Ab initio tight-binding Models for Mono- and Bilayer Hexagonal Boron Nitride (h-BN)

Hexagonal boron nitride ($\it h$-BN) exhibits dominant $\pi$-bands near the Fermi level, similar to graphene. However, unlike graphene, where tight-binding (TB) models accurately reproduce band edges near the $K$ and $K^{\prime}$ points in the Brillouin zone, a wider bandgap in $\it h$-BN necessitates capturing the band edges at both the $K$ and $M$ points for precise bandgap calculations. We present effective TB models derived from $\it ab initio$ calculations using maximally localized Wannier functions (MLWFs) centered on boron and nitrogen sites. These models consider hopping terms of up to four distant neighbors and achieve excellent agreement with $\it ab initio$ results near the $K$ and $M$ points. Furthermore, we compare the band structures from our simplified models with those obtained from $\it ab initio$ calculations and the full tight-binding model to assess their accuracy. To account for the effects of strains, we introduce fitting parametrizations that relate the hopping parameters of the effective TB model to the lattice constant and interlayer distance. Additionally, we utilize the two-center approximation to calculate the interlayer hopping energies based on the relative distances between sublattices to generalize the interlayer hopping parameters across different stacking configurations. We demonstrate the effectiveness of this method by comparing the electronic structure of zero-twist and twisted $\it h$-BN systems with $\it ab initio$ calculations.