Abstract
The topological defects, vortices in bulk superconductors (SCs) and phase slips in lowdimensional SCs are known to lead to the occurrence of a finite resistance. We report on a topological transition between the both types of topological defects under a strong transport current in an open SC nanotube with a submicronscale inhomogeneity of the normaltothesurface component of the applied magnetic field. When the magnetic field is orthogonal to the axis of the nanotube, which carries the transport current in the azimuthal direction, the phaseslip regime is characterized by the vortex/antivortex lifetime ∼ 10^{−14} s versus the vortex lifetime ∼ 10^{−11} s for vortex chains in the halftubes, and the induced voltage shows a pulse as a function of the magnetic field. The topological transition between the vortexchain and phaseslip regimes determines the magneticfield–voltage and current–voltage characteristics of curved SC nanomembranes to pursue highperformance applications in advanced electronics and quantum computing.
Similar content being viewed by others
Introduction
Topological defects in a superconductor, where the order parameter locally vanishes and its phase has no definite value, lead to the emergence of a finite resistance: “the superconductor is no longer ‘superconducting’ in a practical sense: it offers resistivity to the current!”^{1}. The most wellknown topological defects are vortices (antivortices) in a bulk superconductor, where the superconducting order parameter decreases from a certain value in the region far from a defect to zero on a line called vortex core^{2}. The phase of the order parameter is not defined on the vortex core. When encircling a vortex core, the phase of the order parameter changes by ±2π for ordinary vortices/antivortices or by an integer multiple of ±2πn (n = 2, 3…) for giant vortices/antivortices.
Under certain circumstances, topological defects occur in confined superconductor structures. The total phase advance by a multiple of 2π around a topological defect is called phase slip. The concept of the phase slippage in the resistive state of the narrow (quasi1D) superconductor filaments was introduced in ref. ^{3}. At the phaseslip event, both real and imaginary parts of the superconducting order parameter vanish at a point along the filament. The analysis in ref. ^{3} was based on the thermalfluctuationdominated regime of the occurrence of phase slips near the critical temperature. Fluctuationdriven phase slips may be considered as a sequence of the events of local vanishing and local reappearance of the order parameter. Following^{4}, let us assume, that in a quasi1D superconductor carrying transport current just below the critical temperature, there occurs a weak link (a domain of weakened superconductivity). In case if the transport current is slightly increased, the superconducting state vanishes in the weaklink domain. In the next instance, the weaklink domain reveals normal conductivity mediated by unpaired electrons, whose drift velocity is reduced by inelastic scattering. As long as there are unpaired electrons below the critical temperature with a drift velocity smaller than the critical velocity of the condensate, those electrons condense into Cooper pairs, which are accelerated to carry the current as a supercurrent, and the process repeats again.
Later, other mechanisms of phase slips were developed. In particular, inhomogeneity of a superconductor filament^{5} makes its critical current inhomogeneous, so that an increase in the transport current leads to a local collapse of the order parameter. The current must be then entirely carried as the normal current. This allows superconductivity to reappear, and the cycle repeats due to the relaxation oscillation of the order parameter^{5}. For low temperatures <0.2 K, the behavior of a 1D superconductor was shown to be dominated by quantum mechanical tunneling through the freeenergy barrier: quantum phase slips^{6}. An experimental evidence for a quantum phase transition driven by the quantum fluctuations in a twodimensional (2D) superconductor has been recently obtained through magnetotransport measurements^{7}.
A further insight in the phaseslip phenomenon was gained for 2D and quasi2D superconductors. For nanowires of 100 nm width (nanostripes), it was shown^{8,9} that at relatively low current densities, the 1D Langer–Ambegaokar–McCumber–Halperin mechanism^{3,10} based on thermally induced phaseslip centers dominated over the 2D mechanism related to unbinding of vortex–antivortex pairs below the Berezinskii–Kosterlitz–Thouless (BKT) transition in 2D^{11,12,13}. At temperatures above the BKT critical temperature up until the critical temperature, thermal fluctuations are sufficient to unbind vortex–antivortex pairs, while below the BKT critical temperature all vortices are tightly bound into vortex–antivortex pairs^{14}. The effect of the magneticfieldinduced BKT scenario was observed in a 2D spindimer system by using a multilayer magnet^{15}. Another example of a phase transition in superconducting vortex systems is a liquid–solid transition experimentally found in 2D geometry^{16}. A BKTtype crossover was detected in a trapped quantum degenerate gas of rubidium atoms^{17}. The experimental signatures of quantum phase slips were the observation of temperatureindependent dissipation for a Bose–Einstein condensate in a 3D optical lattice^{18} and of velocitydependent dissipation for an ultracold quantum gas in a 1D optical lattice^{19}. The crossover from thermal to quantum phase slips controlled by the velocity was detected in 1D superfluid tubes^{20}.
At weaker transport currents, dissipation below the critical temperature is attributed to thermally activated motion of the order parameter over the freeenergy barriers, which separate metastable states. At stronger transport currents, the free energy required for unbinding a vortex–antivortex pair is reduced, and the corresponding thermally activated contribution to the resistivity is expected to be dominant. In this regime, the resistivity due to vortex flow is substantially affected by electron heating. In an external magnetic field, the situation changes drastically. Even in a weak magnetic field, the contribution of moving vortices dominates over the thermally activated phaseslip mechanism in the resistivity in superconducting wires of submicron width^{8}.
In novel superconductor nanostructured microarchitectures, e.g., open nanotubes^{21,22} and nanocoils^{23,24,25} in a magnetic field orthogonal to their axes, distribution of the order parameter over the surface is highly inhomogeneous. This opens up new possibilities for manifestations of phase slips as compared to quasi1D filaments and quasi2D stripes. In the nanohelices as small as 100 nm in diameter and aspect ratio up to 65, fingerprints of vortex and phaseslip patterns have been experimentally identified, and supported by numerical simulations based on the timedependent Ginzburg–Landau (TDGL) equation^{25}.
In the present paper, we demonstrate that a transition between different patterns of vortices is accompanied by occurrence of a weak link at the side of a nanotube, which is opposite to the slit, giving rise to novel phaseslip events. We identify the structure of superconducting screening currents (SSCs) induced by an external magnetic field and show that curved superconductor nanoarchitectures provide a unique possibility to study the interplay of multiple disconnected regions of SSCs. The dynamics of topological defects (superconducting vortices and phase slips) and the topological transitions between different configurations of SSCs manifest themselves through current–voltage or magneticfield–voltage characteristics. The induced voltage increases more than threefold on certain intervals of the magnetic field in Nb and Sn open nanotubes.
Results
The simplest example of multiple disconnected loops of SSC is the two loops of SSC (Fig. 1a), which can be realized experimentally in an open superconducting tube placed in a homogeneous magnetic field orthogonal to the tube’s axis (Fig. 1b). The 2Dcylindrical structure is convenient for the numerical study of the order parameter dynamics, since the mathematical model can be formulated in a way equivalent to the planar structure in an inhomogeneous magnetic field. The tube has a cut in the paraxial direction, which will hereinafter be referred to as “slit”. Two electrodes are attached to both edges of the slit (Fig. 1b), in order to support an azimuthal transport current. The width of the slit is supposed to be much smaller than the circumference 2πR, and the electrodes extend along the entire length of the edges. The whole system is placed in the magnetic field B = Be_{z}, which induces SSCs (Meissner currents) circulating in each halftube. Above the first critical magnetic field, the vortex pattern typical for the open tube is shown in Fig. 1c. The SSCs (Fig. 1d) in the planar structure (Fig. 1e) and the vortex pattern (Fig. 1f) are cardinally different from the SSCs (Fig. 1a) and the vortex pattern (Fig. 1c) in the open tube, correspondingly. The superconducting state is described by the TDGL equation for the complexvalued order parameter ψ coupled with the Poisson’s equation (see “Methods” section). A rolledup superconducting nanomembrane can be embedded fully and conformally in a polymer (Supplementary Fig. 1), i.e., varnish GE7031, that has a thermal conductivity, which is about one order of magnitude higher than that of helium gas at low temperatures. This experimental concept can be used for efficient cooling of the superconductor and would then allow for higher current flows through the structure without overheating effects (Supplementary Note 1). The effects of the induced magnetic field are considered to be negligible^{26} (Supplementary Figs. 2–4, Supplementary Note 2).
The modulus and phase of the order parameter, and the SSC patterns for the planar structure are represented in Fig. 2a–f, where the transport current is applied in the ydirection. The superconducting current after subtraction of the transport current is presented in Fig. 2c, f. Each vortex is a topoligical defect identified through the phase of the order parameter, while the phaseslip appearance is accompanied by the topological transition of SSCs. For the planar structure, the spaciotemporal analysis of the order parameter reveals a topological transition from one loop of SSCs (Fig. 2c) to three loops (Fig. 2f), when the phase slip appears. However, no topological changes are detected in the vortex dynamics, when the phase slip has appeared. The phenomenon of fluxflow instability that governs the occurrence of localized regions with suppressed superconductivity was described theoretically in a disordered thin superconducting strip in a homogeneous magnetic field^{27}. In an open nanotube, we observe that a region of the fastmoving vortices (shown in Fig. 2g, h) near an edge transforms into a phaseslip region, which propagates to the opposite edge of the structure. Two loops of SSCs are demonstrated in Fig. 2i.
The phaseslipinduced topological transition of SSCs in a Nb open nanotube at T/T_{c} = 0.95 is shown in Fig. 3a–f, where two loops of SSCs transform into four loops. As distinct from the planar structure, this transition is associated with the topological changes in the vortex dynamics: vortex–antivortex pairs start to be generated at a frequency, which is about three orders of magnitude higher than that for vortex chains. The manifestation of topological nature of vortices is shown in Fig. 3a, d, which depict the phase of the order parameter. When encircling each vortex/antivortex core in a specific direction (for example, counterclockwise), the phase changes by +2π for a vortex and by −2π for an antivortex (see the black and white circles in Fig. 3d). The spaciotemporal structure of the phase slip is thus topologically different from the trivial topology of the initial two loops of SSCs. The vortex–antivortex pairs generation has a striking similarity with the BKT transtion observed in 2D systems (XYmodels). Our simulation shows that when the magnetic field increases, the superconducting state without a phase slip reenters the structure with a chain of moving vortices, which are then transformed into channels of fastmoving vortices (Fig. 3b, c). Such channels are supposed to be observed experimentally^{28}. As a result of the sequence of transitions: vortexchain state–phaseslip state–vortexchain state, a voltage pulse (Fig. 3b) is expected to be observed in the open tube, while in the planar structure, a monotonic growth of the voltage as a function of the magnetic field occurs. Three regimes of the order parameter dynamics illustrated by Supplementary Movies 1–7: the vortexchain motion before the phase slip (the vortex lifetime is ~1.1 × 10^{−11} s), the phase slip, and the vortexchain motion after the phase slip (see also Supplementary Fig. 6, Supplementary Note 4).
In Fig. 4a–c, the modulus and phase of the order parameter, as well as the magneticfield–voltage characteristic are represented for a Sn open nanotube at T/T_{c} = 0.77. The voltage derivative as a function of the magnetic field (Fig. 4d) emphasizes the beginning and end of the voltage peak.
Discussion
The obtained dynamics of the phase of the order parameter reflects the dynamics of kinematic vortex–antivortex pairs, first theoretically proposed in ref. ^{29} and experimentally detected in ref. ^{30} in planar microstructures. Kinematic vortex–antivortex pairs were shown^{30} to have velocities higher than the Abrikosov vortex velocities. In refs. ^{29,31}, kinematic vortex–antivortex pairs were numerically described under strong transport current (close to the critical value) at zero and weak magnetic fields. In the tubular structure (Fig. 1b), multiple kinematic vortex–antivortex pairs nucleate under strong transport current at the regions with zero and weak normaltosurface component of the magnetic field. Remarkably, the green loops of SSCs in Fig. 3e connect front and rear halftubes, thus introducing a new topology of the superconducting current. A topological transition from two disconnected regions of SSCs (red) to four (red and green) and then back to two (red) such regions occurs when increasing the magnetic field for certain intervals of values of the transport current. Under these conditions in open tubes, the nucleation locus connects both halves of the tube (Fig. 3e), as distinct from the previously known cases when kinematic vortex–antivortex pairs do not occur, and the both halves of the tube are strictly disconnected (Fig. 1a). Generation of the vortex–antivortex pairs marked with black and white circles in Fig. 3d (Supplementary Movie 4) results from (i) their unbinding due to the high transport current and (ii) motion due to the Magnus force caused by the magnetic field. Nucleation and separation of vortex–antivortex pairs at the side of the nanotube, which is opposite to the slit, are followed by their motion till (i) their denuncleation at the sides of the tube or (ii) annihilation of a vortex from the pair with an antivortex from a neighboring pair (respectively, an antivortex from the pair with a vortex from another neighboring pair), when there exist two or more vortex–antivortex pairs. The relatively fast motion of the vortices and antivortices (with the vortex/antivortex lifetime ~9.6 × 10^{−15} s) on the side of the nanotube, which is opposite to the slit, leads to an apparent picture of an extended static phase slip (Fig. 3f). As follows from the comparison with the calculated vortex dynamics in an open tube (Supplementary Movie 1), the ratio of velocities for kinematic vortex–antivortex pairs and Abrikosov vortices is as large as about two orders of magnitude, what is close to that obtained for a Sn planar film^{30}.
The voltage generated in dc transport represents different order parameter states: the pure superconducting state, the mixed state (vortices and the superconducting state), the phaseslips states, and the normal state. A switching between those states corresponds to jumps in voltage as a function of the current or the magnetic field. The voltage pulse shown in Fig. 3b is calculated for the currents close to the critical ones (i.e., higher than 0.5 J_{c}). Within these values of parameters, the state of the order parameter is estimated to be very sensitive to the temperature.
The order parameter state in a Nb open nanotube is more sensitive to temperature variations in comparison with a Sn open nanotube (as follows from the comparison between Supplementary Figs. 7 and 8, see Supplementary Note 5). The temperature range, where the phaseslip regime exists, is wider for the Sn nanotube than for the Nb one: the voltage pulse in Nb occurs for temperature varying within 1% of T_{c}, while for Sn this temperature range is increased up to ∼7% of T_{c}. The voltage derivative as a function of the magnetic field (Fig. 4c) demonstrates a relatively small peak for the case, when vortices with a characteristic dimension ~4ξ each form a chain, with a length just equal to the length L of the tube.
A nontrivial superconducting currents topology induces the phaseslip dynamics, which determine the voltage–magnetic field and voltage–current characteristics in nanoarchitectures, with multiple loops of SSCs. The spaciotemporal structure of the phase slips in an open tube and a planar structure is unveiled. The crucial difference between them lies in the vortex–antivortex pair generation in the open tube. The nonmonotonous magneticfield–voltage and current–voltage characteristics imply a possibility to efficiently tailor the superconducting properties of nanostructured materials by inducing a nontrivial topology of SSCs. The topological transition between the vortexchain and phaseslip regimes opens up new perspectives for the advanced technological applications of rolledup superconductor nanoarchitectures, such as for highperformance detectors and sensors, energystorage components, quantum computing, and microwave radiation detection. It is also of heuristic importance for understanding of the phaseslip phenomena in confined lowdimensional superfluids.
Methods
The system is placed in the applied magnetic field B = Be_{z}, which induces SSCs (Meissner currents) circulating in each halftube. The superconducting state is described by the TDGL equation for the complexvalued order parameter ψ in the dimensionless form (Tables 1 and 2):
where A is the vector potential; φ the scalar potential and κ = λ/ξ the Ginzburg–Landau parameter with the London penetration depth λ and the coherence length ξ. Boundary conditions follow from the absence of the normal component of the superconducting current at the free boundaries of the superconductor:
On the site of the electrodes, the boundary condition ψ = 0 can be used at the superconductor–normalmetal boundary (see, for example, ref. ^{32}). In order to check how boundary conditions may influence the order parameter dynamics, we considered several possibilities (Supplementary Fig. 9, Supplementary Note 6). We found that the voltage peak occurs in all cases.
The scalar potential φ is found as a solution of the Poisson’s equation coupled with Eqs. (1) and (2):
where the superconducting current density is \({\mathbf{j}}_{{\mathrm{sc}}} = \frac{1}{{2i\kappa }}\left( {\psi ^ \ast \nabla \psi  \psi \nabla \psi ^ \ast } \right)  {\mathbf{A}}\psi ^2\) and σ is the normal conductivity. The transport current density \(j_{{\mathrm{tr}}}(y) = {\mathrm{const}} = j_{{\mathrm{tr}}}\) is imposed via the boundary conditions for Eq. (3) at the edges of the slit, to which the electrodes are attached:
The vector potential with components A_{s}(s, y) and A_{y}(s, y) (where s ≡ Rθ) is chosen in the Coulomb gauge (the influence of the gauge choice on the vortex dynamics is discussed in Supplementary Fig. 5 and Supplementary Note 3):
In order to guarantee the gauge invariance of the solution, we use link variables^{33}:
Then the set of Eqs. (1)–(4) takes the form:
We solve the set of Eqs. (7) and (8) numerically, using the relaxation method with a random (ψ from the range [0, 1]) initial distribution of the order parameter ψ(s, y). In the presence of the magnetic field (B > B_{c1}) and the transport current, it evolves toward a quasistationary state (Supplementary Movie 2). In the singlevortexchain regime, the quasistationaly state is characterized by a periodic vortex nucleation and denucleation at the points on the edges, with the highest or the lowest component of the magnetic field normaltothesurface (see ref. ^{21}). Evaluations of temperature (the disorder) effects on the voltage peak survival are represented in Supplementary Figs. 7 and 8, and Supplementary Note 5 (Supplementary Fig. 10, Supplementary Note 7).
Vortices move paraxially along the tube. The moving vortices generate an electric field, which is opposite to the transport current density. We evaluate this field in terms of the voltage between the two electrodes shown in Fig. 1b, ΔΦ = φ_{1} − φ_{2}. The scalar potential φ is found from the Poisson Eq. (3) with the boundary conditions Eq. (4). Finally, the following spaciotemporal averaging of the voltage is applied in line with the typical experimental situation (Supplementary Fig. 6):
where s_{1} and s_{2} are δ/2 and \(2\pi R  \frac{\delta }{2}\), correspondingly, δ is the slit width, L_{Φ} is the length of the averaging area (its maximal value is the length L of the tube), \(T \gg \Delta t_1\) is the time of averaging, Δt_{1} is the time required for a vortex to reach the opposite edge of a tube after nucleation^{21}.
Data availability
The video data that support the findings of this study are available in “figshare” repository with the identifier https://doi.org/10.6084/m9.figshare.12330986.v3.
Code availability
The codes that support the findings of this study are available from the corresponding author upon reasonable request.
References
Kopnin, N. B. Theory of Nonequilibrium Superconductivity (Clarendon Press, Oxford, 2001).
Tinkham, M. Introduction to Superconductivity (McGrawHill, New York, 1996).
Langer, J. S. & Ambegaokar, V. Intrinsic resistive transition in narrow superconducting. Channels Phys. Rev. 164, 498–510 (1967).
Tidecks, R. CurrentInduced Nonequilibrium Phenomena in QuasiOneDimensional Superconductors (Springer, BerlinHeidelberg, 1990).
Skocpol, J., Beasley, M. R. & Tinkham, M. Phaseslip centers and nonequilibrium processes in superconducting tin microbridges. J. Low Temp. Phys. 16, 145–167 (1974).
Giordano, N. Evidence for macroscopic quantum tunneling in onedimensional superconductors. Phys. Rev. Lett. 61, 2137–2140 (1988).
Saito, Y., Nojima, T. & Iwasa, Y. Quantum phase transitions in highly crystalline twodimensional superconductors. Nat. Commun. 9, 778 (2018). 1–7.
Bell, M. et al. On the nature of resistive transition in disordered superconducting nanowires. IEEE Trans. Appl. Sup. 17, 267–270 (2007).
Bell, M. et al. Onedimensional resistive states in quasitwodimensional superconductors: experiment and theory. Phys. Rev. B 76, 094521 (2007). 1–5.
McCumber, D. E. & Halperin, B. I. Time scale of intrinsic resistive fluctuations in thin superconducting wires. Phys. Rev. B 1, 1054–1070 (1970).
Berezinskii, V. S. Destruction of longrange order in onedimensional and twodimensional systems having a continuous symmetry group I. Classical systems. Sov. Phys. JETP 32, 493–500 (1971).
Berezinskii, V. S. Destruction of longrange order in onedimensional and twodimensional systems having a continuous symmetry group II. Quantum systems. Sov. Phys. JETP 34, 610–616 (1971).
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in twodimensional systems. J. Phys. C 6, 1181–1203 (1973).
Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low Temp. Phys. 36, 599–616 (1979).
Tutsch, U. et al. Evidence of a fieldinduced BerezinskiiKosterlitzThouless scenario in a twodimensional spindimer system. Nat. Commun. 5, 5169 (2014). 1–9.
Chen, B. et al. Twodimensional vortices in superconductors. Nat. Phys. 3, 239–242 (2007).
Hadzibabic, Z. et al. BerezinskiiKosterlitzThouless crossover in a trapped atomic gas. Nature 441, 1118–1121 (2006).
McKay, D., White, M., Pasienski, M. & DeMarco, B. Phaseslipinduced dissipation in an atomic Bose–Hubbard system. Nature 453, 76–80 (2008).
Tanzi, L. et al. Velocitydependent quantum phase slips in 1D atomic superfluids. Sci. Rep. 6, 25965 (2016).
Scaffidi, S. A. et al. Exploring quantum phase slips in 1D bosonic systems. Eur. Phys. J. Spec. Top. 226, 2815–2827 (2017).
Fomin, V. M., Rezaev, R. O. & Schmidt, O. G. Tunable generation of correlated vortices in open superconductor tubes. Nano Lett. 12, 1282–1287 (2012).
Rezaev, R. O. et al. Voltage induced by superconducting vortices in open nanostructured microtubes. Phys. Stat. Sol. RRL 13, 1–12 (2019).
Fomin, V. M. et al. Superconducting nanostructured microhelices. J. Phys. Cond. Mat. 29, 395301 (2017). 1–9.
Lösch, S. et al. Microwave radiation detection with an ultrathin freestanding superconducting niobium nanohelix. ASC Nano 13, 2948–2955 (2019).
Córdoba, R. et al. Threedimensional superconducting nanohelices grown by He^{+}focusedionbeam direct writing. Nano Lett. 19, 8597–8604 (2019).
Smirnova, E. I., Rezaev, R. O. & Fomin, V. M. Simulation of dynamics of the order parameter in superconducting nanostructured materials: effect of the magnetic field renormalization. Low Temp. Phys. 46, 325–331 (2020).
Vodolazov, D. Y. Fluxflow instability in a strongly disordered superconducting strip with an edge barrier for vortex entry. Supercond. Sci. Technol. 32, 115013 (2019). 1–7.
Embon, L. et al. Imaging of superfast dynamics and flow instabilities of superconducting vortices. Nat. Commun. 8, 85 (2017). 1–10.
Andronov, A. et al. Kinematic vortices and phase slip lines in the dynamics of the resistive state of narrow superconductive thin film channels. Phys. C 213, 193–199 (1993).
Sivakov, A. G. et al. Josephson behavior of phaseslip lines in wide superconducting strips. Phys. Rev. Lett. 91, 267001 (2003). 1–4.
Berdiyorov, G. R., Milosevic, M. V. & Peeters, F. M. Kinematic vortexantivortex lines in strongly driven superconducting stripes. Phys. Rev. B 79, 184506 (2009). 1–8.
Córdoba, R. et al. Longrange vortex transfer in superconducting nanowires. Sci. Rep. 9, 12386 (2019). 1–10.
Kato, R., Enomoto, Y. & Maekawa, S. Effects on the surface boundary on the magnetization process in typeII superconductors. Phys. Rev. B 47, 8016–8024 (1993).
Parks, R. D. Superconductivity 1 (Marcel Dekker, New YorkBasel, 1969).
Nakamura, Y. et al. Intrinsic charge transport in stanene: roles of bucklings and electronphonon couplings. Adv. Electron. Mater. 3, 1700143 (2017). 1–9.
Tanuma, S., Powell, C. J. & Penn, D. R. Calculations of electron inelastic mean free paths. IX. Data for 41 elemental solids over the 50 eV to 30 keV range. Surf. Interface Anal. 43, 689–713 (2011).
Huth, M., Porrati, F. & Dobrovolskiy, O. V. Focused electron beam induced deposition meets materials science. Microelectron. Eng. 185186, 9–28 (2018).
Acknowledgements
The authors thank A. Bezryadin, J. Lorenzana, D. Roditchev, V. A. Shklovskij, H. Suderow, F. Tafuri, R. Tidecks, V. M. Vinokour, A. D. Zaikin, and E. Zeldov for fruitful discussions. The authors are grateful to DFG (Germany) for support under the Projects # FO 956/51, # FO 956/61, and to ZIH TU Dresden for providing its facilities for highthroughput calculations. V.M.F. acknowledges partial support within the framework of the COST Action CA16218 (NANOCOHYBRI) of the European Cooperation in Science and Technology, and through the MEPhI Academic Excellence Project (Contract # 02.a03.21.0005). R.O.R. thanks RFBR (Russia) and Tomsk Region for support through the research Project # 1941700004. Open access funding provided by Projekt DEAL.
Author information
Authors and Affiliations
Contributions
V.M.F. led the project and conceived the investigations. R.O.R. and E.I.S. performed numerical simulations. O.G.S. and V.M.F. developed key conceptual ingredients for the physical interpretation. V.M.F. and R.O.R. wrote the manuscript. All authors contributed to discussions about the numerical modeling, the analysis of the obtained results, and the preparation of the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rezaev, R.O., Smirnova, E.I., Schmidt, O.G. et al. Topological transitions in superconductor nanomembranes under a strong transport current. Commun Phys 3, 144 (2020). https://doi.org/10.1038/s42005020004114
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005020004114
This article is cited by

Topological transitions in ac/dcdriven superconductor nanotubes
Scientific Reports (2022)

Magnetic field enhanced critical current and subharmonic structures in dissipative superconducting gold nanowires
Quantum Frontiers (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.