Doping fingerprints of spin and lattice fluctuations in moir\'e superlattice systems

Twisted Van der Waals systems offer the unprecedented possibility to tune different states of correlated quantum matter with an external non-invasive electrostatic doping. The nature of the superconducting order presents a recurring open question in this context. In this work, we quantitatively assess the case of spin-fluctuation-mediated pairing for $\Gamma$-valley twisted transition metal dichalcogenide homobilayers. We self-consistently and dynamically calculate the doping dependent superconducting transition temperature $T_{\mathrm{c}}$ revealing a superconducting dome with a maximal $T_{\mathrm{c}}\approx 0.1-1$ K depending on twist angle. We compare our results with conventional phonon-mediated superconductivity and identify clear fingerprints in the doping dependence of $T_{\mathrm{c}}$, which allow experiments to distinguish between different pairing mechanisms.Comment: 9 pages, 3 figures + SM with 18 pages, 11 figure