Misalignment instability in magic-angle twisted bilayer graphene on hexagonal boron nitride

We study the stability and electronic structure of magic-angle twisted bilayer graphene on hexagonal boron nitride (TBG/BN). Full relaxation has been performed for commensurate supercells heterostructures with different twist angles ($\theta'$) stackings between TBG BN. find that slightly misaligned configuration $\theta' = 0.54^\circ$ AA/AA stacking globally lowest total energy due to constructive interference moir\'{e} interlayer potentials thus greatly enhanced in its $1 \times 1$ supercell. Gaps are opened at Fermi level ($E_F$) small enable strong breaking $C_2$ symmetry atomic TBG. For large $\theta'$ close those supercells, broadened flat bands can still be resolved from spectral functions. The is also identified as a critical angle evolution $\theta'$, which range mini-bands around $E_F$ begins become narrower increasing their gaps dispersive wider. discovered stablest TBG/BN finite about $0.54^\circ$ gapped agree recent experimental observations.