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Spin Polarized Metastable Helium De-excitation Processes on Metal Surfaces S. Achilli,† G. P. Brivio,† G. Fratesi,† and M. I. Trioni*,‡ † ‡
ETSF and Dipartimento di Scienza dei Materiali, Universita di Milano-Bicocca, Via Cozzi 53, I-20125 Milano, Italy CNR - National Research Council of Italy, ISTM, Via Golgi 19, 20133 Milano, Italy ABSTRACT: Spin polarized de-excitation of a metastable helium atom interacting with metal surfaces is treated within density functional theory. The method is based on a selfconsistent calculation of the spin dependent electronic properties of the system, such as the surface density of states and the localized surface states, to compute the transition rate. On the high work function Ag(100) and Ag(111) surfaces, the helium 2s electron is delocalized in the metal and hence the transition rate is weakly spin dependent. The existence of a Shockley surface state in Ag(111) determines a neutralization rate that is about 59% larger than that from Ag(100). On a low work function metal, namely Na(100), the rate is of smaller magnitude than those on silver because the 2s triplet resonance is found to be more occupied. Consequently, emitted electrons can display a strong spin dependence also for a paramagnetic surface.
1. INTRODUCTION Helium scattering has proven to be a fundamental experimental technique to investigate a variety of surface properties: periodic structures,1 phonons,2 diffusion,3 and charge profiles4 are just some examples. But helium atom spectroscopy (HAS) with the gas atom in the ground state cannot provide information on the spectral electronic properties of surfaces. To elucidate these properties, one can resort to helium atom metastable de-excitation spectroscopy (MDS) (for a review see ref 5). This technique is sensitive to electronic features just at the sample surface more effectively than other spectroscopies, such as ultraviolet photoelectron spectroscopy (UPS), owing to the lack of penetration of the helium probe.6 In practice, the helium atom in an excited metastable state impinges the surface where it is de-excited by an Auger process: the dependence on the kinetic energy of the emitted electrons contains relevant information on the electronic states protruding from the solid, including the so-called surface states whose amplitude decays into the bulk. The physics underlying de-excitation of metastable He, or rare gases in general, impinging a solid surface is nowadays well understood.5 The generally accepted picture of this phenomenon divides it into two different processes: on relatively high work function substrates, a resonant ionization (RI) of the excited helium atom (e.g., He* in the 23S triplet state) by transfer of one electron to the solid, followed by an Auger neutralization (AN) with an electron ejection, represents essentially the de-excitation process. Instead, on low work function metal substrates where resonant ionization is mostly suppressed, de-excitation mainly occurs as a direct Auger (AD) event.7 The total de-excitation rate may decrease by up to 2 orders of magnitude going from the first mechanism to the second one.8 In principle, however, both these channels can participate in the de-excitation of metastable helium r 2011 American Chemical Society
on a metal surface. The prevailing de-excitation route is determined by the relative position of the 2s triplet atomic level with respect to the Fermi energy EF of the substrate. Indeed, a sharp separation of the two previously described processes is only schematic because the hybridization of the 2s triplet atomic level with the metal states allows for a fractional occupancy of the atomic orbital, and a proper treatment should yield a total electron flux comprising both contributions. An important feature of MDS recently investigated is the spin polarization of the emitted electron.914 In fact, because of its intrinsic spin asymmetry, MDS is a very useful technique to investigate spin dependent electronic properties at magnetic surfaces.10,15,16 Furthermore, there is also experimental evidence of the spin polarization of the emitted electron from paramagnetic surfaces.17 The experimental results clearly show that it is possible to obtain a highly polarized electron beam (up to around 70%) or a poorly polarized one for low and high work function substrates, respectively.18 This effect is related to the two different de-excitation mechanisms previously described: in fact, on the low work function surfaces, the emitted electron essentially belongs to the 2s level of He* and, consequently, its spin is well-defined, being He* in its triplet state. Conversely, on low work function substrates, the emitted electron originates from the states at the surface, slightly modified by the interaction with the He+ ion. Several theoretical approaches have been proposed in the past.7,1921 A recent one,22 which introduces the presence of the surface via its unperturbed electronic response functions, includes a model potential23 capable of describing the relevant features of the electronic structure of the surfaces. Hence this method provided a Received: January 19, 2011 Revised: June 5, 2011 Published: July 08, 2011 8498
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2. MODEL The Auger transition rates are calculated via the Fermi golden rule (atomic units are used hereafter): P ¼ 2π
∑F jÆFjV jIæj2 δðEF EIÞ
ð1Þ
The final and initial states, |Fæ and |Iæ, are built up by single Slater determinants of KS orbitals. These are worked out using the embedding Green’s function method for the single adatom within DFT,26 previously applied to MDS on jellium surfaces.25 The matrix elements of the Coulomb potential V driving the Auger decays can be expressed as follows: ÆFjV jIæ ¼ Æaσ a ;bσ b jV jcσ c ;AσA æ ¼ δσc σa δσA σb DA;c;a;b δσc σb δσA σa EA;c;a;b
ð2Þ
where c and A label core and Auger states, respectively, while a and b are states in the valence band. The spatial part of the matrix elements, D and E, is given by DA;c;a;b EA;c;b;a Z ψ ðr1 Þ ψc ðr1 Þ ψb ðr2 Þ ψA ðr2 Þ ¼ dr1 dr2 a jr1 r2 j
ð3Þ
where the spin indices σi are implicit. We now express the Auger transition rate by assuming that a 1sV electron was initially removed from the helium atom. This can decay with the emission of a spin up (v) or spin down (V) Auger electron, with kinetic energy E. Consequently P in eq 1 contains two contributions of opposite spin, P v and P V. Making explicit the sum over the final hole states (1/2∑aσa ,bσb) involved in the transition and summing over the spin indices yields Z v d^k jDA v ;1s V ;a V ;bv j2 δðE + E1s Ea Eb Þ P ðEÞ ¼ 2π
∑ a;b
ð4Þ
V
P ðEÞ ¼ π
∑ a;b
Z
d^k jDA V ;1s V ;a V ;bV EA V ;1s V ;a V ;bV j2
δðE + E1s Ea Eb Þ
ð5Þ
Then, the spin polarization of the emitted electrons can be defined as Z dE ½P v ðEÞ P V ðEÞ Z ð6Þ dE ½P v ðEÞ + P V ðEÞ It is important to note that the DFT approaches, as that used in this paper, provide the correct electronic structure of the metastable helium atom interacting with the metal surface, in particular concerning the width and filling of the 2s resonance. Hence, the possibility of resonant ionization, mainly depending on the work function of the substrate, is automatically considered by the ab initio calculation. The substrate is described by a phenomenological pseudopotential, modulated only in the direction normal to the surface,23 entering the semi-infinite bulk, and constant parallel to the surface. Such a potential is capable of determining a realistic surface density of states including surface states. Details for the parametrization of the potential for the surfaces considered here can be found in ref 23. The single helium atom is accounted for by a full potential treatment. The DFT calculation is performed by the embedding method26 in which the spin dependent exchange correlation potential is worked out in the PerdewZunger approximation.27 In all the cases considered (He, He* on all substrates), the radius of the embedded sphere in which the KS equation is solved is equal to 7 a0 (a0 being the Bohr radius), which we verified includes all the relevant perturbation induced by the He atom. In the embedded region a linearized augmented wave basis set is used. All relevant parameters have been verified to produce full convergent results: we used a real space logarithmic radial grid of 401 points up to 4.3 a0 from the nucleus position and a kinetic energy cutoff of 8.7 Ry for the plane wave expansion in the remaining interstitial region (from 4.3 to 7 a0); the maximum angular quantum number for the spherical part of the basis has been fixed equal to l = 12. Density of states for He* interacting with the metal are calculated by using an imaginary part of the energy η = 0.2 eV. Such a choice is also adopted to simulate the Auger transition rates and accounts for the experimental broadening (experimental resolution, core lifetime, phonons etc.).
3. RESULTS 3.1. Helium De-excitation on Ag(100) and Ag(111). The analysis of the electronic properties of the clean surface and their modification due to the interaction with the He* atom is basic to characterize the MDS de-excitation process. We analyze first the clean Ag(100) and Ag(111) surfaces. Those have relatively high work function, i.e., 4.43 and 4.56 eV, respectively. In Figure 1 we present the density of states (DOS) of the semi-infinite systems at Γ(i.e., with k = 0), and integrated in the full surface Brillouin zone, calculated in the surface region (for the sake of later comparison, we have chosen to report here the DOS in a slab just containing the embedded sphere we adopted for the calculations with the He atoms). Both DOS’s at Γshow a )
first estimate of the role of surface states in the Auger neutralization rate, though the spin dependent modification of such states induced by the metastable helium probe is missing. However, studies within the self-consistent density functional theory (DFT), were able to take into account such an effect, induced either by an He+ ion in jellium bulk24 or by He* on a jellium surface.25 In this work we contribute to the description of the MDS process by presenting a DFT spin-polarized approach in which the He*/metal surface interacting system is treated explicitly in the KohnSham (KS) equation. The spin polarization of emitted electrons is accounted for by the Auger transition terms,25 while the surface potential is able to describe surface states and the surface projected gap.23 So the role of the occupied surface states, modified by the interaction with the probe, is taken into account in the de-excitation process . This model has a “natural” reference energy level represented by the far vacuum potential, which allows one to consider consistently the work function of the substrate and the ionization level of He*. After introducing the model in the next section, we present the spin dependent electronic properties and the de-excitation electron spectra of He* on two relatively high work function metal surfaces, namely Ag(100) and Ag(111), the latter one supporting a Shockley surface state, and on a relatively low work function surface, namely Na(100). The spin dependence and the effect of the surface states in the neutralization rates are quantitatively determined in all cases.
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Figure 3. Induced charge for He*/Ag(100) (upper panels) and He*/ Ag(111) (lower panels) for the majority (left side) and minority (right side) spin components. The thick dotted white lines mark the first atomic surface layer. The isoline values differ by δF = 0.0002 electrons/ a03. Brighter areas report excess electrons.
Figure 2. Induced DOS of He* on Ag(100) (upper panel) and Ag(111) (lower panel). The vertical lines mark the Fermi and vacuum levels of each substrate.
surface projected bulk band, partially unoccupied for Ag(100), and a surface projected energy gap extending from the upper edge of the band almost up to the vacuum level. Here the series of image states accumulate, looking similar for the two surfaces. The main difference between these two DOS’s is the presence
of a sharp peak just below EF in the Ag(111) case, due to the discrete and occupied Shockley surface state. Such a state determines the step in the k -integrated DOS of Ag(111), which can be seen close to EF: noticeably, its contribution to the DOS at these energies is of the same order of magnitude as that of the bulk band. Differently, Ag(100) features a broad surface resonance mainly above EF. Second, we introduce the metastable helium atom located at d = 5 a0 from the first layer, a characteristic distance at which de-excitation is most likely,22,25,28 and analyze its electronic properties. In Figure 2 we plot the induced density of states (IDOS), i.e., the total density of states of the majority (v) and minority spin (V) components of the He*/surface system, minus that of the corresponding clean surfaces, computed in the embedded sphere. The two components differ due to the removal of a spin V electron from the 1s state of helium, which induces a polarization in the potential. In the IDOS’s of both systems we note a large and fairly narrow peak above EF that can be attributed mainly to the 2pz level of helium, broadened following hybridization with substrate states. Such a feature is peaked at different energies for the two spin populations and does not participate in de-excitation being unoccupied. The former 2s level of He* is delocalized over all the substrate band (contributing also to the unoccupied peak). Hence, resonant ionization of He* is the main de-excitation mechanism in this case, as expected from the rather large work function of the metal surface. Finally, it is important to point out that the IDOS of He*/Ag(111) shows a peak below EF, which is absent in He*/Ag(100). We verified that it is determined by the Shockley surface state interacting with the He* atom. This induced localized resonance is spatially confined in the atomic region and is shifted down in energy with respect to the Shockley state of the clean surface, owing to the attractive He+ ion potential. Figure 3 shows the induced charge, i.e., the total charge of the interacting He*/metal system minus that of the unperturbed substrate. Results for Ag(100) and Ag(111) are reported in the two upper and lower panels, respectively, while the majority and minority spin components are plotted on the left and right side. We note that the surface response to the perturbation induced by )
Figure 1. DOS at Γ (solid red line) and total DOS (dashed blu line) for Ag(100) and Ag(111) in the upper and lower panels, respectively. Spatial integration is performed in a slab 14 a0 thick (vacuum plus the first surface layer). The vertical thin lines mark the Fermi and vacuum levels.
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Figure 4. Local DOS of He and He* interacting with Ag(100) and Ag(111) in the upper and lower panels, respectively: Solid red line, majority spin component of He*; dashed blue line, minority spin component of He*; dot-dashed violet line, He; dotted green line, clean surface.
the ionized He+ atom results in a screening charge that accumulates just outside the surface, with a more pronounced maximum in front of the atom for the majority spin component. The tail of the 2s resonant state of helium that extends below EF also contributes to this screening charge. We observe a spin dependence of the induced charge that is more intense and extends over a wider area for the majority spin component. The screening charges are 10% larger on the Ag(111) due to the additional contribution of the Shockley surface state. In Figure 4 we show the local density of states (LDOS) calculated in a spherical volume of radius 5 a0 around the atom, for neutral and metastable He on both the Ag(100) and Ag(111) surfaces (upper and lower panels, respectively) in comparison with the LDOS of the corresponding clean surface. At variance with the structureless LDOS of He*/Ag(100), that of He*/ Ag(111) exhibits a marked peak just below EF, due to the localized Shockley surface state. As we mentioned above, this state contributes a step increase in the clean surface LDOS (dotted green line) at about EF and is shifted down in energy by the attractive He+ ion potential, which acts differently on the two spin components. We also point out that the presence of He* creates LDOS’s larger than those of the clean surfaces, given the attractive nature of the He* induced potential. On the other hand, the nonexited helium atom in front of the surface affects the metal LDOS very weakly, owing to its complete shell configuration. It repels electronic charge, hence decreasing the silver LDOS slightly, differently from He*. The spin polarizations of the LDOS are almost equivalent on the two surfaces. We finally examine the total Auger de-excitation rate, plotted in Figure 5 for Ag(100) and Ag(111), together with the spin up and spin down contributions taken individually. The total rate (solid line) displays an asymmetric peak that is similar to those
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Figure 5. Auger de-excitation rates on Ag(100) and Ag(111) in the upper and lower panels, respectively.
Figure 6. Left panel: difference between the induced charges of He*/ Ag(111) and He*/Ag(100). The isoline values differ by δF = 0.00005 electrons/a03. Lighter and darker colors describe excesses and deficits of electron charge, respectively. Right panel: local DOS difference between the same systems.
already measured29 and calculated on Na, Cs, and Al jellium-like surfaces.28 Note first that P ðEÞ is larger for He*/Ag(111) than for He*/Ag(100) owing to the relevant role of the Shockley state of Ag(111) in the Auger process, in agreement with previous theoretical and experimental studies.22,30 In particular, we found a total neutralization rate for He*/Ag(111) that exceeds that of the He*/Ag(100) system by 59%. This percentage is larger than that reported in ref 22, namely 30%. The residual difference may be due to a higher occupancy of the localized surface state in our self-consistent approach. We detail how the MDS rates in Figure 5 reflect the different electronic properties of He* on Ag(100) and on Ag(111). For this reason in Figure 6 we report the difference between the total induced charges (left panel) and the LDOS’s (right panel), for the two surfaces. Observed variations can be ascribed mainly to the localized Shockley surface state of He*/Ag(111). In fact, more charge is found in the atomic region and the LDOS difference is essentially peaked just below EF, where such a surface state is localized. 8501
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Figure 9. Induced charge, as in Figure 3, for He*/Na(110). Figure 7. DOS, as in Figure 1, for Na(110).
Figure 10. Auger de-excitation rate, as in Figure 5, for He*/Na(110). Figure 8. Induced DOS, as in Figure 2, for He*/Na(110).
The spin polarization of the de-excitation rate is negligible for both systems (absolute value lower than 2%), because P ðEÞ f P ðEÞ. This reflects the low spin dependence of the occupied LDOS’s (see Figure 2) and is characteristic of the RI+AN process, where the spin polarized 2s electron has been transferred to the metal. 3.2. Helium De-excitation on Na(110). We now present a similar analysis for He* on Na(110). This surface has a much lower work function, namely 2.9 eV. In Figure 7 the surface DOS of Na(110) at Γ, calculated as before, is plotted together with the total DOS. Observe that the metal band is much narrower than that of silver and there are no occupied surface states different from Ag(111); the sharp surface resonance above EF is of no relevance to the Auger rates. We also note that the image states lying below the vacuum level are hybridized with a surface projected band forming a series of resonances. The low work function of this system plays against the ionization of the He* through electron transfer to the metal. Hence, an increase in the relative contribution of the direct Auger de-excitation (AD) process to the total rate can be foreseen. At variance with the IDOS of He*/Ag systems (discussed in the previous section), the IDOS’s of the He*/ Na(110) system shown in Figure 8 display a broad feature below EF, more evident for the majority spin population, which is easily attributed to the 2s resonant state of He* (we still observe the unoccupied resonant peak above EF as in Figure 2). Consequently, the occupation of the 2s majority spin resonance increases significantly and the spin asymmetry of the IDOS is more pronounced than that for silver. Similarly, the induced charge reported in Figure 9 for the two spin components exhibits a marked spin dependence. Further,
notice that the majority spin electronic charge induced in the neighborhood of the He* contains a relevant spherical contribution due to the atomic-like wave function of the 2s resonance. Overall, the spin polarization of the induced charge is about 50% larger than that in the silver case. Finally, the Auger rates are reported in Figure 10 for the two spin populations and look similar to those previously computed on the Na-like jellium surface.28 With respect to the silver case, the line shape of P ðEÞ for He* on Na(110) is narrower, following the smaller width of the surface projected valence band (Figure 7). But the most important differences are represented by the smaller intensity (total rate about 50% lower than that of He*/Ag(100)) and by a large spin polarization of P ðEÞ (43%) of this system with respect to those previously analyzed. Both these results are a consequence of the suppression of the RI+AN process, leaving with the less intense and more spin polarized AD.
4. DISCUSSION The theoretical approach used in this paper together with the modeling of both the surface and the helium probe allowed us to consider all the key ingredients needed for the description of deexcitation of metastable helium in front of a metal surface. First, the substrate potential23 is capable of describing the main electronic surface features, i.e., surface states, work function, and surface projected band gap. Second, the fully self-consistent treatment of the He*/metal interacting system worked out within DFT is capable of including the spin dependent modification of surface electronic properties in the de-excitation phenomenon, which are especially important for the surface localized states. In this way the role of surface states and resonances is properly considered. Third, the calculated Auger matrix elements fully 8502
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The Journal of Physical Chemistry A account for the spin degree of freedom and properly include exchange interaction and particle indistinguishability.24,25,31 We recall that all results presented here were determined for an He-surface distance of 5.0 a0. The dependence of the He* deexcitation on distance has been previously studied,22,25 suggesting the following points. As the distance increases, the resonances in the induced DOS narrow and consequently also the Auger line shape narrows. The transition rate decreases exponentially owing to the reduced overlap between metal and atom wave functions. So, the measured spectrum is dominated by the de-excitation contribution at the closest distance in the scattering trajectory (for which our value is a reasonable estimate), provided that a significant fraction of He atoms is still in the metastable state. This allows us to consider the previously reported results as representative of experimental conditions. In this work we highlighted the role of an occupied surface state in the de-excitation rate of He* by comparing (100) and (111) silver surfaces. While the charge densities and work functions of the two systems are nearly equivalent, the deexcitation rate is larger by 59% on Ag(111), which supports a Shockley surface state, than on Ag(100), where such a state is not present. Indeed, this state provides an additional decay channel that is particularly efficient because of its localization in the surface region, further enhanced around the atom by the attractive potential of He+. The effect of the surface work function was pointed out by analyzing He* interacting with Ag(100) and Ag(111) and Na(110) surfaces. For relatively high work function surfaces such as silver ones, the 2s resonance is mostly unoccupied. Hence, resonant ionization takes place, followed by Auger neutralization. Such a process leads to a very small spin polarization of the de-excitation rate. This result is in agreement with previous ones for substrates where RI+AN is the relevant mechanism.18 Conversely, on the low work function Na(110) surface de-excitation mainly occurs by a direct Auger process, in agreement with the experimental evidence,18 because the 2s resonance is at least partially filled. Missing the very efficient RI+AN process,32 the total rate lowers with respect to that on high work function surfaces, while a much larger spin polarization of the de-excitation rate follows from that of the electronic properties of the interacting system (as we have seen in the density of states and induced charges).
5. CONCLUSIONS In conclusion, in this paper we have calculated spin dependent de-excitation rates for metastable helium atoms interacting with a paramagnetic metal surface. Additionally, we have discussed the dependence of the Auger rate and its spin polarization on the surface work function and on surface states. The unique surface sensitivity of MDS makes it a suitable probe also to analyze electronic properties of magnetic monolayers and nanostructures at surfaces. Theoretical studies of MDS of such systems are now accessible to our approach. In such cases, the spin polarized Auger rates will further display a particularly interesting dependence on the relative orientation of the He* magnetic moment and that of the system under investigation. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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’ ACKNOWLEDGMENT We acknowledge financial support by Fondazione Cariplo through the project IMMAGINA (Grant No. Rif. 20085.2412) and by the Italian Ministry of University (MIUR) (Grant No. 2008AKZXY-004). ’ ADDITIONAL NOTE Originally submitted for the “J. Peter Toennies Festschrift”, published as the June 30, 2011, issue of J. Phys. Chem. A (Vol. 115, No. 25). ’ REFERENCES (1) Boato, G.; Cantini, P.; Mattera, L. Surf. Sci. 1976, 55, 141. (2) Brusdeylins, G.; Doak, R.; Toennies, J. Phys. Rev. Lett. 1981, 46, 1437. (3) Ellis, J.; Toennies, J. Phys. Rev. Lett. 1993, 70, 2118. (4) Rieder, K.; Parschau, G.; Burg, B. Phys. Rev. Lett. 1993, 71, 1059. (5) Harada, Y.; Masuda, S.; Ozaki, H. Chem. Rev. 1997, 97, 1897. (6) Conrad, H.; Ertl, G.; K€uppers, J.; Wang, S. W.; Gerard, K.; Haberland, H. Phys. Rev. Lett. 1979, 42, 1082. (7) Dunning, F. B.; Nordlander, P. Nucl. Instrum. Methods B 1995, 100, 245. (8) Fouquet, P.; Witte, G. Phys. Rev. Lett. 1999, 83, 360. (9) Bixler, D. L.; Lancaster, J. C.; Kontur, F. J.; Nordlander, P.; Walters, G. K.; Dunning, F. B. Phys. Rev. B 1999, 60, 9082. (10) Kurahashi, M.; Suzuki, T.; Ju, X.; Yamauchi, Y. Phys. Rev. Lett. 2003, 91, 267203. (11) Alducin, M.; Juaristi, J. I.; Diez Mui~ no, R.; R€osler, M.; Echenique, P. M. Phys. Rev. A 2005, 72, 24901. (12) Sun, X.; F€orster, S.; Li, Q. X.; Kurahashi, M.; Zhang, J. W.; Yamauchi, Y.; Baum, G.; Steidl, H. Phys. Rev. B 2007, 75, 035419. (13) Kurahashi, M.; Sun, X.; Yamauchi, M. W. Phys. Rev. B 2010, 81, 193402. (14) Pratt, A.; Wolffinden, C.; Kr€ oger, R.; Tear, S. P.; Binns, C. IEEE Trans. Magn. 2010, 46, 1660. (15) Moroni, R.; Bisio, F.; Canepa, M.; Mattera, L. Nucl. Instrum. Methods B 2002, 193, 480. (16) Moroni, R.; Olivieri, E.; Mattera, L. Nucl. Instrum. Methods B 2003, 203, 29. (17) Hart, M. W.; Hammond, M. S.; Dunning, F. B.; Walters, G. K. Phys. Rev. B 1989, 39, 5488. (18) Lancaster, J. C.; Kontur, F. J.; Walters, G. K.; Dunning, F. B. Phys. Rev. B 2003, 67, 115413. (19) Penn, D. R.; Apell, P. Phys. Rev. B 1990, 41, 3303. (20) Salmi, L. A. Phys. Rev. B 1992, 46, 4180. (21) Kantorovich, L. N.; Shluger, A. L.; Sushko, P. V.; Stoneham, A. M. Surf. Sci. 2000, 444, 31. (22) Sarasola, A.; Silkin, V. M.; Arnau, A. Phys. Rev. B 2007, 75, 045104. (23) Chulkov, E. V.; Silkin, V. M.; Echenique, P. M. Surf. Sci. 1999, 437, 330. (24) Alducin, M.; Diez Mui~ no, R.; Juaristi, J. I. Phys. Rev. A 2004, 70, 12901. (25) Bonini, N.; Brivio, G. P.; Trioni, M. I. Phys. Rev. B 2003, 68, 035408. (26) Trioni, M. I.; Brivio, G. P.; Crampin, S.; Inglesfield, J. E. Phys. Rev. B 1996, 53, 8052. (27) Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, 5048. (28) Trioni, M. I.; Butti, G.; Bonini, N.; Brivio, G. P. Surf. Sci. 2005, 587, 121. (29) Woratschek, B.; Sesselman, W.; K€uppers, J.; Ertl, G.; Haberland, H. Surf. Sci. 1987, 180, 187. (30) Bandurin, Y.; Esaulov, V. A.; Guillemot, L.; Monreal, R. C. Phys. Rev. Lett. 2004, 92, 017601. (31) Feibelman, P. J.; McGuire, E. J.; Pandey, K. C. Phys. Rev. B 1977, 15, 2202. (32) Fouquet, P.; Witte, G. Surf. Sci. 2000, 454456, 256. 8503
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