Operation of a superconducting nanowire quantum interference device with mesoscopic leads

A theory describing the operation of a superconducting nanowire quantum interference device (NQUID) is presented. The device consists of a pair of thin-film superconducting leads connected by a pair of topologically parallel ultranarrow superconducting wires. It exhibits intrinsic electrical resistance, due to thermally activated dissipative fluctuations of the superconducting order parameter. Attention is given to the dependence of this resistance on the strength of an externally applied magnetic field aligned perpendicular to the leads, for lead dimensions such that there is essentially complete and uniform penetration of the leads by the magnetic field. This regime, in which at least one of the lead dimensions—length or width—lies between the superconducting coherence and penetration lengths, is referred to as the mesoscopic regime. The magnetic field causes a pronounced oscillation of the device resistance, with a period not dominated by the Aharonov-Bohm effect through the area enclosed by the wires and the film edges but, rather, in terms of the geometry of the leads, in contrast to the well-known Little-Parks resistance of thin-walled superconducting cylinders. A detailed theory, encompassing this phenomenology quantitatively, is developed through extensions, to the setting of parallel superconducting wires, of the Ivanchenko-Zil’berman-Ambegaokar-Halperin theory of intrinsic resistive fluctuations in a current-biased Josephson junction and the Langer-Ambegaokar-McCumber-Halperin theory of intrinsic resistive fluctuations in a superconducting wire. In particular, it is demonstrated that via the resistance of the NQUID, the wires act as a probe of spatial variations in the superconducting order parameter along the perimeter of each lead: in essence, a superconducting phase gradiometer.

Figure 1

(Color online) Schematic depiction of the superconducting phase gradiometer. A current $J$ is passed through the bridges in the presence of a perpendicular magnetic field of strength $B$.

Figure 2

Geometry of the two-wire device, showing the dimensions. The coordinate system used for the right lead (with the origin in the center of the lead) is also shown. The coordinates of the four corners of the right lead, as well as the coordinates of the points at which the two wires are connected to the right lead, are indicated. As shown, we always assume that the wires are attached near the center of the short edges of the leads.

Figure 3

(Color online) (a) Close-up of the two nanowires and the leads. The top (bottom) thick arrow represents the integration contour for determining the phase accumulation ${\theta }_{1,L\leftarrow R}\left({\theta }_{2,L\leftarrow R}\right)$ in the first (second) wire. The dotted arrow in the left (right) lead indicates a possible choice of integration contour for determining the phase accumulation ${\delta }_{2\leftarrow 1,L}\left({\delta }_{2\leftarrow 1,R}\right)$. These contours may be deformed without affecting the values of the various phase accumulations, as long as no vortices are crossed. (b) Sketch of the corresponding superconducting phase at different points along the AB contour when one vortex is located inside the contour.

Figure 4

Current profile in a long superconducting strip, calculated for a finite-length strip by summing the series for $\mathbf{\nabla }\varphi -(2\pi ∕{\Phi }_0)\mathit{A}$ [from Eq. (15)] numerically. Note that there is no vortex in the center of the lead.

Figure 5

(Color online) Phase profile in the leads in the vicinity of the trench, generated by numerically summing the series for $\varphi$ for a finite-length strip. Arrows indicate phases connected by nanowires.

Figure 6

Phase profile on the $x=-L$ (i.e., short) edge of the strip. Numerical summation [Eq. (18) with 100 terms] for $L=2$ and $L=\infty$ $(l=1)$ as well as the linear form from Eq. (20). Note that the linear fit is good near the origin (e.g., for $a\lesssim 0.25l$), and the curves for $L=2$ and $L=\infty$ coincide.

Figure 7

(Color online) Sample 219-4: Experimental data (solid lines) and theoretical fits using the Josephson junction model (dashed lines). (a) Resistance vs temperature curves. (1) Zero magnetic field and low total current. (2) Magnetic field set to maximize the resistance and low total current. (3) Zero magnetic field and $70\mathrm{nA}$ total current. (b) Resistance as a function of magnetic field at various temperatures from $1.2\text{to}2.0\mathrm{K}$ in $0.1\mathrm{K}$ increments. The fitting parameters used were $J_{\mathrm{c}1}=639\mathrm{nA}$, $J_{\mathrm{c}2}=330\mathrm{nA}$, $T_{\mathrm{c}1}=2.98\mathrm{K}$, and $T_{\mathrm{c}2}=2.00\mathrm{K}$, with corresponding coherence lengths ${\xi }_1\left(0\right)=23\mathrm{nm}$ and ${\xi }_2\left(0\right)=30\mathrm{nm}$. Only one set of fitting parameters [derived from curves (1) and (2)] was used to produce all the theoretical curves.

Figure 8

Thermally activated phase slip processes under consideration. (a) Parallel phase slips. (b) Sequential phase slips.

Figure 9

Diagram representing the ground state (central filled circle), the two lowest-energy metastable states (left and right filled circles), and the saddle-point states connecting them (open circles) for the case of magnetoresistance minimum (i.e., $\delta =0,\pi ,\dots$). The saddle-point states at the top of the graph correspond to phase slips in the top wire (i.e., wire 1); those at the bottom correspond to the bottom wire (i.e., wire 2). (Saddle-point states for parallel phase slips are not shown.) For each state ${\kappa }_1$ and ${\kappa }_2$ are listed. For each barrier (represented by an edge and labeled by $a$ through $h$), the table at the right lists the gain in phase across the trench, the gain in Helmholtz free energy, the barrier height (i.e., the gain in Gibbs free energy), and the amount of phase that would effectively be generated at the end of the phase slip event (i.e., upon completion of a closed loop, the amount of phase generated is the sum of the effective phases).

Figure 10

Diagram and corresponding table for the case of magnetoresistance maxima (i.e., $\delta =\pi ∕2,3\pi ∕2,\dots$). See the caption to Fig. 9 for explanation of the diagram.

Figure 11

(Color online) Squared amplitude $u(b∕2)$ of the order parameter at the end of a wire, as a function of its value $u_0=u\left(0\right)$ at the midpoint of the wire, computed using the JacobiSN function [see Eq. (C20b)], for the case $b=16$, $J=0.235$. The black line corresponds to trajectories that do not go through a pole; the gray line corresponds to trajectories that do pass through at least one pole. The intersection of the dashed and black lines represents those trajectories that satisfy the boundary condition $u(\pm b∕2)=1$. [The intersection of the dashed-dotted and black lines represents trajectories that start and stop at the same point, i.e., $u(b∕2)=u_0$].

Figure 12

Current (in units of the critical current) vs end-to-end phase accumulation for superconducting wires of various lengths: $0\xi$ (solid line), $1.88\xi$, $5.96\xi$, $14.4\xi$ (dotted line). The transition from LAMH to Josephson junction behavior is evident from the loss of multivaluedness of the current, as the wire length is reduced.

Figure 13

(Color online) Sample 930-1: Resistance vs temperature curves. Experimental data (full lines) and theoretical fits using the LAMH-type model in the intermediate regime (dashed lines). Theoretical curves terminate when the short-wire regime is reached, i.e., $5\xi \left(T\right)\sim b$. (1) Zero magnetic field and low total current. (2) Magnetic field set to maximize the magnetoresistance and low total current. (3) Zero magnetic field and $80\mathrm{nA}$ total current. (4) Magnetic field set to maximize the resistance and $80\mathrm{nA}$ total current. The fit on the left was optimized numerically, and the one on the right was obtained by hand, showing that a more realistic value of ${\xi }_{01}$ remains reasonably consistent.

Figure 14

Effective single junction critical current for a multijunction array, as a function of $\delta$. The critical current has been rescaled so that $J_c(\delta =0)=1$. Note the similarity with a multislit interference pattern.

Figure 15

“Mechanical potential” $U[u=f^2]$ at an intermediate value of the dimensionless current, plotted as a function of amplitude squared to make comparison with Fig. 11 more convenient.

Figure 16

Squared amplitude $u$ of the order parameter as a function of position along the wire for the two types of solution: metastable and saddle point.