Electronic origin of the negligible magnetostriction of an electric steel Fe1-xSix alloy: Adensity-functional studyDorj Odkhuu, Won Seok Yun, and Soon Cheol Hong Citation: Journal of Applied Physics 111, 063911 (2012); doi: 10.1063/1.3694744 View online: http://dx.doi.org/10.1063/1.3694744 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/111/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ab initio studies of disorder in the full Heusler alloy Co2FexMn1−xSi J. Appl. Phys. 113, 17B106 (2013); 10.1063/1.4801745 Effect of magnetic field annealing on magnetic properties for nanocrystalline (Fe1− x Co x )78.4Si9B9Nb2.6Cu1alloys J. Appl. Phys. 113, 17A320 (2013); 10.1063/1.4795620 Enhanced magneto-impedance in Fe73.5Cu1Nb3Si13.5B9 ribbons from laminating with magnetostrictiveterfenol-D alloy plate Appl. Phys. Lett. 101, 251914 (2012); 10.1063/1.4773237 Structural, magnetic, and magnetostriction behaviors during the nanocrystallization of the amorphous Ni 5 Fe68.5 Si 13.5 B 9 Nb 3 Cu 1 alloy J. Appl. Phys. 99, 08F104 (2006); 10.1063/1.2162810 Origin of large magnetostriction in FeGa alloys J. Appl. Phys. 91, 7358 (2002); 10.1063/1.1450791

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Electronic origin of the negligible magnetostriction of an electric steelFe1-xSix alloy: A density-functional study

Dorj Odkhuu, Won Seok Yun, and Soon Cheol Honga)

Department of Physics and Energy Harvest-Storage Research Center, University of Ulsan, Ulsan 680-749,South Korea

(Received 4 November 2011; accepted 17 February 2012; published online 23 March 2012)

To understand the negligible magnetostriction of Fe-Si alloys used as cores in electric transformers or

motors, density-functional calculations were performed on the tetragonal magnetostriction coefficient,

k001, of Fe1-xSix (x� 0.25) alloys as a function of x using the highly precise all-electron full-potential

linearized augmented plane-wave method. It was found that the calculated magnetostriction

coefficients closely reproduced experimental trends and the negligible magnetostriction near x¼ 0.111

originated from the mixed phases of A2, B2, and D03; each different atomic type of Fe of Fe0.889Si0.111

has a small and opposite contribution to the magnetostriction. VC 2012 American Institute of Physics.

[http://dx.doi.org/10.1063/1.3694744]

I. INTRODUCTION

Minimizing energy losses is an important part of making

many systems more environmentally friendly. Effective

transformers are indispensable in the efficient transmission

and distribution of electric power. Efficiency is significantly

improved by the addition of Si to the transformer core steels.

One of the main sources of energy loss is the vibration of

transformers caused by magnetostriction of the core. A mate-

rial with low magnetostriction is required to minimize

energy loss by reducing transformer vibrations.1

Experiments have shown that the magnetostriction of

Fe-Si alloys with low Si concentrations is small,2–5 and the

magnetostriction coefficients of k001 and k111 cross each

other at around 11-12 at. % Si and both are nearly zero at

these concentrations. Since the Fe-Si alloys have high elec-

tric resistivity, high permeability, and high saturation mag-

netization in addition to extremely low magnetostriction, the

alloys are highly desirable for soft magnetic cores. However,

the application of an alloy with Si concentrations higher than

6.5 at. % is limited because of the difficulties in machining

or shaping due to its brittleness. Fortunately, recent techni-

ques, such as rapid solidification, melt-spinning, and

powder-metallurgy have been successful in synthesizing

alloys that have good ductility.6–9

Even though magnetic properties, including the magneto-

striction and magneto-crystalline anisotropy energy (EMCA)

of Fe-rich Fe-Si alloys, have been intensively investigated by

a number of experimental groups,2–9 a microscopic under-

standing of the origin of the significantly reduced magneto-

striction of Fe-Si alloys at certain Si concentrations is needed

in order to develop new magnetostrictive materials. Estimat-

ing the magnetostriction of a 3d-transition alloy by a first-

principles calculation method remains challenging because

the spin-orbit coupling (SOC) to determine the intrinsic mag-

netostriction is quite weak in 3d-transition metals compared

to the other terms of the Kohn-Sham equation. Despite these

challenges, such first-principles calculations using the full-

potential linearized augmented plane-wave (FLAPW) method

have been successful for explaining the magnetostriction of

Fe-based alloys enhanced by the addition of non-magnetic

elements (Al, Ga, Ge, and Be) in terms of their electronic

structures.10–13 Experimental studies14–20 of single crystals

without obvious phase mixtures recently detected enhanced

magnetostrictions, supporting intrinsic origins of the

enhanced magnetostriction rather than extrinsic ones,21–23

including the rotation of nanoprecipitates induced by mag-

netic fields. This proposal has also been confirmed by very

recent first-principles predictions of the binary and ternary

Fe-rich alloys with 5d Pt and Ir, which have a very large mag-

netostriction due to the large SOC of the 5d-orbital.24–26

In this work, the tetragonal magnetostriction coefficient

k001 of Fe1-xSix alloys was calculated with different Si concen-

trations (x � 0.25) using the FLAPW method and confirmed

the experimental observations2–5 that the magnetostriction of

Fe1-xSix becomes negligible around x¼ 0.111. Effects of the

atomic structure on the nearly zero magnetostriction of

Fe0.889Si0.111 were investigated by adopting several different

atomic structures because Si atoms in a real sample might be

randomly positioned in a mixed phase.

This paper is organized as follows: Sec. II presents the

computational approach and relevant structural models. In

Secs. III A and III B, the fundamental properties of Fe1-xSixalloys are presented including lattice constants, magnetic

moments, and elastic coefficients. Calculated magnetostriction

coefficients are provided and compared with the experimental

and previous theoretical data in Sec. III C In Secs. III D and

III E, the origin of the zero magnetostriction of Fe0.889Si0.111

is discussed. Electronic structures are given to provide physi-

cal background on the variation in magnetostriction with Si

concentration in Sec. III F. Section IV summarizes our results.

II. DETAILS OF THE CALCULATIONS

Different phases of A2 (a-Fe), B2 (CsCl-type), and D03

(BiF3-type) are available for Fe-rich Fe-Si alloys, and the

phase evolves from A2 to B2 to D03 as the Si concentration

a)Author to whom correspondence should be addressed. Electronic mail:

schong@mail.ulsan.ac.kr.

0021-8979/2012/111(6)/063911/7/$30.00 VC 2012 American Institute of Physics111, 063911-1

JOURNAL OF APPLIED PHYSICS 111, 063911 (2012)

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increases up to 25 at. % Si.27 The different ordered phases of

A2, B2, and D03 are known to coexist at concentrations

between 10 and 22 at. % Si. Accordingly, a large supercell

must be employed to generate a realistic model to simulate

experimental situations. Ordered 2� 2� 2 supercells were

used for Fe0.9375Si0.0625 [Fig. 1(a)], Fe0.8125Si0.1875 [Fig.

1(e)], and Fe0.75Si0.25 [Fig. 1(f), the D03 structure]. In order

to more realistically simulate mixed phases of A2, B2,

and D03 structures of 10-13 at. % Si alloys having near

zero magnetostriction, 3� 3� 3 supercells were adopted for

Fe0.9075Si0.0925 [Fig. 1(b)], Fe0.889Si0.111 [Fig. 1(c)], and

Fe0.871Si0.129 [Fig. 1(d)]. The magnetostriction of bcc Fe was

also calculated for a reference. Structural optimization with

respect to volume and tetragonal distortion was performed

by total energy minimization.

The density-functional Kohn-Sham equations were

solved in a self-consistent manner using the highly precise

FLAPW method.28 The FLAPW method deals with both

core and valence electrons very accurately, and is one of the

most suitable methods for the study of magnetic systems. A

spin-polarized generalized gradient approximation (GGA)

(Ref. 29) was used to take into account the exchange-

correlation interaction among electrons. Augmented plane

waves with an energy cutoff of 12.25 Ry (256 Ry) were used

to expand the wave functions (charge and potential) in the

interstitial region. For calculations, wave functions, charge

density, and potential inside muffin-tin spheres with radii of

2.0 a.u. for Si and 2.2 a.u. for Fe were expanded with l � 8

lattice harmonics. We used 2176, 936, and 288 k-points in

the irreducible Brillouin zone for bcc Fe, the 2� 2� 2, and

the 3� 3� 3 supercells, respectively. Self-consistency was

assumed when the root mean-square differences between the

input and the output spin and charge densities were less than

1.0� 10�5 e/(a.u.).3

The tetragonal magnetostriction coefficient, k001, is pro-

portional to the change in length (Dl/l) along the (001) direc-

tion due to an applied magnetic field. For the calculation of

k001, we elongated or contracted the systems along the z axis,

keeping the unit cell volume fixed. The k001 of a system can

be obtained from the strain (c/c0) dependence of its EMCA

and total energy (Etot).30 EMCA originates from the SOC term

in the Dirac equation, Hsl ¼ nð~r �~LÞ, where n, ~r, and ~L are

the SOC strength constant, spin, and orbital angular momen-

tum operators, respectively. The SOC was treated in a per-

turbed way and the lowest-order contribution to the SOC is

given by31

Esl ¼ �ðnÞ2X

o;u

jhoj~r �~Ljuij2

eu � eo

; (1)

where o and u represent the sets of occupied and unoccupied

states, respectively.

III. RESULTS AND DISCUSSION

A number of theoretical calculations of the structural,

electronic, and magnetic properties of Fe-rich Fe1-xSix alloys,

especially Fe3Si, have been reported.32–36 Since the magneto-

striction of a material is related to both its magnetic and elas-

tic properties, the magnetism and elasticity of Fe1-xSix alloys

were evaluated.

A. Lattice constant and magnetic moment

Calculated results of the equilibrium lattice constant and

magnetic moment of Fe1-xSix alloys are presented in Figs. 2(a)

and 2(b) as solid squares, respectively. Experimental (open

symbols)36–39 and other theoretical values (solid symbols)32,33,36

FIG. 1. The crystal structures of Fe1-xSixalloys at (a) x¼ 0.0625, (b) x¼ 0.0925, (c)

x¼ 0.111, (d) x¼ 0.129, (e) x¼ 0.1875, and

(f) x¼ 0.25. The structures were determined

by total energy minimization to be the most

stable of the possible geometries. The light

and dark balls represent Fe and Si atoms,

respectively.

063911-2 Odkhuu, Seok Yun, and Cheol Hong J. Appl. Phys. 111, 063911 (2012)

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are also plotted for comparison. The lattice constant decreased

linearly with Si concentration, which is expected due to the

smaller atomic radius of Si compared to Fe. The present results

were found to be more consistent with experimental results,

compared to previous calculations.

The presence of Si atoms significantly affected the mag-

netic moments of their first and second nearest neighbor Fe

to Si due to hybridization between the Fe-d and Si-s,p orbi-

tals. The local magnetic moment of the first nearest neighbor

Fe was calculated to be reduced to 1.35 lB, whereas the

moment of the second nearest neighbor Fe atom was

enhanced to 2.56 lB in the Fe0.75Si0.25 alloy, consistent with

previous experiments.40 The average magnetic moment of

Fe1-xSix was found to decrease with the number of Si atoms

[see Fig. 2(b)] due to the enhanced number of Fe atoms hav-

ing Si atoms as nearest neighbors. The present calculated

magnetic moments are also more consistent (less than 1%

over the whole range of Si concentrations) with the experi-

mental results38,39 than the other theoretical results.32,33,36

B. Bulk modulus and elastic constant

The elasticity of Fe1-xSix alloys was investigated as a

function of Si concentration. The bulk modulus B and tetrag-

onal shear constant C0 were determined by calculating Etot as

a function of volume (V) and tetragonal distortion (c/c0),

respectively. The calculated results are presented in Fig. 3,

together with currently available experimental results2,41–43

for comparison. Experiments show that B and C0 decrease

slightly with Si concentration up to 12 at. % and then

increase with further increases in x. The agreement between

theoretical and experimental results is reasonable, even

though there are small deviations for some Si concentrations.

The inverted triangle in Fig. 3 was obtained from a measure-

ment43 on an ordered single crystal under low temperature

and is more consistent with the present calculated value than

the other previous results. This indicates that the small devia-

tions may arise from the finite temperatures and the disorder

of the samples in the experiments.

C. Magnetostriction

The curvature of Etot (elastic property) and the slope of

EMCA (magnetic property) determine the magnetostric-

tion.11,25 Calculated magnetostriction coefficients k001 for Fe1-

xSix alloys are presented as solid squares in Fig. 4 and com-

pared with experimental results (denoted as open symbols)2–4

and other theoretical results (denoted as solid inverted trian-

gles).44 As shown in Fig. 4, the magnetostriction measured in

the experiments starts from a positive value (þ20 ppm) for

pure Fe and increases initially with Si concentration up to

FIG. 2. Concentration dependence of (a) lattice constants normalized by the

size of the bcc Fe unit cell and (b) magnetic moments per atom of Fe1-xSixalloys. Experimental (open symbols) and other theoretical data (solid sym-

bols) are presented for comparison with the present results (solid squares).

FIG. 3. (a) Bulk modulus B and (b) tetragonal shear constants C0 of Fe1-xSixalloys. Solid squares and open symbols represent the present and the experi-

mental results, respectively.

063911-3 Odkhuu, Seok Yun, and Cheol Hong J. Appl. Phys. 111, 063911 (2012)

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about 6 at. %. After reaching a maximum, it decreases rapidly

with Si concentration and ends at a saturated negative value

(about�20 ppm) for Si concentrations higher than about 20 at.

%. The calculated magnetostriction coefficients for pure bcc

Fe and D03 Fe0.75Si0.25 wereþ 21 and �18 ppm, respectively,

which was quite consistent with the experimental results.

Furthermore, very small magnetostrictions (0.8 and 2.5 ppm

for x¼ 0.111 and 0.129, respectively) were reproduced and

experimentally observed around an Si concentration of 12 at.

%. Thus, the present calculated magnetostrictions of Fe1-xSixalloys with different Si concentrations seem to represent the

experimental trend quite well.

As expected, due to the nature of the isoelectricity of Si

and Ge, the general behavior of the magnetostriction of Fe1-

xSix is very similar to that of Fe1-xGex (Ref. 12) with similar

Ge concentrations. However, close examination reveals that

the following differences do exist: (i) the maximum and zero

magnetostrictions are located at different Si (6 and 11 at. %)

and Ge concentrations (11 and 14 at. %) and (ii) the magni-

tude of the negative magnetostrictions are quite different

(-18 ppm for D03 Fe3Si and -730 ppm for D03 Fe3Ge) at

x¼ 0.25. It is evident from the phase diagrams that the D03

phase of Fe1-xSix, which contributes to its negative magneto-

striction, starts to form at a relatively smaller x value com-

pared to that of Fe1-xGex. Therefore, it is reasonable to

ascribe the first difference to the early formation of the D03

phase of Fe3Si. The second difference is partially due to the

large C0 (48 GPa) of Fe3Si in contrast to that (6.8 GPa) (Ref.

12) of Fe3Ge. The origins will be discussed later in terms of

their electronic structures.

D. The effects of atomic configuration onmagnetostriction

As mentioned in the previous section, the calculated

magnetostriction of Fe0.889Si0.111 (labeled A-type) shown in

Fig. 1(c) was nearly zero. However, it was still necessary to

take some other structures into account for Fe0.889Si0.111 in

order to investigate atomic structural effects on the magneto-

striction. The total energies and magnetostrictions were cal-

culated for some plausible structures and are presented in

Figs. 5(a)–5(c), labeled the B-, C-, and D-types, respectively.

The calculated Etot and EMCA of Fe0.889Si0.111 for the four

different atomic configurations are plotted in Figs. 6(a) and

6(b) as functions of tetragonal distortion c/c0, respectively.

The optimized lattice constants were calculated to be 2.843

A, regardless of atomic configuration. As shown in Fig. 6(a),

the A-type atomic structure was confirmed to be the most

stable. The B-type had nearly degenerated to the A-type,

with just a slightly higher energy of 1.5 meV/Fe, but the D-

type structure, where some Si atoms have a Si atom as their

nearest neighbor, had a relatively high total energy. This

indicates that Si atoms tend to be distributed uniformly in

the alloy. The stability of the D03-like A-type structure is

consistent with previous experimental observations of D03

structures.27 The A-, B-, D-types were found to be stable in

the cubic symmetry against tetragonal distortion and the

EMCA values at c/c0¼ 1.00 were very small. The C-type, on

the other hand, was stabilized such that it was slightly tetrag-

onally distorted (c/c0¼ 1.005), as expected from the

symmetry.

As can be seen in Fig. 6(b), the slopes of the EMCA values

are extremely low compared to those of other magnetostric-

tive Fe-based alloys.10–13,24–26 Furthermore, the slopes of the

relatively stable structures (A- and B-types) are lower than

those of the unstable structures (C- and D-types). As listed in

Table I, the stable structures have small magnetostriction

coefficients and the dependency on atomic configuration is

not very significant. Hence, near zero magnetostriction is

expected around 11 at. % Si, regardless of the degree of the

alloy.

FIG. 4. Magnetostriction of Fe1-xSix alloys as a function of Si concentration.

Solid squares, open symbols, and solid reverse triangles denote the present,

experimental, and other theoretical values, respectively.

FIG. 5. The plausible atomic configurations

of (a) B-type, (b) C-type, and (c) D-type

structures of Fe0.889Si0.111. The light and

dark balls represent Fe and Si atoms,

respectively.

063911-4 Odkhuu, Seok Yun, and Cheol Hong J. Appl. Phys. 111, 063911 (2012)

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Total energy differences, magnetic moments of the near-

est neighbor Fe atoms, and average magnetic moments per

Fe atom of the A-D configurations are also summarized in

Table I. It is noteworthy that the average magnetic moment

and tetragonal shear constant C0 are noticeably insensitive to

atomic configuration (except the D-type structure), although

as expected, the moment of the nearest neighbor Fe atom is

closely related to the number of nearest neighbor Si atoms.

Figure 7 shows the spin density contour of the A-type

structure of Fe0.889Si0.111 plotted on the (110) plane. Solid

lines denote positive spin polarization while the dotted lines

denote negative spin polarization. The negative spin polar-

ization of the Si atom is clearly seen, and the magnetic

moments of Si were calculated to be �0.06 to �0.09 lB. The

spin density distribution of Fe far away from the Si atom is

spherical, like that of bcc Fe, whereas an anisotropic distri-

bution toward a Si atom is noted for the first and second

nearest neighbor Fe atoms. Charge transfer between the Si

and its neighbor Fe atoms occurs in addition to charge rear-

rangement within the Fe atoms. As a result, the moment of

the Fe atom increases from 1.68 to 1.98 to 2.32 lB as the dis-

tance from the Si atom increases.

E. Origin of zero magnetostriction

In order to determine the physical origin of the nearly

zero magnetostriction of Fe0.889Si0.111, the individual atomic

contribution to the EMCA was calculated. There are ten dif-

ferent Fe atom types in the A-type structure, as seen in Fig.

8(a). The separate contributions of the ten different Fe atoms

to EMCA are plotted as functions of the tetragonal distortion

in Fig. 8(b). Interestingly, five atom types have positive

slopes while the other five types are negative. From the

slopes, it can be concluded that none of the Fe atoms played

dominant roles in determining k001 in contrast to most of the

other Fe alloy systems studied previously11,12 where it was

found that the nearest Fe atoms to nonmagnetic elements

contributed more than 80%. An Fe atom having Si as the first

nearest neighbor [denoted by Fe(2), Fe(4), and Fe(8)] nega-

tively contributes to the magnetostriction, whereas Fe atoms

in bcc-like environments [denoted by Fe(1), Fe(3), Fe(6),

Fe(7), and Fe(10)] that lack an Si atom as a first nearest

neighbor contribute positively to magnetostriction, even

though the Fe(5) and Fe(9) atom have negative contributions.

The minimal magnetostriction of Fe0.889Si0.111 originates

from the balance between these opposing contributions.

F. Density of states

In order to provide a physical background for the behav-

ior of Fe1-xSix magnetostrictions and to elucidate the subtle

FIG. 6. Strain dependence of (a) total energy Etot and (b) magneto-

crystalline anisotropy energy EMCA of A-, B-, C-, and D-type structures of

Fe0.889Si0.111. The total energy of the A-type structure at c/c0¼ 1 is set to

zero for reference.

TABLE I. The calculated total energy difference Ediff (meV/Fe), magnetic

moment of the Fe atoms that are nearest neighbors to the Si atoms, and the

average magnetic moment per atom MFe/Mave (lB), tetragonal shear con-

stant C0 (GPa), dEMCA/d(c/c0) (meV/Fe), and magnetostriction coefficient

k001 (10-6) for different atomic configurations of Fe0.889Si0.111. The total

energy of the A-type structure is set to zero for reference.

Configurations Ediff MFe Mave C0 dEMCA/d(c/c0) k001(k100)

A-type 0 1.71 1.91 56 0.83 0.8

B-type 1.5 1.72 1.91 54 �2.95 �3.3

C-type 9.1 1.59 1.91 53 5.08 6.1 (7.2)

D-type 37.5 1.94 1.98 45 3.82 5.2

FIG. 7. Spin density contour plots on the (110) plane of Fe0.889Si0.111. The

solid and dotted lines represent positive and negative spin densities, respec-

tively. The units are e/(a.u.).3

063911-5 Odkhuu, Seok Yun, and Cheol Hong J. Appl. Phys. 111, 063911 (2012)

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difference between the behaviors of Fe1-xSix and Fe1-xGex,

we plotted the spin-resolved density of states (DOS) of the

nearest neighbor Fe atom at x¼ 0.0625, 0.111, 0.1875, and

0.25 as shown in Fig. 9. The dotted and solid lines represent

the t2g and eg states, respectively. Their electronic structures

were found to be considerably influenced by the number of

first nearest neighbor Si atoms. The general trend for the

change in the DOS of Fe1-xSix with Si concentration is quite

similar to that of Fe1-xGex [cf. Fig. 4 of Ref. 12], which

results in similar magnetostriction behaviors. One major dif-

ference was noted, however. At x¼ 0.25, the majority eg

peak, which is positioned just at the Fermi level for

Fe0.75Ge0.25 and generates a large negative magnetostriction

through the contribution of <z2jLxjxz,yz>,12 shifts by about

0.3 eV above the Fermi level. Hence, the absolute value of

the magnetostriction of Fe0.75Si0.25 was significantly reduced

compared to that of Fe0.75Ge0.25.

To understand the correlation between the electronic

structures and the magnetostrictions, the SOC contributions

were calculated through different spin channels of spin up-

up (UU), up-down (UD), and down-down (DD) to EMCA for

Fe1-xSix. The changes in EMCA under tetragonal distortion,

DEMCA¼EMCA(c/c0¼ 1.02)-EMCA(c/c0¼ 0.98), were calcu-

lated. The calculated DEMCA’s were 5.8 (-12.6), -15.1 (-1.9),

and 21.3 (-2.1) leV/Fe for the UU, UD, and DD channels at

x¼ 0 (x¼ 0.25), respectively. The DD contribution of bcc

Fe is dominant, but the UU contribution becomes significant

at x¼ 0.25.45 Therefore, the majority eg state [see Fig. 9(d)]

that develops above the Fermi level must be responsible for

the significant UU contribution. Interestingly, for

Fe0.889Si0.111, the UD and DD contributions play an equiva-

lent role, whereas the UU channel is negligible due to a com-

pletely filled spin up band.

IV. SUMMARY AND CONCLUSIONS

The tetragonal magnetostriction coefficients, k001,were

calculated for Fe1-xSix alloys by adopting some different

plausible atomic structures for different Si concentrations

(x � 0.25) and using the first-principles FLAPW method.

The present study provides a comprehensive atomic-scale

understanding of the negligible magnetostriction of silicon

steel used in electric transforms or motors, which will be in-

structive in the further development of electric core steels.

Fundamental physical properties, such as equilibrium

lattice constants, magnetic moments, and elastic properties

were calculated and were in agreement with experimental

values. The calculated k001 of Fe1-xSix initially increased

with respect to x starting from aboutþ 20 ppm of pure bulk

Fe, but decreased rapidly after reaching a maximum at

approximatelyþ 40 ppm near x¼ 0.06 and became negligi-

ble near x¼ 0.12. Eventually, D03 Fe0.75Si0.25 was calculated

to have a negative magnetostriction of -18 ppm. This Si

concentration-dependent variation in magnetostriction was

consistent with experimental observations.

Based on total energy calculations, it was concluded that

the mixed phase of A2, B2, and D03 for the 11-13 at. % Si

alloy was energetically stable. Calculations of individual

atomic contributions to the magnetostriction indicated that

the negligible magnetostriction of the 11-13 at. % Si alloy

originated from opposing contributions made by the different

atomic types of the mixed phase of the alloy.

FIG. 8. (a) Different atomic types of Fe in Fe0.889Si0.111 and (b) magneto-

crystalline anisotropy energy (EMCA) of each Fe atomic type as a function of

tetragonal distortion c/c0.

FIG. 9. Atom- and spin-projected density of states of the nearest neighbor

Fe atom to the Si atom in Fe1-xSix alloys with (a) x¼ 0.0625, (b) x¼ 0.111,

(c) x¼ 0.1875, and (d) x¼ 0.25. The negative values represent the spin

down states and the Fermi level is set to zero.

063911-6 Odkhuu, Seok Yun, and Cheol Hong J. Appl. Phys. 111, 063911 (2012)

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ACKNOWLEDGMENTS

This work was supported by the Priority Research Cen-

ters Program (2009-0093818) and the Basic Science

Research Program (2009-0088216) through the National

Research Foundation funded by the Ministry of Education,

Science and Technology of Korea.

1S. V. Kulkarni and S. A. Khaparde, Transformer Engineering, Design andPractice (Marcel-Dekker, Inc., New York, 2004).

2G. Bertotti, Magnetic Alloys for Technical Applications, Soft MagneticAlloys, Invar and Elinvar Alloys (Springer-Verlag, Berlin, 1994).

3W. J. Carr and R. Smoluchowski, Phys. Rev. 83, 1236 (1951).4Q. Xing, D. Wu, and T. A. Lograsso, J. Appl. Phys. 107, 09A911

(2010).5R. C. Hall, J. Appl. Phys. 31, 1037 (1960).6A. J. Moses, J. Magn. Magn. Mater. 112, 150 (1992).7Y. Takada, J. Appl. Phys. 64, 5367 (1988).8R. K. Roy, A. K. Panda, M. Ghosh, A. Mitra, and R. N. Ghosh, J. Magn.

Magn. Mater. 321, 2865 (2009).9T. P. P. Phway and A. J. Moses, J. Magn. Magn. Mater. 320, e611 (2008).

10R. Q. Wu, J. Appl. Phys. 91, 7358 (2002).11S. C. Hong, W. S. Yun, and R. Q. Wu, Phys. Rev. B 79, 054419 (2009).12J. X. Cao, Y. N. Zhang, W. J. Ouyang, and R. Q. Wu, Phys. Rev. B 80,

104414 (2009).13Y. N. Zhang, J. X. Cao, and R. Q. Wu, Appl. Phys. Lett. 96, 062508

(2010).14Q. Xing and T. A. Lograsso, Appl. Phys. Lett. 93, 182501 (2008).15Q. Xing, Y. Du, R. J. McQueeney, and T. A. Lograsso, Acta Mater. 56,

4536 (2008).16M. Huang and T. A. Lograsso, Appl. Phys. Lett. 95, 171907 (2009).17Q. Xing, T. A. Lograsso, M. P. Ruffoni, C. Azimonte, S. Pascarelli, and D.

J. Miller, Appl. Phys. Lett. 97, 072508 (2010).18D. Wu, Q. Xing, R. W. McCallum, and T. A. Lograsso, J. Appl. Phys. 103,

07B307 (2008).19S. Thuanboon, G. Garside, and S. Guruswamya, J. Appl. Phys. 104,

013912 (2008).20M. Huang, A. O. Mandru, G. Petculescu, A. E. Clark, M. Wun-Fogle, and

T. A. Lograsso, J. Appl. Phys. 107, 09A920 (2010).21A. G. Khachaturyan and D. Viehland, Metall. Mater. Trans. A 38, 2308

(2007).

22S. Bhattacharyya, J. R. Jinschek, A. Khachaturyan, H. Cao, J. F. Li, and D.

Viehland, Phys. Rev. B 77, 104107 (2008).23H. Cao, P. M. Gehring, C. P. Devreugd, J. A. Rodriguez-Rivera, J. Li, and

D. Viehland, Phys. Rev. Lett. 102, 127201 (2009).24D. Odkhuu and S. C. Hong, J. Appl. Phys. 107, 09A945 (2010).25D. Odkhuu, W. S. Yun, S. H. Rhim, and S. C. Hong, Appl. Phys. Lett. 98,

152502 (2011).26Y. N. Zhang and R. Q. Wu, Phys. Rev. B 82, 224415 (2010).27P. Villars and L. D. Calvert, Pearson’s Handbook of Crystallographic

Data for International Phase (American Society for Metals, Materials

Park, OH, 1985).28E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Phys. Rev. B

24, 864 (1981).29J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865

(1996).30R. Q. Wu, L. J. Chen, A. Shick, and A. J. Freeman, J. Magn. Magn. Mater.

177–181, 1216 (1998).31D. S. Wang, R. Q. Wu, and A. J. Freeman, Phys. Rev. B 47, 14932

(1993).32N. I. Kulikov, D. Fristot, J. Hugel, and A. V. Postnikov, Phys. Rev. B 66,

014206 (2002).33J. Kudrnovsky, N. E. Christensen, and K. Andersen, Phys. Rev. B 43,

5924 (1990).34E. G. Moroni, W. Wolf, J. Hafner, and R. Podloucky, Phys. Rev. B 59,

12860 (1999).35T. Khmelevska, S. Khmelevskyi, A. V. Ruban, and P. Mohn, J. Phys.:

Condens. Matter. 18, 6677 (2006).36A. K. Arzhnikov, L. V. Dobysheva, G. N. Konygin, E. V. Voronina, and

E. P. Yelsukov, Phys. Rev. B 65, 024419 (2001).37I. A. Kuznetsov, G. A. Dorogina, E. S. Gorkunov, N. P. Antenorova, S.

M. Zadvorkin, and A. A. Pankratov, Phys. Met. Metallogr. 101, 247

(2006).38M. Fallot, Ann. Phys. 6, 305 (1936).39L. K. Varga, F. Mazaleyrat, J. Kovac, and J. M. Greneche, J. Phys.: Con-

dens. Matter 14, 1985 (2002).40W. A. Hines, A. H. Menotti, J. I. Budnick, T. J. Burch, T. Litrenta, V.

Niculescu, and K. Raj, Phys. Rev. B 13, 4060 (1976).41A. Machova and A. Kadeckova, Czech. J. Phys., Sect. B 27, 555 (1977).42H. L. Alberts and P. T. Wedepohl, Physica 53, 571 (1971).43J. B. Rausch and F. X. Kayser, J. Appl. Phys. 48, 487 (1977).44Y. N. Zhang, J. X. Cao, I. Barsukov, J. Lindner, B. Krumme, H. Wende,

and R. Q. Wu, Phys. Rev. B 81, 144418 (2010).45D. Odkhuu, and S. C. Hong, IEEE Trans. on Mag. 47, 2920 (2011).

063911-7 Odkhuu, Seok Yun, and Cheol Hong J. Appl. Phys. 111, 063911 (2012)

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