Electron-hole asymmetry and band gaps of commensurate double moiré 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 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. 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 moire period ratios indicate it is possible to achieve narrow width W ≤ 10meV isolated folded band bundles for tBG angles θ ≤ 1°.