Vortex Counting and Velocimetry for Slitted Superconducting Thin Strips

The maximal speed $v^{\ast }$ for magnetic flux quanta is determined by the energy relaxation of unpaired electrons and is thus essential for superconducting microstrip single-photon detectors (SMSPDs). However, the deduction of $v^{\ast }$ from the current-voltage ($I$-$V$) curves at zero magnetic field is hindered by the unknown number of vortices, $n_v$, as a small number of fast-moving vortices can induce the same voltage as a large number of slow-moving ones. Here, we introduce an approach for the quantitative determination of $n_v$ and $v^{\ast }$. The idea is based on the Aslamazov and Larkin prediction of kinks in the $I$-$V$ curves of wide and short superconducting constrictions when the number of fluxons crossing the constriction is increased by one. We realize such conditions in wide $\mathrm{Mo}\mathrm{Si}$ thin strips with slits milled by a focused ion beam and reveal quantum effects in a macroscopic system. By observing kinks in the $I$-$V$ curves with increase of the transport current, we evidence a crossover from a single- to multifluxon dynamics and deduce $v^{\ast }\simeq 12$ km/s. Our experimental observations are augmented with numerical modeling results, which reveal a transition from a vortex chain over a vortex jet to a vortex river with increase of $n_v$ and $v$. Our findings are essential for the development of one-dimensional and two-dimensional few-fluxon devices and provide a demanded approach for the deduction of maximal vortex velocities at the SMSPD operation conditions.

Figure 1

(a) Schematic of a superconducting strip with a slit causing a nonuniform current-density distribution. Vortices enter the strip at the slit apex, cross it under the action of the transport current, and exit via the opposite strip side. $W$, strip width; $w$, isthmus width. (b) SEM image of a part of the slitted strip $\mathrm{Mo}\mathrm{Si}214$.

Figure 2

Voltage kinks in the $I$-$V$ curves of the slitted strips $\mathrm{Mo}\mathrm{Si}218$ (a) and MoSi214 (c) at $T=5$ K. Inset: the same $I$-$V$ curves in a broader range of currents. (b),(d) Current dependence of the differential resistance for the same samples. (e),(f) Smearing of the minima (indicated by the arrows) in the normalized differential resistance $(dV/dI)/(dV/dI){|}_{6\mathrm{K}}$ with increase of temperature.

Figure 3

TDGL modeling results. (a) $I$-$V$ curve for a superconducting strip with a slit at $T/T_c=0.8$. Inset: evolution of the number of vortices (vorticity) and vortex velocity with increase of the transport current. Voltage is in units of $V_0=10k_B{T}_c/(2e)$, velocity in units of $10V_0/n_v$, and current in units of the Ginzburg-Landau depairing current $I_{\mathrm{dep}}$. (b) Current dependence of the differential resistivity. (c) Exemplary snapshots (at a given instant of time) of the superconducting order parameter distribution at the $I$-$V$ points indicated in (a). Panel (i) corresponds to a transient regime between (h),(j). In the case of strong heating effects, the hatched area in (i) is turned into a normal domain, which then grows in size until the entire sample transits to the normal state.

Figure 4

(a) Kink voltage versus number of vortices in the slitted $\mathrm{Mo}\mathrm{Si}$ strips. Open symbols: TDGL and AL model calculations. Solid symbols: experimental data. (b) Vortex velocity versus number of vortices and the $v^{\ast }$ values (larger symbols) deduced by using Eqs. (2) and (3). In both panels, dashed lines are guides for the eye.