We study the transversal magnetoconductivity and magnetoresistance of a massive Dirac fermion gas. This can be used as a simple model for gapped Dirac materials. In the zero mass limit the case of gapless Dirac semimetals is also studied. In the case of Weyl semimetals, to reproduce the non-saturating linear magnetoresistance seen in experiments, the use of screened charged impurities is inevitable. In this paper these are included using the first Born approximation for the self-energy. The screening wavenumber is calculated using the random phase approximation with the polarization function taking into account the electron-electron interaction. The Hall conductivity is calculated analytically in the case of no impurities and is shown to be perfectly inversely proportional to the magnetic field. Thus, the magnetic field dependence of the magnetoresistance is mainly determined by σxx. We show that in the extreme quantum limit at very high magnetic fields the gapped Dirac materials are expected to have σxx ∝ B −3 leading to xx ∝ B −1 , in contrast with the gapless case where σxx ∝ B −1 and xx ∝ B. At lower fields we find that the effect of the mass term is negligible and in the region of the Shubnikov-de Haas oscillations the two systems behave almost identically. We suggest a phenomenological scattering rate that is able to reproduce the linear behavior at the oscillating region. We show that in the case of the scattering rate calculated using the Born approximation the strength of the relative permittivity and the density of impurities affects the magnetic field dependence of the conductivity significantly.
Using a microscopic theory for the magnetoconductivity at low magnetic fields we show how the Hall and longitudinal conductivity can be calculated in the low scattering rate limit. In the lowest order of the scattering rate, we recover the result of the semiclassical Boltzmann transport theory. At higher order, we get quantum corrections containing the Berry curvature and the orbital magnetic moment. The quantum corrections vanish if time reversal symmetry holds.
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