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

Abstract

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 /J , with 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 /J . The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, 0 , π , and π . The presence of the 0-and 0 -junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the π -and π -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 − π transition. The model allows us to obtain easily a phase diagram /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.

Quantum-Dot Parity Effects in Trivial and Topological Josephson Junctions

“…Opening the reference junction, we observed a 0-π transition as a function of dot occupancy in the zero lobe, consistent with previous stud-ies [22,[28][29][30], while in the first lobe, the 0-π transition was absent, consistent with recent predictions that hybridization of the dot spin with zero-energy states in the topological-superconducting leads favors a conventional current-phase relation, that is, a 0-junction [31][32][33][34][35]. We note that hybridization of odd-occupied dot spin with discrete zero-energy states in the leads is distinct from Kondo hybridization [36], the latter also favoring a 0junction, as discussed below [13,22,25,27,37,38].…”

Catalog of Andreev spectra and Josephson effects in structures with time-reversal-invariant topological superconductor wires

“…This junction has been analyzed in several works. [59][60][61][62][63][64][65][66][67] The main characteristic of the Andreev levels of this type of junction is the existence of an energy gap, which leads to a smooth behavior of the Josephson current. By comparing the results for the Josephson current obtained by the direct diagonalization of the Hamiltonian for the connected wires using Eq.…”

“…The description of the leads -in which only one site is considered-in the effective Hamiltonian corresponds to the atomic limit, 59,66 in which the superconducting gap |∆ α | is assumed to be much larger then the hopping term |t α |. In spite of its simplicity, as we briefly discuss below this approximation is able to describe qualitatively the 0 − π transition .The occurrence of the 0 − π transition is, precisely, main characteristic of this junction.…”

“…This transition has been widely discussed in the literature. [59][60][61][62][63][64][65][66][67] . In Fig.…”

SU(4) Kondo entanglement in double quantum dot devices

“…L = 1 in Eq. ( 5)] which can be interpreted as the narrow conduction band limit [39,40] of the full model (L → ∞).…”