The paper proposes a self-consistent Green function description of the induced surface superconductivity in a disordered three-dimensional topological insulator (TI) coupled to an s-wave superconductor. We recover earlier results regarding the induced spin-triplet p-wave pairing, showing that a mixture of p- and s-wave pair correlations appears as a result of broken spin-rotation symmetry on the helical surface of the TI. Unlike the s-wave pairing, the p-wave component is found to be suppressed in dirty TIs in which the elastic mean-free path is much smaller than the superconducting coherence length. The suppression is due to the generic nonlocality of the spin-triplet correlations, which makes them strongly dependent on the mean-free path in a disordered system. In dirty TIs the induced superconductivity is predicted to be predominantly s-wave like. In cleaner TIs, however, the p-wave component may reach a magnitude comparable with (but not larger than) the s-wave pairing.
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