Angular dependence of Josephson currents in unconventional superconducting junctions

Josephson effect in junctions between unconventional superconductors is studied theoretically within the model describing the effects of interface roughness. The particularly important issue of applicability of the frequently used Sigrist-Rice (SR) formula for Josephson current in $d$-wave superconductor/insulator/$d$-wave superconductor junctions is addressed. We show that although the SR formula is not applicable in the ballistic case, it works well for rough interfaces when the diffusive normal metal regions exist between the $d$-wave superconductor and the insulator. It is shown that the SR approach only takes into account the component of the $d$-wave pair potential symmetric with respect to an inversion around the plane perpendicular to the interface. Similar formula can be derived for general unconventional superconductors with arbitrary angular momentum $l$.

Figure 1

(Color online) Schematic illustration of the models of (a) $D∕DN∕I∕DN∕D$ and (b) $P∕DN∕I∕DN∕P$ junctions.

Figure 2

(Color online) Maximum Josephson current for $d$-wave junctions. (a) and (b) $DID$ junctions. (c) and (d) $D∕DN∕I∕DN∕D$ junctions. Solid lines in (c), (d) are proportional to $\cos 2\alpha \cos 2\beta$.

Figure 3

(Color online) Maximum Josephson current for $p$-wave junctions. (a) and (b) $PIP$ junctions. (c) and (d) $P∕DN∕I∕DN∕P$ junctions with solid lines that are proportional to $\cos \alpha \cos \beta$.

Figure 4

(Color online) Schematic illustration of the inversion symmetry of the pair potential around the plane normal to the interface. (a) $d_{x^2-y^2}$ wave, (b) $p_x$ wave, (c) $d_{xy}$ wave, and (d) $p_y$ wave. The $\varphi$ dependences are given by $\cos 2\varphi$, $\cos \varphi$, $\sin 2\varphi$, and $\sin \varphi$.