Atom–surface band structure and bound-state resonances

Surface Science 409 (1998) 130–136

Atom–surface band structure and bound-state resonances A theoretical study M. Cristina Vargas a,*, W. Luis Mocha´n b a Departamento de Fı´sica, Cinvestav del IPN Unidad Me´rida, Apdo. Postal 73, Cordemex, 97310 Me´rida, Mexico b Instituto de Fı´sica, Universidad Nacional Auto´noma de Me´xico, Apdo. Postal 48-3, 62251 Cuernavaca, Mexico Received 23 January 1998; accepted for publication 20 March 1998

Abstract The energy bands of an atom adsorbed on a crystalline surface have been calculated for representative hypothetical systems. The corrugation corresponding to the atom–surface interaction potential of several of the studied systems is similar to the corrugation reported for systems in which the surface holds a layer of adsorbed particles. Characteristic features of the band structure, that depend on the corrugation of the atom–surface interaction potential and on the depth of the associated bound-state, have been found. The bands of all the systems evolve to parabolic shaped bands displaced downwards with respect to the corresponding freeatom parabolic bands. Theoretical studies of atom–surface elastic scattering revealed that bound-state resonances do not appear if the energy of the incident atoms is not high enough to reach the region where the band is parabolic. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Adatoms; Band structure; Bound state resonances; He scattering, diffraction – elastic

1. Introduction A recent study of band structure corresponding to an atom adsorbed on a model corrugated surface showed the importance of considering band structure effects in studies of bound-state resonances in atom–surface diffraction, for strongly corrugated systems [1]. Among other things, it has been seen that the corrugation corresponding to an atom–surface interaction potential causes the difference between the energy bands and the free particle parabolic bands. The corrugation of an atom–surface system is defined as the difference between the maximum and the minimum values * Corresponding author. Fax: (+52) 99 81 2917; e-mail: [email protected] 0039-6028/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 0 3 9- 6 0 28 ( 98 ) 0 02 9 2 -1

of z on the level surface defined by V(r)=l; the value of l in this paper is taken to be 0. Some characteristics of the atom–surface band structure have already been studied for systems of corruga˚ ; however, it is important to consider tion up to 1 A that nowadays there is great interest in the study of monolayers of atoms or molecules physisorbed on clean crystalline surfaces (for noble gases adsorbed on crystalline surfaces, see for instance Refs. [2–4]; for compound molecules adsorbed on crystalline surfaces, see for instance Refs. [5,6 ]). The reported corrugations for systems with a noble ˚ [7,8], and systems gas monolayer are around 1.5 A with a molecule monolayer are expected to have higher corrugations. Adsorbed particles are bound to surfaces with energies of the order of hundreds of millielectronvolts or even less [9]. Low energy

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He atoms may leave unperturbed the adsorbed monolayer and do not penetrate the surface, therefore He–surface diffraction techniques are very suitable to study these compound surfaces. Besides other things, the position of the He–surface boundstate energies could be obtained from bound-state resonance experiments, as in fact has been done with precision for weakly corrugated systems (see for example Refs. [10–13]). Nevertheless, there are only a few reported studies on He–(compound surface) bound-state resonances [7,14–18]. These few works focused on bound-state resonances; it has been accepted that, due mainly to the strong corrugation of the He–adsorbate-surface interaction potential, there is a lack of understanding of the position where the resonances appear. Besides, it has been stated that some deep boundstates corresponding to strongly corrugated systems may not produce resonance peaks [19,20]. Since the position where the resonance appears comes from the position of the energy bands, the band structure corresponding to such highly corrugated systems should then be studied. The aim of this work is to study the energy band structure corresponding to atoms adsorbed on model corrugated surfaces and the way these characteristics can manifest themselves through bound-state resonances in the diffraction experiments. Specifically, we want to focus on those characteristics related to the corrugation of the atom–surface interaction potential and to the depth of the ground-state corresponding to the surface-averaged potential. The band structure for an atom–surface system is calculated by solving an eigenvalue matrix equation, deduced elsewhere [1,21], obtained from the 3D Schro¨dinger equation for an atom in the presence of a crystalline surface: (H−E (K ))|Yl =0 (1) l K by writing its solutions as a superposition of 2D plane waves:

r|Yl = ∑ bl

r|n, K+G (2) K n, K+G n, G where r¬(R, z) is the position vector of the particle of mass m, z and R are its projections normal and along the surface, respectively; K is the parallel to

131

the surface component of the atom wavevector; {G} is the 2D reciprocal lattice of the surface; l is an index to enumerate the eigenfunctions; and

r|n, K+G=W (z) exp[i(K+G) · R] n is a product of an eigenfunction of the 1D Schro¨dinger equation:

C

−

B2 d2 2m dz2

D

+V (z) W (z)=e W (z) n n n 0

whose spectrum is given by {e }, and a plane wave n moving along the surface, whose wavevector is the sum of K and the reciprocal vector G. Multiplying the 3D Schro¨dinger Eq. (1) by n, K+G|, the following equation is obtained: ∑ n∞, G∞

GC

D

B2 (K+G)2 d ∞ d ∞ e + n 2m G, G n, n

+ n|V

G−G∞

H

(z)|n∞ bl∞ n , K+G∞

(3)

=E (K )bl . l n, K+G The eigenvalues E (K ) constitute the energy bands; l being

n|V

G−G∞

P

|n∞=

2

(z)W ∞ (z) dz W* (z)V n n G−G∞

−2 where V (z) is the coefficient corresponding to the k reciprocal vector k of the Fourier expansion for the atom–surface interaction potential: V(r)=V (z)+ ∑ V (z)eiG · R. 0 G G≠0 If we take a number N of reciprocal lattice vectors G in the wave function expansion (2), and there are M eigenstates W (z), corresponding to n the surface-averaged potential, V (z), the solution 0 of Eq. (3) for a certain wavevector K yields a number NM of eigenvalues E (K ). According to l Eq. (2), information about the states that are mixed to give rise to the eigenvalue E (K ) can be l found in the corresponding eigenvector bl of magnitude 1. Each component bl is associated to n, K+G a reciprocal lattice vector G and to an eigenstate W (z) of the surface-averaged potential V (z). As n 0 the band structure is periodic, we have found that

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the best way to compare the calculated bands with the corresponding free-particle parabolic bands is to focus on the eigenvalues E (K ) associl ated with eigenvectors whose maximum component is bl , that is, eigenvalues associated with n, K+0 state mixtures where the state for a given n and G=0, predominates. According to our interest, we classify the atom–surface systems into three types: (1) those with a ‘‘shallow’’ ground state; (2) those with a ‘‘deep’’ ground state; and (3) those with a ‘‘very deep’’ ground state. Then, we form a representative set of hypothetical systems for each one of the three types mentioned. All the systems belonging to the same set are characterized by the same surface-averaged potential, but each has a different corrugation. The range of corrugations includes ˚ and higher. values of 1.5 A The systems are described through the atom– surface interaction potential, which we model through an exponential corrugated potential: V(r)=Wea(u(R)−z)−C (1−e−z/f)3/z3 3 where

e =−3.03 meV, e =−1.01 meV and e = 1 2 3 −0.24 meV. The value of W for each system, member of a specific set, is determined by the corrugation parameter c=2aQ, being W=A/I 2 (c). 0 Band structures for different systems have been calculated along the 110 crystal direction ( 10 surface direction). The difference between the calculated band structure and the corresponding freeparticle parabola, Ef (K )=B2K2/2m+e , for n=0 n n and G=0, is presented in Fig. 1, as a function of the parallel wavevector K, for three elements (three different corrugations) of set 1. Figs. 2 and 3 correspond to set 2 and 3, respectively. The value of bl is also plotted in the figures. From these 0, K+0 results, the following general characteristics of band structures can be inferred. $ For small wavevectors, there are clear disconti-

(4)

u(R)=2Q(cos G x+cos G y) 0 0 is the corrugation function. The surface-averaged potential, V (z), is: 0 V (z)=Ae−az−C (1−e−z/f)3/z3 (5) 0 3 where A= Weau =WI 2 (2aQ), I being the modis 0 0 fied Bessel function of order zero. The parameters have been set to the following values [related to the He–NaCl(001) system]: surface lattice constant ˚ ˚ −1), l=3.99 A (and therefore G =1.5747 A 0 ˚ ˚ 3, a=4 A ˚ −1, f=0.75 C =106 meV A A, leaving A 3 as the characteristic parameter of each set. In this way, we take: $ set 1 (representative of type 1 systems) A= 17.12 eV; eigenstate energies: e =−3.71 meV, 0 e =−1.25 meV and e =−0.31 meV. 1 2 $ set 2 (representative of type 2 systems) A= 6.2 eV; eigenstate energies: e =−6.15 meV, 0 e =−2.45 meV, e =−0.76 meV and e = 1 2 3 −0.17 meV. $ set 3 (representative of type 3 systems) A= 4.59 eV, eigenstate energies: e =−7.24 meV, 0

Fig. 1. Displacement of the ‘‘lowest’’ calculated band (stars) with respect to the corresponding free-atom parabolic band, as a function of the parallel wavevector, for systems of set 1. (a) Corrugation parameter c=0.3 corresponding to a corrugation ˚ . (b) Corrugation parameter c=0.76; corrugation of 0.44 A ˚ . (c) Corrugation parameter c=1.5; corrugation 2.4 A ˚. 1.1 A Dashed marks correspond to bl . 0, K+0

M.C. Vargas, W.L. Mocha´n / Surface Science 409 (1998) 130–136

133

Fig. 2. Displacement of the ‘‘lowest’’ calculated band (stars) with respect to the corresponding free-atom parabolic band, as a function of the parallel wavevector, for systems of set 2. (a) Corrugation parameter c=0.3; corresponding to a corrugation ˚ . (b) Corrugation parameter c=0.76; corrugation of 0.47 A ˚ . (c) Corrugation parameter c=1.1; corrugation 1.8 A ˚. 1.2 A Dashed marks correspond to bl . 0, K+0

Fig. 3. Displacement of the ‘‘lowest’’ calculated band (stars) with respect to the corresponding free-atom parabolic band, as a function of the parallel wavevector, for systems of set 3. (a) Corrugation parameter c=0.3; corresponding to a corrugation ˚ . (b) Corrugation parameter c=0.76; corrugation of 0.47 A ˚ . (c) Corrugation parameter c=1.1; corrugation 1.8 A ˚. 1.2 A Dashed marks correspond to bl . 0, K+0

nuities around the boundaries of the Brillouin zones corresponding to band splittings. For systems with low corrugations, the discontinuities can be appreciable only for the first three or four Brillouin zones; but for those with high corrugation, the discontinuities remain appreciable for higher order Brillouin zones. For large wavevectors, all the calculated bands recuperate the parabolic shape, but they remain below the free-particle band, at an approximated constant distance, as found previously [1]. The deeper the bound-state or the higher the corrugation, the larger the wavevector at which the bands recuperate the parabolic shape. The value of bl which presents great varia0, K+0 tions for small wavevectors reaches an almost constant value after the band recovers the parabolic shape.

The displacement, both at the bottom of the band (K=0) and at large wavevectors ˚ −1), depends upon the depth of the (K~8 A bound-state corresponding to the surfaceaveraged potential and upon the corrugation of the system. The deeper the bound-state or the higher the corrugation, the larger the displacement. The last feature can be seen better in Figs. 4 and 5. The former shows, for each of the three formed sets, the displacement at the bottom of the lower band relative to the free-particle band, Ef (0)−E (0), as a function of the corrugation; 0 l the latter shows the same comparison, but this time the displacement, Ef (K )−E (K ), is plotted ˚ −1. It can be 0 seen in lFig. 5 that the for K=8 A displacement of the band for weakly corrugated systems is very small, in agreement with the fact

$

$

$

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M.C. Vargas, W.L. Mocha´n / Surface Science 409 (1998) 130–136

Fig. 4. Displacement at the bottom of the lower band relative to the free-particle band, Ef (0)−E (0), as a function of the 0 l corrugation. Plots (a), (b) and (c) correspond to systems of set 1, 2 and 3, respectively.

Fig. 5. Displacement of the lower band relative to the free˚ −1, as a function of particle band, Ef (K )−E (K ), for K=8 A 0 l the corrugation. Again, plots (a), (b) and (c) correspond to systems of set 1, 2 and 3, respectively.

that free-particle approximation has been successfully used to study that kind of system [10–13]. In order to study how band structure features manifest themselves through bound-state resonances, we have performed close coupling diffraction calculations [22] for several of the representative systems, with incident atoms of different energy. We have found that the region where the band is not parabolic yet is a critical region since a correspondence between the close coupling calculated diffraction peaks and the expected resonance position according to the calculated band structure cannot be found. That is, if incident atom energies fall in this critical region, the expected resonances do not show up as peaks of intensity in the diffracted beams. On the other hand, if the incident atom energy falls in the region where the bands have parabolic shape, bound-state resonances seems to appear in the same way for strongly and weakly corrugated systems. It is important to

remark, though, that for very weakly corrugated systems, the free-particle approximation (assuming free-particle parabolic bands) would be suitable to get, from selective adsorption experiments, the energy corresponding to the bound-states of the atom onto the surface, while for more corrugated systems, the bound-states obtained in the same way should be corrected through a consistent analysis as proposed in ref. [1]. These results are illustrated in Fig. 6, where we present close coupling calculation results for the more corrugated studied system of set 1, whose bands are presented in Fig. 1c and again in Fig. 7, where the displacement of the calculated band with respect to the free-particle band is now plotted as a function of the energy of the calculated band. In Fig. 6, we present, for different energies of the incident beam, a plot of the relative intensity of the specular beam versus the incident polar angle, in the proximity of the expected resonance with the ‘‘lowest’’ band through the reciprocal vector

M.C. Vargas, W.L. Mocha´n / Surface Science 409 (1998) 130–136

Fig. 6. Intensity of the specular beam (in arbitrary units) of He atoms of different energies, versus incident polar angle h. Azimuthal incidence is in the 110 crystal direction. Vertical marks indicate the expected position for the resonance, through the reciprocal vector G=(1, 0)G . The numbers at the left indi0 cate the energy of the incident atoms given in millielectronvolts.

Fig. 7. The data plotted in Fig. 1c, displacement of the ‘‘lowest’’ calculated band, DE=Ef (K )−E (K ), are now plotted as a 0 l function of the energy of the calculated band itself, E (K ). l System belonging to set 1, corrugation parameter c=1.5 corre˚. sponding to a corrugation of 2.4 A

135

G=(1, 0)G . The incident beam is set along the 0

110 crystal direction ( 10 surface direction). Vertical marks indicate the expected position for the resonance, according to the calculated band position. It can be seen that for energies lower than 19 meV, the expected resonance peaks do not appear, even when the band has recovered the parabolic shape by the region of energies around 17 meV. It is interesting to notice that the peaks start to appear when the value of bl reaches 0, K+0 an almost constant value, which still changes locally each time there are band-crossings. Therefore, the critical zone boundary seems to be around the point where the mixing of the states that give rise to the ‘‘lowest’’ band reaches some kind of stability. These observations yield the conclusion that the selective adsorption phenomenon for highly corrugated systems may be studied in the same way as for not so corrugated systems, provided that the incident particle energies lie beyond the critical region and the analysis of the experimental data is done by considering band structure effects. We understand, however, that for high energies the number of diffracted beams grows, giving rise to technical difficulties, for both experimental and theoretical work. Besides, for some energies outside the critical region (20, 22, 24 and 25 meV ), the expected resonance does not show; and for others (21 meV for example), the resonant peak is hardly visible and therefore may be very difficult to observe experimentally; although theoretical data show relative changes of order 10 or higher. These findings are consistent with the arguments of Benedek et al. [20] about the fact that some bound-state resonances may not be observed in the elastic scattering. As the critical region becomes wider due to two factors, the corrugation of the system interaction potential and the depth of the bound-state, it is important to remark that it is not the depth of the bound-state alone that makes ‘‘invisible’’ a resonance: there could be observable deep bound-states, associated with weakly corrugated potentials; and not observable shallow bound-states, associated with strongly corrugated potentials. In conclusion, we have found that band structure of atom–surface systems evolves to parabolic

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M.C. Vargas, W.L. Mocha´n / Surface Science 409 (1998) 130–136

shaped bands, even for highly corrugated systems, and that bound-state resonances may appear in the elastic scattering only when the energy of the incident atoms falls within the region where the bands are parabolic.

Acknowledgements We acknowledge several useful discussions with Professor J.C. Ruiz-Sua´rez.

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