Josephson Transport through a Hubbard Impurity Center

We investigate the Josephson transport through a thin semiconductor barrier containing impurity centers with the on-site Hubbard interaction $u$ of an arbitrary sign and strength. We find that in the case of the repulsive interaction the Josephson current changes sign with the temperature increase if the energy of the impurity level $\epsilon$ (measured from the Fermi energy of superconductors) falls in the interval $(-u,0)$. We predict strong temporal fluctuations of the current if only a few centers are present within the junction. In the case of the attractive impurity potential ($u<0$) and at low temperatures, the model is reduced to the effective two level Hamiltonian allowing thus a simple description of the nonstationary Josephson effect in terms of pair tunneling processes.

Figure 1

The Josephson current $I^{\prime }=I/I_0(\varphi )$ as a function of the energy of the single-occupied impurity state $\epsilon$ for $u/\Delta =0.5$ and different temperatures: $T^{\prime }=T/\Delta =0,0.02,0.04,0.1,0.2$ for plots 1–5, respectively.

Figure 2

(a) The Josephson current $I^{\prime }=I/I_0(\varphi )$ as a function of temperature $T^{\prime }=T/\Delta$ for different positive interactions $u$ and $\epsilon =-u/2.$ (b) Dependence of the Josephson current on ${\epsilon }^{\prime }=\epsilon +u/2$ for $u/\Delta =-0.5$ and different temperatures: $T^{\prime }=0,0.05,0.1,0.2$ (top to bottom).

Figure 3

Impurity energy $E^{\prime }=(E-{\epsilon }_2/2)/E_0$ as a function of the phase $\varphi ={\varphi }_1-{\varphi }_2$ (solid lines) and the corresponding Josephson currents $I_{\pm }^{\prime }=I_{\pm }/(e{E}_0)$ (dashed lines).

Figure 4

Voltage-current dependence of the shunted Josephson contact for receptivities $R{e}^2/\hslash =1.6,1.2,0.8$ (top to bottom). Upper inset shows the setup while the lower one shows the differential resistance ($dV/dI$) as a function of the applied voltage corresponding to the lower graph ($R{e}^2/\hslash =0.8$).