New analysis of He scattering data from Ag(110) and Cu(110)

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kurface science

Surface Science 276 (1992) 333-340 North-Holland

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New analysis of He scattering data from Ag( 110) and Cu( 110) P. Cortona Dipartimento

di Fisica deN’Universitd

di Genova,

Unitri INFM and Centro FSBT/CNR,

Via Dodecaneso

33, 16146 Genova, Italy

M.G. Dondi Istituto di Fisica di Ingegneria dell’Universitci 16146 Geneva, Italy

di Geneva,

Unitci INFM and Centro FSBT/

CNR, T/is Dodecaneso

33,

A. Lausi Laboratorio

INFM-TAX,

Padriciano

99, 34012 Trieste, Italy

and F. Tommasini Dipartimento

di Fisica dell’llniuersitci

di Trento and Laboratorio

INFM-TASC,

Padriciano

99, 34012 Trieste, Italy

Received 26 March 1992; accepted for publication 26 June 1992

A method for the analysis of the He-surface scattering data using a model potential based on the superposition of pseudo-pairwise anisotropic terms and degenaracy-dependent self-interaction-corrected (D-SIC) calculations of the atomic electron densities is presented and applied to the study of the electron density of Ag(ll0) and Cu(ll0). Rearrangements of the electron clouds around the surface atoms with respect to those of the free atoms, leaving unchanged the lateral average, are observed in both cases.

1. Introduction The He beam scattering method is widely used for the study of the static and dynamic properties of solid surfaces [l-3]. In particular the He diffraction patterns yield quite accurate pictures of the surface electron density [1,4,5] and allow the study of surface phenomena such as reconstructions [6,71, phase transitions [6,8,91, physisorption and chemisorption [4,10]. Nevertheless, the development of methods for the analysis of the experimental data still requires some work. In fact, when the study deals with highly corrugated surfaces such as, for instance, (1 x 2)PtCllO) [7], 0039-6028/92/$05.00

(1 x 2)Au(llOl 19,111 or oxygen overlayers on metals [12,13], the experimental data present structures with rainbow and bound state resonance oscillations which strongly overlap and are smeared out by multiple scattering effects [14]. In these cases the semiclassical approach to the scattering problem may lead to ambiguous or even wrong conclusions. On the other hand the full quantum mechanical calculation of the diffraction probabilities, for a given He-surface interaction potential, requires much computer work due to the large number of diffractive channels which may be involved in the scattering process. Therefore, suitable representations of

0 1992 - Elsevier Science Publishers B.V. All rights reserved

P. Cortona et al. / He scattering from Ag(llO) and Cu(ll0)

334

the interaction potential leaving only a few parameters to be determined by the experiments, and allowing easy modelling of surface reconstructions and chemisorption, are needed. From this point of view the description of the interaction potential as a superposition of pairwise terms is highly appealing. Analysis of He scattering data from metal surfaces based on the summation of pairwise potentials was previously reported by Garcia et al. [15] for Cu, by Eichenauer et al. [16] for Ag and Cu and by Dondi et al. [17] for Ag. These studies have shown that discrepancies between experimental data and theoretical results appear when the repulsive part of the potential is determined from Hartree-Fock (HF) calculations [16,18] or even when, according to the effective medium theory [19,20], it is assumed proportional to the HF charge densities [21]. In the same framework the choice of atomic electron densities calculated by Herman and Skillman (HS) [22] gives rise to better results [16,17]. However, agreement between theory and experiment can only be obtained by introducing different values of the Esbjerg and Norskov [19] proportionality constant (Y for Ag and Cu. Furthermore, these HS charge densities were obtained by using the Slater approximation which is known to overestimate the exchange potential. In a recent paper [21] we have reconsidered the subject and we have shown that these discrepancies may be removed by starting from atomic electron densities calculated self-consistently in the local density approximation with degeneracydependent self-interaction-correction (D-SIC) [23]. According to Tang and Toennies [241 (hereafter IT) we have written the pairwise potential as u(r)

= (YN exp( -pr)

(1) where (Y has been set to the theoretical value proposed by Manninen et al. [20] and N and p have been determined by fitting the D-SIC atomic

electron densities. Furthermore, a simple rule connecting all the multipolar coefficients C,, to only one free parameter was given. The resulting atom-surface laterally averaged interaction potentials accounted very well for the bound state resonance data available for Ag(ll0) [25], Cu (113), Cu(115) and Cu(117) surfaces [26]. This suggests that the electron density of a metal surface, in the region explored by the He atom, may be much closer to the superposition of free atomic densities than previously expected. It is noticed that the role of the D-SIC electron densities is substantial in removing the inconsistencies previously found by using HF densities; this is possibly due to the fact that the D-SIC method properly accounts for the electronic correlation, while this is not the case in the HF approximation. Clearly, the fact that the lateral average of the superposition of free atomic densities is close to the laterally averaged electron density does not mean that no charge rearrangement occurs when the metal atom is embedded in the surface. The atomic electron clouds could in fact spread in the surface plane and smooth the isoelectron density surfaces without affecting the laterally averaged density. As shown in ref. [17] these lateral rearrangements of the atomic electron clouds, evidenced by He beam diffraction measurements from Ag(llO), can be phenomenologically accounted for by writing the surface electron density as a summation of anisotropic pseudo-pairwise terms. In the present paper the subject is considered in more detail and extended to the He-Cu(ll0) diffraction experiments reported in ref. [27]. Moreover, a systematic way of dealing with the analysis of the He-surface scattering data is presented. The quite good agreement obtained in the wide ranges of angles of incidence and beam energies explored by the experiments, strongly supports the model and provides a solid ground for the study of the surface electron density by He scattering experiments. The paper is organized as follows: section 2 deals with the pseudo-pairwise anisotropic potentials introduced by starting from eq. (1); section 3 presents the results for the Fourier components of the He-surface interaction potential; closecoupled calculations of the He-diffraction proba-

P. Cortona et al. / He scattering from Ag(ll0)

bilities for Ag(ll0) and Cu(ll0) are compared to the available scattering data in section 4; finally a discussion and a summary are given in section 5.

2. Pseudo-pairwise

=aN

exp( -pr)

-B

exp( -y’),

(2) in the whole range of distances, r I u, for which U(T) 2 0. The value of y in eq. (2) may be set freely in the range 3p/5 I y < 2/I/3, provided that N is slightly adjusted with respect to eq. (1) and B is properly chosen. The choice y = 0.64p, allowing N to be left unchanged, is hereafter taken. With this assumption, eq. (2) fits the repulsive wall of eq. (1) with an accuracy close to 1% for the whole set of atomic systems. By taking the zero cross distance (7 as a free parameter, the short range potential is then written as u(r) = aN exp( -pr){l

- exp[0.360(

r -a)]}, (3)

and eq. (1) is replaced by u(r) =u(r)

-W(r),

count for the rearrangements of the electron density which may take place at the surface and/or for the spatial resolution of the He probe, is introduced by generalizing the short range potential in eq. (3) as

potential

The potential given in eq. (0, was introduced by TT and was shown to provide a very accurate representation for the interaction energy of a wide class of diatomic systems. Here it is noted that, for all the atomic systems considered by TT and for the He-noble-gas pairs reviewed by Aziz [28], eq. (1) is equivalent to u(r)

335

and Cu(ll0)

(4) where W(T) accounts for the long range tail of the dispersive energy. The advantages of splitting the potential as indicated in eq. (4) appear when the summation of pairwise terms extended to the He-metal pairs is considered. First, the summation of the exponential functions in eq. (3) is easily carried out analytically. Second, W(T) is negligible at short range, thus its summation yields a contribution W(z) to the laterally averaged atom-surface potential, but does not affect the periodic Fourier components. Third, the summation of W(T) may be approximated within a few percent accuracy by eq. (12) given in the next section. Before proceeding to the calculation of the He-surface potential, a phenomenological ac-

Xexp(

-Pp){l -

exp[O.36P(p -a)]), (5)

where 77, and q,, are anisotropy parameters

and

p=

(6)

z2+(77*X)*+(TyY)*.

Eqs. (5) and (6) are written in such a way as to make the laterally averaged atom-surface potential independent on the parameters 77, and n,,. Following the usual surface notation, the x axis is chosen parallel to the rows of close-packed atoms (I - X direction). As a matter of fact the pseudo-pairwise potential u(r) =u(r)

-W(T)

(7) provides a very flexible representation for the contribution of a surface atom to the He-surface interaction energy. Moreover, the values of the parameters N and /3 may be evaluated by starting from D-SIC calculations [23] of the electron densities of the free atoms, and by setting the Esbjerg and Norskov constant, according to Manninen et al. [20], as (Y= 30 eV0A3. In this way, the values aN = 7,2 eV, p = 2.61 A-’ and aN = 67.5 eV, p = 2.71 A-’ were obtained for Ag and Cu, respectively [21]. The parameter q used in ref. [21] is replaced in the present paper by U, which has a more direct physical meaning. Consistency with ref. [21] is ensured for P = 3.823 A and u = 3.67 A for Ag and Cu, respectively. It is observed that the product pa = 10 is nearly the same for the two systems, a result which may help in dealing with similar systems. In summary, the pseudo-pairwise potential defined by eqs. (5)-(7) has been introduced. It takes into account the dispersive energy at all order in the multipolar expansion, and, by the parameters nx and v,,, it accounts for the rearrangements of the atomic electron clouds occurring at the surface and/or for the spatial resolution of the He probe [29,30].

336

P. Cortona et al. / He scattering from Ag(llOj

3. He-surface

interaction

The He-surface ten as

potential

interaction

and Cu(ll0) 1

potential

P-mn

is writ-

10

V(R, z) - XV,(z)

exp(iG-R),

G

-1

(8) IO

where G = (G,, G,) are the vectors of the surface reciprocal lattice and (R, 0) =(x, y, 0) is the topmost nuclear plane. According to the discussion given in the previous section, the Fourier components V,(z) may be written as v,(z) = U,(z) - %+W)7 (9) where U,(z) are the Fourier components of the summation of the short range pseudo-pairwise potential defined by eqs. (5) and (6). These are given by

10

-. 2

-3 20

40 60 @,(deg)

80

Fig. 1. Comparison of the calculated diffraction probabilities Pm: (full lines) with the experimental data for He-Ag(llUf reported in ref. [17]. The measurements have been done along the (100) azimuth with beam energy E = 17.8 meV. The diffraction peaks are labeiled by: (0) (0,O); i A) (O,l); (A)

(0,ik (0)to,%cm) (03.

U,(z) =

21rA R

Fexp(iG*L),)

Next, $ ’ +PG(Z-zk)averaged G ew(PG(z-zk))

- exp( -0.36&r)

70 1+YG(z-zk)

3

7% exp(dz

- +I>

9 (10)

where dk = CL?,, z,) is the displacement of the kth atomic layer with respect to the outermost atomic layer, 0 is the area of the surface unit cell and

the contribution W(z) to the laterally potential arising from the summation of the tails W(Y) is considered. It is expected to have, at large distance from the surface, the long range behaviour C/(z - 2,13, predicted by Zaremba and Kohn [31], with Z, related to the interlayer separation d as Z, = d/2 and C related to C, as C = rrC,/(6LM> [321. By performing the numerical summation of the terms w(r) = U(I) - U(T), with u(r) and L’(T) defined by eq. (3) 1 COlC

P m”

YG= /(0.64/3)2+

($)‘+

(2)‘.

IO

(‘1)

Due to the exponential decay of the terms, the summation in eq. (10) may be limited to very few layers. Furthermore, the surface relaxation can be taken into account provided that the values zk are known; nevertheless, observation of surface relaxation effects by He beam scattering appear to be out of the question, unless very open surfaces are considered. In fact, the short range potential is dominated by the contribution of the topmost layer and small changes in the parameters of the pseudo-pairwise potential give the same effect as large surface relaxations.

IO

-1

-2

1 o--

3 20

40 60 Q(deg)

80

Fig. 2. Comparison of the calculated diffraction probabilities (full lines) with the experimental data for He-Gg(llO) reported in ref. [17]. The measurements have been done along the (100) azimuth with beam energy E = 64.6 meV. The diffraction peaks are labelled by: (0) (O,O); (A) (0,l); (A)

Pz

co& (0)(0,x); w (0,2f; (of (0,3); m) co,?).

P. Cortona et al. / He scattering from Ag(l10)

337

and Cu(l10)

Ll

0

Q(deg) Fig. 3. Comparison of the calculated diffraction probabilities P$ (full lines) with the experimental data for He-Ag(llO) reported in ref. [17]. The measurements have been done along the (110) azimuth with beam energy E = 64.6 meV. The diffraction peaks are labelled by: (0) (0,O); (A) (0,l); (A)

Fig. 5. Comparison of the calculated diffraction probabilities (full lines) with the experimental data for He-Cu(ll0) reported in ref. 1271.The measurements have been done along the (100) azimuth with beam energy E = 21 meV. The diffraction peaks are labelled by: (0) (O,O);(A 1 (OJ); (A CO&.

Pzd

(03.

and eq. (11, respectively, it appears that, choosing (Y,N, /3, (T, C,, as in diatomic systems, K’(z) may be approximated as W(z) c

= (z-.&)~+c~~

exp (-~K~-~,,/@l’)’ (12)

with an accuracy close to 1%.

In summary, the He-surface interaction potential is described by eqs. @O-(12). The main advantage of the model is that the whole set of Fourier components entering the calculations of the scattering probabilities, is given in terms of a few parameters presenting a direct physical meaning. When the anisotropy parameters n, and nY are set to one, the model is equivalent to the summation of the IT potentials given by eq. (1). Values of the anisotropy parameters smaller

1

P

EdC

Inn

IO

-1

,o-32~ 1o-2

10-5-

20

40

60

80

@,(deg) Fig. 4. Comparison of the calculated diffraction probabilities data for He-Ag(ll0) reported in ref. (171. The measurements have been done along the (111) azimuth with beam energy E = 64.6 meV. The diffraction peaks are labelled by: (0) (0,O); (A) (1,l); (A)

Pz”,’ (full lines) with the experimental

(hi).

40

60

80

Q(deg)

Fig. 6. Comparison of the calculated diffraction probabilities data for He-Cu(ll0) remparted in ref. [27]. The measurements have been done along the (100) azimuth with beam energy E= 63 meV. The diffraction peaks are labelled by: (0) (0,O); (A 1 (0,l); (A)

Pea’ (full lines) with the experimental

(03); (0)(0,2); (0) fo2f; ( n ) (0,3x

P. Cortona et al.

20

40

60

He scattering from Ag(lIO)

80

40

of the spreads surface of the

4. Analysis High resolution He diffraction measurements for Ag(ll0) [17] and Cu(ll0) [271 have been re-

CDIC P rn" lo-

-1

1

o--2

1

o--3

I 20

T ti,, 40

/ 8

60

Fig. 9. Comparison of the calculated diffraction probabilities Pzi (full lines) with the experimental data for He-Cu(ll0) reported in ref. [27]. The measurements have been done along the (100) azimuth with beam energy E = 241 meV. The diffraction peaks are labelled by: (0) (0,O); (A) (0,l); (A) (o,i); (0) (0,2); (0) co,% ( n ) (0,5).

(0) (o,Z); ( n ) (OS).

than one damp the Fourier components atom-surface potential and account for of the atomic electron densities in the plane as well as for the limited resolution He probe [29,30].

1

,A, 60

@,(deg)

Q(de9) Fig. 7. Comparison of the calculated diffraction probabilities Pzd (full lines) with the experimental data for He-Cu(ll0) reported in ref. 1271. The measurements have been done along the (100) azimuth with beam energy E= 77 meV. The diffraction peaks are labelled by: (0) (0,O); (A) (0,l); (A)

(03; (0) (0,2);

and Cu(ll0)

80

O,(deg)

Fig. 8. Comparison of the calculated diffraction probabilities PzA (full lines) with the experimental data for He-Cu(ll0) reported in ref. [27]. The measurements have been done along the (100) azimuth with beam energy E = 125 meV. The diffraction peaks are labelled by: (0) (0,O); (A) (0,l); (A) (0,i); (0) (0,2); (0) (o,Z); (W 1 to,%.

ported in the last few years. For Ag(ll0) several diffraction patterns at angles of incidence ranging between 20” and 80” were measured along the (loo), (110) and (111) azimuths, with beam energies of 17.8 and 64.6 meV [17]. For Cu(llO), diffraction patterns taken at several angles of incidence along the (100) azimuth with beam energies of 21, 63, 77, 125 and 241 meV are available [27]. With CYN, p, u set as given in section 2, C = 249 meV A3 and Z, = 0.755 A for Ag, C = 226 meV A3 and Z, = 0.673 A for Cu, i.e., for the laterally averaged potentials which were shown to provide quite accurate fits of the bound state resonance data [21], the anisotropy parameters 7, and rlY are determined by fitting the diffraction data to the diffraction probabilities as calculated by the close-coupling method [33,341. As shown in figs. l-4 for Ag(l10) and in figs. 5-9 for Cu(llO), quite good fits of the experimental data are obtained at all angles and at all energies by setting r), = 0.71, qY = 0.78 for Ag and nY = 0.70 for Cu. Note that only diffraction patterns along (100) are available for Cu, so that the value of 7, cannot be obtained from the experiments. The calculation results reported in figs. 5-9 refer to the case vX = 0.64, scaled with respect to vY as in the Ag case; however, no substantial changes of

P. Cortona et al. / He scattering from Ag(I IO) and Cu(f IO)

the diffraction intensities along (100) appear, at least for nI ranging between 0.6 and 0.7.

5. Discussion The fact that the periodic Fourier components of the dispersive energy play a substantial role in determining the He diffraction intensities from Ag(ll0) was already shown in a previous paper [17]. There only the leading term of the pairwise multipolar expansion, damped according to TT, was taken into account. In the present case, the dispersive energy is phenomenologically dealt with at all order in the multipolar expansion and the whole description of the He-surface interaction potential results to be substantially improved, allowing the He-Cu(ll0) and HeAg(ll0) scattering data to be accounted for in a unique scheme. The analysis of the present paper is consistent with the assumption that the Fourier components and the lateral average of the repulsive potential are proportional, with the same proportionality constant, to the corresponding components of the electron density. The damping of the periodic Fourier components of the potential, phenomenologically described by the anisotropy parameters in the pairwise potentials, is interpreted as due to a real spread of the atomic electron clouds in the surface plane. A consequence of this is a reduced corrugation of the isoelectric surfaces. For instance, by starting from the values of the parameters nX and n,, resulting from the analysis of the He-Ag data, the peak to peak corrugation of the 2.10e3 A-” isoelectron-densi~ surface is 0.22 A, while a direct supe~osition of D-SIC densities (nX = nY = I), gives a peak to peak cor~gation of about 0.6 A. We note, however, that a similar damping effect on the Fourier components of the potential can also be due to the finite spatial resolution of the He probe in sampling the electron density [291. The analysis of the scattering results from more open surfaces, for instance from (1 x 2) Au(llO), could provide further insight on this point. In summary, a model of the He-surface interaction potential has been presented and applied

339

the study of the electron density of the Ag(ll0) and Cu(lI0) surfaces. The model may easily be extended in order to account for surface reconstruction and chemisorption. Preliminary results for (1 x 2)Au(llO) [ill and (2 X l)O-Ag(ll0) 1121 seem to be very promising.

to

References [I] T. Engel and K.H. Rieder, in: Structural Studies of Surfaces with Atomic and Molecular Beam Diffraction, Vol. 91 of Springer Tracts in Modem Physics (Springer, Berlin, 1982). [2] G. Comsa and B. Poelsema, in: Scattering of Thermal Energy Atoms from Disordered Surfaces, Vol. 115 of Springer Tracts in Modern Physics (Sprjnger, Berlin, 1989). [3] R.B. Doak, in: Atomic and Molecular Beam Methods, Ed. G. Scales (Oxford University Press, New York, 1992). [4] K.H. Rieder and T. Engel, Phys. Rev. Lett. 45 (1980) 824. [.5] J. Lapujoulade and Y. Lejay, Surf. Sci. 69 (1977) 3.54. 161 D. Cvetko, A. Lausi, A. Morgante, F. Tommasini and KC. Prince, Surf. Sci. 269/270 (1992) 68. [7] E. Kirsten, G. Parschau and K.H. Rieder, Surf. Sci. Lett. 236 (1990) L365. [S] G. Bracco, C. Malo and R. Tatarek, Vuoto 21 (1991) 19.5. 191 J. Sprosser, B. Salanon and J. Lapujouiade, Europhys. Lett. 16 (1991) 283. [lo] G. Vidali, G. Ihm, H.-Y. Kim and M.W. Cole, Surf. Sci. Rep. 12 (1991) 13.5. [ll] P. Cortona, D. Cvetko, M.G. Dondi, A. Lausi, A. Morgante, K.C. Prince and F. Tommasini, Phys. Rev. B, submitted. 1121 P. Cortona, M.G. Dondi, A. Lausi and F. Tommasini, to be published. [13] J. Lapujoulade, Y. Le CruEr, M. LeFort, Y. Lejay and E. Maurel, Surf. Sci. 118 (1982) 103. 1141 V. Celli and D. Evans, in: Dynamics of Gas-Surface Interaction, Vol. 21 of Springer Series in Chemical Physics, Eds. G. Benedek and U. Valbusa (Springer, Berlin, 1982). [l-5] N. Garcia, J.A. Barker and I.P. Batra, Solid State Commun. 47 (1983) 485. [16] D. Eichenauer, U. Harten, J.P. Toennies and V. Celli, J. Chem. Phys. 86 (1987) 3693. 1171 M.G. Dondi, S. Terreni, F. Tommasini and U. Linke, Phys. Rev. B 37 (1988) 8034. [18] I.P. Batra, P.S. Bagus and J.A. Barker, Phys. Rev. B 31 (1985) 1737. ]19] N. Esbjerg and J.K. Norskov, Phys. Rev. Lett. 45 (1980) 807. 1201 M. Manninen, J.K. Norskov, M.J. Puska and C. Umrigar, Phys. Rev. B 29 (1984) 2314.

340

P. Cortona et al. / He scattering from Ag(ll0)

[21] P. Cortona, M.G. Dondi and F. Tommasini, Surf. Sci. Lett. 261 (1992) L35. [22] F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs, NJ, 1963). [23] P. Cortona, Phys. Rev. A 34 (1986) 769. [24] K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [25] M.G. Dondi, L. Mattera, S. Terreni, F. Tommasini and U. Linke, Phys. Rev. B 34 (1986) 5897. [26] J. Perreau and J. Lapujoulade, Surf. Sci. 122 (1982) 341. [27] B. Salanon, G. Armand, J. Perreau and J. Lapujoulade, Surf. Sci. 127 (1983) 13.5.

and Cu(ll0)

[28] R.A. Aziz, in: Inert Gases, Vol. 34 of Springer Series in Chemical Physics, Ed. M.L. Klein (Springer, Berlin, 1984) p. 5. [29] Y. Takada and W. Kohn, Phys. Rev. B 37 (1988) 826. [30] J. Tersoff, Phys. Rev. Lett. 55 (1985) 140. [31] E. Zaremba and W. Kohn, Phys. Rev. B 13 (1976) 2270. [32] W.A. Steele, in: The Interaction of Gases with Solid Surfaces, Ed. D.H. Everett (Pergamon, Oxford, 1974) ch. 2, p. 13. [33] G. Wolken, J. Chem. Phys. 58 (1973) 3047. [34] R.J. Blake, Comput. Phys. Commun. 33 (1984) 425.