We theoretically investigate the bilinear current, scaling as $j\sim EB$, in two- and three-dimensional systems. Based on the extended semiclassical theory, this bilinear current, including both longitudinal and transverse response, is classified according to their different scaling relations with the relaxation time. We reveal the distinct contributions to the ordinary Hall effect, the planar Hall effect, and magnetoresistance. We further highlight the spin contribution of an intrinsic ordinary Hall current, which has a geometric origin. Our theory is explicitly applied to study massive Dirac models and $\mathcal{PT}$-symmetric systems. The results demonstrate the capacity of the bilinear current to probe the band structure, specifically its ability to distinguish Weyl/Dirac points with different dispersions. Specifically, a tilt is necessary for the planar Hall effect (PHE) because of a symmetry constraint, and the orbital PHE only exist in the three-dimensional system. The $\tau$-scaling current vanishes due to $\mathcal{PT}$ symmetry. Finally, ${\tau }^0$- and $\tau$-scaling reveal their band-geometric nature, and they are enhanced near the small-gap region. Our work presents a theory of magnetotransport, potentially laying the groundwork for future experimental studies or device fabrications.

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• Anomalous Hall effect
• Quantum transport
• Boltzmann theory
• k dot p method