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

We develop a general perturbative framework based on a superconducting atomic limit for the description of Andreev bound states (ABS) in interacting quantum dots connected to superconducting leads. A local effective Hamiltonian for dressed ABS, including both the atomic (or molecular) 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.

Figure 1

Phase diagram of a simple dot with Coulomb interaction $U$, energy level ${\xi }_d$ and hybridization $\Gamma$ to superconducting electrodes in the effective local limit. The transition line corresponds to $E_{\sigma }^0=E_{-}^0$.

Figure 2

(Color online) Phase diagram of a simple dot with Coulomb interaction $U$, tunnel coupling $\Gamma$ to superconducting electrodes with gap $\Delta$ for $\phi =0$ and $\pi \Gamma =0.2D$. The symbols indicate NRG data from Ref. 2 and the various lines our results.

Figure 3

(Color online) Transition line between a doublet state and the BCS-type state at particle-hole symmetry ${\xi }_d=0$ (solid curve) for $\phi =0$ and $\pi \Gamma =0.2D$. The vertical dotted line corresponds to the transition asymptote in the effective local limit at $\Delta \to \infty$. The dots indicate NRG data from Ref. 2 and the solid line our result. The inset displays the same curves on a logarithmic scale.

Figure 4

(Color online) Renormalization of the Andreev bound-state energies as a function of $T_K/\Delta$ (the Kondo temperature is given in the text). The dashed curves correspond to the high-energy bound state $b\left(\Delta \right)$, the solid curves correspond to $a\left(\Delta \right)$. All curves have been calculated for $U=0.005D$ and ${\xi }_d=0$ with several hybridization values $\pi \Gamma /D=0.001,0.002,0.005$ (from left to right). Symbols are the NRG results obtained in Ref. 30.

Figure 5

(Color online) Same data as in Fig. 4 but normalized by the gap.

Figure 6

(Color online) Renormalization of the Andreev bound-state energies outside particle-hole symmetry. The dotted curves correspond to the high-energy bound state $b\left(\Delta \right)$, the solid curves correspond to $a\left(\Delta \right)$. All curves have been calculated for $U=0.5D$ and $\pi \Gamma =0.05D$ with several level shifts ${\xi }_d/U=0.3,0.375,0.4,0.425,0.45$. Quantitatively similar results were obtained by the NRG in Ref. 30.

Figure 7

(Color online) Evolution of the Andreev bound-state energies as a function of the dot’s energy level for $U=0.005D$ and $\Delta =U$. The hybridization takes several values $\pi \Gamma /D=0.001,0.002,0.005$.

Figure 8

(Color online) Superconducting correlations as a function of the gap $\Delta$ (for $U=D/200$ and ${\xi }_d=0$).

Figure 9

(Color online) Superconducting correlations as a function of the Coulomb interaction $U$, for $\Delta /\left(\pi \Gamma \right)=0.3,1.0$ and at particle-hole symmetry ${\xi }_d=0$. Solid lines are the results of our self-consistent equations, diamonds correspond to NRG data from Ref. 2, and crosses are perturbation theory in $U$ at second order.

Figure 10

(Color online) Superconducting correlations outside particle-hole symmetry (for $\pi \Gamma =0.2D$, $U=6\Gamma$, and $\Delta =0.1D$).

Figure 11

(Color online) Josephson current through the bound states for $U=3\Gamma$ and $\Delta =0.1D$.