Frequency-domain measurement of the spin-imbalance lifetime in superconductors

We have measured the lifetime of spin imbalances in the quasiparticle population of a superconductor $\left({\tau }_s\right)$ in the frequency domain. A time-dependent spin imbalance is created by injecting spin-polarized electrons at finite excitation frequencies into a thin-film mesoscopic superconductor (Al) in an in-plane magnetic field (in the Pauli limit). The time-averaged value of the spin-imbalance signal as a function of excitation frequency $f_{\text{rf}}$ shows a cutoff at $f_{\text{rf}}\approx 1/\left(2\pi {\tau }_s\right)$. The spin-imbalance lifetime is relatively constant in the accessible ranges of temperatures, with perhaps a slight increase with increasing magnetic field. Taking into account sample thickness effects, ${\tau }_s$ is consistent with previous measurements and of the order of the electron-electron scattering time ${\tau }_{ee}$. Our data are qualitatively well described by a theoretical model taking into account all quasiparticle tunneling processes from a normal metal into a superconductor.

Figure 1

(a) Scanning electron micrograph of a typical device (scale $\text{bar}=1\mu \mathrm{m}$) and schematic drawing of the measurement setup. $\text{S}=\text{superconductor}$ ($\sim 8.5$ nm thick Al film with a native oxide), $\text{N}=\text{normal}$ metal (100 nm Al), $\text{F}=\text{ferromagnet}$ (40 nm Co, with a 4.5 nm Al capping layer). The native oxide on S constitutes a tunnel barrier between it and any other given electrode. Quasiparticles are injected into S across a tunnel barrier by applying a voltage $V_{\text{dc}}$ across J1 or J2. These are spin polarized because of the Zeeman field in S. The nonlocal voltage $V_{\text{nl}}$ and differential nonlocal signal $d{V}_{\text{nl}}/d{V}_{\text{dc}}$ are measured between F and S (at J3) as a function of magnetic field and temperature, as well as a function of the amplitude $V_{\text{rf}}$ and frequency $f_{\text{rf}}=\omega /2\pi$ of high-frequency (1–50 MHz) voltages applied to the injection electrode. The local conductance $dI/d{V}_{\text{dc}}$ is measured simultaneously at the injection electrode. (b) The conductance $dI/d{V}_{\text{dc}}$ across J2, which is proportional to the quasiparticle density of states in the superconductor, as a function of $V_{\text{dc}}$ at different magnetic fields. (c) The nonlocal voltage $d{V}_{\text{nl}}$ measured at J1 as a function of $V_{\text{dc}}$ at the same fields. (d) The nonlocal voltage $d{V}_{\text{nl}}$ measured at J3 as a function of $V_{\text{dc}}$ at the same fields. (e) Theoretical fit to one of the traces in (b), which yields an estimate of ${\tau }_s$ at $H=680$ mT of 2.3 ns. We also obtained 1.2, 1.6, and 2.6 ns for 425, 510, and 936 mT, respectively. The error is 10%–20% based on the fits and could be larger if uncertainties in the spin-resolved DOS are considered.

Figure 2

(a) Measured local conductance $dI/d{V}_{\text{dc}}$ across J2 as a function of $V_{\text{rf}}$ at $f_{\text{rf}}=1$ MHz and $H=680$ mT. (b) Measured differential nonlocal signal $d{V}_{\text{nl}}/d{V}_{\text{dc}}$ at J3 as a function of $V_{\text{rf}}$ at $f_{\text{rf}}=1$ MHz and $H=680$ mT. Classical rectification is the dominant rf effect. The $V_{\text{rf}}$ given here is the value at the output of the generator. As noted in the main text, $V_{\text{rf}}$ at the device can be estimated from the classical rectification of features in the $V_{\text{rf}}=0$ trace. (c), (d) Two slices of (a), (b) plotted as dotted lines together with numerical calculations (solid lines) based on the superconducting DOS extracted from the measured local conductance at $V_{\text{rf}}=0$. Small “mismatches” in the conductance can lead to large differences in the nonlocal signal; however, the theory qualitatively agrees with the data.

Figure 3

(a) Differential nonlocal signal $d{V}_{\text{nl}}/d{V}_{\text{dc}}$ measured at J3 with injection at J2 as a function of $V_{\text{dc}}$ with constant-power excitations (of $\sim 250 \mu \mathrm{V}$ at the device) at $f_{\text{rf}}=1$ and 50 MHz. (b) Numerical calculation of $d{V}_{\text{nl}}/d{V}_{\text{dc}}$ as a function of $V_{\text{dc}}$ with constant-power excitations at $f_{\text{rf}}=0.02/2\pi {\tau }_s,1/2\pi {\tau }_s$, based on the superconducting DOS extracted from the measured local conductance at $V_{\text{rf}}=0$. (c), (d) $d{V}_{\text{nl}}/d{V}_{\text{dc}}$ at the $V_{\text{dc}}$ values indicated in (a) as a function of $f_{\text{rf}}$. We subtract “opposing” peaks to obtain the antisymmetric part of the signal, which is due to spin. Fits to numerical calculations yield ${\tau }_s=3.2$ and 6.4 ns with a fitting error of 10%–20%. Note that the cutoff does not occur exactly at $1/\left(2\pi {\tau }_s\right)$.

Figure 4

Differential nonlocal signal $d{V}_{\text{nl}}/d{V}_{\text{dc}}$ as a function of rf frequency at different fields for “inner” and “outer” peaks, both (a), (c) experimental data and (b), (d) theory. Both show a decrease in amplitude at high fields. Fits of theory to data suggest a slight rise of ${\tau }_s$ with magnetic field, as well as a difference between inner and outer peaks. (See text.)