Coulomb disorder in three-dimensional Dirac systems

In three-dimensional materials with a Dirac spectrum, weak short-ranged disorder is essentially irrelevant near the Dirac point. This is manifestly not the case for Coulomb disorder, where the long-ranged nature of the potential produced by charged impurities implies large fluctuations of the disorder potential even when impurities are sparse, and these fluctuations are screened by the formation of electron/hole puddles. In this paper, I present a theory of such nonlinear screening of Coulomb disorder in three-dimensional Dirac systems, and I derive the typical magnitude of the disorder potential, the corresponding density of states, and the size and density of electron/hole puddles. The resulting conductivity is also discussed.

Figure 1

Schematic illustration of electron/hole puddles in a 3DDS at the Dirac point. The Coulomb potential energy $e\varphi$ is shown as a function of some coordinate $r$. Coulomb impurities create potential fluctuations with typical size $r_s$ and amplitude $\Gamma$. Regions of positive $e\varphi$ correspond to electron puddles, while negative $e\varphi$ implies hole puddles.

Figure 2

(a) The variance in the disorder potential as a function of the chemical potential $\mu$. The dashed line corresponds to the linear screening regime of Eq. (12). (b) The spatially averaged density of states as a function of the chemical potential. The dashed line corresponds to Eq. (13), which is the DOS for a nondisordered system. ${\Gamma }_0$ and ${\nu }_0$ are given by Eqs. (7) and (8), respectively.