Doping fingerprints of spin and lattice fluctuations in moiré superlattice systems

Twisted Van der Waals systems offer the unprecedented possibility to tune different states of correlated quantum matter with external noninvasive electrostatic doping. The nature of the superconducting order presents a recurring open question in this context. In this work, we assess quantitatively the case of spin-fluctuation-mediated pairing for $\mathrm{\ensuremath{\Gamma}}$-valley twisted transition metal dichalcogenide homobilayers. We calculate self-consistently and dynamically the doping-dependent superconducting transition temperature ${T}_{\mathrm{c}}$ revealing a superconducting dome with a maximal ${T}_{\mathrm{c}}\ensuremath{\approx}0.1--1\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ depending on twist angle. We compare our results with conventional phonon-mediated superconductivity, and we identify clear fingerprints in the doping dependence of ${T}_{\mathrm{c}}$, which enable experiments to distinguish between different pairing mechanisms.