We show that gapped triplet superconductivity, i.e., a triplet superconductor with a triplet order parameter, can be realized in strong spin-orbit-coupled (100) quantum wells in proximity to an s-wave superconductor. It is revealed that in quantum wells with the singlet order parameter induced from the superconducting proximity effect, not only can the triplet pairings arise due to spin-orbit coupling, but the triplet order parameter can also 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 normal metal in proximity to an s-wave superconductor [Rev. Mod. Phys. 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 (px±ipy) one with the d vector parallel to the effective magnetic field due to the spin-orbit coupling. Finally, we perform a 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 is shown that with proper electron density, the minimum of the renormalized singlet and the maximum of the induced triplet order parameters are comparable, and hence they can be experimentally distinguished.
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