Localized Plasmons in Noble Metal Nanospheroids - The Journal of

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Localized Plasmons in Noble Metal Nanospheroids E. Stefan Kooij,* Waqqar Ahmed,† Harold J. W. Zandvliet, and Bene Poelsema Physics of Interfaces and Nanomaterials, MESAþ Institute for Nanotechnology, University of Twente, P.O. Box 217, NL-7500AE Enschede, The Netherlands

bS Supporting Information ABSTRACT: For the entire range of spheroidal noble metal nanoparticles, including spheres, rods, and disks, the optical properties are investigated as a function of their geometry and size by means of extinction spectra obtained by numerical methods. For spherical silver and gold particles, Mie theory is used to identify multipole plasmon resonances up to hexadecupole order as a function of particle radius. With increasing particle dimensions, higher-order multipole modes become more important. Moreover, the similarity in dielectric functions of silver and gold gives rise to almost identical optical properties at wavelengths exceeding 700 nm. For oblate and prolate nanoparticles, the dipole, quadrupole, and octupole plasmon resonances for aspect ratios up to 8 are considered. As with the nanospheres, extinction spectra for small nanodisks and nanorods are in agreement with the quasi-static approximation, whereas for larger dimensions, a red shift is observed. The plasmon peak positions are analyzed in terms of a material-independent factor that only depends on the geometry. We attempt to obtain a generalized description of the plasmon energies as a function of particle geometry and size. Finally, the size-dependent red shift is compared to surface plasmon dispersion relations and discussed in terms of surface curvature of the nanoparticle edge.

I. INTRODUCTION A. Plasmon Resonances in Nanoparticles. It is well known that the optical properties of noble metal nanoparticles are dominated by strong features in the extinction spectra.1,2 These plasmon resonances are generally in the visible and infrared range of the spectrum. Collective exctitations of the valence electrons by incident electromagnetic radiation give rise to charge density fluctuations at the nanoparticle surface, also referred to as surface plasmon polaritons.35 The wavelength position and strength of the extinction peaks in the optical spectra are very sensitive to the actual geometry of the nanoparticles.611 Many different shapes have been considered over the past decade. Complex branched shapes, such as nanostars1214 and nanoflowers,15,16 have recently received more research attention owing to their increasing availability and their potential application in surface-enchanced Raman scattering (SERS). The optical properties of these multibranched nanoparticles have been shown to be highly tunable over a wide spectral range by varying their precise morphology. With increasing branch length, the most pronounced plasmon resonance red shifts well into the near-infrared. The optical response has been discussed in terms of plasmon hybridization between the core and the branches of the star-shaped entities.12,14 Halas and co-workers17,18 have shown that noble metal nanoshells enable a wide tuning range of the plasmon resonances, even into the mid-infrared range. Changing the ratio of the inner and outer radii of the metallic shells gives rise to a large variation r 2011 American Chemical Society

of the localized plasmon resonances. Experimental results agree very well with theoretical work. The pronounced red shift has been discussed to a high degree of accuracy in terms of hybridization of plasmon modes of a sphere and a cavity, the latter with a slightly smaller radius. Anisotropically shaped nanoparticles with two or three different orthogonal dimensions exhibit multiple plasmon resonance modes. Oblate shapes, including nanoplatelets19 and nanodisks,20,21 are characterized by one short “thickness” and two more or less similar, larger “widths”. The two distinctly different dimensions give rise to transverse and longitudinal plasmon resonances, which strongly depend on the aspect ratio of the nanoparticles. Most pronounced is the red shift of the longitudinal plasmon resonance with increasing aspect ratio. Similarly, elongated metallic nanoparticles characterized by a relatively small diameter and a longer length possess two distinct plasmon resonance modes. Most work has been devoted to nanorods,11,2231 but more recently, also “nanorice”32 has been investigated in this respect. Both the transverse and the longitudinal modes of the prolate nanoparticles depend strongly on the aspect ratio. Especially, the longitudinal plasmon mode shows a strong red shift with increasing length, which is relevant for many different applications. Also, composite particles consisting of different constituent metals, as well as a large variety of coreshell nanoentities, have Received: November 17, 2010 Revised: April 22, 2011 Published: May 10, 2011 10321

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The Journal of Physical Chemistry C been synthesized combining the specific materials properties of the different composing elements.3336 Coupling of the (optical) properties of the composing elements enables design and manufacture of a completely new class of materials. Finally, two- and three-dimensional superstructures fabricated through self-assembly of nanoparticles have attracted considerable attention, owing to their often unique optical properties and potential applications.28,3743 For example, two-dimensional close-packed arrays of silver nanoparticles exhibit two distinct plasmon resonance modes, which can be ascribed to in-plane and out-of-plane collective resonances of the assembled nanospheres.38,41 The electromagnetic coupling of the response of neighboring nanoparticles is strongly dependent on the polarization of the incident light; light polarized in the plane of the 2D array gives rise to a red shifted collective mode, whereas a polarization perpendicular to the array plane leads to a blue shifted response.28,38,4143 B. Optical Response beyond the Quasi-Static Approximation. In most articles on optical properties of noble metal nanoparticles, the interaction with electromagnetic radiation is considered in terms of dipolar resonance modes.7,8,19,23,2731,4245 Moreover, frequently the quasi-static approximation is employed owing to its mathematical simplicity, therewith enabling straightforward analysis of the relatively simple equations.1,2 An introductory overview of the quasi-static approximation for oblate and prolate nanoparticles is provided in the Supporting Information. For nonspherical nanoparticles having a high degree of symmetry, such as rods and disks, the quasi-static dipole approximation has proven to reproduce the trends observed experimentally.68,23,44 The plasmon resonance of these anisotropic nanoparticles exhibits a splitting into two different modes depending on the precise geometry. Unfortunately, the quasistatic approximation is only valid for particles considerably smaller than the wavelength of the incident light. Most often, this is not the case for one or more dimensions of the particles. More recently, the progress toward well-defined and controlled synthesis procedures allows production of almost monodisperse nanoparticle suspensions. Combined with the availability of extensive computational power allowing simulation of increasingly larger particles, this has led to an extension of the field to include investigation of the optical characteristics of much larger particles, including multipolar resonances.22,2426,32,4649 For spherical particles, Mie theory has been implemented to easily determine accurate absorption and scattering spectra. For anisotropic nanoparticles, experimental spectra reveal multiple resonances that are sensitive for both the aspect ratio as well as the absolute dimension. On the basis of numerical simulations, using, for example, the discrete dipole approximation (DDA), the finite difference time domain (FDTD) method, the modified long wavelength approximation (MLWA), or the T-matrix method, the different multipolar resonance modes have been identified. In recent years, near-field optical microscopy and high-resolution cathodoluminescence spectroscopy have been used to spatially resolve the various resonance modes.5052 Most work has been focused on the plasmon resonances in nanorods, owing to their possible application as nanoantennas, for example, as light-harvesting entities. C. Outline of the Paper. Here, we present a comprehensive overview of the plasmon resonance modes in the optical properties of spheroidal nanoparticles, that is, spherical and rod- and disk-shaped particles. As far as we are aware, such a detailed comparison between prolate and oblate nanoparticles is not

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Figure 1. Imaginary (top) and real (bottom) parts of the dielectric functions of gold and silver (solid and dashed lines, respectively) as a function of wavelength in air.

available. Optical extinction spectra for varying geometries are obtained from exact Mie calculations (nanospheres) and using the DDA (anisotropic nanoparticles). Phase retardation as well as multipole resonance modes are considered, which both arise from the fact that one or more particle dimensions are comparable to or larger than the wavelength of the light. Furthermore, we attempt to analyze the simulation results in terms of optical dispersion relations. We also discuss the effect of radius of curvature on the plasmon resonance. To provide some more insight into the effect of particle shape, we not only consider ellipsoidal nanooblates and nanoprolates but also analyze the intermediate geometry comprising particles with three different axes. One major advantage of the DDA is that calculating the optical response of such geometries is straightforward. The results will ultimately serve as a benchmark for experimental results, as well as provide design rules for novel applications including these nanoscale building blocks.

II. CALCULATION OF OPTICAL PROPERTIES For all calculations presented in this work, we used the bulk dielectric functions, εAu and εAg, for gold and silver, respectively, as collected and tabulated by Palik;53 plots of the real and imaginary parts of the dielectric functions are shown in Figure 1. At short wavelengths, the interband transitions are most pronounced in the ε1 spectra below 300 and 500 nm for silver and gold, respectively. At longer wavelengths, the contributions from the free s electrons dominate the spectra. Recently, Ungureanu et al.54 performed a systematic analysis of the effect of different tabulated dielectric functions on calculated extinction spectra. They concluded that minor differences are observed and that generally good agreement with experimental data is obtained. Because we only investigate the plasmon resonance peak position, size effects on the dielectric function, that is, surface scattering, do not play a role and are not taken into account.23 In all spectra, the wavelength λ is taken to be that in air/vacuum. A. Spherical Nanoparticles: Mie Theory. Gustaf Mie provided the first theoretical description of the optical properties of spherical particles.55 Scattering and extinction efficiencies Qsca and Qext in terms of the size parameter x = ka, where a is the sphere radius, are obtained from an exact solution of Maxwell’s equations in spherical coordinates. The wave vector k = 2π(εm)1/2/λ is 10322

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The Journal of Physical Chemistry C

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Figure 3. Wavelength position (a) and extinction efficiency (b) of the dipole (l = 1, squares), quadrupole (l = 2, circles), octupole (l = 3, triangles), and hexadecupole (l = 4, diamonds) plasmon resonance modes for silver (open symbols) and gold (closed symbols) as a function of nanosphere radius.

Figure 2. Extinction efficiency spectra as a function of radius calculated using Mie theory for spherical nanoparticles dispersed in water. The gray scale for silver (a) nanospheres corresponds to values in the range of 011 (black-white), whereas for gold (b), the gray scale corresponds to values between 0 and 8. The dashed lines indicate the peak position of the different resonance modes as a function of the radius of the particles.

related to the wavelength λ/(εm)1/2 in the medium surrounding the particles, that is, water in the present case. The absorption efficiency follows from energy conservation, that is, Qabs = Qext Qsca. The efficiencies Q are equal to the corresponding cross sections, σ, normalized to the effective particle cross section, πa2eff, with aeff the effective particle radius. For calculation of the Mie coefficients and efficiencies, the Matlab routines by M€atzler56,57 are used. B. Anisotropic Nanorods and Nanodisks: Discrete Dipole Approximation. Generally, analytical expressions describing the optical response of (nano)particles with arbitrary size and shape do not exist. The discrete dipole approximation (DDA) is a numerical technique that enables calculation of the optical absorption and scattering of targets, that is, (nano)particles of arbitrary geometry and orientation.58 The DDA is, in fact, an approximation of the continuum target by an array of N polarizable points on a square lattice. In response to the local electric fields, these points acquire a dipole moment. The local field experienced by each individual dipole is constituted by both the external field arising from the incident illumination and the internal field generated by all other dipoles. To solve this problem of 3N-complex linear equations in the DDA, we use the software package DDSCAT.7.0 by Draine and Flatau,59 which is freely available.60 For details on the DDA, the software package, and/or its implementation, the reader is referred to the available literature, for example, refs 6, 7, and 61. The number of dipoles N within an array, required to provide an accurate description of the continuum macroscopic target, is

subject to optimization. On one hand, N should be large to prevent spurious surface effects.23,58,61 Surface dipoles of the target generally give rise to an overestimation of the extinction cross section. The ratio of surface to bulk dipoles decreases with increasing N, which would suggest taking a very large number of dipoles.23 On the other hand, however,58 computational time increases approximately proportional to N3, therewith imposing an upper limit to N. More details on the accuracy and number of dipoles needed to ensure convergence of the results are provided in the Supporting Information. The wavelength spacing in all spectra obtained using the DDA amounts to 10 nm.

III. NANOSPHERES The variation of optical extinction spectra for silver and gold spherical particles is shown in Figure 2. For both types of particles, only a single resonance is observed in the limit of small radii; for silver, the plasmon resonance is near 400 nm, whereas for gold, it is at 520 nm. This single extinction peak for the smallest particles corresponds to the dipolar mode. With increasing size of the nanoparticles, a number of changes are observed in the spectra. The dipolar resonance starts to shift toward longer wavelengths due to phase retardation. The electromagnetic wave incident on the particles can no longer be considered to have a negligible phase difference across the particle dimension. Moreover, for the smallest particle dimensions, the extinction is dominated by absorption, while scattering is negligible. With increasing particle size, scattering becomes more pronounced. Most important is the fact that a significant shift of the dipolar plasmon resonance already occurs for radii below 50 nm. In many cases reported in the literature, erroneously, the quasi-static approximation is considered to be valid in this range of particle dimensions. For considerably larger particles, the dipolar resonance shifts further toward the infrared, but also a second extinction feature develops. For silver, this occurs for particle radii above 50 nm, 10323



Figure 4. (top) Schematic representation of the charge density distribution for the lowest four multipole plasmon resonance modes in spherical nanoparticles. (bottom) Dispersion relation for localized plasmon resonance modes in silver (open symbols) and gold (closed symbols) nanospheres, obtained from the data in Figure 3. The solid line represents the light line in water, whereas the dashed lines are calculated dispersion curves, as described in the text.

whereas for gold, this can only be discerned for radii exceeding approximately 75 nm. This second extinction peak is due to the quadrupole plasmon resonance within the noble metal nanoparticles. As with the dipole resonance, initially, the quadrupole resonance is dominated by light absorption, whereas for larger particles, scattering becomes more dominant, which is accompanied by a shift to longer wavelengths. For even larger particles, higher-order plasmon resonances develop, which all shift toward the infrared with increasing particle dimensions. The magnitude of the multipole plasmon resonance, as well as their spectral width, is different for gold and silver; for the latter, the peaks are more narrow and higher as compared with those of gold (note the different gray scales in Figure 2). Despite the similar electronic configuration of the two materials, this is due to the fact that the interband transitions in silver are located at considerably shorter wavelengths, thereby interfering less with the behavior of the free electrons involved in the plasmon resonances. The shift of the multipole plasmons as a function of particle size for gold and silver is compared in Figure 3. The dipole, quadrupole, octupole, and hexadecupole resonance modes are indicated by l = 1, l = 2, l = 3, and l = 4, respectively. As already mentioned, the dipolar plasmons for silver and gold in the small particle limit are located near 400 and 520 nm. With increasing particle size, these resonances shift to longer wavelengths, while their maximum extinction efficiency decreases after a maximum. For larger particles, higher-order multipole resonances appear, which also shift toward the infrared and exhibit a decrease of the extinction efficiency. More importantly, the wavelength position of the plasmon resonances for gold and silver becomes very similar at longer wavelengths, as can be seen in Figure 3. At wavelengths exceeding 700 nm, the position (and strength) of the various resonances is almost identical for gold and silver. The fact that the localized surface plasmon contribution to the optical response is similar for gold and silver arises from the

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similarities in electronic structure. The dielectric function above 700 nm (see also Figure 1) is dominated by Drude-like free electron behavior. Relevant parameters are the density of s electrons and their relaxation rate, which are nearly the same. Interband transitions, which are at different energy levels for both materials, do not play a major role. As such, the dielectric functions of gold and silver are almost identical when approaching the infrared. As already described in the Introduction, surface plasmon polaritons give rise to charge density waves at the metaldielectric interface. For nanoparticles, these modes are localized and have to “fit” on the circumference of the nanospheres. For the modes considered in Figure 3, the charge density on the sphere surface is schematically represented in the top part of Figure 4. For the dipole resonance, all electrons collectively respond to the electromagnetic radiation, giving rise to excess and depletion of electrons at opposite sides of the nanoparticle. For higher-order resonances, the charge density fluctuations are different, as indicated in Figure 4. Considering that the resonance mode has to “fit” the particle circumference, the wave vector q can directly be derived from the particle radius a and the resonance mode number l: q¼

2π l ¼ 2πa=l a

ð1Þ

In Figure 4, the data of Figure 3 are replotted; now the peak energy62 is plotted versus plasmon wave vector. For gold (closed symbols), the data collapse to a single curve, whereas for silver (open symbols), a similar scaling is observed. For the latter, the scaling is not optimal, but for both types of nanospheres, the plasmon peaks at lower energy, that is, longer wavelengths, follow the light line in water (shown by the solid line). We can compare the trends in Figure 4 for both materials to the theoretical dispersion curve for goldwater and silverwater interfaces, similar to previous work by Schider et al.46 The dispersion curve is given by E R ðεÞεm 1=2 q¼ ð2Þ cp R ðεÞ þ εm where εm is the refractive index of the dielectric, that is, water in the present case, and ε is the metal dielectric function.3 For large wave vectors, that is, short plasmon wavelengths, the dispersion curves (dashed lines in Figure 4) saturate at an energy value corresponding to the localized plasmon resonance energy for nanoparticles in the quasi-static approximation. At small wave vectors, the dispersion curves approach the light line. For both silver and gold, the simulated plasmon resonances agree nicely with the dispersion relation.

IV. ANISOTROPIC NANOPARTICLES A. Oblate Nanodisks. Optical extinction spectra for diskshaped gold and silver nanoparticles for aspect ratios of 3 and 7 are shown in Figure 5 for various sizes. Spectra were obtained for randomly oriented particles. The optical response of the nanodisks exhibits two distinct plasmon resonance peaks corresponding to one transverse mode (short axis) and two degenerate longitudinal modes (long axes). In Figure 5, only the evolution of the longitudinal peaks with size is depicted. For all oblate nanoparticles, the smallest particles have a short diameter of 5 nm, corresponding to a short radius of 2.5 nm. In all cases, the longitudinal resonance shifts toward longer wavelengths with increasing size. Comparing the extinction 10324



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Figure 5. Extinction spectra for silver (a, b) and gold (c, d) oblate nanoparticles. Simulation results for aspect ratios of 3 (a, c) and 7 (b, d) are obtained for randomly oriented nanodisks. The dimensions of the disk-shaped nanoparticles are varied; spectra are shown for nanodisks with a short radius of 2.5, 5.0, 7.5 nm, etc.

Figure 6. Wavelength position of the extinction peaks in Figure 5 corresponding to the longitudinal plasmon resonance modes for gold (closed symbols) and silver (open symbols) nanodisks with an aspect ratio of 3 (a) and an aspect ratio of 7 (b), as a function of their short radius. Peak positions for the dipole (l = 1, squares) and quadrupole (l = 2, circles) resonance modes are shown.

efficiency values for the aspect ratio 3 and 7 nanodisks (left and right panels in Figure 5, respectively), the latter are obviously higher owing to the larger volume of the nanoparticles. Moreover, the red shift with increasing size is much larger for the high aspect ratio disks. The silver (top panels) and gold (bottom panels) oblates exhibit very similar behavior, although the plasmon peaks for the gold nanodisks are located at longer wavelengths as compared with those for silver. Another difference between both metals pertains to the extinction efficiency values for the smallest nanodisks. For silver, the peak values amount to 4.5 and 11, whereas for gold, these

values are 2 and 9, for aspect ratios of 3 and 7, respectively. The difference arises from the fact that interband transitions in silver are at considerably higher energies and, as such, do not interfere with the silver plasmon peaks, whereas for gold, the interband transitions lead to a slight damping of the resonance maxima. Furthermore, for larger dimensions of the oblate particles, the spectra for silver and gold are almost identical, owing to the similarity in the dielectric functions of both materials at low energies, that is, at long wavelengths. Above, we only focused on the most dominant peak in the spectra corresponding to the longitudinal dipole l = 1 plasmon mode. With increasing dimensions of the nanodisks, not only do the spectra in Figure 5 exhibit a red shift but also a second peak develops. This extinction maximum is due to the quadrupole l = 2 resonance, similar to that described for spherical nanoparticles in the previous section. With increasing size, the quadrupole peak also shifts toward longer wavelengths. The wavelength positions of the plasmon resonance peaks in Figure 5 are summarized in Figure 6a,b for aspect ratios of 3 and 7, respectively. The peak positions are plotted as a function of the short radius a of the nanodisks. Both the dipolar (l = 1, squares) and the quadrupolar (l = 2, circles) resonance mode maxima are shown for silver and gold (open and closed symbols, respectively). From the results in Figure 6, the aforementioned observations are apparent. For small dimensions, the resonances are near the quasi-static values (see also the Supporting Information, Figure S2), whereas for the largest dimensions, an obvious red shift is observed. Moreover, the similarity of the dielectric functions of silver and gold at wavelengths above approximately 700 nm gives rise to nearly identical plasmon peak positions. When the dipole l = 1 peaks are compared with the quadrupole l = 2 peaks, the former exhibit a stronger increase with size, similar to the observations for nanospheres in Figure 3. Additionally, as can be expected, the plasmon resonances for the oblate particles with an aspect ratio of 7 show a much stronger increase than those for an aspect ratio of 3. 10325



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Figure 7. Resonance factor f as a function of the squared short radius a of silver (open symbols) and gold (closed symbols) nanodisks of varying aspect ratios as indicated. Results for the dipole mode with l = 1 (a) and the quadrupole mode with l = 2 (b) are shown.

As mentioned in the Introduction, the quasi-static approximation derived from classical electrodynamics has provided a valuable tool to analyze the optical response of spheroidal particles in the limit of infinitely small dimensions.2,23 As first described by Encina and Coronado,24 from the quasi-static approximation, a resonance condition can be derived in terms of the wavelength (or energy) at which the plasmon resonance can be expected. Considering the dielectric functions of the particle ε and the surrounding medium εm, the resonance condition is expressed by ε ¼ f εm

ð3Þ

The factor f is a geometrical factor that only depends on the nanoparticle particle size and shape, that is, the aspect ratio of prolate or oblate nanoentities. For spheres, the factor f is equal to 2, but for prolate and oblate spheroids, different f values are obtained. More generally, for the dipolar plasmon modes, the geometrical factor f is related to the depolarization factor L through f = (1 L)/L; for spherical particles, L = 1/3. In the case where the incident light is polarized along the major axis, therewith inducing a longitudinal resonance mode, f is larger than 2 and increases with the aspect ratio. Consequently, because the dielectric function of noble metals becomes more negative with increasing wavelength, a red shift of the longitudinal plasmon resonance is expected for these higher aspect ratio particles with larger f values. In other words, the factor f represents a material-independent measure for the red shift. To further analyze the results for nanodisks, as those shown in Figure 6, we use eq 3 to express the plasmon peak positions in terms of the resonance factor f. A detailed description of the procedure is presented by Encina and Coronado.24 The results for the dipole and quadrupole plasmon modes for all aspect ratios are shown in Figure 7. The f values are plotted as a function of the square of the short radii a of the nanodisks, as this yields approximately linear relations. As expected, all values for f in Figure 7 are larger than 2. Moreover, very similar results are obtained for silver and gold, which is most pronounced for the dipole resonance. For the quadrupole plasmon, there is a larger discrepancy between silver and gold. Also, the red shift with aspect ratio as well as size is

Figure 8. Dipole (l = 1) and quadrupole (l = 2) plasmon resonance energies for silver and gold (open and closed symbols, respectively) nanodisks as a function of wave vector, as described in the text. Results for different particle short radii a, as indicated in the figure, are shown, while the aspect ratio is varied, increasing from right to left, as indicated by the arrow; data points for the same short radius and plasmon resonance order are connected by solid lines. The surface plasmon dispersion relation of silver and gold (eq 2) is represented by the dashed and dotted lines, respectively. The straight solid line corresponds to the light line in water.

expressed by increasing f values. Further analysis of the resonance factors of oblate (and prolate) nanoparticles will be described in the Discussion section. Similar to what we have done in the previous section for nanospheres, we can relate the plasmon energy (or wavelength) for every nanodisk to a wave vector as determined from the geometry of the oblate particle. For this purpose, we have to consider the perimeter of the particle as “felt” by the longitudinal resonance, that is, an ellipse defined by the short and long axes of the nanodisk (see the Supporting Information). We assume that the dipolar plasmon mode wavelength “fits” the circumference of the nanodisk. For the quadrupole mode, the plasmon wavelength is equal to half of the circumference. The results for different sizes and varying aspect ratios are depicted in Figure 8. Considering, first, nanodisks with a fixed short radius (for example a = 10 nm, the up/down triangles in Figure 8), variation of the aspect ratio corresponds to a change of the long diameter of the disk-shaped particles. When the results are compared with the surface plasmon dispersion relation described by eq 2, all data points are below and/or to the right of the dispersion curve. Also, with increasing length (or aspect ratio; direction of the arrow in Figure 8), the plasmon resonances approach the light line. In agreement with the quasi-static approximation, small particle dimensions are far right of the light line, approaching the saturation value of the plasmon resonance position for spherical nanoparticles. From the results in Figure 8, it may also be deduced that, for a specific resonance mode (dipole or quadrupole), the resonance wave vector moves to smaller values for larger values of the short diameter. Alternatively, one may consider the radius of curvature of the nanodisks to increase, giving rise to resonance energies closer to the surface plasmon dispersion relation. Moreover, comparing the results for the dipole and quadrupole modes, it is obvious that the latter l = 2 modes are at larger wave vectors. A more detailed discussion will follow in the next section. B. Prolate Nanorods. Similar to the case of the nanodisks in the previous sections, we also performed detailed simulations of the optical properties of gold and silver nanorods. The resulting extinction spectra for aspect ratio 3 and 7 nanorods as a function 10326



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Figure 9. Extinction spectra for silver (a, b) and gold (c, d) prolate nanoparticles. Simulation results for aspect ratios of 3 (a, c) and 7 (b, d) are shown; calculations are performed for randomly oriented silver nanorods, whereas the gold nanorods had a fixed orientation of 30° with respect to the polarization direction of the incident light. The dimensions of the rod-shaped nanoparticles are varied; spectra are shown for nanorods with a short radius of 2.5, 5.0, 7.5 nm, etc.

of size are shown in Figure 9. In Figure 10, the plasmon peak positions for the dipolar (l = 1), quadrupolar (l = 2), and octupolar (l = 3) longitudinal resonance modes are plotted as a function of the short radius a of the nanorods. The spectra for both silver (Figure 9a,b) and gold (Figure 9c,d) nanorods reveal the double degenerate transverse resonance at approximately 360 and 500 nm for the two materials, respectively. In addition, several longitudinal peaks are observed. With increasing dimension of the prolate nanoparticles, all resonance peaks exhibit a pronounced red shift. As expected and similar to the case of the nanodisks, the shift to longer wavelengths with size is more pronounced for the higher aspect ratio nanorods. Moreover, with increasing dimension of the nanorods, the higher-order multipole resonances become increasingly more important. Owing to the small shift of the transverse peaks, as a result of considerably smaller dimensions in this direction, only the dipole plasmon is visible. Spectra for silver nanorods were simulated for randomly oriented nanorods. However, for gold, the orientation of the nanorods was fixed with the long axis oriented 30° with respect to the polarization direction of the incident light. This enables a more accurate determination of both the even and the odd resonance longitudinal modes. For rods oriented perpendicular to the E-field of the electromagnetic radiation, only the transverse mode is excited and the longitudinal modes are absent. When the long axis of the prolate nanoparticles coincides with the polarization direction, the transverse mode is absent. For symmetry reasons, only the odd modes (l = 1, l = 3) are excited in this case; the even modes (l = 2 in this case) are not excited. For gold nanorods, an orientation angle of 30° appears to be the optimal angle, exciting both even and odd longitudinal modes as well as the transverse dipole mode. The peak positions of the prolate nanorods are at markedly larger wavelengths as compared with their oblate counterparts. A similar conclusion can also be drawn in the quasi-static approximation in the limit of small particle dimensions (see also Figure S2 in the Supporting Information). Moreover, when the results in Figures 9

Figure 10. Wavelength position of the extinction peaks in Figure 9 corresponding to the longitudinal plasmon resonance modes for gold (closed symbols) and silver (open symbols) nanorods with an aspect ratio of 3 (a) and an aspect ratio of 7 (b), as a function of their short radius. Peak positions for the dipole (l = 1, squares), quadrupole (l = 2, circles), and octupole (l = 3, triangles) resonance modes are shown.

and 10 for silver and gold are compared, very similar results are obtained at wavelengths exceeding approximately 700 nm. Both the height and the position of the peaks are nearly identical. These observations are a result of the similarity of the silver and gold dielectric functions at larger wavelengths, as described in the previous section. The red shift of the longitudinal plasmon peaks is much larger as with the nanodisks. This is even more obvious when the peak positions, such as those in Figure 10, are expressed in terms of the 10327



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Figure 12. Dipole (l = 1), quadrupole (l = 2), and octupole (l = 3) plasmon resonance energies for silver and gold (open and closed symbols, respectively) nanorods as a function of wave vector, as described in the text. Results for different particle short radii a, as indicated in the figure, are shown, while the aspect ratio is varied, increasing from right to left, as indicated by the arrow; data points for the same short radius and plasmon resonance order are connected by solid lines. The surface plasmon dispersion relation of silver and gold (eq 2) is represented by the dashed and dotted lines, respectively. The straight solid line corresponds to the light line in water. Figure 11. Resonance factor f as a function of the squared short radius a of silver (open symbols) and gold (closed symbols) nanorods of varying aspect ratios as indicated. Results for the dipole mode with l = 1 (a), the quadrupole mode with l = 2 (b), and the octupole mode with l = 3 (c) are shown.

resonance factor f (eq 3). For all aspect ratios in the range of 28, the resulting f values for dipole, quadrupole, and octupole modes are shown in Figure 11. Comparing gold and silver (closed and open symbols, respectively), it is confirmed that, for both materials, approximately the same results are obtained. This is most pronounced for the dipolar plasmon resonance (l = 1 in Figure 11a). This confirms that f is indeed a material-independent parameter describing the plasmon red shift. Again, the f values seem to exhibit a linear behavior when plotted as a function of the square of the short radius a of the nanorods. This is similar to the observations in Figure 7 for oblate nanodisks. However, a difference lies in the fact that the larger red shift for nanorods (Figure 9) gives rise to considerably higher values for f. A further analysis of the resonance factor in terms of particle dimensions is described and discussed in the next section. Finally, as with the nanospheres and nanodisks (see Figures 4 and 8, respectively) the energy of the plasmon resonances can be compared to the surface plasmon dispersion curve for the respective materials. The results are shown in Figure 12. Similar to the case of the nanodisks in Figure 8, all data points are to the right and/or below the bulk surface plasmon dispersion curves for silver and gold (indicated by the dashed and dotted lines, respectively). Generally, the plasmon energies are at lower values as compared with those of their oblate counterparts. Nevertheless, similar trends are observed. Considering nanorods with a fixed short diameter (these are represented in Figure 12 by the connected symbols), an increase of the aspect ratio corresponds to an increase of the nanorod length. For longer nanorods with a fixed short axis, the plasmon resonances shift more toward the light line (indicated by the arrow). The same trend is observed for nanorods with larger absolute dimensions. Comparing the various multipole resonances, it is observed that the lowest-order plasmon modes are closest to the

plasmon dispersion curve (compare, for example, the nanorods with a = 25 nm in Figure 12), again similar to the case for the oblate nanodisks in the previous section.

V. DISCUSSION A. Geometric Resonance Factor. In the previous sections, the optical response of oblate and prolate nanoparticles was investigated using simulated extinction spectra. The red shift of the longitudinal plasmon resonance peaks was analyzed using a geometrical parameter f, which does not depend on the specific material properties but only on the size and shape of the nanodisks and nanorods. From the plots in Figures 7 and 11, it was found that, for all aspect ratios, the f values are proportional to the square a2 of the short radii of the nanoparticles

f ¼ Aðl; ηÞ þ a2 Bðl; ηÞ

ð4Þ

where A(l, η) and B(l, η) are parameters, which, in principle, only depend on resonance mode (l = 1, 2, 3) and aspect ratio η. We fitted all the data in Figures 7 and 11 to eq 4. In Figure 13, the resulting values for both parameters are plotted on double logarithmic scales. As already described, the results in Figure 13 confirm the similarities between silver and gold particles. Moreover, the linear behavior observed in the loglog plots (solid lines in Figure 13 are a guide to the eye) indicate that all values for A and B seem to depend exponentially on the aspect ratio η. This seems to hold for all multipole resonance modes considered in this work. The results in this work pertain to the lowest three resonance modes and for limited aspect ratios in the range of η = 29. Moreover, we consider relatively small nanoparticles, with short radii up to 35 and 45 nm for oblate and prolate nanoparticles, respectively, and long radii up to 125 and 200 nm. When the behavior of the parameter B for oblate and prolate nanoparticles is compared, very similar behavior is observed. For both types of particles, B is proportional to the aspect ratio squared, that is, B µ η2. The proportionality constants are summarized in Table 1. The behavior for gold and silver nanodisks is very similar, as is also the case for nanorods of gold and silver. The B parameters for the dipolar modes of both types of particles are very similar. The quadrupolar resonance values 10328



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Figure 13. Parameters A(l, η) (a, b) and B(l, η) (c, d) as a function of nanoparticle aspect ratio, describing the linear relation between the resonance factor f and the squared short radius a2 as expressed by eq 4 (see also Figures 7 and 11). Results are shown for oblate (a, c) and prolate (b, d) nanospheroids exhibiting dipole (l = 1, squares), quadrupole (l = 2, circles), and octupole (l = 3, triangles) resonance modes. Open and closed symbols represent data for silver and gold, respectively. Solid lines are a guide to the eye, indicating power law behavior with slope = 1 (a), slope = 1.5 (b), or slope = 2 (c, d).

Table 1. Summary of the Exponential Aspect Ratio Dependence of the Fit Parameters A(l, η) and B(l, η) As Defined by eq 4 gold (Au)

A

oblate

prolate

oblate

prolate

µ η1

µ η1.5

µ η1

µ η1.5

l=1

1.42

1.71

1.47

1.69

l=2

0.954

0.698

0.976

0.699

l=3

0.400

l=1 l=2 l=3

0.403

µη

µη

2

B (nm2)

silver (Ag)

µη

2

2

1.13 3 103

0.97 3 103

1.11 3 103

3.00 3 10

2.08 3 10

3.45 10

4

4 5

8.46 3 10

4

µ η2 1.05 3 103 2.28 3 104 9.81 3 105

are of the same order of magnitude, but somewhat lower in the case of nanoprolates. So far, we have no explanation for this difference yet. The parameter A (also listed in Table 1) shows distinctively different behavior when comparing the nanodisks (Figure 13a) to the nanorods (Figure 13b). For the oblate nanoparticles, we obtain a linear relationship, that is, A µ η, whereas for the nanorods, we find that A µ η1.5. Again, there is a strong similarity between the proportionality constants for silver and gold in the case of the nanodisks. A similar comparison holds for the nanorods in the case of silver and gold. As with the proportionality constants for the parameter B mentioned above, we cannot compare the results for A of oblates and prolates because their dependence on the aspect ratio η is markedly different. In previous reports, the plasmon resonance modes in prolate nanoparticles has been analyzed in terms of

standing waves on a nanoantenna. In these cases, it was shown that the different multipole resonance modes were scalable by considering that the l = 1 dipole mode corresponds to half a wavelength on the length of the nanorods, whereas for the l = 2 quadrupole mode, the length of the nanoantenna defines a full wavelength.2426,45 Our results do not confirm this scaling, most likely owing to the relatively low aspect ratios considered in this work. B. Comparison to Previous Work. In two recent papers, a similar approach, as we have presented here, was used to analyze the optical properties of rod-shaped nanoparticles. Schmucker et al.45 performed a detailed comparison between experimental spectra of relatively large nanorods, electrochemically manufactured within an anodic aluminum oxide template, and theoretical calculations using the DDA, MLWA, and the quasi-static approximation. Encina and Coronada24 carried out an extensive DDA study of long nanorods. In both reports, a cylindrical nanorod shape was considered. From the results, an expression for the geometrical factor f was derived, from which the longitudinal plasmon resonance peaks of rod-shaped (nano)particles could be predicted. We can compare our results for nanorods, represented by eq 4 and the parameters in Table 1, to the results presented in these reports. The results of Schmucker et al.45 prove to be substantially different from the analysis by Encina and Coronado.24 The discrepancies are ascribed to the different size ranges considered in the different studies. We attempted to reproduce the analysis of Schmucker et al. but obtained markedly different results with typical f values a factor of 23 larger as compared with those of the Encina and Coronado work. In fact, this is also mentioned in their Supporting Information. We ascribe the differences to a numerical error in the model as described. Owing to the similar dimensions of the nanoparticles considered in our present work and that of Encina and Coronado,24 we compare the two models in Figure 14 for different particle 10329



Figure 14. Comparison of f values of nanorods developed in this work (eq 4 and Table 1) to the results as derived by Encina and Coronado.24 Results from the present work (solid lines) and those of Encina and Coronado (dashed lines) are plotted as a function of nanorod length for three values of the rod diameter, as indicated in the panels. The dipole, quadrupole, and cotupole resonances are considered. The f values are compared for aspect ratios up to 20.

diameters and the lowest three resonance modes. Although there are differences, the range of f values in both models is comparable. Deviations are observed at large aspect ratios, typically larger than 10. These values were not considered in our work; the results for lower aspect ratios agree reasonably well. Also, for small aspect ratios, the two models differ slightly. The former is not very obvious from the curves in Figure 14. For the smallest particles (bottom panel in Figure 14), the discrepancies become more pronounced. Both aforementioned deviations can be accounted for by the fact that, in our analysis, we used exponential relations, which become relatively insensitive in the limit of small particle dimensions. Moreover, Encina and Coronado modeled the nanorod shape using a cylindrical target, whereas we used an ellipsoidal shape. As has been shown, ellipsoidal particles exhibit longitudinal resonance peaks at somewhat shorter wavelengths as compared with cylindrical particles with the same aspect ratio.23,54 C. Radius of Curvature. The origin of the different exponential behavior for prolate and oblate nanoparticles is not obvious. Qualitatively, it can be rationalized considering that the lateral dimensions of nanorods and nanodisks with identical aspect ratios are different. In other words, the radius of curvature at the outer ends of the nanoparticles, as compared to the polarization direction of the incident light, seems to have a pronounced influence. This is wellknown from the literature, where the field enhancement at noble metal extremities depends strongly on the curvature.3 For sharper features, the field enhancement, and therewith the wavelength shift of the resonance modes, is more pronounced. We drew a similar conclusion based on the representation of the nanodisk and nanorod plasmon resonances in Figures 8 and 12. With increasing radius of curvature, the plasmon resonance energies

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Figure 15. Evolution of extinction spectra for gold nanoparticles with long axes of 140 (a) and 35 nm (b), oriented at an angle of 45° with respect to the polarization direction of the light. One short axis is fixed at 20 and 5 nm, respectively, while the third axis is varied, inducing a change from prolate to oblate shape of the nanoparticles. (c) Shift of the longitudinal resonance peaks as a function of ratio of the axis, which is varied to that of the short axis. In other words, a prolate particle has a ratio of 1, whereas the oblate shape corresponds to a ratio of 7.

approach the bulk surface plasmon dispersion relation. Recently, Zhu described a similar dependence of the plasmon resonances on the particle curvature using the quasi-static approximation for nanospheroids and nanoshells.44 The simulations, such as we have presented in the previous section, enable a more elaborate discussion, because we consider particles of finite dimensions. Simulations enable a simple change of geometry of the nanoparticle, which is often hard to realize by simple synthesis procedures. In Figure 15, extinction spectra are shown for gold nanoparticles of which the shape is changed from prolate to oblate. The overall aspect ratio of all particles amounts to 7. Starting with the prolate nanorods, spectra for particles with a long axis of 140 (a) and 35 nm (b) are shown. The small nanorods exhibit only a pronounced dipole resonance peak, whereas for the larger nanorods, both the dipolar (l = 1) and the quadrupole (l = 2) plasmon peaks are observed. The transition toward an oblate shape is achieved by increasing one of the short axes of the prolate nanoparticles. In Figure 15, spectra are shown with increasing size of the aforementioned short radius, until finally a nanodisk is “formed”. With the transition from prolate to oblate, all resonance peaks shift toward shorter wavelengths. In fact, this is expected based on the extinction spectra in Figures 5 and 9, where plasmon peaks for nanodisks are also observed at shorter wavelengths as compared with those of nanorods. The shift of the plasmon peaks is most apparent from Figure 15c, in which the peak positions are plotted versus the ratio of the short axis, which is varied, and the longest axis. These results are in good qualitative agreement with the aforementioned importance of the radius of curvature. For prolate nanoparticles with two short axes, the curvature is strongest; that is, the overall radius of curvature is smallest. Upon going from a prolate to an oblate shape, the radius of curvature increases in one 10330



Figure 16. Evolution of extinction spectra for gold nanoparticles with a long axis of 35 nm (a), oriented at an angle of 45° with respect to the polarization direction of the light. The oblate shape of the particle is varied toward a sphere by varying the short axis from 5 to 35 nm. (b) Shift of the longitudinal resonance peak as a function of the length of the short radius.

direction, giving rise to a blue shift of the plasmon peaks. For the smallest nanospheroids (closed symbols in Figure 15c), the magnitude of the decrease of the plasmon peak wavelength is more pronounced as compared with that of the large particles. This is expected because the change in curvature is most pronounced for the smallest nanoparticles. Similar to the transition from a nanorod to a nanodisk, we can simulate the response during a transition from disk to sphere. The result of these simulations is summarized in Figure 16. Now we start with a nanodisk with an aspect ratio of 7 and increase the short axis until eventually a spherical shape is reached. The longitudinal resonance peak exhibits a further shift toward shorter wavelengths. Once again, this is in agreement with the fact that the radius of curvature increases upon going from a disk to a sphere. Above, we have only considered the influence of the radius of curvature on the extinction characteristics of nanospheroids of a single dimension and concluded that larger radii of curvature, that is, “flatter” surfaces, give rise to plasmon peaks at shorter wavelengths. When we consider the effect of absolute nanoparticle size, there is obviously also an effect related to the overall dimension of the nanospheroids. If only the radius of curvature would be relevant, increasing the size of a particularly shaped nanoparticle, that is, having larger radii of curvature, would give rise to a blue shift. However, comparing nanospheres, rods, and disks (Figures 3, 6, 10, and 15) reveals that, for particles with identical geometries, the spectral features red shift with absolute size. Apparently, both the absolute size as well as the radius of curvature are relevant for the optical behavior arising from localized surface plasmon resonances.

VI. CONCLUSIONS Using numerical methods based on Mie theory and also employing the discrete dipole approximation, we have performed a systematic investigation of the optical response of noble metal

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nanospheroids. Both the size and the geometry are considered in the analysis of extinction spectra in relation to the orientation of the (anisotropic) nanoparticles with respect to the polarization direction of the incident light. For spherical nanoparticles, the first four multipole plasmon resonances are considered as a function of particle size. The red shift of the extinction peaks is analyzed in relation to the plasmon wavelength derived from the geometry, that is, the circumference of the nanoparticles, enabling the construction of a plasmon dispersion curve. The results are compared to the bulk surface plasmon dispersion for gold and silver. The optical response of anisotropic nanoparticles, comprising multipole plasmon resonances, is analyzed for prolate rods and oblate disks in terms of a material-independent geometric resonance factor. It turns out that, for a specific particle shape, defined by the aspect ratio, the resonance factors for silver and gold are identical and depend quadratically on the absolute size of the nanospheroids. Furthermore, the dependence of the resonance factor on the aspect ratio is compared for oblate and prolate nanospheroids, providing a general description of the plasmon resonance position as a function of particle shape and size. The results are discussed in terms of particle size and degree of surface curvature of the nano-objects.

’ ASSOCIATED CONTENT

bS

Supporting Information. Brief description of the quasistatic approximation, which can be used to calculate the optical response of small ellipsoidal nanoparticles in the dipole approximation. Additional details on the accuracy and convergence of the DDA calculations are provided. An approximation for the circumference of an elliptical nanoparticle is reviewed in terms of the geometry. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Present Addresses †

Department of Physics, COMSATS, Institute of Information Technology, Park Road Campus, Islamabad, Pakistan.

’ ACKNOWLEDGMENT W.A. acknowledges support from the Higher Education Commission in Pakistan. This work was sponsored by the Stichting Nationale Computerfaciliteiten (National Computing Facilities Foundation, NCF) for the use of supercomputer facilities, with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organisation for Scientific Research, NWO). ’ REFERENCES (1) van de Hulst, H. C. Light Scattering by Small Particles; Dover Publications: New York, 1981. (2) Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer-Verlag: Berlin, 1995. (3) Raether, H. Surface Plasmons on Smooth and Rough Surfaces and on Gratings; Springer Verlag: Berlin, 1988. (4) Murray, W. A.; Barnes, W. L. Adv. Mater. 2007, 19, 3771–3782. 10331


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Localized Plasmons in Noble Metal Nanospheroids - The Journal of

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