Importance of magnetic shape anisotropy in determining magnetic and electronic properties of monolayer $\mathrm{VSi_2P_4}$

Abstract: Two-dimensional (2D) ferromagnets have been a fascinating subject of research, and magnetic anisotropy (MA) is indispensable for stabilizing the 2D magnetic order. Here, we investigate magnetic anisotropy energy (MAE), magnetic and electronic properties of $\mathrm{VSi_2P_4}$ by using the generalized gradient approximation plus $U$ (GGA+$U$) approach. For large $U$, the magnetic shape anisotropy (MSA) energy has a more pronounced contribution to the MAE, which can overcome the magnetocrystalline anisotropy (MCA) energy to evince an easy-plane. For fixed out-of-plane MA, monolayer $\mathrm{VSi_2P_4}$ undergoes ferrovalley (FV), half-valley-metal (HVM), valley-polarized quantum anomalous Hall insulator (VQAHI), HVM and FV states with increasing $U$. However, for assumptive in-plane MA, there is no special quantum anomalous Hall (QAH) state and spontaneous valley polarization within considered $U$ range. According to the MAE and electronic structure with fixed out-of-plane or in-plane MA, the intrinsic phase diagram shows common magnetic semiconductor (CMS), FV and VQAHI in monolayer $\mathrm{VSi_2P_4}$. At representative $U$$=$3 eV widely used in references, $\mathrm{VSi_2P_4}$ can be regarded as a 2D-$XY$ magnet, not Ising-like 2D long-range order magnets predicted in previous works with only considering MCA energy. Our findings shed light on importance of MSA in determining magnetic and electronic properties of monolayer $\mathrm{VSi_2P_4}$.