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Strain effects on topological and valley properties of Janus monolayer $\mathrm{VSiGeN_4}$

Strain is an effective method to tune the electronic properties of two-dimension (2D) materials, and can induce novel phase transition. Recently, 2D $\mathrm{MA_2Z_4}$ family materials are of interest because of their emerging topological, magnetic and superconducting properties. Here, we investigate the impact of strain effects ($a/a_0$:0.96$\sim$1.04) on the physical properties of Janus monolayer $\mathrm{VSiGeN_4}$ as a derivative of $\mathrm{VSi_2N_4}$ or $\mathrm{VGe_2N_4}$, which possesses dynamical, mechanical and thermal stabilities. For out-of-plane magnetic anisotropy, with increasing strain, $\mathrm{VSiGeN_4}$ undergoes transition between ferrovalley semiconductor (FVS), half-valley-metal (HVM), valley-polarized quantum anomalous Hall insulator (VQAHI), HVM and FVS. These imply twice topological phase transitions, which are related with sign-reversible Berry curvature and band inversion between $d_{xy}$+$d_{x^2-y^2}$ and $d_{z^2}$ orbitals for K or -K valley. The band inversion also leads to transformation of valley splitting strength between valence and conduction bands. However, for in-plane magnetic anisotropy, no special quantum anomalous Hall (QAH) states and valley polarization exist within the considered strain range. The actual magnetic anisotropy energy (MAE) shows no special QAH and HVM states in monolayer $\mathrm{VSiGeN_4}$. Fortunately, these can be easily achieved by external magnetic field, which adjusts the easy magnetization axis of $\mathrm{VSiGeN_4}$ from in-plane one to out-of-plane one. Our findings shed light on how strain can be employed to engineer the electronic states of $\mathrm{VSiGeN_4}$, which may open new perspectives for multifunctional quantum devices in valleytronics and spintronics.