Ph.D Thesis

THEORETICAL STUDIES OF STRUCTURAL, MECHANICAL,

ELECTRONIC AND MAGNETIC PROPERTIES OF RARE-

EARTH MONOAURIDES

By

SARDAR AHMAD

DEPARTMENT OF CHEMISTRY

UNIVERSITY OF MALAKAND

2016

Ph.D Thesis

THEORETICAL STUDIES OF STRUCTURAL, MECHANICAL,

ELECTRONIC AND MAGNETIC PROPERTIES OF RARE-

EARTH MONOAURIDES

BY

SARDAR AHMAD

Thesis submitted to the Department of Chemistry,

University of Malakand for the partial fulfillment of the

requirements for the degree of

DOCTOR OF PHILOSOPHY (Ph.D)

IN

CHEMISTRY

DEPARTMENT OF CHEMISTRY

UNIVERSITY OF MALAKAND

2016

i

Declaration I declare that the thesis entitled “THEORETICAL STUDIES OF STRUCTURAL,

MECHANICAL, ELECTRONIC AND MAGNETIC PROPERTIES OF RARE-EARTH

MONOAURIDES” is my original work and has never been presented for the award of any

degree at any other University before and that all the information sources cited and or quoted

have been indicated and acknowledged with complete references.

Sardar Ahmad

ii

SUBMISSION ATHURIZATION CERTIFICATE

Certified that the thesis of Mr. Sardar Ahmad entitled “THEORETICAL STUDIES OF

STRUCTURAL, MECHANICAL, ELECTRONIC AND MAGNETIC PROPERTIES OF

RARE-EARTH MONOAURIDES” has been reviewed in the final form. Permission as

indicated by the signatures given below is granted to submit the copies to the University of

Malakand for final evaluation process.

Supervisor

Prof. Rashid Ahmad _____________________ Chairman Department of Chemistry

Co-Supervisor

Prof. Iftikhar Ahmad _____________________ Vice-chancellor Abbottabad University of Science and Technology

Chairman Department of Chemistry

Prof. Rashid Ahmad _____________________

iii

Dedicated

To

My

Respectable Parents

iv

List of Publications

During this research work, the following articles were published or are under-review.

1. Sardar Ahmad, Rashid Ahmad, S. Jalali-Asadabadi, Zahid Ali, Iftikhar Ahmad “First

principle studies of electronic and magnetic properties of Lanthanide-Gold (R-Au) binary

Intermetallics” J. Magn. Magn. Mater., 422(5), 458‐463 (2017).

2. Sardar Ahmad, Rashid Ahmad, Hamide Vaizie, H. A. Rahnamaye Aliabad, Imad Khan, Zahid

Ali, S. Jalali-Asadabadi, Iftikhar Ahmad and Amir Abdullah Khan “First-principles studies of

pure and fluorine substituted alanines” Int. J. Mod. Phys. B, 30(14), 1650079 (2016).

3. Sardar Ahmad, Rashid Ahmad, Muhammad Bilal, Najeeb ur Rahman “First principle studies

of Thermoelectric properties of R-Au compounds (R=Tb, Ho, Er, Tm and Yb)” (under-review).

4. Sardar Ahmad, Rashid Ahmad, M. Shafiq, S. Jalali-Asadabadi, Iftikhar Ahmad “Strongly

correlated intermetallic rare-earth monoaurides (R-Au): ab-initio study” (under-review).

5. Sardar Ahmad, Rashid Ahmad, S. Jalali-Asadabadi, Zahid Ali, Iftikhar Ahmad “First

principle studies of structural, magnetic and elastic properties of orthorhombic rare-earth

diaurides intermetallics R-Au2 (R=La, Ce, Pr and Eu)” (under-review).

6. Sardar Ahmad, Rashid Ahmad, Iftikhar Ahmad “Structural, Thermal, Magnetic, Elastic and

Thermoelectric properties of Cubic binary Rare-earth monoauride (R-Au) intermetallics”

(under-review).

v

Acknowledgements

I wish to express my appreciation and sincere gratitude to my supervisor Prof. Rashid Ahmad,

Chairman, Department of Chemistry, whose friendly discussion and cooperative attitude proved

immensely valuable to me at every step. His guidance helped me in all moments of research and

thesis writes up.

I would also like to express my deep thanks and great appreciation to my co-supervisor

Prof. Iftikhar Ahmad, Vice-chancellor Abbottabad University of Science and Technology, whose

enthusiasm for the Center for Computational Materials Science hovered over me for the entire

period of my studies.

I am also thankful to all the faculty members of the Department of Chemistry, faculty

members of the Department of Physics Dr. Zahid Ali, Dr. Imad Khan, all the members of the

Center for Computational Material Science (CCMS) or rather I would say CCMS family

University of Malakand, Dr. Hazrat Ali, Dr. Muhammad Bilal, Dr. Muhammad Shafiq, Mr.

Sajid Khan, Mr. Saifullah, Mr. Banaras Khan, Mr. Rashid Iqbal, Mr. Fawad Khan, Mr. Abdul

Ali, Mr. Amin Khan, Mr. Adnan Ali Khan, Mr. Ziad Ahmad, Mr. Farooq Usman, Mr. Shafiq

Ahmad, Mr. Abdur Rehman, Mr. Mazhar Rehman, Mr. Raham Zeb and Mr. Gul Rehman for

facilitating my research work and providing a friendly environment. Most importantly, my

deepest love and gratitude go to my family members, for their love, patience and support

throughout my studies.

I also admire the financial support from HEC of Pakistan for project No. 20-

3959/NRPU/R & D/HEC2014/119 which will enhance the research facility in the CCMS,

University of Malakand.

Special thanks to my wife and children, Masroor Ahmad, Mahwish Sardar, Ammar

Ahmad and Abdullah without their encouragement, love and moral support. I could not have

finished this research work.

Finally, I would like to thank all those who supported me directly or indirectly in completing my

thesis in time. ALLAH may bless them all.

SARDAR AHMAD

vi

Table of Contents

Chapter 1 ............................................................................................................................................. 1

Introduction ......................................................................................................................................... 1

1.1 Intermetallics ............................................................................................................................... 1

1.2 Structural properties .................................................................................................................... 3

1.3 Mechanical properties ................................................................................................................. 3

1.4 Electronic properties ................................................................................................................... 3

1.5 Thermoelectric properties ........................................................................................................... 4

1.6 Magnetic properties ..................................................................................................................... 4

1.7 Thermodynamic properties ......................................................................................................... 5

1.8 Literature survey of R-Au intermetallics .................................................................................... 5

1.8.1 Structural properties of R-Au intermetallics ........................................................................ 5

1.8.2 Thermal properties of R-Au intermetallics ......................................................................... 10

1.8.3 Magnetic properties of R-Au intermetallics ....................................................................... 14

1.8.4 Electronic and Thermoelectric properties of R-Au ............................................................ 18

1.9 R-Au2 (R=La, Ce, Pr, Eu) compounds ...................................................................................... 18

1.9.1 Structural study of R-Au2 compounds ................................................................................ 18

Present work ................................................................................................................................ 20

Chapter 2 ........................................................................................................................................... 21

Theoretical Background and Computational Methods ................................................................. 21

2.1 Density Functional Theory “DFT” ............................................................................................ 21

2.2 Kohn and Sham (KS) Equation ................................................................................................. 21

2.3 Exchange Correlation Potentials ............................................................................................... 22

2.3.1 Local Density Approximation “LDA” ............................................................................... 22

2.3.2 Generalized Gradient Approximation (GGA) .................................................................... 23

vii

2.3.3 Generalized Gradient Approximation with Hubbard (GGA+U) ........................................ 24

2.4 Full Potential Linearized Augmented Plane Wave (FP-LAPW) method ................................. 24

2.5 Cubic-elastic Package ............................................................................................................... 25

2.6 BoltzTraP Package .................................................................................................................... 27

2.7 WIEN2k code ............................................................................................................................ 28

Chapter 3 ........................................................................................................................................... 29

Results and Discussion ...................................................................................................................... 29

3.1 Rare-earth monoaurides (R-Au) ................................................................................................ 29

3.1.1 Structural Properties of rare-earth monoaurides R-Au ...................................................... 29

3.1.2 Elastic and mechanical properties of R-Au ........................................................................ 33

3.1.3 Thermodynamic properties of R-Au ................................................................................... 39

3.1.4 Chemical bonding of R-Au compounds ............................................................................. 41

3.1.5 Cohesive energy of R-Au compounds ............................................................................... 43

3.1.6 Electronic properties of R-Au compounds ......................................................................... 46

3.1.7 Magnetic properties of R-Au .............................................................................................. 50

3.1.8 Thermoelectric properties of R-Au compounds (R=Tb, Ho, Er, Tm and Yb) at low ........ 53

temperature .................................................................................................................................. 53

3.2 Rare-earth diaurides (R-Au2) (R= La, Ce, Pr and Eu) Orthorhombic intermetallics. ............... 63

3.2.1 Structural properties R-Au2 ................................................................................................ 63

3.2.2 Magnetic properties R-Au2 compounds ............................................................................. 65

3.2.3 Elastic constants and mechanical properties of orthorhombic R-Au2 intermetallics ......... 68

Conclusions ........................................................................................................................................ 75

References .......................................................................................................................................... 77

viii

List of Figures

Figure1.1: Unit cell of cubic R-Au compounds. ............................................................................. 9

Figure1. 2: Melting temperatures of the R-Au compounds .......................................................... 13

Figure3. 1: Comparison of the calculated lattice constants by PBEsol, PBE and Hubbard U with

experimental values ...................................................................................................................... 32

Figure 3. 2: Spin polarized electronic charge density of R-Au compounds in (100) and (110)

planes ............................................................................................................................................ 42

Figure 3. 3: Cohesive energy (Ry) and melting points (o C) of R-Au intermetallics .................... 45

Figure 3. 4: Spin polarized band structures of R-Au intermetallics ............................................. 47

Figure 3.5: Total density of states of R-Au compounds by GGA, GGA+U, HF and HF+SOC

potentials ....................................................................................................................................... 48

Figure 3.6: Partial Density of State of R-Au intermetallics .......................................................... 49

Figure 3.7: Band structures of R-Au compounds ......................................................................... 54

Figure 3. 8: Seebeck coefficient of R-Au compounds. ................................................................. 56

Figure 3. 9: Thermal conductivities of R-Au compounds ............................................................ 58

Figure 3. 10: Electrical conductivities of R-Au compounds......................................................... 60

Figure 3. 11: Figure of merit of R-Au (R=Tb, Ho, Er, Tm and Yb) compounds. ........................ 62

ix

List of Tables

Table1.1: Lattice constants of cubic R-Au compounds .................................................................. 8

Table1. 2: Melting points, Bond energies using Pauling model, Dissociation energy and Standard

heats of formation of cubic R-Au compounds .............................................................................. 12

Table1. 3: Total magnetic moments of R-Au intermetallics......................................................... 15

Table1. 4: Neel temperatures (TN), Magnetic field (H), Curie temperature (θ) and per ion

magnetic moment of cubic R-Au compounds .............................................................................. 16

Table1. 5: Calculated total magnetic moments and stable ground state energies per unit cell of R-

Au using HF potential ................................................................................................................... 17

Table1. 6: Lattice constants of R-Au2 (R= La, Ce, Pr and Eu) ..................................................... 19

Table3. 1: Calculated lattice constants (Å) compared with experimental and other theoretical

results ............................................................................................................................................ 31

Table 3. 2: Calculated values of elastic constants (GPa), Bulk modulus B0 (GPa), Voigt’s shear

modulus GV, Reuss’s shear modulus GR, Hill’s shear modulus GH, Young’s modulus Y (GPa),

B/G ratio, Cauchy pressureC , Poission ratio ν, Internal strain parameter , Anisotropy

A, Lame’s first parameter λ and Lame’s second parameter μ and Shear constant C ................. 38

Table 3. 3: Calculated values of density ( ), sound velocity of transverse, longitudinal and

average sound velocity (νs, νl and νm) and Debye temperature ( ) of R-Au compounds ............ 40

Table 3. 4: Cohesive energies, total ground state energies and ground state energies of free atoms

of R and Au. .................................................................................................................................. 44

Table 3.5: Stable ground state energies per unit cell of R-Au intermetallics ............................... 51

Table 3.6: Calculated and experimental effective / total magnetic moments (µB) of R-Au

intermetallics by PBEsol, GGA+U, HF-B3LYP and HF-B3PW91 ............................................. 52

x

Table 3.7: Lattice constants, (Å) by PBEsol compared with experimental results of Orthorhombic

R-Au2 ............................................................................................................................................ 64

Table 3.8: Ground state energies calculated for Orthorhombic R-Au2 compounds ..................... 66

Table 3. 9: Magnetic moments (µB) of orthorhombic R-Au2 ....................................................... 67

Table 3.10: Values of elastic constants (GPa), Bulk modulus B0 (GPa), Shear modulus G

(GPa),Young’s modulus Y (GPa), Poission ratio ν (GPa) and B/G ratio of R-Au2 ..................... 72

Table 3.11: Values of Cauchy pressure (GPa) of Orthorhombic R-Au2 in x, y and z directions . 73

Table 3.12: The anisotropic factors A1, A2 and A3 of orthorhombic R-Au2 compounds............... 74

xi

Abstract

In this thesis the structural, elastic, mechanical, magnetic, electronic, thermodynamic and

thermoelectric properties as well as the bonding nature and cohesive energies of the cubic

intermetallics, R-Au (R= Ce, Pr, Nd, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu) and the structural,

magnetic, elastic and mechanical characteristics of the orthorhombic compounds, R-Au2 (R= La,

Ce, Pr and Eu) have been investigated by using the full potential linearized augmented plane

wave (FP-LAPW) method in the outline of the density functional theory (DFT).

The structural properties of the cubic R-Au compounds are calculated by PBE and

PBEsol potentials. The computed lattice constants are found consistent with the experimental

results. Our results show a decreasing trend in lattice constants from Ce to Lu with the exception

of the Yb that its divalency confirms. The elastic constants of these compounds are calculated.

The Cubic-elastic software and PBE-sol exchange correlation are used to determine the elastic

properties of the cubic R-Au intermetallics. The mechanical properties of the compounds like

Young modulus, Bulk modulus, Poisson ratio, Anisotropy, Shear modulus, Kleinman

parameters, Lame's coefficient, Cauchy pressure and B/G ratio are computed. The positive values

of the Cauchy pressure reveal that the bonding nature in these compounds is predominantly

metallic/ ionic. Furthermore, the large values of the B0/G ratio for these compounds show their

ductile nature and the most ductile is CeAu with the highest B0/G value of 2.82. The Poisson’s

ratio shows that these compounds are less compressible and the deviation from unity of the

isotropic ratio reveals that they are anisotropic except DyAu and NdAu.

The sound velocities for shear and longitudinal waves and Debye temperature are also

investigated on the basis of the mechanical properties of R-Au intermetallics. The electronic and

magnetic properties of the cubic R-Au intermetallics are calculated by applying Generalized

Gradient Approximation (GGA), GGA+U and hybrid density functional theory (HF). Our

computation provides best and consistent to experimental results for HF. The magnetic studies

indicate that HoAu and LuAu are non-magnetic, PrAu is ferromagnetic and CeAu, NdAu,

SmAu, GdAu, TbAu, DyAu, ErAu, TmAu, and YbAu are anti-ferromagnetic materials. The

calculated cohesive energies of R-Au intermetallics exhibit direct relation with the melting

points. Spin-dependent electronic band structure demonstrates that all these compounds are

metallic in nature.

xii

Thermoelectric properties of R-Au (R=Tb, Ho, Er, Tm, Yb) compounds are calculated

using post-DFT Boltzmann’s theory (BoltzTraP code). Thermoelectric parameters such as

Seebeck coefficient, thermal conductivity and electrical conductivity and their dependence on

chemical potential are evaluated. All the calculations are executed at a temperature of 300 K.

The highest value of Figure of merit is found for YbAu compound, equal to 0.29.

The structural, elastic and magnetic properties of orthorhombic R-Au2 (R= La, Ce, Pr and

Eu) intermetallics are calculated by applying the FP- LAPW method within the density

functional theory (DFT). The structural properties of these compounds are investigated by the

PBE and PBEsol. The calculated lattice parameters are found consistent with the experimentally

reported results. Our calculated results indicate that these compounds are stable in orthorhombic

CeCu2 type structure. Orthorhombic elastic software and PBEsol exchange correlation potential

are applied to determine the elastic properties of R-Au2 compounds. The mechanical properties

of these compounds like Young modulus, Bulk modulus, Shear modulus, Anisotropic ratio,

Poisson ratio, Cauchy pressure and B/G ratio are explored using the PBEsol to evaluate the

importance of these compounds in various types of engineering applications. One of the striking

features of the mechanical properties of these compounds is their ductile nature and all these

compounds are anisotropic in all direction except CeAu2 and the highest B/G value is for CeAu2

which is 3. The magnetic properties for single cell calculation were carried out using HF-

B3PW91 potential and double cell AFM calculations were carried out with PBE sol potential.

For HF-B3PW91 calculations we used the correct exchange potential of α = 0.15 to 0.25 and

U=2eV in all cases.

Chapter 1 Introduction

1

Chapter 1

Introduction

1.1 Intermetallics

The lanthanides are f-block elements having 4f valance electrons and comprise the

elements from La to Lu. They are also known as rare-earth elements because the scientists once

thought that these elements were present in earth's crust in very small quantities [1, 2]. They

have same electronic configurations and similar properties [3]. These metals react with nearly

all the elements in the periodic table and have extensive industrial applications. They are used as

green component in trichromatic lamps and in Micro-Electro-Mechanical Systems (MEMS)

packaging for the purpose to strengthen the bonding [4-6]. And used as alternate for zinc,

chromate, nitrate and cadmium as a non-toxic corrosion resistant materials [7].

The gold has many interesting chemical and physical properties such as very high density

(19.3g/cm3) and high melting temperature (1064.18°C), good electrical and thermal

conductivities, very high corrosion resistance, negligible magnetic moment and interesting

electronic structure [8-10] .

Intermetallics are compounds which consist of two or more than two different metals

bonded in a specific ratio having mixed ionic, covalent and metallic bonding. No other materials

have added more to the progress of man over the millennia than Intermetallics. It is not possible

to imagine a world without Intermetallics.

Their physical, chemical, magnetic, electrical, and mechanical properties are often better

than those of constituent metals [11, 12], they have appealing mechanical and thermal properties

like high tensile strength, good ductility, good corrosion resistance, high melting points,

resistance to oxidation and thermal stability [13-19]. These unique properties make them

unaltered candidates for aviation, automobile, aerospace and other high-tech applications [11,

20-22].

Lanthanide intermetallics are the first group of atoms containing 4f-orbitals [23]. They

have greatly varying crystal structures, magnetic and electrical properties[24, 25]. These striking

properties are caused by the presence of f-electrons [26]. One of the interesting subgroup of

lanthanide intermetallics is the noble metals (Cu, Ag and Au) based intermetallics. The bond

stabilities of lanthanides with these noble metals are in the order of R-Au≫ RCu> RAg (R= Ce,

Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu) [21]. The most stable compounds

Chapter 1 Introduction

2

among them are the compounds with gold and they form compounds because of their

electronegativity difference [22]. Similarly, very high relativistic effects in lanthanide-gold (R-

Au) bond is also responsible for its great stability [27, 28]. This high stability of the R-Au bond

is explained by the Pauling model as given,

2

AN Q+] B)-D(B+A)-[D(A 2

1 =B)-D(A (1.1)

Where D (A-B) represents the dissociation energy of diatomic bond, ENA is the Pauling

electronegativity of atom A, while Q is constant. This model shows that the 2.4 electronegativity

of gold is mainly due to its relativistic effect, that shows high stability of gold compounds [27].

The dissociation energies of these compounds are in the range of 290 to 335 kJ/mol with minor

irregularities [29, 30]. Their high dissociation energies are due to its greater thermal stability,

high heat of formation and high melting points [22, 30-32]. The bond between Au and R is

metallic and partly ionic in nature, in which the metallic character is 95% and ionic is only 5%

[29, 30, 33].

The R-Au compounds are synthesized by using direct reaction method at the temperature

of about 1373 K, sealed the reactants in the powder form under argon atmosphere in a boron

nitride container to prevent oxidation [34]. The X-ray diffraction confirmed the cubic CsCl type

structure for these compounds [34, 35].

These binary rare-earth with gold alloys, R-Au are generally characterized by the cubic

CsCl crystal structure (B2, Pm3m, space group no. 221) [36-38]. In the CsCl-type structure,

lanthanide (R) atoms occupy the corners and gold (Au) atom is in the body centered position of

the cubic unit cell [39]. The lattice constant values of R-Au compounds show the lanthanide

contraction, with the exception of the compounds containing Yb atom, in which the rare earth

ions are divalent [40].

These intermetallics have attracted considerable interest in recent years. The thermally

stable and good corrosion resistant nature of these materials is very useful for technologies where

high melting temperature and corrosion resistant materials are required. Such technologies are

high temperature gas turbine hardware of power plants, heat exchangers and internal combustion

engines operating at high temperatures [20-22, 41-45].

Chapter 1 Introduction

3

The lanthanide diaurides, R-Au2 (R=La, Ce, Pr and Eu) compounds have the highest

dissociation energies, very high melting points, high dissociation energies, corrosion resistant,

antiferromagnetic and high heat of formation.

These compounds were synthesized by arc-melting method in argon atmosphere and

structures were determined by XRD [46]. The phase stability of LaAu2 is 1214 Co [47], PrAu2 is

1180 Co [28, 48] and EuAu2 is 1085 Co [49]. The melting point of LaAu2 is 1487.15 K [50] and

that of CeAu2 is 1479 K [51]. The enthalpy of formation of CeAu2 at 1123 K is -71.3 ± 2.0

kJ/mole of atoms [51] and that of PrAu2 is −97.2 kJ/mol of atoms [48]. The Debye temperature

of LaAu2 is about 100 K [46].

Intermetallics are found in seven crystalline systems i.e. cubic, orthorhombic, tetragonal,

triclinic, monoclinic, hexagonal and rhombohedral. The R-Au and R-Au2 (R= La, Ce, Pr and Eu)

intermetallics have cubic CsCl and orthorhombic CeCu2- type structure respectively.

1.2 Structural properties

Structural properties shows the arrangement of atoms in space, the type of bonding, the

bond length, the number and type of electrons in a given shell and bond angles in a crystal. They

determine the properties of a material like strength, toughness, ductility, hardness, corrosion

resistance, thermal and electrical conductivity, magnetic and optical properties etc. Therefore,

structural properties are very important in the sense of their proper applications.

1.3 Mechanical properties

The mechanical properties are the relationship between stress and strain. These

properties include hardness, bending, durability, brittleness, elongation, strength, ductility etc.

They are very essential for high-tech use. Elastic constants are used for determining the

mechanical properties of the materials in terms of Poisson ratio and Young, Bulk and Shear

moduli etc. They are also useful in predicting the aging behavior of the materials [52-54]. These

constants are also associated with the thermodynamic behavior that is thermal expansion,

specific heat, melting point and Debye temperature [55, 56].

1.4 Electronic properties

These are the properties which depend on the number, type and location of electrons in an

atom particularly in valence and conduction states. Such properties are band structure, band gap,

Density of States (DOS) and charge density etc.

Chapter 1 Introduction

4

Band gap is the energy difference between the minima of conduction band and the maxima of the

valence band where no electrons are found. In direct band gap the valence band maxima and the

conduction band minima lies in the same direction otherwise indirect band gap. Conductors have

large band gap, semiconductors have small and insulators have no band gap. The Fermi level is

that energy level where the probability of finding electron is 50%. In the same way the DOS is

the electronic states per unit volume. On the basis of the electronic properties we can categorize

the materials as conductors, semiconductors and nonconductors. These properties determine the

thermal, electrical conductivities and the type of forces present in the materials. So once the

nature of material is identified, its relevant technological applications are possible.

1.5 Thermoelectric properties

Thermoelectricity is the conversion of thermal energy into electrical energy which is used

for power generation and refrigeration [57]. The conventional sources of energy are near to

depletion and have a lot of environmental concerns. Therefore, thermoelectric energy received

tremendous attention due to the process in which waste heat is directly converted into useful

electrical energy and eco-friendly nature [58-63]. The parameter used for measuring the

usefulness of a material for thermoelectric uses is the Figure of merit, given as, ZT = σS2T/k,

where k, S and σ shows thermal conductivity, Seebeck coefficient and electrical conductivity

respectively [64-68]. Greater values of Figure of merit are obtained in those substances which

have greater electrical conductivity, small thermal conductivity and high Seebeck coefficients

[69-71]. In general TE materials having a ZT value close to / more than one [72], stable at high

temperature, non-toxic and easy to synthesize are considered suitable for thermoelectric

applications [73-75].

1.6 Magnetic properties

It is the response of a material to the applied magnetic field. The materials are

categorized into diamagnetic or paramagnetic on the basis of its response to the applied magnetic

field. The paramagnetic are associated with electron spin and therefore attracted by a magnet

while the diamagnetic have no unpaired electrons therefore not attracted by a magnet. To use a

material for technology applications, the knowledge of magnetic properties is necessary. The

total magnetic moment in binary lanthanides intermetallics is due to two metallic atoms and the

interstitial regions [76]. In which the major contributor is the lanthanide atom because its 4f-

Chapter 1 Introduction

5

orbitals contain unpaired electrons, while the interstitials regions have negligible magnetic

moments [77].

1.7 Thermodynamic properties

Thermal properties are the properties of a material which is related to its conductivity of

heat. These are related to electron moment or particle vibration of the material. These properties

include Debye temperature, specific heat, thermal stability, thermal conductivity and melting

temperature etc. The knowledge of thermal properties of materials is very important in the

technologies where temperature is involved.

1.8 Literature survey of R-Au intermetallics

1.8.1 Structural properties of R-Au intermetallics

The gold can form compounds with rare earth metals in thirteen different stoichiometric

ratios i.e. 1:6, 1:5, 1:4, 14:51 , 1:3, 1:2, 7: 10, 6:7, 1:1, 5:4, 5:3, 2:1 and 7:3 for example RAu6,

RAu5, RAu4, R14Au51 ,RAu3,RAu2, R7Au10, R6Au7, RAu, R5Au4, R5Au3, R 2Au andR7Au3 [20].

Although earlier it was reported seven stoichiometries, twelve structures and eighty phases of

twelve R-Au (R = La, Ce, Pr, Nd, Gd, Tb, Dy, Ho, Er, Tm, Lu) systems [78, 79]. The 1:1

stoichiometric ratio rare earth auride (R-Au) are their important compounds. They exist in three

polymorphic forms i.e. cubic CsCl (B2), orthorhombic FeB (B27) and orthorhombic CrB (B33)

[37, 79]. The R-Au (R = La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm) (1: 1) are found in CrB

structure while the R-Au (1: 1) (R= Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm and Lu) compounds

exist in CsCl crystal structure [20].

Ionic and metallic bond is present between gold and lanthanide atoms [30, 33, 80].

Electron transfer occur from rare-earth metal to gold as electronegativity difference of these

atoms are favorable for ionic bond formation [81]. The ionic and metallic bonding in these

intermetallics can be confirmed from the fact that these compounds have very high melting

points [30-32], high dissociation energies [30, 80, 82] and high bond energies [83]. The stable

nature of these compounds is due to the large electronegativity difference (>1.13) and equiatomic

stoichiometry of gold and lanthanides [21, 31, 84]. Lanthanides in these compounds are trivalent

[80, 84, 85] except Yb which is divalent [86-88]. Its divalency is confirmed from its abnormally

high lattice constant than the other members of the series. Mostly these compounds are anti-

ferromagnetic in nature [89].

Chapter 1 Introduction

6

According to Chao et al [83], McMasters et al [37, 79] and Gschneidner et al [90] the R-

Au (1: 1) compounds exist in three polymorphic forms, Cubic CsCl (B2), Orthorhombic FeB

(B27) and Orthorhombic CrB (B33) in which the most stable phase is the cubic CsCl (B2) phase.

Kimball et a1 also confirmed that all R-Au (1: 1) compounds except CeAu, exists in CsCl type

structure and stable at room temperature [91]. Their parameter data is given in Table 1.1. The

crystal system in R-Au in cubic phase, CsCl structure and (1:1) stoichiometries are also

confirmed by X-ray diffraction. Their lattice constant are summarized in Table 1.1 which are

nearly identical [35, 92]. These intermetallics have the space group Pm3m (No. 221) [38, 79,

93], where R atoms occupy the corners and Au atom is present at the body centered position of

the cubic unit cell [39]. The unit cell structure is shown in the Fig 1.1.

Chao et al [38, 83] explored a relation between the lattice constant and the trivalent radii

of R-Au compounds. The lattice parameter decreases linearly with the atomic number of the

lanthanide element. Their findings are confirmed by Iandelli and Palenzona and also proved the

divalency of Yb in YbAu compound [94]. The lattice parameters determined by them are given

in Table 1.1

Gschneidner confirmed the divalent nature of Europium (Eu) and Ytterbium (Yb) in R-

Au compounds from their comparatively high heat of formation. In order to form compounds in

which Eu and Yb are trivalent, the extra 4f-electrons in the metals must be promoted to outer or

valence orbitals. The energy to promote the electrons is 23 kcal/g and 9 kcal/g for Europium and

Ytterbium respectively. It is this energy of promotion which plays a vital role in determining the

valence state assumed by Europium and Ytterbium in their compounds. From the magnitude of

these values it is obvious that Yb has more trivalent compounds than Europium [95].

Keeping in view the anomalous behavior of the valency of Cerium, Europium and

Ytterbium they are explored in detail [45, 86, 97, 99-107] and it is concluded that Ce is

tetravalent and Eu and Yb are divalent in (1:1) R-Au compounds. Pottgen and Johrendt [96],

Gschneiner and Eyring [97] and Boer et al [98] explain that these discrepancies are due to either

empty (4f 0), half-filled (4f7) or filled (4f14) 4f shells of Ce4+, Eu2+, and Yb2+ respectively have a

slightly better stability.

Similarly Saccone et al also worked on the valency of Ytterbium and confirmed the

divalency of Yb in Yb-Au phases containing Au in O-50 % and trivalency in Yb-Au phases

containing Au from 50- 100 %. The behavior of Yb lies between the neighboring elements

Chapter 1 Introduction

7

Thulium (Tm) and Lutetium (Lu). Similarly Europium (Eu) possess divalency in the whole

composition range [99]. Rider is the only researcher who claimed the trivalency of Yb in cubic

YbAu [100].

Saccone et al worked on Praseodymium-Gold system [99] and on Neodymium-Gold

system [93] in the composition range of Au in 0-100 % by electron probe microanalysis

(EPMA), X-ray diffraction (XRD), differential thermal analysis (DTA), scanning electron

microscopy (SEM) and optical microscopy (LOM) methods and found that the stable structure of

PrAu and NdAu is cubic and CsCl phase. They also concluded that Gold forms CsCl type

structure compounds with lanthanides except Ce and Eu. These two elements in their compounds

show some similarities in their behavior to those of group II A (Ca, Sr, and Ba) elements.

Gschneidner determined from the heat of formation that the most stable structure for

GdAu at room temperature is the CrB phase [101, 102].

Villars is the only researcher who determined the lattice constant of CeAu in the cubic

phase. However he did not work on cubic phases of TmAu, YbAu and LuAu compounds[103].

His reported lattice constants are given in Table 1.1.

Chapter 1 Introduction

8

Table1.1: Lattice constants of cubic R-Au compounds

Compd Chao McMasters et al Iandelli and

Palenzona

Villars et al Duwez et al

CeAu -------- ------- -------- 3.7000 j --------

PrAu 3.680±1e 3.68 ± 0.01f 3.680g,h,i 3.6800 j --------

NdAu 3.659±4e

3.659a

3.659 ± 0.004f

3.86b

3.659 g,h,i

--------

3.6590 j

--------

3.593k

-------

GdAu 3.593±2e

-------

3.6009±0.0008f

-------

3.601 g,h,i

3.588c

3.6609 j

--------

-------

-------

TbAu 3.576±2e

--------

3.576±0.002f

---------

3.576 g,h,i

3.59d

3.5760 j

--------

3.576 k

-------

DyAu 3.555±2e 3.555 ± 0.002f 3.555 g,h,i 3.5550 j 3.555 k

HoAu 3.541±1e 3.541±0.001f 3.541 g,h,i 3.5410 j 3.541 k

ErAu 3.527±2e 3.5346±0.0002f 3.535 g,h,i 3.5346 j 3.527 k

TmAu 3.516±3e 3.5196±0.0002f 3.520 g,h,i -------- 3.516 k

YbAu -------- 3.5840f 3.584 g,h,i -------- 3.584 k

LuAu 3.488e 3.4955±0.0002f 3.496 g,h,i -------- --------

Refa [93], Refb [99], Refc [104], Refd [105], Refe[38] ,Reff[79] ,Refg62 ,Refh64, Refi65 ,Refj[103],

Refk[83]

Chapter 1 Introduction

9

Figure1.1: Unit cell of cubic R-Au compounds.

Chapter 1 Introduction

10

1.8.2 Thermal properties of R-Au intermetallics

Cubic (1:1) R-Au compounds are the high melting and thermodynamically stable phase

among the different rare-earth and gold phases [106]. The melting points and thermal stabilities

correspond to the equiatomic stoichiometry [39, 84]. They are the highest melting points among

the known lanthanides intermetallics and high as compared to the corresponding lanthanide-

silver (RAg) and lanthanide-copper (RCu) intermetallics.

McMasters et al [79] determined their melting points which are given in Table 1.2 by

single crystal X-ray diffraction methods. The melting points increases smoothly as the atomic

number of lanthanide element increases from 1372°C to 1780°C for CeAu and LuAu

respectively, with the exception of YbAu which is 1292°C [79]. This anomaly is due to the

divalent nature of Yb in this compound. The melting temperature of these intermetallics is very

high and the CsCl to CrB transformations occur around at 100°C below their respective melting

points.

Saccone et al has contributed alot in the determination of melting points of cubic (1:1)

R-Au system and worked on DyAu [107], GdAu [104], PrAu [99], NdAu [93], TbAu [105]

HoAu, ErAu and TmAu [84] systems. Their determined results are given in Table 1.2. They

used differential thermal and X-ray diffraction analyses for the determination of melting points

and concluded that the cubic (1:1) CsCl structure of PrAu and DyAu [99] and GdAu [104,

108]are the highest melting phases.

Wang et al determined the melting points of NdAu and DyAu systems [109] using

Calculation of Phase Diagram (CALPHAD) method based on the available experimental data.

Their result of melting point of DyAu is close to that of McMasters et al but quite different i.e.

20 °C in case of NdAu.

Gingerich and Finkbeiner [82] compared their experimental dissociation energies of

CeAu, PrAu and NdAu with the calculated values based on the Pauling concept of a polar bond

and found the two values are close to each other. They also worked on dissociation energies,

standard heats of formation and bond energies of the gaseous lanthanide auride molecules CeAu,

PrAu and NdAu and predicted their stabilities using Knudsen effusion method in combination

with mass spectrometric analysis [31, 79] and are given in Table 1.2. They concluded that the

dissociation energy is related to strength of the lanthanide (R) and gold (Au) bond. Thus the high

dissociation energies mean the strong bond between R and Au atoms and are highly stable

Chapter 1 Introduction

11

compounds. The high bond energies are due to the high electronegativity difference between the

lanthanides and gold [108, 110, 111]. The lowest bond energy of YbAu is contributed by the

divalent nature of Yb.

The standard heat of formation of CeAu, PrAu, NdAu, GdAu, HoAu and LuAu were

determined experimentally by Fitzner and Kleppa [112-114]. Their results are given in Table 1.2.

They compared their results with the data calculated from Miedema Semi-empirical Model

(MSM) [115] and predicted that the Model exactly predicts overall trends in the data and

generally overestimates the negative enthalpies of formation.

Ferro et al [116] determined the heat of formation of GdAu, TbAu, DyAu, HoAu and

YbAu compounds as shown in Table 1.2 by direct reaction calorimetry method at 300K. It is

concluded that heat of formation increases as we move from CeAu to LuAu, with exception of

YbAu. The low value for YbAu is due to its divalency of Yb. The exothermicity of these alloys

increases with the increase in atomic number and decrease in ionic radii of lanthanide atom (R)

[106]. They [108] also worked on the relationship between the melting points and atomic

number of lanthanides-gold intermetallics (R-Au) and find out a regular increasing trend with

the increasing atomic number of the trivalent lanthanides. The Figure 1.2 indicates greater

deviation occur for the divalent lanthanide metals in YbAu, due to weaker bonding (divalent

nature) in these compounds.

Similarly, according to Gschneidner [101] it is assumed that greater the bonding strength

for a compound, the greater the melting point of the compound.

Chapter 1 Introduction

12

Table1. 2: Melting points, Bond energies using Pauling model, Dissociation energy and

Standard heats of formation of cubic R-Au compounds

Compd Melting point

(°C)

Dissociation

energy

(kcal/mol)

Standard heats

of formation

(kJ/mol atom)

Bond energy

(eV)

CeAu 1372c 75.0±3.5b -62.8±4.2g 3.4a, 3.3b

PrAu 1415c

1420i

72.5±5b

--------

-72.2±4.4h

---------

3.1a, 3.2b

-------

NdAu 1450c, 1470e,l 69±6b -70.263.7h 3.0a, 3.0b

GdAu 1585c,j,q

3.2a

-------

-------

-82.6 ± 2.7m

-82.0±2.0n

------

------

TbAu 1623c

1590n,o

-------

-------

-82.0±2.0n

-86.5 ± 2.2r

3.2a

------

DyAu 1658c

1660l,p,s

-------

-------

-82.0±3.0n

-83.4 ± 1.7r

3.0a

------

HoAu 1698c 1700k

-------

-------

-------

-86.6±3.6m

-83.0±4.0n

3.0a

------

ErAu 1710c,k,s ------- -90.2 ± 2.0r 3.0a

TmAu 1720c,k,s ------- -92.8 ± 1.9r 2.9a

YbAu 1292c,d ------- -75.0±2.0n 2.7a

LuAu 1780c 78.636f -86.0 ± 2.1m 3.3a

Refa [82], Refb [31], Refc [79], Refd [117],Refe [93], Reff [97], Refg [112], Refh [113], Refi [99],

Refj [104], Refk [84], Refl [109], Refm[114], Refn [116], Refo [105], Refp [107], Refq [108], Refr

[34], Refs [34]

Chapter 1 Introduction

13

CeAu PrAu NdAu GdAu TbAu DyAu HoAu ErAu TmAu YbAu LuAu

1250

1300

1350

1400

1450

1500

1550

1600

1650

1700

1750

1800

1850

Melti

ng

poin

ts (

Co)

Compounds

Melting points

Figure1. 2: Melting temperatures of the R-Au compounds

Chapter 1 Introduction

14

1.8.3 Magnetic properties of R-Au intermetallics

Very limited work on the magnetic properties of these compounds has been reported in

literature.

Kissel and Wallace [118] has great contribution in determining the total and per

lanthanide ion (R+3) magnetic moments and also the Neel temperature of GdAu, TbAu, DyAu

ErAu, TmAu and YbAu compounds which are given in Table 1.4. According to them the

compounds GdAu, TbAu, DyAu, ErAu, TmAu and YbAu are antiferromagnetic while HoAu is

paramagnetic in nature as shown in Table 1.3. They determined the magnetic properties of these

compounds at the temperature of 2°K to 300°K.

Buschow [119] reported the magnetic moments of GdAu, TbAu, DyAu, HoAu, ErAu and

TmAu compounds as given in Table 1.4. According to him the magnetic moments in these

compounds are almost due to localized 4f electrons in lanthanide atoms. Durand[120]

determined Curie temperature, antiferromagnetic nature and magnetic moment of GdAu by splat

cooling process as listed in Table 1.4.

No experimental work on magnetic moments of CeAu, PrAu, NdAu and LuAu compounds

reported yet.

Recently Ahmad et al [121] computed the magnetic properties of R-Au intermetallics using

PBEsol, GGA+U, HF (B3LYP) and HF (B3PW91) potentials within the outline of DFT. The

calculated effective magnetic moments and the magnetic nature of all these intermetallics are

presented in Table 1.5. the calculated ground state energies of these compounds indicates that

CeAu, NdAu, GdAu, TbAu, DyAu, ErAu, TmAu and YbAu are stable in the anti-ferromagnetic,

PrAu is in ferromagnetic and HoAu and LuAu in nonmagnetic states.

Chapter 1 Introduction

15

Table1. 3: Total magnetic moments of R-Au intermetallics

Compound Magnetic moment (µB) Magnetic nature

CeAu -------- ------

PrAu -------- ------

NdAu -------- ------

GdAu 7.29 a, b, f AFM a, c, g

TbAu 9.54 a, b AFM a, d

DyAu 10.22 a, b AFM a, g

HoAu 10.50 a, b, 10.1f PM a

ErAu 9.42 a, b, 9.6 f AFM a, d

TmAu 7.32 a, b AFM a, d

YbAu 0.81a AFM a, e

LuAu -------- --------

Refa 98, Refb 99, Refc 101, Refd 102, Refe 103, Reff 104, Refg 105

Chapter 1 Introduction

16

Table1. 4: Neel temperatures (TN), Magnetic field (H), Curie temperature (θ) and per ion

magnetic moment of cubic R-Au compounds

Ref a [118], Ref b [119]

Compd Neel Temp, (K)

AFM to PM

Magnetic field,

(kOe)

Paramagnetic

Curie Temp, (K)

Effective per ion

moment (µB)

CeAu ------ ------- ------- -------

PrAu ------- ------- ------- -------

NdAu -------- ------- ------- -------

GdAu 31a 19.3a 29a 7.92a

TbAu 40a 19.3a 23a 9.54a

DyAu 24-34b 2.2b 7b 10.9b

HoAu 10a 2.3a 0a 10.6a

ErAu 13-19b 2.0b -4b 9.6b

TmAu 8-19b 19.3b -5b 7.32b

YbAu ------- ------- -150a 4.5a

LuAu ------- ------- ------- --------

Chapter 1 Introduction

17

Table1. 5: Calculated total magnetic moments and stable ground state energies per unit cell

of R-Au using HF potential

Compds E Para (Ry) E FM (Ry) E AFM (Ry) Total magnetic

moments (µB)a

Magnetic

nature a

CeAu -55809.429208 -55806.430843 -55809.4308 2.96 AFM

PrAu -56564.050659 -56564.106317 -56564.0509 2.58 FM

NdAu -57338.368179 -57338.492453 -57356.0517 3.9 AFM

SmAu -58947.224525 -58947.574402 -58961.4513 6.21 AFM

GdAu -60657.027263 -60638.830690 -60639.1927 7.30 AFM

TbAu -61515.356380 -61515.755411 -61518.0633 7.93 AFM

DyAu -62413.873565 -62414.147566 -62414.6129 7.2 AFM

HoAu -63334.330012 -63334.310010 -63317.4075 8.4 NM

ErAu -64276.534961 -64276.533794 -64276.5352 5.0 AFM

TmAu -65241.123356 -65241.142392 -65241.1442 3.92 AFM

YbAu -66228.501665 -66228.501460 -66228.5017 2.6 AFM

LuAu -67238.771436 -67238.769915 -67238.7706 0.0011 NM

Refa

Chapter 1 Introduction

18

1.8.4 Electronic and Thermoelectric properties of R-Au

According to our literature study no experimental or theoretical work on electronic and

thermoelectric properties has been reported to date on R-Au (R= Ce, Pr, Nd, Gd, Tb, Dy, Ho, Er,

Tm, Yb and Lu) alloys.

1.9 R-Au2 (R=La, Ce, Pr, Eu) compounds

1.9.1 Structural study of R-Au2 compounds

Limited study is available in literature on the structural properties of orthorhombic R-Au2

(R= La, Ce, Pr and Eu) compounds. Yurong Wu et al determined lattice constant of CeAu2 and

PrAu2 compounds theoretically by modified analytical embedded atom method (EAM) [39].

Palenzona studied the crystal structure of EuAu2 compound by using differential thermal, X-ray

and metallographic analyses and determined its structure and lattice constants [122]. Iandelli et

al determine experimentally the lattice constants of LaAu2, CeAu2 and EuAu2 compounds [123].

McMasters et al investigated crystal structure of LaAu2, CeAu2, PrAu2 and EuAu2 compounds

applying both powder and single crystal X-ray diffraction methods [124].

Similarly Saccone et al reported lattice constants for PrAu2 compound by differential thermal

analysis, X-ray diffraction, optical and scanning electron microscopy and electron probe

microanalysis [99]. All of them confirmed orthorhombic CeCu2-type structure for these

compounds. Their findings of lattice constants are very close to each other and are listed in Table

1.6.

Chapter 1 Introduction

19

Table1. 6: Lattice constants of R-Au2 (R= La, Ce, Pr and Eu)

Compd Yurong Wu

et ala

Palenzonab Iandelli

et alc

McMasters

et ald

Sacconee

LaAu2 ----------

----------

----------

----------

----------

----------

a=4.700

b=7.295

c=8.155

a=4.700

b=7.295

c= 8.155

----------

----------

----------

CeAu2 a=4.655

b=6.958

c=7.923

----------

----------

----------

a=4.528

b=7.203

c=8.068

a=4.528

b=7.203

c= 8.068

----------

----------

----------

PrAu2 a=5.018

b=7.364

c=8.587

----------

----------

----------

----------

----------

----------

a=4.672

b=7.040

c= 8.178

a=4.672

b=7.040

c=8.178

EuAu2 ----------

----------

----------

a=4.715

b=7.331

c=8.171

a=4.670

b=7.330

c=8.140

a= 4.67

b=7.330

a= 8.140

----------

----------

----------

Refa [39] Refb [122] Refc [123] Refd [124] Refe [99]

Chapter 1 Introduction

20

Present work

This thesis is arranged as, Chapter 1 which contains introduction and literature review

and Chapter 2 consists of information about DFT, Wien2k code, orthorhombic elastic package,

BoltzTraP Package and Cubic-elastic package. The Chapter 3rd focuses on our contributions

which include results of our density functional theory computation of structural, elastic,

mechanical and thermoelectric properties as well as magnetic calculations of cubic R-Au

compounds and structural, magnetic and elastic properties of orthorhombic R-Au2 compounds.

Furthermore, the results are also discussed in the same chapter. The overall work is summarized

in the last section under the title of conclusions.

Chapter 2 Theoretical Background and Computational Methods

21

Chapter 2

Theoretical Background and Computational Methods

2.1 Density Functional Theory “DFT”

DFT is an effective computational quantum mechanical modeling method used to explore

the electronic structure of many body systems. Density functional theory has strong predictive role

and successful results with broad applicability in science. It is the most useful and reliable

method in computing the ground state electronic, magnetic, elastic, thermoelectric, optical and

other physical properties of insulators, conductors, metalloids and superconductors [125].

Nowadays DFT has gained tremendous popularity in material science community, chemists,

physicists, engineers and biologists due to its reliable results, low computational costs, less time

consuming, easy access, closer to experimental ones and wide applicability. The whole building

block of DFT lies on Hohenberg and W. Kohn-Sham’s (KS) theorems which explain the ground

state physical properties of many body systems [126].

2.2 Kohn and Sham (KS) Equation

The basic principle of Kohn-Sham theory is to change many body problems of the

interacting electrons in an atom, into a non interacting electrons system [127]. The kinetic

energies (KE) of these non interacting electrons are calculated by applying wave function of the

single particle .

i iis n |2/1| 2

(2.1)

where Ts is the kinetic energy of a non interacting electrons of the atom. The electronic density

of this non interacting Kohan-Sham system is calculated by:

2

rrn i (2.2)

The ground state energy of HK in Kohan-Sham formula is given by:

ndrdrrr

rnrndrrnrVnn xcextsKS

/

/

/

2

1 (2.3)

The lowering of density n(r) is attained by resolving the single particle

Schrodinger wave Equation as:

rrrrV iiieff

2

2

1

(2.4)

Chapter 2 Theoretical Background and Computational Methods

22

where i is the “ith” Kohn and Sham Eigenvalue as

rVdr

rr

rnrVrV xcexteff

/

/ (2.5)

Where Vxc is given by

The exchange-correlation “ Exc[n(r)]”, energy is the total of correlation and exchange energies

which is measured by

N

iiir rrn

1

* (2.7)

where is wave functions, which indicates the smallest energy solution of Kohn and Sham

equation,

rH iirKS n

(2.8)

2.3 Exchange Correlation Potentials

Kohn and Sham (KS) Equations fails to provide details of the exchange-correlation

functional (Exc). To overcome this problem and to take advantage of the basic Kohn and Sham

formulae one has to approximate the exchange correlation functional. There are a number of

approximations, which are Local Density Approximation “LDA”, Generalized Gradient

Approximation “GGA”, Generalized Gradient Approximation with Spin Orbit Coupling

“GGA+SOC”, modified Beck-Jhonson “mBJ”, GGA with Hubbard U “GGA+U” etc.

2.3.1 Local Density Approximation “LDA”

Local Density Approximation [127] is very commonly used exchange correlation

functional. It is simple and a practical approximation to calculate different properties of solids. It

gives an accuracy of about 10-20 % in ionization energy and cohesive energies of atoms. The

LDA calculate bond lengths and lattice constants of a compound with about 1 % accuracy. But

the LDA underestimate the surface energies and band gaps. In Local Density Approximation

potential, exchange correlation is calculated by applying the uniform electron gas relating to the

local density n(r),

rn

rnV xc

xc

(2.6)

Chapter 2 Theoretical Background and Computational Methods

23

rdrnrnn LDAxc

LDAxc

3000 (2.9)

where rnE LDAxc is the exchange-correlation energy density of homogeneous gas at the density

n(r). The exchange and correlation terms constitute the exchange and correlation energy using

the relation

cxxc (2.10)

Failure of Local Density Approximation is that it consider uniform density at each point

[128].

2.3.2 Generalized Gradient Approximation (GGA)

In many cases Local Density Approximation fails to give good results, because it treats

electrons locally and it does not consider the interaction of electrons with each other. To

overcome this short coming, the gradient correction is considered where the Exc is a function of

electron density and its gradient. Due to this reason this approximation is called Generalized

Gradient Approximation (GGA). This Approximation is given by the equation,

drrrrr xcGGAxc (2.11)

GGA calculations are generally considered better than LDA. The most common

potentials (flavors) of GGA used in DFT are given below:

(i) B88; Axel Becke exchange functional [129].

(ii) PBE96; Perdew Burke Ernzerh of exchange correlation functional [128].

(iii) LYP88; Lee Yang Parr’s correlation functional [130].

(iv) PW86; Perdew’s correlation functional [131, 132].

(v) PW91; Perdew and Wang correlation functional [133].

(vi) EV; Engel Vosko correlation functional [134].

(vii) Wu C; Wu Cohen density gradient functional [135].

The most commonly used and reliable GGAs are LYP88 [130] and PBE96 [128, 136].

Generalized Gradient Approximation is considered best than Local Density Approximation for

the systems having the charge density slowly changing. On the basis of ground state properties of

smaller atoms, molecules and solids the Generalized Gradient Approximation results are good as

compared to Local Density Approximation. Beside this it provides good results almost for all

Chapter 2 Theoretical Background and Computational Methods

24

kinds of chemical bonding. Due to the above facts it is very common in computational chemistry

and physics.

2.3.3 Generalized Gradient Approximation with Hubbard (GGA+U)

GGA and LDA gives good results in many cases but not so effective to treat the highly

correlated systems with d or f states compounds. They do not localize the d or f orbitals

electrons, so DFT+U approach is introduced to solve this problem. The approach comes from the

scheme called “Hubbard model Hamiltonian”. The scheme Hubbard model Hamiltonian is

utilized for studying structural, magnetic and electronic properties of very high correlated

electron system [137].

The short coming of Density Functional Theory can be solved by various techniques, the

important one is the self interaction of electrons (SIC) concept. The general equations for this

technique is given as under,

UGGAUGGA

(2.12)

and

,

,2

2 m

mU nNJU

(2.13)

In which “U” is the Hubbard Columbic potential “J” is the exchange-interaction “N” is the total

number of electrons “m” and “δ” are the orbitals occupancy with spin [138].

2.4 Full Potential Linearized Augmented Plane Wave (FP-LAPW) method

The Full Potential Linearized Augmented Plane Wave (FP-LAPW) method [139, 140] is

highly reliable tool for solving Kohn and Sham equations for the electronic structure of

materials. It is based on the Augmented Plane Wave (APW) method [141], where the space is

divided into two regions, interstitial region and Muffin-tin (MT) spheres. The electrons which

are at a greater distance from the nucleus are loosely bonded and act like free electrons are

described by plane waves (interstitial region) and the non-overlapping spheres (MT region) are

described by spherical harmonic radial functions. The augmented plane wave method cannot

consider the Eigen states exactly, which makes Augmented Plane Wave method inadequate for

resolving density functional equations. For solution of this problem, Andersen [139] made

changes in Augmented Plane Wave method and incorporated Linearized Augmented Plane Wave

Chapter 2 Theoretical Background and Computational Methods

25

(LAPW) method. Due to the flexible basis the Linearized Augmented Plane Wave method

properly explains the eigenstates and produces the eigenenergies with single diagonalization.

The core electrons the atom does not contribute to the bonding as they are tightly bound

to the nucleus. The core electrons in an atom are localized in the Muffin-tin (MT) regions. The

valence electrons are leaking out of the Muffin-tin (MT) spheres are valence electrons and takes

part in bonding. In some atoms there are states which lie neither in the core region, nor in the

valence region are called semi-core states. The angular momentum number of semi-core state is

the same but its principle quantum number “n” is lower to that of valence states. The semi core

states are found in f and d block elements.

The drawback of Linearized Augmented Plane Wave method is the treatment of

electrons in the semi core states. This problem was resolved by Singh [142] by incorporating

local orbitals implemented in the linearized plane wave (LAPW+lo) method. In (LAPW+lo)

method the extra basis function is localized in the Muffin-tin (MT) spheres.

In our calculations the muffin-tin (MT) regions radii have been taken as 2.50 a.u. for

Lanthanide (R) as well as for Gold (Au) atoms. For good convergence, the plane waves RMTKmax

is set at 7. The RMT represent the radius of the atom in the unit cell while Kmax is for the

maximum value of k-vector in the plane wave expansion. For the valence wave function inside a

muffin-tin (MT) spheres the maximum value of angular momentum, lmax = 10 is taken. In

interstitial region the charge density in Fourier expansion Gmax is selected up to 12. For elastic

properties calculation k-points were taken as 5000 with a dense k-mesh of 130 k-points in the

irreducible wedge of the Brillouin zone [143].

2.5 Cubic-elastic Package

Different methods are present in the literature for computing the elastic constants of

cubic structure substances [144-146]. In this work we used Cubic Elastic Package [146] for the

computation of elastic properties of cubic R-Au compounds and Orthorhombic-elastic software

[147] for R-Au2 compounds. To get consistent results, the energy approach [148] is applied

through WIEN2k code [149]. The elastic constants “Cij” are computed by using small strain “σ”

to the solids. The change in internal energy of the system is expanded in the form of Taylor

series and is represented by the following equation:

3

6

,,000

2

10,, i

jijii OCVVPVV

ji (2.14)

Chapter 2 Theoretical Background and Computational Methods

26

where V0 represent the volume, P(V0) represent the pressure of the undistorted lattice at the

volume V0 and Cij are taken for the elastic constants. For simplifying Equation 2.14 it is

important to neglect the higher power term O [εi3]. By utilizing Voigt notations, xx is substituted

for 1, yy for 2, zz for 3, yz for 4, zx for 5 and xy for 6 and considering additional symmetries

imposed by the cubic crystal symmetry, the number of elastic constants are decreased. Only three

elastic constants i.e. C11, C12 and C44 are left for a cubic crystal structure. In matrix notations the

Taylor expansion of the cubic elastic constants may be written as below:

44

44

44

111212

121112

121211

00000

00000

00000

000

000

000

C

C

C

CCC

CCC

CCC

C

(2.15)

Bulk modulus “B0” is correlated to elastic constants “C11 and C12” by the following relation

[55],

3

2 12110

CC

(2.16)

The original cubic system can be distorted by applying the deformation matrix D. The following

deformation matrices are applied to calculate elastic constants C11, C12 and C44 [150] .

21

100

010

001

orthoD (2.17)

100

010

001

cubicD

(2.18)

Chapter 2 Theoretical Background and Computational Methods

27

21

100

01

01

monocD

(2.19)

By considering the second order derivative of the energy of the orthorhombic

distortional deformation “Dortho”, volumetric cubic deformation “Dcubic” and monoclinic

distortional deformations “Dmonoc”, the values of elastic constants C11, C12 and C44 may be

evaluated. The second order derivative of energy for “Dortho” is:

121102

2

2 CCVd

d

(2.20)

The second order derivative of energy for “Dcubic” is:

2

2

d

d 12110 23 CCV

(2.21)

and the second order derivative of energy for “Dmonoc” is:

4402

2

4 CVd

d

(2.22)

Besides elastic constants, Young’s modulus, Voigt-shear modulus, Shear constant, Reuss

shear modulus, Hill shear modulus, Poisson ratio, Cauchy pressure, Lame coefficients, Kleinman

parameter and Anisotropy constant are also computed to determined the mechanical properties

and elastic stabilities of the materials under study.

2.6 BoltzTraP Package

The Boltzmann’s equation defines the change of carrier distribution functions caused by

lattice phonon scattering, external fields or different types of defects scattering. For

thermoelectric properties Boltzmann's transport equations are used, which are compiled in

BoltzTraP code by Madsen and Singh [151-154]. Transport properties like Seebeck coefficients

and electrical conductivity are determined by using time relaxation approximation concept. This

code first expands the band energies by utilizing a smooth Fourier expansion while the symmetry

of space group remained unchanged. It then determines the transport properties with some

analytical illustration of the bands build on Boltzmann transport theorem. By using computers,

Chapter 2 Theoretical Background and Computational Methods

28

this theory is helpful in calculating thermoelectric transport coefficients. This code is useful in

calculating the transport properties of high temperature intermetallics and super conductor’s

thermoelectric materials [152-154]. The high values of Seebeck coefficient and electrical

conductivity and small value of thermal conductivity are good for thermoelectric efficiency [154,

155]. This correlation is known as thermoelectric “Figure of Merit” (ZT), and is given by the

relation as [71].

Tk

SZT

2

(2.23)

Here “σ” denotes electrical conductivity, “S” represents Seebeck coefficient, “T” is for

temperature, “k” is thermal conductivity and “S2σ” is known as thermoelectric power.

2.7 WIEN2k code

The WIEN2k code is a package written in FORTRAN 90 language, first developed by P.

Blaha and Karlhienz Schwarz in1990 [148]. This code is supported by Linux operating system

and is used for calculating ground state physical properties of solids by using Density Functional

Theory. The code is based on full potential linearized augmented plane wave method to solve the

Kohn-Sham equation. Numerous improvements have been made in this code. The other versions

are WIEN93, WIEN95 and WIEN97. The WIEN2k code has two parts. In the first part

initialization is done, which determine Muffin-tin (MT) spheres overlapping, identify symmetry

operations, create structure, generate k-mesh in the Brillouin Zone (BZ) and to predict electronic

density. In the second part the self consistency cycle is executed iteratively where Kohn- Sham

equations are applied to obtain convergency and calculate the output parameter (new density,

energies, stresses, forces etc). The WIEN2k code is applied for a prediction of a large number of

properties of a material such as band structures, magnetic nature, thermoelectric properties,

density of states, X-ray spectra, charge density, electron density and electric field gradient etc. In

the present work WIEN2k code which is based on FP-LAPW method is used for computation.

Chapter 3 Results and Discussions

29

Chapter 3

Results and Discussion

Different physical properties like structural, elastic, thermoelectric, electronic and

magnetic properties of cubic (CsCl) R-Au (R= Ce, Pr, Nd, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu)

intermetallics and structural, elastic and magnetic properties of orthorhombic (CeCu2) R-Au2 (R=

La, Ce, Pr, Eu) compounds are computed by using the DFT based theoretical tools. The

exchange correlation potentials are treated with the LDA, GGA, GGA+U.

3.1 Rare-earth monoaurides (R-Au)

3.1.1 Structural Properties of rare-earth monoaurides R-Au

Structural properties of R-Au intermetallics are computed by evaluating the optimized

total energy of the unit cell of each compound with respect to volume by using Birch–

Murnaghan equation of state [156]. The lattice constants for all the intermetallics are obtained by

GGA-PBEsol, GGA-PBE and GGA with Hubbard U exchange correlation potentials. The

computed results are compared with the available experimental data in Table 3.1 and depicted in

Fig.3.1. It is clear from the data presented in the table as well as in Fig. 3.1 that PBE over

estimates and PBEsol under estimate the lattice constants of the compounds. As these are

strongly correlated systems, therefore the application of U is effective for the localization of

electrons and hence we obtain closer values for the lattice constants. The table reveals that as we

move from CeAu to LuAu, the lattice constant of the crystal decreases. This decrease in lattice

constants is caused by lanthanide contraction. This contraction is caused by the poor shielding

effect of 4f-electrons, with the exception of Ytterbium.

Our theoretical calculations as well as experimental results presented in Fig.3.1 clearly

demonstrate unusual increase in the lattice parameter of YbAu, unlike other lanthanides. This

unique behavior of the Yb is because of its divalent nature as compared to the other lanthanides,

where they are trivalent. As the radius of the divalent is larger than trivalent, therefore its overall

unit cell size is also larger. The divalency of Yb is due to the fact that it attains completely filled

configuration of 4f14 by losing 6s2 electrons. The completely filled configuration is more stable

than partially filled configuration. In the divalent state, charge is smaller and hence the force of

attraction is also smaller which naturally results in low melting point, least ionic character and

smaller cohesive energy of YbAu [157]. The figure 3.1 also demonstrates comparison between

different theoretical approaches with the experimental measurements. It is also obvious that PBE

Chapter 3 Results and Discussions

30

shows over estimation and PBEsol shows under estimation for the lattice constant of all the

compounds. The data presented in Table 3.1 clearly indicates that for most of the compounds

under study the PBEsol provides much more reasonable theoretical results, closer to the

experimental results, as compared to the previous theoretical results as well as our PBE

calculations.

Chapter 3 Results and Discussions

31

Table3. 1: Calculated lattice constants (Å) compared with experimental and other

theoretical results

aRef.[37]bRef.[39]cRef.[38]dRef.[158]

Compd PBEsol PBE Other U Expt

CeAu 3.66 3.75 3.59b 3.6777 3.70b

PrAu 3.68 3.69 3.75b 3.6868 3.68a

NdAu 3.62 3.70 3.72b 3.6984 3.66a

GdAu 3.57 3.64 3.63b 3.6250 3.59c

TbAu 3.55 3.62 3.62b 3.6110 3.58a

DyAu 3.55 3.59 3.56b 3.5862 3.56a

HoAu 3.53 3.59 3.58b 3.5510 3.54a

ErAu 3.51 3.59 3.57b 3.5674 3.53a

TmAu 3.50 3.54 - 3.5459 3.52c

YbAu 5.53 3.59 - 3.5831 3.58a

LuAu 3.47 3.52 3.53d 3.5173 3.49c

Chapter 3 Results and Discussions

32

CeAu PrAu NdAuGdAuTbAu DyAuHoAu ErAuTmAuYbAuLuAu3.45

3.50

3.55

3.60

3.65

3.70

3.75

Latt

ice c

on

sta

nt

Compounds

BPEsol PBE Expt U

Figure3. 1: Comparison of the calculated lattice constants by PBEsol, PBE and Hubbard U

with experimental values

Chapter 3 Results and Discussions

33

3.1.2 Elastic and mechanical properties of R-Au

The elastic properties of a material explain that how a material undergoes stress

deformation and recovers to its original form after stress is released. Elastic properties are

essential because they provide important information about the mechanical properties of a

material such as structural stability, type of bonding, hardness, and ductility etc [56]. From

elastic constants we can also determined the thermodynamic properties of a material like,

thermal expansion, specific heat, Debye temperature and Gruneisen parameter. Therefore, the

understanding of elastic constants for most of the materials especially engineering materials is

very essential for their useful applications linked to their mechanical properties such as

thermoelastic stress, load deflection, sound velocities, internal strain and fracture toughness etc.

[159-161]. The independent elastic constants for cubic crystal are three, i.e.C11, C12 and C44 [150,

162, 163]. The stability for a cubic crystal structure is easily inferred by the well known Born

criteria (C11 - C12> 0, C44> 0, C11 + 2C12> 0 and C12< B < C11) [164].

The elastic constants in the table are used to find out the mechanical properties, like

Young modulus (Y), Bulk modulus (B0), Poisson ratio (ν), Shear modulus (G) and Anisotropic

ratio (A) as these properties are crucial for the industrial applications of engineering materials

[165, 166]. These parameters are calculated using PBEsol to evaluate their possible high-tech

engineering applications. The calculated results are presented in Table 3.2.

One of the most important parameters of the engineering materials is the bulk modulus.

Bulk modulus (Bo) is the measure of hardness of the material. It is computed from the elastic

constants using the following relation:

3

2 1211 CCBO

The Bulk modulus is also used for the measurement of average bond strength because it has a

strong relation with the cohesive or binding energy of atoms in a crystal [167]. The calculated

values of the bulk moduli for R-Au compounds are given in Table 3.2. These values clearly

indicate that the bond nature of these intermetallics is ionic/metallic. The large values of these

compounds also confirm that with high bulk moduli they have high melting points.

Another important mechanical property is the average shear modulus, GH. It is the

measure of resistance of a material to the reversible deformations by applying stress [159, 168].

The shear modulus G accurately determines the hardness of a material. The Hill average shear

(3.1)

Chapter 3 Results and Discussions

34

modulus G [168] is the arithmetic average of Voigt (GV) [150] and Reuss (GR) values, which

may be given in terms of elastic constants,

1211443

5

1CCCGV

(3.2)

121144

441211

34

5

CCC

CCCGR

(3.3)

2RV

H

GGG

(3.4)

The computed values of GH are provided in Table 3.2. The table reveals that LuAu

exhibits the largest value of GH (36.046 GPa) being the hardest of all the compounds, while

DyAu is the least hard with lowest value of shear modulus (21.488 GPa). This point is also

supported by their melting points. The remaining alloys lie between these two values.

Another similar mechanical property that measures the stiffness of an elastic material is

called Young’s modulus(Y). The compound will be stiffer when the value of Y is greater. The

Young’s modulus for the compounds is calculated by using the Voigt shear modulus, GV and

bulk modulus, B0 values in the relation:

VO

VO

GB

GBY

3

9

(3.5)

The Young modulus, bulk modulus and shear modulus values are shown in Table 3.2.The

intermediate values of Young modulus of these materials shows that these materials are

intermediate in stiffness [161]. The table shows that all these compounds have greater Young

modulus values as compared to bulk modulus and shear modulus values, so these materials are

hard to break.

The ratio of Bulk modulus to Shear modulus (B0/G) is an important parameter which

explains the ductile/brittle nature of the compound. The material is considered ductile if this ratio

is larger than 1.75, otherwise it is brittle [165]. The brittle into ductile changing in intermetallics

by first principles computations has been represented earlier [35, 169] and verified that the

higher value of B0/G, for a material means its ductile nature[35, 165]. The B0/G ratio for all the

compounds is given in Table 3.2 which confirms that these values are larger than 1.75 so all the

compounds are ductile. Although the B0/G value for most of these compounds is quite large, but

Chapter 3 Results and Discussions

35

among these compounds the highest value of 2.82 observed for CeAu confirms that this

compound is highly ductile in nature.

Cauchy pressure is also an important physical property applied to explain the nature of

bonding in a material [35, 170]. This parameter is calculated by applying the following equation.

C= C12 C44

(3.6)

The computed values of the Cauchy pressures are provided in Table 3.2. The positive

values of Cauchy pressure reveals ionic [159, 162, 171] and metallic [35, 167, 170] nature of

bonding, while the negative value of this parameter indicates covalent bonding. Similarly, a

material with larger negative Cauchy pressure have stronger brittle behavior [172]. The table

clearly indicates that all the compounds under consideration have positive values of Cauchy

pressure and are therefore ionic/ metallic and ductile in nature. The most ionic/metallic

compound among the compounds under study is CeAu and the least ionic/metallic compound is

LuAu.

Poisson’s ratio (ν) can be defined as the ratio of lateral to longitudinal strain in uniaxial

tensile stress. The Poisson’s ratio value can predict the ability of compressibility of a material. Its

value ranges from 0 to 1/2 and is around 0.3 for most of the materials [173]. Low value means

the material is compressible while ν= 1/2 means the material is incompressible, i.e. ν→1/2

means the material tends to incompressible [174, 175]. Its volume remains unchanged no matter

how it is deformed. If ν = 0, then stretching a specimen causes no lateral contraction. The

Poisson’s ratio for a material is calculated by using the relation,

OB

Y

62

1 (3.7)

The calculated values of the Poisson ratio for the compounds under study are provided in

Table 3.2. The table indicates that the value ranges between 0.266 to 0.34 which demonstrates

that these compounds are difficult to compress and resist against external stress. Poisson ratio (ν)

also describes the nature of bonding. The central forces in a material have lower limit of ν is 0.25

and the upper limit is 0.50 [161]. The Table 3.2 indicates that the Poisson ratio values for these

compounds falls within this limit and this indicates that the inter atomic forces present in these

intermetallics are central forces [161].

Anisotropic ratio A, is an important parameter and is the measure of value of anisotropy in

the solid crystal. For the isotropic system, it is equal to one and any deviation from unity

Chapter 3 Results and Discussions

36

provides the magnitude of anisotropy. The elastic anisotropy (A) of a crystal has key importance

in engineering materials for their applications because it is linked with the probability of the

induced micro cracks in a material [161, 176]. The anisotropic ratio is computed from the elastic

constants by the equation,

(3.8)

The calculated anisotropic values in Table 3.2 reveals that the anisotropic ratio for most of

the R-Au compounds deviate from 1. The deviation from unity confirms that these intermetallics

are elastically anisotropic and various properties are different in different directions. Anisotropic

ratio for DyAu and NdAu compounds is almost unity, which demonstrates the isotropic nature of

these compounds. These two compounds also satisfy the other conditions for isotropy that is

λ=C12, μ=C44, and ν must be –1 ≤ ν ≤ ½ [177].

Kleinman [144] introduced an important mechanical property called internal strain

parameter , which explains relative tendency of bond bending to bond stretching. Its value

generally lies between 0 and 1, where 0 shows minimum bond bending or bond angle distortion,

and 1 shows minimum bond stretching [162]. The Kleinman’s parameter is linked to the cubic

elastic constants by the mathematical expression:

1211

1211

27

8

CC

CC

(3.9)

Our calculated values of the Kleinman’s parameters for PrAu, NdAu, GdAu, HoAu,

ErAu, YbAu, TmAu, YbAu, and LuAu compounds, lie between 0.5 to 0.9 and hence bond

bending is dominated in these compounds, while the very small value of this parameter for HoAu

show that the bond stretching is dominated in this compound.

Another mechanical property that determines the isotropic nature of a material is called

Lame’s constants (λ, μ). Any material is isotropic when λ=C12 and μ= C . Lame constant is

calculated from Young modulus and Poisson ratio by applying the relation,

211

Y

and

12 (3.10)

The equation shows that the higher value of Young modulus means greater the value of

Lame coefficient. The calculated values of Lame coefficient for R-Au compounds are given in

Chapter 3 Results and Discussions

37

Table 3.2. From the table it is clear that only NdAu and DyAu compounds are isotropic in nature

which is consistent with our previous results.

Shear modulus ( C ), which is also known as tetragonal shear modulus, is very essential

property of the material that determine the dynamical stability of a material and may be

determined as:

1211

2

1CCC

(3.11)

The dynamic stability of a material needs that C > 0, and negative value of C shows

the instabilities with respect to the tetragonal distortion as reported in experimental work [178].

The computed values for the shear constants of R-Au alloys under study are shown in the Table

3.2. The calculated positive values of R-Au compounds in the table show the mechanical

stability of these compounds.

Chapter 3 Results and Discussions

38

Table 3. 2: Calculated values of elastic constants (GPa), Bulk modulus B0 (GPa), Voigt’s

shear modulus GV, Reuss’s shear modulus GR, Hill’s shear modulus GH, Young’s modulus Y

(GPa), B/G ratio, Cauchy pressureC , Poission ratio ν, Internal strain parameter ,

Anisotropy A, Lame’s first parameter λ and Lame’s second parameter μ and Shear

constant C

Compd CeAu PrAu NdAu GdAu TbAu DyAu HoAu ErAu TmAu YbAu LuAu

C11 98.078 89.408 84.700 87.852 83.500 84.451 83.936 81.732 91.207 89.901 107.596

C12 55.798 35.916 39.500 45.921 34.751 41.445 39.902 39.366 51.758 29.418 44.436

C44 27.519 22.219 23.400 27.453 21.281 21.479 24.834 31.024 30.800 23.578 39.374

Bo 69.80 53.747 54.567 59.898 51.001 55.780 54.580 53.488 64.908 49.579 65.489

Gv 24.967 24.030 23.080 24.858 22.518 21.489 23.707 27.088 26.370 26.243 36.256

GR 24.555 23.833 23.073 24.429 22.419 21.489 23.625 26.162 25.151 25.857 35.836

GH 24.761 23.931 23.077 24.644 22.469 21.489 23.666 26.625 25.760 26.050 36.046

Y 66.92 62.739 60.684 65.511 58.888 57.130 62.127 69.526 69.674 66.922 91.824

B/G 2.820 2.246 2.365 2.431 2.270 2.596 2.306 2.009 2.520 1.903 1.817

C 28.279 13.697 16.100 18.468 13.470 19.966 15.068 8.342 20.958 5.840 5.062

0.34 0.305 0.315 0.318 0.308 0.329 0.310 0.283 0.321 0.275 0.266

0.947 0.680 0.780 0.870 0.702 0.818 0.165 0.804 0.945 0.570 0.697

A 1.302 0.831 1.035 1.309 0.873 0.999 1.128 1.465 1.562 0.780 1.247

λ 53.16 37.727 39.180 43.326 35.988 41.455 38.775 35.430 47.328 32.083 41.318

µ 24.967 24.030 23.080 24.858 22.518 21.489 23.707 27.088 26.370 26.243 36.256

C 21.140 26.746 22.600 20.966 24.375 21.503 22.017 21.183 19.725 30.242 31.580

Chapter 3 Results and Discussions

39

3.1.3 Thermodynamic properties of R-Au

The Debye temperature is a valuable parameter which is closely linked to many

characteristics such as melting points and specific heat. From Bulk modulus, Young modulus

and shear modulus, Debye temperature can be determined by applying the relationship [179].

m

A

M

Nn

k

hD

3

1

4

3)(

(3.12)

The h is plank constant, NA is the Avogadro’s number, kB is Boltzmann constant, “M” is the

molecular mass, m is the average sound velocity, “n” denotes the number of atoms in a

molecule and ρ is the theoretical density which can be calculated as

Ac N

Mn

(3.13)

Here “n” means number of atoms in a formula unit. For CsCl crystal structure, n = 1, “M” is for

molecular mass of the compound, “ c ” is the unit cell volume and NA is the Avogadro’s number.

The average sound velocity, m can be obtained as [159]

3

1

33

12

3

1

ls

m

(3.14)

The “ l ” and “ s ” are the longitudinal and transverse sound velocities, calculated by applying

shear modulus “G” and Bulk modulus “B0”from Navier equations [56]

2

1

34

GB

l

and 2

1

Gs

(3.15)

The computed values of sound velocity, for longitudinal ( l ) and shear waves ( s ), and

Debye average velocity ( m ) for R-Au are presented in Table 3.3 and to the best of our

knowledge no literature is present to be compared. Sound velocities of a material depend on the

elastic moduli via shear modulus (G) and bulk modulus (B0). So materials having greater elastic

moduli will have greater sound velocity.

Chapter 3 Results and Discussions

40

Table 3. 3: Calculated values of density ( ), sound velocity of transverse, longitudinal and

average sound velocity (νs, νl and νm) and Debye temperature ( ) of R-Au compounds

Compd.

(g/cm3)

lv

(m/s)

sv

(m/s)

mv

(m/s)

D

(K)

CeAu 11.049 2953.41 1340.91 1511.74 121.65

PrAu 11.258 2812.53 1325.47 1491.70 120.70

NdAu 11.566 2716.25 1412.51 1580.71 128.63

GdAu 12.596 2601.77 1229.00 1382.97 114.35

TbAu 12.924 2502.85 1318.53 1474.26 122.76

DyAu 13.286 2520.86 1271.74 1425.90 119.43

HoAu 13.536 2618.54 1246.74 1402.37 117.92

ErAu 13.691 2536.96 1375.61 1534.93 129.29

TmAu 13.930 2643.85 1322.92 1484.00 125.53

YbAu 13.346 2513.43 1397.09 1555.95 129.27

LuAu 14.455 2504.73 1503.70 1663.36 141.67

Chapter 3 Results and Discussions

41

3.1.4 Chemical bonding of R-Au compounds

Charge distribution around the atom determines the nature of chemical bonding and we

calculated the electronic charge density for R-Au. The Contour-plots of charge density for R-Au

compounds are shown in Fig.3.2. It is obvious from the plots that there is not much bonding

charge that may link the R and Au atoms covalently. The charge density distribution is

spherically symmetric about each atom that shows that these compounds have strong ionic

character. This can be confirmed from the electronegativity difference of the two atoms, R and

Au, which is in the range of 1.42-1.37, suggesting more ionic character. The Figure also

indicates that the gold atom lies at the center and eight R atoms are positioned at the corners of

the cube. The spin dependent plots also show that a metallic character between R and Au atoms.

Hence, the overall bonding in these compounds is predominantly ionic and metallic.

Chapter 3 Results and Discussions

42

Figure 3. 2: Spin polarized electronic charge density of R-Au compounds in (100) and (110)

planes

Chapter 3 Results and Discussions

43

3.1.5 Cohesive energy of R-Au compounds

Cohesive energy is that energy which is required to decompose a crystal into its

constituent atoms that shows that greater the amount of cohesive energy of a compound greater

will be its stability .The cohesive energies (Ecoh) of the bimetallic materials (R-Au) are calculated

by using the equation [180, 181].

(3.16)

Where “Ecoh” is the cohesive energy of the crystal, “Etotal” is the total energy of the crystal while

“ER” and “EAu” are the ground state energies of the free atoms of the lanthanide and gold

respectively calculated by GGA. As we move from CeAu to LuAu the lattice constant of these

compounds decreases which means increase in the rigidity of a crystal. Consequently, cohesive

energy will increase and this trend is clearly seen in Table 3.4.

High melting point means high cohesive energy [182] and the melting points of R-Au

compounds increases from CeAu to LuAu except YbAu which has the lowest melting point

because of its divalency. In divalent state, the ionic radius is comparatively greater than the

trivalent state. Therefore, the charge is low attraction is lesser which results low melting point,

least ionic character and smaller cohesive energy. The cohesive energies of these compounds

increases in the same fashion as their melting points and are presented in Fig. 3.3.

Chapter 3 Results and Discussions

44

Table 3. 4: Cohesive energies, total ground state energies and ground state energies of free

atoms of R and Au.

Compound ECoh (Ry) Etotal (Ry) ER (Ry) EAu (Ry)

CeAu -18.449 -55809 -17717 -38074

PrAu -18.524 -56564 -18471 -38074

NdAu -18.579 -57338 -19246 -38074

SmAu -18.800 -58948 -20855 -38074

GdAu -18.900 -60638 -22545 -38074

TbAu -19.156 -61515 -23422 -38074

DyAu -19.674 -62414 -24320 -38074

HoAu -20.130 -63334 -25240 -38074

ErAu -20.335 -64277 -26182 -38074

TmAu -20.400 -65241 -27147 -38074

YbAu -18.051 -66226 -28134 -38074

LuAu -20.990 -67239 -29144 -38074

Chapter 3 Results and Discussions

45

Figure 3. 3: Cohesive energy (Ry) and melting points (o C) of R-Au intermetallics

Chapter 3 Results and Discussions

46

3.1.6 Electronic properties of R-Au compounds

The study of band structure is very significant for understanding the electronic nature of a

material and its possible technological applications. Bandgap of a compound is defined as the

energy differences between the top of valence band and the bottom of the conduction band. The

computed band structure along the high symmetry directions Г, Δ, Η, N, Σ, ᴧ and P in the

Brillouin zone for both the spins up and down channels for R-Au (R = Ce, Pr, Nd, Sm, Gd, Tb,

Dy, Ho, Er, Tm, Yb and Lu) compounds have been shown in Fig.3.4. No bandgap is observed in

both spins up and down band structures for these compounds; hence all of them are metals. The

results of their electronic behavior in terms of energy bands, Total Density of States (TDOS) and

Partial Density of States (PDOS) are presented in Figs. 3.5 and 3.6 respectively. In case of spin-

up and spin-down configurations, the overlapping of valance and conduction bands occurs

significantly at Fermi level indicating their metallic behavior. The charge density and band

structure of four representative compounds are shown in Figs. 3.2 and 3.4 respectively and rest

of the compounds are similar to them. The effects of four different applied potentials i.e. GGA,

GGA+U, HF and HF+SOC on the Total Density of States are given in Fig. 3.5. The figure

indicates that GGA+U have greater effect on the localization of the density of states as they have

d and f orbitals, while the HF+SOC have caused splitting of orbitals. The HF alone has no effect

on the compounds under consideration.

The main energy bands of R-Au intermetallics are located at two energy ranges between -

20 eV to -19 eV and -8 eV to 6 eV. The low lying energy bands between -20 eV to -19 eV are

Au-p and R-p and are separated from the bands that participate in the conduction process. The

group of bands between -5 eV to 0 eV just below the Fermi level is mainly due to 5d states of Au

and part of 4f of R atom. The 4f of R atoms lies around the Fermi level. The conduction bands

above the Fermi level are largely due to the R-4f state, which hybridizes with the Au-5d orbital.

It is evident that the low lying bands for these compounds are due to the Au-p and R-p

orbitals. The energy bands around the Fermi level for these compounds between -6.5 and 6 eV

are mainly dominated by the hybridized states of Au-d, R-f and R-d. Some of the R-f, Au-d and

R-d crosses the Fermi level this means that these compounds have typical metallic character.

Chapter 3 Results and Discussions

47

Figure 3. 4: Spin polarized band structures of R-Au intermetallics

Chapter 3 Results and Discussions

48

Figure 3.5: Total density of states of R-Au compounds by GGA, GGA+U, HF and HF+SOC

potentials

Chapter 3 Results and Discussions

49

Figure 3.6: Partial Density of State of R-Au intermetallics

Chapter 3 Results and Discussions

50

3.1.7 Magnetic properties of R-Au

To explore the ground state magnetic order of R-Au compounds, we optimized the

double cell of each material ferromagnetically, anti-ferromagnetically and paramagnetically

(nonmagnetic). The energy difference, for R-Au per unit cell are given in Table 3.5, which show

that CeAu, NdAu, GdAu, TbAu, DyAu, ErAu, TmAu and YbAu are stable in the anti-

ferromagnetic, PrAu in ferromagnetic and HoAu and LuAu in nonmagnetic states. All the

computed results are in conformity with the available experimental results [118] as indicated in

Table 3.5. To investigate the origin of magnetism in these compounds spin polarized single cell

calculations are performed by applying GGA PBEsol, GGA+U, HF (B3LYP) and HF (B3PW91)

potentials within the outline of DFT. The calculated effective magnetic moments are given in the

Table 3.6. The Table.3.5 indicates that the HF (B3PW91) results are closer to the experimental

results` than other potentials used. This shows that HF (B3PW91) is effective for calculating

magnetic properties of these compounds. This is because of these compounds are strongly

correlated and hence needs extra potential like “U” to treat correlation effect in these

compounds.

The Magnetic moments of GdAu, TbAu, HoAu, ErAu, DyAu, TmAu and YbAu are

shown in Table 3.6, our calculations are consistent with the experimental results [118]. The

difference in the calculated results is due to the electron exchange correlation effect. DFT

generally underestimates the physical properties.

All the similar compounds of coinage metals should have the same magnetic moments

[183] as obvious from the case of TmCu, TmAg and TmAu whose experimental magnetic

moments are 7.56, 7.15 and 7.32 µB [118] respectively. That shows that our theoretically

determined magnetic moment of TmAu (3.92 µB ) is comparatively more acceptable than the

calculated values 1.34652 µB and 1.3 µB for TmCu and TmAg respectively reported by Chand et

al [184].

The total magnetic moment of binary intermetallics is the contribution of two metals and

their interstitial regions. In this case Au atom and the interstitials regions have negligible

magnetic moments. Therefore, the total spin magnetic moment is due to the 4f-orbitals electrons

of the lanthanide atoms. In brief the incomplete 4f sub shell of rare earths is the origin of the

magnetic moments [185]. This can be verified by the fact that as we move from left to right in

the period, the number of unpaired electrons increases up to Gd and then decreases. Thus the

Chapter 3 Results and Discussions

51

magnetic moment increase up to GdAu and then decrease till the LuAu become nonmagnetic

[186].

Table 3.5: Stable ground state energies per unit cell of R-Au intermetallics

Compd E Para (Ry) E FM (Ry) E AFM (Ry) Expt. This Work

CeAu -55809.429208 -55806.430843 -55809.4308 - AFM

PrAu -56564.050659 -56564.106317 -56564.0509 - FM

NdAu -57338.368179 -57338.492453 -57356.0517 - AFM

SmAu -58947.224525 -58947.574402 -58961.4513 - AFM

GdAu -60657.027263 -60638.830690 -60639.1927 AFM [118, 187] AFM

TbAu -61515.356380 -61515.755411 -61518.0633 AFM [118, 188] AFM

DyAu -62413.873565 -62414.147566 -62414.6129 AFM [118, 187] AFM

HoAu -63334.330012 -63334.310010 -63317.4075 NM [118] NM

ErAu -64276.534961 -64276.533794 -64276.5352 AFM [118, 188] AFM

TmAu -65241.123356 -65241.142392 -65241.1442 AFM [118, 188] AFM

YbAu -66228.501665 -66228.501460 -66228.5017 AFM [118, 189] AFM

LuAu -67238.771436 -67238.769915 -67238.7706 - NM

Chapter 3 Results and Discussions

52

Table 3.6: Calculated and experimental effective / total magnetic moments (µB) of R-Au

intermetallics by PBEsol, GGA+U, HF-B3LYP and HF-B3PW91

Compd PBEsol GGA+U (U) HF-B3LYP (α)

HF-B3PW91(α) Expt [118,

119]

CeAu 1.95 2.94 (4eV) 2.94 (0.35) 2.96 (0.35) _

PrAu 2.26 2.30 (4eV) 2.40 (0.55) 2.58 (0.55) _

NdAu 3.53 3.67 (4eV) 3.8 (0.35) 3.9 (0.35) _

SmAu 5.80 6.18 (5eV) 6.20 (0.60) 6.21 (0.60) _

GdAu 7.10 7.27 (6eV) 7.28 (0.95) 7.30 (0.95) 7.29

TbAu 5.90 7.8 (7eV) 7.82 (0.92) 7.93 (0.92) 9.54

DyAu 4.80 5.2 (5eV) 6.32 (0.90) 7.2 (0.90) 10.22

HoAu 3.70 4.0 (5eV) 4.12 (0.80) 8.4 (0.80) 10.50

ErAu 2.50 2.8 (8eV) 3.7 (0.65) 5.0 (0.65) 9.42

TmAu 1.20 2.0 (8eV) 3.4 (0.25) 3.92 (0.25) 7.32

YbAu 0.015 0.81 (7eV) 1.8 (0.65) 2.6 (0.65) 0.81

LuAu 0.0009 0.0009 (3eV) 0.00094 (0.35) 0.0011 (0.35) _

Chapter 3 Results and Discussions

53

3.1.8 Thermoelectric properties of R-Au compounds (R=Tb, Ho, Er, Tm and Yb) at low

temperature

3.1.8.1The band structure of R-Au compounds.

Electronic structure of a material determines the magnetic, electrical, optical and

thermoelectric behavior of the material. That is why the electronic structure of a material is

essential for its technological uses. Self-consistent field (SCF) computations are carried out for

the study of electronic band structures of these compounds as described earlier [190, 191]. The

band structures of R-Au (R=Tb, Ho, Er, Tm, Yb) compounds are shown in Fig.3.7 which

indicates that the conduction band and valance band have crossed the fermi level and hence the

states concentration is around the fermi level.

Chapter 3 Results and Discussions

54

TbAu HoAu ErAu

YbAu TmAu

Figure 3.7: Band structures of R-Au compounds

Chapter 3 Results and Discussions

55

3.1.8.2 Seebeck coefficient of R-Au compounds

The materials with high Seebeck coefficient will have high Figure of merit value that

means greater capability to converts waste heat into useful electrical energy [59]. Figure 3.8

presents Seebeck coefficients versus chemical potential for R-Au compounds at room

temperature (300K). The peak values of Seebeck coefficients in p-type region for TbAu , HoAu ,

ErAu , TmAu and YbAu are 80 μV/K, 90 μV/K, 98 μV/K , 90 μV/K and 105 μV/K, respectively

at -0.2 eV chemical potential. This clearly indicates that these materials show better

thermoelectric response in p-type region and larger values of Seebeck coefficient are attained in

the range of 0.3-0.3 eV chemical potentials and ahead of this the Seebeck coefficient decreased.

Therefore, we can get good thermoelectric response of these compounds in this region. In

compounds under consideration YbAu shows highest value of the Seebeck coefficient.

Chapter 3 Results and Discussions

56

Figure 3. 8: Seebeck coefficient of R-Au compounds.

Chapter 3 Results and Discussions

57

3.1.8.3 Electronic Thermal conductivity (κ) of R-Au compounds

Thermal Conductivity is the measure of efficiency of a substance to conduct heat. In

crystals, heat is conducted through free electrons and lattice vibrations. It occurs through lattice

vibrations in semiconductors and free electrons conducts heat in metals [192]. Computation of

thermal conductivity is very important because of its role in the performance of thermoelectric

(TE) materials. The thermal conductivity of the materials used in thermoelectric generators must

be small to retain the temperature gradient along the material. The electronic thermal

conductivity of R-Au (R= Tb, Ho, Er, Tm and Yb) are plotted against chemical potential at 300K

(Fig. 3.9), where at zero chemical potential the thermal conductivity is minimum. Higher Figure

of merit for a material requires smaller values of thermal conductivity. From Fig. 3.9, it is clear

that these compounds will give good thermoelectric response in -0.25- 0.25 eV chemical

potential region because in this range thermal conductivities are at minimum. The smallest

thermal conductivity is found for ErAu which is 44.0 KmsW /10/ 14

Chapter 3 Results and Discussions

58

Figure 3. 9: Thermal conductivities of R-Au compounds

Chapter 3 Results and Discussions

59

3.1.8.4 Electrical conductivity (σ) of R-Au compounds

Electrical conductivity is defined as the movement of electric charge in a substance.

Materials having large electrical conductivity will be the best thermoelectric performers. The

calculated electrical conductivities for R-Au (R=Tb, Ho, Er, Tm and Yb) against chemical

potential at room temperature are shown in Fig.3.10. The figure shows that the electrical

conductivity is at minimum for these compounds around zero chemical potential and enhances as

chemical potential is enhanced. The peak values of electrical conductivities for TbAu, HoAu,

ErAu, TmAu and YbAu are 7.4 × 1020, 6.8 × 1020, 6.0 × 1020, 6.5 × 1020 and 7.4 × 1020 /mΩs

respectively.

Chapter 3 Results and Discussions

60

Figure 3. 10: Electrical conductivities of R-Au compounds

Chapter 3 Results and Discussions

61

3.1.8.5 Figure of merit of R-Au compounds

The plots for Figure of merit against chemical potential at room temperature (300K) for

R-Au (R= Tb, Ho, Er, Tm, Yb) compounds are shown in Fig.3.11. We can see that the highest

rates of Figure of merit are 0.20, 0.245, 0.27, 0.28 and 0.285 for TbAu, HoAu, ErAu, TmAu and

YbAu respectively. The peaks are around -0.25 to 0.25 eV chemical potential, because of the

highest values of Seebeck coefficient and lowest values of thermal conductivities. Large values

of ZT show that these materials can be used for thermoelectric applications. YbAu shows the

highest value of ZT because this compound has the largest value of Seebeck coefficient.

Chapter 3 Results and Discussions

62

Figure 3. 11: Figure of merit of R-Au (R=Tb, Ho, Er, Tm and Yb) compounds.

Chapter 3 Results and Discussions

63

3.2 Rare-earth diaurides (R-Au2) (R= La, Ce, Pr and Eu) Orthorhombic intermetallics.

3.2.1 Structural properties R-Au2

Structural properties of R-Au2 (R= La, Ce, Pr, Eu) compounds are computed by

evaluating the optimized total energy of the unit cell of each compound with respect to volume

by applying Birch Murnaghan Equation of State [193]. The lattice constants for all the

compounds are obtained by PBEsol exchange correlation. The calculated results are presented in

Table 3.7 and compared with the available experimental results. All these compounds, LaAu2,

CeAu2, PrAu2 and EuAu2 are found in orthorhombic, CeCu2 type structure having space group,

Imma, No 74, and atomic positions are [39, 46, 49, 99, 194-196].

x y z

R= 0.00 0.250 0.3577

Au=0.00 0.051 0.1648

Our computed lattice constant results are consistent with the experimental results [195].

However, no theoretical work about their structural properties is available in literature for

comparison.

Chapter 3 Results and Discussions

64

Table 3.7: Lattice constants, (Å) by PBEsol compared with experimental results of

Orthorhombic R-Au2

Compd Lattice constant, (Exptal) [195]

(Å)

This work

(Å)

LaAu2 a=4.700

b=7.295

c=8.155

a=4.813

b=7.121

c=8.007

CeAu2 a=4.528

b=7.203

c=8.068

a=4.709

b=6.971

c=7.906

PrAu2 a=4.672

b=7.040

c=8.178

a=4.655

b=7.007

c=8.081

EuAu2 a=4.670

b=7.330

c=8.140

a=4.670

b=7.208

c=8.072

Chapter 3 Results and Discussions

65

3.2.2 Magnetic properties R-Au2 compounds

To study the ground state magnetic order of orthorhombic R-Au2 (R= La, Ce, Pr and Eu),

we optimized the double cell of each compound ferromagnetically, anti-ferromagnetically and

non-magnetically. The energy difference, for R-Au2 compounds per unit cell is given in Table

3.8 which indicates that all these materials are stable in the anti-ferromagnetic (AFM) state. Only

experimental result of magnetic nature of EuAu2 compound is available for comparison to which

our result is consistent. To investigate the origin of magnetism in these intermetallics; spin

polarized single cell calculations are performed for all these compounds by using HF potential

within the DFT frame work and the calculated magnetic moments per lanthanide atom (R= La,

Ce, Pr and Eu) of all these intermetallics are presented in Table 3.9.The double cell AFM

calculation is carried out with PBEsol potential which shows that all these materials have zero

total magnetic moment as presented in Table 3.9. To the best of our information only the

magnetic moment of Eu in EuAu2 compound is available to which our result are comparable. We

believe our calculation of magnetic moment of these compounds will motivate the

experimentalist because we have carried out the calculation by four different reliable potentials

and concluded that the net spin magnetic moments in binary intermetallics are due to three

contributions, one from the lanthanide atoms (R), the other from Au atom and the third from

interstitial regions. Au atom and the interstitial regions have negligible magnetic moments. In

conclusion the total spin magnetic moment is due to 4-f unpaired electrons of the lanthanide

atom [197, 198]. The lanthanides can be considered as a cluster of ions, mostly trivalent, with

incomplete 4f orbitals, imbedded in a cloud of free electron. The deep 4f shell electrons play a

secondary role in their mechanical and chemical properties but they are the origin of the

magnetic moments[199]. As we go from left to right in a period the number of unpaired electrons

increases that results in the increase of magnetic moment. The Lanthanum (La) has no ‘f’

electrons thus have no magnetic moment while the EuAu2 has the highest magnetic moment

among the investigated compounds because Eu of six unpaired electrons.

Chapter 3 Results and Discussions

66

Table 3.8: Ground state energies calculated for Orthorhombic R-Au2 compounds

Compd EPara (Ry) EFM(Ry) EAFM (Ry) Expt. This

Work

LaAu2 -186319.06729 - 186319.09363 -745276.370 - AFM

CeAu2 -187789.55083 -187789.55077 -751158.204 - AFM

PrAu2 -189298.80417 -189298.89155 -757195.558 - AFM

EuAu2 -195735.24454 -195736.38415 -758954.153 - AFM

Chapter 3 Results and Discussions

67

Table 3. 9: Magnetic moments (µB) of orthorhombic R-Au2

Compd GGA PBE-

sol

GGA+U

(U)

HF-B3LYP

(α)

HF-B3PW91

(α)

Expt

LaAu2 0.00106 0.0076

(2eV)

0.0076

(0.25)

0.00821

(0.25)

-

CeAu2 1.62087 2.2377

(2eV)

2.65

(0.15)

2.66

(0.15)

-

PrAu2 5.21000 5.8122

(2eV)

5.808

(0.15)

5.96656

(0.25)

-

EuAu2 13.916 14.0799

(2eV)

14.1995

(0.25)

14.239

(0.25)

-

Chapter 3 Results and Discussions

68

3.2.3 Elastic constants and mechanical properties of orthorhombic R-Au2 intermetallics

Elastic constants are indispensible for describing the mechanical properties of the

compounds. They give us information regarding the nature and presence of different forces,

stiffness and stability of the compounds [52, 53]. They are also useful in predicting the aging

behavior of the materials [54]. They have the ability to decide the response of the materials to an

external forces described by Poisson ratio and Young, Bulk and Shear moduli. Therefore, they

have a great role in determining the strength of the crystals. Elastic properties are associated with

thermodynamic behavior that is thermal expansion, specific heat, melting point and Debye

temperature. This shows its worth for understanding the mechanical properties of materials.

For orthorhombic type crystals exists 09 independent elastic constants denoted by C11, C22, C33,

C44, C55, C66, C12, C13, and C23 [55, 56]. Due to the special importance of the Young, Shear and

Bulk moduli and Poisson’s ratio for high-tech applications, we computed them from elastic

constants. Our calculated results given in Table 3.10 obey the Born Stability Criteria for

orthorhombic structures[200] reveals the mechanically stable nature of orthorhombic R-Au2.

The ratio of stress to strain that determine the stiffness of materials is called Young’s modulus,

greater its value stiffer will be the material and vice versa. It is given by the equation [201].

(3.17)

The Young’s moduli are high for PrAu2, LaAu2 and EuAu2 which indicates that these

compounds are comparatively stiffer than CeAu2. The CeAu2, EuAu2, LaAu2 and PrAu2 are in

the order of increasing stiffness and decreasing ductility as indicated from their Young’s

modulus and Bo/G ratio respectively. The CeAu2 is the least stiff and good ductile, while PrAu2 is

the least ductile and good stiffer.

Poisson ratio measures the compressibility of a solid [54]. The materials will be

incompressible when 0.5 and will be compressible if =0.2 to 0.49 [202]. In our study, the

measured Poisson’s ratio of R-Au2 compounds is from 0.265 to 0.353, presented in Table 3.10

which indicates that these materials are compressible. Poisson’s ratio difference in different

materials depends upon the nature of the bond present. The typical Poisson’s ratio values for

covalent, ionic and metallic bonds in a crystal are 0.1, 0.25 and 0.33 respectively [203]. In this

context, our compounds are ionic/metallic crystals according to the calculated Poisson’s ratio

and there is no covalent character at all. Poisson’s ratio may also be used for determining the

Chapter 3 Results and Discussions

69

central forces in solids. For an isotropic compound Poisson’s ratio ν, is calculated from the

following relation [201].

(3.18)

For central forces in crystals, the limits of Poisson’s ratio are 0.5 to 0.25 [54]. The value of

calculated Poisson’s ratio in compounds under consideration remain between these two limits

reveals the predominance of central inter atomic forces.

Bulk modulus (Bo) is one of the most important parameters of the engineering materials.

The Bulk modulus of a material is the measure of its hardness [204]. It gives information about

the nature of bond and cohesive and binding energies of the material [167]. Bulk modulus for

orthorhombic compounds is determined from the elastic constants by using the relation [54],

(3.19)

The computed values of the Bulk moduli for these compounds are shown in Table 3.10. That

clearly indicates that the bonding type in these materials is ionic/ metallic. The large values for

these compounds also confirm that the R-Au2 compounds with high Bulk moduli have high

melting points.

The criterion for brittleness or ductility of a material is the value of the Pugh’s ratio,

Bo/G. lesser and higher value of the Bo/G ratio than 1.75, reflects the brittle and ductile nature of

the material respectively [205]. The Bo/G ratio calculated for CeAu2, EuAu2 and LaAu2 are 3, 2,

and 1.8 respectively. So all these compounds are ductile in nature, while Bo/G is almost equal to

1.75 for PrAu2 compound therefore its ductility is comparatively lower (Table 3.10).

Shear modulus determines the resistance toward reversible deformations resulting from shear

stress. It demonstrates better correlation than Bulk modulus with hardness of a material. Voigt

shear modulus is given by this relation; [54]

(3.20)

The computed Shear modulus for PrAu2 is the highest (50.2 GPa) and lowest for CeAu2

(20.7 GPa). Among our compounds the PrAu2 is the hardest and CeAu2 is the soft one in our

compounds (Table 3.10).

The knowledge of elastic anisotropy has important applications in high-tech materials

because it predicts the chance of micro cracks in crystals. In different crystallographic planes,

Chapter 3 Results and Discussions

70

shear anisotropy gives a measure of the level of Anisotropy in atomic bonding in different

planes. Further it gives information regarding stiffness, mechanical stability, hardness,

brittleness, ductile and binding behavior between neighboring atomic planes. Therefore, the

determination of elastic anisotropy of a material is very important to understand its behaviors and

to find means for improving their hardness and durability. That’s why we predicted, elastic

anisotropy of R-Au2 by using the Anisotropic factor which is defined below [200, 206] :

For the shear planes (1 0 0) that is x-axis between <01 1> and <010> directions, it is

(3.21)

For shear planes (0 1 0) that is y-axis between <101> and <001> directions, it is

(3.22)

And for shear planes (0 0 1) that is z-axis between <110> and <010> directions, it is

(3.23)

The obtained anisotropic factors for R-Au2 compounds are given in Table 3.12. The

values of A1, A2 and A3 for an isotropic material is equal to 1, otherwise it is an anisotropic

material [55, 200]. Our result for LaAu2, PrAu2 and EuAu2 show strong anisotropic behavior in

all directions while CeAu2 show isotropic behavior on x-axis and anisotropic behavior on y and

z-axes. The anisotropic values for all the four compounds indicates that the elastic anisotropy for

LaAu2, PrAu2 and EuAu2 in [010] shear planes between (101) and (001) directions is higher than

that of [100] shear plane between (011) and (010) directions and [001] shear plane between the

(110) and (010) directions. In case of CeAu2, it is isotropic in [100] shear plane between (011)

and (010) directions, because A1 is equal to unity and the anisotropy along the [010] shear plane

between (101) and (001) directions is higher than in [001] shear plan between (110) and (010)

directions.

Resistance to linear compression in directions a-, b-, and c- are measured from elastic

constants C11, C22, and C33 respectively. The C22 calculations for LaAu2, CeAu2 and EuAu2 are

lower than the C11 and C33. Therefore, these compounds are more compressible across the b-axis

in comparison to a-axis and c-axis while PrAu2 is more compressible through a-axis than b- and

c-axes.

Chapter 3 Results and Discussions

71

In orthorhombic compound we define the Cauchy pressure in three different directions:

PxCauchy = C23 – C44, Py

Cauchy = C13 – C55 and PzCauchy = C12 – C66 as given in Table 3.11. The

Cauchy pressure gives information about the type of bonding and the ductile behavior of a

material. The compounds with positive or negative Cauchy pressure will have metallic bonds

and ductile nature and directional bonds (covalent) and brittle nature respectively. The ionic

crystals have a large Cauchy pressure either positive or negative. All of our compounds have

positive Cauchy pressure in x and z-axes while negative values in y-axis. It means these

compounds are harder in x and z directions than in y direction.

No experimental or theoretical studies of mechanical properties of these compounds are

available in literature for comparison. However, our calculation can serve as a tool for future

research work. In future experimental work if any will verify our investigations.

Chapter 3 Results and Discussions

72

Table 3.10: Values of elastic constants (GPa), Bulk modulus B0 (GPa), Shear modulus G

(GPa),Young’s modulus Y (GPa), Poission ratio ν (GPa) and B/G ratio of R-Au2

compounds LaAu2 CeAu2 PrAu2 EuAu2

C11 314.69 171 65.8 133.4

C22 94.318 55 172.7 154.2

C33 143.69 113.6 106.6 193

C44 11.7 5.60 11.3 69.0

C55 173.3 109.5 280.7 108.95

C66 18.6 7.75 27.0 22.0

C12 48.65 42 41.4 89.8

C13 87.2 131.8 28 7.67

C23 29.7 32 88.1 108.8

Bo 84.433 63.722 63.42 88.2

G 46.845 20.7 50.24 43.4

Y 118.6 56 119 111.84

ν 0.265 0.353 0.186 0.288

B/G 1.80 3.00 1.26 2.00

Chapter 3 Results and Discussions

73

Table 3.11: Values of Cauchy pressure (GPa) of Orthorhombic R-Au2 in x, y and z

directions

Compounds LaAu2 CeAu2 PrAu2 EuAu2

PxCauchy=C23-C44 18 26.4 77 40

PyCauchy=C13-C55 -86 22 -252.7 -101

PzCauchy=C12-C66 30 34 14.4 68

Chapter 3 Results and Discussions

74

Table 3.12: The anisotropic factors A1, A2 and A3 of orthorhombic R-Au2 compounds

Compounds A1 A2 A3

LaAu2 0.165 3.88 0.239

CeAu2 1.00 4.20 0.22

PrAu2 0.388 10.89 0.694

EuAu2 0.887 3.36 0.814

Conclusion

75

Conclusions

The first principle DFT calculations have been carried out to investigate the structural,

electronic, elastic, magnetic and mechanical properties of R-Au (R= Ce, Pr, Nd, Gd, Tb, Dy, Ho,

Er, Tm, Yb and Lu) compounds, as well as thermoelectric properties of R-Au (R= Tm, Tb, Er,

Ho and Yb) and structural, magnetic, elastic and mechanical properties of the orthorhombic

phase of R-Au2 intermetallics (R= La, Ce, Pr and Eu) by using the PF-LAPW method within the

density functional theory. The structural properties of these intermetallics are computed with

various exchange and correlation functionals and the calculated values are in good agreement

with the experimentally reported results. It is concluded that these compounds are strongly

correlated systems because their lattice constants calculated with the Hubbard potential U are

better. Our calculated results confirm the divalency of YbAu and nicely explain the lanthanides

contraction in these compounds.

The density of states (DOS) and band structures indicate the metallic behavior of these

materials. It can also be summarized that all these materials are elastically stable because they

obey Born stability criteria. They are hard and their melting points are very high which is

inferred from their high bulk and shear’s moduli. The values of Young’s moduli of the

compounds is indicative of their intermediate stiffness, however the positive values of their

Cauchy pressure reveals that the bonding nature is predominantly metallic/ionic. Furthermore,

the large values of the B0/G ratio for these compounds reveal their ductile nature and the most

ductile is CeAu with the highest B0/G value of 2.82. The Poisson’s ratio shows that these

intermetallics are less compressible and the deviation from unity of the isotropic ratio reveals

that they are anisotropic except DyAu and NdAu. Furthermore, the internal strain parameter

confirms bond bending for all the compounds except HoAu which shows bond stretching. The

magnetic properties of R-Au intermetallics are calculated by applying GGA, GGA+U and HF

based on density functional theory. The computational results of the magnetic properties indicate

that the HF results are much closer to the reported experimental results as compared to GGA and

GGA+U approaches. The electronic charge densities plots explain the bonding nature in these

materials. The chemical bond between R and Au is ionic and metallic in nature. The Cohesive

energies show that the stability of these compounds increases as we move from CeAu to LuAu

with the exception of YbAu which is least stable due to its divalent nature. The electronic band

Conclusion

76

structures show that all these intermetallics are metallic in nature. The Boltzmann’s transport

calculations are performed at 300K temperature to investigate Seebeck coefficient, electrical

conductivities and thermal conductivities for R-Au (R= Tb, Ho, Er, Tm and Yb) compounds. The

highest value of the figure of merit is found for YbAu, is 0.29.

The R-Au2 (R=La, Ce, Pr and Eu) orthorhombic compounds are elastically stable because

they obey Born stability criteria. They are hard and their melting points are very high which can

be inferred from their high bulk and shear’s moduli. CeAu2, EuAu2 and LaAu2 are ductile while

PrAu2 is stiffer, which is revealed from their Pugh ratio and Young’s moduli. The compounds,

LaAu2, PrAu2 and EuAu2 show strong anisotropic behavior while CeAu2 shows isotropic

behavior. The positive values of their Cauchy pressure reveal that the bond type in these

intermetallics is metallic/ ionic.

The magnetic properties of these intermetallics show that all these compounds are

antiferromagnetic.

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