An $O(N)$ $Ab~initio$ Calculation Scheme for Large-Scale Moir\'{e} Structures

TL;DR

This paper introduces an efficient $O(N)$ ab initio calculation scheme for large-scale moiré structures, enabling accurate band structure computations for materials with tens of thousands of atoms in the unit cell.

Contribution

The paper presents a novel two-step $O(N)$ method combining Krylov subspace and shift-invert techniques for large-scale moiré materials, improving computational efficiency and accuracy.

Findings

Successfully computed band structures for twisted bilayer graphene at the first magic angle.

Achieved good agreement with existing models and experimental data.

Reduced computational costs while maintaining accuracy.

Abstract

We present a two-step method specifically tailored for band structure calculation of the small-angle moir\'{e}-pattern materials which contain tens of thousands of atoms in a unit cell. In the first step, the self-consistent field calculation for ground state is performed with $O(N)$ Krylov subspace method implemented in OpenMX. Secondly, the crystal momentum dependent Bloch Hamiltonian and overlap matrix are constructed from the results obtained in the first step and only a small number of eigenvalues near the Fermi energy are solved with shift-invert and Lanczos techniques. 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 the first magic angle ($\theta=1.08^\circ$) and other three larger ones with satisfied accuracy on affordable costs. The band structures are in good agreement with those from tight binding models, continuum models, plane-wave pseudo-potential based $ab~initio$ calculations, and the experimental observations. This efficient two-step method is to play a crucial role in other twisted two-dimensional materials, where the band structures are much more complex than graphene and the effective model is hard to be constructed.