Physics Letters A 371 (2007) 504–507
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Magnetic phase transition in ferromagnet Co 100 − x Er x (x = 55, 65) with random anisotropy
E. Loudghiri
a, A. Belayachi
a,∗, N. Hassanain
a, O. Touraghe
b, A. Hassini
b, H. Lassri
baLaboratoire de Physique des Matériaux, Faculté des Sciences Université Mohammed V, B.P. 1014 Rabat, Morocco
bLaboratoire de Physique des Matériaux, Micro-électronique, Automatique et Thermique, Faculté 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 Co45Er55and Co35Er65, prepared by the liquid quenching technique. From an analysis of the approach to saturation magnetization some fundamental parameters have been extracted. Magnetic data taken in the critical region were analyzed using the modified Arrott plot and the critical isotherm. The data satisfy the magnetic equation of state characteristic of second order phase transition over the entire temperature range of the present investigation. The exponents values obtained are in very good agreement 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 TM–RE, where TM is a transition metal and RE a rare earth magnetic ion, are of interest due to their industrial applications[1]. From a fundamental viewpoint, the investigations of the TM–RE systems lead to a great vari- ety of magnetic structures[2]. Due to the competition between exchange interactions and the large local random crystal fields acting 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 TM–RE systems with RMA, the non-linear susceptibility follows a spin glass scaling equa- tion of state [4]. From theoretical side, it has been shown that for space dimensionalityd4 and for spin components ms2 no long-range order exists in zero applied field [5].
This has been confirmed in a number of TM–RE amorphous alloys 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).
Co100−xHRx (HR=Gd, Dy, Er; 40x 70) the magnetic moment inherent in the Co ion disappears aroundx=60[7].
It has been reported that the critical exponents found for some TM–RE alloys cannot be related definitively to any theoretical prediction[8].
In the present study we have performed a detailed analy- sis of the magnetic phase transitions in amorphous Co45Er55
and Co35Er65 alloys. Such study is important and interesting because the nature of phase transition in amorphous alloys with random magnetic anisotropy RMA is less well under- stood.
2. Experimental
The amorphous Co100−xErx alloys, with x=55 and 65, were prepared by melt spinning technique and the amorphous state was checked by X-ray diffraction[9]. The exact chemical composition of the samples was determined by electron probe microanalysis. The magnetization was measured by the extrac- tion method with applied field up to 200 kOe.
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doi:10.1016/j.physleta.2007.06.066
E. Loudghiri et al. / Physics Letters A 371 (2007) 504–507 505
3. Results and discussion
3.1. Magnetization measurements and exchange constant InFig. 1we present the temperature dependence of the mag- netizationM(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 the magnetization with respect to temperature and this is at the inflexion point of M vs. T data. The numerical fit leads the values ofTC=17 K and 10 K for Co45Er55and Co35Er65, re- spectively. Moreover, theM(T )curves do not show the distinct features of a sharp kink as typical for ferromagnets. When de- creasing the temperature a slow increase of the magnetization is observed, indicating a metastable domain arrangement due to restricted domain-wall mobility.
The field dependence of the magnetization was measured up toH=200 kOe in the temperature range 4.2–150 K on all the samples. Some selected features are presented inFig. 2. It can be seen that even for H=200 kOe, the saturation is not reached, only a non-linear continuous increase of the magneti- zation with increasing field is observed at all temperatures. The approach to saturation in the magnetization, for amorphous al- loys, can be described by the following formula[10–16]:
(1) M(H )=M0
1− a1/2
(H+Hu+Hex)1/2− a2
(H+Hu+Hex)2
,
whereHis the magnetic field in kOe,M0is the saturation mag- netization in emu g−1,Hexis the exchange field andHu is the coherent anisotropy field. Theai coefficients (i=1/2,2) de- pend upon the amount of various structural defects and intrinsic fluctuations. The first terma1/2/(H+Hu+Hex)1/2can arise from point-like defects, from intrinsic magnetostatic fluctua- tions and from randomly distributed magnetic anisotropy[10].
The second terma2/(H+Hu+Hex)2is attributed to the mag- netoelastic interaction of quasidislocation dipoles. The magne- tization curves at 4.2 K for the two samples are found to fit Eq.(1) well as shown in Fig. 3. According to the mean field predictions the exchange constant is given approximately by
(2) A= 3kBTCJEr
zEr–Er(1+JEr)rEr–Er
wherekB is the Boltzmann constant,TC is the Curie tempera- ture,JEris the total angular momentum andrEr–Eris the inter- atomic distance between Er atoms taken to be 3.5 Å[17]and zEr–Er is the average number of Er nearest neighbours of the Er atom taken to be 6 and 7 forx=55 and 65, respectively[18].
The results obtained for the parametersM0,a1/2,a2,Hex+Hu
andAare listed inTable 1.
3.2. Random anisotropy parameters
From the results of the approach to saturation the random magnetic anisotropy parameters can be deduced. The factors
Fig. 1. Temperature dependence of the magnetization with applied field of 10 kOe.
Fig. 2. Field dependence of the magnetization for Co45Er55.
Fig. 3.M(H )at 4.2 K, adjusted with formula(1)(see text).
506 E. Loudghiri et al. / Physics Letters A 371 (2007) 504–507
Table 1
The values of saturation magnetizationM0, factorsa1/2,a2, fieldHex+Hu and exchange constantAfor the amorphous CoEr alloys at 4.2 K
Sample M0 (emu g−1)
a1/2 (kOe1/2)
a2 (kOe2)
Hex+Hu (kOe)
A
(108erg cm−1)
Co45Er55 213 3.2 114 35 3
Co35Er65 240 3.6 120 40 1.5
Table 2
The values ofHex,Hr,KL,Raandλfor the amorphous CoEr alloys at 4.2 K
Sample Hex
(kOe)
Hr (kOe)
KL
(107erg cm−3) Ra (Å)
λ
Co45Er55 10.8 41.3 4.0 28 1.4
Co35Er65 10.4 42.4 4.6 23 1.5
a1/2anda2 are related to the anisotropy fieldHr and the ex- change fieldHexby the relations[19–21]:
(3) a1/2= Hr2
15Hex3/2
,
(4) a2=Hr2
15 = 1 15
2KL
M0
2
,
whereKL is the random local anisotropy constant. From the same model,Hexcan be expressed as
(5) Hex=
a2 a1/2
2/3
= 2A M0Ra2.
Knowing all the parametersa1/2,a2, andM0, one can evaluate Hex,Hr, andKL. From the exchange fieldHex and the con- stant exchangeAmentioned above, it is possible to calculate the important structural parameterRa[21]:
(6) Ra=
2A
M0Hex 1/2
.
Table 2 exhibits the values of Hex, Hr, KL and Ra. There is no long-range correlation in the direction of the easy axis along the samples, as the relatively small value of the easy axes correlation length,Ra, indicates. The anisotropy directions are assumed to be randomly distributed beyond the characteristic length scale Ra where atomic short range order takes place.
The dimensionless parameterλ, which plays an important role in distinguishing between strong anisotropy(λ >1)and weak anisotropy(λ <1), was calculated
(7) λ=
2
15 1/2
Hr Hex
.
It is found thatλ >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 strength of the external magnetic field where the systems passe trough different magnetic regimes with increasing field. Spin-glass and
Fig. 4. Modified Arrott plotM1/βvs.(H /M)1/γ (MAP) for Co45Er55. The units ofMandHare emu g−1and 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 type of systems[4]. Using the modified Arrott plot and the critical isotherm we do find, however, that the high-field magnetization follows the standard ferromagnetic–paramagnetic transition in Co45Er55and Co35Er65even though a strong RMA is present.
InFig. 4, theM(H, T )data were used to construct the modi- fied Arrott plot[22](MAP)M1/βvs.(H /M)1/γ. An important advantage of this method is that the exponents can be optimized for a very small temperature range, in principle for one isother- mal curve, above and belowTC, respectively[23]. ForT < TC
it was impossible to get all the data for one isothermal curve on a straight line, because the data for H <30 kOe deviate from linearity. Generally, the low field anomalies occur both for ordered and disordered systems[24]and even for the case of monocrystalline Ni[25]. An interpretation in terms of inho- mogeneities is not possible. On the other hand strong relaxation effects have been observed when taking the measurements of the isotherms at low temperature.
The values of the exponents obtained are β =0.4 and γ=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 is much larger that in ordered systems as previously reported for many disordered systems[26]. The exponentδ, which describes the field dependence of the magnetization forT =TC can be obtained from Ln(M) vs. Ln(H ) plot (Fig. 5). Usually, the isotherms forT < TCare convex and forT > TCconcave. The curvature in such isotherms becomes more pronounced as the temperature at which a given isotherm is taken deviates more and more from TC. However, in our case the isotherms just below and aboveTC exhibit the same curvature, as previously observed for semi-disordered systems[27], indicating that na- ture of the phase transition is slightly different form a classical ferromagnet. Because we do not measure exactly the isotherm forT =TCwe obtain the value ofδby interpolating the slopes
E. Loudghiri et al. / Physics Letters A 371 (2007) 504–507 507
Fig. 5. The Ln(H )vs. Ln(M)isotherms at few temperatures around the Curie temperature for Co0.45Er0.55.
Table 3
Critical exponentsβ, γandδfor MT–RE alloys
Sample β γ δ Ref.
Co45Er55 0.4 1.35 4.8 This work
Co35Er65 0.4 1.35 4.8 This work
Fe30Tb70 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 withΦ=90◦(Φdenotes the angle between easy axis and applied field).
** Mean field theory.
of the approximately straight parts of the near-critical isotherms for large Ln(H )[28].
The exponents values obtained with the results determined previously for rare earth alloys and prediction of various models are given in Table 3. The results are in very good agreement with 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 amorphous Co100−xErx. The mean field theory allowed us to determine the exchange parameterA. Using the modified Arrott plot and the critical isotherm, we have obtained the critical exponents describing 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 transition from paramagnetic to ferromagnetic state in such alloys is not affected 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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