Charge Exchange Processes in Low Energy Ion–Metal Collisions

Charge Exchange Processes in Low Energy Ion– Metal Collisions R. C. Monreal and F. Flores Departamento de Fı´sica Teo´rica de la Materia Condensada C-V, Universidad Auto´noma de Madrid, E-28049 Madrid, Spain

Abstract Charge exchange processes in low energy ion scattering are analyzed and some case study systems (H and He scattered off metals) are discussed in detail. For the Heþ/Al system, we show that Hagstrum’s neutralization model holds if the ion perpendicular energy is smaller than 100 eV. For larger energies, Heþ penetrates the metal surface layer and resonant processes become operative; then charge exchange processes have to be analyzed combining the Auger and resonant mechanisms. For the H/Al system, charge exchange can be understood considering only the resonant mechanism. The ion levels and the ion– metal interactions necessary for calculating the charge exchange processes in the He/Al and H/Al systems, are also presented in this chapter. Contents 1. Introduction 2. Ion – metal interaction: ion levels and linewidths 2.1. He– metal interaction 2.2. H– metal interaction 3. Resonant processes: dynamic solution of the Newns –Anderson Hamiltonian 4. Auger processes 5. Results 5.1. H2 on Al 5.2. Heþ on metal surfaces: glancing incidence 5.3. Heþ on metals: normal incidence 6. Conclusions Acknowledgements References

175 179 181 183 184 186 191 192 193 194 196 197 197

1. INTRODUCTION The pioneering work of Bohr [1] on the slowing down of swift alpha particles in matter defines one of the main events opening the way to modern physics. The ion –matter interaction depends crucially on the ion velocity; in the high ADVANCES IN QUANTUM CHEMISTRY, VOLUME 45 ISSN: 0065-3276 DOI 10.1016/S0065-3276(04)45008-4

q 2004 Elsevier Inc. All rights reserved

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speed limit, v q v0 Z1 (v0 is the Bohr velocity and Z1 the ion atomic number), the ion is stripped of its electron charge and the ion stopping power can be described using linear response theory for the target electrons [2– 4]; this approach enabled a unified description of single-particle and collective excitations of the electron gas. At projectile speeds such that v0 , v , 2=3 v0 Z1 ; an ion can lose electrons to, or capture electrons from, the medium. Bohr proposed a useful criterion for estimating the number of electrons bound to the projectile [5]. Brandt generalized this approach [6] and proposed the following equation: 
 v 2=3 Z1p ¼ Z1 1 2 exp 2 Z1 ; ð1Þ v0 for the effective charge Z1p of the ion. An ab initio theory of the stopping power of ions at intermediate and low 2=3 velocities, v , Z1 v0 ; needs a complete description of the ion charge states as a function of v: For an ion moving in a metal, Fig. 1 shows the different capture and loss processes associated with these ion charge states [7]. In the Auger process illustrated in Fig. 1(a), an electron is captured (or lost) by the ion to (or from) a bound state; this is assisted by the excitation of an electron – hole pair or a plasmon in the solid. The coherent resonant mechanism is illustrated in Fig. 1(b): electron exchange processes are induced by the time-dependent crystal potential as seen from the fixed ion. A third mechanism is the shell process shown in Fig. 1(c); in this case an inner electron of the target can be captured by the moving ion. These three mechanisms have been analyzed for calculating the charge states of different ions [8– 10]. For H and He, these charge states have been used for

Fig. 1. (a) Capture and loss Auger processes for an ion moving in an electron gas; (b) coherent resonant processes; (c) capture shell process.

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Fig. 2. Stopping powers for He moving in Al (LT, linear theory). The total stopping power is separated into the ion charge states contribution and the capture and loss contribution.

determining their stopping powers in the range of low and intermediate velocities [9,10]. Figure 2 shows the case of He on Al: the total stopping power is calculated as the sum of several contributions, three associated with the charge states Heþþ ; Heþ and He0 , the fourth one being due to the capture and loss processes. Similar arguments can be applied to ions interacting with surfaces. In this chapter, we concentrate our discussion on low energy ion scattering (LEIS) [11], when ions of low velocity v p v0 are reflected off the first metal layer. Then, the different charge exchange mechanisms defining the ion charge are shown in Fig. 3. Figure 3(a) shows the Auger process [14], similar to the one shown in Fig. 1(a) for atoms moving inside matter. Figure 3(b) shows a resonant process [12,13] whereby electrons are transferred between the metal and the atom. Notice that, for this process to be operative, the atomic level should resonate with the metal band, a condition that depends on the atom– metal distance as the atomic levels change with the atom – metal interaction. Notice that for LEIS, the equivalent of the shell process of Fig. 1(c) can be neglected [7]. Calculating the stopping powers of ions near surfaces needs a full calculation of their charge states [15]. In this chapter we review the work

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Fig. 3. Charge exchange mechanisms near a metal surface: (a) Auger process, (b) resonant process. The atomic energy level is shown as a function of the distance to the surface.

done in our group for analyzing the resonant and Auger processes associated with different ion states of H and He. As in bulk, ab initio description of the stopping power of ions near surfaces should combine the description of their charge states with a calculation of the stopping power of different species, say Heþþ ; Heþ ; He0 and Hþ or H but this is a subject beyond the scope of this chapter. In Section 2 we analyze the ion – metal interaction and show how to calculate the atomic levels and their linewidths as a function of the atom– metal distance. An important message of our work is that the energy position of these levels play a crucial role in understanding how the resonant and Auger charge exchange mechanisms operate. In Section 3, we discuss how to obtain a full quantum mechanical solution for resonant processes and in Section 4 we consider Auger processes. Section 5 is devoted to discussing how to combine in a full solution both the resonant and Auger mechanisms and analyzing some specific results for H and He interacting with metal surfaces. Finally our conclusions are presented in Section 6.

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2. ION –METAL INTERACTION: ION LEVELS AND LINEWIDTHS For calculating the probability for resonant processes between a metal and the ion states of H or He it is important to determine carefully the position of the ion levels as a function of the ion – metal distance. In the following discussion, for the sake of clarity, we will refer to the case of Heþ ; unless specified otherwise, and consider the charge exchange between Heþ and He0 associated with the interaction of the He-1s level with the metal. In a first approach to this problem one can consider using a conventional density functional (DF) method and calculate, for each He – metal distance, the He-1s level and its linewidth. The difficulty with this approach is that DF calculations do not necessarily yield a good description of the energy levels; in particular, image potential effects are not easily included in these calculations. A way of obtaining an accurate description of the ion levels is to calculate their energies as the differences of the ground states energies of two systems defined by their charges, say, Heþ and He0 : This implies calculating the He-1s level by the following equation: E1s 2 EF ¼ E½He0  2 E½Heþ ;

ð2Þ

where E½Heþ  (E½He0 ) represents the energy of a system for which the ion charge is Heþ ðHe0 Þ: In equation (2) the He-1s level is measured with respect to the metal Fermi level EF : Applying equation (2) to a conventional plane wave-DF calculation is not simple, the reason being that, if the He-1s level is resonating with the metal band, one cannot fix an ion charge state due to the ion – metal charge transfer included in the calculation. We can use, however, equation (2) changing this point of view and using a local orbital-DF approach [16] instead of the more conventional plane wave-DF methods. In a local orbital-DF approach we use an orthogonal local basis, fia ; and introduce the total electron energy of the system in the following way: E½nias  ¼ T½nias  þ EH ½nias  þ Exc ½nias ;

ð3Þ

where the total energy E is written as a function of nias ; the ias electron occupancy which, in this approach, plays the role of rðrÞ in the conventional DF method [17]. In equation (3), T; EH and Exc are the kinetic, Hartree and exchange-correlation energies of the electron gas, respectively. In this approach, the occupancies nias are obtained minimizing the energy E½nias ;

›E½nias  ¼ m: ›nias

ð4Þ

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Using Kohn– Sham theorem one can solve equation (4) by obtaining the self-consistent solution of the following Hamiltonian: " # X X X ›EH ½nias  ›Exc ½nias  † ^ H ¼ 1ias n^ ias þ n^ ias ; Tia;jb c^ ias c^ jbs þ þ ›nias ›nias ias ias ia;jbs ð5Þ where the energy levels, 1ias ; and the hopping integrals, Tia;jb ; define the one-electron part of the total Hamiltonian. Using this local orbital DF approach, we define the different ion charge states by the orbital occupancy, nias (for example, for He0 we take nias ¼ 1). Moreover, to be consistent, at the same time we have to take zero the hopping parameters between the atomic orbitals and the metal [19]. With these assumptions, we have to solve Hamiltonian (5) and calculate, say, E½Heþ  and E½He0 : In the next step, to analyze the resonant processes associated with charge exchange between Heþ and He0 ; we consider a spin-less Newns– Anderson Hamiltonian [18] where the level E1s and the hopping terms T1s;ia that have been neglected to calculate E½Heþ  and E½He0 ; are introduced. This Hamiltonian reads X X X ^ N – A ¼ E1s n^ 1s þ H 1i n^ i þ ½T1s;i c^ †i c^ 1s þ h:c: þ Ti;j c^ †i c^ j ; ð6Þ i

i

i–j

where 1i and Ti;j are parameters associated with the electronic band structure of the metal. Similar arguments can be applied to resonant charge exchange processes between Hþ and H0 or H0 and H2 : In the last case, we define the H-affinity level, A; by means of the equation A 2 EF ¼ E½H2  2 E½H0 ;

ð7Þ

while the ionization level, I; is defined by I 2 EF ¼ E½H0  2 E½Hþ :

ð8Þ

Using equations (7) and (8) we can introduce the following many-body Hamiltonian X X X ^ m:b: ¼ H I n^ 1s;s þ ðA 2 IÞ^n1s" n^ 1s# þ T1s;i ½^c†is c^ 1ss þ h:c: þ 1is n^ is s

þ

X

is

Ti;j c^ †is c^ js

i

ð9Þ

i–j

where we have defined the effective electron – electron interaction within the H-atom by ðA 2 IÞ^n1s" n^ 1s# : One can treat approximately the affinity and the ionization levels independently by considering a spin-less Hamiltonian,

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similar to the one given by equation (6), where the 1s-atomic level is replaced by either the ionization or the affinity levels of H. 2.1. He – metal interaction Calculations for the case of a He – Al interaction are shown in Figs. 4 and 5 [18]. Figure 4 shows T1s;i for different Al-orbitals as a function of the He– Al distance, where i stands for the 2s, 2p, 3s and 3p orbitals of Al. Notice that the He-interaction with the Al-2s or 2p orbitals is practically zero for d larger than 2.5 a.u.; however, at small distances (say, less than 1 a.u.) their values are very large, reflecting the strong interaction between the He-1s level and the Al-core orbitals.

Fig. 4. The hopping parameters T1s;i for the He-1s level and different atomic orbitals of Al as a function of the He –metal distance; (a) continuous line: 3s, dotted line: 3pz ; (b) dot-dashed line: 2s, dashed line: 2pz :

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Fig. 5. He-1s energy level, E1s ; as a function of the atom – surface distance (continuous line). The E~ 1s -level resulting from the interaction with the Al-core orbitals is also shown (dashed line). The top and the bottom of the Al-conduction band are drawn as two horizontal straight lines.

Figure 5 shows E1s as a function the atom – surface distance [18]. As the metal band is described using only 3s and 3p-orbitals, its density of states is limited to a finite range of energy. At large distances, the E1s -level follows the image potential limit (the ionization level of He is located 20.5 eV below EF ), while at small distances it bends down, basically due to the electrostatic interaction between the He-1s level and the Al-atomic charge. At short distances (d smaller than , 1 a.u.), as mentioned above, the hopping interaction between the He-1s level and the Al-core levels is very large; this is reflected in the very strong repulsion of the He-1s level to high energies as shown by the E~ 1s -level of Fig. 5 that is calculated modifying the E1s -level by its interaction with the Al-core orbitals. The E~ 1s -level can be considered as the effective level for calculating the interaction between He and the conduction band of Al, embodying all the Al-core level effects in E~ 1s : Notice the general behavior of E~ 1s : at long distances follows the image potential interaction; at d , 3 a.u., it bends down due to the electrostatic interaction between the 1s-level and the Al-atomic charge; at d , 2 a.u., the E~ 1s -level is promoted to high energies due to the repulsion between the He-1s level and the Al-core orbitals. Due to this repulsion, the He-1s level enters the metal band for distances smaller than , 1:1 a.u., making the resonant processes operative.

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2.2. H– metal interaction The case of H interacting with an Al-surface is shown in Fig. 6 and Table 1 [19]. Figure 6(a) and (b) shows the 1s-affinity and ionization levels for H approaching the Al(100)-top and center positions. For the affinity level, we see how it follows the image potential at large distances, bends slightly down between 5 and 7 a.u. due to the attraction of the Al-charge and, eventually, it is shifted to high energies due to the Pauli repulsion between the 1s-level and the metal charge. The ionization level also follows the image potential at long distances and it is shifted to high energies for distances smaller than 5 a.u. due to the Pauli repulsion with the metal charge [20]. In these calculations we have only included the 3s and 3p-orbitals of Al. As seen in the He-case, the effect of the Al-core orbitals starts to be important for d , 2:0 a.u.; for smaller distances, the effect of the core-levels can be expected to be a strong repulsion on both the ionization and affinity levels. Finally, in Table 1 we show the hopping interactions between H-1s level and the Al-3s and 3p orbitals, as a function of the H-metal distance [19].

Fig. 6. Affinity (A) and ionization (I) levels of H as a function of the ion – metal distance, for center and top sites of Al(100).

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Table 1. Absolute value of the hopping integrals for the H-1s state with the Al-3s and Al-3p states Top d (a.u.) 2.0 3.0 4.0 5.0 6.0 7.0

Center

ts – s (eV)

ts – p (eV)

ts – s (eV)

ts – p (eV)

7.54 3.98 2.19 1.18 0.59 0.27

9.55 6.36 3.72 2.21 1.11 0.44

2.79 2.12 1.51 0.78 0.41 0.21

4.05 3.01 2.03 1.04 0.52 0.25

In the center case, the hopping has been projected onto the line joining one Al atom with the H.

Figure 6 and Table 1 provide the necessary information for calculating the H– Al resonant processes that we discuss in Section 5. 3. RESONANT PROCESSES: DYNAMIC SOLUTION OF THE NEWNS– ANDERSON HAMILTONIAN One can use different methods for calculating the population of the ionic species from equation (6) or (9), as the particles evolve towards and from the surfaces. A common method is to derive semiclassical master equations [21, 22]; these equations are a set of coupled equations in which the populations of the different species depend on time via the rates of the available channels for charge transfer. For instance, for charge transfer between H2 and H0 ; these master equations involve the time derivative of the occupation numbers nðH2 Þ and nðH0 Þ: A semiclassical approach such as this can only be valid if the ion level is far from the edge of the metal band; this is the case of H2 ; where one can safely use these rate equations [19,22]. In other cases, one should analyze the ion occupation number using full quantum mechanical techniques for obtaining the dynamic solution of the Newns– Anderson Hamiltonian. We have followed this approach by using Keldysh–Green function techniques [23]. In this method we calculate the Green-function Gr1s;1s ðt; t0 Þ associated with the 1s-ion level [22,24,25]. This Green-function satisfies the following integro-differential equation idGr1s;1s ðt; t0 Þ=dt 2 E~ 1s ðRðtÞÞGr1s;1s ðt; t0 Þ ¼ dðt 2 t0 Þ þ

ðt t0

dt1 Sr1s;1s ½RðtÞ; Rðt1 Þ; t; t1 Gr1s;1s ðt1 ; t0 Þ;

ð10Þ

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where the self-energy Sr1s;1s is given by Sr1s;1s ½RðtÞ; Rðt1 Þ; t; t1  ¼

ðþ1 dv X Qðt 2 t1 Þe2ivðt2t1 Þ T1s;i ðRðtÞÞgri;j ðvÞTj;1s ðRðt1 ÞÞ; 21 2p i;j

ð11Þ

gri;j ðvÞ being the retarded Green-function of the unperturbed metal surface. In equations (10) and (11) E~ 1s ; and T1s;i depend on the time-dependent ion position RðtÞ: Notice that for R independent of time, equations (10) and (11) reduce to h i v 2 E~ 1s 2 Sr1s;1s ðvÞ Gr1s;1s ¼ dðt 2 t0 Þ ð12Þ and Sr1s;1s ðvÞ ¼

X

lT1s;i l2 gri;j ðvÞ

ð13Þ

i;j

which have often been used to analyze the chemisorption of a level on a surface characterized by gri;j ðvÞ: Once we calculate Gr1s;1s ðt; t0 Þ; the occupancy n1s ðtÞ is given by n1s ðtÞ ¼ n1s ðt0 ÞlGr1s;1s ðt; t0 Þl2 þ

X ðþ1 i;j



ðt t0

21

dt1 Gr1s;1s ðt; t1 ÞT1s;i ðRðt1 ÞÞ

dvnF ðvÞri;j ðvÞ

ðt t0

2ivðt1 2t2 Þ dt2 Grp ; 1s;1s ðt; t2 ÞT1s;j ðRðt2 ÞÞe

ð14Þ where nF ðvÞ is the Fermi– Dirac distribution function of the metal electrons and ri;j ðvÞ the density matrix for the metal surface 1 ri;j ðvÞ ¼ 2 Im gri;j ðvÞ: p

ð15Þ

In equation (14), t0 is the initial time and the first term of this equation represents the ‘memory’ contribution associated with the initial condition, n1s ðt0 Þ; for the occupancy n1s ðtÞ: The second term of this equation takes into account all the back and forth electron resonant jumps between the ion and the metal. For an incident ion, say Heþ ; n1s ðt0 Þ ¼ 0; while for an incident neutral atom, like He0 ; n1s ðt0 Þ ¼ 1: The initial time is defined by the initial

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ion – metal distance d0 large enough to have the atom decoupled from the metal. Notice that in the static limit only the second term of equation (14) contributes to n1s (there is no ‘memory’ contribution) and equation (14) reads n1s ¼

X ðþ1 i;j

21

dvnF ðvÞT1s;i ri;j ðvÞTj;1s lGr1s;1s ðvÞl2 ;

ð16Þ

or n1s ¼ 2

ðEF 1 dv Im ~ 1s 2 Sr1s;1s ðvÞ p 21 v 2 E

ð17Þ

as it should be.

4. AUGER PROCESSES This section is devoted to the analysis of Auger processes, treated independently of resonant processes. Such an analysis is relevant in the case of low-lying atomic energy levels that cannot be resonantly neutralized, like He on Al at distances larger than ,1 a.u. Auger processes have been poorly described in the theoretical literature mainly because they involve at least two metal electrons, the one that neutralizes the ion and the one that is excited. Actually, the process involves many electrons because the Coulomb repulsion between the two electrons giving rise to the Auger process will be screened by the rest of the system. The most striking effect of the screening properties of a many-electron system is the existence of collective excitations known as plasmons. One possible channel for ion neutralization at metals consists of the capture of a metal electron by the ion with the released energy and momentum being taken by a plasmon. This channel, termed in the literature as plasmonassisted neutralization, has been assumed to be important in theoretical calculations [26,27] but has not been properly taken into account [27]. Recently, the excitation of surface and bulk plasmons during ion neutralization at surfaces have been reported experimentally [28,29]. The model presented here is the generalization of the theory of ion neutralization in bulk [7,9] to a surface system [30]. The key quantity is the Auger transition rate given by Fermi’s golden rule X 1 ^ 2 dðEf 2 Ei Þ: ¼ 2p lk f lVlill ð18Þ t i;f

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We consider the interaction of the neutralizing electron with N 2 1 metal electrons and write for the initial, lil and the final, lf l; states appearing in equation (18) lil ¼ l0l^lkor l;

lf l ¼ lnl^lal;

ð19Þ

where l0l and lnl are the ground and an excited state of the many-body electron gas with energies E0 and En ; respectively, lal is the final state of one electron in the atomic ion core with energy Ea and lkor l is a one-electron state representing the initial state of the electron that neutralizes the ion core. Since lkor l and lal should be solutions of the same Hamiltonian, we take for lkor l a one-electron metallic state, lkl; orthogonalized to lal [7,8]. The interaction potential V^ is the Coulomb interaction between the charge densities induced in the metal, dn and in the atom, dr; associated to the lkor l ! lal transition ð ð dnðr1 Þrðr2 Þ V^ ¼ dr1 dr2 : ð20Þ lr1 2 r2 l Then, upon Fourier transforming in the coordinates x parallel to the surface, the matrix elements of equation (18) read ð dq ð ^ ¼ k f lVlil dz1 knldnðq; z1 Þl0lFðk; q; z1 Þ; ð21Þ ð2pÞ2 where we have defined

Fðk; q; z1 Þ ¼

2p kaleiq·x2 e2qlz1 2z2 l lkor l: q

ð22Þ

Finally, by substituting equation (21) into equation (18) and making use of the standard expression relating the imaginary part of the susceptibility xðv; q; z; z0 Þ for interacting electrons (screened susceptibility) to the density operators dnðq; zÞ and dnðq; z0 Þ; the final expression for the Auger capture rate is obtained as ð dq ð ð X ð1 1 dz dz0 ½2Im xðv; q; z; z0 Þ ðza Þ ¼ 2 dv 2 t ð2pÞ k,kF 0

Fðk; q; zÞFp ðk; q; z0 Þdðv þ Ea 2 Ek Þ:

ð23Þ

In equation (23) kF is the Fermi wave vector and the transition rate depends on the distance za between ion and surface through the wave function of the state lal in equation (22). The physics behind equation (23) is clear. Fðk; q; zÞexp½iðEa 2 Ek Þt; where F is given by equation (22), is the effective potential associated with the transition from a metal-electron state lklexpð2iEk tÞ to the core-ion state lalexpð2iEa tÞ: This potential, which oscillates in time with frequency

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v ¼ Ea 2 Ek ; produces fluctuations of charge density in the many-electron system and the imaginary part of the induced charge density dnðk; q; v; zÞ given by ð dnðk; q; v; zÞ ¼ dz0 xðq; v; z; z0 ÞFðk; q; z0 Þ ð24Þ is related to energy losses. In fact the quantity, ð 2Im dz dnðk; q; v; zÞFp ðk; q; zÞ

ð25Þ

gives the rate at which the oscillating potential produces metal excitations of energy v and momentum q: Then, when adding the contributions of all possible electrons that can neutralize the ion, the Auger rate of equation (23) gives the total rate at which metal excitations are produced in the Auger neutralization process. The screened susceptibility xðq; v; z; z0 Þ appearing in equation (23) is a smooth function of the spatial coordinates that interpolates from zero very far from the surface to the bulk value. It contains the spectrum of the singleparticle and collective modes and their coupling. This coupling, known as Landau-damping, depends on distance and it is the only channel that allows decaying of a collective mode in the jellium theory. Surface plasmons of parallel momentum q different from zero are Landau-damped but bulk plasmons cannot decay into electron – hole pairs below a critical frequency in this linear theory. Collective modes can show up in electron emission spectra [28,29] because they are coupled to electron– hole pairs. The theory in which the susceptibility is formally defined for jellium surfaces is the time-dependent density functional theory (TDDFT). In this theory, the susceptibility for interacting electrons (also called screened susceptibility) xðq; v; z; z0 Þ is related to the susceptibility for non-interacting (independent) electrons x0 ðq; v; q; z; z0 Þ via the integral equation

xðq; v; z; z0 Þ ¼ x0 ðq; v; z; z0 Þ ð ð þ dz1 dz2 x0 ðq; v; z; z1 ÞRðq; v; z1 ; z2 Þxðq; v; z2 ; z0 Þ; ð26Þ with Rðq; v; z1 ; z2 Þ ¼

2p 2qlz1 2z2 l e þ m0xc ðz1 Þdðz1 2 z2 Þ q

ð27Þ

with m0xc being the derivative of the exchange-correlation potential. When the effective interaction is simply the Coulomb potential ðm0xc ¼ 0Þ; x is said to be evaluated in the random phase approximation (RPA). Excellent agreement between theory and experiments of electron energy loss [31] and

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photoyield [32] has been achieved if x0 is calculated for a Lang – Kohn jellium surface [33] and then x is evaluated either in the TDDFT or the RPA approximations. However, the computation of the Auger rate of equation (23) is very involved because it is necessary to evaluate x for as many values as required to perform the eight-dimensional integral. This is one of the reasons why the problem of calculating a realistic Auger transition rate at surfaces has been a long-standing one and many approximations to the matrix elements and/or interaction potentials can be found in the literature. One common approximation is the calculation of the matrix elements using the bare (unscreened) Coulomb potential [34] (or, equivalently, the use of x0 instead of x in equation (23)). This approximation is a good one when the energy transfer v is much larger than the plasma frequency vP ; only in this case do electrons behave like independent particles. If the energy v is of the order of vP ; the independent-particle calculation ignores the existence of collective excitations while if v is much smaller than vP the independentparticle calculation does not take into account the strong screening of a nearly static Coulomb potential. This is clearly seen in Fig. 7 (taken from Ref. [35]), where we have plotted the Auger rate per unit frequency 1t ðvÞ; which is defined from equation (23) as ð1 1 1 ðza Þ ¼ dv ðv; za Þ ð28Þ t t 0 for neutralization of Heþ at a distance of za ¼ 5 a.u. from a Lang – Kohn jellium surface of rs ¼ 2:0 a.u. representing Al [33].

Fig. 7. Neutralization rate of Heþ on Al per unit of transferred energy at 5 a.u. from the jellium edge. Squares: independent-particle calculation. Crosses: interactingparticle calculation.

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The independent-particle calculation, represented by the line with squares in Fig. 7, yields a smooth function of v; while the interacting-particle calculation, in which x is evaluated in the TDDFT, raises sharply above v , 0:65vP : This is the range of frequencies of the collective surface modes of Al [31] which are obviously absent in the independent-particle calculation. Then the comparison between both calculations shows that the plasmon-assisted neutralization channel is indeed the dominant one at these energies. The opposite happens in the limit of small v: In this case the independent-particle calculation overestimates the electron – hole pair channel because it does not take into account the screening of the Coulomb potential at very low frequencies. The overall effect on the total rate of Auger neutralization of Heþ on Al is not so dramatic, however, as is shown in Fig. 8. This is due to the compensation of the electron – hole channel with the plasmon-assisted channel when integrating in v (equation (24)), but one must remember that such a compensation does not appear in other systems. For Arþ scattered off Mg, the energy transfer v is in the range between 0:46vP and 1:1vP and the plasmon-assisted channel dominates [28]. Hþ scattered off Al belongs to the opposite case because the energy transfer is below 0:6vP : Another possible simplification to the full calculation of the Auger capture rate consist of the use of the asymptotic long-distance limit of

Fig. 8. Neutralization rate of Heþ on Al as a function of the distance to the jellium edge. Continuous line: interacting-particle calculation. Dashed line: independentparticle calculation.

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Fig. 9. The long-distance limit for neutralization of He on Al (continuous line) is compared with the interacting particle calculation of Fig. 8 (here shown by a dashed line).

equation (22) [35]. In Fig. 9 we present the results of this long-distance calculation together with the exact results of equation (23). We can appreciate that the long-distance approximation overestimates the Auger rate for distances larger than 10 a.u.

5. RESULTS This section is devoted to discussing results for some specific cases where we can illustrate the analysis presented above. Three different examples will be analyzed: (a) H2 scattering off an Al-surface; here, the resonant processes analyzed using master equations yield a good description of the charge exchange mechanism. (b) Heþ scattering off metal surfaces at glancing incidence; in this case, the ion perpendicular energy is so small that the incoming ion does not penetrate too much the surface electron charge; then, the He-1s level does not overlap with the metal band states and the only operative charge exchange process is the Auger one. (c) Heþ scattering off metal surfaces at large incidence angles. Contrary to case (b), Heþ does penetrate the surface electron charge and, for small ion– metal distances, the He-1s level overlap with the metal conduction band. We find that, in this case,

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both resonant and Auger mechanisms play a crucial role in determining the He-charge states. 5.1. H2 on Al This case is analyzed using the H2 -levels and the H–Al interactions given in Fig. 6 and Table 1, respectively. As the H2 -level is always above the metal Fermi level, we can use semiclassical master equations for solving the Newns– Anderson Hamiltonian of this problem [21,22]. Figure 10 shows our results for the total fraction of H2 as a function of the detection angle for incident protons of either 4 or 1 keV [36]. In our calculations we assumed that the ion penetrates the surface up to 3 a.u. away from the first atomic layer. This assumption is not, however, crucial because the population of H2 is determined by the charge exchange processes occurring around the freezing distance characterizing the process; for instance, for exit angles smaller than 208, that freezing distance is larger than 6 a.u. We should mention that for freezing distances larger than 6 a.u. the 2sp levels also play an important role in the resonant process and, in our calculations, we have included them following Ref. [37]. We conclude this section by mentioning that the results shown in Fig. 10 only depend on the H-1s level for exit angles larger than 208; then the freezing distance is smaller than 6 a.u. and the effect of the 2sp-levels can be

Fig. 10. Total fraction of H2 scattered off Al(100). Squares: incident protons of 4 keV; crosses: incident protons of 1 keV. Experiments from Ref. [36].

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neglected. It is remarkable the good agreement found between theory and experiments in this limit, when we consider the large angles of Fig. 10. 5.2. He1 on metal surfaces: glancing incidence When a beam of Heþ ions with total energy of 1– 20 keV is incident on metal surfaces at glancing angles (here the angle of the beam with the surface is typically smaller than 108) the component of the ion velocity perpendicular to the surface is so small that the ions never get closer than ,1.5 a.u. from the first atomic layer. For Al, as we showed in Section 2.2, the He-1s level never resonates with the conduction band of the metal, this fact making Auger processes the only ones responsible for neutralization of the incoming Heþ ion. The pioneering work by Hagstrum [14] settled down both experiments and theory in this range of small perpendicular energies and his ideas have formed a kind of established understanding until very recently. Hagstrum’s measurements of electron spectra revealed that the 1s-level of He was shifted up in energies by about 2 eV when Heþ was neutralized. Equating 2 eV to the classical image potential, he obtained a distance of neutralization of about 4 a.u. from the image plane (7 a.u. from the first atomic layer). Then assuming that the Auger rate decreases exponentially with the distance to the surface, za ; as 1t ðza Þ ¼ A expð2aza Þ; one can fix A and a to get Heþ neutralized at 4 a.u. from the image plane. More recent experiments of mean energy gain of Heþ incident on Al(111) [38] were analyzed in the same way and from that kind of analysis an Auger rate was retrieved that was larger by several orders of magnitude than the ones obtained by the best and more sophisticated theoretical calculations of Refs. [35,39]. However, the origin of the discrepancy between calculated and retrieved Auger rates can be traced back to the incorrect assumption of the image shift of the He-1s level at the distances where neutralization occurs. It was pointed out in Ref. [40] that calculated Auger rates and measured energy gains are not in discrepancy when one uses a He-1s level that is accurately calculated as a function of the distance to the surface. Using the results of Fig. 5, one can see that the mean energy gain of 1.8 eV measured for Heþ on Al in Ref. [38] is compatible with neutralization distances of about 3 and 7 a.u. from the first atomic layer. The latter is located in the asymptotic image potential region and corresponds to the general belief. The former is, however, also consistent with the calculated Auger rates, as it was shown in Ref. [40]. Identical conclusion was reached in Ref. [41] when analyzing electron emission spectra. Very recently it has been possible to measure the very small fraction of ions that survive Auger neutralization in collisions of Heþ with Ag surfaces at grazing incidence [42– 44]. These experiments reveal a number of survivals of 1023 – 1024 ; which are many orders of magnitude larger than the

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Fig. 11. Experimental ion fractions for random scattering direction for the Ag(110) (open dots) and Ag(111) (full dots) surfaces. The lines labelled ‘ion’ and ‘neutral’ correspond to calculated Auger survival probabilities for two types of trajectories.

number it would be obtained under the assumption that neutralization takes place at distances of ,7 a.u. from the metal surface. However, the correct number of survivals is obtained by using the theoretical Auger rates, as shown in Fig. 11 [44]. Therefore, we conclude that the theories presented in Sections 2 and 4 for calculating energy level variation of ions in front of surfaces and Auger neutralization rates are able to account for current experimental findings in the range of very low incident kinetic energies. 5.3. He1 on metals: normal incidence In this section we consider the case of Heþ scattering off metal surfaces with incident and scattering angles close to the surface perpendicular direction. Heþ ; having a kinetic energy of around 1 keV, penetrates the electron surface charge and collides with the metal atoms; typically, the ion suffers a binary collision and recedes back to vacuum. We analyze the charge exchange processes assuming the ion to move along two straight trajectories, before and after the binary collision.

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In order to understand the charge exchange processes of this Heþ – metal collision, we take into account the different mechanisms discussed in this paper, resonant and Auger processes. In our approach to this problem, one has to distinguish two clearly separated regimes: in one of them only Auger neutralization is operative, this mechanism being active for Heþ – metal distances larger than ,1 a.u. As shown in Fig. 5, the He-1s level overlaps the metal band states for Heþ – metal distances smaller than ,1 a.u.; this is the range of distances for which the second regime appears. In this regime, Auger processes are negligible and resonant processes are fully operative up to the turning point of the trajectory, which we find to be ,0.3 a.u. As Auger and resonant processes are spatially separated, we can calculate independently their contributions [18,45]. Then, we calculate the fraction of positive ions, Pþ ; by introducing the Auger survival probabilities for the incoming and þ outgoing paths, Pþ A;in and PA;out ; respectively, as well as the probabilities for þ 0 He to survive and He to be reionized in the close collision Pþ surv and Preion ; respectively. From these quantities we obtain the total ion fraction after the whole ion –surface interaction, Pþ ; by means of the following equation: þ þ þ þ Pþ ¼ Pþ A;in Psurv PA;out þ ð1 2 PA;in ÞPreion PA;out ;

ð29Þ

where the first and second parts of equation (29) represent the survival channel and the reionization channel, respectively [18,45]. In equation (29), the Auger þ probabilities Pþ A;in and PA;out are calculated by means of a classical master equation while the resonant probabilities Pþ surv and Preion are obtained from the dynamic solution of the Newns– Anderson model presented in Section 3. It was shown in [46] that the semiclassical master equations provide an excellent approximation to the full dynamic solution when only Auger processes are operative. Also, the use of equation (29) was justified in Ref. [47]. Figure 12 shows the fraction of Heþ backscattered from Al, Pþ ; as a function of the incident kinetic energy. Survival and reionization ion fractions are also shown in this figure. Notice that for ion energies smaller than 100 eV (distances of closest approach larger than 1 a.u.) the resonant channel is practically closed and the ion fraction is determined by Auger processes. For larger energies, the ion approaches the metal surface and the resonant mechanism starts to be operative. In the calculations of Fig. 12, Pþ shows an unusual behavior with two minima around 150 and 400 eV. Although the appearance of the one at the lowest energy might depend on a more accurate calculation of the Auger and resonant processes for these energies [47], these results doubtlessly show that the energy regime 120– 260 eV represents a transition regime where resonant processes become important, leading to an increase of Pþ once that the resonant channel is fully open. The case of Heþ backscattered from Pd has also been analyzed in Ref. [18]. Important differences have been found with respect to the case of Al; while for Al the He-1s level resonates with the metal conduction band at

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Fig. 12. Ion fraction, Pþ ; of Heþ backscattered off Al. Dashed line: Auger survival probability. Full circles: survival probability including resonant processes only. Full triangles: survival channel. Full squares: our full calculation (equation (29)). The experimental data of Ref. [48] are shown by a continuous line.

distances smaller than ,1 a.u., this resonance appears for Pd at distances of 2 a.u., making the resonant charge transfer between Heþ and Pd less effective. This explains why the ion fraction for the Heþ/Pd collision is basically determined by Auger neutralization processes.

6. CONCLUSIONS Charge exchange processes in LEIS have been analyzed and some case study systems, like H and He scattered off metals, have been discussed in detail. It has been commonly accepted that, for Heþ ; Hagstrum’s Auger neutralization model holds in the LEIS regime, namely for ion kinetic energies of the order of 1 keV. Our analysis for the He/Al system

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confirms this point of view if the ion perpendicular energy is smaller than 100 eV. For larger perpendicular energies, however, our discussion shows that He penetrates the metal surface layer and that the He-1s energy level resonates with the metal conduction band, making resonant processes fully operative. We conclude that, for He, Auger processes are the only ones responsible for charge transfer at low perpendicular energies, while the resonant mechanism becomes also operative at high energies (.100 eV). In the low energy regime, Heþ is neutralized very efficiently at the image plane, which is located about 3 a.u. in front of the first metallic layer. We have also shown how the high energy limit can be analyzed combining the Auger and resonant mechanisms; the Auger process is dominant at long ion– metal distances (.1.2 a.u.), while the resonant one also contributes at closer distances. In the case of H, the 1s-energy level is always resonating with the Alconduction band and resonant charge transfer occurs for large ion – metal distances. In particular, we have seen how charge exchange processes for the H2 /Al system can be understood considering only the resonant mechanism. We also conclude that a detailed calculation of the different mechanisms operating in the ion – metal charge exchange processes depends on a careful description of the ion levels as a function of the ion– metal distance. In this paper, these levels as well as the ion – metal hopping interactions and the Auger capture rates, have been presented for the H/Al and He/Al systems. ACKNOWLEDGEMENTS We gratefully acknowledge financial support by the Spanish Comisio´n Interministerial de Ciencia y Tecnologı´a under projects MAT-2001-0665 and BFM 2001-0150. We also acknowledge many fruitful discussions with P. Bauer, H. H. Brongersma, R. A. Baragiola, V. A. Esaulov, Evelina A. Garcı´a, E. C. Goldberg, N. Lorente, J. Merino, W. More, P. Pou and N. P. Wang. REFERENCES [1] [2] [3] [4] [5] [6]

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