From Atoms to Materials: Predictive Theory and Simulations Week 3 Lecture 4 Potentials for metals and semiconductors Ale Strachan strachan@purdue.edu School of Materials Engineering & Birck anotechnology Center Purdue University West Lafayette, Indiana USA Structure of a minimalist MD code Initial conditions [R i (0), V i (0)] Calculate forces at current time [F i (t)] from R i (t) Integrate equations of motion R(t) R(t+Δt) V(t) V(t+Δt) t t+δt Save properties Output files Done? Y End o Ale Strachan - Atoms Materials
Pair-wise potentials Simplest, non-trivial, function Lennard-Jones (6-) V ( ) ({ R i }) = φ R i R j i< j Sum of pair-wise terms σ φlj ( r) = 4ε r φ Morse Morse ( r) Exponential-6 6 σ r φ exp 6 (r) = ε" # e γr Ar 6 $ % Lattice parameter & cohesive energy r γ r ε exp γ exp r0 r = 0 energy Morse Lennard-Jones (6-) distance Lattice parameter, cohesive energy and bulk moduli Ale Strachan - Atoms Materials 3 Two-body potentials: limitations Pair-wise interactions: E { R i } ( ) = φ ( R ij ) We will calculate the vacancy formation energy i< j Vacancy energy in Perfect crystal Coordination number Z Interactions: only first nearest neighbors E xtal ( ) = ZE 0 ( ) E coh = E xtal = ZE 0 ow let s create a vacancy: E vac ( )= ZE ZE 0 0 Ale Strachan - Atoms Materials 4
Two-body potentials: limitations ZE ( ) 0 Extal = E vac ( )= ZE 0 ZE 0 Vacancy formation energy: ε vac = E vac ( ) Extal ( ) = ZE0 ZE0 ( ) ZE0 = ZE0 Vacancy formation energy is equal (in magnitude) to cohesive energy Ale Strachan - Atoms Materials 5 Two-body potentials: limitations Two-body potentials: vacancy formation ~ cohesive energy Metals: ε vac ~/3 E coh Ecoh (ev) Evac (ev) Evac/Ecoh Al 3.39 0.75 0. i 3.56.6 0.46 Cu 3.65. 0.33 Ag 4.086.5 0.8 Pt 3.94.4 0.36 Au 4.079 0.95 0.3 Two-body potentials and elastic constants: c =c 44 Metals: c >c 44 Pair potentials can only describe simple non-bond interactions Closed-shell atoms (e, Ar, Kr, etc.) Atoms that are fully coordinated (e.g. inter-molecular) Pauli repulsion at short distances London dispersion at longer distances van der Waals Ale Strachan - Atoms Materials 6 3
Capturing many-body effects in metals Embedded atom model (EAM) Mike Baskes and collaborators 980 s + other groups V = φ(r ij ) + F ρ i i< j Embedding energy Local electronic density at the location of atom i Accurate description of: Environment dependence of bonding in metals (vacancy and surfaces) Elastic constants and plastic deformation Phase transformations (melting & solid-solid) Alloys ot very accurate for: Cases where bonding is highly directional i ( ) ρ i = j i f ( R ij ) Ale Strachan - Atoms Materials 7 Interatomic potentials Closed packed metals Semiconductors Molecular materials Ale Strachan - Atoms Materials 8 4
Embedded atom model Ale Strachan - Atoms Materials 9 EAM: Motivation Ale Strachan - Atoms Materials 0 5
Motivation Embedded atom model (EAM) Mike Baskes and collaborators 980 s + other groups V = φ(r ij ) + F ρ i i< j i ( ) ρ i = j i f ( R ij ) Embedding energy Local electronic density at the location of atom i Ale Strachan - Atoms Materials EAM potentials: parameterization Fit parameters based on experimental data Lattice parameter, elastic constants, vacancy & surface energy Use of ab initiodata enables the incorporation of more information Energy (kcal/mol) EAM potential for Tantalum bcc phase Circles: QM Lines: FF (qeam) Volume (Å 3 ) Strachan et al. MSMSE (004) Energy (kcal/mol) Energy (kcal/mol) fcc phase A5 phase 6
EAM potentials: parameterization Large deformations Shear transformation in the twinning mode (ideal strength) ()[--] a = /[-] + s[--] b = /[-] + s[--] c = /[-] qeam FF DFT-GGA vol E (ev) τ(gpa) E (ev) τ(gpa) 8.36 0.88 7.4 - - 7.68 0. 8 0.94 7.37 5.44 0.6.05 0.76.4 0.9 0.43 8. 0.566 36. DFT-data from: Soderlind and Moriarty, Phys. Rev. B 57, 0340 (998) Vacancy and surface formation energies EAM potentials: validation & application Thermal expansion (a-a0)/a0 (%) Experiment qeam FF Temperature (K) From isothermal and isobaric MD runs 7
EAM potentials: validation and application -phase MD simulation Melting at ambient pressure Simulation: 350±50 K (4%) Experiment: 390±50 K Band electrons Experiment shock melting Brown and Shaner (984) Temperature for Hugoniot Pressure (GPa) Cohen ab initio Hugoniot Using exper. pressure -phase MD simulation Free electrons Strachan et al. MSMSE (004) Temperature (K) Directional bonding in semiconductors Diamond, zincblende and wurtzite structures ({ r }) = V f( ) + i r ij i< j i< j< k f (,, ) 3 ri rj rk Ale Strachan - Atoms Materials 6 8
Summary & additional reading Many-body and angular components important in the description of many materials When doing MD simulations always check the applicability and limitations of the interatomic potential Embedded atom model: M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50, 85 (983) M. S. Daw and M. I. Baskes, Phys. Rev. B 9 6443 6453 (984) A. P. Sutton, J. Chen, Phil. Mag. Lett. 6, 39-46 (990) Bond-order potentials for covalent systems J. Tersoff, Phys. Rev. B 37, 699 7000 (988) Ale Strachan - Atoms Materials 7 9