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https://doi.org/10.1088/1361-6595/aac61ehttp://iopscience.iop.org/article/10.1088/1361-6595/aac620http://iopscience.iop.org/article/10.1088/1361-6595/aac620http://iopscience.iop.org/article/10.1088/1361-6595/aac620http://iopscience.iop.org/article/10.1088/1361-6595/aaa86dhttp://iopscience.iop.org/article/10.1088/1361-6595/aaa86dhttp://iopscience.iop.org/article/10.1088/0034-4885/58/12/001http://iopscience.iop.org/article/10.1088/0034-4885/58/12/001

Microscopic modeling of gas-surfacescattering. I. A combined moleculardynamics-rate equation approach

A Filinov1,2,3 , M Bonitz1 and D Loffhagen2

1 Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität, Leibnizstr. 15,D-24098 Kiel, Germany2 INP Greifswald e.V., Felix-Hausdorff-Str.2, D-17489 Greifswald, Germany3 Joint Institute for High Temperatures RAS, Izhorskaya Str.13, 125412 Moscow, Russia

E-mail: [email protected]

Received 9 February 2018, revised 27 April 2018Accepted for publication 18 May 2018Published 11 June 2018

AbstractA combination of first principle molecular dynamics (MD) simulations with a rate equationmodel (MD-RE approach) is presented to study the trapping and the scattering of rare gas atomsfrom metal surfaces. The temporal evolution of the atom fractions that are either adsorbed orscattered into the continuum is investigated in detail. We demonstrate that for this descriptionone has to consider trapped, quasi-trapped and scattering states, and present an energeticdefinition of these states. The rate equations contain the transition probabilities between thestates. We demonstrate how these rate equations can be derived from kinetic theory. Moreover,we present a rigorous way to determine the transition probabilities from a microscopic analysisof the particle trajectories generated by MD simulations. Once the system reaches quasi-equilibrium, the rates converge to stationary values, and the subsequent thermal adsorption/desorption dynamics is completely described by the rate equations without the need to performfurther time-consuming MD simulations. As a proof of concept of our approach, MD simulationsfor argon atoms interacting with a platinum (111) surface are presented. A detailed deterministictrajectory analysis is performed, and the transition rates are constructed. The dependence of therates on the incidence conditions and the lattice temperature is analyzed. Based on this example,we analyze the time scale of the gas-surface system to approach the quasi-stationary state. TheMD-RE model has great relevance for the plasma-surface modeling as it makes an extension ofaccurate simulations to long, experimentally relevant time scales possible. Its application to thecomputation of atomic sticking probabilities is given in the second part (paper II).

Keywords: plasma-surface modeling, low-temperature plasma, gas-surface interaction,adsorption and scattering of neutral particles, molecular dynamics, rate equations, transition rates

1. Introduction

Low-temperature plasma physics has seen remarkable pro-gress over the last decade. This concerns both fundamentalscience studies and technological applications ranging frometching of solid surfaces to plasma chemistry and plasmamedicine. In each of these applications, the contact betweenparticles of the plasma and a solid plays a crucial role. Thiscontact is very complex and includes a large variety of fun-damental physical processes such as secondary electron

emission, sputtering, neutralization and stopping of ions andadsorption and scattering of neutral particles as well as che-mical reactions. In the majority of previous studies in low-temperature plasma physics, these processes have beenomitted or treated phenomenologically. For example, in manystate of the art kinetic simulations based on the Boltzmannequation, e.g. [1, 2] or particle in cell (PIC) simulations, e.g.[3, 4] neutrals are treated as a homogeneous background, andtheir interaction with surfaces is not included in the descrip-tion. The fact that energetic neutrals maybe crucial for

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https://orcid.org/0000-0001-6978-6310https://orcid.org/0000-0001-6978-6310https://orcid.org/0000-0001-7911-0656https://orcid.org/0000-0001-7911-0656https://orcid.org/0000-0002-3798-0773https://orcid.org/0000-0002-3798-0773mailto:[email protected]://doi.org/10.1088/1361-6595/aac61ehttp://crossmark.crossref.org/dialog/?doi=10.1088/1361-6595/aac61e&domain=pdf&date_stamp=2018-06-11http://crossmark.crossref.org/dialog/?doi=10.1088/1361-6595/aac61e&domain=pdf&date_stamp=2018-06-11

secondary electron emission was demonstrated in PIC simu-lations of Derszi et al where neutrals above a threshold energyof 23 eV were traced [5]. They also took into account theeffect of the plasma-induced surface modification by usingmodified cross sections [6]. Similarly, Li et al studied theeffect of surface roughness on the field emission by includinga phenomenological geometric enhancement factor [7]. Notsurprisingly, a better understanding of plasma-surface inter-action, which would lead to predictive capability, has beenrecognized to be a major bottleneck for further progress in thefield [8, 9].

The goal of the present work is to present a microscopictheory-based approach to a specific problem of plasma-sur-face interaction: the scattering, adsorption and sticking of raregas atoms from a plasma at a metal surface. This is expectedto be particularly important at low pressures where the neutralfraction of heavy particles is by far dominating over theionized component. Our approach presents a combination ofmicroscopic modeling within a Langevin molecular dynamics(MD) approach with an analytical model, formulated as rateequations for relevant surface states. This coupled approachhas the advantage of opening the way towards ab initio basedlong-time simulations, as we will explain in detail below.

We start with a brief overview on previous theoreticalworks treating the interaction of gas atoms with solid surfacesand summarize their strengths and limitations. The energytransfer and the scattering of rarefied gases from differentsurfaces have been the subject of multiple studies. Mucheffort has been devoted to determine an accurate scatteringmodel for gas atoms at a solid surface and a realisticdescription of the involved collision processes. AlreadyMaxwell [10] proposed a simple model in his studies of gas-surface interactions, where the scattered gas atoms are sepa-rated into two fractions: one that is reflected specularly andexchanges no energy, and a second one that is accommodatedcompletely and, eventually, desorbs with an equilibriumdistribution. Other studies have been focused on the evalua-tion of the thermal accommodation coefficients [11].

The angular distributions for different gas atoms (helium,neon, argon, krypton, xenon, and deuterium) scattered by asingle-crystal tungsten (110) surface has been intensivelyanalyzed by Weinberg and Merrill [12]. A direct measure-ment of the velocity distribution of argon atoms scatteredfrom poly-crystalline surfaces has been in the focus of thework of Janda et al [13]. These measurements allowed toreveal a dependence of the average kinetic energy of scatteredargon atoms on the average incident kinetic energy and sur-face temperature.

A theoretical treatment of the interaction of Ar atomswith a self-assembled monolayer on Ag(111) in terms of thestochastic scattering theory, including direct scattering, trap-ping, and desorption, has been reported by Fan and Manson[14]. Gibson et al [15] presented a detailed study of Arscattered from an ordered 1-decanethiolAu(111) monolayer.

All these detailed experimental and theoretical studieshave proven that the use of atomic probes as scattering pro-jectiles can be a useful experimental tool for studies ofstructure and dynamical properties of surfaces. The scattered

intensities can be measured as a function of the final trans-lation energy, scattered and incident angles. Furthermore,such analyses provide important information on surface cor-rugation and temperature effects.

Presently, there exist three main theoretical researchdirections in this area. The first one is based on kinetic theorywhere important contributions are due to Kreuzer andTeshima [16] and Brenig [17, 18]. More recently Bronoldet al studied the sticking of electrons at a dielectric surface[19] and the neutralization of ions at a gold surface [20].Overall, the resulting kinetic equations provide a powerfulsemi-analytical tool to compute (mainly) stationary propertiesof gas atoms or charged particles scattering from a surfacewithin suitable many-body approximations.

The second direction is based on ab initio quantumsimulations, most importantly Born–Oppenheimer densityfunctional theory (DFT) or time-dependent DFT (TDDFT).The main advantage of the latter is that the dynamics of theelectrons and the internal excitation of adsorbed atoms andmolecules can be incorporated on a full quantum level.Among recent applications of TDDFT, we mention the ana-lysis of the chemical reaction dynamics of hydrogen on asilicon surface [21] and the energy loss (stopping power) ofions on graphene and boron nitride sheets [22]. An alternativeab initio approach is based on non-equilibrium Green func-tions (NEGF) and is advantageous when electronic correla-tions in the surface are of importance, e.g. [23, 24]. RecentlyNEGF simulations of the stopping power of hydrogen andhelium ions on correlated two-dimensional materials werepresented [25].

The third approach is semiclassical MD simulations thatare extensively used in surface science. Here, the qualitydepends on the accuracy of effective pair potentials (or forcefields) whereas details of the electron dynamics are notresolved. As examples of recent applications, we mention thesimulation of metal cluster growth and diffusion [26, 27] andof the dynamics of bi-metallic clusters [28].

The temporal evolution of the atoms trapped near thesurface is of particular interest for the understanding of theadsorption and scattering of neutral atoms. Their equilibrationkinetics is of fundamental importance for the understandingof the dependence of the sticking probabilities on the energy,incident angle and lattice temperature. However, these timedependences are typically out of the range of both kinetictheories and ab initio quantum simulations. While the formerusually concentrate on stationary properties, TDDFT andNEGF simulations are computationally extremely expensiveand are limited to very short times of the order of a picose-cond and small spatial scales. Therefore, MD simulationsrepresent the only approach capable of resolving thedynamics on sufficiently long-time and spatial scales being ofexperimental relevance. Although the typically required timestep in these simulations is well below one femtosecond,recent progress in computer power made it possible toinvestigate the relevant surface effects on a microscopic levelreaching simulation times of several hundreds of picoseconds.

In the present work (paper I and paper II) we performextensive MD simulations to study the scattering and sticking

2

Plasma Sources Sci. Technol. 27 (2018) 064003 A Filinov et al

of argon atoms at a platinum (111) surface in a time-resolvedfashion. In order to analyze the sticking and thermalizationbehavior, we introduce a novel approach: the trajectories ofgas atoms that are near the surface are sub-divided into threeclasses: trapped (T), quasi-trapped (Q) and continuum (C)states. We demonstrate that these states are the relevantobservables to analyze the sticking problem with excellentstatistics, high accuracy and temporal resolution. Further-more, we demonstrate that the fractions of atoms in trapped,quasi-trapped and continuum states obey a simple system ofcoupled rate equations. Its solution allows us to significantlyreduce the computational cost that is otherwise spent on thetemporal resolution of individual particle trajectories. More-over, we demonstrate how the transition rates between thethree states can be accurately extracted from our MD simu-lations transforming the rate equations into an, in principle,exact description. The accuracy of the latter is only limited bythe accuracy of the used pair potentials. Finally, our analysisof the temporal evolution reveals that after a characteristicequilibration time these transition rates become stationary.This means that the rate equations become sufficient to studythe dynamics of the systems for longer times without furtherneed of MD simulations. This provides the potential to sig-nificantly extend the temporal and spatial scales of thesimulations without compromising the accuracy.

The main goal of the present paper is to introduce thisnew combined MD-rate equation approach in detail and totest it thoroughly on a representative example: the scatteringof argon atoms at a platinum (111) surface. Even though forlow-temperature plasma applications other metals are morecommon, we chose platinum as a test case. Here extensiveexperimental and theoretical data are available for compar-ison, which allow us to critically assess the validity andlimitations of our model. A detailed analysis of the argonsticking probability, its dependence on temperature, incidentenergy and angle is presented in a separate paper (‘paper II’,see [29]).

The paper is organized as follows. In section 2 weintroduce the microscopic model of the gas-surface interac-tion. The setup of the MD simulations is explained insection 3. The tracking of the particle trajectories provides adirect access to time dependencies of the energy loss dis-tribution functions and the average momentum (kineticenergy). The time-resolved evolution of the trapped, quasi-trapped, and continuum states and its dependence on thelattice temperature and incident angle is discussed in detail insection 4. In sections 5 and 6 we introduce the rate equationmodel, which allows us to reproduce the MD results ofsection 4, and, moreover, has a potential to extrapolate thesedata to much longer times being not accessible by usual MDsimulations (see section 6.4). The conclusions are given insection 7.

2. Effective gas-surface interaction

In a common approach, the global interaction potential Vig of

the ith gas atom (a) with the surface (s) is decomposed into a

sum of pair potentials Vas according to

V Z V r a, 1ig

j

Nas

ij1

s

å==

( ) ( ) ( )

V rC

r

C

rbe , 1as r0

66

88

n= - -a-( ) ( )

where r rrij ia

js= -∣ ∣ is the distance between the ith gas atom

and the jth atom of the surface, Ns denotes the number ofsurface atoms, and the set of coefficients C C, , ,0 6 8n a{ } usedfor the parametrization of the Ar–Pt pair potential is specifiedin table 1. The distance to the surface, Z=zi, in equation (1a)is treated as a parameter

r Z x y Z z , 2ij ij ij j2 2 2= + + -( ) ( ) ( )

while the lateral atom position (xi, yi) is kept fixed to analyzethe dependence of the gas-surface interaction on the adsorp-tion site (see below). However, this lateral position can bevaried for the calculation of the potential energy sur-face (PES).

Such type of global interaction potential and its decom-position into pairwise terms for the Ar–Pt(111) system hasbeen recently reconstructed using the periodic DFT approach[30]. Similar analyses have been performed for the interactionbetween an argon atom and gold surfaces [31]. In general, thecomputation of such interaction potentials is still a challen-ging task, but it follows a standard scheme. Ab initio calcu-lations are performed for relatively small surface clusters,where the reconstructed pair potential(1b) can be cross-checked to compare their performance with state-of-the-artvan der Waals-corrected periodic DFT approaches. As aresult, accurate pair potentials suitable for MD simulationsare derived. The parametrization values used for the presentAr–Pt(111) system are listed in table 1.

The obtained global Ar–Pt surface interaction potential ispresented in figure 1 as a function of the height z above thesurface plane z=0. For the unrelaxed lattice the uppermostlayer is placed at z=0. The lateral position of the Ar atomhas been varied in the x–y-plane to analyze the corrugation ofthe PES. Several sites according to the lattice symmetry(‘atop’, ‘bridge’, ‘fcc’ and ‘hcp’) have been chosen, as alsospecified in figure 1 for the fcc(111) lattice. The

Table 1. Parameters of the Ar–Pt (111) interaction potential used inequation (1b), taken from [30]: 3 485.40 eV0n = , α=3.30 Å−1,C6=64.92 eV Å

6, C8=0.0 eV Å8. The equilibrium distance to the

plane z=0 and the interaction energy E V rg0 0= ( ) at the differentadsorption sites (see figure 1) are evaluated for a (7×4×2) slab(number of unit cells in the X, Y and Z directions, each with 6 Ptatoms) at the surface temperature Ts=0. The values r0

and E0

correspond to the relaxed lattice when the positions of a few upperlayers are optimized (see text).

hcp fcc atop Bridge

r0 (Å) 3.3576 3.3576 3.5058 3.3788E0 (meV) −76.8951 −76.8876 −68.8091 −75.7618

r0 (Å) 3.2412 3.2412 3.3894 3.2624

E0 (meV) −78.1302 −78.1246 −69.8872 −76.9785

3


corresponding potential energy minima and distance to thesurface are given in table 1. Their difference with respect tothe energy of the fcc site is additionally plotted in figure 1. Itcharacterizes the corrugation of the PES.

In figure 1 (top) we compare the results obtained for the‘ideal’ (unrelaxed) lattice with the optimized (‘relaxed’) lat-tice. The latter was reconstructed at the surface temperatureTs=0 by minimization of the total energy of the(7×4×2) Pt slab. It accounts for a shift of a few upperlayers to more negative values of the z-coordinate, where thebottom three layers are kept fixed at the same position as inthe unrelaxed lattice. This fact can be seen in figure 1 by ashift of the potential energy minima (see the curves labeled‘rel’) to smaller heights relative to the plane z=0. Theestimated binding energy of −78 meV is in good agreementwith the total adsorption energy of about −80 meV reportedin the experimental work of Head-Gordon et al [32]. Thisvalue is also used as the reference for the optimization of theempirical potentials aimed to reproduce the experimentalresults on the vertical Pt(111)–Ar harmonic vibrational fre-quency of r 50w ~( ) meV and on the trapping, desorptionand scattering data [33–35]. For an overview of the mostcommonly used empirical potentials we refer to the recentwork of Léonard et al [30].

In order to perform a diffusive motion on the frozensurface, i.e., lattice atoms are fixed in their equilibriumposition, the kinetic energy of the adsorbate atom shouldexceed the energy barrier ΔE at the bridge site. However, nosuch restriction applies at finite temperatures as the adsorbateatoms can continuously exchange energy with the latticeatoms. In the following, three different lattice temperaturesare simulated: Ts=80, 190, and 300 K, corresponding to6.90, 16.38, and 25.7 meV, respectively. Hence, the thermalenergy supplied from the lattice is significant to overcome theenergy barrier for the inter-site hops in all cases.

3. MD simulation of the lattice at finite temperature

We perform deterministic MD simulations to study the Ar–Pt(111) system. A (7×4×2) platinum crystal slab of 336atoms (consisting of 6 atomic layers) is used. The sample isdivided into three parts. Three lower layers form a staticcrystal. The atoms in this part are frozen at their equilibriumpositions and serve as a basement for the dynamical upperlayers. The three uppermost layers are an active zone of thecrystal. They interact dynamically with the incoming gasatoms. Two bottom layers in the active zone consist of theatoms which are used as a boundary thermostat to realisticallytreat the removal of energy from the active zone. In this waythe excess kinetic energy can dissipate from the active zone.This removal of excess kinetic energy is simulated byrestoring the temperature of the heat bath atoms using Lan-gevin dynamics [36]. The Langevin term keeps the kineticenergy of the lattice atoms at the value specified by the latticetemperature Ts. Finally, the Langevin term was switched offfor the uppermost layer, and only dynamical correlations dueto the gas-lattice (Ar–Pt) and lattice atoms (Pt–Pt) binaryinteractions are retained.

The system of N=Na+Ns atoms is described by thepotential energy

r r r rV V V , 3i

N

j

Nas

i ji j

Nss

i j1 1

a s s

åå å= - + -= = <

(∣ ∣) (∣ ∣) ( )

where Na is the number of gas atoms. The interactionsbetween the gas atoms are neglected, i.e., V r 0aa =( ) ,assuming sufficiently low gas density. This situation is quitetypical for low-temperature and low-pressure plasmas. Also,in our analysis we concentrate on the influence of the inci-dence conditions and lattice temperature on the adsorptionprocess. Effects due to pre-existing adsorbed atoms, i.e.related to a finite coverage of the surface are a topic on itsown and will be studied elsewhere. This will also require anextension of the rate equation model presented in section 5 toan inhomogeneous system.

Effective atom–atom pair potentials are used for both Ar–Pt (see equation (1b)) and Pt–Pt interactions. The interactionsbetween the Pt atoms are modeled using the modifiedembedded atom potential, which quite accurately reproducesthe spectrum of the transverse and longitudinal phonons [37].

Figure 1. Top: global interaction potential V Zg ( ) for Ar–Pt (111) fora set of adsorption sites specified by the lattice symmetry. Unrelaxed(‘unrel’) and relaxed (‘rel’) lattices are compared (see text). Bottom:corrugation of the global potential ΔE plotted with respect to theenergy of the hcp-site.

4


In the directions parallel to the surface (x–y plane) periodicboundary conditions are applied.

At the beginning of the simulation, i.e., before introdu-cing the Ar atom, the crystal is allowed to equilibrate to thesurface temperature Ts. This procedure takes about 6 ps. Theequilibration is monitored by the instantaneous lattice kineticenergy. After the equilibration phase (t>6 ps) the distribu-tion functions of the tangential (Ek

) and normal (Ek^) com-

ponents of the kinetic energy for the atoms in the threedynamical layers have been evaluated. They quite accuratelyreproduce the expected (one-dimensional and two-dimen-sional) Boltzmann distribution at the temperature Ts

P EE T

E T1

exp , 4kk s

k sp

= -^^

^( ) ( ) ( )

P ET

E T1

exp . 5ks

k s= - ( ) ( ) ( )

This procedure confirms the correct implementation of theheat bath via Langevin MD.

After the lattice has approached steady state, an Ar atomis introduced at a height z=20Åabove the surface outsidethe cut-off radius of the potential Vas. The trajectories areobtained by integrating the classical equations of motionusing a fourth-order propagator algorithm [36], to accuratelyaccount for the effect of the random force (the Langevinterm). The integration time step is fixed at 0.06 fs. Trappingprobabilities, energy exchange calculations and energy dis-tribution functions are evaluated based on samples of1000–5000 trajectories. During an ‘elementary’ event, theimpinging gas atom interacts with all atoms within the cut-offradius r 10cAr Pt ~- Å. A simulated trajectory is stopped when(i) the gas atom leaves the surface after undergoing one ormore collisions with the surface (called ‘bounces’) and attainsa distance above the surface greater than z rc cAr Pt= - and (ii)the gas atom experiences more than nb=40 bounces.

The value nb=40 of the number of bounces was chosenempirically. By tracking the temporal evolution of trajectorieswith nb40, we observe in most cases the thermalization ofthe adsorbate atoms to the lattice temperature and conv-ergence of the energy distribution functions to their quasi-stationary form on the corresponding time scale. In particular,we analyzed the accommodation of the parallel andperpendicular momentum components. Our MD simulationsat low and high temperatures reproduce the general trend of aslower accommodation of the parallel momentum component,as it was first pointed out by Hurst et al [38] and confirmed inmany other analyses [30, 39–41]. This thermalization analysisfor the system Ar on Pt (111) is presented in detail inpaperII [29].

4. Time-resolved trapped, quasi-trapped andscattered fractions

In many physical systems a strong chemical bonding to thesurface is the dominant trapping mechanism. In contrast, inthe present system the trapping process occurs due to the

physisorption potential well described by a van der Waals-type potential, seeequation (1b). Depending on the latticetemperature, the trapped particles desorb after a finite resi-dence time (see section 6.4). The central question is whetherthis time is sufficient for the adsorbate atoms to equilibratewith the surface so that they eventually desorb with a (quasi)equilibrium energy distribution function. We underline thatthis is not an academic question but one of practical impor-tance. Indeed, the theoretical description is expected to sim-plify significantly in cases where thermalization is observed.

This problem was first put forward by Maxwell in hisstudies of gas-surface interactions [10] and was further takenup by Knudsen [42]. The basic concept is based on thethermal accommodation and the efficiency of energyexchange at the gas-surface interface. The latter cruciallydepends on both the incident gas parameters and surfacecharacteristics. In the special case of complete accommoda-tion the desorbed particles leave the surface with a distribu-tion function specified by the Knudsen flux [14]. This type ofassumption is frequently used to explain experimental data forthe sticking probabilities of the adsorbate and for the dis-tribution functions of energy, momentum and flux vector ofthe thermally desorbed atoms.

One of our goals is to test this assumption by micro-scopic modeling of the gas-surface interactions and analyzethe equilibration kinetics. Using realistic calculations basedon the microscopic model introduced in sections 2 and 3, weaim at reproducing the available experimental data for thesticking probabilities [43] and at providing a general frame-work for the analysis of the temporal evolution of the atomstates localized near the surface. While we treat the scatteringclassically, quantum–mechanical effects are included bymeans effective interaction potentials, which are constructedfrom ground-state DFT calculations [30]. Inelastic effectsobserved in the scattering events at large energies and highsurface temperature are fully taken into account by the Lan-gevin MD scheme [36].

4.1. Classification of particle trajectories: trapped, quasi-trapped, and scattering states

The scattering from the surface is modeled by using mono-energetic gas atoms with fixed values of incident kinetic energyEi and angle θ. A single collision with the surface introduces atransition from an initial momentum, pi, to a new momentumstate pf . The final states ‘f’ are distinguished by the surface

normal, pf^, and parallel, pf

, components, i.e. p p pf f f= +^ .

Correspondingly, the kinetic energy Ek of a particle withmomentum k is split into two orthogonal contributions,

E E E a, 6k k k= +^ ( )

Ep

mb

2, 6k

2=^

^( ) ( )

Ep

mc

2. 6k

2=

( ) ( )

In addition, every particle moves in the potential landscape ofthe surface atoms that is characterized by the local surface

5


binding (physisorption) potential, V, giving rise to the totalenergy

r r rE E V . 7k= +( ) ( ) ( ) ( )

In the following, the dependence of {E, Ek, V} on the localatom position r is not explicitly specified. In addition, allenergies and their corresponding distribution functions areevaluated at a minimum distance r from the surface, wherethe total energy of a gas atom is approximately conserved.The temporal evolution of the total energy E(t) is shown infigure 2. Here and in the following, we use the valuet a m E10 6.53 ps0 3 0 Ar h= =· as a time unit, where a0 isthe Bohr radius, mAr denotes the atomic mass of the argonatoms, and Eh=27. 211 eV is the Hartree energy. It becomesclear that plateau regions of E(t) are well separated from theenergy ‘jumps’ corresponding to the inelastic collision pro-cesses. During this inelastic process, there is a strong energyexchange with the surface atoms and, therefore, the evaluationof equation (7) becomes meaningless.

After interaction with the surface a fraction of particles isscattered back, whereas another fraction is (temporally)trapped. A classification of the states is straightforward byanalyzing the particle energy.

I. Scattering states: the condition to leave the surface dueto the momentum exchange is E V 0k + >

^ . The back-scattered (unbound) particles are referred to as ‘con-tinuum’ (‘C’) states.

II. Bound states: particles with E V 0k + <^ remain

localized near the surface. The localization dependssolely on the normal component, while the parallelcomponent can be arbitrary. Therefore, depending onthe sign of the total energy(7), such states can befurther sub-divided into

a. Trapped (‘T’) states: these are particles with E

^( ) , where both energies are evaluated at the atomposition r. The corresponding trajectories have no turningpoint and leave the surface region. In between, such trajec-tories can experience multiple bounces and transitionsbetween the trapped (n(t)=−1) and quasi-trapped (n(t)=0)states depending on the sign of the total energy E(t).

By analyzing a statistical ensemble of the trajectories of n(t), a first ‘physically’ relevant observation can be made.

Figure 2. Temporal evolution of the surface states n(t) (see maintext) along three dynamical trajectories based on the sign of the totalenergy E(t) and the trapping condition, E V 0k + <

^ . The curve z(t)shows the distance of the particle from the surface. The latticetemperature equals Ts=300 K, and the incidence parameters of theatoms are Ei=36.34 meV and θ=60°.

6


Before a particle is desorbed, it typically gets excited to aquasi-trapped state. The direct excitation probability from atrapped to a continuum state is significantly reduced at lowlattice temperatures (e.g. Ts=80 K), but it steadily increaseswith Ts, as will be shown in detail in section 6.

For large statistical ensembles of the trajectory states n(t)we can explicitly evaluate the three fractions N t t Nj jn=( ) ( )with j=T, Q, and C, where tjn ( ) is the number of trajectorieswith n(t)=j at a given instant of time. They satisfy thenormalization condition NQ(t)+NT(t)+NC(t)=1.

Figure 3 shows the temporal evolution of these threefractions of Ar atoms influencing a Pt(111) surface atTs=80 K for two conditions of incidence as well as thecorresponding average number of bounces and their variance,

nbs , on the same time scale. The value Rst(t)=NQ(t)+NT(t)=1−NC(t) taken at 0.4t/t00.6 defines theinitial sticking probability. Detailed results on the presentsystem can be found in paperII [29]. This quantity remainsunchanged until a second bounce takes place. On average,this second bound occurs around t∼0.7t0.

At low lattice temperature and incident atom energy, theinitial sticking fraction remains quite large: Rst≈0.75(0.65)for θ=30°(60)°. The continuum states NC(t) show a satur-ation already for t1.5t0 corresponding to 9.8 ps. Withinthis period the trajectories experience on average nb∼5bounces with the surface.

In contrast, the ‘pure’ trapped states show a convergentbehavior at much later times, mainly due to an exponential-like decay of the quasi-trapped states. Within the time interval0.4

linear dependence of the transition probability on τ is a resultof the perturbation theory and is valid for small time differ-ences τ. Third, the fact that the probability depends only on τ(and is independent of t) implicitly assumes that the system isstationary. Finally, the appearance of the delta function is aconsequence of the Markov approximation which assumesthat the correlation time has passed, and the energy spectrumhas become stationary, see e.g. [23, 44]. All these assump-tions are justified in case of a macroscopically stationarysurface which is only weakly perturbed by the scatteringprocess of the adsorbate atom.

In particular, we can safely assume for collisions ofatoms with thermal and subthermal energy that the initial andfinal states of the substrate remain close to thermal equili-brium. The recoil energy, initially transferred to the surfacelayer, is rapidly dissipated to the bulk due to fast atomicvibrations taking place with the Debye frequency and a strongcoupling between the substrate atoms when compared to thecoupling with the adsorbate atoms. As a result, a saturation tothe thermal equilibrium within the substrate is expected ontime scales much shorter than the adsorbate thermal accom-modation time.

Hence, as a further simplification, we can introduce thetransition probabilities averaged over the initial states of thesubstrate (specified by the Boltzmann factor rn)

T f T i . 10fi,

2å m t n rá ñ = á ñm n

n∣ ∣ ∣ ∣ ˆ ( )∣ ∣ ( )

This allows us to obtain the temporal evolution of the system ofgas atoms alone, which is obtained with the Pauli-Ansatz [17]

n t T n t , 11fi

fi iåt+ = á ñ( ) ∣ ∣ ( ) ( )

which assumes low gas atom density so that all scatteringevents can be treated as independent of each other. Usingthe definition of the kinetic coefficients according toT Tfi fi t= á ñ¯ ∣ ∣ , we end up with the system of coupled kineticequations between the initial and final states

n t T n t , 12fi

fi iå=˙ ( ) ¯ ( ) ( )

where the limit τ → 0 has been taken. This result is wellknown and directly expresses the connection between micro-scopic calculations [45, 46] and kinetic theory [47, 48].

As a next step, we explicitly specify the quantum num-bers {f, i} of the gas atoms moving near the surface. Sincequantum diffraction effects can be neglected due to the largemass of the gas atoms, a quasi-classical treatment can be used.Thus, it is sufficient to specify the states by the initial andfinal momenta p p,i f{ }, which are additionally split into atangential and parallel component relative to the surface.Consequently, we introduce the following notations

p pn t n p t n t n p t, , , , , , 13i i i f f f= =^ ^ ( ) ( ) ( ) ( ) ( )

p pT T p p, , , 14fi f f i i=^ ^ ¯ ¯ ( ) ( )

in the kinetic equations. Note that the averaged transition rate T̄(14) depends solely on the incoming and outgoing momentum.

As a next step, we assume that the transition rate Tfi¯has only a weak directional dependence on the anglebetween the vectors pi

and pf . Its main dependence results

from two scalars, namely the perpendicular and parallelkinetic energy components of the adsorbate atom, i.e., Tfi »¯

T , , ,fi f f i i ^ ^ ¯ ( ). This assumption is important for the

derivation of the rate equations presented below. It can bejustified by the theoretical treatment of a classical collisionfrom vibrating surfaces [49–51].

In particular, in [52, 53] an explicit expression for thezeroth-order reflection coefficient R0 has been derived for anatomic projectile colliding with a surface consisting of dis-crete scattering centers (of mass M) whose initial momentaare given by an equilibrium distribution at the lattice temp-erature Ts. This expression is given by

p p pR

E

m

p k T E

E E E

k T E

d ,

d d 8

exp4

, 15

f i

f f

f

i

fiB s r

f i r

B s r

0 2

3 42

1 2

2

pt

pW

=D

´ -- + D

D

^

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) ∣ ∣∣ ∣

( )( )

and shows a dependence on several key parameters. Here, pi^

is the z-component of the incident momentum, fi 2t∣ ∣ is theform factor of the scattering center, which depends on theinteraction potential, and Ef (i) is the kinetic energy after(before) the collision. The only parameter, which contains adirectional dependence, is the recoil energy expressed as

p p pE

M

p p

M

p p

M M2 2 2.r

fi f i f i f i2 2 2 2 2

D =D

=-

+-

-^ ^ ( ) ( )

Due to the inelastic and Brownian-like character of the scat-tering processes, the contribution of the first two terms shoulddominate. The directional dependence (third term) is expectedto be weak. Thus, we can use E E , , ,r r f f i i D » D

^ ^ ( ).Finally, it is assumed that the scattering amplitude fit∣ ∣ is aconstant, with a value derived for hard sphere scattering.

Now we can proceed and explicitly define the populationof surface states and the inter-state transition probabilities bythe dependence on the kinetic energy components

n t n t n t n t, , , , , , 16i i i f f f = =^ ^ ( ) ( ) ( ) ( ) ( )

T T , , , . 17fi f f i i =^ ^ ¯ ¯ ( ) ( )

In the following we omit the subscripts ‘i’ and ‘f’ and indicatethe initial and final states, instead, by using different energysymbols:

, , 18i i =^ { } ( )

, . 19f f ¢ =^ { } ( )

8


Following the detailed discussion given in section 4, allpossible surface and scattering states can be classified intothree categories using energy criteria. Now, the trapped,quasi-trapped and continuous particle fractions introduced insection 4 can be explicitly defined via integration of the time-dependent energy distribution functions, which define thepopulation of different surface states, according to

N t n t n t

a

d d , , d , ,

20

T

V V

0 0 T

ò ò ò= =^-

^

W

^

( ) ( ) ( )

( )

∣ ∣

N t n t n t

b

d d , , d , ,

20

Q

V

V0 Q

ò ò ò= =^

-

¥^

W^ ( ) ( ) ( )

( )

∣ ∣

N t n t n t

c

d d , , d , .

20

CV 0 C

ò ò ò= =¥

^¥

^

W

( ) ( ) ( )

( )∣ ∣

Here, we introduced the shorthand notation dsòW with

s=T, Q, C for the distinction of different states and theintegration limits. This energy integral always comprises adouble integration over the tangential and normal kineticenergy components. The upper integration limit for the tan-gential component specifies that the trapped and quasi-trap-ped states stay localized (bound) near the surface due to thecondition V 0- +

Hence, the energy distribution function is non-stationary aswell until the system reaches quasi-equilibrium.

6.1. Reconstruction of the transition rates from the MDsimulations

Now we demonstrate how the rates Tαβ(t) can be extractedfrom the MD simulation data. In figure 4 we plot the changein the population of three states (Q, T and C) due to the nettransition fluxes

N t T t N t t ad d , 27QT QT T=( ) ( ) ( ) ( )

N t T t N t t bd d , 27TQ TQ Q=( ) ( ) ( ) ( )

N t T t N t t cd d . 27CQ CQ Q=( ) ( ) ( ) ( )

Two incident conditions are compared, which correspond tothe energy Ei=15.93 meV at the angle θ=30° andEi=36.34 meV at θ=60°, respectively, for the latticetemperature Ts of 80 K. Each of the decay channels can beuniquely identified by analyzing the initial and final statealong the trajectory for every bounce event shown in figure 2for several examples. The transition rate can be extracted byperforming the numerical differentiation

T tN t

t N t

d

d

128=ab

ab

b( )

( )( )

( )

for the curves shown in figure 4.

The following procedure has been used. First, the MDdata have been smoothed by a Gaussian kernel

f tG t

G tf t

G t G t G t

,

e , , 29

Gj

Nj

j

jj

N

j

1

1

t t j

h

2

2 2

å

å

=

= =

=

-

=

-

( )( )( )

( )

( ) ( ) ( ) ( )( )

where h tn= D· ( 2 10n = ¼ ), and N is the total number ofpoints on the curve. The new values are evaluated using theweighted contribution on neighboring points, and, hence, thestatistical fluctuations at each point tj are suppressed. Thedifferentiation of fG with respect to time leads to

tf t

G t

G t

t t

hf t f t

d

d. 30G

j

Nj j

j G1

2å= --

-=

( )( )( )

[ ( ) ( )] ( )

The smoothness of the derivative can be controlled by theparameter h and adjusted to give better agreement with theMD data. Typically, the value 6n ~ was found to be a rea-sonable choice.

Alternatively, the transition rates can be expressed via theintegral form

N t T N T N

N t T N t t

d d d ,

d 0, d , .

31

t t

tE

t

t

E E E

0E

E

Eò òt t t t t= += +

ab ab b ab b

ab ab b

>( )∣ ( ) ( ) ( )

( ) · ( )( )

Here, we used the assumption that the system state b hasreached a quasi-equilibrium state for tt E. Then, thetransition rate has only a weak time dependence, i.e.,T Tt

EEt »ab t ab( )∣ . For the example shown in figure 3, the

equilibration time t E is about 3t0. Finally, the quasi-equili-brium transition rate can be determined from equation (31) asthe ratio of the integrated population of states

T tN t N t

N t t

d d 0,

d ,. 32E

E

E=

-ab

ab ab

b( )

( ) ( )( )

( )

By a proper choice of the equilibration time t E, the estimatedrate T tEab ( ) should exhibit only a weak time dependence andrepresent the asymptotic limit of the more general time-dependent rate in equation (28).

The data N Nd , dab b{ } are provided by the MD simula-tions. To determine the transition rates and the population ofstates more accurately, we have used the statistical averagesover several thousand trajectories. We performed the analysissimilar to figure 2 for each set of initial parameters (Ts, Ei, θ)and determined the fluxes between different states.

6.2. Test of the approach

The comparison of the transition rates given by equations (28)and (32) is demonstrated in figure 4. We choose the time ofequilibration t t4E 0= for the rate TQT and t t1.2E 0= for TTQ,TCQ. The results of equation (28) are represented by the dots,and those from equation (32) by the solid lines. Both shouldmatch at t=t E. The choice of t E in each case needs someadjustments, such that T Eab should be nearly constant for

Figure 4. Left: time dependence of the transition rates Tαβ (t). Themono-energetic beam has the energy Ei=15.93(36.34)meV for theincident angle θ=30°(60)°. Lattice temperature: Ts=80 K. Right:change in the population N td ab ( ) of the three states (a = Q, T, C)with the normalization N t N t N t 1Q T C+ + =( ) ( ) ( ) due to the decaychannel b a . The dashed line, deviating from the MD data(dotted curve) at small times, is the prediction of the rates at quasi-equilibrium extrapolated to t tE .

10


t tE> . The quality of the quasi-equilibrium approximationcan be checked on the right panel. The rates Tαβ(t), recon-structed from equation (28), accurately fit the MD data (pre-sented by the dotted curves) for all simulation times and thechanges in state populations ( Nd QT , Nd TQ and Nd CQ).

Due to a statistical noise present in the MD data, theextracted rates exhibit artificial oscillations, see e.g. TCQ(t). Atlonger times they can be successfully removed using theequilibrium estimator equation (32), which is much lessinfluenced by the statistical noise. The corresponding equili-brium rate T Eab is shown by the horizontal dashed curves onthe left panels. In particular, the rate TEQT for the incident angleθ=30°(60°) can reproduce the change in the population ofquasi-trapped states, N td QT ( ), for t t3 0> (t t4 0> ). The leftpanel in figure 4 shows that the corresponding non-equili-brium rate TQT(t) actually converges slowly to TQT

E on a timescale, which depends on the incident angle.

A similar analysis can be performed for TTQE and TCQ

E .Here, we observe that the quasi-equilirium assumption can beintroduced at a significantly earlier moment. Using t t1.2E 0=we can nearly exactly reproduce the MD data (see the rightpanel) for the conversion of the quasi-trapped states to thetrapped state, i.e., N td TQ ( ), and with some deviationsobserved at t t2 0< the desorption of the quasi-trapped statesto the continuum, i.e., N td CQ ( ). The rates TTQ(t) and TCQ(t)stay practically constant over the entire simulation period,starting at t∼t0. However, at t

approximately holds. Indeed, this condition can be satisfiedfor the case shown in figure 5 when the ratio of both popu-lations increases to NT(t)/ NQ(t)4 for t t2 0> . The proofwhy the relation(33) should hold in general will be given insection 6.4.

One can use the rates shown in figure 6 to analyze howfast the system reaches quasi-equilibrium. Using the data forTQT(t) and TTQ(t), we estimate t t1.5E 0» for Ts=300 K andt t3E 0» for Ts=80 K. Hence, a higher lattice temperaturefavors a faster adsorbate equilibration. A more quantitativediscussion of the convergence to quasi-equilibrium is pre-sented in paperII [29].

The corresponding analysis of the temperature effects forthe larger incident angle θ=60° is shown in figures 7 and 8for the two temperatures Ts=190 and 300 K. Due to thelarger incident angle, the initial population of quasi-trappedstates is above 40% and slightly decreases with increasingtemperature. The population of trapped states is around 20%for both the surface temperatures. While the values specifiedat the time of the first bounce (t t0.5 0 ) are similar, theirtemporal evolution is quite different due to an enhancedthermal desorption at the larger temperature Ts=300 K. Thiscan be clearly resolved from the temporal evolution of thecontinuum states, given by the curves NC(t). The trappedfraction NT(t) seems to saturate during the time interval

t t1.5 20 , and it subsequently decreases for t t2 0> dueto thermal desorption going faster at Ts=300 K. This canalso be identified by a faster increase of the continuum frac-tion NC(t). During the period t t2 60 the NT(t) fractionis reduced by a factor of 1.5 from 30% to 20%, while it isreduced only by 2% at Ts=190 K.

In contrast, the Q states show only a weak dependence onTs. They decay with a similar slope for both lattice tem-peratures. As in figure 5, we observe a fast conversion via thetwo channels Q T and Q C at first. The first channeldominates due to a high rate TTQ (see figure 8). Its valuedecreases only slightly with increasing temperature, whereasthe asymptotic value of the two other rates TEQT and T

ECQ at

quasi-equilibrium drops by a factor of 2 by lowering the

temperature from Ts=300 to 190 K. This finding seemsquite reasonable. The de-excitation transition takes place,when an excited state (a quasi-trapped trajectory) releases anenergy to go into a lower energy state (a trapped trajectory).

In contrast, the two transitions T Q and Q Crequire that a finite portion of energy should be supplied fromthe lattice. In this case, the excitation probability should scalewith the population of phonon modes and depend on multi-phonon excitations. Here, a strong temperature dependence isexpected. Indeed, the data for TQT and TCQ presented infigures 6 and 8 confirm this expectation and demonstrate aclear temperature dependence. However, if the rates arecompared at the same lattice temperature, but different inci-dent angles (see θ=30° and 60° in figure 4), the differencein the rates does not exceed 10%.

We can conclude that the rate equations provide a veryuseful tool to analyze the non-equilibrium kinetics during thefirst few picoseconds. The temporal evolution of the popu-lation of different states can be successfully described by thenet fluxes in terms of a set of statistically averaged parameters—the transition probabilities Tαβ(t). These transition rates canbe accurately extracted from the MD data by analyzing thetemporal behavior of the particle trajectories. Typically, weobserve that the saturation of the transition rates at theirequilibrium values can be reached within t t t3 6E 0 0~ - , i.e.,20–40 ps, for incident energies below 100 meV. The equili-bration takes longer for larger incident angles/energies andlower lattice temperatures.

6.3. Temperature dependence of the transition rates

Results for the transition rates at quasi-equilibrium for variousincidence conditions, i.e., different incident energies Ei,incident angles θ and surface temperatures Ts, are summarizedin figure 9.

First, we discuss the de-excitation transition from Q to Tstates (Q T ). The corresponding rate TETQ exhibits only aweak dependence on temperature and incident angle/energy.

Figure 7. (a) Time dependence of the continuum (NC), trapped (NT)and quasi-trapped (NQ) fractions for Pt (111). The mono-energeticbeam has the energy E 36.34 meVi = and the incident angleθ=60°. Two lattice temperatures are compared: Ts=190 K and300 K. (b) Average number of bounces n tbá ñ( ). Pairs of solid (dash)lines correspond to the upper and lower bounds given by thevariance: nb nbsá ñ .

Figure 8. Left: time dependence of the transition rates Tαβ (t) for thecase shown in figure 7. Lattice temperatures: Ts=190 and 300 K.Right: change in the population N td ab ( ) for both temperatures.

12


It stays practically unchanged for incident energiesE 130i meV and θ=30° when the lattice temperature isvaried in the range K T K80 300s . Hence, the binaryatom–atom collisions play a major role here, while surfacetemperature effects are secondary.

The value TETQ increases when the incident angle is closerto the surface normal. It is about 20%–30% larger for θ=30°than for θ=60°. However, this comparison is performed atdifferent incident energies to guarantee that the initial stickingprobability is the same for both the incident angles. If therates TETQ are compared at similar incident energies, theobserved difference is reduced. This becomes obvious e.g.when comparing the cases Ei=49 meV at 30° versus36 meV at 60° or 131 meV at 30° versus 141 meV at 60° infigure 9. A more detailed analysis of the (θ, Ei)-dependencewould require a larger set of data.

Now, we consider the transition rates to a higher energystate: T Q , T C and Q C . They reveal a strong,partially linear dependence on the lattice temperature. Inaddition, the rate TEQT shows a clear dependence on the initialenergy Ei, whereas the rate TCT is almost independent of Ei.Such behavior originates from a different portion of energytransferred from the lattice and required for each type ofexcitation. Much less energy is required for the T Qtransition, when only the parallel component of kinetic energyneeds to be changed to make the total energy positive. Themost probable contribution is expected from the trapped statesin the high-energy tail of the distribution function. Here, somecorrelations with Ek

in the incident beam should be present. Incontrast, during the T C excitation the normal componentEk

^ needs to be changed significantly by an amount com-parable with the depth of the physisorption well E0∣ ∣ beingabout 80 meV for Ar on Pt(111) to bring a particle to thecontinuum. Here, the correlations with the incident energiesbelow E0∣ ∣ should be small. Note also that the transition

probability TECT is a factor of 3–4 smaller than that of TEQT and

TECQ on the average. Therefore, the most probable excitationto the continuum is a two stage process: T Q followedby Q C .

The transition rates presented in figure 9 have a directpractical application. In combination with the rate equations,they can be used to extrapolate the temporal evolution of thestate populations to longer time scales being not accessible byusual MD simulations. In particular, this is important for theanalysis of thermal desorption at low lattice temperatures,such as Ts=80 K, when a significant depletion of the trappedstates can be observed only on the time scales exceedingthose that are used in the present simulations, i.e., for t t6 0(40 ps). Some applications of this idea will be discussed morein detail below.

6.4. Analytical solution of the rate equations

In the present section we present the analytical solution of therate equations introduced in section 6.1 and demonstrate itsefficiency to predict the temporal evolution for longer times.In the end, we derive the estimator of the average residencetime of the adsorbate atoms trapped on the surface prior totheir thermal desorption.

In case the transition rates are time-independent, the rateequation model reduces to a system of homogeneous lineardifferential equations of first order and can be solved analy-tically. Therefore, we rewrite equations (26a)–(26b) in matrixnotation according to

NR N

t

tt

d

d. 34=

( ) · ( ) ( )

Here N t N t N t,1 2=( ) { ( ) ( )} is a column vector with the ele-ments N t N t tT E1 = +( ) ( ), N t N t tQ E2 = +( ) ( ), and R is a2×2 matrix of the transition rates with the elements

R T T R T

R T T R T

, ,

, . 35CT QT TQ

CQ TQ QT

11 12

22 21

=- + ==- + =

( )( ) ( )

Notice that the fraction of continuum states is not consideredhere as it follows directly from particle number conservation.

Once the matrix R is diagonalized, the eigenvalues {λi}and the eigenvectors ni{ } with i=1, 2 define the completesolution which can be written in the form

N nt C e . 36i

it

i1

2iå= l

=

( ) ( )

The solution of the eigenvalue problem is given by

n n

R R

R R R R

R R R R

1

2

4 ,

, , , . 37

1 2 11 22

22 112

12 21

1 1 22 21 2 2 22 21

l

l l

=- +

- +

= - = -

⎡⎣⎢

⎤⎦⎥

∣ ∣ ∣ ∣

(∣ ∣ ∣ ∣) )

( ) ( ) ( )

( )

Here, we used explicitly the fact that the diagonal elementsare negative, R R11 11= -∣ ∣ and R R22 22= -∣ ∣, as it followsdirectly from the definition(35).

Figure 9. Equilibrium transition rates TEab between different states(trapped (T), quasi-trapped (Q) and continuum (C)) for differentincident energies Ei at the three lattice temperatures Ts=80, 190and 300 K and two incident angles θ=30° (left panel) and 60°(right panel).

13


The expansion coefficients {Ci} in (36) can be found byinverting the initial conditions

N nC

N N t N N t

0 ,

0 , 0 . 38i

i i

TE

QE

1

2

1 2

å=

= ==

( )

( ) ( ) ( ) ( ) ( )

They depend on the population of the trapped and quasi-trapped states at the quasi-equilibration time t E and read

C N RN

R

10

0.1 2

1 21 2 1 22

2

21l ll=

-- -

⎛⎝⎜

⎞⎠⎟( ) ( )

( )( ) ( )

We summarize the result by writing the complete temporalevolution in the form

N t C R C R

N t C R C R

e e ,

e e . 39

t t

t t1 1 1 22 2 2 22

2 1 21 2 21

1 2

1 2

l l= - + -= +

l l

l l

( ) ( ) ( )( ) ( )

These results can be further simplified by taking into accountthat the transition rate TTQ has the largest value (see figure 9).The relation T T T T, ,TQ CQ CT QT , in its turn, leads toR R R R, ,22 11 12 21∣ ∣ ∣ ∣ which allow us to further simplify thesolution by an expansion in the small parameterg R Rij 22= ∣ ∣. When considering only the leading terms, theeigenvalues and eigenvectors become

R O R g a, 401 2 11 22 22 4l = - D +(∣ ∣ ) ( · ) ( )( ) ( )

R RR R

Rb2 , , 4022 11

12 21

22lD = - + D D =∣ ∣ ∣ ∣

∣ ∣( )

n nR R c, , , , 401 21 2 21l= D - D = -D( ) ( ) ( )

C NN

Rd

10

0, 401 1

2

21l=

D+ D

⎡⎣⎢

⎤⎦⎥( )

( ) ( )

C NN

Re

10

0402 1

2

21ll= -

D- D - D

⎡⎣⎢

⎤⎦⎥( ) ( )

( ) ( )

and the dynamics of N1,2(t) are given by

N t NR

RN

N t NR

N

0 1 e e

1 0 e e ,

0 e 1 e

0 e e . 41

t t

t t

t t

t t

1 1

12

222

2 2

211

1 2

1 2

1 2

1 2

g g

g

g g

l

= - +

+ - -

= + -

+D

-

l l

l l

l l

l l

( ) ( )[( ) ]

( )∣ ∣

( )[ ]

( ) ( )[ ( ) ]

( )[ ] ( )

The leading terms are easily identified by the smallness of theparameter γ=(Δ/Δλ)=1.

Using this representation (41), we analyze both the short-time and the long-time behavior. The expansion around theinitial time t=0 yields

N t N R N t R N tN t N R N t R N t

0 0 0 ,0 0 0 , 42

1 1 11 1 12 2

2 2 22 2 21 1

= + += + +

( ) ( ) ( ) · ( ) ·( ) ( ) ( ) · ( ) · ( )

showing that the populations increase linearly both with timeand the initial rate values.

To analyze the long-time behavior, we order the eigen-values, 1 2l l

limit of both concentrations(43), the residence time can bedefined solely in terms of the initial concentrations and thestationary transition rates

tN N

Na

1ln

0 0, 47R

1

1 2

lg

=+⎜ ⎟⎛⎝

⎞⎠

[ ( ) ( )] ( )

N NR

RN

NR

N b

1 0 0

0 0 . 47

112

222

221

1

g

l

= - +

+ +D

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( ) ( )∣ ∣

( )

( ) ( ) ( )

Note that the ratio of concentrations in the logarithmic term istypically of the order of one so that the estimate(46) resultsagain from (47a) with e 1g = - .

Finally, we summarize our results for t R, and itsdependence on the incidence conditions and the latticetemperature in table 2. As expected, t R depends only weaklyon the incidence conditions for the same lattice temperature,where the variations are within the statistical errors. Thisconfirms that any memory of the incidence conditions is lostwithin t t3 20E 0= » ps, and the analytical solutions(43),constructed for t tE , accurately describe both the decayof the adsorbate and its characteristic residence time.Our simulations predicted that the residence times aret R=(170–180) ps for Ts=190 K and 50 55-( ) ps forTs=300 K.

For Ts=80 K, we need to extend our simulationsbeyond t t6 400= » ps to provide a more accurate estima-tion of the transition rates Tij

E and t R. A noticeable decay ofthe trapped fraction due to thermal desorption just starts at40 ps and, therefore, the simulations need to be extended to atleast 100 ps to ensure that the constructed analytical solutionfits well the MD simulation data similar to the cases presented

in figure 10. Still some estimate can be given based on theparametrization of the experimentally determined desorptiontimes by a Frenkel–Arrhenius formula

t t e . 48R p U k TB= ( )( )

Here the prefactor tp typically varies for physisorbed gasesfrom 105 ps for helium desorbing from constantan [54] to10−2 ps for xenon desorbing from tungsten [55]. By expres-sing the adatom desorption frequency as t1 Rn = , the pre-factor tp can be interpreted as an average time between thesuccessive bounces on the surface, and the Boltzmann factore U k TB- ( ) as the static desorption probability, with U beingcomparable with the depth of the surface potential E0. Wefound that such interpretation applies very reasonable to oursystem. We estimated tp=1.52 ps and U=61.7 meV, byapplying the fit(48) to our data for t R at Ts=190 and 300 K(see table 2). The temperature T was chosen to be the effectiveadsorbate temperature T [29], where we have usedT 150 K = for Ts=190 and 200 K for Ts=300 K. Theobtained fit parameters well agree with the depth of physi-sorption potential, E 780 ~∣ ∣ meV (see table 1), and theaverage time between the bounces, which varies in therange from 0.93 to 1.10 ps for the lattice temperature

T80 K 300 Ks . Here, we have used the time dependenceof average bounce number n tbá ñ( ), presented in figure 5and 7.

Finally, the extracted fit parameters, tp and U, allow toestimate the residence time, t R∼11000 ps, at the latticetemperature Ts=80 K (using T 80 K = ). This value agreesquite well with the lower bound for t R presented in table 2.

7. Conclusion

The studied kinetics of adsorption and desorption of atomicprojectiles physisorbed on solid metallic surfaces is the mostelementary process serving as a starting point for a detailedunderstanding of more complex processes, i.e. for the che-misorbed species which can undergo substantial structuraland electronic modifications. In the present study, werestricted ourselves to physisorption at low coverage so thatthe interaction between gas particles in the adsorbate can beneglected.

Our main motivation was to explore the capabilities ofaccurate MD simulations for the sticking of argon atoms on ametal surface. However, the main obstacle is the enormousdifference in the time scales of the atomic motion being about10−13 s which has to resolved in the simulations and of thedesorption processes ranging from 10−6 to few seconds inexperiments. These scales cannot be reached with MDsimulations, even on supercomputing hardware, without fur-ther approximations. Therefore, we developed a newapproach that couples MD simulations to an analytical rateequations model which has allowed us to extend the calcu-lations to several hundreds of picoseconds, and furtherextensions are possible as well. The rate equations are nottrivial, as one first has to realize that atoms near the surfaceafter the first scattering event have to be classified into three

Table 2. Residence time of Ar on Pt (111) estimated fromequation (47a) (with e 1g = - ) for different incidence conditions(θ and Ei) and lattice temperature Ts. The last column shows thedeviation from a more simple estimate(46).

Ts θ Ei (meV) tR (ps) t tR R1l

80 K 30° 15.9 >7900 0.9960° 36.3 >8000 0.98

190 K 0° 12.8 184(10) 0.9530° 15.9 179(10) 0.95

49.4 165(10) 0.92105.3 182(10) 0.88

60° 36.3 163(10) 0.91112.7 169(10) 0.80

300 K 0° 12.8 57(5) 0.9030° 15.9 56(5) 0.90

49.4 52(5) 0.87105.3 50(5) 0.85131.1 49(5) 0.85

45° 21.6 55(5) 0.90100.7 46(5) 0.82

60° 36.3 56(5) 0.86112.7 50(5) 0.78

15


possible categories: continuum (desorbed), quasi-trappedstate (moving in the surface plain) and trapped ones.

Most importantly, we have demonstrated that thiscombination of MD and rate equations can be performedsuccessfully without loss of accuracy, for times exceedingthe equilibration time t E. The key is that the time t E andall relevant input parameters to the rate equations–thetransition rates between the particle categories—are directlyextracted from the MD data for which a reliable procedurehas been developed. The rates are found to have a strongtime dependence, during the initial period, until theysaturate for times t around the equilibration time t E,after which they remain constant. i.e. T t Tt t

EE »ab ab>( )∣ .

These stationary values are determined by the shape of thequasi-equilibrium energy distribution function of the atomsin contact with the surface which is discussed in detail inpaperII [29].

Let us now critically discuss limitations and possibleimprovements. First, our statistical approach, of course, doesnot contain a microscopic treatment of the individual quantumscattering events. The dynamics were treated semi-classicallyby adopting binary interaction potentials that reconstruct thePES precalculated by state-of-the-art DFT calculations [30].Further improvements are possible by using more accurateforce-fields in the MD simulations.

Second, the derivation of the rate equations in section 5was based on several assumptions: our starting equation (9)for the transition probability was based on a standard Mar-kovian approximation for the interaction with the dissipativesubsystem. Specifically, we ignored possible correlationsbetween different adsorbate states, i.e., the memory effects forinter-state transitions. Next, we have assumed that the pho-non-induced transitions dominate, while thermal excitationsof the solid are rapidly dissipated due to fast vibrations andthe coupling of the substrate atoms. At the same time, theexcellent agreement with the MD simulations provides strongsupport for these assumptions. On the other hand, weunderline, that our rate equations are valid only at low cov-erage with adsorbate atoms. At higher coverage, surface stateswill be blocked by adsorbed atoms. These effects can bestraightforwardly included into the rate equations which thenbecome nonlinear in the concentration. Such extensions willbe presented in a future study.

Third, the rate equations description becomes very effi-cient once the total system of gas plus surface has reachedthermal equilibrium. The main requirement is that therelaxation time for an atom on the surface to reach localequilibrium is much less than the typical residence time of anatom on the surface. In this regime, the gas atoms losememory of their initial state and become randomized withrespect to energy and momentum. Interestingly, this situationis particularly well fulfilled at low lattice temperatures,Ts80 K, when the thermal desorption is extremely slowpresenting a challenge to MD simulations. The particle tra-jectories must then be integrated during a very long residencetime. Moreover, to obtain a good statistics over the desorptionrates, the angular and velocity distributions, thousands oftrajectories must be sampled. To make the problem tractable,

techniques for treating ‘rare events’ have been proposed[56, 57], for an overview see [58]. In particular, studies of thethermal desorption of Ar and Xe from Pt(111) have beenconducted by means of the stochastic classical trajectoryapproach [59, 60] when the residence time exceeds 1 s.

Finally, the present simulations did only consider thescattering of single atoms, one at a time. In the case ofplasmas in contact with a surface this is justified at suffi-ciently low pressures and particle fluxes to the surface. As aconsequence, at long times the fraction of trapped and quasi-trapped atoms is slowly decreasing, see figure 10. For thecomputation of the sticking probability [29] and of the resi-dence time these time scales are not essential. However, forother applications such as the growth dynamics of an adsor-bate layer the long-time behavior is of direct interest. In thatcase it is expected that the decay of the adsorbed fractions iscompensated by the continuous influx of atoms from theplasma leading to a quasi-stationary state. The present rateequations model can be straightforwardly extended to includethis flux as a source term. This will enable one to study theseprocesses systematically in dependence on the plasmaconditions.

ORCID iDs

A Filinov https://orcid.org/0000-0001-6978-6310M Bonitz https://orcid.org/0000-0001-7911-0656D Loffhagen https://orcid.org/0000-0002-3798-0773

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1. Introduction2. Effective gas-surface interaction3. MD simulation of the lattice at finite temperature4. Time-resolved trapped, quasi-trapped and scattered fractions4.1. Classification of particle trajectories: trapped, quasi-trapped, and scattering states4.2. Transitions between T, Q and C states

5. Derivation of rate equations from kinetic theory6. Rate equation model6.1. Reconstruction of the transition rates from the MD simulations6.2. Test of the approach6.3. Temperature dependence of the transition rates6.4. Analytical solution of the rate equations

7. ConclusionReferences

Microscopic modeling of gas-surface scattering. I. A ... · All these detailed experimental and...

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