  It is well known that the critical temperature of multi-gap superconducting
3D heterostructures at atomic limit (HAL) made of a superlattice of atomic
layers with an electron spectrum made of several quantum subbands can be
amplified by a shape resonance driven by the contact exchange interaction
between different gaps. The $T_C$ amplification is achieved tuning the Fermi
level near the singular nodal point at a Lifshitz transition for opening a
neck. Recently high interest has been addressed to the breaking of inversion
symmetry which leads to a linear-in-momentum spin-orbit induced spin splitting,
universally referred to as Rashba spin-orbit coupling (RSOC) also in 3D layered
metals. However the physics of multi-gap superconductivity near unconventional
Lifshitz transitions in 3D HAL with RSOC, being in a non-BCS regime, is not
known. The key result of this work getting the superconducting gaps by
Bogoliubov theory and the 3D electron wave functions by solution of the Dirac
equation is the feasibility of tuning multi-gap superconductivity by suitably
matching the spin-orbit length with the 3D superlattice period. It is found
that the presence of the RSOC amplifies both the k dependent anisotropic gap
function and the critical temperature when the Fermi energy is tuned near the
circular nodal line. Our results suggest a method to effectively vary the
effect of RSOC on macroscopic superconductor condensates via the tuning of the
superlattice modulation parameter in a way potentially relevant for spintronics
functionalities in several existing experimental platforms and tunable
materials needed for quantum devices for quantum computing.
