Numerical study of the phase slips in ultrathin doubly connected superconducting cylinders

In the framework of Ginzburg-Landau theory, we numerically investigate the thermally activated phase slips which are responsible for the current dissipation in ultrathin doubly connected superconducting cylinders in the presence of transport current and external magnetic field along the cylinder axis. A hollow cylinder of radius $R$ is mathematically transformed into a two-dimensional (2D) superconducting strip of width $w=2\pi R$ with periodic boundary condition. The phase slips may occur via free-energy saddle points of two distinct kinds. The saddle points of the first kind exhibit a one-dimensional (1D) variation of order parameter described by the (extended) Langer-Ambegaokar-McCumber-Halperin (LAMH) theory [Phys. Rev. 164, 498 (1967); Phys. Rev. B 1, 1054 (1970)]. The saddle points of the second kind exhibit a 2D variation of order parameter, showing that each phase slip is realized through a thermally activated process of vortex-antivortex pair creation and annihilation. In particular, there exists a critical radius $R_c$ separating the 1D LAMH behavior (below $R_c$) and the 2D vortex-antivortex behavior (above $R_c$). The effects of external magnetic field on these saddle points are presented. The critical radius $R_c$ is found to decrease with increasing field strength, and hence applying a magnetic field may induce a transition in the phase-slip characteristics.

Figure 1

(a) Schematic illustration of an ultrathin DCSC of radius $R$, length $l$, and thickness $d$, with the external magnetic field ${\mathbf{B}}_e$ and transport current of density ${\mathbf{j}}_e$ applied along the cylinder axis, i.e., the $z$ direction. (b) Schematic illustration of the corresponding superconducting strip of width $w=2\pi R$, with $x$ and $y$ coordinates corresponding to $z$ and $R\theta$ in (a), respectively.

Figure 2

(Color online) (a) Dimensionless free energy $F$ evaluated along the PSS MEP from ${\psi }_3$ to ${\psi }_2$ plotted as a function of the arc length $s$ in the $\psi \left(x,y\right)$-function space, for $l=200$ and $w=16$, with the black solid and red dashed lines representing the $\mathbf{\Phi }=0$ and $\mathbf{\Phi }=0.2$ cases, respectively. The ${\psi }_3$ state is taken as the reference point at which $s=0$. The arc length measured along the MEP is normalized by that from ${\psi }_3$ to ${\psi }_2$; hence, $s$ runs from 0 to 1. In each set of the data, the corresponding value at ${\psi }_3$ has been subtracted to let the curve start from zero. (b) Distributions of order-parameter magnitude (left) and current density (right) for the PSS solution at $\mathbf{\Phi }=0$. The grayscale varies from black for $\left|\psi \right|=0$ to white for $\left|\psi \right|=1$. Here only the segments of noticeable spatial variations are shown for clear illustration. (c) The same as (b) obtained for $\mathbf{\Phi }=0.2$.

Figure 3

(Color online) (a) Dimensionless free energy $F$ evaluated along the PSVAP MEP from ${\psi }_3$ to ${\psi }_2$ plotted as a function of the arc length $s$ in the $\psi \left(x,y\right)$-function space, with the black solid and red dashed lines representing the $\mathbf{\Phi }=0$ and $\mathbf{\Phi }=0.2$ cases, respectively. Here $F$ and $s$ are defined in the same way as for Fig. 2a. (b) Distributions of order-parameter magnitude (left) and current density (right) for a sequence of states labeled along the curve for $\mathbf{\Phi }=0$ in (a). The grayscale varies from black for $\left|\psi \right|=0$ to white for $\left|\psi \right|=1$. Only the segments of noticeable spatial variations are shown for clear illustration. (c) The same as (b) obtained for $\mathbf{\Phi }=0.2$.

Figure 4

(Color online) The passage for the motion of the vortex relative to the antivortex at $\mathbf{\Phi }=0.2$. The black solid line represents the result obtained in the framework of GL theory and the red dashed line represents the result obtained in the framework of London theory. Here the dotted lines represent an extrapolation of the GL result because the positions of vortex and antivortex become indistinguishable as they are close enough. The squares denote the position of the vortex relative to the antivortex in the PSVAP solution.

Figure 5

(Color online) (a) Contour plot of the phase distribution $\phi \left(x,y\right)$ (in the unit of $2\pi$) in the PSVAP solution for $\mathbf{\Phi }=0.2$ [corresponding to state II in Fig. 3c]. The reference point of $\phi =0$ is taken at $x=-l/2$ and the color scale varies from blue for $\phi =1$ to red for $\phi =2$. (b) The phase variation in the $x$ direction, plotted for the three $y$ levels, marked by the horizontal lines in (a): $y=-6$ (black solid line), $y=0$ (red dashed line), and $y=+6$ (blue dotted line). Only the segment of $-20\le x\le 20$ is shown for clear illustration.

Figure 6

(Color online) Free-energy barrier plotted as a function of $\mathbf{\Phi }$, evaluated for a sample of width $w=8$ and length $l=800$, with the initial winding numbers $n=1$ (black) and $n=20$ (red). The solid lines represent $\Delta F^{\text{PSS}}$ associated with the PSS solution calculated according to the (extended) LAMH theory, while the symbols represent $\Delta F^{\text{PSVAP}}$ associated with the PSVAP solution numerically obtained in the GL description (squares) and the London description (circles) by employing the string method.

Figure 7

(Color online) The critical width $w_c$ plotted as a function of $\mathbf{\Phi }$, evaluated for weak (black squares) and strong (red circles) transport current densities, realized by using the initial winding numbers $n=1$ and $n=20$ for the system of length $l=800$.