A Perspective on superconductivity in curved 3D nanoarchitectures

Since its discovery in 1911 by Heike Kamerlingh Onnes,¹ superconductivity has enabled numerous applications across various branches of science and technology² such as energy (cables),³ transportation (magnetic levitation),⁴ medical care (magnetic resonance imaging),⁵ particle accelerators (high-field magnets),⁶ magnetic (quantum interferometry)⁷ and electromagnetic (bolometry)⁸ field sensing, quantum optics (single-photon counting),⁹ and information technology (fluxonic devices and quantum computing).^10,11 The latter three domains urge the miniaturization of superconductors to the nanoscale, their operation in a broad (dc to THz) frequency range, and the integrability with other technologies. In this regard, extending superconductors into the third dimension allows for the footprint reduction of a device, its thermal decoupling from the substrate, and the development of multi-terminal devices and circuits with complex interconnectivity. Herewith, curved geometries bring about the smoothness of conjunctions, which allows for avoiding undesired weak links at sharp turns,¹² minimizing current-crowding effects,¹³ and facilitates theoretical modeling.

Recent reviews on 3D and curved superconductors and ferro- and antiferromagnets are given in Refs. 14 and 15. The effects of topology and geometry on various properties of self-rolled micro- and nanoarchitectures are discussed in the recent book.¹⁶ In this Perspective, with the focus on the conceptual clarity, we share our vision of the development of the field of superconductivity in curved 3D nanoarchitectures and outline the challenges to be met as well as the fundamental knowledge to be gained on the road toward applications, see Fig. 1.

The most essential effects originating from the extension of a planar superconductor into a curved 3D structure are exemplified in Fig. 2(a) for a 2D film rolled up into an open 3D tube. First, as the direction of the external magnetic field B varies with respect to the normal to the tube surface, the normal (B_n) and tangential (B_t) field components become strongly inhomogeneous. Second, the pattern of the Meissner screening currents I_M evolves from a singly connected loop to a complex pattern with a multiply connected topology. Third, since the majority of technologically relevant superconductors at the nanoscale are penetrated by magnetic flux quanta (Abrikosov vortices)¹⁷ at moderate magnetic fields, the nontrivial topology of the screening currents leads to emerging phenomena in the dynamics of topological defects (vortices and phase slips) of the superconducting order parameter $Ψ$⁠. Finally, in the presence of even rather high magnetic fields, some parts of a 3D nanostructure can be in vortex-free or surface superconducting states.¹⁸ In this way, the combination of the curved geometry of the structure with the nontrivial topology of the Meissner currents gives rise to phenomena that do not occur in planar superconductor films. In particular, the interplay among the dynamic patterns of the screening currents, vortices, and phase slips opens up a possibility to efficiently tailor the magnetotransport properties of 3D nanoarchitectures, e.g., the shape of the magnetic field–voltage [V(B)] and current–voltage (I–V) characteristics toward advanced technological applications such as high-performance bolometers,¹⁹ fluxonic circuits,²⁰ and sensors of the magnetic field.²¹ 

From the viewpoint of basic research, curved superconductor 3D nanostructures represent a valuable platform of topological matter. Most commonly, topological matter is mentioned in the context of topologically protected surface/edge states governed by Dirac physics and/or topologically nontrivial electronic structure in the momentum space.²² These effects underlay, e.g., the quantum Hall effect and the properties of topological insulators, superconductors, and semimetals. A further essential realm of phenomena is associated with systems in which nontrivial topology occurs due to a special geometry of the structures or fields in real space, e.g., in quantum rings, Möbius strips, multiterminal Josephson junctions, and skyrmions.^16,23,24 The topological transitions in superconductor 3D nanoarchitectures considered in what follows are caused by a curved geometry in real space, too.

We note that the three-dimensionality of nanoarchitectures in real space does not necessarily imply the 3D regime of correlations of the superconducting charge carriers therein. For the latter, all dimensions of the system should be larger than the coherence length ξ,²⁵ while the dimensions of structures considered here range from few nm to a few hundreds of micrometers. Furthermore, 3D nanoarchitectures considered in what follows are distinct from traditional bulk systems as they represent low-dimensional structures expanding in a designed way into the third dimension in real space. In the majority of cases, we consider 2D manifolds (thin-walled objects) expanded into a 3D geometry in real space while charge transport therein occurs in the quasi-1D or -2D regime.

The article is organized as follows: we first outline experimental techniques for fabrication and characterization of curved 3D superconductors at the nanoscale and then review selected representative effects in the dynamics of topological defects therein with a particular focus on thin-walled open tubes and nanohelices. Next, we outline current challenges in this rapidly developing research domain and highlight some prospect directions of fundamental and applied research in the field of superconductivity in curved 3D nanoarchitectures, as illustrated in Fig. 1.

Nowadays, roll-up technology,²⁶ 3D nanoprinting,³² and self-controlled growth of nanotubes of low-dimensional materials (LDMs)^27,33 have reached a high level of maturity for fabrication of curved superconductor 3D nanoarchitectures. The major principles of the former two techniques are illustrated in Figs. 2(b) and 2(c), while an exemplary geometry of gate-controlled magneto-transport measurements on LDM nanotubes (which is adaptable to other types of nanoscale superconducting systems) is shown in Fig. 2(d). The capabilities of the fabrication techniques regarding materials, geometries, and their typical dimensions are outlined in Table I.

Roll-up technology²⁶ implies the use of two materials with different lattice constants, which are deposited onto a substrate covered with an etchant-sensitive (sacrificial) layer [Fig. 2(b)]. After the lithography step, the substrate is immersed into a solvent for area-selective etching of the sacrificial layer. Once this layer is etched away, each material tends to acquire its inherent lattice constant, resulting in a mechanical strain. The release of the mechanical strain causes the bilayer to bend in the direction of the material with the smaller lattice constant. After one revolution, an open tube can be formed, while multiple revolutions resulting from longer etching times yield Swiss-roll-like structures. By defining the geometrical pattern at some tilt angle with respect to the roll-up direction, helical structures can be fabricated.¹⁹ 

This additive manufacturing technology is based on direct-writing by focused ion or electron beam induced deposition (FIBID or FEBID, respectively).^37,38 FIBID and FEBID rely upon the beam-induced dissociation of precursor molecules into a permanent deposit and volatile molecular fragments, see Fig. 2(c). The focused beam is provided by a scanning electron microscope, and the shape and size of the deposited structure are controlled by the beam movement. A vertical resolution better than 1 nm can be achieved routinely with both techniques. The demonstrated smallest nanowire diameters for superconductor 3D nano-architectures deposited with a Ga⁺ FIB are about 300 nm for WC²⁹ and 170 nm for NbC,¹² and these are about 30 nm for WC grown with a He⁺ FIB.^30,31 FIBID and FEBID can be used for fabrication of superconductor (S) and ferromagnet (F) 3D nanoarchitectures^14,32,39 and S/F hybrids.^40,41 Furthermore, they can be combined with post-growth area-selective irradiation with ions and electron beams for geometry refinements^42,43 and on-demand engineering of the conducting^44,45 and magnetic^46–48 properties of the S and F layers.

An interesting class of superconducting materials occurring in a curved 3D geometry is offered by LDMs and, in particular, carbon and transition metal dichalcogenide (TMD) nanotubes.^27,28 TMDs are attracting great attention as 2D materials beyond graphene with application potential for electronics, spintronics, photonics, and valleytronics.^49–51 Many TMDs, which are semiconductors without carrier doping, exhibit superconductivity under ionic gating^52,53 and can form tubular shapes with noncentrosymmetric chiral structures,^54,55 in which superconductivity harbors exotic quantum phenomena and nontrivial Cooper pairing.^56,57 One of the manifestations of the chiral structure in the electronic transport is the unidirectional resistance.³⁶ As shown in Fig. 2(d), the two directions of current flow are not identical due to the chiral nature of the conducting medium, when the magnetic field is applied parallel to the tube axis. In this geometry, the first-harmonic ac resistance exhibits quantum magnetoresistance oscillations⁵⁸ originating from the interference of supercurrent along the circumference of the nanotube, while the finite second-harmonic ac resistance indicates the unidirectional electrical transport due to the chiral symmetry.³⁶ 

Generally, the superconducting and resistive properties of micro- and nanostructures can be characterized by integrated-response and local-probe techniques. The former include conventional electrical resistance measurements, measurements of microwave power absorption,^39,59,60 detection of microwave generation from phase slips, vortex motion^42,43 and other excitations (e.g., spin waves in S/F hybrid structures⁶¹), and for large enough material volumes, ac susceptibility and magnetic moment measurements.^62,63 For micrometer-scale 3D structures, we anticipate that micro-Hall magnetometry⁶⁴ can be applied. In general, while there have been much less experiments than theoretical studies of curved 3D superconductor nanoarchitectures, the adaptation of integrated-response techniques appears less challenging than that of local-probe techniques.

The local probes include those sensitive to magnetic fields [scanning Hall microscopy (SHM),^65–67 magnetic force microscopy (MFM),^68,69 and scanning SQUID-on-tip (SOT) microscopy^21,70], localized thermal excitation [low-temperature scanning electron microscopy⁷¹ and laser scanning microscopy (LSM)⁷²], and surface density of states [scanning tunneling microscopy (STM)^73–75]. SHM, MFM, and SOT require the probe rastering over the superconductor surface at a distance adjusted to conform the shape of the superconductor, which is challenging for 3D nanoarchitectures. Though the diffraction-limited size of the laser probe in LSM restricts its use to microarchitectures, the “far-field” nature of the laser probe should, in principle, allow for mapping of surfaces of 3D nanostructures via xy-scanning in conjunction with focus adjustment.⁷² 

The use of transmission techniques, such as Lorentz microscopy,⁷⁶ is complicated because of the shadow effects in 3D nanostructures. The application of reflection techniques, such as magneto-optical imaging (MOI),⁷⁷ x-ray magnetic circular dichroism (XMCD),⁷⁸ and magneto-optical Kerr effect (MOKE),⁷⁹ is additionally complicated by the requirement of placing a magnetic indicator film on top of the superconductor. The severe requirements regarding the tunneling conditions in STM make its implementation for 3D nanoarchitectures barely feasible. At the same time, 3D nanoprinting of local probes⁸⁰ appears as a potentially viable approach for the development of point-contact spectroscopy⁸¹ of 3D superconductor nanoarchitectures.

Superconductivity and vortex matter in curved 3D micro- and nanoarchitectures have been a subject of research of many groups around the world, but primarily theoretically so far. Among the systems investigated in the last two decades, the most extensively studied ones are cylindrical^{30,34–36,82–95} and spherical^96–111 shells, as well as helical^{19,31,112,113} and Möbius strips.^114–118 The most widely used approach for modeling the spatiotemporal dynamics of the superconducting order parameter therein is based on the time-dependent Ginzburg–Landau (TDGL) equation; further approaches are mentioned in Ref. 14. Since our studies of curved 3D superconductors were primarily concerned with thin-walled open nanotubes^{34,93,95,119–122} and helices,^19,31,113 we next outline the most essential findings related to these nanoarchitectures.

The key finding of the recent decade is that a plethora of properties emerge for 3D structures having curvature radii R on the (sub-)μm length scale. Such radii are comparable with the magnetic penetration depth λ, and they are much larger than the coherence length ξ in superconductors like Nb,^19,123,124 WC,^31,125 and NbC.^12,126 The latter circumstance implies that the considered radii are not yet small enough to ensure an influence of the curvature on the kinetic energy of the superconducting charge carriers. Instead, it is the (sub-)μm-scale inhomogeneity of B_n, which leads to a nontrivial topology of the superconducting screening currents.¹⁶ In particular, for modeling based on the TDGL equation, we have revealed that the patterns of the screening currents in thin-wall open superconductor nanotubes consist of two disconnected loops at weak transport currents [Fig. 2(a)] and four disconnected loops at strong transport currents [panel 2 in Fig. 3(a)]. This multiple connectedness of the screening currents in curved 3D superconductors is in drastic contrast with the single connectedness of one Meissner current loop in planar superconductor thin films [Fig. 2(a)].

The dynamics of the order parameter patterns in open nanotubes can be controlled by their radius,⁹³ see Fig. 3(a). In equilibrium, in the presence of a B-field applied perpendicular to the tube axis, fragments of a hexagonal vortex lattice appear in the tube areas where B_n ≈ B, and there are no vortices in the areas where B_t ≈ B. The curvature-induced inhomogeneity of $Bn(φ)=−B sin φ$ (⁠$φ$ is the polar angle) leads to a nonuniform deformation of the vortex lattice. Equilibrium patterns of $|Ψ|2$ reveal the coexistence of vortices and the Meissner currents in the tubes. At relatively small magnetic fluxes $Φ/Φ0$ (⁠$Φ0$ is the magnetic flux quantum), the Meissner currents flow in vortex-free regions of the tube. At higher magnetic fluxes, the Meissner currents begin to penetrate into the regions with vortices. The induced Magnus force pushes the vortices away from the regions where $Bt≈B$⁠. At yet higher fluxes, the sample becomes strongly inhomogeneous as the superconducting state is expelled from the regions where $Bt≈B$⁠, while vortices form one or more chains in the regions where $Bn≈B$⁠. The particular vortex configuration is determined by the competition between the triangular vortex lattice and the boundary region on the cylindrical surface where vortices occur. Distinct from planar films, this boundary region in tubes is not a geometric boundary, but rather results from the interplay between the magnetic field and the geometry of the cylindrical surface [Fig. 2(a)].

If a transport current I is applied through the electrodes attached to the banks of the slit (azimuthal current flow), vortices are driven by the Magnus force $FM$ in the direction defined by $I×B$⁠. At small B, vortices periodically nucleate at the tube edges, move along the tube, and denucleate at the opposite edges. Due to opposite directions of $FM$ in the upper and bottom half-tubes, the vortices move there in opposite directions. This is distinct from 2D thin films, where the oppositely directed motion occurs for vortices and antivortices. Interestingly, the vortices in both half-tubes nucleate, move, and denucleate synchronously. As a result, superconductor tubes of ∼1 μm radius act as generators of correlated vortex pairs in the frequency range $∼0.1$–10 GHz, and the generation frequency can be tuned via B and R variation.

Different regimes of the vortex dynamics are effectively controlled by varying the positions of pinning centers in open tubes.¹¹⁹ The vortex nucleation period reveals branching due to an inhomogeneous distribution of the transport current in microtubes with multiple electrodes.⁹⁵ This allows for vortex removal from certain regions of the superconductor, which is of practical interest, e.g., for suppressing the $1/f$-noise due to hopping of trapped vortices and, thus, for extending the operation regime of superconductor-based sensors to lower frequencies.⁹⁵ With an increase in B, a bifurcation occurs at a certain B value, which corresponds to the appearance of two possible vortex trajectories.¹²⁰ One of the trajectories is characterized by a much faster growth of the time of flight $Δt1$ with the magnetic field, than the other one. The bifurcation of vortex trajectories originates from the complex interplay among the nucleation rate, Magnus force, and intervortex interaction. In summary, open superconductor nanotubes provide highly controllable vortex transmission lines as 3D fluxonic frequency generators with the application potential for quantum computing.¹²⁷ 

At larger B, the vortex pattern consisting of one or a few vortex chains is getting deformed as is more prominent for tubes of larger R. After the nucleation of another vortex at one tube's edge, its motion leads to a shift of all other vortices in the closest-to-slit row until the last vortex in the row denucleates at the opposite edge. At $B<Bcr$⁠, as indicated in Fig. 3(a), there are no vortices in the tubes. At $B>Bcr$⁠, two sets of paths correspond to vortices moving in opposite directions in the two half-tubes. The time of flight $Δt1$ and the period of nucleation of vortices at a given edge point $Δt2$ are rising and decaying functions of B, respectively, effectively controlled by R, see panel 2 in Fig. 3(a). When $Δt1$ is smaller (larger) than $Δt2$⁠, a sparse (dense) vortex dynamics regime occurs [panel 3(4) in Fig. 3(a)]. A transition from the sparse to dense regime takes place at $Btr$ [panel 2 in Fig. 3(a)]. In the case of a few chains, an active chain of paraxially moving vortices is the closest one to the slit edge, while remote-from-slit chains are quasi-static, since the transport current is significantly perturbed there by the circulating currents of the closest-to-slit vortices. This behavior is peculiar to superconductor open tubes and does not occur in planar membranes.

For a given value of the transport current I, the V(B) curve for an open tube is a piecewise-linear function, which consists of two (approximately linear) sections with different slopes. The sixfold decrease in the V(B) slope implies a transition in the vortex dynamics from one vortex chain to many chains.¹²¹ Thus, the shape of the V(B) curve suggests a possibility for an indirect analysis of vortex patterns in open nanostructured microtubes via electrical resistance measurements.¹²¹ 

At larger I, a topological transition between vortex dynamics patterns and phase-slip regimes may occur,³⁴ leading to a “pulse” in the V(B) curve [panel 1 in Fig. 3(b)], corresponding to a wide phase-slip area in panel 4 of Fig. 3(b). With an increase in B, the phase-slip area is followed by the reentrance of chains of moving vortices. The topological transition consists in the occurrence of two further loops [indicated with green in panel 2 of Fig. 3(b)] of weak superconducting currents connecting two regions of the screening currents (which are disconnected in the case of vortex-chain dynamics). The topological transition from two disconnected regions of the screening currents (red) to four (red and green) and then back to two (red) occurs with increasing B for certain I intervals. Under these conditions, the vortex nucleation locus connects both halves of the tube [panel 3 in Fig. 3(b)]. This is distinct from the case of weak currents, at which the half-tubes are strictly disconnected [Fig. 2(a)]. The generation of vortex–antivortex pairs, as indicated by white and black circles in panel 3 in Fig. 3(b), results from their unbinding due to the strong transport current. The nucleation and separation of vortex–antivortex pairs on the opposite-to-slit microtube side is followed by their motion until they denucleate at the sides of the tube or annihilation of a vortex from the pair with an antivortex from a neighboring pair. The motion of vortices and antivortices on the opposite-to-slit microtube side is much faster than the vortex lifetime for the disconnected vortex dynamics in the half-tubes. It is this fast dynamics of vortices, which leads to the extended phase slip in panel 4 in Fig. 3(b).

The application of FIBID has recently allowed for fabrication of WC nanohelices with diameters down to 100 nm,³¹ whose resistance–current R(I) characteristics exhibit voltage steps [panels 1 and 2 in Fig. 3(c)]. TDGL simulations reveal several patterns of $|Ψ|2$ corresponding to the inhomogeneous spatial distribution of $Bn$ over the surface of the helical structure [panel 3 in Fig. 3(c)]. The experimentally observed resistance jumps (>100 Ω) are about two orders of magnitude larger than the resistance induced by an individual vortex (⁠$≃1 Ω$⁠). The resistance jumps are, therefore, attributed to the occurrence of phase slips, which start to appear at the I values indicated in panel 2 in Fig. 3(c). The transition of any two half-turns into the full phase-slip regime causes the resistance to increase by about 1 kΩ, while the presence of a phase slip in all half-turns without vortex dynamics results in $R≃2.25$ kΩ.

From an arbitrary initial state, the order parameter evolves to one of the three following quasi-stationary patterns: a pure vortex state, vortices coexisting with regions with a depressed order parameter (phase-slip regime), and a pure phase-slip regime. It is assumed that the whole half-winding, at which the order-parameter depression appears, switches to the normal state due to Joule heating. In the regions with minimal $Bn$⁠, the order-parameter depression extent is suppressed. Thus, the experimentally observed plateaus in R(I)³¹ are attributed to the development of the phase-slip regime in some half-windings, whose number increases with increase in I. The complex 3D geometry of nanohelices determines topologically nontrivial patterns of the screening currents and confinement potentials that depend on the helix radius and pitch (or, equivalently, curvature and torsion) and stipulate the occurrence of different patterns of topological defects. The helical geometry cardinally changes the resistive properties of superconductors and, thus, paves the way for a family of future electronic components such as sensors, energy storage elements, resonators, and nanoantennas based on 3D superconductor nanoarchitectures.

Superconducting Nb helices with diameters down to $6 μ$m were recently fabricated by roll-up technology from a 50 nm-thick Nb film¹⁹ [Fig. 4(a)]. The very weak thermal contact of the helices with the substrate suggests their use as sensitive transition-edge sensors (TESs) for detection of microwave radiation. In the measurement setup shown in Fig. 4(a), the lock-in amplifier provides a modulation for the microwave signal (120 GHz) whose power (300 mW) is reduced to 1% by a filter and then split by the polarizer into two paths: one going to the nanohelix and the other illuminating a commercially available QMC bolometer. An analysis of the experimental data¹⁹ reveals that although the nanohelix has a smaller signal-to-noise ratio compared to the QMC detector, its noise-equivalent-power is about four orders of magnitude smaller than that for the QMC sensor. In addition, the timescale of current fluctuations, which are needed to heat the nanohelix, is at least one order of magnitude shorter than 10–$20 μ$s response times of the TES used in sub-millimeter wave astronomy.¹⁹ This experimental finding renders curved 3D superconductor nanoarchitectures as interesting objects of study at microwave frequencies, superimposed dc and ac drives, and an abrupt switching on/off of the dc transport current.

An essential step was taken in Ref. 122 toward understanding of the (dc + ac)-driven dynamics of topological defects in open superconductor nanotubes. The key theoretical finding¹²² is a transition between two regimes in the superconducting dynamics. The first regime is characterized by a dominant first harmonic in the Fourier spectrum of the induced voltage at the frequencies of the ac. Such a voltage response is typical of the two limiting cases, when the open tube is mainly superconducting or normally conducting. The second regime features a rich Fourier spectrum of the induced voltage because of a complex interplay between the internal (dc-driven) dynamics of superconducting vortices or phase slips and the dynamics induced by the superimposed ac drive. Figure 4(b) illustrates the evolution of the modulus and phase of the order parameter in an open superconductor nanotube with R = 400 nm at the perpendicular-to-axis magnetic field B = 4 mT for the ac frequency f = 0.6 GHz and a series of ac modulation depths $j1/j0=0.3$ (1), 0.5 (2), and 0.8 (3).

Topological transitions in open nanotubes have been analyzed theoretically under gradual and abrupt switch-on of the current and magnetic field.¹³⁰ An abrupt switch-on triggers the transition from the vortex patterns to the phase-slip regime, increasing the dissipation. The dependence of the superconducting regimes on the switch-on speed and the stability of such regimes implies that there is a barrier between them. As a result, a hysteresis effect has been predicted for the I-V curve of open nanotubes.¹³⁰ With increasing B, the hysteresis loop widens in current and narrows in voltage.

The induced voltage in open nanotubes has been explored numerically under the transport current modulated with the “external” frequency $fj$⁠.¹²⁸ If $fj$ is close to the “internal” frequency $fU$ associated with vortex nucleation events, which appears in the Fourier spectrum of the induced voltage, then $fU$ locks to $fj$ [between 14.2 and 15.3 GHz in Fig. 4(c)], since vortices nucleate at the same frequency as the external one. In addition, higher harmonics of $fj$ and $fU$ and their combinatorial frequencies occur in the Fourier spectrum of the induced voltage, as expressed in terms of the voltage signal intensity $Uf2/2$ in panels 1 and 2 in Fig. 4(c). The average voltage exhibits a resonance-like feature at $fU=fj$ [panel 3 in Fig. 4(c)].

Numerical simulations for superconductor nanomembranes of nontrivial geometry require complicated algorithms, which significantly increase computational time. In order to circumvent this difficulty, an algorithm based on the differential-geometry formalism¹²⁹ has been developed for the GL equations in a curved space. Effectively, a 3D GL model for an ultrathin 2D membrane embedded into a 3D flat space in a homogeneous external magnetic field can be reduced to a 2D GL model with a magnetic field normal to its surface as a scalar field. This opens the way to manipulate the effective $Bn$⁠, engineer the superconducting properties of the membrane, and control the geometric regions in the normal and superconducting states. The occurrence and behavior of vortices and phase slips are efficiently governed by the geometry of the curved membrane. In particular, a superconductor membrane shaped as a “bowl” is predicted to harbor curved phase-slip events [encircled in Fig. 4(d)].

Most importantly, in the studies so far, global effects of curvature are mediated by the geometry leading to a nontrivial topology of the superconducting screening currents over a large size scale of nanostructures. By contrast, the local effects of curvature require curvature radii on the scale of the coherence length ξ, which is still a severe experimental challenge. In particular, the minimal diameters of Ga⁺-FIBID 3D superconductor nanowire structures (⁠$≃170$ nm¹²) are much larger than $ξ≃5$ nm in dirty low-temperature superconductors. Much smaller feature sizes can be achieved in superconductor 3D nanostructures fabricated by He⁺-FIBID (nanowire wall thicknesses down to $≃20$ nm,³⁰ helix curvature radii down to $≃50$ nm). Hence, even these feature sizes are an order of magnitude larger than ξ to cause accessible curvature effects on the kinetic energy of Cooper pairs in these materials.

At present, 3D superconductor nanoarchitectures fabricated by FEBID are not yet available, requiring the synthesis of precursor gases yielding enough high growth rates for 3D fabrication.¹³¹ In addition, superconductor FIBID nanostructures have only been demonstrated for WC³¹ and NbC¹² so far, limiting the choice of materials to these two dirty-limit superconductors. Finally, 3D nanoprinting by FIBID and FEBID is a serial process, which suits best rapid prototyping. While direct-write methods can be applied to some extent for fabrication of structures composed of many individual elements, the applications requiring large arrays of 3D nanoarchitectures should appeal for parallel fabrication processes.

An advantage of rolled-up technology²⁶ is that it is a parallel process, yielding multiple structures in one fabrication step. However, this technology yields restricted classes of geometry, being currently limited to open tubes, multi-walled tubes,³⁵ and helical structures.¹⁹ 

Superconductivity in carbon nanotubes has been investigated primarily in the assembled form of single, double, or multi-walled nanotubes.^132,133 Investigations of superconductivity and vortex matter in individual tubes are very interesting but remain elusive. In this regard, TMDs represent a viable alternative for studying the superconducting transport in individual LDM nanotubes. In addition, the chiral structures occurring in tubular geometries of TMDs bring about non-reciprocity of superconductivity. This non-reciprocity of charge transport is dual¹³⁴ to the non-reciprocity of magnetic flux transport, which finds application in vortex rectifiers (vortex diodes, or ratchets).^41,135–137 The gate voltage control^36,138 brings about a further degree of freedom for the design of vortex triodes,¹³⁹ multiterminal hybrid circuits,¹⁴⁰ and electric field-controlled fluxonic devices.¹⁴¹ 

Superconducting spherical and toroidal shells, as well as Möbius strips largely remain beyond the reach of current experimental capabilities. There were reports on the millimeter-sized spherically shaped YBCO granules by seeded-melt growth¹⁴² or by binding millions of μm-sized particles in strong electric fields.¹⁴³ However, much less experimental work has so far been concerned with hollow, topologically nontrivial micro- and nano-sized superconducting systems. Thus, multi-crystalline isotropic Pb hollow sphere-like samples were produced by a solvothermal synthesis.¹¹⁰ Magnetic measurements revealed that the produced samples with an inner diameter of 400 nm are superconducting with a transition temperature $Tc$ of 11.05 K.¹¹⁰ For the characterization of such submicrometer-large samples, micro-Hall magnetometry,⁶⁴ nano-SQUID,¹⁴⁴ and microwave inductive sensing^39,145 should, in principle, be applicable. However, reports on the characterization of micro- and nanoscale curved superconductors remain rather rare.

In nonorientable systems, such as a Möbius strip, unusual vortex states were predicted.¹¹⁶ Though synthesizing rings, Möbius strips, and figure-of-eight strips of NbSe₃ single crystals by chemical-vapour transport was demonstrated back in 2002,¹¹⁵ experimental realization of Möbius nanostrips represents a challenge yet to be met. Very recently, ferromagnetic Möbius nanostrips have been successfully fabricated by FEBID,¹⁴⁶ suggesting that direct-write nanofabrication technologies are likely to become the techniques of choice for fabrication of superconducting spheres and Möbius nanostrips. With a further development of 3D direct-write technologies, we anticipate possibilities for fabrication of core-shell semispherical structures. At the same time, further advancements in materials science are required to make accessible superconductor hollow toroidal nanoshells.

C. Topological, unconventional and hybrid superconductors in curved 3D geometriesGreat interest is now attracted by topological superconductors, see recent reviews^147–149 and original works.^150–155 These studies are primarily directed at the generation of Majorana edge modes for decoherence-immune and fault-tolerant quantum computing.¹⁵⁶ In 2D superconductors, the presence of such Majorana modes is determined by the amount of flux piercing the superconductor at a vortex: an odd number of fluxons gives a Majorana, while an even number does not. Recently, it has been shown that, in the absence of any flux, the ground state on the annulus does not support Majorana modes, while the one on the cylinder does.¹¹⁸ A nonorientable Möbius strip, which has only one edge, has been shown to necessarily have a defect line along the centerline to support edge Majorana modes.¹¹⁸ 

Recent proposals render the possibility for realization of topological superconductivity by hybridizing ordinary superconductors with helical materials with the help of magnetic perturbations. The curved geometric profile also allows for tuning the spin correlations of the superconducting state via the induced inhomogeneity of the spin–orbit coupling that affects the Josephson critical current¹⁵⁷ and, in curved nanostructures with Rashba spin–orbit coupling, leads to nontrivial textures of spin-triplet pairs.¹⁵⁸ Due to the geometric Meissner effect, 2D chiral superconductors on curved surfaces spontaneously develop a magnetic flux.¹⁵⁹ This effect has been employed to shed light on the location of zero-energy Majorana modes, and it provides an unequivocal signature of chiral superconductivity.¹⁵⁹ As a next step, we anticipate that the fascinating interplay of topological superconductivity (in the momentum space) with the nontrivial topology of the Meissner currents induced by the 3D geometry (in real space) will be investigated in the years to come.

A few challenges can be identified regarding theoretical modeling of curved 3D superconductor nanoarchitectures as follows: first, to intensify and deepen their exploration, analytical numerical instruments should be elaborated for simulations of feasible structures and arrays of realistic dimensions (of the order of tens to hundreds micrometers) with inherent imperfections and inhomogeneities in the presence of various external fields. So far, only simplified geometries have been successfully tackled. TDGL equation-based modeling of complex curved 3D nanoarchitectures requires advanced computational capabilities and the use of high-performance computing. Since simulations are very time-consuming, alternative approaches seem to be needed such as deep learning with its capability to empirically recognize patterns of the complex order parameter for complex-shaped systems via supervised training and deep neural networks.¹⁶⁰ Second, in order to establish a link with the quantum fields and gravity theories¹⁶ and to enlarge the scope of nontrivial geometries, which could be tackled within the differential-geometry-based algorithm,¹²⁹ numerical calculations should be implemented on GPUs. This would allow for understanding the impact of the membrane's geometry on the inhomogeneity of $Bn$ and the superconducting screening currents, as well as for the development of a geometry-driven control over the superconducting properties of curved 3D nanoarchitectures. Third, in realistic superconductor nanoarchitectures, the occurrence of imperfections (mechanical defects in self-rolled films and impurity atoms in 3D-written structures) must be taken into consideration. Aside from that, the real edge barriers for nucleation of superconducting vortices are not perfect (e.g., notches^126,161), and material composition variations should be accounted for in order to examine how the quality of the structure edges affects the evolution of the order parameter in the entire superconductor nanoarchitecture. Fourth, as a major challenge, the (dc + ac)-driven escape of quasiparticles (heated electrons) from the vortex cores leading to a flux-flow instability^40,162–164 at large transport currents should be addressed both theoretically and experimentally. Finally, the manipulation by individual topological defects in curved 3D nanostructures should be extended toward the quantum regime at mK temperatures. Such studies may be essential for the paradigm underlying quantum information and quantum computing.¹⁶⁵ Overall, the impact of curvature should be analyzed from a more general viewpoint with the transfer of concepts from/to other disciplines such as soft-matter physics, chemistry, biology, and mathematics.

At the applications facet, the particular prospective directions can be outlined as follows: (i) Enhancement of the sensitivity of superconductor magnetic-field nanosensors by orders of magnitude and design of innovative superconductor quantum-interference filters and switchers based, e.g., on the controlled vortex dynamics and topological transition between vortices and phase slips in superconductor open nanotubes and hybrid nanostructures, (ii) development of robust elements for fluxon-based quantum information processing, such as self-assembled networks of Josephson junctions (JJs), parametric amplifiers, memory elements, and frequency generators, e.g., based on fluxon transmission lines in superconductor open nanotubes, (iii) development of superconductor bolometers and THz-detectors, which possess a significant advancement in sensitivity and noise reduction as compared to the available ones, e.g., using superconductor nanohelices, and (iv) exploration of the potentialities of curved 3D nanoarchitectures as platforms and interfaces for hybrid superconducting circuits with complex interconnectivity.

Superconductivity and vortex matter in curved 3D nanoarchitectures have turned into a rapidly developing domain of modern superconductivity being an essential topical area of curved condensed matter. Remarkably, for already more than a century, curvature has primarily been at the heart of general relativity with the associated phenomena at the astrophysics length scales. By contrast, at the nanometer length scale, curvature is naturally occurring in molecular biology with a well-known example of double-stranded molecules of nucleic acids such as DNA. Thus, curved condensed-matter systems are between these extreme length scales, and at the same time, they are cross-inspired by the current interest in the impact of curvature shown in other disciplines, ranging from soft-matter physics to chemistry, biology, and mathematics.

As emphasized in this Perspective, over the last two decades, theoretical predictions in relation to superconductivity in curved 3D nanoarchitectures have reached the “critical mass” and, on the experimental side, there are already a few techniques capable of fabrication of various curved 3D geometries. Though further method developments are required for fabrication of nanoscale spherical shells, Möbius strips, and toroidal structures, a complex interplay of the patterns of superconducting screening currents with the 3D geometry in tubular and helical structures has already unveiled a plethora of emerging effects in the dynamics of topological defects of the superconducting order parameter in curved 3D nanorachitectures. Hence, we are now entering a decade when these and not-yet-foreseen effects should be explored experimentally, giving rise to emerging applications in magnetic field sensing, bolometry, fluxonic devices, and information processing.

The authors gratefully acknowledge fruitful collaborations with I. A. Bogush, R. Córdoba, J. M. De Teresa, V. N. Gladilin, M. Huth, W. Lang, R. O. Rezaev, V. A. Shklovskij, and O. G. Schmidt. V.M.F. acknowledges the German Research Foundation (DFG) for support via Project No. FO 956/6-1 and ZIH TU Dresden for providing its facilities for high throughput calculations. O.V.D. acknowledges the German Research Foundation (DFG) for support via Project No. DO 1511/3–1 and the Austrian Science Fund (FWF) for support through Grant No. I 4889 “CurviMag.” Support from the European Cooperation in Science and Technology via COST Action Nos. CA16218 (NANOCOHYBRI) and CA19108 (HiSCALE) is acknowledged.

The authors have no conflicts of interest to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.