Weak Dimensionality Dependence of the Charge Density Wave Transition in NbSe$_2$

Contradictory experiments have been reported about the dimensionality effect on the charge density wave transition in 2H NbSe$_2$. While scanning tunnelling experiments on single layers grown by molecular beam epitaxy measure a charge density wave transition temperature in the monolayer similar to the bulk, around 33 K, Raman experiments on exfoliated samples observe a large enhancement of the transition temperature up to 145 K. By calculating from first principles the charge density wave temperature, we determine that the intrinsic charge density wave is barely affected by dimensionality as suggested by the scanning tunnelling experiments. The transition temperature is estimated by calculating the temperature dependence of the phonon spectra within a non-perturbative approach to deal with anharmonicity and determining at which temperature the phonon energy of the mode driving the instability vanishes. The obtained transition temperature in the bulk is around 59 K, in rather good agreement with experiments, and it is just slightly increased in the single-layer limit to 73 K, showing the weak dependence of the transition on dimensionality. Our results demonstrate that the charge density wave melts due to the ionic contribution to the entropy, not the electronic one, and underline that environmental factors, such as sample preparation or the substrate, have a large impact on the transition temperatures.