- Email: [email protected]
Coulomb bound states and ion neutralization in ESD
L590
Surface Science 173 (1986) L590-L596 North-Holland. Amsterdam
SURFACE COULOMB
SCIENCE BOUND
LETTERS STATES
AND ION NEUTRALIZATION
J. RUBIO, J.M. LOPEZ SANCHO, M.P. L6PEZ SANCHO
M.C. REFOLIO
lnstituto de Fisica de Materiales, CSIC, Serrano 144, 28006Madrid. Received
29 October
1985; accepted
for publication
19 March
IN ESD
and Spain
1986
Excited states with two holes and one electron (2h-le states) are known to be crucial in electron stimulated desorption. Starting with a positively ionized adsorbate (lh state). we show in this note that two-hole one-electron states (2h-le) can be generated by means of an Auger-like mechanism. Explicitly, when the Coulomb interaction between the adsorbate hole and the d-band of a transition metal surface is stronger than a critical value, a bound state is formed between the adsorbed hole and electron-hole pairs. Charge transfer to the adsorbate affinity level is strongly suppressed.
The last years have witnessed a considerable impact of Auger-type ideas on the field of electron stimulated desorption (ESD), specially after the work of Knotek and Feibelman [l] Ramaker et al. [2] and Jennison et al. [3]. The emphasis has, thereby, shifted from primary excitation processes to processes which may enhance the lifetime of intermediate states so that the excited adsorbate is given enough time of desorb. Although it is fairly clear that the essential ingredients of the Menzel-Gomer and Redhead model (MGR) are still present [4], one may feel that one-hole (lh) primary excitations have lost somewhat of their role in low-threshold electron stimulated desorption (ESD). In this note we show that lh excitations may be rather effective for low-threshold ESD of ions from transition metal surfaces provided the Coulomb interaction U of the adsorbate hole with the metal d-band is stronger than a critical value U,. For U > U,, a two-hole bound state develops. The adsorbate-surface bond is thus broken while the repulsion between the two holes increases the lifetime of this configuration. The ion then starts to move off the Franck-Condon region, thereby decreasing both U and the hopping strength, say v. When U c UC the bound state and, therefore, the metal hole both disappear. If, then, V is still large enough, the ion is neutralized; otherwise, the ion escapes. The question we address is how a 2h-le state develops starting from a state. Let us now start by showing that a hole placed in front of a metal surface may bind an electron state. Take the Hamiltonian H=x
Zknk
+
%%
+
$Fc;c,+h.c.,
k
0 Elsevier Science Publishers B.V. Physics Publishing Division)
0039-6028/86/$03.50
(North-Holland
J. Rubio et at. / Coulomb bound states and ion neutralization
in ESD
L591
which describes the interaction of an adsorbate hole (ccc,*) with the surface layer of a semi-infinite crystal characterized by the first term. The dispersion relation and occupation number of the metal surface band are denoted, as usual, by ck and rrk = ckfck, k being the parallel crystal momentum. Spin is ignored throughout. With U independent of k (see below) and V = 0, this Hamiltonian is exactly solvable for the metal one-electron Green function [5] G,,,(t)
= -i{~c~(~)c~(O))
(T is Wick’s chronological operator). Its energy (w) Fourier transform is given bY
4
for any c,c,+ . Here gk( w) is the Green function of the clean metal surface, e.g. in the retarded form
in terms of the corresponding spectral density of states, nk(c). Since U has been taken as k-independent here, its repulsive strength is concentrated on just one metal atom, say the 0th atom. This is not necessarily a bad assumption, as U intends to represent a Coulomb repulsion strongly screened by the metallic electrons within an interatomic distance. We of course loose the partial wave structure: U produces just s-wave scattering on the 0th atom, which is the metal atom closest to the adsorbate [S]. We must therefore calculate
which can be written as
Eq. (4) is a Hubbard-type Green function. If the where g, = (l/N)C,g,,,. adorbital is full, G, is just the clean metal Green function; if it is empty, Gw feels the full effect of the interaction. In any intermediate situation, Go0 is a mixture of both. Bound states and/or resonances are given by l+UReg,(r)=O,
(5)
as usual in scattering theory. Fig. 1 gives the differential density of states, &N(C) = -(l/n)
Im[G,(c+)
- g,(e’)],
c+ = 6 + iq,
L592
J. Rubio et al. / Coulomb bound states and ion neutralization
in ESD
a)
Fig. 1. Differential
density
of states for (a) U = W/8 and (b) U = 3W/4.
for a half-occupied semi-elliptical DOS (clean metal surface) and two values of U, U, < U, (no roots of eq. (5)) and U, > U,, (two roots). With a semi-elliptical DOS U, = W/4 ( W = metal bandwidth). For U = U, the metal DOS is simply skewed towards low energies, but for U = U, a delta function splits off from the bottom of the band with a corresponding drop inside the band. The lowering in energy in the metal is given, as is well known, by the integral up to the Fermi level of the phase shift associated with eq. (5). This energy can be either transferred to the exiting electron (UPS relaxation) or used to excite an electron-hole pair (an Auger-like process). Usually, one would expect the electron-hole pair to delocalize and nothing special would happen. However, a configuration interaction analysis shows [6] that this process leads to a bound 2h-le state provided U > UC, where u,=
w[l-:(V/2w)2],
W being the bandwidth. If this condition similar to others derived before [7], is fulfilled, one has a stable hole in the metal band, in the sense that hopping to neighbouring metal atoms is hindered. There is still the possibility, however, of neutralization of the ion by transferring charge to an unfilled level (the affinity level) which may have dropped below the Fermi level upon hole creation. This is just the charge transfer mechanism of relaxation studied by many [8,9] in connection with XPS, UPS, etc. ending up with a neutralized adsorbate. But, noticing that
J. Rubio ei al. / Coulomb bound states and ion neutralization
a)
in ESD
L593
b) ---O--_&a
I
-IL EF
2.
UaC
Uac
Uab
Fig. 2. The unfilled orbital model for (a) U < UCand (b) U > UC.
Uc > U,, the affinity level will also feel the repulsion from the bound metal electron and will raise again above the Fermi level, thus remaining essentially unfilled. Everything depends on the relative values of the Coulomb integrals acting on the affinity level. The situation is shown in fig. 2a for U ( U, and fig. 2b for U > UC. For U c UC the electron-hole delocalizes and we have the normal situation of the charge transfer to the unfilled orbital model: E, drops to ea - U,,, below er, and is subsequently filled. For U > UC. however, E, drops only by U,, - Uabrand will usually stay above the Fermi level. The discussion just made is embodied in the following Hamiltonian for the situation (n,) = 0, U > UC: H=c,n,+
c yz,+(E,k(scatt)
u,,+
v.,)n.+~
c (ca’ck+c:c,). k(scatt)
(6)
The first two terms describe the bound and scattering states of the metal surface, whereas the third term accounts for the affinity level attracted by the c-hole (U,,) and repelled by the metal bound electron (U,,). The last term describes charge transfer between the affinity level and the scattering states of the metal, controlled by the hopping strength V. Now we want to find q, = (n,) in terms of the system’s parameters. This is obtained, as usual, by integrating up to the Fermi level the affinity density of states N,(E) = -(l/?r)G,,(e + in), where G,,(w) is given by
The 0th atom Green function G,, eq. (4), appears here because we have ignored all the possible k-dependence of V, just as in eq. (3). The scattering part of.G, is obtained by the simple expedient of taking the Schmidt-Hilbert
L594
J. Rubio et al. / Coulomb bound states and ion neutralization I
I
I
I
I
I
I
I
5
6
7
6
in ESD
-
9-
----__
0-*-.-._* 76-
--
-\
\. ‘.
‘\
\
‘. g 5-
4, \\
L32-
‘\, \
l-
I 1
I
I
I
2
3
L
U(N) Fig. 3. The affinity charge, qa, as a function of U for three values of the hopping line, V = 1; dashed, V = 1.5, and dashed-dotted curve. V = 2 (in eV).
strength,
P’. Full
transform of the corresponding density of states, but integrating ouer the band only, i.e., excluding the bound state. For U < U, one simply has H=~r,n,+(r,-U,,)n,+~~(c:~~+~~~~). k
k
as usual [5], and G,, is given by [a - (~a-
v,,) - I”gc&)]%(w)
= I,
in terms of the clean metal Green function. Fig. 3 shows the calculated qa = (n,) as a function of U for several values of T/. We have taken a half-filled semi-elliptical DOS of width W = 8 eV, which- is a rough representation of the surface DOS of middle row transition metals [lO,ll] but reasonable enough for qualitative purposes. The affinity level after ionization, c, - U,, has been placed at 3 eV below cr, which is appropriate for CO+ [6]. As to the values of Uab, no convincing criterium was found and we finally opted for U,, = U (although it should be somewhat smaller, as the 277 orbital in CO+ is more delocalized than 5a in CO). As the figure shows, qa drops by an order of magnitude between U = 3.5 and 4, this effect being somewhat less drastic as V increases (compare the full curve, Y= 1, with the dashed-dotted one, V= 2). In contrast, qa remains practically constant up to U = 3, and slowly goes to zero above U = 4. A hole placed in front of a metal surface is, therefore, quickly neutralized for U < UC. For U > UC, however, the bound metal electron repels any electron charge that might, otherwise, be put into the affinity level. As U increases further, it is becoming more and more difficult to neutralize the ion. The effect of V is just the opposite: the larger V, the easier to transfer charge into the affinity level.
J. Rubio et al. / Coulomb bound states and ion neutralization
in ESD
L595
Now suppose the ion starts leaving the surface. All the ion-metal interactions will decrease. The affinity level, fig. 2b, will go down and, eventually, will cross the Fermi level; the bound electron level, ft,, will approach the bottom of the band and, eventually, will merge into the band when any of these things happens, the ion will be neutralized provided V is still large enough to allow for charge transfer. If, on the contrary, V has already dropped to vanishingly small values (and it decreases exponentially with increasing distance to the surface), the ion will escape as ion. A final comment. Everything we have discussed above depends rather critically on the existence of a metal bound state due to an external hole as well as on the extend of localization of the defect of charge left behind inside the metal band. Although the existence of this bound state has been recognized for a long time [12], its physical reality has been put into question, at least for a nearly-free metal, on the basis that it would require an electronic density so low that the system would no longer be metallic [13]. This objection does not apply to the surface of a transition metal with relatively localized d-orbitals, some of them protruding outside the surface. Density functional calculations in the local density approximation [14] have shown rather shallow and quite extended bound states just below a jellium band. The situation is again quite different for a narrow d-band. At any rate, it has been the main point of the present note to stress that this kind of bound state provides a mechanism to generate 2h-le states, thereby emptying metal group orbitals, breaking adsorbate-metal bonds, etc. It might therefore have some relevance in ESD phenomena. References [l] M.L. Knotek and P.J. Feibelman, Phys. Rev. Letters 40 (1978) 964; P.J. Feibelman and M.L. Knotek, Phys. Rev. B18 (1978) 6531. [2] D.E. Ramaker, C.T. White and J.S. Murday, J. Vacuum Sci. Technol. 18 (1981) 748; J. Chem. Phys. 78 (1983) 2998. [3] D.R. Jennison, J.A. Kelber and R.R. Rye, J. Vacuum Sci. Technol. 18 (1981) 466; Phys. Rev. B25 (1982) 1384; H.H. Madden, D.R. Jennison, M.M. Traum, G. Margaritondo and N.G. Stoffel, Phys. Rev. B26 (1982) 896. [4] D. Menzel, in: Desorption Induced by Electronic Transitions (DIET I), Eds. N.H. Talk, M.H. Traum, J.C. Tully and T.E. Madey (Springer, Berlin, 1983) p. 53. [5] Actually, the metal Green function can be found exactly for any separable U,,, = UfkfkT.Our calculation corresponds to fk = fk, = 1, but one can use this function to describe a variety of situations (see ref. [12]). [6] J. Rubio, J.M. Lopez Sancho, M.C. Refolio and M.P. Lopez Sancho, to be published. [7] M. Cini, Solid State Commun. 24 (1977) 681; G.A. Sawatzky, Phys. Rev. Letters 39 (1977) 504. [S] For a review, see: B. Gumhalter, Progr. Surface Sci. 15 (1984) 1. [9] J. Rubio and J.M. Lopez Sancho, Surface Sci. 95 (1980) L267; M.C. Refolio, J.M. Lopez Sancho and J. Rubio, Surface Sci. 117 (1982) 459.
L596 [lo]
J. Rubio et al. / Coulomb bound states and ion neutralization
in ESD
S.L. Weng, E.W. Plummer and T. Gustafsson, Phys. Rev. B18 (1978) 1718; W.R. Grise, D.G. Dempsey, L. Kleinman and K. Mednick. Phys. Rev. B20 (1979) 3045. [ll] M.P. Lopez Sancho, J.M. Lopez Sancho and J. Rubio, J. Phys. F14 (1984) 1205; J. Phys. Cl8 (1985) 1803; M.P. Lopez Sancho, Solid State Commun. 50 (1984) 629. [12] B. Roulet, J. Gavoret and P. Nor&es, Phys. Rev. 178 (1969) 1072; P. Nor&es and C.T. De Dominicis, Phys. Rev. 178 (1969) 1097. [13] N.F. Mott, Phil. Mag. 20 (1969) 1. [14] J.K. Nsrskov. Phys. Rev. B26 (1979) 446; B.I. Lundqvist, 0. Gunnarsson, H. Hjelmberg and J.K. Nsrskov. Surface Sci. 89 (1979) 196.
Surface Science 173 (1986) L590-L596 North-Holland. Amsterdam
SURFACE COULOMB
SCIENCE BOUND
LETTERS STATES
AND ION NEUTRALIZATION
J. RUBIO, J.M. LOPEZ SANCHO, M.P. L6PEZ SANCHO
M.C. REFOLIO
lnstituto de Fisica de Materiales, CSIC, Serrano 144, 28006Madrid. Received
29 October
1985; accepted
for publication
19 March
IN ESD
and Spain
1986
Excited states with two holes and one electron (2h-le states) are known to be crucial in electron stimulated desorption. Starting with a positively ionized adsorbate (lh state). we show in this note that two-hole one-electron states (2h-le) can be generated by means of an Auger-like mechanism. Explicitly, when the Coulomb interaction between the adsorbate hole and the d-band of a transition metal surface is stronger than a critical value, a bound state is formed between the adsorbed hole and electron-hole pairs. Charge transfer to the adsorbate affinity level is strongly suppressed.
The last years have witnessed a considerable impact of Auger-type ideas on the field of electron stimulated desorption (ESD), specially after the work of Knotek and Feibelman [l] Ramaker et al. [2] and Jennison et al. [3]. The emphasis has, thereby, shifted from primary excitation processes to processes which may enhance the lifetime of intermediate states so that the excited adsorbate is given enough time of desorb. Although it is fairly clear that the essential ingredients of the Menzel-Gomer and Redhead model (MGR) are still present [4], one may feel that one-hole (lh) primary excitations have lost somewhat of their role in low-threshold electron stimulated desorption (ESD). In this note we show that lh excitations may be rather effective for low-threshold ESD of ions from transition metal surfaces provided the Coulomb interaction U of the adsorbate hole with the metal d-band is stronger than a critical value U,. For U > U,, a two-hole bound state develops. The adsorbate-surface bond is thus broken while the repulsion between the two holes increases the lifetime of this configuration. The ion then starts to move off the Franck-Condon region, thereby decreasing both U and the hopping strength, say v. When U c UC the bound state and, therefore, the metal hole both disappear. If, then, V is still large enough, the ion is neutralized; otherwise, the ion escapes. The question we address is how a 2h-le state develops starting from a state. Let us now start by showing that a hole placed in front of a metal surface may bind an electron state. Take the Hamiltonian H=x
Zknk
+
%%
+
$Fc;c,+h.c.,
k
0 Elsevier Science Publishers B.V. Physics Publishing Division)
0039-6028/86/$03.50
(North-Holland
J. Rubio et at. / Coulomb bound states and ion neutralization
in ESD
L591
which describes the interaction of an adsorbate hole (ccc,*) with the surface layer of a semi-infinite crystal characterized by the first term. The dispersion relation and occupation number of the metal surface band are denoted, as usual, by ck and rrk = ckfck, k being the parallel crystal momentum. Spin is ignored throughout. With U independent of k (see below) and V = 0, this Hamiltonian is exactly solvable for the metal one-electron Green function [5] G,,,(t)
= -i{~c~(~)c~(O))
(T is Wick’s chronological operator). Its energy (w) Fourier transform is given bY
4
for any c,c,+ . Here gk( w) is the Green function of the clean metal surface, e.g. in the retarded form
in terms of the corresponding spectral density of states, nk(c). Since U has been taken as k-independent here, its repulsive strength is concentrated on just one metal atom, say the 0th atom. This is not necessarily a bad assumption, as U intends to represent a Coulomb repulsion strongly screened by the metallic electrons within an interatomic distance. We of course loose the partial wave structure: U produces just s-wave scattering on the 0th atom, which is the metal atom closest to the adsorbate [S]. We must therefore calculate
which can be written as
Eq. (4) is a Hubbard-type Green function. If the where g, = (l/N)C,g,,,. adorbital is full, G, is just the clean metal Green function; if it is empty, Gw feels the full effect of the interaction. In any intermediate situation, Go0 is a mixture of both. Bound states and/or resonances are given by l+UReg,(r)=O,
(5)
as usual in scattering theory. Fig. 1 gives the differential density of states, &N(C) = -(l/n)
Im[G,(c+)
- g,(e’)],
c+ = 6 + iq,
L592
J. Rubio et al. / Coulomb bound states and ion neutralization
in ESD
a)
Fig. 1. Differential
density
of states for (a) U = W/8 and (b) U = 3W/4.
for a half-occupied semi-elliptical DOS (clean metal surface) and two values of U, U, < U, (no roots of eq. (5)) and U, > U,, (two roots). With a semi-elliptical DOS U, = W/4 ( W = metal bandwidth). For U = U, the metal DOS is simply skewed towards low energies, but for U = U, a delta function splits off from the bottom of the band with a corresponding drop inside the band. The lowering in energy in the metal is given, as is well known, by the integral up to the Fermi level of the phase shift associated with eq. (5). This energy can be either transferred to the exiting electron (UPS relaxation) or used to excite an electron-hole pair (an Auger-like process). Usually, one would expect the electron-hole pair to delocalize and nothing special would happen. However, a configuration interaction analysis shows [6] that this process leads to a bound 2h-le state provided U > UC, where u,=
w[l-:(V/2w)2],
W being the bandwidth. If this condition similar to others derived before [7], is fulfilled, one has a stable hole in the metal band, in the sense that hopping to neighbouring metal atoms is hindered. There is still the possibility, however, of neutralization of the ion by transferring charge to an unfilled level (the affinity level) which may have dropped below the Fermi level upon hole creation. This is just the charge transfer mechanism of relaxation studied by many [8,9] in connection with XPS, UPS, etc. ending up with a neutralized adsorbate. But, noticing that
J. Rubio ei al. / Coulomb bound states and ion neutralization
a)
in ESD
L593
b) ---O--_&a
I
-IL EF
2.
UaC
Uac
Uab
Fig. 2. The unfilled orbital model for (a) U < UCand (b) U > UC.
Uc > U,, the affinity level will also feel the repulsion from the bound metal electron and will raise again above the Fermi level, thus remaining essentially unfilled. Everything depends on the relative values of the Coulomb integrals acting on the affinity level. The situation is shown in fig. 2a for U ( U, and fig. 2b for U > UC. For U c UC the electron-hole delocalizes and we have the normal situation of the charge transfer to the unfilled orbital model: E, drops to ea - U,,, below er, and is subsequently filled. For U > UC. however, E, drops only by U,, - Uabrand will usually stay above the Fermi level. The discussion just made is embodied in the following Hamiltonian for the situation (n,) = 0, U > UC: H=c,n,+
c yz,+(E,k(scatt)
u,,+
v.,)n.+~
c (ca’ck+c:c,). k(scatt)
(6)
The first two terms describe the bound and scattering states of the metal surface, whereas the third term accounts for the affinity level attracted by the c-hole (U,,) and repelled by the metal bound electron (U,,). The last term describes charge transfer between the affinity level and the scattering states of the metal, controlled by the hopping strength V. Now we want to find q, = (n,) in terms of the system’s parameters. This is obtained, as usual, by integrating up to the Fermi level the affinity density of states N,(E) = -(l/?r)G,,(e + in), where G,,(w) is given by
The 0th atom Green function G,, eq. (4), appears here because we have ignored all the possible k-dependence of V, just as in eq. (3). The scattering part of.G, is obtained by the simple expedient of taking the Schmidt-Hilbert
L594
J. Rubio et al. / Coulomb bound states and ion neutralization I
I
I
I
I
I
I
I
5
6
7
6
in ESD
-
9-
----__
0-*-.-._* 76-
--
-\
\. ‘.
‘\
\
‘. g 5-
4, \\
L32-
‘\, \
l-
I 1
I
I
I
2
3
L
U(N) Fig. 3. The affinity charge, qa, as a function of U for three values of the hopping line, V = 1; dashed, V = 1.5, and dashed-dotted curve. V = 2 (in eV).
strength,
P’. Full
transform of the corresponding density of states, but integrating ouer the band only, i.e., excluding the bound state. For U < U, one simply has H=~r,n,+(r,-U,,)n,+~~(c:~~+~~~~). k
k
as usual [5], and G,, is given by [a - (~a-
v,,) - I”gc&)]%(w)
= I,
in terms of the clean metal Green function. Fig. 3 shows the calculated qa = (n,) as a function of U for several values of T/. We have taken a half-filled semi-elliptical DOS of width W = 8 eV, which- is a rough representation of the surface DOS of middle row transition metals [lO,ll] but reasonable enough for qualitative purposes. The affinity level after ionization, c, - U,, has been placed at 3 eV below cr, which is appropriate for CO+ [6]. As to the values of Uab, no convincing criterium was found and we finally opted for U,, = U (although it should be somewhat smaller, as the 277 orbital in CO+ is more delocalized than 5a in CO). As the figure shows, qa drops by an order of magnitude between U = 3.5 and 4, this effect being somewhat less drastic as V increases (compare the full curve, Y= 1, with the dashed-dotted one, V= 2). In contrast, qa remains practically constant up to U = 3, and slowly goes to zero above U = 4. A hole placed in front of a metal surface is, therefore, quickly neutralized for U < UC. For U > UC, however, the bound metal electron repels any electron charge that might, otherwise, be put into the affinity level. As U increases further, it is becoming more and more difficult to neutralize the ion. The effect of V is just the opposite: the larger V, the easier to transfer charge into the affinity level.
J. Rubio et al. / Coulomb bound states and ion neutralization
in ESD
L595
Now suppose the ion starts leaving the surface. All the ion-metal interactions will decrease. The affinity level, fig. 2b, will go down and, eventually, will cross the Fermi level; the bound electron level, ft,, will approach the bottom of the band and, eventually, will merge into the band when any of these things happens, the ion will be neutralized provided V is still large enough to allow for charge transfer. If, on the contrary, V has already dropped to vanishingly small values (and it decreases exponentially with increasing distance to the surface), the ion will escape as ion. A final comment. Everything we have discussed above depends rather critically on the existence of a metal bound state due to an external hole as well as on the extend of localization of the defect of charge left behind inside the metal band. Although the existence of this bound state has been recognized for a long time [12], its physical reality has been put into question, at least for a nearly-free metal, on the basis that it would require an electronic density so low that the system would no longer be metallic [13]. This objection does not apply to the surface of a transition metal with relatively localized d-orbitals, some of them protruding outside the surface. Density functional calculations in the local density approximation [14] have shown rather shallow and quite extended bound states just below a jellium band. The situation is again quite different for a narrow d-band. At any rate, it has been the main point of the present note to stress that this kind of bound state provides a mechanism to generate 2h-le states, thereby emptying metal group orbitals, breaking adsorbate-metal bonds, etc. It might therefore have some relevance in ESD phenomena. References [l] M.L. Knotek and P.J. Feibelman, Phys. Rev. Letters 40 (1978) 964; P.J. Feibelman and M.L. Knotek, Phys. Rev. B18 (1978) 6531. [2] D.E. Ramaker, C.T. White and J.S. Murday, J. Vacuum Sci. Technol. 18 (1981) 748; J. Chem. Phys. 78 (1983) 2998. [3] D.R. Jennison, J.A. Kelber and R.R. Rye, J. Vacuum Sci. Technol. 18 (1981) 466; Phys. Rev. B25 (1982) 1384; H.H. Madden, D.R. Jennison, M.M. Traum, G. Margaritondo and N.G. Stoffel, Phys. Rev. B26 (1982) 896. [4] D. Menzel, in: Desorption Induced by Electronic Transitions (DIET I), Eds. N.H. Talk, M.H. Traum, J.C. Tully and T.E. Madey (Springer, Berlin, 1983) p. 53. [5] Actually, the metal Green function can be found exactly for any separable U,,, = UfkfkT.Our calculation corresponds to fk = fk, = 1, but one can use this function to describe a variety of situations (see ref. [12]). [6] J. Rubio, J.M. Lopez Sancho, M.C. Refolio and M.P. Lopez Sancho, to be published. [7] M. Cini, Solid State Commun. 24 (1977) 681; G.A. Sawatzky, Phys. Rev. Letters 39 (1977) 504. [S] For a review, see: B. Gumhalter, Progr. Surface Sci. 15 (1984) 1. [9] J. Rubio and J.M. Lopez Sancho, Surface Sci. 95 (1980) L267; M.C. Refolio, J.M. Lopez Sancho and J. Rubio, Surface Sci. 117 (1982) 459.
L596 [lo]
J. Rubio et al. / Coulomb bound states and ion neutralization
in ESD
S.L. Weng, E.W. Plummer and T. Gustafsson, Phys. Rev. B18 (1978) 1718; W.R. Grise, D.G. Dempsey, L. Kleinman and K. Mednick. Phys. Rev. B20 (1979) 3045. [ll] M.P. Lopez Sancho, J.M. Lopez Sancho and J. Rubio, J. Phys. F14 (1984) 1205; J. Phys. Cl8 (1985) 1803; M.P. Lopez Sancho, Solid State Commun. 50 (1984) 629. [12] B. Roulet, J. Gavoret and P. Nor&es, Phys. Rev. 178 (1969) 1072; P. Nor&es and C.T. De Dominicis, Phys. Rev. 178 (1969) 1097. [13] N.F. Mott, Phil. Mag. 20 (1969) 1. [14] J.K. Nsrskov. Phys. Rev. B26 (1979) 446; B.I. Lundqvist, 0. Gunnarsson, H. Hjelmberg and J.K. Nsrskov. Surface Sci. 89 (1979) 196.