SMALL PARTICLES-STRUCTURES APPLICATIONS GROWTHEFFECT OF SIZE ON METAL MICROCLUSTER PROPERTIES

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SMALL PARTICLES-STRUCTURES APPLICATIONS

GROWTHEFFECT OF SIZE ON METAL

MICROCLUSTER PROPERTIES

R. Baetzold

To cite this version:

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SMALL PARTICLES-STRUCTURE APPLICATIONS GROWTH.

EFFECT OF SIZE ON METAL MICROCLUSTER PROPERTIES

R. C. BAETZOLD

Research Laboratories, Eastrnan Kodak Company Rochester, New York 14650, U.S.A.

R & u d . - Nous passons en revue les propriBtBs klectroniques des microagrkgats d'argent et de palladium. Des mBthodes d'orbitales molBculaires semi-empiriques, comme la mBthode de Huckel 6tendue et la mkthode CNDO, prkdisent des effets Blectroniques qui diffbrent des propriBtBs du matBriau massif. Les Bnergies de liaison et les largeurs de bande des microagrkgats sont infkrieures 2 celles du matBriau massif. L'inverse est vrai pour les potentiels d'ionisation et l'intervalle entre niveaux d'knergie. Des oscillations du potentiel d'ionisation et de 1'Bnergie de liaison suivant la paritk du nombre d'atomes des microagrkgats, sont observkes pour des chaines de sodium ou d'argent, mais pas pour des agrkgats de palladium.

Abstract.

-

Electronic properties of silver and palladium microclusters are reviewed. Semi-empirical molecular orbital theories including extended Huckel and CNDO predict electronic effects differing from the bulk properties. Binding energies and bandwidths are smaller for microclusters than the bulk. The reverse is true for ionization potential and gap between energy levels. Oscillations in the ionization potential and binding energy between odd and even sized microclusters are found for Na and Ag chains, but not for Pd clusters.

1. Introduction.

-

Semi-empirical molecular orbital calculations using extended Hiickel (EH)

and complete neglect of differential overlap (CNDO) have been employed to examine the electronic properties of metal clusters. These methods which have been used successfully in quantum chemistry have been applied to transition metal clusters [I, 21' of Ag and Pd. We are primarily interested in the trends in electronic properties with cluster size as a guide to understanding catalytic behaviour. Parameters used in these calculations are determined by experimental data for the single atom or diatomic homonuclear cluster as a means of calibrating the calculation.

We will be primarily interested in the binding energies, ionization potential and band structure of small metal clusters. Knowledge of how these properties differ from bulk would be useful in helping to understand how such particles function as catalysts. This work, as with previous work, differs in conclusion from the recent work of Messmer, et al. [3] using the X, method. They found bulk-like electronic structure at metal clusters containing 8-13 atoms. The findings here for most properties are in variance with this view. We also note that a paper presented in these proceed- ings by Cyrot-Lackrnann, et al. [4] shows that calculated local density of states for 38-atom clusters is quite unlike the bulk, but that the shape of the density of states tends toward the bulk with increasing size.

2. Methods.

-

The semi-empirical methods have been explained [I, 51 and will not be reviewed.

Various parameterization schemes can be employed with the extended Hiickel method. In scheme A we employ exponents of the Slater orbitals as given by Clementi [6] and a K in the Wolfsberg-Helmholz formula designed to fit homonuclear binding energies at the potential minima. In scheme B, double zeta type d orbitals were employed with exponents given by Basch and Gray [7]. The value K = 1.75 is used in this case. All parameters used in the present work are listed in table 1. Throughout this paper we will refer to method A as an extended Huckel calculation using parameter set A, method B as an extended Huckel calculation using parameter set B and method C as a CNDO calculation.

3. Results. - 3 - 1 BINDING ENERGIES.

-

Bind- ing energies are calculated from a potential energy minima by comparison of the energy with that of the separated atoms. Figure 1 shows the calculated data for silver atoms using method A and the geometric form of straight chains (1-d), square planar (2-d) and three dimensional (3-d) clusters. The geometries considered were : 3-triangle, 4- tetrahedron, 5-bipyramid, 6-square bipyramid, 7-

bicapped pentagon, &cube, 9 and larger are for the face-centered-cubic (f .c.c.) structure. We find the order of stability 1 d

>

2 d

>

3 d which can be traced to repulsive interactions occurring in the occupied antibonding molecular orbitals. It is observed that clusters with chain structure and even numbers of electrons have greater stability than neighbouring clusters with one greater or lesser number of electrons. This behaviour is also observed for the small 2-d clusters, but not for the 3-d

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R. C . BAETZOLD

Parameters of the calculation

s P d

Element Exponent - Hii Exponent - Hii Exponent - Hii

-

- -

-

Method A Ag 1.35 7.56 1.35 3.83 3.69 11.58 Pd 1.57 7.32 1.57 2.00 3 -40 8.33 Na 0.84 5.14 1.49 3.04 Method B (*) Ag 2.24 7.56 2.24 3.83 6.07(.539) 1 1.58 2.66(.637) Pd 2.15 7.32 2.15 2.00 5.93(.526) 8.33 2.61(.637) Method C Ag 1.35 4.26 1.35 2.39 1.90 6.27

p =

- 1 N a p = - 1 0.84 2.61 0.84 1.56

(*) d orbitals for type B calculation have two exponents, with coefficients given in parenthesis.

Ag Clusters

0.01 I I I 1

0 10 30 50

Number of atoms

FIG. 1. - Binding energy per atom for linear (I-d), square (2-d) and three-dimensional (3-d) silver clusters calculated by EH,

K = 1.30 vs. number of atoms.

structures. This odd-even relationship is not observed for Pd clusters where the highest occupied molecular orbital (HOMO) is primarily of d character rather than the s character observed here for Ag clusters.

Figure 2 depicts binding energy results for clusters with different numbers of atoms in three- dimensional geometries. A reduced binding energy is plotted by dividing the calculated binding energy per atom by the bulk experimental cohesive energy. While there is considerable disagreement in absolute value between the various calculations, each curve has a value considerably less than 1. Also, each curve is increazing as a function of size suggesting that even larger clusters would be more

Number of atoms

FIG. 2. - Binding energy per atom calculated for three- dimensional Ag clusters divided by bulk experimental cohesive energy versus size. Curves A, B, C refer t o method of

calculation discussed in the text.

stable. Changes in binding energy per atom are greatest for small clusters and then become

smaller as the cluster size increases. Hence, while choice of parameters does influence the magnitude of the binding energies it does not change the fact that each curve is far from the bulk value. The calculated binding energies for Ag, compared to the experimental value [8] 1.63 eV are : method A

1.74 eV, method B 2.77 eV, method C 3.0 eV. The calculations using method A predict an elongation of the bond length of Ag clusters with increasing size. The calculated value for Ag2 is

2.1

A

and the corresponding value for Ag13 is

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EFFECT OF SIZE ON METAL MICROCLUSTER PROPERTIES C2-177

calculated effect is overestimated. The experimental bond length of Ag2 is 2.5

A,

taken as the sum of atomic radii. If this value had been used for the diatomic species in figure 2, curve A would be

decreased by 8 % and curve B would be decreased by 17 % at this point. This procedure is frequently used in extended Hiickel calculations and would make curves A and B more realistic since then the larger clusters would have a greater binding energy per atom relative to the diatomic case. This correction is not needed for curve C.

It is of interest to compare calculated values of the cluster binding energies with calculated values for the bulk. Using method A, a binding energy per atom of 1.5 eV has been calculated for the periodic cubic structure [2]. Scaling the calculated binding energies of curve A in figure 1 by this number would move this data up by a multiplicative factor of 2. Thus, even at 55 atoms, this dves only 60 %

of the bulk energy.

We have made comparisons between the calculated stability of icosahedral and f .c.c. clusters. Figure 3 shows the results for Ag clusters using

i

Ag CLUSTERS lcosohedral

:

m

O 2 10 14 18 Number of olorns

FIG. 3.

-

Binding energy per atom calculated for icosohedral and face centered cubic Ag clusters using method B.

method B. The f.c.c. structure is more stable although the occupied bandwith for the icosahedral structure is greater. The bandwidth is determined by the average number of nearest neighbours and reflects the greater packing density on the surface of icosahedral structures 191. The order of stability apparently reflects the greater destabilization of occupied antibonding molecular orbitals for the icosahedral structure.

3 . 2 IONIZATION POTENTIAL.

-

Ionization poten- tials (IP) calculated for the three-dimensional Ag

clusters described here show decreasing values with increasing size. Figure 4 shows IP values calculated using methods A and B with Koopmans' theorem

[lo]. There is some irregularity in the trend versus

size, depending upon structure, observed for the small clusters until the trend settles down near

5 5 t I I

I

10 30 50

Number of atoms

FIG. 4. - Ionization Potential for three-dimensional Ag clusters

versus size calculated with method A and B.

20-30 atoms. This behaviour is reminiscent of the odd-even oscillations found in IP values for linear Ag clusters. The calculated IP at 55 atoms is 6.1 eV which is larger than the bulk work function of Ag

(4.5 eV). The calculated IP values shift 2.3 eV towards the vacuum on progressing from Ag2 to Ag,,. This shift agrees with the corresponding e x p e r i m e n t a l v a l u e of 2 . 5 e V m e a s u r e d recently [ll] using the ESCA technique. Since electrostatic effects and relaxation energies are not considered in this IP calculation, it is not surprising that the work func,tion value is not attained for the larger clusters. These effects would reduce the calculated IP and move it closer to the work function value. It is significant that about 20- 30 atoms are required for the trends in figure 4 to become nearly saturated.

The effect of oscillation of IP values between clusters containing even and odd numbers of electrons is observed for linear chain clusters. We have reported this behaviour for Ag, and figure 5

shows similar calculated and experimental [I21 data for Na clusters. Larger IP values are observed for the clusters with even numbers of electrons. The

I '

LL l i

Number of ofoms

FIG. 5. - Ionization Potential versus size for linear Na

clusters : method C, method A, o experiment

1 2

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C2-178 R. C. BAETZOLD

results of the EH calculations with method A follow the trend in the experimental data as a function of size. The method C calculations overes- timate the odd-even effect, but absolute IP values for odd size clusters agree better with experiment. Odd-even oscillations in the intensity of anionic and cationic metal clusters have been observed using secondary ionic emission techniques coupled with mass spectrometry [13a]. This effect is observed f o r A g k ( 4 S N S 30), Agk(1 S N s ~ ) ,

Cu&.(l=s N s 5 ) and H & ( S < N S 11) as well as

several other clusters. Assuming the intensity of each sized cluster species to be a measure of its stability, we find a correlation with the calculated oscillations between odd and even sized clusters. Larger binding energies and ionization potentials were calculated for systems with even numbers of electrons which should indicate greater stability. Greater intensities in the mass spectrum were found for species with even numbers of electrons. This effect has also been examined theoretically by Joyes [13b].

3.3 BAND STRUCTURE. - The buildup of the Ag band structure towards bulk features as a function of size continues well beyond 13 atoms. In the case of the 5s band of Ag using method A, band spreading continues up to 55 atoms. The width of, the occupied part of this band calculated using 5s and 5p orbitals is 3.0 eV at 13 atoms, 3.4 eV at 19 atoms, 4.8 eV at 43 atoms and 5.3 eV at 55 atoms. In addition there is a gap between occupied and unoccupied levels which decreases in this size range. Even at 55 atoms the gap is 0.2 eV, which is far from the spacing kT found in bulk

metals., Values of this gap as a function of size are shown for three dimensional Ag and Cu clusters as calculated by method B in figure 6. This gap also represents a qualitative measure of the difference in ionization potential and electron affinity of the cluster.

The shape of the density of states of the 5s band in Ag~s clusters is shown in figure 7 for straight

Number of otoms

FIG. 6 .

-

Energy gap between highest occupied and lowest

unoccupied molecular orbitals for three-dimensional Ag 0 and Cu

A clusters calculated using method B.

Energy (eV)

FIG. 7. - Density of 5s molecular orbitals per 0.2 eV interval

for linear and face-centered-cuvic A ~ S X clusters.

chain and f .c.c. geometries. The width is much greater for the f.c.c. case and a dependence of density upon energy similar to the free electron model is observed.

-'/2

,

one-dimensional

pa& ' I 2 , three-dimensional.

The HOMO is 7.10 eV for the linear chain and 6.17 eV for the f.c.c. geometry.

The width of the d band in metal clusters has been examined for Pd clusters. Bandwidths are determined to a large extent by the degree of overlap between orbitals on adjacent centers and the percentage of nearest neighbour bonds in a given structure. It has been shown that, with method A, very little spreading of the d band of Pd takes place for 19' atoms (0.3 eV width) [2]. Using a more diffuse basis set, as with method B, it is shown in figure 8 that PdI9 has a d bandwidth of

FIG. 8 . energy 2 21

-

.- _", & I -12

. - Density of electrons per 0.12 eV interval versus

for bulk and surface atom in Pd19 f.c.c. cluster calculated using method B. I r I I Bulk

I

Pd[9

-Z-

Surface

-

- - - I n ,I I I

,

I I ,I 11 W I I ,A,

fi

-10 -8 -6 - 4

1.8 eV. This value is found for the local density of states of a bulk atom, but the corresponding quantity for the surface atom is much different. The HOMO value is - 7.9 eV and comes within the manifold of d states. We find 0.53 as the average number of d holes per Pd atom which corresponds well with the bulk experimental [14] value 0.6. The

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EFFECT OF SIZE ON METAL MICROCLUSTER PROPERTIES C2- 179

width of the bulk d band is reported as 4 eV [15] molecule tends to remain linear. In the absence of which is much greater than the calculated d this restriction, the adsorbed molecule distorts 40"

bandwidth for Pd19. from its gas phase CCH bond angle.

3 .4 CHEMISORPTION. - Calculations of acetylene interacting with

Pt,

have been recently reported [l6]. An extended Hiickel procedure requires a type of charge self-consistency to treat this problem adequately. When electrons are allow- ed to flow with no adjustment of atomic potential large charge transfers are observed between the acetylene and Pt9 surface. Self-consistency is achie- ved by requiring zero electron transfer between acetylene and Pt9 model. The latter is achieved by shifting all orbitals on C and H uniformly. When this is performed minor rehybridization occurs upon adsorption of the acetylene molecule and the

4. Summary. - We have examined electronic properties of silver and palladium microclusters within the framework of molecular orbital theory. We find electronic properties unlike the bulk. The s bandwidth continues broadening up to 55 atoms of Ag, while for 19 atoms of Pd the d bandwidth is about half the experimental value. The d band has many peaks and an atomic-like character. Binding energies calculated are smaller than the bulk value. Oscillations between odd and even number of electron systems are found for chains of silver and sodium particles. The ionization potential and binding energy are greater for even numbers of electrons.

References

[I] BAETZOLD, R. C., J . Chem. Phys. 55 (1971) 4363.

[2] BAETZOLD, R. C. and MACK, R. E., J. Chem. Phys. 62 (1975) 1513.

[3] MESSMER, R. P., KNUDSON, S. K., JOHNSON, K. H., DIA-

MOND, J. B. and YANG, C. Y., Phys. Rev. B. 13 (1976) 13%.

[41 CYROT-LACKMANN, F., DESJONQUERES, M. C. and GORDON, M. B., J. Physique Colloq. 38 (1977) C2-57.

[5] HOFFMANN, R., J. Chem. Phys. 39 (1963) 137.

[6] CLEMENTI, E., RAIMONDI, D. L. and REINHARD, W. P., J.

Chem. Phys. 47 (1967) 1300.

[7] BASCH, H. and GRAY, H. B., Theor. Chim. Acta 4 (1966) 367.

[8] VERHAEGEN, G., STAFFORD, F. E., GOLDFINGER, P. and ACKERMAN, M., Trans. Faraday Soc. 58 (1962) 1926.

[9] BURTON, J. J., J. Chem. Phys. 52 (1970) 345. [lo] KOOPMANS, T. A., Physica 1 (1933) 104.

Ill] MASON, M. G. and BAETZO~D, R. C., J. Chem. Phys. 64 (1976) 271.

[12] ROBBINS, E. J., LECKENBY, R. E. and WILLIS, P., Adv.

Phys. 16 (1967) 739 ; FOSTER, P. J., LECKENBY, R. E and ROBBINS, E. J., Proc. Phys. Soc. London (At. Mol. Phys.) 2 (1969) 478.

[13a] LELEYTER, M., N Proprietes Electroniques de Petits Agre-

gats Metalliques

,,.

Thesis submitted University of Paris, Orsay (1975).

[13b] JOYES, P., J. Phys. Chem. Solids 32 (1971) 1269. [I41 WOLFARTK, E. P., J . Phys. Chem. Solids 1 (1956) 35. [I51 NORRIS, C. and MEYERS, H. P., J. Phys. F. 1 (1971) 62. [16] BAETZOLD, R. C., KESMODEL, L. L. and FALICOV, L. M.,