Interatomic Potentials for Ionic Systems Byeong-Joo Lee POSTECH-CMSE

Background Importance of Ionic Materials Sensor, Battery, Devices, Metal Surfaces, etc. Need to handle “ionic + covalent + metallic” materials Interfacial Reaction between metals and SiO2 substrate Diffusion of metallic atoms in amorphous SiO2 Atomistic simulation on “ionic + covalent + metallic” materials ???

Purpose and Scope Development of Interatomic Potential Model that covers “ionic + covalent + metallic” materials, simultaneously. Review interatomic potentials for ionic and hybrid materials Propose possible form of an interatomic potential formalism

Outline Interatomic Potential for Ionic Materials Many-Body Potentials Point Charge Model Polarization (Shell Model) Many-Body Potentials Tersoff EAM – MEAM – 2NN MEAM Many-body potentials used for ionic systems Many-Body Potentials for Ionic Materials Charge Equilibration Model EAM + Qeq Tersoff + Qeq Proposal of New Interatomic Potential Form

Interatomic Potential for Ionic Materials Fixed Point Charge Born-Mayer-Huggins TTAM BKS Initially applied to liquid or glass, not crystals : probably, unable to reproduce crystal structures 1st MD on SiO2 glass Woodcock [5], 1976 More information available with upgraded measuring techniques for crystal structures and dynamics 1988: BMH + many-body interaction to reproduce O-Si-O bonding angle: (cosθjik - cosθojik)2 [6] 1988: BMH + modified Coulomb interaction considering excess charge distribution in oxygen + Ab Initio on SiO2 model clusters → α-quartz, α-cristobalite, Coesite, Stishovite, for the first time → TTAM [7] 1990: BMH + Ab Initio + Experimental Information on α-quartz → better description than TTAM → BKS [8] TTAM&BKS: representative Point Charge Potential for SiO2 during 1990s. (qSi = +2.4, qO = -1.2) Limitation: use of Point Charge, pair-wise potentials not applicable to pure Si or Si/SiO2 1994: Jiang & Brown: SW Si – BKS SiO2, ionization energy, charge variation, bond-softening function → behavior of O atom in Si [11] 2010: Soulairol & Cleri: SW Si – BKS SiO2 + different q for interface

Interatomic Potential for Ionic Materials Fixed Point Charge + electronic polarization Include dipole-charge, dipole-dipole interaction due to electronic polarization Shell Model by Dick & Overhauser [13], 1958 Ion = core electron core + valence electron shell Deviation of Center of mass of Shell causes a dipole Shell connected to core by an artificial spring and interact through harmonic restoring force Shell Model has been successful for diatomic molecule, alkali halides and also for Al2O3 [14] BMH + polarization : representative approach during 1980s for alkali halides, binary, mixed oxides [15] Shell model: leading model for ionic materials in GULP [19] 2002: Morse-Stretch pp + fixed point charge Coulomb + dipole polarization for Oxygen ions [17] fitting (force-matching) on liquid SiO2 → better description for polymorphs than BKS Limitation: not applicable to pure Si or Si/SiO2, not describing variable charge Next Step: Many-body + variable charge

Many-Body Potential : EAM – 2NN MEAM Embedding energy of impurity atoms is determined by the electron density of the host (from first-principles) → individual atoms are impurity atoms → EAM concept [29,30] How to compute F and Ф ? No specific function form was given in initial EAM → reason for so many EAMs Rose universal equation of state [23] gives a guide [31] EAM : linear supposition for computation of electron density of a site → mainly for fcc Introduction of bonding directionality → Modified EAM (1nn interaction only) → applied to Si [32], bcc [33] and hcp [34], but stability problem Need to consider 2NN interactions to solve critical shortcomings in MEAM → 2NN MEAM [36,37] → applicable to both metallic and covalent systems: metals, carbides, nitrides, Si, Ge, etc. [38-40]

Many-Body Potential : Tersoff 1985 Abell : Close relation between Morse-type pair potential and Rose universal behavior → replacement of Born-Mayer by Morse-Stretch Tersoff potential [24-26] bij : bond order – 1nn interaction, bond length and angle, effect of local environments, etc. applied to C [27] and SiC [28] and extended to Brenner-REBO [87-89] for alloys : arithmetic mean to λ, μ and geometric mean to A, B, R, S

Many-Body Potential for Ionic Materials Umeno [14] : using Tersoff for SiO2 Independent fitting to λ, μ, A, B instead of mean values applicable to β-cristobalite, β-quartz which was difficult by BKS Kuo [15] : using MEAM for SiO2 applicable to α, β-quartz, α, β-cristobalite, β-tridymite

Charge Equilibration Model 1991 Rappe & Goddard [48] : based on previous concepts on electronegativity, equilibration [49-57]. - equilibrium charge in molecules considering Coulomb interaction and penalty energy for charged isolated atoms (atomic self-energy) IP & EA : ionization potential과 electron affinity χ0 : electronegativity J0 : atomic hardness representing Coulomb repulsion between two electrons in an orbital JAB : Coulomb interaction between A & B computed by a Coulomb integral on atomic charge density expressed for a Slater-type orbital Basic idea in Qeq model is to equalize the atomic chemical potential of all individual atoms (χ1 = χ2 = … = χN) First applied to SiO2 in 1999 [58] : Morse-Stretch pair potential + charge equilibration - Quartz-Stishovite phase transition & Silica glass Swamy & Gale [59] in 2000 : Titanium oxide system including rutile, anatase, brookite, TiO2-II, Ti2O3, monoclinic high- and low temp forms of Ti3O5, TiO, ramsdellite-type TiO2, g-Ti3O5, two Magneli phases: Ti4O7 and Ti6O11

EAM + Charge Equilibration 1994 Streitz & Mintmire [60] : first Qeq approach for crystalline materials, EAM + Qeq for Al2O3 2004 Zhou [70] : solving charge stability problem, - extened to multicomponent oxides, O-Al-Ni-Co-Fe system [71] 2007 Lazic [74] : MEAM (different from Baskes) + Qeq, not much is published Oxidation of Al nano cluster [61,62]  

Tersoff + Charge Equilibration 1996 Yasukawa [76] : introduce atomic energy ΣiΦi & Coulomb energy ½ΣiΣjEIONij - effective point charge with cutoff function in Coulomb potential, not with Ewald summation - Considering changes in ionic radius and short range interaction due to charge Crack propagation behavior of SiO2 with or without H2O Adhesion strength on Al,Cu/TiN,W,SiO2 thin film interface [77] Upgrade in parameter [78] & Formalism for Coulomb interaction [79] 2007 Sinnott & Phillpot group [80] : confirm application to α, β-quartz, α, β-cristobalite, but stability problem. - atomic self-energy up to 4th order & introduction of bond-bending energy, (cosθOSiS - cosθoOSiO)2, - COMB (Optimized Many-Body Potential), but cannot generate amorphous SiO2 & bad results for α-quartz 2010 modified version for SiO2 [81] : Slater 1s orbital type Coulomb integral & Ewald + another penalty term - applicable to α, β-quartz, α, β-cristobalite, β-tridymite, Coesite, Stishovite, generally worse than TTAM 2010 Hf/HfO2, Cu/Cu2O [83,84] : different bond-bending form depending on cation element

Others : ReaxFF Bond-Order : based on correlation between bond order & bond distance or bond energy describe bond dissociation → chemical reaction - including bonding angle, torsion, charge equilibration, van der Waals interaction, etc. - mainly for hydrocarbon system [85], but also to oxides, Si/SiO2 system [86] Most powerful : covering Hydrocarbon system like Brenner-REBO [87-89] and charge equilibration like COMB Number of parameters for Carbon, for example : 90s - how to determine the parameter values ? → 10 ~ 15 systems during up to now - retirement of Prof. Goddard → Dr. van Duin @ Penn State

Summary Up to now no interatomic potential for ionic + covalent + metallic alloy systems

Potential for Ionic+Covalent+Metallic Materials Charge Effect ? Correct physics : easy parameterization and good trasferability Point Charge vs. Charge Distribution ? TTAM that considered charge distribution could describe the SiO2 polymorphs for the first time Shell Model ? No publication for shell model + many-body potential Variable charge can be superior to fixed charge, for bond dissociation, surface, interface, and other defects Coulomb Integral ? COMB10 [81] is generally worse than BKS or TTAM for SiO2 polymorphs Coulomb intergral (COMB10 [81]) vs. effective point charge (Yasukawa 2010 [79]) ? Summation of Long Range Potential (1/r radial behavior) ? Ewald method [70], PPPM [75], direct summation method [82] Charge Equilibration Method ? Inverse matrix [60], Conjugate gradient method [70], Lagrangian dynamics [80] Manybody Potential ? - COMB had to change the functional form for bond-bending term, probably due to the limitation of Tersoff. [Tersoff potential has never been applied to metallic alloy systems] - MEAM is also a kind of bond order potential, 2NN MEAM has been applied to both covalent and metallic alloy systems Conclusion 2NN MEAM + Qeq = Tersoff+Qeq + EAM+Qeq Paying attention to charge stability and extension to multicomponent systems, and searching for the best solution for Coulomb integral, long range potential and charge equilibraion

References A. Rahman, R. H. Fowler and A. H. Narten, ”Structure and Motion in Liquid BeF2, LiBeF3, and LiF from Molecular Dynamics Calculations,” J. Chem. Phys. 57, 3010 (1972). L. V. Woodcock, “Isothermal Molecular Dynamics Calculations for Liquid Salts,” Chem. Phys. Lett. 10, 257 (1971). B. J. Alder and T. E. Wainwright, ”Studies in Molecular Dynamics. I. General Method,” J. Chem. Phys. 31, 459 (1959). B.P. Feuston and S.H. Garofalini, “Empirical Three-Body Potential for Vitreous Silica,” J. Chem. Phys. 89, 5818 (1988). L.V. Woodcock, C.A. Angell and P. Cheeseman, “Molecular Dynamics Studies of the Vitreous State: Simple Ionic Systems and Silica,” J. Chem. Phys. 65, 1565 (1976). L. V. Woodcock, Advances in Molten Salts Chemistry, Vol. 3 Chap. 1, pp.1-75, Plenum, New York (1975). C. R. A. Catlow and A.M. Stoneham, “Ionicity in Solids,” J. Phys. C: Solid State Phys. 16, 4321 (1983). B.W.H. van Beest, G.J. Kramer and R.A. van Santen, “Force Fields for Silicas and Aluminophosphates Based on Ab Initio Calculations,” Phys. Rev. Lett. 64, 1955 (1990). S. Tsuneyuki, M. Tsukada, H. Aoki and Y. Matsui, “First-Principles Interatomic Potential of Silica Applied to Molecular Dynamics,” Phys. Rev. Lett. 61, 869 (1988). R. Soulairol and F. Cleri, “Interface Structure of Silicon Nanocrystals Embedded in an Amorphous Silica Matrix,” Solid State Sciences 12, 163 (2010). Z. Jiang and R. A. Brown, “Modeling Oxygen Defects in Silicon Crystals using an Empirical Interatomic Potential,” Chem. Engin. Sci. 49, 2991 (1994). J. R. Tessman, A. H. Kahn and W. Shockley, “Electronic Polarizabilities of Ions in Crystals,” Phys. Rev. 92, 890 (1953). G. J. Dienes, D. O. Welch, C. R. Fischer, R. D. Hatcher, O. Lazareth and M. Samberg, “Shell-Model Calculation of Some Point-Defect Properties in α-Al2O3,” Phys. Rev. B 11, 3060 (1975). B .G. Dick and A. W. Overhauser, “Theory of the Dielectric Constants of Alkali Halide Crystals,” Phys. Rev. 112, 90 (1958). P. Tangney and S. Scandolo, “An Ab Initio Parametrized Interatomic Force Field for Silica,” J. Chem. Phys. 117, 8898 (2002). P. Vashishta, R. K. Kalia, J. P. Rino and I. Ebbsjö, “Interaction Potential for SiO2: A Molecular-Dynamics Study of Structural Correlations,” Phys. Rev. B 41, 12197 (1990). G. V. Lewis and C. R. A. Catlow, “Potential Models for Ionic Oxides,” J. Phys. C: Solid State Phys. 18, 1149-1161 (1985). J. H. Rose, F. Ferrante and J. R. Smith, “Universal Binding Energy Curves for Metals and Bimetallic Interfaces,” Phys. Rev. Lett. 47, 675 (1981). J. D. Gale and A.L. Rohl, “The General Utility Lattice Program (GULP),” Mol. Simul. 29, 291 (2003). D. Herzbach, K. Binder and M.H. Müser, “Comparison of Model Potentials for Molecular-Dynamics Simulations of Silica,” J. Chem. Phys. 123, 124711 (2005). J. H. Rose, J. R. Smith, F. Guinea and F. Ferrante, “Universal Features of the Equation of State of Metals,” Phys. Rev. B 29, 2963 (1984). J. H. Rose, F. Ferrante and J. R. Smith, “Universal features of bonding in metals,” Phys. Rev. B 28, 1835 (1983). J. R. Smith, F. Ferrante and J. H. Rose, “Universal Binding-Energy Relation in Chemisorptions,” Phys. Rev. B 25, 1419 (1982). J. Tersoff, “New Empirical Approach for the Structure and Energy of Covalent Systems,” Phys. Rev. B 37, 6991 (1988). J. Tersoff, “New Empirical Model for the Structural Properties of Silicon,” Phys. Rev. Lett. 56, 632 (1986) J. Tersoff, ”Modeling Solid-State Chemistry: Interatomic Potentials for Multicomponent Systems,” Phys. Rev. B 39, 5566 (1989). J. Tersoff, “Empirical Interatomic Potential for Carbon with Applications to Amorphous Carbon,” Phys. Rev. Lett. 61, 2879 (1988). J. Tersoff, “Empirical Interatomic Potential for silicon with improved elastic properties,” Phys. Rev. B 38, 9902 (1988). S. M. Foiles, M. I. Baskes and M. S. Daw, “Embedded-Atom Method Functions for the fcc Metals Cu, Ag, Au, Ni, Pd, Pt and Their Alloys,” Phys. Rev. B 33, 7983 (1986). M. S. Daw and M. I. Baskes, “Embedded-Atom Method: Derivation and Application to Impurities, Surfaces, and Other Defects in Metals,” Phys. Rev. B 29, 6443 (1984). M. S. Daw and M. I. Baskes, “Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals,” Phys. Rev. Lett. 50, 1285 (1983). M. I. Baskes and R. A. Johnson, “Modified Embedded Atom Method Potentials for HCP Metals,” Modelling Simul. Mater. Sci. Eng. 2, 147 (1994). M. I. Baskes, “Modified Embedded-Atom Method Potentials for Cubic Materials and Impurities,” Phys. Rev. B 46, 2727 (1992). M. I. Baskes, J. S. Nelson and A. F. Wright, “Semiempirical Modified Embedded-Atom Potentials for Silicon and Germanium,” Phys. Rev. B 40, 6085 (1989). B.-J. Lee, M. I. Baskes, H. Kim, Y. K. Cho, “Second Nearest-Neighbor Modified Embedded Atom Method Potentials for BCC Transition Metals,” Phys. Rev. B 64, 184102 (2001). B.-J. Lee, M. I. Baskes, “Second Nearest-Neighbor Modified Embedded-Atom Method Potential,” Phys. Rev. B 62, 8564 (2000). M. I. Baskes, “Determination of Modified Embedded Atom Method Parameters for Nickel,” Mater. Chem. Phys. 50, 152 (1997). B.-J. Lee, W.-S. Ko, H.-K. Kim, E.-H. Kim, “Overview: The modified embedded-atom method Interatomic Potentials and recent progress in atomistic Simulations,” CALPHAD 34, 510 (2010). B.-J. Lee, “A Semi-Empirical Atomistic Approach in Materials Research,” J. Phase Equilib. Diff. 30, 509 (2009). H.-K. Kim, W.-S. Jung, B.-J. Lee, “Modified Embedded-Atom Method Interatomic Potentials for the Nb-C, Nb-N, Fe-Nb-C and Fe-Nb-N systems,” J. Mater. Res. 25, 1288 (2010). T. Watanabe, D. Yamasaki, K. Tatsumura and I. Ohdomari, “Improved Interatomic Potential for stressed Si, O mixed systems,” Appl. Surf. Sci. 234, 207 (2004). T. Watanabe, H. Fujiwara, H. Noguchi, T. Hoshino and I. Ohdomari, “Novel Interatomic Potential Energy Function for Si, O Mixed Systems,” Jpn. J. Appl. Phys. 38, L366 (1999). F. H. Stillinger and A. Weber, “Computer simulation of local order in condensed phases of silicon,” Phys. Rev. B 31, 5262 (1985). M. I. Baskes, “Modified Embedded Atom Method Calculations of Interfaces,” Report number: SAND--96-8484C, Sandia National Laboratories, Livermore, 1996. S.R. Billeter, A. Curioni, D. Fischer and W. Andreoni, “Ab Initio Derived Augmented Tersoff Potential for Silicon Oxynitride Compounds and Their Interfaces with Silicon,” Phys. Rev. B 73, 155329 (2006). Y. Umeno, T. Kitamura, K. Date, M. Hayashi, and T. Iwasaki, “Optimization of Interatomic Potential for Si/SiO2 system based on force matching,” Comput. Mater. Sci. 25, 447 (2002). A. K. Rappe and W. A. Goddard III, “Charge Equilibration for Molecular Dynamics Simulations,” J. Phys. Chem. 95, 3358 (1991). C.-L. Kuo and P. Clancy, “Development of Atomistic MEAM Potentials for the Silicon–Oxygen–Gold Ternary System,” Modelling Simul. Mater. Sci. Eng. 13, 1309 (2005). L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, New York, (1960). R. T. Sanderson, “Partial Charges on Atoms in Organic Compounds,” Science 11, 207 (1955). R. S. Mulliken, “A New Electroaffinity Scale; Together with Data on Valence States and on Valence Ionization Potentials and Electron Affinities,” J. Chem. Phys. 2, 782 (1934). P. Politzer and H. Weinstein, “Some Relations between Electronic Distribution and Electronegativity,” J. Chem. Phys. 71, 4218 (1979). R. G. Parr, R. A. Donnelly, M. Levy and W. E. Palke, “Electronegativity: The Density Functional Viewpoint,” J. Chem. Phys. 68, 3801 (1978). R. P. Iczkowsky and J. L. Margrave, “Electronegativity,” J. Am. Chem. Soc. 83, 3547 (1961). W. J. Mortier, S. K. Ghosh and S. Shankar, “Electronegativity-Equalization Method for the Calculation of Atomic Charges in Molecules”, J. Am. Chem. Soc. 108, 4315 (1986). W. J. Mortier, K. van Genechten and J. Gasteiger, “Electronegativity Equalization: Application and Parametrization,” J. Am. Chem. Soc. 107, 829 (1985). R. G. Parr and R. G. Pearson, “Absolute Hardness: Companion Parameter to Absolute Electronegativity,” J. Am. Chem. Soc. 105, 7512 (1983). V. Swamy, J. D. Gale, “Transferable Variable-Charge Interatomic Potential for Atomistic Simulation of Titanium Oxides”, Phys. Rev. B 62, 5406 (2000). E. Demiralp, T. Cagin and W.A. Goddard, “Morse Stretch Potential Charge Equilibrium Force Field for Ceramics: Application to the Quartz-Stishovite Phase Transition and to Silica Amorphous”, Phys. Rev. Lett. 82, 1708 (1999). T. Campbell, G. Aral, S. Ogata, R. K. Kalia, A. Nakano and P. Vashishta, “Oxidation of aluminum Nanoclusters,” Phys. Rev. B 71, 205413 (2005). T. Campbell, R. K. Kalia, A. Nakano, P. Vashishta, S. Ogata and S. Rodgers, “Dynamics of Oxidation of Aluminum Nanoclusters using Variable Charge Molecular-Dynamics Simulations on Parallel Computers,” Phys. Rev. Lett. 82, 4866 (1999). F. H. Streitz and J. W. Mintmire, “Electrostatic Potentials for Metal-Oxide Surfaces and Interfaces,” Phys. Rev. B 50, 11996 (1994). S. W. de Leeuw, J. W. Perram and E. R. Smith, “Simulation of Electrostatic Systems in Periodic Boundary Conditions. I. Lattice Sums and Dielectric Constants,” Proc. R. Soc. London Ser. A 373, 27 (1980). C. C. J. Roothaan, “A Study of Two-Center Integrals Useful in Calculations on Molecular Structure. I,” J. Chem. Phys. 19, 1445 (1951). N. Rosen, “Calculation of Interaction between Atoms with s-Electrons,” Phys. Rev. 38, 255 (1931). J. Hautman and M. L. Klein, “An Ewald Summation Method for Planar Surfaces and Interfaces,” Mol. Phys. 75, 379 (1992). D. E. Parry, “The Electrostatic Potential in the Surface Region of an Ionic Crystal,” Surf. Sci. 49, 433 (1975). E. R. Smith, “Electrostatic Energy in Ionic Crystals,” Proc. R. Soc. London Ser. A 375, 475 (1981). X. W. Zhou, H. N. G. Wadley, J.-S. Filhol, M. N. Neurock, “Modified Charge Transfer–Embedded Atom Method Potential for Metal/Metal Oxide Systems,” Phys. Rev. B 69, 035402 (2004). D. M. Heyes, “Surface Stress of Point Charge Lattices,” Surf. Sci. Lett. 293, L857 (1993). J. Ferrante, H. Schlosser and J. H. Rose, “Global expression for representing diatomic Potential-energy curves,” Phys. Rev. A 43, 3487 (1991). J. R. Smith, H. Schlosser, W. Leaf, J. Ferrante and J. H. Rose, “Connection between Energy Relations of Solids and Molecules,” Phys. Rev. A 39, 514 (1989). X. W. Zhou, H. N. G. Wadley, “A Charge Transfer Ionic–Embedded Atom Method Potential for the O–Al–Ni–Co–Fe System,” J. Phys.: Condens. Matter 17, 3619 (2005). A. Yasukawa, “Using an Extended Tersoff Interatomic Potential to Analyze the Static-Fatigue Strength of SiO2 under Atmospheric Influence,” JSME Int. J., Ser. A 39, 313 (1996). R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York, 1981. I. Lazic, M. Ernst, B. Thijsse, “Atomistic Simulation Methods for Studying Self Healing Mechanisms in Al/Al2O3,” Proceedings of the First International Conference on Self Healing Materials, 18-20 April 2007, Noordwijk aan Zee, The Netherlands A. Yasukawa, “Atomistic Simulation of Environment-Assisted Crack Propagation Behavior of SiO2, ” J. Solid Mech. Mater. Engin. 4, 599 (2010). A. Yasukawa, in Japan Society of Mechanical Engineers, p.71, Sept. 19, 2003, Hitachi City, Ibaraki, Japan. T. Iwasaki and H. Miura, “Molecular dynamics analysis of adhesion strength of interfaces between thin films,” J. Mater. Res. 16, 1789 (2001). D. Wolf, P. Keblinski, S. R. Phillpot, and J. Eggebrecht, “Exact Method for the Simulation of Coulombic Systems by Spherically Truncated, Pairwise r-1 Summation,” J. Chem. Phys. 110, 8254 (1999). T.-R. Shan, B. D. Devine, M. Hawkins, A. Asthagiri, S. R. Phillpot and S. B. Sinnott, “Second-Generation Charge-Optimized Many-Body Potential for Si/SiO2 and amorphous Silica,” Phys. Rev. B 82, 235302 (2010). J. Yu, S. B. Sinnott, S. R. Phillpot, “Charge optimized many-body Potential for the Si/SiO2 system,” Phys. Rev. B 75, 085311 (2007) A. C. T. van Duin, S. Dasgupta, F. Lorant and W. A. Goddard III, “ReaxFF: A Reactive Force Field for Hydrocarbons,” J. Phys. Chem. A 105, 9396 (2001). B. D. Devine, T.-R. Shan, S. B. Sinnott, and S. R. Phillpot, “Charge Optimized Many-Body Potential for the Copper/Copper Oxide System,” 2011 (unpublished) T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott and S. R. Phillpot, “Charge Optimized Many-Body Potential for the Hafnium/Hafnium Oxide System,” Phys. Rev. B 81, 125328 (2010). S. J. Stuart, A. B. Tutein and J. A. Harrison, “A reactive Potential for Hydrocarbons with Intermolecular Interactions,” J. Chem. Phys. 112, 6472 (2000). D. W. Brenner, ”Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films,” Phys. Rev. B 42, 9458 (1990). A. C. T. van Duin, A. Strachan, S. Stewman, Q. Zhang, X. Xu and W. A. Goddard III, “ReaxFFSiO Reactive Force Field for Silicon and Silicon Oxide Systems,” J. Phys. Chem. A 107, 3803 (2003). D. W Brenner, O. A Shenderova, J. A Harrison, S. J. Stuart, B. Ni and S. B. Sinnott, “A second-generation reactive empirical bond order (REBO) Potential energy expression for Hydrocarbons,” J. Phys.: Condens. Matter 14, 783 (2002).

Atomistic Simulations - MEAM & Applications Byeong-Joo Lee Dept. of MSE Pohang University of Science and Technology (POSTECH) calphad@postech.ac.kr

Semi-Empirical Atomic Potentials - Historical Background Pair Potentials (~1980) ▷ Elastic Constants are NOT correctly reproduced Many Body Potentials (1980's) ▷ Embedded Atom Method (EAM: 1983) ▷ Finnis and Sinclair Potential (1984) ▷ Glue Model (1986) ▷ Equilivalent-Crystal Model (1987)

Semi-Empirical Atomic Potentials – History of Development EAM Potentials (1983, M.S. Daw and M.I. Baskes) ▷ Successful mainly for FCC elements - many other many-body potentials show similar performance 1NN MEAM Potentials (1987,1992, M.I. Baskes) ▷ Show Possibility for description of various structures - important to be able to describe multi-component system 2NN MEAM Potentials (2000, B.-J. Lee & M.I. Baskes) ▷ Applicable to fcc, bcc, hcp, diamond structures and their alloys

EAM/MEAM – General E : Total Potential Energy F : Embedding Energy  : Electron Density (Considering Bonding Directionality)  : Pair Interaction Energy

EAM/MEAM – Embedding Function M.I. Baskes et al., Phys. Rev. B, 40, 6085 (1989)

EAM/MEAM – Universal EOS J.H. Rose et al., Phys. Rev. B, 29, 2963 (1984)

EAM/MEAM – Electron Density for EAM

EAM/MEAM – Electron Density for MEAM + Angular contribution

EAM/MEAM – Electron Density for MEAM + Angular contribution with ti(0) =1

EAM/MEAM – 1st Nearest Neighbor MEAM

1NN MEAM vs. 2NN MEAM – Many-Body Screening Xik=(Rik/Rij)2 and Xkj=(Rkj/Rij)2 Cmax Cmin i j fc(x) = 1 x  1 0  x  1 0 x  0

2NN MEAM – Computation of pair-wise potential

Evaluation of MEAM Potential Parameters for Elements Ec, Re, B, A, d,  (0),  (1),  (2),  (3), t(1), t(2), t(3), Cmax, Cmin ▷ Cohesive Energy of Stable and Metastable Structure ▷ Nearest Neighbor Distance ▷ Bulk Modulus, Elastic Constants (C11, C12, C44) ▷ Stacking Fault Energy ▷ Vacancy Formation Energy ▷ Surface Energy

Semi-Empirical Atomic Potentials - Performance Elastic Constants ▷ B, C11, C12, C44, ... Defect Energy ▷ Surface Energy ▷ Heat of Vacancy Formation, … Structural Energy ▷ Energy and Lattice Parameters in Different Structures Thermal Property ▷ Specific Heat ▷ Thermal Expansion Coefficient ▷ Melting Temperature, ...

MEAM for BCC Transition Metals – B.-J. Lee et al., PRB, 2001 Elem. C11 C12 C44 E(100) E(110) E(111) Evf Ebcc/fcc Efcc/hcp Fe 2.430 1.380 1.219 2510 2356 2668 1.75 0.069 -0.023 2.431 1.381 1.219 2360* 1.79 0.082 -0.023 Cr 3.909 0.897 1.034 2300 2198 2501 1.91 0.070 -0.02 3.910 0.896 1.032 2200* 1.80 0.075 -0.029 Mo 4.649 1.655 1.088 3130 2885 3373 3.09 0.167 -0.038 4.647 1.615 1.089 2900* 3.10 0.158 -0.038 W 5.326 2.050 1.631 3900 3427 4341 3.95 0.263 -0.047 5.326 2.050 1.631 2990* 3.95 0.200 -0.047 V 2.323 1.194 0.460 2778 2636 2931 2.09 0.084 -0.011 2.324 1.194 0.460 2600* 2.10 0.078 -0.036 Nb 2.527 1.331 0.319 2715 2490 2923 2.75 0.176 -0.012 2.527 1.332 0.310 2300* 2.75 0.140 -0.036 Ta 2.664 1.581 0.875 3035 2778 3247 2.95 0.148 -0.023 2.663 1.582 0.874 2780* 2.95 0.166 -0.041

MEAM for FCC Transition Metals – B.-J. Lee et al., PRB, 2003 Elem. C11 C12 C44 E(100) E(110) E(111) Evf Ebcc/fcc Efcc/hcp ε (0-100oC) Cu 1.762  1.249  0.818  1382  1451  1185 1.11 -0.08    0.007   17.0          1.762  1.249  0.818             1770        1.03-1.30   -0.04    0.006 17.0   Ag    1.315  0.973  0.511       983  1010   842    0.94    -0.08     0.005   18.9          1.315  0.973  0.511             1320         1.1      -0.04     0.003    19.1   Au     2.015  1.697  0.454      1138  1179   928   0.90    -0.06     0.009    14.2         2.016  1.697  0.454             1540         0.9     -0.04     0.003    14.1   Ni    2.612  1.508  1.317      1943  2057  1606   1.51     -0.16     0.02     12.6          2.612  1.508  1.317              2240        1.6     -0.09     0.02     13.3 Pd    2.342  1.761  0.712      1743  1786  1435    1.50     -0.17     0.02     11.0         2.341  1.761  0.712             2043        1.4,1.7   -0.11     0.02     11.0 Pt     3.581  2.535  0.775      2288  2328  1710   1.50     -0.28     0.02     9.2          3.580  2.536 0.774              2691      1.35,1.5   -0.16     0.03     9.0   Al     1.143  0.619  0.316       848   948   629  0.68   -0.12     0.03     22.0          1.143  0.619  0.316             1085          0.68    -0.10     0.06     23.5   Pb     0.556  0.454  0.194       426    440   375    0.58     -0.04     0.003    30.1          0.555  0.454  0.194               534         0.58    -0.02     0.003    29.0

MEAM for Silicon C11 C12 C44 E(100) E(110) E(111) Evf Edia/fcc Edia/hcp Edia/bcc ε (1012dyne/cm2) (erg/cm2) (eV) (eV) (0-100oC) 1.67 0.65 0.80 2631 1766 1442 3.67 0.57 0.55 0.52 2.65 1.68 0.65 0.80 1135* 3.3-4.3 0.57 0.55 0.53 2.69

2NN MEAM Interatomic Potentials – for Al and Fe Property MEAM-Al (exp.) MEAM-Fe (exp.) C11 (1012 dyne/cm2) C12 (1012 dyne/cm2) C44 (1012 dyne/cm2) Evf (eV) QD (eV) EIf (eV) 1.143 (1.143) 0.619 (0.619) 0.316 (0.316) 0.68 (0.68) 1.33 (1.33) 2.49 (-) 2.430 (2.431) 1.380 (1.381) 1.219 (1.219) 1.75 (1.79) 2.28 (2.5) 4.20 (-) E(100) (mJ/m2) E(110) (mJ/m2) E(111) (mJ/m2) d(100) (%) d(110) (%) d(111) (%) 848 (1085a) 948 (1085a) 629 (1085a) +1.8 (+1.8) -8.9 (-8.5±1.0) +1.0 (0.9±0.5) 2510 (2360a) 2356 (2360a) 2668 (2360a) -1.1 (-0.2, -1.5) -1.5 (0) -10.5 (-16.9) Ebcc/fcc (eV/atom) Efcc/hcp (eV/atom) 0.12 (0.10b) 0.03 (0.06b) 0.048 (0.082b) -0.018 (-0.023b) (0-100oC) (10-6/K) Cp (0-100oC) (J/mol·K) m.p. (K) Hm (KJ/mol) Vm (%) 22.0 (23.5) 26.2 (24.7) 937 (933) 11.0 (10.7) 6.7 (6.5) 12.4 (12.1) 26.1 (25.5) 2000 (1811) 13.2 (13.8) 4.0 (3.5)

2NN MEAM – 2NNMEAM for Alloy Systems

2NN MEAM for Alloy Systems – Optimization of Potential Parameter, Fe-Pt Selected value Procedure for the determination ΔEc -0.4600 Fitting to ΔH or Ttr re 2.7181 Fitting to lattice parameter B 2.6201 Fitting to bulk modulus d 0.25dFe+0.75dPt Assumption Cmin(Fe-Pt-Fe) 0.36 ( = CminFe) Cmin(Pt-Fe-Pt) 1.53 ( = CminPt) Cmin(Fe-Fe-Pt) [0.5(CminFe)1/2 + 0.5(CminPt) 1/2 ]2 Cmin(Fe-Pt-Pt) ρ0 ρ0Fe = ρ0Pt = 1.0 A temporary assumption

2NN MEAM for Fe-Cr Binary System – B.-J. Lee et al., CALPHAD, 2001 200K 850K 1000K

MEAM for Cu-Ni Binary System – B.-J. Lee and J.-H. Shim, CALPHAD, 2004

MEAM for Ni-Si Binary System Dilute Heat of Solution (eV/atom) Si in (Ni) -1.50 (-1.37) Ni in (Si) +0.50 Ni3Si 0.36 (0.36) 3.504 (3.504) 2.64 3.67 (3.63-3.75) 2.13 (2.00-2.05) 1.54 1.96 (1.67-1.72) 5.3 (7.2) NiSi2 0.28 (0.28) 5.391 (5.406) 1.93 (1.60) 2.39 1.69 0.70 (0.58) 0.32 8.0 Enthalpy of Formation (eV/atom) Lattice constant (Å) Bulk Modulus (100 GPa) C11 (100 GPa) C12 (100 GPa) C11-C12 (100 GPa) C44 (100 GPa) (100) fracture energy (J·m-2)

MEAM for Co-Pt Binary System - S.I. Park et al., Scripta Mater., 2001. Property Pt3Co PtCo PtCo3 Cohesive Energy 5.500 5.215 4.873 (eV/atom) 5.555±0.017 5.228±0.005 Lattice Constant a=3.833 3.754, c/a=.98 3.625 (Å) a=3.831 3.745, c/a=.98 3.668 Transition 1070-1080 970-980 760-770 Temperature (K) 1000 1100 840

MEAM for Ni-W Binary System – J.-H. Shim et al., J. Mater. Res., 2003 Property fcc (XW=0.11) Ni4W Cohesive Energy 4.922 5.36 (fcc, 5.27) (eV/atom) 4.925 5.40 Lattice Constant a=3.57 a=5.73, c=3.553 (Å) a=3.56 a=5.73, c=3.553

MEAM for Ni-W Binary System a (Å) c (Å) Ec(eV) B(Gpa) Ni4W (D1a) 5.73 3.553 5.36 292 5.73 3.553 5.40 293 Ni3W (L12) 3.62 - 5.58 319 3.58 - 5.65 287 Ni3W (D019) 2.56 4.05 5.59 316 2.53 - 5.42 289 NiW3 (L12) 3.86 - 7.29 316 3.84 - 7.55 283 NiW3 (D019) 2.76 4.44 7.36 321 2.76 - 7.70 304

Empirical Potentials for Multicomponent Systems Fe ▷ Finnis-Sinclair – modified by Calder and Bacon (1993) Fe-Cu ▷ Osetsky (1996) Fe: Pair-Potential, Osetsky (1995) Cu: Pair-Potential, Osetsky (1995) ▷ Ackland, Bacon, Calder (1997) Fe: F-S type, Ackland et al. (1997) Cu: F-S type, Ackland, Tichy, Vitek, Finnis (1987) ▷ Ludwig, Farkas,.. (1998) → C.S. Becquart, C. Domain, Fe: EAM, Simonelli, Pasianot, Savino (1993) Cu: EAM, Voter (1993)

History of Fe-C Alloy Potential R.A. Johnson, G.J. Dienes, A.C. Damask, Acta Metall. 12, 1215 (1964). metal-metal: pairwise interaction metal-carbon: pairwise interaction can consider only one carbon atoms, not applicable to carbides V. Rosato, Acta Metall. 37, 2759 (1989). metal-metal: many-body interaction M. Ruda, D. Farkas, and J. Abriata, Scr. Mater. 46, 349 (2002). metal-metal: many-body interaction (EAM) metal-carbon: many-body interaction (EAM) carbon-carbon: many-body interaction (EAM) unacceptable results

History of Carbon Potential J. Tersoff, Phys. Rev. Lett. 61 (1988) 2879. structural properties (cohesive energies, bond lengths of various polytypes) elastic properties (elastic constants of diamond) point defect properties (vacancy formation and migration energies, and interstitial formation energies in diamond and graphite) applicable to monolayer of graphite applicable to only Diamond Structures (C, Si, Ge, SiC, …) D.W. Brenner, Phys. Rev. B 42 (1990) 9458; J. Phys.: Condens. Matter 14 (2002) 783. modification of Tersoff formalism to better describe hydrocarbons M.I. Heggie, J. Phys.: Condens. Matter 3 (1991) 3065. E.P. Andribet et al., Nucl. Instr. & Meth. in Phys. Res. B 115 (1996) 501. To better describe graphite structure than Tersoff Only for graphite

(2NN) MEAM for Fe, C, N, Fe-C and Fe-N systems Fe, Cr, Mo, W, V, Nb, Ta Second Nearest-Neighbor Modified Embedded Atom Method Potentials for BCC Transition Metals Byeong-Joo Lee, M.I. Baskes, Hanchul Kim and Yang Koo Cho, Phys. Rev. B. 64, 184102 (2001). C A Modified Embedded Atom Method Interatomic Potential for Carbon Byeong-Joo Lee and Jin Wook Lee, CALPHAD 29, 7-16 (2005). Fe-C A Modified Embedded Atom Method Interatomic Potential for the Fe-C System Byeong-Joo Lee, Acta Materialia 54, 701-711 (2006). Fe-N A Modified Embedded-Atom Method Interatomic Potential for the Fe-N System: A Comparative Study with the Fe-C system Byeong-Joo Lee, T-H Lee and S-J Kim, Acta Materialia 4597-4607 (2006).

2NN MEAM for pure Fe - PRB 64, 184102 (2001); 71, 184205 (2005) MEAM F-S type EAM F-S type pair potential expt. This work Calder-Bacon Simonelli Ackland Osetsky C11 C12 C44 Evf ∆Vvf/Ω Evm EIf ∆VIf/Ω E(100) E(110) E(111) ∆d(100) ∆d(110) ∆d(111) ∆Ebcc/fcc ∆Ehcp/fcc a(bcc) a(fcc) ε(0-100oC) Cp (0-100oC) m.p. ∆Hm ∆Vm 2.431a 1.381a 1.219a < 2b < -.4c 0.55d - <110>due 2360f -0.2, -1.5g 0g -16.9g -0.082h -0.023h 2.8665i 12.1j 25.5j 1811h 13.8h 3.5j 2.430 1.380 1.219 1.75 -.41 0.53 4.20 <110>du 1.70 2510 2356 2668 -1.1 -1.5 -10.5 -0.048 -0.018 2.8637 3.611 12.5 26.1 2000 12.9 3.3 2.434k 1.381k 1.221k 1.83 -.21 0.91k 4.85 1.33 1920k -0.054m 0.0m 2.866 2.42 1.47 1.12 1.63 0.66 3.54 <111>cr -0.027 0.0 2.8664 2200 2.43 1.45 1.16 -.18 0.78 4.87 1.76 1812n 1585n 2269n -0.054 2.8665 3.690 2358 21.0 1.19 2.05 -.29 3.92 0.69 -0.052 0.005 2.867 3.612 11.6

MEAM for Carbon – Physical Property of Diamond MEAM Tersoff exp./calc. C11 C12 C44 Evf Evsplit E I(T) f E I(H) f E I(110)db f E I(100)db f 10.79 1.27 6.23 3.35 7.23 unstable 12.7 9.3 10.9a 1.2a 6.4a 4.3a 9.7a 19.6a 20.9a - 10.0a 10.80c 1.27c 5.77c 7.2d 9.1d 23.6d 16.7d Eideal(100) Eideal(110) Eideal(111) E1×1(100) E2×1(100) E1×1(111) Δd1-2(111) 8811 5715 4666 7124 5720 2069 -17.9 +9.5 -52.5 +21.1 7565b 4949b 4040b 6639b 2772b -15.9b +2.0b -39. 8b +4.3b 9850e, 9250g 6540g 7960f, 5340g 9190e 5370e 6270f -49f +9f ε (300-1200 K) Cp (300-1200 K) 8 25.5 1~6h 5~22h

MEAM for Carbon – Physical Property of Graphite MEAM + LJ Tersoff exp./calc. Biso C11 C12 C33 C44 C13 Evf Evsplit E InterLayer f 2.38 10.99 -0.45 0.38 0.0003 6.2 9.5 4.9 - 12.1a -1.9a 7.1a 10.8a 2.86b 10.60c 1.80c 0.365c 0.04c 0.15c, -0.12d, -0.005e 7.6f 9.2f 7.0g E (0001) 84 rhombohedral graphite simple graphite hexagonal graphite exp. Egra/dia -0.01 +0.01 -0.003 -0.02a Lattice parameter, a 2.45 2.45 2.45 2.46~2.47b Lattice parameter, c 6.63 6.95 6.66 6.71~6.93b

MEAM for Carbon – for several structures

MEAM for Carbon – Nanotubes and Fullerenes MEAM (+ LJ) exp./calc. ΔE of graphene ΔE of (10,10) CNT ΔE of (17,0) CNT ΔE of C60 bucky ball Young’s Modulus of (10,10) CNT Vacancy formation energy in (8,0) CNT 0.037 0.05 0.59 10.4 5.71 0.02a, 0.045b 0.086a, 0.10b 0.088a, 0.10b 0.46a, 0.47b 10.02c 5.59d

MEAM for N2 N2 Re (Å) Ec (eV/atom) Exp. 1.10 -4.88 Bond-order 1.11 ▪ Bond length and Cohesive energy for N2 N2 Re (Å) Ec (eV/atom) Exp. 1.10 -4.88 Bond-order 1.11 -4.96 2NN MEAM ▪ Bond length, Bond angle and cohesive Energy for N3 N3 Re (Å) Angle (degree) Ec (eV/atom) Bond-order 1.272 180 -3.712 2NN MEAM 1.116 -3.45

2NN MEAM for Fe-C & Fe-N – in BCC Fe MEAM expt./calc. Dilute Heat of Solution of Carbon (eV) Migration Energy Barrier of Carbon (eV) Vacancy-Carbon Binding Energy (eV) Vacancy-Carbon Binding Distance (ao) Dilute Heat of Solution of Nitrogen (eV) Migration Energy Barrier of Nitrogen (eV) Vacancy- Nitrogen Binding Energy (eV) Vacancy- Nitrogen Binding Distance (ao) 1.22 0.82 0.90 0.43 0.33 0.78 0.64 0.42 1.1a 0.88b, 0.86c,0.81-0.83d 0.41e, 0.85f, 1.05g, 1.1b, 0.44j 0.365k, 0.40j 0.32 0.76~0.80 0.67j 0.45j Carbon in O site vacancy-carbon carbon-carbon SIA-carbon vacancy-two carbon

2NN MEAM for Fe-C & Fe-N – in FCC Fe MEAM expt./calc. Dilute Heat of Solution of Carbon (eV) Migration Energy Barrier of Carbon (eV) Vacancy-Carbon Binding Energy (eV) Carbon-Carbon Binding Energy (eV) Dilute Heat of Solution of Nitrogen (eV) Migration Energy Barrier of Nitrogen (eV) Vacancy- Nitrogen Binding Energy (eV) Nitrogen - Nitrogen Binding Energy (eV) 0.30 1.52 0.67 -0.12 <110> -0.35 <100> -0.48 1.36 0.23 -0.31 <110> -0.10 <100> 0.36a, 0.12b 1.4c,1.53d 0.37 ~ 0.41e <110> alignment is less repulsive -0.53 1.75 <100> alignment

2NN MEAM for Fe-N – in Fe4N ΔHf = +3.1 ~ -40 kJ/mol (-10.5) MEAM: -6.8 kJ/mol a = 3.80 Å MEAM: 3.80 Å

2NN MEAM for Fe-N – in Fe2N Identification of the most stable atomic structure of Fe2N ΔHf = - 5.7 kJ/mol MEAM: -20.9 kJ/mol a = 2.76 Å, c = 4.42 Å MEAM: a = 2.81 Å, c = 4.32 Å

Atomistic Simulation – Interatomic Potentials and Applications Performance of 2NN MEAM for Elements and Alloys Fundamental Properties of Structural Materials Elastic Property Defect (Point, Dislocation, Grain Bd./Interface) Property Phase Transformations Deformation/Fracture Mechanism Fundamental Properties of Nano Materials Thermodynamic Property Atomic/Nano Structural Evolution Fundamental Properties of Amorphous Materials Irradiation Defects, etc.

Second Nearest Neighbor Modified EAM (2NN MEAM) Pure Elements Fe, Cr, Mo, W, V, Nb, Ta, Li Phys. Rev. B. 64, 184102 (2001); MSMSE 20, 035005 (2012) . Cu, Ag, Au, Ni, Pd, Pt, Al, Pb Phys. Rev. B. 68, 144112 (2003). Ti, Zr & Mg Phys. Rev. B. 74, 014101 (2006); CALPHAD 33, 650-57 (2009). Mn, P Acta Materialia 57, 474-482 (2009).; J. Phys.: Condensed Matters (2012), in press. C, Si, Ge, In CALPHAD 29, 7-16 (2005); 31, 95-104 (2007); 32, 34-42 (2008); 32, 82-88 (2008) Multicomponent Systems Fe-C, Fe-N, Fe-H Acta Materialia 54, 701-711 (2006); 54, 4597-4607 (2006); 55, 6779-6788 (2007). Fe-Ti & Fe-Nb Scripta Materialia 59, 595-598 (2008). Fe-Ti-C & Fe-Ti-N Acta Materialia 56 , 3481-3489 (2008); Acta Materialia 57 , 3140-3147 (2009). Fe-Nb-C & Fe-Nb-N J. Materials Research 25, 1288-1297 (2010). Al-H & Ni-H, V-H J. Materials Research 26, 1552-1560 (2011); CALPHAD 35, 302-307 (2011). Fe-Mn Acta Materialia 57, 474-482 (2009). Fe-Cr CALPHAD 25, 527-534 (2001). Fe-Cu Phys. Rev. B. 71, 184205 (2005). Fe-Pt J. Materials Research 21, 199-208 (2006). Fe-Al J. Phys.: Condensed Matters 22, 175702 (2010) Fe-P J. Phys.: Condensed Matters (2012), in press. Al-Ni CALPHAD 31, 53 (2007). Co-Cu J. Materials Research 17, 925-928 (2002). Co-Pt Scripta Materialia 45, 495-502 (2001). Cu-Ni CALPHAD 28, 125-132 (2004). Ni-W J. Materials Research 18, 1863-1867 (2003). Cu-Ti Mater. Sci. and Eng. A 449-451, 733 (2007). Cu-Zr J. Materials Research 23, 1095 (2008). Cu-Zr-Ag Scripta Materialia 61, 801 (2009). Mg-Al , Mg-Li CALPHAD 33, 650-57 (2009); MSMSE 20, 035005 (2012) . Ga-In-N J. Phys.: Condensed Matter 21, 325801 (2009).