Hybrid quantum dot–superconducting systems: Josephson current and Kondo effect in the narrow-band limit

The case of a quantum dot connected to two superconducting leads is studied by using the narrow-band limit to describe the superconducting degrees of freedom. The model provides a simple theoretical framework, almost analytical, to analyze the interplay between the Kondo effect, superconductivity, and finite temperature. In the quantum dot Kondo regime, the model is completely characterized by the ratio $\Delta /J$, with $\Delta$ the superconducting gap and $J$ an effective antiferromagnetic exchange coupling between the dot and the leads. The model allows us to calculate, at any temperature $T$, the equilibrium Josephson current through the dot in a very straightforward way as a function of $\Delta /J$. The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, $0^{\prime },{\pi }^{\prime }$, and $\pi$. The presence of the 0- and $0^{\prime }$-junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the $\pi$- and ${\pi }^{\prime }$-junction configurations are driven by the superconductivity in the leads. The Josephson critical current has a non-monotonic behavior with temperature, that may be used for the experimental characterization of the fundamental $0-\pi$ transition. The model allows us to obtain easily a phase diagram $\Delta /J$ vs temperature, from where we can obtain an overall picture on the stability of the different types of junctions. From the explicit analytical expressions for the ground-state, low-energy excitations, free energy, and Josephson current, it is easy to understand the physical nature of the main features of the critical current and the phase diagram. The results, obtained with a minimum of numerical effort, are in a good qualitative agreement with more demanding calculational approaches aimed to solve the full model.

Figure 1

$\phi$ dependence of eigenvalues ${\lambda }_{S,i}\left(\phi \right)$, with $S=0,S=1/2$, and $S=1$, for $\Delta /J=0.64$. For each $\Delta$, the point of intersection between the two lowest eigenvalues, ${\lambda }_{0,1}\left({\phi }^{*}\right)={\lambda }_{1/2,1}\left({\phi }^{*}\right)$, defines ${\phi }^{*}\left(\Delta \right)$. For this particular case, ${\phi }^{*}(\Delta ={\Delta }_c)\sim 0.45\pi$. For $\Delta /J=0.75$ (not shown), ${\phi }^{*}=0$, and ${\lambda }_{0,1}\left(0\right)={\lambda }_{1/2,1}\left(0\right)$; for $\Delta /J>0.75$ (not shown), ${\lambda }_{0,1}\left(\phi \right)>{\lambda }_{1/2,1}\left(\phi \right)$ for all $\phi$.

Figure 2

Lowest energy levels of $\tilde{H}$ as a function of $\Delta /J$. For $\Delta <{\Delta }_c\simeq 0.64J$ we can observe the singlet (solid line) ground-state energy ${\lambda }_{0,1}\left(0\right)$, and for $\Delta >{\Delta }_c$ the doublet (dashed line) ground-state energy ${\lambda }_{1/2,1}\left(\pi \right)$ is observed. The thin vertical line denotes the transition point.

Figure 3

Zero-temperature Josephson current $I\left(\phi \right)/I_0$ as a function of $\phi$, for different values of $\Delta /J$. All curves, except the ones for $\Delta /J=0.75$ and 1, have a discontinuity at ${\phi }^{*}\left(\Delta \right)$, indicated by the vertical thin straight line. The size of the discontinuity is given by Eq. (27). $\tau =k_{B}T$.

Figure 4

Free energy $F\left(\phi \right)$ as a function of $\phi$ [measured from $F(\phi =0)\text{]}$, for $\tau /J=0.01$ and different values of $\Delta /J$. $F\left(\phi \right)$ enters with zero slope at the limiting values $\phi =0,\pi$.

Figure 5

Josephson current $I\left(\phi \right)/I_0$ as a function of $\phi$, for $\tau /J=0.01$ and different values of $\Delta /J$.

Figure 6

Josephson current $I\left(\phi \right)/I_0$ as a function of $\phi$, for $\tau /J=0.1$ and different values of $\Delta /J$. $I\left(\phi \right)/I_0$ has a nonmonotonic behavior both in the regime of small and intermediate values of $\Delta /J$.

Figure 7

Critical current $I_c/I_0={\text{max}}_{\phi }\left|I\left(\phi \right)\right|/I_0$ as a function of $\Delta /J$ for different values of $\tau /J$. In all cases $I_c/I_0\to 0$ for $\Delta \to 0$.

Figure 8

Critical current $I_c/I_0$ as a function of temperature, for several values of the superconducting order parameter. All curves have a finite value at $\tau =0$, smaller than 1.

Figure 9

Phase diagram in $\Delta /J,\tau /J$ space. The analytical zero-temperature intercepts of the $0\text{−}{0}^{\prime },0^{\prime }\text{−}{\pi }^{\prime }$, and $\pi \text{−}{\pi }^{\prime }$ boundaries are 0, ${\Delta }_c/J=(1+\sqrt{17})/8\simeq 0.64$, and 0.75, respectively. The thin full straight line corresponds to $\Delta =10\tau$, while the thin dashed straight line represents $\Delta =\tau$.