Extreme Sensitivity of the Optical Properties of Metal Nanostructures to

Dec 23, 2010 - They are highly localized at the sharp edges and sides and strongly affected by minor shape variations such as edge rounding. This work...
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Extreme Sensitivity of the Optical Properties of Metal Nanostructures to Minor Variations in Geometry Is Due to Highly Localized Electromagnetic Field Modes Ewa M. Goldys,* Nils Calander, and Krystyna Drozdowicz-Tomsia MQ Photonics, Department of Physics, Macquarie UniVersity, NSW 2109, Sydney, Australia ReceiVed: NoVember 11, 2010; ReVised Manuscript ReceiVed: NoVember 19, 2010

Optical extinction of silver nanostructures produced by electron beam lithography has been modeled by a finite element method. While the simulations of the design geometry produce results that are markedly at odds with the experiment, subtle variations of the shape, such as rounding of structure edges, are able to bring the simulations in agreement with experiment. We present the effect of various aspects of structure geometry on the evolution of electromagnetic field modes. They are highly localized at the sharp edges and sides and strongly affected by minor shape variations such as edge rounding. This work demonstrates that optical absorption/extinction provides a simple and sensitive indication of subtle morphology features in metal nanostructures. Accurate design of plasmonic metal nanostructures with specific optical characteristics remains a challenge despite over a decade of effort.1-3 The modern nanostructures are becoming more and more complex, with an ever-increasing number of design elements and sophisticated arrangement of various metal and dielectric three-dimensional shapes and layers. Computational models capable of incorporating such complex nanostructure geometries include the discrete source method (DSM), the boundary element method (BEM), and the popular discrete dipole approximation (DDA).4-7 General, ab initio methods of numerically solving Maxwell’s equations for the actual geometry on a discrete spatial grid or element functions are gaining increasing acceptance compared to various approximations.1,8,9 Importantly, as the structure quality improves, sharp edges and corners in such nanostructures, of great interest in plasmonic applications, are becoming more frequent, again reinforcing the need for accurate finite difference time-domain (FDTD) and finite element method (FEM) simulations and moving beyond approximate methods. However, despite the availability of commercial solver packages, accurate modeling remains a challenge, and one of the reasons why FDTD and FEM have not yet been uniformly adopted is the frequently reported “spurious” modes. We show in this paper that the EM modes properly calculated in such structures by modern software are not in fact “spurious”, in the sense that they are genuine physical solutions of Maxwell’s equations for the idealized geometry under consideration, but this idealized geometry is difficult or impossible to reproduce experimentally due to metal granularity or fabrication constraints. Some departures between the design and the actual shape would normally be expected, and they tend to be ignored. We report here that these seemingly minor differences have a dramatic impact on the optical properties of the nanostructures. This paper demonstrates that the results of rigorous FEM modeling of metal nanostructures are very sensitively influenced by very small variations in the structure geometry, and we explain that this sensitivity is coming from the high degree of localization of the electromagnetic field at corners, edges, and * To whom correspondence should be addressed. E-mail: ewa.goldys@ .mq.edu.au.

sides of the structures. Through these insights, our analysis can facilitate the design of highly sophisticated plasmonic structures with the desired optical properties. Results and Discussion This work is based on our experimental studies of metal nanostructures produced by e-beam lithography. A typical nanostructure (Figure 1) examined here consists of Ag cylinders with uniform size systematically varied diameters in the range 30-100 nm and different in different arrays. These nanostructures were deposited on a quartz substrate without any metal adhesion layer and coated with a 10 nm Al2O3 layer for protection. A regular two-dimensional array pattern with a fixed period of 150 nm in both directions was produced by a lift-off technique with increasing electron-beam exposure dose. The structures have been characterized optically by measuring their extinction in the 380-1000 nm range. Figure 1c shows the extinction spectra for arrays containing cylinders of various sizes ranging from ∼30 to ∼70 nm. Their shape can be fitted well by Lorentzian curves, indicating a uniform size and distribution of Ag nanoparticles with peaks in the range 430-500 nm accompanied by a small side feature at ∼370 nm. The main peak shifts toward longer wavelengths with the increase of the size/spacing ratio. Similar spectra were observed in other publications concerned with EBL- produced nanostructures with wider spacings.10 The three-dimensional FEM modeling was carried out by using a commercial FEM package Comsol Multiphysics 3.5 a with the RF module. The Maxwell (curl) equations were numerically simulated by using tetragonal elements. Special precautions have been taken at the boundaries to prevent scattering of plane waves by using absorbing layers; such layers ensure that the radiation leaving the nanostructure is not reflected back.11 We tested the accuracy of our simulations by carrying out the modeling of the electric field distribution and the absorption coefficient in a sphere with a diameter of 200 nm, the convergence was verified as well (see Supporting Information). We emphasize here that our simulations were carried out for a specific set of boundary conditions (plane wave arriving from infinity); so, this work can only be compared with other

10.1021/jp1107679  2011 American Chemical Society Published on Web 12/23/2010

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Figure 1. (a) SEM image of selected nanostructures used in this study showing an array of cylinders, approximately 50 nm in diameter with an array period 150 nm × 150 nm, (b) structure cross-section as designed, and (c) experimental extinction spectra of the nanostructures with varying diameter D as indicated.

Figure 2. FEM simulations of an isolated silver cylinder D ) 60 nm, h ) 40 nm with excitation wave arriving from the top, at angles 0° (black), 30° (blue dotted), and 45° (red dotted line) to surface normal. (b and d) Cross-section view of electric field distributions at both extinction peaks 407.5 (b) and 354.5 nm (d). (c and e) Corresponding top view of electric field distributions at the top of the cylinder at both extinction peaks 407.5 (c) and 354.5 nm (e). The intensity and direction of the electric field, in arbitrary units, is indicated by the color scale. Top scale refers to b and c and bottom scale to d and e.

work that uses the same boundary conditions. The nonlocal effects have not been taken into account, as they become visible at length scales on the order of a single nanometer and below.12,13 Maxwell’s equations solved in this work have been applied to cylindrical metal structures, which from a mathematical point of view are regarded as “nonsmooth” domains.14 The edges and/ or corners in such structures produce electromagnetic (EM) field singularities, currently intensely studied, and it is only recently that it has become possible to model them accurately.15 While such singularities are well known in electrostatics (referred to as the “lightning rod” effect), their presence in the optical frequency domain and their connection with the spectroscopic fingerprints have been under intense current investigation. It is also worth noting that the typical dimensions of the investigated nanostructures are such that even a single standing wave cannot be accommodated within; thus, the familiar standing wave patterns will not be formed. Moreover, the structural dimensions, especially their thickness, are close to the skin depth (12 nm for silver16), and thus, the nanostructure behaves in many respects like a lossy dielectric (but with a negative real part of the permittivity), as opposed to a perfect conductor. All these features combined challenge our established notions of how the structure resonances should look like. The geometry of the nanostructures under consideration is complicated, and in order to understand their extinction spectra, we carried out a series of FEM simulations for a range of related structures with features present in our own with increasing complexity. This approach makes it possible to see the con-

nection between specific variations of geometry and the properties of the spectra and provides improved physical insights. Thus, we first discuss the extinction of metal cylinders suspended in free space and then locate these cylinders on a substrate and separately coat them with an ultrathin dielectric layer. Subsequent to that we make minute modifications to the cylinder geometry by rounding top, bottom, or both edges. Further, we explore how the rounding radius of curvature affects the extinction spectra. Finally, we bring together the insights provided by previous simulations and demonstrate an accurate fit to our experimentally measured extinction data. First, we show the simulations of a main building block of our nanostructure arrays, an isolated cylinder in free space (Figure 2). Although the simulations for this structure were carried out in free space, without any support, similar conditions are experimentally achievable by using an index matching fluid. Figure 2a shows the extinction coefficient in such structures, where two features are clearly observed (407.5 and 354.5 nm for the diameter D ) 60 nm and height h ) 40 nm). In sharp contrast to the experimental data in Figure 1, both these features are clearly apparent at various incidence angles of the exciting EM wave, between normal incidence and 45°. Consequently, these two peaks should be experimentally observed, even in experiments where light is collected over a range of angles by a microscope objective with the numerical aperture of 0.7 in air. Figure 2b-e shows the cross-section and top view of the electric field distributions at the two peaks (407.5 and 354.5

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Figure 3. FEM simulations of a cylinder on a substrate with excitation wave arriving along the normal to the surface from the top. (a) Extinction coefficient for D ) 60 nm, h ) 40 nm. (b and c) Cross-section view of electric field distributions for D ) 60 nm, h ) 40 nm at both extinction peaks 477.0 (b) and 398.0 nm (c). (d) Extinction coefficient for a structure on a substrate with a 10 nm Al2O3 capping layer as a function of diameter for D ) 30, 45, 60, and 75 nm, h ) 40 nm. (Insert) Evolution of peak wavelengths with structure diameter. (e-h) Cross-section view of electric field distributions at both extinction peaks for D ) 30 and 75 nm, h ) 40 nm for all figures: (e) long-wavelength peak D ) 30 nm, (f) short-wavelength peak D ) 30 nm, (g) long-wavelength peak, D ) 75 nm, (h) short-wavelength peak, D ) 75 nm. The intensity and direction of the electric field, in arbitrary units, is indicated by the color scale.

nm). Figure 2b shows that at the main extinction peak near 407.5 nm the electric field concentrates very closely near the structure edges. At the second peak (354.5 nm, Figure 2d) the electric field pattern evolves, the concentration near the edges is now much less pronounced, while a much stronger field develops on all structure sides than observed previously. Such EM field distributions are in general agreement with those published on other structures with sharp corners and edges such as triangular nanoplates and nanocubes.17-19 In particular, ref 17 demonstrates three distinctive EM modes, which peaked at the corners, edges, and bulk volume of the triangular nanoplate, respectively. Correspondingly, in our geometry we observe the 407.5 nm mode concentrated at the edges and the second 354.5 nm mode with high amplitude on the structure sides. We were unable to identify the third mode peaking in the nanostructure center; however, we emphasize different boundary conditions in ref 17 (point excitation by an electron beam in ref 17, plane wave at infinity in our work). The EM modes in a nanocube show a more complex pattern, and various combinations of modes peaking at the corners and edges are observed as shown in ref 18. Figure 2a and 2b also shows that the 407.5 nm mode has a well-defined dipolar character, while the 354.5 nm mode has a higher multipolar (hexapolar) component (see Supporting Information for explanation of the assignment). This is somewhat contrary to our established notions about EM resonances in a sphere, where a dipolar resonance is followed by a quadrupolar resonance at shorter wavelengths before higher multipoles appear. In our case a quadrupolar resonance could not be observed. Having established that metal nanocylinders show two wellresolved EM modes in the spectral region of interest we extended our simulations to a more realistic situation where metal nanostructures are placed on a dielectric substrate. The

simulations show that the extinction coefficient for a cylinder on a substrate has a very similar character to a cylinder without a substrate, and the two modes can be very clearly observed. Due to a different refractive index of the substrate, the extinction peaks were significantly shifted to the red (to 477.0 and 398.0 nm), in agreement with earlier reports.21 The separation between these two features has also increased somewhat, and there is a small subsidiary feature at 348.5 nm. Upon close inspection the EM field distributions (Figure 3b and 3c) show similarities with those for an isolated cylinder (Figure 2b and 2d). The electric field at the longer wavelength mode is still concentrated mostly near the edges, but now the amplitude at the edges close to the substrate is greatly enhanced compared to the top edge. Conversely, at the shorter wavelength mode, the electric field is still concentrated near the structure sides but some rebalancing occurs: the field at the bottom side adjacent to the substrate increases, and the field at the circumference near the top decreases, consistent with loss of symmetry upon introduction of a substrate. The electric field at the top edge is increased as well, compared with the isolated cylinder. Thus, the overall character of the electric field distribution for the cylinder on the substrate is such that at the long-wavelength mode the electric field distribution is weighted more heavily toward the lower edge near the substrate and at the short wavelength edge toward the top edge, away from the substrate. This is in full agreement with the study in ref 19 for a nanocube on a substrate. Interestingly, the multipolar character of the modes is now more complex with the dominant, lower wavelength mode (477.0 nm) showing a mixed quadrupolar character and the shorter wavelength (398.0 nm) mode where, again, a mixed quadrupolar character can be seen but with more clearly pronounced higher multipoles. In the following series of simulations we added a 10 nm Al2O3 protective coating layer over the structures (Figure 3d-h). This

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Figure 4. FEM simulations for an isolated cylinder D ) 60 nm, h ) 40 nm. (a) Evolution of the extinction spectra for varying radii of curvature at the edges R ) 0, 2, and 8 nm. (b) Position of the main extinction peak with varying radius of curvature R. (b-d) Side view of the electric field distribution at the main peak for R ) 0 (b), 2 (c), and 8 nm (d).

layer did not change the character of the extinction spectra, and two peaks were still clearly observed, with some additional minor features that will not be discussed here. The two-mode character is still clearly observed, and the two peaks are well separated. They are now somewhat rebalanced with the longer wavelength peak (quadrupolar) dominating the spectrum. Both peaks experienced a surprisingly major overall shift due to the presence of the Al2O3 capping layer of over 50 nm compared to the case of a cylinder on a substrate alone (to 531.0 and 441.0 nm for D ) 60 nm) and some change of the integrated intensity. The conclusion from these considerations is that due to the extremely localized character of the modes concentrated near the edges, even minuscule changes to the dielectric environment of the nanostructures in the proximity of these edges have a profound effect on its optical characteristics. The presence of the protective Al2O3 layer does not alter the character of electric field distributions in a significant way, and the EM field is still concentrated in the same regions as for a cylinder on a substrate. As before, due to the high refractive index of the substrate the field at the edges near the substrate is more intense than at the top edge this is somewhat less pronounced now due to the presence of the high refractive index Al2O3 layer at the top (Figure 3e). At the high-wavelength peak, the Al2O3 layer enhances the mode with high amplitude on the sides, especially at the top surface and somewhat less at the circumference. The electric field becomes particularly strong at the bottom side of the cylinder, and both these changes occur at the expense of the electric field at the structure edges, which are now much less distinct. Overall, despite the complicated geometry, the modes clearly retain their predominant concentration at the edge for the low-wavelength and at the sides for the short-wavelength mode. We also examined how the extinction and electric field distributions change with the structure diameter (between 30 and 75 nm), while the height was kept constant at 40 nm. Both extinction peaks persisted for all these structures, and they shifted with diameter to longer wavelengths (Figure 3d insert). The longer wavelength peak shifted from 515.5 (for D ) 30 nm) to 551.5 nm (D ) 75 nm), while the shorter wavelength changed somewhat less, from 400 (D ) 30 nm) to 435 nm (D ) 75 nm). The evolution of the electric field distribution is particularly interesting (Figure 3e-h). It shows that increasing the structure diameter exacerbates the effect of the substrate and the capping layer. While for a 30 nm diameter structure most of the electric field is concentrated at the bottom edge, at 75 nm this effect is much more pronounced and there is practically no electric field at the upper edge. At the same time, at the high-wavelength peak (400.5 nm), the 30 nm diameter structure shows clear maxima of electric field near the sides

and much less significant concentration near the edges. The 75 nm structure shows the EM field concentrated mostly near the sides, especially at the structure top and bottom. We therefore established that the simulations of an ideal cylinder with sharp edges either isolated or combined with dielectric layers as in the actual experiment show the presence of EM modes concentrated near the edges and sides of the structure. However, such sharp edges are an idealization, and some rounding is expected in real experiments. The literature suggests that nanolithography typically produces structures with rounded corners.22,23 Typical radii of curvature are related to metal grain size, the level of photoresist exposure, and the e-beam spread, all on the order of a single nanometer. In order to take this factor into account we explored the shape variation of the structure edges with different radii of curvature. Again, we returned to the basic building block of the structure, the isolated metal cylinder (Figure 4). As the edges become more rounded, the key change is a blue shift of the low-energy peak and strong attenuation of the high-energy peak that comes closer to the main feature (Figure 4a and 4b). Consequently, even for a moderate rounding of the edges of 2 nm the extinction spectrum quickly becomes dominated by what is effectively a single peak. The blue spectral shift of the main peak with edge rounding is quite pronounced, with shifts of about 1.62 nm per 1 nm change of curvature radius R (Figure 4b). The electric field distribution also varies very quickly with edge rounding, and already at the radius of curvature of 2 nm the electric field at the edges becomes comparable to that at the sides, and thus, the distribution starts to increasingly resemble the mode with high amplitude at the sides. The clear dipolar character of the modes concentrated near sharp edges changes as well, and the mode starts to show a more complex, mixed dipolar, and higher multipolar character (Figure 4c-e). Finally, we extended these simulations to structures on the substrate with a protective capping layer of the same 10 nm thickness as used previously. First, we separately rounded off the top, the bottom, and both edges simultaneously (Figure 5a) by using the curvature radius R ) 10 nm. Among these three options, the rounding off of both edges had the most pronounced effect on the extinction coefficient followed by the rounding off of the edges adjacent to the substrate. With the rounding off of both edges the extinction spectrum formed a single peak at 488.0 nm, while for the bottom-rounded structure a clear doublet has been observed. The top-rounded geometry produced an extinction spectrum of a very similar shape to a structure with sharp edges. Thus, the structure with both edges rounded shows the closest similarity to the experimentally observed single-peak spectra. We further explored in more detail the evolution of the extinction spectra for such structures with the

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Figure 5. FEM Simulations of a cylinder on a substrate with a 10 nm Al2O3 capping layer D ) 60 nm, h ) 40 nm with rounded edges. (a) Extinction coefficient for structures with top rounded, bottom rounded, and both edges rounded, curvature radius R ) 10 nm. (b-f) Side view of EM field distributions at both extinction peaks for these structures: (b) both edges rounded at the main peak 488.0 nm; (c) bottom-rounded structure, electric field at peak at 508.5 nm, (d) top-rounded structure, electric field at peak at 532.0 nm; (e) bottom-rounded structure, electric field at peak at 479.0 nm; (f) top-rounded structure, electric field at peak at 444.5 nm; (g) extinction spectra for both edges rounded, different radii of curvature R ) 0, 2, and 8 nm as shown. (Insert) Evolution of peak features with varying radius of curvature. (h) Evolution of the main peak with structure diameter (30, 45, 60, and 75 nm), both edges rounded with a radius of curvature R ) 5 (continuous line) and 10 nm (dotted line).

radius of curvature R. The results (Figure 5g) show the main peak undergoing a systematic blue shift as the edges become more rounded. At the same time the lower wavelength feature rapidly disappears. Figure 5h follows the variation of the extinction coefficient with the structure diameter D between 30 and 75 nm for both edges rounded with two values of R ) 5 and 10 nm, where, practically, a single peak is observed. As expected, the main peak shifts to shorter wavelengths for smaller structure diameters in an approximately linear fashion (Figure 5h insert). Other shape variations of a similar kind have also been explored, but their simulations were highly inconsistent with the data and are not shown here. By carrying out the presented series of simulations we identified the role of individual elements of nanostructure geometry and the effect they have on the extinction spectra. The key parameter is the radius of curvature as it eliminates the two-peak character of the observed spectra. It must be introduced at both edges, and it has an additional effect of a blue shift. The substrate and capping layer have a similar effect of a red shift due to their high refractive index, while the reducing nanostructure diameter shifts the peak(s) to the blue. When carrying out the fitting one must also be conscious of the limits of validity of the SEM dimensions. In the present case, with no conductive layers on the substrate the SEM measurements are affected by charging, which makes it difficult to accurately measure the structure diameter (and, at this level of resolution, the alternative technique of environmental SEM is not accurate enough). We highlight that any conductive layers, even as thin as a single nanometer (data not shown), have a major effect on the character of the extinction spectra. On the basis of the presented considerations and using a realistic structure geometry accounting for all the structure layers we were able to accurately simulate the extinction in a complex nanolithography-defined nanostructure (Figure 6). The key to

Figure 6. Experimental extinction coefficient for the nanostructure array 150 nm ×150 nm period, with D ) 65 nm, h ) 40 nm, 10 nm capping layer thickness (black) and the calculated spectrum for D ) 60 nm diameter, h ) 40 nm, 10 nm capping layer and curvature radius R ) 10 nm (red).

a successful fit was the adoption of the nonzero radius of curvature; however, other parameters such as structure diameter and its height also needed to be slightly varied with respect to the design values, within the accuracy limit of the SEM measurement. Of course, in principle, one must also include inhomogeneous particle size and shape distributions to the extent that they are present in experiments. This effect has not been accounted for, but it could be responsible for the slight departures observed in Figure 6 at the long-wavelength edge of the peak. In summary, this paper demonstrates how the dipolar and higher multipolar modes with their electric field highly concentrated at the edges and sides can be used to identify fine features of nanostructure shape. This result is significant because it provides a very sensitive method to detect minor changes in

Optical Properties of Metal Nanostructures structure geometry such as rounding of the structure edges or the presence of ultrathin (single nanometer) layers. The edge rounding not only removes the double peak in extinction, which is replaced by a single peak, but also amplifies the spectral shift with nanostructure size. These two pieces of experimental evidence can be simultaneously explained only by the fact that the structures have rounded edges. Thus, a simple extinction measurement can provide a very sensitive indication of whether a cylindrical structure has sharp edges with a radius of curvature less than a couple of nanometers. This diagnostic method is especially significant for plasmonic applications where corners and edges that are sharp on a nanoscale are highly desired. Although this paper concentrates on cylindrical nanostructures, similar ideas are applicable to other shapes, such as cubes, triangular plates, pyramids etc., where a high level of sensitivity to the details of structure geometry has also been reported.19,24,25 Our result also helps understand why earlier efforts to model various regular metal nanostructures on a substrate by using radical approximations such as replacing the actual shapes by oblate spheroids18 or DDA with a limited number of dipoles7 have been so successful. Our work provides evidence that the actual shape of these nanostructures can be closely approximated by ellipsoids. We have thus given here a rigorous justification of this approach, which is based on the actual numerical solution of Maxwell’s equations with a full account of structure geometry. The ellipsoid-based models are highly convenient, as analytical solutions are available, also for an array of such structures, thus making it possible to analyze the coupling, which will be discussed in our subsequent publication. Materials and Methods Experiment. The cylindrical nanostructure samples were prepared by using an electron-beam lithography (EBL) system in the Cornell Nanofabrication Facility on quartz substrates. A JEOL JBX-9600 FS (JEOL, Japan) system was used to expose the pattern with 20 nm resolution achieved at 100 kV accelerating voltage. Two layers of positive photoresist, 2% poly methyl methacrylate (PMMA) in anisole was used followed by 1% of PMMA in methyl isobutyl ketone (MBIK), were spin coated on silica substrate to a total thickness of 90 nm and each baked at 170 °C for 15 min. A regular silver two-dimensional array pattern with a fixed period of 150 nm in both directions and thickness of 40 nm was produced by a lift-off technique with increasing electron-beam exposure dose. The structure contained individual Ag cylinders with varying size in the 30-100 nm range. The period of the pattern was 150 nm in both directions, and the array size was 100 µm × 100 µm. In single-particle arrays, the particles remained separated as the spacing decreased with increasing size of the particles. The SEM images were obtained using Zeiss Ultra SEM without any sample preparation, and the height of the nanostructures was determined by atomic force microscope (AFM) measurements to be 30 ( 10 nm. The structures were deposited on quartz and coated with a 10 nm Al2O3 layer to protect against oxidation. The extinction spectra were obtained by using the SEE2100 microspectrometer in transmission mode, across the spectral range of 380-1000 nm. The spectra were collected by a 50× objective from small regions containing only a single type of pattern, with a numerical aperture of 0.7. The unpolarized and polarized illumination from a mercury lamp was used. Simulations. The calculations were performed at a supercomputer cluster with 15 nodes, with each node based on an Intel Xeon chip E5530 at 2.4 GHz, with 8MB cache and 16 GB of RAM. The Comsol program version 3.5a, with the RF

J. Phys. Chem. C, Vol. 115, No. 3, 2011 681 module, in its three-dimensional version was used for all simulations. The Comsol program is based on the finite element method (FEM), and the RF module applies it to electromagnetic calculations. The FEM scheme used in this work employs nonorthogonal grids which resolve the problem of “spurious” solutions, frequent in calculations with rectangular grids. A typical geometry and mesh in the simulations are shown in Figure S1 (Supporting Information). In the simulations a cube 150 × 150 × 150 nm was first defined. Within that cube a silver cylinder was located, with rounded or sharp edges (blue), and a protecting layer of Al2O3 (red), surrounding the cylinder and covering the substrate below. The refractive index of the silver used in this work was taken from ref 26. The refractive index of the protecting layer was taken as 1.77, and the value of 1.55 was used for the substrate and 1 for air above the sample. The calculations were carried out for the boundary condition of a plane-polarized electromagnetic wave at normal incidence arriving from above, and the upper and lower boundaries of the simulated regions were set to absorbing. Periodic or absorbing boundaries were also set at the cube sides, periodic for arrays of cylinders as in real nanolithography structures, and absorbing for single cylinders simulated in the present work. A typical simulation of a single structure took about 2 h. Acknowledgment. This work was partly supported by the Australian Research Council awards DP0880876 and DP0770902. Supporting Information Available: Validation of the presented FEM simulations, and assignment of multipolar character of the observed EM modes. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Wu, Y.; Nordlander, P. Finite Difference Time-Domain Modeling of the Optical Properties of Nanoparticles near Dielectric Substrates. J. Phys. Chem. C 2010, 114, 7302-7307. (2) Millstone, J. E.; Park, S.; Shuford, K. L.; Quin, L.; Schatz, G. C.; Mirkin, C. A. Observation of a quadrupole plasmon mode for a colloidal solution of gold nanoprisms. J. Am. Chem. Soc. 2005, 127, 5312–5313. (3) Kelly, K. L.; Coronado, E.; Zhao, L. L.; Schatz, G. C. The optical properties of metal nanoparticles, the influence of size, shape and dielectric environment. J. Phys. Chem. B 2003, 107, 668–677. (4) Eremina, E.; Eremin, Y.; Wriedt, T. Opt. Commun. 2006, 267, 524– 529. (5) Myroschynchenko, V.; Rodriguez-Fernandez, J.; Pastoriza-Santos, I.; Funston, A. M.; Novo, C.; Mulvaney, P.; Liz-Marzan, L. M.; Garcia de Abajo, F. J. J. Chem. Soc. ReV. 2008, 37, 1792–1805. (6) Myroschynchenko, V. V.; Carbo-Argibay, E.; Pastoriza -Santos, I.; Perez-Juste, J.; Liz-Marzan, L. M.; Javier Garcia de Abajo, F. Modeling the optical response of highly faceted metal nanoparticles with a fully 3D Boundary Element Method. AdV. Mater. 2008, 20, 4288–4293. (7) Sherry, L. J.; Jin, R.; Mirkin, C. A.; Schatz, G. C.; Van Duyne, R. P. Localised surface plasmon spectroscopy of single silver triangular nanoprisms. Nano Lett. 2006, 6, 9, 2060–2065. (8) Oubre, C. H.; Nordlander, P. Optical properties of metallodielectric nanostructures calculated using the finite difference time-domain method. J. Phys. Chem B 2004, 108, 17740–17747. (9) McMahon, J. M.; Henry, A. I.; Wustholz, K. L.; Natan, M. J.; Freeman, R. G.; Van Duyne, R. P.; Schatz, G. C. Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy. Anal. Bioanal. Chem. 2009, 394, 1819–1825. (10) Hicks, E. M.; Zou, S.; Schatz, G. C.; Spears, K. G.; Van Duyne, R. P.; Gunnarson, L.; Rindziewicius, T.; Kasemo, B.; Kall, M. Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by e-beam lithography. Nano Lett. 2005, 5, 6, 1065–1070. (11) Basu, U.; Chopra, A. K. Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comput. Methods Appl. Mech. Eng. 2003, 192 (11-12), 1337– 1375.

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