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Determination of the magnetocaloric effect from thermophysical parameters and their relationships near magnetic phase transition in doped manganites

We present the results of a comparative analysis of the magnetocaloric effect (MCE) in Pr0.7Sr0.2Ca0.1MnO3, through direct and indirect measurements, using experimentally measured magnetization, specific heat, magnetostriction, resistivity, thermal diffusivity and thermal conductivity parameters. We have demonstrated that the change in each parameter in response to a magnetic field near the ferromagnetic-paramagnetic phase transition temperature of the material correlates with the change in magnetic entropy. These findings allow us to interrelate these parameters and provide an alternative, effective approach for accessing the usefulness of magnetocaloric materials.


Introduction
Doped manganites with a general formula RE 1-x AE x MnO 3 (RE = La, Pr, Nd; AE = Ca, Sr, Ba, etc.) represent strongly correlated electron systems in which electron, magnetic and lattice subsystems are interrelated [1][2][3][4][5]. This means that the parameters characterizing the state of one of the subsystems will correlate with those of the other two. Application of external stimuli, such magnetic field, electrical field and pressure, can alter the parameters describing these subsystems (e.g. resistivity, magnetization, specific heat, magnetostriction, etc.), which appear to occur strongly in the vicinity of the magnetic phase transition of the material [1,2,4]. An excellent example of this includes the colossal magnetoresistive (CMR) effect [6], the large magnetocaloric effect (MCE) [7], and the giant magnetostriction (GMS) effect [8], which even co-exist in a doped manganite system.
Recently, an interrelation of the electrical resistivity with the magnetic entropy near a magnetic phase transition has been discussed in doped manganites [9][10][11][12][13][14]. A simple linear relationship between the magnetic entropy change (S M ) and resistivity with a temperatureindependent proportionality coefficient has been established based on the s-d exchange models [15] and the double exchange model [16]. The temperature-dependent magnetotransport data were used to determine the magnitude of S M with good accuracy, as well as to simulate its temperature dependence around the magnetic phase transition temperature (T C ) [9][10][11][12][13]. However, a clear understanding of their association with other parameters, such as magnetostriction and strain, was missing. Our recent study has shown a direct link between the S M and the change in volume (the magnetostriction, ) of Pr 0.7 Sr 0.2 Ca 0.1 MnO 3 subject to a magnetic field; S M ~  [17].
From the aforementioned studies, it emerges that the change of a spin state in an external magnetic field near Т С not only leads to the appearance of the CMR, MCE, and GMS effects, but also alters electric and thermal transfer processes due to changes in the scattering 3 mechanisms. This appears to be related to the change in spin-phonon interaction affected by a magnetic field near a ferromagnetic-paramagnetic (FM-PM) transition. This thus suggests that changes in resistivity (), thermal conductivity (), and thermal diffusivity () in response to the applied magnetic field can be used to qualitatively establish the relationship between the parameters describing the magnetic subsystem and kinetic coefficients. A comparative analysis of these parameters will allow for an evaluation of contributions of different subsystems to the MCE [18]. As Pr 0.7 Sr 0.2 Ca 0.1 MnO 3 (PSCMO) undergoes a firstorder structural/magnetic transition and possesses the largest S M among Pr 0.7 Sr 0.3-x Ca x MnO 3 compositions investigated [19,20], it is an excellent model system for this purpose of research. Figure 1 shows the temperature dependence of the specific heat of PSCMO in the temperature range of 80 -310 K in magnetic fields of 0, 18, and 80 kOe. At room temperature, the specific heat of PSCMO is 106 J/mol K, which is 4.2R. This value agrees well with the high-temperature specific heat values reported previously for analogous manganites [21] and does not contradict with the Dulong -Petit law for a system containing five atoms, including three "light" oxygen atoms. The specific heat obeys the Debye concepts of the specific heat of solids, with exception of a pronounced anomaly at T C = 196 K, which is associated with a FM-PM phase transition. A sharp peak of the specific heat in a zero field can signal the nature of a first-order transition, but there are no clear signs of thermal hysteresis. In a magnetic field, the peak of the specific heat is suppressed and shifted toward higher temperatures; temperature shift of 17 and 44 K in magnetic fields of 18 and 80 kOe, respectively. This behavior is not characteristic for a second-order FM-PM transition but for a first-order FM-PM transition [22,23]. 4 The inset in Fig. 1 shows the temperature dependence of the magnetic contribution to the heat capacities ΔС р (Т)=C P -C Ph (C Phbackground contribution/lattice contribution) for PSCMO in magnetic fields of 0, 18, and 80 kOe. The magnitude of the specific heat jump in the FM-PM phase transition region is ΔС р ≈ 41 J/mol K, 19 J/mol K, and 5 J/mol K at H = 0, 18 and 80 kOe, respectively. The inset in Fig.1 also shows the temperature dependence of the magnetic entropy change associated with the disordering of the magnetic system during the FM-PM phase transition, determined using the formula:

Results and discussion
The value of ΔS' in a zero magnetic field is equal to 2.75 J/mol K, which is much smaller than the ideal values for Ising (ΔS' = Rln2 = 5.7 J/mol K) and Heisenberg (ΔS' = Rln4 = 11.52 J/mol K) systems [20]. The difference in the experimentally and theoretically ΔS' values can arise from an existence of two-phase magnetic state in the material, and therefore, the sample was not completely transformed into a magnetically ordered state, as also discussed in previous studies [21,[23][24][25]. In a magnetic field, the magnitude of ΔS' decreases, so the ratio ΔS'(H=0)/ΔS'(H=80 kOe) ≈ 3. We should pay attention to the temperature dependence of the entropy change at the magnetic phase transition in a zero magnetic field. Approaching to the transition, starting from T = 181 K up to 200 K (a temperature interval: 19 K), the magnitude of ΔS' increases very rapidly (jump wise) by 8.7 times, and from T = 200 K to T = 253 K (a temperature interval: 53 K) ΔS' changes by 1.5 times. In the second temperature regime, a little ΔS'(T) dependence trend is observed. As a rule, the presence of additional anomalies in the ΔS'(T) behavior indicates the presence of magnetic or structural inhomogeneities in doped manganites [26]. In this case, however, such an extended transition is likely related to the fact that we are dealing with two combined phase transitions: the structural phase transition (SPT) and the magnetic phase transition (MPT), in which one precedes the other (in this case, SPT + MPT). This is also observed in the ΔС р (Т) dependence. 5 Figure 2(a) shows the ƞ(T) dependence at H = 0, 18 and 80 kOe. Let us recall that the thermal diffusivity (ƞ) is a physical quantity characterizing the rate of establishment of equilibrium temperature distribution in thermal processes [27]. In materials with the prevailing phonon mechanism of thermal conductivity, ƞ is related to the phonon mean free path (l ph ) as = 1 3 ℎ , where is the speed of sound. Assuming that is weakly temperature-dependent [28] and taking into account that in doped manganites  ph >>  e [29], variation in ƞ(T) can be determined by l ph (T). As can be seen from Fig. 2(a) is the phonon scattering rate on spin fluctuations). This is the reason for the observed sharp decrease in thermal diffusivity near the T C . The application of an external magnetic field could suppress the phonon scattering related processes, leading to an increase of ƞ, which is equivalent to an increase of thermal conductivity ( Fig. 2 (b)). This can be visualized by constructing the ( − 0 ) 0 ⁄ = ∆ A similar picture can be drawn out for the magnetic field and temperature dependences of thermal conductivity (Fig. 2b). ⁄ . This is most likely due to the fact that the thermal diffusivity is more sensitive to the magnetostructural change than the thermal conductivity. temperatures [11][12][13][14]. In the present study, the MCE magnitude can also be estimated from the MR data, following the method [13,14], and the results of which are presented and discussed below.
The ∆S M of PSCMO has been estimated using both direct and indirect measurements. From Maxwell's relation 0 ( ⁄ ) = ( ⁄ ) [33], the ∆S M can be determined as With the adiabatic temperature change, ΔT ad , obtained from direct measurements and the experimentally measured specific heat C P (T, H), the ∆S M can be determined as From the temperature-dependent specific heat at H = 0 and H  0, the ∆S M can also be determined as ∆ = ∫ (( ( , 0 ) − ( , 1 ))⁄ ) , In addition to the aforementioned formulas for assessing ∆S M , there are other indirect methods. As was stated in the introduction, ∆S M can also be estimated from the resistivity or magnetoresistive data using the following expressions [12,13]: where parameters A and  determine the magnetic property of a substance (A  14-15 J/kg K [13,14],   21 emu/g [12], and and are the total and tunneling magnetoresistance, respectively. It is generally known that the occurrence of a first-order magnetic phase transition is often accompanied by a considerable change in the lattice volume, which in turn leads to the change in magnetization of the sample, thus contributing to the total entropy change [18]. It has been shown that for PSCMO, the ∆S M can be estimated from the magnetostriction data (in the case magnetostriction near the T C shows an isotropic behavior, i.e., the volume magnetostriction       ) using a simple ratio [17]: where the parameter is independent of the magnetic field and is 2⋅10 4 J/kg K for PSCMO.
∆ is the magnetostriction. 8 In one hand, the MCE is due to a change in the degree of ordering of the magnetic subsystem when the applied magnetic field is changed. On the other hand, if the material dissipates the heat carriers on the spin system, then the change in the degree of ordering of the magnetic subsystem will lead to changes in the thermal diffusivity (see Fig. 2) and the thermal conductivity. By considering the relation κ = dC P η (where κ is the thermal conductivity, d is the density), i.e. C P~1 /η, which relates the thermal diffusivity to the specific heat, we can qualitatively estimate the ∆S M of the system from the η(T, H) data using the following expression [20]: where B is a coefficient independent of temperature and magnetic field.
Paying attention to the /(T) dependence at H = 18 and 80 kOe (see inset in Fig. 2 (b)), one can see that temperature dependences of / and ∆S M [17] are quite identical in nature. This means that in doped manganites, the change in thermal conductivity in response to a magnetic field near T C can also be related to the change in magnetic entropy: /(T)~S M . Accordingly, another indirect method can be used for estimating the MCE in manganites using: where q  25.71 J/kg K is the proportionality coefficient independent of temperature and magnetic field. The values of ∆S M calculated using Eqns. (1)-(7) for a magnetic field change of 80 kOe are presented in Fig. 4. The inset in Fig. 4 shows an estimate of ∆ using Eqns.
We have used the following Eqn. (4a) to estimate ∆ using the resistivity data. As is known, in polycrystalline manganites, in addition to the classical MR effect (Δρ/ρ) sd , another TMR effect, based on the intergranular tunneling of the charge carriers through the grain 9 boundaries, may become significant, especially at low temperatures, thus contributing to the total magnetoresistance (Δρ/ρ) total . The use of Eqns. (4a)-(4b) for estimating the MCE from the resistivity and magnetoresistance data without taking these factors into account could be erroneous. Therefore, to separate the various contributions to the total MR, we will adhere to the ideas proposed in [13,14]. As for the TMR description, we will use the formula proposed in works [34,35]: where P is polarization, L(x) is the Langevin function ( ( ) = coth( ) − 1/ (x)), and b is the fitting parameters associated with the magnetic moment and the size of the granules, respectively. In Fig. 3, the dotted line denominates Eqn.  Fig. 4). The difference in the MCE maximum temperatures obtained from these two different estimation methods (see Table 1 and Fig. 4) can be related to the existence of magnetic inhomogeneities within the sample with a firstorder phase transition; the T C distribution of which may be of Gaussian or other nature [36]. (1)- (3)). With large magnetic fields applied, other mechanisms of heat scattering already dominate, changes in the magnetic field of which do not greatly affect the magnitude of ∆ .
As a result of the changes in the magnetic field-influenced scattering mechanisms, the ∆  ( ) dependence at T > T C (at H = 80 kOe) does not fit the standard behavior of ∆ ( ), but the ∆ ≈ 8 J/kg K is consistent with the data obtained from caloric measurements (see Table 1). In low magnetic fields (< 20 kOe), other indirect estimates of ∆ ( ) from Eqns. (4) -(7) also provide a good agreement with those obtained from the standard caloric measurements (see Table 1).