pubs.acs.org/NanoLett
Optical Properties of Nanowire Dimers with a Spatially Nonlocal Dielectric Function Jeffrey M. McMahon,*,†,‡ Stephen K. Gray,‡ and George C. Schatz† †
Department of Chemistry, Northwestern University, Evanston, Illinois 60208 and ‡ Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439 ABSTRACT We study the optical spectra and electromagnetic field enhancements around cylindrical and triangular Ag nanowire dimers, allowing for a spatially nonlocal dielectric function that partially accounts for quantum mechanical effects. For the triangular structures, we pay particular attention to how these properties depend on the sharpness of the nanowire’s tips. We demonstrate that significant differences exist from classical electrodynamics that employs a more common, spatially local dielectric function. These differences are shown to arise from the optical excitation of volume plasmons inside of the structures, analogous to one-particle quantum mechanical states, which lead to complex and striking patterns of material polarization. These results are important for further understanding the optical properties of structures at the nanoscale and have implications for numerous physical processes, such as surface-enhanced Raman scattering. KEYWORDS FDTD, field enhancement, nanowire, nonlocal dielectric
I
nterest in metallic nanostructures has been increasing as experimental fabrication techniques have become more sophisticated.1 A major driving force for this lies in the novel optical properties that such structures exhibit as a result of surface plasmon (SP) excitations. For isolated nanoparticles or clusters, these can result in localized surface plasmon resonances (LSPRs). One consequence of such excitations is that large electric field enhancements relative to the incident field (|E|2/|E0|2), hereon referred to as |E|2 enhancements, can occur.2-6 These are particularly important for surface-enhanced Raman scattering (SERS), which has an electromagnetic contribution proportional to |E|4. SERS enhancements can be sufficiently high that even single molecule detection is possible.7 It is important to note that there is also a chemical enhancement to SERS, which based on previous work is presumed to be minor. We will return to this point below. |E|2 enhancements in nanostructures have been thoroughly studied in the past, from isolated nanoparticles3,4 and dimers2,3,5,6 to other structures. From these studies, values as large as 105 have been found for isolated nanoparticles (specifically triangular nanowires)4 and 107 for junction structures (e.g., dimers of cylinders or spheres).2,6 However, these studies have all assumed that the polarization of the material at a given point is locally related to the electric field, which is termed classical local electrodynamics. Nonlocal electrodynamics, on the other hand, is more rigorous in principle because quantum mechanical effects are partially accounted for through the use of a spatially nonlocal dielectric function. This, as the name implies, leads to a spatially nonlocal relationship between the material polarization and E.8 Such
effects have been known for some time to be necessary to correctly describe the optical properties of some systems.9 Of particular importance for this work is that it has been found that |E|2 enhancements are diminished relative to local predictions when such effects are included.10-12 The quantitative accuracy of the large body of work concerning |E|2 enhancements, as well as the interpretation of effects that rely on them (e.g., SERS), may therefore be questionable.13-15 Nonlocal effects for simple systems have been studied in the past, including thin films,9 cylindrical nanowires,16 isolated nanoparticles17-19 or aggregates thereof,12,20-23 and cavities.24,25 However, due to challenges in describing nonlocal effects, the impact of them on systems with complex geometries (i.e., not spherical or cylindrical) is largely unknown.10 It is the purpose of this Letter to use our previously described nonlocal electrodynamics finite-difference time-domain method10,11 to provide insight into these issues. Computational details for the calculations herein can be found in the Supporting Information. We begin by overviewing the nature of quantum mechanical effects within the present context. The model that we presented in refs 10 and 11, which we apply here, is still within the continuum limit of classical electrodynamics. However, quantum mechanical effects are partially accounted for in the constitutive relationship between E and the electric displacement field D through the use of a spatially dependent and frequency-dispersive relative dielectric function ε(x, x′, ω)8
D(x, ω) ) ε0
* To whom correspondence should be addressed, jeffrey-mcmahon@ northwestern.edu. Received for review: 05/5/2010 Published on Web: 08/17/2010 © 2010 American Chemical Society
∫ dx′ε(x, x′, ω)E(x′, ω)
(1)
where ε0 is the permittivity of free space. In anticipation of the results that we present below, note that D is essentially a 3473
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generalized E that contains information on both the incident electric field and material polarization. Local electrodynamics corresponds to ε(x, x′, ω) ) ε(ω)δ(x - x′), where only the frequency-dispersive portion is variable and is often taken from experimental bulk measurements.26 Even in this limit, some quantum mechanical effects are included. For example, the ω dependence of the bulk dielectric data reflects interband electron transitions. Additional quantum mechanical effects can also be accounted for, such as the reduced electron mean free path due to surface scattering that occurs in small structures,27 which can be described phenomenologically by increasing damping parameters. Both of these effects are accounted for in our model herein,10 although they are not the most important ones in the present study. We now consider what happens when the local approximation is not made. In a homogeneous environment (which we assume the nanowire structures herein can be approximated as), ε(x, x′, ω) only spatially depends on |x - x′|. Therefore, in wavevector-space (k-space) eq 1 is more simply expressed as
D(k, ω) ) ε0ε(k, ω)E(k, ω)
contains terms representing the permittivity at infinite frequency and the interband electron transition. For the full dielectric function, including the value of its parameters, see the Supporting Information. The implementation of eq 3 involves converting it to an equation of motion and solving it via finite-difference techniques.10,11 However, before this can be done an additional boundary condition (ABC) must be enforced on surface(s) of the nonlocal structure(s).18,30 This is an essential aspect of any nonlocal electrodynamics treatment, and herein we employ the Pekar ABC,11 where the total material polarization vector vanishes. It is important to realize that nonlocal effects arise from a combination of the dielectric model used and the ABC. The Pekar ABC is known to give large deviations from local theory relative to others,18,30 but it is easiest to implement numerically.11 Before ending this discussion, we should mention some limitations and errors associated with the method used for the calculations herein. One source of error is related to surface effects. In local electrodynamics, a sharp boundary is assumed at a metallic surface. For a real structure, however, this surface is diffuse.31,32 Self-consistent theories are able to describe this and have an outward relaxation of induced charge relative to local electrodynamics.33 Non-selfconsistent theories, on the other hand, including the hydrodynamic Drude model, have an inward relaxation.34 What this means is that the effective surface is inward relative to the physical (as well as local) one. Fortunately, this error should be relatively minor as long as we consider structures where the features are much larger than the distance of inward relaxation. In the present case this is approximately 1 Å, which is smaller than the uncertainty imposed by the grid spacing anyway (see the Supporting Information). Moving on, one effect unaccounted for in the present work is consideration of the atomic structure, where nuclei correspond to localized positions of electrostatic force. However, for large structures (many hundreds of atoms, or more), such as those considered in this work, this results in a uniform positive background (to a good approximation) and is accurately accounted for in eq 3. Another unaccounted for effect is the “quantum size effect”, where the conductivity of the material depends on the geometry of the structure.35 As indicated above, most of the structures considered in this work are rather large (e.g., 50 nm), where this effect should be relatively unimportant. With the preliminaries out of the way, we now focus on systems known to give high |E|2 enhancements, dimers of cylindrical and triangular Ag nanowires.3,5,6 Schematic diagrams of these are shown in Figure 1. The cylindrical nanowire dimer, Figure 1a, consists of two cylindrical nanowires with diameters of 50 nm separated by a distance h. The other system (Figure 1b) is analogous and consists of two triangular nanowires with vertical side lengths of 50 nm and variable horizontal side lengths l. With variation of l, the sharpness of the tips at the junction can be controlled. The
(2)
The importance of the k dependence in ε can be understood by considering the effects of momentum in small structures. When light interacts with a structure of size d (which could correspond to a nanoparticle size, junction gap distance, etc.), k components (proportional to the momentum) are generated with magnitude k ) 2π/d. These in turn impart an energy of E ) (pk)2/2me, where p is Planck’s constant and me is the mass of an electron, to (relatively) free electrons in the metal. For small d these energies can correspond to the optical range (1-5 eV). This analysis suggests that such effects should come into play for d less than approximately 2 nm. In metals, however, somewhat larger d values also exhibit these effects because electrons in motion at the Fermi velocity can be excited by the same energy with a smaller momentum increase, due to dispersion effects.28 In order to use eq 2 in electrodynamics calculations, a form for ε(k, ω) is needed. On the basis of quantum mechanical considerations using a density functional theory formulation, the hydrodynamic Drude model accurately describes the collective electron motion of a free electron gas8
ε(k, ω) ) -
ωD2 ω(ω + iγ) - β2k2
(3)
where ωD is the plasma frequency, γ is the collision frequency, and in 2D at high frequencies β2 ) 3vF2/4 where vF is the Fermi velocity (1.39 × 106 m/s for Ag).29 The full dielectric function used for the calculations herein also © 2010 American Chemical Society
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were realistic, it is unlikely that such small separations would be experimentally achievable due to limitations in fabrication techniques and surface imperfections. We first consider the extinction cross sections36 (i.e., the total amount of power absorbed and scattered relative to the incident light) for the cylindrical nanowires, Figure 2. The optical responses are found to be nearly identical for both local and nonlocal calculations when the separation is h ) 5 nm (Figure 2a). There are, however, some minor differences in that there are small reductions in the magnitudes of the LSPR peaks and the dipolar LSPRs (the lowest energy peaks) are slightly blue-shifted in the nonlocal results, effects both characteristic of spatial nonlocality in the dielectric response.17,18 As the nanowire separation is reduced to h ) 2 nm (Figure 2b), for example, is an overall red shift in the LSPRs relative to h ) 5 nm. This is understandable, as the LSPR dispersion relation in such structures red shifts as the junction size is decreased.37 Furthermore, it is seen that the nonlocal relative blue shift is even greater than that at h ) 5 nm, and this trend continues for h ) 1 and 0.5 nm (parts c and d of Figure 2, respectively). For example, the dipolar LSPR blue shifts by 0.11 eV when h ) 0.5 nm, whereas it only does so by 0.05 eV when h ) 5 nm. These results are consistent with those that have been found previously for dimers of (nonlocal) spherical nanoparticles.21 The increase in strength of nonlocal effects with decreasing junction size can be understood by examination of eq 1. As the junction size is reduced, larger magnitude k components are generated, which in turn cause LSPRs to occur at higher energies relative to k ) 0. In addition to blue shifting, the nonlocal peaks become progressively broader compared to the local ones. Presumably this occurs because the lifetimes of
FIGURE 1. Schematic diagrams of the (a) cylindrical and (b) triangular nanowires considered in this work.
sizes of these structures were chosen because they are representative of those often used in experiments and are also large enough to exhibit multipolar optical responses and large |E|2 enhancements.2 The dimers are illuminated by an incident field with the E components in-plane and polarized along their long axes with k being perpendicular (i.e., normal incidence). As is well-known, |E|2 enhancements are very high for small junction sizes. We therefore consider h values of 5, 2, 1, and 0.5 nm. Although smaller separations would likely lead to even larger |E|2 enhancements than we find below,2 we do not consider them for two reasons. First of all, as discussed above, surface boundaries are not perfectly sharp. Therefore, electron spill-out would likely blur boundaries of smaller size, and even for a semiquantitative analysis, more information about the physical surface positions would be necessary. In addition, even if sharp boundaries
FIGURE 2. Local (dashed red lines) and nonlocal (solid blue lines) extinction cross sections for 50 nm diameter cylindrical nanowires separated by h ) (a) 5, (b) 2, (c) 1, and (d) 0.5 nm. © 2010 American Chemical Society
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the LSPR modes are decreased through couplings to internal (volume) plasmon modes that only occur when nonlocal effects are included. This claim is supported from calculations which have shown that the damping rate from nonlocal dielectric effects in plasmonic dimer systems can be great for large k components.15 We will pay particular attention to this issue below. Additional peak structures also appear with decreasing separation, which can be attributed to multipolar (higherorder) LSPRs (quadrupolar, octupolar, etc.). However, because of the increase in relative LSPR blue shift with decreasing separation, these resonances are less apparent (not as easily excited) in the nonlocal results.23 Therefore, it is possible that much smaller junctions than previously demonstrated2 are realistically needed to excite them. Nonetheless, they are still apparent, and are also blue-shifted relative to the local results, much like the dipolar LSPR. Focusing on the h ) 1 nm results, for example, the relative blue shift of the quadrupolar LSPR (the second lowest energy peak) is 0.18 eV. This suggests that higher-order LSPRs are more blue-shifted than those of lower order, which has recently been demonstrated for core-shell nanoparticles,19 but opposite to what we have found for core-shell nanowires.38 Given the similarities between nanoparticles and nanowires, it is possible that this trend is system dependent, as we further discuss below. Although, since high-order LSPRs have the largest magnitude k components, it is suspected that in general such resonances will be affected the most by nonlocal effects.19 Also similar to the dipolar LSPR, the nonlocal higher-order LSPRs are broadened relative to the local results. Qualitatively similar trends are found for the optical responses of l ) 50 nm triangular nanowires, Figure 3. Quantitatively, however, much greater differences between the local and nonlocal results exist. For example, the relative blue shift of the dipolar LSPRs are 0.27 and 0.35 eV for h ) 5 and 0.5 nm, respectively, compared to 0.05 and 0.11 eV for the analogous cylindrical nanowires (Figure 2). In addition, the nonlocal peaks are even more (relatively) broadened. The differences between the cylindrical and triangular nanowires can be attributed to the fact that larger magnitude k components are generated by the sharp tips of the latter ones. This is due to rapid variations in the electromagnetic fields across the apexes of the triangular nanowires, whereas the cylindrical ones look locally flat. (This is also part of the reason why the |E|2 enhancements in such structures are often high relative to other shapes.4) Because of this, the nonlocal effects are much stronger compared to the cylindrical nanowires. Another difference is that in the local results the large k components also lead to easier excitation of highorder LSPRs, where such resonances are seen even for h ) 5 nm. In the nonlocal results, however, there appears to be competing effects. The triangular nanowire tips that generate high magnitude k components and should lead to highorder LSPRs are also very efficient for sustaining low-order © 2010 American Chemical Society
FIGURE 3. Extinction cross sections for l ) 50 nm triangular nanowires separated by h ) 5, 2, 1, and 0.5 nm. Local and nonlocal calculations are shown on the bottom and top, respectively.
volume plasmon modes,10 which are responsible for strongly damping LSPRs.15 See Figure 4 in ref 10, for example. This situation is analogous to that discussed before for the relative impact of nonlocal effects on dipolar and multipolar LSPRs.19 By looking at the results in Figure 3, it is seen that high-order LSPRs are not prominent in any of the nonlocal results, except for a slight indication for h ) 0.5 nm. It can therefore be inferred that the latter is the dominant effect. In order to look at these effects further, the optical responses of nonequilateral triangular nanowires with horizontal side lengths of l ) 50, 25, 10, and 5 nm for h ) 0.5 nm were calculated and are shown in Figure 4. As the apexes of the nanowires become sharper (i.e., as l decreases), higher magnitude k components are generated. This leads to a rapid red shift of the LSPRs, as can be seen in the local results. In fact, the dipolar LSPRs are so red-shifted that they are not apparent on the scale in Figure 4, occurring at 1.35 and 1.06 eV for l ) 10 and 5 nm, respectively. It is important to note that as the (local) LSPRs are red-shifted, their magnitudes also increase, which can be seen in the l ) 25 nm results, for example, and also in Figure 3. However, in the nonlocal results, as the apexes of the nanowires become sharper the LSPRs red shift, much like the local results, but instead of increasing in magnitude, they decrease. These 3476
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results are surprising but not unreasonable given the discussion above. As the tip sharpness is increased, the efficiency of sustaining volume plasmons is as well, leading to stronger nonlocal effects. Also as this occurs the nanowires become more narrow, which means that volume plasmons can be sustained not only near the apexes but also over more of each nanowire’s body. This is apparent in the l ) 10 and 5 nm results, where a number of distinct peaks are seen, which are characteristic of (discrete) volume plasmon excitations.10,11 We will discuss these in more detail below. We now turn our attention to the junction |E|2 enhancements (the position where these are maximized) as a function of nanowire separation, Figure 5 and Figure 6. In the local results (dashed red curves), the |E|2 enhancements are qualitatively similar to the extinction cross sections (i.e., the peaks and valleys are alike), a trend that is often,2 but not always,39 the case. This typically occurs because the LSPRs increase multiple scattering events at the junction, which in turn lead to large |E|2 enhancements. The nonlocal results (solid blue curves), on the other hand, show that while all LSPRs (dipolar and multipolar) have clearly distinguishable peaks in the cross sections, only the dipolar ones show significant |E|2 enhancements (see the h ) 0.5 nm cylindrical results, Figure 5d, for example). However, it is possible that systems with more prominent high-order LSPRs (likely not in junction structures, but perhaps core-shell nanowires)38 will show higher corresponding |E|2 enhancements. Even in the absence of distinct peaks in the enhancements for the multipolar LSPRs, at high energies they are still relatively large and are slowly decreasing functions of energy. These results are in contrast to the local ones where distinct peaks
FIGURE 4. Extinction cross sections for triangular nanowires with horizontal side lengths of l ) 50, 25, 10, and 5 nm separated by h ) 0.5 nm. Local and nonlocal calculations are shown on the bottom and top, respectively.
FIGURE 5. Local (dashed red lines) and nonlocal (solid blue lines) |E|2 enhancements at the junction of 50 nm diameter cylindrical nanowires separated by h ) (a) 5, (b) 2, (c) 1, and (d) 0.5 nm. © 2010 American Chemical Society
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nm (Figure 6) represent possibly the most extreme example of this. In this case, the maximum |E|2 enhancements are 2.3 × 105 in the local results but are only 1.4 × 104 when nonlocal effects are included. Considering that the nonlocal results are more rigorous than the local ones in principle, it is possible that previously reported |E|2 enhancements are over an order of magnitude too high. As mentioned previously, it is important to keep in mind that there is some uncertainty in these results due to a lack of complete knowledge about the physical surface positions. But assuming that this error is not too significant, these results could have significant implications to the interpretations of plasmon-enhanced spectroscopy results.13,14 For example, if the actual electromagnetic contribution to SERS1 is smaller than expected on the basis of local theory (which given this discussion could be up to 3 orders of magnitude), it is possible that chemical effects play a more important role than has been considered in the past.40 In addition to the dependence on nanowire separation, it is also interesting to look at the |E|2 enhancements for triangular nanowires as a function of their horizontal side lengths l (tip sharpness) for a fixed h ) 0.5 nm, Figure 7. The |E|2 enhancements are again found to be qualitatively similar to the extinction cross sections (Figure 4), which is expected based on the results presented above. What is more interesting is that the nonlocal |E|2 enhancements show a progressive decrease as the sharpness of the tips are increased, going from l ) 10 to 5 nm for example, whereas the local results appear to increase without bound. Also notice the numerous positions of anomalous absorption in the nonlocal results, which become more prominent as the |E|2 enhancements drop in intensity. We will revisit these points below. In order to further understand the results presented up to this point and to characterize the qualitative behavior of the |E|2 enhancements, we can look at intensity profiles of |E|2. For example, those for cylindrical nanowires separated by h ) 0.5 nm and at the energies of the maximum |E|2 enhancements (Figure 5d) are shown in Figure 8. Besides a noticeable decrease in intensity in the nonlocal result, the enhancements are also found to be distributed around the dimer’s surfaces. For example, there are noticeable enhancements at the poles of the dimer in the nonlocal result, whereas in the local one the major enhancements occur only near the junction. Note that in both cases |E|2 is not symmetric about the dimer’s long axis, which is more noticeable in the nonlocal results because the enhancements are lower at the junction, leading to a higher contrast in the field profile there and elsewhere. The lack of symmetry occurs because of a shadowing of the incident light on the side of the dimer opposite to the incident field, leading to an overall lower distribution of the |E|2 enhancement there. The differences that occur in the field profiles can be understood by looking at intensity profiles of |D|2, Figure 9. The nonlocal result shows discrete (standing wave) volume
FIGURE 6. |E|2 enhancements at the junction of l ) 50 nm triangular nanowires for various separations h. Local and nonlocal calculations are shown on the bottom and top, respectively.
and valleys are seen. By inspection of the broadened spectral shapes in Figure 2 and Figure 5, it can be inferred that the two effects are related. However, they are not identical, as can be seen by comparison of the relative |E|2 enhancements between the dipolar and higher-order LSPRs in relation to the relative cross section magnitudes (in both the local and nonlocal results). These points will be revisited below. Figure 5 and Figure 6 also demonstrate that much like for the cross sections, the nonlocal |E|2 enhancements are blue-shifted relative to the local results, which becomes more prominent with decreasing nanowire separation. For example, while the local and nonlocal results are similar for the h ) 5 nm cylindrical nanowires (Figure 5a), there are significant differences for separations of less than approximately 2 nm, where at h ) 0.5 nm there is a 0.14 eV relative blue shift (Figure 5d). It is interesting to note that the amount of blue shift is different than that for the cross sections (Figure 2) but not unreasonable considering that the two are not necessarily correlated.39 Perhaps the biggest and most important difference between local and nonlocal results is that the |E|2 enhancements are significantly lower when nonlocal effects are included.10,12 The triangular nanowires separated by h ) 0.5 © 2010 American Chemical Society
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These profiles are in sharp contrast to the local result, where |D|2 is a smooth function (left side of Figure 9). In both the local and nonlocal results, |D|2 is concentrated near the junction. This is expected, since this is where most of the scattering events occur that can lead to high localizations of both |E|2 and |D|2. In comparison to longitudinal plasmons in isolated nanostructures10,11 which are longitudinal to the incident k, those in this case show structure along both the incident k direction and the dimer’s long axis. This can be seen in the (slight) cylindrical symmetry of the volume plasmons. Furthermore, the direction along the dimer’s long axis shows the greatest intensity. Therefore, at least in these structures, while the k (momentum) from the incident field excites volume plasmons, larger magnitude (dominant) k components are generated by scattering at the junction. This is in contrast to isolated nanowires, where the dominant excitation mechanism is from the incident field.10,11 By considering the nanowire SPs and volume plasmons as distinct modes, most of the qualitative differences in the optical responses and |E|2 enhancements relative to the local results can be explained. In this picture, these modes will overlap, leading to a damping of the SPs. Since SPs are responsible for the scattering that results in high |E|2 enhancements, it thus makes sense that they are lower when nonlocal effects are included. Because the intensity of the volume modes is highest near the junction, it is also understandable that the enhancements related to the higher-order LSPRs, which are oriented along other directions, are more significantly damped than the dipolar one. The spectral broadening of the LSPRs can also be understood, because this damping will lead to decreased lifetimes. With this understanding, we now look at |E|2 profiles for triangular nanowires with various horizontal side lengths l separated by h ) 0.5 nm, Figure 10. In all cases, the profiles are normalized such that only relative intensity comparisons between a given set of local (top panels) and nonlocal (bottom panels) results should be made. In the local results, the |E|2 enhancements are concentrated near the junction as expected. Furthermore, as the sharpness of the tips is
FIGURE 7. |E|2 enhancements at the junction of triangular nanowires separated by h ) 0.5 nm for various horizontal side lengths l. Local and nonlocal calculations are shown on the bottom and top, respectively.
plasmons inside of the nanowires, also called longitudinal plasmons because they are longitudinal to the k components that excited them. Although difficult to discern from their complex patterns, these can be considered analogous to oneparticle quantum mechanical states. (This is easier to visualize in more simple structures, such as isolated nanowires.)10
FIGURE 8. Intensity profiles of |E|2 for 50 nm diameter cylindrical nanowires separated by h ) 0.5 nm. Local and nonlocal calculations are shown on the left and right, respectively. The nanowires are outlined in white. © 2010 American Chemical Society
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FIGURE 9. Intensity profiles of |D|2 for 50 nm diameter cylindrical nanowires separated by h ) 0.5 nm. Local and nonlocal calculations are shown on the left and right, respectively. The nanowires are outlined in white.
FIGURE 10. Intensity profiles of |E|2 at the dipolar LSPR energies for triangular nanowires separated by h ) 0.5 nm for horizontal side lengths of l ) 50, 25, 10, and 5 nm (from left to right, respectively). Local and nonlocal calculations are shown on the top and bottom, respectively. The nanowires are outlined in white. Normalized intensity scales are used, as discussed in the text.
corresponding |D|2 profiles for l ) 50 nm triangular nanowires; see the Supporting Information.) In summary, we have shown that significant differences can exist between local and nonlocal electrodynamics when used to model the optical properties of Ag dimer structures. Specifically, LSPRs can be significantly blue-shifted in energy and broadened, affecting both cross sections and |E|2 enhancements. We studied both cylindrical and triangular structures. In the latter case we showed that while high magnitude k components can be generated by their sharp tips, they can also be efficiently damped by volume plasmons. Considering that nonlocal calculations are in principle more rigorous than local ones, the results presented are important for better understanding the optical properties of systems at the nanoscale.
increased, larger k components are generated, leading to higher |E|2 enhancements that are also distributed over a larger area. While both of these effects are seen in the nonlocal results (albeit with lower enhancements), there appears to be an additional effect. Recall from Figure 4 and Figure 7 that as the sharpness of the tips is increased, loworder volume plasmon modes are more efficiently sustained, giving anomalous absorption. These occur primarily at the apexes10 but can also occur more toward the middle or even the poles of the nanowires if they become more narrow. The overlap of these modes with the LSPRs cause the regions of high |E|2 enhancement to be distributed over a much greater surface area in the nonlocal results, which is most apparent for l ) 10 and 5 nm in Figure 10. This is analogous to the effect discussed for the cylindrical nanowires in Figure 8, except in this case the volume plasmons under discussion are excited by the incident field and not scattering at the junction. (Volume plasmons can be excited by such scattering though, as shown by © 2010 American Chemical Society
Acknowledgment. J.M.M. and G.C.S. were supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0004752. 3480
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Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC0206CH11357.
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Supporting Information Available. Computational details, the full Ag dielectric model, and |D|2 profiles for l ) 50 nm triangular nanowire dimers. This material is available free of charge via the Internet at http://pubs.acs.org.
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DOI: 10.1021/nl101606j | Nano Lett. 2010, 10, 3473-–3481
