Quasiclassical Green’s function theory of the Josephson effect in chiral $p$-wave superconductor/diffusive normal metal/chiral $p$-wave superconductor junctions

We study the Josephson effect in chiral $p$-wave superconductor/diffusive normal metal (DN)/chiral $p$-wave superconductor (CP/DN/CP) junctions using quasiclassical Green’s function formalism with proper boundary conditions. The $p_x+i{p}_y$-wave symmetry of superconducting order parameter is chosen which is believed to be the pairing state in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We show that the superconducting pair amplitude induced in DN has an odd-frequency spin-triplet $s$-wave symmetry and is an odd function of Matsubara frequency. Despite the peculiar symmetry properties of the Cooper pairs, the behavior of the Josephson current is rather conventional. We have found that the current-phase relation is almost sinusoidal and the Josephson current is proportional to$\exp (-L∕\xi )$, where $\xi$ is the coherence length in DN and $L$ is the thickness of the DN layer. We also calculate the Josephson current in CP/diffusive ferromagnet metal/CP junctions and show that the 0-$\pi$ transition can be realized by varying temperature or thickness $L$ similar to the case of conventional $s$-wave junctions. The obtained results may help to explore the properties of superconducting state in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

(Color online) Schematic illustration of the model.

Figure 2

(Color online) Temperature dependence of the maximum Josephson current for $Z=10$. The solid lines are the results for $R_d∕R_b=0.1$ and broken lines are the results for $R_d∕R_b=1$. (a) $E_{\mathrm{Th}}∕{\Delta }_0=0.1$ and (b) $E_{\mathrm{Th}}∕{\Delta }_0=1$.

Figure 3

(Color online) Temperature dependence of the maximum Josephson current for $R_d∕R_b=1$. The solid lines are the results for $Z=1$ and the broken lines are the results for $Z=10$. (a) $E_{\mathrm{Th}}∕{\Delta }_0=0.1$ and (b) $E_{\mathrm{Th}}∕{\Delta }_0=1$.

Figure 4

(Color online) The current-phase relation for $Z=10$ and $R_d∕R_b=1$. The solid lines are the results for $T∕T_{\mathrm{C}}=0.01$ and the broken lines are the results for $T∕T_{\mathrm{C}}=0.1$. (a) $E_{\mathrm{Th}}∕{\Delta }_0=0.1$ and (b) $E_{\mathrm{Th}}∕{\Delta }_0=1$.

Figure 5

(Color online) The critical current as a function of DN thickness for $Z=10$. The solid lines are the results for $T∕T_{\mathrm{C}}=0.1$ and the broken lines are the results for $T∕T_{\mathrm{C}}=0.9$. (a) $R_d∕R_b=0.1$ and (b) $R_d∕R_b=1$.

Figure 6

(Color online) Temperature dependence of the critical current for $Z=10$ and $R_d∕R_b=1$. The solid lines are the results for CP/DN/CP junctions, the broken lines are the results for P/DN/P junctions, and the dotted lines are the results for S/DN/S junctions. (a) $E_{\mathrm{Th}}∕{\Delta }_0=0.1$ and (b) $E_{\mathrm{Th}}∕{\Delta }_0=1$.

Figure 7

(Color online) Temperature dependence of the critical current for $Z=10$ and $R_d∕R_b=1$. The solid lines are the results for $h∕{\Delta }_0=0$, the broken lines are the results for $h∕{\Delta }_0=0.5$, and the dotted lines are the results for $h∕{\Delta }_0=1$. (a) $E_{\mathrm{Th}}∕{\Delta }_0=0.1$ and (b) $E_{\mathrm{Th}}∕{\Delta }_0=1$.

Figure 8

(Color online) Current-phase relation for $Z=10$ and $R_d∕R_b=1$ at $T∕T_{\mathrm{C}}=0.1$. The solid lines are the results for $h∕{\Delta }_0=0$, the broken lines are the results for $h∕{\Delta }_0=0.5$, and the dotted line is the result for $h∕{\Delta }_0=1$. (a) $E_{\mathrm{Th}}∕{\Delta }_0=0.1$ and (b) $E_{\mathrm{Th}}∕{\Delta }_0=1$.

Figure 9

(Color online) The critical current as a function of DF thickness for $Z=10$ and $R_d∕R_b=1$. (a) $T∕T_{\mathrm{C}}=0.9$ and (b) $T∕T_{\mathrm{C}}=0.1$. The solid lines are the results for $h∕{\Delta }_0=0.5$ and the broken lines are the results for $h∕{\Delta }_0=1$.