Quantum Diffusion in the Lowest Landau Level of Disordered Graphene

Abstract

Electronic transport in the lowest Landau level of disordered graphene sheets placed in a homogeneous perpendicular magnetic field is a long-standing and cumbersome problem which defies a conclusive solution for several years. Because the modeled system lacks an intrinsic small parameter, the theoretical picture is infested with singularities and anomalies. We propose an analytical approach to the conductivity based on the analysis of the diffusive processes, and we calculate the density of states, the diffusion coefficient and the static conductivity. The obtained results are not only interesting from the purely theoretical point of view but have a practical significance as well, especially for the development of the novel high-precision calibration devices.

Keywords: electronic transport in graphene; low-dimensional semimetals; quantum hall effect.

Figure 1

Spectrum of the tight-binding model along the line $k_1^{}=0$ with two Dirac cones at the corners of the Brilloun zone. The energy axis is scaled in units of the hopping parameter between nearest-neighbors t.

Figure 2

The circulation of the Berry vector potential corresponding to the occupied band of the full half filled tight-binding model in the reciprocal space with visible vortex-like structures around the position of the nodal points.

Figure 3

Perturbative processes contributing to the dressing of the single-particle propagator due to the disorder to order $g^1$ (one diagram), $g^2$ (three diagrams), and $g^3$ (fifteen diagrams). Some of the diagrams of order $g^3$ should be counted twice because of the degeneracy due to the mirror symmetry with respect to the imaginable vertical axis, which is accounted for by the factors 2 in front of them.

Figure 4

Evolution of the DOS of both Landau sublevels defined in Equation (11) (a–d) plotted in units of the DOS at each suband center $\frac{1}{{\pi }^{5/2}}\frac{k^2}{E_g^{}}$ with increasing disorder strength as a function of the dimensionless energy $\nu$. The following quantities are used: ${ϵ}_0^{}/t=0.15$, ${\Delta }_0^{}/t=0.1$ and $E_g^{}/t=0.01,0.045,0.073,$ and $0.1$ in units of the hopping amplitude. Dashed lines emphasize the position of each eigenvalue.

Figure 5

Perturbative processes contributing to the dressing of the two-particles propagator up to the third order in disorder strength. Solid lines denote the fully dressed Wegner’s propagators and the dashed lines denote the disorder correlators.