Josephson Current in Strongly Correlated Double Quantum Dots

We study the Josephson current through a serial double quantum dot and the associated $0-\pi$ transitions which result from the subtle interplay between the superconductivity, the Kondo physics, and the interdot superexchange interaction. The competition between them is examined by tuning the relative strength $\Delta /T_K$ of the superconducting gap and the Kondo temperature, for different strengths of the superexchange coupling determined by the interdot tunneling $t$ relative to the level broadening $\Gamma$. We find strong renormalization of $t$, a significant role of the superexchange coupling $J$, and a rich phase diagram of the 0 and $\pi$-junction regimes. In particular, when both the superconductivity and the exchange interaction compete with the Kondo physics ($\Delta \sim J\sim T_K$), there appears an island of ${\pi }^{\prime }$ phase at large values of the superconducting phase difference.

Figure 1

Normal-state conductance [(a),(b)] and the critical Josephson current in the superconducting state [(c),(d)] as a function of $t/\Gamma$ for $\Gamma =0.014D$ (a),(c) and $\Gamma =0.02D$ (b),(d). The results are expressed in the units of the conductance quantum $G_0=2e^2/h$ and the supercurrent quantum $I_c^s$. In the superconducting case [(c),(d)], the different curves are for $\Delta /T_K=0.1$ (black circles), 3.6 (red squares), and 10 (blue diamonds). The dashed line is from an effective noninteracting theory, Eq. (4).

Figure 2

Phase boundaries between the 0 and $\pi$ states for $\Gamma =0.014D$ (left panels) and $\Gamma =0.02D$ (right panels). When both the interdot superexchange coupling and the superconductivity are in close competition with the Kondo effect between the superconducting leads and adjacent dots ($t/\Gamma \sim 0.2$, $\Delta \sim T_K$), there appears an island of ${\pi }^{\prime }$ phase at larger phase difference.

Figure 3

Left panels: Josephson current vs phase difference near the ${\pi }^{\prime }$ phase ($\Delta /T_K=3.6$, $t/\Gamma =0.1$, 0.2, 0.3, 0.4, and $\Gamma =0.02D$ from top to bottom). Right panels: Corresponding spin-state-resolved Andreev bound states inside the superconducting gap. Spin doublet states are depicted by (red) circles and singlets by (black) squares. The changes in the spin states are closely related to the island of ${\pi }^{\prime }$ phase in Fig. 2c [similar spin-state-resolved Andreev states for corresponding parameters will also explain the island in Fig. 2d].