Superconductor to normal-metal transition in finite-length nanowires: Phenomenological model

In this paper we discuss the interplay of quantum fluctuations and dissipation in uniform superconducting nanowires. We consider a phenomenological model with superconducting and normal components and a finite equilibration rate between these two fluids. We find that phase-slip dipoles proliferate in the wire and decouple the two fluids within its bulk. This implies that the normal fluid only couples to the superconductor fluid through the leads at the edges of the wire, and the local dissipation is unimportant. Therefore, while long wires have a superconductor-metal transition tuned by local properties of the superconducting fluid, short wires have a transition when the total resistance is $R_{\text{tot}}=R_Q=h/4e^2$.

Figure 1

(Color online) A two-fluid model of a superconducting nanowire with a normal part, depicted as a separate region. In order for electrons to change from normal to superconducting, they need to pass through the conversion layer, which has a finite conductivity. Proliferated phase-slip dipoles inhibit two-fluid relaxation and make the conversion layer insulating, which renders the normal part as an effective single shunt resistor.

Figure 2

(Color online) (a) Single two-fluid grain (pink ellipse) between two leads: dipoles produce two opposing voltage spikes on the two junctions, which oppose super-to-normal conversion currents leaving the superconducting part of the grain (bottom circle in ellipse) and entering the normal part (top circle). When dipoles proliferate, the superfluid-normal conversion resistance (and time) effectively diverges, $r\to \infty$, and the normal and superfluid are completely decoupled. (b) We begin our study with a chain of two fluid grains, which is a discretized version of the nanowire in Fig. 1. Note that $1/r={\rm Y}{a}_x$ and $R=a_x\rho$.

Figure 3

(Color online) To describe a continuous wire, we take the limit $a_x\to 0$ while keeping the coherence length and size of phase slip, $\xi$, fixed. This implies that phase slips are spread over $\xi /a_x$ junctions and can form dipoles with separation $x<\xi$. The voltage signs symbolize the voltage drop caused by a phase slip.

Figure 4

(Color online) (a) Phase diagram of the two-fluid chain and wire. The black line describes (roughly) the SC-normal boundary of an infinite two-fluid chain [Fig. 2b], with $r<R_Q$; both $K=2\pi \sqrt{E_J/E_C}$ and $R$ are local quantities. When $r>R_Q$, normal-superfluid conversion is cut off, the local dissipation becomes unimportant, and only the horizontal line applies. In finite continuous wires (gray line), the transition takes a dissipative nature when the wire is short and occurs when $R_{\text{tot}}=R_Q$; longer wires have a local crossover—dashed line—tuned by the superconducting stiffness $\sim K$. (b) Phase diagram for MoGe nanowires in Refs. 14, 15, 16. The $y$ axis, $L/R_{\text{tot}}\propto A/{\rho }_{\text{MoGe}}$, is proportional to the surface area, which is proportional to $K$. The $x$ axis is the length. The blue dots are insulating samples, whereas the red dots are superconducting. The horizontal dashed line marks the long-wire crossover, while the diagonal black line marks the transition line $R_{\text{tot}}=R_Q$.