JOURNAL OF APPLIED PHYSICS 103, 073911 共2008兲
Magnetism and structural ordering on a bcc lattice for highly
magnetostrictive Fe–Ga alloys: A coherent potential approximation study
T. Khmelevska, S. Khmelevskyi,a兲 and P. Mohn
Center for Computational Material Science, Vienna University of Technology, Getreidemarkt 9/134 Vienna,
Austria
共Received 21 November 2007; accepted 23 January 2008; published online 7 April 2008兲
We calculate the electronic structure, magnetic moments, and ordering energies of highly
magnetostrictive Fe1−xGax alloys from first-principles in the composition range up to x = 0.25. The
coherent potential approximation was used to treat effects of chemical disorder. Given an underlying
bcc lattice in whole range compositions investigated, the DO3 type of ordering is found to have a
lower energy than A2- and B2-type structures. We find that ordering energies strongly depend on the
state of magnetic order such that thermal magnetic disorder strongly favors chemical ordering 共DO3
and B2兲. The values of the magnetic moments of Fe on different sublattices of ordered structures are
found to have a distinctive dependency on the Ga concentration. By taking into account the results
of earlier fully relativistic and full potential calculations of magnetostriction for ordered
stoichiometric Fe3Ga compounds and available experimental phenomenology, our results for
disordered alloys suggest an eventually more complex origin of the giant magnetostriction in Fe–Ga
alloys than it would appear from a simple electronic structure analysis of ordered stoichiometric
compounds. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2903071兴
I. INTRODUCTION
Until very recently, the magnetic properties of Fe–Ga
alloys have attracted much less experimental and theoretical
interest compared to their Fe–Al or Fe–Si counterparts,
which have been thoroughly studied due to both their fundamental and application challenges.1 The situation dramatically changed after the discovery of a huge increase of the
magnetostriction upon Ga doping compared to its value in
pure bcc Fe,2,3 making these alloys to become promising
materials for applications in sensor and actuator technology.4
Intensive studies of the structural5 and magnetic6–11 properties of bcc Fe–Ga alloys have been carried during recent
years by various groups, including also Mössbauer
spectroscopy12,13 and neutron diffraction studies.14,15 It has
been found that the size of the magnetostriction in Fe–Ga is
very sensitive to the thermal treatment or to the state of the
chemical ordering in these alloys.8,10,16 However, the origin
of the anomalously large magnetostriction value ␭100 and its
relation to chemical ordering on the underlying bcc lattice
are still under debate.
In the region of Fe rich compositions in Fe1−xGax alloys,
two long-range ordered phases, B2 and DO3, on a bcc lattice
and a fully chemically disordered A2 phase are observed.5
These types of long-range ordering are presented in Fig. 1
where it can be seen that B2 corresponds to a Ga ordering on
the second nearest neighbor 共NN兲 sites of a bcc lattice,
whereas DO3 is ordering on the third NN sites. This, in particular, illustrates the fact that interatomic interactions favoring B2 order indeed favor directional 具100典 short-range ordering of Ga in Ga–Ga pairs. It has been suggested3 that
these short-range directional ordering of Ga atoms leads to a
Author to whom correspondence should be addressed. Tel.: ⫹43 1 58801
15838. Electronic mail: [email protected].
a兲
0021-8979/2008/103共7兲/073911/5/$23.00
huge increase in the magnetostriction and the observed
strong elastic softening in these alloys.17 Strong additional
arguments toward this hypothesis were further added by
Dunlop et al.12 and Lograsso et al.18 Recently, another point
of view was advocated by Khachaturyan and Viehland,19
who suggested that the increase in the magnetostriction is a
nonsingle domain property, being a rather extrinsic phenomenon coming from DO3 precipitates in the A2 matrix. These
DO3 precipitates are tetragonally distorted and the domains
of the resulting face-centered tetragonal phase may be reoriented in an external field leading to a large pseudoelastic
strain and consequently to the large value of the magnetostriction. Indeed, the mixture of the DO3 + A2 phases is the
most stable in the metastable bcc region of the Fe–Ga phase
diagram5 where the maximum magnetostriction is detected.18
It is interesting that the observed Ga concentration dependence of the magnetostriction and the thermal treatment of
the samples can be accounted for in both phenomenological
models mentioned above.
Up to now, first-principles investigations have only been
performed for the ordered alloys. By employing the full potential linearized augmented plane wave and his torque technique, Wu20 calculated the magnetostrictive coefficients ␭100
FIG. 1. 共Color online兲 Three types of ordering on a bcc sublattice. For the
DO3 structure, only part of the unit cell is shown. The dotted lines are the
unit vectors of the primitive fcc cell of DO3.
103, 073911-1
© 2008 American Institute of Physics
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073911-2
Khmelevska, Khmelevskyi, and Mohn
in stoichiometrically ordered Fe3Ga compounds in the DO3
and L60 structure. The “B2-like” L60 structure has been
used21 to model disordered B2 ordering in Fe3Ga. These calculations predicted a large value of ␭100 only for the L60
structure, whereas the calculated value for DO3 was found to
be small and even having opposite sign20 as compared to
experiment. This observation supports the “intrinsic” model
based on 具100典 Ga–Ga short-range ordering since L60 has
具100典 ordered chains of Ga atoms. Recent measurements10
performed on Fe3Ga samples in the A2, B2, and DO3 phases
suggested that the disordered A2 phase has the largest magnetostriction coefficient and DO3 has the smallest. However,
Fe3Ga with DO3 structure has a positive and, although three
times smaller than for the corresponding A2 type, a still very
high value of ␭100 compared to bcc Fe. In addition, in the
calculations the L60 structure has been found to be elastically
unstable. We find it important that in this paper the qualitative explanation of the origin of the enhanced magnetostriction was based on an analysis of the calculated density of
states 共DOS兲. It was noticed that the increase in the magnetoelastic coupling occurs due to a splitting of the d-states in
the minority spin band near the Fermi level caused by the
low symmetry of some Fe sites in the B2-like structure.
Since in ordered Fe3Ga the symmetry remains high, the magnetostriction value becomes low. Very recently, the elastic
constants in some Fe rich Fe1−xGax compounds with x
⬍ 0.25 have been calculated for several ordered supercell
configurations and the experimentally observed softness of
the tetragonal shear modulus has been reproduced.22
In the calculations mentioned above, the effects of
chemical disorder on the electronic structure were not taken
into account. In this paper, we perform coherent potential
approximation 共CPA兲 studies from first principles. We calculate the electronic structure, ordering energies, and magnetic
properties and their dependence on the Ga concentration in
Fe1−xGax alloys, with x = 0.0– 0.25. In the framework of the
CPA, the chemical disorder is taken into account a in meanfield-like manner.23 Several CPA studies for disorder effects
on the electronic structure have been carried out for similar
Fe–Si alloys, showing a strong interplay between magnetism
and atomic order24–26 but, up to our best knowledge, never
for Fe–Ga. Our study is organized as follows: After a description of the method of calculation, in Sec. III, we investigate the relative stability of three ordered structures of
Fe–Ga alloys on a bcc lattice 共Fig. 1兲. In Sec. III, we present
the calculated DOS and give a semiqualitative discussion of
the huge magnetostriction based on the models described
during the introduction. It is shown that a more detailed explanation of the giant magnetostriction and its concentration
dependence in Fe–Ga alloys would need to go beyond a
simple Néel-like hypothesis by taking into the account local
environment effects more thoroughly.
II. METHOD OF CALCULATION
The calculation of electronic structure and total energies
has been performed in the framework of the local spin density approximations 共LSDAs兲.27 We employ the bulk
Korringa–Kohn–Rostoker 共KKR兲 method in the atomic
J. Appl. Phys. 103, 073911 共2008兲
FIG. 2. Calculated ordering energies for ferromagnetic 共open symbols兲 and
disordered local moment state 共closed symbols兲. The energies are given per
atom and relative to the A2 state with corresponding magnetic order.
sphere approximation28,29 共ASA兲 to calculate the electronic
band structure. The partial waves in the KKR-ASA calculations have been expanded up to lmax = 3 inside the atomic
spheres. The total energy was calculated using multipole
screening electrostatic corrections to the electrostatic potential and energy 共up to l = 6兲, as described in Ref. 30. All
calculations were performed within the scalar relativistic approximation, which contains all relativistic effects with the
exception of spin-orbit coupling. The calculations where
converged with uniform 29⫻ 29⫻ 29 mesh of k-points in the
cubic A2, fcc B2, and bcc DO3 first Brillouin zone, providing
an energy convergence better than 0.01 mRy/atom. A singlesite CPA was used for the description of substitutional atomic
disorder. In Sec. III, the paramagnetic state above the magnetic ordering temperature has been conventionally modeled
by the disordered local moments 共DLM兲 state, as described
by Györrfy et al.31 The total energies of all structures are
compared to the LSDA equilibrium lattice constants calculated for any structure and alloy composition considered.
III. CHEMICAL ORDERING AND MAGNETISM ON FE
SUBLATTICES
According to the latest experimental ambient temperature equilibrium phase diagram5 of the Fe–Ga alloy, the DO3
phase becomes stable as the Ga concentration increases,
whereas the B2 phase appears only as a high-temperature
phase at elevated Ga concentrations. The calculated CPA
ground state energies show the relative phase stabilities at
zero temperature. They are presented in Fig. 2 共open symbols兲, where energy is given per atom and is being measured
with respect to the fully disordered A2 structure. In agreement with experimental estimates, the DO3 structure has the
lowest energy for all Ga concentrations. The energy differences are somewhat smaller than the respective values calculated for Fe–Si alloys.32 However, these differences in ordering energies calculated for the paramagnetic state with
disordered local moments 共black symbols, Fig. 2兲 are about
two times smaller than respective differences in Fe–Si. It
thus appears that thermal magnetic disorder strongly favors
chemical order. Moreover, one notices a significant increase
in the stability of the B2 phase. The combination of these
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073911-3
Khmelevska, Khmelevskyi, and Mohn
FIG. 3. Calculated magnetic moments in the ferromagnetic state for the A2,
B2, and DO3 structure. Upper panel: average moment per Fe atom and lower
panel: local atomic moments on different Fe sublattices.
factors may be the reason why the variation of the heat treatment above the magnetic ordering temperature allows producing a sample with different chemical order at low temperatures relatively more easy in Fe–Ga than in Fe–Si. For
example, the magnetostriction in Fe3Ga has been investigated for all three types of ordering.10
The values of the magnetic moments of the Fe atoms on
different sublattices show a different dependence on the Ga
concentration 共see Fig. 3 where the calculated moments are
shown兲. The dependence of the average value of the Fe moments, however, has one interesting feature common for all
three ordering types 共upper panel of Fig. 3兲. The average Fe
moment reaches its maximal value at some intermediate Ga
concentrations, namely, at ⬃10 at. % Ga for DO3 ordering,
at ⬃12 at. % Ga for B2, and at ⬃15– 17 at. % Ga for disordered A2. It is interesting that the latter value approximately
coincides with the maximum of the magnetostriction experimentally observed in quenched Fe–Ga single crystals,18
whereas disordered A2 Fe3Ga has the largest magnetostriction compared to the ordered samples.10 The existence of
maximum in the concentration dependence of the average Fe
moment has been also observed in recent Mössbauer spectrometry studies,13 although shifted toward lower concentrations compare to our calculated one.
Currently, we cannot provide any logical link between
the peculiar dependence of the Fe moment value on the alloy
composition and the observed phenomena of giant magnetostriction in these alloys. However, some additional remarks
J. Appl. Phys. 103, 073911 共2008兲
can be made. As it was described in some length in the
Introduction, one of the common explanations of giant magnetostriction as intrinsic property of a single crystal phase is
the strong dependence on the oriented Ga–Ga pair ordering.
In the original paper by Neel,33 a similar ordering is suggested to be responsible for the strong magnetic softening in
Fe–Si alloys. An essential magnetic softening also occurs in
Fe–Ga alloys and is believed to have the same kind of
reason.34 The Fe–Si alloys, however, in contrast to Fe–Ga,
are not giant magnetostrictive and at the same time they are
exhibiting a monotonous decreasing of the average Fe moments with Si concentration.26 We thus may speculate that
the understanding of this difference in the Fe moment behavior in Fe–Ga and Fe–Si 共also Fe–Ge or Sn兲 can be key for
understanding of the onset of giant magnetostriction in Fe–
Ga.
Let us now turn to the behavior of the calculated Fe
moments on different sublattices in the ordered alloys 共lower
panel of Fig. 3兲. At low Ga concentrations, the moments on
all sublattices grow; then, at about 10 at. % Ga, the values of
the moments on one sublattice in B2 and DO3 start to
decrease—these are those sublattices that have the largest
number of Ga neighbors 共see Fig. 1兲. Such maximum occurs
in the disordered A2 structure at higher Ga concentrations
since the number of NN Ga atoms increases more slowly
than on the corresponding sublattices in ordered structures.
There is a combination of two effects on these sublattices—a
tendency of moment increase due to the lattice expansion
caused by the Ga substitution and a chemical tendency to
reduce the value of the moment due to p-d bond formations.
A more detailed discussion on the mechanisms of magnetic
moment formation on Fe sublattices can be found in Ref. 25.
We would like to mention here that in Fe–Si, the Fe moments
on sublattices, which have no Si neighbors, increase with the
Si concentration26 compared to our calculated moments in
Fe–Ga. However, the moments on the other sublattices monotonously decrease upon alloying starting immediately
from the diluted regime in Fe–Si.
IV. ELECTRONIC STRUCTURE
In this section, we analyze some features of the DOS in
Fe–Ga alloys with different chemical orderings which has
previously been claimed20 to be important for the appearance
a strong magnetoelastic coupling in Fe–Ga. Relativistic calculations by Wu20 performed for ordered Fe3Ga compounds
with DO3 and L60 structures have shown that only in the
latter structure large magnetostrictive effects of proper sign
occur. Its appearance has been linked to the symmetry splitting of a single peak in the spin-down band near the Fermi
level into two peaks for Fe sites belonging to the sublattice
neighboring to the Ga one. Such a splitting was found to be
absent DO3. This transparent result has added a much insight
to the phenomenological explanation of giant magnetostriction based on the Ga 具100典 pair ordering since the L60 structure is composed of crystallographic lines with such Ga–Ga
pairs. However, the L60 structure may be considered only as
a rough approximate model of the B2 order even in Fe3Ga.
The reason is that the real B2 structure has an essential dis-
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073911-4
Khmelevska, Khmelevskyi, and Mohn
FIG. 4. The calculated atom-projected and symmetry projected DOSs for
Fe0.9Ga0.1. Only the DOS of the d-states is shown: full line, t2g and dotted
line eg.
order on the Ga sublattice: each site of which are populated
by Ga and Fe atoms with equal probability in Fe3Ga. One
can even wonder in this respect why L60 should be considered as a better model for B2 then DO3. Thus, our goal here
is to trace an evolution of the spin-down band splitting in
realistic disordered Fe–Ga alloys. Then, by taken as an established fact, Wu’s observation that the giant magnetostric-
J. Appl. Phys. 103, 073911 共2008兲
FIG. 5. The calculated atom-projected and symmetry projected DOSs for
Fe3Ga. Only the DOS of the d-states is shown: full line, t2g and dotted line,
e g.
tion is a consequence of the presence of atoms with such
splitting in their spin-down DOS, one can hope to arrive to
certain further conclusions concerning the magnetostriction
effect in realistic partially ordered alloys.
In Fig. 4, we show the calculated atom-projected DOS of
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073911-5
J. Appl. Phys. 103, 073911 共2008兲
Khmelevska, Khmelevskyi, and Mohn
the d-states of Fe for the alloy Fe0.9Ga0.1. The d-DOS is
divided into states with t2g and eg symmetries. There occur
no special features in the A2 DOS 共upper panel兲 and it is
very similar to that of bcc Fe, except for a disorder induced
smoothing due to the CPA averaging. A similar DOS is found
for Fe on sites, which have no Ga neighbors such as Fe1 in
B2 共second panel in Fig. 4兲 or Fe1 and Fe2 sites in DO3 共not
shown兲 in the first NN atomic shell 共see also Fig. 1兲. The
nontrivial changes in the DOS occur on Fe sites, which are
neighboring to the sublattice populated by Ga atoms. The
additional peak appears in the spin-down t2g manifold exactly near the Fermi level 共marked by arrows, see two lower
panels in Fig. 4兲. A further splitting of this peak is exactly the
reason for large magnetoelastic coupling according to Wu’s
model for the L60 structure. An interesting result is that CPA
does not predict any such splitting for the alloys considered.
Moreover, such a splitting does also not appear in the Fe3Ga
composition 共see Fig. 5兲. The point is that this splitting is not
an averaged property of the particular chemical ordering, but
rather a local environment effect and a result of the lowering
of the local symmetry around the Fe atom. It is not quite
clear, however, how this symmetry lowering should be
caused entirely by the existence of 具100典 Ga–Ga pair ordering. One can imagine that a similar splitting on Fe can also
be induced by other configurations of surrounding Ga atoms.
In order to fully investigate the situation, one needs to go
beyond the CPA and take into the account local environment
effects, for example, using the local self-consistent Green’s
function technique35 for disordered alloys. For example, by
looking at the DOS calculated for selected large supercells
for Fe–Ga in Ref. 22, one can see very peculiar DOS features
at the Fermi level for some Fe positions. One can thus conclude that before a careful study of local environment effects
in realistic disordered alloys would be done, there is no reason to connect the appearance of huge magnetostriction to
the development of the 具100典 directional ordering. Indeed,
the experimental results of Kumagai et al.,10 who found the
largest magnetostriction in the fully disordered A2 Fe3Ga
alloy and not in the B2 phase, are also pointing into this
direction.
V. CONCLUSION
From our investigation, we can conclude that on the average the splitting of the d-states which, according to Wu’s
first-principles model, leads to the giant magnetostriction in
Fe–Ga does not occur in realistic disordered alloys. Thus, it
is not a direct consequence of the particular long-range
chemical ordering, but rather a local symmetry effect. If
there is a way to explain the giant magnetostriction in Fe–Ga
alloys in any kind of an intrinsic model, i.e., based on the
electronic properties of a single domain crystal, in particular,
using Wu’s model, it is not necessarily based on the assumption of the 具100典 directional ordering. A crucial assessment
could be done only with the application of methods beyond
CPA to study realistic disordered alloys and a consequent
statistical analysis.
We also find that in Fe–Ga alloys, there is a strong interplay between thermal magnetic and chemical order. It is
more pronounced than in the well-investigated Fe–Si alloys.
It is found that the magnetic disorder strongly favors a particular chemical order. The latter may deserve further experimental and theoretical attention, also concerning an eventual
peculiar dependence of the chemical ordering on a combined
magnetic and thermal treatment.
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