- Email: [email protected]
The tight-binding ising model for surface segregation in binary alloys: formalism and applications
ELSEVIER
Materials Scienceand EngineeringB37 (1996) 127-130
The tight-binding Ising model for surface segregation in binary alloys: formalism and applications S. Ouannassera, H. Dreysska, L.T. Willeb$* aIPCMS-GEMME, bDepartment
Uniuersitd Lords Pasteur, 23 rue du Loess, 67037 Strasbourg, of Physics, Florida Atlantic University, Boca Raton, FL 33431,
France USA
Abstract We discuss the tight-binding Ising model as a tool for studying surface segregation in binary alloys. An extension of the formalism is outlined and applied to two interesting cases: NiCu(OO1) and MoW(001). Keywords:
Ising model; Surface segregation; Monte Carlo simulations; Metallic alloys
1. Introduction
The field of computational materials science is by now sufficiently mature that its predictions are becoming increasingly accurate and reliable. Empirical or phenomenological approaches have given way to calculations that are based on first principles, i.e. in which no adjustable or experimentally fitted parameters are used. Whereas advances in high-speed computing have given great impetus to the flourishing of this field by making many large scale studies feasible, equal credit must be given to recent breakthroughs in our understanding of the quantum mechanics and statistical physics of solids. In many cases classical thermodynamics is more than adequate to describe a material’s behaviour, but the underlying energetics must be firmly based in the electronic structure of the system. It is in this realm, where the quantum and classical domains meet, that rapid progress has been made. Molecular dynamics and Monte Carlo simulations can now be performed based on potentials extracted from solutions to SchrBdinger’s equation. It is the purpose of the present paper to elucidate this methodology and to employ it in a study of surface segregation for two interesting and contrasting metallic alloy systems, NiCu and MOW. * Corresponding author. Tel.: (407) 367 3379; fax: (407) 367 2662; e-mail: [email protected]. 0921-5107/96/$15.000 1996- Elsevier ScienceS.A. All rights reserved
The entichment of an alloy surface by one of the constituent species is an ubiquitous phenomenon, which follows as a direct consequence of the most basic principles of statistical physics, as anticipated by Gibbs over a century ago. Macroscopically one can argue that the element with the lowest surface tension should preferentially occupy the surface. Microscopically one can attempt to interpret segregation as being due to bond breaking: the component with the weakest bonds will tend to move to the vacuum interface. However, in the case of size mismatch this may be counteracted by elastic effects since simple entropic considerations show that in such cases the minority species will segregate. Thus one can already see that the issue is not as clear cut as one might think at first. Ultimately, the only satisfactory answer to this complex issue can be provided by atomistic calculations based on first principles. 2. Methodology
Ignoring displacive effects, the most natural description of an alloy surface appears to be in terms of a semi-infinite Ising model. This approach has indeed been successfully used, first in semi-empirical models and later in ab initio studies [l-6]. In most of the latter works the electronic structure is described by means of a tightibmdmg Hamlltoman, leading to tke tlght-bmding Ising model (TBIM) pioneered by TrCglia and
128
S. Ouannasser
et al. 1 Materials
Science
coworkers [I -31, although recently more sophisticated band structure methods have also been used [7,8]. The crucial statement that justifies this approach is a profound result by Sanchez et al. [9] which shows that the total energy of an alloy may be expanded in a complete orthonormal set of correlation functions. The expansion coefficients are called effective cluster interactions (ECI), starting from point energies (akin to a chemical potential), they include effective pair interactions (EPI) as well as higher order terms. The explicit expression for the point energies reads V, = :[,!?,A - EF], while for the EPI one has V,, = i[E,A,A + E;f - E,“;” - Eit]. Here p and 4 denote lattice sites and EL(EK) is the total energy of a system consisting of an atom of type 1 on site p (and one of type J at site q) embedded in the completely disordered medium. The total energy expansion may be performed over all 2” possible configurations of the system (assumed to have N sites), an approach known as the unrestricted scheme and leading to concentration independent ECI. Alternatively, one can perform the expansion in the restricted scheme, i.e. at fixed concentration, which yields concentration dependent ECI. The relation between the two classes of EC1 was elucidated by Asta et al. [lo] for the bulk and by Ouannasser [l I] for the surface. Specifically, one can show that the unrestricted EC1 for a surface can be obtained from the restricted EC1 by performing all calculations with an equiatomic composition in each layer parallel to the surface. The unrestricted scheme leads to a great simplification in the calculations. In the restricted scheme one needs to iterate the concentration profile and the EC1 to convergence simultaneously, since they depend on each other. In the unrestricted scheme one bypasses this self-consistency procedure completely. Next the question arises of how to determine the EC1 from electronic structure information. The work of TrCglia and coworkers is based on the so-called generalized perturbation method (GPM) in which the electronic grand potential is expanded in terms of fluctuations relative to some medium, usually taken to be described by the coherent potential approximation (CPA). An alternative approach is to start directly from the formal definition for the EC1 and to perform the averaging in real space. This direct configurational averaging (DCA) technique is ideally suited for implementation in conjunction with the tight-binding recursion method and avoids the complications associated with off-diagonal disorder. It has been employed in numerous studies, both for bulk [12] and for surfaces [4-61, and generally exhibits rapid convergence with the number of configurations over which the averaging is performed. Finally, a number of possible choices exist to solve the statistical physics problem. In the present work (as in most other theoretical studies) the main interest is in
and Engineering
B37 (1996)
127-130
temperatures above the bulk critical temperature T,. Interesting conceptual problems arise for T < T,, (see Refs. [2,5]), but their resolution will have to await further studies. For T > T,, the layers parallel to the surface may be taken to be completely disordered and the problem becomes essentially one-dimensional, the various planes being labeled by their index 11 @ = 1 being the surface plane). Under these assumptions the Bragg-Williams method is known to produce adequate results and has been widely used [l-6]. However, more sophisticated approaches such as the cluster variation method (CVM) or Monte Carlo simulations may also be used.
3. Results and discussion In the present work the (001) surfaces of two disordered alloy systems are studied, f.c.c. NiCu and b.c.c. MOW. The pure element tight-binding Hamiltonians are based on Papaconstantopoulos’s first principles values [13] for NiCu and on Harrison’s prescription [14] for MOW. As usual, local charge neutrality is imposed by a rigid shift along the Fermi levels and the Shiba approximation is invoked to determine the AB hopping integrals. The Hamiltonian is diagonalized by means of the recursion method with ten levels in the continued fraction and a Beer-Pettifor quadratic terminator. The DCA is performed in the unrestricted scheme over 30 configurations to determine the EC1 with convergence to within 1%. In all cases, clusters beyond the pair produced negligible contributions. Consequently, only the point energies and EPI are considered here. Segregation profiles were determined by Monte Carlo simulations on slabs consisting of 12 layers, the bottom layer being kept at the bulk concentration but providing a reservoir of atoms that may diffuse into the other layers. The layers contained 32 x 64 sites for f.c.c. NiCu and 32 x 32 sites for b.c.c. MOW. The first 10 000 Monte Carlo steps (MCS) were rejected, to allow the system to equilibrate, and thermodynamic averages were performed over the next 10 000 MCS. The NiCu system is one of the most frequently studied (see Refs. [5,7,8] and references therein) both theoretically and experimentally. While general consensus exists that Cu segregates strongly at all concentrations, surfaces, and temperatures, much disagreement exists about the approach to the bulk concentration. In a previous study [5] with an approach similar to the present methodology, but in the restricted scheme, the NiCu system was investigated. Strong Cu segregation was found, with a monotonic approach to the bulk limit. The reason for this behavior was the negative sign of the point energies as well as the EPI. Subsequently however, Ruban et a1.[8], using linearmuffin-tin-orbital (LMTO) calculations and the Con-
S. Ouannasser
et al. 1 Materials
Science
nolly-Williams method obtained a depletion in Cu in the second layer due to a positive point energy V,. Here, we revisit this problem using the more refined approach outlined in Section 2. Once again we find all point energies to be negative, which will lead to a monotonic profile with strong Cu enrichment at the surface. To confirm this, Fig. 1 shows in solid line the segregation profile at 750 K for c = 0.25, calculated by Monte Carlo simulation, which shows the expected behavior. Certainly, it is very encouraging that an improved treatment qf the same Hamiltonian as in Ref. [5] yields essentially identical results. Conversely, the discrepancy between the TBIM results and those of Ruban et al. remains unsettling. It is possible, as suggested by Ruban et al., that the approximation of the total energy as a sum of one-electron energies inherent in the TBIM is responsible for the modified sign of V,. However, the ConnollyWilliams method itself may be questioned. For example, it is not based on a rigorous (infinite) expansion [9], but instead fits exactly a series of total energy calculations to a finite expansion. As a consequence, the physical meaning of the expansion coefficients is unclear. We note that the triplet and quadruplet interactions in Ref. [8] are considerable, whereas they are negligible in the TBIM approach. The reason for the latter effect is the renormalization of higher-order terms [lo,1 11. However, Ruban et al. also present a direct calculation of surface energies showing that Ni prefers to occupy the second layer. Clearly, the issue deserves further study. The MoW(OO1) surface has been studied experimentally by electron microprobe and Auger electron spectroscopy [15]. Strong MO segregation to the surface plane was found, but the subsurface composition was not determined. Our calculated EC1 show an oscillation in sign of the relative point energies and small,
T=
1.00
0.75
fi8
0.50
\\ \ \\
‘\
\
\
\
and Engineering
3
5
layer Fig. 1. Segregation profile for Ni,Cu, -,(OOl) surface at T= 750 K, c = 0.25 (-) and Mo,W, -,(OOl) surface at T= 500 K, c = 0.50 (---).
129
127-130
but negative, EPI. The fact that V,- V, is large and negative leads to strong MO segregation, in agreement with experiment. More interestingly, one notes a competition between the oscillation in sign of Q-V,, which will tend to produce an oscillating profile, and the negative EPI, which will push towards a monotonic profile. Monte Carlo simulations for c = 0.50 at 500 K, produce the dependence shown in Fig. 1 in dashed line: the weak EPI are overwhelmed by the much stronger point energies leading to an oscillating profile. This is a very interesting phenomenon: the negative EPI imply phase separation in the bulk (with very low critical temperature, 150 K), but the segregation is oscillatory. Few systems exhibit this apparently conflicting behaviour. It cannot be understood by theories that do not take properly into account the point energies in the subsurface layers. It is also worth pointing out that the low critical temperature is outside the reach of experiment, but that a prior theoretical study [16], based on a clusterBethe-lattice study with the CVM, yielded a of 80 K. In view of the approximation inherent in replacing the actual lattice with a Bethe-tree and the other numerical uncertainties in both methods, this must be considered quite compelling evidence for the presence of phase separation at very low temperatures. Experimental observation of this phenomenon (perhaps via indirect evidence) and measurement of MO depletion in the subsurface layers would be strong confirmation of the present calculations.
T=
T,z
T,
4. Conclusions Monte Carlo simulations of a TBIM for NiCu(OO1) and MoW(OO1) have produced a monotonic segregation profile for the former and an oscillating profile for the latter. In both cases, the point energies are the dominant terms in determining the segregating species and the subsurface composition. In the MOW case they overwhelm the clustering tendency of the EPI. The interactions have been determined in an unrestricted averaging scheme,’ which yields renormalized concentration independent ECI. The expansion of the internal energy of the system is formally justified and rapid convergence is numerically established. The main drawback of this formalism is the use of the tight-binding approximation, which specifically entails that the total energy is computed as a sum of one-electron energies. This approach is potentially severe, especially at a surface where charge redistribution may be pronounced. Nevertheless, numerous studies with the TBIM have shown it to be remarkably accurate and versatile. On the plus side, this formalism is based on a rigorous expansion in which the coefficients may be given a precise, physical mean-
\\ \\ /’/’+---w--e ______/’ ‘kf.
0.25
B37 (1996)
130
S. Ouannasser et al. / Materials Science and Engineering B37 (1996) 127-130
ing. This is in contrast to the Connolly-Williams method in which the coefficients are merely fitting parameters which have all sorts of effects lumped together. Other advantages of the TBIM include its realspace formulation, a more natural setting for problems with broken symmetry than reciprocal space methods, and the use of DCA, which avoids the limitations of the CPA. Thus, the TBIM’s lack of rigor in certain aspects of the electronic structure appears to be more than offset by a higher level of reliability in the treatment of the disorder. The result is a balanced, powerful tool well suited for a wide range of problems. Certainly, compared with the phenomenological theories of a decade ago, progress on the surface segregation problem has been most impressive.
Etrrophys. Mt., 7(1988) 111G. Treglia,B.LegrandandF. Ducastelle, 575.
319. 121G. Treglia,B.LegrandandP.Maugain,Sut$Sci.,225(1990) B. Legrandand G. TrCglia,Prog. Theor. P/y. [31 F. Ducastelle, Suppl,. 101 (1990)159. L.T. WilleandD. deFontaine,SolidState Commtrr~., [41 H. Dreysse, 78 (1991)355. L.T. WilleandD. deFontaine,Phys. Ret<. B, 47(1993) [51 H. Dreysse, 62.
J. Eugene,H. Dreysse, C. Wolvertonand D. de 161S. Ouannasser, Fontaine,Surf. Sci., 307 (1994)826. V. Drchal,J. KudrnovskjlandP.Weinberger, P/+x [71 A. Pasturel, Rev. B, 48 (1993)2704.
PI A.V. Ruban,LA. Abrikosov,D. Ya. Kats,D. Gorelikov,K.W. Jacobsen andH.L. Skriver,Pltys.Ret). B, 49 (1994)11383.
F. Ducastelle andD. Gmtias,Pltysica A, 128(1984) 191J.M. Sanchez, 334.
Phys. Rev. [lOI M. Asta,C.Wolverton,D. deFontaineandH. Dreysse,
Acknowledgements This work was supported by grant No. CRG.940331 from the North Atlantic Treaty Organization. The authors gratefully acknowledge support by Florida State University through the allocation of supercomputer resources.
B, 44 (1991)4907. T/&e, Universitede Nancy-I, 1994. El11S. Ouannasser, A. Berera,L.T. WilleandD. deFontaine,Phys. Reu, WI H. Dreysse, B, 39 (1989)2442. Handbook of the Bared Structure of u31 D.A. Papaconstantopoulos, Elemental Solids, Plenum,NewYork, 1986. v41 W.A. Harrison,Electronic Structwe and the Properties of Solids: The Physics of the Chemical Bond, Freeman, SanFrancisco,CA, 1980. andS.A.Petrone,J. ~‘CIC. Sci. Tech&, 18(1981) 529. [I51 P.T.Dawson andA. Pasturcl,J. Phys. F, 18 (1988) 903. V61 C. Colinet,A. Bessoud
Materials Scienceand EngineeringB37 (1996) 127-130
The tight-binding Ising model for surface segregation in binary alloys: formalism and applications S. Ouannassera, H. Dreysska, L.T. Willeb$* aIPCMS-GEMME, bDepartment
Uniuersitd Lords Pasteur, 23 rue du Loess, 67037 Strasbourg, of Physics, Florida Atlantic University, Boca Raton, FL 33431,
France USA
Abstract We discuss the tight-binding Ising model as a tool for studying surface segregation in binary alloys. An extension of the formalism is outlined and applied to two interesting cases: NiCu(OO1) and MoW(001). Keywords:
Ising model; Surface segregation; Monte Carlo simulations; Metallic alloys
1. Introduction
The field of computational materials science is by now sufficiently mature that its predictions are becoming increasingly accurate and reliable. Empirical or phenomenological approaches have given way to calculations that are based on first principles, i.e. in which no adjustable or experimentally fitted parameters are used. Whereas advances in high-speed computing have given great impetus to the flourishing of this field by making many large scale studies feasible, equal credit must be given to recent breakthroughs in our understanding of the quantum mechanics and statistical physics of solids. In many cases classical thermodynamics is more than adequate to describe a material’s behaviour, but the underlying energetics must be firmly based in the electronic structure of the system. It is in this realm, where the quantum and classical domains meet, that rapid progress has been made. Molecular dynamics and Monte Carlo simulations can now be performed based on potentials extracted from solutions to SchrBdinger’s equation. It is the purpose of the present paper to elucidate this methodology and to employ it in a study of surface segregation for two interesting and contrasting metallic alloy systems, NiCu and MOW. * Corresponding author. Tel.: (407) 367 3379; fax: (407) 367 2662; e-mail: [email protected]. 0921-5107/96/$15.000 1996- Elsevier ScienceS.A. All rights reserved
The entichment of an alloy surface by one of the constituent species is an ubiquitous phenomenon, which follows as a direct consequence of the most basic principles of statistical physics, as anticipated by Gibbs over a century ago. Macroscopically one can argue that the element with the lowest surface tension should preferentially occupy the surface. Microscopically one can attempt to interpret segregation as being due to bond breaking: the component with the weakest bonds will tend to move to the vacuum interface. However, in the case of size mismatch this may be counteracted by elastic effects since simple entropic considerations show that in such cases the minority species will segregate. Thus one can already see that the issue is not as clear cut as one might think at first. Ultimately, the only satisfactory answer to this complex issue can be provided by atomistic calculations based on first principles. 2. Methodology
Ignoring displacive effects, the most natural description of an alloy surface appears to be in terms of a semi-infinite Ising model. This approach has indeed been successfully used, first in semi-empirical models and later in ab initio studies [l-6]. In most of the latter works the electronic structure is described by means of a tightibmdmg Hamlltoman, leading to tke tlght-bmding Ising model (TBIM) pioneered by TrCglia and
128
S. Ouannasser
et al. 1 Materials
Science
coworkers [I -31, although recently more sophisticated band structure methods have also been used [7,8]. The crucial statement that justifies this approach is a profound result by Sanchez et al. [9] which shows that the total energy of an alloy may be expanded in a complete orthonormal set of correlation functions. The expansion coefficients are called effective cluster interactions (ECI), starting from point energies (akin to a chemical potential), they include effective pair interactions (EPI) as well as higher order terms. The explicit expression for the point energies reads V, = :[,!?,A - EF], while for the EPI one has V,, = i[E,A,A + E;f - E,“;” - Eit]. Here p and 4 denote lattice sites and EL(EK) is the total energy of a system consisting of an atom of type 1 on site p (and one of type J at site q) embedded in the completely disordered medium. The total energy expansion may be performed over all 2” possible configurations of the system (assumed to have N sites), an approach known as the unrestricted scheme and leading to concentration independent ECI. Alternatively, one can perform the expansion in the restricted scheme, i.e. at fixed concentration, which yields concentration dependent ECI. The relation between the two classes of EC1 was elucidated by Asta et al. [lo] for the bulk and by Ouannasser [l I] for the surface. Specifically, one can show that the unrestricted EC1 for a surface can be obtained from the restricted EC1 by performing all calculations with an equiatomic composition in each layer parallel to the surface. The unrestricted scheme leads to a great simplification in the calculations. In the restricted scheme one needs to iterate the concentration profile and the EC1 to convergence simultaneously, since they depend on each other. In the unrestricted scheme one bypasses this self-consistency procedure completely. Next the question arises of how to determine the EC1 from electronic structure information. The work of TrCglia and coworkers is based on the so-called generalized perturbation method (GPM) in which the electronic grand potential is expanded in terms of fluctuations relative to some medium, usually taken to be described by the coherent potential approximation (CPA). An alternative approach is to start directly from the formal definition for the EC1 and to perform the averaging in real space. This direct configurational averaging (DCA) technique is ideally suited for implementation in conjunction with the tight-binding recursion method and avoids the complications associated with off-diagonal disorder. It has been employed in numerous studies, both for bulk [12] and for surfaces [4-61, and generally exhibits rapid convergence with the number of configurations over which the averaging is performed. Finally, a number of possible choices exist to solve the statistical physics problem. In the present work (as in most other theoretical studies) the main interest is in
and Engineering
B37 (1996)
127-130
temperatures above the bulk critical temperature T,. Interesting conceptual problems arise for T < T,, (see Refs. [2,5]), but their resolution will have to await further studies. For T > T,, the layers parallel to the surface may be taken to be completely disordered and the problem becomes essentially one-dimensional, the various planes being labeled by their index 11 @ = 1 being the surface plane). Under these assumptions the Bragg-Williams method is known to produce adequate results and has been widely used [l-6]. However, more sophisticated approaches such as the cluster variation method (CVM) or Monte Carlo simulations may also be used.
3. Results and discussion In the present work the (001) surfaces of two disordered alloy systems are studied, f.c.c. NiCu and b.c.c. MOW. The pure element tight-binding Hamiltonians are based on Papaconstantopoulos’s first principles values [13] for NiCu and on Harrison’s prescription [14] for MOW. As usual, local charge neutrality is imposed by a rigid shift along the Fermi levels and the Shiba approximation is invoked to determine the AB hopping integrals. The Hamiltonian is diagonalized by means of the recursion method with ten levels in the continued fraction and a Beer-Pettifor quadratic terminator. The DCA is performed in the unrestricted scheme over 30 configurations to determine the EC1 with convergence to within 1%. In all cases, clusters beyond the pair produced negligible contributions. Consequently, only the point energies and EPI are considered here. Segregation profiles were determined by Monte Carlo simulations on slabs consisting of 12 layers, the bottom layer being kept at the bulk concentration but providing a reservoir of atoms that may diffuse into the other layers. The layers contained 32 x 64 sites for f.c.c. NiCu and 32 x 32 sites for b.c.c. MOW. The first 10 000 Monte Carlo steps (MCS) were rejected, to allow the system to equilibrate, and thermodynamic averages were performed over the next 10 000 MCS. The NiCu system is one of the most frequently studied (see Refs. [5,7,8] and references therein) both theoretically and experimentally. While general consensus exists that Cu segregates strongly at all concentrations, surfaces, and temperatures, much disagreement exists about the approach to the bulk concentration. In a previous study [5] with an approach similar to the present methodology, but in the restricted scheme, the NiCu system was investigated. Strong Cu segregation was found, with a monotonic approach to the bulk limit. The reason for this behavior was the negative sign of the point energies as well as the EPI. Subsequently however, Ruban et a1.[8], using linearmuffin-tin-orbital (LMTO) calculations and the Con-
S. Ouannasser
et al. 1 Materials
Science
nolly-Williams method obtained a depletion in Cu in the second layer due to a positive point energy V,. Here, we revisit this problem using the more refined approach outlined in Section 2. Once again we find all point energies to be negative, which will lead to a monotonic profile with strong Cu enrichment at the surface. To confirm this, Fig. 1 shows in solid line the segregation profile at 750 K for c = 0.25, calculated by Monte Carlo simulation, which shows the expected behavior. Certainly, it is very encouraging that an improved treatment qf the same Hamiltonian as in Ref. [5] yields essentially identical results. Conversely, the discrepancy between the TBIM results and those of Ruban et al. remains unsettling. It is possible, as suggested by Ruban et al., that the approximation of the total energy as a sum of one-electron energies inherent in the TBIM is responsible for the modified sign of V,. However, the ConnollyWilliams method itself may be questioned. For example, it is not based on a rigorous (infinite) expansion [9], but instead fits exactly a series of total energy calculations to a finite expansion. As a consequence, the physical meaning of the expansion coefficients is unclear. We note that the triplet and quadruplet interactions in Ref. [8] are considerable, whereas they are negligible in the TBIM approach. The reason for the latter effect is the renormalization of higher-order terms [lo,1 11. However, Ruban et al. also present a direct calculation of surface energies showing that Ni prefers to occupy the second layer. Clearly, the issue deserves further study. The MoW(OO1) surface has been studied experimentally by electron microprobe and Auger electron spectroscopy [15]. Strong MO segregation to the surface plane was found, but the subsurface composition was not determined. Our calculated EC1 show an oscillation in sign of the relative point energies and small,
T=
1.00
0.75
fi8
0.50
\\ \ \\
‘\
\
\
\
and Engineering
3
5
layer Fig. 1. Segregation profile for Ni,Cu, -,(OOl) surface at T= 750 K, c = 0.25 (-) and Mo,W, -,(OOl) surface at T= 500 K, c = 0.50 (---).
129
127-130
but negative, EPI. The fact that V,- V, is large and negative leads to strong MO segregation, in agreement with experiment. More interestingly, one notes a competition between the oscillation in sign of Q-V,, which will tend to produce an oscillating profile, and the negative EPI, which will push towards a monotonic profile. Monte Carlo simulations for c = 0.50 at 500 K, produce the dependence shown in Fig. 1 in dashed line: the weak EPI are overwhelmed by the much stronger point energies leading to an oscillating profile. This is a very interesting phenomenon: the negative EPI imply phase separation in the bulk (with very low critical temperature, 150 K), but the segregation is oscillatory. Few systems exhibit this apparently conflicting behaviour. It cannot be understood by theories that do not take properly into account the point energies in the subsurface layers. It is also worth pointing out that the low critical temperature is outside the reach of experiment, but that a prior theoretical study [16], based on a clusterBethe-lattice study with the CVM, yielded a of 80 K. In view of the approximation inherent in replacing the actual lattice with a Bethe-tree and the other numerical uncertainties in both methods, this must be considered quite compelling evidence for the presence of phase separation at very low temperatures. Experimental observation of this phenomenon (perhaps via indirect evidence) and measurement of MO depletion in the subsurface layers would be strong confirmation of the present calculations.
T=
T,z
T,
4. Conclusions Monte Carlo simulations of a TBIM for NiCu(OO1) and MoW(OO1) have produced a monotonic segregation profile for the former and an oscillating profile for the latter. In both cases, the point energies are the dominant terms in determining the segregating species and the subsurface composition. In the MOW case they overwhelm the clustering tendency of the EPI. The interactions have been determined in an unrestricted averaging scheme,’ which yields renormalized concentration independent ECI. The expansion of the internal energy of the system is formally justified and rapid convergence is numerically established. The main drawback of this formalism is the use of the tight-binding approximation, which specifically entails that the total energy is computed as a sum of one-electron energies. This approach is potentially severe, especially at a surface where charge redistribution may be pronounced. Nevertheless, numerous studies with the TBIM have shown it to be remarkably accurate and versatile. On the plus side, this formalism is based on a rigorous expansion in which the coefficients may be given a precise, physical mean-
\\ \\ /’/’+---w--e ______/’ ‘kf.
0.25
B37 (1996)
130
S. Ouannasser et al. / Materials Science and Engineering B37 (1996) 127-130
ing. This is in contrast to the Connolly-Williams method in which the coefficients are merely fitting parameters which have all sorts of effects lumped together. Other advantages of the TBIM include its realspace formulation, a more natural setting for problems with broken symmetry than reciprocal space methods, and the use of DCA, which avoids the limitations of the CPA. Thus, the TBIM’s lack of rigor in certain aspects of the electronic structure appears to be more than offset by a higher level of reliability in the treatment of the disorder. The result is a balanced, powerful tool well suited for a wide range of problems. Certainly, compared with the phenomenological theories of a decade ago, progress on the surface segregation problem has been most impressive.
Etrrophys. Mt., 7(1988) 111G. Treglia,B.LegrandandF. Ducastelle, 575.
319. 121G. Treglia,B.LegrandandP.Maugain,Sut$Sci.,225(1990) B. Legrandand G. TrCglia,Prog. Theor. P/y. [31 F. Ducastelle, Suppl,. 101 (1990)159. L.T. WilleandD. deFontaine,SolidState Commtrr~., [41 H. Dreysse, 78 (1991)355. L.T. WilleandD. deFontaine,Phys. Ret<. B, 47(1993) [51 H. Dreysse, 62.
J. Eugene,H. Dreysse, C. Wolvertonand D. de 161S. Ouannasser, Fontaine,Surf. Sci., 307 (1994)826. V. Drchal,J. KudrnovskjlandP.Weinberger, P/+x [71 A. Pasturel, Rev. B, 48 (1993)2704.
PI A.V. Ruban,LA. Abrikosov,D. Ya. Kats,D. Gorelikov,K.W. Jacobsen andH.L. Skriver,Pltys.Ret). B, 49 (1994)11383.
F. Ducastelle andD. Gmtias,Pltysica A, 128(1984) 191J.M. Sanchez, 334.
Phys. Rev. [lOI M. Asta,C.Wolverton,D. deFontaineandH. Dreysse,
Acknowledgements This work was supported by grant No. CRG.940331 from the North Atlantic Treaty Organization. The authors gratefully acknowledge support by Florida State University through the allocation of supercomputer resources.
B, 44 (1991)4907. T/&e, Universitede Nancy-I, 1994. El11S. Ouannasser, A. Berera,L.T. WilleandD. deFontaine,Phys. Reu, WI H. Dreysse, B, 39 (1989)2442. Handbook of the Bared Structure of u31 D.A. Papaconstantopoulos, Elemental Solids, Plenum,NewYork, 1986. v41 W.A. Harrison,Electronic Structwe and the Properties of Solids: The Physics of the Chemical Bond, Freeman, SanFrancisco,CA, 1980. andS.A.Petrone,J. ~‘CIC. Sci. Tech&, 18(1981) 529. [I51 P.T.Dawson andA. Pasturcl,J. Phys. F, 18 (1988) 903. V61 C. Colinet,A. Bessoud