## 4 Conclusions  
In summary, the structural, electronic, and magnetic properties of the Janus MXene MnCrNO\({}_{2}\) monolayer under various biaxial strains were proposed and investigated in detail using density functional theory calculations. It was found that the optimized MnCrNO\({}_{2}\) monolayer possesses a hexagonal structure with a \(P3m1\) space group and exhibits good dynamical stability. The intrinsic Janus monolayer MnCrNO\({}_{2}\) is semiconductive and adopts the ferromagnetic ground state with an out-of-plane easy axis. The phonon dispersion spectra show that this monolayer structure can sustain strain effects within a wide range of strains from \(-10\%\) to \(+8\%\). By applying biaxial tensile strain, we can induce a decline in the bandgap in the 2D semiconductive MnCrNO\({}_{2}\), attributed to the distinct roles played by Mn and Cr in the VBM or CBM bands. Furthermore, the magnetic anisotropy undergoes a transition from an out-of-plane easy axis to an in-plane easy axis when the applied compressive strain reaches nearly \(-8\%\). This in-plane magnetization is primarily influenced by the efficient hybridization of the d orbitals, particularly in Mn atoms. We anticipate that the prediction of the Janus MnCrNO\({}_{2}\) monolayer will offer a new avenue for exploring 2D magnetic materials with out-of-plane symmetry breaking and for future electronic and spintronic devices.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14188427/s1](https://www.mdpi.com/article/10.3390/app14188427/s1)]([https://www.mdpi.com/article/10.3390/app14188427/s1](https://www.mdpi.com/article/10.3390/app14188427/s1)), Figure S1: Spin-resolved total density of states and projected density-of-states of each element of monolayer MnCrNO\({}_{2}\) when stain is applied at \(-10\%\), 0 and 10%; Figure S2: Spin-resolved projected density-of-states of each element of monolayer MnCrNO\({}_{2}\) for different U values for Mn and Cr; Figure S3: Phonon spectrum of the monolayer MnCrNO\({}_{2}\) under strain; Figure S4: Band structures of Janus monolayer MnCrNO\({}_{2}\) based upon PBE under strain; Figure S5: Band structures of Janus monolayer MnCrNO\({}_{2}\) based upon HSE06 under strain; Figure S6: The 3d-orbital-resolved MAE of Cr and Mn atoms in the Janus monolayer MnCrNO\({}_{2}\) under strains; Figure S7: The p-orbital-resolved MAE of N, O1 and O2 atoms in the monolayer MnCrNO\({}_{2}\); Table S1: The Bader charge transfer of X atom (X = Cr, Mn, N, O1, O2) under different strains from \(-10\%\) to \(+10\%\), at \(2\%\) intervals.  
W.Y.: funding acquisition, conceptualization, data curation, investigation, writing--original draft; J.S.: investigation; R.J.: investigation; L.Z.: conceptualization, supervision; Y.C.: funding acquisition, resources, supervision; D.H.: conceptualization, investigation, writing--review and editing, supervision. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Natural Science Foundation of China (Grant No. 12174157 and 12347156), the Natural Science Foundation of Jiangsu Province (No. BK20230516), the Scientific Research Project of Jiangsu University (Grant No. 4111190003), the College Students' Innovation Training Program of Jiangsu Province of China (Grant No. 202310299023Z), and the College Students' Research Topics of Jiangsu Province of China (Grant No. Y21A075).