Solid State Communications, Vol. 78, No. 5, pp. 429-432, 1991. Printed in Great Britain.

0038-1098191 $3.00 + .00 Pergamon Press plc

SURFACE S E G R E G A T I O N OF Pt-Ni ALLOYS

S. Modak and S. Gangopadhyay

Saha Institute of Nuclear Physics, 92, Acharya Prafulla Chandra Road, Calcutta 700 009, India

(Rece ived 27 December 1990 by C .N .R . Rao)

Surface segregation of Pt-Ni alloys has been studied with the help of an Anderson-Hubbard Hamiltonian and a mixed Bethe lattice formal- ism. Results for the surface segregation at the (1 I 1) surface of the alloy are presented. It is found that the Pt atoms segregate to the surface and that the magnetic moments of Ni atoms are reduced at the surface of the alloy.

1. I N T R O D U C T I O N

IT IS WELL known that Pt-Ni alloys are exceptions from simple segregation criteria [1-4], which have so far been successful in explaining the segregation behaviour of a large number of other alloys. Conven- tional theoretical models predict the segregation of Ni atoms at both the (1 1 1) and (1 1 0) surfaces of the Pt-Ni alloy. But experimentally it is well established that while in the case o f ( ! 1 1) surface Pt segregates to the surface, Ni segregates in the (1 I 0) surface of the alloy [5-7]. Treglia and Legrand [8] have used the bond-breaking model and have calculated the size- mismatch energy from a simplified tight-binding approach. Their results show that Pt segregates to both the (I 1 1) and (1 1 0) surface. Lundberg [9] has used the embedded-atom method, and by using a statistical approach to simulate the equilibrium state of the Pt-Ni alloys showed Pt enrichment at the (1 1 1) surface. Recently, Legrand et al. [10] have used a tight binding Ising model (TBIM) to explain the experimen- tal observation of face-related segregation reversal of this Pt-Ni alloy. Here we study the surface segregation behaviour of Pt-Ni alloy system from the basis of Anderson-Hubbard Hamiltonian by using the mixed- Bethe-Lattice model [11].

In Section 2 we outline the model. In Section 3 we present our results for the surface segregation of Pt-Ni alloy system. Conclusions are drawn in Section 4.

2. THE MODEL

The Anderson-Hubbard Hamiltonian for the sys- tem A,B~ ~ is written as

n = Z ~,'~o + Z tijc~c,o + Zui.i~nii (1) i,a i,i,a i

where c~ and ci~ respectively creates and destroys an electron of spin a at site i, and n~. = cSc,., e~ is the

energy level at site i, i.e. the single site energy for the pure metal, e~ can have values eA or e8 if the site is occupied by an A - or B - atom respectively, t~/is the hopping integrals for electronic transitions between lattice sites i a n d j and can have values tAA, lAB, IBB. U i is the onsite Coulomb interaction.

This Hamiltonian as expressed in equation (1), in the unrestricted Hart ree-Fock (UHF) approximation is written as

H = ~ e.ini. + ~ '+ ' <ni .>n~. ti)(~ia~i~ -{- E Ui i,a i,j,a i,a

- - ~ . ui(ni} >(ni,L >. (2) i

Now magnetic moment Iti at site i can be expressed as

tti = <niT> -- <ni~> (3)

and the number of electrons ni at site i is expressed as

ni = <niT> + <nil>. (4)

Using the relations (3) and (4) the Hamiltonian expression (2) can be written in the form [12]

n = I~ i q.- ~ (hi-{- Iti) nia q'- E IuCiaCJa i c j

1 .,(n~ - ltD. (5)

4

We assume that on each lattice site i there exists magnetic moment #~ and that they point up ( + ) or down ( - ) along a specified direction. Then the alloy system A,.B~ ~ is equivalent to the system of atoms with moments ItA + ]itA- and Its+/itB- • And hence this system can be treated as an alloy of four components where the lattice sites are occupied either with ItA+/ It~ or ItB+ /it8 - We further assume that the magnetic moments at lattice sites are aligned randomly along or opposite to the specified direction i.e. no ordering in

429

430

the alignment is involved. This treatment essentially leads to the fact that I~A+I = I ~ I and I~+1 = Ipn I. By this assumption we effectively have identified any site i as follows: First, a site i is occupied either by A - or by B - a tom and that probability is deter- mined by the alloy concentration; once a site is occu- pied by an A - or B - a tom then the question comes about the direction of alignment of the magnetic moment . Now the probabili ty that an A - site is occupied by PA+ or PA is 0.5 since no ordering is involved. Same is the situation for a B site.

Internal energy E of the alloy is now calculated by using this tight-binding Anderson-Hubbard Hamil- tonian. For a particular temperature T and bulk con- centration x, alloy concentration at the surface is determined by minimizing the free energy F(Xl. xz, . . . . xh) of the system with respect to the concen- trations x~ . . . . . etc. of the first, second . . . . etc. crystal layers parallel to the surface subject to the constraints that the number of atoms and the number of electrons are fixed.

Mathematically, we have to minimize an effective free energy F ' given by [11]

F ' = ~" Ei (.x', ,xe ,x 3 . . . . . x h ) - T S - e ~ x ~ . 2-1 2 -1

- /3 ~ (n~) (6)

where 2 is the layer index, 2 = 1 indicating the first surface layer, and so on. E~ denotes the internal energy of an a tom in the 2th layer; T and S are the tempera- ture and entropy of the system respectively; ~ and/3 are the Lagrange multipliers arising from the two con- straints conserving the total number of atoms and electrons in the system. ( n ; ) denotes the average number of electrons on an a tom in the 2th layer.

Internal energy E~ is calculated as

EF

E~ = E f Ep~,~(E) d E - E,~.. (7) a

Here p~..,(E) is the average LDOS of a spin electrons and is given by

x A (1 -- X) S p~,o(E) = 2 (p;+" + m.,o ) + - - 2 - - - (pf.+~ + m~,o )

S U R F A C E S E G R E G A T I O N OF Pt -Ni ALLOYS

(8)

and

1 E, = -~ ~ u,(n~ -- #2). (9)

The configurational entropy of the random alloy system is given by

Vol. 78, No. 5

S = - k n ~ [ x ~ l n x x + (1 - xx) ln(1 - x~)].

(10)

The LDOS for e spin electrons at a site with magnetic moment l is determined from the diagonal element of the one-particle Green's function G ~ ii, a

1 i,I p~.~ - Im G t ii, a,2 i = A, B. (11) 7~

The diagonal Green's function G[~,~,~(E) is evaluated by the mixed Bethe-Lattice formalism where the energy levels of the electrons are essentially given by

UA

UA

UB

(12)

UB

3. RESULTS

The essential parameters needed for the calcu- lation are the band widths of the pure elements of the alloy, the mean energies of the d band, the number of d electrons and the value of intra atomic Coulomb correlation parameter U. The d bandwidths of the two metals Pt and Ni are taken to be 7.3 eV and 3.7 eV [13] respectively. The difference between the mean energies ofdelect rons is taken as ept - eN~ = 2.3 [13]. The number of d electrons are taken to be 9.7 [14] for Pt and 9.4 [15] for Ni. The absolute value of magnetic moment at Pt site is 0.3 Bohr magneton and at Ni site 0.6 Bohr magneton. U for Ni has been taken as 2eV [16] and for Pt it is 3eV. In absence of any experimental value of U (i.e. Uaa) for Pt we have estimated this value of U by solving the equations (3), (12a) and (12b) for Pt numerically with the constraint that ni = (ni t ) + (n~l). The same procedure when applied for Ni gives U = 2 eV in agreement with the experimental value [16].

In this calculation the surface electronic bands have been properly shifted so as to minimize the charge transfer between the surface and the bulk. The Lagrange parameter ~ has been estimated from the

expression ~ = OF/Oxb and/3 is simply the Fermi level. Our results for segregation at the (1 1 1) surface of the PtxNi~ x alloy is presented in Fig. 1. The magnetic moment o f a Ni a tom at the (1 1 1) surface of the said alloy has also been estimated and presented (Fig. 2). From the figure it is seen that the magnetic moment of

Vol. 78, No. 5 SURFACE SEGREGATION OF Pt-Ni ALLOYS

1 , 0

0.8

0.6

Xs

0-4

0.2

I I I I

0.0 0.2 0.4 0.6 0.8 1.0 Ni x Pt

Fig. 1. Segregation behaviour of (111) surface of Pt~Ni~ ~ alloy.

Ni atoms at the surface is much reduced. The mag- netic moment of the atoms at the surface has been estimated from the relation

er

/~ = f ( P ~ , T - p ~ ) d E .

Here s is to denote the surface.

4. CONCLUSIONS

Pt-Ni alloy is particularly important because it is a magnetic alloy and hence the interplay of Coulomb interaction and disorder is important for this type of alloy. Here we have first studied the system starting

0.6

"E o

g 0.4

o m

~. 0.2 :a.,,

|

0.2 |

0.4 X

0.G

Fig. 2. Magnetic moment of a Ni atom at the (1 1 I) surface of PLNi~_ ~ alloy vs alloy concentration.

431

from the Hubbard Hamiltonian. Our results show the tendency of Pt atoms to segregate to the (1 1 !) surface of the alloy. In our calculation we have taken only the concentration of the first layer to be different from the bulk. But experiments show that there is an oscillation in the composition in the first three layers of the system. If we take the average of these concentrations of the first three surface layer and compare it with our calculated results - we shall see that our results compare well with the experimental observations.

In these calculations we have neglected the spin correlation effect and have taken the long range order to be zero. This model can be improved by taking the effect of spin correlations and long range order. Cal- culations considering both the spin correlations and finite long range order is in progress. This model, however, in the present form cannot explain the Ni enrichment of the (1 1 0) surface of the alloy because of the surface relaxation in top atomic layers. From LEED data it is found that there is 19% contraction between the first and the second surface layer and 11% expansion between the second and the third surface layer with respect to the bulk layer spacing.

"This effect of relaxation has to be considered properly for the study of (1 10) surface composition of the PtNi alloy. To study the surface segregation of (1 ! 0) surface of this alloy it is thus necessary to perform the three layer calculation with suitable interlayer hopping parameters.

Acknowledgements - The authors wish to thank Dr B.C. Khanra and Dr R.K. Moitra for useful discussions.

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SURFACE SEGREGATION OF Pt-Ni ALLOYS Vol. 78, No. 5 432

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