Spin-precession-assisted supercurrent in a superconducting quantum point contact coupled to a single-molecule magnet

The supercurrent through a quantum point contact coupled to a nanomagnet strongly depends on the dynamics of the nanomagnet's spin. We employ a fully microscopic model to calculate the transport properties of a junction coupled to a spin whose dynamics is modeled as Larmor precession brought about by an external magnetic field and find that the dynamics affects the charge and spin currents by inducing transitions between the continuum states outside the superconducting gap region and the Andreev levels. This redistribution of the quasiparticles leads to a nonequilibrium population of the Andreev levels and an enhancement of the supercurrent which is visible as a modified current-phase relation as well as a nonmonotonous critical current as function of temperature. The nonmonotonous behavior is accompanied by a corresponding change in spin-transfer torques acting on the precessing spin and leads to the possibility of using temperature as a means to tune the back-action on the spin.

Figure 1

(a) Two superconducting leads are coupled over the spin of a nanomagnet. The hopping amplitude for spin-independent tunneling is given by $V_0$, while $V_S$ is the coupling between the tunneling electrons and the spin of the nanomagnet. (b) The spin of the nanomagnet, $\mathbf{S}$, precesses with the Larmor frequency, ${\omega }_L$, due to an external effective magnetic field, $\mathbf{H}$, that is applied at an angle $\vartheta$ with respect to the orientation of the spin. The density of states in a rotating frame (see definition in the text) is plotted for a transparent junction ($v_T=1$) with transmission probabilities (c) $({\mathcal{D}}_0=0.999,{\mathcal{D}}_S=0.025)$, given by the hopping amplitudes $[v_0=v_T\cos (0.1\pi /2),v_S=v_T\sin (0.1\pi /2)]$, and (d) $({\mathcal{D}}_0=0,{\mathcal{D}}_S=1)$ given by the hopping amplitudes $[v_0=v_T\cos (\pi /2)=0,v_S=v_T\sin (\pi /2)=1]$ at temperature $T/T_c\to 0$. The Andreev levels, which can be seen as sharp states inside the gap, have here for visibility reasons been given an artificial broadening, $0^{+}$, and the continuum density of states is normalized to 1 away from the gap edge. The upper (lower) Andreev levels, ${\varepsilon }^{+}$ (${\varepsilon }^{-}$), are given an effective Zeeman splitting (${\varepsilon }^{+(-)}\to {\varepsilon }_{\uparrow ,\downarrow }^{+(-)}$) by the spin precession. The spin precession couples the Andreev levels and the continuum states as well as the states ${\varepsilon }_{\uparrow }^{+(-)}$ and ${\varepsilon }_{\downarrow }^{+(-)}$. (e) The current-phase relation (CPR) is plotted for $({\mathcal{D}}_0=0.999,{\mathcal{D}}_S=0.025)$ at temperatures (solid lines from top to bottom) $T/T_c=7.5\times {10}^{-4}$, $0.25$, $0.50$, $0.75,$ and $0.85$. The dashed line shows the CPR for $({\mathcal{D}}_0=1,{\mathcal{D}}_S=0)$ at temperature $T/T_c\to 0$. (f) The CPR for a junction with $({\mathcal{D}}_0=0,{\mathcal{D}}_S=1)$ at temperatures (from bottom to top) $T/T_c=7.5\times {10}^{-4}$, $0.25$, $0.50$, $0.75$, and $0.85$. In (e) and (f), the current is given in units of $j_0^{\mathrm{ch}}=e\Delta /\hslash$, where $\Delta$ is the superconducting gap at $T=0$. Figures (c)–(f) are calculated for precession angle $\vartheta =\pi /4$ and precession frequency ${\omega }_L/\Delta =0.2$.

Figure 2

(a) Population of the lower Andreev level ${\varepsilon }_{\uparrow }^{-}$ (black lines) and the upper Andreev level ${\varepsilon }_{\downarrow }^{+}$ (red lines) as function of phase difference, $\varphi$, for a 0 junction with transmission probabilities ${\mathcal{D}}_0=0.999$ and ${\mathcal{D}}_S=0.025$, precession frequency ${\omega }_L/\Delta =0.2$ and tilt angle $\vartheta =\pi /4$. At temperature $T/T_c=7.5\times {10}^{-4}$, the population of the Andreev levels is unaffected by the phase difference. At higher temperatures, $T/T_c=0.25$, $0.50$, $0.75,$ and $0.85$, the population changes abruptly at a phase difference ${\varphi }_p$ leading to a jump in the CPRs shown in Fig. 1. (b) Density of states for the junction in (a) at temperature $T/T_c=0.85$. The dashed lines mark ${\Delta }^{\pm }\left(T\right)=\pm \Delta \left(T\right)$. The dotted lines denote ${\Delta }_{\sigma }^{\pm }\left(T\right)={\Delta }^{\pm }\left(T\right)-\sigma {\omega }_L/2$.

Figure 3

Enhanced critical current as a function of temperature, $T/T_c$, for $({\mathcal{D}}_0=0.999,{\mathcal{D}}_S=0.025)$ (dashed lines) and $({\mathcal{D}}_0=0,{\mathcal{D}}_S=1)$ (solid lines). The enhancement of the spin-precession-assisted critical current $j_c^{\text{ch}}\left({\omega }_L\right)$ is calculated with respect to the critical current through a junction with a static spin and the respective hopping amplitudes specified above, viz. $j_c^{\text{ch}}({\omega }_L/\Delta =0),$ which is the critical current obtained from Eq. (10). The precession angle is $\vartheta =\pi /8$.

Figure 4

Critical current as function of precession frequency, ${\omega }_L$, for (a) (${\mathcal{D}}_0=0.999,{\mathcal{D}}_S=0.025$) and (b) (${\mathcal{D}}_0=0,{\mathcal{D}}_S=1$), where the precession angle is taken to be $\vartheta =\pi /8$ and the temperature is $T/T_c=7.5\times {10}^{-4}$ (black), $0.25$ (red), $0.50$ (green), $0.75$ (blue), and $0.85$ (violet).

Figure 5

(a) and (b) The superconducting phase difference, ${\varphi }_c$, corresponding to the critical charge current, i.e., $j_c^{\mathrm{ch}}=j^{\mathrm{ch}}\left({\varphi }_c\right)$, is plotted as a function of temperature, $T/T_c$, for (a) (${\mathcal{D}}_0=0.999,{\mathcal{D}}_S=0.025$) and (b) (${\mathcal{D}}_0=0,{\mathcal{D}}_S=1$) for precession frequencies ${\omega }_L/\Delta =0.1$, $0.2,$ and $0.3$. The abrupt jump in the phase difference occurs when the critical current is given by the spin-precession-enhanced current. The abrupt change in ${\varphi }_c$ leads to an abrupt change in the spin-current component $j_H^s\left({\varphi }_c\right)$, which is plotted for hopping strengths (c) (${\mathcal{D}}_0=0.999,{\mathcal{D}}_S=0.025$) and (d) (${\mathcal{D}}_0=0,{\mathcal{D}}_S=1$). The precession angle is $\vartheta =\pi /8$ in all panels.