Quantum corrections in the Boltzmann conductivity of graphene and their sensitivity to the choice of formalism

Abstract

Semiclassical spin-coherent kinetic equations can be derived from quantum theory with many different approaches (Liouville equation based approaches, nonequilibrium Green's functions techniques, etc.). The collision integrals turn out to be formally different, but coincide in textbook examples as well as for systems where the spin-orbit coupling is only a small part of the kinetic energy like in related studies on the spin Hall effect. In Dirac cone physics (graphene, surface states of topological insulators like Bi1−xSbx, Bi2Te3 etc.), where this coupling constitutes the entire kinetic energy, the difference manifests itself in the precise value of the electron-hole coherence originated quantum correction to the Drude conductivity σ0 ∼ e 2 h kF. The leading correction is derived analytically for single and multilayer graphene with general scalar impurities. The often neglected principal value terms in the collision integral are important. Neglecting them yields a leading correction of order ( kF) −1 , whereas including them can give a correction of order ( kF) 0 . The latter opens up a counterintuitive scenario with finite electron-hole coherent effects at Fermi energies arbitrarily far above the neutrality point regime, for example in the form of a shift δσ ∼ e 2 h that only depends on the dielectric constant. This residual conductivity, possibly related to the one observed in recent experiments, depends crucially on the approach and could offer a setting for experimentally singling out one of the candidates. Concerning the different formalisms we notice that the discrepancy between a density matrix approach and a Green's function approach is removed if the Generalized Kadanoff-Baym Ansatz in the latter is replaced by an anti-ordered version. This issue of Ansatz may also be important for Boltzmann type treatments of graphene beyond linear response.

Finite conductivity minimum in bilayer graphene without charge inhomogeneities

“…This value agrees with Ref. 30, where a related modified Boltzmann approach is combined with a four-band effective Hamiltonian for the carriers, as well as with recent theoretical predictions 36 using other closely related approaches. Our min differs from the one obtained for ballistic bilayer graphene, 24,25,37 where the min is attributed to evanescent modes penetrating the sample from contacts.…”

“…When the collision integral is evaluated to leading ͑second͒ order in the ͑configuration-averaged͒ impurity potential, the collision term ͑including its off-shell terms 34,36 ͒ reduces to the simple matrix relation-time form, I͓f͑k͔͒ → −f ͑1͒ ͑k͒ / , where f ͑1͒ is the deviation from equilibrium. This is a remarkable property of the two-band bilayer model with ␦-function scatterers.…”

Aspects of the theory of graphene

“…A recent review paper by Das Sarma et al (2010) provides a well-documented review of the subject. In addition, other very recent reviews of theoretical approaches in graphene quantum transport are now available (Kailasvuori & Lüffe 2010;Peres 2010). It should also be borne in mind that relativistic theory also provides for pair production that, in the case of massless graphene, supplies electron and hole charge carriers for conduction (Schwinger 1951;Schmidt et al 1998;Lewkowicz & Rosenstein 2009).…”

Low-temperature linear transport of two-dimensional massive Dirac fermions in silicene: Residual conductivity and spin/valley Hall effects

“…In previous studies, two such mechanisms were proposed. One is the interband correlation: Notwithstanding the vanishing of equilibrium electron density, dc electric field excites an electron from the valence to conduction band, resulting in a nonvanishing conductivity [53][54][55][56][57][58][59][60]. The other one is associated with the low-energy states at the edges of samples.…”