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

Understanding the flow of spins in magnetic layered structures has enabled an increase in data storage density in hard drives over the past decade of more than two orders of magnitude [1]. Following this remarkable success, the field of 'spintronics' or spin-based electronics [1,2,3] is moving beyond effects based on local spin polarisation and is turning its attention to spin-orbit interaction (SOI) effects, which hold promise for the production, detection and manipulation of spin currents, allowing coherent transmission of information within a device [1,2]. While SOIinduced spin transport effects have been observed in two-and three-dimensional samples, these have been subtle and elusive, often detected only indirectly in electrical transport or else with more sophisticated techniques [4,5,6,7,8,9]. Here we present the first observation of a predicted 'spinorbit gap' in a one-dimensional sample, where counter-propagating spins, constituting a spin current, are accompanied by a clear signal in the easily-measured linear conductance of the system [10,11].We first introduce the class of phenomena we dub 'the one-dimensional spin-orbit gap' using a simple example adapted from Ref. [10], then describe our experiment in detail, and finally present a more elaborate model which captures most of the features seen in our data.The spin-orbit interaction is a relativistic effect where a charged particle moving in an electric field experiences an effective magnetic field which couples to its spin [12]. In semiconductor heterostructures, the electric field can arise as a result of either the lack of an inversion centre in the crystal (bulk inversion asymmetry, BIA) or a lack of symmetry in an external confining potential (structural inversion asymmetry, SIA) due to crystal interfaces or additional structures such as metallic gates [13]. The strength of the resulting effective magnetic field is proportional to both the particle's momentum and the original electric field.Consider a spin-degenerate one-dimensional subband with a Hamiltonian H 0 = 2 k 2 2m , where is Planck's constant, k the particle's momentum and m its mass ( Figure 1a). The leading order SO contribution to the Hamiltonian iswhere − → σ is the particle's spin, V the electrostatic potential and β a material-dependent parameter [14]. This term breaks the spin-degeneracy of the system and results in two spinful subbands separated by a lateral (wave vector) shift as shown in Figure 1b.Despite this rather striking change in the band structure, measurements of conductance through the system cannot distinguish the situation shown in Figure 1b from the case where the spins are degenerate. In both cases the edges of the two spin subbands occur at the same energy, so in both cases the conductance rises by G 0 = 2e 2 /h when the Fermi level of the system is tuned through this energy (e.g. by applying a voltage to a nearby gate) [15,16,17]. (Figure 1d.) To detect the SO shift in a transport measurement, a different approach is needed.Note that the spins as shown in Figure ...

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...

Tunnel junctions, an established platform for high resolution spectroscopy of superconductors, require defect-free insulating barriers; however, oxides, the most common barrier, can only grow on a limited selection of materials. We show that van der Waals tunnel barriers, fabricated by exfoliation and transfer of layered semiconductors, sustain stable currents with strong suppression of sub-gap tunneling. This allows us to measure the spectra of bulk (20 nm) and ultrathin (3- and 4-layer) NbSe2 devices at 70 mK. These exhibit two distinct superconducting gaps, the larger of which decreases monotonically with thickness and critical temperature. The spectra are analyzed using a two-band model incorporating depairing. In the bulk, the smaller gap exhibits strong depairing in in-plane magnetic fields, consistent with high out-of-plane Fermi velocity. In the few-layer devices, the large gap exhibits negligible depairing, consistent with out-of-plane spin locking due to Ising spin–orbit coupling. In the 3-layer device, the large gap persists beyond the Pauli limit.

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We have performed device-based tunnelling spectroscopy of NbSe 2 in the vortex state with a magnetic field applied both parallel and perpendicular to the a − b plane. Our devices consist of layered semiconductors placed on top of exfoliated NbSe 2 using the van der Waals transfer technique. At zero field, the spectrum exhibits a hard gap, and the quasiparticle peak is split into low and high energy features. The two features, associated with the effective two-band nature of superconductivity in NbSe 2 , exhibit markedly distinct responses to the application of magnetic field, suggesting an order-of-magnitude difference in the spatial extent of the vortex cores of the two bands. At energies below the superconducting gap, the hard gap gives way to vortex-bound Caroli-de Gennes-Matricon states, allowing the detection of individual vortices as they enter and exit the junction. Analysis of the sub-gap spectra upon application of parallel magnetic field allows us to track the process of vortex surface formation and spatial rearrangement in the bulk.

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