13 - Low-Dimensional Quantum Systems

Abstract

In low dimensional quantum systems, electrons (and holes and excitons) are confined to move in a plane, along a wire, or within a quantum dot. This chapter begins with an introduction to mesoscopic systems, and to low dimensional structures that has been experimentally fabricated (quantum wells, quantum wires, quantum dots, heterostructures, superlattices and quantum point contacts). The Landauer formula for conductance and its multi-port extension is introduced. Quantum dot topics, including the Coulomb blockade, disorder effects and their description within random matrix theory and the quantum dot Kondo effect, are discussed. Graphene and its electronic properties are discussed. The 2D Dirac Hamiltonian is introduced, beginning with the Dirac electron theory and then its application to the two dimensional case relevant for graphene. The quantum Hall effect and Klein tunneling phenomena in graphene are considered. A brief survey is given of some novel phenomena that occur in low-dimensional systems, including weak localization, dephasing, shot noise, strong correlations, Wigner crystals, exotic concepts associated with the fractional quantum Hall effect (composite fermions and fractional statistics), high temperature superconductivity, 1D spin systems, and the spin Hall effect.