Pauli limit violation, finite-momentum pairing, superconducting diode effect in moire Ising superconductors

In this work, we propose a new type of noncentrosymmetric superconductor in moir'e materials, in which the Ising spin-orbital coupling is much larger than the moir'e band bandwidth. We call such kind of Ising superconductor as a moir'e Ising superconductor. Using a realistic continuum model of twisted homobilayer transition metal dichalcogenides, we show that moir'e Ising superconductors possess the following novel properties: (i) The spin-singlet paring and spin-triplet pairing are guaranteed to be equally mixing; (ii) The in-plane critical magnetic field shows a violation of Pauli limit as the conventional Ising superconductor. But the suppression of the superconductivity is caused by orbital effects, while the Zeeman effects are neglectable; (iii) The orbital effects of in-plane magnetic fields can further drive finite momentum pairings. In particular, due to the noncentrosymmetric behaviour and finite interlayer tunneling, we find a $2q_B$-Fulde–Ferrell (FF) pairing ($2q_B=eBd$ is a momentum shift caused by the magnetic field $B$, $d$ denotes the layer separation) is more favourable, in which the cooper pairs at two layers both carry $2q_B$ momentum; (iv) Remarkably, we further find that the $2q_B$-Fulde–Ferrell pairing would result in a gaint ($Delta j_csim 50\%$) and gate-tunable superconducting diode effect. Our theory establishes the concept of moir'e Ising superconductors and motives the studies of superconducting moir'e materials with strong Ising spin-orbital coupling.