AC/DC spin current in ferromagnet/superconductor/normal metal trilayer systems

TL;DR

The paper develops a microscopic theory of spin pumping through FMI/SC/NM trilayers using the Keldysh Green’s function formalism to derive both AC (classical FMI spin) and DC (quantum magnon) spin currents in the NM layer. It couples a detailed microscopic model with Quantics Tensor Cross Interpolation (QTCI) to perform high-dimensional momentum–frequency integrals efficiently. Key findings include a coherence peak in both AC and DC currents near the superconducting transition and a thickness-dependent crossover from exponential to non-exponential decay at certain microwave frequencies due to quasiparticle interference within the SC layer. The work provides a unified framework for superconducting spin pumping and demonstrates the viability of QTCI in complex spin-transport calculations, with implications for spintronic devices and extensions to unconventional superconductors.

Abstract

Spin pumping with superconductors has been extensively studied, particularly in double-layer systems. In this study, we investigate spin pumping in a trilayer system comprising a ferromagnetic insulator (FMI), a superconductor (SC), and a normal metal (NM). We derive the AC and DC spin currents in the NM layer induced by spin motion in the FMI under circularly polarized microwave irradiation. If we treat the spin motion as classical, the AC spin current is expressed. On the other hand, if we treat the spin motion as quantum quasiparticles, the DC spin current is derived. After these derivations, while the computational cost of evaluating the spin current is extremely high, we mitigate this using the Quantics Tensor Cross Interpolation (QTCI) method. We present numerical results showing the dependence of the spin current on temperature, microwave frequency, and superconductor layer thickness. Notably, the temperature dependence of AC and DC spin currents exhibits a coherence peak. Furthermore, we have discovered a transition structure in the dependence of the spin current on the thickness of the superconductor layer, where the dependence changes after a particular frequency.

Figures (9)

• Figure 1: Schematic illustration of the trilayer system considered in this study. The spin current, denoted by $\vec{J}_{\rm s}(L,t)$, is injected into the top of the NM layer at time $t$ by a circularly polarized microwave field with frequency $\Omega$. The spin-space coordinate axes $(x,y, z)$ are defined such that the $z$-axis is aligned with the direction of microwave propagation. The interfaces between the FMI and SC, and between the SC and NM, are located at $z = 0$ and $z = L$, respectively.
• Figure 2: The normalized AC spin current defined on Eq. (\ref{['J-AC-norm']}). The parameters are set as $t=4.0 \Delta_{0}(0)$, $T_{\rm C} = 1.0$ K and $\mu = -0.2 t$. Each panels correspond to $L=12$, $13$, and $14$. (a) shows the $x$-component of the AC spin current. The solid (red), dashed (blue), and dotted (green) lines without markers correspond to $\Omega/ k_{\rm B}T_{\rm C} = 0.1$, $0.3$, and $0.5$, respectively. The dashed (circle), dotted (square) and dotted (triangle) with markers correspond to $\Omega/ k_{\rm B}T_{\rm C} = 1.0$, $1.5$, and $2.5$, respectively. (b) shows the $y$-component of the AC spin current. The solid (red), dashed (blue), and dotted (green) lines without markers correspond to $\Omega/ k_{\rm B}T_{\rm C} = 0.1$, $0.3$, and $0.5$, respectively. The dashed (circle), dotted (square) and dotted (triangle) with markers correspond to $\Omega/ k_{\rm B}T_{\rm C} = 1.0$, $1.5$, and $2.0$, respectively. The $z$-component of the AC spin current is always zero.
• Figure 3: Frequency dependence of the AC spin current. The parameters are set as $t=4.0 \Delta_{0}(0)$, $T_{\rm C} = 1.0$ K and $\mu = -0.2 t$. The solid (red), dashed (blue), and dotted (green) lines correspond to $L = 12$, $13$, and $14$, respectively. (a) shows the $x$-component of the AC spin current. (b) shows the $y$-component of the AC spin current.
• Figure 4: Thickness dependence of the AC spin current. The parameters are set as $t=4.0 \Delta_{0}(0)$, $T_{\rm C} = 1.0$ K and $\mu = -0.2 t$. The $L$ is thickness of the superconducting layer, and the legend shows the microwave frequencies normalized by the critical temperature $T_{\rm C}$. The solid (red), dashed (blue), dotted (green) and dashdot (black) lines correspond to $T/T_{\rm C} = 1.1$, $0.9$, $0.5$ and $0.1$, respectively. (a) shows the $x$-component of the AC spin current. The panels shown the frequency $\Omega/k_{\rm B}T_{\rm C} = 0.1$, $0.3$, $0.5$, $1.0$, $2.0$, and $2.5$. (b) shows the $y$-component of the AC spin current. The panels shown the frequency $\Omega/k_{\rm B}T_{\rm C} = 0.1$, $0.3$, $0.5$, $1.0$, $1.5$, and $2.0$.
• Figure 5: The normalized DC spin current defined on Eq. (\ref{['J_renom-T']}). The parameters are set as $t=4.0 \Delta_{0}(0)$, $T_{\rm C} = 1.0$ K and $\mu = -0.2 t$. The solid (red), dashed (blue), and dotted (green) lines without markers correspond to $\Omega/ k_{\rm B}T_{\rm C} = 0.1$, $0.5$, and $1.0$, respectively. The lines with circles (yellow), squares (purple), and triangles (black) correspond to $\Omega/ k_{\rm B}T_{\rm C} = 1.5$, $2.0$, and $2.5$, respectively. The $L$ is thickness of the superconducting layer, and the legend shows the microwave frequencies normalized by the critical temperature $T_{\rm C}$. In numerical, we get $\Delta J_{x} = \Delta {J}_{y} = 0$.