- Email: [email protected]
Surface-plasmon mediated proton neutralization at metal surfaces
surface science ELSEVIER
Surface Science 370 (1997) 77-84
Surface-plasmon mediated proton neutralization at metal surfaces F.A. G u t i e r r e z * Departamento de Fisica, Universidadde Concepci6n, Casilla 4009, Concepci6n, Chile Received 13 May 1996; accepted for publication 9 July 1996
Abstract Proton neutralization at gold surfaces by surface-plasmon emission is analyzed. Transition rates and neutralization fractions are evaluated within the fixed-ion approximation and compared with those for the usual Auger channel. We found that for protonsurface distances larger than one bohr radius the collective transition rates are larger than the Auger rates, while the neutralization fractions are very similar for both channels. Therefore in those situations in which the collective mode is energetically allowed, which is the case for several metals, it has to be considered simultaneously with the Auger and resonance tunneling modes.
Keywords: Auger transfer; Charge exchange at surfaces; Gold; Ion-surface scattering; Low energy ion scattering
I. Introduction
Resonance tunneling and Auger capture are generally considered to be the relevant channels for ion neutralization at metal surfaces [ 1-6]. For the static case and when these processes are analyzed separately Snowdon and co-workers [1,2] found that they are of comparable magnitude, with the conclusion that both mechanisms should be considered simultaneously for a correct quantitative description of ion neutralization. They also found that theoretical neutralization fractions for each mode are much lower than the experimental ones. Later, Zimny et al. [3] considered resonant and Auger electron capture on an equal footing, and also resonance and Auger ionization for proton neutralization after grazing reflection at an A1 surface within a rate-equation approach. They * Corresponding author. Fax: 56 41 240280; e-mail: [email protected]
found that for low parallel velocities vii, resonance tunneling seems to be predominant, while for high v,, Auger neutralization takes over. At intermediate velocities both channels appear to be of comparable importance. However, the discrepancy between theory and experiment still remains. Meanwhile Almulhem and Girardeau (AG) [7] proposed another mechanism of ion neutralization for the H(ls)-A1 system, whereby the energy released in the capture is taken away by a surface plasmon. This mechanism might be a good candidate to reduce the discrepancy between theory and experiments. Unfortunately the results of AG are incorrect, as indicated earlier by Zimny et al. [3], due to errors in the evaluation of the transition rates. In the first place the collective process is energetically forbidden for A1 when vii= 0 (the case considered by them), becoming possible only for vii larger than a certain threshold velocity. Secondly, their transition rates were expressed in terms of two (instead of four)-dimensional integrals
0039-6028/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0039-6028 ( 9 6 ) 00951-X
78
F.A. Gutierrez/Surface Science370 (1997) 77-84
based on the incorrect assumption that the matrix elements for the plasmonic neutralization channel depends only on the magnitude of q (plasmon wavevector) and kll (component of the electron wavevector parallel to the surface) but not on their orientation. Therefore the role of surface plasmons on proton neutralization at surfaces is unknown at the moment. Besides the H(ls)-A1 system, Snowdon and co-workers [1,2,8] (and earlier Horiguchi et al. [ 9]) considered Auger neutralization for H (ls)-Au, for which the experimental results of Verbeek et al. [10] also exist. As we shall see for gold, the plasmon mode of proton neutralization is energetically allowed even in the static case when vii=0. Thus by application of the orthogonalized Born approximation developed by AG [7], we obtain transition rates and neutralization fractions for H(ls)-Au and compare the results with those obtained by Snowdon et al. [1] for the Auger process for a similar model of the proton-metal collision. We found that for proton-surface distances beyond a bohr radius, transition rates for the collective mode are larger than those reported for the Auger capture, with the difference getting larger for increasing ion-surface distance. This result is in qualitative agreement with recent results by Monreal and Lorente [ 11 ], for He ÷ interacting with A1 surfaces, where the inclusion of dynamical screening (surface-plasmon excitation) enhances their Auger transition rate in such a way that it has a much slower decrease than their unscreened Auger rate with increasing ion-surface distance. We also found that the neutralization fractions for the collective channel and for the Auger channel are very similar. Therefore it appears that in those situations in which the surface-plasmon mode is allowed, one should include it simultaneously with the resonant tunneling and Auger channels for a correct description of proton neutralization on surfaces. Atomic units are used throughout this paper unless otherwise stated.
2. Theory The matrix elements for the plasmon-mediated neutralizing transition are given in the ortho-
gonalized Born approximation by [7]
(v'qlHintlk)= X/ qA/~°J~{fdar~,(r_S)~k(r)e_qlz I × e-lq'rll-- f d3r dar'~*(r-s) x A(r-
s, r'- s)qbk (r')e-ql~le - * ' ' t
'
(1) with the hydrogen-bound state kernel
A(r-s, r'-s)= ~v(r-s)~*(r'-s).
(2)
V
The metal is modeled by a jellium filling the halfspace z < 0 [1-3], the proton is taken at rest (at a distance s from the surface) during the evaluation of the transition rate in the so-called fixed ion approximation (FIA), with the neutral fraction being obtained by integration over a classical ion trajectory [ 1,2,9]. The first term within brackets in Eq.(1) represents the plane wave Born approximation for the transition rate while the second term takes into account the orthogonalization of the metal orbitals to the bound atomic states [7,11]. In this way spurious contributions from bound-bound channels are eliminated automatically. In Eq. (1), A is the surface area, q is the surface-plasmon wavevector with q = [ql
F.A. Gutierrez/Surface Science370 (1997) 77-84 and W is the corresponding work function, and with the energies measured from the bottom of the conduction band. Their explicit form is
79
where
f=q+kz-d,
g=q+k=+d,
(8)
d = ~/1 + (q--kll) 2, kz,2 = 2Vo-k~, 2
~k(r) =
and with
(eik,t"ll [(k'~+ ikz)e ik;z+ ( k'~-- ik,)e-~'z],
k: ,f4e -~k= e-~h'[ m2 = ~-7 [ ~ +[(l+hs)F-h] --~-j (3)
{
x e-~q--q[2(sl+l)+l 2] la j ,
~z>O,
(9)
where where (kz) ,2 =2Ek,t k2=2(Vo-E'k), k~-kz+(k~)2-2 , 2__ ' g 1L2 2Vo, and E k - - k--2-r~ll with kll the magnitude of kll, the component of k paralM to the surface. The step potential appears as a reasonable approximation for the surface potential of a metal in the presence of an ion as argued by Gadzuk [12]. In particular for Au there is good agreement between the electron density obtained from the selfconsistent local density functional calculation of Lang and Kohn [ 13] with that obtained with the step potential model [8]. The use of the unperturbed wavefunction for the ls state of hydrogen considered by AG and other authors [ 1,2,7],
1 x/~
F=h-kz,
G = h + k z , h=~/l +k~l, /= 4 ~
(10) In Eq.(5) the metal orbital is orthogonalized respect to the ground state of hydrogen, as in Ref. [11]. We have checked that contributions from the other terms of the orthogonalizing series (which are very cumbersome) contribute only a few percent for s > 1 the range we are interested in. We also note that the function m~ defined by (7) remains finite even when f vanishes for certain values of q, k= and kll. In fact when q + k= ~ d , so that f ~ 0 , one finds that ml ~ m °, with m° = ~
el,-sl,
(4)
makes it possible to integrate Eq. (1) in analytical form to obtain the matrix elements for plasmon capture into the ground state of hydrogen. The final expression is [-7]
(ls, qlHimlk)=
M-
/____e.M, "V qA
(5)
47z
(6)
~ (ml - m 2 ) , x/ zn V Vo
2.
+ ~ + ~
.
(11)
The transition rate for the plasmonic mode is P=2n
~
[(ls, qlHi,tlk)12O(cl-ey)
(12)
k,q < qc
with the initial energy ei = ½k2 and the final energy ey = E(ls) + ~o~, where E(ls) = Vo - ½ since the energies are measured (in atomic units) from the bottom of the conduction band. Changing the sums in Eq. (12) into integrals as usual and use of Eq. (5), leads to
P(s) = - 87~4 dk<-kF
d3k
d2q ~qe
with
k• {4e-; +k-)"
ml= ~
"
--
+[(l+ds)f-d]
e-~a'[
d3 j ,
x ~ qA (7)
~r).
(13)
Consider now cylindrical coordinates for the
80
F.A. Gutierrez/Surface Science 370 (1997) 77-84
0.05
I O0 -% N
4
0,04
0
q=0.93
.~ ('~ 0.03 ~ '
A
=0.75 =0.52
x
3
\ \
.2
\
\
1 0 -1
(,9
\
,,
4
,,. ,,.
(3_
(3_ 0,02
10-'3
o
4
2
6
8
10
,
0.0
2
2,5
,5.0
electron wavevector k =(kll, (Pk, kz) and polar coordinates for the plasmon wavevector q=(q, ~oq)so that
ws P(s) = 8~ 3 J 0
z
10.0
12,5
Fig. 2. N e u t r a l i z a t i o n rates for a p r o t o n o n gold for the collective p l a s m o n m o d e for qc= 0.93. The d a s h e d line corres p o n d to the n o n - o r t h o g o n a l i z e d case in w h i c h only the first
term in Eq.(6) is considered. The solid line contains the orthogonalization corrections contained in the second term of Eq. (6).
which after integration gives
IM(kz, ~Ok;q, ~q)[20(½k2-e:)O(e:-½k2),
f f c dq
fo k~4"i~-~ dkllklL IMl2~(½kll2
_[_ ~k~: 1 2 -
e:).
(14)
The change of variable u = ½k~ and the introduction of the step function O(z) to account for the integration limits allows us to rewrite the last integral in Eq. (14) as
f:oou
(16)
with the condition ~1 k 2l p - e f - 2 k1 2z. Therefore the transition rate is:
P(s) = O(½k2 - e:)Po (s), x
15,0
S
S Fig. 1. Neutralization rates for a proton on gold. The solid lines correspond to the three cases for the collective plasmon mode (qc= 0.93, 0.75 and 0.52), while the dashed lines are those from Ref. I-1] for the Auger transition with the labels corresponding to the value of the screening parameter 2.
7.5
1
(17)
with
×
f?;?
X 2 kll2
d~ok
= e: -- ½k 2 .
d@~lM(kz, ~Ok;q, ~0q)12; (18)
The expression we obtain for the transition rate
P(s) differs from that of Ref. [7] (their Eq. (47)) in several aspects: (i) we obtain the extra factor O(½k2v--e:) which takes into account the conservation of the energy (ei=½k2=e:) together with the constraint k
F..A. Gutierrez/ Surface Science 370 (1997) 77-84
81
1.0 C" O +--
o o L,. LL. C
0.8 '%\
x N
0.6
o O N
o_
0.4
U L. q)
Z
0.2
0.0
1 0-2
" , , ,, _ ""
q =0.93 =0.75 =0.52
0 •~ x I
34
I
I
I
I
",x,,, -. "" -
llI
I
1 0 -1
2
I
34
I
I
I
I I I ]
100
I
I
2
34
I
I
I
I I
1 01
Energy (keY) Fig. 3. Neutralization fractions for a proton on gold. The solid lines correspond to the three cases for the collective plasmon mode (qc=0.93, 0.75 and 0.52) while the dashed lines are those from Ref. I-l] for the Auger transition with the labels corresponding to the value of the screening parameter 2. The line labeled "e" represents the experimental results of Verbeek et al. [10].
they considered) one has [7] Vo=0.5862, ~os= 0.4106 and kF=0.9261. Then ey=E(ls)+og~= Vo-½+co~=0.4968 and ½ k 2 - e I < O so that P(s) vanishes for any s. Thus the transition rates of AG for H(ls)-A1 are incorrect, a fact already noted by Zimny et al. [3]. (ii) they assumed that the matrix elements M in Eq.(18) do not depend on the angles {Ok and (O~ to reduce the four-dimensional integral to a two-dimensional one. That M depends on the angles (0k and (O+ is clear from our Eqs.(6)-(8) (see also eqs. (A10) and (A15) in Ref. [7]). (iii) The range of integration for the variable kz is in our case [0, 2 ~ ¢ ] instead of their larger range [0, kF] which is incorrect, since for k, > 2 ~ y , kit would take complex values. Finally the neutral fraction f o for a straight-line constant-velocity classical ion trajectory with specular reflection is [1,2,9] f o = 1- e x p l
o.(S)fo+P(s)ds] 2
(19)
where v±(s) is the constant component of proton velocity perpendicular to the surface and P(s) is
the neutralization rate, both being functions of the surface proton distance s.
3. Results
The surface-plasmon energy can be evaluated from the well known expression o~s=c%/x/2 with c% = x/4nne2/m the bulk-plasmon frequency, where n is the electron density and m the electron mass. For the cut-off plasmon wavevector qc a rigorous expression does not exist.. To check the importance of the cut-off wavevector in the evaluation of the collective transition rates we shall consider a range of values, which go from the simple estimate [14] qc "~ COp/VF(VFis the Fermi velocity) up to qc ,~ 1.5kv. Therefore the values of the relevant parameters for gold are [ 15] (in atomic units) ~s = 0.2349, O~p/VV= 0.52, Vo= 0.38 and kF =0.62. The calculations were performed using standard Gaussian integration methods with different number of points. These calculations were supplemented with the accelerating Aitken's 52-process [ 16].
F.A. Gutierrez/ Surface Science370 (1997) 77-84
82
energy (related to the perpendicular velocity v±) are plotted in Fig. 3 for qc=0.52, 0.75 and 0.93. We also include the Auger neutralization fractions of Snowdon et al. [1] (for 2=0.45 and 0.90) together with the experimental results of Verbeek et al. [ 10]. The three curves for collective neutralization fractions which go close to each other fall between the two curves for the Auger mode being much closer to that with 2 = 0.45. For energies of the order of or larger than 1 keV all the theoretical curves are significantly lower than the experimental one.
We first note that the collective process is energetically allowed for the H ( l s ) - A u system. In fact e y = E ( l s ) + ~ o s = Vo-½ + cos=0.1149 and 1 2 :kF--e I >0, so that the transition rates P(s) given by Eq.(17) do not vanish. They are shown in Fig. 1 for the three cases qc=0.52, 0.75 and 0.93. For purposes of comparison we also show the Auger neutralization rates reported by Snowdon et al. [ 1 ] for two values of their screening parameter 2 for the electron-electron interaction. We find that: (i) the collective transition rates have a maxim u m at s,~l, after which they decay with the distance (for qc = 0.93 at s = 5 the transition rate is one order of magnitude smaller than its m a x i m u m value). (ii) Larger values of qc lead to larger transition rates; the strongest differences appear within the range s < 1.5 where the transition rate for q~= 0.93 is more than twice that for q~ = 0.52. However, beyond the point at which they reach their maxim u m value the transition rates for larger q~ decay faster than those for small q~ so that they eventually merge at some s. In fact for s > 5 the differences are negligible. (iii) For s > 1 the plasmonic transition rates for the three values of q¢ are larger than the Auger transition rates for the two values of 2. Since the latter decay much faster than the former, the differences increase for larger s in such a way that for s > 4 the Auger transition rates become negligible compared to those for the collective mode. In Fig. 2 we show that the effect of the orthogonalization correction for the case q~ = 0.93 is to reduce the plasmon-mediated transition rate by one order of magnitude or more. This situation also occurs for q~ = 0.75 and 0.52. Finally the plasmon-mediated neutralization fractions as functions of the projectile kinetic
4. Discussion and conclusions We have applied the orthogonalized Born matrix elements for surface-plasmon mediated ion neutralization on metal surfaces [7] to the case of a proton on gold and compared the results with those for the Auger mode reported by Snowdon et al. [1]. The calculations for both mechanisms consider a semi-infinite jellium metal and the steppotential model for the surface, which lead to electron densities in good agreement with those from the self-consistent local density functional calculations of Lang and K o h n [ 13] for gold. For the unperturbed atom case, the condition for the collective transition (to the ground state of the incoming ion) to be allowed ½ k ~ - e ; > O can be rewritten as ½> W+Ogs. Raether [17] has tabulated the values of surface-plasmon energies (theoretical and experimental) for several metals. These are shown in Table 1, together with the work functions [15] and their sums for A1, Mg, Li, Na, K, In and Ag. Only for A1 is the collective mode
Table 1 Experimental (~o~(exp))and theoretical (ogs(theor)plasmon energies for several metals, as given in Ref. [ 14]; the work functions (W) are taken from Ref. [-13]; when the sum of the plasmon energy and the work function is smaller than the energy of the ground state of hydrogen (13.6 eV) the collectivetransition is allowed (energies are given in eV) Element
A1
Mg
Li
Na
K
In
Ag
ogs(exp)(eV) ~os(theor) (eV) W (eV) W + ~o~(exp)(eV) W + ~os(theor)(eV)
10.6 10.58 4.25 14.85 15.35
7.15 7.34 3.64 10.79 10.98
3.95 5.01 2.38 6.33 7.39
3.85 3.96 2.35 6.20 6.31
2.60 2.62 2.22 4.82 4.84
8.7 8.9 3.8 12.5 12.7
3.6 6.3 4.3 7.9 10.6
F.A. Gutierrez/Surface Science 370 (1997) 77-84
not allowed, due to the large surface-plasmon energy of this metal as a consequence of its high electron density. The perturbation of the atomic state by the surface and its effect on the Auger transition rates for proton on gold was analyzed by Snowdon et al. [1]. They found that the ionization energy Eo(s) for the perturbed hydrogen atom was quite close to the unperturbed value E0(oe) when s > 8 but got reduced for smaller distances. In particular Eo(1)=0.8Eo(oe)=0.4 (reduction of 20%). The decrease of the Auger transition rate by the perturbation was not negligible; however the most important effect is the appearance of a threshold distance at which the shifted atomic energy crosses the Fermi level to a region where the transition is forbidden, making the Auger transition rate vanish. For the plasmon mode (within the FIA) the threshold will appear when Eo(s)= W+og~. For gold the sum W+o~= 0.39 is still smaller than Eo(1)=0.4, so that the collective transition rate for this metal will be finite (although lower than for the unperturbed case) up to distances of the order of one bohr radius. It is important to note, however, that at this distance our simplified model for the collective transition might not be appropriate even if we consider the perturbation of the atomic wavefunctions since in this case the contribution from the bulk plasmons might not be negligible. The above considerations are also valid for the metals shown in Table 1 if the energy shifts are not too drastic. For instance for Li, Na and K it is clear that the threshold will be reached only if the ionization energy is reduced by more than 50% (compared to the 20% for gold). Finally we mention that even if the collective mode is closed for a proton at rest it becomes open for certain parallel ion velocities as seems to be the case for aluminum [3]. The transition rates were evaluated in the FIA (proton at rest) with the neutral fraction being obtained by integration over a classical ion trajectory. It would be rather desirable to have a calculation which avoids this approximation so that velocity effects are considered from the beginning, and at the same time one avoids the use of the classical ion trajectory. Consideration of the general Fock-Tani transformation [18] (instead of the simpler transformation for a proton at rest) and
83
proceeding along similar lines as in Ref. [7], one obtains the generalized matrix elements (cq qlHi~tIK, k)
x e iq'''l eiX'"Ok (r) --
fdR
d3r d3R' d3r'~b*(r, R)e-ql~l e-lq',H
× A(r- R, r' - R')eiX'R'C/)k (r')~, )
(20)
where R represents the proton's coordinates and K the corresponding wavevector. A short derivation of Eq. (20) is given in Ref. [ 19]. More details about the generalized matrix elements and their applications to ion-metal neutralization will be given in a forthcoming publication. Very recently Monreal and Lorente [ 11 ] developed a theory for Auger processes (Vion~0) at metal surfaces, in which the excitation of both electron-hole pairs and surface plasmons are taken into account through the dielectric surface response function of the many-electron system. They applied it to a simple model of the He + +A1 inelastic collision, which is similar to the one we consider here. They found that the contribution of surface plasmon excitation causes a much slower decrease of the Auger rate with increasing distance between the ion and surface than predicted by essentially non-screened calculations. Their findings are in qualitative agreement with our results in the sense that the Auger transition rate reported by Snowdon et al. [1] decays much faster with the ion-surface distance than the collective transition rate, in such a way that the total transition rate for both mechanism simultaneously should decay much slower than the Auger rate alone. In conclusion, the transition rates for the surfaceplasmon mode of proton neutralization on gold surfaces are larger than those for the Auger mode, while the neutralization fractions for both processes are similar. Therefore a correct calculation of neutralization fractions should consider not only
84
F.A. Gutierrez/Surface Science 370 (1997) 77-84
the r e s o n a n t a n d the Auger mode, b u t also the collective p l a s m o n c o n t r i b u t i o n .
Acknowledgements This work was s u p p o r t e d in part by F O N D E C Y T - C H I L E (project 1951167) a n d by Direcci6n de Investigaci6n de la U n i v e r s i d a d de Concepci6n. I a m very m u c h i n d e b t e d to M. Riquelme for his assistance with the n u m e r i c a l calculations a n d to N. Arista a n d V.H. P o n c e for fruitful discussions a n d k i n d hospitality while the a u t h o r was at the C e n t r o A t 6 m i c o de Bariloche, where the final part of this work was written.
References [1] K.J. Snowdon, R. Hentschke, A. N~irmann and W. Heiland, Surf. Sci. 173 (1986) 581. [2] K.J. Snowdon, R. Hentschke, A. N~rmann, W. Heiland, E. Muhling and W. Eckstein, Nucl. Instrum. Methods B 23 (1987) 309. [3] R. Zimny, Z.L. Miskovir, N.N. Nedeljkovi6 and Lj.D. Nedeljkovir, Surf. Sci. 255 (1991) 135, and references therein. [4] A.T. Amos, B.L. Burrows and S.G. Davison, Surf. Sci 277 (1992) MOO. [5] U. Wille, Nucl. Instrum. Methods B 79 (1993) 132; A.A. Almulhem, Surf. Sci. 304 (1994) 191; R. Souda, K. Yamamoto, W. Hayami, B. Tilley,T. Aizawa and Y. Ishizawa, Surf. Sci. 324 (1995) L349.
[6] N. Lorente and R. Monreal, Nucl. Instrum. Methods B 78 (1993) 44; Surf. Sci. 303 (1994) 253; Phys. Rev. A 49 (1994) 4716. [71 A.A. Almulhem and M.D. Girardeau, Surf. Sci. 210 (1989) 138. [8"1 R. Hentschke, K.J. Snowdon, P. Hertel and W. Heiland, Surf. Sci. 173 (1986) 565. [9] S. Horiguchi, K. Koyama and Y.H. Ohtsuki, Phys. Status Solidi B 87 (1978) 757. 110"1 H. Verbeek, W. Eckstein and R.S. Bhattacharya, Surf. Sci. 95 (1980) 380. [11] R. Monreal and N. Lorente, Phys. Rev. B 52 (1995) 4760. [12] J.W. Gadzuk, Surf. Sci. 6 (1967) 133. [13] N.D. Lang and W. Kohn, Phys. Rev. B 1 (1970) 4555. [14] J. Schmit and A.A. Lucas, Solid State Commun. 11 (1972) 415; 419; A.A. Lucas, Phys. Rev. B 20 (1979) 4990. A simple experimental demonstration of surface plasmons on gold (at senior undergraduate level) appears in A.S. Barker, Am. J. Phys. 42 (1974) 1123. Up-to-date references on surface plasmons appear in the two recent reviews by W. Plummer, K. Tsuei and B. Kim, Nucl. Instrum. Methods B 96 (1995) 448, and M. Rocca, Surf. Sci. Rep. 22 (1995) 71. [15] N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976). [16] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge University Press, New York, 1986). [17] H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer, Berlin, 1980) p. 139. [18] M.D. Girardeau, J. Math. Phys. 16 (1975) 1901; Phys. Rev. A 26 (1982) 217, and references therein. [19] F.A. Gutierrez and J. Diaz-Valdrs, Proc. 8th LatinAmerican Congr. Surface Science and its Applications, Cancun, Mexico, September 1994 (American Institute of Physics) to be published.
Surface Science 370 (1997) 77-84
Surface-plasmon mediated proton neutralization at metal surfaces F.A. G u t i e r r e z * Departamento de Fisica, Universidadde Concepci6n, Casilla 4009, Concepci6n, Chile Received 13 May 1996; accepted for publication 9 July 1996
Abstract Proton neutralization at gold surfaces by surface-plasmon emission is analyzed. Transition rates and neutralization fractions are evaluated within the fixed-ion approximation and compared with those for the usual Auger channel. We found that for protonsurface distances larger than one bohr radius the collective transition rates are larger than the Auger rates, while the neutralization fractions are very similar for both channels. Therefore in those situations in which the collective mode is energetically allowed, which is the case for several metals, it has to be considered simultaneously with the Auger and resonance tunneling modes.
Keywords: Auger transfer; Charge exchange at surfaces; Gold; Ion-surface scattering; Low energy ion scattering
I. Introduction
Resonance tunneling and Auger capture are generally considered to be the relevant channels for ion neutralization at metal surfaces [ 1-6]. For the static case and when these processes are analyzed separately Snowdon and co-workers [1,2] found that they are of comparable magnitude, with the conclusion that both mechanisms should be considered simultaneously for a correct quantitative description of ion neutralization. They also found that theoretical neutralization fractions for each mode are much lower than the experimental ones. Later, Zimny et al. [3] considered resonant and Auger electron capture on an equal footing, and also resonance and Auger ionization for proton neutralization after grazing reflection at an A1 surface within a rate-equation approach. They * Corresponding author. Fax: 56 41 240280; e-mail: [email protected]
found that for low parallel velocities vii, resonance tunneling seems to be predominant, while for high v,, Auger neutralization takes over. At intermediate velocities both channels appear to be of comparable importance. However, the discrepancy between theory and experiment still remains. Meanwhile Almulhem and Girardeau (AG) [7] proposed another mechanism of ion neutralization for the H(ls)-A1 system, whereby the energy released in the capture is taken away by a surface plasmon. This mechanism might be a good candidate to reduce the discrepancy between theory and experiments. Unfortunately the results of AG are incorrect, as indicated earlier by Zimny et al. [3], due to errors in the evaluation of the transition rates. In the first place the collective process is energetically forbidden for A1 when vii= 0 (the case considered by them), becoming possible only for vii larger than a certain threshold velocity. Secondly, their transition rates were expressed in terms of two (instead of four)-dimensional integrals
0039-6028/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0039-6028 ( 9 6 ) 00951-X
78
F.A. Gutierrez/Surface Science370 (1997) 77-84
based on the incorrect assumption that the matrix elements for the plasmonic neutralization channel depends only on the magnitude of q (plasmon wavevector) and kll (component of the electron wavevector parallel to the surface) but not on their orientation. Therefore the role of surface plasmons on proton neutralization at surfaces is unknown at the moment. Besides the H(ls)-A1 system, Snowdon and co-workers [1,2,8] (and earlier Horiguchi et al. [ 9]) considered Auger neutralization for H (ls)-Au, for which the experimental results of Verbeek et al. [10] also exist. As we shall see for gold, the plasmon mode of proton neutralization is energetically allowed even in the static case when vii=0. Thus by application of the orthogonalized Born approximation developed by AG [7], we obtain transition rates and neutralization fractions for H(ls)-Au and compare the results with those obtained by Snowdon et al. [1] for the Auger process for a similar model of the proton-metal collision. We found that for proton-surface distances beyond a bohr radius, transition rates for the collective mode are larger than those reported for the Auger capture, with the difference getting larger for increasing ion-surface distance. This result is in qualitative agreement with recent results by Monreal and Lorente [ 11 ], for He ÷ interacting with A1 surfaces, where the inclusion of dynamical screening (surface-plasmon excitation) enhances their Auger transition rate in such a way that it has a much slower decrease than their unscreened Auger rate with increasing ion-surface distance. We also found that the neutralization fractions for the collective channel and for the Auger channel are very similar. Therefore it appears that in those situations in which the surface-plasmon mode is allowed, one should include it simultaneously with the resonant tunneling and Auger channels for a correct description of proton neutralization on surfaces. Atomic units are used throughout this paper unless otherwise stated.
2. Theory The matrix elements for the plasmon-mediated neutralizing transition are given in the ortho-
gonalized Born approximation by [7]
(v'qlHintlk)= X/ qA/~°J~{fdar~,(r_S)~k(r)e_qlz I × e-lq'rll-- f d3r dar'~*(r-s) x A(r-
s, r'- s)qbk (r')e-ql~le - * ' ' t
'
(1) with the hydrogen-bound state kernel
A(r-s, r'-s)= ~v(r-s)~*(r'-s).
(2)
V
The metal is modeled by a jellium filling the halfspace z < 0 [1-3], the proton is taken at rest (at a distance s from the surface) during the evaluation of the transition rate in the so-called fixed ion approximation (FIA), with the neutral fraction being obtained by integration over a classical ion trajectory [ 1,2,9]. The first term within brackets in Eq.(1) represents the plane wave Born approximation for the transition rate while the second term takes into account the orthogonalization of the metal orbitals to the bound atomic states [7,11]. In this way spurious contributions from bound-bound channels are eliminated automatically. In Eq. (1), A is the surface area, q is the surface-plasmon wavevector with q = [ql
F.A. Gutierrez/Surface Science370 (1997) 77-84 and W is the corresponding work function, and with the energies measured from the bottom of the conduction band. Their explicit form is
79
where
f=q+kz-d,
g=q+k=+d,
(8)
d = ~/1 + (q--kll) 2, kz,2 = 2Vo-k~, 2
~k(r) =
and with
(eik,t"ll [(k'~+ ikz)e ik;z+ ( k'~-- ik,)e-~'z],
k: ,f4e -~k= e-~h'[ m2 = ~-7 [ ~ +[(l+hs)F-h] --~-j (3)
{
x e-~q--q[2(sl+l)+l 2] la j ,
~z>O,
(9)
where where (kz) ,2 =2Ek,t k2=2(Vo-E'k), k~-kz+(k~)2-2 , 2__ ' g 1L2 2Vo, and E k - - k--2-r~ll with kll the magnitude of kll, the component of k paralM to the surface. The step potential appears as a reasonable approximation for the surface potential of a metal in the presence of an ion as argued by Gadzuk [12]. In particular for Au there is good agreement between the electron density obtained from the selfconsistent local density functional calculation of Lang and Kohn [ 13] with that obtained with the step potential model [8]. The use of the unperturbed wavefunction for the ls state of hydrogen considered by AG and other authors [ 1,2,7],
1 x/~
F=h-kz,
G = h + k z , h=~/l +k~l, /= 4 ~
(10) In Eq.(5) the metal orbital is orthogonalized respect to the ground state of hydrogen, as in Ref. [11]. We have checked that contributions from the other terms of the orthogonalizing series (which are very cumbersome) contribute only a few percent for s > 1 the range we are interested in. We also note that the function m~ defined by (7) remains finite even when f vanishes for certain values of q, k= and kll. In fact when q + k= ~ d , so that f ~ 0 , one finds that ml ~ m °, with m° = ~
el,-sl,
(4)
makes it possible to integrate Eq. (1) in analytical form to obtain the matrix elements for plasmon capture into the ground state of hydrogen. The final expression is [-7]
(ls, qlHimlk)=
M-
/____e.M, "V qA
(5)
47z
(6)
~ (ml - m 2 ) , x/ zn V Vo
2.
+ ~ + ~
.
(11)
The transition rate for the plasmonic mode is P=2n
~
[(ls, qlHi,tlk)12O(cl-ey)
(12)
k,q < qc
with the initial energy ei = ½k2 and the final energy ey = E(ls) + ~o~, where E(ls) = Vo - ½ since the energies are measured (in atomic units) from the bottom of the conduction band. Changing the sums in Eq. (12) into integrals as usual and use of Eq. (5), leads to
P(s) = - 87~4 dk<-kF
d3k
d2q ~qe
with
k• {4e-; +k-)"
ml= ~
"
--
+[(l+ds)f-d]
e-~a'[
d3 j ,
x ~ qA (7)
~r).
(13)
Consider now cylindrical coordinates for the
80
F.A. Gutierrez/Surface Science 370 (1997) 77-84
0.05
I O0 -% N
4
0,04
0
q=0.93
.~ ('~ 0.03 ~ '
A
=0.75 =0.52
x
3
\ \
.2
\
\
1 0 -1
(,9
\
,,
4
,,. ,,.
(3_
(3_ 0,02
10-'3
o
4
2
6
8
10
,
0.0
2
2,5
,5.0
electron wavevector k =(kll, (Pk, kz) and polar coordinates for the plasmon wavevector q=(q, ~oq)so that
ws P(s) = 8~ 3 J 0
z
10.0
12,5
Fig. 2. N e u t r a l i z a t i o n rates for a p r o t o n o n gold for the collective p l a s m o n m o d e for qc= 0.93. The d a s h e d line corres p o n d to the n o n - o r t h o g o n a l i z e d case in w h i c h only the first
term in Eq.(6) is considered. The solid line contains the orthogonalization corrections contained in the second term of Eq. (6).
which after integration gives
IM(kz, ~Ok;q, ~q)[20(½k2-e:)O(e:-½k2),
f f c dq
fo k~4"i~-~ dkllklL IMl2~(½kll2
_[_ ~k~: 1 2 -
e:).
(14)
The change of variable u = ½k~ and the introduction of the step function O(z) to account for the integration limits allows us to rewrite the last integral in Eq. (14) as
f:oou
(16)
with the condition ~1 k 2l p - e f - 2 k1 2z. Therefore the transition rate is:
P(s) = O(½k2 - e:)Po (s), x
15,0
S
S Fig. 1. Neutralization rates for a proton on gold. The solid lines correspond to the three cases for the collective plasmon mode (qc= 0.93, 0.75 and 0.52), while the dashed lines are those from Ref. I-1] for the Auger transition with the labels corresponding to the value of the screening parameter 2.
7.5
1
(17)
with
×
f?;?
X 2 kll2
d~ok
= e: -- ½k 2 .
d@~lM(kz, ~Ok;q, ~0q)12; (18)
The expression we obtain for the transition rate
P(s) differs from that of Ref. [7] (their Eq. (47)) in several aspects: (i) we obtain the extra factor O(½k2v--e:) which takes into account the conservation of the energy (ei=½k2=e:) together with the constraint k
F..A. Gutierrez/ Surface Science 370 (1997) 77-84
81
1.0 C" O +--
o o L,. LL. C
0.8 '%\
x N
0.6
o O N
o_
0.4
U L. q)
Z
0.2
0.0
1 0-2
" , , ,, _ ""
q =0.93 =0.75 =0.52
0 •~ x I
34
I
I
I
I
",x,,, -. "" -
llI
I
1 0 -1
2
I
34
I
I
I
I I I ]
100
I
I
2
34
I
I
I
I I
1 01
Energy (keY) Fig. 3. Neutralization fractions for a proton on gold. The solid lines correspond to the three cases for the collective plasmon mode (qc=0.93, 0.75 and 0.52) while the dashed lines are those from Ref. I-l] for the Auger transition with the labels corresponding to the value of the screening parameter 2. The line labeled "e" represents the experimental results of Verbeek et al. [10].
they considered) one has [7] Vo=0.5862, ~os= 0.4106 and kF=0.9261. Then ey=E(ls)+og~= Vo-½+co~=0.4968 and ½ k 2 - e I < O so that P(s) vanishes for any s. Thus the transition rates of AG for H(ls)-A1 are incorrect, a fact already noted by Zimny et al. [3]. (ii) they assumed that the matrix elements M in Eq.(18) do not depend on the angles {Ok and (O~ to reduce the four-dimensional integral to a two-dimensional one. That M depends on the angles (0k and (O+ is clear from our Eqs.(6)-(8) (see also eqs. (A10) and (A15) in Ref. [7]). (iii) The range of integration for the variable kz is in our case [0, 2 ~ ¢ ] instead of their larger range [0, kF] which is incorrect, since for k, > 2 ~ y , kit would take complex values. Finally the neutral fraction f o for a straight-line constant-velocity classical ion trajectory with specular reflection is [1,2,9] f o = 1- e x p l
o.(S)fo+P(s)ds] 2
(19)
where v±(s) is the constant component of proton velocity perpendicular to the surface and P(s) is
the neutralization rate, both being functions of the surface proton distance s.
3. Results
The surface-plasmon energy can be evaluated from the well known expression o~s=c%/x/2 with c% = x/4nne2/m the bulk-plasmon frequency, where n is the electron density and m the electron mass. For the cut-off plasmon wavevector qc a rigorous expression does not exist.. To check the importance of the cut-off wavevector in the evaluation of the collective transition rates we shall consider a range of values, which go from the simple estimate [14] qc "~ COp/VF(VFis the Fermi velocity) up to qc ,~ 1.5kv. Therefore the values of the relevant parameters for gold are [ 15] (in atomic units) ~s = 0.2349, O~p/VV= 0.52, Vo= 0.38 and kF =0.62. The calculations were performed using standard Gaussian integration methods with different number of points. These calculations were supplemented with the accelerating Aitken's 52-process [ 16].
F.A. Gutierrez/ Surface Science370 (1997) 77-84
82
energy (related to the perpendicular velocity v±) are plotted in Fig. 3 for qc=0.52, 0.75 and 0.93. We also include the Auger neutralization fractions of Snowdon et al. [1] (for 2=0.45 and 0.90) together with the experimental results of Verbeek et al. [ 10]. The three curves for collective neutralization fractions which go close to each other fall between the two curves for the Auger mode being much closer to that with 2 = 0.45. For energies of the order of or larger than 1 keV all the theoretical curves are significantly lower than the experimental one.
We first note that the collective process is energetically allowed for the H ( l s ) - A u system. In fact e y = E ( l s ) + ~ o s = Vo-½ + cos=0.1149 and 1 2 :kF--e I >0, so that the transition rates P(s) given by Eq.(17) do not vanish. They are shown in Fig. 1 for the three cases qc=0.52, 0.75 and 0.93. For purposes of comparison we also show the Auger neutralization rates reported by Snowdon et al. [ 1 ] for two values of their screening parameter 2 for the electron-electron interaction. We find that: (i) the collective transition rates have a maxim u m at s,~l, after which they decay with the distance (for qc = 0.93 at s = 5 the transition rate is one order of magnitude smaller than its m a x i m u m value). (ii) Larger values of qc lead to larger transition rates; the strongest differences appear within the range s < 1.5 where the transition rate for q~= 0.93 is more than twice that for q~ = 0.52. However, beyond the point at which they reach their maxim u m value the transition rates for larger q~ decay faster than those for small q~ so that they eventually merge at some s. In fact for s > 5 the differences are negligible. (iii) For s > 1 the plasmonic transition rates for the three values of q¢ are larger than the Auger transition rates for the two values of 2. Since the latter decay much faster than the former, the differences increase for larger s in such a way that for s > 4 the Auger transition rates become negligible compared to those for the collective mode. In Fig. 2 we show that the effect of the orthogonalization correction for the case q~ = 0.93 is to reduce the plasmon-mediated transition rate by one order of magnitude or more. This situation also occurs for q~ = 0.75 and 0.52. Finally the plasmon-mediated neutralization fractions as functions of the projectile kinetic
4. Discussion and conclusions We have applied the orthogonalized Born matrix elements for surface-plasmon mediated ion neutralization on metal surfaces [7] to the case of a proton on gold and compared the results with those for the Auger mode reported by Snowdon et al. [1]. The calculations for both mechanisms consider a semi-infinite jellium metal and the steppotential model for the surface, which lead to electron densities in good agreement with those from the self-consistent local density functional calculations of Lang and K o h n [ 13] for gold. For the unperturbed atom case, the condition for the collective transition (to the ground state of the incoming ion) to be allowed ½ k ~ - e ; > O can be rewritten as ½> W+Ogs. Raether [17] has tabulated the values of surface-plasmon energies (theoretical and experimental) for several metals. These are shown in Table 1, together with the work functions [15] and their sums for A1, Mg, Li, Na, K, In and Ag. Only for A1 is the collective mode
Table 1 Experimental (~o~(exp))and theoretical (ogs(theor)plasmon energies for several metals, as given in Ref. [ 14]; the work functions (W) are taken from Ref. [-13]; when the sum of the plasmon energy and the work function is smaller than the energy of the ground state of hydrogen (13.6 eV) the collectivetransition is allowed (energies are given in eV) Element
A1
Mg
Li
Na
K
In
Ag
ogs(exp)(eV) ~os(theor) (eV) W (eV) W + ~o~(exp)(eV) W + ~os(theor)(eV)
10.6 10.58 4.25 14.85 15.35
7.15 7.34 3.64 10.79 10.98
3.95 5.01 2.38 6.33 7.39
3.85 3.96 2.35 6.20 6.31
2.60 2.62 2.22 4.82 4.84
8.7 8.9 3.8 12.5 12.7
3.6 6.3 4.3 7.9 10.6
F.A. Gutierrez/Surface Science 370 (1997) 77-84
not allowed, due to the large surface-plasmon energy of this metal as a consequence of its high electron density. The perturbation of the atomic state by the surface and its effect on the Auger transition rates for proton on gold was analyzed by Snowdon et al. [1]. They found that the ionization energy Eo(s) for the perturbed hydrogen atom was quite close to the unperturbed value E0(oe) when s > 8 but got reduced for smaller distances. In particular Eo(1)=0.8Eo(oe)=0.4 (reduction of 20%). The decrease of the Auger transition rate by the perturbation was not negligible; however the most important effect is the appearance of a threshold distance at which the shifted atomic energy crosses the Fermi level to a region where the transition is forbidden, making the Auger transition rate vanish. For the plasmon mode (within the FIA) the threshold will appear when Eo(s)= W+og~. For gold the sum W+o~= 0.39 is still smaller than Eo(1)=0.4, so that the collective transition rate for this metal will be finite (although lower than for the unperturbed case) up to distances of the order of one bohr radius. It is important to note, however, that at this distance our simplified model for the collective transition might not be appropriate even if we consider the perturbation of the atomic wavefunctions since in this case the contribution from the bulk plasmons might not be negligible. The above considerations are also valid for the metals shown in Table 1 if the energy shifts are not too drastic. For instance for Li, Na and K it is clear that the threshold will be reached only if the ionization energy is reduced by more than 50% (compared to the 20% for gold). Finally we mention that even if the collective mode is closed for a proton at rest it becomes open for certain parallel ion velocities as seems to be the case for aluminum [3]. The transition rates were evaluated in the FIA (proton at rest) with the neutral fraction being obtained by integration over a classical ion trajectory. It would be rather desirable to have a calculation which avoids this approximation so that velocity effects are considered from the beginning, and at the same time one avoids the use of the classical ion trajectory. Consideration of the general Fock-Tani transformation [18] (instead of the simpler transformation for a proton at rest) and
83
proceeding along similar lines as in Ref. [7], one obtains the generalized matrix elements (cq qlHi~tIK, k)
x e iq'''l eiX'"Ok (r) --
fdR
d3r d3R' d3r'~b*(r, R)e-ql~l e-lq',H
× A(r- R, r' - R')eiX'R'C/)k (r')~, )
(20)
where R represents the proton's coordinates and K the corresponding wavevector. A short derivation of Eq. (20) is given in Ref. [ 19]. More details about the generalized matrix elements and their applications to ion-metal neutralization will be given in a forthcoming publication. Very recently Monreal and Lorente [ 11 ] developed a theory for Auger processes (Vion~0) at metal surfaces, in which the excitation of both electron-hole pairs and surface plasmons are taken into account through the dielectric surface response function of the many-electron system. They applied it to a simple model of the He + +A1 inelastic collision, which is similar to the one we consider here. They found that the contribution of surface plasmon excitation causes a much slower decrease of the Auger rate with increasing distance between the ion and surface than predicted by essentially non-screened calculations. Their findings are in qualitative agreement with our results in the sense that the Auger transition rate reported by Snowdon et al. [1] decays much faster with the ion-surface distance than the collective transition rate, in such a way that the total transition rate for both mechanism simultaneously should decay much slower than the Auger rate alone. In conclusion, the transition rates for the surfaceplasmon mode of proton neutralization on gold surfaces are larger than those for the Auger mode, while the neutralization fractions for both processes are similar. Therefore a correct calculation of neutralization fractions should consider not only
84
F.A. Gutierrez/Surface Science 370 (1997) 77-84
the r e s o n a n t a n d the Auger mode, b u t also the collective p l a s m o n c o n t r i b u t i o n .
Acknowledgements This work was s u p p o r t e d in part by F O N D E C Y T - C H I L E (project 1951167) a n d by Direcci6n de Investigaci6n de la U n i v e r s i d a d de Concepci6n. I a m very m u c h i n d e b t e d to M. Riquelme for his assistance with the n u m e r i c a l calculations a n d to N. Arista a n d V.H. P o n c e for fruitful discussions a n d k i n d hospitality while the a u t h o r was at the C e n t r o A t 6 m i c o de Bariloche, where the final part of this work was written.
References [1] K.J. Snowdon, R. Hentschke, A. N~irmann and W. Heiland, Surf. Sci. 173 (1986) 581. [2] K.J. Snowdon, R. Hentschke, A. N~rmann, W. Heiland, E. Muhling and W. Eckstein, Nucl. Instrum. Methods B 23 (1987) 309. [3] R. Zimny, Z.L. Miskovir, N.N. Nedeljkovi6 and Lj.D. Nedeljkovir, Surf. Sci. 255 (1991) 135, and references therein. [4] A.T. Amos, B.L. Burrows and S.G. Davison, Surf. Sci 277 (1992) MOO. [5] U. Wille, Nucl. Instrum. Methods B 79 (1993) 132; A.A. Almulhem, Surf. Sci. 304 (1994) 191; R. Souda, K. Yamamoto, W. Hayami, B. Tilley,T. Aizawa and Y. Ishizawa, Surf. Sci. 324 (1995) L349.
[6] N. Lorente and R. Monreal, Nucl. Instrum. Methods B 78 (1993) 44; Surf. Sci. 303 (1994) 253; Phys. Rev. A 49 (1994) 4716. [71 A.A. Almulhem and M.D. Girardeau, Surf. Sci. 210 (1989) 138. [8"1 R. Hentschke, K.J. Snowdon, P. Hertel and W. Heiland, Surf. Sci. 173 (1986) 565. [9] S. Horiguchi, K. Koyama and Y.H. Ohtsuki, Phys. Status Solidi B 87 (1978) 757. 110"1 H. Verbeek, W. Eckstein and R.S. Bhattacharya, Surf. Sci. 95 (1980) 380. [11] R. Monreal and N. Lorente, Phys. Rev. B 52 (1995) 4760. [12] J.W. Gadzuk, Surf. Sci. 6 (1967) 133. [13] N.D. Lang and W. Kohn, Phys. Rev. B 1 (1970) 4555. [14] J. Schmit and A.A. Lucas, Solid State Commun. 11 (1972) 415; 419; A.A. Lucas, Phys. Rev. B 20 (1979) 4990. A simple experimental demonstration of surface plasmons on gold (at senior undergraduate level) appears in A.S. Barker, Am. J. Phys. 42 (1974) 1123. Up-to-date references on surface plasmons appear in the two recent reviews by W. Plummer, K. Tsuei and B. Kim, Nucl. Instrum. Methods B 96 (1995) 448, and M. Rocca, Surf. Sci. Rep. 22 (1995) 71. [15] N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976). [16] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge University Press, New York, 1986). [17] H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer, Berlin, 1980) p. 139. [18] M.D. Girardeau, J. Math. Phys. 16 (1975) 1901; Phys. Rev. A 26 (1982) 217, and references therein. [19] F.A. Gutierrez and J. Diaz-Valdrs, Proc. 8th LatinAmerican Congr. Surface Science and its Applications, Cancun, Mexico, September 1994 (American Institute of Physics) to be published.