Change in entropy at a first-order magnetoelastic phase transition : Case study of Gd 5 „ SixGe 1 À x ... 4 giant magnetocaloric alloys

The change in entropy, DS, at the first-order magnetoelastic phase transition in Gd 5(SixGe12x)4 alloys for x<0.5 has been measured with a high-sensitivity differential scanning calorimeter with built-in magnetic field,H. Scaling ofDS is achieved by changing the transition temperature, Tt , with x and H from 70 to 310 K.Tt is thus the relevant parameter in determining the giant magnetocaloric effect in these alloys. The calorimetric determination of the change in entropy is also in agreement with the indirect calculation obtained from the magnetization curves measured up to 23 T using both the Clausius–Clapeyron equation and the Maxwell relation. A simple phenomenological model based on the magnetization curves accounts for these results. © 2003 American Institute of Physics. @DOI: 10.1063/1.1556274 #

͑Presented on 15 November 2002͒ The change in entropy, ⌬S, at the first-order magnetoelastic phase transition in Gd 5 (Si x Ge 1Ϫx ) 4 alloys for xр0.5 has been measured with a high-sensitivity differential scanning calorimeter with built-in magnetic field, H. Scaling of ⌬S is achieved by changing the transition temperature, T t , with x and H from 70 to 310 K. T t is thus the relevant parameter in determining the giant magnetocaloric effect in these alloys.The calorimetric determination of the change in entropy is also in agreement with the indirect calculation obtained from the magnetization curves measured up to 23 T using both the Clausius-Clapeyron equation and the Maxwell relation.A simple phenomenological model based on the magnetization curves accounts for these results.© 2003 American Institute of Physics.͓DOI: 10.1063/1.1556274͔ The magnetocaloric effect ͑MCE͒ may be defined as the adiabatic change in temperature or the isothermal change in entropy that arises from the application/removal of a magnetic field, H. Recently, a great deal of interest has been devoted to searching for systems that show first-order magnetoelastic phase transitions, since they are expected to display giant MCE.Among these materials, Gd 5 (Si x Ge 1Ϫx ) 4 ͑Refs. 1 and 2͒ and MnAs-based 3 alloys are the most promising.The aim of this article is to study the change in entropy in Gd 5 (Si x Ge 1Ϫx ) 4 alloys, which is a controvertial issue.The use of the Maxwell relation at the nonideal first-order transition 1,4 has been opposed to the use of the Clausius-Clapeyron equation. 5In order to clarify this controversy, in this article we discuss the origin of the difference between the change in entropy related to latent heat at the first-order transition ⌬S, and the total change in entropy due to variation of the field from H 1 to H 2 at a given T, ⌬S(H 1 →H 2 ,T).
The giant MCE in Gd 5 (Si x Ge 1Ϫx ) 4 originates from the first-order transition that appears in two compositional ranges.For 0.24рxр0.5, the transition occurs from a hightemperature paramagnetic ͑PM͒, monoclinic ͑M͒ phase to a low-temperature ferromagnetic ͑FM͒, Gd 5 Si 4 -type orthorombic (O -I) phase, at temperatures ranging from 130 (x ϭ0.24) to 276 K (xϭ0.5). 1,2For xр0.2, the transition takes place from a high-temperature antiferromagnetic ͑AFM͒, Gd 5 Ge 4 -type orthorombic (O -II) phase to the lowtemperature FM/O-I phase, whose temperature varies linearly from 20 (xϭ0) to 120 K (xϭ0.2). 1,2 A second-order PM-AFM transition occurs at T N ͑from ϳ125 K for xϭ0 to ϳ135 K for xϭ0.2) in the O -II phase.Differential scanning calorimetry ͑DSC͒ under H is the most suitable method by which to obtain the H dependence of latent heat and change in entropy at a first-order phase transition, since DSC measures the heat flow, in contrast to quasiadiabatic calorimetry, where determination of the heat capacity is uncertain due to the release of latent heat.In this article, DSC measurements of ⌬S as a function of T and H are reported for Gd 5 (Si x Ge 1Ϫx ) 4 alloys.Scaling of ⌬S was suggested, where the scaling variable, T t , is the temperature of the first-order magnetoelastic transition. 6New DSC data under H are given in order to confirm the scaling plot.We also show that DSC values of ⌬S are in agreement with the indirect values obtained from the magnetization curves M (H) using the Clausius-Clapeyron equation and the Maxwell relation. 6Both indirect methods for increasing and decreasing H are analyzed.
Gd 5 (Si x Ge 1Ϫx ) 4 alloys were prepared by arc melting under argon.As-cast buttons were cut into slices and some were annealed for 4 h at 950 °C under 10 Ϫ5 Torr vacuum.M (H) curves were recorded up to 230 kOe for xϭ0.18 and 0.45 from 4.2 to 310 K. Calorimetric data were recorded using a high-sensitivity DSC. 6Heating and cooling runs were performed in 4.2-300 K under fields up to 50 kOe.
The M (H) isotherms measured for xϭ0.45 and 0.18 exhibit the field-induced nature of the transition that spreads over a field range, ⌬H t , which is ϳ4 T for our sample x ϭ0.45.The transition field H t is defined at each T as the inflection point of the M (H) curve.A linear relation between H t and T is obtained for xϭ0.45, which yields ␣ ϵdT/d( 0 H t )ϭ4.5Ϯ0.2K/T.For xϭ0.18 two linear ranges are observed: ␣ϭ3.66Ϯ0.07K/T for Tр120 K and ␣ϭ2.28Ϯ0.02K/T for Tу120 K.
DSC data for xϭ0.18 ͑Fig.1͒ also reveal the first-order nature of the AFM-FM transition and the second-order nature of the PM-AFM transition.The first-order transition a͒ Electronic mail: xavier@ffn.ub.es shows a large peak in Q ˙/T ˙ϵdQ/dT (Q ˙is the recorded heat flow and T ˙is the heating/cooling rate͒ and significant field dependence of T t , which is estimated as the temperature at the maximum of the peak.DSC data confirm the linear relation between H and T t and yield ␣ϭ4.8Ϯ0.1 K/T for x ϭ0.45 and ␣ϭ3.64Ϯ0.05K/T for xϭ0.18, in agreement with values obtained from M (H).The second-order transition is observed as a small -type jump in the dQ/dT baseline.
The absolute value of ⌬S as a function of T t is shown in Fig. 2. Since T t corresponds to the transition temperature of the first-order transition for each x and H, this allows one to sweep T t from ϳ70 to ϳ310 K. ⌬S was calculated by numerical integration of (dQ/dT)/T throughout the first-order DSC peaks, and from the M (H) isotherms using the Clausius-Clapeyron equation ⌬SϭϪ⌬M (dH t /dT t ). 5,7M is determined from the jump in magnetization at the transition.⌬S for xϭ0.5 taken from Ref. 5 is also displayed.Because T t is tuned by both x and H, this enables one to derive a scaling of ͉⌬S͉ with T t for compositions xр0.5, thus proving the equivalence of magnetovolume and substitution-related effects.Three different trends are shown in Fig. 2. For 0.24рx р0.5, ͉⌬S͉ associated with the PM-FM transition monotonically decreases with T t , while, for xр0.2, ͉⌬S͉ either decreases or increases depending on T t .As H shifts T t , it is possible to observe both the AFM/O-II→FM/O-I transition at T t and, when the first-order transition overlaps the secondorder one at high enough H ͓T t (H)уT N ͔, a PM/O-II →FM/O-I transition.For that reason, xϭ0.18 has two different values for ␣, depending on T t .For the AFM-FM transition, ͉⌬S͉ increases monotonically with T t , while for the PM-FM transition, ͉⌬S͉ decreases with T t .Consequently, ͉⌬S͉ is maximum for each composition at T t ϭT N .The fact that T N slightly decreases with H and increases with x gives rise to different maxima ͑labeled in Fig. 2͒.
⌬S values obtained at each temperature from DSC and from the Clausius-Clapeyron equation are coincident within experimental error for xϭ0.45 and 0.5, and for xϭ0.18 in the temperature range where the AFM-FM transformation takes place ͑Fig.2͒.Deeper inside, Fig. 3͑a͒ shows these values of ⌬S upon heating and upon a decrease in H for x ϭ0.45 and 0.5 ͑scattered symbols͒, and also the change in entropy for xϭ0.45 ͑dashed lines͒ obtained from M (H) using the Maxwell relation, upon a decrease in H, ⌬S(H max →0,T)ϭ͐ H max 0 (‫ץ‬M/‫ץ‬T) H 0 dH.These curves are evaluated at different temperatures and for different maximum applied fields, H max .They display the typical plateau-like behavior previously reported, 1,5  Ϫ⌬H t /2, the first and the third integrals account for the change in entropy related to the H and T dependence of M in each phase.Only the second term gives the contribution to the change in entropy at the magnetoelastic transition.This is indicated by the fact that the plateau-like behavior of the solid lines in Fig. 3͑a͒, computed using only the second integral, matches the ⌬S vs T t curve.Note also that when 0 H max is less than 0 ⌬H t Ϸ4 T, which is the minimum field needed to complete the transition, the values of ⌬S(H max →0,T) are lower than the ⌬S values ͓see the curve corresponding to 0 H max ϭ2 T in Fig. 3͑a͔͒.Moreover, for H max у⌬H t , the plateau-like region extends over the temperature range in which H max уH b (T).Hence, as H b (T) increases with T, the abrupt decrease from the plateau-like region at higher T is due to truncation of the second integral at H max .
A phenomenological model is presented in order to compare the Maxwell and Clausius-Clapeyron approaches.The magnetization curves are considered to be of the form M (T,H)ϭM 0 ϩ⌬M F((TϪT t (H))/), 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 of →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 T t , the change in entropy is given analytically by ⌬S(H max →0)ϭ⌬S"F͕͓T ϪT t (H max )͔/͖ϪF͕͓TϪT t (Hϭ0)͔/͖…, where ⌬SϭϪ⌬M /␣ is the value in the Clausius-Clapeyron approach.In general, ⌬S(H max →0) is a fraction of ⌬S, which depends on the magnitude of the shift of T t with H, and reaches its maximum value ⌬S for high enough H.The results are even valid in the limit →0, for which ⌬S(H max →0)ϭ⌬S for all H max .A simple analytical picture is provided by assuming that F is a linear function of the temperature which extends in the temperature range ⌬T t ϭ␣⌬H t ϭ.The results are shown in Fig. 3͑b͒.The general trends compare very well with experimental results in Fig. 3͑a͒ obtained by integrating the Maxwell relation only within the transition range.It is observed that when H max is not high enough to complete the transition (H max Ͻ⌬H t ), then ⌬S(H max →0)ϭ(H max /⌬H t )⌬S is smaller than ⌬S, and (H max /⌬H t ) is the transformed fraction of the sample.
In summary, DSC under H was used successfully to measure the change in entropy at the first-order magnetoelastic phase transition for Gd 5 (Si x Ge 1Ϫx ) 4 ,xр0.5.The change in entropy at the transition scales with T t , since T t is tuned by x and H, and the scaling is thus expected to be universal for any material showing strong magnetoelastic effects.The scaling proves that the magnetovolume effects due to H are of the same nature as the volume effects caused by substitution.Calorimetric values of ⌬S match those from the Clausius-Clapeyron equation and the Maxwell relation provided the latter is evaluated only within the range of field in which the transition takes place, and the maximum H is high enough to complete the transition.The T and H dependences of M in each phase outside the transition region yield an additional change in entropy, also accounting for the giant MCE.
The Spanish CICYT ͑MAT2000-0858 and MAT2001-3251͒ and Catalan DURSI ͑2001SGR00066͒ are thanked.The Grenoble High Magnetic Field Laboratory ͑IHPP, European Union͒ is acknowledged.Two of the authors ͑F.C. and J.M.͒ acknowledge Departament d'Universitats, Recerca i Societat de la Informacio for Ph.D. grants.
which can be above or below the ⌬S vs T t curve depending on the value of H max .If we consider that the Maxwell relation has three contributions, ⌬S(H max →0,T)ϭ͐ H max H b (‫ץ‬M/‫ץ‬T) H 0 dHϩ͐ H b H a (‫ץ‬M/‫ץ‬T) H 0 dH ϩ͐ H a 0 (‫ץ‬M/‫ץ‬T) H 0 dH, with H b ϭH t ϩ⌬H t /2 and H a ϭH t