Effects of hybridization and spin-orbit coupling to induce odd-frequency pairing in two-band superconductors

The effect of spin-independent hybridization potential and spin-orbit coupling on two-band superconductor with equal-time s-wave interband pairing order parameter is investigated theoretically. To study symmetry classes in two-band superconductors, the Gor'kov equations are solved analytically. By defining spin-singlet and spin-triplet s-wave order parameter due to two-band degree of freedom, the symmetry classes of Cooper pair are studied. For spin singlet case, it is shown that spin-independent hybridization generates Cooper pair that belongs to even-frequency spin singlet even-momentum even-band parity (ESEE) symmetry class for both intraband and interband pairing correlations. For spin-triplet order parameter, intraband pairing correlation generates odd-frequency spin triplet even-momentum even-band parity (OTEE) symmetry class, whereas interband pairing correlation generates even-frequency spin-triplet even-momentum odd-band parity (ETEO) class. For the spin singlet, spin-orbit coupling generates pairing correlation that belongs to odd-frequency spin-singlet odd-momentum even-band parity (OSOE) symmetry class and even-frequency spin-singlet even-momentum even-band parity (ESEE) for intraband and interband pairing correlation, respectively. In the spin-triplet case for intraband and interband correlation, spin-orbit coupling generates even-frequency spin-triplet odd-momentum even-band parity (ETOE) and even-frequency spin-triplet even-momentum odd-band parity (ETEO), respectively. Our results are in agreement with previous experimental and theoretical studies.