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.
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