Nonequilibrium spin-splitter effect in altermagnet superconductor hybrids

TL;DR

This work addresses nonequilibrium spin-splitter effects in superconducting altermagnets and altermagnet–superconductor hybrids under an AC drive. The authors employ the Usadel framework within the Keldysh formalism to compute spin currents and edge spin densities, revealing a strong, tunable response for frequencies on the order of the superconducting gap $Δ_0$ while establishing that no equilibrium spin-splitter exists in altermagnets. A key finding is that the out-of-phase spin density does not diverge in the adiabatic limit, in contrast to spin-Hall-type responses, and the in-phase component persists below the gap due to quasiparticle dynamics; spin accumulation at edges remains finite as $ω→0$ even when the spin current vanishes. The results show that the nonequilibrium spin-splitter effect can serve as both a diagnostic of altermagnetism and a tunable mechanism for spin control, with strong temperature dependence and potential applications in superconducting spintronics.

Abstract

We study the nonequilibrium spin-splitter effect in superconducting altermagnets and superconductor altermagnet hybrids by computing the alternating spin current and edge the spin density in the presence of an alternating electric field. We show that while in the normal state the effect is not sensitive to the field frequency, in the superconducting state, there is a strong effect for frequencies on the scale of $Δ_0$ or lower. We contrast the effect to the spin accumulation induced by the spin-Hall effect, by showing that for the altermagnet spin-splitter effect the out-of-phase spin density does not diverge in the adiabatic limit. This difference is attributed to the absence of any equilibrium spin-splitter effect in altermagnets. In fact, the out-of-phase component vanishes below the gap excitation frequency $2Δ_0$, because below this frequency the absence of dissipation and the behavior of the system under time-reversal directly determine the relative phase between the charge current, spin current, and spin accumulation. The nonequilibrium effect can be tuned by external parameters like temperature. In fact, it has a nonmonotonic temperature dependence, taking its largest value for temperatures around $0.8T_{c}$. The value at this temperature can be significantly larger than the normal state spin density or the low temperature spin density. Thus, besides using the nonequilibrium spin-splitter effect to identify altermagnets, its tunability makes it also suitable for applications.

Figures (6)

• Figure 1: The studied geometry. In the presence of an alternating electric field, an alternating spin polarization is generated at the transverse edges of an altermagnet unless the electric field direction is aligned with the lobes of the spin splitting.
• Figure 2: The real (a) and the imaginary (b) parts of the zero temperature spin-splitter conductivity and the longitudinal conductivity. The longitudinal conductivity follows the usual Mattis-Bardeen theory. The spin-splitter conductivity behaves similarly to the longitudinal charge conductivity, but it goes up much faster near $\omega = 2\Delta_0$, while there longitudinal component vanishes rather than diverges as $\omega\xrightarrow{}0$. The dashed line indicates the excitation gap, below which dissipation only appears due to the Dynes parameter $\eta = 0.002\Delta_0$. Inset: Normalized ratio of the spin-splitter conductivity and longitudinal charge conductivity $\sigma_{SSE}/(\sigma_{xx}P T_{xy})$.
• Figure 3: Real and imaginary, i.e., in- and out-of-phase parts of $S(\omega)$ upon application of an electric field $E(\omega)$, normalized to the normal-state spin accumulation $S_{z,N} = \nu_0 PT_{xy}dE(\omega)$ in the limit $d\ll \xi$ and zero temperature. The dashed line indicates the excitation gap, below which any dissipative signal is induced by $\eta = 0.002\Delta_0$.
• Figure 4: Nonequilibrium spin-splitter effects for different temperatures. As $T$ approaches $T_{c}$, the features induced by superconductivity appear for smaller $\omega$, reflecting the smaller gap, and are more pronounced, reflecting that the Green's function changes faster with energy in the limit $d\ll \xi$. Inset: The adiabatic response as a function of temperature. There is a peak close to $T=T_{c}$, in which the adiabatic response exceeds the normal state value significantly. We used $\eta = 0.002\Delta_0$. There are quasiparticles for each frequency because of the nonzero population of states above the gap.
• Figure 5: The dependence of the generated spin in the limit $d\ll \xi, l_s$ on the Dynes parameter at $T = 0.1T_c$. The results are normalized using the spin accumulation that appears in the normal state. There is dissipative signal for all frequencies because of the residual density of states induced by the Dynes parameter