Self-consistent description of Andreev bound states in Josephson quantum dot devices

Abstract

We develop a general perturbative framework based on a superconducting atomic limit for the description of Andreev bound states ͑ABS͒ in interacting quantum dots connected to superconducting leads. A local effective Hamiltonian for dressed ABS, including both the atomic ͑or molecular͒ levels and the induced proximity effect on the dot is argued to be a natural starting point. A self-consistent expansion in single-particle tunneling events is shown to provide accurate results even in regimes where the superconducting gap is smaller than the atomic energies, as demonstrated by a comparison to recent numerical renormalization group calculations. This simple formulation may have bearings for interpreting Andreev spectroscopic experiments in superconducting devices, such as scanning tunnel microscope measurements on carbon nanotubes or radiative emission in optical quantum dots.

The Andreev states of a superconducting quantum dot: mean field versus exact numerical results

“…This correspondence based on the separation of electron and hole-like excitations is specially clear in the large ∆ limit which we discuss in what follows. ∆ → ∞ case: As shown in previous works [10,16,19] the problem can be exactly diagonalized in this limit which already illustrates in a simple way the 0-π transition. The states can be classified according to the total spin S = 0 or S = 1/2.…”

“…Thus, in a method like the non-crossing approximation (NCA), which is able to describe the 0-π transition in the large charging energy regime, a single subgap resonance appears whose crossing of the Fermi level signals the transition [15]. On the other hand, other approximations like Hartree-Fock (HFA) or exact diagonalizations in the infinite ∆ limit, where ∆ denotes the superconducting gap parameter in the leads, point to the existence of up to 4 levels symmetrically located around the Fermi energy in the π-phase [10,16,17]. The bound state spectrum has also been analyzed using the numerical renormalization group (NRG) method [18][19][20] although in these works only up to two ABS were identified.…”

Nonequilibrium transport through a Josephson quantum dot

“…25,29,30,36,37 In the case of T = 0, a first order quantum phase transition leads to a sharp change of sign and amplitude of the current as a function of the complex phase difference of the superconductors φ = φ L − φ R . The phase of φ smaller than the position of the phase transition φ c is called singlet (or 0) phase.…”

“…[16][17][18][19][20][21][22][23][28][29][30][31][32][34][35][36]38 As later on we will mainly consider regimes in which the Kondo effect does not develop, we do not describe the details of these physics. Beautiful spectroscopic data [49][50][51][52][53] triggered much current effort to study the parameter dependencies of the energy of the Andreev bound states (ABS) 1,35,37 which mainly carry the current.…”

“…Recent theoretical [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] and experimental progress [40][41][42][43][44][45][46] added a quantitative understanding for mesoscopic systems including the effect of finite but equal temperatures (∆T = 0). 29,36,38 It was even possible to achieve a satisfying agreement between experimental data for the critical current obtained with a carbon nanotube as the quantum dot 44 and model calculations.…”