Research On The Confined Electron States In Topological Insulators

Topological insulators are new kinds of materials in condensed matter physics which are characterized by nontrivial band topology and conducting surface states.With a series of novel electron properties and their application prospect in fields such as low energyconsumption devices,spintronics and quantum computing,topological insulators have been the frontier and hot fields in the condensed matter physics.Developing a series expansion method for high-order ordinary differential equation arrays to solve the lowenergy effective model of topological insulators,we have studied the energy spectra,charge currents and optical absorption spectra of the confined structures,including Coulomb impurities,magnetic quantum dots and quantum dots and rings formed by structure inversion asymmetry in the two-dimensional topological insulators or thin films of the three-dimensional topological insulators.Firstly,the Coulomb impurity problem in infinite and finite two-dimensional topological insulators is studied.For the infinite case,it is found that the energy level structure and the level degeneracy are quite different for the topological trivial and nontrivial phases,and that the levels are discontinuous near the phase transition point.For the finite case of quantum dots,we find the edge states can be strongly coupled with the Coulomb states when the Coulomb potential is strong and the dot size is small.The level order of the Coulomb and edge states changes aside the coupling region,and they turn into mixtures in the coupling region.The modulations of magnetic fields to the Coulomb,edge states and their coupling are also studied.Next,magnetic quantum dots formed by non-uniform magnetic fields is studied.The energy spectra are compared for the topological trivial and nontrivial phases,and we find the ground states of the magnetic quantum dots can oscillate with the dot size and magnetic field when the system is topological nontrivial,on condition that the magnetic fields exceed a critical value.The charge currents of the ground states which are locked with the real spins are found to distribute near the dot boundary,and the charge current density changes directions across the dot boundary due to the highly polarization of the isospin and the existence of the modified mass term in the model.Then,quantum dots and rings formed by structure inversion asymmetry in threedimensional topological insulator thin films are studied.Based on the solutions of special functions,we find the anomalous size effects and the resonant tunneling effect in the dots and rings which are caused by the complex band structures.We show that the vertical transition between the edge states can occur in the dot with help of the inversion asymmetry potentials,and the characteristics of the optical absorption spectra are obtained for different kinds of transition.Finally,the series expansion method for high-order ordinary differential equation arrays is developed for dealing with the above problems.The developed method can be easily used for exact solutions of other multi-band systems.