Interplay of magnetic field and magnetic impurities in Ising superconductors

Phonon-driven $s$-wave superconductivity is fundamentally antagonistic to uniform magnetism, and field-induced suppression of the critical temperature is one of its canonical signatures. Examples of the opposite are unique and require fortuitous cancellations and very fine parameter tuning. The recently discovered Ising superconductors violate this rule: An external magnetic field applied in a certain direction does not suppress superconductivity in an ideal, impurity-free material. We propose a simple and experimentally accessible system where the effects of spin-conserving and spin-flip scattering can be studied in a controlled way, namely ${\mathrm{NbSe}}_{2}$ monolayers dosed with magnetic $3d$ atoms. We predict that the critical temperature is slightly increased by an in-plane magnetic field in ${\mathrm{NbSe}}_{2}$ dosed with Cr. Due to the band spin splitting, magnetic spin-flip scattering requires a finite momentum transfer, while spin-conserving scattering does not. If the magnetic anisotropy is easy-axis, an in-plane field reorients the impurity spins and transforms spin-conserving scattering into spin-flip. The critical temperature is enhanced if the induced magnetization of ${\mathrm{NbSe}}_{2}$ has a substantial long-range component, as is the case for Cr ions.