Supercurrent through a multilevel quantum dot close to singlet-triplet degeneracy

Abstract

We investigate two serially aligned quantum dots in the molecular regime of large tunnel couplings $t$. A Zeeman field $B$ is used to tune the energy difference of singlet and triplet spin configurations. Attaching this geometry to BCS source and drain leads with gap $\ensuremath{\Delta}$ and phase difference $\ensuremath{\varphi}$ gives rise to an equilibrium supercurrent $J$. To compute $J$ in the presence of Coulomb interactions $U$ between the dot electrons, we employ the functional renormalization group (FRG). For $B\ensuremath{\approx}t$, where the singlet and (one out of a) triplet spin states are equal in energy, the current exhibits characteristics of a 0-$\ensuremath{\pi}$ transition similar to a single impurity. Its magnitude in the $\ensuremath{\pi}$ phase, however, jumps discontinuously at $B=t$, being smaller on the triplet side. By exploiting the flexibility of the FRG, we demonstrate that this effect is generic and calculate $J$ for realistic experimental parameters $\ensuremath{\Delta}$, $U$, and gate voltages $\ensuremath{\epsilon}$. To obtain a more thorough understanding of the discontinuity, we analytically treat the limit $\ensuremath{\Delta}=\ensuremath{\infty}$, where one can access the exact many-particle states. Finally, carrying out perturbation theory in the dot-lead couplings substantiates the intuitive picture that Cooper-pair tunneling is favored by a singlet spin configuration while inhibited by a triplet one.