Multimode Resonances in Silver Nanocuboids - Langmuir (ACS

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Multimode Resonances in Silver Nanocuboids Michael B. Cortie,*,† Fengguo Liu,‡ Matthew D. Arnold,† and Yasuro Niidome‡,§ †

Institute for Nanoscale Technology, University of Technology Sydney, PO Box 123, Broadway NSW 2007, Australia Department of Applied Chemistry, Kyushu University, Fukuoka 819-0395, Japan § International Institute for Carbon-Neutral Energy Research, Kyushu University, Fukuoka 819-0395, Japan ‡

S Supporting Information *

ABSTRACT: A rich variety of dipolar and higher order plasmon resonances have been predicted for nanoscale cubes and parallopipeds of silver, in contrast to the simple dipolar modes found on silver nanospheres or nanorods. However, in general, these multimode resonances are not readily detected in experimental colloidal ensembles, due primarily to the usual variation of size and shape of the particles obscuring or blending the individual extinction peaks. Recently, methods have been found to prepare silver parallopipeds with unprecedented shape control by nucleating the silver onto a tightly controlled suspension of gold nanorods (Okuno, Y.; Nishioka, K.; Kiya, A.; Nakashima, N.; Ishibashi, A.; Niidome, Y. Uniform and Controllable Preparation of Au-Ag Core-Shell Nanorods Using Anisotropic Silver Shell Formation on Gold Nanorods. Nanoscale 2010, 2, 1489−1493). The optical extinction spectra of suspensions of such monodisperse particles are found to contain multiple extinction peaks, which we show here to be due to the multimode resonances predicted by theoretical studies. Control of the radius of the nanoparticle edges is found to be an effective way to turn some of these modes on or off. These nanoparticles provide a flexible platform for the excitation, manipulation, and exploration of higher order plasmon resonances.

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when silver14−24 or some other element such as Ni or Pt is deposited onto a gold nanorod core.25−27 However under the appropriate conditions cuboids of Pd or Ag have been obtained on an Au nanorod core.28,29 Nanoscale cubes of silver are interesting because they have the potential to show a more complex set of plasmon resonances than are possible with spheres and rods.30−34 For example, simulations show that extinction spectra become more complex and are red-shifted as a silver nanosphere is morphed to a cube though a series of intermediate shapes35 or as a cube is increased in size.32 The main radiative resonances possible in a cube are shown in Figure 1 and were originally identified by Fuchs.30 We will denote these prototypical cube modes as C1 to C6 using the numbering system of Fuchs. The resonance designated here as C6 appears to have been incorrectly drawn in Fuch’s original Figure 1. We assert that the correct depiction for this resonance is as shown here and by others.31 (Note that a different numbering system is normally used for sphere modes, which are numbered l = 1, 2, 3..., where l is the multipole order.) The wavelengths at which the cube resonances occur in the quasistatic limit can be determined for various candidate materials (Table 1). It is clear that, of the four elements considered, only

INTRODUCTION A plasmon resonance can be excited on a metallic nanostructure when it is illuminated with light of an appropriate wavelength and polarization. Such resonances take the form of a coupling between the free electrons of the nanostructure and the electromagnetic oscillation of the light. This phenomenon is responsible for a wide range of interesting optical effects because it is associated with enhanced absorption, scattering, and extinction of the light. Plasmon resonances in gold nanospheres, nanoshells, and nanorods have been extensively investigated, and have already been exploited in diverse technological contexts.1−8 Unlike gold, silver readily crystallizes as cube-shaped nanoparticles.9,10 Silver nanostructures have very strong plasmon resonances in their own right, with the peak extinction for a suspension of silver nanospheres in water, for example, at about 390 nm, compared to that of gold nanospheres at about 520 nm. Furthermore, under the right conditions, silver can be deposited onto an existing gold nanostructure to produce a gold@silver core−shell structure. Deposition of silver onto gold nanospheres has been well-studied in the past,11−13 and the resulting particle generally follows the spherical shape of the gold core. On the other hand, deposition of silver, or some other element, onto a gold nanorod core offers increased geometric flexibility, because gold nanorods can readily be produced with a controlled size and aspect ratio. Most studies report a rodlike symmetry in the resulting hybrid nanoparticle © 2012 American Chemical Society

Special Issue: Colloidal Nanoplasmonics Received: January 27, 2012 Revised: March 25, 2012 Published: March 26, 2012 9103

dx.doi.org/10.1021/la300407u | Langmuir 2012, 28, 9103−9112

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(i.e., peak splitting) can sometimes be discerned in single particle measurements.37,40 An even greater number of resonance modes is possible in a right cuboid, that is, by decreasing the symmetry by allowing one of the orthogonal cube dimensions to be greater in length than the other two.45 (These shapes may also be described as nanoprisms, although that family of shapes is more general and includes, for example, nanotriangles.46) However, experimental resolution of these additional modes requires the preparation of nanocuboids of very reproducible shape, failing which the peaks merge and become indistinguishable as for the cube. Here we show that a sufficient degree of monodispersity can be obtained under recently discovered synthesis conditions, and this allows the interesting spectral complexity of silver right cuboids to be readily revealed in colloidal ensembles. Using a combination of experiment and calculation, we identify the plasmon oscillations of typical silver nanocuboids. We reproduce the spectral features observed and show that the presence or absence of individual peaks can be controlled by the refractive index of the surrounding medium and the geometry of the particle.

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EXPERIMENTAL SECTION

Synthesis. The Au−Ag core−shell nanorods were prepared according to a previously published procedure.29 Gold nanorods were obtained from Dai-Nippon Toryo Co. Ltd. In a typical case, the gold nanorods were dispersed in a reaction solution (13 μM Au, 11.6 mL) that contained 4.76 mM ascorbic acid, 4.72 mM NaOH, and 75.5 mM hexadecyltrimethylammonium chloride (CTAC). A suspended solution of silver chloride (10 mM, 0.25 mL) was added to the reaction solution. The temperature of the reaction solution was kept at 30 °C. The thickness of the silver shells depended on the relative quantities of silver ions and gold nanorods. In this work, the gold and silver contents in the reaction solution were 0.15 and 2.5 μmol, respectively. Extinction spectra were obtained using a conventional spectrophotometer (V-570, JASCO). An optical cell, 1 cm optical path length, was used for the measurements. Transmission Electron Microscopy (TEM). The nanorod suspensions were centrifuged twice to eliminate the surfactant in the solutions. After each centrifugation, the supernatant was removed, and the precipitate was redispersed in distilled water. A small portion of the colloid was sampled and cast onto a TEM grid (elastic carbon-support film grids, Okenshoji Co., Ltd.). Subsequently, the TEM grid was dried in vacuum for 3 h. TEM images were obtained using a JEM-2010 transmission electron microscope (JEOL. Ltd., performed at 120 KV). The TEM images were analyzed using ImageJ software (free from http://rsb.info.nih.gov/ij/). A plugin “Measure_Roi” (http://www. optinav.com/Measure-Roi.htm) was also utilized to measure the length and diameter of nanorods. This plugin measures the maximum distance between any two points in the region of interest as the length. The width is defined as the maximum distance in the perpendicular direction to the defined length. Simulation of Fully Retarded Nanostructures. The optical properties of retarded (i.e., finite sized) nanostructures were simulated using the discrete dipole approximation (DDA) as implemented in the DDSCAT code of Draine and Flatau.47,48 Version 7.1 of the code was used. The target is represented by an array of polarizable points and the electromagnetic scattering of an incident periodic wave on the target is solved numerically. The effective radius, aeff, is an important parameter in these simulations and is defined as the radius of a sphere with the same volume as that of all the dielectric materials in the target. DDSCAT provides accurate simulations of electromagnetic scattering provided that 2πaeff/λ < 25 and the dipole spacing is sufficiently small so that (2πd|m|)/λ < 1. The dipole spacing d is given by d = (V/N)1/a, and m is the complex refractive index.49−53 However, convergence is slow unless (|m − 1| 3, require very long computation times and more closely spaced dipoles.

Figure 1. Six strongest radiative cube resonances recalculated by the present authors using a quasistatic analysis, shown as colored maps of surface charge distribution, and (bottom) the associated optical extinction spectrum simulated using Johnson and Christy36 dielectric data for Ag in vacuum. The edges of the cube have been rounded slightly (p = 0.1, see later) to ensure numerical convergence and to approximate the effect of the limited number of surface patches used by Fuchs. The modes are numbered using the same system as that originally used by Fuchs.30 Each mode of the cube has 3-fold degeneracy due to cubic symmetry; only modes which couple to an electric field along the indicated direction have been drawn. The color scale has been truncated to enhance the visibility of the charge symmetry of the faces which have weak excitation compared to the corners and edges.

Ag has the capability of showing the full set of cube resonances, with damping in some or all of the resonances in cubes made of the other three elements. Note that various, slightly different, tabulations for the dielectric function of Ag exist, and the value of resonance does depend rather sensitively on the values of ε1(ω) used (Supporting Information, Figure S1). Therefore, three commonly used dielectric functions for Ag are compared in Table 1. In theory, the resonances in a silver cube can be quite sharp, suggesting that they might be a very suitable basis for developing a refractrometric-type sensing system.37 However, even in Ag, the sharp spectral features associated with the cube resonances are generally washed out in practice by the polydisperse nature of real samples, and by surface scattering (damping) of the resonance, for example,9,28,30,38−41 although a second mode 9104


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Table 1. Free Space Wavelengths of Light at Which the Six Strongest Cube Eigenmodes Could Be Expected in Various Candidate Materials under Quasistatic Conditionsa free space wavelength of mode, nm Ag43

Ag36

Ag44

mode designation

ε′(ω)

Cu43 vac

Au43 vac

Pd43 vac

vac

H2O

vac

H2O

vac

H2O

C1 C2 C3 C4 C5 C6

−3.68 −2.37 −1.90 −1.27 −0.78 −0.42

(421) (372) (351) (314) (299) *

524 504 (497) (487) * *

(308) (237) (219) 198 183 173

387 361 353 345 339 334

442 398 382 361 346 338

386 361 353 342 333 329

442 398 383 365 344 334

398 363 352 341 334 328

460 412 392 362 344 334

The numbering of the modes and the ε′(ω) at which each occurs is from Fuchs.30 (Comparable values for the cube eigenmodes may also be found in the paper by Langbein.42) The bulk dielectric data given by Weaver and Fredericks43 was used, except for silver, where three different dielectric functions are compared. (“*” indicates that the value cannot be matched due to interband transitions. A value in parentheses indicates that the value of ε1 can be matched but resonance will occur in a region of strong interband damping, defined here as ε2 > 2.0.). a

provide a better or more conservative fit.56−59 According to Stahrenberg et al., the differences between the various tabulations of n and k for Ag are caused by (1) the original investigators using different experimental and data reduction techniques to one another and (2) the original samples having different defect structures and surface characteristics anyway.60 For water, we used the complex refractive index of Palik.44 Another important consideration was to establish an appropriate value of dipole spacing for the calculations to follow, as too great a spacing causes numerical instability while too small a spacing needlessly protracts the calculations. We found that a dipole spacing of 0.6 nm or less gave acceptable results. The effect of the “tolerance” parameter of DDSCAT, which determines when a sufficient accuracy has been attained, was also investigated and a value of 1 × 10−4 was found to be adequate for our needs. Also, it makes little difference to the results whether Au cores of the typical sizes reported here are included in the simulation or not. The reason is that the Ag exterior shields small Au cores from electromagnetic radiation, a finding that is in agreement with the prior literature.13 (These points are illustrated in Figure S3 of the Supporting Information.) Note, however, that Au cores of greater size, for example, occupying a volume fraction of 15% or greater, would be expected to cause red-shifting of the resonances. Electric field intensities were extracted using the DDFIELD utility of DDSCAT, and processed using our own software to produce bitmaps and vector fields. Simulation of Quasistatic Optical Eigenmodes. The eigenmodes of the individual radiative resonances were calculated using a surface integral eigenmode framework similar to that of Fuchs,30 by diagonalizing a geometry-dependent interaction matrix to determine the mode position and strength, which can then be applied to particular materials to deduce the optical extinction.61,62 To increase accuracy, the diagonal matrix elements were integrated over the source coordinate63 using subsampling. As noted by Hohenester and Krenn,64 it is also important to round the edges of the structure to match that observed in experiment, because the modes are particularly sensitive to sharp edges, which also greatly increase the sampling required for convergence. The same superellipsoid model as described earlier was used, and this was meshed from a structured grid to yield N triangular elements. Symmetry was used to calculate the modes on just one octant to ameliorate mode degeneracy, and to reduce the time for diagonalization which goes as N3. Modes are assumed to have antisymmetric charge along the excitation axis, hence only those modes that are radiating in the quasistatic limit are found (this is an important point of difference from the DDA technique which in principle incorporates all modes, or from the scheme of Zhou et al.32 which extracts the contributions of only dipole and quadrupole resonances from DDA polarizations). Each axis was excited in turn and the results averaged to simulate random orientation of the particles in solution. An example of an ensemble average was also generated by averaging the spectra of 100 shapes with lengths and weighting factors drawn from a Gaussian fit to the distribution measured for the experimental sample.

The targets were generated using a custom-written program and were based on a “superellipse” (one of the superquartic shapes54) given by ⎤(p / q) ⎡ (2/ p) ⎛ x ⎞(2/ q) ⎛ z ⎞(2/ p)⎥ ⎢⎛⎜ y ⎞⎟ ⎜ ⎟ + +⎜ ⎟ =1 ⎝a⎠ ⎝c⎠ ⎥⎦ ⎢⎣⎝ b ⎠ Here we set q = p. The parameters p and q can be used to control the sharpness of the edges and corners in this geometric model (Figure 2).

Figure 2. Examples of cuboidal targets generated for use in the DDA calculations with varying degrees of edge and corner sharpness by varying the p parameter as shown. A cross section through the target with p = 0.5 shows its optional internal gold nanorod seed. The aspect ratio is given by L/W. This is an important consideration because it is already known that the radii of the corners and edges have a significant effect on the optical extinction.55 Corner radii of targets were estimated with the aid of the ThreePointCircularROI plugin for ImageJ (http://www.dentistry. bham.ac.uk/landinig/software/software.html and http://rsb.info.nih. gov/ij/ respectively) in order to generate a correlation between p and the corner radii measured on the experimental material (see Figure S2 of the Supporting Information). For convenience x, y, and z are chosen here so that the axis of 4-fold symmetry of the superellipse lies along the x axis of the DDSCAT coordinate system. We used the complex dielectric functions of Johnson and Christy36 for silver and Weaver and Frederikse43 for gold, noting that, in the case of silver, the results of simulations are quite sensitive to which data is used for dielectric function. In agreement with the recent report of McMahon et al.,34 we find that the use of the Johnson and Christy data provides a better match to the measured spectra of silver nanocuboids because it provides sharper resonance peaks than those predicted by the use of the other data sets. This is an indication of a highly crystalline and relatively defect-free lattice structure in the present silver nanocuboids. In cases where the Ag nanostructures are less crystalline, then use of the Johnson and Christy data would overestimate the quality of the resonances and use of the alternate data sets may 9105


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RESULTS AND DISCUSSION

and diameter of the particles can be measured directly using ImageJ. Figure 4 shows the size distribution of the Au−Ag

Shape and Size of Nanoparticle Ensemble. The TEM images show that Au@Ag nanoparticles are paralleopipeds with rounded edges and corners (Figure 3). Therefore, the length

Figure 4. Size distribution of Au−Ag core−shell nanorods. There is more variability in the length dimension than the width direction.

core−shell particles that we prepared. (Histograms of their lengths and diameters are provided in the Supporting Information as Figure S4.) The radius of curvature of the edges or corners of the particles is (as will be shown later) an important parameter. Therefore, the corner radii were measured and analyzed statistically (Figure 5).

Figure 5. Corner radius distribution of the Au@Ag nanocuboids described in the text. The p values necessary to produce these radii in a superellipse are shown along the top.

The mean sizes of the nanoparticles in the samples studied are given in Table 2. Table 2. Mean Sizes of Au−Ag Core−Shell Nanorodsa sample size, n

length, nm

width, nm

aspect ratio

corner radius, nm

148

52.8 ± 9.4

35.6 ± 5.7

1.51 ± 0.30

10.1 ± 1.6

a

Ranges given correspond to one standard deviation.

Extinction Spectrum of Experimental CTAC-Stabilized Sample. The extinction spectrum is complex, clearly showing the presence of higher order plasmon modes (Figure 6). The peaks are surprisingly sharp and well-differentiated from one another compared to previous reports for silver nanocuboids.40 We consider that the reason is that the present particles are much smaller than those of the previous report cited (typical particles in the present instance being only 15 to 20% of the volume of those of that paper). The much smaller size of the present particles means that peak broadening due to scattering of light is considerably diminished. Another point to note is that the present particles are “wrapped” with surfactant molecules,

Figure 3. TEM images of nanorods at various magnifications. The Au cores can be seen as the darker (more opaque) regions within the Ag cuboids. 9106


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Figure 6. Extinction spectrum of a suspension of Au−Ag core−shell nanocuboids wrapped in CTAC. The increase in absorption at 300 nm is due to the aqueous medium.

a factor which will red-shift their plasmon resonances relative to a naked particle. However, even though the individual resonance peaks are clearly visible, the oscillation responsible for causing each of these modes is not immediately apparent. In principle, reduction of the nanoparticle symmetry from cubic to tetragonal should have resulted in splitting of each of the modes shown in Figure 1 into a lower energy longitudinal and a higher energy transverse resonance which we will differentiate here by adding the subscript L or T, as for example in C1L and C1T. However, as we will show later, rounding of the edges and corners also eliminates some of the cube modes, which complicates analysis. Therefore, it is not clear from the experimental data alone whether, for example, the peak at 400 is due to a C2T resonance or perhaps a C2L resonance. In addition, the modes shown in Figure 1 are only a subset of the possibilities for a noncubic shape anyway, with other more complex modes, both radiative and nonradiative, also being possible in principle. Simulated Extinction Spectra of Cubes and Right Cuboids. The data of Table 2 indicated that an Ag nanocuboid of 53 × 36 × 36 nm3 should be used to simulate the nominal shape. A value of the p parameter of 0.52, which generated a corner radius of ∼11 nm, was chosen as the baseline for this work. Similar results would have been obtained for any value of p between about 0.48 and 0.55 (see Supporting Information, Figure S2). The actual refractive index of the medium in the “near-field” region immediately surrounding the particles was not known (it would be greater than that of pure water due to the presence of dissolved salts in the solution as well as the formation of an adsorbed shell of surfactant molecules in the near-field of the particle). In general, the effect of increasing the near-field refractive index of the medium surrounding the nanocuboid is to red-shift the resonances. An estimate of the effective near-field refractive index of the medium was obtained by calculating the extinction spectra of the above cuboid for various values of the refractive index of an infinite surrounding medium, and comparing the results to the measured spectra, Figure 7. (We concede that this approach does not directly address the question of how thick the adsorbed shell of surfactant molecules would need to be to produce the result.) When plotted against free space wavelength of the light, the extinction spectrum is stretched out in the red-direction by an increase in the refractive index of the medium, with high wavelength (low energy) peaks moving further than low wavelength (high energy) peaks. This stretching phenomenon also provides a useful tool for the separation and identification of the individual multimode resonances (see later). This analysis

Figure 7. Simulations showing the effect of the near-field refractive index of the surrounding medium on the calculated extinction spectrum of a 53 × 36 × 36 nm3 silver nanocuboid generated with p = 0.52.

indicates that positions of the experimental extinction peaks are matched if the refractive index of that part of the medium adjacent to the particles is taken as 1.63 ± 0.01. In this case, the simulated peaks are at 548, 455, 397, and 366 nm, in reasonable agreement with the 549, 447, 400, and 346 nm of experimental data of Figure 6. It would be useful in the future to disperse nanocuboids of this type in a medium of different refractive index to that of the growth solution, for example in glycerol− water, to confirm this approach. The width and relative heights of the peaks are, however, not well matched with a simulation that uses a single shape of nanocuboid. This is not surprising as the measurements of the dimensions of real particles showed some spread, especially in the length parameter. In general, the position of the longitudinal peak of elongated nanostructures is more sensitive to dimensions (particular aspect ratio) than the transverse peaks (Supporting Information Figure S5). Therefore, the averaging effect of having an ensemble of nanostructures of varying dimensions will attenuate and broaden the longitudinal peak more than a transverse one, and may reduce the amplitude of the longitudinal peak below that of the transverse one. This effect is found in ensembles of gold nanorods, for example.65 Construction of such an ensemble using the DDA method would have required a several hundred core-days of computation and was not attempted. However, the quasistatic method is orders of magnitude quicker, and so an ensemble average (Figure 8) could be constructed using 100 particles of various lengths drawn from the spread in lengths determined in the TEM analysis (Table 2). It can be seen that because the position of the longitudinal resonance is more sensitive to the dimensions of the rod than that of the other resonances, the net result of creating an ensemble average of cuboids of varying lengths is to attenuate the average longitudinal peak, relative to the other peaks. This explains the relatively low amplitude of the longitudinal extinction peak in the experimental data. Varying the transverse dimensions would only have improved the ensemble average slightly, for the reasons given earlier. Note that the quasistatic peaks are both blue-shifted and sharpened relative to those of the DDA method because retardation of light is ignored in the quasistatic analysis. This exercise also highlights the potential gain in sharpness of resonances to be had if colloidal 9107


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quasistatic results would be blue-shifted relative to the retarded ones and so in this case use of this refractive index does not provide a perfect match to the peak positions of the experimental data. Due to the relatively curved shapes, the modes for the chosen superellipsoid (p = 0.52) are somewhat closer to an ellipsoid (p = 1) than a hard cuboid (e.g., p = 0.1). The lowest energy mode is a longitudinal dipole, then the transverse dipole (similar to l = 1 of a sphere or C1 of the cube), and two higher order modes (one a hybrid between l = 3 and C2, and another matching C6) (Figure 10). Although other modes may be present, they are masked. Figure 8. Quasistatic simulation of the extinction spectrum of an ensemble of silver nanocuboids with the observed distribution of lengths. The background refractive index is n = 1.63. Averaging the spectra in this way illustrates how the peak due to the longitudinal plasmon resonance in a single shape can become diminished in the ensemble relative to the others.

suspensions of an even tighter distribution of even smaller sizes could be prepared. The effect of varying the p parameter of the superellipsoid, which controls the sharpness of the edges and corners of the solid, is shown in Figure 9. In general, the effect of decreasing

Figure 9. Effect of increasing the sharpness of corners and edges on the calculated extinction spectra of silver nanocuboids of 53 × 36 × 36 nm3 in a transparent medium with near-field refractive index of n = 1.500. The p-values of 0.1, 0.3, 0.5, and 1.0 correspond to radii of about 2, 6, 11, and 22 nm, respectively. A p value of 0.01 gives a sharpedged cuboid, while p = 1.0 produces a shape similar to an ellipsoid.

the edge and corner radii is to red-shift the resonances, as well as stretch out the spectrum so that previously degenerate (overlapping) peaks become resolvable. This is consistent with reports in the literature in which a increase in edge radius caused by annealing should cause a blue shift.37 It is clear that most of the higher energy resonances actually require sharp edges in order to be developed above the background extinction. Note also that targets generated with a constant length and width, but increasing edge sharpness, will increase slightly in volume across the series. An increase in volume generally causes redshifting of resonances so a small part of the red-shifting effect seen here is due to that source. Nature of the Resonances. The radiative quasistatic resonances were calculated using the eigenmode method for the standard shape. These calculations were for superellipsoids immersed in a medium of greater refractive index than pure water to account for the effect of the CTAC surfactant. A refractive index of 1.63 was used in order to provide consistency with the fully retarded calculations, however it would be expected that the

Figure 10. Quasistatic eigenmodes developed on three Ag shapes, each with nominal dimensions of 53 × 36 × 36, and background index n = 1.63 shown as colored maps of surface charge distribution. The calculated extinction spectra (blue) are compared to the experimental ensemble spectrum (red dotted) in the lower panels. (a) p = 0.1 representing a hard-edged cuboid (from left to right, modes C6T, C2T, C1T, and C1L) for comparison to Figure 1, (b) p = 0.52 representing the best estimate of the experimental geometry, and (c) p = 1 an ellipsoid for comparison (from left to right, transverse spherical modes l = 5, 3, 1 and longitudinal l = 1). The two shortest wavelength modes have zero amplitude in this case.

We observed that convergence and speed in these calculations were adversely affected by having sharp edges and corners, due to the need to mesh such features finely. However, the calculations converged acceptably for geometries with p > 0.1. The dimensions assumed for the targets also exerted a strong effect on the results. Increasing the length red shifts the longitudinal dipole as expected but has little effect on the other modes. However, the 9108


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short wavelength modes are still not really captured unless the edges are sharper than reported. The resonances chiefly responsible for producing the four most prominent peaks in the simulations can also be analyzed by examining the nature of their associated electric fields as revealed in the fully retarded calculations performed using DDCAT. In this case, the effect of applying single polarizations of the electric field in either the longitudinal or transverse directions was examined so that the modes could be better separated (Figure 11). Here, for convenience, we have labeled

electric field distributions are shown in Figure 12. The La and Ta oscillations correspond closely to the longitudinal and

Figure 11. Decomposition of the calculated extinction spectrum of the standard shape (p = 0.52, 53 × 36 × 36 nm3, nmedium = 1.625) into extinction peaks generated by longitudinal and transverse polarizations of the light.

Figure 12. Electric field components and vectors for the six of the dipole resonances identified in Table 3, visualized on a plane through the midsection of the nanocuboid. The electric field lines are the projection of the 3D vectors onto the relevant plane. A plus sign shows a region of positive surface charge, and vice versa for a minus sign. The direction of propagation of the light is from top to bottom in the longitudinal simulations, and from left to right in the transverse simulations. The component of the field (x-left to right on page or y-vertical on page) which was color coded is shown in an annotation in the top right-hand corner of each diagram. The La resonance was captured at ϕ = 90° (where ϕ is the phase angle which varies from 0 to 360° as the incoming wave passes through the target), the Ta resonance was captured at ϕ = 90°, the Lb at 280°, the Tb at ϕ = 280°, the Tc at ϕ = 98°, and the Ld at 0°.

the extinction peaks in longitudinal polarization in increasing order of energy as La, Lb, Lc, and Ld and analogously for the transverse peaks. We caution, however, that no specific correlation of these extinction peaks with the numbered resonance modes of a cube or sphere (introduced earlier) is implied by this labeling system. (It does not necessarily follow, for example, that the Ld peak need correspond to any of the modes shown in Figure 1 and, indeed, any of the extinction peaks in Figure 6 or Figure 11 could in principle be the result of the superposition of more than one resonance mode.) For example, it is clear from the skewed shape that the peaks labeled Ta and Tb are produced by more than one resonance mode, however these were too close in frequency for them to be separated by visual examination of the electric field distributions. It would be interesting to explore this bifurcation of the transverse peaks further but the exercise lies outside of the scope of the present paper. The modes that could be identified are summarized in Table 3. Their corresponding

transverse dipole modes identified by the quasistatic analysis but are of course somewhat red-shifted.64 The Lb peak is produced by a quadrupole mode, in which each charged region of the dipole has bifurcated, so that there are four poles in total. This is not a radiant mode in the quasistatic limit so is not found by a purely quasistatic analysis, or indeed by a DDA calculation of a particle that is too small to induce appreciable retardation of the incident light (see Supporting Information, Figure S6). This peak was also found by Zhou et al.32 in silver

Table 3. Resonance Peaks Identified in Simulations of a Standard Nanocuboid (p = 0.52, 53 × 36 × 36 nm3, nmedium = 1.625)a nanocuboid resonance

free space λ at which calculated mode peaks, nm

polarization at which excited

correspondence to standard cube modes of Figure 1

suggested correspondence to peaks in experimental ensemble, nm

La Lb Lc Ld Ta Tb Tc

547 439 395 366 455 406 366

longitudinal longitudinal longitudinal longitudinal transverse transverse transverse

C1L (dipolar) no match C3L (multipolar) C4L (multipolar) C1T (dipolar) l = 3 sphere mode C5T (multipolar)

549 overlaps with Ta overlaps with Tb overlaps with Tc 447 400 346

a

For convenience, each one is designated with a letter of the alphabet in order of decreasing wavelength and with L or T to designate which polarization excited it. 9109


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resonances on these shapes. The gold core is shielded by the silver cuboid surrounding it and can be neglected in these calculations. Due to the combination of rounded corners and the particular aspect of the cuboids, there is a degree of hybridization of the resonance modes. Nevertheless, the calculations show that at least seven component resonances are likely to occur, with some overlap in wavelength, which produces the four resonances observed experimentally. The two strongest extinction peaks in the experimental ensemble are caused by the longitudinal and transverse dipole modes. The calculations suggest that there should also be a longitudinal quadupole mode (although this would be obscured by the dipolar transverse mode in colloidal suspensions), an l = 3 spherical-type transverse mode, and examples of higher order modes similar to the fundamental C3, C4, C5, and C6 modes of a cube.

cubes of 50 nm or bigger and corresponds to their quadrupole “Peak 2”. As was the case here, these workers found this peak to be missing in smaller targets. The transverse version of this quadrupole mode is obscured by the Ta oscillation but does produce a shoulder on the short wavelength side of the Ta peak. The Tb oscillation matched the spherical l = 3 mode in Figure 10c. The Tc mode matched the C5 mode of Figure 1 (see also Figure S7 in the Supporting Information). The Lc oscillation (Supporting Information Figure S8) is a C3L mode but the charge intensity on the eight cuboid corners is not particularly well developed due to their being rounded. The remaining oscillation (Ld) has the C4L symmetry. Identification of the higher order modes was facilitated and verified by examining a series of sections through the electric fields at different z values, and by stepping through animations of them. Snapshots of the electric field distributions at various times in the oscillation cycle are given in the Supporting Information as Figures S9 and S10. Animations of these modes have been uploaded to YouTube on the Internet (search on “plasmon cortie”). It can be seen from the diagrams of electric field distribution that, in contrast to the dipole oscillations, the multimode resonances are relatively nonradiating, because the close presence of oppositely charged regions on the sample surface pulls field lines in toward the surface of the target. Actually, strictly nonradiative multimode resonances in an isolated nanoparticle would normally not be excited by an incoming plane wave in a purely quasistatic analysis. Here, however, the finite size of the sample means that it takes the wave a reasonable fraction of the cycle to cross from one side of the target to the other (the socalled “retardation”). This breaks the symmetry of the waveparticle interaction and enables excitation of a multimode resonance. Note that symmetry can also be broken by placing the nanoparticle onto a dielectric substrate, for example, refs 31 and 34. The simulations also include information on the behavior of the resonances in the time domain. In the simulation of the multipolar Tb resonance, for example, the electric field at the various nodes is out of phase, due both to the multimodal nature of the resonances and the finite size effect as light traverses the target. (See Supporting Information Figure S11 for examples.) In the most general case, the optical spectra of orthorhombic nanocrystals could contain over 150 absorptive modes, but as symmetry is increased to that of a cubic crystal the resonances merge into up to six distinct modes.45 Spheres and ellipsoids on the other hand have relatively fewer resonance modes due to their greater symmetry. The tetragonal silver nanocuboids described here are an intermediate case, with at least seven resonances being clearly resolved in the simulations and four in the experimental data.

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ASSOCIATED CONTENT

S Supporting Information *

Plot of three dielectric functions for silver. Plot of corner radius of targets versus p parameter. Optical extinction of a Ag nanocube calculated using three dielectric functions. Graphs showing effect of various parameters on DDSCAT simulations. Histograms of dimensions of experimental particles. Graph showing effect of width of particle on resonances. Graphs showing effect of size on calculated optical extinction spectra. Additional charts of electric field distribution including animation sequences showing the electric fields at various moments in their cycles. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: +61-2-95142208. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.B.C. thanks Dr. Carlos Pecharromán Garciá of Instituto de Cienca de Materiales de Madrid, Spain, for useful insights on the optical resonances of cubes and the Australian Research Council for financial support. Y.N. thanks KAKENHI (Grantin-Aid for Scientific Research) on the Priority Area “Strong Photon−Molecule Coupling Fields (No. 470)”, Grant-in-Aid for Scientific Research (B) (No. 22350037), and a Grant-in-Aid for the Global COE Program, “Science for Future Molecular Systems” from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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CONCLUSIONS Although the multimode resonances in colloidal silver nanoparticles are usually obscured by peak broadening caused by a variation of particle sizes in the ensemble, they will become visible if the particle size and shape distribution is tight enough. A technique to achieve this with silver nanocuboids is described. The nucleation and growth of the cuboids is performed with the aid of a seed suspension of gold nanorods. Under the conditions described, it becomes possible to grow well formed and relatively similar nanoparticles. The experimental optical extinction spectrum of these cuboids reveals at least four pronounced extinction peaks. We have used both quasistatic and retarded electromagnetic analyses to analyze the possible

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