Journal of the Physical Society of Japan Vol. 74, No. 6, June, 2005, pp. 1766–1771 #2005 The Physical Society of Japan

Orientation and Composition Dependences of the Surface Energy and Work Function Observed by First-Principles Calculation for the Mo–Hf System L. T. K ONG and B. X. LIU
Advanced Materials Laboratory, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China (Received February 1, 2005; accepted April 12, 2005)

First-principles calculations were employed firstly to identify the D03 , B2 (and/or L2a ) and D03 structures are relatively stable for the possible non-equilibrium Mo3 Hf, MoHf and MoHf3 alloys, respectively, and then to determine the surface energies and work functions of three low index crystalline surfaces for the identified Mo–Hf alloys as well as for the pure Mo and Hf metals. In the present calculations, two atomic configurations are considered for each crystalline surface, i.e., an ideal as-set surface and a relaxed surface, which corresponds to a more or less real situation. It is found that with increasing the openness of the crystalline surface, the surface energy increases while the work function decreases, suggesting an anisotropic behavior, and the influence of relaxation on these surface properties is enhanced considerably. Besides, based on an analytical analysis, an approximately linear dependence between the surface property and Mo–Hf alloy composition is found that the surface energy as well as the work function of the alloys decreases with increasing of the Hf concentration. KEYWORDS: Mo–Hf alloy, first principles calculation, surface energy, work function DOI: 10.1143/JPSJ.74.1766

1.

Introduction

The properties of a solid can be substantially modified as its physical dimensions are reduced to the nanoscale. Such quantum size effect is well known and has been studied extensively in connection with the modification of various properties of the solids.1) Much less familiar but of equal significance is the modification of the surface properties, among which the surface energy and the work function are the two most fundamental properties of a metallic surface. The determination of the surface energy and work function is therefore of great importance in the understanding of a wide range of surface phenomena. Most of the documented experimental surface energy data stem from the surface tension measurements in the liquid phase and then extrapolated to zero temperature.2) Although these data are the most comprehensive source of the surface energies, they are for the isotropic crystals and include some uncertainties of unknown magnitude. Hence, they do not show information as to the surface energy of a particular surface facet. Similarly, the documented work functions, e.g., compiled by Michaelson,3) are derived mainly from the measurements on the polycrystalline samples and in many cases have not been confirmed by measurements under an ultra-high-vacuum condition. In the past decades, there have been several theoretical studies of the surface properties, beginning with the pioneering jellium calculations by Lang and Kohn,4,5) and later by Perdew and co-workers with improved calculation methods.6,7) Since then, the first-principles approaches have also been proposed and employed to study the surface energies and work functions of metals.8–11) Concerning the work function anisotropy of the metals observed in experiments, Smoluchowski has ever proposed an empirical model in 1940s to attribute such anisotropy behavior to the


Corresponding author. E-mail: [email protected]

smoothing of the surface electron density.12) It is noted, however, that the forgoing calculations/theory mainly concentrated on the pure metals and/or some absorption issues, while little attention has ever been paid to the intermetallic alloys, especially the non-equilibrium alloys. We therefore dedicated, in the present study, to investigate the surface energies and the work functions in a selected binary metal system, i.e., the Mo–Hf system, by means of first-principles calculations. We firstly identify the structural stability of some possible non-equilibrium intermetallic compounds with relatively simple structures at three alloy compositions, i.e., Mo3 Hf, MoHf, and MoHf3 , and then study the surface energies and work functions of these Mo– Hf alloys as well as of the pure Mo and Hf metals for their low index single crystalline surfaces. Incidentally, one may find that in the equilibrium phase diagram of the Mo–Hf system, there stands a Mo2 Hf intermetallic compound, which has a rather complicated hexagonal C36 type structure for computing its surface property by currently available computational resources. In the present study, the calculated surface properties of the Mo–Hf alloys are therefore only compared with those of pure Mo and Hf metals. 2.

Calculation Methods

The calculations were performed with the Vienna ab initio simulation package (VASP)13–15) based on the density functional theory and the plane-wave basis pseudopotential method. The exchange-correlation energy was calculated within the generalized gradient approximation.16,17) The interaction between the valance electrons and ionic cores was represented by projector-augmented wave (PAW) pseudopotentials.18,19) The electronic wave functions were expanded in a plane-wave basis set with a kinetic energy cutoff equal to 280.7 eV. The k-space integrations were performed by employing the Monkhorst–Pack scheme.20) For calculations of the bulk properties, an 11  11  11 kmesh was used for the cubic lattices, and an 11  11  5

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L. T. KONG and B. X. LIU

mesh was employed for the hexagonal ones. In the calculations, five compositions (Mo, Mo3 Hf, MoHf, MoHf3 , and Hf) and some relatively simple structures at each composition were selected. For the details concerning the choosing of the alloy compositions as well as the possible structures, the readers are referred to some of our previous publications.21,22) In the calculations, the c=a ratios of the
NiAs, D019 , L60 , L10 , L2a and HCP structures are fully relaxed. The calculated energy–volume correlations of the bulk phases were then fitted to the Murnaghan equation of state (EOS).23) The obtained lattice constants were used as input parameters for the calculations of the surface properties. The supercell geometry and in most cases a Monkhorst– Pack mesh of 11  11  1 are adopted to calculate the surface properties. The crystalline surface was represented by a slab consisting of several atomic layers separated by a vacuum region. The calculations were fulfilled by three steps. First, a static calculation was carried out for the as-set supercell and its total energy Eslab-unrel and total local potential were obtained. Second, the positions of the atoms in the three topmost layers on one side of the slab were fully relaxed in the x-, y- and z-directions. Third, another round of static calculation was conducted for the relaxed configuration, and its total energy Eslab-rel and total local potential were acquired. The surface energy  means the surface excess free energy per unit area of a particular crystal facet, for the asset supercell, it can be expressed as: unrel ¼

1 ðEslab-unrel  nEbulk Þ 2A

ð1Þ

where Eslab-unrel is the total energy of the slab, Ebulk is the total energy per atom of the bulk crystal, n is the number of atoms in the supercell, and A is the cross-section area of the slab. After the relaxation, the total energy of the supercell was changed with an amount of:
Erel ¼ Eslab-rel  Eslab-unrel

ð2Þ

Because the energy change was induced by the relaxation of one side of the slab, the surface energy after the relaxation is therefore: rel ¼ unrel þ
Erel =A

ð3Þ

Apparently, the above defined formulae are generally reasonable and applicable for the pure metals. When dealing with alloys and/or compounds, however, eq. (1) would be questionable. Once a supercell is constructed to model a surface structure, the stoichiometry of its bulk phase could not always be reserved. For example, in the case of B2 MoHf, if five layers of atoms are considered for the surfaces of (100) and/or (200), the stoichiometry of the supercell would be 3 : 2 and/or 2 : 3, while no longer 1 : 1 as in its bulk phase. As a result, the term Ebulk in eq. (1) can not be explicitly accounted for and the resultant surface energy would therefore be questionable. One may argue that if six (or just even) layers are considered for the supercell, the stoichiometry could well be reserved, yet the two surfaces of the slab are no longer identical and their respective contributions to the surface energy could not be well defined either. Because of the limitation of the current computational

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ability, it is impossible and/or unnecessary to construct a supercell consisting of millions of atoms to eliminate the stoichiometry difference between the supercell and its bulk phase. We therefore propose a feasible solution to pursue further, i.e., to study an average surface energy of a specific orientation, instead of studying an explicit surface energy of a specific surface. For example, in the case of B2 MoHf, we simply calculate the surface energies of the (100) and (200) surfaces according to formulae (1)–(3) and their average values could be obtained, which are then denoted as the surface energy of the surfaces in the orientation of (100). The work function is defined as the minimum energy required for removing an electron from the interior of a solid to a position outside the solid. In the electronic structure calculations, such removing of the electron corresponds to the energy difference between the Fermi level and the vacuum level:  ¼ Vvac  EF

ð4Þ

which could be evaluated from the distribution of the total local potential along the z-direction of the supercell. Corresponding to an average surface energy, an average work function is also adopted for the orientations which consist of non-identical crystal surfaces. 3.

Results and Discussion

3.1 Structural stability of the Mo–Hf alloys Prior to studying the surface properties of the Mo–Hf alloys, the structural stability of the Mo–Hf alloys should firstly be identified, i.e., the energetically preferred structures at the respective compositions should be determined. Accordingly, the energy–volume correlations of the selected Mo–Hf phases were calculated and then fitted to the Murnaghan’s EOS. Table I lists the fitted cohesive properties. Inspecting Table I, one can retrieve the following information. First, the present calculations correctly reproduced the lattice constants and bulk moduli of the pure Mo and Hf metals. As is seen, the lattice constant and bulk  (3.195 A,  modulus of Mo (Hf) are calculated to be 3.152 A c=a ¼ 1:586) and 2.655 Mbar (1.089 Mbar), matching well  (3.19 A,  c=a ¼ 1:583) with the documented data of 3.15 A 24) and 2.73 Mbar (1.09 Mbar), respectively. Besides, the BCC Hf is indeed predicted to have a higher total energy than the HCP one, matching well with the fact that the equilibrium structure of Hf is HCP. Such agreements validated the reliability of the present calculation methods. Second, one finds that most of the alloy phases are predicted to be of positive heat of formation, except that at the composition of Mo3 Hf, the L60 and D03 alloy phases are calculated to have negative heat of formation. These results reflect the unique characteristics of the Mo–Hf system, i.e., although the system features a negative heat of formation according to Miedema’s theory,2) it only has one equilibrium compound at the Mo-rich side. Besides, the positive heats of formation are mostly calculated to be of small values, indicating that most of the considered alloy phases are possibly obtainable by using some appropriate non-equilibrium materials processing techniques. Third, one can easily learn about the structural stability of the possible nonequilibrium Mo–Hf alloys. At the composition of Mo3 Hf,

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L. T. KONG and B. X. LIU

Table I. Calculated lattice parameters (a, c=a), minimum total energy (E0 ), bulk modulus (B0 ), equilibrium atomic volume (V0 ), and heat of formation (
H) of Mo–Hf alloys. Available experimental data are given in parenthesis. a  (A)

Alloys Mo

BCC

c=a

3.152

E0 (eV/atom) 10:910

(3.15)a) Mo3 Hf

MoHf

MoHf3

Hf

L12

4.110

D019

2.804

A15

5.161

L60 D03

4.553 6.445

B0 (Mbar)

V0  3 /atom) (A


H (eV/atom)

2.655

15.664

0.000

(2.725)a) 10:207

1.959

17.353

0.433

10:269

2.027

17.345

0.371

10:509

2.041

17.184

0.131

0.711

10:659 10:676

2.134 2.140

16.773 16.733

0:020 0:036

1.818

B1

5.465

9:452

1.212

20.402

0.917


-NiAs L10

3.169 3.963

2.736 1.192

10:067 10:072

1.537 1.563

18.853 18.551

0.302 0.297

L2a

3.318

1.000

10:359

1.605

18.264

0.010

B2

3.318

10:359

1.659

18.258

0.010

L12

4.335

9:899

1.282

20.362

0.200

A15

5.448

9:914

1.271

20.217

0.185

D019

3.147

1.524

9:930

1.272

20.583

0.169

L60

4.821

0.715

9:973

1.319

20.021

0.126

D03

6.840

9:977

1.317

20.001

0.122

HCP

3.195

1.586

9:828

1.089

22.410

0.000

(3.19)a)

(1.583)a)

22.136

0.175

BCC

3.538

(1.09)a) 9:653

1.025

a) Ref. 24

the D03 phase has the lowest minimum total energy, while the L12 one is predicted to have the largest heat of formation among the five calculated structures. The same is also true for the composition of MoHf3 , except that the D03 MoHf3 has a positive heat of formation. For MoHf, one sees that all the alloy phases considered have a positive heat of formation, and among which, the B1 phase has the largest heat of formation. The L2a and the B2 phases share a same lowest energy level, as well as the same lattice constant. Actually, the two phases are identical at their relatively stable state, and while deviating from the relatively stable state, the c=a ratio of the L2a phase will no longer be 1.000. Incidentally, one may also notice that the predicted bulk moduli of the Mo–Hf alloys decrease with the increasing of the Hf concentration, which might be attributed to the fact that Hf has a smaller bulk modulus than Mo. 3.2

Surface energies and work functions of the Mo–Hf alloys Having identified the structural stability, we now turn to study the surface properties of the Mo–Hf alloys. Firstly, we take the Mo(110) surface as an example to seek for the suitable number of the atomic layers and the proper thickness of the vacuum region. Firstly, a series of supercells consisting of a slab of 3 to 10 atomic layers and a vacuum of  in thickness were computed. It turned out that the 10 A differences between the calculated surface energy and work function of the supercell with 5 atomic layers and those of the supercell with 10 atomic layers were already less than 1.0%, suggesting that a 5-layer slab may probably be sufficient for the calculations. In the following calculations, a 5-layer slab was therefore adopted in most cases, with an exception that 13 atomic layers are considered for the (111)

orientations. Having determined the suitable thickness of the slab, the surface energy and work function of the Mo(110) surface [consisting of five layers Mo(110)] were calculated with a series of vacuum thicknesses to find out the proper thickness of the vacuum. It is found that when the thickness  neither the surface of the vacuum region is greater than 10 A, energies nor the work functions show prominent variation,  is suggesting that a vacuum thickness of around 10 A adequate to eliminate the effect of the slab periodicity.  is adopted for all Consequently, a vacuum thickness of 10 A the following calculations as a compromise between the calculation precision and computational efficiency. Table II lists the calculated surface energies and work functions of three low index single crystal surfaces of pure BCC Mo and HCP Hf, together with some available experimental and/or theoretical values. It is seen that the calculated surface energies and work functions both show an apparent anisotropic behavior. For Mo, the close packed surface (110) has a smallest surface energy and a highest work function. The same is also true for Hf. Generally, the surface energy and the work function of a given metal depend on the crystallographic orientation, i.e., the surface facet, of the surface under consideration. If one assumes that the surface energy is proportional to the number of atomic bonds being cut in the formation of the surface, one finds for a bcc crystal that the surface energy  ð111Þ > ð100Þ > ð110Þ

ð5Þ

which has ever been verified by using the modified embedded-atom method.28) While the anisotropy of the work function is usually found to follow ð111Þ < ð100Þ < ð110Þ

ð6Þ

J. Phys. Soc. Jpn., Vol. 74, No. 6, June, 2005

L. T. KONG and B. X. LIU

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Table II. Surface energies and work functions calculated for the relaxed and (as-set) surfaces of Mo and Hf, together with some available theoretical (Theo.) and experimental (Expr.) values. Surface energy (J/m2 )

BCC Mo

HCP Hf

Work function (eV)

This work

Theo.

Expr.

This work

Theo.

Expr.

(100)

3.21(3.49)

3.52a)

3.00b)

4.01(4.06)

4.05a)

4.53d)

(110)

2.77(2.87)

3.14a)

4.59(4.64)

4.94a)

(111)

3.00(3.37)

(001)

1.67(1.80)

(110) (11 0)

1.80(1.95)

3.21(3.21)

1.54(2.10)

3.91(3.63)

1.75c)

2.15b)

4.58(4.47)

4.95d) 4.55d)

4.00(3.86) 4.26c)

3.90e)

a) Ref. 10; b) Average value, ref. 2; c) Ref. 25; d) Ref. 26; e) For polycrystalline, ref. 27

In short, one expects the surface energy to increase and the work function to decrease with increasing of the openness of the surface. In other words, the surface energy is expected to decrease and the work function to increase with increasing the atomic packing/density of the surface, which has ever been attributed to the smoothing of the surface electron density.12) In general, the calculated surface energy and work function of the pure metals Mo and Hf follow the above described trends. It is undoubted that the relaxation would substantially change the atomic structure and surface condition of the asset supercell, which is expected to subsequently result in some change in the surface energy and work function. Table III lists the calculated atomic relaxations of BCC Mo, which are given as the percentage change of the interlayer spacing for various orientations. Available theoretical and experimental results are also compiled for comparison. In the table i j denotes the percentage change for the interlayer distance between the ith and the jth layers. It is observed that, firstly, the dominant relaxations of the (100), (110) surfaces of BCC Mo are the contraction of the first interlayer spacing and that the relaxations of the deeper layers are rather small. Things are a little different for the (111) Table III. Layer relaxation of BCC Mo and B2 MoHf with respect to their respective as-set interlayer distance for various orientations. (In unit of %) Available data from other studies are also listed. (NCP, normconserving pseudopotential; GPT, generalized pseudopotential theory.
, the apex layer is of Mo atoms; y , the apex layer is of Hf atoms; z , distance change between Mo layers; x distance change between Hf layers.)
34

Orientation

Study

BCC Mo

(100)

This work

10:07

2.49

2:70

NCP a) GPT b)

11:1 10:2

2.3 1.3

1:7

9:5  3:0

1:0  2:0

Expt. (110)

(111)

This work NCP

a)

GPT Expt.

2:65

2.14

4:3

0:2

b)

5:8

1.8

d)

1:6  2:0

This work NCP

B2 MoHf

c)


12


23

Phase

a)

0.36 0:4

22:85

24:21

13.95

18:7

20:3

13.7

(100)


This work

13:87

1:71

(200)y

This work

14:24

6.79

0.41

(110)z (110)x

This work This work

9:11 1:04

3.75 0:85

0.05 0.55

a) Ref. 29; b) Ref. 30; c) Ref. 31; d) Ref. 32

1.77

surface, who exhibits substantial atomic displacements up to the third layer. Secondly, the relaxations of the surface also show clear anisotropy. As expected, the relaxation of the interlayer spacing is in proportional to the openness of the surface. The most significant relaxation occurs in the (111) surface, while only a small relaxation is observed in the close packed (110) surface. Thirdly, although in the present study only 5 atomic layers (except for the case of (111) surface, for which 13 atomic layers are considered.) are comprised in the slabs and only the top three layers are relaxed, the obtained layer relaxations are quite compatible with those reported by previous calculations which considered 11 atomic layers and all the 11 layers of slabs are relaxed and those from experimental observations. Referring again to Table II, one finds that the influence of relaxation on the surface energy and the work function is proportional to the magnitude of the relaxation, which also exhibits anisotropic behavior. For example, the surface energy change after relaxation for (110) surface of BCC Mo is only about 3%, while it is nearly 11% for (111) surface, in accordance with the fact that the atoms in the (111) surfaces changed their positions more significantly than in the (110) surfaces. Moreover, one may also notice that the presently predicted values are quite compatible with the previous theoretical and experimental observations. For example, the arithmetic average work function of the (001), (110) and (11 0) surfaces is 3.90 eV, which agrees well with the experimental polycrystalline value of 3.90 eV. For the possible intermetallic compounds, the predicted relative most stable bcc structured D03 Mo3 Hf, B2 MoHf, and D03 MoHf3 phases were selected to study the surface energy and work function. Three low index single crystalline surfaces of (100), (110), and (111) were studied. Table IV compiles the calculated results. From Table IV, one observes that the calculated surface energies and work functions for the low index single crystal surfaces of the Mo–Hf alloys also show the same trends as those observed in the pure metals of Mo and Hf. First, the surface energy of the close packed surface is lower than that of the open surface, while the work function shows a reverse ordering. For instance, the relaxed surface energy for the (110) and (111) surfaces of B2 MoHf are 1.96 and 2.37 J/m2 , respectively, while the relaxed work function for these two surfaces are 4.53 and 3.67 eV, respectively. Second, the influence of relaxations on the surface energy and the work function increases with the openness of the surface. For example, the changes of surface energy for the (110) and

L. T. KONG and B. X. LIU

Alloy Mo

BCC

Mo3 Hf D03

MoHf

B2

MoHf3 D03

Hf

BCC

Surface

Surface Energy (J/m2 )

Work Function (eV)

As-set

Relaxed

As-set

Relaxed

(100)

3.49

3.21

4.06

4.01

(110)

2.87

2.77

4.64

4.59

(111)

3.37

3.00

3.86

4.00

(100)

3.17

2.78

3.93

3.89

(110)

2.52

2.36

4.53

4.47

(111)

3.08

2.67

3.60

3.82

(100)

2.43

2.19

3.73

3.69

(110)

2.01

1.96

4.58

4.53

(111)

2.62

2.37

3.46

3.67

(100)

1.86

1.63

3.69

3.60

(110)

1.88

1.62

4.48

4.38

(111)

2.31

1.88

3.40

3.55

(100)

1.53

1.50

3.30

3.30

(110) (111)

1.68 2.02

1.60 1.68

4.31 3.39

4.31 3.35

4.60

3.00

2

Table IV. Surface energies and work functions calculated for low index single crystal surfaces of Mo–Hf alloys, together with those for pure BCC Mo and non-equilibrium BCC Hf.

Work Function (eV)

J. Phys. Soc. Jpn., Vol. 74, No. 6, June, 2005

Surface Energy (J/m )

1770

4.40 4.20

As-set Relaxed

4.00 3.80

As-set Relaxed 2.50

2.00

1.50 0.00

0.25

0.50

0.75

1.00

Concentration of Hafnium

Work Function (eV)

Fig. 1. Variation of the calculated surface energies and work functions of the close packed surfaces for the Mo–Hf alloys [(110) of BCC Mo, D03 Mo3 Hf, B2 MoHf, D03 MoHf3 ; and BCC Hf] against the concentration of Hafnium.

As-set Relaxed

4.00

3.80

3.60

3.40 2

Surface Energy (J/m )

(111) surfaces of B2 MoHf are about 2.5 and 9.5%, respectively, and the changes of work functions are about 1.1 and 6:1%, respectively. Referring to the distance change due to relaxation listed in Table III, one observes that the relaxations in the Mo–Hf intermetallic compounds are a little complicated. The B2 MoHf is also taken as an example. It is seen that for (100) and (200) surfaces, whose top most layers are occupied by Mo and Hf atoms, respectively, the resultant atomic relaxations are a little different. While for the (110) surface, whose top most layer is occupied by different sized Mo and Hf atoms, the changes of the interlayer distance of Mo atoms and Hf atoms caused by atomic relaxation are therefore different from each other. As a result, the atoms of different type in the same plane of the as-set supercell will no longer reside in the same plane. Nonetheless, it is also observed that the general tendency is still followed, that is, the relaxation is greater in the open surface than in its close packed counterpart. We now turn to study the influence of the Hf concentration on the surface energy and work function of the Mo– Hf alloys. The predicted relatively most stable D03 Mo3 Hf, B2 MoHf, D03 MoHf3 at their respective compositions and the pure BCC Mo and non-equilibrium BCC Hf were selected as models. Figure 1 illustrates the variations of the surface energy and the work function of the respective close packed surfaces [(110) of BCC Mo, D03 Mo3 Hf, B2 MoHf, D03 MoHf3 ; and BCC Hf] against the Hf concentration, while Fig. 2 shows that for the respective most open surfaces [(111) of BCC Mo, D03 Mo3 Hf, B2 MoHf, D03 MoHf3 ; and BCC Hf] among the surfaces considered. These figures intuitively illustrate the influence of relaxation on the surface energy and the work function. It is clearly observed that, as expected, the relaxed surface energies are lower than their respective as-set counterparts. The average difference between the relaxed and as-set surface energies is on the order of 7% for the close packed surfaces, while it is on the order of around 16% for the open surfaces. As for the work functions, it is also observed that

As-set Relaxed

3.00 2.50 2.00 1.50 0.00

0.25

0.50

0.75

1.00

Concentration of Hafnium Fig. 2. Variation of the calculated surface energies and work functions of the most open packed surfaces among the surface considered for the Mo– Hf alloys [(111) of BCC Mo, D03 Mo3 Hf, B2 MoHf, D03 MoHf3 ; and BCC Hf] against the concentration of Hf.

J. Phys. Soc. Jpn., Vol. 74, No. 6, June, 2005

L. T. KONG and B. X. LIU

the difference between the work functions of the as-set supercell and the relaxed one is much smaller in the close packed surfaces than in the open surfaces. These phenomena predicted by the present calculations are in accordance with the fact that a larger relaxation would take place in an open surface than that occurring in a close packed surface. Inspecting these figures, one sees clearly the dependence of the surface energy and work function on the Hf concentration. It is seen that generally, both the surface energy and the work function decrease with the increasing of the Hf concentration for respective surfaces, although the extent of variation is not the same. In particular, the work functions of the close packed surfaces only exhibit a weak decreasing tendency. More interestingly, it is observed that both the surface energy and the work function of a specific surface seem to decrease linearly with the increasing of the Hf concentration. In other words, for a specific surface of the BCC structured Mo–Hf intermetallic compound, there seems a linear correlation for its surface energy as well as its work function versus the Hf concentration, compðhklÞ ¼ MoðhklÞ þ ðHfðhklÞ  MoðhklÞ Þ  cHf compðhklÞ ¼ MoðhklÞ þ ðHfðhklÞ  MoðhklÞ Þ  cHf

ð7Þ ð8Þ

where compðhklÞ (compðhklÞ ), MoðhklÞ (MoðhklÞ ), and HfðhklÞ (HfðhklÞ ) are the surface energy (work function) of the (hkl) surface of the Mo–Hf intermetallic compound, the pure BCC Mo and BCC Hf, respectively. cHf is the atomic concentration of Hf in the compounds. Following the rule, it is readily understandable that the surface energy varies more significant with the increasing of the Hf concentration than the work function does for the close packed surfaces, as the difference of the surface energies between the (110) surfaces of BCC Mo and BCC Hf is much greater that that of their respective work functions. Examining the surface energies and work functions of the (100) orientations of the Mo–Hf alloys, one can find that eqs. (7) and (8) are also valid. Based on the above analysis, it is reasonably concluded that for a binary intermetallic compound, its surface energy and work function depend apparently on the orientation of the surface concerned and that these properties also depend on the composition of the surface. Provided the other conditions are the same (bulk structure, surface orientation, etc.), the surface energy and the work function could be estimated according to a linear correlation similar to eqs. (7) and (8). 4.

Concluding Remarks

We have shown that based on VASP, first-principles calculations are capable of identifying the structural stability of the possible non-equilibrium Mo–Hf alloys, and capable of determining the surface energy and work function of the

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Mo–Hf alloys as well as the pure Mo and Hf metals. For the pure metals and intermetallic Mo–Hf compounds, the present calculations indicate an apparent anisotropic behavior for the surface energy and work function and reveal a significant influence of relaxation on the surface properties and the influence is also anisotropic. Besides, the surface energy and work function for a specific orientation of the intermetallic Mo–Hf compounds seem to depend linearly on the atomic concentration of the constituents. Acknowledgement The authors are grateful to the financial support from the National Natural Science Foundation of China, The Ministry of Science and Technology of China (G20000672), and the Administration of Tsinghua University. 1) See, for example, A. D. Yoffe: Adv. Phys. 42 (1993) 173. 2) F. R. de Boer, R. Room, W. C. M. Mattens, A. R. Miedema and A. K. Niessen: Cohesion in Metals (North-Holland, Amsterdam, 1988). 3) H. B. Michaelson: J. Appl. Phys. 48 (1977) 4729. 4) N. D. Lang and W. Kohn: Phys. Rev. B 1 (1970) 4555. 5) N. D. Lang and W. Kohn: Phys. Rev. B 3 (1971) 1215. 6) R. Monnier and J. P. Perdew: Phys. Rev. B 17 (1978) 2595. 7) R. Monnier and J. P. Perdew: Phys. Rev. B 22 (1980) 1124. 8) K. M. Ho and K. P. Bohnen: Phys. Rev. Lett. 59 (1987) 1833. 9) H. L. Skriver and N. M. Rosengaard: Phys. Rev. B 43 (1991) 9538. 10) M. Methfessel, D. Hennig and M. Scheffler: Phys. Rev. B 46 (1992) 4816. 11) T. C. Leung, C. L. Kao, W. S. Su, Y. J. Feng and C. T. Chan: Phys. Rev. B 68 (2003) 195408. 12) R. Smoluchowski: Phys. Rev. 60 (1941) 661. 13) G. Kresse and J. Hafner: Phys. Rev. B 47 (1993) 558. 14) G. Kresse and J. Furthmu¨ller: Comput. Mater. Sci. 6 (1996) 15. 15) G. Kresse and J. Furthmu¨ller: Phys. Rev. B 54 (1996) 11169. 16) J. P. Perdew and A. Zunger: Phys. Rev. B 23 (1981) 5048. 17) J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais: Phys. Rev. B 46 (1992) 6671. 18) P. E. Blo¨chl: Phys. Rev. B 50 (1994) 17953. 19) G. Kresse and D. Joubert: Phys. Rev. B 59 (1999) 1758. 20) H. J. Monkhorst and J. D. Pack: Phys. Rev. B 13 (1976) 5188. 21) J. B. Liu, Z. F. Li, J. X. Zhang, B. X. Liu, G. Kresse and J. Hafner: Phys. Rev. B 64 (2001) 054102. 22) L. T. Kong, J. B. Liu and B. X. Liu: J. Phys. Soc. Jpn. 71 (2002) 141. 23) F. D. Murnaghan: Proc. Natl. Acad. Sci. U.S.A. 3 (1944) 244. 24) C. Kittel: Introduction to Solid State Physics (John Wiley & Sons, New York, 1996) 7th ed. 25) H. L. Skriver and N. M. Rosengaard: Phys. Rev. B 46 (1992) 7157. 26) S. Berge, P. O. Gartland and B. J. Slagsvold: Surf. Sci. 43 (1974) 275. 27) D. E. Eastman: Phys. Rev. B 2 (1970) 1. 28) J. M. Zhang, F. Ma and K. W. Xu: Surf. Interface Anal. 35 (2003) 662. 29) J. G. Che, C. T. Chan, W.-E. Jian and T. C. Leung: Phys. Rev. B 57 (1998) 1875. 30) J. A. Moriarty and R. Phillips: Phys. Rev. Lett. 66 (1991) 3036. 31) L. J. Clarke: Surf. Sci. 91 (1980) 131. 32) L. Morales de la Garza and L. J. Clarke: J. Phys. C 14 (1981) 5391.