Insight into the formation mechanism and driving forces of one-dimensional flat bands in twisted bilayer black phosphorene

Using first-principles calculations, this study explores the formation mechanism of one-dimensional (1D) flat bands in twisted bilayer black phosphorene (BP) and investigates the underlying driving forces behind this phenomenon. We show that 1D flat bands arise from the unique distribution of the four high-symmetry stacking configurations of untwisted BP in the twisted BP and the alignment of their band edges along specific directions, a phenomenon we term the anisotropic interlayer stacking effect. Further analysis reveals that interlayer quasi-bonding (QB) interactions and quantum confinement effects drive band edge alignment, with interlayer QB interactions among second-neighbor atomic pairs playing a key role in certain high-symmetry configurations. Moreover, multi-two-orbital interactions are essential for accurately capturing the complexity of interlayer QB interactions, rather than relying on simple two-orbital interactions. These findings enhance our understanding of 1D flat band physics and interlayer interactions.