These calculations are similar to those for T 1 in NMR [ 51 ]. Calculations within the quasi-classical approximation pointed to an enhanced spin-flip scattering at the gap edge leading to a strong suppression of the spin imbalance at low temperature [ 52 ]. Early experiments on superconducting Al [ 53 ] and Nb [ 54 ] junctions of macroscopic dimensions area 1 mm 2 showed that spin accumulation produces an excess of quasi-particles.

This was interpreted as a longer recombination time due to a very long quasi-particle spin lifetime i. In other words, spin relaxation was proposed as the bottleneck for quasi-particle relaxation.

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The long spin relaxation time measured in light metals at low temperatures compared to that in heavy metals , and also indications of an increase of the spin lifetime in the superconducting state, suggest that the superconducting spin relaxation length could be of the order of micrometres, a length scale accessible to standard electron lithography. The spatial decay of spin imbalances can thus be probed directly with detector electrodes placed at different distances from the injection point. Spin-polarized electrons diffuse from the injection point or area and relax over a length x s.

In superconductors, as mentioned above, charge and spin imbalances can diffuse from the injection point and relax over different length scales.

## Citations: “Electric fields and collective oscillations in superconductors”

Spin imbalance in lateral spin valves depends on the distance x of the detector from the injection point. The spin resistance decays exponentially with x and is given by.

The system is assumed to be quasi-one-dimensional. Generalizing the Johnson technique to lateral spin valves, Jedema et al. They obtained characteristic decay lengths of nm at 4. This corresponds to spin imbalance relaxation times of ps and 45 ps, respectively, in good agreement with previous experiments.

A measurement of the spin resistance in a lateral spin valve and its x dependence is shown in figure 1 , with the Al in the normal state. From Quay et al. Inset The dependence of the magnetoresistance signal on device length distance between F1 and F2 follows the exponential decay expected from equation 3.

Online version in colour. Following Johnson [ 58 ], Jedema et al. P t is the probability that, once injected at the injector F1 cf. For a diffusive wire. Unfortunately, the Hanle effect cannot be measured in superconducting thin films, as superfluid screening the Meissner effect prevents perpendicular magnetic fields from uniformly penetrating the film.

In addition, non-uniform perpendicular magnetic fields strongly modify the density of states in the film, complicating data interpretation. Field penetration would be uniform in superconducting nanowires whose diameters are much smaller than the penetration depth, regardless of field orientation; however, we are not aware of spin injection experiments in these systems. Measurements on superconducting lateral spin valves also suffer from stray fields due to the magnetization of the ferromagnetic electrodes; this can strongly affect superconductivity in a spatially non-uniform manner.

As spin resistance is defined as the difference in the non-local signal between the parallel and anti-parallel detector—injector magnetization configurations, it is of the essence to be able to distinguish between out-of-equilibrium spin-dependent effects and equilibrium magnetic landscape effects. Designed to minimize stray field effects while switching between anti-parallel and parallel configurations, the experiment performed by Poli et al.

The spin relaxation length was observed to decrease in the superconducting state with respect to the normal state, consistent with spin relaxation dominated by spin-flip scattering from magnetic impurities cf. By contrast, a recent experiment by Wakamura et al. An external magnetic field can be used not only to control the magnetization orientation of ferromagnetic electrodes but also to change the quasi-particle excitation spectrum in the superconductor.

In thin films, whose thickness d is smaller than [ 64 ], orbital screening effects due to an in-plane magnetic field H can be neglected.

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This condition on d , known as the Pauli limit, comes from imposing that the diamagnetic energy be smaller that the Zeeman energy. Building on the concept introduced by Huertas-Hernando et al. Two experiments have followed this idea [ 57 , 69 ]. Introducing a Zeeman-split BCS density of states in equations 2. This expression is particularly suggestive: the spin imbalance can be seen here to depend clearly on the polarization of the injector electrode first term as well as on the Zeeman splitting of the density of states in the superconductor second term.

Physically this corresponds to different relaxation mechanisms for spin imbalances in the presence or absence of a Zeeman-induced splitting of the BCS density of states. The shape of the signal measured in the experiment figure 2 and its amplitude as a function of the applied magnetic field in the limit of small field agree well with equation 4. Anti-symmetric peaks due to spin imbalance are seen on a field-independent symmetric background due to charge imbalance.

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Inset Peak height as a function of magnetic field from anti-symmetrized data. The straight line is a guide to the eye. The blue line is a fit to our theory, yielding a spin imbalance relaxation time of about 25 ns. The red line is the symmetric part of the trace at G a , due primarily to charge imbalance.

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The spinimbalance signal changes sign while the charge imbalance signal remains the same. The difference in amplitudes between the two spin signals is due to a residual magnetic field. See electronic supplementary material of Quay et al. Note that, as spin polarization is provided mainly by the Zeeman splitting of the superconducting density of states, the injector electrode no longer needs to be a ferromagnet. This can be seen in figure 3 , where we present data from a device, similar to that shown in figure 1 , but in which the injector is normal. The thickness of the superconducting Al wire is reduced to 8.

While the spin accumulation first increases with increasing field figure 3 c , the charge imbalance monotonically vanishes with field as expected for increasing orbital depairing figure 3 d. Note also that the signals from spin and charge imbalance have different symmetries, even and odd, respectively, in agreement with equation 2.

The Zeeman splitting is visible in G L , which is proportional to the density of states in S. The next two panels show the non-local voltage V NL measured at a ferromagnetic detector c and at a normal metal detector d as a function of the local bias voltage V L across N2—S for different H.

Here the injector N2 is Al nm in the normal state. The amplitude of the signal from spin imbalance even signal initially grows with increasing field, saturates and then gradually vanishes at the critical field. The amplitude of the charge imbalance odd signal instead decreases monotonically with increasing field. Spin currents carried by quasi-particles can be converted into transverse charge currents by the inverse spin Hall effect [ 72 ], resulting in charge accumulation at the transverse edges of the superconductor [ 73 ].

## Relaxation of nonequilibrium quasiparticles in mesoscopic size superconductors.

The spin Hall effect originates from side jump and skew scattering induced by spin—orbit coupling [ 63 , 74 , 75 ]. Very recently an increase of three orders of magnitude in the non-local inverse spin Hall resistance of a NbN wire well below T c compared to the normal state was reported [ 76 ]. The inverse Hall signal was observed only when injector and detector electrodes were closer together than the charge relaxation length.

The experiments by Quay et al. In both cases, a finite voltage will be measured between the superconductor and a ferromagnetic detector with which it is in contact via a tunnel barrier. The latter scenario, which corresponds to a spin-dependent thermoelectric effect, is a result of the broken electron—hole symmetry of the density of states of each spin in the superconductor due to the Zeeman field.

Spin and charge imbalances. A more general description would be based on spin-dependent chemical potentials and distribution functions. A complete description of the non-equilibrium excitations requires the knowledge of the spinful distribution functions and , where the need not be Fermi—Dirac nor share the same chemical potential.

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Indeed, by solving Boltzmann diffusion equations, and taking into account local modification of superconductivity due to the out-of-equilibrium distribution functions, Krishtop et al. Silaev et al. The former relaxes via elastic spin-flip processes; the latter relaxes via inelastic processes electron—electron or electron—phonon scattering and is unaffected by elastic scattering. All of this recent theoretical work points to the electron—phonon relaxation time as an upper bound for the spin imbalance relaxation time.

This is a natural explanation for the long times measured by Quay et al. A reduced gap results in a diminished inelastic scattering rate and thus an increased lifetime. The upper limit for the spin relaxation time becomes the normal state energy relaxation time at that temperature. However, this characteristic time scale corresponds to the lifetime of the out-of-equilibrium magnetization in the superconductor rather than the spin-flip time of each quasi-particle. This is similar to T 1 in nuclear magnetic resonance terminology [ 51 ]. Spin resonance experiments allow a direct measurement of the spin coherence time, T 2 [ 83 , 84 ].

In a typical electron spin resonance ESR experiment, electrons are immersed in an external homogeneous static magnetic field, H.