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Gapped triplet $p$-wave superconductivity in strong spin-orbit-coupled semiconductor quantum wells in proximity to $s$-wave superconductor

We show that the {\it gapped} triplet superconductivity, i.e., a triplet superconductor with triplet order parameter, can be realized in strong spin-orbit-coupled quantum wells in proximity to $s$-wave superconductor. It is revealed that with the singlet order parameter induced from the superconducting proximity effect, in quantum wells, not only can the triplet pairings arise due to the spin-orbit coupling, but also the triplet order parameter can be induced due to the repulsive effective electron-electron interaction, including the electron-electron Coulomb and electron-phonon interactions. This is a natural extension of the work of de Gennes, in which the repulsive-interaction-induced singlet order parameter arises in the normal metal in proximity to $s$-wave superconductor [Rev. Mod. Phys. {\bf 36}, 225 (1964)]. Specifically, we derive the effective Bogoliubov-de Gennes equation, in which the self-energies due to the effective electron-electron interactions contribute to the singlet and triplet order parameters. It is further shown that for the singlet order parameter, it is efficiently suppressed due to this self-energy renormalization; whereas for the triplet order parameter, it is the $p$-wave ($p_x\pm ip_y$) one with the ${\bf d}$-vector parallel to the effective magnetic field due to the spin-orbit coupling. Finally, we perform the numerical calculation in InSb (100) quantum wells. Specifically, we reveal that the Coulomb interaction is much more important than the electron-phonon interaction at low temperature. Moreover, it shows that with proper electron density, the minimum of the renormalized singlet and the maximum of the induced triplet order parameters are comparable, and hence can be experimentally distinguished.