Influence of next-nearest-neighbor couplings on the electronic properties of blue phosphorus: a multi-orbital tight-binding investigation

Abstract This work is a theoretical investigation of the influence of next-nearest-neighbor (NNN) interactions on the electronic features of blue phosphorus (buk-P) along the Г – M – K direction within the multi-orbital tight-binding (MOTB) model. The objective is to enhance the quantitative accuracy of the TB representation of the multilayer buk-P system. On the other hand, we intend to identify and discuss how distant hopping parameters up to the fifth in-plane nearest-neighbors (5NN) and second out-of-plane nearest-neighbors (2NN) affect the energy dispersion bands of two-dimensional configurations with a hexagonal lattice. We demonstrate that the inclusion of more long-range neighbors is necessary to accurately describe the electronic band levels throughout the Brillouin zone. In the monolayer, the maximal deviation of the 5NN in-plane interaction MOTB result from the 2NN in-plane hopping integral MOTB computation is approximately ≈100 meV for the conduction band at the Г – K direction and approximately ≈80 meV for the valence band at the Г point, a trend that is also observed in the AA bilayer. This consistency strongly suggests that the 5NN intralayer couplings are the primary contributors to this transition in these geometries. However, in both systems, the introduction of the 5NN in-plane couplings significantly improves the overall valence band (conduction band) picture with first-principles interpretations (the conduction band minimum shifts from the Γ – K in the 2NN description to the Γ – M region in the 5NN model), and their energies occur at the Γ – K ( Γ – M ) region. Our models accurately reproduce the most effective self-energy computations based on the GW approach, showing an indirect energy band gap of approximately E g ≈3.00 (1.44) eV for the monolayer (AA bilayer). To our knowledge, this study is the first to explore the influence of NNN interactions on the energy dispersion bands of buk-P modeled by several valence-state 5NN intralayer and 2NN interlayer hopping MOTB approximations. These insights are important for exploring the potential of buk-P in advanced applications, particularly in fields such as quantum computing and optoelectronic devices, where precise control over electronic states is essential.