Surface green function matching approach to the surface dynamics of ionic crystals

253

Surface Science 143 (1984) 253-266 North-Holland, Amsterdam

SURFACE GREEN FUNCTION MATCHING APPROACH TO THE SURFACE DYNAMICS OF IONIC CRYSTALS II. Theoretical analysis of the inelastic scattering of He from NaF(OO1) in the eikonal approximation A.C. LEVI Gruppo Nazionale di Struttura de/la Mate& de1 Consiglio Narinnale de& Ricerche, D~p~rtjmento di Fzsica deli’ University Via Dodecaneso 33, I- 16146. Genovo, ita!v

G. BENEDEK

and L. MIGLIO

Gruppo Narionale di Struttura della Materia de! Consiglio Nazionale delle Ricerche, Dipartimento Fisica dell’ Unioersii& Via Celoria 16, I - 20133 Milano, Italy

di

and G. PLATERO, Institute de F&a Madrid, Spain Received

V.R. VELASCO

and F. GARCIA-MOLINER

de1 Estado Sbhdo (CSIC-UAM),

22 December

1983; accepted

Unrvrrsidad Autbnoma

for publieation

26 March

de Madrid, Cantoblanco,

1984

A theoretical analysis of recent scattering time of flight (TOF) data for NaF(OOl)/He is done by means of a new version of the eikonal approximation to the inelastic processes, which allows for the use of atom-surface potentials obtained by summing over two-body potentials. A Morse potential for He-ion interactions is adopted in the present work. A good agreement between theory and experiment is found with the present scheme.

1. Introduction The recent experiments on the inelastic scattering of atoms from ionic crystal surfaces have disclosed for the first time the dispersion relations of surface phonons [l-3] and raised new interesting problems. As pointed out long ago [4], the scattering intensities as deduced from time-of-flight (TOF) spectra reflect the surface-phonon densities in a distorted way due to the rapid variation of the coupling constants with momentum and energy transfers. The centrai problem in the interpretation of TOF spectra is actually the evaluation of the inelastic coupling constants. For strongly corrugated surfaces, the exact solution of the scattering problem using numerical techniques such as utilized 0039~6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

by Garcia et al. (GR method) [5] was considered so far more adequate than the distorted-wave Born approximation (DWBA) [6] or the eikonal approximation {7], even at the cost of an oversimplified picture for the atom-surface potential, such as the hard corrugated surface (HCS) model [8]. However, a systematic analysis of He inelastic scattering from LiF(OO1) in the HCS scheme [9], while very satisfactory with regard to the general spectral features, does not fully account for a few but significant details such as the intensities of optical surface modes. In this respect the nature of the atom-surface potential plays a crucial role, and should be incorporated in the theory at least in an approximate way. Moreover, monitoring various surface-atom potentials on the inelastic intensities provides a deeper physical insight into the coupling mechanisms. fn this work, as an application of the dynamical theory illustrated in the previous paper IlO] (hereafter referred to as Part I), we analyze the He scattering TOF data for NaF(OO1) recently reported by Brusdeylins, Doak and Toermies [2,3], by means of the eikonal approximation (EA) extended by one of us [7] to the inelastic processes. The present version of the EA, illustrated in section 2, allows for the use of atom-surface potentials obtained by summing over two-body interactions. A Morse potential (MP) is adopted for He-ion interactions. The anaIysis of selected experimental data, corresponding to resonance-free conditions, is presented in section 3.

Inelastic scattering of atoms from surfaces is a matter of considerable theoretical complexity, so that an exact treatment is usually out of question, and approximations are unavoidable. The interaction between an atom and the surface of a solid occupying the half-space z < 0 may be considered to consist of three contributions: (1) the surface-averaged, z-dependent potential causing the atoms impinging on the surface to be reflected back into the vacuum, but also containing the Van der Waals or possibly chemical attraction of the atom to the solid; (2) the corrugation, i.e., the periodic x--v-dependent interaction due to the atomicity of the crystal, and (3) the dissipative interaction due to the displacements of the ions from their equilibrium positions. ff both (2) and (3) are small, a perturbative approach is applicable and the appropriate approximation is the DWBA. Otherwise different approximate methods are to be used and the eikonai approximation, although elementary, is an interesting example of them [73. Quite generally, the inelastic scattering probability for an atom to go into the solid angle dS1 losing the energy )1w may be written dP =$$$

fi

dSE d(Aw) Li

/i” exp( -iwt) -CQ

(TL.&, (0)TkI.,,(t)) dt, (2.1)

A.C. Levr et al. / SGFM approach to surface dynamics.

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255

where L is a quantization length, m the atomic mass, and ttki and hk, the initial and final momenta, respectively; Tk,,k is the exact T matrix for the scattering process labelled by the atomic wave vectors (still an operator over the quantum states of the crystal); the time evolution is driven by the crystal Hamiltonian and the average is over a canonical ensemble of initial crystal states. Approximations can then be obtained by giving convenient approximate forms of T. In particular, the eikonal approach may be obtained in the following steps: (i) First of all, the exact, inelastic, T matrix is replaced by the elastic T matrix of the potential produced by the vibrating solid at time t. Such approximation is reasonable, provided the atom is definitely faster than the ions in the crystal, which is usually the case for He scattering. (ii) In the second place, the T matrix is written as an integral over the surface coordinate R (i.e., over the impact parameters in the beam). This representation does not involve any approximation:

(2.2) where 77is the phase shift and S is the source function. (iii) Thirdly, the source function is assumed to be S(R)

= -ik,,/m,

(2.3)

where k,, is the projection of ki onto the local normal to the surface. The above three steps lead to the EA, which is further completed by evaluating the phase shift q(R). This may be considered to be a sum of three terms: ~(R)=v,

(2.4)

+1)2+77x.

9, is the phase plane 77, =Q.R,

shift (with respect

to a reference

trajectory)

for a reflecting

(2.5)

where Q is the parallel component of the momentum transfer, n2 is the contribution of the static potential and n3 contains all the information about the crystal vibrations. In the case of a hard wall potential v2 and q3 would be q2=qzl(R),

(2.6a)

73 = 4; K(R,t)

(2.6b)

(c(R) being the shape function of the surface, 6l( R, t) its variation due to the thermal vibrations and qz the perpendicular component of the momentum transfer). In a more general case for q2, (2.6a) may still be used provided l(R) indicates the surface of classical turning points (of course l(R) depends on

energy

in this case). As far as q)13is concerned,

73 = C%(R)

the expression (2.7)

.u,(r)

will be used. Formula (2.7) has quite genera1 validity (see appendix A), although in the case of a hard wall it may be derived from (2.6b) (see appendix B). In the genera1 case, Aq,( R) has the physical meaning of momentum transfer from the impinging atom to the Ith ion in the crystal when the atom hits the surface at R, while u,(t) is the displacement of the Ith ion at the time t. In turn q, may be taken to be proportional to F,, the force that the Ith ion exerts on the atom at (R, l(R)):

q/(R) = [q,/E(R)l F,(R),

(2.8)

where Fz is the total force in the z direction on the atom at the same point. Eq. (2.8) descends from the assumption that the interaction time At is short and is the same for all the ions involved. In this case I

I

and therefore q,=A@,=

(q,/‘F-)F,

Eq. (2.8) has the advantage that the right-hand side can be explicitly evaluated in terms of a specific pair potential mode1 for, say, the He-Nat and He-F-~ interactions, so that the evaluation of the quantities q, requires no further assumptions. This means that, although the static part of the potential is still given by a corrugation function c(R), in the present version of the EA the coupling to the phonon modes goes beyond the hard wall mode1 and is effected in terms of the gradient of a realistic continuous potential as in the DWBA, with, however, the correct static phase shift incorporated (compare formula (2.10) below: the DWBA result would be obtained when neglecting q2 in the exponent). In the form of the present EA the framework is quite simple, hence the computer programs are easy to handle; also the potential shape is incorporated in a simple and physically understandable way. However, the approximation is rather inaccurate, especially for large corrugations; bound state resonances are difficult to incorporate, so that they are not included. In the present work Morse two-body potentials are used, the relevant parameters being c(He-Na+) c(He-F-)

= 15.7 K, = 24.5 K,

a(He-Na’) a(He-FP)

= 2.36 A, = 2.98 A.

For this potential the surface is rather flat, the largest corrugation occurring along (llO), where l,, z O.lA. The total force F is finally obtained by a simple

A.C. Levi et al. / SGFM approach to surfuce dynamics. II

summation of such pairwise potentials. The single phonon scattering probability

251

is then given by (2.9)

where the subscripts coordinates.

s and s’ label ionic

species

and (Y and /3 are Cartesian

He/NaF(OOlt e,=24.0”

ki= 6.01 A-’

I

-2

0 0

2

I lOI3 rad s“ I

Fig. 1. Experimental (above) and theoretical (below) inelastic scattering intensity for He from a NaF(OO1) surface at 300 K in the plane along (100) for a 90 o scattering geometry, incidence angle k, = 6.01 A-‘. The experimental spectrum is taken from 8, = 24.0 O, and incident momentum Doak et al. [2,3]. E labels the position of the elastic peak; S, are delta peaks corresponding to Rayleigh waves.

258

A.C. Lear et ul. / SGFM approach to surjuce dynmws.

II

The vector c, is r,=xexp(-iQ.R,I

I+‘)!

q,(R)

exp[iQ.R+iq,(R)j

dZR.

(2.10)

unit cell

Here W is the Debye-Waller is proportional to a spectral

exponent and q,(R) is given by (2.8) while A ,,, density tensor p,,, of vibrations:

P,,,(Q>~) (hf,fb~,~)~/’ w[ew(~w/k,T)

A.,,(Q.u) =

47T3A

- 11

cos[Q.(R,-R,,)].

(2.11)

where R, is the position of the ion of species .r within the unit cell. Considering the equivalence proved in Part I, between the surface Green function matching and the invariant Green function (IGF) methods [11.12].

w

I

1Ol3 rad s-l

I

Fig. 2. As fig. 1 for 0, = 30.5 o and Ir, = 6.00 A a Kayleigh mode at the zone boundary.

‘. The S, peak at 2.4~

10”

rad s-l

corresponds tu

A.C.

Levi et al. / SGFM

approach

to surface dynamics.

II

259

the evaluation of n,_, matrix elements is essentially based on the technique used for the IGF method in the framework of the breathing shell model [13]. Results for the scattering amplitude from the NaF(OO1) surface are discussed in the following section.

3. Analysis of He scattering data from NaF(OO1) surface In this section the results of these calculations based on the Morse potential and the eikonal approximation are compared to room temperature TOF spectra of He scattering from the NaF(OO1) surface. A large set of data has been recently obtained by Doak, Brusdeyhns and Toennies with low-energy (k, = 6 A-‘) and high energy (k, = 11 A-‘) beams. A selection of TOF

He/NaFcOOl) ki= 6.OOii-’

ei- 31.5”

0

2 0

4

6

I 1Ol3 rad s-l I

Fig. 3. As fig. 1 for 0, = 31.5 o and k, = 6.00 A-‘. occurring at the zone boundary.

The peak labelled

S, is the crossing

mode

260

and ki = 6.32 A-‘. The sharp experimenfal peak at w =O Fig. 4. As fig. 1 for Bi = 43.0° corresponds to the elastic diffuse scattering. S, denotes the optical Lucas mode with sagittal potarization.

spectra for a low-energy beam scattered along the (100) direction is shown in figs. I-5 (above) as function of the transferred frequency o together with the calcufated spectra (below). The values of the incidence angle Bi, are chosen in order to avoid inelastic resonances, since no account of their effect is taken in the present theory. Figs. 1 (Bi = 24O ), 2 (8, = 30.5 CJ) and 3 (B, = 31.5 “) refer to processes occurring around the diffraction peak for G = (%,I). The dominant features, consisting in ail cases in the Rayleigh wave peaks (S,), occur for positive w (phonon annihilation}. In fig. 1 the S, peak on the creation side because of (w- - 1.7 x 10’” rad/s) is not detected in experiment presumably the large noise &vet in this region. The intensity of the other two S, peaks are in remarkable agreement with those of the experimentat spectrum. Here the weaker peak corresponds to an~~ihi~ation of a backward phonon, the stronger to a forward phonon. For 8; = 30S0 (fig. 2), the backward S, phonon becomes the strongest one, in fairly good agreement with theory. Here we have two S, peaks on the

A.C. Levi et al. / SGFM

approach to surface dymmcs.

II

261

HelNaFfOOl)

0

2

4

o I 1013 rad s-’

6

1

Fig. 5. Same as fig. 4 for 19, = 46.70 and k, = 6.33 A-‘. experimental spectrum is also shown magnified 8 times.

The high

frequency

region

of the

forward annihilation side. The one at 2.4 X 1013 rad s-‘, corresponding to a zone-boundary Rayleigh wave, has apparently a too large theoretical intensity. With a slight increase of 8, to 31.5’ (fig. 3) a kinematical focussing condition [14] is approached and two S, peaks become closer one to another. Moreover, theory shows an additional peak for the crossing mode S, [15] which corresponds, together with S,, to the experimental bump at 2.2 X 10” rad SC’. The two following spectra (figs. 4 and 5) are chosen slightly below (ai = 43.0 ’ ) and above ( Bi = 46.7 o ) the specular condition and show intense scattering from Rayleigh phonon creation and annihilation processes, respectively. Note the position of the peaks with respect to the elastic peak at w = 0 in the two cases. The theory gives correctly the relative intensities of the S, peaks, the stronger one being close to the elastic peak. In this effect the Bose-Einstein factor plays the main role. The theoretical intensities on the high frequency side are magnified ten times; in the measured spectra some feature correspond-

262

A. C. Levi et crl. / SGFM upprmch

to sur@e

dynumm.

II

ing to S, is seen, while no structure is visible for the sharp optical phonon S, (fig. 4). A global inspection of figs. l--5 demonstrates a good agreement between theoretical and experimental structures also with regard to the intensities and frequency distributions originating from bulk modes. They are well reproduced by the theoretical calculation indicating that the one-phonon processes are dominant. However, the few misfits occurring in the amplitudes of surface localized modes have to be considered very attentively, since their analysis can provide a guideline for improving the atom-surface potential model. This is due to the strong dependence of the scattering amplitude from surface localized modes upon the expression for atom-surface coupling. The spatial behaviour of the interaction potential between the noble atom and the array of ions is here approximated by some model, but a realistic form is needed in order to achieve a fully satisfactory result for the scattering probabilities, particularly for

1

r

He/NaF(OOl)

---.-_I 0 0 I 1Ol3 rad s-l

4

I

Fig. 6. Same as fig. 1 for 6’ = 66.9” and a high energy beam with k, =11.18 A-‘. Here the multiphonon process are dominant. The S, peak on the annihilation side is so much broadened that it cannot be resolved from the elastic peak. On the other hand creation processes in the optical region become more intense. S, denotes a surface longitudinal optical mode.

optical surface modes. Indeed, the theoreticai inelastic intensities present large oscillations for a varying momentum transfer; such oscillations are related to the potentials parameters such as the corrugation and the softness. Another reason which makes the comparison of local mode intensities difficult, is the sensitivity of these latter to the change in incidence angle 8,. Thus small incertitudes in the experimental determination of Bi may produce strong ~uctuations in the local mode peak height. We display in fig. 6 ( Oi = ~36.9~) one experimental spectrum for a high energy incident beam (ki = 11.8 A-‘) where the multiphonon processes are dominant [16], so that the one phonon line-shapes are distorted. The incidence -angle is large and the structure refers to the region of G = (2,2) processes; we do not display any spectrum for the G = (i,i) region because of the presence of strong resonances as in the LiF case. It is interesting to observe the light appearance of some optical modes, which are fairly well predicted by the calculated scattering amplitude (S, and S, modes). However, much better experimental results have been obtained recently for a cold NaF(OO1) surface;

o ( lOi

rad/s)

K (t@cm-1) Fig. 7. Real and imaginary parts of the coupling coefficients, eq. (2.10), for ion index K = + (Na+ ) and - (F- ), and Cartesian index n = x and L, as functions of momentum transfer (lower abscissa scale) and energy transfer (upper abscissa scale).

264

A.C. Leclr et al. / SGFM approach to surf&e &wumic:\. II

they will be presented in a forthcoming paper together with the theoretical interpretation [17]. One final observation is made about the coefficients I’, (eq. (2.13)) as a function of Q and w. As stated above, the oscillations of the coupling coefficients produce a considerable distortion of the scattering amplitude from the shape of the surface phonon density of states p,,, (Q.o). An example is shown in fig. 7 for the parameters corresponding to the high-energy scattering data of fig. 6. This and the preceding paper demonstrate the usefulness of embodying the surface lattice dynamics in the inverse surface projected Green function as a suitable way of introducing in the analysis the first theoretical ingredient. i.e.. surface phonons. This must be supplemented by a reliable approximation to construct the scattering amplitudes, which, as seen above, the MP-EA does rather satisfactorily.

Acknowledgements This work has been performed in the framework of the Scientific Cooperation Agreement between the Consejo Superior de Investigaciones Cientificas (CSIC), Spain, and the Consiglio Nazionale delle Ricerche (CNR), Italy. Dr. R.B. Doak (Bell Laboratories, Murray Hill, NJ), Dr. G. Brusdeylins and Professor J.P. Toennies (Max-Planck-Institut fir Str~mungforschung, Gbttingen, FRG) are gratefuily acknowledged for several discussions and detailed information on their experimental data quoted in refs. [l-3].

Appendix A In the EA (or in any semiclassical theory) the relevant quantity ing is exp(in). It is natural to take n3, the vibration-dependent phase, to be a uuriution 677= (l/h)

a(R),

where 5’ is the classical action for a trajectory be written in terms of ionic displacements

But the variation 66p= cr;,

for scatterpart of the

* Uf,

of the Lagrangian

is a virtual

impinging

at R. 6s in turn may

work (A.2)

A.C. Levi et al. / SGFM approach lo surface dynamics. II

265

thus 6~(R)/Su,(t’)=F,(R,t’). Substituting

(A.3)

(A.3) into (A.l),

&~=+~j.,(t’)+,(R,r’) I

the general

result

dt’

(A.4)

is

obtained. In the case of a fast collision, the force F,vanishes unless t’ is very close to the time t, where the collision takes place. Thus u,( 1’) may be replaced by u,(t) and taken out of the integral, which then can be performed immediately, giving: v3

=

h

=

C~,CO

(A.51

%W

Appendix B We introduce a surface may be expressed as GS(R,t)

vector u(

R,t),in terms of which 8Y

= S(R).u(R,t),

where the vector S(R) (-&Vsx,Further

displacement

(B-1) has components

KPYJ). u is expressed

in terms of displacements

of the individual

ions:

u(R,t)=~X(R-R,)Yu,(~). Using Q+Qz

the classical

reflection

(B-2)

condition

vS=Q

(B-3)

and eq. (B.l), nJ is written: 83=4’~=cq,‘~,,

(B.4)

where q,=q.X(R-R,) can be interpreted

as having

the same physical

meaning

as in (AS).

References [I] G. Brusdeyiins, R.B. Doak and J.P. Toennies, Phys. Rev. Letters 16 (1981) 437. [2] R.B. Doak, PhD Thesis, MIT, Cambridge, MA (1981), unpublished.

266

[3] [4] [5] [6] [7] [S] [9] [lo] [II] [12] [13] 1141 [15] [16] [17]

A.C. Leuc et al. / SGFM apprmch

to sur&~

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G. Brusdeylins, R.B. Doak and J.P. Toennies, Phys. Rev. 827 (1983) 3662. G. Benedek and G. Seriani. Japan. J. Appt. Phys. Suppl. 2. Part 2 (1974) 545. N. Garcia. J. fbaiiez, J. Solana and N. Cabrera, Surface Sci. 60 (1976) 385. N. Cabrera, V. Celli. F.O. Goodman and R. Manson, Surface Sci. 19 (1970) 67. A.C. Levi, Nuovo Cimento 548 (1979) 357. G. Benedek and N. Garcia, Surface Sci. 103 (1981) L143. G. Benedek. R.B. Doak and J.P. Toennies, Phys. Rev. B28 (1983) 7277. G. Platero, V.R. Velasco, F. Garcia-Moliner, G. Benedek and L. Miglio. Surface Sci. 143 (1984) 244. G. Benedek, Surface Sci. 61 (1976) 603. G. Benedek and L. Miglio, in: Ab fnitio Calculation of Phonon Spectra, Eds. J. Devreese. V.E. van Doren and P.E. van Camp (Plenum. New York. 1982) p. 215. U. Schroder, Solid State Commun. 4 (1966) 347; U. Schroder and V. Niisslein. Phys. Status Solidi 21 (1967) 309. G. Benedek. G. Brusdeylins, R.B. Doak and J.P. Toennies, Phys. Rev. B27 (1983) 2488. G. Benedek, G.P. Brivio, L. Miglio and V.R. Velasco, Phys. Rev. B26 (1982) 497. H.D. Meyer, Surface Sci. 104 (1981) 117. and references therein. G. Brusdeyhns, R.B. Doak. J.G. Skofronick, J.P. Toennies, G. Benedek and L. Miglio. to be published.