Fig. 2: Electric field modulation effects on different magnetic order.

a Schematic of the electric field-controlled switchable anomalous Hall conductivity(middle) in 3MnB₂T₄·2B₂T₃ along with spin-resolved electronic band structures corresponding to the UUD (left) and UDD (right) magnetic states. The blue or brown rectangular shape in the diagram symbolizes the MnB₂T₄ layer, exhibiting an upward or downward magnetic moment. Conversely, the light blue shape denotes the incorporated B₂T₃ layer. The interlayer magnetic coupling could be modulated by varying electric polarization (−E and +E) between the top and bottom gate. The up (↑)-up (↑)-down (↓) (UUD) and up (↑) -down (↓)- down (↓) (UDD) magnetic order is connected by \(\hat{P}\hat{T}\) symmetry, and the direction of the anomalous Hall current (the curved arrows) is flipped by the switching of electric polarization. b Side view of atomic structure of 3MnB₂Se_4·2B₂Se₃ (B = Bi, Sb). The yellow arrows indicate magnetic orientation in each magnetic layer, and the blue ones show the direction of electric field. c, d Energy difference between up (↑)-up (↑)-down (↓) and up (↑)-down (↓)-down (↓) magnetic states under different electric field in 3MnSb₂Se₄·2Sb₂Se₃ (MSS-5L), 3MnSb₂Se₄ (MSS-3L), 3MnBi₂Te₄·2Bi₂Te₃ (MBT-5L) and 3MnBi₂Te₄ (MBT-3L). Positive (negative) values indicate that the electric field is oriented in the upward (downward) direction.