Using JPF1 format

JOURNAL OF APPLIED PHYSICS

VOLUME 89, NUMBER 12

15 JUNE 2001

Model of ferromagnetic clusters in amorphous rare earth and transition metal alloys L. Wang Department of Physics, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore

J. Dinga) and Y. Li Department of Material science, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore

Y. P. Feng Department of Physics, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore

N. X. Phuc and N. H. Dan Institute of Material Science, NCST, Hanoi, Vietnam

共Received 24 July 2000; accepted for publication 17 March 2001兲 Experimental results on amorphous rare earth and transition metal alloys suggest the presence of Fe-rich clusters. A model is proposed in which the magnetic units are magnetic clusters. The magnetization of the clusters decreases with the increase of temperature. In this model, there are two critical temperatures, T system and T cluster . T cluster is the Curie temperature of the magnetic clusters, c c c is the measurement of the strength of which is also the Curie temperature of the sample. T system c and T system , the system exhibits superparamagnetism interactions between clusters. Between T cluster c c with strong cluster interactions. The strong cluster interactions result in the ferromagnetic state below the critical temperature (T system ), which is called a cluster ferromagnetism. Our experimental c data 共magnetization curves and susceptibility values of amorphous Y60Fe30Al10 and Nd60Fe30Al10 ribbons兲 support the cluster ferromagnetic model. The zero temperature coercivity and the are also discussed in this article. © 2001 American Institute relationship between T block and T system c of Physics. 关DOI: 10.1063/1.1371005兴

I. INTRODUCTION

the magnetization curves could be well described with Langevin function of a cluster system with strong cluster interactions.12 Studies on magnetic viscosity measurements supported the formation of clusters.11 All these studies have shown that inhomogeneous amorphous structure is present in some amorphous rare earth and transition metal alloys. The Curie temperature of inhomogeneous magnetic materials have been discussed recently.17,18 Intergranular interaction is a very important factor in determining the Curie temperature. In this article, we have proposed a phenomenological model for amorphous rare earth transition metal alloys in which the magnetic units are magnetic clusters. The magnetization of clusters decreases with the increase of temperature. There are two critical temperatures T cluster and c cluster in our model. T is the Curie temperature that T system c c describes the paramagnetic/superparamagnetic transition of the magnetic clusters. T system corresponds to the c superparamagnetic/ferromagnetic transition of the system due to strong interactions between the clusters. The magnetic anisotropy of clusters and the interaction between the clusters determine the magnetic properties of the amorphous materials. When T⬎T cluster , the system is paramagnetic. When c T cluster ⬍T⬍T system , the system is superparamagnetic with c c strong interaction of clusters. When T⬍T system , the system is c ferromagnetic. The magnetic structures of the clusters and the system in different temperature regions are illustrated in

Amorphous rare earth and transition metal alloys possess large magnetic anisotropy, magnetostriction, and high Curie temperature.1–3 These properties are promising for many applications, such as magnetic and magneto-optic recording. These materials have been intensively investigated since the 1970s.1–3 However, the mechanism of magnetization and coercivity is not well understood. Many theoretical and experimental works4–8 described the hard magnetic properties of those amorphous materials based on random magnetic anisotropy 共RMA兲.9 RMA is a consequence of the existence of strong electrostatic fields. RMA affects the f-electron spins in a similar way as the magneto-crystalline anisotropy in crystalline materials. The theoretical basis of RMA is discussed within the context of conventional crystal field calculation.10 It is noteworthy that the RMA model is based on homogeneous amorphous systems. Recently, many research reports have shown the presence of ferromagnetic clusters in magnetic metal alloys, thin ferromagnetic film, and double-exchange systems.11–14 Many theoretical works have discussed the magnetic properties of ferromagnetic clusters.15,16 Our previous studies on amorphous rare earth and transition metal alloys have shown that a兲

Author to whom all correspondence should be addressed; Electronic mail: [email protected]

0021-8979/2001/89(12)/8046/8/$18.00

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J. Appl. Phys., Vol. 89, No. 12, 15 June 2001

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FIG. 1. Schematic illustration of magnetic structure of the magnetic cluster model in different temperature.

Fig. 1. In order to support our model, magnetization data of amorphous Y60Fe30Al10 and Nd60Fe30Al10 ribbons have been studied in this work. II. EXPERIMENTAL DETAILS

FIG. 2. Hysteresis loops of Y60Fe30Al10 ribbons melt spun at 共a兲 5 m/s and 共b兲 30 m/s taken at different temperatures.

A. Sample preparation

Ingots of Nd60Fe30Al10 and Y60Fe30Al10, respectively, were prepared by arc melting of the following metal elements 共99.9% Y, 99.9% Nd, 99.99% Fe, and 99.99% Al兲. In order to study the effect of quenching rate on structure and magnetic properties, melt-spun ribbons were fabricated using a chill block melt spinner with wheel surface speeds varying between 5 and 30 m/s. The experimental details have been published elsewhere.19–21 The samples were characterized by x-ray diffraction 共XRD兲 with Cu K ␣ radiation. Transmission 57Fe Mo¨ssbauer spectrometry analysis was performed between 4.2 K and room temperature using a conventional spectrometer 共Ranger company兲 with a source of 25 m Ci– 57Co in a rhodium matrix. Magnetization curves, hysteresis loops, and susceptibility of all specimens were measured using a vibrating sample magnetometer 共VSM兲 with the maximum applied field of 7162 kA/m 共9 T兲 from 290 to 4.2 K. High-temperature magnetization curves and susceptibility were obtained using a conventional VSM magnetometer with a maximum applied field of 945 kA/m 共1.2 T兲 for Nd60Fe30Al10 ribbons. The susceptibility values of a Y60Fe30Al10 ribbon were measured using a Faraday balance in the temperature range of 290 and 423 K. B. Experimental results

According to our XRD studies, ribbons melt spun at 30 m/s were amorphous. Melt spinning at lower speeds resulted in the formation of a mixture of crystalline rare earth RE 共Nd or Y兲 and amorphous. The amount of crystalline rare earth RE increased with decreasing melt-spinning speed. In the

ribbons melt spun at 5 m/s, a large amount of crystalline RE coexisted with the amorphous phase. The result was confirmed by our transmission electron microscopic examinations. The detail of the microstructural results has been published previously.21 Figure 2 shows the hysteresis loops of Y60Fe30Al10 melt spun at 30 m/s and 5 m/s, respectively, at different temperatures. The hysteresis loops of the Y60Fe30Al10 ribbon melt spun at 30 m/s taken at 20 K and above show typical characteristic of superparamagnetism. Coercivity of 60 kA/m was measure at 4.2 K, indicating the presence of ferromagnetism. In the magnetic susceptibility measurement with a Faraday balance, paramagnetic characteristic was shown at a temperature of 355 K and above. Therefore, the magnetism of the Y60Fe30Al10 ribbon melt spun at 30 m/s could be described with paramagnetism in T⬎355 K, superparamagnetism in T⬍355 K and ferromagnetism at temperatures around 4.2 K 共Table I兲. The magnetization curve of the ribbon melt spun at 5 m/s taken at 290 K could be described with superparamagnetism, while ferromagnetism and coercivity exist at 150 K and below 关Fig. 2共a兲兴. This result indicates that a lower wheel surface speed during melt spinning leads to the ferromagnetic state appearing at a higher temperature. Figure 3 shows the hysteresis loops measured at 4.2–290 K and superparamagnetic curves measured at 343–503 K for Nd60Fe30Al10 melt spun at 30 m/s. The magnetization curves taken at 343 K and above could be well described with superparamagnetism 关Fig. 3共b兲兴. In lower temperatures 共⬍290 K兲, ferromagnetism and hard magnetic behavior were ob-


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J. Appl. Phys., Vol. 89, No. 12, 15 June 2001 TABLE I. Molecular field coefficient ␣, T ccluster , average magnetic moment of a cluster m s ,T csystem , and the cluster density N derived from the analysis of the universal function fitting 关 M z (T,H)/ f (T/T ccluster)vs f (T/T ccluster)(H⫹ ␣ M )/T 兴 for Y60Fe30Al10 and Nd60Fe30Al10 ribbons melt spun at different wheel surface speeds v 共5,15, and 30 m/s兲. The T ccluster and T csystem obtained by the extrapolation of 1/␹ vs temperature curve is also listed. Y60Fe30Al10 Composition

␣ m s( ␮ B) N共1025m⫺3)

Nd60Fe30Al10

30 m/s

15 m/s

5 m/s

30 m/s

5 m/s

3 150 27

8 500 8

61 600 6.7

475 15 360

¯ ¯ ¯

Fitting

T csystem共K) T ccluster共K)

8 360

72 520

135 500

397 520

¯ ¯

Extrapolation

T csystem共K) T ccluster共K)

6 355

¯ ¯

215 ¯

390 545

460 570

served. Coercivity increased with decreasing temperature. These results show that the magnetism of the Nd60Fe30Al10 ribbon melt spun at 30 m/s could also be described with paramagnetism at T⬎550 K, superparamagnetism 共390–550 K兲 and ferromagnetism at T⭐390 K, as shown in Table I. Similar results were obtained for the ribbon melt spun at 5 m/s 共Table I兲. Figure 4 shows Mo¨ssbauer spectra of Y60Fe30Al10 ribbon melt spun at 5 m/s and Nd60Fe30Al10 ribbons melt spun at 30 m/s. All Mo¨ssbauer spectra of the Y60Fe30Al10 ribbon melt spun at 30 m/s were nonmagnetic doublets, except the spectrum taken at 4.2 K in our previous work.12 Only a small magnetic splitting appeared in the spectrum taken at 4.2 K,

confirming the presence of ferromagnetism at very low temperatures 关Fig. 2共b兲兴. On the other hand, the onset of the magnetic splitting appeared at 220 K for the Y60Fe30Al10 ribbons melt spun at 5 m/s, as shown in Fig. 4. The Mo¨ssbauer spectra could be considered as a mixture of a nonmagnetic doublet and a magnetic sextet with a broad hyperfine field distribution.16 The average hyperfine field was not strongly dependent on temperature. With decreasing temperature, only the amount of the magnetic part increased. This behavior is typical for cluster glass. For Nd60Fe30Al10 ribbons, all the Mo¨ssbauer spectra taken between 290 and 4.2 K could also be well considered as a mixture of a nonmagnetic doublet and a broad magnetic sextet. The superparamagnetism in magnetic measurements and cluster glass behavior in Mo¨ssbauer spectroscopy suggested the presence of clusters. The possibility of presence of clusters in amorphous rare earth and transition metal alloys has

FIG. 3. Hysteresis loops of the Nd60Fe30Al10 ribbon melt spun at 30 m/s taken at 共a兲 280, 200, and 120 K, and magnetization curves taken at a temperature between 共b兲 343 to 503 K.

FIG. 4. Mo¨ssbauer spectra of 共a兲 Nd60Fe30Al10 ribbon melt spun at 30 m/s and 共b兲 Y60Fe30Al10 ribbons melt spun at 5 m/s taken at different temperature.


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pressed as m s f (T/T cluster ), where m s is the saturation c magnetization of the cluster phase at zero temperature, and T cluster is the Curie temperature of the magnetic clusters, c which is also the Curie temperature of the sample. The tem) can be experimenperature dependent function f (T/T cluster c tally estimated or solved according to different models. Molecular field approximation for ␮ spin⫽2 ␮ B is used for the ) in this paper. description of f (T/T cluster c as the measureWe define a critical temperature T system c ment of the strength of the interaction between clusters. And, we also suppose that the critical temperature of T block , as the measurement of anisotropy energy, is much lower than the . These assumptions are suitable for our sample, T system c which can be demonstrated by our latter analysis of the experimental data. No parts in the Hamiltonian 共1兲 can be omitted at low temperature that is below T system . Our system is c cluster to T . Because T block is superparamagnetism from T system c c , the anisotropy energy term much lower than T system c ៝ i (T) 兴 2 can be omitted in Hamiltonian 共1兲 in the ⫺D 兺 i 关 n៝ i •m temperature region of superparamagnetism. In the molecular field theory approximation, it is easy to obtain the analytical formula for magnetization M: M 共 T,H 兲 ⫽Nm s f 共 T/T cluster 兲L c

冉

m s f 共 T/T cluster 兲关 H⫹ ␣ M 共 T,H 兲兴 c k BT

冊

,

共2兲

FIG. 5. Zero-field-cooling 共ZFC兲 and field-cooling 共FC兲 curves of the 共a兲 Y60Fe30Al10 ribbon melt spun at 5 m/s, 共b兲 Y60Fe30Al10 ribbon melt spun at 30 m/s, and 共c兲 Nd60Fe30Al10 ribbons melt spun at 30 m/s. The operation field is 12 kA/m (150⬎Oe) for 共a兲 and 共b兲 and 4 kA/m 共50 Oe兲 for 共c兲.

been reported.11,12 Figure 5 shows the zero-field cooling 共ZFC兲 and field cooling 共FC兲 curves of the both Y60Fe30Al10 and Nd60Fe30Al10 ribbons melt spun at 30 m/s. The FC and ZFC curves are typical for magnetic spin or cluster glasses.22,23

m s⫽

III. THEORETICAL MODEL

According to the experimental results, the Hamiltonian of E of a inhomogeneous amorphous rare earth and transition metal alloy is given as: E⫽⫺D ⫺H

1

៝ i 共 T 兲 •m ៝ j共 T 兲 Ji jm 兺i 关 n៝ i •m៝ i共 T 兲兴 2 ⫺ 2 兺 i⫽ j

兺i m៝ i共 T 兲 ,

where L is the Langevin function, N is the total number of clusters in a unit volume, and ␣ is the molecular field parameter. ) is a universal funcFrom Eq. 共2兲, M (T/H)/ f (T/T cluster c ) H⫹ ␣ M (T,H) /T. Thus, the universal tion of f (T/T cluster 关兴 c ) versus f (T/T cluster )关H curve M (T/H)/ f (T/T cluster c c ⫹ ␣ M (T,H) 兴 /T is acquired by two suitable parameters; the and the molecular field Curie temperature of cluster T cluster c coefficient ␣. So, these two parameters can be obtained from the experimental M versus H curve at a different temperature in the superparamagnetic region by the universal curve fitting process. The total magnetic moment m s of a cluster is:

共1兲

where m(T) is the magnetic moment of a cluster. D is the magnetic anisotropy of a cluster. n i denotes a random unit vector representing the local axis of easy magnetization direction. J i j is the exchange interaction between the ith cluster and jth cluster. H is the applied field. m(T) can be ex-

3k B M 共 T,H 兲 T , 2 M 0 f 共 T/T cluster 兲共 H⫹ ␣ M 兲 c

共3兲

)(H⫹ ␣ M ) is the initial slope of where M (T,H)/ f 2 (T,T cluster c the universal curve. M 0 is the saturation magnetization at 0 K. The initial susceptibility ␹ at a low field can be expressed as:

冦

␹⫽ ⌰⫽

Nm s2 f 2 共 T/T cluster 兲 c 3k B 共 T⫺⌰ 兲 Nm s2 ␣ f 2 共 T/T cluster 兲 c

.

共4兲

3k B

The temperature T⫽⌰ defines the critical temperature that gives the strength of the magnetic interactions T system c between clusters. Therefore,


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FIG. 6. f (T/T ccluster) and f 2 (T/T ccluster) vs T/T ccluster curves together with the estimated M s (T)/M s (4.2 K) values for Y60Fe30Al10 and Nd60Fe30Al10.

T system ⫽ c

/T cluster Nm s2 ␣ f 2 共 T system 兲 c c 3k B

.

FIG. 7. M / f (T/T ccluster) vs f (T/T ccluster)(H⫹ ␣ M )/T universal curves for the Y60Fe30Al10 ribbons melt spun at 共a兲 5 m/s, 共b兲 15 m/s, 共c兲 30 m/s, and 共d兲 Nd60Fe30Al10 melt spun at 30 m/s.

共5兲

) is not strongly dependent on temperature 共Fig. f 2 (T/T cluster c . If T system is 6兲 in the temperature region of T⬍0.5T cluster c c cluster cluster , when T⬍0.5T c , we expect much lower than 0.5T c a linear relationship between the inverse susceptibility and can be determined from the inverse temperature and T system c susceptibility versus temperature plot extrapolated to zero. If T system ⬎0.5T cluster , the 1/␹ versus T curve is not a linear c c can still be estimated approximately curve, however, T system c by the extrapolation.

IV. MAGNETIC ANALYSIS A. Universal curve fitting results and analysis

The temperature dependence f (T/T cluster ) in the ferroc magnetic region can be experimentally estimated by measuring saturation magnetization as a function of the temperature. However, the highest magnetic field of 1.2 T of the hightemperature magnetometer VSM was not sufficiently high ) from for the accurate estimation. We calculate f (T/T cluster c the molecular field approximation,17 assuming the magnetic ) and moment of a spin ␮ spin⫽2 ␮ B . The plot of f (T/T cluster c cluster ) versus T/T is shown in Fig. 6. The estif 2 (T/T cluster c c mated relative saturation magnetization M s (T)/M s (4.2 K) is plotted in Fig. 6 in the temperature range of 290 and 4.2 K for Y60Fe30Al10 and Nd60Fe30Al10. For both samples, the cal) can be accepted with a certain degree of culated f (T/T cluster c error. The two parameters T cluster and ␣ were obtained from the c fitting in Fig. 7. The fitting results are listed in Table I for Y60Fe30Al10 ribbons melt spun at 30, 15, and 5 m/s, respectively, and the Nd60Fe30Al10 ribbon melt spun at 30 m/s. For the estimation of ms in Eq. 共3兲, saturation magnetization was calculated measured at 4.2 K was used for M 0 . T system c by Eq. 共5兲.

It can be seen that the molecular field coefficient ␣ is relatively small for the Y60Fe30Al10 ribbon melt spun at 30 m/s. The low ␣ value corresponded to weak cluster interac共only 8 K兲. The calculated magnetitions and a low T system c zation of clusters is approximately 150 ␮ B , which corresponds to 75 Fe atoms assuming ␮ Fe⫽2 ␮ B . After melt spinning with lower wheel surface speeds, the magnetic moment increased to 500 ␮ B and 520 ␮ B for the ribbons melt spun at 15 and 5 m/s, respectively. The cluster interactions became much stronger with the molecular field coefficient ␣ of 8 and 61, respectively. The increase of the cluster moment and stronger cluster interactions could be attributed to the formation of crystalline yttrium, so that the iron concentration in the amorphous phase is much higher. The increase of the cluster moment, molecular field coefficient and T cluster c resulted in the increase of T system . c was estimated to be 360 K for Y60Fe30Al10 ribbon T cluster c melt spun at 30 m/s and about 500 K after melt spinning at 15 and 5 m/s 共Table I兲. Our Mo¨ssbauer spectra showed that the average hyperfine field of the magnetic part 共Fig. 4兲 was significantly lower than the hyperfine field of pure iron, confirming the presence of Y and Al in the cluster phase. The composition of the cluster phase may be varied with different wheel surface speeds, because of the formation of crystalline yttrium at lower melt-spinning speeds. T system and T cluster of Nd60Fe30Al10 were much higher c c than those of Y60Fe30Al10 ribbons. And the cluster moment was fairly low, only about 15 ␮ B per cluster. The low value of m s is partly due to the relatively large error during the fitting and calculation, and is also associated with the possibility of a spherimagnetic structure of the cluster phase.1 The and T system are probably attributed to the maghigher T cluster c c netic Nd atoms.



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FIG. 9. Calculated susceptibility ␹ in comparison with the measured data points for the Y60Fe30Al10 ribbon melt spun at 30 m/s.

FIG. 8. Inverse susceptibility 1/␹ as a function of temperature for Y60Fe30Al10 ribbons melt spun at 5 and 30 m/s. The operation field is 12 kA/m 共150 Oe兲.

B. Initial susceptibility ␹

According to our analysis 关Eq. 共4兲兴, the 1/␹ versus temperature curve should have a linear part in the superparamagnetic region and paramagnetic region, respectively, if T system c is much lower than 0.5 T cluster . The extrapolation of the two c and T cluster , respectively. linear parts can determine T system c c Figure 8 shows the inverse of susceptibility versus temperature for the Y60Fe30Al10 ribbons melt spun at 30 and 5 m/s, respectively at relatively low temperatures. For the ribbon melt spun at 30 m/s, the extrapolation to zero of the inverse value of ⬃6 K. The other susceptibility, yielded a T system c linear part of 1/␹ versus T in the temperature range of 350– 420 K indicates the paramagnetic state. The extrapolation about 355 K. The resulted in a Curie temperature T cluster c susceptibility was measured with Faraday balance in the and temperature region. The two critical temperatures, T system c , estimated from the extrapolation were very close to T cluster c those obtained from fitting and calculation. This result indicates that the model of cluster ferromagnetism can describe the amorphous Y60Fe30Al10 ribbons well. Figure 9 shows the ⬍T calculated susceptibility ␹ from Eq. 共4兲 for T system c cluster and paramagnetic ␹ for T⬎T in comparison ⬍T cluster c c with the measured ␹ values. Figure 9 shows a good agreement between the theoretical calculation and the experimental data. For the Y60Fe30Al10 ribbon melt spun at 5 m/s, The obtained by extrapolation of the 1/␹ versus temperaT system c ture curve is about 215 K. From our Mo¨ssbauer study, magnetic splitting started at a temperature around 220 K, as shown in Fig. 4. 1/␹ decreased continuously and entered into

a flat part at T⬍100 K. This result suggests that the superparamagnetic/ferromagnetic transition occurs in a wide temperature range 共100–215 K兲. The calculated T system value c in Table I was 135 K, which is much lower than the T system c obtained by extrapolation method. The reason is the fitting result is not very good for the ribbon melt spun at 5 m/s, as shown in Fig. 7共a兲. Figure 10 shows the 1/␹ vs temperature of the Nd60Fe30Al10 ribbon melt spun at 30 m/s. The operation field was 240 kA/m 共3 kOe兲. For T⬎540 K, 1/␹ increased rapidly and showed a clear linear relationship with temperature. was estimated to be 545 K by extrapolation. A low T cluster c operation field 关 H⫽4 kA/m(50 Oe) 兴 was used for the sus, because Eq. 共4兲 is only ceptibility study to obtain the T system c suitable at very low field in superparamagnetic region. As was 390 K estimated shown in the inset in Fig. 10, T system c from extrapolation. The 1/␹ versus T curve is not linear in is higher than the superparamagnetic region, because T system c 关Eq. 共4兲兴. However, T system can still be determined 0.5 T cluster c c approximately by extrapolation in this condition.

FIG. 10. The inverse susceptibility 1/␹ as a function of temperature for Nd60Fe30Al10 ribbon melt spun at 30 m/s under the operation fields of 240 kA/m and 4 kA/m 共in the inset兲.


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Similar results were obtained for the Nd60Fe30Al10 ribbon melt spun at 5 m/s. The two critical temperatures were and T cluster , estimated to be ⬃460 and ⬃570 K for T system c c respectively. This result has shown that the superparamagnetic region is narrow 共between 460 and 570 K兲. The narrowing of superparamagnetism region is due to the strong cluster interactions.

strated the peak position of ZFC curve is related with the interaction between the clusters. The comparison of the estimated T block and T system might demonstrate that the interacc tion between clusters is one of the dominating factors determining the magnetic character of the melt spun ribbons, which is an important assumption in our model.

C. Blocking temperature

In the ferromagnetic region, we must consider the contribution of anisotropy energy to estimate the coercivity. It is difficult to calculate the value of coercivity from Eq. 共1兲. Here, we use a much easy way to estimate the value of coercivity. According to Alben and Chi’s Monte Carlo simulation work5 based on the random magnetic anisotropy theory, H c ⬇0.05 J␮ B2 /ms when D/J⬍4 and H c ⬇2 J␮ B2 /ms when DⰇJ. From the fitted molecular field coefficient ␣ and m s , Jm s2 is ⬃0.12⫻10⫺21 J for Y60Fe30Al10 ribbon melt spun at 30 m/s and ⬃3.6⫻10⫺20 J for the Nd60Fe30Al10 ribbon melt spun at 30 m/s. The estimated H c is ⬃0.05 T for amorphous Nd60Fe30Al10 ribbons and is ⬃1.5⫻10⫺7 for amorphous Y60Fe30Al10 ribbons. The estimated H c values are much smaller than the coercivities measured in experiments 共Figs. 2 and 3兲. Therefore, Alben and Chi’s simulation work is not suitable for our materials. Besides Chi and Alben’s simulation,5 Callen, et al.6 have used the molecular field theory to estimate the coercivity based on the random anisotropy model. Their result suggested that coercivity increases with increasing anisotropy energy. According to their results, the higher coercivity in amorphous Nd60Fe30Al10 could be explained by the higher anisotropy energy of Nd-rich clusters. The coercivity mechanism is still not well understood. Our future work will be focused on the coercivity mechanism based of the cluster ferromagnetism proposed in this work.

The description of the magnetic properties of single domain clusters or particles in general for nonzero temperature is based on the superparamagnetic theory. In this theory, the relaxation time ␶ is determined by this formula:

␶ ⫺1 ⫽ f 0 exp共 ⫺E barrier /k B T 兲 ,

共6兲

where E barrier corresponds to the total energy barrier and f 0 is a characteristic constant frequency. If there are no interactions between clusters, E barrier⫽E anisotropy⫽K 1 V where K 1 is the anisotropy energy constant of the cluster phase and V is the volume of a cluster. When the time scale of the relaxation time determined by Eq. 共6兲 is much larger than the observation time, the system appears ferromagnetic. The temperature that determined from Eq. 共6兲 with ␶ being set equal to the experimental observation time t defines the blocking temperature. E anisotropy⫽25⫻k⫻T block ,

共7兲

where E anisotropy is the magnetic anisotropy energy of a cluster. The calculated T block corresponds to the temperature where the peak of ZFC curve appears in the experiment. If there are interactions between the magnetic clusters, E barrier is not equal to E anisotropy and the calculated T block is lower than the temperature where the peak of ZFC appears.24 It is interesting to compare the T block values with the values in Table I. The magnetic moment of a cluster T system c is about 150– 600 ␮ B , thus the total number of atoms in a cluster should not excess 500 including Y 共or Nd兲 and Al. The estimated cluster size is around 4⫻10⫺27 m3. It is a reasonable number considering the cluster density of the order of 1⫻1026 m⫺3 共Table I兲. For the soft magnetic Y60Fe30Al10 ribbons, the anisotropy constant K 1 is expected in the range of around 0.02⫻106 J/m3, which is a typical value for iron–metalloid alloy.25 E is calculated to be 0.7 ⫻10⫺22 J. The blocking temperature T block is calculated to values. be ⬃0.2 K, which is far below the T system c For the Nd60Fe30Al10 ribbons, hard magnetic properties were found during our magnetic measurements. The magnetic anisotropy K 1 should be much higher than that of the soft magnetic Y60Fe30Al10 ribbons. Taking K 1 ⫽1 ⫻106 J/m3 which is a typical value for the iron alloys with rare earths,25 E is calculated to be 0.4⫻10⫺20 J. T block is estimated to be 11.6 K, which is also well below the T system c values for the Nd60Fe30Al10 ribbons as shown in Table I. The estimation of T block is in disagreement with the experimental temperature where the peak of ZFC appears. This could be due to two reasons: one is the use of small value for the anisotropy constant of the clusters; the other is the peak movement caused by the interactions between the clusters.23 Our recent Monte Carlo simulation work26 has also demon-

D. Coercivity at zero temperature

V. CONCLUSION

The magnetization loops, ZFC and FC curves and Mo¨ssbauer spectra of amorphous Y60Fe30Al10 and Nd60Fe30Al10 ribbons was measured. All the results suggest the presence of small Fe-rich clusters. According to the experimental results, a model that is called as cluster ferromagnetism has been proposed for amorphous rare earth and transition metal alloys. The model is based on the molecular field approximation. The magnetic behavior can be described with the three regions, paramagnetism, superparamagnetism, and ferromagnetism. The ferromagnetic state mainly results from the strong interactions between clusters. The system has two is the Curie temperature and critical temperatures: T cluster c is the measurement of the strength between magnetic T system c clusters, which separate the three temperature regions. Based on this model, the molecular field coefficient ␣ and T cluster was obtained by the universal curve fitting. The c , and the cluster size magnetization of clusters ms , T system c . The can be calculated after we obtain the ␣ and T cluster c strength of interactions and the cluster size are strongly dependent on the melt-spinning condition. The ribbons prepared by a relatively low quenching rate 共low wheel surface


Wang et al.


speed兲 might have larger cluster size, stronger interactions, and higher Curie temperatures. The temperature dependence of the inverse of susceptibility confirms the presence of two critical temperatures for the amorphous ribbons. The measured 1/␹ versus temperature curves are in good agreement with those calculated from the ferromagnetic cluster model. The blocking temperatures estimated using the obtained pavalues, confirming that rameters are well below the T system c the strong interactions between clusters maybe the main reason for the ferromagnetic state in the amorphous rare earth and transition metal alloys. 1

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Using JPF1 format - NUS Physics

Using JPF1 format - NUS Physics

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