Band structures of solids composed of metal clusters

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0038-1098/9356.00+.00 Pergamon Press Ltd

Solid State Communications, Vol. 85, No. 1, PP. 11-14, 1993. Printed in Great Britain.

BAND STRUCTURES

OF SOLIDS COMPOSED

OF METAL CLUSTERS

M. Mxnnlnen and J. Mansikka-aho

Department of Physics, University of Jyv~skyl~ P.O. Boz 35, SF-~0351 JyvS~kyld S. N. KbA.nna and P. Jena Physics Department, Virginia Commonwealth University

Richmond, Virginia ~3~8~-~000 (Received 10 October 1992, accepted for publication 30 October 1992 by B. Lundqvist) Spherical metal clusters with compact atomic geometry and a magic number of valence electrons axe expected to form weakly bonded materials where the clusters retain their identity. A simple model potential is used to study the band structure of such materiaJs. The results show that materials assembled of metal clusters are expected to be semiconductors.

clusters. To this end we have performed band structure calculations for cluster materials. We assume that the simple metal clusters with a closed electronic shell is nearly spherical and can be adequately described with the spherical jellium model. The selfconsistent potential is then nearly constant inside the cluster and be reasonably approximated with a square well

The properties of small metal clusters axe governed by the shell structure of the valence electrons[I]. In alkali metals the geometry of the cluster is coupled to the filling of the electronic shells[2 - 4] and only in large clusters an ordered structure (crystalline or icosahedral) is favoured[5]. Owing to the large energy gap at the Fermi level, clusters with magic numbers of valence electrons axe expected to be less reactive than nonmagic clusters, and behave like giant inert gas molecules. Recently, it was suggested[6] that a crystalline structure could be assembled from the magic metal clusters. The clusters would be weakly bound to each other and form a closed packed structure like fcc. A nonmetallic example of this kind material is solid fullerene[7, 8] where the fnllerene molecules are bound together with van der Waals interactions. In the case of simple metals the smallest electronic magic numbers are 2, 8, 20, and 40. If the magic number of valence electrons in a cluster could be accompanied by a spherical packing of atoms, the cluster would derive additional stability. Jena and Khanna[6] suggested that A112Si is such a cluster which could be used in assembling new materials. A112Si has a compact ieosahedrai geometry and a closed electronic shell due to its 40 valence electrons. In the case of monovalent metals, the 8 atomic Mkali clusters have a high symmetry, whereas in the case of 20 atom cluster the situation is still unclear[9, 14]. That cluster assembled materials could have properties quite different from those assembled from atoms with the same composition could certainly give materials science a new dimension. The key requirement for the possibility of weakly bonded cluster material is the existence of narrow energy bands, i.e. the characteristics of the closed electronic shell should remain in a solid assembled from

V(r) = - V : ( R - r),

O)

where V0 = --'I' -- eF, @ being the work function of the metal and eF the Fermi energy measured from the bottom of the conduction band. R is the cluster radius, R = rsN x/a , where r , is the electron density parameter and N the number of valence electrons in the cluster. The cluster assembled material is approximated by making a lattice of the potentials (Eq. (1)) of the individual clusters. Since the van der Waals bonded materials have a tendency to form closed packed structures we consider only the fcc lattice. The electronic band structure is solved using a plane wave basis. Due to the simplicity of the potential a good accuracy was obtained with about 2000 plane waves. The AI12Si cluster was approximated by choosing ra = 2.07 and V0 = -0.586 a.u. appropriate to metallic A1. Figure 1 shows the development of the energy band.' from the discrete cluster energy levels when the lattice constant becomes smaller. The Figure is obtained by computing the energy eigenvalues for five high symmetry points (I', X, W, L and K ) in the Brillouin zone. A t large distances the discrete energy levels of an isolated cluster are recovered. The Fermi energy lies in the gap between the 2p and lg states. The 2p and lg bands start to overlap when the lattice constant is about 21.2 a.u. The distance between the neighbouring potential 11

STRUCTURES OF SOLIDS

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the intracluster A1-AI bond (5.2 a.u.). For example, a 6 a.u. A1-A1 distance would give a lattice constant of a = 23 a.u. and a 5.2 a.u. A1-A1 distance would give a = 22 a.u. The corresponding band gaps at the Fermi energy would be 0.8 eV and 0.5 eV, respectively•

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Vol. 85 No, 1

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Fig. 1 Energy eigenvalues of AlnSi cluster material as a function of the lattice constant. The different energy bands are denoted by the single electron states in a single potential well. The Fermi energy is between 2p and lg bands.

Figure 2 shows the band structure for AI,2Si crystal with the lattice constant of 22 a.u. The figure confirms that the band minima and maxima are obtained at the high symmetry points of the Brillouin zone. At the Fermi level (between 2p and lg shells) the smallest gap is at the F-point, but the smallest gap can be also at other symmetry points (L-point between If and 2p, X-point between 2s and lf) or there can be an indirect gap as between l d and 2s shel-ls. Theeffe--ctive masses of electrons at the bottom of the 'conduction band' (1gelectrons) axe 0.42 and 0.26 and those of the holes of 'valence band' (2p-electrons) are 0.47 and 0.20. These values, as also the energy gap, compare well with those of the common semiconductors• The overall bandstructure is insensitive to the details of the potential parameters. Figures 3 and 4 show the results for Na20 cluster material (r, = 3.93 and V0 = -0•205 a.u,). Again the energy gap at the Fermi level (between 2s and If states) disappears only when the potential wells nearly touch each other. The band structures axe qualitatively similar to those obtained for the model potential mimicking A l n S i as seen in Fig. 4. Similar results were also obtained for Nas cluster material. The use of the self-consistent effective potentials of the jellium model could slightly change

wells is at that point about 1.0 a.u• At so close distance the A1-A1 distance between the neighbouring clusters would be smaller than the A1-A1 distance within an A112Si cluster• This means that if an fcc crystal is formed from Alz2Si clusters it will be insulating. It is expected that the intercluster A1-A1 distance in a weakly bonded cluster material is clearly larger than

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Fig. 2 Energy bands for fcc AlnSi. The lattice constart is 22 a.u.

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Fig. 3 Energy eigenvalues of Na20 cluster material as a function of the lattice constant. The Fermi energy is between 2s and If bands. The complicated crossing of the bands at small lattice constants (32.. •28) is not analyzed, only the energy eigenvalues of the high symmtery points (r, X, W, L and K ) are shown for a = 30 and a=28.

B A N D S T R U C T U R E S O F SOLIDS

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Fig. 4 Energy bands for fcc Nae0. The lattice constant is 32 a.u. the size of the band gap but it would not change the qualitative features obtained with the model potentials. The above band structure calculations show that if an fee crystal can be formed from nearly spherical metallic clusters it will have a band gap at the Fermi level• This gives a possibility of van der Waais bonded materials. The total energy of such a material would probably be larger than that of a homogeneous bulk metal or alloy. Total energy calculations are needed to study whether cluster assembled materials can exist as stable local minima in the total energy functional. Khanna and Jena[6] have shown that two Mg4 clusters form a weakly bound cluster molecule where the individual four atomic clusters are only slightly disturbed from their isolated structures. Saito and Olmishi[10] have used the jellium model to study the interaction of two 19 atom sodium clusters. They also found a cluster molecule but due to the open electron shell the two clusters were strongly bound together at a relatively short distance. The jellium model could be used to study the physisorption between two Na20 clusters. The local density approximation indicates the existence of such a ' minimum[Ill but so far no accurate calculations have been made. Pachero and Ekardt[12] have studied the van der Waals interaction between small sodium clusters in t h e jellium model• The asymptotic formula for the attractive potential would give for Na20 a binding energy of about 66 meV at a contact distance. This gives an estimate of the possible binding energy between the clusters• However, sodium clusters are known to be fairly soft and the atoms are mobile already well below the melting temperature[13, 14]. It is then unlikely that a van der Waals bonded cluster m a t e r i a l consisting of magic sodium (or other alkali metal) clusters could be made in practice, since the

13

clusters would melt together in trying to assemble the d u s t e r material. On the other hand, experiments on gold clusters supported on a tungsten tip show that even in the contact with the metal surface the clusters can retain their electronic shell structure[15]. A better candidate for a cluster assembled material would be A112Si where the electronic magic number is accompanied with a compact highly symmetric icosahedral geometry. The cohesive energy of A112Si is comparable to that of bulk A1. Unfortunately, the jellium model is not suitable in calculating the interaction between A1 clusters. Due to the negative surface energy the jeUium model would always favour a cluster material as compared to a homogeneous jellium. It should be stressed that the repulsive interaction coming form the dosed electron shells are needed for making the physisorbed state possible between two A112Si icosahedra. Computer simulations for 13 atom icosahedrai dusters (Cu) with many-atom interactions show that without the electronic shell effects two icosahedra brought close to each other immediately melt together forming one larger cluster[16]. In conclusion, we have calculated the shell structures of fcc materials assembled from metallic clusters. The results show that A112Si is a possible candidate of van der Waals bonded cluster materials with a band gap of about 0.5 eV at the Fermi energy• Also other materials made of spherical metal clusters with a magic number of valence electrons would have a band gap at the Fermi energy. Ab initio total energy calculations axe needed to confirm the possible existence of such materials.

References

[1] W. A. de Heer, W. D. Knight, M. Y. Chou, and M. L. Cohen, Solid State Physics 40, 93 (1987). [2] J. Martins, J. Buffet, and R. Car, Phys• Rev. B 31, 1804 (1985). [3] B. K. Rao, P. Jena, and M. Manninen, Phys. Rev. B. 32, 477 (1985). [4] M. Marminen, Phys. Rev. B 34, 6886 (1986). [5] T. P. Martin, T. Bergmann, H. GShlich, and T. Lange, Chem• Phys. Left. 172, 209 (1990). [6] S. N. Khanna and P. Jena, Phys. Rev. Lett. 6 9 , 1664 (1992). [7] W. KrKtschmer, L. D. Lamb, K. Fostiropoulos, and D. It. Huffman, Nature 347, 354 (1990). [8] A. F. Hebard, in Physic8 and Cheraistry of Finite Systems: From Cluster8 to Crystals, eds. P. Jena, S. N. Khanna, and B. K. Rao (Kluwer, Dortrecht 1992). [9] V. Bona~i~-Kouteck~', P. Fantucci, and J. Kouteck~,, Chem. Rev. 91, 91 (1991)• [10] S. Saito and S. Ohnishi, Phys. Rev. Lett 59, 190 (1987)• [11] R. Schmidt, Private communication. [12] J. M. Pachero and W. Ekardt, Phys• Rev. Lett. 6 8 , 3694 (1992). [13] W. Andreoni, G. Galli, and M. Tosi, Phys. Rev.

14

BAND STRUCTURES OF SOLIDS

Left. 55, 1734 (1985). [14] U. RSthlisberger and W. Andreoni, J. Chem. Phys. 94, 8129 (1991).

Vol. 85 No. 1

[15] M. E. Lin, R. P. Andres, and R. Reifenberger, Phys. Rev. Left. 67, 477 (1991). [16] S. Valkealahti, unpublished.