1. Úvod do tímového projektu Peter Ballo Katedra fyziky Fakulta elektrotechniky a informatiky

5. Interatomic Potentials Before we can start a simulation, we need the model! Interactions between atoms, molecules,… are determined by quantum mechanics: –Schrödinger Equation + Born-Oppenheimer (BO) approximation –BO: Because electrons T is so much higher (1eV=10,000 K) than true T and they move so fast, we can get rid of electrons and consider interaction of nuclei in an effective potential “surface.” V(R). –Approach does not work during chemical reactions. Crucial since V(R) determines the quality of result. But we don’t know V(R). –Semi-empirical approach: make a good guess and use experimental data to fix it up –Quantum chemistry approach: works in a real space. –Ab initio approach: it works really excellent but…

6. Semi-empirical potentials Assume a functional form, e.g. 2-body form. Find some data: theory + experiment Use theory + simulation to fit form to data. What data? –Atom-atom scattering in gas phase –Virial coefficients, transport in gas phase –Low-T properties of the solid, cohesive energy, lattice constant, bulk modulus. –Melting temperature, critical point, triple point, surface tension,…. Interpolation versus extrapolation. Are results predictive?

7. Some tests -Lattice constant -Bulk modulus -Cohesive energy -Vacancy formation energy -Property of an impurity

8. Lennard-Jones potential V(R) =  i<j v(r i -r j ) v(r) = 4  [(  /r) 12 - (  /r) 6 ]  = minimum  = wall of potential Reduced units: –Energy in  –Lengths in  Good model for rare gas atoms Phase diagram is universal! (for rare gas systems). 

9. Silicon potential Solid silicon is NOT well described by a pair potential. Tetrahedral bonding structure caused by the partially filled p-shell: sp 3 hybrids (s+p x +p y +p z, s-p x +p y +p z, s+p x -p y +p z, s+p x +p y -p z ) Stiff, short-ranged potential caused by localized electrons. Stillinger-Weber (1985) potential fit to: Lattice constant,cohesive energy, melting point, structure of liquid Si for r<a Minimum at 109 o riri rkrk rjrj ii

10. Metallic potentials Have a inner core + valence electrons Valence electrons are delocalized. Pair potentials do not work very well. Strength of bonds decreases as density increases because of Pauli principle. EXAMPLE: at a surface, LJ potential predicts expansion but metals contract Embedded Atom Method (EAM) or glue models better. Daw and Baskes, PRB 29, 6443 (1984). Embedding function electron density pair potential Good for spherically, closed-packed, symmetric atoms: FCC Cu, Al, Pb Not so good for BCC.

11. BALLO, P., KIOUSSIS, N., LU, G.Materials Research Society Proceedings, Vol.634. : MRS, 2001, s. B3.14.1-7.Boston. USA, 27.11.-1.12.2000.

13. BALLO, P., KIOUSSIS, M., LU, G. Phys. Rev. B, 64, 024104 (2001).

14. Vacancy formation energy as a function of the layer number from the interface. BALLO, P., SLUGEN, V. Phys. Rev. B, 65, 012107 (2002).

15. BALLO, P., HARMATHA, L. Phys. Rev. B, 68,153201 (2003).