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Charge-transfer processes in atom-surface collisions at low energies: a Green's function approach
surface science ELSEVIER
Surface Science 325 (1995) 311-322
Charge-transfer processes in atom-surface collisions at low energies: a Green's function approach Evelina A. Garcia *, E.C. Goldberg, M.C.G. Passeggi INTEC (UNL-CONICET), Gftemes3450, CC91, 3000 Santa Fe, Argentina Received 20 July 1994; accepted for publication 28 October 1994
Abstract
A Green's functions analysis of the charge transfer probability is performed within the one particle Anderson-Newns model and used to calculate the probabilities of the different charge-states of a scattered or sputtered atom by finite and semi-infinite linear chains. This formalism has the advantage that it allows one to handle simultaneously continuous band states, localized surface states and core-like states of the substrate without a significative increase in the complexity of the numerical calculation. It also allows one to incorporate, at least within a semi-classical level, the effects on the ionization probability of variations in the velocity of the projectile along the trajectory. As expected, these become important at low velocities, where velocity changes largely affect the phase interferences taking place within the double transit of the projectile through the collision region in the scattering process. Keywords: Atom-solid interactions, scattering, diffraction; Green's function methods; Ion-solid interactions, scattering, channeling
1. Introduction
Electron-exchange processes between atoms (ions) and solid surfaces play a substantial role in surface analysis techniques such as ion-beam scattering spectroscopy (ISS), neutral-beam scattering spectroscopy (NSS), or secondary-ion mass spectroscopy (SIMS). Two processes may be identified in the inelastic ion scattering: one in which a net charge-exchange takes place between the projectile and the surface, while the other corresponds to the collision-induced excitation of surface electrons, where the charge state of the projectile is conserved after the impact. In charge-exchange processes, neutralization can occur
* Corresponding author. Fax: + 5 4 42 550944.
due to Auger as well as resonant processes [1,2]. The latter involves resonant or quasi-resonant charge exchange with the valence band electrons, both causing little changes in the kinetic energy of the projectile. By contrast, re-ionization, in which an electron of a neutralized projectile is transferred to an empty valence band state of the surface, is of particular interest since it implies a large energy loss A E comparable to the ionization energy of the projectile [3,4]. On the other hand, the excitation of surface valence electrons appears as a shift of the binary-collision peak towards a lower energy position. The details of the surface-electronic structure on inelastic ion scattering are essential to obtain information on the mechanisms causing the electron transitions. These effects have been investigated by Souda et al. using projectiles with simple structures
0039-6028/95/$09.50 © 1995 Elsevier Science B.V. All fights reserved SSDI 0 0 3 9 - 6 0 2 8 ( 9 4 ) 0 0 7 5 9 - 4
312
E.A. Garcia et al./ Surface Science 325 (1995) 311-322
as beams of He + ions and He atoms in its ground state with kinetic energies ranging from 0.1 to 1.0 keV, for a variety of surface-samples either in the form of pure elements or in compounds [5]. These authors found that excitation of surface valence electrons is usually accompanied by reionization of He; that the ionization probability of He is minimized if target atoms have filled d levels at an energy shallower than the He ls level. Also, excitation of one or two electrons from the surface p-bands to the conduction band occurs for He + scattering from ionic compounds, provided the p-bands are located shallower than the He ls level. Re-ionization of He is thought to be caused by the promotion of He ls electrons with a binding energy of 24.6 eV to a vacant valence level of the surface, being the ls orbital promotion due to the interaction with a core orbital of the target atoms [6]. The well-known oscillations against the kinetic energy of the scattered-ion yield are explained in terms of the quasi-resonant charge-exchange between the d orbitals of the group IIIb elements such as Ga and In, and the He ls orbital [7,8]. In the study of neutralization of low energy He + ions on alkali-metal covered surfaces, Hagstrum et al. [9] have concluded that the mechanism underlying the strong electron emission is due to a resonant neutralization to the triplet aS He* state, followed by Auger de-excitation. However, Souda et al. [10] found that the electron emission takes place along the incoming trajectory before Auger de-excitation of He + occurs suggesting that the ejected electron comes from the decay of the core-excited He state. This is consistent with results obtained by metastable de-excitation spectroscopy [111. The study of the charge-transfer and its relationship with the surface electronic structure has also attracted considerable attention over the past years in connection with reactive events occurring at the surface. One of the simplest reactive system is the scattering of H + (D +) from solid surfaces [12-16]. The chemical effects observed in the spectra of scattered D ÷ ions [15,16] can be related to the fact that the D l s level is located close to the valence band making the interaction with the band states more likely to occur. In this case, two steps are essential for determining the neutralization probability of D * : The first step involves the transition of the
D Is hole to an electronic state of the target, being followed by the diffusion of the hole in the solid. The hole diffusion is favoured by covalent and metallic bonds, while this is largely suppressed in the case of ionic compounds. On the other hand, a quantitative understanding of the neutralization process offers a new measure of the local electronic structure at the surface, e.g., the charge state of alkali adsorbates on metal and semiconductor surfaces studied by low-energy D* scattering [17]. In the case of alkali-ion scattering used as a probe for real space surface structure determination, there are conflicting views related to whether the macroscopic work function or the local electrostatic potential is important in resonant neutralization processes [18,19]. All these experimental data indicate that the charge-exchange process between atoms or ions and a solid surface is a complex phenomenon whose description involves several ingredients characterizing the surface electronic structure which may give rise to mutually competing mechanisms. Several theoretical descriptions for ion-surface-neutralization process at low energies have been proposed [20-28], with most of them being based on the time-dependent Anderson model. Indeed, this model provides an adequate framework for studying these processes, since it includes: (i) the electronic band-structure of the solid, allowing for the presence of both extended and localized surface states; (ii) the atomic state perturbed by the solid; and (iii) the atom-solid hopping interaction. A fourth term, which accounts for the electron-electron repulsion at the atom site, may also be included and treated within a Hartree-Fock approximation corrected by perturbative methods [29]. The time-dependence of the Anderson-Hamiltonian is entirely defined once the assumption of an atomic trajectory is made. To calculate the probabilities of the different charge-states of a scattered atom within the Anderson description, by using parameters for the different terms defined consistently with reference to some realistic atom-surface interaction mode [30] avoiding approximations like the wide-band limit and solving the quantum-mechanical equations of motion, is still an objective. In this paper we will follow the work by Muda and Newns [24] in their calculation, at a semi-ab-initio level, of the charge-exchange
E.A. Garcia et al. / Surface Science 325 (1995) 311-322 process at the surfaces. We propose the use of the time-dependent Green's functions formalism which has the advantage that one can handle simultaneously continuous band states, localized surface states and core-like states of the substrate, without a significative increase in the complexity of the numerical calculation. Other important features of this formalism are: (a) it allows one to introduce an effective potential which determines self-consistently the atom velocity along the trajectory, taking into account the different electronic channels involved, and their occupation probabilities [31,32] and (b) it provides the zeroth-order Green functions necessary to perform perturbative corrections due to the intra-atomic Coulomb repulsion term [29]. In Section 2, a brief description of the general formalism is presented. Section 3 deals with the model system, and the comparison between results obtained by considering finite and semi-infinite chains for different situations of the scattering process. The presence of localized surface states is analyzed in the case of ion scattering. Section 3 is also devoted to the study of the effect on the ionization probability, of an atomic velocity varying consistently with an effective potential obtained from the eikonal approximation for collisional processes [31,32]. Electronic correlation on the projectile site will not be considered in the present work. Using the same criterium as Muda and Newns [24] to test the model we compare our results with the experimental data obtained by Souda and Aono [3]. In Section 4 we resume our conclusions.
2. Theory 2.1. The Green's function formalism The Anderson-Newns model Hamiltonian which describes the atom-surface interaction is given by: ,,~g(t) = E (ea(/)na,~ + E e , n~,~
o" x
k
-'~ E [VakC+aerCkcr+ VkaC;~rCao'] k
313
where ~a(t) and e, are the atom and substrate energy levels, respectively, Vka(t) is the hopping between the atom state and the states of the solid and U is the Coulomb repulsion energy at the site of the moving atom. In the Hartree-Fock (HF) approximation, Eq. (1) is reduced to
HF(t)
= ~{[Ea(t)+U(na_~(t))]na~+
~-~,,nk~ k
-t- E [Vak( t)C+crCkcr q- Vka( t)Ck+ Ca~] I 1 k -- U(nar (t))(na~ (t)).
(2)
A formalism adequate to solve the time-dependent SchrSdinger equation for the one particle Hamiltonian of Eq. (2), is based on the following Green's functions:
Gq~( t, to) = -iO( t - to)(O ]c+~( to)Cq~( t) + Cq~(t)c+~(to)[0),
(3)
where [0) is the initial state at t = to; c+~(to) creates an electron in a one-particle eigenstate g,~ of ,,~nF(t 0) at the initial time to, and Cq,r(t) destroys one on a q-state with q running over the whole set of solid and atom states. Within this independent particle description, the time-dependent wave function can be written as: 1 I ~0(t)) = ~ d e t
[U(t, to)Xa[a=oceupied ,
(4)
where U(t, t o) is the evolution operator. Then, it is straightforward to see that (Olc+~a(to)Cq~(t)]O) gives the weight of the q-state in the one-particle time-dependent state which evolves from the initially occupied eigenstate X,~; while (O[cq~(t)c+~(to)[0) gives the weight of q-state in the one-particle time-dependent state which evolves from the initially empty eigenstate X,~. It may be shown [9,20] that the two times correlation function (c+~(t)Cq~(t')) can be expressed in terms of these functions Gq~(t, to), with the only requirement being that of knowing the eigenstates a of this system at the initial time:
(c+~(t)Cq~(t')) = E G'~ (t, o)Gq~(t, to) , o-*
t
o"
!
O~ = OCC
+ lUnao, na_o.),
(1)
(5)
E.A. Garciaet al./ SurfaceScience325 (1995)311-322
314
where the sum is performed over the states initially occupied X~(to). Therefore, the average occupation number of the atom as a function of time is given by (nag(t)> =
)-". IGa~(t, OL ~
t0)l 2,
(6)
the final form for the equation of motion of ga~(t, to) can be written as
-~ ga~( t, to)
OCC
and the different charge-states probabilities are P - ( t ) = (na~ (/)>(na~ ( t ) ) ,
= - - i ( 6 ( t -- to)(Ol{Cag(to), C~+g(t0)} i0)
P°( t) = ,~<(1 - nag( t) )>(na_g( t)> ,
+ft,~¢(t, "r)gab(r, t0) dr
P + ( t ) = ((1 - nat ( t ) ) } ( ( 1 - na~ ( t ) ) } = 1 _p0 _p-, for the negative, neutral and positive charge states, respectively. These are strictly consistent with the effective one-electron Hamiltonian expression (2). It is well known that the time-dependent HF approximation (TDHF) works rather well for small values of the correlation parameter U. More precisely one expects that TDHF leads to good results when the uncertainty in the atomic level energy introduced by the particle motion is of the order of U [33]. Nevertheless, for values of U not so large the TDHF calculation can be considered as the starting point of a perturbative treatment of the correlation term [29]. For large values of U, it is more appropriate to consider the spinless version of (2), which implies restricting the atomic number to values between 0 and 1. In this case we have
P°(t) =(na(t)}, P+(t) = 1 - (na(t)>The negative ionization probability can be calculated independently by interchanging the ionization level with the affinity one [34]. In this work, the charge-state probabilities are calculated by using this spinless picture. It is important to notice that for large values of U, if there is initially one electron with a given spin in the atomic orbital, the TDHF evolution works better than the spinless formulation
[351. By introducing the following phase transformations: J
Ga~(t, to' =ga%(/, t o ) e x p ( - - i f t i [ ea('r) + U(n~_ g(z)}] d z ) ,
G~%(t, t o ) = g ~ ( t , to) e x p [ - i e k ( t - / 0 ) ] ,
(7a) (7b)
to
k
Vak( t)gk~( to , to) , )
with {Caa( to) , c ~+ ( t 0 ) }
= Cag(to)C+g(to) + c~+(to)Cag(to), l'~a~(t) = Vak(/) ( -- ifti[ e, -- "a(1") --U(na_ g(7")> ] d z ) , and the self-energy Z~(t, ~-) defined as
2f~(t, r) = - i O ( t - r) E l,~a~(/) l~k~( r ) . k oThus, only the values of gq~(t 0+ , t 0) are required to completely define our problem. For scattering processes, the eigenstates { X,~(t0)} correspond to the k-states of the solid plus the localized atomic state ~ba, then: o" gq~(t o+ , t o ) = - i 6 q ~ ,
with
q=k,a.
In the sputtering case, the set { X,,(t0)} is given by the molecular states of the compound atom-substrate system. Therefore,
gq~(t~,to)=-iCq(to),
with
q=k,a,
(8)
where in Eq. (8) Cq(t o) is the weight of the state ~bq in the LCAO expansion of the molecular state X~(to). The interesting aspects of this formalism can be summarized as: (i) It provides a unified treatment for the sputtering and scattering processes with clear differences imposed only by the initial conditions in each case. (ii) It allows one to introduce the features of a continuous band structure for the substrate without a
E,4. Garcia et al. / Surface Science 325 (1995) 311-322
more expensive calculation than that required for an N-atom discrete system, with large N. It also gives the possibility of describing the interaction atomsurface as a superposition of pair-interactions between the surface atoms and the moving one [30]. This can be seen clearly by supposing that in the substrate k-state can be written as:
315
ansatz based on the "eikonal approximation" leads to very satisfactory results. This approximation consists in writing the wave function for the complete system (nuclei + electrons) as follows:
~ ( R , r, t) = EXi(R, i
t)Oi(r , t) e x p [ - i S ( R , t ) / h ] ,
dPnk(r) = Ecnk( ot, R i ) q ~ ( r - R i ) ,
(9)
ot,i
where n is the band index, k is the wave vector and a denotes the atomic state at each lattice site of coordinate R i. In this form the electron hopping between the atom and the solid is expressed in terms of the coefficients cnk(a, Ri), which in turn define the local density of states of the substrate. Assuming a strongly localized process the sum over the surface atoms can be reduced to the nearest atom neighbors. This is a complete general formulation which allows on one side, to describe the atom-solid interaction as superposition of atom-atom interactions, and on the other to include details of the band structure of the solid surface. The presence of localized surface states can be naturally incorporated within this formalism. (iii) Another important feature is that, this Green's functions formalism allows one to include the effects of a time-dependent atom velocity which is self-consistently adjusted with the charge-states probabilities. This is discussed in Section 2.2. (iv) Finally, these time-dependent one electron Green's functions can be used advantageously as starting functions to perform perturbative corrections due to the correlation term [29].
where X i are the "nuclear functions", and qbi are the many-electron states. The coordinate r labels the set of electronic coordinates ( r l , . . . , rN); and R is the nuclear coordinate of the moving atom measured with reference to the surface. The Schr/Jdinger equation can be written as: [(-h2/2M)
V2 + X~(r, R) - (iO/O/)]
× ~ ( R , r, t ) = 0 ,
(10)
with ,~e(r, R) the electronic part of the total Hamiltonian. The electronic configurations are chosen as eigenstates of the non-interacting atom-solid system: X e ( r , R ) =¢TC~a+ X s + ~ r ( r , R) with ( ¢q~Ca"~- ~ s ) ~A)i = E i ~ i .
By replacing ~ ( R , r, t) of Eq. (9) into Eq. (10), projecting over an electronic state @j and taking the classical limit h ~ 0, we obtain
-Xj(OS/at) = [( VRS)2/2M]Xj
+ Exi(
j I ro I i>.
i
2.2. Formalism involving a variable atom-velocity The motion of an atom near a surface is a complex problem in which nuclear and electronic coordinates are coupled. By assuming a classical trajectory for the atom, this problem can be reduced to one described by an electronic time-dependent Hamiltonian. The time-dependence is provided by the assumed atom trajectory. One expects that the velocity associated with this trajectory, is determined by some "average potential" which takes into account the different electronic channels involved in the dynamical process. This point has been widely discussed in previous works [31,32] and it was found that the
Multiplying both sides by Xj*, summing over j and taking into account that EIXjl2=I, J
one arrives at -
(OS/Ot) = ( VRS)2/EM + EXj* (t)Xi(t) i,j X [ E i ~ i j at- <¢~j I ~ ' ( r ,
R) I q~,>] = E . (11)
This is a Hamilton-Jacobi equation where it is possible to identify - ( a S / O t ) = E as the total energy
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
316
which is conserved, (VRS)/M is the classical velocity of the particle and
V( R, t) = ~,Xj* ( t)Xi( t)[ Eirij i,j + (q~/[ ~ r ( r , R)I
(12)
¢~i>]
is the potential which provides the velocity variations due to the electronic transitions. According to Eq. (12), the potential V(R, t) is determined by the mean value of the electronic Hamiltonian in the dynamical state of the system, that is:
V(R, t) = (~0(t) I Ye [ ~O(t)), where
~b(t) = EXi( t)c19i(R , r). i
Eqs. (11) and (12), allow one to define the semiclassical variable atom velocity as
v( t) = [ 2 ( E - V( R, t)/M)] 1/z
(13)
In the case of the Anderson-Newns description for a linear trajectory, the potential V(R, t) adopts a very simple form in terms of the time-dependent Green's functions gq~( t, to).
V(R, t) = (~b(t) I • Eknk tr,k
+ E [ Ca(l) + U(na-~(t))]nacr o"
+ E (V~:a(t)C-k~Cacr+ h.c.) I ~b(t))
forwardly calculated from Eq. (6). Eq. (14) provides an effective potential V(R, t), which determines the atom velocity according to expression (13), and it also depends on the velocity v through R = R(t, v), thus leading to a self-consistent calculation of both V(R, t) and v(t).
3. Model system We use a model system in which the substrate is represented by a tight-binding linear chain of atoms. In this form we may use either a cluster or a semi-infinite system to describe the solid with a surface. The Hamiltonian matrix elements in the nearest-neighbor approximation are:
Hij= fl for i=j + / - 1 , Hij=O for i 4 : j + / - 1 , where i and j indicate atomic site in the substrate, and {~bi} is the set of orthonormal orbitals on each atomic site i. Note that by allowing H l l ~ ~s ::j/::0 we include the possibility of generating a surface state. The exact eigenvalue Ek and eigenfunctions q~k of this Hamiltonian for an N-atoms linear chain are given by: Case (a): Es=0,
ek=2[3cos[k~r/(N+l)],
ty,k 1
~-~U(nac~(t))(na-a( t)) + Vn_n.
2
~0k =
~i sin(ikTr/N+ 1) ~bi(r),
o,
Expressing (Cq~Cp~) in terms of the Green's functions (Eq. 5) and after some algebra, one arrives at
l
V(R,t)= ~(E,k(nkcr(t))+,a(naa(t ) )
~s ~ 0
k
and
Izl
,k = ,s + 213 cos[k
-½U(na~( t) )(na-cr( t) ) )
with
z=-EJ[3,
/(N+ 1)],
~k = ~-, qbi(r) { ~
[sin(ikTr/N + 1)
i
+2Re(i
E a = occup
Ega~*( t,
+z sin((i-
to)
o~
/{1
a
×(-~ga~(t, t0)) } + V._n,
(14)
where V._. is the nuclear potential interaction, and the mean values (nk~(t)) and (n.~(t)) are straight-
1)kTr/N+ 1)]}
+ z cos(k~r/N + 1)};
Case (c):
Izl>l, ek = ~ + 2[3 cos(kTr/N+ 1),
¢~4:0
and
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
~o~= ~., q b i ( r ) { ~ [ s i n ( i k r r / N
have considered a straight-line trajectory with constant velocity,
+ 1)
i
+ z s i n ( ( i - 1)klr/N+ 1)]} /{1
+ z
cos(kit/N+ 1)};
and those for the localized states which occur in case (c), given by
q~t= EA(--L)iqbi(r),
L=l/z,
where
i
1/z2).
e, = ,s(1 +
In the limit of large N ( N ~ ~), x = k / N + 1. The coefficients giving the weight of the first atom site of the chain in the k-state, are given by: Gsin(7rx) ci~ = (1 + z 2 + 2z cos(rrx) '
V~-sin(~-x)
cl=-
,
ci~ = ~/1 + z 2 + 2 z cos('trx)
EC~Vai(t). i
If only the interaction between the adatom state and the orbital centred on the first atom of the chain is retained, Vak(t) results to be
CklV( t).
In case (c), k refers either to the localized state or to the extended states. To describe the atom motion, we i
.
i
,
i
•
i
.
,
,
,
,
r
,50
_
(15a)
R = vt ~ sputtering trajectory,
(15b)
where R c is the turning point which depends on the kinetic energy. Two different R-dependences of the electron hopping have been considered:
(i) V(R) = Vo e x p ( - A R ) ,
(16a)
(ii) V(R) = VoR e x p ( - A R ) .
(16b)
The second model of V(R) looks more realistic since it reproduces approximately the behaviour expected for the one-electron contribution to the nondiagonal matrix elements between atomic states [24].
-t2
The hopping integral Vak(t) can be written as:
Vak = Vka( t ) =
R = R e + v I t ] ~ scattering trajectory,
3.1. Substrate-size effects and the presence of a localized surface state
while in case (c)
Vak(t) =
317
40
.',z_
We are interested in the analysis of the effects originated by the discrete nature of the substrate levels, and in the comparison of these results with those obtained by assuming a continuous-limit description of the substrate. Here we have only considered the exponential model for V(R) given by Eq. (16a), with V0 = 0.25 a.u. and A = 1 a.u. In Fig. 1 we present the results for the ionization probability as a function of the inverse of velocity (Tc), for an atom scattered by a 3-atom, a 25-atom and by semiinfinite substrates for an atomic energy level resonant with respect to the Fermi level (c a = E F = 0 a.u.), and a hopping between substrate atoms equal to /3 = 0.1 a.u. In the limit of large /3-values and for an atomic energy resonant with the Fermi level, the following expression for the ionization probability is found:
& c o
P+= ½[1-exp(-2Vo2Tc//3)].
20
o
O I
I
I
I
0
5
10
15
/
20
[
i
Z5
30
inverse of velocity (o.u.)
Fig. 1. Scattering ionization probability as a function of the inverse of atom velocity for: (©) analytical approximation of wide-band limit; ( ~ , ) continuous band; ( [ ] ) 25-atom chain and ( 0 ) 3-atom chain. Ea = 0. a.u. and fl = 0 . 1 a.u. Neutral atom as initial condition.
This analytical behaviour of the wide-band limit approximation [20,22] is also shown in Fig. 1. The general features of the results shown in Fig. 1 may be understood taking into account the energy-uncertainty (AE) associated with the atomic level due to the atom motion. As we are using atomic units, this AE is given approximately by the atom velocity value (v). At large velocities, when v is comparable to the bandwidth, the behaviour suggested is an
E.A. Garcia et aL / Surface Science 325 (1995) 311-322
318 100
~
90
a~.
80
o N
70
'
'
I 0
,
'
-:
-'
l
I
,
10
20
:
'-
-'
i
g
a L
,
50
I
*
40
I 50
inverse of velocity (o.u.) I0O
60
I
i
I
I0
i
l
20
,
I
30
i
I
40
,
I
50
inverse of velocity (o.u.)
Fig. 2. Sputtering ionization probability as a function of the inverse of atom velocity. (a) ( 0 ) 3-atom chain and ( n ) 25-atom chain, for /3 = 0.01 a.u. (b) ( 0 ) 3-atom chain and ( n ) 25-atom chain, for/3 = 0.2 a.u. ~a = 0.1 a.u.
oscillatory one for the narrow-band limit situation. When the atom velocity is smaller than the energy separation between substrate levels, the finite cluster description shows important differences with reference to the semi-infinite substrate. For velocities much smaller than the bandwidth, the wide-band limit situation is reached provided that these velocities are larger than the energy separation between substrate levels. Finally, the electron diffusion within the solid associated with quasicontinuous band-states explains why the ionization probability caused by the semi-infinite substrate is always larger than that of a 25-atom chain, and why this last one is always larger than that for the 3-atom substrate. In Figs. 2a and 2b we observe that the ionization probability of an atom sputtered from a surface has a smoother dependence on T¢ than that corresponding to the scattering situation, and besides, it is not strongly dependent on the substrate size. In the sputtering case, we are assuming that the initial conditions are fixed from the knowledge of the eigenstates of the atom-substrate complex in equilibrium, and also that only the outgoing trajectory is
involved. The behaviour of the charge transfer probability in sputtering is qualitatively similar to the wide-band limit situation for scattering, where we know that the incoming trajectory can be ignored. Therefore, we can suppose that the phase interferences occurring between the incoming and outgoing trajectories are mainly responsible for the structure of the ionization probability behaviour [36]. In the case of the 25-atom substrate with /3 = 0.2 a.u., a significative decrease in the ionization probability with the velocity is observed as compared with the 3-atom case. This can be understood on the basis of an increasing number of substrate states which contribute to the atom neutralization. The pronounced changes in slopes disappear by including more atoms in the substrate description, although the same behaviour is basically maintained. To analyze the behaviour of the ionization probability for atoms scattered in the presence of a localized surface state, we take the R-dependence on the hopping as that of Eq. (16b), with V0 = 3 a.u. and A = 1.5 a.u. The turning point R~ was determined by considering a Molliere potential for the interaction between the atom and the solid, with parameters chosen for the He-Si system. The low-energy approximation for Re is [24] Rc = 12.92(E 0) -0.5319,
(17)
with R c in atomic units and E0, the incidence energy, in eV. The nodal form of the electronic hopping leads to a significative increase in the ionization probability at energies lower than 1 keV, compared with the values obtained by using an exponential R-dependence [24,36]. The behaviour of the ionization probability when a surface state exists (~s = - 0 . 1 5 a.u., /3 = 0.1 a.u.) as compared with the case with no surface state is shown in Fig. 3. We observe that the velocity dependence changes drastically when the surface state is present. At large velocities the ionization occurs with small probability, while in the low velocity regime the probability grows to values in the order of 50%. Taking into account that the atom energy level is 0.1 a.u. below the Fermi level and is initially occupied, this increase in the ionization probability may be explained in terms of bonding and antibonding states formed by the hybridization between the atomic state and the
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
localized surface state. Since the surface state is also occupied, the bonding and antibonding states are initially occupied, but the presence of empty band states gives the possibility of atom ionization. At very low velocities the ionization probability is given by the weight of the atom site in the antibonding state. At large velocities the coupling with the band states is predominant, although the behaviour of the ionization probability reflects the changes in the band structure due to the presence of the surface state. This hypothetical case shows an alternative mechanism to explain an increase of the ionization probability at low velocities.
8
8
a @
i0 o
o
0 0 •
0 O
~
•
I0 -I
0 0 @
o N C
0 0
.o
10-~
@@
I 5
i 7
i 9
lIT
l T3
I 15
i 17
l 19
I
[
3.2. Effects of the varying velocity on the charge exchange probability in atom-surface scattering
I
I
e
I
I
I
I
exp(]
'
/
b
I0 °
~o ••°00o •
zl
i0.1
0o
•
°0
o 0
v( n) =
21
inverse of velocity (a.u.)
101
Selecting the nodal form proposed by Muda and Newns [24] for the electronic hopping:
8
10'
~
319
-
0
where I
I0 -3
~=(g/Ro)
q,
5
and
I
I
7
I
L
9
[
11
, inverse
R0=l.6a.u.,
V0 = 0 . 9 a . u .
and
q=l.3,
the behaviour of V(R) is quite similar to those of the matrix elements of the Hartree-Fock Hamiltonian between the He Is state and the 3s and 3pz Si states
~1
=.o
I
'
'
'
•
'
'
'
'
'
'
'
'
'
'
'
,
I 12
,
t 14
,
6 I
'
'
'
'
d 18
,
I 20
40-
//
30o=
/
zoI0-
0 0
2
I 4
,
~ 6
,
l 8
inverse
,
I 10 of
velocity
'
(O.u.)
Fig. 3. Scattering ionization probability as a function o f the inverse o f a t o m v e l o c i t y . ( ) a b s e n c e o f a l o c a l i z e d state; (------) p r e s e n c e o f a l o c a l i z e d state, ea = - 0 . 1 a.u., /3 = 0.1 a.u. and the turning point g i v e n b y e x p r e s s i o n ( 1 9 ) . Neutral a t o m as initial condition.
J
I
I
13 of velocity
I
15
[
L
17
I
~
*
211
19
(o.u.)
Fig. 4. Scattering ionization probability as a function o f the i n v e r s e o f a t o m v e l o c i t y . ( @ ) variable v e l o c i t y ; ( © ) constant v e l o c i t y for /3 = - 0 . 1 8 a.u. (a) Ea = - 0 . 3 6 a.u. and (b) ea = -0.72
a.u. Neutral a t o m as initial condition.
[24]. We keep the assumption of a straight-line trajectory (Eq. (15a)) and the low energy approximation (Eq. (17)) for the turning point. The linear-chain hopping parameter is taken as: fl = - 0 . 1 8 a.u. The atom energy level ls is assumed constant along the atom trajectory since there is not an energy-shift model consistent with the one assumed for the electronic hopping. We performed calculations for two different atom energy values, one corresponds approximately to the ionization energy of He (Ea = - 0 . 7 2 a.u.), and the second value is ~a = - 0 . 3 6 a.u., which coincides with the bottom band energy. In Figs. 4a and 4b, we observe the ionization probability for a scattered atom as a function of the inverse of the initial atom velocity, at E~ = - 0 . 3 6 and - 0 . 7 2 a.u. respectively. Here we compare the results obtained by using a constant velocity with those including the effects of a velocity which varies
320
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
consistently with the potential energy expression in Eq. (14). As is expected [31,32] the differences between them are important at low velocities (v < 0.1 a.u.). Figs. 5a and 5b show the effective potential (Eq. (14)) along the particle trajectory (negative and positive times correspond to the incoming and outgoing trajectories, respectively) for two initial atom velocities, Tc = 12 a.u. and Tc = 18 a.u. respectively, in the e~ = - 0 . 3 6 a.u. case. The electronic contribution is also plotted separately. The nuclear interaction potential is related only with the atom-surface atoms interaction, since we may ignore the constant contribution from the solid. The differences observed in the potential V~ff(R, t) for the incoming and outgoing trajectories are just due to the dynamical process which involves an excitation of the atom-surface system. We can see that these differences are smaller as the initial velocity decreases, showing the ten-
0.5
I
I
i
i
I
Tc = 18. a.u. 04
/5
e
"~
03
~
02
.~
0.1
o
c 9
0.0 I
-300
i
-20(1
I 0
I
-100
0.5
I
I
200
i
300
(a~.)
time
-g
I
I00
I
i
b
T c = 1 8 . a.u. 0.4
.= 03
!
02
0.1
I Tc=12. a.u.
0.0 -300
I
-200
i
I
i
- 100 time
o
,,
I
I O0
i
I
200
i
5O0
(a.u.)
Fig. 6. Scattering ionization probability ( ) and atomic velocity ( - - - - - ) as functions of time for Tc = 18. a.u. (a) with constant velocity and (b) with variable velocity. Neutral atom as initial condition.
//
; \
I
0
-1 i
-150
I
I
i
I
I
I
100
i
150
(o.u.)
time
10
I
50
I
0
-50
-I00
i
i
i
b
Tc = 18. au. 8 6 vo
=:
4
2 0 -2 I
-300
-200
i
I
,
i
-I00
,
0 time
I
I
I00
200
300
(a.u.)
Fig. 5. Effective potential as a function of time; ( nuclear plus electronic contributions; ( - - - - - ) only electronic contribution, for (a) Tc = 12. a.u. and (b) Tc = 18. a.u.
dency towards an adiabatic process. The ionization probability along the atom trajectory for an initial velocity value (Tc = 18 a.u.) is shown in Figs. 6a and 6b for constant and variable velocity calculations, respectively. The velocity variation along the trajectory is also shown. The main effect of the self-consistent calculation using the effective potential shown in Fig. 5b, is a decrease in the velocity with respect to its initial value, leading to smaller values in the ionization probability for low velocities observed in Fig. 4a. In order to test the low-energy dependence of the ionization probability on the atomic velocity when this is calculated self-consistently, we use the experimental results of Souda and Aono [3] of I B / I A for the H e - T i on the TIC(100) surface, where IA and I B are the intensities of the elastic and energy loss
E.A. Garcia et al. / Surface Science 325 (1995) 311-322 6
1
i
'
i
I
i
i
i
'
~
i
a
0 5
o o
4
o
o
o o
3
o
o
o 2
off •
•
o"
321
situations we want to compare. According to these results we can say that in low-energy charge-exchange processes, it is necessary to take into account the variation of the velocity along the trajectory in a self-consistent way with the potential which includes the transition probabilities.
o
0
l
4. Conclusions o I
0
i
I
t
, I , I
i
212
18
8 kinetic
6
i
,
,
2~4
I
26
[a.u.)
enerqy
n
5
4
o o
o o
o
3
o
o •o
2 •
[2
0
0
O
o
o• [
10
,
I
12
I
14
,
I
16 kinetic
,
I
*
I
18
20
energy
(a.u.)
212 '
21,4
z~6
Fig. 7. I B / I A as functions of kinetic energy. ( O ) calculation with a self-consistent velocity, (©) calculation with a constant atomic velocity; and ( D ) experimental values. (a) e a = - 0 . 3 6 a.u. and (b) ea = - 0 . 7 2 a.u.
peaks respectively. Following the steps of Muda and Newns [24] we assume: IB/IA ~ PI/P~
where P[ is the ionization probability on collision, which is the one we calculate, and p i is the survival probability of He ion on the incoming trajectory. Assuming an exponential form for both Auger and resonance neutralization processes: PsI = exp( -
volvo),
with vc the characteristic velocity of the process. The value of vc is determined by fitting the theoretical value of IBJIA at E 0 = 23.5 a.u. with the experimental value. In Figs. 7a and 7b the calculated and the experimental values of IBJIa versus E 0 are shown for the e a = - 0 . 3 6 and - 0 . 7 2 a.u. cases, respectively. The calculated values correspond to the two
A formalism describing the dynamical charge-exchange between either scattered or sputtered atoms from solid surfaces is presented. It gives the possibility to deal simultaneously with several important ingredients participating in these pocesses: (i) the interaction model for atomic energy level and atom-surface hopping; (ii) the different features of the solid band-structure, depending on whether it is a pure covalent or ionic one; (iii) the calculation of an effective potential which determines self-consistently the atom velocity along its trajectory; and finally (iv) the effects of the electronic repulsion in the atom site. We have shown that the presence of a surface localized states may lead to drastic changes on the velocity dependence of the ionization probability for scattered atoms; and that a variable velocity consistent with the collision process has to be taken into account in charge exchange at low energies.
Acknowledgements The Instituto de Desarrollo Tecnol6gico para la Industria Qulmica (INTEC) belongs to the Consejo Nacional de Investigaciones Cientificas y T6cnicas (CONICET) and Universidad Nacional del Litoral (UNL). Support from CONICET through grant 75300 is gratefully acknowledged.
References [1] H.D. Hagstrum, Phys. Rev. 94 (1954) 336. [2] T.W. Rusch and R.L. Erickson, J. Vac. Sci. Technol. 13 (1976) 374. [3] R. Souda and M. Aono, Nucl. Instrum. Methods B 15 (1986) 114. [4] R. Souda, M. Aono, C. Oshima, S. Otani and Y. Ishizawa, Surf. Sci. 179 (1987) 199.
322
E~4. Garcia et al. / Surface Science 325 (1995) 311-322
[5] R. Souda, T. Aizawa, C. Oshima, S. Otani and Y. Ishizawa, Phys. Rev. B 40 (1989) 4119. [6] S. Tsuneyuki and M. Tsukada, Phys. Rev. B 34 (1986) 5758. [7] R.L. Erickson and D.P. Smith, Phys. Rev. Lett. 34 (1975) 297. [8] J.C. Tully, Phys. Rev. B 16 (1977) 4324. [9] H.D. Hagstrum, P. Petrie and E.E. Chaban, Phys. Rev. B 38 (1988) 10264. [10] R. Souda, W. Hayami, T. Aizawa, S. Otani and Y. Ishizawa, Phys. Rev. B 46 (1992) 7315. [11] R. Hemmen and H. Conrad, Phys. Rev. Lett. 67 (1991) 1314. [12] R.S. Bhattacharya, W. Eckstein and H. Verbeek, Surf. Sci. 93 (1980) 563. [13] H. Verbeek, W. Eckstein and R.S. Bhattacharya, Surf. Sci. 95 (1980) 380. [14] J.N.M. Van Wunnik, J.C. Geedings, E.H. Grannemann and J. Loss, Surf. Sci. 131 (1983) 17. [15] R. Souda, T. Aizawa, W. Hayami, S. Otani and Y. Ishizawa, Phys. Rev. B 42 (1990) 7761. [16] R. Souda, W. Hayami, T. Aizawa, S. Otani and Y. Ishizawa, Phys. Rev. B 45 (1992) 14358. [17] R. Souda, W. Hayami, T. Aizawa, S. Otani and Y. Ishizawa, Phys. Rev. Lett. 69 (1992) 192. [18] G.A. Kimmel, D.M. Goodstein, Z.H. Levine and B.H. Cooper, Phys. Rev. B 43 (1991) 9403. [19] L.Q. Jiang, Y.D. Li and B.E. Koel, Phys. Rev. Lett. 70 (1993) 2649. [20] A. Blandin, A. Nourtier and D. Hone, J. Phys. (Paris) 37 (1976) 369.
[21] [22] [23] [24] [25] [26] [27]
[28] [29] [30] [31] [32] [33] [34] [35] [36]
W. Bloss and D. Hone, Surf. Sci. 72 (1978) 277. R. Brako and D.M. Newns, Surf. Sci. 108 (1981) 253. H.K. McDowell, J. Chem. Phys. 77 (1982) 3263. Y. Muda and D.M. Newns, Phys. Rev. B 37 (1988) 7048. P. Nordlander, Scanning Microscopy Supplement 4 (1990) 353. B.L. Burrows, A.T. Amos, S.G. Davison and K.W. Sulston, Phys. Rev. B 45 (1992) 13407. J.B. Marston, D.R. Anderson, E.R. Behringer, B.H. Cooper, C.A. DiRubio, G.A. Kimmel and C. Richardson, Phys. Rev. B 48 (1993) 7809. D.C. Langreth and P. Nordlander, Phys. Rev B 43 (1991) 2541. E.C. Goldberg and F. Flores, Phys. Rev. B 45 (1992) 8657. E.C. Goldberg, R. Monreal, A. Martin-Rodero and F. Fiores, Phys. Rev. B 39 (1988) 5684. S. Sawada, A. Nitzan and H. Metiu, Phys. Rev. B 32 (1985) 851. E.C. Goldberg, J. Ferron and M.C.G. Passeggi, Phys. Rev. B 40 (1989) 8666. E.C. Goldberg and M.C.G. Passeggi, J. Phys.: Condens. Matter 5 (1993) 4259. E.C. Goldberg, E.R. Gagliano and M.C.G. Passeggi, Phys. Rev. B 32 (1985) 4375. K.L. Sebastian, Phys. Rev. B 31 (1985) 6976. P.G. Bolcatto and E.C. Goldberg, Surf. Sci. 289 (1993) 187.
Surface Science 325 (1995) 311-322
Charge-transfer processes in atom-surface collisions at low energies: a Green's function approach Evelina A. Garcia *, E.C. Goldberg, M.C.G. Passeggi INTEC (UNL-CONICET), Gftemes3450, CC91, 3000 Santa Fe, Argentina Received 20 July 1994; accepted for publication 28 October 1994
Abstract
A Green's functions analysis of the charge transfer probability is performed within the one particle Anderson-Newns model and used to calculate the probabilities of the different charge-states of a scattered or sputtered atom by finite and semi-infinite linear chains. This formalism has the advantage that it allows one to handle simultaneously continuous band states, localized surface states and core-like states of the substrate without a significative increase in the complexity of the numerical calculation. It also allows one to incorporate, at least within a semi-classical level, the effects on the ionization probability of variations in the velocity of the projectile along the trajectory. As expected, these become important at low velocities, where velocity changes largely affect the phase interferences taking place within the double transit of the projectile through the collision region in the scattering process. Keywords: Atom-solid interactions, scattering, diffraction; Green's function methods; Ion-solid interactions, scattering, channeling
1. Introduction
Electron-exchange processes between atoms (ions) and solid surfaces play a substantial role in surface analysis techniques such as ion-beam scattering spectroscopy (ISS), neutral-beam scattering spectroscopy (NSS), or secondary-ion mass spectroscopy (SIMS). Two processes may be identified in the inelastic ion scattering: one in which a net charge-exchange takes place between the projectile and the surface, while the other corresponds to the collision-induced excitation of surface electrons, where the charge state of the projectile is conserved after the impact. In charge-exchange processes, neutralization can occur
* Corresponding author. Fax: + 5 4 42 550944.
due to Auger as well as resonant processes [1,2]. The latter involves resonant or quasi-resonant charge exchange with the valence band electrons, both causing little changes in the kinetic energy of the projectile. By contrast, re-ionization, in which an electron of a neutralized projectile is transferred to an empty valence band state of the surface, is of particular interest since it implies a large energy loss A E comparable to the ionization energy of the projectile [3,4]. On the other hand, the excitation of surface valence electrons appears as a shift of the binary-collision peak towards a lower energy position. The details of the surface-electronic structure on inelastic ion scattering are essential to obtain information on the mechanisms causing the electron transitions. These effects have been investigated by Souda et al. using projectiles with simple structures
0039-6028/95/$09.50 © 1995 Elsevier Science B.V. All fights reserved SSDI 0 0 3 9 - 6 0 2 8 ( 9 4 ) 0 0 7 5 9 - 4
312
E.A. Garcia et al./ Surface Science 325 (1995) 311-322
as beams of He + ions and He atoms in its ground state with kinetic energies ranging from 0.1 to 1.0 keV, for a variety of surface-samples either in the form of pure elements or in compounds [5]. These authors found that excitation of surface valence electrons is usually accompanied by reionization of He; that the ionization probability of He is minimized if target atoms have filled d levels at an energy shallower than the He ls level. Also, excitation of one or two electrons from the surface p-bands to the conduction band occurs for He + scattering from ionic compounds, provided the p-bands are located shallower than the He ls level. Re-ionization of He is thought to be caused by the promotion of He ls electrons with a binding energy of 24.6 eV to a vacant valence level of the surface, being the ls orbital promotion due to the interaction with a core orbital of the target atoms [6]. The well-known oscillations against the kinetic energy of the scattered-ion yield are explained in terms of the quasi-resonant charge-exchange between the d orbitals of the group IIIb elements such as Ga and In, and the He ls orbital [7,8]. In the study of neutralization of low energy He + ions on alkali-metal covered surfaces, Hagstrum et al. [9] have concluded that the mechanism underlying the strong electron emission is due to a resonant neutralization to the triplet aS He* state, followed by Auger de-excitation. However, Souda et al. [10] found that the electron emission takes place along the incoming trajectory before Auger de-excitation of He + occurs suggesting that the ejected electron comes from the decay of the core-excited He state. This is consistent with results obtained by metastable de-excitation spectroscopy [111. The study of the charge-transfer and its relationship with the surface electronic structure has also attracted considerable attention over the past years in connection with reactive events occurring at the surface. One of the simplest reactive system is the scattering of H + (D +) from solid surfaces [12-16]. The chemical effects observed in the spectra of scattered D ÷ ions [15,16] can be related to the fact that the D l s level is located close to the valence band making the interaction with the band states more likely to occur. In this case, two steps are essential for determining the neutralization probability of D * : The first step involves the transition of the
D Is hole to an electronic state of the target, being followed by the diffusion of the hole in the solid. The hole diffusion is favoured by covalent and metallic bonds, while this is largely suppressed in the case of ionic compounds. On the other hand, a quantitative understanding of the neutralization process offers a new measure of the local electronic structure at the surface, e.g., the charge state of alkali adsorbates on metal and semiconductor surfaces studied by low-energy D* scattering [17]. In the case of alkali-ion scattering used as a probe for real space surface structure determination, there are conflicting views related to whether the macroscopic work function or the local electrostatic potential is important in resonant neutralization processes [18,19]. All these experimental data indicate that the charge-exchange process between atoms or ions and a solid surface is a complex phenomenon whose description involves several ingredients characterizing the surface electronic structure which may give rise to mutually competing mechanisms. Several theoretical descriptions for ion-surface-neutralization process at low energies have been proposed [20-28], with most of them being based on the time-dependent Anderson model. Indeed, this model provides an adequate framework for studying these processes, since it includes: (i) the electronic band-structure of the solid, allowing for the presence of both extended and localized surface states; (ii) the atomic state perturbed by the solid; and (iii) the atom-solid hopping interaction. A fourth term, which accounts for the electron-electron repulsion at the atom site, may also be included and treated within a Hartree-Fock approximation corrected by perturbative methods [29]. The time-dependence of the Anderson-Hamiltonian is entirely defined once the assumption of an atomic trajectory is made. To calculate the probabilities of the different charge-states of a scattered atom within the Anderson description, by using parameters for the different terms defined consistently with reference to some realistic atom-surface interaction mode [30] avoiding approximations like the wide-band limit and solving the quantum-mechanical equations of motion, is still an objective. In this paper we will follow the work by Muda and Newns [24] in their calculation, at a semi-ab-initio level, of the charge-exchange
E.A. Garcia et al. / Surface Science 325 (1995) 311-322 process at the surfaces. We propose the use of the time-dependent Green's functions formalism which has the advantage that one can handle simultaneously continuous band states, localized surface states and core-like states of the substrate, without a significative increase in the complexity of the numerical calculation. Other important features of this formalism are: (a) it allows one to introduce an effective potential which determines self-consistently the atom velocity along the trajectory, taking into account the different electronic channels involved, and their occupation probabilities [31,32] and (b) it provides the zeroth-order Green functions necessary to perform perturbative corrections due to the intra-atomic Coulomb repulsion term [29]. In Section 2, a brief description of the general formalism is presented. Section 3 deals with the model system, and the comparison between results obtained by considering finite and semi-infinite chains for different situations of the scattering process. The presence of localized surface states is analyzed in the case of ion scattering. Section 3 is also devoted to the study of the effect on the ionization probability, of an atomic velocity varying consistently with an effective potential obtained from the eikonal approximation for collisional processes [31,32]. Electronic correlation on the projectile site will not be considered in the present work. Using the same criterium as Muda and Newns [24] to test the model we compare our results with the experimental data obtained by Souda and Aono [3]. In Section 4 we resume our conclusions.
2. Theory 2.1. The Green's function formalism The Anderson-Newns model Hamiltonian which describes the atom-surface interaction is given by: ,,~g(t) = E (ea(/)na,~ + E e , n~,~
o" x
k
-'~ E [VakC+aerCkcr+ VkaC;~rCao'] k
313
where ~a(t) and e, are the atom and substrate energy levels, respectively, Vka(t) is the hopping between the atom state and the states of the solid and U is the Coulomb repulsion energy at the site of the moving atom. In the Hartree-Fock (HF) approximation, Eq. (1) is reduced to
HF(t)
= ~{[Ea(t)+U(na_~(t))]na~+
~-~,,nk~ k
-t- E [Vak( t)C+crCkcr q- Vka( t)Ck+ Ca~] I 1 k -- U(nar (t))(na~ (t)).
(2)
A formalism adequate to solve the time-dependent SchrSdinger equation for the one particle Hamiltonian of Eq. (2), is based on the following Green's functions:
Gq~( t, to) = -iO( t - to)(O ]c+~( to)Cq~( t) + Cq~(t)c+~(to)[0),
(3)
where [0) is the initial state at t = to; c+~(to) creates an electron in a one-particle eigenstate g,~ of ,,~nF(t 0) at the initial time to, and Cq,r(t) destroys one on a q-state with q running over the whole set of solid and atom states. Within this independent particle description, the time-dependent wave function can be written as: 1 I ~0(t)) = ~ d e t
[U(t, to)Xa[a=oceupied ,
(4)
where U(t, t o) is the evolution operator. Then, it is straightforward to see that (Olc+~a(to)Cq~(t)]O) gives the weight of the q-state in the one-particle time-dependent state which evolves from the initially occupied eigenstate X,~; while (O[cq~(t)c+~(to)[0) gives the weight of q-state in the one-particle time-dependent state which evolves from the initially empty eigenstate X,~. It may be shown [9,20] that the two times correlation function (c+~(t)Cq~(t')) can be expressed in terms of these functions Gq~(t, to), with the only requirement being that of knowing the eigenstates a of this system at the initial time:
(c+~(t)Cq~(t')) = E G'~ (t, o)Gq~(t, to) , o-*
t
o"
!
O~ = OCC
+ lUnao, na_o.),
(1)
(5)
E.A. Garciaet al./ SurfaceScience325 (1995)311-322
314
where the sum is performed over the states initially occupied X~(to). Therefore, the average occupation number of the atom as a function of time is given by (nag(t)> =
)-". IGa~(t, OL ~
t0)l 2,
(6)
the final form for the equation of motion of ga~(t, to) can be written as
-~ ga~( t, to)
OCC
and the different charge-states probabilities are P - ( t ) = (na~ (/)>(na~ ( t ) ) ,
= - - i ( 6 ( t -- to)(Ol{Cag(to), C~+g(t0)} i0)
P°( t) = ,~<(1 - nag( t) )>(na_g( t)> ,
+ft,~¢(t, "r)gab(r, t0) dr
P + ( t ) = ((1 - nat ( t ) ) } ( ( 1 - na~ ( t ) ) } = 1 _p0 _p-, for the negative, neutral and positive charge states, respectively. These are strictly consistent with the effective one-electron Hamiltonian expression (2). It is well known that the time-dependent HF approximation (TDHF) works rather well for small values of the correlation parameter U. More precisely one expects that TDHF leads to good results when the uncertainty in the atomic level energy introduced by the particle motion is of the order of U [33]. Nevertheless, for values of U not so large the TDHF calculation can be considered as the starting point of a perturbative treatment of the correlation term [29]. For large values of U, it is more appropriate to consider the spinless version of (2), which implies restricting the atomic number to values between 0 and 1. In this case we have
P°(t) =(na(t)}, P+(t) = 1 - (na(t)>The negative ionization probability can be calculated independently by interchanging the ionization level with the affinity one [34]. In this work, the charge-state probabilities are calculated by using this spinless picture. It is important to notice that for large values of U, if there is initially one electron with a given spin in the atomic orbital, the TDHF evolution works better than the spinless formulation
[351. By introducing the following phase transformations: J
Ga~(t, to' =ga%(/, t o ) e x p ( - - i f t i [ ea('r) + U(n~_ g(z)}] d z ) ,
G~%(t, t o ) = g ~ ( t , to) e x p [ - i e k ( t - / 0 ) ] ,
(7a) (7b)
to
k
Vak( t)gk~( to , to) , )
with {Caa( to) , c ~+ ( t 0 ) }
= Cag(to)C+g(to) + c~+(to)Cag(to), l'~a~(t) = Vak(/) ( -- ifti[ e, -- "a(1") --U(na_ g(7")> ] d z ) , and the self-energy Z~(t, ~-) defined as
2f~(t, r) = - i O ( t - r) E l,~a~(/) l~k~( r ) . k oThus, only the values of gq~(t 0+ , t 0) are required to completely define our problem. For scattering processes, the eigenstates { X,~(t0)} correspond to the k-states of the solid plus the localized atomic state ~ba, then: o" gq~(t o+ , t o ) = - i 6 q ~ ,
with
q=k,a.
In the sputtering case, the set { X,,(t0)} is given by the molecular states of the compound atom-substrate system. Therefore,
gq~(t~,to)=-iCq(to),
with
q=k,a,
(8)
where in Eq. (8) Cq(t o) is the weight of the state ~bq in the LCAO expansion of the molecular state X~(to). The interesting aspects of this formalism can be summarized as: (i) It provides a unified treatment for the sputtering and scattering processes with clear differences imposed only by the initial conditions in each case. (ii) It allows one to introduce the features of a continuous band structure for the substrate without a
E,4. Garcia et al. / Surface Science 325 (1995) 311-322
more expensive calculation than that required for an N-atom discrete system, with large N. It also gives the possibility of describing the interaction atomsurface as a superposition of pair-interactions between the surface atoms and the moving one [30]. This can be seen clearly by supposing that in the substrate k-state can be written as:
315
ansatz based on the "eikonal approximation" leads to very satisfactory results. This approximation consists in writing the wave function for the complete system (nuclei + electrons) as follows:
~ ( R , r, t) = EXi(R, i
t)Oi(r , t) e x p [ - i S ( R , t ) / h ] ,
dPnk(r) = Ecnk( ot, R i ) q ~ ( r - R i ) ,
(9)
ot,i
where n is the band index, k is the wave vector and a denotes the atomic state at each lattice site of coordinate R i. In this form the electron hopping between the atom and the solid is expressed in terms of the coefficients cnk(a, Ri), which in turn define the local density of states of the substrate. Assuming a strongly localized process the sum over the surface atoms can be reduced to the nearest atom neighbors. This is a complete general formulation which allows on one side, to describe the atom-solid interaction as superposition of atom-atom interactions, and on the other to include details of the band structure of the solid surface. The presence of localized surface states can be naturally incorporated within this formalism. (iii) Another important feature is that, this Green's functions formalism allows one to include the effects of a time-dependent atom velocity which is self-consistently adjusted with the charge-states probabilities. This is discussed in Section 2.2. (iv) Finally, these time-dependent one electron Green's functions can be used advantageously as starting functions to perform perturbative corrections due to the correlation term [29].
where X i are the "nuclear functions", and qbi are the many-electron states. The coordinate r labels the set of electronic coordinates ( r l , . . . , rN); and R is the nuclear coordinate of the moving atom measured with reference to the surface. The Schr/Jdinger equation can be written as: [(-h2/2M)
V2 + X~(r, R) - (iO/O/)]
× ~ ( R , r, t ) = 0 ,
(10)
with ,~e(r, R) the electronic part of the total Hamiltonian. The electronic configurations are chosen as eigenstates of the non-interacting atom-solid system: X e ( r , R ) =¢TC~a+ X s + ~ r ( r , R) with ( ¢q~Ca"~- ~ s ) ~A)i = E i ~ i .
By replacing ~ ( R , r, t) of Eq. (9) into Eq. (10), projecting over an electronic state @j and taking the classical limit h ~ 0, we obtain
-Xj(OS/at) = [( VRS)2/2M]Xj
+ Exi(
j I ro I i>.
i
2.2. Formalism involving a variable atom-velocity The motion of an atom near a surface is a complex problem in which nuclear and electronic coordinates are coupled. By assuming a classical trajectory for the atom, this problem can be reduced to one described by an electronic time-dependent Hamiltonian. The time-dependence is provided by the assumed atom trajectory. One expects that the velocity associated with this trajectory, is determined by some "average potential" which takes into account the different electronic channels involved in the dynamical process. This point has been widely discussed in previous works [31,32] and it was found that the
Multiplying both sides by Xj*, summing over j and taking into account that EIXjl2=I, J
one arrives at -
(OS/Ot) = ( VRS)2/EM + EXj* (t)Xi(t) i,j X [ E i ~ i j at- <¢~j I ~ ' ( r ,
R) I q~,>] = E . (11)
This is a Hamilton-Jacobi equation where it is possible to identify - ( a S / O t ) = E as the total energy
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
316
which is conserved, (VRS)/M is the classical velocity of the particle and
V( R, t) = ~,Xj* ( t)Xi( t)[ Eirij i,j + (q~/[ ~ r ( r , R)I
(12)
¢~i>]
is the potential which provides the velocity variations due to the electronic transitions. According to Eq. (12), the potential V(R, t) is determined by the mean value of the electronic Hamiltonian in the dynamical state of the system, that is:
V(R, t) = (~0(t) I Ye [ ~O(t)), where
~b(t) = EXi( t)c19i(R , r). i
Eqs. (11) and (12), allow one to define the semiclassical variable atom velocity as
v( t) = [ 2 ( E - V( R, t)/M)] 1/z
(13)
In the case of the Anderson-Newns description for a linear trajectory, the potential V(R, t) adopts a very simple form in terms of the time-dependent Green's functions gq~( t, to).
V(R, t) = (~b(t) I • Eknk tr,k
+ E [ Ca(l) + U(na-~(t))]nacr o"
+ E (V~:a(t)C-k~Cacr+ h.c.) I ~b(t))
forwardly calculated from Eq. (6). Eq. (14) provides an effective potential V(R, t), which determines the atom velocity according to expression (13), and it also depends on the velocity v through R = R(t, v), thus leading to a self-consistent calculation of both V(R, t) and v(t).
3. Model system We use a model system in which the substrate is represented by a tight-binding linear chain of atoms. In this form we may use either a cluster or a semi-infinite system to describe the solid with a surface. The Hamiltonian matrix elements in the nearest-neighbor approximation are:
Hij= fl for i=j + / - 1 , Hij=O for i 4 : j + / - 1 , where i and j indicate atomic site in the substrate, and {~bi} is the set of orthonormal orbitals on each atomic site i. Note that by allowing H l l ~ ~s ::j/::0 we include the possibility of generating a surface state. The exact eigenvalue Ek and eigenfunctions q~k of this Hamiltonian for an N-atoms linear chain are given by: Case (a): Es=0,
ek=2[3cos[k~r/(N+l)],
ty,k 1
~-~U(nac~(t))(na-a( t)) + Vn_n.
2
~0k =
~i sin(ikTr/N+ 1) ~bi(r),
o,
Expressing (Cq~Cp~) in terms of the Green's functions (Eq. 5) and after some algebra, one arrives at
l
V(R,t)= ~(E,k(nkcr(t))+,a(naa(t ) )
~s ~ 0
k
and
Izl
,k = ,s + 213 cos[k
-½U(na~( t) )(na-cr( t) ) )
with
z=-EJ[3,
/(N+ 1)],
~k = ~-, qbi(r) { ~
[sin(ikTr/N + 1)
i
+2Re(i
E a = occup
Ega~*( t,
+z sin((i-
to)
o~
/{1
a
×(-~ga~(t, t0)) } + V._n,
(14)
where V._. is the nuclear potential interaction, and the mean values (nk~(t)) and (n.~(t)) are straight-
1)kTr/N+ 1)]}
+ z cos(k~r/N + 1)};
Case (c):
Izl>l, ek = ~ + 2[3 cos(kTr/N+ 1),
¢~4:0
and
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
~o~= ~., q b i ( r ) { ~ [ s i n ( i k r r / N
have considered a straight-line trajectory with constant velocity,
+ 1)
i
+ z s i n ( ( i - 1)klr/N+ 1)]} /{1
+ z
cos(kit/N+ 1)};
and those for the localized states which occur in case (c), given by
q~t= EA(--L)iqbi(r),
L=l/z,
where
i
1/z2).
e, = ,s(1 +
In the limit of large N ( N ~ ~), x = k / N + 1. The coefficients giving the weight of the first atom site of the chain in the k-state, are given by: Gsin(7rx) ci~ = (1 + z 2 + 2z cos(rrx) '
V~-sin(~-x)
cl=-
,
ci~ = ~/1 + z 2 + 2 z cos('trx)
EC~Vai(t). i
If only the interaction between the adatom state and the orbital centred on the first atom of the chain is retained, Vak(t) results to be
CklV( t).
In case (c), k refers either to the localized state or to the extended states. To describe the atom motion, we i
.
i
,
i
•
i
.
,
,
,
,
r
,50
_
(15a)
R = vt ~ sputtering trajectory,
(15b)
where R c is the turning point which depends on the kinetic energy. Two different R-dependences of the electron hopping have been considered:
(i) V(R) = Vo e x p ( - A R ) ,
(16a)
(ii) V(R) = VoR e x p ( - A R ) .
(16b)
The second model of V(R) looks more realistic since it reproduces approximately the behaviour expected for the one-electron contribution to the nondiagonal matrix elements between atomic states [24].
-t2
The hopping integral Vak(t) can be written as:
Vak = Vka( t ) =
R = R e + v I t ] ~ scattering trajectory,
3.1. Substrate-size effects and the presence of a localized surface state
while in case (c)
Vak(t) =
317
40
.',z_
We are interested in the analysis of the effects originated by the discrete nature of the substrate levels, and in the comparison of these results with those obtained by assuming a continuous-limit description of the substrate. Here we have only considered the exponential model for V(R) given by Eq. (16a), with V0 = 0.25 a.u. and A = 1 a.u. In Fig. 1 we present the results for the ionization probability as a function of the inverse of velocity (Tc), for an atom scattered by a 3-atom, a 25-atom and by semiinfinite substrates for an atomic energy level resonant with respect to the Fermi level (c a = E F = 0 a.u.), and a hopping between substrate atoms equal to /3 = 0.1 a.u. In the limit of large /3-values and for an atomic energy resonant with the Fermi level, the following expression for the ionization probability is found:
& c o
P+= ½[1-exp(-2Vo2Tc//3)].
20
o
O I
I
I
I
0
5
10
15
/
20
[
i
Z5
30
inverse of velocity (o.u.)
Fig. 1. Scattering ionization probability as a function of the inverse of atom velocity for: (©) analytical approximation of wide-band limit; ( ~ , ) continuous band; ( [ ] ) 25-atom chain and ( 0 ) 3-atom chain. Ea = 0. a.u. and fl = 0 . 1 a.u. Neutral atom as initial condition.
This analytical behaviour of the wide-band limit approximation [20,22] is also shown in Fig. 1. The general features of the results shown in Fig. 1 may be understood taking into account the energy-uncertainty (AE) associated with the atomic level due to the atom motion. As we are using atomic units, this AE is given approximately by the atom velocity value (v). At large velocities, when v is comparable to the bandwidth, the behaviour suggested is an
E.A. Garcia et aL / Surface Science 325 (1995) 311-322
318 100
~
90
a~.
80
o N
70
'
'
I 0
,
'
-:
-'
l
I
,
10
20
:
'-
-'
i
g
a L
,
50
I
*
40
I 50
inverse of velocity (o.u.) I0O
60
I
i
I
I0
i
l
20
,
I
30
i
I
40
,
I
50
inverse of velocity (o.u.)
Fig. 2. Sputtering ionization probability as a function of the inverse of atom velocity. (a) ( 0 ) 3-atom chain and ( n ) 25-atom chain, for /3 = 0.01 a.u. (b) ( 0 ) 3-atom chain and ( n ) 25-atom chain, for/3 = 0.2 a.u. ~a = 0.1 a.u.
oscillatory one for the narrow-band limit situation. When the atom velocity is smaller than the energy separation between substrate levels, the finite cluster description shows important differences with reference to the semi-infinite substrate. For velocities much smaller than the bandwidth, the wide-band limit situation is reached provided that these velocities are larger than the energy separation between substrate levels. Finally, the electron diffusion within the solid associated with quasicontinuous band-states explains why the ionization probability caused by the semi-infinite substrate is always larger than that of a 25-atom chain, and why this last one is always larger than that for the 3-atom substrate. In Figs. 2a and 2b we observe that the ionization probability of an atom sputtered from a surface has a smoother dependence on T¢ than that corresponding to the scattering situation, and besides, it is not strongly dependent on the substrate size. In the sputtering case, we are assuming that the initial conditions are fixed from the knowledge of the eigenstates of the atom-substrate complex in equilibrium, and also that only the outgoing trajectory is
involved. The behaviour of the charge transfer probability in sputtering is qualitatively similar to the wide-band limit situation for scattering, where we know that the incoming trajectory can be ignored. Therefore, we can suppose that the phase interferences occurring between the incoming and outgoing trajectories are mainly responsible for the structure of the ionization probability behaviour [36]. In the case of the 25-atom substrate with /3 = 0.2 a.u., a significative decrease in the ionization probability with the velocity is observed as compared with the 3-atom case. This can be understood on the basis of an increasing number of substrate states which contribute to the atom neutralization. The pronounced changes in slopes disappear by including more atoms in the substrate description, although the same behaviour is basically maintained. To analyze the behaviour of the ionization probability for atoms scattered in the presence of a localized surface state, we take the R-dependence on the hopping as that of Eq. (16b), with V0 = 3 a.u. and A = 1.5 a.u. The turning point R~ was determined by considering a Molliere potential for the interaction between the atom and the solid, with parameters chosen for the He-Si system. The low-energy approximation for Re is [24] Rc = 12.92(E 0) -0.5319,
(17)
with R c in atomic units and E0, the incidence energy, in eV. The nodal form of the electronic hopping leads to a significative increase in the ionization probability at energies lower than 1 keV, compared with the values obtained by using an exponential R-dependence [24,36]. The behaviour of the ionization probability when a surface state exists (~s = - 0 . 1 5 a.u., /3 = 0.1 a.u.) as compared with the case with no surface state is shown in Fig. 3. We observe that the velocity dependence changes drastically when the surface state is present. At large velocities the ionization occurs with small probability, while in the low velocity regime the probability grows to values in the order of 50%. Taking into account that the atom energy level is 0.1 a.u. below the Fermi level and is initially occupied, this increase in the ionization probability may be explained in terms of bonding and antibonding states formed by the hybridization between the atomic state and the
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
localized surface state. Since the surface state is also occupied, the bonding and antibonding states are initially occupied, but the presence of empty band states gives the possibility of atom ionization. At very low velocities the ionization probability is given by the weight of the atom site in the antibonding state. At large velocities the coupling with the band states is predominant, although the behaviour of the ionization probability reflects the changes in the band structure due to the presence of the surface state. This hypothetical case shows an alternative mechanism to explain an increase of the ionization probability at low velocities.
8
8
a @
i0 o
o
0 0 •
0 O
~
•
I0 -I
0 0 @
o N C
0 0
.o
10-~
@@
I 5
i 7
i 9
lIT
l T3
I 15
i 17
l 19
I
[
3.2. Effects of the varying velocity on the charge exchange probability in atom-surface scattering
I
I
e
I
I
I
I
exp(]
'
/
b
I0 °
~o ••°00o •
zl
i0.1
0o
•
°0
o 0
v( n) =
21
inverse of velocity (a.u.)
101
Selecting the nodal form proposed by Muda and Newns [24] for the electronic hopping:
8
10'
~
319
-
0
where I
I0 -3
~=(g/Ro)
q,
5
and
I
I
7
I
L
9
[
11
, inverse
R0=l.6a.u.,
V0 = 0 . 9 a . u .
and
q=l.3,
the behaviour of V(R) is quite similar to those of the matrix elements of the Hartree-Fock Hamiltonian between the He Is state and the 3s and 3pz Si states
~1
=.o
I
'
'
'
•
'
'
'
'
'
'
'
'
'
'
'
,
I 12
,
t 14
,
6 I
'
'
'
'
d 18
,
I 20
40-
//
30o=
/
zoI0-
0 0
2
I 4
,
~ 6
,
l 8
inverse
,
I 10 of
velocity
'
(O.u.)
Fig. 3. Scattering ionization probability as a function o f the inverse o f a t o m v e l o c i t y . ( ) a b s e n c e o f a l o c a l i z e d state; (------) p r e s e n c e o f a l o c a l i z e d state, ea = - 0 . 1 a.u., /3 = 0.1 a.u. and the turning point g i v e n b y e x p r e s s i o n ( 1 9 ) . Neutral a t o m as initial condition.
J
I
I
13 of velocity
I
15
[
L
17
I
~
*
211
19
(o.u.)
Fig. 4. Scattering ionization probability as a function o f the i n v e r s e o f a t o m v e l o c i t y . ( @ ) variable v e l o c i t y ; ( © ) constant v e l o c i t y for /3 = - 0 . 1 8 a.u. (a) Ea = - 0 . 3 6 a.u. and (b) ea = -0.72
a.u. Neutral a t o m as initial condition.
[24]. We keep the assumption of a straight-line trajectory (Eq. (15a)) and the low energy approximation (Eq. (17)) for the turning point. The linear-chain hopping parameter is taken as: fl = - 0 . 1 8 a.u. The atom energy level ls is assumed constant along the atom trajectory since there is not an energy-shift model consistent with the one assumed for the electronic hopping. We performed calculations for two different atom energy values, one corresponds approximately to the ionization energy of He (Ea = - 0 . 7 2 a.u.), and the second value is ~a = - 0 . 3 6 a.u., which coincides with the bottom band energy. In Figs. 4a and 4b, we observe the ionization probability for a scattered atom as a function of the inverse of the initial atom velocity, at E~ = - 0 . 3 6 and - 0 . 7 2 a.u. respectively. Here we compare the results obtained by using a constant velocity with those including the effects of a velocity which varies
320
E.A. Garcia et al. / Surface Science 325 (1995) 311-322
consistently with the potential energy expression in Eq. (14). As is expected [31,32] the differences between them are important at low velocities (v < 0.1 a.u.). Figs. 5a and 5b show the effective potential (Eq. (14)) along the particle trajectory (negative and positive times correspond to the incoming and outgoing trajectories, respectively) for two initial atom velocities, Tc = 12 a.u. and Tc = 18 a.u. respectively, in the e~ = - 0 . 3 6 a.u. case. The electronic contribution is also plotted separately. The nuclear interaction potential is related only with the atom-surface atoms interaction, since we may ignore the constant contribution from the solid. The differences observed in the potential V~ff(R, t) for the incoming and outgoing trajectories are just due to the dynamical process which involves an excitation of the atom-surface system. We can see that these differences are smaller as the initial velocity decreases, showing the ten-
0.5
I
I
i
i
I
Tc = 18. a.u. 04
/5
e
"~
03
~
02
.~
0.1
o
c 9
0.0 I
-300
i
-20(1
I 0
I
-100
0.5
I
I
200
i
300
(a~.)
time
-g
I
I00
I
i
b
T c = 1 8 . a.u. 0.4
.= 03
!
02
0.1
I Tc=12. a.u.
0.0 -300
I
-200
i
I
i
- 100 time
o
,,
I
I O0
i
I
200
i
5O0
(a.u.)
Fig. 6. Scattering ionization probability ( ) and atomic velocity ( - - - - - ) as functions of time for Tc = 18. a.u. (a) with constant velocity and (b) with variable velocity. Neutral atom as initial condition.
//
; \
I
0
-1 i
-150
I
I
i
I
I
I
100
i
150
(o.u.)
time
10
I
50
I
0
-50
-I00
i
i
i
b
Tc = 18. au. 8 6 vo
=:
4
2 0 -2 I
-300
-200
i
I
,
i
-I00
,
0 time
I
I
I00
200
300
(a.u.)
Fig. 5. Effective potential as a function of time; ( nuclear plus electronic contributions; ( - - - - - ) only electronic contribution, for (a) Tc = 12. a.u. and (b) Tc = 18. a.u.
dency towards an adiabatic process. The ionization probability along the atom trajectory for an initial velocity value (Tc = 18 a.u.) is shown in Figs. 6a and 6b for constant and variable velocity calculations, respectively. The velocity variation along the trajectory is also shown. The main effect of the self-consistent calculation using the effective potential shown in Fig. 5b, is a decrease in the velocity with respect to its initial value, leading to smaller values in the ionization probability for low velocities observed in Fig. 4a. In order to test the low-energy dependence of the ionization probability on the atomic velocity when this is calculated self-consistently, we use the experimental results of Souda and Aono [3] of I B / I A for the H e - T i on the TIC(100) surface, where IA and I B are the intensities of the elastic and energy loss
E.A. Garcia et al. / Surface Science 325 (1995) 311-322 6
1
i
'
i
I
i
i
i
'
~
i
a
0 5
o o
4
o
o
o o
3
o
o
o 2
off •
•
o"
321
situations we want to compare. According to these results we can say that in low-energy charge-exchange processes, it is necessary to take into account the variation of the velocity along the trajectory in a self-consistent way with the potential which includes the transition probabilities.
o
0
l
4. Conclusions o I
0
i
I
t
, I , I
i
212
18
8 kinetic
6
i
,
,
2~4
I
26
[a.u.)
enerqy
n
5
4
o o
o o
o
3
o
o •o
2 •
[2
0
0
O
o
o• [
10
,
I
12
I
14
,
I
16 kinetic
,
I
*
I
18
20
energy
(a.u.)
212 '
21,4
z~6
Fig. 7. I B / I A as functions of kinetic energy. ( O ) calculation with a self-consistent velocity, (©) calculation with a constant atomic velocity; and ( D ) experimental values. (a) e a = - 0 . 3 6 a.u. and (b) ea = - 0 . 7 2 a.u.
peaks respectively. Following the steps of Muda and Newns [24] we assume: IB/IA ~ PI/P~
where P[ is the ionization probability on collision, which is the one we calculate, and p i is the survival probability of He ion on the incoming trajectory. Assuming an exponential form for both Auger and resonance neutralization processes: PsI = exp( -
volvo),
with vc the characteristic velocity of the process. The value of vc is determined by fitting the theoretical value of IBJIA at E 0 = 23.5 a.u. with the experimental value. In Figs. 7a and 7b the calculated and the experimental values of IBJIa versus E 0 are shown for the e a = - 0 . 3 6 and - 0 . 7 2 a.u. cases, respectively. The calculated values correspond to the two
A formalism describing the dynamical charge-exchange between either scattered or sputtered atoms from solid surfaces is presented. It gives the possibility to deal simultaneously with several important ingredients participating in these pocesses: (i) the interaction model for atomic energy level and atom-surface hopping; (ii) the different features of the solid band-structure, depending on whether it is a pure covalent or ionic one; (iii) the calculation of an effective potential which determines self-consistently the atom velocity along its trajectory; and finally (iv) the effects of the electronic repulsion in the atom site. We have shown that the presence of a surface localized states may lead to drastic changes on the velocity dependence of the ionization probability for scattered atoms; and that a variable velocity consistent with the collision process has to be taken into account in charge exchange at low energies.
Acknowledgements The Instituto de Desarrollo Tecnol6gico para la Industria Qulmica (INTEC) belongs to the Consejo Nacional de Investigaciones Cientificas y T6cnicas (CONICET) and Universidad Nacional del Litoral (UNL). Support from CONICET through grant 75300 is gratefully acknowledged.
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E~4. Garcia et al. / Surface Science 325 (1995) 311-322
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