The theory of planar ballistic SNS junctions at $T=0$

Abstract

The Letter presents the theory of planar ballistic SNS junctions at $T=0$ for any normal layer thickness $L$ taking into account phase gradients in superconducting leads. The current-phase relation was derived in the model of the steplike pairing potential analytically and is exact in the limit of large ratio of the Fermi energy to the superconducting gap. At small $L$ (short junction) the obtained current-phase relation is essentially different from that in the previous theory neglecting phase gradients. It was confirmed by recent numerical calculations and was observed in the experiment on short InAs nanowire Josephson junctions. The analysis resolves the problem with the charge conservation law in the steplike pairing potential model.

Figures (3)

• 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: Saw-tooth CPR ($T=0$, $L\to \infty$).
• Figure 3: Current-phase relations at $T=0$. (a) $L=0$. The solid line shows the current phase relation 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 $\zeta_0$.