Weak momentum scattering and the conductivity of graphene

Electrical transport in graphene offers a fascinating parallel to spin transport in semiconductors including the spin-Hall effect. In the weak momentum scattering regime the steady-state density matrix contains two contributions, one linear in the carrier number density $n$ and characteristic scattering time $\tau$, the other independent of either. In this paper we take the Liouville equation as our starting point and demonstrate that these two contributions can be identified with pseudospin conservation and non-conservation respectively, and are connected in a non-trivial manner by scattering processes. The scattering term has a distinct form, which is peculiar to graphene and has important consequences in transport. The contribution linear in $\tau$ is analogous to the part of the spin density matrix which yields a steady state spin density, while the contribution independent of $\tau$, is analogous to the part of the spin density matrix which yields a steady state spin current. Unlike in systems with spin-orbit interactions, the $n$ and $\tau$-independent part of the conductivity is reinforced in the weak momentum scattering regime by scattering between the conserved and non-conserved pseudospin distributions.