New model to study the magnetic anisotropy in ternary amorphous alloys

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A new model to study the magnetic anisotropy in ternary amorphous alloys

A. Itri *, H. Lassri, M. El Yamani

Laboratoire de Physique des Mate&riaux, Faculte& des Sciences, avenue Ibn Batouta, B.P. 1014, Rabat, Morocco

Laboratoire de Physique des Mate&riaux et de Microe&lectronique, Universite&Hassan II, Faculte&des Sciences, B.P 5366 MaaLrif, An(n ChocK, Route d'El Jadida km-8, Casablanca, Morocco

Received 3 December 1998

Abstract

We have prepared Fe Er

VB

\Vand Fe

\VAl VEr

B

amorphous ribbons by melt spinning device. We performed some magnetic measurements (magnetization versus temperature, magnetization versus magnetic"eld,2) on these samples and we introduced theoretical models to determine the origin of their magnetic properties concerning exchange and anisotropy energies. Then using mean-"eld theory, we have explained temperature dependence of magnetization and evaluated exchange interactions (J

$^}$, J

$^}#andJ

#^}#). Applying the random anisotropy model to magnetization versus"eld curves at the approach to the magnetic saturation, we determined some fundamental parameters such as exchange (H

#) and random anisotropy (H

0)"elds, random anisotropy constantK

, and the ferromagnetic correlation length (R

$). We have also evaluated a new model to study the magnetic anisotropy of a ferrimagnetic and ferromagnetic systems, we distinguish between weak and strong anisotropy systems and we obtain good results in agreement with previous works on intermetallic compounds. Using this model we explain the strange behaviour of our alloy's magnetic anisotropy versus temperature. All results show that our systems are ferrimagnets with wandering axes. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Magnetism; Amorphous ribbons; Random anisotropy; E!ective anisotropy model

1. Introduction

The rare-earth (R) transition metal (T) intermetallic compounds are useful because they com- bine the large magnetic interaction and high Curie temperature of the transition metal with the large anisotropy and magnetostriction of rare-earth ele-

*Corresponding author.

E-mail address:elyamani@fsr.ac.ma (A. Itri)

ments. In 1982 Sarkis et al. [1] gave an expression to the e!ective anisotropy of the intermetallic compounds (RT) having a hexagonal structure. They considered all directions where the sub-network magnetic moment can point, their study was comp- lemented by Rinaldi and Pareti [2]. All of them di!erentiated between weak and strong magnetic anisotropy systems. In 1994 Barbara [3] studied the ferromagnetically coupled R

T intermetallics and succeeded in relating the e!ective anisotropy

"elds of etch sub-network to the e!ective

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

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anisotropy "eld of the system. In the same year Kronmuller et al. [4] working on those compounds'microstructures announced an expression close to that obtained by Sarkis et al. Amorphous transition metal-metalloid (T}Me) alloys have brought about considerable activity. However, studies of rare-earth doped T}Me metallic glasses are recent [5}8] and have revealed soft magnetic properties for Fe}B-based amorphous alloys [9]

and also Co}B-based amorphous alloys [10]. In order to study the in#uence of the addition of Er and Al atoms addition on the magnetic properties of Fe}B alloys, we prepared Fe

\VAl VEr

B and FeEr

VB

\Vribbons. We will evaluate some important magnetic parameters using a theoretical model such as random local anisotropy model [11], and mean-"eld theory [12]. After that we will focus on a new model to determine the e!ective magnetic anisotropy and we will compare it with previous works on the intermetallics.

2. Experimentals

Using the well-known melt spinning device, we prepared FeErVB\V and Fe\VAlVErB, amorphous ribbons with concentration range from x"0% Er (% Al) tox"20% Er (% Al) in an inert Argon atmosphere. The samples were 30}40lm thick with about 1}2 mm wide. The amorphous state was detected by X-ray di!raction. The com- position was determined by electron probe micro- analysis. The magnetization curves were measured in temperature range 4.2}300 K by extraction method, under applied"elds up to 16 T. The Curie temperature was also determined by thermomag- netic curves using a vibrating sample magneto- meter.

3. Results and discussions

3.1. Magnetization versus temperature and applied magneticxeld study

The saturation magnetization decreases linearly with increasing Er content at both ¹"4.2 and 300 K. That is due to the antiferromagnetic coup-

ling between Er and Fe magnetic moments. In order to calculate the magnetic moment of Er (l#) at 4 K, we followed the procedure shown in Ref.

[7]. Taking the magnetic momentum of Fe as constant (l$"2.06l ) forx)4% Er [13], we"nd the mean value of the Er magnetic momentum to be aboutl#"8.3l which is less than the Er theoretical value (l#"9l ), approximately the same value was obtained for similar amorphous alloys [14]. This means that the Er magnetic moments are asperomagnetically distributed with a value of the canting angle of about 663. We note that we obtained a Fe mean magnetic moment of the order of 1.9l for the Fe

\VAl VEr

B

series which means that the Fe orbital moment is unfrozen in that state.

The magnetization versus temperature shows the same behaviour, for all studied compositions (x(20). Indeed, as the temperature increases, the magnetization rises to reach a maximum, then decreases and cancels each other in the vicinity of the Curie temperature ¹

. However, the sample x"

20% Er curve shows a compensation approximately at 4.2 K. The variation of magnetization versus temperature can be well described in terms of the mean-"eld theory for both amorphous alloys [12,15}17], and intermetallic compounds [18,19].

The studied amorphous alloys are composed of two magnetic sub-networks: The transition metal (Fe) and the heavy rare earth (Er). The saturation magnetization is expressed by

M(¹)""M

$(¹)!M

#(¹)", (1) where M

$ is the Fe sub-network magnetization andM

#is the Er sub-network magnetization. Ac- cording to the mean-"eld theory the magnetization of these sub-networks is expressed in terms of the Brillouin functions [13] which are functions of the exchange interactions Fe}Fe (J$^}$), Fe}Er (J$^}#), and Er}Er (J#^}#). Fitting the experimental data numerically, we adjusted the exchange parameters, and also the Curie temperature (¹

) (Figs. 1 and 2). We noted that J

#^}# is about 2;10\ J, J

$^}# arises while J

$^}$ and ¹ de- crease when the Er content increases (Table 1), whileJ

$^}$andJ

$^}# increase and¹

diminishes when the Al concentration rises (Table 2). This is probably due to charge transfer from the SP band

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Fig. 1. The temperature dependence of the magnetization for the Fe Er

VB

\V.

of the metalloid to the 3d band of the transition metal and to 3d}5d hybridization e!ects between Fe and Er atoms which increases when adding both Er and Al atoms [20}22]. Using the values of JG\Hobtained above, the exchange constantAwas calculated in terms of Hasegawa's model [23]. The Curie temperatures calculated numerically are in good agreement at about 2% with those measured

experimentally using the VSM. The magnitude of A decreases gradually with increasing Er and Al content at various temperatures. The exchange constantA decreases with increasing temperature (Figs. 3 and 4), which is due to thermal agitation of the magnetic moments.

According to Chudnovsky et al. [11,24}26], the approach to saturation in the magnetization curves

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Fig. 2. The temperature dependence of the magnetization for the Fe

\VAl VEr

B .

Table 1

Mean-"eld theory results for the Fe Er

VB

x# J

$^}$(10\J) J

$^}#(10\J) J

#^}#(10\J) ¹ (K) ¹ !(K)

1.7 6. 1.2 0.2 585 585.3

2.7 5.7 1.5 0.2 557 558.3

3.8 5.5 1.8 0.2 530 533

20.0 3.0 2.0 0.2 290 321.5

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Table 2

Mean-"eld theory results for the Fe

\VAl VEr

B

x J

$^}$(10\J) J

$^}#(10\J) J

#^}#(10\J) ¹ (K) ¹ !(K)

0 7.01 2.1 0.2 540 545.5

4 7.235 2.0 0.2 527 532.3

6 7.236 2.1 0.2 512 518.1

10 7.32 2.2 0.2 488 493.8

15 7.36 2.3 0.2 452 458.8

Fig. 3. Exchange constant versus temperature for the FeEr

VB

\V.

can be described in the following two ways. For an applied "eld H'H

# the "eld dependence is ex-

pected to follow an H\ law. Whereas when H(H

#, which is appropriate for our study, the dependence is best described by anH\law. So if one plotsMas a function ofH\one gets a linear dependence between these two quantities. The following equation describes this situation:

M!M M

"1

15

^H^H

⁽²⁾

whereHis the"eld transition between the corre- lated spin glass state and the ferrimagnet with wandering axes state:

H"H0

H#. (3)

Fig. 4. Exchange constant versus temperature for the Fe\VAl

VEr B

.

The exchange"eldH

#can also be expressed by:

H#" 2A

MR, (4)

whereR is the correlation length of the local anisotropy axes. We assumedR to be equal to 10 As, which was determined experimentally on similar alloys [27}29].M

is the saturation magnetization and H

0is the random anisotropy "eld and is re- lated to the anisotropy constantK

by the relation H0"2K

M

. (5)

Figs. 5 and 6 show the temperature dependence of K for Fe

Er VB

\V and Fe

\VAl VEr

B , re- spectively. As the temperature is decreased, the random anisotropy constant "rst increases

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Fig. 5. Random local anisotropy constant versus temperature for the Fe

Er VB

\V.

Fig. 6. Random local anisotropy constant versus temperature for the Fe

\VAl VEr

B .

showing a broad maximum and it starts decreasing as the temperature is lowered further. This strange behaviour will be discussed in detail in the last paragraph.

Finally fromK

,AandR and with the help of the relation:

j"

¹⁵²

^K^A^R^, ⁽⁶⁾

jwas calculated. It is known [25] that for j(1 (weak anisotropy) the ferromagnetic correlation length (R$) becomes greater than R and can be written as

R$"R

j. (7)

It is found that in our alloysj(1, which corresponds to a ferrimagnet with wandering axis system. Experimental data show that R

$ decreases with increasingx. As the temperature is increased the ferromagnetic correlation length"rst decreases and then starts increasing (Figs. 7 and 8). This R$ behaviour can be understood in terms of the temperature dependence of the variables on which R$depends according to Eqs. (6) and (7).

3.2. Ewective anisotropy model(EAM)

We have just calculated the random local anisotropy, using the magnetization curves at various temperatures, in terms of the phenomenological model proposed by Chudnovsky et al. [11]. The thermal variation of that entity shows that it passes through a maximum when the temperature is lowered. In our case, we have the contribution of two sub-networks at the magnetic anisotropy on the one hand the Er which is a rare earth possessing an important magnetic anisotropy, and on the other hand the Fe for which the mean magnetic momentum is lower than that of the metallic counterpart. This situation shows that the Fe orbital momentum is unfrozen now, then we will"nd a spin}orbit interaction which will give rise to a local magnetic anisotropy in the Fe sub-network.

Finally, the magnetic random local anisotropy constant evaluated by Chudnovsky et al. [11] is a function of the inter-sub-lattice exchange interactions (J0^}2) and the local anisotropies of the sub-networks (K

0andK 2).

Having been inspired by the work of Sarkis et al.

[1] and Barbara [3], we consider the following

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Fig. 7. Ferromagnetic correlation length versus temperature for the Fe

Er VB

\V.

Fig. 8. Ferromagnetic correlation length versus temperature for the Fe

\VAl VEr

B .

energy densities for both sub-networks to evaluate this e!ective anisotropy for a ferrimagnetic system:

e0"!M

0Hcosh!K 0sinh

#n 0^}2M

0M

2cos(t#h), (8)

e2"!M

2Hcost!K 2sint

#n 0^}2M

0M

2cos(t#h), (9) whereHis the applied magnetic"eld,MGthe magnetization of the i sub-network, n0^}2 the macro- scopic exchange interactions,K

G the random local anisotropy of theisub-network (i"R, T),t(re- spectively h) the angle between M

2 (M

0) and the applied magnetic"eld direction (Fig. 9).

We minimize these energy densities with respect totandh, which gives

M0Hsinh!K0sin 2h

!n 0^}2M

0M

2sin(t#h)"0, (10) M2Hsint!K

2sin 2t

!n0^}2M0M2sin(t#h)"0. (11) Then we introduce the small angles to examine the small "elds and small angles behaviour: m"

h!p/2 (respectivelyg"p/2!t) theM 0(M

2) de- viation from the anti-alignment plan.

Up to linear terms in these small angles, the minimization equations become

g" M 2(2K

0/n 0^}2M

0M

2#1)!M 2[2K 0

0K 2/n

0^}2M 0M

2#K 0#K

2]H, (12) m" M

2!M 0(2K

2/n 0^}2M

0M 2#1)

2[2K0K2/n0^}2M0M2#K0#K2]H. (13) We see that the canting angle is thus linear inH.

Fig. 9. Ferrimagnetic system in an applied"eld (H) withpthe anti-alignment plan.

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Fig. 10. EAM results for the random local anisotropy in FeErVB\Vcompared to those of Chudnovsky et al. model.

Miller et al. [30] used a standard method to determine the anisotropy constants of a crystalline system. In an applied magnetic "eld (H), the free energy is then

F"K !K

sinh!K

sinh!MHcosh. (14) Minimizing with respect to the angle between the magnetization and the magnetic "eld for small

"elds and small deviations from the easy direction, they found the component of the magnetization along the magnetic"eld direction to be

Mcosh+MH

2K#2. (15)

We adapt that result to the amorphous state which permits us to determine the e!ective anisotropy for both sub-networks separately. Then for the tran-

sition metal sub-network we"nd

K2 "M2 n 0^}2M

2M 0(K

0#K

2)#2K 0K (M2!M 2

0M 2)n

0^}2M 2M

0#2K 0M2

(16) and for the rare-earth sub-network we obtain K0"M0 n0^}2M2M0(K0#K2)#2K0K2

(M0!M 0M

2)n 0^}2M

2M 0#2K

2M0. (17) Knowing that the component of the magnetization of the alloy along the applied magnetic "eld is found to be

M&"M

2cost#M

0cosh+M 2g!M

0m (18)

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Fig. 11. EAM results for the random local anisotropy in Fe

\VAl VEr

B

compared to those of Chudnovsky et al. model.

and using the result obtained by Miller et al. the alloy's e!ective anisotropy is

K"M n 0^}2M

2M 0(K

0#K 2)#2K

0K Mn 2

0^}2M 2M

0#2(K

0M2#K 2M0).

(19) We obtain thus an explicit form of the e!ective anisotropy which is a combination of exchange interactions and magnetic anisotropy of both sub- networks. When comparing di!erent expressions we obtained, we found a direct relation between sub-networks magnetization and the alloy's e!ective anisotropy:

M K"M2

K2

#M0 K0

. (20)

This relation is in good agreement with that obtained by Barbara for the intermetallic ferromagnets.

We did the same calculations for a ferromagnetic system and we found the same results taking into account that the resultant magnetization is given by M"M

2#M

0, and that the angle between both sub-networks'magnetization ist!hand not t#h. As it can be seen we obtain the same result as that obtained by Sarkis et al. for the intermetallics. In the case of weak magnetic anisotropy we obtain the following approximation:

K+(K 0#K

2)

;¹^#²^;^(K₀^M₂^$^K₂^M₀⁾¹^/n₀^}₂^M₀^M₂^M^(K₀^#^K₂⁾^,

(21) where the sign plus corresponds to a ferromagnetic coupling and the sign minus corresponds to an antiferromagnetic coupling of the magnetic moments.

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For the thermal variation of the alloy's e!ective anisotropy we use the fact that the magnetic anisotropy is proportional to the cube of the magnetization [31] for each sub-network and we introduce it in formula (21):

KG(¹)

KG(0)+

^M^M^GG⁽(0)^¹⁾

^. ⁽²²⁾

3.3. Application of the EAM to the Fe80ErVB20\Vand Fe74\VAlVEr6B20ribbons

To apply this model to the amorphous ribbons we consider that the sub-lattice anisotropy constants found in the formula (21) are as per the contents in the alloy. Thus for the transition metal we can write the anisotropy constant as K

2"

x2K

2 where K

2 is the anisotropy constant per ion for the ¹, we follow the same way with the rare-earth anisotropy constant. We take the rare earth's anisotropy per ion to be constant, equal to 2.5;10(J m\) which is the value evaluated for the Co

Er

amorphous thin"lm [13], and with the use of relation (21) we calculate the transition metal's anisotropy per ion at 4 K. The mean value of the magnetic anisotropy constant of Fe is of the order of 2.8;10 (J m\) for the Fe

Er VB alloy and of the order of 4.6;10(J m\) for\V Fe\VAl

VEr B

. We see that to reduce the"eld of the antiferromagnetic coupling break one has to increase the magnetic anisotropy per ion of the transition metal by addition of aluminium. We use these values to evaluate numerically the e!ective anisotropy variation versus temperature. As seen in Figs. 10 and 11, we reproduce the results obtained by the phenomenological model of Chudnovsky et al. (K"K

) and we con"rm the strange behaviour of this magnetic entity [32]. Thus, one can speculate that this result could be attributed to the increase in the anisotropy of Er atoms competing with those of the Fe atoms and, on the other hand, to the competition between anisotropy and exchange"elds. As the anisotropy is expected to increase with the sublattice magnetization the e!ect from Er atoms is expected to become more and more important at lower temperatures. After that we calculated the thermal variation of the ferromagnetic correlation length using the relation

Fig. 12. EAM results for the ferromagnetic correlation length in FeErVB\Vcompared to those of Chudnovsky et al. model.

Fig. 13. EAM results for the ferromagnetic correlation length in Fe\VAl

VEr B

compared to those of Chudnovsky et al.

model.

given by the phenomenological model cited in Sec- tion 3.1 and we found a minimum at the vicinity of the maximum observed in the anisotropy curves (Figs. 12 and 13). Then we con"rm that the behaviour of that entity is a function of the variables on which it depends.

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4. Conclusion

In conclusion, we have analysed magnetization curves of the amorphous alloys series in the frame- work of the random anisotropy model. The results show several features (exchange constant, local random anisotropy, ferromagnetic correlation length) all consistent with each other and in agreement with theoretical predictions. We have also evaluated a new model to study the magnetic anisotropy of ferrimagnetic and ferromagnetic systems, we dis- tinguished between weak and strong anisotropy systems and we obtained good results in agreement with previous works on intermetallic compounds.

Using this model we explained the strange behaviour of our alloy's magnetic anisotropy versus temperature. The anisotropy studies show that these alloys are ferrimagnets with wandering axes.

Acknowledgements

I am grateful to Professor Abdelillah Benyoussef for his encouragement and his help in performing the analytical part of this work.

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