Recent experiments have observed bulk superconductivity in doped topological insulators. Here we ask whether vortex Majorana zero modes, previously predicted to occur when s-wave superconductivity is induced on the surface of topological insulators, survive in these doped systems with metallic normal states. Assuming inversion symmetry, we find that they do but only below a critical doping. The critical doping is tied to a topological phase transition of the vortex line, at which it supports gapless excitations along its length. The critical point depends only on the vortex orientation and a suitably defined SU (2) Majorana fermions, defined as fermions that are their own antiparticles unlike conventional Dirac fermions such as electrons, have long been sought by high energy physicists, but so far in vain. Of late, the search for Majorana fermions has remarkably shifted to condensed matter systems [1-3], especially, to superconductors (SCs), where states appear in conjugate pairs with equal and opposite energies. Then, a single state at zero energy is its own conjugate and hence, a Majorana state or a Majorana zero mode (MZM). These states are immune to local noise and hence, considered strong candidates for storing quantum information and performing fault tolerant quantum computation [4]. Moreover, they show non-Abelian rather than Bose or Fermi statistics which leads to a number of extraordinary phenomenon [5].Despite many proposals direct experimental evidence for a MZM is still lacking. While initial proposals involved the ¼ 5=2 quantum Hall state and SCs with unconventional pairing such as p x þ ip y [3], a recent breakthrough occurred with the discovery of topological insulators (TIs) [6], which feature topologically protected metallic surface bands. When a conventional s-wave SC is brought near this metallic surface, a single MZM is trapped in the vortex core [7]. Since then, several TIs were found to exhibit bulk superconductivity on doping [8] or under pressure [9,10]. The normal phase of these SCs is now metallic, which raises the question: can a SC vortex host a surface MZM even when the bulk is not insulating?In this Letter, we answer this question in the affirmative and in the process, discover a convenient way to obtain a MZM, which allows us to conclude that some existing experimental systems should possess these states. Our proposal involves simply passing a magnetic field through a TI-based SC, such as superconducting Bi 2 Te 3 , in which the doping is below a certain threshold value. We also find general criteria for SCs to host vortex MZMs and show that some non-TI-based SCs satisfy them too.A heuristic rule often applied to answer the above question is to examine whether the normal state bulk Fermi surface (FS) is well separated from surface states in the Brillouin zone. If it is, MZMs are assumed to persist in the bulk SC. While this may indicate the presence of low energy states, it is not a topological criterion since it depends on nonuniversal details of surface band structure, and canno...