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 of a double-loop Andreev interferometer where two Josephson currents in the normal metal wire may be controlled independently. The amplitude and symmetries of the observed thermopower oscillations may help to illuminate the unexplained dependence of oscillation symmetry on sample geometry.
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…
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 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…
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.
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 vi…
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…
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…
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…
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…
Superconductivity is characterized by a nonvanishing superconducting pair amplitude. It has a definite symmetry in spin, momentum and frequency (time). While the spin and momentum symmetry have been probed experimentally for different classes of superconductivity, the odd-frequency nature of certain superconducting correlations has not been demonstrated yet in a direct way. Here we propose the thermopower as an unambiguous way to assess odd-frequency superconductivity. This is possible since the thermoelectric coefficient given by Andreev-like processes is only finite in the presence of odd-fr…
I discuss thermoelectric properties of a quantum dot coupled to one normal and one superconducting lead in the presence of Kondo effect and Andreev scattering. I will focus on conductance, thermal conductance, thermopower and related quantities like thermoelectric figure of merit which is a direct measure of the usefulness of the system for applications and Wiedemann-Franz ratio which indicates if the system is in the Fermi liquid state. I will show that the superconductivity strongly modifies the thermal properties of the system. In particular, the thermopower is strongly enhanced near the su…
We study the electric and thermoelectric transport properties of correlated quantum dots coupled to two ferromagnetic leads and one superconducting electrode. Transport through such hybrid devices depends on the interplay of ferromagnetic-contact induced exchange field, superconducting proximity effect and correlations leading to the Kondo effect. We consider the limit of large superconducting gap. The system can be then modeled by an effective Hamiltonian with a particle-non-conserving term describing the creation and annihilation of Cooper pairs. By means of the full density-matrix numerical…
Nonlocal tunneling spectroscopy of multiterminal semiconductor-superconductor hybrid devices is a powerful tool to investigate the Andreev bound states below the parent superconducting gap. We examine how to exploit both microscopic and geometrical symmetries of the system to extract information on the normal and Andreev transmission probabilities from the multiterminal electric or thermoelectric differential conductance matrix under the assumption of an electrostatic potential landscape independent of the bias voltages, as well as the absence of leakage currents. These assumptions lead to sev…
A detailed investigation of the non-equilibrium steady-state electric and thermoelectric transport properties of a quantum dot coupled to the normal metallic and s-wave superconducting reservoirs (N-QD-S) are provided within the Coulomb blockade regime. Using non-equilibrium Keldysh Green's function formalism, initially, various model parameter dependence of thermoelectric transport properties are analysed within the linear response regime. It is observed that the single-particle tunnelling close to the superconducting gap edge can generate a relatively large thermopower and figure of merit. M…
Two-terminal conductance spectroscopy of superconducting devices is a common tool for probing Andreev and Majorana bound states. Here, we study theoretically a three-terminal setup, with two normal leads coupled to a grounded superconducting terminal. Using a single-electron scattering matrix, we derive the subgap conductance matrix for the normal leads and discuss its symmetries. In particular, we show that the local and the nonlocal elements of the conductance matrix have pairwise identical antisymmetric components. Moreover, we find that the nonlocal elements are directly related to the loc…
We discuss the nonlinear current of an interacting quantum dot coupled to normal and superconducting reservoirs with applied voltage and temperature differences. Due to the particle-hole symmetry introduced by the superconducting lead, the pure (subgap) thermoelectric response vanishes. However, we show that the Andreev bound states shift as the thermal gradient increases. As a consequence, the $I$--$V$ characteristic can be tuned with a temperature bias if the system is simultaneously voltage biased. This is a cross effect that occurs beyond linear response only. Furthermore, we emphasize the…
We present conductance-matrix measurements of a three-terminal superconductor-semiconductor hybrid device consisting of two normal leads and one superconducting lead. Using a symmetry decomposition of the conductance, we find that the antisymmetric components of pairs of local and nonlocal conductances match at energies below the superconducting gap, consistent with expectations based on a non-interacting scattering matrix approach. Further, the local charge character of Andreev bound states is extracted from the symmetry-decomposed conductance data and is found to be similar at both ends of t…
We study the spectrum of Andreev bound states and Josephson currents across a junction of $N$ superconducting wires which may have $s$- or $p$-wave pairing symmetries and develop a scattering matrix based formalism which allows us to address transport across such junctions. For $N \ge 3$, it is well known that Berry curvature terms contribute to the Josephson currents; we chart out situations where such terms can have relatively large effects. For a system of three $s$- or three $p$-wave superconductors, we provide analytic expressions for the Andreev bound state energies and study the Josephs…
