Inelastic diffraction of He atoms from Xe overlayer adsorbed on the graphite (0001 )

Surface Science 496 (2002) L13±L17

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Surface Science Letters

Inelastic di€raction of He atoms from Xe overlayer adsorbed on the graphite (0 0 0 1) A.Kh. Khokonov *, Z.A. Kokov, B.S. Karamurzov Kabardino±Balkarian State University, str. Chernyshevskogo 173, Nalchik 360004, Russia Received 8 September 2000; accepted for publication 24 September 2001

Abstract A simple model for the normal vibrations of noble gases adsorbed on graphite in the commensurate phase is presented. The external layer is presented as a lattice of atoms bonded to each other by threads with their tension de®ned by lateral interaction. The proposed model provides analytical expressions for phonon dispersion curves. It also allows one to construct the one-phonon dynamic structure factor, which is used in expressions for single-phonon inelastic surface-scattering probabilities. A quantitative comparison with experiments at the normal angle of incidence has also been completed. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Molecule±solid scattering and di€raction ± inelastic; Atom±solid interactions; Noble gases; Adatoms; Phonons; Graphite; Surface thermodynamics (including phase transitions); Physical adsorption

1. Introduction The scattering of a supersonic beam of helium atoms is one of the most useful techniques of surface structure studies on the atomic scale. Atomic beam di€raction experiments are used to study the incommensurate±commensurate transition of Xe atoms adsorbed on a graphite surface [1]. X-ray synchrotron radiation [2] and transmission high-energy electron di€raction [3] is also used for these purposes. However, we consider the inelastic scattering of low-energy molecular beams (E < 0:1 eV) as the most perspective method [4,5] for the following reasons. This technique is not

* Corresponding author. Tel.: +7-866-2222087; fax: +7-0959563504. E-mail address: [email protected] (A.Kh. Khokonov).

destructive to the physisorbed layers. The incident atoms with de Broglie wavelengths on the order of  have energies comparable with the phonon 1 A energies. The momentum transfer of the scattered molecule is of the same magnitude as the Brillouin zone. Rare gases adsorbed on graphite basal planes proved to be a bene®cial experimental system for the study of two-dimensional (2D) phase transitions. It has been found that p experimentally p xenon forms the … 3  3†R30° 2D phase at temperatures < 50±60 K which is stable with respect to phonon±adatom interaction. A further increase in temperature would result in its melting. Therefore the construction of the adequate vibrational model for the overlayer±substrate system seems to be of a high priority. Evaluation of the surface vibratory spectrum modi®ed by the adsorbed monolayer is usually performed by numerical methods within the

0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 1 ) 0 1 6 3 6 - 3

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framework of a speci®c lattice model. The crystal is simulated as a slab with a limited number of atomic layers. The in¯uence of adsorbed layers on the long wavelength surface vibrations are then studied according to the macroscopic theory of Ref. [6]. Two types of adsorbed monolayers are considered with regard to the value of the interaction force with the substrate: (1) a ``strong-coupled'' impure layer, in which the elastic substrate force interaction constant is of the same order as in the bulk of a crystal; (2) a ``weak-coupled'' impure layer, in which the force constant of the interaction with the substrate is much less than the characteristic bulk force constants. For a description of noble gas monolayers weakly coupled with a substrate, an analytical model for lattice vibrations can be developed.

coordinated position on the top of the hexagon of the graphite lattice. In the commensurate phase, the adsorbed atoms are locked into a registry with the honeycomb structure and occupy every third hexagon center. The adsorbed atoms form an equilateral triangular lattice with nearest-neighbor  A primitive unit cell is de®ned distance a ˆ 4:26 A. p by the translational 3=2† p vectors a1 ˆ a…1=2; and a2 ˆ a…1=2; 3=2†, as shown in Fig. 1. The correspondingpreciprocal lattice vectors p are   b1 ˆ …2p=a†…1; 1= 3† and b2 ˆ …2p=a†…1; 1= 3†. We denote the static potential seen by an adsorbate atom due to its attraction by the graphite substrate as Vs …r†, which is usually expanded in the Fourier series X VG …z† expfiGRg; …1† Vs …r† ˆ V0 …z† ‡ G6ˆ0

2. Overlayer dynamics We adopt a notation in which the z-direction is normal to the surface with positive z values above the surface. Vectors are resolved into components parallel and perpendicular to the surface as r ˆ …R; z†. On the (0 0 0 1) face of graphite, the carbon atoms form a 2D honeycomb structure. The adatoms have minimum energy in a six-fold

where G ˆ n1 b1 ‡ n2 b2 are surface reciprocal lattice vectors and z is the equilibrium height of the adatom above the substrate. Next, we consider the transverse vibration of Xe monolayer. The external layer is assumed to be a lattice of atoms interconnected by the threads with tension T. The latter is determined by the sum of the pairwise adatom's potentials and the lateral part of the adatom±substrate potential (1), for which G 6ˆ 0. The e€ect of the remaining substrate

p p Fig. 1. The … 3  3†R30° Xe monolayer structure in registry with the graphite (0 0 0 1) surface. ai and bj are the direct and reciprocal unit cell vectors, respectively.

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potential V0 …z† is manifested by the shift of the vibrational branches. The magnitude of the shift is equal to the vibrational frequency X0 of the external plane as a whole in the e€ective potential of the substrate [7]. In addition we assume that the transverse vibrational modes of the overlayer are independent from those of the substrate. Such an assumption is justi®ed by the fact that the force constant for the adatom±surface potential is much smaller than the force constant of the pairwise potential of atoms within the substrate. The meansquare amplitudes of the carbon atoms' normal vibrations in the surface layer are small enough to be neglected [8]. Thus, the rigid-substrate approximation can be used safely. The instant con®guration of the overlayer adatoms is described by the displacement ul of the lth atom from its equilibrium position Rl . The resultant of all forces applied to the adatom leads to the following equation of motion M ul ˆ

6 X

Tl0 l

rV0 …z ‡ ulz †:

…2†

l0 ˆ1

The sum taken over the nearest six adatoms shown in Fig. 1, M is the mass of the adatom, Tl0 l is the tension of the thread interconnecting the adatoms. We are primarily interested in vibratory modes vertical to the surface. These modes are the main contributors to the inelastic molecular surfaces scattering process. The projection of the vector Tl0 l along the vertical direction is …Tl0 l †z ˆ T al0 l , where T is the absolute value of the tension and al0 l is the angle between the vector Tl0 l and the x±y plane. This angle is de®ned by the vertical displacement of the adjacent adatom by the relation ul0 z ulz al0 l ˆ : a

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justable parameters and ®tted to the experimental data or more sophisticated calculations. The solution of Eq. (3) is of the form ul ˆ u exp i…QRl xt†. Using the fact that X exp…iQRl0 † ˆ 2…cos…Qa1 † ‡ cos…Qa2 † l0

‡ cos…Q…a1 ‡ a2 ††† and expanding Q ˆ p1 b1 ‡ p2 b2 , the dispersion relation can be expressed as x2 …Q† ˆ X20 ‡ 2x20 ‰3

…cos…2pp1 †

‡ cos…2pp2 † ‡ cos…2p…p1 ‡ p2 †††Š;

…4†

where X20 ˆ cs =M and x20 ˆ ct =M. In this paper we use the ®tting parameters hx0 ˆ 1:57 meV and hX0 ˆ 0:63 meV from the phonon dispersion curves of Hakim et al. [7]. Calculated dispersion curves of the transverse vibrations are shown in Fig. 2. The corresponding density of phonon states G…x† and mean-square amplitudes are given in Fig. 3.

3. Inelastic scattering in the di€raction region An explanation will now be outlined of the observed intensities [1] within the framework of the eikonal-like semiclassical scattering approximation

Hereafter the letter z is dropped from notations for vertical displacements. The equation of motion for the normal displacements of lth adatom becomes ! 6 X M ul ˆ c t ul0 6ul …3† cs u l ; l0

where cl ˆ T =a and cs ˆ o2 V0 =ou2l are the force constants, which in general can be treated as ad-

Fig. 2. Phonon dispersion curves along the KC and CM directions of reduced Brillouin zone of the xenon overlayer.

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Fig. 3. (a) Phonon spectrum for the vertical vibrations of xenon atoms, (b) temperature dependence of the mean-square amplitudes of xenon atoms.

[9±12]. This approach allows one to obtain a simple Gaussian approximation [11] to the dynamic structural factor of the inelastic phonon scattering and the intensities of the di€raction peaks. A scattering amplitude calculation is attempted, which only takes into account the thermal displacements vertical to the surface. In this case it is convenient to follow Ref. [10]. We denote by x ˆ …Ef Ei †= h and K ˆ Kf Ki the energy and the parallel wave vector exchange, respectively. Then Z X 2 _Pfi ˆ 1 jTG j exp… 2W …qz †† ds exp ixs …2p h† 2 G Z   dR exp i…K G†R ‡ q2z J …R; s† ; …5† where J …R; s† ˆ hg…R; s†g…0; 0†i is the correlation function of the time-dependent part of corrugation function, W …qz † ˆ …1=2†q2z hg2 i is the Debye±Waller factor and qz ˆ jkiz j ‡ kfz . The construction of the correlation function J …R; s† is based upon the solution for transverse overlayer modes calculated above. If such a solution is found, it is convenient to use the vertical displacements of the separate atoms ul …s† instead of the function g…R; s† following the relation X g…R; s† ˆ s ul …s†d…R Rl †; …6† l

where s is the surface area of the unit cell.

The amplitudes can be estimated by using the hard corrugated wall model with a shape function taken in the form of [1] f0 …R† ˆ 2f10 f cos‰G0 xŠ ‡ cos‰G0 yŠ ‡ cos‰G0 …y ‡ 2f11 f cos‰G0 …y ‡ x†Š ‡ cos‰G0 …x ‡ cos‰G0 …2x

y†Šg;

x†Šg

2y†Š …7†

 f11 ˆ 0:0081 A,  G0 ˆ 2p=a. with f10 ˆ 0:0098 A, The results of the di€raction patterns' shape calculation for the one-phonon inelastic helium  1 by xenon overlayer scattering with ki ˆ 11:05 A are shown in Fig. 4. The calculation has been carried out at a normal incidence of the molecular beam hi ˆ 0° [1]. In the case considered, the multiphonon processes can be neglected, since its conditions satisfy the relation to [5] lg Ez Ts < 0:01; M kB h2D

…8†

where Ez is the characteristic energy of gas atoms, Ts is the surface temperature, hD is the Debye temperature, lg is the mass of the gas atom. Corresponding data agreement of the calculated di€raction patterns with the experimental data [1] indicates the adequacy of the threads model for the description of weak-coupled overlayer dynamics. Accordingly, the discussed model turns out to be a

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Acknowledgements We are grateful to professor M. Khokonov for fruitful discussions and we are indebted to L. Gerrard for assistance.

References

Fig. 4. Angular distribution for the inelastic scattering of hep p  lium by a … 3  3†R30°. Xe overlayer structure on graphite at the normal angle of incidence. The surface temperature is 17 K. The dashed line is the numerical calculation, the solid line is the experimental data [1].

convenient starting point in the research of inelastic di€raction scattering in the case of weakly coupled, adsorbed overlayers.

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