Main

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 μQP rises whereas that of the Cooper pairs μP drops by the same amount to conserve particle number. This charge imbalance was first observed in a pioneering experiment, where μQP−μP was measured1,2,14. (Hereafter μP≡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 μQP↑≠μQP↓≠μP, we can define a charge imbalance μC≡(μQP↑+μQP↓)/2 and spin imbalance μS≡(μQP↑−μQP↓)/2 (ref. 13). If charge relaxes faster than spin, a situation may arise in which μC = 0 while μS≠0. This is our chargeless spin imbalance. (See Supplementary Information for more details.) In the experiment, μQP↑−μP and μQP↓−μP 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 λS of the injection point and all out-of-equilibrium signals are enhanced by the small injection volume. In diffusive transport, λS = (D τS2)1/2, where τS2 is the spin relaxation time and D the diffusion constant (∼5×10−3 m2 s−1 in our samples15). Our samples are FISIF lateral spin valves16, where the F are ferromagnets (Co), the I are insulators (Al2O3) and S is the superconductor (Al), as shown in Fig. 1a. The SIF junctions have sheet resistances of ∼1.6×10−6 Ω cm2 (corresponding to a barrier transparency T∼5×10−5) and tunnelling is the main transport mechanism through the insulator. By sweeping an external magnetic field parallel to the ferromagnetic electrodes, F1 and F2, we can align or anti-align their magnetizations because of their different magnetic shape anisotropies (Fig. 1c). We simultaneously perform local and non-local transport measurements using standard lock-in techniques at low temperature (70 mK–4 K): we apply a current across the junction J1, between F1 and S, so that spin-polarized electrons are injected into the superconductor, and we measure the voltage drops and differential resistances across J1 (‘local’, between F1 and S) and across J2 (‘non-local’, between F2 and S; Fig. 1b). The distance between J1 and J2 varies between 200 nm and 500 nm, within the Al spin relaxation length16. The non-local voltage drop at J2 is proportional to either μQP↑−μP or μQP↓−μP, depending on the relative alignments of F1, F2 and the magnetic field. (See ref. 17 and Supplementary Information.)

Figure 1: Device characterization and measurement set-up.
figure 1

a, Scanning electron micrograph of a typical device (8B3, scale bar, 1 μm). b, Schematic drawing of the same and of the measurement set-up. A current I is injected from a ferromagnet (F1, Co/Pd) into a superconductor (S, Al) through a tunnel barrier. The nonlocal voltage V NL and nonlocal differential resistance RNL = dV NL/dI are measured between a distant ferromagnetic electrode (F2, Co/Pd) and S as a function of the magnetic field (applied parallel to F1 and F2) and temperature; this probes the chemical potential of the spin up or down electrons with respect to the Cooper pairs, depending on the relative orientations of F1, F2 and the magnetic field. The local voltage and local differential resistance R = dV/dI (between S and F1) are measured simultaneously. The schematic drawing also illustrates a spin imbalance which survives longer in time and over a longer distance than the associated charge imbalance. c, Nonlocal magnetoresistance measurements at 4 K (where the aluminium is in its normal state) allow us to identify the relative alignments of F1 and F2 (Inset). The dependence of the magnetoresistance signal on device length (distance between F1 and F2) yields a spin flip length of 450±50 nm and a spin relaxation time τsf = 48±10 ps. Data from devices represented by black triangles have been normalized to account for a larger Al width (300 nm instead of 200 nm, see equations in main text).

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We first measure the nonlocal magneto-resistance at 4 K, with the aluminium in its normal (non-superconducting) state, to identify the switching fields of the ferromagnets (Fig. 1c). The amplitude of the nonlocal magneto-resistance signal as a function of the distance between the ferromagnetic electrodes (in different samples) also allows us to extract the spin relaxation length in the normal state of Al, assuming an exponential spatial decay (Fig. 1c, inset) as expected in diffusive metallic spin valves16. This yields λS = 450±50 nm, τS2 = 40±10 ps and a spin polarization of PCo∼10%. All these results are consistent with previous experiments16,18. In particular, the low Co polarization often observed in highly transparent planar tunnel junctions such as ours19 is due to the barrier strength dependence of the relative contributions of the s and d bands to the tunnelling current20.

At the base temperature of our dilution refrigerator (70 mK), at which the Al is superconducting (we measure TC∼1.23 K), we first study our device with the ferromagnets aligned with each other and with the magnetic field. Figure 2a shows the non-local differential resistance across J2 as a function of the voltage across J1. Between 0 and 1,500 G, we notice a double-peak structure (that is, a dip then a peak), which intensifies with increasing magnetic field, on a relatively field-independent smooth background.

Figure 2: Spin imbalance.
figure 2

(Device 8B3) a, Differential nonlocal resistance as a function of local voltage at different magnetic fields from −1,418 G (red) to 0 G (blue). Anti-symmetric peaks due to spin imbalance are seen on a field-independent symmetric background due to charge imbalance (Inset). Peak height as a function of magnetic field (from anti-symmetrized data, Fig. 3). The straight line is a guide to the eye. b, A schematic representation of the theoretical model showing densities of states and distribution functions of various populations in F1 and S. Owing to both the polarization of the ferromagnet and the Zeeman-split density of states in the superconductor, there is net spin accumulation in the superconductor and a shift in the chemical potentials of spin-up and spin-down quasiparticles. We measure the chemical potential of the spin-down and spin-up quasiparticles with respect to the Cooper pair condensate chemical potential.

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We show below that this double-peak structure results from spin accumulation in S whereas the background results from charge imbalance. We treat spin and charge imbalances independently as, to a first approximation, they are not coupled.

We first discuss spin imbalance. In F1, spin-up and spin-down electrons have densities of states (DOS) n↑ and n↓, constant within the energy range of interest. (The polarization is P = (n↑−n↓)/(n↑+n↓) and the total DOS NF = n↑+n↓.) In S, as has been observed in tunnelling spectroscopy experiments21,22, an external in-plane magnetic field splits the quasiparticle Bardeen–Cooper–Schrieffer (BCS) DOS through the Zeeman effect: nQP↓(↑)(E) = (E±μBH)/((E±μBH)2−Δ2)1/2. In our experiment, this BCS DOS is also smoothed out through orbital pair-breaking, which provides a depairing mechanism for the Cooper pairs23,24. (See Supplementary Information for details on the causes and effects of pair-breaking in our experiment; we use the measured DOS in our calculations and fits to data.) At equilibrium, the occupation of all states is described by the Fermi–Dirac distribution function f(E) = 1/(eβE+1), where β = 1/kBT with kB Boltzmann’s constant and T the temperature.

As discussed above, applying a current I (and voltage V) between F1 and S gives rise to a finite spin accumulation in S and shifts the chemical potentials of spin-up and spin-down quasiparticles in the superconductor so that fQP↓(↑)(E) = f(E±μS) (Fig. 2b).

Assuming μS≪eV, and using Fermi’s golden rule13, we obtain for the spin current:

Here NN is the normal aluminium DOS at the Fermi level, M the tunnelling matrix element (assumed to be constant in E) and A a constant. The total current I = I↑+I↓ is given by a similar expression.

The voltage drop detected at J2 due to the spin imbalance is refs 1, 17:

where Pd is the detector polarization, τS the spin relaxation time, e the electron charge, Ω the injection volume (the volume of the Al strip beneath the injection electrode) and gNS the normalized detection junction conductance. We measure the spin differential resistance, RS(V) = dμS/dI (Fig. 2a).

Assuming further that kBT, μBH≪V, we can expand in μBH and replace the Fermi–Dirac functions with step functions to obtain

where Pi is the injector polarization and nQP(E) is the quasiparticle density of states at zero field. (This helpful approximation is not made in the theoretical fits presented here. It makes little difference to the numerical results; see Supplementary Information.)

This expression is particularly suggestive: the spin imbalance can be seen here to depend clearly on the polarization of the injector electrode (first term)5,25 and on the Zeeman splitting of the DOS in the superconductor (second term)26,27. Therefore, in the presence of a magnetic field the injection electrode does not need to be polarized to create a spin imbalance; in principle a non-magnetic electrode would also work. From equation (1), we expect RS(V) to have a constant component (first term) and a component anti-symmetric in V which grows linearly with magnetic field (second term). This is precisely what we observe in Fig. 2a, on a parabolic background.

To extract the spin signal, we note that RS(V) is proportional to Pd. In other words, if Pd changes sign then RS(V) should do the same, whereas non-spin signals should remain unchanged. Therefore, in Fig. 3a, we measure RS(V) with the detector oriented first one way (blue) then the other (red) at the same magnetic field. (A slight difference in amplitude is due to residual magnetic fields; see Fig. 3b and Supplementary Information.) Indeed, part of the signal changes sign whereas the parabolic background remains constant. We note that the sign-reversing part of the signal is essentially odd in V, with no constant offset (the red and blue curves cross at zero); this means that our spin signal comes primarily from the Zeeman-induced term above.

Figure 3: Spin versus charge imbalance.
figure 3

(Device 8B3) a, Differential nonlocal resistance as a function of local voltage at 496 G with the detector electrode aligned (blue line) then anti-aligned (red line) with the injector electrode and the magnetic field. The spin imbalance signal changes sign whereas the charge imbalance signal remains the same. The difference in amplitudes between the two spin signals is due to a residual magnetic field. (See Supplementary Information.) b, The sum and difference between the two traces (divided by two), giving approximately the charge and spin signals respectively. Note that the sum is almost identical to a trace taken at zero applied field. The effect of the residual field can be seen here. c, The anti-symmetric part of the trace at 1,418 G from Fig. 2a, due primarily to spin imbalance. The blue line is a fit to our theory, yielding a spin relaxation time of about 25 ns. d, The symmetric part of the trace at 1,418 G from Fig. 2a, due primarily to charge imbalance.

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We can understand the dominance of the Zeeman-induced spin imbalance over the polarization-induced imbalance if they relax via different mechanisms—respectively elastic and inelastic—and thus over different timescales. In this case, we can write:

with τS1≫τS2 at low temperature. The fact that the normal state magnetoresistance (Fig. 1c) is much smaller than the low-temperature Zeeman-induced spin signal is consistent with this interpretation and suggests a relatively temperature-independent τS2.

We now turn to the charge imbalance signal, which we can distinguish from the spin imbalance signal through symmetry considerations: RC(V) = dμC/dI is an even function of V (and of H), whereas the Zeeman-induced RS(V) is odd in V (and in H). (We have seen that the polarization-induced RS(V), even in V and H, is negligible.) This means that, for RNL(V) = RC(V)+RS(V), the (anti-)symmetric component corresponds to the (spin) charge imbalance signal. Figure 3c,d show these signals separately for the B = 1,418 G trace in Fig. 2a.

We see here that the spin signal is maximal at a voltage at which the charge signal is negligibly small, and that conversely the charge imbalance becomes non-negligible at higher voltages where there is no spin imbalance. Spin and charge signals are thus well separated in energy.

To obtain the spin relaxation time τS1, we fit our theory to the spin signal (Fig. 3c, blue line) with τS1 as the only free parameter. This yields τS1∼25 ns. (The DOS used in the fit is that measured across J1.) Furthermore, the magnetic-field dependence of the spin relaxation time (Fig. 4c) τS1∼sinh(g μBH/kBTeff)/(g μBH/kBTeff) for fields less than about half the critical field confirms that spin relaxation occurs primarily through inelastic scattering processes28, which are ‘frozen out’ by the magnetic field. (Fig. 4b)

Figure 4: High magnetic fields and temperature depedence.
figure 4

(Device 15A4) a, Differential nonlocal resistance as a function of local voltage and magnetic field up to the critical magnetic field and beyond. The disappearance of superconductivity at a critical field of about 6 kG can be clearly observed. b, A horizontal slice of a at 430 μV. The green line is a fit to theory, following ref. 31, yielding a charge relaxation time of 3±1 ps. c, Estimated spin flip times obtained for fits to data in a. The solid line is a fit to τS1∼sinh(g μBH/kBTeff)/(g μBH/kBTeff) with Teff = 300 mK. This could be interpreted as a higher effective temperature for the quasiparticles. d,e, Temperature dependence of the spin imbalance signal. In d, data are shown similar to those in Fig. 3a (here anti-symmetrized) at different temperatures, and in e, data are similar to those in Fig. 3c (legend in e also applies to d). All traces, except for the one at 70 mK, are normalized by the relative detector gNS (with respect to 70 mK; refs 1, 2). Solid lines are theoretical fits.

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To compare τS1 to the charge relaxation time τQ, we measure the charge imbalance signal at high bias voltage (V = 430 μV) as the magnetic field is increased. As pointed out in ref. 29, there is a one-to-one correspondence between (field-induced) depairing and charge relaxation. Therefore, RC initially decreases owing to the field-induced pair-breaking 30, then diverges at the critical field HC∼6 kG as the superconducting gap goes to zero, before dropping abruptly to zero in the normal state31. A theoretical fit (Fig. 4b, green line, see Supplementary Section SB for details) yields a charge relaxation time of τQ = 3±1 ps≪τS1.

Figure 4a shows the nonlocal resistance as a function of local voltage over a larger range of magnetic fields and summarizes our main results: the asymmetric (red and blue) spin signal grows with magnetic field then diminishes and becomes narrower in V as the superconducting gap decreases. In the background, the charge signal decreases, then diverges with magnetic field. Both disappear at the critical field.

We also investigate the temperature evolution of the spin imbalance signal at B = 296 G for both aligned and anti-aligned states (Fig. 4d) and at B = 1,418 G (Fig. 4e). The magnitude of the spin signal diminishes with increasing temperature, owing primarily to temperature broadening of all distribution functions in the system. Theoretical fits to the data in Fig. 4e tell us that τS1 decreases with increasing temperature: τS1 = 14.2, 14.1, 12.3, 7.9 ns at T = 70, 200, 400 and 600 mK respectively32. This temperature dependence is consistent with theoretical predictions of spin-lattice relaxation times of conduction electrons in the superconducting state33.

After submitting our manuscript we became aware of similar work, described in ref. 34.

Methods

We fabricated our samples with standard electron-beam lithography and angle evaporation techniques in an electron-beam evaporator with a base pressure of 5×10−9 mbar. We first evaporate 20 nm of Al, which is then oxidized at 10−2 mbar for 10 min to produce a tunnel barrier, then 50 nm of Co and finally 20 nm of Pd as a capping layer. The Pd capping layer reduces the overall device resistance; prevents oxidation of the Co; and smooths out magnetic textures in the Co. All transport measurements were done in a 3He–4He dilution refrigerator with a base temperature of 70 mK. The a.c. excitation current was modulated at 37 Hz; its amplitude was 1 μA at 4 K and 10 nA at all other temperatures. Sample fabrication and measurement circuit details are provided in Supplementary Sections SC and SD.