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We show that the {\it gapped} triplet superconductivity, i.e., a tripletsuperconductor with triplet order parameter, can be realized in strongspin-orbit-coupled quantum wells in proximity to $s$-wave superconductor. It isrevealed that with the singlet order parameter induced from the superconductingproximity effect, in quantum wells, not only can the triplet pairings arise dueto the spin-orbit coupling, but also the triplet order parameter can be induceddue to the repulsive effective electron-electron interaction, including theelectron-electron Coulomb and electron-phonon interactions. This is a naturalextension of the work of de Gennes, in which the repulsive-interaction-inducedsinglet order parameter arises in the normal metal in proximity to $s$-wavesuperconductor [Rev. Mod. Phys. {\bf 36}, 225 (1964)]. Specifically, we derivethe effective Bogoliubov-de Gennes equation, in which the self-energies due tothe effective electron-electron interactions contribute to the singlet andtriplet order parameters. It is further shown that for the singlet orderparameter, it is efficiently suppressed due to this self-energyrenormalization; 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 effectivemagnetic field due to the spin-orbit coupling. Finally, we perform thenumerical calculation in InSb (100) quantum wells. Specifically, we reveal thatthe Coulomb interaction is much more important than the electron-phononinteraction at low temperature. Moreover, it shows that with proper electrondensity, the minimum of the renormalized singlet and the maximum of the inducedtriplet order parameters are comparable, and hence can be experimentallydistinguished.

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