Position- and momentum-dependent interlayer couplings in two-dimensional semiconductor moiré superlattices
Authors: Guo Rui-Ping , Yu Hong-Yi ,   Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China,   State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China
Published: December 2022

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 an insight into the underlying physical mechanisms of these intriguing phenomena, it is essential to understand the interlayer coupling form of the electrons in moiré systems. In this work, 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 form localized 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 <inline-formula><tex-math id="M8">\begin{document}$ \rm{S} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M8.png"/></alternatives></inline-formula>-type and the excited-state <inline-formula><tex-math id="M9">\begin{document}$ {\rm{P}}^{\pm } $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M9.png"/></alternatives></inline-formula>-type wavepackets. It is shown that at a position where the interlayer coupling between two <inline-formula><tex-math id="M10">\begin{document}$ \rm{S} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M10.png"/></alternatives></inline-formula>-type wavepackets vanishes, the coupling between an <inline-formula><tex-math id="M11">\begin{document}$ \rm{S} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M11.png"/></alternatives></inline-formula>-type wavepacket and a <inline-formula><tex-math id="M12">\begin{document}$ {\rm{P}}^{+} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M12.png"/></alternatives></inline-formula>-type wavepacket (or between an <inline-formula><tex-math id="M13">\begin{document}$ \rm{S} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M13.png"/></alternatives></inline-formula>- type wavepacket and a <inline-formula><tex-math id="M14">\begin{document}$ {\rm{P}}^{-} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20222046_M14.png"/></alternatives></inline-formula>-type wavepacket) reaches a 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 accord with the experimental observations.