Spin susceptibility and electron-phonon coupling of two-dimensional materials by range-separated hybrid density functionals: Case study of Li$_x$ZrNCl

We investigate the capability of density functional theory (DFT) to appropriately describe the spin susceptibility, $\chi_s$, and the intervalley electron-phonon coupling in Li$_x$ZrNCl. At low doping, Li$_x$ZrNCl behaves as a two-dimensional two-valley electron gas, with parabolic bands. In such a system, $\chi_s$ increases with decreasing doping because of the electron-electron interaction. We show that DFT with local functionals (LDA/GGA) is not capable of reproducing this behavior. The use of exact exchange in Hartree-Fock (HF) or in DFT hybrid functionals enhances $\chi_s$. HF, B3LYP, and PBE0 approaches overestimate $\chi_s$, whereas the range-separated HSE06 functional leads to results similar to those obtained in the random phase approximation (RPA) applied to a two-valley two-spin electron gas. Within HF, Li$_x$ZrNCl is even unstable towards a ferromagnetic state for $x<0.16$. The intervalley phonons induce an imbalance in the valley occupation that can be viewed as the effect of a pseudomagnetic field. Thus, similarly to what happens for $\chi_s$, the electron-phonon coupling of intervalley phonons is enhanced by the electron-electron interaction. Only hybrid DFT functionals capture such an enhancement and the HSE06 functional reproduces the RPA results presented in M. Calandra {\it et al.} [Phys. Rev. Lett. {\bf 114}, 077001 (2015)]. These results imply that the description of the susceptibility and electron-phonon coupling with a range-separated hybrid functional would be important also in other two-dimensional weakly doped semiconductors, such as transition-metal dichalcogenides and graphene.