Flake Size Limits for Growth of Vertically Stacked Two-Dimensional Materials by Analytical Diffusion-Based Kinetic Model

Unveiling the growth mechanism of monolayer and multilayer two-dimensional (2D) materials is crucial for the controllable synthesis of 2D homo- and heterostructures. Here, we construct a general diffusion-based kinetic model to explore the growth of vertically stacked 2D materials. The diffusion equations with high generality are solved analytically with linearized boundary conditions based on mass conservation and thermodynamics. The growth velocity of subsequent layer is demonstrated to strongly rely on the instantaneous sizes of both flakes (denoted as initial Layer1 and subsequent Layer2). Due to the adsorption flux on flakes, the growth velocity of Layer2 decreases first and then rises with the increase in the size of Layer1. In such a case, two limit sizes of Layer1 can be observed. Beyond the lower (upper) limit size of Layer1, the growth of Layer2 is prohibited (allowed). Meanwhile, Layer2 grows without limitation if Layer2 reaches a critical size. Moreover, we explore the large-scale size evolution of 2D flakes within a quasi-static framework. It is shown that whether Layer2 can grow indefinitely is affected by the initial sizes of flakes due to the kinetic competition. This analytical kinetic growth model is expected to develop recipes for controlled vertically stacked 2D materials in desired applications.