Simple derivation of moiré-scale continuous models for twisted bilayer graphene

Abstract

We provide a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. Our derivation is based on a variational approximation of the TBG Kohn-Sham Hamiltonian and asymptotic limit techniques. In contrast with other approaches, it does not require the introduction of an intermediate tight-binding model. The so-obtained model is similar to that of the Bistritzer-MacDonald (BM) model but contains additional terms. Its parameters can be easily computed from Kohn-Sham calculations on single-layer graphene and untwisted bilayer graphene with different stackings. It allows one in particular to estimate the parameters ${w}_{\mathrm{AA}}$ and ${w}_{\mathrm{AB}}$ of the BM model from first principles. The resulting numerical values, namely ${w}_{\mathrm{AA}}={w}_{\mathrm{AB}}\ensuremath{\simeq}126\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$ for the experimental interlayer mean distance are in good agreement with the empirical values ${w}_{\mathrm{AA}}={w}_{\mathrm{AB}}=110\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$ obtained by fitting to experimental data. We also show that if the BM parameters are set to ${w}_{\mathrm{AA}}={w}_{\mathrm{AB}}\ensuremath{\simeq}126\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$, the BM model is an accurate approximation of our reduced model.