Composition dependence of exchange stiffness in Fe
and M. Farle
Fakultät für Physik and Center for Nanointegration Duisburg-Essen (CeNIDE), Universität Duisburg-Essen, Lotharstr. 1,
D-47048 Duisburg, Germany
Experimentelle Physik IV, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Research and Technology Development, Seagate Technology, 47010 Kato Road, Fremont, California 94538, USA
Department Chemie und Biochemie, Ludwig-Maximilians-Universität München, Butenandtstr. 11, D-81377 München, Germany
共Received 18 March 2010; revised manuscript received 5 July 2010; published 4 August 2010
The exchange stiffness constants of chemically disordered Fe
ﬁlms with thickness around 50 nm were
determined by means of ferromagnetic resonance. It was found to increase with increasing Fe content from
6⫾4pJ/ m for x =0.27 to 15⫾4pJ/ m for x =0.67. Theoretical results from fully relativistic and scalar-
relativistic band-structure calculations using the Korringa-Kohn-Rostoker method conﬁrm the experimentally
obtained values. In addition, determination of the magnetocrystalline anisotropy by angular-dependent mea-
surements of the ferromagnetic resonance gave the possibility to estimate the exchange length that was found
to be 40–50 nm for all compositions investigated in this work.
DOI: 10.1103/PhysRevB.82.064403 PACS number共s兲: 76.50.⫹g, 71.70.Gm, 75.30.Gw
The systematic investigation of spin waves and exchange
stiffness gives the possibility, e.g., to gain more insight into
spin torque and domain-wall 共de兲pinning which are recently
discussed intensively especially relating to new magneto-
logic or data storage devices. For example, the recently pre-
sented domain-wall logic
uses the magnetic domain wall in
nanowires made of soft-magnetic materials like Permalloy as
transition edge in a changing signal. However, reducing the
dimensionality of such components to the nanometer scale
leads to the FePt system as a promising candidate
high magnetocrystalline anisotropy in the chemically ordered
state to avoid thermally activated ﬂuctuations of the mag-
netic moments, the so-called superparamagnetism,
cussed in the literature.
The intense research activities on nanoparticles of
alloys over the last decades did not only lead to new
results on the structural and magnetic properties 共see, e.g.,
Refs. 4–8兲 but also reveals some lack of knowledge about
the bulk system. Driven by the on-going discussion about
spin canting effects that may occur in Fe
we examined the exchange stiffness in the
corresponding bulk material which is connected to the length
scale of dominating exchange coupling 共exchange length兲
which usually suppresses spin canting. Ferromagnetic reso-
nance 共FMR兲 was used as a powerful tool for the determina-
tion of 共i兲 the magnetocrystalline anisotropy by angular-
dependent measurements and 共ii兲 the exchange stiffness
constant A by the analysis of standing spin waves excited in
the material. The results are supported by theoretical calcu-
lations using the spin-polarized relativistic Korringa-Kohn-
Rostoker 共SPR-KKR兲 method.
II. EXPERIMENTAL DETAILS
ﬁlms with thickness around 50 nm were
grown on MgO共001兲 substrates at room temperature by mag-
netron cosputtering from Fe and Pt targets in a vacuum sys-
tem with a base pressure of about 10
Pa. The deposition
rate was about 0.1 nm/s. X-ray diffraction indicates a high
degree of structural order, the mosaic spread is below 1°. The
layer thickness determined by Rutherford backscattering was
found to be 46⫾ 6 nm and therefore, the ﬁlms are expected
to exhibit bulk properties.
Room-temperature FMR experiments were performed us-
ing a constant microwave frequency of
⬇10 GHz with a
power of P ⬇5 mW. The sample was centered in a cylindric
microwave cavity operated in the TE
mode and a quasi-
static external magnetic ﬁeld was swept up to
=1.8 T. For this setup, the magnetic part of the microwave
coupled into the cavity is maximum in the center of the
cavity and aligned parallel to the axis of rotational symmetry
of the cylindric cavity which is perpendicular to the external
quasistatic magnetic ﬁeld. The electric ﬁeld component of
the microwave vanishes in the center of the cavity. However,
a sample with ﬁnite dimensions may short the electric ﬁeld
lines off-center. In order to minimize these effects, the
sample was cut into small pieces, about 2⫻ 2mm
. For this
size, also inhomogeneities of the magnetic ﬁeld component
are negligible. In general, microwave absorption of the
sample can be detected if the precession frequency of mag-
netization equals the frequency of the irradiated microwave.
In the ground state of the system, all spins of a ferromagnet
are aligned parallel due to the exchange interaction while
precessing around the effective magnetic ﬁeld H
of the external magnetic ﬁeld, anisotropy ﬁelds, exchange
ﬁeld, and the magnetic component of the microwave. This is
the so-called uniform mode of precession.
Spin waves 共magnons兲 may be excited by the microwave.
A schematic example of a spin wave is shown in Fig. 1.In
this case, the effective magnetic ﬁeld is parallel to the z
direction and the spin wave propagates along the y direction.
All magnetic moments precess around the ﬁeld direction in-
cluding the same angle
but with a constant angular differ-
between neighboring moments. Additional surface or
interface pinning of the spins may lead to the occurrence of
standing spin waves. In the case of a magnetic ﬁeld pointing
PHYSICAL REVIEW B 82, 064403 共2010兲
1098-0121/2010/82共6兲/064403共6兲 ©2010 The American Physical Society064403-1
along the sample normal, their possible wave vectors are
given by the condition k
/ t, where t denotes the sample
thickness. For small
, the frequency of the
precession around the exchange ﬁeld can be written as
/ ប is the magnetogyric ratio depending on the
spectroscopic splitting factor g and D is the spin-wave stiff-
ness related to the exchange stiffness via A =
D/ 2. In a
FMR experiment, standing spin waves yield additional reso-
shifted relatively to the one of the uniform pre-
In this work, only spin waves with n =1 could be observed.
For this case, the exchange stiffness can be determined from
the experimental data using the following equation:
The magnetocrystalline anisotropy as well as the effective
magnetization were determined by polar and azimuth
angular-dependent measurements at room temperature. Since
the anisotropy may strongly depend on the temperature,
FMR spectra were taken also at 20 K for the sample with the
lowest Curie temperature,
. No shift in the
resonance ﬁeld compared to room temperature measure-
ments was obtained within experimental errors. Therefore,
FMR measurements at room temperature seemed to be suf-
ﬁcient for all samples investigated in this work.
III. SPR-KKR CALCULATIONS
Band-structure calculations for chemically disordered
alloys were performed by means of the fully rela-
tivistic spin-polarized version of the KKR band-structure
method 共SPR-KKR兲 within the framework of spin density-
As structural input for the SPR-KKR cal-
culations, lattice constants of the single-crystalline Fe
ﬁlms experimentally investigated in this work were used.
The values obtained by x-ray diffraction can be found
and are in agreement to other values reported in
the literature for this system.
The SPR-KKR method rep-
resents the electronic structure in terms of the Green’s func-
tion evaluated by means of multiple-scattering theory. This
feature allows to deal with the chemical disorder by using
the coherent potential approximation alloy theory as done in
this work for the chemically disordered Fe
and angular momentum resolved density of states at the Fe
and Pt sites as well as element-speciﬁc magnetic moments
have been published elsewhere.
In the scalar-relativistic mode, ab initio Heisenberg pair
exchange parameters were calculated using the formulation
of Liechtenstein et al.
The exchange constant was calcu-
lated for all Fe-Fe, Fe-Pt, and Pt-Pt pairs as a function of
distance. In the case of dominating exchange coupling be-
tween nearest-neighbor atoms, the exchange stiffness can be
written in general for a single-element system as
where a denotes the lattice constant, J the exchange coupling
constant, and S and
the spin moment and the spin mag-
netic moment, respectively.
To achieve higher accuracy, in
this work not only nearest-neighbor atoms but contributions
from all atoms within a cluster of radius R =3a were in-
cluded. The values of J for all considered Fe-Fe, Fe-Pt, and
Pt-Pt pairs were weighted by their probability and summed
up assuming complete chemical disorder. For instance, the
probability to ﬁnd an Fe-Fe pair in an Fe
, an Fe-Pt pair P
=x共1−x兲 and a Pt-Pt pair
. Thus J can be written for all nearest-neighbor
contributions as J=x
there is a signiﬁcant difference in the calculated and experi-
mentally determined values of the Fe spin magnetic moment
for the Fe-rich alloys, experimental values
were taken to
determine the value of the exchange stiffness.
IV. RESULTS AND DISCUSSION
A. Spin waves and exchange stiffness
As an example, experimental FMR data for Fe
presented in Fig. 2共a兲 for two different polar angles
tween the external magnetic ﬁeld and the sample normal, i.e.,
=90°. In this graph, the ﬁrst derivative of the
absorption signal is shown as a function of the external mag-
netic ﬁeld. Note that the sharp resonance lines around
=0.33 T are caused by paramagnetic impurities in the
MgO substrate. The strong shift in the resonance ﬁeld that
can be assigned to the resonant microwave absorption of the
FePt ﬁlm is mainly due to the shape anisotropy that favors a
magnetization direction in the sample plane 共
=0° a second resonance line is visible at
with a lower intensity compared to the resonance of the uni-
form mode at
⬇1.28 T. The full angular dependence
is shown in Fig. 2共b兲 as contour plot of the ﬁrst derivative of
the absorption signal as a function of external ﬁeld value and
polar angle. The absolute values of the gray scale intensities
describe the amplitude of the ﬁrst derivative of absorbed mi-
crowave power according to the gray scale shown in Fig.
2共a兲. From this plot it can be seen that the spin-wave reso-
nance line is clearly detectable for −10° ⬍
⬍10°. From the
resonance position at
=0° the exchange stiffness was cal-
culated according to Eq. 共3兲 for all Fe
ﬁlms with their
FIG. 1. Example of a spin wave propagating along the y direc-
tion while the effective ﬁeld is parallel to the z direction.
ANTONIAK et al. PHYSICAL REVIEW B 82, 064403 共2010兲
For comparison the exchange stiffness was also deter-
mined by means of the SPR-KKR method. For this purpose,
the exchange constant of the coupling for Fe-Fe, Fe-Pt, and
Pt-Pt pairs was calculated as a function of distance. This is
shown in Fig. 3 for one example, i.e., Fe
. It is clearly
visible that the exchange is dominated by the coupling be-
tween two nearest-neighbor Fe atoms. The coupling constant
is about 14 meV yielding a ferromagnetic coupling. This
value exhibits only a weak dependence on the composition in
the range investigated in this work 共not shown here兲.Inthe
case of two neighboring Pt atoms or an Fe-Pt pair the ex-
change coupling is about one order of magnitude smaller.
For next-nearest-neighbor Fe atoms the coupling prefers an
antiferromagnetic spin arrangement, for third-nearest neigh-
bors the coupling is ferromagnetic again. At distances above
twice the lattice constant, the coupling constant 共almost兲 van-
ishes. For the case of bcc Fe as a reference, an exchange
stiffness of 24 pJ/m was calculated which is in good agree-
ment to the experimentally obtained values of 21 and 25
pJ/m reported in the literature.
The values for the
system are summarized in Table I. Since there are
various deﬁnitions of the exchange stiffness, for a better
comparison the sum A
in millielectron volt per
angstrom is given in addition which is also sometimes called
exchange stiffness in the literature. Again, the values of A
for all considered Fe-Fe, Fe-Pt, and Pt-Pt pairs were
weighted by their probability and summed up. Its value is
related to the exchange stiffness A as deﬁned in this work by
the inverse volume of the unit cell. Both experimentally and
theoretically obtained values of the exchange stiffness are
shown in Fig. 4 as a function of Fe content. In the experi-
mental data, there is an increase in the exchange stiffness
with increasing Fe content from 6⫾ 4pJ/ m for x =0.27 to
15⫾ 4pJ/ m for x=0.67. The results from SPR-KKR calcu-
lations are in good agreement to these values. The trend of
increasing exchange coupling with increasing Fe content can
be qualitatively understood in terms of both structural and
compositional changes: the higher Fe content leads to a
smaller lattice constant and therefore the exchange stiffness
increases as can be seen from Eq. 共4兲. However, this is only
true for moderate changes in the lattice constants since large
changes may change the value of the exchange coupling con-
stant J signiﬁcantly and may even yield an antiferromagnetic
coupling. With respect to the compositional changes, by
summing over all contributions of nearest-neighbor Fe and
Pt atoms, the fraction of Fe-Fe contributions increases with
increasing Fe content. Since these contributions are the ones
with the highest exchange coupling, the exchange stiffness
increases according to Eq. 共4兲. For all compositions, the ex-
change stiffness is smaller than in bcc-Fe bulk material.
Comparison with the value for FePt in the chemically or-
dered fct state reported in the literature, i.e., 10 pJ/m,
gests that there is no measurable inﬂuence on the exchange
TABLE I. Calculated and experimentally obtained values of the
exchange stiffness. Note that the Fe contents of the measured
samples slightly differ from the nominal values 共cf. Fig. 4兲.
0.32 170 4.5 6.2⫾ 4.0
0.40 320 8.7 11.4⫾ 4.0
0.48 370 10.2 11.9⫾ 4.0
0.60 400 11.5 10.3⫾ 4.0
0.68 430 13.0 15.0⫾ 4.0
0.72 470 13.9
FIG. 2. 共a兲 FMR spectra of epitaxial Fe
at room temperature and
⬇10 GHz for two different angles between the external
magnetic ﬁeld and the sample normal. The ﬁrst derivative of absorbed microwave power is plotted against the external magnetic ﬁeld. 共b兲
Contour plot of the ﬁrst derivative of absorbed microwave power as a function of external magnetic ﬁeld value and polar angle.
FIG. 3. Exchange coupling for Fe-Fe, Fe-Pt, and Pt-Pt pairs as a
function of radial distance r in units of the lattice constant a.
COMPOSITION DEPENDENCE OF EXCHANGE STIFFNESS… PHYSICAL REVIEW B 82, 064403 共2010兲
stiffness of the crystal symmetry in this case.
B. Magnetic anisotropies and exchange lengths
In order to calculate the exchange length, i.e., the width of
a 180° domain wall, the magnetocrystalline anisotropy con-
stant has to be known since the anisotropy energy and ex-
change coupling are competing values in this case according
to the following equation for the exchange length assuming a
Bloch-type domain wall:
is the cubic fourth-order term of the magnetocrys-
talline anisotropy density 共sometimes denoted as K
literature兲. In the case of our Fe
ﬁlms, the anisotropy
constant was extracted from angular-dependent FMR mea-
surements. Both polar and azimuthal angles were varied. In
Fig. 5 the experimental FMR spectra are shown depending
on the external magnetic ﬁeld and azimuth angle as contour
plot with the gray scale relating to the amplitude of the ﬁrst
derivative of absorbed microwave power similar to Fig. 2共b兲.
The Fe content of the samples is increasing from the left to
the right 共27 at. %, 46 at. %, 58 at. %, and 67 at. % Fe兲.
The decrease in the mean resonance ﬁeld with increasing Fe
content indicates the increase in the effective magnetization.
In addition, the linewidth becomes smaller for higher Fe con-
tents that may indicate an increase in relaxation times. How-
ever, a discussion of relaxation in FMR is beyond the scope
of the paper. The magnetic anisotropy has been analyzed by
the angular dependence of the resonance ﬁeld which is plot-
ted in Fig. 5 共lower panel兲. It is clearly indicating the four-
fold anisotropy. Additionally, a twofold anisotropy contribu-
tion is visible. The easy direction of magnetization changes
between x =0.46 and x =0.58 from the 具111典 directions to the
具100典 directions as it is known, e.g., for the composition-
dependent magnetocrystalline anisotropy of Fe
Also in the case of Fe
alloys, changes in the easy di-
rection of magnetization as a function of composition
In our case, the transition can be seen in Fig. 5 since for
共a兲 and 共b兲 maximum resonance ﬁelds are obtained for
=0°, 90°, 180°, and 270° whereas in 共c兲 and 共d兲 minimum
resonance ﬁelds are obtained at these angles. Note that the
twofold anisotropy contribution does not follow this trend: in
all cases it is along a direction including an angle of 5° with
the 共100兲 direction of the substrate leading to a slight asym-
metry in the angular-dependent resonance ﬁelds. This indi-
cates an anisotropy due to steps of the substrate or may be
growth induced. The latter seems to be a more likely expla-
nation since the ﬁlms are quite thick and therefore, substrate-
FIG. 4. Exchange stiffness in fcc-Fe
ﬁlms as a function of
Fe content obtained by KKR calculations 共open symbols兲 and FMR
experiments at room temperature 共ﬁlled symbols兲. The solid line is
a guide to the eye. Experimental value for bcc Fe is taken from Ref.
FIG. 5. Upper graphics: contour plot of the ﬁrst derivative of absorbed microwave power as a function of external magnetic ﬁeld value
and azimuth angle of 共a兲 Fe
, 共b兲 Fe
, 共c兲 Fe
, and 共d兲 Fe
at room temperature and
⬇10 GHz. Lower
graphics: extracted azimuthal dependence of the FMR resonance ﬁeld. Symbols refer to experimental data and lines refer to simulations.
ANTONIAK et al. PHYSICAL REVIEW B 82, 064403 共2010兲
induced anisotropies at the interface should not be measur-
able. However, its origin is not clear up to now but is of less
importance since we will only roughly estimate the values of
the exchange length.
The magnetocrystalline anisotropies were quantiﬁed by
simulation of the azimuth and polar angle dependence of the
resonance ﬁeld using a program developed by Anisimov
based on the Landau-Lifshitz-Gilbert formalism.
software, the resonance ﬁeld is described in terms of mini-
mization of the free-energy density including second- and
fourth-order anisotropy contributions and the Zeeman
The resonance ﬁeld is calculated for any chosen
pairs of polar and azimuthal angle,
according to Ref. 32,
兲 denotes the ﬁrst 共second兲 derivative of the
free-energy density to the angle x共xy兲. Both experimental
polar and azimuthal angular dependence of the resonance
ﬁeld were ﬁtted using the same set of ﬁtting parameters, i.e.,
an effective magnetization, the spectroscopic splitting factor
and in addition, a uniaxial anisotropy in the sample
plane as discussed before. 共The dependence on the polar
angle is not shown here.兲 We obtained values of K
between 2.8⫾1.0 and 7.1⫾ 0.5 kJ/ m
for all compositions
. For the latter case, a smaller value of
1.6⫾ 0.5 kJ/ m
was found by simulation of the experimen-
tal data. This may be related to the transition of the easy
direction of magnetization near that composition as men-
tioned above. All these values are rather small compared,
e.g., to bulk Fe in the bcc state but of the same order of
magnitude as in the case of pure Fe in the fcc state.
Using the values of K
and the corresponding exchange
stiffnesses, the exchange length is found to range between 40
and 50 nm for all compositions investigated in this work.
This value is about twice the value of bulk bcc Fe 关
=23.3 nm 共Ref. 27兲兴.
By analyses of angular-dependent FMR and spin-wave
resonance, the composition dependence of exchange stiff-
ness, magnetocrystalline anisotropy, and exchange length in
ﬁlms with compositions 0.27⬍x ⬍ 0.67 were deter-
mined. As the main result, the exchange stiffness constant
was found to increase with increasing Fe content from 6⫾4
to 15⫾ 4pJ/ m. These values are in good agreement to the
SPR-KKR results presented here. In addition, we found a
clear indication of a transition of the in-plane easy direction
of magnetization from 具111典 directions for Fe contents below
the equiatomic composition to 具100典 for Fe-rich composi-
tions. The exchange length calculated from exchange stiff-
ness and magnetocrystalline anisotropy was about 40–50 nm
and does not show any composition dependence within ex-
Concerning the question raised in the introduction one
may conclude that spin canting effects in chemically disor-
nanoparticles with diameters around 5 nm and
below are unlikely since this diameter is only about a tenth
of the bulk exchange length. In order to induce spin canting
effects, the magnetic anisotropy of the nanoparticles would
have to be 100 times larger than in the corresponding bulk
material which was never observed.
We thank M. Acet and H. C. Herper 共U. Duisburg-Essen兲
for helpful discussions. This work was ﬁnancially supported
by the DFG within the framework of SFB 445.
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