  We present a comparative theoretical study of the effects of standard
Anderson and magnetic disorders on the topological phases of two-dimensional
Rashba spin-orbit coupled superconductors, with the initial state to be either
topologically trivial or nontrivial. Using the self-consistent Born
approximation approach, we show that the presence of Anderson disorders will
drive a topological superconductor into a topologically trivial superconductor
in the weak coupling limit. Even more strikingly, a topologically trivial
superconductor can be driven into a topological superconductor upon diluted
doping of independent magnetic disorders, which gradually narrow, close, and
reopen the quasi-particle gap in a nontrivial manner. These topological phase
transitions are distinctly characterized by the changes in the corresponding
topological invariants. The central findings made here are also confirmed using
a complementary numerical approach by solving the Bogoliubov-de Gennes
equations self-consistently within a tight-binding model. The present study
offers appealing new schemes for potential experimental realization of
topological superconductors.
