Phonon resonances in adsorbed thin layers

Vacuum/volume

314Ipages 323 to 32711997 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0042-207x/97 $17.00+.00

Pergamon

48/number

PII: SOO42-207X(96)00282-6

Phonon resonances

in adsorbed

thin layers

P Zieliriski,* E Oumghar and L Dobrzyriski, Laboratoire de Dynamique etstructure URA CNRS 801, Universitk de Lille I, 59655 Villeneuve d’Ascq, France

des Mat&iaux

Mol&ulaires,

The explicit formulae for the Green functions are given for the bulk crystal, for a crystal with surface and for finite-thickness thin layers adsorbeb’on a surface as a function of microscopic interaction parameters. The case of the (0Ol)Fe surface and of epitaxial layers of Won the same surface is studied on the basis of the experimental dispersion curves. The vibrations of atoms at any distance from the surface are characterised by the resulting local density of states. A number of resonances and of antiresonances are found for different depths into the crystal. 0 1997 Elsevier Science Ltd. All rights reserved

Introduction

Bulk Green functions

The presence of surfaces and of adsorbed thin layers influences the dynamical properties of the crystal by introducing localiscd phonon states of frequencies lying outside the bulk bands and by modifying the vibrational motions of the atoms at frequencies within the bulk bands. In the harmonic approximation the latter effect is mainly due to the mixing of waves of different wave vectors, which results from breaking the translational periodicity of the system in the direction perpendicular to the surface. As a consequence, the local density of states (LDOS) for the layers close to the surface becomes different from the corresponding density of states in the bulk. These features are now studied by scattering methods such as inelastic scattering of electrons,’ He atoms,2 etc., which make use of radiation of a limited penetration depth. In analogy to the localised states, the maxima of the LDOS within the bulk band are called resonances and the minima are called antiresonances. In this paper we present a Green function method of calculation of the LDOS for atoms lying at any depth into the crystal. As an realistic example we study the surface (OO1)Fe and thin epitaxial layers of W adsorbed on the same surface using the microscopic force constants known from the bulk dispersion curves for these materials.“ Since no experimental data exist at the moment on the interaction parameters between the Fe and the W atoms in epitaxial systems, we have chosen to put the corresponding force constants equal to the arithmetic means of the force constants for both materials. The epitaxy of both isostructural bee crystals is assumed to be perfect, which amounts to the neglect of expected surface discommensurations which can arise due to a difference in the lattice constants. Within a purely harmonic approximation the resulting strain in the W layer does not affect the force constants, which are the only microscopic parameters needed in the present theory. -.. ._-. -. .-. *Permanent address: Institute of Nuclear Physics, ul. Radzikowskiego 152, 3 l-342 Krakbw, Poland.

The bulk Green function G&, I,,l,‘;w) for a system with surfaces perpendicular to the spatial direction I,, is the inverse of its k,Fourier-transformed dynamical matrix H,,(k,,, /,,&‘;a), where k, is the wave vector parallel to the surface. The indices I’ and i’ = 1,2,3 correspond to the components of the atomic displacements. If the range of the atomic interactions is finite, one can derive an explicit formula for the bulk Green functionsV6

(1) where z,~are the roots of the determinant det(D(k, ,z;w)) = 0, of the (kli,k3) Fourier-transformed dynamical matrix D in which the quantity e’/‘i, is replaced by z. Only the roots with Iz,I _<1 should be taken in eqn (1). The constant B is the coefficient which multiplies the highest power of z in the expression for the determinant det(D(k,,,z;o)). Th-e number M of the roots is finite for any finite range of the atomic interactions. The quantities A,,(k,:z;w) are the cofactors of the dynamical matrix D(k ,z;w). The condition Iz,,I< I for every n = l,...M defines the frequency region outside the bulk bands. The number of the roots with Iz,I = 1 counts the number of the bulk bands for the given frequency 0. The quantity

#(k;

,l,;n)) = -$I

G$;‘(k,,,l,,f,,cti)

for every Iz,,] = 1 represents

the contribution

(2) to the LDOS due to 323

PZielihski

et a/: Phonon

resonances

in adsorbed

thin layers

j

_---_

Frequency

,

I

I

6

6

:

,t,..j .’

10

x 1012 Hz

(b) 0.20

f

2

0.15

i i I f 3 :: :: :: ; ;, :: : :

f d 6 .G I 8

0.10

0.05

0.00 2

j : .::.

I

I

I

4

6

8

10

x 1 012 Hz

Frequency

(c) 0.20

ij . ;;:

f

,#

0.15

% 5 z

0.10

B c g 0

0.06

0.00

I i:! :;: .5 :: :: i;

;i : : : : :: : : : :

2

:. .’

/

--,j ,......”

./

...~.~~.~..~.“~...~.~~~~~

:

.\x

I

I

I

I

I

I

3

4

5

6

7

6

Frequency

(. ;

9

10

x 1 012 Hz

Figure 1. LDOS q&,/p) for i = x (a), i = y (b) and i = z (c) for the two first layers 1, = l(-on the clean (OO1)Fe surface.

324

‘: ; :

f

-)andf,=2(---)mdforthebulkcrystal(.

.)

PZielirkki

et al: Phonon resonances in adsorbed thin layers

(a)

6 Frequency

x 1012 Hz

(b) 0.20

E

0.15

,fi f $2 ul

z

0.10

i i : :

p 8

0.05

~~ 0.00

I 6

I

2

. ... ... .... ...... .... ...

4 Frequency

2 ii :: :: .... i ; I

a

10

x 1 012 Hz

03 0.20

=

2

0.15

1 4

0.10

z

E d

0.05

0.00 4

6 Frequency

Figure2. Same as in Figure

1 but for a one epitwial

8

ld

x 1012 Hz

layer of W on the (001)Fe surface

325

P Zieliriski et a/: Phonon resonances in adsorbed thin layers

2

4

a

6 Frequency

x 10”

Hz

Frequency

x lOi

Hz

10

(b) 0.20

A 3 z d rn z

0.15

0.10

0.

.G E 8

0.05 ! 0.00

-I2

10

@I 0.20

2

4

6 Frequency

Figure 3. Same as in Figure

326

1 but for three cpitaxial

adlayers

x 1012 Hz

of W on the (001)Fe surface.

PZieliriski et al: Phonon resonances in adsorbed thin layers the corresponding band for the given direction i of the atomic displacement. The LDOS related with the same direction are identical for every atomic plane I3 as long as the crystal is infinite. Surface Green functions In the case of a crystal with surface the indices I, and 13’ take only positive values &,I,’ = 1,2,3,... This range of indices will be denoted by D. Moreover, some elements of the k,-Fouriertransformed dynamical matrix of the system are perturbed by the cleavage and by the replacement of the substrate atoms with those of the adsorbed material in a finite region of f, = l...,N atomic planes. The total dimension of the dynamical matrix perturbed by the surface and/or by the adsorbate is therefore 3N. Using the above definitions of the index regions one can write the following schematic formula for the Green function in the crystal with surface or with an adsorbed thin layer’ g(D,D)=

G(D,D)-

G(D,3N)A-'(3N,3N)

x V(3N,6N)G(6N,D),

(3)

where G(D,D) is the bulk Green function (eqn( 1)) limited to the range of positive indices 1, and I,‘. The expression V(3N, 6N) is the matrix that should be added to the initial k,,-Fouriertransformed dynamical matrix of the bulk system so as to obtain the semi-infinite dynamical matrix of the system under consideration. One should remark that there are always some nonvanishing elements of the matrix V in the region I,> 1 and I,‘< I, which should be taken into account. That is why the second dimension of the matrix V has been indicated as 6N. Finally, the matrix A(3N,3N) is defined as follows

A(3N,3N)= I+ V(3N,6N)G(6N,3N) where I stands for the 3N x 3 N identity density of states

(4) matrix.

Now the local

two-dimensional Brillouin zone, where a is the lattice constant. Figure 1 shows LDOS ~,,($,/+J) for i = x, y and z and for I3 = I,2 on the clean surface (OO1)Fe with no variation of the force constants near the surface. The latter assumption does not preclude a geometrical relaxation of the surface structure, provided that one rests within the harmonic approximation in which the force constants are independent of the atoms’ positions. For comparison we show the corresponding LDOS in the bulk material. There is a noticeable difference in the LDOS in the first and in the second lattice plane. A maximum in the q,,&,l,;w) close to the lower border of the bulk band is a trace of the pseudolocalised shear-horizontal acoustic wave. A very characteristic antiresonance behaviour of q,&,f,;o) is visible about 4.2 x lOI Hz. The analogous picture for one layer of W (Figure 2) adsorbed on the same surface shows generally similar features with, however, a tendency to shift the LDOS towards lower frequencies, which is a result of a greater atomic mass of the Tungsten atoms. A richer structure of resonances comes about for three epitaxial W adlayers (Figure 3). The results of the study of the obtained resonances through the whole two-dimensional Brillouin zone will be presented elsewhere.8 Discussion The present method of analysis of atomic motions in the systems of thin layers on crystalline substrates is relatively simple. It does not involve the kind of lengthy calculations needed when the Kramers-Kronig relation is used. It provides characteristics of the atomic motions at any desired depth into the crystal. The examples studied show that the atomic motions at different depths can vary significantly, which is a very important piece of information when analysing the results of scattering experiments with the use of radiations of different penetration depths. Acknowledgements P. Z. thanks Conseil Regional Nord-Pas de Calais (France) granting his stay at the University of Lille 1.

will become a function

of the depth [, into the crystal.

Application to the (0Ol)Fe surface with epitaxial W layers The structure of the bulk bands and the dispersion of the surface localised phonons at (OO1)Fe has been obtained6 with the use of the force constants fitted to the experimental bulk dispersion curves for the bee Fe crystaL4 To illustrate the method described above, we present the I, dependence of the LDOS (eqn (5)) in the section of the bulk band at the point k = (0.7/u, 0.05/u) of the

for

References I. Ibach, H., Surf. Sci., 1994, 299/300, I 16. 2. Benedek, G. and Toennies, J., SurJ Sci., 1994, 299, 5X7. 3. Masri. P. and Dobrzyriski, L.. J. Whys. Chem. Solids, 1973,34,847. 4. Castiel, D., Dobrzytiski, L. and Spanjaard, D., Surj: Sci., 1976, 59, 252. 5. Zicliriski, P., Php Rer:, 1988, B38, 12338. 6. Zielitiski, P. and Dobrzyybski, L., Phys. Rec., 1990, B41, 10377. 7. Dobrzyriski, L., Surj: Sci. Reps, 1986, 6, 119. 8. Zielitiski, P. et al., to be published.

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