Collective and single-particle dynamics in time-resolved two-photon photoemission

M. Merschdorf, C. Kennerknecht, and W. PfeifferPhysikalisches Institut, Universität Würzburg, 97074 Würzburg, Germany

(Received 3 August 2004; published 3 November 2004)

A general model for time-resolved two-photon photoemission from solids is presented comprising bothcollective and single-electron dynamics. In combination with interferometric time-resolved two-photon pho-toemission, this allows one to determine the shape of the collective response function across the excitationspectrum and the single-particle lifetime of excited electrons. Ag nanoparticles supported on graphite exhibit astrong collective resonance and serve as model system to demonstrate experimentally this separation of col-lective and single-particle dynamics in two-photon photoemission.

DOI: 10.1103/PhysRevB.70.193401 PACS number(s): 78.47.1p, 71.45.2d, 78.67.2n, 79.60.Jv

In recent years time-resolved two-photon photoemission(TR 2PPES) has substantially contributed to a better under-standing of ultrafast electronic relaxation processes inmetals,1–3 surface states,4 and adsorbates.5 It is based on thepump-probe principle: A pump pulse excites electrons intoan intermediate state and a time-delayed probe pulse thenmaps its decaying population. Energy-resolved detection ofthe photoelectrons reveals the single-particle dynamics; i.e.,the lifetime of distinct intermediate states in the unoccupiedband structure. This is in contrast to purely optical methodslike transient absorption and reflection measurements ortime-resolved second and third harmonic generation spec-troscopy(SHG, THG). These techniques probe the collectiveresponse of the electrons. It has, for example, recently beendemonstrated that TR THG spectroscopy directly yields theplasmon polariton lifetime in metal nanoparticles.6 On firstview, one might therefore assume that 2PPES and opticalspectroscopy yield complementary information, i.e., single-particle dynamics are determined in 2PPES experiments,whereas the collective response is seen in optical spectros-copy. This is implicitly assumed in the common interpreta-tion of TR 2PPES results. However, as we will demonstratehere, the TR 2PPE signal is also influenced by the collectiveresponse of the electrons. The relationship between collec-tive and single-particle dynamics in 2PPES is therefore morecomplex and can alter the quantitative information drawnfrom 2PPE experiments. First evidence for the intricate rela-tion of collective and single-particle dynamics became evi-dent in recent 2PPE experiments on metal nanoparticles.7,8

Metal nanoparticles exhibit a strong collective resonance—the Mie plasmon—making them an ideal model system forour investigation.

The separation of collective and single-particle dynamicsas it is presented here is based(i) on a model for the 2PPEprocess that accounts for collective and single-particles dy-namics and(ii ) on TR 2PPES with interferometric resolution.Interferometric TR 2PPE was introduced by Ogawaet al. toinvestigate the coherence associated with the optical excita-tion of the occupied surface state on Cu(111).3 In contrast, inthe present work the interferometric resolution provides in-formation about the collective response. The interferometricTR 2PPE signal contains information on the local excitationfield that is modified by the collective electron response. Thismodification and the separation of collective and single-

particle dynamics are demonstrated for TR 2PPE spectros-copy using supported Ag nanoparticles on graphite as amodel system.

In the following, a general model for a 2PPE process ispresented that comprises the collective as well as the singleparticle response of the electrons(see also Fig. 1). The inci-dent light fieldEext interacts with an ensemble of electronsand induces a collective response; i.e., a polarizationP. Theresulting internal fieldEint acts on the electrons as a time-dependent perturbation and is therefore responsible for thephotoemission. The internal field amplitude may be derivedas follows: The induced polarization is given byPsnd=«0xsndEextsnd with the susceptibilityxsnd. It can be linkedto Eint via «0Eint=s1+xd«0Eext=«0Eext+P which in turn canbe solved for a relationship of the formEintsnd=GsndEextsnd. The internal field is thus described in the fre-quency domain by a complex response functionGsnd thatacts multiplicative onto the incident light spectrumEextsnd.

This has far-reaching consequences for time-resolved ex-periments. In the frequency domain the resulting pulse ex-hibits an altered spectral amplitude and phase, correspondingin the time domain to a modified pulse shape, center fre-quencyn0

int, and durationtPulseint . A modified pulse duration

tPulseint directly affects the width of a TR 2PPE signal and

consequently also the extracted lifetime of the intermediatestate. For a planar metal surfaceGsnd is known from thedielectric function«snd. However, e.g., for a nanostructured

FIG. 1. Schematic representation of the 2PPE. The incident lightfield Eext is modified by the response functionGsnd of the manyelectron system as indicated in the square vertex. The resultinginternal field Eint causes the 2PPE via an intermediate state. Thegraph in the collective response box sketches the excitation spec-trum (black line) andGsnd for a resonance with a Lorentzian line-shape(gray line).

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surface,Gsnd is determined by«snd and the boundary con-ditions for the fields leading to distinct spectral features inthe response function. Accordingly, in practiceGsnd is notknown and we therefore representGsnd in a most generalway using a Taylor expansion for the amplitudeuGsndu=oiAishi / i ! dsn−n0di and the phaseFGsnd=oiwifs2pdi / i ! g3sn−n0di of Gsnd with n0 as the center frequency of theincident laser pulse, and the coefficients for amplitudeAi andphasewi. The model assumes an isotropicGsnd. However,this model can be generalized for an anisotropic and for anonlinear collective response by using a tensorial responsefunction and higher order susceptibilitiesxnsnd, respectively.

The internal fieldEint is responsible for the 2PPE process.This process is phenomenologically modeled using the den-sity matrix propagation for a three-level system. This ap-proach is commonly used to describe the 2PPE process.9,10

The TR 2PPE signal is governed by the inelastic lifetimeTsEd of the electrons in the intermediate state. For isolatedstates, also coherent effects associated with the transitionscan be observed.3,11 However, the corresponding dephasingtime in the density matrix formalism scales with the inverselinewidth of the transition. In the limit of continuous bandswith a constant density of states, all coherent effects in theTR 2PPE signal cancel,10 and the propagation of the three-level system is reduced to a rate equation. For the Ag nano-particles the assumption of a continuous band with a constantdensity of states for the nanoparticles is supported by theobservation that the two- and three-photon photoemissionspectrum exhibits no spectral features beside the Fermi levelonset.12

We use Ag nanoparticles supported on graphite to demon-strate the influence of the collective response on 2PPE andthe separation of collective and single-particle dynamics inTR 2PPE experiments. The supported metal nanoparticlesexhibit a strong collective resonance—the plasmon polaritonor Mie plasmon—giving rise to a strong enhancement of the2PPE yield.7 For isolated elliptical particles, Mie theory inthe quasistatic limit yieldsGisnd=f1+s«snd−1dLig−1 for thethree plasmon modes oriented along the principal axes13 withthe depolarization factorsLi. For supported oblate particles,the two (1,1) modes are oriented parallel to the surface,whereas the(1,0) mode is oriented perpendicular to the sur-face. For graphite as a substrate, the(1,1) modes are shiftedto lower energy and strongly damped because of the stronginteraction with the electrons in the graphite layers. As esti-mated from the polarization dependence of the 2PPE yield,the excitation of the(1,1) mode contributes only to less than1% to the total yield forp-polarized incident light. Hence,for p-polarization the 2PPE yield is dominated by the exci-tation of the(1,0) mode that is located at about 3.4 eV.12 Theincident pulse spectrum at a photon energy of 3.2 eV is lo-cated in the wing of this plasmon polariton resonance. Fors-polarized excitation the(1,1) modes dominate the photo-emission yield. The polarization dependence also shows thatphotoemission due to SHG photons is negligible in thepresent case. Accordingly, our model for the collective re-sponse can be restricted to the linear term.

The nanoparticles were prepared following the procedurein Ref. 14. Cleaved highly oriented pyrolytic graphite is

sputtered with Ar+ ions (1 keV, 1011 cm−2) and oxidized inair (520 °C, 20 min), thereby forming pits in the topmostlayer of the graphite. Silver is evaporated onto the samples0.1 Å s−1d, condenses into the pits, and forms particles withsizes of several nanometers. The height distribution of theinvestigated sample is 2.2±1.0 nm as determined using aninsitu scanning tunneling microscope. The frequency-doubledoutput of an amplified Ti:sapphire laser system(390 nm,pulse duration 45 fs, intensity ,108 W cm−2 forp-polarization, intensity,109 W cm−2 for s-polarization) il-luminates the sample at an incidence angle of 55°. The ki-netic energy of the emitted photoelectrons is analyzed by atime-of-flight spectrometer with a resolution better than50 meV. Two identical pulses are generated in a Mach-Zehnder interferometer and the delay between the two pulsesis controlled with a piezostage. The interferometric two-pulse-correlation signal as a function of the delay betweenthe pulses is accumulated in successive scans by a phase-corrected data acquisition with a repeatability ofl /75 at l=390 nm. The linear autocorrelation of the laser pulses isrecorded by a photodiode(Silicon Sensor,pin photodiodeSSO-PD50-7TO8S) yielding the spectrum of the incidentlight via a Fourier transform. The calibration of the spectralresponse of all optical components as well as of the photo-diode allows the quantitative comparison between linear au-tocorrelation and interferometric TR 2PPE signal.

In a two-photon process, intermediate states in the rangefrom 1.6 to 3.1 eV aboveEF contribute to the photoemis-sion yield. Note that the internal field is the same for all2PPE excitations, whereas the single-particle lifetime varieswith the energy of the intermediate state. The normalizedinterferometric two-pulse-correlation for an intermediatestate energy of 1.6 eV aboveEF is shown in Fig. 2(b), as arepresentative of the signals for all other intermediate statesthat are recorded in parallel. The signal exhibits the expected1:8 signal enhancement that is typical for a two-photon pro-cess. The fast modulation of the signal reflects the interfer-ence between the two pulses. The lifetime of the intermediatestate is not obvious in the interferometric signal. However,the low-pass filtered signals for two different intermediatestates shown in Fig. 2(a) exhibit different widths and there-fore reflect that the lifetime of the intermediate state at2.8 eV aboveEF is smaller than for the state 1.6 eV aboveEF. To determine the lifetime, the model developed above isimplemented into a fitting routine: The internal fieldEint iscalculated for every delay between the two pulses using theincident laser pulseEext andGsnd in the given parametriza-tion. The resulting field acts on the three-level system andthe time integral of the population in the uppermost state isused as a measure for the 2PPE yield. A nonlinear optimiza-tion then yields the parameters forGsnd and the intermediatestate lifetimeTsEd. The agreement of data and fit is illus-trated in the inset in Fig. 2(b). The best fit is obtained for theTaylor expansion coefficientsA1=1.45 eV−1, w2=−200 fs2,andw3=93103 fs3 [Fig. 2(c)]. We found that no higher or-der coefficients are necessary to reproduce the measured sig-nal within noise limits. However, neglecting the collectiveresponse, i.e.,uGsndu=1, leads to a significant increase of theresidual[compare Fig. 2(d)]. The phase ofGsnd is still opti-

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mized for reasons explained later. The comparison of theresidual for both cases shown in Figs. 2(b) and 2(d) demon-strates that the internal field correction is essential to repro-duce the experimental data. The positive slopeA1 of uGsnducauses a slight blue shift of the internal spectrum towards thecenter frequency of the resonance. A shift of the center fre-quency leads to an increasing dephasing of the interferencefringes with increasing delay and thus the residual is maxi-mal in the wings and vanishes for small delay.

The optimized coefficientsw2 andw3 depend on both thespectral phase of the response functionGsnd and the incidentlaser pulse. This ambiguity originates from the fact that onlythe spectral amplitude of the incident laser pulse is experi-mentally determined but not its spectral phase. Therefore, wecannot determine the spectral phase ofGsnd in the presentexperiment. However, the expansion coefficientsw2=−6 fs2

andw3=−50 fs3 for the spectral phase ofGsnd are estimatedusing a harmonic oscillator model for a resonance ofGsnd at3.4 eV and a linewidth of full width at half-maximum=200 meV, which corresponds to a plasmon polariton life-time of 10 fs. This indicates that the spectral phase inducedby the resonance is negligible compared to the uncompresseddispersion of the incident laser pulses. Consequently, in thepresent case the collective response leads primarily to amodification of the spectral amplitude.

In addition to the coefficients characterizingGsnd, ourmodel contains the lifetimeTsEd of an excited electron. This

lifetime causes the broadening of the TR 2PPE signal shownin Fig. 2(a). A fit of our model to the experimental datayields TsEd, shown in Fig. 3. Qualitatively, the decreasinglifetime with excess energy above the Fermi energyEF re-sults from the growing phase space for inelastic scatteringevents.15 The comparison of the electron lifetimes deter-mined with s- and p-polarized excitation confirms the pre-sented strategy to disentangle collective and single-particledynamics in TR 2PPE experiments. WhereasGsnd forp-polarized light is dominated by the coupling to the strong(1,0) mode, it is exclusively determined by the(1,1) modefor s-polarized light. This leads to a differentGsnd fors-polarized(best fitA1=0.8 eV−1) andp-polarized excitation(best fit A1=1.45 eV−1). The single electron lifetimesTsEddetermined for the different excitation conditions are identi-cal (see Fig. 3). The fact that the collective response varieswith the incident polarization demonstrates that in general amodel for TR 2PPE has to include the collective response ofthe electrons. In addition, the observation of a finite lifetimeof the electrons in intermediate states confirms the restrictionof our model to linear response, since a large second ordersusceptibility would allow a single-photon excitation of thephotoemission state leading to a TR 2PPES signal that ex-hibits no lifetime broadening.

The experimental values for the lifetime of excited elec-trons in supported Ag nanoparticles agree well with a recenttheoretical prediction for bulk silver using the self-energyformalism.16 Other theoretical studies also yield similar re-sults for the electron lifetime.17,18 Experiments performedwith TR 2PPES on bulk Ag or Ag films yield significantlylower lifetimes.17,19 This deviation between theory and ex-periment is attributed to electron transport effects that appar-ently reduce the inelastic electron lifetime.17 Qualitatively,this agrees well with the observed lifetime in supported Agnanoparticles, for which transport plays an even smaller rolethan in Ag films. This seems to indicate that quantum sizeeffects, which are expected for nanoparticles of only a fewnanometers in diameter, do not significantly influence thesingle-particle lifetime. However, this seemingly absentquantum size effect might be attributed to size effects that

FIG. 2. (a) Low-pass filtered interferometric TR 2PPE for inter-mediate state energies of 1.6 and 2.5 eV aboveEF. The constantsignal contribution is subtracted.(b) Interferometric 2PPE for anintermediate state energy of 1.6 eV aboveEF. The inset shows theagreement of data and fit.(c) Residual(data minus fit) for the bestparameter set that yields values ofA1=1.45 eV−1, w2=−200 fs2,w3=9000 fs3, T=20 fs. (d) Residual of the best fit withA1 fixed tozero. For clarity the data in(c) and (d) have been noise filtered.

FIG. 3. Inelastic electron lifetimeTsEd as a function of theintermediate state energy aboveEF. The two different datasets wereacquired withp-polarized ands-polarized excitation. Theoreticalpredictions(Ref. 16) and experimental results for a 15 nm thick Agfilm (Ref. 17) are shown for comparison.

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are partially counterbalanced. For example, a reduced den-sity of states might be opposed by an increase of theelectron-electron scattering probability.

The presented model allows one to include both collectiveand single-particle dynamics into the interpretation of TRphotoemission signals and thus closes the gap betweenpurely optical methods and 2PPE. The interferometric TR2PPE signal allows the determination of the inelastic elec-tron lifetime and the shape of the collective response func-tion. Its applicability is not restricted to nanoparticles butalso extends to, e.g., bulk material, large molecules with acollective resonance(e.g., C60), and nanostructured materialsin general. As a rule of thumb, the collective response mustbe considered in a TR 2PPE experiment whenever the re-sponse function exhibits a significant variation over the spec-tral range covered by the applied laser pulse. Accordingly,laser pulses of only a few femtoseconds duration20 or evenattosecond pulses that exhibit broad spectra21 are increas-

ingly prone to dispersion effects related to the collective re-sponse of the system under investigation. A recently pro-posed scheme for the determination of the carrier-envelopephase of ultrashort laser pulses is based on the field effect onthe multiphoton photoemission from metal surfaces.22 Thepresent work demonstrates that the collective response has tobe considered carefully in such an application since it di-rectly affects the carrier-envelope phase.

Summarizing, a general model for two-photon photoemis-sion has been presented that implements both collective andsingle-particle dynamics. In combination with interferomet-ric time-resolved two-photon photoemission, this model al-lows one to determine the shape of the collective responsefunction across the excitation spectrum and the single-particle lifetime of excited electrons. Supported Ag nanopar-ticles were used as a model system to demonstrate the influ-ence of the collective response in TR 2PPE.

1C.A. Schmuttenmaer, M. Aeschlimann, H.E. Elsayed-Ali, R.J.D.Miller, D.A. Mantell, J. Cao, and Y. Gao, Phys. Rev. B50, 8957(1994).

2T. Hertel, E. Knoesel, M. Wolf, and G. Ertl, Phys. Rev. Lett.76,535 (1996).

3S. Ogawa, H. Nagano, H. Petek, and A.P. Heberle, Phys. Rev.Lett. 78, 1339(1997).

4U. Höfer, I.L. Shumay, C. Reuss, U. Thomann, W. Wallauer, andT. Fauster, Science277, 1480(1997).

5A.D. Miller, I. Bezel, K.J. Gaffney, S. Carrett-Roe, P. Szymanski,and C.B. Harris, Science297, 1163(2002).

6B. Lamprecht, J.R. Krenn, A. Leitner, and F.R. Aussenegg, Phys.Rev. Lett. 83, 4421(1999).

7J. Lehmann, M. Merschdorf, W. Pfeiffer, A. Thon, S. Voll, and G.Gerber, Phys. Rev. Lett.85, 2921(2000).

8M. Scharte, R. Porath, T. Ohms, M. Aeschlimann, J.R. Krenn, H.Ditlbacher, F.R. Aussenegg, and A. Liebsch, Appl. Phys. B: La-sers Opt.73, 305 (2001).

9H. Petek and S. Ogawa, Prog. Surf. Sci.56, 239 (1997).10M.J. Weida, S. Ogawa, H. Nagano, and H. Petek, Appl. Phys. A:

Mater. Sci. Process.71, 553 (2000).11U. Höfer, Appl. Phys. B: Lasers Opt.68, 383 (1999).

12M. Merschdorf, W. Pfeiffer, A. Thon, S. Voll, and G. Gerber,Appl. Phys. A: Mater. Sci. Process.71, 547 (2000).

13U. Kreibig and L. Genzel, Surf. Sci.156, 678 (1985).14H. Hövel, T. Becker, A. Bettac, B. Reihl, M. Tschudy, and E.J.

Williams, J. Appl. Phys.81, 154 (1997).15J.J. Quinn, Phys. Rev.126, 1453(1962).16V.P. Zhukov, F. Aryasetiawan, E.V. Chulkov, and P.M. Ech-

enique, Phys. Rev. B65, 115116(2002).17M. Aeschlimann, M. Bauer, S. Pawlik, R. Knorren, G. Bouzerar,

and K.H. Bennemann, Appl. Phys. A: Mater. Sci. Process.71,485 (2000).

18W. Ekardt, W.D. Schöne, and R. Keyling, Appl. Phys. A: Mater.Sci. Process.71, 529 (2000).

19M. Aeschlimann, M. Bauer, and S. Pawlik, Chem. Phys.205, 127(1996).

20T. Brabec and F. Krausz, Rev. Mod. Phys.72, 545 (2000).21M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, C. Spiel-

mann, G.A. Reider, P.B. Corkum, and F. Krausz, Science291,1923 (2001).

22C. Lemell, X.M. Tong, F. Krausz, and J. Burgdörfer, Phys. Rev.Lett. 90, 076403(2003).

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