Title: Low-energy nine-layer rhombohedral stacking of transition metal dichalcogenides

Abstract: Transition-metal dichalcogenides (TMDs) show unique physical, optical, and electronic properties. The known phases of TMDs are 2H and 3R in bulk form, 1T and associated reconstructions, and 1H in monolayer form. This paper reports a hypothetical phase, 9R, that may exist in TMDs (Mo, W)(S, Se, Te)$_2$, meeting both dynamical stability and elastic stability criteria. 9R phase has the same space group as 3R, $i.e.$ rhombohedral $R3m$ without inversion symmetry, and has 9 layers in a conventional unit cell. We find that 9R has an energy within 1 meV per formula unit of 3R and can be energetically favored by a particular strain condition. We further calculate the electronic, elastic, piezoelectric, Raman, and second-harmonic generation signatures of 9R TMDs and compare them with the corresponding 2H and 3R phases. 9R has similar properties to 3R but shows distinctive Raman peaks in the low-frequency regime, improved piezoelectric properties, and unique band splitting arising from layer coupling at the conduction band minimum. These distinct properties make 9R an attractive candidate for applications in piezotronics and valleytronics.