Josephson Current through a Nanoscale Magnetic Quantum Dot

We present theoretical results for the equilibrium Josephson current through an Anderson dot tuned into the magnetic regime, using Hirsch-Fye Monte Carlo simulations covering the complete crossover from Kondo-dominated physics to $\pi$ junction behavior in a numerically exact way. Within the “magnetic” regime, $U/\Gamma \gg 1$ and ${ϵ}_0/\Gamma \le 1$, the Josephson current is found to depend only on $\Delta /T_K$, where $\Delta$ is the BCS gap and $T_K$ the Kondo temperature. The junction behavior can be classified into four different quantum phases. We describe these behaviors, specify the associated three transition points, and identify a local minimum in the critical current of the junction as a function of $\Delta /T_K$.

Figure 1

Schematic setup of an Anderson dot with local energy ${ϵ}_0$ and charging energy $U>0$ coupled to BCS leads. For simplicity, we assume identical BCS gap $\Delta$ on both sides, with phase difference $\varphi$ across the dot and hybridization $\Gamma$.

Figure 2

MC results for the Josephson current in units of $I_0=e\Delta /\hslash$ for two parameter sets with $\Delta /T_K\approx 23$. Squares are for $U/\Delta =20$, $\Gamma /\Delta =3.44$, and circles for $U/\Delta =4$, $\Gamma /\Delta =1$. Unless noted otherwise, in all figures, curves are guides to the eye only. Stochastic MC errors are always smaller than the symbol size or indicated by vertical bars. Inset: Current-phase relation in the deep $\pi$ junction regime. MC results for $\Delta /T_K=6.5\times {10}^4$ coincide with the analytical result (dashed curve) [6].

Figure 3

MC data for the current-phase relation at small-to-intermediate $\Delta /T_K$.

Figure 4

Same as Fig. 3 for intermediate-to-large $\Delta /T_K$.

Figure 5

MC results for the critical current (2) as a function of $\Delta /T_K$. The arrows mark the boundaries between different phases (see text). The inset shows the region around the minimum on a magnified scale. Error bars are always smaller than symbol size.