DEVELOPMENT OF SEMI-EMPIRICAL ATOMISTIC POTENTIALS MS-MEAM

1. DEVELOPMENT OF SEMI-EMPIRICAL ATOMISTIC POTENTIALS MS-MEAM M. I. Baskes Los Alamos National Laboratory and University of California, San Diego

2. OUTLINE • The Challenge • The Concepts • Bond energy • Many body effects • Transferability • Reference state • Screening • The Models • Pair potentials • Embedded atom method (EAM) • Modified EAM (MEAM) • Multi-state MEAM (MS-MEAM

3. ATOMISTIC MODELS HAVE TWO PURPOSES (I) • Obtain understanding of physical processes • Model system (empirical) potentials • how specific properties affect collective behavior • dependence of yield strength on stacking fault energy • Semi-empirical potentials (fit to experimental data) • plasticity • phase transformations • First principles • diffusion This is meant to be a short list of the many physical process that can be examined by atomistics

4. ATOMISTIC MODELS HAVE TWO PURPOSES (II) • Obtain quantitative properties of specific materials • First principles • lattice constant • structural stability • elastic moduli • Semi-empirical potentials (fit to experimental data) • thermal expansion • melting point • yield strength • thermodynamics / free energy / phase diagrams Many more examples exist

5. IN ORDER TO ACHIEVE THE SECOND PURPOSE WE NEED A METHOD THAT ENCOMPASSES • Accuracy • thermodynamic properties must be known accurately to be useful (a few tenths of a percent of the cohesive energy) • Computational speed • analytic or tabular model • scales linearly with the number of atoms • parallel architecture We only consider here models that have appropriate speed and then try to improve accuracy

6. CONCEPT: BOND ENERGY • Every pair of atoms is connected by a bond (spring) • The bond energy depends on the separation of the atoms • The energy of a material is the sum of the bond energies

7. CONCEPT: MANY BODY EFFECTS • All bonds are not equal • The bond energy also depends on the local environment (coordination) • Coordination / bond length / bond energy are correlated (Pauling)

8. CONCEPT: TRANSFERABILITY • The model will be accurate for all atomic environments • Volume (Nearest neighbor (NN) distance) • Coordination (crystal structure - number of NN) • Defects or strain (loss of symmetry)

9. CONCEPT: REFERENCE STATE (I) • Reference structure • A specific crystal structure • Properties of the reference structure can be obtained from experiment or first principles calculations • Energy vs. volume (NN distance) • Elastic constants • Defect energies • Reference structures have high symmetry • Scaling • energy per atom of the equilibrium reference structure is -1 • distance is scaled by the equilibrium NN distance

10. CONCEPT: REFERENCE STATE (II) • Reference path • A specific path connecting 2 reference structures • Properties along the reference path can be obtained from first principles calculations • Energy vs. distance along path • Reference paths encompass low symmetry states • Coordination changes along a reference path • Incorporation of many reference states will facilitate transferability

11. CONCEPT: SCREENING • Atomic interactions have a finite range • Radial screening • at a cutoff distance the interactions go to zero (smoothly) • dependant on distance • independent of local geometry • Angular screening • intervening atoms reduce interactions to zero • dependent on local geometry • high compression • Necessary for computational scaling with the number of atoms Fellows 11/18/2005

12. A PAIR POTENTIAL REPRESENTS ONLY DISTANCE DEPENDENT BONDING Different Strength Same Strength

13. MODEL: PAIR POTENTIAL • Computation • Analytic or tabular • Scales with number of atoms • Parallel architecture Accuracy • Transferable • Volume • Coordination • Defects/strain i: all atoms j: neighbors of atom i independent of environment radial screening

14. _ ρis obtained from a linear superposition of atomic densities Fandϕare obtained by fitting to the following properties: Universal Binding Energy Relationship(UBER) (lattice constant, bulk modulus, cohesive energy) Shear moduli Vacancy formation energy Structural energy differences (hcp/fcc, bcc/fcc) THE EMBEDDED ATOM METHOD IS SEMI-EMPIRICAL embedding energy host electron density pair interaction UBER

15. MODEL: EAM • Computation • Analytic or tabular • Scales with number of atoms • Parallel architecture Accuracy • Transferable • Volume • Coordination • Defects/strain i: all atoms j: neighbors of atom i radial screening depends on environment

16.   θ        COMPLEX MATERIALS REQUIRETHE ADDITION OF ANGULAR FORCES • EAM uses a linear superposition of spherically averaged electron densities • MEAM allows the background electron density to depend on the local symmetry 

17. MODEL: MEAM • Computation • Analytic or tabular • Scales with number of atoms • Parallel architecture Accuracy • Transferable • Volume • Coordination • Defects/strain • Environmental dependence of bonding • Angular screening • Assumed functional forms • embedding function • electron density • background electron density • screening

18. Background Electron Density Universal Binding Energy Relationship UBER Embedding Function Pair Potential MODIFIED EMBEDDED ATOM METHOD (MEAM) 12 parameters + angular screening for the pair potential and electron densities

19. CONCEPT OF THE SCREENING ELLIPSE LEADS TO A SIMPLE SCREENING MODEL screening ellipse defined by C 2y/rik Cmin and Cmax set limits of screening 2x/rik goes from 0 to 1 smoothly

20. MULTI-STATE MEAM (MS-MEAM) • Same Functional Form as MEAM • Multiple Reference States • Environmental Dependence of Bonds • Angular Screening • Assumed Functional Forms • Asymptotic embedding function • Background electron density

21. MODEL: MS-MEAM • Computation • Analytic or tabular • Scales with number of atoms • Parallel architecture Accuracy • Transferable • Volume • Coordination • Defects (we hope!) • Same functional form as MEAM • Multiple reference states • Environmental dependence of bonds • Angular screening • Assumed functional forms • asymptotic embedding function • background electron density

22. Screening MULTI-STATE MODIFIED EMBEDDED ATOM METHOD (MS-MEAM) Basic Ansatz Embedding Function Background Electron Density

23. FIRST APPLICATION OF MS-MEAM HAS BEEN COMPLETED • Cu Chosen as Model Material • VASP/GGA-PW Used for First Principles Energy Calculations • ~1000 E/V points calculated • M. I. Baskes, S. G. Srinivasan, S. M. Valone, and R. G. Hoagland, Multistate modified embedded atom method, PHYSICAL REVIEW B 75, 094113 2007

24. MS-MEAM EMBEDDING FUNCTION fcc equilibrium density

25. MS-MEAM ELECTRON DENSITIES • simple smooth functions • negative square densities

26. NEED TO HAVE TWO SETS OF ELECTRON DENSITIES • magnetism • electronic states • charges

27. MS-MEAM SCREENING FUNCTIONS screening ellipse  k   i j Fellows 11/18/2005

28. MS-MEAM IS PREDICTIVE FORENERGY vs. NN DISTANCE * used in development of functions coordination 1-12

29. ENERGY DIFFERENCES FOR EQUAL COORDINATION STRUCTURES ARE SMALL * used in development of functions Fellows 11/18/2005

30. ELASTIC CONSTANTS PREDICTED BY MS-MEAM SHOW SHARP INCREASE AT HIGH COMPRESSION fcc C44

31. BCC  SC  FCC (trigonal) 2D-HEX  2D-SQ FCC  BCC (Bain) HOMOGENEOUS TRANSFORMATIONS USED TO DETERMINE SCREENING FUNCTIONS

32. TRANSFORMATIONS ARE A SERIOUS TEST OF TRANSFERABILITY * * used in development of functions

33. CONCLUSIONS • MS-MEAM Has The Potential to be a Fast, Accurate Method of Calculating Atomistic Interactions • Consider MS-MEAM to be a Method for Interpolation/Extrapolation of a FP Data Base • There is No Fitting – Just Direct Calculation From the Data Base • This Method Could Enable Quantitative Thermodynamic Predictions of Multi-component, Multi-phase Materials