Abstract Phase Transitions and Associated Magnetic

ABSTRACT

PHASE TRANSITIONS AND ASSOCIATED MAGNETIC AND TRANSPORT PROPERTIES IN SELECTED NI-MN-GA BASED HEUSLER ALLOYS

by Sunday Arome Agbo

The phase transitions and associated magnetic and transport properties of Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 have been investigated via x-ray diffraction, scanning electron microscopy, magnetic, calorimetric, and electrical resistivity measurements. While Ni2Mn0.70Cu0.30Ga exhibited a tetragonal structure at room temperature, cubic and orthorhombic phases coexisted in the other two samples. The scanning electron microscope micrographs indicated that no impurity phases existed in the samples. All three samples exhibited the first order martensitic phase transformation upon cooling and heating. The phase transitions were accompanied by large magnetic entropy changes and anomalies in the electrical resistivity data. For a field change of 50 kOe, peak magnetic entropy changes of -17 J kg-1K-1, -39 J kg-1K-1, -1 -1 and -26 J kg K were observed for Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 respectively, when the measurements were done while heating. When the measurements were done while cooling from 400 K to lower temperatures, the -1 -1 -1 -1 -1 -1 peak values were -33 Jkg K , -17 Jkg K , and -15 Jkg K for Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 respectively. The experimental results are discussed taking the decoupling and coupling of the phase transitions into consideration.

PHASE TRANSITIONS AND ASSOCIATED MAGNETIC AND TRANSPORT PROPERTIES IN SELECTED NI-MN-GA BASED HEUSLER ALLOYS

A Thesis

Submitted to the

Faculty of Miami University

in partial fulfillment of

the requirements for the degree of

Master of Science

Sunday Arome Agbo

Miami University

Oxford, Ohio

2020

Advisor: Dr. Mahmud Khan

Reader: Dr. Herbert Jaeger

Reader: Dr. Stephen Alexander

This Thesis titled

PHASE TRANSITIONS AND ASSOCIATED MAGNETIC AND TRANSPORT PROPERTIES IN SELECTED NI-MN-GA BASED HEUSLER ALLOYS

Sunday Arome Agbo

has been approved for publication by

The College of Arts and Science

and

Department of Physics

______Dr. Mahmud Khan

______Dr. Herbert Jaeger

______Dr. Stephen Alexander

Table of Contents

Chapter 1: Introduction ……………………………………………………....1 Chapter 2: Theoretical Background……………………………………….….5 2.1 Basics of Magnetism……………………………………….……..5 2.2 Classification of Magnetic Materials……………………………..6 2.2.1 Diamagnetism…………………………………………………..…7 2.2.2 Paramagnetism………………………………………………….....9 2.2.3 Ferromagnetism………………………………………………….11 2.2.4 Antiferromagnetism……………………………………………...15 2.2.5 Ferrimagnetism………………………………………………...... 15 2.3 Exchange Interactions…………….………………..…………….15 2.3.1 Direct Exchange Interaction……………………………………..16 2.3.2 Indirect Exchange Interaction……………………………………17 2.3.3 Superexchange Interaction……………….……………….……...17 2.3.4 Double Exchange Interaction……………………………………18 2.4 The Magnetocaloric Effect…………….…….……………….….18 2.4.1 Magnetocaloric Effect Thermodynamics………….………….….19 2.5 Phase Transition…………….…….……………….………….….23 2.5.1 The Martensitic Phase Transition ……………………………….23 Chapter 3: Experimental Methods………………………………………...... 26 3.1 Sample Fabrication…………………………………….…….…..26 3.2 Structural Characterization………………………………….…...27 3.2.1 Theory and Principle of X-Ray Diffraction Methods…….….…..27 3.2.2 X-Ray Diffraction Measurements...... 28 3.2.3 Analysis of X-Ray Diffraction Data……………………….…….30 3.3 Compositional Characterization………………………………....31 3.3.1 Scanning Electron Microscope………………………………..…31 3.3.2 Energy Dispersive X-Ray Spectroscopy………………………...34 3.4 Electrical and Magnetic Characterization…………………….…34 3.4.1 The Physical Property Measurement…………………………….36

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3.4.2 Vibrating Sample Magnetometer (VSM)…………………….....39 3.5 Calorimetric Characterization…………………………………..41 Chapter 4: Results and Discussion………………………………………….43 4.1 Temperature dependence of magnetization……………….….....43 4.2 Structural Analysis……………………….……………….….....47 4.3 Compositional Analysis……………………….………………..48 4.4 Electronic Transport Property Analysis……………...…………50 4.5 Field dependence of magnetization …………………………….54 Chapter 5: Summary and Conclusion………………………………………67 References …………………………………………………………………..68

List of Figures

Figure 1.1 The X, Y, and Z elements that constitute that constitute a Heusler Alloy are highlighted in the periodic table…………………………………………………….…1 Figure 2.1 Schematic diagram of the magnetic dipole moment………………………6 Figure 2.2 Pictorial representations of a diamagnetic material displaying, a) the alignment of net magnetic moments vs. external field H, b) magnetization vs. external field, and c) magnetization vs. temperature T…………………………………………..…9 Figure 2.3 Pictorial representations of a paramagnetic material displaying a) the alignment of magnetic dipole moments for zero and positive field strength, b) magnetization as a function of H and c) magnetization as a function of T……………..11 Figure 2.4 Schematic diagram showing Magnetic domains in a ferromagnetic material in the absence of external magnetic field…………………………………….…12 Figure 2.5 Pictorial representation of a magnetic hysteresis loop for a generic ferromagnetic material showing the basic parameters…………………………………...14 Figure 2.6 Schematic depiction of superexchange for MnO………………………...18 Figure 2.7 Schematic diagram showing isothermal and adiabatic processes of the MCE on the application and removal of magnetic field in a system………………….…21

Figure 2.8 Schematic diagram of L21 and C1b cubic structures…………….……….24 Figure 2.9 Schematic diagram showing the cubic, tetragonal, and orthorhombic crystal structures of Ni2MnGa …………………………………………………..………25 Figure 3.1 Schematic representation of Bragg’s condition………………………….28 Figure 3.2 Pictorial representation of an X-ray diffractometer ……………….…….29 Figure 3.3 Schematic diagram of an X-ray diffractometer………………………….30 Figure 3.4 Schematic diagram of SEM showing the main components…………….32 Figure 3.5 A depiction of a resistivity puck …………………………………….…..35 Figure 3.6 schematic diagram of the main components in the PPMS system……….36 Figure 3.7 Schematic diagram of the PPMS probe assembly showing the major components………………………………………………………………………...…….38 Figure 3.8 A depiction of a VSM-PPMS system showing its components………….40

Figure 3.9 Schematic diagram of a heat flux DSC………………………………...... 41

Figure 4.1 Temperature dependence of magnetization for Ni2Mn0.70Cu0.30Ga measured at H = 1 kOe…………………………………………………………………..44

Figure 4.2 Temperature dependence of magnetization for Ni2Mn0.70Cu0.25Cr0.05Ga measured at H = 1 kOe…………………………………………………………….…….45

Figure 4.3 Temperature dependence of magnetization for Ni2Mn0.70Cu0.30Ga0.95In0.05 measured at H = 1 kOe…………………………………………………………….…….46

Figure 4.4 Room temperature XRD pattern for Ni2Mn0.70Cu0.30Ga,

Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 ………………...... 48

Figure 4.5 Room temperature SEM micrographs of Ni2Mn0.70Cu0.30Ga,

Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 ……………………………….49

Figure 4.6 Temperature dependence of normalized resistance for Ni2Mn0.70Cu0.30Ga measured at zero magnetic field…………………………………………………………51 Figure 4.7 Temperature dependence of normalized resistance for

Ni2Mn0.70Cu0.25Cr0.05Ga measured at zero magnetic field……………………………….52 Figure 4.8 Temperature dependence of normalized resistance for

Ni2Mn0.70Cu0.30Ga0.95In0.05 measured at zero magnetic field……………………………53

Figure 4.9 Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.30Ga measured isothermally (warming and cooling) at temperatures near the martensitic phase transition...... 55

Figure 4.10 Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.25Cr0.05Ga measured isothermally (warming and cooling) at temperatures near the martensitic phase transition…………………………………………………………………………………56 Figure 4.11 Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.30Ga0.95In0.05 measured isothermally (warming and cooling) at temperatures near the martensitic phase transition……………………………………………………………………………...….57

Figure 4.12 Temperature dependence of magnetic entropy changes, ΔSM(T), for

Ni2Mn0.70Cu0.30Ga measured while (a) warming and (b) cooling………………………..58

Figure 4.13 Temperature dependence of magnetic entropy changes, ΔSM(T), for

Ni2Mn0.70Cu0.25Cr0.05Ga measured while (a) warming and (b) cooling………………….59

Figure 4.14 Temperature dependence of magnetic entropy changes, ΔSM(T), for

Ni2Mn0.70Cu0.30Ga0.95In0.05 measured while (a) warming and (b) cooling……………….60

Figure 4.15 Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.30Ga measured while warming and cooling at T = 345 K……………………………………..61 Figure 4.16 The Arrott plot, M2 versus H/M in the vicinity of phase transition temperature for Ni2Mn0.70Cu0.30Ga………………………………………………….…...62 Figure 4.17 The Arrott plot, M2 versus H/M in the vicinity of phase transition temperature for Ni2Mn0.70Cu0.25Cr0.05Ga…………………………………….…………..63 Figure 4.18 The Arrott plot, M2 versus H/M in the vicinity of phase transition temperature for Ni2Mn0.70Cu0.30Ga0.95In0.05………………………………………………63

Figure 4.19 DSC heat flow curves upon heating and cooling for Ni2Mn0.70Cu0.30Ga...64 Figure 4.20 DSC heat flow curves upon heating and cooling for

Ni2Mn0.70Cu0.25Cr0.05Ga………………………………………………………………….65 Figure 4.21 DSC heat flow curves upon heating and cooling for

Ni2Mn0.70Cu0.30Ga0.95In0.05……………………………………………………………….66

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Dedication

To the memory of my father, Shaibu Agbo

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Acknowledgements

I am eternally indebted to my advisor, Dr. Mahmud Khan for his sincere guidance and tireless support for the completion of this thesis and throughout my stay at Miami University. His patience, critiques and detailed explanations of concepts have really helped me in completing this research work. I would also like to use this medium to thank my committee members Dr. Herbert Jaeger and Dr. Stephen Alexander for taking their time to read my thesis and for their useful comments and suggestions.

Finally, I would like to thank my wife, Virginia and son Noah for their moral support and motivation. Love you both.

Chapter 1: Introduction

The current conventional refrigeration technologies have energy efficiencies of around 20% and use ozone depleting or greenhouse effect related gasses as refrigerant, which make them both economically and environmentally unfriendly [1-6]. The problems associated with these technologies have led to a search for alternative cooling technologies, such as magnetic refrigeration technology (MRT). The magnetocaloric effect (MCE) is a phenomenon, which is exploited in MRT. The process is described by the heating (or cooling) of a magnetic material when subjected to an external magnetic field [3-5, 7]. The magnetic moments in the material align with the field causing a reduction in the magnetic component of the total entropy [1]. In order to compensate for the decrease in this magnetic entropy (since the total entropy of a system is constant), there is an equivalent increase in the other components of the total entropy. For magnetocaloric materials, these other components are the electronic entropy and lattice entropy or heat. Magnetic refrigerators therefore utilize magnetocaloric materials as refrigerants. Therefore, the discovery of new materials having large tunable MCEs near room temperature plays a major role in the advancements of magnetic cooling technology.

In the past decades, research on MCE has focused on materials that exhibit first-order coupled magnetic and structural phase transitions. A first order phase transition is a transition that has discontinuities in quantities associated with the first derivative of Gibb’s free energy. During the phase transition, a discontinuity may be observed in the volume, magnetization, and entropy of a magnetic material [8]. The MCE associated with the first order phase transition has some peculiar characteristics including a large isothermal entropy change with sharp peak, small temperature operating range, and thermal and magnetic hysteresis. The second order transition has discontinuities in the second or higher derivative of Gibb’s free energy. The MCE characteristics associated with a second order phase transition include broad peak, wide temperature operating range, no thermal hysteresis, minimum magnetic hysteresis and small isothermal entropy.

The FOMPT materials that are extensively studied are Gd5Si2Ge2, La(Fe,Si)13, MnFePAs, and selected Ni-Mn based intermetallic alloys [9-12]. These materials typically exhibit large

1 magnetic entropy changes (a parameter that quantifies MCE) near the phase transition; and therefore, are considered potential candidates as refrigerants for use in a magnetic refrigerator [4]. However, a primary setback associated with these materials is the thermal and magnetic hysteresis (example is La (Fe,Si)13) which can dramatically reduce their refrigeration capacity by decreasing the refrigeration cycle. More so, some of these materials are too expensive as in the case of Gd5Si2Ge2 while others are poisonous (example is MnFePAs). Reduction of these setbacks while preserving the magnetocaloric properties of these materials is challenging, and is an extensively researched topic. In line with this discussion, we have studied the phase transitions and associated magnetic properties of three Heusler alloys namely, Ni2Mn0.70Cu0.3Ga,

Ni2Mn0.70Cu0.30Ga0.95In0.05, and Ni2Mn0.70Cu0.25Cr0.05Ga.

Heusler alloys are intermetallic compounds with two general stoichiometric forms: X2YZ (full Heusler) and XYZ (half Heusler). While X and Y are transition metal elements, Z is generally a main-group element. Y can occasionally be a rare earth lanthanide, as shown in Fig.1. [13, 14]. The discovery of these compounds was made in 1903 by Friedrich Heusler. He synthesized

Cu2MnSn and found it to exhibit ferromagnetic (FM) properties despite the fact that the compound did not contain any ferromagnetic elements. The magnetic properties observed in

Cu2MnSn and later in Heusler alloys in general were reported to be connected with their chemical composition, crystalline structure, and method of fabrication [15]. The research interest in these compounds is due to their interesting properties which include, but are not limited to, the magnetocaloric effect (MCE) [16-19], exchange bias effects (EB) [20], FM shape memory effects (FSME) [21], and large magnetoresistance (MR) [11].

Many of these properties are associated with the temperature-induced first order magnetostructural phase transitions (FOMPT): a coupled ferromagnetic to paramagnetic transition and a structural transition from martensitic phase to the high-temperature austenitic (cubic) phase. The temperature at which ferromagnetic to paramagnetic transition take place is known as the Curie temperature (푇퐶) while martensite temperature (푇푀) is the temperature at which the structural transition occurs.

Fig. 1.1. A periodic table showing the regions containing elements belonging to X, Y, and Z in Heusler Alloys [14].

Among the ferromagnetic Heusler alloys, Ni2MnGa-derivative materials have received considerable research interest. These materials exhibit multiple functional properties that are primarily linked with the first order martensitic phase transformation (MPT). The MPT is regarded as a non-diffuse structural transition from cubic (austenite) phase to a low-symmetry non-cubic (martensite) phase. Depending on the material, the martensite phase could be orthorhombic, tetragonal, or monoclinic [7], [22]. The MPT in Ni-Mn-Ga based Heusler alloys are often attributed to several intrinsic properties including, the valence electron concentration, spacing between the atoms in the crystalline lattice, and particularly the development of hybrid states between Ga p and Ni d orbitals. Heusler alloys also experience ferromagnetism; a phenomenon experience as a result of indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions (a concept discussed in detail in chapter 2 of this thesis) between neighboring Mn atoms in the crystal [22-23].

An interesting property of Ni2MnGa Heusler alloys is that their TM’s and TC’s can be precisely controlled by varying their stoichiometry and by atomic doping [24], [25]. For example, when

Mn is partially replaced by Cu in Ni2Mn1-xCuxGa, the martensitic transition shifts from 220 K to higher temperatures while the second order ferromagnetic transition shifts from 380 K towards lower temperatures [26-28]. For x = 0.25, the two transitions occur at nearly the same temperature resulting in a coupled first order phase transition [1]. As of now, the structural and magnetic properties of Ni2Mn1-xCuxGa with x > 0.25 have not yet been reported in literature. Since a first order phase transition exhibits a thermal hysteresis while a second order phase

3 transition does not, it is interesting to study the phase transitions in Ni2Mn1-xCuxGa with x slightly larger than 0.25. Coupling (structural and magnetic transitions occurring simultaneously) and decoupling effects may be expected in such a material. In line with this discussion, we have studied the structural and magnetic properties of Ni2Mn1-xCuxGa with x > 0.25. In the same fashion, we also studied the structural, electronic, magnetic, and magnetocaloric properties of

Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30 Ga0.95In0.05 compounds. The theoretical considerations of this thesis are discussed in chapter 2. The experimental methods employed are described in chapter 3. The results and discussion section of this thesis is presented in chapter 4. The concluding remarks of this project are presented in chapter 5.

Chapter 2: Theoretical Background

2.1 Basics of Magnetism

The discovery of magnetism is traced to the ancient world when people found out that lodestones could attract Fe [29]. The term magnet originates from a Greek word magnētis lithos [30], meaning the magnesian stone, lodestone [31]. While the ancient Greek and the ancient Chinese were credited with the discovery of magnetism and using it for navigation [32, 33], rational scientific treatment of magnetism was made by William Gilbert in 1600. William Gilbert, in his publication “On the Magnet and Magnetic Bodies, and on the Great Magnet the Earth”, he used terrella to describe many of his experiments. From his experiments, he arrived at a conclusion, which indicated that the Earth was itself magnetic; evident in the fact that the compasses pointed north (some people had earlier believed the compass was attracted by a huge magnetic island on the North Pole or a Polaris). He further reported that if lodestone is placed in contact with steel needles, the steel needle could be magnetized [34]. The experiment on the effect of a current on the magnetic needle published by Hans Christian Oersted was considered by some as the origin of “modern” magnetism research [35]. From his findings, he concluded that when brought close to a current-carrying wire, a compass needle would deflect. This submission was the first to suggest that electricity and magnetism could possibly have a connection. This suggestion was later confirmed in the early 1900s, during the quantum mechanics heyday. Further discoveries were made by Ampere (Ampere’s law), Biot-Savart and Faraday [36-38]. The deduction that magnetism comes from the motion of electrons and the spin of the electrons and nuclei which describe atoms was made from all the treatments of magnetism by the above-listed contributors and others not mentioned in this thesis.

Magnetism is regarded as one part of the joint phenomenon of electromagnetism. There are two sources of magnetism, namely, electric currents and spin-orbital interactions in atoms [39]. The spin-orbital interactions in atoms govern the creation/fabrication of new magnetic materials (which is the subject of this thesis) and this will be the focus of this subsection. The magnetic moment can be regarded as the magnitude and orientation of a magnetic dipole or other object

(loops of electric current, moving elementary particles, various molecules, and many astronomical objects) that produces a magnetic field [40]. It can be expressed mathematically by the equation given below:

휏⃗ = 휇⃗ × 퐵⃗⃗ 2.1 where 휏⃗ represents the torque of alignment produced by 퐵⃗⃗ (applied magnetic field) and 휇⃗ represents the magnetic moment of an atom. The net magnetic moment is linked only with atoms that have unfilled electron shells. This is because completely filled shell will have zero net spins. A schematic diagram of the magnetic dipole moment is shown in Fig. 2.1.

Fig. 2.1. Schematic diagram of the magnetic dipole moment.

The magnetic moment of an atom can be expressed mathematically as:

휇⃗ = 훾ℏ(퐿⃗⃗ + 푆⃗) 2.2 where 퐿⃗⃗ is orbital momentum, 푆⃗ is the spin angular momentum, 훾 represents gyromagnetic ratio

2.2 Classification of Magnetic Materials

As pointed out in section 2.1, the spin associated magnetism is generally characterized by the net magnetic moment, the ordering of the moment, and the temperature at which the ordering takes place. Based on the magnetic ordering of a material, magnetism may be classified as diamagnetism, paramagnetism, ferrimagnetism, ferromagnetism, and anti-ferromagnetism [41].

This classification is in terms of the magnetic behavior of the materials and is dependent on their bulk magnetic susceptibility [41, 42]. Magnetic susceptibility (χ) is a dimensionless (in Gauss unit) quantity that defines the degree of magnetization of a material when an external magnetic field is applied. It can be expressed mathematically as the ratio of magnetization M to the external magnetic field strength B. This is expressed by the equation below:

푀휇 푀 χ = 0 = 2.3 퐵 퐻 where 휇0 is the electromagnetic permeability of free space and H denotes the magnitude of the external magnetic field.

2.2.1 Diamagnetism

In diamagnetic materials, the atoms do not possess net magnetic moment in the absence of an applied field. In other words, there are no unfilled electronic shells of the constituent atoms. All materials have a diamagnetic effect but the diamagnetic effect may be obscured by the bigger paramagnetic or ferromagnetic term. The susceptibility does not dependent on temperature. Diamagnetic material has a negative susceptibility (χ< 0). The negative susceptibility is due to the fact that whether or not a magnetic field is present, there is always a permanent moment linked with the spin of the electron. However, the magnetic moment linked with the orbital movement of the electrons is proportional to the external magnetic field and is known as induced moment. The induced moment is often -ve (Lenz’s law), leading to diamagnetic effect [43]. Now, describing the diamagnetism phenomenon using Langev’s theory. When an atom is positioned in a magnetic field, the orbital electrons undergo Larmor precession about their orbit with a frequency given by:

푒퐵 휔 = 2.4 퐿 2푚 In the above Equation, e is the charge of an electron, B represents the external magnetic field, and m represents electronic mass. The Larmor procession acts opposite to the magnetic flux and creates a current (I) whose strength is given by the expression:

푍푒 푒퐵 퐼 = − 2.5 2휋 2푚

In the above Equation Z represents the number of electrons experiencing Larmor precession. That one may relate the current in Equation 2.5 to a magnetic moment, there is a need to define the radius of the loop denoted by r, that way one would be able to calculate the area. The radius, r is the mean square of the distance that runs perpendicular from the orbital electrons to the field axis that passes through the center of the nucleus. Therefore, taking the cross product of the area and the current, the expression for the induced magnetic moment is determined and given by:

푍푒2퐵 휇⃗ = − 〈푟2〉 2.6 4푚

Now, when μ is multiplied by N (taken to be the number of atoms in the system, this gives magnetization, dividing the result by B, we have:

휇 푁푍푒2 휇 푁휇 χ = − 0 〈푟2 〉 = 0 2.7 6푚 퐵

The Equation (2.7) above confirms that susceptibility does not depend on temperature for diamagnetic materials. A Pictorial representation of a diamagnetic material displaying the alignment of net magnetic moments as a function of external field H, magnetization as a function of external field, and magnetization as a function of temperature T is shown in Fig. 2.2.

Fig 2.2. Pictorial representations of a diamagnetic material displaying, a) the alignment of net magnetic moments vs. external field H, b) magnetization vs. external field, and c) magnetization vs. temperature T. [44]

2.2.2 Paramagnetism

Paramagnetism is a collective process that takes place in a group of atoms (in a crystal) having a permanent net magnetic moment in which the resultant magnetic moment aligns in the direction of the external magnetic field as shown in Fig. 2.3a [44]. There have been numerous theories of paramagnetism which hold true for particular classes of materials. One of these theories is Langevin theory that describes paramagnetism in materials with non-interacting localized electrons. Langevin mode theory postulates that the magnetic moment of each atom in a material has a random orientation due to thermal agitation. [44]. When a magnetic field is applied, these moments are slightly aligned in such a way that a low magnetization appears in the direction of the external field. Thermal agitation enhances with temperature, making it difficult to align the moments and as a result, the susceptibility decreases. This behavior is called the Curie law and it is mathematically expressed below in Equation 2.8

퐶 χ = 2.8 푇 In the above Equation C represents a constant known as the Curie constant and is dependent on the type of material.

Varieties of materials show paramagnetic behavior. Some of these materials include aluminum, oxygen, titanium, and iron oxide (FeO). These materials show +ve susceptibility in the presence of an applied field. Using the Langevin Theory of Diamagnetism, the source of the +ve χ for

9 paramagnetic material can be explained [8, 45-47]. From equation 2.2 we know that the magnetic moment per atom is given as:

휇⃗ = 훾ℏ퐽⃗ = −g 휇퐵퐽⃗ 2.9 where 퐽⃗ = (퐿⃗⃗ + 푆⃗), 휇퐵 is the Bohr magneton. It is expressed as: 푒ℏ 휇 = 2.10 퐵 2푚

The energy linked with individual magnetic dipole moment is given by:

U = − 휇⃗. 퐵⃗⃗ = 푚퐽g휇퐵B 2.11 where 푚퐽 denotes the azimuthal quantum number, g denotes spectroscopic splitting factor. The allowed values of the azimuthal quantum numbers are obtained using the total angular momentum value, J.

There is a restriction on the total angular momentum J value. It is restricted to ±1/2 for a system of a free atom with a single spin and zero orbital angular momentum. In such case, g is taken to be 2. The two probable values of energy of the same magnitude but opposite sign ±휇퐵B are obtained when the value of g and J are substituted in Equation 2.11. The expression for the magnetization (derived using Boltzmannian statistical method and power series) is given as:

푁휇 휇2퐻 푀⃗⃗⃗ = 0 2.12 3퐾퐵푇

In Equation 2.12 퐾퐵 represents the Boltzmann constant, and T denotes the temperature in Kelvin. Substituting Equation 2.12 in 2.3 gives the expression for the Curie’s law, given by the equation below.

푀⃗⃗⃗ 푁휇 휇2퐻 퐶 χ = = 0 = 2.13 퐻⃗⃗⃗ 3퐾퐵푇 푇

For magnetic materials experiencing ferromagnetic-to-paramagnetic transition at the Curie temperature,푇푐, equation 2.13 can be adjusted to Curie-Weiss law, expressed as: 퐶 χ= 2.14 (푇−푇푐) It can be inferred from equations 2.13 and 2.14 that paramagnetism is temperature dependent. A pictorial representation of the moment arrangements in a paramagnetic material in the presence

10 of zero and applied magnetic field is shown in Fig.2.3. The figure also shows the variation of the magnetization and susceptibility with H and T, respectively [44].

Fig 2.3. Pictorial representations of a paramagnetic material displaying a) the alignment of magnetic dipole moments for zero and positive field strength, b) magnetization as a function of H and c) magnetization as a function of T [44].

2.2.3 Ferromagnetism

The term ferromagnetism describes materials that have permanent magnetization. In other words, it describes materials that exhibit a net magnetic moment even in the absence of an external magnetic field [48]. The classical theory of ferromagnetism was introduced by Weiss in 1907 who described the phenomenon in terms of a molecular field within the ferromagnetic material. The Weiss molecular field arises from the interactions between the atoms that make up the 3 material and can be significantly large (not far from 10 Tesla) [49]. Therefore, Weiss field can act as the external magnetic field. It is worthy to note that the magnetization depends on the magnitude of the Weiss field.

Ferromagnetism can be divided into two types according to the nature of the coupling interaction within the material. The true ferromagnetism which is a consequence of the direct electron exchange between the neighboring atoms that is appropriately close together. This is well described in quantum mechanics by the Heisenberg model of ferromagnetism. This model talks about the magnetic moment being aligned parallel with regard to exchange interaction between neighboring moments. Weiss proposed the existence of magnetic domains within the material; described as regions of the alignment of atomic magnetic moments. The response of the material to a magnetic field is determined by the motion of these domains. It implies that the

11 susceptibility is dependent on external magnetic field. Thus, ferromagnetic materials are normally compared with regard to saturation magnetization (magnetization when there is an alignment of all domains) instead of susceptibility [44, 50, 51]. The idea of magnetic domains is demonstrated in Fig. 2.4.

Iron, nickel, cobalt, and gadolinium are the only elements in the periodic table that are ferromagnetic above and near room temperature. When a ferromagnetic material is heated the atomic magnetic moments lose their alignments and this loss enhances with increasing temperature. Consequently the saturation magnetization also decreases. Eventually the material undergoes a magnetic phase transition and becomes paramagnetic. The temperature of this transition is known as the Curie temperature, TC, and varies with materials. TC is 770°C for Fe,

1131°C for Co, and 358°C for Ni. Above TC, the susceptibility varies according to the Curie- Weiss law.

The second category of ferromagnetism results from the superexchange coupling between the neighboring atoms. This interaction occurs through the electron shell of an intermediate anion, e.g., oxygen ions in oxide materials. A good example of a ferromagnetic material in this category is magnetite, which has a Curie temperature of 580°C and exhibits the largest magnetic moment among all the naturally found minerals [50, 51].

Fig. 2.4. Schematic diagram showing Magnetic domains in a ferromagnetic material in the absence of external magnetic field [50].

From Fig. 2.4 above, it is important to note that though the individual domains in ferromagnetism can have magnetization (which is not zero) but the net magnetization of the bulk material is nearly zero due to the domains invalidating each other. There are two microscopic reorganizations that will take place when a ferromagnetic material is placed within an applied magnetic field. For a small nonzero magnetic field, the domains parallel to the external magnetic field (preferable domains) will start to grow whereas the oriented domain that is less preferable will reduce in size via a method called boundary displacement [8, 45]. As a result, there will be an increase in the net magnetization of the specimen as compared to the situation where the field is zero. Increasing the external magnetic field steadily, a point will reach when the energetically- preferable domains will be unable to expand further, causing the magnetization of the domains antiparallel to the external field to be rotated.

Whenever ferromagnetic material is placed in a varying magnetic field, it responds to the changing magnetic the field by exhibiting a magnetic hysteresis. It is a closed-loop which shows that the magnetization lags behind the varying applied magnetic field through a complete cycle. The interpretation of the bulk magnetization is possible using the magnetic hysteresis. This interpretation uses four basic parameters. These include the saturation magnetization (푀푆), which is defined as the maximum magnetization that does not increase with any further increase in magnetic field H; saturation remanent magnetization푀푅푆, defined as the magnetization residual at zero field; coercivity 퐻푐, defined as magnetic field required to get rid of the induced

13 magnetization; and remanent coercivity 퐻퐶푅, the magnetic field applied to get rid of remanent magnetization 푀푅푆 [44]. A schematic diagram of the hysteresis loop for a generic ferromagnetic material showing basic parameters is shown in Fig. 2.5.

Fig. 2.5. Pictorial representation of a magnetic hysteresis loop for a generic ferromagnetic material showing the basic parameters [44].

2.2.4 Antiferromagnetism

In antiferromagnetic materials the magnetic moments aligns antiparallel due to the exchange interaction between neighboring atoms. As a result, there is a cancellation of the magnetic field and the material seems to behave in similar manner as a paramagnetic material [52, 53]. Above a transition temperature, antiferromagnetic materials just like ferromagnetic materials become paramagnetic. The temperature at which this transformation happens is known as the Néel temperature, 푇푁. The Néel temperature for chromium is 37ºC.

2.2.5 Ferrimagnetism

Ferrimagnetism is a phenomenon that generally occurs in compounds whose crystal structures are more complex than pure elements. In ferrimagnetic materials, the exchange interactions results in the atoms being aligned parallel in some of the crystal sites while in others the alignment is anti-parallel. Just like in a ferromagnetic material, ferrimagnetism material breaks down into magnetic domains. There is also a similarity in the magnetic behavior of both materials. Ferrimagnetic materials exhibit relatively lower saturation magnetizations than ferromagnetic materials [45].

2.3 Exchange Interactions

When two neighboring particles in a crystal are identical, exchange interactions may occur between them through a quantum mechanical process. The wave functions of the indistinguishable particles exhibit exchange symmetry. In other words, when the two particles are exchanged, their wave function remains unchanged or changes sign (symmetric or antisymmetric, respectively). The exchange interaction can be experienced by both bosons and fermions. The electrons which bring about most magnetic behaviors are fermions. This interaction is occasionally known as Pauli’s repulsion for fermions and it is related to the Pauli Exclusion Principle that states, “no two electrons can occupy the same quantum state” [31, 54]. There are four major exchange interactions that are associated with the magnetic materials being

15 studied in this project. These exchange interactions which will be discussed shortly include direct exchange, indirect exchange, double exchange, and superexchange.

2.3.1 Direct Exchange Interactions

Normally, exchange interactions are short-ranged; the direct exchange interactions are confined to electrons in nearest neighbor atoms. For crystals, the Heisenberg Hamiltonian for the exchange interaction between all the (i,j) pairs of atoms of the many-electron system can be expressed as follows:

1 퐻 = (−2퐽 ∑ (푆⃗ . 푆⃗ )) = − ∑ 퐽〈푆⃗ . 푆⃗ 〉 2.15 퐻푒푖푠 2 푖,푗 푖 푗 푖,푗 푖 푗

The 1/2 factor in Equation 2.15 is incorporated to explain the fact that the interaction between the same two atoms is totaled two times in carrying out the sums. The J represents the exchange constant. The relationship depends on the crystal structure. An example is when considering a simple cubic lattice having a as its lattice parameter, A, the exchange stiffness constant which functions as a characteristic of a ferromagnetic material is given as [55]:

퐽 〈푆2〉 퐴 = 푒푥 2.16 푠푐 푎 where Jex the exchange integral. For a body-centered cubic lattice, we have

2퐽 〈푆2〉 퐴 = 푒푥 2.17 푏푐 푎 4퐽 〈푆2〉 퐴 = 푒푥 2.18 푠푐 푎

2.3.2 Indirect (RKKY) Exchange Interaction

Kittel and Ruderman were the first to propose the Indirect Exchange Interaction. This was proposed in 1954. Kasuya and Yosida later improved on the proposed Indirect Exchange Interaction by Kittel and Ruderman. The interaction is often referred to as the RKKY interaction where the nuclear magnetic moments due to the localized inner d- or f-shell electron spins in metals interacts via the conduction electrons [55]. This phenomenon occurs as a result of a material having a magnetic moment being positioned in a sea of conduction electrons, and as such does not have any restriction as regard to being spin polarized by the “impurity” sites. An indirect exchange interaction with other sites within a particular radius can then result due to the spin-polarization of these electrons.

2.3.3 Superexchange Interaction

Unlike direct interaction which involves coupling between nearest-neighbor cations without requiring an intermediary anion, Superexchange, involves strong coupling between two next-to- nearest neighbor cation via a non-magnetic anion. Superexchange occurs as a consequence of the electrons originating from the same donor atom and being coupled with the receiving ions' spins. The coupling can be either ferromagnetic or antiferromagnetic but mostly antiferromagnetic. Hendrik Kramers proposed the phenomenon (superexchange) in 1934 when he found that crystals interact with one another. He noticed this behavior in MnO. He found that Mn atoms in MnO interact with one another even though they have nonmagnetic oxygen atoms between them (see Fig. 2.6) [56]. Kramer’s model was improved upon by Phillip Anderson in 1950 [57]. In the 1950s, John B. Goodenough and Junjiro Kanamori established a set of semi-empirical rules which are now called the Goodenough-Kanamori rules [58- 60].

Fig. 2.6. Schematic depiction of superexchange for MnO [56].

2.3.4 Double Exchange Mechanism

The double exchanged mechanism, a magnetic coupling interaction was first proposed in 1951 by Clarence Zener to account for electrical transport properties [61]. It is a phenomenon that predicts the relative ease with which electrons from one magnetic atom may be exchanged with another magnetic atom with a non-magnetic atom acting as an intermediary in this exchange. This mechanism has significant implications for whether materials show spiral magnetism. It also has a significant implication as to whether materials are ferromagnetic or antiferromagnetic [62]. The double exchanged mechanism differs from superexchange in that in the latter, the occupancy of the shell of the two metal ions is the same or varies by two, and the electrons are localized whereas, for double exchange mechanism, the electrons are delocalized resulting in the material exhibiting magnetic exchange coupling along with metallic conductivity. In other words, while a ferromagnetic or antiferromagnetic alignment occurs between two atoms with the same number of the electron in superexchange, in double exchange mechanism, one atom needs to have an extra electron compared to the other for the interaction to take place [63].

2.4 The Magnetocaloric Effect

The magnetocaloric effect (MCE) was discovered in Fe in 1881 by Warburg [64] and the theoretical explanation of the phenomenon was made by Debye and Giauque in 1926 and 1927, respectively [65-67]. The phenomenon is the thermal response of a magnetic material upon its

18 exposure to a changing magnetic field. The thermal response is generally quantified by two parameters; the isothermal magnetic entropy change, ΔSM(T), and the adiabatic temperature change, ΔTad(T).

A prototype magnetic refrigerator that can work at room temperature was constructed in 1976 by G.V. Brown [68]. This breakthrough was achieved after nearly 100 years after the first observation of the MCE. The constructed gadget used pure Gd as the coolant and used magnets so as to prompt a huge temperature variation. One of the limitations of this device is the use of expensive superconductors and the fact that pure Gd is expensive. Research geared towards overcoming challenges associated with the prototypical magnetic refrigeration unit continued to intensify. In 1997, there was a discovery of abnormally large MCE known as the giant MCE in

Gd5(Si2Ge2) by Gschneidner and Pecharsky Pecharsky [10]. MCE is currently being utilized in refrigeration technologies for hydrogen and helium liquefaction. These refrigerators work between 4.2 K and 20 K. Recent discoveries indicated that MCE may be used to design room temperature refrigerators as well [69, 70]. An extensive thermodynamic explanation of the MCE is given in the subsequent sections.

2.4.1 Magnetocaloric Effect Thermodynamics

The origin of the MCE can best be explained using thermodynamics, which establishes the relation among the magnetic variables, entropy, and temperature. Though all magnetic materials intrinsically exhibit MCE, the degree of this effect is dependent on the properties of each material. A diagram showing the two basic processes (isothermal and adiabatic) of the MCE on the application and removal of magnetic field in a system is presented in Fig. 2.7.

For a ferromagnetic material at constant pressure, the total entropy S (H, T) is a function of an applied magnetic field H and temperature T. It is comprised of the lattice entropy Sl (T), which is dependent on temperature, electronic entropy Se (T), a function of temperature, and magnetic entropy Sm (H, T), a function of temperature and applied magnetic field. The expression for S (H, T) is given by the Equation below

S (H, T) =Sm (H, T) + Sl (T) + Se (T) 2.19 Let’s consider a case where there is an application of a magnetic field adiabatically in a process that is reversible, there will be a decrease in the magnetic entropy but since the total entropy of the system remains constant, there will be an increase in temperature. This implies,

S (Hi, Ti) = S (Hf, Tf) 2.20 The adiabatic temperature increase can be viewed as the measurement of the magnetocaloric effect in the material. This can be expressed as:

∆Tad= 푇푓 − 푇푓 2.21

Let’s consider a case where there is an application of the magnetic field in an isothermal process, this will lead to a decrease in the total entropy of the system as a result of the decrease in the magnetic contribution as such, the entropy change is therefore given by the Equation below.

∆푆푚 = S (Ti, Hi) − S (Ti, Hf) 2.22

H =0 H≠ 0 T = const. H ∆푆 ≠ 0

disorder

S2 S1

disorder order

Isothermal process ∆S ≠ 0 (푆2 > 푆1) .

H≠ 0 H = 0

H S = const. ∆푇푎푑 ≠ 0

푇1 푇2

order disorder

Adiabatic process ∆푇푎푑 ≠ 0 (푇1 > 푇2)

Fig. 2.7. Schematic diagram showing isothermal and adiabatic processes of the MCE on the application and removal of magnetic field in a system [70].

The two quantities (adiabatic temperature change (∆Tad) and isothermal entropy change (∆Sm) characterize MCE. Magnetic field H, magnetization M and temperature are related to the

21 magnetocaloric effect values ∆Tad (T, ∆H) and ∆Sm (T, ∆H), by a Maxwell Equation expressed as:

휕푆(푇,퐻) 휕푀(푇,퐻) ( ) = ( ) 2.23 휕퐻 푇 휕푇 퐻

When Equation 2.23 is integrated for a process occurring at constant temperature (and constant pressure) then Equation 2.24 below is obtained

퐻푓 휕푀(푇,퐻) ∆Sm(푇, ∆퐻) = ∫ ( ) dH 2.24 퐻푖 휕푇 퐻

where the sign of ∆SM is determined by the sign of the derivatives. If ∆SM is negative, it is denoted as direct MCE. While if it is positive, it is known as inverse MCE. Equation 2.24 explains the proportionality relation between the magnetic entropy change and both the deferential of magnetization with respect to temperature at constant magnetic field and to the magnetic field change. Employing the following thermodynamic relation

휕푇 휕푆 휕푇 ( ) = − ( ) ( ) 2.25 휕퐻 푆 휕퐻 푇 휕푆 퐻

휕푆 퐶퐻 = 푇 ( ) 2.26 휕퐻 퐻 where 퐶퐻 in Equation 2.26 is the specific heat capacity at constant magnetic field. The infinitesimal adiabatic temperature change can be expressed as (considering Equation 2.23)

푇 휕푀(푇,퐻) 푑Tad = − ( ) ( ) dH 2.27 퐶(푇,퐻) 퐻 휕푇 퐻

When Equation 2.27 is integrated, adiabatic temperature change (∆Tad), the second quantity that characterizes MCE is determined and given by Equation 2.28 below

퐻 푇 휕푀(푇,퐻) ∆T (푇, ∆퐻) = − 푓 − ( ) ( ) dH 2.28 ad ∫퐻 푖 퐶(푇,퐻) 퐻 휕푇 퐻

where Hf and Hi are the final and initial magnetic fields respectively. The -ve sign is incorporated into the above equation to justify the point that the magnetization of a material decreases as the temperature of the material increases, magnetic field magnitude notwithstanding.

2.5 Phase Transitions

A phase transition is regarded as a change in state from one phase to another. A sudden change in one or more physical properties with an insignificant variation in temperature is the characteristic that defines phase transition [71]. The phase transition is not limited to the transition from gas to liquid to solid or vice versa. Similar to the phase transitions which separate the states of matter, a phase transition is said to occur in a magnetic material when there is a transformation in the state of magnetic ordering. It is worthy to note that a temperature-induced phase transition is proportionate to a rapid change in the magnetization for magnetic materials. This can result in enhancing magnetocaloric effect properties. In this current project, the phase transitions applicable are second-order phase transitions and first-order magneto-structural phase transitions.

2.5.1 The Martensitic Phase Transition

The martensitic phase transition (MPT), named after Adolf Martens [72], is a diffusionless structural transition from austenite (highly-symmetric cubic phases) to a low-symmetry (orthorhombic, tetragonal, or monoclinic) martensite phase [73, 74]. The austenite is named after Sir William Chandler Roberts-Austen due to his revolutionary work with iron allotropes. Heusler alloys crystallize in either one of the two cubic phrases (L21 or C1b cubic structure) at their specific high temperature. A schematic diagram of L21 and C1b cubic structures is displayed in Fig. 2.8.

Fig. 2.8. Schematic diagram of L21 and C1b cubic structures [75].

During the MPT, the atomic positions of the constituent elements do not change much and is generally less than the interatomic distance between the atoms. The MPT in Heusler alloys particularly in Ni-Mn-Ga based are often attributed to several intrinsic properties including, the valence electron concentration, interatomic spacing, and the development of hybrid states. The crystalline structure of the martensitic phase is dependent on some factors which include elements in making the alloys and the relative concentration of the elements. As such, the structure can be tetragonal, orthorhombic, or monoclinic. A schematic diagram showing the cubic, tetragonal, and orthorhombic crystal structures of Ni2MnGa is presented in Fig. 2.9.

Fig. 2.9. Schematic depiction of the crystal structure of Ni2MnGa showing (a) cubic L21 structure (b) a body centered tetragonal L10 structure (c) the 3M premartensitic structure of Pnmn orthorhombic sructure (d) the martensitic 7M superstructure of Immm orthorhombic structure (e) modulated orthorhombic unite cell [74].

The MPT normally starts with nucleation and develops up to when the transformation is done, as such; it is not a continuous phase transition. Since the intrinsic properties such electronic structure, conduction electrons among others depend on the structure of the compound, a noticeable changes of these properties are generally observed during the MPT. The change in the latent heat of the material during MPT caused by the nucleation and growth of competing structural phases is the most significant property change related to the work in this Thesis [8].

Chapter 3: Experimental Methods

3.1 Sample Fabrication

The Heusler alloys Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30 Ga0.95In0.05 of

~2 g each (a digital balance was employed to measure the mass) were prepared using the standard arc melting technique under argon atmosphere using Ni, Mn, Cu, Cr, In and Ga virgin metals of more than 3N (> 99.9%) purity gotten from Alfa Aesar and Materion. The appropriate mass of each of the elements was calculated using stoichiometric calculations and measured using a digital balance. Pliers set were used to cut each of the elements to a desired mass. An excess 2.5% Mn was added to each sample to mitigate mass loss during melting. In order to make sure that there is homogeneity of the samples after melting; the samples were flipped and melted again. This was done a minimum of three times. For each sample, mass loss was found to be less than 0.4 %, an appropriate margin of error.

In the arc melting method, there is sealed vacuum chamber where an electric arc is produced between a needle-like cathode and a high purity copper hearth, this is made possible with a help of a high-current and low-voltage current source. Sample constituents and titanium are normally placed in cavities in copper hearth inside the chamber. The chamber is then evacuated to less than 70 milliTorr and purged three times by argon gas. Afterwards, the electric arc is turned on, producing a blue jet of high energy argon plasma of up to 2200 ℃ capable of melting sample elements thereby creating intermetallic polycrystalline button. It is worthy to note that in order to avoid contamination from any possible impurities, the titanium getter was melted before melting the sample. Cold water was circulated through the system during melting to prevent cathode and anode copper hearth from melting.

To ensure homogeneity of the crystal that will be formed, the arc-melted samples were wrapped in Ta foil (to prevent the possible diffusion of samples), sealed in a Vycor tube that is partially filled with argon gas, and annealed in an electric furnace at 850 °C for 72h. Immediately the samples were taken out from the chamber, they were quenched in cold water after which they were then cut into pieces by a low-speed diamond saw for all the required measurements.

3.2 Structural Characterization

The crystal structures of the investigated Heusler alloys were characterized by X-ray diffraction technique. The method is widely adopted for various reasons which include, but are not limited to its uniqueness (it is very often the available method to analyze material properties when these properties depend on the crystal state), non-destructive nature (samples do not change during and after measurement), speed of analysis, accurate analysis (relative accuracies of 0.1% to 0.3 0.1% are achievable) and the wavelength of x-rays is comparable to the inter-atomic distance of a crystal [76]. The theory and principle underlying this technique is discussed.

3.2.1 Theory and Principles of X-Ray Diffraction Methods

X-rays are regarded as a type of electromagnetic radiation. The theory of X-rays is therefore based on the interaction of electromagnetic radiation with matter. The diffraction effects associated with X-rays were observed by W.L. and W.H. Bragg in 1913. They noticed that X- rays reflect at crystal specific angles rather than the classical reflection by visible light. Their observation led to the derivation of the Bragg condition given by

2푑sin휃 = 푛휆 3.1

Where d is the inter-planar spacing of the crystal (in Angstroms), 휃 is the incident angle relative to diffracting plane, λ is the wavelength of radiation used, and n is the order of reflection, which signifies the penetration of x-rays into deeper crystal planes. The schematic representation of the Bragg’s condition is given in Fig. 3.1 below.

Fig. 3.1. Schematic representation of Bragg’s condition [77].

3.2.2 X-Ray Diffraction Measurement

In order to determine the phase purity and identify the crystal structures of the Heusler alloy systems (Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05), X-ray diffraction (XRD) measurements were carried out on a Scintag PAD-X Powder X-ray Diffractometer. The device comprises of an x-ray source, a monochromator to get the desired wavelength, a high voltage power supply, a sample holder, and a detector. A pictorial representation of the device is displayed in Fig. 3.2.

Fig. 3.2. Pictorial representation of an X-ray diffractometer.

The measurement using this device involves grinding the sample to fine powder with a mortar and pestle. The fine powder is then smeared on a rectangular glass. The glass is then placed in the sample holder. Radiations from the x-ray are then shined on the surface of the powder sample placed in the sample holder. The detector collects the radiations diffracted from the powder sample. This instrument employs theta-theta geometry, similar to the diagram depicted in Fig. 3.1 to measure the diffraction patterns at room temperature. The intensity of reflected x-rays was collected every 0.01° for 2 theta ranging from 20° to 80°. These measurements take

29 approximately 1 hour and 20 minutes to complete. A diagram depicting the x-ray Diffractometer is displayed in Fig. 3.3.

Fig 3.3. Schematic diagram of an X-ray diffractometer [77].

3.2.3 Analysis of X-Ray Diffraction Data

The goal of this project is to investigate the structural and magnetic properties of selected Heusler alloy systems and see how these properties correlate. In order to achieve this, there is a need to get the Information on crystal structure, structural parameters, and crystalline phases of the alloys. This information can be gotten from x-ray diffraction pattern. The diffraction pattern is determined by the type of the unit cell forming the crystal structure. The crystalline phase of the samples is usually identified by matching the x-ray diffraction patterns with the expected patterns provided by the database given by the International Centre for Diffraction Data (ICDD). The lattice plane spacing (d) and the lattice parameters (a) are related by the equation given by

푎 푑 = 3.2 ℎ푘푙 √ℎ2+ 푘2+ 푙2

In Equation 3.2 h, k, and l represent the Miller indices of the crystal lattice. Equation 3.2 holds true for cubic systems alone. For sample with a tetragonal structure (as in the case of this project), d and the 2 lattice parameters detailing the tetragonal martensite structure are connected via the Equation below

1 ℎ2+푘2 푙2 = + 3.3 푑2 푎2 푐2

For this thesis, we employed diffraction analysis software, Powdercell [78], to index the diffraction data.

3.3 Compositional characterization

3.3.1 Scanning Electron Microscope

The scanning electron microscope (SEM) is a multipurpose instrument employed to examine and analyze the microstructure morphology and composition characterizations [79]. The electronic console and electron column are the two major components that comprise the SEM instrument. The electronic console is comprised of control knobs and switches. These components are used for instrument adjustments: these include accelerating voltage, contrast, focus, magnification, filament current, and brightness. There are SEMs that use a computer system in connection with the electronic console. This makes it needless to have bulky console that contains control knobs, Cathode Ray Tubes (CRTs), and an image capture device. Examples of such SEMs are the FEI Quanta 200 and the one employed in this project. The mouse and the keyboard are used to navigate the computer system through which all the primary controls are accessed. In the setup, the user has to only be acquainted with the graphic user interface or software that controls the instrument in preference to control knobs and switches usually found on older style SEMs. The CRTs located on the electronic console display the image produced by the SEM whereas with FEI the image is displayed on the computer monitor [80-82].

Fig. 3.4. Schematic diagram of SEM showing the main components [77].

The second component of the SEM, the electron column, which happens to be the place the electron beam is produced under vacuum, fixated to a lesser diameter, and scanned across the surface of a sample using electromagnetic deflection coils. The lower part of the column is known as the sample chamber. Inside this chamber slightly above the sample, is where the secondary detector is located. Samples are mounted and secured onto the stage, controlled using a goniometer. [80-82]. The schematic diagram of a scanning electron microscope showing its main components is displayed in Fig. 3.4.

Images are produced using SEM by scanning the sample (placed on the stage) with a high- energy beam of electrons. The interaction of the electrons with the sample results in the productions of secondary electrons (SE), backscattered electrons (BSE), and characteristic x- rays. Signals produced by SE and BSE are normally analyzed in SEM to understand the surface topology and composition of the sample respectively. As such, compositional and topographical information about the sample are derived from these signals. The intensity of BSEs is used to determine the composition of the sample, whereas topographical information is derived from SEs. The images which are formed and displayed on the computer screen are the signals collected by one or more detectors. When the electron beam comes in contact with the surface of the specimen, it can pierce into the specimen to a depth of a few microns. The penetration is dependent on two major factors; accelerating voltage and the density of the specimen [77].

The samples considered in this thesis are Heusler alloys. The fabrication of these samples was not designed to have reasonable topographic features. As a result, there was no striking information from the SEs. Since one of the goals of this project is to confirm the correct composition of the samples and to search for phase segregation, the signals produced by BSEs give more information for this purpose. A charge signals from the BSEs is created by placing the quadrant silicon detector photo-multiplier directly above the sample.

For this project, we used a Zeiss Supra 35 VP FEG microscope to carry out the SEM investigations. In order to obtain high quality SEM data, each sample was carefully polished to a mirror-like surface by using a polisher [77].

3.3.2 Energy-Dispersive X-ray Spectroscopy

Energy dispersive x -ray spectroscopy (EDS) is a standard analytical method employed for identifying the chemical elements in a sample (qualitative analysis), and also for estimating the relative abundance of these elements (quantitative analysis) [83, 84]. This technique is typically carried out in conjunction with an SEM and uses x -rays that are radiated from the specimen when the electron beam is bombarded to characterize the elemental composition of the specimen [84]. As previously explained in section 3.3.1, in SEM, the instrument scans focused electron beam across the surface of the specimen and produces x -ray fluorescence from the atoms in its path. The EDS x -ray detector detects the number of x -rays emitted versus their energy. In qualitative analysis, the elements present in the sample are determined by determining the energies of the x -rays emitted from the area being excited by the electron beam. In quantitative analysis, the concentration of the element present is measured by the rate of detection of these x - rays. The Zeiss Supra 35 VP FEG microscope, we employed to carry out the SEM investigations in this thesis is equipped with a Bruker EDS, which has the capability of detecting characteristic x - rays stemming from elements with higher atomic mass than sodium. In order to make sure that all the transition elements are visible, imaging and EDS were carried out using 15 KeV. Software called ESPRIT was used to quantity the EDS data.

3.4 Electrical and Magnetic characterization

All the electronic and magnetic property measurements for this project were carried out employing a Physical property measurement system (PPMS) manufactured by Quantum Design Inc. The PPMS is a multipurpose device which is capable of carrying out various measurements, including but are not limited to torque magnetization, ac and dc magnetization, electronic transport property measurements. This is possible by installing suitable experimental options alongside an automated sequence written for specific measurements [77]. In order to take the electronic properties measurements, the resistivity option in the PPMS was used. Resistivity measurements versus temperature (ρ-T) were performed for the electrical characterization of the samples. The steps for the measurements involved cutting the samples

34 fabricated based on the steps discussed in section 3.1 followed by polishing to form thin rectangular pieces on which four electrical contacts were made. These pieces were then mounted on the resistivity puck (see Fig. 3.5). This puck consists of four leads, two of these leads are current leads, I+ and I-, via which current flows and the remaining two are voltage lead, V+ and V-, across which voltage is measured. This technique employs standard four-probe method by connecting four leads, I+ and I-, through which current flows and two are voltage lead, V+ and V-, across which voltage is measured. Resistivity was measured with temperature ranging from 5 K to 400 K.

Fig 3.5. A depiction of a resistivity puck.

For the magnetic property measurements, magnetization (M) measurements vs temperature [M(T)] and applied magnetic field [M(H)] were conducted employing the Vibrating Sample Magnetometer (VSM) option in PPMS. In this research, the measurements were carried out in the temperature range of 5–400 K and in applied magnetic fields of up to 50 kOe. Magnetization under Zero field cooling (ZFC) condition was performed by cooling sample from room temperature to 5 K in a zero magnetic field. When the temperature became steady at 5 K, a field of 1000 Oe was then applied and ZFC data was obtained. Field cooling (FC) data were obtained while cooling down the sample to 5 K in the presence of the 1000 Oe field.

3.4.1 The Physical Property Measurement System

The PPMS, as earlier discussed in section 3.4, is a multipurpose device which is capable of carrying out various measurements. This is possible by installing suitable experimental options

35 alongside an automated sequence written for specific measurements. The temperature range for which the PPMS is able to carry out measurements is from 1.7 – 400 K and in an external magnetic field of up to 9 T in magnitude. It is worth to note that the detailed description of the PPMS, VSM and other components discussed in this section and section 3.42 are based on the contents presented in PPMS Hardware Manual [85]. A schematic representation of the PPMS showing its major components is displayed in Fig. 3.6.

Fig 3.6. Schematic diagram of the main components in the PPMS system [85].

As can be seen from the figure above (Fig. 3.6), the main components of the PPMS include Dewar, sample probe, controller, vacuum Pump, electronic cabinet, and sample puck and assorted tools. The PPMS uses software called MultiVu which is windows-based control software. It enables the PPMS to create an automated sequence command and control the

36 system parameters like temperature and magnetic field.

The dewar contains liquid helium (He). It has a capacity of 30 L. The liquid helium is transferred from time to time into the system from an outside He recovery and liquefaction plant. There is a jacket space that surrounds the He dewar. It is occupied with liquid N of 40 L capacity from time to time. This jacket is required because liquid He boils at 4.22 K. As such when the He dewar is surrounded with 77 K liquid N, it help to reduce the boiling rate. Such reduction is substantial. This helps preserve helium in liquid state inside the dewar. The ruminant is then evacuated from the dewar by the help of a dry scroll pump.

The probe is comprised of various components. These components are helium impedance tube, electronic connections, the sample puck connectors, the helium level meter, the superconducting magnet, the gas lines, and temperature control hardware. The probe is immersed in a helium bath which allows its components to be cooled at all time. In order to enable the apparatus relate with the Model 6000 Controller and functions as a relay for any experimental data, a 12-pin is connected at the base of the probe. Direct-drive vacuum pump is used to draw He via the cooling annulus. This helps to control gas flow. This makes certain that the sample chamber is cooled to an even temperature. This is obtained by taking the average of the readings from two platinum resistance and negative temperature coefficient thermometers positioned in the sample region. The chamber is heated using a nichrome heater located beneath the sample puck. The diagram depicting the PPMS probe assembly showing the major components is shown in Fig. 3.7.

Fig 3.7. Schematic diagram of the PPMS probe assembly showing the major components [85].

The electrical connections to the magnet and impedance assembly are contained in the baffled rods. These rods run through the length of the probe. It is required that the coil stays in the superconducting state all the time. To achieve this, the niobium-titanium superconducting coil is immersed fully in the liquid N bath. The orientation of the superconducting coil is such that the magnetic field produced whenever a current flow through the circuit is along the long axis of the sample region. In order to disconnect the exterior current source once the desirable field is established by the means of the persistent current, a persistent switch is used.

The electronics cabinet comprises of a vacuum pump, a Model 6000 controller, and a power strip. In addition, the electronics cabinet also has some space for additional hardware and other 38 electronics that are needed for some PPMS options.

The direct drive vacuum pump controls the sample chamber pressure and temperature. This pump is installed in the bottom of electronics cabinet.

Model 6000 PPMS Controller is an integrated user interface which is made up of gas valves and gas lines (which are used to control temperature), CPU board, a motherboard, and a system bridgeboard. The valves help to regulate gas flow rate and the vacuum.

The sample puck is used to hold the sample which can then be inserted into the sample chamber for measurements. Insertion rod, adjustment tool and wiring testing tool are used for handling and testing of the puck function.

3.4.2 Vibrating Sample Magnetometer (VSM)

The VSM is one of the measurement options existing in PPMS. It is manufactured by Quantum Design. It is a very sensitive device employed for measuring the magnetic properties of the sample. The device (VSM) is capable of taking a precise measurement of the magnetic moment. It is very sensitive to a significantly small moment changes which can detect as low as 1 millionth emu of magnetic moment change. When the sample to be measured is placed inside the VSM, the sample vibrates perpendicular to the external magnetic field which is provided by the superconducting coil. The sample travels between the center points of a two-segment pickup coil which is found in the sample region. The sample is magnetized by the external field based on its state. It becomes magnetized when in a ferromagnetic state. Consequently, the motion of the sample produces a magnetic flux. An emf is exerted on the pickup coils by the flux based on Faraday’s law given by the equation below:

푑∅ 푑∅ 푑z 푉 = = 3.4 푖푛푑푢푐푒푑 푑푡 푑푧 푑푡

39 where, ∅ is the magnetic flux induced in the pickup coil, z is the vertical distance from the sample to the coil, and t is the time. The pickup coils will experience a voltage induced by the emf. The induced voltage can then be converted into a magnetization value. The sample oscillates at a frequency of 40 Hz by default in the VSM. The PPMS has the capability of controlling the oscillatory behavior in a precise manner and as such will be able to detect variations in magnetization. The voltage across sinusoidally oscillating sample is expressed as:

푉푐표푖푙 =2휋푓퐶푚퐴sin(2휋푓푡) 3.5

where, f is the frequency of sample oscillation, C is the coupling constant, m is the dc magnetic moment, and A is the amplitude of sample oscillation.

The magnetic moment measurement is acquired through the measurement of the coefficient of the sinusoidal voltage response from the detection coil. The VSM motor module controls the position and amplitude of oscillation. A depiction of a VSM-PPMS system showing its components is presented in figure 3.8.

Fig. 3.8. A depiction of a VSM-PPMS system showing its components [85]

3.5. Calorimetric Characterization

Differential scanning calorimetry (DSC) is a thermoanalytical method that simultaneously measures the heat needed to raise the temperature of a sample and a reference. The difference in the heat energy for the sample and reference is then collected as a function of temperature [86]. There are two types of DSC instruments. These are Heat Flux Type and Power Compensation Type. The former was employed for the DSC measurements conducted in this project. A schematic diagram of heat flux DSC is shown in Fig. 3.9

Fig. 3.9. Schematic diagram of a heat flux DSC [87].

The DSC instrument used in conducting the DSC measurements in this thesis was a TA Instruments DSC2000Q unit, provided by Miami University Department of Chemical, Biological and Paper Engineering. The instrument consists of two chambers. The working principle of TA Instruments DSC2000Q unit (other DSC instruments have similar working principle) involves positioning the reference material (an empty aluminum pan in our case) in contact with a heater and a thermocouple. This set up is done in one of the chambers. In the second chamber, a similar

41 aluminum pan which holds the sample to be investigated is positioned. A specified computer program is used to cause the DSC to heat each chamber at the same rate. During the process, the power output is adjusted for each chamber’s heater. The difference in power output between the two heaters is the actual heat flow, which is plotted against temperature [86]. Since the focus of this thesis is phase transition, the basic principle underlying this technique relating to phase transition is that when the sample experiences a phase transition, on average heat will need to flow to it than the reference to keep both at identical temperature. The quantity of heat that will be transferred to the sample is dependent on the type of process taking place (exothermic or endothermic). More heat needs to be transferred to the sample as it melts from solid to liquid to raise its temperature at the same rate as the reference material. This is the case since the sample absorbs heat as it experiences the endothermic phase transition from solid to liquid. In the same manner, as the sample experiences exothermic process such as crystallization, less heat is needed to increase the sample temperature. The quantity of heat absorbed or given off during the transitions is therefore measured by the DSC instrument by detecting the difference in heat flow between the sample and reference material [86].

Chapter 4: Results and Discussion

The results of the experimental study performed on Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 are presented in this section.

4.1 Temperature dependence (warming and cooling) of magnetization

Figure 4.1 shows the magnetization M(T) data measured while warming and cooling the

Ni2Mn0.70Cu0.30Ga sample in a magnetic field of 1 kOe. The measurement was done in the temperature range of 5 - 400 K. The inset shows the behavior of the magnetization around the transition temperature. The phase transition temperatures were determined from the derivative of the M(T) data. The data obtained while warming showed a ferromagnetic transition at T  345 K. While cooling the transition was observed at T  338 K, demonstrating a thermal hysteresis of 7 K. This behavior suggested that the transition at 345 K is a first order phase transition. As can be seen from the inset of the graph, in the high temperature region, the M(T) data obtained while cooling demonstrated an enhanced discontinuity. In this region a thermal hysteresis of 7 K was observed, while at lower temperatures, the thermal hysteresis was only 2 K. The M(T) data indicated that Ni2Mn0.70Cu0.30Ga exhibited the first order MPT coupled with the second order ferromagnetic transition. The data also suggested that the coupling is stronger while cooling.

Fig. 4.1. Temperature dependence (warming and cooling) of magnetization for

Ni2Mn0.70Cu0.30Ga measured at H = 1 kOe. The inset shows the behavior of the magnetization around the transition temperature.

The M(T) data for the Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 systems are presented in Figures 4.2 and 4.3, respectively. The data were measured using the same protocol mentioned above for Ni2Mn0.70Cu0.30Ga. The behavior of the magnetization around the transition temperature is represented by the insets in both figures (Figures 4.2 and 4.3). For the

Ni2Mn0.70Cu0.25Cr0.05Ga sample, as can be seen from the Fig. 4.2, there is a ferromagnetic transition at T  296 K upon warming. While cooling, the transition occured at T  295 K showing a thermal hysteresis of 1 K, suggesting a first order phase transition at T  296.

Fig. 4.2. Temperature dependence (warming and cooling) of magnetization for

Ni2Mn0.70Cu0.25Cr0.05Ga measured at H = 1 kOe. The inset shows the behavior of the magnetization around the transition temperature.

The transition at T  296 K is the first-order martensitic phase transition the transition at T  295 K corresponds to a second-order ferromagnetic transition of austenite. The data shows a stronger coupling while warming.

For the Ni2Mn0.70Cu0.30Ga0.95In0.05 sample, the data obtained while warming as presented by Fig. 4.2 and represented by the red curve, showed a ferromagnetic transition at T  277 K. While cooling two transitions were observed at T  276 K and T  261 K. The hysteresis observed was an indication that transition at T  277 K is a first order phase transition.

Fig. 4.3. Temperature dependence (warming and cooling) of magnetization for

Ni2Mn0.70Cu0.30Ga0.95In0.05 measured at H = 1 kOe. The inset shows the behavior of the magnetization around the transition temperature.

Comparing the temperatures at which the phase transitions occur for these systems,

Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05, for the

Ni2Mn0.70Cu0.30Ga the data obtained while warming showed a ferromagnetic transition at T 

345 K. While cooling the transition was observed at T  338 K, for the Ni2Mn0.70Cu0.25Cr0.05Ga System, ferromagnetic transition occurs at T  296 K upon warming while cooling the transition was observed at T  295 K, which is near room temperature, a requirement for solid-state cooling applications. For Ni2Mn0.70Cu0.30Ga0.95In0.05 alloy, the transitions temperatures were T  277 K and T  276 K for warming and cooling process respectively.

It can be deduced that the introduction of Cr into the Ni2Mn0.70Cu0.30Ga system has reduced the transition temperatures from higher temperatures to nearly room temperature. More so, the thermal hysteresis is observed to reduce appreciably (from 7 K to 1 K). It is worthy to note that partially replacing Ga with In in the Ni2Mn0.70Cu0.30Ga system resulted in two transitions during cooling. It was also observed that the substitution led to a reasonable decrease in the transition temperatures and a reduction in the thermal hysteresis.

4.2 Structural Analysis

Considering the discussion in section 4.1, a non-cubic martensitic structure was expected for

Ni2Mn0.70Cu0.30Ga at room temperature. As shown in Fig. 4.4a, the room temperature XRD patterns for the Ni2Mn0.70Cu0.30Ga sample showed a tetragonal structure belonging to the

I4/mmm space group, as expected. For a cubic L21 structure a single diffraction peak is typically observed near 2 = 450. For a tetragonal structure, the peak splits into two distinct peaks, as evident in Fig. 4.4a. The lattice parameters for Ni2Mn0.70Cu0.30Ga were a = 3.918(4) Å and c = 6.621(4) Å.

The M(T) data showed that Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 exhibited the martensitic transition near 296 K and 277 K, respectively. The temperatures are fairly close to room temperature (298 K). Therefore, at room temperature both samples are expected to exhibit intermediate crystalline structures between cubic and tetragonal/orthorhombic. This expected behavior is evident in Fig. 4.4b and 4.4c, where a partial splitting of the 220 peak near 430 is observed. This behavior is indicative of an intermediate structure for Ni2Mn0.70Cu0.25Cr0.05Ga and

Ni2Mn0.70Cu0.30Ga0.95In0.05 near room temperature. Typically the intermediate phase comprises of the cubic and non-cubic (in our case orthorhombic) phases.

Fig. 4.4. Room temperature XRD pattern for (a) Ni2Mn0.70Cu0.30Ga, (b) Ni2Mn0.70Cu0.25Cr0.05Ga

and (c) Ni2Mn0.70Cu0.30Ga0.95In0.05.

4.3 Compositional Analysis

In order to search for phase segregation, backscatter SEM images of Ni2Mn0.70Cu0.30Ga,

Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 samples were collected, as shown in Fig.

4.5. As can been seen from Fig. 4.5a, the SEM micrographs indicated that Ni2Mn0.70Cu0.30Ga sample was single phase. Introducing Cr into the Ni2Mn0.70Cu0.30Ga system did not affect the phase as the SEM micrographs of Ni2Mn0.70Cu0.25Cr0.05Ga sample (Fig. 4.5b) suggested that the sample was also single phase. The SEM micrograph for the Ni2Mn0.70Cu0.30Ga0.95In0.05 sample also showed a single phase. However, a single precipitate spot was observed on the sample

48 which most likely was a residue from polishing. This spot was not observed in any other places in the sample (see Fig. 4.5c).

a b

Fig. 4.5. Room temperature SEM micrographs of (a) Ni2Mn0.70Cu0.30Ga, (b)

Ni2Mn0.70Cu0.25Cr0.05Ga (c) Ni2Mn0.70Cu0.30Ga0.95In0.05.

The compositional distribution of the constituent elements of the compounds (Ni2Mn0.70Cu0.30Ga,

Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05) was investigated using the Energy Dispersive X-ray (EDX) analysis This analysis is aimed at checking the type of constituent elements forming the compound, the atomic percentage of each of the constituent elements and of course the homogeneity of the sample [88]. Based on the EDS analysis, it was observed that the actual compositions of the compounds are reasonably close to the targeted compositions taking errors due to measurements into account. An EDS analysis of the precipitate spot seen in

49 the Ni2Mn0.70Cu0.30Ga0.95In0.05 sample showed presences of Carbon, which is consistent with our assumption of residuals from polishing.

4.4 Electronic Transport Property Analysis

In order to explore the electronic properties of the Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 Heusler alloys, electrical resistivity measurements were conducted on the PPMS using the four-probe technique. These measurements also serve as a means of confirming the TM, and TC obtained from the M(T) data (Figs. 4.1, 4.2 and 4.3). Fig

4.6. shows the temperature dependence of the normalized resistivity (/400K) for

Ni2Mn0.70Cu0.30Ga. The sample was cooled down from room temperature to 200 K and the resistance was measured as a function of increasing temperature (curve 1 in Fig.4.6). A small yet vivid step-change was observed near 345 K, signifying the MPT. The /400K(T) obtained while cooling (curve 2) showed a significantly larger step change near 338 K suggesting that the phase transition while cooling was relatively more discontinuous than the transition during warming. When the sample was warmed for the 2nd time (curve 3) a step change similar to curve 2 was nd observed near 345 K. The /400K(T) data obtained during the 2 cooling (curve 4) nearly overlapped with the data obtained during 1st cooling (curve 2).

Fig. 4.6. Temperature dependence (warming and cooling) of normalized resistance for

Ni2Mn0.70Cu0.30Ga measured at zero magnetic field.

The curves of the temperature dependence of the normalized resistivity (/400K) for

Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 samples are displayed in Figs. 4.7 and 4.8 respectively. For the Ni2Mn0.70Cu0.25Cr0.05Ga sample, it was cooled down from room temperature to 5 K and the resistance was measured as a function of increasing temperature from 5 K to 400 K(curve 1 in Fig.4.7). This is then followed by cooling down from 400 K to 5 K (curve 2 in Fig.4.7). The second round of warming is represented by curve 3 while curve 4 represents the second round of cooling. The values of transition temperature obtained from the M(T) curves (Fig. 4.2), is in good agreement with those estimated from the resistivity data (as indicated by the curves in Fig. 4.7).

Fig. 4.7. Temperature dependence (warming and cooling) of normalized resistance for

Ni2Mn0.70Cu0.25Cr0.05Ga measured at zero magnetic field.

As can be seen from curve 1 (first warming) in Fig 4.7, a minor step-change was observed near the transition (T  296 K) suggesting the MPT. When the sample was warmed for the 2nd time (curve 3), it followed the same path and trend as the first warming (the data completely overlapped as shown in the Fig). More so, the step change in the first warming is consistent with the second warming curve 2 and was observed at the same temperature, confirming the MPT. The data obtained while cooling as indicated by curve 2 showed a much larger step change near the transition (T  295 K) suggesting that the phase transition while cooling is distinct from the transition during warming. A small bend is observed slightly after the larger step change which could be attributed to change in the electronic properties and an indication that the transition during warming is different from that during cooling. The same changes observed during the first cooling were observed during the second cooling and at the same temperatures. More so, the data for both first and second cooling overlapped.

Fig. 4.8. Temperature dependence (warming and cooling) of normalized resistance for

Ni2Mn0.70Cu0.30Ga0.95In0.05 measured at zero magnetic field.

As can be observed from the ρ(T) graph (Fig. 4.8) for the Ni2Mn0.70Cu0.30Ga0.95In0.05, looking at the warming process represented by the red curve, the martensite phase seems to have a lower resistivity as compared to the austensite phase. The martensitic transition is signified by the sharp increase in resistivity. This discontinuity relates to a transition from a martensite state to an austenite state which is consistent with the trend from magnetization data. The sharp increase of resistivity observed near the transition temperature (T  277 K) indicated a change in the electronic properties of the material at the phase transition. More so, a small step-change was observed at the transition temperature (T  277 K) confirming the value of the transition temperature obtained from the M(T) data (Fig. 4). Similar trends were observed during cooling. A small step-change at the transition temperature (T  276K) and then followed by a sharp increase at TM.

It is worthy to note that for all the samples (Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and

Ni2Mn0.70Cu0.30Ga0.95In0.05); during cooling (represented by the blue curves in Figs. 4.6, 4.7 and 4.8) sharp jumps were observed in the resistivity at the transition as compared to during warming (see red curves in Figs. 4.6, 4.7 and 4.8). These sharp jumps were also observed during the

53 second cycles seen in the figures (Figs. 4.6, 4.7 and 4.8). This could be attributed to the fact that during the transition, there was a change in the structure of the samples. Due to the brittle nature of the samples, the change in the structure caused the samples to crack. This crack is an irreversible process unlike the phase transition which is a reversible process. The crack consequently resulted in the jumps in the observed resistivity.

4.5 Field dependence (warming and cooling) of magnetization

Isothermal M(H) data were collected at various temperatures for each of the sample (260 K- 370

K for Ni2Mn0.70Cu0.30Ga, 260 K- 350 for Ni2Mn0.70Cu0.25Cr0.05Ga,and 220 K- 320 K for

Ni2Mn0.70Cu0.30Ga0.95In0.05) near the MPT as shown in Figs. 4.9, 4.10, and 4.11. Both M(H) data obtained while warming (Figs. 4.9a, 4.10a, and 4.11a.) and cooling (Figs. 4.9b, 4.10b, and 4.11b.) showed the behavior typical for a ferromagnetic material (magnetization decreased with increasing temperature). However, near the MPT the M(H) data obtained for the

Ni2Mn0.70Cu0.30Ga alloy while cooling indicated a field induced phase transition, which was not observed in the data obtained while warming (see the regions marked by red lines in Fig.4.9). This behavior agreed with the M(T) data (Fig. 4.1) and showed that the coupling of the MPT with the ferromagnetic transition is different while warming and cooling the sample. A similar trend was observed for the Ni2Mn0.70Cu0.25Cr0.05Ga system as shown in Fig.4.10.

Fig. 4.9. Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.30Ga measured isothermally (warming and cooling) at temperatures near the martensitic phase transition.

Fig. 4.10. Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.25Cr0.05Ga measured isothermally (warming and cooling) at temperatures near the martensitic phase transition.

Fig. 4.11. Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.30Ga0.95In0.05 measured isothermally (warming and cooling) at temperatures near the martensitic phase transition.

The temperature dependence of the magnetic entropy changes, ∆푆푀 (T), for various field change was determined by measuring the isothermal M(H) data as a function of increasing field near the phase transition, and applying the following expression [89]:

푀푖+1−푀푖 ∆푆푀 = ∑푖 4.1 푇푖+1−푇푖

The change in magnetic entropy (ΔSM(T)) data for different magnetic field changes for warming and cooling processes for all the samples (see Figs. 4.12, 4.13 and 4.14) were evaluated from the

M(H) data shown in Figs. 4.9, 4.10 and 4.11. For the Ni2Mn0.70Cu0.30Ga sample, interestingly, a -1 -1 peak ΔSM of -17 J kg K (for a field change of 50 kOe) was observed during the warming process, which is nearly half the peak value of -33 Jkg-1K-1 observed during cooling as presented

57 in Fig. 4.12. It was also interesting to note, that the both peak values as shown in Fig. 4.12 were observed at nearly the same temperature (~343 K) even though a 7 K thermal hysteresis was nd observed in the M(T) data. Additionally, the ΔSM(T) data for warming exhibited a 2 anomaly at 348.5 K, which was not observed in the data obtained while cooling. These observations strongly demonstrated that two phase transitions (apparently partially coupled) occurred while warming. The coupling between the two transitions became stronger when the sample was cooled down from high to low temperature.

Fig. 4.12. Temperature dependence of magnetic entropy changes, ΔSM(T), for Ni2Mn0.70Cu0.30Ga measured while (a) warming and (b) cooling.

-1 -1 For the Ni2Mn0.70Cu0.25Cr0.05Ga sample, a peak ΔSM of -39 J kg K (for a field change of 50 kOe) was observed during the warming process. This value is more than twice the peak value of -17 Jkg-1K-1 observed during cooling as presented in Fig. 4.13. It was also interesting to note,

58 that the peak value during warming was observed at T = ~296.5 K as shown in Fig. 4.13, which is consistent with the transition temperature (T = ~296 K) obtained from the M(T) data in Fig. 4.2. The temperature (T = ~295.5 K) at which the peak value was observed during cooling also agrees with the estimated value (T = ~295 K) obtained from the M(T) data in Fig. 4.2.

Fig. 4.13. Temperature dependence of magnetic entropy changes, ΔSM(T), for

Ni2Mn0.70Cu0.25Cr0.05Ga measured while (a) warming and (b) cooling.

-1 -1 For the Ni2Mn0.70Cu0.30Ga0.95In0.05 sample, a peak ΔSM of -26 J kg K (for a field change of 50 kOe) while a peak value of -15 Jkg-1K-1 was observed during cooling as presented in Fig. 4.14. The temperatures at which the peak values were observed during warming and cooling, T = ~276.5 K and ~275.5 K respectively compared well with the values ( T = ~277 K and ~276 K) obtained from the M(T) data in Fig. 4.3.

Fig. 4.14. Temperature dependence of magnetic entropy changes, ΔSM(T), for

Ni2Mn0.70Cu0.30Ga0.95In0.05 measured while (a) warming and (b) cooling.

The obtained values of ΔSM(T) for both heating and cooling processes for all the samples studied are very large as compared to the values reported for pure gadolinium −5.84 J/(kg K), which is the bench mark [90]. It is worthy to note that when Cu was partially replaced by Cr in the

Ni2Mn0.70Cu0.30Ga alloy, the peak value during warming increased from −17 J/(kg K) to −39 J/(kg K), while the peak value during cooling decreased from −33 J/(kg K) to −17 J/(kg K) suggesting that the coupling while warming was stronger than while cooling as compared to

Ni2Mn0.70Cu0.30Ga alloy where coupling was stronger while cooling.

To determine the hysteretic nature of the isothermal magnetization near the MPT for the

Ni2Mn0.70Cu0.30Ga sample, hysteresis loops at various temperatures near the MPT were measured while warming and cooling the sample. For demonstration, the loops measured at 345 K are shown in Fig. 4.15. It is clear in the figure no hysteresis was observed in the M(H) data obtained while warming. On the contrary, a large hysteresis (typical for a first order metamagnetic phase transition) was observed at the same temperature when the data was obtained while cooling. This behavior confirmed the partial decoupling (warming) and nearly completes coupling (cooling) of the phase transitions in Ni2Mn0.70Cu0.30Ga.

Fig. 4.15. Field dependence of magnetization, M(H), for Ni2Mn0.70Cu0.30Ga measured while warming and cooling at T = 345 K.

In order to get more understanding about the order of the phase transition, M2 (T) data were collected at various temperatures for each of the samples (322 K to 350 K for Ni2Mn0.70Cu0.30Ga,

275 K to 300 K for Ni2Mn0.70Cu0.25Cr0.05Ga, and 265 K to 283 K for Ni2Mn0.70Cu0.30Ga0.95In0.05) near the MPT and the data were used to plot an Arrott plot for each samples (see Figs. 4.16, 4.17 and 4.18).

Fig. 4.16. The Arrott plot, M2 versus H/M in the vicinity of phase transition temperature for

Ni2Mn0.70Cu0.30Ga.

From the figures, it can be seen that the M2 vs. H/M curves is S shaped for all the samples around the transition temperatures (T  345 K for Ni2Mn0.70Cu0.30Ga, T  296 K for

Ni2Mn0.70Cu0.25Cr0.05Ga, and T  277 K for Ni2Mn0.70Cu0.30Ga0.95In0.05). This is an indication that the alloys undergo a first order metamagnetic transition [91]. However, it should be noted that the

S shape was more visible in the Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 samples.

For the Ni2Mn0.70Cu0.30Ga alloy, the curves appeared to be nonlinear but did not really display the same S shaped behavior. This possibly indicated that the metamagnetic behavior in the sample

62 was weakened, causing a suppression in the discontinuity in magnetization during the reverse martensitic transition. .

Fig. 4.17. The Arrott plot, M2 versus H/M in the vicinity of phase transition temperature for

Ni2Mn0.70Cu0.25Cr0.05Ga.

Fig. 4.18. The Arrott plot, M2 versus H/M in the vicinity of phase transition temperature for

Ni2Mn0.70Cu0.30Ga0.95In0.05.

The DSC measurements were performed upon cooling and heating with a ramp rate of 10 K/min for all the three samples following the steps detailed in section 3.5 of this thesis. The Heat flow data was collected for both heating and cooling processes. The results of the measurements for the Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 samples are shown in Figs. 4.19, 4.20 and 4.21, respectively. The transition temperatures can be estimated from the endothermic (warming) and exothermic (cooling) peaks.

Fig. 4.19. DSC heat flow curves upon heating and cooling for Ni2Mn0.70Cu0.30Ga

For the Ni2Mn0.70Cu0.30Ga alloy, as presented in Fig. 4.19, the values of transition temperatures; T  355 K and T  332 K for heating and cooling processes respectively estimated from the DSC data compare fairly well with for T  345 K for warming process and T  338 K for cooling process obtained from the M(T) data, defined as the temperature where the dM/dT is the maximum. It is worthy to note that the coupling and decoupling effects that were observed in the nd M(T) data which was confirmed by the ΔSM(T) data evident by a 2 anomaly at 348.5 K, which was not observed in the data obtained while cooling (see Fig 4.12) were also observed in the DSC data evident by the two peaks observed during the warming process (see the red curve in

Fig 4.19) which was not the case during cooling (see the blue curve in Fig 4.19). The noticeable thermal hysteresis indicated by the difference in the exothermic/endothermic peaks positions confirms the first-order magnetostructural transition. The endothermic and exothermic peaks observed during heating and cooling cycles, respectively, are primarily due to the latent heat of the first-order phase transition.

Fig. 4.20. DSC heat flow curves upon heating and cooling for Ni2Mn0.70Cu0.25Cr0.05Ga

For the Ni2Mn0.70Cu0.25Cr0.05Ga sample, as can be seen in Fig. 4.20, the values of transition temperature for warming (T  298 K) and cooling (T  296 K) estimated from the DSC curves are consistent with T  296 K and T  295 K for warming and cooling processes respectively obtained from the M(T) data. The thermal hysteresis observed confirms that first-order structural transitions have occurred.

Fig. 4.21. DSC heat flow curves upon heating and cooling for Ni2Mn0.70Cu0.30Ga0.95In0.05

For Ni2Mn0.70Cu0.30Ga0.95In0.05 alloys, the transition temperature for the warming process estimated from the endothermic peak (see the red curve in Fig. 4.21) T  278 K while it was T  275 K for the cooling process estimated from the exothermic peak (see the blue curve in Fig. 4.21). These values are consistent with T  277 K and T  276 K for warming and cooling processes respectively obtained from the M(T) data. The difference in the endothermic and exothermic peaks is the thermal hysteresis suggesting a first-order structural transition.

Chapter 5: Summary and conclusion

The magnetic and magnetocaloric properties of Ni2Mn0.70Cu0.30Ga, Ni2Mn0.70Cu0.25Cr0.05Ga and

Ni2Mn0.70Cu0.30Ga0.95In0.05 have been extensively studied. Ni2Mn0.70Cu0.30Ga sample exhibited a tetragonal structure at room temperature while cubic and orthorhombic phases coexisted in

Ni2Mn0.70Cu0.25Cr0.05Ga and Ni2Mn0.70Cu0.30Ga0.95In0.05 samples. The EDS analysis indicated the actual compositions of the compounds were reasonably close to the targeted compositions. For the Ni2Mn0.70Cu0.30Ga system, it was observed that upon heating, a first order partially-coupled magnetostructural phase transformation was observed in the material near ~345 K. A coupling of the phase transitions was enhanced during cooling causing a large increase in the magnetocaloric properties of the material. For a field change of 50 kOe, the peak magnetic entropy change obtained while heating was -17 J kg-1K-1. The peak value was -33 Jkg-1K-1 when the measurements were done while cooling from 400 K to lower temperatures. When Cu in the

Ni2Mn0.70Cu0.30Ga system was partially replaced by Cr forming the Ni2Mn0.70Cu0.25Cr0.05Ga system, it was observed that the coupling effect was stronger while heating than cooling a reverse of the effect observed in the Ni2Mn0.70Cu0.30Ga alloys. More so, the introduction of Cr into the Ni2Mn0.70Cu0.30Ga system reduced the transition temperature from higher temperature to ~296 K, a temperature ideal for near-room temperature magnetic refrigerators. Thermal hysteresis was also observed to reduce appreciably (from 7 K to 1 K). For a field change of 50 kOe, the peak magnetic entropy change obtained while heating was -39 J kg-1K-1. The peak value was -17 Jkg-1K-1 when the measurements were done while cooling from 400 K to lower temperatures. When Ga in the Ni2Mn0.70Cu0.30Ga alloy was partially replaced by In forming

Ni2Mn0.70Cu0.30Ga0.95In0.05 compound, the transition temperature decrease appreciably from ~345 K to ~277 K. For a field change of 50 kOe, the peak magnetic entropy change obtained while heating was -26 J kg-1K-1. When the measurements were done while cooling from 400 K to lower temperatures, the peak values were -15 Jkg-1K-1. Thermal hysteresis in this case reduced from 7 K to 1 K. This work in part demonstrated that the magnetocaloric properties of first order materials are strongly dependent on the coupling of their structural and magnetic phase transitions.

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