Size Evolution of the Surface Plasmon Resonance Damping in Silver

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Size Evolution of the Surface Plasmon Resonance Damping in Silver Nanoparticles: Confinement and Dielectric Effects Jean Lerme* Laboratoire de Spectrometrie Ionique et Moleculaire (UMR 5579), Universite de Lyon, Universite Lyon I, CNRS, Bât. A. Kastler, 43 Bld du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France

bS Supporting Information ABSTRACT: A key parameter for optimizing nanosized optical devices involving small metal particles is the spectral width of their localized surface plasmon resonances (LSPR), which is intrinsically limited by the confinement-induced broadening (quantum finite size effects). I have investigated the size evolution of the LSPR width induced by quantum confinement in silver nanoparticles isolated in vacuum or embedded in transparent matrixes. Calculations have been performed within the time-dependent local density approximation in an extended size range, up to 40000 conduction electrons (diameter D ≈ 11 nm). The slope characterizing the 1/D linear evolution predicted by the simple classical “limited mean free path” model is found to depend noticeably on the surrounding matrix. The confinement-induced damping of the collective LSPR excitation is shown to be intimately related to the departure of the electron-background interaction from a pure harmonic law. In jellium-type models the damping is governed by the electronic spillout tail, which leads to the decay of the coherent excitation of the electronic center-of-mass coordinate into incoherent intrinsic electronic motions, that is, single particle-hole excitations (Landau damping). The computed linear slope is found roughly two times smaller than the one measured in experiment on single silver particles, indicating that part of the size dependence of the fast plasmon damping results from the contribution of the granular ionic structure, especially the electronphonon contribution in the large size domain. The strong sensitivity of the LSPR damping to the surface profile of the confining potential, which depends on numerous structural surface parameters, in particular those related to the ionic background modeling, is emphasized. A short analysis of the quantum box model and of the classical approach is also provided.

1. INTRODUCTION The optical properties of noble metal nanoparticles are of great importance in many technological, imaging, medical, and sensing applications at the nanoscale.14 The interest for these species is due essentially to the collective excitations of their conduction electrons, known as localized surface plasmon resonances (LSPR).5 These resonances give rise to large absorption and scattering cross sections in narrow spectral ranges and can be tuned continuously over the entire near UVvisible spectral range, through size, shape, and dielectric effects.6,7 The homogeneous bandwidth of the dipolar LSPR band, which is directly correlated to the lifetime of the coherent collective excitation, is a key parameter from a fundamental point of view, as well as for technological applications since it sets an upper limit for the quality factor of the resonance. It is intimately related to the field enhancement accompanying the LSPR excitation and is thus of central importance in many applications, such as, for instance, surface-enhanced fluorescence or Raman scattering.13,5 However, the line width is—relative to the other characteristics of the LSPR—much less documented with reliability. Actually most experimental determinations are extracted in the spectral domain from LSPR bandwidths measured on ensembles, and are thus affected by inhomogeneous line broadening effects arising from the size, shape and local environment distributions in the probed r 2011 American Chemical Society

sample. Nowadays, thanks to various sensitive near/far field techniques, these drawbacks can be overcome and measurements on individual particles have been successfully achieved.822 Numerous mechanisms may contribute to the damping of the coherent collective excitation, and consequently to the LSPR bandwidth.23 Let us quote for instance the radiation damping, which is the leading contribution in the large size range, or the electronelectron, ion (phonon) or defects scattering processes in which the energy of the coherent electronic motion is transformed into heat in the electronic system and the ionic background.5 An important additional mechanism in the case of gold and copper is the coupling with interband transitions, where the LSPR is damped via excitations of d-electrons above the Fermi level in the conduction band.24 Surface effects, directly related to the finite size of the particle, as the chemical interface damping for matrix-embedded particles,25 shape distortions, surface roughness, and electron scattering off the surface,5 contribute to the LSPR broadening. Contrary to the other surface effects, which are not easy to control and may give rise to varying and often unpredictable Received: April 14, 2011 Revised: June 8, 2011 Published: June 30, 2011 14098

dx.doi.org/10.1021/jp203481m | J. Phys. Chem. C 2011, 115, 14098–14110

The Journal of Physical Chemistry C damping efficiencies, the electron-surface scattering related to the finite size of the potential well confining the electrons is unavoidable and intrinsic by nature. This effect, which will be addressed in this paper, was for a long time empirically included in the modeling of the optical properties of metal particles, because a simple classical “billiard” picture, the so-called “limitedmean-free-path effect”,5,26 accounts quite well for the size evolution of the LSPR bandwidth pΔω [defined throughout this paper as the full width at half-maximum (fwhm) on the energy scale]. Within this classical approach, rooted in the classical Drude theory of metals, the size evolution of the fwhm obeys the following scaling law for spherical particles vF ð1Þ pΔω ¼ pΓðRÞ ¼ pΓ0 + gp R where pΓ0 is the intrinsic bulk contribution to the bandwidth,27 vF the Fermi velocity, R the particle radius, and g a dimensionless parameter of the order of unity. In the frame of this “ballistic” approach several works have been reported to estimate the value of g, depending whether the scattering of the electron off the surface is specular, diffuse, etc., for various particle shapes.5,2830 It should be emphasized that all finite size- and surface-induced effects are expected to lead to a similar scaling law (that reflects the surface to volume ratio), and the experimentally determined g-value lumps indeed all the various contributions. In particular, for ensembles of embedded particles, g-values ranging roughly from 0 to 1.5 have been reported,5,31 depending on the surrounding matrix or the experiment, although the particle sizes and thus the mean free paths are identical. Later on, more refined models were developed to quantify the size dependence of the LSPR-broadening induced by the electron confinement. Following the pioneering paper by Kawabata and Kubo32 several simplified quantum-mechanical models, based on various approaches, have been reported to determine the “effective dielectric function of the particle”, taking into account explicitly the quantization of the electronic level spectrum in different geometries. Whatever the approach involved, the imaginary part of the particle-dielectric function is calculated from the whole set of dipolar transitions between occupied and unoccupied single electron states in a flat-bottom hard-walled potential well of infinite depth. From these models, referred to as “the quantum box model” (QBM) in this paper, various g-values have been calculated for specific geometries for which quasianalytical expressions may be obtained (see ref 5, where most versions of the QBM are listed). However, these calculations involve numerous approximations, at the origin of the dispersed g-values computed by different authors for a given particle shape.33,34 In the frame of self-consistent quantum-mechanical approaches, the decay of the coherent plasmon excitation is ascribed to its coupling, via the particle surface, with one-electron excitations [the so-called single particle-hole (p-h) states], mechanism usually referred to as “Landau damping mechanism”. Indeed Landau damping is the main decay mechanism in the small size range. This dissipative exchange of energy and momentum between collective and intrinsic degrees of freedom is responsible for the plasmon band fragmentation, namely, the structured multipeak LSPR pattern obtained in random-phase approximation (RPA) or time-dependent local density approximation (TDLDA) calculations on small clusters.3538 The present theoretical work addresses the contribution of the quantum confinement to the size evolution of the LSPR

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bandwidth. In a previous short letter (referred to as “Paper I” in the following) TDLDA calculations on spherical silver particles embedded in various host-matrices over a large size range, permitting to quantify the 1/R linear law predicted by the phenomenological classical model, have been reported.39 Results had been also compared with a single-particle experiment involving Ag-core/SiO2-shell particles.40 The purpose of the present paper is 4-fold. First, much more detailed information about the theoretical background [Supporting Information] and a more thorough analysis of the results reported in Paper I are provided. The reader will be referred to Paper I in some places, to avoid redundant material. Moreover, additional computations exemplifying the strong dependence of the confinement-induced LSPR broadening on the surface profile of the potential well, as well as on the ionic background modeling near the surface, are reported on. Second the tight relationship between the LSPR damping and the departure of the electron-background from a pure harmonic law [generalized Kohn’s theorem (GKT)] is emphasized and exemplified. More precisely, as long as a structureless jellium-like description of the ionic background is involved, the LSPR broadening is shown to be governed by the electronic density tail probing the nonharmonic part of the interaction in the vicinity of, and beyond, the classical particle surface. Third the background-induced dielectric effects are investigated through TDLDA calculations and analyzed in the frame of the GKT. Fourth, both the classical “ballistic” model and the popular QBM, where the damping is mainly ruled by the extent of the potential well confining the electrons, are analyzed in view of the self-consistent TDLDA calculations and the implications of the GKT.

2. OUTLINE OF THE MODELING In this work and Paper I large silver nanoparticles, up to 40000 atoms, are involved. In this respect, a structureless jellium-type background description3538 is quite appropriate for investigating mean size trends over such a wide size range and for comparison purpose between various embedding matrixes or surface features. Since our interest is focused on the electron confinement contribution alone, in relation with the classical and QBM approaches, this simplified background modeling is quite justified. Effects from the “bumps” in the effective potential well confining the electrons, which would originate from the granular ionic structure, are patently more related to the electron-ion/ phonon scattering contributions. Therefore, spherically symmetric jellium-like ionic distributions, disregarding the discrete 3D ionic lattice, are involved throughout this work. The computations are carried out thanks to a home-developed computer code based on the general equations of the KohnSham (KS) density functional theory (DFT) TDLDA formalism. The dielectric effects (polarization, screening, and absorption) of the underlying backgrounds (core electrons and surrounding matrix) on the conduction electron response are taken into account through their respective classical bulk dielectric functions [ε1(ω) and ε2(ω), respectively] according to the generalized theory developed in previous papers.41,42 The reader is referred to section A of the Supporting Information for the details of the theoretical background and the notations. The infinitesimal parameter δ entering the Green function (eq A10 in the Supporting Information) deserves to be carefully discussed in the present work. In practical calculations a finite value is used, and this parameter turns out to act as a smoothing 14099

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parameter. Actually δ may be considered as introducing phenomenologically line broadening effects that are not included in the modeling, such as, for instance, the additional line broadening resulting from the removal of the 2 (2l + 1) single electron level degeneracy by either the 3D structure of the ionic lattice or the surface roughness and faceting (loss of the spherical symmetry). In standard jellium-type models, involving no dielectric media, this parameter amounts to attributing an intrinsic width 2δ to each p-h excitation line (Lorentzian-shaped peak). Therefore, in the absence of surface-induced Landau damping, that is, in the absence of coupling between collective and intrinsic degrees of freedom, the plasmon bandwidth in such models is expected to be equal to 2δ. In TDLDA computations carried out in the frame of such jellium-type models, this asymptotic LSPR bandwidth 2δ is indeed recovered for very large sizes. In the context of the present work, the value 2δ has thus to be identified with the bulk contribution pΓ0 in eq 1. 2δ has been set to the value 120 meV, consistent with the experimental data,40 in most calculations. It turns out that this value is small enough to resolve the fragmentation pattern of the LSPR band in the small and medium size ranges but sufficiently large to ensure reasonable computational times (a photon energy step equal to 6.8 meV is used for δ = 60 meV).

3. RESULTS 3.1. Size Evolution of the LSPR Bandwidth: Estimation of the g-Factor. Many-body TDLDA calculations, beyond the

independent-electron approximation, have been thus carried out to estimate the g-factor characterizing the confinement contribution to the LSPR bandwidth of silver nanoparticles. A large size range, up to R ≈ 5.5 nm, has been investigated for silver particles embedded in silicon dioxide (SiO2 glass), aluminum oxide (Al2O3), and vacuum (ε2 = 1). The matrix dielectric constants are taken from the experimental data given in a previous work.43 In the spectral range of interest ε2 is on the order of 2.16 and 3.2 for SiO2 and Al2O3, respectively. A homogeneous spherical hard-walled ionic distribution of radius R = rsN1/3 (standard jellium density, with rs = 3.02 bohr) and a single dielectric interface located at R have been assumed. Additional calculations involving either a soft background wall or two concentric dielectric interfaces have been also performed. Depending on the size, the plasmon band is more or less fragmented (see Figure 1 in Paper I and Figure 1 in the present manuscript). For small sizes the extent of the oscillator-strength distribution is much larger than the intrinsic line broadening 2δ. The structured LSPR pattern reflects the surface-induced Landau damping mechanism, which yields a well-developed fragmentation into single p-h excitations owing to the large energy spacing of the quantized KS-orbitals involved in the various allowed dipolar transitions. These figures clearly show that numerous single p-h excitations, in quasi-degeneracy with the collective excitation, underlie the Lorentzian curve-shaped LSPR band computed with large δ-values, i.e., are responsible for the plasmon decay in the time domain. These Figures point out also that the fragmentation into p-h excitations (i.e., the LSPR broadening) is strongly matrix dependent. The matrix dependence results essentially from (i) the change of the KS-confining potential and consequently of the set and ordering of the occupied and unoccupied single electron levels (n,l) and (ii) the spectral and matrix dependence of the dynamical screening. As the size increases, the LSPR

Figure 1. (a) Absorption cross sections of Al2O3-embedded Ag5000 silver particles (D ≈ 5.5 nm), assuming a homogeneous jellium density, for three values of the intrinsic line broadening, namely, 2δ = 20 meV (black curve), 2δ = 120 meV (thick gray curve), and 2δ = 4 meV (thin gray curve). For 2δ = 4 meV the spectrum has been computed in the energy range 2.22.8 eV and divided by 10. The dashed black curve is a Lorentzian-shaped curve fitting of the spectrum with 2δ = 120 meV. The vertical bar indicates the classical plasmon resonance energy pω0*. (b) Same as (a) for Ag832 (D ≈ 3 nm).

fragmentation is less conspicuous and is often reflected only through tiny superimposed shoulders on the surface plasmon band. Obviously the fragmentation of the absorption spectra into distinguishable p-h excitation lines depends critically on the value of the intrinsic line broadening entering the Green function (δ parameter in eq A10 in the Supporting Information). In both figures the vertical bars are located at the classical LSPR energies pω0*. These asymptotic values, toward which converge the TDLDA predictions in the large size range, have been determined from simple classical quasistatic calculations involving the same background dielectric functions. The appropriate Drudelike parametrization with pωP = 8.98 eV and pΓ0 = 2δ has been taken for modeling the dielectric function associated with the conduction electrons. Since the LSPR band in silver is located above the interband threshold Ωib and Γ0 , ω*, 0 one has, with a great accuracy

ω0 ωp =½Re½ε1 ðω0 Þ + 2ε2 ðω0 Þ1=2

ð2Þ

From these spectra one can see that the centroid of the fragmented collective plasmon band [pωLSPR(R)] is noticeably red-shifted relative to the classical LSPR energy pω*, 0 and strongly broadened and damped. In the standard jellium model (ε1,2(ω) = 1), part of the red-shift can be estimated thanks to sum-rule formulas and is related to the amount of electrons spilling out the jellium surface (the so-called spillout 14100



Figure 2. Evolution of the LSPR bandwidth at half-maximum as a function of the inverse of the particle diameter, for SiO2-embedded silver particles, assuming an intrinsic line broadening equal to 2δ = 120 meV (black triangles). The empty squares and stars correspond to models involving either a steplike jellium density at the surface or surface skins of ineffective background polarizability, respectively (see text). The data have been obtained from a systematic least-squares Lorentzian curve fitting of the computed absorption cross-section spectra (see Figure 1). The straight lines are linear fits of the data for SiO2-embedded (full line), Al2O3-embedded (upper dotted-dashed line), and free (lower dotteddashed line) silver particles. The black squares are the experimental data.40 The straight dashed line fitting the experimental data points in the small size range is obtained using eq 1 with pΓ0 = 0.12 eV, vF = 1.39 108 cm s1, and g = 0.73. In the inset are shown the sizedependences of the LSPR bandwidths, for SiO2-embedded silver particles, when using a line width parameter 2δ equal to 20 meV (stars), 120 meV (triangles), and 180 meV (squares). In the inset the asymptotic values (at 1/D = 0), for the three δ-values, are also indicated (17, 104, and 156 meV, respectively).

correction).36 The sum-rule red-shift underestimates however noticeably the LSPR frequency red-shift obtained from selfconsistent RPA or TDLDA calculations. Physically, the decrease of the plasmon frequency in this specific case (standard jellium model) is due to the fact that the restoring force by the background, i.e., dVjel(r)/dr, is lower beyond the jellium edge (eqs A12 and A13, with ε1,2(ω) = 1, in the Supporting Information). The same statement (that is, the correlation between the magnitudes of the frequency shift ω0* ωLSPR(R) and of the spillout electron density tail) can be inferred in the presence of dielectric media, but to my knowledge, no explicit analytical formula exists for quantifying this correlation in this case. Nevertheless, both figures indicate that the red-shift is more pronounced when the matrix refractive index is larger, i.e., the screening effects larger. The matrix effects are reflected in the self-consistent KS ground-state calculations by an enlargement of both the softness of the surface profile of the KS effective potential Veff[r, F0] and, concomitantly, of the electronic spillout tail, with increasing matrix dielectric index (see Figure 2 in Paper I). It may be conjectured that the broadening of the LSPR, as the red-shift, is directly related to the electronic density tail experiencing the Coulombic part of the electron-background interaction potential (eq A13 in the Supporting Information). This statement is supported by noting that the extent of the oscillatorstrength distribution (see Figure 1 in Paper I) is directly correlated with the magnitude of the matrix dielectric index

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and therefore with the softness of the KS-confining potential edge and concomitantly of the electron density tail (see Figure 2 in Paper I, corresponding to the specific size N = 440).44 The correlation between the LSPR broadening and the extent of the electronic density tail beyond the background radius R will be assessed in section 4. In the large size domain the optical absorption is found to converge steadily toward the classical prediction through a Lorentzian curve-shaped bunching of the dipole oscillatorstrength distribution in a narrow spectral range close to pω*0 (see Figure 1 in Paper I, corresponding to the large size N = 8000). In the large size range, an unambiguous determination of the size evolution of the LSPR broadening, in particular of the parameter g in eq 1, can thus be achieved. To minimize biased errors, which could depend on the selected method of data analysis, an identical procedure over the entire size range have been applied, consisting in a direct least-squares Lorentzian curve fitting of the computed absorption cross sections in the LSPR spectral range. The theoretical results obtained in the size range N = 83240000 (black triangles) and the experimental data (black squares),40 for SiO2-embedded silver particles, are shown in Figure 2. The full straight line is a linear 6-points fit of the theoretical results, involving the six largest sizes (range N = 500040000) and crossing the corresponding asymptotic value ω0* determined by classical calculations, that is, 0.104 eV for SiO2embedded particles. It should be pointed out that, in the presence of dielectric media, the asymptotic value pΔω(∞) obtained for very large sizes is not equal to 2δ and depends on the first derivative of the real component of ε1(ω) at the resonance frequency ω0*.45 From simple classical calculations it can be shown that the asymptotic value is given by (expression evaluated at ω = ω*) 0 " #1 ω2 + Γ20 ðω2 + Γ20 Þ2 ∂Re½ε1 ðωÞ 1+ Δωð∞Þ ¼ Γ0 ∂ω ω2 2ωω2p "

ω3 ∂Re½ε1 ðωÞ Γ0 1 + 2 ∂ω 2ωp

#1 ð3Þ

where Γ0 is the damping parameter entering the dielectric function corresponding to the conduction electrons (Drude model). The linear fits of the theoretical results for silver particles in vacuum and embedded in alumina are also shown for completeness (lower and upper dotteddashed lines, with pΔω(∞) = 0.086 eV and pΔω(∞) = 0.11 eV, respectively; see Figure 3 in Paper I where are plotted the theoretical data points for both matrixes). The slopes in Figure 2 correspond to g-values equal to approximately 0.09 (vacuum), 0.32 (SiO2), and 0.42 (Al2O3) (vF = 1.39 108 cm s1 for bulk silver).46 The magnitude of the g-parameter is consistent with that of the corresponding matrix index, and, as conjectured before, is correlated with the extent of the electronic spillout tail (see Figure 2 in Paper I). For small sizes the electron level spectrum involves far separated discrete levels or level bunches (due to the spherical symmetry) and results in a strongly size-dependent fragmentation of the LSPR band. This explains the large departure of the theoretical data from the asymptotic 1/R law when the data analysis procedure is applied in this size range. This feature is consistent with a semiclassical study of the size-evolution of the LSPR bandwidth, where a damped oscillatory size-dependence in the small size range, due to electronic shell effects, is 14101


The Journal of Physical Chemistry C predicted.47,48 I have also verified that the estimation of the g-parameter from the size evolution of the TDLDA spectra does not depend on the δ-value that is chosen for setting the intrinsic line width. This was achieved in calculating absorption spectra over the entire size range for SiO2-embedded silver clusters with various δ values (2δ = 20, 120, and 180 meV) and plotting the fwhm bandwidths as a function of 1/D. The three slopes differ by less than five percent, confirming that the g-value extracted from the TDLDA spectra by the selected data analysis procedure is almost independent of the δ parameter (see the inset in Figure 2). The experiment, based on the spatial modulation spectroscopy (SMS) technique,1517,19,22 gives access to the absolute extinction cross sections of single nanoparticles down to a few nanometers size. It has been used to determine the LSPR bandwidth of single silver nanoparticles in the size range 2R ≈ 10100 nm, coated with 15 nm thick silica shells in order to ensure a well-controlled dielectric environment.40 Classical Mie theory calculations have been performed to on the one hand show that the comparison between theory and the available experimental data is quite justified and on the other hand to estimate the magnitude of the systematic errors made (see section B in the Supporting Information). The experimental data set exhibits an almost linear increase with 1/D in the small size range where absorption dominates. The strong increase of the experimental LSPR bandwidth for large sizes (typically for D > 25 nm; see Figure 2) is explained by multipolar effects,5 which are not included in the theoretical approach since the spatial dependence of the applied electric field is disregarded (no retardation effects), and radiation damping. Figure 2 shows that the computed confinement-induced contribution to the LSPR bandwidth in small SiO2-embedded particles is insufficient for explaining the overall finite size-induced broadening observed in experiment. From the experimental data the value gexp ≈ 0.7 ( 0.1 has been estimated from eq 1. This suggests that the quantum confinement, that is, the surface-induced Landau damping, is not the only mechanism ruling the size-evolution of the overall LSPR broadening in this size range. In ref 40, an additional contribution, arising from the size-dependence of the electronphonon scattering rate, has been invoked. A corrected value for the pure quantum confinement effect of g ≈ 0.45 ( 0.1, in better agreement with the one computed here, was then estimated. Additional surface effects at the Ag/SiO2 interface, lattice contraction in the outermost atomic shells, as well as the corrugation of the potential well and defects might be responsible for additional size dependencies which are not discussed here (see hereafter). 3.2. Dependence of the Damping on the Surface Background Modeling. The confinement-induced contribution has been estimated in the frame of a specific background modeling, namely, in assuming a hard-walled jellium density and a single dielectric interface located at R. To estimate to what extent these specific model assumptions may be relaxed, two additional series of TDLDA calculations have been performed in modifying the modeling of the particle background and of the metal/matrix interface, for SiO2-embedded silver particles. In the first one a steplike surface jellium density has been assumed, that is, F+(r) = 0 0 RF0+ in the radial R range R-d < r 3< R+d (d being given, d is set by the equation F + (r)dr = 4πR F0+/3 = Q+). In this radial range the background dielectric function was assumed to be proportional to the jellium density [that is, ε(ω) = Rε1(ω)]. The second series corresponds to the model introduced in order to interpret

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the size evolution of the resonance frequency in small noble metal clusters.49 This model involves (i) a skin of ineffective ion polarizability over the radial range R-dc < r < R, which is rooted in the spatial localization of the d-electron wave functions and (ii) a vacuum rind extending over the radial range R < r < R+dm mimicking the local matrix porosity and/or defects at the metal/ matrix interface. In the radial region R-dc < r < R+dm the background dielectric function is thus equal to 1. Both models involve thus two concentric dielectric interfaces. The formula giving the effective interaction between two elementary charges in the presence of two concentric spherical dielectric interfaces Vc(r1,r2,ω) (see the Supporting Information), given in the Appendix B of ref,50 has been used to modify appropriately the expression of the electron background interaction Vjel(r). I present here the results obtained with the parameter set R = 0.5, d = 2, dc = 3.5, dm = 2 (lengths in Bohr radius unit a0). Computations show that, in a first approximation, only the surface profiles of the self-consistent confining potential, and concomitantly the electron density tail at the particle surface, are (slightly) modified. Nevertheless, these tiny changes, located in a very narrow (relative to R) radial range, modify noticeably the LSPR bandwidth, especially in the small size range. In the first model the g-parameter is slightly increased (empty squares in Figure 2), whereas it is decreased in the second one (empty stars), confirming that the confinement-induced contribution to the LSPR broadening is intimately related to the electronic spillout tail probing the surface region of the particle, or equivalently to the surface profile of the KS-confining potential. These additional TDLDA calculations indicate however that, as long as reasonable structureless jellium-like models are considered (d, d0 , dc, dm of the order of a few Bohr), the confinementinduced LSPR damping contribution remains insufficient to interpret the experimental data.

4. ANALYSIS OF THE CONFINEMENT-INDUCED LSPR BROADENING 4.1. The GKT. It is often stated that the LSPR broadening in small metal particles is the direct consequence of the finite extent of the potential well confining the electrons and correlatively of the electronic spectrum quantization. The above theoretical results, which involve similar confining potential radii but computed for different matrixes, suggest rather that a major role is played by the surface profile of the ground-state effective potential Veff[r,F0] and concomitantly by the electronic spillout. Actually the confinement is not a sufficient condition, in contradiction with the classical billiard picture and the QBM predictions. The LSPR broadening is in fact intimately related to the external (i.e., the interaction with the background) potential energy experienced by the electrons in the ground state. This property is a direct consequence of the collective nature of the plasmon excitation: the motion of the electronic system as a whole, that is, of the center-of-mass (CM) coordinate, is governed by the external forces acting on the conduction electrons. From early works on quantum dots involving finite electron systems confined in semiconductor nanostructures, it was established that the damping of the collective oscillatory motion of the electron gas results to a great extent from the departure of the external potential confining the electrons from an exact quadratic form.5155 When the external potential is purely harmonic it was shown that, on the one hand, the CM motion of the electron gas separates completely from the 14102



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of the harmonic-oscillator state. In the quasistatic limit the interaction with a monochromatic field linearly polarized along the e direction is WðtÞ ¼ ½

Figure 3. Normalized ground-state densities (a) and self-consistent KS potentials (b) of the particle Ag1314 (D = 3.5 nm), corresponding to the Hjel (solid black curves), Hhar (gray curves), and Hstep (dashed curves) jellium-type model Hamiltonians. The three Hamiltonians differ only in the electron-background interaction (see text). The dashed curves are related to a model involving a step-like inhomogeneous jellium density at the surface. The short horizontal lines in (b) indicate the Fermi levels in the three systems. The solid vertical bars indicate the location of the classical particle radius R = rsN1/3(homogeneous jellium edge). The dashed vertical bars in (a) show the extent of the surface skin of lower jellium density (F0+/2) in the Hstep model Hamiltonian.

intrinsic internal motions and, on the other hand, the dipole excitation in the nonretarded limit occurs at the eigenfrequency ω0 characterizing the curvature of the external potential (GKT). In the present context this means that, for small sizes, the surface plasmon excitation has an infinite lifetime if the electron positively charged background interaction is purely harmonic along the x-, y-, and z-directions. For instance, by assumption of an isotropic quadratic external potential of the form Vhar(r) = meω02r2/2, the Hamiltonian representing Ne interacting electrons of effective mass me, namely Ne Ne 1 2 1 1 ð4Þ H ¼ ∇i + me ω20 ri2 + 2m 2 jr rj j e i i¼1 i R+d0 . With such a positive charge density the external electron-background potential is only harmonic in the radial range r < R-d. The results for the cluster size N = 1314 and the parameter set (R = 0.5, d = 2a0, d+d0 = 3.78a0) are displayed in Figure 3 (dashed curves) and Figure 4c. The shell thickness (d+d0 ) is very small relative to the overall cluster radius R in this illustrative example, noticeably smaller than the distance between two neighboring atoms in bulk silver. As expected, reducing the jellium density at the surface results in an increase of the softness of the confining potential wall and an enlargement of the “number of electrons” spilling out beyond the classical particle radius R. Additional calculations involving other sizes and/or shell thicknesses have confirmed these general features. With respect to the absorption cross-section noticeable red-shift and broadening of the LSPR band are induced, with a magnitude directly related to the extent of the surface skin of smaller jellium density. One has to mention that the dependence of the LSPR-characteristics on the diffuseness of the electron density at the surface was reported a long time ago by several authors within classical61 or hydrodynamic theories.62,63 However, owing to the crudeness of the theoretical approaches and above all because no external electron-background interaction enters the modeling (the electron density F0(r) is taken as an input data), the results reported in these papers provide only qualitative information about the correlation. In particular, even in prescribing the appropriate electron density (gray curve in Figure 3a), the LSPR bandwidth predicted within these approaches would be large and thus inconsistent with the GKT. Such trends are not specific of changes of the ionic density at the surface, as exemplified previously. In fact similar effects on the optical response can be likewise reproduced in assuming a homogeneous jellium, spherical at large scale, with a rough surface. Support for this statement can be found in another work.64 In this work it was shown that when the surface roughness of the confining potential (of spherical overall shape) involves spatial periods on the order or smaller than the Fermi wavelength [λF ≈ 3.3rs (∼0.5 nm for silver)] the electrons are only sensitive to the spherical average of the rough potential. Corrugation at the atomic scale is basically of this kind. For instance, the spherical average of a hard-walled finite-depth flat potential well, involving a short-scale corrugation (of radial

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thickness A), exhibits a soft surface profile over a radial range of thickness A. This argument can be in the same way applied either to a rough homogeneous jellium or to the external electron-rough jellium interaction. Therefore a homogeneous jellium with a rough surface is expected to induce a large red-shift and broadening of the LSPR band, as those induced by spherically symmetric jellium distributions with soft or stepped edges. The case of a large-scale surface corrugation has to be discussed specifically. The following comments also concern particles with specific structured boundaries, for instance, involving small plane facets (that are favored owing to the underlying periodic crystalline structure), and to a lesser extent very strongly distorted particles. In this case, the lost of the spherical symmetry and the large-scale “corrugation” of the particle surface is effectively experienced by the conduction electrons (the Fermi wavelength is noticeably smaller than the surface corrugation length scales). First, as in the case of a short-scale surface corrugation, the radial range where the electron-background interaction deviates strongly from a strict quadratic law is enlarged, implying, in view of the GKT, an increase of the LSPR damping. Let us emphasize that, when the spherical symmetry of the jellium density is lost (and in contrast to spherically symmetric homogeneous jellium involving a soft or step-like surface density) the inner pure quadratic interaction is broken in the entire particle volume but remains however approximately preserved in the largest part of the particle volume. The magnitude of the departure from harmonicity strongly depends on the overall particle shape and on the length scales of the corrugation. Second, considering the Landau damping mechanism picture, the removal of the 2 (2l + 1) single electron level degeneracy results in a broadening of each p-h excitation line and therefore of the fragmented LSPR band. So, it can be conjectured that, as in the case of a spherical homogeneous jellium with a soft or steplike surface density, an enlargement of the LSPR bandwidth will systematically result from any shape deformation from a perfect sphere (corrugation, defects, faceting). Finally, let us conclude that, in the extreme case of a strongly anisotropic particle shape, for instance, an ellipsoidal one, the optical response will be anisotropic as well, and the LSPR bandwidth is expected to be dependent on the direction of the collective electronic oscillatory motion, that is, on the polarization of the applied electric field. This dependence can have a 2-fold origin, extrinsic and intrinsic. The first one (referred to as the “extrinsic” factor) is a simple dielectric effect (see eq 3) and is due to the fact that the LSPR frequencies corresponding to the various polarizations lie in most cases in different spectral ranges. The second one (“intrinsic” factor) results from the possible dependence of the damping parameter Γ0 (see eq 3) on the direction of the oscillatory collective motion. From a quantum-mechanical point of view (Landau damping mechanism), and in the frame of the GKT approach, this dependence would result from that of the coupling between the intrinsic motions (Hamiltonian Hint in eq 5) and the CM motion along the specific direction that is involved (in the general case, three harmonic frequencies ω0,55 associated with the axes x, y, and z are involved in eqs 4 and 5. Three one-dimensional harmonic oscillator CM Hamiltonians are thus involved, giving rise to three distinct coupling matrix element sets when the departure of the electron-background interaction from harmonicity is introduced). These model calculations, and the above comments, indicate that the damping of the plasmon is strongly dependent on the ionic structure and/or density near the surface. Noticeable 14105



Figure 5. (a) Solid line curves: absorption cross sections (intrinsic line width 2δ = 120 meV) of matrix-embedded Ag440 particles computed within a model involving a harmonic electron-background interaction over the entire radial range for defining the ground-state properties [see eq A12 in the Supporting Information; ε1,2(0) = 3.73]. The inner dielectric function is assumed to be frequency independent (ε1(ω) = 3.73). ε2 is the frequency-independent matrix dielectric function. Dashed line curve: absorption cross-section computed with ε1(ω > 0) = 3.73 + 3i and ε2 = 6. (b) Solid line curves as in part a. Dashed line curves: absorption cross sections computed taking into account the spectral dependence of the interband dielectric function of silver. (c) Solid line curves as in (b) (dashed line curves) except that the harmonic electronbackground interaction has been replaced by the Coulombic one beyond R for calculating the ground state properties (eq A13 in the Supporting Information). Dashed line curve: absorption cross-section computed with ε1(ω > 0) = 3.73 + i and ε2 = 3.73. In the three figures the vertical bars indicate the plasmon resonance energies pω*, 0 where ω* 0 obeys the equation ω = ωp/[Re[ε1 (ω) ] + 2ε2]1/2. In (c) these values are the classical Mie predictions toward which the LSPR frequencies converge in the large size limit.

changes of the LSPR lifetimes may be induced by slight modifications of the ionic distribution at the particle surface. In experiments, when large metal particles are involved (typically N > 100), numerous isomers of low energy or partially faceted particle shapes are produced. Each of these structures is probably characterized by a specific damping of its own plasmon resonance, despite the overall shapes can be considered as quasi identical. This probably explains why LSPR bandwidth measurements are indeed very difficult to rationalize or compare with reliability if different experimental conditions prevail and if no detailed information about the surface structure is available. On the other hand, from a theoretical point of view, predictions are extremely sensitive to the assumptions concerning either the ionic structure modeling or the prescribed effective confining potential. This fact was overlooked in early works based on the

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non-self-consistent QBM approach in which the confining potential is conveniently imposed for ensuring quasi-analytical calculations. To my opinion, since practical calculations on large particles involve necessary assumptions about the ionic background (as, for instance, in smoothing the granular inner structure or particle surface), only approximate low-bound LSPR-bandwidth estimates are provided by any model calculations aiming at tackling the confinement contribution to the overall LSPR damping. In this context single-particle experiments combined with high-resolution imaging technique seem indispensable in providing structural information and relevant constraints with regard to modeling. It should be noticed that, fortunately, the damping and the redshift of the LSPR band are strongly correlated, restricting therefore additional assumptions that could be invoked. 4.4. Dielectric Effects. The consequences of the GKT have been illustrated in sections 4.2 and 4.3 in the frame of standard jellium-type models (the backgrounds are optically inert). In the presence of polarizable dielectric backgrounds the theoretical analysis is much more complex. First, the electronelectron Coulomb interaction is no longer translation invariant because the effective interaction between two elementary charges depends now on their locations relative to the dielectric interface, owing to induced surface polarization charges. Second, the spectral dependence of the background dielectric functions has to be taken into account. In consequence, a formulation of the problem with an Hamiltonian cast in standard form, namely, H = Hcm + Hint + W, cannot be rigorously derived in the presence of dielectric media. So I have resorted to numerical calculations for analyzing this more complex situation. I summarize the major results obtained through this investigation (see Figure 5). I have first carried out a series of TDLDA calculations assuming ε1 = ε2 = ε (ε real), with a harmonic electronbackground interaction over the entire radial range for defining the ground-state properties and the independent-electron correlation function χ0(r,r0 ,ω) (model referred to as the Hhar-model), namely, Vhar(r) = (1/2ε)m0ω02[r2 3R2] [eq A12 of Supporting Information, with ε = 3.73 (static interband index in silver)]. As expected, as long as the spectral dependence of the dielectric functions is disregarded, the absorption cross section consists of a single Lorentzian-shaped peak centered at the classical frequency ω00 = ωp/(3ε)1/2 = ω0/(ε)1/2, the fwhm of which is equal to 2δ (gray curve in Figure 5a). These findings are indeed obvious in view of the Hamiltonian formulation (eqs 4 and 5) since changing the dielectric constant in the full space amounts merely to multiplying the electronelectron Coulomb interaction by 1/ε and thus only Hint(ri0 ,pi0 ) is affected (equivalently, this amounts merely to changing the value of the vacuum permittivity). Modifying the inner background and/or matrix dielectric functions for positive frequencies [ε1(ω > 0) 6¼ ε and/or ε2(ω > 0) 6¼ ε] results in a change of the LSPR frequency ωmax, which is no longer equal to the curvature ω00 characterizing the quadratic electron-background interaction, owing to dynamical frequency-dependent screening effects (black full line curves in Figure 5a, computed for ε2(ω > 0) = 1 and 6). It can be noticed also that for this specific electron-background interaction in the ground state and as long as real ε1,2(ω > 0) values are involved, the dynamical screening-induced shift of the LSPR band occurs without broadening. This feature strongly suggests that, for nonabsorbing backgrounds in the relevant spectral range, the fragmentation of the LSPR band into p-h excitations is intimately 14106


The Journal of Physical Chemistry C related to the “number of electrons” experiencing the non harmonic part of the screened electron-background interaction in the ground state, as in standard jellium-type models (no dielectric media) for which the GKT can be directly applied. By adding an imaginary component in the dielectric function of the inner background in the spectral range of the LSPR band [ε1(ω > 0)complex] a strong damping and broadening of the resonance band is induced, without noticeable change of the band maximum frequency (dashed curve in Figure 5a; ε1(ω > 0) = 3.73 + 3i, ε2(ω > 0) = 6). Physically the decay mechanism corresponds to the coupling of the n = 1 plasmon state with the continuum of p-h excited states involving the background core electrons, namely, the states corresponding to excitations of valence-band core electrons into unoccupied levels above the Fermi energy in the conduction band.65 This efficient decay mechanism is actually the main factor ruling the LSPR broadening in gold and copper particles embedded in low-index matrixes. This explains why the plasmon band emerges hardly from the rising edge of the interband transitions in these systems.66 In Figure 5b the dashed curves have been obtained by taking into account the spectral dependence of the interband dielectric function of silver (real below about 4 eV) and are compared with those plotted in Figure 5a (full line curves). As seen previously, no broadening of the LSPR band by the Landau damping mechanism is observed. The very tiny width change observed for high matrix indexes is a simple classical dielectric effect (see eq 3). The noticeable narrowing of the LSPR band for ε2(ω > 0) = 1 results from the large increase of ∂Re[ε1(ω)]/∂ω below the interband threshold Ωib. It should be pointed out that the frequency at the band maximum, ωmax, is not the solution (noted ω*) 0 of eq 2 (except in the specific case corresponding to the gray curve in Figure 5a) but may be, either red-shifted or blueshifted, relative to ω0* (values indicated by vertical bars in Figure 5). In the second series of calculations the harmonic interaction Vhar(r) has been replaced by the screened Coulombic interaction Vjel(r) = Q+/(εr) (eq A13 in the Supporting Information) beyond the jellium radius R for defining the ground-state properties and the independent-electron correlation function χ0(r,r0 ,ω) (Hjel model). In Figure 5c are plotted the absorption cross sections in taking into account the spectral dependence of the interband dielectric function of silver (full line curves). The dashed curve corresponds to a model calculation involving a frequency-independent complex inner dielectric function [ε1(ω > 0) = 3.73 + i] illustrating a fictitious additional damping induced by interband transitions. Direct comparison with the dashed curves in Figure 5b shows that the nonharmonic part of the external potential probed by the spillout electrons in the ground state is, as in standard jellium-type models, directly responsible for (i) the broadening of the LSPR band through the Landau damping mechanism and (ii) the systematic red-shift of the LSPR frequency ωmax relative to the classical value ω*. 0 These model calculations provide strong support to the discussion reported before concerning the direct relationship between on the one hand the LSPR broadening magnitude and on the other hand the electronic spillout tail probing the Coulombic part of the electron-background interaction.

5. COMMENTS ON THE CLASSICAL “BILLIARD” PICTURE AND THE QBM As exemplified throughout this work the classical picture is unsuitable for quantifying the confinement-induced size

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dependence of the LSPR bandwidth, in particular the g-parameter characterizing the linear evolution in eq 1. Without ad hoc assumptions this phenomenological (of geometrical nature) approach leads to finite-size corrections that are basically only dependent on the size (and shape) of the particle, whereas the TDLDA investigations have pointed out the striking sensitiveness of the LSPR bandwidth to the surface properties, the matrix refractive index, the particle charge, and the profile of the ionic background edge. Depending on the system, g-values ranging from 0 (harmonic electron-background interaction) to unity (matrix of very high refractive index, soft or rough surface ionic density) can be obtained. More fundamentally it has been shown that the confinement-induced contribution is not directly correlated to the size of the confining potential but rather ruled by the electronic spillout tail experiencing the nonharmonic part of the electron-background interaction. The QBM suffers from the same deficiencies since the surface details, thought to be of no importance, are disregarded from the outset. As in the classical picture the finite size corrections in the QBM depend essentially on the overall radius of the confining potential only. However, it should be stressed that the approximation consisting in involving for numerical convenience an infinite-depth hard-walled potential is not the central point to be called into question. Indeed, even in involving the self-consistent confinement potentials appropriate to each background modeling, the QBM approach yields quasi similar LSPR broadenings and size dependencies and is thus unable to (i) fulfill the physical consequences rooted in the GKT and (ii) ensure the strong sensitiveness of the LSPR damping to the surfaces properties. Briefly the QBM67 is actually grounded on the following classical equations Γ ≈ 2εim/|∂εre/∂ω| and ε = 1 + (R0/ε0V) evaluated at the resonance frequency ω*, 0 where ε = εre + iεim is the complex dielectric function of the metal, V the particle volume, and R0/V the polarizability per unit volume computed in the independent electron approximation (eq A11 in the Supporting Information), “suggesting” that the size dependence of Γ is given by that of the imaginary part of R0. In fact, except for very small sizes, the oneelectron level spectrum is weakly dependent on the profile of the confining potential wall. In consequence the finite size-induced damping contribution in the QBM will be to a large extent independent of the surface details and will be, for example, almost identical for the three potential wells plotted in Figure 3b. To support this statement free, i.e., independent-electron, absorption cross sections (eq A1 in the Supporting Information, with R replaced by R0) are displayed in Figure 4 (gray curves), for the model Hamiltonians Hhar, Hjel, and Hstep [the corresponding confining potential wells are plotted in Figure 3b]. The free responses are very similar proving the weak influence of the surface profile of the confining potential on the one-electron spectrum and on the free polarizability, which are both mainly determined by the radius of the potential well. In consequence, in contrast with what is stated in the QBM, the imaginary component Im[R0(ω)] evaluated at the resonance frequency is probably unrelated with the magnitude of the LSPR damping. Detailed analysis of the failure of the QBM, which has to be traced back to the irrelevancy of the two previous classical equations in the present context, is out the scope of this work. It should be noted that the free responses, characterized by oscillator strength distributions located in the low-energy range, are dramatically different from the corresponding full TDLDA responses. This stresses that the screening effects and the correlations induced by the long-range Coulomb interactions 14107


The Journal of Physical Chemistry C are essential to concentrate the oscillator strength distribution in a narrow spectral range close to the classical frequency ω0*. In the TDLDA formalism these screening effects are entirely described through the dynamical (time varying at the frequency ω) selfconsistent modification of the effective potential confining the electrons (eq A6 in the Supporting Information). Contrary to the free response, the full TDLDA response, in particular the LSPR bandwidth, is strongly dependent on the surface profile of the self-consistent KS-potential and fulfill the GKT.

6. CONCLUSION In this work the confinement-induced LSPR damping contribution in silver clusters, free or embedded in transparent matrixes, has been investigated over a large size range within the self-consistent TDLDA formalism, appropriately modified to take into account the dielectric/screening effects induced by the ionic background and the embedding matrix. The size evolution of the LSPR bandwidth resulting from the quantum confinement obeys a matrix-dependent linear law as a function of the inverse particle diameter, which is characteristic of surface-induced effects. In the small size range the LSPR band is fragmented into well-resolved particle-hole excitations (Landau damping mechanism) that reflect the coupling of the coherent electronic oscillation with the intrinsic electronic motions. In this size range the quantized electronic level spectrum involves rather large gaps and strong quantum size effects are observed in the absorption cross sections. As the size increases, the oscillator strength distribution concentrates, through a Lorentzian-shaped profile, into a narrow spectral range, slightly red-shifted relative to the classical Mie frequency, the fwhm of which decreases linearly as a function of the inverse particle size. In the frame of the GKT picture, and as long as the ionic degrees of freedom are frozen, the damping of the plasmon excitation is shown to be rooted in the departure of the electronbackground interaction from a pure quadratic law. In structureless jellium-type models the damping of the collective excitation, or equivalently the broadening of the LSPR band, is ruled by the relative amount of electrons experiencing the nonharmonic part of the interaction which extends close to and beyond the particle surface. The model calculations reported throughout this work emphasize the strong sensitivity of the damping on features characterizing the particle surface, as for instance the surface ionic density and roughness, the profile of the confining potential wall, and the electronic spillout. Concerning the matrix dependency, this paper shows that, in addition to the chemical interface damping mechanism,25 which requires an atomistic description of the metal/matrix interface, simple dielectric/screening effects are responsible for part of this dependency. These features are not taken into account in the popular “ballistic” model nor in the QBM where, in particular, the infinite-depth hard-walled potential well excludes any matrix effect and sensitivity to the details close to the surface. In the case of SiO2-embedded silver clusters the theoretical parameter g characterizing the size-dependence of the LSPR bandwidth is consistent but noticeably lower than the experimental one. Besides the size-dependency of the electronphonon coupling, additional surface effects at the interface of the silver core-silica shell of the chemically synthesized nanoparticles might be also at the origin of part of the discrepancy. In the frame of the previous analysis, in relation with the GKT, the discrepancy has to be ascribed to the breaking of the quadratic

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electron-background interaction resulting from the nonhomogeneous granular ionic structure. In a complete modeling, including both the electronic and ionic dynamical variables, transfer of momentum and energy occurs between the electronic and ionic systems. However, in view of the very short lifetime of the collective plasmon excitation (typically on the order of 10 fs or less) and the low phonon energies, it can be conjectured that essentially momentum exchange between both systems may take place during this time scale, the plasmon energy being transferred into incoherent intrinsic electronic excitations (Hint in eq 5) via the Landau damping mechanism induced by the granular structure. It should be stressed that, at very low temperature and in the size range studied in this work, the subsequent energy transfer toward the ionic lattice, which can be followed in the time domain using the technique of pumpprobe spectroscopy,6870 is expected to be mainly mediated by the presence of the surface (finite size effects) since extended electronic Bloch states are not scattered by an infinite perfect periodic crystal. In any case, for such finite systems, the thermal fluctuations of the periodic ionic lattice (giving rise to the usual electronphonon coupling in crystalline bulk matter), the finite extent of the confined Bloch states (relaxing the usual solid-state physics selection rules) as well as the noncrystalline granular structure near the surface (structural disorder, defects, ...), are responsible for this transfer occurring on longer time scales. Concerning the fast energy transfer between the collective and intrinsic electronic degrees of freedom in small particles, estimating the respective contributions arising from on the one hand the equilibrium ionic structure and on the other hand the temperature effects is a difficult task but would be of particular interest from a fundamental point of view in the context of the nanoplasmonics.

’ ASSOCIATED CONTENT

bS

Supporting Information. Theory and relevance of the comparison between the TDLDA results (absorption spectra of matrix-embedded silver particles) and the experiment (extinction spectra of coreshell Ag@SiO2 particles) and analysis of previous works. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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Size Evolution of the Surface Plasmon Resonance Damping in Silver

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