Intrinsic magnetoresistance in three-dimensional Dirac materials with low carrier density

Negative longitudinal and positive in-plane transverse magnetoresistance have been observed in most topological Dirac/Weyl semimetals and some other topological materials. Here, we present a quantum theory of intrinsic magnetoresistance for three-dimensional Dirac fermions at a finite and uniform magnetic field B . In a semiclassical regime, it is shown that the longitudinal magnetoresistance is negative and quadratic of a weak field B while the in-plane transverse magnetoresistance is positive and quadratic of B . The relative magnetoresistance is inversely quartic of the Fermi wave vector and only determined by carrier density, irrelevant to the external scatterings in the weak scattering limit. This intrinsic anisotropic magnetoresistance is measurable in systems with low carrier density and high mobility. In the quantum oscillation regime a formula for the phase shift in Shubnikov-de Haas oscillation is present as a function of the mobility and the magnetic field, which is helpful for experimental data analysis.