  Materials realization of chiral topological superconductivity is a crucial
condition for observing and manipulating Majorana fermions in condensed matter
physics. Here we develop a tight-binding description of Pb$_3$Bi/Ge(111),
identified recently as an appealing candidate system for realizing chiral
$p$-wave topological superconductivity [Nat. Phys. 15, 796 (2019)]. We first
show that our phenomenological model can capture the two main features of the
electronic band structures obtained from first-principles calculations, namely,
the giant Rashba splitting and type-II van Hove singularity. Next, when the
$s$-wave superconducting property of the parent Pb system is explicitly
considered, we find the alloyed system can be tuned into a chiral topological
superconductor with Chern number $\mathcal{C} = -2$, resulting from the
synergistic effect of a sufficiently strong Zeeman field and the inherently
large Rashba spin-orbit coupling. The nontrivial topology with $\mathcal{C} =
-2$ is further shown to be detectable as two chiral Majorana edge modes
propagating along the same direction of the system with proper boundaries. We
finally discuss the physically realistic conditions to establish the predicted
topological superconductivity and observe the corresponding Majorana edge
modes, including the influence of the superconducting gap, Land\'{e}
$g$-factor, and critical magnetic field. The present study provides useful
guides in searching for effective $p$-wave superconductivity and Majorana
fermions in two-dimensional or related interfacial systems.

Superconductivity;Materials Science