Moiré band structures of twisted phosphorene bilayers

Abstract

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 $\Gamma$-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 $\Gamma$-point electrons and holes at different twist angle ranges: a Hubbard regime for small twist angles $\theta <2^{\circ }$, where the electronic states form arrays of quantum-dot-like states, one per moiré supercell; a Tomonaga-Luttinger regime at intermediate twist angles $2^{\circ }<\theta \lesssim 10^{\circ }$, characterized by the appearance of arrays of quasi-1D states; and finally, a ballistic regime at large twist angles $\theta \gtrsim 10^{\circ }$, 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.

Journal accepted version. 20 pages, including 11 figures and 5 appendices

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