3D C-paired Spin-Valley Locking materials in collinear antiferromagnetic order

C-paired spin-valley locking(SVL) refers that the spin and valley degree of freedoms are locked by the real space crystal symmetries. Materials with C-paired SVL can generate piezomagnetism(PZM) and transverse spin current. In this letter we first propose the general theory of C-paired SVL and conclude 47 magnetic point groups compatible with collinear antiferromagnetic(AFM) order that one C-paired SVL material must belong to. Depending on how the spin orientations of locked valleys coupled, we classified C-paired SVL into three types: collinear, non-collinear, and collinear + non-collinear. From these 3D collinear AFM materials fabricated in experiments, we predict a series of materials realizing C-paired SVL in the assistance of our package that classifies all momenta in every specific magnetic space group. Among them, CoF₂ as the representative material of PZM shows large PZM and spin Hall angle by first-principles calculations. Our study and package build the foundation for identifying and classifying the C-paired SVL phenomenon in 3D magnetic materials and provide numerous materials that own desired controllable magnetic and electronic properties.