Electronic Transport in Low-Dimensional Systems Quantum Dots, Quantum Wires And Topological Insulators
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This thesis presents the work done on electronic transport in various interacting and non-interacting systems in one and two dimensions. The systems under study are: an interacting quantum dot , a non-interacting quantum wire and a ring in which time-dependent potentials are applied , an interacting quantum wire and networks of multiple quantum wires with resistive regions [3, 4], one-dimensional edge stages of a two-dimensional topological insulator , and a hybrid system of two-dimensional surface states of a three-dimensional topological insulator and a superconductor . In the first chapter, we introduce a number of concepts which are used in the rest of the thesis, such as scattering theory, Landauer conductance formula, quantum wires, bosonization, topological insulators and superconductor. In the second chapter, we study transport through a quantum dot with interacting electrons which is connected to two reservoirs. The quantum dot is modeled by two sites within a tight-binding model with spinless electrons. Using the Lippman-Schwinger method, we write down an exact two-particle wave function for the dot-reservoir system with the interaction localized in the region of the dot. We discuss the phenomena of two-particle resonance and rectification. In the third chapter, we study pumping in two kinds of one-dimensional systems: (i) an infinite line connected to reservoirs at the two ends, and (ii) an isolated ring. The infinite line is modeled by the Dirac equation with two time-independent point-like backscatterers that create a resonant barrier. We demonstrate that even if the reservoirs are at the same chemical potential, a net current can be driven through the channel by the application of one or more time-dependent point-like potentials. When the left-right symmetry is broken, a net current can be pumped from one reservoir to the other by applying a time-varying potential at only one site. For a finite ring, we model the system by a tight-binding model. The ring is isolated in the sense that it is not connected to any reservoir or environment. The system is driven by one or more time-varying on-site potentials. We develop an exact method to calculate the current averaged over an infinite amount of time by converting it to the calculation of the current carried by certain states averaged over just one time period. Using this method, we demonstrate that an oscillating potential at only one site cannot pump charge, and oscillating potentials at two or more sites are necessary to pump charge. Further we study the dependence of the pumped current on the phases and the amplitudes of the oscillating potentials at two sites. In the fourth chapter, we study the effect of resistances present in an extended region in a one-dimensional quantum wire described by a Tomonaga-Luttinger liquid model. We combine the concept of a Rayleigh dissipation function with the technique of bosonization to model the dissipative region. In the DC limit, we find that the resistance of the dissipative patch adds in series to the contact resistance. Using a current splitting matrix M to describe junctions, we study in detail the conductances of: a three-wire junction with resistances and a parallel combination of resistances. The conductance and power dissipated in these networks depend in general on the resistances and the current splitting matrices that make up the network. We also show that the idea of a Rayleigh dissipation function can be extended to couple two wires; this gives rise to a finite transconductance analogous to the Coulomb drag. In the fifth chapter, we study the effect of a Zeeman field coupled to the edge states of a two-dimensional topological insulator. These edge states form two one-dimensional channels with spin-momentum locking which are protected by time-reversal symmetry. We study what happens when time-reversal symmetry is broken by a magnetic field which is Zeeman-coupled to the edge states. We show that a magnetic field over a finite region leads to Fabry-P´erot type resonances and the conductance can be controlled by changing the direction of the magnetic field. We also study the effect of a static impurity in the patch that can backscatter electrons in the presence of a magnetic field. In the sixth chapter, we use the Blonder-Tinkham-Klapwijk formalism to study trans-port across a line junction lying between two orthogonal topological insulator surfaces and a superconductor (which can have either s-wave or p-wave pairing). The charge and spin conductances across such a junction and their behaviors as a function of the bias voltage applied across the junction and various junction parameters are studied. Our study reveals that in addition to the zero conductance bias peak, there is a non-zero spin conductance for some particular spin states of the triplet Cooper pairs. We also find an unusual satellite peak (in addition to the usual zero bias peak) in the spin conductance for a p-wave symmetry of the superconductor order parameter.