A large scale tight-binding study of marginally twisted trilayer two-dimensional crystals

Twisted bilayer two-dimensional (2D) crystals show the moiré length-scale domains of the stable stacking orders. When a number of layers increases to three, more diverse stable stacking orders are possible, thus allowing for the twisted trilayers to demonstrate much more complex patterns of the domains than the bilayers. For example, marginally twisted trilayer graphenes (mTTG), where the twist angle is smaller than ~0.1˚, can exhibit various domain patterns such as triangle, kagome, and hexagram [1]. On the other hand, multi-layered hexagonal boron nitride (hBN) can be ferroelectric depending on its stacking order. Hence, marginally twisted trilayer hBN (mTTBN) can exhibit the characteristic patterns of the ferroelectric domains, which are shown to be controlled by an external electric field. Here, we studied low-energy electronic structures of the mTTG and mTTBN using tight-binding theory calculations employing a Lanczos algorithm-based eigenproblem solver. We find that several distinct electronic states appear at vertices and boundaries between different stacking domains, which are electronically controllable.