  We further investigate a class of time-reversal-invariant two-band s-wave
topological superconductors introduced in Phys. Rev. Lett. 108, 036803 (2012).
We show how, in the presence of time-reversal symmetry, Z_2 invariants that
distinguish between trivial and non-trivial quantum phases can be constructed
by considering only one of the Kramers' sectors in which the Hamiltonian
decouples into. We find that the main features identified in our original 2D
setting remain qualitatively unchanged in 1D and 3D, with non-trivial
topological superconducting phases supporting an odd number of Kramers' pairs
of helical Majorana modes on each boundary, as long as the required $\pi$ phase
difference between gaps is maintained. We also analyze the consequences of
time-reversal symmetry-breaking either due to the presence of an applied or
impurity magnetic field or to a deviation from the intended phase matching
between the superconducting gaps. We demonstrate how the relevant notion of
topological invariance must be modified when time-reversal symmetry is broken,
and how both the persistence of gapless Majorana modes and their robustness
properties depend in general upon the way in which the original Hamiltonian is
perturbed. Interestingly, a topological quantum phase transition between
helical and chiral superconducting phases can be induced by suitably tuning a
Zeeman field in conjunction with a phase mismatch between the gaps. Recent
experiments in doped semiconducting crystals, of potential relevance to the
proposed model, and possible candidate material realizations in superconductors
with $s_\pm$ pairing symmetry are discussed.
