We compute using envelope function calculations the energy and the oscillator strength of excitons in zinc-blende/wurtzite quantum wells (QWs), such as those that appear in many examples of semiconductor nanowires, and in basal plane stacking faults (BSFs). We address specifically the model-case of GaN. In addition to the electron-hole Coulomb interaction, we account for the quantum-confined Stark effect. We demonstrate that despite the type-II band alignment at the zinc-blende/wurtzite interfaces, a significant binding and a rather strong oscillator strength are preserved by excitonic effects. When adjacent crystal phase QWs are coupled together, we compute increased as well as decreased exciton oscillator strength with respect to the single QW case, depending on the QW-QW coupling scheme. Comparing the results of our calculations with available data, we finally conclude in favor of the absence of built-in electric fields perpendicular to the BSF planes.

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