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Journal of Magnetism and Magnetic Materials 309 (2007) 244–250

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Thermal phonons affecting the long-time evolution of Ni–Mn–Gamartensite under magnetic field

N. Glavatskaa, V.A. L’vovb,�, I. Glavatskyya

aInstitute for Metal Physics NASU, Vernadskiy blvd. 36, Kiev-142, 03680, UkrainebRadiophysics Department, Taras Shevchenko University, Glushkov str. 2, Kiev, 03022, Ukraine

Received 21 April 2006; received in revised form 26 July 2006

Available online 28 July 2006

Abstract

The experimental data illustrating a slow evolution of the microstructure, deformation and magnetization of Ni–Mn–Ga martensite

under the stationary magnetic field have been analyzed. The thermal phonons that effectively transform the twin structure of martensite

have been considered. A strong interdependence between the twin width and the effectiveness of thermal phonons transforming the twin

structure has been predicted theoretically. The temperature dependencies of the square average values of the thermally induced random

stress and strain have been computed for the different values of twin width. The twin width providing the availability of time-dependent

effects in the Ni–Mn–Ga martensite has been estimated by comparison of the effective random stress with the stress, which is sufficient

for the transformation of twin structure. The estimated twin width proved to be of the order of 0.1 mm. It was concluded, therefore, that

the experimentally observed time evolution of the elastic twins of 10mm width is a secondary effect accompanying a thermally induced

transformation of the submicron elements of martensite microstructure.

r 2006 Elsevier B.V. All rights reserved.

PACS: 75.50.Cc; 75.60.Lr; 81.30.Kf

Keywords: Ferromagnetic martensite; Microstructure; Time evolution; Thermal phonons

1. Introduction

A giant magnetically or mechanically induced deforma-tion of the Ni–Mn–Ga martensites affected by thecomparatively weak magnetic field or mechanical stresshas been observed not long ago [1–5]. It was shown that thedeformation process is caused by the twinning/detwinningof the single crystalline experimental specimen [2,3].A description of the magneto-mechanical effects observedin Ni–Mn–Ga alloys and a detailed review of the relevantexperimental and theoretical researches are presented inRefs. [6,7]. Since these alloys are expected to exhibit a greatpotential as the magneto-mechanical actuators and sen-sors, their deformation under the magnetic field isintensively studied now in both static and dynamic regimes(see e.g. Refs. [8–12] and references therein).

- see front matter r 2006 Elsevier B.V. All rights reserved.

/j.jmmm.2006.07.007

onding author. Tel.: +380 44 4520134.

ddress: victorlvov@univ.kiev.ua (V.A. L’vov).

According to the modern experimental and theoreticalstudies, the magneto-mechanical response of the Ni–Mn–Ga martensite to the external magnetic field applicationcan be conventionally subdivided into the quick and slowparts, the latter is referred to as the time-dependentmagnetostrain effect (TDMSE). As far as the TDMSEcan exhibit itself during the operation of different magneto-mechanical devices, its thorough investigation is carriedout now with the increasing pace [8–10,12–15].A cooperative (experimental and theoretical) study of

TDMSE disclosed a decisive role of the fluctuatingmagneto-mechanical stress in the time evolution ofmagnetically induced deformation [13,14]. It was shown,that an effectiveness of the fluctuating stress can becharacterized by the mean-square value of its fluctuationsx0, which, in turn, can be formally introduced as aparameter of statistical distribution of the random stressvalues [14]. It was clearly emphasized, however, that theobtained results have a preliminary character in view of the

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Fig. 1. Optical microscopy images illustrating the slow evolution of the

morphology of 5M-martensite in the Ni49.5Mn28.5Ga22 single crystal,

which was exposed to the stationary magnetic field m0H ¼ 0.3T during the

time period of 0.37 h (a) and 17.37 h (b). Magnetic field was applied

perpendicularly to the imaged surface and easy magnetization axis.

N. Glavatska et al. / Journal of Magnetism and Magnetic Materials 309 (2007) 244–250 245

phenomenological approach to the problem solution, thearbitrariness of x0 value and the uncertainty in the values ofsome other physical parameters.

The main goal of the present paper is a study of the timeevolution of martensitic structure at the microscopic level.To this end the experimental micrographs illustrating thetime evolution of twinned Ni–Mn–Ga alloy will bepresented and the concomitant physical effects will beanalyzed. An effect of the thermal phonons on the periodictwin structure will be studied theoretically in order toestimate the mean-square value x0 of those stress fluctua-tions, which effectively transform the microstructure ofmartensite. The considered problem is rather difficult andintriguing due to the multilevel character of Ni–Mn–Gamartensitic structure: the crystal lattice of these alloyscombines a short-period (five-layered) periodic modulationwith two or even more levels of quasiperiodic twinning inthe submicron and micron scales (for more information seee.g. Refs. [16–19]).

2. Experimental

The single crystals of two alloys with close to stoichio-metric chemical compositions Ni49.8Mn28.5Ga21.7 (alloy 1)and Ni49.5Mn28.5Ga22 (alloy 2) were produced by theBridgman method, homogenized at 1273K during 72hand ordered at 1070K during 48 h. The specimens with thedimensions 5� 5� 5mm were cut parallel to {1 0 0} planesfor both alloys and, in addition, a specimen of the alloy 2with the dimensions 6� 6� 8mm was cut parallel to {1 1 0}plane. A modulated 5-layered martensitic structure isdetermined in the studied crystals using the neutrondiffraction [20]. Ignoring the weak modulation the unit cellof martensitic phase can be approximated by the tetragonalcell with the lattice parameters a ¼ b ¼ 0.595 nm (measuredfor alloy 1 at T ¼ 293K) and orthorhombic cell witha ¼ 590nm, b ¼ 0.583 nm and c ¼ 0.558nm (measured foralloy 2 at T ¼ 290K). The Curie temperature TC and theforward martensitic transformation temperature TM areTC ¼ 368K, TM ¼ 299K (for alloy 1), TC ¼ 373K, andTM ¼ 303K (for alloy 2).

The optical microscope CARL ZEISS was used to studythe time evolution of twins under the stationary magneticfield.

The time-dependent strain was measured in perpendi-cular to the applied magnetic field direction using theoriginal high-sensitive magnetic dilatometer [21] with thedilatation resolution of about 0.1 mm, which results inthe precision of the strain measurement of about of 0.002%(for the 5mm specimen length). This automated dilat-ometer controlled the specimen length, magnetic fieldand temperature variation with the time resolution of0.025 s.

Time-dependent magnetization was measured at roomtemperature by vibration AeroSonic VSM 3001 magnet-ometer. The single crystalline specimen was preliminarilymagnetized in the field m0H ¼ 0.9 T applied in /0 0 1S

crystallographic direction, to bring the specimen to thequasi-single-variant state. The magnetic field m0H ¼ 0.3 Twas applied then in the perpendicular direction and thetime-dependent magnetization was measured.All measurements related to the time-dependent beha-

vior of martensite were performed under the stationarymagnetic field.

3. Slow evolution of the twin structure and the concomitant

physical effects

For the study of slow structural evolution of theNi–Mn–Ga martensite a monocrystalline specimen wasbrought to the single-variant state and then affected by thestationary magnetic field m0H ¼ 0.3 T, which is substan-tially smaller than the saturating field for magnetostrain.The time evolution of two martensitic domains, which arevisible at the surface of the specimen, is shown in Fig. 1.The domains are separated by the straight boundary. Thedomain situated to the right of the twin boundary isoccupied by the alternating twins with the width d0�10 mm.

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(a)

(b)

Fig. 2. Time-dependent deformation of the Ni49.8Mn28.5Ga21.7 single

crystal under the fixed magnetic field m0H ¼ 0.3 T at constant temperature

296K: (a) elongation of the specimen measured under the stationary

magnetic field and after its switching off; (b) fine details of the time

dependence of deformation in the stationary magnetic field.

N. Glavatska et al. / Journal of Magnetism and Magnetic Materials 309 (2007) 244–250246

It may be assumed that the twinning is originated by themagnetic field, because the field is applied perpendicularlyto the ‘‘easy magnetization axis’’ of the single-variant state,and therefore, the appearance of the second martensitevariant, whose easy axis is aligned with the field, isenergetically favorable. Due to the influence of thestationary magnetic field the domain occupying the rightarea of the Fig. 1(a) slowly spreads through the experi-mental specimen in the direction marked by the arrow. Thespread is accompanied by the growth of fine twins alongtheir boundaries and the appropriate increase of thevolume fraction of the favorable martensite variant.

It will be argued below, that the slow twinning/detwinning of martensite in the micrometer scale is causedby the interaction of thermal phonons with the fineelements of martensitic structure, which are not visible inFig. 1 but are quite typical for Ni–Mn–Ga martensite[18,19,22]. A motive for this argumentation follows fromFig. 1: the twins, that are visible in this figure, mayhypothetically be considered as internally twinned in thesubmicron range; in this case a direction of spread of thetwinned martensite domain (marked by arrow) may beidentified with the perpendicular dropped to the plane ofinternal twin boundaries.

The assumption resulting from the micrographs issupported by the analysis of physical effects, whichaccompany the time evolution of martensite microstruc-ture. Fig. 2(a) illustrates a time-dependent deformation ofthe Ni–Mn–Ga single crystal in the stationary magneticfield, which was applied to the experimental specimen att ¼ 0 and switched off after 2min. A deformation of thespecimen (characterized by its contraction DL) wasmeasured every 0.1 s. The deformation curve has a stepwisecharacter. The smallest step is shown in the inset. This stepis caused by the abrupt change of the twin structure in thespatial region with the width l0, which can be estimatedfrom the height of the step DLmin�0.5 mm and tetragonallattice distortion of the crystal lattice 1�c/a ¼ 0.06. Theresult of estimation is l0�DLmin/(1�c/a)E10 mm.

As it follows from the inset in Fig. 2(a), a random noiseproduces the changes in the specimen dimensionsDLrand�0.1 mm. The evaluation presented above showsthat these changes correspond to the changes of twinstructure in the spatial regions with the dimensions of theorder of l0�2 mm. This estimation is even more obviousfrom the Fig. 2(b), which also shows that the minimalchange of the specimen length is close to 0.1 mm. It can beconcluded, hence, that the time-dependent deformation iscaused by the variation of structural elements whosedimensions do not exceed 2 mm, at least.

It may be of practical importance that the time evolutionof the martensitic structure is accompanied not only by thedeformation of the specimen but also by the slow variationof the magnetization value (see Fig. 3). As well as the time-dependent deformation, the magnetization value exhibitsboth the smooth and jump-like changes. The timedependencies of the magnetic field and temperature are

presented in Fig. 3(a), to prove that the observed variationof magnetization value 30 soto120 s cannot be causedneither by the temperature instability nor by the variationof the applied field.The variation of magnetization M continues during the

long time, and hence, it follows the slow evolution of themartensite. Abrupt change of the magnetization valueobserved at tE85 s indicates the presence of the effectivecenter of pinning of twinning dislocations in the some pointof the studied specimen. This center hampers the evolutionof the twin structure during the time interval from 30 to85 s and causes the internal stressing of martensite. Thejump of magnetization occurs in the moment when thetwinning dislocation overcomes the pinning force andthe stress relaxes. A careful analysis of the small fragmentsof the magnetization graph shows that the random valuesDM/M are of about of 10�4.Thus, the available experimental data suggest that the

slow evolution of martensite structure is originated inthe micron or submicron scale and then manifests itself asthe twinning/detwinning of the specimen in the 10-mmrange. This suggestion can be substantiated theoretically.

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(a)

(b)

Fig. 3. The time dependence of magnetization of the single crystal

Ni49.8Mn28.5Ga21.7 under the stationary magnetic field: (a) the magnetiza-

tion values (circles) obtained under the field 0.24T applied in the moment

t ¼ 3 s (curve) at the fixed temperature 297K; (b) the final fragment of the

time dependence of magnetization value illustrating its thermal jumps.

x

c

x-variant y-variant

y a

c

a

d0

x-variant . . . etc.

λ

Fig. 4. The twinned tetragonal martensite and the effective phonon with

the minimal wave length (schematically).

N. Glavatska et al. / Journal of Magnetism and Magnetic Materials 309 (2007) 244–250 247

4. Thermal phonons affecting the martensite microstructure

The fluctuating elastic stress consists of the regular andrandom parts. The regular part is caused by the mechanicalloading or applied magnetic field, while the random part isinduced by the thermal phonons, which affect the micro-structure of martensite. As a first approximation, astandard expression for the density of elastic energy ofthe cubic crystal

U ¼1

2C11 �2xx þ �

2yy þ �

2zz

� �þ C12 �xx�yy þ �xx�zz þ �yy�zz

� �þ 2C44 �2xy þ �

2xz þ �

2yz

� �ð1Þ

will be used below for the evaluation of the random part offluctuating stress, i.e. a small tetragonal or orthorhombicdistortion of the crystal lattice will be disregarded. More-over, the velocities s?1 ¼ (C44/r)

1/2 and s?2 ¼ [(C11–C12)/2r]1/2 of the transversal sound waves running in [1 0 0] and[1 1 0] directions will be replaced with the ‘‘average’’velocity of the transversal sound st ¼ ðC=rÞ

1=2, where C

will be reckoned equal to the soft elastic modulus measuredin Ref. [23]. Strictly speaking, the latter simplification is not

justified for the Ni–Mn–Ga alloy [24] but it may be usedfor the rough estimation of random stress. The sound wavepropagating in the crystal is a periodic displacement ofatoms

~uðpÞðk; tÞ ¼ uðpÞðkÞ cosðkr� otÞ (2)

resulting in the variable elastic strain, which is described bythe tensor

~�ðpÞij ðk; tÞ ¼ �1

2kiuðpÞj ðkÞ þ kju

ðpÞi ðkÞ

h isin ðkr� otÞ, (3)

where the superscript p ¼ ||,? marks the longitudinal ortransversal polarization of the wave. While the temperatureof the experimental specimen exceeds the characteristicDebye temperature, the energy E of the every thermalphonon is approximately equal to kBT. On the other hand,a substitution of the strain Eq. (3) into the Eq. (1) with asubsequent averaging in time and integration in coordi-nates results in the expression E ¼ VC(p)k2|u(p)|2/4, whereC(p)¼ C11 for the longitudinal wave and CðpÞ ¼ C for the

transversal one. Thus, the amplitude of the thermallyinduced elastic wave is approximately expressed as

juðpÞj ¼ BpðTÞ=k, (4)

where Bp(T) ¼ 2[kBT/VC(p)]1/2.The microstructure of Ni–Mn–Ga martensite is tradi-

tionally modeled by the alternating domains (variants)of tetragonal lattice [2], which are in the twin relation (seeFig. 4). The coordinates x and y are reckoned parallel tothe principal axes of the neighboring twin componentsbecause c/a ratio is close to unit.The deformation of crystal is caused mainly by its

twinning/detwinning under the action of external magneticfield or/and axial mechanical loading. In this case, thedeformation process is promoted by those thermalphonons which create the effective elastic strain

~zðpÞðk; tÞ � ~�ðpÞxxðk; tÞ � ~�

ðpÞyy ðk; tÞ (5)

because this strain breaks the equivalence of x- and y- twincomponents. With the use of the Eqs. (3) and (4) the

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(a)

(b)

(c)

Fig. 5. Experimental temperature dependence of the soft elastic modulus

of Ni51.5Mn23.6Ga24.9 martensite [23] (a); the fluctuations of strain (b) and

stress (c) computed using this dependence.

N. Glavatska et al. / Journal of Magnetism and Magnetic Materials 309 (2007) 244–250248

effective strain can be expressed as

~zðpÞðk; tÞ ¼ BpðTÞðnyeðpÞy � nxeðpÞx Þ sin ðkr� otÞ, (6)

where n and e(p) are the unit vectors aligned with k and u(p),respectively.

The total mean-square value of the elastic strain, whichis induced by all thermal phonons is

z20 ¼Xk;k0 ;p

~zðpÞðk; tÞ~z

ðpÞðk0; tÞ

* +t

. (7)

The time averaging results in the disappearance of theterms with k6¼k0 from the sum in the Eq. (7). Moreover,the short-wave phonons cannot effectively participate inthe twinning/detwinning process, because the local strainscreated inside the twin by the short-wave phonons differin sign, and hence, compensate each other. To promotethe expansion of one of the twin components andcontraction of another one the strain must be of fixed signall over the twin. In this case the effect of the thermalphonons on the twin is essentially similar to the effect ofthe axial mechanical loading. Thus, the lengths of thewave vectors in the Eq. (7) are limited by the maximalvalue k0 ¼ 2p/l0, where the characteristic (minimal) wave-length l0 is of the order of double twin width: l0�2d0(see Fig. 4).

Substituting the effective stress for its expression Eq. (6),performing the time averaging, and replacing the summa-tion in k by integration, one can reduce Eq. (7) to the form

z20 ¼8p9

1

C11þ

I

C

� �kBT

l30, (8)

where

I ¼3

16p2

ZZ4p

ðnxe?x � nye?y Þ2 don doe,

on and oe are the solid angles in the space of the vectors nand e?, respectively. The integration accomplished withaccount of perpendicularity of these vectors results in thevalue I ¼ 3/10.

According to the Hooke’s law the thermal phononinduces the effective internal stress

~xðpÞðk; tÞ ¼ 2C ~z

ðpÞðk; tÞ, (9)

and therefore, the mean-square value of the stressfluctuations can be expressed as

x20 ¼32p9

C I þC

C11

� �kBT

l30. (10)

The Eqs. (8) and (10) show that the fluctuations of strainand stress depend on the twin width d0�l0/2 and thetemperature of the specimen. It is of importance, that,firstly, the smaller is twin width, the larger are fluctuations,and secondly, the temperature dependence of fluctuationsis caused not only by the factor kBT, but by thetemperature dependence of the elastic modules, as well.

These deductions can be strictly confirmed by computa-tions, when the temperature dependencies of the elasticmodules are known.

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�(S)

�(f )

�(f )�(s)

�(s)

�(f )

6.00

5.95

5.90

Str

ain

(%)

2 3 4Stress (MPa)

(a)

(b)

Fig. 6. An evaluation of the characteristic stresses and strains from the

experimental stress–strain dependence measured for the single-variant

Ni–Mn–Ga martensite: (a) experimental stress–strain curve [25] and its

final segment (inset); (b) elastic strain determined from the final segment of

the stress–strain curve.

N. Glavatska et al. / Journal of Magnetism and Magnetic Materials 309 (2007) 244–250 249

The experimental values of the soft modulus measuredin Ref. [23] for the Ni51.5Mn23.6Ga24.9 alloy are shown inFig. 5(a) . The figure demonstrates, that the soft modulusCðTÞ is small in comparison with C11E200Gpa, andtherefore, the fluctuations may be evaluated from theEqs. (8) and (10) with C�111 ¼ 0. The functions z0(T) andx0(T) computed for three different values of limitingwavelength l0 ¼ 20,70,200 nm, are shown in Figs. 5(b)and (c), respectively.

The softening of elastic modulus (Fig. 5(a)) causes asubstantial intensification of fluctuations of the elasticstrain (Fig. 5(b)) and attenuation of the random stress(Fig. 5(b)) in the vicinity of MT temperature TME275K.This effect illustrates the instability of the crystal lattice inthe temperature range of MT.

5. Discussion

The thermal phonon effectively promotes the reorienta-tion of twinned martensite only if the phonon wave-lengthis larger than twin width. Therefore, the number of‘‘effective’’ phonons induced by these phonons randomstress, which transforms the twin structure, drasticallydepends on the twin width. It is worthwhile to compare theeffective random stress with the critical value of axialmechanical stress needed for the reorientation (twinning/detwinning cycle) of the single variant Ni–Mn–Ga mar-tensite exhibiting a giant magnetically induced deforma-tion. To this end the experimental stress–strain dependencereported in Ref. [25] can be used. A processing of thisdependence indicates, that the reorientation process startswhen s ¼ s(s)E0.9MPa, e ¼ e(s)E4.5� 10�4, and finisheswhen s ¼ s(f)E1.75MPa, e ¼ e(f)E9.2� 10�4 (see Fig. 6).

A comparison of the obtained critical values with theaverage amplitudes of the random strain and stressdemonstrates that:

(i)

The thermal phonons, which are able to transform amesoscale (super-fine) twin structure with the twinwidth d0�10 nm, induce the random stress whoseaverage amplitude is comparable in value with thecritical stress needed for the complete reorientation ofmartensite or even exceeding it. However, thisamplitude is considerably smaller than the stressneeded for the intermartensitic transformation of theNi–Mn–Ga alloys [16]. The manifestations of thismicrostress in martensites are not studied till now, itseems.

(ii)

The thermal phonons, which are able to transform afine twin structure with d0�100 nm, induce the randomstress with the average amplitude x0�0.5MPa. Thisamplitude is comparable in value with the ‘‘switch on’’stress s(s), and therefore, is sufficient [14,22] for theslow deformation of the specimen in the staticmagnetic field or under the stationary mechanicalloading. This random stress causes the deformatione(t0)�e(0)�1% during the time t0�1min [14,22].

(iii)

The random stress, which affects the large-scale(optically observable) twin structure with d0�1mkm,is of the order of 0.01MPa, and therefore, it cannotinduce the appreciable change of the structure and aconcomitant deformation of the specimen.

It should be emphasized, that the points (i–iii) involveonly the orders of magnitudes, and therefore, they areessentially applicable not only to the regular twinstructures and twinning dislocations, but also to the otherstructural elements of martensite (self-accommodatinggroups, spatial domains bounded by the stacking faults,etc), which are sensitive to the axial stressing or magneticfield application.Point (iii) supports an idea that the experimentally

observed (Fig. 1) slow transformation of the large-scalespatial domains of the specimen is a secondary effectaccompanying the primary transformation of martensite inthe submicron or mesa-scale range. The idea arises from

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the following feature of Fig. 1: the process of formation ofthe second martensite variant is not accompanied bythe noticeable change of the distance d0�10 mm betweenthe twin boundaries. Actually, this process looks likethe elongation of the boundaries of the large-scale twins.The direction of the spread of the twinned domain ofmartensite is normal to the boundaries of the submicrontwins, which hypothetically exist in the specimen andare quite usual for the Ni–Mn–Ga martensites, inparticular, for those exhibiting giant magnetostrain (seee.g. Refs. [18,19]).

In conclusion, it can be argued that the drastictemperature dependence of magnetostrain, which makesthis effect practically observable only in the vicinity of MTtemperature (for more details see Refs. [20,26]), is causedmainly by two tendencies taking place on approaching tothis temperature. The first tendency is a softening of thecrystal lattice and an appropriate increase of magnetostric-tion, which is observed for both austenitic [27] andmartensitic [28] phases. The second tendency is an increaseof the thermally induced strain (Fig. 5(b)), which issuperimposed on the magnetostrictive strain. It may beassumed, that the external magnetic field ‘‘switches on’’ thereorientation of martensite when the sum of the magnetos-trictive and random strains exceeds certain critical value,which is weakly dependent on the temperature. The resultsof experimental and theoretical verification of this assump-tion will be reported in our next publication.

Acknowledgments

This work is done in the frame of the STCU-EOARDpartner project P-137 with the funding support ofEuropean Office of Aerospace Research & Development(EOARD), USA. The authors would like to thankMr. Aleksiy Rudenko (Institute for Metal Physics, Kiev,Ukraine) and Mr. O. Heczko (Helsinki University ofTechnology, Finland) for participation in the experimentalstudies.

References

[1] K. Ullakko, J.K. Huang, C. Kantner, R.C. O’Handley, V.V.

Kokorin, J. Appl. Phys. 69 (1996) 1966.

[2] R.D. James, M. Wuttig, Phil. Mag. A 77 (1998) 1273.

[3] R.D. James, R. Tickle, M. Wuttig, Mat. Sci. Eng. A 273–275 (1999)

320.

[4] R.C. O’Handley, S.J. Murray, M. Marioni, H. Nembach, M.S. Allen,

J. Appl. Phys. 87 (2000) 4712.

[5] C. Jiang, T. Liang, H. Xu, M. Zhang, G. Wu, Appl. Phys. Lett. 81

(2002) 2818.

[6] A.N. Vasil’ev, V.D. Buchel’nikov, T. Takagi, et al., Physics-Uspekhi

46 (6) (2003) 559.

[7] J. Kiang, L. Tong, JMMM 292 (2005) 394.

[8] N.I. Glavatskaya, K. Ullakko, JMMM 218 (2000) 256.

[9] N. Glavatska, I. Glavatskiy, Y. Ge, V.K. Lindros, J. Phys IV 112

(2003) 1009.

[10] M. Marioni, D. Bono, A.B. Banful, M. del Rosario, E. Rodriguez,

B.W. Peterson, S.M. Allen, R.C. O’Handley, J. Phys. IV France 112

(2003) 1001.

[11] M.A. Marioni, S.M. Allen, R.C. O’Handley, J. Appl. Phys. Lett 84

(2004) 4071.

[12] N. Glavatska, J. Ferroelectrics 290–292 (2003) 93.

[13] N.I. Glavatska, A.A. Rudenko, V.A. L’vov, JMMM 241 (2002) 287.

[14] V. L’vov, A. Rudenko, N. Glavatska, Phys. Rev. B 71 (2005) 024421.

[15] S.I. Bosko, V.D. Buchelnikov, S.V. Taskaev, T. Takagi, A.N.

Vasiliev, Int. J. Appl. Electr. Mech. 21 (2005) 11.

[16] V.V. Kokorin, V.V. Martinov, V.A. Chernenko, Scripta Met. Mat.

26 (1992) 175.

[17] H.D. Chopra, C. Ji, V.V. Kokorin, Phys. Rev. B 61 (2000) R14913.

[18] H. Xu, Y. Ma, C. Jiang, Appl. Phys. Lett. 82 (2003) 3206.

[19] P. Mullner, V.A. Chernenko, G. Kostorz, JMMM 267 (2003) 325.

[20] N. Glavatska, I. Glavatskiy, G. Mogylny, S. Danilkin, D. Hohlwein,

O. Soderberg, V. Lindroos, A. Beskrovnij, J. Phys. IV 112 (2003) 963.

[21] V.T. Cherepin, N.I. Glavatska, I.N. Glavatskiy, V.G. Gavriljuk,

Measure Sci Technol 13, 174.

[22] V. A. L’vov, N. Glavatska, I. Glavatskiy , Y. Ge, O. Heczko, O.

Soderberg, S.-P. Hannula, Int. J. Appl. Electromagn. Mech. 23 (2006)

75.

[23] V.A. Chernenko, J. Pons, C. Seguı, E. Cesari, Acta Mater. 50 (2002)

53.

[24] L. Dai, J. Cullen, M. Wuttig, J. Appl. Phys. 95 (2004) 6957.

[25] V.A. Chernenko, V.A. L’vov, P. Mullner, G. Kostorz, T. Takagi,

Phys. Rev. B 69 (2004) 134410.

[26] O. Heczko, L. Straka, J. Appl. Phys. 94 (2003) 7139.

[27] N.A. Vasil’ev, V.V. Kokorin, Yu.L. Savchenko, V.A. Chernenko,

Sov. Phys. JETP 71 (1990) 803.

[28] O. Heczko, JMMM 290–291 (2005) 846.