Bifurcation Diagram and Pattern Formation of Phase Slip Centers in Superconducting Wires Driven with Electric Currents

We provide here new insights into the classical problem of a one-dimensional superconducting wire exposed to an applied electric current using the time-dependent Ginzburg-Landau model. The most striking feature of this system is the well-known appearance of oscillatory solutions exhibiting phase slip centers (PSC’s) where the order parameter vanishes. Retaining temperature and applied current as parameters, we present a simple yet definitive explanation of the mechanism within this nonlinear model that leads to the PSC phenomenon and we establish where in parameter space these oscillatory solutions can be found. One of the most interesting features of the analysis is the evident collision of real eigenvalues of the associated $PT$-symmetric linearization, leading as it does to the emergence of complex elements of the spectrum.

Figure 1

The real parts of the first six eigenvalues of the $PT$-symmetric spectral problem (3).

Figure 2

The phase diagram of the different stable states in the temperature-current plane. The parameter $\Gamma$ is proportional to $T_c-T$. The curves ${\Gamma }_1(I)$, ${\Gamma }_2(I)$, ${\Gamma }_3(I)$ are drawn with solid line, dashed line, and dotted line, respectively. The meaning of the different curves and regimes is explained in detail in the text.

Figure 3

The I–V curve for $\Gamma =6.3$ and $I$ increasing. Notice that, although the periodic state does not exist for such temperature, and thus there is no PSC here, the I–V curve does exhibit a jump discontinuity at $I\approx 12.57$.