Anomalously large thermopower of mesoscopic normal-metal/superconductor interferometers has been investigated by Chandrasekhar et al. It was shown that, depending on the geometry of the interferometer, the thermopower is either symmetric or antisymmetric periodic function of the magnetic flux. We develop a detailed theory of the observed thermoelectric phenomena in the framework of the non-equilibrium quasiclassical approach. In particular we provide, for the first time, a possible explanation of the symmetric thermopower oscillations. This effect is attributed to the electron-hole symmetry violation that originates in the steady-state charge imbalance between different arms of the interferometer. Our theory can be tested by an additional control over the charge imbalance in a modified setup geometry. We also predict a sign reversal behavior of the thermopower with increasing temperature that is consistent with the experiments by Parsons et al.
Using a new second derivative technique and thermometers which enable us to determine the local electron temperature in a mesoscopic metallic sample, we have obtained quantitative measurements of the low temperature field and temperature dependent thermopower of Andreev interferometers. As in previous experiments, the thermopower is found to oscillate as a function of magnetic field. The temperature dependence of the thermopower is nonmonotonic, with a minimum at a temperature of $\simeq0.5$ K. These results are discussed from the perspective of Andreev reflection at the normal-metal/supercond…
We report measurements of thermopower oscillations $vs$ magnetic field in a diffusive Andreev interferometer. Upon the increase of the dc current applied to the heater electrodes, the amplitude of these oscillations first increases then goes to zero as one would expect. Surprisingly, the oscillations reappear at yet higher heater currents with their phase being $π$-shifted compared to low current values. From direct measurements of the temperature gradient we estimate the amplitude of the oscillations to be orders of magnitude smaller than predicted by theory.
We report measurements of the thermopower S of mesoscopic Andreev interferometers, which are hybrid loops with one arm fabricated from a superconductor (Al), and one arm from a normal metal (Au). S depends on the phase of electrons in the interferometer, oscillating as a function of magnetic flux with a period of one flux quantum (= h/2e). The magnitude of S increases as the temperature T is lowered, reaching a maximum around T = 0.14 K, and decreases at lower temperatures. The symmetry of S oscillations with respect to magnetic flux depends on the topology of the sample.
We examine the thermopower Q of a mesoscopic normal-metal (N) wire in contact to superconducting (S) segments and show that even with electron-hole symmetry, Q may become finite due to the presence of supercurrents. Moreover, we show how the dominant part of Q can be directly related to the equilibrium supercurrents in the structure. We also discuss the thermopower arising due to an anomalous kinetic coefficient which is finite in the presence of supercurrent and in some situations gives the dominant contribution. In general, a finite thermopower appears both between the N reservoirs and the s…
The thermopower of mesoscopic normal metal/superconductor structures has been measured at low temperatures. Effect of supercurrent present in normal part of the structure was studied in two cases: when it was created by applied external magnetic field and when it was applied directly using extra superconducting electrodes. Temperature and magnetic field dependencies of thermopower are compared to the numerical simulations based on the quasiclassical theory of the superconducting proximity effect.
We theoretically investigate coherent oscillations of the thermopower $\mathcal{S}$ as a function of the magnetic flux $Φ$ 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…
We study the thermopower of diffusive Andreev interferometers, which are hybrid loops with one normal-metal arm and one superconducting arm. The thermopower oscillates as a function of the magnetic flux through the loop with a fundamental period corresponding to one flux quantum $Φ_0=h/2e$. Unlike the electrical resistance oscillations and the thermal resistance oscillations, which are always symmetric with respect to the magnetic field, the symmetry of the thermopower oscillations can be either symmetric or antisymmetric depending on the geometry of the sample. We also observe that the symmet…
Andreev interferometers, normal metal wires coupled to superconducting loops, display phase coherent changes as the magnetic flux through the superconducting loops is altered. Properties such as the electronic and thermal conductance of these devices have been shown to oscillate symmetrically about zero with a period equal to one superconducting flux quantum, $Φ_o = h/2e$. However, the thermopower of these devices can oscillate symmetrically or antisymmetrically depending on the geometry of the sample, a phenomenon not well understood theoretically. Here we report on thermopower measurements o…
We investigate thermoelectric transport through Andreev interferometers. We show that the ratio of the thermal and the charge conductance exhibits large oscillations with the phase difference $ϕ$ between the two superconducting contacts, and that the Wiedemann-Franz law holds only when $ϕ=π$. A large average thermopower furthermore emerges whenever there is an asymmetry in the dwell times to reach the superconducting contacts. When this is the case, the thermopower is odd in $ϕ$. In contrast, when the average times to reach either superconducting contact are the same, the average thermopower i…
The diffusion of energy that is locally deposited into two-dimensional electron gases by Joule heating generates transverse voltages across devices with broken symmetry. For mesoscopic structures characterized by device dimensions comparable to the energy diffusion length, the resulting thermopower strongly depends on details of the potential profile defined by electric gates. We discuss these mesoscopic features within a diffusion thermopower model and propose schemes to measure the energy diffusion length and its dependence on gate voltage.
The energy-dependent Andreev reflection eigenvalues determine the transport properties of normal-superconducting systems. We evaluate the eigenvalue density to get an insight into formation of resonant electron-hole transport channels. The circuit-theory-like method developed can be applied to any generic mesoscopic conductor or combinations thereof. We present the results for experimentally relevant cases of a diffusive wire and a double tunnel junction.
The Adriatico Research Conference on Superconductivity, Andreev Reflection, and Proximity Effect in Mesoscopic Structures took place at the International Center for Theoretical Physics in Trieste, Italy, July 8-11, 1997. The organizers were Elias Burstein, Leonid Glazman, Teun Klapwijk, and Subodh R. Shenoy. We describe some of the central issues discussed at the conference, along with more personal reflections prompted by new developments.
We show that irradiation of a voltage-biased superconducting quantum point contact at frequencies of the order of the gap energy can remove the suppression of subgap dc transport through Andreev levels. Quantum interference among resonant scattering events involving photon absorption is furthermore shown to make microwave spectroscopy of the Andreev levels feasible. We also discuss how the same interference effect can be applied for detecting weak electromagnetic signals up to the gap frequency, and how it is affected by dephasing and relaxation.
We study shot noise in mesoscopic diffusive wires between a normal and a superconducting terminal. We particularly focus on the regime, in which the proximity-induced reentrance effect is important. We will examine the difference between a simple Boltzmann-Langevin description, which neglects induced correlations beyond the simple conductivity correction, and a full quantum calculation. In the latter approach, it turns out that two Andreev pairs propagating coherently into the normal metal are anti-correlated for E<E_c, where E_c=D/L^2 is the Thouless energy. In a fork geometry the flux-sensit…
We derive the statistics of the time-delay matrix (energy derivative of the scattering matrix) in an ensemble of superconducting quantum dots with chaotic scattering (Andreev billiards), coupled ballistically to $M$ conducting modes (electron-hole modes in a normal metal or Majorana edge modes in a superconductor). As a first application we calculate the density of states $ρ_0$ at the Fermi level. The ensemble average $\langleρ_0\rangle=δ_0^{-1}M[\max(0,M+2α/β)]^{-1}$ deviates from the bulk value $1/δ_0$ by an amount depending on the Altland-Zirnbauer symmetry indices $α,β$. The divergent aver…
Current noise is measured with a SQUID in low impedance and transparent Nb-Al-Nb j unctions of length comparable to the phase breaking length and much longer than the thermal length. The shot noise amplitude is compared with theoretical predictions of doubled shot noise in diffusive normal/superconductor (NS) junctions due to the Andreev reflections. We discuss the heat dissipation away from the normal part through the NS interfaces. A weak applied magnetic field reduces the amplitude of the 1/f noise by a factor of two, showing that even far from equilibrium the sample is in the mesoscopic re…
When coupling a superconductor to a normal conducting region the physical properties of the system are highly affected by the superconductor. We will investigate the effect of one or two superconductors on the conductance of a ballistic chaotic quantum dot to leading order in the total channel number using trajectory based semiclassics. The results show that the effect of one superconductor on the conductance is of the order of the number of channels and that the sign of the correction from the Drude conductance depends on the particular ratios of the numbers of channels of the superconducting…
We study the thermoelectric transport of a small metallic island weakly coupled to two electrodes by tunnel junctions. In the Coulomb blockade regime, in the case when the ground state of the system corresponds to an even number of electrons on the island, the main mechanism of electron transport at lowest temperatures is elastic cotunneling. In this regime, the transport coefficients strongly depend on the realization of the random impurity potential or the shape of the island. Using the random-matrix theory, we calculate the thermopower and the thermoelectric kinetic coefficient and study th…
We present thermopower $S$ and resistance $R$ measurements on GaAs-based mesoscopic two-dimensional electron gases (2DEGs) as functions of the electron density $n_s$. At high $n_s$ we observe good agreement between the measured $S$ and $S_{\rm{MOTT}}$, the Mott prediction for a non-interacting metal. As $n_s$ is lowered, we observe a crossover from Mott-like behaviour to that where $S$ shows strong oscillations and even sign changes. Remarkably, there are absolutely no features in $R$ corresponding to those in $S$. In fact, $R$ is devoid of even any universal conductance fluctuations. A statis…
In an unconventional superconductor, the interplay of scattering off impurities and Andreev processes may lead to different scattering times for electronlike and holelike quasiparticles. Such electron-hole asymmetry appears when the impurity scattering phase shift is intermediate between the Born and unitary limits and leads to an expectation for large thermoelectric effects. Here, we examine the thermoelectric response of a $d$-wave superconductor connected to normal-metal reservoirs under a temperature bias using a fully self-consistent quasiclassical theory. The thermoelectrically induced q…
Low temperature transport measurements on superconducting film - normal metal wire - superconducting film (SNS) junctions fabricated on the basis of 6 nm thick superconducting polycrystalline PtSi films are reported. The structures with the normal metal wires of two different lengths L=1.5 $μ$m and L=6$μ$m and the same widths W=0.3$μ$m are studied. Zero bias resistance dip related to pair current proximity effect is observed for all junctions whereas the subharmonic energy gap structure originating from phase coherent multiple Andreev reflections have occurs only in the SNS junctions with shor…
An inherent difficulty in studying mesoscopic effects in semiconductor--superconductor hybrid structures is the large Schottky barrier which often forms at the interface. A large technological effort has been invested in in improving the contact between the superconductor and the two-dimensional electron gas (2DEG) of a semiconductor heterostructure, and in recent years this has become possible for e.g. GaAs-Al, GaAs-In, and InAs-Nb junctions. This development motivates quantitative theoretical modeling of sample-specific transport properties. The aim of our work is to model the conducting pro…
The general scattering-matrix formalism is suggested for the description of shot noise in single-channel quantum NS contacts in the presence of smeared Andreev levels in the $eV<Δ$ regime. It is shown that the noise spectral density as a function of frequency shows a number of characteristic features due to the presence of the Andreev resonances in the system. It turns out that, by analyzing these singularities, one can determine the relative phases of the Andreev reflection amplitudes corresponding to the neighboring resonances.
Mesoscopic circuit elements such as quantum dots and quantum point contacts (QPCs) offer a uniquely controllable platform for engineering complex quantum devices, whose tunability makes them ideal for generating and investigating interacting quantum systems. However, the conductance measurements commonly employed in mesoscopics experiments are poorly suited to discerning correlated phenomena from those of single-particle origin. Here, we introduce non-equilibrium thermopower measurements as a novel approach to probing the local density of states (LDOS), offering an energy-resolved readout of m…
We investigate the Majorana physics and its effect on the electron transport in the non-topological superconductor(NS)-topological superconductor(TS) double junctions of a ring geometry. We find that, depending on the ratio between the lengths of two topologically different regions and the localization lengths of the Majorana fermions formed between them, two completely different transport mechanisms are working: perfect crossed Andreev reflection (CAR) for the short NS region and perfect normal Andreev reflection for the short TS region. The difference is explained in terms of the topological…
We study transport mediated by Andreev bound states formed in InSb nanowire quantum dots. Two kinds of superconducting source and drain contacts are used: epitaxial Al/InSb devices exhibit a doubling of tunneling resonances, while in NbTiN/InSb devices Andreev spectra of the dot appear to be replicated multiple times at increasing source-drain bias voltages. In both devices, a mirage of a crowded spectrum is created. To describe the observations a model is developed that combines the effects of a soft induced gap and of additional Andreev bound states both in the quantum dot and in the finite …
Electronic properties of mesoscopic systems made of normal metal in contact with normal and superconducting reservoirs are determined by Andreev Reflection. We address both experimentally and theoretically how Andreev Reflection manifests itself in shot noise experiments and what physics can be deduced from such measurements. We report high frequency measurements in which Andreev Reflection affects the distribution statistics of the electronic excitations, resulting in a shift of the classical-to-quantum regime or equilibrium-to-shot noise crossovers. We also demonstrate that the effective cha…
We investigate basic properties of the thermopower (Seebeck coefficient) of phase-coherent conductors under the influence of dephasing and inelastic processes. Transport across the system is caused by a voltage bias or a thermal gradient applied between two terminals. Inelastic scattering is modeled with the aid of an additional probe acting as an ideal potentiometer and thermometer. We find that inelastic scattering reduces the conductor's thermopower and, more unexpectedly, generates a magnetic-field asymmetry in the Seebeck coefficient. The latter effect is shown to be a higher-order effect…
We investigate the properties of complex mesoscopic superconducting-normal hybrid devices, Andreev-Interferometers in the case, where the current is proped through a superconducting tunneling contact whereas the proximity effect is generated by a transparent SN-interface. We show within the quasiclassical Green's functions technique, how the fundamental SNIS-element of the such structures can be mapped onto an effective S$^\prime$IS-junction, where S$^\prime$ is the proximised material with an effective energy gap $E_g<Δ$. We propose the Andreev-Interferometer, where $E_g$ can be tuned by an e…
Peaks associated to multiple Andreev reflections have been used to identify the number of conduction channels, and their transmitivity in mesoscopic superconducting junctions. We analyze the influence of the detailed shape of each channel, and the number of weak links within it, on the final results. Connexions with the statistical interpretation of conductance histograms is also made.
We report on the measurement of multiple Andreev resonances at atomic size point contacts between two superconducting nanostructures of Pb under magnetic fields higher than the bulk critical field, where superconductivity is restricted to a mesoscopic region near the contact. The small number of conduction channels in this type of contacts permits a quantitative comparison with theory through the whole field range. We discuss in detail the physical properties of our structure, in which the normal bulk electrodes induce a proximity effect into the mesoscopic superconducting part.
We study the Andreev reflection of a hybrid mesoscopic ring in the presence of a crown-like magnetic texture. By calculating the linear-response conductance as a function of the Zeeman splitting and the magnetic flux through the ring, we are able to identify signatures of the Berry phase acquired by the electrons during transport. This is proposed as a novel detection scheme of the spin-related Berry phase, having the advantage of a larger signal contrast and robustness against ensemble averaging.
We report our investigation of the sample to sample fluctuation in transport properties of phase coherent normal metal-superconductor hybrid systems. Extensive numerical simulations were carried out for quasi-one dimensional and two dimensional systems in both square lattice (Fermi electron) as well as honeycomb lattice (Dirac electron). Our results show that when the Fermi energy is within the superconducting energy gap $Δ$, the Andreev conductance fluctuation exhibits a universal value (UCF) which is approximately two times larger than that in the normal systems. According to the random matr…
We investigate ac Josephson effect in the resonant tunneling through mesoscopic superconducting junctions. In the presence of microwave irradiation, we show that the trajectory of multiple Andreev reflections can be closed by emitting or absorbing photons. Consequently, photon-assisted Andreev states are formed and play the role of carrying supercurrent. On the Shapiro steps, dc component appears when the resonant level is near a series of positions with spacing of half of the microwave frequency. Analytical result is derived in the limit of infinite superconducting gap, based on which new fea…
We derive a general scattering-matrix formula for the pumped current through a mesoscopic region attached to a normal and a superconducting lead. As applications of this result we calculate the current pumped through (i) a pump in a wire, (ii) a quantum dot in the Coulomb blockade regime, and (iii) a ballistic double-barrier junction, all coupled to a superconducting lead. Andreev reflection is shown to enhance the pumped current by up to a factor of 4 in case of equal coupling to the leads. We find that this enhancement can still be further increased for slightly asymmetric coupling.
We study the subgap transport properties of a small capacitance normal metal-superconductor tunnel junction coupled to an external electromagnetic environment. Mesoscopic interference between the electrons in the normal metal strongly enhances the subgap conductance with decreasing bias voltage. On the other hand, quantum fluctuations of the environment destroy electronic phase coherence and suppress the subgap conductance at low bias (Coulomb blockade).The competition between charging effects and mesoscopic interference leads to a non-monotonic dependence of the differential subgap conductanc…
We present a model of the thermopower in a mesoscopic tunnel junction between two ferromagnetic metals based upon magnon-assisted tunneling processes. In our model, the thermopower is generated in the course of thermal equilibration between two baths of magnons, mediated by electrons. We predict a particularly large thermopower effect in the case of a junction between two half-metallic ferromagnets with antiparallel polarizations, $S_{AP} \sim - (k_B/e)$, in contrast to $S_{P} \approx 0$ for a parallel configuration.
We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding insight into the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magneti…
Motivated by recent experiments with proximitized nanowires, we study a mesoscopic s-wave superconductor connected via point contacts to normal-state leads. We demonstrate that at energies below the charging energy the system is described by the two-channel Kondo model, which can be brought to the quantum critical regime by varying the gate potential and conductances of the contacts.
We develop a theoretical proposal for the charge Kondo effect in mesoscopic normal-superconductor-normal heterostructures, where the superconducting gap exceeds the electrostatic charging energy. Charge-Kondo correlations in these devices alter the conventional temperature-dependence of Andreev reflection and electron cotunneling. We predict typical Kondo temperatures of $\gtrsim 10 {\rm mK}$, and suggest experimental architectures that combine superconducting charge-qubits with semiconducting nanowires at cryogenic temperatures.
The transport through a metal-superconductor interface is governed by a special charge conversion process, the Andreev reflection, where each incident electron drags another electron with itself to form a Cooper pair. At the normal side a hole is left behind dressed by superconducting correlations. For a low transparency interface the simultaneous transfer of two charges is strongly suppressed leading to a reduced conductance. Here we demonstrate that this reduced conductance can be turned to an infinite one by tuning the nanoscale geometry. Creating variable size nanojunctions between a thin …
We study a Cooper pair transistor realized by a mesoscopic superconductor island that couples to a pair of $s$-wave superconducting leads. For a trivial island, the critical supercurrent between the leads exhibits a well-known $2e$-periodicity in the island-gate charge. Here, we show that for an island with spatially separated zero-energy Majorana or Andreev bound states the periodicity of the magnitude of the critical supercurrent transitions to $1e$ in the island-gate charge. Moreover, for Andreev bound states the current-phase relation displays a sign reversal when the parity of the charge …
We study the quantum interference (QI) effects in three-terminal Andreev interferometers based on polyaromatic hydrocarbons (PAH's) under non-equilibrium conditions. The Andreev interferometer consists of a PAH coupled to two superconducting and one normal conducting terminals. We calculate the current measured in the normal lead as well as the current between the superconducting terminals under non-equilibrium conditions. We show that both the QI arising in the PAH cores and the bias voltage applied to a normal contact have a fundamental effect on the charge distribution associated with the A…
We experimentally study the influence of 1-40 GHz radiation on the resistance of normal (N) mesoscopic conductors coupled to superconducting (S) loops (Andreev interferometers). At low RF amplitudes we observe the usual h/2e superconducting-phase-periodic resistance oscillations as a function of applied magnetic flux. We find that the oscillations acquire a pi-shift with increasing RF amplitude, and consistent with this result the resistance at fixed phase is an oscillating function of the RF amplitude. The results are explained qualitatively as a consequence of two processes. The first is the…
We experimentally study the influence of 1-40 GHz radiation on the resistance of normal (N) mesoscopic conductors coupled to superconducting (S) loops (Andreev interferometers). At low RF amplitudes we observe the usual h/2e superconducting-phase-periodic resistance oscillations as a function of applied magnetic flux. We find that the oscillations acquire a pi-shift with increasing RF amplitude, and consistent with this result the resistance at fixed phase is an oscillating function of the RF amplitude. The results are explained qualitatively as a consequence of two processes. The first is the…
The controlled growth of nanowires (NWs) with dimensions comparable to the Fermi wavelengths of the charge carriers allows fundamental investigations of quantum confinement phenomena. Here, we present studies of proximity-induced superconductivity in undoped Ge/Si core/shell NW heterostructures contacted by superconducting leads. By using a top gate electrode to modulate the carrier density in the NW, the critical supercurrent can be tuned from zero to greater than 100 nA. Furthermore, discrete sub-bands form in the NW due to confinement in the radial direction, which results in stepwise incre…
Mesoscopic superconductivity deals with various quasiparticle excitation modes, only one of them -- the charge-mode -- being directly accessible for conductance measurements due to the imbalance in populations of quasi-electron and quasihole excitation branches. Other modes carrying heat or even spin, valley etc. currents populate the branches equally and are charge-neutral, which makes them much harder to control. This noticeable gap in the experimental studies of mesoscopic non-equilibrium superconductivity can be filled by going beyond the conventional DC transport measurements and exploiti…
We explore charge imbalance in mesoscopic normal-metal/superconductor multiterminal structures at very low temperatures. The investigated samples, fabricated by e-beam lithography and shadow evaporation, consist of a superconducting aluminum bar with several copper wires forming tunnel contacts at different distances from each other. We have measured in detail the local and non-local conductance of these structures as a function of the applied bias voltage V, the applied magnetic field B, the temperature T and the contact distance d. From these data the charge-imbalance relaxation length lambd…
