Multipolar plasmon resonances in supported silver particles: The case of Ag / α − Al 2 O 3 ( 0001 )

PHYSICAL REVIEW B, VOLUME 65, 235424

Multipolar plasmon resonances in supported silver particles: The case of AgÕa-Al2O3„0001…

Remi Lazzari,1,2,* Stephane Roux,1,† Ingve Simonsen,1,3,‡ Jacques Jupille,1,§ Dick Bedeaux,4,i and Jan Vlieger41Laboratoire Mixte CNRS/Saint-Gobain (UMR 125) ‘‘Surface du Verre et Interfaces,’’

39, Quai Lucien Lefranc, BP 135, F-93303 Aubervilliers Cedex, France2CEA Grenoble, De´partement de Recherche Fondamentale sur la Matie`re Condenseé, Service de Physique des Mate´riaux

et Microstructures–Interfaces et Rayonnement Synchrotron, 17, Rue des Martyrs, F-38054 Grenoble Cedex 9, France3Department of Physics, Theoretical Physics Group, The Norwegian University of Science and Technology (NTNU),

N-7034 Trondheim, Norway4Leiden Institute of Chemistry, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands

~Received 26 February 2002; published 10 June 2002!

The present paper is devoted to the description of optical absorptions in supported metallic particles smallerthan the wavelength of light. The breaking of symmetry brought by the presence of the substrate and by thefinite contact angle for the particle leads to a nonhomogeneous local field and allows excitation of multipolarabsorption modes. Their energetic positions, oscillator strengths, and widths are calculated in the limit of zerodamping by use of a multipolar expansion of the potential inside and around the island. The charge vibrationpattern and the location of the field enhancements are clearly identified for each mode. Even if the dipolarmode is the most intense, the highlighting of the secondary modes in optical measurements is possible inpeculiar experimental conditions. Some experiments of surface differential reflectivity spectroscopy for silverdeposits on oxide substrates are given as examples.

DOI: 10.1103/PhysRevB.65.235424 PACS number~s!: 78.20.Bh, 61.46.1w, 68.55.Ac, 73.20.Mf

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I. INTRODUCTION

Over the last 100 years, there has been a tremendousoretical and experimental work on the optical propertiesmetallic particles1 as they represent an intermediate statetween molecules and solid. The starting point was the wof Maxwell Garnett2 at the turn of the century. He gave thfirst explanation of the coloration of glasses doped bymetal precursor. The link between the optical propertiessuch glasses and the presence of metallic particles wasobvious since the bulk dielectric constant of metal doesshow such absorptions. A more complete derivation of liscattering by a spherical particle was given by Mie in 1903

He solved the complete Maxwell equations for an incomplane wave on a sphere by the method of Rayleigh pawaves and calculated the scattering and absorption crosstions. Of course, in the size regime where the particle radis much smaller than the wavelength of light (x5R/l,few %), the quasistatic approximation can be assumedsolving only the Laplace equation for the potential. In thcase, the main effect of the electric field is to polarizeelectronic gas of the metal and to excite collective motionelectrons: surface plasmon polaritons. The term surface fiits origin in the fact that, despite a collective oscillationthe electronic gas with respect to the positive backgroundions in a jellium picture, the restoring force comes from tsurface polarization. The excited quasiparticles are a resuthe mixing of plasmon and photons. In dielectric particlexactly the same phenomenon occurs in the infrared wexcitation of phonons~surface phonon polaritons!. For a me-tallic sphere, the polarization process leads simply to apole behavior with a surface charge distribution equal tocosine of the angle between the homogeneous excitaelectric field and the current point on the surface. Inquasistatic size regime, the extinction cross sectionsext ismainly given by absorptionsabs as the scatteringssca

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;(R/l)4 is of three orders of magnitude lower than absotion sabs;(R/l). More precisely one has:

sext59v

cem

3/2V0

e2~v!

@e1~v!12em#21e2~v!2, ~1!

wheree(v)5e1(v)1 i e2(v) is the frequency-dependent delectric constant of the metal particle,em the dielectric func-tion of the surrounding medium, andV05(4p/3)R3 the vol-ume of the cluster. A resonance occurs at the minimum ofdenominator@e1(v)12em#21e2(v)2. If e2(v) is small ordoes not vary in the considered frequency region, one recers the classical relatione1522em for the Frohlich mode ofa sphere. With a Drude metal, the absorption is found av5vp /A3 with vp the bulk plasmon frequency. For an ellipsoidal particle, the same type of dipolar vibrations are salong the main axis. However, by breaking the symmetrythe surrounding, high-order multipoles can be excited.

The first example is when the ratio between the sphradius and the wavelength increases. For such particles sthe incoming fieldE0 develops oscillations inside the particin such a way that neglecting the retardation effects beconot valid. The nonhomogeneous fieldE0 allows the appear-ance of electric and magnetic multipoles. Far away fromcluster, the associated waves are identical to those comfrom equivalent point multipoles. As underlined above, telectric multipolar modes are linked to surface polarizatwhile magnetic multipoles are due to eddy currents. Clexperimental evidence of these modes is scarce inliterature.4,5 The great difficulty in experimental spectra isovercome the size distribution of the particle collection athe interface damping6,7 which smear out all the structuresMoreover, the mixing between absorption and scattermodes implies the use of many experimental techniquebe separated. Most of the time, noble metals like silveralkali are chosen for their free-electron gas behavior. Th

©2002 The American Physical Society24-1

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REMI LAZZARI et al. PHYSICAL REVIEW B 65 235424

plasma oscillations are not damped by interband transitand the absorption structures are sharp.

Another way to put forward this phenomenon is to stuirregularly shaped particles. In this case, the inhomogendoes not come from the exciting field but from the particsusceptibility itself. Even in the quasistatic regime, a grvariety of charge vibration patterns can be excited partilarly in the presence of sharp corners. These type of geetries are encountered in the faceting processes of nanotallites. The Wulff construction at thermodynamequilibrium8 gives the relative spreads of the crystallograpplanes and the overall crystal shape. Fuchs9 tried to illustratethis phenomenon by computing the phonon-polariton abstions of small dielectric cubes like NaCl or MgO. He undelined the importance of corners in the trapping of polariztion charges. Furthermore, optical studies are ofperformed on supported particles. Their final shape arfrom a complex balance between thermodynamic and kiics factors but, at least, the notion of contact angle has tointroduced to describe the geometry of the system andwetting properties. The influence on the optical absorptiof such a line of contact between three dielectric mediapoorly known. Moreover, the presence of the substrate cplicates greatly the situation as it brings a new breakingsymmetry with the image field, even for a supportsphere.10,11 In the quasistatic regime, the optical behaviora particle on a substrate can be seen as a cascade of pization processes. The incident field field creates a dipinside the particle. The substrate reacts by creating an imdipole which in turn modifies the local field around theland. This image dipolar field excites the quadrupolar poizability of the island and as a consequence a quadruimage appears in the substrate and so on. This mechanisimage interaction is in a complicated way linked to the mtipolar polarizabilities of the particle. But even if Beitet al.12 and Chauvaux and Meesen13 detected and underlinethis phenomenon on supported alkali clusters, clear evideof such multipolar absorptions is still lacking because ofperimental difficulties and because rather crude simple mels are often used to interpret the optical response of sported particles.14–16

The aim of the present paper is to demonstrate the ocrence of multipolar absorption modes in experimental spefrom small supported particles and to give a descriptionthese modes in the nonretarded limit for supported truncaspherical particles. The paper is organized as follows. Aftedescription of the experimental setup in Sec. II, surfaceferential reflectivity~SDR! spectra from silver nanoparticlesupported by alumina substrates are presented in Sec. Ithe discussion of Sec. IV, it is explained that the multipomodes are found in the zero damping limit. The role of sustrate in their appearance and of the way to reveal themincreasing the coupling between the particles are studOscillator strengths and broadenings of these modesevaluated to first order in the imaginary part of the dielecconstant. The model is then applied in Sec. V to the expments ofin situ SDR for silver on alumina. Clear evidencegiven of the occurrence of multipolar modes.

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II. EXPERIMENTAL SETUP

Experiments have been performed in a ultra high-vacuchamber equipped with the classical surface science t~base pressure 331028 Pa). The sapphirea-Al2O3(0001)samples purchased from Matek Gmbh were epitaxially pished with a miscut less than 0.2° as checked byex situatomic force microscopy. Once mounted on a tantalum pin UHV, thanks to an electronic bombardment system,substrates were cleaned by a strong annealing under paoxygen pressure provided by a gas dosing pipe.17 The sur-face composition was checked by x-ray photoemissionthe C 1s line was below the detection limit. This proceduallows us to obtain a (131) sharp low-energy electron diffraction ~LEED! pattern. Silver was evaporated fromKnudsen cell and the flux calibrated by a quartz microbance~0.07 ML/s with 1 ML given by the spacing betweetwo atomic plane along the@111# direction of the fcc struc-ture!. The sample temperature was regulated atT5600 Kby a Chromel-Alumel thermocouple force wedged to tsubstrate surface.

The optical experiment used to monitorin situ and in realtime the growth is surface differential reflectivity spectrocopy. The recorded quantity is the relative variation of tsample reflectivity:

DR

R5

R2R0

R0. ~2!

The reference is the reflectivity of the bare substrate,R0. Thelight emitted by a deuterium lamp in the UV-visible rang~1.5–6 eV! is collimated on the sample with an incideangle of 45°, collected inp or s polarization and dispersed ba grating spectrograph~300 grooves/mm! on a Peltier-cooledphotodiode array. The environment is carefully controllduring the recording of the optical spectra. This advice iscourse essential with highly reactive metals like alkali.18–20

But more generally, the surface plasmon even with nometal is highly sensitive to the adsorbed species. The pnomenon of interface damping put forward by Kreibig aco-workers and Personn6,7,21 induces a strong broadening othe absorptions peaks. Thus, it is really difficult to identthe multipolar resonances for sample submitted to ambatmosphere or elaborated by a chemical way.

It is well known that the low affinity of noble metal withdielectric substrates leads to a cluster growth mode22 withtypical size in the nanomeric range. Upon deposition,morphology of the thin film is dictated by both thermodnamics and kinetics factors, leading in most situations tbroad distribution of clusters shapes and sizes22,23~from 20%to 100%) which mixes up all the features. The substrinhomogeneities worsen the situation by creating nucleacenters and paths of diffusion of different energies. The wto reduce the width of the distribution up to a few tens opercent is to activate the diffusion process and to get cloto the thermodynamic equilibrium by increasing the tempeture. Thus, in the present case, the sample temperatureregulated atT5600 K.

III. REFLECTIVITY OF SILVER THIN FILMS

SDR spectra collected during the deposition of silvera-Al2O3(0001) are shown in Fig. 1. Spectra collected inp

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MULTIPOLAR PLASMON RESONANCES IN SUPPORTED . . . PHYSICAL REVIEW B65 235424

FIG. 1. Evolution of the SDR spectra recorded during the growth of a thin silver film atT5600 K on Al2O3(0001) inp ~left panel! ors ~right panel! polarization. The dispersion of the high-energy resonance with deposited thickness for thep polarization is given in the insetThe evolution of the main structures ins polarization is displayed with broken lines to guide the eye.

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polarization involve three features lying above 3.6 earound 3.5 eV, and below 3.2 eV, respectively. The dispearance of the 3.6-eV peak ins polarization indicates that iis dominated by excitations normal to the surface. At vaance, the other features mostly come from excitations palel to the surface. The positions and shapes of the 3.65–eV and 3.2–2.6 eV peaks are characteristic of a thrdimensional growth mode.24,25

Redshifted at the beginning of the silver deposition, the hienergy feature of the SDR spectra shifts toward higherergy at a coverage higher than 0.3 nm@inset of Fig. 1~a!#. Itis after the onset of this blueshift that the 3.5-eV featubecomes sizable in both polarizations. The two main surfplasmon-polariton peaks inp polarization and the main feature in s polarization are qualitatively understood in thframework of the dipole model14,15 and can be modeled bmeans of multipolar approaches.24,25However, the origin andbehavior of the shoulder are to be explained.

IV. DISCUSSION

Because the two mains features observed in absorpspectra are assigned to dipolar resonances, the new mlying around 3.5 eV can be suggested to arise from a mupolar response. In addition, since these modes appear aonset of the blueshift of the high-energy resonance, theylikely perturbed by particle-particle interactions. A first goin this discussion is therefore to show how multipolar modcan be identified in the zero-damping limit.

A. Revealing the multipolar modes

The thin film is modeled by a distribution of particlewhich are represented by truncated spheres~medium 3! sup-ported on a substrate~medium 2!, embedded in a medium 1and excited by a uniform fieldE0. The buried part of thesphere is introduced as a different medium 4. The corponding frequency-v-dependent dielectric functions are d

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notede i with i 51,2,3,4 (e45e2). The supported sphere idefined by means of a truncation parametert r . The Appen-dix A presents a detailed derivation of the polarization pcess of such particles through a multipolar expansion ofpotential. The polarizability tensor results from the solutiof a linear system of equations for the multipolar coefcients.

For particles smaller than the wavelength, the absorpcross sectionsabs is mainly given by the imaginary part othe particle polarizability Im(am) (m50,61).1 As for a freesphere in vacuum@see Eq.~1!#, the multipolar absorptionsare defined by the poles of the polarizabilities in the zedamping limit which is found when the imaginary partsthe dielectric constants of the particle and the substrate vish, that is to saye i5e i

R1 ike iI with k→0. To be clearer, let

us introduce the following notation for the linear systeEq. ~A5!:

M~v,k!X~v,k!5E~v,k!. ~3!

The matrices are denoted by script letters, the vector by itletters, and the indexm50,61 is most of the time droppedM(v,k) stands for matrix ~A8! which describes thefrequency-dependent response of the system up to multiporderM. HereX(v,k) is the vector for the multipolar components of the potential expansion:Xlm(v,k)5(Alm ,Blm)with l 51, . . . ,2M . The source field which excites the sytem either parallelm561 or normal to the surfacem50 isE(v,k). Thus, as formally,

am;A1m5X1m5@M~v,k!#1,l ,m21 El ,m~v,k!, ~4!

the resonances occur at frequencies where the maM(v,k) eigenvalue finds its lowest moduli. In the zerdamping limit k→0, it corresponds to frequencies whethese eigenvalues are zero. Another way to understthese resonant modes is to see them as undamped ostions of the potential which need no external field to

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maintained, which corresponds toM(v,k50)X50; thesolution is found when the determinant of the matrDet@M ( v,k50 # ), is zero. This condition gives the resonant frequenciesv r and the associated eigenvectorsX(v5v r) and matrixM(v5v r ,k50). The corresponding maof the potential gives information on the self-polarizaticharges since the equipotential lines surround charges.

The oscillator strengths~the integrated intensities! andbroadenings of the absorption modes are determined bytroducing the damping in the excitating field at resonanE(v5v r ,k50) via a limited development of the matriand of the vector around the point (v5v r ,k50) up to firstorder in the differencev2v r and in the dampingk:

M~v,k!5M~v r ,0!1S ]M]v D ~v2v r !1 i S ]M

]k Dk, ~5!

X~v,k!5xX~v r !1S ]X

]v D ~v2v r !1 i S ]X

]k Dk. ~6!

Partial derivatives are taken atv5v r ,k50. By usingM(v r ,0)X(v r)50, the linear system, Eq.~3!, becomes tofirst order ink andv2v r :

M~v r ,0!F S ]X

]v D ~v2v r !1 i S ]X

]k DkG1xF S ]M

]v D ~v2v r !1 i S ]M]k DkG X~v r !5E~v r ,0!.

~7!

To determine the amplitude termx, the rotation terms for theeigenvector are eliminated by multiplying this equationthe left eigenvector tY(v r) of M(v r ,0). Indeed, asM(v r ,0), whose determinant is zero, is nonsymmetric,left X(v r) and rightY(v r) eigenvectors defined by

M~v r ,0!X~v r !50, tM~v r ,0!Y~v r !50 ~8!

are not equal. Thus, one has

x5tY~v r !E~v r ,0!

tY~v r !F S ]M]v D ~v2v r !1 i S ]M

]k DkG X~v r !

. ~9!

Finally, the polarizability is found by multiplying this amplitude x by the first term of the right eigenvectorX(v r) inthe basis of (Alm ,Blm) @Eqs.~A6! and~A7!#. At the limit ofsmall damping by summing on all modes, the polarizabiis written in the spectral representation:

a5(r

Fr

v r2v2 ig r1a0~v,k!, ~10!

whereFr andg r are the oscillator strength and the widththe absorption peak, respectively:

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]k Uk50

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]v Uk50

vr Di , j

Xir

, ~11!

with 1/E0inc the right normalization by the incident electr

field E0 as given by Eqs.~A6! and~A7!. The terma0(v,k)stands for the difference between the approximated sperepresentation@sum in Eq.~10!# and the exact polarizabilitycomputed by solving the linear system, Eq.~3!. A discussionon the validity of such a spectral decomposition is giventhe Appendix and the numerical implementation of the prolem is presented in the Appendix.

B. Nature of the modes

The oscillator strength and energy of a series of surfplasmon-polariton modes are shown in Fig. 2 in the casehemispherical particle of silver supported on a substrate wa dielectric constante253 which is supposed to be nonabsorbing. The bulk dielectric constant of silver is extractfrom Palik’s compilation.26 The equipotential lines assocated with the four more intense normal and parallel moare represented in Figs. 3 and 4, respectively. Bearingmind that the equipotential lines surround the polarizatcharges, these maps give a visualization of the charge ltion. The two dipolar modes, which are labeled 0-A and 1in Figs. 3 and 4, dominate the optical response. Their oslator strengthsFr are one order of magnitude greater than tother modes of similar polarization. The~1-A! mode presentsa strong pining of the charge at the triple-contact line btween vacuum, substrate, and island, which can be unstood in terms of the tip effect in electrostatics. The~0-A!mode corresponds to a dipole with a accumulation of charat the interface with the substrate and the top of the clus

FIG. 2. Absorption peaks defined by the imaginary part ofpolarizability normalized by the particle volume Im(a i1a')/V@Eq. ~A17!# for a silver hemisphere (t r50, M528, e253).

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All the others modes displayed in Figs. 3 and 4 are of qdrupolar nature. Although these are optically inactive forisolated sphere, they show up in the present conditionscause the breaking of symmetry due to the particle truncaand the presence of the substrate increase drasticallyoscillator strengths. Roughly, one order of magnitude isbetween each type of mode: dipolar, quadrupolar, . . . . Aconsequence the shape of the absorption given by the imnary part of the polarizability~Fig. 2! is mainly governed bythe few first mode and in some situations only by the dipoones~0-A,1-A!.

C. Substrate-induced depolarization field

When it is brought in contact with a particle, a dielectrsubstrate creates a depolarization field which perturbs b

FIG. 3. Maps of potential at orderM528 along a plane perpendicular to the substrate (y50) for the four most intense norma(m50) modes~from A to D! for a hemisphere of silver on a dielectric substrate (e253).

FIG. 4. Same plots as for Fig. 3 for the parallel mod(m51).

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the oscillator strength and the energy of each mode. Atrend, the modes represented in Figs. 3 and 4 are redshwhen the real part of the dielectric constante2 increases~Fig.5!. This can be explained in the framework of the dipointeraction model by replacing a particle of polarizabilityaby a dipole at a distanced from the surface. Accounting fothe substrate-induced local field on the dipole, the polaability is renormalized by this depolarization field in the folowing way:

a i* 5a i

11Aa i, a'

* 5a'

112Aa'

, ~12!

where

A51

32pe1d3

e12e2

e11e2.

Here A characterizes the image dipole strength. Indeed,frequency of a Lorentz oscillator,

a5Fr

v r2v2 ig r, ~13!

is redshifted proportionally to its oscillator strength, sinceAis negative. One has

a i* 5Fr

v r2v1AFr2 ig r, a'

* 5Fr

v r2v12AFr2 ig r.

~14!

In other words, the image dipole creates a depolarizafield which acts in the opposite direction as the force btween the electronic gas and the positive background of iothus reducing its effective restoring force. However, thmodel does not end up with a modification either of toscillator strength~observed in Figs. 3 and 4! or of thebroadening. Moreover, the divergence of the dipole intertion whend→0 indicates that the high-order coupling mube accounted for.

After a careful analysis, modes can be classified into tcategories: those arising from the field singularity at the trcation angle of the supported particle~0-B, 0-C, 1-A, and1-B!, which are mildly perturbed by the substrate, and thodue to charges at the particle/substrate interface, whichvery sensitive to the contact with the substrate~0-A, 0-D,1-C, and 1-D!. For instance, the oscillator strengths of tmodes denoted~0-A! and~0-D! in Fig. 3 and~1-C! in Fig. 4increase by four orders of magnitude upon increasinge2 withthe substrate. Therefore, because the most intense modedergo the strongest shifts upon increasinge2, the presence ofthe substrate may help revealing features of higher multilar order.

However, contact with the substrate also results indamping of the oscillations through dissipative procesand, consequently, in an increase of the width of the mo~see Fig. 6! with increasing the imaginary part of the dieletric function e2. As expected, Fig. 7 demonstrates that thigher the vibrating charge at the interface, the higher isdamping ~broadening! of the modes. Thus, as always, aincrease of Re(e2) implies an increase of Im(e2), there is anexperimental limit in the use of the substrate interactionisolate multipolar features.

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FIG. 5. Influence of the substrate-induced dpolarization field on the oscillator strengths anfrequencies of the most active modes for a silvhemisphere (M520). The lines with symbolscorrespond to the~A-B-C-D! modes of Figs. 3and 4! ~vertical dotted linee253). The other dot-ted lines correspond to less important modes.

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D. Particle-particle interactions: A way to shift the resonances

Another important point is how these multipolar modbehave upon interactions between particles. In the frawork of the quasistatic approximation, it has been shownit is sufficient, up to a rather high coverage (50%), to tainto account particle interactions only up to dipolar order

FIG. 6. Effect of the damping in the substrate on the imaginpart of the polarizability@Re(e2)53,M520#. Note the disappearance of the intermediate quadrupolar feature upon broadening ostructures.

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a renormalization of the polarizability in the followinway:27–31

a'I 5

a'

122a'I'20

, a iI5

a i

11a iI i20

. ~15!

Here,a' or a i stands for the polarizabilities of a clusterinteraction with the substrate as calculated from Eqs.~A6!and ~A7!. The interactions functionsI 20 are defined by:

FIG. 7. Influence of the imaginary part of the substrate dielecconstant on the widths of the modes@M520, Re(e253)#. Thelines with symbols correspond to the four modes seen in Figs. 34 with the same convention as in Fig. 5.

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A20pe1L3 FS202S e12e2

e11e2D S20

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2051

A20pe1L3 FS201S e12e2

e11e2D S20

r G . ~16!

The higher the substrate coupling and the coverage,larger theseI 20 functions are. The directS20 and S20

r imageslattices sums by:

S205(iÞ0

S L

r D 3

Y20~u,f!ur5Ri

,

S20r 5(

iÞ0S L

r D 3

Y20~u,f!ur5R

ir. ~17!

Here S20 describes the interaction with the others directpoles placed atr5Ri whereas the effect of image dipolesr5Ri

r is included inS20r . L is the lattice parameter of th

dipoles network where are placed the particles. No strdiscrepancies were found up to rather high coverage~up to50%) between regular and random arrays of dipoles.31 No-tice that the termi 50 is excluded as the polarizabilitya isthat of an interacting island. These formulas are valid ofor the island case. For the cap particle, one has to invere1ande2.

By putting the expression~10! into Eq. ~15! and by im-plicitly making a development to order zero around eamode one finds:

a.(r

Fr

v rI 2v2 ig r

, ~18!

with

~v rI !'5~v r !'22I'

20~Fr !' , ~v rI ! i5~v r ! i1I i

20~Fr ! i .~19!

These expressions demonstrate that particle-particle intetions only induce a frequency shift of the oscillator, proptional to its strength, but its strength and width remain costant. Parallel and normal modes shifts have opposite sigthe termI 20 keeps constant over the spectral range whichthe case with dielectric substrate in the UV visible. Notithat the negative sign of theI 20 terms justifies the blueshifand redshift of them50 andm51 modes, respectively.

V. MULTIPOLAR RESONANCES FOR SILVER ONALUMINA

The formalism of the surface susceptibilities developby Bedeaux and Vlieger32–35allowed us to derive the formulas of differential reflectivity on a nonabsorbing dielectrsubstratee2:

DRs

Rs54

v

c

n1cosu

e22e1Im~g!,

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~e22e1!~e1sin2u2e2cos2u!

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~20!

with g5ra iI andb5ra'

I /(e* )2 @Eqs.~15!#. e* 5e1 ,e2 foran island and a cap, respectively, andu is the angle of inci-dence.r is the surface density of clusters. Equations~20!demonstrate clearly that the SDR spectra arise fromimaginary part of the particle susceptibilities, i.e., from tabsorption. It has to be noticed that the same type of formlas were derived by Bagchiet al.36 for the reflectivity of asurface with a nonlocal surface dielectric tensor. By puttthe spectral representation, Eq.~A18!, into Eqs.~20!, as ex-pected, the parallel polarizability determines the SDR spein s polarization whereas inp polarization, the normal andparallel modes appear with opposite sign.

For small particles, it was established several yearsthat the bulk mean free path of the conduction electronincreased by the scattering at the surface of the particle.silver, a classical finite-size correction is often used foselectrons:1,6

e~v!5eB~v!1vp

2

v21 ivtB21

2vp

2

v21 ivt21, ~21!

whereeB is the bulk dielectric function,\vp59.17 eV isthe plasma frequency ofs electrons alone, and\/tB50.018 eV is the bulk relaxation time. The main effectsuch a correction is to modify, quasiuniformly, the widthg rof the absorption modes. The key point is the energeticsition of the modes as the increase of\/t results in an in-crease of Im(e3) in the low-energy part of the spectrum.

The present model of a truncated sphere was used tthe experimental data with a finite-size correction for tsilver dielectric constant, Eq.~21!. The correction for therelaxation time was

\

t5

\

tB1

vF

R, ~22!

wherevF50.91 eV nm is the Fermi velocity andR the par-ticle radius. The particles were placed on an hexagonaltice. The best agreement with the 4-nm spectrum shownFig. 8 was obtained with a radiusR56.4 nm, a densityr53.231011 part cm22, and and a truncation parametert r50.57. The obtained thicknesse52.8 nm is a bit lower thanthe nominal thickness because of the sticking coefficienting lower than 1.37 Here t r50.57 leads to an effective contact angle ofuc5125° close to that expected for the epected equilibrium shape. All the features are accountedby the present model, in particular the shoulder inp polar-ization. This shoulder is revealed upon the increase ofdeposited thickness by the electromagnetic coupling betwthe growing particles. Indeed, this feature appears whendispersion with the deposited thickness of the high-eneresonance inp polarization~inset in Fig. 1! changes sign. Atthe beginning of the growth, the size of the particles is su

4-7

r-

e

r


FIG. 8. Comparison betweenthe truncated sphere model at oder M524 and the 4-nm SDRspectrum of Fig. 1 inp polariza-tion ~left panel! ands polarization~right panel!.

FIG. 9. Computed absorptioncoefficient at orderM524 of the4-nm layer of Ag/Al2O3(0001) byreducing the imaginary part of themetal dielectric constant (0.01e3).The interparticle coupling up todipolar order is accounted for inthe dotted line plot whereas it isswitched off for the solid line plot.The locations of the modes armarked with vertical bars in thelower panel according to theiactivity.

rtithw

e

isaee

ore

m

llit

ethan

pli-

all

elas-

se,c-nsi-be-ism-in

-tralare

heling

.a.

m

that the surface plasma resonance is shifted towablue.38–40This quantum size effect disappears when the rasurface over volume increases, leading to a redshift inabove-mentioned energy shift. Then a new blueshift shoup because of particle coupling~see Sec. IV D!. The inter-particle coupling is essential to the appearing of the intermdiate shoulder inp polarization whereas ins polarization thepeak is present even at the beginning of the growth. Tolate the absorption structures in the simulation, the imaginpart of the silver dielectric constant was drastically reducto 1% of its value in the absorption coefficient of the lay~Fig. 9!. Five main modes are isolated with roughly the crespondence given in Table I with the above mentionmodes.

Obviously, the quadrupolar modes 3-4-5 modify copletely the overall shape of the SDR spectra. Inp polariza-tion ~Fig. 9!, the situation is complicated by the fact that athe m50,61 modes are active and, as demonstrated wEq. ~20!, act with opposite signs as shown in Fig. 10~upperpanels!. As a consequence, the structures are isolated inperiments only when the particle interaction pushes awaymode frequencies according to their oscillator strengths

TABLE I. Correspondence with the modes depicted in Figsand 4 and the isolated structures for silver deposited on aluminDandQ mean dipole and quadrupole, respectively.

Peaks 1 2 3 4 5

1-A 0-A 1-C 1-D 0-BNature D i D' Qi Qi Q'

23542

dsoes

-

o-rydr-d

-

h

x-ed

to their activity m50,61 @Eq. ~19!#. In s polarization, them50 modes are inactive and the spectrum is greatly simfied, allowing to appear only the structure 3-4~Fig. 10, lowerpanels, and Fig. 1!. It is interesting to notice that the spectrdecomposition@Eq. ~A18!# is unable to describe the opticabehavior beyondE53.8 eV. Indeed, the interband 5s-5ptransitions in bulk silver41 are not accounted for in this typapproach as the absorption is only described in terms of pmon oscillations which are artificially broadened. Of courthe full solution of the linear system in the Appendix acounts for this phenomenon. This onset of interband trations appears, like the plasma oscillation, from the earlyginning of the growth, implying that the clusters sizesufficient enough to develop the band structure. To copletely assess this decomposition, the 4-nm spectrumspolarization was fitted with three Lorentzians~Fig. 11! assuggested by formula~20!. A linear background was subtracted and to get rid of the interband transition the specrange was limited below 4 eV. The obtained parameterspresented in Table II.

The agreement is fairly good for the peak position. Tcalculated intensities do not take into account the coup

3 TABLE II. Experimental and theoretical values for the 4-nSDR-spectrum multipolar decomposition.

Peak 1 Peak 3 Peak 4

Position~eV! 2.75–2.78 3.54–3.55 3.05–3.19Width ~eV! 0.27–0.16 0.10–0.14 0.14–0.17Intensity ~a.u.! 1.04–0.46 0.06–0.16 0.06–0.03

4-8

Fith

ex

elanth

eionrthiccethacle

io-

e

re

rei-

ned

ear


between modes due to the particle interaction@Eqs. ~18!#which seems to modify the oscillator strength as seen in10. For the quadrupole modes, the order of magnitude forwidths is correct but the dipolar mode is broader thanpected, which may be because of a size distribution.

V. CONCLUSION

A formalism of multipolar absorptions has been devoped to understand the optical properties of supported nparticles. Even if the size of the particle is smaller thanoptical wavelength, the nonhomogenous field generatedthe image term inside the substrate and the realistic shapthe cluster modify drastically the problem of the absorptof light by allowing the excitation of high-multipolar-ordemodes. The multipolar expansion of the potential gaveopportunity to derive a spectral representation of the partpolarizability in terms of damped oscillators. The influenof the various parameters on the oscillator strength, widand frequency has been carefully examined. By takingvantage of the polarization activity and the particle-particoupling, quadrupolar absorptions were clearly isolatedsome experiments of differential reflectivity for silver depsits on oxide substrate.

APPENDIX

Expansion of the potential

The chosen particle shape is that of a truncated spherradiusR ~medium 3! supported on a substrate~medium 2!and embedded in medium 1. The buried part of the spheintroduced as a different medium~medium 4!. The origin ofthe coordinates systemO is taken at the center of the spheand thez axis is oriented downwards, pointing into the d

23542

g.e-

-o-ebyof

ele

,d-

n

of

is

rection of the substrate. The substrate plane is defithrough its altitudez5d with 2R<d<R and the imagepoint O8 is introduced as the symmetric point (0,0,2d) withrespect to the surface. A pointP(x,y,z) is marked by itsspherical coordinates (r ,u,f) in the O system and(r,u r ,f r) in the O8 system.24 The signed quantity

t r5d

Rwith 21<t r<1 ~A1!

is introduced as a truncation parameter.t r51 is a spheretouching the substrate in only one point, andt r50 corre-sponds to a hemisphere whereast r521 is associated with a

FIG. 11. Fitting of the 4-nm SDR spectrum ins polarizationwith the three main Lorentzian peaks of Fig. 9. Notice that a linbackground was subtracted.

es. The
FIG. 10. Calculated SDR spectra for the fitted morphology with the location and integrated intensities of the absorption modcurves are computed either directly by solving the linear system, Eq.~A5! ~solid line!, or by the spectral representation of Eq.~A18! ~dottedline! at orderM524.
4-9

,s

o

eimdrt-

onhethfoeenfo

eis

e

nd

ne

are

-eak

n-

capde-tsnd.

scy.-al

of

leiz-

ce.


completely buried particle. As a language conventiont r.0 particle is called an island while a cap designatet r,0 particle. The system is excited by a uniform fieldE0with polar coordinates (u0 ,f0). The dielectric functions ofthe various media are denotede i with i 51,2,3,4 (e45e2).

The Laplace equation has been already solved for a ptive truncation ratiot r>0 ~Refs. 24, 25, 27, and 42! by usingspherical multipoles43 Yl

m(u,f) ~see Ref. 24 for the precisdefinition! and treating the substrate plane by the chargeage technique. Fort r,0, this approach no longer hold. Winand Vlieger27,42 suggested to use a trick consisting of inveing the axis system to derive the cap caset r,0 from thefinal formulas of the island one,t r.0. In this context, Simo-nsenet al.24 allowed the expansion center to freely movethe revolution axis in order to always keep it inside tphysical domain. But as the spherical symmetry is lost,computational burden greatly increased. Even if the finalmulas of the Wind-Vliegeret al. approach are correct, thunderlying expansion of the potential used is not transparHere, we give the complete derivation of the potentialt r,0, in a completely analogous way as fort r.0:

C1~r !5C0~r !1(lm

lÞ0

Almt r 2 l 21Yl

m~u,f!,

C2~r !5C0t ~r !1(

lm

lÞ0

Almr 2 l 21Ylm~u,f!

1(lm

lÞ0

Almr r2 l 21Yl

m~u r ,f r !,

C3~r !5c01(lm

lÞ0

Blmt r lYl

m~u,f!,

C4~r !5c01(lm

lÞ0

Blmr lYlm~u,f!1(

lm

lÞ0

Blmr r lYl

m~u r ,f r !.

~A2!

C0(r ) and C0t (r ) ~Refs. 24 and 25! are the potentials in-

duced by the uniform incident and transmitted fields, resptively. c0 is a matching constant. In medium 1, the fieldtransmitted from pointO to the surface, giving rise to thterm Alm

t .The boundary conditions, i.e., continuity of potential a

of normal displacement field,24,25 are the following at eachinterfacer s between mediai and j:

C i~r s!5C j~r s!, ]nC i~r s!5]nC j~r s!. ~A3!

These are used to get the unknown multipolar coefficieAlm ,Blm , . . . . In comparison to thet r.0 case, those at thsubstrate surface are modified in the following way~valid fort r,0):

23542

aa

si-

-

er-

t.r

c-

ts

Almr 5~21! l 1m

e22e1

e21e1Alm , Alm

t 52e2

e21e1Alm ,

Blmr 5~21! l 1m

e42e3

e41e3Blm , Blm

t 52e4

e41e3Blm . ~A4!

To get the multipole unknown independent coefficientsAlmandBlm , the boundary conditions on the sphere surfaceused. The boundary conditions, Eqs.~A3!, are projected on aspherical harmonicYl

m(u,f) using the fact that these functions form a complete basis on the sphere surface. This wformulation of the boundary conditions gives an infinite liear set of equations for the multipolar coefficients forl51,2,3, . . . andm50,61:

(l 151

`

Cll 1

m R2 l 122Al 1m1 (l 151

`

Dll 1

m Rl 121Bl 1m5Hlm ,

(l 151

`

Fll 1

m R2 l 122Al 1m1 (l 151

`

Gll 1

m Rl 121Bl 1m5Jlm . ~A5!

The equations with the matrix elements for an island or aare given hereafter. The right-hand side of this systemscribes the source fieldE0 while the matrix system representhe geometric and dielectric response function of the islaThe potential is fully obtained with the constant termc0

given by the equations containingH00. Its expression is given

in the Appendix. The system, Eqs.~A5!, is solved numeri-cally by truncating it at an arbitrary multipole orderM. HereM is in principle chosen24 such that the boundary conditionand thus the potential converged with the desired accura

The effective polarizability tensor links the dipole moment p of the charge distribution and the applied extern

field E0 : p5 aE0. Because of the symmetry constraints

the system, only two components ofa parallel and normal tothe surface (a i ,a') are nonzero. The terms inr 22 in themultipolar expansion in medium 1 give the far-field dipobehavior with the following relations between the polarabilities and the multipole coefficients for an island (t r.0):

a'52pe1A10/@Ap/3E0cosu0#,

a i524pe1A11/@A2p/3E0sinu0exp~2 if0!#. ~A6!

In the cap case (t r,0), the dividing electric field is(e1 /e2)E0cosu0 instead ofE0cosu0 for the z component,whereas the parallel one is kept atE0sinu0exp(2if0), be-cause the direct multipoles are located below the surfaMoreover,e1 should be replaced bye2:

a'52pe2A10/@~e1 /e2!Ap/3E0cosu0#,

a i524pe2A11/@A2p/3E0sinu0exp~2 if0!#. ~A7!

Matrix elements and the constant potentialc0

The boundary equations~A3! lead after a straightforwardalgebra to the linear system, Eqs.~A7!, whose matrix ele-ments are given by:

4-10

s

y.

e is

lla-

with

. It


Cll 1

m 5u21S 2e1

e11e2D d l l 1

2vS e12e2

e11e2D

3z l l 1

m $Qll 1

m ~ t0!2~21! l 11mSll 1

m ~ t0!%,

Dll 1

m 52u43S 2e3

e31e4D d l l 1

1vS e32e4

e31e4D

3z l l 1

m $Qll 1

m ~ t0!2~21! l 11mTll 1

m ~ t0!%,

Fll 1

m 52S 2e1e2

e11e2D ~ l 111!d l l 1

2vu21e1S e12e2

e11e2D

3z l l 1

m H ~ l 111!Qll 1

m ~ t0!2~21! l 11mF ]

]tSll 1

m ~ t,t0!Gt51

J ,

Gll 1

m 52S 2e3e4

e31e4D l 1d l l 1

2vu43e3S e32e4

e31e4D

3z l l 1

m H l 1Qll 1

m ~ t0!1~21! l 11mF ]

]tTll 1

m ~ t,t0!Gt51

J ,

~A8!

with t05ut r u, u215u435v51 for an island, andu215e2 /e1 , u235e4 /e3 , v5(21)l 1 l 111 for a cap. Here thefollowing notation has been introduced:

z l l 1

m 51

2 F ~2l 11!~2l 111!~ l 2m!! ~ l 12m!!

~ l 1m!! ~ l 11m!! G1/2

. ~A9!

In the previous system, theQll 1

m are defined by the integral

Qll 1

m ~ t0!5E21

t0Pl

m~x!Pl 1m~x!dx. ~A10!

The matricesSll 1

m (t0) andTll 1

m (t0) are defined by:

Sll 1

m ~ t0!5@Sll 1

m ~ t,t0!# t51 , Tll 1

m ~ t0!5@Tll 1

m ~ t,t0!# t51 ,

~A11!

where

Sll 1

m ~ t,t0!5E21

t0Pl

m~x!Pl 1m~@ tx22t0#@ t224t0tx14t0

2#21/2!

3~ t224t0tx14t02!2( l 111)/2dx, ~A12!

Tll 1

m ~ t,t0!5E21

t0Pl

m~x!Pl 1m~@ tx22t0#@ t224t0tx14t0

2#21/2!

3~ t224t0tx14t02! l 1/2dx, ~A13!

with

t[r

R, x[cosu. ~A14!

One can recognize the Legendre functionsPlm(x) defined in

Refs. 24 and 25. The source field is given by:

23542

Hl05A4p/3E0cosu0H u21

e1

e2d l11vS e12e2

e2D

3@A3t0z l00 Ql0

0 ~ t0!2z l10 Ql1

0 ~ t0!#J ,

Jl05A4p/3E0cosu0e1d l1 ,

Hl152A2p/3E0sinu0exp~2 if0!d l1 ,

Jl152A2p/3E0sinu0exp~2 if0!@u12e2d l1

1v~e12e2!z l11 Ql1

1 ~ t0!#, ~A15!

with v51 and (21)l 12 for an island and a cap, respectivelTo conclude the constant termc0 @Eq. ~A2!# which permitsone to match the potential inside and outside the spherfound in terms of the multipole coefficients thanks to thel50 equation of the linear system:

c05RS e1

e221D H 1

A3z01

0 Q010 ~ t0!1t0@p2z00

0 Q000 ~ t0!#J

3E0cosu021

2Ap(l 51

`

Al0R2 l 21qS e12e2

e11e2D

3z0l0 @Q0l

0 ~ t0!2~21! lS0l0 ~ t0!#1

1

2Ap

3(l 51

`

Bl0RlqS e32e4

e31e4D z0l

0 @Q0l0 ~ t0!2~21! lT0l

0 ~ t0!#,

~A16!

with p5q51 for an island andp50, q5(21)l 11 for a cap.

Spectral representation of the polarizability

Being decomposed in a sum of damped Lorentz oscitors, in the limitv→v r , the polarizability is written in Eq.~10! in a local Kramers-Heisenberg form. In the limitk50,the system behaves as a sum of undamped oscillatorsDirac absorptions:

a5(r

FrPS 1

v r2v D1 ip(r

Frd~v2v r !, ~A17!

whereP is the principal part. Of course, in the form~10!, thepolarizability does not fulfill the Kramers-Kronig relation44

as it results from a local development around each modeis possible to artificially impose this condition by writing

a5(r

2v rFr

v r22v22 i2vg r

1a0KK~v,k!. ~A18!

4-11

nd

t


FIG. 12. Comparison betweereduced polarizabilities computedirectly by solving Eq.~3! ~boldcurves! or with the spectral repre-sentation @sum in Eq. ~10!#(k51) ~thin curves! for a hemi-sphere (t r50) of silver on a sub-strate with a dielectric constane253 (M524).

d

ess i

itlv

innctce

ncbil

t

ne.

ric, to

tem

heforare.ots

m-

In this approach, the modes are supposed to be uncouplethe fact that one can choose arbitrary small dampingk anddeviation from the resonant pointv r . This model gives onlyan estimate of the broadening to first order ink. Upon ap-proaching the limitk51, the interactions between modcan greatly modify the widths and the oscillators strengthsuch a way that the notion itself of eigenmodes losesmeaning. However, polarizability curves computed by soing exactly the system, Eqs.~3!, compare quite well withabsorptions curves calculated by takingk51. An examplefor the polarizabilities of a hemisphere of silver is givenFig. 12. The graph in Fig. 12 shows that the main differe@term a0(v,k) in Eq. ~10!# comes from a mainly constanshift of the real part of the polarizability. The consequenon the optical properties of islands layers is negligible sithey depend mainly on the imaginary part of the polarizaities. Notice that the spectral representation is not abledescribe the increase of absorption beyondE53.8 eV; thisphenomenon is linked to interband transitions in silver anot to collective damped oscillations of the electric charg

A careful analysis of the matrix, Eqs.~A5!, shows that itsdeterminant is a rational function of orderM in the particledielectric constante3. Thus, for a substrate whose dielectconstant presents a small dependence on frequencymaximum number of modes is given by the size of multip

.

-

23542

by

ns-

e

e-o

d

he-

lar basisM. Obviously, their optical activityFr depends onthe dielectric constants of the materials and on the sysgeometry.

Numerical implementation of the oscillator strength andbroadening

The Legendre polynomialsPlm(x) are computed through

stable recurrence relations.45,46 The integrals Qll 1

m (t r),

Sll 1

m (t r), Tll 1

m (t r) @Eqs. ~A10!–~A13!# are computed bynumerical integration with the algorithm of Piessenset al.47

which handles well the strong oscillating behavior of tLegendre polynomials. However, recurrence relationsthese integrals derived in Ref. 27 have shown that theyregular functions~polynomials! of the truncation parameterFor finding the multipolar absorptions peaks, first the roof the determinant of the matrix systemM(v,0), Eq.~A5!,are bracketed by dichotomy. The matrix determinant is coputed by theLAPACK LU-decomposition scheme.48 The asso-ciated left X(v,0) and rightY(v,0) eigenvectors are thenfound by an inverse iteration process.45 The development ofterms in (e i2e j )/(e i1e j ) aroundv5v r andk50 gives thederivative matrices]M/]vuvr ,,k50 and ]M/]vuvr ,k50.Thus using the formulas~11!, one is able to compute theoscillator strength Fr and broadeningg r for parallel(m561) and perpendicular (m50) modes.

v.

a,

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ah-ra-

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4-13

Multipolar plasmon resonances in supported silver particles: The case of Ag / α − Al 2 O 3 ( 0001 )

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