Full text of DOI:10.1557/JMR.2000.0097

Magnetic moment and atomic volume in
supersaturated Fe–Cu solid solutions:
Ab initio calculations compared with experiments
Wenqing Zhang
Division of Applied Sciences and Engineering, Harvard University, Cambridge, Massachusetts 02138
E. Ma
Department of Materials Science and Engineering, The Johns Hopkins University,
Baltimore, Maryland 21218
(Received 28 July 1999; accepted 7 December 1999)
The properties of nonequilibrium face-centered-cubic (fcc) and body-centered-cubic
(bcc) Fe–Cu alloys were studied using the first-principles full-potential linearized
augmented plane wave method within the generalized gradient approximation. The ab
initio calculation results are compared quantitatively with the magnetic moment and
atomic volume observed for mechanically alloyed FexCu100−x (x ס 0 to 100)
supersaturated bcc and fcc solid solutions. The calculations show that Cu alloying
leads to a small enhancement of the magnetic moment of bcc Fe. The fcc Fe moment,
on the other hand, experiences a more pronounced increase into a high-spin state upon
alloying with Cu. It reaches approximately the same value as that in the bcc alloys for
all Cu concentrations where fcc solutions are obtained in experiments, corroborating
previous ab initio calculations using different methods. The magnetic moment
increases are accompanied by an atomic volume expansion. Both the calculated
moment and volume behavior are in good agreement with those measured for fcc and
bcc Fe–Cu solutions. The magnetovolume expansion upon magnetic interaction
between the alloyed Fe and Cu, rather than the positive heat of mixing, constitutes the
primary reason for the atomic volume increase observed.
I. INTRODUCTION
alloys is almost composition independent. This is differ-
ent from the behavior of some binary alloys of Fe with an
early transition metal element, where the Fe moment is
substantially reduced above a critical alloy concentra-
tion,13 and also different from the case of amorphous
Au–Fe, where the Fe moment quickly jumps to a much
higher value at a very small addition of the alloying
element.14 The experimental data in Fig. 1 also hint that
the moment is not sensitive to crystal structures: Fe at-
oms in the face-centered-cubic (fcc) alloys can have a
moment very much the same as that in the Fe-rich body-
centered-cubic (bcc) solutions. In addition, the measured
average atomic volume of the Fe–Cu solid solution
phases exhibits an obvious positive deviation from that
given by the Vegard’s Law for an ideal solid solution.2–4
In fact, adding the smaller Fe atoms into fcc Cu expands
rather than contracts the lattice. Such interesting mag-
netic and volumetric behavior must have their origin in
the electronic/magnetic interactions in these nonequilib-
rium phases.
The equilibrium phase diagram of Fe–Cu, a system
characterized by a positive heat of mixing, shows virtu-
ally no mutual solid solubility between the two elements
at ambient temperatures. However, using the nonequilib-
rium processing technique of mechanical alloying in a
mixture of elemental Cu and Fe powders, supersaturated
FexCu100−x (x ס 0 to 100) solid solutions can be syn-
thesized over the entire composition range at room tem-
perature.1–11 On the Fe-rich side (x > 80), single-phase
bcc solutions form, while for x < 60, single-phase fcc
solutions are the products. The two crystal structures co-
exist in the composition range of approximately 80 > x >
60.2–4,10,11 The formation of such nonequilibrium phases
makes it possible to study the effects of Cu alloying on
the magnetism of Fe,12 as well as the magnetic properties
of these ferromagnetic solid solution alloys.4–8,10 Inter-
estingly, it has been observed that the atom-averaged
magnetic moment measured for these solution alloys fol-
lows a nearly simple dilution behavior,1,4,8,12 as seen in
the average moment versus composition plot of Fig. 1
(open circles). This observation suggests that the indi-
vidual moment of the Fe atoms inside the solid solution
Complicated multiple magnetic states and magneto-
volume instabilities have been studied extensively in fcc
Fe15,16 and Invar (Fe–Ni, Fe–Pt) alloys.16,17 In the Fe–Cu
system, very recently James et al.18 reported first-
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W. Zhang et al.: Magnetic moment and atomic volume in supersaturated Fe–Cu solid solutions
Almost concurrent with the work reported in Refs. 18
and 19, recently we have also performed ab initio calcu-
lations of magnetic moment as well as atomic volume for
both the bcc and fcc Fe–Cu alloys. In the following, these
calculation results will be presented and compared with
reproducible experimental data obtained previously in
our lab for Fe–Cu solutions of both types of crystal struc-
tures.4 Note that our calculations employed the self-
consistent full-potential linearized augmented plane
wave (FP-LAPW) method within the generalized-
gradient approximation (GGA). The use of this different
ab initio scheme is useful in order to compare the merits
of different calculation methods. For example, the accu-
racy and range of applicability of CPA in concentrated
alloys, in comparison with other methods, has in fact
been a subject under continuous investigation.20–24 Cal-
culations through a different route may allow us to assess
the reliability of the predictions and reach a better agree-
ment with, and explanation of, the experimental moment
and lattice parameter (atomic volume) data.
II. METHOD OF CALCULATION
FIG. 1. Calculated and measured atomic magnetic moment in fcc and
bcc Fe–Cu alloys versus Fe concentration. Solid symbols are for in-
dividual Fe moment, and open symbols for average moment per metal
atom in the alloy, M. Experimental data are shown with open circles.
The solid line is the Vegard’s Law interpolating Cu and bcc Fe.
Dashed and dotted lines are used to connect the data points obtained in
the present calculation to guide the eye.
In this work we choose to simulate the alloys using
ordered Fe–Cu intermetallics with bcc and fcc-based
structures. This method has been proved to be able to
provide quite reliable information about the heat of mix-
ing and phase stability of alloys.24–27 The validity of this
approach as applied to our random solutions will be dis-
cussed later in this paper.
principles calculations of the magnetic properties of solid
solutions using the Korringa–Kohn–Rostoker method
within the coherent-potential approximation (CPA).
They showed that the Fe magnetic moment increases
slowly with Cu addition from 2.2 to around 2.5 ␮B in bcc
alloys, and a moment in the same range was also found
in fcc alloys when the Cu content is above approximately
17 at.% (x ס 83) at which an alloying induced low spin
to high spin transition occurs. Tatarchenko et al.,19 em-
ploying the linear-muffin-tin-orbital method with CPA in
their calculation for only the fcc Fe–Cu solutions, simi-
larly obtained an abrupt nonmagnetic to high-spin state
transition at x ס 82 and a somewhat higher fcc Fe mo-
ment (2.5–2.7 ␮B). These calculations18,19 predict a com-
position dependence of the moment that is similar to the
experimentally observed nearly linear simple dilution be-
havior in Fig. 1, although with a slightly higher slope.
Tatarchenko et al.19 further reported the lattice param-
eters of the fcc alloys and found that for Cu above
18 at.% (x < 82) the lattice parameter is almost constant
at the same value as that of pure Cu. This result does not
fully explain the composition-dependent lattice param-
eter observed in mechanically alloyed fcc Fe–Cu solid
solutions.2–4 No calculations and comparisons have been
made for bcc solutions in Ref. 19.
The FP-LAPW method28 is employed to carry out all
electronic structure and total energy calculations. The
method divides the cell into nonoverlapping atomic
spheres, i.e., muffin-tin area (MT) and interstitial (INT)
area, and a different basis is used for MT and INT areas.
Since no shape approximations are made for the potential
and charge density in both the INT and MT, this ap-
proach has been proven to be the most reliable method to
study electronic structure and energetics of solid crys-
tals.26,28 The calculations are spin-polarized, and the
exchange-correlation contributions to the total energy are
included using the GGA of Perdew et al.29 This choice of
using GGA, and not just the local spin density approxi-
mation (LSDA), was made because it is well known that
LSDA cannot correctly predict the phase stability of
Fe,30–32 a problem that can be rectified using GGA for Fe
and Fe-containing complex systems.30–34
The unit cell structure we used is shown in Fig. 2. The
atoms occupy the corner and fcc sites. The fcc or bcc
lattice can be obtained for c/a ס 2 or c/a ס √2, respec-
tively. Different crystal structures can be obtained with
different atomic occupancy of the eight sites in the cell.
If A atoms occupy only the corner sites and all other sites
are left for B atoms, a structure of AB7 is obtained. One
more A atom at (1/2, 1/2/, 1/2) will lead to D022 or D03,
654
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W. Zhang et al.: Magnetic moment and atomic volume in supersaturated Fe–Cu solid solutions
the fcc and bcc structures are listed in Table I. The results
from experiments37 or other theoretical calculations30–32
are also included in the same table for comparison. Our
results are in excellent agreement with those obtained by
other groups. The calculated atomic volume of fcc Cu is
slightly bigger than the experimental value, a known re-
sult due to the adoption of the GGA.29 The same proce-
dure is employed to obtain the parameters for alloys.
III. RESULTS AND DISCUSSION
The magnetic moment as a function of Fe concentra-
tion, x, calculated for both the fcc and bcc alloys, has
been displayed in Fig. 1. Also included are experimental
data for these ferromagnetic alloys we measured at
10 K.4 As shown in the figure, ab initio calculations in
general predict some increases of the individual Fe mo-
ment (solid squares and triangles) in both the fcc and the
bcc Fe–Cu alloys compared with elemental Fe, in agree-
ment with other recent calculations using different ap-
proaches.18,19 When alloyed with Cu, the fcc Fe moment
is elevated to a level similar to that in bcc alloys, which
has been predicted recently for all compositions above
approximately 17–18 at.% Cu,18,19 and observed in epi-
taxial fcc Fe/Cu thin films.38 Cu remains largely unpo-
larized. When converted into average magnetic moment
per metal atom in the alloy, M, versus x (open squares
and triangles in Fig. 1), all the calculated data fall close
to the solid line in Fig. 1 interpolating the two end points
for fcc Cu and bcc Fe, which would be the result for a
simple magnetic dilution. The latter is what has been
observed experimentally in mechanically alloyed Fe–Cu
solid solutions4 (open circles). The reasonable agreement
between the calculated and measured moments confirms
that indeed the magnetic behavior observed in experi-
ments can be largely rationalized in terms of the elec-
tronic structure of the alloys. A small positive deviation
is seen in Fig. 1, similar to what was reported in Refs. 18
and 19. This deviation in M arises from the small in-
crease of the individual Fe moment in alloys relative to
that in pure bcc Fe, and is not very pronounced and
visible in Fig. 1 because of the high Cu concentration in
these alloys. This discrepancy will be further discussed at
the end of this section.
FIG. 2. Unit cell structure of ordered alloys AnB8−n employed in the
present calculations. Only one structure AB7 is shown as an example.
Some atoms are not shown for clarity. The cell in thin lines shows how
the structure can be viewed as bcc lattice when c/a ס √2.
and another two atoms at (1/2, 1/2, 0) or (0, 0, 1/2)
correspond to L10 or B2 structures, respectively.35 The
advantage of this method is that we can get the fcc- and
bcc-based lattices in a consistent and parallel way. This
allows a direct comparison of the results between the two
types of lattice.
In our computations, the MT radius for Fe and Cu
atoms were set to 2.0 au and 2.1 au, respectively. Thirty
special k points in the irreducible part of the first Bril-
louin Zone (BZ) were used for the integral over BZ to
generate the charge density. The product of the MT ra-
dius, RMT, and the maximum reciprocal lattice vector
kcut to determine the LAPWs, i.e., RMT kcut, was equal to
8.0. The cutoff of lattice vector G in the Fourier expan-
sion of charge density and potentials was set to 10.0 au−1.
Convergence was assumed when the total energy varia-
tion is <0.1 mRyd/atom. These parameters have been
adopted for all the structures studied.
We first checked the ground-state properties of para-
magnetic (PM) fcc and bcc Cu, and ferromagnetic (FM)
fcc and bcc Fe. The total energies, ET, are calculated for
several volumes, V, around the equilibrium one, V0. To
obtain the exact minimum energy E0 and V0 for a given
magnetic state, the ET versus V data obtained are fitted
using the Murnaghan’s equation of state.36 The obtained
equilibrium volume V0 and energy difference between
In Fig. 1, Fe is calculated to be also in high-spin state
in fcc alloys even when Cu content is below 18 at.%.
This result is different from previous first-principles cal-
TABLE I. Ground state properties of ferromagnetic Fe and paramagnetic Cu.
Fe
Cu
fcc
bcc
fcc
bcc
V0 (Å3/atom)
M (␮B/atom)
Ebcc − Efcc (kJ/mol)
11.14
2.00
иии
11.80 (11.82[37])
2.29 (2.22[37])
−19.00 (−18.24[30],−30.18[31],−19.55[32])
12.24 (11.76[37])
12.23
иии
иии
иии
3.56
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W. Zhang et al.: Magnetic moment and atomic volume in supersaturated Fe–Cu solid solutions
culations that predict the ground state of fcc Fe to be
nonmagnetic.15,19 Antiferromagnetic ordering16 and low-
spin state15–19 have also been seen before the high-spin
states are reached at larger atomic volumes.15–19 We are
aware of this complicated picture. In our calculations,
only high-spin ferromagnetic states are considered, be-
cause the experimental data suggest that they are the only
states existing in the solution alloys. Due to the adoption
of GGA, a high-spin state of close to 2.0 ␮B is predicted
to have a slightly lower energy compared with the para-
magnetic state of fcc pure Fe. We note that the exact state
pure fcc Fe should be in is not the focus of this work, and
moreover it does not affect our comparison with experi-
mental data in Fig. 1. This is because in experiments only
bcc Fe and bcc solid solutions are obtained for Cu con-
tent below 20% (x > 80). The first two points for ferro-
magnetic fcc alloys on the Fe-rich side are thus
inconsequential. For x < 80, our moment results for fcc
alloys agree with those calculated using CPA and other
methods.18,19
The small change in Fe moment with Cu alloying
was noticed before by Chien et al.,12 who pointed out
that in pure metals the position of the Cu d band is lower
than that of the Fe d band by about 2 eV, such that upon
alloying, the Cu d band would not interact very strongly
with the Fe d band. Increasing Cu in Fe–Cu alloy would
decrease the number of interacting magnetic neigh-
bors around Fe atoms, resulting in reduced Fe d
bandwidths, which implies larger moments. Also, alloy-
ing enables fcc Fe to achieve the atomic volume condu-
cive to the transition to higher moment states. The
magnetovolume expansion effects will be further dis-
cussed next.
The calculated average atomic volumes in both the fcc
and bcc Fe–Cu alloys are shown in Fig. 3 as a function of
Fe concentration. The calculations were made for both
the ferromagnetic and paramagnetic states, which allows
a comparison with the experimental data2–4 as well as an
assessment of the effects of ferromagnetic interaction.
Several points are worth noting in this plot. First of all,
when paramagnetic (full symbols), both the fcc and bcc
alloys follow fairly closely a Vegard’s Law interpolating
Cu and paramagnetic Fe (but will not be anywhere close
to the Vegard’s Law using the experimental materials Cu
and ferromagnetic bcc Fe as the end points). Secondly,
by comparing the atomic volume behavior between the
paramagnetic and ferromagnetic (open symbols) Fe–Cu
alloys, with a positive heat of mixing in both cases,4 it
can be concluded that the magnetic interaction rather
than the positive heat of mixing would be primarily re-
sponsible for any obvious positive deviation in atomic
volume away from the Vegard’s Law observed in this
system.2–4 In other words, the connection between the
heat of mixing and atomic volume is subtle and a system
with a positive heat of mixing may not necessarily ex-
FIG. 3. Calculated average atomic volume, for both fcc and bcc struc-
tures, of FM and PM Fe–Cu alloys versus Fe concentration.
hibit significant atomic volume expansions from that for
an ideal solid solution. Thirdly, in both the fcc and bcc
forms, our calculations show that ferromagnetic Fe has a
larger atomic volume than paramagnetic Fe, which is
consistent with a magnetovolume effect.15 Finally, alloy-
ing ferromagnetic Fe with Cu further dilates the lattice.
This result indicates that the Fe atom, in ferromagnetic
state, further increases its apparent/effective volume
upon alloying with Cu. Such a behavior is especially
pronounced in the case of fcc ferromagnetic Fe. The
volume expansion observed in Fig. 3 can be seen to be
directly correlated with the Fe moment increase upon
alloying in Fig. 1. In other words, alloying leads to an
increase in the magnetic moment of bcc Fe, which is
accompanied by a small volume expansion. Alloying
also promotes fcc Fe to such a high-spin state that it has
a moment similar to that in bcc alloys, leading to a more
pronounced increase in fcc Fe volume (Fig. 3). As a con-
sequence, Fe in bcc alloys, and more importantly in fcc
alloys as well, will have volumes similar to but higher
than that of ferromagnetic bcc pure Fe. This will cause
the volumes observed to be not only close to but also
above that given by the Vegard’s Law interpolating the
atomic volumes of fcc Cu and ferromagnetic bcc Fe, the
two starting materials used in experiments.
A relative volume expansion ratio can be defined as
V0 − V
␥ =
,
(1)
V
656
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W. Zhang et al.: Magnetic moment and atomic volume in supersaturated Fe–Cu solid solutions
where V0 is the atomic volume of the alloy, and V is that
The primary physical origin of the volume behavior is
captured in our calculations despite the approximations
neglecting the atomic vibrations and the difference be-
tween ordered and random solutions. The use of only
ordered alloys in our calculations requires a few more
comments. As seen in the figures, the calculation results
are similar for fcc and bcc structures. Additional tests
showed that the results are also not sensitive to different
crystal structures in the fcc geometry at the same com-
position (e.g., L12 versus D022 at Fe75Cu25 or Fe25Cu75,
not shown). Also, it has been shown before that the
magnetic moments calculated ab initio for ordered al-
loys are only slightly larger than those for random
solutions.18 As such, it is expected that the general con-
clusions drawn from the calculations presented would
also hold for random solid solutions. More detailed in-
formation about the alloys and the effects of the local
environment may be obtained by calculations for many
different configurations at each composition, and at
more compositions. This, however, is beyond the scope
of the present work aimed at elucidating the general
trend in the moment and volume behavior observed in
experiments.
Finally we comment again on the small discrepancy
between the calculated and measured moments,
Fig. 1. One possible reason for this difference is that
our calculations, being made for ordered alloys, slightly
overestimated the moments of random alloys, as
observed previously in Ref. 18 and mentioned in the
last paragraph. On the other hand, it is also likely
that the real moments in the alloys are larger than
those reported in Fig. 1 and closer to the calculated
values. This is because our samples, as all the unstable
alloys formed by nonequilibrium processing, have very
small grain sizes of the order of 10 nm. Ten percent
or so of the alloy atoms will reside in grain boundary
regions. Some of these grain boundary atoms may be
in an amorphous-like structure. It has been shown
before that Fe atoms in some amorphous alloys may
dramatically loose magnetic ordering, thus contributing
no appreciable moment to the measurement.41,42
Moreover, a small amount of Fe atoms (of the order
of 1% for the sample at the equiatomic composition)
can be left unalloyed and in the nonmagnetic fcc state
(␥–Fe) in the as-milled powder samples.5,8,43 Since
the experimental data in Fig. 1 have been averaged
over all atoms present in the alloy, these effects
would result in a slightly smaller apparent M. The lat-
tice parameter (atomic volume) data, on the other
hand, are not affected because they were calculated
from the positions of the Bragg x-ray diffraction
(XRD) peaks originated from the solution crystallites.
The small amount of “grain boundary” amorphous-like
atoms and ␥–Fe crystals are not detectable in the XRD
experiments.
predicted by the Vegard’s Law interpolating the atomic
volumes of fcc Cu and ferromagnetic bcc Fe. Figure 4
shows ␥ as a function of x, for both the ferromagnetic and
paramagnetic Fe–Cu alloys, in comparison with experi-
mental average atomic volume data based on lattice pa-
rameters measured in our previous work.4 The baseline at
zero represents the Vegard’s Law using the two experi-
mental elements as the end points. The relative volume
expansion ratio calculated for ferromagnetic alloys, as
well as its composition dependence, are in good quanti-
tative agreement with experimental data (circles) for me-
chanically alloyed Fe–Cu solid solutions. This indicates
that magnetism, i.e., the magnetic interaction between Cu
and Fe leading to magnetovolume expansion as dis-
cussed in the preceding paragraph for the calculated re-
sults, can indeed be used to explain the atomic volume
behavior observed in experiments.4 The system gets to
lower its total energy due to this magnetic interaction,
although at the expense of an energy increase due to
volume expansion.39 As mentioned above, the experi-
mentally observed positive deviation from the Vegard’s
Law, although not uncommon in solution alloys,40 is
in this case a consequence of the magnetism of the al-
loys rather than the positive heat of mixing between
Fe and Cu.
FIG. 4. Calculated and measured atomic volume deviation from the
Vegard’s Law averages (using fcc Cu and ferromagnetic bcc Fe as
references), in terms of ␥, versus Fe concentration.
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W. Zhang et al.: Magnetic moment and atomic volume in supersaturated Fe–Cu solid solutions
11. P.J. Schilling, J.H. He, R. Tittsworth, and E. Ma, Acta Mater. 47,
IV. CONCLUSIONS
2525 (1999).
The first-principles calculations in this work using the
FP-LAPW-GGA method corroborate recent calculation
results18,19 that, different from some other transition
metal elements, Cu alloying causes enhancement of the
magnetic moment of Fe. The bcc Fe moment is enhanced
slightly, whereas the Fe moment in the fcc alloys reaches
a high-spin state at the same level as that in bcc alloys at
all compositions where fcc Fe–Cu solutions can actually
be obtained in experiments. The moment increase is
likely to be associated with the decrease in Fe band-
widths upon alloying. The magnitude of the calculated Fe
moment explains fairly well the average moments and
the nearly simple dilution behavior measured for me-
chanically alloyed fcc and bcc Fe–Cu solid solutions.
Our calculations also indicate that an increase in the Fe
atomic volume is associated with the moment increase,
especially in fcc alloys. The magnitude and composition
dependence of this volume expansion are in agreement
with the experimentally observed average atomic volume
data for both the bcc and fcc solid solutions. It is mainly
the magnetovolume effect, not the positive heat of mix-
ing of the Fe–Cu system, that is responsible for the ob-
served atomic volume and its positive deviation from the
Vegard’s Law averages of Cu and ferromagnetic bcc Fe.
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