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:
J. Giner, F. Gautier. ORDERING ENERGY OF A MODEL TERNARY ALLOY WITH CsClTYPE STRUCTURE IN RELATION WITH ITS ELECTRONIC STRUCTURE. Journal de
Physique Colloques, 1977, 38 (C7), pp.C7-301-C7-305. <10.1051/jphyscol:1977760>. <jpa00217263>
HAL Id: jpa-00217263
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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 different 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 qualitatively 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 competition 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 q2 variation of the ordering energy indicates
that it is possible to define pair interactions by analogy
with the phenomenological theories of the orderdisorder 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 stimulated (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
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1977760
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 distributed 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 /Isublattice (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 equivalent 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 occupation probabilities are 0, = 1 - x)
The tight binding hamiltonian is the same as that
used in I for the binaries : the band structure is characterized 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 neglected. 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 intraatomic 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 electronelectron 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 q2. 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 :
The results of this section can be summarized as
follows :(i) the curves giving q1 and q , as a function of
temperature have the same trends as in the binary
alloys. In particular, the stable configuration at low
temperature is the completely ordered state, (ii) the
antistructurc atoms in the a sublattice are in majority
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 corresponding binary alloys. In V(Mn,Fe,-,) kTc is
monotonously raised from VMn to VFe.
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. 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. The broken lines correspond
to the BW solution with constant pair interaction parameters as
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;
A#A'
Y I ZA; - A') x
~ 1 ,
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 contribution 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
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 physically meaningless, are composition dependent.
(iii) The ratio
C
Wij(Rap)I C
aP
aa
or SP
Wij(Raa)
PP '
lies between 3 and 4 for VMn and VFe.
.I. G l N r R AN11
F. G A U T I E R
(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 compared 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 vicinity 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 investigation.
Enthalpy of mixing (in cV) as deduced from the band
1i11. \ . I ! IOII\ .1111'\\
of the system V(Mn,Fe,-,).
FIG. 4.
rn,r<l,.l
0K
:I(
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
+
+
+
+
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.
um
arrangement at
in thc ~ y ~ ~ i l i b r i atomic
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 boundan binary alloys. This is shown for
Ti(Fe,Co, -,) in the insert of figure 5 ; the continuous
and broken lines correspond to the equilibrium configuration at kT = 0 and kT = 0.266 eV respectively.
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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