Electron-hole asymmetry and band gaps of commensurate double moire patterns in twisted bilayer graphene on hexagonal boron nitride

Spontaneous orbital magnetism observed in twisted bilayer graphene (tBG) on nearly aligned hexagonal boron nitride (BN) substrate builds on top of the electronic structure resulting from combined G/G and G/BN double moiré interfaces. Here we show that tBG/BN commensurate double moiré patterns can be classified into two types, each favoring the narrowing of either the conduction or the valence bands on average, and obtain the evolution of the bands as a function of the interlayer sliding vectors and electric fields. Finite valley Chern numbers ±1 are found in a wide range of parameter space when the moiré bands are isolated through gaps, whereas the local density of states associated with the flatbands are weakly affected by the BN substrate invariably concentrating around the AA-stacked regions of tBG. We illustrate the impact of the BN substrate for a particularly pronounced electron-hole asymmetric band structure by calculating the optical conductivities of twisted bilayer graphene near the magic angle as a function of carrier density. The band structures corresponding to other N-multiple commensurate moiré period ratios indicate it is possible to achieve a narrow width of W≲30-meV isolated folded band bundles for tBG angles θ≲1∘.