Abstract
The ultrafast dynamics of superconducting vortices harbors rich physics generic to nonequilibrium collective systems. The phenomenon of fluxflow instability (FFI), however, prevents its exploration and sets practical limits for the use of vortices in various applications. To suppress the FFI, a superconductor should exhibit a rarely achieved combination of properties: weak volume pinning, closetodepairing critical current, and fast heat removal from heated electrons. Here, we demonstrate experimentally ultrafast vortex motion at velocities of 10–15 km s^{−1} in a directly written NbC superconductor with a closetoperfect edge barrier. The spatial evolution of the FFI is described using the edgecontrolled FFI model, implying a chain of FFI nucleation points along the sample edge and their development into selforganized Josephsonlike junctions (vortex rivers). In addition, our results offer insights into the applicability of widely used FFI models and suggest NbC to be a good candidate material for fast singlephoton detectors.
Introduction
The dynamics of vortices at large transport currents is of major importance for the comprehension of vortex matter under farfromequilibrium conditions and it sets practical limits for the use of superconductors in various applications^{1,2,3,4,5,6,7,8,9}. The physics of currentdriven vortex matter is getting especially interesting when the vortex velocity exceeds the velocity v ≈ 3 km s^{−1} of other possible excitations in the system, allowing for the Cherenkovlike generation of sound^{10,11} and spin^{12,13} waves by moving fluxons, which opens up novel routes to excite waves in magnon spintronics^{14,15}. Furthermore, there is currently great interest in the interplay of Meissner currents and magnetic flux quanta with spin waves in the rapidly developing domain of magnon fluxonics^{16,17}.
The maximal current a superconductor can carry without dissipation is determined by the pairbreaking (depairing) current I_{dep}. However, a highly resistive state in real systems is usually attained at much smaller currents due to the presence of regions in which superconductivity breaks down long before I_{dep} is reached. Namely, in a vortexfree state, the earlier breakdown of superconductivity is due to spatial variations of the order parameter caused by structural imperfectnesses and the sample geometry^{18,19}. In the vortex state, fastmoving vortices are known to lead to a quench of the lowdissipative state at I^{*} ≪ I_{dep} as a consequence of the fluxflow instability (FFI) associated with the escape of quasiparticles (normal electrons) from the vortex cores^{20,21}. Accordingly, to achieve I_{c} ≲ I_{dep} and high vortex velocities v ≳ 5 km s^{−1}, a high structural homogeneity and fast cooling of quasiparticles (governed by the quasiparticles’ energy relaxation time τ_{ϵ} and the escape time of nonequilibrium phonons to the substrate τ_{esc}) are both required. However, while short τ_{ϵ} is inherent to disordered superconducting systems^{22,23}, few of them have I_{c} ≲ I_{dep} in conjunction with weak volume pinning needed to maintain longrange order in the fastmoving vortex lattice. Variation in the local pinning forces induced by uncorrelated disorder (volume pinning) leads to a broader distribution of v and thereby prevents the exploration of vortex matter at high velocities^{24,25,26,27}.
Recently, two approaches were used to demonstrate ultrafast vortex motion at v ≳ 5 km s^{−1}. In the first case, a clean Pb bridge with both, an edge barrier for vortex entry and a high demagnetization factor (socalled geometrical barrier) was studied^{6}. In the used geometry there was a strongly nonuniform current distribution both across and along the bridge due to a small Pearl length 2λ^{2}/d ≪ w, where d and w are the film thickness and width, respectively. A weak pinning and a short electron–phonon relaxation time τ_{ep} in Pb^{28} allowed one to diminish nonequilibrium effects and achieve the regime with ultrafast Abrikosov–Josephson vortices^{6}. In the second case, an array of ferromagnetic Co nanostripes on top of a superconducting Nb film led to a dynamic ordering of flux quanta guided by the nanostripes and allowed to achieve a narrow distribution of their velocities^{29}. In both of these approaches, specially designed, locally nonuniform structures were used. At the same time, a closetoideal uniform system where the fast heat removal from electrons rather than the finite width of the v distribution becomes the limiting factor for ultrafast vortex dynamics was never investigated experimentally. Theoretically, however, it was recently predicted that dirty superconductors with weak volume pinning and strong edge barrier for vortex entry should also allow for ultrafast vortex dynamics^{30}. Extremely dirty superconductors are known to have a short electron–electron inelastic scattering time τ_{ee} which leads to a decrease of τ_{ep}^{31}. This diminishes nonequilibrium effects and may lead to an increase of the critical velocity of vortices. One of the most important requirements for the observation of an edgecontrolled FFI is a spatially homogenous edge in conjunction with a weak pinning in the superconductor’s volume^{30}. The presence of a strong edge barrier in such superconductors leads to a current gradient near the edge where vortices enter the superconductor and where FFI is actually nucleating.
Here, we demonstrate experimentally the phenomenon of edgebarriercontrolled FFI in directwrite superconductors with a closetoperfect edge barrier and deduce vortex velocities up to 15 km s^{−1} from their current–voltage curves (I–V). The investigated system is the recently synthesized NbC superconductor fabricated by focused ion beam induced deposition (FIBID)^{32}, with a very high resistivity ρ = 572 μΩcm. This implies a large effect of the inelastic electron–electron scattering with the characteristic times τ_{ee} ≲ τ_{ep} which speeds up the relaxation of disequilibrium. The NbC microstrips have a rather low depinning current and their critical current is controlled by the edge barrier for vortex entry. In contrast to ref. ^{6}, in our system λ^{2}/d ≫ w, which means a negligible demagnetization factor (no geometrical barrier) and a uniform current distribution across the strip at zero magnetic field. The spatial evolution of the FFI is described in terms of the edgebarriercontrolled FFI model recently developed by one of the authors^{30}, implying a chain of FFI nucleation points along the sample edge and their development into selforganized Josephsonlike junctions (vortex rivers) evolving to normal domains which expand along the entire sample. In addition, our results offer insights into the applicability of widely used FFI models and render NbC to be a good candidate material for fast singlephoton detectors.
Results
System under investigations
We study the vortex dynamics in a directwrite NbC superconducting microstrip fabricated by FIBID^{32}. The microstrip is characterized by a transition temperature of T_{c} = 5.6 K and closetodepairing values of the zerofield critical current I_{c} ≈ 0.7 − 0.74I_{dep} above 0.5T_{c}. The dimensions of the microstrip are: thickness d = 15 nm, width w = 1 μm, and length l = 6.6 μm, see Fig. 1 for the geometry. The microstrip is characterized by the coherence length at zero temperature ξ(0) ≈ 6.5 nm, the penetration depth λ(0) ≈ 1060 nm, and the Pearl length 2λ^{2}(0)/d ≈ 150 μm, that is 2λ^{2}(0)/d ≫ w ≫ ξ(0). The perpendiculartofilmplane magnetic field with induction B = μ_{0}H populates the microstrip with a lattice of Abrikosov vortices. The applied dc current exerts a Lorentz force on the vortices that causes their motion with velocity v across the microstrip. The associated voltage drop V along the microstrip is recorded as a function of the applied current I in the currentbiased mode. The microstrip is capped with an insulating NbC layer fabricated by focused electron beam induced deposition (FEBID)^{32,33}. Further details on the sample fabrication and its structural properties are given in the “Methods” section.
Current–voltage characteristics
Figure 2 displays the I–V curves measured at 4.2 K (0.75T_{c}) and 5.04 K (0.9T_{c}) for a series of magnetic fields between 30 and 240 mT. With increase of the current, a series of different resistive regimes can be identified, as indicated in the I–V curves: the pinned regime (I), the nonlinear fluxflow regime (II), and the FFI (III) causing abrupt onsets of the normal state (IV). Of especial interest for the following is the regime of high vortex velocities just before the FFI (III) with the I–V sections enlarged in Fig. 2c, d.
From the last points before the voltage jumps, referring to Fig. 2c, d, the vortex instability velocity v^{*} is deduced by the relation v^{*} = V^{*}/(BL). The resulting dependence v^{*}(B) is presented in Fig. 3a. Remarkably, v^{*} is between 5 and 10 km s^{−1} at larger fields B ≳ 100 mT and it is between 10 and 15 km s^{−1} at B < 100 mT. The temperature dependence v^{*}(T) is presented in Fig. 3b for two magnetic field values. The field 50 mT is exemplary for a relatively sparse vortex lattice (vortex lattice spacing a ≈ 220 nm) while a ≈ 110 nm at 200 mT for the assumed triangular vortex lattice with \(a=\sqrt{2{\Phi }_{0}/\sqrt{3}H}\), where Φ_{0} is the magnetic flux quantum. At both fields, the experimental data nicely fit the law v^{*} ~ (1 − t)^{1/4}, where t = T/T_{c}, with v^{*}(0.6T_{c}, 50 mT) = 12 km s^{−1} and v^{*}(0.6T_{c}, 200 mT) = 7.7 km s^{−1}, while a deviation of v^{*}(B) from the B^{−1/2} dependence is observed at B ≲ 50 mT in Fig. 3a. The decreasing dependence of v^{*}(B) below about 10 mT due to the decreasing vortex density (the socalled lowfield crossover in the v^{*}(B) dependence^{34}) is beyond our consideration, as we are especially interested in the regime of very high vortex velocities.
Influence of the edge barrier on the vortex dynamics
The magnetic field dependence of the critical current at 4.20 K is presented in Fig. 3c. At smaller fields, I_{c}(B) decreases linearly with B, while at larger fields the decrease of I_{c} becomes nonlinear and slower. This behavior can be explained by the presence of some threshold field B_{stop}, which demarcates the Meissner (vortex free) and the mixed states of a superconducting stripe^{35}. Namely, the dependence I_{c}(B) in the Meissner state (B < B_{stop}) is linear and it is described by the expression I_{c}(B) = I_{c}(0 T)(1 − B/2B_{stop}), where B_{stop} in the Ginzburg–Landau model^{36} is given by \({B}_{{\rm{stop}}}={B}_{{\rm{s}}}/2={\Phi }_{0}/(2\sqrt{3}\pi \xi (T)w)\). Here, B_{s} is the field value at which the surface barrier for vortex entry is suppressed at I = 0, ξ is the superconducting coherence length, and w is the microstrip width. The definition of B_{stop} following from I_{c}(2B_{stop}) = 0 is illustrated in Fig. 3c. For 10 mT ≲ B ≲ 100 mT, the dependence of the critical current is described well by the dependence I_{c}(B) = I_{c}(0 T)B_{stop}/2B, and I_{c}(B) exhibits a linear decrease at low fields. At larger fields, B ≳ 100 mT, a further crossover at B^{*} to a slower decrease of I_{c}(B) as B^{−0.5} is observed. The totality of our experimental data indicates the dominating role of the edge mechanism^{37} of vortex pinning in the studied sample at B ≲ 100 mT, as is further commented in Supplementary Note 1.
Influence of an edge defect on the vortex dynamics
An additional reference measurement has been made for a microstrip with an artificially fabricated edge defect. The defect (notch) was milled by focused Ga ion beam at one edge of the microstrip and it has a shape of an equilateral triangle with a side of about 100 nm, see the inset in Fig. 4a. For a direct comparison of the edgebarrier effects on the vortex entry from different sides of the microstrip, the I_{c}(B) and I^{*}(B) curves are presented for both field and current polarities in Fig. 4. For the microstrip with even edges, the I^{*}(±B) curves fall onto one another in the entire range of magnetic fields in Fig. 4a and the I_{c}(B) dependence in Fig. 4b is symmetric with respect to the B reversal. By contrast, for the microstrip with the notch, the maximum in I_{c}(B) in Fig. 4b is shifted to +8 mT, in agreement with previous experiments on microstrips with defects (holes) close to one of their edges^{38}. At negative fields, the notch locally suppresses the edge barrier and thereby facilitates the entry of (anti)vortices. This leads to a small reduction of I_{c}(B) up to larger field magnitudes at which the role of the volume pinning increases. At positive fields, when vortices enter the microstrip from the opposite side, the notch does not affect the vortex entry and this is why I_{c} is not affected by the presence of the notch at B ≳ 15 mT. Remarkably, when vortices enter the microstrip via the edge with the notch, I^{*}(B) at 20 mT ≲ B ≲ 100 mT decreases by up to about 10% in comparison with I^{*}(B) when vortices enter from the opposite side, which is in line with the calculations^{39}. Importantly, due to the nonlinear upturns of the I–V curves just before the instability jump, a decrease of I^{*} by about 10% leads to a stronger decrease of the instability velocity v^{*}. This provides a direct evidence of the decisive role of the edge barrier on the FFI, as will be detailed next.
Discussion
We first compare the experimental results with the widely used Larkin–Ovchinnikov (LO) FFI model^{21,40} with the modifications introduced by Bezuglyj and Shklovskij (BS)^{41} and Doettinger et al.^{42}. Although edgebarrier effects are not considered in these models^{21,40,41,42}, it is still interesting to check what quasiparticle energy relaxation time τ_{ϵ} values, related to the instability velocity, can be deduced from fitting of the experimental data to these models.
Within the framework of the LO theory^{21,40}, the microscopic mechanism of FFI is the following. When the electric field induced by vortex motion raises the quasiparticle energy above the potential barrier associated with the order parameter around the vortex core, quasiparticles leave it and the core shrinks. The shrinkage of the vortex cores leads to a reduction of the viscous drag coefficient and a further avalanchelike acceleration of the vortex, eventually quenching the lowresistive state. The original LO theory was developed in the dirty limit near T_{c} and in neglect of heating of the superconductor. To account for quasiparticle heating due to the finite heatremoval rate of the power dissipated in the sample, the LO theory was extended by BS^{41}. In the BS generalization, the latter effect was considered in the framework of the kinetic equation LO approach, which assumes a nonthermal (nonFermi–Dirac) electron distribution function, while Joule heating was taken into account using a thermal distribution function and the electron temperature T_{e} was determined from the heat conductance equation. In contrast to the Bindependent instability velocity v^{*} in the LO model, a v^{*}(B) variation is expected in the BS model^{41} and takes the form:
where h is the heat removal coefficient. While the magnetic field dependence v^{*}(B) nicely fits Eq. (1) at B ≳ 50 mT, a notable deviation of v^{*}(B) toward smaller values is observed in Fig. 3a at B ≲ 50 mT. This deviation will be commented in what follows. In all, the complete set of the instability parameters deduced from Fig. 2 nicely fits the BS scaling law, see Supplementary Fig. 1. However, if one associates τ_{ϵ} with the electron–phonon scattering time τ_{ep} in the LO model, the deduced τ_{ϵ} is at least one order of magnitude smaller than one could expect from τ_{ϵ} found in similar lowT_{c} highly disordered superconductors^{43,44,45}, see Supplementary Discussion.
In the LO model modified by Doettinger et al.^{42}^{[,46}, the quasiparticle energy relaxation time can be found from the following equation:
In Eq. (2), the term \(a/\sqrt{D{\tau }_{\epsilon }}\), where a is the intervortex distance, has been added to incorporate the necessary condition of spatial homogeneity of the nonequilibrium quasiparticle distribution between vortices at relatively small magnetic fields. The calculation results by Eq. (2) are shown by solid lines in Fig. 3a where the energy relaxation time has been varied as the only fitting parameter. The best fits are achieved with τ_{ϵ} = 16 ps which could be considered as a more accurate estimate for the energy relaxation time in the NbCFIBID superconductor. We note that with this τ_{ϵ} estimate, the quasiparticle diffusion length \({l}_{\epsilon }=\sqrt{D{\tau }_{\epsilon }}\) ≈ 28 nm is much smaller than the intervortex distance a at all used magnetic fields and, importantly, l_{ϵ} ≲ 2ξ(T) with 2ξ(0.75T_{c}) ≈ 25 nm and 2ξ(0.9T_{c}) ≈ 38 nm.
The edgebarriercontrolled FFI scenario^{30} is different from the FFI scenario of LO and BS. Indeed, LO and BS considered a moving periodic vortex lattice in an infinite superconductor in the Wigner–Seitz approximation and hence could not take into account the collective effects related to the transformation of the vortex lattice and edgebarrier effects. In contrast, in the edgebarriercontrolled FFI model^{30} a nonuniform distribution of vortices is taken into account, as well as the local Joule heating and cooling (due to the time variation of the magnitude of the superconducting order parameter ∣Δ∣) depending on the vortex position. The edgebarriercontrolled FFI model allows for studying a “local” instability and collective effects in the vortex dynamics relying upon the solution of a heat conductance equation for the electrons and a modified timedependent Ginzburg–Landau equation for Δ(r, t). In this model, it was shown that, in the lowresistive state, there is a temperature gradient across the width of the microstrip with maximal local temperature near the edge where vortices enter the sample^{30}. The higher temperature at the edge is caused by the larger current density in the nearedge area due to the presence of the edge barrier for vortex entry and, hence, the locally larger Joule dissipation. With increase of the current, there is a series of transformations of the moving vortex lattice. In Fig. 5, we show examples of the calculated I–V curves and snapshots of ∣Δ∣(r) for the parameters of the superconductor as in ref. ^{30}. Similar transformations connected with reorientations of the moving vortex lattice in the insets 1–2 in Fig. 5b were experimentally observed^{47} and theoretically analyzed^{48} previously.
At currents just below I^{*}, localized areas with strongly suppressed superconductivity and closely spaced vortices appear near the hottest edge (left edge in the insets in Fig. 5). Upon reaching I^{*}, these areas begin to grow in the direction of the opposite edge and form a highly resistive Josephson SNSlike link (vortex river) along which vortices move^{3,6,30,49}. These vortices are of the Abrikosov–Josephson type, as they are moving in areas with suppressed order parameter. Due to the increasing dissipation, vortex rivers evolve into normal domains which than expand along the microstrip. In consequence of this, a jump to the highly resistive state occurs at I^{*}. In all, the simulation results demonstrate that transformation of the moving vortex array is a collective phenomenon, which involves correlated changes in the motion of many vortices with increase of the current and, at I^{*}, results in the appearance of Josephsonlike SNS links known as vortex rivers^{3,6,49}.
In the edgecontrolled FFI model^{30}, the current I^{*} increases linearly with the width of the strip, while V^{*} does not depend on w as it does in the LO and BS models. This result holds at B ≫ B_{stop} when a is much smaller than the microstrip width w and a becomes smaller than the width of the vortexfree region near the edge of the microstrip. This means that despite the nucleation of FFI points occurs near the edge where the local temperature and the current densities are maximal, far from the edge where the current density is uniform, the vortices should move at relatively high velocities. Otherwise the FFI will not develop across the whole microstrip and one has only origins of the vortex rivers, as it can be seen from Fig. 5 in^{30} at I ≲ I^{*}. The linear scaling of I^{*}(w) with the microstrip width w is corroborated by the experimental observation in Fig. 6a, where the I–V curves for two microstrips with the widths w = 1 μm and 500 nm are shown at T = 4.2 K and B = 50 mT.
In the edgebarriercontrolled FFI model^{30}, the energy relaxation time depends not only on the electron–phonon relaxation time τ_{ep} (as in the LO model) but also on the escape time of nonequilibrium phonons to the substrate τ_{esc} and the ratio of the electron and phonon heat capacities, C_{e} and C_{p}, respectively. At T ≃ T_{c} and for a small deviation from equilibrium one has:
where τ_{E} ≃ τ_{ep}/4.5 is the electron–phonon relaxation time renormalized due to fast electron–electron inelastic scattering. Here, τ_{ep} is the electron–phonon relaxation time used in the LO model. Following the arguments of ref. ^{42}, one can claim that the instability occurs at the velocity v^{*} ~ a/τ_{ϵ} when the intervortex distance is \(a\,\lesssim\, \sqrt{D{\tau }_{{\rm{\epsilon }}}}\). This condition leads to a dependence of v^{*}(B), which was revealed in numerical calculations^{30}. One important difference between the modified LO model^{42} and the edgecontrolled FFI model is that in the latter^{30}, a ~ B^{−1/2} only at relatively large magnetic fields, when the intervortex distance at I ~ I_{c} and I ~ I^{*} is almost the same despite the change in the structure of the moving vortex lattice. At relatively small magnetic fields, a in the vortex rows is smaller than \({(2{\Phi }_{0}/B\sqrt{3})}^{1/2}\) at I ~ I^{*} and, thus, the number of vortices is smaller than follows from the simple estimate nΦ_{0} = BS, see Fig. 5a. Altogether, this leads to a weaker experimental dependence v^{*}(B) than follows from the “global” instability model with v^{*} ~ B^{−1/2}^{42}. Qualitatively, it is this behavior which is observed in the experiment, see Fig. 3a.
The large v^{*} values observed in our system should be attributed not only to τ_{E} < τ_{ep} but, also, to a small τ_{esc} in Eq. (3). Indeed, due to the insulating NbCFEBID layer on top of the microstrip, there seems to be no phonon bottleneck which could exist due to an acoustic mismatch between a thin dirty superconductor and a substrate^{44}. As an estimate, for our system we deduce τ_{esc} ~ 4d/u ≈ 24 ps, where u ~ 2.5 km s^{−1} is the mean sound velocity. This value is larger than τ_{ε} ~ 16 ps deduced from the experimental data using the modified LO model. We have to stress that numerical coefficients in the LO model are strictly valid only rather close to T_{c} (when Δ(T) ≪ k_{B}T_{c}, i.e., at T ≳ 0.9T_{c}) and in the case when τ_{ee} ≫ τ_{ep} and τ_{esc} = 0. Therefore these coefficients may be different in our dirty system with τ_{ϵ} ~ τ_{esc} and at temperatures further away from T_{c}.
Finally, we would like to note that, unfortunately, there is no analytical relation between v^{*} and τ_{ϵ} in the edgebarriercontrolled FFI model^{30}. Accordingly, a discussion of the relation between v^{*} and τ_{ϵ} has to remain on a qualitative level. From Eq. (3) it follows that a change of τ_{E}, τ_{esc}, and C_{e}/C_{p} leads to a change of the relaxation time τ_{ϵ}. To illustrate this, in Fig. 6b we present a series of calculated I–V curves at different τ_{esc} values, while the other parameters are kept fixed. Indeed, with increasing τ_{esc} the critical velocity v^{*} ~ E^{*} decreases, but it decreases slower than \({\tau }_{\epsilon }^{1}\) or \({\tau }_{\epsilon }^{1/2}\). Qualitatively, the same tendency is found if one increases the ratio C_{e}/C_{p} for a given τ_{esc} value. Specifically, with an increase of τ_{esc}/τ_{E} by two orders of magnitude, E/E_{0} decreases by only about a factor of three. In the inset of Fig. 6b, one can also see that with the increase of τ_{esc}, the timeaveraged temperature in the center of the superconducting microstrip increases, which indicates an increased contribution of Joule dissipation to the FFI. The increased temperature also affects v^{*} because of the temperature dependence τ_{E} ~ 1/T^{3} and C_{e}/C_{p} ~ 1/T^{2} in the used model^{30}.
We would like to outline an applicationsrelated aspect of the superconducting properties of the studied NbCFIBID microstrip. Namely, the small diffusivity D ≈ 0.49 cm^{2} s^{−1} and the low transition temperature T_{c} = 5.6 K suggest that NbCFIBID may be a candidate material for superconducting singlephoton detectors (SSPDs). We refer to Table 1 for a comparison with parameters of some typical SSPDs and to ref. ^{31} for a further discussion. In this regard, it should be mentioned that for about a decade SSPDs were made of meandering nanostrips with widths in the range 50–150 nm as it was empirically found that the use of wider strips leads either to the loss of the singlephoton nature of the response or to a decrease of the detection efficiency^{50}. This observation was in line with a “geometrichotspot” detection model, in which the width of the supercurrentcarrying strip should be comparable with the diameter of the normal region where the superconducting state is suppressed due to the absorption of the photon.
Recently, a “photongenerated superconducting vortex model” was proposed^{31,51}. It was revealed that the efficiency of the photon detection is not determined by the geometry, as long as the initial current density is uniform and close to the critical pairbreaking current I_{dep}. It was emphasized that even several micron wide dirty superconducting stripes should be suitable to detect single nearinfrared or optical photons if their critical current I_{c} ≳ 0.7I_{dep}^{31}. The only requirement for the width of the strip is that it should be smaller than the Pearl length Λ = 2λ^{2}/d that ensures the uniformity of the supercurrent across the superconductor width. Recently, this condition was satisfied in wide and short NbN^{52} and MoSi^{53} bridges, whose photon response was consistent with the vortexassisted mechanism of initial dissipation^{51}. In this way, given the superconducting properties of our samples, which drastically differ from much cleaner NbC films prepared by pulsed laser ablation in ref. ^{54}, NbCFIBID appears to be a good candidate for fast singlephoton detection. A further enhancement of the critical current in NbCFIBID can be expected for tapered current leads^{52,53} which should minimize the reduction of I_{c} in consequence of undesired currentcrowding effects^{19}, and additional advantages of easy onchip^{55} or onfiber^{56} integration are provided by the directwrite nanofabrication technology. Furthermore, the ability to control the thickness of individual FIBID/FEBID layers with an accuracy better than 1 nm^{57,58} should allow for the fabrication of superconductor/insulator superlattices for studying quantum interference and commensurability effects^{59} as well as photonic crystals with superconducting layers^{60}.
To summarize, we have experimentally demonstrated ultrafast vortex dynamics at velocities up to 15 km s^{−1} in a uniform superconducting microstrip fabricated by FIBID. A stable flux flow at such high velocities is a consequence of the combined effects of a strong edge barrier against a background of weak volume pinning, closetodepairing critical currents, and fast quasiparticles relaxation in the investigated system. The distinctive feature of the directwrite NbC superconductor is a closetoperfect edge barrier which orders the vortex motion at large current values and allows for the description of the spatial evolution of the FFI relying upon the edgebarriercontrolled FFI model. The observed high vortex velocities in NbCFIBID make accessible studies of farfromequilibrium superconductivity^{61} and vortex matter driven by large currents, opening prospects for Cherenkovlike generation of other excitations by the fastmoving vortex lattice in ferromagnet/superconductor hybrid structures. In addition, the small electron diffusion coefficient D ≈ 0.5 cm^{2} s^{−1}, the low superconducting transition temperature T_{c} = 5.6 K, and high I_{c} values exceeding 70% of the depairing current render NbCFIBID to be an interesting candidate material for fast singlephoton detectors.
Methods
Sample fabrication and its structural properties
Superconducting microstrips were fabricated by FIBID in a dualbeam scanning electron microscope (FEI Nova Nanolab 600). The substrates are Si (100, pdoped)/SiO_{2} (200 nm) with lithographically defined Au/Cr contacts for electrical transport measurements^{62}. FIBID was done at 30 kV/10 pA, 30 nm pitch and 200 ns dwell time employing Nb(NMe_{2})_{3}(NtBu) as precursor gas. The asdeposited NbCFIBID microstrips have welldefined smooth edges and an rms surface roughness of <0.3 nm, as deduced from atomic force microscopy scans in the range 1 × 1 μm. Right after the deposition, without breaking the vacuum, the microstrips were covered with a 10nmthick insulating NbC layer prepared focused by FEBID^{33,63}, see Fig. 1 for the geometry. While NbCFEBID structures are amorphous, NbCFIBID deposits have an fcc NbC polycrystalline structure, with grains about 15 nm in diameter^{32}. The typical elemental composition in the NbCFIBID microstrips is 43% at. C, 29% at. Nb, 15% at. Ga, and 13% at. N, as inferred from energydispersive Xray spectroscopy on thicker replica of the fabricated structures. Experiments were done on a series of four samples. In the manuscript, we report typical data for one microstrip. An additional reference measurement has been made for a microstrip with an artificially fabricated edge defect. The defect (notch) was milled by focused Ga ion beam at a beam voltage of 30 kV and a beam current of 10 pA^{64}.
Superconducting properties of the NbCFIBID microstrip
The resistive properties of the microstrip are summarized in Fig. 7. The resistivity temperature dependence ρ(T) is shown in Fig. 7a, where the ρ(T) curve exhibits a transition from weak localization^{65} to superconductivity at T_{c} = 5.6 K. Here, the transition temperature T_{c} is determined using the 50% resistance drop criterion, as illustrated in Fig. 7b. The resistivity at 7 K is ρ_{7K} = 572 μΩcm and the width of the superconducting transition, defined as the temperature difference between the 10 and 90% resistivity values at the transition, amounts to ΔT_{c} ≈ 0.6 K. Application of a magnetic field B leads to a decrease of T_{c} and a transition broadening, and we use the same 50% resistance drop criterion to deduce the temperature dependence of the upper critical field B_{c2}(T) shown in Fig. 7c. Near T_{c}, the critical field slope \(d{B}_{{\rm{c}}2}/dT{ }_{{T}_{{\rm{c}}}}=2.24\) T K^{−1} corresponds, in the dirty superconductor, to the electron diffusivity \(D=4{k}_{{\rm{B}}}/[\pi e(d{B}_{{\rm{c}}2}/dT{ }_{{T}_{{\rm{c}}}})]\approx 0.49\) cm^{2} s^{−1}. The coherence length and the penetration depth at zero temperature are estimated^{52} as \(\xi (0)=\sqrt{\hslash D/\Delta (0)}=6.5\) nm and \(\lambda (0)=1.05\cdot 1{0}^{3}\sqrt{{\rho }_{{\rm{7K}}}/T_{\rm{c}}}\approx 1060\) nm. By employing the 100 μV voltage drop criterion, from the I–V curves, we deduce the critical currents at zero field I_{c}(0.75T_{c}) = 58 μA and I_{c}(0.9T_{c}) = 16 μA. We assume that the temperature dependence of the depairing current can be described by the expression \({{I}_{{\mathrm{dep}}}(T)=I_{{\mathrm{dep}}}(0)(1{(T/T_{c})}^2)}^{3/2}\) with the prefactor I_{dep}(0) = 0.74w[Δ(0)]^{3/2}/(eR_{□}ℏD), which is justified for dirty superconductors^{52,66,67}. Here, Δ(0) is the superconducting energy gap at zero temperature, e the electron charge, and R_{□} the sheet resistance. With the assumed BCS ratio Δ(0) ≈ 1.76k_{B}T_{c}, we obtain I_{dep}(0) ≈ 268 μA. The calculated dependence I_{dep}(T) is compared with the experimentally measured I_{c}(T) in Fig. 7d. We note that I_{c} varies between 0.7I_{dep} ≲ I_{c} ≲ 0.74I_{dep} in the temperature range 0.5 < t < 1, where τ = T/T_{c} is the reduced temperature.
Timedependent Ginzburg–Landau simulations
To study the evolution of the superconducting order parameter, we numerically solve the modified TDGL equation^{31}:
where \({\xi }_{{\rm{mod}}}^{2}=\pi \sqrt{2}\hslash D/(8{k}_{{\rm{B}}}{T}_{{\rm{c}}}\sqrt{1+{T}_{{\rm{e}}}/{T}_{{\rm{c}}}})\), \({\Delta }_{{\rm{mod}}}^{2}=x({\Delta }_{0}\tanh (1.74{\scriptstyle\sqrt{{T}_{{\rm{c}}}/{T}_{{\rm{e}}}1}}))^{2}/ (1{T}_{{\rm{e}}}/{T}_{{\rm{c}}})\), A is the vector potential, φ is the electrostatic potential, D is the diffusion coefficient, σ_{n} = 2e^{2}DN(0) is the normalstate conductivity with N(0) being the singlespin density of states at the Fermi level, and \({{\bf{j}}}_{{\rm{s}}}^{{\rm{Us}}}\) and \({{\bf{j}}}_{s}^{{\rm{GL}}}\) are the superconducting current densities in the Usadel and Ginzburg–Landau models:
where q_{s} = ∇ϕ − 2eA/ℏc, ϕ is a phase of Δ = ∣Δ∣e^{iϕ}, and \({{\bf{j}}}_{{\rm{s}}}^{{\rm{GL}}}=\frac{\pi {\sigma }_{{\rm{n}}} \Delta { }^{2}}{4e\hslash {k}_{{\rm{B}}}{T}_{{\rm{c}}}}{{\bf{q}}}_{{\rm{s}}}\). It should be noted that at T_{e} not very close to T_{c} the Ginzburg–Landau expression for the superconducting current is not valid quantitatively and one needs to use the Usadel expression for \({{\bf{j}}}_{{\rm{s}}}^{{\rm{Us}}}\). In this case, one should also modify the TDGL equation since the ordinary TDGL equation leads to \({\rm{div}}{{\bf{j}}}_{{\rm{s}}}^{{\rm{GL}}}=0\) in the stationary case, while one needs \({\rm{div}}{{\bf{j}}}_{{\rm{s}}}^{{\rm{Us}}}=0\). Accordingly, by adding the term \({\rm{div}}({{\bf{j}}}_{{\rm{s}}}^{{\rm{Us}}}{{\bf{j}}}_{{\rm{s}}}^{{\rm{GL}}})\) in the TDGL equation we provide \({\rm{div}}{{\bf{j}}}_{{\rm{s}}}^{{\rm{Us}}}=0\). At T_{e} → T_{c} the modified TDGL equation reduces to the ordinary TDGL equation and \({\rm{div}}({{\bf{j}}}_{{\rm{s}}}^{{\rm{Us}}}{{\bf{j}}}_{{\rm{s}}}^{{\rm{GL}}})\) goes to 0.
The electron and phonon temperatures, T_{e} and T_{p}, respectively, are found from the solution of following equations:
where \({{\mathcal{E}}}_{0}=4N(0){({k}_{{\rm{B}}}{T}_{{\rm{c}}})}^{2}\), \({{\mathcal{E}}}_{0}{{\mathcal{E}}}_{{\rm{s}}}({T}_{{\rm{e}}}, \Delta  )\) is the change in the energy of electrons due to the transition to the superconducting state, k_{s} is the heat conductivity in the superconducting state:
\({k}_{{\rm{n}}}=2D{\pi }^{2}{k}_{{\rm{B}}}^{2}N(0){T}_{{\rm{e}}}/3\) is the heat conductivity in the normal state, the term jE describes Joule dissipation, and τ_{esc} is the escape time of nonequilibrium phonons to the substrate. The parameter γ is defined as \(\gamma =\frac{8{\pi }^{2}}{5}\frac{{C}_{{\rm{e}}}({T}_{{\rm{c}}})}{{C}_{{\rm{p}}}({T}_{{\rm{c}}})}\), where C_{e}(T_{c}) and C_{p}(T_{c}) are the heat capacities of electrons and phonons at T = T_{c}, and the characteristic time τ_{0} controls the strength of the electron–phonon and phonon–electron scattering^{31}. It should be noted that the electron–photon scattering time enters the TDGL equation indirectly via the electron temperature T_{e} whose dynamics is governed by τ_{e–ph} ~ τ_{0} in the heat conductance equation. This is rather similar to the LO approach, where τ_{e–ph} enters the kinetic equation for the electron distribution function f(E) (in our case this is the heat conductance equation for T_{e}) and f(E) enters the GL equation in the LO model^{20,21}.
To find the electrostatic potential φ, we also solve the current continuity equation:
where j_{n} = −σ_{n}∇φ is the normal current density.
Values of the parameters γ = 9 and τ_{0} = 925 ns used in the calculations are estimates for NbN. Their variation only leads to quantitative changes in the I–V curves.
At the edges where vortices enter and exit the microstrip, we use the boundary conditions j_{n}∣_{n} = j_{s}∣_{n} = 0 and ∂T_{e}/∂n = 0, ∂∣Δ∣/∂n = 0 while at the edges along the current direction T_{e} = T, ∣Δ∣ = 0, j_{s}∣_{n} = 0, j_{n}∣_{n} = I/wd. The latter boundary conditions model the contact of the superconducting strip with a normal reservoir being in equilibrium. This choice provides a way "to inject” the current into the superconducting microstrip in the modeling. The modeled length of the microstrip is L = 4w.
In the considered model, the penetration length of the electric field L_{E} is about the coherence length ξ(T), which is a consequence of τ_{ee} ≪ τ_{ep}. If τ_{ee} ≳ τ_{ep}, then L_{E} can be considerably larger than ξ(T). In general, L_{E} stipulates the stability of the phase slip process in 1D superconducting wires at larger currents^{68}. In the case of vortex rivers (phase slip lines with vortices) it should also lead to their stability at larger currents, providing a critical velocity of Abrikosov vortices close to the velocity of Josephson vortices, which could explain the experimentally observed v^{*} ≳ 10 km s^{−1}. Within the framework of the considered model, a larger L_{E} can be modeled by a smaller numerical coefficient at the time derivative ∂Δ/∂t. This simultaneously leads to a decrease of the relaxation time of ∣Δ∣, which also leads to an increase of v^{*}. For instance, a fivefold decrease of this coefficient (that corresponds to an increase of L_{E} by a factor of \(\approx \sqrt{5}\)) results in a twofold increase of V^{*} and v^{*} and a small decrease of I^{*} at B = 0.1B_{0}. One can also see that in this case vortex rivers are well formed at I = I^{*} and Abrikosov vortices are closer to Abrikosov–Josephson vortices because of the stronger suppression of the order parameter along the vortex river, leading to higher instability velocities.
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Acknowledgements
O.V.D. acknowledges the German Research Foundation (DFG) for support through Grant No. 374052683 (DO1511/31). D.Y.V. acknowledges the Russian Science Foundation for support through Grant No. 177230036. A.V.C. acknowledges support within the ERC Starting Grant No. 678309 MagnonCircuits. Furthermore, this work was supported by the European Cooperation in Science and Technology via COST Action CA16218 (NANOCOHYBRI). Support through the Frankfurt Center of Electron Microscopy (FCEM) is gratefully acknowledged. Open access funding has been provided by the University of Vienna.
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O.V.D. conceived the experiment and performed the measurements. O.V.D. and M.Y.M. designed the samples. F.P. and R.S. fabricated the samples under the supervision of M.H. R.S. automated the data acquisition. O.V.D. and V.M.B. evaluated the data. D.Y.V. provided theoretical support and performed simulations. O.V.D., D.Y.V., A.V.C., and M.H. discussed the interpretation and the relevance of the results. O.V.D. and D.Y.V. wrote the paper. All authors discussed the results and contributed to the paper writing.
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Dobrovolskiy, O.V., Vodolazov, D.Y., Porrati, F. et al. Ultrafast vortex motion in a directwrite NbC superconductor. Nat Commun 11, 3291 (2020). https://doi.org/10.1038/s4146702016987y
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DOI: https://doi.org/10.1038/s4146702016987y
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