arXiv:cond-mat/9704137v1 [cond-mat.mtrl-sci] 16 Apr 1997 e faosadtm clso 10-100 of scales time and atoms of ber irsoi ecito fsml structures simple provide of to description used successfully microscopic and a intensively been have theory scription. fitrs eur fodr10 order of problems require ac- many are interest process, that of physical atoms a of in number features participating small tively key a by the described of be many can While atomic underlying dynamics. the and of structure description microscopic a on ing pdadapidt ubro ieetsses and systems, different of number devel- a been to have applied silicon more and for years, oped potentials recent empirical In 30 po- exploited. than empirical be the can ideal extent approach an what tential to it exploring make for material candidate covalent representative interest the intrinsic its as and available, technologi- experimental studies relevant great theoretical of and amount Its vast the importance, materials. cal covalent for potentials of possibly and interest rewards. great of great issue po- an empirical remains reliable data tentials Developing fitting the them. construct from to to far used ability structures unproven of em- physics their the in is capture difficulty potentials The empirical ploying expensive. less com- much are which putationally potentials interatomic cases empirical these be for might alternative possible A untenable. scription ubro tm srqie,mkn an making required, is atoms large of a transition, number crystal amorphous–to–crystal the boundaries, and grain growth or dislocations, disordered instance surfaces, for stepped including cases, complex more h td fmtraspoete sicesnl rely- increasingly is properties materials of study The iio sats aefrtedvlpeto empirical of development the for case test a is Silicon 1 n h oa est prxmto (DFT/LDA) approximation density local the and binitio Ab ‡ sgvn n xeln efrac o odne hssan phases condensed com for a paper, performance requires companion excellent and a and In given, parameters is models. fitting re existing of is efficient number most form the small functional a this fo contains Although useful be it defects. to c and promises of phases model features bulk this form, essential functional sm the the a into Because grou and the (iii) atoms. of atoms; overcoordinated properties undercoordinated elastic of and re-hybridization energetics the This (i) number. ter fully: coordination three-body effective and an two-body through environment includes model interaction The risfrtedaodadgahtcsrcue n inversi and structures graphitic model and the diamond underlying the results for theoretical erties The silicon. bulk eateto ula niern,MsahstsInstitu Massachusetts Engineering, Nuclear of Department euercn hoeia dacst eeo e function new a develop to advances theoretical recent use We niomn eedn neaoi oeta o ukSili Bulk for Potential Interatomic Dependent Environment .INTRODUCTION I. ehd ae ndniyfunctional density on based methods † eateto hsc,HradUiest,Cmrde A0 MA Cambridge, University, Harvard Physics, of Department 3 atnZ Bazant Z. Martin –10 6 ree ihrnum- higher even or ps o rprde- proper a for binitio ab † 2 fhmo Kaxiras Efthimios , For . Arl1,1997) 16, (April de- 1 yia aeo iio nacmainarticle companion proto- a in the silicon to of applied case is typical form semi- functional tetrahedral This bulk in conductors. forces interatomic of form tional problematic. still diffu- are deformation, crystallization plastic and as sion impor- such of phenomena simulations Realistic bulk descrip- tant transferable potential. single a a resisted by has tion phase) liquid defects the solid and phases, amorphous handle and to (crystalline difficult material most small the model and are Surfaces clusters accurately can configurations. each atomic special and various interaction, strategy of fitting range form, and functional sophistication, other of each degree to compared recently more eiigormdlfrom in- in model successful used our a advances deriving theoretical of Recent features potential. desirable teratomic con- the important about extract and clusions silicon for approximations models and II, quantum potentials section of In existing review follows: as briefly organized we is model the of opment to,adte oajs aaeest fit intu- to physical parameters adjust by to motivated then form, and functional ition, a guess to V ial,scinVcnan oecnldn remarks. section concluding some in contains V of discussed section number and Finally, minimal presented IV. incorpo- a is that using parameters form results fitting functional theoretical A the rates III. section in outlined I EIWO MIIA OETASAND POTENTIALS EMPIRICAL OF REVIEW II. nti ril,w eieagnrlmdlfrtefunc- the for model general a derive we article, this In h sa prahfrdrvn miia oetasis potentials empirical deriving for approach usual The dsaedaodltie i)tecovalent the (ii) lattice; diamond state nd nld oe nlsso lsi prop- elastic of analysis novel a include eciigitrtmcfre nsilicon in forces interatomic describing r n of ons eo ehooy abig,M 02139 MA Cambridge, Technology, of te ohtasto omtli odn for bonding metallic to transition ooth ukdfcsi demonstrated. is defects bulk d eia odn ntebl r built are bulk the in bonding hemical † akbyraitcb sa standards, usual by realistic markably omlto sal ocpuesuccess- capture to able is formulation swihdpn ntelclatomic local the on depend which ms n .F Justo F. J. and opttoa ffr oprbeto comparable effort computational lt aaeeiaino h model the of parameterization plete lfr o neaoi ocsin forces interatomic for form al binitio ab .EprclPotentials Empirical A. APPROXIMATIONS oeieeeg curves. energy cohesive ‡ binitio ab 2138 oa nrydt are data energy total con 3 3 , 4 , 5 hydffrin differ They . u vnbulk even but , binitio ab 6 h devel- The . total 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. The potential of Pearson et. al.27, as hybridization, the SW potential seems to give a reason- the authors emphasize, is not physically motivated, but able description of many states experimentally relevant, rather results from an exercise in fitting. Their use of such as point defects, certain surface structures, and the Lennard-Jones two-body terms and Axilrod-Teller three- liquid and amorphous states3. The SW potential contin- body terms, characteristic of Van der Waals forces, has no ues to be a favorite choice in the literature, due in large justification for covalent materials. The potential of Mis-
2 triotis, Flytzanis and Farantos (MFF)28 is an interesting and small clusters, in this work we restrict ourselves to attempt to include four-body interactions. Although the bulk material and thus use a simpler, scalar environment importance of four-body terms is certainly worth explor- description. Our goal is to obtain the best possible de- ing, the inclusion of a four-body term in a linear cluster scription of condensed phases and defects with a simple, expansion is not unique, and theoretical analysis tends theoretically justified functional form. to favor nonlinear functionals7,13,14. A natural strategy to improve on the SW and Ter- soff models is to replace simple functional forms with more flexible ones and complement them with more elab- B. Approximation of Quantum Models orate fitting schemes. The Bolding and Andersen (BA) potential29 generalizes the Tersoff format with over 30 adjustable parameters fit to an unusually wide range of An alternative to fitting guessed functional forms structures. Although it has not been thoroughly tested, is to derive potentials by systematic approximation of the BA potential appears to describe simultaneously bulk quantum-mechanical models. So far, this approach has phases, defects, surfaces and small clusters, a claim that failed to produce superior potentials, but important con- no other potential can make3. However, its complexity nections between electronic structure and effective inter- makes it difficult to interpret physically, and since a large atomic potentials have been revealed. Although attempts fitting database was used, it is unclear whether the po- are being made to directly approximate Density Func- 33 tential can reliably describe structures to which it was tional Theory , the most useful contributions involve ap- not explicitly fit. In this vein, the spline-fitted potentials proximating various Tight Binding (TB) models, which of the Force Matching Method30 represent the opposite can themselves be derived as approximations of first prin- 34 extreme of the SW and Tersoff approaches: physical mo- ciples theories . These methods are based on low or- tivation is bypassed in favor of elaborate fitting. These der moment approximations of the TB local density of potentials involve complex combinations of cubic splines, states (LDOS), which is used to express the average band 7,14,35–39 which have effectively hundreds of adjustable parame- energy as the sum of occupied bonding states . ters, and the strategy of matching forces on all atoms in Pettifor has derived a many-body potential, similar in various defect structures is the most elaborate attempted form to the Tersoff potential, by approximation of the 14 thus far. Although the method may be worth pursuing TB bond order . More recently, an angular dependence as an alternative, it has not yet produced competitive remarkably close to the T3 angular function has been de- potentials31. Moreover, even if a reliable potential could rived for σ bonding from the lowest order two-site term 35 result from such fitting strategies, it would make it hard in the Bond Order Potential expansion , but the analyt- to interpret the results of atomistic simulations in terms ically derived function has a flat minimum around 130o of simple principles of chemical bonding. Such interpre- and thus differs qualitatively with the T1 and T2 poten- tation is essential, in our view, if physical insight is to be tials. With hindsight, a simple physical principle explains gained from computer simulations. these results: a σ bond is most weakened (desaturated) In spite of relentless efforts, no potential has demon- by the presence of an another atom when the resulting strated a transferable description of silicon in all its angle is small (θ< 100o) because in such cases the atom forms3 leading us to another important conclusion: it lies near the bond axis, thus interfering with the σ orbital may be too ambitious to attempt a simultaneous fit of where it is most concentrated. Working within the same all of the important atomic structures (bulk crystalline, framework of the TB LDOS, Carlsson and coworkers have amorphous and liquid phases, surfaces, and clusters) derived potentials with the Generalized Embedded Atom 36–38 since qualitatively different aspects of bonding are at Method . Harrison has arrived at a similar model by work in different types of structures. Theory and gen- expanding the average band energy in the ratio of the eral experience suggest that the main ingredient needed width of the bonding band to the bond-antibond split- 39 to differentiate between surface and bulk bonding pref- ting, the relevant small parameter in semiconductors . erences is a more sophisticated description of the local These potentials resemble the SW potential in its descrip- atomic environment. A notable example in this respect is tion of angular forces with an additive three-body term, the innovative Thermodynamic Interatomic Force Field particularly for small distortions of the diamond lattice. (TIFF) potential of Chelikowsky et. al.32, which includes The transition to metallic behavior in overcoordinated a quantity called the “dangling bond vector” that is a structures involves interbond interactions similar to the weighted average of the vectors pointing to the neighbors Tersoff and embedded atom potentials. of an atom. For symmetric configurations characteris- Many-body potentials can be derived from quantum- tic of the ideal (or slightly distorted) bulk material, the mechanical models if we restrict our attention to impor- dangling bond vector vanishes (or is exceedingly small). tant small sets of configurations. Using a basis of sp3 Conversely, a nonzero value of the dangling bond vector hybrid orbitals in a TB model, Carlsson et. al.7,36 have indicates an asymmetric distribution of neighbors. While shown that a generalization of the SW format, in which the TIFF dangling bond vector description appears to be Eq. (2) is replaced by a form similar to that used by very useful for undercoordinated structures like surfaces Biswas and Hamann (BH)15,
3 2 m efforts to extract features of interatomic forces directly V3(~r1, ~r2)= gm(r1)gm(r2) l12, (5) from ab initio total energy data. In order to investigate mX=0 the global trends in bonding across bulk structures pre- is valid in the vicinity of the equilibrium diamond lattice. dicted by quantum theories, we first perform inversions of In general, the fourth moment controls the essential band ab initio cohesive energy curves in part IIIA. By analyz- gap of a semiconductor, implying four-body interactions, ing elastic properties of covalent solids in part III B, we but the separable, three-body SW/BH terms are a conse- then explore the cohesive forces in certain special (high quence of the open topology of the diamond lattice: the symmetry) bonding states, which can be viewed as an only four-atom hopping circuit between first neighbors is inversion of ab initio energies restricted to selected im- the self-retracing path i j i k i7. portant configurations. We can make analogous→ arguments→ → → for the graphitic lattice to draw conclusions about sp2 hybrid bonds. Ig- noring the weak, long-range interaction between hexag- A. Inversion of Cohesive Energy Curves onal planes, we can assume a TB basis of sp2 hybrid orbitals and follow Carlsson’s derivation. Because the self-retracing path is also the only first neighbor hopping We have recently shown that it is possible to derive ef- circuit in a graphitic plane, a cluster expansion with the fective interatomic potentials for covalent solids directly 41 42 generic BH three-body interaction is also valid for hexag- from ab initio data , . The inversion procedure gener- onal configurations, with the functions in Eqs. (1) and alizes the “ab initio pair potential” of Carlsson, Gelatt 43 (5) differing from their diamond sp3 counterparts, as de- and Ehrenreich to many-body interactions and for ar- 44 scribed below. These calculations also suggest that a bitrary strains beyond uniform volume expansion . For locally valid cluster expansion should acquire strong en- the case of silicon, this work provides first principles evi- vironment dependence for large distortions from the ref- dence in favor of the generic bond order form of the pair erence configuration7. interaction, These studies provide theoretical evidence that the lin- ear three-body SW/BH format is appropriate near equi- V2(r, Z)= φR(r)+ p(Z)φA(r), (6) librium structures, while the nonlinear many-body Ter- soff format describes general trends across different bulk where φR(r) represents the short-range repulsion of structures. For the asymmetric configurations found in atoms due to Pauli exclusion of their electrons, φA(r) surfaces and small clusters, these theories also suggest represents the attractive force of bond formation, and that a more complicated environment dependence than p(Z) is the bond order, which determines the strength Tersoff’s is needed, like the dangling bond vector of the of the attraction as a function of the atomic environ- TIFF potential14,36. In conclusion, direct approxima- ment, measured by the coordination Z. The theoreti- tion of quantum models can provide insight into the ori- cal behavior of the bond order is as follows7,13,14,37,38: gins of interatomic forces, but apparently cannot pro- The ideal coordination for Si is Z0 = 4, due to its va- duce improved potentials. The reason may be that the lence. As an atom becomes increasingly overcoordinated long chain of approximations connecting first principles (Z>Z0), nearby bonds become more metallic, charac- and empirical theories is uncontrolled, in the sense that terized by delocalized electrons. In terms of electronic there is no small parameter which can provide an asymp- structure, the LDOS for overcoordinated atoms can be totic bound for the neglected terms for a wide range of reasonably well described by its scalar second moment. It configurations40. is a well established result that the leading order behavior of the bond order is p(Z) Z−1/2 in the second moment 7 14 38 ∼ approximation , , . For Z Z0 on the other hand, a III. INVERSION OF AB INITIO ENERGY DATA matrix second moment treatment≤ predicts a roughly con- stant bond order (additive bond strengths)36. For small There are very few hard facts concerning the nature coordinations higher moments are needed to incorporate of interatomic forces. Although there has been a pro- important features of band shape characteristic of co- liferation of ab initio energy and force calculations for valent bonding, primarily the formation of a gap in the 7 14 36 37 a wide range of atomic structures, it has proven diffi- LDOS , , , . Thus, the bond order should depart from 1 2 cult to discover any concrete information regarding the the divergent Z− / behavior at lower coordinations with functional form of interatomic potentials. With the ubiq- a shoulder at the ideal coordination of Z = Z0 where the 1 2 uitous fitting approach, it is never clear whether discrep- transition to metallic Z− / dependence begins. ancies with ab initio data result from an incorrect func- Inversion of ab initio cohesive energy curves verifies tional form or simply suboptimal fitting3. Thus, in addi- that trends in chemical bonding across various bulk tion to the practical problem of designing potentials, it bonding arrangements are indeed consistent with these is also difficult to build a simple conceptual framework theoretical predictions41. Previously, the only evidence within which to understand the complexities of chemi- in support of the bond order formalism came from equi- cal bonding. In this section, we summarize our recent librium bond lengths and energies for a small set of ideal
4 crystal structures9–11,13,19. The inversion approach has which materials can be described by a pair potential48,49. revealed for the first time that the bond order decompo- Once it was realized that the Cauchy relations are not sition expressed by Eq. (6) is actually valid for a wide satisfied by the experimental data for semiconductors, a range of volumes away from equilibrium and for a repre- number of authors in this century, led by Born50,51, de- sentative set of low energy crystal structures. In addition rived generalized Cauchy relations for noncentral forces to selecting the generic form of the pair interaction, in- in the diamond structure52,48. Building upon this body version provides a precise measure of the relative bond of work, we have recently analyzed the elastic proper- orders in various local atomic configurations. For ex- ties of several general classes of many-body potentials ample, the bond order of sp2 bonds involving three-fold in the diamond and graphitic crystal structures in order coordinated atoms is about 5% greater than that of four- to gain insight into the mechanical behavior of sp3 and fold coordinated sp3 bonds in silicon. sp2 hybrid covalent bonds, respectively44. These high These results have immediate implications for em- symmetry atomic configurations must be accurately de- pirical potentials. The main conclusion is that the scribed by any realistic model of interatomic forces in a generic Tersoff format is much more realistic than the tetravalent solid. Here we will only outline results di- SW format for highly distorted configurations. However, rectly related to the model presented in the next section. 3 the inversion results also indicate that a coordination- sp Hybrids: Consider one of the simplest many-body dependent pair interaction can provide a fair description interaction models for a tetrahedral solid, that is the of high-symmetry crystal structures without requiring generic SW format defined in Eqs. (1) and (2), with near- additional many-body interactions. In particular, angu- est neighbor interactions and an angular function having lar forces are only needed to stabilize these structures a minimum of zero at the tetrahedral angle (h = h′ = 48 53 under symmetry-breaking distortions, primarily for small 0,h′′ > 0). In that case, first considered by Harrison , , coordinations. In order to make a quantitative connec- the functional form of the potential has only two degrees ′′ ′′ tion between Tersoff’s functional form and our inverted of freedom for elastic behavior, V2 and h , the curva- ab initio data, angular contributions to the bond order tures of the pair interaction and of the angular function 45 must somehow be suppressed for ideal crystal structures. at their respective minima . Since cubic symmetry al- The inversion procedure applied to explicit three-body lows for three independent elastic moduli, there is an interactions has also led to some useful conclusions. Al- implied relation, due to Harrison, though it is not always the case42, inverted three-body (7C11 +2C12)C44 = 3(C11 +2C12)(C11 C12). (7) radial functions g(r) tend to be strictly decreasing func- − tions (like SW), especially when an overdetermined set 54 41 Using the experimental data shown in Table I, the ratio of input structures is used . Inverted angular functions of the two sides of the Harrison relation is 1.16, indicating h(l) also tend to penalize small angles (θ < π/2) less a reasonable description by a simple SW model. In con- than most existing models, in agreement with a compar- 3 trast, the potentials with the Tersoff format, T2, T3 and ative study of empirical potentials . We must emphasize, Dodson (DOD)18, are far from satisfying this relation. however, that the results of this section concern general This does not imply rejection of the Tersoff format, be- trends in chemical bonding, and have little to offer in cause the functional form has more than enough degrees terms of the precise nature of interatomic forces in spe- of freedom to exactly reproduce all the elastic constants. cial atomic configurations, such as the low-energy states However, as such, the inability of Tersoff potentials to of hybrid covalent bonds. To understand better these accurately describe elastic behavior when constrained to critical cases, we employ a related inversion strategy. fit other important properties does suggest a potential shortcoming in the functional form. A more compelling reason to select the SW format over B. Analysis of Elastic Properties others in the literature comes from the unrelaxed shear 0 modulus C44 which does not include relaxation of the internal degrees of freedom in the crystal unit cell. In A useful theoretical approach to guide the develop- the early literature on elastic forces, unrelaxed elastic ment of potentials, which has been pursued recently only 45 moduli were ignored, because they are not experimen- by Cowley , is to predict elastic properties implied by tally accessible. With the advent of ab initio calculations generic functional forms and compare with experimen- that predict elastic constants within a few percent of ex- tal or ab initio data. This tool for understanding inter- perimental values, we can now analyze unrelaxed elastic atomic forces dates back to the 19th century, when St. properties as well. Considering again the simple SW for- Venant showed that the assumption of central pairwise mat, with its two degrees of freedom, we have discovered forces supported by Cauchy and Poisson implied a reduc- another relation for the unrelaxed moduli, tion in the number of independent elastic constants from 46 0 21 to 15 . The corresponding six dependencies, given 4C11 +5C12 =9C44. (8) by the single equation C12 = C44 if atoms are at cen- ters of cubic symmetry, are commonly called the Cauchy As shown in Table I, the experimental and ab initio elas- relations46,47. They provide a simple test for selecting tic moduli satisfy this relation within experimental and
5 computational error. On the other hand, more general sp2 Hybrids: We have also obtained useful informa- cluster potentials and functionals, including the Tersoff tion about interatomic forces due to sp2 hybrid bonds format, BH and PTHT, do not require this relation, and from the elastic moduli of the graphitic structure44. In in fact cannot satisfy it under the usual circumstances. this analysis we neglect interplanar interactions, which This is demonstrated in Table I and explains why it has are insignificant compared to the covalent bonds within proven difficult to obtain good elastic properties with the a single, hexagonal plane. Our goal is to understand the Tersoff potential56. These results unambiguously select elastic properties of sp2 hybrids appearing around three- the SW format with first neighbor interactions for de- fold coordinated atoms in a bulk environment, such as scribing small homogeneous strains of the diamond lat- a dislocation core or a grain boundary58. An isolated tice. Although imperfect, internal relaxation with the hexagonal plane embedded in three-dimensional space SW format is also much better than with other models. has three independent (unrelaxed) elastic constants, C11, 0 59 Combining Eqs. (7) and (8), we arrive at a relation in- C12, and C44 with units of energy per unit area . It can 0 0 volving all four moduli, C11, C12, C44 and C44, be shown that C44 = 0 for any three-body cluster poten- tial or functional, in perfect agreement with the vanishing 2 60 0 (C11 +8C12) ab initio value . There is no relation for the remaining C44 C44 = , (9) − 9(7C11 +2C12) constants, C11 and C12, implied by empirical models be- cause each functional form possesses at least two degrees that expresses the effect of internal relaxation. If the two of freedom. degrees of freedom in the SW format are used to repro- Drawing on the TB approximations described above, duce the experimental values of C11 and C12, and thus 0 which correctly predict the general form of interactions also C44 by Eq. (8), then the predicted value of C44 from mediated by sp3 hybrids, we proceed by assuming a sep- Eq. (9) is 0.71 Mbar, which is only 12% smaller than arate three-body cluster potential for sp2 hybrids given the experimental value of 0.81 Mbar. The elastic behav- by Eqs. (1) and (5). By analogy with the sp3 case, ior of the SW format is quite remarkable considering it we further assume the simpler SW form of Eq. (2) for has only half of the necessary degrees of freedom, while the three-body interaction, with the important difference most other models are overdetermined for elastic behav- 3 that the angular function has a minimum of zero at the ior. This explains the surprising fact that the SW poten- hexagonal angle of 2π/3 rather than at the tetrahedral tial gives one of the best descriptions of elastic properties angle. We again restrict the interaction range to nearest in spite of not having been fit to any elastic constants. neighbors engaged in the covalent bonds that dominate We conclude that it is the superiority of the simple SW cohesion. These are not the only possible choices, but functional form that gives the desirable properties, not a we can evaluate their validity through analysis of elastic complex fitting procedure. moduli. Using analytic expressions for the elastic constants it With such a functional form61, which differs from all is possible to devise a simple prescription to achieve good existing empirical potentials62, stability considerations elastic properties with the SW format. As a simple con- imply C11 > 3C12, which is indeed satisfied by the ab sequence of h( 1/3) = 0, the curvature of the pair po- 60 − initio values, C11 = 1.79 Mbar and C12 = 0.51 Mbar . tential is given by, More importantly, we can relate the mechanical proper- ties of sp2 and sp3 hybrids. The relative radial stiffness ′′ 3Vd φ (rd)= 2 (C11 +2C12). (10) is given by a simple ratio of elastic constants, 4rd ′′ 2 φh(rh) 8rd Ah(C11 + C12)h The curvature of the angular function can be related to = 2 , (12) 57 ′′ the second shear modulus , φd (rd) 9rh Vd(C11 +2C12)d 9V where the subscript h refers to the equilibrium hexag- 2 ′′ d 2 g(rd) h ( 1/3) = (C11 C12), (11) √ − 32 − onal plane with area per atom Ah = ah 3/4, and d refers to the diamond lattice. Using the ab initio re- 3 ˚ where rd, ad and Vd = ad/8 denote the equilibrium first sult, rh = 2.23A, the prefactor in Eq. (12) is 0.99, so neighbor distance, lattice constant and atomic volume. the elastic constant ratio in parentheses provides a direct Using the ab initio data in Table I, the right hand sides comparison of sp2 and sp3 radial forces. The ab initio of Eqs. (10) and (11) evaluate to 8.1 eV/A˚2 and 5.7 eV, value of that ratio is 1.4 0.1, implying that sp2 bonds respectively. This provides a simple two-step procedure have 40% greater radial± stiffness than sp3 bonds. The to maintain good elastic behavior while fitting any model same result also follows directly from inverted pair po- with the SW format near the diamond lattice: (i) scale tentials for the graphitic and diamond structures41. the pair interaction V2(r) to obtain the correct bulk mod- A similar elastic analysis yields an expression for the 2 3 ulus K = (C11 +2C12)/3, and (ii) scale the three-body relative angular stiffness of sp and sp hybrid bonds, energy to set the second shear modulus. As shown above, ′′ 2 this will lead to perfect unrelaxed elastic constants and hh( 1/2) 256gd(rd) Ah(C11 3C12)h − = 2 − , (13) only a 12% error in C44. h′′( 1/3) 243g (r ) V (C11 C12) d − h h d − d
6 Using the ab initio data, we have the general result, a reliable potential for bulk properties while keeping the g (r )2h′′( 1/2)/g (r )2h′′( 1/3) = 0.46 0.15. As- functional form simple enough to allow for efficient com- h h h − d d d − ± suming gd(r) gh(r) with each function decreasing in putation of forces as well as intuitive understanding of accordance with≈ inversion results41, then the product of chemical bonding in covalent solids. prefactors in Eq. (13) is nearly unity. In that case the ra- tio of elastic constants in parentheses allows us to quan- tify the relative bending strength of the hybrid bonds. A. Scalar Environment Description The ab initio value for the ratio of 0.44 0.15 indicates that the angular stiffness of sp2 bonds± is smaller than The simplest description of the local environment of that of sp3 bonds by about a factor of two, in spite of an atom is the number of nearest neighbors, determined the greater radial stiffness of sp2 bonds. Our conclusion by an effective coordination number Z for atom i, for the relative bending strength of sp2 and sp3 hybrids i would be reversed only if g (r ) were smaller than g (r ) g g d d Z = f(R ) (14) by at least a factor of two, which seems unlikely in light i im mX6=i of the bond orders. Elastic constant analysis suggests that a hybrid co- where f(Rim) is a cutoff function that measures the con- valent bond is well represented by a separable, first- tribution of neighbor m to the coordination of i in terms neighbor, three-body cluster potential whose angular 2 3 of the bond length Rim. The special sp and sp bonding function has a minimum of zero at the appropriate angle. geometries can be uniquely specified by their coordina- This may seem to contradict the ample evidence we have tions due to their high symmetry. Since environment cited in favor of the Tersoff format for large distortions of dependence is not needed in those cases, it is natural to the diamond lattice, particularly those involving changes take the coordination number to be a constant, except in coordination. These findings are consistent, however, when large distortions from equilibrium occur. Moreover, in light of Carlsson’s argument that cluster potentials like covalent bonds tend to involve only first neighbors, as in- SW can accurately fit narrow ranges of configurations dicated by ab initio charge density calculations of open while cluster functionals like Tersoff’s provide a less ac- structures like the diamond lattice65. Thus, we choose curate but physically acceptable fit to a much broader 63 the neighbor function to be exactly unity for typical co- set of configurations . valent bond lengths, r
7 However, the analysis of elastic properties discussed ear- (with coordinations given in parentheses): graphitic (3), lier favors the present approach for environment depen- diamond (4), BC-8 (4), BCT-5 (5), β-tin (6), SC (6) and dence near the diamond lattice. Another important dif- BCC (8). These structures span the full range from three ference between our model and Tersoff’s is the separa- and four-fold coordinated covalent bonding in sp2 and tion of angular dependence from the bond order. As sp3 arrangements, to overcoordinated atoms in metallic we shall see, this allows us to control independently the phases. The inverted ab initio bond order versus coordi- preferences for bond strengths, bond angles, and angu- nation is shown in Fig. 1, along with two additional data lar forces in a way that the Tersoff potential cannot. By points. Since we have only first neighbor interactions in keeping the bond order simple, we can also directly use the diamond lattice, we can obtain another bond order the important theoretical results that motivated the Ter- for three-fold coordination from the ab initio formation soff potential in the first place. energy (3.3 eV) for an unrelaxed vacancy. An additional data point for unit coordination comes from the experi- mental binding energy (3.24 eV) and bond length (2.246 66 B. Coordination-Dependent Chemical Bonding A)˚ of the Si2 molecule . The bond order data has a clear shoulder at Z = Z0 = Our potential consists of coordination-dependent two- 4 where the predicted transition from covalent to metallic and three-body interactions corresponding to the defin- bonding occurs. For overcoordinated atoms with Z>Z0, ing features of covalent materials: pair bonding and an- the bond order approaches its rough asymptotic behav- 1 2 gular forces. The energy of a configuration R~ is a sum ior, p Z− / , characteristic of metallic band struc- { i} ∝ over single-atom energies, E = Ei, each expressed as ture. For coordinations Z Z0, the bond order departs i −1/2 ≤ a sum of pair and three-body interactionsP from the Z divergence, due to the formation of a band gap in the LDOS associated with covalent bonds. 2 3 ~ ~ A natural choice to capture this shape is a Gaussian, Ei = V (Rij ,Zi)+ V (Rij , Rik,Zi), (16) 2 −βZ Xj Xjk p(Z)=e . In Fig. 1, we see that the bond or- der function we obtain from fitting6 is fairly close to the depending on the coordination Zi of the central inversion data. It is intentionally somewhat too large atom. The pair functional V2(Rij ,Zi) represents the for coordinations 5–8 to compensate for the small, but strength of bond (ij), while the three-body functional nonvanishing many-body energy for those structures, as V3(R~ ij , R~ ik,Zi) represents preferences for special bond described below. The collapse of the attractive functions angles, due to hybridization, as well as the angular forces φA(r) = (V2(r, Z) VA(r))/p(Z) with this choice of bond that resist bending away from those angles. From our order shown in Fig.− 2 is reasonably good, thus justifying atomic perspective, the pair interaction is broken into a the bond order formalism across a wide range of volumes. sum of contributions from each atom, and similarly the Our potential is the first to have a bond order in such three-body interaction is broken into a sum over the three close agreement with theory, which is a direct result of angles in each triangle of atoms. Note that due to the our novel treatment of angular forces. environment dependence, the contributions to the bond Angular Terms: In a thorough comparative study of Si strength from each pair of atoms are not symmetric in potentials, Balamane et. al. attribute the limitations of general, V2(Rij ,Zi) = V2(Rji,Zj ). empirical models to the inadequate description of angular 6 Pair Bonding: We adopt the well-established bond or- forces3. Our potential contains a number of innovations der format of Eq. (6) for the pair interaction. Drawing in handling angular forces, leading to a significant im- on the popularity of the SW potential, we use those func- provement over existing models in reproducing ab initio tional forms for the attractive and repulsive interactions, data. Analysis of elastic properties shows that, at least near equilibrium, the three-body functional should be ex- B ρ σ V2(r, Z)= A p(Z) exp , (17) pressed as a single, separable product of a radial function r − r a g(r) for both bonds and an angular function h(θ,Z), − which go to zero at the cutoff r = a with all deriva- ~ ~ tives continuous. This choice can reproduce the shapes V3(Rij , Rik,Zi)= g(Rij )g(Rik)h(lijk,Zi). (18) of inverted pair potentials for silicon41. Because we have constructed Z, and hence p(Z), to be constant near the Although the radial functions could vary with coordina- diamond lattice, our pair interaction reduces exactly to tion, in the interest of simplicity we have focused on the the SW form for configurations near equilibrium, thus angular function as the most important source of coordi- allowing us to obtain excellent elastic properties as ex- nation dependence. Inversion of ab initio cohesive energy 41 plained above. Making this choice of repulsive term with curves suggests that a consistent choice for the radial the parameters obtained by fitting to defect structures6, functions is the monotonic SW form, we can follow the procedure of Bazant and Kaxiras41 to extract the implied bond order p(Z) from ab initio co- γ g(r) = exp , (19) hesive energy curves for the following crystal structures r b −
8 which also goes to zero smoothly at a cutoff distance b, two-fold coordination, this choice reproduces the prefer- a value different from the two body cutoff a. Having ence for bonding along two orthogonal p-states with the separate cutoffs for two and three-body interactions is low energy, nonbonding s state fully occupied. For six- reasonable because they describe fundamentally differ- fold coordination, the choice θ0(6) = π/2 also reflects ent features of bonding. Although the pair interaction the p character of the bonds. However, structures with might extend considerably beyond the equilibrium first Z = 6 like SC and β-tin are metallic, with delocalized neighbor distance, the angular forces should not be al- electrons that tend to invalidate the concept of bond- lowed to extend beyond first neighbors, if they are to be bending underlying the angular function, a crucial point interpreted as representing the resistance to bending of we shall address shortly. The vanishing many-body en- covalent bonds. ergies for the graphitic plane and diamond structures al- Much of the new physics contained in our potential low fitting of the pair interactions V2(r, 3) and V2(r, 4) ′′ comes from the angular function h(l,Z). Theoretical to be guided by Eq. (10), which determines V2 (rd, 4) considerations lead us to postulate the following general from the bulk modulus, and Eq. (12), which requires ′′ ′′ form: V2 (rh, 3)/V2 (rd, 4) 1.4. Moreover, the shifting of the minimum of the angular≈ function in our model incorpo- l + τ(Z) h(l,Z)= H , (20) rates coordination-dependent hybridization in a way that w(Z) other potentials cannot. where H(x), w(Z) and τ(Z) are generic functions whose Through the function w(Z), our angular function has essential properties we now describe. The overall shape of another novel coordination dependence to represent the the angular function is given by H(x), a nonnegative38,7 covalent to metallic transition. The width of the min- function with a quadratic minimum of zero at the origin, imum w(Z) is broadened with increasing coordination, H(0) = H′(0) = 0 and H′′(0) > 0. The function H(x) thus reducing the angular stiffness of the bonds as they should also become flat away from the minimum well at become more metallic. Similarly, as coordination is de- the origin, resulting in zero angular force for large distor- creased from 4 to 3, the width of the minimum is in- creased to reproduce the smaller angular stiffness of sp2 tions away from equilibrium. This feature, which is ab- 3 sent in most potentials including SW, is essential for the bonds compared to that of sp bonds. Thus, the function angular term to have physical meaning far from equilib- w(Z) should have a minimum at Z0 = 4 and diverge with rium. When covalent bonds are strongly bent from their increasing Z. Fitting of the model can be guided by Eq. equilibrium angle they are weakened and replaced by new (11), which determines w(4) from the second shear mod- electronic states. Thus, for large angular distortions it is ulus, and by Eq.(13), which requires w(3)/w(4) √2. ≈ not possible to define a restoring force that drives atoms The softening of the angular function is important be- back towards an equilibrium bond angle. These proper- cause it allows the decrease in cohesive energy per atom ties of H(x) can be satisfied by the following choice, concomitant with overcoordination to be modeled by a weakening of pair interactions. In contrast, cluster po- 2 H(x)= λ 1 e−x , (21) tentials like SW penalize overcoordinated structures with − increased three-body energy that overcomes the decrease 28 in pair bonding energy. This is an unphysical feature, which is similar in shape to the MFF angular function . since overcoordinated structures do not even have cova- However, our angular dependence is considerably more lent bonds, and the many-body energy cannot be viewed sophisticated than MFF due to its environment depen- as a consequence of stretching sp3 bonds far from the dence. tetrahedral geometry. In this sense, the reasonably good Motivated by theory, we choose the function τ(Z) to description of liquid Si (a metal with about 6 neighbors control the coordination-dependent minimum of the an- per atom) with the SW potential appears to be fortu- gular function, l0(Z) = cos(θ0(Z)) = τ(Z), with the itous. following form61,62, − The coordination dependence of our angular function −u4Z −2u4Z τ(Z)= u1 + u2(u3e e ). (22) makes it possible for the first time to reproduce the well- − known behavior of the bond order. The reason is that the The parameters, u1 = 0.165799 , u2 = 32.557, u3 = contribution of the three-body functional to the total en- − 0.286198, and u4 = 0.66, were chosen to make the pre- ergy is suppressed for ideal crystals and overcoordinated −1 ferred angle θ0(Z) = cos [ τ(Z)] interpolate smoothly structures. The shifting of the minimum makes the three- between several theoretically− motivated values, as shown body energy vanish identically for sp2 and sp3 hybrids, in Fig. 3: We have already argued that τ(4) = 1/3 and the variable width greatly reduces the three-body and τ(3) = 1/2 (so that sp3 and sp2 bonding corre- energy in metallic structures. With the three-body en- spond to the diamond and graphitic structures respec- ergy suppressed, we can use our knowledge of the bond tively), which determines two of the four parameters in order for the graphitic, diamond, β-tin and other lattices τ(z). The remaining two parameters were selected so from inversion of cohesive energy curves to capture the that τ(2) = τ(6) = 0 or θ0(2) = θ0(6) = π/2. For energetics of these structures in the pair interaction, as
9 described above. Several other potentials have tried to sp2 and sp3 hybridization and some metallic states, but incorporate the bond order predicted from theory, but might also include more general situations in which atoms the uncontrolled many-body energy makes it impossible are more or less symmetrically distributed, like the liq- to connect directly with theory. Our treatment of an- uid and amorphous phases and reconstructed dislocation gular forces is intuitively appealing because the forces cores and grain boundaries. The theory behind the model primarily model the bending of covalent bonds, with the begins to break down for noninteger coordinations, since control of global energetics left to the pair interactions. our effective coordination number is a way of smoothly Although our model contains a complicated environ- interpolating between well-understood local structures. ment dependence, forces can still be evaluated with com- More seriously, no attempt is made to handle asymmet- putational speed comparable to much simpler existing ric distributions of neighbors, which are abundant in sur- potentials. The coordination dependence introduces an faces and small clusters. Theory suggests that our model extra loop into each force calculation. For the three- may be fitted to provide a good description of condensed body functional, this introduces a fourth nested loop over phases and defects in bulk tetrahedral semiconductors, atoms m outside each triplet (ijk) that contribute to co- such as Si, Ge and with minor extensions perhaps alloys ordination of atoms i, which would make force evaluation such as SiGe, that can be understood in terms of simple much slower than the typical three-body cluster expan- principles of covalent bonding. sions used in most other potentials. However, our choice of f(r) greatly reduces the frequency of four body com- putations because nonzero forces result if the fourth atom ACKNOWLEDGMENTS lies in the range of being a partial neighbor, c < rim V. CONCLUSION In summary, we have used recent theoretical innova- 1 P. Hohenberg and W. Kohn, Phys. Rev. 136, 864 (1964); tions to arrive at a functional form that describes the de- W. Kohn and L. J. Sham, Phys. Rev. 140, 1133 (1965). pendence of chemical bonding on the local coordination 2 M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. number. Bond order, hybridization, metalization and an- D. Joannopoulos, Rev. Mod. Phys. 64, 1045 (1992). gular stiffness are all described in qualitative agreement 3 H. Balamane, T. Halicioglu, and W. A. Tiller, Phys. Rev. B with theory. Consistent with our motivation, we have 46, 2250 (1992); T. Halicioglu, H. O. Pamuk and S. Erkoc, kept the form as simple as possible, reproducing the es- Phys. Stat. Sol. B 149, 81 (1988). 4 sential physics with little more complexity than exist- S. J. Cook and P. Clancy, Phys. Rev. B 47, 7686 (1993). 5 ing potentials. The fitted implementation of the model E. Kaxiras, Comp. Mater. Sci. 6, 158 (1996). 6 described in the companion paper6 involves only 13 ad- J. F. Justo, M. Z. Bazant, E. Kaxiras, V. V. Bulatov, and justable parameters. Using the results of the present arti- S. Yip, in preparation. 7 A. E. Carlsson, Solid State Physics 43, 1 (1990). cle, we provide theoretical estimates of almost half of the 8 parameters, thus greatly narrowing the region of param- F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 eter space to be explored during fitting. The remaining (1985). 9 56 parameters are chosen to fit important bulk defect struc- J. Tersoff, Phys. Rev. Lett. , 632 (1986). 10 J. Tersoff, Phys. Rev. B 37, 6991 (1988). tures. 11 J. Tersoff, Phys. Rev. B 38, 9902 (1988). Considering the theory behind our model, we can an- 12 J. Ferrante, J. R. Smith, and J. H. Rose, Phys. Rev. Lett. ticipate its range of applicability. We have shown that 50, 1385 (1983); J. H. Rose, J. R. Smith and J. Ferrante, the structure and energetics of the diamond lattice can be Phys. Rev. B 28, 1835 (1983). almost perfectly reproduced. Because small distortions 13 3 G. C. Abell, Phys. Rev. B 31, 6184 (1985). of sp hybrids are accurately modeled, we would also 14 D. G. Pettifor, Springer Proc. in Physics 48, 64 (1990). expect a good description of the amorphous phase. De- 15 R. Biswas and D. R. Hamann, Phys. Rev. Lett. 55, 2001 2 fect structures involving sp hybridization should also be (1985); Phys. Rev. B 36, 6434 (1987). well described. In general, the model should perform best 16 E. Kaxiras and K. Pandey, Phys. Rev. B 38, 12736 (1988). whenever the coordination number can adequately spec- 17 J. R. Chelikowsky, Phys. Rev. Lett. 60, 2669 (1988). ify the local atomic environment. This certainly includes 18 B. W. Dodson, Phys. Rev. B 35, 2795 (1987). 10 19 K. E. Khor and S. Das Sarma, Phys. Rev. B 38, 3318 search Society Symposia Proceedings, 408 (M. R. S., Pitts- (1988); 39, 1188 (1989); 40, 1319 (1989). burgh, 1996), 79. 20 D. W. Brenner, Phys. Rev. Lett. 63, 1022 (1989). Brenner 43 A. E. Carlsson, C. Gelatt, and H. Ehrenreich, Phil. Mag. shows how the Tersoff format is equivalent to the Embed- A 41 (1980). ded Atom Method of Ref.21. 44 M. Z. Bazant and E. Kaxiras, in preparation. 21 M. I. Baskes, Phys. Rev. Lett. 59, 2666 (1987); M. I. 45 E. R. Cowley, Phys. Rev. Lett. 60, 2379 (1988). Baskes, J. S. Nelson and A. F. Wright, Phys. Rev. B 40, 46 A. E. H. Love, Mathematical Theory of Elasticity, 4th ed. 6085 (1989). (Cambridge University Press, 1927). 22 J. Wang and A. Rockett, Phys. Rev. B 43, 12571 (1991). 47 M. Born and K. Huang, Dynamical Theory of Crystal Lat- 23 P. N. Keating, Phys. Rev. 145, 637 (1966). tices (Clarendon Press, Oxford, 1954). 24 D. W. Brenner and B. J. Garrison, Phys. Rev. B 34, 1304 48 H. B. Huntington, Solid State Physics 7, 213 (1958). (1986). 49 A. H. Cottrell, The Mechanical Properties of Matter 25 A. M. Stoneham, V. T. B. Torres, P. M. Masri and H. R. (Kreiger, New York, 1981). Schober, Phil. Mag. A 58, 93 (1988). 50 M. Born, Ann. Physik 44, 605 (1914). 26 J. N. Murrell and J. A. Rodrigues-Ruiz, Mol. Phys. 71, 823 51 M. Born, in Lattice Dynamics, ed. by R. F. Wallis (Perga- (1990); A. R. Al-derzi, R. L. Johnston, J. N. Murrell and mon, New York, 1965), 5. J. A. Rodrigues-Ruiz, Mol. Phys. 73, 265 (1991). 52 J. de Launay, Solid State Physics 2, 220 (1956). 27 E. M. Pearson, T. Takai, T. Halicioglu and W. A. Tiller, 53 W. A. Harrison, Ph.D. Dissertation, University of Illinois, J. Crystal Growth 70, 33 (1984). Urbana, Illinois (1956). 28 A. D. Mistriotis, N. Flytzanis, and S. C. Farantos, Phys. 54 G. Simmons and H. Wang, Single Crystal Elastic Constants Rev. B 39, 1212 (1989). and Calculated Aggregate Properties: A Handbook (MIT, 29 B. Bolding and H. Anderson, Phys. Rev. B 41, 10568 Cambridge, MA, 1971). (1990). 55 N. Bernstein and E. Kaxiras, submitted to Phys. Rev. B; 30 D. F. Richards and J. B. Adams, in Grand Challenges in Materials Theory, Simulations and Parallel Algorithms, in Computer Simulation, Proceedings High Performance ed. by E. Kaxiras, J. Joannopoulos, P. Vashista, and R. Computing ‘95, ed. by A. Tentner, 218 (1995). Kalia, Materials Research Society Symposia Proceedings, 31 D. F. Richards, J. B. Adams, J. Zhu, L. Yang, and C. 408 (M. R. S., Pittsburgh, 1996), 55. Mailhiot, Bull. Am. Phys. Soc. 41, 264 (1996). 56 Note that the angular function of the third Tersoff 32 J. R. Chelikowsky J. C. Phillips, M. Kamal and M. Strauss, potential11, which was specifically optimized for elastic Phys. Rev. Lett. 62, 292 (1989); J. R. Chelikowsky and properties, resembles the SW angular function while the J. C. Phillips, Phys. Rev. B 41 5735 (1990); J. R. Che- other versions do not. likowsky, K. M. Glassford and J. C. Phillips, 44 1538 57 W. A. Harrison, Electronic Structure and the Properties of (1991). Solids (Dover, N.Y. 1980). 33 J. S. McCarly and S. T. Pantelides, Bull. Am. Phys. Soc. 58 Although the covalent bonds in a graphitic plane resemble 41, 264 (1996). those around three-fold coordinated atoms in the bulk, we 34 J. Harris, Phys. Rev. B 31, 1770 (1985); A. P. Sutton, M. note that the out-of-plane elastic properties of the graphitic W. Finnis, D. G. Pettifor and Y. Ohta, J. Phys. C 21, 35 structure (e.g. C44) are affected by nearly free π electrons (1988). which do not participate in the covalent sp2 hybrid states57. 35 P. Alinaghian, S. R. Nishitani and D. G. Pettifor, Phil. These extra electrons, one per atom, try to stay in perpen- Mag. B 69, 889 (1994); A. P. Horsfield, A. M. Bratkovsky, dicular p-like orbitals, thus contributing to the rigidity of M. Fearn, D. G. Pettifor and M. Aoki, Phys. Rev. B 53, the plane. Nevertheless, we expect the dominant mechani- 12694 (1996). cal behavior of three-fold coordinated bulk defect atoms to 36 A. E. Carlsson, P. A. Fedders and C. W. Myles, Phys. Rev. be quite similar to atoms in a graphitic plane. B 41, 1247 (1990). 59 J. F. Nye, Physical Properties of Crystals (Clarendon Press, 37 A. E. Carlsson, Springer Proc. in Phys. 48, 257 (1990). Oxford, 1957). 38 A. E. Carlsson, Phys. Rev. B 32, 4866 (1985); A. E. Carls- 60 We calculated the DFT/LDA elastic moduli for an isolated son and N. W. Ashcroft, 27, 2101 (1983). hexagonal plane using a plane wave basis with a 12 Ry cut- 39 W. A. Harrison, Phys. Rev. B, 41, 6008 (1990). off and 1296 points in the full Brillouin zone for reciprocal 40 The expansion parameters in these studies are the dimen- space integrations. These choices guarantee sufficient accu- sionless ratios of high to low order moments of the TB racy. In order to preserve periodic boundary conditions, the LDOS, which may not be small, especially for defect struc- out-of-plane lattice parameter was fixed at at c = 5.5 A,˚ tures with states in the band gap or for metallic states like which is large enough to ensure neglible interplanar forces. the liquid, the β-tin crystal structure, and certain surfaces. The parabolic regime of the energy versus strain data (typ- Low order moments do capture general trends in energy, ically, strains less than 3%) was isolated by considering the but cannot be expected to maintain quantitative accuracy. goodness of parabolic fit, as measured by the χ2 statistic. 41 M. Z. Bazant and E. Kaxiras, Phys. Rev. Lett., 77, 4370 61 S. Ismail-Beigi and E. Kaxiras, private communication (1996). (1993). 42 M. Z. Bazant and E. Kaxiras, in Materials Theory, Sim- 62 Khor and Das Sarma used a shifted equilibrium bond angle ulations and Parallel Algorithms, ed. by E. Kaxiras, J. within the Tersoff format, but they did not specify exactly Joannopoulos, P. Vashista, and R. Kalia, Materials Re- how the equilibrium angle should depend on the local en- 11 vironment in a general configuration19. Eng. 1, 91 (1992). 63 See Fig. 2 of Carlsson7. 66 K. P. Huber and G. Herzberg, Constants of Diatomic 64 D. W. Brenner, M.R.S. Bulletin, 21, 36 (1996). Molecules (Van Norstrand, New York, 1979). 65 E. Kaxiras and L. L. Boyer, Modeling Simul. Mater. Sci. 12 TABLE I. Comparison of elastic constants (in units of Mbar) for diamond cubic silicon computed from empirical models with experimental or ab initio (LDA) values. The values for experiment (EXPT) are from Simmons and Wang54, for tight-binding (TB) from Bernstein and Kaxiras55 and for the empirical potentials Biswas-Haman (BH), Tersoff (T2, T3), Dodson (DOD) and Pearson-Takai-Halicioglu-Tiller (PTHT) from Balamane3 . The Stillinger-Weber (SW) values were calculated with the analytic formulae of Cowley45 and scaled to set the binding energy to 4.63 eV3. In the lower half of the table, we test the elastic constant relations discussed in the text by calculating the ratios αH (7C11 + 2C12)C44/3(C11 + 2C12)(C11 C12) and 0 ≡ − αB (4C11 + 5C12)/9C44. ≡ EXPT LDA SW BH T2 T3 DOD PTHT TB C11 1.67 1.617 2.042 1.217 1.425 1.206 2.969 1.45 C12 0.65 0.816 1.517 0.858 0.754 0.722 2.697 0.845 C44 0.81 0.603 0.451 0.103 0.690 0.659 0.446 0.534 0 C44 1.11 1.172 1.049 0.923 1.188 3.475 2.190 1.35 αH 1.16 1.00 0.98 2.99 2.31 1.69 1.71 2.80 αB 0.99 1.00 1.67 1.10 0.89 0.27 1.29 0.82 13 1.2 Si 2 1.1 GRA DIA 1 VAC p BC8 0.9 SC BCT5 0.8 βΤΙΝ 0.7 BCC 0.6 0 1 2 3 4 5 6 7 8 9 Z FIG. 1. Ab initio values for the bond order as a function of coordination, obtained from the inversion of cohesive energy curves for the graphitic (GRA), cubic diamond (DIA), BC8, BCT5, SC, β-tin and BCC bulk structures and with additional points for the unrelaxed vacancy (VAC) and the dimer molecule (Si2), as explained in the text. For comparison the solid line shows the Gaussian p(Z) obtained from fitting to defect structures. The dotted line shows the 1/√Z dependence, the theoretically predicted approximate behavior for large coordinations. 0 -0.5 -1 -1.5 φ -2 A -2.5 -3 -3.5 -4 -4.5 2.2 2.4 2.6 2.8 3 3.2 3.4 r FIG. 2. Attractive pair interactions from inversion of ab initio cohesive energy curves for the structures in Fig. 1 using the bond order and repulsive pair potential of our model. The solid lines are for the covalent structures with coordinations 3 and 4, while the dotted lines are for the overcoordinated metallic structures. The reasonable collapse of the attractive pair potentials indicates the validity of the bond order functional form of the pair interaction across a wide range of volumes and crystal structures. 14 130 120 110 θ o 100 90 80 70 0 2 4 6 8 10 12 Z FIG. 3. The coordination dependence of the preferred bond angle θo(Z) (in degrees), which interpolates the theoretically motivated points for Z = 2, 3, 4, 6, indicated by diamonds. 15