DOI 10.1007/s10948-014-2542-z
REVIEW PAPER
Theoretical Work in Magnetocaloric Effect of LaFe 13 − x Si x Compounds
A. Boutahar·M. Phejar·V. Paul Boncour·L. Bessais· H. Lassri
Received: 17 February 2014 / Accepted: 11 March 2014
© Springer Science+Business Media New York 2014
Abstract In this paper, the magnetocaloric effect (MCE) of LaFe13−xSixcompounds with 1.2≤x≤2.2 has been inves- tigated. For this purpose, the magnetization dependence on the temperature and magnetic field were measured. Mag- netic entropy change(−SM)allowing estimation of the MCE was determined based on thermodynamic Maxwell’s relation. The experimental results show that TC increases with the Si content, whereas the magnetic entropy varia- tion decreases. A large magnetic entropy change has been observed. The maximum(−SmaxM )of LaFe10.8Si2.2occur- ring close to TC = 240 K is about 2.3 Jkg−1K−1 for an applied field change of 0–2 T. In addition, a magne- tocaloric effect of LaFe13−xSix compounds (x =1.2 and 1.6) has been also carried out using phenomenological model. Dependence of the magnetization on temperature variation for LaFe13−xSix compounds (x=1.2 and 1.6) in different applied magnetic fields was simulated. The values of maximum entropy change, full width at half maximum, and relative cooling power (RCP) for the LaFe11.8Si1.2
and LaFe11.4Si1.6compounds in different applied magnetic fields were calculated.
Keywords Magnetization·Magnetocaloric effect· Simulation and modeling
A. Boutahar ()·H. Lassri·
LPMMAT, Facult´e des Sciences Ain Chock, Universit´e Hassan II-Casablanca, BP 5366, Mˆaarif, Casablanca, Morocco e-mail: boutahar.fsac@gmail.com
M. Phejar·V. P. Boncour·L. Bessais
ICMPE-CMTR, UMR CNRS 7182, 2-8, rue H. Dunant, 94320, Thiais, Paris, France
1 Introduction
Extensive research about magnetic refrigeration has been carried out over the last years, since it has many advantages over gas refrigeration, such as higher efficiency, low noise, softer vibration, longer using time, and environment pro- tection [1,2]. Therefore, increasing attention has been paid to find magnetic refrigerants with a large magnetocaloric effect near room temperature.
MCE was first discovered by Warburg [3] in 1881. It is an intrinsic property of magnetic materials, so the application of this effect can be used in many magnetic energy con- version machines or magnetic refrigerators [4]. In the latter, the magnetocaloric effect (MCE) is the tendency of certain materials, such as paramagnetic salts or ferromagnetic sub- stances, to heat up when placed in a magnetic field and to cool down when removed from it.
Due to the low cost and their giant magnetocaloric effect, much interest has been carried on LaFe13−xSixcompounds with cubic NaZn13-type structure [5,6]. For a Si content in the range of x = 1.2 to 2.2, these compounds present important magnetocaloric properties with significant mag- netic entropy change values and a high relative cooling power (RCP) factor around 200 K which is caused by an itinerant electron metamagnetic (IEM) transition [7,8]. In order to increase the Curie temperature of these materials, many studies have been carried out by substituting another rare-earth or transition metal such as Co for La and Fe, respectively, or by the insertion of interstitial atoms like hydrogen or carbon [8].
Recently, the transition metal-based compounds attracted much attention after the discovery of giant MCE in MnFeP1−xAsx [9] and MnAs1−xSbx [10] compounds. In this paper, we shall report on the analysis of the magnetic and magnetocaloric properties of LaFe13−xSixcompounds,
and a theoretical work on magnetization versus temperature in 2-T magnetic field has been done. A phenomenological model is used for simulation of magnetization dependence on temperature variation to predict magnetocaloric proper- ties such as magnetic entropy change, heat capacity change, and relative cooling power.
2 Experimental
LaFe13−xSixcompounds with 1.2≤x ≤ 2.2 were synthe- sized by high-energy ball-milling and subsequent annealing by using LaSi as a precursor to prevent the oxidation of lan- thanum. The precursor was prepared by arc melting in a water-cooling copper hearth with an unconsumable tungsten electrode in a purified argon atmosphere. The sample was turned over and remelted five times to ensure good homo- geneity. The weight losses did not exceed 1 %. The starting materials were pure elements (≥99.9 %). To prevent any loss of lanthanum caused by the large amount of powder stuck on the vial wall and its possible oxidation, an excess of 20 % of La was systematically added. The samples were annealed for 30 min at 1,373 K. The composition and homo- geneity of the alloys were investigated by X-ray diffraction (XRD). The magnetic and magnetocaloric properties were determined using a physical property measurement system (PPMS) magnetometer from Quantum Design operating up to 9 T and from 2 to 350 K.
3 Results and Discussion
3.1 Structural Characterization
LaFe13−xSix compounds were crystallized with a cubic NaZn13-type structure. This structure presents three
20 30 40 50 60 70 80 90 100
0 1000 2000
Intensity (arb. units)
Measured (Iobs) Calculated (Icalc) Bragg Position I obs - I calc
-(Fe,Si) NaZn13
La2O3
Fig. 1 Example of fitting of the XRD patterns of LaFe13−xSix compounds by use of the Rietveld method. LaFe11.7Si1.3
Table 1 A unit cell parameter, the atomic position, and the Bragg reliability factor deduced from the Rietveld refinement
Refined compositions a ( ˚A) y96i z96i RB
LaFe11.8Si1.2 11.4695 0.1171 0.1791 2.0
LaFe11.7Si1.3 11.4676 0.1171 0.1792 3.2
LaFe11.5Si1.5 11.4643 0.1172 0.1792 3.5
LaFe10.8Si2.2 11.4635 0.1173 0.1791 2.5
nonequivalent crystallographic sites: 8a, 8b, and 96i. La atoms occupy the 8a sites, while Fe atoms occupy the 8b and 96i sites. In as-milled alloys, the resulting phases are α-(Fe,Si), LaFeSi, and a small amount of the NaZn13phase due to a nonequilibrium solidification behavior. A very short annealing treatment of 30 min at high temperature is sufficient to reduceα-(Fe,Si) and (LaFeSi) content. The distribution of Fe and Si atoms in LaFe13−xSix series was pointed out by XRD.
The structure refinement for the X-ray pattern was car- ried out using a FULLPROF computer code based on the Rietveld analysis which gives the unit cell parameter and the auto-coherent domain size in the assumption of a Thompson-Cox-Hastings line profile. Experimental and Rietveld refined X-ray diffraction patterns for LaFe11.7Si1.3
as an example is displayed in Fig.1. For all samples, the mean diffraction crystallite sizes are found less than 120 nm.
The results of the structure refinement performed for the LaFe13−xSixseries, forx=1.2, 1.3, 1.5, and 2.2, are listed in Table1. For the compound annealed at 1,273 K, a small amount of LaFeSi was also present in the samples. In the case of compounds annealed at temperature ≥1,373 K, a small amount ofα-(Fe,Si) (≤1.9 %) and La2O3 (≤2.0 %) phases was observed in addition to the main NaZn13phase.
0 4 8
80 120 160 200
M (emu/g)
µ0H (T)
1.6 1.8 2.0 2.2
LaFe13-xSix
Fig. 2 The magnetization versus magnetic field at 2 K for the LaFe13−xSixcompounds
Table 2 Results of the magnetic characterization of the LaFe13−xSixcompounds
Refined compositions TC(K) Order of transition TFWHM(K) −SmaxM (J kg−1K−1) RCP (J kg−1)
LaFe11.8Si1.2 192.5 First 9.86 11 106.4
LaFe11.6Si1.4 200.2 First 5.08 24.4 124
LaFe11.5Si1.5 201.0 First 7.4 17.3 128
LaFe11.4Si1.6 202.5 First 9.4 16.5 155
LaFe11.2Si1.8 216.5 First 21.84 6.5 142
LaFe11.0Si2.0 228.5 First 32.9 3.8 125
LaFe10.8Si2.2 240.0 Second 59.9 2.3 137.7
The Curie temperature is given for magnetic field of 0.01 and 0.2 T forx=1.2, 1.4, 1.5, 1.6, 1.8, 2.0, and 2.2, respectively. The full width at half maximum in the temperature dependence of the magnetic entropy change (−SMmax), the magnetic entropy change, and the relative cooling power are given for a field of 0–2 T
3.2 Magnetic Measurements
To study the effect of Si content on the magnetic properties, magnetization M versus an applied magnetic field at 2 K is measured, forx = 1.6,1.8, 2.0, and 2.2 and reported in Fig.2. It is found that the magnetic moment decreases with the addition of Si. This is probably due to the charge transfer from sp band of the metalloid to the 3d band of the transition metal and the 3d–5d hybridization effects between Fe and La atoms which increase when adding Si atoms.
The Curie temperature was determined by measuring the magnetization of LaFe13−xSixcompounds as a function of temperature. Table 2 shows the variation of the Curie temperature as a function of Si content. The transition tem- perature of the annealed ball-milled compounds increases from 192.5 to 240 K when x is shifted from 1.2 to 2.2.
The increase in Curie temperature in the La(Fe,Si)13
compound should be explained not only on the basis of
a geometrical effect involving unit cell volume evolu- tion, but also on the concept of preferential occupancy of nonmagnetic Si atoms at Fe sites. The increase in Curie temperature in La(Fe,Si)13 compound might be attributed to the reduction in the number of Fe–Fe antiferromagnetic pairs (dFe−Fe < 2.45 ˚A) due to a magnetic dilution effect [11,12]. However, a better understanding of the interaction mechanism regarding Curie temperature variation is pos- sible only if one involves electronic effects. These effects can be explained by considering that positive ferromagnetic interactions dominate in an almost full band, whereas a competition between positive and negative interactions may occur for partial filling. The Si substitution induces a filling of the Fe 3d band with the sp electrons that favors the direct 3d–3d interactions.
The isothermal magnetization curves of different temper- atures near the vicinity of the Curie temperature are shown in Fig. 3 for different concentrations. The M–H curves
Fig. 3 Magnetization versus applied magnetic field isotherms measured for LaFe13−xSix compounds
0 1 2 3 4 5
0 30 60 90 120 150
M (emu/g)
3 K
250 K
x = 1.5
M(emu/g)
x = 1.2
0 1 2 3 4 5
0 30 60 90 120 150
146 K
296 K 130 K
3 K
0 1 2 3 4 5
0 30 60 90 120 150
x = 2.2
340 K 169 K
3 K
µ0H (T)
160 200 240 0
10 20
160 200 240 280 180 240 300 x=1.2
x=1.5
-ΔS (J/kgK)
x=2.2
T (K)
Fig. 4 Temperature dependence of the magnetic entropy change under a magnetic field change of 0–2 T (open symbols) and 0–5 T (closed symbols) for LaFe13−xSixcompounds
below TC exhibit a characteristic ferromagnetic behavior;
with the temperature increasing aboveTC, there is a field- induced metamagnetic transition from the ferromagnetic to the paramagnetic state characterized by a sharp change in the magnetization within a narrow temperature range.
Based on the thermodynamically theory, the isothermal magnetic entropy changeSM associated with a magnetic field variation is given by
SM(T , H )H =
HF
HI
∂M(T , H )
∂T
H
dH. (1)
In practice, an alternative formula is usually used for numerical calculation:
SM(T , H )H =μ0
i
Mi+1−Mi
Ti+1−Ti Hi, (2) where μ0 is the vacuum permeability, and Mi and Mi+1
are the magnetization values measured at temperatures Ti
and Ti+1 in a field change Hi [13]. The accuracy of the calculated SM depends on the accuracy of the mea- surements of magnetic moment, temperature, and magnetic field. According to the literature, the estimated accuracy ofSM using this technique is about 20–30 % above the TC [14]. Nevertheless, the Maxwell relation (1) must be used carefully since the adjacent isotherms reflect the tem- perature dependence of the iso-field magnetization [15].
180 210
60 90 120 150
0 4 8 M (T) 12
M (emu/g)
T (K)
(a)
SM (J/kg K) SM(T)
µ0H=2T x=1.2
90 120 150
160 200 2400
10 20
M (emu/g) SM (J/kg K)
M (T) µ
0H=5T
x=1.2
(b)
SM (T)
T (K)
Fig. 5 Magnetization and magnetic entropy change in 2 T (a) and 5 T (b) magnetic field for the LaFe11.8Si1.2versus temperature. The dashed curves are modeled results, and symbols represent experimen- tal data
Figure4shows the temperature dependence of the magnetic entropy change measured for an applied magnetic field of 0–2 and 0–5 T for LaFe13−xSix (x = 1.2, 1.5, and 2.2) compounds. The maximum peak value of (−SM) for the LaFe11.8Si2.2 compound is 4.3 J kg−1 K under a magnetic field change of 0–5 T. The maximum value of (−SM) decreases with the Si content by increasing the Si con- tent from 1.4 to 2.2. This decrease of (−SM) is due to the change from the first-order transition toward the sec- ond order when increasing the Si content. This decrease of (−SM) versus the Curie temperature has also been observed in compounds prepared by arc melting [16].
On the other hand, we have calculated the RCP values for all samples using the following equation [17–19]:
RCP= −SMmax×δTFWHM, (3)
where (−SMmax) and δTFWHM are the maximum of the entropy variation and the full width at half maximum in
Table 3 Model parameters for LaFe13−xSix(x=1.2 and 1.6) in different applied magnetic fields
Sample μ0H (T) Mi(emu/g) Mf(emu/g) TC(K) B (emu/g K) SC(emu/g K)
LaFe11.8Si1.2 2 135.5 81.6 193 −0.435 −5.4
LaFe11.8Si1.2 5 135 91 205 −0.44 −3.71
LaFe11.4Si1.6 2 110 26 207 −0.51 −8.21
190 200 210 220 230 0
50 100 150
0 8 M(T) 16
µ0H=2T x=1.6
M (emu/g)
T (K)
SM (J/kg K) SM(T)
Fig. 6 Magnetization and magnetic entropy change in 2-T magnetic field for the LaFe11.4Si1.6versus temperature. The dashed curves are modeled results, and symbols represent experimental data
the temperature dependence of the magnetic entropy change (−SM).
The value of the saturation magnetization was improved by decreasing the Si content (Table 2). This leads to an increase in the magnetic entropy variation when the amount of Si is shifted from 2.2 to 1.2. For the intermetallic compound LaFe10.8Si2.2, the RCP values are 137.7 and 339.4 J kg−1 in magnetic field change of 0–2 and 0–5 T, respectively. The RCP increases when the magnetic field is increased.
3.3 Theoretical Considerations
According to phenomenological model in Refs. [20,21], the dependence of magnetization on variation of temperature and Curie temperatureTCis presented by
M(T )=
Mi−Mf
2
[Tanh(A(T −TC))]+BT +C, (4)
where Mi is an initial value of magnetization at ferromagnetic-paramagnetic transition, and Mf is a final value of magnetization at ferromagnetic-paramagnetic tran- sition, whereA = 2M(B−Si−MCf), B is magnetization sensitivity
dM
dT at ferromagnetic state before transition, SCis magne- tization sensitivity dMdT at Curie temperature TC, andC = Mi+Mf
2
−BTC.
A magnetic entropy change of a magnetic system under adiabatic magnetic field variation from 0 to final value Hmax
is available by SM=
−A
Mi−Mf
2
sech2(A (TC−T ))+B
Hmax. (5) The foundation of large magnetic entropy change is attributed to high magnetic moment and rapid change of magnetization at TC. A result of (2) is a maximum mag- netic entropy changeSmax (whereT = TC)that can be evaluated as the following equation:
Smax=
−A
Mi−Mf
2
+B
Hmax. (6) Equation (3) is an important equation in taking into consideration the value of the magnetic entropy change to evaluate the magnetic cooling efficiency with its full width at half maximum.
A determination of full width at half maximumδTFWHM can be carried out as follows:
δTFWHM= 2
Acosh−1 2A (Mi−Mf) A (Mi−Mf)+2B
. (7) This equation gives a full-width at half-maximum mag- netic entropy change contributing for the estimation of magnetic cooling efficiency as follows.
A magnetic cooling efficiency is estimated by consider- ing the magnitude of magnetic entropy change,SM, and its full width at half maximum δTFWHM [22]. A product of(−Smax)andδTFWHMis called relative cooling power based on magnetic entropy change.
RCP = −SM×δTFWHM
=
Mi−Mf−2B A
Hmax
×cosh−1 2A (Mi−Mf) A (Mi−Mf)+2B
. (8)
From this phenomenological model, it can easily access to the values of δTFWHM, |SMax|, and RCP for the LaFe13−xSix alloys (x=1.2 and 1.6) under magnetic field variation.
Table 4 The predicted values of applied magnetocaloric properties LaFe13−xSix (x=1.2 and 1.6) in different applied magnetic fields
Sample μ0H (T) −Smax(J kg−1K−1) δTFWHM(K) RCP (J kg−1)
LaFe11.8Si1.2 2 10.87 10.53 114.46
LaFe11.8Si1.2 5 18.2 14.09 256.43
LaFe11.4Si1.6 2 18.2 8.5 154.7
3.4 Theoretical Work
Numerical calculation was made with parameters as dis- played in Table3. These parameters were determined from the experimental data. Figures 5 and 6 show magnetiza- tion versus temperature in the 2- and 5-T magnetic field of LaFe13−xSix compounds (x = 1.2 and 1.6), respectively.
The symbols represent experimental data, while the dashed curves represent modeled data using the model parameters given in Table3. It is seen that the results of calculation are in a good agreement with the experimental results. Figures5 and 6show predicted values for the change of magnetic entropy. The values of maximum magnetic entropy change, full width at half maximum, and relative cooling power for different concentrations (x = 1.2 and 1.6) in different applied magnetic fields were calculated by using (3)–(5), respectively, and tabulated in Table 4. The peak values of (−SM) are 10.87 and 18.2 J kg−1 K−1 at 2 T for LaFe13−xSix compounds (x = 1.2 and 1.6), respectively.
These values are in good agreement with the experimental results.
4 Conclusion
In conclusion, we have studied the MCE of LaFe13−xSix compounds with 1.2 ≤ x ≤ 2.2. The magnetic entropy changes were obtained from isothermal magnetization curves. This indirect technique is fast and gives crucial information for determining the quality of a magnetic refrig- erator material. The results show that this series exhibits interesting levels of the MCE. Under a magnetic field change of 0–2 T, the maximum of the entropy variation is about 2.3 J kg−1K−1at T=243 K, and the relative cool- ing power is estimated to be more than 137.7 J kg−1 for LaFe10.8Si2.2. Nevertheless, the Curie temperature can be tuned easily by changing the Si concentration. On the other hand, dependence of the magnetization on temperature vari- ation for LaFe13−xSix compounds (x = 1.2 and 1.6) in
different applied magnetic fields was simulated. The model gives a good fit of the experimental M(T). Besides, we have also determined some fundamental parameters such as the values of maximum magnetic entropy change, full width at half maximum, and relative cooling power.
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