Interplay of ferroelectricity and interlayer superconductivity in van der Waals bilayers

TL;DR

This work develops a quasiclassical Eilenberger framework for interlayer superconductivity in van der Waals bilayers with sliding ferroelectricity, modeling a tunnel-coupled bilayer with a relative band shift $U_{12}$ and analyzing both spin-singlet and spin-triplet interlayer pairing. It shows that superconductivity nucleation near ferroelectric domain walls can surpass the homogeneous-state $T_c$, with the Tc shift governed by the interlayer tunneling $t$ and band offset $oldsymbol{χ}$, and that domain-wall superconductivity is well captured by a delta-well approach validated by numerics. The in-plane magnetic field reveals contrasting orbital and paramagnetic effects: orbital renormalizes tunneling to suppress singlet pairing but can enhance triplet pairing, while paramagnetic effects drive reentrant phases and anisotropic behavior, potentially yielding nonmonotonic $T_c(H)$ for the spin-triplet case. The results align with recent experiments on vdW bilayers and provide predictions for domain-wall–tuned interlayer superconductivity and magnetic-field phase diagrams dependent on the spin structure of the pairs.

Abstract

We study the distinctive features of the interplay between the interlayer superconductivity and ferroelectricity in van der Waals heterostructures. Corresponding analysis is carried out within the framework of the quasiclassical Eilenberger equations for a tunnel coupled bilayer with inhomogeneous relative shift of the conduction bands between the layers, which describes the net charge transfer in sliding ferroelectrics. It is shown that the critical temperature of the interlayer superconductivity can be significantly enhanced for superconducting nuclei localized in the vicinity of ferroelectric domain walls. We demonstrate that the increase in the tunneling amplitude leads to the decrease (increase) in the difference between the critical temperatures for localized and homogeneous superconducting states for the spin-singlet (spin-triplet) interlayer superconductivity. We also perform an extensive analysis of the effects of the in-plane magnetic field on the interlayer superconductivity. It is shown that the orbital effect can result in the suppression of the spin-singlet interlayer superconductivity and to the enhancement of the spin-triplet one. We find that possible manifestations of the paramagnetic effect include the suppression of the interlayer superconductivity by rather weak Zeeman fields, the two-fold anisotropy of the critical magnetic field for the spin-triplet states as well as the appearance of the reentrant superconducting phases. It is shown that the joint influence of the orbital and paramagnetic mechanisms on the spin-triplet interlayer superconductivity can even lead to a nonmonotonic behavior of the superconducting critical temperature as a function of the external magnetic field. The obtained results are discussed in the context of recent experimental data on van der Waals structures with coexisting superconductivity and sliding ferroelectricity.

Figures (9)

• Figure 1: Schematic picture of the considered tunnel coupled bilayer system with the interlayer electronic attraction. Dashed lines show the position of the ferroelectric domain wall, which separates the region with homogeneous polarization $\mathbf{P}$ directed perpendicular to the plane of the layers. Upper plots: schematic pictures of the Fermi surfaces (FS) far from the domain wall.
• Figure 2: Typical dependencies of the critical temperature of the interlayer superconductivity $T_c$ for homogeneous spin-singlet (a) and spin-triplet (b) states vs. the relative band shift $U_1 - U_2 = -2\chi$. Here $T_{c0}$ is the critical temperature for $t = 0$ and $\chi = 0$.
• Figure 3: Phase diagrams for different tunneling amplitudes in the case of a spin-singlet order parameter. Green solid and blue dashed lines correspond to domain wall superconductivity, but were obtained in different ways: by direct numerical solution of the Eqs. (\ref{['self_cons_domain_wall_main_text']}) or by numerical solution of Eq. (\ref{['Tc_localized_equation']}), obtained in delta well approximation, respectively. Red dashed-dotted lines are critical lines of the transition into uniform superconducting state. Panels (a), (b), (c) and (d) correspond to $t=0$, $t = 0.5T_{c0}$, $t = T_{c0}$ and $t = 2T_{c0}$ respectively.
• Figure 4: Phase diagrams for different tunneling amplitudes in the case of a spin-triplet order parameter. Green solid and blue dashed lines correspond to domain wall superconductivity, but were obtained in different ways: by direct numerical solution of the Eq. (\ref{['self_cons_domain_wall_triplet_main_text']}) or by numerical solution of Eq. (\ref{['Tc_localized_triplet_equation']}), obtained in delta well approximation. Red dash-dotted lines are critical lines of the transition into uniform superconducting state. Panels (a), (b) correspond to $t = 0.5T_{c0}$ and $t = 0.8T_{c0}$, respectively.
• Figure 5: Typical dependencies of the critical temperature $T_c$ on the external magnetic field $H$ for the spin-singlet case (only the orbital mechanism of the superconductivity suppression is taken into account). The plots are obtained on the basis of Eq. (\ref{['self_cons_orbital_main_text']}) for $t = T_{c0}$, and different band offsets: (a) $\chi/T_{c0} = 1.25$, 1.3, 1.4, 1.5, (b) $\chi/T_{c0} = 1$, 1.05, 1.1, 1.15, 1.2.