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### Out-of-equilibrium superconductors’ spinful excitation modes

The ground state of conventional (Bardeen–Cooper–Schrieffer) superconductors is composed of Cooper pairs of electrons in a spin singlet configuration. In equilibrium, this macroscopic quantum state can carry a dissipationless charge current (known as a supercurrent), but not spin or energy currents. In contrast, the single particle excitations, or QPs, are spin-1/2 fermions, which can carry spin, energy and charge currents. The density of states of these QPs (*ρ*(*E*)) is zero in an energy range ±Δ about the Fermi energy (*E*_{F}), and has coherence peaks just above this gap (Fig. 1a).

Out-of-equilibrium QP populations in superconductors can be described by the particle energy distribution function *f*(*E*). Neglecting the QP spin, *f*(*E*) can be decomposed based on symmetry into energy *f*_{L}(*E*) = *f*(−*E*) − *f*(*E*) and charge *f*_{T}(*E*) = 1 − *f*(*E*) − *f*(−*E*) modes^{20,21}. The simplest *f*(*E*) which excites these modes are, respectively, an effective temperature (different from the lattice temperature) and a charge imbalance. In the presence of a charge imbalance, the number of electron-like and hole-like QPs are non-identical, and the QP chemical potential is different from the Fermi energy. Extensive experimental and theoretical work has been done on both charge and energy modes (see ref. ^{22} and Chapter 11 of ref. ^{23}).

In the spinful case, the decomposition above can be generalised by the addition of spin and spin energy modes, *f*_{T3}(*E*) = [*f*_{T↑}(*E*) − *f*_{T↓}(*E*)]/2 and *f*_{L3}(*E*) = [*f*_{L↑}(*E*) − *f*_{L↓}(*E*)]/2^{15,16,18}. *f*_{L3} is most simply excited by a spin-dependent temperature and *f*_{T3} by a spin-dependent chemical potential. The spin and spin energy modes only exist if spin up and down QPs have different distribution functions, i.e. if *f*_{↑}(*E*) ≠ *f*_{↓}(*E*). By construction, *f*_{L} and *f*_{L3} are odd in energy, while *f*_{T} and *f*_{T3} are even in energy. In the following, we focus mainly on *f*_{L3}, the spin energy mode.

To generate different spin up and down distribution functions, it is necessary to preferentially excite QPs of one spin species. In thin superconducting films, this can be done by applying an in-plane magnetic field (*H*), which lowers (raises) the energy of spin down (up) QPs by the Zeeman energy (*E*_{Z}) and splits the DOS so that only spin down excitations (spin down electron-like and spin up hole-like QPs) are allowed in the energy range Δ − *E*_{Z} ≤ ∣*E*∣ ≤ Δ + *E*_{Z} (Fig. 1b) (*E*_{Z} = *μ*_{B}*H*, with *μ*_{B} the Bohr magneton). Current injection in this energy range thus creates spin-polarised QPs regardless of the magnetic properties of the tunnel barrier or the injector electrode.

For our experiments, we use thin-film superconducting (S) aluminium wires, with a native insulating (I) oxide layer, across which lie normal metal (N) and superconducting (S′) electrodes. The former is used as an injector and the latter as detectors (Fig. 1f). S is terminated on both sides by reservoirs at a distance of about 5 μm from the NIS junction. The magnetic field (*H*) is applied in the plane, parallel to S.

Our basic spectroscopy measurement consists of injecting a constant current (*I*_{inj}) at the injector (*J*_{inj}), and measuring the current (({I}_{det })) and/or the differential conductance (({G}_{det }={mathrm{{d}}}{I}_{det }/{mathrm{{d}}}{V}_{det })) as a function of the applied voltage (({V}_{det })) at one of the detectors (J_{det1}, J_{det2} and J_{det3} in Fig. 1f). Measurements were performed in a dilution refrigerator with a base temperature of 90 mK. J_{det1} lies within both an electron–electron interaction length (*λ*_{e–e} ≈ 1 μm^{24,25}) and a spin–flip length (*λ*_{sf} ≈ 300 nm^{26,27}) of the injector.

We model our system using the Keldysh–Usadel equations, which describe out-of-equilibrium diffusive superconductors (see Supplementary Methods 1.1.1 for details). Following refs. ^{16,18,28}, we solve these numerically in one dimension, assuming negligible (inelastic) electron–electron and electron–phonon interactions, and include a Zeeman magnetic field. Experimental parameters are used in the model: the normal state diffusion constant *D* ≈ 10 cm^{2} s^{−1}, *L* = 10 μm, *R*(*J*_{inj}) = 13 kΩ. The diffusion time from the injector to the reservoirs is ({tau }_{{rm{diff}}}={l}_{{rm{inj-res}}}^{2}/Dapprox 20,{rm{ns}}), where ({l}_{{rm{inj}}-{rm{res}}}) is the injector–reservoir distance ≈*L*/2. As *τ*_{diff} is much smaller than the QP recombination time (*τ*_{rec} ≳ 1 μs^{29}), QPs relax and recombine at the reservoirs. At the interface with the injector, the boundary conditions are given by spectral current continuity and the injector distribution function *f*_{inj}(*E* − *e**V*_{inj}), assumed to be Fermi–Dirac.

In our numerical results for the closest detector (Fig. 1c), we see that the QP distribution function bears signatures of both the density of states in S (Fig. 1b) as well as the distribution function in the injector: It has a peak at *E* = Δ and goes sharply to zero at *E* = *e**V*_{inj}, with *e* the electron charge. The distribution function is also spin-dependent.

To interpret our experimental results, it is also helpful to understand the link between the spin energy mode *f*_{L3} and charge imbalance by considering the non-equilibrium QP number as a function of energy:

$$N(E)= , {N}_{uparrow }(E)+{N}_{downarrow }(E)\ = ,frac{1}{2}[{f}_{uparrow }(E)-{f}_{0}(E)]{rho }_{uparrow }(E) +frac{1}{2}[{f}_{downarrow }(E)-{f}_{0}(E)]{rho }_{downarrow }(E)$$

(1)

$$,,=-!frac{1}{2}[{rho }_{-}{f}_{{rm{L3}}}+{rho }_{+}{f}_{{rm{T}}}]+frac{1}{2}[{rho }_{+}({f}_{{rm{L}}}^{0}-{f}_{{rm{L}}})-{rho }_{-}{f}_{{rm{T3}}}]$$

(2)

Here *ρ*_{↑}(*E*) and *ρ*_{↓}(*E*) are the DOS of spin up and spin down QPs, respectively; ({rho }_{+}(E)equiv frac{1}{2}[{rho }_{uparrow }(E)+{rho }_{downarrow }(E)]=rho (E)); ({rho }_{-}(E)equiv frac{1}{2}[{rho }_{uparrow }(E)-{rho }_{downarrow }(E)]); and *f*_{0}(*E*) and ({f}_{{rm{L}}}^{0}(E)) are, respectively, *f*(*E*) and *f*_{L}(*E*) at equilibrium.

In Eq. (2) we notice that the term *ρ*_{−}(*E*)*f*_{L3}(*E*) is even in energy, which means that the spin energy mode *f*_{L3} adds particles at both positive and negative energies, and raises the overall QP chemical potential, thus creating a charge imbalance. (The first (last) term in Eq. (2) is even (odd) in energy and creates a charge (energy) imbalance.) (Fig. 1b) In addition, the factor *ρ*_{−}(*E*) means that *f*_{L3} add particles in the energy range Δ − *E*_{Z} ≤ ∣*E*∣ ≤ Δ + *E*_{Z}, regardless of the injection voltage or other experimental parameters (Fig. 1a and b). *f*_{T} also creates a charge imbalance, which however appears at low magnetic fields and high energies (Supplementary Fig. 6 and Supplementary Discussion 2.5). Our spectroscopic technique allows us distinguish between *f*_{L3} and *f*_{T}, based on their different energy dependences. We refer the reader to refs. ^{16,18} and Supplementary Methods 1.2 for further theoretical details.

### Spectroscopic spin-sensitive QP detection

We first characterise both injector and detector junctions, and explain our spectroscopy technique. Figure 2a shows the differential conductance of the injector (*G*_{inj} = d*I*_{inj}/d*V*_{inj}) as a function of the applied voltage (*V*_{inj}) at different magnetic fields (*H*). At the temperatures of our experiment, *G*_{inj} is almost exactly proportional to the density of states in S^{23}. We can see that *H* induces Zeeman splitting of the QP density of states. *H* also couples to the orbital degree of freedom, inducing screening supercurrents and hence a rounding of the QP coherence peak due to orbital depairing^{23,30}. From fits to the data, we obtain an Abrikosov–Gor’kov orbital depairing parameter of *α* = *R*_{ORB}*H*^{2}, with *R*_{ORB} ≈ 6.5 μeV T^{−2}. The critical field of S is ≈2.7 T. In the results shown here, the Zeeman energy is always greater than the depairing parameter (see Supplementary Section 2.1 for details).

If the detector temperature is much smaller than the superconducting energy gap in S′ (({k}_{{rm{B}}}{T}_{det }ll {Delta }_{det }), with *k*_{B} Boltzmann’s constant), the differential conductance of SIS′ junctions as a function of the applied voltage in the subgap region (({V}_{det }, < , (Delta +{Delta }_{det })/e)) is given by

$${G}_{det }({V}_{det })=frac{1}{e{R}_{{rm{N}}}}int N(E)frac{partial {rho }_{det }(E+e{V}_{det })}{partial {V}_{det }}{mathrm{{d}}}E$$

(3)

where ({rho }_{det }(E)) the density of states in S′, *N*(*E*) the QP number from Eqs. (1) and (2), and *R*_{N} the normal state resistance of the detector junction.

Most of the integral in Eq. (3) comes from the coherence peak in ({rho }_{det }(E)) at (E={Delta }_{det }). This peak picks out *N*(*E*) the number of QPs in S, shifted by ({Delta }_{det }). In other words, ({G}_{det }({V}_{det }-{Delta }_{det }/e)) gives the number of QPs at energy (E=e{V}_{det }), while ({I}_{det }({V}_{det }-{Delta }_{det }/e)) gives the total number of QPs for (Ele e{V}_{det }). Our measurements thus give us spectroscopic information on the QPs (see Supplementary Methods 1.2.2 for details).

Charge imbalances in the QP population lead to an even-in-energy *N*(*E*). As (frac{partial {rho }_{det }(E+e{V}_{det })}{partial {V}_{det }}) is odd in *E*, and the convolution of an even and an odd function gives an odd function, we expect a charge imbalance to lead to a component of ({G}_{det }({V}_{det })) which is odd in ({V}_{det }).

At finite magnetic fields, these spectroscopic measurements become spin-sensitive if Zeeman spin-splitting occurs in S but not in S′; the unsplit coherence peak in S′ separately probes the number of excitations in S at the two gap edges for spins up and down, respectively, at ({V}_{det }^{uparrow (downarrow )}=| Delta pm {E}_{{rm{Z}}}-{Delta }_{det }| /e).

We suppress the spin-splitting in S′ through the strong spin–orbit coupling of sprinkled Pt, which acts as a spin-mixer (see “Methods” section, Supplementary Methods 1.2.3 and refs. ^{30,31,32,33}). Figure 2b shows ({G}_{det }({V}_{det })) at different *H* and *I*_{inj} = 0. At *H* = 1 T, we see two peaks, as expected for a non-spin-split detector. (Were there a Zeeman splitting in S′ equal to that in S, the situation would be equivalent to two spinless SIS′ junctions in parallel, one for each spin, and there would be a single peak in ({G}_{det }({V}_{det })) instead of two. The asymmetrical signal in Fig. 5 would remain in the data, but we would be unable to differentiate the contribution from the two spins and clearly identify *f*_{L3}—see Supplementary Fig. 16.) We note also that the detector current is typically 0.1–1 nA ≪ *I*_{inj} ~ 10–100 nA throughout the subgap region: the detector is close to equilibrium.

### Non-Fermi–Dirac QP energy distributions

Measurements at zero magnetic field already reveal non-Fermi–Dirac distributions. Figure 3a shows the current–voltage characteristics of the closest detector junction at two injection currents: 0 nA (black trace) and 120 nA (red trace). We focus on the low-voltage range (the ‘subgap region’) before the abrupt rise of ({I}_{det }) at ({V}_{det }=(Delta +{Delta }_{det })/e), where the opposite-energy coherence peaks of S and S’ align. We see that the red trace is higher than the black. This indicates the presence of additional QPs created by injection. (Such measurements of ‘excess QPs’ have been made in extended junctions, but because of the spatial averaging, the spectroscopic information was lost. For a review, see ref. ^{22}.)

This creation of QPs by current injection can also be seen in the differential conductance measurement (({G}_{det }({V}_{det }))) at three values of *I*_{inj}: 0, ≈13, and 120 nA (Fig. 3b). Here, we see more clearly that most of the QPs are at the gap edge ((e{V}_{det}=Delta – Delta_{det})). If we try to fit the trace at *I*_{inj} ≈ 13 nA with a thermal QP distribution, it is clear that this grossly over-estimates the number of QPs at high energies (Fig. 3b, dotted line). The QPs do not thermalise.

Instead, as shown in our calculations (Fig. 1) and discussed earlier, the QP states in S are filled up to *V*_{inj}: the electron distribution function in *N* is ‘imprinted’ onto the QPs in S. This can be seen by overlaying the *I*_{inj}(*V*_{inj}) measurement in Fig. 3a, shifted by ({Delta }_{det }/e), on a plot of ({G}_{det }) as a function of (({V}_{det })) and *I*_{inj} (Fig. 3c). We see that, at each current, the injector voltage falls exactly at the location of a step in ({G}_{det }) (seen here as a change in colour). The accumulation of QPs at the gap edge in S can also be seen on this colour scale as a yellow horizontal feature.

Our calculations reproduce both the step-like feature corresponding to ({I}_{{rm{inj}}}({V}_{{rm{inj}}}+{Delta }_{det }/e)), as well as the horizontal feature (Fig. 3d). Thus, at a distance of about 300 nm ≪ *λ*_{e−e} from the injector (i.e. at J_{det1}) and in the energy range of interest for the detection of the *f*_{L3} mode, the QPs have not yet thermalised, and it is reasonable to neglect electron–electron interactions.

### Spin energy mode

At finite magnetic fields, current injection at low energies becomes spin-polarised: we expect different distribution functions for spin up and down QPs, and in particular to excite the spin energy mode. We show in Fig. 4a calculations of ({G}_{det }) as a function of ({V}_{det }) (in the sub-gap region) and of *I*_{inj}, at 1 T where the density of states in S is well spin-split (Fig. 2a). Following features from low to high energies, we expect peaks in ({G}_{det }({V}_{det })) at (e{V}_{det }=(pm | Delta -{Delta }_{det }-{E}_{{rm{Z}}}| )) which we shall call *P*_{2} and *P*_{3}, corresponding to the coherence peaks of spin down excitations (spin down electron-like or spin up hole-like QPs). Peaks at ({V}_{det }=pm | Delta -{Delta }_{det }+{E}_{{rm{Z}}}|) (*P*_{1} and *P*_{4}), corresponding to the coherence peaks of spin up excitations, appear when *I*_{inj} is increased and spin up excitations are also injected.

Comparing this to the data (Fig. 4c), we see *P*_{2} and *P*_{3} clearly, but *P*_{1} and *P*_{4} are less prominent. This is due to the increased electron–electron interaction at high energies and QP number. (For clarity, the Josephson or supercurrent contribution has been subtracted from ({G}_{det }). See Supplementary Methods 1.2.2 for details.)

Next, we compare the number of electron-like and hole-like QPs by taking two slices of Fig. 4c at (e{V}_{det }=+| Delta -{Delta }_{det }-{E}_{{rm{Z}}}|) (Fig. 4d). The traces are not identical. The difference between them, which is the charge imbalance, is maximal at *I*_{inj} ≈ 8 nA, corresponding to maximal spin polarisation of the injection current, i.e. when the injection voltage is just below the coherence peak associated with spin down excitations. This charge imbalance is also reproduced in the calculation (Fig. 4b).

The charge imbalance associated with *f*_{L3} has particular energy and magnetic field signatures: it is expected to appear in the energy range Δ − *E*_{Z} ≤ ∣*E*∣ ≤ Δ + *E*_{Z}. In Fig. 5a, we plot the component of the data in Fig. 4a which is odd in ({V}_{det }), which gives the charge imbalance. The odd component is indeed largest in the expected energy range. As a function of magnetic field, the charge imbalance first becomes visible when *E*_{Z} > 3.5*k*_{B}*T*. It then continues to increase with magnetic field, as expected, then starts going down. The decrease is caused mainly by smearing of both injector and detector densities of states, due to orbital depairing. Our calculations reproduce the data at 1 T well (Fig. 5b, dash-dotted line).

The odd-in-({V}_{det }) component of the data in Fig. 4b, d, which comes from *f*_{L3}, is small compared to the even-in-({V}_{det }) component, which comes from either *f*_{L} or *f*_{T3}. The QPs from *f*_{L} or *f*_{T3} contribute to a finite magnetisation in the superconductor, previously detected by other methods^{16,27,34,35,36}. At *H* = 0, we recover the previously observed charge imbalance signal^{37,38,39,40,41}, associated with the *f*_{T} mode, which occurs at high energies and low magnetic fields (see Supplementary Discussion 2.5).

Beyond a spin–flip length from the injector, spin up and down QP distribution functions become identical, leading to the disappearance of *f*_{L3}. Indeed, we do not observe *f*_{L3} at J_{det2} or J_{det3} (see Supplementary Discussion 2.3).

Compared to normal metals and semiconductors, the spin energy mode in superconductors has the advantage of being excitable by using the spin-split DOS. Its association with an energy-localised charge imbalance make it easy to distinguish from other modes. Using superconductors as detectors allowed us to have spectroscopic information on the QPs, by using the coherence peak in the detector density of states. This work paves the way for new spin-dependent heat transport experiments, as well as the generation of spin supercurrents by out-of-equilibrium distribution functions in conventional superconductors^{18,42}.

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