J Supercond Nov Magn DOI 10.1007/s10948-016-3443-0
ORIGINAL PAPER
Theoretical Investigation of the Magnetocaloric Effect on Stochiometric and Deficient La0.7Ca0.3MnO3 Manganite at Room Temperature J. Makni-Chakroun1 · H. Omrani1 · W. Cheikhrouhou-Koubaa1,2 · M. Koubaa1 · A. Cheikhrouhou1
Received: 5 January 2016 / Accepted: 28 January 2016 © Springer Science+Business Media New York 2016
Abstract In this paper, the magnetic and magnetocaloric properties of La0.7 Ca0.3 MnO3 (LCMO), La0.69 0.01 Ca0.3 MnO3 (L1 ) and La0.7 Ca0.29 0.01 MnO3 (L2 ) systems near a second-order phase transition from a ferromagnetic to a paramagnetic state, have been studied, using a phenomenological model. Based on this model, we are getting the better fits to magnetic transition and we can predict the values of the magnetocaloric effect such as magnetic entropy change, full-width at half-maximum, relative cooling power, and magnetic specific heat change from the calculation of magnetization as a function of temperature under different external magnetic fields. The maximum magnetic entropy change (−S max M )shifts to higher values with one percent of both calcium and lanthanum deficiencies, while the relative cooling power (RCP) and the full-width at half-maximum are reduced. According to the master curve behavior for the temperature dependence of SM predicted for different maximum fields, this work has confirmed that the paramagnetic–ferromagnetic phase transition observed for our sample is of second order.
J. Makni-Chakroun
[email protected] H. Omrani
[email protected] 1
Materials Physics Laboratory, Faculty of Sciences of Sfax, Sfax University, B. P. 1171, 3000 Sfax, Tunisia
2
Digital Research Center of Sfax, Sfax Technopark, BP 275, 3021 Sakiet-ezzit, Tunisia
Keywords Phenomenological model · Magnetocaloric effect · Specific heat change · Universal curve
1 Introduction The magnetocaloric effect (MCE) is the working principle of the magnetic refrigeration technology, which offers a competitive alternative to conventional gaz compressionexpansion refrigeration systems due to its high energy efficiency and environment-friendly solution for replacing the current refrigerators using greenhouse gases that are harmful to the environment and contributing to global warming. In this context, the development of materials exhibiting massive magnetocaloric effect around room temperature at fairly low fields is the key to the realization of such technology for room temperature applications. Recently, La-based manganites with perovskite structure materials exhibiting the so-called magnetocaloric effect in the vicinity of room temperature have been of great interest, considering their possible practical applications [1–3]. Perovskite-like manganites La1−x Cax MnO3 exhibit a variety of physical properties depending on the Ca concentration x. The strong correlation among magnetic, electronic, orbital, and transport properties of manganites makes these systems particularly sensitive to external perturbations, such as temperature, application of magnetic field, or high pressure [4]. This paper is about the theoretical work on magnetization versus temperature in different magnetic fields for the La0.7 Ca0.3 MnO3 and its lacunars in both lanthanum and calcium with (1 %) manganites. It used phenomenological model for simulation of magnetization dependence on temperature variation to investigate magnetocaloric properties, such as magnetic entropy change, heat capacity change,
J Supercond Nov Magn
temperature change, refrigerant capacity, and relative cooling power.
2 Theoretical Considerations According to phenomenological model, described in [5], the dependence of magnetization on the variation of temperature and Curie temperature TC is presented by: Mi − Mf ) tanh [A(TC − T )] (1) 2 To reach a full agreement between the experimental and the theoretical results for the variation of magnetization versus temperature at different magnetic fields (µ0 H ) under adiabatic conditions, we made several attempts and analyses in the context of phenomenological modeling dependence of magnetization on temperature variation. Two more terms BT and C are added to explain the variation of magnetization of several types of magnetic materials. So, the dependence of magnetization versus temperature and TC (Curie temperature) is presented by: M(T , H = Hmax ) = (
M(T , H = Hmax ) = (
Mi − Mf ) tanh [A(TC −T )]+BT +C 2 (2)
where • • • • • • •
Hmax is the maximum external field. TC is the Curie temperature. (Mi /Mf ) is an initial/final value of magnetization at ferromagnetic–paramagnetic transition as shown in Fig. 1. B is magnetization sensitivity (dM/dT) at ferromagnetic state before transition. C) A = 2(B−S Mi −Mf SC is the magnetization sensitivity dM/dT at Curie temperature TC . (M +M ) C = i 2 f − BTC
Differentiating Eq. (2) gives the following: Mi − Mf dM = −A sech2 (A (TC − T )) + B dT 2
(3)
where sec h(x) = cosh1 (x) Figure 1 show the curve temperature dependence of magnetization in constant applied field was retraced by Eq. (2) which is determined by the physical mechanism that the magnetic moments can be increased by decreasing temperature. At temperatures well below Curie point, the electronic magnetic moments of a ferromagnetic specimen are essentially all lined up, when regarded on a microscopic scale.
Fig. 1 Dependence of magnetization as a function of temperature
The magnetic entropy change (SM ) can be obtained through the adiabatic change of temperature by the application of a magnetic field. In order to evaluate the MCE, they can be calculated the variation of SM with temperature, when the magnetic field is varied from 0 to µ0 Hmax using the following expression [6]: μ0 H2 ∂S dμ0 H (4) SB = ∂H T μ0 H1 From Maxwell’s thermodynamic relationship [6], ∂M ∂S = ∂T H ∂H T Equation (4) can be rewritten as follows: H2 ∂M dH S H = ∂T H H1
(5)
(6)
Numerical evaluation of magnetic entropy change can be carried out from Eq. (6) using isothermal magnetization measurements. In spite of magnetization measurements at small discrete field and temperature intervals, SH can be computed approximately from Eq. (6) by: Mi − Mi+1 |S H | = H (7) Ti − Ti+1 i
Thus, magnetic entropy changes associated with applied field variations can be calculated from Eq. (7). From Eqs. (3) and (6), SM can be rewritten as follows: Mi − Mf sech2 (A (TC − T )) + B dH 2 H1 H2 Mi − M f = −A sech2 (A (TC − T )) + B dH 2 H1 Mi − Mf = −A sech2 (A (TC − T )) + B H (8) 2
S H =
H2
−A
The finding of a large magnetic entropy change is accredited to high magnetic moment and rapid change of magnetization near TC . A result of Eq. (8), which is a maximum
J Supercond Nov Magn Table 1 Model parameters for La0.7 Ca0.3 MnO3 ,La0.69 0.01 Ca0.3 MnO3 , and La0.7 Ca0.29 0.01 MnO3 systems at several magnetic fields Compounds
μ0 H (T )
Mi (emu/g)
Mf (emu/g)
TC (K)
B(emu g−1 K−1 )
SC (emu g−1 K−1 )
LCMO L1 L2 LCMO L1 L2 LCMO L1 L2 LCMO L1 L2 LCMO L1 L2 LCMO L1 L2
0.5
67.83 70.44 66.03 67.04 68.93 65.17 65.28 63.57 62.13 63.63 58.77 58.29 62.60 56.27 55.29 60.24 55.45 53.67
4.16 1.55 5.96 4.95 3.06 6.82 6.71 8.42 9.86 8.36 13.16 13.70 9.39 15.72 16.65 11.75 16.54 18.32
151 191 190 157 196 195 168 205 203 177 213 211 185 220 217 193 226 224
−0.023 −0.008 −0.009 −0.037 −0.032 −0.024 −0.005 −0.088 −0.056 −0.067 −0.130 −0.093 −0.076 −0.151 −0.12 −0.094 −0.153 −0.13
−0.9462 −0.897 −1 −0.801 −0.921 −0.928 −0.756 −0.832 −0.787 −0.674 −0.739 −0.694 −0.602 −0.678 −0.622 −0.5546 −0.619 −0.571
1
2
3
4
5
magnetic entropy change Smax (where T = TC ), can be evaluated by the following equation [7]:
Mi − Mf S Max = −A + B H 2 Mi − Mf + B Hmax S max = −A 2
δT FWHM = T2∗ − T1∗ −1
ATC + cosh =
Mi −Mf Mi −Mf −A +B = −A sech2 (A (TC −T ∗ ))+B H 2 2
A Mi − Mf + 2B = 2A Mi − Mf sech2 (A (TC − T ∗ )) A M −Mf )+2B sech2 (A (TC − T ∗ )) = (2A i M −M ( i f) A(Mi −Mf )+2B sec h(A (TC − T ∗ )) = ± 2A(Mi −Mf ) 2A(Mi −Mf ) ∗ cosh (A (TC − T )) = ± A M −M +2B , ( ) i f 2A(Mi −Mf ) −1 ∗ A (TC − T ) = ± cosh A(Mi −Mf )+2B
2A(Mi −Mf ) −1 ATC ∓cosh A(Mi −Mf )+2B T∗ = A (10)
ATC − cosh − δT FWHM =
2A(Mi −Mf ) A(Mi −Mf )+2B
A
(9)
Equation (1) is important because the value of the magnetic entropy change allows evaluating the magnetic cooling efficiency. The relative cooling power is the negative value of the peak. The full-width at half-maximum magnetic entropy change and the full-width at half-maxima (δTFWHM) can be carried out as follows [6, 7]: A full-width at half-maximum δT FWHM can be deduced as follows. Put T = T ∗ in Eq. (8) when S M = S2Max ; taking Eq. (1) into account, it gives the following:
H 2
From Eq. (10), δT FWHM can be deduced as follows:
2 cosh−1 A
−1
2A(Mi −Mf ) A(Mi −Mf )+2B
A
2A Mi − Mf
A Mi − Mf + 2B
(11)
The most meaningful parameter that provides a measure of the effectiveness of magnetic materials for the applications in magnetic refrigeration is the relative cooling power (RCP) [8–10]. RCP is a measure of the quantity of heat transferred by the magnetic refrigerant in one ideal cycle and it is expressed as follows:
RCP = S M (T , Hmax )∗ δT FWHM = Mi −Mf − 2 B A 2A(Mi −Mf ) (12) ×Hmax ∗ sec h A(Mi −Mf )+2B The RCP corresponds to the amount of heat that can be transferred between the cold and hot parts of a refrigerator in one ideal thermodynamic cycle. This parameter allows an easy comparison of different magnetic materials for applications in magnetic refrigeration; therefore, larger RCP values lead to better magnetocaloric materials. Another figure of merit which is used to compare the magnetic refrigerant materials is the refrigerant capacity
J Supercond Nov Magn Fig. 2 Temperature dependence of magnetization for La0.7 Ca0.3 MnO3 (a), La0.69 0.01 Ca0.3 MnO3 (b), and La0.7 Ca0.29 0.01 MnO3 (c) compounds, in different applied magnetic field shifts. The solid curves are modeled results and symbols represent the experimental data
(RC). The RC can be determinate by numerically integrating the area under the SM (T ) curve using the temperatures at half-maximum of the peak as integration limits [11]. Here, RC value can be obtained as follows: TC −
δT FWHM 2
n=
where (a) is a constant and the (n) exponent depends on the magnetic state of the sample. It can be locally calculated as follows [14]:
S (T ) .dT = Hmax − (Mi − Mf ) TC +
δT FWHM 2
× tanh
AδT FWHM 2
+ BδT FWHM
(13)
The change of specific heat associated with a magnetic field variation from zero to μ0 H is given by [12]: C P ,H = T
δS M δT
(14)
According to this model, C P ,H can be rewritten as follows:
C P ,H = −T A2 Mi − Mf ×sech2 [A (TC −T )] tanh[A (TC −T )] Hmax (15) Numerous works pertaining to the field dependence of the magnetic entropy change (SM ) of manganites at the ferromagnetic–paramagnetic transition have been conducted. The field dependence of SM can be expressed as follows [13]: S M = a(μ0 H )n
(16)
dln |S M | d ln |μH H |
(17)
In recent years, Franco and co-workers suggested that the SM (T ) curves modeled with different maximum applied fields should collapse onto a single universal curve in the case of a second-order phase transition [15]. The construction of this phenomenological master curve necessitates the normalizing of all the SM (T , μ0 H ) curves with their respective peak entropy change S max M and to rescale the temperature axis as follows:
(TC − T )/(Tr1 − TC ) ; (TC − T )/(Tr2 − TC );
T ≤ TC T > TC
(18)
where θ is the rescaled temperature and Tr1 and Tr2 are the temperature values of the two reference points that have been selected as those corresponding to 0.5 S max M . From this phenomenological model, it can easily , δTFWHM , RCP, RC, and assess the values of S max M min C max P /C P for LCMO, L1 , and L2 under magnetic field variation. Moreover, the order of phase transition in the present systems will be clarified by utilizing this model.
J Supercond Nov Magn
3 Simulation In order to apply the proposed phenomenological model, numerical calculations were carried out with parameters as displayed in Table 1. Figure 2 shows the temperature dependence of magnetization M (T ) in different applied magnetic fields for LCMO, L1 , and L2 compounds. While the solid lines represent simulated data using model parameters given in Table 1, the symbols represent the experimental points. It is noteworthy to mention that there is a good agreement between the experimental and the calculated results. It is seen that for the given parameters, the results of calculation are in good agreement with the experimental data. The M(T ) curves reveal that the all samples (0.5 ≤ μ0 H ≤ 5T) exhibits a magnetic transition from ferromagnetic state to paramagnetic one, without any detected anomaly, when increasing temperature. In addition, it can be reported that the Curie temperature significantly increases with the increase of magnetic field. Figure 3 shows the predicted results of the magnetic entropy change |S M | data as a function of temperature at several magnetic applied fields. Furthermore, the magnetic entropy change depends on the applied magnetic field change. The position of the peak shifts to higher temperatures with increasing µ0 H , from 151, 191, and 190 K at (µ0 H ) = 0.1 T to 193, 226, and 224 K at (µ0 H ) = 5 T for LCMO, L1 , and L2 , respectively. On the other hand, the maximum magnetic entropy change S max M exhibits a linear rise with the increase in the field, as shown in Table 2. This
Fig. 3 Temperature dependence of the magnetic-entropy change for different magnetic field intervals for La0.7 Ca0.3 MnO3 (a), La0.69 0.01 Ca0.3 MnO3 (b), and La0.7 Ca0.29 0.01 MnO3 (c) samples
indicates a much larger entropy change to be expected at higher magnetic field, thus signifying that the effect of spin– lattice coupling is associated to the changes in the magnetic ordering process in the sample [12]. In addition to the magnitude of the SM , other important parameters are used to characterize the refrigerant efficiency of the material: the relative cooling power (RCP) and the refrigerant capacity (RC) defined in Eqs. (7) and (8). RCP and RC give an estimate of quantity of the heat transfer between the hot (Thot ) and cold (Tcold ) ends during one refrigeration cycle, and it is the area under the SM versus T curve between two temperatures (T = Thot -Tcold ) of the FWHM of the curve. The variation of the RCP and RC factors as a function of the applied magnetic field, for the LCMO, L1 , and L2 compounds, exhibits a linear increase, as shown in Fig. 4 and Table 2. It is to be noted that our compounds shows an interesting RCP associated with the ferromagnetic– paramagnetic transition as expected by the strong variation of magnetization around TC . Using Eq. (10), the specific heat change (CP ) of the sample versus temperature at various magnetic fields is shown in Fig. 5. In this figure, it can be seen that CP goes through an unexpected change of sign around TC with a positive value above TC and a negative one below TC . Since ∂M/∂T < 0, S M < 0 results, and accordingly, the total entropy decreases upon magnetization. Furthermore, C P ,H < 0 for T < TC and C P ,H > 0 for T > TC [16, 17]. The sum of the two parts is the magnetic contribution
J Supercond Nov Magn Table 2 Model parameters for La0.7 Ca0.3 MnO3 ,La0.69 0.01 Ca0.3 MnO3 , and La0.7 Ca0.29 0.01 MnO3 systems at different applied magnetic fields Compounds
μ0 H (T )
−1 −1 −S max M (J kg K )
δTFWHM (K)
RCP (J kg−1 )
RC (J kg−1 )
−1 −1 C max K ) p,H (J kg
−1 K−1 ) C min p,H (J kg
LCMO
0.5 1 2 3 4 5 0.5 1 2 3 4 5 0.5 1 2 3 4 5
0.47 0.88 1.51 2.01 2.42 2.77 0.50 0.91 1.67 2.22 2.69 3.09 0.49 0.92 1.58 2.08 2.48 2.86
62.80 65.71 77.34 87.92 98.57 109.23 59.19 67.09 73.60 74.98 86.56 95.96 53.40 59.19 63.47 74.32 82.94 92.40
29.77 57.93 117.05 177.59 238.63 303.33 30.08 61.38 123.15 166.51 233.50 296.54 26.68 55.00 100.32 154.89 206.45 265.16
23.68 46.51 93.82 141.60 190.77 239.71 24.18 49.51 96.65 134.14 183.56 232.60 21.47 43.70 81.40 122.79 162.30 204.88
1.76 3.22 5.02 6.30 7.15 7.54 2.39 4.13 7.13 8.78 10.29 10.37 2.66 4.69 7.02 8.72 9.77 10.66
−1.31 −2.33 −3.56 −4.46 −5.02 −5.24 −1.86 −3.20 −5.57 −7.00 −8.10 −8.02 −2.16 −3.70 −5.55 −6.93 −7.73 −8.46
L1
L2
to the total specific heat which affects the cooling or heating power of the magnetic refrigerator [18]. The values of maximum magnetic entropy change, fullwidth at half-maximum, relative cooling power, refrigerant Fig. 4 Field dependence of RCP and RC for La0.7 Ca0.3 MnO3 (a), La0.69 0.01 Ca0.3 MnO3 (b), and La0.7 Ca0.29 0.01 MnO3 (c) samples
capacity, and maximum/minimum values of the change of specific heat at different magnetic fields for LCMO, L1 , and L2 are calculated (Eqs. (5), (6), (7), (8), and (10)), respectively, and listed in Table 2.
J Supercond Nov Magn Fig. 5 Temperature dependence of CP under different magnetic field variations for La0.7 Ca0.3 MnO3 (a), La0.69 0.01 Ca0.3 MnO3 (b), and La0.7 Ca0.29 0.01 MnO3 (c) samples
Using Eq. (11), according to Oesterreicher et al. [19], the field dependence of the magnetic entropy change of materials with a second order phase transition can be expressed as follows: SM αH n . Fig. 6 Ln (S max M ) versus Ln (μ0 H ) plots for La0.7 Ca0.3 MnO3 (a), La0.69 0.01 Ca0.3 MnO3 (b), and La0.7 Ca0.29 0.01 MnO3 (c) compounds
where the exponent n can be calculated by: n = dln SM /dln H . To determine the exponent n, a linear plot of SM vs. H is constructed at the transition temperature of the peak of the magnetic entropy (Fig. 6). The values of n are found to be 0.76 ± 0.024, 0.79 ± 0.062, and 0.75 ± 0.025 for LCMO, L1 , and L2 , respectively. The
J Supercond Nov Magn Fig. 7 Normalized entropy change (S M (T μ0 H )/S max M ) versus rescaled temperature (θ ) for La0.7 Ca0.3 MnO3 (a), La0.69 0.01 Ca0.3 MnO3 (b), and La0.7 Ca0.29 0.01 MnO3 (c) compounds at different applied magnetic field change intervals
deviation observed in the n values, from that predicted by the mean field model (n = 2/3) [19, 20], confirms the critical behavior studied previously for our samples and the invalidity of the mean field model in the description of our materials near the Curie temperature for all samples. The deviation from the mean field behavior is due to the presence of local inhomogeneities in the vicinity of transition temperature [21]. This value is similar to those obtained for soft magnetic alloys, gadolinium (Gd), and other magnetic materials containing rare earth metals [22]. The normalized entropy change S M /S max as a M function of the rescaled temperature (θ) for the magnetic ordering transitions of LCMO, L1 , and L2 compound is shown in Fig. 7. It is evident that all normalized entropy change curves collapse into a single curve, which proves that the paramagnetic–ferromagnetic phase transition observed for our sample is of a second order [23].
that this phenomenological model is useful for the prediction of magnetocaloric properties. Building on this model, we can calculate the values of the magnetic entropy change, full-width at half-maximum, relative cooling power, and magnetic specific heat change from the data of magnetization as a function of temperature under different external magnetic fields. Magnetocaloric effect makes the sample promising for room-temperature magnetic cooling applications. According to the master curve behavior for the temperature dependence of SM predicted for different maximum fields, the present work has confirmed that the paramagnetic–ferromagnetic phase transition observed for our samples is of a second order. Finally, the field dependence of the magnetic entropy change analysis shows a power law dependence, S = a(μ0 H )n , with n(TC ) = (0.76, 0.79, and 0.75) for LCMO, L1 , and L2 , respectively.
Acknowledgments This work was supported by the Tunisian Ministry of Higher Education and Scientific Research.
4 Conclusion With the help of the phenomenological model, a detailed investigation of magnetic and magnetocaloric properties of the polycrystalline sample LCMO, L1 , and L2 synthesized by Sol-Gel method has been conducted. The variation of the magnetization versus temperature, under different external magnetic fields, reveals a ferromagnetic–paramagnetic transition at room temperature. The extracted data confirm
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