Gapped superconductivity with all symmetries in InSb (110) quantum wells in proximity tos-wave superconductor in Fulde-Ferrell-Larkin-Ovchinnikov phase or with a supercurrent

We show that all the singlet even-frequency, singlet odd-frequency, triplet even-frequency, and triplet odd-frequency pairings, together with the corresponding order parameters (gaps), can be realized in InSb (110) spin-orbit-coupled quantum well in proximity to $s$-wave superconductor in Fulde-Ferrell-Larkin-Ovchinnikov phase or with a supercurrent. It is revealed that with the singlet even-frequency order parameter induced by the proximity effect, triplet even-frequency pairing is induced due to the broken spin-rotational symmetry by the spin-orbit coupling. Since the translational symmetry is broken by the center-of-mass momentum of Cooper pair in Fulde-Ferrell-Larkin-Ovchinnikov phase or with a supercurrent, the singlet odd-frequency pairing can be induced. With the translational and spin-rotational asymmetries, the triplet odd-frequency pairing is also realized. Then, we show that the corresponding order parameters can be obtained from the self-energy of the electron-electron Coulomb interaction with the dynamic screening. The singlet and the induced triplet even-frequency order parameters are found to exhibit the conventional $s$- and $p$-wave characters in the momentum space, respectively. Whereas for the induced odd-frequency order parameters in quantum well, the singlet and triplet ones show the $p$- and $d$-wave characters, respectively. Moreover, the $p$-wave character of the singlet odd-frequency order parameter exhibits anisotropy with respect to the direction of the center-of-mass momentum. While for the triplet one, we find that ${d}_{{x}²}$- and ${d}_xy$-wave characters can be obtained with respect to the direction of the center-of-mass momentum. We show that at proper density, the singlet even-frequency order parameter is suppressed and the induced singlet odd-frequency, triplet even-frequency, and triplet odd-frequency ones can be detected experimentally.