Josephson current through a quantum dot coupled to a molecular magnet

Josephson currents are carried by sharp Andreev states within the superconducting energy gap. We theoretically study the electronic transport of a magnetically tunable nanoscale junction consisting of a quantum dot connected to two superconducting leads and coupled to the spin of a molecular magnet. The exchange interaction between the molecular magnet and the quantum dot modifies the Andreev states due to a spin-dependent renormalization of the quantum dot's energy level and the induction of spin flips. A magnetic field applied to the central region of the quantum dot and the molecular magnet further tunes the Josephson current and starts a precession of the molecular magnet's spin. We use a nonequilibrium Green's function approach to evaluate the transport properties of the junction. Our calculations reveal that the energy level of the dot, the magnetic field, and the exchange interaction between the molecular magnet and the electrons occupying the energy level of the quantum dot can trigger transitions from a 0 to a $\pi$ state of the Josephson junction. The redistribution of the occupied states induced by the magnetic field strongly modifies the current-phase relation. The critical current exhibits a sharp increase as a function of either the energy level of the dot, the magnetic field, or the exchange interaction.

Figure 1

Junction with two superconducting leads and a quantum dot coupled to the spin $\mathbf{S}$ of a molecular magnet. In the laboratory frame (a), the magnetic field forming an angle of $\vartheta$ with the spin starts a precession of the spin and splits the energy level of the quantum dot into $E_{0\uparrow }$ and $E_{0\downarrow }$. In the frame of the rotating spin (b), the Fermi energies of the spin-up (spin-down) electrons and the quasiparticle states in the leads are shifted by $-(+){\omega }_L/2$. The transformation cancels the Zeeman splitting of the quantum-dot energy level induced by the magnetic field.

Figure 2

Spin-resolved density of states of the quantum dot. In (a) and (b), $v_s=0.3\Delta$ and $\vartheta =0$. In (b) a magnetic field of ${\omega }_L=\Delta /2$ is applied, shifting the continuum states of the spin-up (spin-down) electrons by $\pm {\omega }_L/2$. In (c) and (d), $\vartheta \ne 0$ and an electron on the quantum dot can change its spin direction such that the states of the spin-up and spin-down electrons are mixed. In (d), the magnetization is precessing in the plane ($\vartheta =\pi /2$). The other parameters in (a)–(d) are $\phi =\pi /2$, $\Gamma =0.1\Delta$, $E_0=0$, and $\eta ={10}^{-3}\Delta$.

Figure 3

Phase dependence of the Andreev states. In (a)–(c), $\Gamma =\Delta /10$ and $E_0=0$. In (a), the exchange coupling $v_s$ splits the Andreev state into spin-up and spin-down states. The effect of a finite $\vartheta$ is shown in (c). Panel (d) shows the Andreev states for $\Gamma =\Delta /2$, $\vartheta =\pi /4$, and a position of the energy level on the dot of $E_0=0$ and $E_0=\Delta /2$. The dashed lines indicate the effective Fermi energies of the spin-up and spin-down quasiparticles at zero temperature.

Figure 4

Current-phase relations for $\Gamma =\Delta /10$ and $E_0=0$ [(a)–(c)]. In (a) ${\omega }_L=0$, $\vartheta =0$, and $v_s$ varies from 0 to $0.3\Delta$. The values correspond to the Andreev states of Fig. 3a, where the current is strongly suppressed due to the shift of the Andreev states across the Fermi energy. In (b), $v_s=0.3\Delta$, $\vartheta =0$, and ${\omega }_L$ varies from 0 to $\Delta$. In (c), $v_s=0.3\Delta$, ${\omega }_L=\Delta /2$, and $\vartheta$ varies from 0 to $\pi /2$. In (d), $v_s=0.3\Delta$, ${\omega }_L=\Delta /2$, $\vartheta =\pi /4$, $\Gamma =\Delta /2$, and $E_0$ is varied. The temperature is set to $k_{B}T={10}^{-4}\Delta$ and $\eta ={10}^{-4}\Delta$.

Figure 5

Current-phase relation for $v_s=0.1\Delta$, $\vartheta =\pi /4$, ${\omega }_L=\Delta /2$ (a). By changing $E_0$, the state changes from a 0 to a $\pi$ state. In (b), $E_0=0$, $v_s=0.3\Delta$, and $\vartheta =0.4\pi$. The magnetic field changes from ${\omega }_L=-\Delta$ to ${\omega }_L=\Delta$. In both panels $\Gamma ={\Gamma }_L={\Gamma }_R=0.1\Delta$, $k_{B}T={10}^{-4}\Delta$, and $\eta ={10}^{-4}\Delta$.

Figure 6

Critical current as a function of $E_0$. The parameters in (a) correspond to Fig. 3c for different exchange couplings at $\vartheta =\pi /4$ and ${\omega }_L=\Delta /2$. In (b), $v_s=0.3\Delta$ and $\vartheta =\pi /4$. In both panels $\Gamma =\Delta /10$ and $k_{B}T={10}^{-4}\Delta$.

Figure 7

Critical current $I_c$ in units of $e\Delta /\hslash$ as a function of $E_0$ and $v_s$ at $\vartheta =\pi /4$ [(a) and (b)]. In (a) ${\omega }_L=0$ and in (b) ${\omega }_L=\Delta /2$. In (c) and (d), the critical current is shown as a function of $E_0$ and ${\omega }_L$ at $v_s=0.3\Delta$. The angle $\vartheta$ varies from $\vartheta =\pi /4$ in (c) to $\vartheta =0.4\pi$ in (d). The other parameters are $\Gamma ={\Gamma }_L={\Gamma }_R=0.1\Delta$ and $k_{B}T={10}^{-4}\Delta$.