Strain-tunable Dzyaloshinskii--Moriya interaction and skyrmions in two-dimensional Janus Cr$_{2}$X$_{3}$Y$_{3}$ (X, Y = Cl, Br, I, X $\neq$ Y) trihalide monolayers

Recently, great effort has been devoted to the search for two-dimensional (2D) ferromagnetic materials with inherent strong Dzyaloshinskii--Moriya interaction (DMI). Here, through a first-principles approach, we systematically investigate the effect of biaxial strain on the DMI, the Heisenberg exchange interaction, and the magnetic anisotropy energy (MAE) of Janus Cr$_{2}$X$_{3}$Y$_{3}$ (X, Y = Cl, Br, I, X $\neq$ Y) monolayers. Both DMI and MAE can be significantly enhanced by tensile strain, while a reversal of the chirality of DMI in Cr$_{2}$Cl$_{3}$Br$_{3}$ and a switch of MAE from off-plane to in-plane in Cr$_{2}$I$_{3}$Cl$_{3}$ are induced by a compressive strain of $2\%$. Microscopically, DMI and MAE are associated mainly with the large spin--orbit coupling of the heavy nonmagnetic halogen atoms rather than that of the magnetic Cr atoms. In particular, the peculiar magnetic transition of Cr$_{2}$I$_{3}$Cl$_{3}$ is explained by competition between direct exchange and superexchange interactions. Micromagnetic simulations show that a small external magnetic field of 65~mT stabilizes a skyrmion with a diameter of 9.8~nm in the Cr$_2$I$_3$Cl$_3$ monolayer. Our results will provide guidance for further research on DMI and skyrmions in 2D Janus materials, as well as a basis for the potential applications in spintronic devices.