Self-consistent theory of transport in superconducting wires

Electrons in superconductors condense to form pairs known as Cooper pairs [1]. The characteristic decay length of the wavefunction of this pair (or the "size" of a Cooper pair) is the superconducting coherence length !. When any of the 3 physical dimensions of a superconducting system become comparable to this characteristic length the effect of fluctuations become important and superconductivity is predicted to be destroyed [2]. In 1D systems (nanowires with diameter < !) the limit at which superconductivity is quenched and the mechanism by which it is quenched is an active field of study. Fluctuations and changes in boundary conditions lead to many novel phenomena in 1D nanowires. Experimental exploration of these novel effects forms the basis of this dissertation. Metallic 1D nanowires have been fabricated using template-based electrodeposition and evaporation. Availability of nanowires of different morphologies helps in performing comparative experiments to isolate effects due to disorder from true 1D physics. The electronic transport properties of superconducting, ferromagnetic and normal nanowires with superconducting and normal electrodes have been studied in various measurement geometries. Experiments on aluminum nanowires studying the counterintuitive anti-proximity effect (APE) [3] have been performed. The results of these experiments appear to bring a complete understanding of the phenomenon and have resolved a number of puzzles in the early experiments. In addition, measurements of a single resistance reading found switching from the superconducting to the normal state close to T c of the wire and at low temperatures in the APE regime. The switching at low temperature is triggered by individual quantum phase slips. These results indicate that the low temperature APE regime offers a clean platform for the observation of individual quantum phase slips, a goal eluded in numerous experiments. iv Systematic studies on crystalline and granular ferromagnetic cobalt and nickel nanowires sandwiched between superconducting electrodes have been performed. A very long-range proximity effect (~ 600 nm) was found. This range is two orders of magnitude larger than that measured for bulk superconductor-ferromagnet systems. The superconducting transition was foreshadowed by a large peak in resistance dubbed the 'critical peak'. Possible explanations of these counterintuitive effects have been discussed. In earlier experiments on crystalline gold nanowires contacted with superconducting tungsten electrodes, a mini-gap state along with magnetoresistance oscillations indicating individual vortex trapping were found [4]. The experiment has been repeated here with different electrodes and different nanowire morphologies. The mini-gap state persists in these samples demonstrating it is a robust state independent of nanowire and electrode morphology. v Contents LIST OF FIGURES ix

1998

We study the transport properties of an NSN structure with an insulating barrier at each NS interface. Coherent quasiparticle scattering is assumed and self-consistency is implemented exactly to guarantee local charge conservation. The presence of a finite condensate flow has a greater influence on the transport properties than either the gap depression near the interfaces or the coherent nature of scattering.

Physical Review B, 2012

We study the non linear response of current transport in a superconducting diffusive nanowire between normal reservoirs. We demonstrate theoretically and experimentally the existence of two different superconducting states appearing when the wire is driven out of equilibrium by an applied bias, called the global and bimodal superconducting states. The different states are identified by using two-probe measurements of the wire, and measurements of the local density of states with tunneling probes. The analysis is performed within the framework of the quasiclassical kinetic equations for diffusive superconductors.

Journal of Physics: Condensed Matter, 1991

When a quasi-partide current passes into II disordered superconductor, from ideal normal leads connected to external reservoirs, a finite electrical resistance Rs arises from scattering processes within the superconductor. A new formula for Rs is obtained, which reduces to the well-known Landauer formula in the absence of superconductivity. It &, k(T0,T.J are reflection (transmksion) coefficients associated with nomnal and Andrcev scattering respectively, one finds, in one dimension at zero temperature, Rs = (h/Ze2)(& + T, t S)/(R. + To-6) where 6 is a small parameter arising from the absence of inversion symmetry. Generalizations of this result to &rite temperat-and higher dimensions are also obtained.

Physical Review B, 2008

We investigate transport properties of a superconducting junction of many ($N \ge 2$) one-dimensional quantum wires. We include the effectofelectron-electron interaction within the one-dimensional quantum wire using a weak interaction renormalization group procedure. Due to the proximity effect, transport across the junction occurs via direct tunneling as well as via the crossed Andreev channel. We find that the fixed point structure of this system is far more rich than the fixed point structure of a normal metal$-$superconductor junction ($N = 1$), where we only have two fixed points - the fully insulating fixed point or the Andreev fixed point. Even a two wire (N=2)system with a superconducting junction i.e. a normalmetal$-$superconductor$-$normal metal structure, has non-trivialfixed points with intermediate transmissions and reflections. We also include electron-electron interaction induced back-scattering in the quantum wires in our study and hence obtain non-Luttinger liquid behaviour. It is interesting to note that {\textsl{(a)}} effects due to inclusion of electron-electron interaction induced back-scattering in the wire, and {\textsl{(b)}} competition between the charge transport via the electron and hole channels across the junction, give rise to a non-monotonic behavior of conductance as a functionof temperature. We also find that transport across the junction depends on two independent interaction parameters. The first one is due to the usual correlations coming from Friedel oscillations for spin-full electrons giving rise to the well-known interaction parameter (${{\alpha = (g_2-2g_1)/2 \pi \hbar v_F}}$). The second one arises due to the scattering induced by the proximity of the superconductor and is given by(${{\alpha^\prime = (g_2 + g_1)/2 \pi \hbar v_F}}$).

Physical Review B, 1999

We have calculated the temperature dependence of the conductance variation (δS(T )) of mesoscopic superconductor normal metal(S/N) structures, in the diffusive regime, analysing both weak and strong proximity effects. We show that in the case of a weak proximity effect there are two peaks in the dependence of δS(T ) on temperature. One of them (known from previous studies) corresponds to a temperature T1 of order of the Thouless energy (ǫ T h ), and another, newly predicted maximum, occurs at a temperature T2 where the energy gap in the superconductor ∆(T2) is of order ǫ T h . In the limit L φ < L the temperature T1 is determined by Dh/L 2 φ (L φ is the phase breaking length), and not ǫ T h . We have also calculated the voltage dependence δS(V ) for a S/F structure (F is a ferromagnet) and predict non-monotonic behaviour at voltages of order the Zeeman splitting.

1997

We present a scattering description of transport in several normal-superconductor structures. We show that the related requirements of self-consistency and current conservation introduce qualitative changes in the transport behavior when the current in the superconductor is not negligible. The energy thresholds for quasiparticle propagation in the superconductor are sensitive to the existence of condensate flow (vs 0).

Eprint Arxiv Cond Mat 9810023, 1998

We investigate the electronic transport coefficients in unconventional superconductors at low temperatures, where charge and heat transport are dominated by electron scattering from random lattice defects. We discuss the features of the pairing symmetry, Fermi surface, and excitation spectrum which are reflected in the low temperature heat transport. For temperatures $k_B T \la \gamma \ll \Delta_0$, where $\gamma$ is the bandwidth of impurity induced Andreev states, certain eigenvalues become {\it universal}, i.e., independent of the impurity concentration and phase shift. Deep in the superconducting phase ($k_B T \la \gamma$) the Wiedemann-Franz law, with Sommerfeld's value of the Lorenz number, is recovered. We compare our results for theoretical models of unconventional superconductivity in high-T$_c$ and heavy fermion superconductors with experiment. Our findings show that impurities are a sensitive probe of the low-energy excitation spectrum, and that the zero-temperature limit of the transport coefficients provides an important test of the order parameter symmetry.

Journal of Physics: Condensed Matter, 1993

After presenting an intuitive picture of quasi-panicle UanspOrf in m e s m p i c superconductors, which emphasizes the intimate relation behueen Andreev scauering and m resistivity. we develop a general Wry of oc transport in mesoscopic.normal~uperconducting shuctures. Generalized multi-pmte conductance formulae are derived, which fake into aCFoun1 MK only Ihe effect of Andreev rcatte~ng on transpon coeffidents. but also the nonsonservation of quasi-panicle charge which arises in the presence of a superconducting mndensate. Experiments on quari-panicl& charge imbalance are described naturally by this approach. ~. 1. Introduction At.a low enough temperature, or for small enough systems, a quasi-particle such as an electron or hole can pass through a sample without scattering inelastically. In this mesoscopic limit, the phase coherence of quasi-particles is preserved and transport pmperties depend in detail on the diffraction pattern produced by elastic scattering from inhomogeneities and boundaries. During the past decade, the study of normal mesoscopic systems has Jed to the .discovery of a range of .new phenomena, including universal conductance fluctuations [1,2], quantized conductance of point contacts [3,4] and the detection of macroscopic changes in transpoa coefficients arising from tunnelling within a single atomic two-level system [5]. Until recently, these advances had centred almost exclusively on normal mesoscopic systems. However during the past few years, the hitherto distinct fields of mesoscopic physics and superconductivity have come together, leading to the possibility of a range of new developments involving hybrid normal-superconducting structures. One class of such systems is typified by mesoscopic Josephson junctions, which arise when a mesoscopic weak link is~fofied between two non-mesoscopic superconducting contacts. For such structures, the critical current I, is predicted to exhibit a range of new quantum phenomena Recent examples are the discretization of the'critical c-nt through a ballistic point contact [SI, the appearance of a universal resonant Josephson current through quantum dots [7] and the prediction of universal supercurrent fluctuations in diffusive point contacts [SI. Another class of hybrid systems arises when the superconductor itself is mesoscopic. This irises when a small superconducting sample is connected to the outside world through normal extemal leads, or when superconducting islands are immersed in a normal mesoscopic background. In such systems, transpon propetties such as electrical or thermal conductances exhibit new effects which are absent from their normal counterparts. For

Physica C: Superconductivity, 2003

We report on the transport properties of single superconducting lead nanowires grown by an electrodeposition technique, embedded in a nanoporous track-etched polymer membrane. The nanowires are granular, have uniform diameter of $40 nm and a very large aspect ratio ($500). The diameter of the nanowire is small enough to ensure a 1D superconducting regime in a wide temperature range below T c . The non-zero resistance in the superconducting state and its variation caused by fluctuations of the superconducting order parameter were measured versus temperature, magnetic field, and applied DC current (or voltage). The current induced breakdowns in the V -I characteristics may be explained by the formation of phase slip centers. Moreover, DC voltage driven measurements reveal the existence of a new S-shape behavior near the formation of these phase slip centers.

Journal of Physics: Condensed Matter, 1998

We calculate the conductance for a diffusive normal wire (N) in contact with a superconductor (S). Using a numerical scattering matrix approach and a quasiclassical Green function technique, we compare the conductance G of the system connected to two normal reservoirs when the superconductor is in its normal state with the conductance G s in the superconducting state and predict that the difference δG = G s − G may be negative or positive depending upon the S/N interface resistance and the interface resistance at the ends of the N wire. As the temperature T is varied, δG(T) may change sign and exhibit two maxima.

Physical Review Letters, 2011

We study the nonlinear cotunneling current through a spinful quantum dot contacted by two superconducting leads. Applying a general nonequilibrium Green function formalism to an effective Kondo model, we study the rich variation in the IV-characteristics with varying asymmetry in the tunnel coupling to source and drain electrodes. The current is found to be carried respectively by multiple Andreev reflections in the symmetric limit, and by spin-induced Yu-Shiba-Russinov bound states in the strongly asymmetric limit. The interplay between these two mechanisms leads to qualitatively different IV-characteristics in the cross-over regime of intermediate symmetry, consistent with recent experimental observations of negative differential conductance and re-positioned conductance peaks in sub-gap cotunneling spectroscopy.

NATO Science for Peace and Security Series B: Physics and Biophysics, 2009

Journal of Physics: Condensed Matter, 1995

We study the nonlinear transport properties of NS (normal-superconductor) and NSN structures by means of a self-consistent microscopic description. A nonzero superfluid velocity causes the various quasiparticle channels within S to open at different voltages. The gap reduction is very sensitive to the details of quasiparticle scattering. At low temperatures, superconductivity, some times in a peculiar gapless form, may survive up to voltages much higher that k B T c /e. The minimum voltage for quasiparticle transmission is shown to decrease strongly with temperature and with the transmittivity of the barrier.

Physical Review B, 2007

Motivated by recent findings of unconventional superconductors exhibiting multiple broken symmetries, we consider a general Hamiltonian describing coexistence of itinerant ferromagnetism, spin-orbit coupling and mixed spin-singlet/triplet superconducting pairing in the context of mean-field theory. The Hamiltonian is diagonalized and exact eigenvalues are obtained, thus allowing us to write down the coupled gap equations for the different order parameters. Our results may then be applied to any model describing coexistence of any combination of these three phenomena. As a specific application of our results, we consider tunneling between a normal metal and a noncentrosymmetric superconductor with mixed singlet and triplet gaps. The conductance spectrum reveals information about these gaps in addition to how the influence of spin-orbit coupling is manifested. Explicitly, we find well-pronounced peaks and bumps in the spectrum at voltages corresponding to the sum and the difference of the magnitude of the singlet and triplet components. Our results may thus be helpful in determining the relative sizes of the singlet and triplet gaps in noncentrosymmetric superconductors. We also consider the coexistence of itinerant ferromagnetism and triplet superconductivity as a model for recently discovered ferromagnetic superconductors. The coupled gap equations are solved self-consistently, and we study the conditions necessary to obtain the coexistent regime of ferromagnetism and superconductivity. Analytical expressions are presented for the order parameters, and we provide an analysis of the free energy to identify the preferred system state. It is found that the uniform coexistence of ferromagnetism and superconductivity is energetically favored compared to both the purely ferromagnetic state and the unitary superconducting state with zero magnetization. Moreover, we make specific predictions concerning the heat capacity for a ferromagnetic superconductor. In particular, we report a nonuniversal relative jump in the specific heat, depending on the magnetization of the system, at the uppermost superconducting phase transition. We propose that this may be exploited to obtain information about both the superconducting pairing symmetry realized in ferromagnetic superconductors in addition to the magnitude of the exchange splitting between majority and minority spin bands.