Ag(111)

Solid State Communications, Vol. 48, No. 12, pp. 1045-1048, 1983. Printed in Great Britain.

0038-1098/83 $3.00 + .00 Pergamon Press Ltd.

THEORY OF ONE-PHONON SCATTERING OF ATOMS FROM METAL SURFACES: APPLICATION TO He/Ag(1 1 1) V. Bortolani, A. Franchini, F. Nizzoli and G. Santoro Dipartimento di Fisica, Universit~i di Modena, 41100 Modena, Italy G. Benedek Gruppo Nazionale Struttura della Materia dell Consiglio Nazionale della Ricerche, Dipartimento di Fisica, Universit~i di Milano, 20123 Milano, Italy V. Celli Department of Physics, University of Virginia, Charlottesville, VA 22901, U.S.A. and N. Garcia Departamento de Fisica Fundamental, Universidad Autonoma, Canto Blanco, Madrid-34, Spain

(Received 2 August 1983 by R. Fieschi) We present explicit expressions for one-phonon scattering of atoms from metal surfaces, within the distorted wave Born approximation by including Van der Waals forces. A comparison is made with the recent highresolution time of flight spectra for He/Ag(1 1 1). We explain in quantitative terms the position and the intensity of the peaks due to scattering from the Rayleigh wave of silver. Also the bulk phonons contribute to the spectra, but the calculated intensities underestimate the data. THE INELASTIC SCATTERING of low-energy atomic beams has proved to be a very successful technique for the determination of the surface phonons in ionic crystals [ 1,2]. In the time of flight (TOF) spectra well defined structures associated with Rayleigh phonons, bulk phonons and resonances have been observed over ,;everal Brillouin zones [2]. The interpretation of the corresponding data for metals is more difficult. In the first experiments on scattering of He from Cu(001) [3] and of Ne from Ni(111) [14] it was not possible to detect surface phonons up to the zone boundary. Feuerbacher and Willis [4] suggested an explanation of this behaviour in terms of a large effective interaction radius of the Ne atoms. Recent high-resolution experiments performed with He atoms on metal surfaces of Au [5] and Ag [6] have shown well defined dispersion relations of surface phonons as well as of bulk phonons over the whole first Brillouin zone (BZ). In this paper we present a theory of the He-surface scattering TOF spectra in the one-phonon approximation in order to analyze the very accurate data available for silver. Our theory explains the different behaviour of the cross-section of metals with respect Io alkali-halides. We identify the mechanisms which are responsible for the rapid decay of the scattering

intensities vs the phonon wavevector. In particular we reproduce quantitatively the Rayleigh wave (RW) dispersion relation over the whole BZ and the experimental decay of its intensity. The contribution of the bulk phonons is also analyzed. However in this case we find discrepancies between the calculated and measured spectra. The observation of very small first order diffraction peaks [6, 7] indicates the smoothness of the (1 1 1) Ag surface. For this reason it is justified to use the distorted wave Born approximation [8]. For one phonon scattering the inelastic coupling interaction is given by

U = ~ u,'V, V(r),

(1)

l

where ul is the displacement of the atom in the site l and V(r) is the atom-surface potential. According to Esbjerg and Norskov [9] V(r) is proportional to the electronic charge density. For the typical energies used in scattering experiments (20 meV) the atoms experience the charge density at large distance from the surface (the turning point is about 7 - 8 a.u.). It has been shown in a previous paper [10] that in this case the surface charge density can be well approximated by a superposition of atomic wave functions and the interaction potential can be written in the separable form

1045

1046

ONE-PHONON SCATTERING OF ATOMS FROM METAL SURFACES

V(r) = Uo exp [--/3(z --D(R))]

l

(2)

The position vector r has been conveniently split in a normal component z and in a parallel component R. The sum over l refers to the atoms of the outermost layer. The position of the lth atom is indicated by z 1 (normal component) and RI (parallel component). The corrugation D(R) is defined by the second equality of equation (2). Since/3D ~ 1 we have /3D = Y~ exp ( [ 3 z , ) e x p [ - - Q ~ ( R - - R a ) 2 / 2 l

l

- 1.

(3)

We found Uo = 546 eV, 13= 2.2A -1 and Qe = 0.73 A-1 for Ag (1 1 1) [10]. The lateral Fourier transform of the potential of equation (2) gives a factor exp [--(Q/Qc)2/2] which produces in the case of metals of cut-off at large momentum transfer Q. In fact for metals the values of Qe are much smaller than the zone boundary wavevector. For the attractive part of the laterally averaged potential we use the asymptotic Van der Waals interaction Vvw (z) = -- Cvw/(Z -- Zvw), where Cvw = 770 meVA a and Zvw = 1.38 A, according to Zaremba and Kohn [ 11 ]. We assume for the Van der Waals threedimensional interaction the form Vvw [z --D(R)], where the corrugation D(R) is taken to be the same as for the attractive part of the potential. The lateral Fourier transform of the gradient (iQ, a/az) of the total potential becomes (27r/A QZe) exp [-- (Q/Qe)2/2] (iQ, (3) x ~[Sz[Uo exp (-- (3z) -- Vvw(Z)],

0~=36*

0=52 o

4

= Uo ~ exp [-- Qe2(R - Ra)Z/2] exp [-- ~(z --zl).

Vol. 48, No. 12

2

d 0

4

0i=55"

m

0 -0.8

[

,,2

-0.6

-0A

~

-0.2

1 0'=38"

0,0

0.2

0.4

0.6

0.8

At(rnsec)

Fig. 1. Experimental (hystograms from [6]) and calculated TOF spectra for Ag(1 1 1). Oi is the incidence angle measured from the surface normal. The angle between the incidence and scattering directions is 90 ° . Q is along the [1 1 2] direction. The shaded area represents the calculated contribution to the spectra due to the Rayleigh wave. The remaining part of the calculated spectra is due to scattering from bulk modes at the surface. 25

20

15 >

@

E 3

10

(4)

where A is the area of the surface unit cell. We notice that here Vvw is computed from the flat surface. Similarly the lateral averaged potential Vo is given by Vo = (2n]AQ2e)[Uo exp (--/3z) -- Vvw(Z)].

(5)

With this separable form of the interaction the crosssection per unit area for an incident atom with initial momentum (KI, kiz), energy E i and final momentum (Kt, krz ) and energy E r reads d2R dEf d~f

2

0.0

0.2

0P,

0.6

0.8

1.0

1.2

Q(A-')

Fig. 2. Calculated dispersion relation of the Rayleigh wave (solid line) together with the continuum of bulk modes projected on the (1 1 1) surface (shaded area). The dashed line represents the longitudinal threshold. Solid and open circles represent the experimental maxima in TOF spectra [6] for creation and annihilation events respectively. The circles are labelled by the value of the incidence angle.

( 2 m E t ) l/z

- N M Q,] ~" ~ ( Q , j ) J k i z l Iktzl 2 x l-- i~ez(Q,]) + a e x ( Q , ] ) l 2 x exp ( - 2W) exp (-aZ/a2e) x t(xi= I~ Vo/~z I×t=)12n [~o(Q,/)] x 6(K t - Ki -- Q)6 [E t - E i -- boo(Q,/)]. (6)

Here ~o(Q, j) is the phonon frequency of the jth-mode, e(Q,j) the polarization vector evaluated at the surface, n the occupation number and exp (-- 2W) the usual

Vol. 48, No. 12

ONE-PHONON SCATTERING OF ATOMS FROM METAL SURFACES q (A")

0.0

0.2

0.4

0.6

0.8

i

|

i

,

1.0 |

1.2 i

1.4

1o-3~ .,,_, \o.\

:

I 0-5

\ '\'~, 'B

~B

lo

\A

Fig. 3. Inelastic scattering amplitude from Rayleigh waves of TOF spectra along [1 1 2]. The experimental results [6], represented by open (annihilation) and solid circles (creation events), are referred to the specular intensity Ioo. Surface temperature 140 K. The calculated curves are labelled by A for creation and by B for anihilation. Solid lines: full calculation by including the Van der Waals interaction. Dashed-dotted lines: without the Van der Waals interaction. Dashed lines: calculation performed with an attractive square well of 5 meV, according to the Beeby approximation [ 14].

Debye-Waller factor. In the matrix element appearing in equation (6) the eigenfunctions X correspond to the laterally averaged potential of equation (5) and are determined numerically. The evaluation of equation (6) requires the knowledge of the eigenvectors e(Q, j) and eigenvalues w(Q, j) of the (1 1 1) surface of Ag. We have evaluated these quantities for a layer of 69 (1 1 1) atomic planes, within a force constants parametrization of the bulk lattice dynamics of silver. The details of the calculation are given elsewhere [12]. The surface force constants have been considered equal to those of the infinite crystal. In Fig. 1 are presented the results of our calculations together with the experimental TOF spectra [6] in four different scattering geometries. The calculated spectra depend on both the longitudinal and the normal (shear vertical) components of the surface displacement field, according to equation (6). We have fbund that the bulk spectrum is usually dominated by the normal component, apart from a small region close to the longitudinal threshold where the longitudinal component becomes important. It can be seen from Fig. 1 that the dispersion of the RW is well reproduced.

1047

The RW dispersion relation and the projected bulk phonons are given in Fig. 2. The observed Rayleigh phonons follow very closely the evaluated dispersion relation over the whole BZ. We have found that in the bulk region the evaluated density of states, although appreciable, is rather low compared with the RW contribution and that increases somewhat at the longitudinal threshold. We cannot reproduce the bulk peaks observed in the experimental data of Figs. 1 and 2. In Fig. 3 it is shown the behaviour of the RW scattering intensity as a function of the momentum transfer. The almost linear decay of the intensity logarithm is well reproduced by our theory, including the asymmetry between phonon creation and annihilation events. The linear behaviour comes from a fine cancellation in equation (6) between the cut-off term in Q and the matrix element of the total potential. We have also performed this calculation neglecting the Van der Waals interaction. As can be seen in Fig. 3, one obtains from equation (6) a very poor fitting of the data when the attractive part of the potential is neglected. Also the introduction of the Beeby [13] correction to simulate the Van der Waals potential does not significantly improve the agreement with the experiment. In fact either the initial or the final energies are comparable with the depth of the attractive well (6 meV). Finally we notice that it would be possible to fit the data without any attractive part in the potential, by using an unphysical value of the softness parameter /~> 4 A-1 and consequently a very large Qe cut-off. Values of/3 greater that 4 A-1 are in contrast with other calculations [14, 15] and do not reproduce the elastic scattering intensities [7]. Acknowledgements - The authors are grateful to R.B. Doak, U. Harten and J.P. Toennies for having communicated their results before publication. Partial financial support from Ministero Pubblica Istruzione and Centro di Calcolo della Universit~ di Modena is acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

G. Brusdeylins, R.B. Doak & J.P. Toennies, Phys. Rev. Lett. 44, 1417 (1980); 46,437 (1981). G. Benedek, J. Electr. Spectr. 30, 71 (1983). B.F. Mason & B.R. Williams, Phys. Rev. Lett. 46, 1138(1981). B. Feuerbacher & R.F. Willis,Phys. Rev. Lett. 47, 526(1981). M. Cates & D.R. Miller, J. Electr. Spectr. 30, 157 (1983). R.B. Doak, U. Harten & J.P, Toennies, Phys. Rev. Lett. 51,578 (1983). G. Boato, P. Cantini & R. Tatarek, Proc. 7th lnter. Vac. Congr. and 3rd lnter. Conf. Solid

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9. 10.

ONE-PHONON SCATTERING OF ATOMS FROM METAL SURFACES

Surfaces, (Edited by R. Dobrozemsky, F. Rudenauer, F.P. Viehboch & A. Breth) p. 1377. Vienna, (1977); J.M. Home, S.C. Yerkes & D.R. Miller, Surf Sci. 93, 47 (1980). N. Cabrera, V. Celli, F.O. Goodman & R. Manson, Surf Sci. 19, 67 (1970). N. Esbjerg & J.K. Norskov, Phys. Rev. Lett. 45, 807 (1980). V. Bortolani, A. Franchini, N. Garcia, F. Nizzoli & G. Santoro, (submitted for publication to Phys. Rev. B RapM Commun.).

11. 12.

13. 14. 15.

Vol. 48, No. 12

E. Zaremba & W. Kohn,Phys. Rev. B15, 1769 (1977). V. Bortolani, A. Franchini, F. Nizzoli & G. Santoro, in l~namics of Gas-Surface Interaction, (Edited by G. Benedeck & U. Valbusa) p. 196. Springer, Berlin (1982). J.L. Beeby, J. Phys. C: Solid State Phys. 5, 3438 (1972). A. Liebsch & J. Harris, (unpublished). N. Garcia, J.A. Barker & I.P. Batra, J. Electra. Spectr. 30, 137 (1983).