Pumping through a quantum dot in the proximity of a superconductor

We study adiabatic pumping through a quantum dot tunnel coupled to one normal and one superconducting lead. We generalize a formula which relates the pumped charge through a quantum dot with Coulomb interaction to the instantaneous local Green’s function of the dot, to systems containing a superconducting lead. First, we apply this formula to the case of a noninteracting, single-level quantum dot in different temperature regimes and for different parameter choices, and we compare the results with the case of a system comprising only normal leads. Then we study the infinite-$U$ Anderson model with a superconducting lead at zero temperature, and we discuss the effect of the proximity of the superconductor on the pumped charge.

Figure 1

Sketch of the dot attached to two leads. We are interested in the N-dot-S system, where the left lead is normal conducting and the right lead is superconducting, which we will compare to the N-dot-N system, where both leads are normal conducting. Left and right leads are attached to the dot via tunnel barriers. The tunneling strength is described by the respective rate ${\Gamma }_{\mathrm{N}}$ and ${\Gamma }_{\mathrm{S}}$. Possible time-dependent parameters are the strength of the tunnel couplings ${\Gamma }_{\mathrm{N}}$ and ${\Gamma }_{\mathrm{S}}$, and the dot-level position $ϵ$.

Figure 2

Ratio between the pumped charge in the N-dot-S and in the N-dot-N systems, $Q_{X,Y}^{\mathrm{S}}∕Q_{X,Y}^{\mathrm{N}}$, as a function of ${\overline{\Gamma }}_{\mathrm{S}}∕{\overline{\Gamma }}_{\mathrm{N}}$. The curve is the same for any couple of pumping parameters $X$, $Y$. Temperature is equal to zero and the bare dot level is at resonance.

Figure 3

Ratio between the pumped charge in N-dot-S and in the N-dot-N system, $Q_{X,Y}^{\mathrm{S}}∕Q_{X,Y}^{\mathrm{N}}$, as a function of ${\overline{\Gamma }}_{\mathrm{S}}∕{\overline{\Gamma }}_{\mathrm{N}}$. Temperature is equal to $k_{B}T=1.5{\overline{\Gamma }}_{\mathrm{N}}$. The pumping parameters are ${\Gamma }_{\mathrm{N}}$ and ${\Gamma }_{\mathrm{S}}$ (solid line), ${\Gamma }_{\mathrm{N}}$ and $ϵ$ (dashed line), and ${\Gamma }_{\mathrm{S}}$ and $ϵ$ (dotted and dashed line). The bare dot level is at resonance.

Figure 4

Pumped charge in the N-dot-S system (a) and in the N-dot-N (b) system as a function of the average level position $\overline{ϵ}$. Temperature is set to $k_{B}T={\overline{\Gamma }}_{\mathrm{N}}$ and ${\overline{\Gamma }}_{\mathrm{N}}={\overline{\Gamma }}_{\mathrm{S}}$. The pumping parameters are $\{{\Gamma }_{\mathrm{N}},{\Gamma }_{\mathrm{S}}\}$ (solid line), $\{{\Gamma }_{\mathrm{N}},ϵ\}$ (dashed line), and $\{{\Gamma }_{\mathrm{S}},ϵ\}$ (dashed-dotted line). The units on the $x$ axes are the ones determining the width of the respective function.

Figure 5

Pumped charge $Q_{{\Gamma }_{\mathrm{N}},{\Gamma }_{\mathrm{S}}}^{\mathrm{S}}$ (a) and $Q_{ϵ,{\Gamma }_{\mathrm{S}}}^{\mathrm{S}}$ (b) as a function of the average level position for different values of $\Delta$ (solid lines) and $Q_{{\Gamma }_{\mathrm{N}},{\Gamma }_{\mathrm{S}}}^{\mathrm{N}}$ and $Q_{{\Gamma }_{\mathrm{N}},ϵ}^{\mathrm{N}}$ (dashed lines). Temperature is zero and we consider symmetric barriers $({\overline{\Gamma }}_{\mathrm{N}}={\overline{\Gamma }}_{\mathrm{S}})$. The cutoff energy is $W=20{\overline{\Gamma }}_{\mathrm{N}}$.

Figure 6

The peak position ${\epsilon }_{\text{peak}}$ is shown as a function of the superconducting gap for different pairs of pumping parameters on a logarithmic scale. The pumping parameters are $\{{\Gamma }_{\mathrm{N}},{\Gamma }_{\mathrm{S}}\}$ (solid line) and $\{{\Gamma }_{\mathrm{N}},ϵ\}$ (dashed line). The temperature is zero and ${\overline{\Gamma }}_{\mathrm{N}}={\overline{\Gamma }}_{\mathrm{S}}=0.05W$.

Figure 7

Peak width $\Delta {\epsilon }_{\text{peak}}$ for the pumped charge plotted as function of the superconducting gap $\Delta$. The pumping parameters are $\{{\Gamma }_{\mathrm{N}},{\Gamma }_{\mathrm{S}}\}$. The temperature is zero and ${\overline{\Gamma }}_{\mathrm{N}}={\overline{\Gamma }}_{\mathrm{S}}=0.05W$.

Figure 8

Spectral density of the dot in the N-dot-N (solid line) and in the N-dot-S system (dashed line) in the Kondo regime ($ϵ=-0.25W$, ${\Gamma }_{\mathrm{N}}={\Gamma }_{\mathrm{S}}=0.05W$). The gap of the superconductor is half as big as the bandwidth cutoff $(\Delta =0.5W)$. The Kondo temperature of the N-dot-N system and the N-dot-S system differ by a huge factor: $T_k^{\mathrm{N}}=7.8\times {10}^{-4}W$ and $T_k^{\mathrm{S}}=1.3\times {10}^{-6}W$.