Understanding the Josephson Current through a Kondo-Correlated Quantum Dot

We study the Josephson current 0-$\pi$ transition of a quantum dot tuned to the Kondo regime. The physics can be quantitatively captured by the numerically exact continuous time quantum Monte Carlo method applied to the single-impurity Anderson model with Bardeen-Cooper-Schrieffer superconducting leads. For a comparison to an experiment, the tunnel couplings are determined by fitting the normal-state linear conductance. Excellent agreement for the dependence of the critical Josephson current on the level energy is achieved. For increased tunnel couplings the Kondo scale becomes comparable to the superconducting gap, and the regime of the strongest competition between superconductivity and Kondo correlations is reached; we predict the gate voltage dependence of the critical current in this regime.

Figure 1

Comparison of experimental data [5] with the numerically exact solution of the superconducting Anderson model. (a) Best fit of the normal-state linear conductance with applied magnetic field used for extracting values of $\Gamma$ and ${\Gamma }_L/{\Gamma }_R$ (for details see the main text). (b) Measured critical current vs theoretically calculated Josephson current at $\varphi =\frac{\pi }{2}$ (CT-INT, symbols with line; self-consistent Hartree-Fock, thin lines). The arrows indicate the level positions for which the current phase relation is presented in Fig. 2. Inset: Normal-state spectral function at $ϵ=0$.

Figure 2

Josephson current-phase relation for the parameters of the experiment [5] at values of $ϵ$ indicated by arrows in Fig. 1. It is rather sinusoidal even very close to the critical value of $ϵ$ (1.1 and 0.8 meV), and the critical current is thus well approximated by $J(\varphi =\frac{\pi }{2})$.

Figure 3

Josephson current at $\varphi =\frac{\pi }{2}$ for the parameters of the experiment (see Fig. 1) but with increased level broadening $\Gamma =0.4$ and 0.5 meV (instead of 0.27 meV) and thus increased $T_{\mathrm{K}}$. Self-consistent Hartree-Fock is obviously unable to describe the strong competition between superconductivity and Kondo correlations in this parameter regime and leads to a spurious $\pi$ phase for parameters for which the numerical exact solution only shows a precursor close to half dot filling $ϵ=0$.