Hot electron mediated desorption rates calculated from excited state potential energy surfaces
aa r X i v : . [ phy s i c s . c h e m - ph ] J a n Hot electron mediated desorption rates calculated from excited state potential energysurfaces
Thomas Olsen, Jeppe Gavnholt, and Jakob Schiøtz ∗ Danish National Research Foundation’s Center of Individual Nanoparticle Functionality (CINF),Department of Physics, Technical University of Denmark, DK–2800 Kongens Lyngby, Denmark (Dated: November 5, 2018)We present a model for Desorption Induced by (Multiple) Electronic Transitions (DIET/DIMET)based on potential energy surfaces calculated with the Delta Self-Consistent Field extension of Den-sity Functional Theory. We calculate potential energy surfaces of CO and NO molecules adsorbedon various transition metal surfaces, and show that classical nuclear dynamics does not suffice forpropagation in the excited state. We present a simple Hamiltonian describing the system, withparameters obtained from the excited state potential energy surface, and show that this model candescribe desorption dynamics in both the DIET and DIMET regime, and reproduce the power lawbehavior observed experimentally. We observe that the internal stretch degree of freedom in themolecules is crucial for the energy transfer between the hot electrons and the molecule when thecoupling to the surface is strong.
PACS numbers: 31.15.xr, 71.15.Qe, 71.38.-k, 82.20.Gk, 82.20.Kh
I. INTRODUCTION
The advent of femtosecond lasers has initiated ma-jor progress in the study of non-adiabatic surface dy-namics on a wide range of systems. Photo-induced des-orption had already been observed for a few adsorbatesystems using low-intensity nanosecond laser pulses,but high-intensity femtosecond laser pulses have beenshown to induce desorption in a large class of adsorbatesystems and induce chemical reactions whichcannot proceed by thermal heating. The mechanism attributed to these reactions is exci-tation of substrate electrons by the laser pulse. A sin-gle hot electrons can then interact with an initially un-occupied adsorbate resonance thus asserting a force onthe adsorbate nuclei which may then lead to DesorptionInduced by Electronic Transitions (DIET). Using fem-tosecond lasers it is possible to reach high densities ofexcited electrons resulting in a new dominating mecha-nism - Desorption Induced by Multiple Electronic Tran-sitions (DIMET) where several hot electrons interactwith the adsorbate.A different method to produce hot electron based ona Metal-Insulator-Metal (MIM) heterostructure was sug-gested by Gadzuk . With an ideal MIM device it is pos-sible to tune hot electrons to any desired resonance of anadsorbate system and the approach thereby suggests thehighly attractive possibility of performing selective chem-istry at surfaces. Such devices have been constructed andcharacterized and comprise a promising candidate forfuture hot electron femtochemistry experiments.The theoretical framework to describe the non-adiabatic dynamics resulting from a hot electron inter-acting with an adsorbate is usually based on the con-cept of Potential Energy Surfaces (PES). In the Born-Oppenheimer approximation the electrons are assumedto remain in their ground state and are thus decoupledfrom the nuclei. This allow one to map out a ground state PES for the nuclei by calculating the electronic en-ergy for each position of the nuclei. Similarly, when aninitially unoccupied resonance becomes occupied a newexcited state PES arises which has its minimum at a dif-ferent location than the ground state PES and a force isexerted on the adsorbate. Several models have emergedto deal with non-adiabatic dynamics at surfaces but theyare usually limited by the difficulty to obtain reliable ex-cited state PESs and most theoretical results are basedon model potentials. An often used method to treat the extreme DIMETregime with many contributing electrons is using an elec-tronic friction model.
The hot electrons are thenassumed to thermalize rapidly and the influence of theelectrons on the adsorbate are treated statistically usingan electronic temperature which can be several thousandKelvin. The conceptual picture is that of a hot Fermi dis-tribution with a tail partially overlapping an adsorbateresonance and thereby exerting a force on the adsorbate.However, correct calculation of the temperature depen-dent friction still requires knowledge of the excited statePES.The subject of this paper will be the application oftwo-dimensional excited state PESs to calculate desorp-tion probabilities. We will be particularly interested inthe DIET regime where the hot electron has a known en-ergy as relevant for the MIM device and the few-electronDIMET regime. Although the friction models have en-joyed some success , there is still a need of a mi-croscopic non-statistical model of DIMET to test theassumption of thermally equilibrated electrons and tobridge the gap to the DIET regime. Furthermore, thehot electron femtochemistry relevant to the MIM devicecan certainly not be described using an electronic tem-perature since all electrons are tuned to a specific energy.We start by summarizing the method of linear expan-sion ∆SCF-DFT used to calculate the excited statePESs and note some qualitative features using CO onPt(111) as an example. We then discuss the models usedto obtain desorption probabilities based on the calcu-lated potential energy surfaces. First an adiabatic modelin which the adsorbate jumps between the ground andexcited state potentials is presented. A general non-adiabatic Newns-Anderson like model is then intro-duced and the connection to potential energy surfaces isexplained. This model with linear coupling has previ-ously been solved and applied to the one-dimensionaldesorption problem with model parameters . We ex-tend these results to a two-dimensional adsorbate andobtain the non-adiabatic coupling parameters from cal-culated excited state potential energy surfaces. In theDIET regime the model will be used to show that forsmall excited state lifetimes the main channel of energytransfer is the internal degree of freedom, and we em-phasize its importance in desorption dynamics. We com-pare the calculated desorption probabilities for CO andNO on four transition metal surfaces and note some gen-eral features of the desorption dynamics. The scatteringprobabilities obtained in the model are then generalizedto include adsorbates in any vibrationally excited statewhich allow us to extend the calculations to include asubstrate temperature and to treat the DIMET regimewithin the model. In the appendix it is shown how to ex-pand excited states within the projector augmented waveformalism and the results and generalizations of scatter-ing amplitude calculations are summarized. II. POTENTIAL ENERGY SURFACES
The potential energy surfaces were obtained usingthe code gpaw which is a real-space Density Func-tional Theory (DFT) code that uses the projector aug-mented wave method.
In all our calculations we usedthe Revised Perdew-Burke-Ernzerhof (RPBE) exchange-correlation functional since this has been designed toperform well for molecules adsorbed on surfaces, and hasbeen shown to perform better than the original PBEfunctional both for isolated molecules and for ad-sorbed moleculeshammer. For each metal we set up aclosed-packed surface consisting of three atomic layerswith the top layer being relaxed. 10 ˚A of vacuum wasthen introduced above the slab and 0.25 monolayer ofadsorbate molecules relaxed at either top or hcp hollowsite. We then mapped out two-dimensional ground statepotential energy surfaces in terms of the internal stretchand the center of mass (COM) to surface distance coor-dinate using 12 irreducible k-points and a grid spacing of0.2 ˚A.To find the excited state potential energy surfaces weuse the method of linear expansion ∆SCF which wehave published in a previous work and implementedin gpaw . In the previous publication we have tested themethod against inverse photo-emission spectroscopy, andfound that it performed well for molecules chemisorbedon surfaces. In each step of the self consistency cycle an electron is removed from the Fermi level, the densityof an excited state is added to the total density, and theband energy of this state is added to the total energy.To get the band energy right we need to expand the ex-cited state on the Kohn-Sham orbitals found in each it-eration. The method is thus a generalization of the usual∆SCF where occupations numbers are changed. Insteadof changing occupation numbers we occupy an orbitalwhich is not an eigenstate of the KS Hamiltonian but asuperposition of eigenstates in such a way that the stateis as close as possible to the original molecular state. Werefer to appendix A for details on how to do this withinthe projector augmented wave formalism. The excitedstates used in this paper are the anti-bonding 2 π orbitalsof NO and CO.In the previous publication, we investigated the influ-ence of the interactions between neighboring super cellsfor different super-cell sizes, and found that the size-dependency of the excitation energy is consistent withan electrostatic dipole-dipole interaction. Already for a(2 ×
2) surface cell, the interaction energy has becomesmall, and furthermore this interaction energy will havelittle influence on the slope of the excited-state PES, andthus little influence on the calculated desorption rates.For this reason, and to keep the calculations manageable,we use a (2 ×
2) surface cell.As an example we show the two-dimensional excitedstate PES superimposed on a ground state PES in thecase of CO on Pt(111) top site in Fig. 1. The moleculesadsorb with the molecular axis perpendicular to the sur-face with carbon closest to the top site. Due to the sym-metry of the 2 π orbital and the geometry at the groundstate minimum we cannot induce forces parallel to thesurface if the molecule is at the ground state minimumwhen excited. The excited state could have unstableextremal points with respect to the degrees of freedomparallel to the surface, but the model we apply in thiswork only depend on the degrees of freedom with non-vanishing derivatives on the excited state PES and wethus assume that the center of mass (COM) and internalstretch degrees of freedom should capture the essentialdesorption dynamics of the considered systems.Since the excited molecule has an extra electron in ananti-bonding orbital the excited molecule is expected tohave a larger equilibrium bond length and this is alsowhat we observe. A popular and conceptually simple wayof explaining desorption in one-dimensional models ofDIET is the Antoniewicz mechanism, where the excitedmolecule induces an image charge on the surface whichresults in an attractive force on the surface. The ex-cited molecule is then accelerated towards the surface andeventually decays to the steep wall of the ground stateMorse potential. From Fig. 1 we observe a qualitativelydifferent behavior: the COM of the excited molecules ex-perience a repulsive force accelerating the COM of themolecule away from the surface. This is due to the effectof the bond length expansion and the fact that the 2 π orbital have a large density in the vicinity of the carbon C O M d i s t a n c e t o s u r f a c e [ Å ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ground stateResonance state
FIG. 1: Ground and excited state (2 π ) potential energy sur-faces for CO adsorbed on Pt(111) - top site. The coverage is0.25 monolayer. atom which gives a repulsion that dominates the imagecharge attraction. It will be shown below that for theconsidered systems it is primarily excitation of the inter-nal degree of freedom which is responsible for the largeenergy transfers leading to desorption.The potential energy surfaces for CO adsorbed on Pd,Rh and Ru show very similar qualitative features. III. MODELS
The timescale at which adsorbates dissipate energy tothe substrate is typically on the order of picoseconds and since the oscillation times for the two modes is ∼ − f s (see section IV A) we will assume thatthe molecule has plenty of time to desorb if it has ab-sorbed the required energy from a hot electron. This isthe major assumption we will impose and thus when werefer to desorption rates in the following it is the rate oftransferring at least of the energy needed for a moleculeto desorb.Assuming a Lorentzian resonance with Full Width atHalf Maximum (FWHM) Γ = ~ /τ centered at ǫ a , theprobability that a hot electron of energy ε desorbs themolecule becomes P AdDes ( ε ) = (Γ / ( ε − ǫ a ) + (Γ / τ Z ∞ P ( t d ) e − t d /τ dt d , (1)where P ( t d ) is the probability of a desorption event whenthe molecule is excited at t = 0 and decays at time t = t d . Using classical dynamics the probability P ( t d ) canbe obtained by propagating the molecule on the excitedstate PES according to the forces, evaluate the energy gain ∆ E after time t d and put P ( t d ) = 1 if ∆ E > E
Des and P ( t d ) = 0 if ∆ E < E
Des . However, the short lifetimeof the excited electron implies that classical molecularpropagation on the excited state PES may not be a goodapproximation.In fact, the classical limit is obtained when the action S = R dtL ( ˙ x ( t ) , x ( t )) on a representative path satisfies | S | ≫ ~ . (2)Assuming a quadratic excited state potential of frequency ω and initial potential energy E we can evaluate the ac-tion on a classical path between initial time t i and finaltime t f . For generic timescales one just obtains the usualcondition of high excitation numbers E ≫ ~ ω , whereasfor ω ∆ t ≪
1, the additional condition of E ∆ t ≫ ~ needsto be satisfied in order to apply classical dynamics. Inthe case of CO on Pt(111) we have E ∼ . eV (Fig.1) and τ ∼ f s (Fig. 8) which gives E ∆ t ∼ ~ . Thusmolecular propagation on the excited state PES is notexpected to follow the classical equations of motion. Be-low we will show an example where a classical analysisunderestimates desorption probabilities by several ordersof magnitude.This scheme could be extended to a quantum dynami-cal treatment of the molecule by propagating the molec-ular wavefunction using a two-PES Hamiltonian. How-ever, the method still rests on the Born-Oppenheimerapproximation and the adiabatic concept of potential en-ergy surfaces and thus cannot be expected to fully cap-ture the non-adiabatic entangled dynamics of the reso-nant electron and adsorbate coordinates.Instead we consider a Newns-Anderson typeHamiltonian with substrate states | k i , a resonant state | a i , adsorbate coordinates x i , an adiabatic adsorbateground state potential V ( x i ), and a non-adiabatic cou-pling of the resonant electron to adsorbate coordinates ε a ( x i ) H = T ( ˙ x i ) + V ( x i ) + ε a ( x i ) c † a c a + X k ǫ k c † k c k + X k (cid:16) V ak c † a c k + V ∗ ak c † k c a (cid:17) . (3)The strength of the electronic coupling is expressedthrough the function:Γ( ε ) = 2 π X k | V ak | δ ( ε − ǫ k ) . (4)The model as such neglects the electron-electron in-teraction, but we assume that the important part ofthe electron-electron interactions is the restructuring ofthe metallic electrons when the resonance is occupiedand that we can capture this effect in an effective non-adiabatic coupling. To do this we note that we can ob-tain ε a ( x i ) as the expectation value differences of (3)with the adsorbate at x i with and without an electronin the state | a i . Applying this to an interacting problemlead us to identify ε a ( x i ) = V ( x i ) − V ( x i ) where V ( x i )and V ( x i ) are the potential energy surfaces of excitedand ground states which we have obtained with linearexpansion ∆SCF-DFT.In the following we will apply the wide band limitwhich means that the individual coupling coefficients V ak are assumed to vary slowly in energy and the densityof states ρ ( ε ) is taken as constant in the vicinity of theresonance. This gives an energy independent couplingΓ = 2 πρ ( ǫ a ) P k | V ak | and the resonance spectral func-tion corresponding to the electronic part of (3) becomesa Lorentzian with FWHM Γ.Even in the wide band limit it is quite difficult to han-dle the model (3) analytically with arbitrary couplingfunction ε a ( x i ). In particular, we would like to calcu-late the probability that an incoming substrate electronof energy ε i scatters inelastically on the resonance andis reflected back into the substrate with final energy ε f .Fortunately, the potential energy surfaces we are consid-ering are close to being quadratic in the region of interest(see Fig. 1) and the ground and excited state potentialshave approximately the same curvature. Taylor expand-ing V ( x i ) to second order and ε a ( x i ) to first order in thevicinity of the ground state equilibrium positions x i thengives H = ǫ a c † a c a + X k ǫ k c † k c k + X k (cid:16) V ak c † a c k + V ∗ ak c † k c a (cid:17) + X i ~ ω i ( a † i a i + 12 ) + X i λ i c † a c a ( a † i + a i ) , (5)with ǫ a = V ( x i ) − V ( x i ) and λ i = l i √ ∂∂x i V (cid:12)(cid:12)(cid:12) x i = x i , l i = r ~ m i ω i , (6)where we have assumed that an appropriate transforma-tion to normal coordinates has been performed. Notethat if the ground and excited state potentials are ex-actly quadratic with equal second derivatives, we can re-late the coupling constants to the positions x i of theexcited state potential minimum as λ i = ~ ω i ∆ V i with∆ V i = m i ω i | x i − x i | . The quantity g i = ( λ i / ~ ω i ) then corresponds to an ”initial quantum number” on theexcited state surface and this becomes the effective di-mensionless coupling constant in the model (see appendixB). The Hamiltonian (5) has previously been subject todetailed analysis in the context of inelastic scattering and applied to desorption dynamics for the case of aone-dimensional adsorbate with model parameters.Below we extend the results of Refs. 16 and 29 to atwo-dimensional adsorbate and calculated the couplingparameters λ i from excited state potential energy sur-faces. We also calculate scattering amplitudes for an ad-sorbate initially in an vibrationally excited state whichenable us to apply the model to the DIMET regime.
1. DIET
In Eq. (B8) we show how to calculate the scatteringprobability P n i ,n j ( ε i ) that an incoming electron of en-ergy ε i excites the ( n i , n j ) mode of a two-dimensionalharmonic oscillator. The probability of transferring E R or more energy to the adsorbate can then be found bycalculating P R ( ε i ) = X n i ,n j P n i n j ( ε i ) θ ( ~ ω i n i + ~ ω j n j − E R ) , (7)where θ ( x ) is the Heaviside step function. The desorptionrate can then be calculated by integrating this expressionwith the current density of incoming hot electrons. Oneshould note that the probability P n i n j of exciting the( n i , n j ) modes in a two-mode model is not just givenby the product of single mode probabilities P n in a one-mode model. This is due to an indirect coupling of thetwo modes through the resonance. The result can begeneralized to include the substrate temperature and wewill examine the consequences of this below.
2. DIMET
If we assume that the time between individual inelasticscattering events is much longer than the scattering timeitself, it is possible to regard multiple-electron desorptionevents as a sequence of single-electron scattering events.Since we have extended the inelastic scattering probabil-ities to include situations where the molecule is initiallyin a vibrationally excited state, it is also possible to treatDIMET events within the model (5). As an example,let us assume a single vibrational mode which is initiallyunoccupied ( n = 0). When a hot electron with energy ε scatters inelastically on the resonance the result willbe a probability distribution P n ( ε ) for all vibrationallyexcited states n of the molecule. If a second electron withenergy ε now scatters on the resonance the probabilitydistribution will change to P n ( ε , ε ) and so forth. Theprobability P n − n ( ε , n ) of exciting the state n giventhat the initial state were n is calculated in Eq. (B6)and we can write P n ( ε , ε ) = ∞ X n =0 P n − n ( ε , n ) P n ( ε ) (8)for a two-electron event and similar expressions formultiple-electron events. Given an initial distributionof hot electrons we may then calculate the probabilityof a desorption event with any number of contributingelectrons. IV. RESULTSA. Parameters
The parameters in the desorption model (5) are thewidth of the resonance Γ, the frequencies of the normalmodes ω i , the excitation energy ǫ a and the non-adiabaticcoupling coefficients λ i . We cannot calculate Γ from firstprinciples but we estimate its value from the Kohn-Shamprojected density of states. It is typically on the orderof 1 eV , but it will be instructive to treat it as a freeparameter and examine how it affects desorption proba-bilities.The frequencies are obtained from a standard normalmode analysis and ǫ a is obtained as the excitation energyat the ground state potential minimum. The coupling co-efficients are determined by mapping out a small area ofthe excited state potential energy surface in the immedi-ate vicinity of the ground state potential. In each of theconsidered systems we optimize the area such that it issmall enough to be linear but large enough to suppressnumerical fluctuations in the excited state energies. Wethen fit a linear function to this area and transform thederivatives to the normal modes.In all the considered systems the calculated normalmodes are similar but not identical to the standard COMand internal stretch modes. For example with CO onPt(111) the internal stretch and COM modes are respec-tively: d = ( − , .
75) and z = (1 ,
1) whereas the cal-culated modes are in the directions d = ( − , .
68) and z = (1 , .
11) with respect to the ( x C , x O ) coordinatesnormal to the surface. Since the desorption probabilitiesare quite sensitive to the value of the non-adiabatic cou-pling constants it is important that we take the deriva-tives on the excited state PES with respect to the correctnormal modes.Tables I and II below display the calculated param-eters. We have only examined CO at on-top sites andNO at hcp hollow sites. NO is seen to have much lowernon-adiabatic coupling coefficients and excitation ener-gies than CO. The low excitation energies is due to thefact that NO already has one electron in the anti-bondingorbital and the resonance thus has to lie close to theFermi level of the metal. The small coupling coefficientscan also be traced to the ground state occupation of the2 π orbital on NO. In the Kohn-Sham picture we canimagine the resonance corresponding to 2 π lying rightat the Fermi level being partially occupied. When an ex-tra electron is put into the orbital the resonance energy isincreased due to Hartree repulsion and the initial partialoccupation is lost. In the true system things are morecomplicated, but the qualitative features are the same:exciting NO results in less charge being transferred tothe molecule than exciting CO and thus a weaker non-adiabatic coupling. Thus it is much harder to transferenergy to adsorbed NO compared to CO in a one elec-tron event, but since the resonance is located much closerto the Fermi level a thermal distribution of hot electrons Metal ǫ a ω z ω d λ z λ d Pt(111) 3.89 0.054 0.255 -0.142 -0.145Pd(111) 3.64 0.061 0.256 -0.082 -0.164Rh(111) 3.80 0.048 0.247 -0.129 -0.132Ru(0001) 3.74 0.054 0.255 -0.134 -0.120TABLE I: Parameters for CO adsorbed at top site on fourtransition metals. All number are eV .Metal ǫ a ω z ω d λ z λ d Pt(111) 1.71 0.039 0.196 -0.050 -0.053Pd(111) 1.48 0.055 0.201 -0.046 -0.053Rh(111) 1.82 0.073 0.277 -0.042 -0.020Ru(0001) 2.14 0.042 0.192 -0.052 -0.006TABLE II: Parameters for NO adsorbed at hcp hollow site onfour transition metals. All number are eV . is likely to result in more frequent scattering events thanfor CO. B. DIET desorption rates
The probability that a single electron with energy ε i scatters inelastically and transfers the energy E R to anadsorbate can be calculated in the model (5) with Eq.(7). Our basic assumption is that rate of energy dissipa-tion to the substrate is much longer than the time of adesorption event and when we refer to desorption ratesin the following it will mean the rates of transferring theenergy needed for a molecule to desorb in a truncatedquadratic potential.In Fig. 2 we display the probability that an incomingelectron will scatter with an energy loss in excess of thedesorption energy (∆ E > . eV ) for three values of theresonance width. When only a single mode is consid-ered we see the appearance of oscillator sidebands withan energy spacing of ~ ω . At larger resonance width thesidebands are washed out and the probability takes theform of a Lorentzian which is detuned by δǫ a ∼ ∆ E/
2. Asimple way to understand this detuning is as a compro-mise where both the incoming and outgoing electrons areclosest to the resonance. Thus we see the emergence of an effective inelastic resonance with a center that is detuneddependent on the desorption energy and a shape whichis highly dependent on the lifetime. Such a probabil-ity distribution could not have been obtained in a modelwhere the transfer of energy to the adsorbate is decou-pled from the probability of capturing the electron, andthe desorption probability would always be a Lorentzian(in the wide band limit) centered at ǫ a and multiplied bya factor dependent on the details of the potential energysurfaces. For Γ > . eV the COM degree of freedombecomes unimportant and the desorption probabilities FIG. 2: Desorption probability for CO adsorbed on Pt(111)for three different values of the resonance width. For Γ > . obtained using both modes and only the internal degreeof freedom become identical.Assuming an energy independent current of hot elec-trons we can integrate the desorption probabilities inFig. 2 to obtain a desorption rate normalized to theincident flux of electrons. In Fig. 3 we show how eachof the two modes contribute to the desorption rate andcompare with a calculation within the classical adiabaticmodel (1). The two single mode rates are obtained bysetting g d and g z to zero in Eq. (B8). It is seen that it isthe internal stretch mode that governs the energy trans-fer completely in the large width regime and the COMmode governs the energy transfer at low width. The rea-son for this partitioning is the timescale associated withthe two different modes. As seen from tables I and II thenon-adiabatic coupling constants have approximately thesame magnitude for the two modes. However, the periodof oscillation is 5 times larger for the COM mode and forsmall lifetimes there is not enough time to transfer energyto the COM mode. From Fig. 3 we see that the maxi-mum rate of energy transfer in each mode occurs whenΓ ∼ ~ ω i . The desorption rate decrease at small reso-nance width, since the hot electron then becomes weaklycoupled to the resonant stateIn Fig. 4 and 5 we show a comparison of CO and NOadsorbed on the different transition metals. Again com-paring with tables I and II it is seen that it is the couplingto the internal mode alone which controls the magnitudeof the desorption rate at large resonance width. Sincethe internal degree of freedom seems to control the rateof energy transfer in the physical range of the resonancewidth (typically 0 . < Γ < .
5) we will ignore the COMdegree of freedom in the following.
FIG. 3: Desorption rate for CO adsorbed on Pt(111) as afunction of resonance width Γ. In the wide resonance (shortlifetime) regime the rate is seen to be completely governed bythe internal stretch excitation whereas the COM excitation isgoverning the desorption rate in the narrow resonance (longlifetime) regime. The classical rate becomes several orders ofmagnitude smaller than the quantum rate at large resonancewidth. The inset shows the same data on a logarithmic scale.FIG. 4: Rates of transferring 1 . eV to CO on 4 transitionmetals.
1. Comparison of CO and NO
So far we have analyzed some general features of des-orption probabilities and their dependence on the non-adiabatic coupling parameters and the lifetime τ = ~ / Γ.Now we will compare the desorption probabilities of COand NO on four transition metal surfaces using experi-mentally determined desorption energies. Although sub-stantial experimental data exist for various systems in-cluding CO and NO, a direct comparison to experimen-tal data is difficult since experimental desorption yieldsare highly dependent on the distribution of hot electronsin the substrate which depends on the detailed physical
FIG. 5: Rates of transferring 1 . eV to NO on 4 transitionmetals. Metal E D Γ δε P MaxD
Pt(111) 1.37 a · − Pd(111) 1.48 a · − Rh(111) 1.45 a · − Ru(0001) 1.49 a · − TABLE III: Desorption energies and calculated maximumdesorption probability for CO adsorbed at top site onfour transition metals. All numbers except P MaxD is eV . a Experimental values taken from Abild-Pedersen andAndersson . properties of the metal and the applied laser pulse. Thedistribution of hot electrons resulting from a given laserpulse could in principle be calculated from first principle,however, we will make no attempt of such a calculationhere but simply compare desorption probabilities of sin-gle electron events as relevant for the MIM device .In tables III and IV we summarize the desorption energy E d , the estimated resonance width Γ, the detuning ofthe energy at which the incoming electron has the max-imum probability of transferring the desorption energy δε = ε Maxi − ǫ a and the maximum desorption proba-bility P MaxD = P D ( ε Maxi ) for the four transition metals(the maximum probability is detuned from ǫ a as shownin Fig. 2). The detuning very nicely follows the rule ofthumb that δε ∼ E D / ǫ a .In general it is easier for a single electron at the rightenergy to mediate a desorption event involving CO thanwith NO from all the considered systems. However, ina femtosecond laser pulse experiment the resulting hotelectron distribution would have much lower occupationnumbers at the CO resonances than at a typical NO reso-nance. For example taking platinum as an example with Metal E D Γ δε P MaxD
Pt(111) 1.29 b · − Pd(111) 1.17 c · − Rh(111) 1.68 c · − Ru(0001) 1.49 d · − TABLE IV: Desorption energies and calculated maximumdesorption probability for NO adsorbed at hcp hollow siteon four transition metals. All numbers except P MaxD is eV . b Croci et al . c Vang et al , d Butler et al . a thermal electron distribution at 5000 K and referringto tables I and II we see that the electronic occupationnumbers at the resonance energy of CO and NO relatesas f ( ǫ NO ) /f ( ǫ NO ) ∼ sincethe 2 π electrons becomes delocalized and quasistation-ary at certain coverages. Furthermore, both CO and NOare known to form adsorbate structures which is moreinvolved than the simple periodic coverage of 0.25monolayer considered here and the dependence of non-adiabatic coupling coefficients on coverage certainly de-serves a study of its own.However, from Figs. 4 and 5 we do observe the gen-eral trends that NO has a much weaker non-adiabaticcoupling to the surfaces than CO and that for both COand NO the coupling to Pt and Pd are similar whereasthe coupling is weaker for Rh and very low for Ru. Thisdecrease in non-adiabatic coupling could hint at a simpledependence on the number of d-band electrons. Investi-gating this will be the subject of future work. C. DIMET desorption rates
To get an idea of desorption probabilities in theDIMET regime we will start by examining how an ini-tial excitation influences the probability of transferring agiven number of vibrational quanta. When the oscillatoris in an excited vibrational state there is also the possi-bility of stimulated emission of vibrational quanta wherethe incoming hot electron gains energy by the scatteringevent.In Fig. 6 the maximum probability of transferring ∆ n quanta is shown for a range of initial quantum numbers n .We treat n as a continuous variable since in the case of athermal ensemble of states the initial quantum number issimply replaced by a Bose distribution. There is a strik- FIG. 6: Maximum probability of transferring ∆ n vibrationalquanta given that the initial state is n with Γ = 1 . eV . ing increase in the probabilities of transferring energy tothe oscillator if the oscillator is already excited. For ex-ample, the probabilities of exciting 0 → → · − and 2 · − respectively although both transitionsinvolve the same energy transfer. Thus if we compare theone-electron event P → = 6 · − with the product ofthe two probabilities P → → = 6 · − we get an orderof magnitude difference and we still need to include theother channels for transferring 6 quanta in a two-electronevent.This also implies that the effect of a finite substratetemperature is two-fold: The occupation numbers of ex-cited vibrational states will be non-zero meaning that lessenergy transfer is needed to desorb the molecule and thelikelihood of a given energy transfer is increased if themolecule is thermally excited. However at room temper-ature the probability that the internal mode is in its firstexcited state is on the order of 10 − and we can safelyneglect the effect of temperature.A hallmark of the DIMET regime is the power lawdependence of the desorption rate on the laser fluence R ∼ F n where n depends on the particular adsor-bate/substrate system considered. It is by no meanstrivial that the desorption rate should follow a powerlaw and calculating the exponent of a particular systemis a major challenge of any DIMET model.It is reasonable to that assume that the laser fluenceis proportional to the flux of hot electrons hitting themolecule, since the desorption rate typically becomeslinear for small fluences corresponding to the DIETregime. As a simple model for the desorption rate wethen consider a given flux J of hot electrons at a fixedenergy ε i hitting the resonance in equally spaced timeintervals ∆ t = 1 /J . We assume that each vibrationalquantum has a fixed lifetime T vib and that desorptionoccurs immediately if the vibrational energy reaches thedesorption energy E D . The probability that one vibra-tional quantum survives the time interval ∆ t is e − ∆ t/T vib and the probability of decay is (1 − e − ∆ t/T vib ). The prob-ability that the first electron excites the n ’th vibrationalstate is then simply the DIET probability Q ( n ) = P n ( ε i , , (9)where P n ( ε i ,
0) is given by Eq. (B6). The probabilityof the adsorbate being in the n ’th vibrational state afterthe second electron has scattered is Q ( n ) = ∞ X m =0 p ( m ) P n − m ( ε i , m ) , (10)where P n − m ( ε i , m ) is the probability of the transition m → n (Eq. (B6)) and p ( m ) is the probability that theadsorbate was initially in the state m given by p ( m ) = ∞ X k = m Q ( k ) (cid:18) km (cid:19) ( e − ∆ t/T vib ) m (1 − e − ∆ t/T vib ) k − m × θ ( E d − ~ ωk ) . (11)Thus we only sum over values of k below the desorptionenergy since states above E D would previously have beendesorbed by assumption. Similarly the probability Q ( n )of being in the n ’th excited state after the third scatteringevent can be expressed in terms of Q ( n ) and so forth.The desorption probability of the N ’th electron is then P DesN = X n Q N ( n ) θ ( ~ ωn − E D ) . (12)When enough time intervals is included the probabilitiesconverge such that P DesN = P DesN − and the desorption rateis R ( J ) = J · P DesN with J = 1 / ∆ t .In Fig. 7 we show the rate for NO on Pt(111) withΓ = 0 . eV . The desorption energy corresponds to 8vibrational quanta. Note that changing the lifetime T vib in this model just corresponds to rescaling the flux. Thesimilarity to similar experimental figures is striking. Atsmall flux the rate is linear whereas it obeys a power law( R ∼ J n with n >
1) at higher fluences. The fit toa power law is very good for fluxes above 0 . T − vib . Forsmall values of the detuning ( − . < δε < . eV ) we findthat 5 . < n < . For largepositive values of the detuning the exponent decreasesdramatically which is probably due the fact that fewertransitions dominate the dynamics in this region. Thismeans that even though the results was obtained usingthe simple electron flux J ( ε i ) = J δ ( ε i − ǫ a − δε ) we wouldmost likely obtain the same exponent if we generalizedthe model to any flux localized within ± . eV of theresonance.Although the correspondence with the experimentallyfound exponent may be fortuitous in such a simple modelthe power law itself is very robust to changes in the pa-rameters and we obtain similar power laws for CO onPt(111). For example, changing the value of Γ resultsin an overall shift of the rates but the exponents are es-sentially unchanged. Indeed the exponents appear to bedetermined mainly by the number of vibrational quantaneeded for desorption. FIG. 7: Desorption rate as a function of electron flux peradsorption site. For small electron flux the rate is linear inthe flux corresponding to the DIET regime whereas for largerelectron flux the rate obeys a power law ( R ∼ J n with n > V. SUMMARY AND DISCUSSION
We have previously presented a method to obtainexcited state potential energy surfaces for moleculeschemisorbed at metal surfaces. In this paper themethod have been applied and combined with a non-adiabatic quantum model to obtain desorption probabil-ities for CO and NO on four transition metal surfaces.The model we have applied allow us to predict theprobability that a hot electron will transfer a givenamount of energy to the different vibrational modes ofan adsorbate. Our main conclusion is the significant roleof the internal degree of freedom and the failure of clas-sical mechanics to describe the excited state adsorbatepropagation.Combining the model with a simple picture of the de-cay and re-excitation of vibrational states reproducesthe characteristic power laws of DIMET experimentsand yields the exponent associated with a given adsor-bate/substrate system.The model we have used for calculating the energytransfer rates obviously represents a very simplified viewof the dynamics. First of all it is a model of non-interacting electrons. We assume that we can includethe important part of the electron-electron interactionsby using non-adiabatic coupling coefficients λ i obtainedfrom the interacting density with linear expansion ∆SCF-DFT. The approximation amounts to assuming ballistichot electrons and instantaneous restructuring of the elec-tronic environment when occupying the resonance. Al-though this may be the case in some metallic systemselectron-electron interactions could have effects which gobeyond a simple renormalization of the non-adiabatic coupling. The linear non-adiabatic coupling regime lead-ing to Eq. (5) corresponds to an assumption of equalcurvature on the ground and excited state PES. This isa good approximation for CO but NO has a very shallowexcited state PES on some of the transition metals andthere the approximation may not be as good.Furthermore the model assumes that the ground statepotential is quadratic and that the excited state potentialis simply a shifted ground state potential. At least in theCOM direction it is clear From Fig. 1 that the groundstate potential deviates significantly from a quadratic po-tential and since we are concerned with high lying vibra-tional excitations this deviation could perhaps have animportant effect. It may be possible to include anhar-monic terms in the Hamiltonian and calculate new scat-tering amplitudes perturbatively but the this will be leftfor future work.We have focused on the molecules CO and NO, sincethey have a conceptually simple structure and a vastamount of experiments have been performed on thesesystems. However, it is well known that GGA-DFT cal-culations of CO adsorbed an Pt(111) predicts CO to bindat a hollow site in contradiction to the experimentallyobserved top site . While the difference in adsorbtionenergy appears to be less with gpaw than in the calcu-lations presented in Ref. 45, possibly due to the use ofthe PAW method instead of ultrasoft pseudopotentials,the difference is still 80 meV and the inability to pre-dict the correct binding site is worrying. On the otherhand, the existence in the calculation of another adsorp-tion site with a slightly lower energy is unlikely to changethe local shape of the potential energy surface enough to qualitatively change the results obtained here. In addi-tion, we see a very similar behavior for CO on Ru(0001),where DFT does predict the right adsorption site (thetop site). We have thus chosen to put CO at the ex-perimentally observed top site as the hollow site wouldlead to a smaller surface molecule distance and thus verydifferent screening and desorption rate.As previously mentioned the value of Γ is estimatedfrom the Kohn-Sham projected density of states, but wedo not know how well this estimate matches the truevalue and as such we have mostly treated Γ as a freeparameter. In fact the object of interest in the problemis the spectral function of the resonant state, but evenif we had a reliable way of determining this function wewould have to make the wide band approximation (wherethe spectral function is a Lorentzian of width Γ) in orderto calculate scattering rates. Nevertheless it would bevery interesting to calculate this function to get an ideaof the validity of the wide band approximation and toobtain a trustworthy value of Γ.We have not made any attempt to predict how the dis-tribution of energy evolves after a molecule returns to itselectronic ground state, but assume that the dissipationof energy is slow enough that the adsorbate will desorbif the desorption energy has been transferred. This is ofcourse a rather crude assumption and the rate of energy0transfer should be accompanied by a detailed molecularpropagation on the full dimensional ground state PESto improve the results. Ground state molecular dynam-ics would also be necessary to obtain branching ratioswhen there is a possibility of different chemical reactionsinduced by hot electrons.However the model we have presented captures someof the essential features of non-adiabatic dynamics. Forexample the appearance of an effective inelastic reso-nance which is detuned from the electronic resonance byan amount depending on the energy transfer is a purenon-adiabatic result and would never have emerged froman adiabatic model. Furthermore the exponents in theDIMET power laws appear to be determined by the num-ber of vibrational quanta needed for desorption and thuscommunicates the quantum nature of the dynamics. APPENDIX A: PROJECTING KS STATES ON AMOLECULAR ORBITAL IN PAW
The Projector Augmented Wave (PAW) method uti-lizes that one can transform single-particle wavefunctions | ψ n i oscillating wildly near the atom core (all-electronwavefunctions), into smooth well-behaved wavefunctions | ˜ ψ n i (pseudo wavefunctions) which are identical to theall-electron wavefunctions outside some augmentationsphere. The idea is to expand the pseudo wavefunctioninside the augmentation sphere on a basis of smooth con-tinuations | ˜ φ ai i of partial waves | φ ai i centered on atom a .The transformation is | ψ n i = | ˜ ψ n i + X i,a (cid:16) | φ ai i − | ˜ φ ai i (cid:17) h ˜ p ai | ˜ ψ n i , (A1)where the projector functions | ˜ p ai i inside the augmenta-tion sphere a fulfills X i h ˜ p ai | ˜ φ ai i = 1 , h ˜ p ai | ˜ φ aj i = δ ij , | r − R a | < r ac . The method of linear expansion ∆SCF involves ex-panding a molecular orbital | ϕ i i in Kohn-Sham states | ψ n i and do a self consistent calculation with an addi-tional density corresponding to the orbital. . The sim-plest way of getting the expansion coefficients is using theprojector overlaps h ψ n | ϕ i i ∼ h ˜ ψ n | ˜ p ai i which is calculatedin each iteration anyway. However, this method turnsout to be too inaccurate in the case of CO on Pt(111)due to non-vanishing projector overlaps for highly ener-getic Kohn-Sham states as shown in Fig. 8. This impliesthat the expansion coefficients depend on the number ofunoccupied bands included in the calculationTo calculate the overlaps h ψ n | ϕ i i exactly, one shouldstart by performing a gas-phase calculation of themolecule or atom which is to be used in the ∆SCF cal-culation. The pseudo wavefunction ˜ ψ i ( x ) corresponding to the orbital to be occupied is then saved along with theprojector overlaps h ˜ p ak | ˜ ψ i i and the ∆SCF calculation is -10 -5 0 5 10 15 20 Energy [eV] P D O S Projector overlapAll-electron overlap
FIG. 8: Ground state calculation of CO adsorbed on Pt(111)top site. The projected density of states of the 2 π orbitalsusing the methods of projector/pseudo wavefunction overlapand all-electron wavefunction overlap are compared. In theprojector overlap method the orbital is defined by | ˜ p π i = √ ` | ˜ p x i C − | ˜ p x i O ´ which is the orbital most similar tothe gas-phase calculation. The long high energy tail of theprojector overlap signals an inaccuracy of the method andmakes excited state calculations dependent on the number ofunoccupied bands. Thus we use the all-electron overlaps todetermine expansion coefficients in this work. initialized. In each step of the calculation we can thendo a numerical integration to obtain the expansion coef-ficients by c ni = h ψ n | ψ i i (A2)= h ˜ ψ n | ˜ ψ i i + X a,j,k h ˜ ψ n | ˜ p aj i (cid:16) h φ aj | φ ak i − h ˜ φ aj | ˜ φ ak i (cid:17) h ˜ p ak | ˜ ψ i i , where (A1) was used. Note that there is only a single sumover atoms (and only the ones in the molecule) and thatthe cross terms of pseudo/all-electron wavefunction doesnot contribute. This can be seen using the argumentsfollowing Eq. 20 in Ref. 33. APPENDIX B: CALCULATING THE INELASTICSCATTERING PROBABILITY
Here we briefly summarize the calculation leading tothe inelastic scattering probabilities in the model (5). An explicit expression for the probability has previouslybeen obtained for a single mode at initially in theground state. Here we will extend the result to an ex-plicit expression for any number of modes initially in athermal ensemble of vibrationally excited states.From the Hamiltonian (3) the differential reflectionmatrix R ( ε i , ε f ) which is defined as the probability perunit final state energy that an incoming hot electron withenergy ε i scatters on the resonance into a final state of ε f ,can be expressed in terms of a four point Green function.The inelastic part is contained in the expression:1 R in ( ε i , ε f ) = Γ( ε f )Γ( ε i ) Z Z Z dτ dsdt π ~ e i [( ε i − ε f ) τ + ε f t − ε i s ] / ~ G ( τ, s, t ) , (B1)where the Green functions is G ( τ, s, t ) = θ ( s ) θ ( t ) h c a ( τ − s ) c † a ( τ ) c a ( t ) c † a (0) i , c ( t ) = e iHt/ ~ c (0) e − iHt/ ~ , (B2)and hi denotes a thermal ensemble of oscillator states. The expression is valid for any non-adiabatic coupling function ε a ( x ), but in general it can be very hard to obtain an expression for the Green function. An exception is the wideband limit with linear coupling corresponding to the Hamiltonian (5). The Green function then becomes G ( τ, s, t ) = θ ( t ) θ ( s ) e − iǫ a ( t − s ) / ~ − Γ( t + s ) / ~ exp (cid:18) X i g i h i ( t − s ) ω i − (1 + n i ) f i − n i f ∗ i i(cid:19) , (B3)where ǫ a is center of the resonance, n i is the Bose distribution, g i = ( λ i / ~ ω i ) is the effective coupling constant ofthe mode i , and f i ( τ, s, t ) = 2 − e − iω i t − e iω i s + e − iω i τ (1 − e iω i t )(1 − e iω i s ) . (B4)The integrals in the scattering matrix (B1) can be evaluated by writing the exponentials in (B3) as Taylor expansionsand performing the τ integral. This leaves the remaining two integrals as complex conjugates which are evaluated bywriting factors such as (1 − e iω i t ) m by their binomial expansions. For a single oscillator with thermal occupation n we obtain the inelastic reflection matrix: R in ( ε i , ε f , n ) = Γ e − g (1+2 n ) ∞ X m =0 ∞ X m =0 g m + m (1 + n ) m n m m ! m ! δ (cid:0) ε i − ε f − ( m − m ) ~ ω (cid:1) × F ( m , m ) , (B5)with F ( m , m ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =0 m X j =0 ( − i + j (cid:18) m i (cid:19)(cid:18) m j (cid:19) ∞ X k =0 ∞ X l =0 g k + l (1 + n ) k n l k ! l ! 1 ε i − ǫ a − ( i − j + k − l − g ) ~ ω + i Γ / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Although the expression looks rather complicated it has a simple interpretation. Integrating over final state energiesin the vicinity of ∆ n = m − m gives the probability of transferring ∆ E = ∆ n ~ ω to the oscillator if the energy ofthe incoming electron is ε i : P ∆ n ( ε i , n ) = Γ e − g (1+2 n ) (cid:16) g ∆ n (1 + n ) ∆ n ∆ n ! F (∆ n,
0) + g ∆ n +1 (1 + n ) ∆ n +1 gn (∆ n + 1)! F (∆ n + 1 ,
1) (B6)+ g ∆ n +2 (1 + n ) ∆ n +2 ( gn ) (∆ n + 2)!2! F (∆ n + 2 ,
2) + . . . (cid:17) , where the first term is the probability of adding ∆ n bosons, the second term is the probability for removing (coupling ng ) one and adding (coupling ( n + 1) g ) ∆ n + 1 bosons and so forth.We can also evaluate the differential reflection matrix for N oscillators initially in the ground state with frequenciesand coupling constants ω i and g i respectively. The result is R in ( ε i , ε f ) = Γ e − P Ni =1 g i ∞ X m =0 . . . ∞ X m N =0 g m . . . g m N N m ! . . . m N ! δ (cid:0) ε i − ε f − N X i =1 m i ~ ω i (cid:1) (B7) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =1 . . . m N X j N =1 ( − P Ni =1 j i (cid:18) m j (cid:19) . . . (cid:18) m N j N (cid:19) ∞ X l =0 . . . ∞ X l N =0 g l . . . g l N N l ! . . . l N ! · ε i − ǫ a + i Γ / − P Ni =1 ( j i + l i − g i ) ~ ω i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It is amusing that the result (B5) for a one mode system with initial excitation number n follows from the result (B7)if we regard (B5) as a two-mode system at T = 0 with energies ~ ω and − ~ ω and coupling constants g ( n + 1) and gn respectively. For convenience we state the probability of exciting the ( m d , m z ) state from the ground state in thetwo-dimensional model with modes d and zP m d m z ( ε i ) = Γ e − g d + g z ) g m d d g m z z m d ! n z ! (B8) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m d X j d =1 m z X j z =1 ( − j d + j z (cid:18) m d j d (cid:19)(cid:18) m z j z (cid:19) ∞ X k =0 ∞ X l =0 g kd g lz k ! l ! · ε i − ǫ a − ( j d + k − g d ) ~ ω d − ( j z + l − g z ) ~ ω z + i Γ / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
1. Elastic scattering
The elastic part of the scattering matrix for a single oscillator with thermal occupation number n is R el ( ε i , ε f , n ) = δ ( ε i − ε f )(1 + 2Im G R ( ε i )) , G R ( ε ) = Z dt ~ e iεt/ ~ G R ( t ) , (B9) G R ( t ) = − iθ ( t ) h n | c a ( t ) c † a (0) | n i . We can use the linked cluster theorem to derive the retarded Green function and get the result G R ( t ) = − iθ ( t ) e − g (1+2 n ) e ( − iǫ a − ig ~ ω − Γ / t/ ~ ∞ X m =0 ∞ X m =0 g m n m g m (1 + n ) m m ! m ! e − i ( m − m ) ωt (B10)We can then calculate the elastic part of the scattering probability and get P el ( ε i , n ) = 1 − Γ e − g (1+2 n ) ∞ X m =0 ∞ X m =0 g m n m g m (1 + n ) m m ! m ! · ε i − ǫ a − [ m − m − g ] ~ ω ) + (Γ / . (B11)When calculating the elastic scattering probability oneshould also remember to include the m = m terms inB5.The n in the expressions above denote the Bose dis-tribution and not a specific state | n i , but in the contextof DIMET our main point of interest is the probabilitythat a oscillator initially in the state | n i i scatters inelas-tically to the state | n f i . However, the expression in thecase of a pure state is very similar to the thermal ensem-ble, the only difference being that we should make thesubstitution e − g i n i ( f i + f ∗ i ) → L n i ( g ( f i + f ∗ i )) (B12)in (B3), where L n ( x ) is the n ’th Laguerre polynomial.The expression involving Laguerre polynomials is some-what more complicated to handle numerically and there-fore we have chosen to work with the thermal ensemble expressions instead. In the range of parameters in thepresent work the thermal ensemble expressions are verygood approximations since L n ( x ) have the same first or-der Taylor expansion as e − nx and for t < ~ / Γ we get g i f i < . ACKNOWLEDGMENTS
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