Entropy change and magnetocaloric effect in Gd

Isothermal magnetization curves up to 23 T have been measured in Gd 5Si1.8Ge2.2. We show that the values of the entropy change at the first-order magnetostructural transition, obtained from the Clausius-Clapeyron equation and the Maxwell relation, are coincident, provided the Maxwell relation is evaluated only within the transition region and the maximum applied field is high enough to complete the transition. These values are also in agreement with the entropy change obtained from differential scanning calorimetry. We also show that a simple phenomenological model based on the temperature and field dependence of the magnetization accounts for these results.

The magnetocaloric effect ͑MCE͒ is the adiabatic temperature change that arises from the application or removal of a magnetic field.MCE is associated with the isothermal entropy change due to the field variation.Recently, a great deal of interest has been devoted to searching for systems showing first-order magnetostructural transitions with large entropy change, since they are expected to display giant MCE.Among these materials, Gd 5 (Si x Ge 1Ϫx ) 4 ͑Refs.1-5͒ and Mn-As-based 6,7 intermetallic alloys are the most promising candidates.
The correct evaluation of the entropy change related to the MCE is a controversial issue and has lately aroused much discussion. 1,8 -12For Gd 5 (Si x Ge 1Ϫx ) 4 , Gigue `re et al. 8 showed that the use of the Maxwell relation to calculate the entropy change overestimates ͑at least ϳ20%͒ the value obtained from the Clausius-Clapeyron equation that the authors 8,11 claimed to be the correct procedure due to the first-order nature of the transition in these alloys.According to them, the entropy change in the magnetostructural transition is not associated with the continuous change of the magnetization as a function of T and H, but rather with the discontinuous change in the magnetization due to the crystallographic transformation.They claimed that Maxwell relations do not hold since magnetization is not a continuous, derivable function in that case.In contrast, Gschneidner, Jr. et al. 9 argued that the Maxwell relation is applicable even in the occurrence of a first-order transition, except when this transition takes place at a fixed T and H, giving rise to a steplike change of the magnetization ͑ideal case͒.Besides, they claimed that Clausius-Clapeyron equation would imply an H-independent adiabatic temperature change, which however, is not consistent with the experimental observations. 8oreover, Sun et al. 10 showed that the entropy change calculated from the Maxwell relation is indeed equivalent to that given by the Clausius-Clapeyron equation, provided the magnetization M is considered T-independent in whichever phase the transition involves, and M is a step function with a finite jump at the transition temperature.They also suggested that the two procedures may yield different results, since the Clausius-Clapeyron method does not take into account the reduction of spin fluctuations by an applied field.Recent measurements using a differential scanning calorimeter under applied magnetic field have shown that the Clausius-Clapeyron equation leads, within the experimental error, to the correct values of the entropy change at the magnetostructural transition in Gd 5 (Si x Ge 1Ϫx ) 4 alloys. 13ere we present a detailed analysis of the different contributions to the entropy change arising from the application of a magnetic field, in order to account for the discrepancies previously discussed.For this purpose, magnetization isotherms on Gd 5 (Si x Ge 1Ϫx ) 4 were measured up to very high fields.The values of the entropy change obtained from Clausius-Clapeyron and Maxwell methods are compared and analyzed within the framework of a simple phenomenological model based on the temperature and field dependence of the magnetization.
Gd 5 Si 1.8 Ge 2.2 (xϭ0.45) was prepared by arc-melting admixtures of the pure elements in the desired stoichiometry under an argon atmosphere.The sample was placed in a water-cooled copper crucible.The weight losses after arcmelting were negligible.As-prepared button was thermally treated for 4 h at 950 °C under a vacuum of 10 Ϫ5 torr, in an electrical resistance furnace, by heating the sample in a quartz tube.After annealing, the quartz tube was quickly taken out of the furnace to room temperature.The quality of the sample and its crystallographic structure were studied by room-temperature x-ray diffraction ͑XRD͒.][4] The material displayed the expected room-temperature monoclinic structure ( P112 1 /a), with unit-cell parameters aϭ7.586(1)Å, bϭ14.809(1)Å, cϭ7.784(1)Å, and ␥ϭ93.14°͑1͒, in agreement with Refs. 3 and 4. Both XRD and ac susceptibility suggested the existence of minor amounts of a secondary orthorhombic phase ( Pnma) for the as-prepared sample.From the small anomaly appearing at Tϭ294.5Ϯ0.5 K in the ac data and the fitting of the unit-cell parameters, the secondary phase mostly corresponded to xϳ0.51Ϫ0.53. 3,4This secondary phase almost disappeared with annealing.The magnetization measurements were performed at the Grenoble High Magnetic Field Laboratory.M (H) curves were recorded up to 23 T, both in increasing and decreasing H, from 4.2 to 310 K with a temperature step of 3 K.
M (H) isotherms are shown in Fig. 1.These curves exhibit a jump ⌬M at the magnetostructural transition that spreads over a field range 0 ⌬H t ϳ4 T for most of the temperatures, increasing to ϳ5-6 T for temperatures above ϳ297 K.The transition field H t is defined as the field corresponding to the inflection point within the transition region.0 H t varies from 0 (Tϭ236 K) to 17 T (Tϭ307 K). ⌬M has been estimated as the difference in the magnetization at H t between the linear extrapolations of M (H) well above and below the transition region.A linear behavior of H t (T) with a slope ␣ϵdT/d( 0 H t )ϭ4.5Ϯ0.2K/T is found, which is in agreement with that obtained from calorimetric data. 13Since ⌬M also shows a linear dependence on T ͑decreasing with increasing temperature͒, it is deduced from the Clausius-Clapeyron equation ⌬SϭϪ⌬M d( 0 H t )/dT ϭϪ⌬M /␣ that the transition entropy change ⌬S must also vary linearly with T ͑see Fig. 2͒.
Figure 2 shows the entropy change for xϭ0.45 ͑dashed lines͒ obtained from the M (H) isotherms using the Maxwell method, ⌬S(0→H max ,T)ϭ͐ 0 H max 0 (‫ץ‬M /‫ץ‬T) H dH.These curves display the typical behavior previously reported. 1,8Note that the maximum value of the entropy change achieved using the Maxwell relation can be above or below ⌬S depending on H max .This can be understood by taking into account the fact that the Maxwell method includes the following contributions: with H a ϭH t Ϫ⌬H t /2 and H b ϭH t ϩ⌬H t /2.The first and the third integrals give the entropy change that arises from the field and temperature dependence of the magnetization in each phase.Only the second term accounts for the contribution to the entropy change of the magnetostructural transition.This is indicated by the fact that the plateaulike behavior of the solid lines in Fig. 2 ͓computed using the second integral in Eq. ͑1͔͒ perfectly matches the ⌬S values given by the Clausius-Clapeyron equation and by calorimetry.Note also that when H max is less than ⌬H t , which is the minimum field needed to complete the transition, the maximum value of ⌬S(0→H max ,T) is lower than ⌬S ͑see, for instance, the curve corresponding to 0 H max ϭ2 T in Fig. 2͒.Moreover, for H max у⌬H t , the plateaulike region extends over the temperature range for which H max уH b (T).Consequently, as H b (T) increases with T, the abrupt decrease from the plateaulike region at higher T is due to the truncation of the second integral at H max .
To account for the behavior described above, we propose a simple phenomenological model.The magnetization curves are considered to be of the form where M 0 and ⌬M are assumed to be T and H independent, and F(T) is a monotonously decreasing function of width such that F→1 for TӶT t (H) and F→0 for TӷT t (H).The case →0 corresponds to the ideal first-order transition (F is then the Heaviside function͒.Using the Maxwell relation and assuming a linear field dependence of the transition temperature, the entropy change is given by It is worth stressing that when the transition temperature is not field dependent, ⌬S(0→H max )ϭ0 irrespective of the value of ⌬S.In general, ⌬S(0→H max ) is a fraction of the transition entropy change ⌬S that depends on the magnitude of the shift of T t with the magnetic field, and reaches its maximum value, ⌬S, for high enough applied field.Results are even valid in the limit →0, for which ⌬S(0→H max ) ϭ⌬S for all H max .Pecharsky et al. 14 recently arrived at basically the same conclusion using a different approach.
A simple analytical picture is provided by assuming that F is a linear function of temperature which extends within the temperature range ⌬T t ϭ␣⌬H t ϭ. Results are shown in Fig. 3.The general trends compare very well with results in Fig. 2 obtained by integrating the Maxwell relation within the transition range ͓second term of Eq. ͑1͔͒.Note that within the scope of the present model, a true plateau is obtained since ⌬M has been assumed to be T independent, in contrast with the experimental results ͑Fig.1͒, where ⌬M decreases linearly with T. It is also observed that when H max is not high enough to complete the transition (H max Ͻ⌬H t ), then ⌬S(0→H max )ϭ(H max /⌬H t )⌬S is smaller than ⌬S.Accordingly, (H max /⌬H t ) is the fraction of the sample that has been transformed.
In conclusion, the magnetocaloric effect arising from a field variation 0→H max can be properly evaluated through the entropy change obtained from the Maxwell method, even when an ideal first-order transition occurs.When the Maxwell relation is evaluated over the whole field range, the T and H dependences of the magnetization in each phase outside the transition region yield an entropy change larger than that of the transition.It has been shown that the Maxwell relation, the Clausius-Clapeyron equation, and the calorimetric measurements yield the entropy change of the first-order magnetostructural transition, provided ͑i͒ the Maxwell relation is evaluated only within the field range over which the transition takes place and ͑ii͒ the maximum applied field is high enough to complete the transition.The transition tem-perature must significantly shift with the applied field, in order to achieve a large MCE taking advantage of the entropy change associated to the first-order transition, as also suggested in Ref.
Figure 2 also shows the values of the entropy change at the transition, ⌬S, obtained from the Clausius-Clapeyron equation for x ϭ0.45 ͑present data͒ and xϭ0.5 ͑taken from Ref. 8͒, and differential scanning calorimetry ͑DSC͒ data.
14.The financial support of the Spanish CICYT ͑Grant Nos.MAT2000-0858 and MAT2001-3251͒ and Catalan DURSI ͑Grant No. 2001SGR00066͒ are acknowledged.The Grenoble High Magnetic Field Laboratory, through the Improving Human Potential Program of the European Community, is acknowledged.F.C. and J.M. acknowledge DURSI for Ph.D. grants.*Electronic address: xavier@ffn.ub.es