An $O(N)$ $Ab~initio$ Calculation Scheme for Large-Scale Moiré Structures

Abstract

We present a two-step method and the implemented code specifically tailored for band structure calculation of the small-angle moiré-pattern materials. These materials contain tens of thousands of atoms in a unit cell and the existing ab initio codes usually fail to do this within a finite amount of time without simplifying approximations. Our method can dramatically reduce the computational cost. Firstly, the self-consistent field calculation for ground state is performed with O(N ) Krylov subspace method implemented in OpenMX, which is an ab initio package based on localized pseudoatomic orbitals (PAOs) and pseudo-potential method. Secondly, the crystal momentum dependent Bloch Hamiltonian is constructed from the Hamiltonian matrix elements in PAO basis set obtained in the first step and only selected eigenvalues of it is solved with Lanczos and shift-invert techniques. The computational cost increases linearly with the number of atoms per unit cell instead of cubically (O(N 3 )) in other generally used ab initio packages. By systematically tuning two key parameters, the cutoff radius for electron hopping interaction and the dimension of Krylov subspace, we obtained the band structures for both rigid and corrugated twisted bilayer graphene structures at θ = 2.00 • , θ = 1.61 • , θ = 1.41 • and θ = 1.08 • with improved accuracy in affordable costs both in time and computer resources. They are in good agreement with those from tight binding models, continuum models, and plane-wave pseudo-potential based ab initio calculations, as well as the experimental observations. Our method has taken the advantages of O(N ) method, local atomic orbital basis method, and Lanczos technique. This will play a crucial role in other twisted two-dimensional materials with much more complex band structures than graphene, when the continuum model or effective tight-binding model is hard to construct.

An effective curved space-time geometric theory of generic twist angle graphene with application to a rotating bilayer configuration

“…Intuitively, we assume there is a tight binding picture and the lattice is a rigid structure. Although previous formulations have suggested that lattice relaxation helps to isolate the flat band from other higher bands [22,26,44,45], for simplicity we do not consider lattice relaxation here. Note, since lattice relaxation can significantly modulate electron-phonon interaction and introduce strain in TBG, there may be nonvanishing torsion under our geometric theory.…”