Entangled Andreev pairs and collective excitations in nanoscale superconductors

Abstract

Nanoscale superconductors connected to normal metallic electrodes provide a potential source of entangled electron pairs 1-5 . Such states would arise from the splitting of Cooper pairs in the superconductor into two electrons with opposite spins, which then tunnel into different leads by means of a process known as crossed Andreev reflection (refs 6-8). In an actual system, the detection of these processes is hindered by the elastic transmission of individual electrons between the leads, which yields an opposite contribution to the non-local conductance. Here we demonstrate that low-energy collective excitations, which appear in superconducting structures of reduced dimensionality 9 , can have a significant influence on the transport properties of this type of hybrid nanostructure. When an electron tunnels into the superconductor it can excite such low-energy excitations that alter the balance between the different electronic processes, leading to a dominance of one over the other depending on the spatial symmetry of these excitations. These findings help to clarify some intriguing experimental results and provide future strategies for the detection of entangled electron pairs in solid-state devices for quantum computation.A generic set-up for the study of non-local transport through a superconductor is shown in Fig. 1a. It represents a superconducting region attached to three normal electrodes. Two of the leads (labelled 1 and 2 in Fig. 1a) are used to inject a current while the voltage drop is measured on the third one. The two basic microscopic processes contributing to the non-local conductance are illustrated in Fig. 1c,d. In the case of elastic cotunnelling (EC) processes, the injected electron tunnels elastically into the third lead, whereas in the case of crossed Andreev reflection (CAR) processes, it combines with an electron emerging from the third lead to form a Cooper pair in the superconductors. The probability of these processes decays exponentially on the scale of the superconducting coherence length, ξ, which can range between 10 and 100 nm for typical superconductors used in experiments 10,11 . On the other hand, the two processes yield opposite contributions to the non-local conductance (conventionally the CAR contribution is taken as positive) and, as demonstrated by previous theoretical studies 12-14 , tend to cancel each other in the case of Bardeen-Cooper-Schrieffer (BCS) superconductors weakly coupled to non-magnetic leads. Surprisingly, recent experiments by Russo et al. 11 have shown that even in this case the subgap non-local conductance can be appreciably large, exhibiting an intriguing behaviour in which either process can dominate depending on the energy of the injected electrons. This behaviour cannot be accounted for by the existing non-interacting theories.The importance of interactions in breaking the balance between EC and CAR processes can be understood by considering the case where the superconducting region is sufficiently small and can be characterized by a finite charging ...

Microscopic theory of Cooper pair beam splitters based on carbon nanotubes

“…The cancellation between the two contributions to the nonlocal conductance for thick tunnel barriers was shown to be removed by introducing ferromagnetic leads, 6 increasing the barrier transparency, 7 or taking into account Coulomb interactions. 8 The importance of nonequilibrium effects at large bias voltages has been also analyzed. 9 More recent experiments are oriented toward tunable double-quantum-dot systems based on either carbon nanotubes 3 or InAs nanowires.…”

Focused crossed Andreev reflection

“…In our scheme AR induces electron-hole correlations on length scales which are only limited by the mean free path l mf rather than ξ. For typical superconductors, ξ lies between 10 nm and 100 nm, [12] while l mf can reach several microns in 2DEGs [16] Very high mobilities have also been reported for graphene [17,18], which is another candidate for focused CAR. The tuning between CAR and ET through the external magnetic field is possible through the orbital degrees of freedom at field strengths that do not introduce a spin selectivity of the contacts.…”

“…The limiting length scale for electron focusing and therefore CAR enhancement, is the mean free path l mf , which can be several orders of magnitude larger than ξ [12,15]. CAR is enhanced at the cost of ET for magnetic fields that are integer multiples of the focusing field in Eq.…”

“…Theoretical papers report that ET typically dominates CAR in linear response [3,9]. CAR can dominate ET beyond linear response or in the presence of interactions [8,10,12,13].…”

Systematic stability analysis of the renormalization group flow for the normal-superconductor-normal junction of Luttinger liquid wires

“…The physical separation between the junctions of the twowires with the superconductor is of the order of the size of the Cooper-pair. This leads to the realization of a normal-superconductor-normal (NSN) junction which allows for direct tunneling of electrons from one wire to the other and also allows a finite amplitude for the crossed Andreev reflection (CAR) process 25,26,27,28,29,30,31,32,33,34,35 in addition to the normal reflection and Andreev reflection (AR) processes.In an earlier study of the NSN junction 23 , we showed that the NSN junction has more than two fixed points unlike the normal two-wire junction (as mentioned above) or the junction of LL with a bulk superconductor (NS junction) which has only two fixed points -(i) the Andreev fixed point where the amplitude for Andreev reflection (AR), r A = 1 and normal reflection amplitude, r = 0 and which is unstable and (ii) the disconnected fixed point where r A = 0 and r = 1, and which is stable 12,13 . We showed that there exists a fixed point with intermediate values of transmission and reflection.…”