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Negative ionization of the secondary ions of silver and gold sputtered from their elemental surfaces
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 256 (2007) 468–473 www.elsevier.com/locate/nimb
Negative ionization of the secondary ions of silver and gold sputtered from their elemental surfaces A. Sindona b
a,b,*
, P. Riccardi
a,b
, S. Maletta a, S.A. Rudi
a,b
, G. Falcone
a,b
a Dipartimento di Fisica, Universita` della Calabria, Via P. Bucci 31C, 87036 Rende (CS), Italy Istituto Nazionale di Fisica Nucleare (INFN), Gruppo Collegato di Cosenza, Via P. Bucci 31C, 87036 Rende (CS), Italy
Available online 17 December 2006
Abstract Calculations of the ionization probabilities of Ag and Au particles, ejected during sputtering of clean Ag(1 0 0) and Au(1 0 0) surfaces, respectively, are reported. An effective one-electron theory is used to describe: the plane metal surface, with a projected band gap, the secondary emitted atom, whose charge state is investigated, and its nearest-neighbor substrate atom, put in motion by the collision cascade generated by the primary ion beam. Suitable rectilinear trajectories are selected to describe the motion of these two atoms outside the solid. A good agreement is found with van Der Heide’s experiments (P.A.W. van Der Heide, Nucl. Instr. and Meth. B 157 (1999) 126). 2006 Elsevier B.V. All rights reserved. PACS: 79.20.m; 71.10.w; 73.20.r Keywords: Impact phenomena (including electron spectra and sputtering); Theories and models of many-electron systems; Electron states at surfaces and interfaces
1. Introduction Electron transfer and excitations processes occurring at the surface of solids, bombarded with ion sources, have been the subject of continuous interest for decades [1]. In spite of this attention and importance, the underlying mechanisms of secondary ion formation and escape are still unclear [2–6]. Some experiments [5,6] have shown that there exist different possible ionization/neutralization mechanisms active during ejection. One of these, predominating at higher emission energies ( J 100 eV), is well described by resonant electron tunnelling between the ejected particle and the target [7–9]. The other, commonly observed at lower emission energies ([40 eV), is suggestive of some form of surface excitation. A generalized model, accounting for both mechanisms, was proposed in [4] and *
Corresponding author. Tel.: +39 (0)984 496059; fax: +39 (0)984 494401. E-mail address: Sindona@fis.unical.it (A. Sindona). 0168-583X/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.12.059
applied to the ionization of secondary Cu± ions sputtered from Cu(1 0 0) jellium samples: surface excitations were explained in terms of transient quasi-molecules (QMs) formed between the secondary emitted and their nearestneighbor substrate atoms, put in motion by the collision cascade. This paper presents a more accurate version of the model, based on the effective Hamiltonian ^h½Z a ; Z b ¼ ^p2 =2 þ ^v½Z a ; Z b
ð1Þ
of a spinless electron, with momentum p^ and position ^r, interacting with a plane metal target and two neutral atoms. The effective potential ^v½Z a ; Z b depends parametrically on the positions, Za and Zb, of the emitted (a) and the substrate atoms (b), along the surface normal (Section 2). The negative ionization probability of a-atoms is calculated using a spectral decomposition method involving: the truncated set of available states, below the vacuum level of the metal conduction band (with a projected band gap), the orthonormalized wave-function for the affinity level of the
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473
469
a-atom, and the orthonormalized molecular orbital adiabatically formed between a and b atoms (Section 3). The ionization probabilities of Au and Ag ions, sputtered from Au(1 0 0) and Ag(1 0 0) surfaces, respectively, are obtained within the straight line trajectory approximation for the diatomic motion outside the collision cascade area. Suitable choice of the parameters of these trajectories leads to a good agreement with the experiments of [6] (Section 4). 2. Theory The one-electron pseudopotential for the double ionmetal system has the cylindrically symmetric form ^v½Z a ; Z b ¼ ^vs þ ^vQM ½Z a ; Z b þ D^vQM ½Z a ; Z b
ð2Þ
with a local part
2pz vs ðzÞ ¼ A10 þ A1 cos ; as
z<0 5p 4b
ð3Þ
Fig. 1. Instantaneous affinity energies, ea and eb, of two Au ions at position Za and Zb, relative to a Au(1 0 0)-surface whose electronic structure is derived from Eq. (3).
denoting the semi-empirical Chulkov’s potential of a (1 0 0) surface, constructed from pseudopotential local density theory [10]. Eq. (3) is completely determined by fixing A10, A1, A2, b – reported in Table 1 for Ag(1 0 0) and Au(1 0 0) – and constraining continuity and differentiability everywhere in the space. The atomic planes are separated by the lattice constant as , so that the topmost layer of lattice points defines the origin of the coordinate system (z = 0) and zIM labels the position of the image plane. The other components in Eq. (2), whose diagonal matrix elements in the basis of coordinates are sketched in Fig. 1, have already been discussed in [4]. In particular, ^vQM ½Z a ; Z b ¼ ^va ½Z a þ ^va ½Z b is a separable quasi molecular potential, where ^va ½Z aðbÞ establishes the electron interaction with each neutral particle, and D^vQM ½Z a ; Z b accounts for the effective action of the image charges of all electrons and protons in the QM. Parametric time dependence is a consequence of the classical motion of a and b atoms, through the kinematic laws Za(b) = Za(b)(t), yielding ^ hðtÞ ^ h½Z a ðtÞ; Z b ðtÞ. In [4], an analytical picture was adopted based on the following assumptions: (i) the two atoms are initially placed outside the collision cascade area; (ii) their interaction is modelled
by a Morse potential; (iii) all interactions with other surface atoms are neglected. Here, we have tested different types of numerical trajectories calculated from Molecular Dynamics (MD) simulations on bidimensional clusters of about 250 atoms covering the region (60 to 60) au, along the surface plane, and (60, 0) au along the surface perpendicular direction. We used both 1–15 keV Ar and 1–15 keV Cs projectiles, at 0–75 incident angles from the surface normal. The interaction potential between two target particles was given the form of the many body tight binding potential of [13], while the projectile-target interaction was modelled by the Moliere potential of [14]. An example of such simulations, computed with the Verlet algorithm [13,14] using a time step of 1 au, is shown in Fig. 2: particles emitted from Au(1 0 0) clusters, bombarded with 2 keV Cs projectiles, mainly come out as monomers and dimers outside the collision cascade area. In either case, when the ejected atoms are at distances of about 5–10 au from the surface, their interactions with other surface atoms are negligible. Furthermore, the binding energy in a dimer is small compared to its center-of-mass kinetic energy, at interatomic distances larger than 6–7 au. Indeed, the kinematic model of [4] predicted nearly straight trajectories for a-atoms. These considerations let us assume that a and b atoms still possess strong electronic interactions when their binding forces tend to zero, moving with
¼ A20 þ A2 cosðbzÞ;
06z<
¼ A3 eaðzz1 Þ ;
5p 6 z < zIM 4b
¼
ekðzzIM Þ 1 ; 4ðz zIM Þ
z > zIM ;
Table 1 Parameters of Eq. (3), in au, for Au(1 0 0) and Ag(1 0 0) Surface
A10
A1
A2
b
zIM
as
e1
e2
Ag(1 0 0) Au(1 0 0)
0.342 0.397
0.185 0.154
0.143 0.223
2.422 3.363
2.06 1.62
3.86 3.85
0.104 0.142
0.072 0.012
Energies are reported relative to the vacuum level.
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A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473 Table 2 Parameters of Eq. (4), in au, for Ag and Au Ion
Ag Au
Fig. 2. (a)-(c): MD simulation of 2 keV Cs projectiles incident at 30 from the perpendicular direction of a bidimensional FCC Au cluster (see text). The trajectories of some of these atoms are drawn in the plot. (d): time laws, Za(t) and Zb(t), of two possible candidates for a and b atoms.
approximately constant velocities from the instant (t = 0) when both particles are outside the image plane (Fig. 2(d)). The final kinetic energy Ea, of the a-atom, is fixed by the experiment under consideration, whereas other trajectory parameters, such as the kinetic energy Eb, of b-atoms, and the initial positions Za(0) and Zb(0), can be used as free parameters. In the present treatment, we set Zb(0) right outside the image plane and let the initial interatomic distance, Za(0)Zb(0), to be of the order of the lattice separation as . Finally, we consider a (gaussian) distribution of b-particles with (mean) kinetic energy Eb < Ea. 3. Spectral decomposition We construct a finite basis that includes the instantaneous affinity states, of a and b atoms, and the projected bulk states of the target lying below the vacuum level. Specifically, the affinity state of an isolated negative ion, at position Za(b), is obtained from the atomic Hamiltonian ^ ha ½Z aðbÞ ¼ ^ p2 =2 þ ^va ½Z aðbÞ . In the separable potential approximation [11], the corresponding affinity wave function, at Za(b) = 0, takes the analytical form ka r e ela r hrjai ¼ N A ; ð4Þ r where the parameters NA, ka, and la are reported in Table 2 for Ag and Au. At Za(b) 5 0, the wavefunction becomes hrja[Za(b)] i = hr Za(b)uzjai and its (unperturbed) energy, A, coincides with the affinity energy of the negative ion state. Then, the adiabatic affinity states of the a-
Na
ka
la
A
0.8413 1.008
0.3094 0.4119
0.4686 0.6134
0.04785 0.0848
and b-atoms are ja(t)ija[Za(t)]i and jb(t)ija[Zb(t)]i, respectively. The (numerical) eigenstates of the isolated metal band, denoted by {jki}, arise from the diagonalization of the surface Hamiltonian ^hs ¼ ^p2 =2 þ ^vs . The latter has a continuous spectrum {ek}, with a minimum at ek = A10, Fermi energy eF , and a projected band gap in the range e1 < ek < e2 (see Table 1). The corresponding wavefunctions read hrjki ¼ wk? ðzÞeikk rk =L, in which L is the thickness of the solid, k ¼ ðkk ; k ? Þ the electron wave-vectors, and wk? the (real) numerical eigenfunction of the potential (3). L is chosen in order to have a maximum energy spacing between two band states – at kk = 0 – smaller than 0.025 eV. In this way about 500 perpendicular wavevectors are used to reproduce the continuum spectrum below e1. Such a truncation is reasonable, in the case of Ag(1 0 0) and Au(1 0 0), where e2 lies outside the vacuum level (see Table 1). Indeed, electron exchanges mostly involve the occupied continuous states fjkigek 6eF and the affinity states ja(t)i and jb(t)i, whose adiabatic energies are always smaller than e2, reaching asymptotically the unperturbed value A from below. Orthonormalization of fjaðtÞi; jbðtÞi; jkigek 6e1 , via the Graham–Schmidth method, yields the finite basis fjaðtÞi; jbðtÞi; jkigek 6e1 , in which: P 1 ek 6e1 jkihkj jaðtÞi ¼ jaðtÞi; ð5Þ K a ðtÞ is sharply peaked at Za(t), tending to ja(t)i in the long time limit, and P 1 jaðtÞihaðtÞj ek 6e1 jkihkj jbðtÞi ¼ jbðtÞi; ð6Þ K b ðtÞ exhibits a strong molecular character with average lifetime larger than 10 fs, at Ea J 10 eV and Eb J 1 eV. In Eqs. (5) and (6) the function Ka(b)(t) ensures normalization. The negative ionization probability of a-atoms may be written as [4,8,9] X 2 jGþ ð7Þ R ðtÞ ¼ ak ðt; t 0 Þj hðeF ek Þ; k where Gþ ak ðt; t 0 Þ labels the probability amplitude that a band electron, occupying the state jki in the remote past (t0 ! 1), will be transferred to the emitted atom at the time t, when Z a ðtÞ zIM . Complementarily, Gþ bk ðt; t 0 Þ denotes the probability amplitude for electron transfer to the orthonormalized molecular state. Using the semiclassical approximation [9], we reduce the þ calculation of Gþ ak ðt; t 0 Þ and Gbk ðt; t 0 Þ to the solution a 2 · 2 matrix equation [4]:
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473
½i@ t H 0QM ðtÞ
Gak ðt; t0 Þ Gbk ðt; t0 Þ
¼ i eiek ðtt0 Þ
V ak ðtÞ V bk ðtÞ
:
ð8Þ
Here, time evolution is governed by the dissipative Hamiltonian H 0QM ðtÞ ¼ H QM ðtÞ þ RQM ðtÞ, in which H QM ðtÞ has the (real) matrix elements of ^ hðtÞ into the states ja(t)i and jb(t)i. Its diagonal components, ea(t) and eb(t), are the instantaneous energies of the orthonormalized affinity states. The dissipative term is the (complex) self-energy matrix 0 Rt 1 i 0 ds½ea ðsÞek 0 X Z t V ak ðtÞ t V ðt Þ e ak A; RQM ðtÞ ¼ i dt0 @ Rt V bk ðtÞ i 0 ds½eb ðsÞek t0 0 k t e V bk ðt Þ ð9Þ in which labels the tensor product of two column vectors and V aðbÞk ðtÞ ¼ haðbÞðtÞj^ hðtÞjki are the coupling integrals of discrete and continuous states. It should be noted that the interaction with a state jki has the form of an external perturbation. The matrix elements appearing in Eq. (8) evolve over two different time scales, related to the different motions of a and b atoms. This is shown, for example, in the study of the renormalized energies of the orthonormalized affinity states, reported in Fig. 3(a) and (b) for Ag/ Ag(1 0 0) and Au/Au(1 0 0), respectively. Indeed, e0a ðtÞ ¼
a
c
b
d
Fig. 3. Instantaneous energies (e±) and broadenings (D±) of the bonding and antibonding orbitals of the QM, made of a- and b-atoms, with the parameters of Ag/Ag(1 0 0) and Au/Au(1 0 0). In panels (a), (b), the renormalized affinity energies ðe0a and e0b Þ are reported for comparison. In panel (b), the unrenormalized atomic energy ea is also shown to visualize the effect of the shift induced by the self-energy matrix (9).
471
Re½H 0QM ðtÞaa and e0b ðtÞ ¼ Re½H 0QM ðtÞbb are corrected by the effect real part of the self energy matrix (9) inducing short time oscillations, which are more clearly observed in Fig. 3(b), where QM interactions are stronger. In general, e0a ðtÞ changes faster than e0b ðtÞ, although both terms are slow on the femtosecond scale of the electronic transitions, which justifies the use of the semiclassical approximation. At short times after the instant of ejection, e0a ðtÞ is shifted negatively, with respect to its unperturbed value, due to the dominant effect of the electron–surface interaction ð^vs Þ: e0b ðtÞ is promoted above the vacuum level, since the substrate atom is initially closer to the image plane where the leading effect is due to the image potential ðD^vQM Þ. At larger times, e0a ðtÞ tends to the unperturbed affinity energy as fast as 1/Za(t), while e0b ðtÞ behave as 1/ Zb(t). Level crossing allows resonant charge transfer processes between ja(t)i and jb(t)i, coupled by the off diagonal terms ½H 0QM ðtÞab and ½H 0QM ðtÞba . A closer insight into quasi-molecular correlations is gained by considering the instantaneous spectrum of H 0QM ðtÞ, yielding the eigenvalues e±(t) iD±(t). The corresponding eigensates, namely j+(t)i and j(t)i, are the antibonding and bonding states of the QM, obtained from linear combinations of ja(t)i and jb(t)i. Their energies, e±(t), and their virtual broadenings, D±(t), are shown in Fig. 3. We see that e+(t) tends to e0a ðtÞ, while e(t) tends to e0b ðtÞ, in the long time limit (Fig. 3(a) and (b)); thus, an electron captured in the antibonding state will be detected in a-atom. In addition, the bonding orbital is further shifted towards eF , compared to e0a ðtÞ and e0b ðtÞ, which enhances the probability for electron capture to the QM. From this perspective, Eq. (8) establishes electron transfer dynamics based on two mechanisms: one is direct tunnelling from each band state to the ejected atom and the other an indirect, two-step process, with an intermediate transition through the molecular level e(t). Another crucial aspect concerns the evolution of the broadenings of the QM orbitals, shown in Fig. 3(c) and (d); both D+(t) and D(t) drop off to zero with an oscillating exponential behavior, which is not surprising since the adiabatic energies of the orthonormalized affinity states take values in the projected band gap during the whole process. It follows that their adiabatic broadenings are virtually zero Rt so that the phase factors expfi t0 ds½eaðbÞ ðsÞ ek g, within the t 0 -integral in Eq. (9), induce oscillations in RQM ðtÞ. On the other hand, the exponential decrease is related to the hopping terms Vak(t) and Vbk(t). The sign of D±(t) modulates electron transfer processes to the state j ± (t)i, whose occupation probability contains the exponential factor Rt exp dsD ðsÞ . 4. Results A numerical solution for Eq. (8) – at positive times – requires a model to estimate the electronic distribution of the metal band at the instant of ejection (t = 0). Recalling that the final charge fraction of the localized levels is
472
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473
independent on their initial occupancy in the remote past [8], we continue the problem analytically to negative times with Gellmann and Low’s adiabatic prescriptions [12]: 2
p þ ^hðt 6 0Þ ¼ ^ þ ^vð0Þe0 t : 2
ð10Þ
In this case, both Gak(0,t0) and Gbk(0,t0) can be calculated exactly, because their evolution from the remote past is stationary. In addition, the one-electron potential is switched þ off at t = t0, implying Gþ ak ðt 0 ; t 0 Þ ¼ Gbk ðt 0 ; t 0 Þ ¼ 0. Then, the analytical solution of Eq. (8) at t 6 0 reads: Gak ð0; t0 Þ V ak ð0Þ T0 ¼ ieit0 ek k0 ; ð11Þ Gbk ð0; t0 Þ T k V bk ð0Þ in which T 0k
¼
ek e0b ð0Þ
V 0ab ð0Þ
V 0ab ð0Þ
ek e0a ð0Þ
! ð12Þ
is a transfer Hamiltonian from the state jki to the affinity states. Finally, Eq. (11) can be used as initial condition to solve numerically Eq. (8), for each k and t > 0, yielding Gak(t,t0) and Gbk(t,t0). This allows to calculate the negative ionization probability of the state ja(t)i, via Eq. (7). Complementary, we also calculated the negative ionization probability of the state jb(t)i, denoted R0 ðtÞ, by replacement of Gak(t,t0) with Gbk(t,t0) in Eq. (7). Fig. 4 shows the negative ionization probabilities, R(t) and R0 ðtÞ, for the orthonormalized affinity states. We observe there exists a time tF such that both R(t) and R0 ðtÞ are independent on time, for t > tF , whereas they depend on the kinetic energies of a- and b-atoms. tF increases with decreasing Ea, taking typical values in the range 10–200 fs at Ea = 10 300 eV. A signature of charge exchanges between ja(t)i and jb(t)i is given by the oscillating behavior of R(t) and R0 ðtÞ, occurring for typical times
a
b
Fig. 4. Ionization probabilities R(t) and R0 ðtÞ for the orthonormalized affinity states, with the parameters of Ag/Ag(1 0 0) and Au/Au(1 0 0) systems (Fig. 3). hR(t)i is obtained by averaging R(t) over a gaussian distribution of b-particles with mean energy Eb and standard deviation rb.
of the order of 102 fs, for kinetic energies Ea [ 15 eV and Eb [ 1 eV. In Fig. 4 we also report the probability hR(t)i, obtained as the average ionization probability of a-atoms for a gaussian distribution of b-atoms, with mean energy Eb and standard deviation rb. Plots of hR ðtF Þi versus ya, the inverse velocity of the emitted atom, are compared with the experimentally derived distributions of Ag and Au ions sputtered from their elemental surfaces [6]. Eb is adjusted to the data, taking values in the range 1–7 eV, in both Ag/Ag(1 0 0) and Au/Au(1 0 0) systems. Fig. 5 shows a good agreement with data, correctly reproducing the increase in the negative ion population with increasing ya, above 106 s/cm. This is explained, in the present context, with the enhancement of the probability for indirect transfers with decreasing Ea, corresponding to a longer duration of quasi molecular interactions. Conversely, the direct resonant mechanism decreases almost exponentially with increasing ya [7–9]. In summary, we have proposed a one-electron model for resonant ionization of negatively charged, single-valence ions sputtered off (1 0 0)-metal surfaces and we have provided an application to the Ag/Ag(1 0 0) and Au/ Au(1 0 0)-systems. Surface effects were considered in the form of transient QM correlations in the diatomic systems formed between secondary emitted atoms and their nearest-neighbor substrate atoms both ejected in the collision cascade. We have formulated the electron problem in terms of two, spatially-correlated discrete states interacting with a projected band of continuous states, reducing the calculation of the ionization probability to the numerical solution of a matrix equation – Eq. (8). The results of Figs. 3–5 suggest that the final ionization probability is weakly influenced either by the band structure of the material or by many body correlations in sputtered atomic species, while it depends on the width of the projected band gap. Another significant question relates to the analytical con-
a
b
Fig. 5. Theoretical ionization probability of negative ions for the Ag/ Ag(1 0 0) and Au/Au(1 0 0) systems, obtained as in Fig. 3. Experiments are taken from [6].
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473
tinuation to negative times – Eq. (11) – used as a mathematical tool to determine the initial distribution of metal electrons. Different models for such a distribution change the calculation of H 0QM ðtÞ at short times when the emitted atom is close to the surface. Thus, a more realistic model is certainly needed to better estimate the initial shifts and the broadenings of the two level system. References [1] M.L. Yu, in: R. Behrisch, K. Wittmaack (Eds.), Sputtering by Particle Bombardment III, Springer, Berlin, 1991, p. 91. [2] Z. Sroubek, Phys. Rev. B 25 (1982) 6046; Z. Sroubek, G. Falcone, D. Aiello, C. Attanasio, Nucl. Instr. and Meth. B 88 (1994) 365; Z. Sroubek, J. Fine, Phys. Rev. B 51 (1995) 5635. [3] N.D. Lang, Phys. Rev. B 27 (1983) 2019; N.D. Lang, J.K. Norskov, Phys Scripta T 6 (1983) 15; N.D. Lang, J.K. Norskov, Rep. Prog. Phys. 52 (1989) 655. [4] A. Sindona, G. Falcone, Surf. Sci. 423 (1999) 99; A. Sindona, G. Falcone, Nucl. Instr. and Meth. B 57 (1999) 37;
[5]
[6] [7] [8] [9]
[10] [11] [12] [13] [14]
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A. Sindona, G. Falcone, Surf. Sci. 529 (2003) 471; A. Sindona, P. Riccardi, G. Falcone, Nucl. Instr. and Meth. B 230 (2005) 449. R.G. Hart, C.B. Cooper, Surf. Sci. 94 (1980) 105; J.M. Vasile, Phis. Rev. B 29 (1984) 3785; V.I. Veksler, Surf. Sci. 397 (1998) 1; C.H.A. Huan, M.H.S. Low, A.T.S. Wee, K.L. Tan, SIMS XI Proceedings, Vol. 903, Wiley, New York, 1998. P.A.W. van Der Heide, Nucl. Instr. and Meth. B 157 (1999) 126. P.W. Anderson, Phys. Rev. 124 (1961) 41; D.M. Newns, Phys. Rev. 178 (1969) 1123. Brako, D.M. Newns, Rep. Prog. Phys. 52 (1989) 655. H. Shao, D.C. Langreth, P. Nordlander, in: J.W. Rabalais (Ed.), Low Energy Ion-Surface Interactions, John Wiley & Sons Ltd., Sussex, 1994. E.V. Chulkov, V.M. Silkin, P.M. Echenique, Surf. Sci. 437 (1999) 330. P. Nordlander, Phys. Rev. B 46 (1992) 2584. M. Gell-Mann, F. Low, Phys. Rev. 84 (1951) 350. M.A. Karolewski, Nucl. Instr. and Meth. B 230 (2005) 402. M.H. Shapiro, E. Trovato, T.A. Tombrello, Nucl. Instr. and Meth. B 180 (2001) 58.
Nuclear Instruments and Methods in Physics Research B 256 (2007) 468–473 www.elsevier.com/locate/nimb
Negative ionization of the secondary ions of silver and gold sputtered from their elemental surfaces A. Sindona b
a,b,*
, P. Riccardi
a,b
, S. Maletta a, S.A. Rudi
a,b
, G. Falcone
a,b
a Dipartimento di Fisica, Universita` della Calabria, Via P. Bucci 31C, 87036 Rende (CS), Italy Istituto Nazionale di Fisica Nucleare (INFN), Gruppo Collegato di Cosenza, Via P. Bucci 31C, 87036 Rende (CS), Italy
Available online 17 December 2006
Abstract Calculations of the ionization probabilities of Ag and Au particles, ejected during sputtering of clean Ag(1 0 0) and Au(1 0 0) surfaces, respectively, are reported. An effective one-electron theory is used to describe: the plane metal surface, with a projected band gap, the secondary emitted atom, whose charge state is investigated, and its nearest-neighbor substrate atom, put in motion by the collision cascade generated by the primary ion beam. Suitable rectilinear trajectories are selected to describe the motion of these two atoms outside the solid. A good agreement is found with van Der Heide’s experiments (P.A.W. van Der Heide, Nucl. Instr. and Meth. B 157 (1999) 126). 2006 Elsevier B.V. All rights reserved. PACS: 79.20.m; 71.10.w; 73.20.r Keywords: Impact phenomena (including electron spectra and sputtering); Theories and models of many-electron systems; Electron states at surfaces and interfaces
1. Introduction Electron transfer and excitations processes occurring at the surface of solids, bombarded with ion sources, have been the subject of continuous interest for decades [1]. In spite of this attention and importance, the underlying mechanisms of secondary ion formation and escape are still unclear [2–6]. Some experiments [5,6] have shown that there exist different possible ionization/neutralization mechanisms active during ejection. One of these, predominating at higher emission energies ( J 100 eV), is well described by resonant electron tunnelling between the ejected particle and the target [7–9]. The other, commonly observed at lower emission energies ([40 eV), is suggestive of some form of surface excitation. A generalized model, accounting for both mechanisms, was proposed in [4] and *
Corresponding author. Tel.: +39 (0)984 496059; fax: +39 (0)984 494401. E-mail address: Sindona@fis.unical.it (A. Sindona). 0168-583X/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.12.059
applied to the ionization of secondary Cu± ions sputtered from Cu(1 0 0) jellium samples: surface excitations were explained in terms of transient quasi-molecules (QMs) formed between the secondary emitted and their nearestneighbor substrate atoms, put in motion by the collision cascade. This paper presents a more accurate version of the model, based on the effective Hamiltonian ^h½Z a ; Z b ¼ ^p2 =2 þ ^v½Z a ; Z b
ð1Þ
of a spinless electron, with momentum p^ and position ^r, interacting with a plane metal target and two neutral atoms. The effective potential ^v½Z a ; Z b depends parametrically on the positions, Za and Zb, of the emitted (a) and the substrate atoms (b), along the surface normal (Section 2). The negative ionization probability of a-atoms is calculated using a spectral decomposition method involving: the truncated set of available states, below the vacuum level of the metal conduction band (with a projected band gap), the orthonormalized wave-function for the affinity level of the
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473
469
a-atom, and the orthonormalized molecular orbital adiabatically formed between a and b atoms (Section 3). The ionization probabilities of Au and Ag ions, sputtered from Au(1 0 0) and Ag(1 0 0) surfaces, respectively, are obtained within the straight line trajectory approximation for the diatomic motion outside the collision cascade area. Suitable choice of the parameters of these trajectories leads to a good agreement with the experiments of [6] (Section 4). 2. Theory The one-electron pseudopotential for the double ionmetal system has the cylindrically symmetric form ^v½Z a ; Z b ¼ ^vs þ ^vQM ½Z a ; Z b þ D^vQM ½Z a ; Z b
ð2Þ
with a local part
2pz vs ðzÞ ¼ A10 þ A1 cos ; as
z<0 5p 4b
ð3Þ
Fig. 1. Instantaneous affinity energies, ea and eb, of two Au ions at position Za and Zb, relative to a Au(1 0 0)-surface whose electronic structure is derived from Eq. (3).
denoting the semi-empirical Chulkov’s potential of a (1 0 0) surface, constructed from pseudopotential local density theory [10]. Eq. (3) is completely determined by fixing A10, A1, A2, b – reported in Table 1 for Ag(1 0 0) and Au(1 0 0) – and constraining continuity and differentiability everywhere in the space. The atomic planes are separated by the lattice constant as , so that the topmost layer of lattice points defines the origin of the coordinate system (z = 0) and zIM labels the position of the image plane. The other components in Eq. (2), whose diagonal matrix elements in the basis of coordinates are sketched in Fig. 1, have already been discussed in [4]. In particular, ^vQM ½Z a ; Z b ¼ ^va ½Z a þ ^va ½Z b is a separable quasi molecular potential, where ^va ½Z aðbÞ establishes the electron interaction with each neutral particle, and D^vQM ½Z a ; Z b accounts for the effective action of the image charges of all electrons and protons in the QM. Parametric time dependence is a consequence of the classical motion of a and b atoms, through the kinematic laws Za(b) = Za(b)(t), yielding ^ hðtÞ ^ h½Z a ðtÞ; Z b ðtÞ. In [4], an analytical picture was adopted based on the following assumptions: (i) the two atoms are initially placed outside the collision cascade area; (ii) their interaction is modelled
by a Morse potential; (iii) all interactions with other surface atoms are neglected. Here, we have tested different types of numerical trajectories calculated from Molecular Dynamics (MD) simulations on bidimensional clusters of about 250 atoms covering the region (60 to 60) au, along the surface plane, and (60, 0) au along the surface perpendicular direction. We used both 1–15 keV Ar and 1–15 keV Cs projectiles, at 0–75 incident angles from the surface normal. The interaction potential between two target particles was given the form of the many body tight binding potential of [13], while the projectile-target interaction was modelled by the Moliere potential of [14]. An example of such simulations, computed with the Verlet algorithm [13,14] using a time step of 1 au, is shown in Fig. 2: particles emitted from Au(1 0 0) clusters, bombarded with 2 keV Cs projectiles, mainly come out as monomers and dimers outside the collision cascade area. In either case, when the ejected atoms are at distances of about 5–10 au from the surface, their interactions with other surface atoms are negligible. Furthermore, the binding energy in a dimer is small compared to its center-of-mass kinetic energy, at interatomic distances larger than 6–7 au. Indeed, the kinematic model of [4] predicted nearly straight trajectories for a-atoms. These considerations let us assume that a and b atoms still possess strong electronic interactions when their binding forces tend to zero, moving with
¼ A20 þ A2 cosðbzÞ;
06z<
¼ A3 eaðzz1 Þ ;
5p 6 z < zIM 4b
¼
ekðzzIM Þ 1 ; 4ðz zIM Þ
z > zIM ;
Table 1 Parameters of Eq. (3), in au, for Au(1 0 0) and Ag(1 0 0) Surface
A10
A1
A2
b
zIM
as
e1
e2
Ag(1 0 0) Au(1 0 0)
0.342 0.397
0.185 0.154
0.143 0.223
2.422 3.363
2.06 1.62
3.86 3.85
0.104 0.142
0.072 0.012
Energies are reported relative to the vacuum level.
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A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473 Table 2 Parameters of Eq. (4), in au, for Ag and Au Ion
Ag Au
Fig. 2. (a)-(c): MD simulation of 2 keV Cs projectiles incident at 30 from the perpendicular direction of a bidimensional FCC Au cluster (see text). The trajectories of some of these atoms are drawn in the plot. (d): time laws, Za(t) and Zb(t), of two possible candidates for a and b atoms.
approximately constant velocities from the instant (t = 0) when both particles are outside the image plane (Fig. 2(d)). The final kinetic energy Ea, of the a-atom, is fixed by the experiment under consideration, whereas other trajectory parameters, such as the kinetic energy Eb, of b-atoms, and the initial positions Za(0) and Zb(0), can be used as free parameters. In the present treatment, we set Zb(0) right outside the image plane and let the initial interatomic distance, Za(0)Zb(0), to be of the order of the lattice separation as . Finally, we consider a (gaussian) distribution of b-particles with (mean) kinetic energy Eb < Ea. 3. Spectral decomposition We construct a finite basis that includes the instantaneous affinity states, of a and b atoms, and the projected bulk states of the target lying below the vacuum level. Specifically, the affinity state of an isolated negative ion, at position Za(b), is obtained from the atomic Hamiltonian ^ ha ½Z aðbÞ ¼ ^ p2 =2 þ ^va ½Z aðbÞ . In the separable potential approximation [11], the corresponding affinity wave function, at Za(b) = 0, takes the analytical form ka r e ela r hrjai ¼ N A ; ð4Þ r where the parameters NA, ka, and la are reported in Table 2 for Ag and Au. At Za(b) 5 0, the wavefunction becomes hrja[Za(b)] i = hr Za(b)uzjai and its (unperturbed) energy, A, coincides with the affinity energy of the negative ion state. Then, the adiabatic affinity states of the a-
Na
ka
la
A
0.8413 1.008
0.3094 0.4119
0.4686 0.6134
0.04785 0.0848
and b-atoms are ja(t)ija[Za(t)]i and jb(t)ija[Zb(t)]i, respectively. The (numerical) eigenstates of the isolated metal band, denoted by {jki}, arise from the diagonalization of the surface Hamiltonian ^hs ¼ ^p2 =2 þ ^vs . The latter has a continuous spectrum {ek}, with a minimum at ek = A10, Fermi energy eF , and a projected band gap in the range e1 < ek < e2 (see Table 1). The corresponding wavefunctions read hrjki ¼ wk? ðzÞeikk rk =L, in which L is the thickness of the solid, k ¼ ðkk ; k ? Þ the electron wave-vectors, and wk? the (real) numerical eigenfunction of the potential (3). L is chosen in order to have a maximum energy spacing between two band states – at kk = 0 – smaller than 0.025 eV. In this way about 500 perpendicular wavevectors are used to reproduce the continuum spectrum below e1. Such a truncation is reasonable, in the case of Ag(1 0 0) and Au(1 0 0), where e2 lies outside the vacuum level (see Table 1). Indeed, electron exchanges mostly involve the occupied continuous states fjkigek 6eF and the affinity states ja(t)i and jb(t)i, whose adiabatic energies are always smaller than e2, reaching asymptotically the unperturbed value A from below. Orthonormalization of fjaðtÞi; jbðtÞi; jkigek 6e1 , via the Graham–Schmidth method, yields the finite basis fjaðtÞi; jbðtÞi; jkigek 6e1 , in which: P 1 ek 6e1 jkihkj jaðtÞi ¼ jaðtÞi; ð5Þ K a ðtÞ is sharply peaked at Za(t), tending to ja(t)i in the long time limit, and P 1 jaðtÞihaðtÞj ek 6e1 jkihkj jbðtÞi ¼ jbðtÞi; ð6Þ K b ðtÞ exhibits a strong molecular character with average lifetime larger than 10 fs, at Ea J 10 eV and Eb J 1 eV. In Eqs. (5) and (6) the function Ka(b)(t) ensures normalization. The negative ionization probability of a-atoms may be written as [4,8,9] X 2 jGþ ð7Þ R ðtÞ ¼ ak ðt; t 0 Þj hðeF ek Þ; k where Gþ ak ðt; t 0 Þ labels the probability amplitude that a band electron, occupying the state jki in the remote past (t0 ! 1), will be transferred to the emitted atom at the time t, when Z a ðtÞ zIM . Complementarily, Gþ bk ðt; t 0 Þ denotes the probability amplitude for electron transfer to the orthonormalized molecular state. Using the semiclassical approximation [9], we reduce the þ calculation of Gþ ak ðt; t 0 Þ and Gbk ðt; t 0 Þ to the solution a 2 · 2 matrix equation [4]:
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473
½i@ t H 0QM ðtÞ
Gak ðt; t0 Þ Gbk ðt; t0 Þ
¼ i eiek ðtt0 Þ
V ak ðtÞ V bk ðtÞ
:
ð8Þ
Here, time evolution is governed by the dissipative Hamiltonian H 0QM ðtÞ ¼ H QM ðtÞ þ RQM ðtÞ, in which H QM ðtÞ has the (real) matrix elements of ^ hðtÞ into the states ja(t)i and jb(t)i. Its diagonal components, ea(t) and eb(t), are the instantaneous energies of the orthonormalized affinity states. The dissipative term is the (complex) self-energy matrix 0 Rt 1 i 0 ds½ea ðsÞek 0 X Z t V ak ðtÞ t V ðt Þ e ak A; RQM ðtÞ ¼ i dt0 @ Rt V bk ðtÞ i 0 ds½eb ðsÞek t0 0 k t e V bk ðt Þ ð9Þ in which labels the tensor product of two column vectors and V aðbÞk ðtÞ ¼ haðbÞðtÞj^ hðtÞjki are the coupling integrals of discrete and continuous states. It should be noted that the interaction with a state jki has the form of an external perturbation. The matrix elements appearing in Eq. (8) evolve over two different time scales, related to the different motions of a and b atoms. This is shown, for example, in the study of the renormalized energies of the orthonormalized affinity states, reported in Fig. 3(a) and (b) for Ag/ Ag(1 0 0) and Au/Au(1 0 0), respectively. Indeed, e0a ðtÞ ¼
a
c
b
d
Fig. 3. Instantaneous energies (e±) and broadenings (D±) of the bonding and antibonding orbitals of the QM, made of a- and b-atoms, with the parameters of Ag/Ag(1 0 0) and Au/Au(1 0 0). In panels (a), (b), the renormalized affinity energies ðe0a and e0b Þ are reported for comparison. In panel (b), the unrenormalized atomic energy ea is also shown to visualize the effect of the shift induced by the self-energy matrix (9).
471
Re½H 0QM ðtÞaa and e0b ðtÞ ¼ Re½H 0QM ðtÞbb are corrected by the effect real part of the self energy matrix (9) inducing short time oscillations, which are more clearly observed in Fig. 3(b), where QM interactions are stronger. In general, e0a ðtÞ changes faster than e0b ðtÞ, although both terms are slow on the femtosecond scale of the electronic transitions, which justifies the use of the semiclassical approximation. At short times after the instant of ejection, e0a ðtÞ is shifted negatively, with respect to its unperturbed value, due to the dominant effect of the electron–surface interaction ð^vs Þ: e0b ðtÞ is promoted above the vacuum level, since the substrate atom is initially closer to the image plane where the leading effect is due to the image potential ðD^vQM Þ. At larger times, e0a ðtÞ tends to the unperturbed affinity energy as fast as 1/Za(t), while e0b ðtÞ behave as 1/ Zb(t). Level crossing allows resonant charge transfer processes between ja(t)i and jb(t)i, coupled by the off diagonal terms ½H 0QM ðtÞab and ½H 0QM ðtÞba . A closer insight into quasi-molecular correlations is gained by considering the instantaneous spectrum of H 0QM ðtÞ, yielding the eigenvalues e±(t) iD±(t). The corresponding eigensates, namely j+(t)i and j(t)i, are the antibonding and bonding states of the QM, obtained from linear combinations of ja(t)i and jb(t)i. Their energies, e±(t), and their virtual broadenings, D±(t), are shown in Fig. 3. We see that e+(t) tends to e0a ðtÞ, while e(t) tends to e0b ðtÞ, in the long time limit (Fig. 3(a) and (b)); thus, an electron captured in the antibonding state will be detected in a-atom. In addition, the bonding orbital is further shifted towards eF , compared to e0a ðtÞ and e0b ðtÞ, which enhances the probability for electron capture to the QM. From this perspective, Eq. (8) establishes electron transfer dynamics based on two mechanisms: one is direct tunnelling from each band state to the ejected atom and the other an indirect, two-step process, with an intermediate transition through the molecular level e(t). Another crucial aspect concerns the evolution of the broadenings of the QM orbitals, shown in Fig. 3(c) and (d); both D+(t) and D(t) drop off to zero with an oscillating exponential behavior, which is not surprising since the adiabatic energies of the orthonormalized affinity states take values in the projected band gap during the whole process. It follows that their adiabatic broadenings are virtually zero Rt so that the phase factors expfi t0 ds½eaðbÞ ðsÞ ek g, within the t 0 -integral in Eq. (9), induce oscillations in RQM ðtÞ. On the other hand, the exponential decrease is related to the hopping terms Vak(t) and Vbk(t). The sign of D±(t) modulates electron transfer processes to the state j ± (t)i, whose occupation probability contains the exponential factor Rt exp dsD ðsÞ . 4. Results A numerical solution for Eq. (8) – at positive times – requires a model to estimate the electronic distribution of the metal band at the instant of ejection (t = 0). Recalling that the final charge fraction of the localized levels is
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independent on their initial occupancy in the remote past [8], we continue the problem analytically to negative times with Gellmann and Low’s adiabatic prescriptions [12]: 2
p þ ^hðt 6 0Þ ¼ ^ þ ^vð0Þe0 t : 2
ð10Þ
In this case, both Gak(0,t0) and Gbk(0,t0) can be calculated exactly, because their evolution from the remote past is stationary. In addition, the one-electron potential is switched þ off at t = t0, implying Gþ ak ðt 0 ; t 0 Þ ¼ Gbk ðt 0 ; t 0 Þ ¼ 0. Then, the analytical solution of Eq. (8) at t 6 0 reads: Gak ð0; t0 Þ V ak ð0Þ T0 ¼ ieit0 ek k0 ; ð11Þ Gbk ð0; t0 Þ T k V bk ð0Þ in which T 0k
¼
ek e0b ð0Þ
V 0ab ð0Þ
V 0ab ð0Þ
ek e0a ð0Þ
! ð12Þ
is a transfer Hamiltonian from the state jki to the affinity states. Finally, Eq. (11) can be used as initial condition to solve numerically Eq. (8), for each k and t > 0, yielding Gak(t,t0) and Gbk(t,t0). This allows to calculate the negative ionization probability of the state ja(t)i, via Eq. (7). Complementary, we also calculated the negative ionization probability of the state jb(t)i, denoted R0 ðtÞ, by replacement of Gak(t,t0) with Gbk(t,t0) in Eq. (7). Fig. 4 shows the negative ionization probabilities, R(t) and R0 ðtÞ, for the orthonormalized affinity states. We observe there exists a time tF such that both R(t) and R0 ðtÞ are independent on time, for t > tF , whereas they depend on the kinetic energies of a- and b-atoms. tF increases with decreasing Ea, taking typical values in the range 10–200 fs at Ea = 10 300 eV. A signature of charge exchanges between ja(t)i and jb(t)i is given by the oscillating behavior of R(t) and R0 ðtÞ, occurring for typical times
a
b
Fig. 4. Ionization probabilities R(t) and R0 ðtÞ for the orthonormalized affinity states, with the parameters of Ag/Ag(1 0 0) and Au/Au(1 0 0) systems (Fig. 3). hR(t)i is obtained by averaging R(t) over a gaussian distribution of b-particles with mean energy Eb and standard deviation rb.
of the order of 102 fs, for kinetic energies Ea [ 15 eV and Eb [ 1 eV. In Fig. 4 we also report the probability hR(t)i, obtained as the average ionization probability of a-atoms for a gaussian distribution of b-atoms, with mean energy Eb and standard deviation rb. Plots of hR ðtF Þi versus ya, the inverse velocity of the emitted atom, are compared with the experimentally derived distributions of Ag and Au ions sputtered from their elemental surfaces [6]. Eb is adjusted to the data, taking values in the range 1–7 eV, in both Ag/Ag(1 0 0) and Au/Au(1 0 0) systems. Fig. 5 shows a good agreement with data, correctly reproducing the increase in the negative ion population with increasing ya, above 106 s/cm. This is explained, in the present context, with the enhancement of the probability for indirect transfers with decreasing Ea, corresponding to a longer duration of quasi molecular interactions. Conversely, the direct resonant mechanism decreases almost exponentially with increasing ya [7–9]. In summary, we have proposed a one-electron model for resonant ionization of negatively charged, single-valence ions sputtered off (1 0 0)-metal surfaces and we have provided an application to the Ag/Ag(1 0 0) and Au/ Au(1 0 0)-systems. Surface effects were considered in the form of transient QM correlations in the diatomic systems formed between secondary emitted atoms and their nearest-neighbor substrate atoms both ejected in the collision cascade. We have formulated the electron problem in terms of two, spatially-correlated discrete states interacting with a projected band of continuous states, reducing the calculation of the ionization probability to the numerical solution of a matrix equation – Eq. (8). The results of Figs. 3–5 suggest that the final ionization probability is weakly influenced either by the band structure of the material or by many body correlations in sputtered atomic species, while it depends on the width of the projected band gap. Another significant question relates to the analytical con-
a
b
Fig. 5. Theoretical ionization probability of negative ions for the Ag/ Ag(1 0 0) and Au/Au(1 0 0) systems, obtained as in Fig. 3. Experiments are taken from [6].
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 468–473
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