  We present microscopic, self-consistent calculations of the superconducting
order parameter and pairing correlations near the interface of an $s$-wave
superconductor and a three-dimensional topological insulator with spin-orbit
coupling. We discuss the suppression of the order parameter by the topological
insulator and show that the equal-time pair correlation functions in the
triplet channel, induced by spin-flip scattering at the interface, are of
$p_x\pm i p_y$ symmetry. We verify that the spectrum at sub-gap energies is
well described by the Fu-Kane model. The sub-gap modes are viewed as interface
states with spectral weight penetrating well into the superconductor. We
extract the phenomenological parameters of the Fu-Kane model from microscopic
calculations, and find they are strongly renormalized from the bulk material
parameters. This is consistent with previous results of Stanescu et al for a
lattice model using perturbation theory in the tunneling limit.
