Researches On Topological Insulators And Theirs Properties

As a new phase of matter, topological insulator has attracted extensive attention in the condensed matter physics and materials science. Completely different from the conventional insulator, topological insulator exhibits gapless edge or surface states which are protected by topology and are robust against any non-magnetic impurities and small perturbations. The topological properties of topological insulator can be characterized by Z2 invariant. Recent researches indict that topological insulator has some novel quantum phenomena, and is supposed to open up a new avenue to many potential applications in spintronics and quantum computing. In this paper, we will invest the topological properties of two and three dimensional topological insulators by means of topological band theory, kernel polynomial method and the first-principle calculations. Our works are as follows:1. With tight-binding method we first demonstrate that the α - graphyne lattice can support topological insulator. By calculating the topological invariant Z2 and the edge states, we identify topological insulators. We present the phase diagrams of α-graphyne with different filling fractions as a function of spin-orbit interaction and the nearest-neighbor hopping energy. We find there exist topological insulators in α - graphyne and we note that for different filling fractions there are only two phases, trivial band insulator and topological insulator. We analyze and discuss the characteristics of topological phases of α - graphyne.2. It has been argued whether the localization properties of graphene in the presence of Anderson disorder are consistent with the one-parameter scaling theory, thus it is meaningful to study the effect of Anderson disorder on the localization properties of graphyne lattice with the same Dirac structure. By means of tight-binding method and variable moment kernel polynomial method, we analyze the localization properties of/β - graphyuesheet subjected to the Anderson disorder. To distinguish a localized state from an extended one, it is convenient to compare two characteristic quantities:the typical DOS and the mean DOS. We find in contrast to graphene sheet, the DOS of β - graphyne is characterized by plenty of Van Hove singularities. With disorder strength increasing, starting from two boundaries of the spectrum, pty is suppressed and when γ> 12t it vanishes, namely, at the critical disorder strength the entire spectrum is localized. Only a single finite size system is insufficient to study the localization properties, We need to consider the normalized typical DOS. By calculating the normalized typical DOS, we can obtain the critical disorder strength. We can find with the system size increasing, the critical disorder strength is decreasing. Then we give the contours of the normalized typical DOS in the energy-disorder plane. From the contour map, we can find the mobility edge and Lifshitz boundaries. At last, in order to understand the internal structure of the extended and localized states, we calculate the LDOS in the band center. When the disorder strength reachesγ=12t, the states are confined to many isolated islands, which implies that these states are totally localized and the metal turns into an insulator, that is to say, the Anderson transition occurs.3. With tight-binding model, we demonstrate the square-octagon lattice can also support the topological insulators. By means of n-field method, we evaluate Z2 topological invariant and present the phase diagrams of different spin-orbit coupling interaction and filling fractions. We find even without intrinsic spin-orbit coupling interaction, there exist topological insulators at 1/4 and 3/4 filling fractions. When the intrinsic spin-orbit coupling interaction is suitable (λ∞= 0.-10),the topological insulator will appear at 1/4 and 3/4 filling fractions no matter how small the strength of Rashba spin-orbit coupling interaction. At the same time, at 1/2 filling fraction, if 0< λR.λ∞< 1, there will also exist topological insulator. We analyze and discuss the edge state characteristics of square-octagon lattice. And in order to understand the model of edge states, we calculate the distribution of density of states and the expectation value of the spin operator.4. We study a tight-binding model of the supercubane-like lattice with spin-orbit coupling. By evaluating the Z2 topological indices, we find that the supercubane-like lattice can support strong topological insulator and the phase diagrams of the lattice with different filling fractions arc present. Strong topological insulators with Z2 invariants (1;000) and (1;111) can be realized for 1/8 filling fraction and semimetals can be obtained for 1/8, quarter and half filling fractions, in which the conduction band touches with valence band only at a few isolated points in the Brillouin zone. We analyze and discuss the characteristics of these topological insulators and their surface states. Spin textures of surface states are also evaluated for lll slab geometry. For the upper and lower cone the surface states have opposite pseudospin helicities. And especially, for the lower cone when the chemical potential approaches to the top of the valence bands, the distortion of the spin texture will occur.5. Based on first-principle calculations the topological properties of the X8(X= C.Si. Ge.Sn. Pb) under hydrostatic strain have been investigated. Most of the materials can be dynamically stable with negative formation energy and no imaginary phonon frequency except Ph8. Due to X8(X= C. Si. Go. Sn) containing only light element, hence the strength of SOC is weaker and SOC has little impact on the band structure of X8(X = C.Si.Ge.Sn). However, except SOC, the strain, which widely exists in material surface and interface, could have significant impact on the electronic properties of semiconductor or semimetal. Then we will discuss the effect of hydrostatic strain. We find that the hydrostatic strain can not induce a quantum phase transition between topological trivial and nontrivial state for both C8 and Si8. While for Gen and Sn8 the tensile strain can play a unique role in tuning the band topology, which will lead to a topological nontrivial state with Z2 invariants (1;000). Although the topological transition occurs above the Fermi level, the Fermi level can be tuned by applying electrostatic gating voltage, and more importantly, our study indicates that elementary substance with light element can also be promising to realize three-dimensional topological insulators.