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ELECTRONIC STRUCTURE OF MATERIALS. From reality to simulation and back A roundtrip ticket. Interatomic Potentials. Before we can start a simulation, we need the model ! Interactions between atoms, molecules,… are determined by quantum mechanics:
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ELECTRONIC STRUCTURE OF MATERIALS From reality to simulation and back A roundtrip ticket
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…
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?
Lennard-Jones potential V(R) = i<jv(ri-rj) 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) .
Morse potential • Like Lennard-Jones • Repulsion is more realistic-but attraction less so. • Minimum neighbor position at r0 • Minimum energy is • Extra parameter “a” can be used to fit a third property: lattice constant, bulk modulus and cohesive energy.
Various Other Potentials simplest: Hard-sphere b) Hard-sphere, square-well c) Coulomb (long-ranged) for plasmas d) 1/r12 potential (short-ranged)
Atom-atom potentials • Total potential is the sum of atom-atom pair potentials • Assumes molecule is rigid, in non-degenerate ground state, interaction is weak so the internal structure is weakly affected. Geometry (steric effect) is important. • Perturbation theory as rij >> core radius • Electrostatic effects: multipole expansion (if molecules are charged or have a permanent dipole, …) • Induction effects (by a charge on a neutral atom) • Dispersion effects: • dipole-induced-dipole (C6/r6) • Short-range effects-repulsion caused by cores: exp(-r/c)
Fit for a Born (1923) potential • Attractive charge-charge interaction • Repulsive part determined by atom core. EXAMPLE: NaCl • Obviously Zi=1 • Use cohesive energy and lattice constant (at T=0) to determine A and n: • EB=ea/d+ er/dn dEB/dr= –ea/d2+ ner/dn-1=0 => n=8.87 A=1500eVǺ8.87 • Now we need a check. The “bulk modulus”. • We get 4.35 x 1011 dy/cm2 experiment is 2.52 x 1011 dy/cm2 • You get to what you fit!
Silicon potential • Solid silicon is NOT well described by a pair potential. • Tetrahedral bonding structure caused by the partially filled p-shell: sp3 hybrids (s+px+py+pz , s-px+py+pz , s+px-py+pz , s+px+py-pz) • 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 109o rk ri i rj
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.
Problems with potentials • Difficult problem because potential is highly dimensional function. Arises from QM so it is not a simple function. • Procedure: fit data relevant to the system you need to simulate, with similar densities and local environment. • Use other experiments to test potential if possible. • Do quantum chemical (SCF or DFT) calculations of clusters. Be aware that these may not be accurate enough. • No empirical potentials work very well in an inhomogenous environment. • This is the main problem with atom-scale simulations--they really are only suggestive since the potential may not be correct. Universality helps.
Some tests • Lattice constant • Bulk modulus • Cohesive energy • Vacancy formation energy • Property of an impurity
What are the forces? • Common examples are Lennard-Jones (6-12 potential), Coulomb, embedded atom potentials. • They are only good for simple materials. • The ab initio philosophy is that potentials are to be determined directly from quantum mechanics as needed. • But computer power is not yet adequate, in general. • But nearing in the future (for some problems). • A powerful approach is to use simulations at the quantum level to determine parameters at the classical level.
GB The interatomic potential used in the simulation is based on the Embedded Atom Method (EAM). For additional simulations we used Ab initio method in combination with the usual copper pseudopotential.
