A Deterministic Method to Construct a Common Supercell Between Two Similar Crystalline Surfaces

Abstract

Here, a deterministic algorithm is proposed, that is capable of constructing a common supercell between two similar crystalline surfaces without scanning all possible cases. Using the complex plane, the 2D lattice is defined as the 2D complex vector. Then, the relationship between two surfaces becomes the eigenvector-eigenvalue relation where an operator corresponds to a transformation matrix. It is shown that this transformation matrix can be directly determined from the lattice parameters and rotation angle of the two given crystalline surfaces with O(log N_max) time complexity, where N_max is the maximum index of repetition matrix elements. This process is much faster than the conventional brute force approach ( $O\left(N_{\max }^4\right)$ ). By implementing the method in Python code, experimental 2D heterostructures and their moiré patterns and additionally find new moiré patterns that have not yet been reported are successfully generated. According to the density functional theory (DFT) calculations, some of the new moiré patterns are expected to be as stable as experimentally-observed moiré patterns. Taken together, it is believed that the method can be widely applied as a useful tool for designing new heterostructures with interesting properties.

Keywords: common supercell; complex plane; eigenvector–eigenvalue relation; heterostructure; moiré pattern.