Weak-Localization Magnetoresistance and Valley Symmetry in Graphene

Because of the chiral nature of electrons in a monolayer of graphite (graphene) one can expect weak antilocalization and a positive weak-field magnetoresistance in it. However, trigonal warping (which breaks $\mathbf{p}\to -\mathbf{p}$ symmetry of the Fermi line in each valley) suppresses antilocalization, while intervalley scattering due to atomically sharp scatterers in a realistic graphene sheet or by edges in a narrow wire tends to restore conventional negative magnetoresistance. We show this by evaluating the dependence of the magnetoresistance of graphene on relaxation rates associated with various possible ways of breaking a “hidden” valley symmetry of the system.

Figure 1

(a) Diagram for the Drude conductivity with (b) the vertex correction. (c) Bethe-Salpeter equation for the Cooperon propagator with valley indices $\xi \mu {\xi }^{\prime }{\mu }^{\prime }$ and $AB$ lattice indices $\alpha \beta {\alpha }^{\prime }{\beta }^{\prime }$. (d) Bare “Hikami box” relating the conductivity correction to the Cooperon propagator with (e) and (f) dressed “Hikami boxes.” Solid lines represent disorder averaged $G^{R/A}$; dashed lines represent disorder.

Figure 2

MR expected in a phase-coherent graphene ${\tau }_{\phi }\gg {\tau }_i$: with ${\tau }_z$, ${\tau }_w\gg {\tau }_i$ (dashed line) and ${\tau }_{*}\ll {\tau }_i$ (solid line). In the case of ${\tau }_{\phi }<{\tau }_i$, $\delta \rho =0$, so that $\Delta \rho (B)=0$.