A matrix formulation of magnetoresistance for an arbitrary J-fold multicarrier semiconductor system via the reduced-conductivity-tensor scheme in the nonquantizing regime

In this article, a matrix formulation of magnetoresistance in semiconductors is presented for an arbitrary $J$ -fold multicarrier system which is based on the reduced-conductivity-tensor scheme and applicable in the nonquantizing regime where Landau orbital quantization is negligible. In the formalism, a unique expression of the magnetoresistance is deduced in terms of two vectors which depend on the carrier densities and mobilities, and three matrices which represent various inter-carrier couplings under the applied magnetic field. In particular, the mobility-difference matrix plays a key role, and its simple form strongly suggests a two-carrier model of magnetoresistance for a narrow continuum distribution. Explicit closed-form formulas of magnetoresistance are derived for the two-carrier $\text{(J=2)}$ and three-carrier $\text{(J=3)}$ systems as special cases of the general formalism. The field dependence and asymptotic behavior of the magnetoresistance are also discussed, and a two-carrier model of magnetoresistance is formally proposed.