π junction behavior and Andreev bound states in Kondo quantum dots with superconducting leads

We investigate the temperature- and coupling-dependent transport through Kondo dot contacts with symmetric superconducting $s$-wave leads. For finite temperature $T$ we use a superconducting extension of a self-consistent auxiliary boson scheme (SNCA), while at $T=0$ a perturbative renormalization group treatment is applied. The finite-temperature phase diagram for the $0\text{−}\pi$ transition of the Josephson current in the junction is established and related to the phase-dependent position of the subgap Kondo resonance with respect to the Fermi energy. The conductance of the contact is evaluated in the zero-bias limit. It approaches zero in the low-temperature regime, however, at finite $T$ its characteristics are changed through the coupling- and temperature-dependent $0\text{−}\pi$ transition.

Figure 1

Quantum dot coupled to two superconductors. ${\Gamma }_L$ and ${\Gamma }_R$ denote the effective couplings to the left and right lead, ${\varphi }_L$ and ${\varphi }_R$ label the phases of the corresponding superconducting order parameter. The BCS gap $\Delta$ is assummed to be equal in both superconductors. In the model, defined by Eq. (1), the local Coulomb repulsion $U$ is set to infinity.

Figure 2

Generating functional for the extension of the NCA to the broken-symmetry state (SNCA). The solid, wavy, and dashed lines represent the conduction electron, slave boson, and pseudofermion propagators, respectively.

Figure 3

(Color online) The spectral function of the impurity $d$-level for weak coupling (upper panel, $T_K∕\Delta =0.125$) and strong coupling (lower panel, $T_K∕\Delta =2.0$). The leads are in the superconducting state at $T=0.5\Delta$. The spectral functions are sensitive to the phase difference $\varphi ={\varphi }_L-{\varphi }_R$ between left and right lead.

Figure 4

(Color online) Phase dependence of the Josephson current for weak, intermediate, and strong coupling values of $T_K∕\Delta$ (left column of panels). In the right panel, the spectral function of the impurity $d$ level is presented for each coupling strength and temperature.

Figure 5

(Color online) Phase diagram for the $0\text{−}\pi$ transitions. As discussed in the text, the lower right area corresponds to the 0-phase and the upper left area to the $\pi$ phase. The SNCA data points refer to the $\pi \text{−}{\pi }^{\prime }$ transition (circles), the ${\pi }^{\prime }\text{−}{0}^{\prime }$ transition (triangles), and the $0^{\prime }\text{−}0$ transition (squares). As seen from these curves, the respective transition points as a function of $T$ scale roughly with ${\log }_{10}{T}_K$. The two symbols shown at $T=0$ represent NRG results for the transition point between the thermodynamically stable $\pi$ and 0 states, i.e., the ${\pi }^{\prime }\text{−}{0}^{\prime }$ transition. Specifically, the open inverse triangle represents the transition point from a spin doublet to a singlet state of a Kondo impurity in a bulk superconductor within the NRG evaluation of Satori et al. (Ref. 56). The star on the horizontal axis is approximately the transition point in the NRG analysis for the symmetric Anderson model of the quantum dot contact [Choi et al. (Ref. 27)]. The fat line at the horizontal axis is the regime where the perturbative RG analysis suggests the ${\pi }^{\prime }\text{−}{0}^{\prime }$ transition at zero temperature.

Figure 6

(Color online) Scaling trajectories for various bare coupling values (a) $T_K∕\Delta =0.1$, (b) $T_K∕\Delta =0.2$, (c) $T_K∕\Delta =0.5$, (d) $T_K∕\Delta =1$, and (e) $T_K∕\Delta =2$. The open circles represent fixed points, the dashed line is the fixed point line. The last fixed point in the perturbative regime is found for $T_K∕\Delta \simeq 0.25$. For $T_K∕\Delta \simeq 0.65$ no downturn of the trajectory is observed. In the limit $T_K∕\Delta \to \infty$ the trajectory follows the $J$ axis towards strong coupling.

Figure 7

(Color online) Zero bias conductance $G$ as a function of coupling strength $T_K∕\Delta$ for fixed temperature.

Figure 8

(Color online) Derivative of the zero bias conductance $G(T_K∕\Delta )$ with respect to the coupling parameter at $T∕\Delta =0.36$. Left frame: linear scales; the ${\pi }^{\prime }\text{−}{0}^{\prime }$ transition is at approximately $T_K∕\Delta \simeq 0.4$, for strong coupling $dG∕d(T_K∕\Delta )$ is constant. Right frame: coupling scale is in logarithmic presentation; for weak coupling $dG∕d(T_K∕\Delta )$ diverges logarithmically.

Figure 9

(Color online) Zero bias conductance $G$ as a function of temperature $T∕\Delta$ for fixed coupling strengths $T_K∕\Delta$.

Figure 10

Pseudofermion self-energy $\Sigma \left(i\omega \right)$ and slave boson self-energy $\Pi \left(i\nu \right)$.

Figure 11

Diagonal element ${\mathcal{G}}_d\left(i\omega \right)$ and off-diagonal element ${\mathcal{F}}_d\left(i\omega \right)$ of the impurity Green’s function.

Figure 12

Diagrams for $G_{\mathbf{k}\sigma a,\sigma }^{<}$. The double line represents the dot electron Green’s function and the single line, the conduction electron Green’s function. The left diagram contributes to the normal current and the second diagram contributes to the supercurrent (B4).