Pumping of spin supercurrent in unitary triplet superconductors

Abstract

One efficient mechanism for generating a charge supercurrent is Andreev reflection, in which the electric current injected from a normal metal into a conventional superconductor is converted into a supercurrent, thereby preserving charge conservation. We here propose a general principle for generating spin supercurrents in triplet superconductors by analogy with such charge transport, i.e., assuming spin conservation. We find a spin torque that is proportional to the triplet superconducting order parameter and, in the spin-conservation scenario, converts the particle spin to that of Cooper pairs. Based on this general principle, we propose an implementation to efficiently generate a spin supercurrent in unitary triplet superconductors, even though Cooper pairs carry no spin polarization at equilibrium, by the magnetization dynamics ${\bf M}(t)$ of a proximity magnetic nanostructure. The efficiency of this spin pumping is not solely limited to the $d{\bf M}/dt\times {\bf M}$ due to the emergent particle-hole symmetry, thereby going beyond the conventional spin pumping of electrons. This general principle provides an efficient approach to generating and manipulating dissipationless spin currents in many unconventional superconductors.

Figures (4)

• Figure 1: Comparison of the generation of charge supercurrent in conventional $s$-wave superconductors [(a)] and the pumping of spin supercurrent in triplet $p$-wave superconductors with unitary triplet Cooper pairing [(b)]. In (a), the electric current injected from a normal metal (NM) to the conventional superconductor (SC) converts to a supercurrent carried by Cooper pairs via the Andreev reflection by retaining charge conservation. In (b), the injection of spin torque from the driven region by a local dynamical exchange field to the non-driven region generates a spin supercurrent by retaining the spin conservation.
• Figure 2: Spin of quasiparticles and illustration of scattering process with quasiparticles scattered from $\{\hat{a}_{{\bf q}1},\hat{a}_{{\bf q}2},\hat{a}_{{\bf q}3},\hat{a}_{{\bf q}4}\}$ to $\hat{b}_{{\bf k}1}$ that includes the intraband and interband processes. The color map on the band indicates the spin of quasiparticles $S_z\in[-\hbar/2,\hbar/2]$.
• Figure 3: Schematic of pumping the spin current carried by the quasiparticles and Cooper pairs in a unitary $p$-wave superconductor, driven by the coherent magnetization dynamics of the ferromagnetic resonance of the magnetic nanowire via an interfacial exchange field. The width and thickness of the wire are $d$ and $w$, respectively. The saturation magnetization $M_s$, denoted by the thick black arrow, is biased to the wire $\pm \hat{\bf z}$-direction. The thick red arrow represents the longitudinal spin pumping that contains both the dissipative and dissipationless spin current.
• Figure 4: Pumping of spin current and spin supercurrent into the $p$-wave triplet superconductors, driven by the dynamical interfacial exchange field due to the ferromagnetic resonance of magnetic nanowires. The shadow region in (a), (c), and (e) indicates the region of the dynamical interfacial exchange field covered by the magnetic nanowire. (a) and (b) plot the spatial and temperature dependencies of the pumped spin current that is polarized along the saturation magnetization $\pm\hat{\bf z}$-direction. (a) is calculated at the temperature ${ T} =1~{\rm K}$, noting the superconducting transition temperature $T_c=2.05$ K. (c) addresses the spatial profile of the spin-torque density when $T=1~{\rm K}$, which is also polarized along the saturation magnetization $\hat{\bf z}$-direction. The temperature dependence of the spin-torque density inside the pumping region is shown in (d). (e) illustrates the spatial dependence of the spin supercurrent by expressing the spin-torque density ${\bf T}_s(y)=-\partial_y{\bf J}_{sc}(y)$ when $T = 1~{\rm K}$, under the assumption ${\bf J}_{sc}(y=0)=0$. (f) illustrates the temperature dependence of the spin supercurrent close to the pumping region. (g) illustrates the spin torque and pumped spin supercurrent when ${\bf M}_s\parallel \hat{\bf z}$. The material parameters used for the calculation are given in the text.