Compensated two-carrier magnetoresistance (${\rho }_{xx}=[1+(\mu B{)}^2]/(2{\sigma }_0)$)

Consider isotropic Electrons and electron holes with equal density $n$, charges $\mp e$, equal positive charge carrier mobility $\mu$, and single-species zero-field electrical conductivity ${\sigma }_0=ne\mu$. In a transverse magnetic field, opposite Hall conductivity components cancel, while diagonal components add to $2{\sigma }_0/[1+(\mu B{)}^2]$. Inverting gives ${\rho }_{xy}=0$ and ${\rho }_{xx}={\rho }_{yy}=[1+(\mu B{)}^2]/(2{\sigma }_0)$. Its large-field growth is ${\sigma }_0{B}^2/(2n^2{e}^2)$. The total zero-field resistivity is $1/(2{\sigma }_0)$, half the single-species resistivity; confusing these quantities introduces a factor-of-two normalization error.