Electronic, transport and optical properties in multilayer phosphorene

Abstract

We theoretically investigate multilayer phosphorene's electronic, transport, and optical properties by numeric and semianalytical methods. First, by combining the tight-binding model and the band unfolding technique, we explore the effects of atomic vacancies on the band structure of multilayer phosphorene. Results for mono-, di-, and tri-vacancies are analyzed by keeping or not the sublattice and/or the sublayer symmetries. A detailed description of the used theoretical framework is illustrated for defect-free structures of a monolayer, bilayer, and trilayer phosphorene with Bernal stacking, which allows us to identify that the stability of the effective unfolded band is dependent on the size of the N-layer phosphorene supercell, revealing as an appropriated supercell with a size of 6 x 6 unit cells. In the presence of vacancies, one emerges in the band structure an almost n-fold degenerate quasi-flat state in a system with n vacancies when the sublattice symmetry or/and the inversion symmetry are broken. Within this combined approach, we also investigate the defect effects due to randomly distributed vacancies in phosphorene nanoribbons with zigzag and armchair edges. Second, we theoretically investigate the effect of a perpendicularly applied electric field and the influence of edge types (armchair and zigzag) on the conductance and collimation of charge carriers dynamics in monolayer-bilayer, monolayer-bilayer-monolayer, and bilayer-monolayer-bilayer phosphorene junctions. By using the Landauer-Büttiker formalism and the tight-binding model, we explore the probability current density and the conductance in relation to the system parameters, such as Fermi energy, electric field magnitude, and domain wall length. Our results show (i) that such physical properties exhibit a strong dependence on the crystallographic orientation of the electronic propagating mode and the interface type due to the high anisotropy on the N-layer phosphorene energy bands, effective masses, and group velocities along the x and y directions, (ii) current modulations and (iii) a pronounced dependence on the collimation of the electron beams at the interface in the investigated multilayer phosphorene junctions, obtained by applying an electric field that breaks the inversion symmetry and causes a bandgap tuning, opening (closing) the gap for monolayer (bilayer) phosphorene. All results are explained in the context of the band structure mismatch/alignment between the portions of the multilayer phosphorene junction. Finally, optic transitions in multilayer phosphorene close to the Fermi level are explored via a semianalytical approach. For that, we use a continuum approximation for multilayer phosphorene, derived from a fifteen hopping parameters tight-binding model, and consider the first-order light-matter interaction. This approach allows us, in a simple way, to calculate the momentum matrix elements associated with polarized light in x and y directions, and then to obtain the total two-dimensional dielectric function of multilayer phosphorene extracted from Fermi's golden rule. The transition peaks in the total two-dimensional dielectric function are analyzed in view of the multilayer phosphorene band structure by taking different numbers of phosphorene layers.