Electron dynamics in metallic nanoparticles

Ž .Chemical Physics 251 2000 181–203www.elsevier.nlrlocaterchemphys

Electron dynamics in metallic nanoparticles

J.-Y. Bigot ), V. Halte, J.-C. Merle, A. DaunoisÍnstitut de Physique et Chimie des Materiaux de Strasbourg, Groupe d’Optique Nonlineaire, Unite Mixte a 7504 CNRS-ULP-ECPM, 23 rue´ ´ ´

du Loess, 67037 Strasbourg Cedex, France

Received 10 March 1999

Abstract

We studied the dynamics of electrons in copper and silver nanoparticles embedded in a transparent matrix, using thetechnique of pump–probe femtosecond spectroscopy. Comparative measurements are made in thin films of the same metals.In the case of the nanoparticles, the electron dynamics is strongly influenced by the surface at the boundary of the metal andthe surrounding dielectric matrix. A detailed study of the pump–probe signals near the plasmon resonance of thenanoparticles reveals the importance of electron–electron scattering during several hundreds of femtoseconds. The influenceof these scattering processes on the real and imaginary parts of the metal dielectric function is compared in the nanoparticlesand thin films. In addition, the non-thermal component of the electrons and the heat transfer to the surrounding dielectric aremeasured. The results are analyzed with a model of effective medium, where the metal dielectric function is described in therandom phase approximation, including the surface effects in a phenomenological way. q 2000 Elsevier Science B.V. Allrights reserved.

1. Introduction

The behavior of the electron dynamics in metallicnanoparticles is different than the one observed inthe corresponding bulk metal. In order to examinethese differences, let us consider the relaxationmechanisms that take place in a metallic particlewhen it is excited with a short optical pulse. In Fig.1, we have sketched the energy relaxation followingthe excitation of the nanoparticle with a femtosecondpulse. Different time scales are involved. Initially,the energy is transferred to the electrons by absorp-tion of photons via interband and intraband transi-tions. During this quasi-instantaneous process, the

) Corresponding author.

phase memory is conserved between the electromag-netic field and the electronic states, and the densityof excited states depends on the spectral shape of thelaser pulse. The corresponding electron distributionis non-thermal as expected in any metallic systemw x1,2 . In this time scale of a few femtoseconds, theelectronic system is strongly correlated and its col-lective character is important.

The next step of the energy relaxation corre-sponds to a thermalization of the electrons. Theoccupied electronic states tend to a Fermi–Diracdistribution with a well defined temperature whichdepends on the laser pulse intensity. The increase oftemperature can easily reach several hundreds ofdegrees. The phase coherence is lost and the collec-tive modes have decayed into quasi-particle pairs.Several time resolved photoemission experiments,

0301-0104r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0301-0104 99 00298-0

( )J.-Y. Bigot et al.rChemical Physics 251 2000 181–203182

Fig. 1. Sketch of the relaxation processes in a metallic nanoparti-cle.

performed in noble metal films, have shown that thetemporal scale of this thermalization process is of a

w xfew hundreds of femtoseconds 1–8 . For small par-ticles, with a diameter typically less than a few tensof nanometers, the scattering time of the electrons atthe particle surface is less than one hundred fem-toseconds. It can be deduced simply by assuming aneffective damping V rR where V is the velocityF eff F

of electrons at the Fermi level and R an effectiveeff

particle radius which takes into account the effect ofquantum confinement on the density of states in the

w xmetal 9 . In this time scale, the electron–electronŽ .e–e scattering is also very efficient to redistributethe energy to thermalize the electrons. Therefore thetwo mechanisms of surface and e–e scattering are incompetition. As we will see in the next sections ofthe present paper, this difference shows up in a time

dependent broadening and shift of the surface plas-mon associated to the nanoparticle.

Another difference between the particles and bulkmetal comes from the surface induced polarization.As it is well known, the dielectric confinement be-comes important when the size of the particles ismuch smaller than the optical wavelength of the

w xexcitation 10–13 . In the nanoparticles, it gives riseto eigenmodes of the electromagnetic field which areanalogous to the cavity modes of a resonator. Forsmall enough particles, the first order mode is pre-dominant and the model of Maxwell–Garnett of the

w xeffective medium applies 14,15 . In a similar man-ner, the Coulomb potential inside the nanoparticle isrenormalized by the surface induced polarization.This results in a modification of the dynamical

w xscreening of the e–e interaction 16 . In the third partof the paper, we will discuss how it influences thedynamics of the electrons in the particle as comparedto the bulk metal.

Another important mechanism in the electron dy-namics, which is shown in Fig. 1, is the energytransfer to the lattice. This process, which has beenstudied with great detail in thin films made of vari-

w xous metals 17–25 , can be described by two coupledŽ .baths the electrons and the lattice . It also displays

differences when comparing the dynamics in metal-lic particles and the corresponding bulk. At themicroscopic level, it is the electron–phonon interac-tion which couples the two baths. Therefore, due tothe reduced dimensionality of the nanoparticles, thesurface modes of the metallic particle influence theenergy transfer between the electrons and the lattice.In addition, the coherent behavior of the latticevibrations shows up when the electrons are in equi-

w xlibrium with the lattice temperature 26–28 . The laststep in the relaxation is the energy transfer to thedielectric matrix. This transfer corresponds to theheat diffusion from the metal to the environment. Itis therefore sensitive to the thermal conductivity ofthe surrounding medium. We will show how thisenergy transfer is affected by the matrix.

One should mention that the above considerationsare relevant for metallic particles with small enoughsizes, so that surface effects are important. On theother hand, they do not apply to clusters smaller thana few tens of angstroms. The discrete nature of the˚ ëlectronic states has then to be considered. Here, the

( )J.-Y. Bigot et al.rChemical Physics 251 2000 181–203 183

metal is still considered as a three-dimensional sys-tem where the surface affects both the many-bodyinteractions between the electrons and the electron–lattice relaxation. It is important to stress that thesurface effects will be added as ad hoc phenomena.A more rigorous way would be to consider theelectron correlation in finite systems. It is an interest-ing and challenging numerical task for the futuresince it requires very large computer capacities whichare not yet available.

In Section 2, we first present experimental resultsobtained in thin films of copper and silver. Theelectron dynamics is then compared to the corre-sponding nanoparticles made of the same metal. Wethen present a model based on the dielectric functiondescribed in the effective medium approximation.The dynamics is incorporated via the time dependenttemperatures of the electrons and the lattice and theinfluence of the surface is incorporated in a phe-nomenological way in the case of the nanoparticles.

2. Experimental results

2.1. Experimental technique

The electron dynamics has been studied in thinfilms and nanoparticles of copper and silver. Thenanoparticles are obtained by a nucleation-growthtechnique. The metal oxide is first introduced in themelted glass where the germination process takesplace. The glass composition is made of SiO , PbO,2

Na O, K O. The lead oxide PbO is added to in-2 2

crease the refractive index of the matrix and Na O,2

K O allow to lower the viscosity and to enlarge the2

temperature range where the glass is processed. Thedoped glass is then annealed in order to increase thesize of the initial germs. It is the Oswald maturation

Ž .process. By adding a reducing agent Sb, Sn , or bya processing under hydrogen atmosphere, the germi-

w xnation and growing processes are promoted 29 . Theaverage diameter of the nanoparticles obtained withthis technique is larger than 5 nm with low concen-

Ž y3 y6.trations in volume typically 10 to 10 . Themicroscopy electronic transmission observationsdemonstrate that the nanoparticles are spherical witha relatively homogeneous size distribution. Let us

notice that such samples have a high chemical stabil-ity. For instance, they can be washed in a solution ofaceton or ethanol without damage. In addition, theypresent a high optical breakdown threshold to the

Ž 10 y2 .laser power more than 10 W cm . In contrast,we noticed that samples made by a co-deposition of

Žthe nanoparticles and the matrix like with the tech-w x.nique of Low Energy Cluster Beam Deposition 30

have a breakdown threshold about ten times lower.Fig. 2 shows typical absorption spectra of copperŽ . Ž .2a and silver 2b nanoparticles. The optical den-sity is represented as a function of wavelength for

Ž .particles with an average diameter of 10 nm CuŽ .and 6.5 nm Ag . Both samples display a plasmon

Ž .resonance situated respectively at 2.22 eV 558 nmŽ .for Cu and 2.85 eV 434.5 nm for Ag. In the case of

copper, the resonance is situated near the interbandŽ .transitions d™p E s2.17 eV or 571 nm fromdp

the filled d band to the unoccupied states in the pconduction band. This spectral degeneracy is at theorigin of the large absorption on the high energy sideof the plasmon resonance. As discussed in Section2.3, it is an ideal situation to study the effect ofcollision broadening of the plasmon by the quasipar-ticle states excited by the femtosecond pulses. In thecase of silver, the resonance is situated well below

Fig. 2. Optical density of nanoparticles of copper and silverembedded in a glass matrix near the plasmon resonance.


Žthe interband transition thresholds d™p E s3.99dp. Ž .eV or 310 nm and p™s E s3.85 eV or 322 nmps

from the occupied p states to the unoccupied s states.The resonance is therefore well defined and, bycontrast to Cu, it allows the observation of thedynamics associated to the heating of the electronpopulations, without probing directly the excitedelectronic population. In the next sections, we willsee that the two types of materials allow to clearlydistinguish the effects of the electron dynamics onthe real and imaginary parts of the dielectric functionof the metal.

The Cu and Ag polycrystalline thin films havebeen evaporated on a glass substrate under highvacuum. Their thicknesses, of a few hundreds ofAngstroms, is determined by X-ray diffraction at¨grazing incidence. The complex linear refractive in-dex NsN y i N , is determined by ellipsometry1 2

and by measuring the linear transmission and reflec-tion. In the case of silver, the real and imaginaryparts of N are almost constant since they have aweak dispersion in the spectral range of interestŽ .2.2–3.5 eV . We obtain the following values: N s1

0.09 and N s1.6. They are comparable to those2w xgenerally reported for silver films 31–33 and the

low N value of our samples is an indication of their2

good crystalline quality. In the case of copper, thereis an important dispersion of the refractive index dueto the interband transitions in the spectral region of

Ž .interest 1.7–2.6 eV . During the experiments thesamples, which are exposed to the ambient air, aresubject to oxidation. However, we have checkedwith ellipsometric measurements that no significantchange of the optical constants occur within a fewdays. The oxidation layer, which rapidly forms whenthe films are taken out of the vacuum chamber,stabilizes after reaching a thickness of a fewAngstroms.¨

Time resolved transmission and reflection mea-surements have been performed using different laserapparatus. For the silver thin films and nanoparticles,the femtosecond pulses are produced by a tunabletitanium sapphire laser. The pulses issued from anoscillator operating at 80 MHz are amplified in a

Žregenerative amplifier pumped by a Nd:YLF yt-.trium lithium fluoride laser with a repetition rate of

5 kHz. The maximum energy per pulse is ;200 mJtunable in the range 760–860 nm and the pulse

duration is ;100 fs. One part of the amplified beamŽis used to generate a broad band continuum 0.35–1

.mm in a 3 mm sapphire plate. The group velocitydispersion of this continuum is only partially com-pensated with a sequence of four prisms. A chirp of;1 ps in the spectral region 360–560 nm has beendetermined from cross-correlation measurements.The second part of the amplified beam is frequencydoubled in a 1 mm thick BBO crystal. Two pumpand probe configurations are used depending on theinformation that is required. In the first one, thepump and probe are degenerate in frequency andtunable in the spectral range 430–380 nm. Thisconfiguration is used either with the amplified laserat 5 kHz repetition rate or with the frequency dou-bled non-amplified beam issued from the oscillatorat 80 MHz. Using the two repetition rates allows theenergy density of the pump beam, absorbed in the

Žsamples, to be varied in a broad range 0.1–5002 .mJrcm . Fig. 3 shows the laser set-up and the

experimental configuration.For the case of copper thin films and particles, the

femtosecond pulses are produced by a collidingpulsed mode locked cavity operating at 620 nm andamplified at 5 kHz with a copper vapor laser. Theamplified pulses of 80 fs duration are divided intotwo parts. One beam corresponds to the pump andthe other one is used to generate pulses of 10 fsduration in a compression line with a fiber, gratingsand prisms. These pulses are used as a broad spec-trum probe. For both types of material, the differen-

Ž . Ž .tial transmission DTrT t s T –T rT is mea-on off off

sured as a function of the temporal delay t betweenŽ .the pump and probe, T T being the normalizedon offŽ .probe transmission with without the pump. In the

case of the thin films, we also measured the differen-Ž .tial reflection D RrR t . The detection scheme is a

synchronous detection using a chopper and a lock-inamplifier for the measurements at a fixed wave-

Ž .length. The spectral measurements DTrT l,t aremade as a function of the probe wavelength l, forvarious pump–probe delay t, using a monochro-mator and a dual array CCD camera. The experi-ments are performed at room temperature. For theglass embedded Ag nanoparticles, additional mea-surements at liquid helium temperature have beenmade in order to compare the electron–lattice relax-ation for different electron and lattice specific heat.


Fig. 3. Laser experimental set-up to study the dynamics of Ag nanoparticles and films.

The ratio of the pump and probe beam diameterson the samples is ;3 and their overlap is monitoredon a video camera. In the following sections, whenwe indicate an energy density of the pump pulse, wewill always refer to the one that is absorbed by thesample. The two beams are cross polarized in orderto minimize the coherent interaction of the probewith the polarization set by the pump beam whenthey overlap in time. However, since the phase relax-ation of electrons is very short in metals, this precau-tion is necessary only when the detailed dynamicsaround zero delay is required. In all measurements,the intensity of the probe is 100 times less than thepump one. Let us stress that the pulse duration of thepump is longer than the electron dephasing timeŽ w x.typically less than 20 fs in noble metals 5–34 sothat all mechanisms related to the phase of the laserfield are ignored in the following.

2.2. Electron dynamics in Cu and Ag thin films

w xSince the early work of Eesley 17 , the electrondynamics in thin films of the noble metals Cu, Au

w xand Ag has been extensively studied 18–25,35–40 .We report here on the dynamics of Cu and Ag filmsin order to compare their behavior with the corre-sponding Cu and Ag nanoparticles, using the sameexperimental apparatus. In addition, the simultaneousmeasurement of the differential transmission and re-flection as a function of pump–probe delay allows toretrieve the time dependent complex dielectric func-

Ž .tion ´ t of the metal. We note D´ and D´ the1 2

corresponding changes of the real and imaginary partŽ .of ´ t , induced by the pump pulses. They are the

physical quantities of interest to understand the elec-tron dynamics. One should keep in mind that, in ametallic film, the important reflectivity does not


allow to interpret the change of the probe transmis-sion as a simple change of the material absorption.Both DTrT and D RrR have to be measured. Inorder to retrieve the dielectric function, we follow

w xthe procedure used by Rosei 41 to analyze staticthermo-modulation measurements. In this procedure,the Fabry–Perot type of transmission Ts tt ) and´reflection Rsrr ) of a thin metallic film of thick-ness ll are functions of N, l and ll . They aredifferentiated with respect to a small variation of therefractive index D NsD N –iD N to give:1 2

BCyAD D´ sB DTrT yD D RrR , 1aŽ . Ž . Ž . Ž .1

BCyAD D´ sC D RrR yA DTrT , 1bŽ . Ž . Ž . Ž .2

where A, B, C and D depend on N, l and ll , andthe dielectric function is obtained from ´s´ –i´1 2

sN 2. In the static thermo-modulation experiments,where the temperature of the lattice Q is increasedl

Ž .via a current applied to the sample, the relations 1correspond to a spectral variation of the dielectric

Ž . Ž .function D´ v and D´ v . Here they correspond1 2Ž .to a spectro-temporal variation D´ v ,t and1

Ž .D´ v,t . In this case, the change of the dielectric2

function comes from a modification of the electronicdistribution induced by the laser pulse, or equiva-lently, induced by a change of the electronic temper-

Ž . Ž .ature DQ sQ t yQ y` when the electrons aree e e

in a well-defined thermal distribution. Let us empha-size that this procedure requires that the inducedchange of dielectric function is small enough in theexplored spectral range. It is the case in the present

Ž .experiments where the highest D´ t is less than5%. In addition, it is essential to have an accurateknowledge of the material parameters. It is importantto characterize them for each sample and not tosimply assume that the refractive index is knownfrom the literature where a large disparity of resultsexist due to the different conditions of film deposi-tion.

Fig. 4 represents the detailed dynamics of thetransmission and reflection of a thin film of copperof 30 nm thickness, obtained at normal incidencewith a pump energy density of 60 mJrcm2. The setof figures on the left hand side corresponds to

Ž . Ž . Ž . Ž .D RrR t full line and DTrT t dotted line mea-sured at different probe wavelengths in the vicinity

ŽFig. 4. Time dependence of the differential reflection D RrR full. Ž .line and transmission DTrT dotted line in a 30 nm thick Cu

Ž .film and the corresponding dielectric function D´ full line and2Ž . 2

D´ dotted line . The pump energy density is 60 mJrcm .1

of the interband transition from the filled d band tothe Fermi level in the p conduction band. The

Ž . Ž . Ž . Žcorresponding D´ t full line and D´ t dotted2 1.line are represented in the right hand side for the

same wavelengths. The differential reflectivityŽ . Ž .changes its sign between the curves 4 b and 4 d .

In a static thermo-modulation experiment, the reflec-tivity is expected to change its sign at the interbandthreshold. It corresponds to the excitation of occu-


pied states below the Fermi level to unoccupiedstates above E . Here, the change of sign of D RrRF

has to be understood in a dynamical context, wherethe hot electrons are cooling by giving their excessof energy to the lattice. This can give rise to a

Ž .complex change of reflection as seen in Fig. 4 cwhere D RrR is positive during the first 200 fs andthen becomes negative. We associate this effect to adynamical shift of the interband transition. The reso-nant aspect of the modification of the dielectricfunction due to the electron dynamics can be seen in

Ž . Ž . ŽFig. 5 a where we have represented D RrR l open. Ž . Ž .circles and DTrT l closed circles for the

pump–probe delay 240 fs. The maximum change ofDTrT and the sign inversion of D RrR occur at 585

Ž .nm. The corresponding spectral variations of D´ l2Ž . Ž . Ž .closed circles and D´ l open circles are shown1

Ž .in Fig. 5 b . It is important to stress that the curvesŽ .of Fig. 5 b are obtained with the convention ´s

Ž . Ž .´ –i´ in Eqs. 1a and 1b . A general feature can1 2Ž . Ž .be understood straightforwardly in Fig. 5 b . D´ l1Ž .has a minimum at ;570 nm and D´ l a maxi-2

mum at ;590 nm. They are both related to theinterband transition and reflect the Kramers–Kronig

Ž .Fig. 5. Spectral dependence of D RrR open circles , DTrTŽ . Ž .closed circles and the corresponding D´ closed circles and2

Ž .D´ open circles for the time delay 240 fs obtained with the 301

nm Cu film.

Ž .Fig. 6. Time dependent transmission spectra DTrT v,t of the30 nm Cu film.

Žrelation between the two quantities D´ has the2.derivative shape of D´ . The spectral positions of1

these extrema shift as a function of time, an effectwhich we attribute to a renormalization of the inter-band transition. It is better seen in Fig. 6 where wehave represented the spectral variation of the trans-mission during the first 1.4 ps. The minimum of

Ž .DTrT l shifts between 570 and 580 nm.The detailed dynamics of the dielectric function at

Ž .different wavelengths is shown in Fig. 7. D´ t and2Ž .yD´ t are represented for the two wavelengths1Ž . Ž .570 nm circles and 611 nm squares . The curves

have been normalized for comparison and the fulllines correspond to exponential curves displayed as aguide to better see the relaxations behavior. Finedetails of the electrons dynamics are present. Firstly,the general relaxation of the imaginary part of ´ issimilar for the two wavelengths and it is of ;0.85ps. In contrast, the relaxation of the real part is muchfaster off resonance at 611 nm. The correspondingrelaxation time is 0.4 ps while it is ;1.3 ps at the

Ž .resonance. Secondly, the relaxation of both D´ t1Ž .at resonance and of D´ t are not exponential.2

A close view of the pump–probe signals at thewavelength 570 nm, reveals that the decays are notexponential during more than one picosecond. Thisdecay, which corresponds to the cooling of the elec-trons to the lattice, contains also the dynamics of thenon-thermal electronic populations. It is known, from

wtime resolved photo-emission experiments 1–x8 , that the non-thermal character of the electron


Fig. 7. Dynamics of the normalized D´ and yD´ for the Cu2 1Žfilm for two probe wavelengths near circles: ls570 nm, 2.175

. Ž .eV and far squares: ls611 nm, 2.03 eV from the interbandoptical transition.

distribution can persist for a long time near theFermi level. It shows up nicely here in the timedependent dielectric function. The effect is stronger

Ž .on D´ t since the two components associated to1

the inter and intraband processes are not probingŽ .equivalently the electronic density see Section 3 .

ŽOne is selective in wavelength the interband pro-. Ž .cess while the other the intraband process is sensi-

tive to the overall energy spectrum. Figs. 4–7 revealthe complexity of the electron dynamics near theinterband transition which we further discuss in Sec-tion 3 with a model dielectric function of the metal.

The situation is somehow simpler when thedynamics is studied far from the interband transition.It is the case for the measurements made in thethin films of silver at the wavelength 400 nm, i.e.,far from the interband transitions d™p and p™s.

Ž . Ž . Ž .Fig. 8 a shows the signals DTrT t and D RrR tobtained on a 33 nm thick Ag film, using the degen-erate pump–probe configuration at 400 nm. Theenergy density of the pump absorbed by the sampleis 60 mJrcm2. Both signals decay exponentially

Ž . Ž .with a time constant of 960"20 fs. Fig. 8 b and crepresents the dynamics of the real and imaginary

Ž .parts of the dielectric function. While D´ t decays1Ž .with a time constant of 1 ps, D´ t increases to2

reach a plateau after ;2 ps. The corresponding risetime is also of 1 ps. These decay and rise times arerelated to the cooling of the electrons to the latticeand to the simultaneous increase of the lattice tem-perature. As discussed in Section 3 it is mostly theintraband part of the dielectric function which affects´ when the sample is probed at this wavelength. It2

is interesting to notice that the relaxation is exponen-tial. The non-thermal component of the electron gasis very fast and hardly observed with the temporalresolution of the laser. It is indeed hidden in thesignal to noise ratio of the experiment. A fine analy-sis, where this noise is de-convoluted, allows toobtain the behavior of Fig. 9. The initial signal in the

Ž .curve D´ t corresponds to the non-thermal elec-2

tronic population which manifests far from the inter-Ž .band threshold see Section 3 .

Ž . Ž .Fig. 8. Time dependence of DTrT , D RrR a , D´ b and D´1 2Ž .c in a 33 nm thick silver film at 400 nm. The pump energydensity is 60 mJrcm2.


Ž . Ž .Fig. 9. D´ t and D´ t as in Fig. 8 showing the initial1 2Ž .relaxation in D´ . The curves are obtained after a fit of DTrT t2

Ž .and D RrR t with spline functions.

2.3. Electron dynamics in Cu and Ag nanoparticles

The optical response of noble metallic clustersembedded in a glass matrix displays a resonance

Ž .when the condition ´ v sy2´ is fulfilled, where1 0

´ is the dielectric constant of the matrix. This0

resonance, which corresponds to the first order modew xof the scattering of light by a metallic sphere 13,42 ,

w xcan also be viewed as a plasmon mode 43,44 since,in its simplest description, it is directly related to theelectronic properties of the collective excitation ofthe metal. This resonance was described as a colloid

w xband in the early work of Doyle 45 . One shouldkeep in mind that both the intraband and interbandoptical processes are involved in the material re-sponse to a light excitation. As a consequence, thequasi-particle aspect of the electronic structure isalways inherent to the dielectric properties of ametallic sphere. Far from being a handicap, thisallows us to use the surface plasmon as a tool toprobe the dielectric properties of the metal, since theusual structureless character of the optical responseof the metal is suppressed. This is particularly truefor the study of the electron dynamics in the metallic

w xnanoparticles 46–49 . The aim of this section is tosee how the dynamical properties reported in thepreceding section are influenced by the existence ofthis surface plasmon mode.

Fig. 10 represents the spectrally and temporallyresolved differential transmission of copper nanopar-ticles with an average diameter of 10 nm. Because of

Ž y4 .the low particle concentration 2P10 in volume ,the reflection is essentially governed by the glassmatrix and no significant D RrR pump–probe signal

Ž .can be observed. The set of curves DTrT l,t dis-Ž .plays a positive resp. negative contribution on the

Ž .high resp. low energy side of the interband transi-tion energy of copper. It corresponds to a modifica-tion of the electronic populations around the Fermilevel, which nicely reflects the heating and coolingprocesses of the electrons. First, the distribution ofoccupied electronic states smears around the Fermienergy when the electrons acquire some kinetic en-ergy due to the pump pulse. The process lasts severalhundreds of femtoseconds during which the electronsare not necessarily in thermal equilibrium. Then, thedistribution spectrally narrows when the electronsrelax their energy via the electron–phonon interac-tion. During this process, the electronic and latticetemperatures tend to equilibrate, a process whichlasts a few picoseconds. The detailed analysis of thepump–probe signals shows that the surface plasmonis spectrally broadened during several hundreds of

Fig. 10. Spectral and temporal dependence of the differentialtransmission of Cu nanoparticles with a diameter of 10 nm. Theabsorbed pump energy density is 200 mJrcm2.


femtoseconds. We have shown in a previous paperw x Ž .47 that it manifests as an enhancement of DTrT l

near the plasmon resonance for the short time delays.Another interesting feature of the electron dynam-

ics shows up when the relaxation is analyzed in thevicinity of the resonance. In Fig. 11 different tempo-ral cross-sections of the pump–probe signal are rep-resented. The signal decays with a longer relaxationtime when the probe energy is at the plasmon reso-nance. This effect involves different mechanisms.The first one involves the interaction of the elec-

w xtronic populations with the plasmon 47 . Due to thelong lived quasi-particle states near the Fermi level,the mechanism of collision broadening of the surfaceplasmon is felt for a longer time by the probe beam.One should keep in mind that, in the pump–probeexperiment, the collective character of the electrons

Žexcited by the pump is lost very quickly a few tens.of femtoseconds . Therefore, it is the plasmon that is

created by the probe which is sensitive to this mech-anism. As an alternative, a second mechanism has

w xbeen proposed recently 16 . It is somehow the re-verse situation where the electron–hole pairs caninteract to emit a plasmon. This process has beenshown to be enhanced by the polarization at thesurface of the metallic sphere. These two mecha-nisms are discussed further in Section 3.

Let us now examine the electron relaxation in thesilver nanoparticles, where the resonance is situatedfar from the interband transitions. In Fig. 12,

Fig. 11. Time dependence of DTrT in Cu nanoparticles of 10 nmdiameter in the vicinity of the surface plasmon resonance. Thedotted line is the cross correlation between the 60 fs pump and 10fs probe pulses.

Ž .DTrT t signals, obtained with Ag nanoparticles ofaverage diameter 6.5 nm, are represented for differ-ent pump–probe conditions. The corresponding opti-

Žcal density of the low concentration sample 1.5y5 .10 in volume is shown in Fig. 2. For each set of

curves the signals have been normalized to the high-est one in order to compare the dynamics. Thenormalization factor is indicated in parenthesis as

Ž .well as the relaxation time t . In Fig. 12 a , the two1

signals correspond to a probe energy at the plasmonŽ .resonance 430 nm and two energy densities of the

Ž y2 .pump 270 and 18 mJ cm . As expected for ametal, t is longer for the highest pump intensity, an1

effect which is due to the temperature dependentŽ .specific heat of the electrons. In Fig. 12 b , the pump

intensity is set at 150 mJcmy2 and the probe ener-Ž .gies are respectively 430 nm 410 nm for the long

Ž .slow decay. A similar behavior is obtained at lowtemperature when the samples are immersed in

Ž .pumped liquid helium at 2 K. In Fig. 12 c the probeis set at 430 nm and the energy densities are 315 and25 mJ cmy2 . The relaxation is also longer for the

Ž .highest pump intensity. In Fig. 12 d the intensity is25 mJ cmy2 and the probe wavelengths are 430 and410 nm. The relaxation time is longer at resonance.

Ž . Ž .Fig. 12 b and d corresponds to a slowing down ofthe electron energy relaxation at the plasmon reso-nance. It is analogous to the observations made inthe copper nanoparticles. The effect, which is presentboth at 2 K and at room temperature, does notdepend on the initial lattice temperature. The elec-tron–lattice interaction can therefore be excluded inthe process. It is also important to stress that, al-though the experiment is made in a degenerate con-

Žfiguration in the case of silver both the pump andprobe are issued from the doubled amplified Ti:Sap-

.phire laser , the effect of the pump beam is only toprovide a hot electron distribution which is not inequilibrium with the lattice. This heating comes fromthe pump absorption via the intraband processes andby weaker two photon interband processes. How-ever, after a few hundreds of femtoseconds, theelectrons have no memory of the way they havebeen excited. In order to further investigate theslowing of the electron relaxation at the plasmonresonance, we have performed measurements over awide range of energy densities of the pump, usingboth the amplified and non-amplified laser beams.


Ž . Ž .Fig. 12. DTrT t in a nanoparticles of silver with an average diameter of 6.5 nm. The parameters for the slow and fast relaxation are: a2 Ž . 2 Ž . 2pump: 270 and 18 mJrcm , ls430 nm, 300 K; b pump: 150 mJrcm , ls430 nm and 410 nm, 300 K; c pump: 315 and 25 mJrcm ,

Ž . 2ls430 nm, 2 K; d pump: 25 mJrcm , ls430 nm and 410 nm, 2 K.

The results are gathered in Fig. 13, where an ex-tended view of t over three orders of magnitude is1

represented for the two wavelengths: at the plasmonŽ .resonance 430 nm open circles and off-resonance

Ž .410 nm closed circles . The measurements below anabsorbed density of energy of 10 mJrcm2 have beenobtained with the non-amplified laser system and thedotted lines connect the two types of measurements.When the intensity decreases, the relaxation time t 1

becomes the same for the two wavelengths. Thethreshold of this nonlinear effect is estimated tocorrespond to an averaged photon flux of one photonper nanoparticle. This estimation is made by takinginto account the size and density of particles andassuming a quantum efficiency of one. Fig. 13 clearlyshows that the mechanism which is at the origin ofthe slower electron relaxation when the probe is at

resonance depends on the density of excited elec-trons. The difference now with copper is that nodirect probing of the electron population is involvedsince the surface plasmon is excited far from theinterband threshold.

As mentioned earlier and shown in Section 3, thespectral behavior of pump–probe signal can be wellexplained in copper nanoparticles by a collisionbroadening of the surface plasmon. This mechanism,which is due to the electron–electron interaction, isalso present in silver nanoparticles. However, it man-ifests differently via a spectral shift of the resonance.

Ž .Fig. 14 shows the spectrum DTrT l for a pump–probe delay of 3 ps. The pump pulse is set at 400 nmand the probe comes from the continuum generatedin the sapphire crystal. The derivative shape of DTrTcorresponds to a frequency shift of the plasmon. It is


Fig. 13. Pump–probe relaxation time t as a function of the pump1

energy density in Ag nanoparticles of 6.5 nm diameter for twoŽ . Žprobe wavelengths: ls430 nm open circles , ls410 nm closed

.circles .

due to a modification of the real part of the metaldielectric function, induced by the electron–electronscattering.

In order to study the non-thermal component ofthe electron gas in the Ag nanoparticles, we haveperformed pump–probe experiments using pulses of30 fs duration at 800 nm. In this case, the surfaceplasmon resonance is not probed resonantly. Fig. 15

Fig. 14. Spectrum of DTrT of the Ag nanoparticles of 6.5 nmdiameter for a pump–probe delay of 3 ps.

Ž .Fig. 15. Differential transmission DTrT t of 6.5 nm Agnanoparticles excited with pulses of 30 fs duration at 800 nm. Thenormalized pulse auto-correlation is shown as a reference. Thesample has a volumic concentration of particles of ;1%.

Ž .represents the differential transmission DTrT tŽ .open circles , during the first 600 fs, of nanoparti-cles with a diameter of 6.5 nm embedded in a 0.25mm thick alumina matrix. The maximum of thesignal is reached after a delay of ;200 fs withrespect to the autocorrelation which is shown as areference. This delay is longer than the correspond-

w xing dynamics in silver thin films 50 . This prelimi-nary observation, which requires more investiga-tions, already shows that the non-thermal electronicpopulations behave differently in the nanoparticles ascompared to the thin films.

3. Model of the electron dynamics

In order to model the electron dynamics in thethin films and nanoparticles, we consider a complex

Ž .dielectric function ´ v,t of the metal, where thespectral and temporal variation is given by a changeof the occupied electronic energy states. The distri-bution of occupied states is assumed to be describedby a Fermi–Dirac statistics at a well-defined electron

Ž .temperature Q t and we suppose that the change ofe

the material response follows adiabatically thechanges of Q . The electron dynamics is governede

by two main mechanisms: the electron–electron scat-tering and the energy relaxation to the lattice. Thissecond process is described by an exchange of tem-perature between the electrons and the lattice which


has a temperature Q . This model is usually referredlw xto as the two temperature model 51,52 . Let us

stress that this procedure is not valid to describe thenon-thermal component of the electron dynamics. Itis however very convenient to grasp the essentialfeatures of the electron relaxation. We discuss belowthe limitations of the model.

In order to describe the dielectric function, thesimplest manner to proceed is to consider the ran-dom phase approximation with intraband and inter-band optical processes. Following Ehrenreich et al.w x53 , we account for the intraband processes by theDrude model of free electrons and the interbandtransitions are described within the RPA. The corre-sponding dielectric functions ´ and ´ are givenf inter

w xby 54 :

4pne2"

2 1´ v s1yŽ .f m "v "vq igŽ .D

2"vŽ .p

s1y , 2aŽ ."v "vq igŽ .f

2 2 < X < 2e " Pll3Xw xŽ . Ž . Ž .´ v sy d k f E y f EHÝinter o k l o k l2 2 2

XXm p Ž ."v l ll , lXl/ l

22 2X XŽ . Ž .Ž .2"v "v y "v yg y2 i "v gw xl l l l ee ee

= ,222 22 2

XŽ . Ž .Ž ."v y "v yg q4 "v gw xl l ee ee

2bŽ .n, e, m, f are the density, the charge, the mass ando

the distribution function of the electrons in the metal."v is the energy of the volume plasmon. P X andp l l

"v X are the matrix element and the energy of thel l

transition between bands l and lX. g and g accountf ee

for the intraband and interband damping processes.w xThey are given by 55–57 :

g sg qaDQ , 3Ž .f b 1

m3

g see 4 516p "

=

2 2W pk Q q EyEŽ . Ž .B e F.¦ ; EyEcos ur2Ž . F

1qexp ž /k QB e

4Ž .The intraband damping is described in a phe-

nomenological way and it is assumed to depend

linearly on the change of lattice temperature Q . Thisl

procedure does not allow to obtain the time depen-dence of the intraband processes. It is a good approx-imation in our case since we are interested either in

Ž .the resonant processes case of copper and in a timescale when the effects of intraband processes are less

Ž .pronounced case of silver . The interband dampingaccounts for the electron–electron scattering. Withinthe Fermi liquid theory it depends quadratically withrespect to the temperature Q of the electrons ande

their energy to the Fermi level E . W is the CoulombF

interaction term and k the Boltzmann constant.BŽ .The dielectric function given by Eqs. 2a and

Ž .2b is static. We then incorporate the time depen-dence in the Fermi–Dirac distribution via the elec-

Ž .tron temperature Q t . The energy exchange be-e

tween the electrons and the lattice is described by thetwo temperature model. In this model, the two bathsare coupled via an electron–phonon constant G andtheir dynamics is given by the following heat equa-tions:

dQeC Q syG Q yQ qP t , 5Ž . Ž . Ž . Ž .e e e 1d t

dQ1C sG Q yQ ,Ž .1 e 1d t

Ž .where P t is the power density of the laser whichacts as a source term for the initial increase oftemperature of the electrons. C and C are thee l

specific heats of the two baths. In addition, Q ande

Q are considered to be spatially uniform so that thel

heat propagation can be neglected. This is a goodapproximation in the thin films where, in the timescale considered, the environment has little influ-ence. As seen below, it is not the case for thenanoparticles. The parameter g, which at the micro-scopic level represents the electron–phonon cou-pling, is assumed to be constant. This approximationis valid when the lattice temperature is higher thanthe Debye temperature associated to the phonon

w xspectrum 37,38 . Let us apply the above model firstto the case of thin films and then to the nanoparti-cles.

3.1. Electron dynamics in the thin films

In the spectral region that we consider, the inter-band dielectric functions of the noble metals are


accurately described by a three-band model, whichw xtakes into account the crystal symmetry 58 . Follow-

w xing Rosei 59,60 , we consider the interband transi-tions d ™ p from the filled d band to the conductionband p, near the Fermi level, and p™s from theconduction band to the empty s band. We assumeparabolic bands with different masses in the direc-tions parallel and perpendicular to the G ™ Ldirection in the Brillouin zone. The bands aresketched in Fig. 16. The imaginary part of the inter-band dielectric function is then given by:

´ "v ,Q tŽ .Ž .2 inter e

2 2 28p e "s DD "v , E f E,Q tŽ . Ž .Ž .H p™ s o e22m "vŽ .

g 2ee

= d E2 2Ey"v qgŽ . ee

q DD "v , E 1y f E,Q tŽ . Ž .Ž .Ž .H d ™ p o e

2gee= d E , 6Ž .2 2Ey"v qgŽ . ee

where DD is the energy distribution of the jointi ™ jw xdensity of states for the transition i™ j 59,60 . In

Fig. 16. Energy bands of silver used in the calculations of thedielectric function.

Ž .Fig. 17. Calculated spectra of D´ closed circles and D´2 1Ž .open circles in bulk copper for conditions similar to the experi-

Ž Ž ..ment Fig. 5 b .

the case of copper, the p™s interband transition isnot effective and we set DD s0. In order to takep™ s

into account the damping due to the electron–elec-tron scattering processes, we have introduced in the

Ž .integral of Eq. 6 a Lorentzian with a line width g .ee

This procedure is not strictly consistent with theenergy conservation assumed in the calculation ofthe joint density of states. It is however a goodapproximation as checked a posteriori by integrating

Ž .Eq. 6 with complex energy levels. In the case ofcopper, since there is only one oscillator to consider,it is not necessary to consider the joint density ofstates and both the real and imaginary parts of thedielectric function can be calculated directly in theRPA description, including the electron–electronscattering. In the case of silver the real part of ´ isdeduced by a Kramers–Kronig transformation.

In Fig. 17, we have represented the spectral varia-tion of the differential dielectric function D´s´ –´ of copper. The following parameters800 K 300 K

w xare used 47 : E s2.17 eV, "v s9.4 eV, g s0.1dp p f

eV. The increase of electronic temperature DQ se

500 K corresponds to the maximum temperaturereached in the sample for an energy density of thepump pulse of ;300 mJ cmy2 . The general behav-ior reproduces quite well the measurements of Fig.Ž . Ž .5 b . The minimum of D´ open circles corre-1

sponds to the interband transition energy. For theŽ .same energy, the imaginary part D´ closed circles2

vanishes. D´ has two extrema on each side of the2

resonance. For the energy scale displayed here, onlythe maximum near 600 nm is seen. There are two


features in Fig. 5 which are not reproduced by themodel. First, the spectral position of the minimumand maximum of D´ and D´ are different in Fig.2 1

17. This is due to the dynamical shift of the inter-band transition. As seen in Fig. 6 the minimum ofthe differential transmission DTrT shifts as a func-tion of time. It is first displaced to the high energyside and then relaxes back to the static value at 580nm. The change of the Fermi level alone cannotexplain the observed 10 nm shift, since the variationof the chemical potential with the increasing electrontemperature is only of a few meV. We interpret it asa renormalization of the d band. It could easily betaken into account with an ad hoc time dependentparameter. However, a more realistic approach wouldbe to take into account a dynamical screening of theCoulomb interaction due to the many body interac-tion in the hot electron gas. Unlike the conduction

electrons the d electrons are more localized. Theymay therefore be more sensitive to a time dependentscreening. The second difference between the modeland the experiment is that D´ is enhanced by a2

constant background in the experimental results. Thisdiscrepancy is due to the fact that the intrabandprocesses are not included in the model. Their contri-

Ž .bution to the dynamics of D´ is important ;50%2

during the first 300 fs. We have observed that itdecreases faster than the component associated to theinterband transitions. This is further confirmed by

Ž .the relaxation of D´ t that is measured off reso-1

nance. It is of the order of 400 fs as seen in Fig. 7. Inaddition, we have checked for several temporal de-lays that it does not exhibit any particular spectral

Ž .feature in the domain of energy displayed in Fig. 5 .For silver, the situation is different. In the spectral

region of interest for the study of the nanoparticles

Ž . Ž .Fig. 18. Spectral variation of the dielectric function of silver ´ v and ´ v , showing the different intra- and interband contributions.1 2Ž . Ž .D´ v and its Kramers–Kronig transform D´ v correspond to a raise of temperature of the Fermi distribution of 300 K.2 1


Ž .plasmon resonance near 430 nm , the dielectricfunction of the bulk metal is essentially sensitive tovariations of its real part. In Fig. 18, we have

Ž .represented ´ and ´ full lines with their differ-1 2Ž .ent intraband ´ and ´ and interband1Drude 2Drude

Ž .´ and ´ contributions. On the right hand1inter 2inter

side, the variations D´ and D´ of the interband1 2

contribution for an increase of temperature of 300 Kare represented. The imaginary part D´ is calcu-2

lated using the following parameters: g s0.1 eV,b

a s 1.2 = 10y5 eV Ky1 and with a constantŽ 3 4 5. Ž² Ž .:. y1m r16p " Wrcos ur2 of 0.01 eV . Theparameters of the energy distributions DD andp™ s

w xDD are identical to those of Ref. 59,60 . Let usd ™ p

stress that, in the case of silver, the differential of theŽ .real part of the dielectric function D´ v,t is deter-1

Ž .mined with D´ v,t and the Kramers–Kronig1DrudeŽ .of D´ v,t . Notice that this procedure is accu-2inter

Ž .rate since, as seen in Fig. 18, D´ v,t is differ-2inter

ent from zero only in a narrow energy range near theinterband transitions d™p and p™s.

Fig. 19 represents the dynamics of a 33 nm thinfilm of silver calculated with the above model. We

Ž Ž .first compute the dielectric function D´ t and2Ž .. Ž Ž .D´ t and then the material response DTrT t1

Ž .. Ž . Ž .and D RrR t by inverting Eqs. 1a and 1b . This

Ž .Fig. 19. Calculated dynamics of D´ and D´ b and c for the1 2Ž .bulk Ag with conditions similar to the experiment Fig. 8 . The

corresponding differential transmission and reflection are shownŽ .in a .

step requires to have an accurate value of ´ . As we1

stressed in the preceding paragraph, unlike forŽ .D´ v,t , the energy range that we consider does1inter

Ž .not allow to compute ´ v with the Kramers–1interŽ .Kronig of ´ v . Therefore, for this step we used2inter

the experimental value of ´ . The parameters used1

are the following: C sgQ , gs65 Jrmy3 Ky2 ,e e

C s2.4=105 Jrm3 K, Gs30 Wrm3 K. Thel

dynamics of the measured and calculated responsesof the thin silver film is very similar as seen in Figs.8 and 19. However, the theoretical relaxation timest are shorter than the experimental one. This dis-1

crepancy may have different origins. It can be shownin the two temperature models that t is propor-1

tional to the ratio grG and to the maximum electrontemperature at zero delay:

g Q ts0 qQ tŽ . Ž .Ž .e it s , 7Ž .1 2G

Ž . Ž 2 .1r2with Q ts0 s Q q2 Prg , Q being the ini-e i i

tial temperature. Therefore, the calculated time t is1

longer when a smaller coupling constant G is usedor by considering the model of Debye for the elec-tron–phonon interaction. Alternatively, the powerdensity P, which is a difficult quantity to determineaccurately in a pump–probe experiment, may beunderestimated in the model. Finally, the polycrys-talline nature of the sample and the thin oxide layerthat forms on the sample surface may also play arole.

Ž . Ž . Ž . Ž .As seen in Fig. 19 b and c D´ t and D´ t1 2

have different temporal behaviors. While D´ fol-1

lows the relaxation of the electron temperature, D´2

follows the increase of lattice temperature. This isdue to the fact that at the probe wavelength ls400nm, far from the d ™ p and p ™ s interbandthresholds, D´ is mainly affected by the intraband2

processes. On the other hand, the Kramers–KronigD´ is sensitive to both interband and intraband1

Ž . Ž .transitions. The raising time of D´ t ;1 ps is2

close to the one of the lattice temperature t , whichl

can be deduced from the two temperature model as:

C Q max yQ1 1 it s , 8Ž .1 G Q ts0 yQŽ .e i

where Q max is the lattice temperature reached when1

the electrons and the phonons are in equilibrium.


3.2. Electron dynamics in the nanoparticles

The optical response of the nanoparticles is de-scribed with a model of an effective medium. Thedielectric function ´ is related to the ones of theeff

metal ´ and of the matrix surrounding the particlesw x´ by 14,15,61 :d

ý´d´ s´ q3´ r , 9Ž .eff d d v

´q2´d

where r is the density of particles in volume. It isv

assumed that there is no interaction between theparticles which is a good approximation for densitiesup to a few percents. The absorption of the effectivemedium is then calculated with:

2 24p ´ q ´ y´Ž . Ž .1eff 2eff 1effas . 10( Ž .

l 2

In addition, the surface scattering at the metalrdi-electric boundary is taken into account. This is al-ready necessary to model the linear optical propertiesof the nanoparticles. In Fig. 20, we have representedthe optical density of Ag particles with a diameter of6.5 nm. The dashed curve is obtained with the bulkparameters. The full line is obtained with a surfacescattering of 0.37 meV. The fit with the experimental

Ž .curve open circles is excellent. The surface scatter-ing is a dominant mechanism here since classically,the mean free path of the electrons in the bulk metal,

Fig. 20. Calculated optical density of silver nanoparticles of 6.5Ž . Ž .nm diameter with full line and without dashed line including

the surface scattering. The experimental one corresponds to theopen circles.

l sV t where V is the Fermi velocity and t` F coll F coll

the collision time, is of the order of 56 nm for silver,that is ten times larger than the nanoparticles diame-ter. On the high energy side, above 3.5 eV, theabsorption of the glass matrix is not negligible. It istaken into account by using a frequency dependent

Ž .complex dielectric function ´ v . One should stressd

that silver is somehow a particular case among thenoble metals. In contrast to gold for instance, thesurface plasmon resonance does not shift as a func-tion of the diameter for particles with sizes of a fewnanometers. This is due to the competition of thedielectric confinement and spill-out of the conduc-tion electrons and it has been studied in detail usingthe technique of the time dependent local density

Ž . w xapproximation TDLDA 62 . As a consequence, theabsorption spectrum is not very sensitive to the sizedistribution of the nanoparticles. When we incorpo-rate a normal logarithmic distribution of sizes, wefind that it only contributes to a slight broadening ofthe absorption. Let us stress that for the particle sizesused here, the approximation of a surface scatteringby a constant damping is justified since the quantum

w xconfinement is negligible. Kawabata et al. 9 haveincluded the confinement by using an effective parti-cle diameter d which accounts for the density ofeff

states of the metal. In practice, these quantum con-finement effects are negligible for particles with adiameter dG2 nm.

The electron dynamics is then taken into accountvia the time dependent dielectric function of the bulkmetal constituting the nanoparticle. Using the twotemperature model and the three band schemes de-

Ž .scribed in Section 3.1, together with Eqs. 9 andŽ .10 , we obtain the time dependent transmission

Ž .spectra DTrT v,t represented in Fig. 21. The dom-Ž .inant feature is the derivative shape of DTrT v for

each temporal delay. It is due to the variation of theŽ .real part of the dielectric function D´ v,t . The1

Ž .little influence of D´ v,t is due to the fact that the2

surface plasmon resonance is situated far from theinterband transitions where the population effects arepredominant. This can be checked by first computingD´ with the Kramers–Kronig transform of D´1 2

and then imposing D´ s0 in the calculation of the2

effective dielectric function D´ . Therefore, in sil-eff

ver nanoparticles, the electron dynamics manifestsessentially via a spectral shift of the surface plasmon


Ž .Fig. 21. Spectral and temporal variation DT v,t rT of silvernanoparticles with 6.5 nm diameters.

resonance, which is due to the electron–electronscattering.

Let us emphasize that the asymmetrical shape ofŽ .the differential spectrum DTrT l , is not due to the

imaginary part of the metal dielectric function. Itsimply reflects the non-trivial frequency dependence

Ž . Ž .of ´ v via ´ v . This is important since, oneeff 1

would be tempted to fit the linear absorption of thenanoparticles with a Lorentzian line shape with awidth G and a peak position l , and to interpret the0

dynamics via a modification of these two parame-Ž . Ž .ters: G t and l t . This procedure may lead to a0

wrong interpretation of the asymmetry in terms of aŽ .time dependent collision broadening G t of the

surface plasmon mode.It is important to notice that the description of the

electron–phonon relaxation, via a frequency–timeŽ .dependent dielectric function ´ v,t , relies on theeff

two temperature model. Unlike for thin films, suchdescription has its drawbacks in the case of nanopar-ticles. First, it does not allow taking into account thecoherent effects associated to the phonon modes ofthe nanoparticles. Such oscillations have been shownto be important on the time scale of a few tens of

w xpicoseconds 26–28,63 . Secondly, it assumes thatthe electron–phonon coupling G does not depend onthe size. This approximation has been shown to be

w xvalid for gold particles of sizes dG12 nm 64 .However, for silver particles of a few nanometers wehave observed a size dependence of the electron–

w xphonon relaxation time 65 . The dynamics ofnanoparticles with a diameter of 6.5 nm, embeddedin a different matrix is represented in Fig. 22. Theabsorbed pump power is the same and the curveshave been normalized and shifted for a display pur-pose. The relaxation time t of the nanoparticles1

embedded in glass is about twice that of the nanopar-ticles embedded in the alumina matrix. The heatconductivities of these two matrices are very differ-

Žent ks0.01 for the type of glass used here andy1 y1 .ks0.3 W cm K for alumina . Therefore, heat

propagation effects have to be considered in thehighly conductive matrices. Although the electrondynamics in the metal is weakly altered by thematrix effects, it is necessary to consider such effectswhen accurate values of the energy relaxation is

w xrequired 66 .In contrast to silver, the effect of electron–elec-

tron scattering in copper nanoparticles, manifestsessentially on both the real and imaginary part of thedielectric function. This is due to the fact that theplasmon resonance, which is close to the interbandtransition, is more sensitive to the population dynam-ics. In Fig. 23, we have represented the spectralvariation of the transmission of nanoparticles with a

Ž .diameter of 10 nm. Fig. 23 a shows the fit of thelinear transmission. It reproduces well the measure-ment except for the low energy side below 1.8 eV.This is due to the fact that, in the samples, there is aresidual concentration of copper ions which absorb

Fig. 22. Pump–probe dynamics of 6.5 nm Ag nanoparticlesembedded in matrices of different thermal conductivity.


Ž .Fig. 23. Calculated optical density and DTrT v of 10 nm Cunanoparticles.

Ž .near 800 nm. Fig. 23 b shows the differential trans-mission for a static variation of the temperaturecorresponding to DQ sDu s15 K. The most strik-l e

ing feature is the anti-symmetric shape with respectto the energy 2.15 eV, i.e., to the interband transi-

Ž .tion. As in the bulk metal, the positive negativeŽ .induced transmission above below 2.15 eV corre-

sponds to the excitation of electronic states above theFermi energy. In contrast, the spectral shape of thedifferential transmission in the dynamical regime,where the electronic temperature reaches several

hundred degrees is asymmetric with a larger contri-bution on the high energy side, at the plasmon

Ž . Ž .resonance. This is seen in Fig. 23 c where DTrT v

is calculated for an increase of temperature DQ se

800 K. These spectral dependences clearly stress thedifference between a static excitation where the elec-trons are in equilibrium with the lattice, a situationwhich is equivalent to the material response after afew picoseconds, and the dynamical regime wherethe electron–electron scattering dominates, leadingto the plasmon broadening.

Let us now analyze the electron dynamics nearthe plasmon resonance of the nanoparticles with themodel dielectric function described above. For silver,we have calculated the differential transmission of asample containing nanoparticles with a diameter of

Ž .6.5 nm. Fig. 24 represents DTrT t , for the probewavelengths 430, 410 and 400 nm. The pump energydensity is 100 mJ cmy2 . In contrast to the experi-

Ž .ment Fig. 13 , the relaxation is the same when thelaser is at the surface plasmon resonance. For cop-per, the model also fails to reproduce the experimen-tal results. At first sight, this result is not obvioussince, for the bulk copper, the dynamics of D´ has1

some spectral dependence as seen in Fig. 7. As wementioned in Section 2, this is due to the differentrelaxation behavior of the electronic populations nearthe interband transition. However, this difference isnot sufficient to explain the observed slowing of therelaxation at the plasmon resonance in the nanoparti-

Ž .cles Fig. 11 . Fig. 25 represents the calculated dy-

Ž .Fig. 24. Calculated DTrT t for Ag nanoparticles of 6.5 nmdiameter for different probe wavelengths in the vicinity of the

Ž .surface plasmon resonance 430, 410 and 400 nm .


Fig. 25. Calculated dynamics of the copper dielectric functionŽ . Ž . Ž .D´ t and D´ t for the wavelengths 584 nm circles near the2 1

Ž .interband transition and 610 nm squares on the low energy sideof the transition. The curves are normalized and the conditions are

Ž .similar to the experiment Fig. 7 .

Ž . Ž .namics of D´ t and D´ t for the wavelengths2 1Ž .584 nm circles near the interband transition andŽ .610 nm squares on the low energy side of the

transition. As in the experiment, the population dy-Žnamics near the Fermi level is slower t s673 and1.917 fs respectively off and on resonance . The differ-

ence is however weaker than in the experiment. Weattribute this discrepancy to the non-thermal compo-nent of the electrons which is not taken into accountin the model. Most importantly, the dynamics calcu-lated on the nanoparticles does not display the ob-

Žserved slowing down at the plasmon resonance Fig..11 . In the inset of Fig. 25, we have represented the

Ž .normalized DTrT t in the nanoparticles at thewavelengths 584 and 610 nm. The relaxation time isidentical. We conclude that the model dielectricfunction developed in the random phase approxima-tion is not appropriate to describe the electron dy-namics in the metallic nanoparticles.

In a first attempt to resolve the discrepancy be-tween the model and the experiment, we have in-cluded the non-conservation of momentum of theelectrons at the metalrmatrix boundary. It does notgive satisfaction. In particular, the nonlinear intensitybehavior as shown in Fig. 13 cannot be interpreted

w xthis way. It has been shown recently 16 that thesurface polarization has to be taken into account inthe modelization of the dielectric function. This sur-face polarization modifies the dynamical screeningof the electron–electron interactions in the nanoparti-cle. In particular, it leads to a resonant scattering of ahole in the d band into the conduction band byemission of a plasmon. When this effect is taken intoaccount in a self-consistent manner in the calculationof the dielectric function, the relaxation dynamics isslower at the plasmon resonance of copper nanopar-ticles. This surface induced damping mechanism in-creases when the particle size decreases accordingto: g ;dy3, where g is the damping of theh h

d-holes. Very recently, we have measured the relax-ation time of copper nanoparticles with larger sizes.Fig. 26 represents the corresponding dynamics ofnanoparticles with an average diameter of 15 nm.

Ž .Ž .The normalized pump–probe signal DTrT t hasŽ . Žthe same dynamics at 2.21 eV and apart 2.25 and

.2.14 eV from the surface plasmon resonance. Theseresults nicely confirm the interpretation of a surfaceenhanced effect. Let us stress that these surfaceinduced effects may not only modify the Coulombscreening but also the electron–phonon coupling.Indeed, the energy transfer between the electrons andthe lattice is already known to have different compo-nents for non-thermal and thermal electronic distri-

w xbutions in bulk metals 50,67 . Therefore, it is very

Fig. 26. Differential transmission of Cu nanoparticles with adiameter of 15 nm, in the vicinity of the surface plasmon reso-nance. Pump energy density: 100 mJrcm2.


likely that it is also different in nanoparticles where,in addition, the surface induced polarization is im-portant.

4. Conclusion

The dynamics of electrons in metallic thin filmsand nanoparticles embedded in a transparent matrixhas been investigated using the technique of fem-tosecond spectroscopy. These comparative measure-ments, using the same experimental apparatus, en-able us to test the validity of the model dielectricfunction described in the random phase approxima-tion when it is applied to the optical response of thenanoparticles. A detailed analysis of the reflectionand transmission pump–probe signals in the thin Cuand Ag films allows the spectral and temporal behav-ior of the metals dielectric functions to be measured.It is shown that the non-thermal and thermal compo-nents of the electronic distribution have differentcontributions when probing near or far from theFermi level via the interband optical transitions. Inthe case of the nanoparticles, the electron dynamicsleads to a plasmon broadening and a spectral shift asa function of time. It is related to the electron–elec-tron scattering which manifests both in the real andimaginary parts of the material response function.The measurements show that the dynamics is differ-ent in the vicinity of the plasmon resonance. Therelaxation time varies as a function of the probewavelength. In addition, this effect depends on thedensity of excitation and disappears in the low den-sity regime or when the particle size increases. Itsuggests that a new mechanism takes place in thenanoparticle as compared to the bulk. These effectsare in agreement with the interpretation of a surfaceinduced modification of the electron–electron inter-action.

In his course of Optical Physics written at thebeginning of the century, Wood mentioned severalexperiments made with ultramicroscopic particles.He then concluded the chapter by saying: ‘‘It seems

Žthat the theory of optical resonance he refers to the.theory of Garnett can be considered, to some extent,

as confirmed by the researches . . . The opticalconstants of the metal have to be taken into ac-

1w xcount PPP ’’ 68 . A large amount of information hascertainly been obtained since then, both on the opti-cal constants and on the scattering of light by metal-lic particles! However, there are still interesting stud-ies to be done in metallic particles, particularlyregarding their dynamical response to a laser pulseexcitation. Many fundamental mechanisms still re-main to be addressed, among which we may pointout the following: the respective role of collectiveand quasi-particle excitations in the electron dynam-ics of a metal cluster in the quantum confined regimeŽa recent theoretical investigation of the photo-ioni-zation in metal cluster stresses the importance of

w x.many body interactions 69 ; the study of the energydissipation between a metallic cluster and its envi-ronment as pointed out in the present work or in the

w xcase of nanoshells 70 ; the spin dynamics in mag-Žnetic nanoparticles several experiments have re-

cently shown the existence of an ultrafast magneticw xresponse in ferromagnetic thin films 71–74 . This

work is being extended to ferromagnetic particlesw x75 where the superparamagnetic relaxation should

w x.modify the spin dynamics 76 ; the dynamics ofindividual clusters studied with near field optical

w xtechniques 77 ; the regime of multi-scattering be-tween nanoparticles is also an interesting aspect to

w xstudy 78 . These challenging aspects of the clustersdynamics certainly require difficult experimental in-vestigations but it is already very encouraging to seethat, using the fine experimental techniques of timeresolved spectroscopy, we start to see the possibilityof understanding the fundamental differences be-tween the electron dynamics in molecular and metal-lic solid systems.

Acknowledgements

We thank Prof. I. Perakis and Dr. T. Shahbazyanfrom Vanderbilt University in Nashville for interest-ing discussions. We are grateful to G. Versini and C.Ulhaq for the elaboration and characterization of theCu and Ag thin films. The experiments on the

1 Wood was the first to observe the optical resonance inpellicles of alkali metals in 1902.


nanoparticles have been made possible thanks tocollaborations with Prof. J. Guille at the UniversiteLouis Pasteur in Strasbourg and the groups of Profs.M. Broer and M. Perez from the Universite de Lyon´ Í. We address special thanks to M. Albrecht, O.Cregut and D. Acker for their technical support in´the past four years during which this work wasdeveloped.

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Electron dynamics in metallic nanoparticles