VOLUME 53, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 16 JULY 1984

Order-Disorder Transitions and Segregation at the (100) Surface of Cu-Au Alloys

Vijay Kumar and K. H. Bennemann Institutfiir Theoretische Physik, Freie Universitdt Berlin, D-1000 Berlin 33, Germany

(Received 21 February 1984)

Using the cluster-variation method in the tetrahedron approximation, we study order-disorder transitions at the (100) surface of Cu-Au alloys. Depending on surface segregation, the order-disorder transition may be first or second order. Disordering of the surface may occur at temperatures above or below the bulk transition temperature. The recent low-energy ion-scattering data on Cu3Au is interpreted as evidence for stronger long-range order at the surface than in the bulk.

PACS numbers: 64.60.Cn, 64.75. + g, 68.40. + e

The atomic structure at surfaces of metallic alloys plays an important role in catalysis. Recently the chemical composition in the first two layers of Cu3Au(100) surface has been measured from low-energy ion-scattering (LEIS) spectroscopy.1 Long-range order (LRO) at this surface was determined from low-energy electron-diffraction (LEED).2 It has been concluded from the LEED data that the surface becomes continuously disordered before the bulk which disorders via first-order transition. Here we present theoretical results on the (100) surface of Cu-Au alloys which indicate that the two experi­ments may contradict each other. To analyze the interplay of surface segregation and order-disorder transition and to understand the effects due to strain in these alloys we performed various calcula­tions which show a systematic trend for surface segregation and LRO.

In the case of complete atomic order, the (100) surface of Cu3Au alloy may contain either equal amounts of Cu and Au or only Cu atoms. Howev­er, recently it has been observed1 that the surface is always of Cu-Au type. Our calculations show that the Cu-Au type surface has lower surface free ener­gy than the Cu-Cu type surface. Therefore, in agreement with the experiement we discuss here mainly this case. Details of our calculations will be published elsewhere.3

The free energy of the system is calculated by use of the cluster-variation method4 in the tetrahedron approximation. This method has so far produced the best results for the bulk phase diagram of Cu-Au alloys5 and has also been used successfully for the interphase and antiphase boundaries.6 As in the bulk the lattice is subdivided into four equivalent sublattices a, /3, y, and 8. The atoms on these sub-lattices are connected with each other through tetrahedra as shown in Fig. 1. Furthermore, the lat­tice planes parallel to the surface are denoted by X. Each tetrahedron then lies between two nearest-neighbor layers. The tetrahedron probabilities are

denoted by t^yb(Xf \ + l ) where /, j r k9 I refer to the type of atoms on the a, 0, y, and 8 sublattices,

<

600 650 700 750

T ( K ) FIG. 1. Results for the long-range order parameter r?x,

short-range order parameter crXx, and surface-layer Au concentrations xAu(\) as functions of temperature for Cu3Au. Inset: The Cu3Au structure with its sublattices a, /3, y, and 8. \ = 0 ,1 , etc., refer to surface, second layer, etc.; 6, to bulk. T§ refers to the transition tempera­ture at the surface.

278 © 1984 The American Physical Society

VOLUME 53, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 16 JULY 1984

respectively. The site and pair probabilities x/*(X) and p^iX^i) can be expressed as usual in terms of the tetrahedron probabilities. The internal energy f/ is calculated with use of the nearest-neighbor pair energies ezy and the four-body energy parameters <?and r. Then,

U =JT X J 3 ( € ^ - eBB){xA (X) +xA (X +1)} + 6eBB

+ !3 ( l+^)^ (X,X + l ) + 4r2(X,X + l ) + 3 ( l + r ) r 1 ( x a + l ) } e ] + ^ X / / ^ ( 0 0 ) e / 7 . (1)

Here, xA (X) = XxX/(X)/2 and tn(X, X + 1) is the sum of all tetrahedron probabilities for n A atoms in the tetrahedron. Jf is the number of atoms in each layer and € = eAB - (eAA 4- eBB)L2, Jht entropy is given by

- ^ ( ^ ( X X + l ) ) - ^ ( ^ f ( X X + l ) ) - ^ ( ^ 8 ( X + lX + l ) )

+ | [ ^ U / a ( X ) ) + ^ U / ( X ) ) + ^ ( x / ( X + l ) ) + ^ U /8 ( X - f l ) ) ] + ( l - m / 2 ) }

+J? (PifHOO)) -}{j?(x«(0)) +J? (xf(0))}\, (2)

where m = 1 if X = 0, otherwise zero, i f (x) = xlnx -x and a sum over /, y, A:, and / is implied. Minimizing the grand potential G = U - TS - \XJV*X\XA (X) with respect to tffi^X, X + 1), we obtain

^ 8 ( 0 , 1 ) = = t$y*(b)

P(b) exp f o - f A(1 + 0 ) "*Cu + euklQ +}bijAE

kT + ti(kl)

xlP(0l)[x?(0)xf(0)]3/s/[Pi?n00)]l/2},

and

^ S ( X , X + 1) = ^ ^ ^

where

P(XX + l ) = U /a(X)x/(X)x^(X + l)x /

8(X + l ) ] - 5 / 8

x [/tf'(X X )/>,?*( X X + l)Pif(X X + 1 ) / ^ ( X X + l)pf(\\ + l)p£H\ + l,\+l)]l/2, and

^ y 8 ( ^ ) = ^ ( « e x p ( [ ^ / { A c - 6 ( e ^ - e ^ ) } / 4 + ^ - 6 / 7 W ] / ^ r ) .

Here, b refers to bulk and Q = bulam. bu and a w , denote the number of A atoms in ij and //A:/. C/y is 1 if the pair (ij) is of AB type, otherwise 0. eijkl is 0 , 3 ( 1 + # )€, 4€ , 3 (1+ /<)€, or 0 if there are 4, 3, 2, 1, or 0 A atoms in the tetrahedron. £x are Lagrange multipliers to account for the condition that

2uUffi^K\ + i)-ti3^(\-i,\)] = o. £x is the grand potential per atom in the Xth layer and Aii = EB — EA is the difference of surface ener­gies of pure elements. \x is the chemical potential. The LRO parameter iqK and the short-range one (SRO) crxx in the X layer are defined as

i?x = j t f ( X ) - x / ( X ) , (7) and

pfi(XX)+p$£(XX)

(3)

(4)

(5)

(6)

= 1 - 2xA(X)xB(X) (8)

For Cu3Au the surface LRO and gold concentra­tions are calculated for the first five layers7 with e = - 6 6 3 K, # = 0 . 0 1 , and r = - 0 . 0 8 . These values give the best results for the bulk phase dia­gram.5 Surface segregation depends8 primarily on A£ which is taken9 to be -3100 cal/mole. The results are shown in Fig. 1. It is seen that the sur­face becomes continuously disordered about five de­grees before the bulk. For such a transition the correlation length diverges and therefore near the surface critical temperature TQ several layers get af­fected.10 Similar results were obtained for anti­phase boundaries6 of Cu3Au alloys. Note that the SRO parameter and the Au concentration change continuously near r§ , but have a discontinuity at T§. This discontinuity decreases if more layers are taken into account and may possibly disappear. The

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VOLUME53, NUMBER3 P H Y S I C A L R E V I E W L E T T E R S 16 JULY 1984

-1 .0

-0.8

-0.6

•0.4

-0 .2

p—J__^T- - i — r"""

! AE=4.75e

1 1 > T 7 j

1 1* 1 1 1 1

< X

1000

T(K) FIG.

A£ = 4 2. Results for 7?x, croo and x A u (^) for Cu3Au with ,75e. In Fig. 1, AE corresponds to ~ 2.353e.

calculated surface LRO agrees closely with the LEED data,2 but the Au concentration is far below the LEIS result which is shown in Fig. 2. Also for T < Tl we get a slight enrichment of Cu instead of Au due to competition of A£ and e terms.11 This disagreement with experiment is likely due to the neglect of strain effects which are thought to play an important role for Cu-Au alloys because of signi­ficantly different atomic sizes.

To simulate strain effects we increase the value of AE to 4.75e. Results are shown in Fig. 2. Note that, while the Au concentration in the first two layers is now in close agreement with the LEIS data, we obtain near Tl contrary to LEED data that the LRO at the surface is stronger than in the bulk. Moreover, we find that the surface disorders abrupt­ly at about 803 K, far beyond12 Tc

b. Since AE acts like an external surface (magnetic) field, it can make the surface remain ordered above Tf;. To clarify this situation we performed more calcula­tions for different A£ values and for other bulk concentrations. We find near T£ that increasing AE increases the Au concentration and LRO at the sur­face until values close to the bulk ones are reached.

1 0

0.8

-< 0.6 cr

0.4

0.2

n n

™ ^

^^X \ - CuAu x=o\ i\

(100) \ ]

— r

v \

A \ V v \

\A

— , —

•*>

H

—j

H

<

1.0

0.8

0.6

0.4

0.2

0.0

[ T I "" 1

1 ^^S*-| AE = -3100 c a l / mole L _ i ... l i

!

^ b • -J

~\

500 550 600

T(K)

650 • 700 T b

750

FIG. 3. Results for r)X and xAu(\) for CuAu (a = y and/3 = 8).

When the surface has a gold concentration nearly equal to 0.5, the stochiometric composition corre­sponding to slight enrichment of Au, then the LRO is maximal. For larger Au enrichment the surface LRO starts decreasing again and ultimately becomes zero for complete wetting by one of the com­ponents at T = 0. Therefore, if we rely on the re­cent LEIS data which show xA u(0) ~ 0.5 up to 7£, we conclude that the surface LRO is stronger than in the bulk. The results shown in Fig. 2 also show that xA u(0) depends only weakly on TJ0. Such behavior is expected if the heat of segregation8 is significantly larger than the ordering energy kTc. The strongest interplay between segregation and or­dering is expected if the two energies are compar­able.

We also performed calculations for CuAu alloy treating the first seven layers differently from the bulk. As shown in Fig. 3, several layers become continuously disordered at Tc

x < Tj;. Obviously a large number of layers should be treated to get ac­curate results for subsurface layers. In contrast to Cu3Au, for CuAu we find significant enrichment of Au both above and below13 T§. As reported be­fore14 there is a competition between segregation and ordering. As the temperature increases the sur­face Au concentration increases because LRO de­creases.15 This trend is maintained even if the sur­face has become disordered as a result of the in­teractions with the ordered bulk. Beyond T§, Au concentration decreases because of increase of dis-

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VOLUME 53, NUMBER 3 PHYSICAL REVIEW LETTERS 16 JULY 1984

order and tends towards the bulk value. Though no experimental data on CuAu(lOO) exist, comparison of Cu3Au and CuAu results indicate that strain ef­fects should be important for Cu3Au. This is con­sistent with the fact that the strain energy decreases as a result of an expansion of the lattice if *Au~~* 0-5. We also studied a dilute alloy [xAu(b) = 0.1], where the bulk is disordered at all tempera­tures. For kE = 5e, the surface is found to become ordered and shows a second-order transition.

In summary we have studied phase transitions at the (100) surface of Cu-Au alloys in order to analyze the LEED and the LEIS data on Cu3Au. Our results cannot support both the experiments simultaneously. We suggest that more experiments on Cu-Au alloys with different compositions and surface orientations should be performed to get a proper understanding of surface phase transitions in these systems. In our calculations we did not con­sider the effect of surface domains which are likely to be present in reality and can influence the sur­face phase transitions. However, the various calcu­lations presented here indicate clearly that the sur­face affects strongly the order-disorder transitions and that the surface segregation plays an important role.

We are grateful to T. M. Buck for communicating his results prior to publication. This work is sup­ported in part by the Deutsche Forschungsgemein-schaft.

1T. M. Buck, G. H. Wheatley, and L. Marchut, Phys. Rev. Lett. 51,43 (1983).

2V. S. Sundaram, B. Farrell, R. S. Alben, and W. D.

Robertson, Phys. Rev. Lett. 31, 1126 (1973). 3V. Kumar and K. H. Bennemann, to be published. 4R. Kikuchi, Phys. Rev. 81, 988 (1951). 5R. Kikuchi and D. de Fontaine, in Applications of

Phase Diagrams in Metallurgy and Ceramics, edited by C. G. Carger, U. S. National Bureau of Standards Special Publication No. 496 (U.S. GPO, Washington, D.C., 1978), p. 917.

6R. Kikuchi and J. W. Cahn, Acta Metall. 27, 1337 (1979).

7The rest of the system is treated like bulk. 8V. Kumar, Phys. Rev. B 23, 3756 (1981). 9We neglect temperature variation of AE and e as it

produces only minor effects. The concentration depen­dence of interaction energies (Ref. 8) will in general af­fect surface concentration and ordering, but is neglected here.

10An increase in the number of surface layers tends to increase disorder and lowers the surface critical tempera­ture Tc

0 slightly. nFor each Cu atom at the surface there are only four

nearest-neighbor unfavorable Cu-Cu bonds while in the bulk there are eight. On the other hand for each Au atom there are twelve favorable Cu-Au bonds in the bulk as compared to eight at the surface. Thus for a small AE Cu is more favorable at the surface.

12For To > T§ we estimate within the point approxima­tion Tg for a square lattice and T§ for A3B. This gives To/T§^ 1.22 as also obtained in Fig. 2.

13In the ordered state either at the (100) surface or in the bulk of CuAu alloy, there are equal numbers of un­favorable bonds for each Cu and Au atom. Therefore a negative A£ leads to segregation of Au.

14J. L. Moran-Lopez and K. H. Bennemann, Phys. Rev. B 15, 4769 (1977).

15A similar behavior was found for the (110) surface of bcc alloys by J. L. Moran-Lopez and L. M. Falicov, Phys. Rev. B 18, 2542 (1978), in a calculation based on the pair approximation.

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