Proximity-induced mixed odd- and even-frequency pairing in monolayerNbSe2

Monolayer superconducting transition-metal dichalcogenide ${\mathrm{NbSe}}₂$ is a candidate for a nodal topological superconductor by magnetic field. Because of the so-called Ising spin-orbit coupling that strongly pins the electron spins to the out-of-plane direction, Cooper pairs in monolayer superconductor ${\mathrm{NbSe}}₂$ are protected against an applied in-plane magnetic field much larger than the Pauli limit. In monolayer ${\mathrm{NbSe}}₂$, in addition to the Fermi pockets at the corners of Brillouin zone with opposite crystal momentum similar to other semiconducting transition-metal dichalcogenids, there is an extra Fermi pocket around the $\mathrm{\ensuremath{\Gamma}}$ point with much smaller spin splitting, which could lead to an alternative strategy for pairing possibilities that are manipulable by a smaller magnetic field. By considering a monolayer ${\mathrm{NbSe}}₂$-ferromagnet substrate junction, we explore the modified pairing correlations on the pocket at $\mathrm{\ensuremath{\Gamma}}$ point in hole-doped monolayer ${\mathrm{NbSe}}₂$. The underlying physics is fascinating as there is a delicate interplay of the induced exchange field and the Ising spin-orbit coupling. We realize a mixed singlet-triplet superconductivity, $s+f$, due to the Ising spin-orbit coupling. Moreover, our results reveal the admixture state including both odd- and even-frequency components, associated with the ferromagnetic proximity effect. Different frequency symmetries of the induced pairing correlations can be realized by manipulating the magnitude and direction of the induced magnetization.