3.320 Lecture 2 (2/3/05)Practical IssuesShow movie of dislocation generationSystem sizes and PeriodicityHow Large Should the Supercell Be ?Example: Calculating the vacancy formation energy in AlLimitations of Pair Potentials: Application to Physical QuantitiesSome data for real systemsSurface RelaxationCauchy ProblemCrystal StructuresHow to Fix Pair Potential Problem ?Effective Medium Theories: The Embedded Atom MethodAtomic Electron DensitiesClementi and Roetti TablesClementi and Roetti TablesThe Embedding FunctionConvexity of the Embedding FunctionThe complete energy expression: Embedding energy + pair potentialEAM: The Physical ConceptEAM is similar to many other effective medium theories.Typical Data to fit EAM parameters toSome results: Linear Thermal Expansion (10-6/K)Some results: Activation Energy for Self Diffusion (in eV)Some results: Surface Energy and RelaxationSome results: Phonon Dispersion for fcc CuSome results:Structure of Liquid Ag at 1270 KSome results: Grain Boundary in AlIssues and Problems with EAMModified Embedded Atom Method (MEAM) to address problem of spherical charge densityResources for Embedded Atom MethodThe other option: Many Body PotentialsExample: SiliconStillinger Webber Potential for SiSurface Reconstruction for SiSurface Reconstruction for SiA multitude of potentials for SiComparison between potentialsReferences for Si Potentials2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariAtomistic Modeling of MaterialsIntroduction to the Course and Pair Potentials3.320 Lecture 2 (2/3/05)2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariPractical Issues E = 12V(r R ii , j ≠iN∑−r R j)Double summation: Number of operations proportional to N2ForceEnergy Fi = r ∇ iE = −Not feasible with million atom simulations -> use neighbor listsMinimizationStandard schemes: Conjugate Gradient, Newton-Raphson, Line Minimizations (Using Force)Typically at least scale as N2∂V (r R i−r R j)∂r R ij ≠ iN∑–2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariShow movie of dislocation generation2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariSystem sizes and PeriodicityFinite System (e.g. molecule or cluster)No problem; -> simply use all the atomsInfinite System (e.g. solids/liquids)Do not approximate as finite -> use Periodic Boundary ConditionsBaTiO32/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariFor defect calculation unit cell becomes a supercellDefectPeriodic Images of Defect2/3/05 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 Al0.90.850.80.750.70.650.60.550.520 30 40 50 60 70 80 90 100 110Vacancy Formation Energy (eV)Number of Atoms in SupercellXXFigure by MIT OCW.Limitations of Pair Potentials: Application to Physical QuantitiesVacancy Formation Energy2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N Marzari2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariSome data for real systemsAfter 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).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.2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariSurface RelaxationWith potentials relaxation of surface plane is usually outwards,for metals experiments find that it is inwardsVacuumSurface PlaneV(r)rNN2NN2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariCauchy Problemσ11σ22σ33σ12σ13σ23⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ • ε11ε22ε33ε12ε13ε23⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ CijFor Potentials C12= C442/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariCrystal StructuresPair Potentials can fundamentally not predict crystal structuresin 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 PotentialsPair FunctionalsCluster FunctionalsCluster PotentialsMany-BodyNon-Linearity2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariEffective Medium Theories: The Embedded Atom MethodCohesive energy depends on number of bonds, but non-linearlyProblem with potentialsSolutionWrite energy per atom as E = f(number of bonds) where f is non-linear functionEnergy FunctionalsHow 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 atom2/3/05 Massachusetts Institute of Technology 3.320: Atomistic Modeling of Materials G. Ceder and N MarzariAtomic Electron Densitiesρi=fja(Ri− Rj)j ≠i∑Electron Density on Site iAtomic electron density of atom jiAtomic 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 TablesClementi and Roetti [At. Data