Konstantin Efetov: Transport in Graphene

Konstantin Efetov started by drawing parallels between high-T_cs (which
dominated earlier discussion) and graphene. He emphasized that both are 2D
systems which have been lauded as materials of the future for energy and
nanoelecronics respectively. Konstantin reminded the audience that grapehene
has 2D honeycomb crystal lattice, and therefore contains two sublattices. The
Brillouin Zone has two valleys with linear, Dirac-like, energy dispersion, so
that the effective hamiltonian for pure graphene is a 4x4 block-diagonal
matrix. Gating the substrate with graphene film on top allows changing the
filling fraction easily.

Konstantin then described the effects of impurity scattering on transport in a
system with such an energy spectrum following his recent work with Igor
Aleiner [PRL 97, 236801 (2006)]. The work was motivated by experimental claims
of delocalized low energy states and universal metallic resistivity in
graphene (which, it seems, is no longer universal), and theoretical analyses
of weak localization corrections.

Efetov and Aleiner considered general purely potential impurity scattering
(spin-orbit interaction is graphene is weak), which replaces zeros in the
block-diagonal Hamiltonian with finite values. Symmetries of the problem
(time-reversal, translation, etc.) dictate that there are 5 independent
parameters that characterize disorder. Konstantin pointed out that the
self-consistent Born approximation does not work for Dirac spectrum as there
are many logarithmic corrections that need to be resummed using the RG
methods. The main conclusion is that all 5 impurity constants grow under RG flow.

The work considers a finite filling fraction and proceeds by looking at the
free energy functional using the
supersymmetry approach. The main conclusions are that, if one neglects the
scattering between bands, the system maps onto a symplectic ensemble,
resulting in antilocalization: increase in conductivity upon lowering the
temperature. However, upon lowering the temperature, intervalley scattering
becomes important, and one finds an orthogonal ensemble for which all states
are known to be localized.

The prediction is for a non-monotonous behavior of the conductivity with
temperature. As T is decreased, first the conductivity is reduced in accord
with the log corrections. At lower T antilocalization kicks in and the
conductivity increases when the temperature is lowered. At yet lower T, the
intervalley scattering takes over, and the conductivity drops to zero as the
states become localized. Konstantin concluded by saying that there is no
chance for minimal metallic conductivity due to generic disorder.

Assa Auerbach asked whether this behavior had been found numerically since
this is a non-interacting theory that lends itself easily to modeling.
Konstantin replied that exploring the phase space of 5 parameters is hard, and
that the localization length is expected to be large.

Claudio Castellani and Andrey Chubukov both asked what happens when graphene
is tuned very close to the Dirac point, i.e. filling fraction is small. The
system is right away in the strong disorder limit, but Konstantin believes the
conductivity still goes to zero at T=0 since the states are almost localized
already.

John Mydosh asked what are the potential impurities, and apparently these are
mostly charged impurities on the substrate.

Victor Galitskii asked what determines the crossover scale between
antilocalization and localization. There is no unique answer, this is related
to how different components of the impurity scattering vary with T.

Catherine Pepin asked whether interaction correction have been considered and
what they do. There seems to be some work done on that, but lunch truck was
about to arrive, and we stopped.