MLM: Multi-Layer Moire -- A Python Package for Generating Commensurate Supercells of Twisted Multilayer Two-Dimensional Materials

TL;DR

MLM is an open-source Python package that efficiently generates commensurate supercells for twisted multilayer 2D materials, enabling advanced computational modeling of moire superlattices with arbitrary layers and twist angles.

Contribution

The package introduces a solve-and-round algorithm that reduces the supercell search to an O(N^2) linear algebra problem, improving efficiency over traditional methods.

Findings

Successfully applied to various materials including graphene, MoS₂, SrTiO₃, and heterostructures.

Scales to supercells with millions of atoms, robust for small twist angles below 1 degree.

Produces simulation-ready structures compatible with VASP and LAMMPS.

Abstract

Moire superlattices formed by stacking atomically thin two-dimensional materials with a relative twist angle have emerged as a versatile platform for engineering quantum electronic, optical, and ferroic properties. Computational modelling of such systems with periodic boundary conditions requires the identification of commensurate supercells in which the moire periodicity is reproduced exactly, or within a prescribed tolerance. While several codes exist for bilayer systems, extension to three or more layers with independently chosen twist angles remains a significant challenge. Here we present MLM (Multi-Layer Moire), an open-source Python package that constructs periodic, PBC-compatible moire supercells for an arbitrary number of twisted layers with any Bravais lattice type. The package employs a solve-and-round algorithm that reduces the coincidence-site search to an $O(N^2)$ linear-algebra problem per twist angle, compared to the O(N^4) brute-force enumeration required by conventional approaches. We demonstrate the package on bilayer graphene, bilayer and trilayer MoS$_2$, bilayer SrTiO$_3$, and a PbTiO$_3$/SrTiO$_3$ oxide heterostructure, producing simulation-ready structure files for both VASP and LAMMPS. The fractional-coordinate atom-selection algorithm scales to supercells containing millions of atoms and is robust across all twist angles including very small angles below 1 degree.