Anomalous superconducting proximity effect and coherent charge transport in semiconducting thin film with spin-orbit interaction

We present a microscopic theory of the superconducting proximity effect in a semiconducting thin film with spin-orbit interaction ($N_{SO}$) in an external magnetic field. We demonstrate that an effective 1D Hamiltonian which describes induced superconductivity in $N_{SO}$ in contact with a usual $s$-wave superconductor possesses not only spin-singlet induced superconducting order parameter term, as commonly adopted, but spin triplet order parameter term also. Using this new effective Hamiltonian we confirm previous results for a normal current across contacts of $N_{SO}$ with a normal metal and for a Josephson current with the same $N_{SO}$ with induced superconductivity, obtained previously in the framework of the phenomenological Hamiltonian without spin-triplet terms. However, a calculated current-phase relation across the transparent contact between $N_{SO}$ with induced superconductivity in magnetic field and usual $s$-wave superconductor differs significantly from previous results. We suggest the experiment which can confirm our theoretical predictions.