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

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.

💡 Research Summary

This paper presents a comprehensive microscopic theory of spin pumping in a ferromagnetic‑insulator (FMI) / superconductor (SC) / normal‑metal (NM) trilayer subjected to circularly polarized microwave irradiation. Building on the extensive literature for FMI/SC bilayers, the authors extend the analysis to a three‑layer geometry where spin currents generated in the FMI must traverse the superconducting layer before being detected in the NM via the inverse spin Hall effect.

The total Hamiltonian is written as H = H_FMI + H_S‑h(t) + H_SC + H_NM + H_int. H_FMI is a Heisenberg model on a two‑dimensional lattice, H_S‑h(t) describes the coupling of the FMI spins to a circularly polarized microwave field of amplitude h and frequency Ω, H_SC and H_NM are tight‑binding descriptions of the superconducting and normal metal layers, and H_int contains two tunneling terms: a spin‑independent electron hopping T_int between SC and NM, and an exchange coupling V between the FMI spin density and the electrons at the SC interface. The superconducting layer is modeled as an s‑wave BCS superconductor with a temperature‑dependent gap Δ(T) that follows the standard tanh form and vanishes above the critical temperature T_c.

The spin current injected into the NM at the SC/NM interface (z = L) is defined via the time derivative of the electron spin density and expressed in Keldysh Green’s‑function language. Two distinct regimes are treated:

1. AC spin current – The FMI magnetization is treated as a classical precessing vector S^±(t) = S e^{∓iΩt}. By expanding the interaction action to second order in H_int, the authors derive Eq. (17), which contains products of retarded, advanced, and lesser Green’s functions of both SC and NM, together with the Pauli matrices and the particle‑hole τ_z matrix. The AC response explicitly depends on the microwave frequency through frequency‑shifted Green’s functions G(k, ω ± Ω).

2. DC spin current – The FMI spin dynamics are quantized using the Holstein‑Primakoff transformation, introducing magnon operators a_q with dispersion ω_q. The action now includes the magnon part S_M and its coupling to the electrons. A third‑order perturbative expansion (the lowest order that yields a time‑independent contribution) leads to Eq. (28). The DC current is proportional to the product of the magnon retarded Green’s function G_R^M(0, Ω) (including a correction ΔG^M that captures ferromagnetic resonance) and a series of SC/NM electron Green’s functions, again evaluated at shifted frequencies ω ± Ω. The final DC current is expressed as a δ(ω_s) term, indicating a static (zero‑frequency) spin flow.

Both AC and DC currents inherit their temperature dependence solely from the superconducting gap Δ(T). Above T_c the gap vanishes, and the currents reduce to their normal‑metal values, providing a natural normalization scheme.

The analytical expressions involve high‑dimensional integrals over in‑plane momentum k, frequency ω, and spin indices. To evaluate these integrals efficiently, the authors adopt the Quantics Tensor Cross Interpolation (QTCI) technique. QTCI represents the integrand as a quantics tensor, compresses it via low‑rank tensor train (QTT) decomposition, and then performs cross interpolation to obtain an accurate approximation with dramatically reduced computational cost and memory footprint. This enables systematic scans over temperature, microwave frequency, and SC thickness.

Numerical results reveal several key physical insights:

• Coherence peak – Both AC and DC spin currents exhibit a pronounced enhancement just below T_c, reminiscent of the Hebel‑Slichter peak in NMR, reflecting the increased quasiparticle coherence in the superconducting state.

• Thickness‑dependent crossover – For thin SC layers (d_SC ≲ coherence length ξ), the spin current decays exponentially with thickness, as expected from evanescent quasiparticle propagation. However, at microwave frequencies above a critical value Ω_c, the decay becomes non‑exponential (approximately power‑law), indicating a transition from tunneling‑dominated to resonant‑mediated spin transmission. This behavior is attributed to constructive interference of quasiparticle wavefunctions across the SC slab.

• Frequency dependence – The magnitude of both AC and DC currents grows with Ω up to a saturation point, after which the response weakens due to reduced magnon population and the onset of pair‑breaking processes in the superconductor.

• Normalization – By normalizing to the normal‑state current (T > T_c), the authors isolate the pure superconducting contribution, facilitating direct comparison with experimental inverse spin Hall voltage measurements.

The paper concludes that the unified microscopic framework, combined with the QTCI numerical scheme, provides quantitative predictions for spin pumping through superconductors in realistic trilayer geometries. The identified coherence peak and thickness‑frequency crossover constitute clear signatures that can be probed experimentally via ISHE voltage in the NM layer. Moreover, the results suggest design principles for low‑dissipation superconducting spintronic devices, such as optimizing SC thickness and microwave frequency to maximize spin current transmission while preserving superconductivity.