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

This section elaborates on the derivation of dispersion relations within the multi-orbital tight-binding (MOTB) model, explicitly accounting for neighbor interactions up to fifth order (5NN in-plane) for the monolayer and up to second order (2NN out-of-plane) for the bilayer blue phosphorene. Taking the central B0 atom as a reference, we define the lattice displacement vectors dnj (n = 1–5) for successive coordination shells as first-nearest-neighbor (A1j, j = 1–3), second-nearest-neighbor (B2j, j = 1–6), third-nearest-neighbor (A3j, j = 1–3), fourth-nearest-neighbor (A4j, j = 1–6), and fifth-nearest-neighbors (B5j, j = 1–6). These vectors dnj systematically incorporate all relevant atomic positions within the cutoff radius of the MOTB Hamiltonian, enabling precise calculation of orbital overlap integrals and hopping parameters that govern the electronic band structure. The inclusion of fifth-neighbor couplings particularly enhances the model’s accuracy in reproducing the curvature of dispersion bands observed in the GW calculations, while maintaining computational efficiency through the Slater-Koster (SK) parameterization scheme.