## 3 Results
### Effect of Biaxial Strains  
The effects of biaxial strain on the 2D supercell are modeled by stretching and compressing along the a-axis and b-axis. We investigate the impact of biaxial strains ranging from \(-10\%\) to \(+10\%\) on the magnetic and electronic properties of monolayer Janus MnCrNO\({}_{2}\). Here, the applied biaxial strain is defined as  
\[\varepsilon=\frac{a-a_{0}}{a_{0}}\times 100\%,\]  
Figure 2: Side view of the three magnetic configurations. The red and green arrows represent spin-up and spin-down, respectively. where \(a\) and \(a_{0}\) are the unstrained and strained lattice constants of the monolayer MnCrNO\({}_{2}\), respectively. The negative and positive values of \(\varepsilon\) correspond to the biaxial compressive and tensile strains, respectively. First, we check the biaxial strain (both tensile and compressive strains) effect on the thermodynamic stability of monolayer Janus MnCrNO\({}_{2}\). It can be seen in Figure S3 that the monolayer structure is thermodynamically stable for both the strain-free condition and the compression strains, without any imaginary frequency for all the wave vectors in the entire Brillouin zone. For the tensile strains from +2% to +8%, the tiny imaginary frequencies, which may correspond to the transverse acoustic (TA) modes, only appear in a narrow region near the \(\Gamma\) point. This kind of stability is generally similar to that of the strain-free sample. When increased to +10%, the compressive strain results in more significant negative frequency modes, as shown in Figure S3e, implying the possible structural instability. We then study the effect of strain on the band structure of the MnCrNO\({}_{2}\) monolayer. The electronic band structures are calculated with applied strain ranging from \(-10\)% to +10%, at 2% intervals. Figure 4 illustrates the bandgaps and band structures of the strained MnCrNO\({}_{2}\) monolayer under different strains. As shown in Figure 4a, the application of various strains can effectively tune the bandgap of the MnCrNO\({}_{2}\) monolayer obtained with the PBE and HSE06 functionals. With strains ranging from \(-8\)% to +8%, the bandgap decreases significantly from 1.33 eV to 0.16 eV within the PBE framework. Although such a decrease may lead to a transition from semiconductor to metal when the tensile strain reaches +10%, the monolayer might be unstable due to imaginary frequencies in the phonon modes. This behavior is distinct from the electronic properties of the MXene counterpart, WCrCO\({}_{2}\)[49], but resembles those of TMDs when they are tuned by applying mechanical strains [50]. The subtle difference lies in the orbital contribution. For TMDs, the subsequent semiconductor-to-metal transition is achieved due to the transition between the d\({}_{x^{2}}\) orbitals, while in the MnCrNO\({}_{2}\) monolayer, several 3d orbitals and O-2p orbitals work together to drive the transition, as can be verified in the Bader charge analysis in Table S1. The band structures under \(-10\)% biaxial strain and those under +8% biaxial strain are shown in Figure 4b-e, respectively. Within the framework of the HSE06 functional, the bandgap of the MnCrNO\({}_{2}\) monolayer reaches a maximum, i.e., 1.68 eV, under \(-4\)% compressive strain. For strains from \(-4\)% to +10%, the bandgaps calculated by the HSE06 functional are larger than those by the PBE functional. The PBE- and HSE06-based band structures under strains of \(\pm 2\)%, \(\pm 4\)%, \(\pm 6\)%, and \(-8\)% are presented in Figures S4 and S5, respectively. What differs from the PBE-based results is that the CBM obtained within HSE06 under compressive strains from \(-4\)% to \(-10\)% is spin-down, as shown in Figure S5b-d.  
Figure 3: Spin-polarized band structures and DOS of the MnCrNO\({}_{2}\) monolayer using (**a**) PBE functional and (**b**) hybrid-functional HSE06, respectively. In panel (**a**), the CBM is situated at the M point, while the VBM is located close to the \(\Gamma\) point along the K–\(\Gamma\) path, indicated in green.