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Relaxation Effects in Twisted Bilayer Graphene: a Multi-Scale Approach

We present a multi-scale density functional theory (DFT) informed molecular dynamics and tight-binding (TB) approach to capture the interdependent atomic and electronic structures of twisted bilayer graphene. We calibrate the flat band magic angle to be at $θ_{\rm M} = 1.08^{\circ}$ by rescaling the interlayer tunneling for different atomic structure relaxation models as a way to resolve the indeterminacy of existing atomic and electronic structure models whose predicted magic angles vary widely between $0.9^\circ \sim 1.3^\circ$. The interatomic force fields are built using input from various stacking and interlayer distance dependent DFT total energies including the exact exchange and random phase approximation (EXX+RPA). We use a Fermi velocity of $\upsilon_{\rm F} \simeq 10^{6}$~m/s for graphene that is enhanced by about $\sim 15\%$ over the local density approximation (LDA) values. Based on this atomic and electronic structure model we obtain high-resolution spectral functions comparable with experimental angle-resolved photoemission spectra (ARPES). Our analysis of the interdependence between the atomic and electronic structures indicates that the intralayer elastic parameters compatible with the DFT-LDA, which are stiffer by $\sim$30\% than widely used reactive empirical bond order force fields, can combine with EXX+RPA interlayer potentials to yield the magic angle at $\sim 1.08^{\circ}$ without further rescaling of the interlayer tunneling.