First Principle Calculations & Inelastic Neutron Scattering on the Single-Crystalline Superconductor LaPt2Si2

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IN

DEGREE PROJECT MATERIALS SCIENCE AND ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2020,

First Principle Calculations &

Inelastic Neutron Scattering on the Single-Crystalline Superconductor LaPt Si ₂ ₂

FEDERICO MAZZA

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

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Master in Engineering Material Science (ITM) Date: June 30, 2020

Supervisor: Prof. Martin Månsson & Dr. Johan Hellsvik Examiner: Prof. Peter Hedström

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Abstract

This work presents a comprehensive study on single crystalline LaPt²Si², in which superconductivity and a charge density wave (CDW) coexist. The usage of density functional theory (DFT) modeling and Inelastic Neutron Scattering has been the primary form of investigation, in order to determine all the characteristic features of the sample taken under consideration. From the results one can observe that the Fermi surface nesting is the primary contributor for the CDW wavevector ~q_CDW = (1/3, 0, 0). In addition, the phonon density of states present two typical energy levels, with soft modes in the Pt3-Pt4 layer coherent with the presence of a CDW. The superconducting temperature has been estimated at T^c= 1.6 K. The experimental data from the inelastic instrument High Resolution Chopper Spectrometer (HRC) at the J-PARC neutron source are in good agreement with the theoretical simulations, showing the same energy levels for the polarization phonon modes (from 4 to 18 meV and from 32 to 42 meV).

Keywords: Superconductivity, LaPt2Si2, Inelastic Neutron Scattering, Charge Density Wave, Band Structure Calculations, DFT, SNIC, J-PARC.

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Sammanfattning

Denna rapport presenterar en omfattande studie av enkristalls LaPt²Si² i vil- ken supraledning och en laddningsdensitetsvåg (CDW) samexisterar. Använ- dandet av DFT-modellering och neutronspridning har varit de huvudsakliga undersökningsmetoderna, för att bestämma alla karakteristiska drag hos det undersökta provet. Från resultaten kan observeras att den inneslutna Fermi- ytan är den huvudsakliga bidragaren till CDW-vågvektorn ~q_CDW= (1/3, 0, 0).

Vidare visar den närvarande fonontillståndsdensiteten två typiska energinivå- er, med mjuka lägen i Pt3-Pt4-skiktet, som stämmer överens med närvaron av en CDW. Den supraledande temperaturen har uppskattats till T^c = 1.6 K.

Experimentella data från det inelastiska instrumentet HRC vid J-PARCs neu- tronkälla stämmer väl överens med teoretiska simuleringar, som visar samma energinivåer för polarisationsfononlägena (från 4 till 18 meV och från 32 till 42 meV).

Nyckelord: Supraledning, LaPt2Si2, neutronspridning, neutronspektrosko- pi, Bandberäkningar, DFT, J-PARC.

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Acknowledgement

First of all, I would like to express my appreciation to Professor Martin Måns- son for the amazing opportunity that he provided me when asked for a Master Thesis project work and for the help throughout the entire second semester. I will always be grateful to him, since he has proven on many occasions to be really skilled in every aspect of both scientific knowledge and personal life.

His attitude and kindness is some of the aspects that I admire the most and for that I will always keep him in the highest regard.

Another person that I would like to address with a special regard is Dr.

Johan Hellsvik, NORDITA, which provided guidance for the DFT modeling and the set-up for the simulations. I believe he has an outstanding talent not only within his field of expertise but also in other branches of science.

Further, Professor Yasmine Sassa of Chalmers is greatly acknowledged for having the original idea for this project and for arranging access to the LaPt2Si2

crystals. Here I would also like to acknowledge Prof. Zakir Hossain of the In- dian Institute of Technology, Kanpur and Dr. Arumugam Thanizhavel of the Tata Institute of Fundamental Research, that synthesized the LaPt2Si2single crystals.

I am very grateful to Professor Takatsugu Masuda and his team from Uni- versity of Tokyo, for the fantastic help provided at ISSP regarding the crystal alignment and for all the expertise at the HRC instrument. I also thank Profes- sor Kim Lefmann, University of Copenhagen, that joined and supervised the experiments in Japan together with his MSc student Irene Sanlorenzo.

Elisabetta Nocerino, PhD student in Prof. Månsson’s group, the principal investigator of the HRC neutron experiments. I could never thank her enough for the help provided throughout the six months that we have been working together. Her passion about science is almost contagious and her ability to explain difficult concepts is clearly an outstanding trait. Ola Kenji Forslund, PhD student in Prof. Månsson’s group, which helped in the paper review, providing good comments and feedback when needed in order to increase the written quality of the manuscript.

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I am also very grateful to the Swedish National Infrastructure for Comput- ing (SNIC), that awarded us the computational time (Dnr. SNIC 2020/5-183) on the supercomputer. Finally, I also greatly appreciate the financial support from the Japanese High Energy Accelerator Research Organization (KEK), which allowed me to join and conduct the neutron experiments at J-PARC / MLF.

Figure 1: The group of Professor T. Masuda of University of Tokyo, ISSP, together with team SMaRT of Professor M. Månsson, KTH Royal Institute of Technology.

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List of Abbreviations

BCS theory Bardeen-Cooper-Schrieffer theory

CDW Charge Density Wave

DFT Density Functional Theory

DOS Density of States

EDX Energy X-Ray Sprectroscopy

EM Electro-Magnetic

eV Electron-volt

FS Fermi Surface

FX Fast Extraction Port

GGA Generalized Gradient Approximation HK theorem Hohenberg-Kohn theorem

HRC High Resolution Chopper Spectrometer INS Inelastic Neutron Scattering

ISSP Institute of Solid State Physics

J-PARC Japan Proton Accelerator Research Complex K-space Reciprocal Space

KS theorem Kohn-Sham theorem

LA Longitudinal Acoustic

LDA Local Density Approximation

LO Longitudinal Optical

MLF Material and Life Science

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MR Main Ring

NBS Neutron Brillouin Scattering RCS Rapid-Cycling Synchrotron

RT Room Temperature

SC Superconductivity

SEM Scanning Electron Microscope

SOC Spin-Orbit Coupling

SX Slow Extraction Port

TA Transverse Acoustic

TAS Triple-Axes

T_c Superconducting Critical Temperature T_CDW Charge Density Wave Temperature

TO Transverse Optical

TOF Time-of-Flight

T_SL Temperature of the superlattice reflections

XRD X-Ray Diffraction

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List of Figures

1 The group of Professor T. Masuda of University of Tokyo, ISSP, together with team SMaRT of Professor M. Månsson, KTH Royal Institute of Technology. . . vi 1.1 Schematic representation of the atomic structure of LaPt2Si2

compound having a tetragonal cell. . . 2 2.1 Overview of the 3-dimensional Bravais lattices, from [4] . . . 5 2.2 Schematic description of displacement vsK vector in (a) lon-~

gitudinal and (b) transverse mode, adapted from [4] . . . 7 2.3 Acoustic phonon dispersion relation in an ideal crystal, adapted

from [4] . . . 9 2.4 Example of a Fermi Sphere in a three-dimensional space, adapted

from [4] . . . 11 2.5 Representation of the nesting vector [15] . . . 16 2.6 X-ray diffraction for two parallel planes, adapted from [4] . . . 17 2.7 Representation of the Edwald’s sphere, adapted from [4] . . . 18 2.8 Schematic representation of an X-ray tube, adapted from [17] . 19 2.9 Typical X-rays energy levels and radiations, adapted from [17] 20 2.10 Focusing Circle in X-ray diffraction, adapted from [17] . . . . 21 2.11 Scattering triangles in INS, on the left side a representation of

the neutron energy loss (the neutron gives energy to the lattice) and on the right side of the energy gain (the neutron receives energy from excitations in the sample), adapted from [19] . . . 22 2.12 Schematic view of neutron production using a spallation process. 24 2.13 (a) Half-filled metallic band without any CDW. (b) Formation

of the 1D CDW dimerize the lattice and a gap opens at the Fermi level. Adapted from [22] . . . 25 2.14 The Kohn anomaly for one dimensional metal at different tem-

peratures, adapted from [15] . . . 26

xi

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xii LIST OF FIGURES

2.15 Experimental result from resistivity measurements of mercury (Hg) revealing the superconducting phase transition T^c= 4.2 K 27 2.16 Overview of different models in relation with different physics

aspects . . . 28 2.17 Cooper pair formation due to the electron-phonon interaction,

adapted from [27] . . . 29 3.1 Large LaPt₂Si₂crystal with the [100] direction along the crystal. 32 4.1 Aerial view of the Japan Proton Accelerator Research Com-

plex (J-PARC), adapted from [29] . . . 34 4.2 (a) Schematic representation of the HRC instrument, adapted

from [31]. (b) 3D-CAD of the HRC instrument. (c) Installa- tion of the massive detector tank of HRC at MLF / J-PARC. . . 35 4.3 Performance of Fermi chopper, Fermi A is the high-intensity

chopper and Fermi B is the high-resolution chopper, adapted from [31] . . . 36 5.1 Overview of the 10 atoms primitive cell, along with all the cell

parameters for LaPt2Si2. . . 39 6.1 Calculated electronic band structure for the (a) Non-magnetic

case (b) Spin-polarized structure and (c) Spin-orbit-coupled system. . . 42 6.2 Plot of the Fermi surface centered around the Γ point along

with the other high symmetry points. The nesting vector is emphasized and it is close to ~q_CDW. White lines constitutes the Brillouin zone. . . 43 6.3 Top panels: the phonon DOS and dispersion relations for the

non-magnetic structure. Bottom panels: the phonon DOS and dispersion relations for the SOC case. The difference of interest among the two cases is highlighted with the red circle at 10 meV . . . 45 6.4 Experimental determination of the T^c, adapted from [33] . . . 47 6.5 (a) preliminary alignment along [110], the luminescent ring

was in good agreement with the results from the Laue simula- tor. (b) The final alignment along [110] . . . 48

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LIST OF FIGURES xiii

6.6 (a) Sample aligned in transmission Laue setup. (b) Sample mounted in XRD machine. (c) Mounting of the sample onto the sample holder. (d) Final alignment of the crystal on holder (e) Complete sample mounting with thin Aluminium canister attached, covering the sample. Final assembly was made in a pure He atmosphere and an indium sealing was used. . . 49 6.7 On the left side, a general overview with the aluminum back-

ground subtracted. On the right side, the raw data with the aluminum background . . . 50 6.8 The phonon modes measured by INS at T = 85 K. (a-b) Show

the TA without and with the dashed white lines as guide to the eye, respectively. (c-d) Show the LA and LO without and with the dashed white lines as guide to the eye, respectively.

Black symbols in all panels are the fitted parameters obtained by analyzing the data with DAVE. . . 51 6.9 Longitudinal phonon modes at 3K . . . 52 6.10 Cut at the elastic (zero energy transfer) line with variable H

to get an overview of the Bragg peaks in the crystal. At H =

±0.33 there are clear superlattice peaks present (circled in red) for different temperatures. (a) Measurement at 3 K. (b) Mea- surement at 85 K. (c) Measurement at 220 K . . . 54 6.11 General overview of intensity differences at different temper-

atures. Top-left T = 3 K, top-right T = 85 K and bottom T = 220 K . . . 55 6.12 The transverse phonon modes of at 220 K in an integrated con-

tour of K and L at H = 2 . . . 55

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List of Tables

3.1 a general overview of the sample information, structure, space group lattice parameters and volume of the primitive cell . . . 32 6.1 Relaxed structure for T=0 K, the value of x, y and z correspond

to the crystal axes a, b and c . . . 41

xiv

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Chapter 1 Introduction

In recent years, the study of complex low dimensional systems revealed many intrinsic characteristics of these material in terms of crystal structure instabilities and phase transitions. The LaPt²Si² compound exhibit a quasi two di- mensionality, crystallizing in a CaBe²Ge²-type tetragonal structure. Different studies have proposed a series of approaches in order to determine the material properties, specifically two possible tetragonal space-groups have been identified for the symmetry of the crystal structure: I4/mmm (No. 139) and P 4/nmm (No. 129). The difference here, is that the former shows a higher level of symmetry within the unit cell, resulting in an infinitesimal change in the analysis of experimental data. Since the prediction of the fundamental properties can be performed using computer simulation throughout the usage of density functional theory (DFT) a higher level of precision is required, so that in this study, the space-group taken under consideration is I4/mmm, an overview of the crystal structure is provided in Fig. 1.1.

Another factor to consider is the fact that at low temperature there is a first order structural transition from tetragonal to orthorhombic, which is believed to be induced by a charge density wave (CDW) at TCDW = 112 K, followed by a superconducting transition at Tc = 1.22 K. The coexistence of both superconductivity and CDW is of major interest, due to the fact that they are both mediated by electron-phonon interactions. The superlattice reflections that arise, due to the presence of the CDW, correspond to (n/3, 0, 0) where n = 1 and 2 so, Nagano et al. suggested that at low temperature (TSL = 10 K) the original unit cell triple its dimensions along [100]. Inelastic neutron scattering (INS), along with density functional theory (DFT) calculation are suitable tools to investigate all such features. Recently, several papers have been published in order to fully characterize and understand this complex compound.

1

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2 CHAPTER 1. INTRODUCTION

Figure 1.1: Schematic representation of the atomic structure of LaPt2Si2com- pound having a tetragonal cell.

However, nobody has studied the phonon density of states (DOS) and the dispersion relations experimentally, as well as the presence of the super-structural peaks coinciding with the CDW [1] [2].

1.1 Scope of the work

The aim of the project, is to specifically characterize the material properties of LaPt2Si2. The general goal is to contribute to the development of fundamental research regarding superconducting materials. First of all, DFT calculations, with the Elk 6.3.2 [3] code have been performed to identify the electronic band structure of the non-magnetic, spin-polarized and spin-polarized spin-orbit- coupling structure, to see if there is any magnetic moment present and how the electronics state would change accordingly. Following with the estimation of the phonon dispersion relations and the phonon DOS with the construction of a supercell 2×2×1 in order to identify any possible variation that can occur based on the system size by comparing the results with previously published papers. In addition, the CDW will be measured based on the estimation of the Fermi surface nesting. In terms of experiments, the LaPt2Si2single crystal has first been aligned along [100], [010] and [110] by using the transmission Laue

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CHAPTER 1. INTRODUCTION 3

monochromatic x-ray diffraction (XRD) at the University of Tokyo, Institute of Solid State Physics (ISSP). Further, through the use of inelastic neutron scattering (INS) it has been possible to measure the phonon dispersion curves in the system in order to have a comparison between simulation and experimental data. The INS has been performed at J-PARC (Japanese Proton Accelerator Research Complex) using the High-Resolution Chopper Spectrometer (HRC).

With the experiment, it was possible to obtain the entire phonon spectra with the acquisition of full 4D S(Q, ω) maps via azimuthal (rotational) scans at three different temperatures.

1.1.1 Limitations

The work does not aim to propose any particular application of this compounds, but it is mainly focused on the fundamental investigations of the structural properties of this novel material. Since the synthesis of the LaPt2Si2sin- gle crystal has been developed only recently, and so far all the characterization has been made from powder samples only, the process is not yet optimized.

Consequently, the resulting crystals do show signs of minor presence of multiple domains as well as impurities. Due to this, the intensity of the superlattice peaks resulted to be very low, when compared with the one of the Bragg’s peaks. Even though HRC is one of the best INS instruments in the world, in terms of resolution and input energy, the determination of the CDW phonon softening was impossible. Further, since the neutron experiments are being performed at large-scale neutron sources, the access to experimental beamtime is highly limited. The experiment conducted for this thesis work was in fact proposed and approved almost a year prior to the start of this project. There was indeed a clear plan and possibility to apply for and conduct additional experiments at J-PARC (e.g. single crystal neutron diffraction), however, the covid-19 situation unfortunately made such extensions impossible.

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Chapter 2 Background

2.1 Crystal structure

The first concept that is imperative to introduce is the definition of a crystal.

Since X-ray diffraction was developed in 1912, it has been possible to observe that these types of solids consist of a periodic arrangement of atoms and, with an appropriate model, it is possible to fully describe their structure. An ideal crystal is defined by an infinite repetition of identical groups of atoms, such group is called the basis. In addition, it is necessary to convey on a set of points to which the basis is connected, this is called the lattice. In a three-dimensional environment one can define the translation vectors ~a₁, ~a₂, ~a₃ which allow to visualize the arrangement of atoms in the crystal from an arbitrary point of view r. By introducing another arbitrary point r⁰, translated as an integral multiple of lattice parameter a, we obtain the relation

r⁰ = r + u₁a~₁+ u₂a~₂+ u₃a~₃ (2.1) where u¹, u2, u3 are integers. So that the set of point expressed by r⁰ is the definition of the lattice. In a three-dimensional space there are 14 different types of lattices defined as the Bravais lattice, which differ in several parameters, such as angles and cell axes, volumes, position of the atoms and so on.

A more detailed overview is provided in Fig. 2.1 [4].

2.1.1 Crystal planes and directions

It is possible to identify planes and directions within the crystal by using a set of integers referred as the Miller Indices. Knowing that a crystal plane orientation is specified by three points, if each one of those lays on a different

4

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CHAPTER 2. BACKGROUND 5

Figure 2.1: Overview of the 3-dimensional Bravais lattices, from [4]

crystal axes is then possible to express the spatial coordinates of the points in relation to the constants. I n order to refer to the Miller indices, it is necessary to obey the following rules:

• Find the intercepts with regards the crystal axes

• Take reciprocals and divide by a common factor

• Find the smallest three integers

The indices obtained are conventionally denominated as (hkl), which state the plane index. Furthermore, in this manner, one can either refer to a single plane or a set of parallel planes. According to the chosen origin, the hkl indices could be negative or positive in respect to the interceptions with the crystal axes, in the former case, one must put a minus sign on top of the index. In referring to crystal directions the idea is the same, however, the set of smaller integers are indicated inside squared brackets, such as [hkl] [4].

2.1.2 Reciprocal space

By analyzing the mathematical laws underlying the construction of a three- dimensional lattice space, it has been found out that one can translate the real-

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6 CHAPTER 2. BACKGROUND

space coordinates into an associated 3D reciprocal space lattice. The relationship

exp(i~G · ~r) = 1 (2.2)

shows the correspondence to a direct translation vector ~r, which consequently defines a reciprocal translation vector G. Since the translation vector has~ [length] as unit, it means thatG will be expressed as [1/length]. The useful-~ ness of introducing a reciprocal space is essential when describing concepts regarding the study of crystalline solids in terms of diffraction properties. In order to construct the reciprocal lattice, one can start by considering a lattice point as origin and then by drawing normal to all possible planes in the real space lattice. Introducing d, as the distance from one plane to the origin, it is possible to represent the same plane at 1/d on its normal. The total collec- tion of these points will lead to the construction of the reciprocal space lattice.

Another approach to define the relationship between real space and reciprocal space can be mathematically obtained by introducing the notation

G = h ~~ b₁+ k ~b₂+ l ~b₃ (2.3) where hkl are the Miller indices. If we re-introduce ~a₁, ~a₂, ~a₃ as primitive vector of any crystal lattice thenb~₁, ~b₂, ~b₃has to be defined as primitive vectors of the reciprocal lattice, consequently leading to [4]

b~₁ = 2π a~₂× ~a₃

~

a₁· ~a₂ × ~a₃; ~b₂ = 2π a~₃× ~a₁

~

a₁ · ~a₂× ~a₃; ~b₃ = 2π a~₁× ~a₃

~

a₁· ~a₂× ~a₃ (2.4)

2.1.3 Phonons

In solid state physics, phonons are usually defined as "quanta of lattice vibrations" and they explain most of the phenomena that cannot be analyzed by a static nuclear lattice theory. In a classical point of view, the crystal lattice has always been assumed to be rigid and so any possible displacement from the ideal position was never considered. In reality, the lattice cannot be assumed rigid, therefore it is necessary to take into account the vibrational energy of the system, in order to give good descriptions of several material’s properties, such as the electronic band structure, the nature of the chemical bonds that keep the crystal together, the heat capacity, the thermal expansion and other transport properties. The starting point is to define the internal energy of the system U as

U = E_static+ E_vib (2.5)

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Figure 2.2: Schematic description of displacement vsK vector in (a) longitu-~ dinal and (b) transverse mode, adapted from [4]

where Estaticis the electronic ground state described in the classical theory of a fixed lattice and Evib is the energy due to the lattice motion. This relation introduced the concept of mobility within the solid, even though the magnitude of the vibrations are very small in amplitude [5]. In the simplest case, let’s consider the elastic vibrations with one atom in the primitive cell. Supposing a wave vector K propagating in the x-direction, this will lead to an in-phase~ displacement of the entire planes of atoms parallel to the direction ofK, so that~ it will be possible to identify with a single coordinate u^s, the displacement of the s plane. For each wave vectorK, there will be one longitudinal polarization~ and two transverse polarization modes, as it is shown in fig. 2.2 [4]. The total force acting on s from s ± 1 is defined as

F_s= C(u_s+1− u_s) + C(u_s−1− u_s) (2.6) where C describes the force constant between the nearest neighbor planes. The equation of motion is

M∂²u_s

∂t² = C(u_s+1+ u_s−1− 2u_s) (2.7)

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where M is the mass of the atom. The solution of equation 2.7 can be found for all the time-dependent displacement exp(−iωt),which also imply that ^∂

2us

∂t² =

−ω²u_s, so that equation 2.7 will be

−M ω²u_s= C(u_s+1+ u_s−1− 2u_s) (2.8) this relation can be solved by using a travelling wave solution and the result will provide a full description of the displacement u

us±1 = uexp(is~Ka)exp(±i~Ka) (2.9) whereK is the wave vector and a is the spacing between the atomic planes.~ By substituting (2.9) into (2.8) and using the identity 2 cosKa = exp(i~~ Ka) + exp(−i~Ka), we will obtain the dispersion relation ω(~K), see Fig. 2.3.

ω² = 2C M

1 − cos~Ka

(2.10) At the boundary of the Brillouin zone, which is defined as the primitive cell of the reciprocal space lattice, that is found at K = ±^π_a it has been observed that the slope of the curve goes to zero, so that for sinKa = sin(±π) = 0.

By trigonometric identity, equation (2.10) can be re-written as ω = 4C

M

1/2

| sin1 2

Ka |~ (2.11)

as a general rule, it is possible to identify how many polarization modes will be present. Assuming that three are the degrees of freedom of any atoms, one for each x, y, z direction, and p as the number of atoms in the primitive cell, the total number of atoms will be pN , where N is the number of primitive cells.

Therefore, 3pN will be the total degrees of freedom of the crystal. Since the allowed numbers of acoustic branches are one longitudinal acoustic (LA) and two transverse acoustic (TA) for a total of 3N degrees of freedom, the ones accommodated for the optical branches will be the remaining (3p − 3)N . In order to understand how many modes will exist in a specific range of energy, it is necessary to calculate the density of the states. Assuming a small number of modes for a given ∂K in a small ∂ω, it is possible to define the density of~ the states as

D(ω) = D(K)∂ ~K

∂ω (2.12)

where D(K) is defined as the density of states in the K-space (a scalar variable).

Assuming a three-dimensional space, with N³ cubical primitive cell of side

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Figure 2.3: Acoustic phonon dispersion relation in an ideal crystal, adapted from [4]

L, the allowed values of K will be

K_x, K_y, K_z = 0; ±2π L; ±4π

L; ±N π

L (2.13)

meaning that there will be only one allowed value of K per volume (^2π_L)³, which will lead to the following relation

L 2π

3

= V

8π³ (2.14)

by integrating the density of the states over a sphere of thickness ∂K and radius K, one will get the density of states D(ω) for each polarization mode [6] [7]

[4]

D(ω) = Z

D(K)∂ ~K

∂ω =

Z V K² 2π²

∂ ~K

∂ω (2.15)

2.1.4 Fermi energy and Fermi surface

In order to describe the energy, as well as the electron charge along with the electron occupancy in a quantum mechanical framework, it is necessary to

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introduce several key principles that diverge from classical theories. The free electron gas is usually introduced to predict the bulk properties of the solids.

The main consideration is that, according to the Pauli’s exclusion principle, electrons can only occupy distinct single particle states, so that they can be considered to move free along the 3D space. Let’s consider a free electron gas in one-dimension and suppose that the total length of the system is defined by L. Assuming an electron of mass m and wave function ψn(x), the energy of the electron in the orbital will by given by

_n= ¯h² 2m

nπ L

2

(2.16) if one denotes as nf the topmost filled energy level, it is possible to consider the case in which electrons will fill the system from the bottom level, defined as n = 1, and continue until reaching the higher level until N electrons will be accommodated. Due to the Pauli’s exclusion principle two fermions cannot have the same set of quantum number, but if they occupy the same energy level, one will have spin up and the other spin down, leading to the relation 2n_f = N . With this consideration it is possible to define the Fermi energy f

as " the energy of the highest filled level in the ground state of N electrons at absolute zero"

_f = ¯h² 2m

N π 2L

2

(2.17) In this case, the density of states in the system D(), for any given , will describe the number of states per interval energy. At T = 0 K, it is possible to relate this two concepts together in order to find the total electronic energy of the solid as it follows

Etot = Z f

0

D()∂ (2.18)

The next step consists into translating those concept in a three-dimensional space, in order to have a complete overview regarding a real-life scenario. By solving the Schrödinger’s equation for a free particle in three dimensions and assuming that the wave-functions will be periodic in x, y and z, the energy of an orbital with wave vectorK will be defined as~

_k= ¯h²

2m(K_x²+ K_y²+ K_z²) (2.19) in the reciprocal space, it is possible to picture the occupied orbitals as points inside a sphere space, as it is shown in Fig. 2.4, which the surface energy

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Figure 2.4: Example of a Fermi Sphere in a three-dimensional space, adapted from [4]

is defined as the Fermi energy. The volume of such sphere is expressed as 4πK_f³/3. Since in every orbital there will be two electrons with opposite spin, the magnitude of the wave vector at the Fermi surface will be [4] [8] [9]

K~_f = 3N π² V

1/3

(2.20)

2.1.5 Electronic band structures

In both solid state physics and material science, the concept of band structure is of paramount importance, since it provides a description of the electronic levels within a crystal structure. Each electronic level is characterized by two quantum numbers, as explained in the previous section, as well as the wavevector K and the band index n. The total energy of the electron ~ ⁿ(K), is assumed to be a continuous function of K. In solids, there are many atoms~ close to each other in a rigid structure, with different orbitals that can only accommodate electrons with different quantum states, resulting in the formation of different energy levels, defined as bands. From here, one can define the band gap as the difference in energy between the valence band, which is the outermost electron orbital filled in the material at absolute zero, and the conduction band, defined as the lowest range of vacant electronic state. The

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location of the Fermi energy plays a major role in the material characterization, as an insulator, semiconductor or metal. Furthermore, the energy gaps are a region in which no wave-like electron orbital can exist. The understanding of the band structures provides a complete description of many electro-magnetic and optical properties of the material. The calculations of those bands can be performed with DFT modeling, in which the energy can be analyzed with respect of the high symmetry points of the Brillouin zone.

By defining a linear crystal with N primitive cells and a as lattice constant, the independent values of allowedK for each primitive cell will be one to each~ energy band. In a three-dimensional space, the result still stands, but since there are two independent orientation of the electronic quantum numbers, it will lead to the conclusion that there are 2N independent orbitals per energy band [4] [10].

2.2 Density Functional Theory

Since the development of quantum mechanics, in the fields of material science and condensed matter physics, the understanding and the exploitation of properties in terms of interactions between atomic nuclei and electrons raised several questions. The feasibility of creating new materials with trial and error methods has become unpractical due to the production costs and their com- plexity. In a theoretical point of view, the electrons-nuclei interactions can be analyzed by solving the Schrödinger’s equation even though, in some occasions, one reaches extreme and impractical propositions. Thus, Dirac in 1929 proposed that "progress depends on the development of sufficiently accurate but tractable, approximate techniques". Consequently, the discovery of the density functional theory (DFT) along with the demonstration of the local density approximation (LDA) opened up new possibility in the design of novel materials. The challenge is to mimic as accurately as possible the complex nature of real compounds. Nowadays, it is possible to determinate the properties based on the calculations of hundreds of atoms in a unit cell, making the density functional theory a key tool in material science [11] [12].

2.2.1 Remarks of DFT

The main principle upon DFT and LDA are based on is the Hohenberg-Kohn (HK) theorem. In a non-spin-polarized system, the total energy E of the in- teracting electrons in an external potential is expressed as a functional of the

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electron density, since the ground state density minimizes E[ρ]

E = E[ρ] (2.21)

In order to extend this concept to a spin-polarized systems, the energy and the properties become a function of the spin density. In the simplest case, in which there are only one spin-up and one spin-down, we will obtain

E = E[ρ↑, ρ↓] (2.22)

the drawback of the HK assumption is that there are no information regarding the energy functional, meaning that the validity of the DFT lies in the accuracy of the approximations used. In order to solve this, it is necessary to express the term E = [ρ] as the Hartee total energy plus the exchange correlation functional Exc[ρ]. By defining the single particle kinetic energy Ts[ρ], the Coulomb interaction energy electron-nuclei Eei[ρ], and the interactions among nuclei Eⁱⁱ[ρ], it is possible to write the energy functional as follows

E[ρ] = T_s[ρ] + E_ei[ρ] + E_H[ρ] + E_ii[ρ] + E_xc[ρ] (2.23) since the exchange correlation functional is also unknown, by using the LDA approximation, it can be written as

E_xc[ρ] = Z

∂³rρ(r)_xc(ρ(r)) (2.24) where xc(ρ) defines the energy of an uniform electron gas and is an approximation of the local function of the density. Another useful approximation is the generalized gradient approximation (GGA), in which in order to have more information about the electron gas, the local gradient and the density are incorporated in the energy exchange term. The correlation between GGA and LDA is now well understood, extensive research has been proposed over the years and the following conclusion can be drawn: they improve signifi- cantly the ground state properties of materials composed by light atoms, as well as structural properties. Kohn and Sham proposed the idea that the electron density can be described as the sum of the single particle density, which was a crucial step that opened up various possibilities in DFT calculations.

For a given Exc, one then can properly define the ground state energy and density, in order to obtain a self-consistent solution for a set of single particle Schrödinger equation. Here, the density is given by the Fermi sum over the occupied orbitals, leading to the Kohn-Sham equation

T + V_ei(r) + V_H(r) + V_xc(r)φ_i(r) = _iφ_i(r) (2.25)

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the number of electrons determines the topmost occupied orbital φi, V is the potential of the different contributions, ⁱis the respective Kohn-Sham eigen- values. The implications of this method are that, instead of solving a single Schrödinger’s equation, one can use DFT calculations to solve single particle equations, resulting in a less computational and man power required as well as a self-consistency. Nevertheless, not always it is possible to determine which is the ground state and which approximation yields the best result in terms of energy minimization, recognizing that is the first step towards the conclusion that each method may result in lower or higher energies in respect to the ground state [11] [12] [13].

2.2.2 The supercell approach and phonon calculations

Phonons dispersion have been of major interest in determining the crystal properties. By adjusting the parameters of the calculations it is possible to achieve a very good accuracy with reasonable computational effort. Although in most of the cases, there are still some major drawbacks that require more intensive analysis. The phonon frequencies can be obtained by calculating the energy differences or can be analyzed by the forces action on the atoms, produced by periodic displacement from the ideal equilibrium positions. As a matter of fact, very low frequency modes are usually associated to phase transformations, while imaginary frequencies give indications regarding the stability of the crystal structure, If the latter are present, it is an indication that the calculated structure is not stable. To obtain a more accurate solution, it is important to simulate a cell that contains multiple cells of the given lattice.

This is known as the supercell approach, where the basis vectors can be defined by linear combinations of the primitive vectors with integer coefficients.

It is possible to mathematically visualize the transition from primitive cell to supercell by the relation



 a_s1 as2

a_s3



=





S₁₁ S₁₂ S₁₃ S21 S22 S23

S₃₁ S₃₂ S₃₃







 a_p1 ap2

a_p3



 (2.26)

where ~a_siare the translation vectors of the supercell, ~a_piare the translation vectors of the primitive cell andS~_ij are integers numbers and it is defined as the supercell matrix. The same principle can be used to convert the translation vectors of the primitive cell in the reciprocal space to the reciprocal superlattice, in order to empower the periodicity of the system. Let’s assume an arbitrary K point, which can be expressed both with the supercell reciprocal

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lattice and the primitive reciprocal lattice multiplied by the supercell matrix



 ks1

k_s2 k_s3



=





S11 S12 S13

S₂₁ S₂₂ S₂₃ S₃₁ S₃₂ S₃₃







 kp1

k_p2 k_p3



 (2.27)

In the case the reciprocal superlattice coordinates are integers, any change from equilibrium position of the atomic positions expressed by the wavevector K,~ will be commensurate with the supercell. Assuming an array N¹× N₂ × N₃ of primitive cells, to set up a calculation of first principle lattice dynamics using an harmonic approximation, it is necessary to calculate the dynamical matrices at eachK point, to represent the lattice vibrations, so that the overall~ model will be as close as possible to a real experiment [14].

2.2.3 Determination of the Fermi surface

Since the development of DFT calculations it has been possible to gather different data set for a preliminary characterization of novel materials, which can be compared with real life experiments. Namely the definition of the Fermi surface, and its ground state characteristics are now available in DFT modeling.

As explained in section 2.1.5, in an ideal case the Fermi sphere is embedded in an equi -energetic surface with constant curvature, with a radius defined by theK~_f vector. Regarding real materials, the Fermi surfaces do not correspond to simple sphere, in fact in most cases they are represented by topologically complicated geometries, and highly connected to the crystalline structure geometry and interactions. A peculiar case is defined as Fermi surface "nesting", which has been defined as: "two parallel pieces of a Fermi surface, such that a single ~q-vector can connect many points" as presented in Fig. 2.5. Usually, it is an indication of the presence of instabilities in low temperature regimes.

Specifically, most of the metallic materials go into a phase of charge density or spin density wave or other topological transitions. For the determination of the Fermi surface it is necessary to use the KS single particle formalism, so that one can obtain a good starting point for the calculation of the electronic band structure and their typical energies. In the K-space, the minimization of the ground-state interaction energies will favour a more stable configuration of the Fermi level, so that the electronic distribution that defines the shape of the Fermi surface, will be approximated into a step-function-like form. Any- way, difficulties arise when considering the particle-particle interactions for any model spectrum. The reason is that it is not sufficient to define the self- energy of the system, leading to a "re-normalization", or even a reconstruction of the Fermi surface throughout the LDA and GGA approximations [8] [13].

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Figure 2.5: Representation of the nesting vector [15]

2.3 Neutron scattering and X-ray diffraction

2.3.1 Bragg’s law and diffraction condition

To understand the structure of a crystal, one can use several diffraction techniques that involve electrons, photons or neutrons. The key principle relies on the utilization of the wavelengths of those particles in relation with the lattice constant. In case the radiation wavelength is comparable with the spacing among the crystal planes, the diffracted beams directions will differ from the incident directions. W. L. Bragg proposed a theory that explains how the radiation diffraction occurs. Let’s consider an incident wave which is reflected by the parallel planes formed by the crystal’s atoms, each of them will reflect only a small fraction of the radiation even though the angles of incident and scattered beams remain the same. During this process, the energy is conserved, and it will result in a diffraction pattern with a constructive interference among the reflections of the parallel planes. Let’s Suppose to have infinite parallel planes in which the distance of the spacing is defined as d, then the path difference of the reflected beams will consequently be 2dsinθ, where θ is the angle , as it is shown in Fig. 2.6. Constructive interference will occur in the case that the path difference is an integer n of the wavelength λ, leading to the relation,

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Figure 2.6: X-ray diffraction for two parallel planes, adapted from [4]

known as the Bragg’s law

2dsinθ = nλ (2.28)

the lattice periodicity is what determines the Bragg’s law, which can be sat- isfied for λ ≤ 2d. Moreover, the possible reflections are determined by the set of the reciprocal lattice vectorsG. By defining the incoming and outgoing~ wave vectors as K and ~~ K⁰, one can identify the difference in phase between beams scattered from different volume elements spaced by r

exp[i(~K − ~K⁰) · ~r] (2.29) the scattered wave is proportional to n(r), which is the local concentration of electrons. By integrating these two principles, one can obtain the scattering amplitude F

F = Z

∂V n(r)exp[i(~K − ~K⁰) · ~r] (2.30) here, one can introduce the following relation

K − ~~ K⁰ = ~∆K (2.31)

where∆K is the scattering vector, defined as the difference between incoming~ and outgoing wave vectors. When the scattering vector is equal to the reciprocal space lattice vector the exponential term of the scattering amplitude van- ishes, so that F will be directly proportional to the volume of the electronic concentration [4] [16].

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For elastic scattering, the energy is conserved and so the frequency of the refracted beam will be equal to the one of the incident beam. In addition the magnitude of the wave vectorsK and ~~ K⁰ can be also expressed as K² = K²⁰. From here, since ∆K = ~~ G, as it is shown in Fig. 2.7, one can define the diffraction condition as

2K · G = G² (2.32)

Figure 2.7: Representation of the Edwald’s sphere, adapted from [4]

2.3.2 Principles and generation of X-rays

In 1895, it has been discovered by Wilhelm Conrad Röntgen, that a penetrat- ing radiation produced by a high voltage discharge between electrodes in a gas has the ability to make materials to fluoresce onto visible light. In case the voltage exceeds 30 kV the resulting radiation will be denoted as X-ray. The primary working principle can be defined as the electrons emitted from a cath- ode could be accelerated by an applied voltage, resulting in a EM radiation that will collide with an anode, a schematic overview of an X-ray tube is provided in Fig. 2.8 [17] [18]. By taking into account, Heartz’s previous studies regard-

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Figure 2.8: Schematic representation of an X-ray tube, adapted from [17]

ing wavelengths of radio waves, it has been possible to generate X-rays. In the following years, major breakthrough discoveries in the fields of material science as well as quantum theory have been brought to light, so that in 1901 Röntgen was awarded with the first Nobel Prize in Physics.

The typical energy range of a photon emitted from a bound or free electron is from 0.1 to 100 keV. If the photon was emitted from the atom nuclei one will produce the so called gamma rays even if the energy lies in the typical X-ray range. When an accelerated electron interacts with the target atom, will eject one electron from one of the inner shells causing a cascade reaction within the ion, in which other electrons from outer shells will fall down to fill the vacancy. Whenever these kind of transitions occur, a photon will be produced.

In addition, when an outer electron falls into a hole at n = 1, 2 or 3 shell, the produced X-ray will be defined as K, L or M respectively, whose characteristic energy will depend on the individual elements, as it is shown in Fig. 2.9.

The choice for the anode material, usually lies between Copper and Molyb- denum. Copper radiation is mostly used for powder diffraction. In addition, it can be suitable in case one has a small size or a large unit cell crystal. How- ever, Molybdenum radiation is most suited for larger crystal or for absorbing

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Figure 2.9: Typical X-rays energy levels and radiations, adapted from [17]

materials due to its very high energy and large penetration. Other sources can be also mentioned, such as Cr, Fe, Ag or W, which are commonly used for very specialized diffraction experiments. A key requirement for X-ray experiment is the fact that the radiation has to have a very narrow wavelength, so that the resolution will be enhanced. When the voltage is applied, both the white radiation, the Kαand Kβ curves will be observed. As a matter of fact, in terms of diffraction experiments the Kα can be defined as the radiation resulted from an electron transition from a 2p orbital of the second shell to the innermost shell (K). It is the most suitable due to its greater intensity. Commonly, filters have the ability to screen the unwanted radiation bands.

Monochromators, provide an alternative way for selecting an X-ray beam with a narrow band distribution. By having a crystal monochromator, one can cut the unwanted radiation wavelength by geometrical elimination, according to the principle of Bragg’s law. A variety of different option is available.

Specifically, they could be either curved at an angle, to provide a smaller diver- gence, or faced parallel to a major set of planes which by orientation can easily diffract the K^αlines. Curved monochromators are usually adopted for special application at synchrotron facilities. In terms of a commercial point of view, graphite crystals are the most commonly used, even though, for specific pur- poses other materials can be taken under consideration, such as germanium or lithium fluoride. Collimators, can be used to shape the X-ray beam, especially in single crystal diffraction, metal tubes are utilized in a way that the inside radius must be a bit larger than the typical sample size. In addition, diffracted beam collimators can also be put in place to avoid that all stray radiation will

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Figure 2.10: Focusing Circle in X-ray diffraction, adapted from [17]

hit the detector.

The key principle regarding X-ray experiments is that the source of the radiation, the sample and the detector, always lies in the so-called focusing circle [17]. While the sample holder remain fixed, the X-ray tube and the detector rotate in unison, even though it is possible to have a fixed tube and a rotating sample and detector. The incident angle ω is the one that lies between the source and the sample, 2θ is the diffracted angle, an overview is provided in Fig. 2.10.

2.3.3 Principles of inelastic neutron scattering

Neutrons are another available option for material characterization. Since those particle have a magnetic moment, they can not only be used to investigate the nuclear crystal structure and phonons, but also the magnetic spin structure, as well as magnetic excitations, due to the fact that they can be polarized. Another advantage is the fact that they interact directly with the atom nuclei, which makes it possible to detect light elements, such as hydrogen, which remains often undetected by X-ray scattering due to its single electron.

The incident neutrons do not stop on the surface but can penetrate deep in the sample under investigation; this ability makes them suitable for bulk characterizations. The β − decay and life time of free neutrons have been measured as 881 seconds, allowing users to conduct robust experiments, even at temperatures as low as superconductive temperatures.

The working principle of elastic neutron scattering has been introduced in sections 2.3.1 and 2.3.2, and it is still valid for inelastic neutron scattering for what concerns the Bragg’s law and the diffraction conditions. The difference between the two techniques lies in the non-conservation of energy during the

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Figure 2.11: Scattering triangles in INS, on the left side a representation of the neutron energy loss (the neutron gives energy to the lattice) and on the right side of the energy gain (the neutron receives energy from excitations in the sample), adapted from [19]

scattering process. In this regard, there is an energy exchange among the incident beam and the atoms of the material. Specifically, one can observe an absorption or emission equal to the quantum phonon energy, defined as ¯hν, giving rise to the energy transfer. The typical frequency in solids is usually on the range of a few THz, which is coherent with the usual neutron energies, which are of a few meV, leading to an accurate determination of the phonon frequencies. Another relevant aspect to consider is the fact that each neutron produced by the source will be characterized by certain velocity and wave vector. In order to define the scattering vectorQ, one has to determine the initial~ and finalK vector ( ~~ K_iandK~_f). An overview of the energy exchange mechanism is provided in Fig. 2.11 [19].

Another way to tackle the problem, would be to analyze the initial and final energies of the neutrons according to the relation

E_i− E_f = ¯hω (2.33)

by defining the neutron mass as Mn, the initial and final energies can be related with the momentum

E_i,f = h¯²K_i,f²

2M_n (2.34)

in addition, the scattering angle is a function of the wave vector Q cosα = K_i²+ K_f²− Q²

2K_iK_f (2.35)

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where Q = Ki− K_f.

Generally, there are two methods used for INS: (1) Triple Axis Spectroscopy (TAS) and (2) Time-of-Flight (TOF). The former method has traditionally been used extensively to study nuclear and spin excitations in mainly single crystalline samples. For TAS measurements a continuous neutron source is ideal, and therefore such instruments are mostly available at reactor sources.

For TOF instruments a pulsed neutron sources is clearly more advantageous and with the modern (and safer) pulsed spallation neutron sources the TOF technique is now becoming far superior. In time of flight experiments, one measures the time the particle needs to reach the detector after hitting the sample, by tuning different chopper speeds and phases one can identify different neutron energies at the same time, in order to obtain the complete 4D data sets of the dynamical structure factor S(Q, ω) that comprise all the momentum (crystal) directions, as well as the energy transfer (Q^h, Q^k, Q^l and ω). This methods comprise both high resolution, that can be used to analyze specific points in the reciprocal space or lower resolution, to get a complete overview of the sample [20]. A schematic overview of the neutron production in a spallation source is provided in Fig. 2.12.

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Figure 2.12: Schematic view of neutron production using a spallation process.

2.4 Charge Density Wave

Charge density wave (CDW) is a phenomenon that was firstly discovered by Fröhlich in 1954 and Peierls in 1955. It is defined as a broken symmetry state that occurs in metals at low temperature due to electron-phonon interactions.

Regarding CDW, two observations can be made, there is a reduction of the number of free electrons, as well as the formation of a superstructure, which can be identified as a lattice superimposed on the basic lattice. As a conse- quence of the presence of the CDW, the resulting ground-state is characterized by a continuous charge density modulation along with a periodic lattice distortion (see Fig. 2.13), both determined by the Fermi wave vectorK~_f, leading to a modification of the phonon and electron spectra [21].

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Figure 2.13: (a) Half-filled metallic band without any CDW. (b) Formation of the 1D CDW dimerize the lattice and a gap opens at the Fermi level. Adapted from [22]

2.4.1 The electron response function

The FS geometry plays a key role in the determination of the dependence of the CDW wavevector ~q_CDWwith the response function ξ(q), which describes the electronic response, in terms of charge arrangement. Having a large ξ(q), means that many electron occupied states, are connected by the same q wave vector to the respective empty state. Hence the wave vector links large parts of the Fermi surface with other parts of the Fermi surface, leading to the conclusion that FS nesting enhance the electron response function. In a 3D case, at T = 0 K, it is possible to define the relation

ξ(q) = −e²n (_f)

1 + 1 − x² 2x ln

| 1 + x 1 − x |

(2.36) where n(^f) is the density of states per spin at the FS and x = q/2k^f. Tem- perature is another factor that has to be considered, for T > 0 the singularity at q = 2kf is cancelled out. Due to electron-phonon coupling, one encounters lattice instabilities for high value of ξ(q), resulting in a gap in the energy spectrum at the Fermi surface. The phonon frequency goes to zero at T = T^CDW, since the static lattice distortion counteracts the phonon mode, resulting in a frozen state. By defining λ as the dimensionless electron phonon coupling constant, one can mathematically define the transition temperature

K_BT_CDW = 1.14₀e⁻^λ¹ (2.37)

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Figure 2.14: The Kohn anomaly for one dimensional metal at different temperatures, adapted from [15]

where KB is the Boltzman constant and 0 is the electronic energy state. The Kohn anomaly for 1D metal at different temperatures is presented in Fig. 2.14, one can observe that the curve is divergent at T = T^CDW and at q = 2K^f leading to the so-called Pierls transition, which induces a static lattice distortion (again, see Fig. 2.13). In 2D and 3D cases the electron response function is also determined by the FS induced anomalies, but they are generally much weaker [15].

2.5 Superconductivity

2.5.1 Zero resistivity

In 1908, with the discovery of how to liquefy helium a new temperature thresh- old was reached by Onnes et al. In 1911, he also discovered that at such low temperature (4 K), mercury (Hg) loses its resistivity. This result led to the realization of a new state of solid matter. Superconductivity is a property of materials that occurs at certain critical temperature defined as Tc and consists in the loss of electrical resistivity. In other words, when a current is sent through the material, the current will be kept even without any applied voltage, resulting in zero power loss. This phenomenon is an electronic phase transition between a normal conducting and a superconductive state, see Fig. 2.15.

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Usually, the temperature ranges in which superconductivity occurs are close to the absolute zero but, nowadays, the research has been focused on finding new materials that can allow this transition at higher temperatures.

The first question that comes to mind when talking about superconductivity is if the resistance of the material goes to zero or only reaches an infinitesimal value. Different experiments have been performed in order to answer the question, to report a few examples, by using the method of decay of persistent currents in superconducting rings, it was possible to estimate a range of resistivity from 2×10⁻¹⁸to 7×10⁻²³Ωcm. Nevertheless, the dependence from the sensitivity of the apparatus, as well as the presence of thermal fluctuations, the in-homogeneity of the material and other factors plays a key role in this kind of studies. Even though the limits are extremely low, it has been stated from W. Buckel:” . . . it is fundamentally impossible to demonstrate in an experiment the assertion that the resistance has fallen to exactly zero. An experiment can only ever deliver an upper limit for the resistance of a superconductor” [23]

[24] [25].

Figure 2.15: Experimental result from resistivity measurements of mercury (Hg) revealing the superconducting phase transition T^c= 4.2 K

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2.5.2 Models for superconductivity

Superconductivity includes many aspects of physics, the zero resistivity is in- cluded in the electromagnetic response, the phase transition below the Tcindi- cates a thermodynamically driven behavior, the quantization of the magnetic flux relies on the field of quantum physics. For those reasons, it is not possible to find a theory that can accommodate all the different factors altogether and at the same time maintaining the accuracy needed. With the purpose of describing the similarities and differences among all the different approaches, it is necessary to introduce the λL and ξ parameters. The former is defined as the London penetration depth, which denotes how much a magnetic field penetrates a superconductor, the typical values go from 50 to 500 nm. The latter is the coherence length, that states the maximum distance in which the superconducting properties disappear upon traveling from a superconducting to a normal region. An overview of different models and their application is given in Fig. 2.16 [26].

Figure 2.16: Overview of different models in relation with different physics aspects

2.5.3 Cooper pairs and BCS theory

Several theories have been proposed to explain the phenomenon of superconductivity, here we will focus on the most accredited of them: the well es- tablished BCS, which is based on the concept of formation of the Cooper Pairs. The principle of Cooper pair has been firstly introduced in 1956 by the American physicist Cooper, which intuited the fact that electrons or even other fermions can bind together at low temperature. Specifically, he introduced the

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Figure 2.17: Cooper pair formation due to the electron-phonon interaction, adapted from [27]

idea of a small interaction that is the cause of a “pairing state”, which is obtained in order to minimize the system’s energy below the Fermi level. In 1972, John Bardeen, Leon Cooper and John Schrieffer won the Nobel Prize, by demonstrating that Cooper pairs are responsible for superconductivity in their description of the BCS theory. The binding mechanism is the result of the electron-phonon interaction. The description of the pairing process can be evaluated from a classical point of view, although this is clearly a quantum ef- fect. Usually, electrons repel each other due to the Coulomb interaction. How- ever, they also have affinity with the positive ions, which distorts the ion lattice, in a way that those positive ions move towards the free electrons, resulting in an increase of the positive charge density. The newly created charge density affects other electrons, resulting in a long-distance electron-electron attraction since the ions displacement overcome the electrons’ repulsive forces. The typical energies are in the order of 10⁻³eV and can easily be overcome by thermal fluctuations. An overview of this principle is shown in Fig. 2.17 [27] [28].

Since electrons have spin S = 1/2, by paring together the total spin will be an integer, so the couple is a composite boson. As a result, multiple Cooper pairs can be in the same quantum state, in conformity with the Pauli exclusion principle. Furthermore, the charge density of the Cooper pairs will be

n = 1

2n_e (2.38)

and the charge of each pair will be defined as

q = −2e (2.39)

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The Cooper pairs can smoothly go through the lattice points without any energy exchange, in fact they do not scatter onto the lattice points. Moreover, there is no energy transfer from the electron pair to the ion lattice, as a result, they do not posses any electrical resistivity.

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Chapter 3 Information about the sample

The single crystalline samples were synthesized by our collaborators Prof. Za- kir Hossain of the Indian Institute of Technology, Kanpur and Dr. Arumugam Thanizhavel of the Tata Institute of Fundamental Research. They have also performed the basic characterizations of the sample quality (structure, superconducting transition, etc.). The single crystals of LaPt2Si2 were grown by using the Czochralski pulling method [1] by using high-purity elements La (99.9 %), Pt (99.9 %) and Si (99.9 %). To optimize the conditions of the growth, a tungsten rod has been utilized and so the crystal was pulled at a constant rate of 10 mm/h. In order to ensure that the crystal structure meet the requirement of a single-phase nature, powder X-ray diffraction (XRD) with Cu-K^α radiation has been performed. To analyze the specimen quality and homogeneity, scanning electron microscopy (SEM) and energy dispersive x- ray spectroscopy (EDX) has been the primary key tools. In addition, with the Laue x-ray diffraction it was possible to check the different crystal directions.

With a spark erosion cutting machine, the initial crystal has been divided into 5 different pieces. An overview of the structure parameters is given in Ta- ble 3.1 [1]. The final crystal used for the INS experiments was Ø = 6 mm, L = 21 mm having a total mass m = 3.9759 ± 0.0002 g. Such high-quality piece was selected and cut out from a much larger single crystal, which is shown below in Fig. 3.1.

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