Geometry-dependent critical currents in superconducting nanocircuits

In this paper, we calculate the critical currents in thin superconducting strips with sharp right-angle turns, 180${}^{\circ }$ turnarounds, and more complicated geometries, where all the line widths are much smaller than the Pearl length $\Lambda =2{\lambda }^2/d$. We define the critical current as the current that reduces the Gibbs-free-energy barrier to zero. We show that current crowding, which occurs whenever the current rounds a sharp turn, tends to reduce the critical current, but we also show that when the radius of curvature is less than the coherence length, this effect is partially compensated by a radius-of-curvature effect. We propose several patterns with rounded corners to avoid critical-current reduction due to current crowding. These results are relevant to superconducting nanowire single-photon detectors, where they suggest a means of improving the bias conditions and reducing dark counts. These results also have relevance to normal-metal nanocircuits, as these patterns can reduce the electrical resistance, electromigration, and hot spots caused by nonuniform heating.

Figure 1

Current flow in a strip carrying average sheet-current density $K_I$ around a circular arc of inner radius $a$ and outer radius $b=2a$, showing current crowding near the inner radius.

Figure 2

(a) Current crowding around a parabolic bend of radius of curvature ${\rho }_c$, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$ [see Eq. (28)], which has the value $S=0$ along the parabolic boundary, $x=-y^2/2{\rho }_c$. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrow shows the current direction. (b) Current flow generated by a vortex interacting with the parabolic boundary, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the value $S_v=0$ for $(x,y)$ along the boundary. The contours, shown here for $(x_v,y_v)=({\rho }_c,0)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current.

Figure 3

Numerical results for (a) ${\delta }_c/\xi$ and (b) $K_{0c}$ (normalized to ${\phi }_0/e\pi {\mu }_0\xi \Lambda$) as functions of the ratio of the radius of curvature ${\rho }_c$ (see Fig. 2) to the coherence length $\xi$. Expansions in powers of ${\rho }_c/\xi$ are shown as dashed curves for ${\delta }_c$ [see Eq. (34)] and $K_{0c}$ [see Eq. (35)]. Expansions in powers of $\xi /{\rho }_c$ are shown as dot-dashed curves for ${\delta }_c$ [see Eq. (36)] and $K_{0c}$ [see Eq. (37)].

Figure 4

(a) Current crowding around a generalized hyperbolic bend of radius of curvature ${\rho }_c$, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$ [see Eq. (40)], which has the value $S=0$ along the boundary. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrow shows the current direction. (b) Current flow generated by a vortex interacting with the boundary, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the value $S_v=0$ for $(x,y)$ along the boundary. The contours, shown here for $(x_v,y_v)=({\rho }_c,0)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current.

Figure 5

Numerical results for (a) ${\delta }_c/\xi$ and (b) $K_{0c}$ (normalized to ${\phi }_0/e\pi {\mu }_0\xi \Lambda$) as functions of the ratio of the radius of curvature ${\rho }_c$ (see Fig. 4) to the coherence length $\xi$. Expansions in powers of ${\rho }_c/\xi$ are shown as dashed curves for ${\delta }_c$ [see Eq. (46)] and $K_{0c}$ [see Eq. (47)]. Expansions in powers of $\xi /{\rho }_c$ are shown as dot-dashed curves for ${\delta }_c$ [see Eq. (48)] and $K_{0c}$ [see Eq. (49)].

Figure 6

(a) Current flow in a strip carrying current $K_{I}a$ around a 180${}^{\circ }$ turn, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, which has the values $S=0$ along the lines $y=\pm a$ at the outer boundaries and $S=K_{I}a$ on either side of the narrow gap along the line $y=0$ for $x<0$. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrows show the direction of the current. The dashed curve, which corresponds to $S=K_{I}a/2$, separates the current-crowding region close to $(x,y)=(0,0)$ from the current-expanding region outside. The magnitude of $\mathit{K}$ is constant ($\mathit{K}=K_I$) along the dashed curve. (b) Vortex-generated current flow, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the values $S_v=0$ for $(x,y)$ along the boundaries. The contours, shown here for $(x_v,y_v)=(0.1a,0)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current.

Figure 7

Critical-current reduction factor $R$ [Eq. (63)] vs $g/2a=(a-W)/a$ for several values of $\xi /a$.

Figure 8

(a) Current flow in a film carrying current $K_{I}W$ from a straight strip of width $W$ around a rounded 180${}^{\circ }$ turn of length CD $=$ $W^{\prime }=W$ into another straight strip of width $W$, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, which has the values $S=0$ along the outer boundaries of the film and $S=K_{I}W$ along the inner boundary. Here, $r^{\prime }=0.500W$ and $r=0.402W$. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrows show the direction of the current. (b) Same as (a), except that the connection is infinite in length (CD $=$ $\infty$), $r^{\prime }=0.500W$, and $r=0.525W$.

Figure 9

Plots of the current-crowding critical-current reduction factor $R$, $\delta b=b-1$, and the ratios $r^{\prime }/W$ and $r^{\prime }/r$ vs the fill factor $f=1/(1+2r^{\prime }/W)$ for $W^{\prime }=W$ (solid curves) and $W^{\prime }=\infty$ (dashed curves). The vertical dotted line marks the parameters for Fig. 8a ($f=0.5,$ $r^{\prime }/W=0.5$, $R=0.599$, $b-1=0.572$, and $r^{\prime }/r=$1.243) and Fig. 8b ($f=0.5,$ $r^{\prime }/W=0.5$, $R=0.707$, $b-1=0.250$, and $r^{\prime }/r=$0.953). The vertical dot-dashed line marks the smallest $f$ for which solutions can be found for $W^{\prime }=\infty$ and the value of $f=1/3$ for which $r^{\prime }/W=1$, $r/W\to \infty$, and the inner boundary takes the optimally rounded shape for a 180${}^{\circ }$ turnaround so that $R=1$.

Figure 10

Current flow around a 180${}^{\circ }$ turnaround with an optimally rounded inner boundary, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, which has the values $S=0$ along the outer boundaries of the film and $S=K_{I}W$ along the inner boundary. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrows show the direction of the current. Since there is no current crowding along the curve BCC${}^{\prime }$B${}^{\prime }$, the critical current is predicted to be the same as for a long straight strip.

Figure 11

(a) Current flow in a film carrying current $K_{I}W$ around a sharp rectangular 180${}^{\circ }$ turn, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, which has the values $S=0$ along the lines $y=\pm a$ at the outer boundary of the film and $S=K_{I}W$ along the inner boundary [$\left|y\right|=h=a-W$ for $x<0$ or $\left|y\right|\le h$ for $x=0$]. (In this figure, $h=a/2=W$ and $\alpha =\sqrt{3}/2$ $=$ 0.866.) The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrows show the direction of the current. (b) Vortex-generated current flow, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the values $S_v=0$ for $(x,y)$ along the boundaries. The contours, shown here for $(x_v,y_v)=(0.05a,-0.55a)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current.

Figure 12

(a) Current flow in a strip carrying current $K_{I}a$ around a 90${}^{\circ }$ turn, shown by the contour plot of the stream function $S(x,y)=\Im \mathcal{G}(x+iy)$ [see Eq. (102)], which has the values $S=0$ along the inner boundary ($y=a$ for $x\ge a$ and $x=a$ for $y\ge a$) and $S=-K_{I}a$ along the outer boundary ($y=0$ for $x\ge 0$ and $x=0$ for $y\ge 0$). The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrows show the direction of the current. The dashed curve, which corresponds to $S=-K_{I}a/2$, separates the current-crowding region near the inner corner from the current-expanding region near the outer corner. The magnitude of $\mathit{K}$ is constant ($\mathit{K}=K_I$) along the dashed curve. (b) Vortex-generated current flow, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, shown here for $(x_v,y_v)=(0.9a,0.9a)$, which has the values $S_v=0$ for $(x,y)$ along the inner and outer boundaries. The contours correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$.

Figure 13

(a) Current flow calculated in Sec. 8c1 for a film carrying current $K_{I}W$ from a straight strip of width $W$ around a rounded 90${}^{\circ }$ turn into another straight strip of width $W$, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, which has the values $S=0$ along the outer boundaries of the film and $S=K_{I}W$ along the inner boundary. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrows show the direction of the current. Here $r/W=1/2$, and current crowding along the arc BC results in a critical-current reduction factor $R=0.654$. (b) Current flow calculated in Sec. 8c2, where the connection is infinite in length and $r/W\to \infty$, a geometry for which there is no current crowding along the arc ABC and no critical-current reduction.

Figure 14

Plots of the current-crowding critical-current reduction factor $R$ [see Eq. (122)], $r/W$ [see Eq. (118)], $b-1$, and $a-1$ vs $b-1$ for the 90${}^{\circ }$ turn discussed in Sec. 8c1. The vertical dotted line marks the parameters for Fig. 13a ($R=0.654$, $r/W=0.5$, $b=1.383$, and $a=1.053$).

Figure 15

(a) Current flow in a strip carrying total current $K_{I}W$ across the top of a T intersection, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, which has the values $S=K_{I}W$ along the top of the T ($y=W$) and $S=0$ along the underside ($\left|x\right|\ge b$, $y=0$) and the sides ($\left|x\right|=b$, $y\le 0$). The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrow shows the current direction. (b) Vortex-generated current flow, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the values $S_v=0$ along the boundaries. The contours, shown here for $(x_v,y_v)=(-b+0.1W,0.1W)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current. The plots show the behavior when $b=W/2$.

Figure 16

Current flow in a strip carrying currents $K_{I}W$ in opposite directions merging at the top of a T intersection, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$ [see Eq. (135)], which has the values $S=2K_{I}W$ along the right boundary ($x>b$, $y=0$ and $x=b$, $y\le 0$), $S=K_{I}W$ along the top of the T ($y=W$), and $S=0$ along the left boundary ($x<-b$, $y=0$ and $x=-b$, $y\le 0$). The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrows show the current direction. The right dashed curve shows $S=K_{I}W/2$ and the left one shows $S=3K_{I}W/2$. Current crowding occurs only for contours near the sharp corners under the dashed curves. The plot shows the behavior when $b=W/2$.

Figure 17

(a) Current flow in the upper half of a strip of total width $W$ carrying total current $K_{I}W$ into a wide contact strip of total width $a$, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, which has the values $S=0$ along the upper boundary and $S=-K_{I}W/2$ along the $x$ axis. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrow shows the current direction. Current crowding leading to $K>K_I$ occurs for all contours above the dashed contour for $S=-K_{I}W/4$. (b) Vortex-generated current flow, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the values $S_v=0$ along the boundaries. The contours, shown here for $(x_v,y_v)=(0.05W,0.45W)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current. The plots show the behavior when $a=2W.$

Figure 18

(a) Current flow around a semicircular notch of radius $a$ near the edge of a strip, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, Eq. (150), which has the values $S=0$ along the boundary. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrow shows the current direction. (b) Vortex-generated current flow, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the values $S_v=0$ along the boundaries. The contours, shown here for $(x_v,y_v)=(0,1.2a)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current.

Figure 19

Numerical results for (a) ${\delta }_c/\xi$ and (b) $R$ as functions of the ratio of the coherence length $\xi$ to the radius $a$ of the semicircular notch. Expansions in powers of $\xi /a$ are shown as dashed curves for ${\delta }_c$ [See Eq. (153)] and $R$ [See Eq. (154)]. Expansions in powers of $a/\xi$ are shown as dot-dashed curves for ${\delta }_c$ [See Eq. (155)] and $K_{0c}$ [Eq. (156)].

Figure 20

(a) Current flow around an isosceles triangular notch (height $a$, base $2b$, and sides $c$) near the edge of a strip, shown by the contour plot of the stream function $S(x,y)=\Im {\mathcal{G}}_{\zeta }(x+iy)$, Eq. (161), which has the values $S=0$ along the boundary. The contours correspond to streamlines of the sheet-current density $\mathit{K}$, and the arrow shows the current direction. (b) Vortex-generated current flow, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the values $S_v=0$ along the boundaries. The contours, shown here for $(x_v,y_v)=(0,0.9c)$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current. The plots show the behavior when $\mu =2/3$ and the vertex angle is ${\theta }_0=\pi /2$.

Figure 21

The notch's critical-current reduction factor $R$, see Eq. (164), vs the vertex angle ${\theta }_0$ for $\xi /a$ $=$ 0.01, 0.03, and 0.1, where $a=c\cos \left(\frac{{\theta }_0}{2}\right)=c\sin \left[\frac{\pi }{2}(\frac{1}{\mu }-1)\right].$

Figure 22

Switching probability distribution due to thermally activated barrier-climbing, $P_{\mathrm{sw}}^{\prime }/P_{\mathrm{sw},\max }^{\prime }\left(I_c\right)$ vs $I/I_c$ [Eqs. (171) and (172)], where $P_{\mathrm{sw},\max }^{\prime }\left(I_c\right)=(N/I_c)\exp [-N/(N+1)],$ shown for $N=1000$ and $r_I/\omega I_c$ $=$ (a) ${10}^{-5}$, (b) $3\times {10}^{-5}$, (c) ${10}^{-4}$, (d) $3\times {10}^{-4}$, (e) ${10}^{-3}$ ($I_{\max }=I_c$), (f) $3\times {10}^{-3}$, (g) ${10}^{-2}$, and (h) $3\times {10}^{-2}$.

Figure 23

Comparison between experimental results for sharp rectangular 180${}^{\circ }$ turnarounds[15] and the corresponding theoretical predictions of Eqs. (175) and (176).

Figure 24

Vortex-generated current flow in a thin-film angular sector of opening angle $\alpha$, shown by the contour plot of the stream function $S_v(x_v,y_v;x,y)$, which has the values $S_v=0$ for $(x,y)$ along the boundaries. The contours, shown here for ${\mathit{r}}_v=(x_v,y_v)=(0,\delta )$ and $\alpha =\pi /2$, correspond to streamlines of the vortex-generated sheet-current density ${\mathit{K}}_v$, and the arrow shows the direction of the current.