An efficient numerical method for constructing heterojunctions between two crystal lattices having the same in-plane lattice parameter ratio

Important physical properties of two-dimensional (2D) electronic devices are mainly determined by interface geometries in heterojunctions. Here we propose a new numerical method for efficiently constructing heterojunctions between two different crystal lattices having the same in-plane lattice parameter ratio such as a hexagonal graphene-hexagonal MoS₂ heterostructure. The bases of two 2D lattices 1, 2 ((α_mx, α_my), m=1, 2, α=a, b) can be regarded as the 2D complex vector l=(z_a z_b) where (z_α=α_mx+iα_my) in the complex plane. Then, the transformation matrix between two lattices and the complex lattice bases satisfy the eigenvalue problem. We find that a ratio of two lattice matrices c=|a_1x/a_2x| should be the same as the ratio of norms (h²+Dk²) of complex quadratic integers p=h+k√(-D) (h, k are integer, D>0) when two lattices form a commensurate heterostructure. Consequently, using the ratio table, the smallest commensurate heterojunction can be efficiently constructed by finding a quadratic integer without scanning all possible supercells. This is significantly more time-efficient (~O(log n) where n is the maximum value of the transformation matrix elements) than the currently widely used algorithm for construing heterojunctions (~O(n⁴)).