the investigation on plasmonic coupling has been focused on the interaction of dipolar modes, or in small structures where higher order modes are difficult to excite. 33 In this work, we present a detailed study of plasmonic coupling of edge modes with a special emphasis on the interaction of high order modes and the formation of gap modes. Through the investigation of the response of a number of nano-squares dimers 40 nm thick with lengths from 420 nm to 1 µm separated by gaps ranging from 50 nm to 100 nm, we study the coupling of these dimers by separating their LSPR into low order and high order modes. To further understand the coupling of edge modes and the formation of gap modes, we also investigate structures composed of two 50 µm long and 40 nm thick silver strips with widths ranging from 470 nm to 1150 nm separated by gaps with sizes from 95 nm to 115 nm (slot waveguides). The dispersion of the plasmon resonances formed by coupled edge modes in the nano-square dimers is also analyzed and compared with the dispersion of modes in planar slot waveguides. The silver structures used in this work are fabricated by electron beam lithography on

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electron transparent Si3 N4 membranes, 34 and their optical response is analyzed by acquiring electron energy loss (EEL) spectrum images with high energy and spatial resolution using a FEI-Titan STEM equipped with an electron monochromator. The effective energy resolution of the acquired spectrum images is further increased by deconvolution. 34 We also perform numerical simulations of EELS using the MNPBEM Matlab toolbox 35,36 that is optimized to simulate metallic nanoparticles using the formalism of the boundary element method. 37 The effect of the silicon nitride substrate is modeled using an effective dielectric function of 2.5 for the surrounding environment, and the silver structures are modeled using a tabulated dielectric function. 38

Coupling of low order modes in nano-square dimers Figure 1 (a) shows the EEL spectra acquired in six different areas of a 420 nm nano-square dimer with 50 nm gap as indicated in the inset of the figure. Due to the high effective energy resolution of the spectra (60 meV), measured from the full-width-at-half-maximum of the zero loss peak, we can identify ten LSPR with peaks in multiple positions. The plasmon resonances present in the nano-square dimer can be separated into two groups: One group formed by the first five LSPR consisting of the lower order modes, and a second group comprising the next five higher order resonances. The measured and simulated EELS maps of the first five resonances are shown in Figure 1(b and d). To extract the EELS maps corresponding to each surface plasmon resonance, we isolate the contribution from each peak in the plasmon maps using the non-linear least squares fitting tool of the “DigitalMicrograph” software, which fits Gaussian peaks to a spectrum image. 39 From the EELS maps, we can identify the nature of the different modes supported by the nano-square dimer. The first three LSPRs correspond to coupling of the dipolar mode of an individual nano-square. Two of these resonances can be identified as bonding (mode 1) at 0.50 eV, where charge carriers of opposite sign are accumulated at

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the dipolar mode with opposite charges at the top and bottom sides of the square and the dipolar mode with opposite charges at the left and right sides. In the dimer, the left-right dipole modes couple creating the aforementioned bonding and anti-bonding modes, while the top-bottom dipoles form this transverse mode. For the structures we analyze, we did not find a strong enough interaction of the top-bottom dipoles that will lift the degeneracy of the transverse bonding and anti-bonding modes. Thus these modes appear as a single transverse mode. Plasmon resonances 4 and 5 are generated by the coupling of the quadrupolar modes in each square. As it is the case in the coupling of the dipolar modes, the quadrupolar mode also splits into a bonding mode at 0.83 eV, where the charges of opposite sign are localized in the corner close to the gap, and an anti-bonding mode at 0.95 eV, where the charges of equal sign are concentrated in the corners close to the gap. Figure S1 in the supplementary information document shows an energy diagram summarizing the coupling of lower order LSPRs and the above explained charge distributions. The formation of the five plasmon resonances by the coupling of the dipolar and quadrupolar modes is an evidence of the wide range of resonances that can be excited by the coupling of plasmonic structures.

Edge modes and the coupling of high order modes in nanosquare dimers Besides the coupling of low order LSPRs, the spectra in Figure 1(a) and (c) also exhibit several peaks corresponding to the coupling of high order resonances. The intensity distribution of these resonances supported by the 420 nm nano-square dimer is displayed in Figure 2. The figure shows the experimental EELS intensity and the simulated EELS probability as a function of electron energy loss (EELS profile) acquired from three different edges: edges on the gap (region “i”); an edge on the side of the dimer (region “ii”); and an edge on the base of the dimer (region “iii”), as illustrated in the annular-dark-field (ADF) image and diagram. The LSPRs formed by the coupling of the nano-square are identified with white dashed lines 6


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tion document edge modes of order two (E2 ), three (E3 ) and four (E4 ) are observed in the nanosquare. When two planar structures, such as the present nano-squares, are in proximity to each other, these edge modes couple. This coupling is responsible for the presence of peaks 6 to 10 in the spectra in Figure 1(a). In the nano-square dimer, this second group of peaks could also be described as edge modes. To explain this premise, we can analyze the resonances by spatial region, as shown in Figure 2(a): In region “iii” (at the base of the dimer), we can distinguish two modes: mode 7 and 10, based on their number of nodes, correspond to second (E2 ) and third (E3 ) order edge modes at 1.25 eV and 1.53 eV respectively. In region “ii” (on the side of the dimer), we can only distinguish clearly one peak (mode 8) at 1.30 eV with two nodes, thus corresponding to a second order edge mode (E2 ). In these two regions (“ii” and “iii”) we observe two second order modes at different energies (mode 7 and 8). To identify this two modes, we used the spectra acquired over the regions along the edges (gray and blue regions of the inset of Figure 1(a)) and not spectra over the corners because the two modes are energetically close and because they share a corner. Therefore, the spectrum integrated over the corner does not represent an accurate value of the resonant energy of these modes. The difference in energy of modes 7 and 8, albeit small, is caused by the coupling of edge mode E2 through the corner of each nano-square on the side of the dimer (region “ii”). This weak coupling causes the apparent shift in energy with respect to the edge mode E2 at the base of the dimer (region “iii”), which does not couple. As we will discuss later in the manuscript and in Figure 3, this shift in energy is an indication of mode splitting due to the coupling of LSPRs confirming that mode 7 and 8 are indeed two different modes. A different behavior is observed in the gap of the dimer (region “i”). In this case, we observe the presence of two modes: mode 6 and 9 with node distributions that corresponds to a second order edge mode at 1.18 eV and 1.46 eV respectively. This similar node distribution indicates coupling of the second order mode of the individual squares that split into these

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two modes. We will designate these plasmon resonances as bonding gap edge (BG−E) mode and antibonding gap edge (AG − E) mode due to the charge distribution similarity with the low order resonances. Charges of opposite sign are accumulated at each edge in the gap in the BG − E mode, and charges of the same sign are localized at the edges of the gap in the AG − E mode. The black line and magenta line spectra, acquired in the gap region of the nano-square dimer and shown in Figure 1 (a) and (c), provide more insight into the nature of these gap edge modes. We notice that resonance number 6, or the bonding gap edge mode order two (BG − E2 ), only appears in the black-line spectra, which indicates that this plasmon resonance can only be excited when the electron beam is positioned over the metal on the edge of the nano-square. On the other hand, mode number 9, or the antibonding gap edge mode order two (AG−E2 ), appears in both the black line and the magenta line spectra. Comparing the LSPRs present in the experimental intensity profile with the ones present in the simulated profile in Figure 2, we note the correspondence of the nodal description of the modes 6 to 10 despite the distortion of the antinodes in the experimental data caused by the considerable roughness of the edges on the 420 nm nano-square dimer. Also, the energy of the simulated modes in Figure 1 and 2 do not show a perfect agreement with the experiments due to the absence of substrate in the simulations, and to imperfections in the fabricated structures. The simulated EELS profile shows the presence of more resonances than the ones identified in the experiments. This is also caused by the edge roughness that has an even more considerable effect on higher order plasmon resonances for which the nodal distance is smaller. In larger square dimers such as the one shown in Figure 4 with a side length of 1 µm, the effect of roughness is lower due to the larger nodal distance. In this case, higher order LSPRs can be detected as shown in the figure. In order to further understand the coupling of edge modes, we simulate the EELS response of the 420 nm nano-square dimer varying the coupling strength by changing the gap spacing. Figure 3 shows the change of the EELS spectra as a function of gap size, for gaps larger than 4 nm, plotted in a temperature scale where the peaks appear as "warmer color" bands, at

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modes, the antibonding modes blueshift slightly as we decrease the separation. Interestingly, the shift saturates and no change in energy is observed for small separations. In Figure 3(a) we also observe the evolution of the antibonding mode (aqua line) formed by the left-right dipole modes in each square as the gap spacing changes in the dimer. We notice that, as in the case of the AG − E2 resonance, this antibonding dipolar mode (mode 3 in Figure 1) only blueshifts slightly as the gap size decreases to finally saturate for gaps below 10 nm. In panel (a) we also observe the crossing of BG − E modes with the antibonding dipolar mode and the AG − E2 mode indicating that these LSPRs do not interact, thus near-field enhancements can be generated. 45 A second coupling regime is observed at the side of the dimers in the position indicated by the blue dot in Figure 1(c) and region “ii” in Figure 2. In this region, the edge modes couple through a corner. As seen in Figure 3(b), this coupling is much weaker than the coupling through the edge. For the 420 nm nano-square, we can only see the effects of this coupling for gap separations under 100 nm. For gap separations above 50 nm, the coupling can only be identified as a shift to higher energies due to the higher EELS probability of the antibonding corner edge (AC − E2 ) mode compared to the bonding corner edge (BC − E2 ) mode, as we noted from our results in Figure 2. Only when we decrease the gap size below 50 nm, we can clearly distinguish the two plasmon resonances, with the bonding mode redshifting and the antibonding mode blueshifting. As we continue decreasing the gap size, the shift of the resonant energy of both modes saturates. This coupling behavior is similar to the coupling of longitudinal LSPRs in nanowires dimers, 16 and the longitudinal antenna plasmon modes in nanorods. 45 In Figure 3(b) we can also observe the anticrossing (white dots) formed by the interaction between mode BC − E2 and high order bonding gap edge modes (BG − E). The interaction weakens as the order of the bonding gap edge modes increases in a manner similar to the anticrossing behavior found in gap plasmons in nanocube dimers where TCP resonances interact with bonding modes. 46 The third regime found in nano-square dimers is identified as “non-coupling”, and it

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through the corner is observed in region “ii” and we can identify the AC −E modes. However, we cannot identify the BC − E modes due to the weak corner coupling in the 100 nm gap of the dimer, and only an energy shift of the AC − E modes is displayed as result of the coupling as we explained above. Regions “iii” and “iv” show the coupling behavior through the gap, and we are able to identify the AG − E and BG − E modes up to order five. The separated regions of the gap show the different localization of the gap edge modes, with the AG − E modes localized over the gap in region “iii”, and the BG − E modes excited over the metal on the edge of the nano-square. Energy filtered maps for all the LSPRs found in this square dimer are shown in Figure S4 in the supplementary information document and display the large number of resonances produced by coupling of the edge modes. Comparing the nodal distribution of the plasmon resonances in this 1 µm square dimer (Figure 4) and the simulations in Figure 2, we can identify the modes that could not be identified for the 420 nm dimer due to their edges roughness. From the nodal distribution, we can recognize the mode at 1.96 eV on the side region in the simulation as mode AC − E3 . Also, the modes at 1.68 and 2.04 eV in the gap region of the simulation are identified as BG−E3 and BG−E4 respectively. The successful application of our coupling scheme to the analysis of the rather complex plasmonic spectra of the 1 µm square dimer shows the power of our approach.

Coupling of edge modes and the formation of gap modes In this subsection, we investigate the coupling of edge modes supported by long silver strips, which form so-called slot waveguides. The spectrum (green curve) of a single 50 µm long, 40 nm thick, and 700 nm wide silver strip displayed in Figure 5 (a) shows the presence of three edge modes. The first resonance is identified as an edge mode of order one (E1 ) or, alternatively the dipole mode as seen in its EELS map in Figure 5 (b). The second and third resonances, which have previously been studied, 42 are identified as edge modes of order two (E2 ) and three (E3 ) as confirmed by the intensity distribution in the EELS profile measured

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with a low intensity across the gap. The map of the second resonance, however, shows a significant intensity across the gap of the slot waveguide between the corners of the strips. This intensity distribution is topologically identical to the one found on the gap edge modes of the nano-square dimers. We can, therefore, assign the first peak as a BG − E1 mode and the second peak as a AG − E1 mode. Similar LSPRs were found in metal-semiconductormetal nanorods and named gap-localized transverse modes. 48 Mode E2 also splits into two modes: 1) mode BG − E2 , with two nodes along the edge, that can only be excited over the surface of the strip edge (area “iii” in Figure 5(c)); and 2) mode AG − E2 that has one very intense antinode at the center of the gap as seen in the EELS profile taken from area “ii” in Figure 5(c). Examining the EELS profile at the same energy of mode AG − E2 over the surface of the strip edge in area “iii” we note a very different intensity distribution, with three nodes, indicating the presence of mode BG − E3 . The two different intensity distributions indicate that two resonances are overlapping at this energy range, and under our experimental conditions (35 meV effective energy resolution and the large background) we cannot separate the two modes. The coupling of mode E3 in the strips, besides mode BG−E3 , also hybridize into mode AG−E3 that can be excited in the gap and exhibits three nodes along the gap shown in the EELS profile from area “ii” in Figure 5(c). The coupling of the edge modes in the slot waveguides and the formation the gap modes is summarized in Figure S5 in the supplementary information document. The results from the slot waveguides, as well as from the square dimer, are consistent with the following trend: The coupling of edge modes of the same order through the edge form two types of hybridized modes. One mode with a higher probability of excitation in the gap region (antibonding) and another with a high probability of excitation in the region over the metallic surface of the structure (bonding). The field distribution of the two types of plasmon resonances, which is proportional to the EELS intensity, resembles the field distribution of the resonances found on infinite slot waveguides with the field concentrated in the gap on the symmetric mode (antibonding mode) and the field localized on the metallic surface on

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from 95 nm to 115 nm. We can clearly observe two distinctive dispersion relations, one for the symmetric or antibonding gap edge modes with the EELS probability confined over the gap (red symbols), and one for the asymmetric or bonding gap edge modes with the EELS probability localized on the edge of the structures (blue symbols). The dispersion of the antibonding gap edge modes is closer to the light line (gray line). Therefore these resonances are less confined and have lower propagation losses than the bonding gap edge modes, which are further away from the light line. The bonding gap modes are, hence, more bound to the surface of the structures exhibiting higher confinement and higher losses due to larger penetration of the field into the metal. Figure 6 also shows the dispersion relation of the plasmon resonances found in the gaps of three nano-square dimers with 630 nm, 850nm, and 1 µm side lengths and gap sizes from 90 nm to 100 nm. As we discussed previously, the square dimers exhibit two types of gap modes similar to the ones found in the slot waveguides: antibonding modes, which are equivalent to the symmetric modes in slot waveguides (black symbols) and bonding modes equivalent to asymmetric modes (yellow symbols) that follow the dispersion of the slot waveguides. This agreement confirms the generality of our proposed coupling scheme.

Conclusions The coupling of plasmon resonances is a promising method for tailoring plasmonic properties of nanostructures. In this work, we have used the high spatial and energy resolution of a STEM-EELS system to analyze, in depth, the coupling mechanism of edge modes in planar nanostructures. We have shown that the coupling can be understood by a simple and intuitive scheme based on three distinct behaviours: 1) A strong coupling through the edge of the nanostructure that forms resonances denominated bonding gap edge modes and that experience a significant redshift as the gap size decreases, and antibonding gap edge modes that blueshift until convergence for small gap separations; 2) a weaker coupling through

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corners, that forms bonding and antibonding corner edge modes similar to the found in coupling of nanowires; 3) a non-coupling behaviour where the edge of the nanostructure behaves independently of the rest of the structure. We further study the formation of gap edge modes by analyzing the coupling in planar slot waveguides, finding that the same coupling mechanism applies. In addition, based on our analysis of the coupling behavior as a function of gap separation, showing that antibonding gap edge modes maintain their wavelength (energy) even at sub 10 nm gaps, we suggest that these antibonding gap edge or symmetric modes can operate as highly confined waveguides. This study and our proposed simple scheme for plasmonic coupling provides simplified design rules for future plasmonic devices compatible with planar industrial fabrication methods.

Acknowledgement E.B.P. and G.A.B. acknowledge support from NSERC under the Discovery Grant Program. Y.Z. and P.N. acknowledges support from the Robert A. Welch Foundation under grant C-1222 and the Army Research Office MURI grant WF911NF-12-0407. A.M. acknowledges financial support from the Department of Physics and Astronomy and the College of Arts and Sciences of the University of New Mexico. The Experimental work was carried out at the Canadian Centre for Electron Microscopy, a National Facility supported by The Canada Foundation for Innovation under the MSI program, NSERC and McMaster University.

Supporting Information Available Additional EELS intensity profiles of a single nano-square showing edge modes; and profiles of higher order resonances similar to the those shown in figure 2, showing bonding and antibonding corner edge modes as well as bonding and antibonding gap edge modes. Energy diagrams summarizing the LSPRs resulting from the plasmonic coupling of edge modes. This material is available free of charge via the Internet at http://pubs.acs.org/. 18


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