Size-Dependent Surface Plasmon Resonance Broadening in

Letter pubs.acs.org/NanoLett

Size-Dependent Surface Plasmon Resonance Broadening in Nonspherical Nanoparticles: Single Gold Nanorods Vincent Juvé,†,⊥,# M. Fernanda Cardinal,‡,⊥,¶ Anna Lombardi,† Aurélien Crut,† Paolo Maioli,† Jorge Pérez-Juste,‡ Luis M. Liz-Marzán,‡,§,∥ Natalia Del Fatti,*,† and Fabrice Vallée† †

FemtoNanoOptics group, Institut Lumière Matière UMR5306, Université Lyon 1-CNRS, 69622 Villeurbanne, France Departamento de Química Física, Universidade de Vigo, 36310 Vigo, Spain § Bionanoplasmonics Laboratory, CIC biomaGUNE, Paseo de Miramón 182, 20009 Donostia-San Sebastián, Spain ∥ Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain ‡

ABSTRACT: The dependence of the spectral width of the longitudinal localized surface plasmon resonance (LSPR) of individual gold nanorods protected by a silica shell is investigated as a function of their size. Experiments were performed using the spatial modulation spectroscopy technique that permits determination of both the spectral characteristics of the LSPR of an individual nanoparticle and its morphology. The measured LSPR is shown to broaden with reduction of both the nanorod length and its diameter, which is in contrast with the predictions of existing classical and quantum theoretical models. This behavior can be reproduced assuming the LSPR width linearly depends on the inverse of an effective length proportional to the square root of the particle surface with the same slope as that recently determined for silica-coated silver nanospheres. KEYWORDS: Single particle spectroscopy, gold nanorods, surface plasmon resonance, surface broadening, quantum confinement, effective length

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(intraband) or different bands (interband) and are thus associated with the intrinsic electronic properties of the nanoparticles.1 Interband damping plays an important role when the LSPR overlaps the interband transitions, leading to large LSPR broadening as in gold or copper nanospheres. When the resonance is away from the interband transitions, as in silver nanospheres and in gold nanorods for the longitudinal LSPR, it is much narrower,12,14 a situation particularly interesting for applications. Its width is then determined by the optical scattering rate of the electrons, involving bulklike electron scattering mechanisms (e.g., electron−lattice, umklapp electron−electron, electron-defect interactions), and electronsurface scattering, or more correctly quantum effects in the confined metal.1,10,15 This surface contribution increases and eventually dominates the LSPR broadening when decreasing the particle size, yielding an intrinsic limit to the LSPR quality factor for small sizes. For spherical nanoparticles, quantum mechanical calculations assuming different electron confinement potentials have shown that the surface contribution to the LSPR broadening is proportional to the inverse of the particle diameter.10,16−18 This behavior has been qualitatively confirmed by experimental measurements on ensembles of nanospheres and, more

nhancement of the optical response of nanometric metallic systems due to their surface plasmon resonances has raised large fundamental and technological interest during the past decade. Such enhancement is associated to the collective resonant electron oscillation driven by an external electromagnetic field, whose manipulation and control has led to the development of the field of plasmonics.1−3 Nanoparticles are here particularly interesting as the spectral position of their localized surface plasmon resonance (LSPR) can be adjusted in the visible−near-infrared range by playing with their shape, size, and composition.4−6 This versatility offers a large potential for applications such as sensing, imaging, local heating for thermal treatment, enhanced Raman scattering, or more generally manipulation of light at nanoscale.7−9 In addition to adjusting the spectral position of the LSPR, a key feature for applications is its quality factor, that is, the LSPR width, as it sets the enhancement of the optical response and of the concomitant local field in and around the particle. Both radiative and dissipative mechanisms contribute to damping of the collective electron oscillation and thus to the LSPR broadening.1,10−13 The former mechanism is due to losses by light reemission (scattering) and becomes negligible for small metal nanoparticles, typically smaller than 20−30 nm for spheres, leading to spectral narrowing of the LSPR with size reduction to a limit set by the dissipative mechanisms.10,11 These reflect Landau damping, that is, decay of the collective charge oscillation into single electron−hole pairs in the same © XXXX American Chemical Society

Received: March 1, 2013 Revised: April 15, 2013

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dx.doi.org/10.1021/nl400777y | Nano Lett. XXXX, XXX, XXX−XXX

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monocrystalline nano-objects32 whose final dimensions were tuned by careful control of the solution pH, seeds, and silver ions concentrations.33,34 When required, preferential reduction of the length was achieved by spatially directed oxidation of the gold nanorods in the presence of cetyl-trimethylammonium bromide and gold salt.35 The largest rods were prepared by homogeneous overgrowth of gold nanorods through the reduction of additional gold salt in the presence of iodide.36 Subsequently, the gold nanorods were stabilized in ethanol with a thiolated polyethylene glycol polymer and silica coating was carried out by means of the Stöber method, as previously reported.37 The nanorods were described as cylinders capped by two hemispheres and coated by a silica shell with uniform thickness (Figure 1a), following TEM characterization (Figure

recently, quantitatively investigated by measuring the size dependence of the LSPR width of single Ag@SiO2 nanospheres.19,20 Much less information is available, both theoretically and experimentally, on the surface contribution in nonspherical nanoparticles. Because of the strong shape dependence of the LSPR wavelength and the relatively large sizes of the nonspherical nano-objects that can be synthesized, LSPR inhomogeneous broadening masks quantum effects when measuring the optical spectrum of an ensemble of particles. Single nanoparticle experiments were previously performed using dark-field microscopy.13,21−26 The optical scattering spectra were usually interpreted by introducing a reduced electron mean free path, using billiard models that aim at classically estimating an effective electron confinement length.27,28 However, although the mean size dependence of the LSPR width was found to be consistent with these ballistic approaches for some shapes (e.g., gold nanorods22), it was in stark contradiction with them for others (e.g., triangular nanoprisms24). Furthermore, for the model system of nanorods, classical and quantum approaches lead to different predictions with a broadening of the longitudinal LSPR mainly scaling with the inverse of the short (classical picture) and long axis length (quantum picture with light polarized parallel to this axis) for large aspect ratios.16,27,28 To quantitatively analyze the impact of the surface contribution to the LSPR width of nonspherical nanoparticles we measured the extinction spectra of single gold nanorods and simultaneously determined their individual dimensions (i.e., long and short axis lengths) using spatial modulation spectroscopy (SMS).29−31 These individual spectral and morphological characterizations permit direct correlation of the LSPR width of a nanorod with its dimensions, a more accurate approach than previous measurements where the morphology of each particle was not individually determined. Furthermore, nanorods coated by a silica shell were used to provide them with well-controlled interface and local environment to avoid spurious broadening effects,19 such as those observed for surfactant-stabilized gold and silver nanospheres.14,20 The results demonstrate dependence of the longitudinal LSPR width of single gold nanorods on both their short and long axis sizes, which is in contrast to the predictions of existing theoretical models. This measured size dependence can be phenomenologically reproduced assuming LSPR width inversely depends on an effective length proportional to the square root of the particle surface. Five different colloidal solutions of silica-coated gold nanorods spanning a wide range of sizes were used for this work (see average dimensions in Table 1). The gold nanorods were prepared by Ag-assisted seeded growth, yielding

Figure 1. (a) Geometry of Au@SiO2 nanorods used in the analysis. The gold core was described as a cylinder capped by two hemispheres. (b) TEM images of Au@SiO2 nanorods from solution a. (c) Ensemble absorbance spectra of solution a (blue), b (purple), c (dashed red), d (black), and e (green).

Table 1. Geometrical characteristics of the five investigated Au@SiO2 nanorod solutions: the mean length L, diameter D, and aspect ratio η = L/D of the Au nanorods deduced from TEM statistics are indicateda sample a b c d e a

L (nm) 32 43.5 47 48.5 70

± ± ± ± ±

3 3 6 4 10

D (nm) 8.5 9 15 17.5 22

± ± ± ± ±

1b). Each nanorod is then described by its full length, L, and diameter, D, its aspect ratio being defined as η = L/D (Figure 1a). Their mean diameter and length range from roughly 8 to 20 nm and 30 to 70 nm, respectively, yielding aspect ratios between 2.5 and 5. The thickness of the silica shell is typically between 15 and 20 nm. As expected, the ensemble spectra of the different solutions show a main resonance in the redinfrared part of the spectrum and a weaker one around 520 nm overlapping the rise of the gold interband absorption (λ ≤ 520 nm) (Figure 1c). The latter is ascribed to both the transverse

aspect ratio η = L/D

1 1 3 3 2

3.8 4.8 3.1 2.8 3.2

± ± ± ± ±

0.8 0.9 1 0.7 0.7

The silica shell thickness is between 15 and 20 nm. B


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cross-section of a single nanoparticle, and thus it permits simultaneous determination of its spectrum and of the nanoobject volume and aspect ratio for simple shapes such as spheres or rods.31,38 Briefly, this is based on sinusoidal modulation at frequency f of the position of a nano-object in the focal plane of a tightly focused laser beam. The presence of a nano-object in the focal spot translates into a modulation of the transmitted light power with an amplitude proportional to the extinction cross-section σext(λ) at the incident wavelength λ. This can thus be quantitatively measured and its spectrum determined using a tunable light source. As we were mostly interested in the gold nanorod longitudinal LSPR, a femtosecond Ti:Sapphire oscillator tunable in the 680−1080 nm range was used. The high spatial quality of the beam permits its focusing close to the diffraction limit, yielding a focal spot size d ≈ 0.7 λ (full width at half-maximum of the light intensity profile) using a 100× microscope objective with a numerical aperture of 0.75. The sample position was modulated at f ≈ 1.5 kHz and the transmitted light power detected by an avalanche photodiode. Demodulation of the transmitted light amplitude was performed at 2f by a lock-in amplifier. The light beam was linearly polarized along a direction that was adjusted using a quarter-wave plate and a polarizer. Polarization-dependent σext spectra were recorded using the SMS technique for 29 individual gold nanorods from the five different colloidal solutions. Measurements were limited to the spectral region of the longitudinal LSPR of interest here. In this spectral range, its amplitude shows the expected dependence on the light polarization direction (Figure 2b), that is, a cos2(α) dependence (α is the angle between the polarization direction and the long axis of the investigated rod, Figure 2a).13,38 For all the measured σext spectra, the longitudinal LSPR shows up as an almost-Lorentzian line in the energy domain. For each rod, the LSPR can thus be described by three independent parameters: its central frequency ΩR, its full width at halfmaximum Γ (with Γ≪ ΩR), and its area Ξ (obtained by integration of the LSPR extinction cross-section in the energy domain)

LSPR of nanorods (i.e., for light linearly polarized along the rod short axis) and to the LSPR of residual quasi-spherical gold nanoparticles.38 The main resonance corresponds to the longitudinal LSPR (i.e., for light polarized along the rod long axis) with a spectral position that is red shifted when the mean aspect ratio of the nanorods is increased (Table 1). In the energy domain, the width of these LSPRs is in the 300−330 meV range, much larger than that typically measured in individual gold nanorods.38 This is a consequence of inhomogeneous broadening in the ensemble measurements, evidencing the need of carrying out single particle experiments for investigating the LSPR homogeneous width. The samples for single particle investigations were prepared by spin-coating on a silica slide one of the colloidal solutions of Au@SiO2 nanorods after appropriate dilution and addition of polyvinyl alcohol (PVA). A surface density of less than one rod per μm2 is then obtained, permitting optical separation of the individual nano-objects. Though the gold nanorod LSPR spectral position is mostly set by the silica shell for a fixed aspect ratio, its thickness (of the order of the gold diameter) is insufficient to fully exclude influence of the external environment for the largest Au@SiO2 rods.39 Addition of PVA permits to embed the deposited Au@SiO2 rods in a thin polymer layer whose refractive index (1.4940) is close to that of the silica shell and the silica substrate (1.45), providing them with a quasiuniform dielectric environment (Figure 2a), as previously demonstrated for gold nanospheres.31 The optical spectra of individual nanorods were measured using the SMS technique.29−31 This has the advantage that it provides access to the absolute value of the polarized extinction

σext(ω) = Ξ

Γ 2π

(ω − Ω R )2 +

2

( Γ2 )

(1)

These parameters are extracted by fitting the measured spectra, as exemplified in Figure 2c. In particular, Γ values varying in the 80−140 meV range were obtained for the individual nanorods. To properly correlate the measured LSPR width with the nanoparticle dimensions, the latter have to be determined for each investigated rod. This can be done fully exploiting the measured spectra, ΩR and Ξ mostly reflecting its aspect ratio and volume, respectively, provided the dielectric constant of its environment is known.39 This is the case here, as the nanorods are coated by a silica shell, further embedded in a polymer film with similar refractive index and deposited on a silica substrate. To precisely determine the dimensions of a nanorod from its optical response, the experimental spectra were fitted using finite-element modeling (FEM) assuming the model shape shown in Figure 1a and a homogeneous environment with a refractive index of 1.45. This approach has been shown to yield reliable results for silica-coated gold nanorods, as demonstrated by directly comparing the optically determined rod dimensions, that is, its length L and diameter D, to those obtained by transmission electron microscopy.41,42 For each investigated

Figure 2. (a) Experimental approach: a single Au@SiO2 nanorod deposited on a silica substrate and embedded in a polymer layer is illuminated with a linearly polarized light beam (polarization angle α with the nanorod long axis). (b,c) Optical response of a single nanorod from sample a. (b) Polarized extinction cross-section at λ = 830 nm (black squares). Red: fit with a cos2(α) function. (c) Extinction crosssection spectrum measured for α = 0° (black squares). Blue dashed line: FEM-computed spectrum using L = 32.9 nm and D = 8.7 nm, γs = 0 meV. Red solid line: FEM-computed spectrum using the same sizes with γs= 54 meV. (d) Sizes of 29 single Au@SiO2 nanorods extracted from interpolation of their extinction spectrum as in panel c. Colors mark the solution used for nanorod spin-coating (same color code as in Figure 1c). Orange dotted lines indicate aspect ratios between 2 and 5. C


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Figure 3. Spectral width of the longitudinal LSPR of single Au@SiO2 nanorods as a function of their length L and diameter D. (a) Measured total width of the extinction spectra Γ. (b) Computed bulk contribution Γ0 = Ω3Rε2(ΩR)/ω2P (using the bulk gold dielectric function and eq 5). (c) Radiative (Γr) contribution computed using finite-element modeling (see main text). (d) Surface (γs) contribution deduced using eq 6. Data associated to actually measured nanorods are shown as open dots (inner color indicates the measured/deduced width). Color maps interpolated to the whole (1/L, 1/D) region covered by the experiments are also presented with contours at fixed widths.

nanorod, L and D (or, equivalently, its aspect ratio η and total volume V) were adjusted in FEM simulations to reproduce the experimental LSPR frequency ΩR and area Ξ using the bulk gold dielectric function.43 Though the computed LSPR width without quantum confinement effects is smaller than the experimental one (Figure 2c), the extracted rod sizes are independent of it (the integrated area being independent of the spectral broadening).39 The extracted sizes of the investigated nanorods span a wide range of diameter and length values (Figure 2d), considerably extending that investigated in a previous study (focused on large sizes, for which surface broadening effect is almost negligible),38 permitting independent analysis of the effects of these two dimensions on the longitudinal LSPR width. The experimental LSPR width Γ of each Au@SiO2 nanorod is shown as a function of its inverse length and diameter in a two-dimensional color plot (Figure 3a, showing both individual data points and their two-dimensional interpolation). It varies in the 80−140 meV range and clearly depends on both size parameters, increasing with reducing either the rod length or diameter (for a fixed diameter or length, respectively). The LSPR spectral width Γ results from both radiative (Γr) and nonradiative (Γnr) damping mechanisms. The latter is due to absorption and related to the imaginary part ε2 of the metal dielectric function in the nanoparticle. For the size range investigated here, the dielectric function ε can be assumed to take a similar form to the bulk metal dielectric function and separated into interband, εib and intraband (Drude-like) contributions

ε(ω) = ε ib −

ωP2 ω(ω + iγ )

(2)

where ωP is the bulk plasma frequency (ℏωP = 9.01 eV for gold). The validity of this expression for spherical nanoparticles has been demonstrated using quantum mechanical models.1,10 The optical scattering rate of the conduction electrons in the particle, γ = γ0 + γs, can be written as the sum of the bulklike optical scattering rate γ0 (due to electron−phonon and umklapp electron−electron scattering) and of a quantum correction term γs (classically associated to electron-surface scattering).10,17−19,44 The latter is inversely proportional to the nanosphere diameter D v γs = 2gs F (3) D where vF is the metal Fermi velocity and gs is a parameter of the order of unity. This parameter is influenced by the particle environment and its value has been experimentally shown to be 2gs ≈ 1.4 for individual silica coated silver nanospheres.18,19 In an attempt to generalize this result for nonspherical nanoparticles, one can assume a similar expression for ε and for the confinement-induced electron scattering rate (eq 3) by introducing an effective confinement length Leff that depends on the particle dimensions21,27 v γs = A F Leff (4) with Leff(D,L) for the nanorods investigated here. Assuming a weakly dispersed dielectric function ε around the LSPR D


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Figure 4. Values of 1/Leff computed for nanorods in the case of (a) a billiard-based model with 1/Leff = S/(4V), and (b) 1/Leff = 1/(L1/2D1/2). (c) Experimental LSPR widths of Au@SiO2 nanorods (Γ, open dots) and nonradiative contribution (Γ − Γr, closed dots) plotted versus the inverse of the effective length computed in (b). The straight line is a fit with slope A = 1.4 (see text). The shaded area indicates the estimated Γ0 bulk value in gold.43,47,48

contribution can be analytically computed, while for a nonspherical particle it can be extracted by numerically computing the LSPR width for its actual size using the bulk metal dielectric function and subtracting the nonradiative contribution Γ0 (Figure 3c). This approach to estimate Γr has the advantage of taking into account the shape and dimension of the nano-object and avoids the use of approximate expressions.13,21 For the investigated sizes, ℏΓr only weakly contributes to the total LSPR width, varying from ∼0 to 15 meV (Figure 3c). Because of the uncertainties in the measured bulk ε2 values in the LSPR spectral range, reflected by the large differences between the tabulated values,43,47,48 estimation of the absolute value of the bulklike broadening contribution may be affected by a large error. However, the weak size dependences of Γ0 and Γr (Figure 3b,c) cannot explain the measured variations of Γ with size, both in terms of amplitude and L and D dependences (Figure 3a), stressing the presence of an additional contribution. This is associated to the surface scattering term, γs, that can be estimated by subtracting the computed Γr and Γ0 contributions from the measured LSPR width Γ (Figure 3d). It can also be directly extracted by fitting the experimental σext spectra using a gold dielectric function corrected for size effects instead of the bulk one, that is, using eq 2 with γ = γ0 + γs, taking γs as a fitting parameter (Figure 2c). This only affects the LSPR width, yielding as stressed above the same values for the rod diameter and length. The two methods yield the same results: γs is found to be almost negligible for the largest nanorods and to largely increase for the smallest ones (up to 60 meV). Note that the absolute value of γs is affected by the large uncertainty on Γ0 and therefore only its size dependence is significant here (as for silver nanosphere investigations19). It depends on both the diameter and length of the nanorods, in a

frequency, quasi-static analytical calculations for nanospheres and nanoellipsoids10,38,45,46and numerical simulations for gold nanorods and bipyramids39 show that the nonradiative LSPR broadening term, Γnr, can thus be written as the sum of an interband contribution, Γib = Ω R3 ε2ib (Ω R )/ωP2 , and the conduction band scattering rate, γ Γnr(Ω R ) =

2ε2(Ω R ) ∂ε1 ∂ω Ω R

( )

≈

Ω3R ωP 2

ε2(Ω R ) = Γib + γ0 + γs (5)

Using the above expression, the full LSPR width can be written as Γ(Ω R ) = Γr(Ω R ) + Γnr(Ω R ) = Γr(Ω R ) + Γ0(Ω R ) + γs (6)

where Γ0 = Γib + γ0 is the bulklike nonradiative broadening term, including both interband and intraband contributions, and γs reflects the size-dependent contribution due to electron quantum confinement in the nanoparticle (its weak frequency dependence being neglected here).15 The bulklike nonradiative term Γ0 is obtained for each nanorod using eq 5 with the tabulated bulk gold dielectric function at its LSPR wavelength43 (Figure 3b). As nanorods have different aspect ratios, ranging from 2.5 to 5, the LSPR wavelength varies from about 650 to 980 nm. This translates into a quasi-constant value of ℏΓ0 ≈ 65 meV for most of the nanorods, as their LSPR is away from the interband transitions (i.e., εib2 (Ωib) ≈ 0 in eq 5). Only for the two nanorods having the smallest aspect ratio, ℏΓ0 increases to ∼80 meV as their LSPR resonance close to 650 nm overlaps the rise of the interband transitions (Figure 3b).13,20 The radiative damping width Γr is connected to light scattering by the particle and increases with the particle volume. For a sphere, this E


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limited precision in the value of Γ0 (i.e., of ε). This surface effect has been previously determined for silver nanospheres embedded in silica shells using the same method. Taking Leff = D for a sphere and comparing eqs 3 and 4, one gets A = 2gs. Interestingly, the measured gs value in these systems, gs ≈ 0.7 ± 0.1, corresponds to the same surface factor A presently measured in silica-coated gold nanorods.19 Using this surface factor and surface length, Leff = (S/π)1/2, a quantitative agreement on the amplitude and size dependence of the LSPR width is also obtained for previous single nanorod experiments as a function of their mean size.21,22 Though a smaller gs value was suggested from photothermal investigations of single gold nanospheres,49 interband-induced damping (leading to an asymmetric LSPR shape) as well as the knowledge of only the mean particle size in the original colloidal solution, complicate the analysis of these results.49 However, when taking into account error bars, the sizedependence of the LSPR half-width of the small (D < 25 nm)19 gold nanoparticles is compatible with the surface values estimated here. Conversely to a billiard model, the expected amplitude of surface effects are here also compatible with other single particle investigations (as for oblate silver nanoprisms).21,22,24 This points out the importance of the geometrical surface area as a relevant parameter for quantitatively describing surface effects in the longitudinal optical response of elongated nano-objects. However, for more complex shapes (such as nonconvex nanoshells26,50,51) this simple approach is not expected to be valid and a full quantum mechanical model taking into account both geometry and light polarization should be employed. In conclusion, quantum size effects affecting the optical response of elongated nanorods have been investigated at a single particle level, as a function of their individual geometrical features. Experiments were performed using the spatial modulation technique on gold nanorods, which were protected by silica coating in order to provide them with a well controlled environment. The quantitative extinction cross-section around the longitudinal localized surface plasmon resonance of individual nanorods was measured, providing the LSPR width and the dimensions for each investigated nano-object. The surface broadening contribution is clearly visible at these sizes and can be separated from the bulklike one and the radiative damping. It reveals a dependence on both the length and diameter of the nanorods, which is in contrast with a simple billiard model that would predict a dependence on the short axis mainly. An effective length related to the surface has been phenomenologically introduced and permits a quantitative explanation of surface effects in these prolate nanoparticles with the same surface factor A = 2gs = 1.4 than that measured for silver nanospheres. Reproduction of the longitudinal LSPR width measured in previous experiments on prolate and oblate shapes was also obtained on the basis of this description. A complete quantum modeling is however needed to understand the origin of the observed dependence of the LSPR width for nonspherical nano-objects.

similar way to the full width Γ, as shown by the ∼45° tilted contour plots that interpolate the single particle experimental data (Figure 3a,d). The measured size dependence of γs and Γ is in contrast to the prediction of the frequently used billiard model estimating the electron-surface scattering probability from a classical trajectory approach.27 The billiard model predicts a surface scattering rate proportional to the surface, S, over volume, V, ratio of the particle, that is, an effective length Leff = 4V/S. For the geometry of our nanorods (Figure 1a, S = πLD and V = πD2(3L − D)/12), one gets Leff = D(1 − 1/(3η)). This is close to their small dimension, Leff ≈ D, due to their relatively large aspect ratios (with η ≥ 2.5) and is thus almost independent of the long dimension, L. This is illustrated by plotting the calculated 1/Leff given by the billiard model as a function of the rod sizes, almost horizontal constant value contours being obtained (Figure 4a) in contrast with the contours of the measured LSPR width (Figure 3a,d). A similar discrepancy was previously pointed out in flat silver nanoprisms, experiments suggesting a size dependence related to their perpendicular bisector length, rather than to their thickness as predicted.24 Conversely, a quantum mechanical model based on a simple infinite confinement potential of cylindrical or rectangular shape indicates a dependence of γs on the nanoparticle length L for light polarized along its long axis,16 which is also inconsistent with our experimental data. The failures of the available classical and quantum simple models to even reproduce the correct size scaling of the LSPR broadening stresses the need to develop a more detailed quantum mechanical modeling, that is, using a realistic self-consistent confinement potential as has been done for nanospheres,18 but this is out of the scope of this paper. The measured L and D dependence of γs and Γ can be quantified using a phenomenological approach, that is, assuming that the impact of size reduction on the LSPR width depends on an effective confinement length Leff(D, L) (eq 4), with Leff = LβD1−β. The value β = 0.5 gives a symmetrical role to L and D (Figure 4b) and reproduces best the measured shape of the contour plot (Figure 3d). This is more clearly shown by plotting the experimental LSPR width Γ (or its nonradiative contribution, Γ − Γr) as a function of the inverse of L1/2D1/2 (Figure 4c). In both cases, a linear dependence is obtained in agreement with eq 4. This dependence suggests that Leff is related to the square root of the particle surface S, Leff = (S/π)1/2, which yields Leff = D for a sphere and Leff = (LD)1/2 = Dη1/2 for a nanorod. This phenomenological expression is a priori valid for aspect ratios from one to few units (at least 5, as investigated here), and experimental investigation of its extension to larger values would be particularly interesting. The experimental slope deduced from fitting the linear dependence of Γ yields a surface coefficient A = 1.4 ± 0.2. The absolute value of Γ0 extracted from the ordinate at the origin of the linear fit, ℏΓ0 = (40 ± 10) meV, is compatible with very recent measurements which yielded a gold electron relaxation scattering rate of ℏΓ0 ≈ (47 ± 10) meV below interband transition threshold.48 It is however smaller than the one computed using Johnson and Christy bulk dielectric constants, confirming possible large uncertainties in their determination.43 Nevertheless, as for nanospheres,19 the amplitude of the surface-induced LSPR broadening effect is deduced from the slope of its dependence on the effective length and is thus precisely determined here as it is only weakly affected by the

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AUTHOR INFORMATION

Corresponding Author

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

# Max-Born-Institut für Nichtlineare Optik and Kurzzeitspektroskopie, D-12489 Berlin, Germany.

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(25) Raschke, G.; Brogl, S.; Susha, A. S.; Rogach, A. L.; Klar, T. A.; Feldmann, J.; Fieres, B.; Petkov, N.; Bein, T.; Nichtl, A.; Kurzinger, K. Nano Lett. 2004, 4, 1853−1857. (26) Nehl, C. L.; Grady, N. K.; Goodrich, G. P.; Tam, F.; Halas, N. J.; Hafner, J. H. Nano Lett. 2004, 4, 2355−2359. (27) Coronado, E. A.; Schatz, G. C. J. Chem. Phys. 2003, 119, 3926− 3934. (28) Moroz, A. J. Phys. Chem. C 2008, 112, 10641−10652. (29) Arbouet, A.; Christofilos, D.; Del Fatti, N.; Vallée, F.; Huntzinger, J.; Arnaud, L.; Billaud, P.; Broyer, M. Phys. Rev. Lett. 2004, 93, 127401. (30) Muskens, O.; Christofilos, D.; Del Fatti, N.; Vallée, F. J. Opt. A: Pure Appl. Opt. 2006, 8, S264−S272. (31) Muskens, O.; Billaud, P.; Broyer, M.; Del Fatti, N.; Vallée, F. Phys. Rev. B 2008, 78, 205410. (32) Goris, B.; Bals, S.; Van den Broek, W.; Carbó-Argibay, E.; Gómez-Graña, S.; Liz-Marzán, L. M.; Van Tendeloo, G. Nat. Mater. 2012, 11, 930−935. (33) Liu, M.; Guyot-Sionnest, P. J. Phys. Chem. B 2005, 109, 22192− 22200. (34) Pérez-Juste, J.; Pastoriza-Santos, I.; Liz-Marzán, L. M.; Mulvaney, P. Coord. Chem. Rev. 2005, 249, 1870−1901. (35) Rodríguez-Fernández, J.; Pérez-Juste, J.; Mulvaney, P.; LizMarzán, L. M. J. Phys. Chem. B 2005, 109, 14257−14261. (36) Grzelczak, M.; Sánchez-Iglesias, A.; Rodríguez-González, B.; Alvarez-Puebla, R.; Pérez-Juste, J.; Liz-Marzán, L. M. Adv. Funct. Mat. 2008, 18, 3780−3786. (37) Fernández-López, C.; Mateo-Mateo, C.; Alvarez-Puebla, R. A.; Pérez-Juste, J.; Pastoriza-Santos, I.; Liz-Marzán, L. M. Langmuir 2009, 25, 13894−13899. (38) Muskens, O. L.; Bachelier, G.; Del Fatti, N.; Vallée, F.; Brioude, A.; Jiang, X.; Pileni, M.-P. J. Phys. Chem. C 2008, 112, 8917−8921. (39) Lombardi, A.; Loumaigne, M.; Crut, A.; Maioli, P.; Del Fatti, N.; Vallée, F.; Spuch-Calvar, M.; Burgin, J.; Majimel, J.; TréguerDelapierre, M. Langmuir 2012, 28, 9027−9033. (40) Mahendia, S.; Kumar Tomar, A.; Goyal, P. K.; Kumar, S. J. Appl. Phys. 2013, 113, 073103. (41) Billaud, P.; Marhaba, S.; Cottancin, E.; Arnaud, L.; Bachelier, G.; Bonnet, C.; Del Fatti, N.; Lermé, J.; Vallée, F.; Vialle, J.-L.; Broyer, M.; Pellarin, M. J. Phys. Chem. C 2008, 112, 978−982. (42) Davletshin, Y. R.; Lombardi, A.; Cardinal, M. F.; Juvé, V.; Crut, A.; Maioli, P.; Liz-Marzán, L. M.; Vallée, F.; Del Fatti, N.; Kumaradas, J. C. ACS Nano 2012, 6, 8183−8193. (43) Johnson, P. B.; Christy, R. W. Phys. Rev. B 1972, 6, 4370−4379. (44) Hartland, G. V. Chem. Rev. 2011, 111, 3858−3887. (45) Kreibig, U. Appl. Phys. 1976, 10, 255−264. (46) Wang, F.; Shen, Y. Phys. Rev. Lett. 2006, 97, 206806. (47) Palik, E. D. Handbook of Optical Constants of Solids; Academic Press: New York, 1985; Vol. 1. (48) Olmon, R. L.; Slovick, B.; Johnson, T. W.; Shelton, D.; Oh, S.; Boreman, G. D.; Raschke, M. B. Phys. Rev. B 2012, 86, 235147. (49) Berciaud, S.; Cognet, L.; Tamarat, P.; Lounis, B. Nano Lett. 2005, 5, 515−518. (50) Averitt, R. D.; Westcott, S. L.; Halas, N. J. J. Opt. Soc. Am. B 1999, 16, 1824−1832. (51) Westcott, S.; Jackson, J.; Radloff, C.; Halas, N. Phys. Rev. B 2002, 66, 155431.

Department of Chemistry, Northwestern University, 2145 Sheridan Rd., Evanston, IL 60208, United States.

Author Contributions ⊥

V.J. and M.F.C. contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge financial support by ANR program P3N2009 under Contract ANR-09-NANO-019-01 Lunaprobe by Ministère des Affaires étrangères et européennes under Project PHC Picasso 22930NE and by the Spanish Acción Integrada FR2009-0034 and MINECO Project No. MAT201015374. M.F.C. acknowledges a Ph.D. student scholarship from INL, Braga. N.D.F. acknowledges Institut Universitaire de France.

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REFERENCES

(1) Kreibig, U.; Vollmer, M. Optical properties of metal clusters; Springer Verlag: Berlin, 1995. (2) Maier, S. A.; Atwater, H. A. J. Appl. Phys. 2005, 98, 011101. (3) Ozbay, E. Science 2006, 311, 189−193. (4) Oldenburg, S. J.; Averitt, R. D.; Westcott, S. L.; Halas, N. J. Chem. Phys. Lett. 1998, 288, 243−247. (5) Kelly, K. L.; Coronado, E.; Zhao, L. L.; Schatz, G. C. J. Phys. Chem. B 2003, 107, 668−677. (6) Liz-Marzán, L. M. Langmuir 2006, 22, 32−41. (7) O’Neal, D. P.; Hirsch, L. R.; Halas, N. J.; Payne, J. D.; West, J. L. Cancer Lett. 2004, 209, 171−176. (8) Aćimović, S. S.; Kreuzer, M. P.; González, M. U.; Quidant, R. ACS Nano 2009, 3, 1231−1237. (9) Hatab, N. A.; Hsueh, C.-H.; Gaddis, A. L.; Retterer, S. T.; Li, J.H.; Eres, G.; Zhang, Z.; Gu, B. Nano Lett. 2010, 10, 4952−4955. (10) Kawabata, A.; Kubo, R. J. Phys. Soc. Jpn. 1966, 21, 1765−1772. (11) Wokaun, A.; Godon, J. P.; Liao, P. F. Phys. Rev. Lett. 1982, 48, 957−960. (12) Klar, T.; Perner, M.; Grosse, S.; Von Plessen, G.; Spirkl, W.; Feldmann, J. Phys. Rev. Lett. 1998, 80, 4249−4252. (13) Sönnichsen, C.; Franzl, T.; Wilk, T.; Von Plessen, G.; Feldmann, J.; Wilson, O.; Mulvaney, P. Phys. Rev. Lett. 2002, 88, 077402. (14) Billaud, P.; Huntzinger, J.-R.; Cottancin, E.; Lermé, J.; Pellarin, M.; Arnaud, L.; Broyer, M.; Del Fatti, N.; Vallée, F. Eur. Phys. J. D 2007, 43, 271−274. (15) Voisin, C.; Del Fatti, N.; Christofilos, D.; Vallée, F. J. Phys. Chem. B 2001, 105, 2264−2280. (16) Kraus, W. A.; Schatz, G. C. J. Chem. Phys. 1983, 79, 6130−6139. (17) Hache, F.; Ricard, D.; Flytzanis, C. J. Opt. Soc. Am. B 1986, 3, 1647−1655. (18) Lermé, J.; Baida, H.; Bonnet, C.; Broyer, M.; Cottancin, E.; Crut, A.; Maioli, P.; Del Fatti, N.; Vallée, F.; Pellarin, M. J. Phys. Chem. Lett. 2010, 1, 2922−2928. (19) Baida, H.; Billaud, P.; Marhaba, S.; Christofilos, D.; Cottancin, E.; Crut, A.; Lermé, J.; Maioli, P.; Pellarin, M.; Broyer, M.; Del Fatti, N.; Vallée, F.; Sánchez-Iglesias, A.; Pastoriza-Santos, I.; Liz-Marzán, L. M. Nano Lett. 2009, 9, 3463−3469. (20) Baida, H.; Christofilos, D.; Maioli, P.; Crut, A.; Del Fatti, N.; Vallée, F. Eur. Phys. J. D 2011, 63, 293−299. (21) Novo, C.; Gomez, D.; Pérez-Juste, J.; Zhang, Z.; Petrova, H.; Reismann, M.; Mulvaney, P.; Hartland, G. V Phys. Chem. Chem. Phys. 2006, 8, 3540−3546. (22) Hu, M.; Novo, C.; Funston, A.; Wang, H.; Staleva, H.; Zou, S.; Mulvaney, P.; Xia, Y.; Hartland, G. V J. Mater. Chem. 2008, 18, 1949− 1960. (23) Munechika, K.; Smith, J. M.; Chen, Y.; Ginger, D. S. J. Phys. Chem. C 2007, 111, 18906−18911. (24) Blaber, M. G.; Henry, A.-I.; Bingham, J. M.; Schatz, G. C.; Van Duyne, R. P. J. Phys. Chem. C 2012, 116, 393−403. G


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