Dynamical torque from Shiba states in $s$-wave superconductors

Magnetic impurities inserted in an $s$-wave superconductor give rise to spin-polarized in-gap states called Shiba states. We study the back-action of these induced states on the dynamics of the classical moments. We show that the Shiba state pertains to both reactive and dissipative torques acting on the precessing classical spin that can be detected through ferromagnetic resonance measurements. Moreover, we highlight the influence of the bulk states as well as the effect of the finite linewidth of the Shiba state on the magnetization dynamics. Finally, we demonstrate that the torques are a direct measure of the even and odd frequency triplet pairings generated by the dynamics of the magnetic impurity. Our approach offers noninvasive alternative to the scanning tunneling microscopy techniques used to probe the Shiba states.

Figure 1

Sketch of the combined system. A classical precessing spin (red) at angle $\theta$ with respect to the $z$ axis is coupled via exchange interaction to an $s$-wave superconductor (grey). A localized Shiba state of size ${\xi }_S$ is formed underneath which is spin polarized and can act back on the classical spin precession. Both reactive (${\mathit{\tau }}_R$) and dissipative (${\mathit{\tau }}_D$) torques are present which affect the dynamics of the classical spin.

Figure 2

Variation of the reactive and dissipative torques. Main: full Shiba ${\tau }_{S,R})$ (blue solid line), approximate Shiba ${\tau }_{a,R}$ (red dotted line) and total ${\tau }_R$ (black dashed line) reactive torques, respectively, as a function of b, $\alpha$, $\mathrm{\Gamma }$, and $\theta$ [shown in $\left(a\right)$, $\left(b\right)$, $\left(c\right)$, and $\left(d\right)$, respectively]. We assumed $\alpha =0.9$ [$\left(a\right)$, $\left(c\right)$, and $\left(d\right)$], $\theta =\pi /6$ [$\left(a\right)$, $\left(b\right)$, and $\left(c\right)$], $\mathrm{\Gamma }=0.01$ [$\left(a\right)$, $\left(b\right)$, and $\left(d\right)$], and $b=0.1$ [$\left(b\right)$, $\left(c\right)$, and $\left(d\right)$]. The insets in the plots show the dependence of the full ${\tau }_D$ (full magenta) and approximate ${\tau }_{a,D}$ (dotted green) dissipative torques, respectively, on the corresponding parameters, with the other values being the same as for the main plots.