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The role of Auger decay in hot electron excitation in copper
Chemical Physics 251 Ž2000. 71–86 www.elsevier.nlrlocaterchemphys
The role of Auger decay in hot electron excitation in copper H. Petek ) , H. Nagano, M.J. Weida, S. Ogawa AdÕanced Research Laboratory, Hitachi, Hatoyama, Saitama 350-0395, Japan Received 4 May 1999
Abstract The role of different excitation mechanisms in two-photon photoemission measurements of the hot electron population dynamics in copper is considered. The effective hot electron lifetimes derived from two-pulse correlation measurements with ; 3.1–3.8 eV, 50 fs laser pulse excitation are different depending on whether the hot electrons are generated by interband d ™ sp or intraband sp ™ sp excitation ŽS. Pawlik, M. Bauer, M. Aeschlimann, Surf. Sci. 377–379 Ž1997. 206.. A proposed explanation is that the latter process actually occurs by the Auger recombination of long-lived d-band holes resulting in complex hot electron population dynamics involving this delayed generation process and decay by the electron–electron scattering wE. Knoesel, A. Hotzel, M. Wolf, Phys. Rev. B 57 Ž1998. 12812x. This proposal is tested by simulation of interferometric two-pulse correlation measurements on the low index surfaces of copper ŽCuŽ111., Ž100., and Ž110.. by the optical Bloch equations. The lower limit for the d-hole lifetime due to the Auger recombination of 24 " 3 fs for modeling of how this generation process affects the hot electron population kinetics is established from the d-hole decoherence measurements at the X 5 point. Optical Bloch equation fits of the data show that at most - 10% of hot electrons at 1.4 eV are generated through a secondary generation mechanism, therefore Auger recombination cannot explain the anomalous hot electron population dynamics. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Hot electron excitation and relaxation is of great fundamental and practical interest in a variety of problems ranging from photo-induced surface reactions to device physics. A significant challenge in the study of hot electron dynamics in metals is to separate the different mechanisms involved in hot electron creation and decay. Hot electrons can be excited by primary absorption of a photon, or as secondary products of energy relaxation of higher energy carriers. Noble metals provide an excellent system for studying hot electron excitation dynamics because of )
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the presence of sp and d-bands near the Fermi level E F . The distinct color, diamagnetism, high electrical conductivity, and low chemical reactivity of noble metals are a consequence of complete occupation of the d-bands, which reach up to y1.6, y2.0 and y3.8 eV Žall energies are relative to E F . for gold, copper, and silver, respectively w1x. The presence of two types of bands opens several channels for direct and indirect excitation of hot electrons. In copper, for example, photons with ) 2 eV energy can create hot electrons via interband d ™ sp excitation, or as a second-order process involving momentum scattering via intraband sp ™ sp excitation ŽDrude absorption. w1,2x. Since the transition moment for interband excitation is larger, a significant fraction of the excitation energy is deposited in d-band holes, which
0301-0104r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 3 0 3 - 1
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can decay via Auger recombination Žh–h scattering. to create secondary hot electrons. Experimentally the observed hot electron population dynamics in copper and gold depend strongly on the energy of the excitation light, which may arise from the influence of holes on the hot electron dynamics w3–5x. The
extent and time scale for the excitation of hot electrons by the Auger decay of d-holes are thus of fundamental interest for understanding the carrier dynamics in noble metals. Linewidths in angle resolved photoemission spectroscopy usually provide the most direct information
Fig. 1. Schematic presentation of an I2PC experiment. An identical pair of ultrafast pulses with a precisely defined delay Žphase. excites 2PP from the sp- or d-bands in the bulk or surface states. The photoemission signal due to either coherent or sequential two-photon absorption is measured for an energy range that is narrower than the homogenous widths of the optically coupled states. The e–h pair decoherence rates and hot electron lifetimes are deduced from the resulting interferometric two-pulse correlation measurements. Carrier phase and energy relaxation occurs through electron–electron and electron–phonon scattering, with rates that can be estimated from Fermi liquid and Debye theories.
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on hole lifetimes w6–9x. The homogenous photoemission linewidth is determined by the optically excited electron–hole pair phase relaxation. It is usually assumed in photoemission spectroscopy that the linewidth can be expressed as a sum of individual electron and hole scattering rates, which are normalized by the carrier group velocities w6,10x. If a photoelectron is excited to a free-electron wave that propagates entirely in the vacuum Žinverse LEED state., the homogenous linewidth is simply given by the sum of elastic and inelastic scattering rates of the hole. Thus the hole inelastic scattering rate can be determined if the quasi-elastic, momentum scattering contribution is known from a separate measurement. Elastic and inelastic scattering rates of both electrons and holes can be studied in the same spirit, but in considerably more detail, directly in the time domain by the time-resolved two-photon photoemission ŽTR-2PP. technique. By combining the band mapping capabilities of photoemission with time-resolution of nonlinear optical spectroscopy it possible to measure energy and momentum resolved electron–hole pair elastic and inelastic scattering rates w11–13x. In a TR-2PP experiment, which is shown schematically in Fig. 1, a pump pulse from an ultrafast laser excites a distribution of e–h pairs that is determined by the optical transition moments and joint distribution of occupied and unoccupied states. The e–h pairs decay either through coherent reemission of the optical field Žreflection., or through momentum scattering into incoherent single particle excitations. The same pump or a time-delayed probe pulse can induce further excitation of the electron above the work function F . The excitation process has both coherent and incoherent contributions that depend on the laser pulse width and phase relaxation rates. The photoemission current is measured as a function of the electron kinetic energy and momentum, and the pump–probe delay D w13–15x. Thus, TR-2PP can measure both the decay of quantum mechanical coherence of the optically created e–h pairs and the hot electron energy relaxation. In order to discuss hot electron excitation mechanisms, it is important to be able to distinguish between coherent and incoherent processes such as photon absorption and Auger recombination. ŽGeneration of hot-electrons by Auger decay can be a coherent process as evidenced by recent coherent
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Auger Raman studies w16x. However, this is expected to be a minor contribution with respect to the incoherent Auger recombination process.. A powerful way to study these processes is the interferometric TR-2PP ŽITR-2PP., where the delay between identical, collinear pump and probe excitation pulses is scanned with sub-optical cycle resolution w13–15x. The resulting interference between the pump and probe-induced linear and nonlinear polarization waves in the metal modulates the two-photon photoemission Ž2PP. current as function of the delay at approximately the frequency of the driving field v and its second harmonic 2 v . If the bandwidth of the monitored 2PP signal is significantly narrower than the homogenous widths of the optically coupled states, then the phase coherent signal corresponds to an energy and momentum resolved free-induction decay of the coherent e–h pairs w14,17x. The test system for studying Auger decay dynamics in this work is copper. With its readily available and simple to prepare single crystal surfaces, extensively characterized band structure, and interesting dynamical properties w4,5,10,13,18–30x, copper has become the de facto model system for ultrafast studies of carrier dynamics in metals. A controversy exists on how to interpret the hot electron population dynamics, because the population decay rates appear to depend on the photoexcitation mechanism Žphoton energy. and do not follow the energy dependence y1 tee ; Ž E y EF . 2 nor the magnitude predicted by Fermi liquid theory ŽFLT. in the free-electron approximation w24,31,32x. The lifetimes of the primary photoexcited hot electrons that are excited by interband dipole d ™ sp excitation follow a different trend with energy compared with higher energy hot electrons in the range of hn ) E1 ) hn y 2 eV, which can only be generated through higher order processes involving elastic or inelastic scattering of carriers. The interpretation of the population kinetics of the latter component should be different if the hot electrons are generated predominantly by Drude absorption Žphase relaxation of the e–h pair. w13,30x, or by the Auger recombination of the photogenerated dholes Ženergy relaxation of the hole. w5x. Since the time scales and kinetic schemes for these two mechanisms are different, with sufficient experimental time resolution it should be possible to distinguish between them.
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In the present work, the phase and energy relaxation processes for the three low index surfaces of copper are measured for holes between y2.4 eV to E F and hot electrons between 0.7–3.1 eV. Auger recombination rates of the d-holes in copper are deduced from the analysis of the d-hole decoherence in the 2PP process. This provides an estimate for the time scale of the delayed rise time of secondary hot electrons due to Auger recombination. A new data analysis scheme based on fitting the pump–probe measurements to the optical Bloch equations ŽOBE. places an upper limit of - 10% for the Auger contribution at 1.4 eV, suggesting that hot electrons in the hn ) E1 ) hn y 2 eV energy range are created mainly through a coherent process, i.e., Drude absorption.
2. Experimental This section presents a brief outline of the ITR2PP experiment, while a detailed description can be found in Ref. w13x. The frequency-doubled output of a Ti:sapphire laser Ž hn s 3.08 eV; 13 fs width of the pulse intensity envelope. is split into equal pump and
probe pulses in a Mach–Zehnder interferometer ŽMZI.. The pump–probe delay D is scanned under feedback control by "150 fs with an accuracy and reproducibility of - 50 as by modulating the length of one arm of the MZI w13–15x. The collinear pulse pair is focused to a ; 80 mm diameter spot on the sample surface within an ultrahigh vacuum chamber Žpressure - 5 = 10y1 1 Torr.. The photoemission current is induced with s- or p-polarized light incident at 308 from the surface normal, and measured for k 5 in an energy resolved fashion with a hemispherical electron energy analyzer Ž58 angular and - 28 meV energy resolution.. Recording of the photoemission current vs. kinetic energy with only pump pulse excitation gives the 2PP spectra. To reduce the noise introduced by laser energy fluctuations, each 2PP spectrum is typically an average of 2 to 5 scans. The ; 75 meV spectral resolution of the 2PP measurements mainly reflects the band width of the excitation light. Interferometric two-pulse correlation ŽI2PC. measurements, where the 2PP current is recorded at a specific energy and parallel momentum as a function of the pump–probe delay, provide time-resolved information on the carrier dynamics. Typically 200 to
Fig. 2. The calculated band structure of copper according to the procedure in Ref. w24x. Thick arrows denote allowed intraband transitions, and thin arrows the likely transitions for excitation from the d-bands on CuŽ111. and Ž110. surfaces. The actual work function is reduced by adsorption of Cs to allow the observation of 2PP from the d-bands w37x.
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2000 scans are averaged at a rate of 1.7 Hz to obtain good counting statistics. To minimize ‘washing out’ of the coherent fringes in I2PC scans, the transmission function of the analyzer is nominally F 20 meV w26x. Single crystal CuŽ111., Ž100., and Ž110. surfaces are prepared by a cyclic procedure of sputtering with 500 eV Arq ions and annealing at 700 K. Since the work function of copper is 4.5–4.9 eV, 2PP cannot be excited from the d-bands Žlocated at F y2 eV. with 3.08 eV light. Therefore, F is reduced by ; 1 eV through adsorption of 0.10 monolayer of Cs from a SAES getter source w33–35x. This does not appear to affect the hot electron lifetimes, but it may contribute to momentum scattering of the photoelectrons w5,36x. The sample temperature is controlled between 33 and 300 K with a closed-cycle He refrigeratorrheater combination.
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3. Results and analysis 3.1. Copper band structure and 2PP spectra The d ™ sp interband and sp ™ sp intraband photoexcitation processes are studied in the frequency and time domains. The calculated band structure of the three low index surfaces of copper is provided in Fig. 2. Fig. 3 shows a 2PP spectrum of CsrCuŽ100. for a sample temperature of 50 K, and hot electron lifetimes that have been determined as described below. Two nearly overlapping peaks at E0 s y2.24 and y2.00 eV correspond to the d-bands at or near the X 5 critical point. The X 5 point is split by 0.10–0.15 eV due to the spin-orbit interaction w29x, as indicated in the figure; however, only one component is partially resolved w37x. Above the X 5 point Žy2 eV - E0 - E F . primary excitation can occur
Fig. 3. The 2PP spectra of CsrCuŽ100. measured with s- and p-polarized light, with an overlay of the hot electron lifetimes from the analysis of I2PC scans. The difference spectrum between s- and p-polarizations allows decomposition of the p-polarized spectrum into three peaks. The inset shows the splitting of the d-bands at the X 5 point due to the spin–orbit interaction w29x. A weak peak at an intermediate state energy of E1 s 2.69 eV is due to the excitation of the Cs anti-bonding state w35x.
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Fig. 4. The 2PP spectra of CsrCuŽ111., Ž110., and Ž100., and an overlay of the measured nonlinear polarization decoherence times T202 from the analysis of I2PC scans. The peaks at E1 s 2.5–3.0 eV are due to the Cs anti-bonding states, which is enhanced for CuŽ111. due to resonant excitation from the occupied surface state Žsee Fig. 2. w34x. An interesting feature of both the 2PP spectra and T202 Ž E0 . measurements is that the latter has a distinct ‘shoulder’ at the X 5 point for both CuŽ111. and Ž110.. 2PP from the X 5 point can only appear ™ ™ ™ by the scattering of the electrons from the G X, into G K and G L directions. This could be a consequence of the e–p scattering, since for 02 CuŽ111. and Ž110. the effective T2 at E0 f y2.0 eV increases with temperature probably due to a larger amplitude of the signal from the X 5 point. The T202 ‘spectrum’ is of higher ‘resolution’ than for 2PP reflecting the homogenous linewidths and analyzer resolution, rather than the laser bandwidth.
only by sp ™ sp Drude absorption, which has a characteristically flat joint density of states. On account of a small transition moment, the Drude component also may include contributions from secondary processes such as Auger recombination w5,13x. Quantitative assessment of the secondary carrier contribution to the Drude component requires a careful analysis of the I2PC scans. Due to differences in the band structure, the 2PPexcitation mechanisms depend on the crystal face. Fig. 4 shows a comparison of the 2PP spectra of the three low index surfaces including the experimental values of the decoherence time T202 for the nonlinear polarization at 50 K, which will be discussed later. Fig. 2 indicates transitions that can be excited from the d-bands by 3.08 eV photons at k 5 s 0. The thick arrows represent d ™ sp transitions that conserve
momentum and should therefore have the largest transition moments. Direct, one-photon resonant transitions in the observed energy range can only occur for CuŽ100. from the D 5 band near the X 5 critical point w38x 1. The 2PP spectrum of CuŽ100. is assigned from energetic considerations to the D 2 ™ D 1 transition for the main peak at E0 s y2.23 eV, and the X 5 ™ surface resonance at 1.1 eV for the shoulder at E0 s y2.00 eV. Further, evidence for the resonance enhancement of the d ™ sp transition is a broad resonance at " n s 3.6 eV in a study of 2PP from CuŽ100. w39x, and the ratio of the interband-to-intraband 2PP intensity in Fig. 4, which is
1
Although the X 2 critical point is close in energy, it should not appear in the 2PP spectrum on account of symmetry.
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G 5 times larger for CuŽ100. than for the other two surfaces. By contrast, the d-band peaks in the 2PP spectra of CuŽ110. and Ž111. must arise from higher order processes involving either non-resonant two-photon absorption or momentum scattering from other parts of the Brillouin zone. The d ™ sp resonant transition in CuŽ110. is too low in energy to contribute to 2PP at E0 ( y2.2 eV, while in CuŽ111. it occurs only for k / 0. The main peaks at E0 s y2.21 and y2.23 eV very likely represent singularities in 1-D density of states corresponding to the K 2 Žy2.10 eV. and L 3 Žy2.24 eV. critical points, which are observed through non-resonant two-photon excitation w29,40x. In the case of CuŽ110. a two-photon resonant S 2 ™ S 1 transition may explain why the d-band peak is ; 0.1 eV below the K 2 point. Even though Fig. 2 suggests that a similar two-photon resonance occurs for CuŽ111., the L 3 ™ L 2X transition at 6.3 eV w28,40x is too high in energy to excite with two 3.08 eV photons. The d-bands peaks at CuŽ111. and Ž110. may also have significant contributions from the momentum scattering of electrons from other parts of the Brillouin zone where one-photon dipole transitions are allowed. Clear evidence for this is the appearance of the X 5 point in the 2PP spectra of CuŽ111. of Ref. w5x, which is more distinct than in Fig. 4 due to smaller excitation band width. These different excitation mechanisms naturally affect the I2PC measurements. 3.2. Interferometric two-pulse correlation measurements The carrier phase and energy relaxation is measured for the three copper surfaces for sample temperatures between 33 and 300 K over a wide energy range corresponding to interband and intraband excitation. Fig. 5 shows two contrasting I2PC scans for interband Ž E0 s y2.24 eV. and intraband Ž E0 s y1.70 eV. excitation of CuŽ100., which are relevant for discussing the 2PP process. Differences in the carrier dynamics can be understood on a semiquantitative level by decomposing the I2PC signal into components consisting of a slowly varying phase-averaged envelope denoted as 0 v , and envelopes of the fast oscillations at approximately the
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excitation frequency 1 v , and its second harmonic v w13,41x. Fig. 5b–d show the component envelopes along with three level excitation schemes, which identify different aspects of two-photon absorption that can be determined from each envelope. The 0 v envelope corresponds to the phase-averaged equalpulse correlation measurement that is commonly obtained with non-collinear pump–probe geometry w5,13,24x. It has contributions from sequential Žincoherent. two-photon excitation through hot electron intermediate states E1 , which varies slowly on time scale of an optical cycle, and coherent two-photon absorption, which dominates near D s 0. The hot electron population dynamics can be determined from the incoherent component, provided it can be distinguished from the coherent one. Information regarding the quantum coherence in photoexcitation is contained in the oscillatory signal. The electric field of the laser induces coherence among E0 , E1 , and E2 through the electric dipole interaction. As long as the coherence persists, interference will occur between the pump- and probe-induced linear and nonlinear polarizations. Linear polarization due to the coupling of E0 l E1 and E1 l E2 oscillates approximately at the excitation frequency v , while the nonlinear polarization between E0 l E2 oscillates at 2 v . The decay of quantum coherence can be deduced from the envelopes of 1 v and 2 v oscillations. If it were to decay instantaneously, these envelopes would correspond to the autocorrelation of the laser pulse w41x. However, the decay of coherence is not instantaneous, and thus the envelope widths depend systematically on the monitoring energy, i.e., the band structure of copper. Analysis of the 1 v envelopes yields the coherence decay times T201 and T212 for the coupled levels E0 l E1 and E1 l E2 , and likewise, the 2 v envelopes provide T202 for the nonlinear polarization, which are given in Fig. 4 w13,41x. In general these measured coherence decay times can be taken as a lower limit of the actual ones, since effects due to the finite analyzer resolution and a continuum of states can reduce the width of the observed 1 v and 2 v envelopes w26x. However, in the case of transitions between discrete states or bands with negligible dispersion, such averaging effects are minimized, and the measured decoherence times correspond to the true ones.
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The hot electron lifetimes T1 , such as shown in Fig. 3, can be deconvoluted from the 0 v envelopes
once the coherent component in I2PC scans is determined independently from the 1 v and 2 v envelopes
Fig. 5. Ža. Measured I2PC scans for E0 s y2.24 and y1.70 eV; and their decomposition into Žb. phase averaged 0 v envelopes; Žc. 1 v envelopes; Žd. and 2 v envelopes. Each component envelope is used to extract the hot electron lifetime, and decoherence of linear and nonlinear polarizations between the optically coupled states, as indicated by the energy level diagrams.
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Fig. 6. Measured I2PC 2 v envelopes near the d-band maxima for Ža. CuŽ100. and Žb. CuŽ110.. The hole energy for each measurement is given on the right axis. The analysis of these data gives the T202 values in Fig. 5. Particularly slow decoherence is observed at the X 5 point, i.e., the top of the d-bands, on CuŽ100.. A weak signal from the X 5 point is also observed on CuŽ110. due to momentum scattering of the photoelectrons Žalso see Fig. 4..
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w13,41x. The measured values of T1 for CuŽ100. are quite similar to those of the other low index copper surfaces w24,30x, and in agreement with the previous measurements from other groups w3,5,21x. The lifetimes do not decrease monotonically with the Ž E y E F .y2 energy dependence predicted by FLT, but follow different trends when hot electrons are generated by interband or Drude absorption, as first reported in Ref. w3x. The comparison of 0 v envelopes in Fig. 5b also shows that the hot electron population dynamics depend strongly on the excitation mechanism. Although the population decay at E1 s 1.4 eV is well described by single exponential kinetics, Knoesel et al. suggest that the unusually long lifetimes can be explained by a more complex kinetic scheme involving a delayed generation of hot electrons by the Auger recombination of long lived d-holes w5x. If hot electron generation takes significantly longer than their decay, then this rate-limiting process would dominate the time evolution of the population at E1. The mechanism of Knoesel et al.
can be tested quantitatively if the Auger recombination rates are known.
4. Discussion 4.1. Auger recombination rate Although the I2PC scans do not measure the hole lifetimes directly in the same manner as for the electrons, hole scattering processes can be deduced from the decoherence rates. Therefore, the dependence of T202 on the band structure, carrier momentum, and sample temperature can provide information on the Auger recombination. Since the Auger recombination makes a contribution to T202 through the scattering of the holes, it can be determined from the measurements in Figs. 4–7 if other phase scattering processes are known or can be
Fig. 7. Ža. The temperature dependence of the 2 v envelope at the X 5 point. Žb. Plot of ŽT202 .y1 derived from the data in Ža. vs. the sample temperature. Fitting of this data with the Debye model gives the electron–phonon mass enhancement factor l s 0.20 " 0.01.
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neglected. Following the usual convention in photoemission spectroscopy, the phase decay rate of the induced polarization is expressed as a sum of electron and hole scattering 1rT202 s Ge q G h . This assumes that exciton–exciton scattering can be ignored. If 2PP occurs to the inverse LEED states, as in the case for CuŽ100., than the electron has a vanishingly small probability in the bulk, and the contribution to ŽT022 .y1 from Ge is negligible. This is supported by measured lifetimes of the high lying Ž; 4.6 eV. image potential states on clean CuŽ100. where electron lifetimes are ) 2 ps w25x. However, according to the band structure in Fig. 2 this approximation may not be valid for CuŽ111. and CuŽ110.. Fig. 6 shows the 2 v envelopes for CuŽ100. and Ž110., which are used to derive T202 in Fig. 4 according to the fitting procedure in Ref. w30x. A larger contribution from Ge can explain the significantly faster decoherence rates for CuŽ110.. The extent of the electron penetration into the bulk, and therefore 2 .y1 the contribution of Ge to ŽT02 at E2 ; 4.1 eV should decrease in the order CuŽ110. ) CuŽ111. ) CuŽ100. w10x, which agrees with the trend in T202 at E0 f y2.2 eV in Fig. 4. The hole contribution to ŽT022 .y1 can be further decomposed assuming Matthiessen’s rule by expressing G h as a sum of h–h, hole–phonon Žh–p. and hole–defect Žh–d. scattering, i.e., G h s G hh q G hp ŽT. q G hd . The Auger recombination rate can be determined if the h–p and h–d contributions are known. The h–d scattering is an extrinsic contribution related to the crystalline quality of the sample, which in the present experiment could be dominated by surface impurities such as Cs atoms. It is unlikely that the bulk defects in well-annealed, high-purity samples could contribute significantly to the G h under present conditions w42x. The effect of surface defects on phase relaxation is known from the freeelectron like surface state linewidth measurements on alkali atom covered CuŽ111. surfaces, which show that ŽT202 .y1 increases linearly with the parallel momentum k 5 of electrons and impurity concentration w43,44x. If the decoherence length, as defined by the product of carrier velocity and T202 , is comparable to the average distance between the defects, then h–d scattering can make a significant contribution. A clear impurity concentration or k 5 dependence of T202 is not observed for the d-bands probably be-
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cause the photoemission can occur from the bulk where the effect of Cs adsorbates is efficiently screened, and they have a smaller group velocity than the surface states. The contribution from h–p scattering to G h is evident from in Fig. 7, which shows the 2 v envelopes for CuŽ100. measured at the X 5 point for several sample temperatures. According to Fig. 7b, G hŽT . has a linear dependence on the surface temperature in the range 33–300 K. This is in line with the Debye model, which predicts that G hp f 2pl k B T, where l is the electron–phonon mass enhancement factor w7,45x. Fit of the data in Fig. 7b gives l s 0.20 " 0.01 and G hy f s 0.0086 fsy1 at 0 K. This is the same value of l as measured by transport measurements for the conduction electrons at E F in the ²100: direction w45x. Having determined G hp ŽT . and argued why Ge and G hd are negligible, the upper limit for the Auger recombination rate at the X 5 point can be determined from G hh s G h y G hp s 1rT202 ŽT . y 2pl k B T. A lower limit for the lifetime of the d-holes at the X 5 point of 24 " 3 fs is determined from the residual, temperature independent contribution to T202 . The Auger recombination time could be even longer if T202 is limited by the finite resolution of the analyzer, or if Ge and G hd cannot be neglected. The above analysis provides the Auger recombination rate at a specific point in the d-bands; however, the optical excitation generates a broad distribution of d-holes. Below the X 5 point the derived values of ŽT202 .y1 increase faster than preŽ . 2 energy scaling, dicted by the ty1 hy h ; E y EF which is expected since the d-bands contribute much larger scattering phase space than predicted by the free-electron model. Also, the d–d scattering may be considerably more efficient than the d–sp Auger recombination. Therefore, a plausible scenario for the d-hole dynamics is a primary relaxation within the d-bands, whereby the holes float up to the X 5 point, followed by secondary process of d–sp Auger recombination near the X 5 point, which generates the hot electrons w46,47x. Thus, assuming that the relaxation from the X 5 point is the rate-limiting step for the secondary hot electron generation, the rate of 0.021 fsy1 provides a lower limit for simulating the contribution of the Auger recombination to the hot electron dynamics.
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4.2. Analysis of the primary and secondary hot electron generation The remaining part of this section presents quantitative assessment and modeling of the primary and secondary hot electron contributions to the measured I2PC data. The tremendous advantage of the I2PC measurements is that it is possible to distinguish hot electron generation processes involving the phase relaxation of the photoinduced e–h pairs Žcoherent polarization. into an incoherent hot carrier distribution Žprimary generation. from the energy relaxation of the primary carriers Žsecondary generation.. This is possible for two reasons: Ži. the two generation processes have distinct kinetics that can be unraveled through an interferometric measurement; and Žii. with 13 fs pulses, secondary generation on longer time scales appears as a distinct delayed rise in the signal. The simplest Žand often completely adequate. way to model the interferometric two-photon correlation data is to use an optical Bloch equation ŽOBE.
formalism for multilevel transitions w48x. In this model the electron is restricted to three energy levels separated by the photon energy of the exciting laser. The three levels are described by a density matrix with diagonal elements representing the population in each level Ž r 00 s ground state, r 11 s hot electron state, r 22 s detected photoelectron., and off-diagonal elements representing coherences between the levels. The OBEs are a set of coupled differential equations that describe how the levels and coherences evolve in the presence of an external electromagnetic perturbation. The system is closed with a total population normalized to unity if there are no source or loss terms outside of the three-level system, i.e., population of the intermediate level by hot electron avalanche relaxation, or removal of electrons due to scattering into other states. To simulate population of the intermediate level by Auger recombination, the OBE formalism needs to be ‘opened’ such that the total population of the system can exceed unity. This is appropriate if
Fig. 8. Fit of the I2PC measurement at E0 s y1.70 eV to the optical Bloch equation model. The excellent fit indicates that most of the hot electrons are generated directly by optical excitation. Several simulations indicate the appearance of the signal if in addition to the primary generation hot electrons also are generated by the Auger recombination. The OBE analysis places an upper limit of 10% generation of hot electrons by a delayed secondary process at this energy Ž E1 s 1.4 eV..
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r 00 Ž t . ; 1 Žweak excitation., and is necessary since a single set of three levels is monitored, while the excitation occurs for a broad continuum of states that can decay into or out of the level of observation. First, the hole population in the ground state is defined as r hole s 1 y r 00 . It is assumed that the ground state holes have a probability per unit time Thole of undergoing Auger recombination, and that this is the only incoherent decay channel that can repopulate the ground state. Therefore, the time evolutions of r 00 and r 11 are modified by adding terms Er 00 Et Er 11 Et
sŽ .... q
1 T hole
r hole ,
a sŽ .... q
Thole
r hole ,
where Ž . . . . represents the usual terms in the OBE. The consequence of Auger recombination is regeneration of the electron in the ground state and secondary hot electron excitation to the intermediate state. a s 1 means that all of the holes created by photoexcitation from the ground state create an Auger electron in the intermediate level, while allowing a ) 1 is a way of including contributions from holes with energies outside the closed three level system. Since the population of holes is equal to the number of hot electrons excited directly via photons Žneglecting two-photon absorption., the fraction of hot electrons created via Auger decay is defined as arŽ a q 1.. Thus a defines the relative amplitudes of the coherent and incoherent components in the I2PC scans w13x. Although a is defined to describe the Auger recombination, other secondary generation, such as the hot electron avalanche relaxation, should have similar effects on I2PC scans, but with possibly different rates. In order to test for the presence of the secondary hot electron generation process an OBE fit of the I2PC trace for E0 s y1.70 eV in Fig. 4a is shown in Fig. 8. The model is an excellent representation of the data according to the residuals, which are determined mainly by the photoelectron counting statistics. The fit gives a hot electron energy relaxation time of 72.5 fs, which agrees well with previous TR-2PP measurements at this energy w3,5,24x. The best fit is obtained when the hot electrons are generated entirely by the primary Drude absorption. If a
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is a variable parameter in the fit and the secondary hot electron generation time is 24 fs, as expected for Auger recombination, the OBE analysis indicates that the maximum secondary contribution is - 10% of the primary carriers. Fitting the data with the delayed rise and amplitude as variable parameters neither significantly improves the fit, nor suggests that the secondary contribution is any larger. Fig. 8 also shows several calculated I2PC traces using the same phase and energy relaxation parameters as above, but with secondary hot electrons contributing from 20 to 60% of the total hot electron population. The secondary contribution increases the incoherent signal amplitude relative to the coherent oscillations and leads to significant deviations from single-exponential decay kinetics that are clearly at odds with the experimental data. However, below E1 s 1.4 eV the secondary contribution acquires a significant amplitude, which will be analyzed in future work w49x. This analysis demonstrates that hot electrons with E1 G 1.4 eV are generated predominantly by the optical excitation and that the measured T1 values represent the population lifetimes. 4.3. I2PC scans in the secondary generation limit The OBE analysis strongly supports the view that the hot electrons above the interband excitation threshold are generated almost exclusively by Drude absorption. Ref. w5x takes the diametrically opposite view, where the anomalously long hot electron lifetimes above the d ™ sp excitation threshold are attributed to the production of hot electrons entirely by delayed Auger recombination of the d-holes. In this section, the present analysis is extended to model these two limiting scenarios in order to emphasize Ži. the importance of measuring hot electron dynamics with ultrashort pulses; and Žii. the power of the ITR-2PP technique for unraveling the complex kinetics of hot electron generation and relaxation. Fig. 9 presents several calculated interferometric and phase averaged two-pulse correlation measurements that simulate the dynamics of hot electrons at E1 s 1.63 eV. In Ref. w5x, a phase averaged two-pulse correlation ŽPA-2PC. is measured at the same energy with a 50 fs pulse width laser. The I2PC traces in Fig. 9a and b are simulated with 13 and 50 fs pulse
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Fig. 9. OBE simulation of interferometric and phase averaged two pulse correlation measurements assuming excitation with Ža. 13 fs and Žb. 50 fs pulses; Ža. and Žc. for entirely direct and Žb. and Žd. delayed generation mechanisms. The phase averaged envelopes for the two models with Že. 13 fs and 50 fs pulse excitation.
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widths in order to contrast the results of the present experiment with those in Ref. w5x. According to the OBE fit of the data, the energy relaxation time at E1 s 1.63 eV is T1 s 51.7 fs. In Ref. w5x, the PA-2PC is modeled by the generation of hot electrons entirely through the Auger recombination with a rise time of 35 fs, and an intrinsic hot electron lifetime of 15 fs, which is in line with the predictions of FLT w5x. Fig. 9c and d show the simulated I2PC scans assuming the Auger recombination kinetic mechanism and laser pulse widths of 13 and 50 fs, respectively. Fig. 9e and f show the phase-averaged 0 v envelopes ŽPA2PCs. for the simulations in Fig. 9a–d. Both the interferometric and phase averaged twopulse correlations show a dramatic contrast between the two excitation mechanisms in the case of 13 fs excitation. However, with 50 fs excitation the PA2PCs are indeed qualitatively similar, and since phase averaged measurements are blind to the nature of the excitation process, it is difficult to provide certain evidence for either mechanism even in the limiting cases. Thus, in order to establish the kinetic mechanism it is imperative to perform measurements with comparable or shorter laser pulse widths than the dynamics under investigation. Since I2PC measurements can distinguish the coherent and incoherent excitation processes, it still may be possible to extract quantitative kinetic rates with 50 fs pulse excitation provided that the fitting model correctly describes the excitation mechanism.
5. Conclusions The conclusions that can be drawn following this detailed analysis of the hot electron and hole relaxation in copper are: Ži. hot electron excitation above ; 0.9 eV proceeds mainly by the direct interband d ™ sp and intraband sp ™ sp excitation, rather than through the incoherent decay of high energy electrons and holes; Žii. the hot electron lifetimes are significantly longer at a given energy when the photoexcitation is through a direct interband process rather than an indirect process; Žiii. the lower limit for the d-hole lifetime at the top of the d-bands is 24 " 3 fs. The conclusion that the excitation process influences the subsequent hot electron energy relaxation is contrary to the sudden approximation in
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photoemission, which assumes that the photoinduced carrier dynamics are independent of the history of their creation. The Auger spectra of core holes in copper provide a precedent for the breakdown of the sudden approximation in observation that the energy of photoelectrons as high as 200 eV above the threshold for a specific channel influences the decay of the corresponding holes w50x. Furthermore, there is a significant disagreement between the highest level theory and photoemission experiments regarding the energy of the d-bands, which is attributed to significant self-energy corrections due to the exchange and correlation, that are not accounted for in calculating the band structure of copper w51x. The imaginary part of the self-energy associated with the interaction of the hot electron with its hole may explain why the hot electron lifetime depends on the nature of the excitation process. The energy and momentum may not uniquely define the carrier lifetime, as assumed in quasiparticle theories, if the lifetime also depends on the history of carrier excitation. Thus, ITR-2PP measurements may provide the means to study the ultrafast many-body Coulomb correlations in photoexcitation and relaxation of carriers in metals.
Acknowledgements The authors thank N. Moriya, S. Matsunami, and S. Saito for technical support; and NEDO International Joint Research Grant for partial funding of this project. M.J. Weida wishes to thank the National Science Foundation and the Center for Global Partnership for support ŽNSF grant INT-9819100..
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w29x R. Courths, S. Hufner, Phys. Reports 112 Ž1984. 53. ¨ w30x H. Petek, H. Nagano, S. Ogawa, Appl. Phys. B 68 Ž1999. 369. w31x J.J. Quinn, Phys. Rev. 126 Ž1962. 1453. w32x J.J. Quinn, Appl. Phys. Lett. 2 Ž1963. 167. ˚ Lindgren, L. Wallden, w33x S.A. ´ Phys. Rev. B 45 Ž1992. 6345. w34x S. Ogawa, H. Nagano, H. Petek, Phys. Rev. Lett. 82 Ž1999. 1931. w35x M. Bauer, S. Pawlik, M. Aeschlimann, Phys. Rev. B 60 Ž1999. 5016. w36x M. Bauer, S. Pawlik, M. Aeschlimann, Phys. Rev. B 55 Ž1997. 10040. w37x H. Petek, H. Nagano, S. Ogawa, Phys. Rev. Lett. 83 Ž1999. 832. w38x D.A. Arena, F.G. Curti, R.A. Bartynski, Phys. Rev. B 56 Ž1997. 15404. w39x T. Wegehaupt, D. Riger, W. Steinmann, Phys. Rev. B 37 Ž1988. 10086. w40x R. Courths, B. Bachelier, B. Cord, S. Hufner, Solid State ¨ Commun. 40 Ž1981. 1059. w41x W. Nessler, S. Ogawa, H. Nagano, H. Petek, J. Shimoyama, Y. Nakayama, K. Kishio, J. Elect. Spect. Rel. Phen. 88–91 Ž1998. 495. w42x V.A. Gasparov, R. Huguenin, Adv. Phys. 42 Ž1993. 393. w43x S.D. Kevan, Phys. Rev. B 33 Ž1986. 4364. w44x X.Y. Wang, R. Paiella, R.M. Osgood, Phys. Rev. B 51 Ž1995. 17035. w45x G. Grimvall, Phys. Scr. 14 Ž1976. 63. w46x C.N. Berglund, W.E. Spicer, Phys. Rev. 136 Ž1964. A1044. w47x C.N. Berglund, W.E. Spicer, Phys. Rev. 136 Ž1964. A1030. w48x J.-C. Diels, W. Rudolph, Ultrashort Laser Phenomena, Academic Press, San Diego, 1996. w49x M.J. Weida, H. Nagano, S. Ogawa, H. Petek, in preparation. w50x D.D. Sarma, C. Carbone, P. Sen, R. Cimino, W. Gudat, Phys. Rev. Lett. 63 Ž1989. 656. w51x V.N. Strocov, R. Claessen, G. Nicolay, S. Hufner, A. Kimura, ¨ A. Harasawa, S. Shin, A. Kakizaki, P.O. Nilsson, H.I. Starnberg, P. Blaha, Phys. Rev. Lett. 81 Ž1998. 4943.
The role of Auger decay in hot electron excitation in copper H. Petek ) , H. Nagano, M.J. Weida, S. Ogawa AdÕanced Research Laboratory, Hitachi, Hatoyama, Saitama 350-0395, Japan Received 4 May 1999
Abstract The role of different excitation mechanisms in two-photon photoemission measurements of the hot electron population dynamics in copper is considered. The effective hot electron lifetimes derived from two-pulse correlation measurements with ; 3.1–3.8 eV, 50 fs laser pulse excitation are different depending on whether the hot electrons are generated by interband d ™ sp or intraband sp ™ sp excitation ŽS. Pawlik, M. Bauer, M. Aeschlimann, Surf. Sci. 377–379 Ž1997. 206.. A proposed explanation is that the latter process actually occurs by the Auger recombination of long-lived d-band holes resulting in complex hot electron population dynamics involving this delayed generation process and decay by the electron–electron scattering wE. Knoesel, A. Hotzel, M. Wolf, Phys. Rev. B 57 Ž1998. 12812x. This proposal is tested by simulation of interferometric two-pulse correlation measurements on the low index surfaces of copper ŽCuŽ111., Ž100., and Ž110.. by the optical Bloch equations. The lower limit for the d-hole lifetime due to the Auger recombination of 24 " 3 fs for modeling of how this generation process affects the hot electron population kinetics is established from the d-hole decoherence measurements at the X 5 point. Optical Bloch equation fits of the data show that at most - 10% of hot electrons at 1.4 eV are generated through a secondary generation mechanism, therefore Auger recombination cannot explain the anomalous hot electron population dynamics. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Hot electron excitation and relaxation is of great fundamental and practical interest in a variety of problems ranging from photo-induced surface reactions to device physics. A significant challenge in the study of hot electron dynamics in metals is to separate the different mechanisms involved in hot electron creation and decay. Hot electrons can be excited by primary absorption of a photon, or as secondary products of energy relaxation of higher energy carriers. Noble metals provide an excellent system for studying hot electron excitation dynamics because of )
Corresponding author. Tel.: q81-492-966111; fax: q81-492966006; e-mail: [email protected]
the presence of sp and d-bands near the Fermi level E F . The distinct color, diamagnetism, high electrical conductivity, and low chemical reactivity of noble metals are a consequence of complete occupation of the d-bands, which reach up to y1.6, y2.0 and y3.8 eV Žall energies are relative to E F . for gold, copper, and silver, respectively w1x. The presence of two types of bands opens several channels for direct and indirect excitation of hot electrons. In copper, for example, photons with ) 2 eV energy can create hot electrons via interband d ™ sp excitation, or as a second-order process involving momentum scattering via intraband sp ™ sp excitation ŽDrude absorption. w1,2x. Since the transition moment for interband excitation is larger, a significant fraction of the excitation energy is deposited in d-band holes, which
0301-0104r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 3 0 3 - 1
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can decay via Auger recombination Žh–h scattering. to create secondary hot electrons. Experimentally the observed hot electron population dynamics in copper and gold depend strongly on the energy of the excitation light, which may arise from the influence of holes on the hot electron dynamics w3–5x. The
extent and time scale for the excitation of hot electrons by the Auger decay of d-holes are thus of fundamental interest for understanding the carrier dynamics in noble metals. Linewidths in angle resolved photoemission spectroscopy usually provide the most direct information
Fig. 1. Schematic presentation of an I2PC experiment. An identical pair of ultrafast pulses with a precisely defined delay Žphase. excites 2PP from the sp- or d-bands in the bulk or surface states. The photoemission signal due to either coherent or sequential two-photon absorption is measured for an energy range that is narrower than the homogenous widths of the optically coupled states. The e–h pair decoherence rates and hot electron lifetimes are deduced from the resulting interferometric two-pulse correlation measurements. Carrier phase and energy relaxation occurs through electron–electron and electron–phonon scattering, with rates that can be estimated from Fermi liquid and Debye theories.
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on hole lifetimes w6–9x. The homogenous photoemission linewidth is determined by the optically excited electron–hole pair phase relaxation. It is usually assumed in photoemission spectroscopy that the linewidth can be expressed as a sum of individual electron and hole scattering rates, which are normalized by the carrier group velocities w6,10x. If a photoelectron is excited to a free-electron wave that propagates entirely in the vacuum Žinverse LEED state., the homogenous linewidth is simply given by the sum of elastic and inelastic scattering rates of the hole. Thus the hole inelastic scattering rate can be determined if the quasi-elastic, momentum scattering contribution is known from a separate measurement. Elastic and inelastic scattering rates of both electrons and holes can be studied in the same spirit, but in considerably more detail, directly in the time domain by the time-resolved two-photon photoemission ŽTR-2PP. technique. By combining the band mapping capabilities of photoemission with time-resolution of nonlinear optical spectroscopy it possible to measure energy and momentum resolved electron–hole pair elastic and inelastic scattering rates w11–13x. In a TR-2PP experiment, which is shown schematically in Fig. 1, a pump pulse from an ultrafast laser excites a distribution of e–h pairs that is determined by the optical transition moments and joint distribution of occupied and unoccupied states. The e–h pairs decay either through coherent reemission of the optical field Žreflection., or through momentum scattering into incoherent single particle excitations. The same pump or a time-delayed probe pulse can induce further excitation of the electron above the work function F . The excitation process has both coherent and incoherent contributions that depend on the laser pulse width and phase relaxation rates. The photoemission current is measured as a function of the electron kinetic energy and momentum, and the pump–probe delay D w13–15x. Thus, TR-2PP can measure both the decay of quantum mechanical coherence of the optically created e–h pairs and the hot electron energy relaxation. In order to discuss hot electron excitation mechanisms, it is important to be able to distinguish between coherent and incoherent processes such as photon absorption and Auger recombination. ŽGeneration of hot-electrons by Auger decay can be a coherent process as evidenced by recent coherent
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Auger Raman studies w16x. However, this is expected to be a minor contribution with respect to the incoherent Auger recombination process.. A powerful way to study these processes is the interferometric TR-2PP ŽITR-2PP., where the delay between identical, collinear pump and probe excitation pulses is scanned with sub-optical cycle resolution w13–15x. The resulting interference between the pump and probe-induced linear and nonlinear polarization waves in the metal modulates the two-photon photoemission Ž2PP. current as function of the delay at approximately the frequency of the driving field v and its second harmonic 2 v . If the bandwidth of the monitored 2PP signal is significantly narrower than the homogenous widths of the optically coupled states, then the phase coherent signal corresponds to an energy and momentum resolved free-induction decay of the coherent e–h pairs w14,17x. The test system for studying Auger decay dynamics in this work is copper. With its readily available and simple to prepare single crystal surfaces, extensively characterized band structure, and interesting dynamical properties w4,5,10,13,18–30x, copper has become the de facto model system for ultrafast studies of carrier dynamics in metals. A controversy exists on how to interpret the hot electron population dynamics, because the population decay rates appear to depend on the photoexcitation mechanism Žphoton energy. and do not follow the energy dependence y1 tee ; Ž E y EF . 2 nor the magnitude predicted by Fermi liquid theory ŽFLT. in the free-electron approximation w24,31,32x. The lifetimes of the primary photoexcited hot electrons that are excited by interband dipole d ™ sp excitation follow a different trend with energy compared with higher energy hot electrons in the range of hn ) E1 ) hn y 2 eV, which can only be generated through higher order processes involving elastic or inelastic scattering of carriers. The interpretation of the population kinetics of the latter component should be different if the hot electrons are generated predominantly by Drude absorption Žphase relaxation of the e–h pair. w13,30x, or by the Auger recombination of the photogenerated dholes Ženergy relaxation of the hole. w5x. Since the time scales and kinetic schemes for these two mechanisms are different, with sufficient experimental time resolution it should be possible to distinguish between them.
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In the present work, the phase and energy relaxation processes for the three low index surfaces of copper are measured for holes between y2.4 eV to E F and hot electrons between 0.7–3.1 eV. Auger recombination rates of the d-holes in copper are deduced from the analysis of the d-hole decoherence in the 2PP process. This provides an estimate for the time scale of the delayed rise time of secondary hot electrons due to Auger recombination. A new data analysis scheme based on fitting the pump–probe measurements to the optical Bloch equations ŽOBE. places an upper limit of - 10% for the Auger contribution at 1.4 eV, suggesting that hot electrons in the hn ) E1 ) hn y 2 eV energy range are created mainly through a coherent process, i.e., Drude absorption.
2. Experimental This section presents a brief outline of the ITR2PP experiment, while a detailed description can be found in Ref. w13x. The frequency-doubled output of a Ti:sapphire laser Ž hn s 3.08 eV; 13 fs width of the pulse intensity envelope. is split into equal pump and
probe pulses in a Mach–Zehnder interferometer ŽMZI.. The pump–probe delay D is scanned under feedback control by "150 fs with an accuracy and reproducibility of - 50 as by modulating the length of one arm of the MZI w13–15x. The collinear pulse pair is focused to a ; 80 mm diameter spot on the sample surface within an ultrahigh vacuum chamber Žpressure - 5 = 10y1 1 Torr.. The photoemission current is induced with s- or p-polarized light incident at 308 from the surface normal, and measured for k 5 in an energy resolved fashion with a hemispherical electron energy analyzer Ž58 angular and - 28 meV energy resolution.. Recording of the photoemission current vs. kinetic energy with only pump pulse excitation gives the 2PP spectra. To reduce the noise introduced by laser energy fluctuations, each 2PP spectrum is typically an average of 2 to 5 scans. The ; 75 meV spectral resolution of the 2PP measurements mainly reflects the band width of the excitation light. Interferometric two-pulse correlation ŽI2PC. measurements, where the 2PP current is recorded at a specific energy and parallel momentum as a function of the pump–probe delay, provide time-resolved information on the carrier dynamics. Typically 200 to
Fig. 2. The calculated band structure of copper according to the procedure in Ref. w24x. Thick arrows denote allowed intraband transitions, and thin arrows the likely transitions for excitation from the d-bands on CuŽ111. and Ž110. surfaces. The actual work function is reduced by adsorption of Cs to allow the observation of 2PP from the d-bands w37x.
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2000 scans are averaged at a rate of 1.7 Hz to obtain good counting statistics. To minimize ‘washing out’ of the coherent fringes in I2PC scans, the transmission function of the analyzer is nominally F 20 meV w26x. Single crystal CuŽ111., Ž100., and Ž110. surfaces are prepared by a cyclic procedure of sputtering with 500 eV Arq ions and annealing at 700 K. Since the work function of copper is 4.5–4.9 eV, 2PP cannot be excited from the d-bands Žlocated at F y2 eV. with 3.08 eV light. Therefore, F is reduced by ; 1 eV through adsorption of 0.10 monolayer of Cs from a SAES getter source w33–35x. This does not appear to affect the hot electron lifetimes, but it may contribute to momentum scattering of the photoelectrons w5,36x. The sample temperature is controlled between 33 and 300 K with a closed-cycle He refrigeratorrheater combination.
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3. Results and analysis 3.1. Copper band structure and 2PP spectra The d ™ sp interband and sp ™ sp intraband photoexcitation processes are studied in the frequency and time domains. The calculated band structure of the three low index surfaces of copper is provided in Fig. 2. Fig. 3 shows a 2PP spectrum of CsrCuŽ100. for a sample temperature of 50 K, and hot electron lifetimes that have been determined as described below. Two nearly overlapping peaks at E0 s y2.24 and y2.00 eV correspond to the d-bands at or near the X 5 critical point. The X 5 point is split by 0.10–0.15 eV due to the spin-orbit interaction w29x, as indicated in the figure; however, only one component is partially resolved w37x. Above the X 5 point Žy2 eV - E0 - E F . primary excitation can occur
Fig. 3. The 2PP spectra of CsrCuŽ100. measured with s- and p-polarized light, with an overlay of the hot electron lifetimes from the analysis of I2PC scans. The difference spectrum between s- and p-polarizations allows decomposition of the p-polarized spectrum into three peaks. The inset shows the splitting of the d-bands at the X 5 point due to the spin–orbit interaction w29x. A weak peak at an intermediate state energy of E1 s 2.69 eV is due to the excitation of the Cs anti-bonding state w35x.
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Fig. 4. The 2PP spectra of CsrCuŽ111., Ž110., and Ž100., and an overlay of the measured nonlinear polarization decoherence times T202 from the analysis of I2PC scans. The peaks at E1 s 2.5–3.0 eV are due to the Cs anti-bonding states, which is enhanced for CuŽ111. due to resonant excitation from the occupied surface state Žsee Fig. 2. w34x. An interesting feature of both the 2PP spectra and T202 Ž E0 . measurements is that the latter has a distinct ‘shoulder’ at the X 5 point for both CuŽ111. and Ž110.. 2PP from the X 5 point can only appear ™ ™ ™ by the scattering of the electrons from the G X, into G K and G L directions. This could be a consequence of the e–p scattering, since for 02 CuŽ111. and Ž110. the effective T2 at E0 f y2.0 eV increases with temperature probably due to a larger amplitude of the signal from the X 5 point. The T202 ‘spectrum’ is of higher ‘resolution’ than for 2PP reflecting the homogenous linewidths and analyzer resolution, rather than the laser bandwidth.
only by sp ™ sp Drude absorption, which has a characteristically flat joint density of states. On account of a small transition moment, the Drude component also may include contributions from secondary processes such as Auger recombination w5,13x. Quantitative assessment of the secondary carrier contribution to the Drude component requires a careful analysis of the I2PC scans. Due to differences in the band structure, the 2PPexcitation mechanisms depend on the crystal face. Fig. 4 shows a comparison of the 2PP spectra of the three low index surfaces including the experimental values of the decoherence time T202 for the nonlinear polarization at 50 K, which will be discussed later. Fig. 2 indicates transitions that can be excited from the d-bands by 3.08 eV photons at k 5 s 0. The thick arrows represent d ™ sp transitions that conserve
momentum and should therefore have the largest transition moments. Direct, one-photon resonant transitions in the observed energy range can only occur for CuŽ100. from the D 5 band near the X 5 critical point w38x 1. The 2PP spectrum of CuŽ100. is assigned from energetic considerations to the D 2 ™ D 1 transition for the main peak at E0 s y2.23 eV, and the X 5 ™ surface resonance at 1.1 eV for the shoulder at E0 s y2.00 eV. Further, evidence for the resonance enhancement of the d ™ sp transition is a broad resonance at " n s 3.6 eV in a study of 2PP from CuŽ100. w39x, and the ratio of the interband-to-intraband 2PP intensity in Fig. 4, which is
1
Although the X 2 critical point is close in energy, it should not appear in the 2PP spectrum on account of symmetry.
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G 5 times larger for CuŽ100. than for the other two surfaces. By contrast, the d-band peaks in the 2PP spectra of CuŽ110. and Ž111. must arise from higher order processes involving either non-resonant two-photon absorption or momentum scattering from other parts of the Brillouin zone. The d ™ sp resonant transition in CuŽ110. is too low in energy to contribute to 2PP at E0 ( y2.2 eV, while in CuŽ111. it occurs only for k / 0. The main peaks at E0 s y2.21 and y2.23 eV very likely represent singularities in 1-D density of states corresponding to the K 2 Žy2.10 eV. and L 3 Žy2.24 eV. critical points, which are observed through non-resonant two-photon excitation w29,40x. In the case of CuŽ110. a two-photon resonant S 2 ™ S 1 transition may explain why the d-band peak is ; 0.1 eV below the K 2 point. Even though Fig. 2 suggests that a similar two-photon resonance occurs for CuŽ111., the L 3 ™ L 2X transition at 6.3 eV w28,40x is too high in energy to excite with two 3.08 eV photons. The d-bands peaks at CuŽ111. and Ž110. may also have significant contributions from the momentum scattering of electrons from other parts of the Brillouin zone where one-photon dipole transitions are allowed. Clear evidence for this is the appearance of the X 5 point in the 2PP spectra of CuŽ111. of Ref. w5x, which is more distinct than in Fig. 4 due to smaller excitation band width. These different excitation mechanisms naturally affect the I2PC measurements. 3.2. Interferometric two-pulse correlation measurements The carrier phase and energy relaxation is measured for the three copper surfaces for sample temperatures between 33 and 300 K over a wide energy range corresponding to interband and intraband excitation. Fig. 5 shows two contrasting I2PC scans for interband Ž E0 s y2.24 eV. and intraband Ž E0 s y1.70 eV. excitation of CuŽ100., which are relevant for discussing the 2PP process. Differences in the carrier dynamics can be understood on a semiquantitative level by decomposing the I2PC signal into components consisting of a slowly varying phase-averaged envelope denoted as 0 v , and envelopes of the fast oscillations at approximately the
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excitation frequency 1 v , and its second harmonic v w13,41x. Fig. 5b–d show the component envelopes along with three level excitation schemes, which identify different aspects of two-photon absorption that can be determined from each envelope. The 0 v envelope corresponds to the phase-averaged equalpulse correlation measurement that is commonly obtained with non-collinear pump–probe geometry w5,13,24x. It has contributions from sequential Žincoherent. two-photon excitation through hot electron intermediate states E1 , which varies slowly on time scale of an optical cycle, and coherent two-photon absorption, which dominates near D s 0. The hot electron population dynamics can be determined from the incoherent component, provided it can be distinguished from the coherent one. Information regarding the quantum coherence in photoexcitation is contained in the oscillatory signal. The electric field of the laser induces coherence among E0 , E1 , and E2 through the electric dipole interaction. As long as the coherence persists, interference will occur between the pump- and probe-induced linear and nonlinear polarizations. Linear polarization due to the coupling of E0 l E1 and E1 l E2 oscillates approximately at the excitation frequency v , while the nonlinear polarization between E0 l E2 oscillates at 2 v . The decay of quantum coherence can be deduced from the envelopes of 1 v and 2 v oscillations. If it were to decay instantaneously, these envelopes would correspond to the autocorrelation of the laser pulse w41x. However, the decay of coherence is not instantaneous, and thus the envelope widths depend systematically on the monitoring energy, i.e., the band structure of copper. Analysis of the 1 v envelopes yields the coherence decay times T201 and T212 for the coupled levels E0 l E1 and E1 l E2 , and likewise, the 2 v envelopes provide T202 for the nonlinear polarization, which are given in Fig. 4 w13,41x. In general these measured coherence decay times can be taken as a lower limit of the actual ones, since effects due to the finite analyzer resolution and a continuum of states can reduce the width of the observed 1 v and 2 v envelopes w26x. However, in the case of transitions between discrete states or bands with negligible dispersion, such averaging effects are minimized, and the measured decoherence times correspond to the true ones.
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The hot electron lifetimes T1 , such as shown in Fig. 3, can be deconvoluted from the 0 v envelopes
once the coherent component in I2PC scans is determined independently from the 1 v and 2 v envelopes
Fig. 5. Ža. Measured I2PC scans for E0 s y2.24 and y1.70 eV; and their decomposition into Žb. phase averaged 0 v envelopes; Žc. 1 v envelopes; Žd. and 2 v envelopes. Each component envelope is used to extract the hot electron lifetime, and decoherence of linear and nonlinear polarizations between the optically coupled states, as indicated by the energy level diagrams.
H. Petek et al.r Chemical Physics 251 (2000) 71–86
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Fig. 6. Measured I2PC 2 v envelopes near the d-band maxima for Ža. CuŽ100. and Žb. CuŽ110.. The hole energy for each measurement is given on the right axis. The analysis of these data gives the T202 values in Fig. 5. Particularly slow decoherence is observed at the X 5 point, i.e., the top of the d-bands, on CuŽ100.. A weak signal from the X 5 point is also observed on CuŽ110. due to momentum scattering of the photoelectrons Žalso see Fig. 4..
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w13,41x. The measured values of T1 for CuŽ100. are quite similar to those of the other low index copper surfaces w24,30x, and in agreement with the previous measurements from other groups w3,5,21x. The lifetimes do not decrease monotonically with the Ž E y E F .y2 energy dependence predicted by FLT, but follow different trends when hot electrons are generated by interband or Drude absorption, as first reported in Ref. w3x. The comparison of 0 v envelopes in Fig. 5b also shows that the hot electron population dynamics depend strongly on the excitation mechanism. Although the population decay at E1 s 1.4 eV is well described by single exponential kinetics, Knoesel et al. suggest that the unusually long lifetimes can be explained by a more complex kinetic scheme involving a delayed generation of hot electrons by the Auger recombination of long lived d-holes w5x. If hot electron generation takes significantly longer than their decay, then this rate-limiting process would dominate the time evolution of the population at E1. The mechanism of Knoesel et al.
can be tested quantitatively if the Auger recombination rates are known.
4. Discussion 4.1. Auger recombination rate Although the I2PC scans do not measure the hole lifetimes directly in the same manner as for the electrons, hole scattering processes can be deduced from the decoherence rates. Therefore, the dependence of T202 on the band structure, carrier momentum, and sample temperature can provide information on the Auger recombination. Since the Auger recombination makes a contribution to T202 through the scattering of the holes, it can be determined from the measurements in Figs. 4–7 if other phase scattering processes are known or can be
Fig. 7. Ža. The temperature dependence of the 2 v envelope at the X 5 point. Žb. Plot of ŽT202 .y1 derived from the data in Ža. vs. the sample temperature. Fitting of this data with the Debye model gives the electron–phonon mass enhancement factor l s 0.20 " 0.01.
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neglected. Following the usual convention in photoemission spectroscopy, the phase decay rate of the induced polarization is expressed as a sum of electron and hole scattering 1rT202 s Ge q G h . This assumes that exciton–exciton scattering can be ignored. If 2PP occurs to the inverse LEED states, as in the case for CuŽ100., than the electron has a vanishingly small probability in the bulk, and the contribution to ŽT022 .y1 from Ge is negligible. This is supported by measured lifetimes of the high lying Ž; 4.6 eV. image potential states on clean CuŽ100. where electron lifetimes are ) 2 ps w25x. However, according to the band structure in Fig. 2 this approximation may not be valid for CuŽ111. and CuŽ110.. Fig. 6 shows the 2 v envelopes for CuŽ100. and Ž110., which are used to derive T202 in Fig. 4 according to the fitting procedure in Ref. w30x. A larger contribution from Ge can explain the significantly faster decoherence rates for CuŽ110.. The extent of the electron penetration into the bulk, and therefore 2 .y1 the contribution of Ge to ŽT02 at E2 ; 4.1 eV should decrease in the order CuŽ110. ) CuŽ111. ) CuŽ100. w10x, which agrees with the trend in T202 at E0 f y2.2 eV in Fig. 4. The hole contribution to ŽT022 .y1 can be further decomposed assuming Matthiessen’s rule by expressing G h as a sum of h–h, hole–phonon Žh–p. and hole–defect Žh–d. scattering, i.e., G h s G hh q G hp ŽT. q G hd . The Auger recombination rate can be determined if the h–p and h–d contributions are known. The h–d scattering is an extrinsic contribution related to the crystalline quality of the sample, which in the present experiment could be dominated by surface impurities such as Cs atoms. It is unlikely that the bulk defects in well-annealed, high-purity samples could contribute significantly to the G h under present conditions w42x. The effect of surface defects on phase relaxation is known from the freeelectron like surface state linewidth measurements on alkali atom covered CuŽ111. surfaces, which show that ŽT202 .y1 increases linearly with the parallel momentum k 5 of electrons and impurity concentration w43,44x. If the decoherence length, as defined by the product of carrier velocity and T202 , is comparable to the average distance between the defects, then h–d scattering can make a significant contribution. A clear impurity concentration or k 5 dependence of T202 is not observed for the d-bands probably be-
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cause the photoemission can occur from the bulk where the effect of Cs adsorbates is efficiently screened, and they have a smaller group velocity than the surface states. The contribution from h–p scattering to G h is evident from in Fig. 7, which shows the 2 v envelopes for CuŽ100. measured at the X 5 point for several sample temperatures. According to Fig. 7b, G hŽT . has a linear dependence on the surface temperature in the range 33–300 K. This is in line with the Debye model, which predicts that G hp f 2pl k B T, where l is the electron–phonon mass enhancement factor w7,45x. Fit of the data in Fig. 7b gives l s 0.20 " 0.01 and G hy f s 0.0086 fsy1 at 0 K. This is the same value of l as measured by transport measurements for the conduction electrons at E F in the ²100: direction w45x. Having determined G hp ŽT . and argued why Ge and G hd are negligible, the upper limit for the Auger recombination rate at the X 5 point can be determined from G hh s G h y G hp s 1rT202 ŽT . y 2pl k B T. A lower limit for the lifetime of the d-holes at the X 5 point of 24 " 3 fs is determined from the residual, temperature independent contribution to T202 . The Auger recombination time could be even longer if T202 is limited by the finite resolution of the analyzer, or if Ge and G hd cannot be neglected. The above analysis provides the Auger recombination rate at a specific point in the d-bands; however, the optical excitation generates a broad distribution of d-holes. Below the X 5 point the derived values of ŽT202 .y1 increase faster than preŽ . 2 energy scaling, dicted by the ty1 hy h ; E y EF which is expected since the d-bands contribute much larger scattering phase space than predicted by the free-electron model. Also, the d–d scattering may be considerably more efficient than the d–sp Auger recombination. Therefore, a plausible scenario for the d-hole dynamics is a primary relaxation within the d-bands, whereby the holes float up to the X 5 point, followed by secondary process of d–sp Auger recombination near the X 5 point, which generates the hot electrons w46,47x. Thus, assuming that the relaxation from the X 5 point is the rate-limiting step for the secondary hot electron generation, the rate of 0.021 fsy1 provides a lower limit for simulating the contribution of the Auger recombination to the hot electron dynamics.
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4.2. Analysis of the primary and secondary hot electron generation The remaining part of this section presents quantitative assessment and modeling of the primary and secondary hot electron contributions to the measured I2PC data. The tremendous advantage of the I2PC measurements is that it is possible to distinguish hot electron generation processes involving the phase relaxation of the photoinduced e–h pairs Žcoherent polarization. into an incoherent hot carrier distribution Žprimary generation. from the energy relaxation of the primary carriers Žsecondary generation.. This is possible for two reasons: Ži. the two generation processes have distinct kinetics that can be unraveled through an interferometric measurement; and Žii. with 13 fs pulses, secondary generation on longer time scales appears as a distinct delayed rise in the signal. The simplest Žand often completely adequate. way to model the interferometric two-photon correlation data is to use an optical Bloch equation ŽOBE.
formalism for multilevel transitions w48x. In this model the electron is restricted to three energy levels separated by the photon energy of the exciting laser. The three levels are described by a density matrix with diagonal elements representing the population in each level Ž r 00 s ground state, r 11 s hot electron state, r 22 s detected photoelectron., and off-diagonal elements representing coherences between the levels. The OBEs are a set of coupled differential equations that describe how the levels and coherences evolve in the presence of an external electromagnetic perturbation. The system is closed with a total population normalized to unity if there are no source or loss terms outside of the three-level system, i.e., population of the intermediate level by hot electron avalanche relaxation, or removal of electrons due to scattering into other states. To simulate population of the intermediate level by Auger recombination, the OBE formalism needs to be ‘opened’ such that the total population of the system can exceed unity. This is appropriate if
Fig. 8. Fit of the I2PC measurement at E0 s y1.70 eV to the optical Bloch equation model. The excellent fit indicates that most of the hot electrons are generated directly by optical excitation. Several simulations indicate the appearance of the signal if in addition to the primary generation hot electrons also are generated by the Auger recombination. The OBE analysis places an upper limit of 10% generation of hot electrons by a delayed secondary process at this energy Ž E1 s 1.4 eV..
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r 00 Ž t . ; 1 Žweak excitation., and is necessary since a single set of three levels is monitored, while the excitation occurs for a broad continuum of states that can decay into or out of the level of observation. First, the hole population in the ground state is defined as r hole s 1 y r 00 . It is assumed that the ground state holes have a probability per unit time Thole of undergoing Auger recombination, and that this is the only incoherent decay channel that can repopulate the ground state. Therefore, the time evolutions of r 00 and r 11 are modified by adding terms Er 00 Et Er 11 Et
sŽ .... q
1 T hole
r hole ,
a sŽ .... q
Thole
r hole ,
where Ž . . . . represents the usual terms in the OBE. The consequence of Auger recombination is regeneration of the electron in the ground state and secondary hot electron excitation to the intermediate state. a s 1 means that all of the holes created by photoexcitation from the ground state create an Auger electron in the intermediate level, while allowing a ) 1 is a way of including contributions from holes with energies outside the closed three level system. Since the population of holes is equal to the number of hot electrons excited directly via photons Žneglecting two-photon absorption., the fraction of hot electrons created via Auger decay is defined as arŽ a q 1.. Thus a defines the relative amplitudes of the coherent and incoherent components in the I2PC scans w13x. Although a is defined to describe the Auger recombination, other secondary generation, such as the hot electron avalanche relaxation, should have similar effects on I2PC scans, but with possibly different rates. In order to test for the presence of the secondary hot electron generation process an OBE fit of the I2PC trace for E0 s y1.70 eV in Fig. 4a is shown in Fig. 8. The model is an excellent representation of the data according to the residuals, which are determined mainly by the photoelectron counting statistics. The fit gives a hot electron energy relaxation time of 72.5 fs, which agrees well with previous TR-2PP measurements at this energy w3,5,24x. The best fit is obtained when the hot electrons are generated entirely by the primary Drude absorption. If a
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is a variable parameter in the fit and the secondary hot electron generation time is 24 fs, as expected for Auger recombination, the OBE analysis indicates that the maximum secondary contribution is - 10% of the primary carriers. Fitting the data with the delayed rise and amplitude as variable parameters neither significantly improves the fit, nor suggests that the secondary contribution is any larger. Fig. 8 also shows several calculated I2PC traces using the same phase and energy relaxation parameters as above, but with secondary hot electrons contributing from 20 to 60% of the total hot electron population. The secondary contribution increases the incoherent signal amplitude relative to the coherent oscillations and leads to significant deviations from single-exponential decay kinetics that are clearly at odds with the experimental data. However, below E1 s 1.4 eV the secondary contribution acquires a significant amplitude, which will be analyzed in future work w49x. This analysis demonstrates that hot electrons with E1 G 1.4 eV are generated predominantly by the optical excitation and that the measured T1 values represent the population lifetimes. 4.3. I2PC scans in the secondary generation limit The OBE analysis strongly supports the view that the hot electrons above the interband excitation threshold are generated almost exclusively by Drude absorption. Ref. w5x takes the diametrically opposite view, where the anomalously long hot electron lifetimes above the d ™ sp excitation threshold are attributed to the production of hot electrons entirely by delayed Auger recombination of the d-holes. In this section, the present analysis is extended to model these two limiting scenarios in order to emphasize Ži. the importance of measuring hot electron dynamics with ultrashort pulses; and Žii. the power of the ITR-2PP technique for unraveling the complex kinetics of hot electron generation and relaxation. Fig. 9 presents several calculated interferometric and phase averaged two-pulse correlation measurements that simulate the dynamics of hot electrons at E1 s 1.63 eV. In Ref. w5x, a phase averaged two-pulse correlation ŽPA-2PC. is measured at the same energy with a 50 fs pulse width laser. The I2PC traces in Fig. 9a and b are simulated with 13 and 50 fs pulse
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Fig. 9. OBE simulation of interferometric and phase averaged two pulse correlation measurements assuming excitation with Ža. 13 fs and Žb. 50 fs pulses; Ža. and Žc. for entirely direct and Žb. and Žd. delayed generation mechanisms. The phase averaged envelopes for the two models with Že. 13 fs and 50 fs pulse excitation.
H. Petek et al.r Chemical Physics 251 (2000) 71–86
widths in order to contrast the results of the present experiment with those in Ref. w5x. According to the OBE fit of the data, the energy relaxation time at E1 s 1.63 eV is T1 s 51.7 fs. In Ref. w5x, the PA-2PC is modeled by the generation of hot electrons entirely through the Auger recombination with a rise time of 35 fs, and an intrinsic hot electron lifetime of 15 fs, which is in line with the predictions of FLT w5x. Fig. 9c and d show the simulated I2PC scans assuming the Auger recombination kinetic mechanism and laser pulse widths of 13 and 50 fs, respectively. Fig. 9e and f show the phase-averaged 0 v envelopes ŽPA2PCs. for the simulations in Fig. 9a–d. Both the interferometric and phase averaged twopulse correlations show a dramatic contrast between the two excitation mechanisms in the case of 13 fs excitation. However, with 50 fs excitation the PA2PCs are indeed qualitatively similar, and since phase averaged measurements are blind to the nature of the excitation process, it is difficult to provide certain evidence for either mechanism even in the limiting cases. Thus, in order to establish the kinetic mechanism it is imperative to perform measurements with comparable or shorter laser pulse widths than the dynamics under investigation. Since I2PC measurements can distinguish the coherent and incoherent excitation processes, it still may be possible to extract quantitative kinetic rates with 50 fs pulse excitation provided that the fitting model correctly describes the excitation mechanism.
5. Conclusions The conclusions that can be drawn following this detailed analysis of the hot electron and hole relaxation in copper are: Ži. hot electron excitation above ; 0.9 eV proceeds mainly by the direct interband d ™ sp and intraband sp ™ sp excitation, rather than through the incoherent decay of high energy electrons and holes; Žii. the hot electron lifetimes are significantly longer at a given energy when the photoexcitation is through a direct interband process rather than an indirect process; Žiii. the lower limit for the d-hole lifetime at the top of the d-bands is 24 " 3 fs. The conclusion that the excitation process influences the subsequent hot electron energy relaxation is contrary to the sudden approximation in
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photoemission, which assumes that the photoinduced carrier dynamics are independent of the history of their creation. The Auger spectra of core holes in copper provide a precedent for the breakdown of the sudden approximation in observation that the energy of photoelectrons as high as 200 eV above the threshold for a specific channel influences the decay of the corresponding holes w50x. Furthermore, there is a significant disagreement between the highest level theory and photoemission experiments regarding the energy of the d-bands, which is attributed to significant self-energy corrections due to the exchange and correlation, that are not accounted for in calculating the band structure of copper w51x. The imaginary part of the self-energy associated with the interaction of the hot electron with its hole may explain why the hot electron lifetime depends on the nature of the excitation process. The energy and momentum may not uniquely define the carrier lifetime, as assumed in quasiparticle theories, if the lifetime also depends on the history of carrier excitation. Thus, ITR-2PP measurements may provide the means to study the ultrafast many-body Coulomb correlations in photoexcitation and relaxation of carriers in metals.
Acknowledgements The authors thank N. Moriya, S. Matsunami, and S. Saito for technical support; and NEDO International Joint Research Grant for partial funding of this project. M.J. Weida wishes to thank the National Science Foundation and the Center for Global Partnership for support ŽNSF grant INT-9819100..
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