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Page 1: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

Physica B 325 (2003) 98–105

Magnetic exchange coupling in amorphousFe82�xHoxB18 alloys

A. Kaala, O. El Marrakechia, S. Sayouria,*, M. Tlem-cania,b, H. Lassric, M. Kellatia

aLaboratoire de Physique Th!eorique et Appliqu!ee D!epartment de Physique, Facult!e des Sciences, B.P. 1796, F "es-Atlas, MoroccobLaboratoire de Physique de la Mati"ere Condens!ee et de l’Environnement (LPMCE), E.N.S. F"es-Bensouda, Morocco

cLaboratoire de Physique des Mat!eriaux et de Micro!electronique, Facult!e des Sciences, Universit!e Hassan II, B.P. 5366, A.ın Chock,

Route d’El Jadida, km-8, Casablanca, Morocco

Received 12 January 2002; received in revised form 21 June 2002

Abstract

Magnetization data of melt-spun amorphous Fe82�xHoxB18 (4pxp16) alloys are analyzed using a two-sublatticemean-field theory (MFT) and the exchange interactions are derived. High-field magnetization measurements performed

on samples with small net magnetizations give evidence of the occurrence of a magnetic transition, characteristic of a

non-collinear rare-earth and transition-metal sublattices structure. Besides, analysis of the non-collinear regime has

permitted the evaluation of the intersublattice exchange coupling parameters which are found to be in good agreement

with those deduced using a mean-field analysis.

r 2002 Elsevier Science B.V. All rights reserved.

Keywords: Amorphous alloys; Magnetization; Mean-field theory; Exchange coupling

1. Introduction

Amorphous alloys made of transition-metal(TM) and rare-earth (RE) elements have arouseda lot of interest because of their interestingphysical properties, which make them suitablefor new magnetic devices applicability. Thesematerials are also of fundamental interest as theypermit the observation of diversified magneticproperties over a wide and continuous concentra-tion range of the constituting elements.Magnetic properties of amorphous alloys are

strongly affected by the lack of structural order,

which causes bond and chemical disorder, result-ing in magnetic moment amplitudes and exchangeinteraction fluctuations. Furthermore, the ran-domly varying electrostatic fields induce, via spinorbit coupling, locally varying single site aniso-tropy, leading to a spreading of the magneticmoment directions, especially for rare-earth mo-ments, and thus destroying the long rangemagnetic order favored by the magnetic exchangecoupling. In amorphous TM–RE–Me, where Me isa metalloid, the magnetic order in the transition-metal sublattice is mainly due to the strongexchange coupling, JTT; between TM atoms whilethat between RE ones is likely to be establishedthrough the inter-sublattice atomic exchange-coupling, JRT; this latter also determines the

*Corresponding author.

E-mail address: [email protected] (S. Sayouri).

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 4 5 7 - 6

Page 2: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

stability of the ferrimagnetic structure, when RE isa heavy rare-earth ion. Usually, these parameterscan be determined by several ways: Curie tem-perature analysis [1], mean-field-theory (MFT)analysis [2–3], antiferromagnetic coupling-break-ing analysis [4], etc. There is now a quite greatamount of works devoted to this subject as well asto random magnetic anisotropy (RMA) effects onamorphous MT80�xRExB20 [5–7]. Recently, high-field magnetization studies on amorphous Fe80�x-

HoxB20 alloys have been performed by Radwanskiet al. [8]. In this paper, we have considered againthese magnetic measurements. We have used themean-field theory to evaluate the exchange inter-actions Fe–Fe, Fe–Ho, and Ho–Ho, and havediscussed the thermal behavior of the magnetictransition in the high field regime.

2. Experiment [9]

Amorphous Fe82�xHoxB18 ribbons with4pxp16 were prepared by the usual melt spinningtechnique in an argon atmosphere. X-ray diffrac-tion was used to check the amorphous state of thealloys. No Bragg peaks were observed during thecharacterization. The exact chemical compositionof the ribbons was determined by electron probemicroanalysis. The magnetization was measuredbetween 8 and 300K using a vibrating samplemagnetometer (VSM), in applied fields up to 1.8 T.The Curie temperature was determined using aVSM with an oven. High-field magnetic isothermsat 4.2K of amorphous Fe82�xHoxB18 ribbons withnominal compositions corresponding to x ¼ 8; 10,and 14 were measured in pulsed fields up to 38T inthe high-field installation of the University ofAmsterdam.The high-field magnetization measurements

were undertaken on finely powdered samples,which are free to orient their moments accordingto the applied field direction [10]. In our case,samples consisted of small pieces with lengthsbetween 1 and 5mm and with a width of 1mm.The individual pieces have a limited, but stillsignificant freedom to rotate within the sampleholder into their minimum-energy direction duringthe measurements. No significant hysteresis has

been observed in increasing and decreasing fields.Step-wise pulses, where the field was kept constantduring 65ms, were used in order to see the effect ofeddy currents. No difference between the twotypes of measurements was observed.

3. Results and discussion

3.1. Thermal behavior of the magnetization

3.1.1. Magnetic moments at low temperature

The concentration dependence of the magneticmoment of the a-Fe–Ho–B alloys at 7.5K and inan applied field of 1.8 T is shown in Fig. 1. Themagnetization decreases rapidly with increasingHo content, indicating the anti-parallel alignmentof the transition-metal and the rare-earth mo-ments. The alloy moment ma can be written [11] as

ma ¼ jð82� xÞmFe � x mHoj=100; ð1Þ

where mFe and mHo denote the magnetic momentsof Fe and Ho, respectively. Assuming a value of

0 4 8 12 16

0.4

0.8

1.2

1.6

Ho

FeHo

Fe

x (at. %)

0.5

1.0

1.5

2.0Fe82-xHoxB18

µ Fe (

µ B )

µ a (µ B

)

Fig. 1. Concentration dependence of the magnetic moment of

the a-Fe–Ho–B alloys at 7.5K.

A. Kaal et al. / Physica B 325 (2003) 98–105 99

Page 3: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

10mB for mHo; as found in Ref. [12], mFe is found tobe of the order of 2mB for x ¼ 4; which is close, tothe experimental accuracy, to 2.04mB found forsimilar alloys with low concentrations on rare-earth element. In what follows, mHo will be takenequal to 10mB, independent of concentration.Inserting this value in Eq. (1), one can deducethe iron magnetic moment for higher concentra-tions on Ho. It is found that the Fe magneticmoment decreases as the Ho concentration in-creases. Such a decrease cannot always be imputedto hybridization effects which mix the transition-metal 3d states with 6s2/5d states of the rare earth,but also to non-collinearity of transition-metalmoments themselves. This fact was recentlyevidenced by Ravach and Teillet [13] throughhigh-field M .ossbauer spectroscopy measurements,carried out on ribbons of amorphous Fe66Dy14B20and Fe62Ho16B18 alloys. These authors showedthat, in Fe66Dy14B20, Fe-moments under appliedfields up to 8T are distributed within a cone andthat the iron-canting angle is as important as in thedysprosium one; furthermore, the Fe-momentmodulus (at 14% in Dy) retains a value thatapproximately corresponds to that obtained forFe80B20, suggesting that hybridization effectsare too weaker than indicated by magneticmoment splitting alone. In Fe62Ho16B18 on theother hand, iron spin structure is nearlycollinear from 2T; in this alloy, Fe semi-angleapex /yS equals 291, then the Fe magneticmoment modulus is only about 1.03 times thatdeduced from moment splitting. This showsthat, at least for this composition in rare-earth,the iron spin structure is nearly collinear infields of about 2 T. In correlation with the valuetaken for the Ho moment, this would indicate thatour alloys are ferrimagnets and that the aniso-tropy/RE–TM exchange ratio is small in thesealloys.

3.1.2. Mean-field analysis

The experimental temperature dependence ofthe magnetization of amorphous Fe82�xHoxB18alloys at 1.8 T is shown in Fig. 2. In thetwo-sublattice mean-field model [3], the experi-mental magnetization curves can be fitted to a setof coupled Brillouin functions, each describing

the thermal behavior of one type of magneticmoment:

miðTÞ ¼ mið0 KÞBJiðxiÞ; xi ¼

giJi mB Hi

kBT; ð2Þ

where BJi; Ji and gi are respectively the Brillouinfunction, the effective total momentum of the ioni; and the Land!e factor of the ion i; mB is the Bohrmagneton, kB is the Boltzmann constant, and Hi isthe molecular field acting on the site i: Themolecular fields HFe and HHo are given by

HFe ¼2 JFe�FezFe�FeSFeðTÞ

gFemB

þ2 JFe�HozFe�HoðgHo � 1ÞJHoðTÞ

gFemB; ð3aÞ

HHo ¼2 JHo�FezHo�FeðgHo � 1ÞSFeðTÞ

gHomB

þ2 JHo�HozHo�HoðgHo � 1Þ

2JHoðTÞgHomB

; ð3bÞ

where JFe2Fe; JFe2Ho and JHo2Ho are the exchangeintegrals for Fe–Fe, Fe–Ho, and Ho–Ho interac-tions, respectively. zij (i; j ¼ Fe;Ho) is the numberof nearest neighbors of the atom j for the atom i:In the fitting procedure, the values found above forthe moments are considered as approximatelyequals to mi ð0 KÞ and are taken as input para-meters, and the exchange integrals JFe2Fe; JFe2Hoand JHo2Ho are considered as adjusted parameters.In the standard mean-field model, the coordina-tion numbers zij are calculated assuming eachatom surrounded by 12 others with equal prob-ability. This assumption is a rough estimate as itdoes not take into account the local structuralorder persisting in the amorphous state, asevidenced experimentally [14,15]. Recently, Ma-chizaud et al. [16] have studied the local order inthe amorphous Fe80�xRExB20 alloys and haveproposed a structural model, which succeeded totake into account the discontinuity found in thevariation of the hyperfine field when the rare earthconcentration is varied around a critical value, xc:In the following, the coordination numbers zij ; ascalculated in Ref. [16], will be used but with aslight modification in order to take into accountthe real composition of our alloys. The valuesfound for JFe2Fe; JFe2Ho; and JHo2Ho; derived

A. Kaal et al. / Physica B 325 (2003) 98–105100

Page 4: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

from the best fitting, as well as the Curietemperature and the magnetic moment mFe; foreach composition, are gathered in Table 1. A goodagreement between the calculated magnetizationcurves and the experimental ones has beenobtained. Krishnan et al. have performed high-field magnetization studies on amorphousFe72�xYxHo8B20 alloys [17]; in particular, they

have deduced for the a-Fe72Ho8B20 alloy (corre-sponding to x ¼ 0) the molecular field coefficientnHo2Fe; which value agrees well with that eval-uated from our study for Fe73.4Ho7.6B19 (Table 2),and the exchange integral JHo2Fe: However, thevalue of the latter parameter differs from the onededuced from our analysis. This discrepancycomes from the value of zHo2Fe ð¼ 16Þ we have

0 150 300 450 600 750

0.3

0.6

0.9

1.2

1.5x = 4

x = 7.6

x = 13.7x = 10

x = 16.2

0 150 300 450 600

0.3

0.6

0.9

1.2

1.5

0 150 300 450

0.3

0.6

0.9

1.2

1.5

0 150 300 450

0.3

0.6

0.9

1.2

1.5

0 150 300 450

0.3

0.6

0.9

1.2

1.5

1.8

Temperature (K)Temperature (K)

Ma

gn

eti

c m

om

en

t (µ

B)

Ma

gn

eti

c m

om

en

t (µ

B)

Ma

gn

eti

c m

om

en

t ( µ

B)

Ma

gn

eti

c m

om

en

t (µ

B)

Ma

gn

eti

c m

om

en

t ( µ

B)

Temperature (K)

Temperature (K)

Temperature (K)

Fig. 2. Temperature dependence of the magnetic moment: (Solid squares) experimental data, (solid lines) calculated alloy moment,

(dashed lines) calculated iron moment, (dotted lines) calculated rare-earth moment.

A. Kaal et al. / Physica B 325 (2003) 98–105 101

Page 5: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

adopted based on the work of Machizaud et al.mentioned above instead of that equal to 8.8(derived from the commonly used expressionzHo2Fe ¼ 12ð802xÞ=100Þ assumed by Krishnanet al. (Table 2).The exchange integral JRT ð¼ JFe2HoÞ increases

as the Ho content increases, following the predic-tions of Brooks et al. [18] that the 4f–5d exchangeinteraction should increase as the 4f–5d hybridiza-tion increases. It is also observed that Hosubstitution for Fe leads to a weakening ofJFe2Fe interaction, in accordance with the decreaseof the Curie temperature. The RE–TM exchangecoupling can also be studied using high-fieldmagnetic measurements, as discussed in the nextsection.

3.2. High-field magnetization behavior

3.2.1. Magnetic moments

Fig. 3 shows the high-field dependence of thealloy magnetic moment at 4.2K for three compo-sitions. The low-field part of the magnetizationcurves has been used to obtain the spontaneousmagnetization, and consequently the magneticmoment of the alloys (Fig. 1). For lower values

Table 1

Magnetic parameters of amorphous Fe82�xHoxB18 alloys obtained from MFT analysis

Alloys mFe (70.05mB) JFe2Fe (10�22 J) JFe2Ho (10

�22 J) JHo2Ho (10�22 J) Tc (75K)

Fe78Ho4B18 2.0 5.8 1.3 0.34 600

Fe73.4Ho7.6B19 1.92 5.66 1.08 0.27 520

Fe73.4Ho10B16.6 1.87 4.70 1.14 0.20 490

Fe68.3Ho13.7B18 1.83 4.69 1.17 0.18 430

Fe67.6Ho16.2B16.3 1.83 4.14 1.22 0.11 380

Tc is the Curie temperature.

Table 2

Magnetic characteristics of amorphous Fe82�xHoxB18 alloys deduced from high-field analysis

Alloys Ms ðmBÞ MFe ðmBÞ mFe=Fe ðmBÞ nHoFe ðT=mBÞ Bc1 ðTÞ J�HoFe (10

�22 J)

Fe72Ho8B20 [17] — — — 38 — 1.5

Fe73.4Ho7.6B19 (this study) 0.69 1.45 1.98 39.9 26 0.85

Fe73.4Ho10B16.6 (this study) 0.42 1.42 1.94 40 17 0.86

Fe68.3Ho13.7B18 (this study) 0.15 1.22 1.78 43.66 7 1.01

Ms is the saturation magnetic moment, MFe the iron magnetization deduced from the splitting of the alloy moment, taking mHo equalto 10mB.

0 10 20 30 40

0.2

0.4

0.6

0.8

1.0

µ a (µ

B)

B (T)

x=10

x=13.7

x=7.6

Fig. 3. High-field dependence of the alloy magnetic moment at

4.2K in fields up to 38T. Symbols are magnetization data

obtained in the step-wise pulses mode. Lines are obtained using

the continuous applied field mode.

A. Kaal et al. / Physica B 325 (2003) 98–105102

Page 6: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

of the Ho concentration, the resultant magnetiza-tion is directed along the Fe magnetization,whereas for higher Ho concentrations, the Ho-magnetization prevails. A compensation of themagnetization occurs for x ¼ 13 after splitting ofthe total magnetization into two sublattice mag-netizations. The splitting is based on the assump-tion that Ho retains its full free ion value, 10mB, inthe trivalent state.As clearly illustrated in Fig. 3, the high-field part

of the magnetization curves for the three samplesexhibit a pronounced magnetic transition thatseparates two regions, which are characterized bydifferential susceptibilities that are at least twoorder of magnitude different. Furthermore, themagnetization curves resemble well those offerrimagnetic compounds. On the basis of mole-cular field description, the total free energy E for aferrimagnetic RE–TM compound, can be writtenas

E ¼ �~BBð ~MMT þ ~MMRÞ þ nRT ~MMR~MMT; ð4Þ

where MR ¼ ðx=100ÞmHo and MT ¼ ð82�x=100Þ mFe denote the magnetization of the twosublattices; the last term is the exchange energy inthe molecular field description and nRT is theintersublattice exchange coupling parameter. Fol-lowing Verhoef’s description of a ferrimagnet in anexternal applied field [19], minimization of theenergy E with respect to the canting angle of theRE and TM moments, a; gives

cos a ¼ � 1 for BpBc1 ¼ nRTjMT � MRj;

¼ þ 1 for BXBc2 ¼ nRTjMT þ MRj ð5Þ

and

cos a ¼B

nRT

� �2�M2

R � M2T

!for Bc1pBpBc2;

where Bc1 is the critical field after which a rotationof RE and TM moments toward each other starts,and Bc2 is the critical field at which the twomoments begin to align ferromagnetically underthe external field effect. This field is often too farabove the accessible range of applied fields and isnot reached in our experiments. A magnetic phasediagram for our alloys is given in Figs. 4 and 5.For field values between the two critical fields, the

magnetization modulus increases linearly with theapplied field as

B ¼ nRTM : ð6Þ

Eq. (6) indicates that the intersublattice exchangecoupling, nRT; can straightforwardly be obtainedby fitting the linear part of the magnetization inthe high-field region. Experimental values of nRTas well as those of the first critical field Bc1 aregathered in Table 2. If only the nearest neighborinteractions are considered and assuming that theRE–TM exchange coupling is spatially isotropicand distance independent in the nearest neighborsphere, the microscopic RE–TM exchange cou-pling constant, JRT; can be related to nRT viaexpression [20]:

nRT ¼2zRTðgR � 1Þ

NTgRgTm2BJRT; ð7Þ

0 4 8 12 16

40

80

120

160

Fe82-xHoxB18

B (

T)

x (at. %)

Bc2

Bc1

CFI

CFF

NCFI

CFI

Fig. 4. Dependence on Ho content of the critical fields Bc1 and

Bc2 at 4.2K, solid circles are experimental data, and open circles

are calculated values. (CFI) collinear ferrimagnetic order,

(NCFI) non-collinear ferrimagnetic order, (CFF) collinear

field-forced ferromagnetic order.

A. Kaal et al. / Physica B 325 (2003) 98–105 103

Page 7: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

where NT is the number of 3d atoms by unit ofmass. The remaining parameters are as definedearlier in this paper. Values of JRT thus calculated,and noted J�

RT; are listed in the last column ofTable 2. It is seen that J�

RT increases as the rare-earth concentration increases. The small discre-pancy between J�

RT (Table 2) and JRT (Table 1)may be due to the neglect of the magneticanisotropy in the mean-field analysis.The thermal behavior of the magnetic transition

in high fields could also be studied by makingmagnetizations in Eqs. (5) temperature dependent.Non-collinear phases are bounded by the tem-peratures TL and TU defined as [21]

kBTU ¼mR BRm � B�� ��

B�1JR

BRm � B�� ��

nRTMRð0 KÞ

� � ð8aÞ

and

kBTL ¼mR BRm þ B�� ��

B�1JR

BRm þ B�� ��

nRTMRð0 KÞ

� �; ð8bÞ

where BRm ¼ nRTMT is the molecular-field actingon the rare-earth sublattice; the intra-sublatticeexchange coupling, nRR; being neglected. A typicalphase diagram, showing the evolution of themagnetic structure in the B2T plane, for thenearly compensated Fe77.3Ho13.7B17 alloy at 4.2K,is given in Fig. 5. At T ¼ Tcomp; MRðTÞ ¼ MTðTÞand TL ¼ TU ¼ Tcomp; and the situation whereB ¼ BRm corresponds to a temperature T ¼ Tcr(Fig. 5).

4. Conclusions

Mean-field-theory analysis of magnetizationdata of amorphous Fe82�xHoxB18 alloys has giventhe JFe2Fe; JHo2Ho and JFe2Ho exchange integrals.A good agreement between the calculated curveand the experimental data has been obtained in thetemperature range from 4.2 to 300K. The magne-tization curves, in high applied fields, have shownthe occurrence of a magnetic transition in sampleswith small net magnetizations. Such a transition isnot expected in systems with a sperimagneticstructure for which a continuous and gradualapproach to saturation, associated with an in-creasing fan angle of the sperimagnetic structure,is expected. The occurrence of the transition canbe understood within a model of two magneticsublattices, formed by the rare earth and 3dtransition-metal moments, as the formation of anon-collinear ferrimagnetic-like structure in highmagnetic fields. This model has allowed accurateevaluation of the intersublattice exchange cou-pling, JRT; which has been found to be close tothat obtained using mean-field-theory analysis,confirming the applicability of the model todescribe the high-field magnetic behavior ofamorphous materials. It can be noticed that themagnetization curves resemble those of ferrimag-netic compounds, which indicates that randommagnetic anisotropy is small in these samples.

30

60

90

0 30 60 90 120 150

B(T

)

T (K)

T cr

B2c

B1c

Tcomp

CFI

NCFF

CFF

Fig. 5. Magnetic phase diagram for the amorphous Fe77.3-Ho13.7B17 alloy in the B2T plane. Tcomp is indicating the

temperature, at zero-field, at which ma ¼ 0:

A. Kaal et al. / Physica B 325 (2003) 98–105104

Page 8: Magnetic exchange coupling in amorphous Fe82−xHoxB18 alloys

Acknowledgements

We are grateful to Professor Frank R. de Boer,Van der Waals Zeeman Institute, University ofAmsterdam, for his valuable comments.

References

[1] N.H. Duc, T.D. Hien, D. Givord, J.J.M. Franse, F.R. de

Boer, J. Magn. Magn. Mater. 124 (1993) 305.

[2] N. Heiman, K. Lee, R.I. Potter, A. I. P. Conf. Proc. 29

(1976) 108.

[3] K. Yano, J. Magn. Magn. Mater. 208 (2000) 207.

[4] F.R. de Boer, K.H.J. Buschow, Physica B 177 (1992) 199.

[5] A. Hassini, M. Slimani, M. Seqqat, H. Oukriss, M.

Hamedoun, A. Bouhdada, H. Lassri, Phys. Stat. Sol. (A)

174 (1999) 239.

[6] A. Hassini, H. Lassri, A. Bouhdada, M. Ayadi,

R. Krishnan, I. Mansouri, B. Chaker, Physica B 275

(2000) 295.

[7] M. Slimani, M. Hamedoun, H. Lassri, S. Sayouri, R.

Krishnan, J. Magn. Magn. Mater. 153 (1996) 132.

[8] R.J. Radwanski, R. Krishnan, J.J.M. Franse, H. Lassri, O.

El Marrakechi, Int. J. Mod. Phys. B 7 (1993) 950.

[9] R. Krishnan, O. El Marrakechi, H. Lassri, P. Rougier,

J. Appl. Phys. 73 (1993) 7599.

[10] Z.G. Zhao, N. Tang, F.R. de Boer, P.F. de Ch#atel,

K.H.J. Buschow, Physica B 193 (1994) 45.

[11] R. Krishnan, H. Lassri, J. Teillet, J. Magn. Magn. Mater.

98 (1991) 155.

[12] R. Krishnan, O. El Marrakechi, H. Lassri, Solid. Stat.

Commun. 7 (1991) 567.

[13] G. Ravach, J. Teillet, J. Phys.: Condens. Matter 10 (1998)

7065.

[14] B. Bouchet-Fabre, A. Kebab, J. Dixmier, H. Lassri, R.

Krishnan, J. Non-Cryst. Solids 192–193 (1995) 355.

[15] G. Ravach, F. Machizaud, J. Teillet, J.M. Le Breton,

A. Fnidiki, J. Phys. Condens.: Matter 12 (2000)

3639.

[16] F. Machizaud, G. Ravach, J. Teillet, J.M. Le Breton,

J. Phys.: Condens. Matter 12 (2000) 8101.

[17] R. Krishnan, H. Lassri, L. Driouch, F.E. Kayzel,

J.J.M. Franse, J. Magn. Magn. Mater. 131 (1994)

L297.

[18] M.M.S. Brooks, L. Nordstr .om, B. Johansson, J. Phys.:

Condens. Matter 3 (1991) 2357.

[19] R. Verhoef, F. RdeB oer, J.J.M. Franse, J. Magn. Magn.

Mater. 89 (1990) 176.

[20] R.J. Radwanski, J.J.M. Franse, S. Sinnema, J. Magn.

Magn. Mater. 51 (1990) 175.

[21] A.E. Clark, E. Callen, J. Appl. Phys. 39 (1968) 5972.

A. Kaal et al. / Physica B 325 (2003) 98–105 105