SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION DOI: /NPHYS2518 Spin imbalance and spin-charge separation in a mesoscopic superconductor C. H. L. Quay, D. Chevallier, C. Bena 1, and M. Aprili 2 Laboratoire de Physique des Solides (CNRS UMR 8502), Bâtiment 510, Université Paris- Sud, Orsay, France. 1 Also at: Institute de Physique Théorique, CEA Saclay Gif-sur-Yvette, France. 2 Also at: Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA A. The Idea of a Pure Spin Imbalance The pure spin imbalance is a generalisation of the pure charge imbalance, which was first explicated and observed by Michael Tinkham and John Clarke in the 1970s [1, 2, 3]. Let us begin by considering a conventional (Bardeen-Cooper-Schrieffer, BCS) superconductor at equilibrium and at zero temperature. (Figures S1a and b.) The ground state is a condensate of paired electrons of opposite spin (Cooper pairs) and the excitations of the system are quasiparticles with energy E Δ, the superconducting gap: : E = (ξ 2 + Δ 2 ) 1/2 with ξ the quasiparticle energy in the normal (Fermi liquid) state. (Figure S1a) The quasiparticle charge varies smoothly from e to +e as ξ varies from to + ; it is zero when E = 0 [3]. Quasiparticles in superconductors can be thus be thought of as being electron-like or hole-like, with hole-like quasiparticles being the absence of electron-like ones. The quasiparticle (BCS) density of states n QP (E) = E/(E 2 Δ 2 ) 1/2 is shown in Figure S1b, with a small phenomenological depairing factor added, i.e. E E + iγ [4] (to make the function more easily plottable) 1. In the absence of non-equilibrium processes, the quasiparticle states are thermally populated and their occupation is described by the Fermi-Dirac function f(e) = 1/(e βe +1) where β = 1/k B T with k B Boltzmann s constant and T the temperature. (Figure 1b) Thus, at zero temperature, there are no hole- or electron-like quasiparticle excitations. Now imagine raising the temperature, but still keeping it much lower than Δ. (Figures S1c and d.) Some quasiparticles appear, as many of them electron-like as hole-like. This is because of the symmetry of the Fermi-Dirac function and the fact that Cooper pairs and quasiparticles have the same chemical potential. Thus, there is no charge imbalance here. (These remarks remain valid if quasiparticles due to pair-breaking rather than thermal excitation.) 1 One must be careful in moving between Figure S1a (the left column of Figure S1) and Figure S1b (the left column of Figure S1): one represents the excitation spectrum of the system while the other represents the available quasiparticle states. In the ground state, all electron-like quasiparticle states up to the Fermi energy are filled (and all the others are empty or, equivalently, filled with holes). This is our vacuum state with no quasiparticle excitations. NATURE PHYSICS 1

2 a c e b d f Figure S1 Idea of a pure spin imbalance. a, b, BCS superconductor at zero temperature. (a) shows the excitation (quasiparticle) spectrum while (b) shows the density of the electron-like quasiparticle states and how they are occupied (their distribution function) at zero temperature. Here E is the quasiparticle energy in the superconducting state and ξ the same in the normal (Fermi liquid) state. There are no quasiparticle excitations at zero temperature. c, d, Superconductor at finite temperature. Some quasiparticle excitations now appear, in equal numbers for electron- and hole-like ones due to the symmetry of the Fermi-Dirac function. There is no charge imbalance. e, f, Charge imbalance with no spin imbalance. Electron- and hole-like quasiparticles exist in unequal numbers but both branches are spin-randomised. The chemical potentials of the quasiparticles and of the Cooper pairs are no longer the same. This dynamic equilibrium or non-equilibrium situation was first observed by J. Clarke [1, 2]. g, h, Charge and spin imbalance. Spin up quasiparticles are in the situation shown in (e) and (f) while spin down quasiparticle states (not shown in (g) for clarity) are thermally populated as in (c) and (d). Thus spin up and down quasiparticles no longer have the same chemical potential. i, j, Spin imbalance with no charge imbalance. Spin up and down chemical potentials are shifted symmetrically about the pair chemical potential so that there is a pure spin imbalance. g h i j

3 In order to have charge imbalance, one must have more electron-like than hole-like quasiparticles (or the reverse). (Figure S1e.) This corresponds to a shift in the chemical potential of the quasiparticles µ QP with respect to that of the pairs µ P. (Figure S1f.) It is this difference µ QP µ P that was measured by John Clarke [1]. (Note that this phenomenon cannot exist in the normal state due to the absence of the condensate.) Note that charge imbalance is a non-equilibrium situation which will relax if not maintained in dynamic equilibrium e.g. by continuous injection of hot quasiparticles, as was the case in the Clarke s work. Note also that a charge imbalance is possible only because of the condensate degree of freedom in superconductors [3, 5, 6], i.e. the fact that neither the number of quasiparticles nor the total quasiparticle charge is conserved. The total number of particles is of course conserved and if we were to dot our i s and cross our t s for Figure S1f we should say that µ QP and µ P shift in opposite directions. In most experiments, including ours, all chemical potentials are measured with respect to µ P which can thus be set to zero. (Cf. Ref. [3].) Up to this point we have implicitly assumed that all the quasiparticles are spin-randomised. Naïvely, one might imagine creating a spin imbalance by starting with Figure S1e and making all the quasiparticles in the figure spin up. (Figure S1g.) (The spin down quasiparticle states would still be thermally populated (as in Figures S1c and d) but are not shown here in order to avoid cluttering the figure.) This would indeed create a spin imbalance: the chemical potentials of the spin up and down quasiparticles are no longer the same, µ QP µ QP. (Figure S1h.) There would, however, also be a charge imbalance: there are still more electron-like than hole-like quasiparticles in Figure S1g. Quantitatively, we can define µ C = (µ QP + µ QP )/2 and µ S = (µ QP µ QP )/2 as measures of the charge and spin imbalances respectively. In order to have a pure spin imbalance (µ C = 0 and µ S 0), we could create a spin down population which is the mirror image (across the vertical axis) of the spin up population in Figure S1g. This situation is shown in Figure S1i. Here it is clear that there is no charge imbalance there are as many electron-like quasiparticles as there are hole-like ones; yet there is a spin imbalance the spin up and spin down potentials are not the same, they are shifted symmetrically about zero. (Figure S1j.) These chemical potentials are what we detect in our experiment. We have done this gedanken experiment with perhaps the minimum number of elements necessary to imagine a pure spin imbalance. Our experiment differs from the one presented here in several ways: First, the quasiparticle density of states is much more depaired than the one shown in Figure S1. (See Figure S7b.) Second, we apply an in-plane magnetic field, which splits the spin up and down quasiparticle densities of states by the Zeeman energy (which is small compared to Δ in our case). Third, the question of how a pure spin imbalance might

4 be created or detected has not been addressed in this reflection. None of these things changes the fundamental nature or ontology of the spin imbalance, even if they do have some technical implications. These are addressed below or in the main text. B. Theoretical Model, Details The theoretical model which has been used to understand the experimental data is the following. The superconductor is described using a BCS Hamiltonian with the electron energy with respect to the chemical potential and the superconducting gap. The tunnelling Hamiltonian, which is responsible for transfer of electrons between the ferromagnetic lead and the superconducting lead can be written as where is the creation operator of an electron in the ferromagnetic lead while corresponds to the creation of an electron in the superconductor. The transmission coefficient for a particle of spin is (we assume that ). The tunnelling Hamiltonian can be rewritten in terms of the quasiparticle operators using the usual Bogoliubov transformation: and, where and are the superconducting coherence factors, and. The quasiparticle energy in the superconductor is given by [7]. Here and throughout the remainder of this section, we work with physical dimensions corresponding to. Using Fermi s golden rule and considering all possible tunnel processes [6, 8], we can calculate the spin and charge currents between the ferromagnet and the superconductor. These currents have been calculated also in Ref. [6] for zero applied Zeeman magnetic field, here we generalize this formalism to include these effects. Thus, the spin up/down electron currents can be written as

5 with, the Fermi-Dirac distribution, and the superconducting density of states for electrons with spin, [9]. The total electron current is given by, while the total spin current is. With respect to the variables used in the main text,, and. Similarly, the spin up and down quasiparticle charge current contributions can be written as where is the excess charge carried by a quasiparticle, given by the difference between the coherence factors and. The total quasiparticle current is. Note that in the above expressions we have neglected the changes in the quasiparticle Fermi functions due to the spin and charge accumulation as the relevant chemical potential shifts are much smaller than the applied voltage difference. The above expressions for, and allow us to calculate = dµ S /di and = dµ C /di that are necessary for fitting the dependence of the measured charge and spin signal on the applied voltage. (As mentioned in the main text, in all of our fits for n QP (E) we ise the quasiparticle density of states measured at the injection electrode, i.e. across the junction J1. The quasiparticle density of states is proportional to the local differential conductance, which is always measured at the same time as the nonlocal signal. Moreover, the coherence factors can also be obtained from the measured zero-magnetic-field density of states of the SC.) Following Ref. [2], we have µ S = S*P d /(2N N Ωeg NS ) and S* = I S τ S /e where N N is the normal aluminium DOS at the Fermi level, P d the detector polarisation (we use 10% as determined from 4K measurements, cf. main text), τ S the spin relaxation time, e the electron charge, Ω the injection volume and g NS the normalised detection junction conductance. Figure S2 Comparison between the experiment and two models Data from Figure 2a of the main text are compared to results using the linear approximation from the main text and the more exact model explicated here. Results are obtained numerically using the measured density of states. The fit is relatively good for the spin signal (see Figures 3c, 4a, and 4c of the main text), and yields a spin relaxation time of the order of nanoseconds, depending on the device.

6 In Figure S2, we plot a measured (from Figure 2a of the main text) and fits using the two different models presented in the main text (linear approximation) and above (exact model). One can see here that the linear approximation is justified in our case. On the other hand, we cannot fit the charge signal using the above model for the charge accumulation. We believe that this is because our model does not take into account several other processes like the crossed Andreev reflection, elastic cotunnelling, and dynamical Coulomb blockade which together with the charge accumulation have been shown to play an important role in describing the measured non-local charge signal [10, 11]. However, in order to extract the charge relaxation time for our system, we can study the dependence of the non-local resistance with respect to the magnetic field. Indeed, the nonlocal resistance due to charge imbalance can be written as [10] where is the fraction of the current which is carried by the injected quasiparticles at a given bias, is the density of states of the SC in the normal state, the injection volume, the normalized zero-bias detector conductance [1, 2], the distance between the two ferromagnetic leads and with the electron diffusion constant which has been previously introduced in the main text. In this model, the charge relaxation time is given by [12, 13] where is the BCS gap parameter exhibiting the usual dependence on temperature and magnetic field, with the orbital pair breaking time and the inelastic scattering time. This model allows us to fit the data presented in Figure 4b of the main text and extract a charge relaxation time on the order of a few picoseconds at low magnetic field. goes to zero at high magnetic fields as the aluminium becomes normal. Our result for (80ps) is similar to that reported in Ref. [10]. Finally, we note that while the expression for above is widely used in the literature, Ref. [14] proposes an alternative, phenomenological expression based on their data where. We also did our fit using this expression and find that it does not significantly change the result for the charge relaxation time.,

7 C. Sample Fabrication, Details a b Figure S3 a, Top view of electron beam lithography pattern written to make our evaporation mask. Shaded regions are where there is no resist after development. b, Coordinate system used to describe evaporation angles in the text. We use a bilayer PMMA 495K/PMMA 950K resist, developed in 1:3 MIBK:IPA in order to have little or no undercut. Our evaporation masks are patterned with standard electron beam lithography. The central region of our masks (which is that relevant to our device) is shown in a to-scale drawing in Figure S3a. The vertical bar is 200nm or 300nm wide, while the horizontal bars are 200nm and 400nm wide. The vertical distance between the horizontal bars is nm. All of the metal deposition is done in a single pump-down. First, we evaporate 20nm of aluminium at θ = -90, φ = 55 at a pressure of mbar. In the central region, this produces only a vertical bar of aluminium. We then oxidise the aluminium for ten minutes at 50mbar. Next, we evaporate 50nm of cobalt at θ = 180, φ = 55 and at a pressure of mbar. In the central region, this produces only horizontal bars of cobalt. The narrower cobalt bar is a little shorter than the corresponding feature in the mask. Finally, we evaporate 20nm of palladium at the same angle and at a pressure of mbar. As mentioned in the main text, our junctions have sheet resistances of ~1.6x 10-6 Ωcm 2. The barrier transparency T, can be estimated following the 1D model of Ref. [15], which yields: where R is the junction resistance, k F the aluminium Fermi vector, T the barrier transparency, d the barrier thickness and A the surface area of the junction. Assuming a barrier thickness of about 1nm (the other quantities are known) we obtain a barrier transparency T ~

8 D. Detailed Measurement Circuit Diagram Figure S4 Detailed diagram of the measurement circuit used in the experiment. Figure S4 shows our measurement circuit in greater detail than was presented in the main text. All amplifiers have input impedances of 100MΩ. All π-filters at low temperature have cutoff frequencies of 1MHz while those at room temperature have cutoff frequencies of 2MHz. The two lockin measurements are synchronized at 37Hz, the AC excitation frequency. Voltages are measured at all four detectors: AC/DC and (non)local. The input impedances of the detection instruments should lead to an offset on the nonlocal signal of the order of 10mΩ; this is negligible compared to the amplitude of the signal we are interested in which is on the order of several Ω. E. Domain Wall Motion and Depairing in the Density of States In Figure S5a we plot the local magnetoresistance of a device at 70mK (well below the critical temperature of aluminium) measured at its wide electrode, i.e. the differential resistance at zero bias current and voltage of the junction between the aluminium and the wide cobalt electrode. Regions of magnetic field where the electrodes are anti-parallel were determined from 4K measurements, where the aluminium was normal, as described in the main text. (Figure 1) As is shown here in Figure S5a, we often notice dramatic increases in the linear resistance of the junction just before the onset of the anti-parallel state (shaded regions, colour-coded according to magnetic field sweep direction). These increased linear resistances correspond to superconducting densities of states that are less depaired, which is to say that the coherence peaks are sharper. This can be seen in measurements of the local differential conductance of the junction (which is proportional to the quasiparticle density of states in the aluminium) at -250G and - 354G on a downward field sweep. (Figure S5b) The trace at -250G (blue dots) can be seen to be much less depaired (smoothed out) than the one at -354 (red line), which is a fairly typical DOS for the low field (up to ±1500G) measurements reported in the main text. We

9 are able to fit the sharper trace (blue dots) reasonably well to a BCS density of states with a phenomenological (Dynes) depairing parameter: Re[(E + iγ)/((e + iγ) 2 Δ 2 ) 1/2 ] with Δ = 120µV and Γ = 6.9µV. The fit is shown plotted as a black line in the figure. Figure S5 Domain wall movement and depairing (Device 8C3) a, local magnetoresistance at 70mK measured at the wide electrode. The red trace was measured as the field was swept from negative to positive values and the blue trace in the other direction. The electrodes are anti-parallel in the red (blue) shaded region during the upfield (downfield) sweep. The anti-parallel state is identified from 4K data such as those in Figure 1c of the main text. b, Local differential conductance (proportional to the density of states) as a function of bias voltage at 70mK, measured at the wide electrode, at -250G (blue dots) and -354G (red line) both on the downward field sweep. (Device 8A4) The black line is a fit to the data at -250G of a BCS density of states with a phenomenological (Dynes) depairing parameter. (See text.) c, local magnetoresistance measured at both electrodes at 70mK. Both show spikes, but at different fields. As in (a), the electrodes are anti-parallel in the shaded regions. We offer a possible explanation for the difference between the red and blue traces in Figure S5b: The quasiparticle density of states in the aluminium is smoothed out mainly by the orbital effects of a background out-of-plane magnetic field [9, 16]. As the external magnetic field is swept, the magnetisation of the cobalt electrodes changes direction or switches through the movement of magnetic domain walls. Just as or just before this happens, the domain walls are close to the junction and their fringing fields can partially cancel out the background magnetic field and restore a less depaired density of states. In the data shown above, this occurs close to the onset of the anti-parallel state because the wide electrode switches first and it is at the junction between this electrode and the aluminium that the measurements were performed. We can see that there is no particular effect when the nar-

10 row electrode switches. We expect the inverse to be true if the local magnetoresistance is measured at the other electrode. To test this, we measured the local magnetoresistance of another device (8A4) at both electrodes at low temperature. (Figure S5c) We observe the same dramatic increase in linear resistance for both electrodes, but at different fields: at the onset of the anti-parallel state for the wide electrode and at the re-entrance of the parallel state for the narrow electrode. This is consistent with our proposed explanation. These results imply that the aluminium quasiparticle density of states can have a gradient across the device (in the space between the electrodes) especially when the electrodes are switching, but also in stable parallel or anti-parallel configurations. In all our calculations we use the density of states measured at the injector electrode, but it should be borne in mind that this is an approximation; a full calculation must take into account the spatial variation of the density of states. F. Background Magnetic Fields In Figure 2 of the main text, it can be seen that the maximum spin signal ( peak height ) is more or less linear with magnetic field. Our theory predicts such a linear dependence as long as the density of states is not significantly modified by the magnetic field. As the Zeemaninduced spin imbalance is dominant in our system, the sign of the signal will depend primarily on the relative orientations of the magnetic field and the detector electrode. In our data (Figure 2), however, the peak height does not seem to extrapolate exactly to zero. We postulate the presence of a small background magnetic field. (See Figure 3 of main text and accompanying legend.) Figure S6 Background magnetic fields (Device 15A4) a, Evolution of the positive voltage maximum spin imbalance signal (cf. Figure 2 of main text) as magnetic field is swept from 1418G to -1418G and back up again. Peak heights were found by fitting parabolas to the tips of peaks such as those in Figure 2 of the main text. b, The same data, with portions (2) and (3) reflected about the horizontal axis. Without hysteresis, all points should fall on the black line.

11 To explore this idea, we track the height of the positive bias voltage peak in Device 15A4 (from data such as those in Figure 2) as the magnetic field is swept 1418G to -1418G then back up again. (Figure S6a) We note that the sign of the signal changes twice: first in passing through zero at a small negative field (~100G) and more abruptly around -400G. We interpret the first sign change as happening at the point at which the effective in-plane field felt by the sample changes sign (it is zero at this point), which means a residual magnetic field of about 100G. The second sign change occurs when the detector electrode switchs, so that it is now pointing in the same direction as the magnetic field. In the sweep from -1418G upwards, we observe the first sign change, but now at ~100G; there appears to be some hysteresis in the system. To see this more clearly, we reflect the second and third portions of the trace about the horizontal axis. (Figure S6b) The hysteresis can be clearly seen in this figure. (Without hysteresis, all points would fall on a straight, diagonal line.) In particular we note that close to -400G, the effective field is quite different for the two detector orientations. This is the origin of the different heights of the spin signals in Figures 3a and 4a. G. Superconducting Density of States as a Function of Applied Magnetic Field Figure S7a shows the density of states of Device 15A4 (measured at the wide electrode) as a function of magnetic field. The critical field, at which the density of states becomes almost flat, can be seen to be about 6kG. Two slices of this figure, at zero field and at a magnetic field greater than the critical field, are plotted in Figure S7b. At 7kG the aluminium is in its normal state and the density of states should be flat; however, a dip can be observed close to zero. This is due to dynamical Coulomb blockade. Figure S7 Density of states as a function of magnetic field. (Device 15A4) a, 3D colour plot of density of states as a function of magnetic field. The colour scale is msiemens. b, Densities of states at zero and high (>H c ) field. In the normal state, a dip in conductance can be seen close to zero bias voltage; this is due to dynamical Coulomb blockade.

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