When macroscopic quantum condensates -- superconductors, superfluids, cold atoms and ions, polaritons etc. -- are put in rotation, a quantum vortex lattice forms inside. In homogeneous type-II superconductors, each vortex has a tiny core where the superconducting gap $Δ(r)$ is known to smoothly vanish towards the core centre on the scale of the coherence length $ξ$. The cores host quantized quasiparticle energy levels known as Caroli-de Gennes-Matricon (CdGM) bound states [Caroli {\it et al.,} Phys. Lett. v. 9, 307 (1964)]. In pure materials, the spectrum of the low-lying CdGM states has the characteristic level spacing $\sim Δ_0^2/E_F$, where $E_F$ is the Fermi energy and $Δ_0$ is the bulk gap. In disordered ones, the CdGM states shift and broaden due to scattering. Here, we show, both experimentally and theoretically, that the situation is completely different in granular Nb films, which are commonly used in superconducting electronics. In these films, in which the grains are smaller than $ξ$, the gap $Δ$ in the quasiparticle spectrum reduces towards the vortex core centres by discrete jumps at the grain boundaries. The bound states adapt to the local environment and appear at unexpectedly high energies. Both $Δ(r)$ and bound states form a puzzle-like spatial structure of the core, elements of which are whole grains. Our discovery shakes up the established understanding of the quantum vortex and encourages a reconsideration of the vortex motion and pinning mechanisms in granular superconductors.

We study the local structure of an Abrikosov vortex centered in a mesoscopic diffusive superconducting granule within the Usadel framework. We show that the spatial dependence of the pair potential, supercurrent, and local density of states near the vortex center is governed by a single parameter, leading to universal behavior in the core region. This parameter also controls discontinuities in these quantities at the interface between the grain and the surrounding superconductor. We further derive an explicit criterion for vortex existence in an isolated grain, yielding a critical radius Rc ≈ 3.47ξs at T → 0 and Rc ≈ 1.84ξ(T) in the Ginzburg–Landau limit. The obtained results enable the extraction of interface parameters from spectroscopic data.

Superconductor-ferromagnet hybrid structures with tunable kinetic inductance are promising elements for neuromorphic and quantum computing circuits. We report the fabrication and microwave characterization of split-ring resonators based on Nb/Co/Nb/Co/Nb/Al spin-trigger multilayers and demonstrate a non-volatile spin-valve effect on their resonant properties. Reversal of the relative magnetization orientation of the cobalt layers produces a reproducible shift of the resonant frequency up to 4 MHz at zero applied magnetic field, corresponding to a change in the kinetic inductance of the structure. The incorporation of a proximitized aluminum overlayer is shown to enhance the inductance contrast between the parallel and antiparallel magnetic states by a factor of approximately three relative to structures without this layer. The experimental results are in quantitative agreement with a microscopic model based on the Usadel equations. The demonstrated magnetic memory of the resonant frequency at zero field establishes spin-trigger multilayers as viable field-programmable inductive elements for superconducting digital and neuromorphic circuits.

Current transport in an SF 1 S 1 F 2 sIS Josephson junction, where one of the superconducting (S) electrodes is implemented as a spin valve (SF 1 S 1 F 2 s) and I is the tunnel insulating barrier, has been studied. In such a valve, the state of the thin s layer is controlled by changing the relative orientation of the magnetization vectors of the ferromagnetic layers F 1 and F 2 . The dependences of the critical current density J c on the temperature, as well as on the thicknesses of the ferromagnetic and superconducting layers, are calculated for two experimentally relevant current supply scenarios: through the outer S electrode and through the thin s film of the spin valve. It has been shown that the current supply through the s film provides higher absolute values of J c and preserves the sign of the first harmonic of the current–phase dependence even in the 0–π transition mode of the magnetic subsystem. The obtained results specify the applicability limits of simplified models of direct current flow and formulate practical criteria for interpreting experiments and designing planar Josephson spin valves.

The miniaturization of superconducting electronics demands precise control over quasiparticle dynamics at the nanoscale. Understanding and controlling quasiparticle dynamics is essential for advancing superconducting electronics, particularly as device dimensions approach the nanoscale. Here we present direct experimental evidence for the quasiparticle-to-supercurrent conversion in planar SN-N-NS Josephson nanobridges with submicron electrodes. By employing a versatile measurement platform, we demonstrate that the geometry of injection through superconducting or normal-metal leads significantly alters the critical current and junction resistance. Using a tunable measurement setup that allows current injection through either superconducting or normal-metal leads, we uncover marked changes in critical current and resistance depending on the injection geometry. These effects become significant when electrode widths fall below the characteristic conversion length of similar to 400 nm. These observations are quantitatively explained by a phenomenological model that accounts for nonequilibrium transport and incomplete quasiparticle conversion in narrow electrodes. These results point to the emergence of nonequilibrium transport governed by an intrinsic conversion length. We develop and apply a method for extracting this length from standard transport measurements, offering a practical diagnostic for nanoscale superconducting circuits. Our work establishes fundamental design rules for minimizing energy dissipation and optimizing performance in superconducting quantum devices. Our findings enable optimized design of compact Josephson junctions and support the integration of proximity-based weak links into scalable superconducting logic and sensing architectures.

We present a comprehensive study of planar Nb-Al-Nb Josephson junctions with submicrometer dimensions (L 100 nm, active area of approximately 5 & times; 104 nm2), where the intrinsic superconductivity of the aluminum weak link plays a crucial role in enhancing device performance. Through a combination of theoretical modeling and experimental characterization, we demonstrate that the aluminum interlayer significantly boosts the critical current Ic 50 & micro;A and the characteristic voltage Vc 1 mV at T = 4 K, while maintaining the nonhysteretic current-voltage characteristics essential for digital applications. Our microscopic model, based on self-consistent solutions of the Usadel equations, reveals that this enhancement originates from the coexistence of proximity-induced superconductivity and intrinsic pairing in aluminum, which is particularly pronounced at an optimal boundary resistance. Structural analysis confirms epitaxial Nb-Al interfaces with minimal interdiffusion, enabling reproducible fabrication of these compact junctions. These results establish Nb-Al-Nb bridges as promising building blocks for high-density superconducting electronics operating at helium temperatures.

In this review, we summarize the foundations underlying a variety of phenomena in superconductor– ferromagnet hybrid structures, with a focus on recent advances in several key areas. These include: (i) the fundamental understanding of proximity effects in superconductor–ferromagnet based systems; (ii) spin-valve effects in superconductor– ferromagnet and superconductor–ferromagnet–superconductor Josephson junctions; and (iii) the design and realization of superconducting memory elements, particularly in hybrid Josephson junctions. We also discuss the experimental progress in fabricating and characterizing spin-valve structures.

Josephson junctions based on topological insulators are expected to host Majorana-type bound states that promote a 4 pi-periodic current-phase relationship. One experimental consequence of this periodicity is the absence of odd Shapiro steps in the current-voltage characteristics under microwave radiation. Here, we experimentally study the frequency-dependent amplitude of Shapiro steps in a ballistic Josephson junction involving a 23 nm-thick single nanocrystal of topological insulator Bi2Te2.3Se0.7 connected to two planar Nb-electrodes. We observe a progressive suppression of the first step while decreasing the frequency below 2 GHz. Using the two-channel thermal resistively shunted junction (tRSJ) model, we show that despite a ballistic regime, the overheating phenomena are enough to account for the observed effect. Our results highlight the complex physics of Josephson junctions and underscore that the missing first Shapiro step alone is not a definitive signature of topological superconductivity.

We present a tunable Josephson generator that enables direct qubit excitation and two-tone spectroscopy while reducing cryogenic wiring complexity and thermal load. The device consists of a current-biased Josephson junction integrated with a superconducting spiral resonator, forming a microwave source that operates at cryogenic temperatures. Unlike conventional qubit drive setups that require coaxial microwave lines from room temperature to the base stage, our system requires only a single direct current (DC) bias line from room temperature. The generated microwave signal is delivered to the qubit via a short coaxial line between the 4 K and 20 mK stages, significantly reducing wiring complexity, thermal load in the cryostat and leveraging unused volume at the 4 K stage. We demonstrate continuous-wave excitation of a superconducting qubit and perform tunable two-tone spectroscopy using the Josephson generator as the drive source, resolving qubit spectra with performance comparable to a standard microwave source. Our generator is characterized by low phase noise and can be phase-locked to an external reference for enhanced stability. Operating at 4 K temperature, this self-contained source offers a low-noise, space-efficient solution for qubit control and spectroscopy, paving the way for scalable cryogenic quantum control architectures.

In this review, we summarize the theoretical foundations underlying a variety of phenomena in superconductor-ferromagnet (SF) hybrid structures, with a focus on recent advances in several key areas. These include: (i) the fundamental understanding of proximity effects in SF-based systems; (ii) spin-valve effects in SF and SFS Josephson junctions; and (iii) the design and realization of superconducting memory elements, particularly in SIsF-type Josephson junctions. We also discuss the experimental progress in fabricating and characterizing spin-valve structures. Our discussion is restricted to junctions incorporating weak, homogeneous metallic ferromagnets with collinear magnetization directions, where the essential physical mechanisms can be analyzed within a relatively tractable theoretical framework.

A technique has been developed for the evaluation of the transparency of grain boundaries in polycrystalline superconducting films. The model is founded upon a numerical algorithm for calculating the density of states of an Abrikosov vortex located in the center of a cylindrical granule, separated from the main superconducting matrix by a boundary with finite transparency. The present study calculates the dependences of the gap difference in the density of states on both sides of the boundary and uses this difference to estimate the transparency of the interface.

The longitudinal electron transport in a multilayer superconducting structure SF1S1F2sN, where S is a superconductor, F is a ferromagnet, s is a thin superconducting layer, and N is a normal metal, has been theoretically studied. Calculations have shown that the rotation of the magnetization of ferromagnetic layers relative to each other makes it possible to smoothly change the kinetic inductance of the structure by several times. A feature of the electronic state of the structure in the region of system parameters corresponding to its transition from a state with the 0 stable Josephson phase to a state with the π stable phase (0–π transition) has been discovered. This feature leads to the decrease in the singlet component of the pairing amplitude and to an increase in the kinetic inductance of the entire structure. The study of the effect of the finite longitudinal current on the charge transport has shown that the destruction of superconductivity in different layers occurs step-by-step, and the dependence of the kinetic inductance Lk on the total transport current J exhibits several plateaus with an almost constant inductance.

The supercurrent in a Josephson SF1S1F2sIS spin valve (“S” is for superconductor, “F” is for ferromagnet, and “I” is for insulator) is studied theoretically. It is found that by rotating the magnetization of one of the ferromagnetic layers, a smooth switching of the system between two states with different critical currents is possible. The operating range of the device can be adjusted by varying the thickness of the intermediate s-layer. The proposed structure is a promising scalable control element for the use in superconducting electronics.

In this paper, we consider a nonlinear one-dimensional problem for the equations of superconductivity theory. A specific feature of the problem is a nonstandard Robin type junction condition on the inner boundary and a discontinuous solution. An optimal homogeneous monotone difference scheme including a condition on the interface is constructed for the problem. By solving a series of elliptic problems and by using Newton’s method, we solve the complete system of Usadel equations, which is the basic mathematical model at the microlevel for describing the currents and fields in superconductors with Josephson junctions. The results of calculations for the problem of a pellet with an Abrikosov vortex are presented.

We study the effect of electrode width on superconducting current transport in Nb-Au-Nb Josephson bridges. The critical current as well as the normal bridge resistance drop with decreasing electrode width on scales of a few mu m, which are orders of magnitude larger than the estimated coherence length of the Au strip. We consider several physical reasons for such an anomalous influence of the width W of the superconducting electrode on the critical current I-c (AIWIc) and provide model fits for the resistive and superconducting properties of the bridges. The smooth dependence of the Nb-Au-Nb bridge parameters on the electrode width can be used to optimize the design of superconducting devices for specific applications.

We have studied the Thouless energy in Josephson superconductor-normal metal-superconductor (SN-N-NS) bridges analytically and numerically, taking into account the influence of the sub-electrode regions. We have found a significant suppression of the Thouless energy with increasing interfacial resistance, in agreement with experimental results. The analysis of the temperature dependence of the critical current in Josephson junctions in comparison with the expressions for the Thouless energy may allow the determination of the interface parameters of S and N-layers.

We have studied the proximity effect in an SF1S1F2s superconducting spin valve consisting of a massive superconducting electrode (S) and a multilayer structure formed by thin ferromagnetic (F1,2) and superconducting (S1, s) layers. Within the framework of the Usadel equations, we have shown that changing the mutual orientation of the magnetization vectors of the F1,2 layers from parallel to antiparallel serves to trigger superconductivity in the outer thin s-film. We studied the changes in the pair potential in the outer s-film and found the regions of parameters with a significant spin-valve effect. The strongest effect occurs in the region of parameters where the pair-potential sign is changed in the parallel state. This feature reveals new ways to design devices with highly tunable inductance and critical current.

The ongoing progress of superconducting logic systems with Josephson junctions as base elements requires the development of compatible cryogenic memory. Long enough junctions subject to magnetic field host quantum phase 2 π -singularities—Josephson vortices. Here, we report the realization of the superconducting memory cell whose state is encoded by the number of present Josephson vortices. By integrating the junction into a coplanar resonator and by applying a microwave excitation well below the critical current, we are able to control the state of the system in an energy-efficient and non-destructive manner. The memory effect arises due to the presence of the natural edge barrier for Josephson vortices. The performance of the device is evaluated, and the routes for creating scalable cryogenic memories directly compatible with superconducting microwave technologies are discussed.

Josephson junctions are currently used as base elements of superconducting logic systems. Long enough junctions subject to magnetic field host quantum phase 2{\pi}-singularities - Josephson vortices. Here we report the realization of the superconducting memory whose state is encoded by the number of present Josephson vortices. By integrating the junction into a coplanar resonator and by applying a microwave excitation well below the critical current, we were able to control the state of the memory in an energy-efficent and non-destructive manner. The performance of the device is evaluated, and the routes for creating scalable cryogenic memories directly compatible with superconducting microwave technologies are discussed.

Superconducting proximity junctions based on topological insulators are widely believed to harbor Majorana-like bound states. The latter serves as a paradigm non-local topological quantum computation protocols. Nowadays, a search for topological phases in different materials, perspective for a realization of topological qubits, is one of the central efforts in quantum physics. It is motivated, in particular, by recent observation of anomalous ac Josephson effect, which being a signature of Majorana physics. Its manifestations, such as a fractional Josephson frequency and the absence of the first (or several odd in more rare cases), Shapiro steps, were reported for different materials. Here we study Shapiro steps in Nb/Bi2Te2.3Se0.7/Nb junctions, based on ultrasmall single crystals of a 3D topological insulator synthesized by a physical vapor deposition (PVD) technique. We present evidence that our junctions are ballistic. When subjected to microwave radiation, the junctions exhibit Shapiro steps, but the first step is missing. Typically it is assumed that the missing first step (MFS) effect cannot be observed in the presence of quasiparticle poisoning due to suppression of the 4{\pi}-periodic component. Our findings within the context of the RSJ-model of Josephson junction dynamics show that such behaviour of samples corresponds to a specific condition, requiring a minimum of 5% of the 4{\pi}-component for disappearance of the first Shapiro step.