# Vortex counting and velocimetry for slitted superconducting thin strips

V. M. Bevz  Physics Department, V. Karazin Kharkiv National University, 61022,
Kharkiv, Ukraine  M. Yu. Mikhailov  B. Verkin Institute for Low Temperature
Physics and Engineering of
the National Academy of Sciences of Ukraine, 61103, Kharkiv, Ukraine  B.
Budinská  University of Vienna, Faculty of Physics, 1090 Vienna, Austria
University of Vienna, Vienna Doctoral School in Physics, 1090 Vienna, Austria
S. Lamb-Camarena  University of Vienna, Faculty of Physics, 1090 Vienna,
Austria  University of Vienna, Vienna Doctoral School in Physics, 1090 Vienna,
Austria  S. O. Shpilinska  University of Vienna, Faculty of Computer Science,
1090 Vienna, Austria  A. V. Chumak  University of Vienna, Faculty of Physics,
1090 Vienna, Austria  M. Urbánek  CEITEC, Brno University of Technology, 61200
Brno, Czech Republic  M. Arndt  University of Vienna, Faculty of Physics, 1090
Vienna, Austria  W. Lang  University of Vienna, Faculty of Physics, 1090
Vienna, Austria  O. V. Dobrovolskiy<EMAIL_ADDRESS>University of Vienna, Faculty of Physics, 1090 Vienna, Austria

###### Abstract

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,
$n_{\mathrm{v}}$, 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 $n_{\mathrm{v}}$ 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 MoSi 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 multi-fluxon 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 $n_{\mathrm{v}}$ and the vortex velocity. Our
findings are essential for the development of 1D and 2D few-fluxon devices and
provide a demanded approach for the deduction of $v^{\ast}$ at the SMSPD
operation conditions.

##  I Introduction

The ultimate speed limit for magnetic flux transport in superconductors via
quantized flux lines (fluxons) is attracting increasing attention Wördenweber
_et al._ (2012); Grimaldi  _et al._ (2015); Jelić  _et al._ (2016); Embon  _et
al._ (2017); Sheikhzada and Gurevich (2017); Shklovskij  _et al._ (2017);
Kogan and Prozorov (2020); Dobrovolskiy  _et al._ (2020a); Pathirana and
Gurevich (2021); Kogan and Nakagawa (2022); Dobrovolskiy  _et al._ (2020b);
Budinská  _et al._ (2022); Dobrovolskiy (2023). This interest is caused by the
fundamental questions regarding the interaction and stability of vortices as
topological objects moving at $>10$ km/s that exceed the maximal drift
velocity of the Cooper-pair condensate Embon  _et al._ (2017); Kogan and
Prozorov (2020). On the applied side, ultrafast vortex dynamics underlies the
vortex-assisted voltage response of superconducting microstrip single-photon
detectors (SMSPDs) Korneeva  _et al._ (2020); Charaev  _et al._ (2020); Chiles
_et al._ (2020). When operated at a close-to-depairing current $I\lesssim
I_{\mathrm{dep}}$, the intrinsic detection efficiency of SMSPDs is predicted
to reach almost unity Vodolazov (2017), opening novel prospects for large-
active-area detectors in free-space quantum cryptography Chen  _et al._
(2021), macromolecule analysis Fein  _et al._ (2019), and other areas
Steinhauer  _et al._ (2021).

The maximal vortex velocities $v^{\ast}$ deduced from current-voltage
($I$-$V$) curves are used for judging whether a material could be potentially
suitable for SMSPDs Lin  _et al._ (2013); Caputo  _et al._ (2017a); Hofer and
Haberkorn (2021); Liu  _et al._ (2021); Cirillo  _et al._ (2021). This
approach is based on the theory of flux-flow instability (FFI) Larkin and
Ovchinnikov (1975); Bezuglyj and Shklovskij (1992), in which $v^{\ast}$
depends on the energy relaxation time of unpaired electrons (quasiparticles)
$\tau_{\varepsilon}$ Peroz and Villard (2005); Leo  _et al._ (2011); Attanasio
and Cirillo (2012); Dobrovolskiy  _et al._ (2019a); Leo  _et al._ (2020).
Thus, FFI studies represent an alternative method for material
characterization, which is complementary to photoresponse, magnetoconductance,
and microwave measurements Wördenweber  _et al._ (2012); Dobrovolskiy and Huth
(2015); Pompeo and Silva (2008); Lösch  _et al._ (2019); Pokusinskyi  _et al._
(2022); Zhang  _et al._ (2016); Sidorova  _et al._ (2021). The temperature
dependence $\tau_{\varepsilon}(T)$ contains information on the dominating
microscopic mechanism of the quasiparticle relaxation Kaplan  _et al._ (1976);
Peroz and Villard (2005); Armenio  _et al._ (2007); Leo  _et al._ (2011);
Attanasio and Cirillo (2012); Caputo  _et al._ (2017b), including electron-
electron Doettinger  _et al._ (1994) and electron-phonon Peroz and Villard
(2005); Embon  _et al._ (2017) scattering, recombination Peroz and Villard
(2005); Leo  _et al._ (2011), and escape of nonequilibrium phonons to the
substrate Vodolazov (2019); Dobrovolskiy  _et al._ (2020b). An accurate
determination of $v^{\ast}$ and $\tau_{\varepsilon}$ is thus essential for
basic research and the development of applications.

However, the deduction of $v^{\ast}$ at low applied magnetic field $B$ is
difficullt since the local magnetic field in a current-carrying conductor can
differ from $B$. Thus, for most SMSPD-suitable materials, such as WSi Zhang
_et al._ (2016); Chiles  _et al._ (2020), MoSi Korneeva  _et al._ (2020);
Charaev  _et al._ (2020); Budinská  _et al._ (2022), MoN Ustavshikov  _et al._
(2020); Ustavschikov  _et al._ (2020), NbC Dobrovolskiy  _et al._ (2020b),
NbRe Ejrnaes  _et al._ (2022), NbReN Cirillo  _et al._ (2021), and NbTiN
Sidorova  _et al._ (2021), the dependence $v^{\ast}(B)$ will likely diverge at
$B\rightarrow 0$ if one uses the standard relation Brandt (1995)

$v^{\ast}=\displaystyle\frac{V^{\ast}}{BL},$ (1)

where $V^{\ast}$ is the voltage measured just before the jump to the highly
resistive state and $L$ is the distance between the voltage leads. Given that
$\tau_{\varepsilon}$ is finite, this divergence is obviously unphysical and it
is caused by the failure of Eq. (1) at $B\rightarrow 0$, as the number of
vortices $n_{\mathrm{v}}$ in the superconductor can be _larger_ than follows
from the estimate $n\Phi_{0}=BS$ Dobrovolskiy  _et al._ (2020b) [$n=1,2,..$;
$\Phi_{0}$: magnetic flux quantum; $S$: area of the sample surface]. The
larger number of vortices is caused by the current flowing through the strip,
which leads to the nucleation of vortices and, thus, the resistive state
associated with the current-driven vortex motion Brandt (1995). While the
current-induced generation of vortices is insignificant at large magnetic
fields, this mechanism becomes dominant at low fields, turning into the only
source of vortices (and antivortices) at $B=0$. In this way, the highest
vortex velocities are expected in the low/zero-$B$ regime and the key
challenge consists in determining the number of fluxons moving in the
superconductor.

For straight strips, the problem of vortex counting and velocimetry is
complicated due to simultaneous vortex entry from many edge defects Budinská
_et al._ (2022). In contrast, an artificially introduced single edge defect
can be used for suppressing the energy barrier locally and letting the
vortices enter at a deliberately chosen edge point Aladyshkin  _et al._
(2001); Bezuglyj  _et al._ (2022). As the current increases, this point turns
into an FFI nucleus Vodolazov (2019), with a vortex river (chain of vortices
with a reduced number of quasiparticles in their cores) spreading across the
superconductor and causing a local FFI Bezuglyj  _et al._ (2019); Budinská
_et al._ (2022). Herein, the average number of vortices in the strip
$n_{\mathrm{v}}$ is determined by the balance between the intervortex
repulsion and the current-vortex and vortex-edge interactions Bezuglyj  _et
al._ (2022); Aslamazov and Larkin (1975). However, if one uses the Josephson
relation $V^{\ast}=n_{\mathrm{v}}\Phi_{0}f^{\ast}$, where
$f^{\ast}=v^{\ast}/w$ is the frequency of the vortex entry into the
superconductor of width $w$, the unknown $n_{\mathrm{v}}$ still restricts the
analysis to finding the proportionality $V^{\ast}\propto f^{\ast}\propto
v^{\ast}$ Embon  _et al._ (2017); Bezuglyj  _et al._ (2022) rather than the
absolute value of $v^{\ast}$. This restriction is associated with the
uncertainty of whether the voltage generated by moving vortices arises due to
a _smaller number_ of _fast-moving_ vortices or due to a _larger number_ of
_slow-moving_ ones. Hence, a technique for controlling the number of vortices
is required for the deduction of $v^{\ast}$ and for the development of
superconducting devices operating in the few- and single-fluxon regime.

Here, we introduce an approach for the quantitative determination of
$n_{\mathrm{v}}$ and $v^{\ast}$. The idea is based on the Aslamazov and Larkin
(AL) prediction Aslamazov and Larkin (1975) of kinks in the $I$-$V$ curves of
wide and short superconducting constrictions when _the number of vortices
crossing the constriction is increased by one_. We realize such conditions in
a-few-micrometer-wide MoSi thin strips in which slits, milled by a focused ion
beam, locally suppress the edge barrier and allow for a controllable vortex
entry. Possessing a high structural uniformity and weak intrinsic pinning,
MoSi is a typical SMSPD material Korneeva  _et al._ (2014); Caloz  _et al._
(2018); Korneeva  _et al._ (2020); Charaev  _et al._ (2020) which also
features high vortex velocities Budinská  _et al._ (2022); Bezuglyj  _et al._
(2022); Haberkorn (2022). While SMSPDs are usually made from $3$–$13$ nm-thick
films Zhang  _et al._ (2016); Chiles  _et al._ (2020); Li  _et al._ (2016);
Caputo  _et al._ (2017a); Gourgues  _et al._ (2019); Polakovic  _et al._
(2020); Korneeva  _et al._ (2020); Charaev  _et al._ (2020); Caloz  _et al._
(2018); Ustavshikov  _et al._ (2020), here we use $15$ nm-thick films to
demonstrate the merit of this technique for resolving the open questions for
MoSi strips of the same thickness, related to the failure of the $v^{\ast}(B)$
fits at $B\rightarrow 0$ Budinská  _et al._ (2022) and the unknown number of
vortices in the strips at $B=0$ Bezuglyj  _et al._ (2022). By measuring the
$I$-$V$ curves at zero magnetic field, we observe voltage kinks (minima in the
differential resistance) of the strips with increase of the transport current.
By associating the kink number $n$ with the number of vortices
$n_{\mathrm{v}}$, we conclude that $n_{\mathrm{v}}$ varies from one to five in
our experiments. From the last point before the FFI jump in the $I$-$V$
curves, at $T=5$ K, we deduce the maximal vortex velocity $v^{\ast}\simeq 12$
km/s yielding the quasiparticle energy relaxation time $\tau_{\epsilon}\simeq
30$ ps. Our experimental observations are augmented with results of the time-
dependent Ginzburg-Landau (TDGL) equation modeling which reveals a vortex-
chain–vortex-jet–vortex-river transition with increase of the number of
vortices and their velocity. An essential result of the TDGL simulations is
that kinks in the $I$-$V$ curve are revealed not only for the 1D vortex-chain
state, as considered by AL Aslamazov and Larkin (1975), but also for the 2D
vortex-jet regime Bezuglyj  _et al._ (2022). In all, our findings provide an
approach for the deduction of maximal vortex velocities in superconductors and
open up a pathway to fluxonic devices in which the number of vortices is
controlled by the transport current.

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 MoSi214.

##  II Experiment

Experiments were carried out on slitted microstrips fabricated from the same
$15$ nm-thick MoSi film. The experimental geometry is shown in Fig. 1(a). Due
to the current-crowding effect Friesen and Gurevich (2001); Clem and Berggren
(2011); Adami  _et al._ (2013), a slit causes a higher current density at its
apex and thus determines the place for the entry of vortices into the strip at
$I>I_{\mathrm{c}}$. Here, $I_{\mathrm{c}}$ is the critical current
corresponding to the first vortex entry. Under the action of the transport
current, vortices cross the strip and exit it via the opposite side. We note
that the significantly stronger edge barrier at the opposite (straight) edge
prevents the penetration of antivortices into the strip. This circumstance
makes the considered problem distinct from the Glazman model Glazman (1986)
dealing with two defects at the opposite strip edges.

Figure 2: Voltage kinks in the $I$-$V$ curves of the slitted strips MoSi218
(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 $(dV/dI)/(dV/dI)|_{\mathrm{6\,K}}$ with
increase of temperature.

The MoSi film was deposited by DC magnetron co-sputtering of molybdenum and
silicon targets in an Ar atmosphere. The deposition was done onto a Si/SiO2
substrate at room temperature. The deposition rates were adjusted to result in
a Mo70Si30 composition. Further details on the film fabrication and
characterization can be found elsewhere Budinská  _et al._ (2022). After the
photolithography step, constrictions of $2\,\mu$m width were formed by a
focused Ga ion beam (FIB) for electrical resistance measurements. Then, slits
with dimensions of $25$ nm$\times 0.2\,\mu$m and $25$ nm$\times 0.6\,\mu$m
(width$\times$length) were milled by FIB in the middle of one side of the
strips labeled MoSi218 and MoSi214, respectively. The slit width was chosen as
a minimal width achievable with the FIB milling, to ensure a sharp bending of
the current-density lines close to the slit apex. The isthmus widths
$w=1.8\,\mu$m and $1.4\,\mu$m were chosen to result in an at least by $10\%$
smaller critical current $I_{\mathrm{c}}$ in the slit area with respect to
$I_{\mathrm{c}}$ in the regions away from it and, thus, to exclude the
penetration of vortices from other possible defects at the strip edges.

A scanning electron microscopy (SEM) image of a part of the slitted strip
MoSi214 is shown in Fig. 1(b). In the absence of a defect, a current flowing
in the $y$ direction would produce a vortex motion in the $x$ direction. The
configurations of vortices in the slitted strips and the associated features
in their $I$-$V$ curves are the subjects of our studies.

MoSi is an amorphous superconductor in the dirty limit Peroz and Villard
(2005); Dobrovolskiy and Huth (2012). It is characterized by a resistivity
$\rho_{8\mathrm{K}}\approx 150\,\mu\Omega$cm, critical temperature
$T_{\mathrm{c}}=6.43$ K, upper critical field $B_{\mathrm{c2}}(0)\approx 10.2$
T, derivative $dB_{\mathrm{c}2}/dT=-2.23$ T/K near $T_{\mathrm{c}}$, electron
diffusion coefficient $D\approx 0.5$ cm2/s, coherence length
$\xi(0)=\sqrt{\hbar D/1.76k_{\mathrm{B}}T_{\mathrm{c}}}=6$ nm (corresponding
to $\xi_{\mathrm{c}}=\sqrt{1.76}\xi(0)=7.8$ nm in the TDGL simulations),
penetration depth $\lambda(0)=1.05\cdot
10^{-3}\sqrt{\rho_{\mathrm{8K}}/T_{\mathrm{c}}}\approx 500\,$nm Kes and Tsuei
(1983), and $\lambda_{\mathrm{eff}}(0)=\lambda^{2}(0)/d\approx 16\,\mu$m. The
inequality $\xi\ll w$ means that the strips are _wide_ while
$w<\lambda_{\mathrm{eff}}$ justifies the applicability of the AL theory
Aslamazov and Larkin (1975). In this theory, effects of the current-induced
magnetic-field variation on the superconducting properties are disregarded.

Electrical transport measurements were done in the temperature range
$T=4.2$-$5.8$ K (0.65-0.9$T_{\mathrm{c}}$) in a He bath cryostat in the
standard four-probe geometry. The $I$-$V$ curves were acquired in the current-
driven regime at zero magnetic field. The differential resistance data were
obtained by differentiating the $I$-$V$ curves with a running average over
nine data points.

##  III Results

###  III.1 Voltage kinks in the $I$-$V$ curves

Figure 2(a) presents the $I$-$V$ curve of the strip MoSi218 at $T=5$ K
($0.78T_{\mathrm{c}}$). The $I$-$V$ curve has a zero-voltage section (I) at
low currents, an extended quasi-linear ohmic branch (II) in the flux-flow
regime, and an abrupt FFI jump (III) to a higher resistive state (see the
inset in Fig. 2(a)). A detailed inspection of the $I$-$V$ curve in regime (II)
reveals _voltage kinks_ which represent our main experimental observation. The
first-kink voltage $V_{1}=9.8\,\mu$V. Up to five voltage kinks can already be
recognized in the $I$-$V$ curve, but even better so in the differential
resistance curve shown in Fig. 2(b). From the rapid onset of the resistance at
low voltages, we deduce the critical current $I_{\mathrm{c}}=451\,\mu$A by
using a $500$ nV voltage criterion. The last voltage point before the FFI jump
to the highly-resistive state, $V^{\ast}=69.2\,\mu$V, corresponds to an
instability current of $I^{\ast}\approx 0.53$ mA$\simeq 1.2I_{\mathrm{c}}$.

The $I$-$V$ curve of the strip MoSi214 looks qualitatively similar, see panels
(c) and (d) in Fig. 2. It exhibits four voltage kinks before the FFI onset at
the instability voltage $V^{\ast}=70.1\,\mu$V at the instability current
$I^{\ast}\approx 0.42$ mA$\simeq 1.2I_{\mathrm{c}}$, with
$I_{\mathrm{c}}=347\,\mu$A and the first-kink voltage $V_{1}=15.1\,\mu$V. The
smaller number of voltage kinks for the strip MoSi214 is attributed to the
narrow isthmus in this sample.

Figure 3: TDGL modeling results. (a) $I$-$V$ curve for a superconducting strip
with a slit at $T/T_{\mathrm{c}}=0.8$. Inset: Evolution of the number of
vortices (vorticity) and vortex velocity with increase of the transport
current. Voltage is in units of $V_{0}=10k_{\mathrm{B}}T_{\mathrm{c}}/(2e)$,
velocity in units of $10V_{0}/n_{\mathrm{v}}$, 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 panel (a). Panel (i) corresponds to a transient regime
between (h) and (j). In the case of strong heating effects, the hatched area
in panel (i) is turned into a normal domain which then grows in size until the
entire sample transits to the normal state.

For both samples, the kinks become more pronounced at lower temperatures. As
the temperature increases, the kinks are getting smeared and eventually vanish
at about $6$ K. If one takes the current dependence of the differential
resistance at 6 K (without kinks) as a curve for normalization,
$(dV/dI)/(dV/dI)|_{\mathrm{6\,K}}$, the evolution of the kinks upon
temperature variation can be seen in panels (e) and (f) in Fig. 2. Namely, as
the temperature increases, the higher-order kinks disappear first and only the
first-order kink remains visible at $5.8$ K.

###  III.2 Numerical modeling

The spatiotemporal evolution of the superconducting order parameter in the
strip was modeled on the basis of the TDGL equation Dobrovolskiy  _et al._
(2020b); Budinská  _et al._ (2022). The considered equations and the major
parameters are detailed in the Appendix. The numerical modeling results are
presented in Fig. 3.

The current dependence of the differential resistance unveils kinks in the
$I$-$V$ curve, see Fig. 3(a,b), just as observed in the experiment in Fig.
2(a,c). The evolution of the number of vortices in the sample,
$n_{\mathrm{v}}$ (see the inset in Fig. 3(a)), with increase of the transport
current can be understood with the aid of contour plots of the absolute value
of the order parameter $|\Delta|$ in Fig. 3(c)-(h). Namely, the kinks in the
flux-flow regime in the $I$-$V$ curve occur when the maximal number of
vortices simultaneously crossing the strip is increased by one.

The evolution of the vortex configurations with increase of the transport
current can be summarized as follows. At low currents, the vortex trajectories
fall onto the same line, forming a vortex chain. When there are more than two
vortices in the strip, the vortex chain evolves to a vortex jet. The vortex
jet emerges because of the repulsive intervortex interaction Bezuglyj  _et
al._ (2022). With increasing vortex velocity, the vortex jet becomes unstable
and evolves to a vortex river, see Fig. 3(i, j), that is formed by fast moving
Josephson-like vortices. At this transition, the number of vortices
$n_{\mathrm{v}}$ in the strip is reduced by a factor of two while the voltage
increases by a factor of three (not shown). These changes in $n_{\mathrm{v}}$
and $V$ imply a six-fold increase of the vortex velocity. However, heating
effects may mask this transition since a normally-conducting domain may appear
in the sample (in the hatched region in Fig. 3(i)) instead of a vortex river.
The origin of the voltage kinks and the vortex configurations in the TDGL and
AL models will be discussed next.

##  IV Discussion

###  IV.1 Vortex configurations in the AL and TDGL models

The central result of our studies is the experimental observation of kinks in
the $I$-$V$ curves of slitted superconducting microstrips (Fig. 2) and their
corroboration by the TDGL equation modeling (Fig. 3). At the same time, our
experiments go essentially beyond the AL model due to the 2D geometry. In the
AL theory, kinks in the $I$-$V$ curve occur when the number of fluxons in the
constriction is increased by one. While the AL theory deals with a 1D chain of
vortices along a straight line, our TDGL simulations suggest that this regime
is only possible for slits narrower than $\lesssim 2\xi_{\mathrm{c}}$.
However, the fabrication of such slits is challenging for superconductors with
$\xi(0)\backsimeq 5$ nm.

The slits in our samples are wide ($\simeq 3\xi_{\mathrm{c}}$) which means
that the nucleation of every vortex may occur at a slightly different point of
the slit apex. As long as there are no more than two vortices in the strip,
the vortex-chain arrangement is maintained, see Fig. 3(d). However, as soon as
a third vortex enters into the strip, fluctuations and inhomogeneities, which
cannot be avoided even in high-quality samples, facilitate the intervortex
repulsion to deflect the trajectories of vortices away from the line $y=0$,
thus leading to the formation of a vortex jet Bezuglyj  _et al._ (2022). With
an increase of the number of vortices, the opening angle of the vortex jet
increases, see Fig. 3(e)-(g).

In a jet, the vortex trajectories are still very regular. If one assigns a
number to each vortex, “odd” and “even” vortices fall on two separate but well
defined lines, see e.g. Fig. 3(g). With a further increase of the transport
current, the vortex velocity component in the transport current direction
($v_{y}$) becomes much smaller than the velocity component $v_{x}$ across the
strip so that the opening angle of the vortex jet is decreasing (compare
panels (g) and (h) in Fig. 3). Accordingly, in not very wide strips ($\xi\ll
w\ll\lambda$) Bezuglyj  _et al._ (2022), the points where vortices exit the
strip at the opposite edge, are displaced toward $y=0$ so that the two vortex
rays approach each other. The confluence of the rays gives rise to a vortex
river in which the superconducting order parameter is suppressed along it.

A distinct feature of the wider strips ($\xi\lll w\lesssim\lambda$) in our
work is that vortex rivers may emerge _in the rays before_ they approach each
other. Indeed, in the strong-current regime, the TDGL model reveals _halving_
of the number of vortices upon a transition to the vortex-river state, see the
inset in Fig. 3(a) where $n_{\mathrm{v}}$ drops from 8 to 4. This abrupt
decrease in the vorticity is indicative of a dynamic topological transition
Fomin  _et al._ (2022) accompanied by a drastic increase of the voltage
(outside the voltage range shown in Fig. 3(a)), which is in line with
Josephson vortex velocities Embon  _et al._ (2017).

Alternatively, the transition to the normal state could be mediated by the
growth of a normal domain Gurevich and Mints (1987). To exclude this
mechanism, we have checked that for a sheet resistance of
$R_{\square}\backsimeq 80\,\Omega$, a $2\xi\sim 10$ nm-wide domain would
result in at least an order of magnitude larger voltage than measured
experimentally and expected for a vortex river. In analogy to phase-slip lines
exhibiting Josephson properties in Ref. Sivakov  _et al._ (2003), the
appearance of Shapiro steps in the microwave-irradiated $I$-$V$ curves of
slitted strips in conjunction with a detection of the Josephson generation
could be suggested for an independent experimental proof of the vortex-river
state in follow-up works.

###  IV.2 Voltage kinks in the AL versus TDGL models

The simulations reveal that the kinks in the $I$-$V$ curves are maintained
even when a 2D vortex jet is formed. The kinks appear as a consequence of the
vortex-vortex repulsion, and the vortices present in the strip hamper the
entrance of a new vortex qualitatively in the same manner as in the 1D vortex
chains considered by AL Aslamazov and Larkin (1975). The quantitative
differences between the AL and TDGL models are the following. First, when
approaching the kink, the second derivative $d^{2}V/dI^{2}$ has a different
sign in both models. Second, while the voltage difference between the kinks
follows $\Delta V=V_{n+1}-V_{n}=const$ in the AL model, $\Delta V$ depends on
$n$ in the TDGL model, see Fig. 4(a). The equidistant voltage spectrum in the
AL theory is based on the assumption $I_{\mathrm{c}}\approx I\approx
I^{\ast}$, which implies a steep $I$-$V$ branch and that the voltage increases
solely due to the increase of $n_{\mathrm{v}}$ while the vortex velocity $v$
is assumed to be constant. By contrast, the TDGL model is free of such a
restriction regarding the transport current. The kink voltage increase in the
TDGL theory is caused by the combined effects of the increase of both the
vortex number $n_{\mathrm{v}}$ and the vortex velocity $v$, see the inset in
Fig. 3(a).

Figure 4(a) suggests that the kink voltage for the strip MoSi 214 better
follows the AL model while for the strip MoSi218 it is better described by the
TDGL model. For both our samples $I_{\mathrm{c}}\gtrsim 0.82I^{\ast}$, i.e.
the comparison with the AL theory is justified. We would like to note that in
a different study Bevz  _et al._ (2023) at $T=4.2$ K, we observed
_equidistant_ voltage kinks for $4\,\mu$m-wide MoSi strips with $25$ nm$\times
2\,\mu$m slits. However, those wider constrictions are characterized by
$I_{\mathrm{c}}\approx 0.5I^{\ast}$, which is rather far from
$I_{\mathrm{c}}\approx I^{\ast}$ assumed in the AL model. At the moment, we
have no explanation for that observation, leaving it for a discussion
elsewhere.

Figure 4: (a) Kink voltage versus number of vortices in the slitted MoSi
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.

Finally, referring to the TDGL simulations at $T=0.8T_{\mathrm{c}}$, we note
that the discontinuities in the modeled $dV(I)/dI$ curve are smoother than in
the experiment. We have checked that these differences are not connected with
the vortex jet in our simulations since they also persist for a long and
narrow ($\lesssim 2\xi_{\mathrm{c}}$) slit in an identical strip in which a
vortex jet appears when the number of vortices in the chain exceeds five. The
smoothness of the minima in the $dV(I)/dI$ curve is attributed to the choice
of the parameter $\alpha=1$ in the TDGL equation (Appendix), whose role is
analyzed by Vodolazov and Ustavschikov in their forthcoming work Vodolazov and
Ustavschikov (2022) where the $I$-$V$ curves for slitted MoN thin strips
exhibit similar voltage kinks.

###  IV.3 Vortex counting and velocimetry

A remarkable feature of the $I$-$V$ curves in Fig. 2(a) is that the kinks are
seen up to the FFI onset. Hence, the kink number $n$ and kink voltage $V_{n}$
can be used both for counting the number of vortices $n_{\mathrm{v}}$ and for
velocimetry. Both quantities can be deduced for all currents in the flux-flow
regime. For example, at $V=V_{1}$, a transition occurs from less than one
vortex to less than two vortices crossing the strip. This means that at any
time at $V=V_{1}$, there is _exactly one vortex_ ($n_{\mathrm{v}}=1$) and the
average vortex velocity is $v=wV_{1}/\Phi_{0}$. Extending this relation to any
$V=V_{n}$, one obtains the generalized expression

$v=\displaystyle\frac{wV_{n}}{n_{\mathrm{v}}\Phi_{0}}.$ (2)

We note that at intermediate voltages $V\neq V_{1},V_{2},..$, the situation is
more complicated because there could be $n$ or $(n+1)$ vortices in the strip.

The instability velocity $v^{\ast}$ can be deduced from the instability
voltage $V^{\ast}\approx V_{n}$,

$v^{\ast}=\displaystyle\frac{wV^{\ast}}{n_{\mathrm{v}}\Phi_{0}}.$ (3)

In this way, knowing the number of vortices $n_{\mathrm{v}}$ at each voltage
kink, we can deduce the vortex velocity $v$ at $V=V_{1},V_{2},..$ and make
estimates for $v=v^{\ast}$ at $V=V^{\ast}$ by using Eqs. (2) and (3). The
results of these calculations at $T=5$ K are presented in Fig. 4(b). With
increase of the current, the vortex velocity increases to $v^{\ast}$. The
deduced values $v^{\ast}_{\mathrm{MoSi218}}\approx 12.1$ km/s and
$v^{\ast}_{\mathrm{MoSi214}}\approx 11.8$ km/s are rather close and agree well
with the $v^{\ast}$ values reported for MoSi Budinská  _et al._ (2022) from
previous FFI studies on straight superconducting strips in the presence of
small perpendicular magnetic fields. If we relate the relaxation time
$\tau_{\varepsilon}$ with the healing of the order-parameter wakes behind the
moving vortices, $\tau_{\varepsilon}=w/(v^{\ast}n_{\mathrm{v}})$, we obtain
$\tau_{\varepsilon}^{\mathrm{MoSi218}}\approx 29$ ps and
$\tau_{\varepsilon}^{\mathrm{MoSi214}}\approx 30$ ps, which also agree well
with the previous estimates Budinská  _et al._ (2022); Bezuglyj  _et al._
(2022).

The merit of the introduced approach can be illustrated as follows. In
principle, one could attempt to estimate the vortex velocity from the self-
field $B_{\mathrm{sf}}$ of the current flowing through the constriction.
However, due to the competing effects of the edge barrier and the nonuniform
current-density distribution, $B_{\mathrm{sf}}$ will not necessarily
correspond to the actual number of vortices in the strip and, hence, will
yield an incorrect vortex velocity. For instance, if one took $I^{\ast}\approx
0.5$ mA and $V^{\ast}\approx 70\,\mu$V at the instability point and modeled
the isthmus as a conductor of width $w$ and thickness $d$, the self-field
would be $B_{\mathrm{sf}}=0.5\,\mu_{0}Iw^{-1}\ln(2w/d)$, yielding
$B_{\mathrm{sf}}\backsimeq 1$ mT. Assuming that the motion of vortices occurs
in a $L\backsimeq 50$ nm-wide isthmus “channel”, Eq. (1) yields an
unrealistically high $v^{\ast}=V/(BL)\thickapprox 1400$ km/s. This
overestimation stems from neglecting the strong barrier suppression and the
current-crowding in the slit apex area. An accurate calculation of the self-
field for geometry-dependent nonuniform current distributions is a complex
problem Clem and Berggren (2011), which is further complicated by the unknown
width of the vortex “channel”.

The presented approach should be applicable to superconductors with weak
vortex pinning, such as NbGe Babic  _et al._ (2004), MoGe Kunchur  _et al._
(2012), WSi Zhang  _et al._ (2016), NbC Dobrovolskiy  _et al._ (2020b), MoN
Vodolazov and Ustavschikov (2022), and epitaxial Nb films Dobrovolskiy  _et
al._ (2018). Qualitatively, intrinsic vortex pinning is expected to lead to
the smearing of the voltage kinks Vodolazov and Ustavschikov (2022) and to
promote the FFI onset at a smaller mean vortex velocity Silhanek  _et al._
(2012); Dobrovolskiy  _et al._ (2017).

Finally, we would like to comment on the kinks observed previously in the
$I$-$V$ curves of Nb weak links Boone  _et al._ (1977) and YBCO constrictions
de Nivelle  _et al._ (1993); De Nivelle  _et al._ (1994), which were
interpreted within the framework of the AL model. By using Eq. (2) one obtains
$v^{\mathrm{Nb}}_{1}=25$ km/s Boone  _et al._ (1977) and
$v^{\mathrm{YBCO}}_{1}=35$ km/s de Nivelle  _et al._ (1993); De Nivelle  _et
al._ (1994) which are by more than an order of magnitude larger than the
$v^{\ast}$ values for these materials Peroz and Villard (2005); Leo  _et al._
(2011); Bezuglyj  _et al._ (2019); Doettinger  _et al._ (1994); Xiao and
Ziemann (1996); Wördenweber  _et al._ (2012). We attribute this discrepancy to
a different origin of the kinks. For instance, it was argued that the kinks
for the YBCO constrictions could be caused by a rearrangement of the moving
vortex ensemble Pedyash  _et al._ (1995). A detailed inspection of the initial
part of the $I$-$V$ curve for those YBCO constrictions (Fig. 5 in Ref. Pedyash
_et al._ (1995)) reveals a kink at $V_{1}\sim 60\,\mu$V which corresponds to
$v_{1}\sim 3$ km/s being in-line with the $v^{\ast}$ values deduced for YBCO
from FFI studies Doettinger  _et al._ (1994); Xiao and Ziemann (1996). In this
way, acquisition of the initial part of the $I$-$V$ curves with a small
current step and their careful inspection are essential for the data
interpretation.

###  IV.4 Implications for superconducting devices

Superconducting microstrips appear as interesting platforms for information
processing and sensing. They can find applications in fluxonics, where one can
conceive vortex gates, couplers and dividers with several inputs and outputs,
as inspired by the vortex configurations in Fig. 3(g). The placement of Hall
voltage leads on both sides of the slit should allow for zero or non-zero
transverse voltage outputs depending on whether the vortices cross or do not
cross the line between the Hall voltage leads Bezuglyj  _et al._ (2022). The
palette of the vortex patterns can be enriched by the fabrication of defects
at the opposite strip edge, allowing for steering of vortex-antivortex pairs
and their annihilation at a given sample point.

Fast moving fluxons can be used for the Cherenkov-like generation of spin
waves Dobrovolskiy  _et al._ (2019b, 2021a) in superconductor/ferromagnet
hybrid systems with prospects for magnon based information processing Chumak
_et al_ (2022). Due to the possible 2D (vortex jet) and 1D (vortex river)
arrangements in the ensemble of fast-moving vortices, excitation of two spin-
wave beams or a single spin-wave beam could be anticipated.

The high sensitivity of flux steering to local perturbations may also be used
to probe the magnetic state of objects placed at the superconductor edge
Dobrovolskiy  _et al._ (2010), which may enable a characteristic entry of
fluxons. In this regard, of particular interest are complex-shaped magnetic 3D
nanoarchitectures Dobrovolskiy  _et al._ (2021b); Makarov  _et al._ (2022),
whose characterization by local-field techniques is challenging.

Compared to other superconducting quantum sensors, such as transition edge
sensors Irwin (1995) and superconducting tunneling junctions Wood and White
(1969), microwire detectors can be operated at higher temperatures and their
response time is orders of magnitude faster than that of kinetic inductance
detectors Day  _et al._ (2003). While superconducting nanowire detectors
Semenov  _et al._ (2001) have already found wide application as highly
sensitive photon detectors for quantum communication, information processing,
infrared astronomy, LIDAR and molecular fluorescence correlation microscopy,
our study shows that the dynamics of a single flux quantum can be detected and
counted in geometries that are more than two orders of magnitude wider than
$\xi$. Relaxation times around 100 ps would clearly comply with the
requirements of fluorescence correlation microscopy, atomic or particle
detection, molecular beam science, as well as mass spectrometry Fein  _et al._
(2019). Microwires are easily integrated to the fabrication of future large-
area detectors Steinhauer  _et al._ (2021).

##  V Conclusion

Summing up, we have introduced an approach for the quantitative determination
of the number of vortices and their velocity in superconducting strips from
the $I$-$V$ curves measured at zero magnetic field. The approach is based on
the Aslamazov-Larkin prediction of kinks in the $I$-$V$ curves of wide and
short constrictions when the number of fluxons crossing the constriction is
increased by one. We have fabricated MoSi microstrips with slits as vortex
nucleation centers and we have observed a crossover from single- to multi-
fluxon dynamics with increase of the transport current. Associating the number
of kinks $n$ with the number of vortices $n_{\mathrm{v}}$ one can calculate
the vortex velocity. If the instability voltage reaches $V^{\ast}\approx
V_{n}$, Eq. (3) can be used to derive the maximal vortex velocity $v^{\ast}$.
If $V^{\ast}$ notably differs from $V_{n}$, $v^{\ast}$ can be estimated by
using Eq. (3) with $n_{\mathrm{v}}=n$ and $nV_{1}=V\backsimeq V^{\ast}$. We
find that our experimental values at $T=5$ K, $v^{\ast}\simeq 12\,$km/s and
$\tau_{\varepsilon}\simeq 30$ ps agree well with those from prior flux-flow
instability studies in the presence of a small perpendicular magnetic field.
The TDGL simulations have additionally unveiled a transition from a vortex
chain over a vortex jets to a vortex river with increase of the number of
vortices. In all, our findings are essential for the development of quantum
technologies based on 1D and 2D few-fluxon devices.

###### Acknowledgements.

The authors are very grateful to Denis Yu. Vodolazov for the TDGL modeling
results and numerous fruitful discussions. V.M.B. acknowledges the European
Cooperation in Science and Technology (E-COST) for support via Grants E-COST-
STSM-CA19108-48969, E-COST-GRANT-CA16218-5759aa9b, and E-COST-GRANT-
CA16218-46e403c7. M.Yu.M. acknowledges the Wolfgang Pauli Institute (WPI)
Vienna for the scholarship within the framework of the Pauli Ukraine Project,
the scholarship from the Krzysztof Skubiszewski Foundation, and the IEEE
Magnetics Society for support via the STCU Project No. 9918. B.B. and S.L.C.
acknowledge financial support by the Vienna Doctoral School in Physics (VDSP).
S.O.S. acknowledges the OeAD (Austria’s Agency for Education and
Internationalisation) for support through the Ernst Mach Grant, EM UKR.
CzechNanoLab project LM2018110 is gratefully acknowledged for financial
support of the sample fabrication at CEITEC Nano Research Infrastructure. This
research is funded in whole, or in part, by the Austrian Science Fund (FWF),
Grant No. I 4865-N. Support by E-COST via COST Actions CA19108 (HiSCALE) and
CA21144 (SuperQuMap) is gratefully acknowledged.

## Appendix

The spatiotemporal evolution of the superconducting order parameter
$\Delta=|\Delta|\exp(i\phi)$ in the strips was modeled numerically on the
basis of the TDGL equation

$\displaystyle\alpha\frac{\pi\hbar}{8k_{\mathrm{B}}T_{\mathrm{c}}}\left(\frac{\partial}{\partial
t}+\frac{2ie\varphi}{\hbar}\right)\Delta=$
$\displaystyle\frac{\pi}{8}\xi_{\mathrm{c}}^{2}\left(\nabla-i\frac{2e}{\hbar
c}A\right)^{2}\Delta+\left(1-\frac{T}{T_{\mathrm{c}}}-\frac{|\Delta|^{2}}{\Delta_{\mathrm{GL}}^{2}}\right)\Delta$
(4)

coupled with the equation for the electrostatic potential $\varphi$

$\mathrm{div}\,\mathbf{j}=\mathrm{div}\,(\mathbf{j}_{\mathrm{s}}+\mathbf{j}_{\mathrm{n}})=\mathrm{div}\left(\frac{\sigma_{\mathrm{n}}}{e}\frac{\pi|\Delta|^{2}q_{\mathrm{s}}}{4k_{\mathrm{B}}T_{\mathrm{c}}}-\sigma_{\mathrm{n}}\nabla\varphi\right)=0,$
(5)

which follows from the continuity of the current lines.

In Eqs. (Appendix) and (5), $\phi$ is the phase of the superconducting order
parameter, the phenomenological coefficient $\alpha$ controls the relaxation
time of the superconducting order parameter, $\tau_{|\Delta|}\sim\alpha$,
$T_{\mathrm{c}}$ is the superconducting transition temperature,
$\xi_{\mathrm{c}}=\sqrt{\hbar D/k_{\mathrm{B}}T_{\mathrm{c}}}$,
$\sigma_{\mathrm{n}}=2e^{2}DN_{0}$ is the normal-state conductivity, $N_{0}$
is the electron density of states on the Fermi surface per electron spin,
$\Delta_{\mathrm{GL}}=3.06k_{\mathrm{B}}T_{\mathrm{c}}$, $A$ is the vector
potential, and $q_{\mathrm{s}}=(\nabla\varphi-2eA/\hbar c)$ is proportional to
the supervelocity.

Details on the procedure employed for solving Eqs. (Appendix) and (5) were
reported in Refs. Budinská  _et al._ (2022); Ustavschikov  _et al._ (2022). In
contrast to Refs. Budinská  _et al._ (2022); Ustavschikov  _et al._ (2022),
the TDGL in the present study was solved in neglect of overheating effects,
since these are only essential very close to the FFI point (regime III). This
simplification is justified since the central observation of our studies
(kinks in the $I$-$V$ curves) is related to the low-dissipative flux-flow
regime (II).

The superconducting strip was modeled as a rectangular polygon with dimensions
$200\xi_{\mathrm{c}}\times 130\xi_{\mathrm{c}}$ ($x\times y$). The slit was
modeled as a rectangular region of size $50\xi_{\mathrm{c}}\times
7\xi_{\mathrm{c}}$ ($x\times y$) with a local suppression of $T_{\mathrm{c}}$.
It was also checked that the modeling of wide slits ($\gtrsim
2\xi_{\mathrm{c}}$) as geometrical defects (i.e. no material in the slit
region) only leads to small quantitative differences in the $I$-$V$ curves but
is much less time-efficient because of the more complicated boundary
conditions around the slit. The calculations were done for $\alpha=1$.

## References

  * Wördenweber  _et al._ (2012) R. Wördenweber, E. Hollmann, J. Schubert, R. Kutzner, and G. Panaitov, “Regimes of flux transport at microwave frequencies in nanostructured high-${T}_{c}$ films,” Phys. Rev. B 85, 064503–1–6 (2012).
  * Grimaldi  _et al._ (2015) G. Grimaldi, A. Leo, P. Sabatino, G. Carapella, A. Nigro, S. Pace, V. V. Moshchalkov, and A. V. Silhanek, “Speed limit to the Abrikosov lattice in mesoscopic superconductors,” Phys. Rev. B 92, 024513 (2015).
  * Jelić  _et al._ (2016) Ž L. Jelić, M. V. Milošević, and A. V. Silhanek, “Velocimetry of superconducting vortices based on stroboscopic resonances,” Sci. Rep. 6, 35687 (2016).
  * Embon  _et al._ (2017) L. Embon, Y. Anahory, Z. L. Jelic, E. O. Lachman, Y. Myasoedov, M. E. Huber, G. P. Mikitik, A. V. Silhanek, M. V. Milosevic, A. Gurevich, and E. Zeldov, “Imaging of super-fast dynamics and flow instabilities of superconducting vortices,” Nat. Commun. 8, 85 (2017).
  * Sheikhzada and Gurevich (2017) A. Sheikhzada and A. Gurevich, “Dynamic transition of vortices into phase slips and generation of vortex-antivortex pairs in thin film josephson junctions under dc and ac currents,” Phys. Rev. B 95, 214507 (2017).
  * Shklovskij  _et al._ (2017) V. A. Shklovskij, A. P. Nazipova, and O. V. Dobrovolskiy, “Pinning effects on self-heating and flux-flow instability in superconducting films near ${T}_{c}$,” Phys. Rev. B 95, 184517 (2017).
  * Kogan and Prozorov (2020) V. G. Kogan and R. Prozorov, “Interaction between moving Abrikosov vortices in type-II superconductors,” Phys. Rev. B 102, 024506 (2020).
  * Dobrovolskiy  _et al._ (2020a) O.V. Dobrovolskiy, E. Begun, V.M. Bevz, R. Sachser, and M. Huth, “Upper frequency limits for vortex guiding and ratchet effects,” Phys. Rev. Appl. 13, 024012 (2020a).
  * Pathirana and Gurevich (2021) W. P. M. R. Pathirana and A. Gurevich, “Effect of random pinning on nonlinear dynamics and dissipation of a vortex driven by a strong microwave current,” Phys. Rev. B 103, 184518 (2021).
  * Kogan and Nakagawa (2022) V. G. Kogan and N. Nakagawa, “Dissipation of moving vortices in thin films,” Phys. Rev. B 105, L020507 (2022).
  * Dobrovolskiy  _et al._ (2020b) O. V. Dobrovolskiy, D. Yu Vodolazov, F. Porrati, R. Sachser, V. M. Bevz, M. Yu Mikhailov, A. V. Chumak, and M. Huth, “Ultra-fast vortex motion in a direct-write Nb-C superconductor,” Nat. Commun. 11, 3291 (2020b).
  * Budinská  _et al._ (2022) B. Budinská, B. Aichner, D. Yu. Vodolazov, M. Yu. Mikhailov, F. Porrati, M. Huth, A.V. Chumak, W. Lang, and O.V. Dobrovolskiy, “Rising speed limits for fluxons via edge-quality improvement in wide MoSi thin films,” Phys. Rev. Appl. 17, 034072 (2022).
  * Dobrovolskiy (2023) O.V. Dobrovolskiy, “Fast dynamics of vortices in superconductors,” (Elsevier, 2023), ISBN 978-0-323-90800-9.00015-9.
  * Korneeva  _et al._ (2020) Yu. P. Korneeva, N.N. Manova, I.N. Florya, M. Yu. Mikhailov, O.V. Dobrovolskiy, A.A. Korneev, and D. Yu. Vodolazov, “Different single-photon response of wide and narrow superconducting ${\mathrm{Mo}}_{x}{\mathrm{Si}}_{1-x}$ strips,” Phys. Rev. Appl. 13, 024011 (2020).
  * Charaev  _et al._ (2020) I. Charaev, Y. Morimoto, A. Dane, A. Agarwal, M. Colangelo, and K. K. Berggren, “Large-area microwire MoSi single-photon detectors at 1550 nm wavelength,” Appl. Phys. Lett. 116, 242603 (2020).
  * Chiles  _et al._ (2020) J. Chiles, S. M. Buckley, A. Lita, V. B. Verma, J. Allmaras, B. Korzh, M. D. Shaw, J. M. Shainline, R. P. Mirin, and S. W. Nam, “Superconducting microwire detectors based on WSi with single-photon sensitivity in the near-infrared,” Appl. Phys. Lett. 116, 242602 (2020).
  * Vodolazov (2017) D. Yu. Vodolazov, “Single-photon detection by a dirty current-carrying superconducting strip based on the kinetic-equation approach,” Phys. Rev. Appl. 7, 034014 (2017).
  * Chen  _et al._ (2021) Y.-A. Chen, Q. Zhang, T.-Y. Chen, W.-Q. Cai, S.-K. Liao, J. Zhang, K. Chen, J. Yin, J.-G. Ren, Z. Chen, S.-L. Han, Q. Yu, K. Liang, F. Zhou, X. Yuan, M.-S. Zhao, T.-Y. Wang, X. Jiang, L. Zhang, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, C.-Y. Lu, R. Shu, J.-Y. Wang, L. Li, N.-L. Liu, F. Xu, X.-B. Wang, C.-Z. Peng, and J.-W. Pan, “An integrated space-to-ground quantum communication network over 4,600 kilometres,” Nature 589, 214–219 (2021).
  * Fein  _et al._ (2019) Y. Y. Fein, P. Geyer, P. Zwick, F. Kiałka, S. Pedalino, M. Mayor, S. Gerlich, and M. Arndt, “Quantum superposition of molecules beyond 25 kDa,” Nat. Phys. 15, 1242–1245 (2019).
  * Steinhauer  _et al._ (2021) S. Steinhauer, S. Gyger, and V. Zwiller, “Progress on large-scale superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 118, 100501 (2021).
  * Lin  _et al._ (2013) S.-Z. Lin, O. Ayala-Valenzuela, R. D. McDonald, L. N. Bulaevskii, T. G. Holesinger, F. Ronning, N. R. Weisse-Bernstein, T. L. Williamson, A. H. Mueller, M. A. Hoffbauer, M. W. Rabin, and M. J. Graf, “Characterization of the thin-film NbN superconductor for single-photon detection by transport measurements,” Phys. Rev. B 87, 184507 (2013).
  * Caputo  _et al._ (2017a) M. Caputo, C. Cirillo, and C. Attanasio, “NbRe as candidate material for fast single photon detection,” Appl. Phys. Lett. 111, 192601 (2017a).
  * Hofer and Haberkorn (2021) J. A. Hofer and N. Haberkorn, “Flux flow velocity instability and quasiparticle relaxation time in nanocrystalline $\beta$-W thin films,” Thin Solid Films 730, 138690 (2021).
  * Liu  _et al._ (2021) Z. Liu, B. Luo, L. Zhang, B. Hou, and D. Wang, “Vortex dynamics in amorphous MoSi superconducting thin films,” Supercond. Sci. Technol. (2021).
  * Cirillo  _et al._ (2021) C. Cirillo, V. Granata, A. Spuri, A. Di Bernardo, and C. Attanasio, “NbReN: A disordered superconductor in thin film form for potential application as superconducting nanowire single photon detector,” Phys. Rev. Mater. 5, 085004 (2021).
  * Larkin and Ovchinnikov (1975) A. I. Larkin and Yu. N. Ovchinnikov, “Nonlinear conductivity of superconductors in the mixed state,” J. Exp. Theor. Phys. 41, 960 (1975).
  * Bezuglyj and Shklovskij (1992) A.I. Bezuglyj and V.A. Shklovskij, “Effect of self-heating on flux flow instability in a superconductor near $T_{c}$,” Physica C 202, 234 (1992).
  * Peroz and Villard (2005) Ch. Peroz and C. Villard, “Flux flow properties of niobium thin films in clean and dirty superconducting limits,” Phys. Rev. B 72, 014515–1–6 (2005).
  * Leo  _et al._ (2011) A. Leo, G. Grimaldi, R. Citro, A. Nigro, S. Pace, and R. P. Huebener, “Quasiparticle scattering time in niobium superconducting films,” Phys. Rev. B 84, 014536–1–7 (2011).
  * Attanasio and Cirillo (2012) C. Attanasio and C. Cirillo, “Quasiparticle relaxation mechanisms in superconductor/ferromagnet bilayers,” J. Phys.: Cond. Matt. 24, 083201 (2012).
  * Dobrovolskiy  _et al._ (2019a) O. V. Dobrovolskiy, V. M. Bevz, E. Begun, R. Sachser, R. V. Vovk, and M. Huth, “Fast dynamics of guided magnetic flux quanta,” Phys. Rev. Appl. 11, 054064 (2019a).
  * Leo  _et al._ (2020) A. Leo, A. Nigro, V. Braccini, G. Sylva, A. Provino, A. Galluzzi, M. Polichetti, C. Ferdeghini, M. Putti, and G. Grimaldi, “Flux flow instability as a probe for quasiparticle energy relaxation time in Fe-chalcogenides,” Supercond. Sci. Technol. 33, 104005 (2020).
  * Dobrovolskiy and Huth (2015) O. V. Dobrovolskiy and M. Huth, “Dual cut-off direct current-tunable microwave low-pass filter on superconducting Nb microstrips with asymmetric nanogrooves,” Appl. Phys. Lett. 106, 142601–1–5 (2015).
  * Pompeo and Silva (2008) N. Pompeo and E. Silva, “Reliable determination of vortex parameters from measurements of the microwave complex resistivity,” Phys. Rev. B 78, 094503–1–10 (2008).
  * Lösch  _et al._ (2019) S. Lösch, A. Alfonsov, O. V. Dobrovolskiy, R. Keil, V. Engemaier, S. Baunack, G. Li, O. G. Schmidt, and D. Bürger, “Microwave radiation detection with an ultra-thin free-standing superconducting niobium nanohelix,” ACS Nano 13, 2948 (2019).
  * Pokusinskyi  _et al._ (2022) A. Pokusinskyi, A. Kasatkin, S. Futimsky, O. Kalenyuk, O. Boliasova, and A. Shapovalov, “Microwave surface resistance in nanostructured high-Tc superconductor films,” J. Appl. Phys. 132, 233904 (2022).
  * Zhang  _et al._ (2016) X. Zhang, A. Engel, Q. Wang, A. Schilling, A. Semenov, M. Sidorova, H.-W. Hübers, I. Charaev, K. Ilin, and M. Siegel, “Characteristics of superconducting tungsten silicide WxSi1-x for single photon detection,” Phys. Rev. B 94, 174509 (2016).
  * Sidorova  _et al._ (2021) M. Sidorova, A. D. Semenov, H.-W. Hübers, S. Gyger, S. Steinhauer, X. Zhang, and A. Schilling, “Magnetoconductance and photoresponse properties of disordered NbTiN films,” Phys. Rev. B 104, 184514 (2021).
  * Kaplan  _et al._ (1976) S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, S. Jafarey, and D. J. Scalapino, “Quasiparticle and phonon lifetimes in superconductors,” Phys. Rev. B 14, 4854–4873 (1976).
  * Armenio  _et al._ (2007) A. A. Armenio, C. Bell, J. Aarts, and C. Attanasio, “High-velocity instabilities in the vortex lattice of Nb/permalloy bilayers,” Phys. Rev. B 76, 054502 (2007).
  * Caputo  _et al._ (2017b) M. Caputo, C. Cirillo, S. Voltan, A. M. Cucolo, J. Aarts, and C. Attanasio, “Influence of the magnetic configuration on the vortex-lattice instability in Nb/permalloy bilayers,” Phys. Rev. B 96, 174519 (2017b).
  * Doettinger  _et al._ (1994) S. G. Doettinger, R. P. Huebener, R. Gerdemann, A. Kühle, S. Anders, T. G. Träuble, and J. C. Villégier, “Electronic instability at high flux-flow velocities in high-${T}_{c}$ superconducting films,” Phys. Rev. Lett. 73, 1691–1694 (1994).
  * Vodolazov (2019) D. Yu. Vodolazov, “Flux-flow instability in a strongly disordered superconducting strip with an edge barrier for vortex entry,” Supercond. Sci. Technol. 32, 115013 (2019).
  * Ustavshikov  _et al._ (2020) S.S. Ustavshikov, Yu.N. Nozdrin, M.Yu Levichev, A.V. Okomel’kov, I.Y. Pashenkin, P.A. Yunin, A.M. Klushin, and D.Yu. Vodolazov, “Photoresponse of current-biased superconductor/normal metal strip with large ratio of resistivities,” J. Phys. D 53, 395301 (2020).
  * Ustavschikov  _et al._ (2020) S. S. Ustavschikov, M. Yu. Levichev, I. Yu. Pashenkin, A. M. Klushin, and D. Yu. Vodolazov, “Approaching depairing current in dirty thin superconducting strip covered by low resistive normal metal,” Supercond. Sci. Technol. 34, 015004 (2020).
  * Ejrnaes  _et al._ (2022) M. Ejrnaes, C. Cirillo, D. Salvoni, F. Chianese, C. Bruscino, P. Ercolano, A. Cassinese, C. Attanasio, G. P. Pepe, and L. Parlato, “Single photon detection in nbre superconducting microstrips,” Applied Physics Letters 121, 262601 (2022).
  * Brandt (1995) E. H. Brandt, “The flux-line lattice in superconductors,” Rep. Progr. Phys. 58, 1465 (1995).
  * Aladyshkin  _et al._ (2001) A.Yu. Aladyshkin, A. S. Mel’nikov, I. A. Shereshevsky, and I. D. Tokman, “What is the best gate for vortex entry into type-II superconductor?” Physica C 361, 67 (2001).
  * Bezuglyj  _et al._ (2022) A. I. Bezuglyj, V. A. Shklovskij, B. Budinská, B. Aichner, V. M. Bevz, M. Yu. Mikhailov, D. Yu. Vodolazov, W. Lang, and O. V. Dobrovolskiy, “Vortex jets generated by edge defects in current-carrying superconductor thin strips,” Phys. Rev. B 105, 214507 (2022).
  * Bezuglyj  _et al._ (2019) A. I. Bezuglyj, V. A. Shklovskij, R. V. Vovk, V. M. Bevz, M. Huth, and O. V. Dobrovolskiy, “Local flux-flow instability in superconducting films near ${T}_{c}$,” Phys. Rev. B 99, 174518 (2019).
  * Aslamazov and Larkin (1975) L. G. Aslamazov and A. I. Larkin, “Josephson effect in wide superconducting bridges,” Sov.-Phys. JETP 41, 381 (1975).
  * Korneeva  _et al._ (2014) Yu. P. Korneeva, M. Yu. Mikhailov, Yu. P. Pershin, N. N. Manova, A. V. Divochiy, Yu. B. Vakhtomin, A. A. Korneev, K. V. Smirnov, A. G. Sivakov, A. Yu. Devizenko, and G. N. Goltsman, “Superconducting single-photon detector made of MoSi film,” Supercond. Sci. Technol. 27, 095012 (2014).
  * Caloz  _et al._ (2018) M. Caloz, M. Perrenoud, C. Autebert, B. Korzh, M. Weiss, Ch. Schönenberger, R. J. Warburton, H. Zbinden, and F. Bussières, “High-detection efficiency and low-timing jitter with amorphous superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 112, 061103 (2018).
  * Haberkorn (2022) N. Haberkorn, “Thickness dependence of the flux-flow velocity and the vortex instability in nanocrystalline $\gamma$-Mo2N thin films,” Thin Solid Films 759, 139475 (2022).
  * Li  _et al._ (2016) J. Li, R. A. Kirkwood, L. J. Baker, D. Bosworth, K. Erotokritou, A. Banerjee, R. M. Heath, C. M. Natarajan, Z. H. Barber, M. Sorel, and R. H. Hadfield, “Nano-optical single-photon response mapping of waveguide integrated molybdenum silicide (MoSi) superconducting nanowires,” Optics Express 24, 13931–13938 (2016).
  * Gourgues  _et al._ (2019) R. Gourgues, J. W. N. Los, J. Zichi, J. Chang, N. Kalhor, G. Bulgarini, S. N. Dorenbos, V. Zwiller, and I. E. Zadeh, “Superconducting nanowire single photon detectors operating at temperature from 4 to 7 K,” Opt. Expr. 27, 24601–24609 (2019).
  * Polakovic  _et al._ (2020) T. Polakovic, W. R. Armstrong, V. Yefremenko, J. E. Pearson, K. Hafidi, G. Karapetrov, Z.-E. Meziani, and V. Novosad, “Superconducting nanowires as high-rate photon detectors in strong magnetic fields,” Nucl. Instr. Meth. Phys. Res. 959, 163543 (2020).
  * Friesen and Gurevich (2001) M. Friesen and A. Gurevich, “Nonlinear current flow in superconductors with restricted geometries,” Phys. Rev. B 63, 064521 (2001).
  * Clem and Berggren (2011) J. R. Clem and K. K. Berggren, “Geometry-dependent critical currents in superconducting nanocircuits,” Phys. Rev. B 84, 174510 (2011).
  * Adami  _et al._ (2013) O.-A. Adami, D. Cerbu, D. Cabosart, M. Motta, J. Cuppens, W. A. Ortiz, V. V. Moshchalkov, B. Hackens, R. Delamare, J. Van de Vondel, and A. V. Silhanek, “Current crowding effects in superconducting corner-shaped Al microstrips,” Appl. Phys. Lett. 102, 052603 (2013).
  * Glazman (1986) L. I. Glazman, “Vortex induced transverse voltage within film,” Sov. J. Low Temp. Phys. 12, 389 (1986).
  * Dobrovolskiy and Huth (2012) O. V. Dobrovolskiy and M. Huth, “Crossover from dirty to clean superconducting limit in dc magnetron-sputtered thin Nb films,” Thin Solid Films 520, 5985–5990 (2012).
  * Kes and Tsuei (1983) P. H. Kes and C. C. Tsuei, “Two-dimensional collective flux pinning, defects, and structural relaxation in amorphous superconducting films,” Phys. Rev. B 28, 5126–5139 (1983).
  * Fomin  _et al._ (2022) V. M. Fomin, R. O. Rezaev, and O. V. Dobrovolskiy, “Topological transitions in ac/dc-driven superconductor nanotubes,” Sci. Rep. 12, 10069 (2022).
  * Gurevich and Mints (1987) A. Vl. Gurevich and R. G. Mints, “Self-heating in normal metals and superconductors,” Rev. Mod. Phys. 59, 941–999 (1987).
  * Sivakov  _et al._ (2003) A. G. Sivakov, A. M. Glukhov, A. N. Omelyanchouk, Y. Koval, P. Müller, and A. V. Ustinov, “Josephson behavior of phase-slip lines in wide superconducting strips,” Phys. Rev. Lett. 91, 267001 (2003).
  * Bevz  _et al._ (2023) V. M. Bevz, B. Budinska, S. Lamb-Camarena, S. O. Shpilinska, C. Schmid, M. Yu. Mikhailov, W. Lang, and O. V. Dobrovolskiy, “Vortex chains and vortex jets in MoSi microbridges,” Phys. Stat. Sol. – Rap. Res. Lett., accepted (2023).
  * Vodolazov and Ustavschikov (2022) D.Yu. Vodolazov and S. S. Ustavschikov, Private communication, (2022).
  * Babic  _et al._ (2004) D. Babic, J. Bentner, C. Sürgers, and C. Strunk, “Flux-flow instabilities in amorphous ${\mathrm{Nb}}_{0.7}{\mathrm{Ge}}_{0.3}$ microbridges,” Phys. Rev. B 69, 092510–1–4 (2004).
  * Kunchur  _et al._ (2012) M. N. Kunchur, M. Liang, and A. Gurevich, “Thermally activated dynamics of spontaneous perpendicular vortices tuned by parallel magnetic fields in thin superconducting films,” Phys. Rev. B 86, 024521 (2012).
  * Dobrovolskiy  _et al._ (2018) O. V. Dobrovolskiy, R. Sachser, M. Huth, V. A. Shklovskij, R. V. Vovk, V. M. Bevz, and M. I. Tsindlekht, “Radiofrequency generation by coherently moving fluxons,” Appl. Phys. Lett. 112, 152601 (2018).
  * Silhanek  _et al._ (2012) A. V. Silhanek, A. Leo, G. Grimaldi, G. R. Berdiyorov, M. V Milosevic, A. Nigro, S. Pace, N. Verellen, W. Gillijns, V. Metlushko, B. Ilić, X. Zhu, and V. V. Moshchalkov, “Influence of artificial pinning on vortex lattice instability in superconducting films,” New J. Phys. 14, 053006 (2012).
  * Dobrovolskiy  _et al._ (2017) O. V. Dobrovolskiy, V. A. Shklovskij, M. Hanefeld, M. Zörb, L. Köhs, and M. Huth, “Pinning effects on flux flow instability in epitaxial Nb thin films,” Supercond. Sci. Technol. 30, 085002 (2017).
  * Boone  _et al._ (1977) B. Boone, C. Arrington, Li-Kong Wang, and B. Deaver, “Properties of new types of niobium weak links,” IEEE Trans. Magnet. 13, 735–738 (1977).
  * de Nivelle  _et al._ (1993) M. J. M. E. de Nivelle, G. J. Gerritsma, and H. Rogalla, “Thermally activated coherent vortex motion in ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta}}$ thin film microbridges,” Phys. Rev. Lett. 70, 1525–1528 (1993).
  * De Nivelle  _et al._ (1994) M. J. M. E. De Nivelle, G. J. Gerritsma, and H. Rogalla, “Coherent vortex motion in YBa2Cu3O7-δ nanobridges prepared by a substrate-etching technique,” Physica C 233, 185–194 (1994).
  * Xiao and Ziemann (1996) Z. L. Xiao and P. Ziemann, “Vortex dynamics in YBCO superconducting films: Experimental evidence for an instability in the vortex system at high current densities,” Phys. Rev. B 53, 15265–15271 (1996).
  * Pedyash  _et al._ (1995) M.V. Pedyash, G.J. Gerritsma, D.M.A. Blank, and H. Rogalla, “Coherent vortex motion in superconducting nanobridges based on YBaCuO thin films,” IEEE Trans. Appl. Supercond. 5, 1387–1390 (1995).
  * Dobrovolskiy  _et al._ (2019b) O. V. Dobrovolskiy, R. Sachser, T. Brächer, T. Böttcher, V. V. Kruglyak, R. V. Vovk, V. A. Shklovskij, M. Huth, B. Hillebrands, and A. V. Chumak, “Magnon-fluxon interaction in a ferromagnet/superconductor heterostructure,” Nat. Phys. 15, 477 (2019b).
  * Dobrovolskiy  _et al._ (2021a) O. V. Dobrovolskiy, Q. Wang, D. Yu. Vodolazov, B. Budinska, R. Sachser, A.V. Chumak, M. Huth, and A. I. Buzdin, “Cherenkov radiation of spin waves by ultra-fast moving magnetic flux quanta,” arXiv:2103.10156 (2021a).
  * Chumak  _et al_ (2022) A. V. Chumak  _et al_ , “Advances in magnetics roadmap on spin-wave computing,” IEEE Trans. Magnet. 58, 1–72 (2022).
  * Dobrovolskiy  _et al._ (2010) O. V. Dobrovolskiy, M. Huth, and V. A. Shklovskij, “Anisotropic magnetoresistive response in thin Nb films decorated by an array of Co stripes,” Supercond. Sci. Technol. 23, 125014–1–5 (2010).
  * Dobrovolskiy  _et al._ (2021b) O. V. Dobrovolskiy, N. R. Vovk, A. V. Bondarenko, S. A. Bunyaev, S. Lamb-Camarena, N. Zenbaa, R. Sachser, S. Barth, K. Y. Guslienko, A. V. Chumak, M. Huth, and G. N. Kakazei, “Spin-wave eigenmodes in direct-write 3D nanovolcanoes,” Appl. Phys. Lett. 118, 132405 (2021b).
  * Makarov  _et al._ (2022) D. Makarov, O. M. Volkov, A. Kakay, O. V. Pylypovskyi, B. Budinska, and O. V. Dobrovolskiy, “New dimension in magnetism and superconductivity: 3D and curvilinear nanoarchitectures,” Adv. Mater. 34, 2101758 (2022).
  * Irwin (1995) K. D. Irwin, “An application of electrothermal feedback for high resolution cryogenic particle detection,” Applied Physics Letters 66, 1998–2000 (1995).
  * Wood and White (1969) G. H. Wood and B. L. White, “Pulses induced in tunneling currents between superconductors by alpha-particle bombardment,” Appl. Phys. Lett. 15, 237–239 (1969).
  * Day  _et al._ (2003) P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, “A broadband superconducting detector suitable for use in large arrays,” Nature 425, 817–821 (2003).
  * Semenov  _et al._ (2001) A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Physica C 351, 349 – 356 (2001).
  * Ustavschikov  _et al._ (2022) S. S. Ustavschikov, M. Yu. Levichev, I. Yu. Pashenkin, N. S. Gusev, S. A. Gusev, and D. Yu. Vodolazov, “Negative differential resistance and shapiro steps in superconducting MoN strip with slit,” JETP Lett. 135, 226 (2022).