Quasiparticle spin resonance and coherence in superconducting aluminium

Conventional superconductors were long thought to be spin inert; however, there is now increasing interest in both (the manipulation of) the internal spin structure of the ground-state condensate, as well as recently observed long-lived, spin-polarized excitations (quasiparticles). We demonstrate spin resonance in the quasiparticle population of a mesoscopic superconductor (aluminium) using novel on-chip microwave detection techniques. The spin decoherence time obtained (∼100 ps), and its dependence on the sample thickness are consistent with Elliott–Yafet spin–orbit scattering as the main decoherence mechanism. The striking divergence between the spin coherence time and the previously measured spin imbalance relaxation time (∼10 ns) suggests that the latter is limited instead by inelastic processes. This work stakes out new ground for the nascent field of spin-based electronics with superconductors or superconducting spintronics.


Supplementary Note 1: Estimates of and in the superconducting Al films
The conductivity of a diffusive metal is related to the diffusion constant by , where is the charge of the electron and the density of states [1]. We obtain S·m -1 for Device A from the blue trace in Figure 1f of the main text, taking the relevant volume to be that of the S bar between the electrodes N1 and N2 from centre to centre. Using states per eV per cm 3 for aluminium, we obtain m 2 ·s -1 .
The penetration depth of the magnetic field into the superconductor is √ 315nm for Device A [2]. Here is the vacuum permittivity, Planck's constant and the zerotemperature superconducting gap. is much greater than the thickness of our Al film, ~8.5nm. If we assume that the mean free path is limited by , then in the Drude model the conductivity should simply scale as and Device B, in which ~6nm, should have 375nm. The orbital energy of electrons in diffusive thin films with a magnetic field applied in the plane of the film is whereas the Zeeman energy is , with the Landé g-factor and the Bohr magneton [3,4]. (Note that Ref. [3] uses cgs units and Ref. [4] is missing a factor of . The expressions given here are correct and in SI units.) At the highest resonant field (0.5T) measured in this work, we have ~0.32 for Device A and ~0.22 for Device B. We are thus always in the 'paramagnetic limit', where the Zeeman effect dominates over orbital depairing.

Supplementary Note 2: Choice of Frequencies
As in our previous work [5], Figures 1b and 1e of the main text are reproducible at any frequency modulo a constant shift in the axis, with being nominal microwave voltage (i.e. the voltage at the output of the generator). This constant shift is due to the frequencydependent attenuation of our microwave line (greater attenuation at higher frequencies) as well as resonances in the line. The conductance of a junction at zero bias is thus a measure of the microwave power arriving at the junction/device.
As noted in our main text, we define (for any given frequency) as the nominal at which the effective voltage at the device is the same as that for = 7.14GHz and = 16.81mV.
To select the frequencies at to search for the quasiparticle spin resonance we measure the conductance of a junction as a function of frequency at various and at ( Figure 2) As can be seen in Figure 1b of the main text, the conductance at has a monotonic dependence on the effective at the device and can be taken as an indication of the latter. The effect of the frequency-dependent transmission of our microwave line is quite apparent in these data.
For the measurements shown in the main text, we selected at which the conductance is at a locally maximal, corresponding also to local maxima in the real microwave voltage at the device as a function of . We do this to avoid any experimental missteps accidentally delivering more power to the device than required, thus possibly blowing it up. In addition, this avoids unnecessary dissipation of energy in the microwave line, which if excessive could lead to a rise in the base temperature of the dilution refrigerator. (We did not notice this in our measurements.)

Supplementary Note 3: Quasiparticle Spin Resonance, Dependence on Microwave Amplitude (Both Detection Schemes) and on (Detection Scheme 1)
As explicated in the main text, the 'operating point' of Detection Scheme 1 (DS1) is defined by the chosen values of and whereas the operating point of Detection Scheme 2 (DS2) is defined by the chosen value of . We show here that our results for resonant field and the resonance linewidth are robust against the choice of operating point in both detection schemes.
We first show that our results from both detections schemes are independent of . Supplementary Figure 3 shows the resonances from Figure 3b  Next, we show that our results from DS1 are independent of the choice of . In Supplementary Figure 4a, we show a measurement of conductance as a function of applied magnetic field at , = -288µV (black trace in 4a, black dot in 4b) together with the same measurement taken at , = -100µV (blue trace in 4a, blue dot in 4b). For the black trace we have = 340mT5mT and = 148mT25mT while for the blue trace we have = 340mT5mT and = 154mT25mT. The resonance appears as a peak in the blue trace and a dip in the black trace. This can in fact be understood by looking at Supplementary Figure 4b: As explicated in the main text, at the resonance, some of the microwave radiation is absorbed by the quasiparticle spins and so less is transmitted to the detectors. We can see from Supplementary Figure 4b that at = -288µV a smaller effective gives a higher conductance, whereas at = -100µV a smaller effective gives a lower conductance, hence the different in the sign of the resonance in the two traces. To optimise sensitivity for this detection scheme, and should be chosen so that is maximal.

Supplementary Note 4: Estimate of the Number of Quasiparticles
The switching current of the Al bar in the absence of microwaves and of quasiparticle injection is ~1800µA (Figure 1f of the main text): we remind the reader that, for the blue trace, current is injected along the length of the S bar. Detection Scheme 1 should be close to this 'equilibrium' situation as the voltages applied across the NIS junctions are of the order of the superconducting gap (at zero temperature).
In contrast, in Detection Scheme 2 (Figure 1f and Figure 3 of the main text), current is injected into the S bar across a tunnel junction and 'removed' via another such junction, e.g. as shown in Figure 1d of the main text. Here, the voltages across the NIS junctions (which typically have resistances of ~5kΩ) at the point where the S bar becomes normal are several mV and we expect the quasiparticles in the S bar to be driven strongly out-of-equilibrium by the injected current. Typical measured are around 500-600nA.
If we assume that the non-equilibrium quasiparticle population can be described by an effective temperature , and that scales with in the same way that does, then in DS2 , with being the critical temperature. Based on Figure 4 of Ref. [6], we can then say that the quasiparticle density in DS2 is at least two orders of magnitude higher than in DS1.