Iron-And Cobalt-Based Heusler Alloy Nanostructures and Their Device Applications

THE CATHOLIC UNIVERSITY OF AMERICA

Iron-and Cobalt-based Heusler Alloy Nanostructures and Their Device Applications

A DISSERTATION

Submitted to the Faculty of the Department of Physics School of Arts and Sciences Of The Catholic University of America In Partial Fulfillment of the Requirements For the Degree Doctor of Philosophy © Copyright All Rights Reserved By Rajendra P. Dulal

Washington, D. C.

2017

Iron- and Cobalt-based Heusler Alloy Nanostructures and Their Device Applications Rajendra P. Dulal, Ph.D. Director: Dr. John Philip

Heusler alloys are compounds with chemical formula X2YZ, where X and Y are transition metals or lanthanides (rare-earth metals) and Z is a main group element. The properties of this remarkable class of materials have been of interest since 1903 when Heusler discovered the ferromagnetic behavior from non-ferromagnetic constituents Cu, Mn, and Al. The spectrum of their possible applications ranges from magnetic and magneto-mechanical materials over semiconductors and thermoelectrics to superconductors and topological insulators. An important feature of the Heusler compounds is the possibility of controlling the valence electron concentration by partial substitution of elements. On the other hand, the properties also depend on the degree of ordering in the crystal structure. They make one of the leading candidate material classes for achieving high spin polarization. A number of Heusler alloys are predicted to be half-metallic ferromagnets that would theoretically provide high spin polarization at the Fermi energy. Furthermore, magnetic

Heusler compounds can exhibit topological semimetal (TSM) behavior. TSM are electronic strong spin -orbit metals/semimetals whose fermi surface arise from crossing between valence band and conduction band, which cannot be avoided due to non-trivial topology. This new state has recently attracted worldwide interest because they may realize the particles that remain elusive in high energy physics. The possible existence of massless chiral fermions known as Weyl fermion exist in a class of TSM known as Weyl semimetals (WSM). Although Weyl fermion systems proposed in condensed matter physics share many similarities with the massless fermionic particle propagating in the vacuum, they have some unusual properties. WSM have unclosed fermi arc which connect two Weyl fermions with opposite chirality. This can lead to show anomalous Hall effect associated with Berry phase. Despite interest, room temperature ferromagnetic semimetals,

regardless of its topological trivial/non-trivial nature, are rare in nature. So, Heusler alloy are the most suitable member of room- temperature topological Weyl semimetal. Magnetic Heusler alloys have several advantages over other compounds where Weyl fermions have been proposed and detected. They usually have considerable electron-electron interaction which helps to study the interplay between the topological semimetal and electronic correlation. The excitement surrounding magnetic Heusler alloys is not only due to the possibility of studying new quantum phenomena, or to the realization of new exotic phases in matter. It also springs from the promise of developing quantum devices working at relatively at high temperature, as the topological protection makes their quantum properties robust against perturbations.

In this thesis, we will explore the properties of iron and cobalt based Heusler alloy. So far, no epitaxial thin films study has been reported. We have grown cobalt and iron based Heusler alloy thin films using ultra-high vacuum electron-beam evaporation. We have used magnesium oxide as a buffer layer to get well ordered thin film. We have fabricated Hall bar devices using metal contact mask method. Our Co2TiGe(CTG) Heusler alloy display semimetallic behavior with band gap of

24 meV. Negative magnetoresistance (MR) has been observed in CTG films. At low magnetic field, weak-localization dominates on the MR behavior. At intermediate magnetic field, chiral anomaly contributes to the magnetoconductivity of thin films. Anomalous Hall conductivity of 25

S/cm have been observed due to Berry curvature. We have theoretically calculated the band structure of CTG using Quantum espresso. The calculated band structure exactly match the reported band structure in the literature. In general, Heusler compounds crystallize in the cubic crystal structure with a space group 퐹푚3̅푚, but in many cases, certain types of disorder are observed. In this thesis, it is also shown that iron -based Heusler alloy exhibit disorder in the crystal lattice. This lattice disorder can lead to different nearest neighbors to Fe, which can result in

remarkable magnetic interactions. Our Fe-based alloy, Fe2CrAl(FCA) exhibit high Curie temperature and large magnetization due to disordering in lattice sites. Semimetallic behavior of

FCA is destroyed by this disorder.

The Dissertation by Rajendra P. Dulal fulfill the dissertation the requirement for the doctoral degree in physics approved by John Philip, PhD, as a director, Ian L. Pegg, PhD, and Lorenzo Resca, PhD, as Readers.

______John Philip, PhD, Director

______Ian L. Pegg, PhD, Reader

______Lorenzo Resca, PhD, Reader

DEDICATON I dedicate my dissertation work to my loving parents Krishna Lal Dulal and Late Rewata Dulal, my wife Sarmila Gautam, my daughter Riya Dulal, my brothers Hari Prasad Dulal, Badri Prasad Dulal, Khada Nanda Dulal, Ganesh Prasad Dulal, Bednath Dulal and my sister Bimala Dulal.

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Contents 1. Introduction ...... 1 1.1 Half-metallic Ferromagnets ...... 1 1.2 Heusler Compounds as Half-metallic Ferromagnets ...... 2 1.3 The Slater-Pauling behavior ...... 2 1.4 Crystal structure of Heusler Compounds ...... 5 1.4.1 Full-Heusler Compounds ...... 5 1.4.2 Half-Heusler Compounds ...... 5 1.4.3 Inverse Heusler Compounds ...... 5 1.4.4 Quaternary Heusler Compounds ...... 6 1.5 Order-Disorder Phenomena in Heusler Compounds ...... 7 1.6 Magnetism of Heusler Alloys Thin Films ...... 8 1.6.1 Exchange Interactions ...... 8 1.6.2 Magnetic Anisotropies ...... 11 1.6.3 Magnetic Domain ...... 12 1.6.4 Domain Wall Formation ...... 13 1.6.5 Magnetization Reversal Process ...... 14 1.7 Application of Heusler alloy ...... 15 1.7.1 Half-Metallic Ferromagnets for Spintronics ...... 15 1.7.2 Heusler Alloys in Thermoelectric...... 16 1.8 Heusler Alloy as the Topological Weyl Semimetals ...... 16 1.9 Magnetoresistance...... 18 1.10 Boltzmann Transport Theory ...... 20 1.11 Temperature Dependence of Resistivity ...... 22 2. Experimental technique and characterization ...... 30 2.1. Vapor Deposition Method...... 30 2.2. Physical Vapor deposition ...... 30 2.3. Electron –beam evaporation ...... 35 2.3.1. Working principle ...... 35 2.3.2. Thin Film Growth Process ...... 37 2.4. Electrospinning ...... 40 2.4.1. Electrospinning Parameters ...... 41

2.5. Nanostructures Characterization Techniques ...... 43 2.5.1. Scanning Electron Microscopy (SEM) ...... 43 2.5.2. Energy Dispersive X-ray Spectroscopy (EDS) ...... 45 2.5.3. X-ray Diffraction (XRD) ...... 46 2.5.4. Vibrating sample Magnetometer (VSM) ...... 48 3. Device Fabrication and Measurement ...... 54 3.1 Resistivity measurements method...... 54 3.2 Hall Bar ...... 56 3.3 Device Measurements ...... 57 4 . Magnetic and Transport Properties of Iron-based Heusler alloy nanoscale thin films ...... 63 4.1 Introduction ...... 63 4.2 Experimental Details ...... 64 4.3 Results and Discussion ...... 65 4.3.1 Morphological and Structural Characterization ...... 65 4.3.2 Magnetic Properties ...... 66 4.3.3 Transport Properties ...... 69 4.4 Conclusions ...... 71 5. Magnetic properties of Iron-based Nanowires ...... 75 5.1 Introduction ...... 75 5.2 Experiment ...... 76 5.3 Morphological characterization ...... 77 5.4 XRD characterization...... 77 5.5 Magnetic Characterization ...... 78 5.6 Conclusion ...... 80 References ...... 81 6 Transport properties of room temperature Weyl Semimetal cobalt-based Heusler alloys ...... 83 6.1 Introduction ...... 83 6.1.1 From Dirac equation to Weyl equation ...... 84 6.1.2 Weyl Nodes ...... 85 6.1.3 Topological signature of Weyl points ...... 86 6.1.4 Berry phase and Berry Curvature ...... 87 6.1.5 Weyl Points and Berry Curvature ...... 88

6.1.6 Quantum Transport, Weak Localization and Weak anti-localization ...... 89 6.1.7 Anomalous Hall Effect ...... 90 6.2 Experimental Details ...... 93 6.3 Morphological and Structural characterization ...... 94 6.4 Magnetic Chracterizations ...... 94 6.5 Electrical Resistivity ...... 95 6.6 Magnetoresistance...... 97 6.7 Weak-Localization ...... 98 6.8 Hall resistivity and Anomalous Hall effect ...... 99 6.9 Conclusions ...... 100

7.Co2TiSn- A Magnetic, Centrosymmetric Weyl Semimetal ...... 109 7.1 Introduction ...... 109 7.2 Experimental details...... 111 7.3 Results and Discussions ...... 112 7.4 Conclusions ...... 114 References ...... 115

List of Figures Figure 1.1: Schematics representation of the density of states for a half-metal compared to a normal metal and a semiconductor [8]...... 1 Figure 1.2 : Slater-Pauling curve for 3d transition metals and their alloys [12]...... 4 Figure 1.3: Different types of Heusler structures. (a) Full Heusler, (b) Half- Heusler, (c) Inverse Heusler, and (d) Quaternary-Heusler structures [49]...... 6 Figure 1.4: Most usual disordered Heusler structure: (a) CsCl-type, (b) BiF3-type, (c) NaTl-type, and (d) W-type [48].Li ...... 7 Figure 1.5: Dependence of the exchange integral 푱풆풙 on the atomic separation 풓풂 normalized to the electron radius d orbital 풓풅 [61]...... 9 Figure 1.6: The dependence of indirect exchange coupling on the thickness of a Cr interlayer separating two ferromagnetic layers [68]...... 10 Figure 1.7: A variety of domain structure for a magnetized sample. a) Uniformly magnetized. b) Two domains. c) Four domains. d) Essentially two domains with two closure domains [71]...... 13 Figure 1.8: Schematic diagram of 180o a) Bloch wall and b) Nѐel wall [61]...... 14 Figure 1.9: Spin-resolved band structure and ferromagnetic half-metallic ground states in Co2TiX. ((a-c) the calculated bulk band structure of the majority spins of Co2TiSI, Co2TiGe and Co2TiSn, respectively. (d-f) Same as panel (a-c) but for minority spin. (g-i) The band structures of both spins...... 17 Figure 1.10: A multilayer system Fe-Cr-Fe with ferromagnetic (left) and antiferromagnetic (right) exchange coupling between iron layers [89]...... 19 Figure 1.11: TMR effect in a magnetic tunnel junction device. When the magnetizations are aligned parallel (left) the device resistance is small and when they aligned antiparallel (right), the device resistance is large [91]...... 20 Figure 2.1: Knudsen cell with directional dependence [4] ...... 33 Figure 2.2: Uniformity of an evaporated film on a flat surface [7]...... 34 Figure 2.3: Schematics of e-beam evaporation [4]...... 36 Figure 2.4: Thin film growth model. [7] ...... 38 Figure 2.5: Schematic diagram of set up of electrospinning apparatus. [15] ...... 39 Figure 2.6: Axisymmetric infinite fluid body kept at potential 흓ퟎat a distance 풂풐 from an equipotential plane [19]...... 42 Figure 2.7: SEM micrograph of PMMA fibers electrospun from toluene solution at (a) 20±4; (b) 40±4; (c) 60±4; and (d) 80±4% relative humidity [24] ...... 43 Figure 2.8: Schematics of SEM [25]...... 44 Figure 2.9: Geometry of X-ray diffraction (Wikipedia)...... 47 Figure 2.10: Schematic diagram of VSM [32]...... 49 Figure 2.11: PPMS equipped with VSM [33] ...... 50 Figure 3.1: Four Probe measurements [1] ...... 54 Figure 3.2: Typical Van der Pauw geometry for measuring resistivity [7] ...... 55 Figure 3.3: Hall effect in a thin film [8]...... 56 Figure 3.4: Schematics of Hall Bar for thin film ...... 58 Figure 3.5: Standard DC resistivity puck from PPMS [10] ...... 59

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Figure 3.6: (a) Horizontal rotator motor, (b) rotor Sample holder probe, (c) User Bridge with Universal puck (d) AC User bridge with thin film device ...... 60 Figure 3.7: different puck for resistivity measurement with horizontal rotator ...... 61 Figure 4.1: SEM image of 80 nm FCA thin film ...... 64 Figure 4.2: (a) XRD patterns of FCA thin films with different thicknesses. The broad peak at 69˚ is from silicon (100) substrate. (b) A typical EDX spectrum of FCA thin film (80 nm)...... 65 Figure 4.3: Magnetic Hysteresis curves of FCA thin films with different thicknesses (a) field applied parallel to the substrate plane and (b) Field applied perpendicular to the substrate plane...... 66 Figure 4.4: Variation of magnetic moment with the film thickness when the field is applied parallel to film plane...... 67 Figure 4.5: Variation of coercivity with film thicknesses for an applied field parallel and perpendicular to the film plane ...... 68 Figure 4.6: (a) M vs. T plot of 100 nm FCA film with zero field. (b) M vs. T of FCA films at different thickness for the field applied parallel to the film plane (field 500 Oe)...... 69 Figure 4.7: ρ vs T of FCA thin films at different thicknesses. (b) ρ vs T for 80 nm thin film. Upper right inset shows resistivity behavior for T < 80 K whereas lower left inset shows resistivity behavior from temperature 80 K to 300 K...... 70 Figure 4.8: ρ Vs T for FCA thin film (100 nm) with field (3 T) and without field...... 71 Figure 5.1: SEM images of as grown nanowires...... 76 Figure 5.2: EdX spectrum of FCA nanowires...... 77 Figure 5.3: XRD pattern of FCA nanowires ...... 78 Figure 5.4: (a) M-H curve of FCA nanowires at 10 K and 300 K. (b) Magnetization vs Temperature curve of FCA wires ...... 79 Figure 6.1: Possible types of Weyl semimetals. In plot a, a type-I Weyl point with a point-like Fermi surface. In plot b, a type-II Weyl point appears as the contact point between electron and hole pockets. The grey plane corresponds to the position of the Fermi level, and the blue (red) lines mark the boundaries of the hole(electron) pockets [41]...... 86 Figure 6.2: Schematic of the structure of the Weyl semimetal in momentum space. Two diabolical points are shown in red, within the bulk 3D Brillouin zone. Each Weyl node is a source or sink of the flux of the Berry connection, as indicated by the blue arrows. The dark grey plane indicates the surface Brillouin zone, which is a projection of the bulk one. The Weyl nodes are connected by a Fermi arc, as shown by the yellow line [9,43]...... 89 Figure 6.3: The vector plot of the Berry curvature in momentum space. The arrows show that the flux of the Berry curvature arrows from one monopole (red) to the other (blue), defining the non-trivial topological properties of a topological semimetal [ 48]...... 91 Figure 6.4: SEM image of 50 nm CTG thin film ...... 92 Figure 6.5: (a) XRD pattern of CTG thin film. (b) EDX spectrum of CTG thin film ...... 93 Figure 6.6: (a) Field variation of magnetization at 300 and 10 K. (b) Magnetization curve as a function of temperature at 1000 Oe field ...... 95 Figure 6.7: Resistivity variation with temperature. Upper left inset shows the resistivity data and fit for T<200 K. Lower inset is resistivity variation and fit for T > 200 K ...... 96 Figure 6.8: (a) Magnetoresistance along longitudinal direction at different temperatures (b) MR versus temperature at 4 T field...... 97

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Figure 6.9: (a) Magnetoconductivity variation at different temperature at small fields. (b) Variation of phase coherence length and conduction channel with temperature. (c) Angle dependent longitudinal MR at 5 K...... 100 Figure 6.10: (a) Hall resistivity as a function of magnetic field. (b) Fitting between hall resistivity and longitudinal resistivity at 200 K. (c) Variation of R0 and SA with temperature (d) Hall conductivity at 5K ...... 101 Figure 7.1: Spin resolved band structure and density of states of a magnetic Weyl semimetal Co2TiSn ...... 110 Figure 7.2: SEM image of 57 nm CTS ...... 111 Figure 7.3: (a) XRD spectrum of a CTS thin film, (b) A typical EDX spectrum of a CTS thin film.... 112 Figure 7.4: M vs H curve of thin film at 300 K ...... 113 Figure 7.5: variation of magnetization with temperature ...... 114

Acknowledgements The six years at Catholic would definitely be one of the best times in my life. I have received solid physics education here, and have evolved from student to researcher. I have met so many nice people during six years, and I would never expect to finish this thesis without their help. First, I would like to thank my advisor Dr. John Philip who was very helpful in guiding my research work and supporting me for over the years. As John’s student, I am always encouraged to think about my own ideas and to discuss with others. I am very thankful to him for his contribution of precious time and expertise during my research. Many thanks to Prof. Ian L. Pegg and Prof. Lorenzo Resca for all of their guidance through the process. Their feedback, ideas and discussions have been absolutely invaluable to me. I am thankful to Prof. Kiran Bhutani and Prof. Otto Wilson for being on my Ph.D. Committee and assisting through the process. I would like to thank my fellow graduate students, post-doctoral fellows. I am grateful to all of you. I acknowledge The Vitreous State Laboratory which provided the research facilities and the staff for their technical support. I am thankful to the Physics Department, Faculty and staff for their supports and guidance. I am especially thankful to The Vitreous State Laboratory for the financial support.

Chapter 1

1. Introduction

1.1 Half-metallic Ferromagnets Half-metallic ferromagnets represent a new class of materials which captivated a lot of attention considering their possible applications in spintronics [1, 2]. The addition of spin degree of freedom to the conventional electronic devices based on semiconductors has several advantages like non-volatility, increased data processing speed, decreased electric power consumption and increased integration densities

[3-5]. In half- metallic materials, the two spin bands show a completely different behavior. The majority spin band shows the typical metallic behavior, whereas the minority band exhibits a semiconducting behavior with a gap at the Fermi level. The existence of the gap leads to 100% spin polarization at the

Fermi level which is supposed to be maximizing the efficiency of magnetoelectronic devices [6, 7]. A schematic representation of the density of states of a normal metal, a semiconductor and a half-metal is shown in Figure 1.1 for comparison [8]. Until now a lot of half- metallic Heusler ferromagnets are known.

Heusler compounds are the most prominent among half-metallic compounds due

Figure 1.1: Schematics representation of the density of states for a half-metal compared to a normal metal and a semiconductor [8].

2 to their relatively high Curie temperature and large magnetization [9-11]. Other known half – metallic materials except Heusler alloys [12-16] are some oxides (e.g. CrO2 and Fe3O4) [17], the magnetites [18], the double perovskite (e.g. Sr2FeReO6) [19], the pyrites (e.g. CoS2) [20], transition metal chalcogenides

(e.g. CrSe) [21-24], the diluted magnetic semiconductors (e.g. Mn impurities in Si or GaAs) [25, 27].

1.2 Heusler Compounds as Half-metallic Ferromagnets Among the half-metallic ferromagnets which are mentioned in part 1.1, although thin films of CrO2 and

La0.7Sr0.3MnO3 have been verified to have 100% spin polarization at the Fermi level at very low temperature [17,28], the Heusler alloys remain attractive for technical applications like spin -injection devices [29], spin-filter [30], tunnel junctions [31], or giant magnetoresistance devices [32,33]. Heusler compounds first attracted interest among scientific community in 1903, when F. Heusler found that the compound Cu2MnAl becomes ferromagnetic, although none of its constituent elements are ferromagnetic by itself [34, 35]. The scientific interest on Heusler compounds faded away until the prediction of half- metallic ferromagnetism in Co2MnSn by Kübbler et al. [36] and in NiMnSb by de Groot et al. [37]. Band structure calculations of Fe-based (Fe2CrX, X= Si, Al, Ge, Sn) alloys have been predicted to exhibit half- metallic behavior [10, 38, 39]. Cobalt based Heusler (Co2FeX, Co2TiX, X=Al, Si, Ge, Sn) alloys have been theoretically predicted to show half -metallicity [13, 40, 41]. In many Heusler compounds the total magnetic moment follows a simple electron- counting rule based on the Slater-Pauling behavior [42, 43].

1.3 The Slater-Pauling behavior Slater [44] and Pauling [45] independently discovered that the magnetic moment (m) of the 3d element and their binary alloys can be estimated on the basis of the average number of valence electrons (nV) per atom. The materials can be divided into two classes depending upon m(nV). The first region is the range of itinerant magnetism and high valence electron concentration (nV ≥8). Systems with closed pack structure (fcc and hcp) are found in this region. The second region is the area of localized magnetism and valence electron concentrations (nV≤8). Mostly bcc and bcc-related structure are found. Iron is a

3 borderline system. Figure 1.2 shows the Slater-Pauling curve for 3d transition metals and some of their alloys. The magnetic moment in unit of Bohr magneton (µB) for the first part of the curve describing itinerant magnetism is given by equation (1.1)

푚 = 2푛 ↑ −푛푣 1.1

= 2{(푛푑 ↑ +푛푠푝 ↑) − 푛푣}휇퐵 1.2 where 푚 = 푚 ↑ −푚 ↓ is total magnetic moment, n↑ is the total number of electrons in the majority states,

푛푑 ↑ the number of spin up electrons in d orbitals, 푛푠푝 ↑ the number of electrons in sp orbitals, and 푛푣 =

푛 ↑ +푛 ↓ is the total number of the valence electrons. For filled majority d bands, one has 푛푑 ↑ = 5 and thus equation (1.2) becomes

푚 = (10 + 2 푛푠푝 ↑ −푛푣)휇퐵 1.3

Pauling gave a value of 2 푛푠푝 ↑ = 0.6 for the second region. It appears from the curve that some of the alloys do not follow the expected curve (Co-Cr and Ni-Cr) in Figure 1.1 [12]. The magnetic moment of

Heusler compounds, especially cobalt based half -metallic ferromagnets, follow the Slater-Pauling rule and are situated in the localized part of this curve. For localized moment systems, an average magnetic moment per atom in the units of Bohr magneton is given by

푚 = (푛푣 − 2푛 ↓ −2푛푠푝)휇퐵 1.4

Where 푛 ↓ denotes the number of electrons in the minority states. In case of half metallic ferromagnets d electrons are constrained in a way that Fermi energy falls into a minimum (or a gap) between occupied and unoccupied d states and therefore minimizes the total energy. The minimum in minority density of states forces the number of electrons in the d bands to be approximately three. In half-metallic ferromagnets with a gap in one of the spin densities all sp electrons are occupied so that 푛푠푝 term vanishes.

Therefore, magnetic moment is given by equation (1.5)

푚 = (푛푣 − 6)휇퐵 1.5

Which means the average magnetic moment per atom is simply given by subtracting 6 from the average number of valence electrons. For ordered compounds with different kinds of atoms it may be more convenient to use all atoms of the unit cell. In case of 1:1:1 Heusler alloy with 3 atoms per unit cell, the

Slater Pauling rule is given by equation (1.6)

푚 = (푁푉 − 18)휇퐵 1.6

It is often more convenient to use valence electron number per formula unit as NV which is accumulated number of valence electrons (s, d electrons for the transition metals and s, p s, p for the main group elements). In 2:1:1 Heusler Compounds there are 4 atoms in the primitive cell and the total magnetic moment equals to

푚 = (푁푉 − 24)휇퐵 1.7

Figure 1.2 : Slater-Pauling curve for 3d transition metals and their alloys [12].

The slater Pauling rule relates the magnetic moment with the number of valence electrons, but is not formulated to predict half metallic ferromagnets. The gap in the minority states of Heusler Compounds has to be explained by the details of the electronic structure [46-48].

1.4 Crystal structure of Heusler Compounds

1.4.1 Full-Heusler Compounds Full Heusler alloys have the form X2YZ where X and Y are transition or noble metals like Fe, Co, Ni, Mn,

Cu etc. and Z is usually IIIB or IVB element such as Al, In. Si, Ge or Sn. These compounds crystallize in cubic structure with space group Fm3̅m (225) with Cu2MnAl prototype [34, 35, 49, 50]. This consists of four interpenetrating face-centered-cubic (fcc) sublattices (Figure 1.3(a)). Two of the sublattices occupied by atoms of element X, one occupied by Y and Z atoms each. The primitive cell of the L21 structure contains four atoms that form the base of the fcc primitive cell. The X atoms occupy the Wyckoff positions

8c (1/4, 1/4, 1/4), the Y and Z atoms are located at 4a (0, 0, 0) and 4b (1/2, 1/2, 1/2).

1.4.2 Half-Heusler Compounds The Half-Heusler compounds have the form XYZ (X and Y are transition metals; Z is main group element)

[51]. They crystallize in a non-centrosymmetric cubic structure with space group F4̅3m (216) and have the CIb structure. This structure can have derived from the tetrahedral ZnS-type structure by filling octahedral lattice sites (Figure 1.3(b)). The Wyckoff positions are 4a (0,0,0) ,4b (1/2,1/2,1/2), and 4c

(1/4,1/4,1/4).

1.4.3 Inverse Heusler Compounds Full Heusler compounds tend to be inverse Heusler compounds if the valence of the X transition metal atom is smaller than the valence of the Y transition metal atom from the same period. They crystallize in the so-called Xα structure with space group F4̅3m. Usually, the element X is more electropositive than

Y [52]. In contrast to the normal Heusler structure where all of the X atoms fill tetrahedral holes, in inverse

Heusler structure, X and Z atoms form a rock salt lattice to achieve octahedral holes with four-fold symmetry. The Wyckoff positions are 4b (1/2, 1/2, 1/2) and 4d (3/4, 3/4, 3/4) for X atoms, while Y and Z atoms are located at 4c (1/4, 1/4, 1/4) and 4a (0, 0, 0) respectively (Figure 1.3(c)). The prototype of this structure is Hg2TiCu. It is possible to emphasize the difference to regular Heusler compounds by expressing the formula as (XY) XZ.

1.4.4 Quaternary Heusler Compounds The ordered quaternary full Heusler compounds are usually named as LiMgPdSn-type which have chemical formula (XX’) YZ where X, X’ and Y are transition metal atoms where the valence of X’ is lower than the valence of X atoms and the valence of the Y elements is lower than the valence of both X and X’[53,54]. This structure exhibits a primitive fcc cell with a basis containing four atoms on the

Figure 1.3: Different types of Heusler structures. (a) Full Heusler, (b) Half- Heusler, (c) Inverse Heusler, and (d) Quaternary-Heusler structures [49].

Wyckoff positions 4a to 4d, which form a larger cubic cell. The exchange of the atoms between 4a and 4b or 4c and 4d positions and between groups (4a, 4b)↔ (4c, 4d) does not change the structure due to symmetry implied by the F4̅3m space group.

1.5 Order-Disorder Phenomena in Heusler Compounds The electrical and magnetic properties of Heusler compounds are strongly dependent on the atomic arrangement of the atoms. Band structure calculations show that small amount of disorder within the distribution of the atoms on the lattice sites cause distinct change in their electronic structure [55-57].

Therefore, a careful analysis of crystal is essential to understand the structure-to-property relation of

Heusler Compounds. The most known disordered structure Heusler structure [51, 54, 58, 59, 60] are shown in Figure 1.4. The most frequent type of disorder for L21 structure is the CsCl-like structure, known as B2-type structure. In this structure, the Y and Z atoms are equally distributed and consequently the 4a and 4b positions are equivalent. As a result, the symmetry is reduced and the resulting space group is

Figure 1.4: Most usual disordered Heusler structure: (a) CsCl-type, (b) BiF3-type, (c) NaTl- type, and (d) W-type [48].

Pm3̅m. On the other hand, the random distribution of X and Y or X and Z leads to a BiF3- type disorder.

(Space group - Fm3̅m). The NaTl-type structure occurs very rarely. In this structure type, the X-atoms, which occupy one of the fcc sublattices, are mixed with the Y atoms, whereas the X atoms on the second sublattices are mixed with the Z atoms. This kind of disorder is known as B32a disorder (Space group -

Fd3̅m). Here, the X atoms are placed at the Wyckoff position 8a (0, 0, 0), while Y and Z are randomly distributed at position 8b (1/2, 1/2, 1/2). In contrast to these partial disorder phenomena all positions become equivalent in the tungsten-type structure leading to bcc lattice with reduced symmetry (Im3̅푚).

This disorder is known as A2 type disorder.

1.6 Magnetism of Heusler Alloys Thin Films

1.6.1 Exchange Interactions

1.6.1.1 Direct Exchange Interactions When the electron spins associated with the neighboring atoms interact by overlapping the quantum wave functions, the mechanism is called direct exchange interaction. The electrons become indistinguishable in this overlapping region and atoms can exchange electrons [61]. The energy associated with this exchange is given by equation (1.8)

퐸푒푥 = −2 퐽푒푥 푺풊. 푺풋 1.8

Where 푺풊 and 푺풋 are spin angular momentum vectors of two atoms iand j. 퐽푒푥 is the exchange integral or the overlap integral. This is basically relative proximity of the electrons to each other and describes the direction and strength of the alignment of the spins. The positive value of 퐽푒푥 gives parallel alignment

(ferromagnetic) while negative value gives antiparallel alignment (antiferromagnetic) [62]. In metallic systems where the electrons form a band structure, the magnetic moment of the material is described by the energy cost of aligning parallel or antiparallel in the specific bands near the Fermi energy (EF). The condition for ferromagnetism is then defined by the Stoner criterion (equation 1.9) [63].

Figure 1.5 : Dependence of the exchange integral 푱풆풙 on the atomic separation 풓풂 normalized to the electron radius d orbital 풓풅 [61].

퐽푒푥(퐸퐹)푛(퐸퐹) > 1 1.9

퐽푒푥 itself is dependent upon the separation of atoms in the material or more importantly the separation of the d orbitals of the localized electrons. The variation in 퐽푒푥 with the atomic separation (ra) is shown in

Figure 1.5 using the Betha-Slater curve. For large atomic separation exchange integral is positive. As ra decreases the exchange integral reaches a peak. At extremely small spacing the electrons can be regarded as having the same spatial coordinates so that Pauli exclusion principle forces the spins to align antiparallel.

1.6.1.2 Indirect Exchange interaction Indirect exchange interaction occurs when localized but separated magnetic moments in metallic system are coupled by conduction electrons. This type of interaction is often called as RKKY interaction due to the proposition by Rudermann and Kittel [64], Kasuya [65] and Yosida [66]. When a localized magnetic impurity is placed in a free electron gas, the wave functions change to accommodate the impurity. This

Figure 1.6: The dependence of indirect exchange coupling on the thickness of a Cr interlayer separating two ferromagnetic layers [68]. effectively adds more possible states for the conduction electrons when their spins are aligned with that of the magnetic moment. Indirect exchange links moments over large distances. The interaction is characterized by a coupling constant j given by equation (1.10)

푗2 1.10 (푅푙 − 푅푚) = 9휋 ( ) 퐹(2푘퐹|푅푙 − 푅푚|) 퐸퐹 where kF is the radius of the conduction electron Fermi Surface, 푅푙 is the lattice position of the point moment. 퐹(푥) is given by 퐹(푥) = (푥 푐표푠푥 − 푠푖푛푥)/푥4 [67].

Figure 1.6 shows the oscillatory nature of coupling of two ferromagnetic layers on their separation [68].

The strength and nature of coupling is represented by magnitude of the applied field at which sample saturates. Antiferromagnetic coupling requires a large field to saturate while ferromagnetic coupling requires a lower saturating field. This coupling can happen across grain boundaries making particularly important in Heusler alloys thin films.

1.6.2 Magnetic Anisotropies

1.6.2.1 Magnetocrystalline Anisotropy Anisotropy is the term used to describe the directional dependence of the properties of a system.

Magnetocrystalline anisotropy is the dependence of the magnetization of the magnetic material with respect to the orientation of specific lattice plane of the crystal [69]. This anisotropy comes from the interaction of spin magnetic moment with the crystal and is controlled by symmetry of the crystal field or the electric field created by the chemical bonding of atoms with the lattice. This quenches the direction of the orbitals along the crystal bonds [63]. Alignment of the electrons spin along these orbitals creates one or set of crystallographic directions. The axis along which it is easier to align magnetization is easy axis whereas hard axis is the hardest to magnetize. The energy necessary to deflect the magnetic moment in the crystal from easy to hard axis is called as magnetocrystalline energy (EK). This external energy breaks the spin- orbit interaction of the crystallographic directions along easy axis. Most of the Heusler alloys exhibit cubic-like magnetocrystalline anisotropy. The energy associated with this anisotropy is given by equation (1.11)

퐸푘 = 퐾0 + 퐾1푓1(훼1, 훼2, 훼3) + 퐾2푓2(훼1, 훼2, 훼3) 1.11

Where퐾0, 퐾1 and 퐾2 are the anisotropy constants for a given material, these coefficients are function of magnetization, as well as temperature of the material and 훼1,훼2, 훼3 are the cosines of the angles made between saturation field direction and [100], [010], and [001] crystal axes respectively. For a cubic lattice, equation (1.11) can be written as [70].

2 2 2 2 2 2 2 2 2 퐸푘 = 퐾0 + 퐾1(훼1 훼2 + 훼1 훼3 + 훼2훼3) + 퐾2훼1 훼2훼3 + ⋯ .. 1.12

For a material with uniaxial anisotropy, easy axis lies along the c-axis, energy is given as

2 4 퐸퐾 = 퐾0 + 퐾1푠푖푛 휃 + 퐾2 sin 휃 +… 1.13

12 where 휃 is the angle between the easy axis and the magnetization of the material. Cubic materials generally have smaller value of 퐾2 as compared to 퐾1 so that it is negligible. For Heusler alloy thin films, anisotropy constants are found to be system dependent and also depend on growth technique as well as nature of the substrate.

1.6.2.2 Shape Anisotropy In the polycrystalline thin films, shape anisotropy is the dominant form of anisotropy because of non- spherical magnetic grains. Shape anisotropy arises from the effect of the demagnetizing field (Hd). Shape anisotropy is also caused by dipole-dipole interaction. Non-spherical objects have demagnetizing field Hd given by equation (1.14) [69, 61].

퐻⃗⃗ 푑 = −퐷. 푀⃗⃗ 1.14

Where D is demagnetizing tensor. For thin films, quantitative treatment uses the magnetostatic energy of the grain and how this changes with the direction of an applied field relative to the magnetization. The demagnetizing energy is given by equation (1.15)

1 1.15 퐸 = 휇 푀⃗⃗ 퐷푀⃗⃗ 푑 2 0

2 퐸푑 = 1/2휇0퐷푀 1.16

Here 1/2휇0퐷 is the shape demagnetizing factor which expresses how demagnetizing field changes with grain shape. We can define shape anisotropy constant as

1 퐾 = 휇 푀2 with anisotropy field 퐵 = 2 퐾 /푀 . 푆ℎ푎푝푒 2 0 푆 퐾 푆ℎ푎푝푒 푆

For the polycrystalline Fe- and Co-based Heusler alloy thin films, shape anisotropy is significantly larger than magnetocrystalline anisotropy.

1.6.3 Magnetic Domain Domain theory is initially proposed by Weiss in 1906 to explain the magnetization process. Domains exist in order to minimize the magnetostatic energy of a magnetic material. When a magnetic material is

saturated in one direction an opposing field is created, called demagnetizing field Hd which stores the

magnetostatic energy. To reduce the demagnetization, a magnetic material breaks into domains [70]. In a

system with cubic anisotropy and shape anisotropy the domain structure is often more complex. For a

single domain specimen as shown in Figure 1.7(a) has large magenetostatic energy associated with leakage

of magnetic flux into the surrounding air space. Figure 1.7(d) shows the formation of domains for material

with anisotropy.

1.6.4 Domain Wall Formation The boundary between two neighboring domains is not abrupt. The change in the direction of moments

occurs over some finite length, typically tens of nm. The magnetization reverses through a series of slight

rotations of localized moments in the boundary region. These boundary regions are known as domain

walls. These walls are divided into Bloch wall and Nѐel Wall [61]. Bloch walls occur when the magnetic

moments rotate perpendicular to the plane of domain wall. Bloch walls occur in bulk samples. In case of

thin films, they create free poles at the sample surface which contribute significant magnetic energy to the

system. Moments rotate in the plane of the film to form Neel walls. Domain walls have a finite width, δ.

The width of domain walls results from the competition between anisotropy energy Ek and the exchange

energy Eex in a magnetic material. The anisotropy energy can be written as

Figure 1.7: A variety of domain structure for a magnetized sample. a) Uniformly magnetized. b) Two domains. c) Four domains. d) Essentially two domains with two closure domains [71].

퐸푘 = 푓(휙) 1.17 where φ is the angle between magnetization vector with easy axis of the system at a specific point. The exchange energy can be written as in equation (1.18)

푑휙 2 1.18 퐸 = 퐴 ( ) 푒푥 푒푥 푑푥 where 퐴푒푥the exchange stiffness constant and x is the position of the moment within the wall. By minimizing the sum of above two equations, domain wall width can be expressed as

1.19 훿 = 휋√퐴푒푥/퐾

If a magnetic material is not a single crystal and made up of grains, then it is possible that these grains will only contain a single domain. If the width of domain wall is greater than the size of the grain, then it is possible to form a single domain wall.

1.6.5 Magnetization Reversal Process When an external magnetic field is applied to a magnetic material the domains aligned with the field grow.

If the material is initially saturated, a field is applied opposite to this saturation field. That field will nucleate a domain with its magnetization aligned with the field. This nucleation occurs through the rotation of small number of moments in the film. At intermediate field strengths, a second mechanism becomes

Figure 1.8: Schematic diagram of 180o a) Bloch wall and b) Nѐel wall [61].

15 significant; this is domain rotation in which the atomic magnetic moments within an unfavorably aligned domain overcome the anisotropy energy and suddenly rotate from their original direction of magnetization into one of the crystallographic easy axes, which is nearest to the field direction. At high fields, coherent rotation occurs, where magnetic moments which are all aligned along an easy axis nearest to the field direction is gradually rotated into the field direction as the magnitude of the field is increased. This results in a single-domain sample. Depending upon the strength of an applied field, domain wall motion could be reversible or irreversible

The growth of domains requires the motion of domain walls. The motion of domain walls is not smooth and the domain walls move through a series of jerky steps known as barkhausen jumps. These jumps are result of continuous pinning and de-pinning of domain wall as it moves through the material. The motion of a domain wall requires an energy input, therefore any regions of the film that increases this energy cost act as domain wall pins. Possible cause of these energy increases includes surface roughness, stresses, vacancies in the lattice sites and impurity within the material. In case of Heusler alloys disordered phase can also aid domain wall motion. Since our film is multilayer, interatomic diffusion between layers can built up of impurities in the magnetic layers, which can also act as a pinning center [61, 73].

1.7 Application of Heusler alloy

1.7.1 Half-Metallic Ferromagnets for Spintronics Magnetoelectronics, also known as spintronics, is an emerging field based on a combination of three conventional information carriers: electron charges, electron spins and phonons. These carriers represent three major fields in information and communication technology; data processing with electron transport, data storage with an assembly of spins and data transfer [74]. Spintronics, based on spin–polarized electron transport is believed to meet the requirements for future technologies, such as low power consumption due to non-volatility of recorded data, enormous increase speed of data processing [75].

In order to make use of the full potential of spintronics, new magnetic materials, magnetic semiconductors and so-called half-metallic ferromagnets (HFM) are needed. Half-metallic ferromagnets seem to be a suitable class of material which meets all requirements of spintronics. The most important reason is their exceptional electronic structure: They behave like metals for majority electrons and like semiconductors for minority electrons. Heusler compounds hold the greatest potential to realize half-metallicity at room temperature. Iron and cobalt based Heusler alloys have been predicted to show half-metallicity by band structure calculation. Fe and Co- based would be best candidates for half-metallic ferromagnets because of their high Curie temperature and low lattice mismatching with major substrates.

1.7.2 Heusler Alloys in Thermoelectric The search for alternative energy technologies has taken accelerated pace in recent years as climate change has become more noticeable. Thermoelectric devices are actively considered as a clean energy for waste heat recovery in automobiles since they operate silently and do not have any moving parts or environmentally harmful fluids. Half Heusler alloys fulfill almost all the requirements for commercial applications [76]. Their mechanical and thermal stability is exceptional in comparison to the commonly used thermoelectric materials. The reported thermoelectric Heusler compounds exhibit high electrical conductivity [77]. MNiSn (M = Ti, Zr, Hf) are the n-type Heusler alloy for possible thermoelectric applications.

1.8 Heusler Alloy as the Topological Weyl Semimetals When Dirac first wrote down his relativistic equation for a fermion field, he had primarily concerned electrons in mind. In his solution, Dirac found the antiparticle, which has the same mass as the electron but has opposite charge. In 1929, Hermann Weyl demonstrated the existence of massless fermion in the

Dirac equation, which was later called the Weyl fermion. In the standard model, all fermions are Dirac fermions, except possibly neutrinos that present the chirality. However, neutrinos were later found to be massive and excluded from Weyl fermions. Weyl fermions have remained undiscovered until very

Figure 1.9: Spin-resolved band structure and ferromagnetic half-metallic ground states in Co2TiX. ((a-c) the calculated bulk band structure of the majority spins of Co2TiSI, Co2TiGe and Co2TiSn, respectively. (d-f) Same as panel (a-c) but for minority spin. (g-i) The band structures of both spins [86]. recently in condensed-matter systems. In solid-state band structures, Weyl fermions exist as low-energy excitations of the Weyl semimetal, in which bands disperse linearly in three-dimensional (3D) momentum space through a node termed a Weyl point [78]. Topological Weyl semimetals are electronic strong spin- orbit metals or semimetals whose Fermi Surfaces arises from crossing between conduction and valence bands, which cannot be avoided due to non-trivial topology [79]. In order to get a Weyl semimetal, the breaking of time-reversal symmetry (TRS) or inversion symmetry (I) is required for the materials.

Although the first inversion breaking Weyl semimetal was recently discovered in TaAs [80], the time- reversal breaking Weyl and nodal line semimetals remain elusive. One possibility might be magnetic topological insulator which can show the quantum anomalous Hall effect [81]. To date, the quantum

anomalous Hall effect has only been observed in magnetically doped insulator thin film samples as Crx

(BiySb1-y)2-xTe3 [82] which required ultra-low (mK) temperatures. By contrast, monolayer samples of magnetic Weyl semimetals, which are natural ferromagnets may realize the quantum anomalous Hall effect at significantly higher temperature. Pyrochlore irridates [83] and HgCr2Se4 [84] have been predicted to exhibit magnetic topological semimetals, but these materials have relatively low transition temperature.

Magnetic Heusler compounds have several advantages over the other compounds where Weyl fermion have been proposed and detected. They are ferromagnetic half-metallic compounds with Curie temperature higher than 300 K. They also exhibit Weyl fermions, with the large associated Berry phase of their Fermi Surfaces [85]. As such, anomalous Hall Effect and spin Hall effect in these materials are theoretically predicted to be large. Since, the structure and position of Weyl fermions depend on the magnetic field, the magnetic Heusler alloy provide us with a realistic and promising platform for manipulating and studying magnetic Weyl physics in experiments. Recently, Co-based Heusler compound

Co2TiX (X= Si, Ge or Sn) have been proposed to exhibit Weyl Semimetals [86].

1.9 Magnetoresistance Magnetoresistance (MR) is the change in electrical resistance of a material when an external magnetic field is applied to it. The MR effect was first discovered by William Thomson (Lord Kelvin) in 1856, but he was unable to lower the electrical resistance of anything more than 5%. This effect was later called ordinary MR. He found experimentally that change in resistance was increasing with the pieces of iron when the current is in same direction as the magnetic force and was decreasing when the current is perpendicular to the magnetic field [87]. This effect is referred to as anisotropic magnetoresistance

(AMR). The effect is due to a larger probability of s-d scattering of electrons in the direction of the magnetic field. Magnetoresistive materials are incorporated in a number of commercially available technologies such as magnetic sensors, magnetic recording heads and magnetic memories [88]. MR is given by equation (1.20)

푅(퐻) 1.20 푀푅 = − 1 = ∆푅/푅(0) 푅(0)

In terms of resistivity,

휌(퐻) 1.21 푀푅 = − 1 = ∆휌/휌(0) 휌(0) where R(H) is the resistance at an applied field H.

A large MR was observed in multilayers consisting of alternating layers of ferromagnetic and nonmagnetic metals. The resistance is largest when the magnetic moments in the alternating layers are oppositely aligned (spin up electrons are scattered by regions of spin down magnetization and vice versa) and smallest when they are aligned parallel (conduction electrons of compatible spin type can move through the heterostructure with minimal scattering). This effect is called giant MR (GMR). These two cases are schematically illustrated in Figure 1.10. GMR was first discovered by Fert et al. [89] and Grünberg et al

[90] in 1986 in multilayers of Fe/Cr and a GMR as large as 50 % was observed at 4.2 K. Both GMR and

AMR tend to be observed in a given device but GMR can be a much larger effect. Colossal MR (CMR) is observed in some materials which enable them to modify their electrical resistance in the presence of a

Figure 1.10: A multilayer system Fe-Cr-Fe with ferromagnetic (left) and antiferromagnetic (right) exchange coupling between iron layers [89].

Figure 1.11: TMR effect in a magnetic tunnel junction device. When the magnetizations are aligned parallel (left) the device resistance is small and when they aligned antiparallel (right), the device resistance is large [91]. magnetic field by orders of magnitude. The CMR effect is observed in manganese based perovskite

materials [91].

When two ferromagnetic layers are separated by a thin insulator layer (as shown in Figure 1.11) the tunnel

magnetoresistance effect based on tunneling of electrons through the insulating barriers occurs. In case of

tunneling magnetoresistance (TMR), the resistance of the multilayer in the perpendicular direction to the

film changes depending on the orientations of the magnetization in ferromagnetic layers. This effect was

first observed by Jullière [92] in a Fe-Ge-Co magnetic tunnel junction (MTJ) in 1975. However, it was

not until 1995 that a large, reproducible TMR of 13.4 % at room temperature (31.6 % at 4.2 K) was

reported by Moodera et al. [93] using an Al2O3 tunnel barrier.

1.10 Boltzmann Transport Theory Boltzmann Transport equation is an important tool for analyzing transport phenomena within systems that

involve density and temperature gradients. The equation is applied to analysis of the general currents

within the system, transport coefficients and the relationships between them. In a state of equilibrium, the

probability of occupation of an energy level E(k) is given by the Fermi-Dirac distribution function

(퐸(풌)−휇)/푘퐵푇 푓0(풌) = 1/푒 + 1 1.22

Where T is the temperature of the sample, 휇 is the chemical potential, kB is Boltzmann constant and 푓0(풌) is a function which depends on energy퐸(풌), k is wave vector. With the application of the external fields such as electric field, magnetic field to the sample, the electron distribution move out of equilibrium distribution. In general, the function 푓(풓, 풌, 푡) depends on position vector 풓, wave vector k and on time t.

Then, number of electrons at time t in phase space volume (풅3풓, 풅3풌) around the point(풓, 풌) is given as

푓(풓, 풌, 푡)/4휋3 풅3풓 풅3풌.

The semi classical dynamics of carriers in a given band, an electron at the point(풓, 풌) at time t evolves toward the point (풓 + 풅풓, 풌 + 풅풌) at time 푡 + 푑푡. Here, we can write 풅풓 = 풗푑푡 = 훿 퐸(풌)/ℏ훿풌 and

푑풌 = 푑(ℏ풌)/풅푡 = F denotes the external force acting on the carriers. During the motion, collision may

휕푓 cause a net rate of change [ ] of the number of electrons in the phase space volume 풅3풓 풅ퟑ풌, Since 푑푡 푐표푙푙 volume in phase are preserved by semiclassical equations of motion using Liouville theorem (equation

1.23)

푭 훿푓 1.23 푓(풓 + 풗 푑푡, 풌 + 푑푡, 푡 + 푑푡 = 푓(풓, 풌, 푡) + [ ] 푑푡 ℏ 훿푡 푐표푙푙

The above equation expresses the detailed balance in each volume 풅3풓 풅ퟑ풌 of the number of carriers, when moving in phase space under the action of external fields and in the presence of collision process.

Expanding equation (1.23) gives Boltzmann equations as [94]

훿푓 훿푓 푭 훿푓 훿푓 1.24 . 풗 + . + = [ ] 훿풓 훿풌 ℏ 훿푡 훿푡 푐표푙푙

훿푓 훿푓 Where and stand for ∇ 푓 and ∇ 푓 repectively. Using relaxation time approximation, the transport 훿푟 훿푘 풓 풓 equation (1.24) becomes

훿푓 훿푓 퐹 훿푓 푓 푓 1.25 . 푣 + ∙ + = − − 표 훿푟 훿푘 ℏ 훿푡 휏 휏

Transport can occur either in form of electron current density 푱 or energy flux density 푼:

3 ퟑ 1.26 푱 = 1/4휋 ∫(−풆)풗풌 풇 풅 풌

3 3 1.27 푼 = 1/4휋 ∫ 퐸(풌)풗풌 푓 푑 풌

where the factor 2/8휋3 takes into spin degeneracy and density of allowed points in k space per unit volume.

By doing some mathematical workout, for isothermal condition, we can write current density as

1 1.28 푱 = 휎[푬 + ∇휇] 푒

푛푒2휏 With 휎 = 푚∗

Where 푬 is electric field, 푒 is electronic charge, 휇 is chemical potential, 푛 is the electron concentration

푚 ∗ is effective mass of electron, 휏 is relaxation time, 휎 is electrical conductivity.

1.11 Temperature Dependence of Resistivity The temperature dependent electrical resistivity, ρ of magnetic materials is a result of electron-phonon scattering (휌푒푝), electron-electron interaction (휌푒푒), and electron-spin wave scattering (휌푒푠). According to

Mattiessen rule all these contributions add up to give total resistivity [95].

휌(푇) = 휌푒푝 + 휌푒푒 + 휌푒푠 1.29

Besides these contributions, the scattering due to lattice defects, called impurity scattering also contribute to the total resistivity of ferromagnetic material. Impurity scattering gives only constant component of the resistivity. Electron–phonon scattering increases with temperature and varies linearly with T at higher

23 temperature range [96]. The resistivity of normal ferromagnetic materials at low temperatures usually has a term proportional to 푇2 because of electron-electron scattering. For magnetic solids, the aligned (or anti aligned) electron spins precess slightly about their ground state, which are in otherwise fixed position.

These are small variations about potential minima so are thus harmonic oscillators obeying Bose-Einstein statistics. This gives spin-waves or magnons. This is also ascribed to one-magnon scattering of conduction electrons where electron undergoes a spin-flip in an inelastic scattering process involving the creation or annihilation of a magnon. This 푇2 dependence is also valid for the d-orbitals electrons which are delocalized [97, 98]. However, in half-metallic ferromagnets (HMF) at low temperature all the states at

퐸퐹 are spin polarized and so spin-flip scattering is not possible. Therefore, for a HMF one expects the absence of a T2 term in the resistivity. The first available magnetic scattering processes involve two magnons, which give rise to a term in the resistivity varying as 푇4.5 (although this term is difficult to disentangle from regular scattering processes involving phonons) [99]. Another feature, which is sometimes observed in HMF, is an upturn in ρ at low temperatures (typically < 40 K). This is attributed to weak localization, which commonly occurs in disordered metallic systems [100]. The conduction electrons have a very short mean free path and so interference between scattered waves may occur. If the electron waves maintain phase coherence along their path, closed loop paths offer the electron two paths of equal phase change (elastic, phononless scattering process). The probability of an electron returning to its starting point is increased and so the material shows an increase in resistivity as the temperature decreases due to this enhanced scattering. However, at higher temperatures, more phonons are excited and so inelastic scattering is more probable and so the probability of back-scattering is reduced, so the resistivity exhibits the expected decrease with decreasing temperature.

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Chapter 2

2. Experimental technique and characterization

In this chapter, the experimental techniques will be discussed which will cover synthesis, characterization and transport measurements. Vapor deposition and electrospinning have been used to grow nanostructures. The synthesized nanostructures were structurally characterized by X-ray diffraction and energy dispersive spectroscopy. The morphology of the samples was analyzed using scanning electron microscopy, and magnetically characterized by vibrating sample magnetometer. Transport measurements were performed by physical property measurement system.

2.1. Vapor Deposition Method

In terms of depositing nanostructures, there are two ways to do so; by chemical vapor deposition (CVD) or by physical vapor deposition (PVD). The process of depositing thin films directly from the source to substrate through vapor phase is called physical vapor method. Chemical vapor deposition is the process of chemically reacting a volatile compound of a material to be deposited, with other gases, to produce nonvolatile solid that deposits atomically on a suitable substrate.

2.2. Physical Vapor deposition

PVD is the process of controllably transfer atoms from a heated source to a substrate located a distance away, where film formation and growth proceed atomistically. The process is purely physical and can be done by different ways such as

• Thermal evaporation

• Pulse laser deposition

• Electron-beam evaporation

• Molecular beam epitaxy

• Ion beam assisted evaporation

• Sputtering

In physical vapor deposition, the material to be deposited is in the form of a solid or liquid phase and requires thermal energy for transformation into the vapor phase. Evaporation thus includes sublimation when a solid directly transforms into the vapor phase and vaporization when the liquid transforms into vapor on thermal treatment. The vapor, by its own nature, expands into an evacuated chamber that contains the substrate. The vapor condenses on the substrate that is lower temperature than the evaporation source.

The formation of a film of a material on solid surfaces involves phase transformation, such as condensation of a vapor onto a solid surface. For a condensed phase to coexist in vapor phase, there exist equilibrium vapor pressure. This pressure (Pe) is the partial pressure of a gas in equilibrium with its condensed phase at a given temperature can be computed from Clasius-Clapyeron equation [1]. If we suppose P is the ambient hydrostatistic pressure acting upon the evaporant in the condensed phase, the rate of evaporation was proportional to (Pe –P) as experimentally found by Hertz [2]. This is consistent with kinetic theory in which the impingement rates are proportional to pressure. The rate at which atoms enter the gas phase can be written as equation (2.1)

푑푁 2.1 푒 = 퐽 퐴 푑푡 푉 푒 where 퐽푉 is the flux of molecules from unit are of the evaporation surface, and A is the area of the evaporating surface. Hertz also found that evaporation rate could not be increased by supplying more heat

32 unless equilibrium vapor pressure was also increased by this action. Thus, maximum evaporation rate can be written as (2.2)

2.2 퐽푉(Max) = (푃푒 − 푃)√1/2휋푚푘퐵푇

Maximum rate can be achieved if P = 0. Knudsen [3] found experimentally that the pressure calculated from weight loss measurements were lower than that pressure measured under proven equilibrium conditions. He introduced the vaporization coefficient 훼푉, to account for the fact that some of the molecule hitting the surface may reflect of do not reach in the equilibrium condition. Therefore 훼푉 is the ratio between an observed vaporization rate and the maximum rate calculated from the kinetic theory of gases and equilibrium vapor pressure of the material. Hence, evaporation rate can be written as (2.3)

푑푁푒 2.3 = 훼푉 (푃푒 − 푃)√1/2휋푚푘퐵푇 푑푡 퐴푒

The above relation is called Hertz-Langmuir-Knudsen equation [4]. The value of 훼푉 lies between 0 to 1.

Mass evaporation rate is given by equation (2.4)

푑푁푒 2.4 Γ = 푚 = 훼푉 (푃푒 − 푃)√푚/2휋푘퐵푇 푑푡 퐴푒 so that mass of the evaporated material is

2.5 푀푒 = ∫ Γ 푑퐴푒푑푡

Evaporation can occur in two ways; free evaporation and effusion. Evaporation from free surface is termed as Langmuir evaporation which is isotropic [5]. Effusion is the escape of molecules through an opening of a container without disturbing the equilibrium condition of a gas in the container. Effusion refers to evaporation through an orifice by which area of the orifice appeared as an evaporation source of the same area. Effusion is somewhat directional and ideally it is Lambertian angular distribution. Maximum

Figure 2.1: Knudsen cell with directional dependence [4] evaporation can be obtained from effusion if the sticking coefficient (αV) is equal to 1. In 1909 Knudsen

[6] invented an arrangement to have αV ~1. This is called a Knudsen cell/effusion cell or K-cell as shown in Figure 2.1. It consists of isothermal enclosure with small orifice of area Ae and the orifice acts like an evaporating surface area at Pe but it cannot reflect incident vapor molecules. Surface area of the evaporant inside the enclosure is large compared to the area of the orifice. Figure shows effusion from Knudsen cell.

Mass deposition rate per unit area from the Knudsen cell obeys cosine law and is given as

2.6 푚 2 푅푚 = 퐶√ 푐표푠휃푐표푠휑 (푃푒 − 푃)/푟 2휋푘퐵푇 where C is a constant, r is the source-substrate distance.

For spherical surfaces with source on its edge: mass evaporation rate is independent of angles and uniform coating can be obtained [7].

Figure 2.2: Uniformity of an evaporated film on a flat surface [7]. For flat surface (as shown in Figure 2.2):

Deposition rate at the center is given as

1 2 푅1 ∝ 2 and at the edge is 푅2 ∝ 푐표푠휃 푐표푠휑/푟2 푟1

2 2.7 푅1 2 푊 2 2 = {푟1 + ( ) /푟1 cos 휃} 푅2 2 assuming 푐표푠휃 = 푐표푠휑

푤 2 2 2.8 ( ) 푅1 2 = {1 + 2 } 푅2 푟1

2 푅 푊 2 2.9 1 = [1 + ( ) ] 푅2 2푟1

푊 2 So, if we define uniformity as 휎 = (푅1 − 푅2)/푅1 then 휎 is proportional to( ) . So, larger r1 means 2푟1 lower deposition rate and higher evaporation waste.

2.3. Electron –beam evaporation

In this evaporation method, the beam of electrons is emitted in such way that it will heat the source and vaporize the material to be deposited. Once the material changes its state of being a liquid to vapor, it is able to condense itself on the substrate. The advantages of this method over other techniques are:

• It is extremely versatile.

• Can achieve temperatures in excess of 3000° C.

• Use evaporation cones or crucibles in a copper hearth.

• Typical emission voltage is 8-10 kV.

• High quality film can be grown.

• Typical deposition rates are 1-100 Å/푠.

2.3.1. Working principle

Figure 2.3 shows the schematics of e-beam evaporation. The electron beam is produced by a hot filament which is represented by the symbol of resistor. The process of the evaporation is taken place in a high vacuum area to allow molecules to move freely in the chamber and hence condense on all surfaces including the substrate, and this follows the basic principle of evaporation [8].

The working principle is simple. It can be divided into three stages. The first stage is a heating process of the hot filament. The hot filament will produce a beam of electrons that is excited due to heating. This process can be done by applying thermionic emission. The second stage is the heating of the material that

Figure 2.3: Schematics of e-beam evaporation [4]. is to be deposited. The material that is to be deposited is shown in Figure 2.3 as evaporant in water-cooled hearth. The source of the heating process is actually the beam of electrons that is produced from stage one.

Now, approaching from stage one, since the electrons are required to be used as the source of boiling of the material in stage two, the beam of electrons must have a certain path in order to be fully used effectively. Excited electrons are able to move randomly in the high vacuum without any force of attraction to it. Hence, to make it to move only towards the material to be evaporated, particularly designed magnets need to be used. In this case, two magnets are required. The first magnet, a focusing magnet is used to attract the electrons towards it right after the electrons are excited. Then, in order to deflect the route of the beam of electrons towards the material to be evaporated, a deflecting magnet will be used.

After the whole process of attracting and deflecting of magnets, now the path towards the material is set and the material can be heated. The material will be heated up to a boiling point, of which governed by the principle of evaporation; once the boiling point is reached, the molecules in the liquid (material

37 evaporant) will collide and transfer energy to each other and thus the liquid would turn into vapor.

Therefore, the material evaporant is now being able to move freely in the high vacuum area. Once the material evaporant is able to move, and then it could move towards the substrate and attach on it, which is the last stage, stage three.

2.3.2. Thin Film Growth Process

Thin film growth through several stages each affecting the resulting microstructure [9]. When an atom impinges on a surface, it must loose enough energy to stay on the surface. An approaching atom can either be absorbed or reflected on the surface of the substrate depending on the incoming flux of atoms, the trapping probability and the sticking coefficient. Such atoms see a potential energy called as barrier defined by the atomic structure and chemistry of the surfaces. These adatoms must overcome local energy barriers to stick on the surface of substrate. Once the atoms stick to the surface, it creates surface energy in the surface. Surface energy is due to the surface atoms stretching their bonds in response to the absorption of the atom. Overall surface energy can be minimized if the atom has enough energy and time to diffuse a low energy sites. The adsorbed atoms diffuse between these adsorption sites randomly given as

Ι = 푣푒−퐸푑/푘퐵푇 2.10

where 푣 is attempt frequency, 퐸푑 is diffusion barrier, T is substrate temperature [10]. If we assume diffusion barrier is same in all directions and that adsorption sites are separated by distance 푎 , we can

1 calculate surface diffusion coefficient as 퐷 = 푎2Ι. A diffusion length can be defined as 퐿 = √퐷휏,휏 is 4 0 diffusion time. At low temperature, the diffusion length increases with temperature. But at high temperature, desorption rate also increases and overtakes absorption. The adatoms will diffuse randomly on the surface of the substrate until they either encounter another atom and form a cluster or desorb from the surface. The process of forming clusters to form a stable thin film is called nucleation. For a stable

Figure 2.4: Thin film growth model. [7] film, nuclei at a critical size are needed. Since adding an atom to the surface creates extra surface energy, atoms or molecule should have a lower chemical potential as a condensate than as a vapor. For nuclei smaller than critical size, the surface energy is too large and overall reaction is thermodynamically unstable. A stable nuclei is only formed when adding molecules decreases the Gibb’s free energy. The nuclei can grow in three different ways as shown in Figure 2.4 [7].

a) Island growth (Volmer-weber): If the total surface energy of the film interface is larger than that

of the substrate-vapor interface then islands growth occurs. This is due to slow diffusion or the

film atoms more strongly bound to each other than the substrate.

b) Layer by Layer growth (Frank-van der Merwe): If the substrate-vapor surface is large than the

other two combined than a smooth layered film with high crystallinity will be formed. In this mode

of growth, the film wets the surface to lower surface energy.

c) Mixed type (stranski-Krastnov): In this type, initially film grows layer by layer and then changes

three dimensional islands. The initial layer is strained to match the substrate.

The growing islands will eventually meet and they will begin restricting to minimize the energy in the systems [11]. This combination of nuclei is called as coalescence. If two nuclei of different sizes are situated near to each other there will be gradient in the adatoms density between the grains. This gradient creates a driving force for diffusion from higher atoms density around the small grain to lower atoms density around larger grain. The result is formation of large grain size called Ostwald ripening [12]. The second mechanism is sintering. In this case, impinging islands will deform elastically, touch, and form a grain boundary thereby trading surface energy for interface and strain energy [13]. A neck forms between two growing nuclei and the curvature of the necks allow faster growth and merging. The third mechanism is called cluster coalescence. In this mechanism, nuclei undergo surface diffusion. When it is chemically favorable for two nuclei to align and stay permanently at a location, they merge to give larger cluster [14].

The nuclei and the subsequent combination of the nuclei lead to a final film microstructure.

Figure 2.5: Schematic diagram of set up of electrospinning apparatus. [15]

2.4. Electrospinning

Electrospinning is an efficient and versatile method to produce continuous nanofibers from submicron diameters down to nanometer diameters by applying high electric field. The term electrospinning was derived from “electrostatic spinning” and it is an old technique that date back to 1897 [15]. The electrospinning technique was not commercially adopted due to competition with mechanical drawing process to form polymeric fibers and it remained an obscure method of making fibers until the mid-1990.

This technique can be considered as a variant of the electrostatic spraying (electrospraying) process, as both methods use high voltage to induce formation of liquid jets [16]. To produce metallic nanowires via electrospinning process, metal compounds should be included in the polymer solution. A typical electrospinning set-up (as shown in Figure 2.5) consists of mainly three components.

• A capillary tube with pipette or needle of small diameter

• A high voltage supplier

• A metal collecting plate

In electrospinning, most of nitrates and polymers are dissolved in a solvent, forming a solution. This

solution is then fed to the capillary or tip of needle for spinning. The process principle involves

subjecting a polymer solution held at its own surface tension at the end of a needle to an electric field.

As the intensity of the electric field is increased, the hemispherical surface of the solution at the tip

elongates and forms a conical shape known as Taylor cone [17]. The behavior of the electrospun jet

can be divided into three main stages: the formation of the Taylor cone, the ejection of the straight jet

and the unstable whipping jet region. A droplet raised to potential Φo can be sustained to hemispherical

shape by surface tension until the electric field exceeds surface tension on the droplet as given by

Rayleigh criteria (as shown in Figure 2.6) [18]:

1 − 2.11 휙0(휋푅0푇) 2 ≤ 4

Where Ro is the volume equivalent droplet radius and T is the surface Tension Coefficient.

If the potential of the droplet is increased above Rayleigh criteria, the hemispherical shape of the liquid starts to deform by the electrostatic into a conical shape known as Taylor cone. If the electric field is further increased, a liquid jet is ejected and stretched from the conical deformation of the drop. Once the jet emerges from the liquid drop, the fluid travels straight path along the direction of the applied field up to a certain distance. This region is the steady region as long as the drop at the tip of the needle can supply fluid to the jet. As the jet travels further, its path becomes unstable and complicated.

2.4.1. Electrospinning Parameters

The ideal targets in electrospinning of a solution into nanofiber are:

• The diameters of the fibers must be consistent and controllable.

• The fibers surface must be defect-free or defect-controllable.

• Continuous single nanofibers must be collectable.

The parameters governing electrospinning process can be divided into three categories [20]. They are solution, process and ambient parameters. Solution parameters depend upon concentration of solution, molecular weight of polymers, viscosity, surface tension and conductivity. A minimum concentration is required for fiber formation. At very low concentrations, electrospray occurs instead of electrospinning.

This is due to low viscosity and high surface tension of the solution. At low concentrations, a mixture of fibers and beads are obtained. As the concentration increases, the shape of beads changes from spherical to spindle-like. Finally, uniform fibers with increased diameters are formed [21]. Optimal solution viscosity is required for electrospinning, as very low viscosity leads to no fibers formation and very high viscosity results in ejection of jets from polymer solution. Viscosity, solution concentration and polymeric

Figure 2.6: Axisymmetric infinite fluid body kept at potential 흓ퟎat a distance 풂풐 from an equipotential plane [19]. molecular weight are related to each other. Another important factor is surface tension. Reducing the surface tension contributes to formation of nanowires without beads. But low surface tension will not always offer ideal electrospinning conditions. It is critical in determining the upper and lower boundaries provided that other parameters are constant [22]. Applied voltage is the crucial factor in electrospinning as the threshold voltage must be exceeded for the charge jets to be ejected from Taylor cone. After threshold voltage is reached, fiber formation occurs, inducing the necessary changes on the solution along with the electric field and starting the process [23]. Ambient parameters such as humidity and temperature also effect wire diameter and morphology. Increased temperature leads to yield of fibers with decreased diameter, while lower humidity may dry the solvent completely. Medeiros et al. have shown that higher humidity causes porous nanofibers as shown in Figure 2.7 [24]. If relative humidity (RH) is higher than

25%, pores start to form on the electrospun fibers, the actual value of RH can vary from polymer to polymer. The RH also affects the skin formation of nanofibers.

Figure 2.7: SEM micrograph of PMMA fibers electrospun from toluene solution at (a) 20±4; (b) 40±4; (c) 60±4; and (d) 80±4% relative humidity [24] 2.5. Nanostructures Characterization Techniques

2.5.1. Scanning Electron Microscopy (SEM)

The history of electron microscopy began with the development of electron optics. In 1926, Busch studied

the trajectories of charged particles in axially symmetric electric and magnetic fields, and showed that

such fields could act as particle lenses, laying the foundations of geometrical electron optics. In 1925, de

Broglie put forward the hypothesis of electron wave. The idea of an electron microscope began to take

shape after the discovery of such electron wave. In 1931, Ruska and his research group in Berlin were

working on electron microscopy. They found using de Broglie equation that electron wavelengths were

almost five orders of magnitude smaller than the wavelength of light used in optical microscopy. It was

thus considered that electron microscopes could prove a better resolution than light instruments, and no

reason existed to abandon this aim. 1932, Knoll and Ruska tried to estimate the resolution limit of the

electron microscope. Knoll built the first ‘‘scanning microscope’’ in 1935. Figure 2.8 shows the schematic

of SEM. SEM consists of two main components, the electronic console and the electron column. The

Figure 2.8: Schematics of SEM [25]. electronic console provides control knobs and switches that allow for instrument adjustments such as filament current, accelerating voltage, focus, magnification, brightness and contrast. The electron column is where the electron beam is generated under vacuum, focused to a small diameter, and scanned across the surface of a specimen by electromagnetic deflection coils. The lower portion of the column is called the specimen chamber. The secondary electron detector is located above the sample stage inside the specimen chamber. Specimens are mounted and secured onto the stage which is controlled by a goniometer [26]. Electron gun is located at the top of the column where free electrons are generated by

45 thermionic emission from a tungsten filament at ~2700 K. Electrons are primarily accelerated toward an anode that is adjustable from 200V to 40 KeV (1keV=1000V). After the beam passes the anode it is influenced by two condenser lenses that cause the beam to converge and pass through a focal point.

Another important part in electron column is apertures. Depending on the microscope one or more apertures may be found in the electron column. The function of these apertures is to reduce and exclude extraneous electrons in the lenses. In the scanning system, images are formed by rastering the electron beam across the specimen using deflection coils inside the objective lens. The stigmator or astigmatism corrector is located in the objective lens and uses a magnetic field in order to reduce aberrations of the electron beam. The electron beam should have a circular cross section when it strikes the specimen however it is usually elliptical thus the stigmator acts to control this problem. Besides these parts, the ability for a SEM to provide a controlled electron beam requires that the electronic column be under vacuum at a pressure of at least 5x10-5 Torr. A high vacuum pressure is required to prevent oxidation of hot tungsten filament and to get clean and dust free environment in electron column. In SEM, electron energy is dissipated as a variety of signals produced by electron-sample interaction when incident electrons are decelerated in solid sample. These signals include secondary electrons that produce SEM images, backscattered electrons, photons that are used for continuum analysis and heat. Secondary electrons are most valuable for showing morphology and topography on samples whereas backscattered electrons are most valuable for illustrating contrasts in composition in multiple samples [27].

2.5.2. Energy Dispersive X-ray Spectroscopy (EDS)

The X-ray signal emitted from the specimen contains characteristic peaks whose energy can be related to an atomic transition and hence to a particular chemical species. There are two electron beam specimen interactions to consider here. There is core scattering, which results in the emission of a continuous background and inner shell ionization, which gives the characteristic peaks. The incident electron has

46 sufficient energy to knock an inner shell electron out to the vacuum. An electron from a higher energy level falls down to the partially filled lower energy level and a photon is emitted. The energy of the photon corresponds to the difference between the two energy levels. Transitions are labelled as K, L or M, which is the energy level from which the electron was ejected and they are also given subscripts such as α, β, γ which indicates from which level the electron that fills the hole has come. The x-ray from the most probable transition is designated α. Therefore, a 퐾훼 x-ray is formed from a transition from the L shell to the K shell whereas a 퐾훽 x-ray results from a transition from the M shell to the K shell [27].

EDS is the X-ray detector system and is based on a p-i-n junction in silicon. An incoming X-ray generates a photoelectron, which leads to the generation of a number of electron-hole pairs. The number of pairs generated is proportional to the energy of the X-ray. The signal is amplified and is then sorted according to voltage amplitude by a multichannel analyzer. Several thousand pulses per second can be processed and so a spectrum can be obtained in a short space of time. The current produced by the X-ray is small compared to the conductivity of the silicon and so the junction is reverse biased [28]. The silicon is doped with Li to increase its resistivity and the detector is cooled to 77 K with liquid nitrogen to keep thermally activated conductivity and electronic noise to a minimum. There is normally a window on the outer surface of the detector, made of a polymer film (or Be) to prevent contamination from condensing on the cold detector.

2.5.3. X-ray Diffraction (XRD)

X-rays photons are form of electromagnetic radiation produced by the ejection of an inner orbital electron and subsequent transitions of atomic orbital electrons from states of high energy to low energy. X-rays were discovered by Roentgen in 1895 and the property of the atomic number dependence of the absorption ox X-rays photons was quickly established and applied for medical diagnostics purposes. For diffraction

Figure 2.9: Geometry of X-ray diffraction (Wikipedia). applications, only short wavelength x-rays (hard x-rays) in the range of a few angstroms to 0.1 angstrom

(1 keV - 120 keV) are used [29]. This is a very widely used characterization technique as it provides a lot of information about the structure of the film. One can determine if there is any texturing, the degree of crystallinity, lattice parameters, and the size of crystallites, if the expected phases are present etc. It is a non-destructive technique and the sample preparation is minimal. However, the large penetration depth of

X-rays means that their path length through the film is too short to produce diffracted beams of sufficient intensity and the substrate tends to dominate the signal. Therefore, long counting times are required and the sample should be as large as possible.

X-rays are produced generally by either x-ray tubes or synchrotron radiation. In an x-ray tube, which is the primary x-ray source used in laboratory x-ray instruments, x-rays are generated when a focused electron beam accelerated across a high voltage field bombards a stationary or rotating solid target. As electrons collide with atoms in the target and slow down, a continuous spectrum of x-rays are emitted, which are termed Bremsstrahlung radiation. The high energy electrons also eject inner shell electrons in atoms through the ionization process. When a free electron fills the shell, an x-ray photon with energy characteristic of the target material is emitted. Common targets used in x-ray tubes include Cu and Mo,

48 which emit 8 keV and 14 keV x-rays with corresponding wavelengths of 1.54 Å and 0.8 Å, respectively

[30].

Figure 2.9 shows schematics of diffraction from a crystal. A crystal lattice consists of a regular arrangement of atoms, with layers of high atomic density existing throughout the crystal. When a monochromatic beam of radiation falls onto high atomic density layers, scattering will occur. For constructive interference, it is necessary that scattered waves originating from the individual atoms will be in phase with one another. Hence, the path difference between two interfering waves must be an integral multiple of wavelengths, and so we find that

2푑푠푖푛휃 = 푛휆 2.12

This is known as the Bragg equation and when this condition is satisfied a diffraction maximum can be observed. At Vitreous State Laboratory, we use Thermo ARL X-ray diffractometer with Cu-Kα radiation, where λ=1.541 Å.

2.5.4. Vibrating sample Magnetometer (VSM)

Vibrating Sample Magnetometer (VSM) systems are used to measure the magnetic properties of materials as a function of magnetic field, temperature, and time. They are ideally suited for research and development, production testing, quality and process control. Powders, solids, liquids, single crystals, and thin films are all readily accommodated in a VSM. In our lab, we have quantum design physical property measurement system which is equipped with VSM. The basic measurement is performed by oscillating sample near a detection (pick up) coil and synchronously detecting voltage induced. By using compact gradiometer pickup coil configuration, relatively large oscillation amplitude (1-3 mm peak) and frequency of 40 Hz, the system is able to resolve magnetization changes of less than 10-6 emu at a data rate of 1 Hz.

The magnetic field range of our VSM is -9 T to 9T and temperature range from 2 -1000 K [31].

Figure 2.10: Schematic diagram of VSM [32]. The basic principle of operation for a VSM is that changing magnetic flux induces a voltage in the sensing coil that is proportional to magnetic moment of the sample (as shown in Figure 2.10). The time dependent induced voltage is given by

푑Φ 2.13 푉 = 푐표푖푙 푑푡

푑휙 푑푧 2.14 푉 = ( ) ( ) 푐표푖푙 푑푡 푑푡 in equation 휙 is the magnetic flux enclosed by the pickup coil, z is vertical position of the sample with respect to the coil, t is time. For a sinusoidally oscillating sample position, the voltage induced is based on following equation.

푉퐶표푖푙 = 2휋푓퐶푚퐴푠푖푛(2휋푓푡) 2.15 where C is coupling constant, m is the DC magnetic moment of the sample, A is the amplitude of the oscillation, and f is the frequency of oscillation. While a VSM measures magnetic moment m, the quantity of interest is the material’s magnetization M. The magnetization M (in cgs units) can be expressed in terms of the mass (emu/g) or volume magnetization (emu/cc), and is the moment m divided by the sample mass or volume, respectively. A VSM is most commonly used to measure a material’s hysteresis or M-H loop.

In some cases, it is preferred to present the magnetization data in terms of the magnetic induction B, whose

CGS unit is the Gauss (G). The relation between M and B is: B(G) = H + 4πM where M is the volume magnetization (emu/cc). In our laboratory, Quantum Design Physical Property measurement system

(PPMS) (Figure 2.11) is used to measure the electrical and magnetic properties of the materials. PPMS is facilitated with several options. It can provide a wide range of temperature ranging from 1.7 to 1000 K and magnetic field can be raised up to 9 tesla. The PPMS is equipped with superconducting solenoid immersed in liquid helium. The system is operated with a 40 Hz frequency and capable to detect the magnetic signal, less than 10-6 [33].

Figure 2.11: PPMS equipped with VSM [33]

References

1) K. S. sree Harsha, Principles of physical vapor deposition of thin films, Science Direct, 11-143

(2006).

2) H. Herz, “on the evaporation of liquids, especially Mercury, in vacuo, Annal. Der Physik 17, 177

(1882).

3) M. Knudsen, Annal. Der Physik 47, 697 (1915).

4) R. B. Darling, EE-527, Microfabrication, 2015.

5) I. Langmuir, J. Am. Chem. Soc. 54, 2798 (1932).

6) P. Atkins, and J de Paula, Physical Chemistry, W.H. freeman (eds.), p. 756(2006).

7) E. Chen, Applied physics 298r, 2004.

8) F. F. Zahari, EEN3016, 2012.

9) T. Michel and J. krug, Islands, Mound and atoms: Patterns and Processes in crystal growth far

from equilibrium, Springer-verlag, Berlin, 2003.

10) H. Brune, Sur. Sci. Rep. 31, 125 (1998).

11) Z. Zhang and M. G. Lagally, Science 276, 377(1997).

12) W. Ostwald, Zeitschrift für Phys. Chemie 37,385(1901).

13) P. W. Voorhees, J. Stat. Phys. 38, 231(1985).

14) C. Ratsch and J. A. Venables, J. Vac. Sci. Technol. A 21, S96 (2003).

15) N. Bhardwaj, and S. C. Kunda, Bitechnol. Adv. 28, 325 (2010).

16) Z. M. Huang,Y. Z. Zhang, M. kotaki, and S. Ramakrishna, Compos. Sci. Technol. 63, 2223

(2003).

17) G. I. Taylor, Proc. R. Soc. London A 258, 383 (1964).

18) L. Rayleigh, Philo. Mag. 14, 184 (1882).

19) A. L. yarin, S. Koombhongse, and D. H. Reneker, J. Appl. Phys. 90, 4836(2001).

20) J. M. Deitzel, J. Kleinmeyer, D. Harris, and N. C. Beck Tan, Polymer 42, 261(2001).

21) S. Sukigara, M. Gandhi, J. Ayutsede, M. Micklus, and F. Ko, Polymer 44, 5721(2003).

22) H. Fong, and D. H. Reneker, J. Poly. Sci. B polym. Phys. 37, 3488 (1999)

23) M. M. Demir, I. Yilgor, E. Yilgor, and B. Erman, Polymer 43, 3303 (2002).

24) E. S. Medeiros, L. H. C. MAttoso, E. N. Uto, K. S. Gregorski, G. H. Robertson, R. D. Offeman,

D. F. Wood, W. J. Orts, and S. H. Imam, J. Biobased Mater. Bioenergy 2, 1 (2008).

25) M. T.Postek, K. S. Howard, A. H. Johnson, and K. L. McMichael, Ladd Research Inc., Williston,

VT, 1980.

26) B. Cheney, Introduction to Scanning electron microscopy.

27) J. I. Goldstein, D. E. Newbury, P. Echlin, D. C. Joy, A. D. Romig, C. E. Lyman, C. Fiori, and E.

Lifshin, Scanning Electron Microscopy and X-ray Microanalysis, 2nd ed. ,Plenum Press, 1992.

28) A. R. Clarke, Microscopy techniques for material science, CRC press, 2002.

29) R. Jenkins, in Encyclopedia of Analytical chemistry, R. A. Meyers (eds.), Wiley and sons,

Chickester, 2000.

30) G. V. Hevesey, Chemical Analysis by X-rays and its Application, McGraw-Hill, New York, 1932.

31) Quantum design operation manual, part no 1096-100, San Diego, CA, 2011.

32) J. C. Toniolo, A. Takimi, m. J. de Andrade, R. Bonadiman, and C. p. Bergmann, J. Mater. Sci.

42, 4785 (2007).

33) Quantum Design, “Vibrating Sample Magnetometer (VSM) Option User’s manual,” in Vibrating

Sample Magnetometer (VSM) Option, no. 1096, 2009.

Chapter 3

3. Device Fabrication and Measurement

Thin films devices can be made from complicated process which involves several steps. Device fabrication of nanostructures involves several steps starting from the synthesis of nanostructures on the suitable substrate followed by coating of the substrate with a suitable resist polymer, then carry out e- beam lithography or photo lithography, development of the exposed patterns and ultrahigh vacuum deposition of electrodes. For a simple device, the process is carried out only one time called single phase lithography while for complex lithography; the process might to carried out multiple times and is called multiphase lithography. Here we have fabricated our thin films device using Hall bar metal contact masks.

3.1 Resistivity measurements method

The resistivity of a metallic or semimetallic material is generally measured using a four-point probe technique. With a four-probe, or Kelvin technique, two of the probes are used to source the current and the other two probes are used to measure the voltage as shown in Figure 3.1[1]. Using four probes eliminates measurement errors due to the probe resistance, the spreading resistance under each probe, and

Figure 3.1: Four Probe measurements [1]

Figure 3.2: Typical Van der Pauw geometry for measuring resistivity [7] the contact resistance between each metal probe and the material. Because a high impedance voltmeter draws little current, the voltage drops across the probe resistance, spreading resistance, and contact resistance are very small. Two common Kelvin techniques for determining the resistivity of a semiconductor material are the four-point collinear probe method as shown in Figure 3.1 and the van der

Pauw method as shown in Figure 3.2 [2]. If the spacing between the probe points is constant, and the conducting film thickness is less than 40%, and the edges of the film are more than 4 times the spacing

푉 distance from the measurement point, the average resistance of the substance is given by 휌 = 푘 where 푆 퐼 k is constant for average resistivity whose value is 4.53. 휌푆 is the resistivity of the substance. For bulk resistivity, the sample is much larger than the spacing of the probes, the sheet resistance is defined as the resistance equal to the bulk resistivity divided by the sample thickness, the resistivity can be written as

푉 휌 = 2휋푠. , where s is distance between voltage probe [3,4].The van der Pauw method involves applying 푆 퐼 a current and measuring voltage using four small contacts on the circumference of a flat, arbitrarily shaped sample of uniform thickness [5,6]. This method is particularly useful for measuring very small samples because geometric spacing of the contacts is unimportant. Effects due to a sample’s size, which is the approximate probe spacing, are irrelevant. Frequently the sample does not have a geometry that is favorable for the above style of measurement, leading to an unknown current distribution. Also, it is often difficult to determine accurately the geometry of the sample, limiting the accuracy of the calculated

Figure 3.3: Hall effect in a thin film [8]. resistivity. In such a case, one often uses the technique of van der Pauw to determine the resistivity of the sample. A common geometry for such a measurement has 4 electrical contacts at the four corners of a roughly square sample as shown in Figure 3.2 [7]. However, the van der Pauw technique is applicable for an arbitrary shaped sample as long as the thickness of the sample is known and is uniform, the contact areas are small, and the contacts are all on the perimeter of the sample. In this case, van der Pauw showed

휋푅 푡 휋푅 푡 that exp (− 퐴퐵,퐶퐷 ) + exp (− 퐵퐶,퐷퐴 ) = 1 where t is the thickness of the sample, ρ is the resistivity, 휌 휌

RAB,CD is the resistance determined by dividing the potential difference VD−VC by the current going from

A to B, and RBC,DA is defined similarly. After measurement of the thickness of the sample and the two resistance values, the resistivity is determined simply by inverting van der Pauw’s relation.

3.2 Hall Bar

The films were grown in Hall-bar geometry due to the convenience of being able to measure longitudinal and transverse voltages simultaneously. As shown in Figure 3.3, current is sourced through the bar-shaped film in the longitudinal direction. When applying a magnetic field in the direction shown, a Lorentz force causes a transient transverse current. As electrons are mutually repulsive, they cannot all cluster at the edge. This results in a concentration gradient of the electron carrier density in the film as electron flow again proceeds in the longitudinal direction. The potential difference that arises from the gradient is known as the Hall-effect voltage.

퐼퐵 푉 = 3.1 퐻 푛푒푡

Thus, the Hall-effect voltage is directly related to the carrier density of the material. Eq. (3.1) expresses this relationship with I the current, B the magnetic field, n the carrier density of the electrons, e the charge of an electron and t the thickness of the film: In addition, the sign of this voltage gives an indication of whether electrons or holes carry the flow of charge. This would indicate the sign of the charges (electron or holes) participating in the current flow in a material under an applied magnetic field. The electrical resistivity can be measured by applying a current between contacts 5 and 6 of the sample shown in Figure

3.4 and measuring the voltage between contacts 1 and 3 or 2 and 4, then using the formula:

푉 푤푡 휌 = 3.2 푆 퐼 퐿 where w is the width of the channel and t is the thickness of the thin film, L is a length of channel, and B is the magnetic flux density at which the measurement is taken [9]. The Hall bar is a good geometry for making resistance measurements since about half of the voltage applied across the sample appears between the voltage measurement contacts. For this reason, Hall bars of similar geometries are commonly used when measuring magnetoresistance or Hall mobility on samples with low resistances.

3.3 Device Measurements

Thin film devices were measured by using PPMS (description in chapter2) resistivity option as well as horizontal rotator. The resistivity option is used to measure DC resistances by applying a current and measuring the resulting voltage. Resistance measurements can also be performed by the AC Transport options which we will discuss later. Which option is more appropriate depends on several factors, including whether an AC signal desirable/acceptable, the resistance of the devices tested and number of devices to be tested. The Resistivity option performs a 4-point resistance measurement on up to 3 channels

1 2 VXY

5 Channel 6

3 4

VXX

Figure 3.4: Schematics of Hall Bar for thin film at a time. The standard resistivity puck is shown in Figure 3.5 which includes pre-mapped pads for + and

– current and voltage measurement pads for Channels 1 – 3. The recessed area measures 11.6 mm x 13.6 mm (with rounded corners), so samples that fit within this area will work best. The base is conducting, so an insulating layer such as Kapton tape will be necessary for samples with conductive back sides.

Electrical contact between the sample and the desired channel input pads can be made via wire bonding, soldered or silver-painted wires or using indium contacts. Alternatively, samples can be mounted on the user puck, which has a blank face and no pre-configured channel set-up. This can be useful for larger samples or if no resistivity pucks are available. The pin-to-channel correspondence for the user puck is posted on the bulletin board. The resistivity or user puck can be used in combination with the user bridge

Figure 3.5: Standard DC resistivity puck from PPMS [10] (the red face plate corresponds to the resistivity option channels) to test sample-to-puck connections in advance of insertion into the PPMS using a hand-held ohm meter or other devices. We have used PPMS

AC transport option for Hall bar measurements. The Horizontal Rotator is not technically an option; it is an accessory which can be used in conjunction with the Resistivity and AC Transport options. Any measurement that can be made with those options can be made with the rotator, with two exceptions: the current limit is reduced from 2 A to 0.5 A in ACT, and only two channels are available in Resistivity. The

Horizontal Rotator consists of a motor as shown in Figure 3.6(a) and the rotator body as shown in Figure

3.6(b). The motor is always connected to the system and active, and simply needs to be mounted on the system and coupled to the rotator body (which is installed in the probe space) in order to be used. It can also be rotated manually outside the system using the sliver motor coupling. The positions are defined such that the sample faces up at 0° and forward at 90°. Slightly more than one full rotation is possible so that the end positions are defined as -10° (going past 0 to 350 on the dial) and 370° (going past 0 to 10).

An on-board thermometer is used for all measurements rather than the main PPMS system thermometers.

The Horizontal Rotator uses special sample holders rather than the pucks which are used with the

Resistivity or AC Transport options. Figure 3.6(c) shows a universal sample puck and Figure 3.6(d) shows our thin film on AC bridge board. Samples can be connected to them in the same manner as the pucks, and connections can be tested in the user bridge via the appropriate adapter. The sample holders are shown in Figure 3.7. The descriptions of the sample holders are as below [11,12].

a) b)

c) d)

Figure 3.6: (a) Horizontal rotator motor, (b) rotor Sample holder probe, (c) User Bridge with Universal puck (d) AC User bridge with thin film device Resistivity – Channel 1 is designated for the thermometer –Channels 2 and 3 are for 4-point resistance measurements.

Resistivity (Wimbush) – Third-party version of the Resistivity holder with several improvements: larger sample area, clear “Resistivity” label, larger contact pads, and no exposed thermometer pads.

Angled Resistivity – Identical connections to the Resistivity holder, but with a second perpendicular board coming out of the main board so that the sample can be rotated within its own plane.

AC Transport – For 5-point Hall measurements, Channel 1 must be used; for other measurements, either

Channel 1 or 2 may be used. Only the voltage pads are associated with the specific channels, and there

Figure 3.7: different puck for resistivity measurement with horizontal rotator are only two current leads which are shared by the channels and used for the active measurement at any given time.

Once the sample is mounted to the holder, the holder must be attached to the rotator.

References

1) M. Lundstrom, “Resistivity and Hall effect measurements, ECE-656, 2011, Purdue University

2) Keithley Application Note Series, four probe resistivity and Hall effect measurements using

Keithley 4200-SCS, Number 2475, 2011.

3) F. M. Smits, Measurements of Sheet resistivity with Four probe, BSTJ, 37, 371(1958).

4) L. B. Valdes, Resistivity measurements on Germanium for transistors, In Proceeding of the IRE,

420 (1954)

5) L. J. V. der Pauw, Philips Technical Review 26, 226 (1958).

6) E. Secula, Resistivity and Hall Measurements, NIST

7) Quantum Design Application Note 1076-304, Performing Van der Pauw Resistivity

Measurements

8) C. Nave, Hyperphysics, (http://hyperphysics.phy-astr.gsu.edu/hphys.html).

9) C. Yang, A Study of Electrical Properties of Bismuth Thin Films, University of Florida 2008.

10) Quantum Design PPMS Resistivity Option User’s Manual 1076-100A.

11) http://www.ccmr.cornell.edu/wp-content/uploads/sites/2/2015/11/PPMS-SOP-Rotator-V3.pdf,

Cornell University

12) Quantum Design PPMS horizontal rotator option User’s Manual 1384-00B

Chapter 4

4 . Magnetic and Transport Properties of Iron-based Heusler alloy nanoscale thin films In this chapter, growth and the transport properties of iron-based (Fe2CrAl) nanostructures are discussed.

The discussion will start from synthesis and characterization of the nanostructures. We have used electron beam evaporation method in an ultra-high vacuum chamber to grow these thin films followed by morphological, structural and magnetic characterizations. We have measured electrical and magnetotransport properties using four probe method.

4.1 Introduction Fe-based Heusler alloys have been extensively studied recently because of their disorder dependent properties. They can exhibit half-metallicity and high temperature magnetism which are useful in spintronics, magnetocalorics and rare-earth free ferromagnets [1-13]. Band structure calculations of iron- based Heusler alloys with L21 structure have shown that they can exhibit half-metallic ferromagnetic behavior [14-16]. Experimental work on bulk samples has shown that the magnetic moment of Fe2CrAl

(FCA) varies from 1.6 – 2.2 µB/formula unit (f.u.), and the Curie temperature varies from 210 – 316 K depending on whether FCA crystallizes in L21, B2, or A2 phases [17-18]. In practice, there is always a certain degree of disorder in Heusler alloys depending on the constituent elements and the temperature of annealing [8]. We have successfully grown thin films of Fe2CrAl with various thicknesses at the nanoscale and systematically studied their structural and magnetic properties. We have observed that lattice disorder in all of our nanoscale FCA thin films, and both a larger magnetic moment and higher Curie temperature

3 µm

Figure 4.1: SEM image of 80 nm FCA thin film than reported bulk samples. The magnetic moment of the thin films varies from 2.5 – 2.8 µB/f.u. and the

Curie temperature is well above 400 K. Our studies clearly indicate that the disorder in the atomic arrangement in the nanoscale films leads to enhanced magnetic exchange interaction, thereby exhibiting large magnetic moment and higher Curie temperature. In a completely disordered FCA lattice, Fe atoms will have Cr or Al as neighbors which can lead to Fe-Cr-Fe or Fe-Al-Fe exchange interaction. Part of this work has been published in peer- review journal [19].

4.2 Experimental Details FCA thin films were grown using ultra-high vacuum electron beam evaporation [20]. The base pressure inside the chamber was below 9 x 10-10 Torr and less than 5 x 10-9 Torr during the film deposition. Polished

Si/SiO2 substrates (500 nm oxide layer) were used for the film deposition. The stoichiometric ratio of Fe,

Cr, and Al was deposited, with film composition and uniformity controlled by a quartz crystal rate monitor and with low deposition rate 0.1- 0.3 Å/s. Films were grown with thicknesses 30, 50, 80 and 100 nm. In order to create lattice disorder, we deposited Fe, Cr, and Al as separate layers, one over the other. In order to attain the required total thickness, each element was deposited in several layers. After growth, the films were annealed in situ at 725 K for 4 h. The morphology of the thin films was studied by scanning electron

65 microscopy (SEM). The crystal structure was determined by x-ray diffraction analysis (XRD) using a

Thermo/ARL X’TRA, (Cu-Kα) diffractometer. The magnetic measurements were carried out using a

Quantum Design vibrating sample magnetometer (VSM). Transport properties were measured using a physical properties measurement system (PPMS).

4.3 Results and Discussion 4.3.1 Morphological and Structural Characterization

Figure 4.1 shows the SEM image of 80 nm FCA thin film. Film shows grains at some part of surface which is due to layer plus island growth. Figure 4.2 (a) shows the XRD patterns of FCA thin films with different thicknesses. The intensity the (220) peak increases with an increase in thickness. The diffraction patterns of the films have (hkl) values (220), (400) and (422). The fully ordered L21 structure is identified by the occurrence of (111) and (200) superlattice reflections 10 µm in the XRD spectrum, which are absent in all XRD spectra shown in Fig. 4.2(a). These (111) and (200) peaks are much weaker for elements from the same period. Intensities of these peaks10 µm could be nearly undetectable if all of the elements of the Heusler compound are from the same period of the periodic table.

Figure 4.2: (a) XRD patterns of FCA thin films with different thicknesses. The broad peak at 69˚ is from silicon (100) substrate. (b) A typical EDX spectrum of FCA thin film (80 nm).

Substrate

Figure 4.3: Magnetic Hysteresis curves of FCA thin films with different thicknesses (a) field

applied parallel to the substrate plane and (b) Field applied perpendicular to the substrate plane. Also, these two peaks will not appear in samples with A2 type crystal structure. Since Al belongs to a different period than either Fe or Cr, our thin films exhibit an A2 phase [21]. Thus, Fe, Cr and Al occupy all four sublattices in the crystal structure. Umetsu et al. have reported the A2 phase of bulk FCA [17]. A more sensitive technique to identity the A2 phase is using Mӧssbauer spectroscopy. Since our films have thicknesses of less than 100 nm, it is difficult to observe the required spectrum for the analysis [22]. We have carried out extensive Energy Dispersive X-ray (EDX) measurements. Figure 4.2(b) displays a typical

EDX spectrum of FCA thin film with 80 nm thickness. From the EDX measurements, we observe an average composition Fe (2.01) Cr (1.0) Al (1.0).

4.3.2 Magnetic Properties

We have measured the magnetic hysteresis curves of the thin films at room temperature with an applied field parallel, Figure 4.3 (a), and perpendicular, 4.3 (b), to the plane of the thin film. The saturation magnetization and the remanence increases with film thickness when the applied field was parallel to the substrate plane. When the field was perpendicular to the plane, a significant increase in coercivity was observed for films with thicknesses 50-100 nm. A large field is required to saturate the magnetization of

Figure 4.4: Variation of magnetic moment with the film thicknesses when the magnetic field is applied parallel to the film plane. the films (50-100 nm) at perpendicular configuration. This implies FCA films have an easy axis parallel to the substrate plane where the hysteresis loop is sharp and square, and a hard axis lying perpendicular to film plane. Figure 4.4 displays the value of the magnetic moment, which increases with the film thickness when the field is applied parallel to the film plane. The magnetic moment varies from 2.5 – 2.8 µB/f. u. as the thickness of the films varies from 30-100 nm. The observed magnetic moment is larger than that reported for the FCA bulk systems [8, 17, 18, 23]. Figure 4.5 shows the variation of coercivity with increasing film thickness. With both parallel and perpendicular applied field, the coercivity initially increases with film thickness, reaches a maximum, and then decreases with further thickness. The coercivity in the parallel configuration is smaller than the perpendicular configuration at all thicknesses.

The variation of the coercivity with the thickness can be explained by the micro magnetism model [24].

The coercivity of a system changes due to the reduction of the nucleation field. by magnetostatic interaction [25]. The coercivity can be expressed as HC = αeffHn(T) – NeffMS(T) where Hn is the nucleation field, αeff is the product of αø and αk, MS is the saturation magnetization, and Neff is the effective demagnetizing factor [26]. The αk depends on the nature and size of the defect region in which nucleation

Figure 4.5: Variation of coercivity with film thicknesses for an applied field parallel and

perpendicular to the film plane of domains or pinning of a domain wall takes place, and αø depends on the alignment of the grains with the field. When the angle between the applied magnetic field and easy magnetization direction increases from zero, the value of αø is increased by the coercivity mechanism of the domain wall pinning, but is decreased by the mechanism of nucleation [27-30]. Min et al. has found similar thickness behavior for

La0.67Ca0.33MnO3 thin films [31]. Besides nucleation and domain wall pinning, strain imposed by substrate, lattice defects and surface roughness might also affect the coercivity [24]. Furthermore, the magnetic hysteresis cycle is partly ruled by the competition between anisotropy energy and exchange interaction energy may reduce the magnetic domains so that anisotropy present in the system might affect the coercivity of the films [32]. The temperature dependence of magnetization for 100 nm film with zero external field is displayed in Figure 4.6 (a). It shows that the Curie temperature is higher than 400 K, which is significantly higher than the bulk FCA [17-18]. The magnetization curves of films with different thickness measured with a magnetic field of 500 Oe are shown in Figure 4.6 (b). All films exhibit similar

M-T behavior and the magnetic moment increases with an increase in the thickness of the film. For Fe- based Heusler alloys, the magnetic properties are sensitive to the atomic position of Fe atoms. In the case of the completely ordered L21 phase, two sublattices are occupied by Fe-atoms, whereas Y and Z

69 sublattices are filled by Cr and Al atoms respectively. For the A2 phase, the disordering is 100% in all four sublattices so that the nearest neighbor of the Fe atom can be either Cr or Al. Strong exchange coupling between Fe-Fe interlayer enhances magnetization and Curie temperature [33]. Isida et al. have calculated the magnetic moment of Fe in the Fe2CrSi alloy with some disorder, including the Fe-Cr and

Fe-Si types [16]. They found that the magnetic moment of Fe in the antisite disorder is larger than in the ordinary site in both cases. This implies disorder induces long range ferromagnetic interactions between

Fe-Fe layers via Fe-Al and Fe-Cr neighboring layers increasing the Curie temperature and magnetic moment of FCA thin films.

4.3.3 Transport Properties

We have measured electrical and magnetic transport properties of FCA thin films by four probe method.

Figure 4.7 (a) shows resistivity versus temperature relation for FCA thin films. The residual resistivity

(RR) ratio increases from 1.06 to 1.68 as the film thickness increases. Such values of RR are not quite uncommon for Heusler alloy thin films [34]. This indicate the polycrystalline nature of thin films. Low value of RR refers that scattering mechanism exist even at low temperature. Figure 4.7 (b) displays

Figure 4.6: (a) M vs. T plot of 100 nm FCA film with zero field. (b) M vs. T of FCA films at different

thickness for the field applied parallel to the film plane (field 500 Oe).

Figure 4.7: ρ vs T of FCA thin films at different thicknesses. (b) ρ vs T for 80 nm thin film.

Upper right inset shows resistivity behavior for T < 80 K whereas lower left inset shows

resistivity behavior from temperature 80 K to 300 K. resistivity as a function of temperature relation for typical 80 nm thin film. Resistivity can be divided into two regions depending upon the scattering mechanism. Above 80 K, phonon scattering becomes dominant which gives power law behavior [35]. We found the relation with temperature as 휌 = 퐴 + 퐵푇1.39. Here

A = 80 µΩ-cm and B = 8.551×10-3 µΩ-cm/K1.39. For the temperature region between 20 to 80 K, the data can be fitted to the expression, 휌 = 퐴 + 퐵푇2 + 퐶푇4.5, where A = 80 µΩ-cm, B = 2.129 ×10-4 µΩ-cm/K2 and C = 2.630× 10-9 µΩ-cm/K4.5. The T2 term is due to the electron-electron scattering process and T4.5 term is due to the two-magnon scattering process which is considered to be half-metallic ferromagnetism.

This indicates even A2 phase of Heusler alloy could exhibit half-metallicity. Temperature variation of electrical resistivity without and with an external magnetic field of 3 Tesla from 10 K to room temperature is shown in Figure 4.8. A maximum negative magnetoresistance (MR) of 18% is observed around 10 K.

Negative contribution to MR reported for other half metallic ferromagnets [36], as well as in conventional ferromagnets such as Co. It is attributed to the suppression of spin disorder scattering [37]. Furthermore,

Figure 4.8: ρ Vs T for FCA thin film (100 nm) with field (3 T) and without field. in multi-domain samples, negative MR can also occur due to the progressive alignment of the different domains along the applied field direction.

4.4 Conclusions We have grown nanoscale A2 type FCA thin films. The magnetic moment varies from 2.5-2.8 per f.u. and the Curie temperature is higher than 400 K. With increasing thickness, the coercivity of the thin films increases, attains a maximum and then decreases. Metallic and half-metallic resistivity have been observed depending upon the temperature range. Thin film has maximum 18% negative MR. The Fe-based Heusler alloys are especially interesting because of the sensitivity of the magnetic as well as transport properties to lattice disorder.

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Chapter 5

5. Magnetic properties of Iron-based Nanowires Synthesis and the magnetic properties of iron-based (Fe2CrAl) nanostructures are discussed in this chapter.

The discussion will start from synthesis and characterization of the nanostructures. We have used electrospinning method to grow polycrystalline nanowires. We then characterize morphology and structure of nanowires followed by magnetic characterization.

5.1 Introduction It is known that magnetic properties of the Fe-based alloys depend on the method of synthesis and its subsequent annealing history and composition. In the previous chapter, we have discussed the magnetic and transport properties of Fe-based thin film. Thin films show larger magnetization and high Curie temperature. Low dimensional nanostructures can show distinctive physical properties compared to their bulk counterpart, even at micro-level. Heusler alloy nanowires are specially interesting due to their simple crystal structure and great chemical flexibility, different physical properties can be realized in Heusler alloys. Recently, Fe-based Heuslar alloy have been predicted to exhibit half metallic ferromagnetism in ordered cubic structure. [1-10]. These calculations showed that inverse cubic structure should be more stable than the ideal structure. However physical properties of Fe-based Heusler alloys are affected by degree of chemical as well as structural disorder present in the system [12-17]. It has been suggested that a small amount of disorder in the lattice site cause rapid change in their electric and magnetic properties

[18]. Some studies on the bulk form of this alloy have done by using arc melting method. Studies on

Fe2CrX nanostructures have not yet carried out [19-20]. The bulk as well as thin film studies report a maximum Curie temperature of 316 K for 100% disordering in lattice sites. Therefore, to understand the

properties of Fe2CrAl (FCA) at the nanoscale, we have synthesized FCA nanowires grown on oxidized silicon (100) substrates. Our group is the first to grow FCA nanowires by electrospinning.

5.2 Experiment For this work, we have used commercially available compounds such as:

• Ferric Nitrate: Fe(NO3)3 .9H2O

• Chromium Nitrate: Cr(NO3)3 .9H2O

• Aluminum Nitrate: Al(NO3)3. 9H2O stoichiometric ratio of Fe, Cr and Al nitrates are mixed in deionized water with a magnetic stirrer at room temperature for 4 hours. As a next step, Polyvinlylalcohol (PVA) and finally beads of polyvinlypyrolidinine (PVP) were added slowly into the nitrate mixture. Resultant viscous solution was vigorously stirred for overnight at room temperature. The viscosity of resulting solution is tailored by right ratio of polymer to nitrates. A high voltage of 20 kV was applied to vertically positioned needle with an inner diameter of 0.4 mm was connected to the source (details of the setup are discussed in chapter 2.4).

In most cases, the needle was simply dipped into the resultant viscous solution. Continuous fibers are collected on Si (100) and Si/SiO2 substrates which was placed at a certain distance from the tip of the needle on a brass cathode. After the electrospinning process, the collected nanofibers were annealed in a

100 n m

Figure 5.1: SEM images of as grown nanowires.

tubular furnace at 873 K for 4 h in ultrahigh purity Ar with 5% H2 gas. The morphology of nanowires is studied by scanning electron microscopy (SEM). X-ray diffraction (XRD) is performed for structural analysis. Magnetic properties are studied using vibrating sample Magnetometry (VSM).

5.3 Morphological characterization Figure 5.1 shows SEM images of FCA nanostructures. SEM image reveals uniform, smooth, continuous nanowires with diameter range from 50 to 300 nm and hundreds of microns in length. Figure 5.2 displays the EDS spectrum of nanowires. The EDS analyses confirm the uniformity of FCA nanowires. Chemical ratio of Fe, Cr and Al is found to be Fe:Cr:Al is 2.01:1:1. We observed that good morphology and surface smoothness of nanowires strongly depend upon the polymers, applied voltage as well as annealing temperature.

5.4 XRD characterization Figure 5.3 Shows the XRD patterns of FCA nanostructure. All diffraction pattern can be indexed with cubic structure with Miller indices (220), (400) and (422). In Heuslar alloys, Bragg reflection with non- zero structural amplitude can occur when Miller indices are either odd or even [8,21,22]. Depending upon the superlattice peak in the XRD spectrum, Full Heusler alloy can be divided into odd, even superlattices and fundamental diffractions [23,24]. For odd superlattice diffraction, peaks are obtained at odd hkl

Figure 5.2: EdX spectrum of FCA nanowires

Figure 5.3: XRD pattern of FCA nanowires values. Only the L21 crystal structure could display odd superlattice diffraction. For even superlattice diffraction, peaks are obtained at those hkl values satisfying h+k+l = 4n+2 where n is a positive integer.

Both B2 and L21 structures display this diffraction. For those hkl values which satisfy the relation h+k+l

= 4n, are known as fundamental diffraction. This type of diffraction can be shown by A2, B2 and L21 structure. The degree of ordering can be measured by comparing the intensities of peaks with fundamental peak. The (200) and (111) peaks are absent in these nanostructures. So, these nanowires display cubic A2 type structure [11]. Lattice constant of FCA nanostructure obtained from XRD analyses is a=0.587 Å.

5.5 Magnetic Characterization Figure 5.4 (a) shows the magnetic (M-H) measurements of as grown nanostructure at different temperatures parallel to the plane of substrate. The hysteresis loops of this alloy shows that they are ferromagnetic. It has been observed that coercivity as well as saturation magnetization of nanowire increases with decrease temperature and saturation magnetization is found to be 8.7×10−4 emu at 10 K.

In addition, the temperature dependence magnetization is measured from 10 to 400 K with zero-field cooling (ZFC) and field-cooling (FC) as shown in Figure 5.4 (b). For ZFC measurement, sample has been cooled down from 300 to 10 K and then magnetization was recorded as function of temperature from 10

Figure 5.4: (a) M-H curve of FCA nanowires at 10 K and 300 K. (b) Magnetization vs Temperature curve of FCA wires to 400 K. For FC, the magnetic field of 1000 Oe is applied parallel to the plane of wires. We observed, a difference in ZFC and FC curves below 50 K. This indicates the thermodynamic irreversibility of nanowires below that temperature. The irreversible magnetic behavior reflects the role of anisotropy in determining the shape of FC and ZFC curves [25]. Magnetic anisotropy aligns the spins in preferred direction. During the process of ZFC measurement, no magnetic field is applied while cooling the sample from 300 K. As a result, the spins are locked in random direction for a polycrystalline sample. When a small magnetic field is applied at the lowest temperature far below the Curie temperature, magnetic moment will depend on the anisotropy of the sample. If the sample is highly anisotropic, the small magnetic field will not sufficient to rotate the spins in the direction of the applied magnetic field and therefore magnetization will be very small. If the system is less anisotropic, the magnetization will increase. During the FC process, the sample is cooled in presence of magnetic field, therefore, spins will be locked in particular direction depending upon the strength of the applied field. The FC magnetization remain almost constant if the anisotropy of the sample is very low or decreases with increasing temperature for highly anisotropic system. From our hysteresis loop, the remanence ratio Mr/Ms was 0.46 at 10 K, but the remanence ratio was 0.35 at room temperature. It is expected that anisotropy field is plays a crucial

80 role in determining the magnetization at a given field strength. Curie temperature of our alloy is observed to be higher than 400 K. This Tc is significantly higher than that was reported from bulk samples [12]. It can be inferred that the magnetic properties of FCA alloy depends upon crystal structure and the grain size at the nanoscale. In case of completely ordered alloy with L21 phase, the neighboring atoms of Fe is either

Cr or Al. In case of completely disordered phase in FCA nanowires, three types of disordering exist that is Fe-Cr type, Fe-Al type and Cr-Al type. Luo et. al have calculated and verified the effect of all these disorders on the band structure of bulk FCA ribbons [ref]. They found the total spin moment increases with increasing Fe-Cr or Fe-Al disorder whereas Cr-Al type weakly effect on magnetic properties. Ishida et.al [17] have calculated the effect of chemical disorder on magnetism in Fe2CrX alloy with disorders Fe-

Cr, Cr-X and Fe-X respectively. They showed the enhancement of magnetic moment in Fe2CrX due to these disorders. Zhang et. al reported the drastic increase of Curie temperature for bulk FCA due to the disordering. Our reports on FCA thin films also showed the enhanced magnetic moment and Curie temperature due to the disordering types. So, disordering in atomic sites boosts the exchange interaction between Fe-Fe pair via Cr or Al which increase Curie temperature significantly. Such exchange interaction effects more in nanowires than bulk because of size of nanostructures.

5.6 Conclusion We have synthesized FCA nanowires for first time and nanowires show disordered structure. Magnetic measurements show ferromagnetic with Curie temperature above 400 K. Thermal fluctuations and anisotropy effect the coercivity dependence with temperature besides size of the nanowires. Disordering and nanoscale size play key role to enhance magnetic behavior.

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Chapter 6

6 . Transport properties of room temperature Weyl Semimetal cobalt- based Heusler alloys Quantum states of matter can have non-trivial topologies which relate to wave function of the electrons.

Such topologies contribute to novel sates of matter such as topological insulators, topological semimetals including Dirac semimetals and Weyl semimetals [1-4]. These topologies play a key role near Fermi level which separates conduction bands and valence bands. Weyl Fermions have been recently interesting due to their ability to violate symmetry principles. Such violation could lead to anomaly like chiral anomaly [5,6]. Weyl semimetals provide the realization of Weyl fermions in solid- state physics despite they were theoretically predicted to exist in high-energy physics [7-9]. Cobalt- based Heusler alloys (Co2TiX, X = Ge, Sn) have been predicted to show Weyl semimetal. First, we introduce about Weyl semimetal and discuss about the Weyl equation and then, the topological signature of Weyl points. After discussing about Weyl fermions, we describe quantum localization effect and anomalous Hall Effect which are vital to understand Weyl physics. We then discuss about growth and characterization of cobalt-based alloy. Finally, we analyze magnetic and magnetotransport properties.

6.1 Introduction Weyl semimetal (WSM) is a solid-state crystal as well as nanostructure materials whose low energy excitations realizes Weyl Fermions in which bands disperse linearly in three-dimensional momentum space through a node. The Weyl semimetal phase can act as an intermediate phase between trivial insulator and topological insulator [10-13]. More recently, relativistic Dirac fermions came into prominence with the discovery of graphene in two dimensions. WSM is a 3-dimensional counterpart of graphene though 2D materials like WTe2 and MoTe2 could exhibit Weyl semimetallic behavior. By breaking crystal centro symmetry or time-reversal symmetry Dirac point of a crystal can be converted

84 into Weyl type crystal [14-17]. Although number of candidates were previously proposed to exhibit topology, Heusler alloys were identified as a potential topological insulator in 2010 [18,19]. More recently, cobalt- based Heusler alloys were predicted to host Weyl fermions [20-21]. One main advantage of such alloy over other many topological materials is their ability to perform multifunctional behavior [22-27]. Co2TiX (X= Ge or Sn) have been interesting class of materials because of their potential application in spintronics [27-32]. They can exhibit half-metallic ferromagnetism and high

Curie temperature. Bulk and thin film experimental studies reported that it has Curie temperature from

360 K to 380 K depending upon the lattice order [33-37]. Here we have found that magnetic moment of

Co2TiGe is 1.9 µB/f.u. and the transition temperature is 380 K which are consistent with Slater-Pauling behavior. Negative magnetoresistance has been observed up to 200 K. Anomalous Hall Coefficient of 25

S/cm has been observed. Such value of anomalous Hall coefficient may be due to intrinsic Berry Phase curvature. Magnetoresistance measurements reveal weak-localization, a quantum interference phenomena.

6.1.1 From Dirac equation to Weyl equation Starting from the Schrödinger equation of the form

휕 ℏ푐 훿 6.1 푖ℏ 휓 = ( ∑ ( 훼푘 + 훽푚푐2)휓 휕푡 푖 훿푥 푘=1,2,3

Where 훼푘 and 훽 are n×n matrices and 휓 is an n-vector. Dirac showed that one can satisfy

휕 2 훿 6.2 (푖ℏ ) 휓 = ( ∑ (−ℏ2푐2 + 푚2푐2)휓 휕푡 훿푥 푘=1,2,3

푘 by choosing 훼 and 훽 as an anti-commuting matrices that square to I2×2. For 4- dimensional space this yields the following matrices.

푖 퐼 0 훼푖 = ( 표 휎 ) and 훽 = ( ) 휎푖 0 0 −퐼

Rewriting in Dirac-representation also known as 훾-matrices.

퐼 0 푖 훾0 = ( ) , 훾푖 = 훽훼푖 = ( 0 휎 ) 0 −퐼 −휎푖 0

In covariant form, Dirac equation becomes,

푚푐 (푖훾휇훿 − ) 휓 = 0 6.3 휇 ℏ

Another representation for 훾-matrices is the chiral representation which is given by changing 훾0 to

퐼 0 훾푐ℎ = ( ) 0 퐼

In this representation, the Dirac equation becomes

푚푐 6.4 − 푖(휕 + 흈. 훁 ℏ 0 ( 푚푐 ) 휓푐ℎ = 0 푖(휕 − 흈. 훁 − 0 ℏ

For massless fermions, it holds that 푚 ⟶ 0. The Dirac equation then decouples into two independent parts, namely Weyl equations, as Weyl had shown in 1929, given by

푖(휕0 + 흈. 훁)ψR = 0, 푖(휕0 + 흈. 훁)ψL = 0 [38-39]. 6.5

Here ψR,L represent two spinors with definite opposite chirality.

6.1.2 Weyl Nodes Weyl Fermions at zero energy correspond to points of bulk band degeneracy, Weyl nodes, which are associated with a chiral charge that protects gapless surface states on the boundary of a sample. These surface states take the form of Fermi arcs connecting the projection of bulk Weyl nodes on the surface

Brillouin zone [40]. Weyl semimetals were previously thought to have a point like Fermi surface at the

Weyl point. This type of Weyl semimetals has been classified as type-I, to distinguish them from the new type-II Weyl semimetals that exist at boundary between electron and hole pockets as shown in

Figure 6.1: Possible types of Weyl semimetals. In plot a, a type-I Weyl point with a point-like Fermi surface. In plot b, a type-II Weyl point appears as the contact point between electron and hole pockets.

The grey plane corresponds to the position of the Fermi level, and the blue (red) lines mark the boundaries of the hole(electron) pockets [41]. Figure 6.1 [41]. The 푲. 푷 Hamiltonian describing the linear behavior of band touching takes the form of a (2 ×2)-Hamiltonian of a chiral Weyl Fermion [42]. Each Weyl point can be regarded with a hedgehog singularity of the Berry curvature. Weyl nodes with opposite chirality are connected by open Fermi arc surface states. The shape of these arcs depends on the boundary conditions of the semimetal and can be engineered. Together with the Fermi surface of bulk states, the Fermi arcs on the opposite surfaces form a closed Fermi surface. A schematic representation of the Weyl nodes and Fermi arcs in momentum space is given in Figure 6.2.

6.1.3 Topological signature of Weyl points Topology and geometry were connected by the Gauss-Bonnet theorem back in the middle of 19th

2 century as ∫ 푑 푥 퐾(푥) = 2휋휒푀 where K is Gaussian curvature of a 2-dimneisional surface of a compact orientable 2- manifold M and 휒푀 is the Euler Characteristics, which is related to genus of the surface.

This remarkable relation evolved through mathematical abstraction to the one of Chern classes is given by

휇 휐 6.6 ∮ 푑푥 푑푥 퐹휇휐(푥) = 2휋퐶1

where 퐹휇휐represents the Berry curvature and 퐶1 is the first Chern class topological invariant. This

topological invariant is a key to the system with broken time-reversal symmetry such as topological

insulators and Weyl semimetals.

6.1.4 Berry phase and Berry Curvature Berry’s “geometric phase” exposed issues in adiabatic quantum mechanics that had been previously

hidden due to implicit gauge fixing [44]. A quantum system at stationary state is described by a

Hamiltonian H(k). If the system is altered adiabatically, then the system will always be in eigenstate of

H (k) [45]. If the Hamiltonian is returned to its original form, the system will return to its original state,

thereby acquiring a phase factor known as the Berry phase [46]. Consider a state |휓 (푡) > prepared in an

initial eigenstate |휓 (푡0) > = |푛(푘(푡0)) >, evolving according to the parameter dependent eigenstate

| 푛(푘(푡)) >, up to a phase. Then at a time t it holds that

푡 −푖 ∫ 푑푡′퐸 (푘(푡′)) 6.7 | 푒푖훾푛(풌(푡)푒 푡표 푡 |푛(푘(푡′)) >

where the second exponential describes the time evolution of the state |푛(푘(푡′)) >. The first phase

represents an extra phase that accommodates for all effect beyond dynamical phases. Making use of

Schrödinger equation for the state |휓(푘(푡′)) >, we get

푡 ′ ′ 푑 ′ 푘(푡) ′ 6.8 훾푛(풌(푡) = 푖 ∫ 푑푡 < 푛(풌(푡 )) | | 푛(풌(푡 )) > = 푖 ∫ 푑푘 < 푛(풌′)|∇푘′| 푛(풌′) > 푡0 푑푡′ 푘(푡0)

in the Brillouin zone, for k(t0) = k(t), this results in a closed path in parameter space, can be written as

훾푛 = ∮ 푑풌 < 푛(풌)|∇푘|푛(풌) > which represents the Berry phase. Using the Kelvin-Stokes theorem, it

can be written as 훾푛 = 푖 ∮ 푑풌 . < 푛(풌)|∇푘|푛(풌) > = ∮ 푑풌. 퐴푛(풌) where An(k) is Berry connection,

representing a vector potential. This quantity is not invariant under gauge transformation |푛(풌) →

푖휙푛(푘) 푒 |푛(풌) since we get: 퐴푛(풌) → 퐴푛(풌) − ∇푘휙푛(풌). Taking curl of this quantity gives famous Berry

curvature Ϝ푛(풌) = ∇풌×퐴푛(풌) and thus Ϝ푛,휇휈(풌) = 휕휇퐴푛,휈(풌) − 휕휈 퐴푛,휇(풌) represents a gauge invariant quantity. A plot of Berry curvature is shown in Figure 6.3.

6.1.5 Weyl Points and Berry Curvature Mathematically, the topological signature of Weyl nodes related to the quantization of Berry flux. For a

th multiband Weyl semimetal, the electrons in n band are described by wave function 휓푛(풌) = <

풌|푛(풌′) > . The Berry connection and curvature of nth band are given by

퐴푛(풌) = 푖 < 푛(풌)|∇k|푛(풌) and ℱ푛(풌) = ∇풌×퐴푛(푘).

Since Berry curvature Ϝ푛 is the curl of the vector field 퐴푛, it can be shown that

∇풌. ℱ푛(풌) = 0 6.9

The above relation holds only if the n bands are non-degenerate. This does not hold for Weyl semimetal at the Weyl node since at least two degenerate bands meet. Therefore, Berry connection and Berry curvature are not well defined at a Weyl node. For a Hamiltonian, describing a Weyl semimetal of the form 퐻 = 푎(풌) 퐼2×2 + 푏(풌). 휎, which is topologically equivalent to a spin in a magnetic field, it can be shown that ith component of Berry curvature is given by [48]

1 1 휕풃 휕풃 6.10 ℱ푖(풌) = − 3 휖푖푗푚풃(풌). × 8휋 |풃(풌) | 휕푘푗 휕푘푚

For right and left Weyl Hamiltonians, where 푏(풌) = ±푣퐹풌, the berry curvature yields

1 풌 1 풌 ℱ (푘) = and ℱ (푘) = − . This leads to 푅퐻 4휋 |풌|3 퐿퐻 4휋 |풌|3

∇풌. ℱ푅퐻(풌) = 훿(풌) and ∇풌. ℱ푅퐻(풌) = −훿(풌). 6.11

The above equation shows that Weyl nodes are indeed monopoles of Berry curvature. They are thus sources of quantized Berry flux in the momentum space.

6.1.6 Quantum Transport, Weak Localization and Weak anti-localization The electronic transport in a solid can be classified by several characteristics length: (i) mean free path 푙,

which measures the average distance that an electron travel before its momentum is changed by elastic

scattering from static scattering centers. (ii) The phase coherence length 푙∅, which measures the average

distanced an electron can maintain its phase coherence. The 푙∅ can be determined by inelastic scattering

from electron-phonon scattering and interaction with other electrons. (iii) The sample size 퐿. If 푙 ≫ 퐿,

electrons can tunnel through the sample without being scattered and diffuse through the sample. On the

contrary, electrons will suffer from scattering and diffuse through the sample, which is called diffusive

transport. If 푙∅ > 푙, electrons will maintain their phase coherence even after being scattered. This leads

to quantum diffusive regime so that localization appears [49].

Weak localization and weak antilocalization are quantum interference effects in quantum transport

phenomena in a disordered electron system. Weak localization suppresses the conductivity and weak

Figure 6.2: Schematic of the structure of the Weyl semimetal in momentum space. Two diabolical points are shown in red, within the bulk 3D Brillouin zone. Each Weyl node is a source or sink of the flux of the Berry connection, as indicated by the blue arrows. The dark grey plane indicates the surface Brillouin zone, which is a projection of the bulk one. The Weyl nodes are connected by a Fermi arc, as shown by the yellow line [9,43].

90 anti-localization enhances the conductivity with decreasing temperature at very low temperature. A magnetic field can destroy the quantum interference effect, giving rise to a cusp like positive (negative) as a signature of weak-localization (weak anti-localization) respectively [50,51]. The surface of topological materials can give rise to a 휋 Berry phase when an electron move adiabatically around the

Fermi surface. This phase can lead to the absence of back scattering [52] and further delocalization of the surface electrons [53]. Such 휋 Berry phase induces a destructive quantum interference between time reversed loops formed by scattering trajectories [ 54]. The destructive interference can suppress backscattering of electrons, then the conductivity is enhanced with decreasing temperature because decoherence mechanism are suppressed at low temperature [55,56]. A magnetic field can destroy interference as well as the enhance conductivity, so that signature of the weak antilocalization is a negative magnetoconductivity. The Berry phase give the time reversal scattering loop which is equivalent to moving an electron on the Fermi surface by one cycle [57]. This phase can be written in terms of eigenfunctions as

2휋 휕 Δ 6.12 휙푏 = −푖 ∫ 푑휓 < 휓푘(풓) | | 휓푘(풓) = 휋(1 − ), 0 휕휓 2퐸퐹 where 휓푘(풓) is the spinor wave functions of the conduction band of Hamiltonian, Δ is a gap between conduction band and valence band, 퐸퐹 is the Fermi energy [58,59]. For massless limit, the Berry phase give antilocalization whereas for acquiring Dirac mass, Berry phase can give weak localization.

6.1.7 Anomalous Hall Effect If an electric current is passed through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers which deflects them to one side of the conductor. A buildup of the charges at the side of conductor creates a voltage. This transverse voltage is called Hall voltage also known as Hall Effect after E. H. Hall who discovered in 1879. The anomalous Hall Effect

(AHE) came into light when Hall reported that his “pressing electricity” effect was considerably larger

Figure 6.3: The vector plot of the Berry curvature in momentum space. The arrows show that the flux of the Berry curvature arrows from one monopole (red) to the other (blue), defining the non- trivial topological properties of a topological semimetal [ 48]. in ferromagnetic iron than in nonmagnetic semiconductors. The AHE had been a one of the biggest problem in the field of condensed matter physics for almost a century because it involves concepts based on topology and geometry that have formulated only in recent times. Very early on, the experimental investigators knew that the dependence of Hall resistivity (휌푥푦) on perpendicular magnetic field was different among ferromagnetic and nonmagnetic conductors [60]. In nonmagnetic conductors, 휌푥푦 depends linearly on an applied field whereas for ferromagnetic conductors it initially increases steeply in weak magnetic field and then saturates at certain field. Pugh and Lippert (1932) established an empirical relation between 휌푥푦, 퐻 and 푀 such that 휌푥푦 = 푅0퐻 + 푅퐴푀, the second term represents the Hall effect contribution due to magnetization.

In 1954, Karplus and Lutinger (KL) proposed a theory for the AHE that provide a crucial step to understand the AHE problem. KL showed that when an electric field is applied to a solid, electrons will acquire an additional contribution to their group velocity. KL’s anomalous velocity was perpendicular to the electric field and Hence could contribute to Hall effect. This contribution depends only on band

92 structure and is independent of scattering. Recently, it has been referred as an intrinsic contribution to the AHE. When the conductivity tensor is inverted, the intrinsic contribution to AHE is proportional to

2 휌푥푥 .The main drawbacks of KL theory is that it neglected contribution due to scattering (Smit, 1955 and

Berger, 1970). So, later people realized that total AHE is sum of intrinsic contribution, skew scattering and side jump scattering. The intrinsic contribution to anomalous Hall conductivity can be calculated as

(Haldane, 2004.)

푒2 푑풌 6.13 휎푖푛푡 = −휀 ∑ 푓(퐸 (풌)푏푙 (풌) 푖푗 푖푗푙 ℏ (2휋)3 푛 푛 푛 where 휀푖푗푙 is the antisymmetric tensor, 푏푛(풌) is the Berry phase curvature corresponding to the states

{|푛, 풌 >} and 퐸푛(풌) is the eigenvalues of a Block Hamiltonian. The skew scattering contribution is due to asymmetric scattering from impurities caused by the spin-orbit interaction. This AHE predicts anomalous Hall coefficient is proportional 휌푥푥 . It can be described within Semi classical Boltzmann transport theory. The side jump experienced by quasi particles upon scattering from spin-orbit coupled impurity can give side jump contribution to the AHE. A natural classification of the contribution to the

AHE, which is guided by experiment and by microscopic theory of metals, is to separate them according

10 μm

Figure 6.4: SEM image of 50 nm CTG thin film

0 to their dependency on the Bloch state transport lifetime 휏. If 휎푥푦 is proportional to 휏 , then that term is

1 due to side-jump scattering or intrinsic. If 휎푥푦 is proportional to 휏 , that is due to skew scattering.

6.2 Experimental Details

Co2TiGe thin films were grown using ultra-high vacuum electron beam evaporation. The base pressure inside the chamber was below 9 x 10-10 Torr and less than 5 x 10-9 Torr during the film deposition. One side polished silicon (100) substrates were used for the film deposition. Prior to deposition, silicon substrate was cleaned with distilled water, isopropyl alcohol and acetone respectively. After cleaning, substrate was etched in 2 % hydrofluoric acid to remove oxide from the surface. Wafers are placed inside high vacuum and preheated at 473 K for 20 minutes. A buffer layer of 2 nm of magnesium oxide was deposited initially and in situ annealed for 1 hour at 573 K for 1 hour. The stoichiometric ratio of

Co, Ti, and Ge was deposited, with film composition and uniformity controlled by a quartz crystal rate monitor and low deposition rate 0.3- 0.5 Å/s. Films were grown with a thickness of 50 nm. After growth, films were annealed in situ at 773 K for 2 h. Hall bar devices that are 50 nm thick were grown using metal contact masks. The morphology of the thin films was studied by scanning electron microscopy (SEM). The crystal structure was determined by x-ray diffraction analysis (XRD) using a

Figure 6.5: (a) XRD pattern of CTG thin film. (b) EDX spectrum of CTG thin film

Thermo/ARL X’TRA, (Cu-Kα) diffractometer. The magnetic measurements were carried out using a

Quantum Design vibrating sample magnetometer (VSM). Transport Properties were measured using a physical property measurement system (PPMS) with horizontal rotator option.

6.3 Morphological and Structural characterization Figure 6.4 shows SEM image of CTG thin film. Surface of the thin film is smooth and continuous.

Figure 6.5 (a) displays x-ray diffraction pattern of CTG film. In general, Co2XZ Heusler alloy can crystallize in chemically ordered L21 phase or in the B2 phase characterized by the disorder between X and Z lattices and regular occupation on the Co atomic sites or in A2 phase with random occupation of the Co, X and Z atomic sites [61]. The ordered L21 structure is characterized by the presence of odd superlattices reflections like (111) or (311) whereas this type of reflection is absent in case of the B2 structure which is characterized by h+k+l = 4n +2 superlattice reflections like (002). The h+k+l = 4n reflections like, (004) or (422), are fundamental types and are unaffected by chemical disordering [62].

The non-zero intensity of (111) and (200) peaks from our thin film indicate that our alloy exhibit the L21 ordered crystal structure with space group Fm3̅m. However, very small intensity of (200) peak indicates there could be L21 structure with B2 phase. The value of lattice constant is 5.81Å which matches per literature. The composition of the thin film was determined using Energy Dispersive Spectroscopy

(EDS). A typical EDS spectrum is shown in Figure 6.5 (b). The ratio of Co:Ti:Ge is equal to

2.01:0.99:1, very close to the stoichiometric ratio. Magnesium peaks in EDS spectrum came from MgO at the bottom layer.

6.4 Magnetic Chracterizations The variation of magnetization versus applied field is displayed in Figure 6.5 (a). The saturation magnetic moment per formula unit calcualted from measurement data was 1.9 µB at 10 K which is close

Figure 6.6: (a) Field variation of magnetization at 300 and 10 K. (b) Magnetization curve as a function of temperature at 1000 Oe field to the therotical value of 2 µB. This value reduces to 1.5 µB when the temperature is increased to 300 K.

The moment value at 10 K follows the Slater-Pauling rule [23]. We have measured magnetization versus temperature with an applied field of 1000 Oe as shown in Figure 6.6 (b). First, sample is cooled in zero magnetic field from 300 down to 10 K and the magnetization values are recorded while warming the sample in presence of a magnetic field. This is known as zero-field cooled (ZFC) magnetization.

Another mode of measuring magnetization is field-cooled (FC) magnetization in which sample is cooled through 300 to 10 K in presence of a magnetic field. The difference between FC and ZFC is much less indicating the soft magnetic system. We find very small thermodynamic irreversibility is observed below 100 K which means the domain wall pinning effects might not be responsible for reversal magnetization. We have observed a Curie temperature of 379 K, while the reported values vary from

380 to 391 K [34-37].

6.5 Electrical Resistivity

Figure 6.7 represents the temperature dependence of the zero field resistivity ρxx(T) ranging from 10 to

300 K. Maximum value of resistivity is observed around 200 K. The temperature dependence of

푑휌 resistivity i.e. is quite small below 200 K. Below the maximum, the behavior of resistivity is 푑푡

Figure 6.7: Resistivity variation with temperature. Upper left inset shows the resistivity data and fit for T<200 K. Lower inset is resistivity variation and fit for T > 200 K metallike and the carrier density is almost temperature independent. The dominance of electron-phonon scattering below the maximum gradually reduces so that resistivity decreases with decrease in temperature [63]. Above the maximum, the resistivity is decreased. Such sign change of resistivity is commonly found in semimetals or narrow gap semiconductors [64-66]. This temperature maximum is observed in many crystalline metals with atomic disorder and is interpreted as due to weak localization or electronic correlations [67,68]. Impurities or defects can give rise to higher carrier concentration which increases conductivity. Unusually high value of resistivity and negative temperature coefficient of resistance (TCR) were observed by Zhang et. al for T < 350 K due to peculiarities of band structure in

Heusler alloy [69]. A similar result has been observed in semimetallic Fe2VAl which has pseudo-gap for the majority and minority bands at Fermi level [70-71]. For temperature less than 200 K, the resistivity

2 can be fitted with 휌 = 휌0 + 퐴푇 + 퐵푇 as shown in upper left inset of figure 6.6. Here 휌0 = 258.8 µΩ cm and A = 3.72 ×10-4 µΩ cm/K2 and B = 1.83×10-6 µΩ cm/K3. The T2 is due to the electron-electron scattering and the T is due to electron- phonon scattering. The resistivity behavior above 200 K can be

퐸푔 − 푘 푇 described by a simple model of conductivity: 휎 (푇) = 휎푎 푒 퐵 ,where 휎푎 is a constant, 퐸푔 is the activation energy [72]. We have fitted this model as displayed in lower right inset of Figure 6.7. From

97 our best fit, we have obtained the value of activation energy as 24 meV. Existence of such gap could be the signature of semimetallic or pseudo-gap semiconducting behavior in Co2TiGe system.

6.6 Magnetoresistance Figure 6.8 (a) shows the longitudinal magnetoresistance (MR) when magnetic field is applied perpendicular to the film plane. The MR curves display positive cusp from 5 to 300 K at low fields.

Unsaturated MR is observed for temperature below 20 K. The negative MR has been found for all temperatures except at 300 K. At 300 K, non -saturating MR tends to be positive above a field of 1 T.

Such crossover from negative to positive MR at 300 K is due to activated carriers by temperature so that

MR curves depends on the magnetic field in a quadratic way [73-74]. The MR vs temperature plot is shown in Figure 6.8 (b). The negative MR below 200 K at higher field is a consequence of suppression of spin disorder scattering caused by the application of magnetic field [75,76]. Since the grown thin film has a Curie temperature around 380 K, spin disorder scattering is not responsible for the longitudinal resistance maximum around 200 K. We observed the maximum negative MR around 100 K as shown in

Figure 6.8 (b) which is at a temperature lower than the temperature at which resistivity maximum appears. Such unusual temperature dependence of MR has also been observed in other Heusler alloy thin

Figure 6.8: (a) Magnetoresistance along longitudinal direction at different temperatures (b) MR versus temperature at 4 T field.

98 films [64,77-79]. The negative MR arising from spin dependent scattering should enhance while lowering the temperature. On the contrary, impurity scattering can compete at low temperature reduces the spin effect. Due to competition between these two scattering, MR possess maximum value at 100 K instead of monotonically decreasing with increasing temperature.

6.7 Weak-Localization The increase in magnetoconductivity at low magnetic field is demonstrated in Figure 6.9 (a). Such positive conductivity can be interpreted due to the effect of weak-localization [80]. Since CTG thin film has a thickness of 50 nm and an electronic mean free path of 103 nm, it can be treated as 2-D system

[80,81]. For a 2-D thin film, weak-antilocalization is studied by Hikami-Larkin-Nagako formula where conductivity decreases with increasing field [82,83]. For weak-localization (WK) in magnetic systems, the conductivity follow a logarithmic dependence on magnetic field which can be written as [84, 85]

훼푒2 훽푒푡2퐵2 6.14 ∆휎 = − 2 ln (1 + ) 2휋 ℏ 4ℏ퐵∅ where 훼 is related to the number of transport channel through surface states, 훽 is related to the ratio of

2 mean free path and film thickness. 퐵∅ = ℏ/ 4푒퐿∅ , 퐿∅ is the effective dephasing length. We have fitted the above equation with the value of 훼 and 퐿∅ shown in Figure 6.9 (b). The value of 훼 decreases as we increase the temperature. A maximum value of 훼 is obtained at 50 K which is around 4. The dephasing length varies from 650 to 320 nm as we increase temperature from 5 to 200 K. The large value of 훼 indicates that WL is responsible for magnetoconductance upturn. This could be due to transport through bulk subbands and surface states. Because of small thickness of the films, the bulk state inside the film is quantized into 2D subbands and WL can be expected in the transport through these sub bands [80,81].

Such a large value of 훼 have reported by Shan et. al [85] and Akiyama et. al. [81]. To confirm, whether

MC is due to geometrically induced or not we have performed angle dependent MR as shown in Figure

6.9 (c) where 900 means field is applied perpendicular to film plane and 00 is parallel to the current flow.

All angle dependent MR nearly overlap indicating it is mixture of 3D bulk effect and 2D surface effect

[86].

6.8 Hall resistivity and Anomalous Hall effect

The Hall resistivity 휌푥푦(퐻) curve in Figure 6.10 (a) present an anomalous hall effect, even hysteresis behavior at low field. The total resistivity for a magnetic system is the sum of H-linear normal Hall

푁 퐴 푁 퐴 resistivity 휌푥푦 and anomalous Hall resistivity 휌푥푦, which is 휌푥푦 = 휌푥푦 + 휌푥푦 [60]. 휌푥푦 = 푅0퐻 + 푅퐴푀

퐴 where 푅0 and 푅퐴 is ordinary and anomalous Hall coefficient respectively. Generally, 휌푥푦 can be scaled

퐴 2 as 휌푥푦 = 푎휌푥푥 + 푏휌푥푥, where 푎 is related to skew-scattering and 푏 is related to the side jump scattering or intrinsic Berry phase contribution [87,88]. We have found that a coefficient for skew-scattering is very small so that it can be negligible. Then, if we divide the total resistivity by H, there is linear relation

휌 휌2 between 푥푦 and 푥푥 [89,90]. In equation form, 퐻 퐻

2 휌푥푦 휌 6.15 = 푅 + 푆 푀 푥푥 퐻 0 퐴 퐻 we have shown the fitting of above equation to get 푅0 and 푆퐴 at 200 K as shown in Figure 6.10 (b). We have found the positive value of 푅0 suggesting hole like conduction of the sample as displayed in Figure

6.10 (c). The change in value of 푅0 supports the resistivity versus temperature relation as in Figure 6.7.

The corresponding effective carrier concentration is 3.8 ×1027 per m3 at low temperature and is found to increase at higher temperature. Such higher value of carrier concentration has been observed in other

Heusler alloy thin films [64,78,91]. The maximum mobility obtained is 6.923 cm2/V.s. which is comparable to other half metallic and semiconducting Heusler alloys [78,92]. Anomalous Hall coefficient is nearly independent of T suggesting that anomalous hall conductivity is proportional to magnetism. We have also computed Hall conductivity at 5 K as displayed in Figure 6.10 (d). The

100 anomalous Hall conductivity has a value of nearly 25 S/cm at 5 K. Such smaller value of conductivity as comparing to those typical ferromagnet is quite surprising [93,94]. This value favors the intrinsic origin of anomalous hall effect arising from Berry curvature.

6.9 Conclusions

We have grown CoTiGe thin film with L21 structure for the first time on silicon (100) substrates with an

MgO buffer layer. The saturation magnetic moment is 1.9 µB/f.u at 10 K, and the Curie temperature of

380 K makes it suitable for room temperature applications. Semimetallic nature has been verified from

101

Figure 6.10: (a) Hall resistivity as a function of magnetic field. (b) Fitting between hall resistivity and longitudinal resistivity at 200 K. (c) Variation of R0 and SA with temperature (d) Hall conductivity at 5K resistivity measurements. The MR presents a maximum at 100 K. Weak localization causes

2 magnetoconductance upturn at lower magnetic field. Hall resistivity is proportional to 휌푥푥 and magnetization, and a small Hall conductivity signals an intrinsic berry phase anomalous hall effect. The carrier concentration is approximately 3.8 ×1027 per m3 at 5 K which is comparable to other Heusler alloy thin films.

102

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Chapter 7

7. Co2TiSn- A Magnetic, Centrosymmetric Weyl Semimetal

Weyl fermions have long been known in quantum field theory but have not been observed in as a fundamental particle in nature. Recently, it has been found that such fermion can be realized in certain material called Weyl semimetal (WSM). A WSM is characterized by topological invariants, extending the classification of topological phases of matter beyond topological insulators. In WSM, the valence and conduction band touch linearly at a pair of discrete points called as Weyl points. Crossing points in the electronic band structure, such as Weyl and Dirac Points are not rare, but very often these points are not at Fermi energy or many other band also cross the Fermi energy. Weyl and Dirac points have a great influence on the transport properties of material. In the context of recent searches for Weyl and Dirac points in the electronic structure of semimetals researchers proposed that electron doped Co2TiSn (CTS) should be a magnetic semimetal. CTS is a family of 26 valence electron ferromagnetic Heusler compound which has centrosymmetric space group Fm3̅m. This chapter describes the growth, morphological, structural and magnetic behavior of CTS.

7.1 Introduction

Cobalt-based Heusler alloys Co2XY, where X = Ti, Zr, Hf, etc. and Y= Sn or Al have been considered to be good candidates for studying itinerant electron ferromagnetism [1]. Among these alloys, CTS is particularly interesting because of its similarity with the prototype half-metallic system NiMnSb [2]. It is predicted to be a half-metallic ferrimagnet with a magnetic moment of 2 μB/f.u. and it has high formation energy of Co-TI site-swap defect [3,4]. For practical purpose, high tolerance of ground state properties against disorder is required. CTS has been studied theoretically as well as experimentally. The

109

110

Figure 7.1: Spin resolved band structure and density of states of a magnetic Weyl semimetal Co2TiSn band structure calculation of CTS has predicted a half-metallic ground state [5]. Recent theoretical prediction suggested the existence of Weyl Fermions in this compound which suggests the presence of time reversal symmetry breaking [6, 7]. The band structure of CTS as shown in Figure 7.1 displays two

Weyl nodes [8]. Weyl semimetals where the bulk conduction and valence bands touch linearly with distinct chirality are of great interest due to open fermi surface arcs and unusual magnetotransport and spectroscopic properties [9]. Studies on bulk CTS have reported a lattice parameter of 6.07 Å, a magnetic moment of 1.95 μB/f.u., and a Curie temperature around 355 K [3,10-11]. It has been reported that the anomalous behavior on resistivity where the temperature coefficient of resistivity becomes negative above Curie temperature. Also, negative magnetoresistance has been observed in this bulk compound [12]. Thermoelectric study showed a large and constant Seeback coefficient of -50μV/K above TC which signaled the possible application in spin caloritronics [12]. There are reports of studies on CTG thin films. Gupta et. al grew CTS films on Si (100) substrate by pulsed laser ablation [13]. They found off-stoichiometric, polycrystalline films with (022) texture. Suharyadi et. al grew L21 ordered films on magnesium oxide substrates with a chromium buffer layer using an alternate deposition technique [14]. Meinert et. al synthesized epitaxial thin films on magnesium oxide substrates using dc magnetron sputtering [15].

111

10 μm

Figure 7.2: SEM image of 57 nm CTS In this study, we have grown and investigated thin films of CTS using ultra high vacuum electron beam evaporation. We have observed that magnetic moment of CTS around 6× 10-5 emu and magnetic transition temperature above 400 K which is higher than the reported value. Our alloy exhibit L21 cubic crystal structure.

7.2 Experimental details CTS thin films were grown using ultra-high vacuum electron beam evaporation. The base pressure inside the chamber was below 9 x 10-10 Torr and less than 5 x 10-9 Torr during the film deposition.

Polished silicon (100) substrates were used for the film deposition. Prior to deposition, silicon substrate was cleaned with distilled water, isopropyl alcohol and acetone respectively. After cleaning, substrate was etched in 2 % hydrofluoric acid to remove the natural oxide from the surface. Wafers are placed inside high vacuum and preheated at 473 K for 20 minutes. A layer of 5 nm of magnesium oxide was deposited as a buffer and annealed at 573 K for 1 hour to stack with substrate. The stoichiometric ratio of Co, Ti, and Sn was deposited, with film composition and uniformity controlled by a quartz crystal rate monitor and low deposition rate 0.3- 0.4 Å/s. Films were grown with thicknesses 57 nm. After growth, the films were annealed in situ at 773 K for 4 h. The morphology of the thin films was studied

112 by scanning electron microscopy (SEM). The crystal structure was determined by x-ray diffraction analysis (XRD) using a Thermo/ARL X’TRA, (Cu-Kα) diffractometer. The magnetic measurements were carried out using a Quantum Design vibrating sample magnetometer (VSM).

7.3 Results and Discussions Figure 7.2 shows the SEM image of CTS thin film. Surface of the thin film is smooth and continuous with bright grains present at the surface. Figure 7.3 (a) displays x-ray diffraction pattern recorded for

CTS film. A fully ordered atomic arrangement in full-Heulser alloys is the L21 structure. However,

Heusler alloy of the form X2YZ may have two kinds of disordering. When the YZ sublattice is randomly occupied by Y and Z atoms, i.e., disordering in the YZ sublattice occurs, the ordering structure is reduced to the B2 type. Furthermore, when disordering between the X and YZ sublattices occurs, the ordering structure is lowered to the A2 type. Y-Z disordering extinguishes odd superlattice diffraction lines (that are defined by the index relation of h, k, and l = odd numbers, e.g., (111)). Furthermore, even superlattice diffraction lines ((h  k  l) 2  2n 1, e.g., (200)) vanish under X-YZ disordering. On the other hand, fundamental diffraction lines ((h  k  l) 2  2n, e.g., (220)) are independent of the ordering structures. The non-zero intensity of (111) and (200) peaks from our thin film indicate that our alloy

(a) (b)

Figure 7.3: (a) XRD spectrum of a CTS thin film, (b) A typical EDX spectrum of a CTS thin film

113

Figure 7.4: M vs H curve of thin film at 300 K exhibit the L21 ordered crystal structure with space group Fm3̅m. The value of lattice constant is 6.065

Å which matches per literature [11,12]. From Figure 7.3 (a), it is observed that our thin film is not pure

CTS as it is mixed with secondary phases like Sn. The composition of the thin film was determined using EDS which is shown in Figure 7.3 (b). The stoichiometric ratio of Co:Ti:Sn is equal to

1.7:1.1:1.2. The EDS results further support the off-stoichiometric ratio of Co, Ti and Sn. The variation of magnetization versus field at 300 K is displayed in Figure 7.4. Magnetic moment of our thin films is very low as compared to previous CTG thin film. We have measured magnetization versus temperature with an applied field of 1000 Oe as shown in Figure 7.5. At first sample is cooled in zero magnetic field from 400 down to 10 K and the magnetization values are recorded while warming the sample in the presence of a magnetic field. This is known as zero-field cooled (ZFC) magnetization. Another mode of measuring magnetization is field-cooled (FC) magnetization in which sample is cooled through 400 to

10 K in the presence of a magnetic field. The values of magnetization are recorded while warming the sample after cooling process. There is a difference between FC and ZFC modes below 100 K which refers to the thermodynamic irreversibility. This means domain wall pinning has great effect on magnetization reversal. We have observed transition from ferromagnetism to paramagnetic around 375

114

Figure 7.5: variation of magnetization with temperature K which is higher than the reported bulk values in the literature [10-11]. The Curie temperature of the system matches with thin film as reported by Meinkart et. al [15].

7.4 Conclusions In summary, we have grown Co2TiSn thin films with L21 structure on silicon (100) substrates with MgO as a buffer layer. We have observed low value of saturation at 300 K and the Curie is around 375 K.

From VSM measurements, we can conclude that our thin film might contain secondary phases of cobalt and tin are mixed though these phases are not clearly visible on XRD spectrum.

115

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