  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 properties 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.
