What happens to spin-polarized electrons when they enter a superconductor? Superconductors at equilibrium and at finite temperature contain both paired particles (of opposite spin) in the condensate phase as well as unpaired, spin-randomized quasiparticles. Injecting spin-polarized electrons into a superconductor (and removing pairs) thus creates both spin and charge imbalances 1-7 , which must relax when the injection stops, but not necessarily over the same time (or length) scale. These different relaxation times can be probed by creating a dynamic equilibrium between continuous injection and relaxation; this leads to constant-in-time spin and charge imbalances, which scale with their respective relaxation times and with the injection current. Whereas charge imbalances in superconductors have been studied in great detail both theoretically 8 and experimentally 9 , spin imbalances have not received much experimental attention 6,10,11 despite intriguing theoretical predictions of spin-charge separation effects 12,13 . Here we present evidence for an almost-chargeless spin imbalance in a mesoscopic superconductor.A pure spin imbalance in a superconductor can be understood in the following manner: imagine injecting spin-randomized electrons continuously into a small superconducting volume and taking out Cooper pairs. The number of electron-like quasiparticles increases, that is, their chemical potential µ QP rises whereas that of the Cooper pairs µ P drops by the same amount to conserve particle number. This charge imbalance was first observed in a pioneering experiment, where µ QP − µ P was measured 1,2,14 . (Hereafter µ P ≡ 0, that is, all chemical potentials are measured with respect to that of the condensate.) If the injected electrons are (or become) spin-polarized, in general µ QP↑ = µ QP↓ = µ P , we can define a charge imbalance µ C ≡ (µ QP↑ + µ QP↓ )/2 and spin imbalance µ S ≡ (µ QP↑ − µ QP↓ )/2 (ref. 13). If charge relaxes faster than spin, a situation may arise in which µ C = 0 while µ S = 0. This is our chargeless spin imbalance. (See Supplementary Information for more details.) In the experiment, µ QP↑ − µ P and µ QP↓ − µ P are measured as a voltage drop between a spin-sensitive electrode and the superconductor.We implement a mesoscopic version of an experiment proposed in refs 12,13; this offers two practical advantages: the detector can be placed within a spin relaxation length λ S of the injection point and all out-of-equilibrium signals are enhanced by the small injection volume. In diffusive transport, λ S = (Dτ S2 ) 1/2 , where τ S2 is the spin relaxation time and D the diffusion constant (∼5 × 10 −3 m 2 s −1 in our samples 15 ). Our samples are FISIF lateral spin valves 16 , where the F are ferromagnets (Co), the I are insulators (Al 2 O 3 ) and S is the superconductor (Al), as shown in Fig. 1a. The SIF junctions have sheet resistances of ∼1.6 × 10 −6 cm 2 (corresponding to a barrier transparency T ∼ 5 × 10 −5 ) and tunnelling is the main transport mechanism through the insulator. By sweeping an...
We study an analytical model of a Rashba nanowire that is partially covered by and coupled to a thin superconducting layer, where the uncovered region of the nanowire forms a quantum dot. We find that, even if there is no topological superconducting phase possible, there is a trivial Andreev bound state that becomes pinned exponentially close to zero energy as a function of magnetic field strength when the length of the quantum dot is tuned with respect to its spin-orbit length such that a resonance condition of Fabry-Perot type is satisfied. In this case, we find that the Andreev bound state remains pinned near zero energy for Zeeman energies that exceed the characteristic spacing between Andreev bound state levels but that are smaller than the spin-orbit energy of the quantum dot. Importantly, as the pinning of the Andreev bound state depends only on properties of the quantum dot, we conclude that this behavior is unrelated to topological superconductivity.To support our analytical model, we also perform a numerical simulation of a hybrid system while explicitly incorporating a thin superconducting layer, showing that all qualitative features of our analytical model are also present in the numerical results. arXiv:1810.09840v2 [cond-mat.mes-hall]
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