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

TL;DR

This paper demonstrates that all types of superconducting pairings, including even- and odd-frequency singlet and triplet, can be realized in InSb (110) quantum wells coupled with an s-wave superconductor under specific conditions, revealing complex pairing symmetries.

Contribution

It introduces a comprehensive theoretical framework showing the coexistence and emergence of all pairing symmetries in InSb (110) quantum wells with proximity-induced superconductivity, including novel odd-frequency pairings.

Findings

All pairing symmetries can be realized in the system.

Induced order parameters exhibit s-wave, p-wave, and d-wave characters.

Odd-frequency pairings show anisotropic and complex momentum-space structures.

Abstract

We show that all the singlet even-frequency, singlet odd-frequency, triplet even-frequency and triplet odd-frequency pairings, and 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 the 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-wave 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-wave 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 $d_{x^2}$-wave and $d_{xy}$-wave characters can be obtained with respect to the direction of the center-of-mass momentum. ......