Physics of Graphene

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Physics of Graphene

• A. M. Tsvelik

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Graphene a sheet of carbon atoms
The spectrum is well described by the
tight- binding Hamiltonian on a hexagonal
lattice
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Lattice effects Ripples in graphene
A typical snapshot of graphene at room
temperature. The size of height 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).
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Crumpling of graphene sheet the main source
of disorder.
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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
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Klein paradox electrons go through potential
barriers
Penetration of particles through
potential barriers. The transmission probability
T is directionally- dependent. For high barriers
(V gtgt E)
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Hopes for applications - spintronics

• The transmission is sensitive to the barrier
  height V.
• If Vs are different for different spin
• orientations (magnetic gates) one can produce
• spin-polarized currents.
• This will allow to manipulate electrons spin.
• One can also create electronic lenses.

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Electronic lenses
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Beam splitter for electrons (Falko, 2007)
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Is it dirty? STM measurements of graphene (Martin
et. Al. 2007)
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.
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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 samples. The spectral
width is smaller than in any material measured
before. Clean substrates?
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Hall effect (Cho and Fuhrer (2007))Conductivity
as a function of the chemical potential.
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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 lt 9T.
• For 20T lt B lt 45T there are plateaus at
• 0, 1 (interactions ?), 2q spin
• degeneracy is lifted.

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Special Landau level n0

• Integer Quantum Hall effect measurements
  (Giesbers et.al. 2007)
• indicate that at B lt 9T the n0 Landau level is
  unusually narrow which increases the T range
  where Hall effect
• is seen.
• Why it is so narrow?

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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.
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Fractional Quantum Hall effect

• n 1 state is pseudospin (valley) ferromagnet
  (McDonald et. al (2006),
• Haldane et. al. (2006))
• 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.

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Interaction

• The strongest interaction in graphene is Coulomb
  interaction it breaks the Lorentz symmetry.

It breaks the Lorentz invariance of the kinetic
energy. It is predicted to make the velocity
energy dependent (Aleiner et.al 2007)
-fine structure constant
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Conclusions

• There are possible technological applications
  related to directional and energy dependence of
  transmission in graphene.
• 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 finalized.
• It is possible that further many-body effects
  will be
• observed in clean samples at low T. Get rid of
  high e substrate!

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Clean or dirty?
Resistor network model by Cheianov et. Al. (2007)