Fig. 4: Ferrielectricity controlled magnetoelectric coupling in thin-layer CCPS.

a Lattice model representation of the stripe-AFE state (first column), the E-enforced heterogenous FiE phase (second column), and energy unfavorable metastable AFE states (third column; grouped in orange dashed lines) for BL and trilayer CCPS, respectively. Here, the vdW monolayer cages are represented by blue/orange rectangles, standing for energy accessible and inaccessible respectively. Within the cages, the ground-state (polarized) Cu⁺ ions are depicted by blue (red) circles. Here, 2L-mAFE-I and 2L-mAFE-II correspond to the flipping of an anti-parallel Cu-ion stripe of within the bottom and top ML, respectively. b Black curve: Energy potential diagram vs Δd (anti-parallel Cu⁺ displacement) of BL CCPS from the AFE ground state to 2L-mAFE-I. Red curve: ΔE_MEC on Δd, which monotonically decreases when Cu⁺ ions move towards the Cr atomic plane. c SHG intensity vs T, showing the AFE transition and the frozen of Cu⁺ ions at around 140 K. d Polarization-dependent SHG at 77 K under different V_b. The distinctive six-fold to two-fold transition in the polarization-SHG patterns proves that the non-linear optical anisotropy of thin-layer CCPS associated with inversion symmetry breaking can be continuously tuned by V_b. e Illustration of the FiE-interlocked MEC mechanism in CCPS, in which lattice distortions introduced by Cu⁺ ion displacements efficiently lower the interlayer superexchange interaction.