Kondo effect and Josephson current through a quantum dot between two superconductors

We investigate the supercurrent through a quantum dot for the whole range of couplings using the numerical renormalization group method. We find that the Josephson current switches abruptly from a $\pi \text{−}$ to a $0$-phase as the coupling increases. At intermediate couplings the total spin in the ground state depends on the phase difference between the two superconductors. Our numerical results can explain the crossover in the conductance observed experimentally by Buitelaar et al. [Phys. Rev. Lett. 89, 256801 (2002)].

Figure 1

The pairing correlation on the quantum dot, ${\Delta }_d\equiv ⟨d_{\uparrow }^{†}{d}_{\downarrow }^{†}⟩$, as a function of $\Delta ∕T_K$. Inset: plot of bare (not normalized) values of ${\Delta }_d$. We have chosen ${ϵ}_d=-U∕2=-0.1D$ and $\Gamma =0.04D$.

Figure 2

(Color online) The single-particle excitation spectrum on the quantum dot. ${ϵ}_d=-U∕2=-0.1D$ and $\Gamma =0.04D\left(T_K=0.0089D\right)$.

Figure 3

(Color online) Josephson current $I_S\left(\varphi \right)$ (in units of $I_c^{\text{short}}\equiv e\Delta ∕\hslash$) as a function of phase different $\varphi$ (a) for $\Delta ∕T_K=10$ and (b) for $\Delta ∕T_K=0.1$. (c) Same curves for $\Delta ∕T_K=1.6$, $1.8$, $2.0$, and $2.2$ (near the $0\text{−}\pi$ junction transition point). (d) Critical current in the Kondo regime. We put ${ϵ}_d=-U∕2=-0.1D$ and $\Gamma =0.04D$. Inset: conductance resulting from the RSJ model (see the text).