Numerical subgap spectroscopy of double quantum dots coupled to superconductors

Double quantum dot nanostructures embedded between two superconducting leads or in a superconducting ring have complex excitation spectra inside the gap which reveal the competition between different many-body phenomena. We study the corresponding two-impurity Anderson model using the nonperturbative numerical renormalization group (NRG) technique and identify the characteristic features in the spectral function in various parameter regimes. At half-filling, the system always has a singlet ground state. For large hybridization, we observe an inversion of excited interdot triplet and singlet states due to the level repulsion between two subgap singlet states. The Shiba doublet states split in two cases: (a) at nonzero superconducting phase difference and (b) away from half-filling. The most complex structure of subgap states is found when one or both dots are in the valence fluctuation regime. Doublet splitting can lead to a parity-changing quantum phase transition to a doublet ground state in some circumstances. In such cases, we observe very different spectral weights for the transitions to singlet or triplet excited Shiba states: the triplet state is best visible on the valence-fluctuating dot, while the singlets are more pronounced on the half-filled dot.

Figure 1

Schematic representation of the system: double quantum dot coupled to superconducting leads forming a ring pierced by the magnetic flux. The tunneling probe is very weakly coupled to either or both quantum dots.

Figure 2

Left-right symmetric DQD system at half-filling. (a) Local spectral function $A_1\left(\omega \right)$. (b) Interdot spectral function $A_{12}\left(\omega \right)$: We plot the positive frequency part on a logarithmic scale.

Figure 3

Expectation values and Shiba state energies in the left-right symmetric DQD system at half-filling. Results, where applicable, are shown for both normal-state and superconducting leads. The arrow indicates the point where the on-site pairing changes sign and the spin-spin correlations in the superconducting and normal-state cases start to deviate.

Figure 4

Spectral function $A_1\left(\omega \right)$ and the diagram of the subgap Shiba states for $\delta =0$ as a function of the superconducting phase difference $\varphi$.

Figure 5

Dependence on the hybridization strength $\Gamma$ in the left-right symmetric case at half-filling. (a) Subgap Shiba states for $\Gamma =0.045$. The arrow brings to attention that the interdot triplet Shiba state is below the interdot singlet state. (b) Spin correlation function $\langle {\mathbf{S}}_1·{\mathbf{S}}_2\rangle \left(t\right)$ for a range of $\Gamma$ (indicated near the corresponding curves) across the singlet-singlet crossover at ${\Gamma }_t\approx 0.04$.

Figure 6

Dependence on the interdot capacitive coupling at half-filling.

Figure 7

Spectral function $A_1\left(\omega \right)$ in the left-right symmetric DQD system away from half-filling. (a) Small interdot tunneling $t={10}^{-3}$. (b) Subgap Shiba state diagram for $t={10}^{-3}$. In the range $\delta \sim 0.1$, the $S=1/2$ and the excited $S=0$ state have small but nonzero energy. (c) and (d) Spectra at stronger interdot coupling.

Figure 8

Expectation values in the left-right symmetric DQD system away from half-filling. The value of $t$ is the same as in Fig. 7. Note the position of the triplet state below the excited singlet for $\delta >0.1$.

Figure 9

Spectral function $A\left(\omega \right)$ and the diagram of the subgap Shiba states for ${\delta }_1={\delta }_2=0.1$ as a function of the superconducting phase difference $\varphi$.

Figure 10

Subgap Shiba state diagram at finite interdot capacitive coupling $V/D=0.1$. Top panel: $t={10}^{-3}$, to be compared with Figs. 7 and 7. Bottom panel: $t={10}^{-2}$, to be compared with Figs. 7 and 8. The arrows indicate the new feature: a region close to the VF point where the interdot coupling induces a phase transition to the doublet state, corresponding to the emergence of a local moment regime with a single electron in the DQD (quarter-filling).

Figure 11

Spectral function on dot 1 (left panels) and dot 2 (right panels) for a range of interdot couplings $t$. The quantum dot 2 is kept at ${\delta }_2=0$, which is the Kondo regime. The arrows mark the different spectroscopically observable excited states (ES).

Figure 12

Subgap spectrum corresponding to the center row in Fig. 11.

Figure 13

Spectral function on dot 1 (left panels) and dot 2 (right panels) in the superconducting state for a range of interdot couplings $t$. The quantum dot 2 is kept at ${\delta }_2=0.1$, which is the valence fluctuation regime. The last row shows the corresponding many-particle Shiba state diagrams.

Figure 14

Two-impurity Kondo effect manifesting through the quantum phase transition between two different many-particle subgap singlet states in the model with exchange coupling $J$ between the dots and no particle transfer ($t=0$).