Spin pumping between noncollinear ferromagnetic insulators through thin superconductors

Dynamical magnets can pump spin currents into superconductors. To understand such a phenomenon, we develop a method utilizing the generalized Usadel equation to describe time-dependent situations in superconductors in contact with dynamical ferromagnets. Our proof-of-concept theory is valid when there is sufficient dephasing at finite temperatures, and when the ferromagnetic insulators are weakly polarized. We derive the effective equation of motion for the Keldysh Green's function focusing on a thin film superconductor sandwiched between two noncollinear ferromagnetic insulators, one of which is dynamical. In turn, we compute the spin currents in the system as a function of the temperature and the magnetizations' relative orientations. When the induced Zeeman splitting is weak, we find that the spin accumulation in the superconducting state is smaller than in the normal states due to the lack of quasiparticle states inside the gap. This feature gives a lower backflow spin current from the superconductor as compared to a normal metal. Furthermore, in superconductors, we find that the ratio between the backflow spin current in the parallel and antiparallel magnetization configuration depends strongly on temperature, in contrast to the constant ratio in normal metals.

Figure 1

FMI|SC|FMI trilayer. The superconductor is a thin film. The large red arrows depict the magnetic moments of localized $d$ electrons in the FMIs. The green cloud illustrates a gas of $s$ electrons with spin up (red) and down (blue). An attractive interaction between the $s$ electrons (red sawtooth-like line) gives rise to superconductivity. The $s\text{−}d$ exchange interaction at the interfaces gives rise to the indirect exchange interaction between the left and right FMI (wiggly gray lines). The precessing magnetization in the left FMI gives rise to spin currents ${\mathit{j}}_{\mathrm{L}}^{\mathrm{s}}$ and ${\mathit{j}}_{\mathrm{R}}^{\mathrm{s}}$ from the FMIs into the SC.

Figure 2

The reactive (red) and dissipative (blue) backflow spin current, normalized to the density of states $N_0$, as a function of $\theta$ through the left interface of a FMI|SC|FMI trilayer for two different temperatures, $T=0.1T_{\mathrm{c}}$ (upper plot) and $T=0.9T_{\mathrm{c}}$ (lower plot). We have used the parameters given in the main text, with $m_{\mathrm{eff}}=0.1{\mathrm{\Delta }}_0$. In the lower plot, we have also plotted the spin current through an analogous FMI|NM|FMI trilayer (dotted lines).

Figure 3

(a) The temperature dependence of the total spin current in the superconducting system for two relative magnetization angles, $\theta =0$ and $\theta =\pi$. The spin current is normalized to the normal metal limit, where the spin current is independent of temperature. (b) The ratio $j_{\mathrm{d}}^{\mathrm{s},\theta =0}/j_{\mathrm{d}}^{\mathrm{s},\theta =\pi }$ plotted as a function of temperature for both the SC and NM systems. We have used the parameters given in the main text. The lowest temperature included is $T=0.03T_{\mathrm{c}}$ in order to ensure that the gradient expansion is justified.

Figure 4

(a) The fluctuations of the gap $\delta \mathrm{\Delta }\left(t\right)$ plotted over one period $2\pi /\omega$ at different temperatures for $\theta =\pi /2$. (b) The detailed temperature dependence of the gap fluctuation amplitude $\max \left|\delta \mathrm{\Delta }\right|$ for different magnetization angles $\theta$. The gap fluctuations are normalized to the gap at zero temperature, ${\mathrm{\Delta }}_0$, and we have used $m_{\mathrm{eff}}=0.1{\mathrm{\Delta }}_0$.