Josephson effect in a multiorbital model for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

We study Josephson currents between $s$-wave/spin-triplet superconductor junctions by taking into account details of the band structures in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, such as three conduction bands and spin-orbit interactions in the bulk and at the interface. We assume five superconducting order parameters in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$: a chiral $p$-wave symmetry and four helical $p$-wave symmetries. We calculate the current-phase relationship $I\left(\phi \right)$ in these junctions, where $\phi$ is the macroscopic phase difference between the two superconductors. The results for a chiral $p$-wave pairing symmetry show that a $cos\left(\phi \right)$ term appears in the current-phase relation because of time-reversal symmetry (TRS) breaking. On the other hand, this $cos\left(\phi \right)$ term is absent in the helical pairing states that preserve TRS. We also study the dependence of the maximum Josephson current $I_c$ on an external magnetic flux $\mathrm{\Phi }$ in a corner junction. The calculated $I_c\left(\mathrm{\Phi }\right)$ obeys $I_c\left(\mathrm{\Phi }\right)\ne I_c(-\mathrm{\Phi })$ in a chiral state and $I_c\left(\mathrm{\Phi }\right)=I_c(-\mathrm{\Phi })$ in a helical state. We calculate $I_c\left(\mathrm{\Phi }\right)$ in a corner superconducting quantum interference device (SQUID) and a symmetric SQUID geometry. In the latter geometry, $I_c\left(\mathrm{\Phi }\right)=I_c(-\mathrm{\Phi })$ is satisfied for all the pairing states and it is impossible to distinguish a chiral state from a helical one. On the other hand, a corner SQUID always gives $I_c\left(\mathrm{\Phi }\right)\ne I_c(-\mathrm{\Phi })$ and $I_c\left(\mathrm{\Phi }\right)=I_c(-\mathrm{\Phi })$ for a chiral and a helical state, respectively. Experimental tests of these relations in corner junctions and SQUIDs may serve as a tool for unambiguously determining the pairing symmetry in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

(a) Lattice model of the junction considered in this paper. (b) Schematic illustrations of an SRO (${\mathrm{Sr}}_2{\mathrm{RuO}}_4$) /NM (normal metal)/$s$-wave superconductor single Josephson junction.

Figure 2

Current-phase relation in the absence of interface Rashba spin-orbit interaction (${\lambda }_R$) for (a) the chiral $p$ wave ($E_u$) in the single-band model, (b) the chiral $p$ wave ($E_u$) in the multiband model, (c) the helical $p$ wave ($A_{1u}$) in the single-band model, and (d) the helical $p$ wave ($A_{1u}$) in the multiband model.

Figure 3

Current-phase relation $I\left(\phi \right)$ in the presence of interface Rashba spin-orbit interaction (${\lambda }_R>0$) for (a) the chiral $p$ wave ($E_u$) in the single-band model, (b) the chiral $p$ wave ($E_u$) in the multiband model, (c) the helical $p$ wave ($A_{1u}$) in the single-band model, and (d) the helical $p$ wave ($A_{1u}$) in the multiband model.

Figure 4

Schematic illustrations of the SRO /NM/$s$-wave superconductor single Josephson junctions considered in this paper. Current-phase relations in junctions (a)–(c) were calculated independently. The results were then combined to calculate the magnetic-field dependence of the corner junction, corner SQUID, and symmetric SQUID.

Figure 5

Schematic illustration of an SRO/NM/$s$-wave corner junction.

Figure 6

Schematic illustration of SRO/NM/$s$-wave SQUIDs: (a) corner SQUID and (b) symmetric SQUID.

Figure 7

Fraunhofer pattern in the SRO/NM/$s$ wave. (a) Schematic illustration of a corner junction, and the corresponding Fraunhofer pattern for (b) chiral ($E_u$) pairing, (c) helical ($A_{1u}$) pairing, and (d) helical ($A_{2u}$) pairing.

Figure 8

(a) Maximum Josephson current $I_c$ in a corner SQUID for (b) chiral ($E_u$), (c) helical ($A_{1u}$), and (d) helical ($A_{2u}$) pairings.

Figure 9

(a) Symmetric SQUID and the corresponding $I_c$ for (b) chiral ($E_u$) and (c) helical ($A_{1u}$) pairings.

Figure 10

$I_1^s$ [the coefficient of $sin\left(\phi \right)$ in the Fourier series of the current-phase relation in the junction] is plotted as a function of $\lambda$ (a), (b), ${\lambda }_R$ (c), (d), and $t_5$ (e), (f). Chiral pairing applies in (a), (c), and (e), and helical pairing with ${\mathrm{A}}_{1u}$ symmetry in (b), (d), and (f). $t_5=0$ and ${\lambda }_R=0$ in (a) and (b). $t_5/t_1=0.15$ and $\lambda =0$ in (c) and (d). $\lambda =0$ and $\lambda /t_1=0.3$ in (e) and (f).