A Framework for Understanding Colossal Magnetoresistance and Complex Resistivity Behaviors in Magnetic Semiconductor

Colossal magnetoresistance (CMR) is commonly observed in magnetic semiconductors when an external magnetic field is applied, usually accompanied by anomalous resistivity peaks or humps which were previously considered as evidence of a metal-insulator transition (MIT). Previous research efforts primarily focused on elucidating the CMR effect in ferromagnetic semiconductor systems, which may be inapplicable to antiferromagnetic or ferrimagnetic systems. In our work, we propose a framework to unravel the mechanisms underlying the CMR related phenomenon in magnetic semiconductor, and apply it to the ferrimagnetic semiconductor Mn$_3$Si$_2$Te$_6$. A two-carrier model, with electron and hole carriers generated by thermal excitation across the magnetization-dependent band gap, is employed, which accurately reproduces the observed $\rho(B, T)$ curves. Additionally, the shift of $T_c$ (or the anomalous resistivity peak) with increasing direct currents, previously attributed to current control of the chiral orbital current (COC) state, is also reproduced within our framework by properly accounting for the Joule heating effects. Our work provides a quantitative methodology for analyzing and calculating the novel resistivity curves in magnetic semiconductors and clarifies the controversy surrounding the origins of CMR and the concomitant complex behaviors.