Atomistic Modeling of Materials Introduction to the Course and Pair Potentials 3.320 Lecture 2 (2/3/05) 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariPractical Issues Energy N r r1 Double summation: Number ofE =∑ V( Ri − Rj ) operations proportional to N22 i , j ≠i rrForce r N ∂V (Ri − Rj )Fi = –∇iE = −∑ r j ≠ i ∂Ri Not feasible with million atom simulations -> use neighbor lists Minimization Standard schemes: Conjugate Gradient, Newton- Raphson, Line Minimizations (Using Force) Typically at least scale as N2 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariShow movie of dislocation generation 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariSystem sizes and Periodicity Finite System (e.g. molecule or cluster) No problem; -> simply use all the atoms Infinite System (e.g. solids/liquids) Do not approximate as finite -> use Periodic Boundary Conditions BaTiO3 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariFor defect calculation unit cell becomes a supercell Periodic Images of Defect Defect 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariHow Large Should the Supercell Be ? Investigate Convergence ! Direct Interactions from Energy Expression (potential) Indirect Interactions due to relaxation (elastic) -> typically much longer range. For charged defects electrostatic interactions are long-ranged and special methods may be necessary. 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariExample: Calculating the vacancy formation energy in Al Vacancy Formation Energy (eV) 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 20 30 40 50 60 70 80 90 100 110 X X Number of Atoms in Supercell Figure by MIT OCW. 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariLimitations of Pair Potentials: Application to Physical Quantities Vacancy Formation Energy 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariSome data for real systems After Daw, M. S., S. M. Foiles, and M. I. Baskes. "The embedded-atom method: a review of theory and applications." Materials Science Reports 9, 251 (1993). 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N Marzari C12/C44Ev/EcohEcoh/kTmfSolidRare GasesArKr1.10.95111.00.6612FCC MetalsNiCuPdAgPtAuPair PotentialLJ 1.0 1.00 131.21.60.310.3730302.5 0.36 252.0 0.39 273.3 0.26 333.7 0.23 34Figure by MIT OCW.Surface Relaxation With potentials relaxation of surface plane is usually outwards, for metals experiments find that it is inwards Surface Plane V(r) NN 2NN r Vacuum 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariCauchy Problem ⎛ σ 11 ⎞ ⎡ ⎤ ⎛ ε 11 ⎞ ⎜σ⎟ ⎢ ⎥ ⎜ε⎟ 22 ⎟ ⎢ 22 ⎟⎜ ⎥ ⎜ ⎜σ⎟ ⎢ ⎥ ⎜ε⎟33 33⎟ = ⎢ ⎥ •⎟ ⎜ ⎜σ 12 ⎟ ⎢ Cij ⎥ ⎜ ⎜ε 12 ⎟ ⎜σ⎟ ⎢ ⎥ ⎜ε⎟ 13 ⎟ 13 ⎟⎜ ⎢ ⎥ ⎜ ⎝σ 23 ⎠ ⎢⎣ ⎥⎦ ⎝ε⎠23 For Potentials C12 = C44 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariCrystal Structures Pair Potentials can fundamentally not predict crystal structures in metals or covalent solids. e.g. fcc - bcc energy difference can be shown to be “fourth moment” effect (i.e. it needs four-body interactions) 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariHow to Fix Pair Potential Problem ? Pair Potentials Pair Functionals Cluster Functionals Cluster Potentials Many-Body Non-Linearity 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariEffective Medium Theories: The Embedded Atom Method Problem with potentials Cohesive energy depends on number of bonds, but non-linearly Solution Write energy per atom as E = f(number of bonds) where f is non-linear function Energy Functionals How to measure “number of bonds” In Embedded Atom Method (EAM) proximity of other atoms is measured by the electron density they project on the central atom 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariAtomic Electron Densities j a (Ri − Rj )iElectron Density on Site i ρ=∑ f j ≠i Atomic electron density of atom j i Atomic densities are tabulated in E. Clementi and C. Roetti, Atomic Data and Nuclear Data Tables, Vol 14, p177 (1974). 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariClementi and Roetti Tables Clementi and Roetti [At. Data Nucl. Data Tables 14, 177 (1974). 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N Marzari Image removed for copyright reasons.Clementi and Roetti Tables Clementi and Roetti [At. Data Nucl. Data Tables 14, 177 (1974). 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N Marzari Image removed for copyright reasons.The Embedding Function Can be represented either analytically or in Table form ⎡ n ⎛ ρ⎞ mm ⎛ ρ⎞ n ⎤ ⎛ ρ⎞F(ρ) = F0 ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ + F1 ⎜ ⎟⎢ n − m ⎝ ρe ⎠ n − m ⎝ ρ⎠ ⎥⎦ ⎝ ρ⎠e e⎣ n ⎤⎛⎡ ⎛ ρ⎞ ρ⎞ n F(ρ) =⎢1 − ln⎜ ⎟ ⎥⎜ ⎟ ⎢ ⎝ ρe ⎠ ⎥⎦⎝ ρ⎠e⎣ More typically, embedding function is tabulated so as to give an exact fit to the equation of state (Energy versus volume) 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariConvexity of the Embedding Function 0 Ni -5 -10 G(eV) 0 0.01 0.02 0.03 0.04 0.05 0.06 .-3 Figure by MIT OCW. ρ ( A ) Source: Daw, M. S., Foiles, S. M. & Baskes, M. I. The embedded-atom method: a review of theory and applications. Materials Science Reports 9, 251 (1993). 2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariThe complete energy expression: Embedding