Electronic Structures And Quantum Transport Properties Of Several Dirac-fermion Materials

Graphene is a truly two-dimensional material which is made up by carbon atoms.In 2004, single layered graphene was successfully prepared experimentally by Geim and Novoselov for the ?rst time. This breakthrough has drawn much attention to the researches on the electronic properties of graphene. Graphene is a zero-gap semiconductor, whose low-energy electrons have a linear dispersion relation and can be described by a massless Dirac equation. Therefore, graphene exhibits many unique electronic properties such as weak antilocalization, Klein paradox and half-integer quantum Hall e?ect, which are not owned by conventional semiconductors. As a typical Dirac-fermion material, graphene can simulate quantum-relativistic phenomena which are hard to observe in the vacuum. Besides graphene, there may exist other Dirac-fermion materials such as graphyne. Graphyne is also a two-dimensional material made up by carbon atoms which contains acetylene bonds. There are multiple structures of graphyne.Some types of them also have massless Dirac-fermions in the low-energy region. Moreover, Dirac-fermions in graphyne exhibit other features, such as the anisotropy of the Fermi velocity. Although the aforementioned Dirac-fermion materials, graphene and graphyne, are carbon materials, there are other materials, say topological insulators,where Dirac-fermions may appear. The topological insulator has a bulk band gap where topologically protected surface or edge states take place. A two-dimensional topological insulator is also referred to as a quantum spin Hall insulator, in which there are two states with opposite spins and momenta localized at each edge. The property that the spin and the momentum are locked together is referred to as helicity which is a typical character of Dirac-fermions. The surface electronic state of a three-dimensional insulator also obeys the massless Dirac equation. However, in contrast to graphene, the number of Dirac cones on each surface of a three-dimensional insulator must be odd.This nondegererate Dirac-fermion is also called a Weyl fermion. Due to their unique electronic properties, Dirac-fermion materials have potential applications in quantum information, quantum calculation and microelectronics.To study the electronic properties of solid materials, one should start from the Hamiltonian of the system. There are mainly two forms of Hamiltonian models of solid systems, i.e. the discrete lattice model and the continuous model. Tight-binding model is a common discrete lattice model, whose basic idea is to expand the eigenstates of the system by atomic orbitals. In addition, a discrete lattice model can also be obtained by expanding a continuous Hamiltonian around some particular point in the Brillouin zone of the system. In numerical calculations based on the discrete lattice model, the Hamiltonian of the system can always be represented by a ?nite-sized matrix, and the band structure, the density of states and the conductivity of the system can be obtained by solving the eigen problem of the matrix or calculating the Green function of the system. The discrete lattice model can conveniently include all kinds of scatterers in the model Hamiltonian of the system, and can obtain high-resolution results by numerical techniques. A continuous Hamiltonian is often obtained from the electronic states near the Fermi level of the system through the k · p method. The advantage of the continuous model is that it can always give analytical results, which can help us to intuitively understand the physical phenomena. In this dissertation, we study the electronic structures and transport properties of several Dirac-fermion materials using these two models. The content of the research in this dissertation is mainly divided into four parts:Firstly, we establish a simple and e?ective tight-binding model for four typical graphyne structures. From the ab initio calculations, we know that the low-energy states in these graphynes are mainly contributed from the pzorbital of the carbon atom.Therefore, a simple tight-binding model which only include the pzorbital is suitable to describe the low-energy electrons in these graphynes. By considering the lattice symmetry, we set at most three parameters in the tight-binding model and determine the values of the parameters by ?tting the calculated band structures with the ab initio results. From the comparison of the energy band results from the tight-binding model and density functional theory, we can conclude that the tight-binding model that we have established can describe the low-energy electrons of these typical graphynes e?ectively. In addition, we reduce the lattices of the four typical graphynes by the renormalization method to the low-energy limit, and derive analytical results about their band structures. In particular, we have determined the positions of the Dirac points of these graphynes in their Brillouin zones. At last, we calculate the projected density of states on di?erent atoms in their unit cells by Lanczos iteration method and discuss some details about the density of states such as the correspondence of the van Hove singularities of the density of states and the saddle points of the band structure. In a word, the simple tight-binding model we have established can quantitatively describe the low-energy electronic properties of the four typical graphyne structures and will provide a simple and e?ective way to further study other problems about graphynes such as transport properties, nano-structures and optical processes.Secondly, we study the transmission spectrums when electrons tunnel through multiple line defects in a graphene lattice by the continuous model. The basic unit of the line defect we study includes two pentagons and one octagon, and this type of line defect has been already prepared in experiments. From the tight-binding model,we can establish the connection condition of the Dirac equation at the line defects.By periodically inserting parallel line defects into a graphene lattice, we can get a superlattice. We derive the eigenenergy equation and the eigenfunctions analytically by the Dirac equation and the connection condition at the line defect. Based on these analytical results, we further study the transmission spectrums when electrons tunnel through multiple line defects in graphene. We ?nd that, due to the line defects,the transmission spectrum is valley polarized. Moreover, because the vector potential induced by the line defects destroys the chirality of the electrons in graphene, we cannot observe the Klein paradox for normal incidence in this system. However, resonant tunneling can occur at other incident angles, and the angle for resonant tunneling depends on the energy of the incident electron and the spacing between the adjacent line defects. It is worthy to note that in some region of the incident energy, a critical angle for total re?ection occurs in the transmission spectrum, which depends on the incident energy and the spacing between the adjacent line defects. Taking advantage of the eigenenergy equation of the superlattice, we derive the equation which must be satis?ed by the critical angle. Moreover, we also calculate the transmission spectrums when a square potential barrier is at present in the scattering region. As a result,both the angle for resonant tunneling and the critical angle for total re?ection depend sensitively on the height of the barrier, which can be simulated by a gate voltage in experiment. Therefore, we can tune the valley polarized electronic transmission properties when electrons tunnel through multiple line defects in graphene by a gate voltage.Thirdly, we study the quantum transport properties of a graphene sheet doped with randomly distributed spin-orbit-coupling impurities by kernel polynomial method.We calculate the conductivity tensor of the system by Kubo-Bastin formula from quantum transport theory based on a tight-binding model. The numerical results indicate that randomly distributed spin-orbit-coupling impurities can drive graphene into a quantum spin Hall state. In graphene, there may exist two di?erent gap opening mechanisms, i.e. the spin-orbit coupling and the sublattice staggered potential. The former one opens a topological nontrivial band gap while the latter one does not. Therefore,the two mechanisms compete with each other. Thus, the system can undergo a quantum phase transition between the topologically trivial and nontrivial phases when the concentration of the spin-orbit-coupling impurities is tuned. We further calculate the conductivity tensor of the system in the presence of a magnetic ?eld. As a result, the system exhibits quantum Hall and quantum spin Hall features at the same time. To be speci?c, in the band gap opened by the spin-orbit-coupling impurities, the total Hall conductivity has a zero value but the spin Hall conductivity exhibits a quantum plateau. On the other hand, the spin Hall conductivity is almost zero but the total Hall conductivity exhibits half-integer quantum plateaus out of the gap. We also study the conductivity tensor of the graphene system doped with randomly distributed spin-orbit-coupling impurities by the continuous model for comparison. We derive the analytical expression of the density of states and the conductivity tensor of the system within the self-consistent Born approximation. The calculated density of states spectrum also exhibits a nontrivial band gap. However, due to the exclusion of higher-order scattering processes within the self-consistent Born approximation, the magnitude of the band gap calculated via the continuous model is di?erent from that calculated from the tight-binding model. To the weak scattering limit, the self-consistent Born approximation can give reasonable results. The Hall conductivity spectrum calculated from Kubo-Stˇreda formula based on the continuous model also exhibits a quantum spin Hall plateau. We also calculate the conductivity tensor of the system using the semi-classical Boltzmann transport theory. As a result, Boltzmann theory gives a constant diagonal conductivity and nonquantized spin Hall conductivities in the band gap opened by the spin-orbit-coupling impurities. Therefore, we conclude that Boltzmann theory is not suitable to study the quantum spin Hall state in graphene induced by spin-orbit-coupling impurities.Finally, we study the band structure of a three-dimensional topological insulator quantum wire in the presence of external electric and magnetic ?elds by the continuous model. We expand the wavefunction of the system in a chosen representation and calculate the band structure of the system using a numerical diagonalization method.As a result, in an ultra-thin ?lm, the coupling between the opposite surface states can lift the degeneracy of the Landau levels, leading to integer quantum Hall e?ect. When the thickness of the ?lm is appropriately tuned, a band inversion between the electrontype and hole-type n = 0 Landau levels occurs, leading to pseudo-spin Hall e?ect. In relatively thick ?lms, a perpendicular electric ?eld can be used to lift the degeneracy of the Landau levels, leading to integer quantum Hall states. Due to the electron-hole asymmetry, the edge states of the hole-type Landau level exhibit peak-like dispersion relations, which destroy the chirality of the edge states, leading to the vanishing of theν <-1 quantum Hall plateaus. In addition, we further study the band structure of a three-dimensional topological insulator with a square cross section in the presence of a tilting magnetic ?eld. As a result, there are well-de?ned Landau levels localized on each surface. In this case, the edge states are localized at the corners of the square cross section. When the tilting angle of the magnetic ?eld is appropriately tuned,a linear-dispersed one-dimensional state occurs at the common edges of the Landau level states from di?erent surfaces. Electrons on this one-dimensional state behave like one-dimensional Weyl quasiparticles whose velocity and eigenenergy can be tuned by rotating the magnetic ?eld within some region.