Proximity effect in a ferromagnetic semiconductor with spin-orbit interactions

We study theoretically the proximity effect in a ferromagnetic semiconductor with the Rashba spin-orbit interaction. The interplay between the exchange potential and the spin-orbit interactions enriches the symmetry variety of Cooper pairs depending on degree of disorder in a ferromagnet. In the ballistic limit, spin-singlet s-wave Cooper pairs are the most dominant in the presence of strong spin-orbit interaction because the spin-momentum locking stabilizes a Cooper pair consisting of two electrons of time-reversal partner to each other. We will show that the spin-orbit interactions generate equal-spin-triplet p-wave pairs. In the dirty regime, on the other hand, equal-spin-triplet s-wave pairs are dominant because random impurity potentials release the locking. The exchange splitting in the conduction band causes the imbalance between two equal-spin pairing components. In a half-metallic ferromagnet, only an equal-spin pairing component survives and carries the spin-polarized supercurrent. We discuss the effects of the spin-orbit interaction on the Josephson current.