Effect of Nanogap Curvature on SERS: A Finite-Difference Time

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Effect of Nanogap Curvature on SERS: A Finite-Difference Time-Domain Study Daedu Lee, and Sangwoon Yoon J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b01453 • Publication Date (Web): 21 Apr 2016 Downloaded from http://pubs.acs.org on April 24, 2016

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The Journal of Physical Chemistry

Effect of Nanogap Curvature on SERS: A FiniteDifference Time-Domain Study Daedu Lee and Sangwoon Yoon* Department of Chemistry, Dankook University, 152 Jukjeon-ro, Suji-gu, Yongin, Gyeonggi 448701, Korea

*Corresponding Author E-mail: [email protected] Phone: +82-31-8005-3152 Fax: +82-31-8021-7197

ACS Paragon Plus Environment

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Abstract

Surface-enhanced Raman scattering (SERS) is one of the most important plasmonic effects, offering a wide range of applications. Strong SERS arises from narrow nanogaps between nanoparticles. The SERS enhancement factor (EF) depends on many parameters that define the nanogap, such as the gap distance, gap geometry, and size and material of the constituent nanoparticles. In this study, we focus on the effect of the curvature of the nanogap on SERS. We perform finite-difference time-domain (FDTD) simulations for Au nanocube–nanosphere dimers, where nanocubes are attached to nanospheres on their vertices with various radii of curvature. The calculations reveal that the induced electric field becomes more localized around the vertex of the nanocube in the nanogap with the decrease in the curvature radius. The EF also drastically increases when the corner of the cube in the dimer sharpens. The EF of the nanocube– nanosphere dimer at ~1013 is far greater than that of nanosphere dimers or nanorod–nanosphere dimers. Through systematic changes of the variables that may affect the SERS EF, we find that in addition to the sharp local structure, a sizable volume is required to obtain the maximum EF. The curvature effect is the dominant contributor to the highest SERS EF for the nanocube– nanosphere dimers, overwhelming radiation damping or plasmon damping by the interband transition. This study identifies the governing factors for SERS and provides a design principle for the best SERS substrates.


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1. Introduction Surface-enhanced Raman scattering (SERS) is one of the most important plasmonic effects arising from noble metal nanoparticles.1,2 Raman scattering from the molecules near noble metal nanoparticles is significantly enhanced compared to those molecules without nanoparticles nearby. The enhancement is sufficiently high to permit the detection of single molecules, which makes SERS comparable to fluorescence spectroscopy.3,4 Because the SERS provides such high sensitivity and molecular fingerprints as a vibrational spectroscopy, it has been widely used in detecting various chemical and biological materials.5 SERS occurs via the excitation of the localized surface plasmon resonance (LSPR) of gold (Au) or silver (Ag) nanoparticles. The resonantly induced oscillation of the conduction electrons in those nanoparticles creates local electromagnetic fields (Eloc), which drastically increases the amplitude of the electromagnetic fields experienced by the molecules in the vicinity of the nanoparticles (local field enhancement).6 The Raman scattering radiation (Esca) from the molecules is also amplified by the oscillatory motion of the surface plasmon (radiation enhancement).6 Because the induced dipole in the Raman scattering process is given by µind = α E(ωL) and the radiated intensity is proportional to |µind|2, the local field enhancement is equal to |Eloc(ωL)/E0|2, where ωL is the excitation frequency and E0 is the incident electromagnetic fields. Similarly, the radiation enhancement is given by |Esca(ωR)/E0|2, where ωR is the Raman frequency. Then, the SERS enhancement for a single molecule is a multiplication of those two factors. Under the reasonable assumptions that the Raman shift is small (ωL ≈ ωR) and the radiation enhancement is identical to the local field enhancement (Eloc(ω) = Esca(ω)), the SERS enhancement factor (EF) leads to the fourth power of the local electromagnetic fields that form around the nanoparticles (|Eloc/E0|4).6 Such a large exponent for the local electric fields is the


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origin of the SERS. The strong local electric fields are particularly formed at sharp structures (“lightning rod effect”). Electrons squeezed into a tip of a structure by the external electromagnetic field increase the charge density and thereby create the significantly enhanced local electric fields around the structure. In this regard, understanding the relation between the structure of the nanoparticles and the local electric fields is the key to designing SERS substrates with the highest possible EF. Numerous nanostructures have been explored for their SERS activity. Isolated Au nanospheres typically generate weak local fields at their resonance surface plasmon frequency, corresponding to EFs on the order of 102-103.7 Many studies have exploited the lightning rod effect to produce stronger electric fields beyond the nanospheres. Sharp edges and tips of nanostars,8 nanocubes,9 and nanoprisms10,11 yield a much higher EF. However, the EF is limited to values below 107 as long as they are isolated monomers. When they are assembled, narrow nanogaps are formed among the nanoparticles, in which electric fields are confined, creating the high electric field intensity.9,12-14 Nanosphere dimers and face-to-face nanocube dimers yield EFs of 106 and 107, which are 3 and 2 orders of magnitude, respectively, higher than their monomer units.7,9,15,16 However, in those prototype homodimers, the lightning rod effect is absent, and the interparticle distance is the only variable that can control the SERS EF. We can additionally address the effect of the local geometry of the nanogaps on SERS by constructing a dimer of nanoparticles with sharp corners and tips. The best candidate for the maximum SERS effect is probably a tip-to-tip dimer of nanostars, fully utilizing the lightning rod effect.17 However, the synthesis of nanostars with tips of a controlled radius of curvature is challenging, and the assembly of a tip-to-tip dimer is even more difficult.18 Therefore, we adopt an experimentally accessible system for this study, i.e., Au


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nanocube–nanosphere dimers. Nanocubes have sharp corners, and the experimental control of their curvature has been realized.19,20 More importantly, the regioselective attachment of the nanocubes onto the nanospheres through one of those vertices is feasible. Recently, we have achieved the high-yield production of Au nanocube–nanosphere dimers, where the sharp corners of the nanocube are linked to the nanosphere surfaces via 1,8-octanedithiol.21 Those nanocube– nanosphere dimers exhibited an unprecedentedly high ensemble-averaged SERS EF of > 1 × 1010. We attributed the high EF to the sharp structures of the nanogaps between the nanocube and nanosphere. Here, we systematically calculate the effect of the sharpness of the curvature that defines the nanogap on the SERS enhancement. We compare this effect with other factors that may affect the SERS, such as the size of the constituent nanoparticles, radiative and interband damping of the plasmons, and interparticle distance. This study will contribute to the identification of the governing factors for the SERS EF and the design of experimentally realizable SERS substrates with the best performance.

2. Simulation Method We used the finite-difference time domain (FDTD) method to calculate the extinction spectra and local electric fields (Lumerical Solutions, Inc.). The Au nanoparticles were set up as shown in each figure in the results and discussion section. Note that the figures are two-dimensional (2D) representations of the three-dimensional (3D) model on which the calculations were actually performed. The dielectric function of Au was modeled from the Johnson and Christy values.22 The total simulation region was set to 1500 × 1500 × 1500 nm3, and perfectly matched-layer (PML) boundary conditions were used. The simulation time was adjusted to 500 fs for the energy field to fully decay. The refractive index of the medium was set at 1.3441 to mimic CH3CN,


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which was the dispersion medium of the Au nanocube–nanosphere dimers in our previous experiment.21 To calculate the extinction spectra, we set up an override region with a fine mesh (0.5 nm) such that it fully enclosed the dimers. Plane waves with a wavelength of 400-1100 nm impinged on the dimers with polarization parallel to the interparticle axis. The total field and scattering field from the nanoparticles were monitored in six directions. To accurately calculate the electric field distribution in the nanogaps, the override region was reduced to a volume of 50 × 50 × 50 nm3 to include the nanogaps with a finer mesh (0.2 nm). The excitation of the dipolar plasmon coupling at the resonance wavelength produced the local electric field. The overall distribution of the local electric field was monitored on a 2D plane that contained the dimer axis. The intensity distribution along the interparticle axis was monitored using a series of planes that were spaced by 0.2 nm and intersected the interparticle axis.

3. Results and Discussion 3.1 Effect of the Curvature Radius of a Nanocube We examined the effect of the local geometry of a nanogap on SERS. We set up an Au nanocube–nanosphere dimer with the cube attached to the sphere on its vertex. In this manner, the nanogap is defined by the round surface of the nanosphere and the sharp corner of the nanocube (Figure 1a). Then, we systematically changed the radius of curvature (rc) of the nanocube and calculated the extinction spectra and localized electric fields at the nanogap for each dimer using the 3D FDTD simulation method. The dimensions of the nanocube (edge length = 84 nm) and nanosphere (diameter = 56 nm) were selected to reproduce the experimentally prepared dimers from our previous publication.21 The gap distance (d) was fixed


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at 1 nm unless noted otherwise. The radius of curvature of the nanocube was varied from 42 nm (corresponding to a sphere with the radius) to 0.5 nm (corresponding to a cube with the sharpest corners). The calculated extinction spectra of the cube–sphere dimers exhibit two resonance peaks upon excitation with light polarized parallel to the dimer axis (Figure 1b). The longer wavelength peaks, which are marked with red circles, correspond to the dipole plasmon coupling mode between the Au nanocubes and Au nanospheres. The dipolar Coulomb interaction along the dimer axis creates a new mode at the redshifted wavelength from the LSPR bands of the individual nanocube or nanosphere.14,23,24 The shorter wavelength peaks, which are marked with blue circles, are attributed to the quadrupole plasmon coupling mode. Plotting the peak positions against the radius of curvature of the nanocube reveals how the plasmon interaction between the nanosphere and the nanocube changes as the curvature develops in the nanogap (Figure 1c). We also recast the peak shifts as the change in energy of the plasmon modes, following the plasmon hybridization model (Figure 1d).23,24 Generally, the LSPR band of a nanocube redshifts as the corner sharpens (open diamonds in Figure 1c). The sharp corners of the nanocube facilitate the charge separation and reduce the restoring force for the plasmon oscillation, which decreases the resonance frequency. As the LSPR energy of the nanocube is lowered (due to the sharpening corners), the hybridized mode of the nanocube with the fixedenergy LSPR mode of the nanosphere shifts toward lower energy accordingly (from red to green in Figure 1d). In the extinction spectra, it is reflected as the redshift of the dipole plasmon coupling band with rc decreasing from 42 to 5 nm. The redshift switches to the blueshift at rc = 5 nm. In this regime, the strong electromagnetic fields generated by the sharp corners of the constituent nanocube dominate the interaction between the nanocube and nanosphere. The


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plasmon coupling mode of the dimer is more like the LSPR mode of the nanocube, resulting in lifting up its energy closely to the LSPR energy of the nanocube. We also find that the crossover of the shift occurs only when the nanocube strongly interacts with the nanosphere. When the interparticle distance between the nanocube and nanosphere is long (d ≥ 3 nm) and, accordingly, the interaction weakens, the plasmon coupling mode continually redshifts without crossovers (Figure 2). We believe that further study is required to unveil the exact origin of the peak shift of the plasmon coupling between nanocubes and nanospheres with changing rc. The peak positions of the dipole plasmon coupling bands (λ = ~770) make the Au nanocube– nanosphere dimers a perfect system for SERS. The resonance wavelengths are far away from the interband transition of the Au nanoparticles, minimizing the possibility of the nonradiative plasmon damping.25 They are also close to the experimental Raman excitation wavelength (λ = 785 nm) often used. At those resonance wavelengths, we calculated the electric field intensity and distribution of the Au nanocube–nanosphere dimers for various radii of curvature of the constituting nanocube (Figure 3a). Our calculations show that the electric fields are strongly confined in the nanogap between the vertex of the nanocube and the round surface of the nanosphere (Figure 3b). With the sharper corner of the Au nanocube, the electric field distribution becomes narrower and is found mostly near the vertex of the nanocube. The intensity of the electric field dramatically increases as the nanocube vertex sharpens (Figure 3c). The maximum SERS EF, obtained from the largest local field intensity in the form of |Emax/E0|4, increases from 1011 to 1013 when the radius of curvature of the nanocube decreases from 42 to 0.5 nm. Notably, the EF on the order of 1013 is one of the largest values attainable from nanoparticles.6


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The local field intensity at a specific point in Figure 3b represents the SERS enhancement felt by a single molecule at the given position. Therefore, the calculated maximum SERS EF is equivalent to the experimental single-molecule (SM) SERS EF.6 For the ensemble SERS measurements, a number of molecules distributed on the surface of the nanostructure must be considered. Thus, the local field intensity should be spatially averaged. Upon averaging, it is legitimate to take into account only the molecules in the “hot spot” that contribute to the SERS signal. Practically, however, identifying the exact hot spots (location, volume, shape, etc.) is impossible. Selectively positioning molecules only in the hot spots is also implausible. Note that in our assembly process published previously, dithiol linkers form self-assembled monolayers (SAMs) all over the nanosphere onto which nanocubes are adsorbed.21 For the conservative estimation of the SERS EF, we count all the molecules probed in the Raman scattering even though they are outside the hot spot.13,21 In this spirit, we set up a fixed-size box (10 nm × 10 nm × 1 nm) inside the nanogap and calculated the volume-averaged EFs for the various radii of curvature. We find that the volume-averaged SERS EF remains largely the same regardless of the sharpness of the Au nanocube (Figure 3c). These calculations suggest that the SERS signal is rather insensitive to the sharpness of the corners of the Au nanocubes that constitute nanocube– nanosphere dimers in ensemble experiments whereas the curvature radius of the Au nanocube significantly influences the enhancement of the Raman signal in SM SERS experiments. We conducted a more detailed analysis on the spatial distribution of the induced local electric field. The electric field is not evenly focused in the nanogap. Instead, a strong electric field is formed closer to the Au nanocube (Figure 4). This inhomogeneity is more pronounced for the Au nanocube–nanosphere dimers with sharply cornered Au nanocubes. The molecules near the Au nanocube more likely experience larger SERS enhancements than those near the Au nanosphere


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surface. This result implies that for SM SERS experiments, a much better approach is to attach a Raman probe molecule to the Au nanocube than to the Au nanosphere. The results discussed so far suggest that the curvature of the nanoparticles defining the nanogap is one of the key factors that determine the SERS EF. The sharp tips in the nanogap produce a more intense electric field and thereby generate a stronger SERS signal than the nanogaps that are formed by smooth structures. This result is unsurprising considering the lightning rod effect. The natural follow-up question is whether the local shape of a nanogap is solely responsible for the Raman enhancement, independent of the other properties. If this is the case, one should be able to achieve the same level of enhancement using nanospheres with a similar radius to the curvature of the nanocube corners, which would make it a lot easier to prepare highly efficient SERS substrates because the size of the nanosphere is more readily controlled than the radius of curvature of the nanocube vertices.13,26 To answer this question, we calculated the SERS EF for the dimers, where the Au nanocubes in Figure 1a were replaced by Au nanospheres with identical radii to the curvature of the nanocubes.

3.2 Effect of the Local Geometry Only We set up an Au nanosphere dimer with a nanogap geometry that resembled the nanocube– nanosphere dimer. The bottom nanosphere has a diameter of 56 nm, and the top one has a radius (r) varying from 42 to 2 nm, which is similar to the radius of curvature of the Au nanocube (Figure 5a). We did not consider Au nanospheres smaller than 2 nm because the surface plasmon is not supported by those small nanoparticles.27,28 Figure 5b shows that the calculated longitudinal plasmon coupling band for the Au nanosphere dimers gradually blueshifts when the size of the top Au nanosphere decreases. For the smallest top Au nanosphere (r = 2 nm), the


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plasmon coupling band of the dimer appears at 540 nm, indicating that the plasmon coupling is dominated by the LSPR of the 56 nm Au nanosphere (λ = 540 nm). The shift of the plasmon coupling band for the nanosphere dimer is accounted for by the contribution of the LSPR bands of the individual Au nanospheres that constitute the dimer. The LSPR of an Au nanosphere redshifts when the size increases, which determines the shift direction of the plasmon coupling band for the dimer.29 We calculated the SERS EF at the resonance wavelengths. The spatial distribution of the electric field for the Au nanosphere dimer induced by the excitation of the plasmon coupling is similar to that of the Au nanocube–nanosphere dimer system. The electric field is confined in the nanogap region and narrows with the decreasing radius of the top Au nanosphere (Figure 4c). The significant difference between the two dimer systems is the change in the SERS EF values when the radius (of curvature) of the top nanoparticle decreases. The SERS EF of the Au nanosphere dimer drastically decreases from 1011 to 107 when the radius of the top Au nanosphere decreases. This result is opposite to the trend of the Au nanocube–nanosphere dimers, where the SERS EF increases as the radius of curvature of the top Au nanocube decreases. The comparison between the cube–sphere and sphere–sphere dimers suggests that the sharp curvature and the large volume must be considered together for the significantly high SERS enhancement. The large volume is required to provide the sufficient number of electrons that can be squeezed into the sharp tips, which leads to the generation of strong local electric fields. However, if the size is too big, radiation damping of the plasmon lowers the SERS EF (vide infra).

3.3 Effect of Size (or Volume)


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We further explored the effect of the size (or volume) of the nanoparticles on the SERS. We again replaced the top nanocubes with nanospheres and increased the size of the nanosphere from R = 40 nm to R = 60 nm, larger than the nanocube (Figure 6a). In this range, the change in the curvature of a nanogap is negligible because the top nanosphere is much larger than the bottom nanosphere, whose radius is fixed at 28 nm. Therefore, we can isolate the effect of the size on the plasmon coupling and SERS from those of other factors. Generally, larger nanoparticles exhibit redshifted and broadened LSPR band because of the electromagnetic retardation.29,30 The extinction cross-section also increases. These properties of an individual nanoparticle are reflected in the properties of the dimer that contains the nanoparticles. Figure 6b shows that the plasmon coupling band of the dimer redshifts as the size of the top nanosphere increases. From Figure 5, we learned that a sizable volume was required to obtain the large SERS effect. The SERS EF increased when the size of the top nanosphere increased from r = 2 to 40 nm. However, when the top Au nanosphere is larger than r = 40 nm, this behavior no longer persists. Figure 6b clearly shows that the SERS EF rather decreases as the size of the Au nanosphere increases from R = 40 to 60 nm. This result is attributed to radiation damping.25,31 The large scattering of light for large nanoparticles dampens the plasmon oscillation, which results in the reduced generation of the local electric fields. Interestingly, an Au nanocube scatters light much more strongly than an Au nanosphere of similar size. Figure 7 shows that the scattering cross-section of an Au nanocube (edge length = 84 nm, rc = 1 nm) is three times larger than that of an Au nanosphere (diameter = 84 nm), which indicates that the radiation damping in the Au nanocube should be far greater than that of the Au nanosphere. Nevertheless, when the Au nanocube forms a dimer with the 56 nm Au nanosphere,


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the SERS EF is two orders of magnitude larger than that when the Au nanosphere forms a dimer. This result suggests that the local field enhancement by the sharp corners of the nanocube overwhelms the loss of field by the large scattering.

3.4 Comparison with Other Nonspherical Nanoparticles Based on the above results, we conclude that the sharp corners and sizable volume make the Au nanocube a unique component that provides the highest SERS EF when it is combined with a nanosphere. We compare it with other nonspherical nanostructures. Nanostars are conceivably one of the best SERS substrates in this regard. However, attaching nanospheres to the tips of the nanostars in a controlled fashion is experimentally challenging.32,33 Among the realistic systems, we select nanorod–nanosphere dimers. Because the surface of the nanorods is stabilized by an identical ligand (cetyltrimethylammonium bilayers) to that of the nanocubes, one may produce nanorod–nanosphere dimers using a similar method to that for nanocube–nanosphere dimers.34,35 In addition, we can further test the effect of the local geometry of a nanogap on the SERS EF without perturbation from the effect of nonradiative plasmon damping by the interband absorption.25 Contrary to nanospheres, nanorods have a longitudinal plasmon mode in the longwavelength region (λ > 600 nm) far away from the interband transition, suppressing the interband damping.25,36 We replaced the top nanocubes with nanorods with the same radii of curvature: rc = 2, 5, 10, and 20 nm (Figure 8a). The aspect ratio (length / width) of the Au nanorods was fixed at 3, so that the longitudinal plasmon coupling band was within the typical experimental detection window (λ ≤ 1100 nm). When the radius of curvature and thereby the nanorod length increase (because of the fixed aspect ratio), the longitudinal plasmon coupling band redshifts from ~750


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nm to ~900 nm (Figure 8b). The Au nanorod–nanosphere dimer yields a SERS EF on the order of ~1010 for the range of rc = 5 – 20 nm. The EF is even lower (4 × 108) when the curvature radius is 2 nm. Even if we increase the aspect ratio to 5 for this pointy rod, the EF increases to only 2 × 109 (Supporting Information). Because the effect of the interband transition is negligible for these nanorod–nanosphere dimers, it is apparently the curvature and the volume that determine the SERS EF. The pointy nanorod–nanosphere dimer exhibits the low SERS EF despite a sharp curvature in the nanogap, because the volume of the nanorod is significantly small. As rc increases, the volume increases (because of the fixed aspect ratio), but the sharpness of the curvature is reduced. The counteraction of these two factors makes the SERS EFs largely the same for rc = 5, 10, and 20 nm (Figure 8c). In this sense, Au nanocube–nanosphere dimers can be regarded as an optimal assembly structure for the highest SERS effect by offering both the sharp corners and the large volume.

3.5 Effect of Interparticle Distance The highest EF (~1013) was obtained from the Au nanocube–nanosphere dimer with the fixed interparticle distance of 1 nm (Figure 3). One may argue that the gap distance of 1 nm is impractical and rather an extreme case because relatively short molecular linkers (e.g., 1,6hexanedithiol or 4-aminobenzenethiol) must be used to obtain this distance.13,37 Furthermore, it is near the boundary between the classical and quantum plasmon coupling regimes.38-40 One should note that the SERS EF decreases when the nanogap narrows in the quantum regime, contrary to the prediction of classical electromagnetism.41 We calculated the SERS EF for Au nanocube–nanosphere dimers as the interparticle distance is increased from 1 to 5 nm (Figure 9). As expected, the EF decreases with the widening gap. We


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note that as long as sharp-cornered nanocubes (rc = 0.5 nm) are used, the SERS EF remains sufficiently high (EF > 1011) until the nanoparticles are separated by 4 nm. In contrast, the same level of enhancement can be achieved only at d = 1 or 2 nm for the other forms of nanodimer. Therefore, Au nanocube–nanosphere dimers are the best choices for single-molecule experiments in which large Raman probe molecules such as rhodamine dyes are typically used.42

4. Conclusions Figure 10 summarizes the effect of the nanogap curvature on the SERS EF for various combinations of nanodimers. For nanocube–nanosphere dimers, the SERS EF increases when the corners of the Au nanocube sharpen. This curvature effect does not hold for other nanosphere dimers and nanorod–nanosphere dimers. From those comparisons, we found that in addition to the sharp curvature, a sizable volume is required to achieve high EF values. Radiation damping should also be considered for large nanosphere dimers. However, Au nanocube–nanosphere dimers produce high electric fields despite the large radiation damping. The SERS EFs of Au nanocube–nanosphere dimers reach as high as 1013, which makes nanocube–nanosphere dimers one of the best SERS substrates that have been theoretically proven and shown to be experimentally realizable.

Supporting Information. SERS EFs of Au nanorod–nanosphere dimers when the aspect ratio of the nanorod increases from 2 to 5. This material is available free of charge via the Internet at http://pubs.acs.org.


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Acknowledgment This research was supported by the Basic Science Research Program through the National Research

Foundation

of

Korea

(NRF)

funded

by

the

Ministry

of

Education

(2013R1A1A2008336).

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Figure Captions Figure 1. (a) Au nanocube (NC)–nanosphere (NS) dimer systems that were explored to study the effect of the local geometry of nanogaps on the plasmon coupling and SERS. (b) Calculated extinction spectra of the Au nanocube–nanosphere dimers for light polarized parallel to the dimer axis. The curvature radius (rc) of the Au nanocube was varied from 42 nm to 0.5 nm. The red and blue circles indicate the peak positions of the dipole and quadrupole plasmon coupling modes, respectively. For comparison, the calculated normalized extinction spectra of an individual Au nanosphere (56 nm) and nanocube (rc = 10 nm) are included in the bottom of the panel. The open squares and open diamonds mark the LSPR peaks of the individual Au nanosphere and Au nanocube, respectively. (c) Resonance wavelengths of the Au nanocube– nanosphere dimers with various curvature radii for nanocubes. Each symbol represents the peak position of the indicated mode in (b). (d) Plasmon hybridization diagram for the dipole plasmon coupling mode of nanocube–nanosphere dimers with varying radii of curvature of nanocubes. With decreasing rc, the LSPR mode of a nanocube redshifts, but the dipole plasmon coupling mode transitions from a redshift to a blueshift. Figure 2. Dipole plasmon coupling wavelengths of the nanocube–nanosphere dimer against the radius of curvature of the constituent nanocube for various interparticle distances (d). Figure 3. (a) Scheme to calculate the electric fields for Au nanocube–nanosphere dimers. (b) Electric field distribution of the Au nanocube–nanosphere dimers for the given radius of curvature (rc) of the Au nanocube. The lower panels display the magnified view of the electric fields near the nanogap region. The scale bars represent 50 and 10 nm for the upper and lower panels, respectively. (c) SERS EFs of the Au nanocube–nanosphere dimers for various radii of curvature Au nanocubes. The red circles represent the EF from the maximum electric field


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intensity, which is equivalent to the single-molecule SERS EF. The blue squares indicate the EF from the volume-averaged electric field intensity in the 10 × 10 × 1 nm3 box. Figure 4. SERS EF along the interparticle axis between the vertex of the Au nanocube and the Au nanosphere for different radii of curvature of the nanocube. Figure 5. (a) Au nanosphere dimer systems explored for the effect of the local shape of the nanogaps on the plasmon coupling and SERS. The radius of the bottom nanosphere was fixed at 28 nm, whereas that of the top nanosphere was varied from 42 to 2 nm. (b) Calculated plasmon coupling resonance wavelengths (blue squares, right axis) and SERS EF (red circles, left axis) of the Au nanosphere dimer as functions of the radius of the top nanosphere. (c) Electric field distribution of the nanosphere dimers for a given radius (r) of the top nanosphere. The lower panels present magnified views of the electric fields near the nanogap region. The scale bars represent 50 and 10 nm for the upper and lower panels, respectively. Figure 6. (a) Au nanosphere dimer systems explored for the effect of the size of the nanoparticles on the plasmon coupling and SERS. The radius of the bottom nanosphere was fixed at 28 nm, whereas that of the top nanosphere (R) was varied from 40 to 60 nm. (b) Calculated plasmon coupling resonance wavelengths (blue squares, right axis) and SERS EF (red circles, left axis) of the Au nanosphere dimer as functions of the radius of the top nanosphere. (c) Electric field distribution of the Au nanosphere dimers for the given radius of the top nanosphere. The lower panels present magnified views of the electric fields near the nanogap region. The scale bars represent 50 and 10 nm for the upper and lower panels, respectively.


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Figure 7. Calculated extinction, scattering, and absorption cross-sections of (a) Au nanosphere with a diameter of 84 nm and (b) Au nanocube with an edge length of 84 nm and a radius of curvature of 1 nm. Figure 8. (a) Au nanorod–nanosphere dimer systems explored for the effect of the local curvature of nanogaps without perturbation by the interband transition. The radius of curvature (rc) of the Au nanorod was adjusted from 2 to 20 nm, while the aspect ratio was maintained at 3. (b) Calculated extinction spectra, (c) maximum SERS EFs, and (d) local electric field distributions of the Au nanorod–nanosphere dimers for different nanorod radii of curvature. The lower panels present magnified views of the electric fields near the nanogap region. The scale bars represent 50 and 10 nm for the upper and lower panels, respectively. Figure 9. SERS EF of Au nanocube–nanosphere dimers as a function of the gap distance (d) for various radius of curvature (rc) nanocubes. Figure 10. Summary of the SERS EFs for various combinations of nanodimers.


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Figure 1. 78x42mm (600 x 600 DPI)


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Figure 2. 71x79mm (600 x 600 DPI)



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Figure 3. 91x47mm (600 x 600 DPI)


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Figure 4. 86x116mm (600 x 600 DPI)



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Figure 5. 91x65mm (600 x 600 DPI)


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Figure 6. 91x55mm (600 x 600 DPI)



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Figure 7. 73x100mm (600 x 600 DPI)


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Figure 8. 91x59mm (600 x 600 DPI)



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Figure 9. 49x35mm (600 x 600 DPI)


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Figure 10 58x56mm (600 x 600 DPI)



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Table of Contents Graphic 44x28mm (600 x 600 DPI)


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Effect of Nanogap Curvature on SERS: A Finite-Difference Time

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