Banerjea, Amitava

Datta, Radhika Prosad

Mookerjee, Abhyit

IACS Institutional Repository

Indian J. Phys. 70A (6), 735-740 (1996)I J P  A— an international journalC a l c u l a t i o n s  o n  C u  c lu s t e r s :  S t r u c t u r e  a n d  E le c t r o n ic  P r o p e r t i e sRadhika Prosad Datta, Amitava Banerjea and Abhyit MookerjeeS. N . B ose N a tio n a l C e n tre  fo r B a ric  Sciences D B - 1 7  S a lt  L a k e  C ity , C a lc u t ta  700064 , In d ia .A b s t r a c t :  The structure and electronic properties of small copper clus­ters are studied using Equivalent Crystal Theory (ECT) and a para­metrised LCAO method. The minimum energy structure is obtained using ECT with Monte Carlo techniques. The LCAO method is then used to study the electronic properties of the optimized clusters.Keywords: Clusters, nanostructures, LCAO, simulationPACS nos: 31.20.Pv, 61.46.-hw, 71.45.NtI .  I n t r o d u c t i o nDuring the past few years much experimental and theoreti :al progress has been made towards the understanding of the physical and chemical properties of several elemental, binary and compound clusters.1,2. Apai et alz car lied out experiments on carbon-supported Cu clusters based on extended X-ray absorption fine structure studies. Balerna et al.4 and Montano et a/.5 have carried out similar experimental studies on Au and Ag clusters. The localised d electrons in noble metals play an important role in the electronic structure through hybridisation with the more ex­tended valence s states which needs to be taken into account in a careful theoretical treatment of these materials. Numerous theoretical studies of the atomic and elec­tronic structure of various elements6""9 have been performed. Recently the ab initio molecular dynamics (MD) approach of Car and Parinello (CP)10 nas been a promising development. However, the CP method is computationally intensive and there are difficulties in handling noble metals like Cu with this method.In this paper we use a semi-empirical method to study the structures of small copper clusters of sizes 10 through 23. The Equivalent Crystal Theory (ECT),11*12 which has been successfully applied to calculate the total energies of metals and semi­conductors, is used to calculate the binding energy of the cluster. A Monte-Carlo technique is used to find the minimum-energy structure. Having obtained the op­timum structure we use the LCAO (linear combination of atomic orbitals) method, particularly the parametrisation scheme due to Harrison,13 to cilculate various elec­tronic properties like the binding energy per atom, the HOMO LUMO gap and the density of states.© 19961ACS736 Radhika Prosad Dattu, Amitava Banerjea and Abhijit MookerjeeI I .  T h e o r yA .  E q u i v a l e n t  C r y s t a l  T h e o r yThe Equivalent Crystal Theory is already well documented.11,12,14 In this method the total energy £  of a solid containing a defect is computed as if each atom were in an equivalent crystal at an appropriate lattice constant which is different from the equilibrium lattice constant. This single crystal is equivalent to the defect solid in the sense that it has the same total energy. The energy of this equivalent crystal (and hence the energy of the solid with defect) can then be determined from the Universal Binding Energy Relation (UBER)15 as£  =  A££*(a*)where AE is the cohesive energy of the elemental crystal, E*(a*) is a universal energy relation (see Fig. 9 of Ref.12),<■* =  ( - = [ a E / t f E / d r l X ,  ‘and rwt is the Wigner-Seitz radius of the equivalent crystal and rwte is the equilib­rium Wigner-Seitz radius. The universal dimensionless function E*(a*) is very well approximated by the Rydberg function:£ * ( 0  = —(1 + a*)e-a*For further details of the theory the reader is referred to the original papers of Smith and Banerjea11,14 and Smith et a/.12B .  L C A O  t h e o r yThe LCAO method used in this paper is described by Harrison.13 The electronic structure Hamiltonian can be constructed from the parameters V,i<7, V,4a, V,4r etc. using the Slater-Koster scheme.16 The d electron parameters for all transition metal systems can be found in Ref. 17 for use in tight binding methods. However, following Menon e t  of.,18,19 we make use of Harrison’s universal scheme13 for the determination of the electronic tight binding parameters. Also these universal parameters have been found to be transferable.13 The Hamiltonian parameters are determined from the dimensionless universal parameters i]u>m in terms of the interatomic distance d by the following prescription^ id m  — Vtdm,3 /2m<P / 2 ’Vddm — Vddi im d 5 'Here r4 is the d-state radius which for Cu has the value13 0.67A. The parameters Vii'm, Vi dm and rjddm are also taken from the solid state table of Ref.. 13 and have the values given by T],ta =  -1.40, v»da =  -3.16, =  -16.20, Vdd* =  8.75.We have chosen a minimal basis set of five d states per atom and the s state. The diagonal matrix elements e, and (.4 of the Hamiltonian are simply the atomic term values given in Ref. 13. We set e, — e4 for reasons given in Refs. 13. and 18.737Calculations on Cu clusters: Structure and Electron,c PropertiesIII. Computational DetailsThe structure optimisation was carried out using the ECT method which has been discussed in Section 2.1. We chose the starting configuration for our 13-atom cluster to  be an icosahedron, since it is generally agreed that an approximately icosa- hedral structure has the minimum energy. We then added or deleted atoms from this structure to obtain the starting configurations for the other clusters. The details will be presented elsewhere.20 The structure of each cluster was then relaxed to a local minimum by a zero-temperature Monte-Carlo method.The binding energy per atom for the optimised clusters was calculated using the LCAO theory as follows. The total energy within a one particle treatment is given byEtot =  ^  fi — Ett +  Eh  iI ,cccwhere c, are the eigenvalues of the single-particle Hamiltonian, Ete refers to the total electron-electron interaction which includes Coulomb repulsion and exchange- correlation energy, and E n  denotes the Coulomb repulsion between the ions. We approximate the difference E j j  — Ett in Eqn. (7) by a short ranged repulsive potential of the Born-Mayer (BM) type given asEn  -  Eee ~  Eb u  =  lT 'A e x p [ - p ( ^ — M  _  i)]2 1 ^  diFollowing Lammers and Borstel21 we have used A = 2.118 x 10“ 2 Ry and p = 14.7238 for Cu.I V .  R e s u l t s  a n d  C o n c lu s io n sWe have obtained the binding energies of copper clusters for N = 10 through 23 using ECT as well as by the parametrised LCAO technique. Fig. 1 shows a comparison of the binding energy per atom for different cluster sizes obtained by the two methods. The dotted lines are the ECT results and the solid lines are from the LCAO calculations. We see from the figure that the values obtained by the two different methods agree quite well for cluster sizes 12 through 23. The agreement is particularly good for the range N =  14 through 20. Given the widely different approach to the energy calculation in the two different methods this agreement is quite remarkable. Significantly, the ECT values show a local minimum at cluster sizes 13, 19 and 21, while a corresponding minimum is not seen in the LCAO values. Between 13 and 19 the ECT values are slightly higher than the LCAO values. We believe the reason for this is that the cluster geometry was optimised with respect to the ECT calculations and not with respect to the LCAO calculations. The values for the binding energy per atom for Cu clusters given by Lammers et a/.21 are slightly larger in magnitude than our calculated values. On the other hand, Datta et al.22 have reported binding energy values, calculated using the TB-LMTO formalism, which are slightly smaller in magnitude than our calculated values over the same range of cluster sizes.738 Radhika Prosad Datta, Amitava Banerjea and Abhijit MookerjeeCluster SizeFig. 1. Binding energy per atom from LCAO (solid line) and ECT (dotted line) plotted vs. cluster size.Figure 2 shows the plot of HOMO-LUMO gap versus cluster size. The graph shows peaks at 13, 17, 20 and 23. The calculations of Lammt-rs ti a/.31 show the HOMO-LUMO gap to have peaks at 13, 20 and 23- However they find a minimum at N = 17. The values of the HOMO-LUMO gap from Ref. 21 ire in general higher in magnitude than our values. Christensen et a/.24 have emphasized the structure dependence of the HOMO-LUMO gap. We attribute the differences between our H O M O - L U M O  gap results and those of other authors to the difference in geometrical structures. Also we see no evidence of the pronounced odd-ev< n alternation in the HOMO-LUMO gap values reported in Ref. 24. Such odd-even a ternations were also not observed by Lammers et a l21 in their TB-LMTO calculatio is.scO13ioSOFig. 2. Plot of HOMO-LUMO gap versus cluster size.Figures 3 and 4 give the total pseudo-density of states (DOS) for clusters of sizes N = 10 and 20. The DOS plots were obtained by convoluting the energy levels of the clusters (as obtained from our LCAO calculations) with a Lorentzian of width .03 eV. The spin degeneracy has not been accounted for explicitly. Some features of the DOS plots deserve comment. The location of the broad peak is around -7 eV for all the cluster sizes. The lowest peak for Cuio is around -16 e\ while for CU20 if *s around -20 eV. So, as the cluster size increases the position of the lowest energy p e a k  seems to move towards the lower energy side. There are peak* around -11 eV and -9.75 eV for Cuio and for CU20 there are two close peaks at -13.75 eV and -13.25 e VCalculations on Cu clusters: Structure and Electronic Properties 739and one peak at -12 eV. This movement of the peaks towards lower energy is observed in general as we move from N =  10 to 23. The height of the broad peak gets enhanced as we move from smaller clusters to larger clusters.p  100.0«/>0.0- 21.0 - 16.0 - 11.0 -6.0Energy (eV)Fig. 3. Plot of total density of states for a 10-atom Copper cluster.C, 100 .0Vice0.0*210 -16.0 -11.0 -6.0 -1.Energy (e\”)Fig. 4. Plot of total density of states for a 20-atom Copper cluster.The binding energy per atom for small copper clusters using the ECT and the parametrised LCAO method are seen to be in good agreement over a wide range of cluster values, though these methods are widely different in principle. The HOMO- LUMO gap and DOS for copper clusters were calculated using the parametrised LCAO method. The computational resources required to perform the semi-empirical calcula­tions in this paper are minimal compared to other computationally intensive ab-initio calculations. We conclude that semi-empirical methods can provide an interface be­tween more accurate and time consuming ab-initio calculations and calculations which assume the form of the interatomic potentials.A c k n o w l e d g e m e n t sThis work was supported by the Solid State and Catalysis Laboratory, Univer­sity of Warwick, U.K.R e f e r e n c e s1 Microclusters, S. Sugano, Y. Nishina and S. Ohnissi eds. (Springer, Berlin 1987).7402 Physics and Chemistry of Finite Systems: From Clusters to Crystals, Vols. 1 and 2,P. Jena, S. N. Khanna and B. K.Rao eds. (Kluwer Academic Publishers, Netherlands1992) .3 G. Apai, J. F. Hamilton, J. Stohr and A. Thompson, Phys. Rev. Lett. 43,165 (1979).4 A. Balerna, E. Bernicri, P. Piccozi, A. Reale, S. Santucci, E. Burrattini, and S. Mobilio, Surf. Sci. 156, 206 (1985).5 P. A. Montano, H. Purdum, G. K. Shenoy, T. I. Morrison, and W. Schultze, Surf. Sci., 156, 206 (1985).6 W. A. Heer, W. D. Knight, M. Y. Chou, M. L. Cohen in Solid State Physics: Advances in Research and Applications, H. Ehrenreich and 0. Turnbull eds., (Academic Press, London 1987), Vol. 40, pp. 93-181.7 Clusters and Fullerenes, Vijay Kumar, T. P. Martin, and E. Tosatti eds., (World Scientific1993) .8 Proceedings of the Fourth International Meeting on Small Particles and Inorganic Clus­ters, Marseille, France (1988), published in Z. Phys. Dl2, (1989).9 Fifth International Meeting on Small Particles and Inorganic Clusters, Konstanz, Ger­many (1988), published in Z. Phys. D19 and 20, (1991).10 R. Car and M. ParineUo, Phys. Rev. Lett. 55, 2471 (1985).11 J. R. Smith and Amitava Banerjea, Phys. Rev. Lett. 59,2451 (1987).12 J. R. Smith, T. Perry, and Amitava Banerjea, Phys. Rev. B 44, 6444 (1991).13 W. Harrison, Electronic Structure and the Properties of Solids, (Dover, New York 1989).14 J. R. Smith and Amitava Banerjea, Phys. Rev. B 37,10411 (1988).15 J. H. Rose, J. R. Smith, and J. Ferrante, Phys. Rev. B 28,1835 (1983).16 J. C. Slater and G. F. Koster, Phys. Rev. 94,1498 (1954).17 D. ArPapaconstantapoulos, Handbook of the Band Structure of Elemented Solids, Plenum Press, New York (1984).18 M. Menon, Connolly, N. Lathiotakis and Andriotis, Phys. Rev. B 50, 8903 (1994).19 M. Menon and K. R. Subbaswamy, Phys. Rev. B 47,12754 (1993).20 Radhika Prosad Datta, Amitava Banerjea, and Abhijit Mookerjee, unpublished.21 U. Lammers and G. Borstel, Phys. Rev. B 49,17360 (1994).22 A. Datta, Abhijit Mookerjee, and A. K. Bhattacharyya, Mod. Phys. Lett. B, 8, 883 (1994).23 C. Kittel Introduction to Solid State Physics, (Wiley Eastern, New Delhi 1976).24 O. B. Christensen, K. W. Jacobsen, J. K. Norskov and M. Manninen, Phys. Rev. Lett.66, 2219 (1991).Radhika Prosed Datta, Amitava Banerjea and Abhijit Mookerjee

Calculations on Cu Clusters : Structure and Electronic Properties

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Calculations on Cu Clusters : Structure and Electronic Properties

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