Superconducting Coherence Length Research Articles

Hysteresis magnetoresistance and edge superconductivity in Nb film with diluted square array of holes

We investigated the transport properties of a superconducting Nb film with a diluted square array of holes, in which one fourth of the sites were removed from the original square lattice. The edge-to-edge separation of the holes in the sample is comparable to the superconducting coherence length, resulting in a system similar to multi-connected superconducting islands. Three regions in magnetoresistance () curves can be identified. In the low field region, fluxoid quantization leads to fine structures in and curves. At higher fields, the entering and leaving of interstitial vortices results in a novel field and temperature dependent hysteresis. At even higher fields, weak oscillations in resistance were observed due to edge superconductivity of inter-hole biconcave thin necks and Little–Parks effect. Moreover, the edge state results in a remarkable field induced narrowing of superconducting transition width.

Revealing the magnetic proximity effect in EuS/Al bilayers through superconducting tunneling spectroscopy

A ferromagnetic insulator attached to a superconductor is known to induce an exchange splitting of the Bardeen-Cooper-Schrieffer (BCS) singularity by a magnitude proportional to the magnetization, and penetrating into the superconductor to a depth comparable with the superconducting coherence length. We study this long-range magnetic proximity effect in EuS/Al bilayers and find that the exchange splitting of the BCS peaks is present already in the unpolarized state of the ferromagnetic insulator (EuS), and is being further enhanced when magnetizing the sample by a magnetic field. The measurement data taken at the lowest temperatures feature a high contrast which has allowed us to relate the line shape of the split BCS conductance peaks to the characteristic magnetic domain structure of the EuS layer in the unpolarized state. These results pave the way to engineering triplet superconducting correlations at domain walls in EuS/Al bilayers. Furthermore, the hard gap and clear splitting observed in our tunneling spectroscopy measurements indicate that EuS/Al bilayers are excellent candidates for substituting strong magnetic fields in experiments studying Majorana bound states.

Effects of nonmagnetic disorder on the energy of Yu-Shiba-Rusinov states

We study the sensitivity of Yu-Shiba-Rusinov states, bound states that form around magnetic scatterers in superconductors, to the presence of nonmagnetic disorder in both two and three dimensional systems. We formulate a scattering approach to this problem and reduce the effects of disorder to two contributions: disorder-induced normal reflection and a random phase of the amplitude for Andreev reflection. We find that both of these are small even for moderate amounts of disorder. In the dirty limit in which the disorder-induced mean free path is smaller than the superconducting coherence length, the variance of the energy of the Yu-Shiba-Rusinov state remains small in the ratio of the Fermi wavelength and the mean free path. This effect is more pronounced in three dimensions, where only impurities within a few Fermi wavelengths of the magnetic scatterer contribute. In two dimensions the energy variance is larger by a logarithmic factor because impurities contribute up to a distance of the order of the superconducting coherence length.

Onset of surface superconductivity beyond the Saint-James–de Gennes limit

We revisit the problem of the surface superconductivity nucleation focusing on the detailed study of the critical field $H_{c3}$ as a function of temperature and disorder. Using the semiclassical Eilenberger formalism we find that away from the Ginzburg-Landau region the ratio between the nucleation critical field $H_{c3}$ and the upper critical field $H_{c2}$ deviates strongly from the Saint-James-de Gennes limit. In particular, the $H_{c3}/H_{c2}$ is found to be a nonmonotonic function of temperature, which reaches the maximum for a set of parameters corresponding to a crossover region from ballistic to diffusive scattering, when the mean free path in a bulk of a superconductor is of the same order as zero-temperature superconducting coherence length. We also analyze the robustness of the nucleated phases with respect to diffusive scattering off the sample boundary by solving exactly corresponding eigenvalue problem of an integral equation for the critical field. The implications of these results for the transport in superconductors of various geometries near $H_{c3}$ are briefly discussed. In particular, we present results for the mechanism of magnetoconductivity oscillations due to surface superconductivity effects.

Quasiparticle relaxation in superconducting nanostructures

We examine energy relaxation of non-equilibrium quasiparticles in "dirty" superconductors with the electron mean free path much shorter than the superconducting coherence length. Relaxation of low-energy non-equilibrium quasiparticles is dominated by phonon emission. We derive the corresponding collision integral and find the quasiparticle relaxation rate. The latter is sensitive to the breaking of time reversal symmetry (TRS) by a magnetic field (or magnetic impurities). As a concrete application of the developed theory, we address quasiparticle trapping by a vortex and a current-biased constriction. We show that trapping of hot quasiparticles may predominantly occur at distances from the vortex core, or the constriction, significantly exceeding the superconducting coherence length.

Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an s -wave superconductor

We consider a system of two one-dimensional nanowires coupled via an $s$-wave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only direct pairing, where one wire is decoupled from the superconductor, or only crossed Andreev pairing, where each nanowire is considered as a spin-polarized edge of a quantum Hall state, we find that the gap is always reduced, by orders of magnitude in certain cases, when both types of pairing are present. Our analytical results are further supported by numerical calculations on a tight-binding lattice. Finally, we show that treating the proximity effect by integrating out the superconductor using the bulk Green's function does not reproduce the results of our exact diagonalization.

Phase-Slip Events in a Wide Superconducting Film Induced by Proximity to a Half-Metallic Strip

Emergence of phase-slip lines is one of the main mechanisms for dissipation in current-carrying superconducting thin films with lateral dimensions larger than the superconducting coherence length. We show that effective control of the wide resistive region between dissipation-free and completely ohmic transport regime can be achieved by evaporating a half-metallic strip, made from a Co2CrAl compound in our case, directly onto a type-II superconducting film, a Mo56Re44 alloy in our case, and tuning the spin-polarized injection current. We have revealed steps-like features in current–voltage characteristics of the Mo–Re films, even without any injection, and explained it within a “self-injection” scenario in contacts formed by ferromagnetic (F) and superconducting (S) layers. At certain injection levels, a set of resistive metastable states following the main criteria for the phase-slip events was observed. We have found that low magnetic fields, even less than the lower critical field of the Mo–Re alloy, have a significant impact on the resistive transition region of the S film probably due to their effect on the magnetic ordering in the F strip, thus producing a strong coupling between transport characteristics of proximized S and F layers.

Modulation of pairing symmetry with bond disorder in unconventional superconductors

We study a two-orbital $t$-$J_1$-$J_2$ model, originally developed to describe iron-based superconductors at low energies, in the presence of bond disorder (via next-nearest-neighbor $J_2$-bond dilution). By using the Bogoliubov--de Gennes approach, we self-consistently calculate the local pairing amplitudes and the corresponding density of states, which demonstrate a change of dominant pairing symmetry from $s_\pm$ wave to $d$ wave when increasing disorder strength as long as $J_1\lesssim J_2$. Moreover, the combined pairing interaction and strong bond disorder lead to the formation of $s_\pm$ wave "islands" with length scale of the superconducting coherence length embedded in a $d$ wave "sea." This picture is further complemented by the disorder-averaged pair-pair correlation functions, distinct from the case with potential disorder, where the "sea" is insulating. Due to this inevitable formation of spatial inhomogeneity, the superconducting $T_c$ determined by the superfluid density $\rho_s(T)$ obviously deviates from the value predicted by the conventional Abrikosov-Gorkov theory, where the pairing amplitudes are viewed as uniformly suppressed as the disorder increases.

Inducing superconducting correlation in quantum Hall edge states

The quantum Hall (QH) effect supports a set of chiral edge states at the boundary of a 2-dimensional electron gas (2DEG) system. A superconductor (SC) contacting these states induces correlation of the quasi-particles in the dissipationless 1D chiral QH edge states. If the superconducting electrode is narrower than the superconducting coherence length, the incoming electron are correlated to outgoing hole along the chiral edge state by the Andreev process. In order to realize this crossed Andreev conversion (CAC), it is necessary to fabricate highly transparent and nanometer-scale superconducting junctions to QH system. Here we report the observation of CAC in a graphene QH system contacted with a nanostructured NbN superconducting electrode. The chemical potential of the edge states across the superconducting electrode exhibits a sign reversal, providing direct evidence of CAC. This hybrid SC/QH system is a novel route to create isolated non-Abelian anyonic zero modes, in resonance with the chiral QH edge.

Single-Photon Detection by a Dirty Current-Carrying Superconducting Strip Based on the Kinetic-Equation Approach

Using kinetic equation approach we study dynamics of electrons and phonons in current-carrying superconducting nanostrips after absorption of single photon of near-infrared or optical range. We find that the larger the ratio $C_e/C_{ph}|_{T_c}$ ($T_c$ is a critical temperature of superconductor, $C_e$ and $C_{ph}$ are specific heat capacities of electrons and phonons, respectively) the larger part of photon's energy goes to electrons, they become stronger heated and, hence, could thermalize faster during initial stage of hot spot formation. Thermalization time $\tau_{th}$ could be less than one picoseconds for superconductors with $C_e/C_{ph}|_{T_c}\gg 1$ and small diffusion coefficient $D\simeq 0.5 cm^2/s$ when thermalization occurs mainly due to electron-phonon and phonon-electron scattering in relatively small volume $\sim \xi^2d$ ($\xi$ is a superconducting coherence length, $d<\xi$ is a thickness of the strip). At larger times due to diffusion of hot electrons effective temperature inside the hot spot decreases, the size of hot spot increases, superconducting state becomes unstable and normal domain spreads in the strip at current larger than so-called detection current. We find dependence of detection current on the photon's energy, place of its absorption in the strip, width of the strip, magnetic field and compare it with existing experiments. Our results demonstrate that materials with $C_e/C_{ph}|_{T_c} \ll 1$ are bad candidates for single photon detectors due to small transfer of photon's energy to electronic system and large $\tau_{th}$. We also predict that even several microns wide dirty superconducting bridge is able to detect single near-infrared or optical photon if its critical current exceeds 70 $\%$ of depairing current and $C_e/C_{ph}|_{T_c} \gtrsim 1$.

Fabrication and Characterization of Epitaxial Films of Superconductor NbN and Highly Spin-Polarized Heusler Alloy Co2Fe0.4Mn0.6Si

We report structural, magnetic, and transport properties of epitaxial films of superconductor NbN and highly spin-polarized Heusler alloy ${\rm{Co}}_{{2}}{\rm{Fe}}_{{\rm{0.4}}}{\rm{Mn}}_{{\rm{0.6}}}{\rm{Si}}$ (CFMS). The saturation magnetization $M_{{\rm{s}}}$ of the film was 870 emu/cm3, which is slightly smaller than values reported earlier. With an increase of CFMS thickness $d_{{\rm{CFMS}}}$ , the sheet resistance $R_{{\rm{sq}}}$ decreased and the upper critical field $\mu _{{0}}H_{{\rm{c2}}}(0)$ decreased monotonically, while the superconducting transition temperature $T_{{\rm{c}}}$ exhibited a minimum around $d_{{\rm{CFMS}}}\,= \,{{3\,{\text{nm}}}}$ . A theoretical analysis of superconducting fluctuations revealed that the high spin polarization of the CFMS layer affected the pair breaking parameter δ , the superconducting coherence length ξ (0), and the diffusion coefficient $D$ .

Superconducting properties of long TiN wires

The low-temperature transport properties of titanium nitride wires with the width comparable with or much larger than the superconducting coherence length are studied experimentally. It is shown that the reduction of the width of wires does not affect the transport properties at the temperatures above the superconducting transition temperature and electron transport in this temperature range is determined by quantum contributions to the conductivity from weak localization and electron–electron interaction. It is established that the reduction of the width of wires does not change the superconducting transition temperature but completely suppresses the topological Berezinskii–Kosterlitz–Thouless transition. It is found that the threshold magnetic field increases with a decrease in the width of wires.

Characteristics of superconducting tungsten silicide WxSi1−x for single photon detection

Superconducting properties of three series of amorphous WxSi1-x films with different thickness and stoichiometry were investigated by dc transport measurements in a magnetic field up to 9 T. These amorphous WxSi1-x films were deposited by magnetron co-sputtering of the elemental source targets onto silicon substrates at room temperature and patterned in form of bridges by optical lithography and reactive ion etching. Analysis of the data on magnetoconductivity allowed us to extract the critical temperature, superconducting coherence length, magnetic penetration depth, and diffusion coefficient of electrons in the normal state as a function of film thickness for each stoichiometry. Two basic time constants were derived from transport and time-resolving measurements. A dynamic process of the formation of a hot-spot was analyzed in the framework of a diffusion-based vortex-entry model. We used the two stage diffusion approach and defined a hotspot size by assuming that the quasi-particles and normal-state electrons have the equal diffusion constant. Our findings are consistent with the most recent results on a hot-spot relaxation time in the WxSi1-x superconducting nanowire single-photon detector. In the 5 nm thick W0.85Si0.15 film the hot-spot has a diameter of 105 nm at the peak of the number of non-equilibrium quasi-particles.

Signatures of time-reversal-invariant topological superconductivity in the Josephson effect

For Josephson junctions based on s-wave superconductors, time-reversal symmetry is known to allow for powerful relations between the normal-state junction properties, the excitation spectrum, and the Josephson current. Here we provide analogous relations for Josephson junctions involving one-dimensional time-reversal-invariant topological superconductors supporting Majorana-Kramers pairs, considering both topological-topological and s-wave-topological junctions. Working in the regime where the junction is much shorter than the superconducting coherence length, we obtain a number of analytical and numerical results that hold for arbitrary normal-state conductance and the most general forms of spin-orbit coupling. The signatures of topological superconductivity we find include the fractional ac Josephson effect, which arises in topological-topological junctions provided that the energy relaxation is sufficiently slow. We also show, for both junction types, that robust signatures of topological superconductivity arise in the dc Josephson effect in the form of switches in the Josephson current due to zero-energy crossings of Andreev levels. The junction spin-orbit coupling enters the Josephson current only in the topological-topological case and in a manner determined by the switch locations, thereby allowing quantitative predictions for experiments with the normal-state conductance, the induced gaps, and the switch locations as inputs.

Vortex dynamics in a thin superconducting film with a non-uniform magnetic field applied at its center with a small coil

This paper models the dynamics of vortices that are generated in the middle of a thin, large-area, superconducting film by a low-frequency magnetic field from a small coil, motivated by a desire to better understand measurements of the superconducting coherence length made with a two-coil apparatus. When the applied field exceeds a critical value, vortices and antivortices originate near the middle of the film at the radius where the Lorentz force of the screening supercurrent is largest. The Lorentz force from the screening supercurrent pushes vortices toward the center of the film and antivortices outward. In an experiment, vortices are detected as an increase in mutual inductance between the drive coil and a coaxial “pickup” coil on the opposite side of the film. The model shows that the essential features of measurements are well described when vortex pinning and the attendant hysteresis are included.

Ultrathin two-dimensional superconductivity with strong spin–orbit coupling

We report on a study of epitaxially grown ultrathin Pb films that are only a few atoms thick and have parallel critical magnetic fields much higher than the expected limit set by the interaction of electron spins with a magnetic field, that is, the Clogston-Chandrasekhar limit. The epitaxial thin films are classified as dirty-limit superconductors because their mean-free paths, which are limited by surface scattering, are smaller than their superconducting coherence lengths. The uniformity of superconductivity in these thin films is established by comparing scanning tunneling spectroscopy, scanning superconducting quantum interference device (SQUID) magnetometry, double-coil mutual inductance, and magneto-transport, data that provide average superfluid rigidity on length scales covering the range from microscopic to macroscopic. We argue that the survival of superconductivity at Zeeman energies much larger than the superconducting gap can be understood only as the consequence of strong spin-orbit coupling that, together with substrate-induced inversion-symmetry breaking, produces spin splitting in the normal-state energy bands that is much larger than the superconductor's energy gap.

Full range of proximity effect probed with superconductor/graphene/superconductor junctions

The high tunability of the density of states of graphene makes it an ideal probe of quantum transport in different regimes. In particular, the supercurrent that can flow through a non-superconducting (N) material connected to two superconducting electrodes, crucially depends on the lenghth of the N relative to the superconducting coherence length. Using graphene as the N material we have investigated the full range of the superconducting proximity effect, from short to long diffusive junctions. By combining several S/graphene/S samples with different contacts and lengths, and measuring their gate-dependent critical currents ($ I_c $) and normal state resistance $ R_N $, we compare the product $eR_NI_c$ to the relevant energies, the Thouless energy in long junctions and the superconducting gap of the contacts in short junctions, over three orders of magnitude of Thouless energy. The experimental variations strikingly follow a universal law, close to the predictions of the proximity effect both in the long and short junction regime, as well as in the crossover region, thereby revealing the interplay of the different energy scales. Differences in the numerical coefficients reveal the crucial role played by the interfacial barrier between graphene and the superconducting electrodes, which reduces the supercurrent in both short and long junctions. Surprisingly the reduction of supercurrent is independent of the gate voltage and of the nature of the electrodes. A reduced induced gap and Thouless energy are extracted, revealing the role played by the dwell time in the barrier in the short junction, and an effective increased diffusion time in the long junction. We compare our results to the theoretical predictions of Usadel equations and numerical simulations which better reproduce experiments with imperfect NS interfaces.

Self-organized topological superconductivity in a Yu-Shiba-Rusinov chain

We study a chain of magnetic moments exchange coupled to a conventional three dimensional superconductor. In the normal state the chain orders into a collinear configuration, while in the superconducting phase we find that ferromagnetism is unstable to the formation of a magnetic spiral state. Beyond weak exchange coupling the spiral wavevector greatly exceeds the inverse superconducting coherence length as a result of the strong spin-spin interaction mediated through the subgap band of Yu-Shiba-Rusinov states. Moreover, the simple spin-spin exchange description breaks down as the subgap band crosses the Fermi energy, wherein the spiral phase becomes stabilized by the spontaneous opening of a $p-$wave superconducting gap within the band. This leads to the possibility of electron-driven topological superconductivity with Majorana boundary modes using magnetic atoms on superconducting surfaces.

Scanning tunneling spectroscopy onSrFe2(As1−xPx)2

We investigated SrFe$\mathrm{_2}$(As$\mathrm{_{1-x}}$P$\mathrm{_x}$)$\mathrm{_2}$ single crystals with four different phosphorus concentrations x in the superconducting phase (x = 0.35, 0.46) and in the magnetic phase (x = 0, 0.2). The superconducting samples display a V-shaped superconducting gap, which suggests nodal superconductivity. Furthermore we determined the superconducting coherence length by measuring the spatially resolved superconducting density of states (DOS). Using inelastic tunneling spectroscopy we investigated excitations in the samples with four different phosphorus concentrations. Inelastic peaks are related to bosonic modes. Phonon and non-phonon mechanism for the origin of these peaks are discussed.

A holographic model of d-wave superconductor vortices with Lifshitz scaling

We study analytically the [Formula: see text]-wave holographic superconductors with Lifshitz scaling in the presence of external magnetic field. The vortex lattice solutions of the model have also been obtained with different Lifshitz scaling. Our results imply that holographic [Formula: see text]-wave superconductor is indeed a type II one even for different Lifshitz scaling. This is the same as the conventional [Formula: see text]-wave superconductors in the Ginzburg–Landau (GL) theory. Our results also indicates that the dynamical exponent [Formula: see text] cannot affect the droplet solutions, and the vortex lattice solutions receive its effects only in the radial part. This naively implies that it does not have direct influence on the shape of vortex lattice even after the higher-order corrections are taken into consideration (away from the phase transition point [Formula: see text]). However, it has effects on the upper critical magnetic field [Formula: see text] through the fact that a larger [Formula: see text] results in a smaller [Formula: see text] and therefore influences the size (characterized by [Formula: see text]) of the vortex lattices. Furthermore, close comparisons between our results and those of the GL theory reveal the fact that the upper critical magnetic field [Formula: see text] is inversely proportional to the square of the superconducting coherence length [Formula: see text], regardless of the anisotropy between space and time.