Strain tunable skyrmions and strong Dzyaloshinskii-Moriya interaction in two-dimensional Janus Cr(X,Y)$_{3}$ trihalides monolayers

Recently, great effort has been poured into sorting out two-dimensional (2D) ferromagnetic materials with inherent strong Dzyaloshinskii-Moriya interaction (DMI). Here, with first-principles approach, we systematically investigate the effect of biaxial strain on DMI and skyrmions in Janus Cr$_{2}$X$_{3}$Y$_{3}$ (X, Y = Cl, Br, I, X $\neq$ Y) monolayers. A large DMI/exchange coupling ratio |$d/J$| reaches or even exceeds 1, suggesting that the DMI is comparable to the Heisenberg exchange interaction, which is beneficial to the creation and stability of skyrmions. We find that DMI is mainly associated with the large spin-orbit coupling of heavier nonmagnetic halogen atoms rather than that of magnetic Cr atoms. Additionally, DMI and magnetic anisotropy energy (MAE) can be significantly enhanced by tensile strain. Interestingly, 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$\%$. Besides, pure skyrmions with diameter less than 10 nm can be modulated by biaxial strain. Our work has guiding significance for the research of DMI and skyrmions in 2D Janus materials as well as the potential applications in spintronic device.