This article was downloaded by: [Stanford University Libraries]On: 19 August 2012, At: 16:21Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Philosophical Magazine Part BPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/tphb20

An embedded atom method potential forthe h.c.p. metal ZrAlexandra S. Goldstein a & Hannes Jónsson aa Department of Chemistry, BG-10, University of Washington, Seattle,WA, 98195, USA

Version of record first published: 27 Sep 2006

To cite this article: Alexandra S. Goldstein & Hannes Jónsson (1995): An embedded atom methodpotential for the h.c.p. metal Zr, Philosophical Magazine Part B, 71:6, 1041-1056

To link to this article: http://dx.doi.org/10.1080/01418639508241894

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantialor systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation thatthe contents will be complete or accurate or up to date. The accuracy of any instructions,formulae, and drug doses should be independently verified with primary sources. Thepublisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs ordamages whatsoever or howsoever caused arising directly or indirectly in connection with orarising out of the use of this material.

PHILOSOPHICAL MAGAZINE B, 1995, VOL. 71, No. 6, 1041-1056

An embedded atom method potential for the h.c.p. metal Zr

By ALEXANDRA S. GOLDSTEIN and HANNES JdNSSON

Department of Chemistry, BG- 10, University of Washington, Seattle, WA 98195, USA

[Received 26 July 1994 and accepted 14 September 1994t1

ABSTRACT The embedded atom method is extended to the h.c.p. metal Zr. The non-ideal

c : a ratio and the elastic responses, including contributions from internal degrees of freedom, are incorporated in the fitting procedure: Simple functional forms are assumed for the pair interaction, atomic electron density and embedding function. The functions are parametrized by fitting to experimental data: cohesive energy, equilibrium lattice constants, single crystal elastic constants and vacancy formation energy. An equation of state of the form proposed by Rose, Smith, Guinea and Ferrante is used to reproduce the pressure dependence of the cohesive energy, taking into account the anisotropic elastic response of the crystal. Dimer data and a high energy sputtering potential are also reproduced to extend the range of validity of the potential into regions of very high and low electron density. Good agreement is obtained between the experimental and calculated properties. The potential is applied to the calculation of stacking fault and self-interstitial formation energies.

9 1. INTRODUCTION Although accurate methods exist for theoretical investigations of metallic systems

efforts are continually made to determine simple, effective ways of performing statistical and dynamical simulations. First principles calculations, though in many cases successful, are difficult and time consuming and these problems are emphasized as systems increase in size or decrease in symmetry. Simulations using empirical potential functions can provide efficient and inexpensive means for studying atomic structure and dynamics and can also elucidate the relevant atcmic-scale interactions.

Of the simpler schemes, pair potentials have had the most widespread use in simulation studies of metals. They can successfully reproduce total energies for many systems but their deficiency lies in the inability to duplicate the metals’ elastic properties, in particular the Cauchy discrepancy (Johnson 1972). This can be remedied by the addition of a volume-dependent term but, in instances where the volume is ambiguous, e.g. fractures or surfaces, the pair potential models tend to fail. A second difficulty in the pair potential scheme arises in the determination of the vacancy formation energy (Johnson 1987). Typically, this energy is found to be approximately one third of the cohesive energy, yet strict pair potential models predict that the two values are equal (excluding the contribution from relaxation which is generally small in comparison). Thus, the lack of many-body interactions has severely limited these potentials in their ability to model metallic properties.

The shortcomings, of the simple pair potential have been addressed in the

Received in final form 30 October 1994.

0141-8637B5 $10.00 0 1995 Taylor & Francis Ltd

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1042 A. S. Goldstein and H. J6nsson

development of empirical many-body potentials. The Finnis-Sinclair (FS) potentials (Finnis and Sinclair 1984), the embedded atom method (EAM) (Daw and Baskes 1984), and the ‘glue model’ (Ercolessi, Tosatti and Parrinello 1988) have been the most widely used of these schemes. Finnis and Sinclair initially developed their potentials for body-centred cubic (b.c.c.) metals, obtaining the many-body interaction with a second-moment approximation in tight-binding theory. In the EAM the many-body term derives from density functional theory and the method was originally developed to study face-centred cubic (f.c.c.) metals. The glue model was developed initially to study gold systems and the many-body ‘glue’ term was introduced to represent the coordination-dependent cohesion. Though the underlying theory for each model differs, the total energy of the system in all cases is written as

where

The pair interaction @v represents the interaction of the ion cores and is often interpreted in terms of a classical electrostatic potential. The many-body term Fi is modelled as the square root of the density in the FS scheme but is left unspecified for EAM potentials. The density function pi is also pair-wise additive and herein lies the simplicity and computational efficiency of these schemes. Though the embedding function introduces the many-body interaction, the evaluations of energies and forces can be made quickly and the calculational effort is only about twice as great as that for a plain pair potential.

These schemes have proven very successful in modelling a wide range of metallic properties. Though the potentials are developed using equilibrium properties of the bulk f.c.c. and b.c.c. metals, they have been successfully used in studies of surface structure and reconstruction (Daw and Baskes 1984, Foiles, Baskes and Daw 1986, Foiles and Daw 1987a, Ackland, Tichy, Vitek and Finnis 1987, Chen, Srolovitz and Voter 1989), phonons (Daw and Hatcher 1985), point defects (Ackland ef al. 1987), grain boundaries (Ackland et al. 1987, Chen et al. 1989), thermodynamic (Foiles and Adams 1989) and liquid properties (Foiles 1985). Unlike those with f.c.c. and b.c.c. structures, fewer potentials have been developed for metals with a hexagonal close packed (h.c.p.) structure. The first, by Oh and Johnson (1988), modelled Mg, Ti and Zr in the EAM framework. Williame and Massobrio (1991) developed an FS-type potential, using different functional forms (Rosato, GuillopC and Legrand 1989) to model Zr. Both potentials were successful in reproducing many of the bulk equilibrium properties but had close to ideal c : a ratios, in disagreement with the measured values. Furthermore, contributions to the elastic constants due to internal degrees of freedom (the ‘inner elastic constants’) were not included in the construction of these potentials. These additional contributions arise in h.c.p. crystals because the unit cell has two atoms and these atoms are not located at points of inversion symmetry (Born and Huang 1954). Another set of FS potentials were generated for Co, Zr, Ti, Ru, Hf, Zn, Mg and Be by Igarashi, Kantha and Vitek (1991). These potentials were constructed to reproduce the experimental c: a ratios, but again the contribution from the inner elastic constants was neglected. Van Midden and Sasse (1992) later showed that the potentials of Igarashi et al. (1991), as well as those of Oh and Johnson and Williame and Massobrio, give rise to sizeable inner elastic constants. Pasianot and Savino (1991) developed a

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

EAM potential for Zr 1043

procedure for fitting potentials for h.c.p. metals which did account for this contribution. They were able to construct potentials for Hf, Ti, Mg and Co, but their method was unable to determine an acceptable potential for Zr. They showed that potentials for Cd, Zn, Be and Y could not be constructed within the EAh4 framework owing to a constraint imposed on the elastic constants by the EAM potential form, which is violated by the experimental values for these metals. Both Igarashi et al. (1991) and Pasianot and Savino (1991) used cubic splines as functional forms in their potential functions. Though well determined at the knot points for the spline functions, erratic behaviour between the knot points is possible when using these highly flexible forms. Finally, Baskes and Johnson (1994) have developed modified embedded atom method (MEAM) potentials for various h.c.p. metals. The MEAM is an extension to the EAM which includes angular-dependent forces introduced through the density function. These have done well at reproducing many of the properties of the metals, but the fitting procedure is difficult and the evaluation of the potential function is costly due to the evaluation of the angular-dependent terms.

Here we extend the EAM approach to the h.c.p. metal zirconium (Zr) using a simple spherical density function. Special considerations have been made for the properties which are a direct result of lower symmetry when compared to f.c.c. structures- specifically, the c : a ratio and the contributions to the elastic constants from internal degrees of freedom. In addition, we include in our fitting procedure energetic information on systems with high and low electron densities, to complement that of the bulk crystal, thereby extending the range of validity. In particular we are interested in modelling high energy sputter deposition onto surfaces, and therefore need potentials which provide a reasonable description of the energetic collision of the incoming atoms with the surface. We first discuss characteristics of the h.c.p. crystal which require consideration when determining the potential function. In $3, we will discuss the experimental properties to which the potential is fitted and our methods for their calculation. The functional forms for our potential will be discussed in $4. In our discussion of results in $ 5, we present calculations of stacking fault and interstitial formation energies. Finally, in $ 6 we summarize the results.

0 2. THE H.C.P. CRYSTAL

Face-centred cubic and hexagonal close packed crystals are close in structure. Both systems are comprised of densely packed planes of atoms, each plane shifted relative to the next such that the atoms fall into the 'holes' created by the three atoms immediately below. In a h.c.p. crystal there is only one relative shift resulting in two types of planes which stack . . .ABABAB . . . . Here, planes denoted A (or B) will be aligned such that the atoms are directly over each other. A second shift, relative to the first plane, is present in the f.c.c. structure and results in . . .ABCABC.. . stacking. In defining the unit cell for the h.c.p. structure, the c axis is established in the direction of stacking and the lattice vector is determined from the distance between successive A planes. If one were to arrange hard spheres in this fashion, it is easily shown that the packing results in the following relationship between the c and Q axes:

(3)

Hexagonal close packed structures with this c : a ratio are often considered to have 'ideal' packing, but this type of behaviour should only be expected if the lattice elements are interacting by purely pairwise potentials. Indeed, the value for the c :a ratio vanes for the naturally occurring h.c.p. metals, several of which exhibit significant

c = ($)"'a = 1.63299~.

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1044 A. S. Goldstein and H. J6nsson

deviations from ideal packing. The deviation from ideal packing is an important structural feature of the h.c.p. metals and accurate potential models should reproduce this behaviour.

This raises the question of the ability of the EAM potential model, with a spherically symmetric electron density, to correctly reproduce a non-ideal c : a ratio. The EAM framework has successfully treated f.c.c. metals, modelling the density function as some combination of s- and spherically averaged d-type electron contributions (Daw and Baskes 1984). For the most part, f.c.c. metals have almost full d shells and very little directional bonding. The manner in which the atoms pack can be well represented by a spherically symmetric atomic electron density function dominated by s electrons. Though not intuitively obvious, a simple spherical representation of the density function can also be used for h.c.p. metals, as demonstrated here for Zr. This might be suspected after considering, for example, Be, which has a c :a ratio deviating strongly from ideal, yet has only s valence electrons. We have found that a non-ideal c:a ratio can be reproduced because of the many-body nature of the embedding energy. It can result from the difference in the local environment of near-neighbour pairs formed by atoms in the close packed plane and pairs formed by atoms in adjacent close packed planes. This can be illustrated by examining the local atomic structure of the h.c.p. crystal using common neighbour (CN) analysis (Faken and J6nsson 1994). Considering near-neigh- bour pairs of atoms as ‘bonded’ pairs, three indices,j, k, I , are used to specify the local environment of pairs of atoms. The number of neighbours common to both atoms in the pair is given by the first indexj. The second index k indicates the number of ‘bonds’ between the common neighbours. Finally, 1 gives the number of bonds in the longest continuous chain of the k bonds, thus distinguishing between topological arrangements of the j neighbours. Using this classification, f.c.c. structures are completely comprised of 421 bonded pairs whereas h.c.p. crystals have a 1 : 1 ratio of 421 and 422 pairs. The geometries of these pairs are pictured in fig. 1, which shows the pairs as they occur in the densely packed planes. Here it is apparent that the local environment is different for inter-plane and intra-plane pairs. This gives rise to the different energetics, different ‘effective’ pair interactions because of the many-body embedding function, and allows the c : a ratio to be reproduced even though the density function is spherically symmetric.

The measured elastic constants contain important information about the forces present when a system is displaced from equilibrium, and about characterization of the mechanical properties. Thus, accurate representation of elastic behaviour is an important consideration when modelling the interactions of the metal atoms, as was evident in the deficiencies of the pair potential models. Born and Huang (1954) analysed the elastic constants in terms of two contributions. The first arises from strain related to the homogeneous displacement of all atoms in the crystal. The second term accounts for internal relaxations which are possible in crystals where atoms are not located at points of inversion symmetry. This second contribution, the inner elastic constants, has been analysed in detail by both Wallace (1972) and Martin (1975a, b). While it can be shown that the sublattice relaxation will have no influence on the elastic constants when only pairwise interactions are present, many-body interactions will lead to a net effect. The h.c.p. lattice can have non-zero contributions to both C1 and C12. Van Midden and Sasse (1992) determined the inner elastic constants for the h.c.p. metal within the EAM framework, and this will be discussed further in Q 3. We have taken into account the contribution from the internal degrees of freedom to the elastic constants when constructing the potential for Zr presented here.

r

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

EAM potential for Zr 1045

Fig. 1

421 Pair

0

422 Pair

421 and 422 pairs as defined by common neighbour (CN) analysis. Atoms in black represent the root pair and grey atoms denote shared neighbours. The different local environments enable a non-ideal c : a ratio even when using a spherically symmetric density function.

Q 3. CALCULATED PROPERTIES In the following discussion, we present the physical properties and relevant

calculations used to parametrize our EAM potential. For the bulk metal, these include the cohesive energy, equilibrium lattice constants, single crystal elastic constants, and vacancy formation energy. We have also included the dimer separation and interaction energy, both at the equilibrium distance and at very short separations. In the expressions which follow, two distinct notations are used for the density; p i and p,(Rij), where p i represents the total electron density at the site of atom i due to its neighbours, while the contribution to the total density from atom j is given by p,(Ru). The subscripts are also used in multi-component systems to distinguish between types of atoms and interactions.

The cohesive energy represents the binding energy of a single atom i within the bulk metal under equilibrium conditions. This energy is then described by eqn. (1) as

1 &oh = - @ij(Rij) -t F(pt) (4)

2 j + i

where Rq = IRj - Ril. A numerical value for the cohesive energy was obtained from simulation by application of a conjugate gradient minimization of the system energy to the h.c.p. crystal configuration. Periodic boundary conditions are applied to model the infinite lattice and the energy is minimized with respect to the length of each simulation box side (i.e. scaling is not isotropic). The cohesive energy Ecoh = EsySlem/N

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1046 A. S. Goldstein and H. J6nsson

is then evaluated. Here, N is the number of atoms in the simulation cell and the equilibrium lattice constants are obtained from the final configuration. The condition of equilibrium requires that no stress exists within the crystal. To ensure this condition, we have required the total residual force in the system to be less than 10 - eV A - ' .

The values of the second order elastic constants Cafiu6 are analytically determined from an expansion of the system energy with respect to the application of an infinitesimal homogeneous strain. Assuming a homogeneous deformation of the system the following expression is obtained

This expression gives the overall homogeneous contribution where the sums range over i unit cell atoms and theirj neighbours. Here, sZc is the equilibrium volume of the unit cell and the total response is an average over atoms in the unit cell.

The inner elastic constants, as determined by Van Midden and Sasse (1992) for h.c.p. metals in the EAM framework, are

where

and

Here, the indices i and j remain as defined in eqn. (5) and the sum over k is over all other atoms, with 6 k j = 1 when k and j are the same atom or atoms at equivalent lattice sites in different unit cells. With the equilibrium lattice information obtained from the above energy minimization, a straightforward evaluation of the elastic constants is afforded by eqns. ( 5 ) 4 8 ) .

To calculate the vacancy formation energy, one atom is removed from the equilibrium configuration obtained from the cohesive energy calculation. The simulation cell is then kept at constant volume and the atoms are allowed to relax until

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

EAM potential for Zr 1047

the system energy converges. The vacancy energy for an N-atom system is then determined as

E + E N - E N - ~ -Ecoh. (9)

Conjugate gradient minimization was used to determine the system energy, with the same requirement for the total residual force previously stated. The above can be written in the familiar terms of the EAM expression by considering the atomic contributions to the overall energy. Essentially, removing a single atom alters the total energy by at least the binding energy for one atom. The embedding energies of the neighbouring atoms will be affected as well due to the change in electron density introduced by the vacancy. Finally, relaxations must be included to account for internal adjustments the atoms will make in response to formation of the vacancy. Thus, the vacancy energy is expressed as

1

2 , j EYv = -- C @(Rj) + 2 { F [ ~ o - p(Rj)l- F(Po)} + Erelax. (10)

For a given potential, the value of the relaxation term is determined from simulation. The relative stability of relaxed h.c.p., f.c.c. and b.c.c. crystal structures was also determined in the fitting procedure by relaxing the crystal configurations using conjugate gradient minimization and then comparing the resulting cohesive energies.

In parametrizing the Zr potential, the dimer properties were included to provide some bounds on the potential for regions of low density. In the EAM scheme, the bond energy of the diatomic molecule is

Ed = @(rd) 2F[p(rd)l. (1 1 )

The energy and bond length rd for the metal dimer were determined by minimization.

0 4 . FUNCTIONAL. FORMS AND ~ I N G PROCEDURE In fitting the properties described above, simple functional forms for the atomic

density, pair interaction and embedding function are assumed and several parameters are adjusted to fit the data. The pair interaction @(r) is chosen as a double exponential allowing a fair degree of flexibility in the fitting

@(r) = A exp ( - ar) + Bexp ( - Br). (12)

In some initial fits, the bulk properties were accurately reproduced but the magnitude of the dimer bond energy was too large and the bond length too small. This was remedied by the addition of a short range term of the form Cexp( - K?). By ensuring the exponential had little contribution at distances relevant to the spacing in the crystal, which are 40% larger than the diatomic bond length, the necessary increase in the dimer energy was provided with little effect on the bulk properties.

The atomic density function was chosen as

pj(r) = P[exp ( - pr) + y exp ( - vr)] + Dexp ( - Er). (13)

At longer distances, such as those found in the bulk material, the main contribution to the density function arises from the valence s- and d-orbital electrons, and this is represented in the first term. The electron density, at relevant distances for the bulk metal fits the Hartree-Fock atomic electron density (Clementi and Roetti 1974) (see fig. 2). The second term in the above expression was included to provide roughly the

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1048 A. S. Goldstein and H. Jdnsson

Fig. 2

3 4 5 6 Atomic Separation (A)

Atomic electron density plotted against atomic separation. At distances relevant in the bulk material, the electron density (solid line) fits the Hartree-Fock (Clementi and Roetti 1974) electron density (points). The density has been scaled, a transformation under which the system energy is invariant (Voter and Chen 1987).

contribution from the inner, core electrons at short distances. The Hartree-Fock density was used as a guide in determining the parameters for this contribution.

The interaction range rcut can also be regarded as an adjustable parameter and is included in the fitting procedure. The pair potential and atomic density functions are truncated at the cut-off distance. The functions are then shifted such that they go continuously to zero at rcut. For the pair potential, a simple switching function was employed near the cut-off to ensure a smooth approach to zero

r < r,,, r,, S r 1 - 6 2 + 15x4 - lox’, rcut.

Here, x = ( r - rsw)/(rcut - r,,) and r,, is the distance at which the switching function turns on.

Given the functional form and parameter values for the pair interaction @(r) and the atomic electron density p ( r ) the embedding function is determined using the universal scaling function proposed by Rose er al. (1984). The function describes the relation between the energy and the lattice constant and this procedure was used by Foiles and Daw (1987b) for f.c.c. crystals. Here a slight modification has been made because of the anisotropic elastic behaviour of h.c.p. metals.

The equation of state is given as (Rose et al. 1984)

&(a*) = - E O( 1 + a*) exp ( - a*). (15) In this expression EO is the equilibrium cohesive energy and a* is the reduced lattice

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

EAM potential for Zr 1049

constant, which is written as a function of the equilibrium lattice constant ao, the bulk modulus B and the equilibrium atomic volume 00:

Assuming @(r) and p(r) have been specified, the embedding function is determined by equating the energies calculated in the EAM and equation of state expressions, eqns. (1) and (13, for a range of values for a*. The application of this method is as follows. The crystal is expanded or compressed to give different values of a. From the new atomic separations, the pair contribution @ and the total density p can be evaluated. The embedding energy is then calculated as

While all atoms within the crystal are not equivalent, the electron density will be the same at all atomic sites. Consequently, this gives a value of the embedding energy per atom, and the corresponding electron density, for any atom within the system. For a range of lattice spacings, here taken from @9a0 to 1.lOa0, an array of embedding energies (using eqn. (1)) and corresponding electron densities (using eqn. (2)) is generated. A least squares fit to a polynomial expression is then applied to express the embedding energy as a simple function of the electron density. We have found that a single polynomial of order eight in the square root of the electron density

n

k = 1

is able to give an adequate representation of the embedding function over the wide range of electron density considered here.

To incorporate the relation between the c and a axes in the function, special considerations were made during the determination of the embedding function for the h.c.p. crystal. Rose's equation of state has been shown to accurately reproduce energies of perfect crystals under hydrostatic pressure. Under this condition, equal stress is applied to each face of the crystal. For simple cubic or f.c.c. structures the compression or expansion will be isotropic. However, for h.c.p. crystals the elastic response in the a and c directions is different, as reflected by the inequality of C11 and C33. Application of equal pressure to each face of the crystal will cause isotropic compression in the densely packed planes, here chosen as the x-y plane, but the c axis will respond differently.

If a point in a material has a displacement r (x ,y ,z ) from an arbitrary origin, the displacement will change as an external force or stress is applied to the material. The normal strains ~ i i , where i = x , y or z, are defined as the change in length of each displacement component divided by its original value

By using the above expression and the relationship between stress and strain, it is now straightforward to extract the relative scaling for the different axes. The stress Sij is

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1050 A. S. Goldstein and H. J6nsson

related to the strain by

Under conditions of hydrostatic pressure, S, = S, = S, = 0, all other So = 0 and, for h.c.p. crystal,

By inverting the matrix of elastic constants and setting o = 1 the values for the normal strains, E,, en and E,, were calculated. It then follows from eqn. (19) that the scaling relationship for the c and a axes for the h.c.p. crystal is

This relationship reduces to isotropic scaling for cubic crystals due to fewer independent elastic constants.

The general equation of state by Rose et al. (1984) has proven to be remarkably accurate and is most applicable, of course, to densities close to that of the equilibrium crystal. However, it is not a reliable method for determining the embedding energy at very high or very low densities. To provide a low density value for the embedding function, the diatomic bond energy and separation are also used in the fitting in a similar manner to the cohesive energy for the bulk crystal. The pair energy and density are determined at the dimer separation and the embedding energy is determined using the diatomic bond energy.

There is great interest in sputtering and sputter deposition of Zr and we have built in a reasonable behaviour of the potential at high energy by using the Biersack (1987) sputtering potential. Using dimer separations corresponding to energies in the region of 1 keV, the density and pair interaction are determined and the embedding energy is obtained from the calculated Biersack potential energy. These density points for the embedding function are combined with those calculated using the equation of state and the overall function for the embedding energy is then determined.

After fixing the parameters in the atomic electron density, there are seven free parameters and thirteen data points (ignoring the fact that a continuous range of points is fitted in the equation of state). To optimize the six parameters in the pair potential @v and the cut-off, we have used a simplex search procedure. The simplex method (Press et al. 1986) is a relatively quick and simple multidimensional minimization scheme, for it involves only function evaluations, and does not require derivatives. To employ this method, an object function to be minimized must be constructed. Here this is taken as the sum of the squared deviation between calculated and experimental values of physical properties described above. The error function is expressed as

where the weight W; is utilized to adjust the relative tolerance in the fitting of the various properties. The weights are also used to guide the fitting routine out of unsatisfactory local minima.

The procedure for fitting the potential is summarized as follows. An initial set of

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

EAM potential for Zr 1051

the parameters is chosen as a starting point in the simplex optimization. For every set of parameters, the embedding energy is determined as described above. The physical properties we have discussed are then calculated and the value of the object function is determined based upon the variation between experimental and calculated values. The error is then minimized by the simplex routine until a fractional tolerance is obtained.

We found that the fitting procedure was facilitated by initial fits to only the bulk properties while neglecting the short range part of the potential, Cexp( - rc?) (i.e. C = 0). Once a satisfactory fit was obtained for the bulk properties, the dimer interaction was then incorporated in the fit. The Biersack data were also introduced at this point and good agreement was obtained over an energy range of 500-700 eV. For example, at a separation of 0.75 A our potential predicts an energy of 549 eV, which agrees well with the Riersack value of 552eV.

9 5. RESULTS AND DISCUSSION Figures 2 and 3 depict the ‘best fit’ functions used to describe the system’s

energetics, with the embedding energy providing the attractive part of the overall potential. Tables 1 and 2 give the parameters and comparison of calculated and experimental properties of Zr. Though the cohesive energy and equilibrium lattice constant are used as input when determining the embedding function, a slight variation is evident in the calculated value. This arises from slight deviations of the polynomial fit from the generated data points. The embedding function is smoothly varying and the polynomial form gives good representation as shown by fig. 3. For the embedding

Fig. 3 10

0

. . 2 .- U U

2 -20 W

-30 t 0 0.1 0.2

Total Electron Density (A-3) Embedding function plotted against total electron density. Polynomial expansion in djj

compared with embedding energy values obtained from Rose’s equation of state (Rose et al. 1984). Points shown are for dimer and crystal, where the crystal data correspond to a scaling of 0.9&1.10q,. The embedding function eventually curves upwards to become repulsive for very high densities. Inset depicts the pair interaction @g as a function of atomic separation. The functions shown here and in fig. 2 constitute the energy for the Zr EAM potential.

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1052 A. S. Goldstein and H. Jdnsson

Table 1. Comparison of calculated and experimental data used in fitting Zr. Lattice constant data is taken from Pearson (1958) and E d is from Smith (1976). Values for the elastic constants were obtained from Simmons and Wang (1 971) and the experimental vacancy energy from Hood (1986). Dimer dak-experimental D, and ab initio R,-were obtained from Arrington et al. (1 994) and M. D. Morse (1994, personal communication).

property Experimental Calculated

3-232 1.593 6-25 1-554 0.672 0.646 1.725 0.363 1-75 3.052 2.3 1 -

3.235 1.596 6.25 1 1.569 0-702 0.663 1.746 0-459 1.43 3-07 2.39 6-2 17 6.239

Table 2. Potential parameters obtained from optimized fits to experimental data for Zr. Units for ai are eVp-UZ.

Parameter Parameter

947.905 - 65 1-969 1723.68 1

1.1409 1.0503 1.5103

9.21 19 279-308

6 36410 7.2324

545.854 6.6229

- 2.4771 9.8703

- 1.5274 8.8601

- 2.6526 4.3 178

- 3.6142 1.2104

energy, the expansion in the square root of the total electron density is a good choice as it allows a large degree of flexibility for fitting the experimental quantities without the magnitude of the embedding energy increasing at an unreasonably rapid rate as the density increases.

For our potential, the contribution to the elastic constants from internal degrees of freedom was small ( - 3%); however, this is no reason to exclude it in the fitting procedure. The magnitude of the contribution varies greatly with the potential parameters, and different sets of parameters were found to give rise to a sizeable contribution. Overall, the agreement for the elastic constants is quite good with the exception of Cu, which is clearly too large. Pasianot and Savino (1991) showed that the EAM form, in eqns. (1) and (2) , imposes the following restriction on the h.c.p. elastic constants

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

EAM potential for Zr 1053

Measured elastic constants of several h.c.p. metals do not satisfy this condition and thus cannot be reproduced with a potential of the EAM form. For Zr, the inequality is slightly violated; a reasonable fit to the elastic constants can be obtained but they cannot be reproduced exactly. Our potential reproduces four of the five independent elastic constants to within 5%, but Cu is distinctly too large by 25%. It is interesting to note that with a different set of parameters we were able to reproduce all five independent elastic constants to within 5%, but under these conditions the most stable lattice was the b.c.c. structure. Willaime and Massobrio (1991) also found that their w M 2 potential which stabilized the b.c.c. structure gave improved values for the elastic constants.

The vacancy formation energy we have calculated is somewhat low: 1.43eV compared with the experimental value of 1.75eV. The unrelaxed vacancy energy is calculated to be 1.79eV, which shows the relaxation contribution is important in our potential. Igarashi etal. (1991) obtained the exact value although fitting as an unrelaxed vacancy formation energy. Willaime and Massobno (1991) did not fit the vacancy energy but we have obtained a value which was too large, 2-07eV with relaxation contributing 0.06eV, using their WMl potential. In both cases, unlike our potential, the relaxation was not a significant contribution ( - 4%).

Of the potentials here discussed, ours is unique in that it uses the dimer properties in the fitting procedure. This is to provide a reasonable description of atoms with low coordination in low density environments in the hope of building in reasonable behaviour of atoms interacting with a crystal surface. We have obtained a diatomic bond energy and separation which are consistent with those predicted from experiment (Arrington, Blume, Morse, Doversa and Sassenberg 1993) and electronic structure calculations (M. D. Morse 1994, personal communication). For comparison, a calculation of the dimer properties using the WM1 potential yields a bond energy of - 4.59 eV and a bond length of 2-73 A. The same calculation for the Igarashi potential results in a bond energy of - 4-77 eV at a separation of 3-39 A. These binding energies are too large and for Igarashi the predicted bond length is larger than the nearest-neighbour distances in the bulk crystal.

The potential must stabilize the h.c.p. structure with regard to other lattices. During the fitting procedure stability with respect to the f.c.c. and b.c.c. structures is ensured. Another test for the structural stability is to calculate the 12 stacking fault energy. The fault occurs in the basal plane and corresponds to a translation in one close-packed plane to give the stacking ... ABABABCACAC ... . The stacking fault energy must be positive and is expected to be large, based on theoretical estimates by Legrand (1984). We have calculated the 12 stacking fault energy using a system of 1584 atoms comprised of 22 planes of 72 atoms. Two equivalent h.c.p. crystals were combined, translating one with respect to the other to give the desired stacking sequence, In effect, this creates two stacking faults as periodic boundaries are used to model the infinite lattice. The size of the system was chosen such that the two faults, separated by 10 planes, would not affect one another with the range of interaction extending over approximately three layers. The system was relaxed using a conjugate gradient minimization of the energy with respect to atomic position. We obtained a stacking fault energy of 121 mJ m-2. This is lower than the theoretical estimate of 340mJ m-’ (Legrand 1984). Igarashi et al. (1991) reported an abnormally low value predicted by their potential, 27 mJ m - 2,

and noted they were unable to obtain a reasonably high stacking fault energy without compromising the lattice stability under deformation.

The self-interstitial formation energies were also calculated to test the potential under non-equilibrium conditions and the results &e presented in table 3. The interstitial

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1054 A. S. Goldstein and H. J6nsson

Table 3. Comparison of self-interstitial formation energies and I2 stacking fault energies for Zr. Self-interstitial energies are in eV and stacking fault energies are in mJ m-’. Bo and C indicate final site and * indicates no available data. Values for WMl from Willaime and Massobrio (1991). OJ from Oh and Johnson (1988) and IKV from Igarashi et al. (1991). GJ refers to the authors’ potential.

Self-interstitial formation energy configuration GJ WM 1 OJ IKV

BC Bo Bt C 0 T S I 2 energy

4-13 4.58 4-15 4.22 4.6 1 4.22 4.22

121

Bo 4-32 BO

4.27 4.48

C C *

Bo 4.73

4.52 462 C

4.92

*

*

8.1 9.2 8.8 7.7 7.5 9.1 8-7

27

sites are as proposed by Johnson and Beeler (198 1) and the calculations were performed on four system sizes, ranging from 1152 to 4800 atoms, to determine possible size effects. The interstitial atom was introduced to the equilibrium configuration and the system was allowed to relax while the system volume was kept constant. Though the calculated energies did differ, the effect due to system size is very small ( < 1 %). The values reported are for a system with 2048 lattice atoms. A comparison with experimental data is not possible for the interstitial formation energies, so our results will be compared with those previously reported. In comparing the values listed in table 3, there are several points to be made. First, the potentials which accurately reproduce the non-ideal c: a ratio find stable minima for each of the proposed sites. The potentials resulting in a near-ideal c : a ratio predict the B,, T (Oh and Johnson 1988, Willaime and Massobrio 1990), B, and S (Willaime and Massobrio 1990) sites to be unstable. Atoms introduced in those positions decay to the C or BO sites. Second, the values reported by Igarashi et al. (1991) are much higher when compared with the other reported values. Combined with the results from the stacking fault energy calculation, this suggests that although the Igarashi potential is well defined at the knot points, it does not extend well beyond those points. Finally, we find the B, configuration to be the most stable. Although the C, T and S configurations relax to the same energies, the sites remain unique with a separation of 0-59A between C and T and 0.49w between C and S.

0 6. CONCLUSION Thus, we have presented an empirical potential for Zr using a simple EAM form

which can be easily implemented and provides good agreement with input experimental data covering a very wide range in electron density. Analytical expressions for the forces required in dynamical simulations are given in the Appendix. The model addresses the issue of the contribution to the elastic constants from internal degrees of freedom and, though found to be negligible in our final potential, it is a contribution that should not be ignored in the fitting process. The non-ideality of the c : Q ratio is also incorporated in the fitting procedure and was successfully reproduced.

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

EAM potential for Zr 1055

ACKNOWLEDGMENTS The authors would like to thank Bruce Berne for helpful insights on the functional

forms. This work was supported by the National Science Foundation, project CHE-92 17774. ASG acknowledges the Pacific Northwest LaboratoriesE3attelle through which this research was supported by the Northwest College and University Association for Science (Washington State University) under Grant DE-FG06-89ER- 75522 or DE-FG06-92RL-1245 I with the US Department of Energy.

A P P E N D I X We give here analytical expressions for the forces acting on the atoms in a system

described by the EAM potential given by eqns. (1) and (2). These are required to perform atomistic classical dynamics simulations as well as the conjugate gradient minimiza- tions used in the fitting procedure. Below, we give analytic expressions for the forces in the EAM framework.

The force on atom i is defined as

Pi = - V ~ E T ~ T = - Vi - 2 @ij(RV) + 2 Fj(pj) = - Vi(E0 + E F ( ~ ) ) . (A 1) " 2 j ( # i ) i I The gradient of the pair potential @ is straightforward and the 01 component (01 = x, y , or z) of the force on atom i is given by

The force associated with the embedding energy is more complex as the many-body effects must be accounted for

The first term represents the change in the embedding energy due to the change in density at the site of atom i, which results from its displacement. When the position of atom i alters, this also changes the density at the sites of its neighbours. The associated energy change is given by the second term. For a system in which all the atoms are equivalent, i.e., a one-component system, the expression reduces to

For constant, isotropic pressure (Andersen 1980) simulations or conjugate gradient minimizations with variable volume, an expression for the system's internal pressure is needed. At zero temperature, the isotropic pressure can be derived from the change in total energy of the system with respect to a change in the volume. This is the virial contribution to the pressure,

Here, the volume is changed isotropically by scaling all distances by a single factor. When relaxing the h.c.p. crystal to determine the cohesive energy and minimum

energy structure, we have performed a conjugate gradient minimization with respect

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2

1056 EAM potential for Z r

to the length of each side of the simulation box. The ‘force’ on each side of the box is derived in a similar way as the pressure for the system resulting in

where a , is the length of simulation box along the x , y or z axis.

REFERENCES ACKLAND, G. J., TICHY, G., VITEK, V., and F m I s , M. W., 1987, Phil. Mag. A, 56, 735. ANDERSEN, H. C., 1980, J. chem. Phys., 72, 2384. ARRINGTON, C. A., BLUME, T., MORSE, M. D., D O V E R S T ~ , M., and SASSENBERG, U., 1994, J. Phys.

BASKES, M. I., and JOHNSON, R. A., 1994, Model. Simul. Muter. Sci. Engng, 2, 147. BIERSACK, J. P., 1987. Nucl. Instrum. Methods Phys. Res. B, 27, 21. BORN, M., and HUANG, K., 1954, Dymmical Theory of Crystal Lattices (Oxford: Clarendon). CHEN, S. P., SROLOVITZ, D. J., and VOTER, A. F., 1989, J. Muter. Res., 4, 62. CLEMENTI, E., and ROETTI. C., 1974, Atom. Data Nucl. Data Tables, 14, 225. DAW, M. S., and BASKES, M. I., 1984, Phys. Rev. B, 29,6443. DAW, M. S., and HATCHER, R. D., 1985, Solid State Commun., 56, 697. ERCOLESSI, F., PAMINELLO, M., andTosArr1, E., 1988, Phil. Mag. A, 58, 213. FAKEN, D.. and J6NSSON. H., 1994, Comput. Muter. Sci., 2, 279. FINNIS. M. W., and SINCLAIR, J. E., 1984, Phil. Mag. A, 50, 45. FOILE~, S. M., 1985, Phys. Rev. B, 32,3409. FOILES, S. M., and ADAMS, J. B., 1989, Phys. Rev B, 40,5909. FOILS. S. M., BASKES, M. I., and DAW, M. S., 1986. Phys. Rev. B, 33, 7983. FOILES, S. M., and DAW, M. S., 1987a, Phys. Rev. Lett., 59,2756; 1987b, J. Muter. Res., 2 ,5 . HOOD, G. M., 1986, J. Nucl. Muter., 139, 179. IGARASHI, M., KHANTHA, and VITEK, V., 1991, Phil. Mag. B, 63, 603. JOHNSON, R. A., 1972, Phys. Rev. B , 6, 2094. JOHNSON, R. A,, 1987, Computer Simulations in Materials Science, edited by R. J. Arsenault,

J. R. Beeler, Jr., and D. M. Esterling (Metals, Ohio Park: American Society for Metals). JOHNSON, R. A., and BEELER, J. R.. 1981, Interatomic Potentials and Crystalline Defects, edited

by J . K. Lee (New York: The Mettalurgical Society of AIME). LEGRAND, P. B., 1984, Phil. Mag. B, 49, 171. MARTIN, J. W., 1975a, J. Phys. C, 8, 2837: 1975b, ibid., 8, 2858. OH, D. J., and JOHNSON, R. A., 1988, J. Muter. Rex , 3, 471. PASIANOT, R., and SAVINO, E. J., 1991, Phys. Rev. B, 45, 12704. FEARSON, W. B., 1958, A Handbook of Lattice Spacings and.Structures of Metals and Alloys

(Oxford: Pergammon). PRESS, W. H., FLANNERY, B. P., TEUKOLSKY, S. A., and VETITERLING, W. T., 1986, Numerical

Recipes: The Art of Scientific Computing (Cambridge: Cambridge University Press). ROSATO, V., GUILLOPB, M., and LEGRAND, B., 1989, Phil. Mag. A. 59, 321. ROSE, J. H., SMITH, J. R., GUINEA, F., and FERRANTE, J., 1984, Phys. Rev. B, 29, 2963. SIMMONS, G., and WANG, H., 197 1, Single Crystal Elastic Constants and Calculated Aggregate

Properties: A Handbook (Cambridge, MA: MIT Press). SMITH, C. J., ed., 1976, Metals Reference Book (London: Butterworth). VAN MIDDEN. H. J. P., and SASSE, A. G. B. M., 1992, Phys. Rev. B, 46,6020. VOTER, A. F., and CHEN, S. P., 1987, Muter. Res. SOC. Sym. Proc., 82, 175. WALLACE, D. C., 1972, Thermodynamics of Crystals (New York: Wiley). WILLAIME, F., and MASSOBRIO, C., 1991, Phys. Rev. B, 43, 11653.

Chem., 98, 1398.

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

16:

21 1

9 A

ugus

t 201

2