Upper critical field in few-layer Ising superconductors

Abstract

The N-layer 2H-stacked transition metal dichalcogenides 2H-NbSe2 and 2H-TaS2 are superconductors in which each quasi-two-dimensional layer breaks inversion symmetry. In this paper, we show that, as for the individual monolayers, it is crucial to include all pockets at the Fermi surface to accurately determine the upper critical field. Furthermore, we propose an experiment where a distinct scaling with a varying displacement field is predicted for an intralayer spin-singlet order in a bilayer. The scaling of the upper critical field with external tuning parameters can thus be used to extract information about the spin-symmetry of the superconducting order. We also explore the possibility of a mixed-parity spin-singlet and -triplet order parameter. In that case, we predict that the experimentally observable scaling would remain that of the spin-singlet component.

Figures (7)

• Figure 1: a) Top and bottom layers of bilayer 2H-NbSe$_2$ have 3 pockets centered around the $\Gamma$-, $K$-, and $K'$-points, which are coupled via the interlayer hopping $t_\perp^j$, $j=\Gamma,K$. The Ising SOC changes sign between the layers $\bm{g}_l \propto l \hat{z}$ , with top and bottom layer $l = \pm 1$ respectively, and its nodal points are gaped out for the degenerate bands in the bilayer. The spin-polarized spin-up (red) and -down (blue) bands are fully degenerate (purple) in the bilayer. b) When inversion symmetry is broken via an interlayer bias potential $\delta \mu$, Eq. \ref{['eq:dmuDef']}, the bands split and the SOC nodes reappear. c) Normal state band structure of a $\Gamma$-pocket around the FS (at $E=0$) close by a nodal line $\lambda_{\bm{k}}^\Gamma (\pi/6) =0$ (dashed lines in (a) & (b)). When inversion symmetry is present, $\delta \mu=0$, the bands remain degenerate for spin-up (dashed lines) and spin-down (solid lines) even when $\theta \neq \pi/6$. The layer polarization $P_{+}^{\zeta l}$, see appendix \ref{['sec:AppEigs']}, is also mixed at $\theta = \pi/6$. In contrast, for $\delta \mu>0$ the bands are only degenerate at the nodal point and have a clear layer-character close to the node.
• Figure 2: In-plane upper critical field for mono- and bi-layer NbSe$_2$Xi2016 and TaS$_2$DeLaBarrera2018Yang2018. For bilayers a fit for the interlayer bias potential $\delta \mu$ is shown. For multi-layers a lower bound of $H_{c2}(T)$ is Eq. \ref{['eq:2LKG']}, when it is suppressed by hopping between the layers at $\delta \mu =0$. The upper bound is achieved when the layers are treated as uncoupled monolayers at the given $T_c$ ($\delta \mu \rightarrow \infty$).
• Figure 3: Prediction for bilayer 2H-NbSe$_2$ with increasing $\delta \mu$ evaluated numerically, for $\lambda_0^\Gamma=17$meV, $t_\perp =15$meV, & $T_c = 5$K, for only Zeeman coupling of the magnetic field to an intralayer singlet pairing. The effective model, section \ref{['sec:dmu']}, shows that including all 3 pockets results in a scaling roughly as $\sqrt{\delta \mu / t_\perp}$ and while for only the $K$-pockets the scaling is $\delta \mu / t_\perp$.
• Figure 4: In a one-pocket model of monolayer 1H-NbSe$_2$ with a mixed parity $s+f$-wave order, the critical field increases along with the proportion of triplet component ($|\Delta_{s,0}| + |\Delta_{t,0}| = |\Delta_{0}|$ is kept constant). Compared to the calculation (dashed line) is that of a purely singlet order (solid line) with the same-size singlet component $H_{c2}/H_p \propto \sqrt{\lambda_0^\Gamma / \Delta_{s,0}}$, where $T_c$ is equal for both curves. The overlap of the curves shows how the increase comes entirely from the decreased singlet component and that all remaining terms, arising from the triplet component, are negligible. In the inset, the susceptibility difference at an angle $\theta$ around the $\Gamma$-pocket (SOC node at $\theta=\pi/6$) is shown with and without the triplet component.
• Figure 5: a) For a 2H-NbSe$_2$ bilayer with inversion symmetry ($\delta \mu =0$) two intralayer mixed parity orders are shown as singlet proportion increases. When the phase $\phi=0$, see b), the critical field is dominated by the interlayer hopping and is not affected by the size of either order parameter. When $\phi=\pi$ the interlayer hopping does not suppress the critical field.