Surface scattering contribution to the plasmon width in embedded Ag nanospheres
SSurface scattering contribution to theplasmon width in embedded Agnanospheres
R. Carmina Monreal, S. Peter Apell, ∗ and Tomasz J. Antosiewicz , Departamento de F´ısica Te´orica de la Materia Condensada C5, and Condensed MatterPhysics Center (IFIMAC), Universidad Aut´onoma de Madrid. E-28049 Madrid. Spain. Department of Applied Physics and Gothenburg Physics Centre,Chalmers University of Technology, SE-41296 G¨oteborg, Sweden Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland ∗ [email protected] Abstract:
Nanometer-sized metal particles exhibit broadening of thelocalized surface plasmon resonance (LSPR) in comparison to its valuepredicted by the classical Mie theory. Using our model for the LSPRdependence on non-local surface screening and size quantization, wequantitatively relate the observed plasmon width to the nanoparticle radius R and the permittivity of the surrounding medium ε m . For Ag nanosphereslarger than 8 nm only the non-local dynamical effects occurring at thesurface are important and, up to a diameter of 25 nm, dominate over thebulk scattering mechanism. Qualitatively, the LSPR width is inverselyproportional to the particle size and has a nonmonotonic dependence on thepermittivity of the host medium, exhibiting for Ag a maximum at ε m ≈ . © 2018 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (290.3700) Linewidth; (240.6648) Surface dynam-ics; (160.4760) Optical properties; (160.4236) Nanomaterials; (290.2200) Extinction.
References and links
1. W. Ekardt, “Dynamical polarizability of small metal particles: Self-consistent spherical jellium backgroundmodel,” Phys. Rev. Lett. , 1925–1928 (1984).2. Y. Borensztein, P. De Andr`es, R. Monreal, T. Lopez-Rios, and F. Flores, “Blue shift of the dipolar plasma reso-nance in small silver particles on an alumina surface,” Phys. Rev. B , 2828–2830 (1986).3. J. Tiggesb¨aumker, L. K¨oller, K.-H. Meiwes-Broer, and A. Liebsch, “Blue shift of the mie plasma frequency inAg clusters and particles,” Phys. Rev. A , R1749–R1752 (1993).4. C. Yannouleas, “Microscopic description of the surface dipole plasmon in large Na N clusters ( (cid:46) N (cid:46) ) ,” Phys. Rev. B , 6748–6751 (1998).5. S. Fedrigo, W. Harbich, and J. Buttet, “Collective dipole oscillations in small silver clusters embedded in rare-gasmatrices,” Phys. Rev. B , 10706–10715 (1993).6. E. Townsend and G. W. Bryant, “Plasmonic properties of metallic nanoparticles: The effects of size quantization,”Nano Letters , 429–434 (2012).7. C. S¨onnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reductionof plasmon damping in gold nanorods,” Phys. Rev. Lett. , 077402 (2002).8. C. S¨onnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, “Plasmon resonances in large noble-metalclusters,” New J. Phys. , 93 (2002).9. T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in singlemetallic nanoparticles,” Phys. Rev. Lett. , 4249–4252 (1998).10. S. Peng, J. M. McMahon, G. C. Schatz, S. K. Gray, and Y. Sun, “Reversing the size-dependence of surfaceplasmon resonances,” Proc. Natl. Acad. Sci. U.S.A. , 14530–14534 (2010). a r X i v : . [ phy s i c s . op ti c s ] S e p
1. C. Cirac`ı, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fern´andez-Dom´ınguez, S. A. Maier, J. B. Pendry, A. Chilkoti,and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science , 1072–1074 (2012).12. A. I. Fern´andez-Dom´ınguez, A. Wiener, F. J. Garc´ıa-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-opticsdescription of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. , 106802 (2012).13. T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescalingof the nonlocal optical response,” Phys. Rev. Lett. , 263901 (2013).14. A. Garcia-Etxarri, P. Apell, M. K¨all, and J. Aizpurua, “A combination of concave/convex surfaces for field-enhancement optimization: the indented nanocone,” Opt. Express , 25201–25212 (2012).15. R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Competition between surface screening and size quantizationfor surface plasmons in nanoparticles,” New J. Phys , 083044 (2013).16. Y. Fan, J. Li, H. Chen, X. Lu, and X. Liu, “Size-dependence of the effective electron-phonon energy relaxationin hollow gold nanospheres,” Opto-Electron. Rev. , 36–40 (2014).17. B. N. J. Persson, “Polarizability of small spherical metal particles: influence of the matix environment,” Surf. Sci. , 153–162 (1993).18. H. Baida, P. Billaud, S. Marhaba, D. Christofilos, E. Cottancin, A. Crut, J. Lerm´e, P. Maioli, M. Pellarin,N. Del Fatti, F. Vall´ee, A. S´anchez-Iglesias, I. Pastoriza-Santos, and L. M. Liz-Marz´an, “Quantitative determi-nation of the size dependence of surface plasmon resonance damping in single Ag@SiO nanoparticles,” NanoLett. , 3463–3469 (2009).19. E. Cottancin, G. Celep, J. Lerm´e, M. Pellarin, J. Huntzinger, J. Vialle, and M. Broyer, “Optical properties of noblemetal clusters as a function of the size: Comparison between experiments and a semi-quantal theory,” TheoreticalChemistry Accounts , 514–523 (2006).20. B. Palpant, B. Pr´evel, J. Lerm´e, E. Cottancin, M. Pellarin, M. Treilleux, A. Perez, J. L. Vialle, and M. Broyer,“Optical properties of gold clusters in the size range 2–4 nm,” Phys. Rev. B , 1963–1970 (1998).21. G. Mie, “Beitr¨age zur Optik tr¨uber Medien, speziell kolloidaler Metall¨osungen,” Ann. Phys. , 377–445 (1908).22. H. H¨ovel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: Bulk dielectricfunctions and chemical interface damping,” Phys. Rev. B , 18178–18188 (1993).23. A. Kawabata and R. Kubo, “Electronic properties of fine metallic particles. ii. plasma resonance absorption,”Journal of the Physical Society of Japan , 1765–1772 (1966).24. P. Ascarelli and M. Cini, “Red shift of the surface plasmon resonance absorption by fine metal particles,” SolidState Commun. , 385–388 (1976).25. R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B , 2871–2876 (1975).26. A. V. Uskov, I. E. Protsenko, N. A. Mortensen, and E. P. O’Reilly, “Broadening of plasmonic resonance due toelectron collisions with nanoparticle boundary: a quantum mechanical consideration,” Plasmonics , 185–192(2014).27. P. Apell and A. Ljungbert, “A general non-local theory for the electromagnetic response of a small metal particle,”Physica Scripta , 113–118 (1982).28. B. N. J. Persson and P. Apell, “Sum rules for surface response functions with application to the van der Waalsinteraction between an atom and a metal,” Phys. Rev. B. , 6058–6065 (1983).29. A. Liebsch, “Dynamical screening at simple-metal surfaces,” Phys. Rev. B , 7378–7388 (1987).30. P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. , 287–407 (1982).31. A. Liebsch, “Surface-plasmon dispersion and size dependence of Mie resonance: Silver versus simple metals,”Phys. Rev. B , 11317–11328 (1993).32. K.-D. Tsuei, E. W. Plummer, A. Liebsch, E. Pehlke, K. Kempa, and P. Bakshi, “The normal modes at the surfaceof simple metals,” Surf. Sci. , 302–326 (1991).33. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B , 4370–4379 (1972).34. K. Kolwas and A. Derkachova, “Damping rates of surface plasmons for particles of size from nano- to microm-eters; reduction of the nonradiative decay,” J. Quant. Spectrocs. Radiat. Transfer , 45–55 (2013).35. J. Lerm´e, H. Baida, C. Bonnet, M. Broyer, E. Cottancin, A. Crut, P. Maioli, N. Del Fatti, F. Vall´ee, and M. Pellarin,“Size dependence of the surface plasmon resonance damping in metal nanospheres,” J. Phys. Chem. Lett. ,2922–2928 (2010).36. J. Lerm´e, “Size evolution of the surface plasmon resonance damping in silver nanoparticles: Confinement anddielectric effects,” J. Phys. Chem. C , 14098–14110 (2011).
1. Introduction
Since the early 1980s calculations and measurements on the optical properties of ultra-smallparticles have seen important progress [1, 2]. Coming from the angstrom end of the size spec-trum we see an equally profound development in the field of clusters and the region wherethe two meet is especially interesting [3-5].Underlying this development are of course boththe advance in computational tools [6] as well as increasingly refined experimental techniques.specially methods which make it possible to produce and study individual particles of a well-defined shape and size underpin the rapid development of the field [7-10]. At the same time thepossible applications of plasmonic resonances in these systems, ranging from medical thera-pies, via biosensing to new devices are but a few examples of what is driving the field rapidlyforward [11-14].Despite the positive development described above there are still crucial aspects of plasmonicresonances in small particles that are not completely understood and hence not utilized to theirfull potential. In a recent paper [15] we presented a theoretical model for analyzing the sizedependence of the surface plasmon resonance of metallic nanospheres in a range of sizes downto a few nanometers. Within this model, we explicitly showed how different microscopic mech-anisms affect the energy of the surface plasmon in quantitative agreement with recent publishedexperimental results for Ag and Au.In this article we address a question related to the one considered in [15]. We investigatethe behavior of the width of the surface plasmon resonance as a function of the size of theparticle and the influence of its surrounding medium, because the understanding of the plas-mon decay rate is essential to control its spectral response. Whereas the energy shift is smallcompared to the peak position in an absorption spectrum of a small particle, the width is amuch more sensitive parameter to what takes place inside and around the particle. Mechanismsaffecting the width include shape changes, vibrations in the particle (with electron-phonon cou-pling dependent on particle geometry [16]) and scattering processes of the conduction electronscommon to bulk materials (electron-electron and electron-defect scattering). Chemical effectsassociated with the bonding of adsorbed molecules from a reactive host matrix to the particlesurface [17] or the presence of ligand molecules also contribute to the plasmon width. Radia-tion damping is not significant for the very small particles we consider in our paper. However,of major concern for us is another damping mechanism, the so called surface scattering relatedto the non-harmonic part of the surface potential where for a perfect harmonic confining po-tential there is no associated damping but now the direct excitation of electron-hole pairs in anelectronically very inhomogeneous region of space (known as spill-out region) plays a majorrole as we will show below. A basic question to address in this context is to what extent is thecollective mode broadening an intrinsic or extrinsic property of the small particle. This is anessential issue because it limits the electromagnetic field enhancement at its surface.We calculate the optical extinction cross-section of Ag spheres of nanometric size embed-ded in different host media, characterized by their permittivity ε m , assumed to be frequencyindependent. We take a top-down approach to the nanoparticles and, starting from a large size,make them smaller and smaller. For metal spheres less than 25 nm in diameter, the wavelengthsof interest are typically much larger than the size so that a quasi-static dipolar approximation isvalid to calculate the optical properties. Within this approximation, the surface scattering mech-anism for the damping of the surface plasmons is introduced in the theory using the frameworkpresented in [15], which includes non-local surface screening and size quantization effects, andwas used to reproduce the LSPR shift in frequency. The surface scattering mechanism gives asurface plasmon linewidth which depends on size and also on the permittivity of the surround-ing medium. In Ag, this width reaches a maximum as a function of the host permittivity, inde-pendently of size. We also analyze the quality factor of the surface plasmon resonance of theseAg spheres. We find that, when surface scattering is the dominant mechanism for the dampingof the surface plasmon, the quality factor shows a trend with size different than when radiativedamping dominates. Our study is motivated by recent quantitative experimental determinationof the size dependence of the surface plasmon resonance in single Ag nanoparticles coated withSiO [18]. In previous experiments [19, 20], the samples contained ensembles of particles witha dispersion in size which made a detailed investigation of this issue difficult. The remarkablegreement we find between the experimental and our calculated plasmon widths affirms ourapproach.
2. Theory
For the small particles we are interested in, the extinction cross-section can be written as afunction of frequency ω as σ ( ω ) = π ω c √ ε m Im [ α ( ω )] , (1)where c is the speed of light, α is the particle polarizability, and the surrounding medium isnon-magnetic. In the classical Mie theory [21], the polarizability of a small metal sphere ofradius R (diameter D ) and dielectric function ε ( ω ) is dominated by the electric dipole termwhich is given by α M ( ω ) = R ε ( ω ) − ε m ε ( ω ) + ε m . (2)Separating ε ( ω ) into its real and imaginary components, ε ( ω ) = ε ( ω ) + i ε ( ω ) , the imagi-nary part of Eq. (2) is Im [ α M ( ω )] = R ε m ε ( ω )( ε ( ω ) + ε m ) + ( ε ( ω )) , (3)which gives to Eq. (1) a nearly Lorentzian line-shape, with a peak at the frequency ω M (theclassical local surface plasmon frequency) a full width at half maximum Γ M fulfilling, ε ( ω M ) + ε m = Γ M = ε ( ω M ) (cid:104) ∂ε ( ω ) ∂ω (cid:105) ω M , (4a,b)respectively.The dielectric function of noble metals is usually written as the sum of the contributions ofthe d electrons ε d ( ω ) and the sp electrons, the second one being of the Drude form, as ε ( ω ) = ε d ( ω ) − ω p ω + i ω / τ b , (5)where ω p is the free-electron plasma frequency and τ b is the bulk relaxation time. Then, theposition and full width at half-maximum of the local surface plasmon resonance are obtainedfrom Eqs. (4a) and (4b) as ( ω M τ b (cid:29) ω M = ω p (cid:112) Re [ ε d ( ω M )] + ε m and Γ M = τ b + Im [ ε d ( ω M )] ω M ω p + ω M ω p (cid:104) ∂ Re [ ε d ( ω )] ∂ω (cid:105) ω M , (6a,b)respectively. Eqs. (6a) and (6b) show that this classical theory produces resonances which areindependent of particle size in the quasistatic approximation. However, it is well known exper-imentally that the frequency and width of the surface plasmons are size-dependent for particlesof radii which, in principle, are within the quasistatic regime. In particular, the width increaseswith decreasing size linearly in R . This fact has been phenomenologically taken into account inthe classical formulation by assuming that a relaxation time τ should be used in Eq. (6b) whichhas contributions from the usual inelastic scattering of the electrons in the bulk, τ b , and alsofrom the scattering at the boundaries, according to the expression τ = τ b + g s v F R , where v F is theermi velocity and g s is usually taken as an adjustable parameter [22]. A surface contributionto the relaxation time of the form expressed by the second term on the right hand side of thepreceeding equation has been computed by several authors using the so-called quantum-boxmodel, in which the electronic states are quantized in an infinite barrier potential at the surfaceof the particle [23-26].In this paper we introduce the surface scattering mechanism for plasmon damping usingthe theory presented in [15]. In this framework, non-local, dynamical effects occurring at thesurface of the metal particle, where electrons spill-out, are incorporated in the theory by meansof the complex length d r ( ω ) , defined as [27] d r ( ω ) R = (cid:82) dr r ( R − r ) δ ρ ( r , ω ) (cid:82) dr r δ ρ ( r , ω ) , (7)where δ ρ ( r , ω ) is the induced electronic density, r is the radial coordinate and the integralsextend to the whole space. The real part of d r ( ω ) is related to the position of the center ofgravity of the screening charge density at the surface and produces red/blue shifts of the surfaceplasmon frequency of spheres depending on whether the screening charge sits outside/insidethe jellium edge. The imaginary part of d r ( ω ) describes the absorption of energy from an ex-ternal probe creating surface excitations (surface plasmons and surface electron-hole pairs) andthe sign criterium used to define Eq. (7) implies that Im [ − d r ( ω )] is positive. The fact thatIm [ − d r ( ω )] is finite at the frequency of the surface plasmon indicates that this collective ex-citation has a finite lifetime because it is coupled and decays into the incoherent excitationof electron-hole pairs (Landau damping). These surface effects can be incorporated into thepolarizability of small spheres, yielding the generalization of the classical Eq. (2) as [27] α ( ω ) = R ( ε ( ω ) − ε m ) (cid:16) − d r ( ω ) R (cid:17) ε ( ω ) + ε m + ( ε ( ω ) − ε m ) d r ( ω ) R . (8)In the classical Mie theory, where the charge density is proportional to δ ( r − R ) , d r ( ω ) iszero and these surface effects are absent. In our approach Eq. (5) is generalized in order toinclude quantum size effects as ε ( ω ) = ε d ( ω ) − ω p ω − ∆ + i ω / τ b , (9)with ∆ playing the role of an energy gap introduced by the quantization of the electronic levelsof the nanoparticle. We should point out that the results we are going to present below are verylittle influenced by the effects of quantization due to size. This is because, as discussed in [15],quantum size effects are only important for spheres smaller than about 5 nm in diameter, whilethe experimental widths we will compare to are measured in spheres of diameter on the orderof and larger than 10 nm.The surface contribution to the absorption cross section can be analyzed in a simple way byassuming that ε ( ω ) is real, which is a good approximation for Ag in the range of frequencies ofinterest below the onset for interband transitions (less than 4 eV) where Re [ ε d ( ω )] (cid:29) Im [ ε d ( ω )] and ωτ b (cid:29)
1. Then the imaginary part of Eq. (8) readsIm [ α ( ω )] = R ε ( ω ) ( ε ( ω ) − ε m ) − Im [ d r ( ω )] R (cid:104) ε ( ω ) + ε m + ( ε ( ω ) − ε m ) Re [ d r ( ω )] R (cid:105) + (cid:104) ( ε ( ω ) − ε m ) Im [ d r ( ω )] R (cid:105) . (10)Therefore, for large values of R , Eq. (10) yields an absorption cross section which has anearly Lorentzian shape, having a maximum at the surface plasmon frequency ω s , and is giveny ε ( ω s ) + ε m + ( ε ( ω s ) − ε m ) Re [ d r ( ω s )] R = , (11)and full width at half-maximum Γ s Γ s (cid:39) ( ε ( ω s ) − ε m ) (cid:104) ∂ε ( ω ) ∂ω (cid:105) ω s Im [ d r ( ω s )] R (cid:39) ε m (cid:104) ∂ε ( ω ) ∂ω (cid:105) ω s Im [ − d r ( ω s )] R , (12)where ε ( ω ) denotes the real part of Eq. (9). Hence, this model gives a width of the surfaceplasmon resonance which increases linearly with R with a slope proportional to Im [ − d r ( ω s )] .A more detailed analysis, using for ε ( ω ) the complex form of Eq. (9), shows that these surfaceeffects will dominate over the classical bulk contribution to the width when 2 Im [ − d r ( ω s )] R ≥ ω s τ b .Using typical values, ¯ h ω s (cid:39) [ − d r ( ω s )] (cid:39) . ¯ h τ b (cid:39) .
02 eV, this relation is wellsatisfied for Ag spheres smaller than 30 nm in diameter, the sizes of interest in this work.It is interesting for the analysis of the results that we will present below to obtain the limitof Eq. (12) for very large values of the permittivity of the surrounding medium. Assume ε m is so large that the surface plasmon frequency is in the region where ε d ( ω s ) (cid:28) ε m and ω s (cid:39) ω p / √ ε m . Then (cid:20) ∂ ε ( ω ) ∂ ω (cid:21) ω s (cid:39) ω p ω s (cid:39) ( ε m ) ω p , (13)and, in the limit ε m (cid:29)
1, eq. (12) readslim ε m → ∞ Γ s = ω p √ ε m Im [ − d r ( ω s )] R . (14)Moreover, since the phase space for exciting electron-hole pairs decreases as ω for ω (cid:28) ω p ,one has lim ω → Im [ − d r ( ω )] −→ . (15)Actually, it is an exact result for a planar surface that Im [ − d ⊥ ( ω )] (cid:39) ω in the low frequencylimit ω (cid:28) ω p [28, 29]. Hence, from Eqs. (14) and (15), the surface scattering (Landau damping)contribution to the linewidth of the surface plasmon has to vanish for large values of ε m .Following [15] we substitute d r ( ω ) by its corresponding counterpart for a planar surface d ⊥ ( ω ) because it has been shown [1] that the induced charge density at the surface of a sphereis very similar to that of a planar surface down to a few nanometers in size. This complex quan-tity has been calculated for surfaces of free-electron like metals in contact with vacuum, usingeither the Time Dependent Local Density Approximation (TDLDA) [29] or the Random PhaseApproximation (RPA) [30] to the non-local self-consistent dielectric response of the metal elec-trons confined by a Lang-Kohn potential barrier. We adapt these calculations to our case of Agby means of the renormalization of the bulk plasma frequency to ω ∗ p given by Re [ ε ( ω ∗ p )] = d r . This approximation is moti-vated by three physical facts: (i) the 4d-electrons of Ag are localized and largely excluded fromthe surface region where the conduction electrons spill-out. This is an essential considerationfor obtaining the observed blue shift of the surface plasmons in planar Ag surfaces and Agspheres [31]. (ii) When plotted versus the reduced frequency ω / ω p , the function d ⊥ ( ω / ω p ) isqualitatively similar for all simple metals, from Al ( r s (cid:39) a ) to Cs ( r s (cid:39) a ) [29, 30]. (iii) Innoble metals, ω ∗ p (instead of the free-electron ω p ) is the frequency at which the metal changesbehavior from surface screening to penetration of an external electromagnetic field. a) energy [eV]0123 3.1 3.2 3.3 3.4 3.5 3.6 no r m a li z ed c r o ss s e c t i on TDLDAon carbon (b)012 energy [eV] TDLDAin SiO Fig. 1. Absorption cross-sections (continuous blue lines) of silver spheres embedded in: (a)a host medium of dielectric function 1.5 (average value for spheres on a carbon substrate)and (b) a host medium of dielectric function 2.16 (SiO ). Calculations have been performedusing the Time-Dependent Local Density Approximation for the dielectric response of theelectronic density of Ag in the surface region. The cross-sections are normalized to thesurface area of the sphere, the radius being 10 nm, 7.5 nm, 5 nm and 4 nm from top tobottom in each subfigure. Both show a blue-shift of the plasmon for decreasing size and abroadening of the peak. Dashed red lines are Lorentzians constructed from the width andpeak position of the full calculation. They are a very good match to the spectra in (a) andnot so good in (b). This reflects the detailed structure of the surface response function whichis the major contribution to the absorption cross-section for these small spheres.
3. Results
The absorption cross-section is calculated from Eqs. (1), (8) and (9), using the TDLDA andRPA calculations of d ⊥ ( ω ) for r s = a (appropriate for Ag) from Refs. [29] (Figure 2 therein)and [30] (Figures 9 and 10 therein), respectively. The RPA values of Im [ − d ⊥ ( ω )] are alsoplotted in Figure 2(b) of Ref. [29]. Here, r s is the one-electron radius and a denotes the Bohrradius. Both approximations can account for the surface plasmon relation of dispersion at planarsurfaces of the free-electron-like metals, even tough TDLDA gives an overall better agreementwith experiment [32]. We find that the results we present and discuss in this section do notdepend on the self-consistent treatment of the dielectric response as long as the electronic spill-out of the metal electrons is include d in the theory. We will see that they are able to accountfor the experimental magnitude of the surface plasmon width of Ag nanospheres in SiO , thusreinforcing the importance of an appropriate treatment of the surface electronic density. ε ( ω ) and τ b are taken from the experimental data of Johnson and Christy [33] from where ¯ h τ b = .
016 eV. The width of the surface plasmon resonance, Γ R , is defined as the full width at halfmaximum of the corresponding absorption curve.Figure 1 shows absorption cross-sections of Ag spheres of decreasing sizes, calculated us-ing the TDLDA values of d ⊥ ( ω ) , embedded in (a) a host medium of ε m = . ( ε m = . π R .Also plotted are Lorentzians (dashed lines) having the same peak position and width as thecalculated cross-sections, and normalized to the same height. The value ε m = . shown in Fig. 1(b), the resonances R ) [nm -1 ] ho s t pe r m i tt i v i t y ε m RPA . . . . . R ) [nm -1 ]TDLDA . . . . . . . full width at half maximum [eV]0.20.30.40.5 0.1 Fig. 2. Color plot of the surface plasmon width as a function of inverse size and host permit-tivity for (a) the Random Phase Approximation and (b) the Time-dependent Local DensityApproximation. The contours and the scale on top are in eV. Results are dominated by thesurface effect we calculate. In both cases the width scales directly with the inverse size.Notice that the width is maximum for a host dielectric medium with permittivity 2.4-2.6. show a smaller blue shift and are much wider than in Fig. 1(a). As the permittivity of the hostmedium increases, the frequency of the surface plasmon decreases moving to a region of fre-quencies where Re [ d ⊥ ( ω )] is smaller (less blue shift) and Im [ − d ⊥ ( ω )] is larger (more width).One should also note that the absorption cross section is less symmetric around the maximumand therefore not so well fitted by a Lorenztian (red dashed lines). This is due to the fact thatIm [ d ⊥ ] is a strongly varying function of frequency in this range of values of ωω ∗ p . In general, wefind that the absorption cross-section becomes more asymmetric with decreasing size in bothRPA (not shown) and TDLDA calculations, as the resonance gets wider and, consequently, thefunctional dependence of Im [ d ⊥ ( ω )] becomes more important. Therefore, the energy depen-dence of the surface excitations is a source of skewness for the optical absorption cross-sectionof perfectly spherical particles with radii typically smaller than 5 nm. Even though the disper-sion in frequencies of ε ( ω ) is another source of asymmetry for the cross-section, we find thatsurface scattering is more important for Ag, were the resonance is below the onset of interbandtransitions. This would be different in the case of Au.Figure 2 displays color maps of the surface plasmon width as function of the permittivity ofthe host medium and inverse of size, using (a) RPA and (b) TDLDA values of d ⊥ ( ω ) . Except atthe largest sizes and lowest permittivities, where the width is on the order of 0.1 eV, the valuesshown in these figures are completely dominated by surface effects (Landau damping). For agiven ε m , the width of the surface plasmons increases almost linearly with R − with a slopewhich increases with ε m up to a maximum, as can be appreciated in this figure. This behavior iscompletely different from the classical theory, where the numerator of Eq. (6b) decreases as ε m increases, yielding decreasing values of Γ M in the range of values of ε m shown in these figures.We should mention here that the experimental results for Ag clusters in different surroundingscompiled in Fig. 3 of Ref. [22] seem to confirm the trend of the present calculations. Then, forall values of R , our model predicts that the plasmon width has a maximum as a function of thehost permittivity at ε m (cid:39) . − .
6, depending on the approximation to d ⊥ ( ω ) , RPA or TDLDA. pa r a m e t e r A host permittivity ε m qua li t y f a c t o r plasmon energy [eV]TDLDA(a) (b) Fig. 3. (a) The linear coefficient of the plasmon width in v F / R , parameter A , as a function ofthe host permittivity for the Random Phase Approximation (dashed red line) and the Time-Dependent Local Density Approximation (solid blue line). The maximum of A leads to themaximum in the surface plasmon damping mechanism present in both approximations. (b)Quality factor of the surface plasmon in silver spheres for the TDLDA. Radius being 10nm, 7.5 nm, 5 nm and 4 nm from top to bottom. We take the quality factor as resonanceenergy divided by width. To obtain the resonance energies, we changed the host dielectricpermittivity in the range 1 - 6. Again we see how the maximum plasmon width comes inaround 3 eV giving a minimum in the quality factor. The existence of this maximum can be understood from our analysis above showing that thesurface scattering mechanism for plasmon damping tends to vanish for sufficiently large valuesof ε m .Our findings can be seen more clearly when we fit our calculated width to the usual form Γ R = Γ + A v F R , (16)where Γ and A are the fitting parameters. The values of Γ we obtain are nearly the same forboth theories and very close to the values of Γ M given by Eq. (6b). These values are in therange 0 . − .
09 eV and, in general, are small in comparison with the full width as can beappreciated in Fig. 2. Figure 3(a) shows the parameter A as a function of ε m for the RPA andTDLDA type of calculations. It shows a pronounced maximum, since A is roughly proportionalto Im [ − d ⊥ ( ω s )] , which yields the maximum of plasmon width with host permittivity seen inFig. 2. We note here that A is larger than 1 for certain values of host dielectric function in sharpcontrast to what is found by box models ignoring the spill-out effect of the surface electronicdensity. It is worth noticing that the values of A extracted from both theories can differ by almosta factor of 2 at some values of ε m and therefore an experimental determination of A could beused as a test for these theories.Figure 3(b) displays the quality factor Q for Ag spheres of decreasing sizes, calculated usingthe TDLDA values of d ⊥ ( ω ) . The quality factor gives a measure of the local field enhancementat the particle surface when the plasmon mode is excited [7] and is defined as Q = ω R / Γ R , where ω R and Γ R are the peak position and full width at half maximum of the plasmonic resonances.The curves are obtained by continuously changing the host permittivity in the range 1 ≤ ε m ≤ Q at ω R (cid:39) PA, ε = 2RPA, ε = 2.1RPA, ε = 2.2RPA, ε = 2.3Nano Lett. 9, 3463 f w h m [ e V ] R ) [nm -1 ]0.02 0.04 0.06 0.08 0.10 0.12(a) TDLDA, ε = 2TDLDA, ε = 2.1TDLDA, ε = 2.2TDLDA, ε = 2.3Nano Lett. 9, 34631/(2 R ) [nm -1 ]0.02 0.04 0.06 0.08 0.10 0.12(b) Fig. 4. The surface plasmon width as a function of inverse size for (a) the Random Phase ap-proximation and (b) the Time-dependent Local Density Approximation. Results are givenfor four different permittivities of the host medium differing from the value for SiO byless than 10%, and compared to the experimental results for Ag spheres coated with thatmaterial reported in [18] (circles). Notice the radiative damping taking over the physics aswe approach particles larger than 25 nm. Then, the smallest damping is obtained for sizesof ca. 20 nm. Q when radiative damping dominates ( R >
15 nm), where Q decreases quickly with ω R [34].Another important difference seen in Q in the regimes of surface scattering versus radiativedamping is its behavior with particle size. When surface scattering dominates ( R <
10 nm) Q decreases with R because Γ R is nearly proportional to R − while ω R depends weakly on R .However, in the radiative damping region, Q decreases with increasing size [34]. The smallfeatures in Q are also present in a calculation neglecting the surface scattering mechanism andare brought about by the experimental values of ε ( ω ) . We should also mention that a calculationof Q excluding the effects of surface scattering will produce quality factors bigger than the onesshown in this figure by factors of 3-8. The same trends are obtained in calculations using theRPA to the dielectric response. This fact points out to the importance of using realistic surfacemodels when calculating optical properties of small metallic systems.Figure 4 shows our calculated widths, using (a) RPA and (b) TDLDA values of d ⊥ ( ω ) ,compared with the experiments in [18] for single Ag nanoparticles coated with a 15 nm shellof SiO . We show results for values of the host permittivity differing by less than 10% from thenominal value for this material, ε m = .
16. This is because it is probable that the thickness ofthe coating is not enough to assume an infinite host in all the cases or that the coating might notbe uniformly distributed around the metal core. The sharp increase in the experimental widthfor 2 R >
25 nm has been shown in [18, 34] to be due to radiative damping, which becomesimportant for increasing sizes, as is clearly illustrated in Fig. 8 of Ref. [34]. The agreement wefind with the experiment, at the sizes where the quasi-static dipolar approximation is valid, isnoticeable. Even though the calculated widths have a stronger variation with ε m when usingthe TDLDA values of d ⊥ ( ω ) [Fig. 4(b)] than when using the RPA values [Fig. 4(a)], one canappreciate that this experiment can be reproduced by both approximations, using reasonablevalues of ε m and without further adjustable parameters.Recently Lerm´e et al. [35, 36] have analyzed the effects of confinement and permittivityof the host matrix on the width of the surface plasmons. These authors take a bottom-up ap-proach to the nanoparticles and perform full TDLDA calculations of the optical absorptioncross-section of Ag clusters of diameters smaller that 11 nm, in vacuum ( ε m =
1) and embed-ded in SiO ( ε m = .
16) and Al O ( ε m = . R , with a slope that in-creases with the permittivity of the host medium, from vacuum to Al O . This is the same trendwe find in our TDLDA calculations. When the results for Ag in SiO in [35, 36] are extrapo-lated linearly up to the region of the experimental sizes, the magnitude of the width so obtainedis much smaller than ours and in poorer agreement with the experiment. Nevertheless, whendecreasing the size below 5 nm, we know that quantum size effects are essential to account forthe color of the energy shift the surface plasmons and, consequently, they would also affect thesurface plasmon width. This is the region of the size spectra where first principles calculationsare indeed necessary.
4. Conclusions
In this work we have theoretically analyzed the evolution with size and host permittivity ofthe linewidth of the local surface plasmon resonances in spherical Ag nanoparticles, using thesame framework which allowed us to successfully reproduce the shift in energy of these dipolarmodes. The physical magnitude controlling that evolution for both energy position and widthof the surface plasmons is the complex ratio d r ( ω s ) / R , whose real and imaginary parts basi-cally fix the energy and width of the surface plasmons respectively. We approximate d r ( ω ) byits counterpart for a planar surface, calculated using two different approximations (RPA andTDLDA) to the non-local self-consistent dielectric response of the metal electrons confinedby a Lang-Kohn potential barrier. We find that the damping of the surface plasmon collectivemode in Ag spheres with diameters smaller than ca. 25 nm, is dominated by the surface scatte-ring mechanism that couples this mode to single electron-hole pairs excitations. The linewidthof the plasmons in these spheres increases with decreasing size in a way approximately linearin 1 / R , with a slope which depends on the permittivity of the host medium, showing a max-imum at ε m (cid:39) . − .
6, depending on the approximation to d r ( ω ) . This fact strongly affectsthe quality factor of the plasmonic resonances which is minimum at resonance energies of ca.3 eV, independent of particle size. From our calculations, we extract the linear coefficient ofthe plasmon width in v F / R , which depends on the surrounding permittivity, and could be usedto account for this increased damping in a phenomenological manner. We compare our resultswith a recent experimental determination of the surface plasmon resonance damping of singleAg spheres coated with SiO , finding a good agreement in the range of sizes 2 R <
25 nm wherethe dipolar approximation is valid. We thus conclude that our approach to surface plasmonsin small particles is a reliable one down to 10 nm in diameter, a range of sizes where sophis-ticated first principles calculations are not feasible yet. However, these kinds of calculationsare necessary when decreasing the particle size below 5 nm, where quantum size effects playan important role in accounting for the color of the energy shift of the surface plasmons andshould be equally important for a determination of their widths. Moreover, their extension tolarger sizes would be highly interesting.