External-field-modulated Energy Spectra And Transport Properties Of Dirac Fermions In Two Dimensions

Since its successful fabrication by Geim and Novoselov using micromechanical cleavage method, graphene quickly becomes the new star of condensed matter physic-s and materials physics. Due to its simple lattice structure and rich physical proper-ties, graphene has potential applications in semiconductor industry. Different from the traditional Schrodinger electrons, the particles in graphene are called Dirac fermions, which obey the relativistic Dirac equation. It opens up a new area. In recent years, many researchers devoted to explore more Dirac fermions systems, and many new two-dimensional materials appear constantly. No matter in lattice structures or band structures, these new materials have obvious differences, so as to results in many different physical properties. All of these provide broad prospects in spin- and valley-electronics, band topology, and optoelectronics.We mainly investigate the energy spectra and transport properties of Dirac fermions under the modulation of external fields. The main objectives are focused on exotic prop-erties of electrons in graphene quasi-periodic superlattice, the transmission and energy spectrum of pseudospin-1 Dirac fermions in Lieb superlattice, striking Andreev reflection at the interface of normal conductor/superconductor based on twisted graphene bilayer, and the anomalous thermomagnetic effects in silicene under the modulation of electric field and off-resonant light. In detail, the dissertation is arranged as follows:In chapter one, we give a brief introduce about lattices and band structures of graphene and some derivative materials including Lieb lattice, twisted graphene bilay-er and silicene. Then we describe the basic topological and transport properties of Dirac Fermi systems. Finally, principal theoretical methods are given.In chapter two, we investigate the Dirac-like electronic behavior in a Thue-Morse graphene superlattice. It is found that unlike conventional Schrodinger electrons, quasi-periodic features such as the striking self-similarity and trifurcation in the transmis-sion spectrum can be manifested only at oblique incidence; In the vicinity of the usu-al Dirac point, extra Dirac points emerge; their locations are dependent merely on the second generation of the Thue-Morse structure and the number is double that in the periodic graphene superlattice; A classification is given about the wavefunctions in the Thue-Morse structure which are transformed from the critical states into extended ones at the Dirac points; The electrons can transmit perfectly at the extra Dirac points, and such a collimation can be used to experimentally detect the numbers and the locations of the extra Dirac points.In chapter three, we study the behavior of pseudospin-1 Dirac fermions in a Lieb optical lattice subjected to an external periodic potential. It is found that there exists a zero-averaged wave-number passband at the incident energy corresponding to half of the potential step, in contrast to zero-averaged wave-number gap in graphene superlattices. By tuning the sublattice site-energy, the passband can be turned into an omnidirectional gap. Consequently, a transformation from omnidirectional transmission to reflection, ac-companied with a switch of conductance from maximum to zero can be realized easily. It is expected that the controllable properties are useful for some applications in optical or electronic devices.In chapter four, by using Dirac-Bogoliubov-de Gennes equation and Blonder-Tinkham-Klapwijk theory, we investigate the Andreev reflection in a normal conduc-tor/superconductor junction based on a twisted graphene bilayer, which gives rise to a fea-sible crossover between linear and quadratic dispersion relations characterized by Berry phases. It is found that in the low-energy linear region, the specular Andreev reflection shows an obvious enhancement with the subgap differential conductance being twice that in a graphene monolayer, demonstrating the chirality of a single layer; however, in high energy region, the special spinor wave functions with 2π Berry phase lead to a remark-able suppression of conductance. The transition of different chiralities on the saddle point corresponding to the van Hove singularity in the electronic density of states can be con-firmed by measurable signal reversal in conductance. These results provide a facile way to control the Andreev reflection in a twisted graphene bilayer.In chapter five, we investigate the anomalous Nernst effect in silicene under the modulation of off-resonant light and perpendicular electric field. It is proposed that the abundant topological phases in silicene can be distinguished by measuring the Nernst conductivity even at room temperature, and their phase boundaries can be determined by differentiating the charge and spin Nernst conductivities. By modulating the electric and light fields, pure spin polarized, valley polarized, and even spin-valley polarized Nernst currents can be generated. Similar investigations can be extended from silicene to ger-manene and stanene, and a comparison is made for the anomalous thermomagnetic figure of merits between them.In the last chapter, we make a summary of our researches and give an outlook about further study in the future.