Time Reversal Effect Of Impurities In The Invariance Of The Topological Insulator

The first examples of topological insulators(TIs),the integer quantum Hall effect and the quantum anomalous Hall effect,have been studied for decades.However,in the last few years,a new time-reversal invariant class of TIs was theoretically proposed in 2d and 3d:This thesis was an effon to solve two outstanding problems in this new field through five parts.The first problem is where to find these new TIs in nature.The second problem is how to physically define and distinguish the characteristics of the time-reversal invafiant TIs.In chapter 1,based on the recent development of study on quantum spin Hall effect and TIs,we introduced in a pedagogical way the background of the field,and the motivations of our study.In the section of quantum spin Hall effect,we focused in particular on the topological properties and its topological stability of the edge states of quantum spin Hall systems,named as "helical liquid".In the section of TI,taken(2+1)-d Dirac fermions for example,we argued that the direct result of the parity symmetry broken is the half-integer quantum Hall effect.In chapter 2,we focused on the spin Hall effect of a 2d hole gas(2DHG) in a perpendicular magnetic field.This study aimed to the status at that time that,in both 2DEG and 2DHG systems,it is difficult to differentiate in a convincing way the intrinsic and extrinsic spin Hall effect(ISHE/ESHE) both theoretically and experimentally.We proposed two phenomena which are instructive to experiments.First,we pointed out that the spin Hall conductance was resonant in 2DHG in a perpendicular magnetic field.This effect occurs only in the ISHE,and helps with the spin accumulation at the boundaries of the systems.Second,it is found that the sign of the ISHE changed periodically with the modulation of magnetic field,spin-orbit coupling(SOC) strength, and hole density.This effect in particular can be used in future information technique. In chapter 3,we studied the spin accumulation from the non-Abelian Aharonov-Bohm effect in a particular kind of SOC systems.It has been shown in the literatures that the SOC of the Dresselhaus type in[110]quantum wells can be mathematically removed by a non-Abelian gauge transformation.In the presence of an additional uniform magnetic field,such a non-Abelian gauge flux leads to a spin accumulation at the edges of the sample,where the relative sign of the spin accumulation between the edges can be tuned by the sign of the Dresselhaus SOC constant,this prediction can be tested by Kerr measurements within the available experimental sensitivities.In chapter 4,we discussed the effects induced by impurities doped in TIs.We first introduced relative facts in angle-resolved photon emission and scanning tunneling microscopy experiments,which not only confirmed completely the theoretical properties of the TIs,but also showed the quasi-particle interference pattern on the surface of TIs. To explain these experiments,we established a general theorem on the law of Friedel oscillations in the presence of either point- or step-like magnetic and/or nonmagnetic impurities with nonspherical Fermi surfaces.We then discussed the behavior of local density of states on the surface of TIs induced by a single magnetic impurity and the RKKY interactions between two magnetic impurities which was proved to be ferromagnetic. A set of renormalization group equations were also given.In the last part of this chapter,bulk impurities in the TIs were focused,where it was shown that there were always localized excited states in the bulk energy gap for arbitrarily strong impurity strength in topological nontrivial region,while the localized excited states vanished for very strong impurity strength in topological trivial region.In chapter 5,we discussed the spontaneous symmetry breakings in 2d Kagome lattice. All the broken phases had infinitesimal instabilities protected by C₆ rotational lattice symmetry,various topological phases,such as quantum anomalous Hall phase, quantum spin Hall phase,and ferromagnetic quantum anomalous Hall phase were generated dynamically.