Magnetic exchange coupling in amorphous Fe82-xHoxB18 alloys

Magnetic exchange coupling in amorphous Fe _82x Ho _x B ₁₈ alloys

A. Kaal

, O. El Marrakechi

, S. Sayouri

*, M. Tlem cani -

, H. Lassri

, M. Kellati

aLaboratoire de Physique Theorique et Appliqu! ee D! epartment de Physique, Facult! e des Sciences, B.P. 1796, F! es-Atlas, Morocco"

bLaboratoire de Physique de la Matiere Condens" ee et de l’Environnement (LPMCE), E.N.S. F! es-Bensouda, Morocco"

cLaboratoire de Physique des Materiaux 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 Fe82xHoxB18(4pxp16) alloys are analyzed using a two-sublattice mean-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.

r2002 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 aroused a lot of interest because of their interesting physical properties, which make them suitable for new magnetic devices applicability. These materials are also of fundamental interest as they permit the observation of diversified magnetic properties 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 exchange interaction fluctuations. Furthermore, the ran- domly varying electrostatic fields induce, via spin orbit coupling, locally varying single site aniso- tropy, leading to a spreading of the magnetic moment directions, especially for rare-earth mo- ments, and thus destroying the long range magnetic order favored by the magnetic exchange coupling. In amorphous TM–RE–Me, where Me is a metalloid, the magnetic order in the transition- metal sublattice is mainly due to the strong exchange coupling, J

between TM atoms while that between RE ones is likely to be established through the inter-sublattice atomic exchange- coupling, J

this latter also determines the

*Corresponding author.

E-mail address:sayouri1@caramail.com (S. Sayouri).

0921-4526/02/$ - see front matterr2002 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

stability of the ferrimagnetic structure, when RE is a heavy rare-earth ion. Usually, these parameters can 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 great amount of works devoted to this subject as well as to random magnetic anisotropy (RMA) effects on amorphous MT

RE

B

[5–7]. Recently, high- field magnetization studies on amorphous Fe

Ho

B

alloys have been performed by Radwanski et al. [8]. In this paper, we have considered again these magnetic measurements. We have used the mean-field theory to evaluate the exchange inter- actions Fe–Fe, Fe–Ho, and Ho–Ho, and have discussed the thermal behavior of the magnetic transition in the high field regime.

2. Experiment [9]

Amorphous Fe

Ho

B

ribbons with 4pxp16 were prepared by the usual melt spinning technique in an argon atmosphere. X-ray diffrac- tion was used to check the amorphous state of the alloys. No Bragg peaks were observed during the characterization. The exact chemical composition of the ribbons was determined by electron probe microanalysis. The magnetization was measured between 8 and 300 K using a vibrating sample magnetometer (VSM), in applied fields up to 1.8 T.

The Curie temperature was determined using a VSM with an oven. High-field magnetic isotherms at 4.2 K of amorphous Fe

Ho

B

ribbons with nominal compositions corresponding to x

8; 10, and 14 were measured in pulsed fields up to 38 T in the high-field installation of the University of Amsterdam.

The high-field magnetization measurements were undertaken on finely powdered samples, which are free to orient their moments according to the applied field direction [10]. In our case, samples consisted of small pieces with lengths between 1 and 5 mm and with a width of 1 mm.

The individual pieces have a limited, but still significant freedom to rotate within the sample holder into their minimum-energy direction during the measurements. No significant hysteresis has

been observed in increasing and decreasing fields.

Step-wise pulses, where the field was kept constant during 65 ms, were used in order to see the effect of eddy currents. No difference between the two types 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 magnetic moment of the a-Fe–Ho–B alloys at 7.5 K and in an applied field of 1.8 T is shown in Fig. 1. The magnetization decreases rapidly with increasing Ho content, indicating the anti-parallel alignment of the transition-metal and the rare-earth mo- ments. The alloy moment

can be written [11] as

xÞm

x

where

and

denote the magnetic moments of Fe and Ho, respectively. Assuming a value of

0 4 8 12 16

0.4 0.8 1.2 1.6

Ho

Ho Fe Fe

x (at. %)

0.5 1.0 1.5 2.0 Fe_82-xHo_xB₁₈

µFe (µ B ) µa (µ B )

Fig. 1. Concentration dependence of the magnetic moment of the a-Fe–Ho–B alloys at 7.5 K.

10m

for

as found in Ref. [12],

is found to be of the order of 2m

for x

4; which is close, to the experimental accuracy, to 2.04m

found for similar alloys with low concentrations on rare- earth element. In what follows,

will be taken equal to 10m

, independent of concentration.

Inserting this value in Eq. (1), one can deduce the iron magnetic moment for higher concentra- tions on Ho. It is found that the Fe magnetic moment decreases as the Ho concentration in- creases. Such a decrease cannot always be imputed to hybridization effects which mix the transition- metal 3d states with 6s

/5d states of the rare earth, but also to non-collinearity of transition-metal moments themselves. This fact was recently evidenced by Ravach and Teillet [13] through high-field M ossbauer spectroscopy measurements,

carried out on ribbons of amorphous Fe

Dy

B

and Fe

Ho

B

alloys. These authors showed that, in Fe

Dy

B

, Fe-moments under applied fields up to 8 T are distributed within a cone and that the iron-canting angle is as important as in the dysprosium one; furthermore, the Fe-moment modulus (at 14% in Dy) retains a value that approximately corresponds to that obtained for Fe

B

, suggesting that hybridization effects are too weaker than indicated by magnetic moment splitting alone. In Fe

Ho

B

on the other hand, iron spin structure is nearly collinear from 2 T; in this alloy, Fe semi-angle apex

equals 29

, then the Fe magnetic moment modulus is only about 1.03 times that deduced from moment splitting. This shows that, at least for this composition in rare-earth, the iron spin structure is nearly collinear in fields of about 2 T. In correlation with the value taken for the Ho moment, this would indicate that our alloys are ferrimagnets and that the aniso- tropy/RE–TM exchange ratio is small in these alloys.

3.1.2. Mean-field analysis

The experimental temperature dependence of the magnetization of amorphous Fe

Ho

B

alloys at 1.8 T is shown in Fig. 2. In the two-sublattice mean-field model [3], the experi- mental magnetization curves can be fitted to a set of coupled Brillouin functions, each describing

the thermal behavior of one type of magnetic moment:

m_iðTÞ ¼m_ið0 KÞB_Jiðx_iÞ;

x

g

J

H

k

T

where B

J

and g

are respectively the Brillouin function, the effective total momentum of the ion i; and the Land! e factor of the ion i;

is the Bohr magneton, k

is the Boltzmann constant, and H

is the molecular field acting on the site i: The molecular fields H

and H

are given by H

2 J

z

S

g

þ

2 J

z

1ÞJ

g

H

2 J

z

1ÞS

g

þ

2 J

z

1Þ

J

g

where J

J

and J

are the exchange integrals for Fe–Fe, Fe–Ho, and Ho–Ho interac- tions, respectively. z

(i; j

Fe; Ho) is the number of nearest neighbors of the atom j for the atom i:

In the fitting procedure, the values found above for the moments are considered as approximately equals to

and are taken as input para- meters, and the exchange integrals J

J

and J

are considered as adjusted parameters.

In the standard mean-field model, the coordina-

tion numbers z

are calculated assuming each

atom surrounded by 12 others with equal prob-

ability. This assumption is a rough estimate as it

does not take into account the local structural

order persisting in the amorphous state, as

evidenced experimentally [14,15]. Recently, Ma-

chizaud et al. [16] have studied the local order in

the amorphous Fe

RE

B

alloys and have

proposed a structural model, which succeeded to

take into account the discontinuity found in the

variation of the hyperfine field when the rare earth

concentration is varied around a critical value, x

In the following, the coordination numbers z

as

calculated in Ref. [16], will be used but with a

slight modification in order to take into account

the real composition of our alloys. The values

found for J

J

and J

derived

from the best fitting, as well as the Curie temperature and the magnetic moment

for each composition, are gathered in Table 1. A good agreement between the calculated magnetization curves and the experimental ones has been obtained. Krishnan et al. have performed high- field magnetization studies on amorphous Fe

Y

Ho

B

alloys [17]; in particular, they

have deduced for the a-Fe

Ho

B

alloy (corre- sponding to x

0) the molecular field coefficient n

which value agrees well with that eval- uated from our study for Fe

Ho

B

(Table 2), and the exchange integral J

However, the value of the latter parameter differs from the one deduced from our analysis. This discrepancy comes from the value of z

16Þ we have

0 150 300 450 600 750

0.3 0.6 0.9 1.2

1.5 x = 4 x = 7.6

x = 13.7 x = 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)

Magnetic moment (µB) Magnetic moment (µB)

Magnetic moment (µB) Magnetic moment (µB)

Magnetic moment (µB)

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.

adopted based on the work of Machizaud et al.

mentioned above instead of that equal to 8.8 (derived from the commonly used expression z

12ð802xÞ=100Þ assumed by Krishnan et al. (Table 2).

The exchange integral J

J

increases as the Ho content increases, following the predic- tions of Brooks et al. [18] that the 4f–5d exchange interaction should increase as the 4f–5d hybridiza- tion increases. It is also observed that Ho substitution for Fe leads to a weakening of J

interaction, in accordance with the decrease of the Curie temperature. The RE–TM exchange coupling can also be studied using high-field magnetic measurements, as discussed in the next section.

3.2. High-field magnetization behavior 3.2.1. Magnetic moments

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

Table 1

Magnetic parameters of amorphous Fe82xHoxB18alloys obtained from MFT analysis

Alloys m_Fe(70.05mB) JFe2Fe(10²²J) JFe2Ho(10²²J) JHo2Ho(10²²J) Tc(75 K)

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

Tcis the Curie temperature.

Table 2

Magnetic characteristics of amorphous Fe82xHoxB18alloys deduced from high-field analysis

Alloys Msðm_BÞ MFeðm_BÞ m_Fe=Feðm_BÞ nHoFeðT=m_BÞ B^c₁ðTÞ J_HoFe (10²²J)

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

M_sis the saturation magnetic moment,M_Fethe iron magnetization deduced from the splitting of the alloy moment, takingm_Hoequal to 10mB.

0 10 20 30 40

0.2 0.4 0.6 0.8 1.0

µ a

(

)

B (T)

x=10

x=13.7 x=7.6

Fig. 3. High-field dependence of the alloy magnetic moment at 4.2 K in fields up to 38 T. Symbols are magnetization data obtained in the step-wise pulses mode. Lines are obtained using the continuous applied field mode.

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 the magnetization occurs for x

13 after splitting of the total magnetization into two sublattice mag- netizations. The splitting is based on the assump- tion that Ho retains its full free ion value, 10m

, in the trivalent state.

As clearly illustrated in Fig. 3, the high-field part of the magnetization curves for the three samples exhibit a pronounced magnetic transition that separates two regions, which are characterized by differential susceptibilities that are at least two order of magnitude different. Furthermore, the magnetization curves resemble well those of ferrimagnetic compounds. On the basis of mole- cular field description, the total free energy E for a ferrimagnetic RE–TM compound, can be written as

E

B Bð

M M

M M

n

M M

M M

where M

and M

x=100Þ

denote the magnetization of the two sublattices; the last term is the exchange energy in the molecular field description and n

is the intersublattice exchange coupling parameter. Fol- lowing Verhoef’s description of a ferrimagnet in an external applied field [19], minimization of the energy E with respect to the canting angle of the RE and TM moments,

gives

cos

1 for B

B

n

M

¼ þ

1 for B

B

n

M

and

cos

B n

M_R²

M

!

for B

B

B

where B

is the critical field after which a rotation of RE and TM moments toward each other starts, and B

is the critical field at which the two moments begin to align ferromagnetically under the external field effect. This field is often too far above the accessible range of applied fields and is not reached in our experiments. A magnetic phase diagram for our alloys is given in Figs. 4 and 5.

For field values between the two critical fields, the

magnetization modulus increases linearly with the applied field as

B

n

M:

Eq. (6) indicates that the intersublattice exchange coupling, n

can straightforwardly be obtained by fitting the linear part of the magnetization in the high-field region. Experimental values of n

as well as those of the first critical field B

are gathered in Table 2. If only the nearest neighbor interactions are considered and assuming that the RE–TM exchange coupling is spatially isotropic and distance independent in the nearest neighbor sphere, the microscopic RE–TM exchange cou- pling constant, J

can be related to n

via expression [20]:

n

2z

1Þ

N

g

g

J

0 4 8 12 16

40 80 120 160

Fe_82-xHo_xB₁₈

B (T )

x (at. %)

B^c₂

B^c₁

CFI CFF

NCFI

CFI

Fig. 4. Dependence on Ho content of the critical fieldsB^c₁and B^c₂at 4.2 K, 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.

where N

is the number of 3d atoms by unit of mass. The remaining parameters are as defined earlier in this paper. Values of J

thus calculated, and noted J

are listed in the last column of Table 2. It is seen that J

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

(Table 2) and J

(Table 1) may be due to the neglect of the magnetic anisotropy in the mean-field analysis.

The thermal behavior of the magnetic transition in high fields could also be studied by making magnetizations in Eqs. (5) temperature dependent.

Non-collinear phases are bounded by the tem- peratures T

and T

defined as [21]

k

T

B

B B

B

B

n

M

ð8aÞ

and

k

T

B

B B

B

B

n

M

; ð8bÞ

where B

n

M

is the molecular-field acting on the rare-earth sublattice; the intra-sublattice exchange coupling, n

being neglected. A typical phase diagram, showing the evolution of the magnetic structure in the B2T plane, for the nearly compensated Fe

Ho

B

alloy at 4.2 K, is given in Fig. 5. At T

T

M

M

and T

T

T

and the situation where B

B

corresponds to a temperature T

T

(Fig. 5).

4. Conclusions

Mean-field-theory analysis of magnetization data of amorphous Fe

Ho

B

alloys has given the J

J

and J

exchange integrals.

A good agreement between the calculated curve and the experimental data has been obtained in the temperature range from 4.2 to 300 K. The magne- tization curves, in high applied fields, have shown the occurrence of a magnetic transition in samples with small net magnetizations. Such a transition is not expected in systems with a sperimagnetic structure for which a continuous and gradual approach to saturation, associated with an in- creasing fan angle of the sperimagnetic structure, is expected. The occurrence of the transition can be understood within a model of two magnetic sublattices, formed by the rare earth and 3d transition-metal moments, as the formation of a non-collinear ferrimagnetic-like structure in high magnetic fields. This model has allowed accurate evaluation of the intersublattice exchange cou- pling, J

which has been found to be close to that obtained using mean-field-theory analysis, confirming the applicability of the model to describe the high-field magnetic behavior of amorphous materials. It can be noticed that the magnetization curves resemble those of ferrimag- netic compounds, which indicates that random magnetic anisotropy is small in these samples.

30 60 90

0 30 60 90 120 150

B (T )

T _(K)

T cr B₂^c

B₁^{c 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 whichm_a¼0:

Acknowledgements

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

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