Moiré band structures of twisted phosphorene bilayers

We report on the theoretical electronic spectra of twisted phosphorene bilayers exhibiting moiré patterns, as computed by means of a continuous approximation to the moiré superlattice Hamiltonian. Our model is constructed by interpolating between effective $Γ$-point conduction- and valence-band Hamiltonians for the different stacking configurations approximately realized across the moiré supercell, formulated on symmetry grounds. We predict the realization of three distinct regimes for $Γ$-point electrons and holes at different twist angle ranges: a Hubbard regime for small twist angles $θ< 2^\circ$, where the electronic states form arrays of quantum-dot-like states, one per moiré supercell; a Tomonaga-Luttinger regime at intermediate twist angles $2^\circ < θ\lesssim 10^\circ$, characterized by the appearance of arrays of quasi-1D states; and finally, a ballistic regime at large twist angles $θ\gtrsim 10^\circ$, where the band-edge states are delocalized, with dispersion anisotropies modulated by the twist angle. Our method correctly reproduces recent results based on large-scale ab initio calculations at a much lower computational cost, and with fewer restrictions on the twist angles considered.

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