垂直堆积二维材料的生长：vdW-BCF解析模型

he mechanism of spontaneous vertical versus in-plane lateral growth of two-dimensional (2D) materials is of great importance to fabricate novel 2D hetero- and homostructures. In this paper, we build a vdW-BCF growth model for vertically stacked 2D materials based on diffusion equations with boundary conditions from thermodynamic equilibrium and mass conservation. General solutions are analytically obtained for bilayer 2D materials (initial Layer1 and subsequent Layer2), taking adsorption and desorption into consideration. As the increase of Layer1 size, the Layer2 growth velocity decreases first and then increases due to the flux adsorption on the Layer1. Two critical sizes of Layer1 may exist for Layer2 to grow. Meanwhile, growth of Layer2 becomes unlimited if Layer2 size reaches a critical value. Moreover, the model can describe the size evolution as the growth process and we demonstrate that initial sizes plays a crucial role in determining the final structure.