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

Multiband s-wave topological superconductors: role of dimensionality and magnetic field response