# Unconventional superconductivity in quasi-2D materials with strong spin-orbit coupling

Raphaël Leriche

► To cite this version:

Raphaël Leriche. Unconventional superconductivity in quasi-2D materials with strong spin-orbit coupling. Superconductivity [cond-mat.supr-con]. Sorbonne Universite, 2019. English. NNT: 2019SORUS577. tel-04006817

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THÈSE

Pour l'obtention du grade de

DOCTEUR DE SORBONNE UNIVERSITÉ

École doctorale Physique en Île de France - ED 564

# Unconventional superconductivity in quasi-2D materials with strong spin-orbit coupling

Auteur :

Raphaël LERICHE

Sous la direction de :

Dr. Tristan CREN

Dr. Christophe BRUN

## *Composition du jury :*

Pr. Christoph RENNER

Rapporteur

Dr. Hermann SUDEROW

Rapporteur

Pr. Andrea GAUZZI

Examinateur

Dr. Marie-Aude MEASSON

Examinatrice

Dr. Nicolas BERGEAL

Examinateur
Abstract

Unconventional superconductivity in quasi-2D materials with strong spin-orbit cou-
pling

The realization of topological superconductors is one of the main current goals of condensed
matter physics. It was indeed predicted that such systems should host Majorana fermions.
These Majorana fermions possess both a non-Abelian statistics and, because of their topological
origin, a certain robustness against local disorder, which makes them attractive for quantum
computing applications. One approach likely to lead to topological superconductivity consists in
considering superconducting systems with strong spin-orbit coupling and with broken inversion
symmetry. It is in this framework that, during this thesis, I performed scanning tunneling
microscopy and spectroscopy measurements on quasi-2D materials : (LaSe)~1,14~(NbSe~2)~2 and
Sr~2~IrO~4.

I first studied the electronic properties of misfit compound LaNb2Se5, which is a parent of transition metal dichalcogenide 2H-NbSe2. (LaSe)1,14(NbSe2)2 is a heterostructure made out of alternations of NbSe2 bilayers with trigonal prismatic geometry and LaSe bilayers with rocksalt structure. (LaSe)1,14(NbSe2)2 is a potential candidate for topological superconductivity because of the presence of both a strong spin-orbit coupling and of broken inversion symmetry in NbSe2 planes. Here, I present spectroscopic results showing that the electronic structure of (LaSe)1,14(NbSe2)2 is very similar to the one of electron-doped monolayer NbSe2 with a shift of the chemical potential of 0,3 eV, priorly never reached. I could also demonstrate the quasi-2D nature of (LaSe)1,14(NbSe2)2 and more particularly the presence of a strong Ising spin-orbit coupling. Moreover, the observed weakness of superconductivity against non-magnetic disorder combined with quasiparticle interferences measurements allowed me to exhibit the unconventional nature of (LaSe)1,14(NbSe2)2 superconducting order parameter. This study opens the possibility to use misfit heterostructures such as (LaSe)1,14(NbSe2)2 to study the physics of transition metal dichalcogenides in the 2D limit, for which many theoretical studies predict topological superconductivity.

In this thesis, I also present a study on the effects of doping on the electronic properties
of iridate compound Sr<sub>2</sub>IrO<sub>4</sub>. Sr<sub>2</sub>IrO<sub>4</sub> is a spin-orbit induced Mott insulator. Because inver-
sion symmetry is locally broken in Sr<sub>2</sub>IrO<sub>4</sub>, some theoretical predictions suggest that Sr<sub>2</sub>IrO<sub>4</sub>
should turn into a topological superconductor once doped. Here, I exhibit a nanometer-scale
inhomogeneous doping-driven Mott insulator to pseudo-metallic phase transition. This work
further justifies the importance of using a local probe such as scanning tunnelling microscopy in
order to complete results on Mott physics obtained by integrative methods like angle-resolved
photoemission spectroscopy.

**Key words :** Correlations (superconductivity, Mott insulator, charge density wave), 2D materials (transition metal dichalcogenides, iridate), scanning tunnelling microscopy and spectroscopy, spin-orbit coupling, topological superconductivity
# Résumé

## Supraconductivité non conventionnelle dans des matériaux quasi-2D à fort couplage spin-orbite

La réalisation de supraconducteurs topologiques constitue un des principaux enjeux actuels de la physique de la matière condensée. Il a en effet été prédit que ces systèmes devaient abriter des fermions de Majorana. Ces fermions de Majorana disposent à la fois d'une statistique non-abélienne et, du fait de leur origine topologique, d'une robustesse face au désordre local, ce qui les rend très attrayants pour des applications en informatique quantique. Une approche susceptible de conduire à de la supraconductivité topologique consiste à considérer des systèmes supraconducteurs à fort couplage spin-orbite et brisant la symétrie d'inversion. C'est dans cette optique que, dans le cadre de cette thèse, j'ai effectué des mesures de microscopie et spectroscopie par effet tunnel sur des matériaux quasi-bidimensionnels : $(LaSe)_{1,14}(NbSe_2)_2$ et $Sr_2IrO_4$.

J'ai tout d'abord étudié les propriétés électroniques du matériau incommensurable $(LaSe)_{1,14}(NbSe_2)_2$, proche parent du composé dichalcogénure de métaux de transition 2H-NbSe$_2$. $(LaSe)_{1,14}(NbSe_2)_2$ est une hétérostructure faite d'alternances de biplans NbSe$_2$ à géométrie prismatique trigonale et de biplans de LaSe avec une structure de sel de roche. Le fort couplage spin-orbite ainsi que la non-centrosymétrie présents dans les plans NbSe$_2$ font de $(LaSe)_{1,14}(NbSe_2)_2$ un potentiel candidat pour de la supraconductivité topologique. Dans cette thèse, je présente des résultats de spectroscopie montrant que la structure électronique de $(LaSe)_{1,14}(NbSe_2)_2$ est similaire à celle de la monocouche de NbSe$_2$ avec un dopage de type électron accompagné par un déplacement du potentiel chimique de 0,3 eV, jusqu'alors inégalé. J'ai également pu démontrer la nature quasi-bidimensionnelle de $(LaSe)_{1,14}(NbSe_2)_2$ et notamment la présence d'un fort couplage spin-orbite de type Ising. De plus, la faible robustesse de la supraconductivité vis à vis du désordre non magnétique couplée à des mesures d'interférences de quasiparticules m'a permis de mettre en avant le caractère non conventionnel du paramètre d'ordre supraconducteur dans $(LaSe)_{1,14}(NbSe_2)_2$. Cette étude permet d'envisager l'utilisation d'hétérostructures incommensurables telles que $(LaSe)_{1,14}(NbSe_2)_2$ pour explorer la physique des dichalcogénures de métaux de transition dans la limite bidimensionnelle, pour laquelle de nombreuses études théoriques ont prédit une supraconductivité topologique.

Dans cette thèse, je présente également une étude des effets du dopage sur les propriétés électroniques de l'oxyde d'iridium $Sr_2IrO_4$. $Sr_2IrO_4$ est un isolant de Mott non conventionnel puisqu'il doit cette propriété à la présence d'un fort couplage spin-orbite. Du fait d'une brisure locale de la symétrie d'inversion, certaines prédictions théoriques ont pu montrer que $Sr_2IrO_4$ devrait devenir un supraconducteur topologique une fois dopé. Ici, je montre qu'avec le dopage, $Sr_2IrO_4$ subit une transition de phase inhomogène à l'échelle nanométrique entre un état isolant de Mott et un état pseudo-métallique. Ce travail justifie la pertinence d'utiliser une sonde locale telle que le microscope à effet tunnel afin de venir compléter des résultats sur la physique de Mott obtenus par des méthodes intégratives comme la spectroscopie électronique résolue en angle.
**Mots clés :** Phase corrélées (supraconductivité, isolant de Mott, onde de densité de charge), matériaux 2D (dichalcogénure de métaux de transition, oxyde d'iridium), microscopie et spectroscopie à effet tunnel, couplage spin-orbite, supraconductivité topologique
Remerciements

Il y a beaucoup de gens que je souhaiterais remercier après ces trois années de thèse.

Tout d'abord, j'aimerais remercier les membres du jury pour avoir accepté de prendre de leur temps pour évaluer mon travail. Je remercie également Sorbonne Université qui a financé cette thèse, et l'EDPIF qui m'a, entre autres, permis de participer à un certain nombre de formations réellement enrichissantes.

Ensuite, j'aimerais bien évidemment remercier tous les membres de l'équipe de Spectroscopie de Nouveaux Etats Quantiques. Tout d'abord, je voudrais remercier Christophe pour m'avoir initié aux méthodes de l'ultravide, de la cryogénie et de la microscopie/spectroscopie à effet tunnel, et de l'avoir toujours fait avec beaucoup de pédagogie. Je remercie également François et Pascal qui ont su nous tirer d'affaire plus d'une fois face à des problèmes techniques, mais aussi pour leur bonne humeur toujours au rendez-vous. Je remercie aussi chaleureusement Alexandra avec qui j'ai passé de nombreuses heures dans les sombres sous-sols de Jussieu et avec qui, à force de persévérance, j'ai pu effectuer plusieurs des expériences présentées dans ce manuscrit. Je voudrais tout particulièrement remercier Tristan pour sa disponibilité et pour les nombreuses discussions que nous avons pu avoir ensemble et qui se sont toujours révélées très enrichissantes autant sur le plan scientifique que sur le plan humain. Je souhaiterais aussi remercier Matteo, Cesare et Marco pour tout le support théorique qu'ils ont fourni et la patience dont ils ont fait preuve durant toutes les discussions animées que nous avons pu avoir. Bien entendu, je remercie tous les autres membres de l'équipe, à savoir, Yves, Marie D., Marie H., Danilo et Mathis avec qui je n'ai peut-être pas collaboré scientifiquement mais qui ont sans le moindre doute contribué à la bonne ambiance de l'équipe.

De façon plus générale je remercie tous les membres du laboratoire avec qui j'ai pu avoir des interactions et en particulier tous les autres thésards, avec qui, même s'il m'est parfois arrivé de rester enfermé trop longtemps dans les sombres sous-sols de Jussieu, j'ai passé de bons moments. Merci aussi à Betty, Laetitia et Isabelle de la gestion financière qui ont toujours su se montrer compréhensives même lorsqu'il fallait organiser des missions à l'étranger dans un trop court délai.

Je remercie David, Richard et Florin du service des basses températures qui ont permis à mes cryostats de s'abreuver régulièrement de doux nectars cryogéniques. Merci également à David H. et Yanling de la plateforme de mesures physique à basses températures de l'Université ainsi qu'à David M. pour nous avoir aidé à caractériser nos échantillons par EDX.

Je voudrais remercier Peter, Tomas, Pavol et Ondrei de l'Institut de Physique des Basses Températures de Kosice en Slovaquie pour la collaboration scientifique que nous avons mise en place et pour l'excellent accueil qu'ils nous ont réservé les deux fois que nous leur avons rendu visite. Un grand merci à Laurent et Shunsuke de l'Institut Jean Rouxel de Nantes qui ont fourni un grand nombre des cristaux mesurés pendant cette thèse.

Parmi tous mes collaborateurs j'aimerais adresser un très grand remerciement à Imad, avec qui j'ai beaucoup appris mais aussi à Lise avec qui j'ai également passé un certain temps dans les sombres sous-sols de Jussieu à me débattre avec des pointes capricieuses et à redouter de fourbes vibrations.

Merci à ma famille, à mes amis et à Pooja pour leur soutien constant et pour avoir feint de
comprendre mes tentatives d'explications de ce qu'est un « supraconducteur topologique ».

Bien entendu cette liste n'est pas exhaustive et je souhaiterais remercier une dernière fois
toute personne ayant contribué de quelque façon que ce soit, de près ou de loin au bon déroule-
ment de cette thèse, qui m'a beaucoup apporté.
# Contents

<table><tr><td>Abstract</td><td>3</td></tr><tr><td>Résumé</td><td>5</td></tr><tr><td>Remerciements</td><td>7</td></tr><tr><td>Introduction</td><td>3</td></tr><tr><td>1 Topological superconductivity</td><td>7</td></tr><tr><td>1.1 BCS theory and the Bogoliubov approach</td><td>7</td></tr><tr><td>1.2 Unconventional superconductivity</td><td>13</td></tr><tr><td>1.3 Inducing triplet superconductivity with spin-orbit coupling</td><td>15</td></tr><tr><td>1.3.1 Origin of spin-orbit coupling</td><td>15</td></tr><tr><td>1.3.2 Rashba spin-orbit coupling</td><td>16</td></tr><tr><td>1.3.3 Ising spin-orbit coupling</td><td>17</td></tr><tr><td>1.4 Topological superconductors</td><td>20</td></tr><tr><td>1.4.1 Introduction to topology in condensed matter systems</td><td>20</td></tr><tr><td>1.4.2 The Kitaev chain</td><td>24</td></tr><tr><td>1.4.3 The 2D p-wave superconductor</td><td>31</td></tr><tr><td>2 Scanning Tunnelling Microscopy/Spectroscopy</td><td>35</td></tr><tr><td>2.1 Topographic constant current mode</td><td>35</td></tr><tr><td>2.2 Probing the local density of states of samples via STS</td><td>37</td></tr><tr><td>2.2.1 Expression of the tunneling current</td><td>37</td></tr><tr><td>2.2.2 Tunneling Spectroscopy</td><td>38</td></tr><tr><td>2.2.3 Data treatment</td><td>40</td></tr><tr><td>2.2.4 Spectroscopic grid experiments</td><td>41</td></tr><tr><td>2.3 Quasiparticle interferences and Fourier transform STM</td><td>42</td></tr><tr><td>2.4 Experimental set-ups</td><td>46</td></tr><tr><td>2.4.1 M3 STM</td><td>46</td></tr><tr><td>2.4.2 LT Omicron STM</td><td>48</td></tr><tr><td>2.4.3 Preparation of the samples</td><td>49</td></tr><tr><td>3 Quasi-2D physics in a misfit heterostructure</td><td>53</td></tr><tr><td>3.1 Tuning the electronic properties of 2H-NbSe<sub>2</sub></td><td>53</td></tr><tr><td>3.1.1 Charge density wave order</td><td>54</td></tr><tr><td>3.1.2 Superconducting order</td><td>55</td></tr><tr><td>3.1.3 Effects of strain</td><td>56</td></tr><tr><td>3.1.4 Effects of disorder</td><td>58</td></tr><tr><td>3.1.5 Effects of doping</td><td>58</td></tr><tr><td>3.1.6 Reducing the number of NbSe<sub>2</sub> layers</td><td>59</td></tr><tr><td>3.2 Structure of misfit compound (LaSe)<sub>1.14</sub>(NbSe<sub>2</sub>)<sub>0.8</sub></td><td>60</td></tr></table>
<table><tbody><tr><td>3.2.1</td><td>Incommensurability probed by STM</td><td>62</td></tr><tr><td>3.3</td><td>(LaSe)<sub>1.14</sub>(NbSe<sub>2</sub>)<sub>2</sub> as electron-doped single-layer NbSe<sub>2</sub></td><td>65</td></tr><tr><td>3.3.1</td><td>Scattering potentials in quasiparticle interferences</td><td>65</td></tr><tr><td>3.3.2</td><td>Energy dispersion from DFT calculations</td><td>66</td></tr><tr><td>3.3.3</td><td>Comparison between the DFT calculations and the quasiparticle interferences</td><td>68</td></tr><tr><td>3.3.4</td><td>2 × 2 charge modulation order</td><td>72</td></tr><tr><td>3.4</td><td>(LaSe)<sub>1.14</sub>(NbSe<sub>2</sub>)<sub>2</sub> : an Ising superconductor</td><td>74</td></tr><tr><td>3.5</td><td>Topological superconductivity in monolayer NbSe<sub>2</sub></td><td>75</td></tr><tr><td>3.6</td><td>(LaSe)<sub>1.14</sub>(NbSe<sub>2</sub>)<sub>2</sub> in the superconducting phase</td><td>79</td></tr><tr><td>3.6.1</td><td>Measurement of the superconducting critical temperature</td><td>79</td></tr><tr><td>3.6.2</td><td>Measured differential conductance in the superconducting phase</td><td>81</td></tr><tr><td>3.6.3</td><td>Exclusion of a proximity effect between the bulk and the surface</td><td>82</td></tr><tr><td>3.6.4</td><td>Other signatures of superconductivity</td><td>83</td></tr><tr><td>3.6.5</td><td>QPIs in the superconducting phase</td><td>84</td></tr><tr><td>3.6.6</td><td>Interpretation of the small value of the superconducting gap in terms of tunnelling selectivity</td><td>88</td></tr><tr><td>3.7</td><td>Example of another misfit compound</td><td>88</td></tr><tr><td>3.8</td><td>Conclusion</td><td>89</td></tr></tbody></table>

<table><tbody><tr><td>4</td><td>Inhomogeneous insulating to pseudo-metallic transition in a doped Mott insulator</td><td>91</td></tr><tr><td>4.1</td><td>Predictions of topological superconductivity in doped Sr<sub>2</sub>IrO<sub>4</sub></td><td>91</td></tr><tr><td>4.2</td><td>Mott insulators</td><td>93</td></tr><tr><td>4.3</td><td>Spin-orbit induced Mott insulator Sr<sub>2</sub>IrO<sub>4</sub></td><td>97</td></tr><tr><td>4.4</td><td>Metal to insulator transition in (Sr<sub>(1-x)</sub>La<sub>x</sub>)IrO<sub>4</sub></td><td>98</td></tr><tr><td>4.4.1</td><td>Topographic measurements</td><td>99</td></tr><tr><td>4.4.2</td><td>Inhomogeneous transition from insulating to pseudo metallic phase induced by La doping</td><td>100</td></tr><tr><td>4.5</td><td>Orientational emergent order in (Sr<sub>(1-x)</sub>La<sub>x</sub>)IrO<sub>4</sub></td><td>109</td></tr><tr><td>4.6</td><td>Pure sample case</td><td>110</td></tr><tr><td>4.7</td><td>Conclusion</td><td>112</td></tr><tr><td colspan="2">Conclusion</td><td>113</td></tr><tr><td colspan="2">Appendices</td><td>115</td></tr><tr><td>.1</td><td>Computation of the tunneling conductance.</td><td>115</td></tr></tbody></table>
Introduction

Superconductivity was discovered more than a hundred years ago but still continues to be
one of the most active areas of investigation in condensed matter physics. The past decades
have indeed known the emergence of topological superconductivity, which generates increasing
interest because of the potential applications it has in the field of quantum computing. The
electron-hole nature of the low energy excitations of superconductors make them a natural and
ideal platform to look for Majorana fermions, whose non-Abelian statistics and robustness to
local disorder could be used in the implementation of some specific quantum algorithms. Such
exotic states are predicted to appear at the edges of topological superconductors.

The first approach historically adopted to realize topological superconductivity consists in considering superconducting systems with strong spin orbit coupling, inversion symmetry breaking and time reversal symmetry breaking (magnetism). This approach was first followed in 2012 by the group of L. P. Kouwenhoven which exhibited zero-bias states at the ends of an InSb superconductor-semiconductor nano-wire devices [1]. In 2014, based on a similar approach, the group of A. Yazdani also observed zero-energy states at the ends of ferromagnetic Fe chains grown on superconducting bulk Pb crystal [2]. In those two studies, the zero-bias states were interpreted as Majorana bound states. This approach was also followed during the PhD theses of the two previous students of our group (but this time in 2D systems). In his thesis, G. Ménard considered the following system : a superconducting Pb monolayer grown on top of a Si(111) substrate with underlying magnetic CoSi islands. The idea was to tune the Pb monolayer into a topologically non-trivial regime right above the magnetic CoSi islands. Depending on the magnetic texture of the CoSi islands, G. Ménard observed states which were interpreted as chiral dispersive Majorana states and Majorana bound states, respectively [3, 4]. In the thesis of D. Longo, magnetic auto-organized manganese phthalocyanine molecules islands were grown on superconducting Pb trilayers [5]. Again, D. Longo observed what was interpreted as dispersive chiral Majorana states at the edges of the magnetic islands.

Another approach consists in inducing superconductivity by proximity into the topological
surface state of a topological insulator. Here, breaking time reversal symmetry is not required to
turn the system into a topologically non-trivial phase. However, applying an external magnetic
field is necessary to point out the topologically non-trivial character of the material. On this
basis, Majorana bound states were observed at the cores of vortices in both Bi₂Te₃/NbSe₂
heterostructures [6] and FeTe₀.₅₅Se₀.₄₅ [7].

In this thesis, we address another approach in which time reversal symmetry is preserved. Superconducting triplet correlations are one of the ingredients needed for the realization of topological superconductivity. Such unconventional triplet correlations can be induced in superconducting systems with strong spin-orbit coupling and in which the inversion symmetry is broken. Recent theoretical studies suggest that topologically non trivial triplet states (possibly p-wave or f-wave) could be realized in few layers transition metal dichalcogenide compounds which possess both strong spin-orbit coupling and inversion symmetry breaking [8]. In it in this framework that we carried out scanning tunnelling microscopy/spectroscopy experiments on superconducting misfit transition metal dichalcogenide (LaSe)₁,₁₄(NbSe₂)₂, which also possesses the strong spin-orbit coupling and inversion symmetry breaking requirements [9]. The other
compound that was studied in this thesis is electron doped $Sr_2IrO_4$, which possesses a strong spin-orbit coupling reminiscent of a Rashba interaction. It was predicted that electron or hole doped $Sr_2IrO_4$ should enter a topological superconducting phase [10]. Here, superconductivity was not observed but we exhibit Mott physics signatures strongly correlated to the presence of intrinsic spin-orbit coupling in the material.

This thesis is organized in four chapters. The first chapter is a general introduction to the notion of topology in condensed matter physics systems, and more particularly in superconductors. The second chapter is dedicated to the description of the experimental method used during this thesis, namely scanning tunnelling microscopy/spectroscopy. Theoretical as well as technical aspects (such as data analysis, sample preparation or principle of the measurements) are presented. The third and fourth chapters present and discuss experimental results obtained in $(LaSe)_{1,14}(NbSe_2)_2$ and electron-doped $Sr_2IrO_4$, respectively.

**Chapter 1:** In this introduction chapter, we develop the theoretical background necessary to address topological superconductivity. Firstly, we present the microscopic description of conventional superconductors through Bardeen-Cooper-Schrieffer theory and the Bogoliubov-de-Gennes approach. On this basis, we propose a generalized mathematical expression for the superconducting order parameter which naturally leads to the notion of unconventional superconductivity. Secondly, we explain the appearance of superconducting triplet correlations in systems with strong spin-orbit coupling and inversion symmetry breaking. Finally, we give a general introduction to the concept of topology in condensed matter physics and more particularly in the case of topological superconductors. The topical examples of the Kitaev chain and the 2D chiral p-wave superconductor are detailed to illustrate the notion of topological classes and topological invariants. Moreover, we show that those systems develop Majorana states at their edges. We discuss the non-Abelian statistics of such exotic states and their possible applications in quantum computing. At the end of the chapter, we present and comment the state-of-the-art potential experimental realizations of topological superconductivity to date.

**Chapter 2:** In this chapter, we first introduce the theoretical background behind Scanning Tunnelling Microscopy and Scanning Tunnelling Spectroscopy (STM/STS). We show that this powerful experimental technique can be used as a local probe to both image the surfaces of materials and access their local density of states. We then present the principle of a spectroscopic grid experiment, often used during this thesis. More particularly, we explain the notion of quasi-particle interferences (QPIs) and how it is possible to take advantage of it *via* Fourier transform of STM/STS images to artificially reconstruct the band structures of materials. Furthermore, we give a short introduction to Ultra-High-Vacuum and very low temperatures experiments. Finally, we present the experimental set-ups on which most of the measurements were carried out during these three years. The advantages and limitations of the technique are discussed.

**Chapter 3:** This chapter is dedicated to the study of superconducting transition metal dichalcogenide misfit compound $(LaSe)_{1,14}(NbSe_2)_2$. Firstly, we present STM/STS measurements performed in the normal phase. The incommensurate potential of misfit compound $(LaSe)_{1,14}(NbSe_2)_2$ acts as non-magnetic disorder and allows for the observation of QPIs. Our data are in very good agreement with Density Functional Theory calculations carried out by M. Calandra, C. Tresca and M. Campetella. Everything suggests that $(LaSe)_{1,14}(NbSe_2)_2$ behaves as electron-doped monolayer $NbSe_2$ with a shift of the chemical potential of approximatively 0,3 eV, which is a level of doping which previous studies on few layers $2H-NbSe_2$ could never attain. Moreover, our QPIs signal presents a splitting which can only be interpreted in the presence of a strong Ising spin-orbit coupling. Such kind of spin-orbit is predicted to induce unconventional superconducting triplet correlations. This is why we secondly present a similar
study of (LaSe)₁,₁₄(NbSe₂)₂, but this time, in the superconducting phase. STM/STS mea-
surements reveal a filling of the superconducting gap which we interpret as a consequence of
the unconventional nature of the superconductivity, which is thus intrinsically weak against
non-magnetic disorder. QPIs patterns obtained in the superconducting phase do not show any
dispersion, which calls for a nodeless order parameter, possibly p-wave or f-wave. This work
opens the possibility to use misfit heterostructures to further investigate quasi-2D physics in
few layers 2H-NbSe₂.

**Chapter 4 :** In this chapter, we present a STM/STS study of electron-doped iridate com-
pound (Sr*(1-x)La*x*)₂IrO₄. The pristine compound Sr₂IrO₄ is a spin-orbit-induced Mott insulator
and is predicted to turn into a topological superconductor once electron or hole doped. Here,
we exhibit a Mott insulator to pseudo-metallic transition driven by La doping. Spectroscopic
grid measurements revealed the inhomogeneous nature of this transition with the apparition of
nanometer-size pseudo-metallic puddles on top of an insulating background. As the La doping
increases, the puddles become more and more metallic but the background remains strongly
insulating. We then confront our results with previous studies performed in similar systems.
Moreover, we discuss the correlations between the position of La dopants and pseudo-metallic
puddles and exhibit of a short-range orientational order in the puddles organization. Our study
justifies the interest of using local probes such as STM in addition to integrative methods like
Angle Resolved PhotoEmission Spectroscopy for which the interpretation of the results can be
delicate. Our work is the first one which explores the full La doping range including pristine
Sr₂IrO₄.
# Chapter 1

## Topological superconductivity

During the 19th century, physicists were racing for the liquefaction of gases, and *a fortiori*, contesting in accessing the lowest temperatures ever observed. In 1877, Louis Paul Cailletet and Raul Pictet independently liquified dioxygen for the first time, thus successfully reaching 90,2 K. In 1899, James Dewar was able to go as low as 20,18 K to liquify dihydrogen, which was previously thought as being a "permanent gas". In 1908, in Leiden, Netherlands, physicist Heike Kamerlingh Onnes was able to first liquify Helium at 4,2 K. This frantic race was motivated by an increasing need of looking at the properties of matter at very low temperatures. It is in this context that in 1911, Heike Kamerlingh Onnes got interested in measuring temperature dependence of the resistivity of mercury. By then, several theories were envisaged as to what should happen once getting closer to absolute zero. The experimental results were highly unexpected: a sharp drop in the resistivity was observed below 4,2 K. This experiment marked the discovery of superconductivity. Below a given critical temperature, superconductors, develop the ability of conducting electric currents without any dissipation. In addition to possess zero electric resistance, Walther Meissner and Robert Ochsenfeld discovered in 1933 that superconductors also have the property of expelling any magnetic field, thus making them perfect diamagnets. This second notable property is called the "Meissner effect" and is responsible for the magnetic levitation of superconductors. By 1933, superconductivity still remained not understood at all. It is only in 1935, that the brothers Fritz and Heinz London proposed an electrodynamical description of superconductivity, their famous equations being able to successfully explain the Meissner effect [11]. In 1950, Vitaly Lazarevich Ginzburg and Lev Landau imagined their empirical "Ginzburg-Landau" theory, still widely used today [12]. It is a second order phase transition theory in which the superconducting gap plays the role of the order parameter. This theory helped describing most of the macroscopic properties of superconductors, but did not provide any microscopic origin for the phenomenon. The first microscopic description of superconductivity was developed seven years later in 1957 by theoreticians John Bardeen, Leon Neil Cooper, and John Robert Schrieffer [13]. It is called the BCS theory, after the names of its inventors. It will be the topic of the following section.

### 1.1 BCS theory and the Bogoliubov approach

In 1956, Leon Cooper showed that if two electrons near the Fermi surface interact attractively, no matter how small the interaction, those two electrons will form a bound state, thus lowering the total energy of the system and leading to a "Fermi surface collapsing" [14]. This is exactly what happens in superconductors, where an attractive interaction bounds the electrons in pairs. This newly formed pair of electrons is called a "Cooper pair". In a conventional superconductor, the interaction between the electrons of these pairs is mediated by the phonons of the lattice they are moving in.
The BCS Hamiltonian.

The Hamiltonian describing the states of non-interacting electrons for a system of chemical potential $\mu$ is the following :

$$ \hat{H}_0 = \sum_{k,\sigma} [\varepsilon_k c_{k,\sigma}^{\dagger} c_{k,\sigma} - \mu c_{k,\sigma}^{\dagger} c_{k,\sigma}] = \sum_{k,\sigma} \xi_k c_{k,\sigma}^{\dagger} c_{k,\sigma}, \quad (1.1) $$

where $k$ is the momentum of an electron and $\sigma$ its spin ($\uparrow\downarrow$). $\varepsilon_k$ is the electrons dispersion relation and $\xi_k = \varepsilon_k - \mu$.

The most general Hamiltonian (which conserve total momentum and does not allow spin flip) describing the phonon-mediated interaction between the electrons is :

$$ \hat{H}_{int} = \frac{1}{N} \sum_{k, k', q, \sigma, \sigma'} V(k, k', q) c_{k+q, \uparrow}^{\dagger} c_{k'-q, \downarrow}^{\dagger} c_{k', \sigma'} c_{k, \sigma}, \quad (1.2) $$

In the original article by Bardeen, Cooper and Schrieffer (see [13]) it is suggested that the exchange interaction energetically favors interacting electrons with opposite spins such that :

$$ \hat{H}_{int} = \frac{1}{N} \sum_{k, k', q} V(k, k', q) c_{k+q, \uparrow}^{\dagger} c_{k'-q, \downarrow}^{\dagger} c_{k', \downarrow} c_{k, \uparrow}, \quad (1.3) $$

A different and more satisfying justification of this simplification will be given shortly.

The arguments that are about to be presented follow the original description given by Bardeen, Cooper and Schrieffer [13]. Since we are using second quantization formalism, we can use a Foch space basis to describe the states. The interaction will tend to reduce the energy of the system if $\hat{H}_{int}$ connects a great number of Foch states via non-zero negative matrix elements. We can notice that the effect of the $V(k, k', q) c_{k+q, \uparrow}^{\dagger} c_{k'-q, \downarrow}^{\dagger} c_{k', \downarrow} c_{k, \uparrow}$ term is to destroy a $(k' \downarrow, k \uparrow)$ pair of states and create a pair of states $(k' - q \downarrow, k + q \uparrow)$ while conserving the total momentum $(k + k')$. Let us introduce the two Foch states :

$$ \begin{cases} |\psi_1\rangle = c_{k', \downarrow}^{\dagger} c_{k, \uparrow}^{\dagger} |k \uparrow: 0, k' \downarrow: 0, (k+q) \uparrow: 1, (k'-q) \downarrow: 1, S\rangle, \\ |\psi_2\rangle = c_{k+q, \uparrow}^{\dagger} c_{k'-q, \downarrow}^{\dagger} |k \uparrow: 1, k' \downarrow: 1, (k+q) \uparrow: 0, (k'-q) \downarrow: 0, S\rangle, \end{cases} \quad (1.4) $$

where 0 means that the corresponding electronic state is unoccupied and 1 that it is occupied. $S$ is a same collection of electronic states occupied in both $|\psi_1\rangle$ and $|\psi_2\rangle$.

The matrix elements between $|\psi_1\rangle$ and $|\psi_2\rangle$ Foch states are non-zero. The first conclusion is the following : $\hat{H}_{int}$ tends to stabilize the formation of $(k' \downarrow, k \uparrow)$ pairs of states. Those pairs of states are closely related to the previously mentioned Cooper pairs. Also, the number of non-zero matrix elements increases if all occupied pairs share the same total momentum $Q = (k + k')$. The second conclusion is that the most stable state is a superposition of paired states with same total momentum $Q = (k + k')$. Finally, the lowest energy configuration is the one where all the pairs have zero total momentum $Q = 0$ and a fortiori, their centers of mass have 0 kinetic energy.

As a summary, in a conventional superconductor, the electrons forming a Cooper pair have opposite momenta and spins. Because the interactive interaction is mediated by the phonons, only electrons of energy inferior to $\hbar\omega_D$ are likely to form Cooper pairs, where $\omega_D$ is the Debye frequency, which is the maximum vibration frequency in a crystal lattice.

Considering all these elements, one can add to $\hat{H}_0$ the following effective Hamiltonian in order to describe the interactions in a superconductor :

$$ \begin{cases} \hat{H}_{int} = \frac{1}{N} \sum_{kk'} V_{kk'} c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger} c_{-k',\downarrow} c_{k',\uparrow}, \\ V_{kk'} = \begin{cases} -V_0 & \text{if } |\xi_{k/k'}| \le \hbar\omega_D \\ 0 & \text{else} \end{cases}, \end{cases} \quad (1.5) $$
$V_{kk'}$ is taken equal to a constant for energies lower than $\hbar\omega_D$ because the electrons of the pair are close to Fermi level, and thus $k \simeq k' \simeq k_F$, where $k_F$ is the momentum of an electron at Fermi level. Written in this way, we see that this **constant** interaction is the Fourier transform of a contact interaction going as $V_0\delta(r-r')$, where $r$ and $r'$ are the positions of the two interacting electrons. Such an interaction can only occurs on a given site with the two electron sharing the same orbital and consequently having opposite spins to respect Pauli's exclusion principle.

The total Hamiltonian describing a conventional superconductor is $\hat{H}_{BCS} = \hat{H}_0 + \hat{H}_{int}$.

## Bogoliubov mean field approach.

We will now present a mean field calculation first justified by Bogoliubov in 1958 (see [15, 16]), which allows to have a quadratic form for the Hamiltonian describing the interactions. Indeed, we have:

$$ \left\{ \begin{array}{l} c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger} = \langle c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger} \rangle + \underbrace{(c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger} - \langle c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger} \rangle)}_{\delta(c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger})}, \\ c_{-k',\downarrow} c_{k',\uparrow} = \langle c_{-k',\downarrow} c_{k',\uparrow} \rangle + \underbrace{(c_{-k',\downarrow} c_{k',\uparrow} - \langle c_{-k',\downarrow} c_{k',\uparrow} \rangle)}_{\delta(c_{-k',\downarrow} c_{k',\uparrow})}, \end{array} \right. \qquad (1.6) $$

where $\langle...\rangle$ is the mean value taken over all the eigenstates of the perturbed system.

The mean field approximation consists in assuming that the action of the operator $\delta(c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger}) \delta(c_{-k',\downarrow} c_{k',\uparrow})$ on the eigenstates of the system is negligible. This approximation works well for systems with many Cooper pairs in a volume of size the typical size of a Cooper pair to the cube. Bogoliubov indeed showed that those terms were under extensive as their contribution grows more slowly that the total volume of the system. After calculation it gives:

$$ \left\{ \begin{array}{l} \hat{H}_{BCS} \approx \hat{H}_{MF} = \sum_k [\xi_k(c_{k,\uparrow}^{\dagger}c_{k,\uparrow} + c_{-k,\downarrow}^{\dagger}c_{-k,\downarrow}) - \Delta_k c_{k,\uparrow}^{\dagger}c_{-k,\downarrow}^{\dagger} \\ \qquad \qquad \qquad \qquad - \Delta_k^* c_{-k,\downarrow} c_{k,\uparrow} + \Delta_k \langle c_{k,\uparrow}^{\dagger}c_{-k,\downarrow}^{\dagger} \rangle + \Delta_k^* \langle c_{-k,\downarrow} c_{k,\uparrow} \rangle], \\ \Delta_k = -\sum_{k'} V_{kk'} \langle c_{-k',\downarrow} c_{k',\uparrow} \rangle, \end{array} \right. \qquad (1.7) $$

We will see in the following that this Hamiltonian can be diagonalized by applying a Bogoliubov transform:

$$ \left\{ \begin{array}{l} \gamma_{k,\uparrow} = u_k c_{k,\uparrow} - v_k c_{-k,\downarrow}^{\dagger}, \\ \gamma_{-k,\downarrow} = v_k c_{k,\uparrow}^{\dagger} + u_k c_{-k,\downarrow}, \end{array} \right. \qquad (1.8) $$

By requiring that $|u_k|^2 + |v_k|^2 = 1$ one can check that $\begin{cases} \{\gamma_\alpha, \gamma_\beta\} = \{\gamma_\alpha^\dagger, \gamma_\beta^\dagger\} = 0 \\ \{\gamma_\alpha, \gamma_\beta^\dagger\} = \delta_{\alpha,\beta} \end{cases}$. This means that the operators $\gamma_\alpha$ are fermionic operators.

The inverse transformation is:

$$ \left\{ \begin{array}{l} c_{k,\uparrow} = u_k^* \gamma_{k,\uparrow} + v_k \gamma_{-k,\downarrow}^{\dagger}, \\ c_{-k,\downarrow}^{\dagger} = -v_k^* \gamma_{k,\uparrow} + u_k \gamma_{-k,\downarrow}^{\dagger}, \end{array} \right. \qquad (1.9) $$

One can replace the $c$ operators in equation 1.7 by their expression as a function of the $\gamma$ operators (we no longer consider the constant term $\sum_k (\Delta_k \langle c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger} \rangle + \Delta_k^* \langle c_{-k,\downarrow} c_{k,\uparrow} \rangle)$ in the mean field Hamiltonian). Besides, if one imposes that the terms in $\gamma_{k,\uparrow}^{\dagger} \gamma_{-k,\downarrow}^{\dagger}$ and in $\gamma_{-k,\downarrow} \gamma_{k,\uparrow}$ are zero (condition for the system to be diagonal in the $\gamma$ basis), then one gets the following relation:

$$ 2\xi_k u_k v_k - \Delta_k^*(-v_k v_k) - \Delta_k(u_k u_k) = 0, \qquad (1.10) $$
By multiplying 1.10 by $\frac{\Delta_k^*}{u_k^2}$, one finds a second degree polynomial equation for $x_k = \frac{\Delta_k^* v_k}{u_k}$ which solutions are: $x_k^\pm = -\xi_k \pm \sqrt{\xi_k^2 + |\Delta_k|^2}$. At high energy, which means far from Fermi level

($\xi_k \to +\infty$), one must have $|v_k| \to 0$. Cooper pairs can, indeed, only be formed of electrons close to the Fermi level, hence, high energy excitations have to behave as electrons and thus $\gamma_{k,\uparrow} \xrightarrow{\xi_k \to +\infty} u_k c_{k,\uparrow} \Leftrightarrow |v_k| \to 0$. The correct solution of the polynomial equation then is $x_k^-$. Finally, by using the fact that $|u_k|^2 + |v_k|^2 = 1$, one finds :

$$ \left\{ \begin{array}{l} |v_k|^2 = \frac{1}{2} \left(1 - \frac{\xi_k}{E_k}\right), \\ |u_k|^2 = \frac{1}{2} \left(1 + \frac{\xi_k}{E_k}\right), \end{array} \right. \qquad (1.11) $$

In the end, the mean field Hamiltonian is the following :

$$ \hat{H}_{MF} = \sum_k E_k (\gamma_{k,\uparrow}^{\dagger} \gamma_{k,\uparrow} + \gamma_{-k,\downarrow}^{\dagger} \gamma_{-k,\downarrow}). \qquad (1.12) $$

One can see after this derivation that $\hat{H}_{MF}$ is diagonal in the $\gamma$ operators basis. The $\gamma^{(\dagger)}$ operators are quasiparticles annihilation (creation) operators. Those quasiparticles of energies $E_k = \pm \sqrt{\xi_k^2 + |\Delta_k|^2}$ are fermions and are sometimes called " bogoliubons". They are the low energy excitations of a superconductor.

<figure><img src="image_2.png" /><figcaption>Figure 1.1: Illustration of the non-conservation of the total Cooper pairs number through low energy excitations of the system as combinations of creation and annihilation operators. a. System with 7 Cooper pairs. b. System a on which the annihilation of an electron of spin up was applied. c. System with 6 Cooper pairs. d. System c on which the creation of an electron of spin down was applied. The b and d configurations are equivalent.</figcaption></figure>

By looking at equation 1.8, one can notice that the bogoliubons are combinations of annihilation and creation fermionic operators. This means that the low energy excitations of a
superconductor are a mixture of both electrons and holes. They consist in adding **and** taking off electrons, or saying differently, it is equivalent to either take or give an electron to the system. The only way to understand this is to admit that the total number of Cooper pairs is not conserved as illustrated in figure 1.1. Also, if we look at the expression of $\Delta_k$ in equation 1.7, we see that is it a combination of terms like $\langle cc \rangle$ which can only be non-zero if we are working in grand canonical ensemble for which the number of particles is not conserved.

In figure 1.1 **a**, the system has seven Cooper pairs and in figure 1.1 **c**, it has six Cooper pairs. In figure 1.1 **b**, the annihilation operator $c_↑$ has been applied to the **a** configuration, and in figure 1.1 **d**, the creation operator $c_↓^†$ has been applied to the **c** configuration. The **b** and **d** configurations are equivalent to one another. The ground state $|\psi_{BCS}\rangle$ of the superconducting state necessarily is a linear combination of states with different Cooper pairs numbers.

## The BCS ground state.

The BCS ansatz for the ground state of the superconducting system is the following:

$$|\psi_{BCS}\rangle = \prod_k (u_k + v_k c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger) |0\rangle, \qquad (1.13)$$

Where $c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger$ is a pair state $(k_↑, -k_↓)$ creation operator and $|0\rangle$ is the fermionic vacuum.

One can notice that $|\psi_{BCS}\rangle$ acts like the vacuum for any bogoliubon annihilation, meaning that $\gamma_{k,\sigma} |\psi_{BCS}\rangle = 0$ (with $\sigma = \uparrow, \downarrow$). The justification of this ground state was given in 1958 by P. W. Anderson [17]. In his article, he defines the operator $B^\dagger = \sum_k \varphi_k c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger$, which can be interpreted as a Cooper pair creation operator. By being the composition of two fermions, the Cooper pairs are what we call composite bosons. This is the reason why Anderson suggests in his article that the ground state can be written in the same way as the coherent macroscopic ground state of a Bose-Einstein condensate:

$$ \begin{aligned} |\phi_{BCS}\rangle &= \sum_{N=0}^{\infty} \frac{1}{N!} B^{\dagger N} |0\rangle = e^{B^\dagger} |0\rangle = \prod_k e^{\varphi_k c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger} |0\rangle \\ &= \prod_k \left(1 + \varphi_k (c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger) + \frac{1}{2} \varphi_k^2 (c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger)^2 + \dots \right) |0\rangle, \end{aligned} \qquad (1.14) $$

Because of the fermionic nature of the $c^\dagger$ operators, $(c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger)^n$ with $n > 1$ terms have zero contribution. Then:

$$|\phi_{BCS}\rangle = \prod_k (1 + \varphi_k c_{k,\uparrow}^\dagger c_{-k,\downarrow}^\dagger) |0\rangle, \qquad (1.15)$$

By posing $\varphi_k = \frac{v_k}{u_k}$ and by normalizing $|\phi_{BCS}\rangle$, on finds back the expression of equation 1.13. This calculation was taken from ref. [18].

$\|v_k\|^2 (\|u_k\|^2)$ is interpreted as the probability that the pair state $(k_↑, -k_↓)$ is occupied (unoccupied). Since superconductivity mixes electrons and holes up, the energy dispersion $E_k$ can be understood in terms of band interaction between the electrons band of dispersion $\xi_k$ and the hole band of dispersion $-\xi_k$ as it is represented in figure 1.2 **a**. This mixing between electrons and holes will be of the utmost importance in the framework of topological superconductivity (see section 1.4).

As it is shown in ref. [19], it is possible to choose $u_k$ real and $v_k$ complex with a $e^{i\varphi}$ phase equal to the phase of the superconducting order parameter, after what the BCS ground reads as:
$$|\psi_{BCS}\rangle = \prod_k (|u_k| + |v_k| e^{i\varphi} c_{k,\uparrow}^{\dagger} c_{-k,\downarrow}^{\dagger}) |0\rangle, \qquad (1.16)$$

Written in this way, we see that the BCS ground state actually is a coherent state of Cooper pairs sharing the same phase and total momentum. The superconducting state is a condensate made out of these composite bosons and shares many similarities with Bose-Einstein condensate of regular bosons. Let us imagine that an electric field has been applied to the superconductor, thus giving a momentum to the centers of mass of the Cooper pairs and inducing the apparition of an electric current. In a normal metal, the finite resistivity comes from the scattering of carriers on any source of disorder, i.e., on anything that breaks the periodicity of the crystal. By encountering disorder, a Cooper pair can potentially scatter from a state of total momentum $Q$ to a state of total momentum $Q' \neq Q$, but, as it was said previously, in a superconductor, all the Cooper pairs must share the same total momentum $Q$. For scattering processes to be efficient and to lead to dissipation, one would thus have to imagine simultaneous scattering of all Cooper pairs, which is very unlikely. This argument, while not being detailed here, helps us explaining the zero-resistivity observed in superconductors. This property highly resembles the superfluidity seen in Bose-Einstein condensates. Moreover, superconduction permits to host very high intensity currents (called supercurrents) which can appear in response to strong magnetic fields, thus leading to perfect diamagnetism and Meissner effect. The two remarkable properties of superconductors can be understood within the BCS theory framework.

## The BCS density of states.

The density of states of states $N(E_k)$ of the low energy excitations can be defined as $N(E_k)dE_k = N_0d\xi_k$, where $N_0$ is the electronic density of states at Fermi level. If $\Delta_k$ is constant and equal to $-\Delta$, where $\Delta$ is called the gap (we will see in an instant why), then after calculation :

$$N(E_k) = N_0 \frac{|E_k|}{\sqrt{E_k^2 - |\Delta|^2}}, \qquad (1.17)$$

This superconducting density of states is represented on figure 1.2 b. We see on the plot that there are no excitations of energies inferior in absolute value to $\Delta$. This can be interpreted as the fact that one has to provide an energy $2\Delta$ in order to break a Cooper pair and thus pass from a superconducting state to a normal one.

This microscopic BCS theory was very successful in describing s-wave superconductors, where the origin of the pairing of the electrons is the electron-phonon interaction, but in some materials, other interaction mechanisms can also lead to superconductivity. Such materials are called unconventional superconductors. Unconventional superconductivity will be the topic of the following section. The next section will be dedicated to the different possible symmetries for the superconducting order parameter and will not focus on the nature of the interactions leading to unconventional superconductivity.
<figure><img src="image_3.png" /><figcaption>Figure 1.2: a. Dispersion a conventional superconductor. b. Density of states of the low-energy excitations of a conventional superconductor.</figcaption></figure>

## 1.2 Unconventional superconductivity

A Cooper pair is a two-identical-particles fermionic state, thus, its wave function $\psi(k, -k, s_1, s_2)$ (where $k$ and $-k$ are the momenta of the electrons and $s_1$ and $s_2$ are the spin of the electrons) must be antisymmetric under the exchange of the two electrons and one has: $\psi(k, -k, s_1, s_2) = -\psi(-k, k, s_2, s_1)$. In the previous section, we have seen that for conventional superconductor the expression of the gap function was $\Delta_{k,\downarrow,\uparrow} = -\sum_{k'} V_{k,k'} \langle c_{-k',\downarrow}c_{k',\uparrow} \rangle$. This notation is a bit different than the one of the previous section since it displays an explicit spin dependence. The reason behind this notation should become clear shortly. The idea is to compare $\Delta_{k,\downarrow,\uparrow}$ to $\Delta_{-k,\uparrow,\downarrow}$ to see if the gap function possesses the same antisymmetric property as the Cooper pair's wave function :

$$ \Delta_{-k,\uparrow,\downarrow} = -\sum_{k'} V_{-k,k'} \langle c_{-k',\uparrow}c_{k',\downarrow} \rangle, \qquad (1.18) $$

$$ \Delta_{-k,\uparrow,\downarrow} = -\sum_{k'} V_{-k,-k'} \langle c_{k',\uparrow}c_{-k',\downarrow} \rangle, \qquad (1.19) $$

Assuming a time reversal symmetric potential $V_{k,k'}$ and because of the anticommutation relation of fermionic operators, one finally has :

$$ \Delta_{-k,\uparrow,\downarrow} = +\sum_{k'} V_{k,k'} \langle c_{-k',\downarrow}c_{k',\uparrow} \rangle = -\Delta_{k,\downarrow,\uparrow}, \qquad (1.20) $$

Interestingly, for a conventional superconductor, the gap function, which is a function of $k$ and of the spins of the electrons, is also antisymmetric under particle exchange. This can be generalized to all types of superconductors and it is why it is often said that the gap function can roughly be assimilated to the Cooper pair's wave function. Given this last statement, one understands that the gap function has both a spatial and a spin part. If one part is antisymmetric, the other is necessarily symmetric (to respect the antisymmetry of the total wave function).
In the case of a conventional superconductor, after recalling that $V_{kk'} = \begin{cases} -V_0 & \text{if } |\xi_{k/k'}| \le \hbar\omega_D \\ 0 & \text{else} \end{cases}$, one clearly sees that $\Delta_{k,\downarrow\uparrow} = \Delta_{-k,\downarrow\uparrow}$. This means that the spatial part of the gap function is even in $k$ (symmetric) implying that the spin part is antisymmetric. It is then possible to write :

$$ \Delta_{\alpha\beta}(k) = \Delta(k)\chi_{\alpha\beta}, \qquad (1.21) $$

where $\Delta_{\alpha\beta}(k)$ is the gap function for any arbitrary type of superconductivity, $\Delta(k)$ is the spatial part of the gap function and $\chi_{\alpha\beta}$ (a $2 \times 2$ spinor matrix) is the spin part of the gap function. The $\alpha$ and $\beta$ indices refer to the spin of the paired state. Starting from this expression one has the two following possibilities (respecting $\Delta_{\alpha\beta}(k) = -\Delta_{\beta\alpha}(-k)$) :

$$ \left\{ \begin{array}{ll} \chi_{\alpha\beta} = -\chi_{\beta\alpha} & \text{and } \Delta(k) = \Delta(-k), \\ \chi_{\alpha\beta} = \chi_{\beta\alpha} & \text{and } \Delta(k) = -\Delta(-k), \end{array} \right. \qquad (1.22) $$

The first case is referred to as spin-singlet pairing because the total spin of a Cooper pair $S$ is equal to 0, while the second case is referred to as spin-triplet pairing because $S = 1$.

In all generality, the gap function can always be written as such :

$$ \Delta_{\alpha\beta}(k) = [\Delta_0(k) + \vec{d}(k) \cdot \vec{\sigma}](i\sigma_2)_{\alpha\beta}, \qquad (1.23) $$

where $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$ and $\sigma_1, \sigma_2$ and $\sigma_3$ are the Pauli matrices. $(i\sigma_2)_{\alpha\beta}$ is in factor because $\sigma_2$ is the only antisymmetric Pauli matrix. Also, the product of any Pauli matrix with $\sigma_2$ is a symmetric matrix. This implies that $\Delta_0(k)$ is a even function of $k$ since $\Delta_0(k)(i\sigma_2)_{\alpha\beta}$ has to by antisymmetric. On the contrary $\vec{d}(k)$ is necessarily odd in $k$. Hence, $\Delta_0(k)(i\sigma_2)_{\alpha\beta}$ and $(\vec{d}(k) \cdot \vec{\sigma})(i\sigma_2)_{\alpha\beta}$ encode the singlet and triplet parts of the superconducting order parameter, respectively.

In conventional superconductors, the Cooper pairs have a total angular momentum $L$ equal to 0 which is why they are referred to as "s-wave". A Cooper pair with angular momentum $L = 1$ is triplet and the corresponding superconductivity is called "p-wave". If $L = 2$, the pairing is singlet, and the superconducting order parameter is d-wave. This kind of pairing is in particular encountered in cuprate-based high-$T_c$ superconductors [20]. $L = 3$ represents a triplet f-wave pairing. More generally, an even angular momentum characterizes a singlet pairing, while an odd angular momentum characterizes a triplet pairing.

In 1961, P.W. Anderson showed that s-wave superconductors are very robust against non-magnetic disorder [21]. The Anderson's theorem tells that only interactions which break time reversal symmetry (like with magnetic impurities) are likely to induce in-gap states in an s-wave superconductor and consequently, potentially kill superconductivity. Things are very different in unconventional superconductors, where the symmetry of the Cooper pair wave function necessarily imposes sign changes in the superconducting order parameter, leading to the apparition of in-gap states, even in presence of non-magnetic disorder. This fragility to non-magnetic disorder is actually a signature of unconventional superconductivity and was especially observed in the framework of this thesis in the study of superconducting misfit compound (LaSe)_{1,14}(NbSe_2)_{2} (see chapter 3).

Triplet superconductivity is one of the possible requirements to engineer topological superconductivity. For instance, the $p_x + ip_y$ 2D superconductor is the most famous example of p-wave pairing. It is characterized by $\Delta_0(k) = 0$ and $\vec{d}(k) = (k_x + ik_y, k_x + ik_y, 0)$ and is known to host Majorana bound states at the interface with superconductors of different topology. This will be addressed in great details in sections 1.4.2 and 1.4.3.

Historically, triplet superconductivity was observed for the first time in organic superconductors [22, 23]. Since then, triplet superconductivity was also discovered in other materials such as the heavy-fermions compounds [24, 25, 26]. In such compounds, it is thought that the superconducting state is due to spin fluctuations pairing mechanism rather than to phonon-mediated
attractive interaction like in conventional superconductors. This kind of exotic mechanism is not necessary to obtain triplet superconductivity. A mixed singlet-triplet superconducting order parameter can indeed be induced in materials with strong spin-orbit coupling and in which inversion symmetry is broken [27]. This will be the topic of the next section.

# 1.3 Inducing triplet superconductivity with spin-orbit coupling

## 1.3.1 Origin of spin-orbit coupling

The Dirac equation is a relativistic quantum wave equation formulated in 1928 by Paul Dirac. It was, at first, an attempt to incorporate special relativity to quantum theory. Schrödinger's equation is not invariant under Lorentz transformations since it presents second order derivative in space while having first order derivative in time. Space and time are thus not on equal footing as they should according to special relativity. The idea of Dirac was to find an equation which presented first order derivatives for both space and time coordinates. This last prerequisite necessarily implies a multi-components wave function. The Dirac equation reads as :

$$i\hbar \frac{\partial |\psi\rangle}{\partial t} = (mc^2\alpha_0 + c\vec{\alpha} \cdot \vec{p}) |\psi\rangle, \qquad (1.24)$$

where $|\psi\rangle$ is a four-components wave function, $m$ is the mass of the particle, $c$ is the velocity of light in vacuum, $\alpha_0$ is 4 × 4 matrix, $\vec{\alpha} = (\alpha_1, \alpha_2, \alpha_3)$ with $\alpha_{1,2,3}$ 4 × 4 matrices and $\vec{p} = -i\hbar\vec{\nabla}$ is the impulsion operator. The different matrices have the following expressions :

$$\alpha_0 = \begin{pmatrix} \sigma_0 & 0 \\ 0 & -\sigma_0 \end{pmatrix} \quad \text{and} \quad \alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix}, \qquad (1.25)$$

where $\sigma_0$ is the 2 × 2 identity matrix and $\sigma_{i=1,2,3}$ are the Pauli matrices.

This equation describes particles with half-integer spins such as the electron. It naturally takes into account the notion of spin, a physical quantity which was experimentally highlighted in the famous Stern and Gerlach experiment carried out a few years earlier in 1922, but also allowed to predict the existence of antiparticles. The stationary Dirac equation :

$$(mc^2\alpha_0 + c\vec{\alpha} \cdot \vec{p}) |\psi\rangle = E |\psi\rangle, \qquad (1.26)$$

If an external potential $V(\vec{r})$ is present, the equation becomes :

$$(mc^2\alpha_0 + c\vec{\alpha} \cdot \vec{p}) |\psi\rangle = (E - V(\vec{r})) |\psi\rangle, \qquad (1.27)$$

The wave function is decomposed into two subsets of components representing the electron and the positron (its antiparticle) respectively : $|\psi\rangle = (|\psi_e\rangle, |\psi_p\rangle)^T$. The two components of the electronic wave function $|\psi_e\rangle$ represent spin up and spin down states. In the following, we will forget about the positronic wave function $|\psi_p\rangle$ and focus on the differential equation in $|\psi_e\rangle$. In the non-relativistic limit, the relevant energy is $\epsilon$ and is defined by $E = mc^2 + \epsilon$. By supposing that $\epsilon \ll mc^2$ and also that $|V| \ll mc^2$, one can develop the equation to the first order in $\frac{\epsilon - V}{mc^2}$. The full calculation is performed in ref. [28] but in the end, we see the apparition of purely relativistic new terms in the expression of the full Hamiltonian, among which is the spin-orbit Hamiltonian (after applying minimal coupling $\vec{p} \rightarrow \vec{p} - e\vec{A}$ with $\vec{A}$ the vector potential) :

$$H = \left( \frac{1}{2m} (\vec{p} - e\vec{A})^2 + V + mc^2 \right) - \frac{e\hbar}{2m} \vec{\sigma} \cdot \vec{B} - \frac{1}{8m^3c^2} (\vec{p} - e\vec{A})^4 \\ + \frac{\hbar^2}{(2mc)^2} \vec{\sigma} \cdot (\vec{\nabla} V \times (\vec{p} - e\vec{A})) + \frac{\hbar^2 e}{8m^2c^2} \nabla^2 V, \qquad (1.28)$$
where the first term is the unrelativistic Hamiltonian, the second is the Zeeman term, the third represents the increasing inertia with increasing speed, the fourth one is the spin-orbit coupling Hamiltonian and the final one is the Darwin term.

If there is no magnetic field, the spin-orbit coupling Hamiltonian is :

$$H_{SOC} = \frac{\hbar}{(2mc)^2} \vec{\sigma} \cdot (\vec{\nabla} V \times \vec{p}), \qquad (1.29)$$

with $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$. As suggested by its name, the spin-orbit interaction couples the motion of a charged particle to its spin. In order to understand the physical origin of this coupling, let us imagine an electron moving at a velocity $\vec{v}$ in a crystal ionic lattice generating an electric field $\vec{E}$. From the point of view of the electron, it is the lattice which is moving, and a special relativity calculation would show that in the reference frame of the electron, the electron feels an effective magnetic field $\vec{B}_{eff} = -\frac{\vec{v}}{c} \times \vec{E}$. The spin $\vec{S} = \frac{\hbar}{2}\vec{\sigma}$ of the electron couples to this magnetic field effectively changing its energy by $-\gamma \vec{S} \cdot \vec{B}$ where $\gamma$ is the gyromagnetic ratio of the electron. The gradient of potential $\vec{\nabla} V$ is proportional to the electric field $\vec{E}$ such that $(\vec{\nabla} V \times \vec{p}) \propto \vec{B}_{eff}$. In the end, $H_{SOC} \propto \vec{S} \cdot \vec{B}_{eff}$, which illustrates qualitatively this phenomenon.

In the work presented in this thesis, mainly three types of spin-orbit couplings were encountered. The first one is the Rashba spin-orbit coupling, which typically appears at the surface of materials where the inversion symmetry is broken. It will be the subject of subsection 1.3.2. The second one is the Ising coupling, present in non-centrosymmetric materials such as in (LaSe)₁,₁₄(NbSe₂)₂, and which will be detailed in subsection 1.3.3. Rashba and Ising spin-orbit couplings are likely to induce superconducting spin triplet correlations. The spin-orbit coupling present in iridate compound Sr₂IrO₄ is a more conventional type of "atomic" spin-orbit coupling [29].

### 1.3.2 Rashba spin-orbit coupling

By definition, inversion symmetry is broken at the surface of a crystal because the surface is the interface between two different semi-infinite spaces (the crystal on one side and the vacuum on the other). This can easily be seen by thinking of the electrostatic potential gradient present at surfaces which prevents electrons to leak into the vacuum (work function). Because of symmetry considerations, this implies that the superficial electric field's direction will always have a component perpendicular to the surface. Moreover, if we consider that the electrons only move parallelly to the surface in a 2D fashion with momentum $\vec{p} = \hbar \vec{k} = \hbar(k_x, k_y)$, then the expression of the spin-orbit coupling Hamiltonian becomes :

$$H_{Rashba} = \alpha(k_x\sigma_y - k_y\sigma_x), \qquad (1.30)$$

With $\alpha$ the strength of the Rashba spin-orbit coupling interaction. It is easy to check that $H_{Rashba}$ does not commute with any component of the spin $\vec{S}$, which means that neither $S_x$, $S_y$ or $S_z$ are conserved. If we consider a 2D electron gas in the presence of Rashba spin-orbit coupling, the eigenenergies of the system are :

$$E_{\pm}(\vec{k}) = \frac{\hbar^2 \vec{k}^2}{2m} \pm \alpha \|\vec{k}\|, \qquad (1.31)$$
<figure><img src="image_4.png" /><figcaption>Figure 1.3: a. 3D representation of the Rashba spin orbit coupling effect on a parabolic dispersion. b. Cut of the same dispersion. c. Energy contour of the dispersion of b at energy E. The black arrows represent the fact that the spin tends to lock in the direction perpendicular to the momentum. The red and blue bands have opposite chiralities.</figcaption></figure>

The dispersion is represented in figure 1.3 a in which $E_{-}(\vec{k})$ is blue and $E_{+}(\vec{k})$ is red. A cut along an arbitrary direction $\theta$ of the $\vec{k}$ vector is displayed in figure 1.3 b. Besides, an isoenergy surface at energy E is shown in figure 1.3 c. Within $E_{+}(\vec{k})$, the spin is not conserved, but the projection of the spin on the direction perpendicular to the momentum of the electron is conserved. Rashba spin-orbit coupling ensure that the spin of an electron remains perpendicular to its momentum. The same can be said for $E_{-}(\vec{k})$, and the + and - set of states have opposite chiralities as suggested by the black arrows of figure 1.3 c.

If $|\vec{k} \uparrow\rangle$ and $|\vec{k} \downarrow\rangle$ were the eigenvectors of the system without spin orbit coupling, then it is possible to show that the eigenvectors of the total system are :

$$|\vec{k} \pm\rangle = \frac{|\vec{k} \uparrow\rangle \pm i e^{i\theta} |\vec{k} \downarrow\rangle}{\sqrt{2}}, \qquad (1.32)$$

Again, it is clear that the eigenstates of the system are a mixture of states with up and down spins. Since the spin is no longer a good quantum number and the inversion symmetry is broken, a two-particles wave function will automatically be a superposition of singlet and triplet states. Because the gap function can be assimilated to the Cooper pair's wave function, s-wave conventional superconductors will undoubtedly develop superconducting spin triplet correlations. This last point has been theoretically described in 1989 by V. M. Edel'shtein [30] and in 2001 by Lev P. Gor'kov and Emmanuel I. Rashba [31]. However, even if triplet correlations can be induced in this way, the ratio between the triplet and singlet parts of the order parameter highly depends on the studied system. Such spin-orbit coupling can induce unconventional superconductivity but not inevitably a topological regime.

### 1.3.3 Ising spin-orbit coupling

Ising spin-orbit coupling is a type of spin-orbit which is observed in 2D systems which possess out of plane mirror symmetry and break in-plane mirror symmetry. On the overall, inversion symmetry is also broken. By taking into consideration what has already been discussed in the previous section, triplet correlations will also necessarily appear in superconducting materials with strong spin-orbit coupling and such geometries. The reference example is monolayer NbSe$_2$, which structure is for instance presented in ref. [32] and exposed in greater details in section
3.1. Monolayer NbSe₂ basically consists in an hexagonal lattice of Nb atoms in which each Nb is at the center of a Se prism. This structure possesses the same symmetry as the cell of figure 1.4 a, where the blue sphere is a Nb atom and the yellow spheres are Se atoms.

<figure><img src="image_5.png" /><figcaption>Figure 1.4: Effect of the geometry of the lattice on the nature of the spin-orbit coupling in monolayer NbSe₂. **a.** Prism which center is a Nb atom (blue) and apexes are Se (yellow) atoms. It has the geometry of the trigonal prismatic lattice of monolayer NbSe₂. **b.** Top view of **a.** **c.** Inversion symmetric of **a.** **d.** Top view of **c.** **e.** The *P* plane is a symmetric plane for the whole structure. This imposes that the spin-orbit coupling magnetic field effectively felt by the electron is oriented along the out-of-plane direction.</figcaption></figure>

A top view of this cell is represented in figure 1.4 b. In figure 1.4 e, one can notice that the structure is unchanged under the *P* plane mirror operation. Figure 1.4 c and d are the inversion transforms of a and b. It is clear that b and d are not superimposable.

In NbSe₂, the physics is borne by the *4d* electrons of Nb, which allows for two things : strong and intrinsic spin-orbit coupling **and** 2D motion of the electrons in the hexagonal lattice of Nb atoms. The velocity $\vec{v}$ of an electron as well as the electric field of the lattice (because of symmetry arguments) are both bound to belong to the *P* plane of figure 1.4 e. An electron will thus effectively feel a magnetic field $\vec{B}_{eff}$ which will be oriented in the out-of-plane direction. This effective magnetic field can be intense, such that it tends to lock the spins of the electrons in the out-of-plane direction. For a superconducting compound, this leads to substantially high in-plane upper magnetic critical fields [33, 34, 35, 36], above the Clogston-Chandrasekhar limit (also called Pauli paramagnetic limit). This will be further developed in section 3.4.
<figure><img src="image_6.png" /></figure>

Figure 1.5: **a.** Illustration of the effective Ising magnetic field felt by the electrons in monolayer NbSe₂ with a top view of the trigonal prismatic lattice of monolayer NbSe₂ in which the blue spheres are Nb atoms and the yellow ones are Se atoms. **b.** Fermi surface of monolayer NbSe₂. **c.** Band structure of monolayer NbSe₂. The Ising splitting between spin up and spin down bands can be seen in both **b** and **c**, and is particularly pronounced around K/K' points.

A microscopic semi-classical origin of this Ising spin-orbit coupling is illustrated in figure 1.5. In this figure, both real and reciprocal spaces (a and b respectively) are correctly oriented with respect to each other such that the $\Gamma M$, $\Gamma K$ and $\Gamma K'$ directions were also superimposed in real space in 1.5 a.

In figure 1.5 **b** and **c**, the orange and purple curves respectively represent spin up and spin down bands. The Ising spin-orbit coupling induces a splitting of spin up and spin down bands around both $\Gamma$ and $K/K'$ points. This splitting is alternated as one turns of $\frac{\pi}{3}$ around $\Gamma$, thus explaining the chosen "Ising" terminology. Moreover, no splitting is observed in the $\Gamma M$ directions.

We will now explain figure 1.5 **a** in great detail. We can notice that the planes parallel to the $\Gamma M$ directions, perpendicular to the monolayer and which contain Nb atoms are symmetry planes for the structure. This means that the local electric field $\vec{E}_{local}$ evaluated at any point of such a plane belongs to this plane. If we consider an electron of momentum $\vec{k}_{\Gamma M}$ belonging to such a plane, the effective spin-orbit magnetic field $\vec{B}_{eff1} \propto -\vec{k}_{\Gamma M} \times \vec{E}_{local}$ is equal to $\vec{0}$, thus explaining why no splitting is expected in the $\Gamma M$ directions. On the contrary, the planes parallel to the $\Gamma K$ or $\Gamma K'$ directions, perpendicular to the monolayer and which contain Nb atoms are not symmetry planes for the structure, and the local electric field $\vec{E}_{local}$ evaluated at any point of such a plane does certainly not belong to this plane. If we consider an electron of momentum $\vec{k}_{\Gamma K}$ belonging to such a plane, but also to a "$\Gamma M$" plane, the local electric field will belong to the "$\Gamma M$" plane and in the end, the effective spin-orbit magnetic field $\vec{B}_{eff2} \propto -\vec{k}_{\Gamma K} \times \vec{E}_{local}$ will be oriented out of the surface. An electron of momentum $\vec{k}_{\Gamma K'}$ (with $\|\vec{k}_{\Gamma K'}\| = \|\vec{k}_{\Gamma K}\|$) belonging to both "$\Gamma K'$" and "$\Gamma M$" planes, will have an effective spin-orbit magnetic field $\vec{B}_{eff3} \propto -\vec{k}_{\Gamma K'} \times \vec{E}_{local}$ equal to $-\vec{B}_{eff2}$, thus explaining the three-fold
alternation of spin splitting in the band structure.

One last remark we can make, is that the magnitude of the splitting rises up as $||\vec{k}||$ becomes bigger, or, saying differently, as we go away from $\Gamma$. This explains why the splitting is more pronounced for the $K/K'$ pockets than for the $\Gamma$ pocket (see figure figure 1.5c). If we electron-dope the system (as it was effectively done in chapter 3), given the hole character of the bands around $K/K'$ and $\Gamma$, the Ising spin splitting will increase around $K/K'$ and decrease around $\Gamma$.

1.4 Topological superconductors

1.4.1 Introduction to topology in condensed matter systems

Topology is the branch of mathematics which studies the properties of space which are preserved under continuous deformations. For example, a mug and a torus, while being completely different objects, share a common topology. One can indeed deform continuously one to obtain the other never applying neither "cutting" or "glueing" of any sorts. This is illustrated in figure 1.6 a.

<figure><img src="image_7.png" /><figcaption>Figure 1.6: **a.** Continuous deformation of a mug into a torus. **b.** Illustration of the concept of topological invariant in polyhedra through the Euler characteristics.</figcaption></figure>

One also understands that it is impossible to continuously change a sphere into a torus. A hole would have to be done into the sphere at some point, and "making a hole" cannot be described mathematically by any continuous transformation. Similarly, a torus cannot be turned continuously into a two-hole torus. It would seem that the number of holes is a quantity fit to discriminate all 3D volumes into separate topology classes. We say that the number of holes is a topological invariant, or so to say, a mathematical quantity shared by all the objects within a given topology class.

To further explore the notion of topological invariant, we will now focus on polyhedra. Again, we intuitively see that a cube can be deformed into a tetrahedron or into an octahedron, but it seems very unlikely to be continuously turned into the holed polyhedron displayed in figure 1.6 **b**. Even though we instinctively understand that the topological invariant will be linked to the total number of holes, it is necessary to properly define it mathematically. For
polyhedra, the topological invariant of interest is called the Euler characteristic and has the following expression: $\chi = V + F - E$. $V$ is the number of vertices (apexes) of the polyhedron, $F$ its number of faces and $E$ its number of edges. For any polyhedron with no hole, one has $\chi = 2$. For the holed polyhedron of figure 1.6, $\chi$ takes a different value and is equal to 0. Any one-hole polyhedron will always have $\chi = 0$.

## The Berry phase.

We now are a bit more familiar with the concepts of topology and topological invariants. In this section, we will see how these notions apply in the framework of quantum physics. The topological nature of a quantum system is related to the constraints imposed by the system's geometry on the phase of wave functions. This statement will become clear after the short calculation we are about to present. Let $H(\vec{R})$ be the Hamiltonian of a quantum system fully described by the set of parameters $\vec{R}$. We call $|n(\vec{R})\rangle$ and $E_n$ the eigenstates and eigenenergies of the system respectively, which explicitly depend on $\vec{R}$. One has:

$$H(\vec{R}) |n(\vec{R})\rangle = E_n(\vec{R}) |n(\vec{R})\rangle,$$

In the following, we will consider that the system was initially in $|n(\vec{R}_{initial})\rangle$ state. If the set of parameters $\vec{R}$ varies slowly enough with time, then, the adiabatic theorem tells us that at each instant $t$ the system is in an eigenstate of the-time dependent Hamiltonian $H(\vec{R}(t))$ and one has the relation:

$$H(\vec{R}(t)) |n(\vec{R}(t))\rangle = E_n(\vec{R}(t)) |n(\vec{R}(t))\rangle, \qquad (1.33)$$

In addition to this, there always is a degree of freedom on the phase of the wave function such that the state $|\psi(t)\rangle$ of the system at time $t$ can be written in a more general way like: $|\psi(t)\rangle = e^{-i\theta(t)} |n(\vec{R}(t))\rangle$. It is important to emphasize that the phase $\theta$ also explicitly depends on time. Its expression can be derived using Schrödinger equation:

$$H(\vec{R}(t)) |\psi(t)\rangle = i\hbar \frac{d}{dt} |\psi(t)\rangle, \qquad (1.34)$$

This gives us a differential equation in $\theta(t)$:

$$E_n(\vec{R}(t)) - i\hbar \langle n(\vec{R}(t)) | \frac{d}{dt} | n(\vec{R}(t)) \rangle = \hbar \frac{d\theta}{dt}, \qquad (1.35)$$

Which can be integrated and finally:

$$\theta(t) = \underbrace{\frac{1}{\hbar} \int_{0}^{t} dt' E_n(\vec{R}(t'))}_{\text{dynamical phase}} - \underbrace{-i \int_{0}^{t} dt' \langle n(\vec{R}(t')) | \frac{d}{dt'} | n(\vec{R}(t')) \rangle}_{\text{-}\gamma_n = \text{geometrical phase}}, \qquad (1.36)$$

$$\gamma_n = i \int_C d\vec{R} \cdot \langle n(\vec{R}) | \nabla_{\vec{R}} | n(\vec{R}) \rangle, \qquad (1.37)$$

In addition to the well-known dynamical phase, the wave function of the system acquires another phase related to the adiabatic transport in $\vec{R}$ parameters space. $\gamma_n$ is a geometrical phase and is often called the Berry phase. Given this expression of $\gamma_n$, it is possible to define the Berry connection: $\vec{A} = i \langle n(\vec{R}) | \nabla_{\vec{R}} | n(\vec{R}) \rangle$. The Berry phase is thus the circulation of the Berry connection over a given path $C$.
<figure><img src="image_8.png" /><figcaption>Figure 1.7: Illustration of the notion of geometrical phase with the example of a spin rotating thanks to an adiabatically varying orientation of the magnetic field.</figcaption></figure>

In order to see a situation is which a quantum system acquires a geometrical phase we will now present the example of a spin under a magnetic field $\vec{B}$, example which is very well detailed in ref. [37]. Here, the set of parameters is $\vec{R} = \vec{B}$. One of the conditions for this Berry phase to show up is adiabatic transport in parameters space. We will consider that the initial state is an eigenstate of the system and that the magnetic field's orientation slowly varies in time while keeping its magnitude constant. We choose it to follow the closed curve in $\vec{B}$ space represented in figure 1.7. The full calculation is performed in ref. [37] but the idea can easily be understood with the help of the cartoon of figure 1.7. When a spin is in presence of a magnetic field, it precesses with a given pulsation $\omega$ around the direction of the magnetic field. After a time $\Delta t$ has passed, the wave function of the particle possessing the spin will have acquired a dynamical phase $\omega\Delta t$. The angular position of the spin is spotted with the help of the system of axes represented in red in figure 1.7. One can notice on the drawing that as the path is travelled in $\vec{B}$ space, the red axes' orientation changes and ends up being rotated of $\frac{\pi}{2}$ in final step 7 compared to initial step 1. This means that in addition to the dynamical phase, the system also acquires a dynamical phase $\gamma_n$ equal to $\frac{\pi}{2}$. This $\frac{\pi}{2}$ phase reflects the solid angle $\Omega$ defined by the travelled closed path. This later results is actually valid for spin 1 particles. But for all spin, the acquired geometrical phase can always be related to the enclosed solid angle. For example, for spin 1/2, the geometrical phase is equal to $\Omega/2$. If $\Omega = 2\pi$, meaning a full rotation in parameters space, one recovers the fact that a spin 1/2 particle acquires a $\pi$ phase when submitted to a $2\pi$ rotation.
Majorana fermions.

In the two following sections, we will present two topical examples of topological superconductivity. The first one is the 1D p-wave superconductor also called the Kitaev chain. The second is the 2D p-wave superconductor. In this later case, the topological nature of the system will be related to the Berry phase. For the Kitaev chain, where one can easily check that the Berry phase is zero, the topological nature of the system rather lies in unavoidable obstruction in the paving of the wave functions' phase across the first Brillouin zone.

The questions that arises now are the following : are there any experimental signatures of the topological nature of a quantum system ? ; could we imagine applications of topology in condensed matter physics ? In order to answer those questions, we will ask another one. What happens at the interface between a superconductor 1 and a superconductor 2 with different topologies ? This is a legitimate question since the interface neither possesses the topology of 1 or 2. We will see in the examples that it is never possible to continuously tune the properties of 1 to obtain the topology of 2 without closing the superconducting gap. This closing of the gap can be assimilated to the "cutting" and "glueing" operations mentioned at the beginning of this chapter. What we will see is that in-gap states are supposed to appear at the interface between the two superconductors, these states often being Majorana fermions.

A Majorana fermion is a fermion (half integer spin) which is its own antiparticle. Majorana fermions were described theoretically in 1937 by physicist Ettore Majorana in the framework of particle physics. They differ from regular Dirac fermions (such as the electron). No elementary particle has yet been identified as a Majorana fermion to date, but neutrinos are possible candidates. It is rather in condensed matter physics that Majorana fermions are likely to be observed as quasiparticle excitations.

Since a Majorana fermion is its own antiparticle, its electrical charge necessarily is 0. As it was shown in section 1.1, the low energy quasiparticle excitations of a superconductor are fermions and are part electron and part hole, or saying differently, part a particle and part its antiparticle. Superconductors are thus the natural playground to look for Majorana quasiparticles.

A Majorana fermion respects the relation $\gamma^\dagger = \gamma$, where $\gamma^\dagger$ and $\gamma$ are the creation and annihilation operators of a Majorana fermion respectively. It also verifies the anticommutation relation : $\{\gamma_a^\dagger, \gamma_b^\dagger\} = \{\gamma_a^\dagger, \gamma_b\} = \{\gamma_a, \gamma_b^\dagger\} = \{\gamma_a, \gamma_b\} = 2\delta_{a,b}$, also meaning that $\gamma^2 = 1$.

**Remark :** The relation $\gamma^\dagger = \gamma$ is actually valid for spinless Majorana bound states. In the case of a $\frac{1}{2}$ spinful Majorana fermion labelled with momentum $k$ one actually has $\gamma_{k,\uparrow}^\dagger = \gamma_{-k,\downarrow}$.

Majorana fermions virtually appear in condensed matter (more precisely in the physics of electrons in solids) systems because they mathematically are a superposition of fermionic states $c^\dagger$ and $c$. One can always write $\gamma = c^\dagger + c$. On the contrary, a Dirac fermion and more particularly an electron can also be decomposed as a superposition of two distinct Majorana fermions which we will label A and B in the following. Hence :

$$ \left\{ \begin{array}{l} \gamma^A = c + c^\dagger \\ \gamma^B = i(c - c^\dagger) \\ c^\dagger = \frac{(\gamma^A + i\gamma^B)}{2} , \\ c = \frac{(\gamma^A - i\gamma^B)}{2} \end{array} \right. , \qquad (1.38) $$

Majorana fermions, unlike bosons or Dirac fermions neither respect Bose-Einstein nor Fermi-Dirac statistics. If we consider a N-particle state of identical bosons, if one swaps the positions of two arbitrary bosons, the wave function of the system remains unchanged. If one again permutes two arbitrary bosons, the wave function is still unchanged. For Dirac fermions, the first
permutation of two arbitrary identical fermions multiplies by − 1 the wave function, and the sec-
ond permutation again multiplies the function by − 1 such that in the overall, the wave function
of the system is unchanged. As a summary, the order in which the two arbitrary permutations
are performed does not matter for both bosons and Dirac fermions. If the permutations are
called σ$_{1}$ and σ$_{2}$ , the actions of σ$_{1}$σ$_{2}$ and σ$_{2}$σ$_{1}$ on the total N-particle state are the same. We
say that Bose-Einstein and Fermi-Dirac statistics are Abelian statistics.

Majorana fermions obey a non-Abelian statistics (see refs.[38, 39]). This could be used in yet to be coming quantum computers. A 2N-Majorana particle system can be written as a superposition of 2$^{N}$-Dirac particle Foch states. Carrying out permutations on a 2N-Majorana particle system allows to encode information in such a superposition. The so called "braiding" operations are needed to carry out some specific quantum computing algorithms [40]. In addition to this, Majorana fermions are very robust against disorder and consequently against decoherence, which is of the utmost importance in quantum computing to guarantee an appropriate calculation progress before the "reading" procedure which reduces the wave packet. This robustness comes from the topological origin of Majorana in-gap states : we say that they are topologically protected [41, 42]. The existence of the Majorana states is dictated by the topology of the system, which is a global property. Such topological edge states are thus insensitive to perturbations such as local disorder. The search for Majorana fermions has been an active field of condensed matter physics for the past decade. Recently, Majorana particle have been claimed to be observed (this will further be discussed in the following sections), but the non-Abelian statistics was never addressed experimentally, which means that a lot of progress still needs to be done, for it is the only way to prove without any ambiguity the Majorana nature of such exotic states of matter.

1.4.2 The Kitaev chain

The topological superconductivity model presented in this section comes from the pioneering
theoretical article of A. Yu Kitaev [43]. In this article, Kitaev showed that Majorana bound
states with non-Abelian statistics could appear at the ends of a one-dimensional spinless p-wave
superconductor.

Theoretical model.

Let us consider a superconducting one-dimensional atomic chain with one **spinless** electron
per site. This spinless fermionic system can be viewed as a system in which all the electrons
have been spin-polarized into a given exclusive spin state (either up or down). Applying a
strong enough external magnetic field is a way of realizing such a configuration. In this 1D
chain, adjacent sites are separated by a distance *a*. We choose a p-wave triplet superconducting
pairing (respecting $\Delta(-k) = -\Delta(k)$). The Bogoliubov-de-Gennes Hamiltonian of the system is
the following:

$$
H_{BdG} = \frac{1}{2} \sum_k \psi_k^\dagger \overbrace{\left( \begin{array}{cc} \varepsilon(k) - \mu & 2i\Delta \sin(ka) \\ -2i\Delta \sin(ka) & -(\varepsilon(k) - \mu) \end{array} \right)}^{\mathcal{H}(k)} \psi_k, \quad (1.39)
$$

where $\psi_k^\dagger = (c_k^\dagger, c_{-k})$, with $c_k^\dagger$ and $c_k$ the creation and annihilation operators of spinless electron of momentum $k$, respectively, $\varepsilon(k)$ is the free electron dispersion, $\mu$ is the chemical potential and $\Delta$ is the superconducting gap amplitude. $\mathcal{H}(k)$ is the Bloch Hamiltonian. This Hamiltonian is different from the total Hamiltonian but it is the one which encodes the topological properties of the system. $\mathcal{H}(k)$ can be written as such :

$$
\mathcal{H}(k) = (\varepsilon(k) - \mu)\tau_z + 2\Delta \sin(ka)\tau_y = \vec{d}(k) \cdot \vec{\tau}, \quad (1.40)
$$
Where $\vec{\tau} = (\tau_x, \tau_y, \tau_z)$, and $\tau_{x,y,z}$ are the Pauli matrices on the particle-hole basis. By considering only a positive first neighbour hopping parameter $t$, the $\vec{d}(k)$ vector has the expression:

$$ \vec{d}(k) = \begin{cases} d_x = 0 \\ d_y = 2\Delta \sin(ka) \\ d_z = -2t \cos(ka) - \mu \end{cases}, \qquad (1.41) $$

Diagonalizing $\mathcal{H}(k)$ leads to the eigenenergies: $E_{\pm}(k) = \pm\sqrt{d_x^2 + d_y^2 + d_z^2} = \pm\sqrt{4\Delta^2 \sin^2(ka) + (-2t \cos(ka) - \mu)^2}$ and to the corresponding eigenstates $|\pm\rangle_k$. In order to give the expression of $|\pm\rangle_k$, we will first introduce the normalized version of $\vec{d}(k)$:

$$ \hat{d}(k) = \frac{\vec{d}(k)}{\|\vec{d}(k)\|} = \frac{1}{\sqrt{d_z^2 + d_y^2}} (0, d_y, d_z) = (0, \hat{d}_y, \hat{d}_z), \qquad (1.42) $$

This later vector lives on a diameter of the Bloch sphere and since $\hat{d}_z^2 + \hat{d}_y^2 = 1$, one can write $\hat{d}_z = \cos(\theta_k)$ and $\hat{d}_y = \sin(\theta_k)$ with $\theta_k$ the angle between $\hat{d}(k)$ and the $z$ direction. One can easily check that the normalized eigenstates of the system are $|+\rangle_k = \begin{pmatrix} \cos(\frac{\theta_k}{2}) \\ i \sin(\frac{\theta_k}{2}) \end{pmatrix}$ and $|-\rangle_k = \begin{pmatrix} \sin(\frac{\theta_k}{2}) \\ -i \cos(\frac{\theta_k}{2}) \end{pmatrix}$.

<figure><img src="image_9.png" /><figcaption>Figure 1.8: **a.** Trivial regime: $\vec{d}(k)$ does not wind around the origin. **b.** Topological regime: $\vec{d}(k)$ winds once around the origin.</figcaption></figure>

We see that $E_{\pm}(k)$ (and also $\vec{d}(k)$) becomes equal to zero, and thus that the gap closes at $k=0$ for $\mu = -2t$ and at $k = \pm\pi$ for $\mu = +2t$. We will show in the following that those gap closings separate regions in parameters space with different topologies. Depending on the value
of $\mu$, the system is either in a trivial or in a topological regime. We will see that for $-2t < \mu < 2t$, the system is topologically non-trivial and for $\mu > 2t$ ($\mu < -2t$) it is topologically trivial.

The topological characteristic of the system depends on the number of rotation $\vec{d}(k)$ performs around the origin while $k$ travels across the whole first Brillouin zone $[-\frac{\pi}{a}, +\frac{\pi}{a}]$. In figure 1.8 a, is represented $\vec{d}(k)$ for $k$ going from $-\frac{\pi}{a}$ to $+\frac{\pi}{a}$ in a case where $\mu > 2t$. In figure 1.8 b, is represented $\vec{d}(k)$ for $k$ going from $-\frac{\pi}{a}$ to $+\frac{\pi}{a}$ in a case where $-2t < \mu < 2t$.

For $\mu > 2t$ (the case $\mu < -2t$ is analogous), one can clearly see that as $k$ goes through the first Brillouin zone, $\vec{d}(k)$ does not perform any rotation around the origin while for $-2t < \mu < 2t$, $\vec{d}(k)$ makes one turn around the origin. Saying differently, in the trivial phase, as $k$ varies continuously from $-\frac{\pi}{a}$ to $+\frac{\pi}{a}$, $\theta_k$ varies continuously from 0 to 0. In the topological phase, as $k$ varies continuously from $-\frac{\pi}{a}$ to $+\frac{\pi}{a}$, $\theta_k$ varies continuously from 0 to $2\pi$. The suited topological invariant can be defined by $\nu = \frac{1}{2\pi} \int_{1BZ} d\theta_k$. For $\mu > 2t$, $\nu = 0$ and for $-2t < \mu < 2t$, $\nu = +1$.

As a summary, we have seen that depending on the region in parameters space, the system enters either a topological or a trivial phase. The different topologies can be understood in terms of winding number $\nu$ of $\vec{d}(k)$ vector around the origin. This winding number is an integer. It is not possible to tune the system from a topological phase to another without closing the superconducting gap. At the set of parameter for which the gap closes, $\nu$ is no longer properly defined. For example, here the gap closes at $\mu = 2t$. When $\mu = 2t$, $\vec{d}(\frac{\pi}{a}) = \vec{0}$, such that for this specific point, $\theta_k$ is ill-defined. By extension, the topological invariant $\nu$ is also ill-defined right at the topological phase transition.

So far, the topological nature of the system was addressed using abstract mathematical quantities but it is possible to get more physical insight by looking at what actually happens to the quasiparticle wave functions. We have seen that in the topological regime, one could continuously vary $\theta_k$ from 0 to $2\pi$ by going from one end of the Brillouin zone to the other. It also means that $|+\rangle_k$ can continuously pave the first Brillouin zone from $k = -\frac{\pi}{a}$ to $k = +\frac{\pi}{a}$.

However, one can notice that $|+\rangle_{k=-\frac{\pi}{a}} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} = -|+\rangle_{k=+\frac{\pi}{a}}$. The wave function at $k = +\frac{\pi}{a}$

is equal to the wave function at $k = -\frac{\pi}{a}$ but necessarily up to a $e^{i\pi}$ phase. It is not possible to make both ends of the first Brillouin zone correspond without any discontinuity. We speak of topological obstruction. Such an obstruction can be prevented in the topologically trivial phase. In this example, the topological nature of the system is related to the existence of an obstruction in the paving of the wave functions over the whole Brillouin zone.

In what follows, we will see what happens in the case of a finite-length chain of spinless 1D p-wave superconductor. The wire is isolated and thus surrounded by vacuum. Vacuum is topologically equivalent to a trivial superconductor (by considering an infinite superconducting gap). If the wire is in a topological regime, we can legitimately wonder what happens at the interface between both wire and vacuum for that interface neither possesses the topology of one or the other. More generally speaking, it is interesting to look at the interface between materials of different topology. In this particular example, we will show that Majorana bound states appear at opposite ends of the 1D chain.

If the chain has $N$ sites, the Bogoliubov de Gennes Hamiltonian can be written in real space as follows:

$$H_{BdG} = \sum_{i=1}^{N-1} (tc_{i+1}^{\dagger}c_i + \Delta c_i^{\dagger}c_{i+1}^{\dagger} + \text{h.c.}) - \mu \sum_{i=1}^{N} c_i^{\dagger}c_i, \quad (1.43)$$
where $c_i^\dagger$ and $c_i$ are the creation and annihilation operators of a spinless electron at site $i$. One can check that $\Delta c_i^\dagger c_{i+1}^\dagger = -\Delta c_{i+1}^\dagger c_i^\dagger$ and hence that $\Delta(k) = -\Delta(-k)$, or so to say, that we are dealing with a triplet order parameter. Following the mathematical procedure described in section 1.4.1, each Dirac fermions can be decomposed as two Majorana fermions :

$$ \left\{ \begin{aligned} c_i^\dagger &= \frac{(\gamma_i^A + i\gamma_i^B)}{2} \\ c_i &= \frac{(\gamma_i^A - i\gamma_i^B)}{2} \\ \gamma_i^A &= c_i + c_i^\dagger \\ \gamma_i^B &= i(c_i - c_i^\dagger) \end{aligned} \right. , \qquad (1.44) $$

After what, the Hamiltonian can be rewritten as :

$$ H_{BdG} = -i \sum_{i=1}^{N-1} (\omega_+ \gamma_{i+1}^A \gamma_i^B - \omega_- \gamma_{i+1}^B \gamma_i^A) + i \frac{\mu}{2} \sum_{i=1}^{N} (\gamma_i^A \gamma_i^B + i), \qquad (1.45) $$

with $\omega_+ = \frac{t+\Delta}{2}$ and $\omega_- = \frac{t-\Delta}{2}$.

If we take $t = \Delta = 0$ and $\mu > 0$, then $\mu > 2t$ and the system is in the trivial regime. In this case the Hamiltonian reads as :

$$ H_{BdG} = i \frac{\mu}{2} \sum_{i=1}^{N} (\gamma_i^A \gamma_i^B + i) = -\mu \sum_{i=1}^{N} c_i^\dagger c_i, \qquad (1.46) $$

We clearly see that the system is diagonal in the $c_i^\dagger$ basis with $i$ going from 1 to $N$. Hence, the system has $N$ eigenstates localized at sites $i$ and of energy $-\mu$. This situation is summarized in the top sketch of figure 1.9.

<figure><img src="image_10.png" /><figcaption>Figure 1.9: Top sketch : Trivial regime. Bottom sketch : Topological regime : different recombination of Dirac fermions leading to the appearance of Majorana bound states at the ends of the chain.</figcaption></figure>

However, if we take $t = \Delta \neq 0$ and $\mu = 0$, then $-2t < \mu < 2t$ and the system is in a topological regime. The Hamiltonian of the system is thus :
$$H_{BdG} = -i\Delta \sum_{i=1}^{N-1} \gamma_{i+1}^{A} \gamma_{i}^{B}, \qquad (1.47)$$

We recall that two combined Majorana fermions form a Dirac fermion. The Majorana operators displayed in equation 1.47 were obtained by writing each Dirac fermion localized at sites $i$ as such a combination. Nevertheless, it is also possible to recombine two Majorana fermions coming from adjacent sites, thus giving rise to new Dirac fermions $a_i^\dagger$:

$$\left\{
\begin{aligned}
a_i^\dagger &= \frac{(\gamma_i^B + i\gamma_{i+1}^A)}{2}, \\
a_i &= \frac{(\gamma_i^B - i\gamma_{i+1}^A)}{2}, \\
\gamma_i^B &= a_i + a_i^\dagger, \\
\gamma_{i+1}^A &= i(a_i - a_i^\dagger)
\end{aligned}
\right.
\qquad (1.48)$$

Following this procedure, the Hamiltonian now has the following expression:

$$H_{BdG} = -2\Delta \sum_{i=1}^{N-1} a_i^\dagger a_i, \qquad (1.49)$$

The Hamiltonian is diagonal in the new $a_i^\dagger$ basis, but contrary to the trivial case, this writing of the Hamiltonian, although correct, only highlight $N-1$ eigenstates of energies $-2\Delta$. This is problematic because the total number of eigenstates must not change from trivial to topological regime: one eigenstate is missing. The only Majorana fermions which were not taken into account in the recombining are $\gamma_1^A$ and $\gamma_N^B$, or saying differently, the two Majorana fermions lying at the two ends of the wire. Those two Majorana fermions can recombine into $a^\dagger = \frac{(\gamma_N^B + i\gamma_1^A)}{2}$ Dirac fermions, after what one can simply add a $a^\dagger a$ term at the Hamiltonian:

$$H_{BdG} = 0a^\dagger a - 2\Delta \sum_{i=1}^{N-1} a_i^\dagger a_i, \qquad (1.50)$$

We now have $N-1$ states at energy $-2\Delta$ and one state at 0 energy: the total number of eigenstates is recovered. The two Majorana end states are combinations of 0 energies electron and hole states, they consequently also lie at 0 energy. If the wire is long enough, the two Majorana states won't recombine efficiently and will be localized at each end of the chain. In view of this last calculation, one can also notice that Majorana states necessarily come by pair. An isolated Majorana fermion cannot be observed in condensed matter systems. This later situation is represented in the bottom sketch of figure 1.9.

The example that was discussed in this section is a toy model for topological superconductivity in one-dimensional systems and is not realistic since spinless electrons do not exist. However, if one finds a way to spin-polarize the electrons condensing into the superconducting state, the required triplet order parameter might be obtained as well as 1D topological superconductivity. Several experimental realizations of the topical Kitaev chain have been proposed since 2012.

## Zero bias states in superconductor-semiconductor nano-wire devices.

The first approach adopted by V. Mourik et al. consisted in considering superconductor-semiconductor nano-wire devices [1]. In their study, they performed transport measurements on an InSb wire electrically contacted to a s-wave superconductor. The presence of a magnetic field in the direction of the wire provoked a Zeeman splitting strong enough to have one band crossing at Fermi level. Moreover, superconductivity was induced in the wire by proximity
to the superconducting electrode. The strong Rashba spin-orbit splitting, due to the absence
of inversion symmetry, gave a triplet part to the superconducting order parameter. Finally,
by mean of gating, the authors claim to have finely tuned the wire into a topological regime
reminiscent of the Kitaev chain and observed 0-bias states which they identified as Majorana
fermions. Their data is represented in figure 1.10.

<figure><img src="image_11.png" /><figcaption>Figure 1.10: **a** $dI/dV$ spectra vs magnetic field with (from 0 to 490 mT). **b** Color scale plot of $dI/dV$ vs magnetic field and bias voltage. Figure taken from ref. [1].</figcaption></figure>

This interpretation is controversial since V. Mourik et al. did not observe any hard superconducting gap. As it can be seen in figure 1.10 **a**, the superconducting gap is filled by many states and the zero bias state is far from being isolated.

A similar study was carried by A. Das et al. this time using semiconducting InAs nano-wires in proximity to superconducting Al [44]. The general idea behind this work is the same than in ref. [1]. In this article, they observe a 0-bias state which however splits with increasing magnetic field (see figure 1.11). This is in contradiction with the expected robustness of Majorana state against magnetic field.

<figure><img src="image_12.png" /><figcaption>Figure 1.11: Color plot of the measured conductance as a function of bias voltage and magnetic field. Figure taken from ref. [44].</figcaption></figure>
In refs. [45] and [46], the group of C. M. Marcus also studied InAs nano-wires in proximity to superconducting Al. In ref. [45], they observed an exponential suppression of the magnetic field-induced energy-splitting with increasing wire length. Moreover, in ref. [46], they could even realize wires where the 0-bias states would remain at 0 energy with magnetic fields high enough to ensure the closing of the superconducting gap (see figure 1.12).

<figure><img src="image_13.png" /><figcaption>Figure 1.12: **a.** Color plot of differential conductance as a function of bias voltage and magnetic field. **b.** Differential conductance spectra as a function of bias voltage and at magnetic fields going from 0 T to 1,5 T. Figure taken from ref. [46].</figcaption></figure>

It would seem that the conceptual difficulties encountered in these systems can be explained in terms of recombination of Majorana end states due to small wire length. However, recent theoretical studies tend to show that the zero-bias states could rather be interpreted as non-topological Andreev bound states [47], or even, that the chosen geometry would mean a significant shift of the chemical potential, turning the system in a topologically trivial regime [48, 49]. The semiconducting nano-wire approach, nevertheless, remains promising. Recently, S. Vaitekenas et al. pursued in this direction, using new geometry, this time considering InAs nano-wires with a **full-shell** of superconducting Al [50]. This study seems to be the most successful since the data is in perfect agreement with theoretical calculations. Nevertheless, the full-shell geometry of the system does not permit the use of gating, on which are based most of the reasonable Majorana fermions braiding procedures.

Exploiting superconducting proximity effect in semiconducting systems presenting strong spin-orbit coupling was the first of several attempts towards topological superconductivity. Yet, from these studies, one can extract the essential ingredients : superconductivity, Zeeman splitting and finally, spin-orbit coupling (to induce a p-wave component in the superconducting order parameter).

## Zero bias states at the ends of magnetic chains.

From 2011 to 2013, many theoretical papers proposed an alternate way of realizing Majorana fermions using chains of magnetic atoms on top of a superconductor[51, 52, 53, 54, 55]. The main message of these articles is the fact that a superconducting spiral chain of magnetic moments is mathematically equivalent to a superconductor in presence of magnetic field and spin-orbit coupling, and thus, is likely to exhibit Majorana bound states at its ends. It is
also mathematically equivalent to a ferromagnetic chain with superconductivity and spin-orbit coupling.

The pioneering work which followed this approach is the one of S. Nadj-Perge et al. in 2014 [2]. In this STM/STS study, ferromagnetic chains of Fe were grown on top of a superconducting bulk Pb crystal in the (110) direction. Zero energy states were observed at the ends of the wires, which were interpreted as Majorana bound states. The Majorana end states are expected to be particle-hole symmetric as well as spin-polarized. These features were claimed to be observed in 2017, when the results were reproduced by the same group with better energy resolution [56] and using spin-polarized STM/STS [57]. In 2015, M. Ruby et al. [58] carried out a STM/STS study on the same system as S. Nadj-Perge et al.. Apparently, the Majorana end states interpretation would imply a topological gap too small (≤ 60 µeV) to be observed at the temperature at which the experiments of ref. [2] were performed (namely 1,4 K). Hence, the existence of topological superconductivity in Fe/Pb(110) is not consensual in the community. M. Ruby et al. also carried out STM/STS studies on a similar system were the Fe atoms were replaced by magnetic Co atoms. In this work, zero-bias modes were delocalized along the chain suggesting a non topological origin. Finally, in 2018, H. Kim et al. built, by atomic manipulation, spin spiral Fe magnetic chains on top of a superconducting Re(0001) substrate [59]. In this work, the existence of Majorana bound states is under debate due to missing well-defined zero bias peaks<fnref n="1" />. Nevertheless, this last study opens an alternative and promising new platform to engineer topological superconductivity.

The following section will be dedicated to two-dimensional superconductivity through the
topical 2D p-wave superconductor.

1.4.3 The 2D p-wave superconductor

Let us consider a square superconducting two-dimensional lattice with one **spinless** electron per site and of lattice parameter *a*. In this example, the electron are again considered to be spinless, like in section 1.4.2. The hamiltonian describing the system is :

$$
H_{BdG} = \frac{1}{2} \sum_{\vec{k}} \psi_{\vec{k}}^{\dagger} \overbrace{ \left( \begin{array}{cc} \mathcal{H}(\vec{k}) & \\ & 2i\Delta(\sin(k_x a) + i\sin(k_y a)) \end{array} \right) }^{\varepsilon(\vec{k}) - \mu} \psi_{\vec{k}} \quad (1.51)
$$

where $\psi_{\vec{k}}^{\dagger} = (c_{\vec{k}}^{\dagger} \ c_{-\vec{k}})$, with $c_{\vec{k}}^{\dagger}$ and $c_{-\vec{k}}$ the creation and annihilation operators respectively,
of spinless electron of momentum $\vec{k} = (k_x, k_y)$, $\varepsilon(\vec{k})$ is the electron dispersion, $\mu$ is the chemical
potential and $\Delta$ is the superconducting gap amplitude. $\mathcal{H}(\vec{k})$ is the Bloch Hamiltonian and
can be written as such :

$$
\mathcal{H}(\vec{k}) = -2\Delta \sin(k_y a)\tau_x - 2\Delta \sin(k_x a)\tau_y + (\varepsilon(\vec{k}) - \mu)\tau_z = \vec{d}(\vec{k}) \cdot \vec{\tau}, \quad (1.52)
$$

where $\vec{\tau} = (\tau_x \ \tau_y \ \tau_z)$, and $\tau_{x,y,z}$ are the Pauli matrices on the particle-hole basis. By considering only a positive first neighbour hopping parameter $t$, the $\vec{d}(k)$ vector has the expression :

$$
\vec{d}(k) = \begin{cases} d_x = -2\Delta \sin(k_y a) \\ d_y = -2\Delta \sin(k_x a) \\ d_z = -2t(\cos(k_x a) + \cos(k_y a)) - \mu \end{cases}, \qquad (1.53)
$$

Similarly to the 1D case, the eigenenergies of the system are function of the components of
$\vec{d}(k)$. One has :

<footnote n="1">¹The observed peaks also develop along the whole chain, which is not is favor of Majorana bound states.</footnote>
$$E_{\pm}(\vec{k}) = \pm \sqrt{d_x^2 + d_y^2 + d_z^2}, \qquad (1.54)$$

$E_{\pm}(\vec{k})$ is equal to zero for four values of $(k_x, k_y, \mu)$ which are : $(0, 0, -4t)$, $(0, \pi, 0)$, $(\pi, 0, 0)$ and $(\pi, \pi, 4t)$. Three topological phases corresponding to $|\mu| > 4t$, $-4t < \mu < 0$ and $0 < \mu < 4t$ appear, since it is impossible to continuously tune the chemical potential to go from one of these phases to an other without closing the superconducting gap. We will now see what is the nature of the topological invariant distinguishing all three cases.

As it was done in 1D, we will introduce the quantity $\hat{d}(\vec{k})$ :

$$\hat{d}(\vec{k}) = \frac{\overrightarrow{d}(\vec{k})}{\| \overrightarrow{d}(\vec{k}) \|} = \frac{1}{\sqrt{d_x^2 + d_y^2 + d_z^2}} (d_x \quad d_y \quad d_z) = (\hat{d}_x \quad \hat{d}_y \quad \hat{d}_z), \qquad (1.55)$$

$\hat{d}(\vec{k})$ is normalized to one and lives on the Bloch sphere. Let us first consider the $\mu > 4t$ topological case. We can notice that $\hat{d}_z$ always is strictly inferior to zero which means that only one hemisphere of the Bloch sphere is explored. On the contrary, in the $0 < \mu < 4t$ topological case, $\hat{d}_z$ explores the whole Bloch sphere. This can easily be checked by taking an example. For instance, if $\Delta = t$ and $\mu = 2t$, then, for $(k_x, k_y) = (-\frac{\pi}{a}, -\frac{\pi}{a})$, $\hat{d} = (0 \quad 0 \quad 1)$ points to the "north" pole while for $(k_x, k_y) = (0, 0)$, $\hat{d} = (0 \quad 0 \quad -1)$ points to the "south" pole. The topological invariant can be defined as the degree of wrapping of the $\hat{d}(\vec{k})$ vector around the Bloch sphere when $\vec{k}$ explores the first Brillouin zone [37]. The associated topological invariant is called the first Chern number. In the $4t < \mu$ case, the Chern number is zero while in the $0 < \mu < 4t$ phase, $\hat{d}(\vec{k})$ wraps one time the Bloch sphere. Formally, this Chern number can be related to the Berry phase introduced in section 1.4.1. In topological physics, there is a property called the "bulk-edge correspondence". This property says that there is a matching between the number of edge states and the value of the topological invariant defined for the bulk material. In this example, the Chern number can take any integer value, which means that topological phases presenting any number of edge states can potentially arise.

We will prove the existence of in-gap edge states at the interface between two p-wave superconductors in different topological regimes. We have seen previously that the superconducting gap was closing at $(k_x, k_y, \mu) = (0, 0, -4t)$. We will thus consider an interface between a $\mu < -4t$ and a $-4t < \mu < 0$ phases with a spatial dependence of the chemical potential along the $x$ direction. The system is taken translationally invariant in the $y$ direction. We propose :

$$\mu(x) = \begin{cases} -\mu_1 & \text{if } x < 0, \\ -\mu_2 & \text{if } x > 0, \end{cases} \qquad (1.56)$$

with $-\mu_1 < -4t$ and $-4t < -\mu_2 < 0$. In addition, since the topological transition occurs at $(k_x, k_y) = (0, 0)$, we can develop $\mathcal{H}(\vec{k})$ to first order in $(k_x, k_y)$ around $(0, 0)$ and :

$$\mathcal{H}(\vec{k}) = \begin{pmatrix} -4t - \mu(x) & 2a\Delta(\frac{\partial}{\partial x} - k_y) \\ 2a\Delta(-\frac{\partial}{\partial x} - k_y) & -(-4t - \mu(x)) \end{pmatrix}, \qquad (1.57)$$

Since the system is invariant under translation in the $y$ direction, $k_y$ is a good quantum number and will label any eigenstates of $\mathcal{H}(\vec{k})$. Also, if we call $|\psi_{k_y}(x, y)\rangle$ such an eigenstate, it can generically be written like $|\psi_{k_y}(x, y)\rangle = e^{ik_y y} |\phi(x)\rangle$, where $|\phi(x)\rangle$ is a spinor which depends on $x$. One can verify that $|\phi(x)\rangle = \frac{1}{\sqrt{2}} e^{\frac{1}{2} \Delta \int_0^x dx' (4t+\mu(x'))} (1, 1)^T$ and that the corresponding eigenenergies are $E(k_y) = -2a\Delta k_y$, which can be arbitrarily close to zero. We see that a band of states fill the superconducting gap. It is shown in ref. [37] that these Dirac fermionic states can
also be decomposed in pairs of Majorana states just as in the Kitaev chain. However, contrary to the one-dimensional case, those Majorana fermions are not zero energy bound states but rather what we call "dispersive Majorana states".

Spinless superconductivity is unrealistic. In order to reproduce the physics that was just presented, sometimes referred to as chiral p-wave superconductivity, one possibility is to have a combination of both Rashba spin orbit-coupling and Zeeman effect [60], just as in the 1D case. The chiral p-wave superconductor has first been claimed to be observed in 2017 by G. Ménard et al. [3]. In this STM/STS work, they grew a superconducting Pb monolayer on top of a Si(111) substrate. Strong Rashba spin-orbit coupling is present at the surface and induced triplet correlations in the superconducting order parameter. Moreover, CoSi magnetic islands likely to provide Zeeman splitting were embedded under the Pb monolayer. The idea is that Zeeman splitting is such that the area of the Pb monolayer which is right above the CoSi islands enters a topologically non trivial regime. At the boundary between this topological "island" and the remaining of the Pb monolayer, they saw dispersive states, which were interpreted as chiral edge states. In 2018, A. Palacio-Morales et al. performed a STM/STS study on a magnet-superconductor hybrid consisting of Fe islands on top of a Re(0001)-O(2×1) surface, and also observed such dispersive edge states suggesting the formation of a topologically non-trivial phase [61].

Although having a topological origin, such chiral edge states cannot be used for the braiding operations that were previously mentioned but could hypothetically lead to quantum computation applications [62]. It does not mean that Majorana bound states cannot be obtained in 2D systems. In refs. [63, 37], it is shown that Majorana bound states are expected in the vortex core of a p-wave superconductor. From 2015 to 2018, several experimental realizations of such states were reported. The first kind was obtained by G. Ménard et al. in Pb/Co/Si(111) [4]. It would seem that some of the topological domains that were already presented in ref. [3] present a symmetry-authorized spin-orbit vortex configuration. Two zero-energy bound states were observed at the center of the domain (the core of the spin-orbit vortex) and at its edge, respectively. In the other realizations, a magnetic field was applied, thus inducing the apparition of regular magnetic vortices in the system. The second type of experimental evidence of Majorana bound states at the cores of vortices was attained in Bi$_2$Te$_3$/NbSe$_2$ heterostructures [6] exploiting the fact that it is possible to generate p-wave correlations at the surface of a topological insulator in proximity to a conventional superconductor as shown theoretically in ref. [64]. Finally, it was shown that Fe-based superconductors can enter a topologically non-trivial regime through doping [65, 66]. On this basis, STM/STS studies performed on FeTe$_{0.55}$Se$_{0.45}$ [7] and on (Li$_{0.84}$Fe$_{0.16}$)OHFeSe [67] highlighted the presence of zero-bias states in the cores of vortices, also in agreement with a Majorana bound state interpretation. We once again stress that none of the zero-bias bound states observations listed in the two last sections, although very promising, can be taken as indisputable evidence of Majorana bound states features since the non-Abelian statistics as yet not been addressed.
# Chapter 2

## Scanning Tunnelling Microscopy/Spectroscopy

Scanning Tunnelling Microscopy (STM) is a technique which can be used to probe the electronic properties of surfaces of materials. STM was invented in 1982 by Gerd Binnig and Heinrich Rohrer [68] and takes advantage of the well-known quantum tunnelling effect. This invention eventually lead both Rohrer and Binnig to win the Nobel Prize in Physics in 1986. In this chapter, we will briefly present the principle of STM as well as what are the different measurements it allows to perform. The great majority of results which are presented in this thesis were obtained using this experimental technique.

First, in section 2.1, we will introduce the procedure to image the surfaces of materials using STM. Second, in section 2.2, we will explain what is Scanning Tunnelling Spectroscopy (STS) technique and how it gives access to the local density of states (LDOS) of samples, making it a very powerful tool. Finally, in section 2.3, we will show how Fourier transform STM and more specifically Quasiparticle Interferences (QPIs) can be used to artificially rebuilt the band structures of materials.

### 2.1 Topographic constant current mode

Let us consider an electron of energy $E$ submitted to an unidimensional rectangular potential barrier of height $W$ and of width $d$ such as represented in figure 2.1. The time-independent Schrödinger equation between the edges of the rectangular barrier is the following :

$$ -\frac{\hbar^2}{2m_e} \frac{d^2}{dx^2} \Psi(x) + (W - E)\Psi(x) = 0. \qquad (2.1) $$

The solution of this equation is trivial and in the end :

$$ \Psi(x) \propto e^{-\frac{\sqrt{2m_e(W-E)}x}{\hbar}} = e^{-\kappa(E)x}. \qquad (2.2) $$

Thus, $|\Psi(d)|^2 \propto e^{-\frac{2\sqrt{2m_e(W-E)}d}{\hbar}}$. One can see that $|\Psi(d)|^2$, the probability that the electron is transmitted through the barrier, decreases exponentially with $d$. The electrons at the surface of a given material have thus a small probability of leaking into the vacuum. In this case, $W$ is called the work function, which is of the order of 4 eV in metals (see any work function table of common metals).
<figure><img src="image_14.png" /><figcaption>Figure 2.1: Unidimensional rectangular potential barrier of height *W* and of width *d*.</figcaption></figure>

If a conducting tip is now brought very close to that very surface (typically at 5 to 10 Å from it), and if a bias voltage *V* is applied between the tip and the surface, electrons have a non-zero probability to hop from the tip to the surface or inversely, depending on the sign of *V*, thus giving rise to a tunnelling current. The tunnelling current and the applied bias voltage are independent parameters. A given set of those parameters however defines a specific tunnelling regime and fixes the tip-sample distance. This technique is highly surface-sensitive since if the tip-sample distance increases of 1 Å, the tunnelling current is divided by a factor 10. Only electrons within few Å from the tip will contribute to the total tunnelling current.

<figure><img src="image_15.png" /><figcaption>Figure 2.2: **a.** Schematic drawing of the principle of STM. **b.** Topographic image of the surface of (LaSe)₁,₁₄(NbSe₂)₂ obtained with STM.</figcaption></figure>

All the topographic imaging performed in this thesis was obtained using constant current mode scanning. This mode works as follows: the tip is moved laterally across the surface while maintaining a constant tunnelling current thanks to a feedback loop which adjusts the height of the tip. Those changes in the vertical position of the tip are recorded as a function of lateral coordinates thus allowing the construction of a topographic image of the studied sample.
Piezoelectric compounds are used so as to obtain the short three-dimensional displacements of the tip which are needed in order to realize atomically resolved topography<fnref n="1" />. By fixing the tip at the top of a piezoelectric tube, and by applying the correct bias voltages between several sides of this tube, it is possible to bend it, contract it or expand it at will (see figure 2.2).

## 2.2 Probing the local density of states of samples via STS

While STM is a very powerful technique to access the crystal structure of surfaces, one must always keep in mind that topographic data also contains electronic spectroscopic features of the material. In this section, we will show how tunnel effect can be exploited to access to LDOS of a material.

### 2.2.1 Expression of the tunneling current

In this part, we will derive the expression of the tunnelling current between two conducting materials when a voltage difference is applied between them.

Let us consider two conducting materials labelled 1 and 2. The Hamiltonian of the system can be decomposed as follows :

$$ \left\{ \begin{array}{l} \hat{H} = \hat{H}_1 + \hat{H}_2 + \hat{H}_T, \\ \hat{H}_T = \sum_{kk',\sigma} (T_{kk'} c_{1k,\sigma}^{\dagger} c_{2k',\sigma} + T_{kk'}^{*} c_{2k',\sigma}^{\dagger} c_{1k,\sigma}) = \underbrace{\sum_{kk',\sigma} T_{kk'} c_{1k,\sigma}^{\dagger} c_{2k',\sigma}}_{T(t)} + \underbrace{\sum_{kk',\sigma} T_{kk'}^{*} c_{2k',\sigma}^{\dagger} c_{1k,\sigma}}_{T^{\dagger}(t)} \end{array} \right. \qquad (2.3) $$

where $\hat{H}_1$ and $\hat{H}_2$ are the Hamiltonians describing respectively systems 1 and 2, $\hat{H}_T$ describes the tunnelling between the two conducting materials, and $T_{kk'}$ is the hopping amplitude between an electronic state of momentum $k$ in one material and an electronic state of momentum $k'$ in the other material.

A Voltage difference $V = V_1 - V_2$ is applied between the two systems, so that the chemical potentials of systems 1 and 2 (respectively $\mu_1$ and $\mu_2$) are different. Moreover, we have :

$$ \left\{ \begin{array}{l} \mu_1 = \mu + eV_1 \\ \mu_2 = \mu + eV_2 \end{array} \right. , \qquad (2.4) $$

The electric current going from 1 to 2 can be defined by : $\hat{I} = -e\langle\hat{I}_p\rangle$ where $\hat{I}_p = -\frac{d\hat{N}_1}{dt}$ with $\hat{N}_1 = \sum_{k,\sigma} c_{1k,\sigma}^{\dagger} c_{1k,\sigma}$, the number operator of the first system. $\langle...\rangle$ is the mean value taken on the perturbed system. Similarly, the number operator of the second system is $\hat{N}_2 = \sum_{k,\sigma} c_{2k,\sigma}^{\dagger} c_{2k,\sigma}$.

A calculation where system 2 is supposed to be a metal (and thus having a constant density of states around Fermi level) is done in Annex.1 and eventually leads to the following equation for the measured differential conductance :

$$ \frac{d\hat{I}}{dV} = e^2 2\pi |T|^2 n_0 \int_{-\infty}^{+\infty} d\omega n_1(\omega) \frac{dn_F}{d\omega}(\omega + eV), \qquad (2.5) $$

where $n_0$ is the density of states at Fermi level of system 2, $n_1(\omega)$ the density of states at energy $\omega$ of system 1 and $n_F$ the Fermi-Dirac function.

At zero temperature, $\frac{dn_F}{d\omega}(\omega + eV) = \delta(\omega + eV)$, so it gives :

<footnote n="1">¹A piezoelectric material is a material which can be electrically polarized when it is subject to mechanical stress and reciprocally can be deformed if a bias voltage is applied at its edges.</footnote>
$$ \frac{d\hat{I}}{dV} = e^2 2\pi |T|^2 n_0 n_1(-eV). \qquad (2.6) $$

The tunnelling conductance $\frac{d\hat{I}}{dV}$ is proportional to $n_1(-eV)$, the density of states of system 1 at energy $-eV$. Experimentally, one can access the density of states of a sample by measuring this conductance using STS (see 2.2.2 for details). At finite temperature, the tunnelling conductance is proportional to the convolution of the density of states of the studied material with the derivative of the Fermi-Dirac function. This leads to an enlargement in energy of $3,5k_BT$ of the states at temperature $T$. At 300 mK, this enlargement in energy is about $\sim 90$ µeV. The effect of thermal broadening on the measured conductance is illustrated in figure 2.3.

<figure><img src="image_16.png" /><figcaption>Figure 2.3: The blue line is the theoretical BCS density of states at zero temperature (see section 1.1). The dotted red line is the Fermi Dirac distribution obtained at temperature $T$. The green line is the measured superconducting $dI/dV$ spectrum obtained at temperature $T$ with a normal tip.</figcaption></figure>

### 2.2.2 Tunneling Spectroscopy

A Scanning Tunnelling Spectroscopy (STS) experiment is similar to a STM one, both method being based on quantum tunnelling effect and permitting to study surfaces. In STS, the tip is brought very close to the sample surface. Here again, a bias voltage $V$ is applied between tip
and sample, thus inducing a tunnelling current. Because of formula 2.6, if one uses a metallic tip whose density of states is known and constant in the voltage range of the measurement, then, by varying the value of V while measuring the tunnelling current (and deriving numerically the conductance as a function of V) it is possible to probe the local density of states at a specific point (x, y) of the studied sample. The local density of states at energy E is defined as :

$$n(E, x, y) = \sum_{n} |\psi_{n}(E_{n}, x, y)|^{2} \delta(E - E_{n}), \qquad (2.7)$$

where the $\psi_n$ are the eigenfunctions of the system associated to eigenvalues $E_n$.

A summary of the principle of the measurement is represented on figure 2.4. In case where the sample is a superconductor (see figure 2.4 b), electrons can hop from tip to sample only when an electronic state is available. Therefore, by scanning the surface, it is possible to obtain a map of the local density of states of a region of the sample, and this, for several values of $-eV$ or equivalently of E.

<figure><img src="image_17.png" /><figcaption>Figure 2.4: a. Principle of STS illustrated by tunnelling processes in a metal. b. Principle of STS illustrated by tunnelling processes in a superconductor.</figcaption></figure>

**Remark:** STM and STS are very often done simultaneously. By doing so, the topography of the studied surface and the spatial dependence of electronic states visible on spectroscopy can be correlated.

### Tunnelling selectivity.

As it was already mentioned in section 2.1, the wave functions of the electrons at the surface of a material decay exponentially into the vacuum. Moreover, in a periodic crystal, the electrons can be described by Bloch wave functions labelled by crystalline momentum $\vec{k}$. The periodicity along the direction perpendicular to the surface is broken, such that at the surface, one can only label the wave functions of the electrons with the component of the momentum parallel to the surface $\vec{k}_{\parallel}$. In ref. [69], J. Tersoff and D. R. Hamann give the expression of such wave functions :

$$\psi_{\vec{k}_{\parallel}, E} = \sum_{\vec{G}} c_{\vec{k}_{\parallel}, \vec{G}} e^{i(\vec{k}_{\parallel} + \vec{G}) \cdot \vec{r}} \exp(-\alpha_{\vec{k}_{\parallel} + \vec{G}} z), \qquad (2.8)$$
where $\alpha_{\vec{k}_{\parallel}+\vec{G}} = \sqrt{|\vec{k}_{\parallel}|^2 + \kappa(E)^2}$, $\vec{G}$ is a vector of the reciprocal lattice, $z$ is the distance to the surface, $\vec{r}$ is the in-plane position and $c_{\vec{k}_{\parallel}, \vec{G}}$ are the components of the Fourier series development of $\psi_{\vec{k}_{\parallel}, E}$.

If the system is in a tip-parallel-to-the-z-direction geometry, one sees from the expression of $\alpha_{\vec{k}_{\parallel}+\vec{G}}$ that the wave functions with small $\vec{k}_{\parallel}$ (momenta close to $\Gamma$) decay over longer distances into the vacuum than the wave functions with bigger $\vec{k}_{\parallel}$ (momenta far from $\Gamma$). This last observation means that, in this geometry, the electrons close to $\Gamma$ in reciprocal space are the ones which mostly contribute to the total tunnelling current. Scanning tunnelling spectroscopy experiments are then intrinsically more sensitive to electronic states with smaller wave vectors $\vec{k}_{\parallel}$. We speak of tunnelling selectivity.

<figure><img src="image_18.png" /><figcaption>Figure 2.5: **a.** $dI/dV$ spectrum taken on $MgB_2$ with the tip aligned along the c-axis. **b.** $dI/dV$ spectrum taken on $MgB_2$ with the tip aligned along the a-axis. Figure taken from [70].</figcaption></figure>

A famous example of such tunnelling selectivity effects is the one of two-band superconductivity in $MgB_2$. In 2002, M. Iavarone et al. observed different $dI/dV$ spectra depending on the orientation of $MgB_2$ crystals with respect to their tip [71]. When the tip is aligned with the c-axis of $MgB_2$, they see $dI/dV$ spectra exhibiting one superconducting gap $\Delta_1$ (see figure 2.5 **a**). However, when the geometry is different and the tip is oriented along the a-axis of $MgB_2$, the $\Delta_1$ is still seen but together with an additional $\Delta_2$ superconducting gap (see figure 2.5 **b**). The calculations confirmed that this effect was actually due the faster decay of the electronic states associated to $\Delta_2$ with respect to the ones responsible for $\Delta_1$ [70].

### 2.2.3 Data treatment

In experiments, $I(V)$ curves are measured. In order to access the local density of states, one has to numerically compute the derivative of $I$ with respect to $V$. However, as illustrated by the red $I(V)$ spectrum in figure 2.6 **a**, experimental data will always present a certain degree of noise. As a consequence, the derivative $dI/dV$ of a raw spectrum will necessarily diverge at many bias voltages. Such divergences are not physical since only related to the level of noise present during the acquisition of the data. Hence, a "smoothing" procedure has to be applied beforehand. During this thesis we applied the following data treatment to all $I(V)$ individual spectra: first, the data is convoluted with a normalized Gaussian of width $3, 5k_BT$ (our energy
resolution is limited by thermal broadening), second, a Savitzky-Golay filter is also applied [72]. Single raw and filtered $I(V)$ spectra are represented in figure 2.6 a. After filtering the data, it is finally possible to obtain the derivative of the experimental single $I(V)$ spectrum (see figure 2.6 b). We see that this spectrum still remains noisy and particularly on the "wings" (which is normal because the noise is proportional to the current). The more time spent for each point during the data acquisition and the finer the energy sampling, the more efficient will be the filtering procedure. Depending on the information one wants to highlight, it might or nor be important to have nice individual spectra. On the one hand, if one is interested in getting a $dI/dV$ spectrum within a given energy range and representative of a given sample, one can simply average a collection of individual $dI/dV$ spectra taken in the same conditions. On the other hand, if one wants to access the local density of states maps at several energies (in a way explained in 2.2.4), then, it is crucial to make sure information can easily be extracted from all individual spectra.

<figure><img src="image_19.png" /><figcaption>Figure 2.6: **a.** The red $I(V)$ spectrum is an individual spectrum taken on a superconducting monolayer of Pb/(Si(111)). In blue is the same $I(V)$ spectrum after performing of gaussian and Savitzky-Golay filtering. **b.** This $dI/dV$ spectrum was obtained by numerically derivating the filtered $I(V)$ spectrum of **a**.</figcaption></figure>

### 2.2.4 Spectroscopic grid experiments

In this subsection, we expose the principle of spectroscopic grid experiments which we have used to obtain most of the results presented in this thesis. We will base our explanation on the schematic representation of figure 2.7. In figure 2.7 **a** is represented an area of a sample illustrated by a topography map. A predefined square grid with $N \times N$ nodes is superimposed (blue crosses) on top of this topography map. The general idea is to take $I(V)$ spectra<fnref n="2" /> with common bias voltage range [$V_{start}, V_{end}$] and sampling at the positions of all the nodes of this

<footnote n="2">²One chooses a given set point corresponding to a couple of current and bias voltage ($I_{setpoint}$, $V_{setpoint}$), thus fixing a given tip-sample distance. Before sweeping the voltage and taking the $I(V)$ curve, one opens the feedback loop, meaning that the spectrum is acquired at constant height.</footnote>
grid. In the end, after performing a similar data treatment as in subsection 2.2.3, one has a
collection of $N \times N \, dI/dV$ spectra which contain the spatial dependence of the density of states
of the sample and this for all energies within $[V_{start}, V_{end}]$. If $V_j$ is an arbitrary bias voltage
belonging to $[V_{start}, V_{end}]$, one can indeed build up a local density of states map at energy $eV_j$
by defining $N \times N$ matrices which all components are the values of all $dI/dV$ spectra evaluated
for $V = V_j$.

<figure><img src="image_20.png" /><figcaption>Figure 2.7: **a.** Grid of STS data. *I(V)* spectra are taken at the nodes of a predefined spatial square grid. **b.** Typical *I(V)* individual spectrum taken at the *i* node position in the [*V*<sub>start</sub>, *V*<sub>end</sub>] bias voltage range. **c.** Derivative of the *I(V)* spectrum of **b.** **d.** Construction of density of states maps for all the energies belonging to [*V*<sub>start</sub>, *V*<sub>end</sub>].</figcaption></figure>

**2.3 Quasiparticle interferences and Fourier transform STM**

In this section, we will see how those spectroscopic grids can be used to access quasiparticle
interferences patterns.

In a crystal, the electrons evolve in the periodic potential of the ionic lattice. Bloch's theorem tells us that the eigenstates of the systems are labelled by crystal momentum **k** and are similar to plane waves however modulated by a complex function $u_{{\bf k}}({\bf r})$ with the periodicity of the lattice. Those eigenstates are called Bloch wave functions and have the following expression: $\psi_{{\bf k}}({\bf r}) = e^{i{\bf k}\cdot{\bf r}}u_{{\bf k}}({\bf r})$. Any source of disorder (defects, steps, dislocations...) breaks the translational symmetry of the periodic crystal such that **k** is no longer conserved and $\psi_{{\bf k}}$ is no longer an
eigenstate. A given electron with incoming momentum $\mathbf{k}_i$ and in $\psi_{\mathbf{k}_i}$ state will be likely to scatter once encountering a defect. If only elastic scattering is considered, this electron might scatter in a final state with momentum $\mathbf{k}_f$ and $\psi_{\mathbf{k}_f}$ state. Since both incoming and final states have the same energy, they can interfere, and the electron is in the superposition of both : $\psi_{\mathbf{tot}} = \psi_{\mathbf{k}_i} + \psi_{\mathbf{k}_f}$. In the following, we will derive the expression of the local density of states or so to say $|\psi_{\mathbf{tot}}|^2$. We recall that the local density of states is the physical quantity probed in STS :

$$ \psi_{\mathbf{tot}} = e^{i\mathbf{k}_i \cdot \mathbf{r}} u_{\mathbf{k}_i}(\mathbf{r}) + e^{i\mathbf{k}_f \cdot \mathbf{r}} u_{\mathbf{k}_f}(\mathbf{r}), \qquad (2.9) $$

$$ |\psi_{\mathbf{tot}}|^2 = (e^{i\mathbf{k}_i \cdot \mathbf{r}} u_{\mathbf{k}_i}(\mathbf{r}) + e^{i\mathbf{k}_f \cdot \mathbf{r}} u_{\mathbf{k}_f}(\mathbf{r}))(e^{-i\mathbf{k}_i \cdot \mathbf{r}} u_{\mathbf{k}_i}^*(\mathbf{r}) + e^{-i\mathbf{k}_f \cdot \mathbf{r}} u_{\mathbf{k}_f}^*(\mathbf{r})), \qquad (2.10) $$

and we define $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i$

$$ |\psi_{\mathbf{tot}}|^2 = |u_{\mathbf{k}_i}(\mathbf{r})|^2 + |u_{\mathbf{k}_f}(\mathbf{r})|^2 + 2\Re(e^{i\mathbf{q} \cdot \mathbf{r}} u_{\mathbf{k}_i}^*(\mathbf{r}) u_{\mathbf{k}_f}(\mathbf{r})), \qquad (2.11) $$

In QPIs measurements, one plots the square modulus of the Fourier transform of the local density of states map, one thus has to calculate $FT(|\psi_{\mathbf{tot}}|^2)(\mathbf{K})$ where FT designs the Fourier transform. We will now look at the first term $FT(|u_{\mathbf{k}_i}|^2)(\mathbf{K})$. We have $u_{\mathbf{k}_i}(\mathbf{r}) = \sum_{\mathbf{G}} c_{\mathbf{k}_i+\mathbf{G}} e^{i\mathbf{G} \cdot \mathbf{r}}$. In this last relation, the sum is on the reciprocal lattice $\mathbf{G}$ vectors. Moreover :

$$ \mathrm{FT}(|u_{\mathbf{k}_i}|^2)(\mathbf{K}) = (\mathrm{FT}(u_{\mathbf{k}_i}) * \mathrm{FT}(u_{\mathbf{k}_i}^*)(\mathbf{K})), \qquad (2.12) $$

Given the fact that the Fourier transform of a product is the convolution product of the Fourier transforms (convolution product is designed by $*$). Because Fourier transform is a linear operation and also that the Fourier transform of $e^{i\mathbf{G} \cdot \mathbf{r}}$ is $\delta(\mathbf{K} - \mathbf{G})$, we have $FT(u_{\mathbf{k}_i})(\mathbf{K}) = \sum_{\mathbf{G}} c_{\mathbf{k}_i+\mathbf{G}} \delta(\mathbf{K} - \mathbf{G})$. In the end, computing $FT(|\psi_{\mathbf{tot}}|^2)(\mathbf{K})$ is just about computing convolution products between $\delta(\mathbf{K} - \mathbf{G})$ and $\delta(\mathbf{K} + \mathbf{G}')$ Dirac functions.

$$ \int \delta(\mathbf{K}' - \mathbf{K} - \mathbf{G})\delta(\mathbf{K} + \mathbf{G}')d\mathbf{K} = \delta(\mathbf{K}' - (\mathbf{G} - \mathbf{G}')), \qquad (2.13) $$

The $FT(|u_{\mathbf{k}_i}|^2)(\mathbf{K})$ term then only gives signal for $\mathbf{K}'$ equal to any reciprocal lattice vector or saying differently at the Bragg peaks positions. The $FT(|u_{\mathbf{k}_f}|^2)(\mathbf{K})$ term is similar and also gives signal at Bragg peaks positions. This results is not really surprising since similar terms arise when there are no defects and thus no QPIs. If one indeed looks at the Fourier transform of a local density of states map of a perfect surface, only signal at the Bragg peaks position are possible. Nevertheless, the $TF(2\Re(e^{i\mathbf{q} \cdot \mathbf{r}} u_{\mathbf{k}_i}^* u_{\mathbf{k}_f})(\mathbf{K}))$ term is only present through interferences. We have :

$$ 2\Re(e^{i\mathbf{q} \cdot \mathbf{r}} u_{\mathbf{k}_i}^*(\mathbf{r}) u_{\mathbf{k}_f}(\mathbf{r})) = 2\Re((\cos(\mathbf{q} \cdot \mathbf{r}) + i \sin(\mathbf{q} \cdot \mathbf{r})(\Re(u_{\mathbf{k}_i}(\mathbf{r}))) -i\Im(u_{\mathbf{k}_i}(\mathbf{r})))\Re(u_{\mathbf{k}_f}(\mathbf{r})) + i\Im(u_{\mathbf{k}_f}(\mathbf{r}))), \qquad (2.14) $$

$$ 2\Re(e^{i\mathbf{q} \cdot \mathbf{r}} u_{\mathbf{k}_i}^*(\mathbf{r}) u_{\mathbf{k}_f}(\mathbf{r})) = 2[\cos(\mathbf{q} \cdot \mathbf{r})\Re(u_{\mathbf{k}_i}(\mathbf{r}))\Re(u_{\mathbf{k}_f}(\mathbf{r})) + \cos(\mathbf{q} \cdot \mathbf{r})\Im(u_{\mathbf{k}_i}(\mathbf{r}))\Im(u_{\mathbf{k}_f}(\mathbf{r})) \\ - \sin(\mathbf{q} \cdot \mathbf{r})\Re(u_{\mathbf{k}_i}(\mathbf{r}))\Im(u_{\mathbf{k}_f}(\mathbf{r})) + \sin(\mathbf{q} \cdot \mathbf{r})\Im(u_{\mathbf{k}_i}(\mathbf{r}))\Re(u_{\mathbf{k}_f}(\mathbf{r}))], \qquad (2.15) $$

It is now time to compute the Fourier transform of the expression that was just derived. The computation of only $TF(2 \cos(\mathbf{q} \cdot \mathbf{r})\Re(u_{\mathbf{k}_i})\Re(u_{\mathbf{k}_f}))(\mathbf{K}^'')$ is similar to the computation of all other three terms left, that is why we will only be focusing on this one. Since the Fourier transform of $2 \cos(\mathbf{q} \cdot \mathbf{r})$ is $\delta(\mathbf{K} - \mathbf{q}) + \delta(\mathbf{K} + \mathbf{q})$. We have :

$$ \mathrm{TF}(2 \cos(\mathbf{q} \cdot \mathbf{r})\Re(u_{\mathbf{k}_i})\Re(u_{\mathbf{k}_f}))(\mathbf{K}^') = ((\delta(\mathbf{K}' - \mathbf{q}) + \delta(\mathbf{K}' + \mathbf{q})) * \mathrm{TF}(\Re(u_{\mathbf{k}_i})\Re(u_{\mathbf{k}_f}))(\mathbf{K}^')), \qquad (2.16) $$
Also, $\Re(u_{\mathbf{k}_i}(\mathbf{r})) = \sum_{\mathbf{G}} a_{\mathbf{G}}^i \cos(\mathbf{G} \cdot \mathbf{r})$ and $\Re(u_{\mathbf{k}_f}(\mathbf{r})) = \sum_{\mathbf{G}} a_{\mathbf{G}}^f \cos(\mathbf{G} \cdot \mathbf{r})$, because both $\Re(u_{\mathbf{k}_i}(\mathbf{r}))$ and $\Re(u_{\mathbf{k}_f}(\mathbf{r}))$ are real periodic functions of $\mathbf{r}$ and can be expanded in Fourier series. Consequently, $TF(\Re(u_{\mathbf{k}_i}))(\mathbf{K}) = \sum_{\mathbf{G}} a_{\mathbf{G}}^i(\delta(\mathbf{K} - \mathbf{G}) + \delta(\mathbf{K} + \mathbf{G}))$ and $TF(\Re(u_{\mathbf{k}_f}))(\mathbf{K}) = \sum_{\mathbf{G}} a_{\mathbf{G}}^f(\delta(\mathbf{K} - \mathbf{G}) + \delta(\mathbf{K} + \mathbf{G}))$. We use again the properties of Fourier transform: $TF(\Re(u_{\mathbf{k}_i})\Re(u_{\mathbf{k}_f}))(\mathbf{K}') = (TF(\Re(u_{\mathbf{k}_i}))*TF(\Re(u_{\mathbf{k}_f}))(\mathbf{K}'))$ and it is again about computing convolution products between $\delta(\mathbf{K}' \pm \mathbf{G})$ and $\delta(\mathbf{K}' \pm \mathbf{G}')$. The Fourier transform of $\Re(u_{\mathbf{k}_i})\Re(u_{\mathbf{k}_f})$ is a sum of Dirac functions $\delta(\mathbf{K}' - \mathbf{G})$ centered on reciprocal lattice vectors $\mathbf{G}$. Lastly, everything results in calculating convolution products between $\delta(\mathbf{K}' - \mathbf{q})$ and $\delta(\mathbf{K}' - \mathbf{G}')$:

$$ \int \delta(\mathbf{K}'' - \mathbf{K}' - \mathbf{q})\delta(\mathbf{K}' - \mathbf{G}')d\mathbf{K}' = \delta(\mathbf{K}' - (\mathbf{q} + \mathbf{G}')), \quad (2.17) $$

The final conclusion is that QPIs give signal at scattering wave vector $\mathbf{q}$ position up to a reciprocal lattice vector $\mathbf{G}'$. The incoming and final states interfere giving rise to standing waves patterns of wave vector $\mathbf{q}$.

According to Fermi's golden rule, and as explained in ref. [73], the amplitude of scattering between states at $(E_i, \mathbf{k}_i)$ and $(E_f, \mathbf{k}_f)$ is:

$$ \omega(i \to f) \propto \frac{2\pi}{\hbar} |V(\mathbf{q})|^2 n_i(E_i, \mathbf{k}_i)n_f(E_f, \mathbf{k}_f), \quad (2.18) $$

where $V(\mathbf{q})$ is Fourier transform of scattering potential at scattering wave-vector $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i$ and $n(E, \mathbf{k})$ is the density of states at energy $E$ and wave-vector $\mathbf{k}$. $E_i$ is considered equal to $E_f$ since we only consider elastic scattering. This expression tells us that the higher the number of incoming and final states and consequently the flatter (the lower the group velocity) the bands between which scattering occurs, the higher the transition probability. Moreover, if two initial and final collections of states are linked by a same scattering wave-vector $\mathbf{q}$, the probability of observing standing-wave at this very vector $\mathbf{q}$ becomes larger too. By gathering all the information that were exposed up to know, we can conclude that QPIs patterns at a given energy $E$ can be obtained by doing a sort of autoconvolution of the energy contour at $E$, taking into account the spectral weight distribution across this energy contour. We speak of joint density of states (JDOS).

The expected QPIs patterns for simple energy contours are discussed in figure 2.8. In all of the examples, we consider systems with hexagonal Brillouin zones. The reason behind this is that the experimental QPIs results of this thesis were obtained on NbSe₂ layers, which have an hexagonal lattice. Figure 2.8 a represents a circular energy contour of radius $r$ and centered on $\Gamma$. The scattering wave-vector $\mathbf{q}$ (of norm $2r$) displayed on the image possesses good nesting properties because the two circular portions of energy contour it links are tangent to one another. The same can be said of all wave-vectors with arbitrary directions but same norm as $\mathbf{q}$. The expected QPIs pattern is represented in figure 2.8 a and is a circle centered on $\Gamma$ and of radius $2r = \|\mathbf{q}\|$, i.e. twice as big as the one of the corresponding energy contour. In figure 2.8 c is represented an hexagonal-shaped energy contour. $\mathbf{q}_1$ scattering wave-vector links two straight and vertical portions of the energy contour which means that a collection of incoming states can scatter into a collection of many final states with the same scattering wave-vector $\mathbf{q}_1$. Hence, it is very likely that QPIs patterns will have a strong contribution at $\mathbf{q}_1$ position in reciprocal space. By symmetry, the same can be said about $\mathbf{q}_2$ and $\mathbf{q}_3$ vectors. The QPIs pattern displayed in figure 2.8 d is thus made out of six spots at $\|\mathbf{q}_1\|$ distance from $\Gamma$ and lying along $\Gamma M$ directions.

The most famous example of the observation of QPIs signal is the case of Cu(111) surface [74]. A parabolic free-electron-like band is present at the surface of Cu(111) (see figure 2.8 e). The Fermi surface is consequently similar to the one of figure 2.8 a. In this STM study, Crommie et al. observed ripples around structural defects due to the interference of quasi-electrons' wave functions (standing-waves). Figure 2.8 f) illustrates how the selectivity of the
scattering channels can be influenced by the geometry of the defect. Here, the considered defect is a step. Let us call $x$ the direction parallel to the step. The lattice remains periodic along $x$, such that $k_x$ is a good quantum number. The consequence of this last remark is that only scattering wave vectors conserving $k_x$ will contribute to the overall QPIs signal.

<figure><img src="image_21.png" /><figcaption>Figure 2.8: **a.** The red line is a circular energy contour of diameter $q$ and centered on $\Gamma$. **b.** The red line represents the expected QPIs pattern for the circular energy contour of **a**. The result is a circle centered on $\Gamma$ and of radius $q$. **c.** The red line is a hexagonal energy contour. $q_1, q_2$ and $q_3$ are good nesting vectors. **d.** The red points lie at $q_1, q_2$ and $q_3$ positions and represent the expected QPIs patterns for the hexagonal energy contour of **c**. **e.** Energy dispersion of the surface state of Cu(111) [74]. This famous example is an illustration of what happens in **a** and **b**. **f.** Standing-wave patterns with periodicity $2\pi/q$ visible along a step in Cu(111) [74].</figcaption></figure>

Scattering channels depend on the symmetry of the defect and also on its spin texture. For example, an incoming electron of spin up might interact with a magnetic impurity and scatter into a spin down state. If spin is conserved, the incoming and scattered states cannot interfere because the spin part of the respective incoming and scattered wave functions are orthogonal and thus, scattering occurs but not QPIs. Nevertheless, if the defect couples the spin of the incoming electron to its momentum (we speak of spin-orbit impurity), spin might no longer be a good quantum number and new channels open regarding QPIs patterns [75]. Materials with strong intrinsic spin-orbit coupling should in principle also open QPIs channels with opposite spins but it is not necessarily observed. The Rashba spin-orbit split surface state of Au(111) is indeed not seen in QPIs measurements or at least not directly. In ref. [76], P. Leicht et al. grew epitaxial graphene flakes on Au(111). In this article, the Rashba band splitting was indirectly exhibited thanks to graphene to gold QPIs scattering channels and not through gold to gold QPIs scattering channels.

All the concepts developed in this section will be necessary to interpret the results presented in chapter 3. In this case, the disorder potential is a priori non-magnetic and not spin-orbit-like.
## 2.4 Experimental set-ups

In this section, we will present the two experimental set-ups on which all the scanning tunnelling microscopy/spectroscopy measurements discussed in this manuscript were performed. First, we will present the M3 microscope on which of the results of chapter 3 were obtained. Second, we will present the Low Temperature Omicron STM, on which some of the results of chapter 3 and all the experimental data of chapter 4 were acquired.

### 2.4.1 M3 STM

#### UHV equipment.

Figures 2.9 a and b represent a photograph and a schematic top view of the M3 STM system, respectively. This set-up is an home-made apparatus allowing to perform scanning tunnelling microscopy/spectroscopy measurements under ultra-high vacuum (UHV), down to 300 mK and under strong magnetic fields (up to 7 T). This equipment is composed of two ultra-high vacuum chambers : a preparation chamber, in which samples can be prepared *in situ*, and the STM chamber, where one can change samples or/and tips. Ii is also in the STM chamber that the samples are cleaved. In both chambers, the pressure is around $10^{-11}$ mbar. An ionic pump as well as titanium sublimators maintain this very low pressure in each chamber<fnref n="3" />

An extra chamber, called the load lock, is attached to the system in order to introduce the samples in UHV. This load lock is linked to a primary pump (which allows at first the load lock to reach a pressure of $10^{-2}$ mbar) and to a turbo pump which eventually leads to a load lock pressure of $\approx 2.10^{-8}$ mbar. A magnetic manipulator allows to transfer the samples from the load lock to the preparation chamber on condition that the load lock pressure is low enough. It is indeed important not to pollute the preparation chamber nor to break the vacuum.

<figure><img src="image_22.png" /><figcaption>Figure 2.9: a Photograph of the M3 setup. b Schematic top view of the M3 setup.</figcaption></figure>

**Remark:** The STM chamber is never put in direct contact with the load lock. This additional security measure ensures the cleanliness of the environment in which the measurements

<footnote n="3">³Such pressures can be reached after performing a bake out procedure which consist in heating the chambers at 150°C for two to four days to degas the internal surfaces.</footnote>
are carried out.

The preparation chamber contains, among other things, a quartz crystal microbalance (QCM) and an evaporator which allowed me to deposit lead on $(LaSe)_{1,14}(NbSe_2)_2$. The QCM is used to calibrate the deposition rate per unit of surface.

The mechanical manipulator allows to move the samples from preparation chamber to STM chamber and inversely. The STM chamber contains a carouse in which one can store several tips and samples. Those tips/samples can be manipulated with the help of a wobble stick in order to be put in or out of the microscope.

The approach of the tip to the surface is done with the help of a camera filming the reflection on a mirror (located in the chamber) of the tip-sample ensemble. It is crucial to approach the tip as close as possible from the surface so as to gain time on the fine approach, which is done *via* automatic piezoelectric displacements<fnref n="4" />. In practice, due to optical lenses and geometrical angle, the tip-sample distance cannot be better controlled than about 20 micrometers.

Moreover, in STM/STS experiment, so as to have atomic resolution, one needs to be properly isolated from external mechanical vibrations. The whole experimental set-up is, hence, mounted on three compressed-air-alimented vibration dampers, which ensure a good isolation from external mechanical noises.

### Low temperature circuit.

In measurement position, the microscope is taken down to the cryostat through a guide rod. The cryostat is the part of the setup in which the microscope will be brought down to very low temperatures. A 60 L hollow cylinder filled with liquid $^4$He at 4, 2 K is separated by an Internal Vacuum Can (IVC) at $\simeq 10^{-8}$ mbar from an internal pot filled with $^3$He$^5$. This later pot is itself in thermal contact with the very bottom of the microscope (this part is called the $^3$He tail). A sorption pump made out of activated carbon is connected through a capillary to the $^3$He pot. Right next to it, a pot drains liquid $^4$He from the $^4$He bath, again, through a capillary. By pumping on this pot filled with liquid $^4$He, it is possible to effectively decrease its temperature down to 1 K. This is why this pot is called the 1K-pot.

### Cooling procedure.

In this short subsection we will explain the condensation process which allows to cool the system down to 300mK. At the beginning of the cooling process, all the $^3$He is cryosorbed on the walls of the charcoal sorption pump. A resistor attached to the sorption pump is heated up to 50 K. This leads to the releasing of all the $^3$He gas into the $^3$He pot. Because the $^3$He is now gaseous and occupies all the $^3$He pot space, it is likely to do heat transfer with the 1K-pot whose temperature can be taken down to $\sim 1,5$ K. The 1K-pot is driving the $^3$He pot temperature descent, and when the sorption pump reaches 39 K, the activated carbon starts to pump $^3$He again. This cryopumping of $^3$He has the effect of decreasing the pressure in the $^3$He pot, and thus, the temperature. Below a given pressure and temperature, the $^3$He liquefies and falls down to the $^3$He tail. At the end of the condensation, the liquid $^3$He present in the $^3$He tail reaches a temperature of 0, 3 K. This condensation can last between 36 and 40 hours. All the different elements involved in the condensation can be seen in the schematic cut of figure 2.10 b. The time dependence of both the temperature of the 1K-pot and the temperature of the $^3$He tail obtained

<footnote n="4">⁴The fine approach procedure consists in alternating the elongation of the *z* piezoelectric tube (in order to fine a tunnelling current) and coarse piezoelectric displacements. Of course the coarse displacements must be smaller than the full *z* piezoelectric tube's extension to avoid any tip crashes. With decreasing temperature, the length of the displacements shortens and, *a fortiori*, the whole procedure duration increases.</footnote>

<footnote n="5">⁵At first, the IVC is filled with gaseous $^4$He which allows for the STM to thermalize with the 4, 2 K cryostat. Then, the IVC is pumped in order to isolate the STM from the cryostat and to go down to lower temperatures by mean of $^3$He condensation.</footnote>
during an actual condensation are represented in figure 2.10 a. This plot is representative of a
typical condensation procedure. During the first 25 minutes of the condensation, the 1K-pot
drives the temperature descent of the ³He tail. Then, when the sorption pump reaches 39 K
and that both 1K-pot and ³He tail are around 1,6 K, ³He is starting to condense until it reaches
a temperature of 300 mK. The peaks one can see on the 1K-pot curve between 50 minutes and
65 minutes are due to the fact that at the corresponding moments, the 1K-pot dried out, and
thus, had to be refilled.

<figure><img src="image_23.png" /><figcaption>Figure 2.10: **a.** Standard temperature versus time plot of a condensation. The blue curve represents the temperature of the 1K-pot whereas the red one represents the temperature of the ³He tail. **b.** Schematic cut of the cryogenics of M3 STM.</figcaption></figure>

**Remark:** The temperature of the 1K-pot can be changed by playing on the flow of liquid $^4$He going through the capillary. Indeed, it is easier to pump on a smaller quantity of liquid $^4$He, and thus possible to reach smaller temperatures by imposing a relatively low flow of liquid $^4$He. Moreover, it is important to refill the 1K-pot for a minute at the end of a condensation and then reduce the liquid $^4$He flow to its minimum value in order to make sure that the 1K-pot will not dry out during the 36 hours the condensation is supposed to last.

2.4.2 LT Omicron STM

Many of the measurements presented in this thesis were performed on a different set-up : namely the LT (for low temperature) Omicron STM. Thanks to our collaboration with Imad Arfaoui, working at MONARIS, a chemistry laboratory of our university, I could access this other equipment. This experimental set-up allows for ultra-high vacuum scanning tunnelling spectroscopy experiments and is composed out of the same parts as M3, namely a load lock, a preparation chamber and a STM chamber. During my thesis, the preparation chamber was not equipped yet, meaning that studies could be carried out only on crystals grown outside the chamber. This LT STM however possesses a cryostat, thus allowing to perform STM/STS measurements at room temperature, 77 K (liquid nitrogen temperature) and 4,2 K (liquid helium temperature). The system also possesses an outer volume which can be filled with liquid nitrogen and plays the role of sheath, thus allowing to increase the time separating two consecutive cryostat fillings of either liquid nitrogen or liquid helium. The autonomy of the
system is approximatively of 30 hours working at 77 K and 18 hours working at 4,2 K (or twice as less as the duration of the condensation in M3 STM). A Foucault current damping system allows to isolate efficiently the LT STM from vibrations. As it will be shown in the next subsection, this set-up has the advantage of permitting a relatively easy tip treatment procedure, absolutely crucial to perform quasiparticle interferences experiments.

<figure><img src="image_24.png" /><figcaption>Figure 2.11: Photograph of the LT Omicron STM setup.</figcaption></figure>

### 2.4.3 Preparation of the samples

#### Cleaving the samples.

In this subsection, we explain how both the samples and the tips we worked with were prepared. During this thesis, we observed crystals which where grown outside the set-up, either at the Institut des Matériaux Jean Rouxel in Nantes for the $(LaSe)_{1,14}(NbSe_2)_2$ crystals, or in Laboratoire de Physique des Solides, Orsay for the $(Sr_{1-x}La_x)IrO_4$ crystals. Most of the time, they did not need any additional *in situ* preparation other that cleaving to ensure that the observed surfaces were not contaminated in any way. In figure 2.12 **a** is shown a sample holder on which was glued a crystal. A cleaver is also glued on top of this crystal. Both sample and cleaver were glued using EPOTEK EE129-4, an epoxy which dries at ambient temperature to prevent heating up potentially fragile crystals.
<figure><img src="image_25.png" /><figcaption>Figure 2.12: **a.** Cleaver glued on top of a (LaSe)₁,₁₄(NbSe₂)₂ crystal. The cleaver can be handled with a wobble-stick and thus can be taken out of the set-up. The cleaving occurs *in situ* and ensure cleanliness of the surface of the sample. **b.** Typical sample holder organization for the experiments that were carried out on LT Omicron STM. **c.** When the measured *I*(*V*) spectra are linear over a big enough bias voltage range, it means that the tip is featureless and good for spectroscopy measurements. If the tip does not present such behavior, it is crashed into the Pt/Ir wire (see **b**). **d.** One checks that the tip presents nice atomic resolution on HOPG sample.</figcaption></figure>

The tips that were used in this thesis were made out of Pt80%Ir20% wire and simply cut
with scissors to make their termination sharp enough. Pt80%Ir20% are indeed known to be
suited to both topography and spectroscopy measurements. In figure 2.12 **b** is shown the typical
organization of a sample holder for experiments performed on LT Omicron STM. A Pt/Ir wire
and a graphite (HOPG) samples are glued next to the samples of interest. Glueing all those
elements together on the same sample holder makes the treatment of the tip much easier. The
Pt80%Ir20% tip is crashed onto the Pt/Ir wire and *I*(*V*) spectroscopy is performed. When the
spectra have the linear shape represented in figure 2.12 **c**, it means that both tip and wire are
metallic and we conclude that the tip is featureless, which is necessary to ensure that the *dI/dV*
spectra are proportional to the local density of states. In addition to being good in spectroscopy,
the tip must also be able to image surfaces with atomic resolution. In order to verify the last
point, one finally performs imaging on HOPG. This procedure was applied prior to all the
following measurements : the whole study of doped iridate compound (Sr(1-x)Lax)₂IrO₄, and
the quasiparticle interferences observed in the normal phase of (LaSe)₁,₁₄(NbSe₂)₂.

Pb deposition.

As is will be shown in 3.6, in a series of experiments, we deposited Pb on top of
(LaSe)₁,₁₄(NbSe₂)₂ in order to see the effect of non-magnetic impurities on superconductiv-
ity. In order to avoid pollution, the standard procedure consists in first degassing the Pb source
without any sample present in the preparation chamber (the power is fixed at 2/3 of the power used during evaporation). This preliminary step allows the potential oxidation of the sources to be eliminated and guarantees a cleaner final deposition. We deposited Pb *via* evaporation. The principle of evaporation is the following : a current flows into a filament in vicinity to a crucible filled with a clean Pb source. A voltage difference applied between the filament and the crucible accelerates the electrons emitted by the filament. Such electrons then collide the crucible and heat it sufficiently for Pb to evaporate (Pb starts to evaporate at 327, 5°C). Since the crucible is oriented towards the sample, Pb atoms are deposited on top of the (LaSe)₁,₁₄(NbSe₂)₂ surface. In the experiment presented at the end of subsection 3.6.5, 0, 47 monolayer of Pb was deposited on the superficial NbSe₂ layer<fnref n="6" /> of by applying a bias voltage of 800 V between the crucible and the filament and flowing a 1, 78 A current in the filament, resulting in a 0, 66 nA measured flux current⁷.

<footnote n="6">⁶Pb and NbSe₂ have approximatively the same lattice constants such that in one monolayer of NbSe₂ there is the same number of Nb atoms than in a Pb monolayer.

⁷This current corresponds to the number of ionized Pb emitted per unit of time. This value can be converted into a Pb depositing rate by performing a calibration with the quartz crystal microbalance.</footnote>
Chapter 3

Quasi-2D physics in a misfit
heterostructure

In this chapter, we present a STM/STS study of misfit transition metal dichalcogenide (TMD) compound (LaSe)₁,₁₄(NbSe₂)₂. This material is a parent compound of TMD 2H-NbSe₂ and shares many physical properties with it. (LaSe)₁,₁₄(NbSe₂)₂ indeed is an heterostructure made out of a stacking of trigonal prismatic NbSe₂ bilayers and rock-salt LaSe layers. Also, just as in bulk 2H-NbSe₂, the NbSe₂ layers are van der Waals bonded. Similarly to 2H-NbSe₂, at low enough temperature, (LaSe)₁,₁₄(NbSe₂)₂ enters a regime where both a lattice charge modulation and superconductivity coexist. Although being a bulk material, (LaSe)₁,₁₄(NbSe₂)₂ presents 2D features, such as a strong Ising spin-orbit coupling, just as in few layers NbSe₂.

In addition to all those shared properties with 2H-NbSe₂, (LaSe)₁,₁₄(NbSe₂)₂ is an incom-
mensurate material. The only symmetry operation which leaves the whole structure unchanged
is the identity, making (LaSe)₁,₁₄(NbSe₂)₂ even more delicate to address. Before presenting
our experimental results, it is necessary to understand the physics at play in these systems.
The first section of this chapter is thus dedicated to the physics of 2H-NbSe₂, which has been
known and intensively studied for decades, while (LaSe)₁,₁₄(NbSe₂)₂ was hardly studied at all
since the 1990's. In the second section, the crystal structure of (LaSe)₁,₁₄(NbSe₂)₂ will be
presented in great details and compared to STM topographic data. In the third part of this
chapter, we present STS measurements performed in the normal phase of (LaSe)₁,₁₄(NbSe₂)₂.
We discuss and compare our data to DFT calculations results. The fourth section focuses
on theoretical predictions of unconventional/topological superconductivity in parent monolayer
TMD compounds. Finally, we discuss experimental data obtained in the superconducting phase
of (LaSe)₁,₁₄(NbSe₂)₂.

## 3.1 Tuning the electronic properties of 2H – NbSe<sub>2</sub>

2H-NbSe₂ is a transition metal dichalcogenide (TMD) known for housing several interesting electronic features. As is will be shown in this section, it is possible to tune the electronic properties of 2H-NbSe₂ by changing a certain number of its intrinsic parameters such as : the strain, the degree of disorder, the number of NbSe₂ layers (or dimensionality) and the doping level. We will see that (LaSe)₁,₁₄(NbSe₂)₂ behaves in many aspects as 2H-NbSe₂ for which those parameters were modified. It is why this chapter is dedicated to the physics of 2H-NbSe₂ submitted to such perturbations.

The structure of 2H-NbSe₂ is represented in figure 3.1. It consists in an alternation of NbSe₂ bilayers with trigonal prismatic structure and separated by van der Waals gaps. The unit cell (a, b, c) is superimposed on the structure.
<figure><img src="image_26.png" /></figure>

Figure 3.1: Bulk 2H-NbSe₂ consists in a stacking of NbSe₂ bilayers (c = 12, 55 Å). Each NbSe₂ layer has a trigonal prismatic structure. The Nb atoms (blue spheres) sit at the nodes of a hexagonal lattice of lattice parameters a = b = 3, 44 Å. Each Nb atom lies at the center of a prism which six apexes are Se atoms (yellow spheres). Figure adapted from ref. [77].

3.1.1 Charge density wave order

<figure><img src="image_27.png" /></figure>

Figure 3.2: STM image of the 3 × 3 CDW in 2H-NbSe₂. The CDW reciprocal wave vectors can be seen in the Fourier transform of this image (taken from ref [78]).

The first noticeable phenomenon encountered in 2H-NbSe₂ is its charge density wave (CDW) order [78]. A CDW is a periodic reorganization of charge in a crystal. It appears below a critical temperature, *T*<sub>CDW</sub>. The period of this additional charge modulation is greater than the period of the lattice in the unperturbed phase (above *T*<sub>CDW</sub>). Historically, the existence of CDWs was predicted Peierls in atomic chains. In the Peierls model, thanks to the nesting properties of the Fermi surface of a half-filled 1D lattice with only first neighbour hopping, one indeed expects an electronic kinetic energy gain in atoms dimerization, and thus, the appearance of
a charge surmodulation with a new periodicity, twice as long as the one of the unperturbed lattice. However, it was shown that this simple picture cannot explain the presence of CDW orders in materials when the dimension of the system is 2 or 3, and most of CDWs can rather be understood in terms of strong coupling between the electrons and the lattice known as electron-phonon coupling [78, 79, 80].

Bulk 2H-NbSe₂ exhibits a 3 × 3 CDW below $T_{CDW} = 33.5$ K. A STM topography map of the surface of 2H-NbSe₂ displaying this 3 × 3 charge modulation is shown in figure 3.2. The modulation occurs in all three directions of the hexagonal lattice of NbSe₂. We thus speak of 3Q charge order. This can also be seen in reciprocal space (the Fourier transform of the topography map is shown in the inset of figure 3.2) where 6 spots are located at one third of the distance of the 6 Bragg peaks (actually the spots do not lie exactly at one third of the Bragg meaning that the charge modulation is incommensurate). If the modulation was only occurring along one direction, we would be speaking of 1Q charge modulation.

### 3.1.2 Superconducting order

The second remarkable electronic phenomenon observed in 2H-NbSe₂ is superconductivity below $T_{SC} = 7.2$ K [77]. The origin of superconductivity has already been discussed in chapter 1. In the case of a conventional superconductor such as 2H-NbSe₂, electron-phonon coupling is also responsible for the formation of Cooper pairs. However, multigap superconductivity was observed in 2H-NbSe₂ leading to deviations from a pure single-gap BCS description [77, 35, 81]. In figure 3.3, one can see two distinct spectroscopic signatures ($\Delta_1$ and $\Delta_2$) giving rise to an overall non-BCS-like $dI/dV$ spectrum.

<figure><img src="image_28.png" /><figcaption>Figure 3.3: $dI/dV$ spectrum taken on bulk 2H-NbSe₂ and displaying two-gap superconductivity (taken from ref [35]).</figcaption></figure>

Such spectroscopic features in the measured differential conductance spectra were first interpreted in terms of a continuous gap distribution [82]. In the STM/STS study of ref. [77], the authors rather speak of two-gap superconductivity, one gap developing around the K Nb pockets of the Fermi surface of 2H-NbSe₂ because of strong electron-phonon coupling and the other developing around the Γ Nb pockets by mean of quasiparticle inter-band scattering. To add even more complexity, I. Guillamon et al. exhibited in a series of STM/STS measurements atomic-scale modulations of the superconducting gap of 2H-NbSe₂, suggesting an anisotropic nature of the gap in reciprocal space [81].

Interestingly, similar tunnelling selectivity effects as the ones observed in $MgB_2$ (see sub-
section 2.2.2) occur in 2H-NbSe₂ [77]. In figure 3.4, one sees that depending on the tip-sample tunneling geometry, the orbital selectivity changes and favors either the observation of the small gap or of the large gap. It is important to keep this last point in mind since, in section 3.6, we will interpret some of our results obtained in parent compound (LaSe)₁,₁₄(NbSe₂)₂ in terms of tunnelling selectivity.

<figure><img src="image_29.png" /><figcaption>Figure 3.4: This figure illustrates the effect of tunnelling selectivity on the measured differential conductance spectra. a. In black is represented an experimental *dI/dV* spectrum taken on bulk 2H-NbSe₂ with tunnelling oriented along the c-axis of 2H-NbSe₂. b. In black is represented an experimental *dI/dV* spectrum taken on bulk 2H-NbSe₂ with tunnelling oriented along the a(b)-axis of 2H-NbSe₂. Depending on the tunnelling configuration, one clearly sees that one gap is observed when the other is not. In green (red) is the theoretical expectation for 100% tunnelling small (big) gap (taken from ref [77]).</figcaption></figure>

**Remark:** The coexistence of both phonon-mediated superconductivity and CDW in 2H-NbSe₂ raises the question of the interplay between the two orders. This problematic is at the center of many of the studies presented in this section.

The origin and nature of both CDW and superconductivity can be addressed by tuning
several NbSe₂ intrinsic parameters. In what follow, we will briefly expose those parameters and
the consequences their modifications can have on the electronic properties of NbSe₂.

**3.1.3 Effects of strain**

Lattice compression or dilation is likely to change the electronic properties of 2H-NbSe₂. In ref. [83], Shang Gao *et al*. pasted a crystal of bulk 2H-NbSe₂ on top of a silica plate. Due to the
different thermal expansion coefficients of 2H-NbSe₂ and silica respectively, cooling down the system from room temperature to 4 K generated strain in the attached 2H-NbSe₂ sample. STM measurements were then performed under those conditions of low temperature and substantial strain going up to 2,5% in some areas. The main result of this study is the observation of a strain-induced phase transition between 3Q CDW of periodicity 3a and a mix of 1Q CDW of periodicity 4a and 3Q CDW of periodicity 2a. Since strain acts on the lattice, it is not surprising that phonon-mediated electronic feature such as CDW is modified. Figure 3.5 illustrates the strain-induced phase transition between different types of CDWs in 2H-NbSe₂. Superconductivity is however unaddressed in this paper.

<figure><img src="image_30.png" /><figcaption>Figure 3.5: Fourier transform of topography maps taken in areas with different strain values. Depending on the strain, the structure either goes a 3Q3a, a 1Q4a or a 3Q2a CDW phase (from ref. [83]).</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Figure 3.6: Modification of normal to superconducting transition induced by controlled tensile and compressive strain. In blue is represented the superconducting critical temperature as a function of strain. The black dotted lines separate different CDW phases (from ref. [84]).</figcaption></figure>

In ref. [84], tensile as well as compressive strain (0% to 0,7%) was generated on bulk 2H-
NbSe₂ in a controlled way with the help of a pair of stacks of shear piezoelectric actuators (their measurements are summarized in figure 3.6). Transport measurements showed that up to a certain critical strain value (0,1% for compressive strain, 0,2% for tensile strain), the superconducting critical temperature is unmodified, suggesting peaceful cohabitation between CDW and superconductivity. Nevertheless, when one exceeds the critical strain value, superconductivity weakens with increasing strain. Theoretical calculations show that this sudden change coincides with a phase transition between a 3Q CDW phase at low strain to a 1Q CDW phase for higher strain. The CDW periodicity yet remains equal to 3a. Depending of the nature of the CDW, superconductivity and CDW might or not be in competition.

### 3.1.4 Effects of disorder

Disorder can also have an impact on the electronic properties of 2H-NbSe₂. In ref. [85], Cho, K. et al. artificially introduced disorder in bulk 2H-NbSe₂ samples through electron radiation. Their conclusion is somehow similar to the one of ref. [84] for no correlations between superconductivity and CDW order are observed under an estimated critical disorder (1C/cm² of residual resistivity), while above this critical value, both superconductivity and charge modulation weaken jointly (see figure 3.7).

<figure><img src="image_32.png" /><figcaption>Figure 3.7: Temperature vs residual resistivity (disorder) phase diagram of electron-radiated disordered 2H-NbSe₂ (from ref. [85]).</figcaption></figure>

### 3.1.5 Effects of doping

Doping is yet another parameter likely to modify the physical properties of 2H-NbSe₂. In ref. [86], electron-donating atoms of Cu were intercalated in between neighbouring NbSe₂ layers. In this work, Huixia Luo et al. observed a decrease of superconducting critical temperature
of CuₓNbSe₂ with increasing level of Cu doping x through transport measurements (see figure 3.8 a). Electron diffraction experiments also clearly support a joint weakening of CDW order because of reduced sharpness of 3 × 3 peaks meaning reduced in-plane coherence length for the charge modulations.

<figure><img src="image_33.png" /><figcaption>Figure 3.8: **a.** Critical superconducting temperature vs doping level in CuₓNbSe₂ (from ref. [86]). T_C goes down with increasing electron doping. **b.** Critical superconducting temperature vs carrier density (doping level) in bilayer 2H-NbSe₂ (from ref. [87]). T_C₀ goes up with increasing electron doping.</figcaption></figure>

One can also achieve doping by ionic liquid gating. This was done in ref. [87], where bilayer NbSe₂ was electron and hole doped by electrostatic gating. In this study, they have been able to reversibly tune the carrier density of 6.10¹⁴ electrons per cm² namely 30% of the intrinsic density of bilayer 2H-NbSe₂, achieving to shift the Fermi level of approximatively +0.07 eV. Once again, the conclusion seems to call for a joint enhancement of superconductivity and CDW since by tuning the Fermi level position from hole-doped to electron-doped regime, superconducting critical temperatures go from 2.8 K to 5.1 K, and the characteristic drop around T_CDW in Hall coefficient vs temperature curves vanishes (see figure 3.8 b).

### 3.1.6 Reducing the number of NbSe₂ layers

Doping is not the only parameter that was changed in ref. [87]. The studied sample was indeed bilayer NbSe₂ and not bulk 2H-NbSe₂. Modifying the dimensionality of the sample, i.e. the number of NbSe₂ layers is also another fruitful approach. For example, by varying NbSe₂ samples thicknesses from bulk to single layer limit, a competition between electron-phonon-driven CDW and superconducting orders was exhibited in ref. [88]. Reducing the number of layers induced a decrease of T<sub>SC</sub> and an increase of T<sub>CDW</sub> up to 145 K in monolayer NbSe₂, or so to say way, higher than the 33.5 K of the bulk (see figure 3.9).

Moreover, approaching the 2D limit also implies breaking inversion symmetry and consequently the appearance of a strong Ising spin-orbit coupling. This exotic spin-orbit coupling gives rise to spin-momentum locking in the out of plane direction and also to unconventional Ising superconducting pairing [33, 34, 35] but this problematic was already mentioned in section 1.3.3 and will be explored in more details in section 3.4.
<figure><img src="image_34.png" /><figcaption>Figure 3.9: Competition between superconducting and CDW orders exhibited across 2H-NbSe₂ samples with thickness going from single layer to bulk (from ref. [88]).</figcaption></figure>

One can wonder whether the multigap physics present in bulk NbSe₂ (which was mentioned previously in this chapter) remains with decreasing number of layer. In their article, T. Dvir et al. measured $dI/dV$ spectra of few-layer superconducting NbSe₂ samples using van der Waals tunnel junctions [35]. They observe multigap superconductivity down to their thinner samples (namely 3 layers NbSe₂). Moreover, they observe a robustness of the large superconducting gap to strong in-plane magnetic fields, which is in favor of Ising spin-orbit coupling physics (see sections 1.3.3 and 3.4), while the small gap is killed. This last example illustrates how superconducting order parameters of different natures can coexist in few layer TMDs.

Interestingly, and as it will be shown in the following sections, $(LaSe)_{1,14}(NbSe_2)_2$ can in first approximation be seen as reduced-dimensionality electron-doped 2H-NbSe₂ with disorder and strain, possibly making it an ideal platform for studying the electronic properties of quasi-2D transition metal dichalcogenides.

## 3.2 Structure of misfit compound $(LaSe)_{1,14}(NbSe_2)_2$

$(LaSe)_{1,14}(NbSe_2)_2$ is a misfit layered compound consisting in alternation of TMD 2H-NbSe₂ bilayers with trigonal prismatic structure and of rock-salt LaSe bilayers. This material was almost not studied at all since the 90's and its structure is known from X-rays experiments carried out in 1993 [89]. Unlike in bulk 2H-NbSe₂ the lattice of NbSe₂ layers is not exactly hexagonal but rather rectangular centered as it is slightly compressed along the **a**₂ direction. The angle represented in figure 3.10 *d* is $\theta = 59,6^\circ$. The lattice vectors of the primitive cell of NbSe₂ layers are **a**₁ ($\| \mathbf{a} \| = a_1 = 3,457$ Å) and **a**₂ ($\| \mathbf{a} \| = a_2 = 3,437$ Å), respectively. The LaSe bilayers possess a square lattice of lattice vectors **b**₁ and **b**₂ ($\| \mathbf{b} \| = \| \mathbf{b} \|$). In what follows, it will be more convenient to adopt a centered cell of lattice vectors **b**₁ ($\| \mathbf{b} \| = b_1 = 6$ Å) and **a**₂ for the NbSe₂ layers. Given this last remark, since both NbSe₂ and LaSe layers share the same **b**₁ lattice vector, the material is commensurate in the **b**₁ direction. Nevertheless, the ratio between the norms of **a**₂ and **b**₂ vectors, which share a common direction, is an irrational number ($\| \mathbf{b} \| / \| \mathbf{a} \| = 1,751...$) making $(LaSe)_{1,14}(NbSe_2)_2$ incommensurate in the **a**₂ direction. However, $\| \mathbf{b} \| / \| \mathbf{a} \| \approx 7/4$ such that $(LaSe)_{1,14}(NbSe_2)_2$ is « almost » periodic in the **a**₂ direction with an approximate commensurate lattice vector **m** ($\| \mathbf{m} \| = 7 \| \mathbf{a} \| \approx 4 \| \mathbf{b} \|$). The lattice parameter in the out-plane-direction is **c** ($\| \mathbf{c} \| = c = 36,5$ Å). The crystal
structure seen from different angles in represented in figure 3.10.

<figure><img src="image_35.png" /><figcaption>Figure 3.10: Crystal structure of (LaSe)1,14(NbSe2)2 seen from different perspectives. a. Side view. b. Side view orthogonal to a. c. Top view. The Nb atoms are represented by blue balls, the Se atoms are represented by yellow balls and the La atoms are represented by red balls. The black rectangle represents the approximate unit cell considered in DFT calculations since the material is not exactly commensurate. d. Cartoon illustrating the compression along the misfit axis for the NbSe2 lattice.</figcaption></figure>

One can write all the lattice vectors on (i, j) orthonormal basis :

$$
\left\{
\begin{aligned}
\mathbf{a}_1 &= a_1(\cos(\theta/2)\mathbf{i} + \sin(\theta/2)\mathbf{j}), \\
\mathbf{a}_2 &= a_2\mathbf{j}, \\
\mathbf{b}_1 &= b_1\mathbf{i}, \\
\mathbf{b}_2 &= b_2\mathbf{j}, \\
\mathbf{m} &= 7\mathbf{a}_2 = 4\mathbf{b}_2,
\end{aligned}
\right.
\qquad (3.1)
$$

Such that in reciprocal space after a short calculation :

$$
\left\{
\begin{aligned}
\mathbf{a}_1^* &= \frac{2\pi}{a_1 \cos(\theta/2)} \mathbf{i}, \\
\mathbf{a}_2^* &= \frac{2\pi}{a_2 \cos(\theta/2)} (-\sin(\theta/2)\mathbf{i} + \cos(\theta/2)\mathbf{j}), \\
\mathbf{b}_1^* &= \frac{2\pi}{b_1} \mathbf{i}, \\
\mathbf{b}_2^* &= \frac{2\pi}{b_2} \mathbf{j}, \\
\mathbf{m}^* &= \frac{2\pi}{7a_2} \mathbf{j} = \frac{2\pi}{4b_2} \mathbf{j},
\end{aligned}
\right.
\qquad (3.2)
$$

To fully explain the spectroscopic results that will be presented in section 3.4, theoreticians
from our group (Matteo Calandra, Marco Campetella and Cesare Tresca) carried out first
principles Density Functional Theory (DFT) calculations using Quantum expresso ab initio
software. In order to perform such calculations, one has to consider a periodic system. Since
(LaSe)1,14(NbSe2) is incommensurate, it is by definition not periodic along the misfit direction
**m.** Nevertheless, the approximate commensurate lattice vector **m** is good enough to consider the system periodic and of unit cell (**m**, **b**₁, **c**). This unit cell is represented by a black rectangle on figure 3.10. Here, we will only present the DFT calculations results regarding the crystal structure of (LaSe)₁,₁₄(NbSe₂)₂. The obtained band structure will be commented and compared to experimental data in section 3.4. In this framework, it is possible to numerically obtain the full structure of (LaSe)₁,₁₄(NbSe₂)₂ and in particular to exhibit the nature of the bonding between different layers. After relaxation, significant deviations from perfect rock-salt structure appear within the LaSe layers (see figure 3.11).

<figure><img src="image_36.png" /><figcaption>Figure 3.11: Focus on one LaSe layer (top view after removal of Nb atoms). Deviations from perfect rock-salt structure can be seen. Some La atoms tend to gather (blue grids) while others tend to move part from each other (green grid).</figcaption></figure>

DFT calculations show that a significant but incomplete charge transfer occurs between LaSe and NbSe₂ layers, effectively doping the NbSe₂ layers, shifting the Fermi level to higher energies. Each Nb atom indeed receives 1,14/2 electrons from the closer LaSe layer. The La atoms thus make a iono-covalent bonding with the closest Se atoms of the neighbouring NbSe₂ layer. Finally, similarly to bulk 2H-NbSe₂, the adjacent NbSe₂ layers are separated by a van der Waals gap. This later bond being the weakest in the whole structure, the samples cleave on a monolayer NbSe₂ terminated surface.

### 3.2.1 Incommensurability probed by STM

Now that we have a good idea of the structure of (LaSe)₁,₁₄(NbSe₂)₂ thanks to X-Rays data, it is important to check that we indeed recover those structural features on STM topographic map. All the observed crystals were grown by Laurent Cario and Shunsuke Sasaki using iodine vapor transport technique in Institut des Matériaux Jean Rouxel (IMN) in Nantes. Their compositions were determined via EDX measurements and are in good agreement with expected stoichiometric ratios. The samples were cleaved at room temperature in ultra-high vacuum under a pressure of approximatively 8.10⁻¹¹mbar or low 10⁻¹¹mbar (depending on if the experiment was done on the LT Omicron STM or on M3 homemade microscope) directly before being loaded into the STM head.
Before actually looking at topographic data let us first present what can be expected beforehand. In figure 3.12 **a.** is represented a simulated hexagonal lattice reflecting the hexagonal lattice of Se at the surface. The modulus square of its Fourier transform is displayed right beneath in figure 3.12 **d.** Without any surprise the reciprocal lattice is also an hexagonal lattice but turned of 30°. In figure 3.12 **b.** is shown a simulated potential of periodicity $|| \mathbf{m} || = 7 || \mathbf{a}_2 ||$ oriented in $\mathbf{a}_2$ direction meant to represent the incommensurate potential felt by the superficial NbSe₂ layer because of its interaction with the underlying LaSe layer. The corresponding modulus square Fourier transform can be seen in figure 3.12 **e.** For realism, the periodic potential was considered to possess several harmonics (namely second, third and sixth harmonics). The visible peaks in reciprocal space are aligned in the misfit direction and are located at $\mathbf{m}^*, 2\mathbf{m}^*, 3\mathbf{m}^*$ and $6\mathbf{m}^*$ wave vectors. Figure 3.12 **c.** represents the hexagonal lattice of **a** multiplied by the "incommensurate potential" of **b**. The corresponding Fourier transform in **f** presents both collections of peaks of **d.** and **e.** but also other peaks developing along the misfit direction starting from the Bragg peaks of hexagonal lattice. Those peaks are replica (or satellites) which lie at linear combinations of q-vectors typical of both signals. This typically appears once two periodic signals are modulated by one another. We will see that this simple cartoon actually describes rather faithfully the observations.

<figure><img src="image_37.png" /><figcaption>Figure 3.12: Simulated triangular lattice (**a.**), incommensurate potential with additional second, third and sixth harmonics (**b.**), and triangular lattice modulated by incommensurate potential (**c.**). **d.**, **e.** and **f.** Modulus square of the Fourier transforms of **a.**, **b.** and **c.** respectively. Relevant lattice vectors and reciprocal lattice vectors are represented on all images.</figcaption></figure>

A typical STM topography map of (LaSe)₁,₁₄(NbSe₂)₂ surface is represented in figure 3.13 **a.** For simplicity, in the following, the rectangular centered lattice of NbSe₂ will be referred to as hexagonal. Since the surface is NbSe₂ terminated<fnref n="1" />, the probed atoms are Se. The Se hexagonal lattice is clearly visible but one can also notice the presence of multiple additional

<footnote n="1">¹It was already said that the samples cleave in between two adjacent van der Waals bonded NbSe₂ layers.</footnote>
spatial modulations across the surface which are due to the interaction with the underlying LaSe layer. In order to properly characterize the origin of those modulations, the modulus of the Fourier transform of a similar but larger scale topographic image taken on the same area with same scanning angle is represented in figure 3.13 b. The Bragg peaks of the reciprocal lattice of NbSe₂ are highlighted by full dark-blue circles. The sharp peak circled in burgundy corresponds to the reciprocal approximate commensurate vector **m***. It also gives the misfit direction represented by a burgundy arrow. The second, third and sixth harmonics of the approximate periodic misfit potential are highlighted by full purple, dark-pink and light-pink circles respectively. Other harmonics (fourth, fifth, seventh...) are never seen thus explaining the chosen signal construction in figure 3.12. The satellites replica starting from NbSe₂ Bragg peaks of both third and sixth harmonics are also observed and accented by dotted dark-pink and light-pink circles respectively. Our interpretation is that the structure factors of the material are such that the replica of **m**\* and 2**m**\* can't be seen.

<figure><img src="image_38.png" /><figcaption>Figure 3.13: **a.** Topographic STM image of (LaSe)₁,₁₄(NbSe₂)₂ surface. **b.** Modulus square of the Fourier transform of a larger scale topographic image. The full blue circles highlight the Bragg peaks of the hexagonal lattice of NbSe₂. The full burgundy circle (and purple, dark pink and light pink full circles) highlight the reciprocal lattice vector (and its second, third and sixth harmonics, respectively). The full green circles highlight the 2 × 2 charge modulation. The orange half circle highlight higher order approximate incommensurate vectors. The dotted circles are replica of the peaks that were just mentioned.</figcaption></figure>

Moreover, large and intense spots encircled in orange very close to Γ and aligned in the misfit axis suggest long-range modulations in the incommensurate direction. Such long-range modulations can be seen (highlighted by yellow oscillations) on figure 3.14 which represents the autocorrelation map of the same STM map that gave the Fourier transform of figure 3.13. These modulations render higher order approximate commensurate vectors. 7/4 is a fairly good incommensurate ratio but other ratios such as 30/17, 33/19, 37/21, 40/23 or 44/25 have a less than one percent difference with the proper ratio|| **b**₂ || / || **a**₂ || = 1,751. The distance separating two adjacent long-range maxima is indeed of the order of 30a₂. The approximate commensurate lattice parameter *m* is also visible on the autocorrelation map since dark lines (stressed by burgundy dotted lines) separated of 7a₂ alternate the misfit direction.
<figure><img src="image_39.png" /><figcaption>Figure 3.14: Autocorrelation map of a topographic STM image displaying long-range modula-
tions interpreted as higher-order approximate commensurate ratios.</figcaption></figure>

The signals presented so far can directly be deducted from the theoretical crystal structure
supporting than we indeed are observing the right material. In addition to those, one can also
observe other contributions. Let us now reconsider the Fourier transform image of figure 3.13 b.
First, diffuse peaks (highlighted by green circles) lie at half the Bragg of the NbSe₂ hexagonal
lattice. They express the presence a 2 × 2 charge modulation in the system. This charge order
will be fully discussed further. Second, the halo surrounding Γ and quasi-reaching half the
Bragg peaks can be interpreted as integrated QPIs spectroscopic signal but this will be the
topic of the following section.

3.3 $(LaSe)_{1,14}(NbSe_2)_2$ as electron-doped single-layer $NbSe_2$

In this section, we will show by mean of quasiparticle interferences measurements compared to
theoretical DFT calculations that $(LaSe)_{1,14}(NbSe_2)_2$ behaves as an electron-doped monolayer
$NbSe_2$. We will also investigate the origin of the 2 × 2 charge modulation.

**3.3.1 Scattering potentials in quasiparticle interferences**

Quasi particle interferences are observed whenever there are scattering centers (see section 2.3).
For example, in ref. [90], very clean samples of bulk 2H-NbSe₂ were studied via STM and
such signal was never observed. On the contrary, in ref. [91], C. J. Arguello and collaborators
intentionally introduced S impurities (S is isovalent to Se, thus no additional charge carriers
are brought to the system), and could consequently performs very nice QPIs measurements.
The STM image of figure 3.13 a does not present any visible defects such as vacancies, steps
or dislocations. Such clean surfaces are representative of all the samples that were measured.
Yet, as it was already mentioned, QPIs spectroscopic signal were even observed in topographic
mode. QPIs are thus due to another kind of scattering potential. Everything suggests that
incommensurability plays the role of disorder potential. Disorder and incommensurability are
indeed compatible as shown by the theoretical study of ref. [92] where incommensurability
leads to localization but also in ref. [93] where specific heat measurements done in many
incommensurate compounds revealed disordered glass-like behaviour.

<figure><img src="image_40.png" /><figcaption>Figure 3.15: Modulus square of the Fourier transform of a topographic STM image performed at -20meV. In first approximation, it can be interpreted as a $dI/dV$ map at Fermi energy.</figcaption></figure>

Figure 3.15 represents the Fourier transform modulus of a topography map (where the misfit axis is vertical) taken at -20mV bias voltage. In figure 3.15, only electrons between 0meV and -20meV contributed to the total tunnelling current, or in other words, electrons reasonably close to Fermi level to consider it to be the Fourier transform modulus of a LDOS map at Fermi level. This will be supported later on since the band dispersion over [0meV,-20meV] is negligible. Given this last argument, the fact that the QPIs signal mostly develops along the misfit direction further supports the incommensurable disorder potential hypothesis. Apart from this observation, it is also possible to see many fine QPIs structures, but before interpreting them, we will first present the spectroscopic DFT results.

**Remark :** In the case of 3.13 **b**, electrons of energies varying from 0 meV to -200 meV contributed to the total tunnelling current, meaning that the spectroscopic halo around Γ is actually a mix between the QPIs patterns for all energies ranging from 0 meV to -200 meV.

3.3.2 Energy dispersion from DFT calculations

In order to numerically access the band structure of (LaSe)₁,₁₄(NbSe₂)₂, the DFT calculations were performed on the last slab of (LaSe)₁,₁₄(NbSe₂)₂ as shown in figure 3.16 **d**. The reason behind this is that such a slab is van der Walls interacting with similar slabs and is consequently almost electronically independent<fnref n="2" /> The spectroscopic data are hence very likely to match faithfully the *ab initio* results. Moreover, given the great number of atoms present in the (**m**, **b**₁, **c**)

<footnote n="2">²The theoreticians verified beforehand that the band structure of the bulk was very similar to the one of the slab.</footnote>
unit cell, dividing it by two can significantly reduce the full calculation time, but, the calculation still represents a quite heavy operation.

The band structure of $(LaSe)_{1,14}(NbSe_2)_2$ is represented in the approximate commensurate Brillouin zone on figure 3.16 **a** in which the red dots represent the La character of the bands<fnref n="3" />. The approximate commensurate Brillouin zone corresponding to the $(m, b_1, c)$ unit cell is represented and compared to the hexagonal Brillouin zone of $NbSe_2$ in figure 3.16 **c**. As it was said previously, a significant charge transfer occurs in $(LaSe)_{1,14}(NbSe_2)_2$, LaSe layers giving electrons to their neighbouring $NbSe_2$ layers. We will see in what follows that the band structure of $(LaSe)_{1,14}(NbSe_2)_2$ is actually similar to the one of monolayer $NbSe_2$ but rigidly shifted towards lower energies. The existing iono-covalent bonding between LaSe and $NbSe_2$ layers tends to decouple even more each $NbSe_2$ layers within a 2H-$NbSe_2$ bilayer. This suggests that the electronic structure of the material could actually be closer to the one electron-doped monolayer $NbSe_2$ than to the one electron-doped bilayer 2H-$NbSe_2$.

<figure><img src="image_41.png" /><figcaption>Figure 3.16: **a.** Band structure of $(LaSe)_{1,14}(NbSe_2)_2$. The bands with a pronounced La character are highlighted with red dots. **b.** Comparison between the band structures of $(LaSe)_{1,14}(NbSe_2)_2$ and of 0, 3 eV e-doped monolayer $NbSe_2$. **c.** Superimposition of the Brillouin zones of $(LaSe)_{1,14}(NbSe_2)_2$ (red) and of $NbSe_2$ (black). **d.** Unit cells on which were performed the DFT calculations presented in **b**.</figcaption></figure>

<footnote n="3">³It is possible to develop the band structure of $(LaSe)_{1,14}(NbSe_2)_2$ in the hexagonal Brillouin zone of $NbSe_2$ by a process called unfolding. The obtained unfolded band structure is represented in 3.17 **a** and resembles the one of $NbSe_2$ but this will be presented in greater details shortly.</footnote>
DFT calculations were also carried out for monolayer NbSe₂. The monolayer was compressed along one direction to match perfectly the compression seen in NbSe₂ layers of (LaSe)₁,₁₄(NbSe₂). The band structure of this constrained NbSe₂ monolayer shifted of 0, 26 eV towards negative energies was superimposed on the band structure of the (LaSe)₁,₁₄(NbSe₂)₂ slab. One can first observe a very good agreement between the two band structures meaning that the rigid energy shift scenario is actually quite convincing. Nevertheless, some bands marked with red spots in figure 3.16 a cannot be seen in the case of monolayer NbSe₂. As is was already said, DFT calculations shows that those are La bands. Interestingly, one of these bands barely crosses Fermi level meaning that the LaSe are not completely insulating. This is a consequence of the iono-covalent bonding between LaSe and NbSe₂ layers, the charge transfer being incomplete. A partial conclusion to this *ab initio* study is that (LaSe)₁,₁₄(NbSe₂)₂ behaves as monolayer NbSe₂ with a chemical potential shifted of ≈ 0,26 eV apart from a small La contribution. This theoretical result now has to be confronted to STS experimental data.

### 3.3.3 Comparison between the DFT calculations and the quasiparticle interferences

In figure 3.10, one can notice that the QPIs signal seems to reflect the periodicity of the hexagonal Brillouin zone of NbSe₂ rather than the one of the reduced Brillouin zone of (LaSe)₁,₁₄(NbSe₂)₂. This might be due to matrix elements involved in scattering processes. This also means that in order to theoretically reproduce the QPIs data, the band structure of (LaSe)₁,₁₄(NbSe₂)₂ should rather be expanded in NbSe₂ Brillouin zone.

<figure><img src="image_42.png" /><figcaption>Figure 3.17: **a.** Comparison between the band structure of (LaSe)₁,₁₄(NbSe₂)₂ and the one of 0, 3 eV e-doped NbSe₂ in the unfolded Brillouin zone. The dot color scale follows the spectral weight of the band. The darkest the higher the weight. **b.** Comparison between the expected La, Nb and total density of states and the one obtained experimentally.</figcaption></figure>

This expansion in NbSe₂ Brillouin zone was done through a procedure called unfolding. The unfolded band structure of (LaSe)₁,₁₄(NbSe₂)₂ is represented on figure 3.17 **a**. The band structure of monolayer NbSe₂ shifted of 0,3 eV is superimposed. From now, we will refer to it