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Physics of Graphene
A. M. Tsvelik
Graphene – a sheet of carbon atoms
The spectrum is well
described by the tightbinding Hamiltonian on
a hexagonal lattice:
Lattice effects: Ripples in graphene
A typical
snapshot of
graphene at
The size of
fluctuations is
comparable to
the lattice size.
2D membranes embedded in 3D space have a tendency to get crumpled.
These dangerous fluctuations can be suppressed by an anharmonic
coupling between bending and stretching modes. Result: the membranes can exist,
but with strong height fluctuations.
Monte Carlo simulations (Katsnelson et. al. (2007)): disordered state with weakly
T-dependent correlation length (70A at 300K and 30A at 3500K).
Crumpling of graphene sheet – the main
source of disorder.
Dirac Hamiltonian for low energy states
The Bloch functions A and B are peaked on the corresponding sublattices. They are
conveniently joined in a vector
V = c/300
Klein paradox – electrons go through potential barriers
Penetration of particles
through potential
barriers. The transmission
probability T is directionallydependent.
For high barriers (V >> E)
Hopes for applications - spintronics
• The transmission is sensitive to the barrier height V.
If V’s are different for different spin
orientations (magnetic gates) one can produce
spin-polarized currents.
This will allow to manipulate electron’s spin.
One can also create electronic lenses.
Electronic lenses
Beam splitter for electrons (Falko, 2007)
Is it dirty? STM measurements of graphene (Martin et. Al.
Histogram of the density
distribution. The energy
width is ~400K
A color map of the spatial density variations in the graphene flake . Blue
regions are holes and gold regions are electrons. The black contour – zero
density. About 100 particles/puddle, k_Fl ~ 10.
They make it dirty, we make it clean!
Angle Resolved Photoemission Spectroscopy
(ARPES) study of the graphene spectrum
done by T. Valla (BNL) on locally grown
 [eV]
The spectral width is smaller than in any
material measured before.
Clean substrates?
0.0 -1
k [Å ]
Hall effect (Cho and Fuhrer (2007))
Conductivity as a function of the chemical potential.
Hall effect
In the absence of disorder the Landau levels are
• Disorder broadens the levels and when the broadening or T exceed
the Zeeman splitting they become 4-fold degenerate.
Filling fractions n = 4(n + ½)
for B < 9T.
For 20T < B < 45T there are plateaus at
n =0, 1 (interactions ?), 2q – spin
degeneracy is lifted.
Special Landau level n=0
• Integer Quantum Hall effect measurements (Giesbers 2007)
indicate that at B < 9T the n=0 Landau level is unusually
narrow which increases the T range where Hall effect
is seen.
Why it is so narrow?
Zero mode and Index theorem
Hamiltonian in one of the valleys.
We neglect the Zeeman splitting.
Vector potential parametrization:
Eigenfunction with zero energy always exists, no matter how non-uniform the
field is:
where f(z) is a polynomial of power
smaller than the magnetic flux.
Fractional Quantum Hall effect
 n =1 state
is pseudospin (valley) ferromagnet (McDonald et. al (2006),
Haldane et. al. (2006))
n = 3 state is the XY pseudospin magnet (Haldane et. al (2006)).
FQHE at these fillings is the only effect observed so far where interactions play
a role.
The strongest interaction in graphene is Coulomb interaction: it breaks the Lorentz
It breaks the Lorentz invariance of the kinetic energy. It is predicted to
make the velocity energy dependent (Aleiner 2007):
-fine structure constant
• There are possible technological applications related to
directional and energy dependence of transmission in
• The problem #1 is manufacturing of clean samples.
• Most of the physics observed so far is a single particle one.
• Many-body effects are observed in FQHE
in strong magnetic fields.
The role of bending fluctuations is not very clear, the theory is not
It is possible that further many-body effects will be
observed in clean samples at low T. Get rid of high e substrate!
Clean or dirty?
Resistor network model
by Cheianov et. Al. (2007)