Surface energies of hcp metals using equivalent crystal theory

Computational Materials Science 46 (2009) 524–530

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Surface energies of hcp metals using equivalent crystal theory E. Aghemenloh *, J.O.A. Idiodi, S.O. Azi Department of Physics, University of Benin, Benin City, P.M.B. 1154, Edo State, Nigeria

a r t i c l e

i n f o

Article history: Received 4 June 2008 Received in revised form 20 January 2009 Accepted 8 April 2009 Available online 10 May 2009

a b s t r a c t The equivalent crystal theory method of Smith et al. [Phys. Rev. B 44 (1991) 6444] originally formulated for fcc and bcc metals, and semiconductors, is here extended to hcp metals and applied to calculate surface energies. The (0 0 1) surface energies obtained for 22 hcp metals are in good agreement with the results of both experiment and ab initio calculations. Ó 2009 Elsevier B.V. All rights reserved.

PACS: 61.43.Dq 31.15.bu 68.35.Md Keywords: hcp metals ECT Surface energy

1. Introduction Surface energy is a physical property of great interest [1–18]. The energy of a free surface plays an important role in several physical and chemical processes such as fracture, catalysis, etc. Experimental measurements of the surface energy are more commonly found for polycrystalline materials, and are also usually subject to errors due to surface-active contaminants and thus have a degree of uncertainty. Early theoretical calculations were based on perturbation theory [1] or non-perturbative variational method [2]. In the last few years, there has been an increasing effort on first-principles calculations [3–7], as well as in the area of semiempirical methods [8–16]. In general, semi-empirical approaches tackle the many-body problem by determining a functional form for the cohesive energy based on some physical model. The functional form often contains one or more parameters which are to be determined by fitting to experimental properties. Once these parameters have been determined, the functional form may then be used to calculate various other properties, such as defect energies, etc. While the various semi-empirical models have been impressively successful in describing a wide variety of metallic properties, a significant short-coming is that the estimate of surface energies from several of these semi-empirical models [8–15] is

* Corresponding author. Tel.: +234 8033611224. E-mail address: [email protected] (E. Aghemenloh). 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.04.011

considerably lower than experiment and first-principles results. The reasons for this short-coming are not yet clear and current research efforts are in the direction of formulating better semiempirical models. A review of recent theoretical efforts dealing with the calculation of properties of materials using semi-empirical methods reveals that four methods are dominant: Finnis–Sinclair method [8], embedded atom method (EAM) [9–14], effective medium theory (EMT) [15], and equivalent crystal theory (ECT) [16,19–24]. The ECT method has been used extensively for both face-centred cubic (fcc) and body-centred cubic (bcc) metals [16,19–24] but has so far not been applied to the hexagonal closed-packed (hcp) metals. The purpose of this study is first, to extend the original ECT method [19] to hcp metals and thereby fill out that gap in the literature, and finally to use the method to study the surface energy problem for these materials. This completes our initial objective of using the ECT method to provide comprehensive surface energy results for fcc [21], bcc [22] and hcp metals. Such extensive results obtained with the same theoretical model, are useful to both theorists and experimentalists. The remainder of this paper is organized as follows. In Section 2, we give a brief discussion of the ECT method and its appropriate extension to hcp metals. In Section 3, we discuss the ECT method of calculating the surface energies of hcp metals. The results of surface energies for 22 hcp metals are reported in Section 4, along with the results obtained by other workers. Concluding remarks are given in Section 5.

E. Aghemenloh et al. / Computational Materials Science 46 (2009) 524–530

2. The ECT method In this section a brief operational description of the ECT method is given. Details of its derivation and the rationale behind its development can be found in Ref. [19]. Equivalent crystal theory (ECT) is based on an exact relationship between the total energy of the constituent atoms in a solid and the atomic locations and it applies to surfaces and defects in both simple and transition metals as well as in covalent solids. Lattice defects and surface energies are determined via perturbation theory on a fictitious, equivalent single crystal whose lattice constant is chosen to minimise the perturbation. The energy of the equivalent crystal, as a function of its lattice constant is given by a universal binding energy relation [25]. Let e be the total energy to form the defect or surface, then

e¼

X

ei

ð1Þ

i

where ei is the contribution from an atom i close to the defect or surface. The linear independence attributed to the terms in Eq. (1) is consistent with the limit of small perturbations which is assumed in the formulation of the ECT. ECT is based on the concept that there exist, for each atom i, a certain perfect, equivalent crystal with its lattice parameter fixed at a value so that the energy of atom i in the equivalent crystal is also ei . This equivalent crystal differs from the ground-state crystal only in that its lattice constant may be different from the ground-state value. We compute ei via perturbation theory, where the perturbation arises from the difference in the ion core electronic potentials of the actual defect solid and those of the effective bulk single crystal. The problem of finding ei , and hence e, is reduced to finding for each atom an effective equivalent single crystal and calculating the energy of the atom i in it. Many body terms contribute to the energy of each atom in real systems. Hence, ei is written as a sort of perturbation series of one-, two-, three-, and four-body terms, each of which is obtained by considering a different effective perfect equivalent single crystal. In this approximation [19], ei takes the form:

"

ei ¼ DE F  ½a1 ðiÞ þ

X

F  ½a2 ði;jÞ þ

j

X i;j

F  ½a3 ði;j;kÞ þ

X

# F  ½a4 ði;p;qÞ

p;q

ð2Þ where DE is the equilibrium cohesive energy per atom of the actual crystal, and F  ½a  is the simple analytic function 







F ½a  ¼ 1  ð1 þ a Þ expða Þ

ð3Þ

The first term in Eq. (2), F  ½a1 ðiÞ, contributes when average neighbour distances are altered via defect or surface formation. It can be thought of as representing local atom density changes. The second term, F  ½a2 ði; jÞ, is a two-body term which accounts for the increase in energy when nearest neighbour bonds are compressed below their equilibrium value. The third term, F  ½a3 ði; j; kÞ, is a three-body term that accounts for the increase in energy which arises when bond angles deviate from their equilibrium values in the undistorted single crystal. Finally, the fourth term, F  ½a4 ði; p; qÞ, describes face diagonal anisotropies. A detailed description, for each lattice type, of the structural effect associated with these four terms can be found in Ref. [19]. Now Eq. (2) can be re-written, in an obvious notation, as

ei ¼ DE

4 X

F  ½ak ðiÞ

Eqs. (1)–(3), (and) (5) correspond respectively to Eqs. (5) and (20)-(22) of Smith et al. (1991). However, the fact that the hcp structure has two lattice constants which convention denotes as a and c makes it mandatory for us to relabel c in Eq. (5) as c1 . Eq. (5) thus becomes

a ¼ ðRec =c1  rWSE Þ=l

ð6Þ

Let r WS be the Wigner–Seitz radius in the actual crystal with the equilibrium value r WSE , then c1 is the ratio between the nearestneighbour distance and rWSE pffiffiffi in the undistorted actual crystal. For the hcp crystal, r WSE ¼ ð3 3a2 c=16pÞ1=3 , and the scaling length l takes the form

l ¼ ðDE=12pBr WSE Þ1=2

ð7Þ

where B is the equilibrium bulk modulus of the actual crystal. By intuition rather than a rigorous proof, our work hangs on the assumption, [26,27], that the universal binding energy curve of Rose et al. [25] describes the hydrostatic compression or expansion of a pure hcp crystal by keeping the ratio c/a constant and using the Wigner–Seitz radius as a reference instead of the lattice constant. Throughout this work it is assumed that the c=a ratio for the equivalent hcp crystal is the same as that for the actual crystal. Then a knowledge of a ðiÞ for the equivalent crystal implies immediately a knowledge of c ðiÞ, and in what follows we shall be focussing attention on how to determine a ðiÞ. Eq. (2) shows clearly that each atom i in the defect region has associated with it four different types of equivalent crystals with lattice parameters a1 , a2 , a3 , and a4 . The value of a , the lattice parameter of the first equivalent crystal associated with atom i, chosen so that the perturbation (the difference in potentials between the solid containing the defect and its bulk, ground-state equivalent crystal) vanishes. Within the frame work of ECT, this requirement translates into the following equation for the determination of the nearest neighbour distance of the first equivalent crystal, Rec :

  1 N1 Rpec expð aRec Þ þ N2 ðC 2 Rec Þp exp  ða þ Þ C 2 Rec k   X X p 1 p Rj expðRj Þ  Rj exp  ða þ Þ Rj ¼ 0  k defect NN defect NNN

ð8Þ

where

  ~ ~ Rj ¼ R j  Ri 

ð9Þ

Eq. (8) is the same as Eq. (26) of Ref. [19] and it is just the mathematical representation for local atom density changes in the defect region. Once Rec has been determined from Eq. (8) then ai ðiÞ can be determined from Eq. (6). In Eq. (8), Rj is the distance be~j and a reference atom located tween the atom located at position R ~i ; N 1 and N 2 are, respectively, the number of nearest at position R neighbours (NN) and next-nearest neighbours (NNN) in the equivalent crystal; and finally, C 2 is the ratio between the NNN distance and NN distance in the undistorted actual crystal. The electronic screening length k in Eq. (8) is chosen, after Smith et al. [19], to be of the form

k ¼ 2:81l

ð10Þ

and the ECT parameter p is defined by

ð4Þ

k¼1

where the scaled lattice constant a is given in terms of the equivalent crystal nearest-neighbour distance Rec as

a ¼ ðRec =c  r WSE Þ=l

525

ð5Þ

p ¼ 2n  2

ð11Þ

where n is the atom principle quantum number. The two summations in Eq. (8) are over the actual defect crystal, the first over nearest neighbours and the second over next-nearest neighbours to atom i.

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E. Aghemenloh et al. / Computational Materials Science 46 (2009) 524–530

The appropriate equations, analogous to Eq. (8), for the determination of Rec for the 2nd, 3rd, and 4th equivalent crystals associated with the defect atom i, are not given here. They can be found in Ref. [19]. The values of the ECT parameters l; k; p and rWSE for each metal can be determined via their defining equations as given in the above presentation. The ECT parameter a in Eq. (8), primarily reflects the structure of the electron density in the overlap region between two neighbouring atoms, and it is determined via the requirement that the energy to form a rigid or monovacancy be equal to the experimental monovacancy formation energy Ef1v . This is actually realized as follows. First, by setting the experimental value of Ef1v =N 1 equal to ei , one determines F  ½a1 ðiÞ from Eq. (2) upon realizing that for the rigid defect F  ½a2  ¼ F  ½a3  ¼ F  ½a4  ¼ 0. Next, from Eq. (3) one determines a1 , and from Eq. (6) one determines the corresponding value of Rec . Finally, upon using this value of Rec for the monovacancy defect in Eq. (8), one determines a for each metal. With a known, one is then in a position to proceed with the surface defect calculations. This is discussed in the next section. 3. Surface energy calculation Here, we illustrate the formalism described in the previous section by providing a detailed application of the ECT method to the calculation of the (0 0 1) surface energy of hcp metals. Before we go through the details we shall first make a few considerations of general interest. It is now well-known that several metallic surfaces differ from the bulk atomic arrangements because of the change in atomic coordination, with or without distortion of the two-dimensional mesh cell, as well as electron density change in the surface region. In this initial study of surface energies of hcp metals, we consider a rigid surface (i.e., no interlayer relaxation or reconstruction), and hence all bond lengths and angles retain their bulk equilibrium values. The surface energy is therefore obtained by solving for the term represented by F  ½a1 ðiÞ only. The neglect of the higher-order anisotropic terms in Eq. (2) is justified from the following. From previous studies [19,22,24,28], the many-atom effects which are represented in the ECT by the inclusion of the two-atom bond length distortion, the three-atom bond angle anisotropy and the four-atom face-diagonal distortion are, in the case of surface energy calculations of low-miller index metallic surfaces, of very little relevance. They introduce a small correction, usually of the order of 1–2% of the leading term in Eq. (2). This is not the case for semiconductors where these anisotropies constitute a significant contribution to the surface energy. In this present implementation the relaxation of the surface atomic positions is neglected. According to the first-principles works by Feibelman [4] and by Mansfield et al. [29], the effect of relaxation on the calculated surface energy of a particular crystal facet may vary from 2% to 5% depending on the roughness. Therefore the neglect of relaxations in this work has little effect on the accuracy of the calculated surface energies. Even with the above simplification, the calculation of the surface energy of hcp metals, within the ECT method, unlike the bcc and fcc metals, is not straight forward. This is mainly due to the fact that the 12 nearest-neighbour atoms of the hcp structure are not all at the same distance. Six of them are at a distance Rnn1 , while the other six are at a slightly different distance Rnn2 . In this study we have adopted the convention that the smaller distance will always be denoted as Rnn1 . If a and c are the lattice constants for the hcp metals, then for all the hcp metals considered in this study, Rnn1 ¼ ½ð4a2 þ 3c2 Þ=121=2 , and Rnn2 ¼ a, with the exception Cd and Zn whose c/a ratio is greater than the ideal hcp ratio pof ffiffiffiffiffiffiffiffi of 8=3. For such metals Rnn1 ¼ a, Rnn2 ¼ ½ð4a2 þ 3c2 Þ=121=2 . For hcp structure, 6 atoms are at the next nearest-neighbour distance Rnnn ¼ ½ð16a2 þ 3c2 Þ=121=2 .

We have organized the surface energy calculations for the hcp metals into three models in order to better illustrate the dependence of results on some fundamental constants of the hcp structure. Results from Model I, will be studied along with two other models of theoretical interest. 3.1. Model I A prescription for determining the ECT parameter a was described in Section 2. Implementing that prescription, we find that for an atom in the bulk that has lost one nearest neighbour, Eq. (8) for determining a becomes

1 12Rpec expðaRec Þ þ 6ðc2 Rec Þp exp½ða þ Þ c2 Rec  k  5ðRnn1 Þp expðaRnn1 Þ  6ðRnn2 Þp expðaRnn2 Þ 1  6ðRnnn Þp exp½ða þ ÞRnnn  ¼ 0 k

ð12aÞ

where c1 ¼ Rnn1 =r WSE and c2 ¼ Rnnn =Rnn1 , and the monovacancy is assumed here and elsewhere to be at a distance Rnn1 from the bulk defect atom. With a known, we are now in a position to determine Rec . Since a surface atom on the (0 0 1) surface has lost three nearestneighbours at a distance ½ð4a2 þ 3c2 Þ=121=2 and also three next nearest-neighbour atoms at a distance Rnnn , Eq. (8) for determining Rec for the surface atom is

1 12Rpec expðaRec Þ þ 6ðc2 Rec Þp exp½ða þ Þ c2 Rec  k  3ðRnn1 Þp expðaRnn1 Þ  6ðRnn2 Þp expðaRnn2 Þ 1  3ðRnnn Þp exp½ða þ ÞRnnn  ¼ 0 k

ð12bÞ

For all hcp metals, with the exception of Cd and Zn, where Eq. (8) takes the form

1 12Rpec expðaRec Þ þ 6ðc2 Rec Þp exp½ða þ Þ c2 Rec  k  6ðRnn1 Þp expðaRnn1 Þ  3ðRnn2 Þp expðaRnn2 Þ 1  3ðRnnn Þp exp½ða þ ÞRnnn  ¼ 0 k

ð12cÞ

3.2. Model II Here we solve Eq. (12a) for the determination of a and Eqs. (12b) and (12c) for the determination of Rec for a (0 0 1) surface atom, but with c1 and c2 given by

c1 ¼ Rnn2 =r WSE

and c2 ¼ Rnnn =Rnn2

That is, the ECT parameters c1 and c2 for this case are to be determined using the larger nearest neighbour distance Rnn2 rather than Rnn1 . The sensitivity of surface energy results upon this alternative parameter choice will be shown by this model. 3.3. Model III Here we ignore the actual difference between Rnn1 and Rnn2 , and assume that the 12 nearest neighbours are at the same distance Rnn1 . This is of course what happens when the c/a ratio takes on the ideal hcp value. Eq. (12a) then becomes

  1 12Rpec expðaRec Þ þ 6ðc2 Rec Þp exp ða þ Þ c2 Rec k     1  11ðRnn1 Þp exp ðaRnn1 Þ  6ðRnnn Þp exp  a þ Rnnn ¼ 0 k ð13aÞ

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E. Aghemenloh et al. / Computational Materials Science 46 (2009) 524–530

while Eqs. (12b) and (12c) for the determination of Rec for a (0 0 1) surface atom become the single equation

Table 2 Computed ECT constants.

1 12Rpec expðaRec Þ þ 6ðc2 Rec Þp exp½ða þ Þ c2 Rec  k 1 p p  9ðRnn1 Þ expðaRnn1 Þ  3ðRnnn Þ exp½ða þ ÞRnnn  ¼ 0 k

Element

P

l (Å)

k (Å)

rWSE (Å)

Be Mg Sc Y Ti Zr Hf Tc Re Ru Os Co Zn Cd Tl Gd Tb Dy Ho Er Tm Lu

2 4 6 8 6 8 10 8 10 8 10 6 6 8 10 10 10 10 10 10 10 10

0.3152 0.3134 0.4583 0.4696 0.3411 0.3926 0.3766 0.2551 0.2458 0.2477 0.2356 0.2631 0.2156 0.2143 0.3320 0.4673 0.4680 0.4005 0.4149 0.3922 0.3663 0.4890

0.8858 0.8807 1.2879 1.3196 0.9586 1.1033 1.0582 0.7169 0.6908 0.6960 0.6620 0.7392 0.6058 0.6023 0.9330 1.3131 1.3152 1.1254 1.1659 1.1022 1.0292 1.3742

1.2412 1.7705 1.8140 1.9909 1.6147 1.7710 1.7449 1.5059 1.5200 1.4811 1.4967 1.3839 1.5355 1.7279 1.8974 1.9894 1.9692 1.9598 1.9527 1.9420 1.9312 1.9155

ð13bÞ

For this case, c1 ¼ Rnn1 =r WSE and c2 ¼ Rnnn =Rnn1 For the sake of simplicity, researchers do oftenp assume ffiffiffiffiffiffiffiffi that the c/a ratio for hcp metals possess the ideal value of 8=3. The effect of this assumption on the surface energy will be shown by this model. Once one knows the Rec value, then a ðiÞ can be calculated directly from Eq. (6) and the surface energy r001 for each case is calculated from the formula

2 DE X

r001 ¼ 2 pffiffiffi a

3

F  ½a ðiÞ

ð14Þ

i

The sum over i includes only one atom per atomic layer, and usually only a few layers need be included for metal low-index planes. In fact, for the (0 0 1) hcp metal surface, only the outermost surface layer is involved.

a (Å1) Model I

Model II

Model III

1.5762 1.6937 2.2467 2.6910 2.7115 3.0724 3.7238 3.7396 4.4088 4.0011 4.8157 3.0921 6.5597 6.9232 3.5568 3.1784 3.2263 3.4002 3.3892 3.4554 3.5385 3.2590

1.3543 1.6572 2.1161 2.4655 2.4646 2.8758 3.4838 3.5191 4.2560 3.5129 4.2248 3.0139 2.4924 2.8296 3.3201 3.0330 3.0480 3.1122 3.1011 3.1391 3.1825 3.1058

1.4539 1.6769 2.1870 2.5801 2.5814 2.9758 3.6108 3.6272 4.3379 3.7172 4.4658 3.0555 3.1599 3.5557 3.4363 3.1173 3.1489 3.2525 3.2433 3.2895 3.3441 3.1979

4. Results and discussion In this section we exhibit the results obtained in this study. First, the input data needed for the calculations are collected together in Table 1 while the ECT constants l; p; a; k and rWSE computed with the help of the input data are exhibited in Table 2. In some cases experimental values of vacancy formation energy, Ef1v , were used for calculations but where they are not available we resorted to using one third of the cohesive energy, Ec/3 which is a common practice in the literature [31–33]. Although, experimental values of Ef1v differs from Ec/3 by more than 10% in cases where they are available, but it suffices to cite the source of each of these values used for calculation. The parameter a is of course model dependent, as shown in Table 2. The determination of electron density parameter (a) followed the traditional practice which relies on the experimental

Table 1 Pure metal properties: Lattice constants a and c, and cohesive energy DE are from Ref. [30]; Bulk modulus B are from Refs. [25,30] and experimental monovacancy formation energies Ef1V for Mg, Ti, Zr, and Zn are from Ref. [31], while Ef1V for other metals are assumed to be approximately 1/3 of cohesive energy [31–33]. Element

DE (eV)

Ef1V (eV)

a (Å)

c (Å)

B (1011 Jm3 )

c/a

Be Mg Sc Y Ti Zr Hf Tc Re Ru Os Co Zn Cd Tl Gd Tb Dy Ho Er Tm Lu

3.32 1.52 3.90 4.37 4.85 6.25 6.44 6.85 8.03 6.74 8.17 4.39 1.35 1.16 1.88 4.14 4.05 3.04 3.14 3.29 2.42 4.43

1.11 0.8 1.30 1.46 1.55 1.75 2.15 2.28 2.68 2.25 2.72 1.46 0.42 0.39 0.63 1.38 1.35 1.01 1.05 1.1 0.81 1.48

2.27 3.21 3.31 3.65 2.95 3.32 3.19 2.74 2.76 2.71 2.74 2.51 2.66 2.98 3.46 3.63 3.60 3.59 3.58 3.56 3.54 3.50

3.59 5.21 5.27 5.73 4.68 5.15 5.05 4.40 4.46 4.28 4.32 4.07 4.95 5.62 5.52 5.78 5.70 5.65 5.62 5.59 5.56 5.55

1.144 0.369 0.435 0.423 1.097 0.973 1.106 2.97 3.715 3.152 4.18 1.948 0.804 0.621 0.382 0.405 0.399 0.411 0.397 0.468 0.397 0.411

1.582 1.623 1.592 1.570 1.586 1.594 1.583 1.606 1.616 1.579 1.577 1.622 1.861 1.886 1.595 1.592 1.583 1.574 1.570 1.570 1.571 1.586

Table 3 Variation of the electron density parameter, ai ðEfiv Þ, (i = l, a, h) for low (l), average (a) and high (h) values; Da = ah  al, while aa is the average value. Efiv is the vacancy formation energy.  ai Efiv Mg Ti

Model I Model II Model III

al

aa

ah

aa

ah

(0.80)

(0.64)

D a / aa (%)

al

(0.96)

(1.86)

(1.55)

(1.24)

D a/ a a (%)

1.6413

1.6937

1.7644

7.2

2.6242

2.7116

2.8348

7.74

1.6109

1.6571

1.7184

6.51

2.4195

2.4646

2.5228

4.46

1.6279

1.7184

1.7423

6.85

2.5219

2.5814

2.6606

5.42

Table 4 Equivalent crystal nearest-neighbour distance Rec (in Å) for hcp surface layer atoms. Element

Be Mg Sc Y Ti Zr Hf Tc Re Ru Os Co Zn Cd Tl Gd Tb Dy Ho Er Tm Lu

Rec Model I

Model II

Model III

2.6324 3.7383 3.8084 4.0875 3.2982 3.6093 3.5539 3.0298 3.0528 2.9389 2.9436 2.8453 2.8173 3.1417 3.8029 4.1029 4.0530 3.9521 3.9491 3.9065 3.8599 3.9779

2.7681 3.7710 3.9272 4.2782 3.4308 3.7214 3.6822 3.1024 3.0970 3.0811 3.0897 2.8743 3.3609 3.7571 3.9215 4.2165 4.1895 4.1287 4.1329 4.0931 4.0505 4.0990

2.6727 3.7468 3.8368 4.1342 3.3345 3.6380 3.5842 3.0496 3.0643 2.9804 2.9863 2.8530 2.9431 3.2696 3.8352 4.1269 4.0824 3.9969 3.9948 3.9547 3.9111 4.0017

528

E. Aghemenloh et al. / Computational Materials Science 46 (2009) 524–530

values of vacancy formation energies. To test the sensitivity of the three models to the computation of a, we assumed a ±20% variation in the experimental vacancy formation energy for two elements Mg and Ti. As listed in Table 3 for Mg, Model II gives a relative error of 6.51% as against 7.2% and 6.85% for Models I and III, respectively. In the case of Ti, Model II also gives a relative error of 4.46% as compared to 7.74% and 5.42% for Models I and III, respectively. These results point to the fact that Model II is least sensitive to variations in input values of vacancy formation energy. Thus, in a general situation Model II is recommended because it is likely to give a lower error in a. Rec , equivalent crystal nearest-neighbour distance, is a vital parameter in the ECT method and was calculated from Eqs. (12)– (13) for the three models considered in this work. The results are

shown in Table 4. The (0 0 1) surface energies for 22 hcp metals are presented in Table 5, and compared with the results from ab initio calculations [7,34], semi-empirical calculations [13,35] and experiment [17,36]. A careful study of Table 5 reveals that surface energy results of Model II are always greater than the results from Model I, while the results of Model III appears to be some average of Models I and II. This is clearly seen in Fig. 1. The same trend is expected from the model parameters (see, for instance, the parameter Rec in Table 3). Our ECT surface energy results for Zn and Cd in Model I, however, appear to be too low compared to the experimental values. We attribute this defect to the fact that of all the hcp metals considered in this study Zn and Cd provide the greatest deviation from the ideal c/a ratio of

Table 5 Surface energies, in Jm2 , for the hcp (0 0 1) crystal face. For comparison we have included the available ab initio, semi-empirical and experimental results. Element

Be Mg Sc Y Ti Zr Hf Tc Re Ru Os Co Zn Cd Tl Gd Tb Dy Ho Er Tm Lu a b c d e f

ECT (Present) Model I

Model II

Model III

1.966 0.673 0.964 0.806 1.467 1.387 1.565 2.613 3.065 2.351 2.666 2.148 0.238 0.173 0.417 0.777 0.756 0.570 0.585 0.623 0.466 0.864

2.555 0.704 1.114 1.008 1.866 1.674 1.885 3.110 3.415 3.362 3.928 2.297 1.057 0.755 0.516 0.881 0.878 0.732 0.749 0.818 0.632 0.974

2.269 0.688 1.043 0.921 1.620 1.539 1.744 2.869 3.241 2.902 3.370 2.221 0.622 0.448 0.471 0.834 0.825 0.663 0.680 0.736 0.561 0.925

Semi-empirical calculationswork

Ab initio calculations work

Experiment

1.650a 0.900a 1.355a 1.001a 1.962a 2.302a 2.041a

2.122c 0.642c 0.820c 0.680c 1.950c 1.530c 1.750c 2.800c 3.270c 3.320c 4.040c 3.180c

2.700e 0.760e 1.275e 1.125e 2.100e 2.000e 2.150e 3.150e 3.600e 3.050e 3.450e 2.550e 0.990e 0.740e 0.575e 1.110e 1.130e 1.140e 1.150e 1.170e <1.180e 1.225e

1.286b

0.864b 1.824b 1.854b

3.940a 3.191a

2.616b

3.056a

1.834d 0.792d 1.834d 1.506d 2.632d 2.260d 2.472d 3.691d 4.214d 3.928d 4.566d 2.775d 0.989d 0.593d 0.297d

0.653a 1.332a 1.515a 2.328a 2.180a 2.318a

1.628f 0.785f

1.989f 1.909f 2.193f 3.626f 3.043f 3.439f 2.522f 0.993f 0.762f 0.602f

Semi-empirical, Ref. [13]. Semi-empirical, Ref. [35]. Ab initio LMTO-ASA, Ref. [7]. FCD-GGA, Ref, [34]. Experimental, Ref. [17]. Experimental, Ref. [36].

4.50 Model I

4.00 Model II

2

Surface energy (J/m )

3.50

Model III

3.00 2.50 2.00 1.50 1.00 0.50 0.00

Be

Mg

Sc

Ti

Co

Zn

Y

Zr

Tc

Ru

Cd

Lu

Hf

Re

Os

Gd

Tb

Dy

Ho

Er

Tm

Fig. 1. Comparison of the calculated surface energies of hcp (0 0 1) surfaces for 3d, 4d and 5d metals. Model I open diamond, Model II open squares and Model III open triangles.

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E. Aghemenloh et al. / Computational Materials Science 46 (2009) 524–530 Experiment

4.50

Model II 4.00

Ab initio 3.50

Semi-empirical

2

Surface energy (J/m )

3.00

2.50

2.00

1.50

1.00

0.50

0.00

Sc

Ti

Co

Zn

Y

Zr

Tc

Ru

Cd

Lu

Hf

Re

Os

Gd

Tb

Dy

Ho

Er

Tm

Fig. 2. The calculated surface energies for the (0 0 1) surfaces for 3d, 4d, and 5d metals from Model II, open squares, compared to ab initio calculations, open triangles[7], experimental values, open diamonds,[17] and semi-empirical calculations, [13], stars.

pffiffiffiffiffiffiffiffi 8=3 = 1.633 (see Table 1). Correspondingly, the percentage deviation of the nearest neighbour distance Rnn1 from Rnn2 is greatest for Zn and Cd. Under these conditions the fundamental ECT equation (Eq. (8)) might require some modification to correctly handle Zn and Cd. Our surface energy results which take full account of the electron density change at the surface do not however include relaxation and reconstruction effects. As already mentioned in the text, the dominant contribution to the surface energy, within the ECT method comes from the electron density change effect. The actual contributions from the effects ignored in this study are currently being studied in the context of other defect properties where their contributions are likely to be more significant, and will be reported in the future. Our results for 3d, 4d, and 5d metals differ from reported experimental values by about 10% and from ab initio values by about 7%. But for Be, Sc, Co and Y computed results in this work are closer to ab initio values. Fig. 2 is a plot of results from Model II, ab inito and experimental values. The figure exhibits the characteristic parabolic variation with respect to valence electrons in accordance with Friedel [37]. However, our results as well as experimental values for the 4f metals showed a somewhat linear relationship. The deviation of experimental values for 4f metals exceeds 20%. To the best of our knowledge, there are no available surface energy values from ab initio calculations for this class of metals. From Table 5, our present results of Model II are in reasonable agreement with the results of semi-empirical calculations of Baskes [13] except largely for the 4f metals and those of Wang et al. [35] but for Be and Ru. It must be pointed out that experimental measurements of the surface energy are more commonly found for polycrystalline materials. In addition, a more consistent comparison of our ECT surface energies with ab initio results would necessitate us to use the same input data as employed in the ab initio work. Finally, the experimental monovacancy formation energy for the hcp metals (with the exception of Mg, Ti, Zr and Zn) has been assumed to be 1/3 of the cohesive energy, due to our inability to obtain the experimental information. These points notwithstanding, our results in Table 5 confirm the conclusions from earlier studies [19,21,22,24] that ECT surface energies are quite comparable to experimental results and first-principles calculations.

5. Conclusions We have successfully extended the ECT method to calculate the surface energies of the (0 0 1) surfaces of 22 hexagonal closepacked metals. Like other ECT works in the literature [19,21,22,24], the surface energies obtained in this work are in good agreement with the results of ab initio calculations and experiment. Model II is recommended as the method of choice in a general situation since its results are in better agreement with experimental results and other published works. Besides, Model II is least sensitive to the variation in experimental values of vacancy formation energy in the calculation of a. Furthermore, the calculated surface energies for 3d, 4d and 5d metals exhibit parabolic variation with valence which is surprisingly linear in the 4f metals. After this work was completed, the recent paper on ECT by Zypman and Ferrante [38] was brought to our notice. We are working earnestly to recast our work in this new ECT framework and the results will be announced as soon as it is completed. Acknowledgement We are grateful to Professors G. Bozzolo, J. Ferrante and T.L. Einstein for useful communication during the course of this work. We thank Prof. D.G. Pettifor and his secretary Mr. Kay Sims for sending us the experimental data used in this work. We remain grateful to an anonymous reviewer for bringing the recent work on ECT by Zypman and Ferrante to our attention. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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