Review of Moiré superconductivity and application of the Roeser-Huber formula

Moiré superconductivity represents a new class of superconducting materials since the discovery of superconductivity in magic-angle (1.1°) twisted bi-layer graphene (MATBG), forming a Moiré lattice with a much bigger crystal parameter as the original lattice constant of graphene. Hence, experimentally changing the Moiré twist angle, 0.93° $⩽\Theta ⩽$1.27, leads to a variation of the superconducting properties and enables a new way of engineering 2D superconducting materials. Details of the robust superconducting state of MATBG as function of charge carrier density, temperature and applied magnetic fields are reviewed. The influence of the top/bottom hexagonal boron nitride layer thickness on the superconducting properties of MATBG was also demonstrated in the literature. In all fabricated MATBG devices, changing of the charge carrier density leads to the appearance of insulating, metallic and even ferromagnetic states, which separate several superconducting domes in the phase diagram (longitudinal resistance, $R_{\mathrm{xx}}$, as function of temperature T and charge carrier density, n). Further works have considered MATBG combined with WSe₂-layers, twisted bi-layer WSe₂, magic-angle tri-layer graphene (MATTG), and most recently, four-layer (MAT4G) and five-layer (MAT5G) stacks. The differences between the layered, cuprate high-$T_c$ superconductors and the Moiré superconductors are compiled together. The collected information is then used to apply the Roeser-Huber formalism to Moiré-type superconductivity to calculate the superconducting transition temperature, $T_c$, using only information of the Moiré lattice and the electronic configuration. To account for the different charge carrier densities in the experimental data sets and the low charge carrier mass demands that a new parameter $\eta$ must be introduced to the Roeser-Huber formalism to enable the description of several superconducting domes found in the phase diagram for a given Moiré angle. Doing so, the calculated data fit well to the correlation curve defined within the Roeser-Huber formalism.