Fig. 2

Spin-polarized and Cr d orbital-decomposed band structures of Cr₂Ge₂Te₆/In₂Se₃ heterostructures. a, b The calculated band structures for the heterostructures with P_up and P_dn In₂Se₃, respectively. For each electric polarization, the heterostructure (in terms of contact registry and interlayer distance) adopted the configuration of global energy minimum (see Supplementary Fig. 1). The majority and minority spin states are indicated by the red and blue circles, respectively, whose size denotes the contribution of Cr d orbitals with certain azimuthal angular momentum |m_z|. In₂Se₃ states are shown by the gray dots, and the band exhibits the overall shift down by ~1 eV while changing In₂Se₃ from P_up and P_dn, which indicates a strong In₂Se₃-polarization dependent interfacial coupling. The pink arrows indicate the SOC elements between empty and filled states, indicated by the red circle for the majority spin or blue for minority, causing the negative or positive value of MAE (see Eq. (1) and text) for (a) or (b), respectively, thus the easy-plane or easy-axis S_Cr