Spin-polarized Shapiro steps and spin-precession-assisted multiple Andreev reflection

We investigate the charge and spin transport of a voltage-biased superconducting point contact coupled to a nanomagnet. The magnetization of the nanomagnet is assumed to precess with the Larmor frequency ${\omega }_L$ when exposed to ferromagnetic resonance conditions. The Larmor precession locally breaks the spin-rotation symmetry of the quasiparticle scattering and generates spin-polarized Shapiro steps for commensurate Josephson and Larmor frequencies that lead to magnetization reversal. This interplay between the ac Josephson current and the magnetization dynamics occurs at voltages $\left|V\right|=\hslash {\omega }_L/2en$ for $n=1,2,...$, and the subharmonic steps with $n>1$ are a consequence of multiple Andreev reflection (MAR). Moreover, the spin-precession-assisted MAR generates quasiparticle scattering amplitudes that, due to interference, lead to current-voltage characteristics of the dc charge and spin currents with subharmonic gap structures displaying an even-odd effect.

Figure 1

(a) An SQPC containing a nanomagnet with spin $\mathit{S}$. A bias voltage $V$ is applied across the junction that is exposed to an effective magnetic field $\mathit{H}$. (b) The field $\mathit{H}$ causes $\mathit{S}$ to rotate with the Larmor frequency ${\omega }_L$. The angle between $\mathit{H}$ and $\mathit{S}$ is $\vartheta$, and $\chi \left(t\right)$ is the in-plane rotation angle of $\mathit{S}$. The first- and second-order tunneling processes are shown in (c) and (d), respectively, in which a spin-$\downarrow (\uparrow )$ electron(hole)-like quasiparticle $e\downarrow (h\uparrow )$ absorbs a precession quantum ${\omega }_L$ and undergoes a spin flip. Shapiro-like resonances between the Larmor and the Josephson frequencies lead to spin supercurrents that are transferred, e.g., by the process shown in (e) in which two Cooper pairs at energies differing with $2eV$ are transferred while one of the quasiparticles in each of the pairs undergoes a spin flip. (f) The Shapiro spin currents may lead to subnanosecond reversal of the precession axis.

Figure 2

Charge background current and conductance $G_B=\partial j_B^c/\partial V$ for a $\pi$ junction with (a) ${\mathcal{D}}_{\uparrow }=0.8$ and (b) ${\mathcal{D}}_{\uparrow }=0.2$. The current is normalized by the normal conductance given by the two spin channels $G_N=[e^2/h][{\mathcal{D}}_{\uparrow }+{\mathcal{D}}_{\downarrow }]$. The conductance curves in panel (a) have been offset with $3({\mathcal{D}}_{\downarrow }=0.8),2({\mathcal{D}}_{\downarrow }=0.5)$, and $1({\mathcal{D}}_{\downarrow }=0.2)$. In panel (b), the conductance is plotted for ${\mathcal{D}}_{\uparrow }=0.2,{\mathcal{D}}_{\downarrow }=0.1$, and ${\omega }_L/\Delta =0.2$. In all plots, $\vartheta =\pi /8$.

Figure 3

Background torque components as functions of bias voltage for a $\pi$ junction. (a)–(c) Dampinglike components describing spin nutations (d). (e)–(g) Fieldlike components describing oscillations of the precession frequency (h). In all panels, $\vartheta =\pi /8$.

Figure 4

(a) Components of the total torque evaluated at the Shapiro resonance voltages ${\gamma }_{L/H,1}^c(eV={\omega }_L/2\Delta )$ and ${\gamma }_{L/H,2}^c(eV={\omega }_L/4\Delta )$ as a function of ${\mathcal{D}}_S$ for a junction with $v_0=0$. The Shapiro steps depend strongly on ${\mathcal{D}}_S$ and are finite only for high transparencies. Inset: Total dc spin current as a function of bias voltage for ${\omega }_L=0.5\Delta ,\vartheta =\pi /8$, and ${\mathcal{D}}_S=0.95$. The in-plane spin current exhibits peaks due to Shapiro resonances at the resonant voltages $eV={\omega }_L/2n$. (b) The torque components as functions of ${\omega }_L$ (left panel) and precession angle $\vartheta$ (right panels) showing that ${\gamma }_{L/H,2}^c$ is sensitive to the precession frequency and that ${\gamma }_{L/H,1/2}^c$ is large even for small precession angles. In all panels, ${\phi }_0=0$ and ${\chi }_0=0$.