Moir\'e band structures of twisted phosphorene bilayers

TL;DR

This paper theoretically investigates the electronic band structures of twisted phosphorene bilayers, revealing three distinct electronic regimes depending on the twist angle, and introduces a computationally efficient model matching ab initio results.

Contribution

The study develops a continuous approximation model for moiré superlattice Hamiltonian in twisted phosphorene bilayers, capturing various electronic regimes with reduced computational cost.

Findings

Identifies three electronic regimes: Hubbard, Tomonaga-Luttinger, and ballistic.

Predicts quantum-dot-like states at small twist angles.

Demonstrates model's agreement with large-scale ab initio calculations.

Abstract

We report on the theoretical electronic spectra of twisted phosphorene bilayers exhibiting moir\'e patterns, as computed by means of a continuous approximation to the moir\'e 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 moir\'e 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 moir\'e 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.