Gap Anisotropy in Layered Superconductors Due to Rashba and Dresselhaus Spin-Orbit Interactions

TL;DR

We extend the theory of layered superconductors to include Rashba and Dresselhaus SOIs within an intralayer BCS framework, deriving temperature Green’s functions via the Gor'kov formalism and obtaining a gap equation that remains valid for finite T. The SOIs render the superconducting gap Δ complex and anisotropic along kz, with Δ(kz) acquiring cos kz and cos^2 kz dependences, and interlayer tunneling J amplifying this anisotropy while also enabling a SOI-assisted contribution to Tc. The zero- and finite-temperature analyses yield analytical expressions for the gap and Tc, reveal a critical SOI strength beyond which superconductivity vanishes, and show that stronger J generally stabilizes superconductivity against SOIs but cannot overcome sufficiently large SOI. These results provide a theoretical foundation for interpreting anisotropic superconductivity in layered and oxide-interface systems and guide experimental exploration of SOI-engineered superconducting states.

Abstract

The theory of layered superconductors is extended in the presence of Rashba and Dresselhaus spin-orbit interactions (SOIs). Using the intralayer BCS-like pairing interaction and employing the Gor'kov formalism, we obtain analytical expressions for the temperature Green's functions and determine the gap function $Δ$ which becomes complex in the presence of SOIs. In the absence of SOIs, $Δ$ is isotropic at both zero and finite temperatures, but it becomes anisotropic even in the presence of a single SOI. This anisotropy is related to the extra $\cos{k_z}$ factors in which the $k_z$ momentum along the $z$ direction contributes to the magnitude of the gap function. It is also found that SOIs suppress $Δ$ at both zero and finite temperatures, and for certain critical values of SOIs and beyond $Δ$ vanishes. Analytical expressions for the critical values of SOIs at zero temperature are obtained. Additionally, how the BCS equation for layered superconductors changes in the presence of SOIs is determined.

Figures (6)

• Figure 1: Variation of the magnitude of the gap function determined by Eq. (\ref{['eq:anistropic_delta']}) with respect to $k_z$ momentum along the $z$ direction. Calculations for all curves are performed with $\tilde{\mu} = 0.5$.
• Figure 2: Variation of the magnitude of the gap function with the strength of spin-orbit interaction at zero temperature. Calculations for all curves are performed with $\tilde{\mu} = 0.5$.
• Figure 3: Variation of the magnitude of the gap function with the strength of spin-orbit interaction at temperatures $T \ll T_C$. Calculations for all curves are performed with $\tilde{J}=0.4, \tilde{\beta}=0.3$, and $\tilde{\mu} = 0.5$.
• Figure 4: The contribution of the $I_1$ integral, determined using Eq. (\ref{['eq:i_1']}), to the gap equation. The calculations are performed with the values $\tilde{\mu} = 0.5$, $\tilde{\omega}_D = 0.05$, $\tilde{J} = 0.4$, and $\tilde{\beta} = 0.8$.
• Figure 5: Fermi surface of layered superconductors in the presence of Rashba and Dresselhaus SOIs. The dimensionless momenta $\tilde{k}_x$ and $\tilde{k}_y$ are calculated using the formulas $\tilde{k}_{x} = k_{x}/\sqrt{2m\mu}$ and $\tilde{k}_{y} = k_{y}/\sqrt{2m\mu}$.