Electronic structures, charge transfer, and charge order in twisted transition metal dichalcogenide bilayers

Moir\'e superlattices of transition metal dichalcogenide (TMD) bilayers have been shown to host correlated electronic states, which arise from the interplay of long wavelength moir\'e potential and long-range Coulomb interaction. Here, we theoretically investigate structural relaxation and single-particle electronic structure of twisted TMD homobilayer. From the large-scale density functional theory calculation and continuum model with layer degrees of freedom, we find that the out-of-plane gating field creates a tunable charge transfer gap at the Dirac point between the first and second moir\'e valence bands. We further study the charge orders at the fractional band fillings. In the flat band limit, we find from Monte Carlo simulations a series of charge-ordered insulating states at various fillings $n=\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},1$. We predict that gating field induces a phase transition between different electron crystals at fixed filling $n=\frac{1}{2}$ or $\frac{2}{3}$. At half-filling $n=1$, the ground state is a Mott insulator with electronically driven ferroelectricity. This work demonstrates that TMD homobilayer provides a powerful platform for the investigation of tunable charge transfer insulator and charge orders.