Recent experiments have provided evidence that one-dimensional (1D) topological superconductivity can be realized experimentally by placing transition metal atoms that form a ferromagnetic chain on a superconducting substrate. We address some propert
ies of this type of systems by using a Slater-Koster tight-binding model. We predict that topological superconductivity is nearly universal when ferromagnetic transition metal chains form straight lines on superconducting substrates and that it is possible for more complex chain structures. The proximity induced superconducting gap is $sim Delta E_{so} / J$ where $Delta$ is the $s$-wave pair-potential on the chain, $E_{so}$ is the spin-orbit splitting energy induced in the normal chain state bands by hybridization with the superconducting substrate, and $J$ is the exchange-splitting of the ferromagnetic chain $d$-bands. Because of the topological character of the 1D superconducting state, Majorana end modes appear within the gaps of finite length chains. We find, in agreement with experiment, that when the chain and substrate orbitals are strongly hybridized, Majorana end modes are substantially reduced in amplitude when separated from the chain end by less than the coherence length defined by the $p$-wave superconducting gap. We conclude that Pb is a particularly favorable substrate material for ferromagnetic chain topological superconductivity because it provides both strong $s-$wave pairing and strong Rashba spin-orbit coupling, but that there is an opportunity to optimize properties by varying the atomic composition and structure of the chain. Finally, we note that in the absence of disorder a new chain magnetic symmetry, one that is also present in the crystalline topological insulators, can stabilize multiple Majorana modes at the end of a single chain.
