Usadel Equations Research Articles

Critical Temperature and Order Parameter in Superconductor/Inhomogeneous Ferromagnet Heterostructures

A method for solving a self-consistent boundary value problem for the linearized Usadel equation is proposed. The method makes it possible to find the sample-normalized distribution of the superconducting order parameter and the critical temperature as functions of the problem parameters and to solve relatively complex spatially inhomogeneous problems, e.g., superconducting heterostructures containing inhomogeneous magnetic layers. Within this approach, layered structures containing superconducting and domain-split ferromagnetic layers are considered. The theory is compared with experiment for the Fe 20 Å/V 340 Å/Fe 8 Å/Cr/Fe 8 Å system.

A Finite-Difference Scheme for Discontinuous Solutions of the Usadel Equations

A Finite-Difference Scheme for Discontinuous Solutions of the Usadel Equations

Spin-Valve-Controlled Triggering of Superconductivity

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.

Ferromagnetic insulator induced inverse proximity effect in superconducting DoS

A ferromagnetic insulator proximate with a superconductor induces some spin asymmetry in superconducting correlations. Density of states (DoS) changes as a result of the induction of magnetization in superconductor (S) under the influence of the inverse proximity effect with the ferromagnetic insulator (FI). The study solves the Usadel equations numerically and calculates characteristic features of the density of states at the interface boundary and at the boundary with vacuum, caused by FI. The electron density of states was analyzed, showing significant singlet-to-triplet conversion, even for small values of the interface parameter (φ). In DoS, the zero bias peak tends to play a larger role than the spin-splitting of the BCS peaks. The features of the density of state obtained in this work can be measured experimentally.

Disorder effects in NbTiN superconducting resonators

Disordered superconducting materials like NbTiN possess a high kinetic inductance fraction and an adjustable critical temperature, making them a good choice for low-temperature detectors. Their energy gap (Δ), critical temperature (T c), and quasiparticle density of states (QDOS) distribution, however, deviate from the classical BCS theory due to the disorder effects. The Usadel equation, which takes account of elastic scattering, non-elastic scattering, and electro–phonon coupling, can be applied to explain and describe these deviations. This paper presents numerical simulations of the disorder effects based on the Usadel equation to investigate their effects on the Δ, T c, QDOS distribution, and complex conductivity of the NbTiN film. Furthermore, NbTiN superconducting resonators with coplanar waveguide (CPW) structures are fabricated and characterized at different temperatures to validate our numerical simulations. The pair-breaking parameter α and the critical temperature in the pure state TcP of our NbTiN film are determined from the experimental results and numerical simulations. This study has significant implications for the development of low-temperature detectors made of disordered superconducting materials.

Electromagnetic Properties of Aluminum-Based Bilayers for Kinetic Inductance Detectors

The complex conductivity of a superconducting thin film is related to the quasiparticle density, which depends on the physical temperature and can also be modified by external pair breaking with photons and phonons. This relationship forms the underlying operating principle of Kinetic Inductance Detectors (KIDs), where the detection threshold is governed by the superconducting energy gap. We investigate the electromagnetic properties of thin-film aluminum that is proximitized with either a normal metal layer of copper or a superconducting layer with a lower $T_C$, such as iridium, in order to extend the operating range of KIDs. Using the Usadel equations along with the Nam expressions for complex conductivity, we calculate the density of states and the complex conductivity of the resulting bilayers to understand the dependence of the pair breaking threshold, surface impedance, and intrinsic quality factor of superconducting bilayers on the relative film thicknesses. The calculations and analyses provide theoretical insights in designing aluminum-based bilayer kinetic inductance detectors for detection of microwave photons and athermal phonons at the frequencies well below the pair breaking threshold of a pure aluminum film.

Gapful electrons in a vortex core in granular superconductors

We calculate the quasiparticle density of states (DoS) inside the vortex core in a granular superconductor, generalizing the classical solution applicable for dirty superconductors. A discrete version of the Usadel equation for a vortex is derived and solved numerically for a broad range of parameters. Electron DoS is found to be gapful when the coherence length \xiξ becomes comparable to the distance between neighboring grains ll. Minigap magnitude E_gEg grows from zero at \xi \approx 1.4 lξ≈1.4l to third of superconducting gap $_0 $ at \xi \approx 0.5 lξ≈0.5l. The absence of low-energy excitations is the main ingredient needed to understand strong suppression of microwave dissipation recently observed in a mixed state of granular Al.

Interplay of superconductivity and localization near a two-dimensional ferromagnetic quantum critical point

We study the superconducting instability of a two-dimensional disordered Fermi liquid weakly coupled to the soft fluctuations associated with proximity to an Ising-ferromagnetic quantum critical point. We derive interaction-induced corrections to the Usadel equation governing the superconducting gap function, and show that diffusion and localization effects drastically modify the interplay between fermionic incoherence and strong pairing interactions. In particular, we obtain the phase diagram, and demonstrate that (i) there is an intermediate range of disorder strength where superconductivity is enhanced, eventually followed by a tendency towards the superconductor-insulator transition at stronger disorder; and (ii) diffusive particle-particle modes (so-called ``Cooperons'') acquire anomalous dynamical scaling $z=4$, indicating strong non-Fermi liquid behavior.

Modeling of the superconducting triplet spin valve with several superconductor layers

Using the matrix method for solving the linearized Usadel equations, the critical temperature and the distribution of singlet pairing components of the superconductor/ferromagnet/superconductor/ferromagnet structure with nonideal boundaries are obtained. A transition from the π- to the 0-phase state between layers of superconductors was obtained with a change in the angle between the magnetizations of ferromagnetic layers in such a structure.

Triplet proximity effects in superconductor/ferromagnet hybrid structures exhibiting intrinsic spin–orbit coupling

The conventional spin–singlets with momentum zero can be transformed into spin-polarized triplet states with non-zero momentum at the superconductor/ferromagnet interface by an inhomogeneous magnetic exchange field. While the decaying spectrum of triplet pairs in ferromagnetic metals has received significant attention, less is known about the decay of these pairs in ferromagnets with intrinsic spin–orbit coupling (SOC) present at the superconductor/ferromagnet interface. We have performed simulations to study the triplet proximity effects in hybrid structures composed of a BCS superconductor and ferromagnet with intrinsic spin orbit coupling (SOC) in the diffusive limit where the quasiclassical boundary conditions have been modified to incorporate the effect of SO coupling in Usadel equations and the self-consistent equation. We laid emphasis on the modification of the density of states inside the ferromagnet near the Fermi energy. The generated short ranged triplets (SRTs) present themselves in the form of zero energy gap in the density of states (DOS) even in the presence of comparatively larger values of the exchange field oriented either in plane or out of plane of the hybrid structure and the Long ranged triplets (LRTs) present themselves in the form of zero energy peak in the DOS for feasible magnitude and orientation of the exchange field. Understanding and modeling these observations is important as it is difficult to model such hybrid structures experimentally. Where earlier research has focused on the scenario of substantial spin orbit coupling in the ballistic threshold/clean limit, in this paper we provide conclusions that are applicable to the diffusive transport/dirty limit regime. Also, the findings reveal that the SO coupling generates a distinct imprint in the DOS, which exhibits extremely nonmonotonic behavior with magnetization orientations.

Spin-projected charge conductance in SNN junctions with noncentrosymmetric superconductors

An SNN-junction in which the superconducting potential is a mixture between s-wave and p-wave potentials is investigated using the Usadel equation equipped with Tanaka-Nazarov boundary conditions. The article provides several ways to distinguish between s + chiral and s + helical p-wave superconductors and a way to determine whether a superconductor has a mixed pair potential. Thus, it is of great importance in the determination of the pair potential of superconductors. It is shown that the different spin sectors satisfy independent equations and can thus be calculated separately even if the d-vector depends on the direction of momentum. This greatly simplifies the equations to be solved. It was found that a difference in conductance for sectors with opposite spins arises if both an s-wave and a p-wave component is present, even in the absence of a magnetic field. The results are confirmed by calculations in the ballistic regime. It is shown that the spin conductance for s + chiral p-wave and s + helical p-wave junctions is qualitatively similar. A setup containing two SN junctions is shown to give a clear difference between the two types of superconductivity.

The Reverse Proximity Effect in Superconductor–Ferromagnetic Insulator Heterostructures

The magnetization induced in a superconductor due to the reverse proximity effect is studied in hybrid structures containing a superconductor and a ferromagnetic insulator. The study was carried out within the method of semiclassical Green’s functions, in which the Usadel equations are solved numerically with boundary conditions suitable for strongly spin-polarized ferromagnetic materials. The conversion of sin-glet superconducting correlations into triplet ones as a result of the proximity effect with a ferromagnet and its manifestation in the features of the electron density of states, induced magnetization, and suppression of the superconducting order parameter have been studied. It is shown that the magnetization can change sign inside the superconducting layer. The magnetization distribution is compared with the data obtained by the authors in previous works.

Field-free anomalous junction and superconducting diode effect in spin-split superconductor/topological insulator junctions

We study the transport properties of a diffusive Josephson junction between two spin-split superconductors (SCs) made of SC-ferromagnetic insulator bilayers on top of a three-dimensional topological insulator. We derive the corresponding Usadel equation describing the quasiclassical Green's functions in these systems and first solve the equation analytically in the weak-proximity case. We demonstrate the appearance of an anomalous phase in the absence of an external magnetic field. We also explore nonreciprocal electronic transport. Specifically, we calculate the diode efficiency of the junction $\ensuremath{\eta}$ by solving the Usadel equation. We obtain a sizable diode effect even at zero applied magnetic field. We discuss how the diode efficiency $\ensuremath{\eta}$ depends on the different parameters and find a nonmonotonic behavior of $\ensuremath{\eta}$ with temperature.

Density of states in the presence of spin-dependent scattering in SF bilayers: a numerical and analytical approach.

We present a quantitative study of the density of states (DOS) in SF bilayers (where S is a bulk superconductor and F is a ferromagnetic metal) in the diffusive limit. We solve the quasiclassical Usadel equations in the structure considering the presence of magnetic and spin-orbit scattering. For practical reasons, we propose the analytical solution for the density of states in SF bilayers in the case of a thin ferromagnet and low transparency of the SF interface. This solution is confirmed by numerical calculations using a self-consistent two-step iterative method. The behavior of DOS dependencies on magnetic and spin-orbit scattering times is discussed.

Multifractally-Enhanced Superconductivity in Two-Dimensional Systems with Spin–Orbit Coupling

The interplay of Anderson localization and electron-electron interactions is known to lead to enhancement of superconductivity due to multifractality of electron wave functions. We develop the theory of multifractally-enhanced superconducting states in two-dimensional systems in the presence of spin-orbit coupling. Using the Finkel'stein nonlinear sigma model, we derive the modified Usadel and gap equations that take into account renormalizations caused by the interplay of disorder and interactions. Multifractal correlations induce energy dependence of the superconducting spectral gap. We determine the superconducting transition temperature and the superconducting spectral gap in the case of Ising and strong spin orbit couplings. In the latter case the energy dependence of superconducting spectral gap is convex whereas in the former case (as well as in the absence of spin-orbit coupling) it is concave. Multifractality enhances not only the transition temperature but, in the same way, the spectral gap at zero temperature. Also we study mesoscopic fluctuations of the local density of states in the superconducting state. Similarly to the case of normal metal, spin-orbit coupling reduce the amplitude of fluctuations.

Effect of the interface transparency and ferromagnetic thickness on the critical temperature of NbN/Gd/NbN hybrid structure

We have investigated the variation of the critical temperature (Tc) in the NbN/Gd/NbN system in the dirty limit under the cumulative effect of ferromagnetic thickness (df) and the SF interface transparency by solving the linearized Usadel equations analytically in the Matsubara Greens function technique and the following trends in the Tc(df) curves were observed. (a) Non-monotonic decrease of Tc with the ferromagnetic thickness df for higher interface transparency. (b) Reentrant behavior of Tc for smaller values of the interface transparency. (c) Monotonous behavior of Tc for extremely low interface transparency. It was also found that the transition into the π−phase in NBN/Gd/NbN system with small interface transparency is unrelated to the oscillations of the superconducting critical temperature in the F-layer. The critical temperature has a non-monotonous dependency when it switches between 0−π phases.

Nonlinear σ model for disordered systems with intrinsic spin-orbit coupling

We derive the nonlinear $\ensuremath{\sigma}$ model to describe diffusive transport in normal metals and superconductors with intrinsic spin-orbit coupling (SOC). The SOC is described via an SU(2) gauge field, and we expand the model to the fourth order in gradients to find the leading non-Abelian field-strength contribution. This contribution generates the spin-charge coupling that is responsible for the spin-Hall effect. We discuss how its symmetry differs from the leading quasiclassical higher-order gradient terms. We also derive the corresponding Usadel equation describing the diffusive spin-charge dynamics in superconducting systems. As an example, we apply the obtained equations to describe the anomalous supercurrent in dirty Rashba superconductors at arbitrary temperatures.

Broadened Yu-Shiba-Rusinov states in dirty superconducting films and heterostructures

The interplay of a potential and magnetic disorder in superconductors has remained an active field of research for decades. Within the framework of the Usadel equation, we study the local density of states (LDOS) near a solitary classical magnetic impurity in a dirty superconducting film. We find that a potential disorder results in broadening of the $\ensuremath{\delta}$-function peak in the LDOS at the Yu-Shiba-Rusinov (YSR) energy. This broadening is proportional to the square root of a normal-state spreading resistance of the film. We demonstrate that modification of multiple scattering on the magnetic impurity due to intermediate scattering on surrounding potential disorder crucially affects a profile of the LDOS in the vicinity of the YSR energy. In addition, we find that a scanning-tunneling-microscopy tip can mask a YSR feature in the LDOS. Also, we study the LDOS near a chain of magnetic impurities situated in the normal region of a dirty superconductor/normal-metal junction. We find a resonance in the LDOS near the YSR energy. The energy scale of the resonant peak is controlled by the square root of the film resistance per square in the normal state.

Quasiparticle density of states and triplet correlations in superconductor/ferromagnetic-insulator structures across a sharp domain wall

A ferromagnetic insulator (FI) in contact with a superconductor (S) is known to induce a spin splitting of the BCS density of states at the FI/S interface. This spin splitting causes the Cooper pairs to reduce their singlet-state correlations and acquire odd-in-frequency triplet correlations. We consider a diffusive FI/S bilayer with a sharp magnetic domain wall in the FI and study the local quasiparticle density of states and triplet superconducting correlations. In the case of collinear alignment of the domains, we obtain analytical results by solving the Usadel equation. For a small enough exchange field or weak superconductivity, we also find an analytical expression for arbitrary magnetic textures, which reveals how the triplet component vector depends on the local magnetization of the FI. For an arbitrary angle between the magnetizations and the strength of the exchange field, we numerically solve the problem of a sharp domain wall. We finally propose two different setups based on FI/S/F stacks, where F is a ferromagnetic layer, to filter out singlet pairs and detect the presence of triplet correlations via tunneling differential conductance measurements.

Theory of disordered superconductors with applications to nonlinear current response

Abstract I present a review of the theory and basic equations for charge transport in superconducting alloys starting from the Keldysh formulation of the quasiclassical transport equations developed by Eilenberger, Larkin and Ovchinnikov, and Eliashberg. This formulation is the natural extension of Landau’s theory of normal Fermi liquids to the superconducting state of strongly correlated metals. For dirty metals the transport equations reduce to equations for charge diffusion, with the current response given by the Drude conductivity at low temperatures. The extension of the diffusion equation for the charge and current response of a strongly disordered normal metal to the superconducting state yields Usadel’s equations for the nonequilibrium quasiclassical Keldysh propagator. The conditions for the applicability of the Usadel equations are discussed, the pair-breaking effect of disorder on the current response, including the nonlinear current response to an electromagnetic field in the dirty limit, τ ≪ ℏ/Δ, are reported. The same nonlinearity is shown to lead to source currents for photon generation and nonlinear Kerr rotation driven by the nonlinear response to excitation of the superconductor by a multi-mode electromagnetic field. The potential relevance of the nonlinear source currents to superconducting radio-frequency cavities as detectors of axion-like dark matter candidates is briefly discussed.