In this thesis, we will theoretically investigate the low-energy electronic properties of monolayer phosphorene using the tight-binding approach. We will specially focus on the topological properties of this system. To investigate these topological features, we will first review the basic theories of systems with and without time reversal symmetry Also, we will introduce efficient numerical methods for computing relevant topological invariants which are very useful for characterization of the topology of complicated band structures. We will numerically implement these methods to well- known models and compare the obtained topological phase diagrams with analytic results to demonstrate the validity of our algorithm. Then, we will start by introducing the tight-binding (TB) model of monolayer phosphorene (MLP). The band structure and effective masses of MLP near the gap are presented based on the TB model, and it is shown that the dispersion is relativistic along the armchair direction. Thereafter, numerical data for this model are presented for zigzag and armchair phosphorene nanoribbons, and the emergence of flat bands in zigzag phosphorene nanoribbons is discussed. The scaling behavior of band gaps with ribbon width is presented and the obtained results are compared with those of the other methods. Next, the behavior of MLP in the presence of a perpendicular magnetic field which breaks the time-reversal symmetry is discussed. Our goal is to address the debating field dependence of Landau levels in MLP. Using a simple approach, we will demonstrate that the Landau levels dependence on magnetic field is indeed as for conventional 2D semiconductor electron gases. We show that the results are valid up to very high field regime. Then, we will discuss the conditions for which such a dependence can continuously evolve into another field dependence. Finally, the effective low-energy TB model Hamiltonian including the SOC terms for MLP is obtained. Then, we will study the effect of axial strains on the band structure produced by this model and our results are compared with DFT results. We will show a topological phase transition in the electronic properties of phosphorene in the presence of particular types of strain and the characteristics of corresponding edge states in zPNRs will be discussed.
