Quasiclassical Theory Of Superconductivity Research Articles

Shaping Graphene Superconductivity with Nanometer Precision.

Graphene holds great potential for superconductivity due to its pure 2D nature, the ability to tune its carrier density through electrostatic gating, and its unique, relativistic-like electronic properties. At present, still far from controlling and understanding graphene superconductivity, mainly because the selective introduction of superconducting properties to graphene is experimentally very challenging. Here, a method is developed that enables shaping at will graphene superconductivity through a precise control of graphene-superconductor junctions. The method combines the proximity effect with scanning tunnelling microscope (STM) manipulation capabilities. Pb nano-islands are first grown that locally induce superconductivity in graphene. Using a STM, Pb nano-islands can be selectively displaced, over different types of graphene surfaces, with nanometre scale precision, in any direction, over distances of hundreds of nanometres. This opens an exciting playground where a large number of predefined graphene-superconductor hybrid structures can be investigated with atomic scale precision. To illustrate the potential, a series of experiments are performed, rationalized by the quasi-classical theory of superconductivity, going from the fundamental understanding of superconductor-graphene-superconductor heterostructures to the construction of superconductor nanocorrals, further used as "portable" experimental probes of local magnetic moments in graphene.

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.

Quasiclassical boundary conditions for spin-orbit coupled interfaces with spin-charge conversion

The quasiclassical theory of superconductivity provides a methodology to study emergent phenomena in hybrid structures comprised of superconductors interfaced with other materials. A key component in this theory is the boundary condition that the Green functions describing the materials must satisfy. Recently, progress has been made toward formulating such a boundary condition for interfaces with spin-orbit coupling, the latter playing an important role for several phenomena in spintronics. Here, we derive a boundary condition for spin-orbit coupled interfaces that includes gradient terms which enables the description of spin-Hall like effects with superconductors due to such interfaces. As an example, we show that the boundary conditions predict that a supercurrent flowing through a superconductor that is coupled to a normal metal via a spin-orbit interface can induce a non-local magnetization in the normal metal.

Breaking time-reversal and translational symmetry at edges of d -wave superconductors: Microscopic theory and comparison with quasiclassical theory

We report results of a microscopic calculation of a second-order phase transition into a state breaking time-reversal and translational invariance along pair-breaking edges of $d$-wave superconductors. By solving a tight-binding model through exact diagonalization with the Bogoliubov-de~Gennes method, we find that such a state with current loops having a diameter of a few coherence lengths is energetically favorable below $T^*$ between 10%-20% of $T_{\mathrm{c}}$ of bulk superconductivity, depending on model parameters. This extends our previous studies of such a phase crystal within the quasiclassical theory of superconductivity, and shows that the instability is not qualitatively different when including a more realistic band structure and the fast oscillations on the scale of the Fermi wavelength. Effects of size quantization and Friedel oscillations are not detrimental. We also report on a comparison with quasiclassical theory with the Fermi surfaces extracted from the tight-binding models used in the microscopic calculation. There are quantitative differences in for instance the value of $T^*$ between the different models, but we can explain the predicted transition temperature within each model as due to the different spectral weights of zero-energy Andreev bound states and the resulting gain in free energy by breaking time-reversal and translational invariance below $T^*$.

Surface Bound States and Spontaneous Edge Currents in Chiral Superconductors: Effect of Spatially Varying Order Parameter

We use the quasiclassical theory of superconductivity and examine the surface bound states and edge currents in two-dimensional chiral superconductors by focusing on the effect of spatially varying...

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.

Phase-driven collapse of the Cooper condensate in a nanosized superconductor

Superconductivity can be understood in terms of a phase transition from an uncorrelated electron gas to a condensate of Cooper pairs in which the relative phases of the constituent electrons are coherent over macroscopic length scales. The degree of correlation is quantified by a complex-valued order parameter, whose amplitude is proportional to the strength of the pairing potential in the condensate. Supercurrent-carrying states are associated with non-zero values of the spatial gradient of the phase. The pairing potential and several physical observables of the Cooper condensate can be manipulated by means of temperature, current bias, dishomogeneities in the chemical composition or application of a magnetic field. Here we show evidence of complete suppression of the energy gap in the local density of quasiparticle states (DOS) of a superconducting nanowire upon establishing a phase difference equal to pi over a length scale comparable to the superconducting coherence length. These observations are consistent with a complete collapse of the pairing potential in the center of the wire, in accordance with theoretical modeling based on the quasiclassical theory of superconductivity in diffusive systems. Our spectroscopic data, fully exploring the phase-biased states of the condensate, highlight the profound effect that extreme phase gradients exert on the amplitude of the pairing potential. Moreover, the sharp magnetic response observed near the onset of the superconducting gap collapse regime can be exploited to realize ultra-low noise magnetic flux detectors.

Josephson coupling across a long single-crystalline Cu nanowire

We report on a fabrication method and electron-transport measurements for submicron Josephson junctions formed by Cu nanowires coupling to superconducting planar Nb electrodes. The Cu nanowires with a resistivity of ρCu≃1 μΩ cm at low temperatures consisting of single-crystalline segments have been obtained by templated electrodeposition using anodic aluminum oxide as a porous matrix. The current-voltage characteristics of the devices have been studied as a function of temperature and magnetic field. For all junctions, the critical current monotonically decreases with a magnetic field. The measured temperature and magnetic field dependencies are consistent with the model for one-dimensional diffusive superconductor/normal metal/superconductor (SNS) Josephson junctions within the quasiclassical theory of superconductivity.

Critical temperature and tunneling spectroscopy of superconductor-ferromagnet hybrids with intrinsic Rashba-Dresselhaus spin-orbit coupling

We investigate theoretically how the proximity effect in superconductor/ferromagnet hybrid structures with intrinsic spin-orbit coupling manifests in two measurable quantities, namely, the density of states and critical temperature. To describe a general scenario, we allow for both Rashba- and Dresselhaus-type spin-orbit coupling. Our results are obtained via the quasiclassical theory of superconductivity, extended to include spin-orbit coupling in the Usadel equation and in the Kupriyanov-Lukichev boundary conditions. Unlike previous works, we have derived a Riccati parametrization of the Usadel equation with spin-orbit coupling which allows us to address the full proximity regime and not only the linearized weak proximity regime. First, we consider the density of states in both SF bilayers and SFS trilayers, where the spectroscopic features in the latter case are sensitive to the phase difference between the two superconductors. We find that the presence of spin-orbit coupling leaves clear spectroscopic fingerprints in the density of states due to its role in creating spin-triplet Cooper pairs. Unlike SF and SFS structures without spin-orbit coupling, the density of states in the present case depends strongly on the direction of magnetization. Moreover, we show that the spin-orbit coupling can stabilize spin-singlet superconductivity even in the presence of a strong exchange field $h\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$. This leads to the possibility of a magnetically tunable minigap: changing the direction of the exchange field opens and closes the minigap. We also determine how the critical temperature ${T}_{c}$ of an SF bilayer is affected by spin-orbit coupling and, interestingly, demonstrate that one can achieve a spin-valve effect with a single ferromagnet. We find that ${T}_{c}$ displays highly nonmonotonic behavior both as a function of the magnetization direction as well as the type and direction of the spin-orbit coupling, offering a new way to exert control over the superconductivity of proximity structures.

Zero-energy bound state at the interface between ans-wave superconductor and a disordered normal metal with repulsive electron-electron interactions

In recent years, there has been a renewed interest in the proximity effect due to its role in the realization of topological superconductivity. Here, we study a superconductor--normal metal proximity system with repulsive electron-electron interactions in the normal layer. It is known that in the absence of disorder or normal reflection at the superconductor--normal metal interface, a zero-energy bound state forms and is localized to the interface [Fauch\`ere et al., Phys. Rev. Lett. 82, 3336 (1999).]. Using the quasiclassical theory of superconductivity, we investigate the low-energy behavior of the density of states in the presence of finite disorder and an interfacial barrier. We find that as the mean free path is decreased, the zero-energy peak in the density of states is broadened and reduced. In the quasiballistic limit, the bound state eliminates the minigap pertinent to a noninteracting normal layer and a distinct peak is observed. When the mean free path becomes comparable to the normal layer width, the zero-energy peak is strongly suppressed and the minigap begins to develop. In the diffusive limit, the minigap is fully restored and all signatures of the bound state are eliminated. We find that an interfacial potential barrier does not change the functional form of the density of states peak but does shift this peak away from zero energy.

Magnetic Josephson junctions with noncentrosymmetric superconductors

We study the dc Josephson effect in a clean noncentrosymmetric superconductor/ferromagnet/noncentro-symmetric superconductor junction within the quasiclassical theory of superconductivity. By considering charge and spin currents, we show that in such junctions an exotic Josephson effect can take place, depending on the superconducting pairing state and spin polarization direction. We focus on the importance of spin-triplet/spin-singlet gaps ratio in such systems showing that its value is related to the existence of even and odd high-order harmonics in the charge and spin current-phase relations, and to the possibility of 0-$\ensuremath{\pi}$ transitions.

Odd-frequency pairing effect on the superfluid density and the Pauli spin susceptibility in spatially nonuniform spin-singlet superconductors

A theoretical study is presented on the odd-frequency spin-singlet pairing that arises in nonuniform even-frequency superconductors as a consequence of broken translation symmetry. The effect of the odd-frequency pairing on the superfluid density and the spin susceptibility is analyzed by using the quasiclassical theory of superconductivity. It is shown that (1) the superfluid density is reduced by the formation of the odd-frequency singlet pairs and (2) the odd-frequency pairing increases the spin susceptibility even though its spin symmetry is singlet. The two unusual phenomena are related to each other through a generalized Yosida formula by taking into account both the even- and odd-frequency pairing effects.

Transport equations for superconductors in the presence of spin interaction

Quasi-classical theory of superconductivity provides a powerful and yet simple description of the superconductivity phenomenology. In particular, the Eilenberger and Usadel equations provide a neat simplification of the description of the superconducting state in the presence of disorder and electromagnetic interaction. However, the modern aspects of superconductivity require a correct description of the spin interaction as well. Here, we generalize the transport equations of superconductivity in order to take into account space-time dependent electromagnetic and spin interactions on equal footing. Using a gauge-covariant Wigner transformation for the Green-Gor'kov correlation functions, we establish the correspondence between the Dyson-Gor'kov equation and the quasi-classical transport equation in the time-dependent phase-space. We give the expressions for the gauge-covariant current and charge densities (quasi-particle, electric and spin) in the transport formulation. The generalized Eilenberger and Usadel limits of the transport equation are given, too. This study is devoted to the formal derivation of the equations of motion in the electromagnetic plus spin plus particle-hole space. The studies of some specific systems are postponed to future works.

Nonlocal correlations in a proximity-coupled normal-metal

We report evidence of large, nonlocal correlations between two spatially separated normal metals in superconductor/normal-metal (SN) heterostructures, which manifest themselves a nonlocal voltage generated in response to a driving current. Unlike prior experiments in SN heterostructures, the nonlocal correlations are mediated not by a superconductor, but by a proximity-coupled normal metal. The nonlocal correlations extend over relatively long length scales in comparison to the superconduncting case. At very low temperatures, we find a reduction in the nonlocal voltage for small applied currents that cannot be explained by the quasiclassical theory of superconductivity. We believe is a signature of new long-range quantum correlations in the system.

Effect of Born and unitary impurity scattering on the Kramer–Pesch shrinkage of a vortex core in a chiral p-wave superconductor

We theoretically investigate a non-magnetic impurity effect on the temperature dependence of the vortex core shrinkage (Kramer–Pesch effect) in a chiral p-wave superconductor. The Born limit and the unitary limit scattering are compared within the framework of the quasiclassical theory of superconductivity. We find that the robustness of the Kramer–Pesch effect against the impurity scattering in the Born limit is lost in the unitary limit.

Proximity effect with noncentrosymmetric superconductors

We describe the superconducting proximity effect taking place in a contact between a noncentrosymmetric superconductor and a diffusive normal/ferromagnetic metal within the quasiclassical theory of superconductivity. By solving numerically the Usadel equation with boundary conditions valid for arbitrary interface transparency, we show that the analysis of the proximity-modified local density of states in the normal side can be used to obtain information about the exotic superconductivity of noncentrosymmetric materials. We point out the signatures of triplet pairing, the coexistence of triplet and singlet pairing, and particular orbital symmetries of the pair potential. Exploiting the directional dependence of the spin polarization pair breaking effect on the triplet correlations, we show how the order relation between triplet and singlet gaps can be discriminated and that an estimation of the specific gap ratio is possible in some cases.

Spin-dependent boundary conditions for isotropic superconducting Green’s functions

The quasiclassical theory of superconductivity provides the most successful description of diffusive heterostructures comprising superconducting elements, namely, the Usadel equations for isotropic Green's functions. Since the quasiclassical and isotropic approximations break down close to interfaces, the Usadel equations have to be supplemented with boundary conditions for isotropic Green's functions (BCIGF), which are not derivable within the quasiclassical description. For a long time, the BCIGF were available only for spin-degenerate tunnel contacts, which posed a serious limitation on the applicability of the Usadel description to modern structures containing ferromagnetic elements. In this paper, we close this gap and derive spin-dependent BCIGF for a contact encompassing superconducting and ferromagnetic correlations. This finally justifies several simplified versions of the spin-dependent BCIGF, which have been used in the literature so far. In the general case, our BCIGF are valid as soon as the quasiclassical isotropic approximation can be performed. However, their use requires the knowledge of the full scattering matrix of the contact, an information usually not available for realistic interfaces. In the case of a weakly polarized tunnel interface, the BCIGF can be expressed in terms of a few parameters, i.e., the tunnel conductance of the interface and five conductancelike parameters accounting for the spin dependence of the interface scattering amplitudes. In the case of a contact with a ferromagnetic insulator, it is possible to find explicit BCIGF also for stronger polarizations. The BCIGF derived in this paper are sufficiently general to describe a variety of physical situations and may serve as a basis for modeling realistic nanostructures.

Temperature-induced0−πcoexistence in clean superconductor-ferromagnet-superconductor Josephson junctions

A clean ballistic superconductor-ferromagnet metal-superconductor (SFS) junction is studied within the quasiclassical theory of superconductivity. We show that the temperature-induced coexistence of the 0 and $\ensuremath{\pi}$ states and the transition $0\text{\ensuremath{-}}\ensuremath{\pi}$ results from the temperature-induced transition between stable and metastable states of the junction. The $\ensuremath{\pi}$ shift occurs in small but finite critical intervals ${{Z}_{c}}$ of the magnetic barrier influence parameter $Z=2dh/\ensuremath{\hbar}{v}_{0}$, where $h$ is the exchange energy and $d$ the barrier thickness. The width of ${{Z}_{c}}$ and the coexistence temperature ${T}_{0\ensuremath{\pi}}$ depend also on the reduced barrier thickness $\overline{d}=d/{\ensuremath{\xi}}_{0}$, where ${\ensuremath{\xi}}_{0}$ is the zero-temperature superconducting coherence length. The interval ${{Z}_{c}}$ increases with $\overline{d}$. For $T\ensuremath{\ne}{T}_{0\ensuremath{\pi}}$ the junction is in the 0 or $\ensuremath{\pi}$ state. In contrast to the dirty case, the critical current ${I}_{c}$ depends monotonously on temperature, changing the sign but not the magnitude at the transition point. The nonvanishing critical current ${I}_{c}({T}_{0\ensuremath{\pi}})$ is related to the existence of the second harmonic in the current-phase relation, which could be observed experimentally.

Theory of nonequilibrium spin transport and spin-transfer torque in superconducting-ferromagnetic nanostructures

Spin transport currents and the spin-transfer torques in voltage-biased superconductingferromagnetic nanopillars (SFNFS point contacts) are computed. We develop and implement an algorithm based on the Ricatti formulation of the quasiclassical theory of superconductivity to solve the time-dependent boundary conditions for the nonequilibrium Green’s functions for spin transport through the ferromagnetic interfaces. A signature of the nonequilibrium torque is a component perpendicular to the plane spanned by the two ferromagnetic moments. The perpendicular component is absent in normal-metal-ferromagnetic nanopillars (NFNFN) contacts, but is shown to have the same order of magnitude as the in-plane torque for non-equilibrium SFNFS contacts. The out-of-plane torque is due to the rotation of quasiparticle spin by the exchange fields of the ferromagnetic layers. In the ballistic limit the equilibrium torque is related to the spectrum of spin-polarized Andreev bound states, while the ac component, for small bias voltages, is determined by the nearly adiabatic dynamics of the Andreev bound states. The nonlinear voltage dependence of the non-equilibrium torque, including the subharmonic gap structure and the high-voltage asymptotics, is attributed to the interplay between multiple Andreev reflections, spin filtering and spin mixing. These properties of spin angular momentum transport may be exploited to control the state of nanomagnets.

Ferromagnetic Josephson Junction with Precessing Magnetization

The Josephson current in a diffusive superconductor-ferromagnet-superconductor junction with precessing magnetization is calculated within the quasiclassical theory of superconductivity. When the junction is phase biased, a stationary current (without ac component) can flow through it despite the nonequilibrium condition. A large critical current is predicted due to a dynamically induced long range triplet proximity effect. Such an effect could be observed in a conventional hybrid device close to the ferromagnetic resonance.