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13

Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS

Søren Raza,1,2,∗ Nicolas Stenger,1,3,∗ Shima Kadkhodazadeh,2 Søren V. Fischer,4 Natalie Kostesha,4

Antti-Pekka Jauho,4,3 Andrew Burrows,2 Martijn Wubs,1 and N. Asger Mortensen1,3,†

1Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark2Center for Electron Nanoscopy, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

3Center for Nanostructured Graphene (CNG), Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark4Department of Micro and Nanotechnology, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

∗Both authors contributed equally

†Corresponding author email: asger@mailaps.org

(Dated: May 25, 2018)

We study the surface plasmon (SP) resonance energy of isolated spherical Ag nanoparticles dis-persed on a silicon nitride substrate in the diameter range 3.5-26 nm with monochromated electronenergy-loss spectroscopy. A significant blueshift of the SP resonance energy of 0.5 eV is measuredwhen the particle size decreases from 26 down to 3.5 nm. We interpret the observed blueshift usingthree models for a metallic sphere embedded in homogeneous background material: a classical Drudemodel with a homogeneous electron density profile in the metal, a semiclassical model corrected foran inhomogeneous electron density associated with quantum confinement, and a semiclassical non-local hydrodynamic description of the electron density. We find that the latter two models providea qualitative explanation for the observed blueshift, but the theoretical predictions show smallerblueshifts than observed experimentally.

INTRODUCTION

Surface plasmons are collective excitations of the elec-tron gas in metallic structures at the metal/dielectricinterface [1]. The ability to concentrate light withSPs [2] and to enhance light-matter interaction on a sub-wavelength scale enables few- and even single-moleculespectroscopy when the size of the metallic structuresis decreased to a few nanometer [3]. These collectiveexcitations are usually well-described by the classicalDrude model for nanoparticles with dimensions of tensof nanometer and larger [1]. In the quasistatic limit,i.e. when the wavelength of the exciting electromagneticwave considerably exceeds the dimensions of the struc-ture, the local-response Drude theory predicts that theresonance energy of localized SPs is independent of thesize of the nanostructure [4], and that the field enhance-ment created in the gap between two metallic nanostruc-tures diverges for vanishing gap size [5]. These predic-tions are however in conflict both with earlier [6–9] andwith more recent experimental results, which have showna size dependency of the localized SP resonance in noblemetal nanoparticles in the size range of 1-10 nm [10] andpronounced deviations for dimer geometries [11, 12].

This dependence of the SP resonance on the size ofnoble metal nanostructures is believed to be a signatureof quantum properties of the free-electron gas. With de-creasing sizes of the nanoparticles, the quantum wave na-ture of the electrons is theoretically expected to manifestitself in the optical response due to the effects of quan-tum confinement [13–17], quantum tunneling [17–20], aswell as nonlocal response [21–27]. Nonlocal effects are adirect consequence of the inhomogeneity of the electrongas, which arises due to the quantum wave nature and

the many-body properties of the electron gas.

The recent developments in analytical scanning trans-mission electron microscopes (STEM) equipped with amonochromator and electron energy-loss spectroscopy(EELS) [28] give the possibility of accessing the near-fieldenergy distribution of the plasmon resonance of individ-ual nanoparticles on a subnanometer scale with an energyresolution better than 0.2 eV. This method has been usedfor the imaging of surface plasmons in many differentmetallic nanostructures [10, 29–32]. With STEM EELSit is possible to correlate the structural and chemical in-formation on the nanometer scale, such as the shape andthe presence of organic ligands, with the spectral infor-mation of the SP resonance of single isolated nanoparti-cles. STEM EELS is thus perfectly suited to probe andaccess plasmonic nanostructures and SP resonances atlength scales where quantum mechanics is anticipated tobecome important.

In this paper we report the experimental study of theSP resonance of chemically grown single Ag nanoparticlesdispersed on 10 nm thick Si3N4 membranes with STEMEELS. Our measurements present a significant blueshiftof the SP resonance energy from 3.2 to 3.7 eV for par-ticle diameters ranging from 26 down to 3.5 nm. Ourresults also confirm very recent experiments made withAg nanoparticles on different substrates using differentSTEM operating conditions [10], thereby strengtheningthe interpretation that the blueshift is predominantly as-sociated with the tight confinement of the plasma andthe intrinsic quantum properties of the electron gas itselfrather than having an extrinsic cause.

We compare our experimental data to three differentmodels: a purely classical local-response Drude modelwhich assumes a constant electron density profile in the

2

metal nanoparticle, a semiclassical local-response Drudemodel where the electron density is determined from thequantum mechanical problem of electrons moving in aninfinite spherical potential well [16], and finally, a semi-classical model based on the hydrodynamic description ofthe motion of the electron gas which takes into accountnonlocal response through the internal quantum kineticsof the electron gas in the Thomas–Fermi (TF) approx-imation [33, 34]. We find good qualitative agreementbetween our experimental data and the two semiclassicalmodels, thus supporting the anticipated nonlocal natureof SPs of Ag nanoparticles in the 1-10 nm size regime.The experimentally observed blueshift is however signifi-cantly larger than the predictions by the two semiclassicalmodels.

MATERIALS AND METHODS

The nanoparticles are grown chemically following themethod described in Ref. [35] and subsequently stabilizedin an aqueous solution with borohydride ions. The meansize of the nanoparticles is 12 nm with a very broad sizedistribution ranging from 3 to 30 nm. The nanoparticlesolution is dispersed on a 10 nm thick commercially avail-able Si3N4 membrane (TEMwindows.com), which has arefractive index of approximately n ≈ 2.1 [36]. To char-acterize our nanoparticles we have used an aberration-corrected STEM FEI Titan operated at 120 kV with aprobe diameter of approximately 0.5 nm, and conver-gence and collection angles of 15 mrads and 17 mrads,respectively. The Titan is equipped with a monochro-mator allowing us to perform EELS with an energy res-olution of 0.15 ± 0.05 eV. We systematically performedEELS measurements at the surface and in the middle ofeach nanoparticle. The EELS spectra were taken with anexposure time of 90 ms to avoid beam damage as muchas possible. To improve the signal-to-noise ratio we ac-cumulated ten to fifteen spectra for each measurementpoint. We observed no evidence of damage after eachmeasurement.The experimental data were analyzed with the aid

of commercially available software (Digital Micrograph)and three different methods were used to reconstruct andremove the zero-loss peak (ZLP): the first method is thereflected tail (RT) method, where the negative-energyhalf part of the ZLP is reflected about the zero-energyaxis to approximate the ZLP at positive energies, whilethe second method is based on fitting the ZLP to thesum of a Gaussian and a Lorentzian functions. The thirdmethod is to pre-record the ZLP prior to each set of EELSmeasurements. All three methods yielded consistent re-sults.The energies of the SP resonance peaks were deter-

mined by using a nonlinear least-squares fit of our datato Gaussian functions. The error in the resonance energy

is given by the 95 % confidence interval for the estimate ofthe position of the center of the Gaussian peak. Nanopar-ticle diameters were determined by calculating the area ofthe imaged particle and assigning to the area an effectivediameter by assuming a perfect circular shape. The er-ror bars in the size therefore correspond to the deviationfrom the assumption of a circular shape, which is esti-mated as the difference between the largest and smallestdiameter of the particle.

THEORY

In the following theoretical analysis our hypothesis isthat the blueshift of the SP resonance energy is relatedto the properties of the electron density profile in themetal nanoparticle. Therefore, we use three different ap-proaches to model the electron density of the Ag nanopar-ticle. In all three approaches, we calculate the opticalresponse and thereby also the resonance energies of thenanoparticle through the quasistatic polarizability α of asphere embedded in a homogeneous background dielec-tric with permittivity ǫB. With this approach, we maketwo implicit assumptions: the first is that we can neglectretardation effects and the second is that we can neglectthe symmetry-breaking effect of the substrate. We havevalidated the quasistatic approach by comparing to fullyretarded calculations [37], which shows excellent agree-ment in the particle size range we consider. The effectof the substrate will be taken into account indirectly bydetermining an effective homogeneous background per-mittivity ǫB using the average resonance frequency of thelargest particles (2R > 20 nm) as the classical limit.The first, and simplest, approach is to assume a con-

stant free-electron density n0 in the metal particle, whichdrops abruptly to zero outside the particle. This assump-tion is the starting point of the classical local-responseDrude model for the response of the Ag nanoparticle,where the polarizability is given by the Clausius–Mossottirelation, which is well-known to be size independent forsubwavelength particles. The classical local-response po-larizability αL is [1]

αL(ω) = 4πR3 ǫD(ω)− ǫBǫD(ω) + 2ǫB

, (1)

where R is the radius of the particle and ǫD(ω) = ǫ∞(ω)−ω2p/(ω

2 + iγω) is the classical Drude permittivity takingadditional frequency-dependent polarization effects suchas interband transitions into account through ǫ∞(ω), notincluded in the plasma response of the free-electron gasitself.The second approach is to correct the standard approx-

imation in local-response theory of a homogeneous elec-tron density profile by using insight from the quantumwave nature of electrons to model the electron densityprofile and take into account the quantum confinement

3

of the electrons. For nanometer-sized spheres, the classi-cal polarizability given by the Clausius–Mossotti relationmust be altered to take into account an inhomogeneouselectron density. In Ref. [16], it is shown that in generalthe local-response polarizability for a sphere embeddedin a homogeneous material is given as

αLQC(ω) = 12π

∫ R

0

r2drǫ(r, ω)− ǫBǫ(r, ω) + 2ǫB

, (2a)

now with a spatially varying Drude permittivity [16, 17]

ǫ(r, ω) = ǫ∞(ω)−ω2p

ω(ω + iγ)

n(r)

n0. (2b)

Here, n(r) is the electron density in the metal nanopar-ticle. Clearly, if n(r) = n0 we arrive at the classicalClausius–Mossotti relation Eq. (1) as expected. To de-termine the density profile in this local-response model,we follow the approach of Ref. [16] and assume that thefree electrons move in an infinite spherical potential well.The approach just outlined of a local-response theorywith an inhomogeneous electron density is very similarto the theoretical model used in Ref. [10] for explainingtheir experimental results. It should be noted that anyeffects due to electron spill-out and quantum tunnelingare neglected in all of the approaches that we consider.The third and final approach is to compare our exper-

imental data with a linearized nonlocal hydrodynamicmodel in which the electron density is allowed to deviateslightly from the constant electron density used in classi-cal local-response theories [22, 38–40]. The dynamics ofthe electron gas is governed by the semiclassical hydro-dynamic equation of motion [25, 26, 34], which results inan inhomogeneous electron density profile. The nonlocalhydrodynamic polarizability αNL(ω) is exactly given as

αNL(ω) = 4πR3 ǫD(ω)− ǫB (1 + δNL)

ǫD(ω) + 2ǫB (1 + δNL), (3a)

δNL =ǫD(ω)− ǫ∞(ω)

ǫ∞(ω)

j1(kLR)

kLRj′1(kLR), (3b)

and these results constitute our nonlocal-response gen-eralization of the Clausius–Mossotti relation of classicaloptics. Here, kL =

√

ω2 + iωγ − ω2p/ǫ∞/β is the wave

vector of the additional longitudinal wave allowed to beexcited in the hydrodynamic nonlocal theory [25, 34], andj1 is the spherical Bessel function of first order. Finally,within TF theory β2 = 3/5 v2F, where vF is the Fermivelocity [34]. We emphasize that for β → 0, the local-response Drude result is retrieved, since δNL → 0 andEq. (3a) simplifies to the classical Clausius–Mossotti re-lation Eq. (1).The SP resonance energy follows theoretically from the

Frohlich condition, i.e. we must consider the poles of

(a) (b) (c)

(d)

(e)

(f)

FIG. 1. Aberration-corrected STEM images of Ag nanopar-ticles with diameters (a) 15.5 nm, (b) 10 nm, and (c)5.5 nm, and normalized raw EELS spectra of similar-sized Agnanoparticles (d-f). The EELS measurements are acquired bydirecting the electron beam to the surface of the particle.

Eq. (3a). For sufficiently small blueshifts and neglectingdamping, the resonance frequency can be approximatedby

ω =ωP

√

Re[ǫ∞(ω)] + 2ǫB+

√

2ǫBRe[ǫ∞(ω)]

β

2R+O

(

1

R2

)

,

(4)where the first term is the common size-independentlocal-response Drude result for the SP resonance that alsofollows from Eq. (1), and the second term gives the size-dependent blueshift due to nonlocal corrections. At thisstage, we note that a 1/(2R) dependence was experimen-tally observed in Refs. [6, 7] using optical spectroscopy.However, Eq. (4) reveals, besides a 1/(2R) dependence,that there is a delicate interplay in the blueshift betweenthe material parameters of the metal, through ǫ∞(ω) andβ, and the background medium ǫB. Furthermore, Eq. (4)shows that the blueshift can be enhanced with a large-

4

permittivity background medium.

RESULTS

Figures 1(a-c) display STEM images of Ag nanoparti-cles with diameters of 15.5, 10.0, and 5.5 nm respectively.The images show that no chemical residue is left from thesynthesis and that the particles are faceted. We find thatapproximately 70% of the studied nanoparticles have arelative size error (i.e. the ratio of the size error bar to theparticle diameter) below 20% (determined from the 2DSTEM images), verifying that the shape of the nanopar-ticles is to a first approximation overall spherical (seeSupplementary Figure 1). On a subset of the particles,thickness measurements using image recordings at dif-ferent tilt angles were performed, revealing informationabout the shape of the nanoparticle in the third dimen-sion. Such 3D investigations confirmed that the shape isoverall spherical, but however could not be completed forall particles due to stability issues: the positions of tinynanoparticles fluctuate under too long exposure of theelectron beam, thus preventing accurate determinationof the shape of the nanoparticle in the third dimensionperpendicular to the substrate.Figures 1(d-f) display raw normalized EELS data, ac-

quired on Ag nanoparticles with diameters 14.1, 9.8, and6.6 nm, respectively. The peaks correspond to the exci-tation of the SP. When the diameter of the nanoparticledecreases, the SP resonance clearly shifts progressivelyto higher energies. Figs. 1(d-f) also display that the am-plitude and linewidth of the SP resonances can vary fromparticle to particle (with the same size) and at times shownarrowing instead of the expected broadening of the res-onance for decreasing nanoparticle sizes [6, 13, 14]. Thisis for example seen in the linewidths in Figs. 1(d-f) whichseem to decrease with size. However, as will be explainedin more detail in the next paragraph, we did not finda systematic trend of the linewidths in our EELS mea-surements probably due to the shape variations in ourensemble of nanoparticles.Figure 2 displays the resonance energy of the SP as

a function of the diameter of the nanoparticles. A sig-nificant blueshift of the SP resonance of 0.5 eV is ob-served when the nanoparticle diameter decreases from26 to 3.5 nm. This trend is in good agreement with theresults shown in Ref. [10], despite the difference in thesubstrate and the STEM operating conditions, a strongindication that the blueshift of Ag nanoparticles is ro-bust to extrinsic variations. Another prominent featurein Fig. 2 is the scatter of resonance energies at a fixedparticle diameter. We mainly attribute the spread inresonance energies at a given particle size to shape vari-ations in our ensemble of nanoparticles (see Supplemen-tary Material). Slight deviations from perfect circularshape in the STEM images will result in a delicate de-

pendency on the location of the electron probe and giverise to splitting of SP resonance energies due to degen-eracy lifting. In this regard, we also note that even aperfectly circular particle on a 2D STEM image may stillpossess some weak prolate or oblate deformation in thethird dimension, resulting in a departure from sphericalshape. Calculations using the local response model showthat a 20% deformation of a sphere into an oblate or pro-late spheroid results in a ∼ 0.4 eV spread in resonanceenergy (see Supplementary Figure 2), which is approx-imately the spread in resonance energy we observe forparticles larger than 10 nm. Furthermore, shape devia-tions may also impact the linewidth of the SP resonance,since the electron probe can excite the closely-spacednon-degenerate resonance energies simultaneously, whichmay appear as a single broadened peak. This broadeningmechanism could explain the apparent linewidth narrow-ing for decreasing particle size seen in Figs. 1(d-f). How-ever, we cannot rule out that other effects beyond shapedeviations contribute to the spread of resonance energiesand impact the SP resonance linewidth. These could forexample be the facets or the particle-to-substrate inter-face [42].

Along with the EELS measurements in Fig. 2, we showEq. (1) for the local-response Drude model (red line) andthe semiclassical local-response model Eq. (2) (blue line).Furthermore, the nonlocal relation of Eq. (3) (green solidline) and the approximate nonlocal relation of Eq. (4)(green dashed line) are also depicted, and we see thatEq. (4) is accurate for particle sizes 2R & 10 nm.

Due to the narrow energy range in consideration

3 6 9 12 15 18 21 24 273

3.2

3.4

3.6

3.8

4 EELS Measurements Local Drude Nonlocal Nonlocal approx. Local inhom.

Res

onan

ce e

nerg

y [e

V]

Particle diameter [nm]

e-

Edge excitation

FIG. 2. Nanoparticle SP resonance energy as a function of theparticle diameter. The dots are EELS measurements taken atthe surface of the particle and analyzed using the RT method,and the lines are theoretical predictions. We use parametersfrom Ref. [41]: ~ωp = 8.282 eV, ~γ = 0.048 eV, n0 = 5.9 ×

1028 m−3 and vF = 1.39 × 106 m/s. From the average large-particle (2R > 20 nm) resonances we determine ǫB = 1.53.

5

(∼ 3.0−3.9 eV), we approximate ǫ∞(ω) as a second-orderTaylor polynomial based on the frequency-dependent val-ues given for Ag in Ref. [41]. We find ǫ∞(ω) = (59.8 +i55.1)(ω/ωp)

2−(40.3+i42.4)(ω/ωp)+(10.5+i8.6). Since

the refractive index of the Si3N4 substrate varies hardly(n ≈ 2.1) in the narrow energy range we consider [36], weassume that the background permittivity ǫB is constantand determine it by approximating the average resonanceenergy of the largest particles (2R > 20 nm) as the clas-sical limit, i.e. the first term of Eq. (4).

It is known that local Drude theory produces size-independent resonance frequencies of subwavelength par-ticles, but this theory is clearly inadequate to describethe measurements of Fig. 2. The nonlocal quasistatic hy-drodynamic model predicts a blueshift in agreement withthe experimental EELS measurements. Interestingly, themeasured blueshift is even larger than predicted. We alsosee that the local-response model with an inhomogeneouselectron density profile shows a similar trend as the non-local hydrodynamic model, indicating that these two dif-ferent models describe very similar physical effects. Theoscillations in the resonance energy in the inhomogeneouslocal-response model seen for small particle diameter aredue to small variations in the density profile with decreas-ing size (i.e. discrete changes in the number of electrons),as also stated in Ref. [10].

The inhomogeneous local-response model and the non-local hydrodynamic model, when applied to a spherein a homogeneous background medium, agree qualita-tively with the EELS measurements. However, they donot provide the full picture. One of the probable is-sues arising is that the substrate is taken into accountindirectly through a homogeneous background medium,a state-of-the-art procedure [10] which however may notbe adequate to describe the effects of the presence of adielectric substrate. It has been shown that the dielec-tric substrate modifies the absorption spectrum of an iso-lated sphere [43] and also the waveguiding properties ofnanowires [31, 44, 45]. In an attempt to include the sym-metry breaking effect of the substrate in our theoreticalanalysis, we apply a simple image charge model. Themain effect of the substrate in this picture stems fromthe interaction of the dipole mode of the nanoparticlewith the induced dipole mode in the substrate [46–48].However, we find that such a dipole-dipole model for thesubstrate is inadequate to describe the large blueshift ob-served experimentally (see Supplementary Material). In-deed, it has been shown that the induced image chargesin the substrate can make the contributions of higher or-der multipoles in the nanoparticle important [49], andit has also been observed theoretically that higher ordermultipoles produce larger blueshifts in the nonlocal hy-drodynamic model (Fig. 2 in Ref. [50]). The impact ofthe substrate on the electron density inhomogeneity andthereby the SP resonance energy depends on the thick-ness and refractive index of the substrate, which may

explain the quantitative agreement between theory andexperiment reported in Ref. [10], since thinner substrateswith smaller refractive indexes were used in their exper-iments. In order to completely address this issue, onewould need to go beyond the dipole-dipole model for thesubstrate, thus future 3D EELS simulations taking non-local effects and/or inhomogeneous electron densities intoaccount would be needed.Another complementary explanation in the context of

the inhomogeneity of the free-electron density could bethe combined contribution of both the inhomogeneousstatic equilibrium electron density and nonlocality. It iswell-known that the static equilibrium electron densityis inhomogeneous, even in a semi-infinite metal [51], dueto Friedel oscillations and the electron spill-out effect atthe metal surface. The Friedel oscillations are modeledin the local quantum-confined model given by Eq. (2)while nonlocality is neglected, and vice versa in the non-local hydrodynamic model given by Eq. (3). As seenin Fig. 2, the two effects separately give rise to similar-sized blueshifts, suggesting that the contribution of botheffects simultaneously could add up to the significantlylarger experimentally observed blueshift. Simply put, anextension of the nonlocal hydrodynamic model to includean inhomogeneous equilibrium free-electron density couldproduce a larger blueshift, which may be in accordancewith the experimental observations. Furthermore, such amodel could also take into account the electron spill-outeffect, which in free-electron models has been argued toproduce a redshift of the SP resonance [21, 50, 52–54],describing adequately simple metals. In contrast, it hasalso been shown that the spill-out effect in combinationwith the screening from the d electrons gives rise to theblueshift seen in Ag nanoparticles [55].Additional size effects such as changes of the electronic

band structure of the smallest nanoparticles, which areconsiderably more difficult to take into account, also im-pact the shift in SP resonance energy [6].

CONCLUSIONS

We have investigated the surface plasmon resonance ofspherical silver nanoparticles ranging from 26 down to3.5 nm in size with STEM EELS and observed a signif-icant blueshift of 0.5 eV of the resonance energy. Wehave compared our experimental data with three differ-ent models based on the quasistatic optical polarizabilityof a sphere embedded in a homogeneous material. Twoof the models, a nonlocal hydrodynamic model and ageneralized local model, incorporate an inhomogeneityof the electron density induced by the quantum wave na-ture of the electrons. These two different models producesimilar results in the SP resonance energy and describequalitatively the blueshift observed in our measurements.Although our exact hydrodynamic generalization of the

6

Clausius–Mossotti relation predicts a nonlocal blueshiftthat grows fast [as 1/(2R)] when decreasing the diam-eter and increases even faster for the smallest particles(2R < 10 nm), the observed blueshifts are neverthelesslarger than predicted.

The quantitative agreement between the two differenttheoretical models and the discrepancy with the largerobserved blueshift suggest that a more detailed theoret-ical description of the system is needed to fully under-stand the influence of the substrate and the effect of theconfinement of free electrons on the SP resonance shiftin silver nanoparticles. On the experimental side, furtherEELS studies of other metallic materials and on differ-ent substrates could unveil the mechanism behind thesize dependency of the SP resonance of nanometer scaleparticles.

Acknowledgments. We thank S. I. Bozhevolnyi for di-recting our attention to the theoretical model in Ref. [16]and G. Toscano for fruitful discussions. The Center forNanostructured Graphene is sponsored by the DanishNational Research Foundation, Project DNRF58. TheA. P. Møller and Chastine Mc-Kinney Møller Foundationis gratefully acknowledged for the contribution towardthe establishment of the Center for Electron Nanoscopy.

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