The primary subject of this thesis is graphene and how the rudimentary attributes of its charge carriers, and local moments on its surface, can be directly manipulated and controlled with electrostatic potentials. We first consider bilayer graphene subject to a spatially varying electrostatic potential that forms two neighbouring regions with opposite interlayer bias. Along the boundary, 1D chiral `kink' states emerge. We find that these 1D modes behave as a strongly interacting Tomonaga-Luttinger liquid whose properties can be tuned via an external gate. Next, we consider superlattices in bilayer graphene. Superlattices are seen to have a more dramatic effect on bilayer graphene than monolayer graphene because the quasiparticles are changed in a fundamental way; the dispersion goes from a quadratic band touching point to linearly dispersing Dirac cones. We illustrate that a 1D superlattice of either the chemical potential or an interlayer bias generates multiple anisotropic Dirac cones. General arguments delineate how certain symmetries protect the Dirac points. We then map the Hamiltonian of an interlayer bias superlattice onto a coupled chain model comprised of `topological' edge modes. We then discuss the relevance of spatially varying potentials to recent transport measurements. This is followed by another study that considers the effect of a magnetic field on graphene superlattices. We show that magnetotransport measurements in a weak perpendicular (orbital) magnetic field probe the number of emergent Dirac points and reveal further details about the dispersion. In the case of bilayer graphene, we also discuss the properties of kink states in an applied magnetic field. We then consider the implications of these results with regards to scanning tunnelling spectroscopy, valley filtering, and impurity induced breakdown of the quantum Hall effect. Finally, we investigate local moment formation of adatoms on bilayer graphene using an Anderson impurity model. We construct various phase diagrams and discuss their many unusual features. We identify regions where the local moments can be turned on or off by applying a external electric fields. Finally, we compute the RKKY interaction between local moments and show how it too can be controlled with electric fields.

The primary subject of this thesis is graphene and how the rudimentary attributes of its charge carriers, and local moments on its surface, can be directly manipulated and controlled with electrostatic potentials. We first consider bilayer graphene subject to a spatially varying electrostatic potential that forms two neighbouring regions with opposite interlayer bias. Along the boundary, 1D chiral `kink' states emerge. We find that these 1D modes behave as a strongly interacting Tomonaga-Luttinger liquid whose properties can be tuned via an external gate. Next, we consider superlattices in bilayer graphene. Superlattices are seen to have a more dramatic effect on bilayer graphene than monolayer graphene because the quasiparticles are changed in a fundamental way; the dispersion goes from a quadratic band touching point to linearly dispersing Dirac cones. We illustrate that a 1D superlattice of either the chemical potential or an interlayer bias generates multiple anisotropic Dirac cones. General arguments delineate how certain symmetries protect the Dirac points. We then map the Hamiltonian of an interlayer bias superlattice onto a coupled chain model comprised of `topological' edge modes. We then discuss the relevance of spatially varying potentials to recent transport measurements. This is followed by another study that considers the effect of a magnetic field on graphene superlattices. We show that magnetotransport measurements in a weak perpendicular (orbital) magnetic field probe the number of emergent Dirac points and reveal further details about the dispersion. In the case of bilayer graphene, we also discuss the properties of kink states in an applied magnetic field. We then consider the implications of these results with regards to scanning tunnelling spectroscopy, valley filtering, and impurity induced breakdown of the quantum Hall effect. Finally, we investigate local moment formation of adatoms on bilayer graphene using an Anderson impurity model. We construct various phase diagrams and discuss their many unusual features. We identify regions where the local moments can be turned on or off by applying a external electric fields. Finally, we compute the RKKY interaction between local moments and show how it too can be controlled with electric fields.

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No. of bitstreams: 1 DissJXP.pdf: 3665347 bytes, checksum: 82164e5257df12677206609fad9335c0 (MD5) Previous issue date: 2015-03-27 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / This work is aimed to theoretically determine, based on the k p method, the energy spectrum of the minimum of conduction band and the top of valence band, in a quantum well heterostructure based on gallium arsenide, in the presence of a tilted magnetic eld. In this work, conduction and valence bands are independently treated. For the conduction band, it is employed the e ective-mass Hamiltonian when the magnetic eld is parallel, perpendicular, and tilted in relation to the con nement direction of the quantum well, de ned here as ^z. The basis that solves the orbital problem for = 0o is also employed to perpendicular ( = 90o) and tilted magnetic eld cases through two approaches: (i) expansion and diagonalization of orbital part of Hamiltonian and (ii) non-degenerate perturbation theory. The problem for = 90o is also analytically treated, through a basis composed of con uent hypergeometric functions. Results obtained by approach (i) are shown to be equivalent to those extracted from the analytical treatment, for a broad range of magnetic eld intensity and quantum well thickness. Such a result motivates the employment of the basis, determined for = 0o, to deal with tilted eld in the conduction band and with the treatment of valence band, based on the Luttinger model. The Luttinger Hamiltonian is expanded and diagonalized, in parallel and tilted magnetic eld con gurations, with respect to ^z. The energy branches as a function of the magnetic eld intensity are computed for the valence band when = 0o, = 35o, and = 70o. The two topmost states are selected, whose spin characters indicate that both predominantly have a heavy hole type with either spin-up or spin-down. From these levels, it is calculated the electron-hole pair spin splitting EZ, for quantum wells of di erent thickness and for angles 0o, 35o and 70o with respect to ^z. The connection with experimental results is performed by using the data of the Zeeman splitting obtained from a sample of multiple quantum wells made of gallium-aluminium arsenide, on which a magnetic eld is applied in = 0o and in = 70o. The calculated EZ show a reasonable agreement with the experimental data in both cases when = 0o and = 70o. / Este trabalho tem como objetivo determinar de maneira teórica, baseando-se no método kp, o espectro de energia do mínimo da banda de condução e do topo da banda de valência de uma heteroestrutura de poço quântico baseada em arseneto de gálio, na presença de um campo magnético inclinado em relação a sua direção de crescimento, definida aqui por z. As bandas de condução e de valência são tratadas de maneira independente neste trabalho. Para a banda de condução, o Hamiltoniano de massa efetiva e utilizado nas configurações de campo magnético paralelo, perpendicular e inclinado em relação à direção de confinamento do poço quântico. A base que soluciona o problema orbital em _ = 0o também _e aplicada aos casos de campo magnético perpendicular (_ = 90o) e de campo inclinado, utilizando duas abordagens: (i) expansão e diagonalização da parte orbital do Hamiltonino do sistema, e (ii) teoria de perturbação não degenerada. O problema em que _ = 90o também _e abordado de maneira analítica, por meio de uma base composta de funções hipergeométricas confluentes. Verifica que os resultados obtidos pela abordagem (i) são equivalentes aos obtidos pelo método analitico para uma faixa ampla de valores de campo magnético e largura do poço quântico. Esta equivalência motiva a utilização da base, obtida para _ = 0o, no caso do campo inclinado na banda de condução, e no tratamento da banda de valência, baseado no modelo de Luttinger. O Hamiltoniano de Luttinger _e expandido e diagonalizado, para as configurações de campo magnético paralelo e inclinado em relação a ^z. Os ramos de energia em função do campo magnético são calculados na banda de valência para _ = 0o, _ = 35º e para _ = 70o. Nesta mesma banda foram selecionados os dois estados mais energéticos, cujos caracteres de spin indicam que ambos são predominantemente do tipo buraco pesado com spin para cima ou para baixo. A partir destes niveis, o desdobramento de spin do par eletron-buraco _EZ _e calculado para poços quânticos de diversas espessuras e para os ângulos de inclinação do campo magnético em relação a ^z de 0o, 35o e 70o. A conexão com a abordagem experimental _e realizada por meio de dados de desdobramento Zeeman determinados para um amostra de poço quântico múltiplo constituído de arseneto de gálio-alumínio, sobre a qual foi aplicado campo magnético em _ = 0o e _ = 70o. As curvas _EZ calculadas neste trabalho mostram uma concordância razoável com os dados experimentais, tanto no caso _ = 0o quanto no caso _ = 70o.

This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.

付記する学位プログラム名: 京都大学卓越大学院プログラム「先端光・電子デバイス創成学」 / 京都大学 / 新制・課程博士 / 博士(理学) / 甲第24402号 / 理博第4901号 / 新制||理||1700(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川上 則雄, 教授 佐々 真一, 教授 高橋 義朗 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We reserve the first part of this thesis to a brief (and by far incomplete, but hopefully self-contained) introduction to the vast subject of quantum Hall physics. We dedicate the first chapter to a discursive broad introduction. The second one is instead used to introduce the integer and fractional quantum Hall effects, with an eye to the synthetic quantum matter platforms for their realization. In the third chapter we present famous Laughlin's wavefunction and discuss its basic features, such as the gapless edge modes and the gapped quasiparticle excitations in the bulk. We close this introductory part with a fourth chapter which presents a brief overview on the chiral Luttinger liquid theory. In the second part of this thesis we instead proceed to present our original results. In the fifth chapter we numerically study the linear and non-linear dynamics of the chiral gapless edge modes of fractional quantum Hall Laughlin droplets -- both fermionic and bosonic -- when confined by anharmonic trapping potentials with model short range interactions; anharmonic traps allow us to study the physics beyond Wen's low-energy/long-wavelength chiral Luttinger liquid paradigm in a regime which we believe is important for synthetic quantum matter systems; indeed, even though very successful, corrections to Wen's theory are expected to occur at higher excitation energies/shorter wavelengths. Theoretical works pointed to a modified hydrodynamic description of the edge modes, with a quadratic correction to Wen's linear dispersion $\omega_k=vk$ of linear waves; even though further works based on conformal field theory techniques casted some doubt on the validity of the theoretical description, the consequences of the modified dispersion are very intriguing. For example, in conjunction with non-linearities in the dynamics, it allowed for the presence of fractionally quantized solitons propagating ballistically along the edge. The strongly correlated nature of fractional quantum Hall liquids poses technical challenges to the theoretical description of its dynamics beyond the chiral Luttinger liquid model; for this reason we developed a numerical approach which allowed us to follow the dynamics of macroscopic fractional quantum Hall clouds, focusing on the neutral edge modes that are excited by applying an external weak time-dependent potential to an incompressible fractional quantum Hall cloud prepared in a Laughlin ground state. By analysing the dynamic structure factor of the edge modes and the semi-classical dynamics we show that the edge density evolves according to a Korteweg-de Vries equation; building on this insight, we quantize the model obtaining an effective chiral Luttinger liquid-like Hamiltonian, with two additional terms, which we believe captures the essential low-energy physics of the edge beyond Wen's highly successful theory. We then move forward by studying -- even though only partially -- some of the physics of this effective model and analyse some of its consequences. In the sixth chapter we look at the spin properties of bulk abelian fractional quantum Hall quasiparticles, which are closely related to their anyonic statistics due to a generalized spin-statistics relation - which we prove on a planar geometry exploiting the fact that when the gauge-invariant generator of rotations is projected onto a Landau level, it fractionalizes among the quasiparticles and the edge. We then show that the spin of Jain's composite fermion quasielectron satisfies the spin-statistics relation and is in agreement with the theory of anyons, so that it is a good anti-anyon for the Laughlin's quasihole. On the other hand, even though we find that the Laughlin’s quasielectron satisfies the spin-statistics relation, it carries the wrong spin to be the anti-anyon of Laughlin’s quasihole. Leveraging on this observation, we show how Laughlin's quasielectron is a non-local object which affects the system's edge and thus affecting the fractionalization of the spin. Finally, in the seventh chapter we draw our conclusions.

Un conducteur est bien caractérisé par sa conductance donnée par la formule de Landauer. Toutefois, le bruit contient davantage d'informations. Il mesure les fluctuations temporelles du courant autour de sa valeur moyenne. De plus, le signe des corrélations croisées est lié à la statistique des porteurs de charge. Cette thèse aborde deux principaux thèmes à savoir le transport dans les liquides de Luttinger et dans les structures hybrides. Dans la première partie, nous commençons par donner une vision détaillée des liquides de Luttinger et des systèmes qu'ils modélisent. Nous parlons également du formalisme de Keldysh permettant de traiter des problèmes hors équilibre. Puis, nous rentrons dans le vif du sujet en étudiant l'effet de la largeur d'un contact ponctuel quantique sur le courant de rétrodiffusion entre les deux états de bords de l'effet Hall quantique. L'augmentation de la largeur du contact ponctuel quantique entraîne une forte diminution du courant de rétrodiffusion. Dans un autre chapitre, nous développons une technique permettant l'utilisation d'un circuit RLC couplé inductivement au circuit mésoscopique pour détecter les corrélations de courant en régime photo-assisté. La mesure de ces corrélations s'effectue à travers la charge aux bornes du condensateur. Dans une deuxième partie, nous consacrons notre étude au transport non-local dans les structures hybrides supraconductrices. L'étude de la réflexion d'Andreev croisée y est détaillée. Finalement, nous étudions une structure en double point quantique reliée à deux électrodes en métal normal et une supraconductrice. Nous mettons en avant la séparation des paires de Cooper en mesurant simultanément les courants de branchement et les corrélations croisées. Nous démontrons que dans le régime antisymétrique, c'est-à-dire lorsque les deux points quantiques ont des niveaux d'énergie opposés par rapport au potentiel chimique du supraconducteur, la réflexion d'Andreev croisée est optimisée. / The conductance is the most natural quantity to characterize a quantum conductor. It is given by the Landauer Formula. However, noise contains more information. It measures the current fluctuations around its average value. Moreover, the sign of the crossed correlations is related to the statistics of carriers. This thesis broaches two main topics which are the transport in the quantum Hall effect and in hybrid circuits.First, we start by introducing the Luttinger liquid and the systems which are modelized by them. Also, we discuss the Keldysh formalism in order to treat nonequilibrium problems. Then, we study the effect of the width of a quantum point contact on the backscattering current between two edge states of the quantum Hall effect. By increasing the width of the quantum point contact, we show that the backscattering current is strongly reduced. In another chapter, we develop a technique to use a RLC circuit inductively coupled to a mesoscopic circuit in order to measure the current correlations in the photo-assisted regime. The measurement of these correlations is performed through the charge on the capacitor plates.Secondly, we present the non-local transport in hybrid structures. The mechanism of Crossed Andreev Reflection is explained. Finally, we study a double quantum dot connected to two normal leads and a superconducting lead. We introduce the separation of the Cooper pair by measuring together the branching currents and the crossed correlations. We demonstrate that in the anti-symmetric regime (the energy level of the two quantum dots have opposite values with respect to the chemical potential of the superconducting lead), crossed Andreev reflection is optimized.

This thesis reports our work on transport related problems in mesoscopic physics using analytical as well as numerical techniques. Some of the problems we studied are: effect of interactions and static impurities on the conductance of a ballistic quantum wire[1], aspects of quantum charge pumping [2, 3, 4], DC and AC conductivity of a (dissipative) quantum Hall (edge) line junctions[5, 6], and junctions of three or more Luttinger liquid (LL)quantum wires[7]. This thesis begins with an introductory chapter which gives a brief glimpse of the underlying physical systems and the ideas and techniques used in our studies. In most of the problems we will look at the physical effects caused by e-e interactions and static scattering processes. In the second chapter we study the effects of a static impurity and interactions on the conductance of a 1D-quantum wire numerically. We use the non-equilibrium Green’s function (NEGF) formalism along with a self-consistent Hartree-Fock approximation to numerically study the effects of a single impurity and interactions between the electrons (with and without spin) on the conductance of a quantum wire [1]. We study the variation of the conductance with the wire length, temperature and the strength of the impurity and electron-electron interactions. We find our numerical results to be in agreement with the results obtained from the weak interaction RG analysis. We also discover that bound states produce large density deviations at short distances and have an appreciable effect on the conductance which is not captured by the renormalization group analysis. In the third chapter we use the equations of motion (EOM) for the density matrix and Floquet scattering theory to study different aspects of charge pumping of non-interacting electrons in a one-dimensional system. We study the effects of the pumping frequency, amplitude, band filling and finite bias on the charge pumped per cycle, and the spectra of the charge and energy currents in the leads[2]. The EOM method works for all values of parameters, and gives the complete time-dependences of the current and charge at any site of the system. In particular we study a system with oscillating impurities at several sites and our results agree with Floquet and adiabatic theory where these are applicable, and provides support for a mechanism proposed elsewhere for charge pumping by a traveling potential wave in such systems. For non-adiabatic and strong pumping, the charge and energy currents are found to have a marked asymmetry between the two leads, and pumping can work even against a substantial bias. We also study one-parameter charge pumping in a system where an oscillating potential is applied at one site while a static potential is applied in a different region [3]. Using Floquet scattering theory, we calculate the current up to second order in the oscillation amplitude and exactly in the oscillation frequency. For low frequency, the charge pumped per cycle is proportional to the frequency and therefore vanishes in the adiabatic limit. If the static potential has a bound state, we find that such a state has a significant effect on the pumped charge if the oscillating potential can excite the bound state into the continuum states or vice versa. In the fourth chapter we study the current produced in a Tomonaga-Luttinger liquid (TLL) by an applied bias and by weak, point-like impurity potentials which are oscillating in time[4]. We use bosonization to perturbatively calculate the current up to second order in the impurity potentials. In the regime of small bias and low pumping frequency, both the DC and AC components of the current have power law dependences on the bias and pumping frequencies with an exponent 2K−1 for spinless electrons, where Kis the interaction parameter. For K<1/2, the current grows large for special values of the bias. For non-interacting electrons with K= 1, our results agree with those obtained using Floquet scattering theory for Dirac fermions. We also discuss the cases of extended impurities and of spin-1/2 electrons. In chapter five, we present a microscopic model for a line junction formed by counter or co-propagating single mode quantum Halledges corresponding to different filling factors and calculate the DC [5] and AC[6] conductivity of the system in the diffusive transport regime. The ends of the line junction can be described by two possible current splitting matrices which are dictated by the conditions of both lack of dissipation and the existence of chiral commutation relations between the outgoing bosonic fields. Tunneling between the two edges of the line junction then leads to a microscopic understanding of a phenomenological description of line junctions introduced by Wen. The effect of density-density interactions between the two edges is considered exactly, and renormalization group (RG) ideas are used to study how the tunneling parameter changes with the length scale. The RG analysis leads to a power law variation of the conductance of the line junction with the temperature (or other energy scales) and the line junction may exhibit metallic or insulating phase depending on the strength of the interactions. Our results can be tested in bent quantum Hall systems fabricated recently. In chapter six, we study a junction of several Luttinger Liquid (LL) wires. We use bosonization with delayed evaluation of boundary conditions for our study. We first study the fixed points of the system and discuss RG flow of various fixed points under switching of different ‘tunneling’ operators at the junction. Then We study the DC conductivity, AC conductivity and noise due to tunneling operators at the junction (perturbative).We also study the tunneling density of states of a junction of three Tomonaga-Luttinger liquid quantum wires[7]. and find an anomalous enhancement in the TDOS for certain fixed points even with repulsive e-e interactions.

<p>In this work, we study two important themes in the physics of the interacting one-dimensional (1D) electron gas: the transition from one-dimensional to higher dimensional behavior, and the role of inhomogeneity. The interplay between interactions, reduced dimensionality, and inhomogeneity drives a rich variety of phenomena in mesoscopic physics. In 1D, interactions fundamentally alter the nature of the electron gas, and the homogeneous 1D electron gas is described by Luttinger Liquid theory. We use Quantum Monte Carlo methods to study two situations that are beyond Luttinger Liquid theory --- the quantum phase transition from a linear 1D electron system to a quasi-1D zigzag arrangement, and electron localization in quantum point contacts. </p><p>Since the interacting electron gas has fundamentally different behavior in one dimension than in higher dimensions, the transition from 1D to higher dimensional behavior is of both practical and theoretical interest. We study the first stage in such a transition; the quantum phase transition from a 1D linear arrangement of electrons in a quantum wire to a quasi-1D zigzag configuration, and then to a liquid-like phase at higher densities. As the density increases from its lowest values, first, the electrons form a linear Wigner crystal; then, the symmetry about the axis of the wire is broken as the electrons order in a quasi-1D zigzag phase; and, finally, the electrons form a disordered liquid-like phase. We show that the linear to zigzag phase transition occurs even in narrow wires with strong quantum fluctuations, and that it has characteristics which are qualitatively different from the classical transition.</p><p>Experiments in quantum point contacts (QPC's) show an unexplained feature in the conductance known as the ``0.7 Effect''. The presence of the 0.7 effect is an indication of the rich physics present in inhomogeneous systems, and we study electron localization in quantum point contacts to evaluate several different proposed mechanisms for the 0.7 effect. We show that electrons form a Wigner crystal in a 1D constriction; for sharp constriction potentials the localized electrons are separated from the leads by a gap in the density, while for smoother potentials, the Wigner crystal is smoothly connected to the leads. Isolated bound states can also form in smooth constrictions if they are sufficiently long. We thus show that localization can occur in QPC's for a variety of potential shapes and at a variety of electron densities. These results are consistent with the idea that the 0.7 effect and bound states observed in quantum point contacts are two distinct phenomena.</p> / Dissertation

Dans cette thèse, nous traitons de l'interaction entre deux états de<br />bord dans le régime de l'effet Hall fractionnaire. Nous nous sommes<br />appuyés sur les réalisations expérimentales récentes de structures<br />(les jonctions quantiques étendues) qui favorisent largement ces<br />interactions.<br /><br />Nous avons d'abord introduit l'effet Hall quantique en mettant<br />l'accent sur la physique des états de bords.<br /><br />Nous avons ensuite étudié le transport tunnel à travers une jonction<br />quantique étendue, en considérant les bords comme des liquides de<br />Luttinger chiraux. Nous exposons différents régimes de courant en<br />fonction de la longueur de la jonction, la force des interactions,<br />le facteur de remplissage et la température. Nous calculons<br />également le bruit associé à ce courant.<br /><br />Nous avons généralisé ces résultats à une jonction en coin, dans<br />laquelle les canaux de bord peuvent être contre ou copropageant.<br />Dans le cas contre-propageant, il apparaît une transition de phase<br />commensurable-incommensurable en fonction des interactions et d'un<br />paramêtre supplémentaire. Dans le cas co-propageant, nous calculons<br />le courant tunnel en perturbation dans un modèle de sine-Gordon<br />chiral.<br /><br />Enfin, nous déterminons les profils de densité des bords et les<br />excitations. Nous mettons en évidence l'apparition d'une instabilité<br />dans ces jonctions quand l'interaction entre les bords est trop<br />forte. La théorie de Chern-Simons montre qu'il se produit alors une<br />reconstruction des bords par les interactions.