Analysis of Single- and Multi-layer Phosphorene Nanoribbons Behavior Under Modulated Electric Fields Using Tight-Binding and Green’s Function Formalism

Black phosphorous is a layered semiconductor material which shows interesting electronic and optical properties. Few-layer black phosphorous called phosphorene is a new, two-dimensional semiconductor material demonstrated in 2014. In this paper, we use the tight-binding method to implement a matrix representation of energy band structure for single-layer and multilayer phosphorene nanoribbon (PNR) structures. In this method, the Hamiltonian of the system is defined to simplify studies of the electronic and optical properties of PNRs. We use our defined matrix representation to study the effect of modulated electric fields (EFs) on the electronic properties of multilayer armchair PNRs. We apply both in-plane linear potential and sine-wave EF across the width and along the length of PNR, respectively. We compare the band gap variation, conductance, and density of states of the armchair PNRs of different widths under no EF, linear potential, sine-wave EF, and simultaneous linear potential and sine-wave EF. Non-equilibrium Green’s function formalism is used in order to calculate the conductance and density of states. The matrix representation of PNRs is very helpful in studying the quantum transport phenomena in large-scale PNRs in field-effect transistors. Linear potential changes the band structure of wider PNRs seriously and reduces the band gap value irrespective of the number of layers. It is found that the critical EF to close the band gap (E_g = 0) is lower in a wide PNR compared to that in a narrow PNR. For sine-wave EFs applied in the transport direction of PNRs, the band gap decreases slowly in short-length PNRs compared to the PNRs with ten-time-longer lengths. Application of a modulated sine-wave EF in the transport direction of PNR structures decreases the band gap of all PNRs by ~ 0.25 eV regardless of the presence of linear potential. By applying simultaneous linear potential and sine-wave EF, and controlling the strengths of fields, we can change the band structure of multilayer PNRs effectively, which can be used to tune the band gap of these materials for particular applications.