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

Here, we propose a deterministic algorithm, that is capable of constructing a common supercell between two similar crystalline surfaces such as square–square or hexagonal–hexagonal lattices without scanning all possible cases. Using the complex plane, we define the two-dimensional (2D) lattice as the 2D complex vector. Then, the relationship between two surfaces becomes the eigenvector–eigenvalue relation where an operator corresponds to a transformation matrix. We show 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 Nmax is the maximum index of repetition matrix elements. This process is much faster than the conventional brute force approach (O(N_max^4)). By implementing our method in Python code, we successfully generate experimental 2D heterostructures and their moiré patterns and additionally find new moiré patterns that have not yet been reported. According to our density functional theory (DFT) calculations, some of the new moiré patterns are expected to be as stable as experimentally-observed moire patterns. Taken together, we believe that our method can be widely applied as a useful tool for designing new heterostructures with interesting properties.