Josephson effect through a quantum dot array

We analyze the ground state properties of an array of quantum dots connected in series between superconducting electrodes. This system is represented by a finite Hubbard chain coupled at both ends to BCS superconductors. The ground state is obtained using the Lanczos algorithm within a low energy theory in which the bulk superconductors are replaced by effective local pairing potentials. We study the conditions for the inversion of the sign of the Josephson coupling ($\pi$-junction behavior) as a function of the model parameters. Results are presented in the form of phase diagrams, which provide a direct overall view of the general trends as the size of the system is increased, exhibiting a strong even-odd effect. The analysis of the spin-spin correlation functions and local charges give further insight into the nature of the ground state and how it is transformed by the presence of superconductivity in the leads. Finally, we study the scaling of the Josephson current with the system size and relate these results with previous calculations of Josephson transport through a Luttinger liquid.

Figure 1

(Color online) $ϵ\text{−}U$ phase diagram of the single quantum dot case in the ZBWM. The grayscale goes from dark gray (blue) corresponding to the 0 state to the lightest gray (light orange) corresponding to the $\pi$ state. The states $0^{\prime }$ and ${\pi }^{\prime }$ are indicated by intermediate shades of gray (light blue and orange). Dashed lines indicate the boundary of the $\pi$ region when calculated using the HF and the SBMF approximations. The inset shows these boundaries when taking into account the full spectrum of the superconducting electrodes with $\Gamma =t$. The parameters are taken in units of $t=\Delta$.

Figure 2

(Color online) $ϵ\text{−}U$ phase diagrams for $N=2,3,4,5$. The color convention is the same as in Fig. 1. For a better visualization, we have added black borderlines to separate the phases.

Figure 3

(Color online) Total charge per spin in the dot region for $U=15$ for $N=3$ (upper panel) and $N=4$ (lower panel).

Figure 4

(Color online) Average first-neighbor spin-spin correlation inside the chain for the same parameters as in Fig. 3. The solid (red) and dashed (blue) lines correspond to the superconducting and normal cases, respectively.

Figure 5

(Color online) Spin correlation between the left lead and the first site of the chain. Same parameters and line convention as in Fig. 4.

Figure 6

(Color online) Spatial distribution of first-neighbor spin correlation in the half-filled case for $U=20$. The inset in the lower panel represents the charge per spin on each site of the system. Site 1 indicates the left electrode. We use the line convention of Fig. 4.

Figure 7

(Color online) Current-phase characteristics for $U=10$ at half-filling. Curves in the positive half-plane correspond to $N=2,4,6$ (from top to bottom). Those in the negative half-plane corresponds to $N=3,5,7$ (from bottom to top). The current is scaled by an exponential $\exp \left(\alpha N\right)$, where $\alpha$ has been fitted to a value $\approx 1.8$. $I\left(\varphi \right)$ is in units of $e\Delta ∕\hslash$.

Figure 8

(Color online) Current-phase characteristics for $U=-10$ at half-filling and $N=1,2,3,4,5,6,7$ from bottom to top. Dashed lines correspond to even $N$. The current is scaled by a factor $N$.