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In this paper, we propose an extended plane wave framework to make theelectronic structure calculations of the twisted bilayer 2D material systemspractically feasible. Based on the foundation in [Y. Zhou, H. Chen, A. Zhou, J.Comput. Phys. 384, 99 (2019)], following extensions take place: (1) antensor-producted basis set, which adopts PWs in the incommensurate dimensions,and localized basis in the interlayer dimension, (2) a practical application ofa novel cutoff techniques we have recently developed, and (3) a quasi-bandstructure picture under the small twisted angles and weak interlayer couplinglimits. With (1) and (2) now the dimensions of Hamiltonian matrix are reducedby about 2 orders of magnitude compared with the original framework. And (3)enables us to better organize the calculations and understand the results. Fornumerical examples, we study the electronic structures of the linear bilayergraphene lattice system with the magic twisted angle ($\sim 1.05^{\circ}$). Thefamous flat bands have been reproduced with their features in quantitativeagreement with those from experiments and other theoretical calculations.Moreover, the extended framework has much less computational cost compared tothe commensurate cell approximations, and is more extendable compared to thetraditional model hamiltonians and tight binding models. Finally this frameworkcan readily accommodate nonlinear models thus will laid the foundations formore effective yet accurate Density Functional Theory (DFT) calculations.

An extended plane wave framework for the electronic structure calculations of twisted bilayer material systems