The theory of planar ballistic SNS junctions

TL;DR

The paper develops a theory of planar ballistic SNS junctions with equal Fermi velocities, explicitly incorporating phase gradients in the superconducting leads via the Josephson phase $\theta=\theta_0+\theta_s$. By solving the Bogoliubov–de Gennes equations in the steplike pairing potential framework, it derives an analytic current-phase relation at $T=0$ for any normal-layer thickness $L$ and resolves a longstanding charge-conservation issue. At high temperatures, where the Andreev level spacing is small, the total current becomes temperature independent and scales as $\sim1/L^4$, contrasting with earlier exponential predictions; this plateau results from cancellations between vacuum and excitation currents. The analysis extends to planar junctions with non-equal Fermi velocities and to non-planar geometries, providing a coherent explanation for backward-skewed CPRs observed in experiments on InAs-based junctions and highlighting broader implications for related weak-link systems.

Abstract

The paper presents the theory of planar ballistic SNS junctions with equal Fermi velocities and effective masses in all layers. The theory takes into account phase gradients in superconducting layers commonly ignored in the past. At $T=0$ the current-phase relation was derived for any thickness $L$ of the normal layer in the model of the steplike pairing potential model analytically. The obtained current-phase relation is essentially different from that in theory neglecting phase gradients, especially in the limit $L\to 0$ (short junction). The analysis resolves the problem with the charge conservation law in the steplike pairing potential model. The current-phase relation at temperatures exceeding the energy spacing between Andreev levels but less than the critical temperature was also calculated numerically. The current at these temperatures is temperature independent and decreases with growing $L$ as $1/L^4$. The previous theory predicted the current exponentially decreasing with growing $T$ and $L$. Possible implications of the analysis for planar junctions with non-equal Fermi velocities and for non-planar junctions (narrow normal bridge between two bulk superconductors) are also discussed.

Figures (6)

• Figure 1: The phase variation across the SNS junction. (a) The vacuum current produced by the vacuum phase $\theta_0$. The current is confined to the normal layer. (b) The superposition of the vacuum current and the condensate current determined by the superfluid phase $\theta_s =L\nabla \varphi$. The phase $\theta=\theta_0+\theta_s$ is the Josephson phase. (c) The condensate current produced by the phase gradient $\nabla \varphi$ in the superconducting layers. In all layers the electric current is equal to $env_s$.
• Figure 2: The saw-tooth CPR at zero temperature. Here $J_0={\pi\hbar\over 2mL}en$ ($={ev_f\over L}$ in the 1D case).
• Figure 3: CPRs at $T=0$. (a) $L=0$. The solid line shows the CPR valid for any dimensionality of the junction. The current phase relation in the theory neglecting phase gradients in leads is shown by the dashed line. (b) $L=\tilde{\zeta}/2$. The curves 1, 2, and 3 are the current phase relations for 1D, 2D, and 3D junctions respectively. In the 1D case the length $\tilde{\zeta}$ coincides with $\zeta_0$.
• Figure 4: The CPRs for 31 Andreev levels at $\alpha = 0.1$ (1) and $\alpha = 0.4$ (2). (a) The vacuum current $J_{vA}$ in Andreev bound states. (b) The reduced vacuum current $\tilde{J}_{vA}$ in continuum states [Eq. (\ref{['JvCr']})]. (c) The total current $J$ at high temperatures.
• Figure 5: Josephson critical current vs. number of Andreev levels $s_m$ for $\alpha=0.4$ (discrete plot). The continuous solid line shows the power law $0.005/s_m^3$.