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We present a detailed theoretical description of quantum coherent electron transport in voltage-biased cross-like Andreev interferometers. Making use of the charge conjugation symmetry encoded in the quasiclassical formalism, we elucidate a crucial role played by geometric and electron-hole asymmetries in these structures. We argue that a non-vanishing Aharonov-Bohm-like contribution to the current $I_S$ flowing in the superconducting contour may develop only in geometrically asymmetric interferometers making their behavior qualitatively different from that of symmetric devices. The current $I_N$ in the normal contour -- along with $I_S$ -- is found to be sensitive to phase-coherent effects thereby also acquiring a $2\pi$-periodic dependence on the Josephson phase. In asymmetric structures this current develops an odd-in-phase contribution originating from electron-hole asymmetry. We demonstrate that both phase dependent currents $I_S$ and $I_N$ can be controlled and manipulated by tuning the applied voltage, temperature and system topology, thus rendering Andreev interferometers particularly important for future applications in modern electronics.

We work out a microscopic theory describing complete statistics of voltage fluctuations generated by quantum phase slips (QPS) in superconducting nanowires. We evaluate the cumulant generating function and demonstrate that shot noise of the voltage as well as the third and all higher voltage cumulants differ from zero only due to the presence of QPS. In the zero-frequency limit voltage fluctuations in superconducting nanowires are described by Poisson statistics just as in a number of other tunneling-like problems. However, at non-zero frequencies quantum voltage fluctuations in superconducting nanowires become much more complicated and are not anymore accounted for by Poisson statistics. In the case of short superconducting nanowires we explicitly evaluate all finite-frequency voltage cumulants and establish a non-trivial relation between these cumulants and the current-voltage characteristics of our system., Comment: 10 pages, 3 figures

We demonstrate that a temperature gradient can strongly stimulate the thermoelectric signal, as well as dc Josephson current, in multiterminal superconducting hybrid nanostructures. At temperatures $T$ sufficiently exceeding the Thouless energy of our device, both the supercurrent and the thermoinduced voltage are dominated by the contribution from nonequilibrium low-energy quasiparticles and are predicted to decay slowly (algebraically rather than exponentially) with increasing $T$. We also predict a nontrivial current-phase relation and a transition to a $\ensuremath{\pi}$-junction state controlled by both the temperature gradient and the system topology. All these features are simultaneously observable in the same experiment.

We propose to study frequency dependent intrinsic dissipation in a highly transparent Josephson junction by means of qubit spectroscopy. The spectral density of the effective dissipative bath may contain significant contributions from Andreev bound states coupled to fluctuations of the Josephson phase. Varying either the bias current applied to the junction or magnetic flux through a superconducting ring in the rf-SQUID setup, one can tune the level splitting value close to the bottom of the Josephson potential well. Monitoring the qubit energy relaxation time one can probe the spectral density of the effective dissipative bath and unambiguously identify the contribution emerging from Andreev levels., 7 pages, 8 figures

We predict a novel $(I_0,\phi_0)$-junction state of multi-terminal Andreev interferometers that emerges from an interplay between long-range quantum coherence and non-equilibrium effects. Under non-zero bias $V$ the current-phase relation $I_S(\phi)$ resembles that of a $\phi_0$-junction differing from the latter due to a non-zero average $I_0(V) = \left< I_S(\phi)\right>_{\phi}$. The flux-dependent thermopower ${\mathcal S}(\Phi)$ of the system exhibits features similar to those of a $(I_0,\phi_0)$-junction and in certain limits it can reduce to either odd or even function of $\Phi$ in the agreement with a number of experimental observations.

We argue that quantum fluctuations of the phase of the order parameter may strongly affect the electron density of states (DOS) in ultrathin superconducting wires. We demonstrate that the effect of such fluctuations is equivalent to that of a quantum dissipative environment formed by sound-like plasma modes propagating along the wire. We derive a non-perturbative expression for the local electron DOS in superconducting nanowires which fully accounts for quantum phase fluctuations. At any non-zero temperature these fluctuations smear out the square-root singularity in DOS near the superconducting gap and generate quasiparticle states at subgap energies. Furthermore, at sufficiently large values of the wire impedance this singularity is suppressed down to $T=0$ in which case DOS tends to zero at subgap energies and exhibits the power-law behavior above the gap. Our predictions can be directly tested in tunneling experiments with superconducting nanowires., 6 pages, 3 figures

Employing quasiclassical theory of superconductivity combined with Keldysh technique we investigate large thermoelectric effect in multiterminal ballistic normal-superconducting (NS) hybrid structures. We argue that this effect is caused by electron-hole asymmetry generated by coherent Andreev reflection of quasiparticles at interfaces of two different superconductors with non-zero phase difference. Within our model we derive a general expression for thermoelectric voltages $V_{T1,2}$ induced in two different normal terminals exposed to a thermal gradient. Our results apply at any temperature difference in the subgap regime and allow to explicitly analyze both temperature and phase dependencies of $V_{T1,2}$ demonstrating that in general there exists no fundamental relation between these voltages and the equilibrium Josephson current in SNS junctions., 8 pages, 1 figure; published version

Combining scattering matrix theory with non-linear $\sigma$-model and Keldysh technique we develop a unified theoretical approach enabling one to non-perturbatively study the effect of electron-electron interactions on weak localization and Aharonov-Bohm oscillations in arbitrary arrays of quantum dots. Our model embraces (i) weakly disordered conductors (ii) strongly disordered conductors and (iii) metallic quantum dots. In all these cases at $T \to 0$ the electron decoherence time is found to saturate to a finite value determined by the universal formula which agrees quantitatively with numerous experimental results. Our analysis provides overwhelming evidence in favor of electron-electron interactions as a universal mechanism for zero temperature electron decoherence in disordered conductors., Comment: 19 pages, 13 figures, invited paper, published in a special issue of Fiz. Nizk. Temp. (Kharkov) dedicated to Prof. Igor Kulik

We theoretically investigate coherent oscillations of the thermopower $\mathcal{S}$ as a function of the magnetic flux $\Phi$ in six-terminal Andreev interferometers. We demonstrate that the thermopower behavior is determined by a number of contributions originating from the Josephson-like and Aharonov-Bohm-like effects as well as from electron-hole asymmetry. The relative weight of these contributions depends on the relation between temperature, voltage bias and an effective Thouless energy of our setup. We particularly emphasize the role of the system topology that may have a dramatic impact on the behavior of $\mathcal{S}(\Phi)$.

Quantum phase slips (QPS) generate voltage fluctuations in superconducting nanowires. Employing Keldysh technique and making use of the phase-charge duality arguments we develop a theory of QPS-induced voltage noise in such nanowires. We demonstrate that quantum tunneling of the magnetic flux quanta across the wire yields quantum shot noise which obeys Poisson statistics and is characterized by a power law dependence of its spectrum $S_\Omega$ on the external bias. In long wires $S_\Omega$ decreases with increasing frequency $\Omega$ and vanishes beyond a threshold value of $\Omega$ at $T \to 0$. Quantum coherent nature of QPS noise yields non-monotonous dependence of $S_\Omega$ on $T$ at small $\Omega$., Comment: 4.5 pages + 2-page supplemental material, 3 figures

We develop a new unified theoretical approach enabling us to non-perturbatively study the effect of electron-electron interactions on weak localization in arbitrary arrays of quantum dots. Our model embraces (i) weakly disordered conductors (ii) strongly disordered conductors and (iii) metallic quantum dots. In all these cases at $T \to 0$ the electron decoherence time is determined by the universal formula $\tau_{\varphi 0}\sim g\tau_D/\ln (E_C/\delta)$, where $g$, $\tau_D$, $E_C$ and $\delta$ are respectively dimensionless conductance, dwell time, charging energy and level spacing of a single dot. In the case (i) this formula yields $\tau_{\varphi 0}\propto D^3/\ln D$ ($D$ is the diffusion coefficient) and matches with our previous quasiclassical results [D.S. Golubev, A.D. Zaikin, Phys. Rev. Lett. 81 (1998) 1074], while in the cases (ii) and (iii) it illustrates new physics not explored earlier. A detailed comparison between our theory and numerous experiments provides an overwhelming evidence that zero temperature electron decoherence in disordered conductors is universally caused by electron-electron interactions rather than by magnetic impurities., Comment: 18 pages, Proceedings

With the aid of the Keldysh effective action technique we develop a microscopic theory describing Andreev level spectroscopy experiments in non-tunnel superconducting contacts. We derive an effective impedance of such contacts which accounts for the presence of Andreev levels in the system. At subgap bias voltages and low temperatures inelastic Cooper pair tunneling is accompanied by transitions between these levels resulting in a set of sharp current peaks. We evaluate the intensities of such peaks, establish their dependence on the external magnetic flux piercing the structure and estimate thermal broadening of these peaks. We also specifically address the effect of capacitance renormalization in a non-tunnel superconducting contact and its impact on both the positions and heights of the current peaks. At overgap bias voltages the $I-V$ curve is determined by quasiparticle tunneling and contains current steps related to the presence of discrete Andreev states in our system., 8 pages, 4 figures

We extend the quasiclassical formalism for diffusive superconductors by deriving anisotropic (gradient) corrections to the Usadel equation. We demonstrate that in a number of physical situations such corrections may play a crucial role being responsible for the effects which cannot be recovered within the standard Usadel approximation. One of them is the so-called photoelectric effect in superconductors and superconducting-normal (SN) hybrid structures. Provided a superconducting part of the system is irradiated by an external ac electromagnetic field the charge imbalance develops and a non-vanishing dc voltage is induced across the SN interface. In the presence of magnetic impurities in a superconductor the magnitude of this effect becomes large and can easily be detected in modern experiments., 8 pages, 2 figures

We demonstrate that thermoelectric currents in superconducting bilayers with a spin-active interface are controlled by the two competing processes. On one hand, spin-sensitive quasiparticle scattering at such interface generates electron-hole imbalance and yields orders-of-magnitude enhancement of the thermoelectric effect in the system. On the other hand, this electron-hole imbalance gets suppressed in the superconductor bulk due to electron scattering on non-magnetic impurities. As a result, large thermoelectric currents can only flow in the vicinity of the spin-active interface and decay away from this interface at a distance exceeding the electron elastic mean free path $\ell$. The magnitude of the thermoelectric effect reaches its maximum provided $\ell$ becomes of order of the total bilayer thickness., Comment: 6 pages, 2 figures

Combining nonperturbative techniques with Monte Carlo simulations we demonstrate that quantum coherence effects for a particle on a ring are suppressed beyond a finite length $L_{\phi}$ even at zero temperature if the particle is coupled to a diffusive electron gas by means of long range Coulomb interaction. This length is consistent with $L_{\phi}$ derived from weak-localization-type of analysis., Comment: 4 revtex pages, 2 figures

We derive a general expression for the conductivity of a disordered conductor with electron–electron interactions (treated within the standard model) and evaluate the weak localization correction δσ wl employing no approximations beyond the accuracy of the definition of δσ wl . Our analysis applies to all orders in the interaction and extends our previous calculation by explicitly taking into account quantum fluctuations around the classical paths for interacting electrons (pre-exponent). We specifically address the most interesting low temperature limit and demonstrate that such fluctuations can only be important in the perturbative regime of short times while they are practically irrelevant for the Cooperon dynamics at longer times. We fully confirm our conclusion about the existence of interaction-induced decoherence of electrons at zero temperature for the problem in question. We also demonstrate irrelevance of a perturbative calculation by Aleiner et al. (AAV) [J. Low Temp. Phys. 126, 1377 (2002)] and refute AAV's critique of our earlier analysis.

The quantum coherence of electrons can be probed by studying weak localizationcorrections to the conductivity. Interaction effects lead to dephasing, with electron-electron interactions being the important intrinsic mechanism. A controversy exists whether or not the dephasing rate, as measured in a weak localization experiment, vanishes at low temperatures. We review the non-perturbative analysis of this question and some of the arguments whichhave been raised against it. The compact form of the presentation shouldmake the derivation more transparent and accessible for discussions. Wealso compare with recent experiments.

We argue that quantum phase slips (QPS) may generate non-equilibrium voltage fluctuations in superconducting nanowires. In the low frequency limit we evaluate all cumulants of the voltage operator which obey Poisson statistics and show a power law dependence on the external bias. We specifically address quantum shot noise which power spectrum $S_\Omega$ may depend non-monotonously on temperature. In the long wire limit $S_\Omega$ decreases with increasing frequency $\Omega$ and vanishes beyond a threshold value of $\Omega$ at $T \to 0$. Our predictions can be directly tested in future experiments with superconducting nanowires.

We demonstrate that in sufficiently long diffusive superconducting-normal-superconducting ($SNS$) junctions dc Josephson current is exponentially suppressed by electron-electron interactions down to zero temperature. This suppression is caused by the effect of {\it Cooper pair dephasing} which occurs in the normal metal and defines a new fundamental length scale $L_\varphi$ in the problem. Provided the temperature length exceeds $L_\varphi$ this dephasing length can be conveniently extracted from equilibrium measurements of the Josephson current., Comment: 11 pages, 3 figures

We demonstrate that electron-electron interactions fundamentally restrict the penetration length of superconducting correlations into a diffusive normal metal (N) attached to a superconductor (S). We evaluate the subgap magnetoconductance $G$ of SN hybrids in the presence of electron-electron interactions and demonstrate that the effect of the magnetic field on $G$ is twofold: It includes ($i$) additional temperature independent dephasing of Cooper pairs and ($ii$) Zeeman splitting between the states with opposite spins. The dephasing length of Cooper pairs can be directly extracted from measurements of the subgap magnetoconductance in SN systems at low temperatures., 8 pages, 5 figures, proceedings of FQMT-13 conference (29 July - 3 August 2013, Prague, Czech Republic)

We predict parametrically strong enhancement of the thermoelectric effect in metallic bilayers consisting of two superconductors separated by a spin-active interface. The physical mechanism for such an enhancement is directly related to electron-hole imbalance generated by spin-sensitive quasiparticle scattering at the interface between superconducting layers. We explicitly evaluate the thermoelectric currents flowing in the system and demonstrate that they can reach maximum values comparable to the critical ones for superconductors under consideration., 6 pages, 1 figure

A theory is developed to study shot noise in superconducting (SAS) and hybrid (SAN) junctions with singly occupied Anderson impurity (A) as a weak link. The zero-frequency DC component of the shot noise spectral density is calculated at zero temperature as a function of the bias at different Coulomb repulsion strengths U, and show a remarkable structure resulting from combination of electron-electron interaction and Andreev reflections., Comment: 4 two column pages including 4 .eps figures

Novel effects emerge from an interplay between multiple Andreev reflections and Coulomb interaction in quantum dot coupled to superconducting leads and subject to a finite potential bias $V$. Combining an intuitive physical picture with rigorous path integral formalism we evaluate the current $I$ through the dot and find that the interaction shifts the subharmonic pattern of the $I-V$ curve is shifted toward higher $V$. For sufficiently ^M strong interaction the subgap current (at $eV < 2 \Delta$) is virtually suppressed., Comment: 4 two column pages including 1 eps figure

Interference of proximity-induced superconducting correlations in mesoscopic metallic rings is sensitive to the magnetic flux Φ inside these rings. This is the reason for magnetoconductance oscillations in such systems. We detected experimentally and explained theoretically a novel effect: the phase of these oscillations can switch between 0 and π depending on the resistance of intermetallic interfaces and temperature. The effect is due to a nontrivial interplay between the proximity-induced enhancement of the local conductivity and the proximity-induced suppression of the density of states at low energies.

Recent experiments on mesoscopic normal-metal--superconductor heterostructures resolve properties on length scales and at low temperatures such that the temperature is below the Thouless energy $k_B T \le E_{Th}$. We describe the properties of these systems within the framework of quasiclassical many-body techniques. Diffusive and ballistic systems are covered, both in equilibrium and nonequilibrium situations. Thereby we demonstrate the common physical basis of various subtopics., 38 pages, LaTeX, sup.sty-style file included, to appear in Superlattices and Microstructures; several minor changes and corrections of typographical errors, two updated figures

We have developed a detailed experimental study of a single-electron transistor in a strong tunneling regime. Although weakened by strong charge fluctuations, Coulomb effects were found to persist in all samples including one with the effective conductance 8 times higher than the quantum value (6.45 k$\Omega$)$^{-1}$. A good agreement between our experimental data and theoretical results for the strong tunneling limit is found. A reliable operation of transistors with conductances 3-4 times larger than the quantum value is demonstrated., Comment: revtex, 4 pages

We develope a theory of a fundamental effect of the interaction-induced decoherence of the electron wave function in a disordered metal. With the aid of the Keldysh technique and the path integral formalism we derive a formally exact equation of motion for the electron density matrix in the presence of interaction. We demonstrate that the effect of interaction of the electron with other electrons and lattice ions in a disordered metal is equivalent to that of an effective dissipative environment. Quantum noise of this environment causes quantum decoherence even at T = 0. Our analysis explicitely accounts for the Pauli principle which plays an important role for inelastic scattering processes but turns out not to affect quantum decoherence. Our results seriously challenge the existence of strong localization in low dimensional disordered metals.

We study strong electron tunneling in the single-electron box, a small metallic island coupled to an electrode by a tunnel junction, by means of quantum Monte Carlo simulations. We obtain results, at arbitrary tunneling strength, for the free energy of this system and the average charge on the island as a function of an external bias voltage. In much of the parameter range an extrapolation to the ground state is possible. Our results for the effective charging energy for strong tunneling are compared to earlier -- in part controversial -- theoretical predictions and Monte Carlo simulations.

Andrei D. Zaikin, Dmitrii S. Golubev, Anne van Otterlo, and Gergely T. Zimanyi 1 Institut fur Theoretische Festkorperphysik, Universitat Karlsruhe, 76128 Karlsruhe, FRG 2 I.E.Tamm Department of Theoretical Physics, P.N.Lebedev Physics Institute, Leninskii pr. 53, 117924 Moscow, Russia 3 Physics Department, Chalmers University of Technology, S-41296 Goteborg, Sweden 4 Physics Department, University of California, Davis, CA 95616, USA

Nonequilibrium effects and their impact on charge transport in superconducting ballistic weak links biased by an ac voltage are investigated microscopically. The microwave field destroys the phase coherence during the multiple Andreev reflection cycle and leads to effective cooling of subgap quasiparticles accelerated due to multiple Andreev reflection. For small bias voltages this effect results in a strong supression of both the excess current and the conductance of the weak link. At higher voltages (of the order of the gap) sideband subharmonic gap structures of the I-V curve are found.

We argue that spin-sensitive quasiparticle scattering may generate electron-hole imbalance in superconducting structures, such as, e.g., superconducting-normal hybrids with spin-active interfaces. We elucidate a transparent physical mechanism for this effect demonstrating that scattering rates for electrons and holes at such interfaces differ from each other. Explicitly evaluating the wave functions of electron-like and hole-like excitations in superconducting-normal bilayers we derive a general expression for the thermoelectric current and show that -- in the presence of electron-hole imbalance -- this current can reach maximum values as high as the critical current of a superconductor., Comment: 7 pages, 2 figures; published version

We present a microscopic study of the quantum fluctuations of the superconducting order parameter in thin homogeneous superconducting wires at all temperatures below T{sub c}. The rate of quantum phase-slip processes determines the resistance R(T) of the wire, which is observable in very thin wires, even at low temperatures. Furthermore, we predict a new low-temperature metallic phase below a critical wire thickness in the 10-nm range, in which quantum phase slips proliferate. {copyright} {ital 1997} {ital The American Physical Society}

We use a detailed microscopic analysis to study electron transport in normal diffusive conductors in the presence of proximity induced superconducting correlation. In the case of transparent barriers, superconducting correlations and electrical field can penetrate causing a whole range of novel nonequilibrium effects. These effects are explained by the introduction of a second, correlated density of states. Our results are fully consistent with recent experimental findings.

We theoretically investigate the behavior of Andreev interferometers with three superconducting electrodes in the current-biased regime. Our analysis allows to predict a number of interesting features of such devices, such as both hysteretic and non-hysteretic behavior, negative magnetoresistance and two different sets of singularities of the differential resistance at subgap voltages. In the non-hysteretic regime we find a pronounced voltage modulation with the magnetic flux which can be used for improving sensitivity of Andreev interferometers., 7 pages, 7 figures

We investigate the effect of interacting quantum phase slips on persistent current and its fluctuations in ultrathin superconducting nanowires and nanorings pierced by the external magnetic flux. We derive the effective action for these systems and map the original problem onto an effective sine-Gordon theory on torus. We evaluate both the flux dependent persistent current and the critical radius of the ring beyond which this current gets exponentially suppressed by quantum fluctuations. We also analyze fluctuations of persistent current caused by quantum phase slips. At low temperatures the supercurrent noise spectrum has the form of coherent peaks which can be tuned by the magnetic flux. Experimental observation of these peaks can directly demonstrate the existence of plasma modes in superconducting nanorings., 11 pages, 5 figures

We have investigated a small-area Josephson tunnel junction in a superconducting quantum interference device (SQUID) geometry, biased through high-resistance thin-film leads. This configuration provides the necessary conditions for the observation of charging effects. The Josephson coupling energy ${\mathit{E}}_{\mathit{J}}$ was varied by an external magnetic flux introduced into a SQUID loop. The current-voltage characteristics of the system were measured at low temperatures T\ensuremath{\sim}80 mK and different values of ${\mathit{E}}_{\mathit{J}}$. At a certain value of the current the Zener tunneling rate exceeds that of the dissipative relaxation and the system ``jumps'' from the lowest to higher Brillouin zones. This corresponds to a crossover on the I-V curve. The crossover current ${\mathit{I}}_{\mathrm{cr}}$ was measured as a function of ${\mathit{E}}_{\mathit{J}}$. Good agreement between our data and a theory of Zener tunneling with dissipation has been found. \textcopyright{} 1996 The American Physical Society.

We report on an experimental and theoretical study of nonlocal transport in superconductor hybrid structures, where two normal-metal leads are attached to a central superconducting wire. As a function of voltage bias applied to both normal-metal electrodes, we find surprisingly large nonlocal conductance signals, almost of the same magnitude as the local conductance. We demonstrate that these signals are the result of strong heating of the superconducting wire, and that under symmetric bias conditions, heating mimics the effect of Cooper pair splitting., Comment: 9 pages, 5 figures

We argue that parametrically strong enhancement of a thermoelectric current can be observed in conventional superconductors doped by magnetic impurities. This effect is caused by the violation of the symmetry between electronlike and holelike excitations due to formation of subgap bound Andreev states in the vicinity of magnetic impurities. We develop a quantitative theory of this effect and demonstrate that it can be detected in modern experiments.

We develop a quasiclassical theory of Andreev interferometers with three superconducting electrodes. Provided tunneling interface resistance between one superconducting electrode and the normal metal strongly exceeds two others, significant current sensitivity to the external magnetic flux is observed only at subgap voltages. If all barrier conductances are comparable, multiple Andreev reflection comes into play and substantial current modulation can be achieved in both subgap and overgap voltage regimes. Our analysis reveals a large variety of interesting features which can be used for performance optimization of Andreev interferometers., Comment: 9 pages, 13 figures

We consider a metallic wire coupled to two metallic electrodes via two junctions placed nearby. A bias voltage applied to one of such junctions alters the electron distribution function in the wire in the vicinity of another junction thus modifying both its noise and the Coulomb blockade correction to its conductance. We evaluate such interaction corrections to both local and non-local conductances demonstrating non-trivial Coulomb anomalies in the system under consideration. Experiments on non-local electron transport with Coulomb effects can be conveniently used to test inelastic electron relaxation in metallic conductors at low temperatures., Published version. 11 pages, 4 figures. New references added, discussion and introduction are extended, appendices added

We argue that electron-electron interactions fundamentally restrict the penetration length of Cooper pairs into a diffusive normal metal (N) from a superconductor (S). At low temperatures this Cooper pair dephasing length $L_\varphi$ remains finite and does not diverge at $T \to 0$. We evaluate the subgap conductance of NS hybrids in the presence of electron-electron interactions and demonstrate that this new length $L_\varphi$ can be directly extracted from conductance measurements in such structures., Comment: 5 pages, 4 figures

In the vicinity of the Coulomb blockade threshold virtual electron tunneling leads to effective screening of the charge on small metallic junctions. We present a nonperturbative calculation of the junction ground state energy and level splitting close to the edges of the Brillouin zone. We show that the electron tunneling rate can be substantially suppressed due to quantum fluctuations of the charge and the junction current-voltage characteristic becomes nonlinear near the Coulomb gap. We also investigate linear transport properties of a single tunnel junction in the presence of Coulomb effects.

Ballistic charge transport through mesoscopic superconductor-normal metal-superconductor (SNS) microbridges is investigated on the basis of the Keldysh technique. At low voltages we find a peak in the conductance which is caused by multiple Andreev reflection of quasiparticles trapped in the pair potential well of the N-layer in the presence of inelastic relaxation there. For larger voltages inelastic relaxation in the N-layer plays no role, quasiparticles are accelerated out of the N-layer and relax in superconducting banks. We demonstrate that even at eV, T ⪡ Δ the system has a finite subgap conductance, i.e. dissipative current can flow through a superconducting constriction both with and without normal metal inclusion. In this case dissipation arises from pairbreaking due to acceleration of virtually created subgap quasiparticles by the electric field. In the voltage range V ∼ Δ/e we analyze the temperature dependence of the subharmonic gap structure which turns out to be most pronounced at T ∼ Δ and vanishes at T → 0 and T → TC. This behavior is the result of a counter current induced by thermally excited quasiparticles.

Single electron tunneling into small superconducting islands is sensitive to the gap energy of the excitations created in the process and, hence, depends on the parity of the electron number in the island. We study these effects by analyzing the kinetics of the system. The tunneling of a single electron, the “odd” one, dominates at low temperatures since in this process the excitation energy can be regained, while the competing regular processes are exponentially suppressed. We discuss the influence of parity effects on single electron tunneling and Andreev reflection in superconducting transistors and find rich structures in the I–V characteristics, which compare well with recent experiments.

We develop a microscopic description of single and double electron tunneling through normal metal-superconductor interfaces in the presence of proximity and charging effects. In both ballistic and diffusive limits we derive the effective action for the system and show that its behavior can essentially depend on the physical boundary conditions in both normal and superconducting electrodes far from the tunnel barrier. We study quantum fluctuations of the charge at the NS interface due to virtual Andreev reflection which can substantially modify the Coulomb blockade for double electron tunneling. We also investigate the charge transfer through mesoscopic NS interfaces and NSN transistors in the presence of Coulomb effects for various physical situations. We describe a novel charge transfer mechanism in NSN structures - double electron cotunneling - and show that this mechanism can play an important role in the limit of relatively small external voltages. The results of our theory agree well with available experimental data.

In the vicinity of the Coulomb-blockade threshold, quantum fluctuations of the charge strongly influence the energy spectrum of small metallic junctions. We develop a nonperturbative calculation of the junction ground-state energy and evaluate the junction charge in the limit of a small junction conductance. Close to the Coulomb-blockade threshold this charge is effectively screened due to intensive virtual electron tunneling. We also describe level splitting close to the edges of the Brillouin zone and show that the electron tunneling rate can be substantially suppressed due to quantum fluctuations of the charge. As a result the junction current-voltage characteristic becomes non-Ohmic near the Coulomb gap.

Ballistic charge transport through mesoscopic superconductor--normal-metal--superconductor (SNS) microbridges is investigated within the framework of the Keldysh technique. At low voltages we find a peak in the conductance G\ensuremath{\sim}(${\mathit{l}}_{\mathrm{in}}$/2a)(1/${\mathit{R}}_{0}$), where ${\mathit{l}}_{\mathrm{in}}$ is the inelastic mean free path, 2a is the normal layer thickness, and ${\mathit{R}}_{0}$ is the Sharvin resistance. In this limit the dissipative current is carried by quasiparticles which are trapped in the pair potential well of height \ensuremath{\Delta}, suffering multiple Andreev reflections and relaxing inside the N layer. For larger voltages eV\ensuremath{\gtrsim}(2a/${\mathit{l}}_{\mathrm{in}}$)\ensuremath{\Delta} quasiparticle are accelerated out of the N layer and relax in superconducting banks. The current-voltage characteristic of the system in this regime is calculated both analytically and numerically. In the intermediate voltage range V\ensuremath{\sim}\ensuremath{\Delta}/e we analyze the subharmonic gap structure on the I-V curve and show that this structure is most pronounced in the temperature range T\ensuremath{\sim}\ensuremath{\Delta}/${\mathit{k}}_{\mathit{B}}$ and vanishes for both T\ensuremath{\rightarrow}0 and T\ensuremath{\rightarrow}${\mathit{T}}_{\mathit{C}}$. We explain this behavior taking into account the contribution of thermally excited quasiparticles with momenta opposite to the direction of total current flow.

Single electron tunneling into small superconducting islands is sensitive to the gap energy of the excitations created in the process and, hence, depends on the electron number parity. At low temperature the properties of the system are 2e-periodic in the applied gate voltage, turning e-periodic at higher temperature. We evaluate the tunneling rates and determine the probabilities for the even and odd state as well as the cross-over condition from the balance between different processes. Our analysis includes excitations in the leads. They are essential if the leads are superconducting. The influence of parity effects on single electron tunneling, Cooper pair tunneling, and the Andreev reflection in superconducting transistors yields rich structures in the I-V characteristic, which explains recent experimental findings., Comment: 8 pages, 1 figure available on request, report KA-TFP-93-2

We derive the effective free energy of a two-dimensional Josephson junction in the presence of an external current and predict that the junction has a phase transition at a temperature TJ below the bulk transition temperature Tc. In the range TJ < T < Tc long-range phase correlation exists in the bulk, but is destroyed in the vicinity of the junction. The critical current Ic is reduced by thermal fluctuations; for a junction of size L, Ic ~ Lb(T) where b(T) < 0 for TJ < T < Tc (i.e. Ic vanishes at L → ∞) while 0 < b(T) < 2 for T < TJ. Our results may account for the absence of an observable supercurrent at temperatures below Tc in YBa2Cu3Ox- and Bi2Sr2CaCu2O8-based junctions.