Scaling of the entropy change at the magnetoelastic transition in Gd

Differential scanning calorimetry under a magnetic field H has been used to measure the entropy change DS at the magnetoelastic transition in Gd 5(SixGe12x)4 alloys, for x<0.5. We show thatDS scales with the transition temperature, Tt , which is tuned byx andH, from 70 to 310 K. Such a scaling demonstrates that Tt is the relevant parameter in determining the giant magnetocaloric effect in these alloys, and proves that the magnetovolume effects due to H are of the same nature as the volume effects caused by substitution.

The magnetocaloric effect ͑MCE͒ has been studied for decades owing to its potential application to magnetic refrigerants. 1The MCE is the isothermal entropy change or the adiabatic temperature change arising from the application or removal of a magnetic field H on a system with magnetic degrees of freedom.Many efforts have been devoted to the analysis of the MCE both in the vicinity of second-order magnetic phase transitions, where Gd is the element that shows the largest effect close to room temperature, 1,2 and in order-disorder blocking processes, e.g., in molecular magnets. 3However, the MCE may be maximized in the vicinity of a first-order magnetoelastic phase transition, when the crystallographic transformation is field induced, resulting in an additional contribution to the entropy change 4,1 : a giant MCE has been discovered in the Gd 5 (Si x Ge 1Ϫx ) 4 compounds with xр0.5, [5][6][7] and recently in MnAs-based materials. 8,91][12][13] Two compositional ranges are of interest.For 0.24рxр0.5, the giant MCE is related to a firstorder magnetoelastic phase transition from a hightemperature paramagnetic ͑PM͒, monoclinic phase ( P112 1 /a) to a low-temperature ferromagnetic ͑FM͒, Gd 5 Si 4 -type orthorombic-I phase ( Pnma), at temperatures ranging from 130 K (xϭ0.24) to 276 K (xϭ0.5). 6,14The structural transition occurs by a shear mechanism 15 and yields a large volume contraction.The field-induced, reversible nature of the magnetostructural transition then results in strong magnetostriction 14 and giant ͑negative͒ magnetoresistance. 16For xр0.2, a second-order PM-toantiferromagnetic ͑AFM͒ transition occurs at T N ͑from ϳ125 K for xϭ0 to ϳ135 K for xϭ0.2). 6 Upon further cooling, a first-order AFM-FM transition takes place, whose temperature ranges linearly from about 20 K (xϭ0) to 120 K (xϭ0.2).MCE is related to such a first-order phase transition.The nature of the AFM phase is currently under discussion, 17 and the magnetic structure may correspond to that of either a canted ferrimagnet, as proposed for Nd 5 Ge 4 ͑Ref.18͒ or a canted antiferromagnet, as for the Ge-rich region of the Tb 5 (Si x Ge 1Ϫx ) 4 alloys. 19,20The AFM-FM tran-sition occurs simultaneously with a first-order structural transition from a high-temperature Gd 5 Ge 4 -type orthorombic-II phase to the low-temperature orthorombic-I phase, a large volume contraction also taking place.No structural anomalies are detected at the second-order PM-AFM transition. 17n the intermediate range 0.2ϽxϽ0.24,orthorombic-II and monoclinic structures coexist. 7ifferential scanning calorimetry ͑DSC͒ is the most suitable method to obtain the latent heat and entropy change at a first-order phase transition. 21In contrast, quasiadiabatic calorimetry, commonly used for the study of the MCE, 22 is designed to measure the heat capacity C p .However, at a firstorder phase transition, the experimental determination of C p is intrinsically uncertain due to the release of latent heat. 23SC under H is thus expected to be the ideal technique for the study of ⌬S at first-order magnetoelastic transitions.We have developed a high-sensitivity DSC with built-in H.In this paper, the calorimetric measurement of ⌬S as a function of T and H is reported for Gd 5 (Si x Ge 1Ϫx ) 4 alloys.A ⌬S scaling plot is obtained, where the scaling variable T t is the temperature of the first-order magnetoelastic phase transition.As T t is shifted with x and H, the scaling of ⌬S thus summarizes the giant MCE in the Gd 5 (Si x Ge 1Ϫx ) 4 alloys.We also unambiguously show that calorimetric values of ⌬S are in agreement with the indirect calculation obtained from the magnetization curves M (H), using the Clausius-Clapeyron equation. 10,24d 5 (Si x Ge 1Ϫx ) 4 alloys with xϭ0.1, 0.18, 0.2, 0.25, 0.3, 0.365, and 0.45 were prepared by arc melting admixtures of the pure elements in the desired stoichiometry under an argon atmosphere.As-cast buttons were cut into slices, and some were thermally treated for four hours at 950 °C under a 10 Ϫ5 -torr vacuum.The quality of the samples and their crystallographic structure were studied by room-temperature x-ray diffraction.ac susceptibility ͑77-300 K, ϭ111-3330 Hz, H ac ϭ1.25 Oe) was used to check that the temperatures of the first-and second-order phase transitions were in agreement with values in the literature. 6,7,17M (H) curves were recorded up to 230 kOe for xϭ0.18 and 0.45, both in increasing and decreasing H, from 4.2 to 310 K with a temperature step of 3 K.Calorimetric measurements were performed using two high-sensitivity differential scanning calorimeters, specifically designed to study solid-solid phase transitions.Heating and cooling runs were performed within 77-350 K for Hϭ0 in a LN 2 cryostat, and within 4.2-300 K under fields up to 50 kOe in a LHe cryostat.The calorimeter with built-in H furnishes a sensitivity ϳ10 times larger at room temperature than that for a conventional DSC, while a reasonably high value is obtained at low T. Neither the thermometry nor the heat flow sensors are affected by H. Data do not depend on the cooling/heating rate.The upper limit of the operating range is 5 K/min.A detailed description will be published elsewhere. 25he measured M (H) isotherms for xϭ0.45 were similar to those previously reported 5,10 for xϭ0.5.The fieldinduced, first-order nature of the magnetoelastic PM-FM transition was evident from the data.It is worth noting that application of a field of 230 kOe shifts T t by an amount of ϳ80 K. From the M (H) curves a transition field H t is defined at each T as the inflection point of the curve.The temperature dependence of H t is shown in Fig. 1, for measurements under increasing and decreasing H.A linear relation between H t and T t is obtained, which yields dT t /d( 0 H t ) ϭ4.5Ϯ0.2K/T.DSC data for xϭ0.18 ͑Fig.2͒ also reveal the first-order nature of the low-temperature AFM-FM transition and the second-order nature of the high-temperature PM-AFM transition.The first-order transition shows ͑i͒ a large peak in Q ˙/T ˙ϵdQ/dT, where Q ˙is the recorded heat flow and T ˙is the heating/cooling rate; ͑ii͒ a hysteresis of 2-3 K between cooling and heating; and ͑iii͒ a significant field dependence of T t , which is estimated as the temperature at the maximum of the peak.The calorimetric data enable us to confirm the linear relation between H t and T t , which is plotted in Fig. 1 for xϭ0.45, and yields dT t /d( 0 H t )ϭ4.8Ϯ0.1 K/T.A slight difference (ϳ5 K) in the determination of the zerofield transition temperature T t (Hϭ0) from M (H) and DSC curves is found, due to the use of different experimental devices.These values are in good agreement with those reported for xϭ0.45, dT t /d( 0 H t )ϭ4.5 K/T ͑Ref.14͒, and xϭ0.43, dT t /d( 0 H t )ϭ4.3 K/T. 6A linear H t (T t ) behavior is found for the other compositions, and the slopes are also consistent with published values. 26,17The second-order transition is observed in DSC as a small -type jump in the dQ/dT baseline ͑Fig.2, inset͒.Although DSC does not give the absolute value of C p , the extrapolation at T t of the baselines at temperatures above and below the first-order transition provides a good estimation of ⌬C p .It is found that ⌬C p is positive for the first-order AFM-FM transition for all compositions with xр0.2 ͓see Fig. 3͑a͒ for xϭ0.1], while a negative ⌬C p is obtained for the first-order PM-FM transition for 0.24рxр0.5 ͓see Fig. 3͑b͒

for xϭ0.3].
The absolute value of ⌬S as a function of T t is shown in Fig. 4. As T t corresponds to the transition temperature of the first-order phase transition for each x and H, this allows us to sweep T t from ϳ70 to ϳ310 K. ⌬S was calculated as follows: ͑i͒ At Hϭ0 (LN 2 cryostat͒ and up to 50 kOe (LHe cryostat͒, by numerical integration of (dQ/dT)/T throughout the first-order calorimetric peaks 27 ; and ͑ii͒ indirect ⌬S evaluation from the M (H) isotherms up to 230 kOe for x ϭ0.45 and 0.18, using the Clausius-Clapeyron equation at a constant pressure, ⌬SϭϪ⌬M (dH t /dT t ). 24,10⌬M is determined from the magnetization jump at the transition.⌬S from the Clausius-Clapeyron equation reported by Gigue `re et al. for xϭ0.5, and obtained up to 70 kOe ͑see Fig. 2 in Ref. 10͒, is also displayed in Fig. 4. As T t is tuned by both x and H, ͉⌬S͉ values scale with T t .This enables us to derive a scaling of ͉⌬S͉ for all T t , i.e., for all compositions with xр0.5.The values given in Ref. 10 also collapse onto this scaling plot.This shows that the relevant parameter in determining ͉⌬S͉ is T t .In addition, the scaling is not a trivial consequence of the scaling of both ⌬M and dH t /dT t , i.e., neither ⌬M nor dH t /dT t scale with T t , which gives further relevance to the scaling of ͉⌬S͉.Notice also that ͉⌬S͉ extrapolates to zero at T t ϭ0, as expected from the third law of thermodynamics.The scaling is a consequence of the firstorder nature of the transition: at a constant H, the Clausius-Clapeyron equation is written as ⌬Sϭ⌬V(dP t /dT t ), where ⌬V stands for the volume jump and P t for the transition pressure.Therefore, ⌬V and ⌬M are related as ⌬V/⌬M ϭ ϪdH t /dP t , and the scaling thus shows the equivalence of magnetovolume and substitution-related effects.
It is worth stressing that ⌬S obtained from DSC measurements and from the Clausius-Clapeyron equation are coincident within the experimental error, for xϭ0.45 and 0.5, and for xϭ0.18 in the temperature range where the AFM-FM transformation takes place.This shows that ⌬S obtained from the Clausius-Clapeyron equation provides an excellent evaluation of the total entropy change at the first-order magnetoelastic transition.A comparision of ⌬S to the entropy change obtained through the Maxwell relation [10][11][12][13] was discussed elsewhere. 28wo diferent trends are shown in Fig. 4. For 0.24рx р0.5, ͉⌬S͉ associated with the PM-FM transition monotonically decreases with T t , which is consistent with ͑i͒ ⌬C p Ͻ0 ͓Fig. 3 ͑b͔͒, as expected from the thermodynamic rela-tion d(⌬S)/dTϭ⌬C p /T; and ͑ii͒ ⌬M decreasing monotonically with T.Moreover, negative ⌬C p may also be estimated from Ref. 26.In contrast, for xр0.2, ͉⌬S͉ either decreases or increases, depending on T t .Due to the magnetoelastic coupling, the application of H shifts T t , so that it is possible to observe both the AFM-FM transition at T t and, at high enough H, a PM-FM transition, when T t (H)уT N .The latter transition is still first-order due to the crystallographic transformation and arises from the PM-AFM transition.For the AFM-FM transition, ͉⌬S͉ increases monotonically with T t , in agreement with ⌬C p Ͼ0 ͓Fig.3͑a͒ and Ref. 26͔.However, for the PM-FM transition, ͉⌬S͉ decreases with T t for x ϭ0.18 and xϭ0.2, in agreement with ͑i͒ ⌬C p Ͻ0 ͓similar to Fig. 3͑b͔͒ and ͑ii͒ ⌬M decreasing monotonically with T.
Consequently, ͉⌬S͉ is maximum for each composition at T t ϭT N , i.e., when, in the FM phase, the applied H is large enough to shift the first-order transition to overlap to the second-order transition at T N .Therefore, the largest value ͉⌬S͉ϭ5.58J/(mol K) occurs at T t Ϸ131 K ͓ϳ the highest value of T N , which corresponds to xϭ0.2 ͑Ref.6͔͒.All the foregoing suggests that ͉⌬S͉, and thus the MCE, will be maximum within the compositional range 0.2ϽxϽ0.24,where the different crystallographic and magnetic phases coexist, and the two branches of ͉⌬S͉ join ͑Fig.4͒.Finally, it seems that the slopes d(͉⌬S͉)/dT t for the PM-FM transition for xϭ0.18 and that for 0.24рxр0.5 are different.We argue that this is due to the high-temperature crystallographic phase being different ͑orthorombic-II and monoclinic phases, respectively, for xϭ0.18 and 0.24рxр0.5).
In conclusion, DSC under H has been used successfully to measure the entropy change at the first-order magnetoelastic phase transition for Gd 5 (Si x Ge 1Ϫx ) 4 , xр0.5.We have shown that the transition entropy change scales with T t and it is in good agreement with the indirect measurements through the Clausius-Clapeyron equation.This is relevant for an understanding of the thermodynamics of first-order magnetoelastic transitions.The scaling of ⌬S is a direct consequence of the fact that T t is tuned by x and H and it is thus expected to be universal for any material showing strong magnetoelastic effects, yielding a field-induced nature of the transition.⌬S is expected to ͑i͒ go to zero at zero temperature, ͑ii͒ tend asymptotically to zero at high temperature since the latent heat is finite, and ͑iii͒ display a maximum at that temperature for which both ⌬M is maximized and T t shows the minimum field dependence.The specific shape of ⌬S vs T t will depend on the details of the phase diagram, T t (x).Finally, the scaling of ⌬S proves that the magnetovolume effects due to H are of the same nature as the volume effects caused by substitution.
The financial support of the Spanish CICYT ͑MAT2000-0858 and MAT2001-3251͒ and Catalonian DURSI ͑2001SGR00066͒ are recognized.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.

FIG. 1 .
FIG. 1. Transition field H t plotted as a function of the transition temperature T t for xϭ0.45, obtained from M (H) ͑increasing and decreasing H) and DSC data ͑cooling and heating͒.Solid lines are linear fits to experimental data.

FIG. 4 .
FIG. 4. Scaling of ͉⌬S͉ at the first-order transition for the Gd 5 (Si x Ge 1Ϫx ) 4 alloys.A variety of applied fields and compositions are represented.Connected symbols correspond to values obtained from M (H).Solid and open diamonds are from Ref. 10. Symbols labeled/not labeled with an H correspond respectively to measurements with the LHe (under H)/LN 2 (Hϭ0) DSC.