Modeling Density Of States And Transport Properties Of Low-dimensional Systems With Numerical Methods

The density of states (DOS) and transport properties of low-dimensional materials areimportant for the characterization of solid state materials and devices. This work consists of twoparts. In the first part，we study the density of states of one dimensional chain and graphenenanoribbons by the time-evolution method. Then，we compare the efficiency of differentalgorithms，and provide a theoretical basis for the future study of the DOS of large size materials.In the second part，we investigate the transport properties of a pair of Majorana bound states inT-shaped junctions by the scattering matrix to characterize Majorana bound states.Traditional method of calculating the DOS depends on the diagonalization of the staticHamiltonian. However，for solid states with huge degrees of freedom，the calculation becomesvery numerically impossible. In this paper，we calculate the time-evolution of wave function tostudy the DOS of low-dimensional materials. Compared to static Hamiltonian method，thecomputational complexity is rapidly decreased，and increases linearly with the degrees of freedom.We apply this method in the one-dimensional atomic chain structure numerically investigate itsDOS and compare the time-evolution of the two algorithms: Chebyshev method andCrank-Nicolson method. The results show that the Chebyshev method is more efficiency. Thesestudies provide a reference for the subsequent analysis DOS of large-scale material，and lay thefoundation for the calculation of low-dimensional structure of photoconductive and other physicalproperties. Then，we present a detailed numerical study of the DOS of graphene nanoribbons. Ourresults give consistent results of the DOS in different bandwidth，length and Anderson disorder.In recent years，as its own antiparticle，Majorana bound states (MBS) have attracted muchattention. MBS can exist in condensed matter materials in the form of quasi-particles. Theoristshave predicted that MBS can exist in one dimensional semiconductor nanowires with largespin-orbit coupling，Zeeman splitting and proximity-induced superconductivity. Signatures of these theoretical results have been observed in recent experiments where zero-bias conductancepeak is observed. However，quantitative values of the zero-bias conductance peak do not agreewith theoretical predictions. As a result，the characteristics transport properties of MBS are ofgreat importance for the search of this elusive particle. In this work，we study the transportproperties of T-Shaped junctions by the scattering matrix method.Our results show that the shot noise Fano factor and the peak value of the differentialconductance in the zero bias satisfy a universal linear relation as，independent of the symmetry and magnitude of the coupling strengths to the leads. This featuremay serve to probe Majorana bound states without the knowledge of the coupling details.