High-$T_c$ superconductivity in weakly electron-doped HfNCl

We investigate the magnetic and superconducting properties in electron-doped ${\mathrm{Li}}_x\mathrm{HfNCl}$. HfNCl is a band insulator that undergoes an insulator to superconductor transition upon doping at $x\approx 0.13$. The persistence of the insulating state for $x<0.13$ is due to an Anderson transition probably related to Li disorder. In the metallic and superconducting phase, ${\mathrm{Li}}_x\mathrm{HfNCl}$ is a prototype two-dimensional two-valley electron gas with parabolic bands. By performing a model random phase approximation approach as well as first-principles range-separated Heyd-Scuseria-Ernzerhof (HSE06) calculations, we find that the spin susceptibility ${\chi }_s$ is strongly enhanced in the low-doping regime by the electron-electron interaction. Furthermore, in the low-doping limit, the exchange interaction renormalizes the intervalley electron-phonon coupling and results in a strong increase of the superconducting critical temperature for $x<0.15$. On the contrary, for $x>0.15, T_c$ is approximately constant, in agreement with experiments. At $x=0.055$ we found that $T_c$ can be as large as 40 K, suggesting that the synthesis of cleaner samples of ${\mathrm{Li}}_x\mathrm{HfNCl}$ could remove the Anderson insulating state competing with superconductivity and generate a high-$T_c$ superconductor.

Figure 1

Electronic structure and density of states (DOS) of ${\mathrm{Li}}_x\mathrm{HfNCl}$ calculated with the PBE functional. The hexagonal structure (H) with $AAA$ stacking is compared to the rhombohedral structure (R) with $ABC$ stacking for the doping $x=0.11$. For the hexagonal structure with $AAA$ stacking, the electronic structure of the doping $x=0.11$ is compared to that of the doping $x=0.31$. The DOS is given in units of states/eV per 2 f.u. of each unit cell.

Figure 2

Spin susceptibility enhancement at different doping with the RPA and HSE06 approximations.

Figure 3

Left: Phonon dispersion along the high symmetry directions of the Brillouin zone for ${\mathrm{Li}}_{0.055}\mathrm{HfNCl}$. Right: The total and intervalley component of the Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ and the electron-phonon coupling $\lambda \left(\omega \right)$.

Figure 4

Superconducting critical temperature $T_c$ as a function of doping calculated with the bare electron-phonon coupling ${\lambda }_{\mathbf{q}\nu }$ as calculated by the PBE functional, and with fully dressed electron-phonon coupling ${\tilde{\lambda }}_{\mathbf{q}\nu }$ using the RPA and HSE06 functional. The experimental data are taken from Ref. [15].

Figure 5

First panel: Average noninteracting electron-phonon coupling $\lambda$ for each doping including the inter- and intravalley components as calculated with the PBE functional. Second panel: ${\omega }_{\log }$ for each doping with inter- and intravalley components, as well as rescaled ${\omega }_{\log }^{\mathrm{RPA}}$. Third panel: Interacting electron-phonon coupling $\tilde{\lambda }$ where the intervalley term is rescaled with RPA electron-electron interaction enhancement. Fourth panel: Superconducting critical temperature $T_c$ as a function of doping, calculated by noninteracting (PBE) and interacting (RPA) electron-phonon coupling, as compared to the experiments from Ref. [15].

Figure 6

Left: Phonon dispersion of ${\mathrm{Li}}_x\mathrm{HfNCl}$ as a function of doping. Right: Total Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ and electron-phonon coupling $\lambda \left(\omega \right)$.