This thesis is composed of three projects including the AC magnetic susceptibility study of high-temperature superconductor YBa$_2$Cu$_3$O$_{7-\delta}$, the ionic-liquid gating study of the Mott insulator Sr$_2$IrO$_4$, and the single-electron study of quantum dot device with self-assembled nanocrystal PbS. Chapter 1 covers a general introduction to all three projects. The basic background and the motivation for each project are presented. Project I is covered in Chapter 2, Chapter 3, and Chapter 4. The first part of Chapter 2 is a theoretical introduction to the Bardeen-Cooper-Schrieffer theory of superconductivity with its main conclusions presented. This chapter builds a basis for the use of high pressure technique to YBa$_2$Cu$_3$O$_{7-\delta}$ in the later chapters. The rest of Chapter 2 reviews the work in the study of high-temperature superconductors, especially on YBa$_2$Cu$_3$O$_{7-\delta}$, on both experiments and theories and the possible applications of high-temperature superconductors. Chapter 3 introduces the YBa$_2$Cu$_3$O$_{7-\delta}$ sample preparation process and the characterisation. A dry cryomagnetic equipment was employed for the measurement. The results and the discussion are presented in Chapter 4. Project II is described in Chapter 5, Chapter 6, and Chapter 7. Chapter 5 firstly introduces the background knowledge of the gated material SrTiO$_3$ and the technical details of the ionic-liquid gating technique. Then the sample growth and the characterisation are presented. The fabrication process of Sr$_2$IrO$_4$ and SrTiO$_3$ (material for a control experiment) are described in Chapter 6. Chapter 7 covers the measurement and the result of the fabricated devices and related discussion. Project III ranges from Chapter 8, and Chapter 9. A literature review of quantum-dot devices and self-assembled nanocrystals is presented in Chapter 8. The experimental design of this nanocrystal quantum dot device is also included. Following it, the fabrication process of quantum-dot devices and the techniques used for fabrication are introduced in the start of Chapter 9. Chapter 9 also gives a description of the probe-station for measurements. The results and discussion of the measurements are covered in the last section of Chapter 9. Chapter 10 summarises and concludes the three projects stated above and gives some suggestions about the directions for future work.

Fermi surface, the locus in momentum space of gapless low-energy excitations, is a concept of fundamental importance in solid state physics. Electronic properties of a material are determined by the long-lived low-energy excitations near the Fermi surface. Conventionally, Fermi surface is understood as a property exclusive to a metallic state, contoured by electronic bands crossed by the Fermi level, although there has been a continuing effort in searching for Fermi surface outside the conventional description. In this thesis, techniques developed to prepare high-quality single crystals of SmB$_6$ and YBa$_2$Cu$_3$O$_{6+x}$ (abbreviated as YBCO$_{6+x}$ hereinafter) are described. By utilising measurement techniques of exceptional sensitivity and exploring a wide range of temperatures, magnetic fields, and electrical currents, we found signatures of unconventional Fermi surfaces beyond the traditional description in these strongly correlated electronic systems. SmB$_6$ is a classic example of Kondo insulators whose insulating behaviour arises due to strong correlation between the itinerant $d$-electrons and localised $f$-electrons. The peculiar resistivity plateau onsets below 4 K has been a decades-long puzzle whose origin has been recently proposed as the manifestation of topological conducting surface states. We found that the insulating behaviour in electrical transport is robust against magnetic fields up to 45 T, while prominent quantum oscillations in magnetisation are observed above 10 T. Angular dependence of the quantum oscillations revealed a three-dimensional characteristics with an absolute amplitude consistent with a bulk origin, and temperature dependence showed a surprising departure from the conventional Lifshitz-Kosevich formalism. Complementary thermodynamic measurements showed results consistent with a Fermi surface originating from neutral itinerant low-energy excitations at low temperatures. Theoretical proposals of the unconventional ground state uncovered by our measurements in SmB$_6$ are discussed. YBCO$_{6+x}$ is a high-temperature superconductor with a maximum $T_{\rm c}$ of 93.5 K and the cleanest member in the family of copper-oxide, or {\it cuprate}, superconductors. The correct description of electronic ground state in the enigmatic pseudogap regime, where the antinodal density of states are suppressed below a characteristic temperature $T^*$ above $T_{\rm c}$, has been a subject of active debates. While the quantum oscillations observed in underdoped YBCO$_{6+x}$ have been predominately interpreted as a property of the normal state where the superconducting parameter is completely suppressed at $\approx$ 23 T, we made the discovery that YBCO$_{6.55}$ exhibits zero resistivity up to 45 T when a low electrical current is used, consistent with the observation of a hysteresis loop in magnetisation. Quantum oscillations in the underdoped YBCO$_{6+x}$ are thus seen to coexist with $d$-wave superconductivity. Characteristics of the quantum oscillations are consistent with an isolated Fermi pocket reconstructed by a charge density wave order parameter and unaccompanied by significant background density of states, suggesting the antinodal density of states is completely gapped out by a strong order parameter involving pairing correlations, potentially in addition to the other order parameters. Transport measurements performed over a wide doping range show signatures consistent with pairing correlations that persist up to the pseudogap temperature $T^*$. The surprising observation of quantum oscillations in insulating SmB$_6$ and superconducting YBCO$_{6+x}$ demonstrates a possible new paradigm of a Fermi surface without a conventional Fermi liquid. A new theoretical framework outside the realm of Fermi liquid theory may be needed to discuss the physics in these strongly correlated materials with enticing electronic properties.

The objective of this thesis is to reach theoretical understanding of the unusual relationship between charge-ordering and superconductivity in correlated-electron systems. The competition between these broken symmetries and magnetism in the cuprate high temperature superconductors has been extensively discussed, but exists also in many other correlated-electron superconductors, including quasi-two-dimensional organic charge-transfer solids. It has been suggested that the same attractive interaction is responsible for both charge-order and superconductivity. We propose that the specific interaction is the tendency in correlated-electron systems to form spin-singlet bonds, which is strongly enhanced at the commensurate carrier density p of ½ a charge carrier per site, characteristic of all superconducting charge-transfer solids. To probe superconductivity driven by electron correlations, a necessary condition is that electron-electron interactions enhance superconducting pair-pair correlations, relative to the non-interacting limit. We have performed state of the art numerical calculations on the two-dimensional Hubbard model on different triangular lattices, as well as other lattices corresponding to K-BEDT-TTF based organic charge transfer solids, for the complete range of carrier densities per site p (0 ≤ p ≤ 1). We have shown that pair-pair correlation for each cluster is enhanced by electron-electron interaction only for p ≃ 0.5, far away from the density range thought to be important for superconductivity. Although initial focus is on charge-transfer solids, the results of the research will impact the field of correlated electrons as a whole. We believe our calculations will provide fundamental and fresh insight to the theory of superconductivity in strongly correlated systems.

La découverte en 2008 des supraconducteurs à base de fer a ouvert un nouveau champ d'investigation de la supraconductivité à haute température critique. En particulier, la phase nématique de ces matériaux pourrait jouer un rôle prépondérant dans le mécanisme de la supraconductivité. Nous avons étudié le composé FeSe par spectroscopie Raman, à pression ambiante et sous pression hydrostatique. Celui-ci ne possède pas d'ordre magnétique statique à pression ambiante, ce qui en fait un composé de choix pour l'étude de l'ordre nématique. Nous avons observé les fluctuations nématiques de charge. Leur évolution dans la phase tétragonale prouve l'existence d'une instabilité nématique d'origine électronique, qui gouverne la transition structurale. Dans la phase orthorhombique, le comportement des phonons souligne le rôle du couplage spin-phonon dans la transition nématique. Par ailleurs, la forme de la réponse Raman supraconductrice est compatible avec l'existence de deux gaps de symétrie s, dont un est anisotrope. Sous pression hydrostatique, les fluctuations nématiques s'atténuent rapidement. Le point critique quantique électronique associé se situe à très basse pression, peu avant l'apparition de l'ordre magnétique. Les fluctuations nématiques disparaissent complètement vers 2 GPa, quand la transition structurale passe de second ordre à premier ordre. C'est également proche de cette pression que se produit une anomalie dans le comportement des phonons, qui indique une modification de la structure électronique du système. Nos mesures révèlent en outre l'existence d'un pseudogap. Sa température d'apparition chute simultanément à la disparition de la phase magnétique, quand la température critique de supraconductivité atteint son maximum. Enfin, la réponse Raman de l'état supraconducteur à 7.8 GPa montre une signature claire d'un gap plein. / The discovery in 2008 of the iron-based superconductors opened a new field of investigation of high-temperature superconductivity. In particular, the nematic phase of these materials may play a major role in the mecanism of superconductivity. We studied the FeSe compound using Raman spectroscopy, at ambient pressure and under hydrostatic pressure. This material does not display any static magnetic order at ambient pressure and is therefore an excellent choice to study the nematic order. We observed the charge nematic fluctuations. Their evolution in the tetragonal phase proves the existence of an electronic nematic instability, which drives the structural transition. In the orthorhombic phase, the behaviour of the phonons underlines the role of the spin-phonon coupling in the nematic transition. Besides, the shape of the superconducting Raman response is compatible with the existence of two s-wave gaps, one of which is anisotropic. Under hydrostatic pressure, the nematic fluctuations reduce rapidly. The associated electronic quantum critical point is situated at very low pressure, just before the appearance of magnetic order. The nematic fluctuations completely disappear around 2 GPa, when the structural transition changes from second order to first order. An anomaly of the phonons also occurs close to this pressure, which indicates a modification of the electronic structure of the system. Our measurements additionally reveal the existence of a pseudogap. Its temperature of appearance reduces significantly simultaneously to the disappearance of magnetic order, when the critical temperature of superconductivity reaches its maximum. Finally, the Raman response in the superconducting state at 7.8 GPa shows a clear signature of a full gap.

Dans cette thèse, nous construisons un modèle de liaisons fortes tridimensionnel minimal pour les cuprates supraconducteurs monocouches à base de lanthane. Celui-ci prend en compte huit orbitales, dont deux d'entre elles impliquent les ions oxygène apicaux. L'optimisation des paramètres microscopiques permet de reproduire presque parfaitement la bande de conduction tridimensionnelle telle qu'elle a été obtenue à partir des calculs DFT. Nous discutons la façon dont chacun des paramètres entrant en jeu dans ce modèle multi-bandes influence la bande de conduction, et nous montrons que la forme particulière de sa dispersion contraint les valeurs des paramètres. Nous mettons alors en évidence que la détermination standard d'un modèle effectif à une bande au travers d'un traitement perturbatif converge lentement en raison de la valeur relativement faible du gap de transfert de charges. A ce stade, cela nous permet, en revanche, de lever le voile sur l'origine microscopique des amplitudes de saut des électrons au sein des plans et en-dehors des plans. Une approche alternative au calcul des paramètres microscopiques de saut du modèle effectif de liaisons fortes est présentée et mise à contribution. Il en résulte que l'accord avec la DFT est préservé à condition que les amplitudes de saut de plus longue portée soient conservées. Une comparaison avec les modèles existants est également effectuée. La surface de Fermi, mettant en exergue des domaines décalés qui alternent en taille et en forme, est comparée à l'expérience. De plus, la densité d'états du modèle est aussi calculée. Une analyse plus approfondie du modèle est réalisée au travers d'une étude en couplage faible des instabilités magnétiques. Les calculs sont effectués sur de grandes cellules et nous avons trouvé une compétition parmi plusieurs instabilités magnétiques tridimensionnelles dans la région d’intérêt du dopage en trous accessible expérimentalement. Bien qu'à notre connaissance cela ne semble pas avoir été évoqué expérimentalement, nous montrons à l'issue de notre étude, que la tendance du modèle à former des ondes de densité de spin incommensurables tridimensionnelles est la plus forte à proximité du dopage 1/8. / In this thesis, we construct a minimal three-dimensional tight-binding model for single-layer La-based cuprate superconductors. It entails eight orbitals, two of them involving apical oxygen ions. Parameter optimization allows to almost perfectly reproduce the three-dimensional conduction band as obtained from DFT. We discuss how each parameter entering this multiband model influences it, and show that the peculiar form of its dispersion severely constraints the parameter values. We then evidence that standard perturbative derivation of an effective one-band model is poorly converging because of the comparatively small value of the charge transfer gap. Yet, this allows us to unravel the microscopical origin of the in-plane and out-of-plane hopping amplitudes. An alternative approach to the computation of the tight-binding parameters of the effective model is presented and worked out. It results that the agreement with DFT is preserved provided longer-ranged hopping amplitudes are retained. A comparison with existing models is performed, too. The Fermi surface, showing staggered pieces alternating in size and shape, is compared to experiment. The density of states is calculated as well. The model is further analyzed through a weak coupling study of magnetic instabilities. It is performed on large clusters and competition between several three-dimensional magnetic instabilities in the hole-doping region of experimental interest is found. We show that the tendency to form a three-dimensional incommensurate spin density wave is strongest in the vicinity of 1/8 doping.