Coulomb Blockade of Proximity Effect at Large Conductance

We consider the proximity effect in a normal dot coupled to a bulk superconducting reservoir by the tunnel contact with large normal conductance. Coulomb interaction in the dot suppresses the proximity minigap induced in the normal part of the system. We find exact expressions for the thermodynamic and tunneling minigaps as functions of the junction’s capacitance. The tunneling minigap interpolates between its proximity-induced value in the regime of weak Coulomb interaction to the Coulomb gap in the regime of strong interaction. In the intermediate case a nonuniversal two-step structure of the tunneling density of states is predicted. The charge quantization in the dot is also studied.

Figure 1

The minigap in the thermodynamic DOS as a function of the Coulomb energy $E_C$. Dashed lines illustrate the asymptotic behaviors (8, 9). The crossover from weak to strong fluctuations regime occurs at $E_C\delta /E_g^2\sim \log (\Delta /E_g)/2$. This figure shows the case $\log (\Delta /E_g)=5$.

Figure 2

The minigap in the tunneling DOS as a function of the Coulomb energy. For small $E_C$ it follows ${\tilde{E}}_g$ given by Eq. (8). In the opposite limit $E_g^{\mathrm{tun}}=E_C$. This figure is plotted assuming $\log (\Delta /E_g)=5$ and $G=40$. Inset: two-step tunneling density of states in the intermediate regime. The charging energy is such that $E_C\delta /E_g^2=2.8$ (as shown by the dot on the main graph) which corresponds to $q=0.95$.

Figure 3

The dependence of the thermodynamic minigap, the tunneling minigap, and the average charge on the gate voltage $V$ (vertical axis) at three different values of Coulomb energy. The parameters are $G=40$, $\log (\Delta /E_g)=5$.