ORDERING ENERGY OF A MODEL TERNARY ALLOY WITH CsCl-TYPE STRUCTURE IN RELATION WITH ITS ELECTRONIC STRUCTURE

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ORDERING ENERGY OF A MODEL TERNARY

ALLOY WITH CsCl-TYPE STRUCTURE IN

RELATION WITH ITS ELECTRONIC STRUCTURE

J. Giner, F. Gautier

To cite this version:

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ORDERING ENERGY OF

A

MODEL TERNARY ALLOY WITH CsCl-TYPE

STRUCTURE

IN

RELATION WITH ITS ELECTRONIC STRUCTURE

(*)

J. GINER

International Center for Theoretical Solid State Physics Institut de Physique, UniversitC de Liege, Sart Tilman/4000 Liege, Belgique

and F. GAUTIER

I.L.L. 156 X, 38042 Grenoble Cedex and

Laboratoire de Structure Electronique des Solides, Universite Louis-Pasteur, 67000 Strasbourg, France

Rhumb. - Nous etudions la transition ordre-desordre d'un alliage ternaire A(B,C, -,) de metaux

de transition de structure CsCl a l'aide d'un modele de bande simple. Nous Ctudions la repartition des differents atomes sur les deux sous-reseaux en fonction de la temperature et de la composition. Nous discutons la validitt de l'approximation des interactions de paires et la forme de la densite d'etats en relation avec les propriktes magnetiques de ces alliages.

Abstract. - We investigate the order-disorder transition of a ternary transition metal alloy

A(B,C,

-3

with CsC1-type structure in a simple band model. We study the distribution of the diffe- rent species of atoms among the two sublattices as a function of temperature and composition. We discuss the validity of the pair interaction approximation and the shape of the density of states in relation with the magnetic properties of these alloys.

1. Introduction. - The determination of ordering effects from the electronic structure of transition metals and alloys has been extensively studied during the last few years [I -41. In a previous paper [ 5 ] , hereafter referred to as 1, we introduced a simple model for the study of the main characteristics of the order-disorder transition from a general point of view. This band model was chosen as simple as possible to allow simple numerical computations and to describe qua- litatively the physical behaviour of some binary alloys with CsC1-type structure (VMn, TiFe, ScCo,

. .

.).

The tight binding hamiltonian was solved using the coherent potential approximation (CPA) generalized in order to deal with the long-range order; we determined self-consistently the energy levels in the Hartree-Fock approximation and we took into account both intra and inter-atomic interactions. The main results of this study can be summarized as follows : (i) the ordering energy varies as the square of the order parameter q and results from a competi- tion between the electrostatic interactions and the distortion of the bands with ordering, (ii) the charge transfer varies linearly with the long range order parameter q and increases with increasing q, (iii) the charge is transferred from elements on the left-hand side of Cr to elements on the right-hand side and, if

(*) Work performed in the framework of the joint project ESlS

of the University of Antwerp and the University of Liege.

the magnitude of the charge transfer depends .on the intra-atomic Coulomb interaction, the ordering energy is not very sensitive to the value of this quantity. The q 2 variation of the ordering energy indicates that it is possible to define pair interactions by analogy with the phenomenological theories of the order- disorder transition. This is the reason why we have developped (16-71; Treglia G . , Ducastelle F. and Gautier F., to be published) a generalized perturbation theory from the completely disordered state and defined effective pair interactions for the long-range order.

The present study of ternary alloys has been sti- mulated (i) by the necessity of a better understanding of the previous general considerations, (ii) by the peculiar magnetic properties of such alloys (TiFe,Co, -,, for example). As far as the first point is concerned, we want to investigate : (i) The validity of the effective pair interaction model and the concentration dependence of such pair interactions. (ii) The distribution of the various atoms among the two sublattices and the relation between such a distribution and the physical properties. (iii) The origin of the ordering and the role of the charge transfer. Moreover, we point out that the effective pair interactions defined for the study of the order-disorder transition cannot be used to represent the enthalpy of mixing of the corresponding disordered alloys. In the present paper, we study the pseudo-binary

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C7- 302 J. GINER AND F. GAUTIER

alloys A(B,Cl-,) with CsCI-type structure like Ti(Fe,Co, -,) and V(Mn,Fel -,). In these alloys and at low temperature, all the A atoms are on the a sublattice whereas the B and C atoms are randomly distri- buted on the /3 sublattice. Up to now no sublattice ordering of the B and C atoms could be observcd from experimental measurements [8-91. The electronic properties of such alloys are known to be composition dependent ; for example the boundary binary alloys (TiFe, TiCo) are non magnetic whereas the ternary

alioys present a ferromagnetic phase (0.2

5

x 5 0.8).

([lo-] I ] ; Hilscher G . , Buis N. and Fraux J. J. M. :

to be published.) Such properties are not exceptional ;

it has been shown [12-131 that ferromagnetism occurs for Co(Til -,Ga,), Co(Til - .Al,) etc., in relation with

partial ordering of the atoms on the /I sublattice (L21 structure). They can be interpreted by homogeneous models or by local environment effects at 0 K [14] but the role of the antistructure atoms for partially ordered states has to be clarified. A priori only a small number of such atoms can carry local moments which in turn determine local moments on the neighbouring atoms ; such clusters are coupled and could be the elements which determine ferromagnetism in the system.

In

9

2, we briefly describe the band model. In

5

3,

we investigate the equilibrium atomic arrangement as a function of temperature and composition and the order-disorder critical temperature. Section 4 is devoted to the discussion of the validity of the pair interaction model. In the last section, we show the variation of the charge transfer with the composition and we discuss the shape of the density of states in relation with the magnetic properties.

The main result of this study is that the effective pair interaction energies are composition dependent so that we have to apply the pair model with care.

2. The band model. - We consider a ternary alloy of composition A(B,Cl -,) whose lattice structure is bcc and CsCl in the disordered and ordered states (respectively). The lattice is separated into two equi- valent S. C. sublattices a and j? ; the atomic arrangement is described by two independent long range order parameters q , and r7, defined so that their values are 1 and 0 for the state of complete order and for the state of complete disorder (respectively). The occu- pation probabilities are

0,

= 1

-

x)

The tight binding hamiltonian is the same as that used in I for the binaries : the band structure is cha- racterized by the values of the energy levels of the atoms occupying the sites a and

fl

and by the values of the hopping integrals between first and second nearest neighbours ; there is no off-diagonal disorder and the k-dependence of the SC dispersion relation is neglect- ed. The energy levels are self-consistently determined in the Hartree-Fock approximation and we take into account both intra and inter atomic interactions. The bandwith of the pure metals is taken equal to 6 eV. the energy level difference between two adjacent elements in the periodic table to 0.75 eV, the intra- atomic Coulomb integral to 15 eV and the charges to 3, 4, 6, 7, 8 electrons for Ti, V, Mn, Fe and Co respectively.

The total configurational energy E,, is written in the point ion approximation as the sum of the ionic Eion and electronic E,, energies. The latter is given by the

band term from which we substract the electron- electron interaction energy. Expressions for these terms are obtained in a straight-forward way from the expressions given in I.

3. The order-disorder transition. - Figure 1 reports the equilibrium values of q , and q2 as a function of temperature for V(Mn,Fel-,) and Ti(Fe,Col-,). These curves have been obtained by minimizing the free energy F(ql, q2) with respect to q1 and q 2 . For

FIG. 1. - Equilibrium value of the long-range order parameters versus kT (in eV). The continuous and broken lines correspond to

the Ti(Feo,,Coo,,) and V(Mn,,,Fe,,,) alloys respectively.

the present qualitative study we used the simple Bragg-Williams's configurational entropy term. The extension to more sophisticated expressions would be straightforward [15]. The total configurational energy as deduced numerically from the generalization of CPA can be approximated by :

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Mn atoms in the V-based alloys, (iii) the concentration of antistructure atoms is always small (about 0.01) in the Ti-based alloys. On the other hand, the critical order-disorder temperature Tc is given by a second order equation in terms of the a,,(x). We have reported in figure 2 the composition dependence of kTc

(k

is the Boltzman constant) for both systems (continuous lines). In Ti(Fe,Co, -,) kTc is higher than in the cor-

responding binary alloys. In V(Mn,Fe,-,) kTc is

monotonously raised from VMn to VFe.

FIG. 2. - k T, (in eV) versus the composition for the Ti(Fe,Co, - ,)

and V(Mn,Fe, -,) alloys. The continuous lines correspond t o the results deduced from the band model. T h e broken lines correspond to the BW solution with constant pair interaction parameters a s deduced from the band model at x = 0.5.

4. Validity of the pair interaction model.

-

The pair interaction (PI) approximation assumes that the total configurational energy per site can be written as

1

utot(x; ~ 1 , ~ 2 ) = 3

.k

X'rij(x; ~ 1 , Y I Z ;

A

-

A')

x

-

A # A ' I . !

x ti,(lk -

i l l )

(4.1) in which the sum extends all over the lattice, Tij is the number of i j pairs of atoms situated at site

A

and

A'

respectively and

A

-

A'

I) the corresponding pair interaction energy assumed to be composition independent. In the Bragg-Williams's model, we neglect all the correlations ; Sij is determined by the contribu- tion of each sublattice so that

U,,

can be described by six binary effective PI parameters

w$.

These

w$

are given by expressions like

in which the Wij(R) are linear combinations of the eij(R). The Wij+ and WiT come into action in the disordered and ordered states respectively. As well known, this approximation leads t o a quadratic form in r ] , and q2 for the ordering energy ; the contact

between the PI model and the present band model is

obtained by identifying both quadratic forms. This leads to

Let us briefly summarize the results we obtained :

(i) The P1 parameters W ; as deduced from eq. (4.3) are composition dependent. This is particularly the case for Wgc0(x) which is negative near to the TiCo composition and positive near to the TiFe composition (see figure 3). This composition dependence is

FIG. 3. . -- Pair interaction energies (in eV) as a function of composition for Ti(Fe,Co, -,).

less marked in the. V(Mn,Fe, -,) alloys (we are not able to predict' the sign of W,,,,(x) because of the smallness of this quantity). As a consequence the departure from the Bragg-Williams' model is larger in Ti(Fe,Co, -,) than in V(MnxFel -,). This can be seen from the composition dependence of kTc in the

PI model (broken lines) as reported in figure 2 ; the corresponding values of Wi7 are determined from eq. (4.3) at x = 0.5.

(ii) As seen from figure 4, the energy of formation at 0 K AH(x) computed in the band model behaves wrighly like x(l - x) only for the binary alloys which have half-filled or quasi half-filled band like for instance V1-,Mn,, Vl-,Fe,. In the case of

Mnl-,Fe,, AH(x) is not symmetric with respect to the 50150 composition. This means that, generaly speaking, the PI parameters Wi;, which are phy- sically meaningless, are composition dependent.

(iii) The ratio

C

Wij(Rap)I

C

Wij(Raa)

aP aa

or SP PP '

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.I. G l N r R AN11 F. G A U T I E R

FIG. 4. Enthalpy of mixing (in cV) as deduced from the band

rn,r<l,.l :I( 0 K 1i11. \ . I ! IOII\ .1111'\\ of the system V(Mn,Fe,-,).

5. The charge transfers and the density of states

(D05). - As explained in I, the self-consistent energy levels do not depend very much on the value of the intra-atomic Coulomb integral Uo for U , > 5 eV. We have chosen Uo = 15 eV in order to obtain the same order of magnitude for the charge transfers as that deduced from the self-consistent band structure calculations in ordered TiFe [17-181. They are equal to about

+

0.14 and

+

0.07 in the ordered and disordered states respectively (+ for Fe and Co,

- for Ti) in Ti(Fe,Co,-,) and they are equal to about

+

0.08 and

+

0.03 in the ordered and disordered states respectively (+ for Mn and Fe, - for V) and increase with decreasing x.

We have reported in figures 5 and 6 the partial densities of states for

in thc ~ y ~ ~ i l i b r i u m atomic arrangement at

kT = 0.266eV and k T = 0.112eV respectively.

The results can be summarized as follows

(i) The avcragc DOS n ( ~ ) in the ordered state has a typical valley in the middle region due to the CsCl symmetry [17-181. The average DOS at the Fermi level n ( ~ ) varies between the values corresponding to the b o u n d a n binary alloys. This is shown for Ti(Fe,Co, -,) in the insert of figure 5 ; the continuous and broken lines correspond to the equilibrium confi- guration at

kT

= 0 and

kT

= 0.266 eV respectively.

(ii) The partial DOS corresponding to the structure atoms (i.e. atoms A on the a sublattice, atoms

B

and C

on the B sublattice) are nearly not modified as compar- ed to the completely ordered state ; they have the same shape as in the corresponding binary alloys.

(iii) The partial DOS corresponding to the antistructure atoms have a well-marked peak in the vici- nity of the Fermi level. The Fermi level lies in the high-density region of the Fe and Co DOS in Ti(Fe,Co, -,) suggesting that these atoms could play an important role in the occurence of ferromagnetism in this system ; this last point is now under investiga- tion.

Flc;. 5. - Conditional densities of states per spin for Ti(Feo,,Coo,,)

at the thermodynamical equilibrium at kT = 0.266 eV. In insert :

average density of states per spin at the Fermi level versus the composition.

FIG. 6. - Conditional densities of states per spin for V(Mno,,Feo,,) at the thermodynamical equilibrium at kT = 0.266 eV.

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Phys. Rev. B 7 (1973) 2430. [6] GAUTIER, F., DUCASTELLE, F. and GINER, J., Phil. Mug.

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ORDERING ENERGY OF CsCI-TYPE ALLOYS C 7 - 3 0

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