Fig. 1

Van der Waals Cr₂Ge₂Te₆/In₂Se₃ heterostructure and magnetoelectric coupling. a Lateral unit cell size is chosen to be equal to that of ferromagnetic Cr₂Ge₂Te₆ experimental lattice constant and commensurate to ferroelectric In₂Se₃-\(\sqrt 3 \, \times \, \sqrt 3\) with −4% strain. The interfacial two atomic layers (one atomic layer from each side of the interface) are shown in the top views of Cr₂Ge₂Te₆-1 × 1 (top) and In₂Se₃-\(\sqrt 3 \, \times \, \sqrt 3\) (bottom). b, c Heterostructure side views with the In₂Se₃ ferroelectric dipole moment directed upward and downward (P_up and P_dn), respectively, and the induced easy-plane and easy-axis Cr spins (S_Cr). The solid (dashed) arrow of S_Cr indicates the allowed (prohibited) finite temperature long-range ferromagnetic ordering in 2D systems by the Mermin-Wagner theorem. For each case, the most stable (hollow) and unstable (top) stacking configurations are given