## 3 Results
### Effect of Biaxial Strains
In contrast, the CBM in (**b**) is located close to the K point, and the VBM is near the \(\Gamma\) point. The Fermi level is denoted by a horizontal dashed line at zero energy. The black and red lines represent spin-up and spin-down, respectively. The analysis above reveals that both the compressive and tensile mechanical strains significantly impact the band structures and interfacial charge transfer. In the following section, we further explore how strain influences the magnetic anisotropy. MAE, defined as the difference between energies corresponding to the magnetization in the in-plane (ab-plane) and out-of-plane direction (c-direction), is calculated using the following equation:  
\[\text{MAE}=E_{\text{in}}-E_{\text{out}},\]  
where \(E_{\text{in}}\) and \(E_{\text{out}}\) represent the total energy of the in-plane and out-of-plane magnetic moment configurations of monolayer Janus MnCrNO\({}_{2}\), respectively. Therefore, a positive (negative) MAE value indicates that the ferromagnets have an out-of-plane (in-plane) spin  
Figure 4: (**a**) Evolution of bandgaps of Janus monolayer MnCrNO\({}_{2}\) under various biaxial strains ranging from \(-10\%\) to \(+10\%\). Black lines and red lines represent the results from the PBE and HSE06 functionals, respectively. The blue dashed line indicates the position of zero bandgap. (**b,c**) The PBE-based band structures of monolayer MnCrNO\({}_{2}\), corresponding to \(-10\%\) compressive strain and \(+8\%\) tensile strain, respectively. Both the CBM and the VBM in (**b**) are located at the M point, forming a direct gap. (**d,e**) The HSE06-based band structures of MnCrNO\({}_{2}\), corresponding to \(-10\%\) compressive strain and \(+8\%\) tensile strain, respectively. The Fermi level is denoted by a horizontal dashed line at zero energy. polarization direction. Since the FM magnetic configuration takes the ground state among the three magnetic configurations, here we focus on the MAE of the FM configuration for simplicity. Figure 5a demonstrates the variation in MAE under applied biaxial strains. The minimum MAE of \(-58\)\(\upmu\)eV is observed under a compressive strain of \(-10\%\) and then gradually increases to \(269\)\(\upmu\)eV under a tensile strain of \(10\%\). An interesting observation is that the MAE becomes negative at around and below \(-8\%\) compressive strain, indicating a switch of the magnetic easy axis from the out-of-plane to the in-plane direction. Thus, the applied strains can regulate both the electronic properties and the magnetization of the monolayer MnCrNO\({}_{2}\) simultaneously. Figure 5b presents the increase in magnetic moment of both the Mn and Cr atoms in monolayer MnCrNO\({}_{2}\) with the FM state under applied biaxial strain. The magnetic moment of Mn ranges from \(3.15\)\(\mu_{\text{B}}\) at \(-10\%\) compressive strain to \(3.90\)\(\mu_{\text{B}}\) at \(+10\%\) tensile strain, whereas the range of magnetic moment for Cr is only one-fifth, i.e., from \(2.99\)\(\mu_{\text{B}}\) to \(3.14\)\(\mu_{\text{B}}\). The probable reason, as can be inferred from Table S1, is that the Cr-3d and O-2p orbitals have more overlap than the Mn-3d and O-2p. The Mn-3d orbitals are mainly more localized near the Fermi level and are thus more influenced by applied strain, resulting in a larger variation in magnetic moment. The magnetic moment of Mn and Cr in the case of the AFM1 and AFM2 configurations is also evaluated and shown in Figure 5c,d, respectively, given that these two configurations can be meta-stable. These variations are almost the same as the results with the FM state. The slight difference lies in the magnetic moment of Cr atoms within the AFM2 configuration, whose values are higher overall than those with the FM and AFM2 state, as shown in Figure 5d. Figure 5: (**a**) Evolution of MAE of Janus monolayer MnCrNO\({}_{2}\) under biaxial strains ranging from \(-10\%\) to \(+10\%\). A switch of the magnetic easy axis occurs at around \(-8\%\) compressive strain, transitioning from an out-of-plane to an in-plane direction.