Critical behavior and its correlation with magnetocaloric effect in amorphous Fe80−xVxB12Si8 (x=8, 10 and 13.7) alloys

Critical behavior and its correlation with magnetocaloric effect in amorphous Fe 80 x V x B 12 Si 8 (x ¼ 8, 10 and 13.7) alloys

A. Boutahar ^a,n , H. Lassri ^a , E.K. Hlil ^b , D. Fruchart ^b

a

LPMMAT,Université Hassan II-Casablanca, Faculté des Sciences Ain Chock, BP 5366 Mâarif–Casablanca, Morocco

b

Institue Néel, CNRS et Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France

a r t i c l e i n f o

Article history:

Received 28 June 2015 Received in revised form 23 August 2015 Accepted 30 August 2015 Available online 2 September 2015 Keywords:

Amorphous alloys

Second order phase transition Critical behavior

Magnetocaloric effect

a b s t r a c t

The critical exponents in Fe

80x

V

x

B

12

Si

8

(x¼8, 10 and 13.7) amorphous alloys were investigated near ferromagnetic to paramagnetic phase transition temperature. All amorphous alloys exhibit a second order ferromagnetic to paramagnetic phase transition (SOMT). The critical exponents ( β ^, γ ^and δ ^{) were}

estimated using the modified Arrott plot technique (MAP), the Widom scaling relation (WSR), and the critical isotherm analysis (CIA). In addition, an independent analysis of the critical behavior is presented in terms of the magnetocaloric effect (MCE). It shows in accordance with conclusion from magnetization data analysis. The estimated critical exponent values are found to be consistent and comparable to those predicted by the mean field model. This result points out to the ferromagnetic exchange interaction of long-range type.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

Amorphous soft magnetic alloy have widely been the subject of intense investigation due to their excellent soft magnetic proper- ties [1 – 3]. In particular, Fe-based amorphous alloys were found to exhibit superior soft magnetic properties (such as lower coercivity, higher permeability, and lower core loss) [4 – 7]. Furthermore, Fe- bases amorphous alloys have been investigated not only for the variety of their physical properties, but also for their potential applications such as magnetic refrigeration (MR) technology which is based on the magnetocaloric effect (MCE) [8 – 11]. In particular, the details of the magnetic phase transition and critical behavior can be obtained through the MCE. Thus, the analysis of the critical behavior in the vicinity of the magnetic phase transi- tion is a powerful tool to investigate in details the mechanisms of the magnetic interaction responsible for the transition [12,13].

Firstly, the critical behavior was described with long-range mean fi eld theory [14]. Afterwards, Motome and Furulawa suggested that the critical behavior should be attributed to short-range Heisenberg model [15,16]. Currently, four different theoretical models such as mean fi eld, three-dimensional (3D) Heisenberg, three-dimensional (3D) Ising, and tricritical mean fi eld were used to investigate the critical properties in amorphous alloys. Recent study reported a correlation between the MCE and critical ex- ponents near the SOMT in magnetocaloric materials [17].

Moreover, Franco et al. [18,19] have introduced a new method to evaluate the critical exponents from the fi eld dependence of the magnetic entropy and the dependence of the exponent n with the critical exponents β ^, γ ^{, and} δ ^.

In this paper, we focus on the critical behavior of amorphous Fe

_80x

V

x

B

12

Si

8

(x ¼ 8, 10 and 13.7) ribbons, where the critical ex- ponents β ^, γ ^{, and} δ have been estimated reliably using different analytical methods. It is found that the values of critical exponents

β ^, γ ^{, and} δ ^{of Fe}

80x

V

x

B

12

Si

8

(x ¼ 8, 10 and 13.7) amorphous ribbons are close to the theoretical prediction of the mean- fi eld model indicating the existence of a long-range ferromagnetic interaction.

2. Experimental methods

Amorphous Fe

_80x

V

x

B

12

Si

8

ribbons (x ¼ 8, 10 and 13.7) was prepared by melt spinning method that details was described in our previous work [8]. The amorphous structure of the melt-spun Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/jmmm

Journal of Magnetism and Magnetic Materials

Fig. 1. X-ray diffraction pattern of the Fe

72

V

8

Si

8

B

12

amorphous ribbon.

http://dx.doi.org/10.1016/j.jmmm.2015.08.121 0304-8853/& 2015 Elsevier B.V. All rights reserved.

n

Corresponding author.

E-mail address: boutahar.fsac@gmail.com (A. Boutahar).

ribbons was con fi rmed by XRD. The magnetic properties were investigated by performing magnetization measurements using a vibrating sample magnetometer.

Fig. 1 shows an example of an XRD pattern of Fe

72

V

8

B

12

Si

8

rib- bon. The presence of wide, diffused maxima around 2 θ angle of 51 ° is typical for amorphous alloys based on iron indicating the mainly amorphous nature of Fe

_80x

V

x

B

12

Si

8

(x ¼ 8, 10 and 13.7) alloys.

Magnetization measurement versus temperature M(T) were recorded under applied magnetic fi eld of 0.5 T, and displayed in Fig. 2. The Curie temperature, de fi ned as the minimum of the dM/

dT curve, were estimated to T

_C

¼ 473.5, 435.5 and 335 K for x ¼ 8, 10 and 13.7, respectively. Plots M

¹

versus T, given in the inset of Fig. 2 indicates that, at high temperature, the linear fi t (absence of a downturn) to the data clearly reveals the absence of the Grif fi ths phase [20] in the paramagnetic region of Fe

_80x

V

x

B

12

Si

8

(x ¼ 8, 10 and 13.7) alloys. This result endorses the good quality of our alloys.

As known, the critical behaviors for a second order phase transition can be investigated in terms of a series of critical ex- ponents. Indeed, in the vicinity of a second-order phase transition, the divergence of correlation length ξ ¼ ξ

0

| (1 (T/T

C

)) |

^υ

leads to universal scaling laws for the spontaneous magnetization M

S

and initial susceptibility χ

0

. In this sense, the mathematic de fi nitions of exponents from magnetization can be expressed as [21 – 23]:

Fig. 3. Modified Arrott plots: isotherms of M

^1/β

vs. (H/M)

^1/γ

with (a) tricritical mean-field model, (b) 3

D

-Heisenberg model, (c) mean-field model and (d) 3D-Ising model for amorphous Fe

80x

V

x

Si

8

B

12

(x¼8, 10 and x¼13.7) alloys.

200 400 600 800

0 70 140

-3 -2 -1

0,0 0,8 1,6

dM /dT (em u /g K )

M ( e mu /g )

T (K)

x=8

x=10

(b)

x=13.7

1/M (g/emu)

Fig. 2. Field-cooled magnetization of the amorphous Fe

_80x

V

x

Si

8

B

12

(x¼8, 10 and

x¼13.7) alloys as a function of temperature and the (b) inset shows the corre-

sponding 1/M versus T.

-4 -2 0 2 4 6 3,0

3,6 4,2

ln (t)

ln (-t)

= 3.1

ln (M )

ln (H)

Experemental data Fitting data

x=8

-4 -3 -2 -1 0 1 2 3

2,5 3,0 3,5

ln (t)

ln (-t)

ln (H)

ln (M ) _x=10

= 3.2 Experemental data Fitting data

-2 -1 0 1 2 3

2,5 3,0 3,5 4,0

ln (t)

ln (-t)

ln (H)

ln (M )

x=13.7

Experemental data Fitting data

=3

Fig. 5. (a) and (b): ln–ln plot used to obtain the

final values ofβ

and

γ. (c) M vs. H on

a ln–ln scale of T¼T

C

for x¼8, 10 and x¼13.7 for amorphous Fe

80x

V

x

Si

8

B

12

alloys.

440 460 480 500

0 1 2 3

Tricritical Mean-field 3D-Ising

3D-Heisenberg Mean-field model

RS

T (K) x=8

400 420 440 460 480

0 1 2 3

T (K)

RS

x=10

Tricritical Mean-field 3D-Ising

3D-Heisenberg Mean-field model

320 340 360

0 1 2

RS

T (K)

x=13.7 Tricritical Mean-field 3D-Ising

3D-Heisenberg Mean-field model

Fig. 4. Relative slope (RS) as a function of temperature for amorphous Fe

80x

V

x

Si

8

B

12

(x¼8, 10 and x¼13.7) alloys.

M T

s

( ) = M

0

( − ) t

^β

, t < 0, T < T

C

, ( ) 1

T h M t / , t 0, T T

C

, 2

0 1

0 0

χ ( )

⁻

= ( )

^ν

> > ( )

M = DH

1/^δ

, t = 0, T = T

C

, ( ) 3 where t ¼ (T T

C

)/T

C

is the reduced temperature, h

0

/M

0

and D are critical amplitudes. The parameters β (associated with M

S

), γ ^(as-

sociated with χ

0

), and δ (associated with T

C

) are the critical ex- ponents. Our aim is to determine the critical parameters and the critical temperature from the magnetization data as a function of the magnetic fi eld at different temperatures.

The Modi fi ed Arrott Plots (MAP) of the Fe

₈₀x

V

x

B

12

Si

8

(x ¼ 8, 10 and 13.7) amorphous ribbons are displayed in Fig. 3. Obviously, they show that all lines are parallel to each other in the high magnetic fi eld region for all models. To select the appropriate line for critical exponent determinations, we calculate the relative slope (RS) de fi ned as RS ¼ S (T)/S (T

_C

) [where S(T) is the slope of M

^1/β

versus ( m

0

H/M)

^1/γ

]. Since the modi fi ed Arrott plots are a parallel lines in the high fi eld, the RS of the most satisfactory model should be close to 1 (unity) irrespective of temperatures [24]. Fig. 4 presents the RS versus temperature for the four models.

One can see that the values of RS are kept to 1 for mean- fi eld model for all amorphous alloys. Consequently, the mean- fi eld model is the best model for the determination of the critical ex- ponents in the present alloys.

The spontaneous magnetization M

S

(T) as well as the inverse of magnetic susceptibility χ

01

(T) were selected from the intersec- tions of the linear extrapolation line (for high-magnetic fi eld parts) with M

^1/β

and the (H/M)

^1/γ

axis, respectively. They are displayed for the three compositions in the inset of Fig. 5. Similarly, these plots are then fi tted with Eqs. (2) and (3), thus yielding values of β

and γ . These new critical exponent values are then used to con- struct new MAP.

According to Eq. (3), the value of critical exponent δ ^{can be}

determined directly from the critical isotherm M(T

C

, H) as seen in

Fig. 5. Thus, new values of the critical exponents for all con- centrations are determined and presented in Table 1.

Another way to estimate the critical component δ ^{, from} β ^and γ ^,

is possible by using the Widom scaling relation shown in Eq. (4):

1 ,

δ γ 4

= + β

( ) The deduced δ values are gathered in Table 1. As consequence, the Widom scaling relation con fi rms the reliability of the critical exponents deduced from the experimental data.

The δ ^, β ^, γ values estimated for the Fe

_80x

V

_x

B

₁₂

Si

₈

(x ¼ 8, 10 and 13.7) amorphous alloys are close to the mean fi eld values, thus indicating that long range interactions dominate the critical be- havior around T

C

. Also, it is stated that the competition between the Fe – Fe ferromagnetic interactions should be responsible for the critical behavior in these systems.

A more accurate method to determine the critical exponents n and δ is possible from MCE. According to the previous work es- tablished in Ref. [8], Fig. 6 shows the temperature dependence on the magnetic entropy change for different applied magnetic fi eld change intervals for amorphous Fe

_80x

V

x

B

12

Si

8

(x ¼ 8, 10 and 13.7) ribbons. The fi eld dependence of ( Δ ^S

Max

) is given by the fol- lowing equation [25,26]:

S H n d S

d H

Leading to ln

ln ,

Max n Max

5

0

Δ α Δ

− = ( − μ )

( ) ( )

Furthermore, exponent n is given by [25,26]:

n Tc 1 1

. 6

β β γ

( ) = + −

+ ( )

Since βδ ¼ ( β þ γ ), the relation (6) can be expressed as:

n Tc 1 1

1 1

7

δ β

( ) = + ( − )

( ) Using the values of β ^and δ obtained above, the values of n are calculated and presented in Table 1. The n values obtained from fi tting of Eq. (5) are shown in Fig. 7. They are in good agreement with those obtained from the Eq. (7), the critical exponents using

Composition Technique T

C

(K)

β γ δ n

Ref

Fe

72

V

8

B

12

Si

8

MAP 472 0.45 1.005

– –

This work

CIA 472

– –

3.1

–

WSR

– – –

3.2 0.62

ΔS^maxα

H

ⁿ

473.5

– – –

0.77

RCP

α

H

^1þ1/δ

473.5

– –

3.04

–

Fe

70

V

10

B

12

Si

8

MAP 434 0.46 1.03

– –

This work

CIA 434

– –

3.2

–

WSR

– – –

3.2 0.63

ΔS^maxα

H

ⁿ

435.5

– – –

0.73

RCP

α

H

^1þ1/δ

435.5

– –

3.08

–

Fe

66.3

V

13.7

B

12

Si

8

MAP 334 0.45 0.91

– –

This work

CIA 334

– –

3

–

WSR

– – –

3.02 0.60

ΔS^maxα

H

ⁿ

335

– – –

0.70

RCP

α

H

^1þ1/δ

335

– –

3.1

–

(Fe

0.9

Nb

0.1

)

76

B

24

Kouvel–Fisher 423.35 0.52 1.07

– –

[28]

CIA

– – –

3.27

–

WSR

– – –

3.05

–

Fe

67

Tb

33

MAP 424.9 0.5 1.0

–

[29]

WSR 424.9 0.40 1.00 3.065

– – –

Mean-field theory Theory

–

0.5 1 3 2/3 [30]

3D-Ising model Theory

–

0.325 1.24 4.82

–

[30]

3D-Heisenberg model Theory

–

0.365 1.336 4.8

–

[30]

Tricitical mean-field Theory

–

0.25 1 5

–

[31]

MAP and critical isotherms methods. The values of n are close than to the 2/3 for mean fi eld systems [25].

Also, the fi eld dependence of RCP is analyzed. It can be ex- pressed as a power law by taking into account the fi eld depen- dence of entropy change Δ ^S

Max

and reference temperature into consideration [26,27]:

RCP α H

^{1 1}+δ

, ( ) 8

where δ being the critical exponent of the magnetic transition.

Field dependence of RCP is illustrated in Fig. 7. The δ values cal- culated for our amorphous ribbons are listed in Table 1. There are in a good accordance with those obtained using the modi fi ed Ar- rott plot and Widom scaling relation.

4. Conclusion

The critical phenomena of amorphous Fe

68

V

12

Si

8

B

12

(x ¼ 8, 10 and 13.7) alloys has been adequately investigated. Reliable critical exponents β ^and γ are deduced from the modi fi ed Arrott plot

400 450 500

0 1 2 3

T (K)

- S M (J/K g K )

1T 2T 3T 4T 5T

x=8

300 350 400

0,0 0,8

1,6 x=13.7

- S M (J/K g K )

T (K)

1T 2T 3T 4T 5T

350 400 450 500

0,0 0,8 1,6 2,4

T (K) - S M (J/K g K )

1T 2T 3T 4T 5T

x=10

Fig. 6. Temperature dependence of the magnetic entropy change

ΔSM

for Fe

80x

V

x

Si

8

B

12

(x¼8, x¼10 and x¼13.7).

0,0 0,6 1,2

0,0 0,7 1,4

4 5 6 x=8

n= 0.77

= 3.04

ln (- S

Max

) ln (- S

Max

) ln (- S

Max

) ln ( RCP)

0,0 0,7 1,4

4,0 4,8 5,6

-0,5 0,0 0,5 1,0

x=10

= 3.08

ln (µ0H) ln (µ0H)

ln (µ0H)

ln ( RCP)

n= 0.76

0,0 0,7 1,4

-0,5 0,0 0,5

3 4 5 x=13.7

n= 0.70 = 3.1

ln (R C P )

Fig. 7. Field dependence of the

ΔSMmax

and the RCP calculated from the magnetic

data, with the calculated n and

δ

for both slops for amorphous Fe

80x

V

x

Si

8

B

12

(x¼8,

x¼10 and x¼13.7) alloys.

exponents was con fi rmed by the Widom scaling theory as well. It is found that the obtained critical exponent values are in ac- cordance with the theoretical prediction of the mean- fi eld model, thus indicating that long range interactions dominate the critical behavior in this system. The agreement between the values of the exponents β ^, γ ^, δ ^and fi eld exponent n obtained from results of the MAP, WSR, CIA methods and the MCE analysis supports the ap- plication of the MCE analysis to determine the critical exponents and the ( Δ ^S

Max

) values can be obtained by extrapolation to higher magnetic fi eld following the power law of − Δ S

Max

α H

ⁿ

.

Acknowledgements

This work is mainly supported by the PHC Maghreb 15MAG07.

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