Ferroelectric polarization manipulates the layer-polarized anomalous Hall effect in bilayers with fully compensated ferrimagnetism

Altermagnets have attracted attention due to their compensating magnetic moments and nonrelativistic spin splitting. However, due to the limitation of the strict symmetry requirements, two-dimensional (2D) altermagnets are still rare, and the regulatory approaches are mostly volatile. Based on the low-energy effective $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ model, a general approach to constructing bilayers with fully compensated ferrimagnetism (fFIM) whose properties are like altermagnets has been proposed, and the achieved layer-polarized anomalous Hall effect (LP-AHE) can be manipulated by the ferroelectric (FE) polarization in them. In the bilayers with fFIM, the coupling between FE polarization and interlayer A-type antiferromagnetism induces layer-locked Berry curvature, which provides advantages of antimagnetic interference and nonrelativistic spin splitting. Furthermore, during the symmetry operating of the bilayers with fFIM, the FE polarizations can be changed, and the reversals of the valley indices and the interlayer coupling energies can be achieved. The obtained tunable multiple couplings of spin, valley, and layer degrees of freedom induce LP-AHE in the bilayers with fFIM. Through first-principles calculations, we confirm this mechanism in the bilayer $\mathrm{C}{\mathrm{o}}_{2}\mathrm{NFCl}$ lattice by performing ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}}_{z}/\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{I}/{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}}_{2z}$ symmetry operations. Manipulatable LP-AHE can be achieved in bilayers with fFIM, which reveals a method for achieving a tunable Hall effect in a broader range of 2D magnets.