Interatomic potentials for monoatomic metals from experimental data and ab initio calculations

Transcription

1 PHYSICAL REVIEW B VOLUME 59, NUMBER 5 1 FEBRUARY 1999-I Interatomic potentials or monoatomic metals rom experimental data and ab initio calculations Y. Mishin and D. Farkas Department o Materials Science and Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia M. J. Mehl and D. A. Papaconstantopoulos Complex Systems Theory Branch, Naval Research Laboratory, Washington, D.C Received 19 November 1997; revised manuscript received 26 August 1998 We demonstrate an approach to the development o many-body interatomic potentials or monoatomic metals with improved accuracy and reliability. The unctional orm o the potentials is that o the embeddedatom method, but the interesting eatures are as ollows: 1 The database used or the development o a potential includes both experimental data and a large set o energies o dierent alternative crystalline structures o the material generated by ab initio calculations. We introduce a rescaling o interatomic distances in an attempt to improve the compatibility between experimental and ab initio data. 2 The optimum parametrization o the potential or the given database is obtained by alternating the itting and testing steps. The testing step includes a comparison between the ab initio structural energies and those predicted by the potential. This strategy allows us to achieve the best accuracy o itting within the intrinsic limitations o the potential model. Using this approach we develop reliable interatomic potentials or Al and Ni. The potentials accurately reproduce basic equilibrium properties o these metals, the elastic constants, the phonon-dispersion curves, the vacancy ormation and migration energies, the stacking ault energies, and the surace energies. They also predict the right relative stability o dierent alternative structures with coordination numbers ranging rom 12 to 4. The potentials are expected to be easily transerable to dierent local environments encountered in atomistic simulations o lattice deects. S I. INTRODUCTION In spite o greatly increased computer speeds, the application o ab initio methods or an atomistic simulation o materials is still limited to relatively small ensembles o atoms and, in molecular dynamics, relatively short simulation times. In contrast, the use o empirical or semiempirical interatomic potentials makes it possible to simulate much larger systems up to atoms or much longer times, and thus to tackle such problems as plastic deormation, racture, or atomic diusion. For this reason there is and will probably always be a demand or realistic interatomic potentials, as there will always be a tendency to simulate as large systems as possible. In this paper we propose an approach to the development o reliable interatomic potentials or monoatomic metals based on a large set o experimental and ab initio data. As an application we develop accurate many-body potentials or Al and Ni, which are intended or atomistic simulations o internal deects in these metals, such as point deects, planar aults, grain boundaries, and dislocations. The potentials can also be used or racture simulations and, with some caution, simulations o surace phenomena. The choice o Al and Ni is dictated by the desire to test our approach or both simple Al and transition Ni metals. Furthermore, this work is a part o our current eort to develop reliable interatomic potentials or ordered intermetallic compounds o the Ni-Al system. The present potentials or Al and Ni will be incorporated into the potentials or the compounds. In Sec. II o the paper we introduce our general approach to the development o interatomic potentials. We discuss the advantages and problems associated with using both experimental data and ab initio structural energies in one database. We also introduce a strategy o parametrization and optimization o interatomic potentials based on the separation o the itting and testing steps. Section III describes our database and urther details o the parametrization, itting, and testing procedures. In Sec. IV we present the results o itting and testing the potentials or Al and Ni. In Sec. V we summarize our results and discuss possible applications o the potentials. II. GENERAL APPROACH The potentials developed in this work are based on the ormalism o the embedded-atom method EAM. 1,2 In this method the total energy o a monoatomic system is represented as E tot 1 2 ij V r ij i F i. 1 Here V(r ij ) is a pair potential as a unction o the distance r ij between atoms i and j, and F is the embedding energy as a unction o the host density i induced at site i by all other atoms in the system. The latter is given by i j i r ij, (r ij ) being the atomic density unction. The second term in Eq. 1 is volume dependent and represents, in an approximate manner, many-body interactions in the system. EAM potentials, together with some other similar potentials, 3,4 are oten reerred to collectively as glue model potentials. All /99/59 5 / /$15.00 PRB The American Physical Society

2 3394 MISHIN, FARKAS, MEHL, AND PAPACONSTANTOPOULOS PRB 59 glue model potentials share the same general orm given by Eqs. 1 and 2, and only dier in the unctional orms o V(r), (r), and F( ). In this work we use very general orms o the potential unctions with no reerence to their original physical meaning. Thus, while we oten use the terminology o the EAM, our potentials could as well be classiied as glue model potentials. Once the general orm o the potential is chosen, the important issues become how to choose the database or itting and how to parametrize and optimize the potential unctions. We shall discuss these issues in order. A. Importance o ab initio data in the development o interatomic potentials Empirical potentials or monoatomic metals are typically itted to the experimental values o the equilibrium lattice parameter a 0, the cohesive energy E 0, three elastic constants, and the vacancy ormation energy E v. For a noncubic material this data set includes additional elastic constants and the equilibrium c/a ratio. This basic set o properties is oten complemented with planar ault energies, low-index surace energies, phonon requencies, and/or other data. Unortunately, reliable experimental inormation on metal properties that can be directly linked to atomic interactions is very limited. Furthermore, most experimental properties represent the behavior o the material in very small regions o coniguration space. For example, the elastic constants and phonon requencies are determined by small atomic displacements rom the equilibrium lattice coniguration. In contrast, in atomistic simulations the system is ree to explore dierent atomic conigurations that can be quite ar away rom the regions represented by the experimental data. The question which then arises is the ollowing: How accurately will the potential represent the energies o such abnormal conigurations? The accuracy o the potential over a range o conigurations, i.e., its transerability, obviously depends on whether the data points used or itting span a wide enough region o coniguration space. A possible way to expand the database is to include a set o atomic conigurations calculated by ab initio methods. For example, Ercolessi and Adams 5 recently proposed developing glue model potentials by itting to both experimental data and ab initio atomic orces calculated or a large set o conigurations including crystals, liquids, suraces, and isolated clusters orce-matching method. Another possibility o incorporating ab initio data is to calculate a set o structural energies, i.e., energies o various crystalline structures o the same material with dierent lattice parameters. Such a set may include not only three-dimensional crystals but also slabs, layers, or even atomic chains. 6,7 In either case the incorporation o ab initio data can improve the accuracy and transerability o the potential dramatically by sampling regions o coniguration space that are not accessible experimentally. This recently emerged approach is very promising, and may serve to bridge the existing gap between ab initio and empirical methods in materials simulations. It should be mentioned, however, that the simultaneous use o experimental and ab initio data in one database entails some problems. In particular, many ab initio methods tend to underestimate the equilibrium lattice parameters o crystals in comparison with experimental data. This tendency to underestimate interatomic distances introduces some degree o incompatibility between the ab initio structural energies and the experimental quantities, and makes one-to-one itting to the structural energies problematic. This applies equally to the orce-matching method 5 where, again, the ab initio orces can be aected by the previous tendency. In the present paper we shall address this problem by introducing a rescaling o interatomic distances during the itting and testing o the potentials. B. Optimization o the itting procedure In some earlier studies the parametrization o the potential unctions was based on simple unctional orms relecting, to some extent, their original physical meaning. 2,8,9 For example, V(r) was represented by a Morse unction and (r) by a combination o power and exponential unctions. An alternative approach, taken in the glue model 3 5 and ollowed in this work, is to use a basis set o cubic splines which, although having no physical oundation, oer plenty o parameters or itting. For the development o an accurate potential it is important to have an optimum number o itting parameters or the chosen database. While the lack o itting parameters, and thus lexibility, may aect the accuracy o the potential, it is not good to have too many itting parameters either. 6,10 As with any model potential, the general orm o Eqs. 1 and 2 has certain physical limitations which cannot be overcome by including more parameters. O course, having a suicient number o parameters, one can it all points in the data set exactly, but the potential thus obtained will perorm badly on conigurations other than those represented by the data set. Robertson, Heine, and Payne 6 recently proposed a strategy which avoids the overitting o the database. They suggested splitting the database into two parts and using one part or itting and the other or testing the potential. I the root-mean-square rms deviation between the desired and predicted properties observed at the testing stage is considerably larger than the rms deviation achieved at the itting step, the database is probably overitted and the number o parameters should be reduced. Robertson et al. 6 and Payne et al. 10 demonstrated that the rms deviation obtained while itting gives no indication o the accuracy o the potential; instead, it is the rms deviation obtained at the testing stage that gives the most meaningul measure o the quality o the potential. The separation o the itting and testing steps suggests an algorithm or inding the optimum number o parameters. Thus one can start with a small number o itting parameters and increase it gradually as long as both rms deviations decrease. Eventually, however, the rms deviation observed at the testing stage will stop decreasing and reach saturation, although the itting rms deviation may continue to decrease. At this point the process can be stopped because the introduction o new parameters will not lead to any urther improvement o the potential. The occurrence o saturation indicates that we have approached the limit o accuracy dictated by the intrinsic shortcomings o the adopted potential model. Robertson, Heine, and Payne 6 illustrated this strategy by itting dierent glue model potentials or Al to a

3 PRB 59 INTERATOMIC POTENTIALS FOR MONOATOMIC large set o structural energies generated by ab initio pseudopotential calculations. In the present work we apply a similar strategy to establish the optimum number o itting parameters or our database. III. PARAMETRIZATION AND FITTING PROCEDURES A. Database or itting and testing 1. Experimental data The experimental part o our database includes the ollowing physical properties o Al and Ni: the equilibrium lattice parameter, 11 the cohesive energy, 12,13 the elastic constants c 11, c 12, and c 44, 14 and the vacancy ormation energy 15,16 see Tables I and II. These experimental values coincide with those employed by Voter and Chen 9 in the development o their EAM potentials or these elements. The potentials o Voter and Chen, which we reer to hereater as VC potentials, have been widely used in atomistic simulations, although other EAM potentials or Al and Ni are also available see, e.g., Res. 8 and We use the VC potentials as a reerence or comparison with our potentials throughout the paper. It should be mentioned that the VC potentials were obtained as a part o the development o EAM potentials or L1 2 Ni 3 Al, and were later incorporated in EAM potentials or B2 NiAl We can thus conveniently extend the comparison with VC potentials to our current work on Ni-Al intermetallics. Additionally, our database includes the experimental values o the vacancy migration energy E m v, 16,31 the intrinsic stacking ault energy SF, 32 and the experimentally measured phonon-dispersion relations The saddle-point coniguration arising during a vacancy jump represents an important region o coniguration space which is not represented by any other properties. The local density at the jumping atom in the saddle-point coniguration is usually lower than that at a regular lattice atom, while the distances to the nearest neighbors are considerably smaller. 36 Thereore the energy o this coniguration, and thus E m v, measure the strength o pairwise repulsion between atoms at short distances. The atomic interaction in this regime is not itted properly in the traditional EAM scheme, with the consequence that EAM potentials typically underestimate E m v. 30 SF represents the relative stability o the hcp phase, and determines the width o the dislocation dissociation on the 111 plane. Realistic values o SF are thus critically important in simulations o plastic deormation and racture. The agreement with experimental phonon requencies is also essential, and, moreover, is considered as a criterion o global reliability o an empirical potential. 37 Two more experimental properties included in the database were the surace energy s and the equation o state EOS, i.e., the crystal energy as a unction o the lattice parameter a. We did not it the potentials exactly to the energies o low-index planes 100, 110, and 111 because their experimental values are not very reliable. Instead, we only required that s (110) s (100) s (111), and that all three energies be close to the surace energy o an average orientation 980 mj/m 2 or Al and 2280 mj/m 2 or Ni Res. 32 and 38. The EOS was taken in the orm o the universal empirical EOS o Rose et al., 39 which uses only a 0, E 0, and the bulk modulus B (c 11 2c 12 )/3 as physical parameters. 2. Ab initio data Ab initio structural energies were generated using the irst-principles linearized augmented plane-wave LAPW method, 40,41 including all electrons and allowing or a general potential. The electronic exchange and correlation was speciied by the Perdew-Wang parametrization 42 o the localspin-density LSD approximation within the Kohn-Sham ormulation o density-unctional theory. 43 Brillouin-zone integrations were perormed using the special k-point set o Monkhorst and Pack, 44 modiied 45 to sample correctly the Brillouin zone o lower-symmetry lattices. To speed convergence, we ollow Gillan 46 and smear out the electronic eigenvalues with a Fermi distribution at T 2 m Ry. We use a rather large basis set and k-point mesh, so that energies are converged to better than 0.5 m Ry/atom. Aluminum-only calculations were carried out in spin-restricted mode, allowing no magnetic moment. Calculations involving Ni were carried out using the spin-polarized LSD by artiicially inducing a magnetic moment on the nickel ions in the starting charge density and iterating to sel-consistency. The comparison o the ab initio structural energies with those predicted by the EAM potential was perormed as ollows. First, because the two types o calculation use dierent reerence levels o energy, only energy dierences between dierent structures could be compared with one another. In order to make this comparison more illustrative, we simply shited all ab initio energies by the amount o E 0 Ẽ 0, where E 0 is the experimental cohesive energy o the cc phase and Ẽ 0 is the ab initio energy per atom o an equilibrium cc crystal. Due to this shit the ab initio calculations predict the right cohesive energy o the cc phase by deinition. I Ẽ(R) is the ab initio energy per atom o any other crystalline structure with a irst-neighbor distance R, then it is the quantity E 0 Ẽ 0 Ẽ(R) that should be compared with the respective structural energy E(R) predicted by the EAM potential: E R E 0 Ẽ 0 Ẽ R. Second, because o the LSD approximation our LAPW calculations tend to underestimate the interatomic distances in the crystal energy versus R dependence. This tendency maniests itsel, in particular, in the underestimation o the equilibrium lattice parameter. Thus, or cc Al and Ni our LAPW calculations predict a and Å, respectively using the approximation by Birch s 47 EOS. Both values are on the lower side o the experimental lattice parameters a and Å, respectively. The ratio o the respective lattice parameters, or the equilibrium irstneighbor distances R 0 a 0 /&, equals or Al and or Ni. This dierence in interatomic distances makes one-to-one comparison o ab initio and EAM-predicted structural energies, as suggested by Eq. 3, essentially inaccurate. In particular, or an equilibrium cc crystal (R R 0 ) we have the true cohesive energy E 0 in the let-hand side o Eq. 3 because the EAM potential is it exactly to E 0 ; see below and 3

4 3396 MISHIN, FARKAS, MEHL, AND PAPACONSTANTOPOULOS PRB 59 TABLE I. Properties o Al predicted by EAM potentials in comparison with experimental and ab initio data. *Fitted with high weight. Fitted with low weight. TABLE II. Properties o Ni predicted by EAM potentials in comparison with experimental and ab initio data. *Fitted with high weight. Fitted with low weight. Experiment or ab initio Present work Voter and Chen Re. 9 Experiment or ab initio Present work Voter and Chen Re. 9 Lattice properties: a 0 (Å)* 4.05 a E 0 (ev/atom)* 3.36 b B (10 11 Pa)* 0.79 c c 11 (10 11 Pa)* 1.14 c c 12 (10 11 Pa)* c c 44 (10 11 Pa)* c Phonon requencies: L (X) (THz) 9.69 d T (X) (THz) 5.80 d L (L) (THz) 9.69 d T (L) (THz) 4.19 d L (K) (THz) 7.59 d T1 (K) (THz) 5.64 d T2 (K) (THz) 8.65 d Other structures: E hcp ev/atom * 3.33 e E bcc ev/atom * 3.25 e E diamond ev/atom 2.36 e Vacancy: E v (ev)* E m v (ev)* 0.65 g Interstitial: E I (O h ) (ev) E I 111 -dumbbell ev E I 110 -dumbbell ev dumbbell ev E I Planar deects: SF mj/m 2 )* 166 h, i us mj/m 2 ) T mj/m 2 ) 75 h gb 210 mj/m 2 ) gb 310 mj/m 2 ) Suraces: s (110) (mj/m 2 ) 980 j s (100) (mj/m 2 ) 980 j s (111) (mj/m 2 ) 980 j a Reerence 11. b Reerence 12. c Reerence 14. d Reerence 33. The results in tabulated orm can be ound in Re. 35. e Calculated in this work assuming the same nearest-neighbor distance as in the equilibrium cc phase. Reerence 15. g Reerence 31. h Reerence 32. i Reerences 51, 52. j For average orientation, Re. 32. Other estimates give 1140 mj/m 2 Re. 38. Lattice properties: a 0 (Å)* 3.52 a E 0 (ev/atom)* 4.45 b B (10 11 Pa)* 1.81 c c 11 (10 11 Pa)* 2.47 c c 12 (10 11 Pa)* 1.47 c c 44 (10 11 Pa)* 1.25 c Phonon requencies: L (X) (THz) 8.55 d T (X) (THz) 6.27 d L (L) (THz) 8.88 d T (L) (THz) 4.24 d L (K) (THz) 7.30 d T1 (K) (THz) 5.78 d T2 (K) (THz) 7.93 d Other structures: E hcp ev/atom * 4.42 e E bcc ev/atom * 4.30 e E diamond ev/atom 2.51 e Vacancy: E v (ev)* E m v (ev)* Interstitial: E I (O h ) (ev) E I 111 -dumbbell ev E I 110 -dumbbell ev dumbbell ev E I Planar deects: SF mj/m 2 )* 125 g us mj/m 2 ) T mj/m 2 ) 43 g gb 210 mj/m 2 ) gb 310 mj/m 2 ) Suraces: s (110) (mj/m 2 ) 2280 h s (100) (mj/m 2 ) 2280 h s (111) (mj/m 2 ) 2280 h a Reerence 11. b Reerence 13. c Reerence 14. d Reerence 34. e Calculated in this work assuming the same nearest-neighbor distance as in the equilibrium cc phase. Reerence 16. g Reerence 32. h For average orientation, see Res. 32 and 38.

5 PRB 59 INTERATOMIC POTENTIALS FOR MONOATOMIC the energy o a uniormly expanded crystal on the right-hand side. For example, or Ni the excess energy associated with this expansion is about 0.04 ev/atom, which is larger than the energy dierence between the cc and hcp phases. In order to remove this inconsistency in a irst approximation, we modiied Eq. 3 by rescaling all distances in the righthand side by a actor o : E R E 0 Ẽ 0 Ẽ R. This relation becomes an identity when applied to the equilibrium cc phase, and is expected to be more accurate than Eq. 3 when applied to other structures and R values. Although based on heuristic arguments rather than solid physical grounds, this rescaling oers a irst step in improving the one-to-one comparison scheme Eq. 3 used in previous studies. 5 7 It should be mentioned that the rescaling o interatomic distances can inluence some properties o the material predicted by ab initio calculations, in particular the elastic constants. It was thereore interesting to evaluate how the rescaling changes such properties in comparison with experimental data. In cases where this comparison was possible, the eect o the rescaling was either insigniicant or avorable. For example, the elastic constants or Al predicted by our LAPW calculations are c , c , and c GPa. These values are systematically higher than the experimental values listed in Table I. Because the elastic constants are proportional to the second spatial derivatives o energy, the rescaling according to Eq. 4 results in multiplying them by actor 2. This slightly reduces the elastic constants c , c 12 62, and c GPa, and makes them closer to the experimental values. Although the eect is not very large, it can be taken as conirmation o the reasonable character o Eq. 4. In accordance with Eq. 4, we used two type o structural energies or itting to or testing against each other: 1 Ab initio energies or a set o dierent structures with a ixed irst-neighbor distance R, which constitutes a certain raction o the ab initio value o R 0. 2 EAM-predicted energies or the same set o structures with a ixed R equal to the same raction o the experimental value o R 0. Our ab initio data set included the cc, hcp, bcc, simple hexagonal sh, simple cubic sc, L1 2 cc with one vacancy per simple cubic unit cell, and diamond structures. The hexagonal structural energies were taken with the ideal c/a ratio. For Al we also included the W A15 structure, and or Ni this structure was not calculated because o computational limitations. The energy o each structure was calculated with three values o R: 0.95R 0, R 0, and 1.1R 0. B. Parametrization o potential unctions Functions V(r) and (r) were represented as V r V s r V s r c r r c, r s r s r c r r c. Here r c is a common cuto radius o both unctions, and (x) x/(1 n x n ) is a cuto unction which serves to guarantee that both the irst and second derivatives o V(r) and (r) tend to zero as r r c. In this work we chose 2 Å 1 and n 4, while r c was treated as a itting parameter. Functions V s (r) and s (r) in Eqs. 5 and 6 are cubic splines through given sets o points r i,v i (i 1,...,N 1 ) and r i, i (i 1,...,N 2 ) with natural boundary conditions. The last point in each set is, o course, (r c,0). The two unctions V(r) and (r) are thus parametrized by a total o N 1 N 2 1 parameters, namely, V i (i 1,...,N 1 1), i (i 1,...,N 2 1), and r c. Likewise, the embedding unction F( ) was represented by a cubic spline through a set o points i,f i (i 1,...,N 3 ). This set includes the points 0,0 and ( 0,F 0 ), where 0 is the density corresponding to the equilibrium cc crystal and F 0 F( 0) is the equilibrium embedding energy. Given the unctions V(r) and (r), the values o 0 and F 0, as well as the derivatives F 0 and F 0 at equilibrium, can be determined uniquely rom the experimental values o a 0, E 0, and B. Indeed, considering the crystal energy per atom, E, and introducing the summation over coordination shells within the cuto sphere, Eqs. 1 and 2 give 0 m N m m, 7 F 0 E m N m V m. 8 Here V m V(R m ), m (R m ), R m is the radius o the mth coordination shell at equilibrium, and N m is the number o atoms at the mth coordination shell. Furthermore, by calculating the irst and second derivatives o E with respect to the irst-neighbor distance, we obtain, respectively, 1 2 N m V m R m F 0 m N m m R m 0, 9 m 1 2 N m V m R 2 2 m F 0 m N m m R m m F 0 m N m m R m 2 9B 0, 10 0 being the equilibrium atomic volume, V m, V m, m, and m the respective derivatives o the unctions at coordination shells. Equation 9 expresses the condition o mechanical equilibrium o the crystal, while Eq. 10 relates the second derivative o E to the bulk modulus B. We can thus determine F 0, F 0, and F 0 rom Eqs. 8, 9, and 10, respectively. Given these values, point ( 0,F 0 ) o the spline turns out to be ixed, while F 0 and F 0 uniquely determine the boundary conditions o the spline. The number o itting parameters associated with the embedding unction is thereore N 3 2. Note that this scheme o parametrization provides an exact it o the potential to a 0, E 0, and B. The basic equations 1 and 2 are known to be invariant under the transormations and r s r, F F /s 11

6 3398 MISHIN, FARKAS, MEHL, AND PAPACONSTANTOPOULOS PRB 59 F F g, V r V r 2g r, 12 where s and g are arbitrary constants. O special interest is the choice o g F 0, called the eective pair scheme; 48 in this case F( ) has a minimum at 0, which greatly simpliies all expressions or the elastic moduli and lattice orce constants. In any case the invariance o the EAM model, expressed by Eqs. 11 and 12, reduces the number o ree itting parameters by two. This reduction can be implemented by ixing one node point in each o the spline unctions V s (r) and s (r) at some arbitrary values. Thus the total number o ree itting parameters in our parametrization scheme equals N p N 1 N 2 N 3 5. C. Fitting procedure We used a computer code designed or itting the potential unctions to a 0, E 0, B, c 11, and c 12 ; phonon requencies at the zone edge point X; unrelaxed values o E v, E v m, SF, s (100), s (110), and s (111); an empirical EOS or any given set o lattice parameters; and the energies o several alternative structures with the same irst-neighbor distance as that in the equilibrium cc phase (R 0 ). In this work the potentials were itted to the ab initio energies o the hcp, bcc, and diamond structures using Eq. 4 or comparison; all other structural energies were let or the testing stage. As mentioned above, our parametrization scheme guarantees an exact it to a 0, E 0, and B. For all other properties we minimized the sum o relative squared deviations rom the desired values with a certain weight assigned to each property. The minimization was perormed using the simplex algorithm o Nelder and Mead 49 with many dierent starting conditions. The weights were used as a tool to control the priority o certain properties over others according to the reliability o the data points, the intended application o the potential, and the intrinsic shortcomings o the EAM model. Thus the highest priority was given to the elastic constants, E v, E v m, and SF and the energies o the hcp and bcc structures. The phonon requencies, the energy o the diamond structure and especially the surace energies were included with the lowest weights. The diamond structure has the lowest coordination number (z 4) and shows the largest deviation rom the ground-state cc structure in comparison with all other structures considered in this work. Although we did ind it necessary to sample this region o coniguration space, the diamond structure was assigned a low weight or two reasons: 1 Equation 4 is unlikely to be reliable at such extreme deviations rom the cc structure. 2 The occurrence o such low coordinations is less probable in simulations o internal deects in metals. The empirical EOS o Rose et al. 39 was itted at 24 lattice parameters in the range rom 0.8a 0 to 1.4a 0, but again with a relatively small weight. We did not expect this universal EOS to be very accurate when applied to the speciic metals in study, particularly ar rom equilibrium. However, the exclusion o the empirical EOS rom the data set resulted in drastic overitting o the database, and the EOS predicted by the potential attained additional inlection points or even local minima. It was thereore helpul to keep the empirical EOS in the database, even though with a small weight, so as to avoid the overitting and suppress the unphysical eatures in the predicted EOS. FIG. 1. EAM potentials or Al a and Ni b in the eective pair ormat. Because the program operated only with unrelaxed quantities, the itting to the relaxed values o E v, E v m, and SF was perormed by trial and error. In this procedure one gains a good eeling or the relaxation energies ater just a ew trials, and can achieve airly good accuracy o itting in a reasonable number o iterations. The optimum number o itting parameters or our database was established by alternating itting and testing as discussed in Sec. II. To implement this strategy we had to generate a large set o potentials with dierent numbers (N p )o itting parameters. Each potential was tested or the rms deviation between EAM-predicted and ab initio values o the structural energies other than those included in the itting database. While the rms deviation o itting decreased with N p, the rms deviation observed at the testing stage irst decreased, then reached a saturation, and inally increased. The potential corresponding approximately to the onset o the saturation was identiied as the optimum potential. IV. INTERATOMIC POTENTIALS FOR Al AND Ni The optimum potentials were ound to be those with N 1 9, N 2 7, and N 3 6 thus N p 17 or Al, and N 1 9, N 2 7, and N 3 5 thus N p 16 or Ni. They are shown in the eective pair ormat in Fig. 1. In this ormat both potentials have two local minima and are, in this respect, similar to the Al potential o Ercolessi and Adams. 5 The cuto radii are

7 PRB 59 INTERATOMIC POTENTIALS FOR MONOATOMIC TABLE III. Tabulated potential unctions or Al and Ni. See Eq. 1 in the text or notation. Al Ni Al Ni r Å V ev r Å V ev F ev F ev r c Å or Al and Å or Ni. While the VC potentials limit the interactions to three coordination shells, our potentials include also the ourth, and the potential or Ni even includes the ith, coordination shell. The eective pair interaction with these coordination shells is repulsive and, o course, very weak. The obtained potential unctions are tabulated in Table III with 25 points per unction. By applying some interpolation between the tabulated points the reader can reproduce the unctions and use them or approximate calculations. For more accurate calculations, like those reported in this paper, the potentials were tabulated with 3000 points per unction and the intermediate values were determined by means o cubic-spline interpolation. These potential iles are available via the World Wide Web 50 or via at mishin@vt.edu. In Tables I and II we list the data included in the itting databases in comparison with the values predicted by the potentials. It is observed that the experimental values o the equilibrium properties, elastic constants, vacancy ormation, and migration energies, and stacking ault energies are reproduced perectly. The right values o E v and E m v are important or the simulation o diusion kinetics, radiation damage, and similar phenomena. While or Ni most o the existing potentials predict reasonable values o both E v and E m v, or Al the agreement is usually poorer, especially with respect to E m v. Taking the most recent Al potentials, 5,25 or example, the Ercolessi-Adams potential predicts E m v 0.61 ev, in good agreement with the experimental data 0.65 ev, while the m potential o Rohrer gives an underestimated value o E v ev. For Al, the earlier experimental value o SF 166 mj/m 2 Re. 32 was later re-interpreted as about 120 mj/m 2 Res. 51 and 52 ; we thus chose to it SF to an intermediate value o 146 mj/m 2. For both Al and Ni, the experimental SF values are signiicantly higher than those predicted by the VC potentials. It should be mentioned that some o the most recent potentials developed by other groups also predict rather high values o SF, particularly 104 mj/m 2 Re. 5 and 126 mj/m 2 Re. 25 or Al. To provide an additional conirmation that the higher SF values represent the right trend, we have evaluated SF in Al by ab initio calculations. We used a ive-layer supercell which realized the stacking sequence...bcabc BCABC BCABC..., with stacking aults separated by just ive 111 layers. The unrelaxed stacking ault energy deduced rom such calculations was mj/m 2, which is comparable with the value o 157 mj/m 2 obtained or the same supercell using our potential. In contrast, the VC potential predicts a relatively low SF value o 87 mj/m 2 or this geometry. An important success o our potentials is that they show good agreement with experimental phonon-dispersion curves Fig. 2. Although only the phonon requencies at point X were included in the itting database, all other requencies are also reproduced with airly good accuracy. The somewhat larger discrepancy observed or Al may have two sources. 1 Al is more diicult or the EAM model than many noble and transition metals. This may be due to the unusually high electron density and thus extreme importance o many-

8 3400 MISHIN, FARKAS, MEHL, AND PAPACONSTANTOPOULOS PRB 59 FIG. 2. Comparison o phonon-dispersion curves or Al a and Ni b predicted by the present EAM potentials, with the experimental values measured by neutron diraction at 80 K Al and 298 K Ni Re. 33 or Al and Re. 34 or Ni. The phonon requencies at point X were included in the itting database with low weight. body interactions, which are accounted or in the EAM model in an oversimpliied manner. 2 Some mismatch between the slopes o the experimental and calculated dispersion curves in the long-wavelength regions especially or the transverse branches in qq0 direction; see Fig. 2 a indicates that there is some disagreement between the elastic constants that can be deduced rom the experimental phonon-dispersion curves, on the one hand, and those obtained by ultrasonic measurements and used in our database, on the other hand. The latter type o discrepancy has nothing in common with the intrinsic limitations o the EAM model. The Al potential o Ercolessi and Adams 5 also gives a good agreement with the experimental phonon requencies. For all other Ni and Al potentials that could be tested in this work, the agreement was considerably poorer. For sel-interstitials, both our potentials and those o Voter and Chen predict the 100 dumbbell to be the lowestenergy coniguration, in agreement with experimental data. 53 The non-split sel-interstitial coniguration in the octahedral position (O h ) turns out to be less avorable than the 100 dumbbell. For large-angle grain boundaries our potentials predict higher energies in comparison with the VC potentials. This trend is illustrated in Tables I and II or the 111 twin and and tilt boundaries. Another important quantity listed in the tables is the unstable stacking ault energy us. The meaning o this quantity is illustrated in Fig. 3, where we show 2 11 sections o so-called suraces 54,55 o Al and Ni on the 111 plane. A surace represents a plot FIG. 3. Calculated sections o the surace o Al a and Ni b on 111 plane along 211 direction. The positions o the stable and unstable stacking aults are indicated. o the planar ault energy as a unction o the ault vector parallel to a certain crystalline plane. In our case, one hal o the cc crystal above a 111 plane was shited rigidly with respect to the other hal in the 2 11 direction. The shit was implemented by small steps, and ater each step the energy o the system was minimized with respect to atomic displacements normal to the 111 plane. Such displacements included both relative rigid-body translations o the two halcrystals and local atomic displacements in the 111 direction. The excess energy associated with the planar ault shows two local minima Fig. 3 : one at the perect lattice position ( 0) and the other at the shit vector corresponding to the ormation o an intrinsic stacking ault ( SF ). The local maximum between the two minima represents an unstable coniguration which is reerred to as an unstable stacking ault. The unstable stacking ault energy us determines the activation barrier or dislocation nucleation, and plays an important role in plastic deormation and racture o metals. 56 The values o us predicted by our potentials are notably higher than those predicted by the VC potentials, and are expected to be more realistic. Qualitatively, however, the behavior o along the shit direction 2 11 is similar or all potentials considered here. The surace energies predicted by our potentials came out to be higher than those predicted by the VC potentials. Both predictions, however, are generally on the lower side o the

9 PRB 59 INTERATOMIC POTENTIALS FOR MONOATOMIC FIG. 4. Energy per atom o cc Al a and cc Ni b as a unction o the lattice parameter. The ab initio values shown in this plot were subject to transormation according to Eq. 4. The empirical EOS o Rose et al. Re. 39 is shown or comparison. experimental values 38 and the results o ab initio calculations see, e.g., Re. 57. As an additional check o this trend, we have evaluated here an unrelaxed value o s (111) or Al using an eight-layer supercell consisting o ive 111 atomic layers and three 111 layers o vacuum. While the result o LAPW calculations was s (111) 974 mj/m 2, our potential and that o Voter and Chen gave 865 and 835 mj/m 2, respectively. Historically, EAM potentials have been much more successul in accounting or the observed surace energies and surace relaxations and reconstructions than the previously used pair potentials. Nevertheless, EAM potentials are well known to underestimate surace energies consistently, the reasons probably being related to the large electron-density gradients occurring at the surace. The recognition o this act was the reason why we assigned surace energies a low weight, and ocused the attention on the properties o internal deects. In Fig. 4 we show the equations o state o Al and Ni, calculated with our potentials, in comparison with the results o ab initio calculations and the empirical EOS o Rose et al. 39 The original ab initio energies were recalculated according to Eq. 4. For Al, the EAM-predicted EOS is very close to that o Rose et al., 39 it is also consistent with the ab initio data, with some small deviations in the region o large expansions. For Ni, however, the ab initio energy values show a signiicant deviation rom the empirical EOS toward higher energies in the expansion region. A remarkable eature o our potential or Ni is that it does predict a very similar deviation in the same region, and is in much better agreement with the ab initio values than the empirical EOS. This discrepancy is not very surprising, as the parameters or Rose s EOS were derived mainly rom experimental data rather than ab initio calculations. Some results o ab initio calculations were used by Rose et al. or comparison. While these usually do ollow a universal behavior, the ab initio values o E 0, R 0, and B oten dier rom the respective experimental data. We, thereore, need not expect our ab initio data to ollow Rose et al. s universal EOS exactly. In Table IV we compare the ab initio structural energies o Al and Ni with those predicted by our potentials. This table represents the ab initio data set used at the testing stage, except or the values marked by the asterisk. The rms deviation obtained at the testing stage was 0.06 ev or Al and 0.15 ev or Ni. These values correspond to the saturation limit discussed in Sec. IV; they measure the limits o accuracy achievable in predicting the structural energies o Al and Ni in the ramework o the EAM model. It should be emphasized, however, that these rms deviations were obtained by averaging over not only dierent structures but also dierent irst-neighbor distances. More importantly, the accuracy in predicting the structural energies depends dramatically on the departure o the structures rom equilibrium. Indeed, Table IV shows that or the irst-neighbor distance ixed at R 0 the agreement between the ab initio and EAM-predicted energies is very good. In this case the potentials successully represent the right dependence o the energy on the local coordination in a wide range o dierent environments. The agreement also remains reasonably good under strong compression (0.95R 0 ) and even stronger expansion (1.1R 0 ), but the discrepancies increase drastically. The variation o R 0 is a very important test o the potentials, because the local atomic conigurations arising in the core regions o crystalline deects may eature not only dierent abnormal coordinations but also distorted interatomic distances. It is noted that or Ni the strain gives rise to a larger discrepancy between ab initio and EAM-predicted energies than it does or Al. The latter eature is quite understandable: due to the higher bulk modulus o Ni, the same strain results in a larger increase in all structural energies o Ni in comparison with those o Al. Figure 5 illustrates all these eatures; it also demonstrates that the scatter o the data points is basically random, i.e., there is no noticeable systematic deviation between the structural energies predicted by the potentials and those obtained by ab initio calculations. Table V summarizes the equilibrium irst-neighbor distances and cohesive energies calculated or dierent alternative crystalline structures o Al and Ni using our potentials. They were obtained by minimizing the crystal energies with respect to a hydrostatic strain. The energies o noncubic structures were also minimized with respect to the c/a ratio. It should be emphasized that such constrained energy minimization does not guarantee that the structures obtained are truly stable or metastable. The calculation o the elastic constants o the structures reveals that some o them are elastically unstable. In Table V, the number o nearest neighbors in each structure, z, is also indicated. Since the A15 struc-

10 3402 MISHIN, FARKAS, MEHL, AND PAPACONSTANTOPOULOS PRB 59 TABLE IV. Energies per atom in ev o dierent crystalline structures o Al and Ni calculated by the LAPW method and by the present EAM potentials. Each structural energy is given or three irst-neighbor distances: 0.95R 0, R 0, and 1.1R 0, where R 0 is the equilibrium irst-neighbor distance in the cc phase. The energies marked by the asterisk were itted during the development o the EAM potentials, all other EAMgenerated energies are predicted by the potentials. *Fit to the ab initio energy as part o the development o the potential. 0.95R 0 R 0 1.1R 0 Element Structure ab initio EAM ab initio EAM ab initio EAM Al cc * hcp a * bcc * sh a L A sc diamond * Ni cc * hcp a * bcc * sh a L sc diamond * a With ideal c/a ratio. ture includes two nonequivalent types o site, we give a value o z averaged over such sites. As usual with the EAM model, the structures with lower coordination tend to be more compact smaller R 0 and less stable larger E 0. The A15 structure, however, demonstrates an exception to this rule, in that it turns out to be anomalously stable and has an unusually small R 0. For Al, the A15 structure is predicted to have the next-lowest energy ater the cc phase, and to be almost as stable as the hcp phase. For Ni, the A15 structure is slightly less stable than the hcp and bcc structures, but again considerably more stable than all other structures listed in the Table V. For Al, we have additionally calculated the energies o the A15 structure at ten dierent lattice parameters around the equilibrium by the LAPW method. The values R Å and E ev/atom evaluated rom these data are in good agreement with our EAM predictions, and conirm the remarkable stability o this structure. It is interesting to compare our cohesive energies o dierent structures o Ni with the results o recent total-energy tight-binding calculations. 58 In Table VI we make this comparison or those structures or which tight-binding cohesive energies are available. In addition to the structures considered previously, Table VI includes A12 -Mn, A13 - Mn, and D0 3 Fe 3 Al structure where Fe sites are occupied by Ni atoms while Al sites are vacant. For relatively simple structures, i.e., structures other than A12, A13, and A15, both types o calculation are consistent with the usual trend to greater stability i.e., smaller E 0 o more compact structures i.e., those with a larger coordination number z, as well as with the bond order concept smaller energy per bond in more compact structures. There is good agreement between our energies and the tight-binding energies, although the latter show a tendency to some underbinding. In contrast, the cohesive energies obtained with the VC potential show considerable deviations rom both previous data sets, with a noticeable tendency to overbinding. The exotic structures A12, A13, and A15 all out o this trend, in that they show anomalously large stability. This behavior is consistent with the known act that the number o irst-neighbor bonds, z, is not always an adequate measure o compactness and stability o crystals, and that urther neighbors should be also taken into account. As a urther test o transerability o our potentials to other, particularly noncubic, environments it was interesting to study the energy behavior along strong deormation paths. In Fig. 6 we show the energies o Al and Ni under a volumeconserving tetragonal strain along the so-called Bain path. 59 In the ab initio and EAM calculations the atomic volume was ixed at the equilibrium value 0 predicted or the cc phase by ab initio and EAM calculations, respectively. The minimum at c/a 1 relects the stability o the cc phase, while the maximum observed at c/a 1/& corresponds to a nonequilibrium bcc phase. Although our potentials and those o Voter and Chen predict qualitatively the same behavior o the energy, our potentials demonstrate much better agreement with ab initio results. Figure 7 shows the calculated energy contour plot or Al along the Bain path, including both hydrostatic and tetragonal distortions. The bcc structure, relaxed with respect to its atomic volume, corresponds to the saddle point in this plot. The elastic constants calculated or this structure satisy the instability criterion c 11 c 12. These observations indicate that the bcc structure o Al is elastically unstable and, i allowed to evolve under the internal orces, it either returns to the ground-state cc structure or develops some urther tetragonal distortion and turns to a metastable body-centered