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Environment Dependent Interatomic Potential for Bulk Silicon

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Environment Dependent Interatomic Potential for Bulk Silicon Environment Dependent Interatomic Potential for Bulk Silicon Martin Z. Bazant†, Efthimios Kaxiras† and J. F. Justo‡ † Department of Physics, Harvard University, Cambridge, MA 02138 ‡ Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (April 16, 1997) We use recent theoretical advances to develop a new functional form for interatomic forces in bulk silicon. The theoretical results underlying the model include a novel analysis of elastic prop- erties for the diamond and graphitic structures and inversions of ab initio cohesive energy curves. The interaction model includes two-body and three-body terms which depend on the local atomic environment through an effective coordination number. This formulation is able to capture success- fully: (i) the energetics and elastic properties of the ground state diamond lattice; (ii) the covalent re-hybridization of undercoordinated atoms; (iii) and a smooth transition to metallic bonding for overcoordinated atoms. Because the essential features of chemical bonding in the bulk are built into the functional form, this model promises to be useful for describing interatomic forces in silicon bulk phases and defects. Although this functional form is remarkably realistic by usual standards, it contains a small number of fitting parameters and requires computational effort comparable to the most efficient existing models. In a companion paper, a complete parameterization of the model is given, and excellent performance for condensed phases and bulk defects is demonstrated. I. INTRODUCTION more recently compared to each other3,4. They differ in degree of sophistication, functional form, fitting strategy The study of materials properties is increasingly rely- and range of interaction, and each can accurately model various special atomic configurations. Surfaces and small ing on a microscopic description of the underlying atomic 3,5 structure and dynamics. While many of the key features clusters are the most difficult to handle , but even bulk can be described by a small number of atoms that are ac- material (crystalline and amorphous phases, solid defects tively participating in a physical process, many problems and the liquid phase) has resisted a transferable descrip- of interest require of order 103–106 or even higher num- tion by a single potential. Realistic simulations of impor- ber of atoms and time scales of 10-100 ps for a proper de- tant bulk phenomena such as plastic deformation, diffu- scription. Ab initio methods based on density functional sion and crystallization are still problematic. theory1 and the local density approximation (DFT/LDA) In this article, we derive a general model for the func- have been intensively and successfully used to provide tional form of interatomic forces in bulk tetrahedral semi- 2 conductors. This functional form is applied to the proto- a microscopic description of simple structures . For 6 more complex cases, including for instance disordered or typical case of silicon in a companion article . The devel- stepped surfaces, dislocations, grain boundaries, crystal opment of the model is organized as follows: In section II, growth and the amorphous–to–crystal transition, a large we briefly review existing potentials and approximations number of atoms is required, making an ab initio de- of quantum models for silicon and extract important con- scription untenable. A possible alternative for these cases clusions about the desirable features of a successful in- might be empirical interatomic potentials which are com- teratomic potential. Recent theoretical advances used in ab initio putationally much less expensive. The difficulty in em- deriving our model from total energy data are ploying empirical potentials is their unproven ability to outlined in section III. A functional form that incorpo- capture the physics of structures far from the fitting data rates the theoretical results using a minimal number of used to construct them. Developing reliable empirical po- fitting parameters is presented and discussed in section tentials remains an issue of great interest and possibly of IV. Finally, section V contains some concluding remarks. great rewards. arXiv:cond-mat/9704137v1 [cond-mat.mtrl-sci] 16 Apr 1997 Silicon is a test case for the development of empirical II. REVIEW OF EMPIRICAL POTENTIALS AND potentials for covalent materials. Its great technologi- APPROXIMATIONS cal importance, the vast amount of relevant experimental and theoretical studies available, and its intrinsic interest as the representative covalent material make it an ideal A. Empirical Potentials candidate for exploring to what extent the empirical po- tential approach can be exploited. In recent years, more The usual approach for deriving empirical potentials is than 30 empirical potentials for silicon have been devel- to guess a functional form, motivated by physical intu- oped and applied to a number of different systems, and ition, and then to adjust parameters to fit ab initio total 1 energy data for various atomic structures. A covalent part to its appealing simplicity and apparent physical material presents a difficult challenge because complex content. quantum-mechanical effects such as chemical bond for- Another popular and innovative empirical model is mation and rupture, hybridization, metalization, charge the Tersoff potential, with three versions generally called transfer and bond bending must be described by an ef- T19, T210, and T311. The original version T1 has only fective interaction between atoms in which the electronic six adjustable parameters, fitted to a small database of degrees of freedom have somehow been “integrated out”7. bulk polytypes. Subsequent versions involve seven more In the case of Si, the abundance of potentials in the liter- parameters to improve elastic properties. The Tersoff ature illustrates the difficulty of the problem and lack functional form is fundamentally different from the SW of specific theoretical guidance. In spite of the wide form in that the strength of individual bonds is affected range of functional forms and fitting strategies, all pro- by the presence of surrounding atoms. Using Carls- posed models possess comparable (and insufficient) over- son’s terminology, the Tersoff potential is a third or- all accuracy3. It has proven almost impossible to at- der cluster functional7 with the cluster sums appearing tribute the successes or failures of a potential to specific in nonlinear combinations. As suggested by theoretical features of its functional form. Nevertheless, much can arguments12–14, the energy is the sum of a repulsive pair be learned from past experience, and it is clear that a interaction φR(r) and an attractive interaction p(ζ)φA(r) well-chosen functional form is more useful than elaborate that depends on the local bonding environment, which is fitting strategies. characterized by a scalar quantity ζ, To appreciate this point we compare and contrast some representative potentials for silicon. The pioneering po- E = [φR(Rij )+ p(ζij )φA(Rij )] (3) tential of Stillinger and Weber (SW) has only eight pa- Xij rameters and was fitted to a few experimental properties ζ = V3(R~ , R~ ), (4) of solid cubic diamond and liquid silicon8. The model ij ij ik Xk takes the form of a third order cluster potential7 in which the total energy of an atomic configuration R~ ij is ex- where the function p(ζ) represents the Pauling bond or- pressed as a linear combination of two- and{ three-body} der. The three-body interaction has the form of Eq. (2) terms, with the important difference that the angular function, although still positive, may not have a minimum at the E = V2(Rij )+ V3(R~ ij , R~ ik), (1) tetrahedral angle. The T1, T2 and T3 angular functions o Xij Xijk are qualitatively different, possessing minima at 180 , 90o and 126.745o, respectively. The Tersoff format has where R~ ij = R~ j R~ i, Rij = R~ ij and we use the conven- greater theoretical justification away from the diamond tion that multiple− summation| is| over all permutations of lattice than SW, but the three versions do not outperform the SW potential overall, perhaps due to their handling distinct indices. The range of the SW potential is just 3 short of the second neighbor distance in the equilibrium of angular forces . Nevertheless, the Tersoff potential is DC lattice, so the pair interaction V2(r) has a deep well another example of a successful potential for bulk prop- at the first neighbor distance to represent the restoring erties with a physically motivated functional form and force against stretching sp3 hybrid covalent bonds. The simple fitting strategy. three-body interaction is expressed as a separable prod- The majority of empirical potentials fall into either the 15–17 18–22 uct of radial functions g(r) and an angular function h(θ) generic SW or Tersoff formats just described, but there are notable exceptions that provide further in- V3(~r1, ~r2)= g(r1)g(r2)h(l12), (2) sight into successful approaches for designing potentials. First, a number of potentials possess functional forms where l12 = cos θ12 = ~r1 ~r2/(r1r2). The angular func- that have either limited validity or no physical motivation tion, h(l) = (l +1/3)2 , has· a minimum of zero at the at all, suggesting that fitting without theoretical guid- tetrahedral angle to represent the angular preference of ance is not the optimal approach. The Valence Force sp3 bonds, and the radial function g(r) decreases with Field23,24 and related potentials25,26 (of which there are distance to reduce this effect when bonds are stretched. over 40 in the literature25) involve scalar products of The SW three-body term captures the directed nature of the vectors connecting atomic positions, an approxima- covalent sp3 bonds in a simple way that selects the dia- tion that is strictly valid only for small departures from mond lattice over close-packed structures. Although the equilibrium. Thus, extending these models to highly various terms lose their physical significance for distor- distorted bonding environments undermines their the- tions of the diamond lattice large enough to destroy sp3 oretical basis.
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