Theory of the Josephson effect in unconventional superconducting junctions with diffusive barriers

We study theoretically the Josephson effect in junctions based on unconventional superconductors with diffusive barriers, using the quasiclassical Green’s function formalism. Generalized boundary conditions at junction interfaces applicable to unconventional superconductors are derived by calculating a matrix current within the circuit transport theory. Applying these boundary conditions, we have calculated the Josephson current in structures with various pairing symmetries. A number of predictions are made: (a) nonmonotonic temperature dependence in $d$-wave superconductor/diffusive normal metal/$d$-wave superconductor (D/DN/D) junctions, (b) anomalous current-phase relations in $p$-wave superconductor/diffusive normal metal/$p$-wave superconductor (P/DN/P) junctions, (c) second harmonics in D/DN/D and P/DN/P junctions, (d) a double-peak structure of the critical current in D/DF/D junctions, and (e) enhanced Josephson current by the exchange field in S/DF/P junctions. We have also investigated peculiarities of the Josephson coupling in D/DF/D, P/DF/P, and S/DF/P junctions. An oscillatory behavior of the supercurrent and the second harmonics in the current-phase relation is studied as a function of the length of the diffusive ferromagnet.

Figure 1

(Color online) Current-phase relation for $T∕T_C=0.1$, $R_d∕R_b=0.1$, and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.2$. (a) $s$-wave junctions. (b) $d$-wave junctions. (c) $p$-wave junctions.

Figure 2

(Color online) Current-phase relation for $T∕T_C=0.02$, $R_d∕R_b=0.1$, and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.2$. (a) $s$-wave junctions. (b) $d$-wave junctions. (c) $p$-wave junctions.

Figure 3

(Color online) Current-phase relation for $T∕T_C=0.01$, $Z=10$, $R_d∕R_b=0.1$, and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.2$. (a) and (c) $d$-wave junctions. (b) and (d) $p$-wave junctions.

Figure 4

(Color online) Temperature dependence of the critical current. (a) $s$-wave junctions. (b) $d$-wave junctions. (c) $p$-wave junctions. Curves $a$. $R_d∕R_b=2$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=1$. Curves $b$. $R_d∕R_b=0.5$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=1$. Curves $c$. $R_d∕R_b=2$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$. Curves $d$. $R_d∕R_b=0.5$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$.

Figure 5

(Color online) (a) Temperature dependence of the critical current for $(\alpha ,\beta )=(\pi ∕8,0)$. Curve $a$. $R_d∕R_b=2$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=1$. Curve $b$. $R_d∕R_b=0.5$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=1$. Curve $c$. $R_d∕R_b=2$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$. Curve $d$. $R_d∕R_b=0.5$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$. (b) real and (c) imaginary parts of anomalous Green’s functions $F$ with $R_d∕R_b=2$ and $E_{\mathit{Th}}∕\Delta \left(0\right)=1$.

Figure 6

(Color online) Temperature dependence of the critical current in S/DN/D junctions with $Z=10$, $R_d∕R_b=1$, and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$. $T_{CD}∕T_{CS}=\left(\mathrm{a}\right)5$ and (b) 10.

Figure 7

(Color online) Length dependence of the critical current with $Z=10$ and $R_d∕R_b=0.1$. (a) $s$-wave junctions. (b) $d$-wave junctions. (c) $p$-wave junctions.

Figure 8

(Color online) Length dependence of the critical current with $T∕T_C=0.01$, $Z=10$, and $R_d∕R_b=0.1$. (a) $d$-wave junctions. (b) $p$-wave junctions.

Figure 9

(Color online) Current-phase relation in D/DF/D junctions for $T∕T_C=0.01$, $Z=10$, $R_d∕R_b=1$, and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$.

Figure 10

(Color online) Temperature dependence of the critical current in D/DF/D junctions. $Z=10$ in (a) and (b). $h∕\Delta \left(0\right)=0.4$ in (c) and (d).

Figure 11

(Color online) Length dependence of the critical current in D/DF/D junctions with $T∕T_C=0.1$, $Z=10$, $R_d∕R_b=1$, and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$. We choose $h∕\Delta \left(0\right)=4$ in (b).

Figure 12

(Color online) Current-phase relation in P/DF/P junctions for $T∕T_C=0.01$, $Z=10$, $R_d∕R_b=1$, and $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$.

Figure 13

(Color online) (a) Temperature and (b) length dependence of the critical current in P/DF/P junctions. $Z=10$, $R_d∕R_b=1$, and $(\alpha ,\beta )=(0,0)$. We choose $E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$ in (a) and $T∕T_C=0.1$ in (b).

Figure 14

(Color online) (a) Current-phase relation at $T∕T_C=0.01$ and (b) temperature dependence of the critical current in S/DF/P junctions for $Z=10$, $R_d∕R_b=1,E_{\mathit{Th}}∕\Delta \left(0\right)=0.1$, and $\beta =0$.