On the solution of the modified Ginzburg-Landau type equation for one-dimensional superconductor in presence of a normal layer

TL;DR

The paper investigates how a thin normal layer modifies the Josephson current–phase relation in a one-dimensional superconductor. It employs a generalized Ginzburg-Landau framework for a SNS-type boundary, derives an exact analytical solution in the thin-layer limit with delta-function potentials, and performs numerical analysis of the current–phase relation using finite-element discretization. Key contributions include an explicit order-parameter profile $R(z)$ and phase offset $Δφ$, a bifurcation-based interpretation of the critical current $j_c$, and evaluation of higher-harmonic content through Fourier coefficients as functions of the layer parameters $g_1$, $g_2$, $g_3$. The work clarifies the crossover between Josephson-like and bulk-flow behavior in 1-D SNS structures and provides analytic and numerical tools for predicting $j_c$ and the current-phase behavior in thin normal layers.

Abstract

We perform an analytical and numerical study of the crossover from the Josephson effect to the bulk superconducting flow for two identical one-dimensional superconductors, co-existing with a layer of normal material. A generalized Ginzburg-Landau (GL) model, proposed by S.J. Chapman, Q. Du and M.D. Gunzburger was used in modeling the whole structure. When the thickness of the normal layer is very small, the introduction of three effective potentials of specified strength leads to an exact analytical solution of the modified stationary GL equation. The resulting current density-phase offset relation is analyzed numerically. We show that the critical Josephson current density corresponds to a bifurcation of the solutions of the nonlinear boundary value problem coupled with the modified GL-equation. The influence of the second term in the Fourier-decomposition of the supercurrent density-phase relation is also investigated. We derive also a simple analytical formula for the critical Josephson current.

Figures (7)

• Figure 1: Some of curves $J(\Delta\varphi)$ for $g_2=0$ and $g_3=0$.
• Figure 2: The solutions for $J = 0.1, g_1=1, g_2=0$ and $g_3=0$.
• Figure 3: The critical current $j_c$ corresponds to a bifurcation point.
• Figure 4: The Influence of the Parameter $g_2$.
• Figure 5: The Influence of the Parameter $g_3$.