1 Journal of Theoretical Physics Founded and Edited by M. Apostol 79 (2002) ISSN 1453-4428 Metallic Clusters Deposited on Surfaces. Puszczykowo talk, 2002 M. Apostol Department of Theoretical Physics, Institute of Atomic Physics, Magurele-Bucharest Mg-6, POBox Mg-35, Romania email: apoma@theory.nipne.ro First, I would like to draw your attention upon a distinction which we should make in connection with the title of our Workshop, betwen the original quantum dots and our quantum dots, we are going to talk about in our sessions. The original quantum dots are two dimensional electron gases, semiconducting or quasi-metallic, restricted in space, of linear size of cca 10 3 10 4 Å, belonging to the mesoscopic world. The environment is essential for them, hence the role of the confining potentials, and their variety as open, closed, lateral or vertical quantum dots. They exhibit quantum effects and size (non-tehrmodynamic) effects, which can most easily be seen in transport, which displays ballistic behaviour, chaotic patterns, phase coherence, charge quantization, Coulomb blockade, etc. In contrast, our dots may be called Atomic Quantum Dots, as they are nanostructures of linear size of cca 10 to 10 3 Å, isolated or under various geometric constraints (like deposited on surfaces), or dynamic constraints (like external applied forces). They exhibit enhanced quantum and size effects, and in addition, we expect an enhanced nano-magnetism. Between 10 and 10 3 Å stretches an entirely new world, the nano-world, populated with a multitude of nano-objects, as the 45-atoms cluster shown here. Our task, the task of the nanoscience and the enabling nanotechnology, would be that of producing knowledge about this new world, and to enhance this knowledge. At this level, the knowledge would consists of describing all the nanoobjects, and drawing a map of the nano-world, with as much physical and chemical information as possible, in much the same manner as the mapping of the humane genome. Consequently, I draw your attention upon the possibility of a far-reaching research project of an enormous scope, which might be called the Nanome Mapping Project, or briefly the NANOME Project, and which might be the general, all-embracing framework of our research, both fundamental, theoretical or experimental, and applied. The main problem with the Atomic Quantum Dots is the computing of the one-electron energy levels. To do that we need the potential and the atomic positions. Bulk solids fall in a finite sequence of symmetry classes, so that we know apriori the atomic positions. For the Atomic Quantum Dots this is no longer so; we need individual knowledge for each of them. This raises again an old problem of the condensed matter, that of matter aggregation, or chemical bond. We are able now to solve satisfactorily this problem, in two or three successive steps. Basically, in a simplified model applicable to some classes of metallic atoms, the ensemble consists of a set of ions, whose effective point-like charges we know in many cases, and the compensating electrons. We put Coulomb attraction and Coulomb repulsions, notice that the problem is describable within the quasi-classical theory, and arrive straightforwardly to the self-consistent potential, and the
2 J. Theor. Phys. inter-atomic effective potentials that ensure binding. Then we minimize the potential energy with respect to the atomic positions and get beautiful geometric forms, like those shown here for 13, 45- or 115-atoms clusters in their ground-states. The potential depend essentially upon the screeening wavevector (the theory being in fact, at this level, a linearized Thomas-Fermi model), which we may know by minimizing the quasi-classical energy, i.e. the potential energy plus the kinetic energy. Then we add the exchange energy to the quasi-classical energy and get the binding energy. This way we are able to get geometric forms, atomic positions, magic numbers, i.e. the extent of stability of the clusters with respect to their neighbours, and other relevant information like vibration spectra. In the second step of the theory we have to add the so-called quantum corrections which give us the one-electron energy levels, and then to check the consistency of the calculations. This program completed we reach our goal of constructing from Quantum Mechanics any atomic aggregate, supra-molecular ensemble, or bit of solid; such clusters are the nuclei of the condensed matter, the smallest part the bulk is constructed of.
J. Theor. Phys. 3 N=13 Ions, electrons, effective charges z i Spherical symmetry, s-orbitals; d, f-orbs, approximately Metallic ions, point-like (z F e = 0.57,z Na = 0.44) Self-consistent potential ϕ = N i=1 z i r R i e q r R i Electron density n = q 2 ϕ/4π
4 J. Theor. Phys. Potential energy, minimizataion N=45 E pot = 3 4 q N i=1 z 2 i + 1 2 N i j=1 Φ(R ij ) Effective inter-ionic potentials Φ(R ij ) = 1 2 qz i z j (1 2 qr ij )e qr ij Quasi-classical energy E q = E kin + E pot, minimization q E kin = (27π 2 /640)q 4 i zi Screening wavevector q = 0.77z 1/3 Average inter-ionic distances a = R ij 2.73/q Binding energy E = E q + E ex, E ex = (9/32)q 2 i z i
J. Theor. Phys. 5 N=115 Apart from ground-states we obtain thereby isomers, i.e. the same number of atoms bind in slightly different froms, separated by small amounts of energy. A table of isomers is shown below, for Fe-clusters. More compelx structures are also produced by theory, like the Fe-CH-cluster below, consisting of a core of 13 Fe atoms and a shell of 6 C 2 H 2 -ligands, or the (unstable) metallic nanowire, or a bit of solid of 855 atoms, preserving a core of an approximate bcc-symmetry.
6 J. Theor. Phys. 0.4 isomers energy (ev) 0.3 0.2 0.1 0.0 0 10 20 30 40 50 60 70 80 N Fe 13 (C 2 H 2 ) 6
J. Theor. Phys. 7 Metallic Nanowire N=855
8 J. Theor. Phys. Two Interacting Clusters The theory enjoys a separability property, this is why we are able to get even two clusters, for instance, weakly interacting, which form up one single cluster after a long while. This property enables the treatment of clusters deposited on surfaces. Indeed, we may assume a semi-infinite solid, and compute according to the present theory the self-consistent potential, within the continum approximation; it has a characteristic exponential decay at the surface, giving rise to a charge double layer, as expected. We may add one atom, and compute the potential energy, which consists of the potential energy of the solid, the self-energy of the added atom and an interaction potential between the solid and the atom. It has an attractive part above the surface and a potential barrier just beneath it; this potential enables us to get the surface energy, i.e. the surface tension, of the semi-infinite solid, as well as the work function of the solid. Now, it is easy to add an ensemble of atoms, and get their interaction energy with the solid; together with their own mutual interaction, it gives the potential energy, which can be minimized in order to get stable atomic positions in the presence of the surface, which means clusters deposited on surfaces. Indeed, such clusters are obtained, some consisting of monolayers, for a smaller number of atoms, some others consisting of superposed terraces. We may compute their binding energy, and the corresponding magic numbers, as shown below. The quantitative computations are made here for Fe-clusters deposited on Na. A few clusters deposited on surfaces are shown here, of a high stability, and intricate geometries. Separability, Solid + ad-atoms, their interaction Surface potential (continuum solid)
J. Theor. Phys. 9 z o y metal x surface n z* z* /2 metal S x metal + n tot + +++ z * /2 + _ S metal x S z*/2 Solid-Ion interaction potential E int = πz z0 qa 3 xe q x E int metal 1/q 1/q x Potential energy Surface tension E pot = 3 4 qz 2 N + πz 2 2q 3 a 6 A
10 J. Theor. Phys. Semi-infinite solid + Atoms E pot = E sol 3q 4 i zi 2 + 1 2 i j Φ(R ij ) πz qa 3 i z i X i e q X i Screening wavevector of the solid Minimization of E pot Quasi-classical energy E q = E kin + E pot E s Binding energy E = E q + E ex Interaction energy, breaking the cluster off the surface Diffusion, interfaces, more-or less-extended contacts
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12 J. Theor. Phys. Magic numbers 0.15 94 7 19 14 23 75 77 85 88 D 0.00-0.15 0 10 20 30 40 50 60 70 80 90 100 N Ground-state energy -4.8 3D-clusters -4.9 2D-clusters E(N)/N (ev) -5.0-5.1-5.2-5.3 0 10 20 30 40 50 60 70 80 90 100 N
J. Theor. Phys. 13
14 J. Theor. Phys.
J. Theor. Phys. 15 This is however not the whole story, because, in some cases, we get one atom just below the surface, or escaped completely into the solid, i.e. diffused, as in figures below; or more atoms beneath the surface, building thus an interface between two solids, and contacts.
16 J. Theor. Phys. One atom beneath the surface One atom escaped into solid 50-atoms cluster diffusing into solid, 100-atoms developing an interface with solid In conclusion, we may say that we are able to construct from first principles atomic clusters deposited on surfaces, or Atomic Quantum Dots, and find out the atomic positions, their equilibrium forms both for the ground state and for isomers, binding energy, and other relevant physical and chemical information; more, we may construct also interfaces and contacts between two solids, or between a solid and a deposited cluster; and the next step, beside other refinements of the theory, would be the computation of the one-electron properties, relevant for the transport, spectroscopy andreactivity of such nano-objects. c J. Theor. Phys. 2002, apoma@theor1.theory.nipne.ro