  Experiments on planar Josephson junction architectures have recently been
shown to provide an alternative way of creating topological superconductors
hosting accessible Majorana modes. These zero-energy modes can be found at the
ends of a one-dimensional channel in the junction of a two-dimensional electron
gas (2DEG) proximitized by two spatially separated superconductors. The
channel, which is below the break between the superconductors, is not in direct
contact with the superconducting leads, so that proximity coupling is expected
to be weaker and less well-controlled than in the simple nanowire configuration
widely discussed in the literature. This provides a strong incentive for this
paper which investigates the nature of proximitization in these Josephson
architectures. At a microscopic level we demonstrate how and when it can lead
to topological phases. We do so by going beyond simple tunneling models through
solving self-consistently the Bogoliubov-de Gennes equations of a
heterostructure multicomponent system involving two spatially separated
$s$-wave superconductors in contact with a normal Rashba spin-orbit-coupled
2DEG. Importantly, within our self-consistent theory we present ways of
maximizing the proximity-induced superconducting gap by studying the effect of
the Rashba spin-orbit coupling, chemical potential mismatch between the
superconductor and 2DEG, and sample geometry on the gap. Finally, we note (as
in experiment) a Fulde-Ferrell-Larkin-Ovchinnikov phase is also found to appear
in the 2DEG channel, albeit under circumstances which are not ideal for
topological superconducting phase.
