## 3 Results
### Effect of Biaxial Strains
(**b–d**) Variations in magnetic moment per Mn or Cr atom under biaxial strain, in units of \(\mu_{\text{B}}\). Insets of (**c**,**d**) are the schematics of the AFM1 and AFM2 magnetic configurations. Since the 3d orbitals of transition metal atoms around the Fermi level mainly determine the magnetic moments in the Janus monolayer MnCrNO\({}_{2}\) and are relevant to the direction of MAE accordingly, we now discuss the 3d-orbital-resolved MAE of Mn and Cr atoms in the MnCrNO\({}_{2}\) monolayer under biaxial strains. Based upon the second order perturbation theory, MAE has the following expression [51,52]:  
\[\text{MAE}=\xi^{2}\left[\sum_{u,\rho}\frac{\langle u|L_{x}|o\rangle^{2}}{E_{u}- E_{o}}-\frac{\langle u|L_{z}|o\rangle^{2}}{E_{u}-E_{o}}\right],\]  
where \(\xi\) and \(u/o\) denote the strength of SOC and the energy eigenstates, and \(E_{u}\) (\(E_{o}\)) is the eigenvalues of the unoccupied (occupied) states. Therefore, MAE originates from the \(L_{x}/L_{z}\) operators and the hybridization between occupied and unoccupied states, as reported in [53]. Figure 6 shows the results of the 3d-orbital-resolved MAE of Mn and Cr atoms under both the strain-free condition and the \(-10\%\) compressive strain, respectively. Obviously, the contribution of MAE mainly comes from the efficient hybridization of the \(\text{d}_{xy}/\text{d}_{x^{2}-y^{2}}\) and \(\text{d}_{yz}/\text{d}_{z^{2}}\) orbitals of both Cr and Mn atoms. It can be seen that each \(\text{d}_{xy}/\text{d}_{x^{2}-y^{2}}\) hybridized orbital leads to a negative difference of matrix elements, and conversely, the \(\text{d}_{xy}/\text{d}_{x^{2}}\) hybridization brings a positive value. As a result, the nature of magnetization, whether in-plane or out-of-plane, primarily depends on the competition between these two dominant hybridization effects. In addition, the matrix element difference between the other 3d orbitals, such as the \(\text{d}_{xz}/\text{d}_{yz}\) orbitals of Cr, \(\text{d}_{xy}/\text{d}_{xz}\), and the \(\text{d}_{yz}/\text{d}_{x^{2}-y^{2}}\) orbitals of Mn, also contributes slightly to the out-of-plane magnetic anisotropy with a positive sign. Figure 6: The 3d-orbital-resolved MAE of Cr and Mn atoms in the monolayer MnCrNO\({}_{2}\) under both strain-free condition (**a**,**b**) and \(-10\%\) compressive strain (**c**,**d**), respectively. Under the strain-free condition, as shown in Figure 6a,b, the out-of-plane magnetic anisotropy is governed by the \(\mathrm{d_{yz}/d_{x^{2}}}\) hybridized orbitals, especially for Cr atoms, resulting in the out-of-plane magnetization of monolayer MnCrNO\({}_{2}\). On the other hand, under \(-10\%\) compressive strain, Figure 6c,d shows that the in-plane \(\mathrm{d_{xy}/d_{x^{2}-y^{2}}}\) hybridization, particularly for Mn, enhances and dominates the in-plane magnetization of monolayer MnCrNO\({}_{2}\). The negative matrix element difference of the Mn \(\mathrm{d_{xy}/d_{x^{2}-y^{2}}}\) orbitals reaches \(-308.9\)\(\mathrm{\SIUnitSymbolMicro eV}\). We also plotted the 3d-orbital-resolved MAE of Cr and Mn atoms under strains of \(-2\%\), \(-4\%\), \(-6\%\), and \(-8\%\), respectively, as shown in Figure S6. All these MAE values undergo a uniform change, and the sum of the negative MAE is superior to that of the positive MAE when the compressive strain is down to \(-8\%\). The switch of the magnetic easy axis in the Janus monolayer MnCrNO\({}_{2}\) is attributed to the hybridized 3d orbitals of Mn and Cr, as the contribution from the p orbitals of the N and O atoms to MAE is relatively weak compared with the 3d orbitals, despite the slight magnetic moments in N and O. This can be confirmed by the p-orbital-resolved MAE of the N and O atoms in Figure S7.