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Physics Letters A 371 (2007) 504507

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Magnetic phase transition in ferromagnet Co100xErx (x = 55, 65)with random anisotropy

E. Loudghiri a, A. Belayachi a,, N. Hassanain a, O. Touraghe b, A. Hassini b, H. Lassri b

a Laboratoire de Physique des Matriaux, Facult des Sciences Universit Mohammed V, B.P. 1014 Rabat, Moroccob Laboratoire de Physique des Matriaux, Micro-lectronique, Automatique et Thermique, Facult des Sciences Ain Chok, Casablanca, Morocco

Received 22 May 2007; received in revised form 19 June 2007; accepted 27 June 2007

Available online 29 June 2007

Communicated by J. Flouquet

Abstract

We have investigated the magnetic phase transitions of amorphous Co45Er55 and Co35Er65, prepared by the liquid quenching technique. Froman analysis of the approach to saturation magnetization some fundamental parameters have been extracted. Magnetic data taken in the criticalregion were analyzed using the modified Arrott plot and the critical isotherm. The data satisfy the magnetic equation of state characteristic ofsecond order phase transition over the entire temperature range of the present investigation. The exponents values obtained are in very goodagreement with the theoretical values calculated for 3D Heisenberg model. 2007 Elsevier B.V. All rights reserved.

PACS: 71.23.Cq; 75.60.Ej; 75.30.Gw

Keywords: Amorphous alloys; Magnetization; Random anisotropy; Phase transition

1. Introduction

The amorphous alloys TMRE, where TM is a transitionmetal and RE a rare earth magnetic ion, are of interest due totheir industrial applications [1]. From a fundamental viewpoint,the investigations of the TMRE systems lead to a great vari-ety of magnetic structures [2]. Due to the competition betweenexchange interactions and the large local random crystal fieldsacting on rare earth magnetic ion, this type of alloys shows non-

collinear magnetic structures and random magnetic anisotropy(RMA) [3]. In the case of some TMRE systems with RMA,the non-linear susceptibility follows a spin glass scaling equa-tion of state [4]. From theoretical side, it has been shownthat for space dimensionality d 4 and for spin componentsms 2 no long-range order exists in zero applied field [5].This has been confirmed in a number of TMRE amorphousalloys by neutron-scattering measurements [6]. For amorphous

* Corresponding author. Tel.: +212 67 07 36 36; fax: +212 37 671119.E-mail address: belayach@fsr.ac.ma (A. Belayachi).

Co100xHRx (HR = Gd, Dy, Er; 40 x 70) the magneticmoment inherent in the Co ion disappears around x = 60 [7].It has been reported that the critical exponents found for someTMRE alloys cannot be related definitively to any theoreticalprediction [8].

In the present study we have performed a detailed analy-sis of the magnetic phase transitions in amorphous Co45Er55and Co35Er65 alloys. Such study is important and interestingbecause the nature of phase transition in amorphous alloys

with random magnetic anisotropy RMA is less well under-stood.

2. Experimental

The amorphous Co100xErx alloys, with x = 55 and 65,were prepared by melt spinning technique and the amorphousstate was checked by X-ray diffraction [9]. The exact chemicalcomposition of the samples was determined by electron probemicroanalysis. The magnetization was measured by the extrac-tion method with applied field up to 200 kOe.

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2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.06.066

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3. Results and discussion

3.1. Magnetization measurements and exchange constant

In Fig. 1 we present the temperature dependence of the mag-netization M(T) under an applied field of 10 kOe. There is an

increase of the magnetization below a transition temperature,which we define via location of inflection point of the curve.The Curie temperature TC has been determined by the maxi-mum of the absolute value of the first-order derivative of themagnetization with respect to temperature and this is at theinflexion point of M vs. T data. The numerical fit leads thevalues ofTC = 17 K and 10 K for Co45Er55 and Co35Er65, re-spectively. Moreover, the M(T) curves do not show the distinctfeatures of a sharp kink as typical for ferromagnets. When de-creasing the temperature a slow increase of the magnetizationis observed, indicating a metastable domain arrangement due torestricted domain-wall mobility.

The field dependence of the magnetization was measuredup to H= 200 kOe in the temperature range 4.2150 K on allthe samples. Some selected features are presented in Fig. 2. Itcan be seen that even for H= 200 kOe, the saturation is notreached, only a non-linear continuous increase of the magneti-zation with increasing field is observed at all temperatures. Theapproach to saturation in the magnetization, for amorphous al-loys, can be described by the following formula [1016]:

(1)

M(H)=M0

1

a1/2

(H+Hu +Hex)1/2

a2

(H+Hu +Hex)2

,

where H is the magnetic field in kOe, M0 is the saturation mag-netization in emug1, Hex is the exchange field and Hu is thecoherent anisotropy field. The ai coefficients (i = 1/2, 2) de-pend upon the amount of various structural defects and intrinsicfluctuations. The first term a1/2/(H+Hu + Hex)1/2 can arisefrom point-like defects, from intrinsic magnetostatic fluctua-tions and from randomly distributed magnetic anisotropy [10].The second term a2/(H+Hu +Hex)2 is attributed to the mag-netoelastic interaction of quasidislocation dipoles. The magne-tization curves at 4.2 K for the two samples are found to fitEq. (1) well as shown in Fig. 3. According to the mean fieldpredictions the exchange constant is given approximately by

(2)A=3kB TCJEr

zErEr(1+ JEr)rErEr

where kB is the Boltzmann constant, TC is the Curie tempera-ture, JEr is the total angular momentum and rErEr is the inter-atomic distance between Er atoms taken to be 3.5 [17] andzErEr is the average number of Er nearest neighbours of theEr atomtaken tobe 6 and7 for x = 55 and 65, respectively [18].The results obtained for the parameters M0, a1/2, a2, Hex+Huand A are listed in Table 1.

3.2. Random anisotropy parameters

From the results of the approach to saturation the randommagnetic anisotropy parameters can be deduced. The factors

Fig. 1. Temperature dependence of the magnetization with applied field of10 kOe.

Fig. 2. Field dependence of the magnetization for Co 45Er55.

Fig. 3. M(H) at 4.2 K, adjusted with formula (1) (see text).

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Table 1The values of saturation magnetization M0, factors a1/2, a2, field Hex + Huand exchange constant A for the amorphous CoEr alloys at 4.2 K

Sample M0(emug1)

a1/2

(kOe1/2)

a2(kOe2)

Hex +Hu(kOe)

A

(108 ergcm1)

Co45Er55 213 3.2 114 35 3

Co35Er65 240 3.6 120 40 1.5

Table 2The values ofHex, Hr , KL, Ra and for the amorphous CoEr alloys at 4.2 K

Sample Hex(kOe)

Hr

(kOe)KL(107 ergcm3)

Ra

()

Co45Er55 10.8 41.3 4.0 28 1.4Co35Er65 10.4 42.4 4.6 23 1.5

a1/2 and a2 are related to the anisotropy field Hr and the ex-change field Hex by the relations [1921]:

(3)a1/2 = H2r

15H3/2ex,

(4)a2 =H2r

15=

115

2KLM0

2,

where KL is the random local anisotropy constant. From thesame model, Hex can be expressed as

(5)Hex =

a2

a1/2

2/3=

2A

M0R2a.

Knowing all the parameters a1/2, a2, and M0, one can evaluateHex, Hr , and KL. From the exchange field Hex and the con-

stant exchange A mentioned above, it is possible to calculatethe important structural parameter Ra [21]:

(6)Ra =

2A

M0Hex

1/2.

Table 2 exhibits the values of Hex, Hr , KL and Ra . Thereis no long-range correlation in the direction of the easy axisalong the samples, as the relatively small value of the easy axescorrelation length, Ra , indicates. The anisotropy directions areassumed to be randomly distributed beyond the characteristiclength scale Ra where atomic short range order takes place.The dimensionless parameter , which plays an important role

in distinguishing between strong anisotropy ( > 1) and weakanisotropy ( < 1), was calculated

(7)=

2

15

1/2Hr

Hex

.

It is found that > 1 (Table 2), which corresponds to a ferro-magnet system with strong random anisotropy.

3.3. Modified Arrott plot and critical isotherm

It is well known that in magnetic systems with random mag-netic anisotropy (RMA), the properties depend on the strengthof the external magnetic field where the systems passe troughdifferent magnetic regimes with increasing field. Spin-glass and

Fig. 4. Modified Arrott plot M1/ vs. (H/M)1/ (MAP) for Co45Er55. Theunits ofM and H are emu g1 and kOe, respectively.

RMA systems exhibit similar magnetic states at low tempera-ture with the spins being frozen in random directions. A transi-tion from spin glass to paramagnetic state can occurs in this typeof systems [4]. Using the modified Arrott plot and the criticalisotherm we do find, however, that the high-field magnetizationfollows the standard ferromagneticparamagnetic transition inCo45Er55 and Co35Er65 even though a strong RMA is present.

In Fig. 4, the M (H,T) data were used to construct the modi-fied Arrott plot [22] (MAP) M1/ vs. (H/M)1/. An importantadvantage of this method is that the exponents can be optimizedfor a very small temperature range, in principle for one isother-

mal curve, above and below TC , respectively [23]. For T < TCit was impossible to get all the data for one isothermal curveon a straight line, because the data for H < 30 kOe deviatefrom linearity. Generally, the low field anomalies occur bothfor ordered and disordered systems [24] and even for the caseof monocrystalline Ni [25]. An interpretation in terms of inho-mogeneities is not possible. On the other hand strong relaxationeffects have been observed when taking the measurements ofthe isotherms at low temperature.

The values of the exponents obtained are = 0.4 and= 1.35 which are in good agreement with those usually de-termined for a 3D Heisenberg system. It should be noted that

for the studied samples the width of the critical regime ismuch larger that in ordered systems as previously reported formany disordered systems [26]. The exponent , which describesthe field dependence of the magnetization for T = TC can beobtained from Ln(M) vs. Ln(H ) plot (Fig. 5). Usually, theisotherms for T < TC are convex and for T > TC concave. Thecurvature in such isotherms becomes more pronounced as thetemperature at which a given isotherm is taken deviates moreand more from TC . However, in our case the isotherms justbelow and above TC exhibit the same curvature, as previouslyobserved for semi-disordered systems [27], indicating that na-ture of the phase transition is slightly different form a classicalferromagnet. Because we do not measure exactly the isothermfor T = TC we obtain the value of by interpolating the slopes

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Fig. 5. The Ln(H ) vs. Ln(M) isotherms at few temperatures around the Curietemperature for Co0.45Er0.55.

Table 3Critical exponents , and for MTRE alloys

Sample Ref.

Co45Er55 0.4 1.35 4.8 This workCo35Er65 0.4 1.35 4.8 This workFe30Tb70 0.365 1.387 4.779 [8]Fe30Tb70 0.400 1.230 4.24 [8]Fe67Tb33

* 0.500 1.00 3.065 [8]Co35Nd65 0.39 1.21 4.10 [29]Co35Tb65 0.38 1.30 4.43 [29]Co33Gd65 0.41 1.16 3.6 [29]MFT** 0.5 1.0 3.0 [30]3D Ising 0.325 1.241 4.82 [30]

3D Heisenberg 0.365 1.336 4.80 [30]* Values of the exponents with = 90 ( denotes the angle between easy

axis and applied field).** Mean field theory.

of the approximately straight parts of the near-critical isothermsfor large Ln(H ) [28].

The exponents values obtained with the results determinedpreviously for rare earth alloys and prediction of various modelsare given in Table 3. The results are in very good agreementwith the three-dimensional Heisenberg model.

4. Conclusions

In this investigation, we have analyzed the approach to satu-ration of the high field magnetization curves of the amorphousCo100xErx . The mean field theory allowed us to determinethe exchange parameter A. Using the modified Arrott plot andthe critical isotherm, we have obtained the critical exponentsdescribing the phase transition in Co45Er55 and Co35Er65 al-loys. The two samples exhibit the critical behavior of the 3D

Heisenberg ferromagnet. This work shows that the transitionfrom paramagnetic to ferromagnetic state in such alloys is notaffected by the presence of the random anisotropy.

Acknowledgements

The high field measurements carried out at the SNCI, Greno-ble are gratefully acknowledged.

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