Discrete Dipole Approximation Calculations of Optical Properties of

Department of Chemistry, Seoul National University, Seoul 151-747, Department of Chemistry, Kyung Hee University, Seoul 130-701, and Department of ...
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J. Phys. Chem. C 2009, 113, 16321–16328

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ARTICLES Discrete Dipole Approximation Calculations of Optical Properties of Silver Nanorod Arrays in Porous Anodic Alumina Soree Kim,† YounJoon Jung,*,† Geun Hoi Gu,† Jung Sang Suh,† Seung Min Park,‡ and Seol Ryu*,§ Department of Chemistry, Seoul National UniVersity, Seoul 151-747, Department of Chemistry, Kyung Hee UniVersity, Seoul 130-701, and Department of Chemistry, Chosun UniVersity, Gwangju 501-759, Republic of Korea ReceiVed: December 31, 2008; ReVised Manuscript ReceiVed: July 26, 2009

The experimental extinction spectrum of 38 nm long silver nanorods grown in the 22 nm pores of an anodic aluminum oxide (AAO) template is theoretically analyzed, using the discrete dipole approximation (DDA) method. A single broadband at 485 nm in the spectrum turns out to be the overlap of the two peaks at 430 and 515 nm, corresponding to the transverse and longitudinal modes of the surface plasmon of the nanorods, respectively, at small angles e75° of incidence. Interestingly, the longitudinal mode in the array is blueshifted by about 50 nm from that of an isolated nanorod while the transverse mode does not shift much. Near the glancing incidence, however, the two modes do not contribute much to the spectrum in the visible range due to significant red-shifting and broadening. We show that the incident angle dependence of the optical properties of the silver array may be utilized in controlling local near-field enhancements for surface-enhanced Raman scattering (SERS) and the maximum enhancements are achievable in the incident angles of 30-45°. I. Introduction In predicting optical properties of arbitrarily shaped metal nanostructures, discrete dipole approximation (DDA)1-4 has served as one of the most useful theoretical tools, along with other standard methods such as the boundary element method (BEM),5,6 multiple multipoles (MMP),7-9 finite-difference timedomain (FDTD),10 and T-matrix methods.11,12 However, what makes DDA particularly interesting and distinguished from the others would be a set of polarization vectors assigned at small units of the material, which provides an intuitive hint on how the conduction electrons are pushed around by the electric field of the incident light. Once the polarization set is obtained, both far-field and near-field properties may be calculated semianalytically, without any serious mathematical efforts. The efficiency and versatility of the DDA method have been proved for calculations of optical properties of a variety of noble metal nanoparticles, such as triangles,13,14 cubes,15 decahedra,16 and octahedra.15,17 Despite these benefits, the inability of DDA to handle either extended or periodic structures has appeared to be a major hurdle to further practical uses. The drawback is particularly unsatisfactory, considering that there has been intense research activity in the fabrication of two-18 or three-dimensional19,20 ordered assemblies or aggregates of nanoparticles, recently. These nanostructured materials are drawing attention as targets for potential applications in surface-enhanced spectroscopies and surface plasmon resonance-based biosensing.21-23 In such cases, * To whom correspondence should be addressed. E-mail: [email protected] (Y.J.) and [email protected] (S.R.). † Seoul National University. ‡ Kyung Hee University. § Chosun University.

uses of more brute-force techniques like FDTD have been deemed inevitable. Traditionally, the optical properties of nanoparticle aggregates have been dealt with by Maxwell-Garnett’s effective medium theory and its variants.24-28 Alternatively, Lazarides and Schatz29 showed that their coupled dipole approximation, in which dispersed nanoparticles are represented as single dipoles, allowed for the calculations of far-field properties as well as Maxwell-Garnetttype theories did. On the other hand, the recent impressive work by Evans et al.30 illustrated that MMP modeling of particle arrays led to not only far-field but also near-field properties that can be compared with experimental data in a quantitative manner. In view of this situation, a recent generalization of DDA by Draine and Flatau31,32 is expected to become a useful new venue for discussing optical properties of periodic nanostructures, because it still offers the same kind of easy interpretation as it did for finite cases, i.e., in terms of polarizations. In particular, one can identify properties genuinely arising from the periodic nature when one compares the results for the periodic system with those of an isolated unit obtained in a separate calculation. In this work, we consider silver nanorods embedded in the highly ordered porous AAO as a model system. We choose this system both to test the efficiency of the DDA method in dealing with the periodic structures and to explain recent experimental results. In their recent work, Suh and co-workers33-35 showed that the growth of silver nanorods in the 22 nm pores of the hexagonal AAO matrix could be controlled (Figure 1, top and middle panels), leading to the tuning of resonance surface plasmon frequencies and that the SERS signals of benzenethiol at the resonance wavelengths of light could be varied with particle lengths or aspect ratios. Interestingly, there, the experimental extinction spectrum of a 38 nm long silver nanorod

10.1021/jp811516s CCC: $40.75  2009 American Chemical Society Published on Web 08/20/2009

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Kim et al. The DDA method assumes that an original unit cell of a periodic structure consists of N small volume elements located at positions r00 j (j ) 1, 2, ..., N) in a cubic lattice. Here, the subscript j denotes the jth volume element while the superscript 00 represents that the volume element is in the original, (0,0)th, unit cell. In the periodically generated (m,n)th unit cell, the ) r00 position of the jth volume element is written as rmn j j + mLu + nLV, where Lu and LV are the lattice vectors. The jth volume element in each cell is assigned a polarizability tensor, Rj. The polarizability is usually prescribed by the lattice dispersion relation, using the dielectric constants of materials.37,38 A monochromic electric field of an incident light is described by Einc ) E0 exp(ik0 · r), where E0 is the electric-field amplitude, k0 is the wavevector of magnitude |k0| ) k0 ) 2πn˜/λ, λ is the wavelength of light, and n˜ is the refractive index of the surrounding medium. The electric field induces the polarization, P00 j , of each element (induced dipole) that can be calculated by

Pj00 ) Rj · Eloc(rj00)

(1)

00 Here, Eloc(r00 j ) is the local electric field at rj , and it has two contributions: one due to the incident electric field and the other due to both the other N - 1 induced dipoles in the original unit cell and induced dipoles in the replicated unit cells, that is,

Eloc(rj00) ) Einc(rj00) + Eother(rj00) ) E0 exp(ik0 · rj00) -

∑ ∑ (1 - δjkδ0mδ0n)Ajk00,mn · Pmn k Figure 1. Scanning electron microscopy images of the silver nanorods in the pores of AAO templates: (top) top view and (middle) side view. (bottom) Extinction spectra in the visible range of the silver nanorods with the average lengths of 38, 51, and 62 nm.

array in AAO exhibits a rather broad, single peak at the wavelength of 485 nm (Figure 1, bottom), while it is widely known that isolated single silver nanorods of the same size show the transverse and longitudinal modes well-separated in the visible regime. We try to theoretically characterize the apparent merge of the two modes using the generalized DDA method. In particular, we discuss changes in the relative intensities and positions of the two modes, associated with incident light directions, and propose possibilities on the origin of peak broadening and shifting. We also perform a detailed analysis on the experimental result based upon the generalized DDA approach. Finally, we make a comment on the validity and efficiency of the generalized DDA method for describing the optical properties of periodic structures, as compared with the results from a finite cluster approach. The paper is organized as follows. In Section II, we will provide a brief overview of the generalized DDA formalism. Calculation results and discussion are presented in Section III. We compare our calculation results with the experimental results in Section IV, and give our conclusions in Section V. II. Discrete Dipole Approximation for Periodic Structures In this section, we present a brief overview of the DDA formalism and its applications to the two-dimensional periodic structures. A more extensive description on the DDA method3,4,36 and its extension to the periodic structures31 have been given elsewhere.

k

(2)

m,n

where δij is the Kronecker delta. The second term can be mn expressed more explicitly, with the interaction matrix, A00, , jk expanded up to the dipolar terms,

Ajk00,mn · Pmn k )

exp(ik0rjk00,mn) (rjk00,mn)3

(1 - ik0rjk00,mn) (rjk00,mn)2

{

k20rjk00,mn × (rjk00,mn × Pmn k ) +

}

00,mn 00,mn · [(rjk00,mn)2Pmn (rjk · Pmn k - 3rjk k )]

(3)

mn 00 00,mn where r00, ) rmn ) | r00,mn |. jk k - rj and rjk jk Now, we set replica polarization, Pjmn, by the periodic conditions

Pjmn ) Pj00 exp[i(mk0 · Lu + nk0 · LV)]

(4)

Then we have 00 ˜ ∑ ∑ (1 - δjkδ0mδ0n)Ajk00,mn · Pmn k ) ∑ Ajk · Pk

k

m,n

k

(5) ˜ jk, is given by where the reduced interaction matrix, A mmax

˜ jk ) A

∑

nmax

∑

(1 - δjkδ0mδ0n)Ajk00,mn ×

m)-mmax n)-nmax

exp[i(mk0 · Lu + nk0 · LV)] (6)

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Note that mmax and nmax values depend on the dimensionality of the structure. Both mmax and nmax must go to infinity in the case of two-dimensional periodic structures. For one-dimensional periodic structures, either mmax or nmax should be set to zero, and the other goes to infinity. For finite objects, mmax ) nmax ) 0. Once we find the polarization P00 j by solving the following equation in a self-consistent manner,

Pj00 ) Rj · (E0 exp(ik0 · rj00) -

∑ A˜jk · P00k )

(7)

k

the extinction and absorption cross sections, Cext and Cabs, can be obtained from the optical theorem,

Cext )

Cabs )

4πk0

4πk0

N

∑ Im(E*inc(rj00) · Pj00)

|E0 | 2 j)1

(8)

∑ {Im[Pj00 · (Rj-1)* · Pj00*] - 32 k30|Pj00|2} N

|E0 | 2 j)1

(9) In addition, the scattering cross section is obtained as Csca ) Cext - Cabs. In practice, the evaluation of the reduced interaction matrix, ˜ jk, which contains the infinite number of terms, is approximated A by the introduction of a damping factor with the interaction cutoff parameter, γ,

˜ jk ≈ A

∑ Ajk00,mn exp[i(mk0 · Lu + nk0 · LV) - (γk0rjk00,mn)4] m,n

(10) For DDA calculations in this article, we found that the use of γ ) 0.01 gave fairly good results without any serious convergence problem. The electron mean free path in bulk silver at room temperature is 52 nm, which is larger than the linear dimensions of nanorods studied in the present work (22 nm along the short axis and 38 or 51 nm along the long axis). In this case, one may expect that the quantum size effect may play some, if not significant, role in the calculations, and consider adjusting the dielectric functions accordingly. For such adjustments, we followed the work of Coronado and Schatz39 that accounts for both the shape and size of nanorods. For instance, we estimated the mean free path of the conduction electron in a 38 nm long nanorod with a 22 nm diameter to be 17.75 nm. We then modified silver dielectric functions reported in ref 40 using the corrected values of the mean free path and derived the polarizabilty by the corrected version of the lattice dispersion relation.38 III. Calculation Results A. Transverse and Longitudinal Modes. Nanorods that are randomly oriented and well separated from each other are wellknown to exhibit an extinction spectrum similar to the average of two peaks: one for only the transverse mode excited and the other for only the longitudinal mode. Thus, one might expect that the same would be the case for the array structure. In principle, however, reproducing an experimental spectrum of a nanoarray system requires the inclusion of anisotropy manifested

Figure 2. (top) Schematic diagram of the two-dimensional silver array structure. Silver nanorods are surrounded by an infinite AAO template of refractive index n˜ ) 1.6. The incident angle, diameter, and length of the nanorod and distance between nanorods are defined as θ, D, l, and d, respectively. The incident light can be of either s- or ppolarization with respect to the plane of incidence. (middle) Dependence of (s,p)-averaged extinction spectra on the lattice geometry when θ ) 90° and d ) 63 nm. Comparison of square and hexagonal lattice cases is shown. (bottom) A series of (s,p)-averaged extinction spectra at θ ) 90°, with interparticle distances of d ) ∞, 140, 98, 63, and 49 nm, respectively.

by the interaction between light and the array system. In this subsection, we first present the results on the spectral change of the two modes as the nanorods come close to each other and make an ordered array. We then analyze the spectra and interpret the experimental data in terms of the two modes. We have shown the model system that we are investigating at the top of Figure 2. For simplicity, the system is projected onto the yz-plane, and the side view of the system is shown. Silver nanorods in their upright positions along the z-direction are infinitely replicated in the xy-plane. The incident angle of light is denoted by θ as shown. When the plane of incidence is defined by the z-axis and the direction of incident light, two polarization directions are possible: s-polarization being perpendicular to the plane of incidence and p-polarization being parallel. A hexagonal lattice of the nanorods may resemble the real lattice of the nanorods fabricated on an AAO template more closely, but our test calculations show that the difference between results for the hexagonal and for the square lattices with their nearest rod-to-rod distances the same is negligible as shown in Figure 2 (middle). Such lattice-type dependence is not very serious at other incident angles as well (Supporting Information, Figure S1). A similar observation has been made in ref 41. Thus, we use the square lattice throughout our calculations for computational efficiency. A unit cell, square column of lateral dimension, d, is repeated horizontally in the homogeneous AAO medium of refractive index, n˜ ) 1.6.42 In

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the center of the unit cell, a nanorod of diameter D ) 22 nm and length l ) 38 nm is built up with 4079 point dipoles. For periodic-structure calculations, the interaction cutoff parameter, γ, is set to 0.01, which leads to a good convergence of spectra. We find that the convergence of data becomes better at smaller incident angles. (Supporting Information, Figure S2). Figure 2 (bottom) shows changes in the spectra when silver nanorods come close to each other and form arrays with d ) 140, 98, 63, and 49 nm. Here, the incident angle is chosen as θ ) 90°.The ordinate label, Extinction, refers to the absolute extinction cross section (Cext, in nm2 per nanorod). We show the spectra averaged over the two polarization directions, i.e., (s,p) (s) (p) (λ,θ) ) 1/2Cext (λ,θ) + 1/2Cext (λ,θ), which is called “(s,p)Cext averaged extinction” in this article. In the case of θ ) 90°, extinction spectra calculated from s- and p-polarized incident lights correspond to the transverse and longitudinal modes of (s) (trans) (p) (long) ) Cext and Cext ) Cext . In excitation, respectively, i.e., Cext case of the isolated nanorod (d ) ∞), the transverse and longitudinal modes are well-separated and so their resonance peaks are found at 430 and 561 nm, respectively. As the nanorods come close to each other, however, the peak intensities are reduced with broadening and red-shifting, which apparently tends to combine the two modes eventually. In the limit of d ) D, the nanorods will make up the two-dimensional film, and the resonance wavelength will go out of the visible range. To study further behaviors in broadening, red-shifting, and merging of peaks in terms of two modes, we decompose the total (s,p)-averaged extinction into the two polarization components for different values of d ) ∞, 98, 63, and 49 nm from top to bottom in Figure 3. We see that both modes suffer broadening and red-shifting. The transverse mode red-shifts more than the longitudinal mode, because as the rod-to-rod distance decreases the near-field coupling becomes stronger in the transverse mode than in the longitudinal mode. This entails a single broad peak at a smaller separation. In fact, such broadening and red-shifting were found in various two- and three-dimensional aggregates, and even in dimers and trimers, as the plasmonic near-field coupling becomes stronger.43 Note that the theoretical spectrum with d ) 63 nm in the third panel of Figure 3, corresponding to the spectrum in Figure 1 (bottom) with l ) 38 nm, shows the peak maximum at 570 nm, but it is too far off to be identified with the maximum at 485 nm observed in the experiment. We will try to resolve this issue in the following sections by focusing on the experimental system with d ) 63 nm and l ) 38 nm. The other experimental system with d ) 63 nm and l ) 51 nm will be considered later when the comparisons between experiments and calculations are made in Section IV. B. Incident Angle Dependence. The angle of glancing incidence, θ ) 90°, considered in the previous subsection is not practically achievable in laboratories. In this subsection we turn to more realistic cases of smaller incident angles that are employed in most optical setups. Figure 4 shows the (s,p)(s,p) (λ,θ) with the value of θ averaged extinction cross section Cext varied by 15°. Note again that, at θ ) 90°, the two modes contribute equally to the total spectrum, while it is only the transverse mode that contributes when θ ) 0°. At this incident angle, both polarizations excite transverse modes only, and (s,p) (trans) (λ,θ ) 0°) ) Cext (λ,θ ) 0°). At the intermediate angles, Cext 15° to 60°, we can see that the spectra undergo significant changes over the whole spectral range. To analyze a spectral transition between these two extremes in Figure 4, we first show, in Figure 5, the dependence of the extinction spectra on the incident angle when the incident light

Kim et al.

Figure 3. A series of (s,p)-averaged extinction spectra are decomposed into the transverse and longitudinal modes with interparticle distances of d ) ∞, 98, 63, and 49 nm from top to bottom. At θ ) 90° transverse and longitudinal modes are selectively excited with s- and p-polarizations of light, respectively.

Figure 4. Dependence of (s,p)-averaged extinction spectra on the incident angle, θ ) 0°, 15°, 30°, 45°, 60°, 75°, and 90°.

is s-polarized (top) or p-polarized (bottom). The s-polarized spectrum in the top of Figure 5 shows the angle-dependence of (s) (trans) (λ,θ) ) Cext (λ,θ), and the peak the transverse mode only, Cext positions remain nearly at the same wavelength of 430 nm, unless θ is greater than 60°. But the p-polarized spectrum (Figure 5, bottom) contains both the transverse and longitudinal mode contributions that depend on the incident angles, and the two corresponding peaks appear at the intermediate incident angles. To decompose the spectra in the bottom of Figure 5 into the transverse and longitudinal components, we use an approximation based on the spectra of the single isolated nanorod. For the isolated nanorod case, a p-polarized spectrum can be exactly

Silver Nanorod Arrays in Porous Anodic Alumina

Figure 5. Dependence of extinction spectra on the incident angle, θ ) 0°, 15°, 30°, 45°, 60°, 75°, and 90°, when (top) s-polarized and (bottom) p-polarized lights are used as excitation sources. Note that, in general, s-polarization excites only the transverse mode, while p-polarization excites both the transverse and longitudinal modes.

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Figure 7. Comparison of the DDA-simulated and experimental extinction spectra of (top) l ) 38 nm and (bottom) l ) 51 nm systems.

60°. When the incident angle becomes even larger, the nearfield interaction becomes important, and the broadening becomes serious, which results in the intensity reduction and the broadening. It is noteworthy that, unless the incident light is almost glancing, the longitudinal mode in the nanorod array looks blue-shifted to 515 nm (Figure 6, bottom) from 561 nm of the isolated nanorod (Figure 3, top), while the transverse mode is still kept at 430 nm (Figure 6, top).

Figure 6. Dependences of (top) the transverse and (bottom) the longitudinal modes on the incident angle, θ ) 0°, 15°, 30°, 45°, 60°, 75°, and 90°. Here we assume that the extinction spectra obtained from (p) p-polarized light can be written as Cext ) (cos2 θ)Ctrans + (sin2 θ)Clong, which becomes exact for the isolated nanorod case. (p) decomposed as Cext ) (cos2 θ)Ctrans + (sin2 θ)Clong. Now, since (s) we know Ctrans ) Cext (Figure 5, top) and C(p) ext (Figure 5, bottom), we can separate out the contribution of longitudinal modes from the p-polarized spectra approximately. Figure 6 exhibits the decomposition of the (s,p)-averaged spectra into the following (s,p) (s) (p) ) 1/2[Cext + (cos2 θ)Ctrans] + 1/2[Cext two contributions, Cext 2 (cos θ)Ctrans], where the first and second terms represent approximately the transverse and longitudinal modes, respectively. In Figure 6, the transverse modes (top) are found near 430 nm, whereas the longitudinal modes (bottom) are found near 561 nm, until the incident angle becomes as large as 90°. The transverse modes can be excited at all angles (top). When the incident angle becomes larger (θ g 60°), the near-field interaction between single rods becomes serious, leading to a significant broadening. In the case of longitudinal modes (bottom), however, a non-monotonic behavior is observed. Note here that, at small incident angles, the chances of the longitudinal mode being excited are minimal, so the extinction is extremely small. As the incident angle increases, so does the intensity until θ )

IV. Comparison with Experiments A. Effects of Incident Angle. We compare the experimental extinction spectra for array systems of two different nanorods (l ) 38 and 51 nm) with our calculation results. The experimental setup does not allow us to determine an accurate value of the incident angle, but we found, after a careful numerical analysis, that the best fits could be obtained with θ ) 75° and so we use this value in the comparison between the experimental spectrum and our calculation results. Figure 7 shows that overall comparison between the DDA calculation and experiment is quite favorable in general, although the agreement is better in the l ) 38 nm (top) than in the l ) 51 nm (bottom) case. Just for comparison, we also performed the rotational average of the (s,p)-averaged extinction over the solid angles shown as dotted lines. Clearly, the experimental results are in better agreement with the calculation result for θ ) 75° than for random incident angles. The discrepancies between theory and experiment may arise due to many factors on both sides, for example, size distributions of the nanorods in the experiment, uncertainties in the dielectric functions used for the calculation, and preferred orientations of the incident light in the experiment. B. Effects of Dielectric Constants. It is possible that some uncertainties in dielectric functions might have influenced our calculation results in view of Figure 7. In our calculations, we used the method of ref 39 to modify the dielectric function of silver by estimating the averaged mean free path of the conduction electron in the nanorod to be 17.75 nm. Since the diameter (22 nm) is smaller than its lengths (38 or 51 nm), however, one may think that use of the two different meanfree-path values along the different axes results in a better agreement between experiment and theory. For instance, an extreme broadening of the transverse mode will remove the dip in the middle of the spectra and so give them the look of a single broad peak.

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Figure 8. (top) The schematic diagram for the unit cell of the nanorod deposited in the pores of AAO templates, with a cylindrical opening in the upper side of the nanorod. The wall height, h, is defined as the distance from the bottom of the nanorod to the top of the AAO template. (middle) The (s,p)-averaged extinction spectra for the unit cell only at θ ) 90°, as the wall height, h, varies. The longitudinal mode resonance wavelength increases with the AAO wall height, while the transverse mode stays almost at the same wavelength. (bottom) Changes in the peak position of the longitudinal mode in the middle-panel spectra with the cylindrical opening as the wall height increases. λ0 ) 561 nm is the resonance wavelength of the longitudinal mode in the AAO case.

To examine this possibility, we performed a separate calculation using the two different mean-free-path values for a cylinder prescribed in the work of Kraus and Schatz,44 but found no significant improvements (see Figure S3 in the Supporting Information for details). This implies that the discrepancies are most likely due to other variables such as the incidence angle and distributions of size and shape. We also consider the effect of anisotropic dielectric environments for the nanorods, with cylindrical openings in the upper side of nanorods as shown in Figures 8 (top) and 1. To model the openings above the nanorods, we consider, in Figure 8 (top), an isolated nanorod structure corresponding to the repeating unit of the nanorod array. Here, a silver nanorod is surrounded by AAO except for the cylindrical opening. A wall height, h, defined as the distance from the bottom of the nanorod to the top of AAO, is to be varied. The middle of Figure 8 shows the (s,p)-averaged spectra at θ ) 90° with varying wall heights as well as that of the nanorod in AAO for comparison. It turns out that the building up of higher walls tends to bring back the spectrum toward that in the AAO case in the middle of Figure 8. The bottom of Figure 8, however, suggests that there is an upper limit on this red-shifting with the wall height. This will

Kim et al. give an additional contribution in the experimental spectrum compared with the theoretical spectrum calculated in the AAO case, although it is hard to quantify the amount of the shift due to other factors such as the array effect. So far, we have used the computational procedure that can handle the periodic structures appropriately by considering a unit cell and replicating it in space. It also will be possible to build up the same kind of structures by adding one particle or a few at a time. Such a bottom-up approach, called the cluster approach, may prove itself very useful when one needs to investigate the finite-size scaling relation in the spectra. We calculated the spectra of clusters made of 1, 3, 7, 19, and 37 nanorods to explore the possibility of estimating the spectra of an infinite array. Our calculation results show that the blueshifting of the longitudinal mode becomes more noticeable with the increase of the cluster size. (See Figure S4 in the Supporting Information for details.) C. Field Enhancements. Finally, we explore the possibility of employing silver nanoarrays in an AAO matrix for SERS applications. For a nanosystem to be useful for SERS study, it is essential to have control over its orientation with respect to light directions, thereby well-defined hot spots are readily available for molecules to be adsorbed. In this regard, silver nanorod arrays fabricated on porous AAO templates seem potentially useful since they can provide well-defined geometry of many nanorod systems compared with randomly dispersed nanorods in solution. In fact, optimizations of the silver nanorods deposited on AAO as templates for SERS are currently being investigated.33-35 (see also Figure S5 in the Supporting Information for preliminary results.) As shown in the middle of Figure 1 and the top of Figure 8, enhanced local fields near the exposed top surface of the silver rods may be used for amplifying SERS signals electromagnetically. To quantitatively characterize the local field enhancement at the top surface of a silver nanorod, we define 〈g〉 ) 〈|Etot|2/ |E0|2〉 as the average of the square of the total scattered field relative to that of the incident field, taken over points about 1.4 nm away from the surface. Now, we check on the dependence of the 〈g〉-factor upon the incident angle in order to find the optimum geometrical conditions for SERS signal amplification, with the p-polarization of light. For the (excitation) wavelengths of incident light, we choose those of lasers we can commonly employ in the visible range, 413, 458, 488, 514, and 632 nm. This way, we can investigate how the 〈g〉-factor changes, as we gradually scan the excitations of transverse to longitudinal modes. Considering the longitudinal mode in the nanorod array located near 515 nm at 0° < θ < 75°, we expect that 514 nm would be best for enhancing the local field at the upper surface of the nanorod. Indeed, the field maximum occurs at the intermediate angle range, 30° to 45°. At a longer excitation wavelength, 632 nm, the maximum moves to a higher angle of 60°, with a considerable decrease in the 〈g〉-factor. At a shorter wavelength, 488 nm, we observe a decrease at higher angles and an increase at lower angles in the 〈g〉-factor. This implies that the transverse mode excitation can also enhance the nearfield intensity in the nanorod array, although not so big as the longitudinal excitation. The field enhancement at the top due to the transverse mode is an interesting and unique array effect, which becomes larger as we move close to the transverse-mode resonance wavelength, such as 413 and 458 nm. The bottom left of Figure 9 shows the best possible electric field enhancements, 〈g〉 near a nanorod in the array, when the incident angle is θ ) 45° and the excitation wavelength is 514 nm. For comparison, we also show the enhancement due to the

Silver Nanorod Arrays in Porous Anodic Alumina

J. Phys. Chem. C, Vol. 113, No. 37, 2009 16327 We have also considered the potential use of enhanced local fields at the upper surface of the nanorods for the SERS application. To achieve the best electromagnetic enhancement, a subtle control of light incident angle is shown to be necessary along with a careful selection of excitation light wavelength. It appears, however, that the maximum field enhancement for the array structure is somewhat smaller than that of the isolated nanorod. To achieve a higher field enhancement at the top surface of nanorods, an array of nanorods with a larger rod-to-rod distance seems preferable. Alternatively, we might use the field enhancement at the side surface of nanorods, as we wish to decrease the rod-to-rod distance. Currently, we are working on experiments of obtaining SERS signals of benzenethiol molecules at the top surface of silver nanorods in the AAO templates, which will follow this theoretical study for publication.35 Acknowledgment. This study was supported by the Korea Science and Engineering Foundation (KOSEF; grant no. R112007-012-03003-0), the BK 21 program funded by the Korea government (MEST), Seoul Science Fellowship (S.K.), and research funds from Chosun University 2009. S.R. is grateful to Prof. Hyunjoon Song for helpful discussions.

Figure 9. (top) Variation of the electric field enhancement, 〈g〉 ) 〈|Etot|2/|E0|2〉, with respect to the incident angle θ, with different excitation wavelengths chosen. (bottom) The best electric field enhancements, g near nanorods in the array (left), compared with the enhancement due to the longitudinal-mode of isolated single nanorod in AAO (right). The incident angle is θ ) 45° and the excitation wavelength is 514 nm, for nanorods in the array.

longitudinal mode of the isolated single nanorod in AAO (right). It suggests that the best field enhancement in the nanorod array is about 5 times smaller than that of the single nanorod case. V. Concluding Remarks In this article, we have studied the spectral properties of 38 nm long silver nanorods grown in the 22 nm pores of AAO matrix, which form the two-dimensional hexagonal lattice with a rod-to-rod distance of 63 nm. The generalized DDA method has been employed to interpret some important features of the experimental extinction spectrum. The broadness observed in the spectrum has been accounted for in terms of the transverse and longitudinal modes of the surface plasmon of the nanorods and their dependence on the angle of incidence, θ. The transverse and longitudinal modes conveniently recognized at the glancing incident angle (θ ) 90°) suffer significant redshifting and broadening due to the near-field couplings between nanorods. When the incident angle is reduced to 75° or smaller, the transverse mode becomes free of any serious red-shifting and closely approaches the resonance wavelength of the isolated nanorod. However, the longitudinal mode in the nanorod array at 75° or smaller exhibits a significant blue-shifting of 50 nm from that of the isolated case. It is interpreted, therefore, that the broad experimental spectrum results mainly from two overlapping plasmonic modes, i.e., transverse and longitudinal modes, though we cannot rule out the possibility that additional broadenings might arise due to the size and shape distribution of the nanorods grown in AAO.

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