Two mechanisms of Josephson phase shift generation by an Abrikosov vortex

Abrikosov vortices contain magnetic fields and circulating currents that decay at a short range $\lambda \sim 100$ nm. However, vortices can induce Josephson phase shifts at a long range $r\sim \mu \mathrm{m}\gg \lambda$. Mechanisms of this puzzling phenomenon are not clearly understood. Here we present a systematic study of vortex-induced phase shifts in planar Josephson junctions. We make two key observations: (i) The cutoff effect: Although vortex-induced phase shift is a long-range phenomenon, it is terminated by the junction and does not persist beyond it. (ii) A linear to superlinear crossover with a rapid upturn of the phase shift occurs upon approaching a vortex to a junction. The crossover occurs at a vortex-junction distance comparable to the penetration depth. Together with theoretical and numerical analysis this allows unambiguous identification of two distinct and independent mechanisms. The short range $r\lesssim \lambda$ mechanism is due to circulating vortex currents inside a superconducting electrode without involvement of magnetic fields. The long range $r\gg \lambda$ mechanism is due to stray magnetic fields outside electrodes without circulating vortex currents. We argue that understanding of controlling parameters of vortex-induced Josephson phase shift can be used for development of novel compact cryoelectronic devices.

Figure 1

Sketches of considered vortex-junction configurations. (a) A top view of planar junction (yellow region) with trapped Abrikosov vortex in the first (top) electrode. Lines indicate current streamlines. (b) A side view of the same configuration. Lines indicate magnetic field lines of vortex stray fields. (c) A top view of Corbino disk geometry, in which the fraction of stray flux entering the junction is proportional to the polar angle of the vortex within the junction segment ${\mathrm{\Theta }}_v$.

Figure 2

(a) Illustration of an image solution for a rectangular electrode with a vortex. Green circles represent primary images from the four edges. Multiple mirror reflections from all edges lead to an infinite series of image vortices and antivortices. (b) Calculated variation of the vortex-induced JPS versus vortex polar angle for a junction with narrow $L_x\ll {\lambda }^{*}$, long $L_z=\infty$ electrodes. Calculations are made for an antivortex approaching the junction at $x_v=0.5L_x$ from $z_v=\infty$ (${\mathrm{\Theta }}_v=0$) to $z_v=0$ (${\mathrm{\Theta }}_v=\pi$), as sketched in the inset. Green line represents the image array solution, which coincides with the solution from Ref. [44] for a Pearl antivortex (dashed magenta line). Black dotted line represents linear dependence $\mathrm{\Delta }{\phi }_v={\mathrm{\Theta }}_v$. (c) Image-array solutions for finite-size rectangular-shape electrodes with $L_z=L_x$ (blue lines) and $L_z=L_x/2$ (red lines). Here we consider a double junction experiment, sketched in the inset, and show a variation of JPS in the bottom (solid lines) and top (dashed lines) junctions made at the upper and lower edges of the electrode, versus the polar angle in the bottom junction, ${\mathrm{\Theta }}_{vb}$. Calculations are made for an antivortex moving at $x_v=0.5L_x$ from the top, $z_v=L_z$ (minimal value of ${\mathrm{\Theta }}_{vb}$), to the bottom, $z_v=0$ (${\mathrm{\Theta }}_{vb}=\pi$), junction, as sketched in the inset.

Figure 3

3D numerical simulations of stray field distribution from a vortex in a system of planar superconducting electrodes. We show side views through a central cross section containing the vortex (red spot with high field). White horizontal bands represent superconducting electrodes with $B=0$. (a) Vortex in the center of a single square-shaped electrode (no junction). (b) For two electrodes, forming a single junction. (c) For three electrodes forming two planar junctions. Here the vortex is at the left edge of the panel and it is placed in the same square shaped electrode as in (a) (left part of the electrode is not shown). From panels (b) and (c) it can be seen that stray fields are predominantly closing through the nearest junction and that only a minor fraction reaches the outmost junction (note the logarithmic scale). This is consistent with the cutoff phenomenon observed in experiment.

Figure 4

Numerical modeling of $I_c\left(H\right)$ patterns for different distances $z_v$ between junction and vortex. Simulations are made for an antivortex at different $z_v$ and $x_v=0.5L_x$ for a long $L_z=\infty$ electrode. (a) Phase shifts calculated from image array solution (thick lines) and Eq. (1), scaled to the same amplitude $\mathrm{\Delta }{\phi }_v$ (thin lines). (b)–(f) Variation of $I_c\left(H\right)$ patterns upon approaching of the antivortex to the junction. Dashed black line in (b) represents the vortex-free Fraunhofer pattern. Thick and thin lines in (b)–(f) are obtained for corresponding lines from (a). The difference is marginal.

Figure 5

Single junction experiment. (a) SEM image of a Nb-Cu-Nb planar junction with a vortex trap in the top electrode at $z_v=1.0\mu \mathrm{m}$ from the junction, $z_v/L_x=0.19$ and ${\mathrm{\Theta }}_v=0.77\pi$. (b) Measured $I_c\left(H\right)$ modulation without vortices (black lines) and with a trapped vortex (magenta lines). (c) $I_c\left(H\right)$ modulation with a trapped antivortex. It is mirror symmetric with respect to the vortex case in (b). (d) Calculated $I_c\left(H\right)$ pattern for the experimental geometry from (c). Inset shows corresponding antivortex-induced Josephson phase shift according to Eq. (1). A good agreement with experimental pattern in (c), without any fitting, indicates that the vortex-induced Josephson phase shift in this case is close to the vortex polar angle, $\mathrm{\Delta }{\phi }_v\simeq {\mathrm{\Theta }}_v$.

Figure 6

Double junction experiment with a vortex trap in the middle electrode, placed closer to the top junction, ${\mathrm{\Theta }}_{vt}=0.92\pi$ and ${\mathrm{\Theta }}_{vb}=0.79\pi$. (a) and (b) Fraunhofer-type vortex free $I_c\left(H\right)$ patterns for both junctions. (c)–(f) Black and blue symbols represent measured $I_c\left(H\right)$ patterns after trapping [(c), (d)] a vortex [(e), (f)] an antivortex. Note that distortion is larger for the top junction [(c), (e)] which is closer to the vortex. Furthermore, vortex-induced JPS is larger than the polar angle for the top junction and smaller for the bottom junction, consistent with simulations in Fig. 2. Red lines represent numerical fits, used for estimation of JPS. Insets show sketches of experiments.

Figure 7

Experimental demonstration of the cutoff effect for a double-junction device with vortex traps in outmost electrodes. (a) SEM image of the device with two nearby Nb-CuNi-Nb junctions. (b) Vortex-free $I_c\left(H\right)$ patterns for both junctions. (c) and (d) $I_c\left(H\right)$ patterns for the top (c) and bottom (d) junctions with a trapped antivortex in the top trap (this is the only trap at this state of the sample, as sketched in the inset). Note that the $I_c\left(H\right)$ pattern for the top junction is strongly distorted, indicating the large total JPS $\mathrm{\Delta }{\phi }_v=1.55\pi$ as seen from the numerical fit (red line) in (c). However, the pattern in the bottom junction is practically unaffected. (e) and (f) Measurements after making a second bottom trap at the same distance to the bottom junction as for the top trap, see the sketch. Symbols represent $I_c\left(H\right)$ patterns with an antivortex in the bottom trap. This time the pattern in the bottom junction is significantly distorted but in the top junction is not. This illustrates that the vortex-induced JPS is terminated at the junction and does not persist beyond it.

Figure 8

Vortex-induced JPS as a function of vortex polar angle. Red symbols are from the present study and blue from Ref. [43]. Solid circles are for single- and double-junction devices with vortices in outmost electrodes, $L_z=\infty$. Open symbols are for double-junction devices with the vortex in the middle electrode with finite $L_z$. Similar symbols correspond to junctions at the same device. Solid and dashed lines represent stray-field, Eq. (1), and current-induced image solutions ($L_z=\infty$, olive and $L_z=0.24L_x$, magenta curve), respectively. A crossover from linear to superlinear dependence occurs at ${\mathrm{\Theta }}_v\gtrsim 0.9\pi$, corresponding to $z_v\lesssim 500$ nm $\sim {\lambda }^{*}$. The crossover indicates transition in the mechanism of the phase shift from stray-field-induced (linear) at large distances $z_v\gg {\lambda }^{*}$ to circulating-current-induced (superlinear) at a short range $z_v\lesssim {\lambda }^{*}$.