Usadel Equations Research Articles

Coherence-enhanced phase-dependent dissipation in long SNS Josephson junctions: Revealing Andreev bound state dynamics

One of the best known causes of dissipation in ac driven quantum systems stems from photon absorption. Dissipation can also be caused by the retarded response to the time-dependent excitation, and in general gives insight into the system's relaxation times and mechanisms. We address the dissipation in a mesoscopic normal wire with superconducting contacts, that sustains a supercurrent at zero frequency and that may be expected to remain dissipationless at frequency lower than the superconducting gap. We probe the high frequency linear response of a Normal/Superconductor ring to a time-dependent flux by coupling it to a highly sensitive multimode microwave resonator. Far from being the simple derivative of the current-phase relation, the ring's ac susceptibility also displays a dissipative component whose phase dependence is a signature of the dynamical processes occurring within the Andreev spectrum. We show how dissipation is driven by the competition between the two aforementioned mechanisms. Depending on the relative strength of those contributions, dissipation can be maximal at $\pi$, when the minigap closes, or can be maximal near $ \pi/2$, when the dc supercurrent is maximal. We also find that the dissipative response increases at low temperature and can even exceed the normal state conductance. The results are confronted with predictions of the Kubo linear response and time-dependent Usadel equations. This experiment shows the power of the ac susceptibility measurement of individual hybrid mesoscopic systems in probing in a controlled way the quantum dynamics of ABS. By spanning different physical regimes, our experiments provide a unique access to inelastic scattering and spectroscopy of an isolated quantum coherent system. This technique should be a tool of choice to investigate topological superconductivity and detect the topological protection of edge states.

Magnetic and superconducting phase diagram of Nb/Gd/Nb trilayers

We report on a study of the structural, magnetic and superconducting properties of Nb(25nm)/Gd($d_f$)/Nb(25nm) hybrid structures of a superconductor/ ferromagnet (S/F) type. The structural characterization of the samples, including careful determination of the layer thickness, was performed using neutron and X-ray scattering with the aid of depth sensitive mass-spectrometry. The magnetization of the samples was determined by SQUID magnetometry and polarized neutron reflectometry and the presence of magnetic ordering for all samples down to the thinnest Gd(0.8nm) layer was shown. The analysis of the neutron spin asymmetry allowed us to prove the absence of magnetically dead layers in junctions with Gd interlayer thickness larger than one monolayer. The measured dependence of the superconducting transition temperature $T_c(d_f)$ has a damped oscillatory behavior with well defined positions of the minimum at $d_f$=3nm and the following maximum at $d_f$=4nm; the behavior, which is in qualitative agreement with the prior work (J.S. Jiang et al, PRB 54, 6119). The analysis of the $T_c(d_f)$ dependence based on Usadel equations showed that the observed minimum at $d_f$=3nm can be described by the so called "$0$" to "$\pi$" phase transition of highly transparent S/F interfaces with the superconducting correlation length $\xi_f \approx 4$nm in Gd. This penetration length is several times higher than for strong ferromagnets like Fe, Co or Ni, simplifying thus preparation of S/F structures with $d_f \sim \xi_f$ which are of topical interest in superconducting spintronics.

Exploring the performance of thin-film superconducting multilayers as kinetic inductance detectors for low-frequency detection

We have solved numerically the diffusive Usadel equations that describe the spatially varying superconducting proximity effect in Ti–Al thin-film bi- and trilayers with thickness values that are suitable for kinetic inductance detectors (KIDs) to operate as photon detectors with detection thresholds in the frequency range of 50–90 GHz. Using Nam’s extension of the Mattis–Bardeen calculation of the superconductor complex conductivity, we show how to calculate the surface impedance for the spatially varying case, and hence the surface impedance quality factor. In addition, we calculate energy-and spatially-averaged quasiparticle lifetimes at temperatures well-below the transition temperature and compare to calculation in Al. Our results for the pair-breaking threshold demonstrate differences between bilayers and trilayers with the same total film thicknesses. We also predict high quality factors and long multilayer-averaged quasiparticle recombination times compared to thin-film Al. Our calculations give a route for designing KIDs to operate in this scientifically-important frequency regime.

Surface impedance and optimum surface resistance of a superconductor with an imperfect surface

We calculate a low-frequency surface impedance of a dirty, s-wave superconductor with an imperfect surface incorporating either a thin layer with a reduced pairing constant or a thin, proximity-coupled normal layer. Such structures model realistic surfaces of superconducting materials which can contain oxide layers, absorbed impurities or nonstoichiometric composition. We solved the Usadel equations self-consistently and obtained spatial distributions of the order parameter and the quasiparticle density of states which then were used to calculate a low-frequency surface resistance $R_s(T)$ and the magnetic penetration depth $\lambda(T)$ as functions of temperature in the limit of local London electrodynamics. It is shown that the imperfect surface in a single-band s-wave superconductor results in a non-exponential temperature dependence of $Z(T)$ at $T\ll T_c$ which can mimic the behavior of multiband or d-wave superconductors. The imperfect surface and the broadening of the gap peaks in the quasiparticle density of states $N(\epsilon)$ in the bulk give rise to a weakly temperature-dependent residual surface resistance. We show that the surface resistance can be optimized and even reduced below its value for an ideal surface by engineering $N(\epsilon)$ at the surface using pairbreaking mechanisms, particularly, by incorporating a small density of magnetic impurities or by tuning the thickness and conductivity of the normal layer and its contact resistance. The results of this work address the limit of $R_s$ in superconductors at $T\ll T_c$, and the ways of engineering the optimal density of states by surface nano-structuring and impurities to reduce losses in superconducting micro-resonators, thin film strip lines, and radio frequency cavities for particle accelerators.

Phase transitions in hybrid SFS structures with thin superconducting layers

Calculations of critical temperature T c of the phase transition to superconducting state of a superconductor/ ferromagnet/superconductor (SFS) hybrid structure with proximity effect is performed on the base of linearized Usadel equations. It is shown that the proximity effect between S and F metals and the exchange interaction can induce an inhomogeneous superconducting state with longitudinal to layers Δ ∝ exp(ipz) modulation of the superconductivity order parameter, which is characterized by nonzero value of the wave number p, describing the Larkin–Ovchinnikov–Fulde–Ferrell instability. Influence of this instability on transitions between 0- and π-states of the SFS structure is studied. It is shown that the 0–π transition is accompanied by a nonmonotonic dependence of both the critical temperature T c and the effective penetration depth Λ of the magnetic field into the hybrid structure on the characteristic size of the ferromagnetic region.

Gauge theory of the long-range proximity effect and spontaneous currents in superconducting heterostructures with strong ferromagnets

We present the generalized quasiclassical theory of the long-range superconducting proximity effect in heterostructures with strong ferromagnets, where the exchange splitting is of the order of Fermi energy. In the ferromagnet the propagation of spin-triplet Cooper pairs residing on the spin-split Fermi surfaces is shown to be governed by the spin-dependent Abelian gauge field which results either from the spin-orbital coupling or from the magnetic texture. The additional gauge field enters into the quasiclassical equations in superposition with the usual electromagnetic vector potential and results in the generation of spontaneous superconducting currents and phase shifts in various geometries which provide the sources of long-range spin-triplet correlations. We derive the Usadel equations and boundary conditions for the strong ferromagnet and consider several generic examples of the Josephson systems supporting spontaneous currents.

Usadel equation in the presence of intrinsic spin-orbit coupling: A unified theory of magnetoelectric effects in normal and superconducting systems

The Usadel equation is the standard theoretical tool for the description of superconducting structures in the diffusive limit. Here I derive the Usadel equation for gyrotropic materials with a generic linear in momentum spin-orbit coupling. It accounts for the spin-charge/singlet-triplet coupling and in the normal state reduces to the system of spin-charge diffusion equations describing various magneto-electric effects, such as the spin Hall effect (SHE), the spin-galvanic effect (SGE) and their inverses. Therefore the derived Usadel equation establishes a direct connection of these effects to their superconducting counterparts. The working power of the present formalism is illustrated on the example of the bulk SGE.

Screening properties of multiply connected ferromagnet–superconductor hybrid structures

Superconducting phase transition temperature T c of a ferromagnet/superconductor (SF) hybrid structure consisting of a hollow superconducting (S) cylinder (shell) with the central part (core) filled with a ferromagnetic (F) metal has been analyzed on the basis of linearized Usadel equations. It has been shown that the proximity effect between the S and F metals, as well as the exchange interaction, may induce an inhomogeneous superconducting state with Δ ~ exp(iLθ + ipz), which is characterized by nonzero circulation of phase L and wavenumber p describing the Larkin–Ovchinnikov–Fulde–Ferrell (LOFF) instability along the cylinder axis. The transitions between the states with different values of L and p, which are accompanied by a nonmonotonic dependence of superconducting transition temperature T c and effective magnetic field penetration depth Λ into the SF structure on the characteristic size of the ferromagnetic region, have been investigated.

Proximity-induced superconductivity in all-silicon superconductor /normal-metal junctions

We have realised laser-doped all-silicon superconducting (S)/ normal metal (N) bilayers of tunable thickness and dopant concentration. We observed a strong reduction of the bilayers critical temperature when increasing the normal metal thickness, a signature of the highly transparent S/N interface associated to the epitaxial sharp laser doping profile. We extracted the interface resistance by fitting with the linearised Usadel equations, demonstrating a reduction of one order of magnitude from previous superconductor/doped Si interfaces. In this well controlled crystalline system we exploited the low resistance S/N interfaces to elaborate all-silicon lateral SNS Josephson junctions with long range proximity effect. Their dc transport properties, such as the critical and retrapping currents, could be well understood in the diffusive regime. Furthermore, this work lead to the estimation of important parameters in ultra-doped superconducting Si, such as the Fermi velocity, the coherence length, or the electron-phonon coupling constant, fundamental to conceive an all-silicon superconducting electronics.

Quasiclassical theory for the superconducting proximity effect in Dirac materials

We derive the quasiclassical non-equilibrium Eilenberger and Usadel equations to first order in quantities small compared to the Fermi energy, valid for Dirac edge and surface electrons with spin-momentum locking, as relevant for topological insulators. We discuss in detail several of the key technical points and assumptions of the derivation, and provide a Riccati-parametrization of the equations. Solving first the equilibrium equations for S/N and S/F bilayers and Josephson junctions, we study the superconducting proximity effect in Dirac materials. Similarly to related works, we find that the effect of an exchange field depends strongly on the direction of the field. Only components normal to the transport direction lead to attenuation of the Cooper pair wavefunction inside the F. Fields parallel to the transport direction lead to phase-shifts in the dependence on the superconducting phase difference for both the charge current and density of states in an S/F/S-junction. Moreover, we compute the differential conductance in S/N and S/F bilayers with an applied voltage bias, and determine the dependence on the length of the N and F regions and the exchange field.

Metastability and bifurcation in superconducting nanorings

We describe an approach, based on direct numerical solution of the Usadel equation, to finding stationary points of the free energy of superconducting nanorings. We consider both uniform (equilibrium) solutions and the critical droplets that mediate activated transitions between them. For the uniform solutions, we compute the critical current as a function of the temperature, thus obtaining a correction factor to Bardeen's 1962 interpolation formula. For the droplets, we present a metastability chart that shows the activation energy as a function of the temperature and current. A comparison of the activation energy for a ring to experimental results for a wire connected to superconducting leads reveals a discrepancy at large currents. We discuss possible reasons for it. We also discuss the nature of the bifurcation point at which the droplet merges with the uniform solution.

Proximity effects in multiband superconductor–ferromagnetic metal structures

Proximity effects in multiband superconductor/ferromagnetic metal (S/F) structures are considered. In the limit when the transport properties of both metals correspond to the diffusion conductivity type, the general form of the Usadel equations and their quasiclassical boundary conditions are formulated. For several limiting cases, analytical solutions of the Usadel equations are obtained. It is shown that induced (proximity) characteristics reflect the multiband structure of the superconductor and significantly depend on the SF boundary parameters and the exchange interaction magnitude in the F layer. Main manifestations of the magnetic proximity effect in the S layer near the SF boundary are considered: additional suppression of the superconducting order parameter by the exchange field, spin splitting of the quasiparticle density of states, the emergence of localized states inside the energy gap, and induced equilibrium magnetization of unpaired electrons in the S layer.

Microwave Admittance of Gold‐Palladium Nanowires with Proximity‐Induced Superconductivity

Quantitative electrical admittance measurements of diffusive superconductor–normal‐metal–superconductor (SNS) junctions at gigahertz frequencies and millikelvin temperatures are reported. The gold‐palladium‐based SNS junctions are arranged into a chain of superconducting quantum interference devices. The chain is coupled strongly to a multimode microwave resonator with a mode spacing of approximately 0.6 GHz. By measuring the resonance frequencies and quality factors of the resonator modes, the dissipative and reactive parts of the admittance of the chain are extracted. The phase and temperature dependence of the admittance near 1 GHz are compared with theory based on the time‐dependent Usadel equations. This comparison allows the identification of important discrepancies between theory and experiment that are not resolved by including inelastic scattering or elastic spin‐flip scattering in the theory.

Analytically determined topological phase diagram of the proximity-induced gap in diffusive n-terminal Josephson junctions

Multiterminal Josephson junctions have recently been proposed as a route to artificially mimic topological matter with the distinct advantage that its properties can be controlled via the superconducting phase difference, giving rise to Weyl points in 4-terminal geometries. A key goal is to accurately determine when the system makes a transition from a gapped to non-gapped state as a function of the phase differences in the system, the latter effectively playing the role of quasiparticle momenta in conventional topological matter. We here determine the proximity gap phase diagram of diffusive n-terminal Josephson junctions (), both numerically and analytically, by identifying a class of solutions to the Usadel equation at zero energy in the full proximity effect regime. We present an analytical equation which provides the phase diagram for an arbitrary number of terminals n. After briefly demonstrating the validity of the analytical approach in the previously studied 2- and 3-terminal cases, we focus on the 4-terminal case and map out the regimes where the electronic excitations in the system are gapped and non-gapped, respectively, demonstrating also in this case full agreement between the analytical and numerical approach.

Abrikosov vortices in SF bilayers

We study the spatial distribution of supercurrent circulated around an Abrikosov vortex in an SF bilayer in perpendicular magnetic field. Within the dirty limit regime and circular cell approximation for the vortex lattice, we derive the conditions when the Usadel equations the F layer can be solved analytically. Using the obtained solutions, we demonstrate the possibility of reversal of direction of proximity induced supercurrents around the vortex in the F layer compared to that in the S-layer. The direction of currents can be controlled either by varying transparency of the SF interface or by changing an exchange field in a ferromagnet. We argue that the origin of this effect is due the phase shift between singlet and triplet order parameter components induced in the F-layer. Possible ways of experimental detection of the predicted effect are discussed.

Manifestation of extrinsic spin Hall effect in superconducting structures: Nondissipative magnetoelectric effects

We present a comprehensive quasiclassical approach for studying transport properties of superconducting diffusive hybrid structures in the presence of extrinsic spin-orbit coupling. We derive a generalized Usadel equation and boundary conditions that in the normal state reduce to the drift-diffusion theory governing the spin-Hall effect in inversion symmetric materials. These equations predict the non-dissipative spin-galvanic effect, that is the generation of supercurrents by a spin-splitting field, and its inverse -- the creation of magnetic moment by a supercurrent. These effects can be seen as counterparts of the spin-Hall, anomalous Hall and their inverse effects in the superconducting state. Our theory opens numerous possibilities for using superconducting structures in magnetoelectronics.

Parametric spin-valve effect in superconductor/ferromagnet structures

The equations of the microscopic theory of superconductivity in the diffusion limit (Usadel equations) are used to interpret experimental data from a study of the spin-valve effect in epitaxial Nb/Ho structures. The cause of the quasimetallic behavior of their critical dependences is determined. The influence of the triplet component of the superconducting condensate on the critical temperature is studied.

The Solitary Superconductivity in Dirty F $$ _1 $$ 1 F $$ _2 $$ 2 S Trilayer with Arbitrary Interfaces

We theoretically study the proximity effect in ferromagnet–superconductor trilayer in the F\(_1\)F\(_2\)S design. Electron–electron pairing interaction in F layers is taken into account. The boundary value problem is considered in the frame of Usadel equations. The conditions of the solitary superconductivity appearance in F\(_1\)F\(_2\)S system are theoretically investigated taking into account of arbitrary transparencies of interfaces. The experimental observation of this effect and its spin-valve application are discussed.

Electrical Conductivity through a Single Atomic Step Measured with the Proximity-Induced Superconducting Pair Correlation.

Local disordered nanostructures in an atomically thick metallic layer on a semiconducting substrate play significant and decisive roles in transport properties of two-dimensional (2D) conductive systems. We measured the electrical conductivity through a step of monoatomic height in a truly microscopic manner by using as a signal the superconducting pair correlation induced by the proximity effect. The transport property across a step of a one-monolayer Pb surface metallic phase, formed on a Si(111) substrate, was evaluated by inducing the pair correlation around the local defect and measuring its response, i.e., the reduced density of states at the Fermi energy using scanning tunneling microscopy. We found that the step resistance has a significant contribution to the total resistance on a nominally flat surface. Our study also revealed that steps in the 2D metallic layer terminate the propagation of the pair correlation. Superconductivity is enhanced between the first surface step and the superconductor-normal-metal interface by reflectionless tunneling when the step is located within a coherence length.

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.