Origin of minimal conductivity in Dirac materials: Momentum-dependent self-energy function from long-ranged disorder scattering

We present a unified understanding of the experimentally observed minimal dc conductivity in Dirac materials. First, based on the linear response theory, we reveal that the momentum-dependent self-energy function induces a tunable minimal conductivity in Dirac electrons, the magnitude of which is directly related to the coefficient of the momentum-dependent part in the self-energy function. Taking the long-ranged Gaussian and Coulomb potentials as examples, the momentum-dependent self-energy function is perturbatively derived using the Born approximation and supplemented by the self-consistent Born approximation and renormalization group analysis. Moreover, we further validate our theory via numerical simulations using the large-scale Lanczos algorithm. The explicit momentum dependence of the self-energy on the intensity, concentration, and range of potential are critically addressed. Therefore, our theory provides a reasonable interpretation of the sample-dependent minimal conductivity observed in experiments.