Extended Data Fig. 10: Effects of strain on exchange pathways and magnetic anisotropic energy in CrSBr.

a, The interlayer magnetic exchange coupling J1, J2, J3 and J4 for the 1^st, 2^nd, 3^rd, and 4^th nearest-neighbor (NN) interlayer Cr pairs. The strain-induced magnetic phase transition is likely driven by a significant enhancement in the 1^st NN interlayer Cr-Cr coupling which favors ferromagnetism. b, Two interlayer magnetic exchange pathways between closest Cr-Cr interlayer pairs in side view. c, Schematics of the first exchange pathway, giving weaker magnetic exchange coupling. Due to the nearly orthogonal Br p orbitals in the super-super exchange pathway, the magnetic coupling is mediated by the weak on-site interaction between the p orbitals (Hund’s rule) in both Br. d, Schematics of the second exchange pathway that favors AFM coupling when α goes closer to 180° (compressive strain along a), and FM coupling when α approaches 90° (tensile strain along a), in agreement with the first principles calculation. Since this pathway allows for direct hopping in one of the Br, it should dominate the interlayer magnetic coupling between closest Cr-Cr interlayer pairs. e, Magnetic anisotropic energies as a function of uniaxial strain along a, obtained from fully-relativistic calculations, with the free boundary condition. The black and red points correspond, respectively, to the energy difference between the hard- and easy-axis AFM phase, and the energy difference between the hard-axis FM phase and the easy-axis AFM phase. The decreasing anisotropic energies agree with the observed decrease of saturating field in the spin canting process upon straining.