Quantum transport theory and phase transition: from Dirac fermions to electrons in semiconductors

I theoretically studied the screening effect and then the Drude conductivity in conducting surfaces of topological insulator ultra-thin (~ 6 nm) films, where the interlayer tunnelling along the thickness is significant. Random phase approximation is assumed here. Dielectric functions show a cusp around two Fermi wavenumbers. The unique carrier density dependence of the Drude conductivity is plotted. These results give clues on extracting the strength of interlayer tunnelling. I also theoretically studied the effect of spin-orbit coupled impurities on the weak antilocalisation in conducting surfaces of topological insulators. I used Keldysh Formalism and preserved the matrix structure of Green's functions and spin-orbit scattering potentials. Nontrivial linear spin-orbit scattering contributions reveal in the elastic scattering time and the Drude conductivity, and weak antilocalisation conductivity has a constant symplectic channel number, which is protected by the time-reversal symmetry. I also theoretically studied the iterant ferromagnetic transition in two-dimensional electronic systems with the Rashba spin-orbit coupling, where exchange interaction is adopted to describe electron-electron correlations. A model based on opposite-displaced Fermi surfaces with vanishing electric current was proposed for searching the ferromagnetic ground state. Analytical calculations qualitatively predict the ferromagnetic transition and argue that the total energy becomes minimised at specific magnetisations depending on exchange interactions. Furthermore, Monte Carlo simulations are utilised to determine the phase diagram of such a system. Two distinctive ferromagnetic phases are identified: in-plane and out-of-plane phases. This phase diagram gives a clearer map for the iterant ferromagnetism in two-dimensional Rashba spin-orbit-coupled electronic systems. I also theoretically studied the weak antilocalisation in Weyl semimetal ultra-thin films where the transverse quantum confinement is remarkable. Here the focus is again the effect of the spin-orbit scattering. When increasing the energy gap due to the quantum confinement, a crossover between the weak antilocalisation and weak localisation is retained in the absence of the spin-orbit scattering, and the spin-orbit scattering only suppresses the weak localisation. Compared with previous studies, the suppression on weak antilocalisation is enhanced after considered the linear contribution of the spin-orbit scattering