Josephson current through a single Anderson impurity coupled to BCS leads

We investigate the Josephson current $⟨J\left(\varphi \right)⟩$ through a quantum dot embedded between two superconductors showing a phase difference $\varphi$. The system is modeled as a single Anderson impurity coupled to BCS leads, and the functional and the numerical renormalization group frameworks are employed to treat the local Coulomb interaction $U$. We reestablish the picture of a quantum phase transition occurring if the ratio between the Kondo temperature $T_K$ and the superconducting energy gap $\Delta$ or, at appropriate $T_K∕\Delta$, the phase difference $\varphi$ or the impurity energy is varied. We present accurate zero- as well as finite-temperature $T$ data for the current itself, thereby settling a dispute raised about its magnitude. For small to intermediate $U$ and at $T=0$ the truncated functional renormalization group is demonstrated to produce reliable results without the need to implement demanding numerics. It thus provides a tool to extract characteristics from experimental current-voltage measurements.

Figure 1

(Color online) The quantum dot Josephson junction considered in this paper.

Figure 2

(Color online) The phase diagram at $ϵ=0$, $\varphi =0$, and ${\Gamma }_L={\Gamma }_R$ characterized by the boundary line between the singlet (upper region) and doublet (lower region) phases. Solid (dashed) lines show FRG results with (without) flow of the static vertex obtained from Eqs. (23, 24, 25, 27) and the definition of the phases via $J(\varphi \to 0)\gtrless 0$. Symbols are NRG data (taken from Refs. 28, 29). The inset shows that near the origin (at large $\Delta$), the slope of the phase boundary is $1∕2$ (dashed line) in accordance with analytical results obtained in the atomic limit [see Eq. (38)].

Figure 3

(Color online) The same as in Fig. 2, but for both symmetric ${\Gamma }_L={\Gamma }_R$ (dashed lines) and asymmetric ${\Gamma }_L=1.44{\Gamma }_R$ (solid lines) at (a) $\varphi \to 0$ and (b) $\varphi \to \pi$. Symbols denote NRG data. The dotted lines are the phase boundaries for $ϵ=-2.5\Delta +U∕2$, ${\Gamma }_L=1.44{\Gamma }_R$, and $\varphi =\pi$.

Figure 4

(Color online) Zero-temperature Josephson current $J$ in units of $J_c=e\Delta ∕\hslash$ as a function of the phase difference $\varphi$ computed with the FRG (solid lines) and NRG (symbols) schemes at $ϵ=0$ and ${\Gamma }_L={\Gamma }_R$. (a) $\Delta$ is varied at fixed $U∕\Gamma =5.2$ $(T_K∕\Gamma =0.209)$. The parameters correspond to $\Delta ∕T_K=0.11$, $\Delta ∕T_K=1.76$, and $\Delta ∕T_K=11$ and were chosen for direct comparison with Fig. 3 of Ref. 38 (note Ref. 46). For clarity, the curves at $\Delta ∕T_K=11$ were scaled up by a factor of $20$. (b) $\Delta ∕\Gamma =0.5$ is fixed at different $\Delta ∕T_K=1.1$, $\Delta ∕T_K=1.7$, and $\Delta ∕T_K=5.8$. The discretization parameter for NRG was chosen to be $\Lambda =4$ (for details see Secs. 3B, 5A and Appendix ).

Figure 5

(Color online) FRG results for the Josephson current $J$ and the average occupation number $n$ at $T=0$ as a function of the dot’s energy $ϵ$ at $U∕\Gamma =3$ (solid line), $U∕\Gamma =5$ (long dashed line), and $U∕\Gamma =7$ (short dashed line). The other parameters are $\Delta ∕\Gamma =1$, $\varphi ∕\pi =0.5$, and ${\Gamma }_L={\Gamma }_R$. These results are very much consistent with recent experiments (see Fig. 4 of Ref. 23).

Figure 6

(Color online) NRG results for the zero-temperature Josephson current through a noninteracting, particle-hole symmetric dot for $\Delta ∕\Gamma =0.023076$ and several sets of the discretization parameter $\Lambda$ and the number of kept states $N_c$. The dashed line denotes the analytic result, Eq. (34). All other NRG results for $J$ in this paper were obtained at $\Lambda =4$.

Figure 7

(Color online) (a) Comparison of fRG (solid line) with the exact result [dashed line, from Eq. (38)] for the critical phase difference ${\varphi }_c$ at $ϵ=0$ and the critical gate voltage ${ϵ}_c$ at $\varphi ∕\pi =0.5$ describing the phase boundary in the atomic limit. FRG results were obtained at $\Delta ∕\Gamma =1000$. (b) The same, but comparing the $U$ and $\varphi$ dependence (at $ϵ∕\Gamma =0.5$, $\varphi ∕\pi =0.5$ and $ϵ∕\Gamma =0.5$, $U∕\Gamma =1$, respectively) of the self-energy components. Only the results in the singlet phase are shown. The end of the lines $\Sigma \left(U\right)$ and ${\Sigma }_{\Delta }\left(U\right)$ indicate the transition into the doublet phase.

Figure 8

(Color online) Josephson current obtained from the FRG scheme (solid lines) in the atomic limit $\Delta =\infty$ at $T=0$. As FRG calculations were performed at $\Delta ∕\Gamma =1000<\infty$, the current in the doublet phase is finite due to $\Gamma ∕\Delta$ corrections [see Eq. (31)]. Note that in this phase $J$ (which was scaled up by a factor of $\Delta ∕\Gamma$) is independent of $U$ and $ϵ$. The dashed lines (which are partly covered by the solid lines on this scale) show the analytical result for the singlet phase [Eq. (39)].

Figure 9

(Color online) NRG results for the Josephson current at finite temperatures $T$. The parameters are the same as in Fig. 10.

Figure 10

(Color online) Energies $E_i$ of the first and second many-body excited states emerging below the edge of the superconducting gap at $U∕\Gamma =5.2$ and $\Delta ∕\Gamma =0.37$. $E_i$ is measured with respect to the ground-state energy. $(Q,S)$ denotes the set of quantum numbers introduced in Sec. 3B. The NRG parameters are taken to be same as in Fig. 4a.

Figure 11

(Color online) The same as in Fig. 4b, but obtained within a self-consistent Hartree-Fock framework.