14
PHYSICAL REVIE% 8 VGLUME 47, NUMBER 1 1 JANUARY 1993-I Self-consistent ordering energies and segregation profiles at binary-alloy surfaces H. Dreysse Laboratoire de Physique du Solide, Uni Uersite de Nancy I, 54506 Vandoeuure-les-Nancy, France L. T. Wille Department of Physics, Florida Atlantic University, Boca Raton, Florida 33431 D. de Fontaine Department of Materials Science and Mineral Engineering, University of California, Berkeley, California 94720 (Received 5 August 1992) We discuss the application of the direct configurational averaging method to the calculation of the thermodynamic properties of alloy surfaces. We analyze critically a number of approaches previously proposed to determine the parameters in different Ising models. We investigate the physical meaning of various interaction energies occurring in those models and compare our technique to phenomenological approaches. The formalism is applied in electronically self-consistent mean-field calculations of the segregation profile at the surface of Ni-Cu and Rh-Ti alloys at temperatures above the bulk disordering temperature. I. INTRODUCTION In recent years major progress has been made towards a longstanding goal in theoretical materials science: the calculation from first principles of alloy phase diagrams and the associated thermodynamic quantities. ' This has necessitated important breakthroughs in ab initio electronic structure calculations, as well as an increased understanding of the applicable tools of statistical phys- ics. Even so, the task is computationally very demanding and has only become feasible through improved comput- er hardware. Although many exciting questions remain, several groups using a range of approaches have now presented first-principles phase diagrams for a variety of systems including metals, semiconductors, ' and su- perconducting oxides. " In contrast, the determination of thermodynamic properties near surfaces or other extend- ed defects has not yet attained the same level of sophisti- cation. It is the purpose of the present paper to unify and review existing formalisms and to present results ob- tained without any adjustable parameters using the direct configurational averaging technique with the tight- binding approximation. Alloy-phase-diagram calculations involve two compu- tationally demanding tasks: an electronic structure cal- culation to determine quantum-mechanical interaction energies and a statistical physics calculation involving the minimization of a free-energy functional constructed us- ing those energies. Both steps must be performed with very high numerical precision. The band-structure deter- mination must be very accurate because interaction ener- gies are formally obtained as very small differences of large cohesive energies. The statistical mechanics prob- lem must be handled at an appropriate level of approxi- mation, by including a sufhcient number of correlation functions in a cluster variation framework' or by careful considerations of equilibration times and finite-size effects in a Monte Carlo simulation. ' The same rigorous demands hold in the case of order-disorder phenomena near surfaces and interfaces, where they are exacerbated because of charge redistributions and the interplay be- tween ordering and electronic structure. In the last century it was recognized by Gibbs on theoretical grounds and it has been confirmed by many experiments since then, that in thermodynamic equilibri- um an alloy surface (or interface) will tend to become en- riched by one of the components. This is the phenomenon of surface (or interface) segregation which has been the subject of intense study over the years. ' A large number of technologically important phenomena occur at or near the solid surface, or near internal inter- faces such as grain boundaries. ' Examples include ca- talysis, corrosion, adsorption, embrittlement, friction, deposition and growth, etc. Thus, a detailed understand- ing of the segregation behavior eventually leading perhaps to its control by doping or heat treatment could have many important applications. On the other hand, from a purely theoretical point of view phase transitions in reduced dimensionalities continue to be of great in- terest and the nature of the different types of segregation behavior has proven to be extremely rich, with the pos- sibility of segregation-induced ordering and wetting, sur- face sandwich formation, reconstruction, etc. Because of the great success in the description of bulk alloys, Ising models have been used extensively to model surface alloy- ing effects as well, although care must be taken in cases where the size difference of the constituent atoms is large and elastic effects become important. Phenomenological models21-24 were able to describe qualitatively the ob- served segregation in various systems in terms of a trun- cated Ising model with the segregating component being determined by the bond-breaking energy (or difference in surface tensions) and the approach to the bulk limit (monotonic or oscillating) by the sign of the ordering pair 1993 The American Physical Society

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PHYSICAL REVIE% 8 VGLUME 47, NUMBER 1 1 JANUARY 1993-I

Self-consistent ordering energies and segregation profiles at binary-alloy surfaces

H. DreysseLaboratoire de Physique du Solide, Uni Uersite de Nancy I, 54506 Vandoeuure-les-Nancy, France

L. T. WilleDepartment of Physics, Florida Atlantic University, Boca Raton, Florida 33431

D. de FontaineDepartment ofMaterials Science and Mineral Engineering, University of California, Berkeley, California 94720

(Received 5 August 1992)

We discuss the application of the direct configurational averaging method to the calculation of thethermodynamic properties of alloy surfaces. We analyze critically a number of approaches previouslyproposed to determine the parameters in different Ising models. We investigate the physical meaning ofvarious interaction energies occurring in those models and compare our technique to phenomenologicalapproaches. The formalism is applied in electronically self-consistent mean-field calculations of thesegregation profile at the surface of Ni-Cu and Rh-Ti alloys at temperatures above the bulk disorderingtemperature.

I. INTRODUCTION

In recent years major progress has been made towardsa longstanding goal in theoretical materials science: thecalculation from first principles of alloy phase diagramsand the associated thermodynamic quantities. ' Thishas necessitated important breakthroughs in ab initioelectronic structure calculations, as well as an increasedunderstanding of the applicable tools of statistical phys-ics. Even so, the task is computationally very demandingand has only become feasible through improved comput-er hardware. Although many exciting questions remain,several groups using a range of approaches have nowpresented first-principles phase diagrams for a variety ofsystems including metals, semiconductors, ' and su-perconducting oxides. " In contrast, the determination ofthermodynamic properties near surfaces or other extend-ed defects has not yet attained the same level of sophisti-cation. It is the purpose of the present paper to unify andreview existing formalisms and to present results ob-tained without any adjustable parameters using the directconfigurational averaging technique with the tight-binding approximation.

Alloy-phase-diagram calculations involve two compu-tationally demanding tasks: an electronic structure cal-culation to determine quantum-mechanical interactionenergies and a statistical physics calculation involving theminimization of a free-energy functional constructed us-ing those energies. Both steps must be performed withvery high numerical precision. The band-structure deter-mination must be very accurate because interaction ener-gies are formally obtained as very small differences oflarge cohesive energies. The statistical mechanics prob-lem must be handled at an appropriate level of approxi-mation, by including a sufhcient number of correlationfunctions in a cluster variation framework' or by carefulconsiderations of equilibration times and finite-size effects

in a Monte Carlo simulation. ' The same rigorousdemands hold in the case of order-disorder phenomenanear surfaces and interfaces, where they are exacerbatedbecause of charge redistributions and the interplay be-tween ordering and electronic structure.

In the last century it was recognized by Gibbs ontheoretical grounds and it has been confirmed by manyexperiments since then, that in thermodynamic equilibri-um an alloy surface (or interface) will tend to become en-riched by one of the components. This is thephenomenon of surface (or interface) segregation whichhas been the subject of intense study over the years. '

A large number of technologically important phenomenaoccur at or near the solid surface, or near internal inter-faces such as grain boundaries. ' Examples include ca-talysis, corrosion, adsorption, embrittlement, friction,deposition and growth, etc. Thus, a detailed understand-ing of the segregation behavior eventually leadingperhaps to its control by doping or heat treatment couldhave many important applications. On the other hand,from a purely theoretical point of view phase transitionsin reduced dimensionalities continue to be of great in-

terest and the nature of the different types of segregationbehavior has proven to be extremely rich, with the pos-sibility of segregation-induced ordering and wetting, sur-face sandwich formation, reconstruction, etc. Because ofthe great success in the description of bulk alloys, Isingmodels have been used extensively to model surface alloy-ing effects as well, although care must be taken in caseswhere the size difference of the constituent atoms is largeand elastic effects become important. Phenomenologicalmodels21-24 were able to describe qualitatively the ob-served segregation in various systems in terms of a trun-cated Ising model with the segregating component beingdetermined by the bond-breaking energy (or difference insurface tensions) and the approach to the bulk limit(monotonic or oscillating) by the sign of the ordering pair

1993 The American Physical Society

47 SELF-CONSISTENT ORDERING ENERGIES AND. . . 63

interaction. While such models may have some pedagog-ical value it will be shown in the present paper that greatcare must be taken in their use and that, in fact, the pointinteractions (normally not considered in Ising model cal-culations) dominate the segregation behavior. Moreover,the phenomenological theories suffer from the fact thatthe interaction energies were introduced in an ad hocfashion ' or in terms of experimentally measured macro-scopic quantities and no attempt was made to relatethem to the underlying electronic structure of the alloy.A great deal of work has been devoted to studying segre-gation in terms of the electronic structure, but inmost of those papers grave difFiculties were encounteredin the incorporation of temperature into a proper statisti-cal mechanical treatment. What is needed is a method tocalculate pair (and point) energies from the electronicstructure and subsequently to use these energies in an ap-propriate statistical model. The theoretical tools to ac-complish this task have only become available in the lastfew years.

The remainder of this paper is organized as follows. InSec. II, the formalism is outlined with particular atten-tion to the choice of reference medium, as this permits usto compare and contrast the present approach to others.To be specific the discussion is cast in terms appropriatefor binary alloys, although the generalization to ternaryalloys is straightforward. Section III presents details ofthe calculations for two systems: Ni Cu, „which showsa miscibility gap in the bulk and Rh Ti, „,which is anordering alloy. This permits us to contrast two extremetypes of behavior and also facilitates comparison withbulk calculations as both systems have been studied re-cently. Concentration profiles are presented and corn-pared with experimental results insofar as the latter areavailable. Section IV contains a more in-depth discussionof the interaction parameters and their dependence onthe concentration profile. The present formalism is com-pared with phenomenological theories by studying thepoint and pair energies and relating them to macroscopicparameters. The paper closes in Sec. V with a summaryof our findings and a critical discussion of the currentstate-of-the-art in alloy surface calculations of order-disorder phenomena.

II. FORMALISM

While much of the first-principles work on alloys andcompounds in the 1980s was done with very sophisticatedand time-consuming band-structure techniques based onthe local-density approximation (LDA), such as theKorringa-Kohn-Rostoker (KKR) or the linearaugmented-plane-wave (LAPW) methods, the last fewyears have seen renewed interest in the simple tight-binding model. Accurate tight-binding parameters cannow be extracted from first-principles band-structure cal-culations, by fitting to the results of LAPW calculationsor by transforming the linear muffin-tin orbitals (LMTO)formalism. These parameters can then be used in situa-tions where the LDA-based techniques become unwieldy,leading to calculations that may not be strictly first prin-ciples, yet have much of the flavor of an ab initio method.

In this spirit, in the present work the semi-infinite alloy isdescribed within the tight-binding model in which theone-electron Hamiltonian for a given configuration o.

takes the form

where n and m run over lattice sites, A, and p label the or-bit als ( A, ,p = I, . . . , 9, if s, p, and d orbitals are con-sidered). Furthermore, s„denotes the on-site energy as-sociated with orbital A, on site n, while P„ is the hoppingterm (overlap integral) between sites n and m and orbitalsA, and p. Except for very small clusters, it is, strictlyspeaking, not feasible to compute the electronic structurecorresponding to the Hamiltonian (I) because the on-siteenergies are functions of the chemical and geometricalenvironment and therefore must be calculated in a self-consistent way. For bulk calculations these effects areusually neglected. In a layer geometry the computationsmay be simplified by assuming that on a given planeparallel to the surface the on-site energies depend only onthe chemical type of the atom considered, but this ap-proach remains to be justified a posteriori.

The coherent potential approximation (CPA) providesan efficient way to study the electronic structure of corn-pletely disordered alloys. ' Developed in this contextin the late 1960s, although similar ideas date back atleast to Lord Rayleigh, this approach has been extendedto the semi-infinite alloy under the assumption of adifferent coherent potential for each plane. However,this conceptually simple idea is not easy to realize and weshall discuss later how to circumvent the di%culties thatthis method entails. In the CPA the physicalconfiguration is replaced by a fictitious effective mediumwhich restores translational symmetry. This medium isdetermined in a self-consistent way such that an atomembedded in it produces no scattering on the average. Ithas been shown that of all effective medium theories theCPA provides the best single-site approximation, whileattempts to go beyond the single-site approximation havebeen beset with great difFiculties. Originally formulatedwithin a tight-binding context (TB-CPA), the method hasfound its greatest successes within the framework ofmultiple-scattering theory (KKR-CPA) and electronicstructure calculations based on it have shown unsur-passed agreement with experiment for a large variety ofsystems. Thus, optimism grew that the CPA might alsobe an excellent starting point of a proper statisticalmechanics theory of order-disorder phase transforma-tions in alloys. Because of the experience gained withthree-dimensional Ising model descriptions of actual sys-tems in which parameters were fit to experiment, theneed naturally arose to determine these interaction pa-rameters from first-principles electronic structure calcu-lations. A formalism to accomplish this was developed inthe pioneering work of Ducastelle and Gautier, inwhich it was shown that the total energy of an alloy maybe obtained by a generalized perturbation method (GPM)relative to a reference medium (for example, the disor-

H. DREYSSE, L. T. WILLE, AND D. de FONTAINE 47

dered state, treated within the CPA, although otherchoices could be made). The resulting expression is not aperturbation series in the usual sense since the energy isexpanded in powers of (p„—c„),where p„ is the occupa-tion variable of site n and c„ the average occupation ofthat site, which are not necessarily small numbers.Nonetheless, convergence is generally rapid because thecoefficients multiplying those terms are rapidly decreas-ing. Thus, this CPA-GPM is well suited for the study ofordering energies and has led to excellent agreement withexperiment for a number of bulk systems. In the case of asemi-infinite binary alloy, NCPA different coherent poten-tials, o.„,must be introduced for each inequivalent planen =1, . . . , NcpA. In the tight-binding framework thesequantities are obtained by the self-consistency equation

ate 0n n n n

where t„' (i = A, B) is the scattering t matrix given by

(2)

c,„o„t.' =1 —( E'„—o.„)F„

with

1F„=—g F„,

(3)

(4)

where HcpA is the CPA Hamiltonian, similar to (1) butwith the on-site energy replaced by the coherent potentialo.„(taken here to be orbital independent):

HcpA g ~n ~&cT &nn, k

+ g ~n, A&P„" &m. ,p~ .num, A, , )M

Thus, the actual Hamiltonian may be broken up into aCPA contribution and a configuration-dependent pertur-bation term:

where F„ is the diagonal matrix element C„„(E)of theCPA Green's function operator G defined through

C(z) = (z Hcp~ )

1 EFV„' = Im f dE g in[1 —(E„—o'„)F„],

9m(10)

and V„'~ is a pair energy,

V„'J = — Im f dE g (G„" ) t„'tjA, , p

Higher-order contributions may be derived in a similarmanner and temperature may be included by multiplyingthe integrand by a Fermi function. It is important tonote that these expressions are exact for a givenconfiguration Ip„' I. As Eqs. (10) and (11) show, the quan-tities V„' and V„'~ must be obtained through an electronicstructure calculation, clearly illustrating that orderingphenomena in alloys are driven by variations in the elec-tron states. Note that in the above derivation it has beenassumed, as is customary, that only the one-electronband-structure term contributes to the energetics.While the derivation in terms of the GPM is the mosttransparent, a number of alternative techniques havebeen proposed to calculate the ordering interactions.These methods include the embedded cluster method(ECM) of Gonis and collaborators which has beenshown to be equivalent to the GPM, ' the method of con-centration waves proposed by Gyorffy and Stocks, theConnolly-Williams method, and the closely related c-Gapproach of Zunger and collaborators. There are minordifferences between these various procedures, dependingmainly on the level of truncation of the series (9), but allhave been successfully used to describe bulk alloys in afirst-principles setting.

In order to better understand the physical meaning ofthe point and pair energies (10) and (11) and to make theconnection with the Ising model calculations it is instruc-tive to look at their interpretation in a statistical physicscontext. To this end we can use an elegant expression forthe internal energy first given by Sanchez, Ducastelle,and Gratias. Introducing pseudospin variables o „(notto be confused with the CPA potentials; these variablesare equal to + 1 if site i is occupied by an A atom, and to—1 otherwise, i.e., o„=2p„"—1), these authors haveshown that the internal energy can be written as a sumover all clusters o.' in the given system

H (0' ) Hcpp +H

whereE=VO+g V g (12)

H =y ~ n, A&(E~ —o.,„)& n, A~

n, A,

is the perturbation term which gives the method itsname.

Introducing the occupation numbers p„', equal to 1 ifsite n is occupied by an atom of type i and equal to 0 oth-erwise, the total energy of the binary alloy with N sitesfor a configuration tp„' I can be written as

E( Ip„' J ) =E,+—g p„' V„'+ g p„'p J V'J +1;; 1

N „,. " " 2N n+~fJ

(9)

where V„' is the point energy,

where g is the correlation function for cluster a defined

by

g =&o.,o, o„&, (13)

where 8'„'J is the total energy of the system consisting ofan atom of type i at site n and an atom of type j at site m

if there are n sites in the cluster considered. Here andin the remainder of this paper the angular bracketsdenote an ensemble average. Within this formalism theV are called effective cluster interactions (ECI). For apair of atoms, one at site n, the other at site m, the ECIreduces to an effective pair interaction (EPI) which reads

(14)

SELF-CONSISTENT ORDERING ENERGIES AND. . . 65

1 g (p„—c„)(p —c )V„+, (15)n, m

where c„ is the concentration on plane n and V„equals

Vnm Vnm + +nm Vnm Vnm (16)

This gives the connection between the EPI defined in thestatistical physics context (14) and the pair energiesdefined in (11) in terms of the electronic structure. Inpractice it has been found that in many cases nearest-neighbor pairs for the fcc structure and nearest- andnext-nearest-neighbor pairs for the bcc structure providean accurate description of the corresponding phase dia-gram. However, for certain alloys higher-order clustersand further-neighbor interactions may be needed, such asin the case of PdV.

In the same manner we can identify in (15) the effectivepoint interaction h„given by

VA VBn n n

This quantity gives the variation of the total energy of thedisordered alloy when at site n a B atom is replaced by an2 atom. For equivalent sites in the bulk, 6„ is unique(i.e., independent of n) but for an alloy with a surfacedifferent values must be computed according to the planeconsidered. Moreover, when all sites are equivalent (as inthe bulk of the fcc lattice) the single-site interaction does

in the fully disordered medium. After transformationof the correlation functions to occupation numbers, a for-mal identification can be realized between the two expres-sions (9) and (12) leading to the conclusion that V„', thepoint energy defined in (10), is equal to the total energy ofthe system containing one atom of type i at site n in thecompletely disordered medium, while V„'~, the pair ener-gy defined in (11),is equal to W„'~ —

( V„'+ V~ ).In addition, a reference level for the energies can be

chosen, since it is always possible to add a constant termto the free energy. Although this is an arbitrary choice itmust be a reasonable one on physical grounds and for thebulk alloy it has been shown that the CPA medium is thebest one. For an alloy surface, however, the situation ismore complex and the following two cases can be con-sidered. In situation 3 examined in the remainder of thispaper the temperature is higher than T„ the criticalorder-disorder temperature for the bulk at the concentra-tion being studied. In that case it is very reasonable totake as the reference medium the completely disorderedstate described by the CPA. At temperatures below T,(situation B) ordering occurs and consequently the disor-dered state is not a good reference point and no referencelevel is explicitly considered. This case has been studiedextensively by Treglia, Legrand, and collaboratorsand we refer the reader to these papers for further details.

For a binary AB alloy and using the CPA medium asthe reference level, Eq. (9) can be rewritten as follows. Ifp„equals p„, then p„= 1 —p„, and using the CPA self-consistency equation the total energy is

E ( (p„' J ) = Vo+ —g (p„—c„)(V„"—V„)1

n

not contribute to the ordering energy and thus there is noneed to calculate it explicitly in a thermodynamic study.Finally, it is important to note that although the total en-ergy of a transition metal cannot be written as a sum ofpair potentials, the ordering energy can be written as arapidly convergent series of point and pair interactions.

For an alloy in which successive layers are completelydisordered the p„'s are uncorrelated so that in such a casewe can write

((p„—c„)(p —c ))=(p„—c„)(p —c )=0,and thus only the point energies V„give a nonzero con-tribution in Eq. (15). In this case the segregation phe-nomena, for the electronic part at least, are only drivenby the values of the point energies. However, when or-dered structures are present (corresponding to situation Babove) the EPI's give an explicit contribution.

Expression (15) for the internal energy may be used ina Monte Carlo simulation or in a mean-field calculationbased on the cluster variation method (CVM). ' ' In thelatter case, in addition to the internal energy one needs toknow an approximate expression for the entropy in orderto perform a phase diagram calculation. Depending onthe temperature under study one may have to includeclusters up to a certain maximum size in the formalism.For temperatures above T, the point-approximation orBragg-Williams method is known to give adequate re-sults, but in other cases larger clusters may be needed.In either case, the self-consistent equilibrium profile is ob-tained by minimizing the resulting free energy relative tothe concentrations in the various planes. A complicationarises because, for a semi-infinite binary alloy, all thequantities V„' and V„'J are functions of the concentrationprofile as can be seen from Eqs. (10) and (11) while theyalso determine the concentration profile. Thus, a fullyself-consistent calculation must be performed: startingfrom an initial guess for the concentrations one iteratesuntil the interactions and concentration profile are con-sistent. This approach corresponds to the canonical en-semble and details of computations based on it will begiven further in this paper. We remind the reader thatthe present derivation assumes that the CPA is a goodreference medium and thus it is expected to be valid fortemperatures above T, (situation A defined above). Inthat case, the segregation is driven by the sign of the vari-ation of the effective point interaction 6&= V&

—V& be-

tween the surface and the bulk (as long as no elastic con-tribution to the free energy is considered) and an elec-tronically self-consistent calculation of the 6's is essen-tial. To simplify this calculation it is tempting to specu-late about alternative approaches. It has previously beendemonstrated that for bulk alloys a grand canonical ap-proach can be used with success. In the case of a semi-infinite alloy this would correspond to a system with thesame concentration of 3 and B atoms on all the planes.By analogy to the bulk case the convergence of expres-sion (9) may be expected to be slower in the grand canoni-cal case than in the canonical case. The viability of thisformalism remains to be established and is currently un-der study. The calculations presented here have been

66 H. DREYSSE, L. T. WILLE, AND D. de FONTAINE

performed for the canonical case.For alloys with a surface, translational symmetry per-

pendicular to the surface of the disordered medium islost, while parallel to the surface it has been restored bythe CPA condition. Thus, for a given separation betweensites n and m, not only one EPI, as in the bulk, must beconsidered, but diff'erent values according to the positionof n and m. For example, if only nearest neighbors on afcc lattice are considered, terms such as V„„and V„„+,must be retained (n labels planes parallel to the surface,n =1 is the top layer). In order to have a systematic no-tation, one can extend a convention introduced previous-

ly according to which V„'„' "' is the ECI with n&

atoms that are neighbors of order n2, and in whichm&, m2, . . . labe1 the planes parallel to the surface. Forexample, V2f denotes the effective pair (subscript 2) in-teraction between nearest (subscript 1) neighbors withone atom in the surface plane (superscript 1) and the oth-er one in the plane immediately below the surface (super-script 2). For point energies the index n, is equal to 1,for pair energies n, equals 2, etc. For higher-order clus-ters a graphical representation must be given.

The main problem is now to compute the point ener-gies and in a general manner the eff'ective cluster interac-tions. As mentioned before, various approaches for bulkbinary alloys have been proposed, ' ' but for asemi-infinite medium a supplementary difficulty arisesdue to the lack of translational symmetry in the directionperpendicular to the surface or interface, which makesuse of the CPA cumbersome. Recently we proposed aconceptually simple method based on taking an averageover a small number of random configurations to obtainthe eff'ective cluster interactions. ' In principle, theECI are diff'erences of total energies of large systems, butsince only localized perturbing potentials are consideredthe effective cluster interaction can be obtained directlyas a global expression, using the generalized phase-shiftapproach. Following Friedel, the generalized phaseshift may be defined as

detG „~det6~~g(z) =ln

detG„~detG~„

where G; is the Green's function corresponding to theHamiltonian (1) but with atoms i and j at sites n and m.(This quantity is not to be confused with the CPAGreen's function. ) We have shown elsewhere that the

I

EPI may be written as

V„= — Im g z dz (20)

The recursion method, introduced in solid-state physicsby Haydock, permits a calculation of the Green's func-tions G, in real space and, by using the "orbital peelingtrick" proposed by Burke ' following the ideas of Ein-stein and Schrieffer, the determination of the eff'ective

pair interaction needs the computation of only 4 X 9 diag-onal elements of the Green's function, while the point in-teraction needs only 2 X9 terms.

In this paper we will focus on the simple but physicallyinteresting case of the fully disordered alloy at tempera-tures T above the bulk order-disorder temperature T, . Inthat case the Bragg-Williams method is known to pro-duce quite accurate results. Thus, only the point corre-lation functions need to be considered and the entropyreduces to

S =g c„inc„+(1—c„)ln(1—c„) . (21)

The equilibrium concentration profile is obtained byminimizing the free energy with respect to the correlationfunctions (i.e., concentrations) and, denoting by c~ thebulk concentration, we find in a straightforward manner

C„cgexp[ —( V„"—V„)jk~T]

1 —c„1—c~(22)

(where k~ is Boltzmann's constant), a result remarkablysimilar to that obtained in phenomenological theories '

but with point energies, rather than EPI's, in the ex-ponential. If elastic contributions to the free energy areconsidered, they appear in the exponential of Eq. (22).This is essential when the atomic sizes are very different.A priori the point energies are functions of the concentra-tion profile and to a good approximation they turn out todepend only on the plane considered.

In more comp1ex situations and below T, ordering mayoccur and the contribution of the EPI's to the free energycan no longer be neglected. In that case a set of coupledtranscendental equations for the concentrations in thelayers is obtained which must be solved numerically.These equations are similar to those in Ref. 21, but withconcentration-dependent EPI's and point interactions,for example, if only nearest-neighbor EPI's are includedfor the (100) surface on the fcc lattice they read

cr„=tanhI [4V„„o„4+ „V„&cr„&+4 „V„&+o „+& b, „]/kz T], n =1,2, . . .—, (23)

where o.„=2c„—1, is the average value of the pseudospinvariable in the nth plane and o.o—=0. In simple situationsthe numerical solution of the system (23) may be accom-plished by straightforward iteration to the fixed point ofthe equations, but this approach has the drawback that itcannot detect whether multiple solutions are present.The same holds for the more rapidly convergentNewton-Raphson scheme. To avoid this shortcoming an

elegant method was proposed by Treglia et al. fol-lowing earlier work by Pandit and Wortis based on theproperties of Aow under area-preserving maps. Thismathematical approach to solving coupled equations al-lows the numerical algorithm to detect multiple solutionsand to describe discontinuous and layering transitions, al-though the method is intrinsically unstable and numeri-cally not very accurate. A complete study should use the

47 SELF-CONSISTENT ORDERING ENERGIES AND. . . 67

area-preserving map approach to describe the transitionsand then utilize a more stable algorithm to compute theprecise location of the phase boundaries. In the followingwe shall consider only cases where the solutions areunique and these complications do not arise.

In this paper we study two alloys which presentdifferent behavior in the bulk: the Ni Cu, system,which is well known to possess a large miscibility gap,and the Rh„Ti, alloy, which exhibits various orderedphases. Bulk calculations for both systems have beenreported recently. ' ' The choice of these systems wasguided by the fact that magnetic effects are negligible (ex-cept perhaps at the Ni-rich end of the Ni-Cu system), rel-ativistic effects may be safely ignored, and atomic sizedifferences are small so that elasticity must not be includ-ed in the calculations. Preliminary results using thedirect configurational averaging approach were brieAypresented elsewhere. ' After the determination of theequilibrium profile we shall concentrate on the propertiesof the ECI, with particular emphasis on the point interac-tions and the EPI.

j:II. RESULTS

Some of the advantages of the direct configurationalaveraging (DCA) method are that it avoids the CPA ap-proximation, that it permits a calculation in real space,and that it allows the possibility to take into account s, p,and d orbitals on the same footing. In addition the recur-sion method also supports a straightforward treatment ofthe off-diagonal disorder, which in the CPA frameworkintroduces complications. Three different sets of hop-ping integrals were used: P""and P are generally tak-en to be the same as for the pure elements and may there-fore be computed with a good level of accuracy, ' areasonable and widely adopted assumption is to takeP" as the geometric mean of P"" and P (i.e.,P" =+P P, the so-called Shiba prescription). Alter-natively, if calculations for the ordered AB compoundare available they might be used to determine the hop-ping integral P" . In the Slater-Koster scheme, '

Pzz~ arerelated to the three- and two-center integrals. Electronicself-consistency is ensured as customary by imposing lo-cal charge neutrality through a rigid shift of the on-siteenergies. In practice, it is found that perturbing poten-tials are only needed for the surface plane and the twofollowing planes. Effective cluster (point, pair, etc.) in-teractions are obtained after a two-level self-consistencyloop. First, for a given concentration profile, the perturb-ing potentials on the planes in the surface region aredetermined. Next the concentration profile is adjusted inorder to obey Eq. (22) or (23). A careful investigation ofthe variation of the point and pair interactions has shownthat in the cases studied here these quantities are sensi-tive only to the irnrnediate neighbors leading to a reason-able computation time.

Formally, the method developed for alloy surfaces byTreglia et al. is similar to our approach. As dis-cussed before, apart from the source of the tight-bindingparameters and other technical details, the main

difference lies in the choice of the reference medium. Asemi-infinite medium with the same concentration andthe same coherent potential (corresponding to the bulkvalue) on every plane was chosen by Treglia et al. As aconsequence these authors must explicitly considereffective pair interactions in addition to the effectivepoint interactions at all temperatures. As already dis-cussed the choice of the reference medium is arbitrary,but it is intuitively clear and borne out by our calcula-tions that our approach, for temperatures above T, (situ-ation A), gives better convergence. In other cases onemust consider renormalized effective cluster interac-tions ' and the problem is again to determine thosecluster interactions that give significant contributions.From previous studies, one may expect that in certaincases triplet interactions may be of the same order ofmagnitude as some pair interactions.

In this work we report results obtained for two semi-infinite alloys: Ni„Cu& „(all x) which presents a cluster-ing trend in the bulk and Rh„Ti, „(near x =0.75)which exhibits various ordered structures. For the com-positions considered both alloys are based on a fcc latticeof which, unless otherwise specified, the (001) surface willbe considered. Preliminary results were briefiy presentedelsewhere. ' All calculations were performed for tern-peratures above the bulk critical temperature, corre-sponding to a completely disordered system in the bulk,but this does, of course, not exclude ordering in the sur-face region. In this temperature regime the Bragg-Williams approximation is known to work quite satisfac-torily. Let us note that for temperatures below T, thephysical processes are more complex because the averageconcentration defined on each plane may differ stronglyfrom the average concentration in the bulk. As a simpleexample, consider an 383 alloy in the LI2 structure. Atlow temperatures, parallel to the (001) surface, the latticeis a succession of planes with compositions close to 100%B and 50% B, although at the surface and in the immedi-ate subsurface layers the concentration may differ fromthese values due to segregation effects. In any case, apartfrom the problems of a proper reference medium dis-cussed above, a simple Bragg-Williams approach, whichneglects correlations, is inconsistent and a generalizationof the cluster variation method appropriate for surfacesmust be used.

In the present work s, p, and d orbitals are considered.If only d orbitals were included the factor 9 in Eqs. (4),(10), and (11) would be replaced by 5. The hopping in-

tegrals for the pure elements are taken fromPapaconstantopoulos's book and were obtained by thisauthor through a fit to very accurate first-principlesLAPW calculations for the pure elements. The recursionmethod was applied using the algorithms developed byNex and collaborators. As usual, local charge neutrali-ty was imposed on each atom through a rigid shift of theon-site energies. " ' Seven levels of the continued frac-tion for the Green's function diagonal elements have beencomputed, with the Beer-Pettifor prescription for thetermination of the continued fraction. In order to obtainsatisfactory convergence it was found for the point ener-gies to be necessary to average over 20 configurations

68 H. DREYSSE, L. T. WILLE, AND D. de FONTAINE 47

while for the pair energies 15 configurations were needed.Higher-order and further-neighbor cluster interactionswere found to be negligible in all cases studied here.These findings are consistent with our earlier studies us-ing the DCA for the bulk.

The Ni Cu& system has been the subject of many ex-perimental and theoretical ' investigations.Even though it is a relatively simple alloy, whose bulkphase diagram is well understood, its surface behaviorhas been the source of some controversy. A large num-ber of experiments using a diversity of surface-sensitiveprobes (catalytic activity measurements, Auger electronspectroscopy, x-ray and ultraviolet photoemission spec-

r oscopy. . . time-of-flight atom probe, ' ' loenergy ion scattering, ' ' ' ' photoemission of ad-sorbed xenon, low-energy electron diffraction, etc.) haveprovided incontrovertible evidence that Cu segregatesstrongly to the surface for all compositions, with the sur-face concentration increasing monotonically as a functionof the bulk concentration, x. It came therefore as a greatsurprise when Sakurai et al. reported evidence for a re-versal in segregating species with Ni segregating for com-positions less than x =0.16 and Cu segregating for con-centrations larger than this value. While there was origi-nally some theoretical evidence that such a phenomenonmight be understood in terms of surface magnetism orcharge transfer, ' more careful studies did not supportthis notion. Moreover, subsequent experiments did notconfirm Ni segregation for small x and several sugges-tions were made to explain the experimental observa-tions. It was pointed out, for example, that a small con-centration of contaminants (specifically, sulfur or oxygen)might lead to precisely the type of effects that were ob-served by Sakurai et al. This illustrates clearly howsensitively the surface concentration may depend on theoverall composition and, in particular, the local electron-ic structure. A second debated observation for the Ni-Cusystem concerns the composition of the subsurface layers,with some experimentalists claiming that the approach tothe bulk limit is monotonic, while others find evidence forCu depletion in the layer immediately below the sur-face. Theoretical analyses of this system have alsogiven mixed results. Because the determination of theequilibrium subsurface concentration is a notoriouslydificult problem reasonable doubt may still be cast on ei-ther observation.

In Fig. 1, we display the concentration profile near aNi„Cu, „(001)alloy surface with the bulk Ni concentra-tion kept at x =0.75 and 0.25, at a temperatureT = 1. 1 T, [corresponding to approximately 225 and425'C, respectively]. As observed experimentally, verystrong Cu segregation occurs towards the surface planewith the composition in the subsurface layers very rapid-ly approaching the bulk concentration. This was foundto be the case over the entire concentration range and, inparticular, no reversal in segregating species was ob-served. It is to be noted that our calculations properlyaccount for charge-transfer effects and therefore do notsupport an earlier suggestion that charge transfer mightbe responsible for a segregation reversal. As can be seenin Fig. 1, the concentration profile is monotonic rejecting

1.00—

~ o.so-O

~ 0.60-

Q)Oc: 0.&o-O

~~Q-—

+ 0.20-

0.00 I

3Plane

I

Bulk

FICx. 1. Self-consistent equilibrium concentration profiles at(001) surface of Ni„Cu& alloy ( T = 1.1T, ): the solid line cor-responds to x =0.25; the dashed line to x =0.7S.

the clustering tendency of the Ni-Cu system, as wouldalso be predicted by the phenomenological theories. Inall cases studied we found the approach to the bulk valueto be monotonic, with no evidence for Ni enrichment justbelow the surface. Thus, we disagree with other experi-mental and theoretical works in which a Cu deficiencywas observed. While it is clear from the sign of the pointinteractions calculated here that such an effect is not ex-pected on the basis of electronic interactions alone, itmay be due to contributions that were neglected here,most likely of elastic or magnetic origin. However, it isalso possible that the experimental observations werebased on nonequilibrium phenomena, for example, due tosputter-induced subsurface segregation. Thus, the issuedeserves further study.

In Fig. 2, we show the equilibrium profile at a tempera-ture T= 1.1T, (approximately 1925 C) near the (001)face of a Rh„Ti& system with a fixed Rh bulk concen-tration x =0.75. The segregation profile is oscillatory, asexpected for an ordering alloy, with a very rapid ap-

1.00-

0.SO—

U~ 0.60-

(9O~ 0.&o-O

0.00 l

Plane

I

Bulk

FIG. 2. Self-consistent equilibrium concentration profile at(001) surface of Rh Ti

&with a bulk Rh concentration

x =0.75 at temperature T=1.1T,.

47 SELF-CONSISTENT ORDERING ENERGIES AND. . . 69

proach of the bulk limit. Our calculations predictmoderate Rh (solvent) enrichment. Unfortunately no ex-perimental results seem to be available for this system. Itis suggested that this would be a good system for furtherstudy, as the Miedema theory ' predicts no segrega-tion, while the current work and a model based on sur-face enthalpies predict Rh segregation. It is interestingto note that the surface enthalpies are directly propor-tional to the point energies included in our model.

IV. DISCUSSION

Because they are key quantities in the formalism andalso because their determination is the computationalbottleneck, we now turn to a more detailed discussion ofthe interaction parameters, specifically the pair and pointinteractions as higher-order terms turned out to be negli-gible in the system studied here. In order to better under-stand the effects of electronic and atomic self-consistencyvarious "artificial" (i.e., non-self-consistent) config-urations will be considered. While the motivation ismainly pedagogical, nonequilibrium profiles may be real-ized experimentally by rapid quenching or ion bombard-ment and thus the nature of the interactions under thoseconditions may have some independent interest. Also ad-sorbed monolayers may alter the equilibrium segregationprofile and lead to concentration gradients similar tothose introduced here.

A. Kft'ective pair interactions

The EPI's play a major role in the ordering processesin bulk binary alloys. These quantities can be measuredexperimentally through diffuse scattering intensities andcan be computed by techniques such as the inverse MonteCarlo method. They also have a simple physical inter-pretation: V„)0 ((0) indicates an ordering (cluster-ing) trend between sites n and m. Of course, the interplaybetween the ordering tendencies of various sites may leadto complicated ordered structures, even when onlyshort-range interactions are considered, one famous ex-ample being the axial next-nearest-neighbor Ising(ANNNI) model which supports a whole range ofmodulated states. A knowledge of the EPI's is alsonecessary to perform a zero-temperature ground-stateanalysis, ' the first step in a full phase diagram calcula-tion. In the case of a semi-infinite alloy below the bulkT„ the variations of the EPI are essential to study segre-gation and ordering phenomena and they are also key in-gredients in phenomenological theories. ' As was dis-cussed before, only for completely disordered systems arethe segregation phenomena driven exclusively by thevariation of the point energies.

First we investigate the variation of the EPI as a func-tion of the concentration profile. Brown and Carlsson' '

performed a model study of the infiuence of defects (lo-calized or extended) on the EPI. The results obtained bythose authors showed an increase of the EPI with de-creasing coordination number. However, the calcula-tions were made under very restrictive assumptions: nei-ther off-diagonal disorder nor electronic self-consistency

EPI (eV)0.000-

—0.010-

—0.020-

—0.030-

—Q.040

11 12 22 23 33Positions

I

V(bLI II&)

FIG. 3. Nearest-neighbor effective pair interactions, V2& (in

eV), at the (001) surface of a Ni„Cu& „alloy with a bulk con-centration x =0.25 (solid line) and x =0.75 (dashed line) for theequilibrium profile given in Fig. 1. For ease of notation V2& isdenoted as V„where the index 1 denotes the surface plane.These curves are obtained through a fully self-consistent calcu-lation.

were considered and the concentration profile was takento be flat (uniform concentration on all the planes). Un-der these conditions, Brown and Carlsson' ' showed thatthe EPI are proportional to Z, where Z is the coordi-nation number. However, binary alloys are well knownto present strong segregation effects on the surface plane.In addition, since the EPI in the bulk may vary stronglyas a function of concentration a priori no general rulefor the value of the EPI on the surface plane can be ex-pected and a complete study must be made.

In Fig. 3 we present the variation of the EPI for twonearest neighbors located in various layers in a semi-infinite Ni Cu& host for the equilibrium profiles(shown in Fig. 1) corresponding to bulk concentrationsx =0.75 and 0.25. It is to be kept in mind that in bothcases the concentration of Ni on the surface plane is verysmall as was found throughout the entire concentrationrange. Calculations for bulk Ni-Cu (Refs. 66 and 102)show a decrease of the nearest-neighbor and next-nearest-neighbor EPI s with decreasing Ni concentration.These results combined with Fig. 3 thus point to a corn-petition between the values in the bulk and those at thesurface. For a bulk concentration x =0.75 we also ob-serve a large variation of the EPI for a point in the sur-face plane and the other point in the plane immediatelybelow it. The strong changes of the EPI in this case are aconsequence of the large variations in composition nearthe surface. The variations in the concentration profileare relatively smaller for a Ni bulk concentrationx =0.25 and the variations of the EPI in that case areconsequently also much smaller.

In Rh„Ti& the EPI have been found to be very sensi-tive to the concentration in the bulk. In Fig. 4 are re-ported the nearest-neighbor EPI's of a Rh„Ti, alloywith a Rh bulk concentration x =0.75. Two electroni-

70 H. DREYSSE, L. T. WILLE, AND D. de FONTAINE 47

EPI (ev)0.080-

EPI (eV)

0.080-

0.060-0.060-

0.040- 0.040-

0.020- 0.020-

0.00011 12 22 23 33 34 44 45 55

Positions

0.000 I

12 V22Positions

I

23I

33I

34

FIG. 4. Nearest-neighbor intralayer (V„„) and interlayer( V„„+&)effective pair interactions (in eV) at the (001) surface ofa Rh Ti, „alloy with a Rh bulk concentration x =0.75 forvarious concentration profiles (solid line: equilibrium profilefrom Fig. 2; dashed line: c

&=0.52, c2 =0.78, c3 =0.74,

c„=0.75, n & 3; dotted line: c„=0.75 for all n). The curves inthe solid and dashed lines were calculated with full electronicself-consistency imposed. The dotted curve corresponding tothe uniform profile was obtained without any electronic self-consistency.

cally self-consistent concentration profiles were con-sidered: one (solid line) corresponding to the equilibriumprofile given in Fig. 2, the other one (dashed line)artificially corresponding to a strong depletion of Rh inthe surface plane (c, =0.52) with an oscillating profile(c2=0.78, c3=0.74, c„=c~=0.75 for n) 3). We ob-serve in both cases a small decrease in surface intralayerEPI relative to the bulk value and a mild increase for thesurface-subsurface interlayer EPI. Let us recall that thebulk EPI of the fcc Rh Ti, „alloy is an increasing func-tion of the Rh concentration. The differences betweenthese two self-consistent profiles can, as in the previouscase of Ni Cu&, be understood as a competition nearthe surface between the EPI bulk values and the localsurface concentration. Of course, this is only a qualita-tive argument that must be backed up by a full calcula-tion. In Fig. 4, we also report by a dotted line the EPIcorresponding to an alloy with a c„=0.75 for all planesand no electronic self-consistency. We notice a sharp in-crease of the EPI near the surface, especially the in-tralayer interactions, in agreement with the calculationsof Treglia et al. and Brown and Carlsson' ' whoalso did not consider electronic self-consistency.

The relative inhuence of the concentration profile andcoordination number can be deduced from Fig. 5, whichshows self-consistent EPI for the (001) surface ofRh Ti, „(assumed to be in the fcc structure) with a Rhbulk concentration x =0.50. A uniform segregationprofile with c„=0.50 on each plane is assumed and thecalculations have been driven to full electronic self-consistency. We observe a sharp increase of the in-tralayer surface EPI V&, and the surface-subsurface

FIG. 5. Nearest-neighbor intralayer ( V„„) and interlayer( V„„+&)effective pair interactions (in eV) at the (001) surface ofa Rh Ti& „alloy with a Rh bulk concentration of 0.50 for auniform concentration profile. This curve was obtained withfull electronic self-consistency imposed and the alloy assumed tobe in a hypothetical fcc structure.

EPI V]2 a result that is to be expected according to thearguments of Brown and Carlsson. ' ' In this particularcase it turns out that the effects are amplified by the sur-face charge redistribution, although this is by no means ageneral trend.

We can conclude that similarly as in the bulk the EPIare functions of the concentration profile. This pointsout the importance of the compositional and electronicself-consistency. Let us note that in the present work thepoint energies and the effective pair interactions havebeen computed directly from the electronic structure andthat the former dominate the segregation behavior (atleast for temperatures above T, ). Simple models ' pre-dict that a positive (negative) EPI in the bulk will drive anonmonotonic (monotonic) concentration profile. In ourcase the EPI for the systems considered here have thesame sign and this simple relation appears to be satisfied.

B. Point energies

For T )T„ in the Bragg-Williams approximation, thesegregation is completely driven by the variation of thepoint energies relative to the bulk values. Special caremust therefore be taken in their computation. As alreadypointed out, the point energies are functions of the con-centration and a full calculation must be performed.However, it would be interesting to obtain general trends.In Ref. 68 we have reported the point energies for a sur-face atom of a Rh Ti& „(001)alloy with a bulk Rh con-centration x =0.75 for varying surface concentrations.The concentration profiles were scaled to present a simi-lar oscillatory behavior. The main results were that forthe non-self-consistent calculations, 6,'exhibits a decreas-ing, nearly linear behavior versus the surface concentra-tion, the extremal values being around 0.55 to —0.20 eV,indicating a very important variation with a possiblesegregation-type change, while for the self-consistent cal-

SISTENT ORDERINQ ENERGIES AND ~ ~ ~ 71

culations, a similar decreaecreasing behavior was found butwit a much smaller slope: the extrema

an .18 eV (see Fig. 3 in Ref. 68).In order to establish if this is a general ty e of behav-ype o e av-

u& alloy with x =0.25, 0.50, and 0.75 in th

suits are ssur ace concentrations fo h hrw ic t'ere-

shown in Fig. 6. For x =0.75 calcula'

performed with and'

hca cu ations were

while for x =0.25 d 0an wit out electronic self-c-consistency,

an .50 only the non-sel~-lt otd Th

monotonic concentratione an t e values for the concentration on the 1

' non epanes

e ig. ~j. First, ita in a cases the point ener ies ver

ncen ration present a nearly uniformmono onic behavior. As can bbe seen the effect of an elec-ronica y self-consistent treatment is to deer

point energies. For a bulkis o ecrease the

or a bulk Ni concentration x =0.75 thereduction is between about 0.10 eV fore or a Cu-rich surface

e or a Ni-rich surface. It is tojn the Rh- T'

is to be noted that,

consistent cale - i case, the ointp

'energies in the self-

is en ca culations vary less than if no sel-consistency is considered. Thisself-consistent calculations feasible because an

'ere . is makes compositionall

„....h. ...,hon e point energies macelerate the ite i erative solution towards a self-consiconcentration profile. Anoth

' ' a,e. not er interesting result is that,in t e non-self-consistent calculations thefor various bulk Ni c

ns, e point energies

slope, the values increasinu i concentrations have roughl they e same

s increasing with an increase of the b lki concentration.

e u

As is known from hf

p enomenological theories ' thesur ace crystallo ra hicg p ic orientation plays an important

~ ~C. l3n the valI sty of the pair-interaction models

a is e t at the totalFor transition metals it is well establish d henergy cannot be written as a sum of isum o pair energies.

wever, for t e ordering energy in the bulk i h

p y n ernonstrated that an expan'

ffe u it as now

pair interactions is feasible and ra iexpansion in effective

p y o g

case of inug is argument does not necessaril h ld

'h

of inequivalent sites such as in the 315y o inte

or near a surin e compounds

a sur ace. However, the pair m d 1'

mo e in view of itsici y is very attractive. The Ising model with onl

pair interactions has been st d' d'

e wi on yn s u ie in great detail and its

properties are well understood. Thereforeterestin to

o . ere ore, it is very in-

g o see under what conditions it is ossible1 p

'gy expansion. We must insist that s h

p s no formal justification but we would like tounderstand its success and limits of a li

mo e, in the absence of elastic contributions, thesegregation is driven by the variat' f thion o e quantity

role in determin'rmining the segregation profile. In Fig. 7 arereported the point energies for a Ni Cbulk Ni concentration x =0.75 for the low-indexorientations. Scaled mwere a

'e -c

ca e monotonic concentration filpro esssumed and no electronic self-ce -consistency was im-

ose . gain a nearly linear behavior was found with

e point energies decrease in going from the (110)(001), and finall 111y ' orientation: this correspondactly to oin fro

spon s ex-

closel acg

'g rom the most open surface t tho e most

els ' It hy pac ed one and is in agreement 'then wi simpler mod-

has been shown already that the Ni Cui a oyis the prototype of an alloy where this sim le h

app ies. As pointed out by Tregliaet al. , t ese point energies are of th e same order ofmagnitude as the difference f th fo e sur ace tensions.

0.00-

—0.20-

v (ev)0.00-

—0.40-

—0.60-

—0.80-

~ ~ ~ ~ ~ \

~ ~

H" ~ ~ ~ ~~ ~ ~ ~

~ ~ ~ ~ ~ ~

-E3~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~

~ ~ ~ ~

—0.20-

—0.40-

—0.60-

—0.80-

........."-.".0~ ~ ~ ~ ~ ~ ~ ~ ".0"

Q~ ~ ~ ~

—1.000.00

II

I I

0.20 0.40 0.60Ni Conc. SLjrfaee Plane

I

0.80I

1.00

FICx. 6. Point energies at the (001) surface of a Niloy, for various Ni b lki u concentrations: x =0.2

- o e ine), and x =0.75 (dashed line.three curves were obtain d

'h se f

Also reported are th 1

aine without electronic self-se f-consistency.

tion x =0.75 h ure e va ues obtained for a ~

wit fulla Ni bulk concentra-

u electronic self-consistenc so iThe concentration profil

'n pro es were taken b a ro r'

ncy (solid line).

the equilibrium fil (Fm pro e (Fig. 1).y ppropriate scaling of

—1.00-

0.00I

II

020 040 060 080Ni conc. Surface Plane

l

1.00

FIG. 7. Point~ . 't energies at the surface of a Ni C

with aa Ni bulk concentration x =0.75 for v

a i~ u& ~ allo

, solid line; (001), dashed line; and (111) dot dline. Electronic self-con

'otte

-consistency has not been imposed. Theconcentration profiles were obtained by a ro rith ilib i fil F'um pro e ig. 1).

72 H. DREYSSE, L. T. WILLE, AND D. de FONTAINE 47

TABLE I. Variation of the bond-breaking term 5;J and thepoint energy 5; (both in eV) relative to the bulk values near the(001) surface of a semi-infinite Nio»Cuo 75 and Ni075CUp 25 al-loy. All quantities have been calculated for the equilibrium con-centration profile at T =1.1T, shown in Fig. 1. The index 1 or2 denotes, respectively, a site on the surface plane or on theplane immediately below, while ao denotes a bulk site.

Ni bulk concentration x =0.25 x =0.75

—0.65—0.15—0.13—0.39—0.04

—1.29—0.72—0.14—0.67—0.13

(24)

V. CONCLUSIONS

We have presented a detailed analysis of the Ising-model approach to surface ordering phenomena in terms

between the surface and the bulk (the "bond-breaking"term). In earlier works these quantities were consideredas parameters and were fit to some macroscopic propertyoften leading to quite good agreement with experiment.In the present work the bond-breaking terms were com-puted for the Ni Cu& system, within the DCA method,for two equilibrium profiles (Fig. 1) corresponding to a Nibulk concentration x =0.25 and 0.75 (Table I). One cannotice, especially for the x =0.75 bulk Ni concentration,that (b, ,

—b ) is approximately equal to(b;+EJ —2b ). These bond-breaking terms are alsofunctions of the concentration profile. Therefore, using abond-breaking term computed from the electronic struc-ture will lead to very similar results as using the exact ex-pressions with the point energies. This result is not com-pletely surprising, if we keep in mind the definition of thepoint and pair energies given in Eqs. (10) and (11). Infact, between say, V„„and V„, the only difference is thatfor the first term one forces a given neighbor of a first 3atom to be an 3 atom whereas for the second term thissite has only a probability x to be occupied by an A atom.However, let us recall that it is this subtle differencewhich determines the value of the EPI. In a bulk alloyusually all the sites are equivalent and thus the phase dia-gram is computed from these EPI and if necessaryhigher-order cluster interactions. At the surface this nolonger holds, but quite frequently the approximate rela-tion mentioned above is satisfied, and thus a pair-interaction model with fitting parameters can provide anaccurate description. However, it must be pointed outthat an examination of other alloys shows that usuallythis relation is not as closely satisfied as for Ni„Cu,with a Ni bulk concentration x =0.75 and consequentlythe predictions of the pair-interaction model are not asaccurate. Nevertheless, in all cases studied the sign ofthe bond-breaking term was the same as that of the segre-gation energy given by the point terms, and thus no qual-itatively incorrect predictions would result.

of the electronic structure. The proper choice of a refer-ence medium has been discussed and the role of the in-teraction parameters entering the formalism has beenclarified. Self-consistent concentration profiles calculatedwithout any adjustable parameters were obtained for theNi Cu& and Rh„Ti& systems. In the former case wefind strong Cu enrichment at all surfaces and throughoutthe concentration range with no evidence for subsurfaceCu depletion, which has been suggested by time-of-Bightexperiments and confirmed in a number of theoreticalstudies. However, there is no universal agreement on thisbehavior and further work needs to be done to settle thisissue. For the Rh-Ti system we predict moderate Rh en-richment in agreement with one previous theoreticalstudy, although other theories find no segregation. Clear-ly this would be a good system for experimental scrutinyand further first-principles work.

Our results indicate that for temperatures above T,and in the absence of elastic effects the segregation isdriven by the variation of the point energies which is themicroscopic analogue of the difference of the surface ten-sions. Analogous effects were found in the Ni„Cu&system by Treglia et al. using an approach similar toours in the Bragg-Williams method and in a recentMonte Carlo study based on the same parameters. Letus note that previous phenomenological models ' have at-tributed to the sign of the EPI the character of the con-centration profile: oscillatory or monotonic. In our exactderivation based on the underlying electronic structure,all the information is contained in the point energies.Clearly, the accurate determination of the point energiesto calculate the equilibrium concentration profile is of theutmost importance. However, we find that the parame-ters used in phenomenological theories very frequentlyhave the same sign and are approximately of the samemagnitude as those entering our formalism, thus leadingto predictions for the segregation behavior that are quali-tatively remarkably accurate.

Future applications of our formalism for systems withatomic size disparities will incorporate a microscopictheory of the elastic contributions to the free energy. Asdiscussed by Treglia et al. this may be accomplishedthrough a second moment expansion, while it is also pos-sible to treat these effects in Monte Carlo simulations.Other lines of inquiry include the feasibility of a grand-canonical description discussed above, and a furtheranalysis of the problem of an appropriate choice of refer-ence medium which is extremely important in the casethat ordered structures are present. The present formal-ism may be expanded to include magnetism which wouldpermit us to study the interplay of magnetic and compo-sitional ordering near surfaces and interfaces. Our corn-puter programs may also be easily modified to handlesegregation at internal interfaces for which exciting ex-perimental results have been reported recently. '

One of the main motivations of the present study wasto relate surface segregation to the electronic band struc-ture without introducing any adjustable parameters. Thiswas accomplished by using the recursion method withtight-binding parameters obtained through a fit to first-principles LAP W calculations. While it would be

47 SELF-CONSISTENT ORDERING ENERGIES AND. . . 73

preposterous to call this a true first-principles calculation,it is an important step towards such a scheme. In this re-gard it is of interest to mention some other recentmethods that are based on a similar philosophy. Thetight-binding LMTO method proposed by Andersen, 3ep-sen, and Sob was recently extended to treat semi-infiniteordered' and disordered' systems. For alloys themethod uses layer Green's functions techniques and im-poses the CPA condition. No segregation profiles werecalculated, but the relative magnitudes of the EPI's werestudied. In agreement with our work it was found thatthe EPI depend strongly on the position and that in cer-tain cases further-neighbor EPI's may dominate over thefirst-neighbor interactions. Applications so far have usedself-consistent bulk potential parameters, but extension toappropriate surface potentials determined from slab orsupercell calculations are envisioned. At present themethod is restricted to the use of the atomic sphere ap-proximation. Another approach based on density func-tional theory uses a generalization of the embedded atommethod' (EAM) to the case of alloy surfaces. ' Asoriginally implemented the method uses Monte Carlosimulations including atomic displacements in thegrand-canonical ensemble keeping volume, total numberof atoms, and chemical potential difterence fixed. Be-cause this is a very time consuming procedure, a recentimplementation has employed a free-energy minimiza-tion technique in the point approximation (similar to thepresent Bragg-Williams method) but with atomic vibra-tions included in the local harmonic model. Both im-plementations have been quite successful, but as they arebased on the EAM potentials which are not unique andinclude fitting parameters they are somewhat less firstprinciples then the present method and that based on theTB-LMTO scheme. '

In keeping with a similar trend for bulk phase stabilitycalculations it may be questioned to what extent anLDA-based first-principles calculation of surface pointand pair interactions may be accomplished without trans-

forming to a tight-binding Hamiltonian. While the for-malism to perform KKR-CPA calculations in a layergeometry has been around for a while, ' it is only recent-ly that the implementation of charge-self-consistent cal-culations has been accomplished. ' Because such calcu-lations are very time consuming, it seems impossible atpresent to perform the compositional self-consistencyloop discussed in the present paper, to ensure that the in-teraction energies are calculated for the equilibrium con-centration profile. This would be true irrespective ofwhether the GPM or the ECM are used to compute theenergies. The only way out of this impasse would be todetermine concentration-independent interactions in thegrand-canonical approach alluded to before, but the via-bility of this scheme remains to be established and iscurrently under study. Because slab and supercell calcu-lations have now reached the same level of accuracy asbulk calculations, one might alternatively want to startfrom first-principles calculations for the ordered corn-pounds and extract ordering energies from those by ap-plication of the Connolly-Williams method. Althoughthe number of interactions to be determined may be rath-er large, and some doubts will remain as to which interac-tions to include and which not, this might be a practicalalternative. In particular, a judicious choice of the struc-tures in the basis set might be possible, perhaps by a gen-eralization to surface structures of the special quasiran-dom structures proposed by Zunger et al. ' for bulk sys-tems.

ACKNOWLEDGMENTS

We would like to thank the scientific committee ofNATO and NSF-CNRS grants for partial financial sup-port. L.T.W. is supported by Grant No. DE-FG05-89ER-45392 from the U.S. Department of Energy.D.d.F. is supported by the Director, Office of EnergyResearch, Materials Science Division, U.S. Departmentof Energy, under Contract No. DE-AC03-76SF00098.

D. de Fontaine, in Electronic Band Structure and its Applica-tions, edited by M. Yussouf (Springer, Berlin, 1987), p. 410.

A. E. Carlsson, Solid State Phys. 43, 1 (1990).3Statics and Dynamics of Alloy Phase Transformations, edited

by P. E. A. Turchi and A. Gonis, 1992 (Kluwer, Dordrecht,in press).

4C. Sigli, M. Kosugi, and J. M. Sanchez, Phys. Rev. Lett. 57,253 (1986)~

5K. Terakura, T. Oguchi, T. Mohri, and K. Watanabe, Phys.Rev. B 35, 2169 (1987); T. Mohri, K. Terakura, T. Oguchi,and K. Watanabe, Acta Metall. 36, 547 (1988).

A. A. Mbaye, L. G. Ferreira, and A. Zunger, Phys. Rev. Lett.58, 49 (1987); S. H. Wei, A. A. Mbaye, L. G. Ferreira, and A.Zunger, Phys. Rev. B 36, 4163 (1987); A. Zunger, S. H. Wei,A. A. Mbaye, and L. G. Ferreira, Acta Metall. 36, 2239(1988).

7M. Sluiter, P. Turchi, F. Zezhong, and D. de Fontaine, Phys.Rev. Lett. 60, 716 (1988).

sM. Sluiter, D. de Fontaine, X. Q. Guo, R. Podloucky, and A. J.

Freeman, Phys. Rev. B 42, 10460 (1990).9A. Qteish and R. Resta, Phys. Rev. B 37, 6483 (1988).

L. G. Ferreira, S. H. Wei, and A. Zunger, Phys. Rev. B 40,3197 (1989);J. E. Bernard, R. G. Dandrea, L. G. Ferreira, S.Froyen, S. H. Wei, and A. Zunger, Appl. Phys. Lett. 56, 731(1990); S. H. Wei, L. G. Ferreira, and A. Zunger, Phys. Rev.B 41, 8240 (1990); K. C. Hass, L. C. Davis, and A. Zunger,ibid. 42, 3757 (1990).

'P. A. Sterne and L. T. Wille, Physica C 162-164, 223 (1989); in

Atomic Scale Calculations of Structure in Materials, edited byM. S. Daw and M. A. Schluter, MRS Symposia ProceedingsNo. 193 (Materials Research Society, Pittsburgh, 1990); G.Ceder, M. Asta, W. C. Carter, M. Kraitchman, D. de Fon-taine, M. E. Mann, and M. Sluiter, Phys. Rev. B 41, 8698(1990);L. Szunyogh and P. Weinberger, ibid. 43, 3768 (1991).R. Kikuchi, Phys. Rev. 81, 998 (1951); D. de Fontaine, SolidState Phys. 34, 73 (1979).

' Monte Carlo Methods in Statistical Physics, edited by K.Binder (Springer, Berlin, 1979).

H. DREYSSE, L. T. WILLE, AND D. de FONTAINE

'4Interfacial Segregation, edited by W. C. Johnson and J. M.Blakely (American Society for Metals, Cleveland, 1979}.

~5W. M. H. Sachtler and R. A. Van Santen, Appl. Surf. Sci. 3,121(1979).

F. F. Abraham and C. R. Brundle, J. Vac. Sci. Technol. 18,506 (1981)~

P. M. Ossi, Surf. Sci. 201, L519 (1988).~sSurface Segregation Phenomena, edited by P. A. Dowben and

A. Miller (CRC, Boca Raton, 1990).D. McLean, Grain Boundaries in Metals (Clarendon, Oxford,1957); M. P. Seah, J. Phys. F 10, 1043 (1980).

20R. Lipowsky, Phys. Rev. Lett. 49, 1575 (1982); R. Lipowskyand W. Speth, Phys. Rev. B 28, 3983 (1983); D. M. Kroll andG. Gompper, ibid. 36, 7078 {1987);G. Gompper and D. M.Kroll, ibid. 38, 459 (1988)~

F. L. Williams and D. Nason, Surf. Sci. 45, 377 (1974); D. Ku-mar, A. Mookerjee, and V. Kumar, J. Phys. F 6, 725 (1976).F. F. Abraham, Phys. Rev. Lett. 46, 546 (1981).A. R. Miedema, Z. Metallk. 69, 455 (1978).J. R. Chelikowsky, Surf. Sci. 139, L197 (1984).

25G. Kerker, J. L. Moran-Lopez, and K. H. Bennemann, Phys.Rev. B 15, 638 (1977); S. Mukherjee, J. L. Moran-Lopez, V.Kumar, and K. H. Bennemann, ibid. 25, 730 (1982}.P. Lambin and J. P. Gaspard, J. Phys. F 10, 2413 (1980);J. P.Muscat, J. Phys. C 15, 867 {1982).R. Riedinger and H. Dreysse, Phys. Rev. B 27, 2073 (1983);H.Dreysse and R. Riedinger, ibid. 28, 5669 (1983); M. C.Desjonqueres and D. Spanjaard, ibid. 35, 952 {1987).S. Mukherjee and J. L. Moran-Lopez, Surf. Sci. 189/190, 1135(1987).

29D. A. Papaconstantopoulos, Handbook of the Band Structureof Elemental Solids iPlenum, New York, 1986).O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571(1984); O. K. Anderen, O. Jepsen, and M. Sob, in ElectronicBand Structure and its Applications, edited by M. Yussouf(Springer, Berlin, 1987), p. 1; J. Kudrnovsky, V. Drchal, andJ. Masek, Phys. Rev. B 35, 2487 (1987);J. Kudrnovsky and V.Drchal, ibid. 41, 7515 (1990).

3~J. S. Faulkner, Prog. Mater. Sci. 27, 1 (1982).F. Ducastelle, in Alloy Phase Stability, edited by G. M. Stocksand A. Gonis (Kluwer, Dordrecht, 1989},p. 293.F. Ducastelle, Order and Phase Stability in Alloys (North-Holland, Amsterdam, 1992).

s4A. Gonis, Green Functions for Ordered and Disordered Systems (North-Holland, Amsterdam, 1992).P. Soven, Phys. Rev. 156, 809 (1967); B. Velicky, S. Kirkpa-trick, and H. Ehrenreich, ibid. 175, 747 (1968);R. J. Elliott, J.A. Krumhansl, and P. L. Leath, Rev. Mod. Phys. 46, 465(1974).N. F. Berk, Surf. Sci. 48, 289 (1975).

s7P. Weinberger, Electron Scattering Theory for Ordered andDisordered Matter (Clarendon, Oxford, 1990}.F. Ducastelle and F. Gautier, J. Phys. F 6, 2039 (1976); G.Treglia, F. Ducastelle, and F. Gautier, ibid. 8, 1437 (1978); F.Ducastelle and G. Treglia, ibid. 10, 2137 (1980); A. Bieber, F.Ducastelle, F. Gautier, G. Treglia, and P. Turchi, Solid StateCommun. 45, 585 (1983). The preceding papers describe ap-plications of the GPM within the TB-CPA formalism. For anextension to the KKR-CPA see P. Turchi, G. M. Stocks, W.H. Butler, D. M. Nicholson, and A. Gonis, Phys. Rev. B 37,5982 (1988).

9M. O. Robbins and L. M. Falicov, Phys. Rev. B 29, 1333(1984).

~0A. Gonis and J. W. Garland, Phys. Rev. B 16, 2424 (1977); A.

Gonis, W. H. Butler, and G. M. Stocks, Phys. Rev. Lett. 50,1482 (1982).

4~A. Gonis, X. G. Zhang, A. J. Freeman, P. Turchi, G. M.Stocks, and D. M. Nicholson, Phys. Rev. B 36, 4630 (1987).

4 B. L. Gyorffy and G. M. Stocks, Phys. Rev. Lett. 50, 374(1983).J. W. D. Connolly and A. R. Williams, Phys. Rev. B 27, 5169(1983);A. E. Carlsson, ibid. 35, 4858 (1987); M. Sluiter and P.E. A. Turchi, ibid. 40, 11 215 {1989).

~J. M. Sanchez, F. Ducastelle, and D. Gratias, Physica A 128,334 (1984).

45H. Dreysse, A. Berera, L. T. Wille, and D. de Fontaine, Phys.Rev. B 39, 2442 (1989).G. Treglia and B.Legrand, Phys. Rev. B 35, 4338 (1987).

~7B. Legrand and G. Treglia, in The Structure of Solid SurfacesII, edited by J. F. Van der Veen and M. A. Van Hove(Springer, Berlin, 1987), p. 167.G. Treglia, B. Legrand, and F. Ducastelle, Europhys. Lett. 7,575(1988).

49G. Treglia, B. Legrand, and P. Maugain, Surf. Sci. 225, 319(1990).B. Legrand, G. Treglia, and F. Ducastelle, Phys. Rev. B 41,4422 (1990).F. Ducastelle, B. Legrand, and G. Treglia, Prog. Theor. Phys.Suppl. 101, 159 {1990).G. Treglia, B. Legrand, J. Eugene, B. Aufray, and F. Cabane,Phys. Rev. B 44, 5842 (1991).

53A. Bieber and F. Gautier, J. Phys. Soc. Jpn. 53, 2061 (1984); Z.Phys. B 57, 335 (1984); Acta Metall. 34, 2291 (1986); 35, 1839(1987).

54M. Asta, C. Wolverton, D. de Fontaine, and H. Dreysse, Phys.Rev. B 44, 4907 (1991);C. Wolverton, M. Asta, H. Dreysse,and D. de Fontaine, ibid. 44, 4914 (1991).F. Solal, R. Caudron, F. Ducastelle, A. Finel, and A. Loiseau,Phys. Rev. Lett. 58, 2245 (1987).V. Kumar and K. H. Bennemann, Phys. Rev. Lett. 53, 278(1984); J. M. Sanchez and J. L. Moran-Lopez, Phys. Rev. B32, 3534 (1985);Surf. Sci. 157, L297 (1985).

57J. L. Moran-Lopez, F. Mejia-Lira, and K. H. Benneman,Phys. Rev. Lett. 54, 1936 (1985); J. M. Sanchez, F. Mejia-Lira, and J. L. Moran-Lopez, ibid. 57, 360 (1986).A. Berera, H. Dreysse, L. T. Wille, and D. de Fontaine, J.Phys. F 18, L49 {1988);A. Berera, Phys. Rev. B 42, 4311(1990).

5sJ. Friedel, in The Physics of Metals, edited by J. M. Ziman

(Cambridge University Press, London, 1969), p. 138.R. Haydock, Solid State Phys. 35, 215 (1980); The RecursionMethod and Its Applications, edited by D. G. Pettifor and D.L. Weaire (Springer, Berlin, 1985).

6 N. R. Burke, Surf. Sci. 58, 349 (1976}.T. L. Einstein and J. R. Schrieffer, Phys. Rev. B 7, 3629(1973}.R. Pandit and M. Wortis, Phys. Rev. B 25, 3226 (1982).

~J. Vrijen and S. Radelaar, Phys. Rev. B 17, 409 (1978); S. an

Mey, Z. Metallk. 78, 502 (1987); S. Srikanth and K. T. Jacob,Mater. Sci. Technol. 5, 427 (1989).

65J. L. Murray, Bull. Alloy Phase Diagrams 2, 335 (1982).M. Sluiter and P. E. A. Turchi, in Alloy Phase Stability andDesign, edited by G. M. Stocks, D. D. Pope, and A. F.Giamei, MRS Symposia Proceedings No. 186 (MaterialsResearch Society, Pittsburgh, 1991).H. Dreysse, G. Ceder, D. de Fontaine, and L. T. Wille, Vacu-um 41, 446 (1990).H. Dreysse, L. T. Wille, and D. de Fontaine, Solid State Com-

47 SELF-CONSISTENT ORDERING ENERGIES AND . . ~ 75

mun. 78, 355 (1991).J. A. Blackman, D. M. Esterling, and N. F. Berk, Phys. Rev.B 4, 2412 (1971); D. A. Papaconstantopoulos, A. Gonis, andP. M. Laufer, ibid. 40, 12 196 (1989).H. Shiba, Prog. Theor. Phys. 46, 77 (1971).J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).P. Turchi, G. Treglia, and F. Ducastelle, J. Phys. F 13, 2543(1983).C. M. M. Nex, Comput. Phys. Commun. 34, 101 (1984); R.Haydock and C. M. M. Nex, J. Phys. C 18, 2235 (1985); M.Foulkes and R. Haydock, ibid. 19, 6573 (1986);M. U. Luchiniand C. M. M. Nex, ibid. 20, 3125 (198?).

~~N. Beer and D. G. Pettifor, in Electronic Structure of ComplexSystems, edited by P. Phariseau and W. M. Temmerman (Ple-num, New York, 1982).P. van der Plank and W. M. H. Sachtler, J. Catal. 7, 300(1967); 12, 35 {1968);W. M. H. Sachtler and P. van der Plank,Surf. Sci. 18, 62 (1969); V. Ponec and W. M. H. Sachtler, J.Catal. 24, 250 (1972).K. Watanabe, M. Hashiba, and T. Yamashina, Surf. Sci. 61,483 (1976).D. T. Ling, J. N. Miller, I. Lindau, W. E. Spicer, and P. M.Stefan, Surf. Sci. 74, 612 (1978).

~8Y. S. Ng, T. T. Tsong, and S. B. McLane, Jr., Phys. Rev. Lett.42, 588 (1979); J. Appl. Phys. 51, 6189 (1980); J. Vac. Sci.Technol. 17, 154 {1980).

~ L. E. Rehn, S. Danyluk, and H. Wiedersich, Phys. Rev. Lett.43, 1764 (1979).K. Wandelt and C. R. Brundle, Phys. Rev. Lett. 46, 1529(1981).P. J. Durham, R. G. Jordan„G. S. Sohal, and L. T. Wille,Phys. Rev. Lett. 53, 2038 (1984).

s2N. Q. Lam, H. A. Hoff, H. Wiedersich, and L. E. Rehn, Surf.Sci. 149, 517 (1985).T. Sakurai, T. Hahizume, A. Jimbo, A. Sakai, and S. Hyodo,Phys. Rev. Lett. 55, 514 (1985).

s4L. E. Rehn, H. A. Hoff, and N. Q. Lam, Phys. Rev. Lett. 57,780 (1986).H. H. Brongersma, P. A. J. Ackermans, and A. D. vanLangeveld, Phys. Rev. B 34, 5974 (1986)~

T. Sakurai, T. Hashizume, A. Kobayashi, A. Sakai, S. Hyodo,Y. Kuk, and H. W. Pickering, Phys. Rev. B 34, 8379 (1986).T. Berghaus, C. Lunau, H. Neddermeyer, and V. Rogge, Surf.Sci. 182, 13 (1987).

8V. Rogge and H. Neddermeyer, Phys. Rev. B 40, 7559 (1989).R. G. Donnelly and T. S. King, Surf. Sci. 74, 89 (1978).

OJ. Urias and J. L. Moran-Lopez, Surf. Sci. 128, L205 (1983).H. F. Lin, C. L. Wang, and C. Y. Cheng, Solid State Commun.59, 253 (1986); C. Y. Cheng, Phys. Rev. B 34, 7400 (1986).L. T. Wille and P. J. Durham, Surf. Sci. 164, 19 (1985).

S. Mukherjee and J. L. Moran-Lopez, Progr. Surf. Sci. 25, 139(1987).

"S.M. Foiles, Phys. Rev. B 32, 7685 (1985);40, 11 502 (1989).J. Eymery and J. C. Joud, Surf. Sci. 231, 419 (1990).R. Najafabadi, H. Y. Wang, D. J. Srolovitz, and R. LeSar,Acta Metall. Mater. 39, 3071 (1991);H. Y. Wang, R. Najafa-badi, D. J. Srolovitz, and R. LeSar, Phys. Rev. B 45, 12028(1992).

9~M. A. Krivoglaz, Theory of X Ray -and Thermal neutronScattering by Real Crystals (Plenum, New York, 1969); P. C.Clapp and S. C. Moss, Phys. Rev. 142, 418 (1966); 171, 754(1968); S. C. Moss and P. C. Clapp, ibid. 171, 764 (1968). Foran extension of the Krivoglaz-Clapp-Moss approach see D.M. Kroll and H. Wagner, ibid. 42, 6531 (1990).V. Gerold and J. Kern, Acta Metall. 35, 393 (1987); W.Schweika and H. G. Haubold, Phys. Rev. B 37, 9240 (1988);W. Schweika and A. E. Carlsson, ibid. 40, 4990 (1989).D. de Fontaine and J. Kulik, Acta Metall. 33, 145 (1985).

~J. Kanamori and Y. Kakehashi, J. Phys. (Paris) Colloq. 38,C7-274 (1977).'R. H. Brown and A. E. Carlsson, Solid State Commun. 61,743 (1987).H. Dreysse, L. T. Wille, and D. de Fontaine (unpublished).S. M. Kuo, A. Seki, Y. Oh, and D. N. Seidman, Phys. Rev.Lett. 65, 199 (1990); J. G. Hu and D. N. Seidman, ibid. 65,1615 (1990).V. M. Tapilin, Surf. Sci. 206, 405 (1988).J. Kudrnovsky, B. Wenzien, V. Drchal, and P. Weinberger,Phys. Rev. B 44, 4068 (1991);J. Kudrnovsky, P. Weinberger,and V. Drchal, ibid. 44, 6410 (1991);V. Drchal, J. Kudrnov-sky, L. Udvardi, P. Weinberger, and A. Pasturel, ibid. 45,14 328 (1992).M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50, 1285 (1983);Phys. Rev. B 29, 643 (1984); S. M. Foiles, M. I. Baskes, andM. S. Daw, ibid. 33, 7983 (1986).

~o~J. B. Pendry, Loto Energy Elec-tron Di+raction (Academic,London, 1974); L. T. Wille and P. J. Durham, J. Phys F 13,L117 (1983); J. H. Kaiser, P. J. Durham, R. J. Blake, and L.T. Wille, J. Phys. C 21, L1159 (1988); J. M. MacLaren, S.Crampin, D. D. Vvedensky, and J. B. Pendry, Phys. Rev. B40, 12 164 (1989).A. Gonis, X. G. Zhang, J. M. MacLaren, and S. Crampin,Phys. Rev. B 42, 3798 (1990);X. G. Zhang, P. J. Rous, J. M.MacLaren, A. Gonis, M. A. Van Hove, and G. A. Somorjai,Surf. Sci. 239, 103 (1990); S. Crampin, R. Monnier, T. Schul-ten, G. H. Schadler, and D. D. Vvedensky, Phys. Rev. B 45,464 (1992).A. Zunger, S. H. Wei, L. G. Ferreira, and J. E. Bernard,Phys. Rev. Lett. 65, 353 (1990).