Methods for the analysis of thermomagnetic phase transitions of magnetocaloric materials

Methods for the Analysis of Thermomagnetic Phase Transitions of Magnetocaloric Materials PhD Thesis Programa de doctorado en Ciencia y Tecnolog´ıa de Nuevos Materiales Luis M. Moreno Ram´ırez Supervisors: Prof. Alejandro Conde Amiano Prof. Victorino Franco Garc´ıa Departamento de F´ısica de la Materia Condensada Universidad de Sevilla A Carmen, Paqui y Miriam Acknowledgements A mis directores Alejandro y Victorino por todo el esfuerzo puesto en mi formaci´on y su trato amable y paciente. Me siento afortunado de haber podido aprender de ellos. Al grupo de S´olidos no Cristalinos por su ayuda durante estos a˜nos. A Javier por ofrecerse a discutir cualquier cuesti´on. A Clara, Pepa y Jhon por su trato siempre cari˜noso. A Jia Yan por todos los buenos ratos en el laboratorio. Al grupo de estudiantes por el buen ambiente: Alejandro, Emanuel, Antonio, Pablo, ´ Alvaro, Jos´e Antonio y Antonio. Quisiera extender el agradecimiento a Carlos, David y Miriam por todas las colaboraciones realizadas. To all the Functional Materials group of the Technical University of Darmstadt for their kind hospitality and fruitful collaboration. To Prof. Gutfleisch for giving me the opportunity to be part of his group. To Dr. Skokov and Dr. Radulov for their scientific guidance. To Fernando for all the support and camaraderie. To Bernd, Leo, Ahmed, Lukas Sch., Lukas Pf. and Juan for all the good moments there. A mi familia y amigos por su sacrificio, apoyo y cari˜no. A Carmen y Paqui, sin las cuales no habr´ıa llegado a este momento. A Miriam por apoyarme en todo momento. Al Ministerio de Educaci´on y Formaci´on Profesional del Gobierno de Espa˜na por financiar esta tesis y las respectivas estancias de investigaci´on a trav´es de las ayudas FPU. Contents Abstract 1 Nomenclature 3 1 Introduction 7 1.1 Magnetocaloric effect . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Thermodynamic functions . . . . . . . . . . . . . . . . . . . 13 1.3 Magnetocaloric materials . . . . . . . . . . . . . . . . . . . 17 1.4 Conventional models and methods . . . . . . . . . . . . . . 23 1.5 Objectives and structure of the thesis . . . . . . . . . . . . 32 2 Experimental techniques 35 2.1 Inductionmelting........................ 36 2.2 Arc melting/Suction casting . . . . . . . . . . . . . . . . . . 36 2.3 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Scanning electron microscopy . . . . . . . . . . . . . . . . . 37 2.5 Vibrating sample magnetometry . . . . . . . . . . . . . . . 38 2.6 Relaxation calorimetry . . . . . . . . . . . . . . . . . . . . . 40 2.7 Direct adiabatic temperature change measurement . . . . . 41 3 Influence of experimental factors on the magnetocaloric effect 43 3.1 Samplegeometry........................ 44 3.2 Compositional inhomogeneities . . . . . . . . . . . . . . . . 51 3.3 Low-temperature limitation of calorimetric measurements . 56 3.4 Measurement protocols for direct adiabatic temperature change.............................. 74 3.5 Conclusions........................... 80 4 Analyses based on the field dependence of the magnetocaloric effect 83 4.1 Curie temperature . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . 92 4.3 Order of the phase transition and critical composition . . . 99 4.4 Conclusions...........................104 5 Studies of Gd-based materials 107 5.1 Gd-GdZn composites . . . . . . . . . . . . . . . . . . . . . . 108 5.2 Gd/Ti multilayers . . . . . . . . . . . . . . . . . . . . . . . 118 5.3 Conclusions...........................126 6 Studies of La(Fe,Si)13-based materials 129 6.1 Ni-dopedseries .........................130 6.2 Cr-dopedseries.........................144 6.3 Conclusions...........................156 7 General conclusions 159 Publication list 163 Bibliography 165 palabra 1 Introduction Magnetic refrigeration at room temperature is an emerging green cooling technology that presents several advantages compared to the conventional gas-based technology. This new technology is based on the magnetocaloric effect which exploits the temperature changes of magnetic materials by magnetic field variations. This chapter briefly describes the basic principles of the magnetocaloric effect, magnetocaloric materials and thermomagnetic phase transitions as well as the conventional models and methods to characterize them. Finally, the objectives and structure of the thesis are also presented. 7 1.1. Magnetocaloric effect 1.1 Magnetocaloric effect The magnetocaloric (MC) effect [1–7] is defined as the reversible temperature change produced in a magnetic material upon a magnetic field change under adiabatic conditions. Based on this definition, the MC effect can be characterized by the induced temperature change, which is known as the adiabatic temperature change (∆TS). This effect was discovered by P. Weiss and A. Picard in 1917 [8], whereby a temperature change of 0.7 K was observed for a nickel sample upon the application of a magnetic field of 1.5 T at temperatures close to its ferro-paramagnetic phase transition (630 K). Some scientific publications attribute the discovery of the MC effect to E. Warburg in 1881 [9] though the temperature change reported is ascribed, in fact, to the energy losses associated to the magnetic hysteresis [10]. In 1921, P. Weiss established a physical description for the MC effect [11] using the different constituent subsystems of a magnetic material, mainly the magnetic and lattice ones. For the magnetic subsystem, the magnetic moments associated to the atoms are considered while for the lattice subsystem, the spacial distribution of those atoms around an equilibrium configuration is considered. These two different subsystems were well studied separately [12]. Based on that, the entropy (S), an extensive magnitude, can be considered as the sum of the entropy contributions of the different subsystems. Considering only the magnetic and lattice subsystems: S=SM+SL,(1.1) where SMand SLare the entropies of the magnetic and lattice subsystems, respectively. For a more detailed analysis, different contributions such as the electronic or coupling terms could be included for consideration [2]. Considering the magnetic material under adiabatic conditions when varying the magnetic field from an initial to a final value (HIand HF, respectively, being ∆H=HF−HI), the total entropy remains constant 8 1. Introduction (∆S(∆H) = S(HF)−S(HI) = 0). Thus, using Eq. (1.1) becomes: ∆SM(∆H)+∆SL(∆H)=0.(1.2) From Eq. (1.2), it can be observed that the magnetic entropy change (∆SM) and the lattice entropy change (∆SL) are compensated. The different entropy changes are not null when the magnetic field modifies the magnetic/structural state of the material. The implication of Eq. (1.2) can be understood by considering a conventional magnetic material (e.g. ferromagnetic or paramagnetic) for which its structure is not modified by magnetic fields. Typically, the magnetic field aligns the magnetic moments along the direction of the applied magnetic field, reducing the magnetic entropy as the number of configurational states is reduced (∆SM<0 when applying a magnetic field). This ∆SMis compensated by a ∆SL (according to Eq. (1.2) ∆SL>0). Lattice entropy change leads to a temperature (T) change in the system as SLmainly depends on the temperature (it was mention that magnetic field does not modify the structure of the material). As the lattice entropy increases with increasing temperature, ∆TS>0. When the application of a magnetic field induces ∆TS>0, the effect is termed as direct MC effect. For a general case, ∆TScan be expressed mathematically as: ∆TS(S, ∆H) = [T(S, HF)−T(S, HI)]S,(1.3) where the index Sdenotes that the difference is at a constant entropy value. It can be observed that the MC effect is induced from the coupling between the lattice and magnetic subsystems of a magnetic material. As the entropy of the system is not an easy experimental parameter to work with, ∆TSis typically expressed as a function of the initial temperature at which the magnetic field is varied. Alternatively, under isothermal conditions when varying the magnetic field, a variation of the entropy of the system (known as the isothermal 9 1.1. Magnetocaloric effect entropy change, ∆ST) is induced. This entropy change leads to a heat exchange between the material and the thermal reservoir. From Eq. (1.1) it can be derived that: ∆ST(T, ∆H)=∆SM(T, ∆H)+∆SL(T, ∆H).(1.4) For a general case, ∆STcan be expressed mathematically as: ∆ST(T, ∆H) = [S(T, HF)−S(T, HI)]T,(1.5) where the index Tdenotes that the difference is at a constant temperature. Considering the previous case of a conventional magnetic material, as SL is a function of temperature, ∆STcoincides with ∆SMaccording to Eq. (1.4) since ∆SL(T, ∆H) = 0. It has to be noted that this is not the case when structural changes are produced by magnetic field. This will be further discussed under Section 1.3. Fig. 1.1 shows ∆STand ∆TSat a particular temperature T0by the application of a magnetic field for a conventional magnetic material using a schematic S−Tdiagram. isothermal H >0 S T ( T 0 , H ) S (a.u.) T (a.u.) T S ( T 0 , H ) H =0 Direct MC effect adiabatic T 0 Figure 1.1: Schematic of MC magnitudes, ∆TSand ∆ST, from SH(T) curves upon the application of a magnetic field for a conventional magnetic material, according to Eqs. (1.5) and (1.3), respectively. 10 1. Introduction ∆STand ∆TSrepresent the two characteristic magnitudes of the MC effect as function of the initial temperature and the magnetic field change. The reversibility of the MC effect as defined depends on the reversibility of the various processes performed. Moreover, when magnetic field is applied/removed under non-adiabatic or non-isothermal conditions, the system would show an entropy change or temperature change smaller than those under adiabatic or isothermal processes. It has to be noted that ∆STis an extensive magnitude (depends on the size of the system) and therefore, it is more useful for the material characterization to use its specific magnitude (∆sT). The interest in the MC effect since the 80s has been mainly ascribed to its potential application in solid state magnetic refrigerators at room temperature [13–16]. It is a solid state cooling technology as the working material is a solid, unlike the traditional system based on gas/liquid. The changes in entropy and temperature produced upon application/removal of a magnetic field enable a cycled magnetic refrigeration system [17]. In general, the refrigerator absorbs heat from a certain region (cold load) and rejects heat to exterior (hot reservoir). In the case of a magnetic refrigerator, it consists of a MC material (exhibiting a large MC effect) as the working material, a variable magnetic field generator and a heat transfer fluid system (gas or liquid) between the working material and the different reservoirs. It operates using the combination of various thermodynamic processes, such as adiabatic, isothermal and isofield types. The principal cycles are the Carnot (adiabatic and isothermal processes), Ericsson (isothermal and isofield processes) and Brayton (adiabatic and isofield processes) cycles [18]. Moreover, there is the AMR (Active Magnetic Regenerator) cycle, for which the MC material functions both as a refrigerant and regenerator (it has been shown that it is the most efficient cycle for magnetic refrigeration at room temperature after the Carnot cycle) [19]. Fig. 1.2 schematically shows the basic operation of a magnetic refrigerator using a Brayton cycle. The different steps are illustrated: (a) the MC material is submitted to a magnetic field under adiabatic conditions, leading to an increase of its temperature, 11 1.1. Magnetocaloric effect (b) the working material exchanges heat with the hot reservoir reducing its temperature (isofield process), (c) the magnetic field is removed and the material temperature decreases and (d) the resulted cooled material absorbs heat from the cold load until recovers its initial temperature (isofield process). These processes continuously cycle during the operation of the refrigerator. Figure 1.2: Schematic of a magnetic refrigerator operating with a Brayton cycle. Image taken from [20]. Magnetic refrigerators have been shown as more energy efficient than the conventional ones, wherein the best commercial refrigerators reach up to efficiency of 40 % while magnetic refrigerator prototypes can attain up to 60 % (an enhancement of 50 %) [14]. In countries like United States, around 60 % of the energy consumption corresponds to energy losses and, considering that 40 % of the total energy consumption arises from temperature control, an increment in the energy efficiency of refrigerators would lead to great benefits, especially for the economy 12 1. Introduction and environment [21]. Moreover, the energy consumption rate due to temperature control will notably increase since developing countries mainly coincide with the hottest geographical regions (e.g. Africa, China, India). In addition, this cooling technology, compared to the conventional one, does not require harmful and toxic gases, such as CFCs (ozone depletion), NH3, HCFCs and HFCs (greenhouse effect) [22]. Nowadays, there are already companies working on the production of commercial magnetic refrigerators. One example would be the commercial wine cooler based on the MC effect, which was presented by HAIER in 2015 (in collaboration with Astronautics of America, BASF and TU Delft) [23]. Besides the interest on magnetic refrigeration, MC materials are also investigated for medical applications [24–26] or energy harvesting devices [17,27,28]. 1.2 Thermodynamic functions The MC effect can be related to certain macroscopic variables of the materials, such as magnetization (M) and heat capacity (C) (or their specific magnitudes, σand c, respectively) through simple thermodynamic relations [2,29]. By basic thermodynamics, it is well-known that a certain magnitude can be expressed in the equilibrium state, for single component systems, as a function of three independent variables, the so-called state variables. Temperature, magnetic field and pressure (p) are the more favorable state variables as they can be easily controlled experimentally. Therefore, the specific entropy can be expressed as s=s(T, H, p), having a differential form as: ds=∂s ∂T H,p dT+∂s ∂(µ0H)T,p d(µ0H) + ∂s ∂pH,T dp, (1.6) where the magnetic permeability of vacuum (µ0) is included. For the study of the MC effect, generally the pressure is kept constant, so dp= 0 (to simplify the notation, the index pis subsequently removed from Eq. (1.6)). 13 1.2. Thermodynamic functions Considering isothermal conditions (i.e. dT= 0) when applying/removing a magnetic field, it can be obtained that: dsT=∂s ∂(µ0H)T µ0dH. (1.7) On the other hand, considering adiabatic conditions (i.e. ds= 0) when applying/removing a magnetic field, Eq. (1.6) becomes: dTS=−∂s ∂T H−1∂s ∂(µ0H)T µ0dH. (1.8) According to Eqs. (1.7) and (1.8), the different derivatives of the entropy (with respect to temperature and magnetic field) have to be expressed as functions of different magnitudes. To do that, cat constant magnetic field is considered: cH=¯dq ∂T H ,(1.9) where ¯d points that the specific heat flux (q) is not a state function. Assuming quasistatic conditions for the heat flux: ¯dq=Tds, (1.10) and combining with Eq. (1.9): ∂s ∂T H =cH T.(1.11) On the other hand, to express the derivative of the entropy with respect to the magnetic field, Maxwell relations have to be considered. These thermodynamic relations arise from the equality of the second derivatives of a certain thermodynamic potential. As Tand Hare the variables, the most useful thermodynamic potential is the Gibbs free energy (G), taking a differential form as: dG=−sdT−σd(µ0H).(1.12) 14 1. Introduction The Maxwell relation for Gleads to: ∂s ∂(µ0H)T =∂σ ∂T H .(1.13) Using Eqs. (1.11) and (1.13), in Eqs. (1.7) and (1.8): dsT=∂σ ∂T H µ0dH, (1.14) dTS=−T cH∂σ ∂T H µ0dH. (1.15) To obtain the entropy and temperature changes produced by a finite magnetic field change, the integration of Eqs. (1.14) and (1.15) have to be performed. For ∆sT, the integration of Eq. (1.14) leads to: ∆sT=µ0ZHF HI∂σ ∂T H dH. (1.16) In the case of ∆TS, the integration is not as trivial as for ∆sT. It can be observed that Eq. (1.15) is a transcendental equation (both cHand σ are functions of Tand H) and a separation of variables is not possible. Assuming that in a temperature interval equals to ∆TS,cHand σare temperature-independent: ZT+∆TS T dT T≈ −µ0ZHF HI 1 cH∂σ ∂T H dH, (1.17) which it can be expressed as: ∆TS≈Texp −µ0ZHF HI 1 cH∂σ ∂T H dH−1.(1.18) Assuming that ∆TS/T << 1, the exponential function can be approximated up to first order, having that: ∆TS≈ −µ0ZHF HI T cH∂σ ∂T H dH. (1.19) 15 1.3. Magnetocaloric materials the effects, which is by the shift of the transition temperature with magnetic field: for direct MC effect, the transition temperature shifts to higher temperatures while for the inverse, it shifts to lower temperatures. Although it is shown that FOPT MC materials have a relatively large response (larger than SOPT MC materials), it is accompained by thermal and magnetic hysteresis, which is susceptible to rate dependence and a decreased cyclic performance. On the other hand, SOPT materials do not exhibit any thermal hysteresis. One goal of MC material research is to combine the pros from both types of materials: large response without associated hysteresis [65–68]. With this goal, many have attempted to develop series of alloys and compounds that can bring us to the intermediate point between two different order of phase transitions (also known as the critical SOPT-FOPT point). Some of the most studied families of FOPT MC materials (e.g. La(Fe,Si)13 [47,69] and MnFe(P,Si) [70] compounds) show a gradual change from FOPT to SOPT by means of composition modification or element additions (e.g. for La(Fe,Si)13, the critical SOPT-FOPT composition was found for a 12.7 at.% of Si content [71]). One of the most promising family of MC materials are La(Fe,Si)13 compounds. They are based on abundant, non-critical and non-contaminant elements. The NaZn13-type structure presented in these compounds (Si incorporation is needed to stabilize the phase) shows a temperature-induced FOPT ferro-paramagnetic transition (as well as field-induced para-ferromagnetic transition) around 200 K (dependent on the Si content). The origin of this magnetic field-dependent transition is ascribed to the itinerant-electron metamagnetism of these compounds [72, 73]. The corresponding FOPT leads to a MC effect with moderate thermal hysteresis (compared to other FOPT MC materials, such as FeRh or Heusler alloys) and with associated volume change of around 1% without structural change (magneto-elastic transition). The transition temperature can be shifted to temperatures closer to room temperature by hydrogenation, whereby the NaZn13-type unit cell expands, facilitating the 22 1. Introduction applications of the material [44]. Still, they exhibit some issues impeding commercialization [74], such as material degradation under cycling [75–77], accompanying hysteresis [78,79], mechanical stability [80–82], etc. One of the aims of this thesis is to use the proposed analysis to explore different optimization routes of MC materials that are based on benchmark Gd-(SOPT) and La(Fe,Si)13-(FOPT) systems. For the former, it is focused on attaining improved RCP and field responsiveness while the latter is dedicated to increasing the FOPT-character or reducing the accompanying hysteresis. 1.4 Conventional models and methods It is already shown the close link between MC effect and thermomagnetic phase transitions. Hence, MC studies allow us to obtain information about the nature of the phase transition. The conventional methods and models used in literature for such purposes (also used for the various analyses in this thesis) will be discussed in the following sections. 1.4.1 Arrott-Noakes equation of state In the critical region of a SOPT (i.e. T≈TCand H≈0, being TC the Curie temperature), the different magnitudes can be expressed as a power law of certain variables, where the exponents of the power laws are called critical exponents [83,84]. These critical exponents depend only on the nature of the phase transition. The origin of this critical behavior can be understood by the meanings of the fluctuations of the systems. A relatively simple analysis was performed by using mean field theories (e.g. the Landau theory) giving a satisfactory explanation (in a qualitatively manner). For a detailed analysis (beyond mean field theories), a more sophisticated mathematical formalism is needed (such as the Wilson’s 23 1.4. Conventional models and methods renormalization group theory). These analyses go beyond the scope of this thesis. In the case of σ, for T < TCand H= 0, it can be expressed in the critical region as: σ(H= 0) ∝(TC−T)β,(1.24) while for T=TC: σ(T=TC)∝H1/δ,(1.25) where βand δare critical exponents. In the case of the magnetic susceptibility (χ) for T > TCand H= 0, it can be expressed as: χ(H= 0) ∝(T−TC)γ,(1.26) being γthe associated critical exponent. These critical exponents are related by: δ−1 = γ β.(1.27) In a general form, it can be established that the magnetization scales at the critical region as: H σδ=fst σ1/β ,(1.28) where fsis a scaling function and t= (T−TC)/TC. Combining Eqs. (1.24), (1.25) and (1.26), A. Arrott and J.E. Noakes proposed the followed phenomenological equation of state (EOS) for Ni close its transition temperature [85]: H σ1/γ =a(T−TC) + bσ1/β,(1.29) where aand bare two introduced parameters. It has to be noted that Eqs. (1.24), (1.25) and (1.26) can be recovered from Eq. (1.29). The Arrott-Noakes (A-N) EOS has been succesfully used to describe the MC response of different SOPT MC materials apart from Ni [86–89]. To 24 1. Introduction determine the A-N parameters, a 3D-fitting was proposed by V. Franco in [87]. This method proposes the temperature as the fitting variable to avoid multivalued solutions, which could complicate the procedure. The A-N EOS was used to numerically describe the MC behavior of the SOPT MC materials studied in this thesis. 1.4.2 Kouvel-Fisher method The Kouvel–Fisher (K-F) method is an iterative procedure to determine the critical exponents of the materials proposed by J.S. Kouvel and M.E. Fisher in 1964 [84]. The proposed method is based on the magnetic behavior close to the critical region (Eqs. (1.24) and (1.25)). For the first step in the method, σT(H) data, using an initial set of critical exponents, are represented in a modified Arrott plot (which represents (H/σ)1/γ vs. σ1/β at a certain temperature). From this plot, the σ(H= 0) and the inverse of the magnetic susceptibility at zero field (χ−1(H= 0)) are obtained from the intercepts of each axis. After that, the obtained σ(H= 0) and χ−1(H= 0) values (as a function of the temperature) are used to obtain a new set of βand γcritical exponents from the slope of the Yand Xaxes vs. temperature plots, respectively, being: Y=σ(H= 0) ∂σ(H= 0) ∂T −1 =1 β(T−TC),(1.30) X=χ−1(H= 0) ∂χ −1(H= 0) ∂T −1 =1 γ(T−TC).(1.31) With the obtained set of critical exponents, the sequence is repeated (starting from the modified Arrott plot) until the obtained critical exponents converge to constant values (which are the final values for the critical exponents). This method is not straightforward and requires time-consuming and careful data analysis. Apart from βand γ,TCis also obtained as the average intercept of both curves in the temperature 25 1.4. Conventional models and methods axis (ideally, they should be the same). These limitations, on top of K-F technique limited to single phase SOPT materials, are overcome by a newly proposed method as described in Chapter 4. 1.4.3 Scaling laws for the magnetocaloric effect Near to the critical region, it is possible to extend the scaling laws to the MC effect [90–92] due to the strong correlation with thermomagnetic phase transitions. To obtain the MC relations, the different scaling relations for the magnetization can be expressed as (among others): σ |t|β=f(0) s,±H |t|∆,(1.32) where ∆ = βγ and f(0) s,±is a scaling function (where + and −signs correspond to the regions of t < 0 and t > 0, respectively). Introducing Eq. (1.32) into Eq. (1.16) and assuming a zero initial magnetic field: ∆sT=±∆s(0) T|t|1−αZ∆H/|t|∆ 0βfs,±(%)−∆∂fs,±(%) ∂% %d%, (1.33) where ∆s(0) Tis a constant, 1 −α=β+ ∆ −1 and %=H/|t|∆. Although f(0) s,±is unknown, it can be observed that the integral of Eq. (1.33) is also a function of %, having: ∆sT=|t|1−αf(1) s,±∆H |t|∆,(1.34) where f(1) s,±is a new defined function. After some algebra, Eq. (1.34) can be converted to: ∆sT ∆H(1−α)/∆=f(2) s,±|t| ∆H1/∆,(1.35) where f(2) s,±is a new defined function. Eq. (1.35) expresses the scaling law for ∆sTin the critical region, which allows much information of the nature of the MC properties to be obtained from SOPT MC materials. As 26 1. Introduction an example, scaling laws of typical ∆sTcurves of amorphous alloys using A-N EOS with Heisenberg critical exponents are illustrated in Fig. 1.6. It can be observed that ∆sTcurves for different magnetic field changes (A) collapse onto a single curve (B). 100 200 300 400 -0.20 -0.15 -0.10 -0.05 0.00 -2 -1 0 1 2 -5 -4 -3 -2 -1 0 0.2 0.4 0.6 0.8 1.0 (B) s T (J kg -1 K -1 ) T (K) T C =300 K, =0.367, =1.388 (A) s T /( 0 H ) (1- )/ (10 2 J kg -1 K -1 T -(1- )/ ) ( T - T C )/( 0 H ) 1/ (10 2 K T -1/ ) 0 H (T) Figure 1.6: Temperature dependence of ∆sTusing A-N EOS with Heisenberg critical exponents and typical parameters for amorphous alloys (A) and scaled curve according to Eq. (1.35) (B). Using Eq. (1.35), it is possible to examine the magnetic field dependence for SOPT MC materials. For T=TC(t= 0), it can be observed that: ∆sT(T=TC)∝∆H(1−α)/∆,(1.36) i.e. ∆sT(T=TC) follows a power law with an exponent that is a function of different critical exponents. This leads to the definition of the field dependence exponent, n, as: n(T, ∆H) = ∂ln ∆sT(T, ∆H) ∂ln ∆H.(1.37) 27 1.4. Conventional models and methods Combining Eqs. (1.36) and (1.37): n(T=TC,∆H) = (1 −α)/∆.(1.38) For non-zero values of t, a certain value of reference temperature (Tr) can be chosen to quantify the width of ∆sTfor a certain ∆H, i.e.: ∆sT(Tr,∆H) = ∆s(1) T·∆sT(T=TC,∆H),(1.39) where ∆s(1) Tis a constant value (normally it is chosen between 0.7 and 0.5 to ensure that Tris chosen near the critical region). Trcorresponds to the temperature for which ∆sTcorresponds to that fraction of the peak. Using Eqs. (1.35) and (1.39), it can be observed that for trTr−TC TC: tr∝∆H1/∆,(1.40) i.e. the width of ∆sTfollows a power law with an exponent that is the inverse of the critical exponent ∆. It has to be noted that, as the different branches for t > 0 or t < 0 are specified in Eq. (1.32), one tris selected for each branch (following the same notation, tr+and tr−, respectively). One important feature is that by determining the field dependences of ∆sT(T=TC) and tr, it is possible to determine the critical exponents of the transitions using Eqs. (1.36) and (1.39) (having a two-equation system with two variables, αand ∆). Once these critical exponents are obtained, other exponents can be determined using different relations. This analysis can be extended to other MC magnitudes, such as ∆spk T, its corresponding temperature (T∆sT pk ) or RCP. Table 1.1 shows some of the critical exponents of the power field dependence for different MC magnitudes [92]. Moreover, according to Eqs. (1.36) and (1.39) with the determination of ∆sT(T=TC) (or ∆spk Tdue to the same field dependence) and tr, a phenomenological universal curve can be constructed without previous assumption or calculus to determine the critical exponents. This phenomenological curve has been widely studied and applied for several 28 1. Introduction SOPT MC materials, finding an excellent agreement between experiments and theory [93–97]. Magnitude Exponent ∆spk T(1 −α)/∆ = 1 + 1/δ(1 −1/β) ∆sT(T=TC) (1 −α)/∆ = 1 + 1/δ(1 −1/β) RCP 1+1/δ tr,±1/∆ T∆sT pk −TC1/∆ (0 for mean field) Table 1.1: Exponents of the power field dependence for different MC magnitudes. Based on these scaling laws, different procedures to determine TCand the critical exponents for SOPT materials were developed in this thesis, which are further found to be field-independent for TCas well as applicable for studying the critical exponents of biphasic systems (which are not possible by the conventional methods). These new methods will be shown in Chapter 4. Furthermore, these analyses, previously not used for studying FOPT MC materials (these are not based on critical exponents), will be shown applicable to identify the phase transition order and the critical SOPT-FOPT (Section 4.3). 1.4.4 Bean-Rodbell model In 1962, C.P. Bean and D.S. Rodbell proposed a model to study the ferro-paramagnetic FOPT with an associated volume change (≈2.5 %) in MnAs [98]. From this, they assumed that magnetic interactions are related to the interatomic distances of the material, proposing a phenomenological dependence for the transition temperature as: Ttrans =TC(1 + Ψ∆v),(1.41) where ∆vis the relative volume change and Ψ is a parameter to express the dependence of the transition temperature with the deformations. 29 1.4. Conventional models and methods Considering Gfor a mean field approach: G=−µ0HM −λM2+p∆v+1 2K∆v2−Ts, (1.42) where Kis the compressibility. The first term arises due to external field, the second to the exchange interaction, the third to the external pressure, the forth to the lattice distortion and the last to the entropy contribution. Both Eqs. (1.41) and (1.42) are related by the exchange constant: λ=3kBTtrans J(J+ 1)Nm(gµB)2µ0 ,(1.43) where Jis the total moment quantum number, Nmis the density of magnetic moments, gis the spectroscopic splitting factor and kBis the Boltzmann constant. According to basic thermodynamics, the equilibrium values of the different magnitudes are those that minimize G. Combining Eqs. (1.42) and (1.43), the relative volume change when Gis minimal is expressed as: ∆v=3 2 J J+ 1NmkBTCKΨM2.(1.44) From Eq. (1.44), it can be observed that ∆vis obtained as a function of M. Introducing Eq. (1.44) into Eq. (1.42), the magnetization when G is minimal is expressed as: M=NmgµBJBµ0gµBJ kBT(H+λ(1)M+λ(3)M3),(1.45) where Bis the Brillouin function and: λ(1) =3kBTC J(J+ 1)Nm(gµB)2µ0 ,(1.46) λ(3) =3 10 2J2+ 2J+ 1 J2(J2+ 1)(NmgµB)2λ0η, (1.47) 30 1. Introduction being η: η=5J(J+ 1) 2J2+ 2J+ 1NmkBTCKΨ2.(1.48) Fig. 1.7 shows the temperature dependence of normalized magnetization calculated using Bean-Rodbell (B-R) model with different ηvalues. 0.8 0.9 1.0 1.1 1.2 0.0 0.2 0.4 0.6 0.8 1.0 T / T C M / M 0 K 0.0 0.5 1.0 1.5 2.0 Figure 1.7: Temperature dependence of normalized magnetization calculated using B-R model with different ηvalues (calculated using the corresponding parameters of Gd). The ηparameter controls the order of the transition, being η > 1 for FOPT (an abrupt Mvariation with Tfrom the ferro- to para-magnetic state is obtained) and η < 1 for SOPT (smooth Mvariation with Tis obtained). In the case of η= 1, it corresponds to the critical SOPT-FOPT point. The B-R model has been successfully applied to describing the MC properties of different materials apart from MnAs as well as Gd5Si2Ge2or La(Fe,Si)13 compounds [99–103]. The magnetoelastic FOPT of the MC compounds was numerically modeled using this model and compared to experimental, which are presented in Chapters 4 and 6. 31 2.5. Vibrating sample magnetometry that the electron beam is degraded, the chamber has to be in high vacuum. Electrons are strongly attenuated inside the sample via elastic or inelastic collisions (the attenuation coefficient depends on the material) and are able to penetrate distances of around µm within the sample (depending on the accelerating voltage). The different products of the interactions between the beam-sample from which information is obtained are: secondary electrons, backscattered electrons (BSE) and characteristic X-rays. BSE are those that collide elastically with the sample, strongly changing the direction of movement (more probable as the atomic number increases). Secondary electrons are those electrons which are knocked out from the atoms and emitted due to inelastic collisions. These electrons have low energy and only escape from the sample if the process has taken place very close to the surface, being strongly dependent on the topography of the sample. With this, X-ray emission is also produced, which is characteristic of the atoms of the sample. From this X-ray emission, by using the energy dispersive espectroscopy (EDX) technique, it is possible to obtain information of the different atoms (and their proportion) on the surface sample. A Philips XL30 FEG scanning electron microscope equipped with energy-dispersive X ray system (SEM/EDX) was used. 2.5 Vibrating sample magnetometry A vibrating sample magnetometer is an instrument that measures the magnetic moment of a sample under an uniform applied magnetic field. The operating principle is based on the Faraday’s law. The magnetic sample is made to oscillate at a fixed frequency which leads to a change in the magnetic flux through a set of detection coils in the surroundings. This flux variation induces an AC electromotive force in the coil that is directly proportional to the magnetic moment of the sample. The signal is amplified by a lock-in amplifier, which rejects all the signals with 38 2. Experimental techniques different frequencies to the sample oscillation, increasing the sensitivity in the measurement. Magnetic moment measurements were performed using a LakeShore 7407 (with oven and cryostat options, in which magnetic measurements were performed from liquid nitrogen temperature up to 1273 K) and in a Quantum Design Physical Properties Measurement System with vibrating sample magnetometer option. ∆sTis determined experimentally from specific moment measurements (as a function of temperature and magnetic field, σ(T, H)) following the discrete form of equation Eq. (1.16): ∆sTTi+1 +Ti 2, HN−H1=µ0 2× N−1 X j=1 σ(Ti+1, Hj+1)−σ(Ti, Hj+1) Ti+1 −Ti +σ(Ti+1, Hj)−σ(Ti, Hj) Ti+1 −Ti× (Hj+1 −Hj),(2.2) where the indexes iand jrepresent different values of Tand H, respectively, Nis the number of measurements and HNand H1are equal to HFand HI, respectively. Eq. (2.2) can be used with either isothermal magnetization curves measured at different temperatures or temperature dependent magnetization curves measured at different applied fields, though the latter is less time-consuming than the former (it is faster to stabilize the magnetic field than the temperature). For SOPT MC materials, both ways can be applied without main drawbacks while for FOPT cases, their measurements should be more tedious as spikes are found when applying Eq. (2.2) to isothermal magnetization curves. This is ascribed to the coexistence of phases during the transition [108, 109]. Hence, it is relevant to account for the effects of partial transformations during each measurement by erasing the thermal and field histories to obtain 39 2.6. Relaxation calorimetry experimental data with physical meaning [110]. In this thesis, this was performed by cooling/heating the sample well below/above its transition with a simultaneous application of an appropriate value of the magnetic field (depending on the particular phase to be stabilized) prior to the measurements at each desired temperature (denoted as discontinuous protocols). Table 2.1 shows a summary of the different existing discontinuous measurements protocols accounting for heating and cooling processes for samples with transition temperatures increasing/decreasing with field (Ttrans/∆H > 0 and Ttrans/∆H < 0, respectively). Ttrans/∆H > 0 (e.g. La(Fe,Si)13) Cooling Heating Set T >> Ttrans at HISet T << Ttrans at HF Set desired TSet desired T Measure MT(HI→HF) Measure MT(HF→HI) Repeat for other TRepeat for other T Ttrans/∆H < 0 (e.g. Ni-Mn-In Heusler alloys) Cooling Heating Set T >> Ttrans at HFSet T << Ttrans at HI Set desired TSet desired T Measure MT(HF→HI) Measure MT(HI→HF) Repeat for other TRepeat for other T Table 2.1: Summary of the different discontinuous measurement protocols assuming that HF> HI. 2.6 Relaxation calorimetry A relaxation calorimeter measures the heat capacity of a sample by controlling the heat while monitoring the resulting change in temperature. The used system consists of a sample platform, in which a heater and a thermometer are attached to its bottom part. Both are electrically connected by four small wires, which also provide the thermal connection and structural support for the platform. Typically, the sample (mass 40 2. Experimental techniques around 1-100 mg) is mounted to the platform using a thin layer of grease (Apiezon N) to ensure a good thermal contact between the sample and the platform. A thermodynamic analysis for the setup leads to a system of two equations with many variables apart from the heat capacity of the sample, for example the heat capacity of the platform and the thermal conductivities of the wires and grease. Some of the parameters can be determined from the addenda measurements such as the heat capacity of the platform. If it is assumed that the heat capacity is kept constant during the heat pulse, it can be obtained (among the rest of parameters as e.g. thermal conductivity of the wires) from the fitting of the temperature evolution to the solution of the two-tau model [111]. For SOPT, the approximation can be applied with good results. In the case of a FOPT, that assumption is not true in the region close to the transition temperature, as a divergence in the heat capacity appears. For this case, fitting procedures cannot be applied, the equation system has to be solved point by point, and in this case, the parameters have to be previously determined. The different assumptions used for determining the different parameters can drastically affect the measurements [112,113]. Heat capacity measurements by the relaxation method were performed using the heat capacity option of a commercial Quantum Design Physical Property Measurement System. 2.7 Direct adiabatic temperature change measurement Direct ∆TSmeasurements were performed using a modified MagEq MMS 902 manufactured by AMT&C Corporation. The sample holder was maintained in vacuum during measurements and samples were thermally isolated to ensure good adiabatic conditions. The magnetic field generator is composed of two concentric Hallbach cylinders, which produce a maximum magnetic field of 1.76 T. The temperature change of the sample 41 2.7. Direct adiabatic temperature change measurement produced by the application/removal of a magnetic field was registered by a Type T thermocouple (with its reference located in the sample holder), while the temperature of the sample holder was controlled by a LakeShore temperature controller. The various measurement protocols on erasing the sample’s prior history were automated using our in-house implemented software with appropriate PID parameters for the temperature controller (to avoid thermal oscillations around a certain temperature). Fig. 2.1 illustrates both continuous (without accounting for the sample’s history) and discontinuous measurement protocols. T 3 T 2 Time (a.u.) Sample holder T (a.u.) T reset T 1 continuous discontinuous Figure 2.1: Time dependence of the sample holder temperature illustrating continuous and discontinuous measurement protocols. 42 3 Influence of experimental factors on the magnetocaloric effect The influence of different experimental factors on the MC response will be covered in this chapter. Both experimental measurements and numerical calculations were used. For macroscopic and microscopic characteristics, the effects of sample geometry and compositional inhomogeneities on ∆sTand its field dependence exponent were analyzed. For calculating the MC magnitudes from heat capacity measurements, the influence of the low temperature limitation on the MC responses was studied. Finally, the influence of continuous and discontinuous protocols on direct ∆TS measurements was discussed. 43 3.1. Sample geometry 3.1 Sample geometry When performing magnetic measurements of a magnetic material, the sample geometry has to be considered to determine the internal magnetic field [114]. This is because the magnetized sample generates magnetic poles at its edges which then create a magnetic field (called demagnetizing field (HD)) that opposes magnetization. Hence, the magnetic field within the material under an applied magnetic field (Happ) is expressed as: H=Happ +HD.(3.1) For ellipsoidal geometries where Mis uniform, HDand Mare related through the demagnetizing factor (ND) as: HD=−NDM, (3.2) where the negative sign indicates that HDhas an opposite direction to M(NDis defined with positive sign). This implies that the internal field is smaller than the one applied. Table 3.1 shows different demagnetizing factors of some typical geometries for certain directions (which can be approximated to the sample geometries studied in this thesis). Generally, the demagnetizing factor is a tensor magnitude (as it relates two vector magnitudes) with unit trace. To obtain the approximate values of the internal field, even for non-ellipsoidal geometries (cylinders, squares...), an effective demagnetizing factor can be used. Sample geometry Direction ND Needle Along the long axis 0 Perpendicular to the long axis 1/2 Sphere Any direction 1/3 Thin film In plane 0 Perpendicular to the plane 1 Table 3.1: NDof typical ideal sample geometries magnetized along certain directions. 44 3. Influence of experimental factors on the magnetocaloric effect For the isothermal magnetic entropy change from magnetization measurements, the demagnetizing factor has to be considered in order to properly calculate Eq. (1.16) [115–117]. For experimental validation, an example of SOPT MC material was selected, whose analysis will be more straightforward than those for FOPT cases. Besides typical SOPT MC material, Gd, amorphous magnetic alloys are well-studied for MC effect. In addition, their negligible magnetocrystalline anisotropy make them an ideal candidate to study the shape anisotropy. For that, amorphous Co62Nb6Zr2B30 materials (known for their excellent soft magnetic behavior [118,119]) were used for investigating the effects of ND on their MC responses [120]. They were synthetized using two alternatives techniques: rapid quenching (RQ) and mechanical alloying (MA) [121]. More information on their preparation can be found in Refs. [122,123]. These two ways of amorphizing the same composition had been reported with differences in their macroscopic and microscopic structures [121]. Macroscopically, MA leads to powder samples while RQ techniques produce micron-thick ribbons or wires. To perform magnetic measurements, the ribbon can be mounted with the applied field contained in its plane, to present a negligible demagnetizing factor (ND≈0) while this is not possible for powder samples, for which the effect of demagnetizing field is unavoidable. For preparing the samples for magnetic characterization, the MA powders (whose geometries are close to spheres) were encapsulated in an Ag capsule, while the ribbons were cut into disks. Fig. 3.1 shows, as solid symbols, the temperature dependence of ∆sT (A) and exponent n(B) for RQ and MA Co62Nb6Zr2B30 considering the applied magnetic field change for 1 T. ∆sTmagnitude was indirectly determined from magnetization curves according to Eq. (1.16) while the field dependence exponent nwas obtained from the calculated ∆sTdata according to Eq. (1.37). Despite that the samples have the same nominal composition, a clear difference in TCfor RQ and MA Co62Nb6Zr2B30 can be observed (RQ alloy exhibits lower TCthan that of MA). This is ascribed to a better homogenization of boron in RQ Co62Nb6Zr2B30 [122]. 45 3.1. Sample geometry Apart from this, exponent nof the MA Co62Nb6Zr2B30 determined when considering the applied magnetic field (solid symbols) shows values deviating from n= 1 for T << TCin the ferromagnetic region, which is in contrast with the theoretical predictions (for a pure ferromagnetic material, independent of the composition and magnetic field, nshould be 1 for T << TCand not exceed it) [92]. -0.3 -0.2 -0.1 0.0 100 200 300 400 500 600 700 0.8 1.2 1.6 2.0 (B) s T (J kg -1 K -1 ) (A) n T (K) RQ MA n =1 Figure 3.1: Temperature dependence of ∆sT(A) and exponent n(B) for both RQ and MA Co62Nb6Zr2B30 considering the applied magnetic field change for 1 T. From the way of preparing the samples for characterization, NDfor RQ ribbon can be assumed to be ND≈0, while ND≈1/3 for MA powder. Using these NDvalues, the internal field can be determined and was used for calculating ∆sT. These results are presented in Fig. 3.2. 46 3. Influence of experimental factors on the magnetocaloric effect 400 500 600 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 400 500 600 0.8 1.0 1.2 1.4 1.6 1.8 w/o N D N D =1/3 T (K) (B) s T (J kg -1 K -1 ) (A) MA n T (K) n =1 1 T Figure 3.2: Temperature dependence of ∆sT(A) and exponent n(B) for MA Co62Nb6Zr2B30 considering the applied (solids symbols) or internal magnetic field change (using ND= 1/3, hollow symbols) for 1 T. It can be observed that there is no significant effect on the ∆sT(T) curves (∼5% reduction in magnitude when determined using the internal field) but it eliminates the earlier observed deviations of nat T << TC, showing the typical field independent behavior predicted by the theory. It has to be noted that in the paramagnetic range, the influence of the demagnetizing factor is negligible (magnetization values are smaller). The corresponding values of n= 2 (as predicted by the theory) are achieved, either using the internal or the applied field (though Fig. 3.1 does not further show the temperature range to observe that nsaturates at 2 ). For expressing the magnetic field dependence, one should note that the magnetic field corresponds to internal field when NDis being used, while it is the applied field if NDis not considered. Fig. 3.3 shows the field dependence of the exponent nat temperatures well below the transition (ferromagnetic region) using the internal (i.e. considering the effect of ND) and the applied fields. 47 3.2. Compositional inhomogeneities 100 150 200 250 300 s T (a.u.) T (K) =25 K =150 K T C =208.6 K 1 T Figure 3.7: Simulated ∆STcurves weighted according to a Gaussian distribution for different TCvalues. 100 150 200 250 300 s T (a.u.) T (K) 0 5 10 20 30 40 50 (K) =150 K T C =208.6 K 1 T Figure 3.8: ∆sTgenerated for different standard deviation values of a Gaussian distribution of TC. Using A-N EOS, the MC behavior of MA alloys with distributed TC was simulated. This EOS allows to accurate fit experimental data in a 54 3. Influence of experimental factors on the magnetocaloric effect temperature range of 100 K near TCfor amorphous alloys with TCclose to room temperature [87]. Thus, for our studies, the Gaussian range was limited to Ω = 150 K, accounting for the previous-mentioned limitation. Contributions near the limit of the range must be assumed negligible to minimize the effect of a finite Gaussian range, which limits the ωrange for exploration. For that, simulated ωdependence of n(T∆sT pk ) for different Ω values shows a saturated deviation of n(T∆sT pk ) for ω > Ω/2.5 (compared to the typical n(T∆sT pk ) vs. ωcurve). Therefore, the reliable simulated values of n(T∆sT pk ) were limited to ω≤60 K for our studies. In addition, field dependence exponent nwas determined from the MC simulations for different standard deviations of the Gaussian TC distribution. Fig. 3.9 shows the experimental and simulated n(T) for both MA and RQ alloys in its main panel. 0 20 40 60 0.7 0.8 0.9 -200 -100 0 100 200 0.5 1.0 1.5 2.0 n T - T s T pk (K) RQ Arrott-Noakes fit (single T C ) MA Arrott-Noakes fit ( T C distribution with =30 K) n ( T s T pk ) (K) Figure 3.9: Experimental and simulated n(T) curves for MA and RQ Co62Nb6Zr2B30. Inset: ω-dependence of n(T∆sT pk ) for the simulated Gaussian distribution of TC. 55 3.3. Low-temperature limitation of calorimetric measurements As a TCshift with respect to RQ ribbons results from poor B incorporation in the amorphous phase, a rescaled temperature axis was used for comparing the behaviors of MA and RQ samples. The ω-dependence of n(T∆sT pk ) is plotted in the inset of Fig. 3.9, wherein n(T∆sT pk ) increases with the standard deviations. A maximum ωvalue of approximately 60 K is used. Hence, the simulated n(T) of MA Co62Nb6Zr2B30 was based on the assumption of a Gaussian Curie temperature distribution with ω= 30 K, which exhibits the best agreement among our analysis. This value of ωcan be related with an approximate 1 at.% B variation. Moreover, the simulated curves reproduce the main features of the experimental data with fairly good agreement: a larger n(T∆sT pk ) value and sharper bend towards the paramagnetic region were observed for RQ ribbons than those for MA powder (due to the distributed TC). Additionally, it is also reproduced that the minimum of n(T) shifts to temperatures below the mean value of TCin the presence of a distribution of TC. 3.3 Low-temperature limitation of calorimetric measurements For the MC determination using heat capacity measurements, first of all, the total entropy of the system has to be calculated according to Eq. (1.21). This requires the knowledge of the zero entropy term and the heat capacity measurements to be extended down to 0 K (to perform the integration). On the one hand, the zero entropy term can be assumed small in a condensed system and field independent [133], having a negligible influence on the total entropy. On the other hand, as temperatures close to 0 K cannot be reached, the calculus of the integral can be considerably affected. This can be ascribed to the limitations of the experimental setup (e.g. liquid N2is used as the coolant) or to the difficulties in measuring the specific heat for a large temperature range (e.g. limited stability in the grease that is typically used for sample mounting to the calorimetric chip). For such cases, some assumptions must be made for the missing 56 3. Influence of experimental factors on the magnetocaloric effect data range. The most commonly used approximation is to assume a linear behavior of cH/T down to 0 K and, therefore, an approximate expression for the entropy is proposed [29]: sap H(T) = 1 2cH(Tini) + ZT Tini cH TdT, (3.5) where Tini is the lowest achievable temperature in cHmeasurements. According to this, the approximated values for both MC magnitudes (∆sap T and ∆Tap S) can be calculated from Eq. (3.5) using Eqs. (1.5) and (1.3). In this section, the influence of this linear cH/T assumption (from Tini down to 0 K) on the MC response is discussed [134–136]. For that, numerical calculations for single phase and multiphasic MC materials were performed. Their results were also further corroborated by experimental work using paradigmatic SOPT and FOPT MC materials: Gd and Gd5Si2Ge2, respectively [134]. To numerically reproduce a single phase MC material, firstly the total heat capacity of the system is considered as the sum of the magnetic, lattice and electronic contributions: C=CL+CE+CM,(3.6) where CL,CEand CMare the heat capacity contributions of the lattice, electronic and magnetic subsystems, respectively. For the lattice and electronic contributions, the Debye and Fermi models have been used, respectively [137]. For the magnetic contribution, the B-R model was used to describe both FOPT and SOPT MC materials [98]. The different heat capacity contributions are calculated as: CL= 9NpkBT ΘDZΘD/T 0 z4ez (ez−1)2dz, (3.7) CE≈τT, (3.8) 57 3.3. Low-temperature limitation of calorimetric measurements CM=NmKBT ∂ lnsinh 2J+1 2J$ sinh 1 2J$−$B($)!/∂T!H ,(3.9) where Npis the number of particles, ΘDis the Debye temperature, τis a constant and $was defined as (Section 1.4.4): $=µ0gµBJ kBTH+λ(1)M+λ(3)M3.(3.10) Gd parameters were used for the simulations (TC= 293 K, g= 2, J= 7/2, ΘD= 163 K, τ= 6.4 mJ mol−1K−2). The simulated specific heat for different magnetic fields for both SOPT (η= 0) and FOPT (η= 1.5) MC materials are shown in Fig. 3.10. 0 100 200 300 400 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 100 200 300 400 0 1 2 3 4 (B) c H (10 2 J kg -1 K -1 ) T (K) =0 (A) c H (10 3 J kg -1 K -1 ) T (K) 0.0 T 2.0 T 5.0 T 7.5 T 10.0 T Debye =1.5 Figure 3.10: Temperature dependence of calculated specific heat for different magnetic fields for η= 0 (SOPT) (A) and 1.5 (FOPT) (B). Lattice contributions are plotted as dashed lines. Only the lattice contribution is plotted separately (dashed line) as the electronic contribution is of too low values to be plotted. It can be observed that the main features of the specific heat for both SOPT and FOPT MC materials are reproduced using the described models: a divergence and a 58 3. Influence of experimental factors on the magnetocaloric effect discontinuity close to Ttrans are obtained for FOPT and SOPT, respetively. Using these data, the total entropy was calculated according to Eq. 3.5, where the data were truncated from different initial temperatures. With the obtained approximated entropy curves, the MC magnitudes were calculated using Eqs. (1.3) and (1.5). For SOPT MC material, its simulated temperature dependence of ∆sap T for 2 T for various initial temperatures using η= 0 is plotted in Fig. 3.11. It is observed that each curve differs from that integrated from 0 K (this is used as the reference) although the shape is preserved. The deviations follow a non-monotonous trend with increasing Tini (e.g. the curve starting from 125 K has a larger deviation than those starting at 75 and 200 K). To quantify this, the differences of the peak values between each ∆sap Tcurve (∆sap,pk T) and that of the reference as a function of Tini are plotted in the inset of Fig. 3.11. For simplicity, these differences (used either for ∆sTor ∆TS) are denoted as Φ from now. 0 100 200 300 400 500 -8 -6 -4 -2 0 0 100 200 -1.5 -1.0 -0.5 0.0 0.5 s ap T (J kg -1 K -1 ) T (K) 0 50 100 150 200 250 =0, 2 T T ini (K) ( s ap,pk T ) (J kg -1 K -1 ) T ini (K) Figure 3.11: Numerical temperature dependence of ∆sap Tfor 2 T and different Tini using η= 0. Inset: Φ differences of ∆sap,pk Tas a function of Tini.Topt ini values are marked with arrows. 59 3.3. Low-temperature limitation of calorimetric measurements There are three cases showing zero Φ difference. Besides a flat region close to 0 K, there are two values of the initial temperature for which the ∆sap Tvalues coincide with those of the complete curve (marked with arrows in the inset of Fig. 3.11). These marked Tini values are denoted as optimal initial temperatures (Topt ini ). For practical purposes, the most useful Topt ini would be the highest, to have an experimental temperature range as narrow as possible. Hence, the largest Topt ini was further used for the remaining studies. Considering the expression in Eq. (3.5), the approximated entropy can be related with the entropy starting from 0 K as: sH=ZT 0 sH TdT=sap H−1 2cH(Tini)+ZTini 0 cH TdT=sap H+ξH(Tini),(3.11) where ξ=−1 2cH(Tini) + RTini 0 cH TdTdepends on Tini and H. With this relation, the deviations of ∆sap Tfrom ∆sTcan be discussed. For a zero initial magnetic field, ∆sTbecomes: ∆sT(T, H) = sH(T)−s0(T) =sap H(T) + ξH(Tini)−(sap 0(T) + ξ0(Tini)) = ∆sap T(T, H)+(ξH(Tini)−ξ0(Tini)),(3.12) where it can be observed that Φ(∆sap T) is: ξH(Tini)−ξ0(Tini).(3.13) It should be noted that this term is temperature independent. Topt ini corresponds to the temperatures that fulfill ξH(Tini) = ξ0(Tini). Fig. 3.12 shows the temperature dependence of Φ(∆sap T) for different Tini. It can be observed that the differences are 0 for Topt ini . A similar analysis was performed for ∆TS. Fig. 3.13 shows the simulated temperature dependence of ∆Tap Sfor different values of Tini in the main panel and Φ(∆Tap,pk S) as a function of Tini in the inset. 60 3. Influence of experimental factors on the magnetocaloric effect 0 100 200 300 400 500 -1.5 -1.0 -0.5 0.0 0.5 =0, 2 T 250 K ( s ap T ) (J kg -1 K -1 ) T (K) 150 K T ini = T opt ini Figure 3.12: Temperature dependence of Φ(∆sap T) for different initial temperatures. The dashed line represents this difference as a function of Tini in a continuous curve. 0 100 200 300 400 500 -2 0 2 4 6 8 10 0 100 200 -1 0 1 2 =0, 2 T T ap S (K) T (K) 0 50 100 150 200 250 T ini (K) ( T ap,pk S ) (K) T ini (K) Figure 3.13: Numerical temperature dependence of ∆Tap Sfor 2 T and different Tini using η= 0. Inset: Φ differences of ∆Tap,pk Sas a function of Tini.Topt ini values are marked with arrows. 61 3.3. Low-temperature limitation of calorimetric measurements It is observed that the different Tini values give ∆Tap Scurves of different peak values and shapes (specially in the high temperature region). As for the difference of each curve to the reference (integrated from 0 K), two Tini with zero difference are observed and marked by the arrows. These observations are similar to those found for delta ∆sap T. Furthermore, the values obtained for the optimal Topt ini are the same as those found for ∆sap T. For ∆TS, the observed deviations were further analyzed. Assuming a certain temperature T∗and defining s∗=s0(T∗) and sap,∗=sap 0(T∗), the expressions for ∆TSand ∆Tap Sare: ∆TS(T∗, H) = TH(s∗)−T0(s∗) (3.14) and ∆Tap S(T∗, H) = Tap H(sap,∗)−Tap 0(sap,∗),(3.15) respectively. The relation between s∗and sap,∗, according to Eq. (3.11), is: sap,∗=s∗−ξ0.(3.16) According to Eq. (3.5), the approximated and integrated from 0 K temperature functions for any magnetic field are related by: Tap H(sap =sH−ξH) = TH(sH).(3.17) Incorporating Eq. (3.16) into Eq. (3.14): ∆TS(T∗) = TH(sap,∗+ξ0)−T0(sap,∗+ξ0),(3.18) that when introducing Eq. (3.17) it gives: ∆TS(T∗) = Tap H(sap,∗+ξ0−ξH)−T0(sap,∗).(3.19) Therefore, when ξH=ξ0(i.e. when Tini =Topt ini ), Eq. (3.19) becomes: ∆TS(T∗)=∆Tap S(T∗).(3.20) 62 3. Influence of experimental factors on the magnetocaloric effect The same condition is also obtained for ∆sT, demonstrating that the optimal initial temperatures are the same for both MC magnitudes. It should be noted that ∆sap Tand ∆Tap Sdeviations show different signs, i.e. when ∆sap Tis underestimated, ∆Tap Sis overestimated and vice versa. Unlike for ∆sap Tdata, the deviations of ∆Tap Sare temperature dependent. For the case of Tini 6=Topt ini ,Tap Hcan be expanded around sap,∗up to first order term as: ∆TS(T∗) = Tap H(sap,∗) + ∂Tap H ∂sap sap,∗ (ξ0−ξH)−T0(sap,∗) ∼ =∆Tap S(T∗)−T∗ cH(T∗)(ξ0−ξH),(3.21) where it was used that for small (ξ0−ξH) values: ∂Tap H ∂sap sap,∗ =∂TH ∂s sap,∗+ξH =∂TH ∂s s∗+ξH−ξ0 ∼ =∂TH ∂s s∗ =T∗ cH(T∗). (3.22) According to this, when Tini 6=Topt ini , ∆Tap Sdeviates with a T/cH(ξ0−ξH) trend. Fig. 3.14 shows the temperature dependence of Φ(∆Tap S) as a function of Tini which illustrates the previously discussed behavior. The field dependence of the deviations (Φ(∆sap,pk T)/∆sap,pk T) is presented in Fig. 3.15 (the earlier discussed studies were performed for a fixed magnetic field change). It is observed that the deviation is field dependent, though Topt ini only shifts slightly to higher temperatures with increasing field, at a rate of 2.1 K T−1for the parameters used. Selecting a single magnetic field of 1 T, the error using its Topt ini for other magnetic fields is relatively small, which is within the error margin of the experimental measurements (e.g. at 2 T is around 1 %). Using Tini values different from Topt ini , larger errors can be obtained (around 8 % at the peak) and anomalous behaviors in the temperature dependencies of ∆sap Tand ∆Tap Sare found (especially evident in the paramagnetic range). 63 3.3. Low-temperature limitation of calorimetric measurements -25 -20 -15 -10 -5 0 150 200 250 300 350 -2 -1 0 1 0 50 100 150 200 250 300 350 0 5 10 15 20 150 200 250 300 350 -1 0 1 2 3 s ap T (J kg -1 K -1 ) (A) ( s ap T ) (J kg -1 K -1 ) T (K) (B) Gd 5 Si 2 Ge 2 , 7.5 T T ap S (K) T (K) 0 150 237.2 250 T ini (K) ( T ap S ) (K) T (K) Figure 3.21: Temperature dependence of ∆sap T(A) and ∆Tap S(B) at 7.5 T for different initial temperatures for Gd5Si2Ge2. The insets show the temperature dependence of the Φ difference of each approximated curve with respect to the curve integrated from 4 K. Φ(∆sap,pk T) as a function of Tini is plotted in Fig. 3.22 for various magnetic fields. It is seen that the approximated MC curves deviate in a non-monotonic trend after the integration from 4 K. Despite the observed noise in the curves, it is clear to see the trend with Tini (the plotted lines are guides to the eye). The observed noise is less evident for Gd compared to Gd5Si2Ge2.Topt ini can be identified from these experimental data for both samples (i.e. both types of MC materials) as mentioned in the previous section. It can be also observed that the role of finite Tini is relatively less significant for Gd5Si2Ge2than for Gd, as was previously predicted by the simulations. 70 3. Influence of experimental factors on the magnetocaloric effect 0 50 100 150 200 250 -4 -3 -2 -1 0 1 2 3 4 0 50 100 150 200 250 -3 -2 -1 0 1 2 3 2.0 T 5.0 T 7.5 T 10.0 T ( s ap,pk T ) (J kg -1 K -1 ) T (K) Gd (A) (B) ( s ap,pk T ) (J kg -1 K -1 ) T (K) Gd 5 Si 2 Ge 2 Figure 3.22: Φ(∆sap,pk T) as a function of Tini for different applied magnetic field changes for Gd (A) and Gd5Si2Ge2(B). Lines are guides to the eye. For Gd5Si2Ge2, the curve corresponding to 2 T has a poor signal to noise ratio due to the relative lower importance of the approximation for FOPT and therefore it is not plotted. For simulations of multiphasic MC material, a non-interacting model for the heat capacity is assumed. Firstly, the simplest case is studied: a biphasic material. With this, the heat capacity of the composite is expressed as: C= (1 −x2)C1+x2C2,(3.23) being x2the phase fraction of phase 2 (secondary phase) and C1and C2 are the heat capacities of phases 1 and 2, respectively. The heat capacity of each phase is obtained as the sum of the contributions of the magnetic, lattice and electronic subsystems of that particular phase (as for single phase materials). The biphasic material is composed of two magnetic phases with different Curie temperatures (TC,1and TC,2for phases 1 and 2, respectively). Its simulations were performed by fixing TC,1to that of the benchmark MC material, Gd (i.e. 293 K), while TC,2was modified. The MC response was 71 3.3. Low-temperature limitation of calorimetric measurements determined using the simulated cHdata and further plotted in Fig. 3.23 as a function of temperature for different Tini values for 1 T. It can be observed that ∆sap Tcurves deviate from that corresponding to Tini = 0 K, whereby the deviations follow a temperature-independent trend. This is also previously observed for single phase case, which indicates that the deviations are not affected by the number of phases. Inset of Fig. 3.23 shows the Φ(∆sap,pk T) differences as a function of Tini, wherein an overall non-monotonic behavior is observed though accompanied by some features different from the case of single phase materials. 0 100 200 300 400 500 600 -5 -4 -3 -2 -1 0 0 100 200 300 -5 0 5 10 s ap T (J kg -1 K -1 ) T (K) 0 50 150 200 250 T ini (K) ( s ap,pk T ) (J kg -1 K -1 ) T ini (K) Figure 3.23: Temperature dependence of ∆sap Tsimulated for a biphasic material with x2= 0.5 and TC,2= 100 K using different Tini for 1 T. Inset: Φ differences at the peak temperature as function of Tini. There are two main temperature values for which zero deviations can be identified: one below TC,2and TC,1, and the other between TC,2and TC,1(both are marked by arrows in the figure). They are related to the Topt ini of each individual phase but not the exact value of the single phase, as it is influenced by the other phase. When Tini is close to TC,2, a discontinuity in Φ is observed at TC,2, implying that the presence of 72 3. Influence of experimental factors on the magnetocaloric effect extra phases at low temperatures can considerably affect the calculated MC response from heat capacity data. Nevertheless, it is important to note that the proposed method for experimentally identifying Topt ini is also applicable for multiphasic cases. A similar analysis was also performed for ∆TS, where the deviations from that integrated from 0 K can be observed to follow a temperature-dependent behavior. As a first approximation, the temperature dependence of these deviations is proportional to T/cH, which was earlier observed for the single phase case. The results show a similar curve shape as that found for ∆sap T(inset of Fig. 3.23), except for the different signs of the deviations (including two points of zero deviation and a discontinuity at TC,2). These results considering different phase fractions are shown in Fig. 3.24. 0 50 100 150 -30 -20 -10 0 10 0 150 300 450 600 -10 -5 0 5 x 2 =0.05 ( T ap,pk S ) (K) T ini (K) 0.50 0.25 0.10 0.05 0.01 x 2 1 T x 2 =0 ( T ap S ) (K) T (K) T ini =100 K Figure 3.24: Φ differences of ∆Tap,pk Sas function of Tini in the temperature region close to the phase transition of the secondary phase (various phase fractions are also considered). Inset: Comparison of the temperature dependence of ∆TSfor a pure phase (x2= 0) and for a system with a secondary phase fraction of 0.05 using Tini = 100 K (coinciding with TC,2). 73 3.4. Measurement protocols for direct adiabatic temperature change In all cases, the Tini dependence of Φ(∆Tap,pk S) shows larger deviations for temperatures above TC,2than for those at lower temperatures, resulting in an asymmetric shape. In addition, when the phase fraction of secondary phase increases, the deviations get more evident, indicating the stronger influence of the extra phase in the system. As an example, the temperature dependence of ∆Tap Sfor a pure phase (x2= 0) is compared to that of a system with a secondary phase fraction of x2= 0.05 in the inset of Fig. 3.24. For both cases, Tini = 100 K (which coincides with TC,2) was used. It is evident that this relatively small fraction of secondary phase led to a clear deviation when compared to the curve for x2= 0. This manifests the significance of accounting for the influence of secondary phases for MC calculations from calorimetric measurements. For the experimental application of the proposed method, Gd-GdZn MC system was selected for investigation (readers should refer to Section 5.1.3 for elaborated descriptions and discussions). 3.4 Measurement protocols for direct adiabatic temperature change For FOPT MC materials, it has been previously mentioned that temperature and field variations of the sample state are irreversible in the region close to their FOPT due to the intrinsic hysteresis associated to the phase transition. This leads to implications when calculating ∆sTfrom isothermal magnetization curves, for which artificial peaks arising from partial transformations or retainment of previous magnetic history states can be found when using continuous measurement protocols [75,108,110]. For direct measurements of ∆TS, the temperature change is measured as a function of the magnetic field and, therefore, the effects of partial transformations during each measurement can also affect these direct MC measurements (also in cyclic conditions [51, 139]). As Heusler alloys exhibit inverse and conventional MC effects, which arise from martensitic 74 3. Influence of experimental factors on the magnetocaloric effect transition (FOPT) and Curie transition of the austenite (SOPT), they were selected for the study of this section: the influence of various protocols to erase the sample’s memory for direct ∆TSmeasurements of SOPT and FOPT MC materials [140]. The Heusler compounds for study were nominally Ni49Mn36In15 and Ni49.5Mn35.5In15 (detailed information on their synthesis and structural characterization can be found in [141]). Their temperature dependences of σfor 0.05 T are shown in Fig. 3.25. It can be observed that they exhibit a martensitic (martensite to austenite) transition followed by a Curie transition of the austenitic phase at higher temperatures. 200 250 300 350 0 10 20 30 Ni 49 Mn 36 In 15 (A m 2 kg -1 ) T (K) Ni 49.5 Mn 35.5 In 15 5 mT Martensitic transition Curie transition Figure 3.25: Temperature dependence of σfor Ni49Mn36In15 and Ni49.5Mn35.5In15 measured at low field. Fig. 3.26 shows the temperature dependence of ∆TSmeasured while cooling for 1.76 T (at a sweep rate of 0.5 T s−1) with continuous and discontinuous measurement protocols for Ni49.5Mn35.5In15. The latter consists of erasing of the memory of the sample, which was performed by subjecting it to field heating up to a reset temperature of 350 K (this 75 3.4. Measurement protocols for direct adiabatic temperature change was selected as it is well above the martensitic transition ∼295 K) prior to measurements at each temperature. 270 280 290 300 310 320 330 340 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 280 300 320 340 -0.4 -0.2 0.0 0.2 T S (K) T (K) discontinuous continuous Cooling Ni 49.5 Mn 35.5 In 15 1.76 T ( T S ) (K) T (K) Figure 3.26: Temperature dependence of ∆TSfor Ni49.5Mn35.5In15 for 1.76 T measured using discontinuous (solid symbols) and continuous (hollow symbols) cooling protocols. Inset: Temperature dependence of Λ(∆TS). For a quick glance, no significant difference in the curves is observed in the studied temperature range (martensitic FOPT and ferro-paramagnetic SOPT) when erasing the history of the sample or not doing it. When taking the difference of both curves, defined as Λ(∆TS) (inset of Fig. 3.26), an observable difference is seen for the region close to the martensitic transition, however, the error bars prevent any further discussion. For Ni49Mn36In15, its temperature dependence of ∆TSfor 1.76 T measured while cooling (A) and heating (B) are presented in Fig. 3.27. In this case, the cooling protocol was performed by subjecting it to field heating up to a reset temperature of 350 K while for heating, a reset temperature of 200 K at zero field was used. In this case, significant differences between the continous and discontinuous measurement curves are clearly 76 3. Influence of experimental factors on the magnetocaloric effect observed (further magnified by Λ(∆TS) in the inset of Fig. 3.27). The MC response associated to the martensitic transition shows an underestimation of ∼20 % when not erasing the thermal and field history of the sample, unlike the case for the conventional MC effect. This is clearly depicted in the insets. -3 -2 -1 0 1 2 225 250 275 300 -1.2 -0.8 -0.4 0.0 0.4 220 240 260 280 300 320 340 360 -5 -4 -3 -2 -1 0 1 2 225 250 275 300 325 -1.2 -0.8 -0.4 0.0 0.4 (B) T S (K) discontinuous continuous (A) ( T S ) (K) T (K) T T S pk 1.76 T Cooling T S (K) T (K) Heating Ni 49 Mn 36 In 15 T T S pk ( T S ) (K) T (K) Figure 3.27: Temperature dependence of ∆TSfor Ni49Mn36In15 for 1.76 T measured using discontinuous (solid symbols) and continuous (hollow symbols) cooling (A) and heating (B) protocols. Insets: Temperature dependence of Λ(∆TS). To compare these results obtained for both compositions, it is important to note that sample masses and shapes used are quite similar. Hence, any 77 3.4. Measurement protocols for direct adiabatic temperature change observed differences in the data are related to the hysteretic behavior when magnetizing/demagnetizing the samples while the measurement procedure register the induced temperature change at a certain initial temperature of the sample. Fig. 3.28 shows the magnetization/demagnetization curves at selected temperatures close to the martensitic transitions of both samples. 0.0 0.5 1.0 1.5 0 20 40 60 80 (A m 2 kg -1 ) 245 K 300 K H (T) Figure 3.28: Magnetic field dependence of σfor Ni49Mn36In15 and Ni49.5Mn35.5In15 at temperatures close to their martensitic transition. The hysteresis can be quantified by the area enclosed between magnetization/demagnetization curves, being 50.2 and 9.6 A m2kg−1T−1 for Ni49Mn36In15 and Ni49.5Mn35.5In15, respectively. According to this, the former exhibits a larger magnetic hysteresis, which attributes to the huge Λ(∆TS) found when accounting for the differences between considering the sample’s history or not considering it. Moreover, the observed thermal hysteresis span between the cooling and heating curves is larger for Ni49Mn36In15 sample (≈12 K) than that for Ni49.5Mn35.5In15 (≈6 K), which agrees with the previous argument. In addition, the observed span of ≈6 K is equivalent to the selected temperature step used for the measurements (i.e. 5 K), hence, further measurements were 78 3. Influence of experimental factors on the magnetocaloric effect performed with a finer step resolution of 2.5 K. Using a finer temperature step resolution of 2.5 K, the influence of the sample’s prior history on its MC response becomes more significant near the martensitic transition, in particular for Ni49Mn36In15, as shown in Fig. 3.29. In this case, the cooling protocol was used with a reset temperature of 350 K and 1.76 T. The underestimation of ∆Tpk Sassociated to the martensitic transition increases to ∼40 % (a two-fold increase from that observed using 5 K). Moreover, it can be observed that the peak temperature shifts to higher temperatures when not accounting for the history of the sample. 210 220 230 240 250 260 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 220 230 240 250 260 -2.0 -1.5 -1.0 -0.5 0.0 0.5 T S (K) T (K) continuous discontinuous Cooling Ni 49 Mn 36 In 15 1.76 T ( T S ) (K) T (K) T T S pk Figure 3.29: Temperature dependence of ∆TSfor Ni49Mn36In15 for 1.76 T using discontinuous (solid symbols) and continuous (hollow symbols) cooling protocols. In contrast to ∆sT, wherein an overestimation is obtained from indirect measurements when the memory of the sample is not accounted for, the effect of using an inappropriate measuring protocol for direct ∆TSgives rise to a decrease in the measured response close to the martensitic transition. This difference is ascribed to the application of the Maxwell relation to different isothermal magnetization measurements 79 4.1. Curie temperature For a more general numerical investigation, Fig. 4.2 shows the field dependence of T∆sT pk and TCobtained after applying the method simulated upon applying the proposed procedure using Heisenberg (main panel) and mean field (inset) exponents. 0 1 2 3 4 5 290 300 310 320 330 340 350 Mean field T (K) 0 H (T) Heisenberg T s T pk T C from MC T C =300 K 0.00 0.25 0.50 0.75 1.00 299.95 300.00 300.05 0 H (T) T (K) Figure 4.2: T∆sT pk and TCobtained after applying the proposed method to the simulated data using Heisenberg (main panel) and mean field (inset) exponents. Using Heisenberg critical exponents, T∆sT pk varies with applied magnetic field and coincides with TCat H= 0, meanwhile TCobtained from the proposed method remains field independent and equal to TC. For the mean field case, Tpk ∆sTand TCfrom MC method are field independent and coincident with TCas shown in the inset of Fig. 4.2. This field independence is an advantage of the proposed method. The proposed procedure was applied to an amorphous alloy of nominal composition Fe62Cr12Cu1Nb3Si15.5B6.5prepared by melt spinning by Vacuumschmelze (details about structure and magnetic properties can be found in [142]). To apply the method, ∆sT(T, ∆H) and n(T, ∆H) 86 4. Analyses based on the field dependence of the magnetocaloric effect near TCwere calculated from magnetization measurements according to Eqs. (1.16) and (1.37), respectively. Gradual nvalues (Fig. 4.3), which monotonically increase from the minimum at values close TCup to 2 for T >> TC, according to their SOPT nature are observed [90]. The obtained T∆sT pk values from ∆sT(T, ∆H) vs. Tcurves are shown for 0.05 and 1.5 T, for which a clear evolution can be observed. It can be observed that up to 0.3 T the n(Tpk ∆sT) is not constant due to the multidomain regime of the sample (as despicted in the inset of Fig. 4.3). Magnetic fields below 0.3 T are not considered in the analysis to avoid erroneous values. 200 220 240 260 280 300 1.0 1.5 2.0 T s T pk (1.5 T) n T (K) T s T pk (0.05 T) 0.05 T 1.5 T 0.0 0.5 1.0 1.5 0.6 0.8 1.0 n ( T s T pk ) 0 H (T) Figure 4.3: Experimental temperature dependence of exponent nfor different magnetic field changes for Fe62Cr12Cu1Nb3Si15.5B6.5amorphous alloy. Inset: Field dependence of nT∆sT pk Using the above data, the proposed procedure to determine TCwas performed and the obtained values of T∆sT pk and TCare plotted against magnetic field values in Fig. 4.4. The same qualitative behavior for the simulated data using Heisenberg critical exponents is observed. With respect to the method proposed though T∆sT pk increases with the applied magnetic field, the calculated TCis field independent, with an average 87 4.1. Curie temperature value of 248.1(5) K. At low fields (below 0.3 T), the measurements do not have a good resolution and the exponent nvaries artificially. It is worth mentioning that this is not a limitation of the proposed method as the result is field independent and the value of TCcan be obtained at higher fields, wherein the signal to noise ratio is enhanced. 0.0 0.5 1.0 1.5 245 250 255 260 T C from MC T (K) 0 H (T) T s T pk 248.1(5) Figure 4.4: Field dependence of T∆sT pk and TCobtained from the proposed procedure for Fe62Cr12Cu1Nb3Si15.5B6.5amorphous alloy. In addition, TCwas further compared to the values obtained using conventional techniques: inflection point of MH(T) curves, Arrott plot and K-F methods. The first approach approximates TCby the inflection point of the magnetization vs. temperature curves (MH(T)). The obtained value is field dependent. The Arrott plot method is based on the assumption of mean field theory and TCapproximation comes from the temperature which corresponds to the isothermal straight line that crosses the origin of H/MT(H) vs. MT(H)2. It can be generalized for any set of critical exponents, thus also known as generalized Arrott plot. The main drawback of this technique is the required knowledge of critical exponents (typical mean field or Heisenberg values are used as the first 88 4. Analyses based on the field dependence of the magnetocaloric effect approximation) as any inappropriate value selection will significantly affect the TCdetermination thereafter. Finally, the K-F method employs an iterative procedure based on the previous modified Arrott plot method (explained in Chapter 1). Fig. 4.5 shows the TCvalues obtained using the inflection point method (A), while (B) presents the Arrott plot of the studied sample. 0.0 0.5 1.0 1.5 245 250 255 260 265 0 1 2 3 4 5 0.0 0.5 1.0 1.5 (B) T C (K) 0 H (T) (A) Inflection Arrot plot 2 (10 3 A 2 m 4 kg -2 ) 0 H / (10 2 T kg A -1 m -2 ) 230 K 270 K T C Figure 4.5: Field dependence of TCusing the inflection point method (A) and Arrott plot (B) for Fe62Cr12Cu1Nb3Si15.5B6.5amorphous alloy. In the first technique, for relatively low fields, a strong field dependence of TCcan be observed (e.g. for 2.5 mT, TC= 248(1) K and for 30 mT, TC= 254(1) K). On the other hand, the Arrot plot method yields to an erroneous value of TC= 263(1) K from the isothermal that intercepts at the origin. In the case of assuming the critical exponents of the Heisenberg model for a generalized Arrott plot, TC= 255(1) K is obtained. This indicates that the selection of the critical exponents strongly affects the value of TCobtained using this method. 89 4.1. Curie temperature Fig. 4.6 shows the temperature plots of Yand Xto illustrate the K-F method. This is obtained after 15 iterations, sufficient to reach convergence, obtaining β= 0.43, γ= 1.67 and TC= 249.6(3) K. It is worth mentioning that it is time consuming to obtain this final value of TCusing this method. On the other hand, our proposed method based on MC effect does not require any iterative or fitting routines (just linear interpolations). 235 240 245 250 255 260 265 0 10 20 30 =1.67 Y (K) T (K) X (K) T C =249.6 K =0.43 0 2 4 6 8 10 Figure 4.6: X,Yvs. Tplots from the K-F method for Fe62Cr12Cu1Nb3Si15.5B6.5amorphous alloy after convergence. As a summary, a good agreement between MC and K-F methods is observed. For the inflection point method, an approximate value is obtained for 2.5 mT though it is strongly affected by magnetic fields (even for relatively low fields as 30 mT). Overestimated values are obtained in the case of Arrott plot method using mean field and Heisenberg critical exponents (as they are quite different from those obtained by K-F). To reproduce the magnetization of real measurements using the A-N EOS, different levels of noise using two different approximations have been 90 4. Analyses based on the field dependence of the magnetocaloric effect included [143]. A constant relative error () for all the measured moment values has been considered: σ(T, H) = σAN (T, H) [1 + 2(0.5−rand(T, H))] ,(4.4) where σAN is the specific moment obtained from the A-N EOS, rand is a random number, between 0 and 1, different for each simulated point. The values of the magnetization can differ from the noise-free data in ±σAN . Well prepared experiments typically contain a signal-to-noise ratio level of 100:1 (=0.01), as typically the moment range is set to auto-range level. To quantify the signal-to-noise ratio, the root mean square (RMS) has been used: RMS(ψ) = v u u t N X i=1 1 N ψi−ψ ψ,(4.5) where ψis a certain magnitude which has Nvalues (where icorrespond to a certain value) and ψis the mean ψvalue. In our case the used mean values are those calculated without noise. With respect to the determination of the Curie temperature, Fig. 4.7 shows the temperature values that can be identified as the TCfor both inflection and MC methods using = 0.01 (the values for the inflection point method without noise are also plotted). It can be observed that while the values obtained by the inflection point method depart from the value of the Curie temperature introduced in the A-N EOS (300 K) as the magnetic field increases, the values obtained by the MC method are independent of the magnetic field, both for the curves with and without noise. The values obtained by the MC method have an upper limit that is the value of the temperature of the peak entropy change, because Tpk ∆sT≥TC. The Curie temperature values obtained by the inflection point method have more noise level than the values obtained by the MC method (RMS(TC)=0.1 and 0.02, respectively). As the obtained values of the Curie temperature from the MC procedure are field independent, the whole curve can be averaged, being the mean value 301.5(9) K (close to the value used for simulation, taking into account that the temperature step is 2 K). 91 4.2. Critical exponents 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 200 250 300 350 0.000 0.025 0.050 295 300 305 310 From MC From inflection T C (K) 0 H (T) T (K) 0 H (T) Figure 4.7: Values of the temperature assumed as the Curie temperature determined by the inflection and MC methods for =0.01 as a function of magnetic field. Inset: magnification of the low magnetic field region. If we use both methods of determination of TCup to low fields, for fields up to 0.02 T, the signal-to-noise ratio is similar for the MC and inflection point methods and the mean values are 298.7(3) K and 302.3(2) K, respectively. For larger fields, the results of the inflection point method quickly deviate from the correct Curie temperature value and, as seen in Fig. 4.7, having a larger noise level. 4.2 Critical exponents From the literature, the most common way to determine the critical exponents is the K-F method. As it has been shown that SOPT MC response is strongly related to the critical exponents of thermomagnetic phase transitions, it is possible to determine the critical exponents using methods based on the magnetic field dependence of the MC effect. In this section, this alternative method will be described and its results will 92 4. Analyses based on the field dependence of the magnetocaloric effect be compared with those determined from K-F for single phase as-spun ribbons. In addition, the latter technique is limited to single phase systems, and it will be shown that this limitation can be overcome by the alternative method based on the field dependence of MC effect [144]. These studies were performed in amorphous ribbons prepared by melt spinning by Vacuumschmelze (compositions Ni74Fe8Si2B16 and Ni68.8Fe13.2Si2B16), which serve as the single phase system. They were then heat treated to form nanocrystals in their amorphous matrix, providing biphasic MC system. More details on sample preparation and their microstructures can be found in [144]. Firstly, the power law fittings to the field dependence of MC response were performed to determine the critical exponents of single phase Ni74Fe8Si2B16 as-spun ribbons. These are illustrated in Fig. 4.8, which shows the field dependences of ∆spk T(A) and Tr(i.e. the temperature above TCthat corresponds to 70 % of the peak) (B). 0.0 0.5 1.0 1.5 0.0 0.1 0.2 0.3 0.4 0.0 0.5 1.0 1.5 175 180 185 190 195 200 (B) T r (K) | s pk T |=-0.02+0.31 H 0.71 0 H (T) | s pk T | (J kg -1 K -1 ) (A) multidomain region 0 H (T) T r =173.3+20.8 H 0.54 multidomain region Figure 4.8: Magnetic field dependence of the ∆spk T(A) and Tr(B) for Ni74Fe8Si2B16 amorphous alloy (symbols). The different fittings to a power law are represented as blue lines. 93 4.2. Critical exponents Data for low fields (below 0.5 T) are not considered to avoid the region of multidomains. From the correlations of the critical exponents to MC magnitudes (refer to Table 1.1), β,γand δwere determined from these two fittings. Next, the critical exponents were determined using the K-F method, Fig. 4.9 shows the Xand Yvs. Tplots after convergence of the values. Table 4.1 shows the comparison of critical exponents β,γand δ obtained for single phase amorphous Ni74Fe8Si2B16 and Ni68.8Fe13.2Si2B16 from the fittings of MC analyses and from the K-F method. An excellent agreement is observed between both methods. 120 140 160 180 200 220 240 260 0 20 40 60 80 =1.35 X (K) Y (K) T (K) =0.47 0 10 20 30 40 50 T C =177.2 K Figure 4.9: Xand Yvs. Tafter applying the K-F method to Ni74Fe8Si2B16 amorphous alloy: fittings (lines) and experimental data (symbols). Sample Ni74Fe8Si2B16 Ni68.8Fe13.2Si2B16 Method K-F MC K-F MC β0.47 0.47 0.42 0.41 γ1.35 1.33 1.51 1.55 δ3.86 3.82 4.60 4.83 Table 4.1: Critical exponents for Ni74Fe8Si2B16 and Ni68.8Fe13.2Si2B16 amorphous alloys obtained by the K-F and MC methods. 94 4. Analyses based on the field dependence of the magnetocaloric effect The reliability of results were further investigated by constructing the universal curves using these critical exponents. There are two possible universal constructions: one presents σH−1/δ vs. tH−1/(β+γ)and the other plots t−βσvs. t−(β+γ)H. Fig. 4.10 shows the two constructions for Ni74Fe8Si2B16 amorphous alloy, wherein the curves constructed using the critical exponents from K-F and MC methods are presented in (A)-(B) and (C)-(D), respectively. 0 10 20 30 10 0 10 1 10 2 10 3 -1.0 -0.5 0.0 0 .5 1.0 1.5 2.0 0 10 20 30 10 -1 10 0 10 1 10 2 10 3 10 4 10 -1 10 0 10 1 10 2 10 3 (A) K-F exponents (D) MC exponents (B) K-F exponents (C) MC exponents ( 0 H ) -1/ (A m 2 kg -1 T -1/ ) t -( + ) 0 H (T) t ( 0 H ) -1/( + ) (T -1/( + ) ) t - (A m 2 kg -1 ) Figure 4.10: Universal constructions for Ni74Fe8Si2B16 amorphous alloy with critical exponents determined by the K-F (A)-(B) and MC (C)-(D) methods. It is observed that experimental points collapse for both methods of critical exponent determination, indicating that a good scaling of the results is obtained independently of the method. This is in agreement to Table 4.1. For the second case study, both samples were partially crystallized by heat treating them in a differential scanning calorimetry to obtain around 20 % of nanocrystalline phase. The formed nanocrystalline phase is γ−(Ni,Fe). More details about the microstructure of the nanocrystallized samples 95 4.3. Order of the phase transition and critical composition Heusler compounds and composite materials [145]. To show the excellent agreement with experimental results, Fig. 4.16 shows the temperature dependence of the exponent nfor a LaFe11.6Si1.4compound for different magnetic field changes. The overshoot is corroborated and, moreover, its field dependence is in an excellent agreement with the simulated results using the B-R model (see Fig. 4.15). 185 190 195 200 205 210 0 2 4 6 8 10 2.0 T n T (K) 0.5 T n =2 Magnetic field Figure 4.16: Temperature dependence of the exponent nfor LaFe11.6Si1.4 for different magnetic field changes. It has to be noted that Heusler compounds cannot be described through the use of the B-R model as they present a magneto-structural transition, however the criterion is shown to be also applicable for these materials. Fig. 4.17 shows the temperature dependence of ∆sTand exponent n for Ni49Mn36In15 compound. It can be observed the overshoot above 2 at temperatures below T∆sT pk , indicating the FOPT character of the martensitic transition. In addition, abrupt changes of nare observed due to the change of sign of ∆sT(marked with cyan areas) though they do not affect the general conclusions. 102 4. Analyses based on the field dependence of the magnetocaloric effect 200 250 300 350 -4 -2 0 2 4 6 8 T (K) s T (J kg -1 K -1 ) -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 n ( T trans ) Ni 49 Mn 36 In 15 1.5 T n =2 s T =0 overshoot Figure 4.17: Temperature dependence of ∆sTand exponent nfor Ni49Mn36In15 Heusler alloy at 1.5 T. Exponent nminima are also observed near the transition temperatures (Fig. 4.14). These minimum values show a clear tendency as ηis modified. Fig. 4.18 shows the nvalues at Ttrans as a function of η. It can be observed that n(Ttrans) values show a monotonous trend with the ηparameter. Two regions and a point can be distinguished: for SOPT region (where η < 1), n(Ttrans)>0.4 while for FOPT (where η > 1), n(Ttrans)<0.4. For η= 1, n(Ttrans) = 0.4. The n(Ttrans) = 0.4 value agrees with the theoretical assumptions for the critical point (β= 0.25 γ= 1 [146], which lead to n= 0.4 according to Eq. (1.38)). Therefore, with the analysis of n(Ttrans) values it is possible to establish another criterion to distinguish between FOPT and SOPT [71]. Moreover, the critical composition of SOPT-FOPT point (FOPT crossovers to SOPT) can be determined when n(Ttrans)=0.4. The current simulations also give support to that criterion and show the compatibility between both methods. However, it has to be noted that the criterion is limited to materials that follow the B-R model, in contrast with the overshoot criterion. 103 4.4. Conclusions 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 FOPT n ( T trans ) critical SOPT-FOPT SOPT n =0.4 Figure 4.18: n(Ttrans) as a function of the ηparameter for 1 T (LaFe11.5Si1.5parameters). These features will be further discussed when analyzing La(Fe,Si)13 compounds in Chapter 6. 4.4 Conclusions In this chapter, several MC analyses based on the field dependence of the MC effect have been shown as an alternative to conventional methods for getting information on thermomagnetic phase transitions. These allow the determination of the Curie temperature and the critical exponents of SOPT MC materials. Furthermore, the field dependence of MC effect is extended to determine the order of the transition and the critical SOPT-FOPT composition. The main results are: •We proposed a method for determining TCbased on the equality of the field dependence exponent nat TCand that at T∆sT pk . The 104 4. Analyses based on the field dependence of the magnetocaloric effect method was successfully applied to a Fe-based amorphous alloy and numerical simulations. The advantages of the proposed method are: –The obtained temperature is field independent. –It is not an iterative or a fitting procedure. –No assumptions on the critical exponents describing the transition are required. •The field dependence analysis of MC effect is shown as a precise technique to determine the critical exponents of the phase transition. For a set of Ni-based amorphous alloys, the values of the critical exponents determined from K-F and MC methods are in excellent agreement. The amorphous alloys were then partially crystallized to a biphase material composed by the residual amorphous and nanocrystalline phases. With the MC method, the critical exponents of the residual amorphous phase were determined when the TCof the crystalline phase is well separated from that of the residual phase (to assume that his MC contribution is negligible) while the K-F method is not applicable. A deconvolution procedure to remove the secondary phase was employed to use the K-F method, which then results in critical exponents that are in good agreement with those from the MC results (that did not require using the deconvolution). •When field dependence MC analyses are extended to study both FOPT and SOPT MC materials, they provide a powerfull criterion to determine the order of the phase transition and the critical SOPT-FOPT composition. The numerical results simulated from the B-R model were found in an excellent agreement with experimental La(Fe,Si)13 results. Two criteria are shown: –For FOPT MC materials, the identification of the order of the phase transition is based on the overshoot of exponent nabove 2 at temperatures near the transition temperatures. For SOPT MC materials, the overshoot is not observed and the exponent nmonotonously increases from n(TC) up to 2. 105 4.4. Conclusions –The values of the exponent nat the transition temperature are below 0.4 for FOPT MC materials, above 0.4 for SOPT and exactly 0.4 at the critical SOPT-FOPT point, allowing for the identification of the critical composition. 106 5 Studies of Gd-based materials The previously proposed methods for characterizing MC materials are now applied with the aim to optimize their response. In this chapter we will focus on SOPT MC materials. Our aim is attained via two ways: (i) optimizing refrigerant capacity in a multiphase systems or (ii) enhancing magnetic field responsiveness by TCdistributions. Gd was used as the base material for these optimization studies. For the first approach, a series of Gd-GdZn were developed with varying phase fractions. The increased refrigerant capacity of the alloy compared to that of pure Gd is achieved. Assuming a non-interacting model for the phases, our model agrees with the experimental results. Moreover, the proposed procedure to minimize deviations in heat capacity measurements that started from finite temperatures is successfully applied. For the second approach for MC optimization, a Gd/Ti multilayer system with tunable TCdistribution due to the layer thickness was developed, for which an increase of the magnetic field responsiveness of the MC effect is achieved. The results are also found to follow a Gaussian distribution of Curie temperatures. 107 5.1. Gd-GdZn composites 5.1 Gd-GdZn composites One approach to optimize the performance of MC materials is to develop composite materials with the appropriate selection of phases and transition temperatures [37, 147, 148]. By appropriately selecting the transition temperatures, these composites can exhibit table-like responses (constant ∆sTvalues in the operational temperature span), which are highly desired for the Ericsson refrigeration cycle. MC composites have been both theoretically and experimentally shown to exhibit enhanced coefficient of refrigerant performance in comparison to those of single phase MC materials [149]. Their numerical reports show that up to 83 % improved RCP can be possible by proper selection of the composite system: optimum TCdifference of the existing phases and their phase proportions. As Gd is the paradigmatic SOPT MC material near room temperature, and with the aim to investigate the MC behavior of a multiphase composite, GdZn phase was selected as the other coexisting phase with Gd for the study. With respect to the TCof Gd (290 K), the GdZn intermetallic compound is relatively near (265 K) and in addition, it is associated with a large MC response. As its binary phase diagram shows that it can coexist with Gd, their composites fulfill the criteria selection of MC composite assembly for RCP optimization. Moreover, unlike others composites, this can be synthesized in a single step. With this, the MC effects of the Gd-GdZn composite materials were studied [150]. 5.1.1 Synthesis Varying phase proportions of Gd-GdZn composites were produced by induction melting Gd50+zZn50−z(where z = 0, 5, 15 and 25) in an Ar-controlled atmosphere. Both commercial Gd and Zn were 99.9 and 99.98 wt.% pure, respectively. The ingot was melted twice to ensure homogeneity. For comparison, MC effect of pure Gd piece was used as the reference for this work. 108 5. Studies of Gd-based materials 5.1.2 Microstructure Micro-XRD experiments were performed using a Bruker D8 Discover diffractometer. This technique allows to characterize small areas (∼1 mm2) of the samples. The micro-XRD patterns of the different produced composites are shown in Fig. 5.1. 30 40 50 60 70 Gd 2 O 3 Gd 50 Zn 50 Gd 55 Zn 45 Gd 65 Zn 35 Gd 75 Zn 25 Gd GdZn Intensity (a.u.) 2 (deg.) Figure 5.1: Micro-XRD patterns of the Gd-GdZn composites. Circular symbols denote the Gd phase detected while triangles correspond to GdZn. Very minor Gd2O3phase is detected (denoted by the grey arrows). As expected, the diffraction patterns of the composites can be mainly indexed to Gd (hcp structure), GdZn (CsCl-type structure) and Gd2O3 (cubic Ia-3 structure), which is very minor phase. To quantify the different phase fractions presented in the composites, a Rietveld refinement of the patterns was performed. Results of the different phase fractions are shown in Table 5.1. Only 4-8 wt.% of Gd2O3was detected, which can be due to surface oxidation related to the Gd phase (as no oxide phase is observed for the single phase GdZn in Gd50Zn50). The refinement results reveal an 109 5.1. Gd-GdZn composites increment in the phase fraction of GdZn with Zn concentration, which is in good agreement with the nominal composition. Nominal comp. xGd (wt.%) xGdZn (wt.%) xGd2O3(wt.%) Gd50Zn50 0.0 100.0 0.0 Gd55Zn45 7.6 84.7 7.7 Gd65Zn35 23.8 70.5 5.7 Gd75Zn25 51.0 45.0 4.0 Table 5.1: Phase contents obtained from the Rietveld refinement analyses of the micro-XRD patterns. 5.1.3 Magnetic and MC properties ∆sTwas indirectly calculated from isothermal magnetization curves according to Eq. (1.16) with an initial magnetic field equal to zero. To avoid a relevant influence of the demagnetizing field on the MC determination, the samples were prepared as thin plates for measurements. The temperature dependence of ∆sTfor Gd-GdZn composites for 2 T is presented in Fig. 5.2. The results for Gd (also in thin plates) are included as green dashed line. It is observed that the composites show two ∆spk T values that correspond to the various phases they exhibit for Zn contents up to 35 at. %. They are associated to the TCof the GdZn and Gd phases at 265 K and 290 K, respectively. Despite that the ∆spk Tof Gd phase for Gd55Zn45 is less evident compared to the GdZn, one can still notice small shoulder bump (at temperatures higher than 265 K), which corresponds to the Gd phase. In the case of the Gd50Zn50, which exhibits only single phase GdZn intermetallic, only a single ∆spk Tis observed, corresponding to 67 % of that observed for pure Gd (3.2 and 4.8 J kg−1K−1, respectively). This can be expected as nonmagnetic Zn dilutes the behavior of Gd in the GdZn intermetallic like trade-offs in alloys compared to pure Gd. Hence, as the GdZn phase fraction increases (see Table 5.1), the MC response decreases though the values at the peak are similar (around 3.2(0.2) J kg−1K−1) in the range of 265-290 K for the composites. For Gd75Zn25 composite 110 5. Studies of Gd-based materials (phase fraction ratio close to 1:1), a table like response is observed as the magnitudes of the ∆spk Tvalues are almost constant in a temperature span of 25 K (3.1(3) J kg−1K−1). 200 225 250 275 300 325 350 -5 -4 -3 -2 -1 0 s T (J kg -1 K -1 ) T (K) Gd 75 Zn 25 Gd 65 Zn 35 Gd 55 Zn 45 Gd 50 Zn 50 2 T Gd Figure 5.2: Temperature dependence of ∆sTfor the Gd-GdZn composites for 2 T (the MC response of pure Gd is shown for comparison). Assuming a non-interacting phase model for the composites, the total ∆sT can be expressed as: ∆sT=xGd∆sT,Gd +xGdZn∆sT,GdZn,(5.1) where xGd and xGdZn are the Gd and GdZn phase fractions, respectively. ∆sT,Gd and ∆sT,GdZn are the isothermal entropy changes of pure Gd and GdZn, respectively. It has to be noted that the contribution of the Gd2O3 phase (∆sT,Gd2O3) is not considered as it is Curie-Weiss paramagnet and ∆sT,Gd2O3<< ∆sT,Gd and ∆sT,GdZn at the studied temperature range. From this, the quantitative phase fractions (or alternatively, the Zn content) can be extracted from the deconvolutions of their various MC responses (∆sT,Gd and ∆sT,GdZn are known). This ∆sTdeconvolution of 111 5.2. Gd/Ti multilayers It can be observed that larger ∆Tpk Svalues are observed when determined using ∆sT-c2Tthan those using ∆sT-c0T. The results show a relatively good agreement between the different approximations though differences of ≈10 % at the peak are clearly observed. It has to be noted that Eq. (1.20) was obtained after strong approximations (mainly that calorimetric data are field independent) and that relatively small differences are acceptable for cases of smooth field dependence of cH data as compared to other techniques. This approximation will not be recommended for analyzing FOPT MC materials. On the other hand, the proposed procedure from truncated calorimetric data does not assume any approximation and has been experimentally and theoretically validated for both SOPT and FOPT MC materials. 5.2 Gd/Ti multilayers Besides the RCP optimization by proper selection of phases coexisting in composite materials, other strategy to improve the performance of MC materials is to increase the magnetic field responsiveness of the MC effect [147]. For all temperature ranges, the MC response increases as the magnetic field increases. However, the application in magnetic refrigerator prototypes, when looking for real commercial applications, use permanent magnet arrays as magnetic field generators that, nowadays, can only generate magnetic fields around 2 T [151]. Increasing the field beyond this value can add complications to the design of the device and the associated cost. Therefore, the search for methods (as well as comprehending them) on how to enhance the magnetic field responsiveness of MC materials could eventually lead to newer possibilities in the application of magnetic refrigeration. The actual problem is that the exponent controlling the field dependence of the MC effect has a minimum at the temperature at which the material has the largest response, the transition temperature. One possible way to increase that minimum can be attained by a distributed TCin the material, which was reported for a Ni-Cu multilayered system 118 5. Studies of Gd-based materials [41]. The TCdistribution arose from the varying composition of each layer, which enhanced its magnetic field dependence of the MC effect as compared to that of the corresponding bulk counterparts. These thin film systems have been studied as MC refrigerants [152] as well as for permanent magnets [153,154], multiferroic materials [155], etc. With the aim of tuning a distributed TCby the layer thickness of a multilayer MC material, the magnetic field responsiveness of Gd/Ti multilayered system was studied [156]. The compositions of the layers was not varied but the thickness of Gd layers were varied with the aid of nonmagnetic Ti spacer layers, which was chosen as spacer due to its low solubility in Gd that minimizes intermixing. 5.2.1 Synthesis and microstructure A Gd/Ti multilayered sample was produced using alternative rf-sputtering deposition of Gd layers and Ti spacers onto glass substrates. Sample fabrication and microstructural characterization were performed by A.V. Svalov and G.V. Kurlyandskaya and more details can be found in [157]. For the sputtering process, a base pressure of 3·10−7mbar and an Ar gas flow with a pressure of 5·10−3mbar were used. The deposition rate was about 0.15 nm s−1for Gd and about 0.07 nm s−1for Ti. The deposition times were varied to obtain Gd and Ti layers with thickness (LGd and LTi, respectively) of 1.5 and 2 nm, respectively. A total number of 50 layers was chosen, being the sample denoted as [Ti(2 nm)/Gd(1.5 nm)]50. The sample has a protective coating Ti(5 nm) layer. It has to be noted that the mass of the Gd layers for [Ti(2 nm)/Gd(1.5 nm)]50 was calculated using the thickness of the layers, the area of the sample and the density of bulk Gd. If the densities of the thin Gd layers are smaller than that of bulk Gd, it is expected to get a smaller error in their mass determination with this method than to take their mass with a mass balance (the substrate has a much larger mass than that of the sample itself). Although this error can produce an 119 5.2. Gd/Ti multilayers underestimated value of ∆sT, it does not alter its field or temperature dependence. As a reference for this work, Gd (99.9 %) was melt spun into ribbons 20 µm thick and 3 mm wide (to avoid demagnetizing field effects) and the results of multilayered sample were compared to those of Gd ribbons. Fig. 5.9 shows the low-angle XRD results (performed using a Rigaku D/max-RB) for [Ti(2 nm)/Gd(1.5 nm)]50, which clearly show a well-defined layered structure with two distinct diffraction peaks at 2θ= 3.77 and 7.16 o. In agreement to the XRD data, its cross-sectional HRTEM image (characterized using a JEOL JEM-2100 TEM) also reveals well-defined layers (inset of Fig. 5.9). 12345678 Intensity (a.u.) 2 (deg.) Figure 5.9: Low-angle X ray pattern and cross-sectional HRTEM image (inset) for [Ti(2 nm)/Gd(1.5 nm)]50. 5.2.2 Magnetic and MC properties Fig. 5.10 shows the temperature dependence of σfor a magnetic field of 0.012 T for [Ti(2 nm)/Gd(1.5 nm)]50 and the inset shows the thickness dependence of TCfor Gd [158]. 120 5. Studies of Gd-based materials 0 50 100 150 200 0 2 4 6 0 5 10 0 100 200 300 (10 -4 A m 2 kg -1 ) T (K) 0.012 T T C (K) L Gd (nm) Figure 5.10: Temperature dependence of σfor 0.012 T for [Ti(2 nm)/Gd(1.5 nm)]50. Inset: TCevolution with respect to the Gd layer thickness: symbols for experimental data while the line represents the power law fit according to [158]. It is observed a smooth decrease of σvalues from low temperatures to almost 0 around 200 K, indicating a notable SOPT character. TCis estimated as 130 K from the inflection point of this σvs. Tcurve (red cross in inset of Fig. 5.10). This value is much lower than the one obtained for Gd bulk (293 K) as ascribed to the finite size scaling. For such system, the value of TCis related to the thickness of the layer (LGd) by a phenomenological power law previously reported for epitaxial Gd [158]: TC,bulk −TC TC,bulk ∝L−φ Gd,(5.2) where TC,bulk is the Curie temperature of the bulk material and φis a constant. Using the previously obtained TCvalue from the inflection point of magnetic data, an estimated LGd of 1.5 nm is observed, in agreement with the expected value from the deposition time. 121 5.2. Gd/Ti multilayers Fig. 5.11 shows the field dependence of σfor Gd bulk material (A) and [Ti(2 nm)/Gd(1.5 nm)]50 (B). Typical magnetic behavior of the Gd bulk ribbons is observed while that of the [Ti(2 nm)/Gd(1.5 nm)]50 is also accompanied by some temperature-independent contributions from the sample holder, glass substrate and Ti spacers (as the sample mass is quite small), which is plotted as a dashed line in Fig. 5.11 (B). As this additional signal does not exhibit a noticeable temperature dependence, the calculation of ∆sTis not affected by this contribution. Furthermore, it is observed that both samples show ferro-paramagnetic behavior with increasing temperatures, wherein this occurs at a lower temperature in the multilayer sample compared to that of the bulk counterpart. 0 50 100 150 200 0.0 0.3 0.6 0.9 1.2 1.5 0 50 100 150 200 250 300 350 (B) 80 K 430 K (A m 2 kg -1 ) (A) Gd [Ti(2 nm)/Gd(1.5 nm)] 50 Holder+substrate+spacer 80 K 280 K (A·m 2 kg -1 ) 0 H (T) Figure 5.11: Magnetic field dependence of σfor Gd bulk (A) and [Ti(2 nm)/Gd(1.5 nm)]50 (B), respectively. Dashed line corresponds to the temperature independent external magnetic contribution. 122 5. Studies of Gd-based materials Fig. 5.12 shows the temperature dependence of ∆sTfor Gd bulk and [Ti(2 nm)/Gd(1.5 nm)]50 for magnetic field changes of 0.75 and 1.5 T. Comparing both samples, it is observed that the curves for Gd bulk look sharper than those of the [Ti(2 nm)/Gd(1.5 nm)]50 (for 1.5 T, ∆spk T of Gd is -3.5 with narrower temperature width while the latter shows -1.0 J kg−1K−1with broader temperature width). 50 100 150 200 250 300 350 400 0 1 2 3 4 - s T (J kg -1 K -1 ) T (K) 0.75 1.5 Gd bulk [Ti(2 nm)/Gd (1.5 nm)] 50 0 H (T) Figure 5.12: Temperature dependence of ∆sTfor Gd bulk and [Ti(2 nm)/Gd(1.5 nm)]50 for magnetic field changes of 0.75 and 1.5 T. In general, larger ∆sTresponsiveness to magnetic fields for SOPT MC materials occurs at the temperature ranges that they are in fully ferro- or para-magnetic states but having small ∆sTvalues. This is represented by the field dependence exponent n(T) behavior, wherein shows n≈1 at the ferromagnetic region and n≈2 at the paramagnetic range. The lowest field responsiveness of ∆sTis found at the region close to the transition temperature, while ∆sThas a maximum value. Consequently, an increase in nin the region close to TCcan improve the applicability of these MC materials (provided that the magnitude of the MC effect is not reduced). Fig. 5.13 shows the field dependence of ∆spk T(in a log-log plot) for Gd bulk and [Ti(2 nm)/Gd(1.5 nm)]50. 123 5.2. Gd/Ti multilayers 0.1 1 0.1 1 Gd Bulk [Ti(2 nm)/Gd(2 nm) ] 50 n=1.006(9) n=0.776(5) s pk T (J kg -1 K -1 ) 0 H (T) Figure 5.13: Magnetic field dependence of ∆spk Tfor Gd bulk and [Ti(2 nm)/Gd(1.5 nm)]50. The value of the slope for the former is 0.776 while that for [Ti(2 nm)/Gd(1.5 nm)]50 its slope increases up to 1.006, which indicates that the MC response of this thin film is more field responsive than those of bulk systems (also in agreement with literature). At low magnetic fields, the field dependence deviates from the expected linear behavior in log-log plots due to the multidomain structure of the samples and, therefore, it is not fitted. For comparing the n(T) behaviors of both samples (shown in Fig. 5.14), a rescaled temperature axis was used since the TCfor both samples are different. The data of Gd bulk have to be corrected for its demagnetizing factor. For bulk Gd, the observed nbehavior corresponds to the typical behavior of a single phase SOPT MC material: n≈1 at T << TC, then decreases to a minimum at TCand eventually to n≈2 at T >> TC. The additional peak observed around 235 K can be ascribed to the spin reorientations [159]. On the other hand, for [Ti(2 nm)/Gd(1.5 nm)]50, its nbehavior differs from that of bulk Gd, in particular the value of n≈1 124 5. Studies of Gd-based materials in a broad temperature span around TC(unlike the deep minimum at TC that is <1 for Gd bulk). This is in agreement to the nbehavior of the Ni-Cu multilayered system with varying compositions in each layer that was reported in literature [41]. In our case this is achieved with a single composition. To describe the observed MC response of Gd multilayered sample (smeared yet wide ∆sTas compared to that of bulk counterparts), it is assumed that it exhibits a distributed TCdue to finite size scaling that can occur for the varying layer thickness for the studied [Ti(2 nm)/Gd(1.5 nm)]50. With this assumption, numerical simulations were also performed. The magnetization of the Gd bulk was modeled using the Brillouin function. For the modeling of the multilayer sample, the previous model of the bulk behavior was modified using a Gaussian distribution of TC(see Section 3.2). The calculated values of the exponent nat TCas a function of ωare represented in the inset of Fig. 5.14. -0.75 -0.50 -0.25 0.00 0.25 0.50 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 n t Brillouin Brillouin+Gaussian 1.5 T Spin reorientation [Ti(2 nm)/Gd(1.5 nm)] 50 Gd 0 10 20 30 40 50 60 0.6 0.7 0.8 0.9 1.0 n ( T C ) (K) Figure 5.14: Temperature dependence of the experimental and simulated (ω= 30 K) exponent nfor Gd bulk and [Ti(2 nm)/Gd(1.5 nm)]50 for 1.5 T. Inset: n(TC) vs. ω. 125 5.3. Conclusions It is observed that the values of n(TC) increase as ωincreases, reaching values close to the experimental observations (1.0 in main panel of Fig. 5.14) at ω≈30 K. This can be ascribed to small differences in the layer thickness, as the addition or subtraction of one atomic layer leads to a variation of TCaround 100 K (see inset of Fig. 5.10), which according to sputtering experiments is reasonable. In the simulations, the obtained n(TC) values are nearly independent of the temperature range (2Ω) near Ω = 2ω(the integrand for values larger than TC+ 2ωis negligible). In addition, the temperature dependence of the simulated nfor Gd bulk and [Ti(2 nm)/Gd(1.5 nm)]50 are also included in the main panel of Fig. 5.14, which shows a good agreement with experimental results. 5.3 Conclusions In this chapter, we have analyzed possible ways for optimization of SOPT MC material systems based on Gd: •For the RCP optimization of MC materials in composite systems, a series of Gd-GdZn composites (using 25 - 50 at.% Zn content) were successfully prepared in a single synthesis step with a good control of the phase fractions and the element contents. This fabrication route overcame several typical drawbacks, such as multiple preparation steps and poor thermal contact between phases. The samples exhibit a similar table-like ∆sT. This characteristic leads to the enhanced RCP values as those compared to the pure materials: 11 % and 45 % improvement with respect to Gd and GdZn pure phases, respectively. It is shown that ∆sTof Gd-GdZn composites can be modeled by considering non-interacting phases, which further enabled the predictions of the MC behavior for the whole compositional range (i.e. extended to compositions not studied experimentally). From their heat capacity data, a ∆Tpk Sof 3-3.5 K for 2 T was obtained for all the composites using different analysis methods described 126 5. Studies of Gd-based materials in earlier chapters. The developed procedure to calculate the MC effect from truncated calorimetric data has been succesfully applied for Gd-GdZn composites, showing a good agreement with those calculated from independent magnetization measurements. •The field responsiveness of SOPT MC materials is enhanced in a Gd/Ti multilayer system prepared by rf sputtering. Its field dependence exponent of ∆sTis shown to be 30 % larger than that of a Gd bulk (micro-thick ribbons), indicating that the former is more responsive to magnetic fields. It is further shown that this observation can be ascribed to a Curie temperature distribution arising from the small variations in the layer thickness by the order of one atomic layer magnitude. This is also observed for their numerical simulations, in which the multilayered material was reproduced using the Brillouin function to describe each contribution of a Gaussian distribution of Curie temperatures. 127 6.1. Ni-doped series be found for samples with higher Ni content. Considering the obtained compositions and the presence of secondary phases, the series shows a solid solutbility limit of ∼0.3, which is a 1.4-fold extension compared to literature [171]. When the incorporation of Ni in the 1:13 matrix becomes challenging, a slight increment of the Si amount is observed. However, it has to be noted that though Si content has increased, it still does not exceed the critical SOPT composition point of 1.65 [71]. Fig. 6.3 shows the XRD spectra for the Ni-doped series. 5 10 15 20 25 30 35 40 45 50 x=0.10 1:13 -Fe La 1 Fe 1 Si 1 x=0.00 x=0.33 x=0.26 x=0.18 Intensity (a.u.) 2 (deg.) Figure 6.3: XRD spectra for the Ni-doped La(Fe,Si)13 series. It is also observed that the samples mainly consist of 1:13 phase despite of minor traces of α-Fe and La1Fe1Si1detected, which are in agreement 134 6. Studies of La(Fe,Si)13-based materials with results obtained from EDX and BSE. The results from the Rietveld refinement of the different XRD spectra, including phase fractions, lattice parameter of 1:13 (l1:13) and the goodness of fit (GoF), are tabulated in Table 6.1, in which GoF values close to 1 (ideal case) for all samples are observed [175]. The amount of α-Fe increases with Ni content, showing a large increment for x = 0.26, which are consistent with the EDX, BSE and XRD results. It is observed that though Ni additions do not modify the NaZn13-type structure, the lattice parameter decreases with increasing Ni content at a rate of -0.0018(6) ˚ A (at.% Ni)−1[107]. Sample 1:13 phase composition x1:13 xα−Fe xLa1Fe1Si1l1:13 GoF (wt.%) (wt.%) (wt.%) (˚ A) x=0.00 La1.00(3)Fe11.2(4)Si1.45(12) 97.9 0.2 1.9 11.49043(13) 1.35 x=0.10 La1.00(3)Fe11.2(2)Ni0.10(2)Si1.42(12) 96.9 0.9 2.3 11.49179(16) 1.17 x=0.18 La1.00(3)Fe11.1(2)Ni0.18(3)Si1.44(12) 96.2 3.8 0.0 11.48836(15) 1.26 x=0.26 La1.00(3)Fe10.9(2)Ni0.26(3)Si1.50(12) 84.9 14.2 0.9 11.4884(3) 1.01 x=0.33 La1.00(3)Fe10.9(2)Ni0.33(3)Si1.54(12) 77.6 21.3 1.1 11.4857(5) 1.10 Table 6.1: 1:13 phase composition determined by EDX, phase fractions, l1:13 and GoF from Rietveld refinement of the XRD spectra of the Ni-doped La(Fe,Si)13 series. To determine the preferential incorporation of Ni in the various phases existing in the samples, the Ni/Fe ratio of the various existent phases for the samples with high Ni content (x = 0.26 and 0.33) are tabulated in Table 6.2. Sample Ni/Fe nominal Ni/Fe (1:13) Ni/Fe (α-Fe) Ni/Fe (La1Fe1Si1) x=0.26 0.0265 0.024 0.012 0.44 x=0.33 0.0357 0.030 0.014 3.55 Table 6.2: Ni/Fe ratio determined by EDX for the different phases presented in the higher Ni-doped La(Fe,Si)13 samples (x = 0.26 and 0.33). The results show that the Ni/Fe ratio is higher in the La1Fe1Si1phase than those in α-Fe and 1:13 phases, indicating a preferential segregation of Ni in La-rich phase. 135 6.1. Ni-doped series 6.1.3 Magnetic and MC properties Fig. 6.4 shows the temperature dependence of the normalized magnetization (with respect to the magnetization value at 150 K) for an applied field of 0.05 T (A) together with Ttrans (B) and thermal hysteresis (∆Thyst) (C) as functions of the Ni content for the set of Ni-doped La(Fe,Si)13 series. 180 190 200 210 220 230 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 0.00 0.10 0.18 0.26 (C) (B) / 150 K T (K) 2 K/min 0 H=0.05 T (A) 0.33 190 195 200 205 210 215 T trans (K) T hyst (K) Ni content, x Figure 6.4: Normalized magnetization (with respect to the value at 150 K) for 0.05 T (A), Ttrans (B) and ∆Thyst (C) as a function of Ni content for the La(Fe,Ni,Si)13 series. A correction for the temperature is needed in order to remove thermal lag at that heating/cooling rates (≈1.25 K for each curve). Ttrans is determined as the temperature for which dσ/dTis maximum. From the normalized representation, it is possible to quickly estimate the amount of α-Fe from the high temperature magnetization, in good agreement with those obtained from the Rietveld refinement of XRD spectra. It can be observed that Ttrans phenomenologically follows a monotonous increase with increasing Ni content (fitted to a power law of the form 136 6. Studies of La(Fe,Si)13-based materials Ttrans =κ(0) +κ(1)xκ(2) with κ(0) = 192.6(5) K, κ(1) = 159(17) K, κ(2) = 2.19(13)). It can be observed that ∆Thyst is reduced with increasing Ni concentration, reaching a minimum value for x = 0.26 (further discussed later in this chapter). Spontaneous magnetization values of the 1:13 phase for the different samples are slightly modified with Ni additions. Removing the contribution of the α-Fe phase from σvs. H(not shown) (from the fitting of the paramagnetic tail, as it was shown previously in Fig. 4.12) a decrease of σSvalues at rates of ≈7Am2kg−1(at.% Ni)−1 is observed, although the presence of α-Fe prevents any further analysis. As FOPT MC materials exhibit thermal and magnetic hysteresis, discontinuous measurement protocols have been applied for both ∆sT, by indirect measurements, and ∆TS, by direct measurements. Fig. 6.5 shows the continuous and discontinuous ∆sTmeasurements from isothermal magnetization curves. The characteristic artifact is observed for the continuous case, while it disappears for the discontinuous. It has to be stressed that the influence is quite notable though the hysteresis of the sample is relatively small. 180 190 200 210 -25 -20 -15 -10 -5 0 s T (J kg -1 K -1 ) T (K) continuous discontinuos LaFe 11.6 Si 1.4 1.9 T Heating Figure 6.5: ∆sTcontinuous and discontinuous measurement of the LaFe11.6Si1.4compound. 137 6.1. Ni-doped series Fig. 6.6 shows the temperature dependence of ∆sTand the Ni-content dependence of |∆spk T|for a magnetic field change of 1.9 T for Ni-doped La(Fe,Si)13 series calculated from σT(H) measurements according to Eq. (1.16) while heating. A monotonic decrease of |∆spk T|with the Ni content is observed, at a rate of -7.0(7) J kg−1K−1(at.% Ni)−1 . In addition, the corresponding temperatures of the peak shift to higher temperatures with Ni addition, while the curves become broader (∆TFWHM varies from 8.6 to 25.7 K for the samples with x = 0.00 and 0.33, respectively). Moreover, ∆STcurves for x = 0.00 and x = 0.10 (despite the slight reduction in magnitude for the latter) maintain an abrupt low temperature increase that change into a gradual type for higher Ni content (x >0.18) although it can be observed that x = 0.18 sample is an intermediate case. 170 180 190 200 210 220 230 240 -20 -15 -10 -5 0 0.0 0.1 0.2 0.3 5 10 15 20 0.00 0.10 0.18 0.26 0.33 T (K) s T (J kg -1 K -1 ) Heating, 1.9 T x | s pk T | (J kg -1 K -1 ) Ni content, x -7.0(7) Jkg -1 K -1 (at.% Ni) -1 Figure 6.6: Temperature dependence of ∆sTfor 1.9 T while heating for the Ni-doped La(Fe,Si)13 series. Inset: |∆spk T|as a function of Ni content. The effect of the demagnetizing field has been checked. The demagnetizing factor has been determined from σvs. Happ curves at low fields and 138 6. Studies of La(Fe,Si)13-based materials temperatures well below the transition. For x = 0.00 sample, NDis estimated as 0.0927(3) which should be similar for all the samples, as the geometry of the different samples is the same. Fig. 6.7 shows the effect of considering the demagnetizing field over ∆sTand exponent n (inset) for parent LaFe11.6Si1.4compound. The recalculation of ∆sTvalues considering the internal field instead the applied one shows a 2 % difference at the peak value, having no influence over the analysis performed. In the case of the exponent n, a notable influence of the demagnetizing field in the ferromagnetic range (as for SOPT MC materials) is observed while the differences are negligibles at the transition temperature. 185 190 195 200 205 210 -20 -15 -10 -5 0 185 190 195 0.0 0.5 1.0 w/o N D N D =0.0927 s T (J kg -1 K -1 ) T (K) LaFe 11.6 Si 1.4 1 T n T (K) T trans Figure 6.7: Temperature dependence of ∆sTand exponent n(inset) for LaFe11.6Si1.4compound considering the applied (solid symbols) and internal (using ND= 0.0927, hollow symbols) field of 1 T. Fig. 6.8 shows the temperature dependence of ∆TSand the Ni-content dependence of the peak (∆Tpk S) for the Ni-doped series while heating. Similar features as those observed for ∆sTare obtained: the shape of ∆TS(T) curves become gradual for x >0.18 (for x = 0.10, a similar shape to that of the x = 0.00 sample is obtained), the ∆Tpk Svalues are reduced at a rate of -1.81(10) K (at.% Ni)−1, the peak is shifted to higher 139 6.1. Ni-doped series temperatures with Ni additions, and the FWHM of the peak increases with Ni content (from 5.3 to 25.7 K for x = 0.00 and 0.33 samples, respectively). 170 180 190 200 210 220 230 240 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 2 3 4 5 6 7 T S (K) T (K) 0.00 0.10 0.18 0.26 0.33 Heating, 1.9 T x T pk S (K) Ni content, x -1.81(10) K (at.% Ni) -1 Figure 6.8: Temperature dependence of ∆TSfor 1.9 T while heating for the Ni-doped La(Fe,Si)13 series. Inset: ∆Tpk Sas a function of the Ni content. To estimate the cooling power of the samples, CRPmin was calculated according to Eq. 1.23. Fig. 6.9 shows CRPmin as a function of the Ni content for the series. The CRPmin value decreases with Ni addition (from CRPmin=3.1 to 0.2 for x = 0.00 and 0.4 samples, respectively). This decrease cannot be ascribed to the presence of α-Fe observed for higher Ni content samples since the α-Fe transition is well above that of 1:13 phase and its mass fraction is not enough to explain the drastic decrease. It should be noted that this tendency is also observed when comparing RCP for the different compositions. In addition, as both experimental and DFT calculations [107] show no drastic modification of magnetization with Ni additions, the alteration of the nature of the phase transition could arise as responsible for this change, which is also compositional dependent. 140 6. Studies of La(Fe,Si)13-based materials 0.0 0.1 0.2 0.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 CRP min Ni content, x 1.9 T Figure 6.9: Ni content dependence of the minimal coefficient of refrigerant performance for the Ni-doped La(Fe,Si)13 series. To analyze the characteristics of the phase transition ascribed to the MC response of the 1:13 phase, the MC field dependence analyses presented in Section 4.3 have been used. On the one hand, the analysis of the overshoot in the temperature dependence of exponent nis performed. Fig. 6.10 shows the temperature dependence of exponent nfor a magnetic field change of 1.9 T. Overshoots of nnear the phase transition for x=0.00, 0.10 and 0.18 samples are observed, which indicate that they undergo a FOPT. It can be also observed that the magnitude of the overshoot is decreasing as the Ni content increases, indicating that the FOPT-character is also decreasing. x = 0.18 should be very close to the composition corresponding to the critical SOPT-FOPT point. On the other hand, for higher Ni-doped samples it is observed a gradual increase from the minimum towards a maximum value at the paramagnetic region, indicating SOPT behavior. It can be noticed that the presence of α-Fe can modify the values of nabove Ttrans, reducing the expected value for the 1:13 single phase [130]. This fact does not modify the procedure as the overshoot above 2 is observed although when impurities are present [145]. 141 6.1. Ni-doped series 170 180 190 200 210 220 230 240 0 1 2 3 4 5 n T (K) 0.00 0.10 0.18 0.26 0.33 1.9 T n =2 x Figure 6.10: Temperature dependence of the field dependence exponent nfor the Ni-doped La(Fe,Si)13 series. In addition, the analysis of the exponent nvalues at the transition is performed. Fig. 6.11 shows the n(Ttrans) as a function of Ni content. It can be observed that x = 0.00, 0.10 and 0.18 samples show n(Ttrans)<0.4 values, indicating they undergo a FOPT. n(Ttrans) values above 0.4 are obtained for x = 0.26 and 0.33 samples, indicating they undergo SOPT. These results are in agreement with the previous MC analysis. Moreover, this procedure allows to calculate the critical SOPT-FOPT composition. For the studied case, it is predicted for a Ni content x = 0.19 (where it is assumed a linear interpolation between the different points). An excellent agreement with numerical calculations using the B-R model can be observed (see Fig. 4.18). For samples with LaFe11.5−xNixSi1.5 compositions, it was reported that their critical SOPT-FOPT composition coincides with a Ni content of x = 0.08 [171], which is a smaller limit compared to our case, though the procedure used by the other authors was less quantitative. Hysteresis results (Fig. 6.4 (C)) follow a complementary trend of those of n(Ttrans), indicating that the reduction of the hysteresis is produced by the reduction of the FOPT character by Ni doping. This is 142 6. Studies of La(Fe,Si)13-based materials produced with a very minor modification of the Ttrans and σin comparison to similar elements as Co [163,164,167]. 0.0 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 FOPT n ( T trans ) Ni content, x x critical 0.19 SOPT Figure 6.11: n(Ttrans) values as a function of Ni content for the Ni-doped La(Fe,Si)13 series. The results obtained by the analysis of the exponent ncan be compared to those of applying the Banerjee’s criterion. The Arrott plots of the studied samples for temperatures close to Ttrans are plotted in Fig. 6.12 (low magnetic fields are not considered, as the samples are in multidomain state). The curves show negative slopes for x = 0.00 and 0.10 samples, indicating that they undergo a FOPT, while only positive slopes are observed for x = 0.18, 0.26 and 0.33 samples, indicating a SOPT behavior. These results are in good agreement with the MC field dependence analysis except for x = 0.18 sample. It should be noticed that this sample exhibits a composition very near to the critical SOPT-FOPT composition for this series. The imposition of a mean field model in this method leads to values of the critical exponents rather different from those of the critical SOPT-FOPT point, which attributes to its failure for this composition. Some discrepancies of the Banerjee’s criterion were also reported for other MC materials [94]. 143