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

OUTLINE The Challenge The Concepts The Models 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

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

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

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

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

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)

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)

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

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

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

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

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

THE EMBEDDED ATOM METHOD IS SEMI-EMPIRICAL embedding energy host electron density pair interaction UBER ρ is obtained from a linear superposition of atomic densities F and ϕ are 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) _

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

COMPLEX MATERIALS REQUIRE THE 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     θ 

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

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

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

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

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

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

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

MS-MEAM EMBEDDING FUNCTION fcc equilibrium density

MS-MEAM ELECTRON DENSITIES simple smooth functions negative square densities

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

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

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

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

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

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

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

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