Stacking of charge-density waves in 2H-NbSe$_2$ bilayers
We employ ab-initio electronic structure calculations to investigate the charge-density waves and periodic lattice distortions in bilayer 2H-NbSe$_2$. We demonstrate that the vertical stacking can give rise to a variety of patterns that may lower the symmetry of the charge-density waves exhibited separately by the two composing 1H-NbSe$_2$ monolayers. The general tendency to a spontaneous symmetry breaking observed in the ground state and the first excited states is shown to originate from a non-negligible inter-layer coupling. Simulated images for scanning tunnelling microscopy (STM) as well as diffraction/scattering patterns show signatures of the different stacking orders. This may not only be useful to reinterpret past experiments on surfaces and thin films, but may also be exploited to devise ad-hoc experiments for the investigation of the stacking order in 2H-NbSe$_2$. We anticipate that our analysis does not only apply to the 2H-NbSe$_2$ bilayer, but is also relevant for thin films and bulk, whose smallest centro-symmetric component is indeed the bilayer. Finally, our results illustrate clearly that the vertical stacking is not only important for 1T structures, as exemplified by the metal-to-insulator transition observed in 1T-TaS$_2$, but seems to be a general feature of metallic layered transition metal dichalcogenides as well.