Fano effect in a Josephson junction with a quantum dot

We theoretically investigate the Fano effect in dc Josephson current at the absolute zero of temperature. The system under consideration is a double-path Josephson junction in which one path is through an insulating barrier and the other one is through a quantum dot (QD). Here the Kondo temperature is assumed to be much smaller than the superconducting gap, and the Coulomb interaction inside the QD is treated by the Hartree-Fock approximation. It is shown that the Josephson critical current exhibits an asymmetric resonance against the QD energy level. This behavior is caused by the interference between the two tunneling processes between the superconductors: the direct tunneling across the insulating barrier and the resonant one through the QD. Moreover, we find that the Josephson critical current changes its sign around the resonance when the Coulomb interaction is sufficiently strong. Our results suggest that $0\text{−}\pi$ transition is induced by the cooperation of the Fano effect and the Coulomb interaction inside the QD.

Figure 1

Schematic illustration of the system under consideration. A S/QD/S junction and a conventional Josephson junction are connected in parallel.

Figure 2

The critical currents in the noninteracting case $\left(U=0\right)$ are plotted against the QD energy level ${ϵ}_d$. The transmission probability $\tau =4\zeta /{\left(1+\zeta \right)}^2$ is changed from 0 (bottom) to 1 (top) with a step of 0.1. The coupling strength is chosen to be $\Gamma /\Delta =0.1$, which is adapted to all figures below.

Figure 3

(a) ABSs below the Fermi level for $\tau =0.3$. The full (dotted) line corresponds to the primary (secondary) ABS, i.e., ${\omega }_A={\omega }_1\left({\omega }_2\right)$. (b) Current contributions from each of the ABSs and the continuous spectrum for $\tau =0.3$. The dashed line indicates the continuous spectrum contribution.

Figure 4

Critical currents as a function of the QD energy level at $U/\Delta =10$ for (a) $\tau =0$ (full line), $\tau =0.0005$ (dashed line), and $\tau =0.0015$ (dotted line); (b) $\tau =0.002$ (full line), $\tau =0.003$ (dashed line), and $\tau =0.005$ (dotted line); and (c) $\tau =0.06$ (full line), $\tau =0.08$ (dashed line), and $\tau =0.1$ (dotted line). The inset of (b) is an extended figure around the resonance ${ϵ}_d=-U$.

Figure 5

Current-phase relation at $U/\Delta =10$ for different ${ϵ}_d$ which are marked by the filled circles in the inset of Fig. 4b. Explicitly ${ϵ}_d/\Delta =-14.5$ (full line), ${ϵ}_d/\Delta =-13.5$ (dashed line), ${ϵ}_d/\Delta =-12.1$ (dotted line), and ${ϵ}_d/\Delta =-11$ (dotted-dashed line).

Figure 6

Critical currents as a function of the QD energy level for $\tau =0.02$ with (a) $U/\Delta =1$ (full line), $U/\Delta =1.25$ (dashed line), $U/\Delta =1.5$ (dotted line), and $U/\Delta =1.75$ (dotted-dashed line); (b) $U/\Delta =2$ (full line), $U/\Delta =3$ (dashed line), $U/\Delta =4$ (dotted line), and $U/\Delta =5$ (dot-dashed line); and (c) $U/\Delta =6$ (full line), $U/\Delta =8$ (dashed line), $U/\Delta =10$ (dotted line), and $U/\Delta =12$ (dotted-dashed line).

Figure 7

Schematic representation of tunneling processes which contribute to $I_{w{t}^2}$ for (a) ${ϵ}_d<-U$ and (b) $-U<{ϵ}_d<-U/2$. Dashed line in the energy diagrams of the QD means the Fermi level chosen as the zero of energy.

Figure 8

Plots of the critical currents versus the QD energy level at $U/\Delta =10$ for (a) $\tau =0.01$ and (b) $\tau =0$. The full (dotted) lines represent the results obtained by the HFA (HIA).