Position- and momentum-dependent interlayer couplings in two-dimensional semiconductor moiré superlattices

In recent years, various novel phenomena have been observed in two-dimensional semiconductor moiré systems, including the moiré excitons, strongly-correlated electronic states and vertical ferroelectricity. To gain insights into the underlying physical mechanism of these interesting phenomena, it is essential to understand the interlayer coupling form of the electrons in moiré systems. In this paper, the position- and momentum-dependent interlayer coupling effects in two-dimensional semiconductor moiré superlattices are investigated. Starting from the monolayer Bloch basis, the interlayer coupling between two Bloch states are treated as a perturbation, and the coupling matrix elements in commensurate and incommensurate bilayer structures are obtained, which are found to depend on the momentum and the interlayer translation between the two layers. Under the effect of an external potential, the Bloch states are localized into wavepackets, and their interlayer couplings are found to depend on the wavepacket width as well as the interlayer translation at the wavepacket center position. Meanwhile the momentum-dependence results in very different interlayer coupling forms for the ground-state S-type and the excited-state P±-type wavepackets. It is shown that at a position where the interlayer coupling between two S-type wavepackets vanishes, the coupling between an S- and a P+-type (or between an S- and a P--type) wavepackets reaches maximum strength. This can be used to manipulate the valley-selective interlayer transport of the ground-state wavepackets through external electric and optical fields. Besides, the vertical ferroelectricity recently discovered in bilayer systems can be attributed to the charge redistribution induced by the coupling between conduction and valence bands in different layers. Using the obtained interlayer coupling form combined with a simplified tight-binding model for the monolayer, the vertical electric dipole density can be calculated whose form and order of magnitude agree with the experimental observation.