Effect of spin splitting of the conduction band on the resistivity and Hall coefficient: Model for the positive magnetoresistance in EuSe

A magnetic field $\overrightarrow{H}$ splits the conduction band of a magnetic semiconductor into two spin subbands separated by an energy $\delta \mathrm{}\left(H\right)\mathrm{}$. This splitting, which is caused by the $s-f$ (or $d-f$) interaction, leads to a redistribution of the conduction electrons between the two subbands. The effects of this electron redistribution on the electrical resistivity and Hall coefficient are calculated assuming that the total number of conduction electrons is constant. Both magnetic scattering (spin-disorder scattering) and nonmagnetic scattering (ionized-impurity and acoustical-phonon scattering) are considered. The resistivity due to spin disorder always decreases with increasing $H$. However, contrary to the usual assumption, the resistivity due to nonmagnetic scattering also changes with $\delta \mathrm{}\left(H\right)\mathrm{}$. Physically, the resistivity due to nonmagnetic scattering depends on a weighted average $\overline{\tau }$ of the electron relaxation time $\tau \mathrm{}\left(E\right)\mathrm{}$ with respect to the electron energy $E$. The redistribution of electrons changes the energies $E$ which dominate $\overline{\tau }$, and also changes the impurity screening radius $r_0$ on which $\tau \mathrm{}\left(E\right)\mathrm{}$ depends. Under certain conditions (apparently satisfied in our EuSe samples) the magnetoresistance due to nonmagnetic scattering alone is positive. In this case, if the ratio of nonmagnetic to magnetic scattering is sufficiently large, the net magnetoresistance is positive at low $H$. At high $H$, when all the electrons are in the lower-energy subband, nonmagnetic scattering is independent of $H$ but magnetic scattering continues to decrease with increasing $H$. This combination of low-$H$ and high-$H$ behaviors leads to a resistivity peak at a field $H_{max}$, as observed in EuSe. The model gives a good estimate for the field $H_{max}$ and its dependence on temperature $T$ and carrier concentration $n$. Moreover, the predicted $T$ dependence of the magnitude of the resistivity peak is in qualitative agreement with experiment.