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A simple theoretical potential for low-energy ion-surface interaction
Surface Science 185 (1987) 269-282 North-Holland, Amsterdam
269
A SIMPLE THEORETICAL POTENTIAL FOR LOW-ENERGY I O N - S U R F A C E INTERACTION
K. MANN, V. CELLI * and J. Peter T O E N N I E S Max-Planck-lnstitut fi~r Str6mungsforschung~ Bunsenstrasse 10, D-3400 GOttingen, Fed. Rep. of Germany Received 24 September 1986, accepted for publication 16 January 1987
We propose a simple way to construct the interaction potential between a low-energy ion and the surface of a monatomic solid, using a sum of pairwise potentials of the Tang-Toennies type. All the parameters of these potentials, except one, are determined by atomic calculations and by the requirement that the asymptotic form of the interaction agrees with the classical image potential. The one adjustable parameter gives the strength of the repulsive part of the potential, assumed to b e proportional to the charge density of the atoms in the solid. The K+-W(100) potential constructed in this way is found to agree with the empirical potential used by Hulpke and Mann to fit their scattering data.
I. Introduction
In their analysis of low-energy (5-50 eV) K + scattering from W surfaces [1,2] Hulpke and Mann use an interaction potential that is almost entirely empirical, for lack of a practical way to construct such a potential from a priori calculations or from other data. Empirical potentials require the adjustment of an undetermined number of parameters, which is time-consuming and leaves open the question of the uniqueness of the fit. For these reasons, as well as in order to progress towards an understanding of ion-surface interactions, we propose here a simple construction that seems to work well for the K + - W case. We consider explicitly the case of monatomic metal surfaces only, but the results are easily extended to monatomic insulators upon multiplying the attractive part of the potential by 2E/(c + 1), where E is the static dielectric constant of the solid. At large distances z, the attraction between a charged particle with charge e and a metal surface is well described by the macroscopic image potential Vim =
-e2/4z.
(1)
The image charge picture, however, loses the validity when the particle * Permanent address: Department of Physics, University of Virginia, Charlottesville, VA 22901, USA.
0039-6028/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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K. Mann et al. / Theoretical potential for low energy ion-surface interaction
approaches to within a few hngstrSms of the surface [3-8]. The validity of eq. (1) can be improved by suitably defining the position of the image plane, i.e. the origin of the coordinate z [3]; physically, however, the singularity at z = 0 must be avoided and a modification of eq. (1) is necessary. In a fast particle-surface collision there is an additional "dynamical" weakening of the attractive interaction since the surface cannot respond fast enough to the rapidly varying field of the particle to allow formation of the full image charge. These dynamical effects have been extensively discussed, for instance, by Sunjic [8] who argues, on the basis of a model where the surface response is dominated by surface plasmons, that the interaction (1) should be multiplied by a factor 1 - exp(-2k~z), where the cutoff wave vector k~ depends on the particle's velocity. We will ignore this correction since dynamical effects are negligible for K + with 10 eV energy. To find a reasonable approximation, we write the particle surface interaction as a superposition of pairwise interactions between the particle and the atoms of the crystal. The z = 0 singularity in eq. (1) is removed by including in the pair interaction a damping function, which has been found to work well for atom-atom interactions [9]. The procedure is described in section 2. At short distances, the other important contribution to the surface interaction comes from the overlap of the electronic charge densities. For a compact, closed shell ion such as K +, it is a reasonable approximation to assume that this overlap interaction is purely repulsive and is simply proportional to the charge density of the surface, according to the prescription that has gained acceptance since the work of Esbjerg and Norskov [10]. Further, we compute the surface charge density profile as a superposition of atomic charge densities, following another practical prescription that is gaining acceptance [11]. All this is done in section 3, and the total potential is computed for the system K+-W(100), which motivated this investigation. The experiments on the scattering of K § ions off W(100) and W(ll0) are described elsewhere [1,2]. The important point for the present discussion is that the rainbow scattering pattern and the energy loss spectra could be explained by classical trajectory calculations, using an empirical potential that we will call the "best fit potential" in this paper. We than compare this best fit potential with the one calculated in section 3, which we call the "semi ab initio potential" because it still contains one adjustable constant, namely the coefficient of proportionality B between the repulsive interaction potential and the surface charge density. As discussed in section 4, the comparison is very encouraging. 2. Representation of the attractive interaction as a sum of pair potentials
We show first that the asymptotic form (1) can be correctly obtained by a sum of pairwise interactions between the charged particle and the atoms of the
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
271
L • Fig. 1. Coordinate system for the summation of pairwise potentials.
crystal that have the induced-dipole form -C4/r 4. This simple mathematical procedure cannot be completely correct since in a metal, such as W, some of the electrons are free and do not belong to polarizable atoms, or atom cores, However, the correct strength of the image potential (1), is obtained by chosing C4 so that there is no E-field in the metal, as discussed after eq. (11) below. Using this C4, the damped induction potential is uniquely obtained as a sum of pair potentials. 2.1. Lattice sums of the image potential The pairwise sum of induced dipole potentials over the crystal atoms that fill the half space z < 0 yields Vind = E E -- C4 p z I r p - R l [ 4"
(2)
Here rp is the distance of the charge from one lattice point in the p t h plane and R t are the two-dimensional lattice vectors of the surface (cf. fig. 1). We assume a monatomic solid for simplicity. The sum over surface lattice vectors R l can be conveniently computed for any potential of the form e x p ( - fl I r - R z I ) / I r - g t l " by transforming it to a sum over the reciprocal lattice vectors G: W , ( r , 13) = ~] e x p ( - / 3 [ r - R t l) = E e x p ( i G . R ) U ~ ( G , z, /3), l Ir - R t l " G
(3)
with g n ( G , z, /3)
a2F,~Lnn1)fB
"~ d u ( R - / 3 )
n 2 e x p ( - z v / G 2 + u 2 ), V/-G2 _{._u 2
(4)
where (R, z) are the components of r and a is the lattice constant of the bcc lattice (a = 3.16 A for W(100)).
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K. Mann et at,. / Theoretical potential for low energy ion-surface interaction
This general formula will be needed later. For the moment we put n = 4, /3 = 0, which gives 7r
~
U4 = ~-~ s
duu 2 exp(-z~-U+u ~/a 2 + u 2
2) = ~~r f o ~ d V V / ~ _ G 2 e x p ( _ 2 v )
(5) For G = 0 the integral is elementary; for G ~ 0 it gives a modified Bessel function of order 1: U4 =
[ Tr/a2z2, G = O, ~ (~rG/a2z)Kl(Gz),
G r 0.
(6)
Insertion of eq. (3) gives for a square surface lattice (e.g. W(100)): W4(X , y, Z ) = E
-C4
t [rp-Rz[ 4
a2
-~ + za
+ za
lt~1
(
1~
cos a
cos
+cos
a
+COS
a
a
+8~K[4~rZ][cos4~rx ~a a~---~--]~ a + c o s 4 tar y ) + . . . .]
(7)
The first term on the right for G = 0 depends only on z and provides the dominant contribution for z > 2 ,~, the other terms are decreasing exponentially with z. According to eq. (2), one must now carry out the summation over lattice planes. For the (100) surface of a bcc lattice, for instance, Find =
~ [W4(x+pa, y , z ) + W 4 ( x + 8 9
y+ 89
z+pa+ 89
(8)
p=0
Convergence occurs after summation over about 100 lattice planes for the G = 0 term, over two or three planes for the terms corresponding to the smallest G vectors, while higher G vectors may be neglected. For z > 2 A the G = 0 term is already two orders of magnitude greater than the others, and it is given by: Vind (G
=
O)
qTC4 a2
(Z + 89
-2 --
p=O
4~rC4 a2
2z + P p=O
a
(9)
The sum in eq. (9) is identical with the series expansion of the trigamma function ~b' [12]: Vind(G = 0 )
- 4 a4qT"C4 ~ ' ( ~ )
z >>a ~
2qTC4 / 1 + a +
a Sz \
-~z
)
....
(10)
K. Mann et aL / Theoreticalpotentialfor low energy ion-surface interaction
273
The leading asymptotic term in eq. (10) is obtained m o r e simply b y converting the sum in eq. (9) into an integral, and the first correction can be obtained b y the E u l e r - M c L a u r i n formula. It is customary to regard this first correction to 1 / z as an effective shift z 0 in the origin of the z coordinate, i.e. 1 / z is replaced b y 1 / ( z - %). I n the present case we find z 0 = a / 4 = 0.79 A. This value is smaller than that c o m p u t e d by L a n g and K o h n [3] in the jellium model: in their language, a / 4 is the position of the jellium edge, and their c o m p u t e d z 0 lies 1.6 a.u. (0.85 A) b e y o n d the jellium edge. By comparing eqs. (1) and (10), we can evaluate the effective induction constant Ca: C4 --
a3 e2 2 4~" - 18.08 eV
.~4.
(11)
The dipole m o m e n t induced on the W atoms b y the external charge is - C 4 / e r 2 and the polarization P is obtained b y dividing b y the bcc atomic volumes Vat = a3/2. Thus P = - e / 4 ~ r r Z = D / 4 ~ r , where we have identified the Maxwell displacement D with the external electric field - e / r 2. Since D = E + 4~rP, we have E = 0, which is the correct constitutive equation for a metal. This argument also shows that for any m o n a t o m i c metal we must have C4 =
Vate2/4~ ".
(12)
Plots of Vi, d according to eq. (8) for the (100) surface of W are given in fig. 2, b o t h for the on top and the four-fold hollow site of the bcc lattice. The asymptotic behavior - e 2 / 4 z is also shown for comparison. 0.0
-2.0
-4
.o
> @J
/;
9
//
-6,o
I
.....
>
(no damping)
Vin d
- - - - v indd (damping included) -8o
-10.0 oo
i i i t f i t
........... continuum integration -macroscopic image potential i
20
i
i
i
4.0
L
60
i
i
8.0
i
10,0
z[~,]
Fig. 2. Representations of the image potential obtained by summation of induced dipole interactions. For both F i n d (section 2.1) and Viand(section 2.2) the lower curve corresponds to the on top position (x = y = 0), the upper curve to the four-fold hollow site (x = y = la) of the bcc lattice, respectively. Also shown is the result of a continuum integration (eq. (27)) and the macroscopic image potential - e2/4z.
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K. Mann et al. / Theoretical potential for low energy ion-surface interaction
As discussed above, V~,d is found to be shifted against the macroscopic image potential to larger z values. The corrugation of V~n~ becomes effective for z _< 3 A, the attraction being stronger for the "on top" rather than for the "hollow" position.
2.2. The damped induction potential As discussed in the introduction, the singularity of the attractive potential at z = 0 is actually not present because the induction force weakens when an external charge penetrates the charge distribution of a surface atom. The situation for the atom-ion induction interaction is more complicated than simple electrostatics would suggest, because of the overlap of the ion's wave function with the atom's wave functions. For the dispersion interaction (r -6) the "damping" of the attraction has been shown to be well described by the Tang-Toennies formula [9], which contains as only (known) parameter the hardness 7 of the Born-Mayer repulsive potential between the two atoms. The same arguments can be appfied to any r -n potentials. In analogy to eq. (2), the damped induction potential is written
Vi~d = E Wd(rp, y),
(13)
P
Wd(rp, y ) = ~
[1 - e x p ( - y x , ) l +(y x , + - ~ x 2 +,2~ . x ~ + ~,3. x ~ ,4 )1, (14)
where x l = [ rp - RI[. The sum over one plane can be written as Wd(x, y, z, 7) = - C 4 [ S ( 0 ) - S ( y ) ] ,
(15)
with
(
,2
,3
.,/4)
S ( , ) = Y"t eXP(x4- yx?) 1 + 7x, + -~--x2 + -~-.x 3 + ~ . x 4 .
(16)
From eq. (3) it follows that
(17)
s ( v ) = ~ e x p ( i G . R ) O ( G , z, , ) ,
G
with
,2 O ( a , Z, , ) = U4 "l- , U3 -}- T
,3
,4
U2 -}- -~. UI -~ -~. Uo ,
(18)
and U~ = U~(G, z, , ) given by eq. (4) for n > 2. For any n and y > 0, U~_I(G, z, 7 ) = -0U~(G, z, y)/OT,
(19)
275
K. Mann et a L / Theoretical potential for low energy ion-sutface interaction
and thus in particular /31 = a 2
~G~ + y2
(20)
'
I q- z~-G~ q- y 2
Uo = ~,u~
(21)
G 2 + .i,2
Using eq. (18), we find
0(r
2~r f~du exp(-zCG~ + u2) [ ( u- 7)2
~, ~,)= - 7 4
,/~+ ~
./3
---7-
+~,(.-~,) +
Z_2]
y4
q'- ~ I U1 Jr- ~ UO,
(22)
from which we can compute S('/), eq. (17). Subtracting S(0) according to eq. (15), we find finally
(23)
Wa(r, v) = -C4~exp(ia.R)Ua(G, z, V), G
with
Ud(e, z~
du uz ,~3 •4 ~[exp(-z~-GS+ua)]--~. U,-~-Uo.
y) = ~ v
(24)
The remaining integral is elementary for G = 0 and is computed numerically for G:#0. As in the case of the undamped potential, eq. (2), it is sufficient to extend the sum over G over the first few members only. The explicit result for the (100) surface of a bcc lattice is
wd(x,y,z,v)=-c4
ud(0, z , ~ ) + 2 u a - a- , z, YI[COS a
(2V~"/r)[ 2, "g
+2Ud - - ,a + 2 Ua
, z, y
COS
2~r(x+y)2~r(x-y) + COS a
cos a
+ cos a
+cosa
+ ....
a ]
(25)
The sum over lattice planes is also carried out analogously to eq. (8), and the terms G =# 0 contribute less than 3% of the sum for z > 2 A in the case of W(100). The explicit form of the laterally averaged induction potential, to be compared with eq. (9), is V;nd(G__0 ) d __
~rC4 ~ a2
1 r l _ e x p r _t T z , r )[ ~ l
' r 2] 2 z- t ~p1y Z+p33 )
p~O Zp
(26)
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
276
where zp = z + 89 Viand has been calculated using the value 2.38 ~ - 1 for the hardness parameter 7 (cf. section 3). It is compared in fig. 2. with the behavior of the undamped potential Vina for the "on top" position (x =y = 0) and for the "hollow" position ( x = y = 89 As in the case of Vina, the damped induction potential is much stronger for the " o n top" rather than for the "hollow" site. The difference is about 16% for z = 1 A. In order to see the effect of damping in comparison with eq. (1) we have integrated the Tang-Toennies potential over the half space by converting the sum in eq. (26) into an integral. As a result we obtain the function Vd = _ 2 ~ r C 4 [ ~ _ e x p ( _ y z ) ( 1 .y ~2 )] a3 z + 2 + -~-z .
(27)
which is also plotted in fig. 2. Here again the error made in replacing the sum by an integral can be corrected for large z by replacing 1/z with 1/(z - Zo), with z 0 = 88 in the present case. The constant C4 is of course the same as in eq. (11). Calculations were also carried out to take account of the additional dispersion potential (van der Waals forces) with Tang-Toennies damping, by using a London formula to estimate the constant C6 for the K + - W system and summing pairwise interactions. The result is a z -3 potential at large z, with damping at small z. In the range of interest, however, the amplitude of this potential is negligible: even at z = 1 A it is about 50 times smaller than the induction potential. 3. Representation of the repulsive interaction as a sum of pair potentials
We use the Esbjerg-Norskov [10] prescription, that the repulsive potential is proportional to the charge density of the surface, p(r)" V~ep = Bp(r).
(28)
Recent work by Takada and Kohn [13] shows that this prescription is asymptotically correct at large z for the laterally averaged (G = 0) part of Vrep, but tends to overestimate the corrugation of the surface profile. Takada and Kohn's theory is formulated for the H e - m e t a l surface interaction, and in that case it gives a way of computing the proportionality constant B in eq. (28) from the continuation to negative energies of the e - H e scattering phase shifts. Unfortunately, this theory has not been extended to ion-metal surface interactions. In particular, such an approach would have the advantage of allowing the explicit calculation of B, which we here take as an adjustable parameter. To proceed, we compute the surface charge density p(r) as a superposition of the charge densities p w ( r ) of individual W atoms: p(r) = ~Pw(lrp-Rt[ p
l
).
(29)
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
pw[a.u.]
t
[
I
i
277
i
10 .2
10 -3
lO-Z.
10-5
10-6
10-7
I
I
I
I
I
I
2
3
':
5
6
7
r[~] Fig. 3. Semi-log plot of the charge density pw(r) of a single W atom, calculated from Hermann-Skillman wave functions [14].
We have used H e r m a n n - S k i l l m a n wave functions [14] to calculate taw(r), which is plotted in fig. 3. To a good approximation we find taw(r) = A e x p ( - y r ) ,
(30)
with V = 2.38 ~,-] and A = 0.432 a.u. With the approximations (28) and (29), and using the same procedure as in section 2, we obtain for the surface lattice geometry of W(100): Vrep = A B
~ [W0(x , y, z + pa, p=0
V) +
Wo(X +
89 y + 89 z + pa + 89 y ) ] ,
(31) with W0 from eq. (3) (for n = 0):
Wo(r, fl)= ~exp(iG.R)Uo(G, z, 7).
(32)
G
Uo(G, z, 7) is given by eq. (21) and the leading terms of the sum over G are as in eq. (25) (with Ud ~ U0). For practical reasons, however, we do not pursue this formal evaluation of Vrep but rather calculate the sum in eq. (29) numerically. Due to the pure
278
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
exponential decay of the atomic charge density 0 w (r) (eq. (30)) convergence is reached very rapidly: Starting with the atom closest to the projection of the ion the surface plane, it is sufficient to include the neighbouring atoms up to the third next neighbours only. Their contribution to the total charge density is already about three orders of magnitude smaller than that of the surface closest to the external charge. 4. Comparison with an empirical potential In order to evaluate the proportionality constant B of eq. (28) we compare the actual shape of o(r) (eq. (29)) in the z direction with the potential derived empirically from low-energy ion scattering data. In the experiment [1,2], angular and energy distributions have been measured for K + ions scattered from W(110) and W(100) surfaces in the energy range from 5 to 50 eV. The rainbow structures observed in the angular distributions and the energy loss spectra could be well reproduced by classical trajectory calculations, assuming a pairwise sum of Born-Mayer potentials with adjustable range T and strength C (corresponding to the product AB, see section 3), plus an attractive part given by a simple analytical formula with two adjustable parameters. The formula for the attractive part was chosen arbitrarily to account qualitatively for the damping of the image potential at small z [1]. It should be noted that in the ad hoc model the attractive potential was assumed to be only z-dependent, whereas the ab initio potential Vind (section 2.2) exhibits a corrugation (of. fig. 2). In fig. 4 the repulsive part of this "best fit" potential is plotted for the " o n top" position of the W(100) lattice, and p(z) from section 3 is scaled to it at a distance of z = 2 A, yielding very good agreement in the shape of the two curves. From the scaling procedure the proportionality constant is found to be B = 894 eV/a.u, which is in reasonable agreement with theoretical values obtained by various authors for the repulsive interaction of rare gas atoms with noble metal surfaces [10]. With this value of B and V'mdof section 2.2 we can construct the total "semi ab initio" potential by adding repulsive and attractive contributions: Vtot =ABe_, Y~,e x p ( - Y l ~ - R, I) + vidd 9 p
(33)
l
In fig. 5 we compare eq. (33) with the best fit potential obtained from rainbow scattering for the "on top" position. The overall agreement is quite satisfying, except in the intermediate region of the well, This is, however, not very surprising, since it has been shown from the computer simulation that rainbow scattering is only very weakly dependent on the shape of the potential well. Thus with a slightly enhanced attractive part the scattering data could also be reproduced accurately by the trajectory calculations [1]. Such a "modified"
K. Mann et aL / Theoretical potential for low energy ion-surface interaction
279
V[eVl 12
1
10
8 6
2 0 -2
i
I
i
Fig. 4. Repulsive part of the experimentally derived best fit potential for the system K+-W(100) (dashed curve) and surface charge density p from eq. (29), scaled to the empirical result [1] at z = 2 ,~ (dash-dotted curve).
best fit potential for which calculations provided equally good agreement with experiment is shown in fig. 5, both for the "on top" and the four-fold "hollow" site of the W(100) lattice. Especially for the "on top" position we obtain very good agreement now with the result of our semi ab initio calculation. Note that the long-range V[eV]
i
t
i
12 10
8 i
i~on-top 0 "2
hollow ~
I
Fig. 5. Comparison between total calculated potential (dashed curves) and total empirical potential for the system K+-W(100); dotted curve: original best fit potential from rainbow scattering; solid curves: modified best fit potential as discussed in the text, for " o n top" and "hollow" site of the bcc lattice.
280
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
attractive regions, the well depths and the positions of the minima are accurately reproduced. It should also be mentioned that the calculated potential exhibits approximately the right corrugation, i.e. the distance in the z direction between the "on top" and the "hollow" curve for fixed value of V. Rainbow scattering is rather sensitive to this quantity, and from the given experiment the corrugation should be most accurately represented in the region from 5 to 10 eV [1]. 5. Discussion
The attractive image potential between a point charge and a single crystal surface is of great importance for a better theoretical understanding of a number of experiments in surface analysis, such as (inverse) photoelectron spectroscopy, SIMS and low energy ion scattering (LEIS). There is, however, still some doubt about the exact shape of this potential at small distances z from the surface, where the simple classical picture must fail. In this paper we demonstrate that the image potential can be regarded as the summation of pairwise attractions between the external charge and the dipoles in the surface induced by this charge. If we assume a Tang-Toennies damping function for the induction potentials, this summation leads to the representation of Fin d in section 2.2, which satisfies the requirements for the image potential near a surface: -- gin d i-s found to be stronger than the classical potential in the region z >__-]a, corresponding to an effective shift of the image plane away from the surface plane, as calculated by Lang and Kohn [3]. - According with other theoretical approaches [5-8], the singularity for z ~ 0 has been avoided. Qualitatively, this behaviour is in good agreement with a result obtained by Manson and Ritchie [5], who calculate the complex self energy of a charged particle in front of the surface. Moreover, our formalism leads to another interesting feature, i.e. the corrugation of the image potential at small z, as discussed in section 2.2. Such a corrugation has been observed experimentally from the examination of image states in inverse photoemission spectra [15]: Even for a flat surface like Ag(100) the corrugation effect is substantial in the interpretation of the energetic levels associated with these image states. The corrugation of the attractive potential calculated here has, of course, an influence on the corrugation of the total potential, which is primarily governed by the "repulsive corrugation". In fact, since the attraction is found to be stronger for an "on top" site of the lattice rather than for the "hollow" site (cf. fig. 2), the total corrugation is effectively reduced. The shape of the total "semi ab initio" potential is in good agreement with the empirical best fit potential obtained from scattering experiments, as
1'2. Mann et al. / Theoretical potential for low energy ion-surface interaction
281
discussed in section 4. The small deviations between calculated and empirical potential, especially in the "hollow" curves, can be attributed to the approximations made in the calculation (e.g. neglect of the effect of the free metal electrons in the summation of V~nd) as well as to imperfections in the model potential used in the computer simulation. Although the agreement is good with the scattering potential the present model predicts binding energies E B for K § on W surfaces that are almost 1 eV smaller than experimental values determined from thermal desorption measurements. For the W(110) surface the desorption experiments give E B = 2.05(_+0.02) eV [16] while the present "semi ab initio" model potential suitably modified for the different surface geometry yields E B = 1.25 eV at R m - ~ 3.4 A. For W(100) the desorption value is E B = 2.25(_+0.05) eV while the model potential predicts E B --1.2 eV at R m = 3.2 A for the " o n top" position (cf. fig. 5) and E B = 1.3 eV at R m = 2.7 A for the "hollow" position which is assumed to be the adsorption site. At the present time it is not clear why the scattering and desorption potentials are so different. The resolution of this discrepancy goes beyond the scope of this paper. Thus we can only conclude that the present model seems to indicate that the former is consistent with the assumptions of the present model. There is on the other hand some evidence that the model would predict the desorption energies much better by accounting for the free electrons in the metal. This evidence comes from the jellium model of Lang and K o h n [3] which treats the electrons as free and predicts gin d
~
- -
e 2 / 4 ( z - z 0).
(34)
Here z 0 is the position of the image plane, computed to be 1.6 a.u. beyond the jellium edge for W. As already mentioned after eq. (10) the pairwise sum of attractive potentials is well approximated by the same formula with z 0 at half the interplanar spacing (i.e. at the jellium edge). Thus the additional shift of 1.6 a.u. can be looked upon as an effect resulting from the free electrons. For K+-W(110) using the shifted image plane Lang and K o h n estimate R m = 3.5 from the experimental value of the dipole moment at low coverage [17] in good agreement with our value. Their binding potential of E B = 2.2 eV at z = R m i s , however, in much better agreement with the desorption value. It is interesting to note that if z 0 is shifted to 1.12 A, which is just the location of the jellium edge, K o h n and Lang predict E B = 1.4 eV much closer to our result. This value is decreased by only about 0.2 eV w h e n the full G = 0 semi ab initio potential is used. The same shift in z o to larger values also brings E B closer to the desorption energy for the W(100) surface, however, in this case too much binding occurs at the hollow site if no damping of Vand is included, as can seen from fig. 2. Finally we note that another way in which to improve the agreement with the desorption energies would be to add quadrupolar and higher order
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K. Mann et al. / Theoretical potential for low energy ion-surface interaction
inductions terms. This, however, does not seem justified at the present time because of the just discussed uncertainties in the electron screening problem. A summation scheme similar to the one predicted in this paper has been successfully applied also to determine the semi ab initio potential for the atom-surface interaction, primarily for the system He-LiF(001) [11]. Diffraction probabilities, well depths and inelastic one-phonon reflection coefficients, calculated from this potential have been shown to reproduce experimental results very well [18]. Good agreement in all three experimental quantities is also found in the case of the C u ( l l l ) and A g ( l l l ) surfaces [19], where the delocalized s-electrons are expected to have a more significant effect than in tungsten.
Acknowledgements We thank A.A. Maradudin for help with the transformations of the lattice sums and J.R. Manson for a critical reading of the manuscript. Several discussions with E. Hulpke are gratefully acknowledged.
References [1] E. Hulpke and K. Mann, Surface Sci. 133 (1983) 171; K. Mann, Dissertation, Max-Planck-Institut fiir Str~Smungsforschung, Bericht 10 (1984). [2] E. Hulpke and K. Mann, Surface Sci. 157 (1985) 245. [3] N.D. Lang and W. Kohn, Phys. Rev. B7 (1973) 3541. [4] D.M. Newns, J. Chem. Phys. 50 (1969) 4572. [5] J.R. Manson and R.W. Ritchie, Phys. Rev. B24 (1981) 4867. [6] J. Harris and R.O. Jones, J. Phys. C6 (1973) 3585. [7] R.F. Kayser and J.B. Hubbard, J. Chem. Phys. 77 (1982) 4704. [8] M. Sunjic, in: Dynamics of Gas-Surface Interaction, Eds. G. Benedek and U. Valbusa (Springer, Berlin, 1982). [9] K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726; to be published. [10] See, for instance, M. Manninen, J.K. Norskov, M.J. Puska and C. Umrigar, Phys. Rev. B29 (1984) 2319; J. Harris and A. Liebsch, J. Phys. C15 (1982) 2775. [11] For a review, see I.P. Batra, Surface Sci. 148 (1984) 1; for a recent application, see V. Celli, D. Eichenauer, A. Kaufhold and J.P. Toennies, J. Chem. Phys. 83 (1985) 2504. [12] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964). [13] Y. Takada and W. Kohn, Phys. Rev. Letters 54 (1985) 470; 55 (1985) 141. [14] F. Hermann and S. SldUman, Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs, NJ, 1963). [15] N. Garcia, B. ReiN, K.H. Frank and A.R. Williams, Phys. Rev. Letters 54 (1985) 591. [16] A. Hurkmans, E.G. Overbosch and J. Los, Surface Sci. 59 (1976) 488. [17] L.D. Schmidt and R. Gomer, J. Chem. Phys. 45 (1966) 1605. [18] D. Eichenauer and J.P. Toennies, J. Chem. Phys. 85 (1986) 532. [191 D. Eichenauer, V. Harten, J.P. Toennies and V. Celli, J. Chem. Phys. 86 (1987).
269
A SIMPLE THEORETICAL POTENTIAL FOR LOW-ENERGY I O N - S U R F A C E INTERACTION
K. MANN, V. CELLI * and J. Peter T O E N N I E S Max-Planck-lnstitut fi~r Str6mungsforschung~ Bunsenstrasse 10, D-3400 GOttingen, Fed. Rep. of Germany Received 24 September 1986, accepted for publication 16 January 1987
We propose a simple way to construct the interaction potential between a low-energy ion and the surface of a monatomic solid, using a sum of pairwise potentials of the Tang-Toennies type. All the parameters of these potentials, except one, are determined by atomic calculations and by the requirement that the asymptotic form of the interaction agrees with the classical image potential. The one adjustable parameter gives the strength of the repulsive part of the potential, assumed to b e proportional to the charge density of the atoms in the solid. The K+-W(100) potential constructed in this way is found to agree with the empirical potential used by Hulpke and Mann to fit their scattering data.
I. Introduction
In their analysis of low-energy (5-50 eV) K + scattering from W surfaces [1,2] Hulpke and Mann use an interaction potential that is almost entirely empirical, for lack of a practical way to construct such a potential from a priori calculations or from other data. Empirical potentials require the adjustment of an undetermined number of parameters, which is time-consuming and leaves open the question of the uniqueness of the fit. For these reasons, as well as in order to progress towards an understanding of ion-surface interactions, we propose here a simple construction that seems to work well for the K + - W case. We consider explicitly the case of monatomic metal surfaces only, but the results are easily extended to monatomic insulators upon multiplying the attractive part of the potential by 2E/(c + 1), where E is the static dielectric constant of the solid. At large distances z, the attraction between a charged particle with charge e and a metal surface is well described by the macroscopic image potential Vim =
-e2/4z.
(1)
The image charge picture, however, loses the validity when the particle * Permanent address: Department of Physics, University of Virginia, Charlottesville, VA 22901, USA.
0039-6028/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
270
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
approaches to within a few hngstrSms of the surface [3-8]. The validity of eq. (1) can be improved by suitably defining the position of the image plane, i.e. the origin of the coordinate z [3]; physically, however, the singularity at z = 0 must be avoided and a modification of eq. (1) is necessary. In a fast particle-surface collision there is an additional "dynamical" weakening of the attractive interaction since the surface cannot respond fast enough to the rapidly varying field of the particle to allow formation of the full image charge. These dynamical effects have been extensively discussed, for instance, by Sunjic [8] who argues, on the basis of a model where the surface response is dominated by surface plasmons, that the interaction (1) should be multiplied by a factor 1 - exp(-2k~z), where the cutoff wave vector k~ depends on the particle's velocity. We will ignore this correction since dynamical effects are negligible for K + with 10 eV energy. To find a reasonable approximation, we write the particle surface interaction as a superposition of pairwise interactions between the particle and the atoms of the crystal. The z = 0 singularity in eq. (1) is removed by including in the pair interaction a damping function, which has been found to work well for atom-atom interactions [9]. The procedure is described in section 2. At short distances, the other important contribution to the surface interaction comes from the overlap of the electronic charge densities. For a compact, closed shell ion such as K +, it is a reasonable approximation to assume that this overlap interaction is purely repulsive and is simply proportional to the charge density of the surface, according to the prescription that has gained acceptance since the work of Esbjerg and Norskov [10]. Further, we compute the surface charge density profile as a superposition of atomic charge densities, following another practical prescription that is gaining acceptance [11]. All this is done in section 3, and the total potential is computed for the system K+-W(100), which motivated this investigation. The experiments on the scattering of K § ions off W(100) and W(ll0) are described elsewhere [1,2]. The important point for the present discussion is that the rainbow scattering pattern and the energy loss spectra could be explained by classical trajectory calculations, using an empirical potential that we will call the "best fit potential" in this paper. We than compare this best fit potential with the one calculated in section 3, which we call the "semi ab initio potential" because it still contains one adjustable constant, namely the coefficient of proportionality B between the repulsive interaction potential and the surface charge density. As discussed in section 4, the comparison is very encouraging. 2. Representation of the attractive interaction as a sum of pair potentials
We show first that the asymptotic form (1) can be correctly obtained by a sum of pairwise interactions between the charged particle and the atoms of the
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
271
L • Fig. 1. Coordinate system for the summation of pairwise potentials.
crystal that have the induced-dipole form -C4/r 4. This simple mathematical procedure cannot be completely correct since in a metal, such as W, some of the electrons are free and do not belong to polarizable atoms, or atom cores, However, the correct strength of the image potential (1), is obtained by chosing C4 so that there is no E-field in the metal, as discussed after eq. (11) below. Using this C4, the damped induction potential is uniquely obtained as a sum of pair potentials. 2.1. Lattice sums of the image potential The pairwise sum of induced dipole potentials over the crystal atoms that fill the half space z < 0 yields Vind = E E -- C4 p z I r p - R l [ 4"
(2)
Here rp is the distance of the charge from one lattice point in the p t h plane and R t are the two-dimensional lattice vectors of the surface (cf. fig. 1). We assume a monatomic solid for simplicity. The sum over surface lattice vectors R l can be conveniently computed for any potential of the form e x p ( - fl I r - R z I ) / I r - g t l " by transforming it to a sum over the reciprocal lattice vectors G: W , ( r , 13) = ~] e x p ( - / 3 [ r - R t l) = E e x p ( i G . R ) U ~ ( G , z, /3), l Ir - R t l " G
(3)
with g n ( G , z, /3)
a2F,~Lnn1)fB
"~ d u ( R - / 3 )
n 2 e x p ( - z v / G 2 + u 2 ), V/-G2 _{._u 2
(4)
where (R, z) are the components of r and a is the lattice constant of the bcc lattice (a = 3.16 A for W(100)).
272
K. Mann et at,. / Theoretical potential for low energy ion-surface interaction
This general formula will be needed later. For the moment we put n = 4, /3 = 0, which gives 7r
~
U4 = ~-~ s
duu 2 exp(-z~-U+u ~/a 2 + u 2
2) = ~~r f o ~ d V V / ~ _ G 2 e x p ( _ 2 v )
(5) For G = 0 the integral is elementary; for G ~ 0 it gives a modified Bessel function of order 1: U4 =
[ Tr/a2z2, G = O, ~ (~rG/a2z)Kl(Gz),
G r 0.
(6)
Insertion of eq. (3) gives for a square surface lattice (e.g. W(100)): W4(X , y, Z ) = E
-C4
t [rp-Rz[ 4
a2
-~ + za
+ za
lt~1
(
1~
cos a
cos
+cos
a
+COS
a
a
+8~K[4~rZ][cos4~rx ~a a~---~--]~ a + c o s 4 tar y ) + . . . .]
(7)
The first term on the right for G = 0 depends only on z and provides the dominant contribution for z > 2 ,~, the other terms are decreasing exponentially with z. According to eq. (2), one must now carry out the summation over lattice planes. For the (100) surface of a bcc lattice, for instance, Find =
~ [W4(x+pa, y , z ) + W 4 ( x + 8 9
y+ 89
z+pa+ 89
(8)
p=0
Convergence occurs after summation over about 100 lattice planes for the G = 0 term, over two or three planes for the terms corresponding to the smallest G vectors, while higher G vectors may be neglected. For z > 2 A the G = 0 term is already two orders of magnitude greater than the others, and it is given by: Vind (G
=
O)
qTC4 a2
(Z + 89
-2 --
p=O
4~rC4 a2
2z + P p=O
a
(9)
The sum in eq. (9) is identical with the series expansion of the trigamma function ~b' [12]: Vind(G = 0 )
- 4 a4qT"C4 ~ ' ( ~ )
z >>a ~
2qTC4 / 1 + a +
a Sz \
-~z
)
....
(10)
K. Mann et aL / Theoreticalpotentialfor low energy ion-surface interaction
273
The leading asymptotic term in eq. (10) is obtained m o r e simply b y converting the sum in eq. (9) into an integral, and the first correction can be obtained b y the E u l e r - M c L a u r i n formula. It is customary to regard this first correction to 1 / z as an effective shift z 0 in the origin of the z coordinate, i.e. 1 / z is replaced b y 1 / ( z - %). I n the present case we find z 0 = a / 4 = 0.79 A. This value is smaller than that c o m p u t e d by L a n g and K o h n [3] in the jellium model: in their language, a / 4 is the position of the jellium edge, and their c o m p u t e d z 0 lies 1.6 a.u. (0.85 A) b e y o n d the jellium edge. By comparing eqs. (1) and (10), we can evaluate the effective induction constant Ca: C4 --
a3 e2 2 4~" - 18.08 eV
.~4.
(11)
The dipole m o m e n t induced on the W atoms b y the external charge is - C 4 / e r 2 and the polarization P is obtained b y dividing b y the bcc atomic volumes Vat = a3/2. Thus P = - e / 4 ~ r r Z = D / 4 ~ r , where we have identified the Maxwell displacement D with the external electric field - e / r 2. Since D = E + 4~rP, we have E = 0, which is the correct constitutive equation for a metal. This argument also shows that for any m o n a t o m i c metal we must have C4 =
Vate2/4~ ".
(12)
Plots of Vi, d according to eq. (8) for the (100) surface of W are given in fig. 2, b o t h for the on top and the four-fold hollow site of the bcc lattice. The asymptotic behavior - e 2 / 4 z is also shown for comparison. 0.0
-2.0
-4
.o
> @J
/;
9
//
-6,o
I
.....
>
(no damping)
Vin d
- - - - v indd (damping included) -8o
-10.0 oo
i i i t f i t
........... continuum integration -macroscopic image potential i
20
i
i
i
4.0
L
60
i
i
8.0
i
10,0
z[~,]
Fig. 2. Representations of the image potential obtained by summation of induced dipole interactions. For both F i n d (section 2.1) and Viand(section 2.2) the lower curve corresponds to the on top position (x = y = 0), the upper curve to the four-fold hollow site (x = y = la) of the bcc lattice, respectively. Also shown is the result of a continuum integration (eq. (27)) and the macroscopic image potential - e2/4z.
274
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
As discussed above, V~,d is found to be shifted against the macroscopic image potential to larger z values. The corrugation of V~n~ becomes effective for z _< 3 A, the attraction being stronger for the "on top" rather than for the "hollow" position.
2.2. The damped induction potential As discussed in the introduction, the singularity of the attractive potential at z = 0 is actually not present because the induction force weakens when an external charge penetrates the charge distribution of a surface atom. The situation for the atom-ion induction interaction is more complicated than simple electrostatics would suggest, because of the overlap of the ion's wave function with the atom's wave functions. For the dispersion interaction (r -6) the "damping" of the attraction has been shown to be well described by the Tang-Toennies formula [9], which contains as only (known) parameter the hardness 7 of the Born-Mayer repulsive potential between the two atoms. The same arguments can be appfied to any r -n potentials. In analogy to eq. (2), the damped induction potential is written
Vi~d = E Wd(rp, y),
(13)
P
Wd(rp, y ) = ~
[1 - e x p ( - y x , ) l +(y x , + - ~ x 2 +,2~ . x ~ + ~,3. x ~ ,4 )1, (14)
where x l = [ rp - RI[. The sum over one plane can be written as Wd(x, y, z, 7) = - C 4 [ S ( 0 ) - S ( y ) ] ,
(15)
with
(
,2
,3
.,/4)
S ( , ) = Y"t eXP(x4- yx?) 1 + 7x, + -~--x2 + -~-.x 3 + ~ . x 4 .
(16)
From eq. (3) it follows that
(17)
s ( v ) = ~ e x p ( i G . R ) O ( G , z, , ) ,
G
with
,2 O ( a , Z, , ) = U4 "l- , U3 -}- T
,3
,4
U2 -}- -~. UI -~ -~. Uo ,
(18)
and U~ = U~(G, z, , ) given by eq. (4) for n > 2. For any n and y > 0, U~_I(G, z, 7 ) = -0U~(G, z, y)/OT,
(19)
275
K. Mann et a L / Theoretical potential for low energy ion-sutface interaction
and thus in particular /31 = a 2
~G~ + y2
(20)
'
I q- z~-G~ q- y 2
Uo = ~,u~
(21)
G 2 + .i,2
Using eq. (18), we find
0(r
2~r f~du exp(-zCG~ + u2) [ ( u- 7)2
~, ~,)= - 7 4
,/~+ ~
./3
---7-
+~,(.-~,) +
Z_2]
y4
q'- ~ I U1 Jr- ~ UO,
(22)
from which we can compute S('/), eq. (17). Subtracting S(0) according to eq. (15), we find finally
(23)
Wa(r, v) = -C4~exp(ia.R)Ua(G, z, V), G
with
Ud(e, z~
du uz ,~3 •4 ~[exp(-z~-GS+ua)]--~. U,-~-Uo.
y) = ~ v
(24)
The remaining integral is elementary for G = 0 and is computed numerically for G:#0. As in the case of the undamped potential, eq. (2), it is sufficient to extend the sum over G over the first few members only. The explicit result for the (100) surface of a bcc lattice is
wd(x,y,z,v)=-c4
ud(0, z , ~ ) + 2 u a - a- , z, YI[COS a
(2V~"/r)[ 2, "g
+2Ud - - ,a + 2 Ua
, z, y
COS
2~r(x+y)2~r(x-y) + COS a
cos a
+ cos a
+cosa
+ ....
a ]
(25)
The sum over lattice planes is also carried out analogously to eq. (8), and the terms G =# 0 contribute less than 3% of the sum for z > 2 A in the case of W(100). The explicit form of the laterally averaged induction potential, to be compared with eq. (9), is V;nd(G__0 ) d __
~rC4 ~ a2
1 r l _ e x p r _t T z , r )[ ~ l
' r 2] 2 z- t ~p1y Z+p33 )
p~O Zp
(26)
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
276
where zp = z + 89 Viand has been calculated using the value 2.38 ~ - 1 for the hardness parameter 7 (cf. section 3). It is compared in fig. 2. with the behavior of the undamped potential Vina for the "on top" position (x =y = 0) and for the "hollow" position ( x = y = 89 As in the case of Vina, the damped induction potential is much stronger for the " o n top" rather than for the "hollow" site. The difference is about 16% for z = 1 A. In order to see the effect of damping in comparison with eq. (1) we have integrated the Tang-Toennies potential over the half space by converting the sum in eq. (26) into an integral. As a result we obtain the function Vd = _ 2 ~ r C 4 [ ~ _ e x p ( _ y z ) ( 1 .y ~2 )] a3 z + 2 + -~-z .
(27)
which is also plotted in fig. 2. Here again the error made in replacing the sum by an integral can be corrected for large z by replacing 1/z with 1/(z - Zo), with z 0 = 88 in the present case. The constant C4 is of course the same as in eq. (11). Calculations were also carried out to take account of the additional dispersion potential (van der Waals forces) with Tang-Toennies damping, by using a London formula to estimate the constant C6 for the K + - W system and summing pairwise interactions. The result is a z -3 potential at large z, with damping at small z. In the range of interest, however, the amplitude of this potential is negligible: even at z = 1 A it is about 50 times smaller than the induction potential. 3. Representation of the repulsive interaction as a sum of pair potentials
We use the Esbjerg-Norskov [10] prescription, that the repulsive potential is proportional to the charge density of the surface, p(r)" V~ep = Bp(r).
(28)
Recent work by Takada and Kohn [13] shows that this prescription is asymptotically correct at large z for the laterally averaged (G = 0) part of Vrep, but tends to overestimate the corrugation of the surface profile. Takada and Kohn's theory is formulated for the H e - m e t a l surface interaction, and in that case it gives a way of computing the proportionality constant B in eq. (28) from the continuation to negative energies of the e - H e scattering phase shifts. Unfortunately, this theory has not been extended to ion-metal surface interactions. In particular, such an approach would have the advantage of allowing the explicit calculation of B, which we here take as an adjustable parameter. To proceed, we compute the surface charge density p(r) as a superposition of the charge densities p w ( r ) of individual W atoms: p(r) = ~Pw(lrp-Rt[ p
l
).
(29)
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
pw[a.u.]
t
[
I
i
277
i
10 .2
10 -3
lO-Z.
10-5
10-6
10-7
I
I
I
I
I
I
2
3
':
5
6
7
r[~] Fig. 3. Semi-log plot of the charge density pw(r) of a single W atom, calculated from Hermann-Skillman wave functions [14].
We have used H e r m a n n - S k i l l m a n wave functions [14] to calculate taw(r), which is plotted in fig. 3. To a good approximation we find taw(r) = A e x p ( - y r ) ,
(30)
with V = 2.38 ~,-] and A = 0.432 a.u. With the approximations (28) and (29), and using the same procedure as in section 2, we obtain for the surface lattice geometry of W(100): Vrep = A B
~ [W0(x , y, z + pa, p=0
V) +
Wo(X +
89 y + 89 z + pa + 89 y ) ] ,
(31) with W0 from eq. (3) (for n = 0):
Wo(r, fl)= ~exp(iG.R)Uo(G, z, 7).
(32)
G
Uo(G, z, 7) is given by eq. (21) and the leading terms of the sum over G are as in eq. (25) (with Ud ~ U0). For practical reasons, however, we do not pursue this formal evaluation of Vrep but rather calculate the sum in eq. (29) numerically. Due to the pure
278
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
exponential decay of the atomic charge density 0 w (r) (eq. (30)) convergence is reached very rapidly: Starting with the atom closest to the projection of the ion the surface plane, it is sufficient to include the neighbouring atoms up to the third next neighbours only. Their contribution to the total charge density is already about three orders of magnitude smaller than that of the surface closest to the external charge. 4. Comparison with an empirical potential In order to evaluate the proportionality constant B of eq. (28) we compare the actual shape of o(r) (eq. (29)) in the z direction with the potential derived empirically from low-energy ion scattering data. In the experiment [1,2], angular and energy distributions have been measured for K + ions scattered from W(110) and W(100) surfaces in the energy range from 5 to 50 eV. The rainbow structures observed in the angular distributions and the energy loss spectra could be well reproduced by classical trajectory calculations, assuming a pairwise sum of Born-Mayer potentials with adjustable range T and strength C (corresponding to the product AB, see section 3), plus an attractive part given by a simple analytical formula with two adjustable parameters. The formula for the attractive part was chosen arbitrarily to account qualitatively for the damping of the image potential at small z [1]. It should be noted that in the ad hoc model the attractive potential was assumed to be only z-dependent, whereas the ab initio potential Vind (section 2.2) exhibits a corrugation (of. fig. 2). In fig. 4 the repulsive part of this "best fit" potential is plotted for the " o n top" position of the W(100) lattice, and p(z) from section 3 is scaled to it at a distance of z = 2 A, yielding very good agreement in the shape of the two curves. From the scaling procedure the proportionality constant is found to be B = 894 eV/a.u, which is in reasonable agreement with theoretical values obtained by various authors for the repulsive interaction of rare gas atoms with noble metal surfaces [10]. With this value of B and V'mdof section 2.2 we can construct the total "semi ab initio" potential by adding repulsive and attractive contributions: Vtot =ABe_, Y~,e x p ( - Y l ~ - R, I) + vidd 9 p
(33)
l
In fig. 5 we compare eq. (33) with the best fit potential obtained from rainbow scattering for the "on top" position. The overall agreement is quite satisfying, except in the intermediate region of the well, This is, however, not very surprising, since it has been shown from the computer simulation that rainbow scattering is only very weakly dependent on the shape of the potential well. Thus with a slightly enhanced attractive part the scattering data could also be reproduced accurately by the trajectory calculations [1]. Such a "modified"
K. Mann et aL / Theoretical potential for low energy ion-surface interaction
279
V[eVl 12
1
10
8 6
2 0 -2
i
I
i
Fig. 4. Repulsive part of the experimentally derived best fit potential for the system K+-W(100) (dashed curve) and surface charge density p from eq. (29), scaled to the empirical result [1] at z = 2 ,~ (dash-dotted curve).
best fit potential for which calculations provided equally good agreement with experiment is shown in fig. 5, both for the "on top" and the four-fold "hollow" site of the W(100) lattice. Especially for the "on top" position we obtain very good agreement now with the result of our semi ab initio calculation. Note that the long-range V[eV]
i
t
i
12 10
8 i
i~on-top 0 "2
hollow ~
I
Fig. 5. Comparison between total calculated potential (dashed curves) and total empirical potential for the system K+-W(100); dotted curve: original best fit potential from rainbow scattering; solid curves: modified best fit potential as discussed in the text, for " o n top" and "hollow" site of the bcc lattice.
280
K. Mann et al. / Theoretical potential for low energy ion-surface interaction
attractive regions, the well depths and the positions of the minima are accurately reproduced. It should also be mentioned that the calculated potential exhibits approximately the right corrugation, i.e. the distance in the z direction between the "on top" and the "hollow" curve for fixed value of V. Rainbow scattering is rather sensitive to this quantity, and from the given experiment the corrugation should be most accurately represented in the region from 5 to 10 eV [1]. 5. Discussion
The attractive image potential between a point charge and a single crystal surface is of great importance for a better theoretical understanding of a number of experiments in surface analysis, such as (inverse) photoelectron spectroscopy, SIMS and low energy ion scattering (LEIS). There is, however, still some doubt about the exact shape of this potential at small distances z from the surface, where the simple classical picture must fail. In this paper we demonstrate that the image potential can be regarded as the summation of pairwise attractions between the external charge and the dipoles in the surface induced by this charge. If we assume a Tang-Toennies damping function for the induction potentials, this summation leads to the representation of Fin d in section 2.2, which satisfies the requirements for the image potential near a surface: -- gin d i-s found to be stronger than the classical potential in the region z >__-]a, corresponding to an effective shift of the image plane away from the surface plane, as calculated by Lang and Kohn [3]. - According with other theoretical approaches [5-8], the singularity for z ~ 0 has been avoided. Qualitatively, this behaviour is in good agreement with a result obtained by Manson and Ritchie [5], who calculate the complex self energy of a charged particle in front of the surface. Moreover, our formalism leads to another interesting feature, i.e. the corrugation of the image potential at small z, as discussed in section 2.2. Such a corrugation has been observed experimentally from the examination of image states in inverse photoemission spectra [15]: Even for a flat surface like Ag(100) the corrugation effect is substantial in the interpretation of the energetic levels associated with these image states. The corrugation of the attractive potential calculated here has, of course, an influence on the corrugation of the total potential, which is primarily governed by the "repulsive corrugation". In fact, since the attraction is found to be stronger for an "on top" site of the lattice rather than for the "hollow" site (cf. fig. 2), the total corrugation is effectively reduced. The shape of the total "semi ab initio" potential is in good agreement with the empirical best fit potential obtained from scattering experiments, as
1'2. Mann et al. / Theoretical potential for low energy ion-surface interaction
281
discussed in section 4. The small deviations between calculated and empirical potential, especially in the "hollow" curves, can be attributed to the approximations made in the calculation (e.g. neglect of the effect of the free metal electrons in the summation of V~nd) as well as to imperfections in the model potential used in the computer simulation. Although the agreement is good with the scattering potential the present model predicts binding energies E B for K § on W surfaces that are almost 1 eV smaller than experimental values determined from thermal desorption measurements. For the W(110) surface the desorption experiments give E B = 2.05(_+0.02) eV [16] while the present "semi ab initio" model potential suitably modified for the different surface geometry yields E B = 1.25 eV at R m - ~ 3.4 A. For W(100) the desorption value is E B = 2.25(_+0.05) eV while the model potential predicts E B --1.2 eV at R m = 3.2 A for the " o n top" position (cf. fig. 5) and E B = 1.3 eV at R m = 2.7 A for the "hollow" position which is assumed to be the adsorption site. At the present time it is not clear why the scattering and desorption potentials are so different. The resolution of this discrepancy goes beyond the scope of this paper. Thus we can only conclude that the present model seems to indicate that the former is consistent with the assumptions of the present model. There is on the other hand some evidence that the model would predict the desorption energies much better by accounting for the free electrons in the metal. This evidence comes from the jellium model of Lang and K o h n [3] which treats the electrons as free and predicts gin d
~
- -
e 2 / 4 ( z - z 0).
(34)
Here z 0 is the position of the image plane, computed to be 1.6 a.u. beyond the jellium edge for W. As already mentioned after eq. (10) the pairwise sum of attractive potentials is well approximated by the same formula with z 0 at half the interplanar spacing (i.e. at the jellium edge). Thus the additional shift of 1.6 a.u. can be looked upon as an effect resulting from the free electrons. For K+-W(110) using the shifted image plane Lang and K o h n estimate R m = 3.5 from the experimental value of the dipole moment at low coverage [17] in good agreement with our value. Their binding potential of E B = 2.2 eV at z = R m i s , however, in much better agreement with the desorption value. It is interesting to note that if z 0 is shifted to 1.12 A, which is just the location of the jellium edge, K o h n and Lang predict E B = 1.4 eV much closer to our result. This value is decreased by only about 0.2 eV w h e n the full G = 0 semi ab initio potential is used. The same shift in z o to larger values also brings E B closer to the desorption energy for the W(100) surface, however, in this case too much binding occurs at the hollow site if no damping of Vand is included, as can seen from fig. 2. Finally we note that another way in which to improve the agreement with the desorption energies would be to add quadrupolar and higher order
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inductions terms. This, however, does not seem justified at the present time because of the just discussed uncertainties in the electron screening problem. A summation scheme similar to the one predicted in this paper has been successfully applied also to determine the semi ab initio potential for the atom-surface interaction, primarily for the system He-LiF(001) [11]. Diffraction probabilities, well depths and inelastic one-phonon reflection coefficients, calculated from this potential have been shown to reproduce experimental results very well [18]. Good agreement in all three experimental quantities is also found in the case of the C u ( l l l ) and A g ( l l l ) surfaces [19], where the delocalized s-electrons are expected to have a more significant effect than in tungsten.
Acknowledgements We thank A.A. Maradudin for help with the transformations of the lattice sums and J.R. Manson for a critical reading of the manuscript. Several discussions with E. Hulpke are gratefully acknowledged.
References [1] E. Hulpke and K. Mann, Surface Sci. 133 (1983) 171; K. Mann, Dissertation, Max-Planck-Institut fiir Str~Smungsforschung, Bericht 10 (1984). [2] E. Hulpke and K. Mann, Surface Sci. 157 (1985) 245. [3] N.D. Lang and W. Kohn, Phys. Rev. B7 (1973) 3541. [4] D.M. Newns, J. Chem. Phys. 50 (1969) 4572. [5] J.R. Manson and R.W. Ritchie, Phys. Rev. B24 (1981) 4867. [6] J. Harris and R.O. Jones, J. Phys. C6 (1973) 3585. [7] R.F. Kayser and J.B. Hubbard, J. Chem. Phys. 77 (1982) 4704. [8] M. Sunjic, in: Dynamics of Gas-Surface Interaction, Eds. G. Benedek and U. Valbusa (Springer, Berlin, 1982). [9] K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726; to be published. [10] See, for instance, M. Manninen, J.K. Norskov, M.J. Puska and C. Umrigar, Phys. Rev. B29 (1984) 2319; J. Harris and A. Liebsch, J. Phys. C15 (1982) 2775. [11] For a review, see I.P. Batra, Surface Sci. 148 (1984) 1; for a recent application, see V. Celli, D. Eichenauer, A. Kaufhold and J.P. Toennies, J. Chem. Phys. 83 (1985) 2504. [12] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964). [13] Y. Takada and W. Kohn, Phys. Rev. Letters 54 (1985) 470; 55 (1985) 141. [14] F. Hermann and S. SldUman, Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs, NJ, 1963). [15] N. Garcia, B. ReiN, K.H. Frank and A.R. Williams, Phys. Rev. Letters 54 (1985) 591. [16] A. Hurkmans, E.G. Overbosch and J. Los, Surface Sci. 59 (1976) 488. [17] L.D. Schmidt and R. Gomer, J. Chem. Phys. 45 (1966) 1605. [18] D. Eichenauer and J.P. Toennies, J. Chem. Phys. 85 (1986) 532. [191 D. Eichenauer, V. Harten, J.P. Toennies and V. Celli, J. Chem. Phys. 86 (1987).