Universal Increase in the Superconducting Critical Temperature of Two-Dimensional Semiconductors at Low Doping by the Electron-Electron Interaction

In two-dimensional multivalley semiconductors, at low doping, even a moderate electron-electron interaction enhances the response to any perturbation inducing a valley polarization. If the valley polarization is due to the electron-phonon coupling, the electron-electron interaction results in an enhancement of the superconducting critical temperature. By performing first-principles calculations beyond density functional theory, we prove that this effect accounts for the unconventional doping dependence of the superconducting transition temperature ($T_c$) and of the magnetic susceptibility measured in ${\mathrm{Li}}_x\mathrm{ZrNCI}$. Finally, we discuss what are the conditions for a maximal $T_c$ enhancement in weakly doped two-dimensional semiconductors.

Figure 1

Electronic structure and density of states of ${\mathrm{Li}}_x\mathrm{ZrNCI}$ for $x=1/18$ (black lines) and $x=2/9$ (red lines). The DOS is in units of states per 6 atoms cell per eV

Figure 2

Top: Spin susceptibility (${\chi }_s$) calculated in the RPA and experimental susceptibility. The experimental data from Ref. [19] have been corrected for an erroneous estimate of the Landau diamagnetic susceptibility ${\chi }_L$ (see Sec. A in Ref. [17] and Supplemental Material in Ref. [19]). Middle: RPA enhancement factor (${\chi }_s/{\chi }_{0s}$). Bottom: Experimental [19] and calculated $T_c$ using different approximations. The superconducting density functional theory (SCDFT) calculation is from Ref. [30].

Figure 3

Phonon dispersion, phonon linewidth (red bars), and Eliashberg function of ${\mathrm{Li}}_{1/18}\mathrm{ZrNCI}$. The Eliashberg function due to intervalley coupling only is shown as the filled region in the right-hand panel (see Ref. [17] for other doping).

Figure 4

(a) Phonon dispersion of the 59 meV modes at $\mathbf{K}$ in a $\sqrt{3}\times \sqrt{3}$ supercell of a ZrNCl bilayer. (b) Effect of the phonon displacements in (a) on the electronic structure of the $\sqrt{3}\times \sqrt{3}$ supercell of a ZrNCl bilayer. The dotted line labels ${\epsilon }_F$. The point $\mathbf{K}$ is folded at $\mathbf{\Gamma }$ in the supercell. The displacements of the Zr and N atoms are $4\times {10}^{-3}$ and $59\times {10}^{-3}\AA$.