Fermi surface incommensurate nestings and phase equilibria in Cu-Pd alloys
Ezio Bruno, Beniamino Ginatempo, and E. Sandro Giuliano
Dipartimento di Fisica and Unita ` INFM, Universita ` di Messina, Salita Sperone 31, 98166 Messina, Italy
Received 20 December 2000; published 5 April 2001
The Cu-Pd phase diagram shows two low-temperature ordered phases corresponding to B 2 and L 1
2
super-
lattices or, regardless of the site occupation, to bcc and fcc geometrical arrangements. We show how the phase
equilibria between these intermetallic compounds can be understood by introducing a hypothetical bcc
random-alloy phase and a three-step transformation from B 2 to random bcc to random fcc to L 1
2
. The two
order-disorder transformations, B 2 to random bcc and random fcc to L 1
2
, analyzed in the framework of the
concentration wave-functional theory, appear to be ruled by the sizes of the relevant Fermi surface FS nesting
vectors in the two random-alloy systems. The transformation between the bcc and fcc solid solution phase is
studied by first-principles Korringa-Kohn-Rostoker and coherent-potential approximation calculations, and it is
shown to be influenced by electronic topological transitions occurring in the same systems. Our analysis points
out the preeminent role played by the lattice-incommensurate FS nesting vectors that in both the random-alloy
systems, give rise to frustrated concentration waves. The location of the B 2-L 1
2
coexistence at T =0 appears
to be determined by the condition that the concentration waves of the fcc and bcc random alloys are equally
frustrated.
DOI: 10.1103/PhysRevB.63.174107 PACS numbers: 64.70.Kb, 71.18.+y, 71.20.Be
I. INTRODUCTION
In the last two decades, the first-principles determination
of the phase diagrams of metallic alloys has stimulated a
considerable amount of theoretical work. While remarkable
progress has been made for order-disorder
1–4
and magnetic
transitions,
5–7
the study of martensitic transitions remains
more difficult. Although first-principles calculations are able
to predict the most stable low-temperature structure, it is not
easy to envisage from these sophisticated calculations which
of the physical mechanisms are responsible for structural
changes. For order-disorder and magnetic transitions, the
Ginzburg-Landau theory allows us to determine, in the high-
T homogeneous phase, the instabilities of the Fermi sea with
respect to a certain infinitesimal charge or magnetization
waves. Thus, perturbation theory can be applied and the
seeds of the transition are identified by studying the response
of the system to certain fluctuations. On the contrary, if the
phase transitions have a peculiar first-order character, as in
the case of most martensitic transitions,
8
perturbation theory
is not applicable.
We have discussed in a previous paper
9
a typical transi-
tion for which perturbative approaches are expected to fail:
the bainitic transition occurring in Cu
c
Pd
1 -c
for concentra-
tions 0.5c 0.7. In this case, the high- and low-T phases
are, respectively, a fcc solid solution and a B 2 ordered su-
perlattice based on a bcc geometrical lattice Fig. 1. The
structural transition of the geometrical lattice can be thought
as the consequence of a tetragonal deformation of the origi-
nal fcc lattice. However, though one can imagine a continu-
ous path,
10
the Bain path, along which the lattice is strained
at constant volume with the tetragonal ratio c / a varying from
2 to 1 Fig. 1, nevertheless the strains involved are too
large for applying perturbation theory, by including e.g., an
elastic energy in the Ginzburg-Landau model. Thus, the only
possible attack to the problem is the brute force total-energy
calculation. This has been done in Ref. 9, where we have
found a significant interplay between concentration fluctua-
tions and lattice deformations. In short local chemical order-
ing increases the charge transfer, which is quite large in or-
dered Cu-Pd alloys,
11
hence, if a thermal fluctuation induces
local ordering, then a considerable energetic gain can be ob-
tained by local deformations that maximize the number of
unlike neighbors.
9
As is well known, this is not possible in a
fcc arrangement but it is in the B 2. The transition is, thus, at
the same time, disorder-order and martensitic, and arises
from the competition between the band energy and the elec-
trostatics. For the present number of valence electrons per
atom, 10.5e / a 10.7, in accord with the classical argu-
ments of Mott and Jones,
12
the former favors the compact fcc
phase, the latter the B 2 sometime called also CsCl super-
lattice, one of the typical structures of ionic crystals. A fur-
ther reason for interest in this system is that another isoelec-
tronic alloy Cu-Pt, at the equiatomic concentration, orders
along a different direction, the 111, forming a L 1
1
superlattice.
13,14
We would like to add that the low-T phase
diagram of a third isoelectronic system AgPd is not yet
known and only a likely metastable fcc solid solution phase
is reported in the literature.
13
The observed different phase
behaviors of the above systems, all characterized by the same
valence electron per atom ratio e / a =10.5, constitute an evi-
dent violation to the Hume-Rothery rule
15,12
about the ‘‘band
filling,’’ and suggest a scenario in which many different ef-
fects compete with one another.
The concentration range in which the Cu-Pd B 2 phase is
observed at low temperatures is 0.5c 0.7, with a critical
concentration c
MAX
0.6.
13
On the other hand, recently pub-
lished embedded-atom Monte Carlo simulations report a co-
existence curve symmetric around c
MAX
=0.5.
16
As recog-
nized by the same authors, their classical simulation fails to
reproduce the experimental value because it cannot incorpo-
rate the band-structure details. The wrong result c
MAX
=0.5
comes out from the inclusion of electrostatics and entropic
effects only, since these are more important at the equi-
PHYSICAL REVIEW B, VOLUME 63, 174107
0163-1829/2001/6317/1741078/$20.00 ©2001 The American Physical Society 63 174107-1