Enhancement of spin susceptibility and electron-phonon coupling by exchange interaction in Li$_x$ZrNCl at low doping: insight from hybrid functional approaches

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 2-dimensional 2-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 the 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 unbalance in the valley occupation that can be viewed as the effect of a pseudo-magnetic 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 Calandra {\it et al.} Phys. Rev. Lett. {\bf 114}, 077001 (2015).