Quantitative model for the $I_{C}R$ product in $d$-wave Josephson junctions

We study theoretically the Josephson effect in $d$-wave superconductor/diffusive normal metal/insulator/diffusive normal metal/$d$-wave superconductor (D/DN/I/DN/D) junctions. This model aims to describe practical junctions in high-$T_C$ cuprate superconductors, in which the product of the critical Josephson current $\left(I_C\right)$ and the normal state resistance $\left(R\right)$ (the so-called $I_{C}R$ product) is very small compared to the prediction of the standard theory for clean $d$-wave superconductor/insulator/$d$-wave superconductor (DID) junctions. We show that the $I_{C}R$ product in D/DN/I/DN/D junctions can be much smaller than that in DID junctions. The proposed theory describes the behavior of $I_{C}R$ products quantitatively in high-$T_C$ cuprate junctions.

Figure 1

(Color online) Schematic illustration of the model for the D/DN/I/DN/D junction.

Figure 2

(Color online) $I_{C}R$ value for $Z^{\prime }=1$, $R_d∕R_b=0.1$, and $(\alpha ,\beta )=(0,0)$.

Figure 3

(Color online) $I_{C}R$ value for $E_{\mathit{Th}}∕\Delta \left(0\right)=1$, $R_d∕R_b=0.1$, and $(\alpha ,\beta )=(0,0)$.

Figure 4

(Color online) $I_{C}R$ value for $E_{\mathit{Th}}∕\Delta \left(0\right)=1$, $Z^{\prime }=1$, and $(\alpha ,\beta )=(0,0)$.

Figure 5

(Color online) $I_{C}R$ value for $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$, $Z^{\prime }=0.1$, $R_d∕R_b=0.1$, and $R_d∕R_b^{\prime }=10$.

Figure 6

(Color online) Comparison between the present theory (solid line), experimental data (Ref. 6) (dotted line), and the TK theory (Ref. 13) (broken line). $I_C$ is plotted as a function of temperature, taking $R=0.375\Omega$ and $\alpha =\beta =0$ for theoretical plots. We choose $E_{\mathit{Th}}∕\Delta \left(0\right)=3$, $Z^{\prime }=0.1$, $R_d∕R_b=0.01$, and $R_d∕R_b^{\prime }=100$ in the present theory, and $Z=10$ in the TK theory.