magneticnanomaterialsub.files.wordpress.com€¦ · Aquesta investigació ha estat nançada per la...

Universitat de BarcelonaDepartament de Física Fonamental

Magnetocaloric effect inGd5(SixGe1−x)4 alloys

Fèlix Casanova i Fernàndez

Barcelona, Desembre 2003

Magnetocaloric effect inGd5(SixGe1−x)4 alloys

Memòria de la tesi presentada perFèlix Casanova i Fernàndez

per optar al grau de Doctor en Ciències Físiques

Director de tesi: Dr. Xavier Batlle i Gelabert.

Programa de doctorat del Departament de Física FonamentalTècniques Instrumentals de la Física

i la Ciència de Materials

Bienni 1999-2001Universitat de Barcelona

Signat: Dr. Xavier Batlle i Gelabert

Aquesta investigació ha estat nançada per la Comisión Interministerial deCiencia y Tecnología (CICYT) a través del projecte MAT2000-0858 i per la Ge-neralitat de Catalunya mitjançant els Grups de Recerca Consolidats, 2001SGR-00066. El treball s'ha dut a terme gràcies a una beca RFI-FIAP concedida pelDepartament d'Universitats, Recerca i Societat de la Informació (DURSI) de laGeneralitat de Catalunya. El programa Improving Human Potential Programmede la Unió Europea ha nançat la recerca feta al Grenoble High Magnetic FieldLaboratory.

A la meva família

AgraïmentsAgrair en primer lloc al meu director de tesi, el Dr. Xavier Batlle, l'interès, la

dedicació i la paciència que ha tingut amb mi. També al Dr. Amílcar Labarta, perl'ajut i els coneixements que m'ha donat. Vull donar les gràcies al Jordi Marcos,al Dr. Lluís Mañosa i al Dr. Antoni Planes per l'excel·lent col·laboració que tanbons fruits ha donat, així com al Dr. Eduard Vives per l'ajut en l'anàlisi d'allaus.Als companys de grup pel seu recolzament i consells: l'Òscar Iglesias, la MontseGarcía del Muro, el Víctor F. Puntes i, sobretot, al Bart Jan Hattink, que m'haensenyat molts secrets del laboratori i de l'ordinador. Agrair a la gent dels ServeisCienticotècnics que m'ha ajudat en la caracterització de les mostres i, en espe-cial, al Dr. Josep Bassas de DRX per la seva inestimable i desinteressada ajuda.Als contactes del GHMFL de Grenoble: el Dr. Gérard Choteau, la Dra. Sophiede Brion i el Sébastien Diaz, per l'ajut i les setmanes que he compartit amb ells.També vull agrair als membres del tribunal per l'honor que m'han atorgat en ac-ceptar formar part d'aquest: els Drs. Javier Tejada, M. Ricardo Ibarra, José CarlosGómez Sal, Antoni Planes i Amílcar Labarta, així com els membres suplents.

Voldria estendre els agraïments als professors, alumnes i PAS del Departamentde Física Fonamental, així com a tots els companys d'altres departaments quem'han acompanyat en aquest camí, com el Jordi Sancho i el Sergi Udina, Mílio.També hi ha vida fora de la facultat, i per això m'agradaria fer menció a la collacastellera dels Arreplegats de la Zona Universitària, on hi he passat tots aquestsanys de vida universitària. Finalment, vull donar les gràcies a la meva família quesempre m'ha fet costat i m'ha permès arribar ns aquí. I, en especial, a la Sònia,que m'ha suportat i recolzat en tot moment.

ix

'But the Mirror will also show things unbidden, and those are often strangerand more protable than things which we wish to behold. What you will see, ifyou leave the Mirror free to work, I cannot tell. For it shows things that were, andthings that are, and things that yet may be. But which it is that he sees, even thewisest cannot always tell. Do you wish to look?'

'The Lord of the Rings', J. R. R. Tolkien

ForewordThe present work is aimed at studying the magnetocaloric effect (MCE) in

Gd5(SixGe1−x)4 series of alloys. The discovery of a giant MCE in these com-pounds has renewed the interest in magnetic refrigeration, which is an energy-efficient and environment friendly alternative to the conventional vapour-cycle re-frigeration. The study of the origin of the giant MCE in Gd5(SixGe1−x)4 -and thesearch for materials with similar MCE- has unveiled a lot of exciting properties,all them related to the rst-order eld-induced magnetostructural transition takingplace in these compounds. This Ph.D. Thesis is devoted to the understanding ofthe MCE and the magnetoelastic coupling in Gd5(SixGe1−x)4 alloys.

In Chapter 1, we provide a general introduction to the MCE and to some of thematerials showing this effect, while Chapter 2 reviews the most relevant propertiesof Gd5(SixGe1−x)4 alloys. After this state-of-the-art, Chapter 3 is devoted to thesample synthesis and annealing strategies used in the present work, as well as tothe characterisation of the samples using conventional experimental techniques.Chapter 4 describes in detail the experimental technique that has been speciallydeveloped for this Ph.D. Thesis: a differential scanning calorimeter (DSC) whichoperates under magnetic eld. Chapter 5 analyses the various contributions tothe magnetocaloric effect and entropy change in Gd5(SixGe1−x)4, by using differ-ent indirect measurements and our DSC. A scaling of the entropy change withthe transition temperature for all Gd5(SixGe1−x)4 alloys is presented in Chapter 6,while the actual origin of this scaling and of the giant MCE in this system -themagnetoelastic coupling- is studied in Chapter 7. Chapter 8 is aimed at the com-plex magnetic behaviour of the Ge-rich compounds, which show a short-rangeantiferromagnetic phase that had not been previously reported in literature. Fi-nally, the study of the dynamics of the rst-order transition is detailed in Chapter9 by analysing the behaviour of the avalanches occurring when the samples arecycled through the rst-order transition.

xi

Contents

Foreword xi

1 The magnetocaloric effect 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Mesurement of the magnetocaloric effect . . . . . . . . . . . . . . 6

1.3.1 Direct measurements . . . . . . . . . . . . . . . . . . . . 61.3.2 Indirect measurements . . . . . . . . . . . . . . . . . . . 7

1.4 Magnetocaloric effect in paramagnets . . . . . . . . . . . . . . . 91.5 MCE in order-disorder magnetic phase transitions . . . . . . . . . 11

1.5.1 MCE in the low-temperature range (∼10-80 K) . . . . . . 111.5.2 MCE in the intermediate-temperature range (∼80-250 K) . 131.5.3 MCE near room temperature . . . . . . . . . . . . . . . . 13

1.6 MCE in rst-order magnetic phase transitions and the giant effect . 141.7 MCE at very low temperature: frustrated magnets and high-spin

molecular magnets . . . . . . . . . . . . . . . . . . . . . . . . . 181.8 Magnetic refrigeration . . . . . . . . . . . . . . . . . . . . . . . 20Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Gd5(SixGe1−x)4 series of alloys 292.1 Discovery of the Gd5(SixGe1−x)4 system and the giant MCE . . . . 292.2 Phase diagram of Gd5(SixGe1−x)4 . . . . . . . . . . . . . . . . . . 312.3 Microstructure and atomic bonds . . . . . . . . . . . . . . . . . . 332.4 Structural and magnetic properties . . . . . . . . . . . . . . . . . 35

2.4.1 The x=0 case . . . . . . . . . . . . . . . . . . . . . . . . 392.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 Giant magnetorresistance . . . . . . . . . . . . . . . . . . 432.5.2 Colossal magnetostriction . . . . . . . . . . . . . . . . . 45

2.6 Characterisation of Gd5(SixGe1−x)4 alloys . . . . . . . . . . . . . 452.6.1 Electronic structure . . . . . . . . . . . . . . . . . . . . . 452.6.2 Structural characterisation . . . . . . . . . . . . . . . . . 46

xiii

CONTENTS

2.7 Evaluation of the MCE at a rst-order transition . . . . . . . . . . 47Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Experimental techniques 533.1 Sample synthesis and thermal treatment . . . . . . . . . . . . . . 53

3.1.1 Synthesis method: arc melting . . . . . . . . . . . . . . . 533.1.2 Heat treatment . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Sample characterisation . . . . . . . . . . . . . . . . . . . . . . . 613.2.1 Magnetisation . . . . . . . . . . . . . . . . . . . . . . . . 613.2.2 Ac susceptibility . . . . . . . . . . . . . . . . . . . . . . 653.2.3 Differential Scanning Calorimetry . . . . . . . . . . . . . 713.2.4 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . 743.2.5 Scanning Electron Microscopy (SEM) and Electron-beam

Microprobe . . . . . . . . . . . . . . . . . . . . . . . . . 823.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 Design and experimental set up of a Differential Scanning Calorime-ter with magnetic eld 914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . 924.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4 Results sweeping T . . . . . . . . . . . . . . . . . . . . . . . . . 984.5 Results sweeping H . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Entropy change at the rst-order magnetostructural transition inGd5(SixGe1−x)4 1055.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Magnetisation measurements . . . . . . . . . . . . . . . . . . . . 1055.3 DSC measurements . . . . . . . . . . . . . . . . . . . . . . . . . 1105.4 Evaluation of the entropy change . . . . . . . . . . . . . . . . . . 114

5.4.1 Magnetic and calorimetric evaluations . . . . . . . . . . . 1145.4.2 Use of the Maxwell relation within the transition region . 118

5.5 Phenomenological models for the entropy change . . . . . . . . . 1195.5.1 First phenomenological approach: M(T ) = const. . . . . . 1195.5.2 Advanced phenomenological approach: M(T ) , const. . . 121

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xiv

Contents

6 Scaling of the transition entropy change in Gd5(SixGe1−x)4 1276.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Calorimetric measurements . . . . . . . . . . . . . . . . . . . . . 1276.3 Scaling of the transition entropy change . . . . . . . . . . . . . . 1306.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 The magnetoelastic coupling in Gd5(SixGe1−x)4 1377.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.2 H − T diagram from magnetisation and DSC measurements . . . 1377.3 dHt/dTt and magnetoelastic coupling . . . . . . . . . . . . . . . 1397.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8 Short-range antiferromagnetism in Ge-rich Gd5(SixGe1−x)4 alloys 1458.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 1458.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9 Dynamics of the rst-order transition in Gd5(SixGe1−x)4: cycling andavalanches 1619.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.2 Comparison of the entropy change induced by temperature and by

eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.3 Cycling through the rst-order transition . . . . . . . . . . . . . . 1689.4 Avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1719.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Conclusions 177

List of publications 181

Resum 183Què és l'efecte magnetocalòric? . . . . . . . . . . . . . . . . . . . . . 183La sèrie d'aliatges Gd5(SixGe1−x)4 . . . . . . . . . . . . . . . . . . . . 189Tècniques experimentals . . . . . . . . . . . . . . . . . . . . . . . . . 192Disseny i muntatge d'un DSC sota camp magnètic . . . . . . . . . . . . 194Variació d'entropia a la transició de primer ordre en els Gd5(SixGe1−x)4 . 200Escalat de la variació d'entropia de la transició en els Gd5(SixGe1−x)4 . . 207L'acoblament magnetoelàstic en els Gd5(SixGe1−x)4 . . . . . . . . . . . 209

xv

CONTENTS

Antiferromagnetisme de curt abast en els Gd5(SixGe1−x)4 rics en Ge . . 211Dinàmica de la transició de primer ordre en els Gd5(SixGe1−x)4 . . . . . 215Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Bibliograa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

xvi

Chapter 1

The magnetocaloric effect

1.1 IntroductionThe magnetocaloric effect (MCE) is dened as the heating or cooling (i.e., thetemperature change) of a magnetic material due to the application of a magneticeld. This effect has been called adiabatic demagnetisation for years, though thisfenomenon is one practical application of the MCE in magnetic materials. Forexcellent reviews on the magnetocaloric effect, see references [1, 2].

The magnetocaloric effect was discovered in 1881, when Warburg observedit in iron [3]. The origin of the MCE was explained independently by Debye[4] and Giauque [5]. They also suggested the rst practical use of the MCE: theadiabatic demagnetisation, used to reach temperatures lower than that of liquidhelium, which had been the lowest achievable experimental temperature.

Nowadays, there is a great deal of interest in using the MCE as an alterna-tive technology for refrigeration, from room temperature to the temperatures ofhydrogen and helium liquefaction (∼20-4.2 K). The magnetic refrigeration offersthe prospect of an energy-efficient and environtment friendly alternative to thecommon vapour-cycle refrigeration technology in use today [6, 7].

1.2 Basic theoryIn order to explain the origin of the magnetocaloric effect, we use thermodynam-ics, which relates the magnetic variables (magnetisation and magnetic eld) to en-tropy and temperature. All magnetic materials intrinsically show MCE, althoughthe intensity of the effect depends on the properties of each material. The phys-ical origin of the MCE is the coupling of the magnetic sublattice to the appliedmagnetic eld, H, which changes the magnetic contribution to the entropy ofthe solid. The equivalence to the thermodynamics of a gas is evident (see Fig.

1

CHAPTER 1. THE MAGNETOCALORIC EFFECT

1.1): the isothermal compression of a gas (we apply pressure and the entropy de-creases) is analogous to the isothermal magnetisation of a paramagnet or a softferromagnet (we apply H and the magnetic entropy decreases), while the subse-quent adiabatic expansion of a gas (we lower pressure at constant entropy andtemperature decreases) is equivalent to adiabatic demagnetisation (we remove H,the total entropy remains constant and temperature decreases since the magneticentropy increases).

The value of the entropy of a ferromagnet (FM) at constant pressure dependson both H and temperature, T , whose contributions are the lattice (S lat) and elec-tronic (S el) entropies, as for any solid, and the magnetic entropy (S m),

S (T,H) = S m(T,H) + S lat(T ) + S el(T ) . (1.1)

Figure 1.2 shows a diagram of the entropy of a FM near its Curie temperature,TC, as a function of T . The total entropy is displayed for an applied external eld,H1, and for zero eld, H0. The magnetic part of the entropy is also shown for eachcase (H1 and H0).

Two relevant processes are shown in the diagram in order to understand thethermodynamics of the MCE:

(i) When the magnetic eld is applied adiabatically (i.e., the total entropyremains constant) in a reversible process, the magnetic entropy decreases, but asthe total entropy does not change, i.e.,

S (T0,H0) = S (T1,H1) , (1.2)

then, the temperature increases. This adiabatic temperature rise can be visualisedas the isentropic difference between the corresponding S (T,H) functions and it isa measurement of the MCE in the material,

∆Tad = T1 − T0 . (1.3)

(ii) When the magnetic eld is applied isothermally (T remains constant),the total entropy decreases due to the decrease in the magnetic contribution, andtherefore the entropy change in the process is dened as

∆S m = S (T0,H0) − S (T0,H1) . (1.4)

Both the adiabatic temperature change, ∆Tad, and the isothermal magneticentropy change, ∆S m, are characteristic values of the MCE. Both quantities arefunctions of the initial temperature, T0, and the magnetic eld variation ∆H =

H1 − H0.Therefore, it is straightforward to see that if rising the eld increases magnetic

order (i.e., decreases magnetic entropy), then ∆Tad(T,∆H) is positive and mag-netic solid heats up, while ∆S m(T,∆H) is negative. But if the eld is reduced,

2

1.2. Basic theory

T=const. ∆S ≠ 0

Isothermal process ⇒ ∆S ≠ 0 ( S2 > S1 )

H = 0

disorder order

H ≠ 0H

S2 S1

S=const.∆Tad ≠ 0

Adiabatic process ⇒ ∆Tad ≠ 0 ( T1 > T2 )

order disorder

H = 0H ≠ 0H

T1T2

Figure 1.1: Schematic picture that shows the two basic processes of the magne-tocaloric effect when a magnetic eld is applied or removed in a magnetic system:the isothermal process, which leads to an entropy change, and the adiabatic pro-cess, which yields a variation in temperature.

3


Figure 1.2: S − T diagram showing the MCE. Solid lines represent the total en-tropy in two different magnetic elds (H0 = 0 and H1 > 0), dotted line showsthe electronic and lattice contributions to the entropy (non-magnetic), and dashedlines show the magnetic entropy in the two elds. The horizontal arrow shows∆Tad and the vertical arrow shows ∆S m, when the magnetic eld is changed fromH0 to H1. Taken from Ref. [2].

4

1.2. Basic theory

magnetic order decreases and ∆Tad(T,−∆H) is thus negative, while ∆S m(T,−∆H)is positive, giving rise to a cooling of the magnetic solid.

The relation between H, the magnetisation of the material, M, and T , to theMCE values, ∆Tad(T,∆H) and ∆S m(T,∆H), is given by one of the Maxwell rela-tions [8], (

∂S (T,H)∂H

)

T=

(∂M(T,H)

∂T

)

H. (1.5)

Integrating Eq. 1.5 for an isothermal (and isobaric) process, we obtain

∆S m(T,∆H) =

∫ H2

H1

(∂M(T,H)

∂T

)

HdH . (1.6)

This equation indicates that the magnetic entropy change is proportional toboth the derivative of magnetisation with respect to temperature at constant eldand to the eld variation. Using the following thermodynamic relations [8]:

(∂T∂H

)

S= −

(∂S∂H

)

T

(∂T∂S

)

H(1.7)

CH = T(∂S∂T

)

H, (1.8)

where CH is the heat capacity at constant eld, and taking into account Eq. 1.5,the innitesimal adiabatic temperature change is given by

dT )ad = −(

TC(T,H)

)

H

(∂M(T,H)

∂T

)

HdH . (1.9)

After integrating this equation, we obtain other expresion that characterisesthe magnetocaloric effect,

∆Tad(T,∆H) = −∫ H2

H1

(T

C(T,H)

)

H

(∂M(T,H)

∂T

)

HdH . (1.10)

By analysing Eqs. 1.6 and 1.10, some information about the behaviour of theMCE in solids can be gained:

1. Magnetisation at constant eld in both paramagnets (PM) and simple FMsdecreases with increasing temperature, i.e., (∂M/∂T )H < 0. Hence ∆Tad(T,∆H)should be positive, while ∆S m(T,∆H) should be negative for positive eld changes,∆H > 0.

2. In FMs, the absolute value of the derivative of magnetisation with respect totemperature, |(∂M/∂T )H |, is maximum at TC, and therefore |∆S m(T,∆H)| shouldshow peak at T = TC.

5


3. Although it is not straightforward from Eq. 1.10 since the heat capacity atconstant eld shows an anomalous behaviour near TC, ∆Tad(T,∆H) in FMs showsa peak at the Curie temperature when ∆H tends to zero [9].

4. For the same |∆S m(T,∆H)| value, the ∆Tad(T,∆H) value will be larger athigher T and lower heat capacity.

5. In PMs, the ∆Tad(T,∆H) value is only signicant at temperatures close toabsolute zero, since |(∂M/∂T )H | is otherwise small. Only when the heat capacityis also very small (same order as |(∂M/∂T )H |), can a relevant ∆Tad(T,∆H) valuebe obtained, which also happens only close to absolute zero. If we are interestedin sizeable ∆Tad(T,∆H) values at higher temperatures, we thus need a solid thatorders spontaneously.

1.3 Mesurement of the magnetocaloric effect1.3.1 Direct measurementsDirect techniques to measure MCE always involve the measurement of the initial(T0) and nal (TF) temperatures of the sample, when the external magnetic eldis changed from an initial (H0) to a nal value (HF). Then the measurement of theadiabatic temperature change is simply given by

∆Tad(T0,HF − H0) = TF − T0 . (1.11)

Direct measurement techniques can be performed using contact and non-con-tact techniques, depending on whether the temperature sensor is directly con-nected to the sample or not.

To perform direct measurements of MCE, a rapid change of the magnetic eldis needed. Therefore, the measurements can be carried out either on immobilisedsamples by changing the eld [10] or by moving the sample in and out of a con-stant magnetic eld region [11]. Using immobilised samples and pulsed mag-netic elds, direct MCE measurements from 1 to 40 Tesla (T) have been reported.When electromagnets are used, the magnetic eld is usually reduced to less than2 T. When the sample or the magnet are moved, permanent or superconductingmagnets are usually employed, with a magnetic eld range of 0.1-10 T.

The accuracy of the direct experimental techniques depends on the errors inthermometry and in eld setting, the quality of thermal insulation of the sample,the possible modication of the reading of temperature sensor due to the appliedeld, etc. Considering all these effects, the accuracy is claimed to be within the5-10% range [2, 10, 11].

At this point, we must mention the new direct measurement of MCE associatedwith rst-order eld-induced magnetic phase transitions, that is presented in this

6

1.3. Mesurement of the magnetocaloric eect

thesis (see Chapter 4): a differential scanning calorimeter (DSC) operating underapplied magnetic eld that measures the enthalpy of transformation (i.e., the latentheat) when the transition is induced by eld. From the latent heat, the entropychange is obtained, being the rst direct measurement of MCE performed throughthe entropy change.

1.3.2 Indirect measurementsUnlike direct measurements, which usually only yield the adiabatic temperaturechange, indirect experiments allow the calculation of both ∆Tad(T,∆H) and ∆S m(T,∆H) in the case of heat capacity measurements, or just ∆S m(T,∆H) in the case ofmagnetisation measurements. In the latter case, magnetisation must be measuredas a function of T and H. This allows to obtain ∆S m(T,∆H) by numerical inte-gration of Eq. 1.6, and it is very useful as a rapid search for potential magneticrefrigerant materials [12]. The accuracy of ∆S m(T,∆H) calculated from magneti-sation data depends on the accuracy of the measurements of the magnetic moment,T and H. It is also affected by the fact that the exact differentials in Eq. 1.6 (dM,dH and dT ) are replaced by the measured variations (∆M, ∆T and ∆H). Takinginto account all these effects, the error in the value of ∆S m(T,∆H) lies within therange of 3-10% [2, 12].

The measurement of the heat capacity as a funcion of temperature in constantmagnetic elds and pressure, C(T )P,H, provides the most complete characterisa-tion of MCE in magnetic materials. The entropy of a solid can be calculated fromthe heat capacity as:

S (T )H=0 =

∫ T

0

C(T )P,H=0

T dT + S 0

S (T )H,0 =

∫ T

0

C(T )P,H

T dT + S 0,H , (1.12)

where S 0 and S 0,H are the zero temperature entropies. In a condensed systemS 0 = S 0,H [14]. Hence, if S (T )H is known, both ∆Tad(T,∆H) and ∆S m(T,∆H)can be obtained [15], see for example Fig. 1.3. However, this evaluation is notvalid if a rst-order transition takes place within the evaluated range, since thevalue of CP is not dened at a rst order transition (see Refs. [16, 17] and Chapter4). In this case, the entropy curves present a discontinuity, which corresponds tothe entropy change of the transition. The entropy discontinuity can be determinedfrom different experimental data, such as magnetisation or DSC, and then theresulting S (T )H functions can be corrected accordingly [16].

The accuracy in the measurements of MCE using heat capacity data dependscritically on the accuracy of C(T )P,H measurements and data processing, since

7


Figure 1.3: Heat capacity of Gd5(Si2Ge2) as a function of temperature under dif-ferent applied elds. The inset displays the total entropy as a function of tempera-ture at different elds, as determined from heat capacity. From these curves, ∆S mand ∆Tad are easily obtained. Taken from Ref. [13].

8

1.4. Magnetocaloric eect in paramagnets

both ∆Tad(T,∆H) and ∆S m(T,∆H) are small differences between two large values(temperatures and total entropies). The error in ∆S m(T,∆H), σ[∆S m(T,∆H)],calculated from heat capacity is given by the expression [15]

σ[∆S m(T,∆H)] = σ[S (T,H = 0)] + σ[S (T,H , 0)] , (1.13)

where σS (T,H = 0) and σS (T,H , 0) are the errors in the calculation of the zeroeld entropy and non-zero eld entropy, respectively. The error in the value of theadiabatic temperature change, σ[∆Tad(T,∆H)], is also proportional to the errorsin the entropy, but it is inversely proportional to the derivative of the entropy withrespect to temperature [15]:

σ[∆Tad(T,∆H)] =σ[S (T,H = 0)]( dS (T,H=0)

dT

) +σ[S (T,H , 0)]( dS (T,H,0)

dT

) . (1.14)

It is worth noting that Eqs. 1.13 and 1.14 yield the absolute error in MCEmeasurements and, therefore, the relative errors strongly increase for small MCEvalues (see Fig. 1.4). Assuming thus that the accuracy of the heat capacitymeasurements is not eld dependent, the relative error in both ∆Tad(T,∆H) and∆S m(T,∆H) is reduced for larger ∆H values.

1.4 Magnetocaloric effect in paramagnetsMCE in PMs was used as the rst practical application, the so-called adiabatic de-magnetisation. With this technique, ultra-low temperatures can be reached (mK-µK). In 1927, the pioneering work of Giauque and MacDougall [5, 18] showedthat using the paramagnetic salt Gd2(SO4)3·8H2O, T lower than 1 K could bereached. Later, MCE at low temperatures was studied in other PM salts, suchas ferric ammonium alum [Fe(NH4)(SO4)·2H2O] [19], chromic potassium alum[20] and cerous magnesium nitrate [21]. The problem for the practical applicationof adiabatic demagnetisation using PM salts lies in its low thermal conductivity.Hence, the next step was the study of PM intermetallic compounds. One of themost studied materials was PrNi5 and it is actually still used in nuclear adiabaticdemagnetisation devices. Using PrNi5 the lowest working temperature has beenreached: 27 µK [22]. Another group of materials that have extensively been stud-ied are PM garnets, because of their high thermal conductivity, low lattice heatcapacity and very low ordering temperature (usually below 1 K). An orderingtemperature so close to absolute zero allows to obtain a large ∆S m and to keepa signicant MCE up to ∼20K. For instance, ∆Tad within 6 and 10 K have beenreached in ytterbium (Y3Fe5O12) and gadolinium (Gd3Fe5O12) iron garnets, withµ0∆H = 11 T, in the 10-30 K T -range [23]. Appreciable MCE values have also

9


Figure 1.4: ∆S m and ∆Tad values in Gd for a magnetic eld change from 0 to 5T and calculated from the experimental heat capacity data measured at 0 and 5 T(open circles). The dotted lines indicate the range of absolute errors and the solidlines show the relative error of the calculated values. Taken from Ref. [2].

10

1.5. MCE in order-disorder magnetic phase transitions

been reached using neodymium gallium garnet (Nd3Ga5O12) at 4.2 K [24] andgadolinium gallium garnet (Gd3Ga5O12) below 15 K [25]. Finally, a large value of∆S m has been observed in magnetic nanocomposites based on the iron-substitutedgadolinium gallium garnets, Gd3Ga5−xFexO12, for x ≤ 2.5 [26].

1.5 MCE in order-disorder magnetic phase transi-tions

Spontaneous magnetic ordering of PM solids below a given temperature is a co-operative phenomenon. The ordering temperature depends on the strenght of ex-change interaction and on the nature of the magnetic sublattice in the material.When spontaneous magnetic ordering occurs, the magnetisation strongly variesin a very narrow temperature range in the vicinity of the transition temperature,i.e., the Néel temperature for antiferromagnets (AFMs) and the Curie temperaturefor FMs. The fact that |(∂M/∂T )H | is large allows these magnetic materials tohave a signicant MCE. Since it is not the absolute value of the magnetisation,but rather its derivative with respect to temperature the one that must be large toobtain a large MCE, rare-earth metals or lanthanides (4 f metals) and their alloyshave been studied much more extensively than 3d transition metals and their al-loys, because the available magnetic entropy in rare earths is considerably largerthan in 3d transition metals: the maximum magnetic entropy for a lanthanide isS m = R ln(2J + 1), where R is the universal gas constant and J is the total angularmomentum.

The MCE in the vicinity of an order-disorder magnetic phase transition iscalculated by using equations 1.6 and 1.10, which arise from the Maxwell relation(Eq. 1.5), since these transitions are second-order and thermodynamic variableschange continuously [1, 17]. The research on these type of materials has beencentered in soft FMs with TC between 4 and 77 K, suitable for applications suchas for example helium and nitrogen liquefaction, and also in materials which ordernear room temperature so as to use their magnetocaloric properties in magneticrefrigeration and air conditioning.

1.5.1 MCE in the low-temperature range (∼10-80 K)The rst evident choise for low-temperature magnetic refrigerant materials aresome pure rare earths such as Nd, Er and Tm, since they order at low tempera-tures. Anyway, the expectations for large MCE are not fullled. MCE in Nd reach∆Tad ∼ 2.5 K at T=10 K for a magnetic eld rise µ0∆H = 10 T [27]. The prob-lem in Er is that several magnetic phase transitions occur between 20 and 80 K,

11


Figure 1.5: The adiabatic temperature change in the best magnetic refrigerantintermetallic materials in the temperature range from ∼10 K to ∼80 K, for a mag-netic eld change between 0 and 7.5 T. Taken from Ref. [2].

which causes the MCE to be constant but small in this overall temperature range:∆Tad ∼ 4 − 5 K for µ0∆H = 7 T [28]. Tm has a peculiar magnetic behaviour: itorders magnetically in a sinusoidally modulated ferromagnetic structure at ∼56 Kand becomes ferrimagnetic at ∼32 K. These features bring about both a restrictedMCE, which barely reaches ∆Tad ∼ 3 K at T =56 K for µ0∆H = 7 T, and a neg-ative ∆Tad between 32 and 56 K for µ0∆H = 1 T [29]. Consequently, the reasonwhy MCE in these pure materials is so small is that most of magnetic phases inNd, Er and Tm are either antiferromagnetic or ferrimagnetic, so that much of theavailable entropy is used in ipping spins to a ferromagnetic order.

The materials which display the largest MCE in the ∼10-80 K range are inter-metallic compounds which contain lanthanide metals. The best of them are REAl2compounds, where RE = Er, Ho, Dy, Dy0.5Ho0.5 [30] and DyxEr1−x (0 ≤ x ≤ 1)[9, 31], GdPd [9, 32], and RENi2, where RE=Gd [33], Dy [34] and Ho[34]. Adia-batic temperature changes for some of them are shown in Fig. 1.5. The maximumMCE peak is reduced as temperature increases from 10 to 80 K, which is asso-ciated with the rapid rise of the lattice heat capacity with temperature in thesealloys. The eld dependence of the MCE in this temperature range varies within∼1 and ∼2 K/T.

12

1.5. MCE in order-disorder magnetic phase transitions

1.5.2 MCE in the intermediate-temperature range (∼80-250 K)This temperature range has not been much studied, mainly for two reasons. First,there are not many applications in this range (it lies above gas liquefactions andbelow room temperature). Second, the T/C fraction (where C is the phononic andelectronic contribution to heat capacity) presents an inherent minimum in metals,as shown in gure 1.6 for a typical metal (Cu). This suggests that the adiabatictemperature variation is minimal in this temperature range (see equations 1.9 and1.10).

One of the best magnetic refrigerant materials in this temperature range ispure Dy [9, 35], with ∆Tad ∼ 12 K at T ∼180 K for a magnetic eld changeµ0∆H = 7 T. As discussed earlier for Tm (section 1.5.1), Dy also presents complexmagnetic structures, which brings about a negative MCE for small eld changes(µ0∆H < 2 T). Recent works [36, 37] have also found a noticeable MCE in amor-phous REx(T1,T2)1−x alloys, where RE is a rare earth metal and T1, T2 are 3dtransition metals, in the range 100-200 K. The eld dependence of the MCE is2 K/T for Dy, but for the rest of the materials, such as those amorphous alloys,rarely reaches 1 K/T. In spite of all these difficulties for the MCE at the interme-diate temperatures, the recently discovered Gd5(SixGe1−x)4 alloys show extremelylarge ∆S m and ∆Tad values, from 2 to 10 times larger than any of the above men-tioned materials [38, 39]. We will later discuss on these alloys, since they are alsoforeseen as excellent magnetic regerant materials at room temperature (see sec-tion 1.5.3). The origin of this giant MCE is explained in section 1.6, while theirproperties are exhaustively discussed in chapter 2.

1.5.3 MCE near room temperatureThe prototype material at room temperature is Gd, a rare earth metal which ordersFM at TC=294 K. This lanthanide has been extensively studied [9, 10, 40, 41], and∆Tad values at TC are ∼6, 12, 16 and 20 K for magnetic eld changes µ0∆H = 2,5, 7.5 and 10 T, respetively, leading to a eld dependence of the MCE of ∼3 K/Tat low elds, which reduces to ∼2 K/T at higher elds. A variety of alloys usingGd and other rare earths have been prepared in order to improve the MCE in Gd.Gd-RE alloys, with RE=lanthanide (Tb, Dy, Er, Ho,...) [42, 43] and/or Y [44]have been studied, but the alloying only decreases TC - which is not desirable,since we depart from room temperature - while the MCE value does not increaseconsiderably with respect to pure Gd. The only exceptions are nanocrystalline Gd-Y alloys, which improve the MCE in Gd for µ0∆H = 1 T [45]. Most intermetalliccompounds that order magnetically near room temperature and above ∼290 Kshow a noticeble lower MCE than that of Gd. For example Y2Fe17, with TC ∼310K, yields a MCE which is about 50% of that in Gd [46]. The same magnitude is

13


Figure 1.6: T/C versus T , where C is the electronic and phononic contribution ofheat capacity in a typical metal (Cu). Taken from Ref. [2].

approximately measured in Nd2Fe17 [46], with TC ∼324 K. It has been suggestedthat (Pr1.5Ce0.5)Fe17 could have a MCE larger than that of Gd, but it has not beenveried experimentally [47]. The only intermetallic compounds that display aMCE as large as that of Gd are Gd5Si4 (with TC ∼335 K) and the germanium-substituted solid solution Gd5(SixGe1−x)4, for 0.5 ≤ x ≤ 1, with TC from ∼290 to∼335 K [38]. In chapter 2, the main features of these alloys will be discussed indetail.

1.6 MCE in rst-order magnetic phase transitionsand the giant effect

In second-order magnetic phase transitions, the existence of short-range order andspin uctuations above the order temperature (TC) brings about a reduction inthe maximum possible |(∂M/∂T )H | value, and the maximum MCE is accordinglyreduced. In contrast, a rst-order phase transition ideally occurs at constant tem-perature (the transition temperature, Tt) and thus the |(∂M/∂T )H | value should beinnitely large. Actually, in an ideal rst-order phase transition, the discontinuityin both magnetisation and entropy causes that the derivatives in the mostly usedMaxwell relation (equations 1.5 and 1.6) must be replaced by the nite increments

14

1.6. MCE in rst-order magnetic phase transitions and the giant eect

of the Clausius-Clapeyron equation for phase transformations. The discontinuityin the entropy is related to the entalphy of transformation, which is also calledlatent heat. The rst-order transition occurs if the two magnetic phases have equalthermodynamic potential [48, 49],[U1 −

n1M21

2

]−ΘS 1+(pV1−HM1) =

[U2 −

n2M22

2

]−ΘS 2+(pV2−HM2) , (1.15)

where Θ is the transition temperature at the eld H, and U1,2, S 1,2, V1,2, M1,2 arethe internal energy, entropy, volume and magnetisation of phases 1 and 2, and nM2

describes the molecular eld contribution. If we assume that the external eld onlytriggers the transition, but does not change the value of the physical parameters(S , M, V , n) in either phase, the difference of the transition temperature for a eldchange of ∆H is given as

∆Θ

∆H = −∆M∆S = const , (1.16)

where ∆M = M2 − M1 is the difference between the magnetisations and ∆S =

S 2 − S 1 the difference between the entropies of the two phases. The sign appearssince a magnetised phase has lower entropy. ∆Θ/∆H is the shift of the transitiontemperature with the transition eld, which is usually evaluated as dTt/dHt froma Tt(Ht) curve. Therefore, the Clausius-Clapyron equation is written as

∆S = −∆M dHt

dTt. (1.17)

Chapter 5 discusses extensively the use of the Clausius-Clapeyron equation. Theexistence of this entropy change associated with the rst-order transition bringsabout an extra contribution to MCE, yielding the so-called giant magnetocaloriceffect. The use of this entropy change can be possible provided that the phasetransition -and thus the entropy change- is induced by magnetic eld. Extensivelysearch for materials with a rst-order eld-induced magnetic phase transition haslately been shown in literature.

The intermetallic compound FeRh was one of the rst materials in which thistype of giant (and negative) MCE was observed. This alloy has a rst-order FM-to-AFM phase transition at Tt ∼316 K, which yields a MCE value as large as -8.4K for µ0∆H = 2.1 T [50]. Unfortunately the giant effect is irreversible, and giantMCE can only be observed in virgin samples.

The recently discovered Gd5(SixGe1−x)4 alloys with 0 ≤ x ≤ 0.5, display a∆S m at least twice larger than that of Gd near room temperature (-18.5 J/(kgK)for µ0∆H = 5 T at T= 276 K) [13], and between 2 and 10 times larger than thebest magnetocaloric materials in the low and intermediate temperature ranges (-26 J/(kgK) at T ∼40 K to -68 J/(kgK) at T ∼145 K for µ0∆H = 5 T, depending

15


on composition, x) [38]. ∆Tad is also very large, reaching for example 15.2 K forµ0∆H = 5 T at T= 276 K and 15 K for µ0∆H = 5 T at T ∼ 70 K [39]. These alloyshave some interesting properties that make them very exciting and candidates tobe used as magnetic refrigerant materials in highly efficient magnetic refrigerators.The rst one is that the transition temperature can be tuned from ∼20 K to ∼276K by just changing the ratio between Si and Ge contents (0 ≤ x ≤ 0.5) [38, 51],and even a Tt ∼305 K can be achieved by adding Ga impurities to Gd5(Si2Ge2)[52]. This allows one to shift at own's will the maximum giant MCE between∼20 and ∼305 K. The second property is that, unlike FeRh, Gd5(SixGe1−x)4 alloysshow a reversible MCE, i.e., MCE does not disappear after the rst applicationof a magnetic eld. The difference in the behaviour of FeRh and Gd5(SixGe1−x)4alloys is associated with the nature of the rst-order phase transition: while theformer has a magnetic order-order transition, the latter plays simultaneously acrystallographic order-order phase transition and a magnetic phase transition, thelatter being order-disorder for 0.24 ≤ x ≤ 0.5 and order-order for 0 ≤ x ≤ 0.2[51, 53, 54, 55]. This magnetoelastic coupling accounts for the rare existence ofa rst-order magnetic order-disorder phase transition and also for the rst-ordermagnetic order-order phase transition. This is exhaustively developed in Chapter2.

Following the outburst caused by the discovery of a giant MCE in Gd5(SixGe1−x)4 intermetallic alloys, extensive research is being undertaken to nd newintermetallic alloys showing rst-order eld-induced phase transitions, which isgenerally associated with a strong magnetoelastic coupling. The rst obvious stephas been to exchange Gd for other rare earth cation in RE5(SixGe1−x)4 alloys, withRE=lanthanide [56]. Some of them, such as for example RE=Tb, seem to showmagnetoelastic similar properties to that of Gd5(SixGe1−x)4, yielding a noticeableMCE (∼ -22 J/(kgK) for µ0∆H=5 T at Tt ∼110 K) [57]. Dy5(SixGe1−x)4 alsoshows a rst-order phase transition for 0.67 ≤ x ≤ 0.78, which yields a MCE of∼ -34 J/(kgK) for µ0∆H=5 T at Tt ∼65 K [58]. The study of the actual mechanismresponsible for the giant MCE in RE5(SixGe1−x)4 alloys makes them interesting,but the low temperature transition that show most of these alloys makes them un-suitable from the point of view of applications near room temperature, in contrastto Gd5(SixGe1−x)4.

MnAs is also well-known for its rst-order magnetoelastic phase transitionfrom FM (with NiAs-type hexagonal crystallographic structure) to PM (with MnP-type orthorhombic structure) order at Tt=318 K, and it might also be a good can-didate since it shows giant MCE (-30 J/(kgK) and 13 K for µ0∆H=5 T at Tt).Unfortunately, it is not very useful for applications due to its large thermal hys-teresis at the transition [59]. However, the partial substitution of As by Sb inMn(AsxSb1−x) reduces both the thermal hysteresis and the transition temperature,which decreases from 318 K for x=0 to 230 K for x=0.3, maintaining rst-order

16

1.6. MCE in rst-order magnetic phase transitions and the giant eect

Figure 1.7: Entropy changes of MnFeP0.45As0.55 (solid circles), Gd5(Si2Ge2) (in-verted solid triangles) and Gd (solid triangles). Data are shown for external eldvariations of 0-2 T (lower curves for each material), and 0-5 T (upper curves),calculated from magnetisation measurements. Taken from Ref. [63].

and magnetoelastic properties. Hence, a competitive material in a wide temper-ature range around room temperature is obtained (-25 to -30 J/(kgK) and ∼10 Kfor µ0∆H=5 T) [59, 60]. Moreover, the addition of Fe and P in the MnFePxAs1−xalloys still maintains the rst-order and eld-induced nature of the phase transi-tion near room temperature, for 0.26≤ x ≤0.66 [61, 62]. For example, x=0.45yields -18 J/(kgK) for µ0∆H=5 T at Tt ∼300 K [63]. A comparison of the entropychange of MnFeP0.45As0.55 to those of Gd5(Si2Ge2) and pure Gd is given in Fig.1.7. The advantage in these alloys is that they are transition-metal-based, whichare much cheaper than rare earths, and the disadvantage is the poissonousness ofthe As content [64].

Finally, it has recently been found that the La(FexSi1−x)13 series of alloys alsoshows a rst-order eld-induced FM-PM transition within x=0.86 (Tt ∼210 K)and x=0.90 (Tt ∼184 K) [65, 66]. However, an itinerant electron metamagnetictransition takes place in this case [66]. That brings about a giant MCE, withan entropy change from -14 to -28 J/(kgK) and a ∆Tad between 6 and 8 K, forµ0∆H=2 T [67, 68, 69]. In order to increase Tt up to room temperature, either Cocan be added [70] or hydrogen can be absorbed [68, 69], maintaining the gianteffect. Figure 1.8 shows the entropy change and the adiabatic temperature change

17


0 - 5 T

0 - 5 T

Figure 1.8: Entropy change and adiabatic temperature change for theLa(FexSi1−x)13 compounds with x=0.88, 0.89 and 0.90, when an external eldvariation 0 to 5 T is provided. Taken from Ref. [69].

obtained for different compositions with a eld variation of 0-5 T.There is another class of materials that also displays a large MCE -the same

order than that of Gd-, although not giant. They are the perovskite-like LaMnO3materials, with Y, Ca, Sr, Li and/or Na substituting for La, and Ti for Mn [71, 72,73, 74]. The main interest of these compounds is that, showing a MCE similarto that of Gd, they are much cheaper. Their disadvantage with respect to theintermetallic alloys is their low density [75].

A comparision between the above-mentioned materials displaying giant MCE(or MCE similar to that of Gd) is showed in Table 1.1.

1.7 MCE at very low temperature: frustrated mag-nets and high-spin molecular magnets

At low temperatures, paramagnetic salts are the standard refrigerant materials formagnetic cooling. The higher the density of the magnetic moments and their spinnumber is, the greater the cooling power of a refrigerant is. With increased densityof spins, however, the strength of interactions leads to an ordering transition. Thetransition temperature thus limits the lowest temperatures achievable with para-magnetic salts. However, in frustrated magnets, the magnetic moments remaindisordered and posses nite entropy at temperatures well below the Curie-Weissconstant. For example, large entropy change at low temperature has recently beendiscovered in TbxY1−xAl2 system [2.4 J/(kgK) at T=12 K for a eld variation of

18

1.7. MCE at very low temperature: frustrated magnets and high-spin molecularmagnets

Material Tt (K) µ0∆H (T) ∆S (J/kgK) ∆Tad (K) ReferenceGd 294 2/5 -5/-9.8 5.7/11.5 [41, 13]

FeRh ∼316 2.1 11.71 -8.4 [50]Gd5(SixGe1−x)4

x = 0.5 276 2/5 -14/-18.5 7.4/15.2 [13]x = 0.25 ∼136 5 -68 12 [38, 39]

Tb5(SixGe1−x)4x = 0.5 ∼110 5 -21.8 - [57]

Dy5(SixGe1−x)4x = 0.75 ∼65 5 -34 - [58]

La(FexSi1−x)13x = 0.877 208 2/5 -14.3/-19.4 - [67]x = 0.880 195 2/5 -20/-23 6.5/8.6 [68, 69]x = 0.890 188 2/5 -24/-26 7.5/10.7 [68, 69]x = 0.900 184 2/5 -28/-30 8.1/12.1 [68, 69]

La(Fe0.88Si0.12)13H1.0 274 2 -19/-23 6.2/11.1 [68, 69]La(Fe0.89Si0.11)13H1.3 291 2 -24/-28 6.9/12.8 [68, 69]La(Fe11.2Co0.7Si1.1) 274 2/5 -12/-20.3 - [70]

MnAs-basedMn(AsxSb1−x)

x = 1 318 2/5 -31/-32 4.7/13 [59]x = 0.1 283 2/5 -24/-30 - [59]

x = 0.25 230 2/5 -18/-23 5.5/10 [60]MnFeP0.45As0.55 ∼300 2/5 -14.5/-18 - [63]

Ceramic manganitesLa0.8Ca0.2MnO3 230 1.5 -5.5 <2.5 [72, 75]La0.6Ca0.4MnO3 263 3 -5.0 <2.4 [73, 75]La0.84Sr0.16MnO3 243.5 2.5/5/8 -3.8/-5.5/-7.9 -/-/<4.1 [74, 75]

Table 1.1: Entropy change, ∆S , and adiabatic temperature change, ∆Tad, ocurringat the transition temperature Tt, at different values of applied eld increase, ∆H,for materials displaying giant magnetocaloric effect. The prototype material atroom temperature, Gd, is also showed for comparision.

19


µ0∆H=2 T and 7.6 J/(kgK) at T=30 K for µ0∆H=2 T], associated with the spin-glass-to-PM (freezing) transition [76]. Moreover, enhanced magnetocaloric effecthas been predicted in geometrically frustrated magnets [77]. This enhancementis related to the presence of a macroscopic number of soft modes associated withgeometrical frustration below the saturation eld.

Another interesting type of materials showing large (and time-dependent) en-tropy change at very low temperature are the high-spin molecular magnets. Molec-ular clusters as Mn12 and Fe8 exhibit extremely high entropy change around theblocking temperature at the Kelvin regime, which is associated with the order-disorder blocking process. Values of 21 J/(kgK) at T '3 K for a eld variation ofµ0∆H=3 T at a sweeping rate of 0.01 Hz are obtained for Mn12 [78]. Therefore,they are potential candidates to magnetic refrigerants in the helium liquefactionregime.

1.8 Magnetic refrigerationCurrently, there is a great deal of interest in utilizing the MCE as an alternatetechnology for refrigeration both in the ambient temperature and in cryogenictemperatures. Magnetic refrigeration is an environmentally friendly cooling tech-nology (see Fig. 1.9 for details). It does not use ozone-depleting chemicals (suchas chlorouorocarbons), hazardous chemicals (such as ammonia), or greenhousegases (hydrochlorouorocarbons and hydrouorocarbons). Most modern refrig-eration systems and air conditioners still use ozone-depleting or global-warmingvolatile liquid refrigerants. Magnetic refrigerators use a solid refrigerant (usuallyin a form of spheres or thin sheets) and common heat transfer uids (e.g. water,water-alcohol solution, air, or helium gas) with no ozone-depleting and/or global-warming effects. Another important difference between vapour-cycle refrigera-tors and magnetic refrigerators is the amount of energy loss incurred during therefrigeration cycle. Even the newest most efficient commercial refrigeration unitsoperate well below the maximum theoretical (Carnot) efficiency, and few, if any,further improvements may be possible with the existing vapor-cycle technology.Magnetic refrigeration, however, is rapidly becoming competitive with conven-tional gas compression technology because it offers considerable operating costsavings by eliminating the most inefficient part of the refrigerator: the compres-sor. The cooling efficiency of magnetic refrigerators working with Gd has beenshown [2, 6, 7, 79] to reach 60% of the Carnot limit, compared to only about 40%in the best gas-compression refrigerators. However, with the currently availablemagnetic materials, this high efficiency is only realised in high magnetic eldsof 5 T. Therefore, research for new magnetic materials displaying larger MCE,which then can be operated in lower elds of about 2 T that can be generated by

20

1.8. Magnetic refrigeration

Figure 1.9: Schematic representation of a magnetic-refrigeration cycle, whichtransports heat from the heat load to its surroundings. Light and dark grey de-pict the magnetic material without and with applied magnetic eld, respectively.Initially disordered magnetic moments are aligned by a magnetic eld, resultingin heating of the magnetic material. This heat is removed from the material toits surroundings by a heat-transfer medium. On removing the eld, the magneticmoments randomize, which leads to cooling of the magnetic material below theambient temperature. Heat from the system to be cooled can then be extractedusing a heat-transfer medium. Taken from Ref. [63].

21


permanent magnets, is very signicant. The heating and cooling that occurs in themagnetic refrigeration technique is proportional to the size of the magnetic mo-ments and to the applied magnetic eld. This is why research in magnetic refrig-eration is at present almost exclusively conducted on superparamagnetic materialsand on rare-earth compounds.

Refrigeration in the subroom temperature (∼250-290 K) range is of partic-ular interest because of potential impact on energy savings and environmentalconcerns. As described along this chapter, materials to be applied in magneticrefrigeration must present a series of properties:

(i) A rst-order eld-induced transition around the working temperature, inorder to utilise the associated entropy change.

(ii) A high refrigerant capacity. Refrigerant capacity, q, is a measure of howmuch heat can be transferred between the cold and hot sinks in one ideal refriger-ation cycle, and it is calculated as:

q =

∫ Thot

Tcold

∆S (T )∆H dT . (1.18)

Therefore, a large entropy change in a temperature range as wide as possibleis needed. Moreover, it is easy to argue that for any practical application it isthe amount of heat energy per unit volume transferred in one refrigeration cycle,which is the important parameter, i.e., the denser the magnetic refrigerant the moreeffective it is [75].

(iii) A low magnetic hysteresis, to avoid magnetic-work losses due to the ro-tation of domains in a magnetic-refrigeration cycle.

(iv) A low heat capacity CP, since a high CP increases the thermal load andmore energy is required to heat the sample itself and causes a loss in entropy, i.e.for a given ∆S , ∆Tad will be lower.

(v) Low cost and harmless. The main problem of the rare-earth-based com-pounds, which are usually the best magnetic refrigerants in the whole temperaturerange (including pure Gd at room temperature) is their high cost. 3d-transition-metal compounds or ceramic manganites are a good alternative concerning thecost of the materials. In particular, the recently reported MnAs-based materialsshow good prospects [59, 63]. However, the presence of As in these compounds,which is poisonous, could make them be useless for commercial applications.Another type of compounds, La(FexSi1−x)13, also presents a large MCE at roomtemperature, has a low cost and in this case all elements are harmless [68, 69].

22

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27


28

Chapter 2

Gd5(SixGe1−x)4 series of alloys

2.1 Discovery of the Gd5(SixGe1−x)4 system and thegiant MCE

Gd5(SixGe1−x)4 alloys were discovered by Holtzberg et al. [1] and Smith et al.[2]. They found that Gd5Si4 orders ferromagnetically at TC=335 K and that asmuch as 50% of Ge could be substituted for Si in the silicide structure (0.5 <x ≤ 1), maintaining the magnetic properties and the orthorhombic structure [1].Gd5Ge4 also had an orthorhombic structure, but different from that of silicide. Thegermanide structure appeared from x=0 to x=0.25 and presented a strange varietyof magnetic phase transitions: a low temperature ordering, which was AFM forGd5Ge4 and FM for the components with added Si, and higher Néel T . Moreover,the paramagnetic Curie temperature (θC) yielded a high positive value for all Ge-rich compounds [1]. The intermediate phase was not identied, but since the endmembers of the solid solution were not isostructural [2, 3] such a discontinuitywas to be expected.

In 1997, Pecharsky and Gschneidner discovered a giant magnetocaloric effectin Gd5(SixGe1−x)4 alloys [4, 5, 6, 7, 8]. For Gd5(Si2Ge2) [4], the calculation ofthe magnetic entropy change, ∆S m, using magnetisation measurements and theMaxwell relation (Eq. 1.6) yielded a value twice larger than that of Gd -the ma-terial with the best MCE at room temperature known until then- at ∼276 K. Withthe measurements of the heat capacity as a function of T and H, the giant value of∆S m was conrmed and the adiabatic temperature change, ∆Tad, as a function ofT was evaluated, giving rise to a narrower and higher (≥ 30%) peak than that ofpure Gd (see Table 1.1).

In order to understand the unclear magnetic properties and phase relationshipin this system, Pecharsky and Gschneidner studied samples in the whole com-positon range 0 ≤ x ≤ 1, leading to the rst phase diagram at zero eld of the

29

CHAPTER 2. GD5(SIXGE1−X)4 SERIES OF ALLOYS

(b)(a)

Figure 2.1: (a) First phase diagram at zero eld of the Gd5(SixGe1−x)4 alloy sys-tem, obtained by Pecharsky and Gschneidner [5, 6, 8]. (b) ∆S m of Gd5(SixGe1−x)4alloys, for x ≤ 0.5, around their rst-order transition, calculated from magnetisa-tion data using the Maxwell relation, for a eld variation from 0 to 5 T [5]. ∆S mvalues for the best magnetic refrigerant materials are displayed as dotted lines.FeRh is also shown (dashed line) for comparision.

30

2.2. Phase diagram of Gd5(SixGe1−x)4

Gd5(SixGe1−x)4 system [5, 6, 8] (see Fig. 2.1 (a)). They identied the interme-diate phase as monoclinic [6], and showed that the lower transitions in both theGe-rich region and the intermediate region were rst-order and reversible, yield-ing the giant MCE in these alloys [5]. From x=0 to x=0.2, MCE increased, whileit decreased from x=0.24 to x=0.5 [5], being larger than any other prototype ma-terial in all temperature ranges (from ∼20 K to ∼276 K) as shown in Fig. 2.1, (b).In the 0.5 < x ≤ 1 composition region, the PM-FM transition was second-order,leading to a 3 to 3.5-fold reduction of the MCE with respect to that of the com-positions showing a rst-order transition. These authors were able to increase thetransition temperature, Tt, in Gd5(Si2Ge2) from 276 K to 286 K by adding 0.33%at. of Ga, without losing the giant MCE [7].

2.2 Phase diagram of Gd5(SixGe1−x)4

The work of Morellon et al. [10, 9] unveiled the actual origin of the rst-ordertransition in the two regions where it appears and showed than the upper transi-tion in the intermediate phase does not exist, but it is rather caused by a resid-ual phase with slightly different x [10]. In summary, the phase diagram of theGd5(SixGe1−x)4 alloys at zero eld shows three compositional ranges (see Fig.2.2). The Si-rich compounds (0.5 < x ≤ 1) display the Gd5Si4-type orthorhom-bic (space group Pnma) structure, O(I), with a second order PM-FM transition[5, 6, 8]. For the intermediate region, 0.24 ≤ x ≤ 0.5, a rst-order magnetostruc-tural phase transition occurs from a high-temperature PM phase (which displaysa monoclinic structure, M, space group P1121/a) to a low-temperature FM phase(with the same O(I) structure than the Si-rich compounds), at temperatures rang-ing linearly from 130 K (x=0.24) to 276 K (x=0.5) [5, 10, 11]. For the Ge-richcompounds (x ≤ 0.2), a second-order PM-AFM transition occurs at TN (from∼125 K for x=0 to ∼135 K for x=0.2) [5]. Upon further cooling, a rst-orderAFM-FM transition takes place, whose temperature ranges linearly from about20 K (x=0) to 120 K (x=0.2). Because of its singularity, the case x=0 is dis-cussed in section 2.4.1. Since neutron scattering cannot be performed in Gd-basedcompounds, the nature of the AFM phase is currently under discussion [9]: themagnetic structure might correspond to that of either a canted ferrimagnet, as pro-posed for Nd5Ge4 [12] or a canted antiferromagnet, as for the Ge-rich region of theTb5(SixGe1−x)4 alloys [13, 14]. The AFM-FM transition occurs simultaneouslywith a rst-order structural transition from a high-temperature Sm5Ge4-type or-thorhombic (space group Pnma) phase, O(II), to the low-temperature O(I) phase[9]. We must note that in this case the symmetry remains unchanged through thetransition, although drastic variations in the cell parameters and atomic positionsalso occur (see section 2.3). In the range 0.2 < x < 0.24, where the second-

31


Figure 2.2: Magnetic and crystallographic phase diagram at zero eld, forGd5(SixGe1−x)4 alloys, as a function of temperature and composition [9]. PMstands for paramagnetic phase, FM for ferromagnetic phase and AFM for anti-ferromagnetic phase. M stands for monoclinic structure, O(I) for Gd5Si4-typeorthorhombic structure and O(II) for Gd5Ge4-type orthorhombic structure. First-order transition is displayed as a solid line.

32

2.3. Microstructure and atomic bonds

order PM-AFM transition disappears, O(II) and M structures coexist [6]. In boththe Ge-rich and intermediate compositional region, the rst-order transition maybe induced reversibly by an applied magnetic eld [5, 10, 9] and by an externalhydrostatic pressure [10, 15, 16], which both linearly shift Tt up to higher temper-atures. The fact that the transition may be eld-induced gives rise to very excitingand novel properties (see below).

A phenomenological description of the rst-order magnetostructural transitiongiven in Ref. [17] derive the H − T magnetic phase diagram and accounts for thethermal and magnetic hysteresis occurring in these alloys.

There are some other details in the phase diagram for Gd5(SixGe1−x)4 alloysthat make it much more complex. Different heat treatments of the samples andthe purity of the components may lead to the stabilisation of different crystallo-graphic structures, which also inuence the magnitude of the MCE [18], speciallyin the boundaries of the compositional regions. Pecharsky et al. [19] published aphase diagram with some differences with respect to that in Fig. 2.2, using alloyswith high-purity Gd. The same authors found a new transition in the compoundsaround x 0.5 [18, 20, 21]. It is an irreversible transition occurring between∼500 K and ∼870 K, from the room-temperature M phase, that yields the O(I)phase again. Then, after cooling down to room temperature again, the compoundbehaves like the Si-rich alloys (with the second-order PM-FM transition with-out structural transformation). If the compound is further heated up to ∼1070-1570 K [18, 20, 21], the M structure reforms, returning to the behaviour of theintermediate-region compounds. Accordingly, a reversible M ↔ O(I) transition-not directly observed- exists between ∼870 K and ∼1070 K.

2.3 Microstructure and atomic bondsIn order to understand the singular behaviour of these alloys [5, 6, 10, 9], theatomic structure has been analysed in detail in literature. The transition is presentlyunderstood by considering the layered crystal structure of Gd5(SixGe1−x)4 [22].Gd atoms (represented as blue spheres in Fig. 2.3; vertex of polyhedra also rep-resent Gd positions) form a two-dimensional (32434) net (Figure 2.3 (a)). Thisquasi-innite layer (slab) is composed of distorted cubes and trigonal prismswhich share common faces. T atoms, which are a mixture of Si and Ge atoms(green spheres in Fig. 2.3), occupy the trigonal prisms, sharing a common rect-angular face to produce a T-T dimer (intraslab bond). The Gd atom at the centerof each cube is surrounded by 4 T and 2 T' atoms (red spheres) [23]. For x=0.5compound, the occupancy in T atoms is 60% Ge and 40% Si, being 60% Si and40% Ge for T' atoms [23]. The T' atoms play a key role in the interslab bondingthus controlling both the crystal structure and properties of the alloys. For the

33


(a)

(b) (c) (d)

Figure 2.3: Crystallographic structures of Gd5(SixGe1−x)4 alloys. (a) Projectionalong b axis, which is common for all structures and emphasise the basic buildingslabs (32434 net), with Gd atom (in blue) inside the cubes and T atoms (mixtureof Si and Ge, in green) inside the trigonal prisms. (b) Sm5Ge4-type orthorhombicstructure [O(II)]. (c) Gd5Si4-type orthorhombic structure [O(I)]. (d) Monoclinicstructure (M). The projection along c axis emphasises T' atoms (mixture of Siand Ge, in red) and the covalent-like bonds between slabs. Note that one half ofthe bonds are broken in the M phase, while all of them are broken in the O(II)phase. Red arrows indicate the shear displacement of the slabs in the O(I) phasewhen the transition to the two other possible phases occurs.

34

2.4. Structural and magnetic properties

O(I) phase (Fig. 2.3 (c)), which is always FM, two-dimensional slabs (layers) areconnected one another through T'-T' covalent-like bonds [22, 23]. The interslabbonds are totally broken when the distance between all T' atoms increases dur-ing the transformation to the O(II) phase [9, 22] (Fig. 2.3 (b)), yielding an AFMphase, while only half of the T'-T' bonds are broken during the transformation tothe M phase [22, 23] for 0.24 ≤ x ≤ 0.5 compounds (Fig. 2.3 (d)), leading to aPM phase. The breaking of the bonds, i.e. the structural transition, occurs by ashear mechanism (also depicted in Fig. 2.3) along the a axis [23] in both composi-tion regions (the distance between T' atoms expands by 32.7% in the intermediatecompounds (0.24 ≤ x ≤ 0.5) [23] and by 34% in the Ge-rich compounds (x ≤ 0.2)[9]), and yields a large volume variation (∼0.4% and ∼0.5%, respectively). To il-lustrate this fact, Fig. 2.4 shows the variation of the lattice parameters and thedistance between T' atoms when the O(I)↔ O(II) transition is thermally inducedin x=0.1 compound. The three structures (O(I), M and O(II)) are thus present atroom temperature by tunning the Si:Ge ratio (x) from Gd5Si4 to Gd5Ge4. Figure2.5 (a) displays the values of the lattice parameters for the whole compositionalrange at room temperature. The shifts in all atomic positions between the threecrystal structures are depicted in Fig. 2.5 (b). As mentioned above, the main dif-ference between the structures is the shift of the atomic positions along the a axis,especially affecting the distance between T' sites.

2.4 Structural and magnetic propertiesCrystallography and magnetism are closely related in Gd5(SixGe1−x)4 [22, 23, 24,25, 26]. On one hand, Choe et al. [23] investigated this relationship by calculat-ing the efective exchange parameter J(R) between Gd sites in the different phasespresent in Gd5(Si2Ge2), where R is the Gd-Gd distance (see Fig. 2.6). They useda nearly free electron model for the conduction band and applied the Ruderman-Kittel-Kasuya-Yosida (RKKY) model. It was found that in the M phase, J(R)>0for short Gd-Gd contacts and J(R)<0 for long Gd-Gd contacts, while in the O(I)phase J(R)>0 for the entire range of Gd-Gd interactions. An hypothetical O(II)phase in Gd5(Si2Ge2) would lead to J(R)<0 for the whole range of R. Therefore,the signicant changes in electronic structure when the crystal structure varies af-fect the exchange interactions, and the values of J(R) may account for the changein the magnetic behaviour of the various crystal structures.

On the other hand, it is worth noting that ferromagnetism only exists in theO(I) structure, i.e. as long as all slabs are connected by T'-T' covalent-like bonds[22]. In order to explain this fact, Levin et al. [24] proposed that ferromagnetismin the O(I) ground state is achieved not only via the indirect RKKY 4 f − 4 f ex-change, but also via a direct Gd-Ge(Si)-Gd superexchange through the interslab

35


T’-T’

x = 0.1

Figure 2.4: Thermal dependence of the lattice parameters (solid symbols) andT ′−T ′ interatomic distance (open symbol) of Gd5(SixGe1−x)4 for x=0.1, obtainedfrom XRD. The transition O(I)↔ O(II) is clearly observed at Tt=81 K. The linesare a guide to the eye. Taken from Ref. [9].

covalent-like bonds. It is well-known that the indirect RKKY exchange between4 f − 4 f localised electrons, through the 6s itinerant electrons, accounts for mostof the phenomenology in lanthanide systems. But in the present system, althoughRKKY is certainly important, it does not account for the abrupt change in mag-netic behaviour (PM↔FM) at the crystallographic transition. Even though therewould be a relevant change in the RKKY interaction along the a axis (where thereis the shear displacement that leads to the 0.8 to 1.1 Å increase of T'-T' dis-tances [10, 22, 23]), the change in the distances along the b and c axes is fairlysmaller. Therefore, the overall RKKY interaction is expected to have a mini-mal variation. However, if in addition to the RKKY interaction, there were aGd-Si(Ge)-Gd superexchange interaction in the low-temperature FM O(I) phasepropagating through the interslab covalent-like bonds, then the breaking of all(or half) of them at the structural transformation would explain the destructionof the ferromagnetism in the system, since the superexchange interaction shoulddisappear. This suggestion is supported by the fact that TC in the Si-rich com-pounds (0.5 < x ≤ 1), which always have the O(I) structure, is higher than thatof pure Gd (by as much as ∼40 K) [1, 5]. Accordingly, the magnetic behavior ofthe Gd5(SixGe1−x)4 compounds can be understood qualitatively in terms of com-petition between intraslabs (conventional indirect 4 f − 4 f RKKY) and interslab

36


(b)(a)

Figure 2.5: (a) Lattice parameters for the Gd5(SixGe1−x)4 system at room tempera-ture. The dotted lines are a guide to the eye. The dashed lines delineate structuralphase regions. (b) Relative atomic shifts along a, b, and c axes, in the crystalstructures of Gd5Ge4 [O(II)] and Gd5(Si2Ge2) (M), with respect the positions inGd5Si4 [O(I)]. M1 and M2 are T-type atoms, while M3 is T'-type atom. Takenfrom Ref. [6].

37


O(I)

O(II)

M

Figure 2.6: Variations in the exchange interactions, J(R), between Gd atoms inGd5(Si2Ge2) with Gd-Gd distance, R, as estimated from the RKKY model usinga nearly free electron model for the conduction electrons. The hatched region be-tween 3.45 and 4.05 Å corresponds to the range of short Gd-Gd distances. Takenfrom Ref. [23].

exchange interactions (Gd-Ge(Si)-Gd superexchange propagated via the covalent-like bonds) [15, 22, 24].

Rao also discussed possible correlations between crystal structure and mag-netism [25]. The author established an almost linear dependence between TC andthe lenght of T'-T' bonds, for the whole compositional range (0 ≤ x ≤ 1). Thelenght of the bonds is thus a crucial structural parameter governing the magneticinteractions in these compounds. It is also suggested that FM interactions sta-bilise the O(I) phase at low temperature in the intermediate region. Moreover, Liuet al. [26] demonstrated that the Gd5(SixGe1−x)4 compounds form a completelymiscible solid-solution crystallised in the O(I) structure at low temperature. Thisfeature is interpretated by considering the competing effects of lattice strain (in-duced by the substitution of Ge for Si) and ferromagnetic exchange interactions,which thus help to stabilise the O(I) phase for x ≤ 0.5 below TC.

This strong coupling between the magnetic and crystallographic sublatticesenables the existence of a eld-induced order-disorder phase transition (PM-FM inthe 0.24 ≤ x ≤ 0.5 range), which is rarely observed. Only MnAs-based alloys [27,28, 29], La(FexSi1−x)13 compounds [30, 31] and the manganite Sm0.65Sr0.35MnO3[32] are some of the few cases. In 0.24 ≤ x ≤ 0.5 compounds both crystallo-graphic phases coexist during the rst-order phase transition, therefore a coexis-

38


Figure 2.7: The magnetic phase diagram of Gd5Ge4 obtained from the heat ca-pacity and magnetisation data. The AFM→FM transition within ∼1.8 and ∼25 Kis completely or partially irreversible (see text for details). The inset shows themagnetisation of Gd5Ge4 cooled in zero magnetic eld. During the rst magneticeld increase, which is shown by open squares in the inset, a metamagnetic-liketransition occurs at ∼18 kOe. During the rst magnetic-eld reduction (closedcircles) and during the second and following magnetic-eld increases (opened tri-angles), the magnetisation behaviour is typical of a soft ferromagnet. Taken fromRef. [35].

tence of ordered (FM) and disordered (PM) magnetic phases is observed [24, 33].Finally, concerning the anisotropy of the magnetic properties, a study in a

x 0.5 single crystal along the three crystallographic directions shows a smallanisotropy in magnetisation, which is thus also observed in MCE [34].

2.4.1 The x=0 caseThe magnetic ground state of the Gd5Ge4 alloy had been reported to be that ofa simple AFM with TN ∼15 K [1], but recent works clearly show that Gd5Ge4orders AFM at TN ∼127 K, and that no FM phase is observed when cooling atzero eld down to ∼1.8 K, the crystallographic structure remaining at the O(II)phase [35, 36, 37, 38]. This fact is in disagreement with the behaviour of the Ge-rich compounds (0<x≤0.2), which order AFM at ∼125-135 K and upon furthercooling undergo a rst-order AFM/O(II)-to-FM/O(I) transition (see sec. 2.2).However, for x=0, the application of a eld of 18 kOe (Hcr) at 4.3 K irreversibly

39


transforms the AFM state into a FM state, with also an irreversible O(II) → O(I)transition, since after the eld is reduced isothermally back to zero, Gd5Ge4 stillremains in the FM/O(I) state. The inverse FM/O(I)→AFM/O(II) transition canbe induced by heating the sample to above ∼25 K, where the well-known rst-order reversible transition occurs, in concordance with the rest of Ge-rich alloys.From 4.3 to ∼10 K, Hcr decreases with temperature, while from ∼10 K to ∼20K becomes nearly constant (∼11 kOe). The latter temperature range shows amore complex behaviour, because the transition in this case is partially reversible.Above ∼25 K, the transition is completely reversible and shifts linearly with themagnetic eld [35, 36, 37, 38]. The same overall behaviour is observed whenthe rst-order AFM/O(II)→FM/O(I) is induced by external hydrostatic pressureinstead of magnetic eld [15]. The magnetic phase diagram for x=0 is displayedin Fig. 2.7.

The antiferromagnetism related to the O(II) phase is believed to be similar tothat of the Ge-rich Tb5(SixGe1−x)4 alloys, which have the same structural phase. Inthe ordered phase of Tb5Ge4, the individual slabs have canted 2D ferromagnetismwhile the interslab ordering is canted antiferromagnetic, leading to 3D ordering[13, 14]. A spin reorientation occurs with T in the AFM state, affecting the in-traslab ferromagnetic canting, but the state remains AFM [14]. In contrast, theground state of Tb5Si4 is canted FM, with also a spin reorientation occuring withT [14]. Figure 2.8 shows the magnetic structures present in Tb5Ge4 and Tb5Si4compounds. In the case of the AFM phase in Gd5Ge4, the extrapolated Curie-Weiss temperature is positive [1, 8, 36, 39], which is an evidence of the existenceof positive (FM) exchange interactions in the AFM phase and it is thus reasonablyto compare both materials. Magen et al. [15] proposed for Gd5Ge4 a model ofmagnetic interactions similar to Tb5Ge4, shown in Fig. 2.9. For Levin et al. [36],the anisotropy in the exchange interactions of Gd atoms may account for the dif-ference between the interslab (AFM) and intraslab (FM) magnetic ordering andtherefore for the variety of probably non-colinear magnetic structures present inGd5Ge4. Szade et al. [39] studied the inverse of the low-eld magnetic suscepti-bility in Gd5Ge4 at high temperature and detected a non-linear behaviour betweenTN and ∼230 K, indicating the onset of some ordering process.

2.5 Other propertiesThe eld-induced, reversible nature of the rst-order magnetostructural transitionin Gd5(SixGe1−x)4 alloys for x ≤ 0.5, results in colossal magnetostriction [10, 9,11, 37], giant magnetoresistance [40, 41, 42, 43], unusual Hall effect [44], andspontaneous generation of voltage [45, 46], besides the giant MCE. Some of themost relevant of these properties are explained briey.

40

2.5. Other properties

Figure 2.8: Magnetic structures of Tb5Ge4 and Tb5Si4 at 2 K and 85 K as deter-mined from neutron powder diffraction data, showing the two distinct phases thateach compound shows at low temperatures. The spheres stand for Tb ions, and adifferent level of shading is used to identify Tb1 (black), Tb2 (medium gray), andTb3 (lightest gray). For Tb5Ge4 at 85 K, the projection in the a-c plane has beenincluded to emphasise the canting of the Tb3 ions. Taken from Ref. [14].

41


O(II ) – AFM O(I) – FM

PH, P

T

Figure 2.9: Crystallographic and magnetic structures of Gd5Ge4 in the a-b plane atlow temperature. Only the Ge atoms participating in the T'-T' covalent-like bondsare depicted as solid spheres. Solid lines stand for formed bonds while dashedlines represent broken bonds. Gray arrows are used to illustrate the change in themagnetic coupling induced by eld, hydrostatic pressure or temperature. Takenfrom Ref. [15].

42

2.5. Other properties

Figure 2.10: Magnetoresistance ratio ∆ρ/ρ = [ρ(H,T ) − ρ(0,T )]/ρ(0,T ) for theGd5(SixGe1−x)4 compound with x=0.45 as a function of the applied magnetic eldat some selected temperatures of (a) 275, (b) 270, (c) 260, (d) 250, (e) 240, and(f) 230 K. Taken from Ref. [40].

2.5.1 Giant magnetorresistance

A remarkable phenomenology in Gd5(SixGe1−x)4 is that of the magnetoresistance,because it also shows a giant effect -giant magnetoresistance (GMR)-, both in0.24 ≤ x ≤ 0.5 compounds [40, 41, 42], see for example Fig. 2.10, and in the0 ≤ x ≤ 0.2 alloys [43]. Since the two structural phases involved in the transitionshow a different electrical resistance, which at Tt is ∼20-25% for 0.24 ≤ x ≤ 0.5and ∼50% for 0 ≤ x ≤ 0.2, a drastic change in the resistance leads to a GMR∼20-50% when the transition to the low-temperature O(I) phase is eld-inducedabove Tt. The application of a eld at the O(I)/FM phase yields a negative butsmall magnetoresistance, corresponding to a FM system with localised moments[40].

The study of GMR in this series of alloys has unveiled that the electrical re-sistance of the high-temperature phase (M or O(II), depending on the composi-tional region) changes when the material is cycled through the structural transition[47, 48]. For clarity, we will call the low-temperature O(I) phase α, while we willcall the high-temperature phase β' or β depending o the resistance value. For avirgin sample, the low-temperature α phase shows a higher resistance than high-temperature β' phase (giving rise to a positive GMR). But when the sample isthermally cycled through the rst-order phase transition, a high-temperature β

43


(a) (b)

Figure 2.11: (a) Temperature dependences of the electrical resistivity ofGd5(SixGe1−x)4 compound with x=0.4875 on heating in zero eld during the 1st,8th, and 20th cycle through the rst-order phase transition. The inset shows thedetailed behaviour between 250 and 300 K during the 8th cycle. (b) Magneticeld dependences of the electrical resistance of the same compound measuredunder increasing eld after a different number of isothermal cycles through therst-order phase transition at 270 K. Taken from Ref. [47].

phase with higher resistance than α phase is obtained (giving rise now to a nega-tive GMR), see Fig. 2.11 (a). The same phenomenon occurs when the sample iscycled through the transition by the application of a magnetic eld, as show in Fig.2.11 (b). Moreover, it is also observed that Tt of the α ↔ β transition is lowerby ∼2-7 K than that of α ↔ β'. The effect is suggested to be originated froma redistribution of the Si/Ge ratio in T and T' atoms [47]: Si content increasesin the intraslab positions (T) and decreases in the interslab positions (T', wherethe covalent-like bonds between slabs take place). The structural transition occursat lower Tt because Ge-rich T'-T' bonds are weaker, as derived from the phasediagram (Tt decreases by ∼11 K for each 1 at. % increase in Ge content in thebulk alloy, see Fig. 2.2). The difference between the behaviour of the resistanceduring the α ↔ β' and α ↔ β transitions is settled by two main contributionsthat arise from the change in the conduction electron concentration and scatter-ing processes during the rst-order phase transition. Another fact observed inresistance studies is that the residual resistivity increases at each cycle due to theincreasing precence of microcracks [40, 41, 42, 47]. This suggests that the largevolume change taking place at the rst-order transition damages irreversibly thesesamples by introducing stress at the grain boundaries.

44

2.6. Characterisation of Gd5(SixGe1−x)4 alloys

The electrical resistance also shows an anomalous behaviour in the second-order AFM-FM transition (for 0 ≤ x ≤ 0.2 compounds). In this case, the resis-tance changes from a metallic behaviour in the FM phase (resistance increaseswith T ) to a semiconductor-like behaviour (resistance smoothly decreases with T )[35, 39, 43, 48].

2.5.2 Colossal magnetostrictionThe change in the crystal structure at Tt leads to a huge linear thermal expan-sion (LTE) of ∆l/l 0.16% for 0 ≤ x ≤ 0.2 (i.e., a volume change ∆V/V =

3(∆l/l) 0.48%) [9, 37] and ∆l/l 0.13% (∆V/V 0.4%) for 0.24 ≤ x ≤ 0.5[10], see for example the inset in Fig. 2.12 for x=0.1. The purity of Gd in thesamples affects the value of the LTE (and therefore the volume expansion) at therst-order transition [49]. LTE measurements carried out in a single crystal withx=0.43 showed that ∆l/l are negative along the b and c axes (-0.20% and -0.21%,respectively), while it is positive along the a axis (+0.68%), leading to a volumechange of ∼0.27% [11]. Similar results were obtained by Han et al. with a x 0.5single crystal [50], who also measured LTE along the a axis as a function of theangle of the applied eld [51]. Since the transition is eld-induced, a strong mag-netostriction is expected at temperatures above Tt, for the corresponding transitioneld. The magnetostriction measurements are independent of the eld direction inpolycrystalline samples (λ‖ = λ⊥), and consequently the volume magnetostrictionmay be evaluated as ω = 3λ‖(⊥). For 0 ≤ x ≤ 0.2 alloys, a large value of the vol-ume magnetostriction ω 0.5% is reached [9, 37] (Fig. 2.12 for x=0.1), while for0.24 ≤ x ≤ 0.5 alloys, ω 0.45% [10]. Accordingly, Gd5(SixGe1−x)4 compoundsfor x ≤ 0.5 are potential candidates as magnetostrictive transducers.

2.6 Characterisation of Gd5(SixGe1−x)4 alloys2.6.1 Electronic structureThe knowledge of the electronic structure of Gd5(SixGe1−x)4 alloys is relevant inorder to understand the properties of these compounds. The electronic structurehas been experimentally investigated by X-ray and ultraviolet photoelectron spec-troscopy (XPS and UPS) on the valence band [39, 52]. Covalent bonding betweenGd and Si or Ge causes a charge redistribution which is observed to be stronger forthe Ge-rich compounds. For Gd atoms the redistributing affects probably 5d elec-trons, therefore this may explain the magnetic properties variation as the indirectexchange would be based on the interaction via 5d electrons more than RKKY-based [39]. The experimental results may be compared with theoretical calcula-

45


Figure 2.12: Magnetostriction (λ) isotherms along the applied eld at some se-lected temperatures for Gd5(SixGe1−x)4 with x=0.1. The inset shows the linearthermal expansion ∆l/l for increasing temperature along the same measurementdirection. Taken from Ref. [9].

tions of the electronic band structure and the density of states [35, 52, 53, 54],which account for the observed resistivity and magnetic behaviour and conrmthat the 5d character of the valence electrons determines the most important prop-erties of the compounds [52].

2.6.2 Structural characterisation

A large variety of works have studied experimentally the main structural featuresof Gd5(SixGe1−x)4 alloys. To mention some of them: surface structure of sin-gle crystals performed with X-ray methods (X-ray powder difraction and Berg-Barret X-ray topography), scanning electron microscopy (SEM) and Auger elec-tron spectroscopy (AES) [55]; observation of the rst-order transition with mag-netic force microscopy (MFM) [56, 57] and transmission electron microscopy(TEM) [58]; determination of phases in as cast samples using SEM, energy dis-persive spectroscopy (EDS) and orientation imaging microscopy [59]; detailedmicrostructure of monoclinic phase using TEM bright eld images and selectedarea electron diffraction (SAED) [60]; and structural differences between the var-ious phases present in the intermediate-range compounds using TEM [58].

46

2.7. Evaluation of the MCE at a rst-order transition

(b)

Figure 2.13: (a) Entropy change for increasing elds, as calculated from theMaxwell relation (circles) -with different eld variations- and the Clausius-Clapeyron equation (triangles), using magnetisation data for Gd5(Si2Ge2) alloy.(b) Direct measurement of the adiabatic temperature change for two Gd5(Si2Ge2)samples, with different eld variations. Taken from Ref. [61].

2.7 Evaluation of the MCE at a rst-order transi-tion

The correct evaluation of the entropy change related to the MCE at a rst-ordertransition is a controversial issue and has lately aroused much discussion [4, 61,62, 63, 64, 65]. For Gd5(SixGe1−x)4, Giguère et al. [61] showed that the use of theMaxwell relation (Eq. 1.5 and 1.6) to calculate the entropy change overestimates(at least ∼20%, see Fig. 2.13 (a)) the value obtained from the Clausius-Clapeyronequation (Eq. 1.17), which the authors [61, 64] claimed to be the correct proceduredue to the rst-order nature of the transition in these alloys. According to them, theentropy change in the magnetostructural transition is not associated with the con-tinuous change of the magnetisation, M, as a function of T and H, but rather withthe discontinuous change in M due to the crystallographic transformation. Theyclaimed that Maxwell relations do not hold since M is not a continuous, derivablefunction in that case. In contrast, Gschneidner et al. [62] argued that the Maxwellrelation is applicable even in the occurrence of a rst-order transition, except whenthis transition takes place at a xed T and H, giving rise to a step-like change of M(ideal case). Besides, they claimed that Clausius-Clapeyron equation would implyan H-independent adiabatic temperature change, which however is not consistent

47


with the experimental observations [61] (see Fig. 2.13 (b)). Moreover, Sun et al.[63] showed that the entropy change calculated from the Maxwell relation is in-deed equivalent to that given by the Clausius-Clapeyron equation, provided M isconsidered T -independent in whichever phase the transition involves, and M is astep function with a nite jump at the transition temperature. They also suggestedthat the two procedures may yield different results since the Clausius-Clapeyronmethod does not take into account the reduction of spin uctuations by an appliedeld.

Furthermore, at a rst-order phase transition, the experimental determinationof the heat capacity CP is intrinsically uncertain due to the release of latent heat(i.e., a discontinuity in the entropy), therefore CP also presents problems in thecalculation of the MCE in the vicinity of a rst-order phase transition [65]. Theentropy discontinuity can be determined from DSC measurements (see sec. 3.2.3and also Ref. [65]), as it will be shown in Chapter 4. The controversy between thevariety of methods to evaluate the MCE at a rst-order phase tansition will alsobe discussed in Chapter 5.

48

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[36] E. M. Levin, K. A. Gschneidner, Jr., and V. K. Pecharsky, Phys. Rev. B 65,214427 (2002).

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52

Chapter 3

Experimental techniques

3.1 Sample synthesis and thermal treatment3.1.1 Synthesis method: arc meltingThe method used in this work to synthesize the samples of the Gd5(SixGe1−x)4 se-ries of alloys is arc melting, a simple but effective method. The process consists inmelting the pure elements of the alloy in the desired stochiometry by dischargingan electrical arc due to the application of a large tension between two electrodes.Our arc-melting furnace was designed and constructed in the mechanical work-shop of Facultat de Física of the Universitat de Barcelona, specically to preparebulk intermetallic alloys, i.e., it is not a commercial furnace (see Figs. 3.1 and3.2). The power (current) supply is a TIG 160 AC/DC (Argon), which may sup-ply up to 160 A. The anode, which lies over a steel platform, is a Cu crusibledesigned to hold the samples of the pure elements. It is cooled by a water owto avoid heating and melting of the Cu. The platform is covered by an hermeticalcylinder of stainless steel, which holds the cathode in its inner top. The cathodeconsists in a sharp rod of W -with 2% of Th-, a refractory material that bears hightemperatures without melting. The cover enables to control the atmosphere of thefurnace chamber by owing high-purity Ar, which evacuates oxygen and act asionizing gas.

The procedure to synthesize a sample is summarised as follows: after the ten-sion is applied, the cathode is approached to the pure elements that are placed onthe Cu crusible, by turning a millimetric screw. Argon gas must ow through thefurnace chamber at a pressure of 1-1.5 bar. When the cathode is close enough, anarc that melts the elements is discharged by ionizing the gas. After some secondsof melting, the pure elements are mixed and the tension can be broken off. Thesynthesis process of a sample includes a number of meltings, with the sample be-ing turned over each time in order to ensure a good homogeneity of the elements

53

CHAPTER 3. EXPERIMENTAL TECHNIQUES

Figure 3.1: Complete view of the arc-melting furnace: chamber, power supplyand Ar gas cylinder.

Figure 3.2: Detail of the inside of the arc-melting furnace chamber: cover with thecathode, on the left, and water-cooled Cu crusible (anode) with a sample above,on the right.

54

3.1. Sample synthesis and thermal treatment

in the alloy. The sample is weighted after each melting to control possible weightlosses, which must be negligible to maintain the desired stochiometry. At leastone of the elements must be metallic, because on the contrary the arc can not bedischarged.

Alloys are easy to obtain with this method, but it also presents a drawback:sample cooling is not homogeneous when the arc is broken off. At the bottom ofthe sample, where there is contact with the water-cooled Cu crusible, the coolingof the sample is faster than at the top of the latter. This fact can be observedin the shape of as-prepared samples of Gd5(SixGe1−x)4 alloys. During the briefcooling process, the top sample crystallises forming characteristic faceted -andbright- sides, like a football ball. At the bottom, the alloy does not cristallise infacets and it simply presents a metallic look.

Samples of Gd5(SixGe1−x)4 obtained in this work generally have a mass be-tween 1 and 2 g, and a current of ∼80 A is needed to melt them. It is worthnoting that these samples are quite brittle and usually display cracks (the moreGe content, the more brittle), so that cutting the samples to obtain fragments fortheir characterisation presents some problems 1. Table 3.1 display all synthesizedsamples with their partitions, and the further heat treatments and measurementscarried out over them. All in all, almost 50 samples have been studied. Since thiswork is aimed at the study of MCE at the rst-order transition of Gd5(SixGe1−x)4alloys, the synthesized samples lies within the compositional range 0 ≤ x ≤ 0.5.

3.1.2 Heat treatmentWhen a sample of Gd5(SixGe1−x)4 with the desired stochiometry is synthesized,some phases with a different value of x may appear due to segregation processes.This is critical for compositions close to the boundary of compositional regions(x 0.5, x 0.2− 0.24), because residual phases with distributed value of x -i.e.,with different structures and magnetic ordering- can be present (see section 2.2and Ref. [1]). Secondary phases as Gd5(Si,Ge)3 (5:3) and Gd(Si,Ge) (1:1) mayappear [2, 3, 4], since they are close to Gd5(Si,Ge)4 (5:4) in composition ratio.This is evident from the Gd-Ge and Gd-Si phase diagrams displayed in Figs. 3.3and 3.4. A proper heat treatment should removes the residual 5:4 phases withdistributed value of x [1], because it would help to homogenise Si and Ge contentall over the sample. However, a high-temperature polymorphic transformationfor x 0.5 compounds has very recently been found (see section 2.2), leadingto the the formation of the O(I) phase from the room-temperature M phase. Thepolymorphism is irreversible between ∼500 K and ∼870 K [5, 6, 7, 8], thereforea heat treatment up to those temperatures would change the structure of the phase

1To avoid breaking of the sample, a low-speed saw has always been used.

55


Figure 3.3: Gd-Si phase diagram as a function of atomic percent silicon (upperpanel) and weight percent silicon (lower panel).

56


Figure 3.4: Gd-Ge phase diagram as a function of atomic percent germanium(upper panel) and weight percent germanium (lower panel).

57


and the MCE. This transformation becomes reversible at ∼1070 K and, at least,up to 1570 K 2 [5, 6, 7]. With all these considerations, we tried to nd the bestheat treatment for as-cast samples. After each heat treatment, samples whereanalised by various magnetic and structural characterisation techniques (which aredescribed in section 3.2). The efficiency of each heat treatment is also discussedin section 3.2. The variety of thermal treatments used are the following:

(i) T1. 4 hours at 1400 ºC under a pure Ar ow, as mentioned in Ref. [1], witha heating/cooling rate of 5º/min. from/to room temperature in a ceramic tubularfurnace. A sample holder of alumina was used in order to bear this temperature.We already noted that the appearance of the samples proved an evident oxidation,with a white powder covering them (gadolinium oxide Gd2O3 shows this aspect).Degasing of alumina may be responsible of the oxidation for the samples.

(ii) T2. We repeated the same treatment (4 hours at 1400 ºC) but adding 0.5%of H2 in the gas ow -which is a reductor gas- to avoid the oxidation. The appear-ance of the samples was much better: although the surface seemed oxidised, theinside of the samples looked completely metallic.

(iii) T3. This heat treatment was the same as T2, but with the annealing tem-perature lowered to 1000 ºC, in order to study the effect of T in the heat-treatedsamples.

(iv) T4. Since it was suspected that the oxidation came from either impuritiesof Ar (unlikely) or degasing of alumina sample holder (likely), we used an elec-trical resistance furnace: samples were placed in a quartz tube (which does notdegas but bears lower temperatures than alumina) under a vacuum of 10−5 mb,and heated at 50º/min up to 950 ºC for 4 hours. Afterwards, the resistance wasswitched off and cooling to room temperature lasted ∼1 hour. The appearance ofthe samples was darker than as-cast ones, but without sign of oxidation.

(iv') T4+Q. This heat treatment is very similar to T4, using the same furnacewith a quartz tube to reach a high vacuum (10−5 mb). In this case, samples wereannealed up to 920 ºC for 8h 45' and, after annealing, the quartz tube was quicklytaken out of the furnace to room temperature (quenching).

(v) T5. To reach higher annealing temperatures, we repeated the heat treatmentin the ceramic furnace used in T1, T2 and T3, by replacing the alumina sampleholder for a platinum wire, which should also bear high temperatures but does notdegas, as alumina does. After annealing at 1400 ºC for 4 hours, Pt was destroyed.Silicon of the samples diffused to Pt, whose structure was completely damaged.Obviously, samples were also damaged.

2A M ↔ O(I) transition thus exists between ∼870 K and ∼1070 K, see section 2.2.

58


x ID heat t. partition XRD SEM MR ac DSC M(H) M(T) dscH

0 #1 NO 12 rod 1 X X X

rod 2 X T

powder X

T5 12

0 #3 NO powder X

0.05 #1 T4+Q 12 rod 1 H

rod 2 H12 rod 1b H

rod 2b X X T / H

0.1 #1 NO 12 rod 1

rod 2 X T / H

powder X

T5 12

0.15 #0 NO pressed X

0.15 #1 NO/T1 piece X/ X/X

NO/T1 rod X/X

0.18 #1 T2 14 rod 1

rod 2

T3 14

NO 14 rod 1 X X X

rod 2

T4 14 rod 1 X X X T

rod 2 X

0.2 #1 NO 12 rod 1 X T

rod 2 X X

powder X

T5 12

0.25 #2 NO 12 rod 1 X T

rod 2

0.3 #2 NO 12 rod 1 X T

rod 2 (powd.) (X)

T4+Q 12 rod 1 X T / H

rod 2

59


x ID heat t. partition XRD SEM MR ac DSC M(H) M(T) dscH

0.365 #1 NO/T1 piece X/ X/X

NO/T1 rod X/X

NO slice Cu X

0.365 #2 T2 13

23 rod X

0.365 #3 NO 12 rod 1 X X

rod 2 T

T5 12

0.45 #0 NO 1 slice A X X

slice B X

0.45 #1 NO slice C X

NO slice D X

NO/T1 piece X/ X/X

0.45 #2 NO/T1 slice E X/

T1 slice F X

0.45 #4 NO 12 X

NO 12 X

0.45 #5 T2 rod 1 X X X

rod 2

powder X

0.45 #7 T2 14

T3 14 rod 1 X

rod 2

NO 14 rod 1 (powd.) (X) X X X T

rod 2 X T

T4 14 rod 1 (powd.) (X) X X X T / H

rod 2 X

T4+Q piece powder X

0.5 #0 NO powder X

Table 3.1: Detail of all samples synthesized. x is the composition of the Gd5(SixGe1−x)4 sample.ID stands for an identication number for each sample with the same x. Heat treatments: noheat treatment (NO), T1, T2, T3, T4, T4+Q and T5 (see text for details). Partition explains thefractions of a given sample and their shapes. Measurements done: X-Ray diffraction (XRD),scanning electron microscopy and microprobe (SEM), magnetoresistance (MR), ac susceptibility(ac), differential scanning calorimetry (DSC), magnetisation (M(H) and M(T)) and DSC undermagnetic eld (dscH). T stands for measurement sweeping T and H sweeping H.

60

3.2. Sample characterisation

0 2 4 6 8 10 12 14 16 18 20 22 240

20

40

60

80

100

120

140

160

180

200

220

T=5 K x=0 x=0.05 x=0.1 x=0.18 x=0.2 x=0.365 x=0.45

µ0H (T)

M (

emu/

g)

0 4 8 12 16 20 24120

140

160

180

200

220

µ0H (T)

M (

emu/

g)

Figure 3.5: Magnetisation isotherms at 5 K for the following as-cast samples: x=

0, 0.05, 0.1, 0.18, 0.2, 0.365 and 0.45. Inset: Detail of the isotherms.

3.2 Sample characterisation

3.2.1 Magnetisation

Magnetisation measurements were mostly carried out at the Grenoble High Mag-netic Field Laboratory (GHMFL), which is managed by CNRS and MPI-FKF.An extraction magnetometer operating from 4.2 to 325 K (using a dynamic Hecryostat) and up to 23 T (using a 10 MW resistive magnet) was used. The rstmeasurement performed for each sample before the systematic study of the rst-order eld-induced magnetic phase transition (see section 5.2) was M(H) at 5 K.These magnetisation curves are displayed in Fig. 3.5 for a variety of as-cast sam-ples (x=0, 0.05, 0.1, 0.18, 0.2, 0.365 and 0.45). All curves show a large high-eldmagnetic susceptibility. The ordered magnetic moment, extrapolated to zero mag-netic eld, is displayed in Table 3.2 along with values obtained from the literature.The theoretical value of the ordered magnetic moment at the saturation for a freeGd3+ ion is 7.0 µB and for metallic Gd is 7.56 µB. Isotherms for x=0, 0.365 and

61


MS (µB/Gd at.)x this work literature0 7.38 7.32 [9], 7.41 [10]

0.05 6.620.0825 7.36 [10]

0.1 6.280.18 6.180.2 6.55

0.2525 7.39 [10]0.365 7.190.375 7.05 [11, 12]0.43 7.46 [10]0.45 7.180.5 7.36 [10]

Table 3.2: Ordered magnetic moment extrapolated at zero eld from the saturationmagnetisation at T=5 K, for as-cast samples with composition x. Some valuesgiven in literature are also included for comparison.

62


Compound TN(K) θC (K) pe f f (µB) Crystal structure ReferenceGd5Ge3 74/74 70/65 8.42/7.82 Mn5Si3-type hexagonal [15]/[14]Gd5Si3 -/55 97/42 8.34/8.5 Mn5Si3-type hexagonal [15]/[14]GdGe 62 -13 8.21 CrB-type orthorhombic [15]GdSi 50/56.2 5/-10.5 8.23/8.63 FeB-type orthorhombic [15]/[13]

Table 3.3: Magnetic and structural properties of possible secondary phases presentin Gd5(SixGe1−x)4 alloys. All phases are AFM. TN stands for the Néel temperature,θC for the paramagnetic Curie temperature and pe f f for the effective magneticmoment.

0.45 show saturation after a change of slope at ∼5, ∼12, and ∼9 T, respectively.Their saturation magnetisation exceeds the theoretical value by 0.2-0.4 µB, prob-ably due to the contribution of 6s and 5d electrons. The rest of the samples donot reach the saturation and also present a change of slope at ∼4 T with a slighthysteresis. This fact evidences the presence of non-FM residual phases, differentfrom the main 5:4 phase. The amount of these secondary phases depends on x, ac-cording to the values of the saturation moment. From the Gd-Si and Gd-Ge phasediagrams (Figs. 3.3 and 3.4), phases with 5:3 and 1:1 appear as the most likely.All these phases, whose properties are listed in Table 3.3, are AFM. Therefore,their magnetic behaviour could account for the slope of the high-eld magneti-sation isotherms, which is very large for both the 1:1 [13] and 5:3 [14] phases.Moreover, a spin-op metamagnetic transition is reported for Gd5Ge3 and Gd5Si3at elds of ∼6.8 and 4.8 T, respectively [14], in concordance with the observedbehaviour in our samples.

Magnetisation isotherms at 5 K can also be used to check possible effects ofthe heat treatments on the samples. Figure 3.6 shows M(H) at 5 K for x=0.45for the variety of heat treatments (as-cast, T2, T3 and T4). Measurement of thesample with T3 was carried out up to 5 T in a Quantum Design SQUID. Inset ofFig. 3.6 displays the same results for x=0.18 (as-cast and T4).

It is worth noting that samples with T2 and T3 treatments show the same curve,proving that annealing within 1000 and 1400 ºC is not a key parameter. Saturationmagnetisation is lowered ∼30% and there is a larger slope in the latter annealedsamples with respect to the as-cast sample, which evidences that a large part ofthe sample shows the appearance of an undesired phase, and/or the growing ofa secondary phase already present in the sample, that saturates at ∼10 T. For T4treatment, the saturation is only reduced ∼8% for x=0.45 (∼12% for x=0.18).Therefore, although all heat treatments homogenise the x value of the 5:4 main

63


0 2 4 6 8 10 12 14 16 18 20 22 240

20

40

60

80

100

120

140

160

180

200

x=0.45T= 5 K

T2 (1400ºC,Ar) T3 (1000ºC,Ar) T4 (950° C,vacuum) as-cast

M (

emu/

g)

µ0H (T)

0 5 10 15 200

50

100

150

200

T4 as-cast

x=0.18T= 5 K

µ0H (T)

M (

emu/

g)

Figure 3.6: Magnetisation isotherms at 5 K for x=0.45 with different heat treat-ments (as-cast, T2, T3 and T4). Inset: Magnetisation isotherms at 5 K for x=0.18(as-cast and T4).

64


phase (see sec. 3.2.2 -ac susceptibility-, sec. 3.2.3 -DSC- and sec. 3.2.4 -XRD-),they also yield the appearance of one or some undesired phases which are not FMat 5 K up to at least ∼10 T ) and this effect is lesser for the T4 treatment. However,these secondary phases do not affect the magnetism of the alloy.

3.2.2 Ac susceptibilityWhen a sample is exposed to an alternating magnetic eld (with an angular fre-quency ω = 2πν, being ν the linear frequency), written in complex notation as

H(t) = Hdc + Hac , eiωt , (3.1)

the magnetisation in the stationary state will also exhibit a periodic time depen-dence with the same frequency ω

M(t) = Hdcχ0 + Hacχac eiωt−iφ , (3.2)

where φ denotes the phase difference between the applied magnetic eld and thesample magnetisation. χ0(=M/Hdc) is the static susceptibility and it is not mea-sured in ac experiments. The complex ac susceptibility χ ≡ χac exp(−iφ) can bedecomposed in an in-phase component χ′ and an out-of-phase component χ′′,

χ ≡ χac e−iφ = χ′ − iχ′′ , (3.3)

where the minus sign arises from the fact that χ′′ is usually dened to be positive.If Hac is sufficiently small, the measured module of the susceptibility χac is to agood degree equal to the so-called dynamic susceptibility ∂M(H)/∂H. At zero dceld, the measured ac susceptibility is approximately equal to the dynamic initialsusceptibility, limH→0 ∂M(H)/∂H. Therefore, ac susceptibility is a suitable tech-nique to observe the different magnetic transitions in a material, since it measuresthe response of the system to a magnetic eld oscillation.

The eld in the bulk of a magnetic sample differs from the applied eld due tothe existence of magnetic dipoles which appear at the sample surface and generatea eld inside the sample that opposes to the external applied eld. This eldis known as the demagnetising eld. Accordingly, the effective eld inside thesample, Hint, is

Hint = Hext − DM , (3.4)where Hext is the external eld, M the sample magnetisation, and D the demag-netisation factor, which depends on the sample's geometry. In SI units, D mayadopt values within 0 and 1, while in CGS units, D varies between 0 and 4π. Fora magnetic eld applied along a cylinder of innite lenght or in parallel to thesurface of an innite plane, then D = 0, whereas D = 4π (or 1 in SI) when the

65


eld is applied perpendicular to an innite plane. In the special case of a sphere,D is equal to 4π/3 (CGS) or 1/3 (SI). For samples with an arbitrary geometrythe calculation of the demagnetisation factors becomes a very complicated task[16, 17].

Susceptibility has to be calculated using Hint rather than Hext. If the correctvalue of the susceptibility (calculated from Hint) is χint and the value that is mea-sured assuming a magnetic eld of Hext is χmeas, then

χint =χmeas

1 − Dχmeas. (3.5)

Since χ = χ′ − iχ′′ is a complex quantity, the above correction takes the followingform:

χ′int =χ′meas(1 − Dχ′meas) − Dχ′′meas(1 − Dχ′meas)2 + (Dχ′′meas)2 ; χ′′int =

χ′′meas(1 − Dχ′meas)2 + (Dχ′′meas)2 . (3.6)

It is useful to know under which conditions χmeas will signicantly differ fromthe real susceptibility χint. For this purpose, we express χmeas in terms of χint byinverting Eq. 3.5, as

χmeas =1

1χint

+ D. (3.7)

It is clear from the above expression that χmeas ≈ χint as long as χint 1/D.On the contrary, if χint is very large (χint 1/D), the measured susceptibilityχmeas ≈ 1/D and is almost insensitive to any real variation of χint. Large χint canoccur in ferromagnetic materials, and in this case, one should try to minimisethe demagnetisation factor D, which can be done by cutting the sample to anappropriate shape, such as in the form of a thin plane or an elongated needle withtheir axis parallel to the applied eld.

The equipment used to measure ac susceptibility in our samples is a LakeShore 7000 series susceptometer/magnetometer, which operates from 77 to 300 K.The high magnetisation of Gd5(SixGe1−x)4 samples (up to 200 emu/g at saturation)gives rise to a high demagnetising eld, hence a very small D value is desirable.

After rst measurements, we observed that D was too large and that the de-magnetising eld overlapped the results, even after the correction. Therefore, thenew samples were cut as long rods, to minimise D. Although the shape of the rodsis similar to a prism, calculation of D was approached by using the formula of aprolate spheroid when the eld is applied along its longer dimension, c [18]:

Dc =4π

r2 − 1

[r√

r2 − 1ln

(r +√

r2 − 1)− 1

](CGS ) , (3.8)

66


140 160 180 200 220 240 260 280 300

0.00.20.40.60.81.01.21.41.61.82.02.22.4

180 190 200 210 220 230 240 250 260 270 280 290 3000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

200 210 220 230 240 250 260 270 280 290 3000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

80 100 120 140 160 180 200 220 240 260 280 3000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

χ ''

(em

u/cm

3 )

x=0.365 #1as castH

ac=1.25 Oe

D=0.966 (cgs)

χ ' (

emu/

cm3 )

T (K)

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

χ ''

(10-3

em

u/cm

3 )x=0.45 #5 T2H

ac=1.25 Oe

D=1.24 (cgs)

χ ' (

emu/

cm3 )

T (K)

-10

-8

-6

-4

-2

0

x=0.45 #1as castH

AC=1.25 Oe

D=1.43 (cgs)

χ '(e

mu/

cm3 )

T(K)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

χ ''(

emu/

cm3 )

x=0.15 #1 as castH

ac=1.25 Oe

D=1.389 (cgs)

111Hz3330Hz χ

' (em

u/cm

3 )

T(K)

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

χ ''

(10

-3 e

mu/

cm3 )

Figure 3.7: Real and imaginary part of the ac susceptibility, for x=0.15 (#1),x=0.365 (#1) and x=0.45 (#1) as-cast samples, and for x=0.45 (#5) T2-heat-treated sample. Frequencies ν=111 Hz (black squares) and 3330 Hz (red spheres)were applied in nule dc eld and ac eld of 1.25 Oe. Demagnetising eld wascorrected using the value of D labeled for each sample and calculated from Eq.3.8.

being a = b < c and r = c/a. Measurements were performed at zero dc eld,ac eld of 1.25 Oe, and frequencies ν=111 and 3330 Hz. First as-cast sam-ples -x=0.15 (#1), 0.365 (#1) and 0.45 (#1)- were measured (see Fig. 3.7) tocheck whether the arc-melting furnace yielded a correct synthesis. The expectedmagnetic transitions in literature were used as the test to check the quality of thesamples. Demagnetisation factors of the samples were D=1.389, 0.966 and 1.43(CGS), respectively.

For x=0.45 (#1) as-cast sample (lower left panel in Fig. 3.7), the FM↔PMtransition is observed in both χ′ and χ′′. The rst-order nature is evident, sincethe transition spreads over a temperature range and thermal hysteresis is observed(Tt calculated at the point of maximum slope yields 245.9 K on heating and 241.3K on cooling, in agreement with other authors [1, 10, 19]). The effect of frequencyis only appreciable in the FM region, where the absolute values of both suscepti-bilities are lower at high frequency, as expected. A slight hysteresis of ∼0.5-0.8K occurs between both frequencies. At high temperatures (∼297 K) an anomalyis observed, probably due to a residual phase with x>0.5, with a second-order

67


FM↔PM transition at that temperature [1]. It is worth noting the negative valueof χ in the FM phase, vanishing to zero at the transition to the PM phase. χ isproportional to the energy absorbed in the system from the ac magnetic eld andis thus dened as positive. This unusual behaviour, observed in all Gd5(SixGe1−x)4samples, is believed to be caused by either the anomalous relaxation processes ofthe domain-wall motion and/or the excitation to a nonequilibrium state due to theexternal eld [11].

For x=0.365 (#1) as-cast sample (upper left panel in Fig. 3.7) the same be-haviour as that for x=0.45 sample is observed. In this case, Tt=188.4 K (193.0K) on cooling (heating), in agreement with other authors [11, 20] and the phasediagram (Fig. 2.2). No anomalies due to residual phases are observed. Hysteresisin frequency is ∼1 K.

For x=0.15 (#1) as-cast sample (upper right panel in Fig. 3.7), two transi-tions are observed, as it is expected for the x ≤ 0.2 compositional region. Therst-order magnetostructural FM↔AFM transition occurs at Tt=90 K (92 K) oncooling (heating), with a negligible hysteresis in frequency. At TN=135.6 K (with-out either thermal or frequency hysteresis), the second-order AFM↔PM magnetictransition takes place, giving rise to a high susceptibility. In this case, the imagi-nary part of the susceptibility is negative in the FM region, it becomes smaller butstill negative in the AFM phase, it changes to a positive value at the second-ordertransition region, and nally tends towards zero at the PM phase.

Therefore, samples within the different compositional regions can be success-fully prepared with our arc-melting furnace. After this rst conclusion, ac sus-ceptibility for a sample x=0.45 (#5) with a T2 thermal treatment was measured(lower right panel in Fig. 3.7) in order to check the effect of the heat treatmentson the transitions. For this sample, D=1.24 (CGS ). Tt appears at 237.5 K (240.0K) on cooling (heating), values very close to those given in Ref. [1], in whichthe sample was similarly heat-treated. The temperature spread in which the tran-sition takes place is reduced, showing a more abrupt jump, which is an indicationthat x distribution around the stochiometric value is narrowed. Thermal hystere-sis is also reduced. The high temperature anomaly disappears, showing that thisheat treatment removes most of 5:4 residual phases with x departuring from nom-inal value, in spite of the damage caused to the sample observed in magnetisationmeasurements (section 3.2.1).

Since the magnetisation measurements already showed that the T4 treatmentwas better than the T2 treatment, a detailed study of ac susceptibility for the as-cast and T4 heat-treated samples was carried out. In order to analyse the 0.24 ≤x ≤ 0.5 region, a new x=0.45 sample (#7) was arc-melted and cut, and a partof it was annealed according to T4. Samples were rod-shaped to minimise thedemagnetising eld, with D=1.82 and 1.13 (CGS) for as-cast and heat-treatedsamples, respectively. In this case, the main differences already observed between

68


200 210 220 230 240 250 260 270 280 290 3000.00.10.20.30.40.50.60.70.80.91.0

D correctedH

ac=1.25 Oe

Hdc

=0

x=0.45 #7

χ' (

emu/

cm3 )

T (K)

T4 @111HzT4 @3330Hzas cast @111Hzas cast @3330Hz

275 280 285 290 295 300

0.004

0.008

0.012

0.016

0.020

0.024

T(K)

χ' (

emu/

cm3 )

Figure 3.8: Real part of the ac susceptibility, for x=0.45 (#7) as-cast and heat-treated (T4) samples. Frequencies ν=111 Hz and 3330 Hz were used in nule dceld and ac eld of 1.25 Oe. Demagnetising eld was corrected. Inset: detail ofthe signal at high temperature, evidencing the anomaly present in as-cast sample,which is reduced in the heat-treated sample.

69


80 100 120 140 160 180 200 220 240 260 280

0.0

0.1

0.2

0.3

0.4

0.5

x=0.18 #1D correctedH

ac=1.25 Oe

Hdc

=0

as-cast @111Hz as-cast @3330Hz T4 rod1 @111Hz T4 rod1 @3330Hz T4 rod2 @111Hz

χ'(e

mu/

cm3 )

T(K)

120 125 130 135 140

0.012

0.014

0.016

T(K)

χ'(e

mu/

cm3 )

Figure 3.9: Real part of the ac susceptibility, for x=0.18 (#1) as-cast and heat-treated (T4) samples. Frequencies ν=111 Hz and 3330 Hz were used in nule dceld and ac eld of 1.25 Oe. Demagnetising eld was corrected. Inset: detail ofthe signal of the heat-treated sample near the second-order transition.

x=0.45 (#1) as-cast and x=0.45 (#5) T2 samples appeared again (see Fig. 3.8):(i) Signal in the FM phase is reduced in the heat-treated sample, evidencing anincrease in the magnetic correlations in the latter; (ii) the transition on cooling(heating) is tuned from 240.7 (245.2) K to 239.3 (241.1) K, being sharper andwith less thermal hysteresis for the T4-treated sample with respect to the as-castone; and (iii) the anomaly which appears in the as-cast sample at high temperature(T=294.5±0.5 K, i.e., a residual phase with x ∼0.51-0.53 [1, 21]) is considerablyreduced in the heat-treated sample (inset in Fig. 3.8).

The latter study was repeated within the x ≤ 0.2 region, using x=0.18 (#1)sample, which was cut in 3 rods and two of them were annealed (T4). Figure 3.9displays ac susceptibility for as-cast and annealed samples, where the demagnetis-ing eld has been corrected (D=1.27, 1.44 and 1.43 (CGS), respectively). The dif-ferences in the rst-order transition (in this case, FM↔AFM) appear again 3. Wenote that the large signal associated with the second-order phase transition in theas-prepared sample (TN=136.5 K), which is also observed in the x=0.15 sample

3Tt=105.2/107.2 K for as-cast sample and Tt=98.0/99.0 K for T4-annealed samples, on cool-ing/heating.

70


(see Fig. 3.7), is strongly reduced when the sample is heat-treated (TN=131.4 K,see inset in Fig. 3.9), in concordance with the fact that a more correlated systemyields a lower ac signal. The large signal of the second-order transition extendsover a range ∼125-∼180 K, it shows differences in the susceptibility depending onthe frequency and whether we are cooling or heating, and it shows more than onepeak. These effects are caused by the presence of FM clusters both in the AFMphase below TN and in the PM phase above TN , fact that is extensively studied inChapter 8.

It is thus conrmed in both compositional regions that the T4 heat treatmentimproves the magnetic structure and sharpens the transitions in all samples, al-though magnetisation measurements at low T showed a slight damage in the latter.

3.2.3 Differential Scanning Calorimetry

Differential scanning calorimetry (DSC)4 is the most suitable method to studyrst-order phase transitions since it measures the heat ow, so that a proper in-tegration of the calibrated signal yields the latent heat and the entropy change atthe transition [22]. In contrast, ac, relaxation and adiabatic calorimetry -the lattercommonly used for the study of MCE [23]- are suitable for determining the heatcapacity CP and they are thus well adapted for studying continuous second-orderphase transitions. It should be noted that in a rst-order transition, the experimen-tal determination of CP is intrinsically uncertain due to the release of latent heat[24]. Moreover, a heat input does not result in a modication of the temperatureof the sample and, accordingly, ac, relaxation and adiabatic techniques are notsuitable for studying rst-order phase transitions.

Therefore, DSC measurements were used to characterise the rst-order tran-sition in Gd5(SixGe1−x)4 alloys for x ≤ 0.5, and check the effect of the heat treat-ments on the transition. DSC measures the heat ow, Q(t), either released orabsorbed by a sample, through a sensor (battery of thermocouples) which fur-nishes an electrical voltage that is proportional to Q. Another sensor, with a sam-ple of reference5 mounted on top, is connected differentially to the former. Thisenables to minimise any drift caused by changes in the temperature of the calori-metric block, which is continuously scanned with time. T (t) of the block is thenmeasured by a carbon-glass resistor. This enables to compute numerically theheating/cooling rate, dT/dt, and dQ/dT= Q (dT/dt)−1 is thus obtained. The T -integration of the peak in dQ/dT , which appears only at the rst-order transition,

4DSC is extensively described in Chapter 4.5A material without transitions in the measuring temperature range.

71


190 200 210 220 230 240 250 260 270-600

-500

-400

-300

-200

-100

0

0

100

200

300

400

500

600

coolingx=0.45

#1 as-cast #1 T1 #5 T2 #7 as-cast #7 T4

T (K)

heatingx=0.45

#1 as-cast #1 T1 #5 T2 #7 as-cast #7 T4

dQ

/dT

(J/

kg·K

)

Figure 3.10: DSC data for different samples of x=0.45 compound, with variousheat treatments.

72


Tt (K) ∆S (J/kgK)x ID Heat T. cool. heat. TN (K) cooling heating

0.45 #1 NO 238.0 248.0 -19.95 16.55T1 224.7 230.5 -8.64 11.72

#5 T2 230.9 237.8 -12.82 12.38#7 NO 238.1 245.0 -17.6 17.0

T4 232.2 240.0 -18.2 16.90.365 #1 NO 195.0 200.0 -30.0 27.42

T1 ∼177 ∼181 Not integrable#2 T2 174.3 177.3 -19.63 21.28#3 NO 198.0 207.9 -27.5 25.5

0.3 #2 NO 165.8 175.1 -32.1 30.90.25 #2 NO 139.3 149.8 -39.5 37.40.2 #1 NO 105.3 119.3 ∼128 -36.3 38.2

0.18* #1 NO 98.3 112.8 ∼127 -29.7 28.8T4 93.0 105.5 ∼128 -25.7 21.4

0.15* #1 NO ∼90 ∼94 ∼128 Not integrableT1 - - ∼133 Not integrable

Table 3.4: Entropy change, ∆S , and temperature of the rst-order transition, Tt,obtained from DSC in all measured samples, on cooling and heating. TN for thecorresponding samples is also displayed. *Tt in these samples is close to LN2 temperature(77 K) and the complete integration of the transition is difficult to obtain.

yields the value of the latent heat (L) and the entropy change (∆S ):

L =

∫ TH

TL

dQdT dT ; ∆S =

∫ TH

TL

1T

dQdT dT , (3.9)

where TH and TL are respectively temperatures above and below the starting andnishing transition temperatures. The temperature of the rst-order phase transi-tion, Tt, may be evaluated as the temperature at the maximum of the dQ/dT peak.Our home-made calorimeter operates over a temperature range from 77 K (LN2)to 340 K (electrical heater) and has an accuracy of 5-10%.

All measurements, carried out for a variety of samples and annealings, fromx=0.15 to x=0.45 (compositions with Tt within the 77-340 K temperature range)are compiled in Table 3.4. Samples with Tt closer to 77 K (x=0.15 and 0.18) showproblems since dQ/dT cannot be entirely integrated. Moreover, the second-order

73


transition, which is shown as a λ-peak in dQ/dT curve, inuences the baseline ofthe rst-order peak in x=0.18.

The behaviour of the rst-order phase transition and the related entropy changeobtained by DSC measurements as a function of heat treatment, can be sum-marised in Fig. 3.10 for the x=0.45 compound, which reproduces the main re-sults obtained for the whole compositional range 0 ≤ x ≤ 0.5. First, two differentas-cast samples (#1 and #7) are displayed in Fig. 3.10 in order to check the re-productivity of the features of x=0.45. Both samples yield the same ∆S valuesand similar Tt, with sligth different hysteresis (see Table 3.4 and Fig. 3.10). T1heat treatment for sample #1 shifts Tt to lower temperatures (by ∼13-18 K) and,although thermal hysteresis is reduced, entropy change is reduced ∼40-60%. Thisproves that a large part of the Gd5(Si,Ge)4 (5:4) structure is transformed into anew phase by T1 annealing. The fact that ∆S largely decreases suggests that theseundesired new phases do not show giant MCE. Besides, for x=0.365 (#1) sample,the rst-order transition is seriously affected and dQ/dT peak cannot be properlyintegrated (see Table 3.4). T2 treatment for x=0.45 shifts Tt by ∼7-11 K to lowerT and the entropy change is reduced ∼30%. The latter value agrees with thatobtained from magnetisation measurements, giving further evidence that T2 an-nealing also transforms part of the 5:4 structure. Finally, the T4 heat treatment forsample x=0.45 (#7) decreases Tt by ∼5-6 K, narrows the width of the rst-orderpeak (i.e., transition is narrowed because the spread in the x value is reduced) andyields the same values of ∆S as the non-treated sample, proving that this anneal-ing procedure is the most suitable for our samples, in agreement with high eldM(H) and ac susceptibility.

3.2.4 X-Ray DiffractionThe detailed determination of the crystallographic structure is relevant for theunderstanding of the magnetic behaviour in Gd5(SixGe1−x)4 alloys (section 2.4).Here we present X-ray diffraction (XRD) for x=0.5, x=0.3, x=0.2, x=0.1 and x=0as examples for the two compositional regions, and x=0.45 samples with differentheat treatments to show the effect of the latter.

x=0.5 (#0) as-cast sample was analysed by XRD and it is shown as an ex-ample of the 0.24 ≤ x ≤ 0.5 compositional range. Diffraction over the uppersides of the original as-cast button (with characteristic plate-like shaped -faceted-crystalls on the surface) showed a strong texture. The sample was cut into slices,one of them was measured, and still showed texture. Therefore, powder XRDwas performed to obtain all reections of the structure of the alloy. The newpowder diffractogram, which was tted using the Rietveld renement programFULLPROF [25], is displayed in Fig. 3.11. The lattice and atomic parametersdetermined in Ref. [21] were used as starting points. Renement with the mono-

74


19 26 33 40 47 54 61 68 75 82 89

-400

-200

0

200

400

600

800

1000

2θ (°)2θ (°)2θ (°)2θ (°)

Inte

nsity

(a.

u.)

x=0.5 (as-cast)

Figure 3.11: XRD data for an as-cast sample of the x=0.5 compound. Open cir-cles correspond to experimental data, while the solid line corresponds to the bestt of the spectra with two phases: one major phase corresponding to the expectedmonoclinic phase of the x=0.5 compound and one minor orthorhombic phase cor-responding to an alloy with x∼0.55. Vertical lines show the Bragg positions (thoseclose to the spectra are for x=0.5, those further apart are for a compound withx∼0.55) and the bottom solid line is the difference between experimental and t-ted data.

75


Gd5(SixGe1−x)4 x=0.5

space group a (Å) b (Å) c (Å) γ (º)P1121/a 7.577(1) 14.790(3) 7.779(1) 93.09(1)

atom x/a y/b z/c occupancyGd1 in 4(e) 0.320(4) 0.251(1) -0.002(3) 1Gd2a in 4(e) -0.001(3) 0.095(1) 0.184(1) 1Gd2b in 4(e) 0.019(3) 0.398(1) 0.181(3) 1Gd3a in 4(e) 0.357(3) 0.886(1) 0.176(3) 1Gd3b in 4(e) 0.330(3) 0.619(1) 0.175(1) 1

M1 in 4(e) (T ) 0.223(6) 0.251(3) 0.379(6) 0.6(1)M2 in 4(e) (T ) 0.97(1) 0.247(6) 0.91(1) 0.6(1)

M3a in 4(e) (T ′) 0.213(7) 0.965(3) 0.514(6) 0.4(1)M3b in 4(e) (T ′) 0.151(7) 0.565(3) 0.502(6) 0.4(1)

Gd5(SixGe1−x)4 x∼0.55

space group a (Å) b (Å) c (Å) γ (º)Pnma 7.510(1) 14.779(3) 7.802(2) 90atom x/a y/b z/c occupancy

Gd1 in 4(c) 0.344(4) 1/4 0.016(3) 0.5Gd2 in 8(d) 0.023(2) 0.0992(8) 0.178(2) 1Gd3 in 8(d) 0.323(2) 0.8775(9) 0.176(2) 1

M1 in 4(c) (T ) 0.204(5) 1/4 0.371(7) 0.02(6)M2 in 4(c) (T ) 0.959(8) 1/4 0.913(8) 0.14(6)M3 in 8(d) (T ′) 0.171(7) 0.955(2) 0.472(6) 0.42(7)

Table 3.5: Space group, cell parameters, atomic sites and occupancy for the twophases present in the as-cast x=0.5 sample. M stands for the atomic sites occupedby a mixture of Si and Ge atoms. T and T ′ sites are explained in section 2.3. Inthis table only Si occupations are listed for M positions.

76


clinic structure (space group P1121/a) did not account for all reections presentin the experimental diffractogram. The existence of a secondary phase with x>0.5,which has an orthorhombic structure (space group Pnma), was thus assumed [1].With this second phase all the remaining reections were tted. Table 3.5 dis-plays all rened structural parameters. The obtained values are consistent withthose given in literature [21, 26] and allow to estimate x of the second phase as∼0.55 (in good agreement with Ref. [21]). Some preferential orientation (∼17%) still exists along the (0 1 0) direction. The percentage of the secondary phasereaches 40% mol. This high value is easily understood by taking into account avery recent result for x0.5 compounds [7, 5, 6]: a polymorphism between Mand O(I) phases occurs depending on the temperature that the sample reaches inthe melting and/or posterior annealing (section 2.2). Therefore, both phases -Mand O(I)- are typical of x0.5 alloys. It is worth noting that occupancy of Si andGe atoms is not equiprobable, but there are preferences depending on the atomicsites, as discussed in Refs. [26, 27]. The study of a monocrystal with x=0.5showed that T sites (intraslab) are preferably occuped by Si (60%), while the oc-cuppancy in T' sites (interslab) is 60% Ge and 40% Si [26]. In our polycrystallinesample, we obtain the same occupancy for the main monoclinic phase (see Table3.5). The other samples with the same compositional region, x=0.3 (#2 as-cast)and x=0.45 (#7 as-cast) also show the monoclinic structure, with a residual phase(∼10% mol.) that we have indentied and indexed as Ge-rich Gd(Si,Ge) (BCr-type orthorhombic structure, Cmcm space group).6 x=0.45 as-cast sample alsopresents a secondary phase corresponding to a 5:4 phase with x>0.5, i.e., withO(I) structure. The cell parameters of the analysed samples are displayed in Table3.7). With these secondary phases all remaining reections were tted.

As a paradigmatic example of the Ge-rich compositional region (0 ≤ x ≤0.2), the x=0.1 compound (#1 as-cast) is presented. The sample was powderedin order to get the diffractogram, which is shown in Fig. 3.12. The expectedPnma space group (orthorhombic structure) was found. The lattice and atomicparameters determined in Ref. [2] were used as starting points. Minor amountsof residual phases were detected and indexed as Ge-rich Gd(Si,Ge) (∼10% mol.)and Gd5(Si,Ge)3 (∼5% mol., P63/mcm space group with Mn5Si3-type hexagonalstructure).7 The rened unit-cell parameters and atomic positions are displayed inTable 3.6. x=0 (#1 and #3) and x=0.2 (#1) as-cast samples yield similar results,although for the latter only Gd(Si,Ge) is detected as residual phase (∼10% mol.).Table 3.7 compiles the space group, cell parameters and residual phases for the

6GdSi shows a BFe-type orthorhombic structure (Pnma), which is different from that of GdGe.The identied secondary 1:1 phase presents the GdGe structure with lower lattice parameters,indicating the presence of Si in the structure.

7In this case, Gd5Si3 and Gd5Ge3 show the same structure, with larger lattice parameters forthe latter compound.

77


21 27 33 39 45 51 57 63 69 75 81

-600

-400

-200

0

200

400

600

800

1000

1200

1400

2θ (°)2θ (°)2θ (°)2θ (°)

Inte

nsity

(a.

u.)

x=0.1 (as-cast)

Figure 3.12: XRD data for an as-cast sample of the x=0.1 compound. Open circlescorrespond to experimental data, while the solid line corresponds to the best t ofthe spectra with three phases: one major phase corresponding to the expectedorthorhombic phase of the x=0.1 compound and two minor phases correspondingto 1:1 [Gd(Si,Ge)] and 5:3 [Gd5(Si,Ge)3] compounds. Vertical lines show theBragg positions (for 5:4, 1:1 and 5:3 phases, from top to bottom) and the bottomsolid line is the difference between experimental and tted data.

78


Gd5(SixGe1−x)4 x=0.1

space group a (Å) b (Å) c (Å) γ (º)Pnma 7.683(2) 14.824(3) 7.778(2) 90atom x/a y/b z/c occupancy

Gd1 in 4(c) 0.296(2) 1/4 -0.005(2) 0.5Gd2 in 8(d) -0.014(1) 0.1009(7) 0.194(2) 1Gd3 in 8(d) 0.382(1) 0.8806(7) 0.164(2) 1

M1 in 4(c) (T ) 0.160(4) 1/4 0.354(4) 0.05M2 in 4(c) (T ) 0.901(5) 1/4 0.909(4) 0.05M3 in 8(d) (T ′) 0.228(3) 0.956(3) 0.485(3) 0.1

Gd(Si,Ge)

space group a (Å) b (Å) c (Å) γ (º)Cmcm 4.330(1) 10.770(3) 3.959(1) 90

Gd5(Si,Ge)3

space group a (Å) b (Å) c (Å) γ (º)P63/mcm 8.547(2) 8.547(2) 6.376(2) 120

Table 3.6: Space group, cell parameters, atomic sites and occupancy for as-castx=0.1 sample. M stands for the atomic sites occuped by a mixture of Si and Geatoms. T and T ′ sites are explained in section 2.3. In this table only Si occupationsare listed for M positions. Space group and cell parameters for secondary phasesare included.

79


Gd5(SixGe1−x)4

x space group a (Å) b (Å) c (Å) γ (º) Sec. phases0 Pnma 7.7010(8) 14.832(2) 7.7896(9) 90 1:1, 5:3

0.1 Pnma 7.683(2) 14.824(3) 7.778(2) 90 1:1, 5:30.2 Pnma 7.676(1) 14.817(2) 7.772(1) 90 1:10.3 P1121/a 7.622(2) 14.826(4) 7.780(2) 92.93(2) 1:1

0.45 P1121/a 7.5896(8) 14.810(2) 7.7846(9) 93.123(7) 1:1, x>0.50.5 P1121/a 7.577(1) 14.790(3) 7.779(1) 93.09(1) x∼0.55

Table 3.7: Space group, cell parameters and secondary phases for variousGd5(SixGe1−x)4 as-cast alloys, obtained from XRD data.

analysed compounds, being in good agreement with Ref. [21]. We note that allcompositions except for x=0.5 present the 1:1 phase. This phase is GdGe-based(a= 4.339(4) Å, b = 10.79(2) Å and c = 3.973(4) Å[28]), but the decreasing ofthe rened cell parameters with x suggests that it is doped with Si, see Table 3.8.With these secondary phases all remaining reections were tted.

Finally, X-ray powder diffraction was carried out in different parts of x=0.45(#7) sample, with no heat treatment, T4 treatment and T4+quenching (see Fig.3.13), in order to prove the effect of annealing on the crystallographic structureof the samples. The three diffractograms showed the same characteristics (mainphase being monoclinic with space group P1121/a) with only slight differencesin the intensity of some reection peaks, which correspond to minor amonts ofresidual phases. Some of these peaks increased from the as-cast sample to thequenched one, corresponding to Gd(Si,Ge) phase and proving that the heat treat-ment favours the segregation of secondary phases. Other peaks, corresponding tothe O(I) structure of the 5:4 phase with x>0.5, decreased with the heat treatments.

To conclude, XRD enables us to show that samples synthesized with the arc-melting furnace present the expected crystallographic structures, which dependson the compositional region of Gd5(SixGe1−x)4 alloys system. T4 heat treatmentand posterior quenching maintains the crystallographic structure of the samplesand homogenises the x value of the 5:4 main phase, although they favour thesegregation of the secondary phases already present in the as-cast samples.

80


Gd(Si,Ge)

Main phase (x) space group a (Å) b (Å) c (Å)0 Cmcm 4.386(2) 10.676(3) 3.981(2)

0.1 Cmcm 4.330(1) 10.770(3) 3.959(1)0.2 Cmcm 4.328(1) 10.758(3) 3.943(1)0.3 Cmcm 4.329(1) 10.735(3) 3.9393(8)

0.45 Cmcm 4.328(1) 10.719(2) 3.916(1)

Table 3.8: Space group and cell parameters for secondary 1:1 phases present invarious Gd5(SixGe1−x)4 as-cast alloys, obtained from XRD data.

21 27 33 39 45 51 57 63 69 75 81

0

200

400

600

800

1000

1200

1400

2θ (°)2θ (°)2θ (°)2θ (°)

Inte

nsity

(a.

u.)

x=0.45 #7

as-cast

T4 treatment

T4 + quenching

Figure 3.13: XRD data for x=0.45 samples with different annealings.

81


Figure 3.14: Secondary electron image, obtained with SEM, of the surface of ax=0 as-cast sample (#1).

3.2.5 Scanning Electron Microscopy (SEM) and Electron-beamMicroprobe

Some samples were analysed by SEM in order to image the different phasespresent. These phases were studied by Energy Dispersive Spectroscopy (EDS),which is incorporated in the SEM system itself. Different phases in the samplewith x=0 were difficult to observe due to the presence of microcracks in the sur-face, since Ge-rich alloys are more brittle than Si-rich ones (see Fig. 3.14). EDSfound only the main 5:4 phase in the various regions of the sample. Heat-treated(T4) sample with x=0.45 showed a more polished surface and a backscatteredelectron image unveiled four different phases (dark grey, middle grey, light greyand middle-grey lines in Fig. 3.15). The various phases were difficult to differen-tiate, since 5:4, 5:3 and 1:1 phases are close in composition.

82


Figure 3.15: Backscattered electron image, obtained with SEM, of the surface ofa x=0.45 T4-treated sample (#7).

83


Figure 3.16: Backscattered electron image, obtained with the SEM of a micro-probe, of the surface of x=0 as-cast sample (#1).

84


Figure 3.17: Backscattered electron image, obtained with the SEM of a micro-probe, of the surface of x=0.2 as-cast sample (#1).

85


Figure 3.18: Backscattered electron image, obtained with the SEM of a micro-probe, of the surface of x=0.45 as-cast sample (#7).

86


For this reason an electron-beam microprobe -with Wavelenght DispersiveSpectroscopy (WDS)- was used due to its higher resolution. A variety of sam-ples (x=0 #1 as-cast, x=0.2 #1 as-cast, x=0.45 #7 as-cast and T4-treated) wereproperly prepared, with a gold deposition on top, to be analysed with the micro-probe. For x=0, althought different tones could be identied in a backscatteredelectron image (see Fig. 3.16), WDS analysis yielded the same 5:4 phase (being55.8% at. Gd for the main light grey phase and 54.2% at. Gd for the dark-greylines). Taking into account the latter result and that the 5 K magnetisation in thissample is saturated at relatively low eld (as compared with the other samples, seeFig. 3.1) -yieding a saturation magnetisation similar to other reported values andhigher than the theoretical 7.0 µB- the secondary phases detected by XRD in thissample are indeed residual. For x=0.2, the main observed phase correspondedto the 5:4 phase with x=0.197, i.e., the nominal stochiometric value. A patternof dark lines is observed in the backscattered electron image of this sample (seeFig. 3.17), which appears to be a 5:4 phase with x=0.158. This secondary phaseis not observed in XRD, probably because it presents the same crystallographicstructure (O(I)) and the x value is close to the value of the main phase, whichjust yields broader peaks in the diffractogram. For x=0.45 as-cast sample, themain observed phase also corresponded to the 5:4 phase, with x=0.413, which isslightly lower than the nominal value. A pattern of dark lines is also observed inthe backscattered electron image of the sample (see Fig. 3.18), and in this casethe WDS analysis yielded a 5:4 phase with x=0.509, in agreement with the valueestimated from the anomaly in the ac susceptibility at T=294.5±0.5 K (∼0.51-0.53). Finally, the same x=0.45 sample, heat-treated, was analysed (Fig. 3.15).The dark-grey phase corresponds to a 2:3 phase, which is present in the Gd-Gephase diagram, close to the 1:1 phase, but not in the Gd-Si one (see Figs. 3.3 and3.4). The middle-grey phase, surrounding the dark-grey phase, corresponds to the1:1 phase, with a Si/Ge ratio of x=0.40. The light-grey phase is the main phase,with 5:4 ratio between Gd and Si/Ge. In this phase, the ratio between Si and Geis x=0.415, which is very close to the value observed in the as-cast sample. Thepattern of dark lines, crossing the rest of phases, is still observed. The WDS anal-ysis of these lines yields a 5:4 phase with x=0.460. This is in agreement with theobservation in ac susceptibility, in which the anomaly related to the residual phasewith x>0.5 disappeared after the heat treatment. The segregation of phases withratio different from 5:4 during the heat treatment is evident from SEM images andelectron-beam Microprobe analysis, as already suggested from other experimentaltechniques. We point out that the 2:3 phase, which was not detected using XRDpatterns, is a very poorly studied phase and almost no literature is available.

87


3.3 ConclusionsAll techniques show that our home-made arc-melting furnace synthesizes Gd5(SixGe1−x)4 samples of the desired stochiometry. SEM and microprobe analyses showthat the main 5:4 phases with the desired x are obtained. Ac susceptibility showsthe magnetic transitions occurring in these alloys, while XRD detects the crystal-lographic structures corresponding to the phases at room temperature. M(H) at 5K shows the presence of secondary amounts of Gd(Si,Ge) (1:1) and Gd5(Si,Ge)3(5:3) phases in all samples. This presence is conrmed by XRD and microprobeanalyses, which also detect residual 5:4 phases with an x value different from thatof the main phase. DSC shows that all samples present the rst-order transition,and that secondary phases do not affect the latter. Heat treatments favour the seg-regation of these secondary phases (M(H), XRD, SEM and microprobe), but alsoreduce the spread in the x value (ac susceptibility and DSC) and removes 5:4 resid-ual phases with very different x values (as susceptibility and microprobe). There-fore, a trade-off between phase segregation and removal of x spread is needed. T4and T4 + quenching treatments enable such a trade-off.

88

Bibliography

Bibliography[1] L. Morellon, P. A. Algarabel, M. R. Ibarra, J. Blasco, B. García-Landa, Z.

Arnold, and F. Albertini, Phys. Rev. B 58, R14721 (1998).


[3] J. S. Meyers, S. Chumbley, F. Laabs, and A. O. Pecharsky, Scripta Mater.47, 509 (2002).


[5] V. K. Pecharsky, A. O. Pecharsky, and K. A. Gschneidner, Jr., J. AlloysComp. 344, 362 (2002).

[6] V. K. Pecharsky, G. D. Samolyuk, V. P. Antropov, A. O. Pecharsky, and K. A.Gschneidner, Jr., J. Solid State Chem. 171, 57 (2003).

[7] A. O. Pecharsky, V. K. Pecharsky, and K. A. Gschneidner, Jr., J. Appl. Phys.93, 4722 (2003).

[8] J. S. Meyers, S. Chumbley, F. Laabs, and A. O. Pecharsky, Acta Mater. 51,61 (2003).





[13] L. D. Tung, K. H. J. Buschow, and J. J. M. F. N. P. Thuy, J. Magn. Magn.Mater. 154, 96 (1996).

[14] F. Canepa, S. Ciraci, and M. Napoletano, J. Alloys Comp. 335, L1 (2002).

[15] K. H. J. Buschow, Rep. Prog. Phys. 42, 1373 (1979).

[16] E. C. Stoner, Philos. Mag. 36, 803 (1945).

89


[17] D. X. Chen, J. A. Brug, and R. B. Goldfarb, IEEE Trans. Magn. 27, 3601(1991).





[22] W. Hemminger and G. Höhne, Calorimetry. Fundamentals and practice.(Verlag Chemie, Weinheim, 1984).

[23] V. K. Pecharsky, J. O. Moorman, and K. A. Gschneidner, Jr., Rev. Sci. In-strum. 68, 4196 (1997).


[25] J. L. Rodríguez-Carvajal, Physica B 192, 55 (1992).



[28] A. G. Tharp, G. S. Smith, and Q. Johnson, Acta Crystallogr. 20, 583 (1966).

90

Chapter 4

Design and experimental set up of aDifferential Scanning Calorimeterwith magnetic eld

4.1 IntroductionCalorimetry has been used to study a number of physical properties of solids formore than a century [1]. Even now, it is considered to be one of the best suitedmethods to determine the nature of phase transitions. Nowadays, a wide varietyof calorimeters exist, which can be broadly classed into two groups. The rstgroup includes those devices which measure the heat ow between the sampleand a thermal block, while the temperature of the calorimeter is continuouslychanged (scanning calorimeters). Most of them use a dummy sample so that theywork differentially (differential scanning calorimeters, DSC). The second groupincludes the calorimeters which are based on the measurement of the temperatureof the sample after a small amount of heat is supplied (adiabatic calorimetry, relax-ation calorimetry and ac calorimetry). In these instruments, the temperature of thecalorimeter is kept constant during the measurement. There are also calorimeterswhich combine the two operating methods, as for instance in case of the modu-lated differential scanning calorimeters which have been recently developed [2].Continuous efforts are devoted to designing calorimeters that are better adapted tothe new materials and with better performances [3, 4].

DSCs are particularly suited to studying rst-order phase transitions since theymeasure the heat ow, and a proper integration of the calibrated signal yields thelatent heat of the transition. In contrast, a.c., relaxation and adiabatic calorime-try are suitable for determining the heat capacity CP and hence are well adaptedfor studying continuous phase transitions. It should be noted that in a rst-order

91

CHAPTER 4. DESIGN OF A DSC WITH MAGNETIC FIELD

transition, a heat input does not result in a modication of the temperature of thesample and, therefore, the latter techniques are not suitable for studying this kindof phase transitions.

The determination of the entropy change associated with the magnetocaloriceffect in Gd5(SixGe1−x)4 series of alloys has been an issue of controversial debate[5, 6, 7, 8, 9], since the effect occurs in the vicinity of the rst-order magne-tostructural transition (see section 2.7). As discussed in section 1.3, the adiabatictemperature change can be evaluated both indirectly through CP(T,H) measure-ments, which are usually obtained with quasiadiabatic calorimeters [10]), and di-rectly with specially designed devices [11, 12]. However, the entropy change canonly be obtained indirectly using CP(T,H) measurements, which are not valid inthe vicinity of rst-order transitions, or magnetisation measurements, and evenin this case the correct approach is not clear (section 2.7). Nevertheless, if theentropy change arises from a rst-order transition, DSC enables us to measure itin a more direct manner. Furthermore, since MCE appears in a rst-order transi-tion only if the latter can be eld-induced, DSC under applied magnetic eld isthus expected to be the ideal technique to study the entropy change at rst-ordermagnetostructural transitions. In this chapter, we describe a high-sensitivity dif-ferential scanning calorimeter specially designed to operate over a temperaturerange from 10 to 300 K and under magnetic elds of up to 5 T. This calorimeterprovides accurate values of the latent heat and entropy change under magneticeld at a rst order phase transition.

4.2 Experimental DetailsThe calorimeter can be adapted to any cryostat equipped with a superconductingmagnet. The apparatus described here has the appropriate size to be used witha Teslatron© (Oxford Instruments) system as a host platform (see Fig. 4.1). Amagnetic eld of up to 5 T is generated along the vertical axis by a superconduct-ing magnet contained in the Teslatron system. Figure 4.2 shows side (a) and top(b) cross sections of the calorimeter. The calorimeter is a copper spool (1) 1. It ismechanically clamped to a long stainless steel tube. All wiring is routed throughthis rod and exits the system via an electrical feedthrough at its far end. The twosensors (2), which are differentially connected, are placed on the attened innersurfaces of the spool. In order to ensure a good thermal contact, the sensors arecoupled to the block with General Electrics Oxford Varnish. These sensors arebatteries of thermocouples (Melcor FCO.45-32-05L) 2 made of P-N- junctions of

1For optimal performance, the mass of the spool has to be large enough so that its specic heatamounts at least 100 times that of the sample (or reference).

2©Melcor Thermoelectrics, 1040 Spruce st., Trenton, NJ; 08648.

92

4.2. Experimental Details

Figure 4.1: Overview of the Teslatron© (Oxford Instruments) cryomagnetic sys-tem operating with LHe from 1.5 to 300 K. The cryostat holds a superconductingcoil which generates magnetic elds up to 5 T. Electronics of the device can beseen at the right-hand side.

Bi2Te3 (32 pairs of junctions on a 6.5 x 6.5 mm2 surface). Sample (3) and inertreference (4) are placed directly on top of each sensor. They are held in place (ingood thermal contact with the sensor) by winding a thin (less than 0.2 mm diame-ter) nylon wire around the assembly. Electrical wires exit the calorimeter through2.5 mm diameter holes, and they are thermally coupled to the upper part of thespool before passing through the stainless steel rod. Such a coupling avoids theexistence of thermal gradients on the wires which could give rise to spurious ther-moelectric voltages. The temperature of the calorimeter is scanned by changingthe temperature of the variable temperature insert (VTI) of the Teslatron cryostat[13]. An accurate reading of the actual temperature of the calorimeter is achievedby monitoring the electrical resistance of a Carbon-glass resistor (LakeShore Cry-otronics INC. CGR-1-500) (5) embedded inside the spool. In order to minimiseconvection of the exchange gas inside the calorimeter, the whole assembly is cov-ered by an external copper cylinder (1 mm thick) (6), which is screwed to the up-per part of the spool. For an optimal operation, the pressure inside the calorimetershould be within the range 200-600 mbar [13]. High purity helium is requiredsince impurities affect the sensitivity of the calorimeter.

The heat released (or absorbed) by the sample (see Fig. 4.2(c)) is measured byreading the voltage furnished by the thermobatteries (electrical output) by usinga nanovoltmeter (Keithley 182). It is worth reminding that, since the two sensors

93


Figure 4.2: Side (a) and top (b) view cross sections of the calorimeter. (1) Cop-per spool, (2) sensors, (3) sample, (4) reference, (5) carbon-glass resistor and (6)cover. The magnetic eld, ~B = µ0 ~H, is applied along the symmetry axis of thecalorimeter. (c) Detail showing the heat ow Q for an exothermal transition.

94

4.3. Calibration

Figure 4.3: View of the probe with the differential scanning calorimeter (copperspool) at the right-hand side, adapted to be used with the Teslatron cryomagneticsystem.

are connected differentially, the drift in the calorimetric output associated withchanges in the temperature of the calorimetric block are minimised by the factthat the heat ow of the reference is substracted from that of the sample, andtherefore, the major contribution to the calorimetric output is the thermal powerreleased (or absorbed) by the sample during the rst order phase transition (latentheat). The resistance of the carbon-glass is read by means of a lock-in amplier(EG & G 7260)3. The whole system (including the electronics of the Teslatronmeasurement system) is controlled by a PC computer. Values of voltage V(t) andtemperature T (t) are acquired at typical rates of 0.25 Hz. A view of the calorimeteris shown in Figs. 4.3 and 4.4.

4.3 CalibrationSince the thermocouples are made of semiconducting elements, the magnetic eldis not expected to affect their thermoelectric output. In contrast, the output voltagewill indeed signicantly depend on temperature and therefore, a proper calibra-tion over the whole operating temperature range is needed. To carry out such acalibration, the sample is replaced by a manganin resistance (50 Ω). A constantpower (W) is dissipated by the Joule effect at the resistance (without the appliedmagnetic eld), and the electrical output at the steady state, Y , is measured. Thesensitivity, K, is then given by: K = Y/W. A typical calorimetric curve for W=18mW is shown in the inset of Fig. 4.5 4. The values obtained for the sensitivityat different temperatures are plotted in Fig. 4.5. Data can be tted by the curve:K(mV/W)=1.4 × 10−8T 4 − 2.0 × 10−5T 3 + 5.1 × 10−3T 2 + 0.86T , which is also

3A microvoltmeter can also be used to read the resistance of the thermometer.4The linearity of the calorimeter has been veried by performing calibrations with different

values of the released power.

95


Figure 4.4: Detail of the differential scanning calorimeter.

Figure 4.5: Sensitivity at zero eld as a function of temperature. The solid line isa t to the data. Inset: example of a typical calibration thermogram at T=173 K.

96

4.3. Calibration

100

200

300

400

500

600 µ0H=0

µ0H=5T

dQ/d

T (

mJ/

K)

220 230 240 250 260-50

0

50

δ (m

J/K

)

T (K)

Figure 4.6: Calorimetric curves recorded at the martensitic transition of aCu2.717Zn0.646Al0.637 single crystal at zero eld (continuous curve) and at µ0H=5 T(discontinuous curve). The difference between these two curves δ is also shown.

plotted in the gure. It is interesting to note that the room temperature value isaround 10 times larger than the sensitivity for a conventional DSC [14], and thatat temperatures as low as 10 K, a reasonably high value is still obtained.

To check the performances of the calorimeter, we have selected a Cu-Zn-Alalloy undergoing a structural (martensitic) transition, as a standard system. Thismaterial is diamagnetic and hence the transition is not affected by a magnetic eld;this will provide a good test of the insensitivity of the sensors to a magnetic eld.On the other hand, the values for the latent heat and entropy change associatedwith the martensitic transition are very well established over a broad temperaturerange by the use of several experimental techniques (the transition temperature canbe modied by slightly tuning the composition) [15, 16]. The calorimetric curvesrecorded during the reverse transition of a Cu-Zn-Al crystal are shown in Fig. 4.6,in the absence of a magnetic eld (continuous line) and for an applied eld of 5T (discontinuous curve). Since the two curves are almost indistinguishable, wehave also plotted the difference between them, δ. No signicant inuence of themagnetic eld is observed.

97


In order to obtain the latent heat and the entropy change in a rst-order phasetransition, we calculate the heat ow as Q(t)=V(t)/K, we compute numericallythe heating/cooling rate dT/dt from the recorded T (t) and nally the calorimetriccurve dQ/dT= Q (dT/dt)−1 is achieved. This calorimetric signal has to be cor-rected from the baseline (details can be found, for instance, in Ref. [17]). Thelatent heat and the entropy change are then given by:

L =

∫ TH

TL

dQdT dT ; ∆S =

∫ TH

TL

1T

dQdT dT , (4.1)

where TH and TL are respectively temperatures above and below the startingand nishing transition temperatures. The values obtained for the latent heat[L(µ0H=0)=336 ± 3 J/mol and L(µ0H=5T)=335 ± 3 J/mol] and for the entropychange [∆S (µ0H=0)=1.40 ± 0.01 J/(mol K) and ∆S (µ0H=5T)=1.39 ± 0.01 J/(molK)] at the martensitic transition of Cu-Zn-Al are in excellent agreement with pub-lished values [15, 16].

4.4 Results sweeping TThe apparatus described in the present chapter is particularly well adapted formeasuring the entropy change at the magnetostructural phase transition under-gone by alloys which exhibit the giant magnetocaloric effect. Proper measure-ment of the entropy change associated with the rst-order transition is expected tocontribute to a better understanding of this interesting phenomenon, in particularfor Gd5(SixGe1−x)4 series of alloys. Figure 4.7 shows an example of the thermalcurves recorded during heating and cooling (i.e., sweeping T ) of a Gd5(Si0.1Ge0.9)4sample under different applied magnetic elds. The small λ-peak at around 130K arises from the second-order PM↔AFM phase transition. The large peak atlower temperatures is due to the rst-order phase transition between two differentorthorhombic structures [O(II) ↔ O(I)], that occurs simultaneously with a mag-netic AFM↔FM transition (see Refs. [18, 19] and section 2.2). The thermal hys-teresis amounts to 2-3 K. The magnetic eld dependence of the rst-order phasetransition is evident from the calorimetric curves. The transition temperature, Tt,linearly increases with the magnetic eld, with a slope (dTt/d(µ0Ht)=4.1 ± 0.1K/T) which is in agreement with that derived from magnetisation measurements[19]. The entropy change at the transition has also been found to increase withmagnetic eld, as shown in the insets of Fig. 4.7. For µ0H=0, ∆S =-24.2 J/(kgK) [-2.85 J/(mol K)] while for µ0H=5 T, ∆S =-33.7 J/(kg K) [-3.96 J/(mol K)],on cooling. Results on heating yield the same behavior, with ∆S absolute valuesslightly lowered with respect on cooling [by ∼0.7 J/(kg K)]. Such a large increaseof the entropy change with T is a consequence of the coupling between structural

98

4.4. Results sweeping T

-25

0

25

50

75 heating 5T4T3T

2T1T

0

dQ/d

T (

mJ/

K)

40 60 80 100 120 140

-75

-50

-25

0

25 cooling

1T

2T3T

4T

0

5T

dQ/d

T (

mJ/

K)

T (K)

0 1 2 3 4 5242628303234

µ0H (T)

∆S (

J/kg

K)

0 1 2 3 4 5242628303234

µ0H (T)

- ∆S

(J/

kg K

)

Figure 4.7: Calorimetric curves recorded sweeping T (on heating and cooling) ina Gd5(Si0.1Ge0.9)4 sample (#1, as-cast) for different applied magnetic elds. Insetsshow the entropy change as a function of magnetic eld.

99


and magnetic degrees of freedom. A thorough discussion of the measurements onthe series of Gd5(SixGe1−x)4 alloys will be given in the next chapters.

4.5 Results sweeping HDSC are usually designed to continuously sweep the temperature while Q(t) andT (t) are recorded. The T sweep induces thermally the rst-order transition in thesample, which releases or absorbes heat. In the particular case of eld-inducedtransitions, the temperature Tt of the peak of the transition in the calorimetriccurve is tuned by the magnetic eld, and consequently the eld dependence of ∆Sis obtained.

Besides, our new differential scanning calorimeter can also work sweepingthe eld. Fixing a temperature above Tt, a high enough magnetic eld also in-duces the rst-order transition. This fact gives rise to a direct measurement ofthe magnetocaloric effect, since the entropy change achieved by the applicationof a magnetic eld can be measured. To our knowledge, this is the rst time that∆S can be measured directly. In this case, the heat ow Q(t) and the increas-ing/decreasing eld H(t) are recorded, leading to the magnetic eld rate, dH/dt,and to the calorimetric curve dQ/dH= Q (dH/dt)−1. L and ∆S are thus given by:

L =

∫ HH

HL

dQdH dH ; ∆S =

1T

∫ HH

HL

dQdH dH =

LT , (4.2)

where HH and HL are, respectively, elds above and below the starting and nish-ing transition elds. As the range of H is more restricted than that of T , thereforethis kind of scannings are limited to a few temperatures above Tt. Figure 4.8 showsan example of the calorimetric curves recorded on increasing and decreasing Hfor a Gd5(Si0.05Ge0.95)4 sample at different xed temperatures above the zero-eldtransition temperature [Tt(H=0)45 K]. The main features of the transition aregiven in section 2.2. The transition eld increases linearly with temperature, witha slope 5.0±0.1 K/T, in excellent agreement with values obtained from both DSCsweeping T and magnetisation. In this sample, only curves at T=50, 55 and 60K show the total completion of the transition and allow the transition peak to beintegrated properly, in contrast with T=65 K, where it is clear that the maximumavailable eld of 5 T is not high enough to complete the transition. The entropychange at the transition increases with T , as shown in the insets in Fig. 4.8, inagreement with the values obtained through DSC sweeping T . Curves at differ-ent eld rates (0.1 and 1 T/min) for the same T yield the same values of L and∆S within the experimental error, showing that measurements do not depend onthe eld rate. However, it is obvious that the actual dynamics of the transition isstrongly affected by the eld rate, and that avalanches are already discernible at

100

4.5. Results sweeping H

0 1 2 3 4 5

-800

-600

-400

-200

0

200

400

600

800decreasing H

increasing H

55 K60 K

65 K

65 K60 K55 K

50 K

50 K

dQ/d

H (

mJ/

T)

µ0H (T)

0.1T/min 1T/min

50 55 60

12

16

20

0.1 T/min 1 T/min

∆S

(J/

kg K

)

T (K)

decreasing H

50 55 60

12

16

20

-∆S

(J/

kg K

)

T (K)

increasing H

0.1 T/min 1 T/min

Figure 4.8: Calorimetric curves recorded sweeping H (increasing and decreasingH) in a Gd5(Si0.05Ge0.95)4 sample (#1, T4+Q heat treatment) at some xed tem-peratures and for two different eld rates. Insets show the entropy change as afunction of temperature for the different rates.

101


0.1 T/min (see Chapter 9). ∆S on increasing H is ∼2-4 J/(kg K) larger than thaton decreasing H. The detailed analysis of these data and those for the rest of thesamples will be undertaken in the following chapters.

4.6 ConclusionsA new differential scanning calorimeter has been developed. The equipment fea-tures a high sensitivity down to 10 K and operates under applied magnetic elds ofup to 5 T and within the temperature range 10-300 K. The device may be used tostudy rst-order solid-solid phase transitions in the presence of magnetic elds. Ithas also been shown that this calorimeter enables an accurate determination of theentropy change in the magnetostructural phase transition of alloys exhibiting giantmagnetocaloric effect, which can be induced sweeping either T or H. Therefore,it is expected that this kind of measurements will clarify the controvertial issue ofthe actual value of the entropy change in the vicinity of a rst-order transition.

102

Bibliography

Bibliography[1] For a recent review on calorimetry see Y. Kraftmakher, Phys. Rep. 356, 1

(2002).

[2] O.S. Gill, S.R Suerbrunn and M. Reading, J. Therm. Anal. 40, 931 (1993).

[3] I.K. Moon, D.H. Jung, K.-B. Lee and Y.H. Jeong, Appl. Phys. Lett 76, 2451(2000).

[4] A.I. Kharkovski, Ch. Binek and W. Kleemann, Appl. Phys. Lett. 77, 2409(2000).

[5] V.K. Pecharsky and K.A. Gschneidner, Jr., Phys. Rev. Lett. 78, 4494 (1997).

[6] A. Giguère, M. Földeàki, B. Ravi Gopal, R. Chahine, T.K. Bose, A. Frydmanand J.A. Barclay, Phys. Rev. Lett. 83, 2262 (1999).

[7] K.A. Gschneider, Jr., V.K. Pecharsky, E. Brück, H.G.M. Duijn, and E. Levin,Phys. Rev. Lett. 85, 4190 (2000).

[8] J.R. Sun, F.X. Hu, and B.G. Shen, Phys. Rev. Lett. 85, 4191 (2000).

[9] M. Földeàki, R. Chahine, T.K. Bose, and J.A. Barclay, Phys. Rev. Lett. 85,4192 (2000).

[10] V.K. Pecharsky, J.O. Moorman, and K.A. Gschneidner, Jr., Rev. Sci. Instr.68, 4196 (1997).

[11] B.R. Gopal, R. Chahine, and T.K. Bose, Rev. Sci. Instr. 68, 1818 (1997).

[12] S. Yu. Dan'kov, A.M. Tishin, V.K. Pecharsky, and K.A. Gschneidner, Jr.,Rev. Sci. Instr. 68, 2432 (1997).

[13] For a detailed explanation on the VTI, which controls the temperature of thesample by tuning high-purity helium pressure inside the sample/calorimeterchamber, see B.J. Hattink, Ph. D. Thesis (chapter 2), Universitat deBarcelona, Catalonia, 2003.

[14] S.M. Sarge, E. Gmelin, G.W.H. Höhne, H.K. Cammenga, W. Hemminger,and W. Eysel, Thermochim. Acta 247, 129 (1994).

[15] R.Romero and J.L. Pelegrina, Phys. Rev. B 50, 9046 (1994).

[16] A. Planes and Ll. Mañosa, Solid State Phys. 55, 159 (2001).

103


[17] J. Ortín, Thermochim. Acta 121, 397 (1987).

[18] V.K. Pecharsky and K.A. Gschneidner, Jr., Appl. Phys. Lett. 70, 3299 (1997).

[19] L. Morellon, J. Blasco, P.A. Algarabel, and M. R. Ibarra, Phys. Rev. B 62,1022 (2000).

104

Chapter 5

Entropy change at the rst-ordermagnetostructural transition inGd5(SixGe1−x)4

5.1 IntroductionIn this chapter we present a detailed analysis of the different contributions tothe entropy change arising from the application of a magnetic eld at a rst-order eld-induced transition, in order to account for the discrepancies discussedin section 2.7 [1, 2, 3, 4, 5, 6]. For this purpose, magnetisation isotherms inGd5(SixGe1−x)4 for 0 ≤ x ≤ 0.5 (compositional range in which the rst-ordermagnetostructural transition takes place) were measured up to very high elds(∼23 T). The values of the entropy change obtained from the Clausius-Clapeyronequation (Eq. 1.17) and the Maxwell relation (Eq. 1.6) are compared and analysedwithin the framework of a simple phenomenological model based on the temper-ature and eld dependence of the magnetisation. Calorimetric measurements ofthe transition entropy change were also carried out on Gd5(SixGe1−x)4 series ofalloys, by using the high-sensitivity differential scanning calorimeter under mag-netic eld described in Chapter 4. Results are compared to those obtained fromindirect approaches through magnetisation measurements.

5.2 Magnetisation measurementsMagnetisation measurements were performed at the Grenoble High MagneticField Laboratory. M(H) curves were recorded up to 23 T, both under increas-ing and decreasing H, from 4.2 to 310 K with a temperature step of 3 to 5 K. Thefollowing samples were measured: x=0 (#1, as-cast), x=0.05 (#1, T4+Q treat-

105

CHAPTER 5. ENTROPY CHANGE IN GD5(SIXGE1−X)4

0

40

80

120

160

39.4, 55.4, 64.7, 75.4, 84.4, 94.8, 102.6 and 109.2 K

x=0.05

M (

emu/

g)

0 2 4 6 8 10 12 14 16 18 20 22 240

40

80

120

160

70.8, 84.3, 97.0,109.8, 115.7, 121.0, 127.2 and 133.0 K

x=0.1

µ0H (T)

0

40

80

120

160

200

5.0, 42.7, 57.9, 69.2, 81.3, 91.9, 96.6 and 105.8 K

x=0

Figure 5.1: Selected magnetisation isotherms of Gd5(SixGe1−x)4 for x=0, 0.05and 0.1 under increasing and decreasing eld. Temperatures labeled for eachcomposition refer to isotherms from top/left to bottom/right.

106

5.2. Magnetisation measurements

0

40

80

120

160

T= 92.2, 101.3, 107.3, 113.1, 118.9,124.8, 130.5, 137.2, 143.0 ,148.6 and 155.0 K

x=0.18

0 2 4 6 8 10 12 14 16 18 20 22 24

0

40

80

120

160

x=0.2

µ0H(T)

M(e

mu/

g)

T= 118.4, 124.3, 130.1, 135.8, 141.3, 144.0, 147.4, 150.6, 153.3, 159.6 and 165.2 K

Figure 5.2: Selected magnetisation isotherms of Gd5(SixGe1−x)4 for x=0.18 and0.2 under increasing and decreasing eld. Temperatures labeled for each compo-sition refer to isotherms from top/left to bottom/right.

107


0 2 4 6 8 10 12 14 16 18 20 22 24

0

40

80

120

0

40

80

120

160

T= 148.5, 162.7, 173.1, 181.9,190.8, 199.9, 208.8, 218.3 and 228.8 K

x=0.3

0

40

80

120

160 x=0.365

M(e

mu/

g)

T= 201.7, 208.8, 215.3, 224.2, 233.4, 239.6, 247.5, 254.3 and 260.9 K

T= 231.0, 239.3, 247.2, 255.0, 262.5, 270.0, 278.5, 286.5, 297.3 and 307.0 K

x=0.45

µ0H (T)

Figure 5.3: Selected magnetisation isotherms of Gd5(SixGe1−x)4 for x=0.3, 0.365and 0.45 under increasing and decreasing eld. Temperatures labeled for eachcomposition refer to isotherms from top/left to bottom/right.

108

5.2. Magnetisation measurements

200 210 220 230 240 250 260 270 280 290 300 3100

20

40

60

80

100

120

140

160

∆=2T

0.5T1T2T

22Tx=0.45

0.1T

M (

em

u/g

)

T (K)

60 70 80 90 100 110 120 1300

20

40

60

80

100

120

140

160

180 22T

∆=2T

2T1T

0.5T

0.1T

x=0.1

T(K)

M(e

mu

/g)

Figure 5.4: Magnetisation as a function of temperature under selected appliedelds (0.5, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 and 22 T) for x=0.1 and x=0.45compounds, taken from M(H) data for increasing eld.

109


ment), x=0.1 (#1, as-cast), x=0.18 (#1, T4 treatment), x=0.2 (#1, as-cast), x=0.3(#2, T4+Q treatment), x=0.365 (#3, as-cast) and x=0.45 (#7, T4 treatment).

M(H) isotherms are shown in Fig. 5.1 (x=0, 0.05 and 0.1), Fig. 5.2 (x=0.18and 0.2) and Fig. 5.3 (x=0.3, 0.365 and 0.45). These curves exhibit a jump ∆M atthe magnetostructural transition [1, 2, 7, 8], which spreads over a eld range ∆Ht(∼2-4 T for 0 ≤ x ≤ 0.2 samples and ∼4-6 T for 0.24 ≤ x ≤ 0.5 compounds).The hysteresis in the transition reveals its rst-order nature. The transition eld,Ht, for increasing and decreasing applied eld is dened for each isotherm as theeld corresponding to the inection point within the transition region. We notethat the application of a eld of 23 T enables the observation of the transition attemperatures of up to ∼80 K above Tt(H = 0). The variation of Ht with T islinear for 0.24 ≤ x ≤ 0.5 compounds, while the slope dHt/dTt changes withintwo limiting values for 0 ≤ x ≤ 0.2 alloys. The detailed study of Ht(T ) for allcompositions is presented in Chapter 7. The change in the slope of the M(H)curves observed in Fig. 5.1 for x=0 and 0.05 above ∼90 K and ∼12 T correspondsto a nonreported magnetic transition appearing at very high elds, which is stud-ied in Chapter 8. For the compounds from x=0 to x=0.2 (Figs. 5.1 and 5.2), therst-order eld-induced transition occurs from AFM to FM phases, while for therest of compounds (from x=0.3 to x=0.45, see Fig. 5.3) it occurs from PM to FMphases. This difference is observed in ∆M, which is more abrupt for the AFM-FMtransition. Figure 5.4 shows the magnetisation data (for increasing eld) displayedas a function of T at constant magnetic eld for x=0.1 and x=0.45 compounds,as paradigmatic examples of Ge-rich and intermediate compostional regions, re-spectively.

5.3 DSC measurementsDSC data under different magnetic elds (0 to 5 T) were measured for the follow-ing samples: x=0 (#1, as-cast), x=0.05 (#1, T4+Q treatment), x=0.1 (#1, as-cast),x=0.18 (#1, T4 treatment), x=0.2 (#1, as-cast), x=0.25 (#2, as-cast), x=0.3 (#2,as-cast), x=0.365 (#3, as-cast) and x=0.45 (#7, T4 treatment). Measurementswere carried out by scanning T at constant magnetic elds, since the availaberange in temperature is larger than the available range in H. In these measure-ments, rst-order transitions give rise to a large peak in thermal curves (dQ/dT ).Second-order transitions are observed as small λ-type jumps in the dQ/dT base-line. The shape of the thermal curves for all compositions with x ≤ 0.2 revealsthe rst-order nature of the low-temperature AFM-FM transition and the second-order nature of the high-temperature PM-AFM transition (see Fig. 5.5 for x=0,0.05 and 0.2, Fig. 5.6 for x=0.18, where the second-order transitions are labeled,and Fig. 4.7 in Chapter 4 for x=0.1). For the rest of compositions (0.24 ≤ x ≤ 0.5),

110

5.3. DSC measurements

20 40 60 80 100 120 140-600

-400

-200

0

200

400

4T3T2T1T

0

5T

second order(c)

x=0.2

T(K)

-600

-400

-200

0

200

400

5T

0

4T3T

2T1T

second order

(b)

x=0.05

dQ/d

T (

J/kg

·K)

-600

-400

-200

0

200

400

second order

(a)

x=04T5T

3T2T1T

Figure 5.5: DSC data on cooling at selected applied elds up to 5 T forGd5(SixGe1−x)4: (a) x=0, (b) x=0.05 and (c) x=0.2. The second-order transitionis labeled for each composition.

111


60 70 80 90 100 110 120 130 140-600

-400

-200

0

200

400

second order

cooling

5T4T3T

2T1T

0

x=0.18

T(K)

dQ/d

T (

J/kg

·K)

-400

-200

0

200

400

600

heating

second order

5T4T3T

2T1T

0

125 130 135

Figure 5.6: DSC data for x=0.18 on heating and cooling the sample under H. In-set: Detail of the second-order transition on heating, from 0 (top) to 5 T (bottom).

112

5.3. DSC measurements

180 200 220 240 260 280-1200

-1000

-800

-600

-400

5T4T

3T

2T1T

0

heating

x=0.45

T(K)

200

400

600

800

1000

0

5T

4T3T

2T1T

cooling

x=0.365

-500

-400

-300

-200

-100

0

100

5T

4T3T2T

1T0

heating

x=0.25

dQ/d

T (J

/kg·

K)

120 140 160 180 200

-200

-100

0

100

200

300

400

1T 2T3T

4T5T

0cooling

x=0.3

T(K)

Figure 5.7: DSC data at selected applied elds up to 5 T for x=0.25 (heating),x=0.3 (cooling), x=0.365 (cooling) and x=0.45 (heating).

113


only one peak is displayed, corresponding to the rst-order PM-FM transition(Fig. 5.7 for x=0.25, 0.3, 0.365 and 0.45). For all samples, a hysteresis of 2-4K between cooling and heating runs is observed, see for example Figs. 5.6 and4.7. Tt is estimated as the temperature at the maximum of the dQ/dT peak andincreases with the applied eld (up to 5 T) in all samples. The detailed study ofH(Tt) for all compositions is presented in Chapter 7. We note that for x=0, thecooling process at zero eld does not show any rst-order peak [hence it is notdisplayed in Fig. 5.5 (a)]. This fact can be explained by taking into account thatthe FM ground state for x=0 can not be achieved by cooling down to low temper-ature at zero eld, since the sample remains AFM, as discussed in section 2.4.1.The application of a eld of ∼1 T is needed in order to stabilize the FM phasethrough an irreversible transition [9, 10, 11]. Fig. 5.5 shows how the rst-ordertransition gets progressively closer to the second-order transition as Si content, x,is increased. In particular, Fig. 5.5 (c) shows how the rst-order peak overlaps thesecond-order jump when a eld of ∼3 T (or larger) is applied for x=0.2.

5.4 Evaluation of the entropy change5.4.1 Magnetic and calorimetric evaluationsThe entropy change as a function of T for each x may be obtained indirectly frommagnetisation data:

(i) On one hand, the entropy change at a rst-order transition, ∆S , can beobtained by using the Clausius-Clapeyron equation (Refs. [2, 12] and section2.7),

∆S = −∆M dHt

dTt,

where ∆M has been estimated as the difference in the magnetisation at Ht betweenthe linear extrapolations of M(H) well above and below the transition region,and dHt/dTt is evaluated from the Ht(T ) curve obtained from the magnetisationisotherms. This estimation of ∆M does not consider the variation of M due to theeld change (since the transition takes place in a eld range ∆Ht), but only due tothe rst-order phase transition.

(ii) On the other hand, the total entropy change due to the variation of themagnetisation by the application of a magnetic eld, ∆S (0 → Hmax), may beevaluated by using the Maxwell relation (Ref. [13] and section 1.2),

∆S (0→ Hmax) =

∫ Hmax

0

(∂M∂T

)

HdH .

114

5.4. Evaluation of the entropy change

This integration is evaluated numerically from magnetisation isotherms. It isstraightforward to note that the entropy change at the transition, ∆S , and the totalentropy change, ∆S (0→ Hmax), do not necessarily yield the same value.

The entropy change can also be obtained by calorimetry. DSC data enables usto obtain the entropy change (and latent heat) at the rst-order transition, after aproper integration of the calorimetric peak (see section 3.2.3 and Chapter 4), as

∆S =

∫ TH

TL

1T

dQdT dT ,

where TH and TL are respectively temperatures above and below the starting andnishing transition temperatures.

A comparision among all three methods to evaluate the entropy change isshown in Fig. 5.8 for x=0.45 (using increasing H and cooling data), as a goodexample of the behaviour of the 0.24 ≤ x ≤ 0.5 alloys and in Fig. 5.9 for x=0and 0.05 (using decreasing H and heating data), as a paradigmatic example of the0 ≤ x ≤ 0.2 compounds.

For x=0.45, the entropy change obtained from the Maxwell method is dis-played as dashed lines. Different curves of entropy change as a function of T areobtained depending on the maximum applied eld, Hmax. These curves displaythe typical behavior previously reported [1, 2]: rst, a rapid increase at low T ,then a maximum value at about Tt(H = 0), followed by a plateau-like behaviour,and nally a sharp decrease at high T . Figure 5.8 also shows the values of theentropy change at the transition, ∆S , obtained from the Clausius-Clapeyron equa-tion for x=0.45 (present data, solid squares) and x=0.5 (taken from Ref. [2], opensquares), and DSC data (open triangles) for x=0.45. We note that ∆S obtainedfrom the Clausius-Clapeyron equation and calorimetry yields the same values,within the experimental error. This suggests that both methods actually evaluatethe entropy change associated with a rst-order transition. The maximum valueof the entropy change achieved using the Maxwell relation can be above or below∆S depending on Hmax.

For x=0 and 0.05 (Fig. 5.9), the comparision between methods is very simi-lar to that in x=0.45. The values obtained using the Clausius-Clapeyron equation(open squares) agree, within the experimental error, with the calorimetric ones(open triangles), although DSC data give slightly higher values, as observed insome other samples (see section 6.3). This small difference may be related to thefact that the coexistence line in the phase diagram is crossed at different direc-tions, i.e., sweeping H in magnetisation and sweeping T in DSC (see Chapter 9).∆S (0 → Hmax) calculated from the Maxwell relation (dashed lines) gives differ-ent values depending on the maximum applied eld, being clearly above ∆S whenHmax > ∆Ht.

115


200 220 240 260 280 300 320

increasing Hx=0.45

20 T15 T10 T7 T5 T2 T

Ent

ropy

cha

nge

(J/k

g·K

)

T(K)

0

-10

-20

-30

Figure 5.8: Entropy change for Gd5Si1.8Ge2.2 (x=0.45) calculated from: (i)Maxwell relation integrating up to Hmax (dashed lines), (ii) Clausius-Clapeyronequation (solid squares this work and open squares for x=0.5 from Ref. [2]), (iii)DSC measurements under eld (open triangles), and (iv) Maxwell relation inte-grating within ∆Ht (solid lines). Hmax is labeled beside each dashed line, and alsostands for the solid lines from left to right increasing the eld.

116

5.4. Evaluation of the entropy change

20 40 60 80 100 1200

10

20

30

(a)

decreasing H

x=0

T(K)

40 60 80 1000

10

20

30

(b)

x=0.05decreasing H

Ent

ropy

cha

nge

(J/k

gK)

Figure 5.9: Entropy change in Gd5(SixGe1−x)4, for (a) x=0.05 and (b) x=0, cal-culated by using: DSC measurements under eld (open triangles); the Clausius-Clapeyron equation (open squares); the Maxwell relation integrating from Hmax(20, 15, 10, 7, 5 and 2 T, from right to left, respectively) to zero (dashed lines);and the Maxwell relation integrating only within the transition region (solid line).

117


Besides, ∆S (0 → Hmax) obtained from the Maxwell relation for x=0 andx=0.05 shows a double peak structure, when integrating from very high elds(15 and 20 T) to zero, which evidences the existence of two magnetic transitionsin the system. The high-T peak is related to the expected AFM-FM transition,while the low-T peak is associated with a transition from the AFM phase to aphase with short-range antiferromagnetic correlations, which appears in the Ge-rich compounds (see Chapter 8). This effect is also evident in ∆S determined fromthe Clausius-Clapeyron equation at high Tt, i.e., at very high Ht.

5.4.2 Use of the Maxwell relation within the transition regionThe difference between the transition entropy change ∆S obtained from bothDSC and the Clausius-Clapeyron approach and ∆S (0 → Hmax) obtained fromthe Maxwell relation, can be understood by taking into account the fact that theMaxwell method includes the following contributions,

∆S (0→ Hmax) =

∫ Ha

0

(∂M∂T

)

HdH +

∫ Hb

Ha

(∂M∂T

)

HdH +

∫ Hmax

Hb

(∂M∂T

)

HdH , (5.1)

with Ha=Ht − ∆Ht/2 and Hb=Ht + ∆Ht/2, being ∆Ht the transition eld range.The rst and the third integrals yield the entropy change that arises from the eldand temperature dependence of the magnetisation in each magnetic phase thatthe transition involves. Only the second term accounts for the contribution to theentropy change of the magnetostructural transition. This is indicated by the factthat, for x=0.45, the plateau-like behaviour of the solid lines in Fig. 5.8 (computedusing the second integral in Eq. 5.1,

∫ Hb

Ha(∂M/∂T )HdH) perfectly matches the ∆S

values given by the Clausius-Clapeyron equation and by calorimetry. A transitioneld region of µ0∆Ht ∼4 T has been used, obtained from the high eld M(H)curves. Note also that when Hmax is less than ∆Ht, which is the minimum eldneeded to complete the transition, the maximum value of ∆S (0→Hmax) is lowerthan ∆S (see for instance, the curve corresponding to µ0Hmax = 2 T in Fig. 5.8).Moreover, for Hmax ≥ ∆Ht, the plateau-like region extends over the temperaturerange for which Hmax ≥ Hb(T ). Consequently, as Hb(T ) increases with T , theabrupt decrease from the plateau-like region at higher T is due to the truncationof the second integral at Hmax.

The same result is plotted as solid lines in Fig. 5.9 for x=0.05 (µ0∆Ht ∼3T from M(H)) and x=0 (µ0∆Ht ∼4 T) samples, showing that ∆S (Ha→Hb) =∫ Hb

Ha(∂M/∂T )HdH matches the Clausius-Clapeyron value. This suggests that the

calculation of the entropy change using the Maxwell relation evaluated within

118

5.5. Phenomenological models for the entropy change

∆Ht and using the Clausius-Clapeyron equation, are equivalent for the rst-ordertransition in the whole compositional range of Gd5(SixGe1−x)4 alloys.

5.5 Phenomenological models for the entropy changeIn order to account for the main features of the entropy change reported in thelast section (5.4), we propose a phenomenological model that takes into accountthe basic features of the magnetisation in a system with a rst-order eld-inducedphase transition. In a rst phenomenological approach, only the basic behaviourof the transition is considered, by assuming that M(T ) is constant outside thetransition region. In the advanced approach, the overall behaviour of the magneti-sation is also taken into account by assuming that M(T ) is not constant outside thetransition region.

5.5.1 First phenomenological approach: M(T ) = const.In this rst model, the magnetisation curves are considered to be of the form:

M(T,H) = M0 + ∆M F(T − Tt(H)

ξ

), (5.2)

where M0 and ∆M are assumed to be T and H independent, and F(T ) is a monoto-nously decreasing function of width ξ such that F → 1 for T Tt(H) and F → 0for T Tt(H). The case ξ → 0 corresponds to the ideal rst-order transition(F is then the Heaviside function). Using the Maxwell relation and assuming alinear eld dependence of the transition temperature (dTt/dHt ≡ α=constant), theentropy change is given by

∆S (0→ Hmax) = ∆S[F

(T − Tt(Hmax)

ξ

)− F

(T − Tt(H = 0)

ξ

)], (5.3)

where ∆S = ∆M/α (the transition entropy change from the Clausius-Clapeyronequation). It is worth stressing that when the transition temperature is not elddependent, ∆S (0 → Hmax) = 0 irrespective of the value of ∆S . In general,∆S (0 → Hmax) is a fraction of the transition entropy change (∆S ), which de-pends on the magnitude of the shift of Tt with the magnetic eld, and reaches itsmaximum value, ∆S , for high enough applied eld. Results are even valid in thelimit ξ → 0, for which ∆S (0→ Hmax) = ∆S for all Hmax.

A simple analytical picture is provided by assuming that F is a linear func-tion of temperature which extends within the temperature range ∆Tt = α∆Ht = ξ.Results are shown in Fig. 5.10. The general trends compare very well with re-sults in Figs. 5.8 and 5.9 obtained by integrating the Maxwell relation within the

119


H1/∆H

t

H2/∆H

t

∆M

0

0

0-H5

0-H40-H

3

0-H2

0-H1

1

Temperature

-∆S

(0→

Hm

ax)/

∆S

H=0

H1

H2

H3 H

4 H5

M

Figure 5.10: Upper panel: temperature dependence of the magnetisation acrossthe transition region at different elds, as described for the rst phenomenologicalmodel. Lower panel: corresponding entropy change ∆S (0→Hmax) calculated fromthe Maxwell relation. In this gure, ∆S stands for the entropy change of thetransition, obtained from the Clausius-Clapeyron equation.

120


transition range (second integral of Eq. 5.1, ∆S (Ha→Hb) =∫ Hb

Ha(∂M/∂T )HdH).

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 experimentalresults (Figs. 5.1, 5.2 and 5.3), where ∆M decreases linearly with T . It is alsoobserved that when Hmax is not high enough to complete the transition (Hmax <∆Ht), then ∆S (0→Hmax) = (Hmax/∆Ht)∆S is smaller than ∆S . Accordingly,(Hmax/∆Ht) is the fraction of the sample that has been transformed. For Hmax ≥∆Ht, ∆S (0→Hmax) reaches its maximum value ∆S (0→Hmax)=∆S , showing theequivalence of the Clausius-Clapeyron equation and the Maxwell relation, pro-vided the latter is only evaluated within the transition eld region.

5.5.2 Advanced phenomenological approach: M(T ) , const.This model is a generalisation of the previous one. This advanced model includesboth the T and H dependences of the magnetisation outside the transition regionand the decrease of ∆M with T . The upper panel in Fig. 5.11 shows the modelledM(T ) curves at different H. The transition temperature is assumed to shift linearlywith the transition eld, dTt/dHt ≡ α=constant. The magnetisation of the low-temperature phase is assumed to decrease linearly with T as M(T ) = ∆M0(1−βT ),being zero at the high-temperature phase. The transition between both phases ex-tends within a temperature range ∆Tt = α∆Ht, which is assumed to be constantaccording to the experimental results (see Figs. 5.4 and 5.12 (a)). In this model,Tt(H) is dened for each curve as the temperature at the center of the transition re-gion. As α is considered to be constant, the model should account for the behaviorof the entropy change for 0.24 ≤ x ≤ 0.5 alloys (see Fig. 5.8 for x=0.45).

The results of the model are compiled in the middle panel in Fig. 5.11. Thebehaviour, which depends on the temperature range and the maximum appliedeld, can be summarised as follows :

(i) For temperatures at which the system is in the low-temperature phase (T ≤TA, with TA ≡ Tt(H = 0) − ∆Tt/2), the entropy change is independent of T andincreases linearly with the maximum applied eld as

∆S (0→ Hmax) = −∆M0βHmax . (5.4)

(ii) In the range TA ≤ T ≤ TB (TB ≡ Tt(H = 0) + ∆Tt/2), which is thetemperature spread of the transition at zero eld (see upper panel in Fig. 5.11),the entropy change increases linearly up to Tt(Hmax)−∆Tt/2 and reaches a plateau,with a value increasing with Hmax (see H1 in Fig. 5.11). The limiting case of thisbehaviour is obtained when the maximum applied eld is strong enough to inducethe whole transition (i.e., Hmax = ∆Ht). Then Tt(Hmax) − ∆Tt/2 equals TB and the

121


-∆M[Tt(H=0)]/α

0-H1 0-H

2 0-H3 0-H

4

0

Ent

ropy

Cha

nge

∆Tt

H4

H3

H2

H1H

0=0

M=∆M0 (1-βT)∆M

0

0

Mag

netiz

atio

n

TBT

A

-∆M0β∆H

t/2

-∆M[Tt(H=0)]/α

0-H1

0-H2 0-H

3 0-H4

0

Ent

ropy

Cha

nge

Temperature

Figure 5.11: Upper panel shows the modelled temperature dependence of themagnetisation across the transition region at different elds, as described for theadvanced phenomenological model in the text. Middle panel shows the cor-responding entropy change ∆S (0→Hmax) calculated from the Maxwell relation.Lower panel: Solid lines stand for the entropy change obtained by integrating theMaxwell relation only within the transition region. Connected squares stand for∆S obtained from the Clausius-Clapeyron equation. The difference between thosevalues is indicated in the Figure.

122


value is

∆S (0→ Hmax) = −∆M0[1 − βTt(H = 0)]α

≡ −∆M[Tt(H = 0)]α

(5.5)

(see case for H2 in Fig. 5.11). For higher elds (Hmax ≥ ∆Ht), the transition isalso completed at TB and there is an additional contribution to ∆S due to the eldand temperature dependences of M of the low-temperature phase (see cases forH3 and H4 in Fig. 5.11). Therefore,

∆S (0→ Hmax) = −∆M[Tt(H = 0)]α

− ∆M0β(Hmax − ∆Ht) . (5.6)

(iii) For temperatures at which the system is in the high-T phase at zero eld(T ≥ TB) and for low elds (see H1 and H2 in Fig. 5.11), ∆S decreases linearly tozero with increasing T , vanishing at Tt(Hmax) + ∆Tt/2, which corresponds to theminimum temperature at which Hmax is not enough to start inducing the transition.For elds where the transition is complete (see H3 and H4 in Fig. 5.11), ∆S showsplateau-like behavior with a slope 2∆M0β/α up to Tt(Hmax) − ∆Tt/2. Above thistemperature, the eld is not enough to complete the transition and ∆S decreaseslinearly to zero, vanishing at Tt(Hmax) + ∆Tt/2.

The lower panel in Fig. 5.11 shows the entropy change (solid lines) calcu-lated by integrating the second term of Eq. 5.1 (∆S (Ha → Hb)). The values of∆S calculated by using the Clausius-Clapeyron equation are also plotted as con-nected squares. Three main features are to be noted: (i) for temperatures at whichthe transition does not occur (T ≤ TA), ∆S (0 → Hmax)=0; (ii) for temperaturesat which the transition can be completely eld-induced (T ≥ TB), and for Hmaxstrong enough to complete it, the plateau-like regions of all curves overlap, yield-ing a slope ∆M0β/α; and (iii) ∆S values obtained from the Clausius-Clapeyronequation decrease with the same slope, but lowered by δ = ∆M0β∆Ht/2. Themodel accounts for the behaviour of the experimental results shown in Fig. 5.12,in Fig. 5.8, and in general for all 0.24 ≤ x ≤ 0.5 compounds. We note that inFig. 5.12 (c) the values of ∆S calculated from the Clausius-Clapeyron equation atlow T increase with T due to the fact that, just above the zero-eld transition tem-perature, a fraction of the sample has not yet been transformed to the PM phaseand still remains FM [8]. We also note that for x ≤ 0.2, although α is not constant,the model accounts for the main features of the PM-FM transition. An extensionof the present model should consider the dependence of α on Ht and Tt.

In order to improve the model, a linear H dependence of the low-T magnetisa-tion outside the transition region is introduced as M(T,H) = ∆M0(1 − βT + γH).This is a more realistic assumption for the magnetisation curves (Figs. 5.4 and5.12 (a)). However, the overall behaviour remains unchanged. In this case, Eq.

123


0

5

10

15

20

25

30

35

0-20 T0-15 T

0-10 T0-7 T

0-5 T

0-2 T

increasing H

-∆S

(J

/kg

K)

140 150 160 170 180 190 200 210 220 230

0

5

10

15

20

25

30

35

0-20 T0-15 T

0-10 T

0-7 T0-5 T

0-2 T

increasing H

-∆S

(J/

kgK

)

T (K)

0

20

40

60

80

100

120

140

160

x=0.3

20 T

6 T4 T2 T1 T

0.5T

M (

em

u/g

)

Figure 5.12: Upper panel shows the magnetisation as a function of temperature atdifferent elds (0.5, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20 T) for x=0.3 compound,taken from M(H) (increasing H) data. Middle panel shows the corresponding en-tropy change ∆S (0→Hmax) calculated from the Maxwell relation. Lower panel:Solid lines stand for the entropy change obtained by integrating the Maxwell re-lation only within the transition region. Connected squares stand for ∆S obtainedfrom the Clausius-Clapeyron equation.

124

5.6. Conclusions

5.5 turns into

∆S (0→ Hmax) = −∆M[Tt(H = 0)]α

− ∆M0γ∆Ht

2α , (5.7)

the slope of the plateau-like region of ∆S values evaluated within the transitionregion from the Maxwell relation is now ∆M0β

′/α, with β′ = β − γ/α, andδ′ = ∆M0β

′∆Ht/2. This shift δ′ between the Clausius-Clapeyron approach andthe Maxwell approach is due to the fact that H heightens Tt, resulting in a reduc-tion of ∆M. Since ∆Tt is assumed to be constant, ∆M/∆Tt in the transition re-gion decreases correspondingly. This H-dependence remains included within thetransition region (as ∂M/∂T ) and still gives an extra term to the entropy changewhen calculated from the Maxwell relation by integrating the second term of Eq.5.1, but does not contribute to the Clausius-Clapeyron equation. Nevertheless,δ′ is small for Gd5(SixGe1−x)4 alloys. For example, for x=0.45 (see Fig. 5.8),∆M0β

′=0.753 emu/(gK) and ∆Ht ∼4 T, yielding δ′ ∼1.5 J/(kgK), which is withinthe experimental error of the entropy change. For x=0.3, ∆M0β

′=1.163 emu/(gK)and ∆Ht ∼7 T, resulting in δ′ ∼4.1 J/(kgK), which may account for the slight dif-ference observed in Fig. 5.12 (c). Generally, δ′ is expected to be small, since itis proportional to the variation of the magnetization outside the transition regionand this variation is small in a FM phase. This may be extended to any othereld-induced transitions that involve a FM phase.

5.6 ConclusionsThe magnetocaloric effect arising from a eld variation 0→Hmax can be properlyevaluated through the entropy change obtained from the Maxwell method, evenwhen an ideal rst-order transition occurs. When the Maxwell relation is evalu-ated over the whole eld range, the T and H dependences of the magnetisation ineach phase outside the transition region yield an additional entropy change to thatof the actual rst-order transition. It has also been shown, from both experimen-tal data and phenomenological models, that the Maxwell relation, the Clausius-Clapeyron equation and the calorimetric measurements yield the entropy changeof the rst-order magnetostructural transition, provided (i) the Maxwell relation isevaluated only within the eld range over which the transition takes place, and (ii)the maximum applied eld is high enough to complete the transition. The transi-tion temperature must signicantly shift with the applied eld, in order to achievea large MCE taking advantage of the entropy change associated to the rst-ordertransition. This is relevant for the understanding of the thermodynamics and MCEof rst-order magnetostructural transitions.

125


Bibliography[1] V. K. Pecharsky and K. A. Gschneidner, Jr., Phys. Rev. Lett. 78, 4494 (1997).













126

Chapter 6

Scaling of the transition entropychange in Gd5(SixGe1−x)4

6.1 Introduction

This chapter is aimed at studying the entropy change associated with the rst-ordermagnetostructural phase transition, ∆S , in Gd5(SixGe1−x)4 alloys, as a function ofboth composition, x, and type of magnetic phase transition, i.e., as a function ofthe phase diagram. The calorimetric measurements of ∆S as a function of T andH are analysed for Gd5(SixGe1−x)4 alloys, within the whole 0 ≤ x ≤ 0.5 range. A∆S scaling plot is obtained, where the scaling variable, Tt, is the temperature ofthe rst-order magnetostructural phase transition. As Tt is shifted with x and H,the scaling of ∆S thus summarises the giant MCE in the Gd5(SixGe1−x)4 alloys.

6.2 Calorimetric measurements

As detailed in Chapter 4, DSC under H is the ideal technique for the study of∆S at rst-order magnetostructural transitions. Calorimetric measurements wereperformed using two high-sensitivity differential scanning calorimeters, speci-cally designed to study solid-solid phase transitions. Heating and cooling runswere performed within 77-350 K for H=0 in a LN2 cryostat with the calorimeterdescribed in section 3.2.3, and within 4.2-300 K under elds up to 5 T in a LHecryostat with the calorimeter with built-in H described in Chapter 4.

127

CHAPTER 6. SCALING OF THE ENTROPY CHANGE IN GD5(SIXGE1−X)4

Tt (K) ∆S (J/kgK)x ID Heat T. µ0H(T) cool. heat. cooling heating0 #1 NO 0 - - - -

1 23.3 28.5 -6.55 7.122 33.2 36.3 -15.06 14.743 40.4 42.9 -21.20 21.264 46.6 48.7 -24.59 23.945 51.2 53.7 -28.81 28.89

0.05 #1 T4+Q 0 43.8 46.5 -14.57 14.291 49.4 51.7 -18.11 17.952 55.1 57.1 -23.04 22.003 60.2 62.1 -25.92 24.594 65.0 66.6 -27.86 26.775 69.1 70.8 -28.32 26.76

0.1 #1 NO 0 70.4 73.1 -24.22 23.521 74.2 76.7 -25.74 25.412 78.9 81.1 -28.03 28.273 83.2 85.4 -30.75 30.264 86.9 89.0 -32.05 31.705 91.0 92.9 -33.65 32.86

0.18 #1 T4 0 98.7 100.9 -36.87 35.121 101.9 104.1 -37.89 36.232 106.1 107.8 -39.62 38.113 110.0 111.8 -40.81 39.474 113.5 115.2 -42.06 40.445 116.8 118.5 -43.70 41.75

0.2 #1 NO 0 113.9 116.6 -41.51 40.831 117.1 119.6 -43.15 42.642 120.6 123.3 -45.31 43.923 124.0 126.4 -46.78 45.974 127.1 129.7 -48.22 47.775 129.6 132.5 -48.12 46.01

128

6.2. Calorimetric measurements

Tt (K) ∆S (J/kgK)x ID Heat T. µ0H(T) cool. heat. cooling heating

0.25 #2 NO 0 143.0 150.5 -42.88 39.981 145.7 152.6 -42.38 38.822 149.1 155.4 -41.90 38.093 152.0 158.3 -40.86 37.534 155.0 160.8 -39.42 35.645 157.8 163.6 -39.42 35.64

0.3 #2 NO 0 169.7 177.5 -36.16 32.971 172.2 179.2 -35.50 32.392 175.2 182.4 -34.55 31.343 177.8 185.4 -33.85 30.664 180.4 188.6 -32.89 29.755 182.4 189.9 -31.89 28.66

0.365 #3 NO 0 200.7 204.5 -29.90 28.781 207.4 211.0 -29.38 28.352 211.6 214.9 -28.61 27.643 215.1 219.2 -27.27 26.044 218.6 222.0 -26.51 25.005 221.8 226.7 -25.93 24.48

0.45 #7 T4 0 243.5 247.1 -21.58 20.301 248.0 251.7 -20.02 17.822 252.8 256.9 -19.11 16.543 257.6 261.7 -17.11 15.164 262.5 266.7 -15.58 13.645 266.6 271.4 -14.01 12.40

Table 6.1: Entropy change and Tt at the rst-order transition obtained from DSCunder magnetic eld in all measured samples, on cooling and heating.

We measured Gd5(SixGe1−x)4 samples with x= 0, 0.05, 0.1, 0.18, 0.2, 0.25,0.3, 0.365 and 0.45, using both calorimeters. For x=0, 0.05, 0.1 and 0.18, the DSCoperating with LN2 cannot reach their transition temperature. Calorimetric curvesunder magnetic eld are described in section 5.3 and shown in Figs. 5.5, 5.6 and5.7. ∆S was calculated by numerical integration of (dQ/dT )/T throughout therst-order calorimetric peaks [1]. The results of ∆S and Tt (which is evaluated asthe temperature at the maximum of the dQ/dT peak) are displayed in Table 6.1as a function of x and H for the calorimeter with built-in H, and also in Table 3.4(Chapter 3) for the calorimeter operating with LN2.

129


Other relevant information can be obtained from the DSC curves, appart fromthe latent heat and transition entropy change: although DSC does not give theabsolute value of Cp, the extrapolation to Tt of the baselines at temperatures aboveand below the rst-order transition provides a good estimation of ∆Cp. It is foundthat ∆Cp is positive for the rst-order AFM-FM transition for all compositionswith x ≤ 0.2 (see Fig. 6.1 (a) for x=0.1), while negative ∆Cp is obtained for therst-order PM-FM transition for 0.24 ≤ x ≤ 0.5 (see Fig. 6.1 (b) for x=0.3). Thecase x=0.2 is very interesting (Fig. 6.1 (c)), since the rst-order peak overlaps thesecond-order one for a high enough eld (∼3 T). For this reason, a change in thesign of ∆Cp is observed in this sample.

6.3 Scaling of the transition entropy changeThe absolute value of ∆S as a function of Tt is shown in Fig. 6.2. As Tt corre-sponds to the transition temperature of the rst-order phase transition for each xand H, this allows us to sweep Tt from ∼20 to ∼310 K. ∆S from the Clausius-Clapeyron equation [∆S = −∆M(dHt/dTt)] reported by Giguère et al. for x=0.5,and obtained up to 7 T (see Fig. 2 in Ref. [2]), is also displayed in Fig. 6.2. AsTt is tuned by both x and H, |∆S | values scale with Tt. This enables us to derive ascaling of |∆S | for all Tt, i.e. for all compositions with x ≤ 0.5. The values givenin Ref. [2] also collapse onto this scaling plot. Values for x=0 are not included,since Gd5Ge4 alloy presents an irreversible transition which makes it differentfrom the rest of Gd5(SixGe1−x)4 alloys (section 2.4.1 and Refs. [3, 4, 5]). Thisscaling shows that the relevant parameter in determining |∆S | is Tt. Besides, thescaling is not a trivial consequence of the scaling of both ∆M and dHt/dTt, i.e.neither ∆M nor dHt/dTt scale with Tt

1, which gives further relevance to the scal-ing of |∆S |. Notice also that |∆S | extrapolates to zero at Tt=0, as expected fromthe third law of thermodynamics. The scaling is a consequence of the rst-ordernature of the transition: at a constant H, the Clausius-Clapeyron equation is writ-ten as ∆S = ∆V(dPt/dTt), where ∆V stands for the volume jump and Pt for thetransition pressure. Therefore, ∆V and ∆M are related as ∆V/∆M = −dHt/dPt,and the scaling thus proves that the magnetovolume effects due to H are of thesame nature as the volume effects caused by substitution.

Two diferent trends are shown in Fig. 6.2. For 0.24 ≤ x ≤ 0.5, |∆S | associ-ated with the PM/M-FM/O(I) transition monotonically decreases with Tt, whichis consistent with ∆Cp < 0 (Fig. 6.1 (b)), as expected from the thermodynamic re-lation d(∆S )/dT = ∆Cp/T . Moreover, negative ∆Cp may also be estimated fromRef. [6]. In contrast, for x ≤ 0.2, |∆S | either decreases or increases depending on

1∆M always decreases with Tt and dHt/dTt presents a particular behaviour which is studied indetail in Chapter 7.

130

6.3. Scaling of the transition entropy change

90 100 110 120 130 140 150 160-600

-400

-200

0

200

400

600µ

0H=5T

µ0H=3T

zero field

∆CP~ 0

∆CP< 0

∆CP> 0

(c) x = 0.2

T(K)

dQ/d

T (J

/kg·

K)

150 180 210

0

200

400

(a)

zero field

x = 0.3

∆CP< 0

T (K)80 100 120

-200

0

200

400

∆CP> 0

(b)

µ0H = 5 T

x = 0.1

dQ

/dT

(J/

kg·K

)

Figure 6.1: DSC data for (a) x=0.1 on heating the sample with µ0H=5 T and (b)x=0.3 on heating the sample without applied eld. The opposite sign of ∆Cp forthe two compositions is shown. DSC data for x=0.2 at different applied elds oncooling is also shown in (c), where the change of the sign of ∆Cp is observed for asame sample. For the sake of clarity, the latter dQ/dT data have the opposite signthan the same data in Fig. 5.5, to enable a comparison with (a) and (b) heatingruns.

131


0 30 60 90 120 150 180 210 240 270 300 3300

10

20

30

40

50

TN

PM-FMM-O(I)

AFM-FMO(II)-O(I)

cool. heat. x 0.05 H0.1 H0.18 H0.2 H=00.2 H0.25 H=00.25 H0.3 H=00.3 H0.365 H=00.365 H0.45 H=00.45 H0.5

|∆S

| (J

/kg

·K)

Tt(K)

Figure 6.2: Scaling of |∆S | at the rst-order transition for the Gd5(SixGe1−x)4alloys. A variety of applied elds and compositions are represented. Solid andopen diamonds are from Ref. [2]. Symbols labeled with an H/H=0 correspondrespectively to measurements with the LHe (under H)/LN2(H=0) DSC.

132

6.3. Scaling of the transition entropy change

0 30 60 90 120 150 180 210 240 270 300 3300

10

20

30

40

50

TN

TN

PM-FMO(II)-O(I)

incr.H decr.H x 0.1

|∆S

| (J

/kg

·K)

Tt(K)

Figure 6.3: Scaling of |∆S | at the rst-order transition for the Gd5(SixGe1−x)4alloys. Values obtained from M(H) up to 23 T for x=0.1 have been added withrespect to Fig. 6.2.

133


Tt. Due to the magnetoelastic coupling, the application of H shifts Tt, so that itis possible to observe both the AFM/O(II)-FM/O(I) transition at Tt and, at highenough H, a PM/O(II)-FM/O(I) transition, when Tt(H) ≥ TN . The latter transi-tion is still rst-order due to the crystallographic transformation and arises fromthe PM-AFM transition. For the AFM/O(II)-FM/O(I) transition, |∆S | increasesmonotonically with Tt, in agreement with ∆Cp > 0 (Figs. 6.1 (a), (c) and Ref. [6]).However, for the PM/O(II)-FM/O(I) transition, |∆S | decreases with Tt for x=0.2,in agreement with ∆Cp < 0 (Fig. 6.1 (c)). Since in calorimetric ∆S measurementsonly a eld of up to 5 T may be applied, ∆S values obtained from magnetisationup to 23 T by using the Clausius-Clapeyron equation have been added in Fig. 6.3.Then, the evolution of ∆S in the PM/O(II)-FM/O(I) transition is clearly observed.The magnetisation measurements are detailed in section 5.2. For the sake of clar-ity, only values for x=0.1 are shown in Fig. 6.3, but all samples with x ≤ 0.2present the same behaviour. The slight difference between calorimetric and mag-netic ∆S values in these samples, as also seen for x=0 and 0.05 in Fig. 5.9, may berelated to the fact that the transition is induced in different directions of the phasediagram (see Chapter 9).

Consequently, |∆S | is maximum for each composition at Tt = TN , i.e. when,in the FM phase, the applied H is large enough to shift the rst-order transition tooverlap to the second-order transition at TN (labeled in Figs. 6.2 and 6.3). There-fore, the largest value |∆S |=48.22 J/(kgK) occurs at Tt ≈130 K (∼ the highestvalue of TN , which corresponds to x=0.2 [7]). All the foregoing suggests that |∆S |,and thus MCE, will be maximum within the compositional range 0.2 < x < 0.24,where the different crystallographic and magnetic phases coexist, and the twobranches of |∆S | join (Figs. 6.2 and 6.3).

6.4 ConclusionsDSC under H has been used successfully to measure the entropy change at therst-order magnetostructural phase transition for Gd5(SixGe1−x)4, x ≤ 0.5. Wehave shown that the transition entropy change scales with Tt. The scaling of ∆S isa direct consequence of the fact that Tt is tuned by x and H and it is thus expectedto be universal for any material showing strong magnetoelastic effects, yieldinga eld-induced nature of the transition. ∆S is expected to (i) go to zero at zerotemperature, (ii) tend asymptotically to zero at high temperature since the latentheat is nite, and (iii) display a maximum at that temperature for which both ∆Mis maximised and Tt shows the minimum eld dependence. The specic shapeof ∆S vs. Tt will depend on the details of the phase diagram, Tt(x). Finally, thescaling of ∆S shows the equivalence of magnetovolume and substitution-relatedeffects in Gd5(SixGe1−x)4 alloys.

134

Bibliography

Bibliography[1] This procedure gives reliable values for ∆S in rst-order phase transitions.

See for instance J. Ortín and A. Planes, Acta Metall. 36, 1873 (1988).

[2] A. Giguère, M. Földeàki, B. Ravi Gopal, R. Chahine, T. K. Bose, A. Frydman,and J. A. Barclay, Phys. Rev. Lett. 83, 2262 (1999).





[7] V. K. Pecharsky and K. A. Gschneidner, Jr., Appl. Phys. Lett. 70, 3299 (1997).

135


136

Chapter 7

The magnetoelastic coupling inGd5(SixGe1−x)4

7.1 IntroductionIn this chapter, we study the effect of the magnetic eld on the magnetostructuraltransition in Gd5(SixGe1−x)4 alloys with x ≤ 0.5. In particular, the variation of thetransition eld, Ht, with the transition temperature, Tt, is discussed as a functionof x. This parameter, dHt/dTt, plays a key role in the scaling of ∆S , showing adifferent behaviour between the two compositional ranges (x ≤ 0.2 and 0.24 ≤x ≤ 0.5) where the magnetostructural transition occurs. Moreover, dHt/dTt isrelated to the strenght of the magnetoelastic coupling: in these compounds, thevalue of ∆S measured when the transition is eld-induced coincides with the valuemeasured when it is induced by the application of pressure [1]. Therefore, throughthe Clausius-Clapeyron equation (Eq. 1.17), it is shown that (see section 6.3)

∆M∆V =

dTt

dHt

dPt

dTt. (7.1)

Accordingly, a strong magnetoelastic coupling yields a small value of dHt/dTt.

7.2 H − T diagram from magnetisation and DSCmeasurements

The systematic measurements of Gd5(SixGe1−x)4 samples (x=0, 0.05, 0.1, 0.18,0.2, 0.25, 0.3, 0.365 and 0.45) are detailed in section 5.2 (magnetisation) and 5.3(DSC under eld).

From both sets of measurements -DSC and M(H)- the dependence of the tran-sition temperature, Tt, on the transition eld, Ht, can be evaluated independently.

137

CHAPTER 7. THE MAGNETOELASTIC COUPLING IN GD5(SIXGE1−X)4

0 40 80 120 160 200 240 2800

4

8

12

16

20 (a)

µ 0H(T

)

T(K)

DSC M(H) x 0 0.05 0.1 0.18 0.2

0.25 0.3 0.365 0.45

70 80 90 100 110 120 130 140 1500

4

8

12

16

20

AFM-FMtransition

PM-FMtransition

DSC M(H) x=0.1 x=0.18

(b)

µ 0H(T

)

T(K)

Figure 7.1: (a) Transition eld, Ht, as a function of the transition temperature, Tt,for Gd5(SixGe1−x)4 (from x=0 to x=0.45) obtained from magnetisation isotherms(increasing and decreasing H) and DSC isoeld data (cooling and heating). (b)Detail of panel (a) showing Ht(Tt) for x=0.1 and x=0.18, on increasing H andcooling.

138

7.3. dHt/dTt and magnetoelastic coupling

From magnetisation isotherms, Ht(T ) is dened at each temperature as the eldcorresponding to the inection point within the transition region. Due to the hys-teresis between increasing and decreasing eld, two different values of Ht areobtained. From DSC, Tt(H) is estimated at each applied eld as the peak positionin the dQ/dT curves. Due to the thermal hysteresis, two different values of Tt areobtained (see Table 6.1). Figure 7.1 (a) displays the transition eld as a functionof the transition temperature obtained from both DSC and M(H) curves. Noticethe good agreement between isoeld and isothermal data. Tt values at zero eldobtained by M(H) (extrapolated) and DSC are displayed in Table 7.1 for all com-positions, being in agreement with the phase diagram (Fig. 2.2). Interestingly, for0.24 ≤ x ≤ 0.5, where only the PM-to-FM transition occurs, Ht(Tt) shows a linearbehaviour over the whole eld range, while for x ≤ 0.2, the slope of Ht(Tt) variesprogressively from a low-eld value (AFM-FM transition) to a high-eld value(PM-FM transition). This effect is illustrated in Fig. 7.1 (b), which shows a detailof Fig. 7.1 (a) for x=0.1 and x=0.18 curves. Such a progressive change in theslope is due to the fact that, at high elds, the magnetostructural transition over-laps the second order PM-AFM transition (Fig. 5.5 (c)), giving rise to a uniquePM-FM transition.

7.3 dHt/dTt and magnetoelastic couplingFigure 7.2 compiles, for all compositions, the values of the slope, dHt/dTt, as afunction of x, determined from the data in Fig. 7.1. For x ≤ 0.2, two limitingvalues of dHt/dTt corresponding to the low and high eld regimes are displayed,while a single value of dHt/dTt is found for 0.24 ≤ x ≤ 0.5. Datum for x=0.5is taken from Ref. [2]. We note the linear dependence of dHt/dTt on x, which isdecreasing for the PM-FM transition (solid line in Fig. 7.2), while it is increasingfor the AFM-FM transition (dashed line in Fig. 7.2). Both lines meet at the com-position range where the second-order transition disappears (0.2 < x < 0.24), inagreement with the phase diagram (Fig. 2.2 and Ref. [3]). The value of dHt/dTtfor x=0 at high elds is lower than expected because a eld higher than 23 T (themaximum available in the present work) must be applied to fully induce the PM-FM transition. Values of dHt/dTt obtained from DSC and M(H) measurementsfor all compositions are displayed in Table 7.2 and compared with values given inliterature.

The strength of the magnetoelastic coupling is associated with the eld depen-dence of Tt (i.e., a strong magnetoelastic coupling yields a small value of dHt/dTt)as demonstrated in the introduction of this chapter. Consequently, the decrease indHt/dTt with increasing x for the PM-FM transition indicates a strengtheningof the magnetoelastic coupling. This may be explained by considering that FM

139


Tt(H = 0) (K)x ID Heat T. DSC M(H)

cool. heat. incr. H decr. H0 #1 NO ∼13* ∼20* ∼15* ∼22*

0.05 #1 T4+Q 43.8 46.5 44.5 45.90.1 #1 NO 70.4 73.1 73.2 76.5

0.18 #1 NO - - 103.7 106.1T4 98.7 100.9 97.8 99.5

0.2 #1 NO 113.9 116.6 114.6 120.50.25 #2 NO 143.0 150.5 - -0.3 #2 NO 169.7 177.5 - -

T4+Q 156.1 157.3 156.6 160.40.365 #3 NO 200.7 204.5 204.3 209.30.45 #7 NO 247.2 252.3 245.1 252.2

T4 243.5 247.1 238.0 244.9

Table 7.1: Transition temperature at zero eld, Tt(H = 0), at the rst-order tran-sition obtained by extrapolating Tt(H) obtained from M(H), and also by DSC atH=0 for all measured samples. *These values are valid after the low-temperature FM phasehas been induced irreversibly in the x=0 compound by the application of a high enough magneticeld (see section 2.4.1).

140

7.3. dHt/dTt and magnetoelastic coupling

0.0 0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

AFM-FMtransition

PM-FMtransition

high field

low field

from M(H), incr. H from M(H), decr. H from DSC, cooling from DSC, heating

d(µ 0H

t)/dT

t (T

/K)

x (Si content)

Figure 7.2: Slope of Ht(Tt) calculated from data in Fig. 7.1. For x=0.25, 0.3,0.365, 0.45 and 0.5 (the latter from Ref. [2]) a single slope is obtained, which cor-responds to the PM-FM transition. For x=0, 0.05, 0.1, 0.18 and 0.2 two limitingslopes are obtained: a low-eld value (associated with the AFM-FM transition)and a high-eld value (associated with the PM-FM transition). Solid and dottedlines are a guide to the eye.

141


d(µ0Ht)/dTt (T/K)DSC M(low H) M(high H)

x cool. heat. inc. H dec. H inc. H dec. H Literature0 0.142 0.158 0.142 0.149 0.458 0.460 0.125 [4]

0.05 0.196 0.204 0.202 0.166 0.591 0.6010.08 0.294 [5]0.1 0.241 0.250 0.273 0.271 0.544 0.525 0.27 [3], 0.26 [3]0.18 0.271 0.278 0.278 0.268 0.441 0.4350.2 0.312 0.311 0.299 0.338 0.437 0.437

M(H)inc. H dec. H

0.25 0.331 0.376 - -0.3 0.294 0.287 0.314 0.321

0.365 0.255 0.277 0.316 0.2960.375 0.28 [6, 7], 0.25 [8],0.43 0.23 [9]0.45 0.213 0.205 0.237 0.241 0.21 [1], 0.22 [10]0.5 0.154 [2], 0.18 [11],

0.14 [12]

Table 7.2: dHt/dTt obtained from M(H) and DSC under magnetic eld for allmeasured samples. For x ≤ 0.2, two limiting values are obtained from magnetisa-tion data. Values from different references are also compiled for comparison.

142

7.4. Conclusions

exchange interactions are stronger for increasing x, as suggested by the magneticphase diagram, where Tt increases linearly with x (Fig. 2.2). The fact that dHt/dTtfor the PM-FM transition has continuous behaviour, although the PM phase ismonoclinic for 0.24 ≤ x ≤ 0.5 and orthorhombic-II for x ≤ 0.2, suggests thatthe magnetoelastic coupling is weakly dependent on the actual crystallographicstructure. Concerning the AFM-FM transition, and taking into account that thestructural transition is the same (for x ≤ 0.2) or very similar (for 0.24 ≤ x ≤ 0.5)to that occurring in the PM-FM case, the increase in dHt/dTt with x may be re-lated to the fact that the transition involves two ordered magnetic phases (FMand AFM). Fig. 7.2 thus summarizes the behavior of the rst-order transition inGd5(SixGe1−x)4 as a function of x, T and H.

The behaviour of dHt/dTt with x is relevant in the scaling of |∆S | whichappears in Gd5(SixGe1−x)4 alloys (Chapter 6): taking into account the Clausius-Clapeyron equation and as ∆M always decreases with T , |∆S | thus increases withTt for x ≤ 0.2 when the AFM-FM transition takes place, due to the larger increaseof dHt/dTt with Tt as compared to the decrease in ∆M with Tt. In contrast, |∆S |decreases with Tt for the PM-FM transition, since the increase in dHt/dTt with Ttis not large enough as to overcome the decrease in ∆M. Therefore, although themain feature of the scaling of |∆S | with Tt is not only determined by dHt/dTt vsTt, the particular dependence of dHt/dTt on x and H enables the scaling.

7.4 ConclusionsThe variation of the transition eld with the transition temperature, dHt/dTt, hasbeen studied in Gd5(SixGe1−x)4 for all the range of compositions where the rst-order transition occurs, 0 ≤ x ≤ 0.5. Taking into account the behaviour of dHt/dTtas a function of x and that ∆M decreases monotonously with Tt, it is shown thatdHt/dTt governs the scaling of ∆S with Tt reported in Chapter 6, giving furtherevidence that the origin of this scaling is the magnetoelastic nature of the transi-tion. Moreover, two distinct behaviors for dHt/dTt have been found on the twocompositional ranges where the magnetostructural transition occurs, thus showingthe difference in the strength of the magnetoelastic coupling of this system.

143


Bibliography[1] L. Morellon, P. A. Algarabel, M. R. Ibarra, J. Blasco, B. García-Landa, Z.

Arnold, and F. Albertini, Phys. Rev. B 58, R14721 (1998).












144

Chapter 8

Short-range antiferromagnetism inGe-rich Gd5(SixGe1−x)4 alloys

8.1 IntroductionThe aim of this chapter is to provide a better understanding of the complex mag-netic structures in the Ge-rich composition region of Gd5(SixGe1−x)4 compounds(0 ≤ x ≤ 0.2). Sections 2.2 and 2.4.1 review the most relevant properties of thesealloys reported in literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11].

We show that, when a high magnetic eld is applied, an unreported magneticphase transition occurs from the antiferromagnetic (AFM) phase to a phase whichpresents short-range magnetic correlations. This is based on the abrupt changein the slope of the high-temperature magnetisation isotherms at high elds (∼14-15 T) and on the behaviour of the inverse of the susceptibility at high tempera-tures. The entropy change as a function of the temperature, obtained by using theMaxwell relation, also supports the existence of this transition. Results suggestthat the transition is due to the breaking of the long-range AFM correlations bythe applied magnetic eld, which leads to short-range AFM correlations and tothe presence of ferromagnetic (FM) clusters. This nding illustrates the rich andcomplex magnetic behaviour of Ge-rich Gd5Si(xGe1−x)4 alloys.

8.2 Results and discussionDSC data at different magnetic elds for x=0, x=0.05 and x=0.1 are shown inFigs. 5.5 (x=0 and x=0.05) and 4.7 (x=0.1). The relation between Tt, which isestimated as the temperature at the maximum of the peak in the dQ/dT curve,and H, on cooling, is plotted in Fig. 8.1. These values are in good agreement withthose reported for x=0 by Levin et al. [7]. The second-order transition is observed

145

CHAPTER 8. SHORT-RANGE ANTIFERROMAGNETISM INGD5(SIXGE1−X)4

0

4

8

12

16

20

Cp (Levin et al.) M(H) DSC M(T)

SRAFMx=0

AFM

FM

0

4

8

12

16

20

SRAFMx=0.05

AFM

FM

µ 0H(T

)

20 40 60 80 100 120 1400

4

8

12

16

20

SRAFM

x=0.1

AFM

FM

T(K)Figure 8.1: The magnetic phase diagrams of some of the Ge-rich Gd5(SixGe1−x)4compounds. Values are obtained from M(H) curves (solid squares), from M(T )curves (open squares) and from DSC under eld (open circles). Data taken fromRef. [7] are also displayed (open triangles and dashed lines). For the sake ofclarity, only values on cooling and increasing eld are displayed. Solid lines are aguide to the eye.

146

8.2. Results and discussion

in DSC as a small λ-type jump in the dQ/dT baseline. The Néel temperature (TN)of this phase transition is also shown in Fig. 8.1, where a decrease in TN withincreasing elds is observed. The same behaviour is reported in Ref. [7] for x=0.

M(H) isotherms for all three compositions are shown in Fig. 5.1. These curvesexhibit a jump, ∆M, at the eld-induced AFM-to-FM magnetostructural transi-tion. The transition eld, Ht, is dened as the eld corresponding to the inectionpoint within the transition region. Ht vs. T , for increasing H, is plotted in Fig. 8.1as well. These values are in agreement with the values obtained from DSC andwith those reported in Ref. [7]. We note that in the AFM zone, the M(H) curvesshow a linear behaviour with a high slope, as corresponds to an antiferromagnetbut not to a ferrimagnet. Besides, for x=0, the rst magnetisation curve at 5 Kshows an irreversible AFM-to-FM transition when a eld of ∼1.5 T is applied(see inset in Fig. 8.2), as already detailed in Refs. [7, 8, 9, 10]. At high tempera-tures and for high enough elds a new transition is observed: a change in the slopeof the M(H) curve suggests a transition from the AFM structure to another mag-netic structure induced by the eld, before the expected rst-order eld-inducedmagnetostructural transition to the O(I)/FM phase takes place. This behaviouris observed for x=0 and x=0.05 and disappears for x=0.1, though the slope inM(H) curves at high T for the latter composition does not correspond completelyto a paramagnet (PM), see Fig. 5.1. In order to study this behaviour, additionalM(H) curves at the temperatures of interest were measured for x=0 and x=0.05(displayed in Fig. 8.2). The elds at which the change of slope takes place areevaluated as the points with maximum curvature in the M(H) isotherms, and theyare represented in Fig. 8.1. It is worth noting that these values match the TN(H)curve for both x=0 and x=0.05. We are thus observing the nominal transition fromthe AFM to the PM phase by increasing the magnetic eld, but since a PM phasecannot be eld-induced, the high-eld phase remains unknown. Since the eldbreaks the AFM ordering, a phase with short-range AFM correlations (SRAFM)is probably formed.

To understand the nature of this high-eld phase, the inverse of the suscepti-bility as a function of T was calculated from M(H) curves (taken from Fig. 5.1) 1,in the range of temperatures and elds where the new phase appears. Figure 8.3shows the results for x=0, 0.05 and 0.1. The Curie-Weiss law,

χ(T ) =cC

T + θC, (8.1)

is obeyed for all x, where cC = (Nat/m)(p2e f fµ

2B/3kB), θC is the paramagnetic Curie

temperature and pe f f =√

g2J(J + 1) is the effective magnetic moment in Bohr1M(H) curves are linear in this high-eld phase (Figs. 5.1 and 8.2), therefore the differential

susceptibility, χ = ∂M/∂H, is obtained by a linear regression.

147


0 2 4 6 8 10 12 14 16 18 20

0

20

40

60

80

100

120

140

160

99.8 K 103.5 K 111.7 K 116.0 K 119.8 K 124.6 K 129.4 K 198.0 K 247.2 K

x=0.05

µ0H(T)

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

M(e

mu

/g)

µ0H(T)

0

20

40

60

80

100

120

140

160

89.0 K 94.0K 98.9K 103.0K 107.0K 109.5K 114.9K 120.6K 125.6K 130.4K 151.7K 195.5K 239.3K

x=0

M(e

mu

/g)

0 2 4 6 80

40

80

120

160

200

1st cycle 2nd cycle

T=5 K

M(e

mu

/g)

µ0H(T)

Figure 8.2: Selected magnetisation isotherms of Gd5(SixGe1−x)4 for x=0 andx=0.05 under increasing and decreasing eld. Inset in upper panel shows therst and second magnetisation curves at 5 K for x=0 after a zero-eld cooling,where the irreversible transition at ∼1.5 T is observed in the rst magnetisationcurve. Inset in lower panel shows a detail of magnetisation curves for x=0.05 atlow elds. The change in the slope of the curves is labeled with an arrow on therst isotherm for x=0 and x=0.05.

148


70 80 90 100 110 120 130 1400

1

2

3

χ-1

(10

-5 k

g/m

3 )

T (K)

x=0 x=0.05 x=0.1

Figure 8.3: Inverse of susceptibility for x=0, 0.05 and 0.1, obtained from M(H)curves in Fig. 5.1, in the range of temperatures and elds for which the new phaseappears. The Curie-Weiss law gives an effective magnetic moment much lowerthan 7.94 µB, which is the value expected per Gd3+ free ion.

149


80 100 120 140 160 180 200 220 240 260 280 300

0

1

2

3

4

5

TN

Tt

cooling heating

x=0.18

T (K)

χ'ac

-1 (

10-5 k

g/m

3 )

120 150 180 210 240-1

0

1

TN

Tt

χ'' ac

(10

-4 e

mu/

cm3 )

T (K)

Figure 8.4: Inverse of ac susceptibility for x=0.18 (#1, T4 treatment), obtainedon cooling and heating. The temperature of the rst-order transition, Tt, and theNéel temperature, TN , are labeled. Inset: detail of the imaginary part of the acsusceptibility, χ, around TN .

magnetons, yielding θC=75.0 K and pe f f =3.99 µB/Gd atom for x=0; θC=80.7 Kand pe f f =3.96 µB/Gd atom for x=0.05; and θC=97.6 K and pe f f =4.61 µB/Gd atomfor x=0.1. Since the Curie-Weiss law is followed with a pe f f much lower than thatexpected for the free Gd3+ ion (7.94 µB), the eld-induced phase is not PM (whichalthough it could be nominally expected, yields no physical meaning) but rather aphase with still short-range AFM correlations (SRAFM phase). FM correlationsshould also be present, because θC is positive, as reported also in Refs. [1, 4, 5, 8].Another evidence of the presence of FM interactions in the AFM phase is the ex-istence of a magnetisation saturation observed for all M(H) curves correspondingto the AFM phase, at low elds (see Figs. 5.1 and 8.2 for the three compositionsand inset in lower panel of Fig. 8.2 for a detail for x=0.05).

The existence of a SRAFM phase allows us to postulate the occurrence of atransition between the latter and the high-temperature PM phase. This hypotheti-cal transition is expected to be smooth and to undergo in a wide temperature range.

150


An evidence of this transition is the non-linear behaviour of the inverse of the sus-ceptibility for x=0 at zero eld between TN and ∼230 K [5], which indicates theexistence of some magnetic correlations within the nominal PM phase. The samebehaviour is observed in the inverse of in-phase component of ac susceptibility,(χ′ac)−1, for x=0.15 and x=0.18 presented in section 3.2.2. Figure 8.4 shows (χ′ac)−1

for a sample with x=0.18 (#1 T4), for cooling and heating, as an example of theseresults, leading to the same shape observed for x=0 [5], with two anomalies at∼175 K and ∼240 K. The same anomalies are clearly observed in the imaginarypart of the ac susceptibility (see inset in Fig. 8.4). The Curie-Weiss law above∼240 K give rise to the same value of 7.33 µB for x=0.15 (#1 as cast), x=0.18 (#1as-cast) and x=0.18 (#1 T4). It is worth noting the presence of hysteresis withinTt (∼100 K) and 175 K, i.e., above and below TN . This fact strongly suggests thatFM clusters are present both in the AFM phase and, above TN , in the SRAFMphase.

After all these observations, M(T ) curves at different applied elds were de-sirable. Therefore, high-eld curves were measured in the GHMFL, while low-eld curves were measured in a commercial extraction magnetometer for x=0 andx=0.05 samples. Figure 8.5 shows M(T ) curves measured at high elds (from 1to 20 T) for x=0 and x=0.05 on heating after a zero-eld cooled (ZFC) process,where the rst- and second-order transitions are observed up to ∼12 T. At higherelds, the rst-order transition overlaps the second-order one. Some M(T ) curvesare also displayed on heating after ZFC and on the subsequent eld-cooling (FC),at selected applied elds for x=0 (inset in Fig. 8.5) and x=0.05. We note thatsome hysteresis is also observed in the whole AFM phase and also above TN upto ∼240 K, indicating that FM correlations are relevant in both the AFM phaseand the magnetic phase above TN(H). Hysteresis in the whole AFM phase and upto ∼230 K is also reported for the temperature dependence of the thermopower inx=0.1 [12].

Figure 8.6 shows M(T ) curves at low eld (0.1 T), together with selected high-eld curves, for x=0. The shape of the second-order AFM-PM transition is verydifferent at low and high elds. As detailed in the inset in Fig. 8.6, the AFM-PM transition at low eld shows the usual shape of a FM-PM transition, whilethe presence of a high eld largely smooths the transition. This proves that FMcorrelations play an active role at the second-order transition, in agreement withthe previous results. This fact together with the hysteresis ocurring along the AFMphase strongly suggest that FM clusters are present in the AFM phase and growin size for decreasing temperatures, down to Tt, where the rst-order transition tothe long-range FM phase takes place due to the percolation of the former. Thissuggestion is in agreement with a recent work of Pecharsky et al. [10], in whichthey observe that the FM phase presents 93% mol. of O(I) structure and the restis O(II), although the magnetisation is at ∼99 % of its saturation value (MS =7.5

151


40 80 120 160 200 240 2800

20

40

60

80

100

120

140

160

FC

ZFC

2 T

1 T

4 T

10 T

20 T

x=0.05

M (

em

u/g

)

T (K)

0

20

40

60

80

100

120

140

160

180

20 T16 T

12 T8 T

4 Theatingx=0

M (

em

u/g

)

50 100 150 200 2500

60

120

180

FC

ZFC

1 T2 T

10 T

M (

em

u/g

)

T (K)

Figure 8.5: Magnetisation of Gd5(SixGe1−x)4 on heating, for x=0 and x=0.05 un-der different applied elds within 1 and 20 T. Some curves in ZFC and FC pro-cesses are also displayed for x=0.05. Inset shows the magnetisation after ZFC andFC processes for x=0 under selected applied elds.

152


0 40 80 120 160 200 240 2800

20

40

60

80

100

120

140

160

180

0.1 T

2 T

8 T

x=0

M (

emu/

g)

T (K)

105 110 115 120 125 130 135

2

4

6

55

60

65

70

0.1 T

8 T

M (

emu/

g)

T (K)

Figure 8.6: Low- and high-eld magnetisation of Gd5(SixGe1−x)4 for x=0. Insetshows the magnetisation around the Néel temperature (labeled with an arrow) atlow and high elds.

µB/Gd atom), being an indication that the fraction of the high-temperature O(II)phase orders ferromagnetically. This behaviour may arise from the competitionbetween the intraslab FM interactions and the interslab AFM interactions presentin the system (section 2.4.1).

In order to study the behaviour of the PM phase and to check a possible tran-sition between the PM and SRAFM phases, the inverse of the susceptibility cal-culated from M(T ) curves at different elds, χ = M/H, is analysed (Fig. 8.7). Inthis case, only χ(T )−1 calculated far from the regions with strong magnetic corre-lations has a physical meaning. For x=0 at 0.1 T, the anomaly at ∼240 K is alreadypresent, as also observed in ac susceptibility for x=0.15 and x=0.18 and in Ref.[5] for x=0. At 1 T this anomaly has vanished. As the eld increases, χ−1(T )heightens near TN , but all curves tend to converge at high temperatures, whereCurie-Weiss law is followed. From this law, pe f f and θC are obtained and they aredisplayed as a function of the magnetic eld in the corresponding inset in Fig. 8.7.Values of pe f f and θC from the literature for Ge-rich compounds are compiled inTable 8.1 for comparison. Our results indicate that the transition from the SRAFMto the PM phase at high elds takes place continuously in a range of temperatures

153


40 60 80 100 120 140 160 180 200 220 240 260 2800

1

2

3

4

(ZFC-FC)1 T

(ZFC-FC)

2 T4 T

10 T

20 T

x=0.05

χ-1

(10

-5 k

g/m

3 )

T (K)

from M(T) from M(H)

0 2 4 6 8 1012141618207.0

7.2

7.4

7.6

7.8

8.0

θC (K)

peff

(µB)

µ0H (T)

70

80

90

100

110

120

0

1

2

3

4

4 T 0.1 T1 T

8 T16 T

20 T

x=0

χ-1

(10

-5 k

g/m

3 ) from M(T) from M(H)

0 2 4 6 8 10 12 14 16 18 206.8

7.0

7.2

7.4

7.6

7.8

8.0

θC (K)

peff

(µB)

µ0H (T)

60

80

100

120

140

Figure 8.7: Inverse of susceptibility calculated from M(T ) curves for x=0 andx=0.05 under different applied elds, on heating (solid lines). Some curves dis-played for x=0.05 show zero-eld cooled and eld-cooled processes. The inverseof the susceptibility calculated from the M(H) curves at the linear region of theSRAFM phase is also displayed (solid circles). Insets show, for each composition,the effective magnetic moment, pe f f , and the paramagnetic Curie temperature, θC,both extrapolated from the linear behaviour at high temperatures by using theCurie-Weiss law, as a function of the applied eld. Solid lines are a guide to theeye.

154


Compound θC (K) pe f f (µB) Referencex=0 94 8.10 [1]

120.7 7.14 [4]94 7.45 [7]92 8.05 [5]

x=0.0825 111.9 7.52 [4]

Table 8.1: Magnetic properties of some Ge-rich Gd5(SixGe1−x)4 alloys from theliterature. θC stands for the paramagnetic Curie temperature and pe f f for the ef-fective magnetic moment.

that lies within TN(H) and ∼240 K, temperature above which χ−1(T ) shows a PMbehaviour for all elds, although the eld dependence of both pe f f and θC (insetsin Fig. 8.7) suggests that even at room temperature there are still magnetic corre-lations due to the presence of the eld. Figure 8.7 also shows χ−1 obtained fromthe M(H) curves displayed in Fig. 8.2, together with the results previously shownin Fig. 8.3. In this case, the susceptibility is evaluated differentially from the slopeof M(H) at the SRAFM phase. The values of χ−1 at high temperatures superim-pose χ−1(T ) obtained from M(T ) for both x=0 and x=0.05, since the behaviour istruly PM and does not depend on the eld. At lower temperatures, the shape ofχ−1 (a concave deviation to higher values of χ−1 with respect to the high temper-ature region, i.e., lower χ) evidences that AFM interactions are dominant. Below∼110 K, χ−1 decreases linearly with decreasing T , which suggests that competingFM and AFM interactions are present in the system.

Figure 8.1 shows the H − T phase diagram for x=0, x=0.05 and x=0.1. Thetransition elds and temperatures are obtained from M(H) isotherms, from M(T )curves and from DSC. Data from Ref. [7] are also plotted for comparison. Forthe sake of clarity, only increasing eld data (from M(H)) and cooling data (fromM(T ) and DSC) are represented. Decreasing eld/heating data have the same be-haviour, with a slight hysteresis at the rst-order transition and no hysteresis at thesecond-order transition. Tt vs Ht and TN vs H curves intersect at a tricritical point(T ∗, H∗), which depends on the composition, being ∼85 K and ∼13.5 T for x=0(90 K and 14 T using Ref. [7]), ∼92 K and ∼12.5 T for x=0.05 and ∼107 K and∼10.5 T for x=0.1. If the magnetic eld is isothermally increased at temperatureT such that T ∗ < T < TN(H = 0), the system undergoes from the AFM to theintermediate SRAFM phase, and nally reaches the FM phase. The range of tem-peratures for which the SRAFM phase can be observed decreases with increasingx. For x=0.1, the temperature range is just ∼20 K and the existence of the SRAFMphase is only evidenced in M(H) curves by a slight curvature. The SRAFM phase

155


is present at zero eld up to at least ∼240 K, as observed in the inverse of sus-ceptibility for various samples and techniques. As the applied eld is increased,the transition to the PM phase is broadened and smoothed. This transition hasnot been indicated in Fig. 8.1 because it spreads in a wide temperature range andcannot be well determined. Hence, susceptibility up to very high temperatures,and at high H, is necessary to complete the SRAFM-PM transition at high elds.

In order to have a deeper sight of the origin of this new transition, we haveanalysed the entropy change as a function of the temperature for each x, obtainedfrom the Maxwell relation,

∆S (0→ Hmax) =

∫ Hmax

0

(∂M∂T

)

HdH.

Figure 8.8 shows the entropy change as a function of temperature for x=0 andx=0.05, obtained with the previous equation. It is worth noting that the entropychange curves obtained from M(H) curves, when integrating from 15 and 20 Tto zero, show a double peak, which suggests that there are two transitions in thesystem (AFM-SRAFM and SRAFM-FM). The peak at the highest temperatureis associated with the rst-order transition and it shifts to higher temperatureswhen the maximum applied eld increases. The low-temperature peak is relatedto the new AFM-SRAFM transition: the change in the slope at the M(H) curves,(∂M/∂H), also involves a change in (∂M/∂T ), which increases the contribution to∆S at the corresponding temperatures and elds. A similar behaviour is observedin Tb5Ge4, where a high-temperature PM-AFM1 and a low temperature AFM1-AFM2 transitions give rise to a double peak structure in the entropy change cal-culated from the Maxwell relation [13].

8.3 ConclusionsWe have shown that a eld-induced magnetic phase transition exists from theAFM phase to a phase which presents short-range AFM correlations (SRAFM).Experimental results suggest that the transition results from the breaking of thelong-range AFM ordering with the application of a magnetic eld, which leadsto competing FM and AFM short-range correlations in the SRAFM phase. FMcorrelations are also relevant in the whole AFM phase and we suggest that theFM clusters already present in the AFM phase grow in size with decreasing Tand percolate at Tt, when the long-range FM ordering takes place. The exis-tence of magnetic correlations in the true PM phase has also been shown. Theexpected transition from the SRAFM to the PM phase takes place at ∼240 Kat zero eld, broadening and smoothing under applied eld. This ndings con-tribute to the understanding of the rich and complex magnetic behaviour of Ge-

156

8.3. Conclusions

40 60 80 100 1200

10

20

30

40

(b)

2 T

5 T

7 T10 T

15 T

20 T

x=0.05

∆S (

J/kg

K)

T (K)

20 40 60 80 100 1200

10

20

30

40

(a)

15 T

20 T

10 T

7 T

5 T

2 T

x=0

∆S(J

/kg

K)

Figure 8.8: Entropy change for Gd5(SixGe1−x)4 (x=0 and x=0.05) calculated byusing the Maxwell relation integrating from Hmax (labeled beside each curve) tozero.

157


rich Gd5(SixGe1−x)4 alloys, which may arise from the competition between theintraslab FM interactions and the interslab AFM interactions (section 2.4.1).

158

Bibliography

Bibliography[1] F. Holtzberg, R. J. Gambino, and T. R. McGuire, J. Phys. Chem. Solids 28,

2283 (1967).










[11] C. Magen, Z. Arnold, L. Morellon, Y. Skorokhod, P. A. Algarabel, M. R.Ibarra, and J. Kamarad, Phys. Rev. Lett. 91, 207202 (2003).


[13] L. Morellon, C. Magen, P. A. Algarabel, M. R. Ibarra, and C. Ritter, Appl.Phys. Lett. 79, 1318 (2001).

159


160

Chapter 9

Dynamics of the rst-ordertransition in Gd5(SixGe1−x)4:cycling and avalanches

9.1 IntroductionIn this chapter, we study the dynamics of the rst-order magnetostructural transi-tion in Gd5(SixGe1−x)4 alloys. Firstly, we study the effect on the entropy change,∆S , of inducing the transition either by T or H. Secondly, we present a systematicstudy of the effect of cycling a sample through the rst-order transition. We showthe evolution of ∆S with the number of cycles and we also analyse the avalanchesbetween metastable states of the system during the transition. All the forego-ing allows to unveil the actual mechanism that drives the rst-order transition inGd5(SixGe1−x)4 alloys.

9.2 Comparison of the entropy change induced bytemperature and by eld

As already explained in Chapter 4, DSC are usually designed to continuouslysweep temperature while Q(t) is measured. The T sweep induces thermally therst-order transition in the sample, while heat is released or absorbed. In the par-ticular case of eld-induced transitions, the temperature Tt of the peak of the tran-sition in the calorimetric curve is tuned by the magnetic eld, and consequentlythe eld dependence of ∆S can be obtained. Besides, our DSC also works sweep-ing H. By xing a temperature above Tt(H = 0) and increasing the magnetic eld,the rst-order transition can also be induced.

161

CHAPTER 9. DYNAMICS OF THE FIRST-ORDER TRANSITION INGD5(SIXGE1−X)4

0 1 2 3 4 5

-300

-200

-100

0

100

200

300 x=0.1

increasing H

decreasing H Rate 1 T/min 0.1 T/min

87.1 K

87.1 K83.2 K

83.2 K78.8 K

78.8 K

74.5 K

74.5 K

dQ/d

H (

mJ/

T)

µ0H (T)

74 76 78 80 82

17181920212223

0.1 T/min,incr.H 0.1 T/min,decr.H 1 T/min,incr.H 1 T/min,decr.H

T (K)

| ∆S

| (J/

kgK

)

Figure 9.1: Calorimetric curves recorded sweeping the eld (increasing and de-creasing H) in a Gd5(Si0.1Ge0.9)4 sample (#1, as-cast) at some xed temperaturesand for two different eld rates. Inset shows the absolute value of the entropychange as a function of temperature for the different rates on increasing and de-creasing H.

162

9.2. Comparison of the entropy change induced by temperature and by eld

In order to compare the values of ∆S obtained from both processes, DSCcalorimetric curves were measured sweeping the temperature and the eld. DSCcuves sweeping the temperature at constant eld -from now on we will call themDSCH(T )-, measured for all compositions, are decribed in section 5.3. DSCdata sweeping the eld at constant temperature -from now on we will call themDSCT (H)- were measured in the following samples: x=0.05 (#1 T4+Q), x=0.1(#1 as-cast), x=0.3 (#2 T4+Q) and x=0.45 (#7 T4). Calorimetric curves forDSCT (H) were recorded on increasing elds up to 5 T and decreasing elds downto zero. Field rates, H, of 1 and 0.1 T/min were applied. The results for x=0.05and x=0.1 are displayed in Figs. 4.8 and 9.1, respectively. We note that ∆S doesnot depend on H. Unfortunately, results for x=0.3 and x=0.45 cannot be used toobtain ∆S : the broadness of the transition region when the eld is sweeped is toolarge as compared to the eld range 0-5 T available in our cryostat (rst-orderpeak cannot be integrated properly).

The values of the entropy change obtained by DSCH(T ) (∆S H) and DSCT (H)(∆S T ) differ in ∼5 J/(kgK) for x=0.05 and ∼7 J/(kgK) for x=0.1 (see Fig. 9.2).These differences are systematic in the whole temperature range in which ∆S T ismeasured, and they are too high to be a consequence of the experimental error(which lies within 5-10 %).

The H − T phase diagram for x=0.05 is displayed in Fig. 9.3 in order to showan example of the thermal- and eld-induced processes in which ∆S H and ∆S Tare measured. Ht and Tt values are evaluated from both DSC measurements, aswell as the beginning and the ending of the transitions.

This difference can be justied using general thermodynamics [1]. The FirstPrinciple in differential form for a magnetic system is

dU = dQ + HdM = TdS + HdM , (9.1)

where dU is the differential internal energy, dQ is the differential transfered heatand HdM is the differential external work needed to magnetise the magnetic sys-tem. The Second Principle for a reversible process, TdS = dQ, has been used. Itis useful to work with the entalphy as the thermodynamic potential rather than theinternal energy. The enthalpy, E, is dened as

E = U − HM , (9.2)

which in differential form becomes

dE = dQ − MdH = TdS − MdH . (9.3)

At constant eld, the change in temperature from the beginning to the end of thetransition leads to

∆EH =

∫TdS = Q ≡ L , (9.4)

163


70 80 90 100 110 120 130 1400

10

20

30

40

20-0T15-0T

10-0T

7-0T5-0T

2-0T

Maxwell ∆Ht

Clausius M(H)/ DSC

T(H)

( 0.1 / 1 T/min) DSC

H(T)

x=0.1

T(K)

∆ S(J

/kgK

)

40 50 60 70 80 90 100 110 120

0

5

10

15

20

25

30

20-0T15-0T

10-0T

7-0T5-0T

2-0T

x=0.05 Maxwell ∆H

t

Clausius M(H) Clausius M(T)/ DSC

T(H)

( 0.1 / 1 T/min) DSC

H(T)

∆ S (J

/kgK

)

T (K)

Figure 9.2: Entropy change in Gd5(SixGe1−x)4, for x=0.05 and x=0.1, calculatedby using: DSCH(T ) on heating (open triangles); DSCT (H) on decreasing H (openand solid circles); the Clausius-Clapeyron equation evaluated from M(H) on de-creasing H (solid squares); and the Maxwell relation integrating from differentvalues of Hmax (labeled for each curve) to zero, and evaluated only within thetransition region (solid lines). For x=0.05, the entropy change calculated by usingthe Clausius-Clapeyron equation obtained from M(T ) on heating (open squares)is also displayed.

164


40 45 50 55 60 65 70 750

1

2

3

4

5

∆ST

∆SH

x=0.05

Tt(K)

µ 0Ht (

T)

incr. H from DSCT(H)

transition limits cool. from DSC

H(T)

transition limits

Figure 9.3: H − T phase diagram for x=0.05, obtained from DSC measurements.Ht(T ) on increasing H (solid squares) and the corresponding starting and nish-ing elds of the transition (dashed lines) are obtained from DSCT (H). Tt(H) oncooling (open squares) and the corresponding starting and nishing temperaturesof the transition (solid lines) are obtained from DSCH(T ). Examples of the pro-cesses in which the entropy change is evaluated, are labeled with arrows.

165


∆S H =

∫ dQT =

∫ dEH

T , (9.5)

i.e., at constant eld, the enthalpy change of the transition is equal to the latentheat, L, and the entropy change of the transition is calculated by integrating thedifferential heat divided by the temperature (Eqs. 9.4 and 9.5 are exactly the sameequations as Eq. 3.10). On the other hand, if the temperature remains constantand the magnetic eld varies from the beginning to the end of the transition, theenthalpy change is written as

∆ET = Q −∫

MdH = T∆S −∫

MdH , (9.6)

and therefore the latent heat and the entropy change, which are the values mea-sured by DSC, have the following expressions:

Q ≡ L = ∆ET +

∫MdH , (9.7)

∆S T =QT =

∆ET

T +1T

∫MdH . (9.8)

In this case, the latent heat has an additional contribution to the enthalpy changedue to the work of the magnetic eld over the system. Since at constant temper-ature, the entropy change is the latent heat divided by the xed temperature, ∆S Thas also this additional contribution.

We note that DSC measures the heat absorbed or released by the sample andtherefore the entropy change is also obtained experimentally. It is straightforwardto see from Eqs. 9.4 and 9.8 that ∆S obtained from DSCH(T ) and DSCT (H) mustbe different. For example, when a eld is applied isothermally (at T=T ′) in asystem, changing from low to high magnetisation, it shows a negative ∆S T . Since(1/T )

∫MdH is positive, the absolute value of ∆S T will be larger than that of ∆S H

associated with a process at constant eld which induces the transition on coolingat the temperature Tt=T ′ 1, provided that ∆ET/T ≈

∫dEH/T 2. The same con-

clusion is valid for decreasing eld and heating processes. For an ideal transition,which occurs at constant eld and temperature, (1/T )

∫MdH vanishes and both

values of the entropy change are the same. We note that differences between theabsolute values of ∆S T and ∆S H are observed in our samples. Moreover, an eval-uation of (1/T )

∫MdH using M(H) curves at the temperatures in which we have

measured both ∆S H and ∆S T values, for x=0.05 and x=0.1, yields ∼6.5 and ∼8J/(kgK), respectively. These values are in good agreement with ∆S H − ∆S T (∼5

1Figure 9.3 provides an schematic view of these two processes.2This consideration is assumed a priori.

166


20 30 40 50 60 70 80 90 100 110

0

5

10

15

20

25

30

20-0T

15-0T10-0T

7-0T

5-0T

2-0T

Maxwell ∆Ht

Clausius M(H) Clausius M(T) DSC

H(T)

x=0

T(K)

∆S(J

/kgK

)

Figure 9.4: Entropy change in Gd5(SixGe1−x)4, for x=0, calculated by using:DSCH(T ) on heating (open triangles); the Clausius-Clapeyron equation evaluatedfrom M(H) on decreasing H (solid squares); the Clausius-Clapeyron equationevaluated from M(T ) on heating (open squares); and the Maxwell relation inte-grating from different values of Hmax (labeled for each curve) to zero, evaluatedonly within the transition region (solid lines).

J/(kgK) for x=0.05 and ∼7 J/(kgK) for x=0.1), proving that these two processesare essentially different due to the work needed to magnetise the system duringthe transition and the validity of the approximation ∆ET/T ≈

∫dEH/T .

An indirect evaluation of the entropy change can be gained by using the Clau-sius-Clapeyron equation. In fact, we demonstrated in Chapter 5 that the Clausius-Clapeyron equation yields the correct value of the entropy change at a rst-ordertransition, and for this purpose we used M(H). Therefore, the entropy changeactually evaluated from the Clausius-Clapeyron equation is ∆S T . This is the rea-son why calorimetric measurements presented in Chapters 5 and 6 [obtained byDSCH(T )] yielded larger values (it measure ∆S H, which have been the DSC val-ues discussed in Chapters 3-8) than those obtained with the Clausius-Clapeyronequation in some samples (see Fig. 5.9). Surprisingly, this difference decreaseswith increasing x. For exampe, for x=0.45 there is no difference between bothvalues (see Fig. 5.8). This is due to the fact that (1/T )

∫MdH strongly decreases

with temperature (both M and 1/T decrease with T ), being ∆S H−∆S T∼1 J/(kgK)for x=0.45, in agreement with the observed behaviour.

We also note that ∆S T obtained from the Clausius-Clapeyron equation matchthe values measured by DSCT (H) (Fig. 9.2), as expected from an equivalent pro-

167


cess. The entropy change calculated from the Maxwell relation by integratingM(H) within the transition region (see section 5.4.2), is in agreement with ∆S Tvalues obtained from DSCT (H) calorimetric curves as well (Fig. 9.2 for x=0.05and x=0.1), since this evaluation arises from a eld-induced isothermal process.Finally, it should also be expected that the application of the Clausius-Clapeyronequation in experimental M(T ) curves, i.e., measurements of M at constant eldsand sweeping the temperature, should yield ∆S H. Hence, M(T ) curves for x=0and x=0.05 presented in Fig. 8.5 are used to calculate indirectly ∆S H. Thesevalues are displayed in Fig. 9.2 for x=0.05 and Fig. 9.4 for x=0. We observethat ∆S H evaluated from M(T ) is larger than ∆S T evaluated from M(H), and ∆S Hvalues matching reasonably well with those obtained from DSCH(T ).

We have shown that the entropy change when the rst-order magnetostructuraltransition is eld-induced is different than when it is thermally-induced. This rele-vant result is a consequence of the non-ideal behaviour of the rst-order transition,since the initial and nal states in the H − T phase diagram (see Fig. 9.3) are dif-ferent in the isothermal and the isoeld processes.

9.3 Cycling through the rst-order transitionAnother relevant effect of the non-equilibrium dynamics of the rst-order magne-tostructural transition is the fact that some properties vary when the transition isrepeatedly induced. In particular, changes in the resistance [2, 3] and thermopower[4] are reported for Gd5(SixGe1−x)4 alloys when they are thermally cycled throughthe transition.

In order to study the effect in the entropy change of cycling the transitionin Gd5(SixGe1−x)4 alloys, we used three different samples (v1, v2 and v3) withx=0.05, taken from the same sample (#1 T4+Q), which were not previously usedfor other measurements (virgin samples). All samples are cut in the shape ofa rod, v2 and v3 being longer than v1. We measured DSC sweeping the eld3

at a constant temperature (T=55 K) following a large number of cycles and fordifferent eld rates (within 0.01 and 1 T/min), which are summarised in Table 9.1.The rst 5 cycles for sample v1, on increasing H, were measured at different H(0.01, 0.05, 0.1 and 1 T/min). Calorimetric curves scale once they are dividedby H, as shown in Fig. 9.5. A low H induces a low signal in the DSC sensors,which is normalised after the signal is divided by the eld rate. However, thelowest H=0.01 T/min appears to yield too a low signal, close to the intrinsic noiselevel of the measuring apparatus (see Fig. 9.5). Rates of 0.05 and 0.1 T/minyield similar shapes and almost perfectly collapse once they are normalised to H,

3In contrast to the previous cycling studies [2, 3, 4], which used thermal cycles.

168

9.3. Cycling through the rst-order transition

H (T/min)Sample cycle # incr. H decr. H

v1 1 0.01 12 0.05 1

3,4,7,17,21,32,43,54,

63,76,87 0.1 1v2 1,3,4,5,

7,10,21,31 0.1 1v3 1-12 0.1 0.1

Table 9.1: Field rates, H, used for the study of the cycling through the rst-ordertransition in the three virgin samples taken from the original x=0.05 (#1 T4+Q)sample. A given number (#) of cycle includes both an increasing and a decreasingH branch. Cycles not described in the table have a eld rate of 1 T/min.

0 1 2 3 4 5-250

-200

-150

-100

-50

0

50

T=55 K

sample v1

x=0.05 0.01T/min (cycle #1) 0.05T/min (cycle #2) 0.1T/min (cycle #3) 1T/min (cycle #5)

dQ/d

H (

mJ/

T)

µ0H (T)

Figure 9.5: DSCT (H) curves recorded on increasing H for sample v1, at differenteld rates.

169


0 1 2 3 4 5

-300

-200

-100

0

0.1 T/min

increasing H

cycle #1 cycle #3 cycle #7 cycle #10

sample v2x=0.05 T=55 K

dQ/d

H (

mJ/

T)

µ0H (T)

1.7 1.8 1.9 2.0190

200

210

220

230

240

250

1 T/min

decreasing H

dQ/d

H (

mJ/

T)

µ0H (T)

Figure 9.6: DSCT (H) curves recorded on increasing H at 0.1 T/min for differentcycles in sample v2. Inset: Detail of the peak in DSCT (H) curves on decreasingH at 1 T/min following succesive cycles for sample v2.

showing an avalanche-type structure of the transition (we understand an avalancheas an irreversible jump tath makes one part of the system to undergo the transitionfrom one phase to the other), which are completely reproducible from cycle tocycle. The fastest rate ( H=1 T/min) does not show avalanches, since not enoughpoints are recorded during the phase transition to observe them.

The shape of the rst DSC measurement (increasing H) for all samples is dif-ferent from subsequent cycles. Even at rst glance it is obvious that the rst mea-surement enclose a lower area than the following measurements (see Fig. 9.5 forsample v1 and Fig. 9.6 for sample v2). Further measurements continue increasingin area, fact which is not directly observed in the low-rate (0.1 T/min) DSC curves,but which is appreciable in the fast-rate curves (1 T/min), as shown in the inset inFig. 9.6. The rst cycle in sample v2 (Fig. 9.6) shows small peaks of a similarsize. Some of them grow, while others diminish in subsequent cycles, reaching areproducible distribution, which is charateristic of athermal transitions (see sec-tion 9.4). The entropy change ∆S T obtained as a function of the cycle number

170

9.4. Avalanches

is displayed for all samples in Fig. 9.7. The slight difference between the valuescorresponding to each sample lies within the experimental error (5-10%), whicharises from the fact that part of the heat released or absorbed by the sample is notdetected by the sensor. The amount of the heat losses depends on the shape of thesample. Three features observed for all samples are to be noted. Firstly, ∆S T valueincreases strongly between the rst and the fourth cycles, then the slope is reducedand it nally saturates before the tenth cycle. Secondly, the ∆S T values do not de-pend on the eld rate, even when H increases by a factor of 10. Third, valuesobtained from the curves on increasing H are systematically larger (in absolutevalue) than those obtained on decreasing H. The rst result can be understoodby considering that both initial and nal states change from cycle to cycle due tothe evolution of the disorder (for example, microcracks associated with the straindue to the continuous expansion/contraction of the crystallographic cell throughthe transition). When the path of the system through the rst-order transition be-comes reproducible, as occurs in our samples (as for example in Fig. 9.6, wherethe curves tend towards a reproducible pattern), then the entropy change associ-ated with the transition reaches a constant value. In this evolution, the low-eldphase disorders (the entropy increases) and/or the high-eld phase orders (the en-tropy decreases). The non-dependence of the entropy change on H indicates thatthe experimental procedure is not affected by the eld rate used, and that the sameinitial and nal states are reached at any eld rate. Finally, differences betweenthe results corresponding to increasing and decreasing H appear because of hys-teresis in the initial and nal states, and therefore the entropy change between thetwo states may show slight differences.

9.4 AvalanchesThe analysis of the avalanche events was also performed. Avalanches are associ-ated with the nucleation and growth of domains of the new phase that take placeduring the rst-order eld-induced phase transition.

First-order transitions can be thermally activated or can be athermal. In theformer, the relaxation from a metastable state may occur at constant external con-ditions due to thermal uctuations, while in athermal transitions it occurs onlyunder the change of an external parameter (magnetic eld, stress, temperature,etc.), which modies the difference of the free energy between the two phases[5, 6].

When a system is externally driven through a rst-order phase transition, itjumps from a given conguration -which is a state corresponding to a local min-imum of the free energy - towards a different conguration, once the local sta-bility limit is reached. The path followed by the system depends on the presence

171


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

6

7

8

9

10

11

12

13

14

15

16

17

T= 55 K

/ inc r. H (0 .1 / 1 T /m in )

decr. H (1 T /m in) sam p le v2x= 0.0 5

|∆S

| (J

/kg

K)

n cyc le s

0 1 2 3 4 5 6 7 8 9 10 11 12 13

10

11

12

13

14

15

16

17

T= 55 KR a te : 0 .1 T/m in

sam p le v3

x=0.05

|∆S

| (J

/kg

K)

n cycles

increas ing H decreasing H

0 10 20 30 40 50 60 70 80 9010

11

12

13

14

15

16

17

18

T= 55 Ksam p le v1

x=0.05

|∆S

| (J

/kg

K)

n cycles

/ inc r. H (0 .1 / 1 T/m in )

decr. H (1 T/m in )

Figure 9.7: Entropy change obtained from DSCT (H) measurements (increasingand decreasing H at different rates) in samples v1, v2 and v3 (x=0.05), as a func-tion of the number of cycle. Field rates, H, are given in the plots. Solid lines area guide to the eye.

172

9.4. Avalanches

of disorder such as dislocations, vacancies or grain boundaries, which controlsthe distribution of energy barriers separating the two phases. As the system isdriven, it passes through a sequence of metastable states with discontinuous stepsor avalanches of the order parameter, which reect the fact that the system jumpsfrom one metastable free energy minimum to another one with an associated en-ergy dissipation, which is responsible for the hysteresis observed in rst-ordertransitions [6]. In the athermal case, the path can be reproduced from cycle tocycle provided that disorder does not evolve [7].

From DSCT (H) curves obtained on increasing and decreasing H, the trans-formed fraction of a sample, y, can be evaluated as a function of H as,

y(H) =1L

∫ H

Hi

dQdH dH , (9.9)

where L =∫ H f

Hi(dQ/dH)dH is the latent heat, and Hi and H f are magnetic elds

above and below the starting and nishing transition elds, respectively. y(H) forincreasing and decreasing H enables us to display the hysteresis loops. In orderto quantify the amplitude of the jumps in the rst-order transition (i.e., structurepeaks) present in x=0.05 samples (see Figs. 4.8, 9.5 and 9.6), we computed thedifference between two consecutive y(H) values from the experimentally mea-sured dQ/dH curve, which was recorded every 4 s. This difference, ∆y, which isa measurement of the size of the avalanches, can vary from 0 (no avalanche eventhas occurred during the measuring time window) to 1 (the whole system under-goes the transition in a single avalanche event). Figure 9.8 shows a distribution of∆y obtained from DSCT (H) measurement at cycle 12 in sample v3.

The distribution of avalanches can be statistically analysed using the followingprobability distribution with two free parameters (λ and α) [8, 9]:

p(∆y) =e−λ∆y(∆y)−α∫ ∆ymax

∆ymine−λ∆y(∆y)−αd(∆y)

. (9.10)

For λ=0, the distribution is a power law [p ∝ (∆y)−α, a critical behaviour wherethere is not a characteristic size], while it is subcritical for λ>0 (the distributiondecays faster than a power law) and supercritical for λ<0 (the distribution decaysslower than a power law) [10]. ∆ymin=10−4 is a value just above the intrinsic noiselevel of the measurements, evaluated by considering ∆y values outside the regionwhere the DSC peak shows structure. ∆ymax=1 is the maximum value.

We have estimated the exponent α and the parameter λ by the maximum like-lihood method [11]. This method is the most reliable since it does not involve thecomputation of histograms, which normally depend on the binning choice. Figure9.8 shows an example of one of such ts. The fact that samples v1 and v2 do

173


1E-4 1E-3 0.01

10

100

1000

sample v3

x=0.05

cycle 12

T = 55 K incr. H

maximum likelihood fit

Cou

nts

∆y

Figure 9.8: Distribution of avalanches obtained from the difference in the trans-formed fraction (∆y) and the corresponding maximum likelihood t for one of themeasurements (cycle 12) in sample v3.

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

350

λ

N cycles

0 10 20 30 40 50 60 70 80 900.5

0.6

0.7

0.8

0.9

1.0

α

N cycles

Figure 9.9: Parameter λ obtained from the distribution of avalanches using thetransformed fraction of the sample v3 (x=0.05), as a function of the cycle. Cycle31 is taken from sample v2 and cycle 89 from sample v1. Inset: Exponent αobtained from the same distribution of avalanches. Solid lines are a guide to theeye.

174

9.5. Conclusions

not present a completely systematic cycling (rst measurements where recordedat different values of H on increasing H) led us to repeat the cycling with sam-ple v3, after we observed that H=0.1 T/min was optimal taking into account theacquisition rate of the calorimeter, which is 0.25 Hz. This H enables us to ob-serve the avalanche structure while the signal of the sensors is large enough to berecorded above noise. Results for increasing H in sample v3 are displayed in Fig.9.9. We note that the parameter λ tends to decrease with the number of cycles,while the exponent α (inset in Fig. 9.9) remains constant (α=0.71±0.05). Wehave also added the values tted for the last cycle of sample v2 (cycle #31) andsample v1 (cycle #89), since those cycles were done with the same eld rate asin sample v3. The latter values are in agreement with the behaviour of the twoparameters for sample v3. λ and α may also be evaluated from DSCT (H) curvesshown in Fig. 4.8 in a sample with x = 0.05 cut from the same original button (#1T4+Q) that samples v1, v2 and v3. The number of cycles previously undergoneby this sample was not controlled, although it can be estimated as ∼15-25. Inthis case, α=0.73±0.05 and λ=209±30, in excellent agreement with the previousresults (Fig. 9.9).

The evolution of the parameter λ indicates that our system evolves from asubcritical distribution towards a power law distribution (where the system doesnot have a preferential avalanche size to undergo the transition), although the valueλ=0 is not reached in the 89th cycle. The characteristic exponent for the powerlaw, α, presents a value (=0.71±0.05) which neither dependends on the evolutionof the system with cycling, nor on the sample. The evolution of the parameters areconsistent with previous observations: when the system has chosen a path whichis optimal to undergo the transition, both the entropy change and the distributionof avalanches tend towards a constant behaviour.

9.5 ConclusionsThe study of dynamics of the rst-order transition in Gd5(SixGe1−x)4 alloys hasunveiled a very interesting behaviour. Our DSC under eld has revealed thatthe entropy change associated with the transition is different when it is eld- orthermally-induced, evidencing that the initial and nal states are different due tothe fact that the transition is not ideal. Cycling through the transition shows thatthe eld-induced entropy change increases for a few cycles, reaching a stationaryvalue. This behaviour is related to the avalanche distribution, which also evolveswith cycling. The structure of avalanches becomes repetitive after a few cyclestending towards a power-law distribution, unveiling the athermal character of themagnetostructural transition.

175


Bibliography[1] M. W. Zemansky and R. H. Dittman, Heat and thermodynamics, 6th ed.

(McGraw-Hill, New York, 1981).




[5] F. J. Pérez-Reche, E. Vives, L. Mañosa, and A. Planes, Phys. Rev. Lett. 87,195701 (2001).

[6] F. J. Pérez-Reche, M. Stipcich, E. Vives, L. Mañosa, A. Planes, and M.Morin, accepted in Phys. Rev. B (unpublished).

[7] J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, andJ. D. Shore, Phys. Rev. Lett. 70, 3347 (1993).

[8] E. Vives, J. Ortín, L. Mañosa, I. Ràfols, R. Pérez-Magrané, and A. Planes,Phys. Rev. Lett. 72, 1694 (1994).

[9] L. Carrillo, L. Mañosa, J. Ortín, A. Planes, and E. Vives, Phys. Rev. Lett. 81,1889 (1998).

[10] E. Vives and A. Planes, Phys. Rev. B 50, 3839 (1994).

[11] R. Barlow, Statistics (Wiley, New York, 1989).

176

Conclusions

This Ph.D. Thesis has been devoted to the preparation and characterisation ofbulk Gd5(SixGe1−x)4 alloys and to the study of both the magnetocaloric effect andthe rst-order magnetostructural transition appearing in these compounds. In thisnal section, we summarise the most relevant results and the main conclusionsobtained from this research. Some recommendations for further work are alsoincluded.

Summary and conclusions• Bulk Gd5(SixGe1−x)4 samples with 0 ≤ x ≤ 0.5 have been successfully

prepared by using our home-made arc-melting furnace. All characterisa-tion techniques show the good quality of the samples. SEM and electron-beam microprobe analyses show that the main 5:4 phases with the desired xare obtained. Ac susceptibility shows the magnetic transitions occurring inthese alloys, while XRD detects the crystallographic structures correspond-ing to the phases at room temperature. M(H) at 5 K show the presenceof secondary Gd(Si,Ge) (1:1) and Gd5(Si,Ge)3 (5:3) phases in all samples.This presence is conrmed by XRD and e-beam microprobe, which alsodetect residual 5:4 phases with an x value different from that of the mainphase. DSC shows that all samples present the rst-order transition, andthat secondary phases do not affect the latter. The heat treatments favourthe segregation of these secondary phases [M(H), XRD, SEM and micro-probe], but also reduce the spread in the x value (ac susceptibility and DSC)and remove 5:4 residual phases with very different x values (as susceptibil-ity and microprobe). Therefore, a trade-off between phase segregation andremoval of x spread is desirable. A treatment at 920 ºC for 4 hours in a 10−5

mb vacuum furnace enables such a trade-off.

• A new differential scanning calorimeter (DSC) has been developed. Theequipment features a high sensitivity down to 10 K and operates under ap-plied magnetic elds of up to 5 T and within the temperature range 10-300

177

CONCLUSIONS

K. The device may be used to study rst-order solid-solid phase transi-tions in the presence of magnetic elds. It has also been shown that thiscalorimeter enables an accurate determination of the entropy change asso-ciated with the magnetostructural phase transition of alloys exhibiting giantmagnetocaloric effect. The transition can be induced by sweeping either Tor H. Therefore, this kind of measurements claries the controvertial issueof the actual value of the entropy change at a rst-order transition.

• The magnetocaloric effect arising from a eld variation 0→Hmax, in a sys-tem which presents a rst-order eld-induced phase transition, can be prop-erly evaluated through the entropy change obtained from the Maxwell re-lation, even when an ideal rst-order transition takes place. When theMaxwell relation is evaluated over the whole eld range, the T and H de-pendences of the magnetisation in each phase outside the transition regionyield an additional entropy change to that associated with that of the ac-tual rst-order transition. It has also been shown, from both experimentaldata and phenomenological models, that the Maxwell relation, the Clausius-Clapeyron equation and the calorimetric measurements yield the entropychange associated with the rst-order magnetostructural transition, ∆S , pro-vided (i) the Maxwell relation is evaluated only within the eld range overwhich the transition takes place, and (ii) the maximum applied eld is ashigh as to complete the transition. The transition temperature must signif-icantly shift with the applied eld, in order to achieve a large MCE takingadvantage of the entropy change associated with the rst-order transition.This is relevant for the understanding of the thermodynamics and MCE ofrst-order magnetostructural transitions.

• DSC under H has been successfully used to measure ∆S associated with therst-order magnetostructural phase transition for Gd5(SixGe1−x)4, x ≤ 0.5.We have shown that the transition entropy change scales with Tt. The scal-ing of ∆S is a direct consequence of the fact that Tt is tuned by x and Hand it is thus expected to be universal for any material showing strong mag-netoelastic effects, yielding a eld-induced nature of the transition. ∆S isexpected to (i) go to zero at zero temperature, (ii) tend asymptotically tozero at high temperature since the latent heat is nite, and (iii) display amaximum at a temperature for which both ∆M is maximised and Tt showsthe minimum eld dependence. The specic shape of ∆S vs. Tt will de-pend on the details of the phase diagram, Tt(x). Finally, the scaling of ∆Sshows the equivalence of magnetovolume and substitution-related effects inGd5(SixGe1−x)4 alloys.

• The variation of the transition eld with the transition temperature, dHt/dTt,

178

has been studied in Gd5(SixGe1−x)4 for all the range of compositions wherethe rst-order transition occurs, 0 ≤ x ≤ 0.5. Taking into account the be-haviour of dHt/dTt as a function of x and ∆M decreasing monotonouslywith Tt, it is shown that dHt/dTt governs the scaling of ∆S with Tt, givingfurther evidence that the origin of this scaling is the magnetoelastic natureof the transition. Moreover, two distinct behaviors for dHt/dTt have beenfound on the two compositional ranges where the magnetostructural transi-tion occurs, thus showing the difference in the strength of the magnetoelasticcoupling in this system.

• It has been shown that an unreported eld-induced magnetic phase tran-sition exists from the AFM phase to a phase which presents short-rangecorrelations (SRAFM). The results suggest that the transition results fromthe breaking of the long-range AFM correlations when a magnetic eld isapplied, which leads to competing FM and AFM short-range correlations.FM correlations are also relevant in the whole long-range AFM phase. Theexpected transition from the SRAFM to the PM phase takes place at ∼240 Kat zero eld, being widened and smoothed under applied eld. This ndingscontribute to the understanding of the rich and complex magnetic behaviourof Ge-rich Gd5(SixGe1−x)4 alloys, which arises from the competition be-tween the intraslab FM interactions and the interslab AFM interactions.

• The study of dynamics of the rst-order transition in Gd5(SixGe1−x)4 alloyshas unveiled a very interesting behaviour. On one hand, our DSC undereld has revealed that the entropy change associated with the transition isdifferent when it is eld- or thermally-induced, evidencing that the initialand nal states are different because the transition is not ideal. On the otherhand, a cycling study shows that the eld-induced entropy change increasesduring the rst cycles, then reaching a stationary value. This behaviouris related to the avalanche distribution, which also evolves with cycling.The structure of avalanches becomes repetitive after a few cycles tendingtowards a power-law distribution, unveiling the athermal character of thetransition.

Future perspectives and recommendationsThe magnetocaloric effect (MCE) has promising applications to magnetic refriger-ation. Magnetic refrigeration, which shows a high efficiency and is environment-friendly, is a serious alternative to the conventional gas-compression technology.Therefore, the search for new materials showing MCE is of great interest.

179

CONCLUSIONS

In this work, we have shown that Gd5(SixGe1−x)4 alloys present a giant MCEdue to a variety of properties, mainly related to the rst-order eld-induced mag-netostructural phase transition occurring in this system. An interesting researchline would be the study of other materials showing a transition with similar prop-erties, in order to be applied to magnetic refrigeration. In fact, after the discoveryof the giant MCE in Gd5(SixGe1−x)4 alloys, new works on MCE have been focusedin MnAs-based and La(Fe,Si)13 intermetallic systems.

Concerning the Gd5(SixGe1−x)4 alloys, a lot of effects remain to be explainedin this exciting system. Although the microscopic mechanisms of the transitionbegin to be understood, other essential questions are still opened. The actual mag-netic structure in the various magnetic phases present in the system is a relevantone. The dynamics of the rst-order transition, which we have just begin to face,is another open question. The competition between AFM and FM interactionsin Ge-rich alloys is another unsolved problem that we have also contributed tounderstand.

Finally, we would like to remark that MCE can be studied in a large varietyof materials. We also remark that Gd5(SixGe1−x)4 alloys offers a very rich andcomplex magnetic and structural behaviour. We hope that the present thesis hashelped to unveil and understand the properties of this system a litte bit more.

180

List of publications

The work of this Ph. D. Thesis has been published in the following articles:

1. Fèlix Casanova, Xavier Batlle, Amílcar Labarta, Jordi Marcos, Lluís Mañosa,and Antoni PlanesEntropy change and magnetocaloric effect in Gd5(SixGe1−x)4Physical Review B 66, 100401(R) (2002).

2. Fèlix Casanova, Xavier Batlle, Amílcar Labarta, Jordi Marcos, Lluís Mañosa,and Antoni PlanesScaling of the entropy change at the magnetoelastic transition in Gd5(SixGe1−x)4Physical Review B 66, 212402 (2002).

3. Fèlix Casanova, Xavier Batlle, Amílcar Labarta, Jordi Marcos, Lluís Mañosa,and Antoni PlanesChange in entropy at a rst-order magnetoelastic phase transition: Casestudy of Gd5(SixGe1−x)4 giant magnetocaloric alloysJournal of Applied Physics 93, 8313 (2003).

4. Jordi Marcos, Fèlix Casanova, Xavier Batlle, Amílcar Labarta, Antoni Planes,and Lluís MañosaA high-sensitivity differential scanning calorimeter with magnetic eld formagnetostructural transitionsReview of Scientic Instruments 74, 4768 (2003).

5. Fèlix Casanova, Amílcar Labarta, Xavier Batlle, Jordi Marcos, Lluís Mañosa,Antoni Planes, and Sophie de BrionEffect of the magnetic eld on the magnetostructural phase transition inGd5(SixGe1−x)4Physical Review B, submitted.

6. Fèlix Casanova, Amílcar Labarta, Xavier Batlle, Jordi Marcos, Lluís Mañosa,Antoni Planes, and Sophie de BrionShort-range antiferromagnetism in Ge-rich magnetocaloric Gd5(SixGe1−x)4

181

LIST OF PUBLICATIONS

alloysPhysical Review B, submitted.

7. Fèlix Casanova, Amílcar Labarta, Xavier Batlle, Eduard Vives, Jordi Mar-cos, Lluís Mañosa, and Antoni PlanesDynamics of the magnetostructural phase transition in Gd5(SixGe1−x)4The European Physical Journal B, invited paper, submitted.

Other articles related to this work are:

1. Jordi Marcos, Antoni Planes, Lluís Mañosa, Fèlix Casanova, Xavier Batlle,Amílcar Labarta, and Benjamín MartínezMagnetic eld induced entropy change and magnetoelasticity in Ni-Mn-GaalloysPhysical Review B 66, 224413 (2002).

2. Jordi Marcos, Lluís Mañosa, Antoni Planes, Fèlix Casanova, Xavier Batlle,and Amílcar LabartaMultiscale origin of the magnetocaloric effect in Ni-Mn-Ga shape-memoryalloysPhysical Review B 68, 094401 (2003).

3. Jordi Marcos, Antoni Planes, Lluís Mañosa, Fèlix Casanova, Xavier Batlle,Amílcar Labarta, and Benjamín MartínezMagnetic eld induced entropy change and magnetoelasticity in Ni-Mn-GaalloysJournal of Magnetism and Magnetic Materials, accepted.

4. Jordi Marcos, Lluís Mañosa, Antoni Planes, Fèlix Casanova, Xavier Batlle,Amílcar Labarta, and Benjamín MartínezMagnetocaloric and shape-memory effects in Ni-Mn-Ga ferromagnetic al-loysJournal de Physique IV, accepted.

182

Resum

L'efecte magnetocalòric en els aliatgesGd5(SixGe1−x)4

Què és l'efecte magnetocalòric?Teoria fonamentalDenim l'efecte magnetocalòric (EMC) com l'escalfament o refredament (o si-gui, una variació de la temperatura) d'un material magnètic degut a l'aplicaciód'un camp magnètic variable [1, 2]. L'origen físic de l'EMC és l'acoblament en-tre la subxarxa magnètica del material i el camp magnètic aplicat, H, doncs aquestcanvia la part magnètica de l'entropia del sòlid. Això és explicat per la termodi-nàmica de sistemes magnètics.

L'entropia d'un ferromagnet (FM) a pressió constant, que depèn d'H i de latemperatura T , és la suma de les entropies electrònica, de la xarxa, i magnètica:

S (T,H) = S el(T ) + S xar(T ) + S m(T,H) . (1)

A l'esquema de la Fig. 1 tenim representats dos processos importants per explicarla termodinàmica de l'EMC:

(i) Apliquem el camp (H0 ns a H1) isotèrmicament (T=cnt.). L'entropia totaldisminueix degut només a la component magnètica, i per tant el procés ve denitper

∆S m = S (T0,H0) − S (T0,H1) . (2)(ii) Apliquem el camp (H0 ns a H1) adiabàticament (S =cnt.) en un procés

reversible. L'entropia magnètica disminueix, però com que la total es manté,

S (T0,H0) = S (T1,H1) , (3)

hi ha un increment de la temperatura, que es pot visualitzar com la diferènciaisentròpica entre les funcions corresponents S (T,H) i és una mesura de l'EMC

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RESUM

T=const. ∆S ≠ 0

Procés isotèrmic ⇒ ∆S ≠ 0 ( S2 > S1 )

H = 0

desordre ordre

H ≠ 0H

S2 S1

S=const.∆Tad ≠ 0

Procés adiabàtic ⇒ ∆Tad ≠ 0 ( T1 > T2 )

ordre desordre

H = 0H ≠ 0H

T1T2

Figura 1: Esquema que mostra els dos processos bàsics de l'EMC quan s'aplicao es treu un camp magnètic en un sistema magnètic: l'isotèrmic, que dóna lloc auna variació d'entropia, i l'adiabàtic, que produeix un canvi de temperatura.

184

Què és l'efecte magnetocalòric?

del material:∆Tad = T1 − T0 . (4)

Totes dues quantitats (∆S m i ∆Tad) són característiques de l'EMC, i són funció dela temperatura inicial T0 i de la variació de camp magnètic ∆H = H1 − H0. Larelació entre les variables termodinàmiques ens la dóna una de les relacions deMaxwell [3]: (

∂S (T,H)∂H

)

T=

(∂M(T,H)

∂T

)

H. (5)

Si integrem aquesta relació, per un procés isotèrmic (i isobàric) tenim que

∆S m(T,∆H) =

∫ H2

H1

(∂M(T,H)

∂T

)

HdH . (6)

Utilitzant algunes identitats termodinàmiques [3] i agafant la relació de Max-well podem expressar l'increment de temperatura adiabàtica com

∆Tad(T,∆H) = −∫ H2

H1

(T

C(T,H)

)

H

(∂M(T,H)

∂T

)

HdH . (7)

on M és la magnetització del material i CH és la capacitat caloríca a camp cons-tant. De les Eqs. 6 i 7 podem deduir com serà l'EMC en materials magnètics:

1. La magnetització a H constant decreix en augmentar la temperatura enparamagnets (PMs) i FMs simples, de manera que (∂M/∂T )H < 0. Per tant∆Tad(T,∆H) serà positiu i ∆S m(T,∆H) negatiu per increments de camp ∆H > 0.

2. |(∂M/∂T )H | és màxim en un FM a la seva temperatura de Curie (TC), demanera que |∆S m(T,∆H)| ha de tenir un pic màxim a T = TC.

3. Tot i no ser evident de les equacions, doncs la capacitat caloríca té uncomportament anòmal entorn de TC, ∆Tad(T,∆H) per FMs a TC té un pic per ∆Htendint a zero [4].

4. Pel mateix valor de |∆S m(T,∆H)|, el valor de ∆Tad(T,∆H) serà major commajor sigui T i menor sigui la capacitat caloríca del sòlid.

5. En PMs el valor de ∆Tad(T,∆H) és només signicatiu per temperaturesproperes al zero absolut, ja que en general el valor de |(∂M/∂T )H | és negligible,i només quan la capacitat caloríca també es fa molt petita (del mateix ordre)obtenim algun valor signicatiu de ∆Tad(T,∆H).

Mesura de l'efecte magnetocalòricL'EMC es pot mesurar de manera directa o indirecta. Les tècniques de mesuradirecta impliquen la mesura de les temperatures inicial (T0) i nal (TF) en variar

185

RESUM

un camp des d'un valor inicial (H0) ns a un valor nal (HF). Llavors la mesurade la variació de temperatura adiabàtica és simplement

∆Tad(T0,HF − H0) = TF − T0 . (8)

Els experiments indirectes es fan a partir de la magnetització o de la capacitatcaloríca. M s'ha de mesurar experimentalment, en funció de T i H. Això permetd'integrar numèricament l'Eq. 6 i obtenir així ∆S m(T,∆H). Aquesta tècnica ésmolt útil per una recerca ràpida de possibles materials magnètics refrigerants [5].La mesura de la capacitat caloríca en funció de T a H constant, C(T )H, dónala informació més completa de l'EMC en els materials magnètics. Sabem quel'entropia d'un sòlid es calcula a partir de la capacitat caloríca com:

S (T )H=0 =

∫ T

0

C(T )H=0

T dT + S 0 ; S (T )H,0 =

∫ T

0

C(T )H

T dT + S 0,H (9)

on S 0=S 0,H és l'entropia a temperatura zero. Per tant, coneixent S (T )H podemcalcular tant ∆Tad(T,∆H) com ∆S m(T,∆H) [6].

Efecte magnetocalòric en paramagnetsL'EMC en sals PMs va ser la primera aplicació pràctica d'aquest fenomen, l'any1927, el que s'anomena desmagnetització adiabàtica [7, 8]. Amb aquest efecte espoden assolir temperatures ultra baixes (mK-µK). Degut a la baixa conductivitattèrmica de les sals, el següent pas va ser estudiar compostos intermetàl·lics PMs.Amb el PrNi5 s'ha assolit la temperatura de treball més baixa: 27 µK [9]. Unaltre grup de PMs molt estudiats són els granats, degut a la seva alta conductivitattèrmica, la seva baixa capacitat caloríca i la seva baixa temperatura d'ordenament(generalment per sota d'1 K) [10, 11, 12].

EMC en una transició de fase magnètica ordre-desordreL'ordenament magnètic espontani dels sòlids PMs per sota de TC fa tenir una|(∂M/∂T )H | gran i per tant un EMC important a aquesta T . El càlcul de l'EMC esfa segons la relació de Maxwell (Eq. 5), doncs aquestes transicions són de segonordre i les magnituds termodinàmiques varien contínuament [1, 13]. La recercade materials ha estat centrada en FMs tous que tinguin TC entre 4 i 77 K per poderaplicar l'EMC en la liquació d'He i N2, o bé TC a l'entorn de T ambient, per ferservir les propietats magnetocalòriques en refrigeració i aires condicionats.

En el rang de temperatures baixes (∼10-80 K), la primera elecció òbvia sónalguns lantànids purs, tals com Nd, Er o Tm, però l'EMC resulta ser molt petit, jaque la majoria de fases magnètiques són antiferromagnètiques i ferrimagnètiques,

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Què és l'efecte magnetocalòric?

de manera que la majoria d'entropia disponible és utilitzada per l'inversió delsspins. Els materials que tenen l'EMC major en aquest rang són els compostosRAl2 on R = Er, Ho, Dy, Dy0.5Ho0.5 [14] i DyxEr1−x (0 ≤ x ≤ 1) [4, 15], el GdPd[4, 16] i l'RNi2 amb R=Gd [17], Dy [18] i Ho[18].

El rang de temperatures intermitges (∼80-250 K) està molt poc estudiat perquèno té moltes aplicacions i la fracció T/C té un mínim inherent en els metalls. Elmillor material en aquest rang és el Dy pur [4, 19], amb una ∆Tad ∼ 12 K a unaT∼180 K per un increment de camp de µ0∆H = 7 T.

El material prototip en el rang de temperatura ambient és el Gd doncs és unaterra rara que s'ordena FM per sota de 294 K. El Gd ha estat molt estudiat [4, 20,21, 22], i els valors de ∆Tad a la seva TC són de ∼6, 12, 16 i 20 K per µ0∆H = 2,5, 7.5 i 10 T. Els únics compostos intermetàl·lics que mantenen la magnitud del'EMC del Gd en una transició de fase ordre-desordre magnètica i tenen TC a Tambient però més alta que el Gd, són el Gd5Si4 (amb una TC∼335 K) i la soluciósòlida amb substitucions de germani Gd5(SixGe1−x)4 per 0.5 ≤ x ≤ 1, que tenenuna TC que va des de ∼290 ns a ∼335 K [23].

EMC en una transició de primer ordre: l'efecte gegantUna transició de fase de primer ordre ocorre idealment a T constant (Tt) i per tant|(∂M/∂T )H | hauria de ser innitament gran. Si la transició és ideal, la discontinui-tat de M i S fa que les derivades de la relació de Maxwell habitualment utilitzada(Eqs. 5 i 6) s'hagi de substituir pels increments nits de l'equació de Clausius-Clapeyron per a canvis de fase (existeix una calor latent de transformació a Tconstant, que té associada una variació d'S )[1, 24, 25]:

∆S = −∆M dHt

dTt. (10)

on ∆S (i ∆M) són les diferències de l'entropia (i la magnetització) entre les fa-ses magnètiques involucrades, a Tt. dHt/dTt és la variació amb la temperatura delcamp al qual ocorre la transició. L'obtenció d'aquesta variació d'entropia de trans-formació és una contribució afegida amb la qual podríem observar el que seria unEMC gegant, sempre que la transició de fase, i per tant la ∆S de transformació,pugui ser induïda per un camp magnètic.

Un dels primers materials on va ser observat aquest tipus d'EMC gegant vaser en el compost intermetàl·lic FeRh, però l'efecte és irreversible en aplicar H.Una sèrie d'aliatges descoberta recentment, els Gd5(SixGe1−x)4 amb 0 ≤ x ≤ 0.5,tenen una ∆S m almenys el doble de gran que el Gd a prop de temperatura ambient(-18.5 J/(kgK) per µ0∆H = 5 T a Tt ∼280 K) [26], i entre 2 i 10 cops major que elsmillors materials magnetocalòrics en les regions de baixa i intermitja temperatura(-26 J/(kgK) a -68 J/(kgK) per µ0∆H = 5 T, depenent de la composició x) [23, 27],

187

RESUM

sent en aquest cas l'efecte reversible. La present tesi està centrada en aquestsaliatges.

Seguint el boom provocat pel descobriment d'un EMC gegant en aquests ali-atges, s'ha iniciat una recerca de nous aliatges intermetàl·lics amb transició defase de primer ordre que pugui ser induïda per H. El MnAs, conegut per la sevatransició de fase magnetoelàstica de primer ordre a Tt=318 K, seria un bon can-didat que presenta EMC gegant (-30 J/(kgK) i 13 K per µ0∆H=5 T), però té unagran histèresi tèrmica que el fa poc útil en aplicacions [28]. En el Mn(AsxSb1−x)es redueix tant la histèresi com Tt (230 K per x=0.3) sent per tant un materialcompetitiu en un ampli rang a l'entorn de T ambient (-25 a -30 J/(kgK) i ∼10 Kper µ0∆H=5 T)[28, 29]. L'aliatge MnFePxAs1−x, també manté una transició defase de primer ordre induïda per H a T ambient [30, 31]: per x=0.45 obtenim -18J/(kgK) amb µ0∆H=5 T a una Tt ∼300 K [32]. L'avantatge d'aquests aliatges ésque estan basats en els metalls de transició, que són molt més barats que les terresrares. La sèrie d'aliatges La(FexSi1−x)13 presenta també una transició de primerordre FM-PM induïda per camp entre x=0.86 (Tt ∼210 K) i x=0.90 (Tt ∼184 K)[33, 34]. L'origen de la transició és en aquest cas una transició metamagnèticad'electrons itinerants [34]. Això fa que tinguin un EMC gegant, amb una ∆Sd'entre -14 i -28 J/(kgK) i un ∆Tad d'entre 6 i 8 K, per un µ0∆H de tan sols 2 T[35, 36, 37]. Una altra classe de materials, basats en l'estructura perovskita delLaMnO3, amb substitucions del La amb Y, Ca, Sr, Li i/o Na i substitucions deMn amb Ti [38, 39, 40, 41], també tenen un EMC de l'ordre del del Gd, però nogegant. L'interès d'aquests materials és que són en canvi molt més barats.

EMC a molt baixes temperatures: magnets frustrats i magnetsmoleculars d'spin granA baixes T , les sals PMs són els materials refrigerants estàndards pel refreda-ment magnètic. Com més alta és la densitat dels moments magnètics, més granés la potencia de refredament. Però si la densitat és massa alta, la força de les in-teraccions porta a una transició d'ordenament, que limita la mínima temperaturaassolible amb sals PMs. Tanmateix, en magnets frustrats, els moments magnèticses mantenen desordenats i posseeixen entropia nita molt per sota de la cons-tant de Curie-Weiss. Per exemple, recentment s'ha descobert una gran variaciód'entropia en el sistema TbxY1−xAl2 [2.4 J/(kgK) a T=12 K per µ0∆H=2 T i 7.6J/(kgK) a T=30 K per µ0∆H=2 T], associat a la transició de vidre d'spin a PM[42]. També s'ha predit un EMC gran en magnets frustrats geomètricament [43].Aquest augment és degut a la presència d'un nombre macroscòpic de modes tousassociats a la frustració geomètrica per sota del camp de saturació.

Un altre tipus interessant de materials que mostren un EMC gran (i depenent

188

La sèrie d'aliatges Gd5(SixGe1−x)4

del temps) a molt baixa T són els magnets moleculars d'spin gran. Els clustersmoleculars com el Mn12 i el Fe8 exhibeixen una variació d'entropia extremada-ment gran entorn de la T de bloqueig -de l'ordre del Kelvin- que està associada alprocés de bloqueig ordre-desordre. Pel Mn12, s'han obtingut valors de 21 J/(kgK)a T '3 K per µ0∆H=3 T variant el camp a 0.01 Hz [44]. Per tant, són candidatspotencials a refrigerants magnètics al règim de la liqüefacció de l'heli.

Refrigeració magnèticaAvui en dia hi ha un gran interès en aprotar l'EMC com a tecnologia alterna-tiva en la refrigeració, tant a T ambient com a temperatures criogèniques. Larefrigeració magnètica és una tecnologia de refredament ecològica, ja que no usaproductes químics nocius per la capa d'ozó (tals com els clorouorocarbons) ogasos d'efecte hivernacle (hidroclorouorocarbons i hidrouorocarbons), com elsque encara es fan servir en els sistemes de refrigeració actuals de cicle de vapor(neveres i ares condicionats). Els refrigeradors magnètics fan servir un refrigerantsòlid i uids de transferència de calor innocus (aigua, solució d'aigua i alcohol,aire o heli gas). Una altra diferència entre els refrigerants magnètics i els de ciclede vapor són les pèrdues energètiques que hi ha en el cicle de refrigeració. Elsdispositius de refrigeració comercials més nous operen molt per sota de l'eci-ència màxima teòrica (Carnot), que està limitada pel compressor. La refrigeraciómagnètica no té aquesta limitació: l'eciència de refredament d'un dispositiu quetreballa amb Gd arriba al 60 % de límit de Carnot [2, 45, 46, 47], comparat amb el40 % dels millor refrigeradors de compressió de gas. Tanmateix, amb els actualsmaterials disponibles, aquesta alta eciència només s'assoleix a camps alts (∼5T). Per tant, la recerca de materials que tinguin un EMC major, operant en campsmenors (∼2 T) que es puguin generar amb imants permanents, és molt important.


El aliatges Gd5(SixGe1−x)4 van ser descoberts per Holtzberg et al. [48], i hanestat extensament estudiats des del descobriment de l'EMC gegant [49]. L'ori-gen d'aquest efecte és la gran variació d'entropia associada a la transició de fasemagnetoestructural de primer ordre induïda per camp magnètic que apareix perx ≤ 0.5 (Refs. [23, 26, 50]), i que és reversible.

El diagrama de fases està representat a la Fig. 2. Per x ≤ 0.5 podem observardos rangs de composició. En la zona 0.24 ≤ x ≤ 0.5, la transició de primer ordreva d'una fase d'alta temperatura PM i d'estructura monoclínica (M) a una fasede baixa temperatura FM i d'estructura ròmbica tipus Gd5Si4 [O(I)], per tem-peratures que varien linealment des de 130 K (x=0.24) ns a 276 K (x=0.5).

189

RESUM

Figura 2: Diagrama de fases magnètic i cristal·logràc dels aliatgesGd5(SixGe1−x)4 en funció de la temperatura i la composició [51]. PM és faseparamagnètica, FM és fase ferromagnètica i AFM antiferro- o ferrimagnètica. Més estructura monoclínica, O(I) estructura ròmbica tipus Gd5Si4 i O(II) estructuraròmbica tipus Sm5Ge4.

190


La temperatura de transició també augmenta linealment amb l'aplicació d'uncamp magnètic [23, 52]. Pels compostos amb x ≤ 0.2, hi ha una transició PM-antiferromagnet(AFM) de segon ordre a TN (de ∼125 K per x=0 a ∼135 K perx=0.2) [23]. A temperatures més baixes, la transició de primer ordre, que enaquesta zona és AFM-FM, té lloc a temperatures que varien linealment des de∼20 K (x=0) ns a 120 K (x=0.2). En el cas concret de x=0, l'estat FM s'assoleixper una transició irreversible induïda per camp des de l'estat AFM [53, 54]. Lanaturalesa de la fase AFM està encara en discussió [51]: l'estructura magnèticapodria correspondre a la d'un ferrimagnet no colineal, com en el Nd5Ge4 [55] o ala d'un AFM no colineal com en els aliatges Tb5(SixGe1−x)4 rics en Ge. [56, 57].La transició AFM-FM va acompanyada alhora d'una transició magnetoestructuraldes d'una fase ròmbica tipus Sm5Ge4 [O(II)] a alta temperatura a la fase O(I) abaixa temperatura. En aquest cas, la temperatura de transició també augmentalinealment amb un camp magnètic [51]. En la zona intermitja 0.2 < x < 0.24, ondesapareix la transició PM-AFM de segon ordre, coexisteixen les estructures M iO(II) [50].

La transició estructural s'esdevé, en ambdues regions de composició, per unmecanisme de cisalla que provoca una gran variació de volum. La transició s'ex-plica considerant l'estructura cristal·lina en capes del Gd5(SixGe1−x)4. En la faseO(I), que és sempre FM, capes bidimensionals estan connectades a través d'enlla-ços covalents Ge(Si)-Ge(Si) [58]. Els enllaços entre capes es trenquen totalmentquan la distància entre àtoms de Gd(Si) creix durant la transformació a l'estruc-tura O(II) [50, 51], donant lloc a una fase AFM, mentre que només la meitat delsenllaços es trenquen en la transició a l'estructura M [50, 58] en els compostos0.24 ≤ x ≤ 0.5, que dóna lloc a una fase PM. Aquestes transicions magnetoes-tructurals, en poder ser induïdes pel camp, provoquen l'aparició de magnetoes-tricció colossal [51, 52] i magnetoresistència gegant [59, 60, 61], a part de l'EMCgegant. La cristal·lograa i el magnetisme estan estretament relacionats en elsGd5(SixGe1−x)4. Levin et al. [62] han proposat que el FM en l'estructura O(I) debaixa T s'assoleix no només a través de l'interacció de bescanvi RKKY entre elec-trons 4 f -4 f , sinó també via un superbescanvi directe Gd-Ge(Si)-Gd a través delsenllaços covalents entre capes. Es creu que l'AFM associat a l'estructura O(II)és similar a l'AFM dels aliatges Tb5(SixGe1−x)4 rics en Ge, que tenen la mateixaestructura. En el Tb5Ge4, les capes individuals presenten un FM 2D no colineal,mentre que l'ordre entre capes és AFM 3D [56, 57]. En el cas de la fase AFM enel Gd5Ge4, la temperatura de Curie-Weiss extrapolada és positiva [27, 48, 54, 63],el qual és una evidència de l'existència d'interaccions de bescanvi positives (FM)en la fase AFM, i per tant és raonable comparar els dos materials. Per Levin et al.[54], l'anisotropia en les interaccions de bescanvi dels àtoms de Gd pot explicarla diferència entre l'ordre magnètic en una capa (FM) i el d'entre capes (AFM) iper tant la varietat d'estructures magnètiques presents en els Gd5(SixGe1−x)4 rics

191

RESUM

en Ge. Szade et al. [63] han estudiat la susceptibilitat inversa del Gd5Ge4 a altatemperatura i camp zero, detectant un comportament no lineal entre TN i ∼230 K,cosa que indica l'inici d'un procés d'ordenament.

La correcta avaluació de la variació d'entropia associada a l'EMC és un aspec-te molt debatut i darrerament ha aixecat molta discussió [25, 26, 64, 65, 66, 67].Pels aliatges Gd5(SixGe1−x)4, Giguère et al. [25] demostren que l'ús de la rela-ció de Maxwell (Eqs. 5 i 6) per calcular la variació d'entropia sobreestima (enalmenys un 20 %) el valor obtingut de l'equació de Clausius-Clapeyron (Eq. 10),que els autors [25, 66] reivindiquen com el procediment correcte degut al caràcterde primer ordre de la transició en aquests aliatges. Segons ells, ∆S a la transi-ció magnetoestructural no està associat al canvi continu de la magnetització enfunció de T i H, sinó al canvi discontinu de la magnetització degut a la transfor-mació cristal·logràca. Armen que les relacions de Maxwell no són aplicablesen aquest cas perquè M no és una funció continua ni derivable. En canvi, Gsch-neidner et al. [64] argumenten que la relació de Maxwell és aplicable ns i toten el supòsit d'una transició de primer ordre, excepte si la transició té lloc a Ti H constant, que originaria un canvi en esglaó a M (cas ideal). A part d'això,diuen que l'equació de Clausius-Clapeyron implicaria un ∆Tad independent d'H,que tanmateix no és consistent amb les observacions experimentals [25]. A més amés, Sun et al. [65] demostren que ∆S calculada de la relació de Maxwell tambéés equivalent a la obtinguda de l'equació de Clausius-Clapeyron en un cas ideal(considerant M independent de T en les dues fases involucrades en la transiciói sent una funció esglaó amb un salt nit a la temperatura de transició. Tambésuggereixen que els dos procediments poden donar resultats diferents doncs elmètode de Clausius-Clapeyron no té en compte la reducció de les uctuacionsd'spin degut a l'aplicació d'un camp.

Tècniques experimentalsLes mostres massives de Gd5(SixGe1−x)4 utilitzades en aquest treball han estatsintetizades amb un forn d'arc no comercial. La tècnica consisteix en fondre elselements en la proporció corresponent mitjançant la descàrrega d'un arc elèctricen atmosfera d'Ar. Les mostres obtingudes han estat tractades tèrmicament se-guint diferents estratègies.

La caracterització de les mostres ha estat feta mitjançant les següents tècni-ques: isotermes de magnetització [M(H)], susceptibilitat ac, calorimetria diferen-cial de ux (de les sigles en anglès, DSC), difracció de raigs X (DRX), micros-còpia electrònica de rastreig (MER) i microsonda electrònica. Totes les tècniquesevidencien que les mostres sintetitzades corresponen als Gd5(SixGe1−x)4. Els anà-lisis amb l'espectroscòpia dispersiva d'energia del MER i l'espectroscòpia disper-

192

Tècniques experimentals

Figura 3: Imatge d'electrons retrodispersats de la superfície d'una mostra ambx=0.45 (nominal) sense tractament tèrmic. Les zones clares corresponen a la faseprincipal, amb x=0.41, i les franges fosques corresponen a una fase amb x=0.51,segons els anàlisis de la microsonda electrònica.

193

RESUM

siva de longitud d'ona de la microsonda demostren que s'obté la fase principal 5:4amb la x desitjada. La susceptibilitat ac detecta les transicions magnètiques quehan de presentar segons la literatura, mentre que la DRX mostra les estructurescristal·logràques de les fases que els correspon a temperatura ambient. Les cor-bes M(H) a 5 K suggereixen la presència de fases secundàries com el Gd(Si,Ge)(1:1) o el Gd5(Si,Ge)3 (5:3), properes a la fase 5:4 en el diagrama de fases. Aques-ta presència és conrmada per la DRX i la microsonda, que també detecten fases5:4 residuals amb x diferent del valor estequiomètric desitjat (vegeu Fig. 3). Lacalorimetria de ux mostra la presència de la transició de primer ordre en totes lesmostres amb x ≤ 0.5, i que les fases secundàries no afecten la transició. Els trac-taments tèrmics afavoreixen la segregació d'aquestes fases secundàries (tal comveiem en M(H), XRD, MER i microsonda), però també redueix la dispersió delvalor d'x (susceptibilitat ac i DSC) i elimina fases 5:4 residuals amb valors de xmolt diferents del nominal (susceptibilitat ac i microsonda). Per tant cal un com-promís entre la segregació de fases secundàries i l'eliminació de la dispersió en x,que assolim amb un tractament de les mostres a 920 ºC entre 4 i 9 hores, en unbuit de 10−5 mb.

Disseny i muntatge d'un calorímetre diferencial deux sota camp magnèticIntroduccióLa calorimetria es considera un dels mètodes més adequats per determinar el ca-ràcter de les transicions de fase [68]. Existeix una gran varietat de calorímetres,que es poden classicar en dos grans grups. El primer inclou els aparells que me-suren el ux de calor entre la mostra i un bloc tèrmic, mentre la temperatura delcalorímetre varia de manera contínua (calorímetres de ux). La majoria fan serviruna mostra de referència per treballar diferencialment (calorímetres diferencialsde ux, de l'anglès DSC). El segon grup inclou els calorímetres basats en la mesu-ra de la temperatura de la mostra després de subministrar certa quantitat de calor(calorimetria adiabàtica, calorimetria de relaxació i calorimetria ac). En aquestsinstruments, la temperatura del calorímetre es manté constant durant la mesura.

Els DSCs són particularment adequats per estudiar transicions de fase de pri-mer ordre, doncs mesuren el ux de calor, i una integració correcta de la senyalcalibrada dóna la calor latent de la transició. En canvi, la calorimetria ac, la de re-laxació i l'adiabàtica són adequades per obtenir la capacitat caloríca CP i per tantaptes per estudiar transicions de fase contínues. Cal remarcar que en una transi-ció de primer ordre, l'emissió de calor latent fa que la determinació experimentalde CP sigui intrínsecament incerta, i per tant aquestes tècniques no són útils per

194

Disseny i muntatge d'un DSC sota camp magnètic

Figura 4: Secció longitudinal (a) i transversal (b) del calorímetre. (1) Bloc decoure, (2) sensors, (3) mostra, (4) referència, (5) resisència de carbó i (6) tapa.El camp magnètic, ~B = µ0 ~H, va al llarge de l'eix de simetria del calorímetre. (c)Detall que mostra el ux de calor Q per una transició exotèrmica.

estudiar aquests tipus de transicions.

Tal com s'ha dit, la determinació de ∆S associada a l'EMC en els Gd5(SixGe1−x)4 ha estat molt discutida. Fins ara, ∆S s'ha extret sempre indirectamentde mesures de CP(T,H) obtingudes amb tècniques calorimètriques que no sónvàlides entorn d'una transició de primer ordre, o de M(T,H), amb la discussióde quina és l'equació correcte (veure més amunt). L'única manera de mesurar∆S d'una transició de primer ordre és doncs amb un DSC. I, com que l'EMCnomés és possible en aquestes transicions si són induïdes per H, un DSC treballantsota camp aplicat apareix com la tècnica ideal per estudiar ∆S en una transiciómagnetostructural de primer ordre, com la que tenim en els Gd5(SixGe1−x)4. Aixídoncs, s'ha dissenyat i construït un DSC d'alta sensibilitat per treballar en un rangde temperatures d'entre 10 i 300 K i sota camps de ns a 5 T, que permet obteniruns valors de la calor latent i la variació d'entropia molt acurats.

195

RESUM

Detalls experimentals i calibracióEl DSC dissenyat i construït s'adapta a un sistema criomagnètic tipus Tesla-tron© (Oxford Instruments), que genera camps de ns a 5 T en l'eix verticalmitjançant una bobina superconductora. Les parts del calorímetre estan senyala-des a la Fig. 4. El calorímetre és una peça de coure (1) agafada a un tub llargd'acer. Tot el lat passa a través del tub i surt del dispositiu per l'extrem superior.Dos sensors (2), conectats diferencialment, estan emplaçats sobre les superfíciesaplanades internes del bloc de coure, amb vernís General Electrics d'Oxford perassegurar un bon contacte tèrmic. Els sensors són piles de termoparells (MelcorFCO.45-32-05L) fetes d'unions P-N de Bi2Te3 (32 parells d'unions en una super-fície 6.5 x 6.5 mm2). La mostra (3) i la referència inert (4) es posen directamentsobre cada sensor, subjectats per un l de nylon ben prim (0.2 mm de diàme-tre). Els ls elèctrics surten del calorímetre per oricis de 2.5 mm de diàmetre,tèrmicament acoblats a la part superior del bloc de coure abans de passar al tubd'acer. Això evita possibles gradients tèrmics en els ls, que podrien generar fal-sos voltatges termoelèctrics. La temperatura en el calorímetre es fa variar a travésde l'espai del criòstat Teslatron on està inserit. La lectura de la T del caloríme-tre es fa mesurant la resistència elèctrica d'una resistència de carbó (LakeShoreCryotronics INC. CGR-1-500) (5) inserida a la peça de coure. Per minimitzar laconvecció del gas d'intercanvi dins del calorímetre, tot el muntatge està cobert perun cilindre de coure (1 mm de gruix) (6), cargolat per la part superior del bloc. Lapressió dins del calorimetre es manté entre 200 i 600 mb amb heli d'alta puresa,perquè les impureses fan variar la sensibilitat dels sensors.

La calor emesa o absorbida per la mostra (vegeu Fig. 4(c)) es mesura llegintel voltatge generat per les termopiles amb un nanovoltímetre (Keithley 182). Lespossibles derives en la senyal associades a canvis de temperatura del bloc calori-mètric es minimitzen pel fet que els dos sensors estan conectats diferencialment,i per tant el ux de calor de la referència es resta del de la mostra, quedant tansols el voltatge degut a la potència tèrmica emesa/absorbida per la mostra durantla transició de primer ordre (calor latent). Tot el sistema (mesura de T , voltatge deles termopiles i electrònica del criòstat) es controla a través d'un PC. L'adquisiciódel voltage V(t) i la temperatura T (t) es fa a 0.25 Hz.

Com que les termopiles estan fetes d'elements semiconductors, no s'esperaque el camp magnètic afecti la seva senyal termoelèctrica, tot i que sí que de-pèn de la temperatura. Per això s'ha fet una calibració posant una resistència demanganina al lloc de la mostra, fent dissipar una potència constant (W) per efec-te Joule. La mesura del voltatge resultant a l'estat estacionari, Y , permet trobarla sensibilitat, donada per K = Y/W. La linealitat es manté per diferents valorsde W. La dependència amb la temperatura pot ésser ajustada segons la corba:K(mV/W)=1.4 × 10−8T 4 − 2.0 × 10−5T 3 + 5.1 × 10−3T 2 + 0.86T .

196


Per comprovar si el calorímetre mesura correctament, s'ha seleccionat un ali-atge de Cu-Zn-Al que té una transició estrucural (martensítica), com a sistemaestàndard. El material és diamagnètic i per tant la transició no varia amb el camp,de manera que es pot testejar la insensibilitat dels sensors al camp. Els resultatsrevelen que les corbes a camp zero i 5 T són iguals, o sigui que les mesures ambcamp poden ser fetes correctament. A més, els valors de la calor latent i la variaciód'entropia associades a la transició martensítica estan molt ben establerts [69, 70].Per obtenir aquests valors, es calcula el ux de calor com Q(t)=V(t)/K, es calculanumèricament el ritme d'escalfament/refredament dT/dt a partir de T (t) registrati, nalment, s'assoleix la corba calorimètrica dQ/dT= Q (dT/dt)−1. La senyal ca-lorimètrica es corregeix extraient la línia de base (pels detalls, vegeu Ref. [71]).La calor latent i la variació d'entropia venen per tant donades per:

L =

∫ TH

TL

dQdT dT ; ∆S =

∫ TH

TL

1T

dQdT dT , (11)

on TH i TL són, respectivament, temperatures per sobre i per sota de les tempera-tures de l'inici i del nal de la transició. Els valors obtinguts per la calor latent[L(µ0H=0)=336 ± 3 J/mol i L(µ0H=5T)=335 ± 3 J/mol] i la variació d'entropia[∆S (µ0H=0)=1.40 ± 0.01 J/mol K i ∆S (µ0H=5T)=1.39 ± 0.01 J/mol K] estan enacord amb els valors publicats [69, 70].

Mesures variant T i variant HL'aparell descrit en aquesta secció està particularment ben adaptat a les mesuresde variació d'entropia associada a transicions magnetoestructurals induïdes percamp presents en aliatges que exhibeixen un EMC gegant. En concret, una mesu-ra adequada de la variació d'entropia associada a la transició de primer ordre ésdesitjable per entendre bé aquest fenomen en la sèrie d'aliatges Gd5(SixGe1−x)4.La Figura 5 mostra un exemple de les corbes calorimètriques enregistrades escal-fant i refredant una mostra amb x=0.1 sota diferents camps aplicats. El pic enforma de λ entorn de 130 K prové de la transició PM↔AFM de segon ordre. Elpic gran a temperatures més baixes és degut a la transició AFM/O(II)↔FM/O(I)de primer ordre. S'observa histèresi tèrmica de 2-3 K i un dependència linealde la temperatura de transició de primer ordre amb el camp aplicat. La variaciód'entropia a la transició augmenta amb el camp en l'exemple representat en lesgràques insertes de la Fig. 5. La discussió d'aquests resultats, i els de les altresmostres de la sèrie, es fa en les següents seccions.

Els DSC estan dissenyats per operar amb rampes de temperatura mentre s'en-registra Q(t) i T (t), tal com hem vist ns ara. La rampa de T indueix tèrmicamentla transició de primer ordre, que absorbeix o emet calor. Quan tenim transicions

197

RESUM

-25

0

25

50

75

escalfant

5T4T3T

2T1T

0

dQ

/dT

(m

J/K

)

40 60 80 100 120 140

-75

-50

-25

0

25 refredant

1T

2T3T

4T

0

5T

dQ/d

T (

mJ/

K)

T (K)

0 1 2 3 4 5242628303234

µ0H (T)

∆S (

J/kg

K)

0 1 2 3 4 5242628303234

µ0H (T)

- ∆S

(J/

kg K

)

Figura 5: Corbes calorimètriques fetes amb rampa de T (escalfant i refredant) enuna mostra amb x=0.1 per diferents camps magnètics aplicats. Les gures insertesmostren la variació d'entropia en funció del camp.

198


0 1 2 3 4 5

-800

-600

-400

-200

0

200

400

600

800H decreixent

H creixent

55 K60 K

65 K

65 K60 K55 K

50 K

50 K

dQ/d

H (

mJ/

T)

µ0H (T)

0.1T/min 1T/min

50 55 60

12

16

20

0.1 T/min 1 T/min

∆S

(J/

kg K

)

T (K)

H decreixent

50 55 60

12

16

20

-∆S

(J/

kg K

)

T (K)

H creixent

0.1 T/min 1 T/min

Figura 6: Corbes calorimètriques fetes amb rampa d'H (creixent i decreixent) enuna mostra amb x=0.05 a diverses temperatures xades i per dos ritmes de camp.Les gràques insertes mostren la variació d'entropia en funció de la temperaturaper diferents ritmes de camp.

199

RESUM

induïdes per camp, la temperatura Tt del pic de la transició en la corba calori-mètrica es mou amb el camp magnètic aplicat, i conseqüentment obtenim la de-pendència de ∆S amb el camp. Tanmateix, aquest DSC també pot operar ambrampes de camp. Fixant una temperatura per sobre de Tt(H = 0), un camp proualt també indueix la transició de primer ordre. Això dóna lloc a una mesura di-recta de l'EMC, doncs es mesura la variació d'entropia assolida per l'aplicaciód'un camp. Pel que sabem, aquesta és la primera vegada que ∆S pot ser mesuratdirectament en aquests materials. En aquest cas, el ux de calor Q(t) i el campcreixent/decreixent H(t) són enregistrats, obtenint el ritme de variació del camp,H, i la corba calorimètrica dQ/dH= Q/ H. L i ∆S vénen doncs donades per:

L =

∫ HH

HL

dQdH dH ; ∆S =

1T

∫ HH

HL

dQdH dH =

LT , (12)

on HH i HL són camps per sobre i per sota dels camps de l'inici i del nal de latransició, respectivament. Com que el rang d'H accesible experimentalment ésmés restringit que el de T , aquestes rampes es limiten a un petit rang de tempera-tures per sobre de Tt. La Figura 6 mostra un exemple de corbes calorimètriquesfetes en H creixent i decreixent per una mostra amb x=0.05 a diferents temperatu-res xades per sobre de la Tt de camp zero (45 K). Les característiques d'aquestatransició s'han explicat anteriorment. Veiem com el camp de transició augmentalinealment amb la temperatura, tal com s'espera. En aquest exemple, la corba deT=65 K ja no té prou línia de base per poder ser integrada correctament. La ∆Sde la transició augmenta amb T , tal com es veu en les gràques insertes de la Fig.6. Cal remarcar que els valors no depenen del ritme de variació del camp utilitzat(0.1 i 1 T/min). Aquestes dades i les de les altres mostres de la sèrie s'analitzenen una secció posterior.

Variació d'entropia a la transició de fase magnetoes-tructural de primer ordre en els aliatges Gd5(SixGe1−x)4

IntroduccióPer fer un anàlisi detallat de les diverses contribucions a la variació d'entropiaresultant d'aplicar un camp magnètic a una transició de primer ordre induïda percamp, s'han estudiat mostres de Gd5(SixGe1−x)4 en el rang on apareix la transiciómagnetoestructural: x=0, 0.05, 0.1, 0.18, 0.2, 0.25, 0.3, 0.365 i 0.45. Per a aquestpropòsit, s'han dut a terme mesures de magnetització a camps alts i mesures calo-rimètriques amb el DSC descrit en la secció anterior.

200

Variació d'entropia en els Gd5(SixGe1−x)4

Mesures de magnetització i DSCLes mesures de magnetització s'han realitzat al Laboratori de Camps MagnèticsIntensos de Grenoble (GHMFL). Les corbes M(H) s'han mesurat ns a 23 T ambcamps creixents i decreixents, des de 4.2 ns a 310 K, amb intèrvals de 3 a 5 K.Algunes M(H) estan dibuixades a la Fig. 7. Les corbes presenten un salt ∆M ala transició magnetoestructural [26, 25, 27, 62], que s'extén en un rang de camps∆Ht (∼2-4 T en les mostres amb 0 ≤ x ≤ 0.2 i ∼4-6 T en les mostres amb 0.24 ≤x ≤ 0.5). La histèresi a la transició revela el seu caràcter de primer ordre. El campde transició, Ht, es deneix a cada isoterma com el camp corresponent al puntd'inexió de la zona de la transició, sent diferent per camp creixent o decreixent.L'estudi detallat d'Ht(T ) i dHt/dTt s'explica més endavant. Pels compostos ambx ≤ 0.2, la transició de primer ordre va d'una fase AFM a una fase FM, mentreque pels aliatges amb 0.24 ≤ x ≤ 0.5 va de PM a FM.

Les mesures calorimètriques s'han fet amb el DSC descrit a la secció anterior,amb rampes de T a camp magnètic constant (de 0 a 5 T). Les transicions de primerordre donen lloc a pics intensos a les corbes dQ/dT , mentre que les de segon ordres'observen com un petit salt en forma de λ a la línia de base. La Figura 5 n'és unexemple. En totes les mostres s'observa una histèresi de 2-4 K entre les rampesde T escalfant i refredant. Tt s'agafa com la temperatura on el pic de dQ/dT ésmàxim. Tt augmenta amb el camp aplicat. L'estudi detallat de Tt(H) s'explicamés endavant.

Avaluació de la variació d'entropiaLa variació d'entropia pot ésser obtinguda indirectament de les dades de magne-tització:

(i) Per una banda, la variació d'entropia a la transició de primer ordre, ∆S escalcula utilitzant l'equació de Clausius-Clapeyron (Eq. 10), on ∆M s'ha avaluatcom la diferència en la magnetització a Ht entre les extraplacions lineals d'M(H)per sobre i per sota de la regió de transició i dHt/dTt s'ha obtingut a partir d'Ht(T )extret de la magnetització.

(ii) Per altra banda, la variació total d'entropia degut al canvi de magnetitzacióen aplicar un camp magnètic, ∆S (0 → Hmax), es calcula fent servir la relació deMaxwell (Eq. 6). Aquesta integració s'avalua numèricament de les isotermes demagnetització. És senzill de veure que ∆S i ∆S (0→ Hmax) no tenen perquè valerigual.

La variació d'entropia també es pot determinar per calorimetria, a partir deles corbes mesurades amb un DSC i després d'una integració apropiada del piccalorimètric associat a la transició de primer ordre (Eq. 11).

Una comparació entre els tres mètodes d'avaluació es mostra, per x=0.45, a

201

RESUM

0 2 4 6 8 10 12 14 16 18 20 22 24

0

40

80

120

0

40

80

120

160

T= 39.4, 55.4,64.7, 75.4, 84.4, 94.8, 102.6 i 109.2 K

x=0.05

M (

emu/

g)

T= 231.0, 239.3, 247.2, 255.0, 262.5, 270.0, 278.5, 286.5, 297.3 i 307.0 K

x=0.45

µ0H (T)

Figura 7: Selecció d'isotermes de magnetització en els Gd5(SixGe1−x)4, perx=0.05 i 0.45, en camps creixents i decreixents. Les temperatures indicades encada composició es refereixen a les corresponents isotermes des de dalt a la dretacap a baix a l'esquerra.

202


200 220 240 260 280 300 320

H creixentx=0.45

20 T15 T10 T7 T5 T2 T

Var

iaci

ó d'

entr

opia

(J/

kg·K

)

T(K)

0

-10

-20

-30

Figura 8: Variació d'entropia per una mostra amb x=0.45 calculada segons: (i)relació de Maxwell integrant ns a Hmax (línies discontínues), (ii) equació deClausius-Clapeyron (quadrats plens per aquest treball i quadrats buits per x=0.5de la Ref. [25]), (iii) mesures de DSC sota camp (triangles buits), i (iv) relacióde Maxwell integrant la regió ∆Ht (línies contínues). Hmax està marcat per cadalínia discontínua, i també marca les línies contínues d'esquerra a dreta per valorscreixents.

203

RESUM

la Fig. 8. La variació d'entropia obtinguda de la relació de Maxwell dóna lloca diferents corbes en funció de T per cada valor del camp màxim aplicat, quemostren el típic comportament descrit anteriorment [26, 25]: primer, un incrementràpid a baixa T , després un valor màxim entorn de Tt(H = 0), seguit per uncomportament aplanat, i nalment un decreixement brusc a T alta. Cal remarcarque ∆S obtinguda amb l'equació de Clausius-Clapeyron i amb el DSC donen igualdins de l'error experimental, cosa que suggereix que els dos mètodes avaluen lavariació d'entropia de la transició de primer ordre. El màxim valor que dóna larelació de Maxwell està per sobre o per sota de ∆S depenent del valor d'Hmax.Per la resta de mostres els resultats són semblants, tot i que, per les x menors,∆S avaluat amb DSC dóna valors lleugerament superiors que amb l'equació deClausius-Clapeyron.

La diferència entre ∆S (DSC i Clausius-Clapeyron) i ∆S (0 → Hmax) (Max-well) es pot entendre considerant que la relació de Maxwell inclou les següentscontribucions:

∆S (0→ Hmax) =

∫ Ha

0

(∂M∂T

)

HdH +

∫ Hb

Ha

(∂M∂T

)

HdH +

∫ Hmax

Hb

(∂M∂T

)

HdH , (13)

amb Ha=Ht −∆Ht/2 i Hb=Ht + ∆Ht/2. La primera i tercera integral dóna la varia-ció d'entropia deguda a la dependència de la magnetització amb T i H en les duesfases implicades fora de la regió de transició. Només el segon terme correspona la contribució de la variació d'entropia associada a la transició magnetoestruc-tural. Això queda pal·lès en el fet que el comportament de les línies contínues(calculades uilitzant només el segon terme de l'Eq. 13) a la Fig. 8 concorda per-fectament amb els valors de ∆S . S'ha fet servir un rang de camps per la zona detransició de µ0∆Ht ∼4 T. Cal ressaltar que quan Hmax és menor que ∆Ht, que ésel camp mínim per completar tota la transició, el valor màxim de ∆S (0 → Hmax)és menor que ∆S (vegeu per exemple la corba corresponent a µ0Hmax = 2 T a laFig. 8). A més, per Hmax ≥ ∆Ht, la regió plana s'extén en tot el rang de T en quèHmax ≥ Hb(T ). Conseqüentment, mentre Hb(T ) creix amb T , la caiguda bruscades de la zona plana a temperatures més altes es deu a que la segona integral estrunca a Hmax.

Models fenomenològicsPer tal de justicar les característiques principals de la variació d'entropia descri-tes en aquesta secció, es proposa un model fenomenològic que té en compte elstrets fonamentals de la magnetització en un sistema amb una transició de primerordre induïda per camp.

En una primera aproximació, només hem considerat el comportament de latransició, imposant M(T ) constant fora de la zona de transició. La forma de les

204


H1/∆H

t

H2/∆H

t

∆M

0

0

0-H5

0-H40-H

3

0-H2

0-H1

1

Temperature

-∆S

(0→

Hm

ax)/

∆S

H=0

H1

H2

H3 H

4 H5

M

Figura 9: Quadre superior: Dependència de la magnetització amb la temperaturaal llarg de la zona de transició a diferents camps, tal com es descriu en el text.Quadre inferior: la variació corresponent d'entropia, ∆S (0→Hmax), calculada dela relació de Maxwell. En aquesta gura ∆S representa la variació d'entropia dela transició, obtinguda de l'equació de Clausius-Clapeyron.

205

RESUM

corbes de magnetització s'han pres com:

M(T,H) = M0 + ∆M F(T − Tt(H)

ξ

), (14)

on M0 i ∆M es consideren independents de T i H, i F(T ) és una funció monòtonadecreixent d'amplada ξ de manera que F → 1 per T Tt(H) i F → 0 perT Tt(H). El cas ξ → 0 correspon a una transició de primer ordre ideal (F ésllavors la funció de Heaviside). Si s'utilitza la relació de Maxwell i s'assumeix unadependència lineal de Tt amb el camp (dTt/dHt ≡ α=cnt.), la variació d'entropiavé donada per:

∆S (0→ Hmax) = ∆S[F

(T − Tt(Hmax)

ξ

)− F

(T − Tt(H = 0)

ξ

)], (15)

on ∆S = ∆M/α (la variació d'entropia a la transició treta de l'equació de Clausius-Clapeyron). És remarcable el fet que si Tt no depèn d'H, llavors ∆S (0→ Hmax)=0independentment del valor de ∆S . En general, ∆S (0 → Hmax) és una fraccióde ∆S , i assoleix el valor màxim, ∆S , per camps aplicats prou grans. Aquestsresultats són vàlids ns i tot en el límit ξ → 0, en el que ∆S (0→ Hmax) = ∆S pertot valor d'Hmax.

Es pot tenir una representació analítica simple assumint que F és una fun-ció lineal amb la temperatura que s'extén en un rang ∆Tt = α∆Ht = ξ. Elsresultats es presenten a la Fig. 9. Els trets generals són semblants al resultatobtingut integrant el segon terme de l'Eq. 13 (línies contínues de la Fig. 8).També s'observa que quan Hmax no és prou gran com per completar la transició(Hmax < ∆Ht), llavors ∆S (0→Hmax) = (Hmax/∆Ht)∆S és menor que ∆S . Con-seqüentment, (Hmax/∆Ht) és la fracció de la mostra que ha estat transformada.Per Hmax ≥ ∆Ht, ∆S (0→Hmax) assoleix el seu màxim valor ∆S (0→Hmax)=∆S ,demostrant l'equivalència entre l'equació de Clausius-Clapeyron i la relació deMaxwell, sempre que aquesta s'avaluï en el rang de camps apropiat a l'entorn dela transició.

La utilització d'un model més complet, que té en compte la dependència de Men T i H fora de la zona de la transició, justica la dependència de ∆S (0→Hmax)amb T i el camp màxim aplicat, donant lloc a valors més grans que ∆S en certescondicions. Amb això es demostra que la relació de Maxwell té en compte la∆S de la transició i a més a més també inclou en la seva avaluació la variaciód'entropia que sorgeix de la dependència de la magnetització amb T i H, forade la regió de transició de primer ordre, que també cal considerar en el càlcul del'EMC.

206

Escalat de la variació d'entropia

Escalat de la variació d'entropia de la transició en elsaliatges Gd5(SixGe1−x)4

L'estudi sistemàtic, en els aliatges Gd5(SixGe1−x)4, de la variació d'entropia asso-ciada a la transició magnetoestructural de primer ordre, ∆S , en funció tant de lacomposició x com del tipus de transició magnètica, s'ha dut a terme amb el DSCsota camp. Aquest estudi ens revela l'existència d'un escalat de ∆S , on la variabled'escala és Tt, que es pot variar amb x o H.

El DSC pot oferir altra informació a part de la calor latent i la variació d'en-tropia a la transició: tot i que no dóna el valor absolut de CP, l'extrapolació a Ttde les línies de base des de temperatures per sobre i per sota de la transició deprimer ordre dóna una bona estimació de ∆Cp. S'observa que ∆Cp és positivaper la transició AFM-FM de primer ordre en les mostres amb x ≤ 0.2, mentreque ∆Cp és negativa per la transició PM-FM de primer ordre en les mostres amb0.24 ≤ x ≤ 0.5. El cas de la mostra x=0.2 és força interesant, doncs el pic de latransició de primer ordre solapa el de segon ordre per un camp de ∼3 T, i per aixòs'observa un canvi en el signe de ∆Cp.

El valor absolut de ∆S en funció de Tt es mostra en la Fig. 10. Com que esrepresenta el valor per cada x i H, la corresponent Tt pot variar des de ∼20 nsa ∼310 K. Això ens permet deduir un escalat de |∆S | per tot Tt, o sigui, per totcompost amb x ≤ 0.5, mostrant que el paràmetre important en determinar |∆S |és Tt. A més, aquest escalat no és una conseqüència trivial de l'escalat de ∆Mo dHt/dTt, com es podria esperar de l'equació de Clausius-Clapeyron (Eq. 10),doncs cap dels dos no escala amb Tt. Això dóna més rellevància a l'escalat de|∆S |. Cal fer notar que |∆S | s'extrapola a zero per Tt=0, tal com s'espera de latercera llei de la termodinàmica. L'escalat és una conseqüència del caràcter deprimer ordre de la transició: a camp constant, l'equació de Clausius-Clapeyrons'escriu com ∆S = ∆V(dPt/dTt), on ∆V representa la variació de volum i Pt és lapressió de transició. Per tant, ∆M i ∆V es relacionen segons

∆M∆V =

dTt

dHt

dPt

dTt, (16)

i l'escalat prova doncs que els efectes de magnetovolum degut a H són de lamateixa naturalesa que les variacions de volum provocats per la substitució de Gepel Si.

En l'escalat es veuen tres comportaments. Per 0.24 ≤ x ≤ 0.5, |∆S | associat ala transició PM/M-FM/O(I) decreix monòtonament amb Tt, sent consistent amb∆Cp < 0 observat de les corbes del DSC, tal com esperem de la relació termodinà-mica d(∆S )/dT = ∆Cp/T . En canvi, per x ≤ 0.2, |∆S | creix o decreix depenent deTt. Degut a l'acoblament magnetoelàstic, l'aplicació d'H mou Tt, de manera que

207

RESUM

0 30 60 90 120 150 180 210 240 270 300 3300

10

20

30

40

50

AFM-FMO(II)-O(I)

PM-FMM-O(I)

refr. escal. x 0.05 H0.1 H0.18 H0.2 H=00.2 H0.25 H=00.25 H0.3 H=00.3 H0.365 H=00.365 H0.45 H=00.45 H0.5

TN

TN

PM-FMO(II)-O(I)

H creix. H decr. x 0.1

| ∆S

| (J/

kg·K

)

Tt(K)

Figura 10: Escalat de |∆S | a la transició de primer ordre pels aliatgesGd5(SixGe1−x)4. Els valors obtinguts amb el DSC es representen per diversoscamps aplicats i composicions. Els símbols de diamant són de la Ref. [25]. Elssímbols marcats amb H/H=0 corresponen respectivament a mesures fetes sotacamp o a camp zero. Els valors de ∆S obtinguts d'M(H) ns a 23 T per x=0.1també estan representats.

208

L'acoblament magnetoelàstic en els Gd5(SixGe1−x)4

és possible observar tant la transició AFM/O(II)-FM/O(I) a Tt com, a H prou alt,una transició PM/O(II)-FM/O(I), quan Tt(H) ≥ TN . Aquesta és encara de primerordre degut a la transformació cristal·logràca i prové de la transició PM-AFM.Per la transició AFM/O(II)-FM/O(I), |∆S | creix monòtonament amb Tt, en acordamb ∆Cp > 0. Tanmateix, per la transició PM/O(II)-FM/O(I), |∆S | decreix ambTt per x=0.2, d'acord amb l'observació descrita anteriorment per aquesta mostra(∆Cp < 0). Per veure millor l'evolució de ∆S per aquesta darrera transició, tam-bé s'han considerat els resultats obtinguts de l'equació de Clausius-Clapeyron,doncs les corbes M(H) arriben a 23 T. Per claredat, només els valors per x=0.1es representen a la Fig. 10, però totes les mostres amb x ≤ 0.2 tenen el mateixcomportament (|∆S | creix amb Tt per despres decréixer).

Conseqüentment, |∆S | és màxim per a cada composició a Tt = TN , o sigui,quan, a la fase FM, el camp aplicat és prou gran com per desplaçar la transició deprimer ordre ns a solapar la de segon ordre a TN (marcat a la Fig. 10). Per tant,el valor més gran |∆S |=48.22 J/(kgK) succeeix a Tt '130 K (∼ el valor més alt deTN , que correspon a x=0.2 [23]). Tot això suggereix que |∆S |, i per tant l'EMC,serà màxim en el rang de composicions 0.2 < x < 0.24, on les diferents fasescristal·logràques i magnètiques coexisteixen, i les dues branques principals del|∆S | s'uneixen (Fig. 10).

L'acoblament magnetoelàstic en els aliatgesGd5(SixGe1−x)4

L'estudi de l'efecte del camp magnètic sobre la transició magnetoestructural enfunció d'x permet veure com el paràmetre dHt/dTt juga un paper clau en l'esca-lat de |∆S | i permet estimar la intensitat de l'acoblament magnetoelàstic: en elsGd5(SixGe1−x)4, el valor de ∆S mesurat quan s'indueix la transició per camp mag-nètic coincideix amb el valor obtingut en induïr-la aplicant pressió [52], obtenintl'Eq. 16, que ens indica que un fort acoblament magnetoelàstic produeix un valorpetit de dHt/dTt.

De les mesures de DSC i M(H) s'ha avaluat, independentment, la dependènciade Tt amb Ht, segons s'ha descrit anteriorment. L'acord entre ambdós càlculs ésbo per totes les mostres. És de ressaltar el fet que per les mostres amb 0.24 ≤x ≤ 0.5, on només tenim la transició PM-FM, Ht(Tt) mostra un comportamentlineal al llarg de tot el rang de camps, mentre que per x ≤ 0.2 la pendent d'Ht(Tt)canvia progressivament d'un valor a camp baix (on tenim la transició AFM-FM)ns a un valor a camp alt (corresponent a la transició PM-FM). Aquesta variacióprogressiva es deu al fet que, a camps alts, la transició magnetoestructural solapala transició PM-AFM de segon ordre, donant lloc a una única transició PM-FM.

209

RESUM

0.0 0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

transicióAFM-FM

transició PM-FM

camp alt

camp baix

de M(H), H creixent de M(H), H decreixent de DSC, refredant de DSC, escalfant

d(µ 0H

t)/dT

t (T

/K)

x (contingut en Si)

Figura 11: Pendent d'Ht(Tt) calculada de les dades de magnetització i DSC. Perx=0.25, 0.3, 0.365, 0.45 i 0.5 (aquesta darrera extreta de la Ref. [25]) s'obté unasola pendent, corresponent a la transició PM-FM. Per x=0, 0.05, 0.1, 0.18 i 0.2s'obtenen dues pendents límit: un valor a camp baix (associat a la transició AFM-FM) i un valor a camp alt (associat a la transició PM-FM). Les línies contínues idiscontínues serveixen només de guia.

210

Antiferromagnetisme de curt abast en els Gd5(SixGe1−x)4

Els valors de la pendent dHt/dTt estan compilats, per tota x, a la Fig. 11. Perx ≤ 0.2, es representen dos valors límit de dHt/dTt, corresponents als règims d'alti baix camp, mentre que per 0.24 ≤ x ≤ 0.5 només tenim un valor. És remarcablela dependència lineal de dHt/dTt amb x, que és decreixent per la transició PM-FM(línia contínua a la Fig. 11) i creixent per la transició AFM-FM (línia discontínuaa la Fig. 11). Ambdues línies s'ajunten a la zona de composicions on la transicióde segon ordre desapareix (0.2 < x < 0.24), en acord amb el diagrama de fases(Fig. 2). En conseqüència, el decreixement de dHt/dTt amb x creixents per latransició PM-FM indica un reforçament de l'acoblament magnetoelàstic. Això espot explicar considerant que les interaccions de bescanvi FM són més fortes enx creixents, tal com suggereix el diagrama de fases, on Tt augmenta linealmentamb x (Fig. 2). El fet que dHt/dTt té un comportament continu per la transicióPM-FM, encara que la fase PM sigui monoclínica per 0.24 ≤ x ≤ 0.5 i ròmbicatipus Sm5Ge4 per x ≤ 0.2, suggereix que l'acoblament magnetoelàstic no depènmolt de quina sigui l'estructura cristal·logràca. Pel que fa a la transició AFM-FM, i tenint en compte que la transició estructural és la mateixa (x ≤ 0.2) omolt semblant (0.24 ≤ x ≤ 0.5) a la que succeeix en la PM-FM, l'augment dedHt/dTt amb x sembla estar relacionat amb el fet que la transició implica duesfases ordenades (FM i AFM). La Figura 11 resumeix doncs el comportament dela transició de primer ordre en els Gd5(SixGe1−x)4 en funció de T , x i H.

Per altra banda, el comportament de dHt/dTt amb x és important en l'esca-lat de |∆S |: tenint en compte l'equació de Clausius-Clapeyron i que ∆M sempredecreix amb T , llavors |∆S | creix amb Tt (per x ≤ 0.2 i quan té lloc la transicióAFM-FM) degut al major increment de dHt/dTt amb Tt en comparació amb ladisminució de ∆M amb Tt. En canvi, |∆S | decreix amb Tt a la transició PM-FM,doncs l'increment de dHt/dTt amb Tt no és prou gran com per superar la dismi-nució de ∆M. Per tant, tot i que el tret principal de l'escalat de |∆S | amb Tt noestà només determinat per dHt/dTt amb respecte Tt, la dependència concreta dedHt/dTt amb x i H permet l'escalat.

Antiferromagnetisme de curt abast en els aliatgesGd5(SixGe1−x)4 rics en GeL'observació d'una transició no descrita en aplicar un camp magnètic molt granen els aliatges Gd5(SixGe1−x)4 rics en Ge (x < 0.2), ens porta a estudiar el dia-grama de fases d'aquests compostos, que presenten unes complexes estructuresmagnètiques tal com s'ha recopilat en la secció que resumeix les característiquesd'aquests aliatges.

Les dades del DSC per x=0, 0.05 i 0.1 (vegeu per exemple la Fig. 5 per aques-

211

RESUM

0

4

8

12

16

20

Cp (Levin et al.) M(H) DSC M(T)

SRAFMx=0

AFM

FM

0

4

8

12

16

20

SRAFMx=0.05

AFM

FM

µ 0H(T

)

20 40 60 80 100 120 1400

4

8

12

16

20

SRAFM

x=0.1

AFM

FM

T(K)Figura 12: Diagrames de fase magnètics d'alguns dels compostos Gd5(SixGe1−x)4rics en Ge. Els valors s'han obtingut de les corbes M(H) (quadrats plens), deles corbes M(T ) (quadrats buits) i del DSC sota camp (cercles buits). Les dadestretes de la Ref. [53] també s'han inclòs (triangles buits i línies discontínues).Per claredat, només s'han representat els valors en camp creixent i refredant. Leslínies contínues serveixen de guia.

212

Antiferromagnetisme de curt abast en els Gd5(SixGe1−x)4

ta darrera composició) ens permeten obtenir Tt(H) com ja s'ha explicat i tambéTN(H), que decreix amb el camp, comportament també descrit a la Ref. [53] perx=0. Aquests resultats es representen a la Fig. 12, i concorden amb els de laRef. [53]. De les isotermes M(H) pels tres aliatges (vegeu Fig. 7 per x=0.05)n'obtenim Ht(T ), que es representa a la Fig. 12 i també quadra amb els altresresultats. Les M(H) mostren una nova transició en la zona AFM: un canvi a lapendent d'M(H) suggereix que hi ha una transició des de l'estuctura AFM a unaestructura magnètica diferent que és induïda pel camp, abans que tingui lloc l'es-perada transició magnetoestructural de primer ordre a la fase O(I)/FM. Aquestcomportament es veu clarament per x=0 i 0.05 (Fig. 7) i es va suavitzant perx=0.1. Els camps en els quals hi ha el canvi de pendent s'avaluen com els puntson la curvatura de M(H) és màxima, i es representen a la Fig 12 per x=0 i 0.05. Ésdestacable que aquests valors concorden amb la corba TN(H), o sigui que el queestem observant és la transició nominal de la fase AFM a la PM, però com queuna fase PM no pot ésser induïda pel camp, la fase d'alt camp ens és desconeguda.Com que el camp trenca l'ordre AFM, el que es forma és probablement una faseamb correlacions AFM de curt abast (de l'anglès, SRAFM).

Per conèixer la nauturalesa d'aquesta fase d'alt camp, s'ha calculat la sus-ceptibilitat inversa en funció de T a partir de les corbes d'M(H), en el rang detemperatures i camps on apareix la nova fase. Com que les corbes són lineals enaquesta fase, la susceptibilitat diferencial, χ = ∂M/∂H, és constant en tot el rangde camps. La llei de Curie-Weiss,

χ(T ) =cC

T + θC, (17)

on cC = (Nat/m)(p2e f fµ

2B/3kB), θC és la temperatura de Curie paramagnètica i

pe f f =√

g2J(J + 1) és el moment magnètic efectiu en magnetons de Bohr, escompleix per les tres mostres, obtenint θC=75.0 K i pe f f =3.99 µB/àtom Gd, perx=0; θC=80.7 K i pe f f =3.96 µB/àtom Gd per x=0.05; i θC=97.6 K i pe f f =4.61µB/àtom Gd per x=0.1. Ja que la llei de Curie-Weiss es segueix amb un pe f f moltmenor que l'esperat pel ió Gd3+ lliure (7.94 µB), la fase induïda pel camp no ésPM sinó que encara presenta correlacions AFM de curt abast (fase SRAFM). Tam-bé hi han d'haver presents correlacions FM, doncs θC és positiva, tal com tambéesmenten les Refs. [48, 27, 54, 63]. Una altra evidència de la presència d'inte-raccions FM en la fase AFM és la saturació que s'observa en les corbes M(H)corresponents a la fase AFM, a camps baixos (vegeu per exemple la Fig. 7 perx=0.05).

L'existència de la fase SRAFM ens permet postular l'existència d'una tran-sició entre la fase SRAFM i la fase PM d'alta temperatura. Aquesta hipotèticatransició s'espera que sigui suau i que tingui lloc en un rang ampli de temperatu-res. Una evidència d'aquesta transició és el comportament no lineal de la suscep-

213

RESUM

tibilitat inversa per x=0 a camp nul entre TN i ∼230 K [63], la qual cosa indical'existència d'algunes correlacions a la fase nominalment PM. El mateix compor-tament s'observa en l'invers de la susceptibilitat ac, (χ′ac)−1, per x=0.15 i x=0.18,amb dues anomalies a ∼175 K i ∼240 K que també son clarament observades en lapart imaginària de la susceptibilitat ac. La llei de Curie-Weiss per sobre de ∼240K dóna el mateix valor de 7.33 µB per x=0.15 i x=0.18. És remarcable la histèresiobservada entre Tt (∼100 K) i 175 K, o sigui, per sobre i per sota de TN . Aixòsuggereix que hi han correlacions FM tant a la fase AFM com, per sobre de TN , ala fase SRAFM.

Seguint amb la recerca d'indicis de la transició entre la fase PM i SRAFMs'han analitzat corbes M(T ) entre 4.2 i 300 K a diferents camps magnètics des de0.1 ns a 20 T, obtingudes amb processos ZFC (escalfant després de refredar ambcamp aplicat nul) i algunes amb el subsegüent procés FC (refredant amb campaplicat), per a les mostres amb x=0 i 0.05. Cal destacar que també s'observa, enaquest cas, histèresi en tota la zona de la fase AFM i també per sobre de TN nsa ∼240 K. Les corbes mostren les transicions de primer i segon ordre (algunesTt i TN corresponents s'han representat a la Fig. 12), però no mostren cap altraanomalia ns a 300 K. La forma de la transició de segon ordre, a camps baixos, ésmés aviat la que mostren les transicions FM-PM (esglaó), i amb camps grans essuavitza degut a la magnetització de les dues fases que prenen part a la transició.Tot plegat fa suposar que tenim clusters FM a la fase AFM i que van creixent enbaixar la temperatura, ns que a Tt té lloc la transició de primer ordre a una faseFM de llarg abast. La presència d'interaccions FM en l'estructura O(II) ha estatrecentment observada per Pecharsky et al. [72].

A partir de les corbes M(T ) s'ha calculat també la susceptibilitat inversa. Perla corba amb H=0.1 T s'observa l'anomalia ja descrita a ∼240 K. Amb H=1 T,aquesta anomalia ha desaparegut completament. Com més alt és el camp aplicat,més es desvia χ−1(T ) de la linealitat, tot i que totes les corbes acaben sent pa-ral·leles per sobre de ∼240 K (a més gran el camp, més alta la temperatura), ones pot aplicar la llei de Curie-Weiss. Els valors extrapolats de pe f f i θC varien de∼7.3µB i ∼130 K per camps propers a zero ns a ∼8µB i ∼70 K per 20 T, variacióque indica que ns i tot a temperatura ambient tenim correlacions magnètiquesdegut a la presència del camp.

Amb aquests dades, la comguració dels diagrames de fases queda comple-tada (Fig. 12). En aquests s'observa que les corbes Tt(H) i TN(H) intersecten aun punt tricrític (T ∗, H∗), que varia amb la composició x. Si el camp magnètic ésaplicat isotèrmicament a una T tal que T ∗ < T < TN(H = 0), el sistema canviades de la fase AFM a la fase SRAFM intermitja, i nalment assoleix la fase FM.Aquest rang de temperatures disminueix amb x creixents. La fase SRAFM apa-reix a camp zero des de TN ns a ∼240 K, tal com s'observa en la susceptibilitatinversa (per diverses mostres i tècniques). A mesura que el camp aplicat augmen-

214

Dinàmica de la transició de primer ordre

ta, aquesta fase s'eixampla i la transició a la fase PM augmenta de temperatura.Aquesta transició no s'ha indica a la Fig. 12 doncs s'extén en un rang ampli detemperatures que no es pot determinar bé. Per això caldria mesurar susceptibilitatsota camp ns a altes temperatures.

Dinàmica de la transició de primer ordre en els aliat-ges Gd5(SixGe1−x)4: ciclats i allausIntroduccióL'estudi de la dinàmica de la transició magnetoestructural de primer ordre presenten els aliatges Gd5(SixGe1−x)4 amb x ≤ 0.5, per entendre els mecanismes que laguien, s'ha dut a terme des de tres punts de vista. S'ha estudiat l'efecte d'induïr latransició per camp o per temperatura en la variació d'entropia (∆S ), s'ha sistema-titzat un ciclat de la transició en mostres verges per veure l'evolució de la variaciód'entropia i, nalment, s'ha analizat les allaus entre estats metastables del sistemadurant la transició.

Comparació de la variació d'entropia induïda per T i per HPer comparar els valors de ∆S obtinguts induïnt la transició per H o per T , s'hamesurat amb el DSC de les dues maneres: variant T a H constant [DSCH(T )],d'on obtenim ∆S H, i variant H a T constant [DSCT (H)], d'on obtenim ∆S T . Lamesura de DSCH(T ) s'ha dut a terme per totes les mostres (vegeu, per exemple, laFig. 5 per x=0.1) i ja han estat comentades més amunt. La mesura de DSCT (H)s'ha fet en les mostres x=0.05 (Fig. 6) i 0.1 amb camps creixents (ns a 5 T) idecreixents i un ritme de variació del camp ( H) de 1 i 0.1 T/min. Els resultats, queno depenen d' H, estan representats a la Fig. 13 per x=0.05.

Els valors de ∆S H són majors en valor absolut que els de ∆S T , diferint ∼5J/(kgK) per x=0.05 (vegeu Fig. 13) i ∼7 J/(kgK) per x=0.1, una diferència massagran per ser degut a l'error experimental (que és del 5-10 %). Aquesta diferènciaes pot explicar fent ús de termodinàmica general [73]. La combinació del primeri el segon principi de forma diferencial és

dU = dQ + HdM = TdS + HdM , (18)

on U és l'energia interna, Q la calor transferida i HdM el treball extern diferencialnecessari per magnetitzar un sistema. Per comoditat s'utilitza l'entalpia, E, coma potencial termodinàmic, denida com

E = U − HM , (19)

215

RESUM

40 50 60 70 80 90 100 110 120

0

5

10

15

20

25

30

20-0T15-0T

10-0T7-0T

5-0T

2-0T

x=0.05 Maxwell ∆Ht

Clausius M(H) Clausius M(T)/ DSC

T(H)

( 0.1 / 1 T/min) DSC

H(T)

∆S (

J/kg

K)

T (K)

Figura 13: Variació d'entropia, per x=0.05, calculada a partir de: DSCH(T ) escal-fant (triangles buits); DSCT (H) amb H decreixent (cercles plens i buits); l'equa-ció de Clausius-Clapeyron avaluada d'M(H) amb H decreixent (quadrats plens) id'M(T ) escalfant (quadrats buits); i la relació de Maxwell integrant des de dife-rents valors d'Hmax (assenyalat per cada corba) a zero, i avaluat només en la regióde transició (línies contínues).

que en forma diferencial és

dE = dQ − MdH = TdS − MdH . (20)

A H constant, si s'indueix la transició variant T , s'obté que

∆EH =

∫TdS = Q ≡ L , (21)

∆S H =

∫ dQT =

∫ dEH

T , (22)

és a dir, la variació d'entalpia a la transició és igual a la calor latent, L, i la variaciód'entropia a la transició es calcula integrant la calor diferencial dividida per latemperatura. Per altra banda, si T es manté constant i el que variem és H, lavariació d'entalpia s'escriu com

∆ET = Q −∫

MdH = T∆S −∫

MdH , (23)

i per tant L i ∆S , que són els valors mesurats pel DSC, tenen les següents expres-sions:

Q ≡ L = ∆ET +

∫MdH , (24)

216


∆S T =QT =

∆ET

T +1T

∫MdH . (25)

En aquest cas, L i ∆S T tenen una contribució addicional a part de la variaciód'entalpia degut al treball del camp magnètic sobre el sistema. Sembla clar, doncs,que ∆S H i ∆S T no tenen perquè donar igual (excepte en el cas d'una transició ideala T i H constant). De fet, l'avaluació del terme (1/T )

∫MdH de les corbes M(H)

correspon aproximadament a la diferència ∆S H-∆S T i amb el signe que li pertoca.Això ens indica que ambdós processos difereixen degut al treball necessari permagnetitzar el sistema i també que ∆ET/T ≈

∫dEH/T .

La variació d'entropia també es pot avaluar indirectament a través de l'equacióde Clausius-Clapeyron. De fet, s'observarà la mateixa diferència si l'avaluem apartir de corbes M(H) o M(T ), doncs els processos son induïts per H o T , respec-tivament. La Figura 13 ensenya com aquests valors concorden amb els respectiusmesurats amb DSC. És de destacar el fet que aquesta diferència disminueix ambx creixents, ja que el terme (1/T )

∫MdH es redueix amb T . Per exemple, per

x=0.45, val ∼1 J/(kgK), i de fet no s'observen diferències (Fig. 8) entre ∆S H (delDSC) i ∆S T (de l'equació de Clausius-Clapeyron). L'avaluació amb la relació deMaxwell integrant M(H) en la regió de transició òbviament quadra amb els valorsde ∆S T (Fig. 13).

Aquest important resultat es deu, doncs, al fet que la transició de primer ordreno és ideal, i per tant els estats nals i inicials en el diagrama de fases H − T sóndiferents en un procés isotèrmic i en un procés a camp constant.

Ciclat a través de la transició de primer ordreEn els Gd5(SixGe1−x)4 s'ha observat que algunes propietats varien cada vegadaque s'indueix la transició magnetoestructural de primer ordre [60, 74, 75]. Perestudiar aquest efecte en la variació d'entropia, s'han utilitzat 3 mostres verges(v1, v2 i v3) procedents de la mateixa mostra original amb x=0.05. S'ha mesuratDSCT (H) a T=55 K, fent diversos cicles i per H diferents1.

La forma de la primera corba de DSC amb camp creixent és diferent de lessegüents, tancant una àrea menor (vegeu la Fig. 14 per v2). De fet l'àrea creix enaugmentar els cicles (es veu clarament en les corbes de camp decreixent a 1 T/mindibuixades a la gràca inserta de la Fig. 14). El primer cicle de la mostra v2 pre-senta pics petits de tamany similar. En els subsegüents cicles alguns pics creixeni altres disminueixen, assolint una distribució reproduïble, fet característic de lestransicions atèrmiques (vegeu més avall). ∆S T obtingut en funció del nombre delcicle (vegeu per exemple la Fig. 14 per v2) ens revela tres trets principals. Primer,

10.1 T/min apareix com el ritme òptim entre obtenir una senyal gran (que és major com majorés H) i un nombre de punts sucient (considerant que l'adquisió de dades es fa a 0.25 Hz)

217

RESUM

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

6

7

8

9

10

11

12

13

14

15

16

17

T=55 K

/ H creix. (0.1 / 1 T/min)

H decr. (1 T/min) mostra v2

x=0.05

|∆S

| (J/

kgK

)

n cicles

0 1 2 3 4 5

-300

-200

-100

0

0.1 T/min

H creixent

cicle 1 cicle 3 cicle 7 cicle 10

mostra v2x=0.05 T=55 K

dQ/d

H (

mJ/

T)

µ0H (T)

1.7 1.8 1.9 2.0190

200

210

220

230

240

250

1 T/min

H decreixent

dQ

/dH

(m

J/T

)µ

0H (T)

Figura 14: Quadre superior: corbes DSCT (H) amb H creixent a 0.1 T/min perdiferents cicles a la mostra v2. La gràca inserta és un detall del pic en les corbesDSCT (H) amb H decreixent a 1 T/min per diferents cicles a la mostra v2. Quadreinferior: Variació d'entropia calculada de les corbes DSCT (H) (H creixent i de-creixent a ritmes diferents) a la mostra v2, en funció del cicle. Les línies contínuesserveixen només de guia.

218


∆S T creix fortament els quatre primers cicles, se suavitza i s'acaba saturant abansdel desè cicle. Això s'entén considerant que els estats inicials i nals varien ambels cicles degut a l'evolució del desordre. Quan el camí del sistema a través dela transició de primer ordre es torna reproduïble, com ocorre en aquestes mostres,llavors ∆S T assoleix un valor constant. En aquesta evolució, la fase de baix camps'ha desordenat i/o la d'alt camp s'ha ordenat. El segon tret és que ∆S T no depènd' H, ns i tot variant aquest un factor 10, indicant que s'assoleixen els mateixosestats inicials i nals amb qualsevol H. El tercer és que els valors obtinguts ambcamp creixent són majors que per camp decreixent. Aquest efecte és degut a quela histèresi en la transició fa que aquesta passi en zones diferents del diagrama defases i per tant entre estats diferents.

AllausEl darrer estudi dut a terme en aquestes mostres és l'anàlisi de les allaus, associ-ades a la nucleació i creixença dels dominis de la nova fase que tenen lloc durantla transició de primer ordre induïda per camp.

Les transicions de primer ordre poden ser activades tèrmicament o bé ser atèr-miques. En les primeres, la relaxació des d'un estat metastable pot ocórrer so-ta condicions externes constants degut a les uctuacions tèrmiques, mentre queen les transicions atèrmiques, ocorre només sota el canvi d'un paràmetre extern(camp magnètic, tensió, temperatura, etc.), que modica la diferència de l'energialliure entre les dues fases [76, 77]. Quan un sistema es fa transitar externament peruna transició de primer ordre, salta d'una conguració amb un mínim a l'energialliure a una altra, un cop s'ha assolit el límit d'estabilitat. El camí que segueixel sistema depèn de la presència de desordre (dislocacions, vacants, fronteres degra...) que controla la distribució de barreres d'energia entre les dues fases. Men-tre el sistema transita, passa a través dels estats metastables d'energia mínima ambsalts discontinus o allaus del paràmetre d'ordre, amb una dissipació d'energia queés responsable de l'histèresi observada en aquestes transicions [77]. En el casatèrmic, el camí es reprodueix d'un cicle al següent sempre que el desordre noevolucioni [78].

De les corbes DSCT (H) obtingudes amb camp creixent i decreixent, podemavaluar la fracció transformada com

y(H) =1L

∫ H

Hi

dQdH dH , (26)

on L =∫ H f

Hi(dQ/dH)dH és la calor latent, i Hi i H f són camps per sobre i per

sota dels camps de transició inicials i nals, respectivament. Això ens dóna elscicles d'histèresi. Per quanticar els salts presents en la transició de primer ordre

219

RESUM

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

350

λ

N cicles

0 10 20 30 40 50 60 70 80 900.5

0.6

0.7

0.8

0.9

1.0

α

N cicles

Figura 15: Paràmetre λ obtingut de la distribució d'allaus fent servir la fracciótransformada de la mostra v3, en funció del cicle. El cicle 31 està tret de la mostrav2 i el cicle 89 de la mostra v1. Figura inserta: exponent α obtingut de la mateixadistribució d'allaus. Les línies contínues són una guia.

(estructura de pics) de les mostres amb x=0.05 (vegeu Figs. 6 i 14) s'ha calculatla diferència entre dos valors d'y(H) consecutius, ∆y, sent una mesura del tamanyde les allaus que pot variar des de 0 (no hi ha allaus en el període de temps entredues mesures) ns a 1 (tota la mostra transita en una sola allau). La distribuciódel tamany de les allaus es pot analitzar estadísticament fent servir la següentdistribució de probabilitats amb dos paràmetres lliures (λ i α) [79, 80]:

p(∆y) =e−λ∆y(∆y)−α∫ ∆ymax

∆ymine−λ∆y(∆y)−αd(∆y)

. (27)

Per λ=0, la distribució és una llei de potències, mentre que és subcrítica per λ>0i supercrítica per λ<0 [81]. ∆ymin=10−4 és un valor just per sobre del nivell desoroll intrínsec de les mesures de DSC, avaluat considerant els valors de ∆y forade la regió amb estructura de pics. ∆ymax=1 és el valor màxim que es pot prendre.

L'exponent α i el paràmetre λ s'han estimat utilitzant el mètode de màximaversemblança [82]. Aquest mètode és el més apropiat ja que no implica el càlculd'histogrames, que faria que hi hagués dependència del resultat amb la mida delsintèrvals que s'agafa. Els resultats per la mostra v3 (que s'ha ciclat sistemàtica-ment a 0.1 T/min en tots els cicles, a diferència de les altres dues mostres, ques'han ciclat a diferents ritmes en els primers cicles) estan representats a la Fig. 15.S'observa que el paràmetre λ tendeix a decréixer amb els cicles, mentre que l'ex-

220

Conclusions

ponent α es manté constant (0.71±0.05). S'han afegit els valors de l'últim ciclede les mostres v2 (nº31) i v1 (nº89), doncs aquests cicles han estat fets al mateixritme de camp que en la mostra v3. λ i α han estat també avaluats a partir de lescorbes DSCT (H) representades a la Fig. 6 de la mostra amb x=0.05 provinent dela mateixa mostra original que v1, v2 i v3. El nombre de cicles en aquesta mostrano ha estat controlat, però es pot estimar en ∼15-25. En aquest cas, α=0.73±0.05i λ=209±30, en acord amb els resultats previs.

L'evolució de λ indica que el sistema passa d'una distribució subcrítica cap auna distribució de llei de potències (on el sistema no té un tamany d'allau preferen-cial en transitar), tot i que el valor λ=0 no s'assoleix en el cicle nº89. L'exponentcaracterístic per la llei de potències, α, presenta un valor (=0.71±0.05) que no de-pèn ni de l'evolució del sistema amb el ciclat ni tampoc de la mostra. L'evoluciódels paràmetres és consistent amb les observacions prèvies: quan el sistema trobaun camí òptim per transitar, tant la distribució d'allaus com la variació d'entropiatendeixen a un comportament constant.

Conclusions• S'han preparat amb èxit mostres massives de Gd5(SixGe1−x)4 amb un forn

d'arc no comercial. Les tècniques de caracterització emprades (magnetitza-ció, susceptibilitat ac, DSC, DRX, MER i microsonda electrònica) indiquenque s'obté la fase principal 5:4 amb la x desitjada. També s'observa la pre-sència de fases secundàries (1:1 i 5:3) i fases 5:4 residuals amb valors de xdiferent del nominal. Els tractaments tèrmics afavoreixen la desaparició deles fases 5:4 amb x diferent, però també la segregació de les fases 1:1 i 5:3 japresents. Un compromís entre els dos efectes s'assoleix amb un tractamenta 920 ºC durant 4-9 hores en un forn sota buit de 10−5 mb.

• S'ha desenvolupat un nou calorímetre diferencial de ux (DSC). L'equip téuna alta sensibilitat i opera sota camp magnètic aplicat de ns a 5 T i en unrang de temperatura d'entre 10 i 300 K. L'aparell estudia transicions de fasesòlid-sòlid de primer ordre en presència de camps magnètics. El calorímetrepermet una acurada determinació de la variació d'entropia associada a unatransició de fase magnetoestructural en aliatges que presenten EMC gegant,transició que pot ser induïda tant variant H com T . Per tant, aquest tipusde mesures han d'aclarir la discussió sobre el valor real de la variació del'entropia a l'entorn d'una transició de primer ordre.

• S'ha demostrat que l'EMC provinent d'una variació de camp en un sistemaamb una transició de fase de primer ordre induïda per camp es pot avaluarcorrectament amb la variació d'entropia calculada a partir de la relació de

221

RESUM

Maxwell, ns i tot quan la transició és ideal. Quan s'avalua amb la relacióde Maxwell considerant tot el rang de camps, la dependència amb T i Hde la magnetització en cada fase fora de la regió de transició produeix unavariació d'entropia addicional a la intrínseca de la transició de primer ordre.També s'ha demostrat, amb dades experimentals i models fenomenològics,que la relació de Maxwell, l'equació de Clausius-Clapeyron i les mesurescalorimètriques donen la variació d'entropia propia de la transició de primerordre, ∆S , sempre que (i) la relació de Maxwell s'avaluï només dins delrang de camps on té lloc la transició, i (ii) el camp màxim aplicat sigui prouintens com per completar la transició. La temperatura de transició s'ha demoure de manera signicativa amb el camp, per tal d'aprotar ∆S i obtenirun EMC gran.

• El DSC sota camp s'ha utilitzat amb èxit per mesurar ∆S pròpia de la transi-ció magnetoestructural de primer ordre en els aliatges Gd5(SixGe1−x)4, ambx ≤ 0.5. S'ha mostrat que ∆S escala amb Tt. L'escalat de ∆S és una conse-qüència directa del fet que Tt varia amb x i H i s'espera que sigui universalper tot material que mostri efectes magnetoelàstics importants, amb unatransició induïda per camp. S'espera que ∆S (i) vagi a zero a temperaturanul·la, (ii) tendeixi assimptòticament a zero a altes temperatures, doncs lacalor latent és nita, i (iii) tingui un màxim a la temperatura en què ∆M esmaximitza i Tt mostra la mínima dependència amb el camp. La forma espe-cíca de ∆S en funció de Tt dependrà del diagrama de fases, Tt(x). L'escalatde ∆S ensenya per tant l'equivalència dels efectes de magnetovolum i elsde substitució en els aliatges Gd5(SixGe1−x)4.

• La variació del camp de transició amb la temperatura de transició, dHt/dTt,s'ha estudiat en tot el rang de composicions dels Gd5(SixGe1−x)4 on apareixla transició de primer ordre, x ≤ 0.5. Tenint en compte el comportamentde dHt/dTt en funció de x i que ∆M decreix monòtonament amb Tt, esveu que dHt/dTt controla l'escalat de ∆S amb Tt, conrmant l'evidènciaque l'origen de l'escalat és el caràcter magnetoelàstic de la transició. Amés, es troben dos comportaments diferents de dHt/dTt en els dos rangs decomposició on hi ha la transició magnetoestructural, mostrant la diferènciaen l'intensitat de l'acoblament magnetoelàstic d'aquest sistema.

• Es descriu una transició de fase induïda per camp ns ara no menciona-da, des de la fase AFM a una fase que presenta correlacions de curt abast(SRAFM). Els resultats suggereixen que la transició apareix per un tren-cament de les correlacions AFM de llarg abast, donant lloc a correlaci-ons AFM i FM de curt abast en competència. Les correlacions FM també

222

Conclusions

són importants al llarg de la fase AFM. La transició esperada entre la fa-se SRAFM i la PM té lloc a ∼240 K a camp zero, i s'eixampla i suavitzaamb camp aplicat. Aquesta troballa ajuda a entendre el complex comporta-ment magnètic en els aliatges Gd5(SixGe1−x)4, que prové de la competicióentre les interaccions FM (dins de les capes que formen la seva estructuramicroscòpica) i AFM (entre capes consecutives).

• L'estudi de la dinàmica de la transició de primer ordre induïda per campen els aliatges Gd5(SixGe1−x)4 revela comportaments molt interesants. Peruna banda, el DSC sota camp aplicat mostra que ∆S associat a la transicióés diferent si aquest s'indueix per camp o per temparatura, evidenciant queels estats inicials i nals són diferents perquè la transició no és ideal. Peraltra banda, un estudi ciclant mostres verges a través de la transició ensenyaque ∆S induïda per camp creix durant els primers cicles, assolint un valorestacionari. Aquest comportament es relaciona amb la distribució d'allaus,que també evoluciona amb el ciclat. L'estructura d'allaus es torna repeti-tiva després d'uns pocs cicles, revelant el caràcter atèrmic de la transició itendint cap a una distribució de llei de potències.

223

RESUM

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