Research On The Property Of Electron In Graphene Superlattices And Topological Materials

Topological insulators are new states of quantum matter with profound physical properties which have their origin in topology, the systems are insulating in the bulk but conductive on the boundary,and they have gapless edges (in a two-dimensional case) or surface (in a three-dimensional case) states due to the protection of the time reversal symmetry which can immune to impurities or geometric perturbations. On a sample edge, two counterpropagating gapless edge modes with opposite spin polarizations ex-ist in the bulk band gap, the existence of edge or surface states in the topological insu-lators does not depend on the geometric condition in the boundary, but comes from the topological characteristic of the bulk. The topological insulators can be distinguished from a trivial insulator by the nonzero topological invariant.Graphene is a kind of hexagonal lattice arranged by monolayer carbon atoms, also the first predicted to exhibit the time reversal invariant topological states and quantum spin-Hall effect. Electrons in graphene have two inequivalent Dirac cones and addition-al twofold degeneracy, obey the two-dimensional Dirac equation for massless particles in the vicinity of the Dirac points. A one-dimensional periodic potential may result in a strong anisotropy for the group velocities of the low-energy charge carriers, which are reduced to zero in one direction but are unchanged in another. Recently, It is ob-served that the new Dirac point will formed in the electronic band and these new Dirac fermion nearby have different landau level features and the quantum hall effect. This dissertation consists of five chapters:In Chapter 1, we give a brief introduction to the development of the series of hall effect, and focus on the basic concepts and theories of topological insulators in recent years, give a topology geometric explain in the perspective of the berry phase. We also briefly review the application of graphene materials in condensed matter physics, describes the nature of the graphene superlattices and the related experimental results. In the end, the research content and meaning of this paper are introduced.In Chapter 2, we investigate Zitterbewegung (ZB) behavior in a graphene superlat-tice with new generations of massless Dirac fermions which have a highly anisotropic group velocities, it results from a graphene subject to one-dimensional periodic poten-tials. An important characteristic of the new-generated Dirac cones in the graphene superlattices is that the group velocity of electrons can be highly anisotropic, i.e., the group velocity along the superlattice direction remains unchanged (vfx=vf) but that along the perpendicular direction can be much lower than vf(vfy<<vf). Such an anisotropy of the group velocity is determined by the magnitude of the 1D period-ic potential. The anisotropic group velocities will lead to an anisotropic energy gap, 2hvf(?)kx2+ky2(vfy/vf)2. Due to vfy/vf<<1, the energy gap is the minimal along the ky direction, and so the frequency of ZB oscillations will be reduced efficiently. It will be shown that for a smaller ratio vfy/vf, there are a larger amplitude of the ZB oscilla-tions and a slower attenuation with time. Thus provide a good system for probing the ZB effect experimentally.In Chapter 3, The properties of the edge states in the topological insulator Ⅰ-nAs/GaSb/AlSb quantum well in the presence of a perpendicular magnetic field are studied numerically. When the material is in the topologically nontrivial state, a pair of degenerate counter propagating spin-polarized edge states exist in the bulk band gap on each edge of the sample, which are gapless in the absence of the magnetic field due to the protection of the time reversal symmetry. Nonzero magnetic field breaks the time reversal symmetry, and leads to Landau levels in the electron energy spectrum. Howev-er, one can still find a pair of counter-propagating spin-polarized edge states in the bulk energy gap near each sample boundary. With the increase of the magnetic field, one edge state remains located near the sample boundary, but the other tends to evolve into the bulk gradually. Furthermore, we study the scattering between the two edge states caused by impurities. We show that the scattering rate is suppressed because of the spatial separation of two edge states, and shows no significant enhancement when the magnetic field increases, which suggests that even though the time reversal symmetry is broken, the quantum spin Hall state remains to be relatively robust.in Chapter 4, we calculate the spin spectrum of the projection operator (PSzP), a phase diagram of the topological insulator thin films is obtained and the result indicates that a strong structure inversion asymmtry always tends to destroy the QSH state, at a critical value of the structure inversion asymmetry, there is a quantum transition from a nontrivial topological phase to trivial one, but if the time reversal symmetry is broken, the tipping point will disappear. Furthermore, we consider the transformation in both the structure inversion asymmtry and magnetic field are present, the former destroys the U(1) symmetry and the latter breaks the time reversal symmetry, consequently, there exists a phase transition distinguish by the closing of spin spectrum. The phase transition are associated with the gap closures of pin spectrum but does not need the energy gap closing. This study enriched the classification of 2d topological insulator phase transition.Finally, Chapter 5 presents a summary of this dissertation, and gives some expec-tation for the investigation.