Upper critical field and pairing symmetry of Ising superconductors

Abstract

Motivated by the fact that the measured critical field $H_{c2}$ in various transition metal dichalcogenide (TMD) superconductors is poorly understood, we reexamine its scaling behavior with temperature and spin-orbit coupling (SOC). By computing the spin-susceptibility in a multipocket system, we find that segments of the Fermi Surface (FS) at which the SOC has nodal points can have a contribution orders of magnitude larger than the remaining FS, hence setting the $H_{c2}$, assuming the presence of a conventional singlet superconducting order parameter. Nodal lines of an Ising SOC in the Brillouin zone are imposed by symmetry, so they cause such nodal points whenever they intersect an FS pocket, which is indeed the case in monolayer NbSe$_2$ and TaS$_2$, but not in gated MoS$_2$ and WS$_2$. Our analysis reinterprets existing measurements, concluding that a dominant singlet-order parameter on pockets with SOC nodes is consistent with the $H_{c2}(T)$ data for all monolayer Ising superconductors, in contrast to previous contradictory pairing assumptions. Finally, we predict a doping-dependent experimental signature of our theory.

Figures (3)

• Figure 1: a) FS of a monolayer Ising superconductor, of the same type as NbSe$_2$. Two fully spin-polarized bands $\xi_\pm = \xi_k \pm \lambda (\boldsymbol{k})$ are split by the Ising SOC $\lambda (\boldsymbol{k})$. We model $\lambda^\Gamma (\theta) = \lambda^\Gamma_0 \cos \left(3 \theta \right)$ and $\lambda^K (\theta) = \lambda^K_0$, where $\lambda^\Gamma_0 \approx \lambda^K_0 /3$. b) The contribution to the difference in susceptibility between a uniform singlet superconducting state and the normal state. At the nodal points of the SOC $\lambda^\Gamma (\theta)=0$ the superconductor is a conventional singlet order. As the SOC is large everywhere else, $\delta \chi (T)$, and consequently the critical field $H_{c2}$, is determined almost entirely by the 6 nodal points. c) & d) The normal and superconducting state susceptibilities $\chi_N (E), \chi_S (E)$ are shown on the $\Gamma$-pocket along radial lines, passing through a nodal point (c), or between nodal points (d). For spin-split bands in (d), the energy range where the combined susceptibility of the bands is non-zero extends to $\xi_{k_\pm}=\pm\lambda_0$ (shaded). In (c) the plots are individually scaled as max$|\chi_N (E)| \gg$max$|\chi_S (E)|$.
• Figure 2: Calculated upper critical field compared to experimental data for monolayer Ising superconductors with a $\Gamma$-pocket (NbSe$_2$Xi2016 & TaS$_2$DeLaBarrera2018) and without (gated MoS$_2$Lu2015 & WS$_2$Lu2018). The $\Gamma$-pocket introduces a $\sqrt{\lambda_0 /\Delta_0}$-scaling of $H_{c2}$ compared to the otherwise linear $\lambda_0 /\Delta_0$-scaling in the other compounds. The ratio $\lambda_0 /\Delta_0$ is taken at the $K$-pockets.
• Figure 3: Prediction for the upper critical field in 1H-NbSe$_2$Rahn2012SupMat with a varying chemical potential $\mu$, corresponding to hole and electron doping in Eq. \ref{['eq:allPocketsN']}. $H_p$ is scaled with $T_c (\mu)$.