Monolayer-gap modulated topological phases in twisted bilayer graphene

We theoretically investigate concomitant topological properties of twisted bilayer graphene by a continuum model when gaps are opened in its two monolayers. An effective Hamiltonian is derived for the flat bands of a twisted bilayer in the vicinity of its Dirac points, then the topological characteristics of these flat bands can be identified for different valleys. Numerical calculations show that topological phases can be induced and modulated by the gaps in the two single layers. A phase diagram is obtained and is divided into three regions with the Chern number $C=0,\ifmmode\pm\else\textpm\fi{}1$, respectively, separated by two straight lines. These observed phenomena can be well explained using simplified analytical treatments. Moreover, we can distinguish the flat bands into four topologically different states, which will bring applications into electronics and valleytronics.