A microscopic theory of the transport properties of quantum point contacts giving a unified description of the normal conductor-superconductor (N-S) and superconductor-superconductor (S-S) cases is presented. It is based on a model Hamiltonian describing charge transfer processes in the contact region and makes use of non-equilibrium Green function techniques for the calculation of the relevant quantities. It is explicitly shown that when calculations are performed up to infinite order in the coupling between the electrodes, the theory contains all known results predicted by the more usual scattering approach for N-S and S-S contacts. For the latter we introduce a specific formulation for dealing with the non-stationary transport properties. An efficient algorithm is developed for obtaining the dc and ac current components, which allows a detailed analysis of the different current-voltage characteristics for all range of parameters. We finally address the less understood small bias limit, for which some analytical results can be obtained within the present formalism. It is shown that four different physical regimes can be reached in this limit depending on the values of the inelastic scattering rate and the contact transmission. The behavior of the system in these regimes is discussed together with the conditions for their experimental observability.
We report on conductance measurements in carbon nanotube based double quantum dots connected to two normal electrodes and a central superconducting finger. By operating our devices as beam splitters, we provide evidence for Crossed Andreev Reflections tunable in situ. This opens an avenue to more sophisticated quantum optics-like experiments with spin entangled electrons.PACS numbers: 73.63.Fg Quantum optics has been an important source of inspiration for many recent experiments in nanoscale electric circuits [1,2]. One of the basic goals is the generation of entangled electronic states in solid state systems. Superconductors have been suggested as a natural source of spin entanglement, due to the singlet pairing state of Cooper pairs. One important building block required for the implementation of entanglement experiments using superconductors is a Cooper pair beam splitter which should split the singlet state into two different electronic orbitals [3,4].The basic mechanism for converting Cooper pairs into quasiparticles is the Andreev reflection in which an originally quantum coherent electron pair in the singlet spin state is produced at an interface between a superconductor and a normal conductor. Conventional Andreev Reflections (AR) are local and cannot readily be used to create bipartite states [5,6]. It has been suggested to make use of electron-electron interactions [6,7, 8,9,10,11,12,13], spin filtering [14] or anomalous scattering in graphene [15] to promote Cooper pair splitting i.e. the Crossed Andreev Reflection (CAR) process.In this letter, we show that Coulomb interactions as well as size quantization can favor the CAR processes in carbon nanotubes. We use a double quantum dot geometry where the nanotube is connected to two normal electrodes and a central superconducting finger. By operating our device as a beam splitter (i.e. biasing the central superconducting electrode), we find that there is a finite current flowing from the superconducting electrode to the left (L) arm and the right (R) arm for a bias voltage smaller than the energy gap of the superconductor, which demonstrates Cooper pair injection. This subgap current is enhanced when we tune the device to the degeneracy * To whom correspondence should be addressed: kontos@lpa.ens.fra. FIG. 1: a. SEM image of a typicalCooper pair splitter device in false colors with the two biasing schemes sketched. The bar is 1µm. A central superconducting electrode is connected to two quantum dots engineered in the same single wall carbon nanotube (in purple) which bridges between electrodes L and R. b. The elementary processes which carry current in the superconducting (S) state. In addition to the conventional local Andreev Reflection process, the Crossed Andreev Reflection can occur in which a Cooper pair is split in the two quantum dots. The relative probability of each of these processes can be inferred from the topology of the beam splitter.
Carbon nanotubes (CNTs) are not intrinsically superconducting but they can carry a supercurrent when connected to superconducting electrodes 1-4. This supercurrent is mainly transmitted by discrete entangled electron-hole states confined to the nanotube, called Andreev bound states (ABS). These states are a key concept in mesoscopic superconductivity as they provide a universal description of Josephson-like effects in quantum-coherent nanostructures (for example molecules, nanowires, magnetic or normal metallic layers) connected to superconducting leads 5. We report here the first tunnelling spectroscopy of individually resolved ABS, in a nanotubesuperconductor device. Analysing the evolution of the ABS spectrum with a gate voltage, we show that the ABS arise from the discrete electronic levels of the molecule and that they reveal detailed information about the energies of these levels, their relative spin orientation and the coupling to the leads. Such measurements hence constitute a powerful new spectroscopic technique capable of elucidating the electronic structure of CNT-based devices, including those with well-coupled leads. This is relevant for conventional applications (for example, superconducting or normal transistors, superconducting quantum interference devices 3 (SQUIDs)) and quantum information processing (for example, entangled electron pair generation 6,7 , ABS-based qubits 8). Finally, our device is a new type of d.c.measurable SQUID. First conceived of four decades ago 9 , ABS are electronic analogues of the resonant states in a Fabry-Pérot resonator. The cavity is here a nanostructure and its interfaces with superconducting leads play the role of the mirrors. Furthermore, these 'mirrors' behave similarly to optical phase-conjugate mirrors: because of the superconducting pairing, electrons in the nanostructure with energies below the superconducting gap are reflected as their time-reversed particle-a process known as Andreev reflection. As a result, the resonant standing waves-the ABS-are entangled pairs of timereversed electronic states, which have opposite spins (Fig. 1a); they form a set of discrete levels within the superconducting gap (Fig. 1b) and have fermionic character. Changing the superconducting phase difference ϕ between the leads is analogous to moving the mirrors and changes the energies E n (ϕ) of the ABS. In response, a populated ABS carries a supercurrent (2e/h)(∂E n (ϕ)/∂ϕ) through the device, whereas states in the continuous spectrum (outside the superconducting gap) have negligible or minor contributions in most common cases 5. Therefore, the finite set of ABS generically determines Josephson-like effects in such systems. As such, ABS
Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in:Advances in Physics 60.6 (2011): 899-958 DOI: http://dx.doi.org/10.1080/00018732.2011.624266 Copyright: © 2011 Taylor & FrancisEl acceso a la versión del editor puede requerir la suscripción del recurso Access to the published version may require subscription Josephson and Andreev transport through quantum dotsA. Martín-Rodero and A. Levy Yeyati Departamento de Física Teórica de la Materia Condensada C-05 Universidad Autónoma de Madrid, E-28049; Madrid, SpainIn this article we review the state of the art on the transport properties of quantum dot systems connected to superconducting and normal electrodes. The review is mainly focused on the theoretical achievements although a summary of the most relevant experimental results is also given. A large part of the discussion is devoted to the single level Anderson type models generalized to include superconductivity in the leads, which already contains most of the interesting physical phenomena. Particular attention is paid to the competition between pairing and Kondo correlations, the emergence of π-junction behavior, the interplay of Andreev and resonant tunneling, and the important role of Andreev bound states which characterized the spectral properties of most of these systems. We give technical details on the several different analytical and numerical methods which have been developed for describing these properties. We further discuss the recent theoretical efforts devoted to extend this analysis to more complex situations like multidot, multilevel or multiterminal configurations in which novel phenomena is expected to emerge. These include control of the localized spin states by a Josephson current and also the possibility of creating entangled electron pairs by means of non-local Andreev processes. Contents VII. Concluding remarks 36Acknowledgements 37References 37 I. INTRODUCTIONThe field of electronic transport in nanoscale devices is experiencing a fast evolution driven both by advances in fabrication techniques and by the interest in potential applications like spintronics or quantum information processing. Within this context quantum dot (QD) systems are playing a central role. These devices have several different physical realizations including semiconducting heterostructures, small metallic particles, carbon nanotubes or other molecules connected to metallic electrodes. In spite of this variety a very attractive feature of these devices is that they can usually be described theoretically by simple "universal-like" models characterized by a few parameters. In addition to their potential applications, these systems provide a unique test-bed for analyzing the interplay of electronic correlations and transport properties in nonequilibrium conditions. Electron transport in semiconducting QDs has been studied since the early 90's when phenomena like Coulomb blockade (CB) was first observed (Kastner, 1993). It soon became clear that QDs could all...
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