Quasiclassical Green Function Research Articles

Superconducting Proximity Effect and Long-Ranged Triplets in Dirty Metallic Antiferromagnets.

Antiferromagnets have no net spin splitting on the scale of the superconducting coherence length. Despite this, antiferromagnets have been observed to suppress superconductivity in a similar way as ferromagnets, a phenomenon that still lacks a clear understanding. We find that this effect can be explained by the role of impurities in antiferromagnets. Using quasiclassical Green's functions, we study the proximity effect and critical temperature in diffusive superconductor-metallic antiferromagnet bilayers. The nonmagnetic impurities acquire an effective magnetic component in the antiferromagnet. This not only reduces the critical temperature but also separates the superconducting correlations into short-ranged and long-ranged components, similar to ferromagnetic proximity systems.

Anisotropic differential conductance of a mixed-parity superconductor/ferromagnet structure

We study the electronic transport properties of a superconductor (S) with a mixed s+p-wave pairing attached to a ferromagnetic metal (F) and a normal electrode (N) in an SFN configuration. Using the quasiclassical Green's function method, we compute the differential conductance $\sigma$ of the junction and demonstrate its dependence on the direction of the exchange field relative to the direction of the d-vector of the pair potential. If the p-wave triplet dominates the pairing, the zero bias conductance depends on the relative direction between the triplet d-vector and the exchange field. In contrast, if the s-wave singlet dominates the pairing, the zero bias conductance is isotropic with respect to the field direction. Furthermore, at zero temperature, the zero bias conductance height can only take two values as a function of $r$, the parameter quantifying the relative amount of s- and p-wave pairing, with an abrupt change at $r=1$ when the superconductor goes from a singlet to triplet dominated ground state. Moreover, we show that the relative amount of s- and p-wave pairing, can be estimated from the dependence of the finite bias conductance on the exchange field direction. Our results provide a way to characterize parity-mixed superconductors performing electrical measurements.

Hybrid helical state and superconducting diode effect in superconductor/ferromagnet/topological insulator heterostructures

It is well-known that the ground state of homogeneous superconducting systems with spin-orbit coupling (SOC) in the presence of the Zeeman field is the so-called helical state, which is characterized by the phase modulation of the order parameter, but zero supercurrent density. In this work we investigate the realization of the helical state in a hybrid system with spatially separated superconductivity and exchange field by considering S/F bilayer on top of a 3D topological insulator. This system is characterized by strong spin-momentum locking and, consequently, provides the most favorable conditions for the helical state generation. The analysis is based on the microscopic theory in terms of the quasiclassical Green's functions. We demonstrate that in the bilayer the helical state survives if the exchange field has non-zero component perpendicular to the S/F interface even in spite of the fact that the superconducting order parameter and the exchange field are spatially separated. At the same time, in this spatially inhomogeneous situation the helical state is accompanied by the spontaneous currents distributed over the bilayer in such a way as to have zero average. Further, we show that this hybrid helical state gives rise to nonreciprocity in the system. We demonstrate the realization of the superconducting diode effect and show that the degree of nonreciprocity is very large in the considered system due to the strongest possible value of the spin-momentum locking.

Effective quasiclassical models for odd-frequency superconductivity: Energy-inversion symmetry, preserved spectral weight, and Meissner response

The odd-frequency superconducting state appears generally in hybrid structures consisting of conventional superconductors and other materials, and features electrons that form temporally nonlocal Cooper pairs. The quasiclassical theory of superconductivity has been extensively used to model such systems, finding in many cases excellent agreement with experimental measurements. Therefore, it is of interest to study effective models of odd-frequency superconductivity to predict new phenomena associated with this form of unconventional pairing. We establish necessary criteria that the quasiclassical Green's functions in odd-frequency superconducting systems in the dirty limit must satisfy in order to be physically reasonable, including conservation of spectral weight. We show that it is possible to write down effective models, which satisfy all the above-mentioned criteria, but which still predict different behavior when it comes to the density of states and the magnetic response of the superconductor. For instance, an effective model for the odd-frequency anomalous Green's function that gives a conserved spectral weight can yield either a peaked or gapped density of states at the Fermi energy, and exhibit conventional, zero, or unconventional Meissner response. This finding demonstrates the importance of carefully considering the properties of effective models describing odd-frequency superconductivity in order to obtain physically reasonable results.

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.

Néel proximity effect at antiferromagnet/superconductor interfaces

Spin-splitting induced in a conventional superconductor weakens superconductivity by destroying spin-singlet and creating spin-triplet Cooper pairs. We demonstrate theoretically that such an effect is also caused by an adjacent compensated antiferromagnet, which yields no net spin-splitting. We find that the antiferromagnet produces N\'eel triplet Cooper pairs, whose pairing amplitude oscillates rapidly in space similar to the antiferromagnet's spin. The emergence of these unconventional Cooper pairs reduces the singlet pairs' amplitude, thereby lowering the superconducting critical temperature. We develop a quasiclassical Green's functions description of the system employing a two-sublattice framework. It successfully captures the rapid oscillations in the Cooper pairs' amplitude at the lattice spacing scale as well as their smooth variation on the larger coherence length scale. Employing the theoretical framework thus developed, we investigate this N\'eel proximity effect in a superconductor/antiferromagnet bilayer as a function of interfacial exchange, disorder, and chemical potential, finding rich physics. Our findings also offer insights into experiments which have found a larger than expected suppression of superconductivity by an adjacent antiferromagnet.

Spectral properties of Andreev crystals

We present an exhaustive study of Andreev crystals (ACs) -- quasi-one-dimensional superconducting wires with a periodic distribution of magnetic regions. The exchange field in these regions is assumed to be much smaller than the Fermi energy. Hence, the transport through the magnetic region can be described within the quasiclassical approximation. In the first part of the paper, by assuming that the separation between the magnetic regions is larger than the coherence length, we derive the effective nearest-neighbour tight-binding equations for ACs with a helical magnetic configuration. The spectrum within the gap of the host superconductor shows a pair of energy-symmetric bands. By increasing the strength of the magnetic impurities in ferromagnetic ACs, these bands cross without interacting. However, in any other helical configuration, there is a value of the magnetic strength at which the bands touch each other, forming a Dirac point. Further increase of the magnetic strength leads to a system with an inverted gap. We study junctions between ACs with inverted spectrum and show that junctions between (anti)ferromagnetic ACs (always) never exhibit bound states at the interface. In the second part, we extend our analysis beyond the nearest-neighbour approximation by solving the Eilenberger equation for infinite and junctions between semi-infinite ACs with collinear magnetization. From the obtained quasiclassical Green functions, we compute the local density of states and the local spin polarization in anti- and ferromagnetic ACs. We show that these junctions may exhibit bound states at the interface and fractionalization of the surface spin polarization.

Spin Hall and inverse spin galvanic effects in graphene with strong interfacial spin-orbit coupling: A quasi-classical Green's function approach

van der Waals heterostructures assembled from atomically thin crystals are ideal model systems to study spin-orbital coupled transport because they exhibit a strong interplay between spin, lattice and valley degrees of freedom that can be manipulated by strain, electric bias and proximity effects. The recently predicted spin-helical regime in graphene on transition metal dichalcogenides, in which spin and pseudospin degrees of freedom are locked together [M. Offidani et al. Phys. Rev. Lett. 119, 196801 (2017)], suggests their potential application in spintronics. Here, by deriving an Eilenberger equation for the quasiclassical Green's function of two-dimensional Dirac fermions in the presence of} spin-orbit coupling\textcolor{black}{{} (SOC) and scalar disorder, we obtain analytical expressions for the dc spin galvanic susceptibility and spin Hall conductivity in the spin-helical regime. Our results disclose a sign change in the spin Hall angle (SHA) when the Fermi energy relative to the Dirac point matches the Bychkov-Rashba energy scale, irrespective of the magnitude of the spin-valley interaction imprinted on the graphene layer. The behavior of the SHA is connected to a reversal of the total internal angular momentum of Bloch electrons that reflects the spin-pseudospin entanglement induced by SOC. We also show that the charge-spin conversion reaches a maximum when the Fermi level lies at the edge of the spin-minority band in agreement with previous findings. Both features are fingerprints of spin-helical Dirac fermions and suggest a direct way to estimate the strength of proximity-induced SOC from transport data. The relevance of these findings for interpreting recent spin-charge conversion measurements in nonlocal spin-valve geometry is also discussed.

ПОДАВЛЕНИЕ СВЕРХПРОВОДИМОСТИ В ГИБРИДНЫХ СТРУКТУРАХ

Materials that possess strong spin-orbit interaction have provoked great interest over the past few years, in particular, in the actively developing field of quantum calculations. A topological insulator is a good example of such a material. The topological insulator has high surface conductivity, whereas in the body it shows the properties of an insulator, this being a purely phenomenological definition of a substance. Superconducting hybrid structures also present a promising elemental basis for quantum calculations and spintronics. This paper considers hybrid nanostructures like superconductor/ ferromagnet/ superconductor (S/F/S) and superconductor/ topological insulator/ superconductor (S/TI/S), where a uniform magnetic field is imposed onto the surface of the topological insulator. The paper investigates the behaviour of critical temperature in the superconducting layer Тс depending on different parameters of the systems, in particular, the dependence of Тс on the thickness of the layer dn in the topological insulator TI (or the ferromagnetic layer F). To solve the problem, we use the formalism of the quasiclassical Green's functions . The model assumes a diffusion mode, which holds true when the electron free path length is much less than the characteristic scale of the system. As a rule, such a limit is easier to be performed, since the commonly made structures have admixtures. To solve the problem on self-consistency of the superconducting energetic slot D we use a unimode approximation. As the results of our calculations, we give curves of the critical temperature behaviour in the systems under consideration. It is demonstrated that while the critical temperature exhibits a nonmonotonic behaviour and can make oscillations because of phase 0 -p transitions in the S/F/S structures, in the S/TI/S structures the temperature of the superconducting transition exhibits a trivial behaviour characterized by the monotonic attenuation.

The inverse proximity effect in strong ferromagnet–superconductor structures

The magnetization in a superconductor induced due to the inverse proximity effect is investigated in hybrid bilayers containing a superconductor and a ferromagnetic insulator or a strongly spin-polarized ferromagnetic metal. The study is performed within a quasiclassical Green function framework, wherein Usadel equations are solved with boundary conditions appropriate for strongly spin-polarized ferromagnetic materials. A comparison with recent experimental data is presented. The singlet to triplet conversion of the superconducting correlations as a result of the proximity effect with a ferromagnet is studied.

Mathematical description data: Spin-resolved electron transport in nanoscale heterojunctions: Theory and applications.

This study demonstrates a mathematical description of a point-like nanocontact model, which is developed to simulate electron transport through a nanoconstriction between magnetic or non-magnetic contact sides. The theory represents a solution to the quasi-(semi)-classical transport equations for charge current, which takes into account second-order derivatives of the related quasi-classical Green functions along the transport direction. The theoretical approach also enables the creation of an I–V model for a heterojunction with embedded objects, where the initial condition, a conduction band minimum profile of the system, is well-defined. The presented spin-resolved current approach covers a complete range of the scales including quantum, ballistic, quasi-ballistic (intermediate), and diffusive classical transport conditions, with a smooth transition between them without residual terms or any empirical variables. The main benefit of the mathematical solution is its novel methodology, which is an alternative candidate to the well-known Boltzmann technique.

Vortex bound state of a Kondo lattice coupled to a compensated metal

We theoretically study physical properties of the low-energy quasiparticle excitations at the vortex core in the full-gap superconducting state of the Kondo lattice coupled to compensated metals. Based on the mean-field description of the superconducting state, we numerically solve the Bogoliubov-de Gennes (BdG) equations for the tight-binding Hamiltonian. The isolated vortex is characterized by a length scale independent of the magnitude of the interaction and the energy level of the core bound state is the same order as the bulk gap. These properties are in strong contrast to the conventional s-wave superconductor. To gain further insights, we also consider the effective Hamiltonian in the continuous limit and construct the theoretical framework of the quasiclassical Green's function of conduction electrons. With the use of the Kramer-Pesch approximation, we analytically derive the spectral function describing the quasiparticle excitations which is consistent with the numerics. It has been revealed that the properties of the vortex bound state are closely connected to the characteristic odd frequency dependence of both the normal and anomalous self-energies which is proportional to the inverse of frequency.

Signatures of the Higgs mode in transport through a normal-metal–superconductor junction

A superconductor subject to electromagnetic irradiation in the terahertz range can show amplitude oscillations of its order parameter. However, coupling this so-called Higgs mode to the charge current is notoriously difficult. We propose to achieve such a coupling in a particle-hole-asymmetric configuration using a DC-voltage-biased normal-metal--superconductor tunnel junction. Using the quasiclassical Green's function formalism, we demonstrate three characteristic signatures of the Higgs mode: (i) The AC charge current exhibits a pronounced resonant behavior and is maximal when the radiation frequency coincides with the order parameter. (ii) The AC charge current amplitude exhibits a characteristic nonmonotonic behavior with increasing voltage bias. (iii) At resonance for large voltage bias, the AC current vanishes inversely proportional to the bias. These signatures provide an electric detection scheme for the Higgs mode.

Charge transport through spin-polarized tunnel junction between two spin-split superconductors

We investigate transport properties of junctions between two spin-split superconductors linked by a spin-polarized tunneling barrier. The spin-splitting fields in the superconductors (S) are induced by adjacent ferromagnetic insulating (FI) layers with arbitrary magnetization. The aim of this study is twofold: On the one hand, we present a theoretical framework based on the quasiclassical Green's functions to calculate the Josephson and quasiparticle current through the junctions in terms of the different parameters characterizing it. Our theory predicts qualitative new results for the tunneling differential conductance, $dI/dV$, when the spin-splitting fields of the two superconductors are non-collinear. We also discuss how junctions based on FI/S can be used to realize anomalous Josephson junctions with a constant geometric phase shift in the current-phase relation. As a result, they may exhibit spontaneous triplet supercurrents in the absence of a phase difference between the S electrodes. On the other hand, we show results of planar tunneling spectroscopy of a EuS/Al/Al$_2$O$_3$/EuS/Al junction and use our theoretical model to reproduce the obtained $dI/dV$ curves. Comparison between theory and experiment reveals information about the intrinsic parameters of the junction, such as the size of the superconducting order parameter, spin-splitting fields and spin relaxation, and also about properties of the two EuS films, as their morphology, domain structure, and magnetic anisotropy.

Nonadiabatic dynamics in strongly driven diffusive Josephson junctions

By measuring the Josephson emission of a diffusive Superconductor-Normal metal-Superconductor (SNS) junction at a finite temperature we reveal a non-trivial sensitivity of the supercurrent to microwave irradiation. We demonstrate that the harmonic content of the current-phase relation is modified due to the energy redistribution of quasiparticles in the normal wire induced by the electromagnetic field. The distortion originates from the phase-dependent out-of-equilibrium distribution function which is strongly affected by the ac-response of the spectral supercurrent. For phases close to $\pi$, transitions accross the Andreev gap are dynamically favored leading to a supercurrent reduction. This finding is supported by a comparison with the quasiclassical Green's function theory of superconductivity in diffusive SNS junctions under microwave irradiation.

Effects of phase coherence on local density of states in superconducting proximity structures

We theoretically study the local density of states in superconducting proximity structure where two superconducting terminals are attached to a side surface of a normal-metal wire. Using the quasiclassical Green's function method, the energy spectrum is obtained for both of spin-singlet $s$-wave and spin-triplet $p$-wave junctions. In both of the cases, the decay length of the proximity effect at the zero temperature is limited by a depairing effect due to inelastic scatterings. In addition to the depairing effect, in $p$-wave junctions, the decay length depends sensitively on the transparency at the junction interfaces, which is a unique property to odd-parity superconductors where the anomalous proximity effect occurs.

Signatures of spin-triplet Cooper pairing in the density of states of spin-textured superconductor-ferromagnet bilayers

The existence of spin-triplet superconductivity in non-collinear magnetic heterostructures with superconductors is by now well established. This observation lays the foundation of superconducting spintronics with the aim to create a low-power consuming devices in order to replace the conventional electronics limited by heating. From a fundamental point of view the investigation of the structure and properties of spin-triplet Cooper pairs continues. Recently, spectroscopic evidence has shown to offer more detailed insights in the structure of triplet Cooper pairs than the supercurrent. Hence, we study here the structure of spin-triplet Cooper pairs through the density of states (DOS) in bilayers of a textured magnetic insulator in proximity to a superconductor. Using quasiclassical Green function methods, we study the local DOS, both spin-resolved and spin-independent. We show that the equal-spin and mixed-spin triplet Cooper pairs leads to different spectroscopic signatures, which can be further enhanced by spin-polarized spectroscopy. Our results show the huge potential spin-polarized tunneling methods offer in characterizing unconventional superconductivity in heterostructures.

Thermodynamic cycles in Josephson junctions

A superconductor/normal metal/superconductor Josephson junction is a coherent electron system where the thermodynamic entropy depends on temperature and difference of phase across the weak-link. Here, exploiting the phase-temperature thermodynamic diagram of a thermally isolated system, we argue that a cooling effect can be achieved when the phase drop across the junction is brought from 0 to π in a iso-entropic process. We show that iso-entropic cooling can be enhanced with proper choice of geometrical and electrical parameters of the junction, i.e. by increasing the ratio between supercurrent and total junction volume. We present extensive numerical calculations using quasi-classical Green function methods for a short junction and we compare them with analytical results. Interestingly, we demonstrate that phase-coherent thermodynamic cycles can be implemented by combining iso-entropic and iso-phasic processes acting on the weak-link, thereby engineering the coherent version of thermal machines such as engines and cooling systems. We therefore evaluate their performances and the minimum temperature achievable in a cooling cycle.

Exact and quasiclassical Green's functions of two-dimensional electron gas with Rashba–Dresselhaus spin–orbit interaction in parallel magnetic field

New exact and asymptotical results for the one particle Green's function of 2D electrons with combined Rashba–Dresselhaus spin–orbit interaction in the presence of in-plane uniform magnetic field are presented. A special case that allows an exact analytical solution is also highlighted. To demonstrate the advantages of our approach we apply the obtained Green's function to calculation of electron density and magnetization.

Influence of diffuse surface scattering on the stability of superconducting phases with spontaneous surface current generated by Andreev bound states

We report a theoretical study on the phase transition between superconducting states with and without spontaneous surface current. The phase transition takes place due to the formation of surface Andreev bound states in unconventional superconductors. Based on the quasiclassical theory of superconductivity, we examine the influence of atomic-scale surface roughness on the surface phase transition temperature $T_s$. To describe the surface effect, the boundary condition for the quasiclassical Green's function is parameterized in terms of specularity (the specular reflection probability in the normal state at the Fermi level). This boundary condition allows systematic study of the surface effect ranging from the specular limit to the diffuse limit. We show that diffuse quasiparticle scattering at a rough surface causes substantial reduction of $T_s$ in the d-wave pairing state of high-$T_c$ cuprate superconductors. We also consider a p-wave pairing state in which Andreev bound states similar to those in the d-wave state are generated. In contrast to the d-wave case, $T_s$ in the p-wave state is insensitive to the specularity. This is because the Andreev bound states in the p-wave superconductor are robust against diffuse scattering, as implied from symmetry consideration for odd-frequency Cooper pairs induced at the surface; the p-wave state has odd-frequency pairs with s-wave symmetry, while the d-wave state does not.