SNS Junctions Research Articles

Andreev reflection, Andreev states, and long ballistic SNS junction

The analysis in the present paper is based on the most known concept introduced by the brilliant physicist Alexander Andreev: Andreev bound states in a normal metal sandwiched between two superconductors. The paper presents results of direct calculations of ab initio expressions for the currents in a long ballistic SNS junction. The expressions are expanded in 1/L (L is the thickness of the normal layer). The main contribution ∝1/L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\propto 1/L$$\\end{document} to the current agrees with the results obtained in the past, but the analysis suggests a new physical picture of the charge transport through the junction free from the problem with the charge conservation law. The saw-tooth current-phase relation at T=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T=0$$\\end{document} directly follows from the Galilean invariance of the Bogolyubov–de Gennes equations proved in the paper. The proof is valid for any variation of the energy gap in space if the Andreev reflection is the only scattering process. The respective roles of the contributions of bound and continuum states to the current are clarified. They depend on the junction dimensionality.

Nonequilibrium Phenomena in Planar Mesoscopic Josephson SNS Structures Based on Superconducting Nb

Nonequilibrium phenomena in planar Josephson SNS nanostructures, where the superconductor (S) is Nb and the normal metal (N) is Cu or Au, have been studied experimentally. Using additional N electrodes attached to the S banks of the Josephson SNS junction, transport measurements have been performed at low temperatures with the injection of quasiparticles with the use of local and nonlocal connection schemes. The charge-imbalance relaxation length in niobium at temperatures much lower than the superconducting transition temperature has been determined experimentally for the first time.

Bound states in nonuniform superconductors

This paper shows that the WKB quantization rule is suitable for the Andreev bound states in nonuniform superconductors. We consider nonhomogeneous superconductivity gap functions $\mathrm{\ensuremath{\Delta}}(x)$ in superconductors with the Bogoliubov quasiparticle energy $E$, the Fermi level ${E}_{F}$, and the total momentum $\mathbf{p}$ at ${E}_{F}+E$. The Andreev bound states in the well of slowly varying $|\mathrm{\ensuremath{\Delta}}(x)|$ are studied, and the well may also be induced by the phase variation of $\mathrm{\ensuremath{\Delta}}(x)$ for massless Dirac fermions. By applying the WKB method to the Bogoliubov--de Gennes equation, we obtain two main results: (i) For ${E}_{F}\ensuremath{\sim}0$, the bound states are determined by ${\ensuremath{\int}}_{{L}_{E}}^{{R}_{E}}|\mathbf{p}\phantom{\rule{0.16em}{0ex}}d\mathbf{x}|=(\frac{1}{2}+n)\ensuremath{\pi}\ensuremath{\hbar}$, where $n\ensuremath{\in}{\mathbb{N}}_{0}$ and ${L}_{E}$ and ${R}_{E}$ are the boundary points between the classically allowed region and forbidden regions, and (ii) for ${E}_{F}\ensuremath{\gg}E$ and $|\mathrm{\ensuremath{\Delta}}(x)|$, the bound states are given by ${\ensuremath{\int}}_{{L}_{E}}^{{R}_{E}}|\mathbf{p}\ifmmode\pm\else\textpm\fi{}{\mathbf{p}}_{F}|dx=(\frac{1}{2}+n)\ensuremath{\pi}\ensuremath{\hbar}$ with small ${p}_{y(z)}$. Empirical quantization conditions are provided for broader parameter regions. In addition to applying the traditional WKB method, we also develop a generalized WKB method to tackle semimetals with parabolic dispersion relationships. The applications of our results are discussed, for example, Dirac $\ensuremath{\pi}$ junctions or nonchiral Majorana wires, SNS junctions, the excitation threshold, and the tunneling rate in NSN junctions. In the $\ensuremath{\pi}$ junction, the Majorana zero modes correspond to the zero-point energy in the WKB formalism. This observation may provide insights into the Majorana bound state in a vortex and the Majorana fermions in high-energy physics.

Effects of spin-orbit coupling and in-plane Zeeman fields on the critical current in two-dimensional hole gas SNS junctions

Superconductor--semiconductor hybrid devices are currently attracting much attention, fueled by the fact that strong spin--orbit interaction in combination with induced superconductivity can lead to exotic physics with potential applications in fault-tolerant quantum computation. The detailed nature of the spin dynamics in such systems is, however, often strongly dependent on device details and hard to access in experiment. In this paper we theoretically investigate a superconductor--normal--superconductor junction based on a two-dimensional hole gas with additional Rashba spin orbit--coupling, and we focus on the dependence of the critical current on the direction and magnitude of an applied in-plane magnetic field. We present a simple model, which allows us to systematically investigate different parameter regimes and obtain both numerical results and analytical expressions for all limiting cases. Our results could serve as a tool for extracting more information about the detailed spin physics in a two-dimensional hole gas based on a measured pattern of critical currents.

Nonvolatile memory cell using a superconducting-ferromagnetic π Josephson junction

Storage of a single magnetic flux quantum in a superconducting loop containing a Josephson junction represents a promising unit cell configuration for construction of a cryogenic memory of superconducting digital circuits. However, application of a DC bias current is required for operation of such a memory cell to maintain trapping of the flux quantum in the storage loop. In this work, we present a superconducting memory cell that uses a superconducting-magnetic π junction. The cell characteristics show flux quantum hysteresis centering at the zero-bias current. We develop a fabrication process that combines superconductor–ferromagnet–superconductor (SFS) junctions with superconductor–normal metal–superconductor (SNS) junctions. The critical current density of the SFS junctions shows a 0–π oscillation as a function of the ferromagnetic layer thickness. The formation of the π junction is confirmed further by the flux modulation curves of a superconducting quantum interference device made from SNS junctions with an additional SFS junction.

Phase tuning of multiple Andreev reflections of Dirac fermions and the Josephson supercurrent in Al–MoTe2–Al junctions

When an electron is incident on a superconductor from a metal, it is reflected as a hole in a process called Andreev reflection. If the metal N is sandwiched between two superconductors S in an SNS junction, multiple Andreev reflections (MARs) occur. We have found that, in SNS junctions with high transparency ([Formula: see text]) based on the Dirac semimetal MoTe2, the MAR features are observed with exceptional resolution. By tuning the phase difference [Formula: see text] between the bracketing Al superconductors, we establish that the MARs coexist with a Josephson supercurrent [Formula: see text]. As we vary the junction voltage V, the supercurrent amplitude [Formula: see text] varies in step with the MAR order n, revealing a direct relation between them. Two successive Andreev reflections serve to shuttle a Cooper pair across the junction. If the pair is shuttled coherently, it contributes to [Formula: see text]. The experiment measures the fraction of pairs shuttled coherently vs. V. Surprisingly, superconductivity in MoTe2 does not affect the MAR features.

Static and dynamic transport properties of multi-terminal, multi-junction microSQUIDs realized with Nb/HfTi/Nb Josephson junctions

The progressive miniaturization of superconducting quantum interference devices (SQUIDs) used, e.g. for magnetic imaging on the nanoscale or for the detection of the magnetic states of individual magnetic nanoparticles causes increasing problems in realizing a proper flux-bias scheme for reading out the device. To overcome the problem, a multi-terminal, multi-junction layout has been proposed and realized recently for the SQUID-on-tip configuration, which uses constriction-type Josephson junctions (JJ). This geometry is also interesting for SQUIDs based on overdamped superconductor—normal metal—superconductor (SNS) JJ. We fabricated four-terminal, four-junction SQUIDs based on a trilayer Nb/HfTi/Nb process and study their static and dynamic transport properties in close comparison with numerical simulations based on the resistively and capacitively shunted junction model. Simulations and measurements are in very good agreement. However, there are large differences to the transport properties of conventional two-junction SQUIDs, including unusual phase-locked and chaotic dynamic states which we describe in detail. We further extract the current-phase relation of our SNS junctions, which turns out to be purely sinusoidal within the experimental error bars.

Multiple Andreev reflections effect spectroscopy of LiFeAs single crystals: three superconducting order parameters and their temperature evolution

The structure of the superconducting order parameter of LiFeAs is studied by incoherent multiple Andreev reflections effect (IMARE) spectroscopy. The high transparent superconductor–thin normal metal–superconductor (SnS) contacts are created by a planar “break-junction” technique. Below T_c approx 17.5 K, the obtained I(V) and dI(V)/dV characteristics of SnS junctions show a presence of at least three bulk superconducting order parameters in LiFeAs. We directly determine the magnitudes, characteristic ratios, and temperature dependences of the superconducting gaps and discuss their symmetry.

Multiband superconductivity in SrFe2−xNixAs2

In this paper, we present a detailed study of the upper critical field and the energy gap structure of SrFe2−xNixAs2 single crystals with different nickel doping levels. Superconducting transitions were measured resistively in longitudinal and transverse magnetic fields up to 16T. The anisotropy of the upper critical field Hc2||ab/Hc2||c decreases monotonically to 1.4 with decreasing temperature for all measured samples. The WHH model was not able to fit measured Hc2(T) temperature dependences well, but the effective two-band model is in good agreement with the data and anisotropy behavior. Values and temperature dependences of superconducting gaps were determined directly by multiple Andreev reflection spectroscopy on symmetrical SnS junctions. Andreev reflections study revealed two superconducting gaps; our data are in line with the experimental results on most iron-based superconductors.Article highlightsWe obtained values and temperature dependencies of two distinct energy gaps in SrFe2−xNixAs2 (Sr-122) superconductor. This is important addition to multiband superconductivity studies in Sr-122 system. Upper critical field temperature dependencies in H||c and H||ab orientations are analyzed in terms of two band model.

Critical temperatures and critical currents of wide and narrow quasi-one-dimensional superconducting aluminum structures in zero magnetic field

We measured the critical temperatures and critical switching and retrapping currents of wide and narrow thin-film quasi-one-dimensional superconducting aluminum structures of the same thickness in zero magnetic field. For the first time, we found that the narrower the structure, the lower the critical temperature and critical current density in the structure. Probably, the influence of depairing centers that are on dirty longitudinal boundaries of the structure, is the stronger than the narrower the structure. It is found for the first time that, in most cases, the temperature-dependent switching critical current in both structures is approximated by two functions. At temperatures below the temperature corresponding to the bottom of the resistive N-S transition of structures, the switching critical current is described by the Kupriyanov–Lukichev theory. At temperatures close to the top of the N-S transition, the switching current is linear with temperature and coincides with the critical Josephson current. At these temperatures, Josephson SNS junctions are formed in structures.

Nonlocal superconducting quantum interference device

Superconducting quantum interference devices (SQUIDs) that incorporate two superconductor/insulator/superconductor (SIS) Josephson junctions in a closed loop form the core of some of the most sensitive detectors of magnetic and electric fields currently available. SQUIDs in these applications are typically operated with a finite voltage which generates microwave radiation through the ac Josephson effect. This radiation may impact the system being measured. We describe here a SQUID in which the Josephson junctions are formed from strips of normal metal (N) in good electrical contact with the superconductor (S). Such SNS SQUIDs can be operated under a finite voltage bias with performance comparable or potentially better than conventional SIS SQUIDs. However, they also permit a novel mode of operation that is based on the unusual interplay of quasiparticle currents and supercurrents in the normal metal of the Josephson junction. The new method allows measurements of the flux dependence of the critical current of the SNS SQUID without applying a finite voltage bias across the SNS junction, enabling sensitive flux detection without generating microwave radiation.

Multiple-Band Andreev Transport in Optimally Doped Superconducting Oxypnictides

Using multiple Andreev reflections (MAR) effect spectroscopy of SnS junctions, we directly determine temperature dependences of the superconducting order parameters, excess Andreev current, and zero-bias conductance (ZBC) in polycrystalline samples of superconducting oxypnictides Sm0.7Th0.3OFeAs and NdO0.6H0.36FeA with almost optimal critical temperatures. It is shown that the results obtained are self-consistent and could be described in the framework of a two-band model. We estimate a dominating contribution 70–85% of the bands with the large gap to the total conductance.

Dynamical voltage-current characteristics of SNS junctions

We have probed the switching dynamics of the Josephson critical current of a superconducting weak link by measuring its voltage/current characteristics while applying an ac current bias in the range 1-200 MHz. The weak link between two Nb reservoirs is formed by an mesoscopic Al wire above its critical temperature. We observe a dynamical phase transition as a function of the frequency and amplitude of the ac current. While at low frequency the transition driven by increasing the current bias is well described by the standard Kramers theory, at high frequency the switching histograms become hysteretic and much narrower than expected by thermal fluctuations. The crossover frequency between the two regimes is set by the electron-phonon interaction rate in the normal metal.

Amplitudes of minima in dynamic conductance spectra of the SNS Andreev contact

Although several theoretical approaches describing multiple Andreev reflection (MAR) effects in the superconductor–normal metal–superconductor (SNS) junction are elaborated, the problem of the comprehensive and adequate description of MAR is highly actual. In particular, a broadening parameter Γ is still unaccounted at all, whereas a ballistic condition (the mean free path for inelastic scattering l to the barrier width d ratio) is considered only in the framework of Kümmel, Gunsenheimer, and Nikolsky (KGN), as well as Gunsenheimer–Zaikin approaches, for an isotropic case and fully transparent constriction. Nonetheless, the influence of the l/d ratio on the dynamic conductance spectrum (dI/dV) features remains disregarded, thus being one of the aims of the current work. Our numerical calculations in the framework of an extended KGN approach develop the l/d variation to determine both the number of the Andreev features and their amplitudes in the dI/dV spectrum. We show, in the spectrum of a diffusive SNS junction (l/d→1), a suppression of the Andreev excess current and a dramatic change in the current voltage I(V)-curve slope at low bias, with only the main harmonic at eV=2Δ bias voltage remaining well-distinguished in the dI/dV-spectrum. Additionally, we attempt to make a first-ever comparison between experimental data for the high-transparency SNS junctions (more than 85%) and theoretical predictions. As a result, we calculate the temperature dependences of amplitudes and areas of Andreev features within the extended KGN approach, which qualitatively agree with our experimental data obtained using a “break-junction” technique.

Local density of States in Two-Dimensional Nano-Structured Superconducting Systems with Superconductor-Normal Metal Interfaces

In nano-structured superconductors, it is important to consider a discreteness of energy levels instead of continuous energy levels due to the quantum confinement effect. This discreteness causes an appearance of many peaks in a density of state (DOS). In this paper, the effect of the discreetness of energy level or the effect of the quantum confinement on the DOS in superconducting systems with superconductor / normal metal (S/N) interfaces are focused. In particular, in nano-sized finite systems, a local density of state (LDOS) becomes strongly spatial dependent because of non-uniform spatial distributions of a gap energy and a pair amplitude. We investigate the pair amplitude and the LDOS by solving the Bogoliubov-de Gennes equations self-consistently. The proximity effect leads to a penetration of the pair amplitude into the normal metal region. Then, the pair amplitude in the normal metal decays non-monotonously because of the effect of the nano-sized finite system. On the other hand, the spatial-averaged LDOS plots as a function of the energy have many peaks in both superconductor and normal metal regions. Also, a contribution of the normal metal to the superconductor causes an appearance of peaks of the LDOS at the energy below the gap energy. In the SNS junction, when width of the normal metal increases, these peaks at the energy below the gap energy appear clearly.

Modeling long imperfect SNS junctions and Andreev bound states using two impurities and the T -matrix formalism

We provide a new analytical tool to calculate the energies of Andreev bound states (ABS) in long imperfect SNS junctions, at present these can only be described by numerical tools. We model an NS junction as a delta-function "Andreev" impurity, i.e., a localized potential which scatters an electron into a hole with opposite spin. We show using the scattering matrix formalism that, quite surprisingly, an "Andreev" impurity is equivalent to an NS junction characterized by both Andreev reflection and a finite amount of normal scattering. The ABS energies are then calculated using the T-matrix formalism applied to a system with two Andreev impurities. Our results lie between those for a perfect long SNS junction limit described by the Andreev approximation (ABS energies depend linearly on the phase and are independent of the chemical potential) and the particle-in-the-box limit (bound state energies are independent of the phase and have a linear dependence on the chemical potential). Moreover, we recover a closed-form expression for the ABS energies by expanding around the particle-in-the-box limit.

Local density of states in clean two-dimensional superconductor–normal metal–superconductor heterostructures

Motivated by recent advances in the fabrication of Josephson junctions in which the weak link is made of a low-dimensional non-superconducting material, we present here a systematic theoretical study of the local density of states (LDOS) in a clean 2D normal metal (N) coupled to two s-wave superconductors (S). To be precise, we employ the quasiclassical theory of superconductivity in the clean limit, based on Eilenberger's equations, to investigate the phase-dependent LDOS as function of factors such as the length or the width of the junction, a finite reflectivity, and a weak magnetic field. We show how the the spectrum of Andeeev bound states that appear inside the gap shape the phase-dependent LDOS in short and long junctions. We discuss the circumstances when a gap appears in the LDOS and when the continuum displays a significant phase-dependence. The presence of a magnetic flux leads to a complex interference behavior, which is also reflected in the supercurrent-phase relation. Our results agree qualitatively with recent experiments on graphene SNS junctions. Finally, we show how the LDOS is connected to the supercurrent that can flow in these superconducting heterostructures and present an analytical relation between these two basic quantities.

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian with PT symmetry: generalized Bohr-Sommerfeld quantization rules

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions; 2) constructing local sections of the fibre bundle of microlocal solutions; 3) normalizing these solutions for the “flux norm” associated to the microlocal Wronskians; 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm; 5) from this we deduce Bohr-Sommerfeld (BS) quantization rules that hold precisely when the fibre bundle of microlocal solutions (depending on the energy parameter E) has trivial holonomy. Such a semi-classical treatement reveals interesting continuous symetries related to monodromy.

Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure

We describe a method for exactly diagonalizing clean $D$-dimensional lattice systems of independent fermions subject to arbitrary boundary conditions in one direction, as well as systems composed of two bulks meeting at a planar interface. Our method builds on the generalized Bloch theorem [A. Alase et al., Phys. Rev. B 96, 195133 (2017)] and the fact that the bulk-boundary separation of the Schrodinger equation is compatible with a partial Fourier transform operation. Bulk equations may display unusual features because they are relative eigenvalue problems for non-Hermitian, bulk-projected Hamiltonians. Nonetheless, they admit a rich symmetry analysis that can simplify considerably the structure of energy eigenstates, often allowing a solution in fully analytical form. We illustrate our extension of the generalized Bloch theorem to multicomponent systems by determining the exact Andreev bound states for a simple SNS junction. We then analyze the Creutz ladder model, by way of a conceptual bridge from one to higher dimensions. Upon introducing a new Gaussian duality transformation that maps the Creutz ladder to a system of two Majorana chains, we show how the model provides a first example of a short-range chiral topological insulator hosting topological zero modes with a power-law profile. Additional applications include the complete analytical diagonalization of graphene ribbons with both zigzag-bearded and armchair boundary conditions, and the analytical determination of the edge modes in a chiral $p+ip$ two-dimensional topological superconductor. Lastly, we revisit the phenomenon of Majorana flat bands and anomalous bulk-boundary correspondence in a two-band gapless $s$-wave topological superconductor. We analyze the equilibrium Josephson response of the system, showing how the presence of Majorana flat bands implies a substantial enhancement in the $4\pi$-periodic supercurrent.

Excess equilibrium noise in a topological SNS junction between chiral Majorana liquids

We consider a Josephson contact mediated by 1D chiral modes on a surface of a 3D topological insulator with superimposed superconducting and magnetic layers. The system represents an interferometer in which 1D chiral Majorana modes on the boundaries of superconducting electrodes are linked by ballistic chiral Dirac channels. We investigate the noise of the Josephson current as a function of the dc phase bias and the Aharonov-Bohm flux. Starting from the scattering formalism, a Majorana representation of the Keldysh generating action for cumulants of the transmitted charge is found. At temperatures higher than the Thouless energy $E_{\rm Th}$, we obtain the usual Johnson-Nyquist noise, $4G_0 k_{\rm B}T$, characteristic for a single-channel wire with $G_0 \equiv e^2/(2\pi \hbar)$. At lower temperatures the behavior is much richer. In particular, the equilibrium noise is strongly enhanced to a temperature-independent value $\sim G_0 E_{\rm Th}$ if the Aharonov-Bohm and superconducting phases are both close to $2\pi n$, which are points of emergent degeneracy in the ground state of the junction. The equilibrium noise is related to the Josephson junction's impedance via the fluctuation-dissipation theorem. In a striking contrast to usual Josephson junctions (tunnel junctions between two $s$-wave superconductors), the real part of the impedance does not vanish, reflecting the gapless character of Majorana modes in the leads.