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Composition dependence of exchange stiffness in Fe

x

Pt

1−x

alloys

C. Antoniak,

1

J. Lindner,

1

K. Fauth,

2

J.-U. Thiele,

3

J. Minár,

4

S. Mankovsky,

4

H. Ebert,

4

H. Wende,

1

and M. Farle

1

1

Fakultät für Physik and Center for Nanointegration Duisburg-Essen (CeNIDE), Universität Duisburg-Essen, Lotharstr. 1,

D-47048 Duisburg, Germany

2

Experimentelle Physik IV, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany

3

Research and Technology Development, Seagate Technology, 47010 Kato Road, Fremont, California 94538, USA

4

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

兲

The exchange stiffness constants of chemically disordered Fe

x

Pt

1−x

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

I. INTRODUCTION

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

1

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

2

with its

high magnetocrystalline anisotropy in the chemically ordered

state to avoid thermally activated ﬂuctuations of the mag-

netic moments, the so-called superparamagnetism,

3

as dis-

cussed in the literature.

The intense research activities on nanoparticles of

Fe

x

Pt

1−x

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

x

Pt

1−x

nanoparticles,

9–11

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.

12

II. EXPERIMENTAL DETAILS

Epitaxial Fe

x

Pt

1−x

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

−6

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

011

mode and a quasi-

static external magnetic ﬁeld was swept up to

0

H

ext

=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

2

. 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

ជ

eff

consisting

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-

ence

␣

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

n

=n

/ t, where t denotes the sample

thickness. For small

and small

␣

, the frequency of the

precession around the exchange ﬁeld can be written as

13

n

=

␥

Dk

n

2

=

␥

D

2

t

2

n

2

, 共1兲

where

␥

=g

B

/ 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 =

0

M

s

D/ 2. In a

FMR experiment, standing spin waves yield additional reso-

nances H

n

shifted relatively to the one of the uniform pre-

cession H

uni

,

H

n

= H

uni

−

2

D

t

2

n

2

. 共2兲

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:

A =

1

2

0

M

s

共H

uni

− H

1

兲

t

2

2

. 共3兲

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,

14–18

FMR spectra were taken also at 20 K for the sample with the

lowest Curie temperature,

19,20

i.e., Fe

0.26

Pt

0.74

. 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

Fe

x

Pt

1−x

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-

functional theory.

12

As structural input for the SPR-KKR cal-

culations, lattice constants of the single-crystalline Fe

x

Pt

1−x

ﬁlms experimentally investigated in this work were used.

The values obtained by x-ray diffraction can be found

elsewhere

21

and are in agreement to other values reported in

the literature for this system.

22

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

x

Pt

1−x

alloys. Spin

and angular momentum resolved density of states at the Fe

and Pt sites as well as element-speciﬁc magnetic moments

have been published elsewhere.

21

In the scalar-relativistic mode, ab initio Heisenberg pair

exchange parameters were calculated using the formulation

of Liechtenstein et al.

23

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

24

A =

JS

2

a

⬇

J

ag

2

s

2

B

2

, 共4兲

where a denotes the lattice constant, J the exchange coupling

constant, and S and

S

the spin moment and the spin mag-

netic moment, respectively.

25

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

x

Pt

1−x

alloy is

P

FeFe

=x

2

, an Fe-Pt pair P

FePt

=P

PtFe

=x共1−x兲 and a Pt-Pt pair

P

PtPt

=共1−x兲

2

. Thus J can be written for all nearest-neighbor

contributions as J=x

2

J

FeFe

+2x共1−x兲J

FePt

+共1−x兲

2

J

PtPt

. Since

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

21

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

0.46

Pt

0.54

are

presented in Fig. 2共a兲 for two different polar angles

be-

tween the external magnetic ﬁeld and the sample normal, i.e.,

=0° and

=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

H

ext

=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

=90°兲. For

=0° a second resonance line is visible at

0

H

ext

⬇1.12 T

with a lower intensity compared to the resonance of the uni-

form mode at

0

H

ext

⬇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

x

Pt

1−x

ﬁlms with their

different compositions.

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

064403-2

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

0.46

Pt

0.54

. 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.

26,27

The values for the

Fe

x

Pt

1−x

system are summarized in Table I. Since there are

various deﬁnitions of the exchange stiffness, for a better

comparison the sum A

⬘

=兺

j

J

0j

r

0j

2

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,

28

sug-

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.

Fe content

A

⬘

共meV/Å兲

A

共pJ/m兲

A

exp

共pJ/m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

0.46

Pt

0.54

at room temperature and

rf

⬇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

064403-3

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:

xc

=

冑

A/K

4

, 共5兲

where K

4

is the cubic fourth-order term of the magnetocrys-

talline anisotropy density 共sometimes denoted as K

1

in the

literature兲. In the case of our Fe

x

Pt

1−x

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

x

Ni

1−x

alloys.

Also in the case of Fe

x

Pt

1−x

alloys, changes in the easy di-

rection of magnetization as a function of composition

20

and

temperature

29

are reported.

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

x

Pt

1−x

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.

26.

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

0.27

Pt

0.73

, 共b兲 Fe

0.46

Pt

0.54

, 共c兲 Fe

0.58

Pt

0.42

, and 共d兲 Fe

0.67

Pt

0.33

at room temperature and

rf

⬇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

064403-4

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.

30

In this

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

energy.

31

The resonance ﬁeld is calculated for any chosen

pairs of polar and azimuthal angle,

and

, respectively,

according to Ref. 32,

冉

␥

冊

2

=

1

M

2

F

冉

F

sin

2

关

兴

+

cos关

兴

sin关

兴

F

冊

−

1

M

2

冉

F

sin关

兴

−

cos关

兴

sin关

兴

F

sin关

兴

冊

2

, 共6兲

where F

x

共F

xy

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

g, K

4

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

4

ranging

between 2.8⫾1.0 and 7.1⫾ 0.5 kJ/ m

3

for all compositions

except Fe

0.58

Pt

0.42

. For the latter case, a smaller value of

1.6⫾ 0.5 kJ/ m

3

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.

33

Using the values of K

4

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

xc

=23.3 nm 共Ref. 27兲兴.

V. CONCLUSION

By analyses of angular-dependent FMR and spin-wave

resonance, the composition dependence of exchange stiff-

ness, magnetocrystalline anisotropy, and exchange length in

Fe

x

Pt

1−x

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-

perimental errors.

Concerning the question raised in the introduction one

may conclude that spin canting effects in chemically disor-

dered Fe

x

Pt

1−x

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.

ACKNOWLEDGMENTS

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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Note that in the SPR-KKR package the deﬁnition of the ex-

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Hamiltonian written as H

xc

=−兺

i⫽ j

J

ij

e

ˆ

i

e

ˆ

j

, where e

ˆ

i

and e

ˆ

j

are

unit vectors pointing into the direction of the local magnetic

moments. Thus, the absolute values of the moments 共in units of

B

兲 are already included in J

ij

.

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