L. %. MEZEY et 81.: Surface Segregation Profiles in Binary Alloys 323

phys. stat. sol. (a) 75, 323 (1983)

Subject classification: 1.3 and 1.G; 21.1; 21.G

Institute of Physics, Budapest Technical University1) ( a ) and Department of Materials Science and Engineering, The Graduate School at Nagatsuta, Tokyo Institute of Technology, Yokohania2) ( b )

Monotonic and Alternating Surface Segregation Profiles in Binary Alloys BY L. Z. MEZEY (a), 0. N~SHIKAWA (b), and cJ. GIBER (a)

A complex calculation method of the characteristics of surface segregation is developed recently. The sufficient conditions of monotonic and alternating surface segregation profiles are established using the method. Then the monotonic profiles are related to the preference for pair formation from like atoms while the alternating ones with that from unlike atoms. The applicability of the conditions deduced here is shown for some binary solid alloys.

Eine komplexe Berechnungsmethode der Charakteristiken der Oberflachensegregation ist neulicli entwickelt worden. Approximationsbedingungen fdr monotone und alternierende Oberflachen- Segregationsprofile werden mit dieser Metliode aufgestellt. Die monotonen Profile werden dann mit der Neigung zur Paarbildung aus einheitlichen Atomen soivie die alternierenden Profile mit der aus verschiedenen Atomen in Verbindung gesetzt. Die Anwendbarkeit der abgeleiteten Be- dingungen wird an binaren Festkorperlegierungen gezeigt.

1. Introduction

It is well known that structural differences, as e.g. those existing between two phases, are associated with compositional differences in therniodynamic equilibrium (and in non-equilibrium) states. The structural difference existing between the topmost surface layer and the bulk phase of solids was shown to be associated with compositional differences, too. This compositional difference extends to several layers below the topmost one, giving characteristic surface segregation profiles.

In view of their large practical importance these profiles are the subject of extended research by the new local analytic techniques (AES, SIMS, AP atom-probe, etc.), especially for binary alloys. The segregation profiles obtained this way were found to be either of the monotonic or the alternating type. In the first case the component enriching in the topmost layer (as compared with the bulk composition) was found to be enriched in the second one as well, to a smaller extent, in the third one be enriched again, to an even smaller extent, etc. In the second case the component enriching in the topmost layer was found to be depleted (as compared with the bulk composition) in the second layer, enriched again in the third one, etc. These local coinpositional changes again were found to be the smaller (in absolute value) the farther the layer is froin the topmost one.

I t is clear that to obtain correct concentration profiles very sensitive analytic methods should be used. The conventional local analytic procedures encounter differ- ent problems : adsorption from the gaseous phase, preferential sputtering, knock-in effects, signal interpretation, etc. For these reasons the objective situation is often hard to reconstruct from the experimental information. It is known, e.g., that it is

l) Budafoki h t 8, H-1521 Budapest, Hungary. 2, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan.

21 *

324 L. Z. MEZEY, 0. NISHIKAWA, and J. GIBER

troublesome to use AES information for exact layer-to-layer descriptions. Using the atom-probe (AP) niethod a layer-to-layer analysis becomes possible [l, 21. By this [l, 21 and by the other methods alternating segregation profileswere observed repeatedly.

There is a large number of interesting theoretical works on surface segregation in solids, however, as is known, the problem of quantitative description has not been solved so far even in the simplest and most investigated case of the binary substitut'io- nal nionophase alloys. This is basically because of two reasons. First, the component surface free energies, absolutely needed in such descriptions, are largely unknown [3] or described with inadequat,e accuracy. Secondly, the question of the unified theore- tical description of the two basic types of real mixture effects is unsolved in regard to their role in multilayer surface segregation. The local ordering [4] and the lattice distortion [5 ] effects are presently described separately, though trial and error at'teiiipts were made for their combination a t least for the topmost surface layer (e.g. in [4]) in some concrete calculations.

In view of these difficulties of the theoretical description, the question of the type of the surface segregation profile was only scarcely investigated. In the fundamental work of Williams and Nason [6] the occurrence of the two basic types was predicted and these were related with the values of the regular mixture parameter. Kuniar [4], on a similar basis, explained the occurrence of the two basic types in numerical cal- culations for two alloy systems. The limited validity of the single or two-parameter description of real alloys is, however, known even for the bulk phases. In the present work, on the basis of a new theory of surface segregation in real alloys, an att,enipt is made to obtain general conditions for the two basic types of segregation profiles.

2. Outline of the Complex Calculation of Surface Segregation (CCSS) Method A new method of description of surface segregation was developed recently. It consists of three main parts, as described in some det'ail in other works [7, 81 and in a more detailed way in [9].

In the first, part the g c free enthalpy of the system c consisting of a bulk phase b and of the surface layers k = 1 , 2 , ..., m is defined as a function of the component mole numbers nf (i = 1, 2, ..., K ) . Using this the general conditions for the stable equilib- rium concentration profile' (described by the component equilibrium mole fractions 2:) are deduced.

In the second part the y;@ component molar surface free enthalpies are calculated for any temperature of interest. Here y; is the surface free enthalpy and @is the molar surface area. The procedure is applicable for all solid chemical elements (see also [lo]) in good agreement with the experimental information.

In the third part further functions, appearing in the general formulae of the first part, are described in good approximation, a t present for binary alloys. These func- tions are the @: component molar surface areas and the p ~ ~ k a n d p ~ ~ b component excess chemical potentials.

The method gave results in good agreement with experimental information for several binary solid and liquid alloys [7 to 91. In the following our problem will be discussed using the results of this method.

3. Discussion of Some Aspects of the CCSS The CCSS method gives the following conditions for the equilibrium segregation profiles (the state of equilibrium is referred to by bars): 1

Monotonic and Alternating Surface Segregation Profiles in Binary Alloys 325

The system of K m equations (1) gives the ( K - 1) m equilibrium values {Zf} and the m-values {jP} as well. These latter are the layer contributions to the total surface free enthalpy 7. I n case of general solutions the stable equilibrium one, with y*, is chosen as the one with the minimum value of the sum of the layer contributions Tk.

The p:ak layer excess chemical potentials are obviously determined not only bv the actual (equilibrium or non-equilibrium) local layer compositions {zt}, but the actual conipositions of the neighbouring layers as well. I n the CCSS this fact is reflected in an approxiniate way by the following equations (valid in the general case of non-equilib- riuni profiles as well) :

p;,k = Lyp.,e(zk--1) + (1 - 20L) p;(zk) + ap;(zk+l) ( i = 1, 2 ; P = 2 , 3, ...) m) . ( 2 )

The equations were given here only for P 2 2 , the mole fraction of one of the coni- ponents (i = 2 ) being denoted by x. For the topniost layer (k = 1 ) the first term on the right-hand side of (2) is substituted by ya@a [7 to 91. The physical ineaning and the determination of the quantities a is discussed elsewhere [8, 91. I n the simplest nearest- neighbour interaction model, neglecting relaxation effects, & is the quotient of the nearest neighbours above or below the layer in question. For our present purposes it suffices, that in any description, by definition, the result is

O < & < l . (3) It was shown by the analysis of several numerical results on real binary solid and

liquid alloys and by theoretical considerations as well [9,11] that the owerwhelming

contribution to Yk was found to be about 0.01 to

0.1 yo of the total value only. Therefore, in a good approximation the conditions

na

th.=2 comes from fr = 1. The sum

(4) may be used.

The use of these conditions and of approximation (4) leads to the following result for the Xk mole fractions of one of the cornponents of a binary mixture (the asterisk will be omitted frorn now on) :

- y k = 0 ( k = 2, 3, ...) m)

+ &[p;(X")

RT

( k = 2 ,3 , ...) m) . ( 5 )

For the topmost layer the quantity y1 may be excluded by using condition ( 1 ) for the other component. This gives

0 0 X I = x b exp ([ @i(X') -1 p n (-) 1 - X I + rg - & -nr @;(z') 1 - z b

RT RT

RT 4- (1 - 201)

326 L. Z. ~ I E Z E Y , 0. P ~ ’ I ~ H I K A W A , and J. GIBER

In the case of ideal mixtures all the terms containing p: are zero and the solution 2’ is determined only by the ternis y@y/ h‘7’. It is easily concluded froni (6) that then in the case §: = @: we obtain

If we have

the result is

I t is obvious that this holds if and only if we hare

21 > 2-b , (10)

that is, if coiiiponent 2 is enriched in the topmost surface layer. This may be com- pared with other results, stating the enrichment of the component with the smaller value of y: [12, 131 in the topmost surface layer.

In substitutional alloys the atomic radii and the structures of the components are not much different and, consequently, the assumption @: 3 @: holds in good first approximation. These alloys are further known to be not much far from the ideal behaviour in the sense that the quantities ,u:IRT are not large in their absolute values as compared e.g. with the case of interstitial ones. It is obvious that in the case of

y v : > y@! (11)

(if the difference is sufficiently large) the result (10) is again obtained, but with the actual value of Z1/xb modified by the other factors of surface segregation.

It is known that in general one of the components is enriched anyway in the top- most layer, be it the one determined by the simplified conditions discussed now or not. The problem is now to determine the character of the segregation profile using this fact.

4. The Criteria of Monotonic and Alternating Segregation in the CCSS

The condition ( 5 ) is easily transformed in the following equivalent froni:

Ol[p@fi-l) - p9(x”] + n[pg(.fi+l) - ,u;(z”] + + (1 - 2~w) [p;(Zt) - ,ug(zb)] + RT ln(Zk/xb) = 0 (2 5 k m) . (12)

Let the minimum value of all the x-values (xb, Z1, 9, ..., Zm) be denoted as %,in and the maximum one as zmax. Let us assume that the following relation holds:

(13)

It is obvious t,hat whether the profile is a monotonic or an alternating one, the same component is enriched in the layers X: - 1 and k + 1. If the mole fraction of this is chosen as x, then two of the terms on the left-hand side of (12) are positive by assump-

Monotonic and Alternating Surface Segregation Profiles in Binarj All03 s 327

tion (13) and by condition (3) . From this we obtain that

(If we had Zh = x”, then two zero terins would be added to the positive ones in (12): if we had xk > XI’, then two positive ones would be added. I n neither of these cases (12) is satisfied.) Relation (14) implies an alternating segregation profile. u hich is the consequence of assumption (13).

Tf u e have

then the first two terms on the left-hand side of (12) are negative. I t is easily clenion- strated, in a similar way as before, that then only in the case of

X k > Xb (16) (12) is satisfied. This condition implies a monotonic segregation profile which is the consequence of assumption (15).

I n conclusion i t niay be stated that conditions (15) and (13) are sufficient for the monotonic and alternating segregation profiles, respectively.

5. Thermodynamic Conditions for Local Clustering and Local Ordering

Let a binary substitutional alloy, consisting of A’ atoms of the components 1 and 2 with a coordination nuniber z and with a composition xk = x be regarded. It consists. obviously, of pairs of the kinds 1-1, 1-2, and 2-2. Let the nuniber of such pairs be denoted by P,, ( i , j = 1 , 2 ; PZ1 = PI, obviously). For these numbers the following conditions, known from the theory of alloys, hold :

(17) (18)

N(1 - 2) z = 2P1, + PI, , N x z = P12 + 2P2, .

The factors two arise because in each P,, pair two of the lattice sites, occupied by the atonis i , are involved. The conditions leave the value of P,, as an independent vari- able, changing between the limits 0 and N x z (in the case x 5 f) and between the limits 0 and X(l -- x) z (in the case x 2 a).

If there is a totally random distribution of the “atoms” on the lattice sites, then t h e P, average number of “atoms” 2 around “atonis” 1 is x and we have

Pi2 = x(l - x) Nx . (19) The nuniber P, (and, thus, PI, = X(1 - 2) zP2) may be determined experimentally. e.g. by X-ray and neutron diffraction methods 1141 and it is found that in general PI, += Pi,. The deviation is nieasared by the local order parameter 01’ defined as

In the case of PI, > Pi2 (a’ < 0) there is a preference for the 1-2 pairs (local ordering) while in the case of Plz < Pi2 (a’ > 0) there is a preference for the 1-1 and 2-2 pairs (local clustering).

The quantity PI, may be obtained in any definite theoretical niodel of the alloy as well, which gives the Ge excess free enthalpy of mixing as a certain function of P12 (a t constant values of teniperature T, pressure P, and of x). The PI, equilibrium value is found by niiniinizing Ge in respect to PI,.

328 L. Z. MEZEY, 0. NISHIKAWA, and J. GIBER

In the Guggenheini description of the quasi-chemical model the following expression is obtained:

- Plz = Piz[l - x(1 - x) (e2rY/rRT - 111 . (21)

Here W is the regular mixture interaction parameter, which is calculated from t,he H e enthalpy of mixing by

It is easy to see that in the case of W < 0 we get c2 > Pf2 and in case of W > 0 the relation p12 < Pi2 holds.

In the general case the quantity fIc/x(l - x) is not constant, as it is assumed in this description. Then, as is easy to see, a function W ( x ) may be defined by ( 2 2 ) . The main reason for i t being not constant is the fact that the components are in general not of equal atomic radii as is assunied in the quasi-chemical approxiniation. This leads to lattice distortion effects. It is easy to see, however, that since F12 was found by ininimalization at constant x, (21) remains valid in this more general case as well. Consequently, the conditions of Iocal ordering and clustering remain the sanie.

6. Tho Connection between the Local Clustering and Ordering Effects and the Types of the Segregation Profiles

In the quasi-chemical model the following form of p i is obtained:

p; = (1 - X)Z ( W + 2gx - 3gx2) , where

The use of (23) gives easily

(23)

(24)

If (13) holds for ull iwlzics of x (0 5 x 1) then it follows that we have U’ < 0 neces- sarily. If (15) holds for all values of x (0 5 x 5 1) then the relation W > 0 follows. We note that the conditions using only W are equal to those obtained in the regular mixture model on a more restricted basis [6]. I n this way the result is obtained that ap$/lax > 0 means an alternating segregation profile and a tendency for local ordering as well, while ap;/ax < 0 means a monotonic segregation profile and a tendency for local clustering as well. This may very easily be given a physical interpretation.

One kind of the atonis is enriched in the topniost surface layer as it was discussed earlier. This same coniponent is enriched in the third layer. The tendency for local clustering obviously leads to enrichment of this coniponent in the second layer, while the one for local ordering leads to its depletion there. This reasoning may be continued further, connecting alternating segregation profiles with local ordering and monotonic ones with local clustering effects. It is clear that this second kind of discussion is valid independent of the model used for the description of the local conipositions around the “atoms” of the mixture.

Monotonic and Alternating Surface Segregation Profiles in Binary Alloys 32 9

7. Discussion of the Numerical Results

Some binary alloys were investigated by us so far [7 to 91. The basic therinodynaniiu information applicable for the description of the functions p t is found in the book of Hultgren et al. [15]. The measurements on these alloys were made by the SEX rriethod and their results were obtained froni the sources indicated in [7 , 81. A detailed con- centration profile was measured for the CuNi alloy a t 873.2 K at several bulk concen- trations. For the AgPd and AuCu alloys inforination on the sampling depth was given. This is 80 nni in the first one and 40 nni a t lower electron energies in the second one. Comparing these data with the average atomic diameters of the components i t inay be concluded that the sampling depth equals about three nionolayers in the first case and about two nionolayers in the second one. Our calculations, the results of which were shown to be in relatively good agreement with the experiniental information [7, 91, gave alternating segregation profiles in the cases of AgPd and AuCu. The cal- culated segregation profiles and the values of the function ,$/RT are shown in Table l.

Table 1 Segregation profiles a.nd the values of ,&‘/RT for some bina.ry alloys

AgPd; 673.2 K iluCu; 883.2 K CuNi; 873.2 I<

X” = 0.3 x” = 0.5 xb = 0.25 d‘ = 0.75 ~b = 0.317

r1 0.0452 0.1493 0.0648 0.6174 0.0868

& ‘ K T -3.9464 --2.8618 - 1.6630 -0.4555 1.0322 .z 0.3883 0.6365 0.2544 0.7720 0.2903

pS/RT - 1.0001 -0.2484 - 1.3674 -0.1675 0.7172 l.3 0.2814 0.4830 0.2496 0.7464 0.3137

,&/RT -1.7530 -0.6780 -1.3781 -0.2065 0.6842 x4 0.3046 0.5033 0.2500 0.7506 0.3166

p:lRT - 1.5939 -0.6044 -1.3772 -0.1999 0.6801 .r5 0.2989 0.4994 0.7499 0.3170

,II$IRT - 1.6321 -0.6181 -0.2010 0.6796 .x6 0.3003 0.5001 0.7500

- 1.6227 -0.6156 0.2999 0.5000

0.2008

pEIRT -1.6253 -0.6160

It, may be concluded from these data that the validity of the conditions (13) and (15) is illustrated well in these cases.

8. On the GaMo and IrSn Alloys

Experimental investigations using the A P method mentioned in the Introduction were made for these alloys [ 2 ] . It was observed that a t a temperature slightly below

3 30 Id. Z. i l l ~ z ~ r , 0. N'ISHIKAWA, and J. GIBER

900 K and at 900 K and above the GaMo surface was covered by Ga and below thio, an alternating segregation profile was observed. The biillr composition (the average composition of the mixed layer) was x') = 0.234 in the first case while it was 0.75 in the other ones, the latter due to the formation of the ordered alloy a t 900 K. The results on IrSn (obtained at temperature slightly above 500 K) showed an Sn layer on the top with an alternating segregation profile below. This result was obtained at zb = 0.369.

Both alloys are known to have limited miscibility and therefore they are not mono- phase. It was shown already that the basic eqnations of surface segregation are valid under any conditions [8, 91. Consequently the general conditions of (13) and (15) obtained from the general equations hold in the case of non-nionophase alloys as well.

A inore serioiis problem is the lack of therniodynanuc inforination on the mixtures. In a first approximation this problein may be treated in the followipg way. It is established that the heats of mixing of binary alloys fornied froiii ''like'' coniponents are positive, while of those fornied from "unlike" components are negative [lcj]. Classification of the metallic components may be made on the basis that they are either non-transitional or transitional ones. Since Ga and Sn are non-transitional metals and 310 and I r are transitional ones, the above rille iniplies that their heats of mixing should be negative. This expectation is somewhat confirmed by the in- vestigation of some alloys of these corriponents. The data on alloys [15] show that there is reliable information from all the possible transitional-non-transitional con)- binations for the alloys of Ga with the transitional nretals Ag, Cu, and Xi and for the alloys of Sn with the transitional metals Ag, L411, Cn, Xi, and Pd. These data are either directly connected with H' or they refer to the GC excess free enthalpy of iiiixing or to the quantities puf directly. The entropy of mixing S' will he assiinied to be negligible, consequently the equality H e = G' will be used in a first approximation. Tnvestigation of the inforination on the alloys mentioned above shows that Re is then negative in all cases when we have thermodynamic information. 1 n a rough approximation equation (22) may he used to describe this quantity in all cases. It is clear that then the fact that the He-vaIues are expected to be negative gives explanation for the alternating profiles.

The fact that Ga and Sn are enriched in the first layers of those alloys can be ex- plained easily by the CCSS method. The basic equations of the segregation profiles, as discussed before, show that in the first layer the coniposition may be determined mainly by the quantities y:@/ RT. This is the case if the difference of these values of the coniponents is relatively large as conipared with the role of the mixing effects expressed by the changes of the quantities pY/RT. Since both alloys are substitutional ones, the latter quantities are relatively sniall in absolute value anyway. The quanti- ties y,"@,"/RT are calculated (for Ga and Mo, respectively) as 7.511 and 15.124 (at 890 K) and as 7.329 and 14.759 (at 910 K). The corresponding values for Sn and Ir are 9.004 and 25.674 (at 510 K). These values explain the results that Ga and Sn are enriched in the first layers of the respective alloys.

ilclmozoledgement

The authors would like to express their gratitude to Prof. Dr. A. Kbnya, &lenlber of the Hungarian Academy of Sciences for his kind help in their work.

Jlonotoniu and Alternating Surface Segregation Profiles in Binary Allojs 331

References

[l I T. T. S. T’. Kimsaswa~iy , S. B. MCLANE and E. W. MULLER, Appl. Phya. Letters

[ 2 ] 0. NISHLKAWA, 11. NACHI, K. KUKIHARA, Y. TSUXASHIXA, and Y. HAHA, phys. stat. sol. (a)

[3] W. M. 13. SX~HTLER and R. A. VAN SANTEN, Appl. Surfarc Sci. 3, 121 (1979). [4] V. K ~ M A R . Phys. Rev. B 23, 3755 (1981). [5] Y. JV. Lers and H. I. AAROSSON, Surface Sci. ‘35, 227 (1980). [6] F. I,. WILLIAMS and D. XASON, Surface Sci. 45, 377 (1974). [7] L. Z. MEZEY and J. GIBER, Proc. VII. Tnternat. Conf. Solid Surface and 111. Enropean Conf.

Surface Sci., Cannes (France), 22 to 26 Sept. 1980, Ed. D. D E G R ~ S and Jf. COSTA, Sociktk Franpaise du Vide, Paris 1980 (p. 38).

[8] L. Z. MEZEY and 5. GIBER, Proc. IV. European Conf. Surface Sci., Munster (FRG), 14 to 16 Sept. 1981, Ed. A. BENRIKGHOVEN, North-Holland Publ. Co.. Amsterdam 1982 ; Surf<xcc Sci. 117, 220 (1982).

23, 1 (1973).

(i7, 461 (1981).

[9] L. Z. MEZEY and J. GIBER, Periodica Polytechnica (Budapest), accepted in 1982. [lo] 1,. Z. MEZEY and J. GIBER, Japan. J. appl. Phys. 21, 1569 (1982). [11] L. z. n l e z E Y and J. GIBER, Surface Sri. 137, L98 (1983). 1121 R. A. Vas S ~ H T E N , R. A. TONEMAN, and R. Bouwn~ix, Surface Sci. 47, 64 (1975). [13] S. M. OVERBURY and G. A. SOMORJAI, J. chem. Phys. 66, 3181 (1977). [14] R. SWALIN. Thermodynamics of Solids, Wiley, Ne\r York 1972. [15] R. HULTGREEK, P. D. DESAI. D. T. HAWKINS, &I. GLEISER. and K. K. KELLKY, Selected

Values of Thrrrnodynamic Properties of Binary Alloys, Amer. Soc. Metals, Metals Park, Ohio 1973.

[l6] J. VERO and itl. K ~ L D O X , Netallurgy, Tanlronyvkiado, Budapest 1975 (in Hnngarian).

(Received March 22, 1983)