Desorption induced by hot electrons: wave packet calculation of CO on Cu surfaces

Surface Science 470 (2001) 293±310

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Desorption induced by hot electrons: wave packet calculation of CO on Cu surfaces D. Bejan a,1, G. Rasßeev a,*, M. Monnerville b b

a Laboratoire de Photophysique Mol eculaire du CNRS, b^ at. 210, F-91405 Orsay Cedex, France 2 Laboratoire de Dynamique Mol eculaire et Photonique, CNRS (URA 779), Centre d'Etudes et de Recherches Lasers et Applications, Universit e des Sciences et Technologies de Lille, b^ at. P5, 59655 Villeneuve d'Ascq, France

Received 17 April 2000; accepted for publication 4 September 2000

Abstract A quantum mechanical photodesorption model, valid for metallic substrates and sub-picosecond laser pulses, is presented. It takes into consideration the photodesorption coordinate and models the metal hot-electron mediated desorption by a three electronic states: an ionic state of the adsorbate and two e€ective states representing the continuum of the metal. This multiple-state picture allows the sharing of the ¯ow of energy injected by the laser between the adsorbate and the substrate. For the ®rst time, the present modeling introduces the hot electrons of the metal through an optical potential based on the kinetic model developed earlier by the authors. This potential, and the resulting desorption yield, depend on the laser ¯uence. For CO on Cu(1 0 0) or Cu(1 1 1), the results are in fair agreement with the experimental ®ndings. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Computer simulations; Desorption induced by photon stimulation; Low index single crystal surfaces; Metallic surfaces; Copper; Carbon monoxide

1. Introduction Desorption dynamics of molecules absorbed on metal surfaces is a ®eld rising many questions to be answered by theory and experiments. Desorption induced by picosecond and femtosecond laser

* Corresponding author. Tel.: +33-16915-8259; fax: +3316915-6777. E-mail address: [email protected] (G. RasËeev). 1 Permanent address: Faculty of Physics, University of Bucharest, P.O. Box MG 11, Bucharest, Magurele, Romania. 2 The Laboratoire de Photophysique Mol eculaire is a ``Laboratoire Associ eal' Univerist e de Paris-Sud''.

pulses has been studied for several systems: NO/ Pd(1 1 1) [1,2], NO/Pt(1 1 1) [3,4], CO/Cu [5,6], O2 / Pd(1 1 1) [7,8], O2 /Pt(1 1 1) [9], NH3 /Cu(1 1 1) [10]. Some general properties have been observed in these studies including a low desorption yield per excitation event, a non-linear dependence of desorption yield with the laser ¯uence and a very short time (less than 1 ps) response of the adsorbate±substrate system. For comparable ¯uences one observes di€erent behavior in picosecond and nanosecond regimes. Namely a picosecond laser enhances signi®cantly the desorption yield and gives rise to a picosecond response to the laser excitation. Also, ®nal state energy distribution of NO desorbed from Pd(1 1 1) [1] exhibits non-thermal

0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 0 ) 0 0 8 6 8 - 2

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D. Bejan et al. / Surface Science 470 (2001) 293±310

rotational population and a high degree of vibrational excitation corresponding to a very high mean rotational and vibrational temperatures. On the contrary, the translational motion is excited only slightly and corresponds to a relatively modest translational temperature. These observations are inconsistent with a conventional thermal or photochemical mechanism for desorption, and have been attributed to a desorption process driven by the high degree of substrate electronic excitation. There are also experimental and theoretical studies on molecules adsorbed on oxidized surfaces [11±14], where, due to a less ecient relaxation to the substrate, the photodesorption yield is much higher. On the theoretical side, many models were developed but have been tested only on few systems: CO/Cu [15±20], NO/Pt(1 1 1) [21±28], NH3 , ND3 / Cu(1 1 1) [29±33] and O2 /Pt(1 1 1) [34,35] where the photodesorption is coupled to the photodissociation. For oxidized substrates the involved theoretical model [11±14] corresponds to a desorption mechanism similar with the one considered for metallic substrates. The recent theoretical models include the substrate electronic excitation as a primary step (®rst suggested by Gadzuk et al. [4]) followed by nonBorn±Oppenheimer coupling between electrons of the metal conduction band and an excited electronic state of the adsorbate. This coupling is facilitated if a resonant state of the adsorbate is present in the energy domain of the excited electrons. The energy ¯ow between the di€erent electronic states mediates the excitation of the nuclear degrees of freedom of the adsorbate and sometimes the photodesorption. Presently these processes are treated by classical, semiclassical and quantum mechanical time independent or time dependent theoretical methods. The classical approach [36] distinguishes two regimes. If the width d of the adsorbate resonance is narrow relative to its energy measured from the Fermi level EF , then the lifetime is long enough to de®ne unambiguously an excited state. This mechanism can be described as multiple excitations between two distinct levels (known as desorption induced by multiple electronic transitions [37,38]). Conversely if the resonance width d is of

the same order of magnitude as the energy di€erence between its maximum and the Fermi level EF (the lifetime is short) then low energy electron± hole pairs dominate the photodesorption process. The mechanism can be described through a temperature independent friction force [15,16,36,39] that models the behavior of these electron±hole pairs. These two models reproduce some of the experimental observations: non-linear yield dependence with the laser ¯uence and internal excitation of the desorbed molecules. However the overall time of the process is slower than the one observed experimentally [16]. The quantum mechanical approach add to the classical models speci®c quantum e€ects such as for example ``wave packet squeezing'' [40]. Commonly the quantum models take into account only two states: a ground and an excited state of the system; they will be referred hereafter as ``two states models''. The time dependent propagation is performed using either the density matrix or wave packet approaches. For open dissipative systems, the reduced density matrix formalism governed by the Liouville±von Neumann equation of Linblad form gives a proper description of excitations and relaxation involved in the desorption process [21± 23,41,42]. But the method is numerically time consuming and is inherently a phenomenological, non-microscopic theory. The wave packet propagation is simpler to implement [24,25,29,30,43] and may provide a microscopic description of the process. There is a connection between the wave function and the density matrix formalism. As shown by Saalfrank [22], one can devise methods, based on stochastic wave packets (i.e. by repeated solutions of Schr odinger equation with non-hermitian Hamiltonians) that in some cases are computer memory saving alternative to density matrix propagation schemes. None of the presently implemented wave packet formulations, calculates the dependence of the desorption yield with the laser ¯uence. The physical problem at hand is complicated due to the presence of two continua one electronic and the other nuclear. In such a situation one can ask the following questions: Does the two-state model reproduce correctly the coupling between the electrons of the metal and the adsorbate?

D. Bejan et al. / Surface Science 470 (2001) 293±310

Should one introduce more elaborate models and what are the extensions one should implement? In the literature there is an attempt to model the electronic and nuclear degrees of freedom by a two-dimensional wave packet approach [44]. This approach gives interesting results but is time consuming because the large di€erence of masses between electrons and nuclei generates the need of very di€erent time grid in the two degrees of freedom. In fact, the non-Born±Oppenheimer electron±nuclei coupling is localized in space and there is no need of such heavy procedure. In the wave packet formalism, the coupling/ dissipation to the electron bath is introduced usually through a complex optical term in the Hamiltonian (see e.g. Refs. [19,25,30]). But in all the published models the in¯uence of the excited (hot) electrons was introduced only by using an e€ective electronic temperature obtained as a solution of the standard two-temperatures model [45]. This model considers the metal composed by two subsystems, the electrons and the phonons. Each subsystem is supposed to be in a local equilibrium and ful®l the corresponding distribution function (Fermi±Dirac for electrons and Bose± Einstein for phonons) with an attributed e€ective temperature. As we already argued in a preceding publication [46], theoretical calculations and experimental results [47,48] show that in the subpicosecond regime the distribution function of the hot electrons system has a form di€erent from the Fermi±Dirac distribution function and therefore the associated temperature is not well de®ned. In that publication [46] we proposed a temperaturefree kinetic model for hot-electrons that models very well the distribution of these electrons at short time scales. In the present paper, we propose a one-dimensional three-electronic state model for photodesorption of molecules from metal surfaces. This model introduces for the ®rst time the hot electrons of the metal using an appropriate nonequilibrium distribution function [46]. Our model is developed following a mechanism of photodesorption close to the ideas of Gadzuk et al. [4]. The laser excites ®rst the electrons of the metal creating a bath of hot, non equilibrium, electrons that will scatter into an unoccupied valence electron reso-

295

nance of the adsorbate forming a temporary negative ion. After the neutralization of the negative ion, the system returns to the ground state of the adsorbate and an excited state of the substrate located in the conduction band of the metal. In the present work this mechanism is modeled by three diabatic electronic states coupled by electrostatic interaction of non-Born±Oppenheimer type. Two of the electronic states correspond to the neutral adsorbate bonded to the excited substrate (the metal electrons are excited at di€erent energies) and one state corresponds to the negative ion of the adsorbate bonded to the unexcited metal. The electrostatic interaction determines the electron hopping between states. Our three-state scenario is interesting because it allows, through an optical potential corresponding to an open system, the laser excitation and the e€ective dissipation of the substrate energy in the metal. The presence of three states (instead of two) allows a realistic microscopic interaction through a non-Born± Oppenheimer coupling between states and a splitting of the ¯ow of the accumulated energy between the adsorbate and the substrate. We apply the present model to CO molecule adsorbed on Cu(1 1 1) or Cu(1 0 0) substrates, systems rather well investigated both experimentally [5,6] and theoretically [15±17,20,49]. The CO/ Cu systems has the following advantages: at low coverage the CO molecules stay mainly in on-top position on Cu(1 1 1) and Cu(1 0 0) surfaces making the system appropriate for a one dimensional study; CO is a stable molecule on nearly all metals so the desorption process do not interfere with the dissociation one; the bonding CO±Cu is weak being at the limit between physisorption and chemisorption and so it is interesting for catalytic modeling. Such systems allow adsorption and desorption at low energy expense thus facilitating the reactions at metallic surfaces.

2. Model In this section we will present the one-dimensional potential energy curves (PECs), the model Hamiltonian including the optical potential, the

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time dependent propagation method and the ®nal state analysis giving the photodesorption yield. 2.1. Potentials The study of photodesorption via excitation of hot electrons necessitates the modeling of two continua: one electronic and the other nuclear. The electronic continuum, considered structureless here, is associated with the conduction band in the metal and two electronic con®gurations (or occupations): either corresponding to neutral metal and a neutral discrete state of the adsorbate or to the metal having a hole and a negative ion resonance of the adsorbate. The nuclear continuum corresponds to the desorption coordinate z, that describes the motion of adsorbate center of mass normal to the surface. An adsorbed diatomic molecule has ®ve other degrees of freedom, namely: one internal vibration, two hindered rotations and two hindered translations. Neglecting these degrees of freedom imply that the adsorbed molecule is structureless, having kinetic energy only in the z-direction. This approximation is valid for CO adsorbed in on-top position normal to the surface that is the most stable adsorption site of CO/Cu systems. As the two electronic states correspond to continuous electronic energy, a representation of the energy relative to the nuclear desorption coordinate z will show two in®nite series of parallel curves. Each series is non-interacting, resulting from a diagonalization of a sub-matrix of electronic states. The interaction between the two series of electronic states is included through a diabatic picture. Following Head-Gordon and Tully [39], we replace these continua of electronic states with a ®nite number of e€ective discrete PECs. Three PECs are retained in the present model and represented in Fig. 1. Two of them, V1 and V3 are parallel and represent the neutral molecule in its ground state bonded to the neutral metal in di€erent excitation states. They are parallel to the ground state potential curve of the adsorbate±substrate system. The third one, V2 , corresponds to the negative molecular ion state bonded to the metal in its ground state. This representation of the electronic continuum corresponds to a discrete sampling and restricts the

Fig. 1. PECs for photodesorption of CO from Cu(1 0 0) or Cu(1 1 1) surfaces. E1 , E2 and E3 are the asymptotic desorption energies.

model to the most important aspects of the desorption mechanism. The ground state PEC, parallel to V1 and V3 , is cut along the desorption coordinate z in an empirical three dimensional (x,y,z) gas±surface interaction potential developed by Tully et al. [50] for CO on Cu(1 0 0). It is an atom±atom potential of the following form: Vground …z† ˆ ‰Vground …r†Šrestricted to " X ˆ Vi …rC ; rO ; ri † i

PEC

#

‡ VCO …jrC ÿ rO j†

;

…1†

restricted to PEC

where rC and rO are vectors de®ning the position of carbon and oxygen atoms and ri is the coordinate of the ith copper atom. The term Vi …rC ; rO ; ri † represents the interaction of CO molecule with the ith Cu atom and VCO …jrC ÿ rO j† describes the direct interaction between the C and O atoms. The analytic expressions and the parameters correspond-

D. Bejan et al. / Surface Science 470 (2001) 293±310

ing to the present potential are given by Tully et al. [50] and will not be reproduced here. The above analytical expression (1) was ®rst used to generate three-dimensional potential energy surfaces with 768 Cu atoms …16  16  3† constructing a (1 0 0) surface with one CO molecule adsorbed on it. Then the same analytical expressions and constants were used to construct a similar CO±Cu system but corresponding to (1 1 1) surface [51]. Locally, the dense Cu(1 1 1) surface has C3 symmetry while the less dense Cu(1 0 0) surface has C4 symmetry. The vibrational energies given by this potential for both surfaces agree well with the experimental ®ndings. From these threedimensional potential surfaces we now generate a cut corresponding to the photodesorption coordinate z. The binding energy of the molecule to the (1 0 0) and (1 1 1) surfaces is 0.59 eV in agreement with calculation by Tully et al. [50] for CO on a 6  6  3 Cu slab, and with the experiment of Tracy [52]. Considering our one coordinate model and the corresponding PEC given in Eq. (1), the di€erence between Cu(1 0 0) and Cu(1 1 1) surfaces is neglected here and we used identical Vground for the two surfaces. The ground state of the adsorbate±substrate system corresponds to the neutral molecule bonded to the neutral metal in its ground state. In this metal ground state the electrons occupy all the energy levels up to the Fermi level EF (at T ˆ 0 K) while in the excited states of the metal some electrons occupy energy levels above EF . The ground state of the adsorbate±substrate system is represented by the potential Vground and to its desorption asymptote we associate the zero energy of our model. Moreover we make a correspondence between this zero energy and EF of the metal. Each curve, parallel to Vground and having the asymptote at some energy E P 0, corresponds to the neutral molecule bonded to the metal having electrons excited on the energy level EF ‡ E. On Fig. 1 we label V1 the potential curve of the neutral CO molecule bonded to the metal having one electron excited on the energy level EF ‡ hm, where hm is the energy of the excitation photon. In the photodissociation literature this V1 state is called dressed state and it is positioned to have hm as asymptotic energy. V3 represents the neutral molecule bonded

297

to the metal with an electron on some level situated between EF and EF ‡ hm. The energy interval DEn ˆ ‰V1 ÿ V3 Šz!1 is the di€erence between the asymptotes E1 and E3 of V1 and V3 PECs. The negative ion resonance PEC V2 corresponds to a charge transfer from the ground state of the metal to the empty 2p orbital of CO molecule. This transfer was veri®ed by an analysis of the charge density obtained from an ab initio calculation [53], using a simple neutral linear cluster made from three atoms (C, O, Cu). The O atom is negatively charged while the C atom and the Cu surface are positively charged. This excited state charge distribution is modeled by adding four charge±charge image Coulomb terms to the ground state potential. With the molecular axis perpendicular to the surface one obtains the following potential (see Fig. 1):  de2 Ce2 1 ‡ ÿ V2 …z† ˆ Vground …z† ÿ 2zeff 4…zeff ‡ zcm † 2  1 2 ‡ ÿ ‡ DEion ; 2…zeff ‡ rCO † 2zeff ‡ rCO …2† where de is the fraction of the electron charge transferred from metal to the adsorbate; C is a constant connected to the dipole moment in the ionic state; zeff ˆ z ÿ z0 , z is the C-surface distance and z0 a correction that takes into account the position of the metal positive charge as described by the jellium model [54]; zcm is the distance between C atom and CO center of mass, rCO is the equilibrium distance between C and O atoms. The ®rst term in the second parenthesis corresponds to C±Cimage interaction, the second one to O±Oimage interaction and the third to the sum of C±Oimage and O±Cimage interactions. We checked that the potential sensitivity to actual values of these parameters is very weak. Compared to the V1 and V3 PECs, the potential V2 has a di€erent z dependence and a displaced equilibrium position (see Fig. 1). The standard procedure positions this state relative to the vacuum level of the metal at W ÿ Ea where W is the metal work function and Ea is the electron anity of CO. In such a model the position of the V2 PEC is ®xed and the width of the resonance is

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D. Bejan et al. / Surface Science 470 (2001) 293±310

completely disregarded. The parameter DEion ˆ ‰V2 …z ˆ z2eq † ÿ Vground …z ˆ z0eq †Š positions the equilibrium value of the V2 potential curve of the negative ion resonance in the electronic continuum of the metal. For CO/Cu(1 1 1) system the 2p resonance was found experimentally [55,56] to have its maximum located at about ER ˆ 3.3 eV above the Fermi level EF . Ab initio calculations using a scattering method (frozen core static exchange [57]) performed for COÿ in the gas phase give for this resonance a width DR of 1.25 eV (3.3 fs) [53]. This is in reasonable agreement with the experimental value of 1.9 eV (2.2 fs), obtained by Rogozik et al. [58] using the inverse photoemission technique for CO/Cu(1 1 0), or with the value of 1.2 eV (3.4 fs) found by Knoesel et al. [59] for CO/Cu(1 1 1) by two-photon photoemission experiment. Experimentally, the lifetime of the excited state (corresponding to the above mentioned width) is estimated between 1 and 20 fs [59]. In our model this lifetime is in¯uenced by two factors: the relative position of the couplings Qij determined by the crossing points between the potential curves V1 , V2 and V2 , V3 (see Fig. 1) and the strength of the couplings Qij . The ®rst is governed by DEn ˆ jV1 ÿ V3 jz!1 while the second measures the eciency of the jump between potential curves governed by the coupling Qij (see Eq. (4) below). To the energy interval DEn we associated a residence time, spent by the molecule on the ionic state V2 , using the harmonic oscillator approximation (see Fig. 2). This time may be roughly considered to be the lifetime of this state. More precisely, suppose that the motion in the ionic state V2 is harmonic with the fundamental frequency x (see Fig. 2). Now let us name zmax the amplitude of this z motion and Dz ˆ z12 ÿ z23 the distance traveled by the system in V2 until its transfer to V3 . Then the time spent in the p  excited state V2 is sR ˆ 2=x arcsin… Dz=zmax †, obtained from the displacement in harmonic motion approximation Dz ˆ zmax sin2 …xt=2†. The excitation energy hm is di€erent in the two experiments on CO/Cu(1 1 1) and CO/Cu(1 0 0) [5,6] that we want to simulate and consequently, our PECs should have di€erent positioning. However, for both systems we use the same DEn and relative positioning of V1 , V2 and V3 . We suppose

Fig. 2. Zoom of the curves crossing region including the parameters used for the calculation of the residence time in the ionic state.

that these three PECs are only globally displaced relative to the ground state, displacement corresponding to the laser excitation energy in a particular experiment. In other words the di€erence between Cu(1 0 0) and Cu(1 1 1) surfaces is neglected and the laser/matter interaction is not strong enough to signi®cantly modify the shape of our potential curves. Consequently in all the calculations described below DEn ˆ 1800 cmÿ1 that corresponds, in the harmonic oscillator approximation, to a residence time (sR ) of about 18 fs. Above we made a correspondence between the asymptote of the ground state and EF . However, as EF does not depend on the desorption coordinate z, each point of Vground corresponds to EF . As all the excitations take place from the equilibrium position of Vground situated at ÿ4700 cmÿ1 (0.59 eV), the potential V2 represented in Fig. 1 should have the equilibrium position in the interval ‰…ER ÿ 0:59 eV† ÿ DR , …ER ÿ 0:59 eV† ‡ DR Š. This interval accounts for the di€erence between the zero energy and the excitation point corresponding to the equilibrium position of Vground and the width of the resonance including the tail.

D. Bejan et al. / Surface Science 470 (2001) 293±310

The V1 , V2 and V3 positions were chosen to ful®l the following supplementary conditions: the intersection between V1 and V2 PECs should be higher in energy than the intersection of the V3 and V2 and close to the energy of the zero vibration level of V1 PEC; the time spent in the ionic state, calculated in the harmonic approximation, should be in agreement with the experimentally measured resonance lifetime.

299

Table 1 summarizes all the parameters of the potential curves and of the interactions used in the next sub-section. 2.2. Nuclear time dependent wave equation Having de®ned above the PECs means that implicitly we have integrated out the electronic

Table 1 Computational parametersa : PECs, Vi int …t† optical potential and Qij interactions, constants of CO/Cu system (EF ,W,D) and time and space propagation parameters Parameter

Value

V2 negative ion potential, (see Eq. (2))

de C z0 re dcm DEion ˆ ‰V2 ÿ Vground Šzˆzeq

e 0.5  0.75 A  1.125 A  0.643 A 1:56 eV

Vi int …z; t† potential

s0 t0 [5] sl [5] t0 [6] sl [6] c1 z0 p0 E0 [5] E0 [6] c1 c2 c3

5:0 fs 50.0 fs 50.0 fs 80.0 fs 80.0 fs ÿ1 3.0 A  3.0 A

‰7:5  10‡8 E20 Š sÿ2 2:3±3:3  108 V mÿ1 1:8±2:8  108 V mÿ1 3.0 5:1  1012 sÿ1 0.2

Interaction parameters between electronic curves (see Eq. (4))

C12 C23 z12 z23 a12 a23

0:2 eV 0.2 eV  1.885 A  1.761 A 2 0.040 A 2 0.025 A

Fermi level, work function, desorption energy, and PECs positioning

EF Work function (W) CO electron anity (Ea ) CO desorption energy (D) DEn ˆ ‰V3 ÿ V1 Šz!1

7.0 eV 4.5 eV 1.5 eV 0.59 eV 0.223 eV

Time and space propagation parameters

dt t1 dz z0 za z1

0.1 fs 819.2 fs  0.011 A  1.000 A  5.000 A  5.000 A

a

The parameters of the ground state potential (Vground ) are listed in the paper by Tully et al. [50].

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D. Bejan et al. / Surface Science 470 (2001) 293±310

degrees of freedom. The system evolves on these three PECs and the hot electrons and photon interact with it through Vi int …z; t†. 2.2.1. Wave function, hamiltonian and wave equation Let W…z; t† denote the nuclear state wave function vector of the system at time t, a column vector built from three components: 0 1 w1 …z; t† …3† W…z; t† ˆ @ w2 …z; t† A: w3 …z; t† Assuming the orthonormality of the electronic eigen states, the probability of ®nding the system in the electronic state ``i'' at time t, within an interval dz around z, is simply proportional to the square of 2 the wave function jwi …z; t†j dz. As explained in the preceding section (see also the Fig. 1) the states j1i …wave function w1 …z; t†† and j3i …wave function w3 …z; t†† represent the neutral CO molecule bonded to the metal in di€erent excitation states while the state j2i …wave function w2 …z; t†† represents the negative ion CO resonance bonded to the metal in its ground state. The states j1i and j3i are noninteracting and discretize continuum of the metal states with a bounded neutral CO molecule. A residual non-Born±Oppenheimer interaction (see Eq. (4)) remains between these states and the negative ion resonance state of CO, j2i. With this de®nition of the wave vector (3) the time dependent Schr odinger equation for our open, dissipative system reads: 0 1 20 1 w1 …z; t† H^1 0 0 dB C 6B C ih @ w2 …z; t† A ˆ 4@ 0 H^2 0 A dt w3 …z; t† 0 0 H^3 0 int 13 Q12 …z† 0 iV1 …z; t† B C7 ‡ @ Q21 …z† iV2int …z; t† Q23 …z† A5 0

0

1

w1 …z; t† B C  @ w2 …z; t† A

Q32 …z†

iV3int …z; t† …4†

w3 …z; t† where in Eq. (4) for purposes of clarity we have written Hamiltonian matrix as a sum of two ma-

trices. The diagonal elements in the ®rst matrix are H^i ˆ ÿ…h2 =2l†Di ‡ Vi , i 2 …1; 3†, describing the motion on each uncoupled diabatic electronic state (see Fig. 1). The two terms correspond to the kinetic and potential energy operators in the Hamiltonian and l is the reduced mass of the system; Vi int …z; t† is an optical potential including the laser excitation and the coupling to electronic bath obtained by analogy to our hot electrons model [46]; Qij …z† describes the electrostatic coupling between the states i and j (in adiabatic representation this corresponds to non-adiabatic coupling). It allows electron hopping between states. If one calculates the non-adiabatic interaction in adiabatic picture by ab initio methods including electron correlation, then one obtains a bell shape interaction [60] that can be modeled using a Gaussian form. One knows that in diabatic picture as the one we use here (see e.g. Ref. [61], Fig. 10) this interaction is not localized and have a smooth variation. However, because out of the interaction region the in¯uence of Qij …z† is negligible and the diabatic and adiabatic states are identical, for simplicity we take here a localized interaction. Precisely this interaction Qij …z† is centered at the intersection point zij between the i and j PECs and have a Gaussian 2 form: Qij …z† ˆ Cij exp‰ÿ…z ÿ zij † =a2ij Š, where Cij and aij are parameters de®ning the strength and the half width of the interaction, respectively (see Table 1). Such a form of the electrostatic interaction was already used by Stromquist and Gao [34]. 2.2.2. The optical interaction potential Vi int The interaction potential Vi int …z; t† models phenomenologically the dynamics of the excited electrons of the metal in the desorption process. Before laser action, the states associated to the PECs V1 and V3 are scarcely populated while that associated to V2 is empty. As V1 and V3 represent states of the excited CO±Cu system, one can take the probabilities to populate these states equal to the occupation probabilities of the E1 , E3 levels of the clean metal (see Fig. 1). When the laser is on, electrons are excited on E1 and E3 levels from levels below EF and, through the electrostatic coupling between the states, the population of the negative ion resonance j2i will then rise also. After laser extinction, the electrons decay from the ex-

D. Bejan et al. / Surface Science 470 (2001) 293±310

cited levels to levels close to EF and the rates of these decreasing populations is related to the lifetime of the hot electrons. In order to describe the behavior of the hot electrons we developed a simple kinetic model [46] that includes an excitation term due to light absorption and a deexcitation term due to electron± electron collisions: df …Ei ; t† ˆ Pex ÿ Pdeex ; dt

…5†

where f …Ei ; t† is the electrons non-equilibrium distribution function, Pex is the probability of electron transition, in the time unit, from the energy level E ˆ Ei ÿ hm ( 6 EF ) to Ei ( P EF ). Considering a Gaussian temporal form of the laser pulse, Pex reads: ! ÿ2…t ÿ t0 †2 …6† …1 ÿ f …Ei ; t††: Pex ˆ p0 t exp s2l Here p0 is the initial transition probability of the electron de®ned by the following equation: p0 ˆ

e2 E20 fFD …E†jMEi ;E j2 ; 4p2 h2

…7†

where e is the charge of the electron, E0 is the modulus of the electric ®eld intensity, fFD …E† is the Fermi±Dirac distribution of the initial level having the energy E ( 6 EF ) and MEi ;E is the transition probability. Eq. (7) was deduced in the approximation of a dipole electromagnetic transition between two resonant discrete levels (see e.g. Ref. [62]). In Eq. (6), t0 and sl are the time parameters of the Gaussian shape of the pulse and …1 ÿ f …Ei ; t†† is the probability that the energy level Ei (where the electron is excited) is unoccupied. In Eq. (5), Pdeex is the probability of electron energy loss in the time unit caused by electron±electron collisions. Pdeex ˆ

f …Ei ; t† : si

…8†

The electron±electron collision time si is given by the Fermi liquid theory [63]: s i ˆ s0

EF2 …Ei ÿ EF †

2

;

…9†

301

where s0 is a characteristic time, particular for each metal and Ei is the corresponding excited level of the electron in the metal, Ei P EF . This simple model does not include secondary electrons (electrons that deexcite and are reexcited by the laser), electron±phonons collisions and electron±surface state interaction. To include the hot electrons of the metal in our photodesorption model, we add an optical potenodinger Eq. (4). Then Vi int acts tial Vi int in the Schr as a population provider constructed much in the same manner as the non-equilibrium distribution function of hot electrons (see Eq. (5)). However in Vi int we try to take into account, in an e€ective way, the neglected interactions discussed above. The explicit expression of Vi int …t† is: Vi int …z; t† ˆ ‰Vi ex …t† ÿ Vi deex …t†Šhz0 …z†;

…10†

where 2

ex

Vi …t† ˆ hc1 …p0 t ‡ c2 † exp

ÿ2…t ÿ t0 † s2l

! …11†

is the excitation part of the interaction potential similar to Pex and p0 , t0 , sl are the parameters used in the Eqs. (6) and (7). Vi ex …t† depends on laser ¯uence through p0 (see Eq. (7)). The deexcitation part of the potential reads: Vi deex …t† ˆ c3

h ; si

…12†

where c3 is a constant. The constants c1 , c2 and c3 in Eqs. (11) and (12) are ®tted to give results in agreement with the experiment. More generally s0 and MEi ;E in Eqs. (9) and (7) where sucient to characterize the interaction only for a simple excitation by laser absorption and deexcitation by electron±electron interaction. But there are other interactions in our system that are taken into account in an e€ective and empirical way through the parameters c1 ; c2 and c3 . For example the multiphoton excitation and the secondary electrons contribute substantially to the excitation term while the electron±phonon, electron±surface state interactions contribute to the deexcitation term. Presently there is no simple working model for these interactions so we introduce them empirically.

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Finally, in Eq. (10), hz0 …z† is a smooth cuto€ function of the Fermi±Dirac type de®ned by: hz0 …z† ˆ

1 ; 1 ‡ exp…c1 …z ÿ z0 ††

…13†

where z0 and c1 are constant parameters chosen to give a hz0 function that becomes zero close to the analysis point za . This cuto€ function hz0 was introduced because the rest of the Vi int …z; t† is entirely built from characteristics of the metal and does not depend on z, the metal adsorbate distance. At analysis point za we consider CO as desorbed and decoupled from the metal. So, an event that takes place in the metal can no longer in¯uence the CO molecule and the hopping probability of the metal electrons on/from the adsorbate becomes negligibly small with limz!za ‰Vi int …z; t†Š ˆ 0. Consequently the neutral CO or negative ion of CO are in the vacuum and have their own gas phase characteristics (see also Section 4). If one compares Vi int …t† with the above expressions of distribution function f …Ei ; t† (Eq. (5)), one immediately sees the proportionality: Vi int …t†  df …Ei ; t†=f …Ei ; t†. Therefore Vi int …t† corresponds to a normalized form of the derivative of the distribution function because only this form can be connected to the Schr odinger equation where the wave function is normalized. Note also that, due to the occurrence of secondary electrons, we have omitted in Eq. (11) the term …1 ÿ f …Ei ; t††. With the time dependent interaction potential Vi int …z; t† included, the Eq. (4) is a time dependent Schr odinger equation in the interaction picture representation [64]. It has to be solved by a propagation scheme able to handle time dependent Hamiltonians like split operator propagator described below. 2.2.3. Initial conditions and propagation of the wave equation We assume that at t0 the system is initially in the state j1i and w1 …z; t0 † is the ground vibrational eigen function of the Hamiltonian H1 …z†. As this state represents the neutral molecule bonded to the metal with electrons excited on the energy level E1 ˆ EF ‡ hm, the initial population of this state equals the probability f …E1 ; t0 † that the metal has

electrons on this energy level. During the time evolution (numerical propagation) of our system this population is recalculated continuously using the formulas of the preceding section. The wave packet in Eq. (4) is discretized on a one-dimensional grid in z. The temporal propagation is performed using the split-operator (SO) method [65,66], following a standard scheme: W…z; t ‡ dt† ˆ exp…ÿiVdt=2h† exp…ÿiTdt=h†  exp…ÿiVdt=2h† W…z; t†:

…14†

Here V and T are the potential and kinetic energy operator matrices appearing in Eq. (4). The operator involving the potential energy part is calculated in coordinate space while the operator containing the kinetic energy part is evaluated in momentum space via fast Fourier transform. 2.3. Final state analysis When performing the asymptotic analysis we can formally write the total wave function as a sum of two contributions: W…z; t† ˆ hza …z†W…z; t† ‡ …1 ÿ hza …z††W…z; t† ˆ WI ‡ WA ;

…15†

where hza …z† is a smooth cuto€ function de®ned in the Eq. (13) above and za is the border between the internal and asymptotic regions. The desorption probability is calculated as the norm of the asymptotic wave function: Z 1 jWA …z; t†j2 dz: …16† Pdes …t† ˆ 0

As we already explained, we start with the system being on the electronic state j1i and the vibrational level v ˆ 0. Then the system evolves and, through the couplings in Eq. (4), will acquire some population in the other two states. Due to the di€erent shapes of the potential curves V1 or V3 and V2 , during time propagation the adsorbate may gain some nuclear kinetic energy for desorption consecutive to conversion of some electronic kinetic into nuclear kinetic energy of the adsorbate (see e.g. Antoniewicz model [67]). In the range of excitation energies considered here, the adsorbate

D. Bejan et al. / Surface Science 470 (2001) 293±310

can leave the surface from any state in the manifold represented by the two e€ective states (j1i and j3i) having its asymptotic energy between the minimum of j1i and the asymptote of the ground state. Our j3i state is an e€ective one and it represents the manifold of all the energy states ful®ling the above condition. Therefore, one should 2 ®rst verify that only jw3 …z; t†j has signi®cant values in the asymptotic region. Secondly, one should compare the kinetic energy distribution of desorbing fragments obtained theoretically from our model with the experimental ®ndings. To calculate the probability that photodesorbed CO acquires a given kinetic energy one relates it, following the method of Balint±Kurti et al. [68] and Monnerville [69], to the half Fourier transform of the desorption function. One starts from the Fourier coecient associated to a kinetic energy E at an analysis point z1 …where z1 2 ‰za ; 1††: Z 1 1 exp…ÿi Et= h† wi …z1 ; t† dt: Ci …z1 ; E† ˆ p 2p 0 …17† In the asymptotic region, all the coupling coecients are zero and the potentials are constant so the Hamiltonian is independent of time and space coordinates. Thus, one can consider odinger wi …z1 ; t†, i 2 …1; 3† as a solution of the Schr equation for a free particle: Z 1 gi …k 0 † exp…ik 0 z1 ÿ iEi0 t= h† dk 0 ; …18† wi …z1 ; t† ˆ 0

where

Ei0

Ei0 ˆ Ei ‡

0

and k are related by: h2 k 02 ; 2l

…19†

Ei0 being the energy corresponding to the plane wave of wave vector k 0 that moves now on the constant potential Ei (see Fig. 1). Inserting Eq. (18) into Eq. (17) and changing the order of the integrals over t and k, one obtains: Z 1 1 gi …k 0 † exp…ik 0 z1 † dk 0 Ci …z1 ; E† ˆ p 2p 0 Z 1  exp…i…E ÿ Ei0 †t= h† dt: …20† 0

303

Now following Balint±Kurti et al. [68] we have: p Ci …z1 ; E† ˆ 2phgi …E† exp…ikz1 †: …21† In the above equation we used the change of integration variable g…k 0 †dk 0 ˆ g…Ei0 †dEi0 :

…22†

Extracting the probability gi …E† from Eq. (21) and using Eq. (17) one obtains: 2 Z   1 1 ÿiEt 2 : exp …z ; t†dt w jgi …E†j ˆ i 1 2 h 2ph 0 …23† In the expression above jgi …E†j2 dE gives the probability that the outgoing molecular fragment has a kinetic energy between E and E ‡ dE. The usual conditions of the experiment and the position of the e€ective potential curves V1 and V3 imply that the open exit channel for photodesorption corresponds only to the potential V3 and wave function w3 . Below, one will actually see that only the jg3 …E†j2 has signi®cant value and the corresponding kinetic energy is closely related to the shape of the intermediate state potential V2 . This result is in agreement with the photodissociation theory [70].

3. Results We have performed calculations on CO adsorbed Cu surfaces corresponding to two experiments performed in sub-picosecond regime. Prybyla et al. [5] studied CO on Cu(1 1 1) system at 0.52 ML coverage, excited by a laser of 2 eV energy, having a pulse duration of 100 fs and 4.5 mJ cmÿ2 absorbed ¯uence. Struck et al. [6] studied CO on Cu(1 0 0) at 0.5 ML coverage, using an excitation laser of 3.1 eV energy, pulse duration of 160 fs with 4.6 mJ cmÿ2 absorbed ¯uence. The one-dimensional z-grid used in our calculations has 1024 equally spaced points ranging  For the SO method employed from 1 to 12.25 A. here (see Eq. (16)), the temporal interval for the propagation is 0.1 fs and the total simulation time 819.2 fs (see Table 1). The temporal and spatial

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steps were chosen to ful®l the stability conditions of the SO scheme [66,71]. As already explained in Section 2.1, for the time independent part of the Hamiltonian, we used the same PECs of Fig. 1 and interactions Qij for both CO/Cu(1 0 0) [5] and CO/Cu(1 1 1) [5] the only di€erence being the global displacement of the three PECs to ®t the di€erent laser excitation energies. The time dependent optical potential Vi int contains the same parameters for Cu(1 1 1) and Cu(1 0 0) surfaces but depends, through p0  E20 , on the particular laser pulse intensity used in each experiment. The p0 value used in Eq. (11) is taken independent on the energy levels of the metal where the excitation takes place. The dipole transition moment in metallic copper was estimated to 2 be jMEi ;E j  10ÿ19 m2 (higher than the 4s ! 4p 2 transition in the atomic copper (jM4s!4p j  10ÿ20 m2 )). The electronic con®guration of copper is [Ar]3d10 4s1 and the present transition takes place in the s±p metal band where the electrons move freely. Consequently we expect an enhanced transition moment in the metal. From the physical point of view, before laser action the system has at time t0 its initial population p1 given by the Fermi±Dirac distribution function for the E1 ˆ 2 eV metal level at a temperature of 100 K and is about 10ÿ100 . When the laser operates, it continuously populates all the three states through non-zero Vi int . To test the model, and particularly the in¯uence of Vi int in Eq. (4), we performed two series of calculations, with Vi int zero and non-zero. For the sake of comparison, in both cases the initial popR 2 ulation p1 ˆ jwi …z; t†j dz was taken to be p1 ˆ 0:005. This population corresponds to the laser excitation probability of the E1 metal level after 1 fs, calculated within our model for the nonequilibrium electron distribution function (see Eq. (5)). We justify this initial time after the laser shine by the fact that the electrons on the E1 ˆ 2 eV level have a lifetime of about 4 fs and one cannot say exactly when the subsequent interactions, giving rise to the evolution of the system, begin to be ecient. First, when Vi int ˆ 0, the initial population p1 ˆ 0.005 corresponds to all the electrons excited in the system. Then the electrostatic coupling

(non-Born±Oppenheimer) of our diabatic curves determines population transfer to the other states and their oscillatory behavior without damping between the three states (see Fig. 3). The populations are very weak and at the end of the simulation time (t1 ˆ 819:2 fs), continue to oscillate, but these oscillations have no physical meaning. Now with both terms non-zero in Vi int and under the laser on experimental conditions of Prybyla et al. (see Fig. 4), the populations of di€erent states grow in time and are 100 times greater than in the preceding case. After the laser extinction the populations begin to diminish due to energy losses through electron±electron collisions and the coupling between the electronic states still determines the transfer of population. One sees that at about 600 fs all the populations are zero. The ®nal test concerns the V2int ˆ V2ex ÿ V2deex interaction (see Eq. (10)) only, which we set to zero or non-zero. A non-zero V2ex means a direct photon excitation of the ionic state. A non-zero V2deex corresponds to V2 state directly relaxing to the ground state of the system without jumping on V3 state, a population that is completely lost for the photodesorption process. Nevertheless as the

Fig. 3. Evolution in time of the populations of the three states of the present model for Vi int ˆ 0.

D. Bejan et al. / Surface Science 470 (2001) 293±310

305

Fig. 4. Evolution in time of the populations of the three states of the present model, for the experimental conditions of Prybyla et al. [5].

In Figs. 5 and 6 we display the desorption probability, Pdes given in Eq. (18) as function of time for the cases when Vi int ˆ 0 and Vi int 6ˆ 0 respectively. As for populations, we see that the desorption probability is 100 times larger when the laser operates. For Vi int ˆ 0 the desorption process begins at about 200 fs and Pdes rises continuously in time but is extremely weak (see Fig. 5). This means that, because of the electrostatic coupling, the population of the third state rises continuously in the asymptotic region. When Vi int is non-zero, Pdes , represented in Fig. 6 for the two experiments discussed in this work, reaches a maximum at about 700 fs and then tends to stabilize. This means that the desorption was over at that time and we have ``monitored'' all the outgoing molecules reaching the asymptotic region beyond za ˆ  5 A. The maximum value of the desorption probability Pdes on Fig. 6 is associated to the desorption yield. The obtained values of 5:7  10ÿ4 for Prybyla et al. [5] and 10ÿ4 for Struck et al. [6] are

modi®cation of the results with the changes in V2int are not signi®cant we performed the calculations presented below with non-zero V2int and for simplicity equal with V1int . The space and time evolution of the wave packet of the three states, corresponding to the 2 square of the wave functions jwi …z; t†j , i 2 …1; 3† may be seen with an appropriate visualization program but is not presented here. It shows almost periodic oscillations between states rises and falls called also recurrences. It has precisely the behavior described in the book by Shinke [70]: ``The wave function oscillates in the inner region, frequently recurs to its place of birth and continuously leaks out''. The mean period is very short, of the order of few femto seconds. During laser action (®rst 100 or 160 fs, in the two experiments discussed here) the wave packet is almost localized around the equilibrium position of the adsorbate on the surface and only after laser extinction it moves to larger z values. The jw1 …z; t†j2 and jw2 …z; t†j2 components of the wave function rapidly 2 diminish after laser extinction and only jw3 …z; t†j at larger times has signi®cant value in the asymptotic region.

Fig. 5. Evolution in time of the theoretical desorption probability Pdes for Vi int ˆ 0.

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D. Bejan et al. / Surface Science 470 (2001) 293±310

Fig. 6. Evolution in time of the theoretical desorption probability Pdes using the laser characteristics of Prybyla et al. [5] experiment (solid line) or Struck et al. [6] experiment (dashed line).

smaller than the experimental ®ndings in both cases: 10ÿ3 for the ®rst [5] and 3  10ÿ4 for the second [6]. It is interesting to note that our model leads (similarly to the experiments) to larger yields for the Prybyla et al. [5] experiment where light, of smaller energy than in Struck et al. [6] experiment, is used. In our model this behavior comes from the proportionality of the deexcitation term in Vi deex (see Eq. (12)) with the energy of light. The time dependence is nearly the same in both cases but, because the laser is on for 100 and 160 fs for Prybyla et al. [5] and Struck et al. [6] experiments respectively, the saturation regime is reached at longer times for Struck et al. [6] experiment. As mentioned above, the theoretical propagation time is of order of 700 fs. In the experimental papers on CO/Cu system two di€erent times are given: Prybyla et al. [5] give a time of 325 fs measured by the second harmonic generation (SHG) method; Struck et al. [6] give a time of 3 ps measured by a pump/probe experimental setup. The SHG method of Prybyla et al. [5] measures in fact the polarizability of the surface atoms and is sensitive to the CO coverage; the signal increases

with the diminishing CO coverage and stabilizes at a constant value afterward. The time needed to arrive at that constant value was identi®ed with the duration of the desorption event. It is not a direct measure of the photodesorption time because, to be counted as photodesorbed, the molecules should leave the interaction region. So, the SHG time measured by Prybyla et al. [5] should be considered as a lower limit of the desorption time. In the experiment of Struck et al. [6] the pump laser was split into two beams orthogonally polarized and applied normal to the surface. Information about the desorption time was obtained from two-pulse correlation data. The desorption yield, as a function of pulse/pulse delay time, is a Gaussian with a FWHM of about 3 ps. But this behavior proves only the non-linearity of the desorption with the absorbed laser ¯uence and the existence of a ®nite time for desorption. The Gaussian width cannot be considered as the desorption time but only an upper limit of the response time of the system. Therefore the desorption times given by the experiments are only indicative of the actual photodesorption time and they are in agreement with our estimation. In Fig. 7, we show the probability that a molecule desorbs with a kinetic energy E, shortly 2 called the kinetic energy distribution, jg3 …E†j , obtained from Eq. (23), where z1 was taken equal to za . Compared to state j3i, the distributions jgi …E†j2 for the states j1i and j2i are negligible small. This result con®rms the above assumption that molecules photodesorb only from the state j3i. The curves on Fig. 7 correspond to Prybyla et al. [5] and to Struck et al. [6] experiments re2 spectively. The kinetic energy distribution, jg3 …E†j has minima and maxima that can be correlated with the recurrences of w3 …z; t†. From that ®gure one also sees that most of the photodesorbed molecules acquired a kinetic energy of about 0.03 eV and the maximum kinetic energy is lower than 0.2 eV. This result compares well with the values obtained by Struck et al. [6] for the desorption of CO from Cu(1 0 0). These authors have found a mean kinetic energy of 0.037 eV, a mean rotational energy of 0.019 eV and a mean vibrational energy of about 0.25 eV. As our model does not consider the internal structure of the desorbed molecule,

D. Bejan et al. / Surface Science 470 (2001) 293±310

Fig. 7. The kinetic energy distribution function g3 …E† of desorbing CO molecules as function of kinetic energy Ekin ˆ E ÿ Easym (see Eqs. (21) and (22)). Solid line corresponds to laser characteristics of the experiment of Prybyla et al. [5]; dashed line corresponds to that of Struck et al. [6].

only comparison with the mean kinetic energies in theory and experiment can be made. Our calculations show that the kinetic energy distribution is closely related to the form of the ionic state potential energy V2 and it is only slightly dependent of the choice of V3 position relative to V1 . This 2 jg3 …E†j distribution increases slightly with laser ¯uence but the maximum kinetic energy is practically the same, independent of laser energy used. Fig. 8 presents the variation of the desorption yield as a function of laser ¯uence (F). For the usually investigated domain of laser ¯uences, we obtain a non-linear dependence yield F 4:5 . For CO/Cu(1 1 1), Prybyla et al. [6] found a F 3:7 dependence, while for CO/Cu(1 0 0), Struck et al. [6] found a F 81 one. We note that the di€erence between these two experiments is signi®cant. One cannot explain these di€erences by the experimental conditions namely by the structure of the surface of the metal (e.g. the presence of a surface state on Cu(1 1 1)), by the position of the adsorbate on it or by the di€erence in the energy, ¯uence or duration of the two lasers. This large discrepancy in the yield behavior rises some questions about at least the way these dependencies are ob-

307

Fig. 8. Desorption yield as function of absorbed laser ¯uence (F ). The dependence scales as F 4:5 .

tained from the experiment and how this compares with our model results. For example is the ¯uence we input in our model comparable with the experimental ¯uence? Struck et al. [6] use a yieldweighted absorbed ¯uence dicult to relate to the theoretical value. So, the fact that our model predicts a non-linear dependence with an exponent that lie between 3.7 and 8 can be considered as a good agreement with the experiments. Note that the other theoretical models used for CO/Cu have given a yield dependence on laser ¯uence of F 5:2 [15], a linear one [49] and F 3:1 [20]. 4. Discussion and conclusions In this paper we proposed a simple, phenomenological model that introduces, for the ®rst time, the in¯uence of hot non-thermal electrons on subpicosecond photodesorption processes. The needed hot electrons distribution is obtained using our kinetic theory model [46] which assumes neither thermodynamic equilibrium nor electronic temperature. The published (two-states) quantum photodesorption models [24,25,29,30,43] introduce the hot

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D. Bejan et al. / Surface Science 470 (2001) 293±310

electrons through a non-diagonal potential term that depends on the electronic temperature. For example, Saalfrank and Koslo€ [21], expressed the quenching rate as a function of the electronic temperature (that depends on the laser ¯uence in the two-temperatures model) and found a photodesorption yield that vary non-linearly with the electronic temperature. Hence, a linear dependence of this temperature on the laser ¯uence (that was not justi®ed) should imply a non-linear dependence of the yield on laser ¯uence. In the literature of two-state quantum photodesorption models a diagonal optical potential term (see e.g. Refs. [20,25,30]), for coupling with the electronic bath and the loss of population from the excited state, has been introduced. Yi et al. [19] used a full matrix optical potential for the laser excitation taken at perturbational level coupling the system to the electronic bath. This treatment originates from the partitioning theory of Feshbach [72,73] as applied to the study of resonances (see e.g. Ref. [74]). Elimination of the open continuum introduces a full matrix of imaginary terms with the prede®ned analytic form. Another way to take into account the coupling with the electronic bath is through a non-symmetric, non-hermitian real potential (see e.g. Refs. [24,30]). Usually the distribution of hot electrons is calculated by the two-temperatures model (electronic and phononic) which is not valid on the short time scale of the present desorption process. Our model uses diagonal optical potential terms that withdraws the assumption of a thermodynamic equilibrium for the distribution of electrons and the need of abstract, without a clear physical meaning, temperature. The optical potential (see Eqs. (4) and (10)) is constructed similar to our kinetic model [46]. The hot electrons distribution depends on the laser ¯uence and so does our interaction potential. The novelty of the present optical potential is related to its form allowing population of the excited state growth when the laser is on and population depletion when the laser is o€. It conserves the usual property of optical potential namely energy dissipation in our open system. Another idea we exploit here is the discretization of the electronic continuum of the metal (band structure) to a three electronic states scheme

instead of two states one widely used. In this way we allow the available energy to be split in two parts: one used for photodesorption process the other being returned to the metallic substrate by coupling with the bath of metallic electrons and phonons. The three states of our model should be viewed as e€ective ones allowing the de®nition of a residence time on the excited state of the adsorbate that then naturally deexcite through an electrostatic coupling between diabatic states. Presently the metal is structureless and does not allow for example the variation of the densities of states with energy and inclusion of dispersion as function of the reciprocal vector. The calculations performed with Vi int ˆ 0 give very weak and irrelevant results (see Figs. 3 and 5), so our optical potentials Vi int play an important role in the dynamics of the system. We performed calculations for many values of our parameters and the model is highly stable when changing the parameters ful®ling the condition jVi int j 6 Qij . On the contrary, if jVi int j P Qij the system becomes unstable, the populations and the desorption probability increase exponentially. After all this situation is comprehensible: essentially Vi int concerns the phenomena in the metal that are taken into account in an e€ective way through Vi int optical potential that acts as a population provider. If these phenomena prevail over the direct coupling Qij we should modify the present model and include these phenomena explicitly over a more elaborate model. Here we have selected the results that correspond to the experimentally studied systems: CO/Cu(1 1 1) [5] and CO/Cu(1 0 0) [6]. The behavior of the desorption probability, showing a desorption process of less then 700 fs, compares well with the estimated lower limit of 325 fs given by Prybyla et al. [5] experiment and upper limit of 3 ps given by Struck et al. [6]. The gain of kinetic energy of the photodesorbed particles is close to the experimental mean value and it is nearly insensitive to the excitation energy of the laser, comprehensible situation because our model changes only slightly with the laser ¯uence. Again this is an argument in favor of the three states model that allows, as explained above, a partial energy ¯ow of the energy back to the metal.

D. Bejan et al. / Surface Science 470 (2001) 293±310

In the present model the photodesorption yield for both experiments is proportional to the power 4.5 of the laser ¯uence. Knowing that we used similar parameters in the two simulations the result is comprehensible. As discussed in the preceding section, this dependence is within the experimental powers of 3.5 and 8.1. Because of large discrepancy in the two experimental values our estimation seems fair. In our model we have disregarded the dispersion of the observables which now will be discussed. When photons are absorbed by the metal electrons, and a momentum transfer is not allowed, the excitations take place at the center of the Brillouin zone C. For CO on Cu(1 1 1), the band gap at C is ÿ0.9 eV below EF and the energy disperses parabolically upward as the parallel momentum, kk increases [75]. At 2 eV above the EF and C point there are no available electronic states with kk ˆ 0. Since the photon momentum is negligibly small, the intra-band photon absorption can occur if some additional scattering with phonons or impurities occurs simultaneously to conserve momentum. Two-photon photoemission analysis [56] at di€erent excitation energies situate the CO …2p† at 3:3  0:2 eV above the Fermi level. At C point a photon of 3.54 eV cannot excite an electron from Ei ˆ EF ÿ 0:9 eV to Ef ˆ EF ‡ 3:3 eV [56]. But such an excitation is observed. The same problem appears for CO on Cu(1 0 0) where there are no available electronic states with kk ˆ 0 for 3.1 eV. So, in both cases one has to consider that the electronic levels above the Fermi level are occupied by electrons with a non-zero parallel momentum (kk 6ˆ 0) that may be excited by photon absorption only through scattering with phonons or impurities. As mentioned our model disregards the microscopic role of phonons, and therefore it can give only a partial description of the desorption process. A necessary improvement in the desorption model is the introduction of detailed interaction between electrons and phonons. The semiclassical model of Springer [16] and the quantum models of Guo [25] and Stromquist and Gao [34] put on evidence a strong dependence of the yield on the number of nuclear degrees of freedom included in the model. As the characteristic frequencies for hindered rotation (282 cmÿ1 )

309

for the CO-surface vibration (305 cmÿ1 ) and for the surface phonons (113 cmÿ1 ) are very close, all these motions may be strongly coupled. Therefore a future extension of our model should take into account the hindered rotation of the CO that will probably enhance the photodesorption yield. In conclusion, we developed the one-dimensional, three-state model of photodesorption that includes the hot electrons of the metal through diagonal time dependent optical potential. This model corresponds to an open system and introduces the hopping probability as a true nonBorn±Oppenheimer interaction between states. The model is able to reproduce the essential ®ndings of the published experimental results, particularly the non-linear yield dependence with the laser ¯uence. The improvements proposed above, particularly the inclusion of phonons and of other degrees of freedom of the adsorbate, will make the model more realistic. Acknowledgements The Laboratoire de Dynamique Moleculaire et Photonique is ``Unite de Recherche Associee au CNRS''. The Centre d'Etudes et de Recherches Lasers et Applications (CERLA) is supported by the Ministere Charge de la Recherche, the Region Nord/Pas de Calais and the Fonds Europeen de Developpement Economique des Regions. D. Bejan acknowledges the scholarship from the French government (``Bourse du gouvernement francßais'') during her Ph.D. studies in Orsay and ``poste rouge'' of CNRS. We acknowledge a computing time grant on a CRAY C from Institut du Developpement et des Ressources en Information Scienti®que IDRIS under the project 960554. The authors are grateful to Drs. K.S. Smirnov, H. Le Rouzo and M. B uchner for discussions and the appropriate subroutine calculating the PEC of the ground state and to Dr. E. Charron for discussions about the strong laser ®eld. References [1] J.A. Prybyla, et al., Phys. Rev. Lett. 64 (1990) 1537. [2] F. Budde, et al., Surf. Sci. 283 (1993) 143.

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