Josephson current through superconductor/diffusive-normal-metal/superconductor junctions: Interference effects governed by pairing symmetry

The Josephson effect in superconductor/diffusive-normal-metal/superconductor junctions is studied numerically by using the recursive Green function method. In superconductors, we consider spin-singlet $s$- and $d$-wave and spin-triplet $p$-wave pairing symmetries. The pairing symmetry governs two interference effects in junctions: the formation of a midgap Andreev resonant state at junction interfaces and the proximity effect in diffusive normal metals. A cooperative effect between the two interference effects causes anomalous Josephson current in a $p$-wave symmetry.

Figure 1

(Color online) A schematic figure of SNS junctions of the tight-binding model is shown in (a). We illustrate the pair potentials in momentum space in (b). In (c), a propagation process in a diffusive normal metal is illustrated, where solid circles represent impurities.

Figure 2

(Color online) Current-phase relations in ballistic SNS junctions are shown for several choices of potential barriers $\left(V_B\right)$, where $T∕T_c=0.001$ and ${\Delta }_0∕t=0.01$. Impurity potentials are fixed at $V_I=0$.

Figure 3

(Color online) Current-phase relations in disordered SNS junctions are shown for several choices of impurity potentials $\left(V_I\right)$, where $T∕T_c=0.001$ and ${\Delta }_0∕t=0.01$. The barrier potential is fixed at $V_B=2t$. At $V_I=2t$, normal metals are in the diffusive transport regime.

Figure 4

(Color online) Ensemble average of Josephson current $⟨J⟩$ and their fluctuations $\delta J$ are plotted as a function of transmission probability of potential barrier in the normal state $T_B$. Results for the $s$- and $p_x$-wave symmetries are shown in (a). In (b), only $\delta J$ for the $d_{xy}$- and $p_y$-wave symmetries are shown because $⟨J⟩$ is much smaller than $\delta J$. Note that the scales of (a) and (b) are identical to each other.

Figure 5

(Color online) The maximum amplitudes of Josephson currents in the $p_x$-wave symmetry $J_c\left(p_x\right)$ are compared with those in the $s$-wave symmetry $J_c\left(s\right)$ in (a), where $T_B$ is the transmission probability of potential barriers in the normal states. In (b), $J_c\left(p_x\right)$ and $J_c\left(s\right)$ are plotted as a function of $T_B$ at $T=0.001T_c$.

Figure 6

(Color online) Current-phase relations for the $p_x$-wave symmetry are shown for several temperatures at $V_B=0$, where ${\Delta }_0=0.01t$ in (a) and ${\Delta }_0=0.0001t$ in (b). For comparison, results in the $s$-wave junctions at $T=0.001T_c$ are shown with a solid line in (a) and (b). The amplitude in the $s$-wave symmetry is multiplied by 5 in (a).

Figure 7

(Color online) Local density of states (LDOS) in normal metals $(0\le j\le L_N=71)$ is shown for the $s$-wave and $p_x$-wave symmetries. The left and right superconductors are attached at $j=0$ and $j=71$, respectively. Note that $E_{Th}$ is about $0.3{\Delta }_0$. In schematic pictures, DNM and S denote a diffusive normal metal and a superconductor, respectively. The local density of states shown here is calculated in the absence of Josephson currents. We have confirmed that the results at $\phi =0.99\pi$ qualitatively show the same behavior as those at $\phi =\pi$.

Figure 8

(Color online) Local density of states at $\phi =0$ in normal metals are shown for the $d_{xy}$-wave and $p_y$-wave symmetries in (a) and (b), respectively. In the $d_{xy}$ symmetry, zero-energy peaks at $j=0$ and 71 correspond to the MARS.

Figure 9

The normal conductance versus the length of disordered region is plotted, where $V_I=2.0t$, $\mu =2.0t$, and $V_B=0$.