Majorana boundary quasiparticles may naturally emerge in a spin-singletsuperconductor with Rashba spin-orbit interactions, when a Zeeman magneticfield breaks time-reversal symmetry. Their existence and robustness againstadiabatic changes is deeply related, via a bulk-edge correspondence, totopological properties of the band structure. The present paper shows that thespin-orbit may be responsible for topological transitions when thesuperconducting system has an underlying sublattice structure, as it appears ina dimerized Peierls chain, graphene, and phosphorene. These systems, whichbelong to the Bogoliubov-de Gennes class D, are found to have an extra symmetrythat plays the role of the parity. It enables the characterization of thetopology of the particle-hole symmetric band structure in terms of bandinversions. The topological phase diagrams this leads to are then obtainedanalytically and exactly. They reveal that, because of the underlyingsublattice structure, the existence of topological superconducting phasesrequires a minimum doping fixed by the strength of the Rashba spin-orbit.Majorana boundary quasiparticles are finally predicted to emerge when the Fermilevel lies in the vicinity of the bottom (top) of the conduction (valence) bandin semiconductors such as the dimerized Peierls chain and phosphorene. In atwo-dimensional topological superconductor based on (stretched) graphene, whichis semimetallic, Majorana quasiparticles cannot emerge at zero and low doping,that is, when the Fermi level is close to the Dirac points. Nevertheless, theyare likely to appear in the vicinity of the van Hove singularities.
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