Strain-induced stripe phase in charge ordered single layer NbSe$_2$

Abstract: Charge density waves are ubiquitous phenomena in metallic transition metal dichalcogenides. In NbSe$_2$, a triangular $3\times3$ structural modulation is coupled to a charge modulation. Recent experiments reported evidence for a triangular-stripe transition at the surface, due to strain or accidental doping and associated to a $4\times4$ modulation. We employ \textit{ab-initio} calculations to investigate the strain-induced structural instabilities in a pristine single layer and analyse the energy hierarchy of the structural and charge modulations. Our results support the observation of phase separation between triangular and stripe phases in 1H-NbSe$_2$, relating the stripe phase to compressive isotropic strain, favouring the $4\times4$ modulation. The observed wavelength of the charge modulation is also reproduced with good accuracy.